Finite Elements Method Finite Variation Method Finite

Finite Volume Method in Heat Conduction

References 1. Yaghi AH, Hyde TH, Becker AA, Sun W, Hilson G, Simandjuntak S, Flewitt PEJ, Pavier MJ, Smith DJ (2010) A comparison between measured and modelled residual stresses in a circumferentially butt-welded P91 steel pipe. ASME J Press Vessel Technol 132:011206-1–011206-10 2. Hibbitt, Karlsson, Sorensen (2007) ABAQUS user manual, version 6.7, Pawtucket 3. Karlsson RI, Josefson BL (1990) Three-dimensional finite element analysis of temperatures and stresses in a single-pass butt-welded pipe. J Press Vessel Technol 112:76–84 4. Brickstad B, Josefson BL (1998) A parametric study of residual stresses in multi-pass butt-welded stainless steel pipes. Int J Press Vessels Pip 75:11–25 5. Deng D, Murakawa H (2006) Prediction of welding residual stress in multi-pass butt-welded modified 9Cr-1Mo steel pipe considering phase transformation effects. Comp Mater Sci 37:209–219 6. Yaghi AH, Hyde TH, Becker AA, Williams JA, Sun W (2005) Residual stress simulation in welded sections of P91 pipes. J Mater Process Technol 167:480–487 7. Yaghi AH, Hyde TH, Becker AA, Sun W (2008) Finite element simulation of welding and residual stresses in a P91 steel pipe incorporating solid-state phase transformation and post-weld heat treatment. J Strain Anal 43:275–293 8. Be´res L, Balogh A, Irmer W (2001) Welding of martensitic creep-resistant steels. Suppl Weld J 80(8):191-s–195-s 9. Yaghi AH, Hyde TH, Becker AA, Sun W (2009) Thermomechanical modelling of weld microstructure and residual stresses in P91 steel pipe. J Energy Mater 4(3):113–123 10. Greenwood GW, Johnson RH (1965) The deformation of metals under small stresses during phase transformation. Proc Roy Soc A 283:403–422 11. Magee CL (1966) Transformation kinetics, microplasticity and ageing of martensite in Fe–3 l–Ni. PhD thesis, Carnegie Mellon University, Pittsburg 12. Karlsson CT (1989) Finite element analysis of temperatures and stresses in a single-pass butt-welded pipe – influence of mesh density and material modeling. Eng Comput 6:133–141 13. Hyde TH, Yaghi AH, Tanner DWJ, Bennett CJ, Becker AA, Williams EJ, Sun W (2009) Current capabilities of the thermo-mechanical modelling of welding processes. J Multiscale Model 1(3&4):451–478 14. Berglund D, Alberg H, Runnemalm H (2003) Simulation of welding and stress relief heat treatment of an aero engine component. Finite Elem Anal Design 39:865–881 15. Yaghi AH, Hyde TH, Becker AA, Sun W (2011) Finite element simulation of a welded P91 steel pipe

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undergoing post-weld heat treatment. Sci Technol Weld Join 16(3):232–238 16. Yaghi AH, Hyde TH, Becker AA, Sun W, Hilson G, Simandjuntak S, Flewitt PEJ, Pavier MJ, and Smith DJ (2012) Comparison of measured and modelled residual stresses in a welded P91 steel pipe undergoing post weld heat treatment. ASME J Press Vessel Technol (Submitted)

Finite Elements Method ▶ Thermal Shock and Modeling of Destruction for Refractory Linings of Metallurgical Installations

Finite Variation Method ▶ Multiple Virtual Crack Extension Technique

Finite Volume Method ▶ Two-Dimensional, Steady-State Conduction

Finite Volume Method in Heat Conduction Artur Cebula1 and Dawid Taler2 1 Institute of Thermal Power Engineering, Faculty of Mechanical Engineering, Cracow University of Technology, Cracow, Poland 2 Institute of Engineering and Air Protection, Faculty of Environmental Engineering, Cracow University of Technology, Cracow, Poland

Overview The finite volume method (FVM) is one of the most popular numerical methods used to solve heat conduction problems [1–9]. The FVM is a more physically oriented approach in comparison to the finite-difference method.

F

F

1646

The FVM can be easily applied to the determination of temperature fields in solids of irregular shape or in solids with variable thermal properties. The first step in the FVM is to formulate the energy conservation in integral form. Then, the integral form is discretized, that is, an energy conservation equation is written for a specific control volume. Consequently, the solution will satisfy the energy conservation equation globally. In the finite volume approach, the formulation is more physically directed, and engineers may well prefer to formulate their problems from a conservation viewpoint. The next step in the FVM, as in the finite element method (FEM), is to divide the whole physical region into a set of nonoverlapping control volumes, which are also called finite volumes or cells. They can form a structured mesh or an unstructured mesh. In the FVM, the terms finite volume, control volume or cell are used in place of the term element used in the FEM [9, 10]. The energy balance equations for control volumes are used to determine the temperature at the nodes, which are situated inside the control volume. Formulating energy balance equations for all the control volumes, we obtain a system of algebraic equations for steady-state problems or a system of ordinary differential equations for node temperatures. Control volumes can be rectangular or polygonal. In contrast to the FEM, in the FVM, interpolation functions (shape functions) are not used to calculate the temperature inside the control volume. A discrete approximation based on the FVM can often be constructed directly from a conservation statement. Such an approach can be used successfully when finite volume imensions, physical properties, boundary conditions, or boundary shape is complex. The discrete equations resulting from finite volume approach are often identical to those obtained from a finite-difference method, which is based on the approximation of the differential equations. However, the finite-difference scheme may not satisfy the conservation equation. On the other hand, accuracy estimates for the FVM are more difficult to obtain than in the finite-difference method. To calculate steady-state temperature at


the nodes, a system of algebraic equations can be solved using the Gauss elimination method when the problem is linear or the iterative procedures like the Gauss-Seidel or over-relaxation method if the problem is linear or nonlinear. In transient problems, a system of ordinary differential equations for the temperature at the nodes can be integrated using the explicit or implicit Euler methods or the Runge-Kutta methods. There are also several other algorithms which can be applied for this purpose. At first, basic concepts of the FVM will be presented, and then examples of the FVM application will be shown to demonstrate the effectiveness of the FVM. The obtained results will be compared with analytical solutions or the solutions acquired by the FVM or FEM using ANSYS FLUENT 13.0 software.

Finite Volume Method The finite volume method (FVM) is also known as the control volume method. To set energy conservation equations for control volumes in the Cartesian and cylindrical coordinate system, a twodimensional transient heat conduction equation will be analyzed. If the thickness of an analyzed region is d and thermal properties – the specific heat c, the density r, and the thermal conductivity k – and heat generation by internal heat sources per unit volume qv are temperature dependent, then the heat conduction equation can be written in the form cðTÞrðTÞ

@T ¼ H q þ qv @t

ð1Þ

where T – temperature, t – time, and H – gradient operator (nabla). The region is divided into control volumes, which have the following dimensions: Dx, Dy, and d in the Cartesian coordinate system (Fig. 1) or Dr, D’, and d (Fig. 2) in the cylindrical coordinate system. Integrating (1) over the control volume (CV) gives the following equation for a single control volume (cell): ð ð ð @T cðTÞrðTÞ dV ¼ H q dV þ qv dV ð2Þ @t CV

CV

CV

where CV stands for the control volume.


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F

Finite Volume Method in Heat Conduction, Fig. 1 Two-dimensional region divided into finite volumes

F a

b Δj

N W

n

w

P rs

rw

e s

E

S

r

rn

j 0

Δr

Finite Volume Method in Heat Conduction, Fig. 2 Two-dimensional cylindrical region divided into finite volumes; (a) finite volume mesh, (b) finite volume

Applying Green-Gauss-Ostrogradsky theorem [11] to the first term on the right-hand side of (2) ð

ð H q dV ¼

CV

q n dS

ð3Þ

CS

gives ð cðTÞrðTÞ CV

@T dV ¼ @t

ð

ð n q dS þ

CS

qv dV CV

ð4Þ

where CS is the control volume surface, while n an outward unit normal vector. Taking into account that n q ¼ 1 q cosðn,qÞ ¼ qn it is evident that when the heat flows into to the control volume, the heat flux vector q is directed to the inside of the control volume and the angle between vector n and q is equal to 180 . The scalar product is then negative, while the surface integral in (4) is positive. If DV denotes the volume of a control cell, then individual terms in (4) can be approximated in the following way:

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1648


ð cðTÞrðTÞ CV

@T dTP dV ffi DVcðTP ÞrðTP Þ ð5Þ @t dt

QSP ¼ ðDxÞ d qSP ¼ ðDxÞ d

ð

n q dS ¼

4 X

Qi

ð6Þ

qv dV ¼ DV qv ðTP Þ

ð7Þ

i¼1

CS

ð

kðTS Þ þ kðTP Þ TS TP 2 Dy

ð12Þ

where the symbols Dx and Dy denote dimensions of the control volume (Fig. 1b). Substituting (9)–(12) into (8), we obtain

dTP dt kðTW Þ þ kðTP Þ TW TP ¼ ðDyÞ d 2 Dx kðTN Þ þ kðTP Þ TN TP þ ðDxÞ d 2 Dy kðTE Þ þ kðTP Þ TE TP þ ðDyÞ d 2 Dx kðTS Þ þ kðTP Þ TS TP þ ðDxÞ d 2 Dy þ ðDxÞ ðDyÞ d qv ðTP Þ

ðDxÞ ðDyÞ d cðTP Þ rðTP Þ

CV

where Qi is the heat flow that is transferred from the neighboring cell. Substituting (5)–(7) into (4), one obtains the following heat balance equation: DV c ðTP Þ rðTP Þ

dTP ¼ dt

4 X

Qi þ DV qv ðTP Þ

i¼1

ð8Þ which will be written in the following Cartesian and cylindrical coordinate systems.

ð13Þ

where

Energy Conservation Equation for the Control Volume in Cartesian Coordinates A division of a region into control volumes and a control volume are shown in Fig. 1. The volume of a single cell is DV ¼ (Dx) (Dy) d. Heat flows from adjacent control cells (volumes) with nodes W, N, E, and S to node P. The heat flow rates are calculated as follows: QWP ¼ ðDyÞ d qWP ¼ ðDyÞ d

kðTN Þ þ kðTP Þ TN TP ð10Þ 2 Dy

QEP ¼ ðDyÞ d qEP ¼ ðDyÞ d

cP ¼ cðTP Þ;

rp ¼ r Tp ;

After transforming, (13) takes the form " dTP kW þ kP TW TP kN þ kP TN TP ¼ kP þ dt 2kP 2kP ðDxÞ2 ðDyÞ2 # kE þ kP TE TP kS þ kP TS TP qv;P þ þ þ 2 2 2kP 2k c ðDxÞ P ðDyÞ P rP

ð14Þ

kðTW Þ þ kðTP Þ TW TP ð9Þ 2 Dx

QNP ¼ ðDxÞ d qNP ¼ ðDxÞ d

kP ¼ kðTP Þ; kP aP ¼ cP rP

kðTE Þ þ kðTP Þ TE TP ð11Þ 2 Dx

In steady state, we have dTp/dt ¼ 0. For a uniform grid Dx ¼ Dy and for constant and temperature-independent thermal properties and heat source power, (14) is TW þ TN þ TE þ TS 4TP þ

qv ðDxÞ2 ¼0 k ð15Þ


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The energy conservation (8) can be transformed into a form similar to (14) after introducing the following notation (Fig. 2): 2 2 rn rs2 D’ 2 D’ d¼ d ð16Þ DV ¼ p rn rs 2p 2 QWP ¼ ðDr Þd

QNP

kW þ kP TW TP 2 rw ðD’Þ

kN þ kP TN TP ¼ rn ðD’Þd 2 Dr

QEP ¼ ðDr Þd

kE þ kP TE TP 2 rw ðD’Þ

QSP ¼ rs ðD’Þd

k S þ k P T S TP 2 Dr

ð17Þ ð18Þ ð19Þ ð20Þ

Substituting (16)–(20) into (8) gives dTP 2 kP kW þ kP ¼ 2 dt 2 lP rn rs2 D’ Dr ð T W TP Þ rw ðD’Þ kN þ kP rn ðD’Þ þ ðTN TP Þ Dr 2 kP kE þ kP Dr þ ðTE TP Þ rw ðD’Þ 2 kP kS þ kP rs D’ ðTS TP Þ þ Dr 2 kP qv;P þ cP rP

