CONVECTION HEAT TRANSFER IN CONCENTRIC

EHT 17(1) #12204

Experimental Heat Transfer, 17:19–29, 2004 Copyright © Taylor & Francis Inc. ISSN: 0891-6152 print/1521-0480 online DOI: 10.1080/08916150490246528

CONVECTION HEAT TRANSFER IN CONCENTRIC ANNULI J. Dirker and H. Van der Vyver Department of Mechanical Engineering, Rand Afrikaans University, Johannesburg, South Africa

J. P. Meyer Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, South Africa

A comparative study is presented of literature involving convective heat transfer in annuli. It is shown that more research is needed in the area of convective heat transfer correlations in concentric annuli, as little agreement is found among existing correlations. A correlation predicting Nusselt numbers in annuli with ratios ranging from 1.7 to 3.2 was developed experimentally for water as fluid. The correlation has an accuracy of 3% in terms of experimental values for a Reynolds number range, based on the hydraulic diameter, of 4,000 to 30,000. The correlation was also compared to numerical predictions.

Many researchers have investigated heat transfer in annuli, particularly in order to obtain correlations that can describe the Nusselt numbers for a wide range of flow conditions and annular diameter ratios. Having direct correlations meant that the timeconsuming process of finding the annular convective heat transfer coefficient by means of, for instance, a linear regression analysis such as the Wilson plot method [1], could be replaced. The Wilson plot analysis method requires a great amount of experimental data from which, by means of an iterative process, a Sieder and Tate type of heat transfer correlation is developed. A summary of some correlations available in the literature, intended for calculating Nusselt numbers in concentric annuli, is given in Table 1. Most of these equations are functions of the annular diameter ratio, the Reynolds number, and the Prandtl number, and correspond with the Dittus-Boelter type of form. The correlations contained in Table 1 and which were developed for water as flow medium were compared for an arbitrary case with an annular diameter ratio of 2 and a Prandtl number of 3.37 and is shown in Figure 1. All correlations predict an almost linear increase in Nusselt number with an increase in the Reynolds number. Compared to the other predictions, the equation by Foust and Christian [2] overpredicts the Nusselt number by approximately a factor of 3. When the predictions of Foust and Christian [2] are omitted, a difference in predicted values of Received 18 December 2002; accepted 21 May 2003. Address correspondence to Prof. J. P. Meyer, Department of Mechanical and Aeronautical Engineering, University of Pretoria, Pretoria, 0002, South Africa. E-mail: [email protected] 19

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J. DIRKER ET AL.

NOMENCLATURE a Ci Co D1 D2 Dh Di h k k Nu n P

annular diameter ratio (= D2 /D1 ) inner tube convective heat transfer correlation coefficient—Wilson plot annulus convective heat transfer correlation coefficient—Wilson plot diameter of outer wall of inner tube, m diameter of inner wall of outer tube, m hydraulic diameter of annulus (= D2 − D1 ), m inner diameter of inner tube, m convective heat transfer coefficient, W/m2 K thermal conductivity, W/m K turbulent kinetic energy Nusselt number exponent of Prandtl number exponent of Reynolds number in Wilson plot function

Pr Re ε ζ µ

Prandtl number Reynolds number dissipation rate correlation function viscosity, N s/m2

Subscripts ave average based on the hydraulic diameter of the Dh annulus f based on film temperature Tf = 21 (Tave + Tw ) i inner tube side o annulus side w wall

