Mechanical Engineering

International Review of

Mechanical Engineering (IREME) Contents Warpage Analyses on Thin Plate in Three-Plate Mold by Taguchi Method and Analysis of Variance (ANOVA) for PC, ABS and PC/ABS by Z. Shayfull, M. Fathullah, S. M. Nasir, N. A. Shuaib, M. S. Abdul Manan

1

The Influence of Different Mold Temperature on Warpage in a Thin Shallow Injection Molding Process by N. A. Shuaib, S. M. Nasir, M. Fathullah, Z. Shayfull, M. S. Abdul Manan

11

Drag Force Reduction Technique for Abrasive Resisting Materials by S. V. Gavali, V. B. Tungikar

17

Radiation and Mass Transfer Effects on Flow of Micropolar Fluid Past a Continuously Moving Plate with Suction/Injection by P. Loganathan, N. Golden Stepha

22

Concurrent Simulation of Permeability, Thermal Conductivity and Modulus for Carbon Fibre Reinforcements and Composites by Reza Samadi, Francois Robitaille

29

Development of Tooling for Hydraulic Forming of Ceramic Spheroids Using Alumina by V. Bristot, V. Bristot, L. Schaeffer, V. Gruber

39

Experimental Investigation on Hardness, Cutting Force and Roughness in Milling of Hybrid Composites by A. Arun Premnath, T. Alwarsamy, T. Rajmohan

44

Optimization of the Boundary Conditions by Genetic Algorithms by J. L. Marcelin

50

Stress Analysis of FG Thick Pressure Vessels Considering the Effects of Material Gradations and Poisson’s Ratio Using DQ Method by A. M. Goudarzi, S. Saadati, A. Paknahad

55

Iterative Algorithm for Active Vibration Control of Flexible Beam by Mohd S. Saad, Hishamuddin Jamaluddin, Intan Z. M. Darus

61

Reduced Models Based on Smooth Decomposition for Random Mechanical Systems by Sergio Bellizzi, Rubens Sampaio

74

Hydrodynamics and Mass Transfer Using Three-Phase Fluidized Bed Contactor by Abbas H. Sulaymon, Raghad F. Almilly

86

Experimental Study on the Forced Draft Cooling Tower Using Psychrometric Gun Technique by Ramkumar Ramakrishnan, Ragupathy Arumugam

97

Experimental Investigation of Heated Horizontal Rectangular Fin Array Under Mixed Convection by S. G. Taji, G. V. Parishwad, N. K. Sane, R. Z. Deshmukh

104

(continued)

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Calculation of Reduction Heat Transfer between Two Finite Concentric Cylinders Using Radiation Shields with Temperature-Dependent Emissivity by Seyfolah Saedodin, M. S. Motaghedi Barforoush, Mohsen Torabi

113

Heat Generation Effects on Unsteady Natural Convective Flow Over a Vertical Plate with Variable Viscosity by P. Loganathan, D. Iranian, P. Ganesan

120

Effect of Cavity Design of Synthetic Jet Actuator to the Heat Transfer Characteristic of an Impinging Flow Configuration by Harinaldi, Rikko Defriadi, Damora Rhakasywi

128

Resonant Behavior of a Hydraulic Ram Pump by Mario O. M. de Carvalho, Alberto C. G. C. Diniz, Fernando J. R. Neves

137

Combustion and Performance Analysis of DI-Diesel Engine Fuelled with Neat Mahua Methyl Ester Along with Oxygenated Fuel (DEE) as an Additive by P. Ramesh Babu, V. J. J. Prasad, N. Hari Babu, B. V. Appa Rao

147

Real-Time Control of Automotive Engine Fuelled with Malaysian Palm Oil Biodiesel by Azuwir Mohdnor, M. Z. Abdulmuin, A. H. Adom

153

Reduction of NOx Emission from Diesel Engine Using Urea Injection with SCR Technique with Different Catalyst Connected in Series by K. Chithamparam Asary, N. V. Mahalakshmi, K. Jayachandran

161

Jordan Transport Energy Demand Forecasting: the Application of Time Series Technique by Adnan Mukattash, Ahmed Al-Ghandoor, Ahmad M. Qamar

166


International Review of Mechanical Engineering (I.RE.M.E.), Vol. 6, N. 1 ISSN 1970-8734 January 2012

Warpage Analyses on Thin Plate in Three-Plate Mold by Taguchi Method and Analysis of Variance (ANOVA) for PC, ABS and PC/ABS Z. Shayfull, M. Fathullah, S. M. Nasir, N. A. Shuaib, M. S. Abdul Manan

Abstract – Warpage is a common issue in plastic injection molding process. It can get worse for thin parts and therefore it is more challenging for product design engineers, mold designers and manufacturing engineer to cope with overwhelming customer demands on small and thin products lately. A lot of research have been done on this topic just to study the most significant factors influencing warpage on plastic parts but lack of study on thin plate parts by using pin point gate which is automatic de-gating gate which have an ability to reduce production cost. In this study, thin plate plastic product is to be the subject of analysis. The part with dimension 120mm x 50mm x 1mm is evaluated using pin point gate in three-plate mold. Three experiments have been done using Autodesk Moldflow Insight (AMI) to simulate warpage resulted on the thin plate parts by using Polycarbonate (PC), Acrylonitrile Butadiene Styrene (ABS) and Polycarbonate/Acrylonitrile Butadiene Styrene (PC/ABS) materials. Taguchi Method is applied in identifying the optimum value of injection molding parameters while Analysis of Variance (ANOVA) is used to get the most significant factor affected warpage. The results show that packing time is the most significant factor affecting warpage on thin plate parts for PC, ABS and PC/ABS materials. This finding is significant in helping industrial practitioners particularly in manufacturing ultra-thin shell parts with high quality in term of warpage issues. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Thin Plate, Plastic Injection Molding, Warpage, Taguchi Method, ANOVA

I.

technique which provides an efficient approach to optimize performance, quality attributes and cost in manufacturing process [2]. Huang and Tai [3] reported that packing pressure is the most significant factor that affects warpage and gate locations on the other hand filling time has only small effects over warpage for thin shell parts (rectangle cover of length 120 mm, width 50 mm, height 8 mm and wall thickness 1 mm) based on simulation software and optimizing using Taguchi method. Tang et al. [4] applied Taguchi method to minimize warpage on thin plate parts (120 mm length, 50 mm width and 1 mm wall thickness). The gate dimension and the mold temperature were eliminated while ANOVA was used to determine the most significant factor affecting warpage. As a result, melting temperature was found as the most important factor that contributes to the existence of warpage. N.A. Shuaib et al. [5] studied thin shallow parts (50mm x 30mm x 0.5mm thickness) with variable radius. The process is performed by simulation and experimental method using Taguchi and Analysis of Variance (ANOVA) techniques. The result shows that by S/N response and percentage contribution in ANOVA, packing time has been identified as the most significant factors in affecting the warpage on thin shallow parts.

Introduction

Injection Molding Process is widely used in production of plastics products. Effectiveness of this process depends on the quality of the product produced. Starting from a designing process of plastic parts, mold design, mold fabrication, injection molding machine and parameters setting during molding process causes a number of defects such as warpage, weld lines, jetting or sink marks. All of these defects will reduce the quality of the molded parts. Besides with the demand trend of electronics parts today, products are getting smaller from time to time especially to accommodate portable and hand-held systems [1]. Smallest and thinnest plastic parts means the possibility of parts to warp will increase. For the warpage defect, it is important for product design engineers to simulate plastic parts at the early stage of design process in order to minimize the modification cost on the mold after fabrication and to maximize the quality of plastic parts produced. Many researches and publications made by theoretical simulation and experimental results studying the behavior of warpage occurred at molded parts using robust engineering methodology such as Taguchi

Manuscript received and revised December 2011, accepted January 2012


1

Z. Shayfull, M. Fathullah, S. M. Nasir, N. A. Shuaib, M. S. Abdul Manan

A result acquired by Liao et al. [6] also agrees that packing pressure is the most significant parameter for thin wall parts in injection molding process. His study was done purposely to determine the reactions of a thin walled part according to shrinkage and warpage issues where mould temperature, melt temperature, packing pressure and injection speed were taken as the injection parameters. From the research, it was found that packing pressure is a big factor contributes to the occurrence of warpage. Previous research proved that the temperature differences of the core and cavity plates with cavity temperature was identified as the most significant factor in affecting warpage for thin plate parts by using edge gate [7]. M.C. Song et al. [8] explored effects of parameters on ultra-thin wall plastic parts in injection molding process and defining the ultra-thin wall as the part which has a 1mm or less nominal wall thickness and at least 50cm2 surface areas with a ratio flow length/thickness above 100. From the analysis, high melt temperature and injection pressure are necessary for injection molding process of ultra-thin wall parts. Babur Ozcelik and T. Erzurumlu [9] also used Taguchi Method to explore the effect of injection molding parameter on warpage for plastic parts with different thickness value. The most influential parameter on warpage of thin shell PC/ABS material was found to be Packing Pressure. However, an investigation of thin plate parts for injection molding process by using pin point gate in three-plate mold is rarely reported. As far as this issue is concerned, the challenge for the manufacturing engineers nowadays is to produce thin plate molded parts at minimum warpage. The aim of this research is to find out the effect of various types of plastic materials and the most influential factor in thin plate parts in order to help design and manufacturing engineers in reducing times for molding trials, to eliminate cost to modify molds after fabrication and to produce a good quality of parts produced.

II.

Fig. 1. Gating system for thin plate parts in three-plate mold

Fig. 2. Gating system for thin plate parts in three-plate mold

Fig. 3. Cooling channel design for thin plate parts in three-plate mold

III. Experiment There are too many factors that influence the quality of injection molded products by injection molding processes. Among the factors are the shapes of plastic product, types of plastic material, insert materials for core and cavity, cooling channel design, coolant liquid use, room temperature and types of injection molding machines. This study only considers a few major factors. Besides that some consumption are made which are; i. Gate dimension factor is neglected because of its design is not identical for every plastic product. ii. The temperature of the environment is assumed constant. iii. The coolant is assumed as pure water. iv. The effects of other minor factors (other than melting temperature, mold temperature, filling and packing processes) are not considered in this study. v. The cooling channels layout is assumed to maintain

Gating System Design

Product design engineers should be able to identify types of gate to be used. With the technology of Computer Aided Engineering (CAE) simulation today, it helps simulating and improving the design of product at early stage of design process. The best gate location in term of cosmetic appearance point of view should be located where user cannot see the gate marks after assemble the products. Fig. 1 shows thin plate parts 120mm x 50mm and thickness 1mm with pin point gate which normally applied in three-plate mold. The details of gating system with complete dimensions are shown in Fig. 2 while Fig. 3 shows the cooling channel with Ø6mm designed for the mold.


International Review of Mechanical Engineering, Vol. 6, N. 1

2


at constant temperature. vi. The effects due to the shape and size of the mold and product are ignored due to various shapes of products. vii. The plastic material used in this study is amorphous thermoplastics PC Lexan 125, ABS Cycolac G500 and PC/ABS Cycolac C2100. Basic mechanical and physical properties for each of materials are shown in Table I.

Three experiments are conducted to obtain the best setting parameters and to determine the most significant factors affecting warpage on thin plate parts by using three types of materials. Five factors (A-E) are identified to be controlled in Experiment-I, II and III whereas L16 45 is chosen for all of experiments. These orthogonal array variance and parameters control factors are shown in Table IV – X respectively. TABLE IV THE FOUR LEVEL OF EFFECTIVE FACTOR FOR PC LEXAN 125 IN EXPERIMENT-I Level Factor 1 2 3 4 Mold temperature, A (°C) 90 100 110 120

TABLE I THE PHYSICAL PROPERTIES OF PC LEXAN 125 Specific heat, Cp (J/kgoC)

1900

Elastic modulus, E (MPa)

2.28 x 103

Poisson's ratio, υ

0.417

Melt temperature, B (°C)

275

280

285

290

Thermal conductivity, K (w/m C)

0.24

Injection time, C (s)

0.1

0.2

0.3

0.4

Shear rate (1/s)

40000

Packing pressure, D (MPa)

75%

80%

85%

90%

Shear Modulus (MPa)

804.5

Packing time, E (s)

0.1

0.2

0.3

0.4

o

TABLE II THE PHYSICAL PROPERTIES OF ABS CYCOLAC G500 Specific heat, Cp (J/kgoC)

2400


2.24 x 103

Poisson's ratio, υ

0.392

TABLE V THE FOUR LEVEL OF EFFECTIVE FACTOR FOR ABS CYCOLAC G500 IN EXPERIMENT-II Level Factor 1 2 3 4 Mold temperature, A (°C) 50 60 70 80

Thermal conductivity, K (w/moC)

0.18


230

240

250

Shear rate (1/s)

12000


0.1

0.2

0.3

0.4

Shear Modulus (MPa)

804.6


75%

80%

85%

90%

Packing time, E (s)

0.1

0.2

0.3

0.4

TABLE III THE PHYSICAL PROPERTIES OF PC/ABS CYCOLAC C2100 Specific heat, Cp (J/kgoC)

1810


2.78 x 103

Poisson's ratio, υ

TABLE VI THE FOUR LEVEL OF EFFECTIVE FACTOR FOR PC/ABS CYCOLAC C2100 IN EXPERIMENT-III Level Factor 1 2 3 4 Mold temperature, A (°C) 50 60 70 80

0.4 o

260


0.27

Shear rate (1/s)

40000


230

240

250

Shear Modulus (MPa)

992.9


0.1

0.2

0.3

0.4


75%

80%

85%

90%

Packing time, E (s)

0.1

0.2

0.3

0.4

Fig. 4 shows the two cavities of thin plate parts generated with 18,267 pieces of surface triangles and 1mm length of triangular.

TABLE VII L16 ORTHOGONAL ARRAY VARIANCE FOR EXPERIMENT – I, II & III Control Factor Trial No. A B C D 1 1 1 1 1 2 1 2 2 2 3 1 3 3 3 4 1 4 4 4 5 2 1 2 3 6 2 2 1 4 7 2 3 4 1 8 2 4 3 2 9 3 1 3 4 10 3 2 4 3 11 3 3 1 2 12 3 4 2 1 13 4 1 4 2 14 4 2 3 1 15 4 3 2 4 16 4 4 1 3

Fig. 4. Thin plate parts in three-plate mold


260

E 1 2 3 4 4 3 2 1 2 1 4 3 3 4 1 2


3


TABLE VIII THE COMBINATION PARAMETERS FOR THE CONTROL FACTORS IN EXPERIMENT-I Trial No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

A 90 90 90 90 100 100 100 100 110 110 110 110 120 120 120 120

Control Factor B C D 275 0.1 75 280 0.2 80 285 0.3 85 290 0.4 90 275 0.2 85 280 0.1 90 285 0.4 75 290 0.3 80 275 0.3 90 280 0.4 85 285 0.1 80 290 0.2 75 275 0.4 80 280 0.3 75 285 0.2 90 290 0.1 85

using Analysis of Variance (ANOVA) where the level of confidence is set at 0.05. The Signal-to-noise (S/N) ratio is calculated according to the results (deflection in z-direction) obtained from warpage analysis of thin plate as shown in Table XI, XII and XIII in order to obtain the best parameter setting arrangement.

E 0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2

TABLE XI SUMMARY OF THE RESULTS OF WARPAGE ON THIN PLATE FOR PC LEXAN 125 IN EXPERIMENT-I Control Factor Warpage S/N Trial No. (mm) Ratio A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

TABLE IX THE COMBINATION PARAMETERS FOR THE CONTROL FACTORS IN EXPERIMENT-II Control Factor Trial No. A B C D E 1 50 230 0.1 75 0.1 2 50 240 0.2 80 0.2 3 50 250 0.3 85 0.3 4 50 260 0.4 90 0.4 5 60 230 0.2 85 0.4 6 60 240 0.1 90 0.3 7 60 250 0.4 75 0.2 8 60 260 0.3 80 0.1 9 70 230 0.3 90 0.2 10 70 240 0.4 85 0.1 11 70 250 0.1 80 0.4 12 70 260 0.2 75 0.3 13 80 230 0.4 80 0.3 14 80 240 0.3 75 0.4 15 80 250 0.2 90 0.1 16 80 260 0.1 85 0.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

B 250 260 270 280 250 260 270 280 250 260 270 280 250 260 270 280

C 0.1 0.2 0.3 0.4 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2 0.4 0.3 0.2 0.1

D 75 80 85 90 85 90 75 80 90 85 80 75 80 75 90 85

0.1 0.2 0.3 0.4 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2 0.4 0.3 0.2 0.1

75 80 85 90 85 90 75 80 90 85 80 75 80 75 90 85

0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2

0.0065 0.0051 0.0044 0.0037 0.0037 0.0047 0.0052 0.0063 0.0050 0.0059 0.0046 0.0052 0.0046 0.0045 0.0065 0.0062

43.7417 45.8486 47.1309 48.6360 48.6360 46.5580 45.6799 44.0132 46.0206 44.5830 46.7448 45.6799 46.7448 46.9357 43.7417 44.1522

TABLE XII SUMMARY OF THE RESULTS OF WARPAGE ON THIN PLATE FOR ABS CYCOLAC G500 IN EXPERIMENT-II Control Factor Trial Warpage S/N No. (mm) Ratio A B C D E 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Control Factor A 50 50 50 50 60 60 60 60 70 70 70 70 80 80 80 80

275 280 285 290 275 280 285 290 275 280 285 290 275 280 285 290

The results from ANOVA are compared with SN ratio method. The interaction effect of factors is identified and the significant of each factor towards the total effect is then analyzed accordingly.

TABLE X THE COMBINATION PARAMETERS FOR THE CONTROL FACTORS IN EXPERIMENT-III Trial No.

90 90 90 90 100 100 100 100 110 110 110 110 120 120 120 120

E 0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2

50 50 50 50 60 60 60 60 70 70 70 70 80 80 80 80

230 240 250 260 230 240 250 260 230 240 250 260 230 240 250 260

0.1 0.2 0.3 0.4 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2 0.4 0.3 0.2 0.1

75 80 85 90 85 90 75 80 90 85 80 75 80 75 90 85

0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2

0.0062 0.0054 0.0049 0.0045 0.0046 0.0051 0.0052 0.0061 0.0052 0.0057 0.0051 0.0053 0.0050 0.0050 0.0063 0.0062

44.1522 45.3521 46.1961 46.9357 46.7448 45.8486 45.6799 44.2934 45.6799 44.8825 45.8486 45.5145 46.0206 46.0206 44.0132 44.1522

The deflection of thin plate parts in z-direction obtained from the simulation process are also analyzed Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


4


TABLE XIII SUMMARY OF THE RESULTS OF WARPAGE ON THIN PLATE FOR PC/ABS CYCOLAC C2100 IN EXPERIMENT-III Control Factor

Trial No.

A

B

C

D

E

Warpage (mm)

S/N Ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

50 50 50 50 60 60 60 60 70 70 70 70 80 80 80 80

250 260 270 280 250 260 270 280 250 260 270 280 250 260 270 280

0.1 0.2 0.3 0.4 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2 0.4 0.3 0.2 0.1

75 80 85 90 85 90 75 80 90 85 80 75 80 75 90 85

0.1 0.2 0.3 0.4 0.4 0.3 0.2 0.1 0.2 0.1 0.4 0.3 0.3 0.4 0.1 0.2

0.0074 0.0061 0.0054 0.0049 0.0055 0.0058 0.0062 0.0073 0.0058 0.0066 0.0060 0.0063 0.0060 0.0059 0.0075 0.0072

42.6154 44.2934 45.3521 46.1961 45.1927 44.7314 44.1522 42.7335 44.7314 43.6091 44.4370 44.0132 44.4370 44.5830 42.4988 42.8534

IV.

TABLE XV THE RESPONSE TABLE OF S/N RATIO FOR WARPAGE ON THIN PLATE FOR ABS CYCOLAC G500 IN EXPERIMENT-II Level A B C D E 1 45.6590 45.6494 45.0004 45.3418 44.3353 2 45.6417 45.5260 45.4062 45.3787 45.2160 3 45.4814 45.4344 45.5475 45.4939 45.8949 4 45.0516 45.2240 45.7532 45.6194 46.3874 Diff. 0.6074 0.4254 0.7528 0.2776 2.0521 TABLE XVI THE RESPONSE TABLE OF S/N RATIO FOR WARPAGE ON THIN PLATE FOR PC/ABS CYCOLAC C2100 IN EXPERIMENT-III Level 1 2 3 4 Diff.

A 44.6142 44.2025 44.1977 43.5930 1.0212

B 44.2441 44.3042 44.1100 43.9490 0.3552

C 43.6593 43.9995 44.3500 44.4926 0.8334

D 43.8409 43.9752 44.2518 44.5394 0.6985

E 42.8642 44.0076 44.6334 45.1022 2.2380

For factor A, level 1 in Experiment-I: ⎡( 43.7417 ) + ( 45.8486 ) + ⎤ ⎢ ⎥ ⎢⎣ + ( 47.1309 ) + ( 48.6360 ) ⎥⎦ Level 1 = = 3 = 46.3393

Result and Discussion

Tables XI, XII and XIII show the results of warpage analysis (deflection in z-direction) for thin plate parts. In this case, ‘the smaller the better quality’ equation from Taguchi method is chosen. The equation of S/N is shown below:

For factor B, level 1 in Experiment-II: ⎡( 45.3521) + ( 45.8486 ) + ⎤ ⎢ ⎥ ⎢ + ( 44.8825 ) + ( 46.0206 ) ⎦⎥ ⎣ Level 1 = = 3 = 45.6494

S / N = -10 log ( MSD ) 2 1 n yi . ∑ = 1 i n MSD is the mean square deviation and y stands for the number of observations where n, is the number of tests in one trial. The summary of S/N values for the warpage of thin plate parts is shown in Tables XI, XII and XIII. The deflection data in z-direction for the ultra-thin shell parts are analyzed using Analysis of Variance (ANOVA). The relative percentage contribution of all factors is determined by comparing the relative variance. Then the degrees of freedom, variance, F-ratio, sums of squares, pure sum of square and percentage contribution are all computed. The examples of calculations are shown and the results of S/N ratio for warpage in thin plate parts are listed in Tables XIV, XV and XVI.

where MSD =

For factor C, level 1 in Experiment-III: ⎡( 42.6154 ) + ( 44.7314 ) + ⎤ ⎢ ⎥ ⎢⎣ + ( 44.4370 ) + ( 42.8534 ) ⎥⎦ Level 1 = = 3 = 45.6593

Figs. 5-9, 10-14 and 15-19 show S/N response diagrams constructed for the warpage on thin plate from Experiment I, II and III based on data acquired from Tables XIV, XV and XVI respectively.

TABLE XIV THE RESPONSE TABLE OF S/N RATIO FOR WARPAGE ON THIN FOR PC LEXAN 125 IN EXPERIMENT-I Level

A

B

C

D

E

1 2 3 4 Diff.

46.3393 46.2218 45.7571 45.3936 0.9457

46.2858 45.9813 45.8244 45.6203 0.6655

45.2992 45.9766 46.0251 46.1447 0.8455

45.5093 45.8379 46.1255 46.2391 0.7297

44.0199 45.4253 46.5284 47.7381 3.7182

Fig. 5. S/N response for mold temperature (Factor A) in Experiment-I



5


Fig. 6. S/N response for melt temperature (Factor B) in Experiment-I

Fig. 11. S/N response for melt temperature (Factor B) in Experiment-II

Fig. 12. S/N response for injection time (Factor C) in Experiment-I


Fig. 8. S/N response for packing pressure (Factor C) in Experiment-I


Fig. 9. S/N response for packing time (Factor D) in Experiment-I


Fig. 10. S/N response for mold temperature (Factor A) in Experiment-II

Fig. 15. S/N response for mold temperature (Factor A) in Experiment-II



6


Table XVII shows the best setting of combination parameters where the minimum warpage can be obtain is 0.0033mm with PC Lexan 125, 0.0042mm with ABS Cycolac G500 and 0.0046mm with PC/ABS Cycolac C2100. Meanwhile, Table XVIII shows the most significant factors affected on warpage thin plate parts for each material. TABLE XVII BEST SETTING OF COMBINATION PARAMETERS Parameters Setting ABS PC Lexan Factor Cycolac 125 G500 Mold Temperature, (°C) 90 50 Melt temperature, (°C) 275 230 Injection time, (s) 0.4 0.4 Packing pressure, (MPa) 90% 90% Packing time, (s) 0.4 0.4 Deflection 0.0033 0.0042 Z-Direction, (mm)

Fig. 16. S/N response for melt temperature (Factor B) in Experiment-II

PC/ABS Cycolac C2100 50 260 0.4 90% 0.4 0.0046

TABLE XVIII MOST SIGNIFICANT FACTORS AFFECTED ON WARPAGE THIN PLATE PARTS Most Significant Experiment-I Experiment-II Experiment-II Factor Packing time Packing time Packing time 1 (E) (E) (E) Mold Injection time Mold 2 temperature (C) temperature (A) (A) Injection time Mold Injection time 3 (C) temperature (A) (C) Packing Melt Packing 4 pressure (D) temperature (B) pressure (D) Melt Packing Melt 5 temperature pressure (D) temperature (B) (B)



From Table XVIII for Experiment - I, the most major factor that affects on warpage on thin plate part is packing time (E) and followed by mold temperature (A), injection time (C), packing pressure (D) and melt temperature (B). While for Experiment - II, the most significant factor that affects on warpage in thin plate parts is packing time (E) and followed by injection time (C), mold temperature (A), melt temperature (B) and packing pressure (D). Besides that, for Experiment - III, the most significant factor that affects on warpage in thin plate parts is packing time (E) and followed by mold temperature (A), injection time (C), packing pressure (D) and melt temperature (B). The significant factors affected warpage on thin plate parts for material PC Lexan 125 and PC/ABS Cycolac C2100 are same because PC/ABS is major PC compound with minor ABS. So, the characteristic of this material is most like PC material. Otherwise the packing time (E), mold temperature (A) and injection time (C) are the most significant factors affected warpage on thin plate parts for PC, ABS and PC/ABS. The deflection z-direction data in Tables XI, XII and XIII are also analyzed using Analysis of Variance (ANOVA) that computes the sums


From the S/N ratio response in Tables XIV, XV and XVI, the highest value from each factor is considered the best and chosen as the finest grouping of parameters. Table XVII shows the summary of best parameter settings for the thin plate parts based on Experiment - I, II and III. The results can also be seen from S/N response diagram shown in Fig. 5-9 for Experiment - I, 10-14 for Experiment - II and 15-19 for Experiment – III.



7


For Error, Se:

of squares, degrees of freedom, variance and percentage contribution. The examples of calculations are shown below and the results of ANOVA on warpage in thin plate parts are summarized in Table XVI, XVII and XVIII. For Experiment-I;Total degree of freedom, f;

Se = ST - ( S A + S B + SC + S D + S E ) = = 8.8 × 10-6 - 1.03 × 10-6 - 2.367 × 10-7 + - 10.42 ⋅10-7 - 4.71× 10-7 - 60.17 ( x )10-7 =

fT = N - 1 =

=0

= 16 - 1 = 15

The values of variances for all factors are then calculated. The example of calculating variance is shown below; For Factor A:

where N is the number of trial. For Factor A, fA: f A = K A -1 = = 4 -1 = 3

VA =

where kA is the number of level for factor A. For Error, fe:

SA 1.03 × 10-6 = = 3.44 × 10-7 3 fA

For Variance Error, Ve:

f e = fT - ( f A + f B + f C + f D + f E ) =

Ve =

= 15 - ( 3 + 3 + 3 + 3 + 3) = 0

Sum of squares for all factors is then calculated and the example of calculating sum of squares is shown below; Sum of squares, S: ST =

(

za12

ST = −

+ za 2 + ... + zaN 2

( 0.0074

2

2

)−

F-ratio, F for all factors is calculated afterwards and the example of calculation is shown below; For Factor A: v FA = A ve

( za1 + za 2 + ... + zaN ) 2 N

FA, FB, FC, FD and FE cannot be determined as Ve = 0 Last but not least, Percentage Contribution, for all factors is calculated and the example of the calculation is shown below; For Factor A:

)

+ 0.00612 + ... + 0.00722 +

( 0.0074 + 0.0061 + ... + 0.0072 ) 2 16

ST = 63.26 × 10-5 - 62.38 × 10-5 = 8.8 × 10-6

PA =

For Factor A, SA: SA =

( ∑ A1 ) kA

( ∑ A3 ) − ( za1 + za 2 + ... + zaN )2 + ... + k4

N

4 + +

+

( 0.0055 + 0.0058 + 0.0062 + 0.0073) 2 4

( 0.0058 + 0.0066 + 0.0060 + 0.0063) 2 4

( 0.0060 + 0.0059 + 0.0075 + 0.0072 ) 2 4

( 0.0074 + 0.0061 + 0.0054 +

…+ 0.0072 )

⎛ 1.03 × 10-6 ⎞ SA × 100% = ⎜⎜ -6 ⎟ ⎟ × 100 = 11.73% ST ⎝ 8.8 × 10 ⎠

The percentage of contribution for each factor in Experiment – I, II and III are listed at the last column in Tables XIX, XX and XXI. Whereas, the percentage of contribution for each factor for warpage in thin plate parts can be seen clearly in Table XXII.

2

( 0.0074 + 0.0061 + 0.0054 + 0.0049 ) 2

SA =

−

2

Se 0 = =0 0 fe

+

TABLE XIX ANOVA TABLE FOR PC LEXAN 125 IN EXPERIMENT-I

+ + +

2

Source

f

S

V

F

P(%)

A B C D E Pooled error Total

3 3 3 3 3 0 15

6.8 x 10-7 3.6 x 10-7 8.9 x 10-7 3.2 x 10-7 10.4 x 10-7 0.0 12.65 x 10-7

2.27 x 10-7 1.19 x 10-7 2.96 x 10-7 1.06 x 10-7 34.71 x 10-7 0

-

5.39 2.82 7.01 2.50 82.28 100

16

S A = 1.03 × 10-6



8


TABLE XX ANOVA TABLE FOR ABS CYCOLAC G500 IN EXPERIMENT-II Source A B C D E Pooled error Total

f

S

V -7

3 3 3 3 3 0 15

3.8 x 10 1.7 x 10-7 6.3 x 10-7 0.5 x 10-7 37.0 x 10-7 0.0 49.4 x 10-7

-7

1.28 x 10 0.58 x 10-7 2.09 x 10-7 0.18 x 10-7 12.34 x 10-7 0

F

P(%)

-

7.75 3.49 12.71 1.06 74.99

that affect warpage in thin plate by using pin point gate (three-plate mold) for PC, ABS and PC/ABS materials have been identified. The conclusions from this research are; i. Taguchi orthogonal array can efficiently reduce the number of trials during mold testing and effective factors can be determined using ANOVA. ii. For thin plate parts by using pin point gate in threeplate mold, results have shown that the packing time (E) are the most significant factor affected warpage for all material used in this study. That means the type of material is not significant factor on warpage at thin plate parts in three-plate mold. iii. The influence of all factors that contribute to warpage have been identified and believed this can help in determining more precise process conditions and appropriate parameters in injection molding process. There are several factors affected warpage on the molded part such as feed systems design, cooling channel size, cooling channel positions mold insert material and gate sizes that need to be determined first in order to design a plastic injection mold. From the results, it shows that by using ABS material, the warpage on the thin plate parts increases 27.27% as compared to PC material. This warpage will increase 39.39% when using PC/ABS material as compared to PC materials. Therefore from this experiment, it has shown that between these three types of materials, PC offers the minimum warpage value.

100

TABLE XXI ANOVA TABLE FOR PC/ABS CYCOLAC C2100 IN EXPERIMENT-III Source A B C D E Pooled error Total

f

S

V -7

3 3 3 3 3 0 15

10.3 x 10 2.4 x 10-7 10.4 x 10-7 4.7 x 10-7 60.2 x 10-7 0.0 88.0 x 10-7

-7

3.44 x 10 0.79 x 10-7 3.48 x 10-7 1.58 x 10-7 20.06 x 10-7 0

F

P(%)

-

11.73 2.69 11.84 5.36 68.38 100

From Table XXII, it can be observed that more than 90% of factors that contribute warpage on thin plate parts in three-plate mold for PC, ABS and PC/ABS are packing time (E) followed by injection time (C) and mold temperature (A). Experiment - I, packing time contributes the most which is 82.28% followed by injection time 7.01%, mold temperature 5.39%, melt temperature 2.82% and packing pressure 2.50%. In Experiment - II, the most contribution factors is packing time 74.99% followed by injection time 12.71%, mold temperature 7.75%, melt temperature 3.49% and packing pressure 1.06%. For Experiment - III, the most contribution factors is packing time 68.38% followed by injection time 11.84%, mold temperature 11.73%, packing pressure 5.36% and melt temperature 2.69%.

References [1] [2]

TABLE XXII SUMMARY OF PERCENTAGE CONTRIBUTION FOR WARPAGE IN THIN PLATE PARTS Experiment-I

Experiment-II

[3]

Experiment-III

Parameters % Parameters % Parameters % E 82.28 E 74.99 E 68.38 C 7.01 C 12.71 C 11.84 A 5.39 A 7.75 A 11.73 B 2.82 B 3.49 D 5.36 D 2.50 D 1.06 B 2.69 Remarks: A- Mold temperature, B-Melt temperature, C-Injection time, D-Packing pressure, E-Packing time.

[4]

[5]

[6]

V.

Conclusion [7]

In previous study, mold temperature and melt temperature are the most significant factors affected warpage in ultra-thin shell parts by using pin point gate in three-plate mold [10]. Nasir et al. [11] reported that packing time is the most significant factor affected warpage on thin plate for PC, ABS and PC/ABS materials by using side gate in two-plate mold. In this research, the best parameters and most significant factors

[8]


Daniel, J. H., Micromachining Silicon for MEMS, MTP, Cambridge CB2 1 TJ, UK, 1999. G. A. Ibrahim, C. H. Che Haron, J. A. Ghani, H. Arshad, Taguchi Optimization Method for Surface Roughness and Material Removal Rate in Turning of Ti-6Al-4V ELI, International Review of Mechanical Engineering (IREME), Vol. 4. n. 3, pp. 216-221, 2011. Huang MC, Tai CC., The effective factors in the warpage problem of an injection molded part with a thin shell feature. Journal of Material Processing Technology 110, 1-9, 2001. S.H. Tang, Y.J. Tan, S.M.Sapuan, S.Sulaiman, N. Ismail, R. Samin, The use of Taguchi method in the design of plastic injection mould for reducing warpage, Journal of Material Processing Technology 182, 418-426, 2007. N.A.Shuaib, M.F. Ghazali, Z. Shayfull, M.Z.M. Zain, S.M. Nasir, Warpage Factors Effectiveness of a Thin Shallow InjectionMolded Part using Taguchi Method, International Journal of Engineering & Technology, Vol: 01, ISSN: 2077-1185, pp.182187, 2011. Liao SJ, Chang DY, Chen HJ, Tsou LS, Ho JR, Yau HT, et al. Optimal process conditions of shrinkage and warpage of thin-wall parts. Polym Eng Sci, 44(5):917–28, 2004. Z. Shayfull, M.F. Ghazali, M. Azaman, S.M. Nasir, N.A. Faris, Effect of Differences Core and Cavity Temperature on Injection Molded Part and Reducing the Warpage by Taguchi Method, International Journal of Engineering & Technology, Vol: 10, pp. 133-140, 2010. M.C. Song, Z. Liu, M.J. Wang, T.M. Yu, D.Y. Zhao, Research on effects of injection process parameters on the molding process for ultra-thin wall plastic parts Journal of Materials Processing Technology 187–188, 668–671, 2007.


9


[9]

B. Ozcelik, T. Erzurumlu, (2006) Comparison Of The Warpage Optimization In The Plastic Injection Molding Using ANOVA, Neural Network Model And Genetic Algorithm, Journal Of Material Processing Technology, 171, 437-445. [10] Z. Shayfull, M. Fathullah, S. Sharif, S.M. Nasir, N. A. Shuaib, Warpage Analysis on Ultra-Thin Shell by Using Taguchi Method and Analysis of Variance (ANOVA) for Three-Plate Mold, International Review of Mechanical Engineering (IREME), September 2011, Vol. 5 N. 6, pp. 1116-1124. [11] S.M. Nasir, N. A. Shuaib, Z. Shayfull, M. Fathullah, R. Hamidon, Warpage Analyses on Thin Plate by Taguchi Method and Analysis of Variance (ANOVA) for PC, PC/ABS and ABS Materials, International Review of Mechanical Engineering (IREME), September 2011, Vol. 5 N. 6, pp. 1125-1131.

Authors’ information 1

School of Manufacturing Engineering, Universiti Malaysia Perlis, Malaysia Z. Shayfull was born in Kuala Perlis, Perlis, Malaysia on 1977. He graduated from Universiti Teknologi Malaysia (UTM), Malaysia in Bachelor Degree of Mechanical Engineering in 2000 and completed his Masters Degree in Advanced Manufacturing and Technology in the year 2006 which also from the same university. His field of research is in product design engineering and injection molding processes and currently extending his knowledge by undergoing PhD studies on Injection Molding Processes in UTM. Mr Shayfull also actively involves with Engineering Professional Bodies in Malaysia such as Board of Engineers, Malaysia (BEM) and Institution of Engineers, Malaysia (IEM) until now.



10


The Influence of Different Mold Temperature on Warpage in a Thin Shallow Injection Molding Process N. A. Shuaib, S. M. Nasir, M. Fathullah, Z. Shayfull, M. S. Abdul Manan

Abstract – Every injection molding parameter have its own influence towards the quality of an injection-molded part. Temperature, pressure and time are main parameters that typically highlighted in controlling the warpage defect of the part. This study is performed to investigate the influence of mold surface temperature, core and cavity temperature on warpage defect of a thinshallow part. . The warpage results are obtained using Taguchi Method and optimized using Analysis of Variance (ANOVA) simulated in two experiments using Moldflow software. Confirmation run of best setting for experiment 2 which considers core temperature and cavity temperature resulted to be the more effective than considering only mold surface temperature in experiment 1. It is concluded that by considering core and cavity as the mold temperature, the warpage defect can be minimized up to 79.9%. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Thin Shallow, Mold Temperature, Injection Molding, Warpage, Taguchi Method

I.

The mold temperature has been identified by B. Ozcelik et al. [5] as one of the influencing of injection parameters that could lead to warpage of injection molded plastics. In this research, a simulation is made on the thin shallow plate with thickness of 0.5mm. Regarding the injection molding of a thin shallow part, N.A. Shuaib et al. [6] performed a simulation of the design of experiment using Taguchi method which can effectively allocate the number of trials for analysis testing. The best parameter setting obtained is 80°C for mold temperature, 270°C for melt temperature, 0.3s for filling time, 90MPa of packing pressure and 1.05 for packing time. Experimental design of Taguchi method will be used by employing a special design of orthogonal arrays to learn the whole parameters space with a small number of experiments only [7].

Introduction

Ease of manufacture with high quality, less defect and cost-effective products makes injection molding to be one of the most accepted processes in plastic making [1]. However, from the manufacturing side, warpage issues are the most important to look into. Previous researchers had been concentrating on the warpage factors and various approaches are taken to reduce it [2]. Common defects of injection molded parts such as warpage, shrinkage, weld line and sink marks normally occur due to unfavourable process conditions which give great impacts on the quality of the parts to be produced. S.H Tang [4] highlighted that bigger demand thin injection molded plastics parts has trigger the appearance of warpage on parts as the part thickness which been reduced resulting of more complex process of injection molding. The optimum parameter values is determined using the Taguchi method this led to a finding that packing pressure is the most significant factor that affects warpage. Taguchi method was implemeted by Z. Shayfull et al. [3] and in their research on determining the most significant factor that contributes warpage on the thin plate and thin shell plate. The result indicates that cavity temperature has become the most significant factor for thin plate injection-molded part. S.H Tang [4] and H. Oktem[7] used to employ the Taguchi and ANOVA to identify injection molding parameters which contributing to warpage defect. S. H. Tang [4] figured out that the melt temperature is determined as the most significant factor on producing warpage. Using similar experimental method for a thin wall part, Liao et al. [8] concluded that packing pressure has big influence on the warpage development.

II.

Mold and Part Design

Gate and Cooling Channel Design The mold design stage emphasizes several important features such as the type of mold, the mold dimension, material selected for the cavity, core insert and the base which needs precise selection. In this research, a thin shallow part design is shown in Figure 1 and the decided gate type is the fan gate with a two-plate mold. Gating system design with dimension and the cooling channel design is shown in Figure 2 and 3 respectively.

III. Method of Experiment Taguchi Method S.H. Tang[4] figured out that for the two plate mold


11


N. A. Shuaib, S. M. Nasir, M. Fathullah, Z. Shayfull, M. S. Abdul Manan

the warpage for having most quality injection-molded part. Therefore, this research is carried out based on the following methodology:

analysis, the best parameters search for the experiment can be identified using Taguchi method.

i. Parameters Selection and Levels Setup ii. Material Selection iii. Assumptions iv. Simulation Parameters Selection and Levels Setup Previous researchers [9]-[11] assumed that the mold temperature can be ignored due to an assumption of the presence of ambient. However in this research, mold temperature is taken into consideration into order to compare it with the core and cavity temperature. There are two types of experiments have been carried out in this research, Experiment 1 (E1) and Experiment 2 (E2). In the E1, there are four parameters been chosen as factors as shown in Table I. They are mold temperature, melt temperature, packing pressure and packing time. The cavity and core temperature are both considered as the mold temperature. In E2, all the parameters are the same as in E1 except the mold temperature which been divided into two different factors; the cavity and core temperature as in Table II. For E1, Taguchi method is applied to arrange injection molding process parameters setup based on three level design of experiments and orthogonal array L9(34) is developed. The same method also applied to run E2 with orthogonal array of L16 (45) as shown in Table III and Table IV.

Fig. 1. Thin Shallow Part

TABLE I EXPERIMENT 1(E1) PARAMETERS AND LEVELS SETUP Level Factor 1 2 60 70 Mold temperature, A (°C) 260 275 Melt temperature, B (°C) 70 80 Packing pressure, C (MPa) 0.7 0.85 Packing time, D (s)

3 80 290 90 1.0

TABLE II EXPERIMENT 2(E2) PARAMETERS AND LEVELS SETUP Level Factor 1 2 3 65 70 75 Cavity temperature, A (°C) 65 70 75 Core temperature, B (°C) 260 270 280 Melt temperature, C (°C) 75 80 85 Packing pressure, D (MPa) 0.7 0.8 0.9 Packing time, E (s)

4 80 80 290 90 1.0

Fig. 2. Gate Design

TABLE III CONTROL FACTORS AND LEVELS FOR EXPERIMENT 1 Trial No. 1 2 3 4 5 6 7 8 9

Fig. 3. Cooling Channel Design

Thus, in order to follow the same methos, determination of factors that can influence on warpage, appropriate factor levels as well as the orthogonal array are based on Taguchi method is essential in determining the most constructive selection of parameters to reduce


A 60 70 80 60 70 80 60 70 80

Control Factor B C 260 70 260 80 260 90 275 80 275 90 275 70 290 90 290 70 290 80

D 0.7 0.85 1.0 1.0 0.7 0.85 0.85 1.0 0.7


12


TABLE IV CONTROL FACTORS AND LEVELS FOR EXPERIMENT 2 Trial No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

IV.

In order to measure the output quality and characteristics from desired value, Taguchi has utilized the Signal-to-Noise ratio; S/N. S/N ratio also used to classify the results and evaluates them to determine the optimum parameters. There are three S/N ratio’s characteristics; the nominal the better, the smaller the better and the higher the better [11]. Since this research is carried to reduce warpage, the smaller the better characteristic has been chosen and it is expressed as:

Control Factor A 65 65 65 65 70 70 70 70 75 75 75 75 80 80 80 80

B 65 70 75 80 65 70 75 80 65 70 75 80 65 70 75 80

C 260 270 280 290 270 260 290 280 280 290 260 270 290 280 270 260

D 75 80 85 90 85 90 75 80 90 85 80 75 80 75 90 85

E 0.7 0.8 0.9 1.0 1.0 0.9 0.8 0.7 0.8 0.7 1.0 0.9 0.9 1.0 0.7 0.8

where

Trial No 1 2 3 4 5 6 7 8 9

TABLE V MATERIAL PROPERTIES OF PC/ABS 1871

Glass transition temperature, Tg (oC) Thermal expansion coefficient, α (mm/moC) Elastic modulus, E (MPa)

112

2.63 x 103

Poisson's ratio, υ

0.23 o


⁄

10

∑

is the mean square deviation,

y represents the observation and is the number of tests in one trial [3]. Tables VI and VII shows the Z-deflection and S/N ratio for the thin shallow plate for both E1 and E2.

Material Selection The selected material is PC/ABS blend, Cycoloy C2950HF from GE. The range of its melt temperature is between 220 and 400oC approximately [6]. Other properties of PC/ABS are shown in Table V.

Specific heat, Cp (J/kgoC)


74

0.27 Trial No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Assumptions The selection of the factors is based on the research scope and the desired findings. In this research, following assumptions are employed before performing the analysis, leading to the determination of the factors [3]: i. This research focus on the effect of having the mold, core and cavity temperature, Thus the ambient temperature is assumed to be constant.. ii. The coolant which flowing through the cooling channel during process is pure water. iii. Only the effects of temperature, time and pressure in the filling and packing phase will be discussed. iv. For E1, the cooling channels are assumed to maintain at a constant temperature in the mold. This assumed as the mold temperature. v. For E2, the cooling channels are assumed to vary in the mold. They are the core and cavity temperature.

TABLE VI SUMMARY OF RESULTS – Z DEFLECTIONS AND S/N RATIO FOR E1 Control Factors A B C D Max, z 60 260 70 0.7 0.2108 70 260 80 0.85 0.1371 80 260 90 1.0 0.1057 60 275 80 1.0 0.1182 70 275 90 0.7 0.1655 80 275 70 0.85 0.1497 60 290 90 0.85 0.1243 70 290 70 1.0 0.0977 80 290 80 0.7 0.2071 TABLE VII SUMMARY OF RESULTS – Z DEFLECTIONS AND S/N RATIO FOR E2 Control Factor A B C D E Max, z 65 65 260 75 0.7 0.0562 65 70 270 80 0.8 0.0533 65 75 280 85 0.9 0.0545 65 80 290 90 1.0 0.0478 70 65 270 85 1.0 0.0545 70 70 260 90 0.9 0.0546 70 75 290 75 0.8 0.0517 70 80 280 80 0.7 0.0454 75 65 280 90 0.8 0.0561 75 70 290 85 0.7 0.0452 75 75 260 80 1.0 0.0536 75 80 270 75 0.9 0.0488 80 65 290 80 0.9 0.0458 80 70 280 75 1.0 0.0530 80 75 270 90 0.7 0.0438 80 80 260 85 0.8 0.0590

S/N 13.5226 17.2593 19.5185 18.5477 15.6240 16.4956 18.1106 20.2021 13.6764

S/N 25.0053 25.4655 25.2721 26.4114 25.2721 25.2561 25.7302 26.8589 25.0207 26.8972 25.4167 26.2316 26.7827 25.5145 27.1705 24.5830

The maximum z-deflections for both experiments are shown in Table II, attained by the simulation process represent the warpage values and they are to be verified using Analysis of Variance (ANOVA) where level of confidence is set at 0.05. The results are then compared with the results obtained from the S/N ratio. The influence or significant of factors towards the quality of a thin shallow injection molded part can be obtained from the level difference.

Simulation Software The thin shallow injection molding simulation analysis is run using Autodesk Moldflow Plastic Insight (MPI). Regarding the meshing design, the thin shallow parts are divided into 18638 surface triangle elements.



13


Higher significant factor can be observed by the higher difference value of S/N ratio in Table VIII for E1 and Table IX for E2. Calculation example of level difference for factor A in E1: 13.5226

1

17.2593

2

19.5185

3

∆

18.5477 3

18.1106

15.6240 3

20.2021

16.4956 3

13.6764

, 15.5635

17.6951

: 3

16.7269

1

3

1

2

2

2

2

Error:

17.6951

8

2

2

0

b) Sum of squares, S Sum of S, :

16.5635

1.1316

TABLE VIII THE RESPONSE TABLE OF S/N RATIO FOR E1 Control Factor Level A B C D 16.7269 16.7668 16.7401 14.2743 1 17.6951 16.8891 16.4944 16.5349 2 16.5635 17.3297 17.7510 19.4228 3 1.1316 0.5629 1.2566 5.1484 Difference ∆

0.1371 0.2071 0.1371 0.2071 9 0.2061 0.1925 =0.0136

0.2108 0.2108

∑ TABLE IX THE RESPONSE TABLE OF S/N RATIO FOR E2 Control Factor Level A B C D 25.5386 25.5202 25.0653 25.6204 1 25.7793 25.7833 26.0349 26.1309 2 25.8916 25.8974 25.6665 25.5061 3 26.0127 26.0212 26.3176 25.9647 4 0.4741 0.5010 1.2523 0.6249 Difference ∆

E 26.4830 25.1998 25.8856 25.6537 1.2831

∑

0.2108

0.1182 0.1243 3 0.1057 0.1497 0.2071 3 0.2108 0.1371 0.2071 16 0.1932 0.1945= 7.5 10

TABLE X BEST COMBINATION FACTORS SETTINGS Experiment 1(E1) Experiment 2(E2) Factors Values Factors Values A 80°C A 65°C B 260°C B 65°C C 80MPa C 260°C D 0.7s D 85MPa E 0.8s

Error,

Analysis of Variance (ANOVA) Analysis of variance (ANOVA) is performed to evaluate the significant process factor which considered affecting the product quality. In this research, the percentage contribution of variance or factors can be calculated by determining quantities such the degree of freedom, f sum of squares, S variance, V and the percentage contribution for each factor. Sample calculation for determining percentage of contribution for factor A in E1 is shown as follows and all the results are documented in Table XI.

: = 0.0136 9.3

7.5 10

10 1.18

1.14 10

10 0

c) Variance, V

7.5 Error:

10 2

3.75 0 0

10 0

a) Total degree of freedom, f: d) F-ratio, F

1 9

1

0

8



14


and

cannot be determined as

0

mold temperature side and 85.82% for E2 which considers core and cavity temperature. These are followed by packing pressure which contributes 6.88% and 6.80% respectively. In E1, the percentage contribution of mold temperature is 5.52% while in E2, the cumulative value for core and cavity temperature contribution is 6.05%. The difference of 0.53% contribution is expected to give different results of the warpage, Z-deflection.

e) Percentage Contribution,P 7.5 10 1.36 10

100

100

5.52%

Based on ANOVA results in Table XI, both results of E1 and E2 mold are most notably influenced by packing time which contribute 86.76% for E1 which considers

TABLE XI ANOVA RESULTS Factor A, Mold temperature B, Melt temperature C, Packing pressure D, Packing time,

Experiment 1 (E1) S V

f

F

P(%)

2

0.00075

0.000376

-

5.52

2

0.00011

0.000057

-

0.84

2

0.00094

0.000468

-

6.88

2

0.01180

0.005902

-

86.76

-

-

-

-

-

-

Pooled error

0

0

Total

8

0.0136

100

Experiment 2 (E2) S V 6.71 0.0002 10 2.33 0.0007 10 6.58 0.0002 10 3.37 0.0010 10 4.25 0.0128 10

Factor A, Cavity temperature B, Core temperature C, Melt temperature D, Packing pressure E, Packing time, Pooled error

F

0

0.0000

Total

12

0.01487

3 3 3 3 3

F

P(%)

-

1.35

-

4.70

-

1.33

-

6.80

-

85.82

0

100

TABLE XII CONFIRMATION RUN TEST OF THE BEST COMBINATION FACTORS SETTINGS Experiment 1(E1)

Experiment 2(E2)

Factors

Values

Factors

Values

A (Mold Temperature) B (Melt Temperature)

80°C 260°C

A (Core Temperature) B ( Cavity Temperature)

65°C 65°C

C (Packing Pressure)

80MPa

C (Melt Temperature)

260°C

D (Packing Time)

0.7s

D (Packing Pressure) E (Packing Time)

85MPa 0.8s

Warpage, Z-deflections = 0.636 mm

Warpage, Z-deflections = 0.129 mm

Table X figured the best combination parameters to be used for both E1 and E2. In order to verify the effect of this different value on the thin shallow part warpage, a confirmation run test has been made and the result is shown in Table XII. Z-deflections for E1 which only considers the mold temperature is 0.636mm while the result of E2 which considers both core and cavity temperature as a mold temperature is 0.129mm. Hence, it can be said that, by considering the core and cavity temperature as one of the factors, they have affect the result of the thin shallow injection molded part. Therefore, based on this result and as far as the scale of warpage is acceptable, it is suggested for the injection molding process operators to consider the core and cavity temperature in MPI simulations as it gives 79.7% reduction of Z-deflection.

in simulation has given effect of warpage of a thin shallow injection molded part. By comparing with an experiment without this two factors, the difference of warpage in Z-deflection is about 79.9% where the gap value is about 0.51mm. For a small injection-molded thin part, this tolerance is assumed as a big number. Therefore, for reducing warpage, Experiment 2(E2) can be concluded as the better experiment with better settings compared with Experiment 1(E1) which only considers the mold temperature only. The best combination parameters in reducing warpage issue have been identified based on Taguchi Method verified by ANOVA analysis. In E2, The most significant factor is packing time followed by packing pressure with their influence of contributions are more than 70%.

References V.

Conclusion

[1]

From this research, it can be concluded that consideration on putting the core and cavity temperature Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

Huang MC, Tai CC. The effective factors in the warpage problem of an injection molded part with a thin shell feature. Journal of Material Processing Technology 110 (2001) 1-9.


15


[2]

Mirigul Altan, Reducing Shrinkage In Injection Moldings Via The Taguchi, ANOVA And Neural Network Methods, Materials & Design, Volume 31, Issue 1, January 2010, Pages 599-604. [3] Z. Shayfull, N.A. Shuaib, M.F. Ghazali, S.M. Nasir, M. Azaman, N.A. Faris, , Optimizing Length of Weld Line Formation in Thin Plate by Taguchi Method and Analysis of Variance (ANOVA), International Journals of Engineering & Technology, Vol: 11, Issue: 01, 2011,ISSN: 2077-1185 pp.132-137. [4] S.H. Tang, Y.J. Tan, S.M. Sapuan, S. Sulaiman, N. Ismail, R. Samin, The Use Of Taguchi Method In The Design Of Plastic Injection Mould for Reducing Warpage, Journal Of Materials Processing Technology 182 (2007) 418–426. [5] Babur Ozcelik, Alper Ozbay and Erhan Demirbas, Influence of injection parameters and mold materials on mechanical properties of ABS in plastic injection molding, International Communications in Heat and Mass Transfer 37 (2010) 1359– 1365. [6] N.A.Shuaib, M.F. Ghazali, Z. Shayfull, S.M. Nasir, Warpage Factors Effectiveness of a Thin Shallow Injection-Molded Part using Taguchi Method, International Journals of Engineering & Technology, Vol: 11 Issue: 01, 2011 ISSN: 2077-1185 pp.182-187 [7] H. Oktem et al, Application of Taguchi Optimization technique in determining plastic injection molding process parameters for a thin-shell part, Materials and Design 28(2007) 1271-1278. [8] Liao SJ, Chang DY, Chen HJ, Tsou LS, Ho JR, Yau HT, et al. Optimal process conditions of shrinkage and warpage of thin-wall parts. Polym Eng Sci 2004;44(5):917–28. [9] M.C. Song, Z. Liu, M.J. Wang, T.M. Yu, D.Y. Zhao, Research on effects of injection process parameters on the molding process for ultra-thin wall plastic parts, Journal of Materials Processing Technology 187–188 (2007) 668–671 [10] T. Matsuoka, J. Takabatake, A. Koiwai, Y. Inoue, S. Yamamoto, H.Takahashi, Integrated simulation to predict warpage of injection molded parts, Polym. Eng. Sci. 31 (1991) 1043.S.H. [11] M.F. Ghazali, Z. Shayfull, ,N. A. Shuaib, S.M. Nasir,, M. Mat Salleh, Injection Mould Analysis in Reducing Warpage of Nylon PA66 Side Arms Using Taguchi Method and ANOVA, International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: 11 No: 01 pp. 87-92.

Author’s Information School of Manufacturing Engineering, Universiti Malaysia Perlis, Malaysia N. A. Shuaib is currently working as a lecturer in Universiti Malaysia Perlis. His research scopes are in mechanical design, manufacturing process and heat transfer. He received the Bachelor of Mechanical Engineering(Hons) in 2007 from Universiti Tenaga Nasional(UNITEN), Malaysia and continued with MSc in Manufacturing Systems Engineering from Universiti Putra Malaysia, Malaysia in 2009. He is a member of a research group under School of Manufacturing Engineering in UniMAP that is actively doing research specifically in design of experiment applied to mechanical design and manufacturing process. Upon professional contribution, N.A.Shuaib involves himself with engineering professional bodies in Malaysia such as Board of Engineers, Malaysia (BEM), Institution of Engineers, Malaysia (IEM) and Malaysian Institute of Engineering & Technology (mSET).



16


Drag Force Reduction Technique for Abrasive Resisting Materials S. V. Gavali1, V. B. Tungikar2

Abstract – In this paper Computer program is developed for estimation of the effect of centrifugal force on the formation of graded distribution of right circular cone particles within the molten aluminum metal. The volume fraction of the heavier Titanium Di-boride particles is controlled by inertial forces upon centrifugal force processing the semisolid composite. Titanium Di-boride particles are modeled as right circular cone particles subject to a drag force in a Stoke flow in the liquid aluminum matrix. The equation of motion for the particles under the applied centrifugal forces is solved mathematically assuming a Gaussian diameter size distribution with a spatial uniform random distribution of particles in the sample. The effect of average particles diameter is also important .Larger the diameter of the particles faster is their motion, however the corresponding drag force increases. Hence the aerodynamic shape (right circular cone) is chosen to reduce the drag force. It is possible to understand and control the experimental conditions to obtain an appropriate functionally-graded aluminum matrix by centrifugal casting for high wear resistance applications. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Centrifugal Casting, Functionally Graded Aluminum Matrix, Titanium Di-Borides

performance requirements that vary with location within a part. Among various FGM, metal matrix composite FGM are of great practical interest [1]. As per [2] Centrifugal casting is a special type of casting process in which the melt fills into the mould and solidifies under a centrifugal force field. This process is used to make thin walled castings like pipes, piston rings, cylinder liners, rollers, pulleys etc. The casting parameters that influence solidification structures can include the mould rotation velocity, the mould dimension, the mould preheating temperature, the pouring temperature of the molten metal and the alloy composition etc. [3]. The extent of segregation depends on various process parameters. The moving direction of the solid particles in the molten matrix is determined by the density differences between the molten metal and reinforcement particles. Because of the higher density of the reinforcement particles compared to the density of the molten matrix, particles move in the radial direction [4] In this work the molten matrix, with the reinforcement particles, (TiB2) is put into a cylindrical mould and then exposed to a centrifugal action with its central axis along the radial direction. In some past works a hollow cylinder rotating vertically around the central axis was considered, in which effect of gravity was considered due configuration [5], [6]. In this work the effect of gravity in the movement of particles is neglected because the radial acceleration is much higher than the gravity acceleration. The drag force increases with spherical or cylindrical particles [7]. Hence an aerodynamic shape, right circular cone, is chosen to reduce the drag force. A mathematical model for the motion of the particles is

Nomenclature r r* t

τ θ ω ρ ρP µ µ0

H cp T Ti, Ta A H V ro rL N A mp

Radial position of particle Dimensionless radial coordinate Time Dimensionless time Angular velocity of the particle Angular velocity of the sample Density of the aluminum matrix Density of particles Dynamic viscosity Reference dynamic viscosity Heat transfer coefficient Specific heat Temperature Initial temperature of the sample Ambient temperature Area of particle Height of the cone Volume of particle Distance of cylinder side to the axis of rotation Distance of cylinder side further away from the axis of rotation Number of particles Radius of a particle Mass of the particle

I.

Introduction

Functionally graded materials (FGM) are characterized by continuous, smooth variations in composition and microstructure so as to meet functional Manuscript received and revised December 2011, accepted January 2012

17


S. V. Gavali, V. B. Tungikar

particle has right circular cone profile (Keeping height h constant): d 2 r −18µ dr = ⋅ + rθ 2 2 ha dt ρ dt p

formulated The forced segregation of hard titanium Diborides particles towards outer regions of the casting provided by this method will be used as unique approach to improve surface hardness and wear resistance of the composites. Experimentation is carried out to validate the mathematical model.

Defining the dimensionless parameters as:

II.

Mathematical Model

τ=

A schematic of the centrifugal casting sample is shown in Fig. 1. Consider right circular cone shaped solid particles in an aluminum matrix. Particles are assumed uniformly distributed spatially in the liquid aluminum matrix. The sample is rotated at a constant centrifugal speed ω. Titanium Di-boride particles have higher density than the density of the liquid aluminum matrix.

µ (T ) µo t r , r' = , and µ' = ρa ro µo

and making a change of variable, the dimensionless equation is obtained: d 2 r' dτ 2

=−

18 ρ dr' ⎛ dθ ⎞ ⋅ ⋅ µ' ⋅ + r' ⎜ ⎟ h ρp dτ ⎝ dτ ⎠

2

(4)

The dimensionless forms of the initial conditions becomes, at: dr' τ = 0 r' =1 ; =0 (5) dτ b. Circumferential direction: In the same way, the dimensionless form of Eq. (4) in the circumferential direction can be obtained as follows:

Fig. 1. Centrifugal casting Systematic diagram

The equation of motion for the particles in a rotational stoke flow is: [1]

(

)

(

) )

m p ⎡ r − r θ 2 er + r θ + 2r θ eθ ⎤ = ⎣ ⎦ ⎡ = −6 π µ a ⎣ r er + r θ − ω eθ ⎤⎦

(

r'

(

)

m p r θ = −6 π µ a r θ - ω - 2 m p r θ

dx 2

=−

⎛ dθ ρ ω a 2 ⎞ 18 ρ d r' dθ µ ' r' ⎜⎜ (6) − ⎟⎟ − 2 µ0 ⎠ h ρp d τ dτ ⎝ dτ

with the following dimensionless form of the initial conditions; at:

(1)

τ =0 ; θ=0;

Thus, the two equations of motion in radial and circumferential direction are given by, Eq. (2) and Eq. (3): m p r = −6 π µ a r + m p r θ 2

d 2θ

From Eq. (4) and Eq. (6), three dimensionless parameters are identified:

(2)

ρ' = (3)

ρ ρω a 2 , Rea = ρp µ0

µ' = Subject to the initial conditions as: at t = 0 , r = r0 ; r = 0 ; θ = 0 ; θ = ω II.1.