ð21Þ

where the symbol kP ¼ kP =ðcP rP Þ stands for the thermal diffusivity. In the case of steady-state problems, one should assume that dTP/dt ¼ 0. Energy conservation equation (14) in the Cartesian coordinates or (21) in cylindrical polar coordinates is written for all nodes, including the nodes at the boundary of the region. Appropriate boundary conditions should be accounted for in the energy equations

100°C

200°C

300°C

400°C

w k= 50 m 140°C

340°C 2

3

1

4

0.15m

Energy Conservation Equation for the Control Volume in Cylindrical Polar Coordinates

F

280°C

180°C

220°C

220°C

220°C

220°C

F

0.15m

Finite Volume Method in Heat Conduction, Fig. 3 Two-dimensional region with prescribed surface temperature divided into finite volumes

for boundary-adjacent control volumes. In order to determine the unsteady-state temperature at the nodes, the system of ordinary differential equations has to be solved. In steady-state problems, the system of algebraic equations resulting from energy conservation equations written for all the control volumes can be solved by direct methods, for example, Gauss elimination method, or by iterative methods like GaussSeidel or over-relaxation method. The application of the FVM to determine steady-state and transient temperature fields will be illustrated by four examples. Example 1. Determining Temperature Distribution in an Infinitely Long Rod with a Square Cross Section An infinitely long rod of square cross section with prescribed temperature at the boundary surfaces will be analyzed at first (Fig. 3). An algebraic equation system will be solved by the iterative Gauss-Seidel method. Equation (15) will be used to write energy conservation equations for all internal nodes. TW þ TN þ TE þ TS 4TP ¼ 0

ð22Þ

Applying (22) for all the internal nodes, we have, respectively, for nodes 1 to 4:

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Finite Volume Method in Heat Conduction, Fig. 4 Temperature distribution in K in two-dimensional region, as shown in Fig. 3, determined using ANSYS FLUENT 13.0

672.35 657.42 642.50 627.57 612.65 597.72 582.80 567.87 552.95 538.02 523.09 508.17 493.24 478.32 463.39 448.47 433.54 418.62 403.69 388.77 373.84

• Node 1 180 þ T2 þ T4 þ 220 4T1 ¼ 0 4T1 T2 T4 ¼ 400

ð23Þ

• Node 2 140 þ 200 þ T3 þ T1 4T2 ¼ 0 T1 þ 4T2 T3 ¼ 340

ð24Þ

• Node 3 T2 þ 300 þ 340 þ T4 4T3 ¼ 0 T2 þ 4T3 T4 ¼ 640

ð25Þ

• Node 4 T1 þ T3 þ 220 þ 280 4T4 ¼ 0 T1 T3 þ 4T4 ¼ 500

ð26Þ

To apply the Gauss-Seidel method to solve equations for Ti, i ¼ 1,. . .,4, (23)–(26) are transformed as follows:

1 T1 ¼ ð400 þ T2 þ T4 Þ 4

ð27Þ

1 T2 ¼ ð340 þ T1 þ T3 Þ 4

ð28Þ

1 T3 ¼ ð640 þ T2 þ T4 Þ 4

ð29Þ

1 T4 ¼ ð500 þ T1 þ T3 Þ 4

ð30Þ

The equation set (27)–(30) was solved by the Gauss-Seidel method using the computer program developed in [5] to give T1 ¼ 213.33 C, T2 ¼ 206.67 C, T3 ¼ 273.33 C, and T4 ¼ 246.67 C. Similar results were obtained using ANSYS FLUENT 13.0: T1 ¼ 213.13 C, T2 ¼ 206.47 C, T3 ¼ 273.12 C, and T4 ¼ 246.47 C. Despite the coarse mesh used in the FVM and very fine mesh applied in the ANSYS FLUENT 13.0 simulation, the differences between the results are quite small. The temperature distribution over the entire cross section of the rod is illustrated in Fig. 4.


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Rearranging (32) gives

Tf h 2

Δx

k 4

6

8

10

12

14

16

Δy

w

Tb

Tb

1

3

5

7 L

9

11

13

Example 2. Determining Two-Dimensional Temperature Distribution in a Straight Fin of Uniform Thickness Calculate temperature distribution in a straight fin. The half of the fin with the finite volume mesh is depicted in Fig. 5. Calculate the fin temperature at nodes shown in Fig. 5. The following data was adopted: w ¼ 0.003 m, L ¼ 0.024 m, h ¼ 100 W/(m2K), Tb ¼ 95 C, Tf ¼ 20 C, Dx ¼ Dy ¼ 0.003 m, and k ¼ 50 W/ (mK). The energy conservation equations (heat balance equations) for control volumes are as follows: • Node 1 Dy Tb T1 Dy T3 T1 T2 T1 þk þ k Dx ¼0 2 Dx 2 Dx Dy ð31Þ This equation is transformed to Dy Dx Dx Dy Dy þ T2 T3 ¼ Tb T1 Dx Dy Dy 2 Dx 2 Dx ð32Þ • Node 2

k

Dx Dy Dx hDx T1 þ þ þ T2 Dy Dx Dy k Dy Dy hDx T4 ¼ Tb þ Tf 2Dx 2Dx k

ð34Þ

15

Finite Volume Method in Heat Conduction, Fig. 5 Half of the straight fin with constant thickness divided into finite volume mesh

k

F

Dy Tb T2 Dy T4 T2 T 1 T2 þk þ k Dx 2 2 Dx Dx Dy þ h Dx Tf T2 ¼ 0 ð33Þ

• Nodes 3, 5, 7, 9, 11, and 13 Energy balance equation for ith node has the form

F Dy Ti2 Ti Dy Tiþ2 Ti Tiþ1 Ti þk þ kDx ¼0 k 2 2 Dx Dx Dy

ð35Þ Hence,

Dy Dy Dx Dx Ti Tiþ1 Ti2 þ þ 2 Dx Dx Dy Dy Dy Tiþ2 ¼ 0; i ¼ 3; 5; 7; 9; 11; 13 2 Dx ð36Þ

• Nodes 4, 6, 8, 10, 12, and 14 Heat balance equation for i-node has the form Dy Ti2 Ti Dy Tiþ2 Ti þk 2 2 Dx Dx Ti1 Ti þ h Dx Tf Ti ¼ 0 þ k Dx Dy

k

ð37Þ

Hence, Dy Dx Dy Dx h Dx Ti2 Ti1 þ þ þ Ti 2 Dx Dy Dx Dy k Dy h Dx Tiþ2 ¼ ; i ¼ 2; 4; 6; 8; 10; 12 2 Dx k ð38Þ • Node 15 k

Dy T13 T15 Dx T16 T15 þk ¼0 2 2 Dx Dy

ð39Þ

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After rearranging (39), we have Dy Dy Dx Dx T13 þ þ T16 ¼ 0 T15 Dx Dx Dy Dy ð40Þ • Node 16 k

Dy T14 T16 Dx T15 T16 Dx Tf T16 ¼ 0 þk þh 2 2 2 Dx Dy

ð41Þ Equation (41) can be written in the form

Dy Dx Dy Dx h Dx h Dx T14 T15 þ þ þ T16 ¼ Tf Dx Dy Dx Dy k k

ð42Þ After substitution of Tb ¼ 95 C, Tf ¼ 20 C, Dx ¼ Dy ¼ 0.003 m, k ¼ 50 W/(mK), and h ¼ 100 W/(m2K) into (32), (34), (36), (38), (40) and (42), we have 2 T1 T2 0:5 T3 ¼ 47:5 T1 þ 2:006T2 0:5 T4 ¼ 47:62 0:5Ti2 þ 2 Ti Tiþ1 0:5 Tiþ2 ¼ 0 i ¼ 3; 5; 7; 9; 11; 13 0:5Ti2 Ti1 þ 2:006Ti 0:5Tiþ2 ¼ 0:12 i ¼ 2; 4; 6; 8; 10; 12; 14 T13 þ 2T15 T16 ¼ 0 T14 T15 þ 2:006 T16 ¼ 0:12

ð43Þ The Gauss-Seidel method was used [5] to solve equation system (43). The temperature distribution in the fin was also determined using the method of separation of variables [5] and the ANSYS FLUENT 13.0 software. The comparison of the results is presented in Table 1. In spite of the coarse mesh used in the FVM method, the accuracy of the results is quite good. The temperature distribution in the half of the fin determined by using ANSYS FLUENT 13.0 is illustrated in Fig. 6. It can be seen from Table 1 and Fig. 6 that temperature difference on the fin thickness is very small.

Finite Volume Method in Heat Conduction, Table 1 Comparison of fin temperature at nodes Node number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Temperature, C Control volume method [5] 92.12 91.94 89.59 89.38 87.46 87.26 85.74 85.54 84.40 84.21 83.46 83.27 82.90 82.71 82.71 82.52

ANSYS FLUENT 13.0 92.15 91.94 89.63 89.42 87.51 87.31 85.80 85.60 84.48 84.29 83.54 83.36 82.98 82.79 82.80 82.61

Analytical method [5] 92.09 91.88 89.55 89.34 87.42 87.22 85.7 85.5 84.37 84.18 83.42 83.23 82.86 82.67 82.67 82.48

Example 3. Lumped-Heat Capacity Model of a Heat Exchanger Tube Consider a heat exchanger tube at an initial temperature Tw0 (Fig. 7). Inner and outer perimeter of the tube that can be of any shape is Uin and Uo, respectively. A fluid stream inside the tube is at temperature T1, while a fluid flowing outside the tube is at temperature T2. If the tube material has high thermal conductivity, then its temperature can be assumed to be uniform across the tube wall and equal to the temperature of the inner and outer tube surface. Performing an energy balance on the control volume (tube of unit length), we can formulate a lumped-heat capacity model of the tube wall [12]: Um dw cw rw

d Tw ¼ h 1 Uin ðT1 Tw Þ þ h 2 Uo ðT2 Tw Þ dt

ð44Þ

where Tw – tube temperature, t – time, Um ¼ ðUin þ Uo Þ=2 – mean tube perimeter, dw – tube wall thickness, cw – specific heat of the tube wall, rw – tube wall density, and h1 and h2 – heat transfer coefficient at the inner and outer tube surface, respectively.