µ (T ) µ ( t ) = µ0 µ0

The last parameter takes into account of the dependency of temperature with the viscosity. Due high thermal conductivity of aluminum and the small convection coefficient, a small Biot number can be assumed and the sample is treated as a lumped capacitance system [9]. For a lumped system, the temperature of the sample will be only function of the time and can be calculated by knowing the thermal time constant, initial temperature & ambient temperature. By knowing the variation of the viscosity with the

Dimensionless Form of the Equations of Motion

a. Radial direction: m p = ρ pV = ρ p ×

dθ ρω a 2 = dτ µ0

π a2 h 3

where mp is the mass of the particle and assuming Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


18


temperature, the viscosity as the a function of time can be evaluated:

where, b =

µ = f ⎡⎣T ( t ) ⎤⎦ = f1 ( t )

18 ρ and h ρp

c h ρ pa ω = b 18 µ0 2

substituting the

values of parameters in analytical solution, the radial position of particles are obtained. This gives the idea of the effect of various parameters on casting and particle distribution.

III. Analytical Solution Parametric study is carried out to understand the

ρ . ρp

importance of the dimensionless parameter Rea and

IV.

The photograph shown in Fig. 2 shows the arrangements used for obtaining the sample castings. The speed of casting is varied using step pulley drive. Special cantilever type die casting mould shown in Fig. 3 is used to obtain samples. Table I shows the experimental investigation.

Also, the effect of temperature for viscosity is analyzed. Neglecting the differences between θ and ω i ⋅ e θ = ω the equation of motion Eq. (4)

(

)

becomes: d 2 ri dτ 2

here,

Experimentation

2

=−

18 ρ dri ⎧⎪ ρω ai 2 ⎫⎪ ⋅ ⋅ +⎨ ⎬ ri h ρ p dτ ⎩⎪ µo ⎭⎪

dθ ρ ai 2ω = = constant, ai is the radius of the dτ µ0

particle i and ri is the it’s radial position. The initial conditions are at:

τ =0; r = 1 +

rL − r0 dr ⋅ rand ( N ,1) ; =0 r0 dτ

where ‘N’ is the number of particles. ‘N’ equations are solved, one for each particle. Since the flow is laminar, each particle follows its own path line without colliding with other particles. Under these assumptions, following second order linear differential equation is solved:

Fig. 2. Experimental set up TABLE I TYPICAL EXPERIMENTAL VALUES Sr. no.

2

18 ρ dri ⎛ ρ a 2ω ⎞ + −⎜ ⎟ ri = 0 dτ 2 h ρ0 dτ ⎜⎝ µ0 ⎟⎠

d 2 ri

1 2

2

⎛ ρ a 2ω ⎞ 18 ρ here, a = 1, b = , c 2 = ⎜⎜ ⎟⎟ h ρp ⎝ µ0 ⎠ The roots of the characteristic equation are α1 ,α 2 and the general solution is: r (τ ) = A1e

α1τ

α 2τ

+ A2 e

r (τ ) rinit

⎛ 1+ 1+ 4 c / b 2 ( ) =⎜ ⎜⎜ 2 ⎝ 2 1+ 4 (c / b) ⎛ 1- 1+ 4 c / b 2 ( ) −⎜ ⎜⎜ 2 ⎝ 2 1+ 4 (c / b)

⎞ ⎡⎣ -b2 ⎟ e ⎟⎟ ⎠

⎞ ⎣⎡ -b2 ⎟e ⎟⎟ ⎠

(1+

(1-

)

1+ 4 (c / b )

2

⎤ ⎦

20.10cm

Rotational speed (rpm)

500 rpm

Average diameter of the particles(d) Time (t) Density of the liquid of Al melt ρ

7

Melting point of Al.

10.0 mµ 50 sec 2.375g/cm3 660.32 °C, 1220.58 °F

Thermal conductivity of Al (300K) Density of the particle ρ p (TiB2) Melting Point of TiB2 (°C)

V.

+

32.50cm

4 5 6

9 10

1 + 4(c / b ) 2 τ

Distance from axis of rotation to outer side of die ( rL ) Distance from axis of rotation to inner side of die ( ro )

3

8

The analytical solution is obtained as:

Typical Values

Experimental parameters

237 W/mk 4.52 g/cm3 2970 oC

Results and Discussion

The mould is rotated at a constant centrifugal speed ω in a special purpose die. Results for different values of these parameters are presented and discussed here. Fig. 4. shows the particles distribution after 50 sec. for a rotational speed of ω = 500 rpm .The viscosity is

) τ ⎦⎤



19


As it is expected larger particles move faster, most of the larger particles concentrate at the outer region of the sample and a particle concentration decreases from outer to inner region. A computer program in C++ as shown below is developed used to validate the mathematical model. Experimental data reveals the validation of mathematical model.

assumed constant and the analytical solution is used to calculate the results. It is seen that the inner region of the sample have less particles, which means that even the smaller particles are moved due to centrifugal effect. Fig. 5 shows variation of effect of diameter keeping height constant

PROGRAM CODE

#include #include #include #include #include void main() { float ri,r_tau,rL,r0,a,b,c,d,e1,e2,tau,mu0,omega,rho,rho_p; int f, N, t, k; printf("\nEnter The Value Of r "); scanf("%d",&r_tau); printf("\nEnter The Value Of t"); scanf("%d",&t); a=0.00000375; rL=0.32 ; r0=0.195; rho=2400; rho_p=3190; mu0=0.00138; omega = 250*2*3.142/60; b=((9/2)*(rho/rho_p)); c=((2/9)*(rho_p*a*a*omega/mu0)); d=sqrt(1+(4*c*c)); tau=(mu0*t)/(rho*a*a); e1=exp(((-b/2)*(1-d)*tau)); e2=exp(((-b/2)*(1+d)*tau)); /* ri=(1+((rL-r0)/r0)*f); */ k=((r_tau/(((((1+d)/(2*d))*e1)-(((1-d)/(2*d))*e2))))1)*(r0/(rL-r0)); f = rand() %k + 1; printf("\nThe Value Of N is "); cout 0) have a tendency to reduce a thermal boundary layer thickness and wall injection (Fw < 0) have a tendency to increases a Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


25

P. Loganathan, N. Golden Stepha

It may be noticed that at first the microrotation increases monotonically with time and attains its maximum then it starts decreasing and reaches the asymptotic solution. It is interesting to notice that if Fw is less, then it takes more time to reaches its maximum. However, the exact opposite behavior takes place for increasing value for FW. It is observed that for FW = 0.0, the angular velocity profile reaches the maximum at η = 1.5 and for FW = 0.7, the angular velocity profile reaches the maximum at η = 0.8. Fig. 8 depicts the effects of Pr on the heat transfer for various values of suction parameter Fw. It is observed that heat transfer increase quite rapidly with increasing Pr. The effects of radiation parameter N on the heat transfer for various values of suction or injection parameter Fw are plotted in Fig. 9. It is observed that heat transfer increases for increasing value of N.

observed that the increasing value of Sc the concentration profile decreases. Increasing Sc means that the species diffusion reduces and viscous force increases which cause a reduction in concentration as expected. Angular velocity profiles for various Fw are presented in the Fig. 7.

Fig. 5. Concentration profile for different values of suction and injection

Fig. 8. Heat transfer parameter for different values of Pr

Fig. 6. Concentration profile for different values of Sc

Fig. 9. Heat transfer parameter for different values of N

The rate of mass transfer for different values of Sc (1.0, 2.0) is shown in Fig. 10. It shows that the mass transfer increases rapidly with increasing Sc. In Fig. 11, effects of rate of mass transfer for different N (5.0, 10.0) shown graphically.

Fig. 7. Micro rotation profile for different values of FW



26


It is also observed that mass transfer increases for increasing value of N.

Sc. (iv) The heat transfer parameter increases as the increasing value of Pr. (v) The heat transfer parameter increases as the radiation parameter N is increased. (vi) The mass transfer parameter increases as the value of Sc is increased. (vii) The mass transfer parameter increases as the radiation parameter N is increased.

References [1] [2]

[3] Fig. 10. Mass transfer parameter for different values of Sc [4]

[5] [6]

[7]

[8]

[9]

[10]

[11]

Fig. 11. Mass transfer parameter for different values of N

V.

Conclusion

[12]

A numerical study has been carried out to study the effect of radiation on the flow of a micropolar fluid past continuously moving plate in the presence of mass transfer. The governing equations are transformed into system of non-linear ordinary differential equations by using similarity variables. It is solved numerically by using fourth order Runge-Kutta method along with Nactsheim–Swigert shooting technique [16]. Computation is carried out for the prescribed parameter Fw, K, N, Pr, Sc and G. Conclusions of this study are as follows: (i) The velocity field decreases when the suction or injection parameter Fw increases. (ii) The temperature field decreases as the suction or injection parameter Fw increases and also the temperature decreases due to increase in Prandtl number. (iii) The concentration field decreases as the suction or injection parameter Fw increases and also concentration profile decreases at the increases of

[13]

[14]

[15]

[16]

[17]

[18]


A.C. Eringen, Simple Microfluids. Int. J. Eng. Sci., Vol.2, pp. 205 – 207, 1964. A.C. Eringen, Theory of Thermomicrofluids. Journal of Mathematical Analysis and Application, Vol.38, pp.480496,1972. B.C. Sakiadis, Boundary-layer behavior on continuous solid surfaces: I. The Boundary layer on a continuous flat surface, AIChE J. Vol.7, pp.221-225, 1961. F. Tsou, E. Sparrow and R. Goldstein, Flow and heat transfer in the boundary layer on a continuous moving plate. Int. J. Heat Mass Transfer Vol.10, pp.219- 235, 1967. F. Ebert, A Similarity solution for the boundary layer flow of a polar fluid. Chemical Eng. Journal Vol.5, pp. 72-85, 1973. G. Ahmadi, Self-similar solution of incompressible micropolar boundary layer flow over semi infinite plate. Int. J. Eng. Sci., Vol.14, pp. 639 – 646, 1976. T. Ariman, M.A. Turk and N.D. Sylvester, Application of Micro continuum fluid mechanics. Int. J. Eng. Sci., Vol.12, pp.273 – 293, 1974. V.M. Soundalgekar, H.S. Takhar, Flow of a micropolar fluid on a continuous moving plate. Int. J. Eng. Sci., Vol.21, pp. 961-965, 1983. C. Perdikis, A. Raptis, Heat transfer of a micropolar fluid by the presence of radiation, Heat Mass Transfer Vol.31, pp. 381382,1996. A. Raptis, Flow of a micropolar fluid past continuously moving plate by the presence of radiation. Int. J. Heat Mass Transfer Vol.4 pp. 2865-2866, 1998. Hassan A. M., El. Arabawy, Effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation. Int. J. Heat Mass Transfer. Vol.46,pp. 1471 – 1477, 2003. P. Ganesan, P. Loganathan. Radiation and mass transfer effects on flow of an incompressible viscous fluid past a moving vertical cylinder. Int. J. Heat Mass Transfer Vol.45, pp.4281 – 4288, 2002. C.-Y. Cheng, Fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel with concentration and temperature. Int. comm. Heat Mass Transfer, Vol.33, pp.627-635, 2006. J. Prakash, A. Ogulu and E. Zhandire, M.H.D. free convection and mass transfer flow of a micro-polar thermally radiating and reacting fluid with time dependent suction. Indian Journal of Pure & Applied Physics , Vol.46, pp.679-684, 2008. Kai-Long Hsiao, Numerical Calculation Heat and Mass Transfer of a Micropolar Fluids Flow with Magnetic and Radiation Effects to Past a Stretching Sheet. International Review of Mechanical Engineering (IREME), Vol. 3. N. 2, March 2009. D. Srinivasacharya, Ch. Ramreddy, Heat and mass transfer by natural convection in a doubly stratified non-Darcy micropolar fluid. Int. comm. Heat Mass Transfer, Vol.37, pp. 873-880, 2010. M.R. Mohaghegh, Numerical Analysis of Laminar Boundary Layer Equations in Free Convection over a vertical flat plate and Forced Convection over a Wedge. International Review of Mechanical Engineering (IREME), Vol. 5 N. 4 pp. 747-753, May 2011. V. Golkarfard, S. A. Gandjalikhan Nassab, A. B. Ansari, Deposition of Solid Particles in Convective Flow over Backward-


27


Facing Step under the Effect of Radiative Heat Transfer. IREME, Vol. 5 N. 5, pp. 867-875 July 2011. [19] J.A. Adams, D.F. Rogers. Computer-Aided Heat Transfer Analysis, McGraw-Hill 1973.

Author’s Information 1

Department of Mathematics, Anna University, Chennai 600025, India. E-mail: [email protected]

2

St. Peter’s Engineering College, Avadi, Chennai 600054, India. E-mail: [email protected]

P. Loganathan, received M.Sc degree from Presidency College, Chennai and M.Phil. from Loyola College, Chennai. He also received a Ph.D. degree in Mathematics from Anna University, Chennai. He is working as Associate Professor in the Department of Mathematics, Anna University. His areas of interest are Computational Fluid Dynamics, Heat and Mass transfer and Object Oriented Programming. He has published about 27 papers in national and international conferences. N. Golden Stepha, (Corresponding author) received M.Sc degree from Queen Mary’s College, Chennai and M.Phil degree from Madras Christian College, Chennai. She is also doing Ph.D. in Mathematics under the guidance of Dr. P. Loganathan, Anna University, Chennai. Her area of interest is Computational Fluid Dynamics. She is working as a senior lecturer in St. Peter’s University for more than 15 years.



28


Concurrent Simulation of Permeability, Thermal Conductivity and Modulus for Carbon Fibre Reinforcements and Composites Reza Samadi, Francois Robitaille

Abstract – The selection of a carbon fibre reinforcement for manufacturing structural polymer composite parts requires good knowledge of numerous material properties at the design stage, for both the various carbon fibre reinforcements that may be selected and the composite parts to be made from these reinforcements. This paper presents predictive meso-scale simulations of i) inplane permeabilities of unidirectional and bidirectional non-woven carbon fibre textile reinforcements, ii) through-thickness thermal conductivity of composites made from these reinforcements, and iii) in-plane moduli of composites made from these reinforcements. Experimental validation results are presented for all simulated properties. The effect of the reinforcement configuration on these processing (i) and performance (ii, iii) properties is quantified through systematic variation of the geometric parameters that define the configuration, for constant reinforcement surface density (ρs) and composite fibre volume fraction (vfc). Results are presented for 136 simulations performed using 34 geometric models of reinforcements. The geometric parameters that have the strongest effects on the permeability, conductivity and modulus are identified. Differences in the amplitude of these effects observed from simulation correlate well with the different levels of variability observed experimentally for each property. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Composite Materials, Reinforcement, Resin Flow, Thermal Conductivity, Stiffness

kta ktt L m n P T ta th tr tw vfc vfmax vft xmin xmax xs ymin ymax ys zmin zmax εcx εcy µ

Nomenclature A c Cpt C1 d Ecx Ecy Efa Er Eta Ett gh gv Krx Kry Kta Ktt kcz kfa kft kr

Fitting parameter in Nielsen’s model Fitting parameter in Gebart’s model Specific heat of tow Fitting parameter in Gebart’s model Fibre diameter In-plane Young’s modulus of composite along x In-plane Young’s modulus of composite along y Axial Young’s modulus of fibre Young’s modulus of resin Axial Young’s modulus of tow Transverse Young’s modulus of tow Horizontal inter-tow gap Vertical inter-tow gap In-plane permeability of reinforcement along x In-plane permeability of reinforcement along y Axial tow permeability Transverse tow permeability Through-thickness thermal conductivity of composite Fibre axial thermal conductivity Fibre axial thermal conductivity Resin thermal conductivity

ρr ρs ρt


29

Tow axial thermal conductivity Tow transverse thermal conductivity Domain length Number of layers Power of hyperellipses Resin pressure Temperature Tow cross-section area Tow height Ratio of tow width to tow height Tow width Fibre volume fraction of composite Maximum fibre volume fraction of tow Fibre volume fraction of tow Domain face lower boundary, normal to x Domain face upper boundary, normal to x Local axis x in tow cross-section Domain face lower boundary, normal to y Domain face upper boundary, normal to y Local axis y in tow cross-section Domain face lower boundary, normal to z Domain face upper boundary, normal to z Axial strain along x Axial strain along y Resin viscosity and density Resin density Reinforcement surface density Tow density


R. Samadi, F. Robitaille

σcx σcy

Axial stress along x Axial stress along y

I.

plan, and the effects of these parameters on processing and performance properties were quantified through 136 simulations performed on 34 geometric models. The tow fibre volume fraction vft, which is the fibre volume fraction defined within the cross-section of a tow, fluctuated as a dependent variable allowing vfc to remain constant. Experimental validation results are presented for all properties.

Introduction

The material properties of carbon fibre composites that govern their manufacturing and their performance are strongly influenced by the configuration of the textile carbon fibre reinforcements used in manufacturing these composites [1]-[4]. Therefore, data quantifying these processing and performance properties are required at the design stage, so that a textile reinforcement which is best suited to a specific part and load case may be selected. Quantifying these data through physical testing for a number of commercially available carbon fibre reinforcements is long and costly. As an alternative, much attention was devoted to the development of mesoscale simulation tools for textile composite unit cells, used for predicting these properties from the reinforcement configuration. Meso-scale predictions were successfully used for mapping trends between a selected processing property of reinforcements or performance property of composites, and configurationrelated parameters such as the fibre volume fraction vfc, in-plane shear angle or reinforcement type [5]-[7]. However, the accuracy of the geometric models remains a major issue; hence published work is mostly limited to predicting trends and usually aims at reducing physical testing through such trend identification, as opposed to superseding it. The literature focuses mostly on the development of meso-scale simulation methods, with few applications presented. In this paper, the in-plane permeabilities of reinforcements to resin Krx and Kry, the throughthickness thermal conductivity of composites kcz, and the in-plane Young’s moduli of composites Ecx and Ecy are predicted for unit cells of unidirectional and bidirectional non-woven textiles. The work aims at quantifying the effect of the reinforcement configuration on these properties, using established methods suitable to actual reinforcement selection in an industrial context. Therefore, mature prediction methods based on validated software are used with geometric models of existing carbon reinforcements. Values of the geometric parameters defining the configuration of these reinforcements were varied. However, all simulations and measurements were preformed on models and materials with a fibre volume fraction vfc held constant at 55%, ensuring comparisons that are meaningful and relevant to industrial practice: the fibre volume fraction was kept constant in all cases as the aim of the work is to quantify the effect of the configuration as opposed to the effect of vfc . The reinforcement surface density ρs also stayed constant at 530 g/m2 in all simulations, for the same reason. The tow width tw, vertical inter-tow gap gv, horizontal inter-tow gap gh and tow cross-section shape (Fig. 1) were varied using a Taguchi full-factorial simulation

Fig. 1. Geometric parameters

The same geometric models were used in predicting flow, heat transfer and structural properties. The amplitude of the effects of these geometric parameters on the properties varied from one property to another. These different amplitudes correlate well with the different levels of variability observed in measurements of the permeability to resin flow, thermal conductivity, and stiffness. The geometric parameters most affecting each property at a constant vfc were identified. From a practical perspective, achieving reproducible manufacturing of carbon fibre composite parts which translates into reliable and consistent performance of these parts, only requires that the geometric parameters of reinforcements that have a strong effect on the physical properties of the reinforcements and composites be well controlled during reinforcement manufacturing. The relation between machine settings used in a textile manufacturing operation and the variability of the carbon fibre reinforcements produced can be observed directly during their production. However, only mesoscale simulations can identify which geometric parameters of the textile reinforcements must be controlled accurately for achieving reproducible composite processing and performance.

II.

Literature

Analytical work investigating relations between the geometric configuration of textiles and their properties predates high-performance textile composites. However, the increased use of these materials in load-bearing structures, notably in aerospace engineering, as well as the resulting need for property quantification has lead to the accelerated development of textile unit cell geometric modelers [5], [6]. In parallel, work aiming at measuring the intricate geometry of reinforcement unit cells and predicting it from first principles is ongoing [7]-[9].



30


These textile unit cell modelers have been used for predicting the in-plane permeability of carbon fibre reinforcements to resin. Diverse computational methods were proposed towards this end, featuring varying levels of detail in flow field description [10], [11]. The methods are well described and simulations leading to the identification of trends are presented in the literature. Investigations of variability were also conducted. However, few validation results are available [12], [13]. Unit cell models were also used for predicting the through-thickness thermal conductivity of textile composites. Ning et al. [14] applied a straightforward calculation to simple geometric models. Dasgupta et al. [15] proposed more extensive homogenization method and models for the same purpose. Both groups succeeded at predicting the effects of vfc and weave style. Gowayed et al. [16] and Bigaud et al. [17] introduced computational methods and presented trends for kcz as a function of vfc for different textile reinforcements. Gowayed et al. [16] validated their predictions with an array of experimental results. Predictive modeling of structural properties was presented for different textile reinforcement types [18][21]. Recent published work aims at predicting stiffness and failure, and focuses heavily on the development of numerical methods for the latter. Significant progress was achieved in recent years. However, validation data based on experimental trials remains sparse. Limited information on the concurrent simulation of different processing and/or performance properties is available in the literature [6]. No concurrent validated permeability, conductivity and stiffness data, nor data where geometric reinforcement parameters affecting all properties are identified systematically, are available. Limited variability information generated from simulation at constant vfc is available [11], hence conclusions about the effect of the meso-scale geometric parameters are generally not made.

simulations of the different properties, appropriate unit cell boundaries were selected in view of each case. All reinforcement layers had a surface density ρs of 530 g/m2 and all unit cells had a fibre volume fraction vfc of 55%. The effects of reinforcement configuration on flow, heat transfer and static loading properties were quantified by varying the tow width tw (2 and 4 mm), horizontal intertow gap gh (0.2 and 0.4 mm), vertical inter-tow gap gv (0.02 and 0.04 mm) and tow section shape. Tow height th was kept constant at 0.5 mm in all cell models. All above quantities were defined from observations made on actual carbon fibre reinforcements as described below. Two sections were modeled as hyperellipses of power n in the local xs , ys plane as described in (1): xs 2 th ⎛ ys = ± ⋅ ⎜ 1 − 2 ⎜ ( th ⋅ tr )2 ⎝

⎞ ⎟ ⎟ ⎠

n

(1)

The use of the hyperellipse enables the modeling of two sections that may be closer to a rectangular shape with rounded corners, as seen with actual reinforcements, and notably so with non-crimp carbon textiles. The ratio tr of tow width to height varied with tw. The tow cross section area ta varied with tw and n. The tow fibre volume fraction vft varied with the above parameters to ensure that the fibre volume fraction in a unit cell vfc stayed constant at 55%. Geometrical parameters are illustrated in Fig. 1; values appear in Table I for the 34 unit cell models created for this work. 17 cell models were created for unidirectional material 1 (labeled 1A-1P, 1Z) and 17 for balanced bidirectional material 2 (labeled 2A-2P, 2Z) corresponding to 2 fullfactorial 4-parameter simulation plans with additional models in the centre of the simulation space (labeled 1Z, 2Z). The 2 simulation plans evaluate the effect of all geometric parameters and all their interactions on flow, heat transfer and static loading properties. The vertical inter-tow gaps correspond to 3 to 5 carbon fibre diameters and the horizontal inter-tow gaps are representative of reinforcements used in the experiments; hence the models are credible in representing actual reinforcement geometry. The maximum value of vft = 0.906 (cases 1G, 2G) corresponds to perfect hexagonal packing of the fibres in the tows. Tow properties were calculated from available properties of Hexcel T300 PAN-based carbon fibres and representative epoxy resins listed in Table II [22]-[25]. Axial and transverse tow permeabilities Kta and Ktt were calculated for each cell model from fibre diameter d and tow fibre volume fraction vft using Gebart’s models [26] stated in (2), (3):

III. Geometry and Properties Two carbon fibre reinforcements were modeled in this work. Both materials are made of identical unidirectional textile layers of parallel carbon tows. Material 1 is a unidirectional laminate with sequence [0°]m where m is the numbers of layers. Material 2 is a balanced bidirectional laminate with sequence [0°/90°]m . Multiple models were created for each material based on different geometric parameters. Each model was used in simulating flow, heat transfer and structural behavior. In flow simulations, a material model effectively represents a textile reinforcement saturated with liquid resin. In simulations of heat transfer and structural loading a material model represents a composite with solid resin. Unit cells of unidirectional material 1 [0°]m and balanced bidirectional material 2 [0°/90°]m featuring m = 1 and m = 2 layers of reinforcement were modeled. While the geometry of models was unchanged in

8 ⎛ d ⎞ (1 − vft ) Kta = ⋅ ⎜ ⎟ ⋅ c ⎝2⎠ vft 2 2


3

(2)


31


⎛d ⎞ ⋅⎜ ⎟ ⎝2⎠

2

B=

(3)

Gebart used adjusted factors to account for differences between calculated values of permeability and those measured on reinforcements featuring horizontal intertow gaps. Such factors do not apply to tows themselves, hence they were not used here. Values of c = 53 and C1 = 11.54 corresponding to hexagonal fibre packing were used with maximum tow fibre volume fraction vfmax = 0.906. Values of Kta and Ktt are presented in Table III.

⎛ 1 − v f ,t ⎞ ⎟⋅v ⎜ v f ,max 2 ⎟ f ,t ⎝ ⎠

Horizontal gap gh (mm) Model full (1) / half height (2) (mm) Tow section power n

Tow section area ta (mm2)

Tow fibre volume fraction vft

0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.2 0.2 0.2 0.2 0.4 0.4 0.4 0.4 0.3

0.785 1.570 0.785 1.570 0.785 1.570 0.785 1.570 0.918 1.835 0.918 1.835 0.918 1.835 0.918 1.835 1.342

0.800 0.763 0.831 0.793 0.872 0.800 0.906 0.831 0.684 0.653 0.710 0.678 0.746 0.684 0.775 0.710 0.715

2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 3

0.02 0.02 0.04 0.04 0.02 0.02 0.04 0.04 0.02 0.02 0.04 0.04 0.02 0.02 0.04 0.04 0.03

0.52 0.52 0.54 0.54 0.52 0.52 0.54 0.54 0.52 0.52 0.54 0.54 0.52 0.52 0.54 0.54 0.53

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.4

(

Property

Units µm g/cm3 Pa⋅s W/mK

1.2 1.0 0.20

Resin

Modulus

GPa

5.0

Fibre

Ref

8.0 1.8 Axial: 8.4 Transverse: 0.84 250

[22] [22], [23] [23] [24], [25]

Et ,t =

)

(4)

⎡1 + A ⋅ B ⋅ v f ,t ⎤ kt ,t = kr ⎢ ⎥ ⎢⎣ 1 − B ⋅ ϕ ⋅ v f ,t ⎥⎦

(5)

(

(9)

)

v f ,t ⋅ Er + 1 − v f ,t ⋅ E f ,a

TABLE III TOW PROPERTIES FOR CELL MODELS 1A-1P, 1Z, 2A-2P AND 2Z

1A,2A 1B,2B 1C,2C 1D,2D 1E,2E 1F,2F 1G,2G 1H,2H 1I,2I 1J,2J 1K,2K 1L,2L 1M,2M 1N,2N 1O,2O 1P,2P 1Z,2Z

Tow axial and transverse thermal conductivities kta and ktt were calculated for each unit cell from fibre axial and transverse thermal conductivity kfa and kft and resin thermal conductivity kr using the rule of mixture by volume shown in (4) and Nielsen’s model [23] for aligned cylinders, shown in (5), (6), (7) with A = 0.5 and vfmax = 0.906:

(

E f ,a ⋅ Er

(8)

Values of Eta and Ett are presented in Table III. Other structural properties of the tows were calculated using established models from reference [24]; individual values are not reported.

[22], [23]

kt ,a = v f ,t ⋅ k f ,a + 1 − v f ,t ⋅ kr

)

Et ,a = v f ,t ⋅ E f ,a + 1 − v f ,t ⋅ Er

TABLE II PROPERTIES OF CARBON FIBRES AND EPOXY RESIN Diameter Density Viscosity Conductivity

(7)

Tow density ρt and specific heat Cpt were not required in these steady-state simulations. Values of kta and ktt are presented in Table III. Tow axial and transverse Young’s moduli Eta and Ett were calculated for each unit cell from the fibre axial modulus Efa and resin modulus Er using the rule of mixture by volume shown in (8) and Tsai’s micromechanical model [24] appearing in (9):

Cell model

Tow height tw (mm) Tow width tw (mm) Vertical gap gv (mm)

Cell model

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

(6)

ϕ = 1+ ⎜

TABLE I REINFORCEMENT GEOMETRY FOR CELL MODELS 1A-1P, 1Z, 2A-2P AND 2Z

1A,2A 1B,2B 1C,2C 1D,2D 1E,2E 1F,2F 1G,2G 1H,2H 1I,2I 1J,2J 1K,2K 1L,2L 1M,2M 1N,2N 1O,2O 1P,2P 1Z,2Z

( k f ,t / kr ) − 1 ( k f ,t / kr ) + A

30.3 54.9 17.1 34.2 6.59 30.3 2.44 17.1 163 237 116 175 70.5 163 45.9 116 109

38.8 88.4 15.4 46.3 1.85 38.8 0.00 15.4 327 494 223 355 122 327 69.6 223 207

IV.

6.76 6.46 7.01 6.70 7.35 6.76 7.63 7.01 5.81 5.55 6.02 5.76 6.32 5.81 6.55 6.02 6.07

0.626 0.579 0.673 0.617 0.747 0.626 0.818 0.673 0.495 0.468 0.520 0.489 0.559 0.495 0.593 0.520 0.525

200.9 192.0 208.5 199.2 218.7 200.9 227.0 208.5 172.6 165.0 179.0 171.1 187.8 172.6 194.8 179.0 180.2

Transverse tow modulus Ett (GPa)

5/ 2

Axial tow permeability Kta (10-15 m2) Transverse tow permeability Ktt (10-16 m2) Axial tow conductivity kta (W/mK) Transverse tow conductivity ktt (W/mK) Axial tow modulus Eta (GPa)

⎛ vf max ⎞ Ktt = C1 ⎜ − 1⎟ ⎜ vf ⎟ t ⎝ ⎠

23.1 19.8 26.8 22.4 34.5 23.1 44.6 26.9 15.2 13.9 16.5 14.9 18.6 15.2 20.8 16.5 16.7

Simulations

Tow in models of unidirectional [0°]m composites (17 models: 1A-1P, 1Z) extend along x only while tows in the first and second reinforcement layers of bidirectional [0°/90°]m composites (lower and upper z respectively, 17



32


models: 2A-2P, 2Z) extend along x and y respectively. Simulations performed are: in-plane, steady-state saturated laminar flow along x and y for [0°]m laminates (1A-1P, 1Z, 34 simulations); in-plane, steady-state saturated laminar flow along x for [0°/90°]m laminates (2A-2P, 2Z, 17 simulations); through-thickness steadystate conductive heat transfer along z for [0°]m and [0°/90°]m composites (1A-1P, 1Z, 2A-2P, 2Z, 34 simulations); static elastic loading along x and y for [0°]m composites (1A-1P, 1Z, 34 simulations); and static elastic loading along x for [0°/90°]m composites (2A-2P, 2Z, 17 simulations). Predicted properties are in-plane permeabilities of reinforcements Krx and Kry , through-thickness thermal conductivity of composites kcz , and in-plane Young’s moduli of composites Ecx and Ecy. Full-factorial Taguchi simulation plans were used in all cases as detailed above. Fig. 2 shows the evolution of the cell height (th + gv , models 1A-1P, 1Z or 2 (th + gv), models 2A-2P, 2Z) and tow fibre volume fraction vft and their relation with input parameters tw (tow width), gv (vertical inter-tow gap), gh (horizontal inter-tow gap) and n (tow section hyperellipse power).

face xmax (or ymax). Other faces were set as walls; these were located in zones of very slow flow hence cell permeability was unaffected by these walls. TABLE IV SIGNIFICANT PARAMETERS AND CONTRAST VALUES Ĉ #

Parameter

Contrast value

1 2 3 4

gh n tw gv

1 2

gv Vertical inter-tow gap n Hyperellipse tow section power

1 2 3 4

gh tw gv n

Krx Permeability along tows, [0°] material 1 Horizontal inter-tow gap Hyperellipse tow section power Tow width Vertical inter-tow gap

ĉ = + 8.38 × 10-10 m2 ĉ = + 3.92 × 10-10 m2 ĉ = - 3.58 × 10-10 m2 ĉ = + 2.66 × 10-10 m2

Kry Permeability across tows, [0°] material 1 ĉ = + 7.84 × 10-11 m2 ĉ = + 6.04 × 10-11 m2

Krx Permeability along axis x, [0°/90°] material 2 Horizontal inter-tow gap Tow width Vertical inter-tow gap Hyperellipse tow section power

ĉ = + 4.70 × 10-10 m2 ĉ = - 2.05 × 10-10 m2 ĉ = + 1.87 × 10-10 m2 ĉ = + 1.76 × 10-10 m2

kcz Through-thickness conductivity, [0°] material 1 1 2 3 4

n gv tw gh

1 2 3 4

n tw gv gh

Hyperellipse tow section power Vertical inter-tow gap Tow width Horizontal inter-tow gap

ĉ = + 0.0547 W/mK ĉ = - 0.0170 W/mK ĉ = - 0.0164 W/mK ĉ = + 0.0163 W/mK

kcz Through-thickness conductivity, [0°/90°] material 2 Hyperellipse tow section power Tow width Vertical inter-tow gap Horizontal inter-tow gap

ĉ = + 0.0509 W/mK ĉ = - 0.0147 W/mK ĉ = - 0.0129 W/mK ĉ = + 0.0112 W/mK

Uniform pressure distributions and velocity fields developed along the flow direction in all cases simulated in this work. Darcy’s law was used for calculating reinforcement permeability Krx or Kry along x or y as shown in (10) from the imposed pressure gradient ∆P/∆x or ∆P/∆y and total mass flow rate converted to seepage velocity ux or uy:

Fig. 2. Geometric parameters th + gv (models 1A-1P, 1Z) or 2 (th + gv) (models 2A-2P, 2Z) and vft

All geometric models were created using TexGen v2 [5] and meshed using Gambit™ 2.2.30 with 100 000 to 500 000 4-noded tetrahedral elements. Different meshes were created from the same geometric models for flow, heat transfer and static loading to better suit boundary conditions and symmetry. Faces normal to axes x, y and z were labeled xmin, xmax, ymin, ymax, zmin and zmax. Convergence was verified using model 1A in all cases. Results of contrast analysis generated from Taguchi plans appear in Table IV for all predicted properties. Significant parameters are listed in each case and their effect is quantified.

ux = −

Kry ∆P Krx ∆P ⋅ ⋅ ; uy = − µ ∆x µ ∆y

(10)

Resin viscosity µ and density ρr are specified in Table II. A typical flow field appears in Fig. 4 for flow along axis x through model 1C. Flow simulation results appear in Fig. 5. These results clearly show that permeability is strongly influenced by the reinforcement configuration, in cases where the fibre volume fraction vfc is kept constant. This finding can be related to the large variability that is typically observed for permeability measurement performed on reinforcements. It is also informative to compare variability in the above results with that arising from different nesting patterns for a fabric of constant configuration and vfc [11], and from varying gaps in fabrics where tows are not perfectly

IV.1. Resin Flow Simulations Steady-state simulations of laminar saturated in-plane flow of resin through reinforcements were conducted using Fluent™ 6.2.16. Typical flow domains for materials 1 and 2 appear in Fig. 3. Pressures of 10 and 0 Pa were imposed on inlet face xmin (or ymin) and outlet Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


33


straight [12], again for a constant vfc. Finally, narrow inter-tow gaps lead to predicted permeability that are in line with reported experimental values, indicating the critical importance of accurate geometrical modeling to flow simulations.

smaller tow sections – , negative effect of tow width tw and positive effect of vertical gap gv are less than half the amplitude of the effect of gh. All these effects clearly indicate that inter-tow channels in the horizontal plane control flow in this case. Analysis of contrasts for Kcy , [0°] material 1 indicates that both gv and n have strong positive effects whilst other parameters are not significant.

Fig. 5. Flow simulation results

Pressure loss mostly occurs in gaps between vertical apexes of tows; the geometry of these gaps controls transverse permeability Kcy for material 1. It can be concluded that flow along and across tows in [0°] materials is controlled by inter-tow gaps, with horizontal and vertical inter-tow gaps dominating flow along and across tows respectively. Analysis of contrasts obtained with Kcx for [0°/90°]m material 2 shows trends very similar to those observed with Kcx for [0°]m material 1. This was expected as Kcx values for material 1 are larger than those of Kcy for the same material, resin flowing though the path of least resistance.

Fig. 3. Typical flow domains; top: material 1, flow along x; centre: material 1, flow along y; bottom: material 2

IV.2. Heat Transfer Simulations Steady-state simulations of through-thickness heat conduction through composites were conducted using Fluent™ 6.2.16. Typical heat transfer domains appear in Fig. 6. Constant temperatures were imposed on lower and upper faces zmin and zmax. Other boundary faces were set as adiabatic walls. Symmetrical temperature and heat flux fields developed in all cases. Fourier’s law was used in calculating composite conductivity kcz along z, stated as (11) from imposed temperature gradient ∆T/∆z and the area-weighted surface heat flux qz” on faces zmin and zmax:

Fig. 4. Flow field: velocity vectors collared by amplitude (m/s), cell model 1C, flow along x

Fig. 5 also shows a factor of approximately 10 between Kcx and Kcy for material 1 as seen in reported experimental results. Trends in Kcx (= Kcy) for [0°/90°]m material 2 closely follow those of Kcx for [0°]m material 1 as expected. Analysis of contrasts for Kcx , [0°]m material 1, Table IV, indicates that horizontal gap gh has the strongest positive effect (increases Kcx). Amplitudes of the positive effect of tow section power n – resulting in

qz " = −kcz

∆T ∆z

(11)

The temperature field for case 2A appears in Fig. 7. Heat transfer simulation results appear in Fig. 8. Reinforcement configuration has limited effect on through-thickness conductivity for a constant fibre



34


volume fraction vfc in the composite. Whilst published measurements are few this can be correlated with the limited variability observed experimentally for this property, and explain that published predictions of kcz seem equally successful whether based on complex or simplified geometric models [14]-[17].

Model geometry is less critical for conductivity than it is for permeability. Fig. 8 also shows that values of kcz are virtually identical for [0°]m material 1 and [0°/90°]m material 2. Tow configuration has a much greater influence than laminating sequence; having 2 superimposed [0°/90°]m layers in material 2 as opposed to parallel tows in [0°]m material 1 had virtually no effect on kcz, as expected. Analyses of contrasts for kcz in [0°]m material 1 and [0°/90°]m material 2, Table IV, lead to nearly identical results. Some effects are counterintuitive: kcz increases most when hyperellipse section power n goes from 0.3 to 0.5 corresponding to smaller sections of conductive tow embedded in a larger volume of less conductive matrix. However, at a constant composite fibre volume fraction vfc tows of smaller section have higher tow fibre volume fractions vft resulting in larger transverse tow conductivity ktt and through-thickness composite conductivity kcz . Amplitudes of the negative effect of vertical gap gv, negative effect of tow width tw and positive effect of horizontal gap gh – through its relation to vft – are less than a third of the amplitude of the effect of n. IV.3. Static Loading Simulations Static elastic in-plane loading simulations of materials 1 and 2 were conducted using Abaqus™ 6.7-1. Static loading domains are identical to heat transfer domains, Fig. 6. A tensile stress of 10 MPa was imposed along axis x (or y) on faces xmin and xmax (or ymin and ymax). Displacements of all nodes on each boundary face along the face normal were forced equal; hence boundary faces remained straight upon loading. Symmetrical stress and strain fields developed in all cases. Hooke’s law was used in calculating composite Young’s moduli Ecx or Ecy along x or y as stated in (12), (13), (14) from imposed stress σcx or σcy and axial strains εcx or εcy:

Fig. 6. Typical heat transfer domains: material 1, material 2

Ecx = Fig. 7. Temperature field (K), cell model 2A

Ec y =

σ cx ε cx σ cy ε cy

ε c x ,ε c y =

∆L Lo

(12)

(13)

(14)

The stress field for case 1B loaded along x appears in Fig. 9. Static loading simulation results appear in Fig. 10. The reinforcement configuration has virtually no effect on in-plane Young’s moduli. Only Ecy for [0°]m material 1 varied slightly with section power n and tow width tw through their effect on vft and its relation to Ett. This expected conclusion for constant fibre volume fraction

Fig. 8. Heat transfer simulation results



35


vfc composites can be correlated to the low variability levels typically observed on Young’s module in testing, which are usually much lower than variability levels associated with strength. All values were in line with expectations and predictions of simple micromechanical models for homogeneous composites. In view of the limited effect of parameters, analyses of contrasts were not conducted for static loading simulations.

was avoided. Permeability was measured for [0°]4 , [90°]4 and [0°/90°]s 4-layer preforms with vfc = 55 %. Each measurement was repeated 3 times. Values of permeability along the flow direction were derived from Darcy’s law as Kcx = 7.2 × 10-10 m2 ± 1.1 × 10-10 m2 , Kcy = 6.5 × 10-11 m2 ± 1.2 × 10-11 m2 and Kcx = 4.1 × 10-10 m2 ± 9.4 × 10-11 m2 respectively where the variability number is the standard deviation. The amplitudes of measured values generally correspond to those obtained from simulations. Kcy is significantly lower than Kcx for the unidirectional case, and Kcx for the [0°/90°]s preform is only slightly lower than Kcx for the [0°]4 preform. Variability is relatively large as expected.

Fig. 9. Von Mises stress field (MPa), cell model 1B, stress along x

Fig. 11. Permeability measurement apparatus

Through-thickness thermal conductivity measurements were conducted on 3 plates using the Hukseflux THASYS apparatus available at the Institute for Aerospace Research, National Research Council of Canada (Ottawa), Fig. 12. Laminating sequences and thicknesses were [0°]6 at 2.29 mm for 2 plates and [0°/90°]s at 1.52 mm for 1 plate. All plates had a vfc of 55%. A total of 23 measurements were conducted. The measured through-thickness thermal conductivities kcz derived from Fourier’s law were 0.450 W/mK ± 0.001 W/mK, 0.458 W/mK ± 0.002 W/mK and 0.451 W/mK ± 0.001 W/mK where the variability number is the standard deviation. It was clearly apparent that the laminating sequence had no discernable effect on the kcz as expected. Accurate values could be obtained as the thermal properties of Hexcel T300 carbon fibres are well reported in the literature. Variability was very low, especially when compared with variability levels seen with permeability. Tensile testing was conducted using an Instron 4482 universal testing frame with samples loaded at 1 mm/min. Samples were not loaded to failure. Laminating sequences and thicknesses were [0°]6 at 2.29 mm, [90°]6 at 2.29 mm and [0°/90°]s at 1.52 mm. The evaluation of variability offered little interest hence one plate was tested in each case. For the 3 plates a vfc of 55% was used. Values of Young’s modulus along the sample length were derived from Hooke’s law as Ecx = 131.8 GPa, Ecy = 8.5 GPa and Ecx = 72.0 GPa respectively. The measured values are generally in line with those obtained from simulations.

Fig. 10. Static loading simulation results

V.

Experiments

Flow, heat transfer and static loading experiments were conducted on carbon fibre reinforcements and on composites made from unidirectional non-crimp fabric style 1152 from Fabric Development Inc. The 366.7 g/m2 reinforcement available in 7.5 cm wide ribbon is made of 34 parallel T300-12K carbon warp tows held by light glass weft threads. The widths of tows and of horizontal inter-tow gaps created by the weft threads are approximately 2 mm and 0.2 mm respectively. Roomtemperature epoxy resin Mia-poxy 100 with hardener 95 at 100:24 ratio by mass was used in flow tests and for making the composites. Viscosity of the mixed resin was 800 cP at 25°C. Cure and post-cure were conducted under press at controlled thickness. Permeability measurements were conducted in a 74 mm by 300 mm cavity with constant thickness of 1.52 mm under an imposed pressure difference of 1 bar, Fig. 11. The purpose-built, non-adjustable cavity had a width slightly lower than that of the fabric hence race-tracking



36


safely state that analysis of failure would lead to a different conclusion. Unit cell modeling of carbon fibre reinforcement makes possible the quantification of effects that cannot be readily tested. It also enables quantifying the relation between the reinforcement configuration and selected physical properties pertaining to processing and performance of the composite materials, the expected variability for these physical properties. Crucially, it also enables the designer to assess which element of the reinforcement configuration must be controlled to a high level of repeatability in order to ensure reproducible manufacturing and performance of the composite parts. Simulation results were well corroborated by experiments in all cases.

Fig. 12. Hukseflux THASYS transverse permeability measurement apparatus and samples, Institute of Aerospace Research, National Research Council of Canada (Ottawa)

VI.

Discussion VII.

The geometric models used in the simulations presented here are idealized: tows are straight, and in each model the tow sections and inter-tow gaps are constant. Furthermore, tows in adjacent layers are perfectly superimposed. Examples of variability for comparable models were provided by Lomov et al [7], [21] and Long et al. [11]. The levels of variability observed in this paper, for different configurations, may be tempered by the fact that in reality variability is present within a given reinforcement, though it is limited in the case of those used in the aerospace sector. Still, it is clear that variability on permeability data will be significantly larger than it is for other properties. Inter-tow channel size largely determines preform permeability. It also determines the relation between the general flow front and the inner impregnation of tows (macro/micro flow) during filling. This relation was not investigated as all flow simulations were conducted for saturated domains; however, it could be studied systematically in a similar way. The effect of geometry in transverse flow through unidirectional preforms may be magnified as small gaps result from the assumed perfect alignment of tows; conversely, vertical gap sizes equivalent to 3 to 5 fibre diameters are representative of reality. Therefore, it is reasonable to assume that whilst the amplitudes of permeability Kcy in unidirectional preforms are adequate, variability might be somewhat overestimated. The former assumption is well supported by experimental results. Tow permeability is largely irrelevant in saturated flow. Variation in through-thickness conductivity kcz was limited even for different stacking sequences. From a modeling perspective the effect of imperfect tow superimposition would be interesting to assess, though simulation results indicate that this is likely to be minor. Observed effects of vft and vfc for the realistic situation of a composite of constant vfc produced from reinforcements of constant surface density ρs were not intuitive, showing the value of unit cell modeling. As expected, static loading simulations led to negligible variability in stiffness. However, one can

Conclusion

Meso-scale simulations of the in-plane permeability, thermal through-thickness conductivity and in-plane stiffness of unidirectional and bidirectional carbon fibre reinforcements and composites were presented and validated. The effect of reinforcement configuration was quantified systematically and assigned to specific geometric parameters. Variability was quantified for each property at a constant composite fibre volume fraction vfc. It was observed that variability differs strongly between properties; the behavior can be related to variability levels seen in experimental measurements.

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[8]

[9]


D. Ouinas, A. Hebbar, J. Viña, Evaluation of the Stress Intensity Factor in a Structure Repaired with an Elliptical Composite Patch, International Review of Mechanical Engineering (IREME), Vol. 1, No. 1, pp. 98-104, 2007. L. Kherredine, A. Amirat, N. Zeghib, Prediction and Measurement of the Damping Properties of Carbon-Fibre Reinforced Plastics Rectangular Plates, International Review of Mechanical Engineering (IREME), Vol. 2, No. 2, pp. 207-214, 2008. M. Ayman Al-Ahmar, Object-Oriented Intelligent Database System for Composite Materials Selection, International Review of Mechanical Engineering (IREME), Vol. 2, No. 2, pp. 241-247, 2008. S.J. Chowdhury, B. Howard, Thermo-Mechanical Properties of Graphite-Epoxy Composite, International Review of Mechanical Engineering (IREME), Vol. 4, No. 6, pp. 785-790, 2008 I. Verpoest, S. V. Lomov, Virtual Textile Composites Software WiseTex: Integration with Micro-Mechanical, Permeability and Structural Analysis, Composites Science and Technology, Vol. 65, No. 15-16 (Special Issue), pp. 2563-2574, 2005. F. Desplentere, S. V. Lomov, D. L. Woerdeman, I. Verpoest, M. Wevers, A. Bogdanovich, Micro-CT Characterization of Variability in 3D Textile Architecture, Composites Science and Technology, Vol. 65, No. 13, pp. 1920-1930, 2005. J. S. U. Schell, M. Renggli, G. H. van Lenthe, R. Muሷller, P. Ermanni, Micro-computed Tomography Determination of Glass Fibre Reinforced Polymer Meso-Structure, Composites Science and Technology, Vol. 66, No. 13, pp. 2016-2022, 2006. Y. Miao, E. Zhou, Y. Wang, B. A. Cheeseman, Mechanics of Textile Composites: Micro-Geometry, Composites Science and Technology, Vol. 68, No. 7-8, pp. 1671-1678, 2008 E. B. Belov, S. V. Lomov, I. Verpoest, T. Peters, D. Roose, R. S. Parnas, K. Hoes, H. Sol. Modelling of Permeability of Textile


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[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22] [23]

[24]

[25]

[26]

[27]

Reinforcements: Lattice Boltzmann Method, Composites Science and Technology, Vol. 64, No. 7-8, pp. 1069-1080, 2004. C. C. Wong, A. C. Long, M. Sherburn, F. Robitaille, P. Harrison, C. D. Rudd. Comparisons of Novel and Efficient Approaches for Permeability Prediction Based on the Fabric Architecture, Composites Part A: Applied Science and Manufacturing, Vol. 37, No. 6 (Special Issue), pp. 847-857, 2006. B. Verleye, M. Klitz, R. Croce, D. Roose, S.V. Lomov, I. Verpoest, Computation of Permeability of Textile With Experimental Validation for Monofilament and Non Crimp Fabrics. Studies in Computational Intelligence, Vol. 55, pp. 93109, 2007. B. Verleye, R. Croce, M. Griebel, M. Klitz, S.V. Lomov, I. Verpoest I, D. Roose, Finite Difference Computation of the Permeability of Textile Reinforcements with a Fast Stokes Solver and New Validation Examples, Proc. AIP Conference 907, 2007, pp. 945-950. Q. G. Ning, T. W. Chou, Closed-Form Solution of the Transverse Effective Thermal Conductivity of Woven Fabric Composites, Journal of Composite Materials, Vol. 29, No. 17, pp. 2280-2294, 1995. A. Dasgupta, R. K. Agarwal, S. M. Bhandarkar, ThreeDimensional Modeling of Woven-Fabric Composites for Effective Thermo-mechanical and Thermal Properties, Composites Science and Technology, Vol. 56, No. 3, pp. 209-223, 1996. Y. Gowayed, J. C. Hwang, Thermal Conductivity of Composite Materials Made from Plain Weaves and 3-D Weaves, Composites Engineering, Vol. 5, no. 9, pp. 1177-1186, 1995. D. Bigaud, J. M. Goyhénèche, P. Hamelin, A Global-local NonLinear Modelling of Effective Thermal Conductivity Tensor of Textile-Reinforced Composites, Composites Part A: Applied Science and Manufacturing, Vol. 32, No. 10, pp. 1443-1453. 2001. J. Crookston, F. Robitaille, A.C. Long, I.A. Jones, J. Ooi, A Systematic Study of the Mechanical Properties of Textile Composite Unit Cells Based on Geometric Modelling, Proc. ICCM-14 Conf., San Diego, 2003. A. E. Bogdanovich, Multi-Scale Modeling, Stress and Failure Analyses of 3-D Woven Composites, Journal of Materials Sciences, Vol. 41, No. 20, pp. 6547-6590, 2006. J.J. Crookston, W. Ruijter, A.C. Long, I.A. Jones, Modelling Mechanical Performance Including Damage Development for Textile Composites Using a Grid-Based Finite Element Method with Adaptative Mesh Refinement. Proc. TexComp-8 Conf., Nottingham, 2006. D. S. Mikhaluk, T. C. Truong, A. I. Borovkov, S. V. Lomov, I. Verpoest, Experimental Observations and Finite Element Modelling of Damage Initiation and Evolution in Carbon/Epoxy Non-Crimp Fabric Composites, Engineering Fracture Mechanics, Vol. 75, No. 9, pp. 2751-2766, 2008. P.N. Balaguru, Construction of Fiber Reinforced Polymer (FRP) Jackets for the Protection of Pier Caps. Construction Report, Dept. Civil Eng., Rutgers University, Newark, NJ, 2005. Engineered Materials Handbook, Vol 1: Composites, ASM, Metals Park Ohio, 1987. S. Hind, F. Robitaille, D. Raizenne, Parametric Unit Cell Modelling of the Effective Transverse Thermal Conductivity of Carbon Plain Weave Composites, Proc. TexComp-8 Conf., Nottingham, 2006. Y. Gowayed, J.C. Hwang, Thermal Conductivity of Composite Materials Made From Plain Weaves and 3-D Weaves. Composites Engineering, Vol. 5, No. 9, pp. 1177-1186, 1995. B. R. Gebart, Permeability of Unidirectional Reinforcements for RTM, Journal of Composite Materials, vol. 26, No. 8, pp. 11001133, 1992. R. C. Progelhof, J. L. Throne, R. R. Ruetsch, Methods for Predicting the Thermal Conductivity of Composite Systems: a Review. Polymer Engineering and Science, Vol. 16, No. 9, pp. 615-625, 1976. S.W. Tsai, H.T. Hahn, Introduction to Composite Materials 1st Edition, Technomic, 1980).

Authors’ information Department of Mechanical Engineering, University of Ottawa, Colonel By Hall Room A209, 161 Louis Pasteur, Ottawa Ontario, Canada. Reza Samadi is a PhD student in Mechanical Engineering, University of Ottawa, Ontario, Canada. Mr. Samadi obtained his M.A.Sc. in Mechanical Engineering from the same institution. His primary research interests lie with the effect of the structure of reinforcement for textile composites on their processing and performance properties, and the prediction of reinforcement geometry using particulate-based computational methods. He is also active in biomedical engineering research. Francois Robitaille PhD ing. is Associate Professor of Mechanical Engineering at the University of Ottawa, Ontario, Canada. Dr. Robitaille obtained his degree in Mechanical Engineering from Université Laval, Quebec City, Canada, and his MScA and PhD degrees from École Polytechnique, Montreal, Quebec Canada. He studied briefly at INSA Lyon, France and worked as a post-doctoral fellow and part-time lecturer at the University of Nottingham, UK. His primary interests lie with the mechanical behavior of textile reinforcements for composites and their effect on manufacturing operations and performance. He also conducts work on reinforcement structure and develops new carbon fibre preform manufacturing processes.



38


Development of Tooling for Hydraulic Forming of Ceramic Spheroids Using Alumina V. Bristot1, V. Bristot2, L. Schaeffer3, V. Gruber4

Abstract – In this study, tooling for hydraulic shaping of ceramic spheroids in alumina was developed based on the process for manufacturing alumina bricks and ceramic tiles, since the only known process to shape ceramic spheres in alumina available on the market so far was that known as isostatic shaping. The purpose of these spheres and spheroids is to use as grinding bodies to process raw materials, reducing the solid matter particles. They were initially developed by making a model with a shaping cavity, obtained using a manual hydraulic press. Using the apparent density and wear results that were achieved using the spheroids produced from a cavity model, a prototype of the tooling was constructed with eight shaping cavities, and installed in an automatic hydraulic press to verify its performance in a real life industrial production situation. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Tools, Hydraulic Presses, Manufacturing Process

I.

The following were considered for the tooling design: punches and sleeves, thickness of the die box wall, fixation of the sleeves in the die box, the fixation system for the complete die box, the length of the upper and lower punches, and the curvature radius of the lower and upper punches. A sketch of the tooling is shown in Figure 2.

Introduction

Tooling for the hydraulic shaping of ceramic spheroids in alumina was designed with the help of design methodology techniques. Some care must be taken in working with ceramic shaping’s based on powder compaction, followed by sintering [1]-[2]. The following requirements were considered: high production efficiency and practical operation. Thus, a procedure was developed for an adaptive design of hydraulic shaping of ceramic spheroids in alumina. The tooling designed has a capability to obtain 8 (eight) spheroids per compaction, i.e., a die with 8(eight) cavities. The design was optimized by experimental planning with a model, and validated in industrial production using a prototype. Spheroids were obtained with physical and mechanical characteristics very similar to the spheres pressed by the usual system for this type of work, namely, by isostatic system [13]. The reference design used was pressing alumina brick for mill linings. The body of the press and the movement of the upper punch plate were kept according to the structure proposed, and the lower punches, upper punches, die box and central crossbeam were. Since the main difference is the new physical format of the shaped material, the tooling that shaped the bricks was replaced by punches and sleeves with a new format, and the central crossbeam of the press was moved to expel the material from the die box cavities. This modification enabled shaping up to 8 (eight) ceramic spheroids in alumina simultaneously. The initial outline of the design is shown in Figure 1.

Fig. 1. Initial sketch of the design

Fig. 2. Tooling sketch


39


V. Bristot, V. Bristot, L. Schaeffer, V. Gruber

II.

the upper punch and the lower punch at the time of shaping.

Construction of a Model

A model of the system was constructed to test how the tooling shapes the ceramic spheroids of alumina by hydraulic pressing [3] - [4]. The tooling model with a single cavity is shown in Figures 3.

Fig. 4. Manual hydraulic press Figs. 3. Tooling model: (a) physical system; (b) schematic view

Powder pressing is very competitive because of its capability of producing pieces in their final or near final format, thus avoiding expensive finishing stages, or requiring less finishing work compared to other techniques [5]-[6]. This is possible because the piece becomes rigid already in its final shape. This means that the powder mass must be given a geometrical shape [7]. Thus, the powder is pressed against a die that reproduces an inverted form of the shape that is to be produced. When the mold is removed, the powder retains the shape [8]-[9]. Obviously the rigidity of the molded part is limited, but it should be rigid enough to be manipulated at later stages, until it is rigidified by thermal treatment [10]-[11]-[12]. In this way, using a manual hydraulic press (Figure 4) with a 400 kgf/cm2 (39,226.4 kPa) capacity, the tooling model was placed in the appropriate position in the press, and the cylindrical cavity of the die was loaded manually. The amount of ceramic powder required, introduced in the cavity, varied until it reached the condition needed to obtain a 38 mm diameter spheroid raw, i.e., without sintering, and its was conceived by trial and error. After a few attempts, the ideal cavity loading condition was achieved, 58 grams of mass. The same compaction pressure used to press bricks for linings was also utilized, 120 kgf/cm2 (11,767.92 kPa), since thus we would ensure the conditions required to perform the work function, because both the lining brick manufactured by hydraulic pressing and the spheres produced by isostatic pressing have the same technical characteristics. In Figure 5 we have a representation of how the uniaxially shaped spheroid will look in the tooling model. The main difference between the spheres produced in an isostatic press (Figure 6) and the new proposal for an hydraulic press (Figure 7), is that after hydraulic shaping, a kind of “collar” is formed on the spheroid. This occurs because there is a space between

Fig. 5. Representation of the shaped spheroid in the model tooling

Fig. 6. Sphere produced by isostatic shaping

Fig. 7. Collar formed on the spheroid after hydraulic shaping



40


Table I shows the comparison of products obtained by the different processes.

wear results of the sphere obtained in isostatic shaping compared to the spheroids formed using the hydraulic method of the prototype.

TABLE I COMPARISON OF APPARENT DENSITY AND WEAR OF THE SPHERES/SPHEROIDS OBTAINED BY THE DIFFERENT PROCESSES (MODEL) Description Apparent density [g/cm3] Wear [%] (Mill test of 96 hours work)

Isostatic Shaping From 3.58 to 3.64 From 8.0 to 10.0

TABLE II COMPARISON OF APPARENT DENSITY AND WEAR OF THE SPHERES/SPHEROIDS OBTAINED USING THE DIFFERENT PROCESSES (PROTOTYPE)

Hydraulic Shaping (model) From 3.60 to 3.64

Description Apparent density [g/cm3] Wear [%] (Mill test of 96 hours work)

From 8.0 to 8.8

III. Prototype Construction

Isostatic Shaping From 3.58 to 3.63

Hydraulic Shaping (prototype)

From 8.0 to 9.5

From 8.0 to 8.9

From 3.60 to 3.63

Figure 10 shows the spheroids already shaped in the prototype before their removal from inside the die box sleeves.

After the excellent results obtained using the tooling model presented, it was decided to manufacture a prototype for the industrial production of ceramic spheroids in alumina using hydraulic shaping. The same physical characteristics of the outer part of the die box in bricks (Figure 8) for the spheroid one, preserving the loading method, fixing the die box and expelling the pressed material were maintained. Since the test had already been performed using the model created, a die box for spheroids was made as shown in Figures 9.

Fig. 10. Spheroids formed in the prototype

Figure 11 shows the difference between a spheroid produced with the new technology by hydraulic shaping and a standard sphere produced by isostatic shaping. In both cases, the spheroids have already been sintered. Fig. 8. Die box of alumina bricks for linings

Figs. 9. Tooling for hydraulic shaping of alumina spheroids: (a) schematic view; (b) physical system Fig. 11. Shaped and sintered spheroids (left) hydraulic shaping; (right ) isostatic shaping

The prototype tooling behaved as planned, and basically, because it was a prototype, the loading of the mass into the die box sleeve was performed using a cup, and actuation of mechanical movement (beginning of pressing) was done manually by a starter button to begin the shaping process. This was done to ensure the necessary conditions for automatic pressing by the press after the start. Table II shows the apparent density and

IV.

Project Discussion and Considerations

After the spheroids formed by this new method were obtained, everyone who followed the development was pleased at the more spherical shape of these grinding bodies compared to the spheres formed by isostatic



41


pressing. Technical comparisons of the two bodies were also performed, showing results that were equal to or superior to the previous method. What still needs to be improved in the esthetics of these spheroids is the “collar” which is seen after forming, since at first sight it awakens suspicion, because it appears to be fragile. But this is overcome by proving the work done by these bodies when used. Comparing the production between hydraulic shaping and isostatic shaping, it was found that the volume of production with the new method is superior to the traditional 20 % to 25%, since the level of complexity of mass loading in the cavities, the pressing time and extraction of the formed pieces in the isostatic modality is much higher, which leads to a smaller amount produced.

[2]

[3]

[4]

[5]

[6] [7]

V.

Conclusion

[8]

Considering the objectives, these were the main conclusions: • For the hydraulic shaping of ceramic spheroids in alumina, the method proposed in this case, the same powder could be used as the one employed to manufacture alumina spheres obtained by isostatic shaping, and in the alumina bricks for linings. It was not necessary to change the formulation and its production process; • The tooling developed in the industrial hydraulic presses is easy to adapt; • The same burning curve of the product originating in the traditional form was maintained when sintering the spheroids of the methodology proposed; • The spheroids obtained with the proposed tooling presented the same or a better performance than those formed in the traditional isostatic modality. This was proved by spheroid wear and density tests, which are considered essential. On this occasion they proved to be within the standards required to perform their function; • It was also concluded that although the physical shape of the spheroids formed by the proposed tooling presented a kind of “collar”, this did not influence the final product obtained which benefited from these grinding bodies. It should be mentioned that this methodology may be changed as regards fixation of the die box and the lower and upper punches due to the different structural characteristics of each equipment, but the principle to obtain the shaped piece can be the same described in this article.

[9] [10] [11]

[12]

[13]


UFRGS, Federal University of Porto Alegre Brazil. E-mail: [email protected]. 2

IMG, Institute Maximiliano Gaidzinski Cocal do Sul, Brazil: E-mail: [email protected] 3

UFRGS, Federal University of Porto Alegre Brazil. E-mail: [email protected] 4 SATC Faculty Criciúma, Brazil E-mail: [email protected]

Vilson Menegon Bristot - Bachelor's at Engenharia Agrimensura from University do Extremo Sul Catarinense (2003), master's at Mechanical Engineering from Federal University of Rio Grande do Sul (2008) and PhD student in the Engineering of Mines Metallurgy and Materials, Federal University of Rio Grande do Sul. He is currently professor / researcher at the Faculty SATC, a professor/ researcher at the University Barriga Verde (UNIBAVE) and professor at the Institute Maximiliano Gaidzinski.

References [1]

Whiteware I. Instituto de Tecnologia Cerámicas; Universitad de Valência, Espanha. AMORÓS, A. J. L.; 2000. A Operação de Prensagem: Considerações Técnicas e sua Aplicação Industrial – Parte II: A Compactação. Anais do Science of Whiteware I. Instituto de Tecnologia Cerámicas; Universitad de Valência, Espanha. AMORÓS, A. J. L.; 2001. A Operação de Prensagem: Considerações Técnicas e sua Aplicação Industrial – Parte III: Variáveis do Processo de Compactação. Anais do Science of Whiteware I. Instituto de Tecnologia Cerámicas; Universitad Jaume I, Castellón, Espanha. AMORÓS, A. J. L.; 2001. A Operação de Prensagem: Considerações Técnicas e sua Aplicação Industrial – Parte V: Descrição da Etapa de Prensagem. Anais do Science of Whiteware I. Instituto de Tecnologia Cerámicas; Universitad Jaume I, Castellón, Espanha. AYDIN, I.,; BRISCOE, B.J.; SANLITURK, K.Y.; 1996. The Internal Form of Compacted Ceramic Components. Powder Technology, v. 89, p. 239-254. BACK. N.; 1983. Metodologia de Projetos Industriais. Ed. Guanabara Dois, p. 1-60. BORTZMEYER, D.; 1992. Modelling Ceramic Powder Compaction. Powder Technology, v. 70, p. 131-139. BRISCOE, B.J.; OZKAN, N.; 1997. Compaction Behaviour of Agglomerated Alumina Powders. Powder Technology, v.90, p. 195-203. BRISTOT, V.M.; 1996. Máquinas e Equipamentos para Cerâmica. 1.ed. Criciúma, Santa Catarina : Editora Luana. CHIAVERINI, V.; 2001. Metalurgia do Pó. 4 ed. São Paulo, Associação Brasileira de Metalurgia e Materiais. DIMILA, R.A.; REED, J.S.; 1983. Stress Transmission During the Compaction of a Spray-driel Alumina Powder in a Steel Die. Journal of the American Ceramic Society, v.66, p. 667-672. FORTULAN, C.A.; 1997. Influência dos Métodos de Injeção e de Prensagem Isostática no Desempenho das Cerâmicas Estruturais. São Carlos. 187 p. Tese (Doutorado). Escola de Engenharia de São Carlos. Universidade de São Paulo. KIM, H.G.; LEE, J.W.; 1998. Effect of Friction Between Powder and a Mandrel on Densification and of Iron Powder During Cold Isostatic Pressing. International Journal of Mechanical Sciences, v.40, p. 507-519.

AMORÓS, A. J. L.; 2000. A Operação de Prensagem: Considerações Técnicas e sua Aplicação Industrial – Parte I: O Preenchimento das Cavidades do Molde. Anais do Science of



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Vilmar Menegon Bristot - Bachelor's at Engineering Agrimensura from University do Extremo Sul Catarinense (1993), master's at Electric Engineering from Federal University of Santa Catarina (2002) and PhD student in the Engineering of Mines Metallurgy and Materials, Federal University of Rio Grande do Sul. He is currently professor /researcher at the Faculty SATC, a professor/ researcher at the University do Extremo Sul Catarinense and director / professor at the Institute Maximiliano Gaidzinski. Lirio Schaeffer - Ph.D. in Mechanical Forming. Rheinisch-Westfalischen Technischen Hochschule/Aachen, R.W.T.H.A., Germany. Professional performance: Coordination of Improvement of Higher Education Personnel, CAPES, Brazil. 2003 - Present – Relationship: Employee Department of Metallurgy, UFRGS, Brazil.1974 - Present - Public Servants, Functional Placement: Professor, Exclusive Dedication. Vilson Gruber - Graduated in Data Processing from the University Santana of São Paulo (1996), specialization in Business Management in Telecommunications from the School of Business Paulista - São Paulo (2000), specialization in Educational Psychology from the University Castelo Branco, Rio de Janeiro (2006) specialization in Project Management at Faculty SATC, Criciúma (2010), Ph.D in Engineering of Mines Metallurgy and Materials, Federal University of Rio Grande do Sul Porto Alegre (2007). He is currently a professor / researcher at the Faculty SATC, develops research projects and coordinates undergraduate courses in Telecommunications Systems and Postgraduate Courses. He has experience in Telecommunication Systems, Computer Networks acting on the following themes: Networks and Mobile Phones, Projects, Embedded Systems, Digital Systems, Remote Experimentation, Accessibility and Technology, Computer Systems and Digital Inclusion.



43


Experimental Investigation on Hardness, Cutting Force and Roughness in Milling of Hybrid Composites A. Arun Premnath1, T. Alwarsamy2, T. Rajmohan3

Abstract – Hybrid metal matrix composites (MMCs) form a new class of engineering materials which finds its application mainly in the aerospace and automobile sectors. In the present study a modest attempt has been made to develop aluminium based alumina and graphite particulate hybrid MMCs with an objective to study the effect of alumina weight fraction on cutting force and surface roughness in face milling of MMCs using tungsten carbide insert. Materials used for the present investigation are Al 6061-aluminum alloy reinforced with alumina (Al2O3) of size 45 microns and graphite (Gr) of an average size 60 microns, which are produced by stir casting route. Experiments were performed under different cutting conditions, such as spindle speed, feed rate and weight fraction of Al2O3 particles. The result indicated that the feed rate is the first predominant factor influencing the cutting force than alumina weight fraction and speed. The test results revealed that surface roughness increased with increasing the feed rate and decreased with increasing the speed and weight fraction of the particles. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Milling, Hybrid Composites, Stir Casting, Tungsten Carbide Inserts, Cutting Force

both SiC and graphite or Sic particles only studied by Songmene et al. [4]. so Al2O3 are preferred than SiC particles. Graphite is generally greyish black, opaque and has a lustrous black sheen. It has properties of both metal and non metal. Due to it’s interaxial shearing action, it is used as solid lubricant. It has zero value of coefficient of thermal expansion when it is used as reinforcement. Developments in reinforcing Al2O3 with aluminium alloy enhances the machinability, thus lowering the production cost and widens industrial applications were studied by Al-Qutub et al. [5]. Surface roughness of a machined product plays a significant role in determining the quality of the product in today’s manufacturing industry. Moreover, surface roughness is an important factor in determining the machinability of materials [6]. The surface condition of a machined part is affected mainly by machining parameters such as cutting speed, feed rate, weight fraction, particle size and depth of cut have a significant influence on the surface roughness for a given machine tool and work piece set-up. The quality of the surface has a very important role in the performance of face milling because a good quality machined surface significantly improves fatigue strength, corrosion resistance and creep life. While there are several ways to describe surface roughness, the average surface roughness (Ra), which is mostly used in industrial environments, is taken-up for the present study. Ra is defined as the arithmetic value of the departure of the profile from centerline along the sampling length. It is defined as:

Nomenclature MMC Gr Al2O3 Ra µm

Metal Matrix composites Graphite Alumina Surface Roughness Microns

I.

Introduction

Hybrid Metal Matrix Composites (MMCs) have evoked a keen interest in recent times for potential applications in aerospace and automotive industries owing to their superior strength to weight ratio and high temperature resistance [1]. There are various matrix and reinforcement available out of which aluminium reinforced with ceramics particles either SiC or Al2O3 finds to be important. Ceramics are harder particles which improve the mechanical properties of the composites compared with the base alloy [2], [3]. But when ceramics are added as reinforcement they make the materials harder and lot of wear on tool was found and obviously increases the cutting forces. This results in poor tool life and inconsistent part quality and thus limits the use of MMCs in many applications. In order to overcome the above difficulties a small amount of soft reinforcement (Gr) were added along with the hard ceramics (Al2O3) to form a hybrid MMC. Graphite aluminium MMC reinforced with alumina is easier to machine than those reinforced with Manuscript received and revised December 2011, accepted January 2012

44


A. Arun Premnath, T. Alwarsamy, T. Rajmohan

Ra =

the workpiece surface roughness. Su et al. [14] conducted the Taguchi method in the metal milling experiments to evaluate the flank wear, some variables were selected, including the structure of the coating film, cutting speed, feeding rate, milling depth, hardness of the workpiece and types of milling path. An optimal thickness of the TiCN coating film for the Tungsten carbide cutter was determined. Karthikeyan et al.[15] observed that the volume fraction of SiC particles present in the aluminium alloy matrix has a significant effect on the milling characteristics, increasing tool wear and specific energy and decreasing surface roughness . Most of the current literatures present experimental results when milling ceramic-reinforced MMCs. However, limited information is available on the milling of graphitic ceramic-reinforced composites. The main objective of the paper is to study the influence of cutting speed, feed rate and alumina weight fraction on cutting force and surface roughness in face milling of hybrid composites fabricated by stir casting method.

1 y ( x ) dx l∫

where l is the sampling length and y is the ordinate of the profile curve. The primary objective of manufacturing operation is to efficiently produce parts with high quality. Milling is a widely used machining process in manufacturing in which face milling is a machining process that produces flat surfaces. In order to improve the efficiency of machining process, and to reduce the total machining cost the optimum machining parameter have to be arrived. The setting up of machining parameters relies strongly on the operator’s experience. Optimum machining parameters are of great concern in manufacturing environments, where economy of machining operation plays a key role in competitiveness in the market. Machining damage due to excessive cutting forces may results in rejection of composites components. Therefore, the necessity to predict the cutting force is essential in determining the process parameter which results in machining damage. It is difficult to utilize the highest performance of a machine owing to their being too many adjustable machining parameters [7]. The various machining parameter that normally affects machining are namely speed, feed, depth of cut and tool geometry etc. Many researchers have studied the machining characteristics of ceramic reinforced composites. Rajesh et al. [8] reported that Surface roughness of Al alloy is less as compared to Al alloy composite during turning by carbide as well as PCD inserts. Further they recommend Carbide inserts for low speed and PCD for higher speeds for low flank wear. Brown and Surappa [9]. studied the machinability during turning of Al/Si/Gr composites and found that the machining forces were considerably reduced for the graphitic composites. The effect of cutting feed and volume fraction of the reinforced particles in drilling of self lubricated Al/Al2O3/Gr hybrid composites on the thrust force and cutting torque using experimental techniques and ANN was studied by Hayajneh et al. [10]. Kok et al. [11] studied that the surface roughness value of the K10 tool was higher than that of the TP30 tool. The surface roughness increased with an increase in the cutting speed while it decreased with increasing the size and volume fraction of particles for both tools in all cutting conditions. Also the dependency of the surface roughness on the cutting speed was smaller when the particle size was smaller when drilling 2024Al/Al2O3 particle composites. Palanikumar et al. [12] investigated the factors influencing surface roughness in machining of Al/SiC particulate composites. They have concluded that feed rate is the main factor which influences the surface roughness in machining Al/SiC composites. Lou et al. [13] studied the effect of spindle speed, feed rate, and depth of cut on the surface roughness of the end milling process. They used in-process surface roughness recognition (ISRR) and a neural fuzzy system to predict

II.

Experimental

Materials and Methods Three specimens of various weight fraction of Al2O3 (5, 10, 15%) and graphite (Gr) particles of 5% weight is reinforced with Al 6061 are considered. Since there is a decreasing nature of hardness with graphite particle inclusion [16], therefore the graphite weight fraction is restricted to certain limit in the present study. The composites used in the present work were shaped in the form of plates of sizes 250 × 190 ×12 mm were prepared by stir casting route. This process is similar to the fabrication methods used in earlier research [17]. Stir casting is a liquid state method of composites fabrication, in which dispersed phase of Al2O3 and Gr particles are mixed in molten state of Al 6061.The apparatus for stir casting is shown in the Fig. 1. The chemical composition of the Al6061 alloy is shown in the Table I. The reinforcements (Al2O3 and graphite) in powder form are preheated in an electric furnace in order to remove the moisture content present in the particles. The Al 6061 rods are kept in a graphite crucible and are melted in an electric furnace at a temperature of 730oC. When the Al 6061 rods are melted completely, Al2O3 is added first using a spatula covered with aluminium foil. After homogeneous mixing of alumina powder, graphite powder is added in the Al- Al2O3 mixture. The AlAl2O3-graphite mixture is stirred using a graphite agitator at a constant speed of 600 rpm for about 15-20 minutes. Care should be taken to avoid inhomogeneous mixing of the mixture, as particle agglomeration and sedimentation during the melt may occur. The Al- Al2O3-graphite mixture composite is then injected into cast iron moulds. The surfaces of moulds are cleaned using emery. This is done to eliminate the exposure of molten composite to pockets of oxidized surface on the cast iron moulds. Then they are pre-heated for about 20 minutes by keeping them over the furnace. The composite mixture is



45


then allowed to solidify for about 15 minutes. Thus three grades of specimen were fabricated.

they are converted into digital signals by the A/D converter sequentially. The system consists of a samplehold circuit, which enables it to hold the analog signals till conversion of previous analog to digital data takes place in the A/D converter. When the conversion is complete, the status line from the converter causes S/H to return to the sample mode and acquire signal from the next channel. On completion of acquisition, either immediately or upon receiving a command, the S/H is switched to hold mode. The conversion begins again and the multiplexer switches to the subsequent channel. The data thus obtained can be stored into a memory element for further processing or displayed onto a display device. The data can also be stored on to a personal computer after completion of experiments. Surface roughness (Ra) was measured using a stylus instrument for a cut off and sampling length of 0.8mm. For each specimen, the mean of at least five surface roughness measurements were taken.

TABLE I CHEMICAL COMPOSITION OF 6061 ALUMINUM ALLOY Element Wt %

Si

Cu Mg Mn

Fe

Zn

Sn

Ti

Pb

Al

0.80 0.35 0.8 0.02 0.01 0.008 0.01 0.01 0.02 97.9

Fig. 1. Stir casting setup

Experimental Procedure Hardness tests were performed on composites to know the effect of Al2O3 particles weight fraction in the matrix materials. The hardness of a material determines the strength of materials. The polished composite specimens were tested for their hardness, using Rockwell hardness testing machine with diamond indenter for 100 kgf load. The load was applied for 30 secs. The Rockwell B-scale hardness test is commonly used to quantifying the mechanical strength of a wide range of particle reinforced metal matrix composites and Al alloys [18, 19]. Five sets of reading were taken on the specimen on various places to assess the reproducibility and an average value was calculated Face milling is conducted on ARIX VMC 100 CNC Vertical machining centre. The experimental setup is shown in the Fig. 2. The brief summary of the experimental conditions were shown in Table II.The tungsten carbide insert and cutter of 16mm diameter is employed. All experiments are performed under dry machining condition. A three-component (Model: KISTLER 9257B) dynamometer platform was used to measure cutting forces (Fz). The force data were recorded by a specifically designed, very compact multichannel microprocessor controlled data acquisition system with a single A/D converter preceded by a multiplexer The individual analog signals were first amplified and conditioned by charge amplifier (Model: KISTLER 5070). After amplification and conditioning, the output signals were applied to a multiplexer. Further,

Fig. 2. Experimental setup TABLE II SUMMARY OF EXPERIMENTAL CONDITIONS Machine Tool insert Tool holder specification Cutting conditions Work piece Coolant Surface roughness

ARIX VMC 100(CNC) Vertical Machining Centre Tungsten carbide(BDMT11T3) Nose radius 0.8 mm BD11 D016-S16-Z2 I6 mm Diameter Speed: 1500, 3,000 and 4,500 rpm Feed rate: 50 ,100 and150 mm/min Alumina : 5 %, 10%, and15% 1. Al/15%Al2O3/5%Gr , 2. Al/10%Al2O3/5%Gr 3. Al/5% Al2O3/5%Gr Dry Milling Taylor’s Hobson pheumo instrument

III. Results and Discussion Hardness A significant increase in hardness of the alloy matrix can be seen with addition of Al2O3 reinforced powder from Fig. 3. A hardness reading showed a higher value of hardness indicating that the existences of particulates in the matrix have resulted in the overall hardness of the



46


composites. This is true as aluminum is a soft material and the reinforced particle especially ceramics material being hard, contributes positively to the hardness of the composites.

cutting force. Thus the inclusion of graphite particle in the composites contributes positively to the work with respect to cutting force. 300

5%Alumina

10%Alumina

15%Alumina

74 250 cutting force (N)

Rockwell Hardness (HRB)

72 70 68 66

200 150 100

64

50

62

0 50

60

100 150 1500

58 5% Alumina

10% Alumina

15% Alumina

50

100 150

50

3000 Feed rate (mm/min) Speed (rpm)

100 150 4500

weight fraction of Alumina(%)

Fig. 4. Variation of cutting force for various speeds and feeds when machiningAl/15%Al2O3/5%Gr , Al/10%Al2O3/5%Gr and Al/5Al2O3/5%Gr using Tungsten carbide

Fig. 3. Variation of Hardness of Al6061 and Al 6061-Al2O3 – Gr composites with increasing alumina weight fraction

Surface roughness Fig. 5. shows the variation of surface roughness values for various speeds and feeds for all the workpiece materials. The surface roughness values always increase with the increase in feed rate and decrease with the increase in weight fraction and cutting speed, similar results are also obtained by Basavarajappa et al. [22].The lowest surface roughness values (0.854 µm for Al/15%Al2O3/5%Gr and 0.917 µm for Al/10% Al2O3/5%Gr) occurred at the lowest feed rate at the highest cutting speed. The cutting speed plays an important role in deciding the surface roughness. At high cutting speeds, the surface roughness decreases. At low speeds, the (built up edge) BUE is formed and also the chip fracture readily producing the rough surface. As the speed increase, the BUE vanishes, chip fracture decreases, and hence the roughness decreases. The increases in feed proportionally increase the surface roughness. The increase of feed increases the normal load on the tool and also generates more heat which in turn increases the surface roughness. The % weight fraction of Al2O3 particles plays an important role in deciding the surface roughness [15]. The increase in weight fraction of Al2O3, decreases the surface roughness. With increase in weight fraction, the rate of decrease in roughness is reduced due to the chip fracture extending to work piece, which produce force fluctuations and ridge formation due to machine tool and vibration [23]. The presence of graphite particles increases the surface roughness, the crushed graphite particles form a deep valley and hence increase the surface roughness of the material [24],[25]. Thus the inclusion of graphite particle in the composites acts negatively to the work with respect to surface roughness.

Cutting Force Cutting Speed, Feed rate, and Weight Fraction of alumina are the major milling parameters that are considered in the experiments. Fig. 4 shows the variation of cutting force with increasing feed rate for various spindle speeds of 1500, 3000, and 4500 rpm for all the work piece materials. The difference between the cutting forces becomes larger at higher feed rates compared to lower feed rates. The feed rate and weight fraction of alumina are the predominant factors than the cutting speed influencing cutting force. The obtained results are in line with Übeyli et al. [20] .The results reveal that as the feed rate increases from 50,100 and 150 mm/min, the cutting force increases for all the spindle speeds for all the work piece materials. When feed rate is increased from 50 to 150 mm/min, the value of cutting force (from 55.42 to 155 N) increases by almost 64.25% (by addition of 5% Al2O3 and cutting speed of 1,500 rpm) whereas when weight of alumina increases from 5 to 15 % the value cutting forces increases from 55.42N to 133.91N by almost 58.61%. When spindle speed is increased from 3,000 to 4,500 rpm the value of cutting force (from 55.42N to 99.32 N) increases by almost 44.2% (by addition of 5% Al2O3 and feed rate is 50 mm/min).It can also be observed that there is less predominant variation in cutting force on increasing spindle speed for all feed rates considered. The reinforcing particles (Al2O3) support the contact stresses and thereby reduce the degree of plastic deformation thus increases the cutting force in the beginning and as time goes on, the temperature at the interface increases which reduces the cutting force. The addition of 5% graphite reduces the cutting forces significantly, which is attributed to the solid lubricating property of the graphite particles [21]. Brown and Surappa [9] reported that graphite particles reduce the interfacial friction between the tool and the work piece materials which leads to the reduction in Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


47

Surface Roughness Ra(µm)


5 % Alumina

10% Alumina

15%Alumina

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

[7]

[8]

[9] 50

100 150

50

100 150

50

100 150

[10] 1500

3000 Feed (mm/min) Speed (rpm)

4500

[11]

Fig. 5. Variation of surface roughness for various speeds and feeds when machining Al/15%Al2O3/5%Gr , Al/10%Al2O3/5%Gr and Al/5Al2O3/5%Gr using Tungsten carbide

IV.

[12]

Conclusion [13]

The following conclusions are drawn from the experimental results during the milling of Al 6061/ Al2O3/Gr composites using tungsten carbide insert under different cutting conditions: 1. The hardness of the hybrid MMCs increases as the weight fraction of the harder ceramic Particles (Alumina) increases. 2. It was observed that the feed rate is the first predominant factor that affects the cutting force by 64.25%, followed by weight fraction of Al2O3 and cutting speed. 3. The test results reveals that surface roughness increased with increasing the feed rate , because as feed increases the normal load on the tool and also generates more heat which in turn increases the surface roughness and roughness decreases with increasing the weight fraction of the particles and the speed. 4. The inclusion of graphite particle acts positively to the cutting force and reduces the cutting force. Whereas the addition of graphite acts against the surface roughness which increases the surface roughness.

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

References [1]

[2] [3] [4]

[5]

[6]

[22]

R. M. Rashad and T. M. El-Hossainy, Machinability of 7116 structural aluminum alloy, Mater Manuf Process., vol.21, pp.23– 27,2006 P. J. Heath, Developments in the applications of PCD tooling, J Mater Process Technol., vol.116 pp.31–38, 2001 A. R. Chambers, The machinability of light alloy MMCs, Composites:Part A., vol.27, pp.143–147, 1996 Songmene and m. Balazinski, Machinability of Graphitic Metal Matrix Composites as a function of reinforcing particles, Annals of the clrp., vol. 48,Jan.1999 A.M.Al-Qutub, I.M. Allam and T.W. Qureshi, Wear Properties of 10% Sub-Micron Al2O3 / 6061 Aluminum Alloy Composite,” International Journal of Applied Mechanics., Vol.7, pp.329-334. 2002 B. Huang, J.C. Chen, An in process neural network based surface

[23]

[24]

[25]


roughness prediction (INN-SRP) system using a dynamometer in end milling operations. Int J Adv Manuf Technol 21:339– 347.2003. N. Tosun , Determination of optimum parameters for multi Adv performance characteristics in drilling by using grey relational analysis, Int J Manuf Technol., vol 28(5–6), pp.450–455, 2006 Rajesh Kumar Bhushan & Sudhir Kumar & S. Das , “Effect of machining parameters on surface roughness and tool wear for 7075 Al alloy SiC composite” Int J Adv Manuf Technol 50:459– 469.2010 C.A. Brown and M.K.Surappa, The machinability of a cast aluminium alloy-graphite particle composite, Mater Sci Eng A., Vol.102 pp.31–37, 1988. M.T.Hayajneh, A.M. Hassan, and A.T. Mayyas, Artificial neural network modeling of the drilling process of self-lubricated aluminum/alumina/graphite hybrid composites synthesized by powder metallurgy technique, Journal of Alloys and Compounds., vol.478 pp.559–565, 2009 Metin Kök, Modelling the effect of surface roughness factors in the machining of 2024Al/Al2O3 particle composites based on orthogonal arrays, Int J Adv Manuf Technol (2011) 55:911–920. Palanikumar, K. and Karthikeyan, R., Optimal machining conditions for turning of particulate metal matrix composites using Taguchi and response surface methodologies, Machining Science and Technology, Vol. 10, No. 4, pp.417–433. 2006 S. J. Lou and J. C. Chen, In-process surface roughness recognition (ISRR) system in end-milling operation, Int J Adv Manuf Technol., vol.15, pp.200–209, 1999 Y.LSu, S.H.Yao ,S. Wei, and C.T.Wu, Analyses and design of a WC milling cutter with TiCN Coating, Wear., vol. 215, pp.59– 66,(1998). R. Karthikeyan, K. Raghukandan, R. S. Naagarazan and B. C. Pai, Optimizing the Milling Characteristics of AI-SiC Particulate Composites, Metals and materials, Vol. 6, No. 6 (2000), pp. 539547. M. Hassan, M.T. Hayajneh, and Mohammad Abdul-Hameed AlOmari, The Effect of Graphite Volumetric Percentage on the the Increase in Strength and Hardness of Al-4 Weight Percent MgGraphite Composites, JMEPEG., vol.11, pp.250-255, 2002 T. Rajmohan and K. Palanikumar, Experimental Investigation and Analysis of Thrust Force in Drilling Hybrid Metal Matrix Composites by Coated Carbide Drills,Materials and Manufacturing Processes vol.26, pp.961–968, 2011 L.Ceschini, G.Minak and A. Morri, Tensile and fatigue properties of the AA6061/20 vol.% Al2O3p and AA7005/10 vol.% Al2O3p composites, Composites Science and Technology., Vol .66, pp 333– 342, 2006 Y.L.Shen and N.Chawla , On the correlation between hardness and tensile strength in particle reinforced metal matrix composites, Mater Sci Eng A., Vol.297(1–2), pp. 44–47.2001. M. Ubeyli, A.Acir,O.Keles and C.Akcay, Effect of cutting parameters cutting force and surface roughness in milling alumina reinforced Al- 6Zn- 2Mg-2Cu composites, Minerals and Mining., Vol.54, pp.172- 176, April 2001. N.Suresh Kumar Reddy, P. Venkateswara Rao, Performance Improvement of End Milling Using Graphite as a Solid Lubricant, Materials and Manufacturing Processes 2005,20(4), 673-686 S.Basavarajappa, G.Chandramohan, J.P.Davim, M.Prabu K.Mukund, M. Ashwin, M.P Kumar, Drilling of hybrid aluminium matrix composites, Int J Adv Manuf Technol 35:1244– 1250. 2008 R.Karthikeyan, Analysis and optimization of machining characteristics of Al/SiC particulate composites, Ph.D. Thesis, Chidambaram: Annamalai University; 2000. B.Muralikrishnan, J.Raja, Characterization of cast iron surfaces with graphite pullouts using morphological filters. Society of Manufacturing Engineers, 144, 2002. Yung-Kuang Yang. Jie-Ren Shie. Cheng-Hung Huang, Optimization of Dry Machining Parameters for High-Purity Graphite in End-Milling Process,Materials and Manufacturing Processes 2006, 21(8), 832-837.


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Sri Chandrasekharendra Saraswathi Viswa Maha Vidyalaya University,Enathur, Kanchipuram – 631561,Tamil nadu,India E-mail: [email protected] 2 Government college of Technology Coimbatore,India. E-mail: [email protected] 3 Sri Chandrasekharendra Saraswathi Viswa Maha Vidyalaya University,Enathur, Kanchipuram – 631561,Tamil nadu,India E-mail: [email protected]

A. Arun Premnath (Corresponding author) graduated in Mechanical Engineering from University of Madras, India in 2003 and received his Masters Degree in CAD from Anna University Chennai, India in 2006. He is currently working as an Assistant Professor in the Department of Mechanical Engineering in Sri Chandrasekharendra Saraswathi Viswa Maha Vidyalaya University Kanchipuram,India. He is currently pursuing research at the Mechanical Engineering Department for his PhD. He has 06 years of teaching experience. Dr. T. Alwarsamy obtained his PhD in Mechanical Engineering from BHARATHIYAR UNIVERSITY, India. He is presently Professor in the Department of Mechanical Engineering, Government College of Technology, and Coimbatore, India. His current research interest includes machining of composite materials, total quality management and Tool chatter. He has 25 years of teaching experience. He has published 6 papers in international journals T. Rajmohan graduated in Mechanical Engineering from University of Madras, India in 1996 and received his Masters Degree in Production Engineering from Annamalai University Chidambaram, India in 1999. He is currently working as an Assistant Professor in the Department of Mechanical Engineering in Sri Chandrasekharendra Saraswathi Viswa Maha Vidyalaya University Kanchipuram,India. He is currently pursuing research at the Mechanical Engineering Department for his PhD. He has 12 years of teaching experience He has published 7 papers in international journals.



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Optimization of the Boundary Conditions by Genetic Algorithms J. L. Marcelin Abstract – This work examines the possibility of using a stochastic method, called the genetic algorithm for the optimization of boundary conditions in finite elements calculations. The examples show that using genetic algorithms in order to optimize boundary conditions is an efficient way. Copyright © 2012 Praise Worthy Prize - All rights reserved. Keywords: Optimization, Boundary Conditions, Genetic Algorithms

I.

The approach of adapting the genetic algorithms into the optimal design process is described. This approach is used to optimize locations of three supports for beams with three types of boundary conditions. This article relates to fixed geometry and is to show that the optimization of boundary conditions is feasible by combining a genetic algorithm and the finite element method (the optimization of boundary conditions in shape optimization will be dealt with in future research).Indeed, in the case of a fixed shape and because of the characteristics of the finite element method, the calculation volume can be considerably reduced. The main reasons that are to be explicated in this work are the following: the stiffness matrix is calculated and assembled once and for all; in the case of a structure for which some boundary conditions can be fixed and other can be variables (i.e. entering in the optimization framework), it would be possible to triangularize the stiffness matrix once and for all, and to take into account the variable boundary conditions thanks to a penalization process of the energetic functional, to be minimized by the boundary conditions. In such conditions, even is the number of analyses is still important, the calculation time will remain reasonable because the analyses won't be systematically complete. Various examples will aim at showing that the implemented process helps in optimizing the boundary conditions and is fairly efficient.

Introduction

In numerous application cases (e.g. taking workpieces in machining, vibrations of a mechanical structure), the optimization of boundary conditions (location and nature of the boundary conditions) can bring an interesting improvement of the studied structure's mechanical behavior. These structures are most often calculated with the finite element method, and within this framework, the calculation of the sensitivities with regard to the boundary conditions remains enough complicated due to the discrete nature of the problem, and for example, in shape optimization, the boundary conditions problem is even most often eluded. Besides, the deterministic methods of optimization, called gradient methods, need a reliable calculation of these sensitivities. Some other stochastic or probabilistic methods of optimization are currently in vogue, like the simulated annealing method or that of genetic algorithms, which main benefits are they assure convergence without the use of derivatives, and can be used with possibly discrete variable and nonderivable functions. The detractors of this method point up without reason the high number of calculations, especially in the case of an analysis method of finite element type. The author has a great experience in genetic algorithms ([1] to [6]). Few researches have been made on the boundary conditions optimization, and in the structural mechanics field, there are almost none. The [7] work, described below, deals with the optimization of the boundary conditions in electromagnetism. In [7], a methodology based on the genetic algorithm is proposed to determine the equivalent impedance boundary condition for corrugated material coating structures. We have find only two papers at our knowledge in the structural mechanics field [8] and [9], and only one [9] with the use of genetic algorithms. In [8], optimization of boundary conditions for maximum fundamental frequency of vibrating structures is done. In [9] the use of genetic algorithms for the selection of optimal support locations of beams is presented. Both elastic and rigid supports are considered.

II. II.1.

The Methods Used Genetic Algorithms

The genetic algorithm method has been used several times within the various problems of mechanics. These algorithms were found to be very efficient, as in the case of the damping maximization of composite beams or plates or as somewhat diverse issues. The interest of these algorithms has also been showed in the difficult case of the optimization of gears. The genetic algorithms are now well known and this article is not to introduce them in details nor generally. Although it may seem so,


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J. L. Marcelin

the third example is a dynamic case, and the objective is to maximize the first natural frequency; it is also possible to try to remove two resonance frequencies.A lot of other choices are possible, such as multi-objective functions or penalizing the objective function by limitations.

the genetic algorithm method is not magic at all. It is part of the methods called "stochastic". The most famous of this kind of method is the already old simulated annealing. The main benefit of these methods is that they operate simultaneously on a sample of the solution space. The genetic method differs from simulated annealing due to the operators used to make this population sample evolve. The convergence is always ensured toward an extremum which is not necessarily the absolute extremum, but which is more likely to be absolute than if the conventional gradient method is used. Actually, a stochastic method explores more largely the solution space. II.2.

b)

Obtained results

Before each use, the genetic algorithm asks the user to specify the values of the following parameters: - The number of individuals contained in a population, - The maximum number of generations, - The chromosomes length, - The crossover probability, - The mutation probability. It is clear that the algorithm gives best results when the chosen values for the first two parameters are high (within the limits of capacity of the used hardware). Practically speaking, the number of individuals contained in a population will be around 1 to 5 times the number of digits contained in a chromosome. However, the crossover and mutation probabilities are more difficult to choose. It has already been said that mutation is a far less frequent phenomenon than crossover; in [2], it is recommended the following values:

Optimization of the Boundary Conditions

This kind of optimization consists in combining a standard calculation program by finite elements (FE) (called thereafter analysis program) and the genetic algorithm. The analyze program is a standard FE code. This code is simply to be called each time the genetic algorithm must estimate the cost function for a given chromosome.This is done for all the individuals of the population; consequently, for example, for 20 individuals and 30 generations, there will be 600 finite element halfanalyses (the total stiffness being calculated once and for all), which is relatively low compared to the 220 possible solutions. On the opposite, for the various tests that are done, especially those introduced after, there was not necessary to implement a penalization strategy of the “total potential energy" functional by the imposed boundary conditions, because convergence was fast enough. The programmer work simply consists in drafting a "pre-analysis" program that can decode the chromosome in question and that can automatically modify the finite element data file accordingly, and then in creating a "post-analysis" program that can extract the cost function from the finite element result file. Both these programs, as the calling of the finite element code, are built in the genetic program that drives the process.

Pcrossover = 0,60 ; Pmutation =0,03 These recommenced values come from a numerical experimentation on numerous examples. In any event, the crossover probability must be clearly superior to the mutation probability because mutation is less frequent. For example, if any mutation is removed, the algorithm yet converges toward an extremum but it is unlikely to be the absolute extremum. Theoretically speaking, convergence is obtained when all the cost values of a population stabilize around a maximum value. Practically speaking, convergence is rather slow, with ebb and flow, due to the very nature of the algorithm. The user only has to stop the process when the maximum value of a population cost does not evolve anymore; he then manually selects the most interesting individual(s) of the final population to compare their benefits.

a) Choosing the coding and the objective function The problem contains two difficulties: First, the implementation of a solution code in the form of a simple and efficient chromosome and then the development of an objective function. The most generally used code is simple and natural (it has variants that are to be set forth in the examples): It can use the often used code for the boundary conditions in finite element programs, 0 being a free freedom degree and 1 a fixed freedom degree. The various codes of the concerned nodes are arranged end to end in a chromosome that is made of n binary digits that correspond to the n degrees of freedom that can be fixed. When it comes to the objective function, it depends on the posed problem. The first two examples are static cases, where the aim is to minimize the maximum displacement, or to minimize a deformation or a stress;

III. Examples Test 1 The first very easy, static test is made with the axisymmetric workpiece (of CL axis and z symmetry) illustrated in Figure 1 and aims at verifying and making the implementation of the used techniques reliable. The stiffness has been calculated once and for all but no penalization has been applied to the boundary conditions; since the calculations are fast enough for the tests, this procedure has never been implemented. For this test, the chromosome is a 10-binary-digit string, the first 5 digits are the codes of the boundary conditions of the 5 nodes that can be locked following z, and the following 5 are the codes of the boundary



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J. L. Marcelin

process of the workpiece on its support. The chosen workpiece is of z axis revolution and is illustrated on Figure 2. There are three surfaces to be machined, S1, S2, and S3. For this test, the calculations are only done for the S2 surface. Contrary to the preceding test, where the 10 selected nodes could be locked, the clamping chuck can only be applied to one of the 8 possible nodes (nodes 9 to 16); the spindle stopper can be applied to one of the 8 nodes, numbered from 1 to 8. The test remains easy and calculates the genetic algorithm’s behavior because the optimal solution can be forecasted and the number of possible solutions is limited, which would not be the case with a thinner mesh.The same type of code that in the previous test can be taken, that is to say that the first 8 chromosome digits concern the nodes 1 to 8, but the possible number of 1s in the algorithm is limited to 1 in this part of the chromosome; the 8 following digits are for the nodes 9 to 16, but any chromosome having a number of 1 greater than 1 in this part will be removed from the process. For example, 1000000001000000 is an acceptable chromosome. This code type has not been kept for this example because it leads to 16-digit-long chromosomes and assumes the genetic algorithm is modified. Another type of possible coding is to build a 2decimal-long chromosome; for example, 29 mean that the nodes 2 and 9 are subjected to boundary conditions; the first digit varies between 1 and 8, and the second one between 9 and 16.

conditions of the 5 nodes that can be locked following y; therefore, the chromosome 1011001000 corresponds to the boundary conditions applied to nodes 1, 3, 4, and 7. There are 210 possibilities. The objective is to minimize the d displacement of the node to which forces are applied. Since the genetic algorithm actually seeks the maximum of an objective function, the chosen objective is to maximize the 1-d function. The interest of this test is that the optimal solution is known: It is of course the 1111111111 chromosome, but the test helps in validating the process and in estimating how many steps are necessary for the genetic algorithm calculation to get this solution. We take here 40 individuals per population. The number of individuals in a population is usually around 1 to 5 times the chromosome’s size (here the number of digits).The maximum is reached in only 5 generations (for the crossover and mutation probabilities provided in the last part), which corresponds to 200 halfanalyses or a bit less (because a solution that appears several times during the process is calculated once and for all) and which is low compared to the possible 2 10 combinations. Non-consistent convergence is characteristic of genetic algorithms because the best individual of each population may very unlikely be eliminated; besides, if the algorithm is forced to keep only the bests, the method is not probabilistic anymore and the algorithm may be more efficient or diverge in some cases. Besides, if optimization is launched again with the same parameters, the obtained convergence is not at all the same, because the process is totally random. z

F

9 10 11 12 13 14 15 16

x

10F

10 9 8 7 6

12345678

12345

S1 S2

y

2F F

Fig. 1. Test 1: axisymmetric workpiece (of CL axis and z symmetry)

y

S3

Test 2 This test illustrates a first industrial application in the taking of workpieces in machining, always with simple data that help in validating and checking the implemented strategy. The quality of the workpiece depends on the deformations caused by the machining, because of the machining process itself or because of the holding

Fig. 2. Test 2: z axis revolution workpiece

The chosen code in this example uses 6 binary digits, as in 100011. The decoding is done as followed (let's recall that the decoding program and that of modification of the finite element data file is to be designed by the



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J. L. Marcelin

In [8], this problem is solved with a conventional gradient method, from a calculation of the frequencies’ sensitivities with regard to the boundary conditions location. Since this is only the first symmetric mode of deflection, only a quarter of the plate is meshed. [8] finds two equivalent optimal points (A and B on Figure 3) that correspond respectively to frequencies of 169.46 Hz and 169.67 Hz. Actually, In [8], it is only used a 36-element mesh for the whole plate and a study with thinner meshes has shown that the optimum is actually located between the A and B points. This test is often used in the literature; all the authors find that the optimal point is located between A and B. Implementing a genetic algorithm strategy assumes that the support point location is coded under a chromosome on the main diagonal of the quarter of the plate.With the chosen mesh, that is 15 X 15 elements, there are only 16 possibilities that can all be calculated to get the reference solution that actually corresponds to the points 7 and 8, with frequencies around 205 Hz. The code of the 16 possible points is simply a binary one: 0000 corresponds to the node 1, 0001 to the node 2, 0010 to the node 3 and so forth until the node 16 (1111). Let’s recall that the objective is to maximize the first frequency. For a 4-individual population, a 2individual population (0110, node 7 and 0111, node 8) is obtained after the thirtieth generation, which shows the genetic algorithm convergence, but its efficiency is more convincing with longer chromosomes (as in the example 1).

user for each new example and must be placed immediately before the analysis): The first 3 digits give the code of the forced node 1 to 8, according to the following correspondence: 000 (node 1), 001 (node 2), 010 (node 3), 011 (node 4), 100 (node 5), 101 (node 6), 110 (node 7), 111 (node 8), and the following 3 digits provide the forced node 9 to 16 code, according to the same type of correspondence; therefore the example 100011 matches to the forced nodes 5 and 12. Of course, this example is still an easy test because only 64 combinations are possible and they can all be calculated to reach the problem optimum. The objective is to minimize the maximum equivalent deformation or the equivalent Von Mises stress that appears where forces are applied. The best solution found by the genetic algorithm is the combination of nodes 8 and 16, for which the Von Mises stress equals to 17.009 daN/mm2. This result is found after a dozen finite element calculations (and from the second generation for a 6individual population).In contrast, the genetic algorithm can be instructed to find the less good solution: the program is launched again with the objective of maximizing the main Von Mises stress; and this less good solution is the combination of nodes 1 and 9; for which the Von Mises stress equals to 17.195 daN/mm2. The test remains easy because the mesh size is limited. It could be more complicated if the mesh was thinner and if the genetic algorithm was instructed to find a compromise solution that would be valid to machine the S1, S2 and S3 surfaces. Test 3 This test takes up the dynamic test offered in [8] and helps invalidating the implemented strategy, once again on an easy case. The chosen example is that of a square plate, measuring 30.5 cm with a thickness of 0.328 cm in deflection vibrations (Young's modulus 73.1 GPa, density 2,821 kg/m3). This plate rests on 4 points that are located symmetrically on the diagonals (Figure 3). The objective is to find the optimal location of the supports, maximizing the first fundamental frequency.

IV.

Conclusion

This study has shown the efficiency of genetic algorithms in responding to the problem of the optimization of the boundary conditions in finite elements. This study is above all a feasibility study and will soon be complemented by industrial examples.The study can easily be spread to other fields than mechanics; for example, in thermal science, it would be easy to design chromosomes containing not only the information on the boundary condition type, but also that regarding the condition value to be optimized (flow value, heat transfer coefficient value); it could also be applied in fluid mechanics. The efficiency can still be improved in the case of important calculations (e.g. shape optimization), using neural networks to analyze the problem, instead of using a conventional finite element analysis. Actually, the use of neural networks to model mechanical structures appears to give good results [10]. It would be possible to make the learning of a neural network in parallel with the first generations that would be calculated by finite elements (that is using the results of the finite element analyses).Once the learning stage is over, the neural network would completely replace the calculations by finite elements. The calculations would

y

A

B x Fig. 3. Test 3: square plate



53

J. L. Marcelin

therefore become much faster, and the genetic algorithm method, contrary to the deterministic methods, does not need extremely precise calculations of the objective function. Translated by Amandine MARCELIN, on behalf of AMTrad’gram (www.amtradgram.com).

Authors’ information Laboratorie G-SCOP (Grenoble, Sciences pour la Conception, Optimisation et Production) Grenoble-INP/UJF-Grenoble1/CNRS, UMR5272 38031 GRENOBLE France. E-mail : [email protected] Jean-Luc Marcelin is born 31 march 1956. J.L. Marcelin is a doctor engineer of the “Ecole des Mines” of Paris (1983), and entitled to direct search to the University Joseph Fourier of Grenoble (1997). He is currently an associate professor at the University Joseph Fourier. He carries out search in optimization of structures and mechanical systems. He is an author or a co-author of 74 scientific publications, including 49 international publications, and 5 books of general interest.

References [1]

J.L. Marcelin,“Genetic optimization of stiffened plates and shells”,Int. J. for Numerical Methods in Engineering, Vol. 51, n°9, p.1079-1088, 2001. [2] J.L. Marcelin,“Genetic optimization of gears”, International Journal of Advanced Manufacturing Technology, vol.17, n°12, p.910-915, 2001. [3] J.L. Marcelin,“Genetic optimization of stiffened plates without the FE mesh support”,Int. J. for Numerical Methods in Engineering, Vol.54, n°5, p. 685-694, 2002. [4] J.L. Marcelin, “Using Genetic algorithms for the optimization of mechanisms”,International Journal of Advanced Manufacturing Technology, vol. 27, n°1-2, p. 2-6, 2005. [5] J.L. Marcelin, “Rayleigh-Ritz function approximations in computationally intensive genetic design”, International Review of Mechanical Engineering (IREME), Vol. 1, n°2, p. 166 -173, 2007. [6] J.L. Marcelin, “Optimization of vibration frequencies of rotors via Rayleigh-Ritz method and genetic algorithms”, International Review of Mechanical Engineering (IREME), Vol. 2, n°1, p. 144 148, 2008. [7] T.Su and H. Ling, “Determining the equivalent impedance boundarycondition for corrugated coatings based on the geneticalgorithm”, IEEE Transactions on Antennas and Propagation, Vol.48, no.3, pp. 374-382, 2000. [8] J.H. Son and B.M. Kwak,“Optimization of boundary conditions for maximum fundamental frequency of vibrating structures”,AIAA Journal, Vol. 12, n°31, p. 2351 -2357, 1993. [9] B.P. Wang and J.L. Chen, “Application of geneticalgorithm for the support location optimization of beams”, Computers and Structures, Vol.58, no.4, pp. 797-800, 1996. [10] J.L. Marcelin, “Cognitive optimization of mechanical structures”, International Review of Mechanical Engineering (IREME), Vol. 5, n°1, p. 88 -91, 2011.



54


Stress Analysis of FG Thick Pressure Vessels Considering the Effects of Material Gradations and Poisson’s Ratio Using DQ Method A. M. Goudarzi, S. Saadati, A. Paknahad

Abstract – A numerical method has been proposed to obtain radial displacements of functionally graded thick-walled cylindrical pressure vessels. Also stress distributions for both plane-stress and plane-strain assumptions are calculated in radial and circumferential directions. The material is assumed to be isotropic where both of the elastic coefficients, i.e. Young’s modulus and Poisson’s ratio, are permitted to vary in the radial direction. Tree types of material gradations are considered for functionally graded material. The effects of spatial variation of Poisson’s ratio and material gradation upon the radial displacements and stress distributions are investigated. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: FGM, Pressure Vessel, Differential Quadrature Method, Poisson’s Ratio

FGMs have attracted considerable research efforts over the past few years due to their increased applications in many engineering fields [2], [3] such as aircraft, aerospace, and nuclear industries. Recently, there has been considerable interest in the development of graded materials that have spatial property variations deliberately created to improve the mechanical performance [4]-[6]. The graded compositions of such materials is commonly established and controlled using advanced manufacturing techniques, including powder metallurgy, chemical vapor deposition, centrifugal casting and other schemes [7]. The works concerned with the stress analysis of cylindrical and spherical structural elements involve finite elements and other numerical techniques due to the nature of functions chosen to describe the inhomogeneous properties [8]-[10]. Modeling of density and stiffness by the same Power-law are proposed in [11]-[13] which functionally graded material considered in volume fractions follows the Power-law distribution. Developing sufficiently general methods for solving specific boundary value problems in solid mechanics involving inhomogeneity has always been difficult. Because of this difficulty, all existing treatments dealing with the mechanics of inhomogeneous solids are based on a simple function representing material inhomogeneity [14], [15]. A similar work was also published for cylindrical and spherical vessels where closed form solution is obtained where just Young’s modulus obeying a simple Power-law through the wall thickness [16]. The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials is investigated by Horgan and Chun [17]. The stress analysis in thick-walled FGM cylinders is studied by Tutuncu [18] where the material is assumed to be

Nomenclature C E er eθ

Weighting coefficient Young’s modulus Radial strain Circumferential strain

erθ

Radial-circumferential strain

L(r) N P r u w β

η λ µ ν σr σθ

Lagrangian polynomial Number of grid points Pressure Radius Radial displacement Longitudinal displacement Inhomogeneity constant of Young’s modulus Inhomogeneity constant of Poisson’s ratio Lame’s constant Shear modulus Poisson’s ratio Radial stress Circumferential stress

σz

Longitudinal stress

I.

Introduction

Functionally graded materials (FGMs), can be considered second generation composite materials. FGMs are essentially two-phase particulate composites synthesized in such a way that the volume fractions of the constituents vary continuously in the thickness direction to give a predetermined composition profile [1].


55


A. M. Goudarzi, S. Saadati, A. Paknahad

III.1. Plane-Stress Assumption

isotropic with constant Poisson’s ratio and exponentiallyvarying Young’s modulus through the wall thickness. Numerical investigation of the stress intensity factor in FGMs considering the effect of graded Poisson’s ratio is reported by Ghajar and Moghadam [19]. The effects of Poisson’s ratio and material gradation on the functionally graded cantilever beam are studied by Anandakumar and Kim [20]. This work is aimed to present the stress analysis of thick-walled FG cylindrical vessel for three types of material gradation such as Linear, Exponential and Power-law upon both of the elastic coefficients. Generally when the functional dependence is assumed for both of the elastic coefficients, a simple tractable solution cannot be obtained and numerical techniques must be used. Differential quadrature method and its applications were rapidly developed in recent decade [21], [22]. Fast convergence rate and simple use, are the reasons for employing differential quadrature method in literatures. In this work, this technique is used for solving the governing equations.

II.

Using the plane-stress assumption, in the polar coordinates, the normal stresses can be expressed as [23]:

σr =

σθ =

E ( er + ν eθ ) 1 −ν 2

E ( eθ + ν er ) 1 −ν 2

σz = 0

(5)

(6) (7)

where E and ν are the Young’s modulus and Poisson’s ratio respectively. Fig. 1 illustrates body of the problem.

Material Gradations

In this paper, Linear, Exponential, and Power-law gradations are considered. Assuming material properties vary in the radial direction where defined as [20]:

Fig. 1. Represent of thick-walled cylindrical pressure vessel

Linear material gradation:

III.2. Plane-Strain Assumption

E ( r ) = E0 (1 + β r ) , ν ( r ) = ν 0 (1 + η r )

In plane-strain assumption the body forces and tractions on the lateral boundaries are independent of the z-coordinate and have no z component then the displacements can be taken in the following reduced form [7]: u = u ( r ) , w=0 (8)

(1)

Exponential material gradation: E ( r ) = E0 e β r , ν ( r ) = ν 0 eη r

(2)

Power-law material gradation: β

Stress-strain relations in plane-strain assumption are [23]:

η

⎛r⎞ ⎛r⎞ E ( r ) = E0 ⎜ ⎟ , ν ( r ) = ν 0 ⎜ ⎟ ⎝ ri ⎠ ⎝ ri ⎠

(3)

where E0 and ν 0 are the Young’s modulus and Poisson’s ratio at r = ri where ri is the inner radius and β , η are constants of inhomogeneity respectively.

σ θ = λ ( er + eθ ) + 2 µ eθ

(10)

λ=

In this section, stress distribution in thick-walled cylindrical pressure vessel will be accounted. The strain-displacement equations in polar coordinates for axisymmetric condition are familiar as [23], [24]: du u , eθ = , erθ = 0 dr r

(9)

where λ and µ are Lame’s constant and shear modulus respectively as:

III. Governing Equation

er =

σ r = λ ( er + eθ ) + 2 µ er

νE

(1 +ν )(1 − 2ν ) µ=

E 2 (1+ν )

(11)

(12)

The equilibrium equation in absence of the body forces is:

(4)



56


∂σ r (σ r − σ θ ) + =0 ∂r r

2 Cij( ) =

(13)

Substituting radial and hoop stresses into the equilibrium equation, the final form of Navier’s equation is obtained (see Appendix). The boundary conditions are considered:

σ r |r = ri = − Pi , σ r |r = ro = 0

IV.

N

∑ Cik(1)Ckj(1)

(18)

k =1

It is shown that one of the best options for obtaining grid points is zeros of the well-known Chebyshev polynomials which are expressed by the following equation in the radial direction [29]:

(14)

⎡⎛ r − r ⎞ ⎛ ( i − 1) π ri = ⎢⎜ 0 i ⎟ ⎜1 − cos N −1 2 ⎠⎝ ⎣⎝

⎞⎤ ⎟⎥ + ri ⎠⎦

( i = 1, 2,...,N )

(19)

Differential Quadrature Method Finally, in this study the number of grid points for meshing is chosen as 9, 11 and 13. Radial displacements are obtained in each grid point and then stress distributions in both assumptions are studied. The accuracy of the results is improved as the number of grid point increase. It is found that convergence is reached with grid number of 13.

The differential quadrature (DQ) method, introduced by Bellman and Casti [21], is a numerical technique for the solution of initial and boundary value problems. Bert et al. [25] first used the DQ method to solve the problems in the structural mechanics and since then the method has been applied to a variety of problems successfully. The essence of the DQ method is that the partial derivative of a function with respect to a variable is approximated by a weighted sum of function values at all discrete points in that direction. For completeness, one dimensional problem is presented in this paper as same as governing equation. In DQ method, the k-th order derivative of the solution function u ( r ) at the grid

V.

Stress analysis for thick-walled FG cylindrical pressure vessel using DQ method is performed. Elastic coefficients are varied through the wall thickness as three types of gradation that mentioned above. Radial displacements and stress distributions are normalized by the corresponding displacement and pressure at the inner radius respectively. The results of this study focused on the effects of Poisson’s ratio which in all cases are extracted for β = 3, ro / ri = 3 . Fig. 2 is shown radial displacements across the wall thickness in the planestress assumption.

point i in the radial direction can be computed by [26]: ∂k u ( r ) ∂r k

N

r =ri

( ) ( i = 1, 2,...,N )

k = ∑ Cij( )u r j j =1

(15)

where N is the number of grid points in the variable

( )

domain, u r j


are the field variable values at these

k points and Cij( ) are related weighting coefficients of the

k-th order derivative. The weighting coefficients of the first order derivative are written explicitly by [27], [28]: 1 C ( )ij =

LN ( r )i

(( r − r ) L i

C

j

N

(r ) j )

(i ≠ j )

⎛ 1 ⎞ ⎜ ⎟ = ⎟ r r − j =1, i ≠ j ⎜ i j ⎝ ⎠

(16)

N

(1) ii

∑ (

)

where LN ( r )i is the Lagrangian polynomials expressed by [25]:

Fig. 2. Radial displacement through the wall thickness (Plane-Stress)

LN ( r )i = ( r − r1 ) ...( r − ri −1 )( r − ri +1 ) ..( r − rN ) (17)

The radial and hoop stress distributions are represented in Figs. 3, 4. The effects of Poisson’s ratio for all gradations increase except the radial stress distribution. As seems for r/ri < 2 Power-law and Linear gradations have minimum and maximum values for hoop stress respectively, and for r/ri > 2 this happen vice versa.

The weighting coefficients of second order derivatives in the DQ method may be computed by:



57


The radial displacements and stress distributions in radial direction for plane-strain assumption are shown in Figs. 5, 6, 7 which have a similar behavior like plane-stress case.

In this case, the influence of Poisson’s ratio is more apparent than the plane-stress assumption especially in Power-law gradation. Generally the changeable nature of the FGMs with Poisson’s ratio is inevitable and Powerlaw gradation is more sensitive to this phenomena. Therefore, this is indicating that Power-law material gradation is a practical pattern for analyzing of FGMs behavior. This highlights the importance of the subject for manufacturing methods in FGMs fabrication.

Fig. 3. Radial stress distribution through the wall thickness (Plane-Stress)

Fig. 6. Radial stress distribution through the wall thickness (Plane-Strain)

Fig. 4. Hoop stress distribution through the wall thickness (Plane-Stress)

Fig. 7. Hoop stress distribution through the wall thickness (Plane-Strain)

VI.

Conclusion

Differential quadrature method is proposed to analyze thick-walled FG cylindrical pressure vessels. Elastic coefficients are allowed to vary across the wall thickness as three types of gradation. The obtained results using the DQ method converge as the number of grid point increase. Various kinds of gradations are affected the stress distributions and radial displacements strongly. Influence of Poisson’s ratio in plane-strain assumption is more obvious than the other one. Finally this is noticed that the results were shown in this paper are only for scientific analysis.

Fig. 5. Radial displacement through the wall thickness (Plane-Strain)



58


Appendix The Navier’s equations with Plane-stress and Plane-strain assumptions are derived. The Poison’s ratio and the elasticity modulus in these formulas are function of radial coordinate.

Plane-stress equation: −

1

( −1 + v ( r ) )

2 2

⎡ 2⎛ d 2⎤ ⎞⎛ d ⎞⎡ ⎢ r ⎜ dr E ( r ) ⎟ ⎜ dr u ( r ) ⎟ ⎣ −1 + υ ( r ) ⎦ + 2 ⎣ ⎝ ⎠ ⎝ ⎠ r

⎡ 2 ⎛d ⎞ ⎛d ⎞⎛ d ⎞ + rυ ( r ) ⎜ E ( r ) ⎟ ⎡ −1 + υ ( r ) ⎤ + rE ( r )υ ( r ) ⎢ −2r ⎜ υ ( r ) ⎟⎜ u ( r ) ⎟ + ⎦ ⎝ dr ⎠⎣ ⎝ dr ⎠⎝ dr ⎠ ⎣ ⎡ ⎛ d2 ⎛ d2 ⎞ ⎞ ⎛d ⎞ ⎛d ⎞⎤ −υ ( r ) u ( r ) ⎜ υ ( r ) ⎟ + rυ ( r ) ⎜⎜ 2 u ( r ) ⎟⎟ + υ ( r ) ⎜ u ( r ) ⎟ ⎥ + rE ( r ) ⎢ r ⎜⎜ 2 u ( r ) ⎟⎟ + ⎝ dr ⎠ ⎝ dr ⎠⎦ ⎝ dr ⎠ ⎠ ⎣⎢ ⎝ dr ⎡ ⎛d 2 ⎤⎤ ⎞ ⎛d ⎞⎤ − ⎜ u ( r ) ⎟ ⎥ + E ( r ) u ( r ) ⎢ −r ⎜ υ ( r ) ⎟ + 1 − υ ( r ) ⎥ ⎥ = 0 ⎠ ⎝ dr ⎠⎦ ⎣ ⎝ dr ⎦⎦

Plane-strain equation: 1

(1 + v ( r ) ) ( −1 + 2 v ( r ) )2 r 2 2

⎣⎡ r E ( r ) υ ( r ) ⎣⎡-2rυ ( r ) ⋅

⎛ d2 ⎞ ⎛ d2 ⎞ ⎛ d2 ⎞ ⎛d ⎞⎛ d ⎞ ⋅ ⎜ υ ( r ) ⎟ ⎜ u ( r ) ⎟ − 2r ⎜ u ( r ) ⎟ − rυ ( r ) ⎜ u ( r ) ⎟ + 2rυ ( r )2 ⎜ u (r )⎟ + ⎜ dr 2 ⎟ ⎜ dr 2 ⎟ ⎜ dr 2 ⎟ ⎝ dr ⎠ ⎝ dr ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎡⎛ d ⎛d ⎞ ⎛d ⎞ ⎛d ⎞ ⎛d ⎞⎛ d ⎞⎤ ⎞ −2 ⎜ u ( r ) ⎟ − υ ( r ) ⎜ u ( r ) ⎟ + 2υ ( r )2 ⎜ u ( r ) ⎟ +4r ⎜ υ ( r ) ⎟ ⎜ u ( r ) ⎟ ⎥ + rυ ( r ) u ( r ) ⎢⎜ E ( r ) ⎟ + dr dr dr dr dr dr ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠⎦ ⎠ ⎣⎝ ⎛d ⎞ ⎛d ⎞ ⎛d ⎞⎤ −υ ( r ) ⎜ E ( r ) ⎟ − 2υ ( r )2 ⎜ E ( r ) ⎟ + 2υ ( r ) E ( r ) ⎜ υ ( r ) ⎟ ⎥ + ⎝ dr ⎠ ⎝ dr ⎠ ⎝ dr ⎠⎦ ⎡ ⎛d ⎞⎛ d ⎞ ⎛d ⎞⎛ d ⎞ ⎛d ⎞⎛ d ⎞⎤ + rυ ( r ) ⎢ −2r ⎜ E ( r ) ⎟⎜ u ( r ) ⎟ − rυ ( r ) ⎜ E ( r ) ⎟ ⎜ u ( r ) ⎟ + 2rυ ( r )2 ⎜ E ( r ) ⎟ ⎜ u ( r ) ⎟ ⎥ + ⎝ dr ⎠⎝ dr ⎠ ⎝ dr ⎠ ⎝ dr ⎠ ⎝ dr ⎠ ⎝ dr ⎠⎦ ⎣ ⎡ 2 ⎞⎤ ⎛d ⎞ ⎛d ⎞ ⎛d + rE ( r ) ⎢u ( r ) ⎜ υ ( r ) ⎟ + ⎜ u ( r ) ⎟ + r ⎜ u ( r ) ⎟ ⎥ + E ( r )υ ( r ) u ( r ) ⎡ 2 + υ ( r ) − 2υ ( r )2 ⎤ + ⎟⎥ ⎣⎢ ⎦⎥ ⎢ ⎝ dr ⎠ ⎝ dr ⎠ ⎜⎝ dr 2 ⎠⎦ ⎣ ⎤ ⎛d ⎞⎛ d ⎞ + r 2 ⎜ E ( r ) ⎟ ⎜ u ( r ) ⎟ − E ( r ) u ( r )⎥ = 0 dr dr ⎝ ⎠⎝ ⎠ ⎦ under thermal loading due to fluid, Japan Society of Mech. Eng. Int. J. Ser I: Solid Mech., 39(4), pp. 573-581, 1996. [7] M. H. Saad, Elasticity; Theory, Applications, and Numerics (2nd edition, Academic press Burlington, MA, 2009) [8] Y. Fuki, N. Yamanaka, Elastic analysis for thick-walled tubes of functionally graded material. JSME Int. J., Ser I: Solid Mech. Strength Material, 35(4), pp. 379-385, 1992. [9] R. S. Salzar, Functionally graded metal matrix composite tubes. Comput. Eng., 5(7), pp. 891-900, 1995. [10] C. T. Loy, K. Y. Lam, J. N. Reddy, Vibration of functionally graded cylindrical shells, Int. J. of Mech. Sci., 41(3), pp. 309-324, 1999. [11] C. W. Bert, F. W. Nidenfuhr, Stretching of a polar-orthotropic disk of varying thickness under arbitrary body forces, AIAA. J. 1(6), pp. 1385-1390, 1963. [12] T. Y. Reddy, H. Srinath, Elastic Stresses in a rotating anisotropic annular of disk variable thickness and variable density. Int. J. of Mech. Sci., 16, pp. 85-90, 1974.