Finite Volume Method in Heat Conduction Finite Volume Method in Heat Conduction, Fig. 6 Temperature distribution in the half of the fin, as shown in Fig. 5, determined using ANSYS FLUENT 13.0

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368.15 367.53 366.91 366.29 365.67 365.05 364.43 363.81 363.19 362.58 361.96 361.34 360.72 360.10 359.48 358.86 358.24 357.62 357.00 356.38 355.76

F Assuming that the tube wall is thin, that is, ro ¼ rin ¼ r, (47) reduces to Tw ¼

Finite Volume Method in Heat Conduction, Fig. 7 Lumped-heat capacity model of a heat exchanger tube

In the steady state when @ Tw =@ t ¼ 0, the wall temperature Tw obtained from (44) is given by Tw ¼

h1 Uin T1 þ h2 Uo T2 h1 Uin þ h2 Uo

ð45Þ

The exact analytical solution for the onedimensional temperature distribution in the wall of the circular tube with finite thermal conductivity kw is [12]

Tw ¼

Bi1 T1 þ Bi2 T2 þ Bi1 Bi2 T2 ln rrin T1 ln rro Bi1 þ Bi2 þ Bi1 Bi2 ln rrino ð46Þ

where the Biot numbers Bi1 and Bi2 are defined as Bi1 ¼

h1 rin h2 r o ; Bi2 ¼ kw kw

ð47Þ

Bi1 T1 þ Bi2 T2 h1 rin T1 þ h2 ro T2 ¼ Bi1 þ Bi2 h1 rin þ h2 ro ð48Þ

To illustrate practical applicability of the present model, the wall temperature of the convective superheater will be calculated [12]. The following data are adopted: ro ¼ 0.019 m, dw ¼ 0.005 m, T1 ¼ 400 C, T2 ¼ 800 C, cw ¼ 612 J/(kgK), rw ¼ 7,724 kg/m3, kw ¼ 40 W/(mK), h1 ¼ 2,500 W/(m2K), and h2 ¼ 120 W/ (m2K). Equation (46) gives the following results: Tw jr¼rin ¼ 424:07 C, Tw jr¼rm ¼ 427:53 C, and Tw jr¼ro ¼ 430:50 C, where the symbol rm ¼ ðrin þ ro Þ=2 ¼ 0:0165 m denotes the mean tube radius. The temperature drop over the tube wall is D Tw ¼ Tw;o Tw;in ¼ 430:50 424:07 ¼ 6:43 K. The tube wall temperature determined from (45) or (48) is Tw ¼ 424.46 C. It can be concluded that (45) may be used to calculate the temperature of the superheater tubes. Next, the Euler explicit finite-difference method will be applied to integrate (44) [12]. Rearranging (44) to the form d Tw h1 Uin þ h2 Uo h1 Uin þ T1 Tw ¼ dt cw rw Um dw cw rw Um dw h2 Uo þ T2 cw rw Um dw ð49Þ

1654


and using the Euler explicit method give h1 Uin þ h2 Uo h1 Uin Tn Twn ¼ cw rw Um dw cw rw Um dw 1 h2 U o þ Tn cw rw Um dw 2

1

2

3

13

5

6

14

8

ho

4

10

7

11

9

12

Dy/ 2

Dt

þ

ð50Þ

b

Twnþ1 Twn

To

Dy

F

a

hin

Solving (50) for Twnþ1 gives Twnþ1 ¼

15

h1 Uin þh2 Uo 1Dt Twn cw rw Um dw h1 Uin h2 Uo n n þDt T þ T ; cw rw Um dw 1 cw rw Um dw 2

Tin y

Dx

Dx / 2

0

x

Finite Volume Method in Heat Conduction, Fig. 8 Division of the quarter of the chimney cross section into control volumes

n ¼0;1;2;... ð51Þ where Tw0 ¼ Tw0 is the initial tube temperature. The finite-difference method is stable if the following condition is satisfied [12]: 1 D t h1 Uin þ h2 Uo 1 c r U d w

w

ð52Þ

m w

Thus, the time step Dt should be chosen that

Dt

2 cw rw Um dw h1 Uin þ h2 Uo

ð53Þ

Example 4. Unsteady-State Temperature Distribution in a Chimney with Square Cross Section A chimney of square cross section is shown in Fig. 8. External dimensions of the chimney are 2b2b. The internal channel has a square cross section with the side length equal to 2a. The flue gas at temperature Tin ¼ 250 C flows through the internal channel, while the outside air temperature is To ¼ 10 C. The following data were adopted for the calculation: b ¼ 0.375 m and a ¼ 0.125 m. The thermal conductivity of the

chimney’s material is k ¼ 1.25 W/(mK). The inner heat transfer coefficient hin on the flue gas side is hin ¼ 60 W/(m2K), while the heat transfer coefficient at the outer surface is ho ¼ 20 W/(m2K). Determine the unsteadystate temperature distribution in the cross section of the chimney, taking into account heat transfer by convection and by radiation. Assume that the equivalent emissivity of the chimney’s interior is ein ¼ 0.9, while the emissivity of the outer surface is eo ¼ 0.8. The unsteady-state temperature distribution will be determined in one eighth of the chimney cross section due to the symmetry of the temperature field. Because of the radiation heat transfer, the problem is nonlinear. Performing energy balances on all the control volumes, as shown in Fig. 8, gives for Dx ¼ Dy the following equation system: • Node 1

1 1 dT1 ¼ ho ðDxÞðTz T1 Þ ðDxÞ ðDxÞcr dt 2 2 þ eo sðDxÞ To4 T14 Dx T13 T1 Dx T2 T1 þk þk Dx 2 2 Dx

ð54Þ


1655

After simple transformations, we have

dT1 4k eo sðDxÞ 4 ¼ To T14 ðDBio ÞTo dt ðDxÞ2 k ð55Þ T2 T13 ð1 þ DBio ÞT1 þ þ 2 2

Because of the symmetry of the temperature distribution, we have T13 ¼ T2: dT1 4k eo sðDxÞ 4 To T14 ¼ ðDBio ÞTo 2 k dt ðDxÞ ð1 þ DBio ÞT1 þ T2 ð56Þ where temperature is expressed in Kelvin, while s ¼ 5.67108 W/(m2K4) is the Stefan-Boltzmann constant. • Node 2 1 dT2 ðDxÞðDxÞcr ¼ ho ðDxÞðTo T2 Þ 2 dt þ eo sðDxÞ To4 T24 Dx T1 T2 Dx T3 T2 þk þk 2 2 Dx Dx T5 T2 þ kðDxÞ Dx

ð57Þ Equation (57) can be written in the form dT2 2k eo sðDxÞ 4 To T24 ¼ ðDBio ÞTo þ 2 k dt ðDxÞ T1 T3 ð2 þ DBio ÞT2 þ þ þ T5 2 2 ð58Þ • Node 3 1 dT3 ðDxÞðDxÞcr ¼ ho ðDxÞðTo T3 Þ 2 dt þ eo sðDxÞ To4 T34 Dx T2 T3 Dx T4 T3 þk þk 2 2 Dx Dx T6 T3 þ k ðDxÞ Dx

ð59Þ

F

or dT3 2k eo sðDxÞ 4 To T34 ¼ ðDBio ÞTo þ 2 k dt ðDxÞ T2 T4 ð2 þ DBio ÞT3 þ þ þ T6 2 2 ð60Þ • Node 4

1 dT4 ðDxÞ ðDxÞcr 2 dt ¼ ho ðDxÞðTo T4 Þ þ eo s ðDxÞ To4 T44 Dx T10 T4 Dx T3 T4 þk þk 2 2 Dx Dx T7 T4 þ kðDxÞ Dx ð61Þ

Due to the symmetry, we have T10 ¼ T3. Transforming (61) gives dT4 2k eo s ðDxÞ 4 To T44 ¼ ðDBio ÞTo þ 2 k dt ðDxÞ ð2 þ DBio ÞT4 þ T3 þ T7 ð62Þ • Node 5

ðDxÞðDxÞcr

dT5 T2 T5 T6 T5 ¼ k ðDxÞ þ k ðDxÞ dt Dx Dx T13 T5 T14 T5 þ k ðDxÞ þ k ðDxÞ Dx Dx

ð63Þ Taking into consideration that T13 ¼ T2 and T14 ¼ T6, one gets dT5 2k ¼ ðT2 þ T6 2T5 Þ dt ðDxÞ2

ð64Þ

F

F

1656


• Node 6 ðDxÞ2 cr

• Node 9

dT6 T3 T6 T5 T6 ¼ k ðDxÞ þ k ðDxÞ dt Dx Dx T7 T6 T8 T6 þ k ðDxÞ þ k ðDxÞ Dx Dx ð65Þ

1 dT9 ¼ ein sðDxÞ Tin4 T94 ðDxÞ2 cr dt 2

T7 T9 Dx Dx T8 T9 Dx T12 T9 þk þk Dx 2 Dx 2 þ hin ðDxÞðTin T9 Þ þ k ðDxÞ

ð71Þ

After simple transformations of (65), we have dT6 2k ¼ ðT3 þ T5 þ T7 þ T8 4T6 Þ ð66Þ dt ðDxÞ2 • Node 7 ðDxÞ2 cr

dT7 T4 T7 T6 T7 ¼ k ðDxÞ þ k ðDxÞ dt Dx Dx T9 T7 T11 T7 þ k ðDxÞ þ k ðDxÞ Dx Dx

ð67Þ Due to the symmetry of the temperature field, we have T11 ¼ T6, and (67) reduces to dT7 k ¼ ðT4 þ 2T6 þ T9 4T7 Þ dt ðDxÞ2

ð68Þ

Considering that T12 ¼ T8, we obtain from (71) dT9 2k ein sðDxÞ 4 Tin T94 ¼ ðDBiin ÞTin þ 2 k dt ðDxÞ þ T7 þ T8 ð2 þ DBiin ÞT9 ð72Þ Taking into account that Dx ¼ Dy ¼ (b–a)/ 2 ¼ (0.375–0.125)/2 ¼ 0.125 m, k ¼ 1.25 W/(mK), c ¼ 835 J/(kgK), r ¼ 2878.9 kg/m3, hin ¼ 60 W/(m2K), ho ¼ 20 W/(m2K), Tin ¼ 250 + 273.15 ¼ 523.15 K, and To ¼ 10 + 273.15 ¼ 283.15 K, one can calculate DBiin and DBio: hin ðDxÞ 60 0:125 ¼ ¼ 6:0 k 1:25 ho ðDxÞ 20 0:125 DBio ¼ ¼ ¼ 2:0 k 1:25

DBiin ¼

• Node 8 3 dT8 ðDxÞ2 cr ¼ ein s ðDxÞ Tin4 T84 4 dt þ hin ðDxÞðTin T8 Þ T6 T8 Dx T9 T8 þk þ k ðDxÞ 2 Dx Dx Dx T18 T8 T14 T8 þ k ðDxÞ þk 2 Dx Dx

ð69Þ

and ein s ðDxÞ 0:9 5:67 108 0:125 ¼ ¼ 5:103 109 1=K3 k 1:25 eo s ðDxÞ 0:8 5:67 108 0:125 ¼ ¼ 4:536 109 1=K3 k 1:25

Since the temperature distribution is symmetric, hence, T14 ¼ T6 and T15 ¼ T9. (69) can be rearranged to dT8 4 k ¼ ½ðDBiin ÞTin þ 2T6 þ T9 3 ðDxÞ2 dt ein s ðDxÞ 4 4 Tin T8 ð3 þ DBiin ÞT8 þ k ð70Þ

At an initial moment when t ¼ 0 s, the chimney is at uniform temperature equal to 10 C; thus, T1(0) ¼ T2(0) ¼ . . . ¼ T9(0) ¼ 283.15 K. The initial conditions take the form T1 ð0Þ ¼ T2 ð0Þ ¼ . . . ¼ T9 ð0Þ ¼ 283:15 K ð73Þ

The equation system was solved using the Runge-Kutta method of the fourth order.

Finite Volume Method in Heat Conduction Finite Volume Method in Heat Conduction, Fig. 9 Temperature distribution in K in one-eighth of the cross section of the chimney; (a) at time t ¼ 7,200 s, (b) in steady state

508.00

1657

F

a

496.75 485.50 474.25 463.00 451.75 440.50 429.25 418.00

F

406.75 395.50 384.25

b

373.00 361.75 350.50 339.25 328.00 316.75 305.50 294.25 283.00

The integration time step in Runge-Kutta method was assumed to be Dt ¼ 60 s. The temperature distribution in K in one eighth of the cross section of the chimney at time t ¼ 7,200 s and in steady state computed using ANSYS FLUENT 13.0 is illustrated in Fig. 9. The comparison of the results obtained by the FVM method and ANSYS FUENT 13.0 for very fine mesh is shown in Fig. 10. It can be seen that at the beginning of the heating process, the differences between temperatures calculated at the same nodes are noticeable. If the mesh in the FVM method were finer, then discrepancies between the results would be smaller. In the steady state, the temperatures at the nodes obtained using the FVM are [5]

T1 ¼ 288:36 K ¼ 15:21 C; T3 ¼ 312:55 K ¼ 39:40 C;

T2 ¼ 300:99 K ¼ 27:84 C; T4 ¼ 315:99 K ¼ 42:84 C;

T5 ¼ 345:27 K ¼ 72:12 C; T6 ¼ 389:56 K ¼ 116:41 C; T7 ¼ 401:19 K ¼ 128:40 C; T8 ¼ 499:22 K ¼ 226:07 C; T9 ¼ 509:66 K ¼ 236:51 C

The temperatures at the nodes obtained by ANSYS FLUENT 13.0 are as follows: T1 ¼ 288:49K ¼ 15:34 C; T3 ¼ 309:67K ¼ 36:52 C; T5 ¼ 342:12K ¼ 68:97 C; T7 ¼ 397:02K ¼ 123:87 C;

T2 ¼ 299:84K ¼ 26:69 C; T4 ¼ 313:60K ¼ 40:45 C; T6 ¼ 381:17K ¼ 108:02 C; T8 ¼ 491:13K ¼ 217:98 C; T9 ¼ 508:88K ¼ 235:73 C

In the steady state, the differences between the calculated temperatures are smaller.