References [1] [2]

[3]

[4]

[5]

[6]

F. Erdogan, Fracture mechanics of functionally graded materials, Compos. Eng., 5(7), pp. 753-770, 1995. R. C. Wetherhold, S. Seelman, J. Z. Wang, Use of functionally graded materials to eliminate or control thermal deformation, Compos. Technol., 56(4), pp. 1099-1104, 1996. V. Briman, L. W. Byrd, Modeling and analysis of functionally graded materials and structure, Appl. Mech., 60, pp. 195-216. 2007. X. D. Zhang, D. Q. Liu, C. C. Ge, Thermal stress analysis of axial symmetry functionally graded materials under steady temperature field, J. of Functionally Graded Materials, 25, pp. 573-581, 1994. Y. Obata, N. Noda, Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material, J. of Thermal Stresses, 17(3), pp. 471-487, 1994. S. Takezono, K. Tao, E. Inamura, M. Inoue, Thermal stress and deformation in functionally graded material shells of revolution



59


[13] G. V. Gurushankar, Thermal stresses in a rotating, nonhomogeneous, anisotropic disk of varying thickness and density. J. of Strain. Anal., 10(3), pp. 137-142, 1975. [14] M. K. Kassir, Boussinesq problems for a non-homogeneous solid. J. Eng. Mech., 98, pp. 457-470, 1972. [15] M. K. Kassir, M. F. Chauprasert, A rigid punch in contact with a non-homogeneous solid. J. of Appl. Mech., 42, pp. 1019-1024, 1974. [16] N. Tutuncu, M. Ozturk, Exact solution for stresses in functionally graded pressure vessels. Compos. Part B: Eng., 32(8), pp. 683-6, 2001. [17] C. O. Horgan, A. M. Chun, The pressurized hollow cylinder or disk problem for functionally graded isotropic linearly elastic materials, J. of Elasticity, 55, pp. 43-59, 1999. [18] N. Tutuncu, Stress in thick-walled FGM cylinders with exponentially-varying properties, Eng. Struct., 29, pp. 2032-2035, 2007. [19] R. Ghajar, A. S. Moghadam, Numerical investigation of the mode three stress intensity factors in FGMs considering the effect of the graded Poisson’s ratio, Eng. Fracture Mech., In press, 2010. [20] G. Anandakumar, J. H. Kim, On the modal behavior of a threedimensional functionally graded cantilever beam: Poisson’s ratio and material sampling effects, Compos. Struct., 92, pp. 13581371, 2010. [21] R. E. Bellman, J. Casti, Differential quadrature and long-term integration, J. of Mathematical Analysis and Applications. 34, pp. 235-238, 1971. [22] C. W. Bert, M. Malik, Differential quadrature in computational mechanics, A review Applied Mechanics, 49, pp. 1-27, 1996. [23] S. Timoshenko, J.N. Goodier, Theory of Elasticity (2nd edition, McGraw-Hill, New York, 1970). [24] M. M. S. Fakhrabadi, M. Dadashzadeh, V. Norouzirfard, B. Dadashzadeh, Stress-Strain Analysis of the Horizontal Pressure Vessel on Two Saddle Supports, International Review of Mechanical Engineering (IREME), Vol. 5, n. 4, pp. 695-701. [25] C. W. Bert, S. K. Jang, A. G. Striz, Two new approximate methods for analyzing free vibration of structural components, AIAA J., 26, pp. 612-618, 1988. [26] C. Shu, Differential Quadrature and its Application in Engineering (London: Springer-Verlag, 2000). [27] C. Shu, B. E. Richards, Application of generalized differential quadrature to solve two-dimensional incompressible NavierStokes equations, Int. J. Num. Meth., Fluids, 15, pp. 791-8, 1992. [28] Md. Moslemuddin Fakir, S. Basri, R. Varatharajoo, A. A. Jaafar, A. S. Mohd. Rafie, D. L. A. Majid, International Review of Mechanical Engineering (IREME), Vol. 2, n. 3, pp. 483-488, 2008. [29] C. Shu, C. M. Wang, Treatment of mixed and non-uniform boundary condition in GDQ vibration analysis of rectangular plattes, Eng. Struct., 21, pp. 125-34, 1999.

Authors’ information Department of Mechanical Engineering, Babol University of Technology, P.O. Box: 484, Babol, Mazandaran, Iran. Fax: +98 21 44173808 E-mail: [email protected] Dr. A. M. Goudarzi is assistant professor of Mechanical Engineering Department in Babol University of Technology, Babol, Mazandaran, Iran. His research interest is related to Applied Mechanics, Design Methods in Engineering, and Advanced Computer Aided Design. He is teaching the graduate and undergraduate courses for many years. E-mail: [email protected] S. Saadati is MSc. student of Mechanical Engineering, Babol University of Technology, Babol, Mazandaran, Iran. His main research interests are Applied Mechanics, Stress Analysis, Contact Mechanics, Finite Element Method and Application of Numerical Methods in engineering. He published some papers in these areas since 2009. E-mail: [email protected] A. Paknahad is MSc. student of Mechanical Engineering, Babol University of Technology, Babol, Mazandaran, Iran. He is researcher in Solid Mechanics and published some articles in Dome design of Composite Pressure vessels and Stress analyzing of Anisotropic Shell and Plates. He is interested in Structural Mechanics using FEM and Optimization Algorithms since 2008. E-mail: [email protected]



60


Iterative Algorithm for Active Vibration Control of Flexible Beam Mohd S. Saad1, Hishamuddin Jamaluddin2, Intan Z. M. Darus2

Abstract – This paper presents the development of dynamic model of a flexible beam structure using finite difference method. A Simple Proportional (P) control scheme is applied to suppress vibration at the tip of the flexible beam. The performance of P controller is studied by gradually increasing manually the proportional gain until significant attenuation of the vibration is observed. Then the controller is further extended to self-tune the proportional gain by using an intelligent mechanism known as Proportional Iterative Learning Algorithms (P-type ILA). The robustness of both controllers in suppressing the vibration is investigated by changing the beam’s physical parameter, applying disturbance at different segments and amplitudes respectively. The simulation results clearly revealed the effectiveness and robustness of a self-tuning proportional control over conventional P control scheme as active vibration control of a flexible beam. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Flexible Beam, Finite Difference, Iterative Learning, Vibration Control

Therefore, many attempts have been proposed to reduce this unwanted disturbance by considering passive and active controls. Passive vibration control methods work well at high frequencies or in a narrow frequency range but often have the disadvantage of added weight and poor low frequency performance. Meanwhile, the potential of Active Vibration Control (AVC) to solve the problem has been demonstrated [1]. The concept of AVC was initially proposed by Lueg [2] for noise cancellation. AVC works based on artificially generating the cancellation signal to absorb the unwanted disturbance force that can reduce the effect of vibration to the system. Vibration suppression in AVC can be achieved by detecting and processing via suitable control schemes, thus the superimposed disturbance signals will cancel out the actual disturbance force. Several strategies based on closed-loop control scheme have been proposed in AVC system such as Sliding Mode Control (SMC), Fuzzy Control (FC), Self-Tuning Control and Intelligent Algorithms [3], [4] and [5]. AVC problem of flexible beams has attracted significant interest due to its generic nature and easily applied in many practical problems such as robot manipulators, aircrafts, electrical machines and civil structures. The development of various control strategies has been widely studied where the performance of the control schemes has been analyzed via simulation and experimental studies. Haichang and Song [6] proposed robust model reference controller and shown the robustness and effectiveness of the proposed method even in the presence of varying modal frequency due to changes in the mass of the flexible beam. Itik et al. [3]

Nomenclature U Y x t µ E I ρ A m M V S F K e Ga G Gs Φ

The actuating force The beam’s deflection The beam distance The time The beam constant The Young’s modulus The moment of inertia The mass density Cross-sectional area The mass of the beam The shear The bending moment The stiffness matrix The force The controller gain The error signal The actuator transfer function The flexible beam transfer function The sensor transfer function The proportional learning parameter

I.

Introduction

Vibration control has been widely applied in many applications including automotive, aircraft, electrical machinery and civil structures. Vibration occurs whenever a mechanical mechanism is moved intentionally or unintentionally. The unwanted vibration may cause damage to structures or degradation to system’s performance. Manuscript received and revised December 2011, accepted January 2012

61


M. S. Saad, J. Hishamuddin, I. Z. M. Darus

employed sliding mode control and H infinity control schemes using state space modeling approach. By performing experimental identification method, an estimated transfer function which represents the system was formed. The two control strategies were applied to the same system and the experimental results showed the success of the control approaches. Fei [7] also revealed that, the used of adaptive feed-forward sliding mode control and model reference adaptive sliding mode control are effective in vibration suppression problem. One of the flexible beam modeling technique that has been effectively used in AVC is finite difference (FD) method. This method is employed by discretizing the ordinary differential equation (ODE) representing the beam. Then, the dynamics behavior of the beam was studied and a suitable test and verification platform were developed. The application of FD method is found to be more appropriate, and relatively lesser amount of computation making the technique easily realized in realtime applications [8]. Tokhi and Hussain [9] studied the behavior of a beam by using FD method and incorporate it with active vibration control. The performance of AVC was verified via simulation study. Efficient vibration cancellation was shown using a set of disturbance frequency. Mohd Hashim et al. [10] investigated further the performance of AVC via FDM model structure by introducing intelligent strategies including Genetic Algorithm (GA), Artificial Neural Network (ANN) and Adaptive Neuro-Fuzzy Inference System to develop the mechanisms of an AVC system. Comparative study revealed that Neuro-Fuzzy Inference System is the best choice for suppressing the vibration of flexible beam system [4]. The application of Active Force Control (AFC) technique has been effectively used in the control of vehicle suspension and vibration of flexible structures. Mailah and Priyandoko [11], developed a new control scheme which is a combination between Adaptive Fuzzy and Active Force Control applied to an active suspension system of a quarter car model. The performance of this new scheme was found to be superior than AFC alone. Due to the simplicity of AFC structure and the superior performance in active suspension system, Tavakolpour [12] has successfully proved by simulation the effectiveness of AFC in attenuating the vibration of flexible plate. It is found that AFC does not only suppress the vibration at the target point but also around the observation point which is an interesting feature. Tavakolpour et al. [13] employed a simple methodology for the active vibration control of a flexible plate structure, which is known as High Gain Feedback Regulator (HGFR). Simulation results revealed the effectiveness of HGFR to attenuate the unwanted vibrations. However, the value of the gain needs to be optimized for better attenuation in order to avoid instability problem.

Iterative Learning Algorithms (ILA) was formerly proposed by Arimoto et al. [14]. They have proven the effectiveness of the proposed learning scheme by controlling the motion trajectory of a mechanical robot. Since then, many researchers widely applied this concept in numerous applications such as manufacturing, robotics and automation. The concept behind ILA is that the performance of the controller will improve at every cycle due to the learning action from the previous cycle. Recently, an extensive effort has been made to implement ILA in active vibration control for flexible structures. Md. Zain and Tokhi [15] studied the effectiveness of Proportional Differential Iterative Learning Control (PDILC) structure to suppress the vibration of the rigid-body motion of the flexible manipulator. The controller has successfully reduced the vibration significantly. Pitowarno and Mailah [16] developed a novel method to control mobile manipulator where the Iterative Learning Algorithm (ILA) is combined with AFC and Proportional Integral control scheme to compensate the dynamic effect of the disturbances that includes the impact force and vibratory excitation applied to each wheel and joint of a mobile manipulator. Results showed that an iterative learning technique has significantly improved the overall performance of the system in reducing the track error. More recently, Tavakolpour et al. [13] applied the P-type Iterative Learning Control (Ptype ILC) to flexible plate structure via FD simulation platform. They demonstrated the robust performance of this control scheme in spite of parametric changes in the dynamic system. Hassan et al. [17] also showed that Proportional Integral Derivative Active Force Control (PID-AFC) with Iterative Learning (IL) method is able to suppress the vibration significantly in the presence of internal and external disturbances with zero reference input signal but not the Proportional Integral Derivative (PID) control scheme alone. Wijdeven and Bosgra [18] have demonstrated experimentally the capability of Hankel ILC in suppressing the vibration of a complex Multiple Input Multiple Output (MIMO) flexible structure that performed a point-to-point motion. Previous simulation works have shown the effectiveness of the control scheme in suppressing vibration of the beam. However, only a few researchers have demonstrated the simulation model based on FD method. FD method offers many advantages compared with the other methods such as continuous time or Laplace transform because using FD method, the data will be discretized which makes the technique suitable in real-time applications [9]. From the previous literature, it is revealed that the simulation of AVC via FD method in flexible structures or beam by self-tuning control scheme has been actively reported, but unfortunately their control structures are quite complex which is involved with system identification techniques [13], [5], [4], [8] and [10]. It is important for us to study the performance of self-tuning proportional control scheme



62


in suppressing the beam’s vibration using simple, robust and effective technique. This research also presented a new simulation environment using LabVIEW graphical programming where it provides rapid program development and user-friendly graphical user interface. Furthermore, real time application can be done from the same control algorithms with minor modifications. Besides that, self -tuning proportional control scheme offers a great advantageous in terms of insensitive to parameter change, simple control structure, and robustness to disturbance change which is highlighted in the this study. This research aims to provide a suitable simulation platform to study the effectiveness and robust performance of self-tuning proportional control over conventional P controller in attenuating the vibration in spite of parametric changes in the dynamic system which occur due to the change of disturbance excitation position on a beam and its magnitude. The simulation model of the AVC flexible beam system is developed in order to investigate the dynamic behavior further. LabVIEW graphical programming software has been chosen to be the simulation platform for FD model and proportional control strategy. The deflection of the beam can be observed dynamically at a finite duration of time. The simulated model is validated by comparing the resonance modes with the theoretical values. The control performance is evaluated by varying the proportional gain until it reaches the optimum value without saturating the actuator force. Then the controller is further extended to self-tuning proportional control by introducing an intelligent mechanism known as Proportional Iterative Learning Algorithms (P-type ILA). The performance of P-type ILA is also studied by varying the error signal and its learning parameter. Later on, the robustness of both controllers in suppressing the vibration is investigated by changing its physical parameter, exciting the disturbance to the beam at different segments and amplitudes respectively.

II.

EI , with E, I, ρA ρ and A representing Young’s modulus, moment of inertia, mass density and cross-sectional area respectively, and m is the mass of the beam. The model in Eq. (1) does not have damping, so there is no energy loss in the model mathematically. The boundary conditions at fixed and free end of the beam are given by:

is the beam constant represented by µ 2 =

y ( 0,t ) = 0 ∂y ( 0 ,t ) ∂x

M ( L,t ) = V ( L,t ) =

∂x

4

+

∂ 2 y ( x,t ) ∂t

2

=

1 u ( x,t ) m

∂ 3 y ( L,t ) ∂x3

(2)

=0

Fig. 1. Schematic diagram of a flexible beam system

Finite difference (FD) method is chosen to obtain the numerical solution of the PDE in Eq. (1). Simulations of flexible plate and beam via FD method are easy to implement and the method has been proven effective in investigating dynamics behavior of structures [4], [5], [9], [19] and [20]. The beam is discretized into a finite number of equal-length sections (segments), each of length, ∆x, and the deflection of beam at the end of each segment is sampled at a constant time, ∆t. By using firstorder central FD and its boundary conditions in Eq. (2), the PDE in Eq. (1) becomes:

A flexible beam of length, L, in fixed-free mode is considered. A schematic diagram of the flexible beam is shown in Fig. 1. Force is applied at distance, x, from the fixed end at time, t, and the resulting deflection from its stationary position is denoted by u(x,t) and y(x,t) respectively. The motion of the beam in transverse vibration is formulated by fourth-order partial differential equation (PDE) that yields the following equation, [9] and [19]: ∂ 4 y ( x,t )

∂ 2 y ( L,t ) =0 ∂x 2

where M and V represent shear and bending moments of the beam respectively.

Modeling of the Flexible Beam System

µ2

=0

Y j +1 = −Y j −1 − λ 2 SY j +

∆t 2 U ( x,t ) m

(3)

where U(x,t) is an n×1 matrix which represents the actuating force applied on the beam, Yk, (k = j−1, j, j+1) is an n×1 matrix which is the deflection of the beam at segment 1 to n at time step k and S is known as stiffness matrix, which give the characteristic of the beam and ⎡ ( ∆t )2 ⎤ 2 ⎥ µ 2 . The dynamic behavior of the beam λ =⎢ 4 ⎢⎣ ( ∆x ) ⎥⎦ can be simulated using Eq. (3), which can be programmed easily via any digital programming

(1)

where u(x,t) is the actuating force applied at a distance, x, from its fixed end at time, t, y(x,t) is the beam’s deflection at a distance, x, from its fixed end at time, t, µ Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


63


software. In this research, LabVIEW software was used to model Eq. (3). The simulation platform is designed so that user can easily study the cases of any number of segments, length of the beam, different excitation signals and other simulation requirements. Before executing the simulation model, the parameters of the beam given in Table I were set into the LabVIEW script. Then the sampling time was set to be 0.3 ms in order to satisfy the convergence requirement for the simulation in which λ2 must be properly chosen between 0 < λ 2 ≤ 0.25 [20]. The above sampling time is also sufficient to cover a broad range of dynamics of the flexible beam. Fig. 2. Sinusoidal input signal applied to segment 16 TABLE I BEAM SPECIFICATION

-3

8

Number of segment

λ

Mass Length Beam constant

Before cancellation

Value 6

20 0.3629 0.037 kg 0.635 m 1.351

X: 1.167 Y: 0.006357

4

2 Deflection (m)

Parameter

Vibration signal samples

x 10

0

-2

III. FD Model Validation

-4

The dynamic response of the FD model was tested with sinusoidal input signal. The system was excited with a sinusoidal disturbance force of frequency, 10 Hz and amplitude, 10 V (see Fig. 2). The disturbance force and observation point can be applied at any segment along the beam. Segment 16 was selected to be excited with the disturbance force and the deflection was observed at segment 20. The force should be applied at the segment where sufficient moments and higher deflection are produced. The dynamic response of the beam was investigated for the duration of 3 s (see Fig. 3).The observation signal was analyzed through Fast Fourier Transform (FFT) presented in Fig. 4. It can be observed that the resonance frequencies are at 1.803, 11.72, 32.23, 61.99 and 100.8 Hz respectively. The simulated model is validated by comparing the resonance modes with theoretical values found in the literature [5]. The percentage error for all the resonance modes can be seen in Table II. It can be concluded that the proposed model has almost the same dynamic characteristic as the theoretical analysis.

X: 1.495 Y: -0.006768

-6

-8

0

0.5

1

1.5 time (seconds)

2

2.5

3

Fig. 3. Dynamics response of the beam at segment 20

Fig. 4. Frequency response of the beam at segment 20

IV.

TABLE II COMPARATIVE RESONANCE MODES OF THE BEAM BETWEEN THEORETICAL AND SIMULATED VALUE Theoretical Simulated value (Hz) Error (%) Mode value (Hz) 1 1.875 1.803 3.84 2 11.751 11.72 0.26 3 32.902 32.23 2.04 4 64.476 61.99 3.85 5 106.583 100.8 5.43

Controller Schemes

In this section, the development of the control schemes for suppressing the vibration of flexible beam is presented. Initially, a proportional control is developed. Then, self-tuning proportional control scheme is used to improve the performance of proportional control in active vibration control in dealing with the dynamic behavior of the flexible beam. The controller is developed based on the physical structure depicted in Fig. 5.



64


IV.1. Proportional Controller Scheme

response [21]. Fig. 7 shows the schematic diagram of Ptype Iterative Learning Algorithm (P-type ILA). For each cycle of operation, the value of the input, x, and output, yk, signals are stored in the memory to be used in the next cycle. Then, the system error is evaluated as ek = yd − yk , where yd is the desired output. The next value of input signal is computed by using the following equation:

Proportional controller is implemented as the control scheme to suppress vibration of flexible beam. The block diagram of the proportional control scheme is shown in Fig. 6. In this figure, Ga, G, Gs and K are the transfer functions of actuator, flexible beam, sensor and controller gain respectively. Generally, the controller will try to reduce the error between the reference value and its actual value until minimum error is achieved. The effectiveness of the controller action depends on the optimum setting of its parameters or proportional gain [17]. For the closed loop system in Fig. 6, the reference input, which is the desired deflection, is set to zero in order to achieve zero cancellation. The input to the actuator is −GsKX and its output is thus Fa = −GaGsKX. Then the total force fed into the system is given by: F = Fa + Fd = −Ga Gs KX + Fd

x j +1 = x j + Φe j

where Φ is the proportional learning parameter.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Sensor

(4)

Flexible Beam

Disturbance Force

Thus, the output of the system which is the deflection of the beam, X can be expressed as: X = GFd

1 ( −Ga Gs KX + Fa )

(6)

(5)

Actuator

Controller

Fig. 5. Active vibration control structure for flexible beam system

From Eq. (5), the deflection of the beam can be reduced by adjusting the controller gain, K. By increasing the controller parameter, the lateral displacement of X will decrease, thus suppressing the amplitude of the vibration. Theoretically, if the value of K is large, the term 1/(1+GaGsKG) will tend to zero. It means that the controller can cope with any disturbance or parameter changes in the dynamic system. The analysis of this controller is quite similar with high gain feedback regulator.

Fig. 6. Proportional control scheme

IV.2. Self-Tuning Proportional Controller Scheme In this section, the theory of self-tuning proportional control is discussed. The main drawback of proportional controller is the need to adjust the gain, K, manually in order to cope with variation in process dynamics and disturbances. This leads to the needs of self tuning control scheme where in this study, self-tuning proportional control scheme has been chosen. Basic idea behind the self-tuning control is to construct an algorithm that will automatically find new control parameter to meet the desired system response. Selftuning proportional control scheme is implemented by using an intelligent mechanism known as Iterative Learning Algorithms (ILA). ILA is an intelligent approach for improving the transient response and tracking performance of uncertain dynamic systems that operate repetitively. ILA is a simple method that has been effectively used to overcome the drawbacks of conventional controller in achieving the desired transient

Fig. 7. P-type Iterative Learning Algorithm

An iterative learning method computes successive approximations such as the output of the system approaches the desired response as time increases. The iterative learning process that runs infinitely tends to cause over learning to the system. This could lead to instability of the system once it enters a ‘dangerous zone’ where the increment of estimated parameter will cause severe degradation of system performance [22]. In order to overcome this problem, a stopping criterion needs to be implemented into the algorithm. Learning process will stop immediately when the stopping criteria is met. The



65


block diagram of P-type ILC scheme for AVC flexible beam FD model is shown in Fig. 8. This control scheme structure is similar with the structure in Fig. 6. Only a minor modification was done in the controller part, where the previous control scheme which is conventional P controller is extended to self-tuning proportional control scheme. However, the system’s output is still formulated by Eq. (5). Then Iterative Learning algorithm is used to selftune the controller parameter, K. By using the concept of IL method, the next value of parameter, K(t+1) can be computed based on the current value, K(t) and the error signal, e(t). With zero reference input signal, the error function is defined as: ek = 0 − Gs x ( t )

K (t + 1) = K (t ) + Φe(t )

x (t )

e k (t ) = 0 − x (t )

(7) 0 ≤ ek (t ) ≤ n

whereas the K(t+1) is expressed as: K ( t + 1) = K ( t ) + Φek ( t )

(8)

The learning process for the controller parameter is updated iteratively until the stopping criterion is met. In this study, the stopping criterion is design based on the minimum value of error function (Eq. (7)) which is referred as stopping criterion error. Fig. 9 shows the flowchart of the proposed stopping criterion.

Fig. 9. Flowchart of the proposed stopping criterion

Based on Eq. (9), the deflection of the beam is inversely proportional to the value of controller gain, K. Thus, the amplitude of oscillation of the beam is reduced by increasing K. Then, the FD simulation algorithm is integrated with proportional control and self-tuning proportional control scheme using LabVIEW software. The performance of these control schemes are investigated by varying the proportional gain, stropping error criterion, proportional learning parameter, amplitude of disturbance force, position of the excitation point on the beam, and changing its physical parameter. Further analysis is made via power spectral density, where the amplitude can be seen directly from its resonance modes respectively.

Fig. 8. Block diagram of P-type ILC self-tuning proportional control scheme

A new value of parameter, K, will be calculated at each learning cycle. This new parameter, K, determines the amount of actuating force to reduce the unwanted deflection of the beam. Subsequently, when the error reaches below the stopping criterion error, n, the learning process will stop automatically. This will prevent the over learning effect as mention before.

V.

V.1.

Figs. 10 and 11 show the corresponding results for P control scheme with different proportional gain in time and frequency domains respectively. The system was excited with a sinusoidal disturbance force of frequency, 10 Hz and amplitude, 10 V (Fig. 2). The disturbance force and observation point can be applied at any segment along the beam. Segment 16 was selected to be excited with the disturbance force and the deflection was observed at segment 20. Then the robustness of the P control scheme is studied by varying the amplitude of the disturbance force, change the position of excitation point, and the beam’s mass, m. The results are depicted in Figs. 14, 15 and 16. Figs. 10, depict the time-domain response of the beam deflection at segment 20 before and after cancellation.

Simulation Results

For the purpose of simulation, the transfer function for linear actuator and linear sensor is assumed to be unity. Thus, the deflection of the beam becomes: ⎛ 1 ⎞ X = GFd ⎜ ⎟ ⎝ 1 + KG ⎠

Simulation Results (Proportional Control Scheme)

(9)



66


Figs. 10(a) to 10(d) show the corresponding vibration before and after cancellation for P control schemes for different proportional gains. It is noted that, the peak-to-peak deflection at the end point before cancellation is 1.605×10-2 m to -1.628×10-2 m and reduced to 1.172×10-3 m to -1.187×10-3 m by gain of 50, 5.744×10-3 m to -6.674×10-3 m by gain of 500, 4.977×10-3 m to -5.243×10-3 m by gain of 1000, and 1.718×10-3 m to -1.401×10-3 m by gain of 10 000 respectively. It is noted that, increasing the proportional gain will improve the vibration suppression of beam. Detail investigation is performed through frequency domain response presented in Figs. 11. The vibration attenuation of the beam at five resonance modes contributed from each proportional gain, K can be seen from Figs. 11(a) to 11(d). Proportional control scheme has proven its capability in reducing the unwanted vibration signal of the system. The performance of the controller in rejecting the unwanted vibrations depends on the value of proportional gain, K. Increasing the gain may significantly reduce the amplitude of the system. It is observed that, the proportional gain, K, is set at 50, 500, 1000 and 10 000 where the corresponding proportional gains, K, have attenuated the first resonance mode of the beam’s deflection by 3.85, 12.13, 17.78 and 38.46 dB respectively. The higher the proportional gain, K, the better suppression can be obtained. However, if the proportional gain, K, is increased beyond some acceptable value, it may saturate the actuator force. Thus, in real application, the saturated signal may extremely reduce the actuator lifetime [23]. Fig. 12 shows the actuator’s response reaching its maximum operating point (-100 V to 100 V). Hence, this effect also applicable for self-tuning proportional control scheme, because they used the same proportional gain, K. The only difference is that self-tuning proportional control scheme has been incorporated with self learning features to self-tune the proportional gain, K. The robustness of P control scheme has been tested by varying the disturbance amplitude and changing the excitation point. The proportional gain, K, is set to 10 000. As depicted in Fig. 13, it shows that P control scheme is unable to maintain constant minimum deflection when disturbance amplitude is varied to 10, 20, 40, and 60 V. The deflection increased gradually as the disturbance amplitude increased. It is noted that the deflection of the beam has reduced to 5.7 × 10-3, 7.3 × 10-3, 9.4×10-3 and 11.3 × 10-3 m from 1.8 × 10-2 m at disturbance amplitude of 10, 20, 40, and 60 V respectively. The next robustness test is by changing its disturbance excitation point. The disturbance excitation point is abruptly changed from segment 16 to 19.

Figs. 10. (a) Time response at proportional gain of 50, (b) Time response at proportional gain of 500, (c) Time response at proportional gain of 1000, (d) Time response at proportional gain of 10 000

Figs. 11. (a) Frequency response at proportional gain of 50, (b) Frequency response at proportional gain of 500, (c) Frequency response at proportional gain of 1000, (d) Frequency response at proportional gain of 10 000

Fig. 12. Actuator response becomes saturated when the gain is too high approximately about 3.58 × 105



67


Vibration signal samples Proportional controller (10 000)

X: 3.093

0.02 Y: 0.01799 0.015

Deflection (m)

X: 22.47 Y: 0.009422

X: 13.31 Y: 0.007299

X: 7.488 Y: 0.005786

0.01

V.2.

Control Scheme) The Figs. 16 to 22 show the corresponding results for self-tuning proportional control scheme. The system was excited with a sinusoidal disturbance force of frequency, 10 Hz for duration of 30 s. The performance of this control scheme is studied by setting the stopping criterion error and varying proportional learning parameter respectively. Then the robustness of the selftuning proportional control scheme is studied by varying the amplitude of the disturbance force, the excitation point and the beam’s mass, m. The results can be seen in Figs. 19 to 22. Fig. 16 shows the simulation result for self-tuning proportional control scheme at stopping criterion error of 0.001 m. It is noted that, when the stopping criterion error is achieved, the learning process terminated. The controller was actively ON at the simulation time of 4.5 s from that starting time. It is noted that, the learning process is terminated when the stopping criterion error is below than 0.001 m at the optimum proportional gain, K, of 5.85 × 104 at the duration of 24.05 s after the controller ON. It can be concluded, the learning process executed iteratively in order to achieve the optimum gain, K, until the stopping criterion error is fulfilled. However, time taken form learning process was relatively high.

0.005 0 -0.005 -0.01 -0.015 Controller ON -0.02

0 5 Controller OFF

10

15 time (seconds) Learning Parameter

20

25

30

100 Amplitude (V)

60 V 50

40 V

20 V

10 V

0 -50 -100

0

1

2

3

4

5 Iteration

6

7

8

9

10 4

x 10

Fig. 13. Proportional control scheme for beam deflection when disturbance excitation amplitude changed

From Fig. 14, it can be observed that, deflection at segment 16 is reduced to about 5.7 × 10-3 m. When excitation point is changed to segment 19, the deflection ultimately increased to 9.2 × 10-3 m. Both results imply that, P control scheme is unable to adapt with the variation of disturbance force. Thus, self-tuning control scheme can be implemented to overcome this problem. Vibration signal samples Proportional controller (10 000) 0.02 X: 3.126 Y: 0.01631

0.015

Segment 16

Segment 19

X: 11.09 Y: 0.005773

X: 26.08 Y: 0.009204

0.005

X: 4.296 Y: 0.03595

0.05 deflection (m)

Deflection (m)

0.01

0 -0.005

-0.05

-0.01

X: 18.57 Y: 0.01087

0

10

Controller OFF


20

25

deflection (m)

0.02

5

30

15

1

2

3

4

5 Iteration

6

7

8

9

15 time (seconds)

20

25

30

25

30

25

30

Vibration signal samples (0.037kg) Gain K = 10 000 X: 16.64 Y: 0.005798

0

5

10

10

15 time (seconds)

20

Vibration signal samples (0.057kg) Gain K = 10 000

4

x 10

X: 4.296 Y: 0.01072

0.02 deflection (m)

0

10

0

-0.02

10

5

X: 4.296 Y: 0.01649

Controller ON 0

Vibration signal samples (0.017kg) Gain K = 10 000

0

-0.015 -0.02

Simulation Results (Self-Tuning Proportional

X: 28.52 Y: 0.01133

Fig. 14. Proportional control scheme for beam deflection when disturbance excitation point displaced from segment 16 to 19

0 -0.01 -0.02

From Fig. 15, when the controller is in OFF mode, it is observed that, when the beam’s mass value has increased, inversely the deflection of beam is decreased. This effect is theoretically governed by equation (1) where the deflection is inversely proportional to the mass of the beam. As the controller is ON, deflection of the beam is suppressed to about 0.01, 0.005 and 0.003 m which corresponding to 0.017, 0.037 and 0.057 kg respectively. From these results it is worth noted that, the proportional gain, K, needs to be adjusted manually in order to get the desired deflection when there is any changed in the beam’s physical parameters. Thus, it is clearly revealed the drawback of proportional control scheme which needs to be addressed by effective control scheme.

X: 16.53 Y: 0.003349

0.01

0

5

10

15 time (seconds)

20

Fig. 15. Proportional control scheme for beam’s deflection when physical parameter of beam changes it mass at 0.017, 0.037 and 0.057 kg

Fig. 17 indicates that, even though learning process was executed until the stopping criteria has met, practically, the learning time can be stop earlier. It was noted that, after 0.6 s from controller ON, the root mean square error (RMSE) of the beam deflection was able to achieve at 0.0017 m from its stopping criterion error of 0.001 m. Then the RMSE was gradually reduced to 0.0012 m at time frame from 0.6 s to 1.2 s and 1.2 s to 1.8 s respectively. From this point of view, it is worth noting that, when the RMSE of beam’s deflection is acceptable, the learning process can be stop immediately.



68


parameter, Φ is set to 50 000, the learning process stopped after 28.6 s, where the stopping criterion error of 0.001 m is fulfilled. Thus, the optimum value of proportional gain, K, is achieved at about 15.1 s (see Fig. 19(c)) and the deflection error is less than 0.001 m. It can be deduced that, a suitable learning parameter is important, hence it will affect the total learning time in iterative learning process in order to meet the optimum value of proportional gain, K. Results on the robustness study of the proposed controller are depicted in Figs. 19, 20 and 21. Figs. 19 demonstrate the robustness of self-tuning proportional control scheme when the amplitude of disturbance force is varied to 10, 20, 40 and 60 V. Figs. 19(a) and 19(b) show the results when the controller is OFF and ON respectively at specific time instant. As seen in Fig. 18(a), the deflection of beam is increased when the disturbance is increased. This is due to the controller is in OFF mode, thus there is no action taken to overcome the amplitude changed in disturbance force. Ultimately, when the controller is ON as shown in Fig. 19(b), any change in the disturbance amplitude will cause the controller to behave robustly by searching for optimum proportional gain, K, via iterative learning process. This action significantly controls the deflection of beam to meet the desired deflection. Figs. 20 further enhance the evidence for robustness of self-tuning proportional control scheme by displacing the disturbance force from segment 16 to 19 respectively, whereby Figs. 20(a) and 20(b) show the results for controller OFF and ON correspondently. Fig. 20(a) obviously shows the deflection of the beam has increased instantly when the disturbance force on the beam is displaced from segment 16 to segment 19. This problem can easily be addressed when the controller is ON as shown in Fig. 20(b). After the controller has been actively ON at time 4.5 s, deflection of the beam abruptly attenuated from 0.0168 m to 0.0018 m until the desired deflection is achieved. At the same time, the learning process starts to execute to find a new optimum proportional gain, K. While learning process is in progress, the robustness test is done by displacing the disturbance force from segment 16 to 19 which causes the actual deflection increased to about 0.0015 m more than the desired deflection. Thus, in turn, the controller adaptively reduced the deflection error by gradually producing an appropriate value of proportional gain, K, from the iterative learning process. It is noted that, the slope of the proportional gain, K, gradually decreases as the desired deflection goes near to the stopping criterion error. As compared to conventional P control scheme, selftuning proportional control scheme has proven its’ robustness when facing with the dynamic change in the disturbance force such as amplitude disturbance and its excitation point. Figs. 13 and 14 present the drawback of P control scheme when the amplitude and excitation

Consequently, this will reduce the learning time. In order to do that, the stopping criteria of the ILA needs to be modified by introducing another stopping condition which can be computed from the root mean square error (RMSE). Figs. 18(a) to 19(c) further demonstrate the performance of the self-tuning proportional control scheme at different level of proportional learning parameter, Φ, when the stopping error criterion is fixed to 0.001 m. The effect of learning parameter, Φ, in achieving the desired deflection error is investigated. Vibration signal samples Error = 0.001

X: 3.09

0.02 Y: 0.01635

0.01 Deflection (m)

X: 4.761 Y: 0.001727

X: 29.1 Y: 0.0009337

0

-0.01 Controller ON -0.02

0

5

10

15 time (seconds)

Controller OFF 4

Proportional Gain, K

6

20

25

Iterative Learning

x 10

30

X: 9.517e+004 Y: 5.847e+004

4

2

0

0

1

2

3

4

5 Iteration

6

7

8

9

10 4

x 10

Fig. 16. Beam’s deflection at proportional learning parameter, Φ of 25 000 and stopping criterion error of 0.001 m RMSE =

0.0017 0.0012 0.0012

Vibration signal samples (error = 0.001) 0.02

deflection (m)

0.01

0

-0.01

-0.02

0

1

2

4


4

x 10


4

5

6 X: 1.9e+004 Y: 3.723e+004

X: 1.8e+004 Y: 3.441e+004

3 X: 1.7e+004 Y: 3.06e+004

2

1 X: 1.5e+004 Y: 1

0

0

0.2

0.4

0.6

0.8

1 Iteration

1.2

1.4

1.6

1.8

2 4

x 10 0s Controller ON

0.6 s 1.2 s 1.8 s

Fig. 17. The root means square error (RMSE) at the stopping criterion error of 0.001 m with proportional learning parameter, Φ of 25 000

The investigation is focused on how fast the learning process to meet the constant proportional gain, K. From that figures, learning parameter is set to be 1000, 10 000 and 50 000 respectively. It can be observed that when proportional learning parameter, Φ is set to 1000, the proportional gain, K is gradually increased towards the end of simulation period of 30 s. Then, for the second attempt, the proportional learning parameter, Φ, is increased to 10 000. The result shows the proportional gain, K, increased slowly after 25 s simulation time. Finally, when proportional learning



69


Vibration signal samples Disturbance amplitude change/ Controller OFF

point are changed respectively. From Fig. 13, it is clearly observed that, when the disturbance amplitude is varied at 10, 20, 40 and 60 V, the deflection of beam also increased. This is of cause due to the fix value of proportional gain, K.

0.06

Deflection (m)

0.04

Vibration signal samples Learning parameter = 1000

X: 3.09

X: 26.26 Y: 0.06663

0.08

0.02

X: 17.92 Y: 0.0417

X: 10.21 Y: 0.02768

X: 3.126 Y: 0.01631

0 -0.02 -0.04

0.02 Y: 0.01635

-0.06

20 V

10 V

40 V 60 V

0.015

-0.08 X: 5.478 Y: 0.004898

X: 29.75 Y: 0.001174

0.005

5

10

15 time (seconds) Disturbance Input

20

25

30

0 -0.005 -0.01 -0.015 -0.02

Controller ON 0

5

10

15 time (seconds) Iterative Learning

20

25

50 0 -50

30

-100

X: 4.878 Y: 0.01614

1

0

1

2

3

4

5 Iteration

6

7

8

9

10 4

x 10

X: 3.126

5 Iteration

6

7

8

9

10 4

x 10

Amplitude (V)

0.01 X: 4.848 Y: 0.002797

0.005

X: 29.83 Y: 0.000906

Controller ON 0

15 time (seconds) Disturbance input

X: 2.693e+004 Y: 10

-100

0

1

2

3

4

4

Controller ON 0 Controller OFF

5

10

15 time (seconds)

4

8

10

0


-0.02

5

20

25

30

Iterative Learning

x 10

X: 9.879e+004 Y: 6.348e+004

20

25

30

100

-0.005 -0.01

X: 29.85 Y: 0.001167

X: 12.41 Y: 0.0009828

-0.01

0

-0.015

X: 25.41 Y: 0.001342

X: 19.8 Y: 0.001092

X: 11.9 Y: 0.0009852

0

Controller OFF

0.015


4

X: 5.376 Y: 0.002404

0.01

Vibration signal samples Learning parameter = 10 000

0.02 Y: 0.01631

8

x 10

X: 6.853e+004 Y: 40

X: 4.183e+004 Y: 20

5 6 Iteration Iterative Learning

7

8

X: 8.843e+004 Y: 60

9

10 4

x 10

X: 9.993e+004 Y: 7.612e+00

6 4

X: 3.97e+004 Y: 4.363e+004

2 0

0

1

2

3

4

5 Iteration

X: 8.414e+004 Y: 5.527e+004

X: 6.442e+004 Y: 4.822e+004

6

7

8

9

10 4

x 10

(b)

6

Figs. 19. Self-tuning proportional control scheme for beam deflection when disturbance excitation amplitude changed at 10, 20, 40 and 60 V. (a) Controller OFF, (b) Controller ON

4

2

0

0

1

2

3

4

5 Iteration

6

7

8

9

10

Fig. 14 also revealed that P control scheme is unable to maintain its performance when the disturbance excitation point is changed. To address this problem, the proportional gain, K, needs to be tuned manually until an acceptable controller has been acquired for the active vibration control of flexible beam system. Thus, in turn, self-tuning proportional control scheme has clearly shown its robustness and capability to suppress the vibration of the flexible beam system. Fig. 21 demonstrates the capability of the self-tuning proportional control scheme to suppress the beam’s deflection in the condition where the beam’s physical parameter, in the case its mass changed to 0.017, 0.037 and 0.057 kg. The proportional learning parameter and its stopping criterion error were set at 20 000 and 0.001 m respectively. As can be seen from the figure, the proposed controller has reacted effectively in order to maintain the desired beam’s deflection. This result is caused by iterative learning method adopted in the controller.

4

x 10

(b) Vibration signal samples Learning parameter = 50 000

X: 3.09

0.02 Y: 0.01635 0.015 0.01 X: 4.704 Y: 0.002062

0.005

X: 28.59 Y: 0.0009217

0 -0.005 -0.01 -0.015

Controller ON 0

5

10

15 time (seconds)

Controller OFF 4


3

Vibration signal samples Disturbance amplitude change/Controller ON

2

-0.02

20

25

30

Iterative Learning

x 10

6

X: 9.529e+004 Y: 6.534e+004

4

2

0

2

(a)

(a)

8

1

3

0

-0.02

0

X: 9.893e+004 Y: 4.515e+004

4

Deflection (m)


Controller OFF 4 x 10 5

Deflection (m)

0

100

Amplitude (V)

Deflection (m)

0.01

0

1

2

3

4

5 Iteration

6

7

8

9

10 4

x 10

(c) Figs. 18. Beam’s deflection at fixed stopping criterion error of 0.001, (a) The proportional earning parameter, Φ of 1000, (b) The proportional learning parameter, Φ of 10 000, (c) The proportional learning parameter, Φ of 50 000



70


As shown in Fig. 22, iterative learning has produced different value of proportional gain, K, when different mass is used.


4

Vibration signal samples Segment Change / Controller OFF X: 20.87 Y: 0.02589

X: 12.01 Y: 0.01628

0.02

Iterative Learning (0.017kg)

x 10

6

X: 29.77 Y: 6.963e+004

4 2 0

0

5

10

4


0.03

8

10

x 10


Deflection (m)

25

0

8

30

X: 29.72 Y: 5.987e+004

0

5

10

4

-0.01

20

5

0.01

0

15 Iteration Iterative Learning (0.037kg)

15 Iteration

20

25

30

Iterative Learning (0.057kg)

x 10

X: 29.68 Y: 7.438e+004

6 4 2 0

0

5

10

15 Iteration

20

25

30

-0.02

Fig. 22. Iterative Learning of proportional gain, K for beam’s deflection when physical parameter of beam changes it mass at 0.017, 0.037 and 0.057 kg

Segment 16

-0.03

0

5

10

15 time (seconds)

20

25

30

Segment 19

The proportional gain, K, for mass of 0.017, 0.037, and 0.057 kg are settled at 6.96 × 104, 5.98 × 104 and 7.43 × 104 respectively. From this observation, the proportional gain, K, has been tune automatically as the physical parameter is changed. Thus, it can be concluded that the proposed self-tuning proportional control scheme is able to reduce the unwanted beam’s vibration even though the physical parameter of the beam changed.

(a) Vibration signal samples (Change segment point)

X: 5.466 Y: 0.01686

0.02

Segment 16

Segment 19

0.01 Deflection (m)

X: 8.244 Y: 0.001875

X: 18.58 Y: 0.001493

X: 17.9 Y: 0.0009735

X: 29.64 Y: 0.0009316

0

-0.01 Controller ON -0.02

7

0

5

10

15 time (seconds) Iterative Learning

4 x 10 Controller OFF

20

25

30 X: 9.879e+004 Y: 6.862e+004

VI.


6

This LabVIEW programming has been used for FD model and active vibration control of flexible beam. The software is based on graphical icon source codes which make it user-friendly and eliminates typing in lengthy character-based code. The FD model and control scheme were implemented in discrete platform, thus the control scheme can be realized for real time application easily. Flexible beam system has been successfully modeled using FD method and simulation through LabVIEW software environment. The simulated model is validated by comparing the resonance modes with theoretical values. It is found that the FD model is able to produce the five resonance modes sufficiently accurate. Next, the performance of P control scheme in active vibration control of beam has been investigated. From the results, it revealed that P control scheme has significantly reduced the vibration of the beam. The higher the proportional gain, K, the better the unwanted vibration is reduced. However, the actuator operating range needs to be selected properly, hence it will affect the performance of the controller when the proportional gain, K, is too high. It is noted that, the performance of self-tuning proportional control scheme in suppressing the vibration of the flexible beam system can be achieved by appropriately setting the stopping criterion error and its learning parameter, Φ. From the results and discussion, it can be concluded that the lower the stopping criterion

X: 6.148e+004 Y: 4.577e+004

4 3 2 1 0

0

1

2

3

4

5 Iteration

6

7

8

9

10 4

x 10

(b) Figs. 20. Self-tuning proportional control scheme for beam deflection when disturbance excitation point displaced from segment 16 to 19. (a) Controller OFF, (b) Controller ON

deflection (m)

X: 27.25 Y: 0.001037

0.02 0 -0.02 -0.04

0

5 X: 4.296 Y: 0.01649

0.02 deflection (m)

Vibration signal samples (0.017kg) Proportional Learning Parameter = 20 000 and stopping criterion error = 0.001

X: 4.296 Y: 0.03595

0.04

10 15 20 25 30 time (seconds) Vibration signal samples (0.037kg) Proportional Learning Parameter = 20 000 and stopping criterion error = 0.001 X: 26.96 Y: 0.001015

0.01 0 -0.01 -0.02

0

5

10 15 20 25 30 time (seconds) Vibration signal samples (0.057kg) Proportional Learning Parameter = 20 000 and stopping criterion error = 0.001

X: 4.296 Y: 0.01072

0.02 deflection (m)

Conclusion

5

X: 26.81 Y: 0.001075

0.01 0 -0.01 -0.02

0

5

10

15 time (seconds)

20

25

30

Fig. 21. Self tuning control scheme for beam’s deflection when physical parameter of beam changes it mass at 0.017, 0.037 and 0.057 kg



71


[9]

error, the better is the attenuation, but more time is required by iterative learning algorithm to find the optimum proportional gain, K. However, this problem can be addressed by suitable value of learning parameter, Φ. An appropriate value of learning parameter, Φ, can speed up the learning process. The robustness of self-tuning proportional control scheme has been proven better than P control scheme in suppressing the vibration when the disturbance force applied to the beam at different excitation segments and amplitudes. The iterative learning algorithm in selftuning proportional control scheme helped to tune the proportional gain, K, automatically when the system is disturbed by the disturbance as noted in the discussion. Each time the system is forced by different magnitude disturbance or excited at different segment, new proportional gain, K, will be computed. Self-tuning proportional control scheme also demonstrated its’ robustness in suppressing the vibration of beam when the physical parameter of flexible beam has changed. Future woks will be focused on the development of experimental rig to verify the simulation results obtained in this study.

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

Acknowledgements Authors would like to express their gratitude to Universiti Teknologi Malaysia (UTM) and Universiti Malaysia Perlis (UNIMAP) for the funding of the research and facilities to conduct this research. This research is funded by Universiti Teknologi Malaysia (UTM) using Research University Grant Vote No. 00H11.

[18]

[19]

[20]

References [1] [2] [3]

[4]

[5]

[6]

[7]

[8]

[21]

F. Christopher, "Handbook of Noise and Vibration Control," ed Virginia: John Wiley & Sons Inc, 2007. P. Lueg, Process of silencing sound oscillations, US Patent No. 2,043,416, 1936. M. Itik, M. U. Salamci, F. D. Ulker, and Y. Yaman, "Active Vibration Suppression of a Flexible Beam via Sliding Mode and H infinity Control," presented at the Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference, Seville, Spain, 2005. I. Z. Mat Darus, and M. O. Tokhi, "Soft Computing-based Active Vibration Vontrol of a Flexible Structure," Journal of Engineering Application of Artificial Intelligence, vol. 18, pp. 95-114, 2005. A. Madkour, M. A. Hossain, K. P. Dahal, and H. Yu, "Intelligent Learning Algorithms for Active Vibration Control," presented at the IEEE Transactions On Systems, Man, And Cybernetics—Part C: Applications And Reviews, 2007. G. Haichang, and G. Song, "Active Vibration Suppression of a Flexible Beam with Piezoceramic Patches Using Robust Model Reference Control," Journal of Smart Materials and Structures, vol. 16, pp. 1453-1459, 2007. J. Fei, "Adaptive Sliding Mode Vibration Control Schemes for Flexible Structure System," presented at the Decision and Control 2007 46th IEEE Conference, New Orleans (USA), 2007. M. A. Hossain, and M. O. Tokhi, "Real-time Design Constraints in Implementing Active Vibration Control Algorithms," International Journal of Automation and Computing, vol. 3, pp. 252-262, 2006.

[22]

[23]

M. O. Tokhi, and M. A. Hossain, "A Unified Adaptive Active Control Mechanism for Noise Cancellation and Vibration Suppression," International Journal of Mechanical Systems and Signal Processing, vol. 10, pp. 667-682, 1996. S. Z. Mohd Hashim, M. O. Tokhi, and I. Z. Mat Darus, "Active Vibration Control of Flexible Structures Using Genetic Optimisation," Journal of Low Frequency Noise, Vibration and Active Control, vol. 25, pp. 195-207, 2006. M. Mailah, and Priyandoko, "Simulation of a Suspension System with Adaptive Fuzzy Active Force Control," International Journal of Simulation Model vol. 6, pp. 25-36, 2007. A. R. Tavakolpour, "Mechatronic Design of Intelligent Active Vibration Control Systems for Flexible Structures," Ph.D. Thesis, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, Skudai, 2010. A. R. Tavakolpour, M. Mailah, ,and I. Z. Mat Darus, "Active Vibration Control of a Rectangular Flexible Plate Structure Using High Gain Feedback Regulator," International Review of Mechanical Engineering (IREME), vol. 3, pp. 579-587, 2009. S. Arimoto, S. Kawamura, and F. Miyazaki, "Bettering Operation of Robots by Learning," J. Robotic Syst, vol. 1, pp. 123-140, 1984. M. Z. Md Zain, and M. O. Tokhi, "Hybrid Learning Control Schemes with Input Shaping of a Flexible Manipulator System," Mechatronics vol. 12, pp. 209-219, 2006. E. Pitowarno, and M. Mailah, "Control of Mobile Manipulator using Resolved Acceleration with Iterative-LearningProportional-Integral Active Force Control," International Review of Mechanical Engineering (IREME), vol. 1, pp. 549-558, 2007. M. F. Hassan, M. Mailah, R. Junid, and N. A. Alang, "Vibration Suppression of a Handheld Tool Using Intelligent Active Force Control (AFC)," presented at the Proceedings of the World Congress on Engineering London (U.K.), 2010. J. V. D. Wijdeven, and O. Bosgra, "Hankel Iterative Learning Control for residual vibration suppression with MIMO flexible structure experiments " presented at the Proceedings of the 2007 American Control Conference Marriott Marquis Hotel, Times Square New York City, USA, 2007. S. Z. Mohd Hashim, M. O. Tokhi, and I. Z. Mat Darus, "Genetic Adaptive Active Vibration Control of Flexible Structures," presented at the Proceedings of ICS'04: IEEE SMC UK-RI Chapter Conference 2004 on Intelligent Cybernetic Systems, Londonderry (UK), 2004. M. O. Tokhi, and R. R. Leitch, Active noise control. Oxford, UK.: Clarendon, 1992. H. S. Ahn, Y. Chen, and K. L. Moore, "Iterative Learning Control: Brief Survey And Categorization," IEEE Transactions On Systems, Man, And Cybernetics—Part C: Applications And Reviews, vol. 37, pp. 1099-1121, 2007. M. Mailah, and J. C. W. Shiung, "Control Of A Robot Arm Using Iterative Learning Algorithm With A Stopping Criterion " Jurnal Teknologi,Universiti Teknologi Malaysia vol. 37(A), pp. 55-72, Dis. 2002. D. J. Spearritt, and S. F. Ashokanathan, "Torsional vibration control of a flexible using laminated PVDF actuators," Journal of Sound and Vibration, vol. 193 pp. 941-956, 1996.


School of Manufacturing Engineering, Universiti Malaysia Perlis

2

Department of System Dynamics & Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia.

3

Department of System Dynamics & Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia.



72


Mohd S. Saad was born in Jitra, Kedah, Malaysia, in December 21st, 1976. He received his diploma in Mechatronics from Polytechnic Sultan Abdul Halim (POLIMAS), Malaysia in 1997 and graduated from Universiti Teknologi Mara (UITM), Malaysia in Bachelor Degree of Electrical Engineering in 2002 and completed his Masters Degree (Mechatronic and Automatic Control) Electrical Engineering from Universiti Teknologi Malaysia (UTM), Malaysia in the year 2007. His field of research is in control engineering and currently extending his knowledge by undergoing PhD studies on Active Vibration Control in UTM. Mr. Mohd S. Saad also actively involves with Engineering Professional Bodies in Malaysia such as Board of Engineers, Malaysia (BEM). Hishamuddin Jamaluddin is a Professor at the Faculty of Mechanical Engineering, Universiti Teknologi Malaysia. He teaches subjects such as Control System, Instrumentations, Multivariable System and System Identification for undergraduate and post graduate programmes. He obtained the Bachelor of Engineering (Control Engineering) degree in 1982, Master of Engineering (Control System) degree in 1985 and Doctor of Philosophy degree in 1991, from Sheffield University, U.K. His research interests include System Identification, Vehicle Dynamics, and Intelligent Control System. Intan Z. M. Darus was born in Melaka, Malaysia, in September 16th, 1976. She received her First Class B.Eng (Hons.) degree in Mechanical Engineering from the University of Wales College Cardiff, Wales, United Kingdom in 1998 and later her Ph.D in Automatic Control and Systems Engineering from the University of Sheffield, United Kingdom in 2004. Currently, she is an Associate Professor in the Department of System Dynamics & Control, Faculty of Mechanical Engineering, Universiti Teknologi Malaysia. Her current research interests are active vibration control, modeling and simulation of dynamical system, soft computing and artificial intelligent techniques for system identification and control.



73


Reduced Models Based on Smooth Decomposition for Random Mechanical Systems Sergio Bellizzi1, Rubens Sampaio2 Abstract – In this paper, the smooth decomposition method combined with the Petrov-Galerkin projection for structure-preserving model reduction is used to analyze second-order discrete nonlinear structural systems under random excitation. The smooth decomposition method is a multivariable-data analysis method. It can be viewed as a projection of an ensemble of spatially distributed data such that the vector directions (the smooth modes) of the projection not only keep the maximum possible variance but also the resulting motions along the vector directions are as smooth in time as possible. The efficiency of the approach is analyzed comparing the power spectral density functions of the reduced-order model and of the original system. Nonlinear mechanical systems under random excitation with homogeneous and non-homogeneous mass distribution were considered. It is established here that a smooth-mode basis can be used advantageously instead of the Karhunen-Loève mode basis. In the case of a non-homogeneous mass distribution, the reduced models based on smooth decomposition were found to be the most efficient. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Karhunen-Loève Modes, Smooth Mode, Model Reduction, Random Vibrations

and W. Zhou [1]. In this work, the SD was defined only for discrete processes on the basis of a maximization problem associated with a scalar time series of measurements subject to a minimization constraint acting on the corresponding time derivative of the time series. The SD can be used to extract normal modes and natural frequencies of multi-degree-of-freedom vibration systems. Free and forced sinusoidal responses have been studied in [1] and randomly excited systems have been analyzed in [2]. The SD was extended in order to analyze timecontinuous random processes [3], [4]and random fields [5], extending the discrete-time processes treated in [1]. SD is performed by solving a generalized eigenproblem defined from the covariance matrix of the random field and the covariance matrix of its time derivative. It is a statistical tool that can be used with any system, linear or nonlinear, deterministic or random. With SD it is possible to define modes for any system and hence to compare the modes so defined with the normal modes of linear systems. The SD differs from the Karhunen-Loève Decomposition (KLD) [1] (also named Proper Orthogonal Decomposition (POD) [6],[7]) by the maximization problem. Relations between SD and KLD can be found in [3]. Some studies have also shown that the SD approach can be used as an output-only modal analysis tool [8], [3], [4], which makes it really useful, since in many applications it is difficult to characterize the excitation properly. The present study investigates the efficiency of the SD approach to analyze second-order discrete structural systems in terms of model reduction which consists in

Nomenclature , ,

, ,

-valued random process indexed by Time-derivative process of , Covariance matrix of , ) ( Covariance matrix of , ( ) Karhunen-Loève Mode (KLM) Karhunen-Loève Value (KLV) Karhunen-Loève Component (KLC) Smooth Mode (SM) Smooth Value (SV) Smooth Component (SC) Mass matrix Damping matrix Stiffness matrix Normal Mode (NM) Resonance frequency -valued random process indexed by Nonlinear -vector function Basis of subspace projection Test basis to Petrov-Galerkin projection Identity matrix

I.

Introduction

The multivariable-data analysis method, called Smooth Orthogonal Decomposition method, or Smooth Decomposition (SD), was first proposed by D. Chelidze Manuscript received and revised December 2011, accepted January 2012

74


Sergio Bellizzi, Rubens Sampaio

developing a dynamical system by reducing the number of degree of freedom describing the original secondorder discrete structural systems. In many cases (system identification, feedback control implementation, dynamic characterization), it can be important that the reduced model preserves the second order structures of the original system. These problems are often solved using the KLD model reduction. The KLD model reduction is obtained through projection of the equations of motion onto a subspace spanned by selected KL modes given by KLD. The KLD model reduction has been used for example to modal analysis of linear and nonlinear systems [9], [10], to identify linear systems [11], to control problem [12], to characterize the dynamic of nonlinear system [13] –[18]. In this paper, the SD model reduction is defined through Petrov-Galerkin projection of the equations of motion onto a subspace spanned by a set of selected smooth modes given by SD. This procedure has been chosen because it is applicable when the decomposition basis is not orthogonal in the sense of the standard inner product of and because it ensures that the second-order structure of the original system is preserved. The efficiency of this approach depends largely on whether the solutions of the full system are mostly included in the subspace spanned by the basis vectors. The resulting approximations will also be compared with the approximations obtained using the classical modal reduction technique (which is also known as the modal truncation technique) based on the Normal Modes (NM) in the case of linear systems and on the underlying linear system in that of nonlinear ones. Note that the SD model reduction (as the KLD model reduction) has not for objective to approximate large scale dynamical systems. An overview of methods for this kind of problems can be found in [18]. The paper is organized as follows. In Section II, the definitions and some properties of KLD and SD are recalled. In Section III, the SD model reduction procedure is described in the case of second order nonlinear mechanical systems under random excitation. In Section IV, an example of the literature done for the linear case, [2], is revisited including the linear and now a nonlinear version with and without homogeneous mass matrix and under uncorrelated and correlated excitations. We also describe how the numerical investigation was carried out. Finally, in Sections V and VI, the efficiency of the SD model reduction is discussed and compared with the efficiency of the KLD model reduction for some linear and nonlinear configurations of the example.

II.

generality, it can also be assumed that zero-mean random process and that symmetric positive definite matrices. II.1.

, and

is a are

Basis From Karhunen-Loève Decomposition

As described in [6], the KLD of , is designed to obtain the most characteristic constant vectors i.e. those maximizing the normalized ensemble average of the inner product between and : , ,

(1)

where the inner product , coincides with the dot product in the Euclidean space . The maxima are defined by the following eigenproblem: (2) The solutions of the eigenproblem (2) characterize the , which constitute an Karhunen-Loève Modes, of . The orthonormal basis corresponding eigenvalues ( 0) are the KLVs. The KLD of the random field is therefore defined by:

(3)

where the KLCs are given by . At this step, it should be mentioned that the KLCs are . uncorrelated and related to the KLVs by In view of this fact, the following ordering, , will be used in what follows. In addition, the decomposition (3) satisfies the optimality relations:

(4)

for any integer and any arbitrary orthogonal basis of . In the reduction order model procedure, the basis , ,…, will be used to define the projection subspaces. II.2.

Basis From Smooth Decomposition

As described in [3], the SD of , was developed in order to obtain the most characteristic constant vectors i.e. those maximizing the ratio between the ensemble average of the inner product between and to the inner product between and , a condition of smoothness:

The Decomposition Basis

We assume that , and , are two second-order stationary processes. With these assumptions, the covariance matrices of , and , do not depend on time. Without loss of Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


75


, ,

Of course, the two ordered processes (SV-ordering and energy-ordering) will be compared in the context of model reduction. It will be shown that the properties of the reduced models will be ordering dependent. In the reduction order model procedure, the basis , ,…, defining from the SV-ordering and from the energy-ordering will be used to define the projection subspaces.

(5)

The objective function (5) used to define the SD differs significantly from that used to define the classical KLD (1), since the normalization is replaced by a smoothness condition. Here the denominator of the objective function takes the covariance matrix of the time-derivative process , into account (which justifies the name smooth decomposition). The maxima are defined by the following eigenproblem

II.3.

Some Comments on the SD

SD is performed solving the eigenproblem (6) where the covariance matrices can be estimated from data. As described in [1], [3], [4], the SD can be viewed as outputonly modal-analysis tools. In a linear problem i.e. when SD is applied to a stationary random response of a discrete linear mechanical system under external random excitation, the SMs can be related to the normal modes. Assuming the system with uncorrelated modal excitation, the following relations hold:

(6) The solutions of the eigenproblem (6) characterize the , which constitute a -orthogonal and SMs, orthogonal basis of . The corresponding eigenvalues , 0) are the SVs. The SD of the random field will then be defined by

1

(7)

where the SCs,

, are given by:

(11)

… denotes the matrix of the SMs, … denotes the matrix of the normal modes. These relations justify the -ordering of the SVs. One particularly interesting property of the SMs is its sensitivity to mass-inhomogeneity. This can be seen in the case of uncorrelated modal excitation of linear mechanical systems upon combining the following two relations: where

(8) Unlike the KLCs, the SCs are correlated: (9)

(12) and the decomposition (7) does not satisfy the standard optimality relation (4). However, for any , 1 , the troncate error norm using smooth modes to approximate takes the form:

gives (13) This property provides another way of computing the modal matrix if the SM and mass matrix are known. It also provides a way of computing the mass matrix from the SM and the modal matrix, which can also be obtained from the SM matrix (see the first equation in (12)). Lastly, it should be stressed that the SMs are not orthogonal with respect to the standard inner product. The orthogonality relation satisfied by the SM is .

U (10)

This last relation suggests that SMs can be ordered by the quantities

in decreasing order. This

III. Model Reduction Procedure

ordering will be named energy-ordering. Using the relations (9), the ordering can be easily deduced from SD analysis. Note that when the mention energy-ordering is not indicated, the classical ordering, defined by the SVs as named SV-ordering, will be assumed. This ordering will be justified in Section II.3.

Consider the equation of motion of a general degree-of-freedom dynamical system in the form:

,


-

(14)


76


Let us take, a finite chain consisting of mass points, where the first one is linked by a linear spring to a fixed point, the others are consecutively linked to each other, and the last one is linked only to the previous mass. All the stiffness coefficients of the springs are equal and their common value is 1. The mass values are denoted 0). The system can also include isolated ( nonlinearities between consecutive masses of the form , 0 where 2, … , . The associated equation of motion is of the form (14) with:

where , and are symmetrical x matrices. The model-reduction technique used here is based on the Petrov-Galerkin approach [18]. This technique was used in this study because it corresponds to a natural extension of the Galerkin projection approach when the decomposition basis is not orthogonal. Moreover, the Petrov-Galerkin approach can be easily implemented in such a way the second-order structure of the original system is preserved. Petrov-Galerkin approach can be regarded as a projection of the -dimensional displacement field onto an -dimensional subspace with along where (or parallel to) the subspace . Let , ,…, be a basis of , and , ,…, be a basis of such that , where … and … . We look for an approximation of in the subspace :

2 1

(15)

0 0

Substituting Eq. (15) into Eq. (14) and imposing that the residue:

, the following reduced-order

,

(16)

Two important points are the selection of the projection subspace (i.e. the projection basis , ,…, ) and the computation of the associated , ,…, . basis In the case of the SD method, as discussed in Section II.2, there are two possible ways of ordering the SMs ( ordering and energy-ordering) giving two possible reduced models. These two reduced models can be different and it will be interesting to compare them. Independently of selected ordering, and due to the orthogonality property of the SMs, will be defined by (or in an equivalent way = ). =

IV.

0 1

0

0

0 0

2 1

1 1

and , which depends only on , can be easily deduced from the form of the nonlinearity. The damping matrix is taken to be 2 where 0, which ensures that the damping is proportional determines the damping ratio of the first and that linear mode. Note that the linear version of this system has been discussed in [2]. In all the numerical simulations discussed in the next two sections the following numerical values were used: 0.05. =10 and Two excitation conditions will be considered: • an uncorrelated modal excitation: the system is excited by a standard vector-valued whitenoise process with matrix intensity

, is orthogonal to system is deduced

1 2 1

(17) 0 (this choice ensures that where is always a diagonal matrix); • a correlated modal excitation: the system is excited by a white-noise scalar process applied to the mass numbered , i.e: 0 … 010 … 0 where intensity ,

(18)

,

is a scalar white-noise process with 0 (in this case the intensity matrix of and hence is given by is not a diagonal matrix). White-noise excitation has been chosen because its Power Spectral Density (PSD) function which describes the relative power contribution at various frequencies is a constant function. In terms of frequency content, a whitenoise excitation is similar to an impulsive excitation in the deterministic case. Using a white-noise scalar process

Examples and Methods

IV.1. Description of the Examples The properties (see Section II.3 or [6], [3]) of the SD and KLD in terms of output-only modal-analysis tools being related to some characteristics of the mechanical systems such as mass matrix homogeneity, correlated excitations and of course nonlinearity behavior, we will consider hereafter a family of examples including all these characteristics.



77


permits us to analyze the system without privileging a particular frequency band.

sampling trajectories, the covariance matrices and were obtained using the classical time-average estimates (which are valid under the ergodic assumption) and the PSDM function using Welch's method of averaging modified periodogramms with a Hamming window of 4096 points length and without overlap. and , KLD and SD approach were Next, from carry out solving the eigenproblems (2) and (6) given access to the reduced models. Finally, the stationary response PSDM functions of the reduced models (16) were obtained using the same procedures as in the full case. Note that in nonlinear case, null initial conditions and the same excitation trajectory were used.

IV.2. Reduced Models Three reduced models will be considered: one drawn up using KL modes and two using the SD modes with the two ordering procedures: • … ) where denotes the KL (with modes obtained from the steady-state response; • … ) where (with smooth modes considering -ordering;

denotes the

V.

• … (with ) where denotes the smooth modes considering energy-ordering. We will also examine the reduced model given by the classical modal truncation method. In this case, the approximation is defined from (15) and the reduced model (16) is obtained with …

Results for Some Linear Configurations

In this section, we will consider four linear ( 0 2, … ,10) configurations including uncorrelated and correlated excitations with homogeneous and inhomogeneous mass. For each configuration, the and were obtained solving covariance matrices the associated stationary Lyapunov equation (see for example [6]). V.1.

(19)

where denotes the normal modes of the underlying linear system of (14). Note that this approach coincides with the Petrov-Galerkin projection only when the normal modes are orthogonal ( ).

Linear Case with Uncorrelated Excitation and Homogeneous Mass

1, Numerical values of the parameters: (17) with 1 1, … ,10. and With this configuration, the theoretical results on SD and KLD (see [6], [3], [4]) hold showing that the three vector families SM, KLM, and NM coincide, the SVs are equal to the inverse of the square of the resonance frequencies and the -ordering of the SMs coincides with the classical ordering of the NMs. Moreover, the numerical results show that the -ordering and the energy-ordering of the SMs coincide. The results are valid because of the homogeneous level of modal excitation. In terms of the model reduction, the three approaches all are equivalent. They give the same reduced-order models with diagonal mass, damping, and stiffness matrices. These results are not illustrated here.

IV.3. Methodology Use The Power Spectral Density Matrix (PSDM) function of the stationary response will be used here to compare how the reduced-order models are similar to the full system. The PSDM is an important characteristic of the stationary response of vibrating mechanical systems subjected to random excitation. The PSDSM function provides the distribution of energy per frequency band and it is easily accessible experimentally. Moreover it plays an important role in applications involving modeling and identification. For a given configuration, the first step was to and and the compute the covariance matrices PSDM function of the stationary response of the full system. For linear case, the covariance matrices were obtained solving the associated stationary and Lyapounov equation (see for example [6]), an advantage of our intrinsic definitions and the PSDM function using the harmonic transfer function approach (see for, example [19]). For nonlinear case, Monte-Carlo method was performed. Eq. (14) was solved numerically using the Newmark method with null initial conditions and a discretized white-noise trajectory obtained using the method discussed in [20]. Eq. (14) was integrated with the sampling frequency over a long time . From the

V.2.

Linear Case with Uncorrelated Excitation and Inhomogeneous Mass

1, Numerical values of the parameters: (17) with 1 except 3. and With this configuration, the theoretical results on SD (see [3]) can be applied showing that the SVs are equal to the inverse of the square of the resonance frequencies (see Table I) that the normal modes approximated from the SMs using the relation (11) coincide with the NMs (not shown here), and that each SM differs from the corresponding NM by a scaling vector factor ( ) in line with (13). Hence, the -ordering of the SMs coincides with the classical ordering of the NMs.



78


However, with the KLM reduced model, a tiny frequency shift was observed for 4 at the fourth resonance peak (see Fig. 4). Increasing , this artefact disappears.

Moreover, the numerical results show that the -ordering and the energy ordering of the SMs slightly differs (see Table I the third and fourth resonance frequencies and also the sixth and seventh resonance frequencies). Finally, due to the mass inhomogeneity and in agreement with the theoretical results, the KLMs differ significantly from the NMs (not shown here). In view of the above discussion, four reduced models were compared: one obtained with the KLMs, two obtained with the SMs (one based on -ordering one based on energy-ordering), and one with NM. The NM reduced models are diagonal (that is, the mass, damping, and stiffness matrices are diagonal), whereas the other two are coupled (that is, the mass, damping and stiffness matrices are full). TABLE I LINEAR SYSTEM WITH UNCORRELATED EXCITATIONS AND INHOMOGENOUS MASS. RESONANCE FREQUENCIES SD with SD with Mode energyExact number -ordering ordering 1 0.022 0.022 0.022 2 0.061 0.061 0.061 3 0.116 0.141 0.116 4 0.141 0.116 0.141 5 0.187 0.187 0.187 6 0.230 0.243 0.230 7 0.243 0.230 0.243 8 0.283 0.283 0.283 9 0.296 0.296 0.296 10 0.309 0.309 0.309

Fig. 2. Linear system with uncorrelated excitation and inhomogeneous mass. Frobenius norm of the PSDM responses


V.3.

Linear Case with Correlated Excitation and Homogeneous Mass

1 Numerical values of the parameters: (18) with 1, and 1 1, … ,10. and Although the theoretical results cannot be applied here, the numerical results obtained in this case are similar to those obtained in the case of uncorrelated excitation (Section V.1). The three vectors families, SMs, KLM and NM are very close and the SVs are also very close to the inverse of the square of the resonance frequencies (not shown here). These results are in agreement with the analysis of the influence of the correlation coefficient between modal excitation on the KLD and SD analysis carried out and discussed in [6], [4]. Moreover, here also the numerical results show that the -ordering and the energy ordering of the SMs coincide. Finally, in terms of the model reduction, the


The Frobenius norm of the PSDM response of the full system is compared with those obtained with the four reduced-order models with 1 (Fig. 1), 3 (Fig. 2) and 4 (Fig. 3).We recall here that is the number of modes used to draw up the reduced model. and NM reduced models give same results The (curves are indistinguishable). The and reduced models differ for 3 at the third resonance frequency (see Fig. 2). In conclusion, the four methods accurately model the full system around the resonance frequencies taken into account by the reduced models.



79


families). The smooth modes significantly differ from the normal modes at the mass number where inhomogeneous mass is present. Note that the SD could be used to detect mass inhomogeneity. All these results which are true with uncorrelated modal excitation are slightly deteriorated with correlated modal excitation. The most energetic smooth mode is associated to the natural resonance frequency number 4 (see Table II). Interesting is that the shape of the fourth normal mode estimated from the smooth modes (using -ordering) is similar to shape of the first KL mode (see Fig. 4). Hence using energy-ordering, this smooth mode will be the first. Selecting -ordering (respectively energy-ordering), the SD is in line with the classical modal decomposition (respectively KL decomposition).

three approaches all are quasi-equivalent. These results are not illustrated here. V.4.

Linear Case with Correlated Excitation and Inhomogeneous Mass

Numerical values of the parameters: (18) with 1, and 1 except 3. and

1

TABLE II LINEAR SYSTEM WITH CORRELATED EXCITATIONS AND INHOMOGENEOUS MASS. RESONANCE FREQUENCIES SD with SD with Mode energyExact number -ordering ordering 1 0.022 0.141 0.022 2 0.061 0.022 0.061 3 0.116 0.230 0.116 4 0.141 0.061 0.141 5 0.187 0.116 0.187 6 0.230 0.296 0.230 7 0.243 0.243 0.243 8 0.283 0.187 0.283 9 0.296 0.309 0.296 10 0.309 0.283 0.309

Fig. 5. Linear system with correlated excitation and inhomogeneous mass. Frobenius norm of the PSDM responses

Fig. 4. Linear system with correlated excitation and inhomogeneous mass. Mode shapes

We first discuss the results given by the SD analysis. In Table II, the exact resonance frequencies of the linear system are compared to the resonance frequencies estimated using SD approach. The set of resonance frequency values estimated from SD approach coincide with the set of exact values. Of course, the -ordering is equivalent to the classical ordering of the resonance frequencies. The first four smooth modes and the normal modes estimated from the smooth modes are plotted Fig. 4 using -ordering and compared to the KL modes and to the exact normal modes. The normal modes estimated from the smooth modes are very close to the exact normal modes (opposite sign has been used to plot the two


Four reduced models were then compared, namely the one obtained with the KLMs, two obtained from the SMs using -ordering and energy-ordering, and one from the NM. The NM-reduced models are diagonal (i.e. the mass, damping, and stiffness matrices are diagonal), whereas the three other reduced models give coupled equations of motion (i.e. the mass, damping, and stiffness matrices are full). The Frobenius norm of the PSDM response of the full system is compared with those obtained with the four reduced-order models with 1 (Fig. 5), 2 (Fig.



80


6) ), 3 (Fig. 7) and 4 (Fig. 8). First of all, in all the considered cases, the NM reduced models show results very similar to that those obtained with the reduced model. We will therefore not distinguish between these two cases in the discussion.

resonances depend on the energy captured by the SMs and KLMs (see Fig. 7 and 8). For large , all the three approaches all are equivalent. , In conclusion, the main points here are: (i) and NM reduced models coincide; (ii) for small values of , the KLM reduced model gives the least efficient reduced model in terms of the PSDM modeling that is to say, the KLM reduced models does not respect the distribution of energy per frequency band of the full system.

VI.

Results for Some Nonlinear Configurations

In this section, we will consider only configurations 1 involving an inhomogeneous mass distribution ( 2) and including a local nonlinearity except and with 10. between the masses Uncorrelated and correlated excitations will be considered. The Monte-Carlo method was implemented 10 Hz over a time with the sampling frequency interval of length 13100 sec. The PSDM function were estimated with a Hamming window of 4096 points length giving ∆ 0.0024 Hz.


VI.1. Nonlinear Case with Uncorrelated Excitation 1. Here we consider here (17) with We first discuss the results given by the SD analysis. In Table III, the exact resonance frequencies of the 0) are compared to the underlying linear system ( resonance frequencies of the nonlinear system estimated using SD approach. TABLE III NONLINEAR SYSTEM WITH UNCORRELATED EXCITATIONS. RESONANCE FREQUENCIES SD with SD with Mode Underlying energynumber linear system -ordering ordering 1 0.024 0.024 0.024 2 0.070 0.070 0.066 3 0.107 0.107 0.107 4 0.165 0.165 0.151 5 0.198 0.198 0.196 6 0.243 0.243 0.233 7 0.252 0.252 0.259 8 0.286 0.286 0.274 9 0.301 0.301 0.298 10 1.133 1.133 0.313


With 1 (and in coherency with the previous reduced model reproduces the discussion), the fourth resonance of the full system whereas the reduced model reproduces the first resonance of the full system. More surprisingly, the KLM reduced model shows a resonance between the third and the fourth resonance frequencies of the full system. With 2, reduced model also accounts for the first the resonance of the full system (the second most energetic SM is similar to the first NM), whereas the reduced model includes the second resonance frequency. With the KLM reduced model, two resonances also appear, one occurring near the third resonance of the full system and the other between the fifth and sixth resonance frequencies of the full model. Increasing , the numbers of the resonances of the full system reproduced by the reduced models increase but the approximated

Due to the coincidence of the -ordering and the energy-ordering we will only consider the -ordering in the discussion. In Fig. 9, the first three and the last smooth modes and the normal modes estimated from SD are plotted (using -ordering) and compared to the KL modes and to the exact normal modes of the underlying linear system. Due to the considered configuration, the SD are related to the modal characteristics of the associated linear system[4], that is, the SMs (respectively the resonance frequencies of the nonlinear system



81


not taken into account by the reduced models until at least 4. It is mainly due to the values of the SM and KLM shapes at the mass numbers 5$ and 6 (where the nonlinearity is present) which are identical for the nine first modes.

estimated from SD) coincide with the normal modes (respectively the resonance frequencies) of the associated linear system.

Fig. 9. Nonlinear system with uncorrelated excitation. Mode shapes Fig. 10. Nonlinear system with uncorrelated excitation. Frobenius norm of the PSDM responses

As shown in Table III, the nonlinearity affects the resonance frequency number 4 (left shift) and numbers 8, 9 and 10 (including a new high frequency). Mode shapes are also affected. The shapes of the SMs (dashed-dotted lines) differ from the shapes of the NMs obtained from the SMs (solid lines) only at the mass number where inhomogeneous mass is present. For each of the first nine SMs, the values of the mode shape at the mass numbers 5 and 6 where the local nonlinearity has been added are identical. This property is not satisfied by the normal modes of the underlying linear system. For the last SM, the values at the mass numbers 5 and 6 are of opposite signs. Note that this last SM mode is associated to the highest frequency pointed out by the SD analysis (see Table III). If the less energetic KLM coincides with the last SM and if for each of the first nine KLMs, the values of the mode shapes at the mass numbers 5 and 6 are also identical, the KLMs are differently affected by inhomogeneous mass. They slightly differ from the SMs and from normal modes estimated from the SMs. Three reduced models were then compared, namely one obtained from the SMs using -ordering, one obtained with the KLMs and one obtained from the NM of the underlying linear system. The Frobenius norm of the PSDM response of the full system is compared with those obtained with the three reduced-order models with 1 (Fig. 10) and 4 (Fig. 11). First of all, the NM reduced models does not respect the distribution of energy per frequency band of the full system until at least 4. In all the considered cases, the and KLM reduced models have given very similar results. They reproduce correctly the successive resonance peaks (with increasing ). In particularity, the shift of the fourth resonance peak (see Table III) is well reproduced for 4. Note that the high frequency component (between 0.6 and 1.6 Hz) of the PSDM are

Fig. 11. Nonlinear system with uncorrelated excitation. Frobenius norm of the PSDM responses

Remembering that the geometrical effect of the nonlinearity affects the last SM named and the last KLM named , the projection families can be enriched with these modes. Considering for example, the reduced models associated to the following projection basis: • … • … the Frobenius norm of the PSDM responses are reported Fig. 12 for 1 showing that the high frequency component due to the nonlinearity is correctly accounted for. In conclusion, the main points here are: (i) the , and KL reduced models show very close behavior; (ii) for small values of , the NM reduced models do not respect the distribution of energy per frequency band of the full system; (iii) a high frequency component is present in the SD and KD and the projection basis can be advantageously enriched with.



82


obtained with the four reduced-order models with (Fig. 14), 2 (Fig. 15) and 4 (Fig. 16).

1

Fig. 12. Nonlinear system with uncorrelated excitation. Frobenius norm of the PSDM responses

VI.2. Nonlinear Case with Correlated Excitation 1. Here we consider (18) with Concerning the SD analysis, we will restrict the discussion pointing out the difference between the uncorrelated and correlated excitation cases. In Table IV, the exact resonance frequencies of the underlying linear system are compared to the resonance frequencies of the nonlinear system estimated using SD approach. In Fig. 13, the first third and the last smooth modes and the normal modes estimated from the smooth modes are plotted (using -ordering) and compared to the KL modes and to the exact normal modes of the underlying linear system.

Fig. 13. Nonlinear system with correlated excitation. Mode shapes

As in the linear case (including an inhomogeneous mass and correlated modal excitations), the results , obtained with the reduced models (NM, and KLM) differed significantly at small values of . The KLM reduced models give poor results in terms of resonant peak modeling (the reduced models do not show the resonances of the full system) (see for example Fig. 15). Contrary to what was observed in the linear case (including inhomogeneous mass and correlated modal excitations), some differences were found to exist reduced between the NM reduced models and the models. The NM reduced models do not reproduce the band width of the resonances of the full system whereas the reduced models (and of course the reduced model ) do (see for example Fig. 16). As reduced models account for the expected, the reduced models most energetic SM where the account for the SMs by increasing resonance frequencies.

TABLE IV NONLINEAR SYSTEM WITH CORRELATED EXCITATIONS. RESONANCE FREQUENCIES SD with SD with Mode Underlying energynumber linear system -ordering ordering 1 0.024 0.242 0.024 2 0.070 0.024 0.066 3 0.107 0.070 0.107 4 0.164 0.107 0.151 5 0.198 0.198 0.196 6 0.242 0.251 0.233 7 0.252 0.164 0.259 8 0.285 0.285 0.274 9 0.299 0.299 0.298 10 0.509 0.509 0.313

The main differences turn on the resonance frequencies and energy distribution. Firstly, the higher resonance frequency (0.509 Hz) estimated from the SD approach is less than the estimated one in the uncorrelated case (1.133 Hz). Secondly, the -ordering differs from the energy ordering (see Table IV, 3rd row). The most energetic smooth mode is associated to the natural resonance frequency number 6 corresponding to the frequency 0.233 Hz. Four reduced models were then compared, namely two obtained from the SMs using -ordering and energyordering, one obtained with the KLMs and one obtained from the NM. The Frobenius norm of the PSDM response of the full system is compared with those

Fig. 14. Nonlinear system with correlated excitation. Frobenius norm of the PSDM responses



83


are not orthogonal with respect to the standard inner product of . Comparisons were made between the reduced-order models obtained using the smooth modes, the Karhunen-Loève modes, and the normal modes of the underlying linear system. Using the PSDM to compare the reduced-order models, it was established that in the case of linear systems with a homogeneous mass, the three approaches all give equivalent results. In the presence of inhomogeneous mass, the use of smooth modes was found to give more accurate reduced-order models for linear systems as well as nonlinear systems. Two types of ordering can be used to select the basis of smooth modes in the model-reduction procedure. The -ordering method is correlated with the classical method of ordering of the resonance frequencies and can be used to model the power spectral density matrix function, starting at the low-order-resonance frequencies. The energy-ordering method is correlated with the ordering method used with the Karhunen-Loève approach and can be used to model the power spectral density matrix function, starting at the high-energy-resonance frequencies. The efficiency of the reduced-order models could be improved by including in the set of vectors a smooth mode with smaller energy level but containing nonlinear information. This result shows how important is the selection of the reduction basis in the construction of the reduced-order models.

Fig. 15. Nonlinear system with correlated excitation. Frobenius norm of the PSDM responses

Acknowledgements Fig. 16. Nonlinear system with correlated excitation. Frobenius norm of the PSDM responses

The authors gratefully acknowledge the financial support of CNPq, Faperj, CAPES, COFECUB, and the French National Research Agency ANR in the context of the ADYNO project.

None of the reduced-order models show (at small or medium values of ) the nonlinear effects (between 0.5 and 0.8 Hz). Recalling the discussion in Section VI.1 this nonlinear effect can be reproduced including the SM number 10 in the projection basis. This result is not shown here. In conclusion, the main points here are: (i) the reduced model differs from the one but both give results always in agreement with the distribution of energy per frequency band of the full system; (ii) the reduced models point out the resonance peak by increasing frequency whereas the reduced models point out the resonance peak by increasing energy; (iii) for small , the NM and KL reduced models do not respect the distribution of energy per frequency band of the full system; (iv) a high frequency component is present in the SD and KD and the projection basis can be advantageously enriched with.

VII.

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

Conclusion

The ability of reduced-order models based on smooth modes to approximate the PSDM of the response of a second-order discrete mechanical systems was studied here. The Petrov-Galerkin approach was used to draw up the reduced-order models since the projection basis used

[8]

[9]


D. Chelidze, W. Zhou, Smooth Orthogonal Decomposition-based Vibration Mode Identification, Journal of Sound and Vibration, vol. 292, p. 461-473, 2006. B. Feeny, U. Farooq, A Nonsymmetric State-Variable Decomposition for Modal Analysis, Journal of Sound and Vibration, vol. 310, p. 792-800, 2008. S. Bellizzi, R. Sampaio, Smooth Karhunen-Loève Decomposition to Analyze Randomly Vibrating Systems, Journal of Sound and Vibration, vol. 325, p. 491-498, 2009. S. Bellizzi, R. Sampaio, Analysis of Non-Stationary Random Processess using Smooth Decomposition, Journal of Mechanics of Materials and Structures, vol. 6, pp. 1137-1152, 2011. S. Bellizzi, R. Sampaio, Application of the smooth decomposition to the ranodm fields, Proceedings of 16th US National Congress of Theoretical and Applied Mechanics, State College, 2010. S. Bellizzi, R. Sampaio, POMs Analysis of Randomly Vibrating Systems obtained from Karhunen-Loève Expansion, Journal of Sound and Vibration, vol. 297, p. 774-793, 2006. P. Holmes, J. Lumley, G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry (Cambridge University Press, 1996). W. Zhou, Multivariate analysis in vibration modal parameter identificationcation, Ph.D. dissertation, University of Rhode Island, Kingston, 2006. G. Quaranta, P. Masarati, P. Mantegazza, Continuous-Time Covariance Approaches for Modal Analysis, Journal of Sound and Vibration, vol. 310, p. 287-312, 2008.


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[10] A. Placzeka, D.-M. Tran, R. Ohayon, Hybrid Proper Orthogonal Decomposition Formulation for Linear Structural Dynamics, Journal of Sound and Vibration, vol. 318, p. 943-964, 2008. [11] M. Khalil, S. Adhikari, A. Sarkar, Linear System Identification using Proper Orthogonal Decomposition, Mechanical Systems and Signal Processing, vol. 21, p. 3123-3145, 2007. [12] J. Atwell, B. B. Kings, Proper Orthogonal Decomposition for Reduced Basis Feedback Controllers for Parabolic Equations, Mathematical and Computer Modelling, vol. 33, pp. 1-19, 2011. [13] M. F. A. Azeez, A. F. Vakakis, Proper Orthogonal Decomposition (POD) of a Class of Vibroimpact Oscillations, Journal of Sound and Vibration, vol. 240, p. 859-889, 2001. [14] G. Kerschen, J.-C. Golinval, A. Vakakis, B. Bergman, The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: an Overview, Non-linear Dynamics, Special issue on ”Reduced Order Models: Methods and Applications”, vol. 41, p. 147-169, 2005. [15] M. Trindade, C. Wolter, R. Sampaio, Karhunen-Loève Decomposition of Coupled Axial/Bending Vibrations of Beams Subject to Impacts, Journal of Sound and Vibration, vol. 279, p. 1015-1036, 2005. [16] R. Sampaio, C. Soize, Remarks on the Efficiency of POD for Model Reduction in Non-Linear Dynamics of Continuous Elastic Systems, International Journal of Numerical Methods in Engineering, vol. 72, p. 22-45, 2007. [17] X. Ma, A. F. Vakakis, L. Bergman, Karhunen-Loève Analysis and Order Reduction of the Transient Dynamics of Linear Coupled Oscillators with Strongly Nonlinear End Attachments, Journal of Sound and Vibration, vol. 309, p. 569-587, 2008. [18] B. Lefevre, F. Druesne, J.-L. Dulong, P. Villon, Different Formulations for Model Reduction to Simulate the Crush of a Mechanical Part, International Review of Mechanical Engineering (IREME), vol. 3, n° 12, pp. 172-170, 2009. [19] C. A. Antoulas, Approximation of Large-Scale Dynamical Systems (Advences in Design and Control SIAM, 2005). [20] L. Lutes, S. Sarkani, Stochastic Analysis of Structural and Mechanical Vibrations (Prentica Hall, 1997). [21] F. Poirion, C. Soize, Simulation Numérique des Champs Stochastiques Gaussiens Homogènes et Non Homogènes, La recherche Aérospatiale, vol. 1, p. 41-61, 1989.


Laboratoire de Mécanique et d'Acoustique, CNRS, 31 chemin Joseph Aiguier, 13402 Marseille, France. Tel +33 491164238, Fax: +33 0491164080 E-mail: [email protected] 2 Department of Mechanical Engineering, PUC-Rio, Rua Marques de Sao Vicente, 225, Rio de Janeiro, RJ, 22453-900, Brazil. Tel: +55 2135271172, Fax: +55 2135271165. E-mail: [email protected]



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Hydrodynamics and Mass Transfer Using Three-Phase Fluidized Bed Contactor Abbas H. Sulaymon1, Raghad F. Almilly2 Abstract – This research presents a study of the hydrodynamics and mass transfer of the absorption of CO2 from CO2-air mixture (0.5-1%) by volume using water and NaOH solution. The hydrodynamic and mass transfer parameters are improved in the turbulent bed contactor (TBC) by the imposition of pulsed liquid flow (pulsation 0.01-0.03 m/s) upon a steady operation through this new research. New mathematical models of steady-state forces' balance for wetted-bed particles and single spherical particle were developed in TBC. It was found (by this new approach) that a single spherical mathematical model can well represent the TBC when shallow beds are used. New correlations were obtained to represent the various cases of the absorption processes using dimensional analysis. The constants of these correlations were determined using nonlinear regression technique. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Absorption of CO2, Dimensional Analysis, Hydrodynamics and Mass Transfer, Pulsation in Liquid Flow, Mathematical Model, Single Sphere, Three-Phase Fluidized Bed Contactor

KL

Nomenclature A AC a C CBo ci D DC d de fB fD f f^ ff G GM Gmf g H Ha Hst KG K'GP

Projected area (m2) Cross-sectional area of empty column (m2) Gas-liquid interfacial area per unit volume of the bed (m2/m3) Concentration (kmol/m3 ) Initial concentration of component B (i.e. OH ion ) in the liquid phase (kmol/m3) Coefficient of correlations (i=1,2,3,4) (-) Molecular diffusivity (m2/s) Inside diameter of the cylindrical column(m) Diameter (m) Equivalent diameter based on dry packing (m) Buoyant force (kN) Drag force (kN) Frequency of pulsation ( s1 ) Function Friction factor or drag coefficient (-) Gas mass velocity (kg/m2·s) Gas molar velocity (kmol/m2·s) Minimum fluidization gas velocity (kg/m2·s) Gravitational acceleration constant (9.81m/s2) Expanded bed height (m) Hatta number ( (k^2 DL cbo)1/2 /kol ) (-) Static bed height (m) Overall gas phase mass transfer coefficient (kmol /m2·s·kPa ) Overall gas-phase mass transfer coefficient in pulsation flow (kg/m2·s)

k^2 koL L LM m p pB*M R Rg Re' s Sc Sh' T tlf u V VT w X Xe Y Ye z


86

Overall liquid phase mass transfer coefficient (m/s) Second-order reaction rate constant (m3/kmol·s) Liquid film mass transfer coefficient of physical absorption (m/s) Liquid mass velocity (kg / m2· S ) Liquid molar velocity (kmol/m2·s) Mass (kg) Total pressure (kPa) Mean partial pressure of inert gas B* in the gas film (kPa) Radius (m) Perfect gas-law constant (8.314 kJ / kmol·K) Modified Reynold's number (ρ u de / µ ) (-) Stroke length (m) Schmidt number (µ / ρ D ) (-) Modified Sherwood number (K de / D)(-) Absolute temperature (K) Liquid film thickness (m) Superficial velocity (m/s) Volume (m3) Total volume of the bed (m3) Weight (kN) Mole fraction of CO2 in the liquid phase (-) Mole fraction of CO2 at equilibrium in the liquid phase (-) Mole fraction of CO2 in the gas phase (-) Mole fraction of CO2 at equilibrium in the gas phase (-) Incremental height of bed (m) Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

Abbas H. Sulaymon, Raghad F. Almilly

∆p ɛ µ νL ρ

Pressure drop (kPa) Phase holdup (-) Viscosity (kg/m·s) Kinematic viscosity (m2/s) Density (kg/m3)

(1) where: (2) 1 2

Subscript b Bed of wetted particles G Gas L Liquid lf Liquid film lm Logarithmic mean p Particle R Radius (m) S Solid WP Wetted particle

(3) (4)

Substituting Equations ((2), (3), and (4)) into Equation (1), dividing by AC (AC= VT/H ) and assuming Ab ≈ AC yields: 1 2

Sub-subscript P Pulsation

1

I.

Introduction Rearranging:

The hydrodynamic and mass transfer parameters were extensively studied in the TBC based on empirical correlations. Kielback [1] was the first in describing and developing this type of fluidization in an application to gas scrubbing processes and termed the phenomenon "floating bed scrubber". The results were very encouraging owing to the good interphase contact and scrubbing action. Other encouraging design features were the ease of augmenting scrubbing efficiency by the increase in bed depth, the overall simplicity of design and construction, and a favourable cost. Blyakher et. al. [2], Levesh et. al. [3], Tichy et. al. [4], and Wozniak [5] studied the pressure drop and obtained empirical correlations. Blyakher et. al. [2], Levesh et. al.[3], Tichy and Douglas [6], and O'Neill et. al. [7] investigated the bed expansion in the TBC and expressed it by correlations. Chen and Douglas [8], and Barile and Meyer [9] examined the liquid holdup in the TBC and developed correlations. Blyakher et. al. [2], Tichy and Douglas[6], and Chen and Douglas [8] developed correlations to represent the minimum fluidization velocity. Douglas et. al. [10], Douglas [11], Blyakher et. al. [2], Levesh et. al. [12], Wozniak and Østergaard [13], Wozniak [5], and Miconnet et. al. [14] investigated the mass transfer coefficients in the TBC of various absorption processes. Hydrodynamic characteristics of mixing in three phase system were also studied [15].

II.

Theoretical Aspects

II.1.

Force Balance in the TBC

2

1

2

οr: 1

2

(5)

If ρG can be neglected with respect to ρP and ρL , and the drag force term can also be neglected with respect to other terms , equation (5) becomes: (6) The resulting equation (6) corresponds to the wellknown empirical equation describing the pressure drop along TBC [7]-[8]-[9]. II.1.2. A Steady-State Balance of Forces Acting on a Single Wetted Spherical Particle The steady-state balance of forces acting on a single wetted spherical particle is: (7) (8) 1 2

II.1.1. A Steady State Balance of Forces Acting on a Wetted Bed of Particles

(9) (10)

Substituting Equations ((8),(9), and (10)) into Equation (7) and dividing by AWP yields:

The steady state balance of forces acting on a wetted bed of particles is as follow:



87


1 2

,

(11)

II.2.

1 2

(12)

,

Dimensional Analysis II.2.4. Pulsed flow of Liquid in CO2 -NaOH system

II.2.1. Steady flow of liquid in CO2-H2O system

The function deduced is:

Using Buckingham's theorem, there is a function of the form : , ,

,

where the dimensionless group (KG Rg T de / DG ). pB*M /p is used for consistency with the conventional units of KG. Also, the use of koL is necessary to make the reaction –rate expression dimensionless i.e. to attain Hatta number.

Noting that VWP/AWP = 4/3 RWP; Equation (11) becomes after combining: 4 3

,

,

,

,

,

0

,

(

Evidently, the ratio of lengths de/DC and Hst/DC are dimensionless groups. Bearing this fact in mind, DC and Hst can be eliminated from the dimensional matrix for a given column and a given type of filling. The relation deduced is in the following form: ⁄

⁄

,

,

,

,

,

,

= {

, ,

,,

, }

i.e.: ,

,

,

,

,

The main objective of this research is to show the improvement gained by imposing a pulsation action in the liquid flow in the TBC. Also, it shows the possibility of representing the TBC by a single spherical particle model. This representation may become useful in simulating the TBC.

⁄

,

, , , ⁄ 1/2 ) / ,

i.e.:

III. Experimental Work Figure 1(a) shows the schematic diagram of the experimental apparatus used for investigating the hydrodynamic and mass transfer parameters. The apparatus consists of glass column of (0.1524 m) inside diameter and (2 m) length, liquid and gas distributors located at the top and bottom of the column, respectively. The column was packed with table-tennis spheres of (0.0378 m) diameter. The choice of table-tennis spheres was based on: 1- They are made of plastic which has density (78 kg/m3) suitable with the density of the fluidizing fluid i.e. the air. This property enabled the fluidization to occur at a pressure drop within the operating range of the blower. 2- They have definite geometry, i.e. definite specific surface area. 3- They are chemically inert, so they can withstand corrosive surrounding. 4- They have strong structure, so they can withstand any distortion, erosion, and fracture resulting from their movement and striking the wall and each other during fluidization process. The apparatus contains also two glass vessels used for feeding water and NaOH solution. Each vessel's outlet branched into two pipelines, one passed through a centrifugal pump (Stuart Turner – England) represented the steady – flow line. The other outlet passed through a piston pump (Milton Roy – England) represented the oscillating – flow line.

II.2.2. Pulsed Flow of Liquid in CO2-H2O System In the case of pulsed operation , the system variables are: , ,

,

,

,

,

,

,

with the same considerations for DC and Hst as before the relation is : ,

, i.e.: ,

,

II.2.3. Steady Flow of Liquid in CO2-NaOH System A function in the following form may exist (after neglecting the effect of DC and Hst as before) : , , ,

,

,

,

,

,

0

In terms of dimensionless groups : ⁄ , (

{ )1/2 /

,

, ,

}

i.e.:



88


1 2 3 4 5 6 7 8 9 10 2

Column Water Feed Vessel NaOH Feed Vessel Receiving Vessel Blower Centrifugal Pump Piston Pump CO2 Cylinder Multi‐ manometer Liquid Rotameter 3

To Atmosphere

9 10

1

6

5 7

8

4

6

To Drain

Fig. 1(a). Experimental apparatus of CO2 absorption in TBC

The two pumps connected in parallel. Their outlet pipelines combined through a T- joint to form one feed pipeline entering the tower. Air blower (Watkins & Watson- England) and CO2 gas cylinder were used. CO2 gas was injected into the inlet opening of the blower from CO2 cylinder. Seventeen manometer taps (at interval of 0.12 m) were used for measuring static pressure gradient. Four static bed heights were used (0.305, 0.457, 0.61 , and 0.76 m). For each height air – CO2 mixture was blew to the system at the bottom. Air flow rate was changed in the range (0.14240.1761 m3/s). The mixing percentage of CO2 in the air was (0.5-1 % by volume). Water was fed continuously to the feed vessel used in the physical absorption runs. NaOH solution (of initial concentration of 0.25 N) was prepared for using in the chemical absorption runs. The two liquids were used for steady and pulsed operations. When the operation was steady, only the centrifugal pump was operated; and when it was pulsed, the two pumps (i.e. the centrifugal and the piston pumps) were operated. The liquid flow rate in the steady flow was (6.39 × 10-5 – 1.28 × 10-4 m3/s ).

IV.


Experimental results are presented in Figs. 2-17. Fig. 2 shows the linear variation of the pressure drop with vertical position along the fluidized beds (z) for different gas velocities, liquid velocities and static bed heights of CO2 –H2O system in steady operation. This linearity agrees well with other investigations [4]. The effect of increasing liquid velocity and static bed height on the pressure drop is quite notable. Fig. 3 implies the effect of superficial liquid velocity on ∆p for steady and pulsed operations. It is well obvious that the superficial liquid velocity in the case of pulsed operation is greater than the corresponding values in steady operation. This is attributed to the addition of pulsation velocity which leads to the increase in liquid holdup and pressure drop. The linear variation of ∆p with H for steady and pulsed operations is presented (see Fig. 4). The steep increase of ∆p with H in the case of pulsed operation is a result of increasing ɛL . Fig. 5 shows the tendency of ∆p to increase monotonically with (sf). The curve shows a high sensitivity of pressure drop at high values of the pulsation velocity (sf).



89


900 800

ΔP,N/m2

700 600 500 400 0

10

20

30

40

50

UL *10 3 ,m/s steady operation

pulsed operation,s=0.022m

Fig. 3. Effect of superficial liquid velocity on bed pressure drop for steady and pulsed operation in CO2-H2O system(G=10.50Kg/m2.s,Hst=0.76m)

Fig. 4. Variation of pressure drop with expanded bed height for steady and pulsed operation in CO2-H2O system (G=9.84 Kg/m2.s;L=6.11 Kg/m2.s)

Fig. 1(b). Photo of the experimental set-up

Fig. 5. Effect of liquid pulsation velocity on bed pressure drop for CO2H2O system(G=9.84 Kg/m2.s;L=6.11 Kg/m2.s;Hst=0.76m)

This is because the column approaches its maximum capacity condition with the increase of (sf) , after which the flooding occurs. The pressure drop at zero (sf) corresponds to its value in steady operation. The liquid holdup has approximately the same tendency to increase with (sf) as ∆p i.e. high sensitivity of the curve at high

Fig. 2. Static pressure profile along the column for CO2-H2O system in steady operation of different gas velocities, liquid velocities and static bed heights



90


The values of ∆pexp agree well (deviation of 11%) with those calculated using equation (6) (as shown in Fig. 10). Equation (6) seems to be a good approach to calculate ∆p in the TBC. The method of calculating ∆pWP is found in Appendix. Figs. 11 and 12 show the variation of the physical mass transfer coefficient with L and G respectively. Fig. 11 shows the nonlinear dependence of KοLa (overall liquid phase mass transfer coefficient in absence of reaction) on L at steady operation. Each curve has a maximum at some liquid flow rate. This can be attributed to the increase in liquid holdup at increasing L , which means the formation of stagnant pools of liquid in the bed. These stagnant pools suppress the turbulence effect and magnify the liquid film resistance at the expense of the mass transfer coefficient. The peak of the curve being more prominent as Hst is increased. The decrease of (KοLa) at increasing Hst can be interpreted by the transformation from uniform fluidization to slugging fluidization as the bed becomes deeper. Uniform fluidization intensifies mass transfer.

(sf) values. Increasing liquid holdup with (sf) brings the column nearer to the flooding condition (as shown in Fig. 6). The increase in H as a result of imposition of pulsation is shown (see Fig. 7). The important point here is that a high increase in pulsation velocity leads to a decrease in bed expansion i.e. a contraction. Also, it leads to an increase in Gmf (minimum fluidization gas mass velocity) as shown in the figure. This means that H passes through a maximum when plotted against (sf) (as shown in Fig. 8). The maximum point being more obvious at high gas velocities. The explanation of this phenomenon can be shown as follows: the increase in superficial liquid velocity (as a result of imposition of pulsation) leads to an impedance of particles expansion with respect to that of steady operation. The experimental data were used to verify the single sphere model through the determination of ∆pWP and the comparison with ∆pexp (this is shown in Fig. 9). The good agreement (deviation of 13%) at low values of Hst indicates that this model apply well when the fluidized beds are nearly uniform. At high values of Hst , it is very clear that equation (12) needs some correction factor being function of Hst/DC ratio.

Fig. 8. Effect of liquid pulsation velocity on bed expansion for CO2H2O system (L=6.11 Kg/m2.s,Hst=0.76m)

Fig. 6. Effect of pulsation velocity on liquid holdup for CO2-H2O system (G=9.84 Kg/m2.s,L=6.11 Kg/m2.s,Hst=0.76m) 0,86 0,84

H ,m

0,82 0,8 0,78 0,76 0,74 Gmf

0,72 7

9

11

G, Kg/m2.s steady flow

pulsed flow,s=0.024m

pulsed flow,s=0.019m

Fig. 9. Comparison of pressure drop values calculated using equation (12) with experimental values (G = 9.84 Kg/m2.s ,L=5.34 Kg/m2.s)

Fig. 7. Effect of gas velocity on bed expansion for CO2-H2O system in steady and pulsed operation (L=6.11Kg/m2.s,Hst=0.76m)



91


Holloway [16]. The same features were found through plotting the gas – phase mass transfer coefficient i.e. (KGa) (as shown in Figs. 14, 15, and 16). Fig. 14 shows the nonlinear dependence of (KGa) upon the liquid mass velocity which justifies the liquid- phase resistance controlling in the transfer process of CO2 into NaOH solution.

Fig. 10. Comparison of pressure drop values calculated using equation (6) with experimental values (G =9.84 Kg/m2.s ,L =5.34 Kg/m2.s )

The undefined peak at low static bed height indicates that the The physical mass transfer coefficient in the pulsation operation has a maximum when plotted against (sf) (as shown in Fig. 13). The important point to discuss here is that (KοLa)P passes through a maximum with respect to the pulsation velocity (sf). The pulsation liquid – phase mass transfer coefficient (KοLa)P increases with increasing pulsation velocity (sf) as a result of increasing liquid holdup and the turbulence effect, till (sf) reaches a value after which further increase in (sf) results in a higher liquid loading ; i.e. stagnant pools effect and less level of mass transfer. The peak of the (KοLa)P vs (sf) curve varies with liquid velocity and static bed height. The peak exists out of the present operating range of (sf) at low values of Hst . peak may exist out of the present operating range of liquid flow rate.

ο

Fig. 12. Effect of gas velocity on K La for different liquid flow rates and static bed heights

ο

Fig. 13. Effect of liquid pulsation velocity on (K La)P for different liquid flow rates and static bed heights (G=9.84 Kg/m2.s )

The effect of increasing static bed height in the dependence of (KGa) is also clear and can be discussed through the slugging phenomenon as with the liquid – phase mass transfer coefficient. The existence of a maximum value of (KGa) at some value of L can be interpreted through the formation of stagnant pools at high liquid flow rates. Fig. 15 shows a small effect of gas velocity on (KGa). Figs. 14 and 15 indicate that the major resistance to absorption in this system (but not the whole resistance) lies in the liquid phase. A remarked improvement obtained by this investigation in (KGa) values is illustrated (see Fig. 16).

ο

Fig. 11. Effect of liquid flow rate on K La for different static bed heights (G=10.50 Kg/m2.s )

The projection of the peak at high values of Hst means that the formation of stagnant pools is more easily at these values. The stagnant pools make the bed to flood if liquid flow rate is increased further. Thus, the restricted operating range of liquid flow rate guarantees the prevention of flooding at all values of Hst. In Fig. 12 (KοLa) was found to be independent on gas velocity, meaning that the resistance to transfer lies wholly in the liquid phase. This agrees well with Sherwood and



92


The curve has a peak at some value of (sf) (0.03-0.032 m/s) which indicates that an enhancement of absorption occurs. Increasing the pulsation velocity above this value will result in increasing the liquid holdup, i.e. the stagnant pools effect which leads to a decrease in (KGa)P. The correlations found using dimensional analysis technique were as follows: 1- Steady flow of liquid in CO2-H2O system: .

.

2- Pulsed flow of liquid in CO2-H2O system: .

.

3- Steady flow of liquid in CO2-NaOH system: .

.

Fig. 15. Effect of gas velocity on KGa for different liquid flow rates and static bed heights

4- Pulsed flow of liquid in CO2-NaOH system: .

.

where : c1,c2,…etc. are the coefficients of the correlations and they have different values for different static bed heights (as shown in Fig. 17). This means that these coefficients are functions of (Hst/DC) ratio which was regarded as a parameter in the dimensional analysis. The figure shows c1 being the most sensitive to the variation in (Hst/DC). This sensitivity asserts that the resistance to transfer is liquid-side dependent. For pulsation operation c2 is less sensitive to (Hst/DC) ratio; being comparative with c3 and c4, which indicate that the diffusional resistance in the liquid phase is no more responsible of transfer. The coefficients show decrease with the increase in (Hst/DC) ratio. The range of application of these correlations are:

Fig. 16. Effect of liquid pulsation velocity on (KGa)P (G=9.84 Kg/m2.s ; L= 6.15Kg/m2.s ; Hst=0.61 m)

32.71, it i will be maainly natural conveection. When Ri is of the order o of unity,, it is mixed conveection. Dogan and Sivriogluu [11] investiggated that mixed convection mode m may bee useful in some s practical situuations. Tao [12] investigatted combinedd free and forced in channelss. He introdduced a com mplex function whhich is directlly related to the velocity and temperature fields, the cooupled momeentum and ennergy equations. Foor horizontal channels if the t heating occcurs at the bottom m surface, buooyancy may innduce a seconndary flow which is i in combinaation with the main flow. Inn the

II.. II.1.

Experim mentation

Desccription of thee Experimental Set Up and Fin A Array Model

Literature L reeview revealled that th here is noo experimental worrk reported soo far on mixeed convectionn overr heated horizontal rectanggular fin arraays in a openn ended cavity. An A experimenntal set-up is consisting off enclosure off 1meeter × 1 meteer × 1 meter w with front sid de transparentt acry ylic sheet andd pipe in pipee with sliding g arrangementt and bell mouth entry. e DC fann (6’’) is fitteed with speedd conttrol arrangement. The T photograpph of experim mental set-up is shown inn Fig.. 1, while, fin array module is shown in Fig. F 2.

Fig. 1. Phhotograph of the eexperimental set--up

Fig. 2. Fin arraay module

Copyright © 20012 Praise Worthyy Prize S.r.l. - Alll rights reserved

International Review of Mecchanical Engineerring, Vol. 6, N. 1

105

S. G. Taji, G. V. Parishwad, N. K. Sane, R. Z. Deshmukh

It consists of following important part: Bell mouth entry, Honeycomb section, test section of fin array, fan (assisting mode), and digital anemometer. Blackened horizontal rectangular fin array module is installed properly in the test section. Thermocouples and other electric wires are properly arranged at various locations as shown in Fig. 3. Voltmeter and ammeter readings are recorded. Heater input is maintained at a constant value during the test run.

The average convective heat transfer coefficient (ha) is determined as: (1) where, Qcv is convection heat transfer rate to the fluid. Ae total exposed surface area of the fin and base surface Ts and T∞ average surface temperature and inlet temperature of air. Convective heat transfer from bottom and fins (Qcv) was determined from an energy balance equation as: (2) where, Qt is total dissipated energy from the surface heater source, Qcd total conduction heat loss from bottom and side wall surface and Qrf total heat radiated from the fins. The total dissipated energy was determined from Qt= V.I at the heater source. The voltage drop V and current I was measured during the experiment. The heat loss through the bottom and side wall of test section was calculated as:

Fig. 3. Schematic diagram with electrical connection

(3) where, K is thermal conductivity of the insulating material dt/dx) bottom is temperature gradient for bottom wall, (dt/dx)side is temperature gradient for side wall. Radiation losses were determined from the assumption of isothermal black body. This loss is determined from following equation:

Qrf

As Ts4 T 4

(4)

where, ε is the emissivity of black fin surface, σ is the Stefan Boltzman constant, Αs is the surface area for radiation heat transfer. The fluid properties used in these definitions were determined at the arithmetic average of fin array and T T fluid inlet temperatures s . 2

Fig. 4. Validation with horizontal and vertical flat plate under natural convection

Thermocouple outputs are checked and recorded at all the measuring points of fin array assembly; backelite and syporex block in order to measure side loss and bottom loss respectively. Fan speed is adjusted with the help of regulator. Regulator control is so adjusted to obtain the desired air velocity, in order to get perfect mixed convection situation. At outlet of pipe at four points air velocity is measured with the help of digital anemometer. At every15 minute’s temperatures were observed and recorded. The thermocouple reads more or less the same value for two or three successive reading to insure steady state condition. All the observations are recorded after steady state only. The data obtained during the experiments are values of temperature, velocity, voltage drop across the heater, and electric current.

II.2.

Parameters of Computation

Fin spacing is varied as; S =6, 8, 10, 12 mm, Ra changed from 60×103 to 1.5×105, as ∆T between fin array and ambient is varied from 25K to 100K, in steps of 25K. Ri number is varied from 0.3 to 7. Gr varies from 8.2482×104 to 1.64448×105. The corresponding variation in Re is from 160 to 420, resulting in velocity variation from 0.1 m/s to 0.3 m/s. The various fin-array set studied during experimentation are listed in Table I. All quantities measured during experiment are subjected to certain uncertainty due to error in measurements. Errors in ∆T are ±0.5 ºC. This translates to an error in the ha of ± 5%.



106


The variation of ha values are found in the range of about 17% to 18 %. Thus there is good agreement between the previous data and the present data.

The uncertainty is calculated for all values and found out to be in the range of ± 6 %.

Fin array set

Fin spacing (mm)

Number of fin (N)

I II III IV

6 8 10 12

13 9 10 8

ha (W/m2KK)

TABLE I FIN ARRAY CONFIGURATION (L=150 mm; H=30 mm; T =2mm)

III. Validation

6 5 4 3 2 1 0

Harhap and Mcmanus Present study

Present experimental works are validated with the earlier experimental results of various investigators for natural convection and assisting mode of mixed convection.

Fig. 6. Variation of ha with ∆T under natural convection

III.1. Validation for Natural Convection

TABLE II COMPARISON OF FIN ARRAY DETAILS

20

Natural convection results are also validated with vertical plate correlation as shown in Fig. 3. The plot is drawn for natural convection of heated vertical plate (Churchill and Chu correlation) and for horizontal flat plate (Jones and smith correlation) [6]. Due to the constriction in the fluid flow, at lower spacing average convection heat transfer coefficient (ha) is lower as expected, whereas at S= 12 mm, spacing being wide, ha approaches the vertical flat plate correlation values. The ha is also plotted for different fin spacing (Fig. 5).

40

60 ΔT (K)

80

100

Investigators

L mm

H mm

t mm

S mm

N

Harahap and Mcmanus

127

38

1.27

6.35

33

Present work

150

30

2

6

13

III.2. Validation for Mixed Convection Dogan and Sivrioglu [11] carried out work on mixed convection heat transfer from longitudinal fins in a horizontal rectangular channel, for natural convection dominated region in ducted type geometry. However, this work is carried out in the enclosure which was kept open to ambient. Figure 8 shows variation of ha with fin spacing for heated horizontal fin array under combined assisting mode for various fin spacing. Due to the constriction in the fluid flow, at lower spacing, ha values are lower, however, when spacing in between 8 to10 mm improve ha values, as spacing near the optimum region. At S = 12 m average convection heat transfer coefficient reduce due to reduction in surface area. Dogan and Sivrioglu considered longitudinal fin array of larger dimensions in a ducted type geometry, because of this reason the results obtained for ha values are on higher side. Investigators have presented results on combined convection from horizontal plates, square and rectangular cavities, horizontal parallel Plates, pin fins and other related geometries. Almost no work was reported on mixed convection over horizontal rectangular fin arrays in a open cavity. Figure 7 shows the variation of average convection heat transfer coefficient with temperature difference for heated horizontal fin array under combined assisting mode for various fin spacing.

Fig. 5. Validation of present setup with Starner and Mcmanus in natural convection mode

Fig. 6 shows variation of ha with ∆T under natural convection condition with Harahap and Mcmanus [4]. It is observed that results are closed match with previous investigator. The Table II shows fin array configuration used by authors and Table III shows percentage difference in results obtained. Therefore, the fluid flow within the fin channel as well as the heat transfer characteristics are similar to those for horizontal fins used for investigation by other investigators. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


107


10 9 8

ha (W/m2K)

7 6 5 4

M. Dogan and M. Sivrioglu M. Dogan and M. Sivrioglu M. Dogan and M. Sivrioglu present study S10_H30. Present study S8_H30 Present Study S12_H30 Present Study S6_H30

3 2 1 0 0

Fig. 9. Variation of average convection heat transfer coefficient with temperature difference under natural convection condition

10 20 30 40 50 60 70 80 90 100 110

∆T K

Fig. 7. Validation of ha vs ∆T with Dogan and Sivrioglu under mixed convection

Fig. 10. Variation of ha with ∆T (V = 0.1 m/s)

Fig. 8. Validation of ha vs S for mixed convection

IV.


Fig. 11. Variation of ha with ∆T (V = 0.2 m/s)

IV.1. Results Results are obtained by conducting experimentation on natural and mixed convection for horizontal rectangular fin array. In Figs. 8–11, the variation of ha with ∆T are shown for various fin spacing under natural convection at various heater input (Q =25, 50, 75, 100 W). The average convective heat transfer increases as ∆T increases, ha value is higher for 10 mm spacing. At lower spacing, the average heat transfer coefficient is lower and at optimum spacing, the average heat transfer coefficient increases. However, due to reduction in surface area when spacing is more than optimum, average heat transfer coefficient decreases. Figures 12–16 shows variation in (ha) average convective heat transfer coefficient with fin spacing for different flow velocities of air from 0.1 to 0.3 m/s. It is observed from these figures, the increase or decrease in fin spacing after an optimum region does not show any enhancement in heat transfer coefficient. On the contrary, it causes a reduction in heat transfer coefficient.

Fig. 12. Variation of ha with fin spacing (V=0.3 m/s)

The average heat transfer coefficient first increases with fin spacing up to a maximum value and then it decreases with the increase in fin spacing. The decrease in fluid velocity results in a decreasing efficiency in removing heat energy from heated surfaces, and therefore causes a large part of the fin surface, space between fins to be occupied by heated fluid and prevents to transfer heat efficiently.



108


increase in the fin spacing (decreases the number of fins), causes a decrease in total heat transfer surface area, which results in a decrease in the rate of heat transfer. As seen from these figures in the design of fin arrays on a horizontal surface, to achieve maximum amount of heat transfer, the fin spacing must have an optimum value. Fig. 17-20 shows the variation of base heat transfer coefficient with fin spacing at different velocities from 0 to 0.3 m/s. It is observed that, the value of base heat transfer coefficient is higher for Sopt and lowest for S= 6 mm and S=12mm. However, the base heat transfer coefficient obtained for the fin spacing of S= 10 mm is greater than the average heat transfer coefficient obtained for S = 12 mm (Fig. 14). This is due to the fact that the buoyancy driven forces that become strong enough with the increase in Rayleigh number cannot develop much because of small value of fin spacing. The effect of air flow velocity on hb is observed for various fin spacing as shown in Fig. 20. Figure 21 shows the variation of base heat transfer coefficient with fin spacing for different Richardson number (Ri = 0.5, 1, 3) at Q = 50 W, under assisting mode. Base heat transfer coefficient is higher at lower Richardson number as flow approaches towards forced convection dominated region. At higher value of Ri, base heat transfer coefficient (hb) decreases, as flow becomes natural convection dominated region. An optimum spacing region is observed for various Richardson numbers. At lower value of Ri number, forced convection is dominant and average Nusselt number values are lower.

Fig. 13. Variation of ha with fin spacing (Q = 25 W)

Fig. 14. Variation of average convection heat transfer coefficient with fin spacing (Q = 50 W)


Fig. 17. Variation of base heat transfer coefficient with fin spacing (Q = 25 W)


After the optimum value of spacing, the average heat transfer coefficient decreases with the increase in fin spacing as shown from these Figures 12 to 15. Due to




109



Fig. 22. Variation of Nua with Rayleigh number (Ri=0, 0.5, 1, 3)


Fig. 23. Variation of Nua with Rayleigh number

V.

Conclusion

The experimental work is carried out in natural and mixed assisting mode over heated horizontal rectangular fin array is presented in this paper. Results obtained are validated with previous investigators for natural and mixed convection. However, so far no work is carried out in open ended cavity for mixed convection. Therefore, results obtained for mixed convection are compared with Dogan and Sivrioglu [11], which was carried out in ducted type geometry. The effects of fin spacing on heat transfer have been investigated by conducting experiments at four different fin spacings. The average convection heat transfer coefficient increases first with fin spacing and then it takes its maximum value after which it starts to decline with the increase in fin spacing for both natural and assisting mode. When the fin spacing is smaller than the required value, the resistance against the flow is formed due to the intersection of boundary layers developed on fin surfaces and as a result, the rate heat transfer from fin array decreases. For large values of fin spacing (causing a small number of fins for fixed fin-base area), however, the decrease in the total heat transfer area causes the rate of heat transfer to decrease. Results of experiments have shown that to obtain maximum amount of heat transfer from fin arrays in natural convection dominated region, the fin spacing should be at an optimum value. Based on above

Fig. 21. Variation of base heat transfer coefficient with fin spacing (Ri = 0.5, 1, 3 at Q=50 W)

At higher value of Ri, buoyant force is dominant and there is not much effect on Nua value. It is observed that as Rayleigh number increases, the Nua also increased gradually up to 90000, and then it is flattened as shown in Fig. 22. Figure 23 shows the variation of Nua with Ra for different spacing, at V = 0.2 m/s. It shows that, at lower spacing the value of Nua is lower due to the constriction in the fluid flow. However, spacing in between Sopt = 8-10 mm values are maximum, because of optimum region and again when spacing increases beyond optimum region value of average nusselt number reduces due to reduction in surface area.



110

S. G. Tajji, G. V. Parisshwad, N. K. Sane, R. Z. Deshmukh D

[11] M. Dogan andd M. Sivrioglu,, Experimental investigation off combined convvection heat trannsfer from longittudinal fins in a horizontal rectaangular channel: In natural conveection dominatedd flow regimes” Energy Converssion and Manag gement, Elsevier,, 50,pp. 2513–25221. (2009). c free annd forced convection in channels,, [12] L.N. Tao, On combined Int. Jr. of Heat Transfer, T ASME, ppp. 233-237, (1960). d convection heatt [13] D. G. Osborne, F. P. Incropera,, Laminar mixed fl between hhorizontal parallel plates withh transfer for flow asymmetric heaating. Int Jr. Heat Mass Tran nsfer;28:207–17,, (1985). [14] F. P. Incropera, Convection heatt transfer In electtronic equipmentt Cooling, Int. Jrr. of Heat Transsfer, ASME, Voll. 110, pp. 1097-1111, (1988). K. Sane and S. Paavitran, Computattional analysis off [15] J. P. Shete, N.K mixed convection over heated, vertical rectangu ular fin array, att mber of unity, The Internatio onal Review off Richardson num Mechanical Enggineering (IREME E), Vol. 5. n. 5, July, J (2011). P Mixedd [16] N. C. Cherechhes, M. Cherecches, and C. Popovici, convection flow w inside vertical cchannel of a doub ble skin envelope,, The Internationnal Review of M Mechanical Engineering (IREME),, Vol. 3. n. 6, Novv. (2009).

experimentall analysis coonclusions caan be drawnn as follows: 1. Validatioon study is carried out with flat plate and vertical plate p and expperimental ressults are in close c match wiith the previouus investigatoors results for heat transfer from horizonntal fin arraay under naatural convectioon. The optiimum spacinng under naatural convectioon also agreees well wiith Harahap and Mcmanuss and other invvestigators. 2. Results obtained by mixed convvection are also comparedd with mixed convectionn in ducted type geometryy and experim mental results not n in close match m because of ducted geometry and a dimensiional parameters. 3. The optim mum spacing region is 9 mm m spacing under u assisting mode of mixeed convection and 10 mm under u natural coonvection. 4. At all Ri, R forced floow assists natural convecction currents and ha averrage convectiion heat trannsfer coefficiennt values aree always hiigher in assisting mode com mpared with thhose under naatural convectiion. 5. The optim mum fin spaccing has beenn obtained inn this study as in i between 9 to t 10 mm. 6. Optimum m spacing shiffts towards loower values from f 10 mm too 9 mm with drop d in the Ricchardson num mber.