F

1658

Fins of Rectangular and Hexagonal Geometry

Finite Volume Method in Heat Conduction, Fig. 10 Temperature at selected nodes of the chimney cross section as a function of time

References 1. Holman JP (2002) Heat transfer, 9th edn. McGraw Hill, New York 2. Kreith F, Black WZ (1980) Basic heat transfer. Harper & Row, New York 3. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing, Washington 4. Jaluria Y, Torrance KE (1986) Computational heat transfer. Hemisphere Publishing/Springer Verlag, Washington/Berlin 5. Taler J, Duda P (2006) Solving direct and inverse heat conduction problems. Springer, Berlin/Heidelberg 6. Versteeg HK, Malalasekera W (2007) An introduction to computational fluid dynamics: the finite volume method, 2nd edn. Pearson Education, Harlow 7. Date AW (2005) Introduction to computational fluid dynamics. Cambridge University Press, Cambridge 8. Dick E (2009) Introduction to finite volume methods in computational fluid dynamics. In: Wendt JF (ed) Computational fluid dynamics, 3rd edn. Springer, Berlin/Heidelberg, pp 275–301 9. Schneider GE (1982) Finite-element methods for conduction. In: Hewitt GF (ed) Handbook of heat exchanger design. Begell House, New York, pp 2.4.8-1–2.4.8.30 10. Lewis RW, Nithiarasu P, Seetharamu KN (2004) Fundamentals of the finite element method for heat and fluid flow. Wiley, Chichester

11. Riley KF, Hobson MP, Bence SJ (2000) Mathematical methods for physics and engineering. Cambridge University Press, Cambridge 12. Taler D (2009) Dynamics of tube heat exchangers. Monograph no.193, AGH University of Science and Technology Press, Cracow (in Polish)

Fins of Rectangular and Hexagonal Geometry Dawid Taler Institute of Engineering and Air Protection, Faculty of Environmental Engineering, Cracow University of Technology, Cracow, Poland

Overview Plate fin and tube heat exchangers are used as economizers in steam power boilers, airconditioning coils, convectors for home heating, waste-heat recovery systems for gas turbines,


a

b

Pl

Ho

t fl

uid

F

L

Flow

M

Pt

Fins of Rectangular and Hexagonal Geometry, Fig. 1 Plate fin and tube heat exchanger with an in-line tube arrangement: (a) cross flow heat exchanger, (b) an in-line tube arrangement

1659

id

ld Co

flu

F a

Flow

b

s

Pl

M

Pt

Pt

r in Flow

do

Fins of Rectangular and Hexagonal Geometry, Fig. 2 Plate fin and tube heat exchanger with a staggered tube arrangement: (a) cross flow heat exchanger, (b) a staggered tube arrangement

δf

cooling towers, air-fin coolers, car radiators, and heater cores, which are used to heat the air within the car passenger compartment [1–9]. Compact heat exchangers with plate fins can be manufactured by electrical or laser welding of the fins to the tube or can also be extruded directly from the tube wall metal. Continuous-plate fins are used extensively in plate fin and tube heat exchangers [1, 3–6]. A recurring component of extended surface is that of a single sheet of metal pierced by round or oval tubes in either an in-line (Fig. 1) or a staggered (Fig. 2) arrangement [4]. In this type of heat exchangers, each fin extends from tube to tube. Marking the symmetry planes between the tubes (Figs. 1a and 2a) gives a rectangle or a hexagon around the tube. It is not possible to find an exact solution for this type of fins. Numerical methods such as the finiteelement method or finite-volume methods can be used to determine the temperature distribution or fin efficiency [8–11].

Pl L

Approximate Determination of Fin Efficiency For the constant heat transfer coefficient over the fin surface, the plate can be divided into imaginary rectangular (Fig. 1b) or hexangular fins (Fig. 2b) [4, 9, 12–15]. Because of the symmetry, the outer circumference of the polygonal fin is thermally insulated. Zabronsky has proposed that the efficiency of the square fin is equal approximately to the efficiency of the circular fin of equal surface area [12]. The accuracy of the method of equivalent circular fin is not always satisfactory, especially for rectangular fins. Carrier and Anderson have shown that the accuracy of the sector method is better [13]. Rich prepared graphs to facilitate the application of the sector method [16]. Schmidt [17, 18] has found mathematical correlations which are accurate and simple. The Schmidt method is based on determining a radius ro,e of a circular fin that has the same

F

1660


Fins of Rectangular and Hexagonal Geometry, Fig. 3 Illustration of the sector method for an in-line tube arrangement: (a) division of the plate fin into imaginary rectangular fins, (b) division of the symmetrical part of the fin into sectors

a

b

Flow

2rin

ro,1

rin

Pl

Pt

ro,n

fin efficiency as the rectangular or hexangular fin. The circular fin efficiency is given by tanhðmrin fÞ mrin f

¼

ð1Þ

where the parameters m and f are defined as

2h kdf

m¼ f¼

rO 1 rin

ð2Þ

r 1 þ 0:35 ln O rin

ð3Þ

The symbols in (1) and (2) denote the following: h, heat transfer coefficient, W/(m2K); k, thermal conductivity of the fin material, W/(mK); rin, ro, inner and outer radius of the fin, m: df, fin thickness, m. For the rectangular fin, Schmidt has developed the following correlation: ro;e ¼ 1:28cðb 0:2Þ1=2 rin

ð4Þ

ro;e ¼ 1:27cðb 0:3Þ1=2 rin

ð6Þ

where L and M are defined in Fig. 2b. The dimensions are selected in such a way that L M. The sector method is much accurate but very laborious [9, 14–16]. It was developed in 1940s [4] and is used to date. Rich prepared graphs included in ASHRAE Handbook [16] which make it easier to determine the efficiency of complex shape fins by the sector method. Rectangular and hexagonal fins may be analyzed by the sector method. In this method, the smallest symmetrical section of the fin (Figs. 3a and 4a) is divided into n sectors (Figs. 3b and 4b). The surface area of the ith sector is Ai. Each sector is considered as a circular fin with the radius ro,i. The outer radius ro,i of each sector is determined by equating the surface area of the sector with the area of the equivalent circular sector. The fin efficiency i of the ith sector is then calculated using the formula for the annular fin efficiency. The fin efficiency for the entire fin is the surface area weighted average of i for each sector

where c¼

M ; rin

b¼

L M

ð5Þ

The dimensions L and M are shown in Fig. 1b, where L is always selected to be greater than or equal to M. The parameter f defined by (3) is evaluated using the equivalent outer radius ro,e instead of ro. Similar correlation has been developed for the hexagonal fin

N P

i Ai ¼ i¼1N P Ai

ð7Þ

i¼1

The fin efficiency determined by the sector method is lower than the actual fin efficiency since only radial heat conduction in the fin is assumed. The circumferential heat flow between


a

b

Flow

Pt

ro,1

rin

Pl

Fins of Rectangular and Hexagonal Geometry, Fig. 4 Illustration of the sector method for a staggered tube arrangement: (a) division of the plate fin into imaginary hexagonal fins, (b) division of the symmetrical part of the fin into sectors

F

1661

ro,n

F a

4.02

b

E

d

g

Fins of Rectangular and Hexagonal Geometry, Fig. 5 Division of plate fin into imaginary hexagonal fins (a) and dimensions of the fin (b)

ϕ

h0AB

Pt /2

B C

4 Tb

1

0

Pt /2

Flow

3 2

D

A

21

0

5 f7.59 h

C 0B

21.38

Pl

the sectors is neglected. Since the heat flow occurs in the direction of the least thermal resistance, then the higher heat flow rate is dissipated from the real fin to the environment. Thus, (7) gives a little smaller fin efficiency in comparison with more sophisticated methods. An example of the application of the of equivalent annulus method and the sector method for determining the efficiency of the hexagonal fin is given in [9]. A plate fin with staggered tube arrangement was divided into hexagonal imaginary fins (Fig. 5a). The dimensions of the analyzed fin are given in Fig. 5b. The efficiency of the fin shown in Fig. 5b was estimated by the sector method. The obtained values of the fin efficiency ¼ 0.9373 agree quite well with the value obtained using the finite element method (FEM) ¼ 0.9380. Also, the value obtained by the method of the equivalent circular radius gave a satisfactory result, ¼ 0.9394, which is also close to the FEM result.

Determining Temperature Distribution and Fin Efficiency Using the CFD Software When the fin geometry is complex, then the temperature distribution and fin efficiency can be determined using the finite element method [9] or the finite control volume method [8, 9]. Commercial software, like ANSYS, FLUENT, or CFX, can be used for this purpose [5, 6, 9, 19–21]. The fin efficiency is defined as the ratio of the heat flow Q_ transferred from the fin to the environment to the heat flow Q_ max dissipated from the isothermal fin with the base temperature Tb to the environment ¼

Q_ Q_ max

ð8Þ

The heat flow Q_ is given by the expression

F

1662


Fins of Rectangular and Hexagonal Geometry, Fig. 6 Rectangular fin used in steam boiler economizers, dimensions are in millimeters

d f = 1.5

rin=19.05mm

h

1

2

38.0

h h

h h Tf

h Tf

r in 3 82.0

ð

Q_ ¼

h T Tf dA

ð9Þ

Al

where the symbol Al denotes the lateral surface area of the fin. When the heat transfer coefficient h and the environment temperature Tf are constant over the fin surface, then (9) can be transformed to Q_ ¼ hAl T Tf

ð10Þ

where T is the area-averaged fin temperature, defined as Ð T ¼

Ne P

TdA

Al Ne P

¼ Ae;i

Te;i Ae;i

i¼1

Al

ð11Þ

i¼1

The symbol T stands for the temperature of the fin surface adjacent to the fluid, Te;i denotes the average temperature of the finite element or finite-volume surface with the area Ae,i exposed to the environment. The fin surface neighboring with the fluid is divided into Ne elements. The maximum heat flow Q_ max , which can be transferred from the fin to the fluid, is Q_ max ¼ hAl Tb Tf

ð12Þ

The procedure developed above can easily be extended to account for the space-dependent heat transfer coefficient. To demonstrate the effectiveness of the presented method, the efficiency of the fin

shown in Fig. 6 was computed using the FLUENT software. The fin base temperature Tb and flue gas temperature Tf (the environment temperature) are 250 C and 650 C, respectively. The fin is made from the low alloy steel 15Mo3 with the thermal conductivity k(T) given by the expression k ¼ 42:773 þ 4:42 102 T 9:59 105 T 2 þ 4:0 108 T 3 where k is expressed in W/(m·K) and T in K. The quarter of the fin was divided into 20,248 finite volumes. First, the threedimensional temperature field was computed, then the fin efficiency was determined. The computed temperature at the characteristic points (Fig. 6) and the fin efficiency are shown in Fig. 7 as a function of the heat transfer coefficient h.

Transient Response of Fins with Complex Shape Transient temperature distributions in continuous fins attached to oval tubes will be calculated using the finite volume – finite element method [9–11]. A system of differential equations of the first order for transient temperature at the nodes will be solved using the Runge–Kutta-Verner method of the fifth order [22, 23]. The developed method can be used in the transient analysis of compact heat exchangers to calculate correctly the heat flow rate transferred from the hot to cold fluid [24].


1663

F

Fins of Rectangular and Hexagonal Geometry, Fig. 7 Fin efficiency , temperatures T1, T2, and T3 at the characteristic points 1, 2, and 3 marked in Fig. 6, and average temperature Tmean ¼ T of the fin as a function of the heat transfer coefficient h

F

Fins are used in heat exchangers to enhance heat transfer on the gas side. First, the mathematical formulation of the FVM-FEM applied for the heat transfer analysis in fins will be presented. Heat balance equations will be set for all interior nodes taking into account heat conduction inside the fin and convective heat transfer on its surface. The boundary conditions of the first, second, and third kind will be accounted for the finite volumes adjacent to the fin boundaries. Subsequently, the transient temperature field in the rectangular fin attached to the oval tube will be computed. The temperature distribution for a step temperature rise in the fin base temperature will be calculated. The fin model will be divided into triangular elements, and then, finite volumes were formed around the nodes by connecting triangle gravity centers with side centers of triangles. A system of differential equations of the first order for temperature at the nodes will be solved using the Runge–Kutta-Verner method of the fifth order. After calculating the temperature distribution, the heat transferred from the fin to the environment and fin efficiency will be

computed and compared with the results obtained by using the commercial software ANSYS 11.0.