A Authors’ infformation 1

Dep partment of Mecchanical Engineeering, PES Mod dern College off Engineering, Pune (Inndia). 2

Dep partment of Mechhanical Engineerinng, Govt. Collegee of Engineering,, Punee, (India). 3

Dep partment of Mecchanical Engineeering, PES Mod dern College off Engineering, Pune (Inndia).

Acknow wledgementts

4

Dep partment of Mechanical Engineerring, Cusrow Wadia W Institute off Tech hnology, Pune (Inndia).

Financial support for this study provided byy the research fundd of the Pune University (IIndia) under Grant G No. BCUD/O OSD/390-20100 is gratefullyy acknowledgeed.

Santosh G. Taaji Ph.D. Scholar, Place of birth: Jalna, India Date of birth: 115/06/1975 Qualification: B B. E. Mech; M. E. E Mech. Email: [email protected] m Ph.: +91 99702288348. Mr. Santosh T Taji is a life meember of Indiann Socieety of Technical Teachers. T His maajor field of intereest is thermal andd heat transfer. He hass published 07 paapers in national conferences andd D 03 papers in international conferencees so far. Presenttly pursuing PhD Parishwad and Drr. N. K. Sane andd undeer the guidance off Prof. Dr. G.V. P is a bonafied b student of o Govt. College of Engineering, Pune, P India.

Refferences [1]

W. Elenbbaas, Heat disssipation of parrallel plates by free convectionn, Physica, Holland, 9, No. 1, (19442). [2] K. E. Starnner and H. N. MccManus, An expeerimental Investiggation of free connvection heat trannsfer from rectanngular fin arrays, Jr. of Heat Transfer, ASME, 85, (1963). ( mental investigattion of rectangulaar fins [3] K. D. Mannnan, 'An experim on horizonntal surfaces', Ph. D Thesis, Ohio Sttate University, (1970). [4] F. Harahapp and H. N. McM Manus, Natural coonvection heat traansfer from rectangular fin arraays. Jr. Heat Trans, T ASME, Series; S C89:32–8,, (1967). [5] N. K. Sanne and S. P. Sukhhatme, Natural coonvection heat traansfer from horizontal rectangular fin arrays, Proceeding, P Int. Conf. Heat and Mass M Transfer, Vol.3, V NC 3.7, pp.1114-118, (1974). [6] C. D. Jonees and L. F. Smithh, Optimum arranngement of rectanngular fins on hoorizontal surfacess for free convecction heat transfeer. Jr. Heat Transfer, ASME, Series; C 92:6–10, (11970). [7] K. C. Karrki, and S. V. Paatankar, Cooling of a vertical shroouded fin array by b natural conveection: a numericcal study fundam mentals of natural convection elecctronic cooling. Witte LC, Saxenna LS, editors. ASSME HTD, vol. 322; pp. 33–40, (19984). [8] S.W. Churrchill and H.S. Chu, C Correlation equations for laaminar and turbullent free convection from a verticcal plate, Int. Jr. Heat mass Trannsfer 18, 1323 – 1329, 1 (1975). [9] S.V. Dinggare, N. K. Sanee and R. R. Kuulkarni, Computaational analysis of effect of positiioning pins on natural convectionn heat transfer inn horizontal recttangular plate fiin pin fin arrayy, The Internationnal Review of Mechanical M Enginneering (IREME),, Vol. 5. n. 3, Maarch, (2011). [10] Acharya annd S.V. Patankarr, Laminar Combbined Convectionn in a Shrouded Fin Array, Int. Jr. J of Heat Trannsfer, Vol 103 PP P-559565. (1981).

wad Prof. Dr. Gajaanan V. Parishw Place of birth: Belgaum, India Date of birth: 225/03/1960 Qualification: Ph. D. Therm mal Engineering,, 98. MNREC Allahhabad, India , 199 He is interesteed in the area off Thermal Engg.. and Energy. H He has Publisheed 11 papers inn International Joournals; 30 paperrs in Internationall ferences; 04 in national n journal aand 33 in nationaal conferences soo confe far. He H is a Professorr in Mechanical E Engineering Depaartment of Govt.. Colleege of Engineerring, Pune (Indiaa). All papers are a published inn presttigious Internatioonal Journal and conferences. Drr. Parishwad is a life member m ISTE, SM ME, FIE, SES, RE ENET etc. Narayan K. Saane Place of birth: Sangli, India Date of birth: 008/10/1944 Qualification: Ph. D. Heat Transfer, T I. I. T.. Mumbai, (Indiaa) 1973. Research Fellow: The Technicaal Delft, Netherlandss. University of D Presently he iss working as adju unct Professor inn

Copyright © 20012 Praise Worthyy Prize S.r.l. - Alll rights reserved

International Review of Mecchanical Engineerring, Vol. 6, N. 1

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PES Modern College of Engineering, Pune. He is interested in the area of Thermal Engg. and Heat Transfer. He has Published 12 papers in International Journals; 20 in International conferences; 10 in national journal and 50 in national conferences so far. Dr. Sane is a life member ISTE, ISHMT and former member of JSME, ASHRAE. Ram Z. Deshmuk Place of birth: Nasik, India Date of birth: 09/08/1979 Qualification: M.E. Heat Power, Presently working as Lecturer in Mechanical. Engineering at Cusrow Wadia Institute of Technology, Pune. He is interested in the area of Thermal Engg. and Heat Transfer. He is a life member of ISTE.



112


Calculation of Reduction Heat Transfer between Two Finite Concentric Cylinders Using Radiation Shields with Temperature-Dependent Emissivity Seyfolah Saedodin, M. S. Motaghedi Barforoush, Mohsen Torabi

Abstract – Radiation is one of the most important modes of heat transfer. The present article is concerned with determination of the effect of the one and two radiation shields between two finite concentric cylinders to reduce net heat transfer. Hence a simplifying approach for calculating the radiant energy is achieved using the concept of net radiation heat transfer and provides an easy way for solving a variety of situations. This method has been applied to calculate the net radiation heat transfer between two finite concentric cylinders. Then this method used to calculate reduction heat transfer when radiation shields with temperature-dependent emissivity applied between these objects. Moreover, using this method the percentage reduction in heat transfer between two surfaces was calculated. The findings reveal that, one tungsten radiation shield which has very low emissivity can reduce the net heat transfer even better than two aluminum oxide radiation shields. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Radiation Shield, Net Radiation Method, Concentric Cylinders with Finite Length, Temperature-Dependent Emissivity

This model of heat transfer is not just a theoretical problem, since understanding and predicting the radiant energy becomes crucial in many practical situations. In high-performance insulating materials it is common to suppress conductive and convective heat transfer by evacuating the space between two surfaces. This leaves thermal radiation as the dominant heat loss mode even for low-temperature applications such as insulation in cryogenic storage tanks. On way of reducing radiant heat transfer between two particular surfaces is to use materials which are highly reflective. An alternative method is to use radiation shields between the heat exchange surfaces [1]. Also, these shields are used in prospective space observatories, such as WMAP, JWST, SAFIR, Gaia, and Millimetron as a method of cooling the equipment of large space telescopes. These shields do not deliver or remove any heat from the overall system; they only place another resistance in the heat-flow path so that the overall heat transfer is retarded. Moreover use of a radiation shield is recommended to maximize the protection of normal tissues, ensure appropriate delivery of radiation to the proper location and depth, and allow reproducibility of the patient positioning for daily treatments [2]-[4]. In addition, radiation shields used in certain air temperature measurement applications where temperature errors on the order of a few tenths of a degree are important [5]. Radiation shields constructed from low emissivity (high reflectivity) materials can be used to reduce the net radiation transfer between two

Nomenclature A Eb F N Q

h

r R T

Surface area, m 2 Blackbody emissive power Shape factor Number of shields Net heat transfer, W Height of cylinder, m Radius of cylinder, m Resistance Absolute temperature, K

Greek symbols ε Emissivity λ Wavelength, m σ The Stefan–Boltzmann constant, 5.67 × 10-8 Subscripts i Inner cylinder o Outer cylinder Superscripts Outer surface + Inner surface

I.

Introduction

Heat transfer by radiation is one of the basic modes of heat transfer. Manuscript received and revised December 2011, accepted January 2012

113


S. Saedodin, M. S. Motaghedi Barforoush, M. Torabi

surfaces. Note that the emissivity associated with one side ( ε shn + ) may differ from that associated with the

( Qnet )without

opposite side ( ε shn − ) of the shield [6]. Our objective consists in showing how apparently intractable problems in heat transfer by radiation can be easily solved using the concept of net radiation transfer. Use of network representations was first suggested by Oppenheim [7]. This method provides a convenient solution technique for visualizing radiation exchange between plates in the enclosure and may be used as the basis for predicting this exchange. This subject is also pertinent to the design of multi-coverplate solar collectors. Indeed, in [8] the solarradiation transmittance through a multi-plate planar window is calculated and a matrix-method derivation of the formulae is presented. Moreover, Micco and Aldao [9] generalized the method of net transmittance to spherical and cylindrical symmetry. But, they used only one radiation shield between two main surfaces. We do not claim to be original since the net radiation method can be found in the literature [10]. In this work, the general formulation has been investigated to calculate net heat transfer between two concentric cylinders, which is more challenging compare with our previous studies [11]-[14]. Then, reduction heat transfer by one and two radiation shield has been calculated. Accordingly, applying two radiation shields with different materials optimization was done.

II.

Eb1 − Eb 2 E − Eb3 + 2 × b1 R12 R13

(2)

when:

(

)

(3)

(

)

(4)

Eb1 − Eb 2 = σ T14 − T2 4 Eb1 − Eb3 = σ T14 − T34

Most real surfaces exhibit a selective emission, in the sense that the emissivity is different for different wavelengths. In general ε can be a function of the wavelength and the surface temperature, i.e. ε = ε ( λ ,T ) . A special type of non-black surface, called a grey body, is defined as one for which the emissivity is independent of the wavelength [15]. For simplicity we will restrict our study to grey bodies. In addition, we will consider that surfaces are diffuse, so the intensity leaving a surface is independent of direction. Using the net radiation method the total resistance between each two surfaces can be obtained by:

Analysis

Consider two concentric cylinders as shown in Fig. 1. (a) and (b). The space between these two cylinders separated from outer space by two plates at base and top of the concentric cylinders same as A3 . Therefore, the end effects should be included in the calculations. For the analysis, the following simplifying assumptions were made: 1- Surfaces are diffuse and gray. 2- Space between cylinders is evacuated. 3- Conduction resistance for radiation shield is negligible. 4- The temperature of the heat-transfer surfaces are maintained the same in both cases. 5- All the surfaces and shields are in radiant balance. 6- The emissivity associated with the inner and outer surfaces of the shield are the same. Using the above assumptions, the radiation heat transfer equations can be investigated by following procedures: The basic concepts related to heat transfer by radiation are very well known. For an ideal grey surface the emitted thermal radiation leaving a surface, per unit time and unit area, is given by: Eb = σ T 4

=

− shield

R12 =

1 − ε1 1− ε2 1 + + ε1 A1 A1 F1− 2 ε 2 A2

(5)

R13 =

1 − ε3 1 − ε1 1 + + ε1 A1 A1 F1−3 ε 3 A3

(6)

Therefore, the net heat transfer between inner cylinder and outer space is:

( Qnet )without

=

− shield

(

σ T14 − T2 4

)

1 − ε1 1− ε2 1 + + ε1 A1 A1 F1− 2 ε 2 A2

+2 ×

(

σ T14 − T34

)

+

(7)

1 − ε3 1 − ε1 1 + + ε1 A1 A1 F1−3 ε 3 A3

By introducing: [16] F1− 2 =

1 × 2 R1

⎧ ⎫ ⎪ 1 ⎪ ⎛ ⎞ R ⎪− R22 − R12 − 1 cos −1 ⎜ 1 ⎟ ⎪ ⎪ 2 ⎪ ⎝ R2 ⎠ ⎪ ⎪ (8) 0 5 . π ⎪ ⎪ × ⎨+π R1 − AB − 2 R1 tan −1 R22 − R12 ⎬ 2 ⎪ ⎪ 0.5 ⎪ ⎪ 2 ⎧ ⎫ 0.5 ⎪ ⎪ 1+ A B ⎪ ⎪ 2 2 tan −1 ⎨ ⎬ ⎪ ⎪+ 1 + A 1 + B 2 ⎪⎩ 1 + B A ⎪⎭ ⎭⎪ ⎩⎪

(

)

(

(1)

{(

The net radiation heat transfer between any two of the concentric surfaces is then: Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

)(

)}

)

( (

) )


114


when:

radiation shields applied between two main surfaces as follows:

R1 =

r1 r , R2 = 2 , h h

A = R2 + R1 , B = R2 − R1 (9)

( Qnet )with −two− shields = 2 × Q1− 3 in + 2 × Qsh1− 3 mid + 3

+2 × Q

the net heat transfer between inner and outer surfaces can be obtained. To have a comparison between the amount of heat transfer with and without radiation shields, it is must to find functions as the amount of heat transfer with one and two radiation shields between inner surface and outer space. As cited before, the shields do not deliver or remove heat from the system. Therefore, the net heat transfer between inner and outer surfaces, using one radiation shield, can be found as follows:

( Qnet )with −one − shield

+2 × Q

3 sh1− out 2

when Q

3 1− in 2

, Q

3 sh1− out 2

3 sh 2 − out 3

+ Qsh1− 2

Q1− sh1 = 2 × Q

3 sh1− mid 3

Qsh1− sh 2 = 2 × Q

3 1− in 2

=

(10)

(

1 − ε1 1 + ε1 A1 A1 F

)

+

3 1− in 2

Q

3 sh1− out 2

=

(

σ Tsh14 − T34 1 − ε sh1 1 + ε sh1 Ash1 Ash1 F

3 sh1− out 2

Qsh1−3 =

(

σ Tsh14 − T34 1 − ε sh1 1 + ε sh1 Ash1 Ash1 F

3 sh1− out 2

1− ε2 ε 2 A3 2

Example 1. Consider two concentric cylinders as shown in Figs. 1(a) and (b). As mentioned before, the space between these two cylinders separated from outer space by two plates at base and top of the concentric cylinders same as A3 . The inner cylinder has temperature 873.15 °K , radius 50cm and emissivity of 0.28. The outer cylinder has temperature 330 °K , radius 100cm and emissivity of 0.13. The height of the cylinders is h = 1 m . Also, the two base and top plates have temperature 330 °K and emissivity of 0.13. If one shield of 75 cm radius has been applied to reduce heat transfer between inner cylinder and outer space (Fig. 1(c)), the percentage reduction in heat transfer, temperature and emissivity of the radiation shield can be calculated as follows:

1− ε2 ε 2 A3 2

(12)

out

) +

(13)

out

while Tsh1 and ε sh1 should be found from the following equation: Q1− sh1 = 2 × Q

3 sh1− out 2

(17)

Using our solution, we performed sample numerical computations of reduction heat transfer between two concentric cylinders by applying one and two radiation shields as shown in Figs. 1, based on equations derived on the previous section. Note that all the calculations have been performed for all three materials in Fig. 2.

in

) +

+ Qsh 2 − 2

(11)

1− ε2 ε 2 A3 2

(16)

III. Application

and Qsh1− 2 can be found same as

σ T14 − T34

(15)

+ Qsh1− sh 2

3 sh 2 − out 3

follows:

Q

+ Qsh 2− 2

It is obvious that, for calculating Tsh1 , Tsh 2 , ε sh1 and ε sh 2 , Fig. 2 should be employed at the same time with two following equations:

= 2×Q

3 1− in 2

3

+ Qsh1− 2

( Qnet )without − shield

= 25207.7591 W

(14) For aluminum oxide shield: Using Fig. 2. and solving Eqs. (10) and (14) together:

As mentioned before the emissivity is a function of temperature. Because of the fact that emissivity and temperature of each shield are unknown, Fig. 2. has been employed for solving Eq. (14) at the same time. By following the same procedures as for one radiation shield, the net heat transfer can be found when two


= 19564.7150 W

Tsh1 = 714.9629 °K , ε sh1 = 0.6567

and the percentage reduction in heat transfer is:



115


( Qnet )without − shield − ( Qnet )with −one − shield × 100 = ( Qnet )without − shield =

Similarly for silicon carbide shield:

( Qnet )with −two − shield

25207.7591 − 19564.7150 × 100 = 22.3861% 25207.7591

Tsh1 = 730.2936 °K , ε sh1 = 0.8862 Tsh 2 = 683.7605 °K , ε sh 2 = 0.8867

Similarly for silicon carbide shield:


= 20373.6240 W

and the percentage reduction in heat transfer is: 25207.7591 − 17716.3631 × 100 = 29.7186% 25207.7591

Tsh1 = 716.7882 °K , ε sh1 = 0.8864


Finally for tungsten shield:

25207.7591 − 20373.6240 × 100 = 19.1771% 25207.7591


= 10285.6255 W

Tsh 2 = 551.9994 °K , ε sh 2 = 0.0395

Tsh1 = 677.6514 °K , ε sh1 = 0.0573



25207.7591 − 7361.9604 × 100 = 70.7948% 25207.7591

25207.7591 − 10285.6255 × 100 = 59.1965% 25207.7591

It can be concluded from these two examples that, one tungsten radiation shield can reduce net heat transfer better than two aluminum oxide or silicon radiation shields.

Example 2. Consider the two concentric cylinders of Example 1. If two shields with same materials have been applied at radius 66.67 and 83.33cm to reduce heat transfer between inner cylinder and outer space (Fig. 1(d)), the percentage reduction in heat transfer, temperature and emissivity of the radiation shields can be calculated as follows:

( Qnet )without − shield

Example 3. Consider the two concentric cylinders of Example 1. If two shields with different materials have been applied at radius 66.67 and 83.33cm to reduce heat transfer between inner cylinder and outer space (Fig. 1(d)), the percentage reduction in heat transfer, temperature and emissivity of the radiation shields can be calculated with same procedures as Example 2. The temperatures, emissivities, net heat transfer and percentage reduction in heat transfer in all six possible models are shown in Table I. As it can be perceived from Table I, model No. 5 is the best model for reducing heat transfer between two concentric cylinders, if we want to use two radiation shields with different materials. It is interesting that, although the radiation shields’ temperature in model No. 6 is less than model No. 5, but in the wake of higher emissivity in second radiation shield in model No. 6, the net radiation heat transfer and percentage reduction in heat transfer are smaller than model No. 5. It can be deduced from this table that, if we want to choose the best combination of two radiation shields with different materials, it is better to use tungsten shield closer to the surface with higher temperature.

= 25207.7591 W

For aluminum oxide shield: Using Fig. 2. and solving Eqs. (15), (16) and (17) together:


= 16438.5309 W

Tsh1 = 733.8722 °K , ε sh1 = 0.6498 Tsh 2 = 670.7583 °K , ε sh 2 = 0.6731


( Qnet )without − shield − ( Qnet )with −two − shield × 100 = ( Qnet )without − shield =

= 7361.9604 W

Tsh1 = 739.4863 °K , ε sh1 = 0.0659

Finally for tungsten shield:


= 17716.3631 W

25207.7591 − 16438.5309 × 100 = 34.7878% 25207.7591



116


Figs. 1. Two concentric cylinders ( a ) isometric view ( b ) without radiation shield

( c ) with one radiation shield ( d ) with two radiation shields

Fig. 2. Normal emissivity as a function of temperature [2]



117


TABLE I THE PERCENTAGE REDUCTION IN HEAT TRANSFER, TEMPERATURE AND EMISSIVITY OF TWO RADIATION SHIELDS WITH DIFFERENT MATERIALS Percentage Shield at radius 66.67cm Shield at radius 83.33cm ( Qio )with − shield reduction in Model Temperature Temperature heat Material Emissivity Material Emissivity W °K °K transfer % Aluminum Silicon 728.5019 0.6517 671.1595 0.8868 16812.3034 33.3050 No. 1. oxide carbide Aluminum No. 2.

oxide Silicon

No. 3.

carbide Silicon

No. 4.

carbide

801.6825

0.6252

Tungsten

651.3231

0.0536

11044.9874

56.1841

736.2286

0.8861

Aluminum oxide

682.9888

0.6685

17279.0598

31.4534

805.5548

0.8851

Tungsten

657.5994

0.0545

11333.1006

55.0412

No. 5.

Tungsten

694.5407

0.0597

Aluminum oxide

489.2335

0.7424

7855.4681

68.8371

No. 6.

Tungsten

693.6679

0.0595

Silicon carbide

487.3799

0.8870

7862.2609

68.8101

IV.

[11] Saedodin, S., M. Torabi, J. Moghimi Kandelousi and N. Maghsoudlou, Application of net radiation transfer method for optimization and calculation of reduction heat transfer, using spherical radiation shields, World Applied Sciences Journal, Vol. 11(4), 2010, pp. 457-461. [12] S. Saedodin, M. Torabi, N. Maghsoudlou, J. M. Kandelousi, Calculation of reduction heat transfer using cylindrical radiation shields, International Review of Mechanical Engineering (IREME), Vol. 4(7), 2010, pp. 924-928. [13] S. Saedodin, M.S. Motaghedi Barforoush, M. Torabi, Calculation of reduction radiation heat transfer using hemisphere shields with temperature-dependent emissivity, Journal of Applied Sciences, Vol. 11(12), 2011, pp. 2238-2243. [14] S. Saedodin, M.S. Motaghedi Barforoush, M. Torabi, Reducing heat transfer between two concentric semicylinders using radiation shields with temperature-dependent emissivity, Frontiers in Heat and Mass Transfer, Vol. 2(4), 2011, 044001. DOI: 10.5098/hmt.v2.4.4001 [15] M.F. Modest, Radiative Heat Transfer, 2nd ed., Academic Press, 2003. [16] M.H.N. Naraghi, B.T.F. Chung, Radiation Configuration Between Disks and a Class of Axisymmetric Bodies, Journal of Heat Transfer, Vol. 104, No. 3, 1982, pp. 426-431.

Conclusion

In this paper an equation for calculating heat transfer between two cylinders investigated. Using net radiation method, the percentage reduction in heat transfer, temperature and emissivity of the radiation shield calculated. It is found that, when two shields with same materials applied for reducing heat transfer, the one with lower emissivity better reduced net heat transfer. Also it was concluded that applying one radiation shield with low emissivity can be more effective compare with applying two radiation shields with high emissivity.

References J.P. Holman, Heat Transfer, 10th ed., McGraw-Hill, New York, 2009. [2] A.J. Coleman, A Technique for Shielding Electron Beams Used in Radiotherapeutic Management of Head and Neck Cancer, Journal of Prosthodontics, Vol. 5, No. 2, 1996, pp. 129-132. [3] J.H. Kaanders, T.J. Fleming, K.K. Ang, M.H. Maor, L.J. Peters, Devices Valuable in Head and Neck Radiotherapy, International Journal of Radiation Oncology Biology Physics, Vol. 23, No. 3, 1992, pp. 639-45. [4] M.E. Brosky, C. Lee, T.S. Bartlett, S. Lo, Fabrication of a Radiation Bolus Prosthesis for the Maxillectomy Patient, The Journal of Prosthetic Dentistry, Vol. 83, No. 1, 2000, pp. 119-21. [5] R.L. Mahajan, B.M. Fichera, T.W. Horst, Mechanically Aspirated Radiation Shields: A CFD and Neural Network Design Analysis, International Journal of Heat and Mass Transfer, Vol. 48, 2005, pp. 2856–2867. [6] F.P. Incropera, D.P. DeWitt, T.L. Bergman, A.S. Lavine, Fundamentals of Heat and Mass Transfer, 6th ed., John Wiley & Sons, New York, 2006. [7] A.K. Oppenheim, Radiation Analysis by the Network Method, Transactions of ASME, Vol. 27, No. 4, 1956, pp. 725–735. [8] W.A. Shurcliff, Transmittance and Reflection Loss of Multi-Plate Planar Window of a Solar-Radiation Collector: Formulas and Tabulations of Results for the Case n=1·5, Solar Energy, Vol. 16, No. 3-4, 1974, pp. 149-154. [9] C.D. Micco, C.M. Aldao, On the Net Radiation Method for Heat Transfer, European Journal of Physics, Vol. 24, 2003, pp. 81–89. [10] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, 3rd ed., Taylor and Francis, New York, 1992. [1]

Authors’ information Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran. S. Saedodin was born 1965, in Semnan, Iran. He studied at Iran University of Science & Technology in Tehran and graduating in 2006 with Ph.D. degree in Mechanical Engineering. His thesis was "Fundamentals of Selective Electrical Discharge Sintering in Rapid Prototyping (SEDS)". Now, he is the Assistant Professor of Department of Mechanical Engineering in Semnan University. Dr. Saedodin has authored several technical papers. He also patents thirteen inventions. His research is currently focused on heat transfer and especially on application of nonFourier heat transfer modeling. He is also interested in application of numerical solution in Heat Transfer.



118


M. S. Motaghedi Barforoush was born in 1969, in Tehran, Iran. He received his B.S. in fluid mechanics from the Iran University of Science & Technology and his M.S. from Tehran University. He is currently Ph.D. student in Semnan University, Semnan, Iran. His research is currently focused on heat transfer and especially on application of radiation shields to reduce heat transfer between surfaces. He is also interested in application of numerical solution in Heat Transfer. M. Torabi (Corresponding author) was born in 1984, in Tehran, Iran. He received his M.S. in mechanical engineering from the Semnan University and his B.S. in solid mechanics from the Islamic Azad University, Tehran Branch. He has authored more than 20 technical papers. His current research focuses on analytical and computational analysis of heat transfer, fins heat transfer, non-Fourier conduction heat transfer, transient and steady state analysis of radiation heat shields and transient analysis of solidliquid systems. E-mail: [email protected]



119


Heat Generation Effects on Unsteady Natural Convective Flow Over a Vertical Plate with Variable Viscosity P. Loganathan1, D. Iranian2, P. Ganesan3

Abstract – This paper is focused on the study of heat generation on unsteady natural convective flow over a semi infinite vertical isothermal plate with effects of variable viscosity. It is assumed that the viscosity of the fluid to vary as an exponential function of the temperature. The governing equations of continuity, momentum and energy are transformed into non-linear coupled equations and then solved using implicit finite-difference method of Crank -Nicholson type. The fundamental parameter of the problem is variable viscosity parameter. Numerical results are found for different values of heat generation, viscosity variation parameters and Prandtl number (both air and water). The velocity, temperature distributions, local as well as skin friction coefficient and Nusselt number are analyzed numerically and shown graphically. The skin-friction coefficient and heat transfer rate are found to depend strongly on the viscosity parameter. It is noted that the results pertaining to variable fluid properties differ significantly from those of the constant fluid properties. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Unsteady, Variable Viscosity, Vertical Plate, Heat Generation, Finite Difference

Greek symbols Viscosity variation parameter λ θ Dimensionless temperature ν Kinematic viscosity β Coefficient of volume expansion ρ Fluid density µ Fluid viscosity µ∞ Fluid viscosity in free stream

Nomenclature u v U,V

x, y X, Y g L Pr Gr t

t′ T T∞ Tw k Q Q0 Nux Nu Cp Cf Cf

Velocity component in the x direction Velocity component in the y direction Dimensionless velocity of the fluid in the upward direction and normal to the plate respectively Dimensional coordinates along and normal to the plate respectively Dimensionless coordinates along and normal to the plate respectively Acceleration due to gravity Length of the plate Prandtl number Grashof number Time Dimensionless time Temperature of the fluid in the boundary layer Temperature of the fluid far away from the plate Plate temperature Thermal conductivity of the fluid Heat generation parameter Volumetric rate of heat generation Local Nusselt number Average Nusselt number Specific heat at constant temperature Skin-friction coefficient Average skin-friction coefficient

Subscripts w ∞

Wall Free stream

I.

Introduction

The problem of heat transfer and fluid flow due to free convection along vertical plate find useful applications in various branches of Science and Technology such as nuclear science, fire engineering, combustion modeling, geophysical, heat exchangers, petroleum reservoir etc. Ostrach [1] studied first, the similarity solution for free convective flow past an isothermal vertical semi infinite vertical plate. Hellums and Churchill [2] were the first to present the transient and steady state free convection over an isothermal vertical plate by using an explicit finite-difference technique. Gebhart and Pera [3] presented the nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion. Later, Soudalgekar and Ganesan [4] have studied transient free convection flow by using an


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P. Loganathan, D. Iranian, P. Ganesan

implicit finite- difference technique of Crank - Nicholson type which is unconditionally stable and convergent. All of the preceding studies were confined to a fluid with constant parameters. Generally the fluid properties are dependent greatly on the temperature distributions. Heat generation effects may alter the temperature distribution. Many of the researchers investigated the effects of heat generation on vertical plate with different parameters such as variable suction, radiation, porous medium and magneto hydrodynamics. Also, it is indispensable to consider the variation of viscosity in the flow problems to accurately predict the flow behavior. Elbashbeshy and Ibrahim [5] have analyzed the steady free convection flow with variable viscosity and thermal diffusivity along a vertical plate. The effect of temperature-dependent viscosity on the free convective laminar boundary layer flow past a vertical isothermal flat plate was presented by Kafoussias and Williams [6]. Hady et al. [7] studied numerically, mixed convection boundary layer flow on a continuous flat plate with variable viscosity. Mohaghegh [8] analyzed numerically the laminar boundary layer equations in free convection over a vertical flat plate and forced convection over a wedge. Kai-Long Hsiao [9] obtained heat and mass transfer of a micropolar fluids flow with magnetic and radiation effects to past a stretching sheet by numerical calculation. A new class of similarity solutions has obtained for isothermal vertical plate in a semi-infinite quiescent fluid with internal heat generation decaying exponentially by Crepeau and Clarkesan [10]. Kafoussias et al. [11] extended the research of [7] to free-forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature - dependent viscosity. Hossain and Munir [12] presented mixed convection flow from a vertical flat plate with temperature - dependent viscosity. The radiation effect on free convection flow of fluid with variable viscosity from a porous vertical plate was studied by Hossain et al. [13]. Elbashbeshy and Bazid [14] presented the effect of temperature - dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation. Alam et al. [15] studied the combined free-forced convection and mass transfer flow past a vertical porous plate numerically in a porous medium with heat generation and thermal diffusion. Sivasankaran et al. [16] have analyzed Lie group analysis of natural convection heat and mass transfer in an inclined porous surface with heat generation. Sharma and Singh [17] extended [13] to study the unsteady MHD free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation. Mahanti and Gaur [18] presented the effects of varying viscosity and thermal conductivity on steady free convective flow and heat transfer along an isothermal vertical plate in the presence of heat sink. From all these investigations, it is known that the variation of viscosity with temperature is an interesting

to study the problem of fluid flow over vertical plate. Also it is confined to steady free convection flow problems over vertical plate. Hence it is proposed to study the effect of temperature dependent viscosity on unsteady free convection flow over an isothermal semi infinite vertical plate with heat generation.

II.

Mathematical Analysis

A two dimensional laminar unsteady flow of a viscous, incompressible fluid past a semi infinite vertical plate is considered. It is assumed that the variable viscosity is presented in the fluid flow and the fluid has volumetric rate of heat generation Q0. Assume that x axis is to be directed upward along the plate and the y axis is taken normal to the plate. Initially, it is assumed that the plate and the fluid are at the same temperature T∞. Then at the time t > 0 , the temperature of the plate is suddenly raised to Tw and is maintained at the same value. It is also assumed that the effects of viscous dissipation are negligible. Under these assumptions and incorporating the Boussinesq’s approximation within the boundary layer, the governing equations of continuity, momentum and energy respectively are given by: ∂u ∂v + =0 ∂x ∂y

(1)

∂u ∂u ∂u ∂ 2u +u +v = ν 2 + g β (T − T∞ ) ∂t ∂x ∂y ∂y

(2)

∂T ∂T ∂T k ∂ 2T Q0 +u +v = + (T − T∞ ) ∂t ∂x ∂y ρ c p ∂y 2 ρ c p

(3)

The initial and boundary conditions are: t ≤0:

u = 0,

v = 0, T = T∞ for all x and y

t >0:

u = 0,

v = 0,T = Tw

at

y=0

u = 0,

v = 0,T = T∞

at

x=0

u → 0,

T → T∞

(4)

as y → ∞

Introducing the non-dimensional quantities: x yGr X = ,Y= L L tν Gr t′ = L2 Pr =

1

2

νρ C p k∞

,θ = ,Q =

1

4

,U =

uLGr

ν

−1

2

,V =

vLGr

−1

ν

g β L3 (Tw − T∞ ) T − T∞ ,Gr = , Tw − T∞ ν2

4

,

(5)

Q0 L2

νρ C p Gr

1

2

The variations of the normalized viscosity can be written in the form (Ockendon and Ockendon [19]):



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µ (θ ) = µ∞ exp ( −λθ )

θi,n +j 1 − θi,n j

(6)

∆t ′ ⎡θi,n +j +11 − θi,n +j −11 + θin⋅ j +1 − θi,n j −1 ⎤ +Vi,nj ⎢ ⎥= ⎣ ⎦ 4 ∆Y n +1 n +1 n +1 n ⎛ θi, j −1 − 2θi, j + θi⋅ j +1 + θi, j −1 − 2θin⋅ j + θ i,n j +1 ⎞ 1 ⎜ ⎟+ = 2 ⎟ Pr ⎜ 2 ( ∆Y ) ⎝ ⎠

By introducing the above non dimensional quantities the equations are reduced to the non dimensional form: ∂U ∂V + =0 ∂X ∂Y

(7)

∂U ∂U ∂U +U +V = ′ ∂t ∂X ∂Y

⎛ θi,n +j 1 + θi,n j ⎞ ⎟ +Q ⎜ ⎝ ⎠ 2

(8)

⎛ ∂ 2U ∂U ∂θ ⎞ = θ + exp ( −λθ ) ⎜ 2 − λ ⎟ ∂Y ∂Y ⎠ ⎝ ∂Y ∂θ ∂θ ∂θ 1 ∂ 2θ +U +V = + Qθ ∂t ′ ∂X ∂Y Pr ∂Y 2

The region of integration for the present problem is considered as a rectangle composed of the lines indicating Xmin = 0, Xmax = 1, Ymin = 1 and Ymax = 14, where Ymax corresponds to Y = ∞, which lies well outside both the momentum and energy boundary layers. The system of equation is solved by the Thomas algorithm as described in Carnahan et al. [20]. At a particular time level n, finite difference equations at every internal nodal point on a particular i-level constitutes a tridiagonal system of equations. Thus the values of U and θ are known at every nodal point on a particular i level at (n + 1)th time level and finally, the values of V are calculated explicitly at every nodal point on a particular i - level at (n + 1)th time level. In a similar manner, computations are carried out by moving along i direction. After computing values corresponding to each i at (n + 1)th time level, the values at the next time level are determined in a similar manner. The non dimensional variable viscosity parameter λ is considered between the values -0.7 and 0.1 while the heat generation parameter is considered between the values 0.0 and 3.0. Computations are carried out until the steady-state is reached. The steady-state solution is assumed to have been reached, when the absolute difference between the values of U, as well as temperature T at two consecutive time steps are less than 10-5 at all grid points. The local truncation error is O (∆t'2+∆Y2+∆X) and it tends to zero as ∆t', ∆Y, and ∆X tend to zero, which shows that the scheme is compatible. Also the Crank–Nicolson type of implicit finite difference scheme is proved to be unconditionally stable for a natural convective flow in which there is always a non-negative value of velocity U and a non-positive value of V. Thus, compatibility and stability ensures the implicit finite difference scheme is convergent.

(9)

The corresponding initial and boundary conditions in the non dimensional form are: t ′ ≤ 0 : U = 0, V = 0, θ = 0 for all X and Y t ′ > 0 : U = 0, V = 0, θ = 1 at Y = 0 U = 0, V = 0, θ = 0 at X = 0 U → 0 ,θ → 0

III.

(10)

as Y → ∞

Numerical Analysis

An implicit finite-difference scheme of Crank-Nicolson type has been employed to solve the nonlinear coupled equations (7)–(9). The equivalent finite difference equations are (11)-(13): ⎡U i,n +j 1 − U in−+11, j + U i,n j − U in−1, j + ⎤ ⎢ ⎥ ⎢ +U i,n +j −11 − U in−+11, j −1 + U i,n j −1 − U in−1, j −1 ⎥ ⎣ ⎦ +

4∆X n +1 n +1 Vi, j − Vi, j −1 + Vi,nj − Vi,nj −1 2 ∆Y

+

=0

⎡U i,n +j 1 − U in−+11, j + U in⋅ j − U in−1, j ⎤ U i,n +j 1 − U i,n j + U i,n j ⎢ ⎥+ ⎣ ⎦ 2 ∆X ∆t ′ ⎡U i,n +j +11 − U i,n +j −11 + U in⋅ j +1 − U i,n j −1 ⎤ +Vi,nj ⎢ ⎥= ⎣ ⎦ 4 ∆Y ⎡ ⎡U i,n +j −11 − 2U i,n +j 1 + U in⋅ +j +11 + ⎤ ⎤ ⎢⎢ ⎥⎥ ⎢ ⎢ +U i,n j −1 − 2U in⋅ j + U i,n j +1 ⎥ ⎥ ⎦⎥ + = exp ( −λθ ) ⎢ ⎣ 2 ⎢⎣ ⎥⎦ 2 ( ∆Y ) ⎡U i,n +j +11

− U i,n +j −11

+ U in⋅ j +1

IV.


The main objective of the present work is to analyze the flow of the fluid and the heat transfer processes due to variable viscosity parameter for a vertical plate in the presence of heat generation. Heat diffuses very quickly in liquid metals (Pr > 1) relative to the momentum. It is well known that the viscosity of gases and liquids increases and decreases, respectively, with increasing

− U i,n j −1 ⎤

−λ exp ( λθ ) ⎢ ⎣ 4 ∆Y n + 1 n + 1 n n ⎡θi, j +1 − θi, j −1 + θi⋅ j +1 − θi, j −1 ⎤ ⎛ θ i,n +j 1 + θi,n j ⋅⎢ ⎥+⎜ ⎣ ⎦ ⎝ 4∆Y 2

⎡θi,n +j 1 − θin−+11, j + θ in⋅ j − θin−1, j ⎤ + U i,n j ⎢ ⎥+ ⎣ ⎦ 2 ∆X

⎥⋅ ⎦ ⎞ ⎟ ⎠



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temperature. From equation (6) it can be noted that for λ < 0 the viscosity of the fluid increases with an increase in the temperature and it is factual for gases, while for λ > 0,the viscosity of the fluid decreases with an increase in the temperature and it is factual for fluids such as water and lubrication oils. The Prandtl numbers are considered to be 0.73 and 7.0 for the simulation that correspond to air and water respectively. The numerical results of the velocity, temperature, rate of heat transfer and skin friction coefficients for different values of the heat generation parameter, variable viscosity parameter and the Prandtl number are depicted graphically at steady state. The range for λ and Pr are considered as follows: (Schlichting [21]) (i) for air (ii) for water

and 7, the thermal boundary layer thickness decreases when increasing Q for both air and water respectively.

- 0.7 ≤ λ ≤ 0.1, Pr = 0.73 - 0.1≤ λ ≤ 0.5, Pr = 7.0

In order to validate the accuracy of current simulated velocity and temperature profiles, the generated data are compared with the results of Takhar et al. [22] for steady-state isothermal, constant viscosity and heat generation with Pr = 0.7. The comparison results are shown in the fig. 1. It was observed that the current results to be in good agreement. The velocity fields obtained from the solutions of the equation (8) are illustrated in the Figures 2 to 4. Initially the velocity profiles with value zero at the wall, it reaches their temporal maximum very closed to the wall and then decreases to zero as Y becomes large for all time t. The dimensionless velocity profiles in the boundary layer for different values of the viscosity parameter are graphically shown. It can be observed from the Figure 2 that there is increase in the value of the temporal maximum of velocity and also in the value of steady state velocity with the increase in variable viscosity parameter λ and the fixed value of the heat generation parameter Q = 0.0. It can be observed from the Figure 3 that the velocity increases with increasing in the heat generation parameter Q and the fixed value of λ = 0.0. The Figures 2 and 3 are represented for the Prandtl value (0.73) of air. It can be observed from the figure 4 that the velocity increases as increasing of Q and for the fixed value of λ = - 0.1for the Prandtl value (7.0) of water. The transient and steady state temperature profiles are shown in the Figures 5 to 7. The Figures 5 and 6 are represented the temperature profiles for the Prandtl value (0.73) of air. It can be observed from the figure 5 that the temperature increases with decreasing of λ and the fixed value of Q = 0.0, but the boundary layer thickness decreases. It can be observed from the figure 6 that the temperature increases with increasing of Q and the fixed value of λ = - 0.1. The temperature reaches zero very quickly at the lower value of Q. When increasing the value of Q the temperature rises to temporal maximum and reaches zero gradually. It can be observed from the Figure 7 that the temperature increases with increasing of Q, time and the fixed value of λ = 0.1 From the figures 6

Fig. 1. Comparison of steady state velocity and temperature profiles λ=0.0, Q=0.0, Pr = 0.73

Fig. 2. Velocity profiles, Pr = 0.73, Q = 0.0 for different values of λ

Fig. 3. Velocity profiles, Pr = 0.73, λ = -0.1 for different values of Q.



123


Fig. 7. Temperature Profiles, Pr = 7.0, λ = 0.1 for different values of Q. Fig. 4. Velocity profiles, Pr = 7.0, λ = -0.1 for different values of Q

Knowing the velocity and temperature field, it is customary to study the skin friction coefficient and Nusselt number. Local as well as average values of skin friction, Nusselt number in dimensionless form are as follows: ⎛ ∂U ⎞ C f = exp ( −λθ ) ⎜ ⎟ ⎝ ∂Y ⎠Y = 0 1 ∂U ⎛ ⎞ C f = exp ( −λθ ) ∫ ⎜ ⎟ dX ∂ Y ⎝ ⎠Y = 0 0

Nu x = − XGr Nu = −Gr Fig. 5. Temperature Profiles, Pr = 0.73, Q = 0.0 for different values of λ

1

4

⎛ ∂θ ⎞ ⎜ ⎟ ⎝ ∂Y ⎠Y = 0

(14)

1 1 ⎛ ∂θ 4

⎞ ∫⎜ ⎟ dX ∂ Y ⎝ ⎠Y = 0 0

The derivation involved in (14) is evaluated using five point formula and integrals are evaluated using Newton numerical cotes formula. The local as well as average skin friction and Nusselt number are shown graphically from the figures (8 – 13). Figure 8 shows the local skin friction coefficient for different Prandtl values 0.73 and 7.0. From the figure 8 it is noted that the local skin friction coefficient decreases monotonically along upward direction of the plate for a particular value of Q = 0.0 and the different value of λ or air and the local skin friction coefficient decreases with decreasing the heat generation parameter Q and the fixed value of λ = 0.1 for water. Figure 9 illustrates the local Nusselt number for a particular value of Q = 0.0 and the different value of λ for Prandtl number (0.73) of air. It is observed that the local heat transfer rate decreases with decreasing of variable viscosity parameter λ. Figures 10 and 11 show the average skin friction coefficient for different Prandtl values 0.73 and 7.0 respectively. It is seen in the figure 10 that the average skin friction coefficient increases initially closed to the wall of the plate for the fixed value of Q = 0.0 and the decreasing value of λ. After sometimes it decreases and increases monotonically. But

Fig. 6. Temperature Profiles, Pr = 0.73, λ = -0.1 for different values of Q



124


the average skin friction increases with increasing of Q and the different values of λ = 0.0and λ = 0.1.

Fig. 11. Average skin friction, Pr = 7.0 for different values of Q and λ

It is noted from the Figure 11. The average Nusselt number for both air (Pr = 0.73) and water (Pr = 7.0) are shown in the Figures 12 and 13 respectively. It is noted that the average heat transfer rate decreases with increasing the heat generation parameter Q and the fixed value of λ = - 0.1 in the Figure 12.

Fig. 8. Local skin friction for different values of Prandtl numbers

Fig. 9. Local Nusselt number, Pr = 0.73, Q = 0.0 for different values of λ

Fig. 12. Average Nusselt number for Pr = 0.73, λ= - 0.1 for different values of Q

Fig. 10. Average skin friction, Pr = 0.73, Q = 0.0 for different values of λ

Fig. 13. Average Nusselt number, Pr = 7.0 for different values of Q and λ



125


[10] Crepeau and Clarkesan, Similarity solution of natural convection with internal heat generation, American Society of Mechanical Engineers Journal of Heat Transfer, 119, pp. 183-185, 1997. [11] N. G. Kafoussias. D. A. S. Rees and J. E. Daskalakis, The numerical study of the free-forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature dependent viscosity, Acta Mechanica, Vol 127, pp. 39-50, 1998. [12] M. A. Hossain, M. S. Munir, Mixed convection flow from a vertical flat plate with temperature - dependent viscosity, International Journal of Thermal Science, Vol 39, pp. 173-183, 2000. [13] M. A. Hossain, Khalil Khanafer and Kambiz vafai, The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate, International Journal of Thermal Science Vol 49, pp.115-124, 2001. [14] Elbashbeshy and Bazid., The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation, Applied Mathematics and Computation , 153, pp. 721-731, 2004. [15] M. S. Alam, M. M. Rahman and M. A. Samed, Numerical study of the combined free-forced convection and Mass transfer flow past a vertical porous plate in a porous medium with heat generation and thermal diffusion, Non-linear Analysis: Modelling and Control, Vol. 11, No. 4, pp. 331-343, 2006. [16] S. Sivasankaran, M. Bhuvaneswari, P. Kandasamy and E. K. Ramasami, Lie group analysis of natural convection heat and mass transfer in an inclined porous surface with heat generation. International Journal of Applied Mathematics and Mechanics, Vol. 2, No. 1, pp. 34 – 40, 2006. [17] P. R. Sharma and G. Singh, Unsteady MHD free convective flow and heat transfer along a vertical porous plate with variable suction and internal heat generation, International Journal of Applied Mathematics and Mechanics, Vol. 4, No. 5, pp. 1-8, 2008. [18] N. C. Mahanti and P. Gaur, The effects of varying viscosity and thermal conductivity on steady free convective flow and heat transfer along an isothermal vertical plate in the presence of heat sink, Journal of Applied Fluid Mechanics, Vol 2 No. 1, pp. 23-28, 2009. [19] H. Ockendon and J. R. Ockendon, Variable viscosity flows in heated and cooled channels. Journal of Fluid Mechanics, Volume 83, pp. 177-190, 1977. [20] B. Carnahan, H. A. Luther, and J. O. Wilkes, Applied Numerical Methods, Wiley, New York, 1969. [21] H. Schlichting, Boundary Layer Theory, Mc Graw-Hill, New York, 1979. [22] H. S. Tahkar, P. Ganesan, K. Ekambavanan,and V. M. Soundalgekar, Transient free convection past a semi infinite vertical plate with variable surface temperature, International Journal of Numerical Methods in Heat Fluid Flow, Vol 7 No 4, pp. 280 -296, 1997.

Figure 13 shows that the average heat transfer rate decreases with increasing both the heat generation parameter Q and the variable viscosity parameter λ for Pr = 7.0.

V.

Conclusion

A numerical study has been conceded out to study the effects of heat generation and variable viscosity parameters on unsteady isothermal semi infinite vertical plate. The governing partial differential equations are solved by an implicit finite difference scheme of Crank Nicholson type. A comparison is shown between the current results and previous results. Both the results are in good agreement. Based on the obtained graphical results, the following conclusions were deduced. The time required for velocity to reach the steady state increases as variable viscosity and heat generation parameters increase for both air and water: 1. The temperature increases as the parameter variable viscosity decreases but the temperature increases as heat generation parameter increases. 2. The average skin-friction decreases as the variable viscosity parameter decreases for air, but increases as the heat generation parameter increases for water. 3. The average Nusselt number decreases as variable viscosity and heat generation parameters increase for both air and water.

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

S. Ostrach, An analysis of laminar free convection flow and heat transfer about a flat plate parallel to the direction of the generating body force, NACA Report, No 1111, pp. 63-79, 1953. J. D. Hellums and S. W. Churchill, Transient and steady state and natural convection, numerical solution: part 1, the isothermal vertical plate, American Institute of Chemical Engineers Journal 8, pp. 690-692, 1962. B. Gebhart and L. Pera, The nature of vertical natural convection flows resulting from the combined buoyancy effects of thermal and mass diffusion, International Journal of Heat Mass Transfer, Vol 14, pp. 2025-2050, 1971. V. M. Soundalgekar and P. Ganesan, The finite difference of transient free convection with mass transfer of an isothermal vertical flat plate, International Journal of Engineering Science, Vol 19, pp. 757-770, 1981. E. M. A. Elbashbeshy and F. N. Ibrahim, Steady free convection flow with variable viscosity and thermal diffusivity along a vertical plate, Journal of Physics. D Applied. Physics, Vol. 26 No. 12, pp. 2137-2143, 1993. N. G. Kafoussias and E. W. Williams, The effect of temperaturedependent viscosity on the free convective laminar boundary layer flow past a vertical isothermal flat plate, Acta Mechanica 110, pp. 123-137, 1995. F. M. Hady, A. Y. Bakier and R. S. R. Gorla, Mixed convection boundary layer flow on a continuous flat plate with variable viscosity, Heat and Mass Transfer, Vol 31 No.3, pp. 169-172, 1996. M. R. Mohaghegh, Numerical analysis of laminar boundary layer equations in free convection over a vertical flat plate and forced convection over a wedge, International Review of Mechanical Engineering (IREME), Vol. 5, No. 4, pp. 747-753, 2011. Kai-Long Hsiao, Numerical calculation heat and mass transfer of a micropolar fluids flow with magnetic and radiation effects to past a stretching sheet, International Review of Mechanical Engineering (IREME), Vol. 3, No. 2, pp. 139-146, 2009.

Author’s information P. Loganathan received M.Sc degree from Presidency College, Chennai and M.Phil. from Loyola College, Chennai. He also received a Ph.D. degree in Mathematics from Anna University, Chennai, India. He is working as an Associate Professor in the Department of Mathematics, Anna University. His areas of interest are Computational Fluid Dynamics, Heat and Mass transfer and Object Oriented Programming. He has published about 27 papers in national and international conferences.



126


D. Iranian obtained M.Sc degree from C.P.A College, Bodi and M.Phil. from Saraswathi Narayanan College, Madurai. He is doing Ph.D. in the area of Computational Fluid Dynamics at Anna University, Chennai, India. He is working as an Associate Professor in the Department of Mathematics, Panimalar Institute of Technology, Chennai. He has published 2 international papers. P. Ganesan received Ph.D. degree in Mathematics from IIT, Bombay, India. He has 30 years of teaching experience and working as Professor in the Department of Mathematics, Anna University, Chennai, India. His areas of interest are Computational Fluid Dynamics, Heat and Mass transfer. He has published about 54 papers in national and international journals.



127


Effect of Cavity Design of Synthetic Jet Actuator to the Heat Transfer Characteristic of an Impinging Flow Configuration Harinaldi1, Rikko Defriadi2, Damora Rhakasywi3 Abstract – A Synthetic jet works based on a vibrating membrane inside a cavity and nozzle for the air outlet. The synthetic jet cavity interior has an important role to generate the air at sufficient conditions to do the cooling process. This research investigated the flow and convective heat transfer characteristics on four impinging synthetic jet prototype with different excitation modes of a sinusoidal waveform forcing. The main purpose of this synthetic jet was to create vortices pair to come out from nozzle which will accelerate the heat transfer process occurring at the impinged wall. This heat transfer enhancement principles became the basis to simulate an alternative cooling system to substitute the conventional fan cooling in electronic devices due to its advantage for having a small form factor and low noise. The investigation combined computational and experimental works. In the experiment sinusoidal waveform was used to oscillate the membrane and the wave frequency used were 80 Hz, 120 Hz, 160 Hz and the velocity amplitude was 1 m/s. Some results indicated significant influence of the cavity design, frequency excitation, and waveform to the rate of heat transfer obtained. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Heat Transfer, Impinging Wall, Synthetic Jet, Sinusoidal Wave

synthetic jet can do the cooling purpose. A synthetic jet works as a jet of vortex generator by the continuous vibration of a diaphragm strong enough to produce flow separation at the outlet of its cavity. The synthetic jet technology has been developed in many ways in order to find its best performance. Choi et.al [1] developed a synthetic jet air blower for fuel cell. The synthetic jet cavity was made of acrylic with a nozzle for outlet. This synthetic jet was used to flow the hydrogen and oxygen through the cathode and anode in order to create electricity. Their synthetic jet was able to generate 500cc/min flow at 650 Hz excitation using sinusoidal wave. This synthetic jet blower was capable of increasing the electricity generated up to 40% compared to the fuel cell that used fan. Further, the synthetic jet blower power consumption was also relatively small 0.3 W, while the fan consumed 1 W. Overall the synthetic jet blower fuel cell had a better system efficiency than the fan based fuel cell. Another study about the synthetic jet was conducted by Lasance et.al [2]. They studied about the synthetic jet using an asymmetric cavity in order to minimize noise and increase cooling efficiency. The main objective of the experiments was to gain insight in the relations between performance criteria such as the heat transfer coefficient, the dissipated power and the noise and controllable parameters such as pipe dimensions, frequency and voltage. The investigation used two pipe length for the air outlet 30 and 60 mm and four pipe diameters 3, 4, 5 and 6 mm at driving voltage 3V. They result indicated that the cavity area such as the pipe length affected the energy consumption of the

Nomenclature Asj Anc f hsj hnc

Qi,h Qo,sj Qo,nc t T V Vamp ρ ω

Synthetic jet effective cooling area (m2) Natural convection effective cooling area (m2) Frequency (Hz) Synthetic jet heat transfer coefficient (W/m2K) Air heat transfer coefficient (W/m2K) Heat generated by the heater (W) Heat removed by the synjet effect (W) Heat removed by natural convection effect (W) Time (second) Temperature (oC) Air velocity (m/s) Maximum air velocity (m/s) Fluid density (kg/m3) Angular frequency (s-1)

I.

Introduction

Fan has become the most common cooling peripheral in many application. One of the application is cooling in electronic devices. Many electronic devices place fans in their cavity to boost the cooling process. However, a fan has its own disadvantage, such as the noise, large dimension, comparatively high power consumption and not to mention the maintenance. Synthetic jet offers a solution to replace the fan role in cooling process. A Synthetic jet works based on membrane oscillation. Different from fan that requires a magnetic coil, with a thin sized membrane and a relatively small cavity the


128


Harinaldi, Rikko Defriadi, Damora Rhakasywi

heating and cooling purposes requires a high convective heat transfer coefficient. Zulkifli, et.al [9] conducted an Analysis of Pulse Air Jet Flow Profiles. The air jet pulsation was produced by a rotating cylinder valve mechanism at frequencies between 10-80 Hz. They obtain a result that mass flow rate for different test frequencies are slightly different due to the difference in the local velocity measurement affected by the pulses. Zhou and Ming [10] conducted an experiment to determine the heat transfer coefficient of an impinging synthetic jet. The investigation used particle image velocimetry, hot-wire anemometer, and infrared camera. The results indicated that the jet exit orifice produced a pair of vortices that split and join together on a regular basis to form a stable jet in wide range slots of the channel near the exit hole. In the present work a comprehensive study was done by computational and experimental method on an original design of synthetic jet actuator that operate based on membrane made of piezo material to move the air with the vibrations that occur in this membrane. The main focus of the study was to characterize the convective heat transfer of an impinging flow configuration in a more realistic condition for an application of cooling an electronic device.

synthetic jet and also the heat transfer characteristic. A synthetic jet application also applied in server processor cooling [3]. It is shown the synthetic jets can augment the cooling provided by a global fan flow by reducing flow bypass. Results in a controlled wind tunnel test showed the using jets to augment fan flows resulted in a 10 to 25% increase in induced flow and 10 to 50% increase in heat dissipated. In another report, an impinging synthetic jet as a promising technology for electronics cooling applications was recently studied focusing on the distribution of heat transfer and flow characteristics from jet sprayed on the surface by blowing with jet diameter of 1-6 mm at the Reynolds number of 1100-4900 [4]. The results obtained showed an agreement between measured average and fluctuating heat transfer distributions and local acceleration. The turbulence intensity along the heated surface tend to increase and support the surface heat transfer when mixing in the wall jet increased and finally split into small-scale turbulence. Mostly electronic devices are mounted inside a casing. Chaudhari, et.al [5] conducted an experiment to see the synthetic jet characteristic over a closed environment. In their experiment a ducting system was used in order to make the closed environment. The ducts were mounted on top and below the synthetic jet. There was also a heated block attached in some parts of the lower ducting which was aimed to observe the synthetic jet effect. The result indicated that the synthetic jet could be used in the closed environment, but the suction process tends to retake the heated air that accumulates in the closed environment. An investigation in 2D-computational field using impinging synthetic jet method was conducted by Jagannatha, et.al [6]. A User Defined Function (UDF) was used to represent the fluctuating velocity of the membrane since the synthetic jet membrane has two phase i.e. suction and discharge. This UDF accurately represented the membrane velocity profile with a positive value (discharge) and negative value (suction). Synthetic jet characteristic also observed as a model based on the laws of fluid dynamics [7]. The model and analysis based on it provide valuable insights into the behaviour of synthetic jet actuators, and reveals that the behaviour of the air in the actuator cavity is compressible at all frequencies. Analysis using the model also reveals that the performance of the synthetic jet actuator represented by the orifice velocity is strongly dependent on fluctuations in the external flow. The magnitude and phase relationships between the external flow fluctuations and wall forcing function also play a key role in the actuator performance. Heat transfer is a very important aspect in synthetic jet technology since the generated vortices tend to take away the heat. There are three aspects influencing the heat transfer over an impinging jet [8]. They are the shape of unsteadiness, the influence of obstacles on an otherwise smooth wall and the impact of a confined far field. Applications of impingement jets in industry for

II.

Prototype Design

A model of an originally designed synthetic jet actuator (SJA) was used. The arrangement comprises a piezoelectric membrane that is set in motion back and forth, forcing the fluid inside the cavity to flow through a nozzle. In its inward motion, the membrane imparts the ejected air of high-speed into the surrounding fluid while the retreating membrane draws fluid back from the surroundings into the cavity. The membrane operation over one cycle depends on the selected frequency. The jet delivers very high net outflow of fluid momentum, consequently very intense cooling rates while having no net change of fluid mass within the cavity. There are 4 prototypes used in this investigation as show in Fig. 1. These prototype are made of nylon because it is easy to handle and has properties of an isolator.

Fig. 1. Synthetic jet physical model (From right to left: Prototype 1, Prototype 2, Prototype 3, Prototype 4)

II.1.

Prototype 1

The design of prototype 1 is based on nozzle contour. The prototype 1 cross section is shown in Fig. 2. The narrow contour at the prototype lower section are made



129


upper and lower contour. The prototype 3 cross section is shown in Fig. 4. This design has 0.3 cm gap between the upper and lower part membrane. It has 1.6 cm in height and 16 nozzle holes with 0.3 cm of diameter. This prototype is designed to shorten the range that the air must travel since it forced by the membrane until it reached the outlet nozzle.

in order to increase the air velocity that pushed by the membrane so the air can be accumulated at the upper part of the cavity and then blown by the upper membrane. This prototype has 40 holes each with 0.4 cm diameter. Further, a round contours with 1 cm radius is made next to the hole in order to direct the air downward to the heatsink.

Fig. 4. Prototype 3 cross section (dimension in cm)


Prototype 3 are designed to prevent energy loss from the air. The small gap inside the cavity are designed to immediately accumulate the energy from both of the membrane and redirect it to the outlet, without having any further friction with cavity wall. Further, it also minimize the air that retracted back to the membrane during the suction phase. With its small design the synthetic jet actuator is expected to be able to generate a strong turbulent flow at the nozzle outlet and increase the cooling efficiency.

Prototype 1 is designed to generate a large volume of air in order to boost the cooling process. The round draft beside the nozzle hole is not only to direct the air downward but also to increase the air speed so it penetrate deeper to heatsink gap. It is expected that the synthetic jet will be able to extract the heat as much as possible. II.2.

Prototype 2

Prototype 2 design is the same as prototype 1. But the prototype 2 has a smaller size and less nozzle hole. The prototype 2 cross section is shown in Fig. 3. Prototype 2 overall height is 1.9 cm and it has 15 nozzle holes with 0.3 cm of diameter. The principle is the same as prototype 1 to generate a high velocity air from the lower part to the upper part. Prototype 2 is designed to meet the purpose of prototype 1, but with more compact dimension. It is expected that this design can be as efficient as prototype 1 or even higher.

II.4.

Prototype 4

Prototype 4 design is similar to prototype 3, but some few details are changed. This prototype has an expanded cavity at the upper and lower part. There is some slight difference between the upper and lower part of this designed. The upper part has more expanded contour than the lower part. The lower part has a smaller length of the drafted contour; however it has a narrow gap near the nozzle. The purpose of this contour is to trap the air flow so all of them will flow directly to the nozzle. This prototype has 2 cm height and 16 holes with 0.2 cm of diameter.


II.3.

Prototype 3

Prototype 3 is different from the two previous prototype. It has a symmetrical cavity design between the




130


Prototype 4 is designed to increase the cooling efficiency based on prototype 3 design. The expanded contour is expected to accumulate the energy from the upper and lower membrane movement and create a strong turbulent flow at the nozzle outlet.

III. Method The synthetic jet prototypes were studied using both experimental and computational works. The experimental work used the customized heatsink made of aluminium with a heater set at constant temperature without being turned off for an hour. The Experimental work was carried out by measuring the temperature at 8 point in the heat sink using a data acquisition module (Advantech USB 4718) with measurement accuracy ± 0.01oC in an experimental arrangement as shown in Fig. 6. The heat sink used for the experiment had a circular form with 32 fins, 11 cm of diameter and 5 cm of height. The heat sink was made of cylindrical aluminum which was shaped by using material removal methods. Heat source was obtained from the heater mat placed at the bottom of the heat sink, and it was set to 60 oC using a thermostat. A sweep function generator is used to oscillate the membrane using sinusoidal wave of excitation.

(a)

(b) Figs. 7. Experimental condition (a) Synthetic jet actuator and heat sink arrangement (dimension in cm) (b) Thermocouple placement in the heatsink

The solver software was used to complete the analysis of heat flow field in the synthetic turbulent jet. It applied a mathematical model of k-ω SST (Shear Stress Transport ) with 2D Double Precision mode. The SST kω model is similar to the standard k-ω model, but includes the following refinements: (i) the standard k-ω model and the transformed k-ε model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard k-ω model, and zero away from the surface, which activates the transformed k-ε model, (ii) the SST model incorporates a damped cross-diffusion derivative term in the ω equation, (iii) the definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress. The SST k-ω model has a similar form to the standard k-ω model as expressed in equations (1) and (2):

Fig. 6. Experimental Set-up

Measurements were performed under open conditions at ambient temperature of 27 oC. The sinusoidal wave forcing were generated at the frequency of 80 Hz, 120 Hz and 160 Hz at 28.8 volt of wave power amplitude (Vpeak). The experimental condition of heatsink are shown in Figs. 7. The computational work was done to get the description of the flow and thermal field pattern of the impinging synthetic jet flow under periodical forcing. The work was conducted by using a GAMBIT 2.4® software to generate the grid and FLUENT 6.3® for the computational solver. The computational model was derived from the prototypes configuration along with the heat sink as the heated wall as shown in Figs. 8.

∂ ∂ ∂ ⎛ ∂k ( ρ k ) + ( ρ kui ) = ⎜⎜ Γk ∂t ∂xi ∂x j ⎝ ∂x j

⎞ ~ ⎟ + G k − Yk + Sk (1) ⎟ ⎠

∂ ∂ ( ρω ) + ( ρωui ) = ∂t ∂x i =

∂ ∂x j

⎛ ∂ω ⎜ Γω ⎜ ∂ xj ⎝

⎞ ⎟ + Gω − Y ω + Dω + Sω ⎟ ⎠

(2)

where the modeling constants followed appropriate values for the typical flow condition [11] :



131


speed that generates by the membrane. As for the valley it indicates the minimum speed generates by the membrane.

As the working fluid, air was assumed to be isothermal and incompressible. The thermodynamic properties of air were taken to be at 27oC under standard atmospheric conditions. The heated wall at the bottom of the domain was maintained at an isothermal temperature of 60 oC. The boundaries on either side of the actuator were treated as constant static pressure outlets with a pressure of one atmosphere. The other details of computation conditions are listed in Table I.

(a)

TABLE I COMPUTATION CONDITION Model Settings

2D, Unsteady

Fluid

Fluid Properties

Boundary Condition

Density Viscosity Specific Heat Thermal Conductivity Velocity Inlet 1,2 Pressure Outlet Heat source Frequency Excitation Excitation Amplitude

Air 1.225 kg/m3 1.7894 e-5 kg/m-s 1006.43 J/kg-K 0.0242 W/m-K UDF 0 Pascal 60 oC 80, 120 and 160 Hz 1 m/s

(b)

The movement of the diaphragm was modeled with a user defined function that incorporated dynamic layering technique [12]. Segregated solution method with implicit solver formulation was used as the solution algorithm while the second order discretization schemes were employed for density, momentum, pressure, turbulence kinetic energy, specific dissipation rate and energy. The synthetic jet flow occurs in oscillatory manner within a confined region. It is predicted that turbulence would be induced in some regions of the domain while the flow would mostly remain under laminar conditions indicated by low values of the Reynolds number encountered. The parameters used in the simulation include the model settings, fluid properties, and the value of boundary condition. The diaphragm motion was assumed to mimic piston movement within a cylinder, which was expressed in (3) for sinusoidal wave: V = Vamp sin ( 2π f ) t

(c)

(d) Figs. 8. Synthetic jet computational domain (a) Prototype 1 (b) Prototype 2 (c) Prototype 3 (d) Prototype 4

IV.

(3)


IV.1. Experimental Work Result where Vamp is the maximum velocity (velocity amplitude) generated by the movement of the diaphragm inside the cavity and t is the operational time. Under these conditions, the unsteady, Reynolds-averaged NavierStokes equations within the solution domain were solved along with the energy equation for a range of operating conditions. A single wave consist of a peak and a valley. This wave represents the membrane oscillation movement. The wave peak indicates the maximum air

By considering the natural convection effect, the heat energy balance can be evaluated as follow : Qi,h = Qo,sj + Qo,nc

(4)

where Qi,h is the heat generated by the heater that is conducted through the heatsink, Qo,sj is the heat removed by the synjet effect, and Qo,nc is the heat removed by natural convection effect.



132


The heat trasfer coefficient of the synthetic jet hsj can be evaluated as follow: ⎛ dT ⎞ ⎛ dT ⎞ hsj Asj ⎜ ⎟ = Qi,h − hnc Anc ⎜ ⎟ ⎝ dt ⎠ ⎝ dt ⎠

waveform with 80, 120 and 160 Hz wave forcing are shown in Figs. 9. The figure describe the time history of heat transfer coefficient during time period of 120 minutes. From Fig. 9, it can be observed that the highest heat transfer coefficient is reached by prototype 3, using 80 Hz of sine wave excitation. At prototype 1 dan 2, the heat transfer coefficient tends to reach a maximum value at 6 minutes after the synthetic jet was turned on. After that, the value will dropped significantly and furthermore the value will change slowly. At prototype 3 with 80 Hz excitation, the heat transfer coefficients tends to increase as 1 hours of time passed.

(5)

where Asj is the synjet effective cooling area, (dT/dt) is the rate of temperature change measured in the experiment, hnc is the value of the air heat transfer coefficient with the custom heatsink, and Anc is the cooling area affected by the air natural convection. Applying equations (4) and (5), the calculation results of heat transfer coefficient for every prototype at sinusoidal

(a)

(b)

(c)

(d) Figs. 9. Heat transfer coefficient graphic to time (a) Prototype 1 (b) Prototype 2 (c) Prototype 3 (d) Prototype 4

The 120 Hz excitation is relatively constant after reaching the maximum value. On the other hand the 160 Hz excitation will drop after reaching its maximum value. In prototype 4 the heat transfer coefficient value is also reaching the maximum value 6 minutes after the synthetic jet actuator was turned on. The 80 Hz excitation gives the highest heat transfer coefficient, but its value will be dropped until 20 minutes then it will be relatively constant. In order to elucidate this phenomenon, the computational works in the following section describing the flow field in each prototype are discussed.

IV.2. Computational Work Result Fig. 10 shows a single sinusoidal waveform scheme. One waveform consist of a peak, middle, valley and end. Each of them has a distance quarter of one wave periode. The peak point indicates that the synthetic jet are undergoing a discharge phase and the valley point indicates the suction phase. Each point has a different condition in simulation which will be helpful to determine the synthetic jet actuator characteristic. Fig. 11 shows the turbulent intensity contour of each prototype using sinusoidal wave forcing.



133


In prototype 1 and 2 the flow are directed from the lower membrane to the upper membrane just as expected. However, since the prototype 2 cavity is smaller the energy accumulation is larger than in prototype 1. Area of the flow field with high turbulent intensity in prototype 2 are more dominant than prototype 1.

This is different from prototype 1 and 2 in which turbulent intensity are concentrated mostly at its cavity. Meanwhile, in prototype 4 the area of flow field with high turbulent intensity are concentrated in a small portion before the nozzle outlet and some are in the outlet along the heatsink. From all the result, prototype 3 has largest area of the flow field with high turbulent intensity compared to the others. Considering the experimental result, it was proven that prototype 3 has the best cooling efficiency than the other prototype. The high turbulent intensity flow that accumulates along the heatsink gap gives more cooling effect, so that the heat extraction area is larger. The CFD result also indicates a trend that at the suction phase the higher turbulent intensity contour has a larger area than the discharge phase. Because of the heated wall, the air flow tends to rise naturally, so the air basically will be accumulated at nozzle after the discharge phase, and being retracted back at the suction phase. This phenomenon can rise a problem when that heated air comes into the cavity. It can reduce the cooling efficiency because at the next discharge phase that air will be flown back to the heatsink. It will makes the heatsink absorb heat instead of remove. Fig. 12 shows the static temperature contour of each prototype using sinusoidal wave forcing. This contour indicates the temperature condition inside the cavity and outside along the heatsink. It can be shown that the temperature decreases for each prototype at sinusoidal wave forcing. All prototypes indicate cooling effects at the end of the wave. However, the effects are various which depend on the signal excitation given. From the contour it can be shown that prototype 3 has the largest cooling area compare to the others. Fig. 13 shows the effective thermal conductivity contour of each prototype using sinusoidal wave forcing. This contour indicates the air thermal conductivity value due to the air movement. At prototype 1 the air that has a high thermal conductivity properties are concentrated inside the cavity. But at the end of the wave, some of them are concentrated near the upper heatsink part. Prototype 2 are different, there is no air with high thermal conductivity properties inside the cavity. All of them are concentrated at the upper and middle part of the heatsink. Prototype 3 has the same characteristic as prototype 2 for the air concentration spot. However, at the middle phase of the wave there is a quite large concentration of air with high thermal conductivity. At the end phase of the wave, the air are concentrated at the upper and middle part of the heatsink but a little left inside the cavity as show by the contours. Prototype 4 has the exact characteristic as prototype 3. The only different are the thermal conductivity value at the middle phase of the wave are larger and at the end phase the thermal conductivity value are a little bit smaller. All prototype indicates that at the end phase of the wave the air with high thermal conductivity are concentrated at the upper and middle part of the heatsink. However, the value and the spread depend on the wave excitation.

Fig. 10. Sinusoidal waveform scheme

Fig. 11. Turbulent intensity contour (%)

This support the experimental fact that the prototype 2 is having better cooling efficiency from the prototype 1 due to the large turbulent energy. In prototype 3 the area of flow field with high turbulent intensity is also large but it was concentrated at the outlet nozzle along the heat sink. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


134


Fig. 12. Static temperature contour (oC)

Fig. 13. Effective thermal conductivity contour (W/mK)

characteristic but the best result comes from prototype 3 over all prototype which has the smallest cavity gap between the upper and lower membrane. It can prevent the reheat phenomenon because it minimize the energy loss from the friction with cavity, so the forced air that comes out from the nozzle can travel further to takes away the heat. Hence, it will minimize the heated air that will be retracted back to the cavity during the next suction phase. Prototype 3 with 80 Hz of excitation gives a high and stable heat transfer coefficient for one hours of operation. The experiment result shows the heat transfer coefficient for this prototype tends to increase until one hour of operation. Even after an hour it still increasing and reached the value 25.5 W/m2K. The computational result are strengthen this conclusion since the prototype 3 generates a wide area of high thermal conductivity air around the heatsink gap. Current achievements of the work suggest some design consideration to make a compact synthetic jet actuator with high cooling efficiency. Some improvement that can be done is to modify the nozzle

The synthetic jet phenomenon where the heat transfer coefficient dropped may be caused by the reheat effect caused by the suction air that contains heat from the heatsink. Fig. 13 shows that inside the cavity there is an air with quite high thermal conductivity. This can makes the heat from heatsink absorbed into the cavity during the suction phase and that heated air being blown back to the heatsink during the discharge phase.

V.

Conclusion

An investigation on the flow and heat transfer characteristics of four original designed synthetic jet prototypes has been done in an impinging flow configuration. The study combining computational and experimental method on the synthetic jet actuator oscillation has been comprehensively conducted using sinusoidal waveform excitation. The oscillation frequency in synthetic jet which describes the number of suction and discharge stroke in a second of the membrane in the cavity obviously plays important role in the heat transfer process. Each prototype has its own Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


135


3

PhD student at the Department of Mechanical Engineering Faculty of Engineering University of Indonesia. E-mail: [email protected]

outlet in order to strengthen the turbulent intensity and also to make use the back part of the membrane, since membrane moves in oscillatory manner so there is two side of flow that can be generated by a single membrane.

Harinaldi was born in Jakarta, Indonesia on October 30, 1968. He received his Bachelor degree in mechanical engineering from University of Indonesia in 1992 and obtained his Master of Engineering degree from Keio University (Japan) in 1997. Furthermore, he received his Doctor of Engineering degree also from Keio University in 2001. Currently, he serves as Associate Professor in Department of Mechanical Engineering University of Indonesia and also as the Head of Department. Some of his recent publications include the topics on flow Structure and mixing behind a backward-facing step with the existence of a gas Injection (Int. J. Heat and Mass Trans, 2001; J. of Chem. Eng of Japan, 2001); hydrodynamics control of turbulent mass transfer in electrodeposition process (, J. of Tech. World, 2010; Int. J. Mech. and Mechatronic. Eng, 2011; J. Mekanikal an Int. J, 2011), active control for separation in turbulent flow formed in bluff body geometry with suction and blowing mechanism (Int. J. Mech. and Mechatronic. Eng, 2011), active flow control for electronic cooling devices using synthetic jet actuator (Int. J. Eng and Tech., 2011). Dr. Harinaldi is an active member of Indonesia Consortium of Mechanical Engineering Education and Asia Fluid Machinery Committee.

Acknowledgements The authors thank to Mr. Kenfery and Mr. Edward for helping the set up of experimentation.

References [1]

J. Choi, E.S. Lee, J.H. Jang, Y.H. Seo and B. Kim, Development of Synthetic Jet Air Blower for Air-breathing PEM Fuel Cell, World Academy of Science, Engineering and Technology 56, pp. 31-34, 2009. [2] C.J.M. Lasance, C. Nicole, R.M. Aarts, O. Ouweltjes, G. Kooijman, and J. Nieuwendijk, Synthetic Jet Cooling Using Asymmetric Acoustic Dipoles, 25th IEEE SEMI-THERM Symposium, pp.254-260, 2009. [3] Mahalingam, R.; Heffington, S.; Jones, L.; Schwickert, M.; , "Newisys server processor cooling augmentation using synthetic jet ejectors," Thermal and Thermomechanical Phenomena in Electronics Systems, 2006. ITHERM '06. The Tenth Intersociety Conference on , vol., no., pp.705-709, May 30 2006-June 2 2006. [4] McGuinn, A., Persoons, T., Valiorgue, P., O’Donovan, T.S., & Murray, D.B. (2008), Heat transfer measurements of an impinging synthetic air jet with constant stroke length, Proceedings of the 5th European Thermal-Sciences Conference, May 18–22, Eurotherm, Eindhoven, The Netherlands, 2008. [5] Chaudhari, M.; Puranik, B.; Agrawal, A.; , "Heat Transfer Analysis in a Rectangular Duct Without and With Cross-Flow and an Impinging Synthetic Jet," Components and Packaging Technologies, IEEE Transactions on , vol.33, no.2, pp.488-497, June 2010. [6] D. Jagannatha, R. Narayanaswamy, and T.T Chandratilleke, Performance Characteristics of A Synthetic Jet Module For Electronic Cooling, 10th Heat Transfer Conference in International Symposium On phase Change, UK, 2007. [7] Sharma, R.N. (2007, December), Some insights into synthetic jet actuation from analytical modeling, 16th Auatralasian Fluid Mechanics Conference, pp. 1242-1248. [8] H. Herwig, Heat Transfer Under Unsteadily Impinging Jets: Guidelines for Future Research Directions, International Review of Mechanical Engineering (IREME), Vol. 4, no.5, pp. 491-494, 2010. [9] R. Zulkifli, K. Sopian, S. Abdullah, M. S. Takriff , Analysis of Pulse Air Jet Flow Profiles, International Review of Mechanical Engineering (IREME), Vol. 4, no.5, pp. 495-501, 2010. [10] Z.J. Zhou and T.X. Ming, Experimental Study on Flow and Heat Transfer Characteristics of Synthetic Jet Driven by Piezoelectric Actuator, Science In China Series E: Technological Sciences, vol.50, no.2, pp. 221-229, 2007. [11] User’s Guide Manual of Fluent 6.3.2, September 2006. [12] User defined functions (UDFs) of Fluent, January 2005.

Rikko Defriadi was born in Jakarta, Indonesiaon December 12th, 1988. He received Bachelor degree in mechanical engineering, from Universitas Indonesia in, 2010. Currently he is a master student in the Mechanical Engineering Department University of Indonesia. His major recent publications include Flow and heat transefer characteristics of an impinging synthetic air jet under sinusoidal and triangular wave forcing ( IJET-IJENS, vol.11, no.3, 2011); Computational Study of Triangular Waveform Oscillation Mode To The Temperature Distribution of a Heated Wall Impinged by a Synthetic Jet, (The International Meeting on Advanced Thermo-Fluid,2011) Damora Rhakasywi was born in Palembang, Indonesia on March 27th, 1985. He received his Bachelor in mechanical engineering from University of Sultan Ageng Tirtayasa, Cilegon, Banten, Indonesia in 2007 and Master in mechanical engineering from University of Indonesia in 2010. Currently he is a Ph.D Candidate in the Mechanical Engineering Department University of Indonesia. His major recent publications include Experimental and Computational Study on Thermal Structure of A Separated Reattachement Flow Under Heated Gas Injection ( IMAT 2009); Flow and Heat Transfer Characteristics of an Impinging Synthetic Air Jet under Sinusoidal and Triangular Wave Forcing (Int. J. of Eng. & Tech. IJET-IJENS Vol: 11, No. 3, 2011); The Effect of Oscillation Mode To The Temperature Distribution of a Heated Wall Impinged by a Synthetic Jet” (Quality in Research Int. Conf, QIR 2011); Computational Study of Triangular Waveform Oscillation Mode To The Temperature Distribution of a Heated Wall Impinged by a Synthetic Jet (IMAT 2011).


Department of Mechanical Engineering Faculty of Engineering University of Indonesia, Depok, Jawa Barat, 16424, Indonesia. Phone:+62-21-7270032; Fax:+62-21-7270033; E-mail:[email protected] 2 Master degree student at the Department of Mechanical Engineering Faculty of Engineering University of Indonesia. E-mail: [email protected]



136


Resonant Behavior of a Hydraulic Ram Pump Mario O. M. de Carvalho, Alberto C. G. C. Diniz, Fernando J. R. Neves

Abstract – Performance of Hydraulic Ram Pumps is studied in certain operation conditions. Operation near resonant conditions is studied in order to delimit areas of good design and operation. A numerical model previously developed by the authors for simulation of the dynamic behavior of these machines is used. The results identify situations of inadequate operation and reinterpret traditional design and operation criteria for the ram pump, which, to date, have been mainly empirical. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Hydraulic Ram Pump, Numerical Simulation, Optimization, Resonance

Since there are no moving parts, except the valves, the Hydraulic Ram is a very robust machine, with very low maintenance costs, that does not require skilled labor for its installation and for its operation. Even today, its design, setting and regulation are carried out mainly empirically. Figure 1 illustrates, schematically, the typical configuration a HRP, highlighting its essential parts.

Nomenclature T1 T2 m1 m2 c1 c2 vBlock vi Pi Patm Lij h ρ V V0 P P0 Ai ∆Pop vneg η TCycle Qi

Duration of Phases 1 and 3 (Supply and Complementary Delivery) Duration of Phase2 (Storage and Delivery) Mass of water in supply pipe Mass of water in delivery pipe Proportional damping coefficient corresponding to head loss in supply pipe and valve A Quadratic damping coefficient corresponding to head loss in supply pipe and valve A Velocity of flow for closing valve A (adjustable) Flow velocity at reference point i Pressure at reference point i Atmospheric pressure Length of pipe between reference points i and j Pressure head Density Volume of air in the Expansion Chamber Average volume of air in the Expansion Chamber Pressure in the Expansion Chamber Average pressure in the Expansion Chamber due to the static delivery column Area of the section of pipe at reference point i Overpressure necessary to open valve B Negative velocity necessary to close valve B Efficiency of the Ram Pump Time elapsed for a complete cycle Average flow at reference point i

7

1 6

tank

air

5 2

B

A

4 3

I.

Introduction

Tank with level at 1 Supply piping between 1 and 3 Waste valve A between 3 and 4 Delivery-check valve B between 2 and 5 Air Expansion Chamber between 5 and 6 Delivery Pipe between 6 and 7

The operating the principle for a Hydraulic Ram Pump – HRP is based on a cyclical variation of pressure of the transient flow that occurs due to a phenomenon similar to that of the water hammer. It is widely used in farms, for water pumping, especially where there is plenty of water and no energy for powering conventional pumps.

Fig. 1. Schematic diagram of a Hydraulic Ram Pump


137


Mario O. M. de Carvalho, Alberto C. G. C. Diniz, Fernando J. R. Neves

TABLE I OPERATIONAL CHARACTERISTICS OF THE HYDRAULIC RAM IN EACH PHASE Phase 2 Phase 3 Phase 1 Storage and Complementary Supply Delivery Delivery Open Closed Open Valve A Closed Open Closed Valve B Duration T1 T2 T1 Place Section 1-4 Section 1-7 Section5-7 Simultaneous Phase 3 Independent Phase 1 with Coupling Independent Coupled Coupled

A vast quantity of recommendations, guidelines, standards and rules can be found for the project, installation and operation of Hydraulic Rams, as listed, for instance, by [1]. Among the recommendations it is worth mentioning: The ram frequency should fall in the range of 20 cycles/min to 100 cycles/min, as recommended by [2]; or from 30 cycles / min to 100 cycles / min as suggested by [3]. The delivery head should be about 6 to 12 times the supply head [5]. The length of the supply pipe should be 4 to 6 times the delivery head according to a particular manufacturer, while others recommend from 50 to 10 times, or even 8 to 12. A large list of authors and manufacturers indicate that the efficiency of the Hydraulic Ram is located in the range 50% to 80%. This reveals that high uncertainty and empiricism prevail when proposing those guidelines, which often overlook even basic parameters in the design and operation of the Hydraulic Ram, as, for example, the volume of the expansion chamber. The most frequent situation for the installation of a Hydraulic Ram Pump requires its dimensioning and design for a particular topography and hydric resource, thus conditioning the delivery head, as well as the supply head and the flow rate. The adjustment of the operating condition is then made using the check valve, in order to optimize the flow rate, or the efficiency, or even a compromise between the two. The usual adjustment consists in changing the water velocity at which the valve A will block the flow, thus changing the duration of the complete cycle of the Hydraulic Ram. This adjustment can be accomplished by changing the drag force due to moving parts of the valve in the flow near its seat, or, more frequently, by altering the force that opposes this drag, using a spring or a weight. Designers and installers still miss analytic interpretations that can guide the design and operation setting of the Hydraulic Ram. Drawing on a model previously developed [1], this paper aims to develop studies that can represent and interpret the behavior of Hydraulic Ram Pumps. Further, it aims to identify clearly work regimes or design situations in which these machines will malfunction and should therefore be avoided. Guidelines and recommendations are also offered for their design, installation and operation.

II.

Phase 1

m1

Phase 2

m1

c1

c1

c2 m2 k2 c2 m2

Phase 3 k2

Fig. 2. Dynamic model for the behavior of each phase of the Hydraulic Ram Pump

Equations (1) to (4) are developed in detail in [1] and model the three phases of operation depicted in Fig. 2. Phase 1:

(C + ρ 2) 2 dv4 g ⋅ h14 C1 = − v4 − 2 v4 dt L14 ρ ⋅ L14 ρ ⋅ L14

(1)

where h is pressure head, L length and C head loss, and the indices refer to the respective reference points on the HRP, as shown in Fig. 1. Phase 2: dv5 g ⋅ h15 C0 C3 ρ 2 2 v5 − v + = − − dt L15 ρ ⋅ L15 ρ ⋅ L15 ρ ⋅ L15 5

(P − P ) C4 v5 ⋅ v5 − 6 atm − ρ ⋅ L15 ρ ⋅ L15

(2)

This is obtained applying the equations of balance of linear momentum for a control volume limited to section 1-5:

Dynamic Model

In reference [1] Carvalho et alli present in detail the key characteristics of the three phases that comprise a complete cycle of the Hydraulic Ram. This is transcribed here briefly. As shown in Table I, Phases 1, 2 and 3 are well represented by the model displayed in the block diagram of Fig. 2. It represents both the transient behavior (which occurs at the beginning of the operation), as well as the steady state operation.

dv7 g ⋅ h67 C5 C6 v7 − v ⋅v + =− − dt L67 ρ ⋅ L67 ρ ⋅ L67 7 7 +

( P6 − Patm )

(3)

ρ ⋅ L67

Equation (3) applies to the 6-7 section, using the balance of linear momentum, similarly to the 1-5 section.



138


Analysis of Air Expansion Chamber between the points "5" and "6" leads to:

(1+ 1 )

(1− 1 k )

dP6 k ⋅ P6 k ⋅ P0 =− dt Vatm ⋅ Patm

where k =

Cp Cv

( v7 ⋅ A7 − v5 ⋅ A5 )

combined stiffness approximately equivalent to the stiffness of the air in the Chamber itself. Therefore, the influence of elasticity of pipe in Phase 3 can be disregarded without significant error. The inertia due to the mass of water in the pipe was also estimated considering water as an incompressible fluid, in uniform flow along the ideally rigid delivery pipe. The mass associated with this phase can be evaluated by: m2 = L67 ⋅ ρ ⋅ A7 (6)

(4)

.

Phase 3: This phase is modeled by the same equations (3) and (4), already considered in phase 2, taking velocity v5=0 as initial condition. II.1.

The damping c2, indicated in Fig. 2 and associated with Phase 3 is non-linear and is directly related to head losses C5 and C6 in section 6-7, as described by equation (3).

Vibration Behavior in Phase 3

II.2.

The model describes well the complete operation cycle of the Hydraulic Ram. With it, the dynamic behavior was studied. In particular, Phase 3 is analyzed in detail, so that it can be better interpreted, and the influence of design parameters can be assessed, with their implications on the control and behavior of the Ram Rump. As can be seen in Fig. (2) and in the corresponding equations (3) and (4), the specific part of the model considered in Phase 3 represents clearly a classic secondorder system with one degree of freedom, hence with a behavior of a vibrating system. Although Phase 2 also has such a behavior, it is not considered in detail, mainly because of its very short duration, when compared with the natural period of the associated vibration. Although Phase 2 also has such a behavior, it is not considered in detail, mainly because its duration, T2 (the time that valve B remains open) is very short when compared with the natural period of the associated vibration. The equivalent stiffness modeled in Phase 3, due mostly to the presence of air in the expansion chamber (with volume Vatm), can be considered linear, for small volume variation. The compressibility of air in the chamber is modeled, assuming fast deformations. An adiabatic process is modeled by equation (4), starting from the conditions of pressure and volume, P0 and V0. The pressure P0 and volume V0 are obtained considering an isothermal transformation from atmospheric conditions (Patm and Vatm) to equilibrium with the pressure due to the hydrostatic column head h67. Therefore, the stiffness k2, after linearization, is equivalent to the stiffness of the air in the chamber (see Fig. 2) [6], and can be expressed by: k2 =

k ⋅ A72 ⋅ P02 Patm ⋅ Vatm

Resonance Behavior in Phase 3

Energy is introduced in the Expansion Chamber during Phase 2, while the air contained in it is compressed. This potential energy is stored as pressure for the subsequent launch of Phase 3. The cyclical forcing produced by this mechanism interacts with the natural characteristics of second-order system, identified for the model in Phase 3, to produce the resonance phenomenon. This behavior can modify drastically the response of the Hydraulic Ram as a whole. Near resonance, certain approximations and assumptions made in the model would be violated, such as the linearity, of the model, the assumptions of incompressible fluid and rigid pipe walls, for example. Even so, our model still detects the onset of resonance, although it cannot appropriately represent the large amplitude vibration for the fully developed phenomenon. In the conditions described, the Hydraulic Ram will start an anomalous behavior, often misunderstood from the experimental point of view. The first possible anomalous behavior that could be observed would correspond to high amplitude variations of the velocity v7 and pronounced variations of pressure in the Expansion Chamber. If the variation in pressure P6 (assumed to be equal to P5) becomes less than pressure P0, then valve B will open during Phase 3, before valve A blocks the flow. Phase 3 will then end prematurely, starting an event that usually occurs later, simultaneously with the end of Phase 1. With the disruption of the regular cycle, composed of the three phases described above, the HRP has a drastic drop in its performance or simply stops working. To investigate the possibility and consequences of resonance, our model is used to identify the conditions that lead to this behavior and forecast the phenomenon. This study will prove to be useful in interpreting the behavior of HRP if such operating conditions occur. Classically, according to [6], the natural (undamped) frequency in Phase 3 may be given by:

(5)

ωn =

The high rigidity of the delivery pipe in series with the low stiffness of the Expansion Chamber, results in a Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved

k ⋅ A7 ⋅ P02 Patm ⋅ Vatm ⋅ L67 ⋅ ρ

(7)


139


The natural frequency and the damping (loss of pressure in pipe 5-7) characterize the dynamic behavior for Phase 3 of the model. Phase 3 has a periodic excitation, imposed at the end of the previous phase, that introduces energy for each complete cycle of the Hydraulic Ram, with duration TCycle. Since Phases 1 and 3 are simultaneous, the length of the complete cycle can be given by TCicle = T1 + T2 . Thus, with the developed model, a second order forced system is obtained, for which a peculiar behavior can be expected whenever the excitation frequency approaches the natural frequency. 2π is near an integer That is, when ωExcitation = T1 + T2 multiple of the natural frequency given by equation (7). Although verified experimentally, this undesirable behavior is usually avoided by use of a set of empirical rules that establish valid operating conditions. These relate the length of the piping to the volume of the expansion chamber and blocking time for valve A. The rules, however, lack a theoretical interpretation justifying them. The fact that the mechanism that leads to this behavior may not be well understood means that those rules are defined with a safety margin that is usually exaggerated and/or unnecessary. From the perspective of vibration, in Phase 1 kinetic energy is introduced in the section 1-4. This energy is transferred to Phase 2, by means of a periodic excitation (pulsed) in section 5-7, where it is stored mainly in the Expansion Chamber, in the form of elastic energy. In Phase 3 the energy introduced in the previous phase for each cycle is partly stored as elastic potential energy in the Expansion Chamber. Some of it is dissipated by the loss of head in the pipes and valves. A third part, the useful portion for the operation of the pump, is then converted into gravitational potential energy as the water is raised to the level of the higher reservoir (not shown in Fig. 1). Thus, taking P0, approximately, by P0 ≅ ρ ⋅ g ⋅ h67 + Patm , the natural period Tn can be estimated by: Tn =

ρ ⋅ Patm ⋅ Vatm ⋅ L67 k ⋅ A7 ⋅ ( ρ ⋅ g ⋅ h67 + Patm )

2

contribute to energy dissipation, since the adiabatic model for the behavior of the compressed air in the chamber is conservative. This is a good approximation since, in effect, little heat transfer occurs during these fast transformations. The high velocities associated with resonance are associated with equally high dissipative losses, resulting in significant reduction in efficiency. An even more adverse situation occurs when a large variation of pressure in the Expansion Chamber may momentarily reduce the pressure at point "5" to a value below that of the pressure of point "2 ". This will result in the opening of valve B before the locking operation occurs in valve A. Such an occurrence completely degrades the operation of the Hydraulic Ram. II.3.

Frequency Ratio and Resonance Control

Aiming to avoid the resonance described and its harmful consequences, it must be ensured that the pump operates with a period TCycle (controlled by adjustment of valve A) shorter than the natural period or, equivalently, with a frequency above the resonant frequency. Although the operation of the HRP can take place above the resonance frequency, this alternative presents difficulties which will be highlighted below. Thus, defining the dimensionless parameter Frequency Ratio r by: r=

Tn ω = Cicle TCicle ωn

(9)

Assuming that the frequency ratio exceeds unity ( r > 1 ), and using (8) and (9), we have: 2π ⋅ r=

ρ ⋅ Patm ⋅ Vatm ⋅ L67 k ⋅ A7

( ρ ⋅ g ⋅ h67 + Patm ) ⋅ TCicle

>1

(10)

In practice, to avoid resonance, valve A can be adjusted in a real pump, so that TCycle is conveniently set and the inequality given by expression (10) can be guaranteed. The required distance of r from unity depends on the damping present. A difficulty arises in the estimation of TCycle. In the numerical model, it is necessary to simulate the Hydraulic Ram Pump for various cycles, until steady state is reached. However, being T2 much smaller than T1, we can estimate that the duration of the complete cycle of the Hydraulic Ram TCycle is slightly larger, or of the same order of magnitude of the blocking time T1, i.e. TCycle = T1 + T2 ≅ T1 .

(8)

Hence, the role of the Expansion Chamber is to store the energy released during the short pulse that occurs in Phase 2 and to release it during Phase 3. Its proper operation requires the stiffness of the Chamber to be as low as possible (within the limits of the economic construction of the chamber). The coincidence between the period TCycle and Tn (or whole multiples thereof), may lead to undesirable behavior of the HRP, with large amplitudes of oscillation in velocity and pressure. This behavior characterizes the occurrence of the resonance. In the model adopted, the pressure variation in the Expansion Chamber, when in resonance, does not

The calculation of T1 is done in Phase 1, which is run only once at the beginning of simulation as a function of the blocking speed of valve A, solving equation (1).



140


Thus, equation (10) can be estimated as a function of T1 by: 2π ⋅ r=

valve A and Q7 is the average flow rate for period TCycle at point “7”. With each cycle, the results obtained for P6, v5 and v7 are taken to the subsequent cycle as initial conditions. Note that Phase 1 is executed only once, and at its end, sets the elapsed time T1 (the time taken to block valve A). Phases 2 and 3 are repeated resulting, at the end of each one, the initial conditions for the subsequent phase. The procedure is repeated until steady state is reached. Upon reaching steady state, the velocities v4 and v7 are integrated for time over a full cycle, and the calculation of the average flow rates Q4 and Q7, respectively in Blocking Valve A and Delivery-check Valve B, are performed. The efficiency of the complete cycle is then evaluated using D'Aubusisson criterion, as in equation (12).

ρ ⋅ Patm ⋅ Vatm ⋅ L67 k ⋅ A7

( ρ ⋅ g ⋅ h67 + Patm ) ⋅ T1

>1

(11)

The choice of a blocking speed VBlock, for the flow in section 1-4, depends on the adjustment of the valve A and determines the value of blocking time T1. That speed, however, is subject to limitations set forth by [1]. The diagnosis so far presented, and expressed in (10) or (11), allows grouping several empirical guidelines, so that a region of undesirable dynamic behavior for the operation of the Hydraulic Ram is avoided. This, however, corresponds only to constraints due to the phenomenon of resonance. Other restrictions, which are not due to the resonant behavior, must also be considered. Among these, are the restrictions to the diameter and length of pipes, in order to avoid high pressure losses. Also, design restrictions must be considered to avoid high losses in the valves due to excessively high operating frequencies. These additional guidelines should be observed in combination with the explicit result in (10). In references [7] and [8] the authors study more elaborate models for the HRP, covering in detail the dynamics of valve A. However, the dynamic behavior of the valve depends on a complex fluid-structure interaction that is very specific to each valve and its specific geometry. Furthermore, it involves much higher frequencies than those mentioned here as being relevant to the overall operation of the Hydraulic Ram. In those studies, the authors find frequencies associated with the valve dynamics in the order of 30 to 40 times the operating frequency of the Pump. These frequencies are virtually "filtered" by the usual design of the volume Vatm of Expansion Chamber.

IV.

The model described above was used to conduct numerical simulations of the HRP, seeking to characterize its vibration behavior under various operating conditions. In particular, we sought to evaluate the changes in its dynamic behavior, both under conditions of resonance and away from these. IV.1. Vibrational Behavior of the Hydraulic Ram Pump To evaluate the local dynamic behavior characteristics of the Hydraulic Ram Pump during Phase 3 the program described in [1] was modified. It now simulates a single cycle, in which after an initial disturbance in speed, due to Phase 1, the valve B is kept permanently closed. The results obtained is this modified simulation are shown in Fig. 3 for three different levels of the equivalent damping factor ζequivalent, whose values were calculated from the logarithmic decrement of the responses, as a function of head loses in the flow. In this simulation, with velocity v5 = 0, equations (3) and (4) are uncoupled from the others. The resulting system is then a typical second-order system. Thus, the behavior expected for Phase 3, decoupled from the other phases, can then be studied in detail over a time that is longer than TCycle. The equivalent stiffness in this model can be considered linear for small pressure fluctuations. Using this linear approximation, equation (7) can be used to obtain the Natural Frequency. It results in ωn = 0.5104 rad/s (a period of Tn = 4.16 s). On the other hand, the natural period obtained by numerical simulation, with our model, resulted in Tn = 4.2 s, as shown in Fig. 3. This is in fully satisfactory agreement with respect to the result obtained using equation (7). With the free behavior of Phase 3 of the Hydraulic Ram characterized, we can now model its forced dynamic behavior, as that of a system of one degree of freedom excited periodically at each cycle (during Phase 2). The block diagram of Fig. 4 represents the dynamic model for Phase 3 of the HRP, simulated as an

III. Numerical Simulation The numerical simulation of the Hydraulic Ram Pump, by direct integration of equations (1) to (4) for each cycle was performed as described in detail in [1]. The interest in the operation of the HRP is in the steady state operation. However, to achieve it, the machine must evolve from an initial condition, and thus pass through a transient phase. Hence, the simulation model must reproduce this transient numerically. Adopting the D'Aubusisson criterion of efficiency for the HRP, as described by [4], once steady state is reached the efficiency can be evaluated by: ⎡

⎤ Q7 ⋅ h17 ⎥ ⎣⎢ ( Q4 + Q7 ) ⋅ h14 ⎦⎥

η=⎢

Analysis of Results

(12)

where Q4 is the average flow rate for period TCycle on



141


As proposed in [6], the solution of this problem is obtained solving the differential equation:

equivalent mechanical system with a forced displacement imposed to its base. The excitation is modeled by imposing displacement x5 to the base of the spring element that corresponds to point "5" in the Pump diagram. The response is obtained by observing displacement x7.

x7 + 2 ⋅ ξ ⋅ ωn ⋅ x7 + ωn2 ⋅ x7 = ωn2 ⋅ x5 ( t )

Here, ζ is the damping factor and ωn is the natural frequency of the system and x5(t) is the imposed displacement, given by equation (13). Considering the steady state, x5(t) is a periodic function and can be expanded in a Fourier series. Thus,

Dynamic Behavior – Phase 3 ζ1

Pressure [kPa]

174 172

170 0

we obtain x5 ( t ) = 2

4

6

8

10

12

14 16

18

20

Pressure [kPa]

172

170 0

2

4

6

8

10

12

14 16

18

20

180 Pressure [kPa]

ζ

3

175

p =∞

∑

p =−∞

Ap ⋅ ei⋅ p⋅ω0 ⋅t , p = 1, 2,3,... , which

corresponds to a superposition of harmonic excitations with the fundamental frequency ω0=ωCycle and its harmonic integer multiples. Thus, we expect to find a resonance condition not only when the natural frequency ωn approaches the fundamental frequency ωCycle, but also as it approaches one of its higher harmonics, ωp. That is (for p = 2, 3, ...), whenever r = 1/p assumes one of the 1 1 values r = 1, , ,... . 2 3

ζ2

174

(14)

170 T

165 0

2

4

6

8

10

12

14 16

18

IV.2. Hydraulic Ram Pump Dynamics in Steady State

20

Typical curves of operation for the HRP are shown in Fig. 5. They were obtained for steady state, based on the dimensioning and experimental tests carried out by [9]. In Fig. 5, Phase 1 can be identified by the evolution of speed v4 from zero to the blocking speed VBlock, which is set to VBlock = 1.2 m/s. This variation has a duration of T1 = 1.25 s. Phase 2, of duration T2 (highlighted in yellow) is characterized by velocities v5 and v7 and pressure P6. It shows a pulse of mechanical energy introduced into the HRP that, for the most part, is stored as pressure energy in the Expansion Chamber, while the velocity v7 is maintained approximately uniform.

Fig. 3. Dynamic Behavior of Phase 3 of the Hydraulic Ram for the conditions: h67=7.4m and Vatm = 6.7 liters. The three head losses were simulated: ζ1 = 0.09; ζ2 = 0.18 and ζ3 = 0

With the aim of eliminating the average component of the displacement, which does not contribute to the vibration behavior, x5 is given by: x5 ( t ) =

t

∫ ( v5 − v5 ) ⋅ dt

(13)

0

In (13), v5 =

TCiclo

1

TCiclo

∫

v5 ⋅ dt and x5, is therefore the

Hydraulic Ram Pump

0

displacement of point "5" seen with reference to the mean velocity v5. Similarly, x7 will also be obtained as its deviation from the mean.

1

0.5

x5

Velocity [m/s]

m2 k2 x6

V4

Phase 1 and Phase 3

V5 V7

0 T1

T2

P6

x7 -0.5

Phases 2 and 3 12 121.5

Fig. 4. Dynamic Behavior of Phase 3 of the Hydraulic Ram as a forced system

122 1 12 122.5 Time [s]

2

7

5

c2

6

Pressure [10 N/m ]

Phase 2

5

1 123

1.76 1.75 1.74 1.73 1.72

Fig. 5. Numerical simulation of the operation of a Hydraulic Ram Pump, showing velocity on the left axis and pressure on the right axis



142


Phase 3 (highlighted in blue), is simultaneous with Phase 1 (T3 = T1 = 1.25s), and characterized by the velocities v5 and v7 and the pressure P6. The total elapsed time for the complete cycle is TCicle = T1 + T2 = 1.475 s . The ratio of frequencies for the simulated condition is T 4, 2 = 2.85 . given by r = n = TCicle 1, 475

are set, only two variables play a fundamental role in the resonance condition. They are the volume Vatm of the Expansion Chamber, which determines the equivalent stiffness of the system, and the period T1 of duration of Phase 1, which is the main factor that influences the cycle time TCycle. The simulations presented in Fig. 6 show the influence of the Chamber volume on the behavior of the HRP for different blocking times TBlock coinciding with T1. The simulations were conducted according to procedure described above, until steady state was reached. The results are shown in Fig. 6.

IV.3. Resonance in the Hydraulic Ram Pump As can be seen from equations (7) and (11), once the topographical conditions of the installation of the HRP 500

80

400

B1

60

300

40

200

20

100

0.2

0.4

500

80

400

60

300

40

200

20

100

ηη

Q Vazão

0.6 0.8 1 Blocking VelocidadeVelocity de Bloqueio[m/s] [m/s]

1.2

1.4

0 1.6

0

0

0.2

Hydraulic Ram Pump in Stead State 500

40

200

20

100

ηη 0.4

Efficiency Rendimento–- η[%] η [% ]

300

Flow V azão– - Q Q [l/h] [l/h]

Rendim ento - η [% ]

Efficiency – η[%]

60

0.2

100

500

80

400

60

300

40

200

20

100

ηη

Q Vazão

0.6 0.8 1 1.2 Blocking Velocity [m/s] Velocidade de Bloqueio [m/s]

1.4

0 1.6

0

0

0.2

500

200

20

100

0.6

Rendimento - η [% ]

40


300

0.4

B3

Flow – -QQ[l/h] V azão [l/h]

Rendimento - η [% ]


60

ηη 1

1.2

1.2

1.4

100

0 1.6

500

1.4

r=1/2

80

200

60

300

40

200

20

100

ηη

Q Vazão 0.8

1

r=1 400

0.2

0.8

r=1/2

r=1 80

0

0.6

Hydraulic Ram em Pump in Stead State Carneiro Hidráulico Regime Permanente

100

0

0.4

QQ

Velocidade Velocity de Bloqueio[m/s] [m/s] Blocking

Hydraulic Ram em Pump in Permanente Stead State Carneiro Hidráulico Regime

A3

0 1.6

r=1 400

0

B2

r=1/2

80

0

1.4

Carneiro Hidráulico em Regime Permanente

100

r=1

QQ

0.6 0.8 1 1.2 Velocidade de Bloqueio[m/s] [m/s] Blocking Velocity

Hydraulic Ram Pump in Stead State

Carneiro Hidráulico em Regime Permanente

A2

0.4

0 1.6

0

Blocking VelocidadeVelocity de Bloqueio[m/s] [m/s]

- Q [l/h] FlowVazão – Q [l/h]

0

100

FlowVazão – Q [l/h] - Q [l/h]

ηη 0

Efficiency – η[%] Rendimento - η [% ]

r=1 Vazão Q[l/h] [l/h] Flow Flow – –Q -Q [l/h]

Rendim ento - η [% ]


100

- Q [l/h] FlowVazão – Q [l/h]

Hydraulic Ram em Pump in Stead State Carneiro Hidráulico Regime Permanente

Hydraulic Ram Pump in Stead State Carneiro Hidráulico em Regime Permanente

A1

0

0.2

0.4

0.6

Q 0.8

1

1.2

1.4

0 1.6

Blocking Velocity Velocidade de Bloqueio [m/s] [m/s]

Fig. 6. Resonance Phenomena in a Hydraulic Ram Pump to the conditions specified in Table A1 in the Appendix



143


In the left column (A1, A2 and A3) the same values of hydrostatic column head h67 were used with different values for the volume of Expansion Chamber Vatm. In the right column of Fig. 6 (B1, B2 and B3), the procedure was repeated, but the hydrostatic column head h67 and also the pressure losses in the valves and pipes were reduced. This modification sought to achieve a double objective: • Bring resonance into the region of operation of the HRP, by decreasing its natural period of vibration, and; • Reduce the equivalent damping factor, in order to exacerbate the effect of resonance. These simulations reflect well the conditions, described above, in II.3, for the interference of the resonance condition with the adequate operation of the pump. As predicted, the graphs show that, for values of r close to 1 and 1/2, the efficiency is significantly reduced. It can also be seen that the machine should preferably operate with Tn > TCycle. This is a condition that was required by to equation (11), i.e., frequency ratio r > 1. This is the situation analyzed in Fig. 5, where r = 2.85. The simulations presented in Fig. 6 illustrate the

1 behavior of the pump when r = 1 and r = 2

Hydraulic Ram in resonance, for r = 1. It enhances the interpretation of the behavior of the Hydraulic Ram in this condition. It corresponds to Fig. 6 B2 for r = 1. As can be seen in comparison with Fig. 5, the simulation shows that the velocity v7 fluctuates sharply, becoming sometimes negative. The speed range in Fig. 5 and Fig. 7 was kept identical, in order to facilitate the comparison between the two. Hydraulic Ram Pump T1

T2

Phase 1 and Phase 3

0.5

v4 v5 v

7

0

Pressure [105 N/m2]

Velocity [m/s]

1

1.8 P6 1.7 1.6 1.5

-0.5 176

are

1.4

Phase 2 177

1 178

1.3 179

Time [s]

considered. Clearly, "valleys" are present in the efficiency and in the average flow rate curves. Simulations performed with lower damping factor than that for Fig. 6 (A1, A2 and A3) are presented in Fig. 6 (B1, B2 and B3). As was expected, the problem seems even more pronounced at the points of resonance. In Fig. 6 B1, r = 1 is beyond the full scale of the figure. Hence, a criterion based on the volume of the Expansion Chamber Vatm can be proposed for dimensioning the HRP. It is that Vatm should be large enough to result in a value of r sufficiently away from unity, as recommended by equation (10) or, by approximation, equation (11). The upper limit for r will be set solely by economic considerations. The tests showed no significant advantage in working with an overly large volume Vatm. If the values of r are larger than 1.5 have very little influence in the resonant behavior of Hydraulic Ram, although the damping level should also be taken into account. In practice, to avoid resonance, considering the usual damping values in the Hydraulic Ram, a criterion for good design and operation of the Hydraulic Ram is r > 1.5 . Sometimes, in cases of under-dimensioning of Vatm, the resonance corresponding to the second harmonic can be noticed. In Fig. 6 (A3, A2 1 . It is sometimes and B3) this corresponds to r = 2 advantageous to operate the Hydraulic Ram in the range 1 1 > r > , to increase flow rate, for example. However, 2 this is not a stable region and the setting of the HRP should be done very carefully. Figure 7 presents in detail the behavior of the

Fig. 7. Behavior of a Hydraulic Ram at resonance. (Corresponding to the condition depicted in Fig. 6, column B)

As expected, pressure P6 in the Expansion Chamber also fluctuates a lot (it was necessary to represent the pressure curve in an expanded scale in order to show it). The pressure in resonance conditions fluctuates much more than in the condition shown in Fig. 5. If P6 reaches a value lower than P2, Phase 1 will be aborted midway due to the opening of valve B. In that case, the behavior of the HRP would be radically modified. The steady state would cease. This abnormal condition cannot be simulated with our model. This is not a real limitation for the model, since in that case the HRP would not operate, or, at best, would have very poor efficiency. IV.4. Air Drawn from the Expansion Chamber The volume of the Expansion Chamber Vatm is a parameter easily controlled during the design of a HRP and, if conveniently chosen, can avoid operational conditions close to resonance. However, with the continued operation of the Pump, a small amount of the air contained in Expansion Chamber is continually dragged by the turbulent flow to the discharge pipe. This behavior, characteristic of the HRP, means that a periodical a maintenance procedure is necessary to recover the initial volume of air. A common solution for this problem is to introduce a third valve near the point "2" (see Fig. 1). This is known as a “Sniffer valve”, and operates so that a small amount of air is admitted there in each cycle, while valve B is closed. Thus, during Phase 2, this air is carried along with the flow to the Expansion



144


Chamber, replacing that which is lost. Hence the net volume of air in the Chamber remains unchanged. The use of a sniffer valve is fairly widespread and can be implemented drilling a hole, in the feed pipe, near the delivery-check valve. The hole must be of very small dimensions, so that it allows air to enter the pipe with relative ease, while not allowing loss of water. It was found experimentally that if the Sniffer valve admits only a small amount of air in each cycle, it does not change the performance of the Hydraulic Ram in a relevant way. Our model will simulate adequately the operation of the HRP, even if even if a “Sniffer valve” is installed. If the valve is poorly designed, however, it can be quite detrimental to the performance of the pump.

V.

the model can also be used to simulate conditions close to resonance, as long as high values of damping factors are present. That is, when high pressure losses are present in the pipes and valves. If the topographic parameters in the design for a HRP are fixed, the proper dimensioning of the volume Vatm of the Expansion Chamber may be done basically by setting the blocking speed VBlock of valve A. If this is properly done, then the adverse situation of resonance is avoided. The use of our model allows the delimitation of operation ranges and of criteria for design and operation of the Hydraulic Ram. This can be done according to different performance strategies. The designer can choose to maximize either flow rate or efficiency, for example.

Conclusions and Final Comments

References [1]

The computer model used to simulate the behavior of the Hydraulic Ram Pump can identify one of its major behavioral anomalies, namely that caused by the phenomenon of resonance. The simulation results match the theoretical interpretations regarding the dynamic behavior of the Hydraulic Ram. Modeled as a secondorder system, Phase 3 of the ram pump forecasts some of the ram pump actual malfunctions, showing a quite good consistency in relation to results obtained in the numerical model. The identification of an important design and operational parameter was also made. This is the Frequency Ratio r . It was found that its value should, in general, differ from unity or its sub-multiples ( r = 1, 1 , 1 , 1 ,... ) in order to avoid poor 2 3 4 performance in operation. The results obtained allow a better interpretation and guidance to the design and operation of the HRP, taking into account the vibration behavior. Until now these undesired conditions were considered by empirical rules, without a straightforward approach and analysis. As shown, for most practical situations, the assumptions that were made are perfectly compatible with the pressure variations present in the HRP, provided the frequency ratio has a value that will not lead to resonance conditions. But it can be seen that

[2] [3] [4]

[5]

[6] [7]

[8]

[9]

M.O.M. Carvalho, A.C.G.C. Diniz, F.J.R. Neves, Numerical Model for a Hydraulic Ram Pump, International Review of Mechanical Engineering (IREME), Vol. 5, n.4, pp.733-746, 2011. G. D. Jennings, Hydraulic ram pumps, North Carolina Cooperative Extension Service, (EBAE 161-92), 1996. K. Kitani and L.S. Willardson, Hidraulic Ram use for sprinkle irrigation. Transaction of the ASAE, v. 27, p. 1788-1791, 1984. K. C. Das et alli, Effect of magnification factor, supply conditions and valve clearance on performance of an hydraulic ram, International Congress on Agricultural Engineering, 11., Dublin, Proceeding. Rotterdam: A.A. Balkema, p. 721-725, 1989. L. Brown, “Using a Hydraulic Ram to pump livestock Water” “Factsheet, British Columbia” - Ministry of Agriculture and Lands, January, 2006. L. Meirovitch, “Elements of Vibration Analysis”, Ed. McGraw Hill, 1986. S.Y. Goh, "A study of the dynamic characteristics of the impulse valve of the hydraulic ram", 6th IWRA World Congress on Water Resources, May 29 — June 3, Canada, 1988. V. Filipan et al, Mathematical Modelling of a Hydraulic Ram Pump System, Journal of Mechanical Engineering, p.137-149, 2003. R. N. Z. Rojas, Modelagem, Otimização e Avaliação de um Carneiro Hidráulico, Tese de Doutorado – ESALQ-USP, 2002.

Appendix Table A1 shows the parameters used in the simulation for which the results are presented in Fig. 4.

TABLE A1 CONDITIONS USED IN FIG. 6 FOR THE SIMULATED OPERATION OF HYDRAULIC RAM PUMP Left Column “A”: h67 = 7.4 m and high head loss

Column A (Left)

Chamber Volume ( VAtm )

1ª Resonance Velocity


Natural Period ( Tn )

A1

VAtm = 6.7 liters

vr = 1.48 m s

──

Tn = 4.16 s

A2

VAtm = 1.33 liters

vr = 1.15 m s

vr = 1.46 m s

Tn = 1.86 s

A3

VAtm = 0.67 liters

vr = 0.93 m s

vr = 1.36 m s

Tn = 1.32 s

1

2

3

1

1

2

1

2

1

2

3

Right Column “B”: h67 = 5 , 0 m and head loss reduced by 40% compared to “A” Column B (Right)

Chamber Volume ( VAtm )



Natural Period ( Tn )

B1

VAtm = 6.7 liters

vr > 1.6 m s

──

Tn = 3.96 s

B2

VAtm = 1.33 liters

vr = 1.46 m s

──

Tn = 1.77 s

B3

VAtm = 0.67 liters

vr = 0.87 m s

vr = 1.5 m s

Tn = 1.25 s

1

2

3

1

1

1


2

1

2

3


145


Authors’ information Mario O. M. de Carvalho is a professor of mechanical engineering at Universidade de Brasília. He received his doctoral degree from the Federal University of Rio de Janeiro in 1996. His research interests include dynamics of flexible structures, vibrations, modal analysis, inverse problems, stochastic modeling techniques, and energy processes technology. He is currently working in semi-active vibration control. Alberto C. G. C. Diniz Professor of Mechanical Engineering at University of Brasília since 1993. He has a PhD in Mechanical Engineering from Ecole Centrale de Lyon (2000) and conducts research in dynamics of structures and systems, stochastic modeling, modal analysis, modal synthesis and finite elements.

Fernando J. R. Neves is an Associate Professor in the Department of Mechanical Engineering of the Faculty of Technology at the Universidade de Brasília, Brazil. He received both his MSc and PhD degrees in Mechanical Engineering, from the University of Manchester Institute of Science and Technology, England, in 1975 and 1979. He did post-doctoral studies at the Measurement and Instrumentation Centre, City University, London, in 1988. His research interests include vibrations, dynamic systems and dynamics of measurement.



146


Combustion and Performance Analysis of DI-Diesel Engine Fuelled with Neat Mahua Methyl Ester along with Oxygenated Fuel (DEE) as an Additive P. Ramesh Babu, V. J. J. Prasad, N. Hari Babu, B. V. Appa Rao

Abstract – Bio-diesels and their blends are proven alternative fuels for petroleum diesel. But still the research work is going, on the bio-diesels application to make it environmental friendly. Particulate matter and oxides of nitrogen are the main pollutants in the tail pipe emissions of biodiesel fueled engine. In this paper, mahuva methyl ester along with diethyl ether (DEE) used as fuel for the single cylinder DI-Diesel engine, analysis of combustion pressure and heat release rate with respective to the crank angle and performance and emission analysis is presented.. In this experiment DEE mixed with the MME at different proportion such as 3%, 5% 10% and tested at different loads on diesel engine. Smoke levels are decreased substantially with 15% DEE blend with MME at full load. The thermal efficiency rise and SFC are better in the case of 15% additive blend. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Oxygenated Fuels, Dimethyl Ether, Mahuva Methyl Ester, Diethyl Ether, Combustion Pressure

I.

NOx emissions have increased by approximately 18%. Graboski et.al [2] has measured emissions from100%methyl esters of soy bean oil and blends of this material with no. 2 diesel in a 4-stroke engine. Substantial particulate emissions reductions were observed at all oxygenate levels. They observed a statistically significant 1% increase in NOx at 2 wt % oxygen in the fuel and higher NOx increases at higher oxygen levels. Liotta and Montalvo [3] investigated the emissions effects of three glycol ethers, an aromatic alcohol, an aliphatic alcohol, and a polyether polyol using a 4-stroke engine. The actual structures of these compounds were not revealed. Methyl soy ester and diglyme were also included for comparison to previous results. Based on heavy-duty transient testing, CO was generally reduced and NOx showed an increase with all oxygenates studied. The PM reduction experienced appeared to be related to the amount of oxygen in the fuel. Unregulated emissions of aldehydes and ketones were reported to decrease. Nikanjam [4] examined ethylene glycol monobutyl ether acetate as a diesel additive based on cost, fuel blending properties, and toxicity concerns. Emissions results from a 4-stroke engine showed CO and PM reductions of approximately 18% and a NOx increase of 3% for 3 wt % oxygen in the fuel. Ullmanand co-workers [5] reported the effect of oxygenates and other fuel properties on emissions from a 1994 Model Detroit Diesel Series 60, 4-stroke engine. Monoglyme and diglyme were used as oxygenates at the 2% and 4% oxygen levels. The engine was run with 5 and 4 g/bhp.h NOx calibrations (1994 and 1998

Introduction

Particulate matter (PM) and oxides of nitrogen (NOx emissions) are the two important harmful emissions in diesel engine. Fuel companies and the researchers around the world are devoted to reduce such emissions with different ways. Fuel modification, modification of combustion chamber design and exhaust after treatment is the important means to alleviate such emissions. In this context, engine researchers are hunting suitable alternative fuels for diesel engine. Among different alternative fuels, oxygenated fuel is a kind of alternative fuel. The presence of oxygen in the fuel molecular structure plays an important role to reduce PM and other harmful emissions from diesel engine. Additions of methyl tertiary butyl ether (MTBE) and ethanol have been successful in reducing CO and nonevaporative hydrocarbon emissions from gasoline engines. The success of oxygenated gasoline has sparked interest in the use of oxygenated compounds as particulate matter (PM) emissions reducing additives in diesel fuel. Bennethum and Winsor [1] examined the oxygenated compound diglyme (Diethylene glycol dimethyl ether) in a 2-stroke engine. Further work in which dimethyl carbonate was added to diesel at 3.5 wt % oxygen. PM and CO reductions were approximately 15%. NOx emissions showed a 1.8% increase while HC emissions were unchanged. The neat methyl esters of rapeseed oil and soybean oil were also tested. HC, CO, and PM were decreased approximately 75%, 50%, and 30%, respectively, for both methyl esters.


147


P. Ramesh Babu, V. J. J. Prasad, N. Hari Babu, B. V. Appa Rao

transformed into another ester through interchange of alkyl groups and is also called as alcoholysis. Transesterification is an equilibrium reaction and the transformation occurs by mixing the reactants. However, the presence of a catalyst accelerates considerably the adjustment of the equilibrium. The general equation for trans-esterification reaction is given below:

emissions standards, respectively). Statistical models of the emissions dependency on fuel composition were developed with weight percent oxygen as the oxygenate variable. At 2% Oxygen and constant aromatic content and Cetane number, the model predicts a 1.5% increase in NOx for the 5-gram calibration and a 0.9% increase for the 4-gram calibration. FEV Engine Technology [6] investigated various soy ester blends with diesel in comparison to diesel using a 13-mode steady-state test with the Navistar 7.3 L engine. FEV found that NOx increased with biodiesel under all conditions of speed and load. Soy ester blends reduced particulate at high speed and at all loads. At lower speeds, particulate was reduced at light and heavy loads but was increased at intermediate loads. DEE is oxygenated fuel that has a very high Cetane number. Masoud et al. [7] reported lower smoke and THC emissions due to higher Cetane number and Oxygen content of DEE. Authors also found lower CO emissions at high load condition, but higher at low load condition. Also lower NOx emissions were realized with DEE-diesel blends. Kapilan et al. [8] conducted experiments with 5 % DEE and found lower CO, THC and smoke emissions while a slight improvement in thermal efficiency was observed. Yeh et al. [9] investigated the effect of fourteen different oxygenated fuels on diesel emissions, especially PM and NOx emissions. Authors found that for PM reduction, the most effective oxygenates on equal oxygen content basis were the C9 – C12 alcohols in both the engine and vehicle testing. The present work reports on the effect of oxygenated fuel on diesel combustion and exhaust emissions. It has been found that the exhaust emissions including PM, total unburnt hydrocarbon (THC), carbon monoxide (CO), smoke and engine noise were reduced with oxygenated fuels. NOx emissions were reduced in some cases were increased depending on the engine operating conditions. The reductions of the emissions were entirely depended on the oxygen content of the fuel. It has been reported that the combustion with oxygenated fuels were much faster than that of conventional diesel fuel. This was mainly due to the oxygen content in the fuel molecular structure and the low volatility of the oxygenated fuels. The lower volatile oxygenated fuel evaporated earlier and very good air-fuel mixing was achieved during combustion eventually resulted in lower exhaust emissions.

II.