Mathematical Formulation of the Finite Volume: Finite Element Method for Determining Transient Temperature in Fins Two-dimensional heat conduction equation for a fin in which temperature difference across the thickness is negligible has the following form: cðTÞ rðTÞ

@T 2h ¼ H q T Tf @t df

ð13Þ

where the heat flux vector q is given by Fourier’s law q ¼ kx ðTÞ

@T @T i ky ðTÞ j @x @y

ð14Þ

The symbols in (13) and (14) denote the following: c, specific heat, J/(kg·K); r, density,

F

1664


kg/m3; T, fin temperature, K or C; Tf, fluid temperature, K or C; t, time, s; h, heat transfer coefficient, W/(m2·K); df, fin thickness, m; kx and ky, thermal conductivity in x and y direction, W/(m·K). The surface integral of (13) over a given region O in 2D space is ð ð @T dA ¼ H q dA cðTÞ rðTÞ @t O

The temperature in the triangle 1-2-3 is approximated by a linear function T ¼ a1 þ a2 x þ a3 y

where the constants a1, a2, and a3 determined from the conditions T ðx1 ; y1 Þ ¼ T1 ; T ðx2 ; y2 Þ ¼ T2 ;

T ðx3 ; y3 Þ ¼ T3 ð20Þ

O

ð

2h T Tf dA df

ð19Þ

are

O

ð15Þ Applying the divergence theorem to (15) gives ð d T VO cðTÞ rðTÞ ¼ q n ds dt G

2h T Tf A O df

ð16Þ

where T is the mean temperature, n - is the outward unit normal on s, ds is the arch length along G, and AO is the surface area of the region O. Equation (16) can be expressed for the finite region O1aoc1 as ðo df A 123 df d T1 cðT1 Þ rðT1 Þ ¼ q n ds 3 2 dt 2 df 2

a

ðc q n ds

A 123 h T Tf 3

1 ½ðx2 y3 x3 y2 ÞT1 2A123 þ ðx3 y1 x1 y3 ÞT2 þ ðx1 y2 x2 y1 ÞT3 1 a2 ¼ ½ðy2 y3 ÞT1 2A 123 þ ðy3 y1 ÞT2 þ ðy1 y2 ÞT3 1 a3 ¼ ½ðx3 x2 ÞT1 2A 123 þ ðx1 x3 ÞT2 þ ðx2 x1 ÞT3 a1 ¼

ð21Þ The integrals in (17) can be evaluated using the Fourier law (14) and approximate temperature distribution (19) ðo q n ds ¼ kx ðTo Þ a

þ ðy3 y1 ÞT2 þ ðy1 y2 ÞT3 xa xo þ ky ðTo Þ 2A 123 ½ðx3 x2 ÞT1 þ ðx1 x3 ÞT2 þ ðx2 x1 ÞT3

o

ð17Þ where the symbol A123 denotes the surface area of the triangle 1-2-3 (Fig. 1). The lateral surfaces 1-a and 1-c are assumed to be thermally insulated. Equation (13) will be written for nonlinear transient heat conduction equation in the medium with anisotropic and temperature-dependent thermal conductivity @T @ @T cðTÞ rðTÞ ¼ kx ðTÞ @t @x @x @ @T 2h ky ðTÞ þ T Tf @y @y df

ya yo ½ðy2 y3 ÞT1 2A 123

ð22Þ ðc yc yo q n ds ¼ kx ðTo Þ ½ðy2 y3 ÞT1 2A 123 o

þ ðy3 y1 ÞT2 þ ðy1 y2 ÞT3 xc xo ky ðTo Þ 2A 123 ½ðx3 x2 ÞT1 þ ðx1 x3 ÞT2 þ ðx2 x1 ÞT3

ð23Þ ð18Þ If the lateral surface 1–3 is heated by the heat flux qs and the surface 1–2 is heated by


1665

convection, then additional integrals must be taken into account in (17):

y

F

3

ðc q n ds ¼ qs s1c

ð24Þ

1

b

and b2

ða T1 þ Ta q n ds ¼ h Tf s1a 2 1

c

b1

b2

! 2 T1 þ T1 þT 2 ¼ h Tf s1a 2 3T1 T2 þ ¼ h Tf s1a ð25Þ 4 4

noc b1 0 a1

a2 a1

qs

nao

2

a2 a α Tf

where the side lengths s1c and s1a are given by (Fig. 8) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1c ¼ ðxc x1 Þ2 þ ðyc y1 Þ2 ð26Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð27Þ s1a ¼ ðxa x1 Þ2 þ ðya y1 Þ2

Fins of Rectangular and Hexagonal Geometry, Fig. 8 Control volume for energy conservation equation in the region 1-a-o-c-1, which is a part of the finite volume associated with the node 1

Substituting integrals (22) and (23) into (17) and considering additional integrals (24) and (25) which account heat transfer on the sides 1-c and 1-a give

The sides 1-a and 1-c are insulated in internal finite volumes, thus the last two terms in (28) should be omitted.

A123 df dT1 y c y a df ¼ kx ðTo Þ 3 2 dt 2A123 2 ½ðy2 y3 ÞT1 þ ðy3 y1 ÞT2 þ ðy1 y2 ÞT3 x c x a df þ ky ðTo Þ ½ðx2 x3 ÞT1 þ ðx3 x1 ÞT2 2A123 2 A 123 h T f To þ ðx1 x2 ÞT3 þ 3 df df 3T1 T2 þ qs s1c þ h Tf þ s1a 2 4 4 2

cðT1 ÞrðT1 Þ

ð28Þ where the symbol To stands for the temperature at the gravity center of the triangle 1-2-3 (Fig. 8): To ¼

T1 þ T2 þ T3 3

ð29Þ

1 0

x

Temperature Distribution in the Rectangular Fin Attached to the Oval Tube The dimensions of the analyzed fin, expressed in millimeters, are shown in Fig. 9. The fin was divided into 19 finite volumes (Fig. 10). The lateral surfaces 4–19, 1–13, 2–3 are thermally insulated, while on the surfaces 4-6-3 and 1-5-2, convection heat transfer occurs. The computations were carried out for the following data: c ¼ 896 J/(kg·K), r ¼ 2707 W/(m·K), k ¼ 207 W/(m·K), df ¼ 0.08 mm, Tf ¼ 0 C, Tb ¼ 100 C, T0 ¼ 0 C, h ¼ 50 W/(m2·K). The temperature distribution was also calculated using ANSYS, v.11.0. The division of the fin

F

F

1666


The temperatures T1 ¼ 96.355 C, T2 ¼ 86.359 C, T3 ¼ 85.337 C, and T4 ¼ 93.328 C at the nodes 1, 2, 3, and 4 are in good agreement with the results obtained using the ANSYS software: T1 ¼ 96.403 C, T2 ¼ 86.903 C, T3 ¼ 84.479 C, and T4 ¼ 93.021 C. The transient response of the fin illustrates Figs. 13 and 14. The coincidence of the results obtained by the developed method and ANSYS is good despite the coarse finite-volume mesh (Fig. 10) used in the FVM-FEM. Using the computed temperatures, the steadystate fin efficiency fin can be evaluated:

into finite elements is depicted in Fig. 11. An inspection of the results presented in Fig. 12 indicates that high temperature gradients are observed in the region close to the fin base.

Ð fin ¼

h T Tf dA

Afin

h Afin Tb Tf

¼

Tfin Tf Tb Tf

ð30Þ

where the symbol Tfin stands for the mean temperature of the fin and Afin, total fin area on which the heat transfer occurs. Based on the FEM computations, the following formula for the fin efficiency was estimated:

Fins of Rectangular and Hexagonal Geometry, Fig. 9 Rectangular fin attached to oval tube

y

3

2

9 10 8

7 16

Fins of Rectangular and Hexagonal Geometry, Fig. 10 Division of the fin model into finite volumes for the FVM-FEM

6 12 4

15

17 18 19

11

5

14 x

13

1


1667

F

Fins of Rectangular and Hexagonal Geometry, Fig. 11 Division of the fin model into finite elements for the FEM

F

Fins of Rectangular and Hexagonal Geometry, Fig. 12 Temperature distribution on the fin surface obtained by the FEM

fin ¼

1 þ 6:45184 104 h ; 1 þ 2:761444 103 h þ 5:16617 107 h2 2 0 h 250 W= m K

ð31Þ

where the heat transfer coefficient h is expressed in W/(m2·K). The fin efficiency was also determined based on the temperature distribution obtained by the developed method

F

1668


Fins of Rectangular and Hexagonal Geometry, Fig. 13 Time changes of fin temperature at nodes 1 and 2

Nl P A123; i To; i Tf þ Al; j Tl; j Tf j¼1 i¼1 ¼ Afin h Tb Tf N P

fin

ð32Þ

where the symbols denote the following: A123,i, surface area of the ith triangle; To,i, fin temperature at the gravity center of the ith triangle; N, number of triangles; Al,j, area of the jth lateral surface of the fin with the thickness df; Tl; j , mean temperature of the jth lateral surface of the fin with the thickness df ; Nl, number of lateral surfaces with the thickness df. The comparison of the fin efficiency calculated from the expressions (31) and (32) is shown in Table 1. The accuracy of the present

method is very satisfactory. In spite of the coarse finite-volume mesh used in present method (Fig. 10), the coincidence of the calculated efficiencies is very good. The developed method for the analysis of the transient response of the rectangular plate fins is very effective. It can be used for determining the transient response and steady-state efficiency of fins attached to the oval as well as to the circular tubes. The agreement between results obtained by the present and the FEM method is very satisfactory, despite a coarse mesh used in the presented finite volume – finite element method. The developed method can be used in the transient analysis of compact heat exchangers to calculate correctly the heat flow rate transferred from the hot to cold fluid.


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F

Fins of Rectangular and Hexagonal Geometry, Fig. 14 Time changes of fin temperature at nodes 3 and 4

F

Fins of Rectangular and Hexagonal Geometry, Table 1 Comparison of the fin efficiency obtained from the FEM (31) and presented FVM-FEM (32) h, W/(m2·K) Present method FEM, (31)

0 1 1

25 0.9491 0.9502

50 0.9081 0.9060

75 0.8712 0.8664

References 1. Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, Hoboken 2. Brandt F (1985) W€arme€ ubertragung in Dampferzeugern und W€armeaustauschern. FDBR Fachverband Dampfkessel, Beh€alterund Rohrleitungsbau E.V., Vulkan Verlag, Essen 3. Web RL (1994) Principles of enhanced heat transfer. Wiley, New York 4. McQuiston FC, Parker JD, Spitler JD (2005) Heating, ventilating, and air conditioning. Analysis and design, 6th edn. Wiley, Hoboken

100 0.8376 0.8308

125 0.8070 0.7986

150 0.7789 0.7692

175 0.7531 0.7424

5. Taler D (2002) Theoretical and experimental analysis of heat exchangers with extended surfaces. Volume 25, Monograph 3, Polish Academy of Sciences, Cracow Branch, Commission of Motorization, Cracow 6. Taler D (2009) Dynamics of tube heat exchangers. Monograph 193, AGH UWND Publishing House, Cracow (in Polish) 7. Taler J, Przybylin´ski P (1982) Heat transfer by round fins of variable conduction and non-uniform heat transfer coefficient. Chem Process Eng 3:659–676 8. Rup K, Taler J (1989) W€arme€ ubergang an Rippenrohren und Membranheizfl€achen. BrennstoffW€arme–Kraft 41:90–95

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9. Taler J, Duda P (2006) Solving direct and inverse heat conduction problems. Springer, Berlin 10. Acharya S, Baliga B, Karki K, Murthy JY, Prakash C, Vanka SP (2007) Pressure-based finite-volume methods in computational fluid dynamics. ASME J Heat Trans 129:407–424 11. Taler D, Korzen´ A, Madejski P (2011) Determining tube temperature in platen superheater tubes in CFB boilers. Rynek Energii 93:56–60 (in Polish) 12. Zabronsky H (1955) Temperature distribution and efficiency of a heat exchanger using square fins on round tubes. ASME Trans J Appl Mech 22:119 13. Carrier WH, Anderson SW (1944) The resistance of heat flow through finned tubing. Heating, Piping, and Air Conditioning 16(5):304 14. Shah RK, Bell JK (1997) Heat exchangers. In: Kreith F (ed) The CRC Handbook of mechanical engineering. CRC Press, Boca Raton, pp 118–164 15. Shah RK, Sekulic´ DP (2003) Fundamentals of heat exchanger design. Wiley, Hoboken 16. Handbook ASHRAE (1997) Fundamentals volume. American Society of Heating, Refrigerating and AirConditioning Engineers, Atlanta 17. Schmidt TE (1949) Heat transfer calculations for extended surfaces. Refrig Eng 4:351–357 18. Schmidt ThE (1950) Die W€armeleistung von berippten Oberfl€achen. Abh. Deutsch. K€altetechn. Verein. Nr. 4, C.F. M€ uller, Karlsruhe 19. Taler D, Cebula A (2004) Modeling of flow and thermal processes in compact heat exchangers. Chemical and Process Engineering 25:2331–2342 (in Polish) 20. Taler D, Cebula A (2010) A new method for determination of thermal contact resistance of a fin-to-tube attachment in plate fin-and-tube heat exchangers. Chem Process Eng 31:839–855 21. Taler J, Taler D, Sobota T, Cebula A (2012) Theoretical and experimental study of flow and heat transfer in a tube bank. In: Petrova VM (ed) Advances in engineering research, vol 1. Nova Science, New York, pp 1–60 22. IMSL Math/Library (1994) International mathematical and scientific library. Visual Numerics. Houston, Texas 23. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipes in Fortran 77, 2nd edn. Cambridge University Press, Cambridge 24. Taler D (2011) Direct and inverse heat transfer problems in dynamics of plate fin and tube heat exchangers. In: Belmiloudi A (ed) Heat transfer, mathematical modelling, numerical methods and information technology. InTech, Rijeka, pp 77–100, free online edition: www. intechopen.com

Fins of Straight and Circular Geometry

Fins of Straight and Circular Geometry Dawid Taler Institute of Engineering and Air Protection, Faculty of Environmental Engineering, Cracow University of Technology, Cracow, Poland

Overview Longitudinal or radial fins are attached to a bare surface to enhance heat flow rate from a surface to an adjacent fluid. Fins are used when the convection heat transfer coefficient h is low, as is frequently the case for gases such as air or flue gas [1–9]. Fins are commonly used to augment heat transfer from electronic components, condensers and evaporators in air-conditioning systems, automobile radiators and air heaters, engine and compressor cylinders. Finned tubes are also used widely in heat recovery steam generators situated after gas turbines in combined-cycle gas and steam turbine power plants and as feed water heaters (economizers) in steam power boilers. Finned tubes are used extensively in air-fin coolers in which hot process liquids, usually water, flow inside tubes, and atmospheric air is circulated outside the finned tubes by natural, forced, or induced draft [10]. Transverse highfin tubular elements are found in such diverse places as dry cooling towers, indirect-fired heaters, waste-heat recovery systems for catalytic reactors, convectors for home heating, and gascooled nuclear reactors. High fins are made of aluminum or copper because of high thermal conductivity and fin efficiencies when exposed to heating or cooling by gas at moderate temperature and atmospheric pressure. Plate fin and tube heat exchangers can be made by inserting the tubes through the sheet metal strips with stamped or drilled holes and expanding the tubes slightly to cause pressure at the tube-to-strip contacts and to reduce thermal


1671

contact resistance. The bond resistance can be neglected if the tubes and strips are brazed together. If hot gases with high temperatures flow over the fins and steam or water flows inside the tubes, the extended surface usually consists of a chromium steel tubes and a ribbon which is helically wound and continuously welded to the tube. Electrical arc or resistance welding is used to attach the fins to the tubes. Finned superheater tubes are fabricated by laser welding recently [11, 12]. If finned steel tubes are manufactured by winding a metal ribbon in tension around the tube without welding, then to provide good thermal contact, they can be galvanized. In the following, heat flow in a fin will be analyzed to determine the temperature distribution along the fin and, hence, to evaluate its efficiency. Because fins are thin, the temperature difference across the fin thickness can be neglected. These assumptions allow the conduction along the fin to be treated as one- or two-dimensional, which significantly simplifies the analysis. Steady and transient heat flow in straight and annular (radial, circular) fins of uniform thickness will be analyzed. Thermal conductivity of the fin material may depend on temperature.

General Equation of Heat Conduction in Fins The aim is to derive a differential equation of transient heat transfer in fins with arbitrary shapes (Fig. 1) under the assumption that temperature across the fin thickness is constant. In other words, one should neglect temperature difference on the fin thickness and derive formulas for temperature distribution and fin efficiency. By assuming that fin temperature remains constant within the fin’s cross section and changes only in the direction of x axis, the heat balance for control volume A(x)Dx has the form Q_ x ¼ Q_ xþDx þ Q_ k

ð1Þ

Tf

F

Qk= hPΔx(T−Tf)

A(x)

Q

Q

x

x+Δx

T(x) Δx

x

Fins of Straight and Circular Geometry, Fig. 1 Heat flow through fins with arbitrary shapes

Heat flows Q_ are expressed by the following formulas: dT dT ; Q_ x ¼ kA ; Q_ xþDx ¼ kA dx x dx xþDx Q_ k ¼ h PDx T Tf ð2Þ By substituting (2) into (1), one obtains dT kAdT dx xþDx kA dx x Dx

h P T Tf ¼ 0

ð3Þ

where A(x) is a cross-section area of a fin perpendicular to the direction of heat flow through a fin, P is the fin circumference at a point with x coordinate. If Dx ! 0, then (3) assumes the form d dT kA hP T Tf ¼ 0 dx dx

ð4Þ

For a constant thermal conductivity k and constant cross-section A, (4) can be written in the form d2 T m 2 T Tf ¼ 0 2 dx

ð5Þ

where m2 ¼

hP kA

ð6Þ

F

F

1672


Two boundary conditions are necessary in order to determine temperature distribution in a fin of height L. The first condition is assigned at the point x ¼ 0 at the base of the fin and the second at the end of the fin at point x ¼ L. In practical computations, it is usually assumed that fin-base temperature Tb is constant and equal to a temperature of a surface on which the fin is mounted; that is, it is assumed that fins do not disturb temperature distribution in a construction element to which they are attached. The fin tip is usually regarded as being thermally insulated, since the surface area of the tip is considerably smaller than the area of fin’s side surfaces; therefore, one can neglect the heat flow transmitted by the tip. Assuming that heat exchange takes place on the tip of the fin, the boundary conditions have the form Tjx¼0 ¼ Tb k

ð7Þ

dT ¼ hw Tjx¼L Tf dx x¼L

ð8Þ

where hw is the heat transfer coefficient from the tip to surroundings, while temperature Tf is the temperature of a fluid that surrounds the fin. It is usually assumed that hw ¼ h or hw ¼ 0, when a fin tip is thermally insulated. _ either Fin efficiency is a ratio of a heat flow Q, absorbed or dissipated by an actual fin, to a maximal heat flow Q_ max , which the fin could either absorb or dissipate. Maximal heat flow Q_ max occurs when temperature of the fin is uniform within its entire volume and is equal Tb. Heat flow Q_ can be calculated as a flow that is transferred through the base of the fin or as a transferred by lateral surfaces of the fin ðL dT _ Q ¼ kA ¼ h T Tf P dx dx x¼0

ð9Þ

Fin efficiency is formulated as ¼

Q_ Q_ max

ð11Þ

Assuming that h, P, and Tf are independent of their position, formulas for efficiency of fins with standard shapes are not very complicated. If the fin tip is not insulated, then additional heat flow through the tip should be allowed for in (9).

Straight Fin of Constant Thickness Temperature distribution and efficiency of a simple fin with constant thickness, under the assumption that fin tip is thermally insulated, will be determined. Next, to consider thermal exchange through the fin tip, its height will be increased by half of the fin’s thickness. Then, formulas for the fin-tip temperature, heat flow transferred by the fin surface, and fin efficiency will be derived. If fin-base temperature is Tb, while fin tip is thermally insulated, boundary conditions have the form Tjx¼0 ¼ Tb

ð12Þ

dT ¼0 dx x¼L

ð13Þ

In the case of a simple fin, shown in Fig. 2, circumference P, on which thermal exchange takes place, measures P ¼ 2(w + df), while the cross-section area with regard to the direction of thermal conduction is A ¼ wdf. Fin parameter m2 defined by (6) has, in the given case, the form

m2 ¼

2hðw þ dt Þ 2h k w dt k dt

ð14Þ

0

ðL

Q_ max ¼ h Tb Tf P dx 0

ð10Þ

since usually, df < < w. Once the new variable is introduced y ¼ T Tf

ð15Þ


P

Tb

1673

dT _ Q ¼ kA dx x¼0 m sinh mðL xÞ ¼ kwdf Tb Tf cosh mL x¼0 sffiffiffiffiffiffiffi 2h Tb Tf tanh mL ¼ kwdf kdf

Tf W

δf

A L

ð23Þ

Lc

Q_ max ¼ h 2wL Tb Tf

0 X

Fins of Straight and Circular Geometry, Fig. 2 Simple fin of constant cross section

Equation (4) and boundary conditions (12), (13) can be written in the following way: d2 y m2 y ¼ 0 dx2 yjx¼0 ¼ yb ;

F

y b ¼ T b Tf

dy ¼0 dx x¼L

ð16Þ ð17Þ ð18Þ

Solution of the homogenous equation (5) has the following form: y ¼ C1 emx þ C2 emx

ð19Þ

From boundary conditions (7) and (8), two algebraic equations are obtained: C1 þ C2 ¼ yb

ð20Þ

mC1 emL mC2 emL ¼ 0

ð21Þ

Once constants C1 and C2 are determined from (20) and (21) and substituted into (19), one obtains the following after transformations: yðxÞ TðxÞ Tf cosh mðL xÞ ¼ ¼ yb cosh mL T b Tf

ð22Þ

The heat flow Q_ and Q_ max will be determined in order to define fin efficiency

ð24Þ

The efficiency of a rectangular fin with constant cross section is qffiffiffiffiffi 2h kwdf kd Tb Tf tanh mL Q_ f ¼ ¼ h 2wL Tb Tf Q_ max ¼

tanh mL mL

ð25Þ

where m is given by (14). Hyperbolic trigonometric functions expressed using the following formulas:

are

ex ex ex þ ex ; cosh x ¼ ; 2 2 sinh x ex ex tanh x ¼ ¼ cosh x ex þ ex sinh x ¼

In formulas (22) and (25), fin-tip heat exchange is not taken into consideration. It can be considered approximately when the fin height is increased by df/2 (Fig. 2). Once the fin height substitute is introduced, Lc Lc ¼ L þ

df 2

temperature distribution and fin efficiency are expressed as yðxÞ TðxÞ Tf cosh mðLc xÞ ¼ ¼ yb cosh mLc Tb Tf ¼ where m ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2h=kdf

tanh mLc mLc

ð26Þ

ð27Þ

F

1674


To illustrate an application of the derived formulas, an example is presented. The following values are assumed for the calculation: fin material – copper with the thermal conductivity k ¼ 390 W/(mK), fin thickness df ¼ 0.5 mm, height L ¼ 7.5 cm, width w ¼ 0.7 m, fin-base temperature Tb ¼ 80 C, air temperature Tf ¼ 20 C, heat transfer coefficient h ¼ 10 W/(m2K). After substitution, one obtains the following: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 10 ¼ 10:12741=m m¼ 390 0:0005 df 1 Lc ¼ L þ ¼ 0:075 þ 0:0005 ¼ 0:07525 m 2 2 Fin-tip temperature T ðLc Þ and fin efficiency are

1 T ðLc Þ ¼ Tb Tf þ Tf cosh mLc 1 þ 20 ¼ ð80 20Þ coshð10:1274 0:07525Þ ¼ 65:97 o C tanh mLc tanhð10:1274 0:07525Þ ¼ 0:8428 ¼ ¼ 10:1274 0:07525 mLc

Heat flow dissipated to surrounding air is Q_ ¼ Q_ max ¼ h 2wLc Tb Tf ¼ 0:8428 10 2 0:7 0:07525 ð80 20Þ ¼ 53:273 W

Temperature Distribution and Efficiency of a Circular Fin of Constant Thickness The aim is to derive a differential equation for a circular fin of constant thickness from a general (5) of heat transfer in fins and to determine formulas for temperature distribution in the circular fin for a heat flow transferred by the fin and for the fin efficiency. In the case of a round fin shown in Fig. 3, surface area of a fin cross section is A ¼ 2p rdt, while circumference, on which thermal exchange occurs, is P ¼ 4p r.