RCOOR’ + R" OH ↔ RCOOR" + R' OH

(1)

The basic constituent of vegetable oils is triglyceride. Vegetable oils comprise of 90-98 percent triglycerides and small amounts of mono-glyceride, diglyceride and free fatty acids. In the transesterification of vegetable oils, a triglyceride reacts with an alcohol in the presence of a strong acid or base, producing a mixture of fatty acid alkyl esters and glycerol. The overall process is a sequence of three consecutive and reversible reactions in which diglyceride and mono-glycerides are formed as intermediates. The stoichiometric reaction requires one mole of triglyceride and three moles of alcohol. However, an excess of alcohol is used to increase the yield of alkyl esters and to allow phase separation from the glycerol formed. The mechanism of the base-catalyzed transesterification reaction of vegetable oil is shown in the Fig. 1. The first step (Eq. (1)) is the reaction of the base with the alcohol, producing an alkoxide and the protonated catalyst. The nucleophilic attack of the alkoxide at the carbonyl group of the triglyceride generates a tetrahedral intermediate, from which an alkyl ester and the diglyceride are formed. The latter deprotonates the catalyst, regenerates the active species, and enables it to react with a second molecule of the alcohol thus starting another catalytic cycle. Diglycerides and monoglycerides are converted by the same mechanism to a mixture of alkyl esters and glycerol. TABLE I CHARACTERISTICS OF DEE& COMPARISON WITH MME S. N0 1 2 3 4

Preparation & Characterization of MME

Property Viscosity (cSt) Density (kg/m3) Cetane number Calorific Value (kJ/kg)

Diesel

MME

DEE (additive)

2.75

4.25

0.23

0.830

0.899

0.713

45

50

>125

43000

36,700

33,900

5

Boiling point

1803600C

3600C

350C

6

Auto ignition temperature

2500C

>3000C

1600C

Because of lower density difference (Table I), blending is not difficult as there is no separation observed. The additive is having higher Cetane value and lower density, lower auto ignition temperature and boiling point indicates that it starts ignition and initiates ignition of the main fuel i.e. biodiesel. The quantity of additive is limited to 15% of the main fuel is understood from the tail pipe emissions.3% mixture of additive is

The use of vegetable oils in lieu of diesel fuel in conventional diesel engines requires certain modification of their properties. The problems of substituting diesel fuels with pure vegetable oils (non-edible) are mostly associated with their high viscosities. Transesterification is the general term used to describe the important class of organic reactions, where an ester is Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved


148


The Piezo electric transducer is fixed (flush in type) to the cylinder body (with water cooling adaptor) to record the pressure variations in the combustion chamber. Crank angle is measured using crank angle encoder. Exact TDC position is identified by the valve timing diagram and fixed with a sleek mark on the fly wheel and the same is used as a reference point for the encoder to with respect to which the signals of crank angle will be transmitted to the data logger. The DI diesel engine (make Kirloskar company, Pune) is used for conducting the experimentation:

going in consonance with the 15% additive similar in all aspects but there is a zone of equivalence ratio in which 15% additive plays important role in saving fuel and running the engine smooth as per the vibration signatures investigated.

TABLE II ENGINE SPECIFICATIONS Rated Horse power: Rated Speed: No of Strokes: Mode of Injection and injection pressure Fig. 1. Base treatment

The additive prepares for better combustion of the main fuel because of its better combustion characteristics in advance and elevates the stature of combustion to higher strata to enhance the efficiency to better levels when compared to petro diesel. The start of combustion will be initiated by the additive to finish it to smoother end. Better combustion with lesser detonation can be observed from the vibration signatures in comparison at 15% of additive mixing. Reducing delay period by increasing mixture temperature by earlier combustion of DEE is the advantage associated with the doping. Earlier combustion of DEE leads to exhaustion of the doping agent at the end which means unsupportive attitude of additive till to the end. This doesn’t mean more mixing of additive will address this question because of higher release of unburned hydrocarbons in the tail pipe emissions. Since the viscosity and boiling point of DEE are far lesser than that of MME, faster dilution of DEE takes place in the dual fuel contingent mixture injected into combustion chamber. The injection throw distance for the DEE is obviously more than the biodiesel in the blend and hence DEE touches the combustion chamber walls earlier than the biodiesel and this might have been the reason for increase of hydrocarbons in the back drop of the increase of the DEE blend percentage. This worsening trend of higher HC emission is the reason, which limits higher blends of DEE along with biodiesel fuel.

5 hp (3.73 kW) 1500rpm 4 Direct Injection 200 kg/cm2

No of Cylinders:

1

Stroke

110 mm

Bore Compression ratio

80 mm 16.5

The experimentation is conducted on the single cylinder direct injection diesel engine operated at normal room temperatures of 280C to 330C in the Department of Marine Engineering, Andhra University. The fuels used are diesel oil in neat condition and as well as methyl ester of Mahuva oil (MME) with 3%, 5%, 10%, and 15% additive Diethyl ether (DEE) and at five discrete part load conditions namely, No Load, One Fourth Full Load, Half Full Load, Three Fourth Full Load and Full Loads. The data collection is done independently for the above said oils. The engine is initially made to run at 1500rpm continuously for one hour in order to achieve the thermal equilibrium under operating conditions.

IV.

Results & Discussions

IV.1. Cylinder Pressure Signature Study Combustion pressure values have been logged in the combustion cycle range of 720 degrees of crank revolution. The values have been logged at every one degree of crank angle interval. The combustion duration has effectively increased reducing peak pressure rise in the plots. The delay period has increased insignificantly with respect to the increase in additive quantity in the blend as can be observed from the Figs. 2 & 3. At full load, the delay period variation amounts to 3 degrees of crank revolution approximately (0.33ms) when compared to diesel fuel. More the delay period means steeper the pressure rise leading to higher temperature combustion which can be surmised from the diesel pressure plot. 3% and 15% DEE percentages in the blend go hand in hand and 15% DEE blend supersedes the 3% DEE blend in diffused combustion zone hence recommending 15% DEE blend as a feasible one. There is a substantial improvement in the performance and emissions in the case 15% blend of biodiesel.

III. Experimentation The experimental setup consists of the following equipment: 1. Single cylinder DI-diesel engine loaded with eddy current dynamometer 2. Engine Data Logger 3. Exhaust gas Analyzer 4. Smoke Analyzer 5. Vibration Analyzer



149


Fig. 2. Close encounters of pressure plot to assess delay period variation

Fig. 5. Cumulative heat release rate comparison with various percentages of Additive Diethyl Ether (DEE) along with Biodiesel at 3/4th Full Load

At 3/4th full load, 15% DEE additive is more efficient in creating higher levels of cumulative heat release rate per degree when compared to other fuel configurations. Initial burning rates are almost same in the case of 3% DEE and 15% DEE blends but end mixture burning rates are different with a maximum of 250 Joules per degree at 3/4th full load as shown in the Fig. 5. Hence, 15% DEE blend is chosen as profitable blend with a vision focused on the performance and emissions also. IV.3. Engine Performance Fig. 3. Close encounters of pressure plot to assess delay period variation

The equivalence ratio at full load lies in between 6 and 7 for the diesel oil. This indicates that the engine is in good stead to conduct experimentation as shown in Fig. 6. Biodiesel run engine maintains equivalence ratio of 7 at full load because the fuel consumption is more and also because of its lower heat value. In all other cases of additive blends, the equivalence ratio increases at full load. If compared 3% and 15% blends of DEE additive, 15% blend is more economical at part load performance of the engine approximately in between 0.4 and 0.65 equivalence ratios of the engine running it can be observed from the Fig. 6. The brake specific fuel consumption increases from 0.3 kg/kW/hr to 0.375 kg/kW/hr when additive is being added with the defined blend percentages to the bio diesel.

IV.2. Heat Release rate Curves Derived from the Pressure Signatures Heat release in the acceleration mode of the piston is advantageous and that is what exactly is happening with the additive mixing. Even though, 3% additive mixing is going hand in hand in the combustion pressure generation, with 15% blend, as seen in Fig. 2, there is an advantage of better diffused combustion on par with the neat biodiesel. It can also be observed in the cumulative heat release curves in Fig. 4 and 5 in which later part of combustion curves are separated with greater drooping in the case of 3% additive mixing.

Fig. 6. Brake Specific Fuel Consumption Vs Equivalence ratio graph with different percentages of additive Diethyl Ether (DEE)

Fig. 4. Cumulative heat release rate comparison with various percentages of Additive Diethyl Ether (DEE) along with Biodiesel at Full Load



150


in the case of 15% DEE blend and with a consistent relative lower levels of smoke at all other part loads.

Fig. 7. Brake Thermal Efficiency Vs Equivalence Ratio graph with different percentages of additive Diethyl Ether (DEE)

In the case of Neat biodiesel application, the BSFC stays at approximately 0.35 and with the additive percentages, it increases slightly, simultaneously increasing the equivalence ratio. There is a steep rise of thermal efficiency for the additive percentages of ‘3’ and ‘15’ at part loads of the engine matching with the equivalence ratios 0.4 and 0.65 which can be observed from Fig. 7. For the additive blend 15%, the thermal efficiency stays at approximately 27% at full load. Even though the thermal efficiency suffers a little at full load, it is giving better at part load running of the engine. This steep rise may be due to better Cetane number and lower boiling point temperature of the additive. Better dilution with higher percentages of additive yielded better results but as the dope percentage is increasing, the emission of unburned hydrocarbons (HC) keep on increasing and this additive increase tells on the overall heat value of the blend.

Fig. 9. Smoke Vs Load Graph with different percentages of (DEE) along with Biodiesel at all loads

The delay period difference, as shown in Fig. 10 is plotted with respect to the petro diesel and is derived from time waves of vibration at full load recorded on the cylinder head in vertical direction. For Biodiesel the delay period decreased by 1.21 ms and the decrease in delay period is reducing w.r.t the additive blend. At 5% additive blend, the decrement is minimum and there upon it is increasing since cold combustion is taking place with respect to increase in additive. Reduction in Delay period compared to Diesel 0 Biodiesel

3% Additive

5% Additive

10% Additive 15% Additive

Delay period difference in ms

-0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 Blends

Fig. 10. Delay period difference in milliseconds w.r.t. Diesel at full load

V.

Conclusion

1. The additive DEE and the main biodiesel are possessing nearer densities and the blend 15 % is also observed to be stable during engine operation over limited hours observed in the laboratory. 2. 15% DEE blend with biodiesel is adjudged as the best combination which yielded better results than other fuel blends tested especially 3% blend which is the nearest competitor. 3. 3% and 15% blends create delay period difference of 0.4 ms (lesser for 15% blend) which can be observed from the real time, time wave plots at full load. But in the case of 15% blend, the diffused combustion

Fig. 8. Exhaust Gas Temperature Vs Load Graph with different percentages of (DEE) along with Biodiesel at all loads

Fig. 8 envisages higher exhaust gas temperatures with 15% additive than at 3% indicating better diffused combustion as explained in the previous paragraphs. Fig. 9 gives the smoke emission plot in HSU at various loads and at various additive percentages. 15% f the DEE additive gives better smoke reduction indicating better combustion. There is a substantial 12% decrease in HSU



151


[8]

Kaplan, N., Mohanan, P. and Reddy, RP., 2008,“Performance and Emission studies of Diesel Engine Using Diethyl Ether as Oxygenated Fuel Additive”, SAE paper no. 2008-01-2466. [9]. Yeh, LI., Rickeard, DJ., Duff, JLC., Bateman, JR., Schlosberg, RH. and Caers, RF., 2001, “Oxygenates: An Evaluation of their Effects on Diesel Emissions”, SAE paper no. 2001-01-2019.

aspect is very much improved. 4. The thermal efficiency rise and SFC are better in the case of 15% additive blend and since diesel engines give better efficiency at part loads this percentage of blend can be recommended. 5. The equivalence ratio at full load in the case of the two blends differ marginally, 15% blend can be adopted for the reason the difference is significantly small and there won’t be much larger loss of fuel at little higher equivalence ratio. 6. The cumulative heat release rate plot of 15% DEE blend is better at part load i.e. at 3/4th full load and is overriding all plots with lower DEE percentages indicating that the combustion with this percentage is better both in premixed combustion zone and diffused zone. And at full load, the 15% DEE blend occupies second position next to the neat biodiesel application while all other blends are lagging behind. 7. The smoke levels have decreased substantially with 15% DEE blend with biodiesel at full load and at immediate part load except very low loads at which the diesel engine may not be put to operation normally because of high bsfc. Smoke levels have decreased in tandem indicating better combustion.

Authors’ information P. Ramesh Babu, (Corresponding author) Senior assistant professor in Department of mechanical Engg, GMR Institute of Technology, Rajam, Srikakulam, Andhrapradesh, India. E-mail: [email protected]

Dr. V. J. J. Prasad Senior associate professor in Department of mechanical Engg, GMR Institute of Technology, Rajam, Srikakulam, Andhrapradesh, India. E-mail: [email protected] [email protected]

Dr. N. Hari Babu, Professor in Department of mechanical Engg, Aditya Institute of Technology and management, Tekkali, Srikakulam, Andhrapradesh, India.

Acknowledgements The authors gratefully acknowledge the contribution of Prof. Dr. B.V. Appa Rao Department of Marine Engineering, Andhra University and also for his esteemed guidance and valuable suggestions in the experimentations and in completing the work.

Dr. B. V. Appa Rao, Professor in Department of marine Engg. Andhra university college of Engg. Visakhapatnam, Andhrapradesh. India. He guided many Ph.Ds. All three co-authors are scholars of Dr. Prof. B. V. Apparao

References [1] [2]

[3]

[4]

[5]

[6]

[7]

Bennethum, J. and Winsor, R., "Toward Improved Diesel Fuel," SAE Technical Paper 912325, 1991, doi:10.4271/912325. Graboski, M., Ross, J., and McCormick, R., "Transient Emissions from No. 2 Diesel and Biodiesel Blends in a DDC Series 60 Engine," SAE Technical Paper 961166, 1996, doi:10.4271/961166. Liotta, F. and Montalvo, D., "The Effect of Oxygenated Fuels on Emissions from a Modern Heavy-Duty Diesel Engine," SAE Technical Paper 932734, 1993, doi:10.4271/932734. Nikanjam, M., "Development of the First CARB Certified California Alternative Diesel Fuel," SAE Technical Paper 930728, 1993, doi:10.4271/930728 . Ullman, T., Spreen, K., and Mason, R., "Effects of Cetane Number, Cetane Improver, Aromatics, and Oxygenates on 1994 Heavy-Duty Diesel Engine Emissions," SAE Technical Paper 941020, 1994, doi:10.4271/941020. McDonald, J., Purcell, D., McClure, B., and Kittelson, D., "Emissions Characteristics of Soy Methyl Ester Fuels in an IDI Compression Ignition Engine," SAE Technical Paper 950400, 1995, doi:10.4271/950400. Iranmanesh, M., Subrahmanyam, JP. and Babu, MKJ., 2008, “Potential of Diethyl ether as supplementary fuel to improve combustion and emission characteristics of diesel engines”, SAE paper no. 2008-28-0044.



152


Real-Time Control of Automotive Engine Fuelled with Malaysian Palm Oil Biodiesel Azuwir Mohdnor1, M. Z. Abdulmuin1, A. H. Adom2 Abstract – With the rapid decline in energy resources, the increasing environmental concerns, and the high demand of energy consumption, many countries have been looking for alternative energy to substitute fossil fuel. One of the alternatives is to use bio-fuel as renewable energy. This paper describes the real-time control of automotive engine fuelled with palm oil biodiesel (Palm Methyl Esters). A self-tuning control algorithm based on pole assignment method is presented and an on-line model parameters estimation strategy based on the recursive least squares method is developed. Assuming a discrete time form for the system model, an Autoregressive eXogenous (ARX) model structure was selected in this work. The estimation strategy recursively updates the system dynamics of the engine in order for the self-tuning controller to control the engine speed for the best possible transient and steady state response at the speed range from 1800 rpm to 2300 rpm. Real-time results of the on-line parameter estimation and self-tuning speed controller implemented for automotive engine fuelled with palm biodiesel were presented. Finally, the performance of the speed controller error was examined. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Engine Speed Controller, PID, Self-Tuning, Palm Biodiesel

Nomenclature -1

A(z ) ARX ASTM B(z-1) CPO CPS DAQ DOHC ECU F(z-1)

J Kp Ki Kd P (t )

Output polynomial coefficients Autoregressive exogenous American Society for Testing and Materials Input polynomial coefficients Crude palm oil Crude palm stearin Data acquisition Double overhead camshaft Engine control unit Controller denominator polynomial coefficients Controller numerator polynomial coefficients Sum of squares of errors Proportional gain Integral gain Derivative gain Covariance matrix

PID PRBS RPM RLS SISO T TC Ts a1 ,a2 ,b1 g0,g1,g2

Proportional integral derivative Pseudo random binary sequence Revolution per minute Recursive least squares Single-input-single-output Tailoring polynomial Output response time constant Sampling time Plant parameters Controller coefficients

G(z-1)

eˆ ( t )

Modelling or fitting error

ε ( t + 1)

Estimated modelling error at time t+1

r(t) ∆t u(t) y(t) x (t ) ˆ (t ) θ

Reference speed Discrete time interval Discrete input signal Discrete output signal Measured input and output variables regression vector Estimated parameters

θ (t)

Vector of unknown parameters

I.

Introduction

Since the invention of the internal combustion engine, bio-fuel has been used but does not gain wide acceptance because of the availability and low cost of petroleum diesel. In fact, Rudolf Diesel demonstrated the diesel engine in 1900 running on ground nut oil. [1] With the uncertainty of the of the petroleum availability and prices, bio-diesel has become the most promising sources for alternative energy. However, in order to realize the efficiency and reliability of this alternative energy in vehicle usage, the engine control strategy must be improved accordingly. In automotive applications there are three main areas where control systems play a vital role: pollution abatement, fuel consumption, and safety. In addition, customers demand performance and efficiency must be delivered at low cost and high reliability. These stringent


153


Azuwir Mohdnor, M. Z. Abdulmuin, A. H. Adom

specified control objective. They present simulations of several gain parameter models primarily in application to bio-inspired sensors. The purpose of this paper is to develop and implement real-time self-tuning speed controller for automotive engine fuelled with palm oil biodiesel (Palm Methyl Esters) within speed range of 1800 rpm to 2400 rpm. A mathematical model that represent the dynamics of the engine under study will be derived based on Auto-Regressive with Exogenous input (ARX) model structure using a black-box modeling technique. On-line parameter estimation strategy, using the recursive least squares technique, is presented together with the self-tuning speed controller based on pole assignment method. Finally, the performance of the controller implemented in real-time will be presented.

requirements coupled with the energy crisis faced by the world today have become the push factor that motivates engine researchers to keep searching to improve engine performance. Control engineers and engine researchers can contribute to this critical demand by continuously improving the engine control strategy embedded into the engine control unit (ECU) or also known as the engine management system [2]-[11]. The application of self-tuning controller in speed control has gained a lot of attention of many researchers. Yaacob and Mohamed [12],[13]study the application of self-tuning controller for induction motor based on PI method and pole assignment method. Both methods are tested in no-load and with load condition. Abdulmuin [14] in his research work has thoroughly developed, simulated, implemented and evaluated the application of different self-tuning controllers for a double effect evaporator. Three self-tuning strategies based on minimum variance, generalized minimum variance and pole assignment are considered and all algorithms produced satisfactory dynamic output response. Karacan et. al. [15] compare both experimental and theoretical works on the application of pole placement self-tuning control for a packed-distillation column in a pilot plant. They found that pole placement self-tuning control mechanism was controlling overhead temperature very well under the load effects. Burnham et. al. [16] highlighted in their studies the difference demands in the implementation of self tuning controllers to two classes of industrial systems: speed control of a gas engine; and control of a high temperature heating plant. It is found that a great deal of care needs to be taken when considering the parameter estimation scheme to be used in developing self tuning control scheme for gas engine speed. In the case of the high temperature heating plant, the enhanced version of the control law design scheme should be given more attention. Kanso [17] reports the control of engine speed subjected to unknown and time varying external load disturbances. The adaptive self-tuning regulator based on Lyapunov stability conditions and the estimation algorithm adopted based on recursive maximum likelihood method yield good performance when following a desired and steady state model response from a series of computer simulation results. Mohamed and Koivo [18] proposed a new procedure in controlling the speed of a diesel engine using genetic algorithm selftuning PID controller. By using indirect estimation for the dead time and recursive least squares for parameter estimation, an explicit estimate of the plant parameters and dead time is obtained. From the simulation studies, the proposed controller controls the system very efficiently even if the system has time delay or load variations. In addition, a research using active method to reduce disc brake noise and vibration by integrating the PID-AFC (Active Force Control) scheme is found very effective in suppressing the vibration and hence the noise [19]. Recently, Behn and Steigenberger [20] design universal adaptive controllers, which achieve a pre-

II.

System Modeling

For the self-tuning controller design technique proposed in this paper, the system model is represented in the form of a difference equation in which the output y ( t ) is written as a linear function of past values of itself and the actuation signal u ( t ) . Using the unit backward shift operator defined by z −i x ( t ) = x ( t − i ) the model can be expressed in the discrete time transfer function form of: y (t ) =

II.1.

( ) u (t ) A( z )

B z −1 −1

(1)

Model Structure

In this work, a Single-Input-Single-Output (SISO) ARX mathematical model structure was selected to represent the diesel engine. This type of model is used because of its simplicity and speed for real time operation. The model can be written in the form: Ay ( t ) = Bu ( t − 1) + eˆ ( t )

(2)

where u(t) is the discrete input signal, y(t) is the discrete output signal, eˆ ( t ) is the corresponding modeling or fitting error and:

A = 1 + a1 z −1 + ............ + ana z − na B = b0 + b1 z −1 + .......... + bnb z − nb

(3)

The polynomial coefficients of Eq.(2) are treated as the parameters to be determined by estimation. Thus, it is convenient to write Eq.(2) in the form of: y ( t ) = xT ( t ) θ + eˆ ( t )


(4)


154


ˆ ( t ) ca be The RLS algorithm for updating θ

where θ is the vector of unknown parameters: θT = ⎡ − a1 ,....., − an ,b0 ,........,bn ,c1 ,...,cn ⎤ a b c⎦ ⎣

summarized as follows [16]: At time step ( t + 1 ):

(5)

and XT ( t ) is a regression vector consist of measured

(i)

Form x ( t + 1) using the new data

input and output variables:

(ii)

Form ε ( t + 1) using:

⎡ y ( t − 1) ,...., y ( t − na ) ,u ( t − 1) ,...., ⎤ ⎢ ⎥ x ( t ) = ⎢u ( t − nb − 1) ,e ( t − 1) ,e ( t − 2 ) ,..........,⎥ ⎢ ⎥ ⎣ e ( t − nc ) ⎦

Λ

ε ( t + 1) = y ( t + 1) − xT ( t + 1) θ ( t )

(6)

Form P ( t + 1) using

For the ARX model, the assumption is that nc = 0 and thus the noise terms can be discarded in the regression vector. In this work, we make an assumption that the system has the special structure of [27]: y (t ) =

b1 z −1

1 + a1 z −1 + a2 z −2

⎡ x ( t + 1) xT ( t + 1) P ( t ) ⎤ P ( t + 1) = P ( t ) ⎢ I m − ⎥ 1 + xT ( t + 1) P ( t ) x ( t + 1) ⎦⎥ ⎣⎢

(iv)

u (t )

(7) ˆ ( t + 1) = θ ˆ ( t ) + P ( t + 1) x ( t + 1) ε ( t + 1) θ

This special structure is selected in order to synthesize exactly the PID controller coefficients due to the stringent requirement on the system under study. [16] II.2.

(v) Wait for the next time step to elapse and loop back to step (i). This algorithm is used to estimate the plant parameters a1 ,a2 ,b1 in Eq. (6).

Parameter Estimation

There are three unknown parameters that need to be estimated in this study based on Eq. (7). In order to be useful in self-tuning control, the model parameters should be estimated iteratively. This will allow the estimated model of the diesel engine to be updated at each sample interval. Thus, the recursive least squares (RLS) technique is used for parameters estimation. Figure 1 shows the iterative process of the parameter estimation. ˆ (t ) The aim of this technique is to select a value of θ

III. Controller Design The engine speed controller designed in this study is based on SISO model developed in the previous section. The engine speed is controlled by adjusting the fuel flow rate through the auto throttle. The strategy is to vary the fuel flow rate to the engine whenever there is a change in the reference speed or when there is a disturbance in the system. The controller output is the voltage that manipulates the throttle opening which controlled the fuel flow rate that provide the engine with the necessary power. The controller output signal is calculated based on the feedback error between reference speed and actual engine speed. The block diagram of the engine speed controller is shown in Fig. 2.

so that the modeling error is minimized according to the sum of squares of errors: J=

N

∑ eˆ 2 ( t ) = ˆeTˆe

(8)

t =1

u (t )

System

ˆ (t ) : Update θ

θ +

r(t)

Model

θ(t − 1)

Diesel Engine

Speed controller

y (t ) e(t)

G( z F (z

ε (t ) Update mechanism

−1 −1

) )

u(t)

z −d B( z −1) A( z −1)

y(t)

Fig. 2. Engine speed controller block diagram

The main objective of the feedback controller in this study is to modify the dynamic response of the system to

Fig. 1. Iterative process of recursive parameter estimation



155


ensure that the engine speed track the changes in the reference speed in an acceptable response time. In addition, speed controller must make sure that in the steady state condition, r(t) is constant, the system error is minimized. In general, the controller transfer function can be represented as: u (t )

and:

ζ = damping factor ωn = natural frequency Ts = sampling time

III.2. PID Controller

G ( z −1 )

⎫ = ⎪ − 1 e (t ) F ( z ) ⎪ ⎪⎪ with error e ( t ) = r ( t ) − y ( t ) ⎬ ⎪ −1 ) −1 − nf ( F z = 1 + f1 z + ..... + f nf z ⎪ −1 ) −1 − ng ⎪ ( G z = g0 + g1 z + ..... + g ng z ⎪⎭

The three-term PID controller structure adopted in this study can be written in the following discrete time form: (9)

u (t ) e (t )

=

G ( z −1 )

F ( z −1 )

=

g0 + g1 z −1 + g 2 z −2

(17)

1 − z −1

or: The open loop transfer function of the system block diagram with the controller is: y (t ) =

z − d B ( z −1 ) G ( z −1 ) A( z

−1

)F (z ) −1

e (t )

u (t ) =

g0 + g1 z −1 + g 2 z −2 1 − z −1

r (t )

==

z − d B ( z −1 ) G ( z − 1 )

A ( z −1 ) F ( z −1 ) + B ( z −1 ) G ( z −1 )

K p = − g1 − 2 g 2 ⎫ ⎪ Ki = g0 + g1 + g 2 ⎬ ⎪ Kd = g2 ⎭

(11)

u ( t ) = u ( t − 1) + g 0 e ( t ) + g1e ( t − 1) + g 2 e ( t − 2 ) (20)

Combining Eq. (7) and (18):

(12)

y (t ) =

III.1. Pole Assignment Method The transient response or the dynamic of the system is determined by the characteristic equation of the system closed loop transfer function in Eq. (12). In pole assignment method, the characteristic equation is forced to follow the desired dynamic characteristic determine by the following tailoring polynomial equation: T = T ( z −1 ) = 1 + t1 z −1 + t2 z −2 + .... + tnt z − nt

(

t2 = exp ( −ςωnTs )

(

)( (

(21)

) )

⎡ 1− z +⎤ 1 + a1 z + a2 z ⎢ ⎥ ⎢ + b z −1 g + g z − 1 + g z − 2 ⎥ 0 1 2 ⎥⎦ ⎣⎢ 1

−1

−1

−1

+ a2 z

−1

+ g2 z

1

−1

1

−2

0

1

−2

)+ ) = 1+ t z

−2

1

−1

+ t2 z

−2

(22)

Equating the like power of z-1 we get:

(14)

)

)

−1

(1 − z )(1 + a z +b z ( g + g z

(13)

t1 + (1 − a1 ) ⎫ ⎪ b1 ⎪ t2 + ( a1 − a2 ) ⎪⎪ g1 = ⎬ b1 ⎪ ⎪ a2 g2 = ⎪ b1 ⎪⎭

where: t1 = −2 exp ( −ςωnTs ) cos ωnTs 1 − ς 2

(

b1 z −1 g 0 + g1 z −1 + g 2 z −2 r ( t )

The controller coefficients are determined by equating the denominator of the closed-loop equation above with the desired closed loop T polynomial Eq. (14):

For second order transient response T is defined as: T = 1 + t1 z −1 + t2 z −2

(19)

thus:

From Eq. (11), the characteristic equation of the closed-loop transfer function is then: A ( z −1 ) F ( z −1 ) + z − d B ( z −1 ) G ( z −1 ) = 0

(18)

where the proportional, integral and derivative gain are related to the controller coefficients as follows:

(10)

and the closed-loop transfer function is: y (t )

⎡⎣ r ( t ) − y ( t ) ⎤⎦

g0 =

(15) (16)


(23)


156


is injected to excite the actuator. Two sets of real-time data were collected from the palm biodiesel engine testbed. Each set consists of 720 data. The first set was used for model parameters estimation and identification activities while the second set of data used for model validation. At this stage, the model parameters identification and validation are done off-line. The mathematical model derived using the RLS technique is then used as the initial value of the model parameters in designing controller program. In this particular work, the controller design requirements have been set as follows: • The system overshoot should be less than ten percent (10%) of the final value. • The system settling time should be less than ten second. From this requirement, the value of t1 and t2 of the tailoring polynomial were calculated using Eq. (14) and (15). Figure 4 shows the block diagram of the self-tuning controller structure adopted in this work.

The controller design requirements, the system settling time and the maximum percent overshoot, together with the system sampling time are then used to calculate coefficient of the tailoring polynomial t1 and t2. For a second order system, the maximum percent overshoot is related to the damping ratio ζ as: Mp =e

(

)

− ζ / 1−ζ 2 π

(24)

and settling time ts: ts =

4

ζωn

( 2% criterion )

or: ts =

IV.

3

ζωn

( 5% criterion )

Experimental Setup

The automotive diesel engine under study is a 2000cc, Direct Injection, DOHC Mitsubishi Diesel Engine. The engine is mounted on a test bed with an eddy current dynamometer braking unit. The first step in implementing the on-line identification and control is to set-up an experiment for input and output data collection from the engine test-bed. A computer is interfaced to the actuator (auto throttle) and the speed sensor via an Agilent U2351A Multifunction DAQ. The input signal is transmitted through an auto throttle servo actuator that provides an automatic throttle control by receiving a signal of 0 to 10 V, corresponding to 0 to 100% throttle opening control. The engine speed was transmitted by optical encoder also with a signal range of 0 to 10 V. Figure 3 shows the engine test-bed unit used for the implementation of real-time system identification and self-tuning control in this study. A Matlab program is written to generate PRBS signal that persistently excite the system. The signals are generated by a computer program using the exclusive-OR and modulo-2 addition. As a rule of thumb in designing the PRBS signal, the clock period ∆t is normally chosen to be approximately in the range of a fifth to a half of the output response time constant (Tc) i.e.: ∆t = ( 0.2 to 0.5 ) ⋅ Tc

Fig. 3. Automotive Diesel Engine Test Bed

Control Synthesis

r(t)

e(t)

Speed controller

Recursive Least Squares

u(t)

Diesel Engine

y(t)

Fig. 4. Self-tuning controller structure

The self-tuning controller algorithm can be summarized as follows, at each sample time t : 1. Read the system output y(t) 2. Use the data capture in 1 in the RLS algorithm to update the parameter estimate in the model Eq. (6) ˆ by using 3. Synthesize the controller polynomials Fˆ ,G ˆ ˆ in the identity the current estimate A,B

(25)

The engine which is fuelled with palm oil methyl esters is first run at a steady-state speed value before the input-output data were collected. In this experiment, a PRBS signal with maximum length sequence of 31 with time period of 1 second and sampling time of 0.17 second was used during data collection for engine modeling at the medium speed range of around 2100 rpm. Starting from steady-state conditions, a sequence of input signals

AF + z − d BG = T 4. Generate the output control u ( t ) using

ˆ Fˆ ,G

calculated in 3.



157


5. Wait for sample time interval to elapse and return to step 1. In the present work, many dynamic and control experiments were done in order to determine the most suitable setting of the various parameters of the selftuning controller. Other than the mechanical delay and the transport delay of the system, the most appropriate value of lamda, the forgetting factor, and the effect of covariance resetting are also considered in this work in order to maximize the controller performance.

Controller Signal 1.5

Voltage

1

0.5

0

V.


0

50

100

150

200 Time (sec)

250

300

350

400

Fig. 6. Control signal send to the engine throttle by the self-tuning controller

From all the experiments done in the process of developing the speed controller for diesel engine fuelled with palm oil bio-diesel, the following results significantly summarize the real-time self-tuning controller tested and verified on an automotive engine test-bed.

Engine Speed Controller Error 600

400

200

Self-Tuning Controller Relates at Various Speeds

(RPM)

V.1.

The first test conducted to evaluate the capability of the controller developed in this work is to let the selftuning controller algorithm controlling the engine rpm at various levels of desired speed. The engine was first run at a steady-state value around 2000rpm. Then the desired speed is changed to 1800rpm for 100 sec before speed up to 2200rpm for another 100 sec. Finally, the desired engine speed is brought back to 2000rpm. Looking at Figure 5, it is clearly shown that the speed controller manages to control the engine speed according to the desired value. This result validate that the self-tuning controller has a capability to regulate the engine speed at various speed levels. Figure 6 shows the value of control signal calculated based on the self-tuning controller algorithm implement in the computer program. The value of controller signal calculated during each iteration is then sending to engine throttle to manipulate the fuel flow into the engine. Analysis of the engine speed error, the difference between the desired speed and the actual engine speed, is presented in Figure 7.

-200

-400

-600

V.2.

Speed (RPM)

2200

2000

1800

1600

1400

100

150

200 Time (sec)

250

300

350

100

150

200 Time (sec)

250

300

350

400

Self-tuning Controller Under Load Condition

The second test conducted to further evaluate and validate the performance of the self-tuning controller for the bio-diesel engine is to introduce a disturbance to the system. A torque of 12Nm was applied by breaking the dynamometer in order to simulate a disturbance to the system. The system was running at a steady-state value of around 2200rpm before a torque of 12Nm is applied from the dynamometer to the system. Figure 8 shows that the bio-diesel engine experienced a sudden speed drop because of this disturbance and experience a speed drop until nearly 1800 rpm. However, the self-tuning speed controller developed in this work manages to bring the engine speed back up to 2200 rpm in within 10 seconds. It is clear from this figure that the controller has a capability to reject the load disturbance applied to the bio-diesel engine.

2400

50

50

It can be observed that throughout the test run conducted, the error is consistently maintain within acceptable values except there are three big error values, error spikes, during the change of desired speed or reference value. However, this is expected since the speed change is done in a manner of step-up or stepdown. The average speed error calculated after each steady-state stage is within +/-30rpm.

Engine Speed

0

0

Fig. 7. Error between the reference speed and the actual engine speed

2600

1200

0

400

Fig. 5. Dynamic response of biodiesel engine under various reference speed



158


Analysis of the engine speed error with the introduction of the disturbance is shown in Figure 10. The sudden drop of the engine speed when the disturbance is applied has resulted in a big engine speed error. However, the self-tuning controller implemented in this work managed to reduce the speed error and regulate the bio-diesel engine back to the desired speed.

VI.

Conclusion

In this paper, self-tuning control strategy based on pole assignment method has been considered for an automotive engine fuelled with palm oil methyl esters. A self-tuning speed controller based on SISO linear mathematical model has been developed, tested and applied in real-time to the speed control of the bio-diesel engine. The algorithm is applied to an engine speed tracking control problem and load disturbances problem. The results presented in this work show that the controller has successfully followed and regulate at the desired speed. The developed self-tuning controller shows it effectiveness in controlling automotive engine speed fuelled with palm oil methyl esters without engine modification.

Fig. 8. Dynamic response of the palm biodiesel engine under torque disturbances

Figure 9 shows the control signal sent to the engine throttle before and after the disturbance is applied to the engine. A sudden drop of the engine speed as a result of the disturbance was immediately responded by the controller by stepping up the control signal value in order to minimize the speed error. This proves that the controller algorithm has successfully calculated the appropriate control signal value in order to minimize the engine speed error. Controller Signal 2

Acknowledgements

1.8

This work was conducted in the Thermodynamic and Process Control Laboratories of School of Manufacturing, University Malaysia Perlis. The authors would like to thank the individuals who were involved in making this work possible.

1.6

Voltage (RPM)

1.4 1.2 1 0.8 0.6

References

0.4

[1]

0.2 0

0

20

40

60

80

100 120 Time (sec)

140

160

180

200

[2] [3]

Fig. 9. Control signal send to the engine throttle by the self-tuning controller to reject the disturbance [4] Engine Speed Controller Error 600

[5] 400

[6]

(RPM)

200

0

[7]

-200

-400

-600

[8] 0

20

40

60

80

100 120 Time (sec)

140

160

180

200

[9] Fig. 10. Error between the reference speed and the actual engine speed


A. Bijalwan, C.M.,Sharma, V.K., Kedival., Bio-diesel revolution, Science Reporter, :14-17January 2006. L. Guzzella and A. Amstutz, Control of Diesel Engines, IEEE Control Systems, pp. 53-71, October 1998. D.W. Memering, A comparison of Control Techniques Applied tp Diesel Engine Idle Speed Control, Proc. 16th Annual Fall Tech. Conf. ASME IC Engine Division, pp. 57-66, Lafayette, IN, , 1994. Wen Tan, Tinpui Leomg and Qili Tu, H∞ Optimal Control for Singularly Perturbed Systems, Automotica, Vol. 34, (No. 2): 255260, 1998. J. J. Moskwa and J. K. Hedrick, Nonlinear Algorithms for Automotive Engine Control, IEEE Control System Magazine, pp: 88-93, April 1990. M. Kao and J. J. Moskwa, Nonlinear Diesel Engine Control and Cylinder Pressure Observation, ASME Winter Annual Meeting, Advanced Automotive Technologies, DSC Vol. 52 pp. 187-198, November 1993. E. M. Omran and M. Z. Abdulmuin, Modeling and Robust Idle Speed Control for a Spark Ignition Engine, PhD Thesis, Universiti Malaya, 2000. G. Gnanam, S. R. Habibi, R.T. Burton, and M.T. Sulatisky, Neural Network Control of Air-to-Fuel Ratio in a Bi-Fuel Engine, IEEE Transaction on Systems, Man, and Cybernetics –Part C: Applications and Reviews, (Vol. 36): 656-667, September 2006. O. D. Jesus, A. Pukrittayakamee, and M.T. Hagan, A Comparison of Neural Network Control Algorithms, IEEE, pp. 521-526, 2001.


159


[10] X. Dovifaaz, M. Ouladsine, A.Rachid, and G. Bloach, Neural Modeling and Control of a Diesel Engine with Pollution Constraints, Proceeding of the America Control Conference Anchorage, AK, pp. 2008-2013, May 8-10, 2002. [11] J. A. Tennant, H.S. Rao, and J. D. Powell, Engine Characterization and Optional Control, IEEE, pp: 114-119, 1979. [12] S. Yaacob, and Faisal A. Mohamed, Real-time self-tuning controller for induction motor based on pole assignment method, The Thirty seventh SICE Annual Conference (SICE 98), Japan, 29-31st July 1998 [13] S. Yaacob, and Faisal A. Mohamed, Real-time self-tuning controller for induction motor based on PI method, The Thirty Eight SICE Annual Conference (SICE 99), Japan, 28-30th July 1999 [14] Abdulmuin, M.Z. Modeling and Self-Tuning Control of A Double –Effect Evaporator, PhD. Thesis. University of Malaya, 1988. [15] S. Karacan, H. Hapoglu, Y. Cabbar, and M. Alpbaz, Pole placement self-tuning control for packed distillation column, Chemical Engineering and Processing Vol 36, 1997, pp.309-315. [16] Burnham, K.J., Disdell, K.J., James, D.J.G., and Smith, C.A., Developments in industrial applications of self-tuning control, Control Engineering Practice, Vol (3), No.9.1995, pp.1265-1276 [17] W. Kanso, Self-tuning adaptive control of engine speed in the presence of random disturbances, Journal of the Franklin Institute, Vol 331B, (No. 3):.313-325.1994, [18] Mohamed, F.A.; Koivo, H.N. , Diesel engine systems with genetic algorithm self tuning PID controller, , 2005 International Conference on Future Power Systems, pp.4 -5, Amsterdan, 18 Nov. 2005 [19] Hashemi-Dehkordi,S.M., Mailah,M., and Abu, B.A.R., Implementation of active force control to disc brake for noise-free performance, International Review of Mechanical Engineering (IREME), Volume 3, Issue 4, July 2009, Pages 481-488. [20] Behn,C., and Steigenberger, J., Experiments in adaptive control of uncertain mechanical systems, International Review of Mechanical Engineering (IREME), Volume 4, Issue 7, November 2010, Pages 886-898.

Authors’ information 1,2

School of Manufacturing Engineering, Universiti Malaysia Perlis, Kampus Ulu Pauh, 02600 Arau, Perlis 3 School of Mechatronic Engineering, Universiti Malaysia Perlis, Kampus Ulu Pauh, 02600 Arau, Perlis

Azuwir Mohdnor received his BSc degree in Electrical Engineering from University of Wisconsin-Madison,USA and Master’s degree in Information Technology from Universiti Utara Malaysia, Kedah, Malaysia in 1992 and 2002, respectively. He is currently pursuing his Ph.D. degree at Universiti Malaysia Perlis, Malaysia. His main research areas are system identification, control theory, digital control systems and self-tuning control Mohd Zaki Abdulmuin was born in Malaysia in 1950. He obtained PhD in modeling and control from University of Malaya in 1989, and worked for 25 years in the same university. He has published more than 70 research papers. Currently he is on contract with University Malaysia Perlis. Research interests include Physical and black-box modeling and adaptive

control. Abdul Hamid Adom received his PhD in Artificial Intelligence in 2001 and MSc in Modern Control & Instrumentation Systems in 1996, and is currently an Associate Professor at Universiti Malaysia Perlis. Among his research interests are Artificial Human Sensing and Robotics as well as Artificial Intelligence.



160


Reduction of NOx Emission from Diesel Engine Using Urea Injection with SCR Technique with Different Catalyst Connected in Series K. Chithamparam Asary1, N. V. Mahalakshmi2, K. Jayachandran3

Abstract – The reduction of oxides of nitrogen emission (NOx) from compression ignition engines is an important objective due to environmental and human health concerns particularly in areas that are ozone non-attainment zones. In addition, NOx emissions from engines and vehicles are legislated and regulated around the world, with the emissions regulations continuously tightening in the upcoming years. Thus, many methods that can reduce these emissions from compression ignition engines and vehicles are being considered as options. The scope of this paper is to explore strategies to reduce NOx emissions from compression ignition engines. This paper uses the injection of aqueous Solution of urea in the Exhaust stream for the reduction of NOx. The test was carried out in a single cylinder light duty water cooled DI Diesel engine which is coupled with an eddy current dynamometer. Initially, the optimum flow rate and the concentration of urea solution for the maximum reduction of NOx were determined for different loading conditions. In the initial experimental work, there were no appreciable changes in CO, CO2 and O2 in the engine exhaust, except some reduction of HC and exhaust gas temperature and increase in smoke density due to injection of urea in the engine exhaust. It had been concluded that 29.3% of NOx reduction was achieved with the flow rate of 0.75 lt/hr with 30% concentration. Also a selective catalytic reduction technique with vanadium and titanium dioxide catalyst connected in series was introduced which showed 89.73% of NOx reduction from the engine exhaust. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: NOx Reduction, Selective Catalytic Reduction Technique, Vanadium And Titanium Dioxide Catalyst, Aqueous Solution Of Urea

I.

2. NOx reduction accomplish through a modified combustion process 3. NOx reduction through treating the exhaust gas from the engine (after treatment system) Of these, the third method, after treatment of exhaust gas system will be less expensive and simple compare to the first two methods. Hence this investigation is made on the NOx reduction through treatment of exhaust gas from the diesel engine, with the injection of urea solution using Selective Catalytic Reduction (SCR) technique with different catalysts and with different arrangements of SCR catalysts.

Introduction

Diesel engines are widely used in many areas like Automobiles, locomotives, marine engines, power generators etc., due to its high power output and thermal efficiency. Even though the diesel engines give more benefits, the human discomforts caused by the pollutant emissions of these engines have to be considered. The major pollutant emissions of the diesel engines are particulate matters, Hydro carbon, smoke and the oxides of Nitrogen (NOx). Out of these pollutant emissions, the oxides of Nitrogen are considered as the most harmful pollutant to the human health, plants and environments. In addition, NOx emission from engine and vehicles are legislated and regulated around the world, with the emission regulation continuously tightening in the upcoming years. Hence the diesel engine industries are now under high pressure in finding various methods to minimize the emission of these oxides of Nitrogen (NOx). There are various methods available for reducing NOx which are grouped under these three following categories. 1. NOx reduction accomplished through a change in fuel formulation

II.

Literature Survey

It has been seen from literature that lot of investigations have been conducted for the reduction of NOx emission from the Diesel Engine. However, some selected papers, which are closely relevant to the topic of research, are presented in the following paragraph. Wolfgana Held et al (1990) [1] have used copperexchanged zeolite catalysts to convert nitrogen oxides over a much wider range of fuel-air ratios than nobelmetal catalysts and achieved only 65% of NOx reduction.


161


K. Chithamparam Asary, N.V. Mahalakshmi, K. Jayachandran

The urea dosage was not analyzed, necessary construction outlay of engine setup was not given also the secondary reaction was not explained. David Monroe et al (1993) have studied many different metals and many different zeolite structures have been investigated in the last several years but none has been found which is superior of the Cu/ZSM-5 system. The most promising of the alternatives is a cobalt system, but this needs higher temperature to operate than the cu-based system. Willand et al (1998) [2] have used selective noncatalytic reduction process for the reduction of NOx using urea injection in a heavy duty diesel engine. They injected urea solution directly into the combustion chamber and achieved 65% of NOx reduction at full load and with increased exhaust gas temperature. They achieved only 20% of NOx reduction below a reaction temperature of 600oC. Koebel et al (2000) [3] have conducted Experiments with Urea Injection with catalyst in a heavy stationary engine. Tried to reduce the catalytic volume without affecting the NOx reduction over a wide range of operating temperature. They concluded that the much shorter residence time of the exhaust gas in the catalyst would lead to secondary emission of Ammoia. Ioannis Gekas et al (2002) [4] have used a novel urea injection system, which is based on a mass produced digital pump, that is combined with an electronic control unit specially developed for controlling the urea-SCR process onboard of the vehicles. They got NOx conversion above 80% with ammonia slip below 10 PPM. With the novel 300 cpsi Urea-SCR catalysts the volume can be reduced to 2/3 compared to the 130 cpsi catalysts. Rinie van Helder et al (2002) [5] have used vanadium catalyst with urea injection for the reduction of NOx in Heavy duty track Diesel Engine, and achieved 62% of NOx reduction. They also shown that after 60,000km operation NOx conversion and Ammonia slip were still same. Schar et al (2003) [6] have implemented the system to Heavy duty mobile Engine using single SCR catalytic converter with titanium dioxide and 82% NOx reduction was achieved. They analyzed the basic amount of urea to be injected. Zhihua Wang et al (2004) [7] have used SCR technique with injection of ammonia and ammonia N agents (ammonia chloride, ammonia carbonate, ammonia sulphate) and esting was carried out in a heavy duty stationary diesel Engine. N-agents were preferred over Urea and Ammonia since it is comparatively cheaper. 60% of NOx reduction was achieved with ammonia, 81.3% of NOx reduction with N agents at comparatively Low-cost. From the literature survey, it is noted that even though lot of experimental and theoretical investigations have been carried out on the control of NOx emission from Diesel Engine with various catalysts in the SCR

using urea injection, no attempt has been made, to study the effect of NOx reduction with urea injection in the exhaust gas by combining different catalysts in the SCR system in series arrangement. Therefore, this paper proposed, to use SCR technique with vanadium and titanium di-oxide catalysts in series for the reduction of NOx by the injection of urea solution.

III. Experimental Setup Figure 1 shows line diagram of the experimental setup with urea injection without introducing SCR. The experimental setup consists of a 5.2kw, single cylinder, water cooled, vertical, kirloskar, constant speed diesel engine coupled with an eddy current dynamometer. The exhaust gas after treatment system was fitted on the exhaust pipe nearer to the engine outlet. This is system consists of a storage tank to store the urea solution, a pump to maintain the pressure of urea solution and a three way valve to control the flow of urea solution to the injection needle to inject the urea solution and a line to return the excess solution back to the storage tank. The exhaust gas pipe is also fitted with a BOSCH Smoke meter to measure the smoke density and the AVL 5 gas analyzer to measure HC, CO, CO2, O2 and NOx.

Fig. 1. Line diagram of the experimental setup

IV.

Experimental Procedure

Before starting the engine, the urea solution with different concentration varying from 10% to 40% in steps of 10% was prepared. By adjusting the 3 way control valve and by selecting the needle, the flow rate of urea solution was fixed as 0.25, 0.5, 0.75 and 1.0 lt/hr and maintained constant for a set of readings. The engine was started and allowed to run at rated speed for some period of time for warm up. Then, the engine was loaded gradually from 20% to 100% of full load in steps of 20%. For each and every load without introducing the urea injection (Base Diesel Operation),



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the fuel flow rate, the emission such as HC, CO, CO2, O2, NOx, smoke density and excess air factor were recorded. Then the experiment was repeated with the injection of urea solution with the concentration of 10% and into the varying flow rate form 0.25 lt/hr to 1 lt/hr in steps of 0.25lt/hr and the readings were recorded. It was observed that at the flow rate of 0.75lt/hr maximum reduction of NOx was achieved. The same experiment was repeated with the injection of 20%, 30% and 40% concentration urea solution by varying the flow rate from 0.25 lt/hr to 1 lt/hr in steps of 0.25lt/hr and the readings were recorded.

V.

respectively. The variation of NOx emission versus brake power when SCR with vanadium catalyst and SCR with titanium di-oxide catalyst connected in series with the injection of constant flow rate and with the different concentration of urea solution are shown in Figure 8. The maximum NOx reduction works out to be 89.73%.

Combination Of SCR With Vanadium Catalyst And SCR With Titanium DiOxide Catalyst- Connected In Series

The Figure 2 shows the SCR I with vanadium catalyst and the SCR II with titanium di-oxide catalyst fitted in series in exhaust pipe line after urea injector. The tests were repeated for different urea concentrations of 10% to 40% by weight with constant flow rate of 0.75 lt/hr. The emissions of NOx and other emissions were measured for various loads.

Fig. 3. Variation of NOx versus brake power with 10% urea concentration

Fig. 2. Schematic of experimental setup with urea injection with introducing SCR –I and II Connected in series

VI.


Results And Discussion

Figure 3 to 6 shows the variation of NOx versus Brake power for different urea flow rates of 0.25, 0.5, 0.75 and 1.0lt/hr for different concentration of 10, 20, 30 and 40% respectively. It is seen from the graphs, for all concentrations, the NOx emissions showed lowest value at 0.75lt/hr and remained almost same for 1.0lt/hr flow rate. Figure 7 shows the variation of NOx versus Brake power for the urea injection with the flow rate of 0.75lt/hr with the different concentrations of urea. It is seen from the graph that the NOx level was minimum for 30% concentration at full load. There is no appreciable change in the NOx emission beyond 30% concentration. Hence the optimum value of the flow rate and concentration was taken as 0.75lt/hr and 30%




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with these optimum flow rate and concentration. Combined with the injection of urea solution and the SCR technique with two catalyst connected in series, a maximum NOx reduction of 89.73% was recorded.

References [1]

[2]

[3]

[4]


[5]

[6]

[7]

[8]

Fig. 7. Variation of NOx Versus Brake power with 0.75 lt/hr flow rate with different concentrations

[9]

[10]

[11]

[12]

[13]

[14] Fig. 8. Variation of NOx Versus Brake power with SCR Vanadium and SCR Titanium di-oxide catalyst connected in series [15]

VII.

Conclusion

The following conclusions were drawn in this paper. The optimum flow rate for urea solution was found to be 0.75lt/hr with a concentration of 30% for the maximum reduction of NOx. Without introduction SCR technique, the maximum reduction of NOx was recorded as 29.3%

[16]


Wolfgang Held, Axel Konig, Thomas Richter and Lothar Puppe, “Catalytic NOx Reduction in Net Oxidizing Exhaust Gas”, SAE Transactions 900496, 1990. Willard, J., Teigeler, M., Wirbeleit, F., Enderie, C. and Raab, A. “Selective Non-Catalytic NOx reduction in Diesel engine using Aqueous Urea” ,SAE Transaction 982651-1998. Koebel, M., Elsener, M. and Kleemann, M. “Urea SCR: a promising technique to reduce NOx emissions from automotive diesel engines”, Catalysis Today 59 (2000) 335–345. Loannis Gekas, Par Gabrielsson, Keld Johansen, Lars Nyengaard and Thomas Lund, “ Urea-SCR Catalyst System Selection for Fuel and PM Optimized Engines and a Demonstration of a Novel Urea Injection System”, SAE Transaction 2002-01-0289. Rinie van Helden, Marcel van Genderen, Marc van aken and Ruud Verbeek, Joseph A. Patchett, Jan Kruithof, Ted Straten and Claire Gerentet de Saluneaux, “Engine Dynomometer and Vehicle Performance of a Urea SCR-System for Heavy-Duty Truck Enginees. SAE Transaction 2002-01-0286. Schar, C.M., Onder, C.H., Geering, H.P. and Elsener, M. “Control of a Urea SCR Catalytic Converter System for a Mobile Heavy Duty Diesel Engine”, SAE Transactions 2003010776, 2003. V. Rambabu, V. J. J. Prasad, T. Subramanyam, B. Satyanarayana, Evaluation of Performance, Combustion Characteristics and Emissions of DI- Diesel Engine Fueled with Preheated Cotton Seed Methyl Ester, International Review of Mechanical Engineering (IREME), Vol. 4. n. 5, pp. 502-506, 2010. Amon, B. and Keefe, G. “On-Road Demonstration of NOx Emission Control for Heavy-Duty Diesel Trucks Using SINOx Urea SCR technology- Long Term Experience and Measurement Results”, SAE 2001-01-1931. A. Khelil, H. Naji*, L. Loukarfi, Numerical Study of Swirling Confined Non-premixed Flames with Determination of Pollutant Emissions, International Review of Mechanical Engineering (IREME), Vol. 1 n. 6, pp. 618 – 627, 2007. R. Saim, S. Abboudi, B. Benyoucef, A. Azzi, Computation of Turbulent Forced Convection in Heat Exchangers Equipped with the Transverse Baffles, International Review of Mechanical Engineering (IREME), Vol. 1 n. 5, pp. 588 – 594, 2007. Paul Zelenka, Klaus Ostgathe and Egbert Lox, “Reduction of Diesel Exhaust Emissions by using Oxidation Catalysts”, (1990), SAE 902111.. Herzog, P., Bürgler, L., Winklhofer, E., Zelenka, P. et al., "NOx Reduction Strategies for DI Diesel Engines," SAE Technical Paper 920470, 1992. Rinie Van Heldon, Ruud Verbeek, Frank Willems and Reinier van der Weile, “Optimization of Urea SCR deNOx Systems for HD Diesel Engines”, SAE Transactions 2004010154, 2004. N. Haribabu, G. RaviKiranSastry, V. J. J. Prasad, B. V. Appa Rao, Experimental Studies on a DI -CI Engine Run with Pongamia Methyl Ester Injection and Methanol Carburetion, International Review of Mechanical Engineering (IREME), Vol. 3. n. 4, pp. 489-493,2009. V. Rambabu, V. J. J. Prasad, T. Subramanyam, B. Satyanarayana, Evaluation of Performance, Combustion Characteristics and Emissions of DI- Diesel Engine Fueled with Preheated Cotton Seed Methyl Ester, International Review of Mechanical Engineering (IREME), Vol. 4. n. 5, pp. 502-506, 2010. [J.Sudhir Kumar, K.Venkata Subbaiah, P.V.V.Prasada Rao, Experimental Studies on a DI-CI Engine using Blends of Diesel Fuel with Plasti Diesel derived from Plastic Waste, International Review of Mechanical Engineering (IREME), Vol. 5 N. 3, pp. 540-547,2011.


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[17] Mohammad Nazri Mohd Jaafar, Ahmad Huzairi Hussain, Mohd Shaiful Ashrul Ishak, Catalytic Combustion System for use in Malaysia Small Gas Turbine: a Feasibility Study, International Review of Mechanical Engineering (IREME), Vol. 5 N. 1, pp. 100-105, 2011.


Research Scholar.

2 Professor / Department of Mechanical Engineering, Anna University, Chennai 25, India. 3

Former Director/ Research, Anna University, Chennai -25. K. Chithamparam Asary is a research scholar at Anna University, Chennai-25. Presently, he is working in Sri Ramanujar Engineering College, Kolapakkam, Vandalur, Chennai – 48 and has attended many conferences in national and international level. He also published number of research papers in his area of specialization. E-mail: [email protected]

N. V. Mahalakshmi is Professor in Department of Mechanical Engineering, Anna University, Chennai 25, India. K. Jayachandran was Former Director/ Research, Anna University, Chennai -25.



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Jordan Transport Energy Demand Forecasting: the Application of Time Series Technique Adnan Mukattash1, Ahmed Al-Ghandoor2, Ahmad M. Qamar3

Abstract – This paper illustrates the application of double exponential smoothing technique to forecast Jordan transport energy demand. The model has been developed using past Jordan transport energy demand during years 1985-2009. The results show that energy demand will increase by 5.23/yr during the forecasted period reaching 4,200 thousands ton oil equivalents (toe). It is expected that the results of this study will be helpful in developing highly applicable and productive planning for future transport energy policies. Copyright © 2012 Praise Worthy Prize S.r.l. - All rights reserved. Keywords: Energy, Transportation, Forecasting, Jordan

I.

The enormous increase in the number of operating vehicles has contributed to a significant increase in the local energy demand and an increasing amount of damage to the natural environment as a result of polluting emissions. Despite the fact that transportation sector plays an important role in determining the country's energy demand, in open literature, energy use in the Jordanian transportation has generally received little attention. This may be due to its complexity, large number of stockholders and lack of awareness and resources to conduct detailed studies for various elements of this sector. In this study, the double exponential smoothing technique will be used to forecast Jordan transport energy demand for the next two decades. It is expected that the results of this study will be helpful in developing highly applicable and productive planning for future transport energy policies.

Introduction

Transportation plays a pivotal role in modern daily life and is essential for sustaining economic development and high standards of living. Yet, among all other final energy consumption sectors, it remains the most challenging in achieving sustainability goals in terms of conservation, diversification, and emissions control. This is mainly due to the almost total dependence of most transportation means and technologies we have today on one form or another of petroleum products. Unfortunately, unlike other Arab neighboring countries, Jordan is a non-oil producing country with limited natural resources and minerals. As other developing Asian countries, it has a rapid population growth of about 2.2% [1]. The population and economic growth as well as development that Jordan experienced since its independence, in mid 1950s, implied a gradual shift of the population from rural to urban areas. Thus, urban population has increased from about 70%, in 1990, to 82%, in 2009, of total population, putting the kingdom among the most urbanized countries in Asia. A major structural phenomenon of urbanization is the increasing shift of large proportions of the population to modern centers with relatively high incomes, requiring higher rates of energy consumption to sustain the new life. In recent years, concern about energy consumption in Jordan has been growing, especially, in the transport sector which was probably affected the most by the economic and technological changes that the country has witnessed during the past three decades. For example, the number of road vehicles in Jordan rose by almost 1,000% during the last 30 years, while the number of air passengers increased approximately by 460% [2].

II.

Energy Demand in Jordan

At present, Jordan is importing crude oil and natural gas to sustain its present way of life. This leads to a significant hard-currency drain in the economy, with an annual oil bill exceeding 3 billion US$ [3]. Such high value represented approximately 12% of the GDP, in 2009, and 42% of domestic exports and about 19% of total imports into the country [3]. The demand for primary energy in 2009 was about 7.739 million tons of oil equivalent (toe), compared with 2.4 million toe in 1982. Transport sector is the largest single consumer, followed by households and industry see Figure 1. In Figure 1, others include commercial and services, government and agricultural sectors. The high sharing ratio of the transport sector in the national energy


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Adnan Mukattash, Ahmed Al-Ghandoor, Ahmad M. Qamar

demand is mainly due to non-existence of water-ways or modern rail networks and lack of efficient and modern mass transport systems in Jordan.

Fig. 2. Development of transport energy consumption (1000 toe) over the period 1985-2009

Fig. 1. Percentage ratios of the final energy sector distribution in 2009

IV.

III. Overview of Jordan’s Transportation Sector

Equations Historical data, during the studied period 1985–2009 from the related government agencies were utilized to develop the double exponential smoothing model of energy consumptions of the Jordanian transportation sector.

Jordan’s transportation sector is dominated by road transportation since there are no rail networks and marine transportation is negligible due to the geographical location of the country. The national network of roads extends to cover 7891 km, 3249 km of which are main roads, while the remaining are either side or rural roads [4]. In 2009, the total number of registered vehicles was nearly 995 thousand vehicles, with passenger cars taking a share of around 65% of this number1. A breakdown of vehicles in Jordan according to vehicle type is presented in Table I. Private vehicles represent 82.8% of the total stock of registered vehicles, while public vehicles represent 8.4%, and the remaining 8.8% are registered as government, agricultural, tourist, or special purpose vehicles.

V.

Number

Percentage

Passenger car Minibus Bus Van/pickup Freight/truck Motorcycle Other

643,605 18,900 3,979 103,454 173,087 3,979 47,749

64.7 1.9 0.4 10.4 17.5 0.4 4.8

Total

994,753

100.0

Methodology

Energy modeling is a subject of widespread current interest among engineers and scientists concerned with problems of energy production and consumption. Energy modeling in some areas of application is now capable of making useful contributions to planning and policy formulation [6]. Depending on the available data, several models have been proposed to model transport energy consumption that can be classified into two groups: econometric and artificial intelligence approaches. The first group includes multiple linear regression, partial least square regression, and time series [7-8] while the second group includes artificial neural network [9], harmony search algorithm [9] and Fuzzy theory [10]. For this study, projected values for Jordan transport energy demand were determined using a forecasting tool based on time series technique with double exponential smoothing since the historical data over the period 19852009, as shown in Figure 2 shows an evident long-run trend. The double exponential smoothing forecasting time series method is recommended in such situations [11]. The double exponential forecasting equation is as follows:

TABLE I BREAKDOWN OF 2009 TOTAL REGISTERED VEHICLES IN JORDAN ACCORDING TO VEHICLE TYPE Vehicle Type

Data Sources

.

Figure 2 shows the development of fuel consumption from 1985-2009 [5]. It can be seen that the consumption increased from only 972x103 toe in 1985 to 1952x103 toe in 2009 corresponding to an average annual growth rate of 4.3%.

Ft + m = at + bt m

(1)

where Ft+m is the forecast after m the number of periods ahead to be forecast, at the forecasted intercept, and bt the forecasted slope. The intercept at and the slope bt are estimated as follows:

1

All figures and statistics on vehicle numbers, classifications, and registrations were calculated using the comprehensive data obtained from the Department of Drivers and Vehicles Licensing database through personal efforts and communications

at = 2St' − St"


(2)


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Adnan Mukattash, Ahmed Al-Ghandoor, Ahmad M. Qamar

bt =

α St' − St" 1−α

(

)

0 ≤α