Tb Tb = T(rin)

Lc L

Tb

δf/2

r δf

F

Tf A(r) rp rin ro roc

Fins of Straight and Circular Geometry, Fig. 3 Round fin with constant thickness

Parameter m is defined as rffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffi hP h4pr 2h ¼ ¼ m¼ kA k2prdf kdf

ð28Þ

Differential (5) assumes the following form: 1 d dT 2h r T Tf ¼ 0 r dr dr kdf

ð29Þ

Once excess temperature y ¼ T – Tf is introduced and an allowance for (28) made, (29) can be written in a form d2 y 1 dy m2 y ¼ 0 þ dr 2 r dr

ð30Þ

It is a modified Bessel equation, for which general solution has the form yðrÞ ¼ C1 I0 ðmr Þ þ C2 K0 ðmr Þ

ð31Þ

Constants C1 and C2 will be determined from boundary conditions yjr¼rin ¼ yb ;

yb ¼ Tb Tf

dy ¼0 dr r¼r2

ð32Þ ð33Þ


1675

F

Once constants are determined and substituted into (31), one obtains a formula for temperature distribution y (r) in a fin TðrÞ Tf y ¼ yb Tb Tf ¼

K0 ðmrÞI1 ðmro Þ þ I0 ðmr ÞK1 ðmro Þ I0 ðmrin ÞK1 ðmro Þ þ K0 ðmrin ÞI1 ðmro Þ ð34Þ

Heat flow Q_ transferred through the fin is

F

dT _ Q ¼ kAb dr r¼rin ¼ 2pkrin df yb m

K1 ðmrin ÞI1 ðmro Þ I1 ðmrin ÞK1 ðmro Þ K0 ðmrin ÞI1 ðmro Þ þ I0 ðmrin ÞK1 ðmro Þ

ð35Þ Since maximal flow Q_ max is Q_ max ¼ hAfin Tb Tf 2 yb ¼ h2p ro2 rin

Fins of Straight and Circular Geometry, Fig. 4 Fin efficiency of circular fin with constant thickness as a function of the parameter mL for following radius ratios ro =rin : 1.2, 1.6, 2.0, 3.0, and 5.0

ð36Þ

fin efficiency, then, can be determined from formula below Q_ _ Qmax 2rin K ðmrin ÞI1 ðmro Þ I1 ðmrin ÞK1 ðmro Þ 1 ¼ 2 2 K ðmr ÞI ðmr Þ þ I ðmr ÞK ðmr Þ m ro rin 0 in 1 o 0 in 1 o mL mLu mL mLu K1 u1 I1 u1 I1 u1 K1 u1 2 mL mLu mL mLu ¼ mLð1 þ uÞ K0 u1 I1 u1 þ I0 u1 K1 u1

¼

ð37Þ where L ¼ ro rin ; u ¼ ro =rin : The heat flow through the tip can be taken into account by replacing the radius ro in formulas (34)–(37) by a slightly larger radius roc ¼ ro + df/2. As in the case of a simple fin, the fin length Lc is larger than L and is equal to Lc ¼ L + df/2. Figures 4 and 5 present the efficiency of circular fins as a function of the product mL and ratio ro/rin. The fin-tip temperature, fin efficiency, and heat flow dissipated by the fin will be calculated

Fins of Straight and Circular Geometry, Fig. 5 Fin efficiency of circular fin with constant thickness as a function of the parameter mL for following radius ratios ro =rin : 1.4, 1.8, 2.5, 4.0, and 6.0

using the following data: fin-base temperature Tb ¼ 90 C, temperature of surroundings Tf ¼ 20 C, rin ¼ 12.5 mm, ro ¼ 28.5 mm, df ¼ 0.4 mm, material of a fin – aluminum with heat conduction coefficient k ¼ 205 W/(mK), heat transfer coefficient on the fin surface

F

1676

h ¼ 70 W/(m2K). Take into account heat exchange on a fin tip by increasing fin height L to Lc ¼ L + df/2. After substitution, one obtains mrin ¼ 41:32 0:0125 ¼ 0:5165 df roc ¼ ro þ ¼ 0:0285 þ 0:0002 ¼ 0:0287 2 mroc ¼ 1:1859 Evaluating the Bessel functions I0 ðmroc Þ ¼ I0 ð1:1859Þ ¼ 1:3837;


Bessel function values, present in formulas for temperature distribution and circular fin efficiency, can be evaluated using the tables of Bessel functions [13] or calculated by means of library procedures [14–16].

Approximated Calculation of a Circular Fin Efficiency The following approximate formulas can be used for determining circular fin efficiency: (a) According to Schmidt [17, 18]

K1 ðmroc Þ ¼ K1 ð1:1859Þ ¼ 0:4443 K0 ðmroc Þ ¼ K0 ð1:1859Þ ¼ 0:3247

S ¼

I1 ðmroc Þ ¼ I1 ð1:1859Þ ¼ 0:7035; I0 ðmrin Þ ¼ I0 ð0:5165Þ ¼ 1:0678; K0 ðmrin Þ ¼ K0 ð0:5165Þ ¼ 0:8977

T ðroc Þ ¼ 73:38o C

In order to determine the fin efficiency using tables of special functions [13], it is necessary to calculate K1 ðmrin Þ ¼ K1 ð0:5165Þ ¼ 1:5887

Lc ’ ¼ 1 þ 0:35 ln 1 þ ; rin sffiffiffiffiffiffiffi 2h Lc ¼ roc rin ; m ¼ kdf

B ¼

Fin efficiency is

2rin tanh mLc 2rin þ Lc mLc tanh mLc ðtanh mLc Þp 1þ C ð40Þ 2mrin ðmrin Þn

where C ¼ 0:071882;

p ¼ 3:7482;

n ¼ 1:4810

1:5887 0:7035 0:2670 0:4443 0:8977 0:7035 þ 1:0678 0:4443

¼ 0:8188 Heat flow transferred through the fin is 2 2 rin Q_ ¼ Q_ max ¼ h2p roc Tb Tf ¼ 0:8188 70 2 p 0:02872 0:01252 ð90 20Þ ¼ 16:82 W

ð39Þ

If > 0.5, then efficiency calculated using formula (38) does not differ more than 1 % from the exact value. (b) According to Brandt [2]

I1 ðmrin Þ ¼ I1 ð0:5165Þ ¼ 0:2670

2 0:0125 ¼ 41:32ð0:02872 0:01252 Þ

ð38Þ

where

one obtains T ðroc Þ 20 0:3247 0:7035 þ 1:3837 0:4443 ¼ 90 20 1:0678 0:4443 þ 0:8977 0:7035 ¼ 0:7624

tanh mLc ’ mLc ’

ð41Þ Maximal error from efficiency determination by means of formula (40) is smaller than 0.6 % from an error made when determining efficiency by means of (37). (c) Formula according to [19] H ¼

1 1þ

2 1 3 ðmLc Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi roc =rin

ð42Þ


1677

Formula (42) gives good results, when > 0.75. Efficiency of the same fin like in previous example will be calculated using the approximate formulas. The following results are obtained: (a) According to Schmidt (38)

m ¼ 41:32 1=m; rin ¼ 0:0125 m;

Lc ¼ roc rin ¼ 0:0162 m; S ¼

roc ¼ 0:0287 m

’ ¼ 1:2909

tanhð41:32 0:0162 1:2909Þ ¼ 0:8082 41:32 0:0162 1:2909

Since a real value of the efficiency is ¼ 0.8188, relative error, then, comes to eS ¼

ð S Þ 0:8082 0:8188 100% ¼ 100% 0:8188

¼ 1:295%

(b) According to Brandt (40)

B ¼

2 0:0125 0:584575 2 0:0125 þ 0:0162 41:32 0:0162 0:584575 0:071882 2 41:32 0:0125 ! ð0:584575Þ3:7482 ¼ 0:8162 ð41:32 0:0125Þ1:4810

1þ

The relative error is eB ¼

B 0:8162 0:8188 100% ¼ 100% 0:8188

¼ 0:312%

(c) According to formula (42) 1 qffiffiffiffiffiffiffiffiffiffi ¼ 0:8155 H ¼ 1 1 þ 3 ð41:32 0:0162Þ2 0:0287 0:0125

F

The relative error is eH ¼

H 0:8155 0:8188 100% ¼ 100% 0:8188

¼ 0:409% From the comparison of the results presented above, one can see that the least accurate result is obtained from Schmidt formula (38). Brandt formula (40) allows for the most accurate calculation of circular fin efficiency; however, the amount of work required to obtain the results is not much smaller than in the case of the exact formula (37). Equation (42) is both simple yet accurate.

Transient Heat Transfer in Fins There are situations in which a study of the transient heat transfer in fins is necessary. This is true for fins used in heat recovery steam generators after gas turbines during start-up and shutdown operations in intermittently operating heat exchangers and in finned heat exchangers with automatic control systems. The transient response of a longitudinal fin of rectangular profile and a radial convection fin has been studied for time-dependent boundary conditions as step change in base temperature or step change in the temperature of the environment using exact or approximate analytical methods [1]. In this entry, the transient heat transfer in fins with temperature-dependent physical properties will be studied using the finite volume method (FVM) [9]. At first, the transient response of the straight fin of rectangular profile is analyzed. Next, the analysis for circular fins that experience time changes in base or fluid temperature is presented.

Straight (Longitudinal) Fin of Constant Thickness The partial differential equation governing the transient heat transfer in the fin can be written as

F

1678


@ @T 2h @T kðTÞ T Tf ¼ cðTÞrðTÞ @x @x df @t

δf/2 N+2

ð43Þ

Δx=L/N

This equation is to be solved subject to the boundary conditions

N+1

Tjx¼0 ¼ Tb ðtÞ

ð44Þ

h

@T jx¼L ¼ 0 @x

ð45Þ

i+1

symetry plane

F

and the initial condition ð46Þ

i L

T jt¼0 ¼ T0 ðxÞ

Tf

Δx

and

The heat balance equation for the ith control volume has the following form:

i−1

df d Ti df kðTi1 Þ þ kðTi Þ D xcðTi ÞrðTi Þ ¼ 2 2 dt 2 Ti1 Ti df kðTiþ1 Þ þ kðTi Þ Tiþ1 Ti þ 2 Dx 2 Dx þ D x hðTf Ti Þ; i ¼ 2; . . . ; N þ 1

2

X

ð47Þ Denoting by ki ¼ kðTi Þ; ci ¼ cðTi Þ; ri ¼ rðTi Þ; and introducing ki ¼ kðTi Þ ¼ ki =ðci ri Þ, (47) can be written in the form " d Ti ki kðTi1 Þ þ kðTi Þ ¼ ðTi1 Ti Þ 2 2kðTi Þ dt ðDxÞ kðTiþ1 Þ þ kðTi Þ ðTiþ1 Ti Þ 2kðTi Þ # 2ðDxÞ2 h ðTf Ti Þ ; i ¼ 2; . . . ; N þ df k i

þ

ð48Þ Introducing a fictional node N + 2 (Fig. 6) and taking into account the boundary condition (45), which can be approximated by the central difference quotient as TNþ2 TN ¼0 2D x

ð49Þ

1 0 Fin base

Fins of Straight and Circular Geometry, Fig. 6 Division of the straight fin into (N + 1) control volumes; D x ¼ L=N

give TNþ2 ¼ TN

ð50Þ

Equation (48) is also valid for i ¼ N + 1. Writing (48) for i ¼ N + 1 and accounting for (50), one obtains " d TNþ1 kNþ1 kðTN Þ þ kðTNþ1 Þ ¼ ðTN TNþ1 Þ kðTNþ1 Þ dt ðDxÞ2 # 2ðDxÞ2 h T TNþ1 þ df kNþ1 f ð51Þ


The temperature T1 is known and equal to the fin-base temperature Tb, which may be time dependent, that is, T1 ¼ Tb ðtÞ

ð52Þ

The initial condition (46) is replaced by the set of initial conditions for the node temperatures Ti ¼ T0 ðxi Þ;

xi ¼ ði 1Þ D x;

i ¼ 1; . . . ; N þ 1

ð53Þ The initial problem for the set of ordinary differential equations, defined by (48) and (51), the boundary condition (52), and the initial conditions (53) can be solved using one of many methods available for solving ordinary differential equations. The system of differential equations of the first order for the transient temperatures at the nodes was solved using the Euler explicit method. Almost the same results were obtained by more accurate method like the Runge–Kutta-Verner method of the fifth order [14]. The heat transfer rate through the surface of the fin and the fin efficiency can be evaluated in terms of the surface temperature distribution

¼0 TðtÞ

0:5T1 þ 0:5TNþ1 þ

L

ffi

N P

1 @ @T 2h @T kðTÞr T Tf ¼ cðTÞrðTÞ r @r @r df @t ð58Þ This equation is to be solved subject to the boundary conditions Tj r¼rin ¼ Tb ðtÞ

ð59Þ

@T jr¼ro ¼ 0 @r

ð60Þ

and

T jt¼0 ¼ T0 ðrÞ

"

Ti ð55Þ

Unlike the steady-state case, the base heat transfer rate @T Q_ ¼ df 1m jx¼0 @x

Assuming one-dimensional conduction and temperature-dependent thermal properties, the differential heat conduction equation can be written as

ð61Þ

The heat balance equation for the ith control volume (Fig. 7) has the following form:

i¼2

N

Circular Fin of Constant Thickness

ð54Þ

The average fin temperature is given by

T dx

Next, the temperature in the circular fin will be analyzed.

and the initial condition

2 h T Tf dx T Tf ¼ ðtÞ ¼ 0 Tb Tf 2h L Tb Tf ÐL

ÐL

F

1679

ð56Þ

does not equal the fin surface heat transfer rate Q_ ¼ 2L 1m h T Tf ð57Þ

p

Dr ri þ 2

2

Dr ri 2

2 #

df d Ti cðTi ÞrðTi Þ 2 dt

df kðTi1 Þ þ kðTi Þ Ti1 Ti 2 2 Dr df kðTiþ1 Þ þ kðTi Þ Tiþ1 Ti þ 2pðri þ D r=2Þ 2 # 2 Dr " 2 2 Dr Dr þ p ri þ ri h ðTf Ti Þ; 2 2

¼ 2pðri D r=2Þ

i ¼ 2; . . . ; N þ 1

ð62Þ Denoting by ki ¼ kðTi Þ; ci ¼ cðTi Þ; ri ¼ rðTi Þ and introducing ki ¼ kðTi Þ ¼ ki =ðci ri Þ, (62) can be written in the form

F

F

1680


d Ti ki kðTi1 Þ þ kðTi Þ Dr ðTi1 Ti Þ ¼ 1 2kðTi Þ 2ri d t ðDrÞ2 kðTiþ1 Þ þ kðTi Þ Dr 1þ ðTiþ1 Ti Þ þ 2kðTi Þ 2ri # 2ðDrÞ2 h þ ðTf Ti Þ ; i ¼ 2; . . . ; N d f ki

N+2 N+1 i+1 Δr

i

h

Introducing a fictional node N + 2 (Figs. 7, 8) and taking into account the boundary condition (60), which can be approximated by the central difference quotient as

ri+Δr/2 ri

ri−1

1

ri−Δr/2

2

rN+1

ð63Þ

ri+1

Tf

i−1

r1=rin

kt

ð64Þ

TNþ2 ¼ TN

ð65Þ

give

rp

hin

TNþ2 TN ¼0 2D r

Fins of Straight and Circular Geometry, Fig. 7 Division of the circular fin into (N + 1) control volumes; D r ¼ ðrNþ1 r1 Þ=N

Fins of Straight and Circular Geometry, Fig. 8 Thermal properties of the fin material-AISI 409 steel

Equation (63) is also valid for i ¼ N + 1. Writing (63) for i ¼ N + 1 and accounting for (50), one obtains


1681

d TNþ1 kNþ1 kðTN Þ þ kðTNþ1 Þ ¼ ðTN TNþ1 Þ kðTNþ1 Þ dt ðDrÞ2 # 2ðDrÞ2 h þ ðT TNþ1 Þ df kNþ1 f ð66Þ The temperature T1 is known and equal to the fin-base temperature Tb, which may be time dependent, that is, T1 ¼ Tb ðtÞ

ð67Þ

The initial condition (46) is replaced by the set of initial conditions for the node temperatures Ti ¼ T0 ðri Þ; ri ¼ r1 þ ði 1Þ D r; i ¼ 1; . . . ; N þ 1

ð68Þ

The fin efficiency is determined as follows:

¼

Dr 2p r1 þ Dr Q_ 4 2 h T1 Tf ¼ 2 p rNþ1 r12 h Tb Tf Q_ max N P 2pri Dr h Ti Tf i¼2 þ 2 p rNþ1 r12 h Tb Tf Dr 2p rNþ1 Dr h TNþ1 Tf 4 2 2 þ p rNþ1 r12 h Tb Tf ð69Þ

¼

r1 þ Dr 4

T1 Tf Dr Q_ ¼ 2 2 rNþ1 r1 Tb Tf Q_ max

2ri Ti Tf þ rNþ1 Dr TNþ1 Tf 4 2 þ i¼2 rNþ1 r12 Tb Tf N P

the boundary condition (67), and the initial conditions (68) can be solved using one of many methods available for solving ordinary differential equations. The system of differential equations of the first order for the transient temperatures at the nodes was solved using the Euler explicit method [16]. Almost the same results were obtained by more accurate method like the Runge–Kutta-Verner method of the fifth order [14]. Transient fin temperature and fin efficiency will be calculated for two fins (Table 1). Thermal properties of the fin material are temperature dependent (Fig. 8). The results of the calculations are shown in Fig. 9. Since the height of the fin B is larger than the height of the fin A (Table 1), the transient response time of the fin B is longer in comparison with the response time of the fin A. The fin temperature and fin efficiency were also calculated using exact analytical formulas (34) and (37) for the constant thermal conductivity evaluated at the mean temperature Tm ¼ (T1 + TN+1)/2 (Table 2). The agreement between numerical and analytical results is excellent.

Fins of Straight and Circular Geometry, Table 1 Data for calculations of fins A and B Fin Material

Rearranging (69) gives

ð70Þ The initial problem for the set of ordinary differential equations, defined by (63), (66),

F

Outer radius ro, mm Inner radius rin, mm Thickness df, mm Heat transfer coefficient h, W/(m2K) Environment temperature Tf, C Fin-base temperature Tb, C Initial temperature T0, C Number of nodes Time step Dt, s

A Stainless steel AISI 409 24.85 16.85 1.2 120

B Stainless steel AISI 409 41.25 22.25 1.0 120

450

450

250

250

250 21 0.01

250 21 0.01

F

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Fins of Straight and Circular Geometry, Fig. 9 Transient response of the fins A and B for the step change in temperature of the environment from 250 C to 450 C

Fins of Straight and Circular Geometry, Table 2 Comparison of the numerical and analytical results for the fins A and B Fin Finite volume method (after 120 s) Tip temperature TN+1, C Efficiency Analytical solution Tip temperature T(ro), C Thermal conductivity km ¼ k [0.5(T1 + TN+1)], W/(mK) Efficiency

A

B

291.58 0.8514

394.48 0.4478

291.60 26.69

394.65 26.92

0.8513

0.4477

References 1. Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, Hoboken 2. Brandt F (1985) W€arme€ ubertragung in Dampferzeugern und W€armeaustauschern. Essen, FDBR Fachverband Dampfkessel, Beh€alterund Rohrleitungsbau E.V., Vulkan Verlag, Essen 3. Web RL (1994) Principles of enhanced heat transfer. Wiley, New York 4. McQuiston FC, Parker JD, Spitler JD (2005) Heating, ventilating, and air conditioning. Analysis and design, 6th edn. Wiley, Hoboken

5. Taler D (2002) Theoretical and experimental analysis of heat exchangers with extended surfaces, vol 25, Monograph 3, Polish Academy of Sciences, Cracow Branch, Commission of Motorization, Cracow 6. Taler D (2009) Dynamics of tube heat exchangers. Monograph 193. AGH UWND Publishing House, Cracow (in Polish) 7. Taler J, Przybylin´ski P (1982) Heat transfer by round fins of variable conduction and non-uniform heat transfer coefficient. Chemical and Process Engineering 3:659–676 (in Polish) 8. Rup K, Taler J (1989) W€arme€ ubergang an Rippenrohren und Membranheizfl€achen. BrennstoffW€arme-Kraft 41:90–95 9. Taler J, Duda P (2006) Solving direct and inverse heat conduction problems. Springer, Berlin 10. Hintzen FJ, Benzing W (1999) Zunehmender Einsatz der Trockenk€ uhlung in internationalen Kraftwerksprojekten. VGB Kraftwerkstechnik 79:125–129 11. Adamiec J, Wie˛cek M, Gawrysiuk W (2010) Manufacturing of boiler components by high power disc laser welding. In: Proceedings of the 11th international conference on boiler technology 2010, 19–22 October 2010, Szczyrk, Poland, Silesian University of Technology, Institute for Power Plant Machinery, Gliwice, Z. 25, vol1, pp 49–70 (in Polish) € 12. Reichel HH (2001) Schaden an einem Uberhitzer einer GuD-Anlage durch unzureichende Phasentrennung in der Trommel. VGB PowerTech 81:80–86

Flow-Thermoelastic Vibration 13. Janke E, Emde F, Lo¨sch F (1960) Tafeln ho¨herer Funktionen. B. G. Teubner, Stuttgart 14. IMSL Math/Library (1994) International mathematical and scientific library. Visual Numerics, Houston 15. MathCAD 7 Professional (1997) MathSoft Inc., Cambridge, MMA 16. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1996) Numerical recipies in Fortran 77, 2nd edn. Cambridge University Press, New York 17. Schmidt ThE (1950) Die W€armeleistung von berippten Oberfl€achen. Abh. Deutsch. K€altetechn. Verein. Nr. 4, C.F. M€ uller, Karlsruhe 18. Schmidt ThE (1949) Heat transfer calculations for extended surfaces. Refrig Eng April, pp 351–357 19. Gnielinski V, Zukauskas A, Skrinska A (1992) Banks of plain and finned tubes. In: Hewitt GF (ed) Handbook of heat exchanger design. Begell House, New York, pp 2.5.3–1–2.5.3–16

First Law of Thermodynamics ▶ Energy and First Law of Thermodynamics

First-Order Shear and Normal Deformable Theory ▶ Thermally Induced Vibration, Circular Plates

First-Order Shear Deformation Theory ▶ Thick Plates, Reissner–Mindlin Theory, Statical Problems

1683

F

Flash Diffusivity Methods ▶ Impulse Method for Determining Thermal Diffusivity of Solids

Flash Method of Determining Thermal Diffusivity ▶ Impulse Method for Determining Thermal Diffusivity of Solids

Flexible Substrates ▶ Thermal Stresses of Thin Films on Flexible Substrates

Flexural Strength (sf) ▶ Thermal Shock Resistance (TSR) and Thermal Fatigue Resistance (TFR) of Refractory Materials. Evaluation Method Based on the Dynamic Elastic Modulus

First-Order Theory

Flexure

▶ Large Plate Deflections, von Ka´rmań Theory, Dynamical Problems ▶ Large Plate Deflections, von Ka´rmań Theory, Statical Problems

▶ Saint-Venant’s Problem for Cosserat Elastic Shells

Five-Elementary-Functions Method ▶ Multiply Connected Bodies, Thermal Stresses

Flow-Thermoelastic Vibration ▶ Fluid-Thermal Structural Coupling in the Modeling of Carbon Nanotubes

F