±20% relative to the average predicted value is found to exist. The same trend was found to be true for a wide range of annular diameter ratios and Prandtl numbers. No indication was found in the literature of the existence of an accurate heat transfer correlation for concentric annuli. The purpose of this investigation was thus to deduce a correlation with which accurate predictions could be made of average Nusselt numbers at the inner annular wall under turbulent flow conditions using water. EXPERIMENTAL FACILITY Eight different concentric tube-in-tube heat exchangers, each with a different annular diameter ratio and cross-sectional area, were used to perform experimental tests on. Refer to Table 2 for some dimensions. Water in the annulus was heated by circulating hot water through the inner tube. The heat exchangers, each with an effective length of about 6 m, were constructed from hard-drawn refrigeration copper tubing and were operated in a horizontal counterflow arrangement. All heat exchangers were thermally insulated form the ambient by using standardized pipe insulators. Concentric annular cross-sectional areas were maintained over the entire lengths of each heat exchanger, by supporting the inner tubes with sets of radial pins producing starlike supporting structures. The size and position of the supporting pins were carefully calculated to minimize possible sagging of the inner tube. Pins, with diameter of 0.6 mm, were placed symmetrically to minimize possible unbalanced flow patterns. In the case of the smallest annulus, the supporting structures occupied at most 6.5% of the crosssectional flow area. Temperature measurements were facilitated by means of K-type thermocouples fixed on the outside surfaces of entry and exit regions of the heat exchangers. At each measuring location either two or three thermocouples were used to obtain a more curate temperature value. Temperature errors were usually less than 0.1 K. Measuring points

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Dh =

Crookston et al. [11]

for a ≤ 5

for a ≥ 5

10, 16, 31

1.232, 1.463, 1.694

Not specified

1–14.3

1–∞

1–10

17,000–100,000

30,000–390,000

Not specified

104 –3 × 105

104 –106

Not specified

12,000–220,000

Not specified

Not specified 1.65, 2.45, 17

3,000–60,000

Not specified

Not specified

Reynolds number range

1.2–1.84

1.18–6,800

1.18–6,800

Diameter ratio range

∗ Original equations were rewritten as to have the Reynolds and Nusselt numbers based on the annular hydraulic diameter: D = D − D . h 2 1

3/4 NuDh ,f = 0.23a 1/4 ReD Pr 1/3 h

1/3

NuDh ,f = 0.0200a 0.5 Re0.8 Dh ,f Pr f

h

Stein and Begell [10]

ζ =1

Dh =

n NuDh = 0.023Re0.8 D Pr

∗ Nu

0.06759a 0.16 ζ Re0.8 Dh (a + 1)0.2 0.6 a−5 with ζ = 1 + 7.5 (a + 1)ReDh

Results listed in tables for various conditions

∗ Nu

0.04a Re0.8 Pr 0.4 (a + 1)0.2 Dh 1 µ 0.14 3 NuDh = 0.023Re0.8 Dh Pr µw 2 ln a − a 2 + 1 n NuDh = 0.023 Re0.8 Dh Pr a − 1/a − 2a ln a µ 0.14 n NuDh = 0.023a 0.45 Re0.8 Dh Pr µw

0.15 (a − 1)0.2 Re0.8 Pr 1/3 Dh = 0.038a Dh

µ 0.14 µw o µ 0.14 1/3 NuDh = 0.03105a 0.15 (a − 1)0.2 Re0.8 Pr Dh µw

∗ Nu

Correlation

Dittus-Boelter [9]

Petukhov and Roizen [8]

Kays and Leung [7]

Wiegand et al. [6]

Monrad and Pelton [5]

McAdams [4]

Foust and Christian [2]

McAdams [4]

Davis [3]

Author(s)

Table 1. Equations available from the literature describing the Nusselt number in a smooth concentric annulus during forced convection

Air

Water

Not specified

Air

Fluids: µmaterial ≤ 2µwater Not specified

Water, air

Not specified

Water

All media

All media

Medium

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J. DIRKER ET AL.

Figure 1. Different predictions of the Nusselt number as a function of the Reynolds number for a = 2 and Pr = 3.37.

were sufficiently insulated from the ambient by means a foam rubber substance. Volumetric flow rates were measured by using semirotary circular piston-type displacement flow meters with a measuring accuracy of greater than 98%. Hot water supplied by an on-site hot-water storage tank (1,000 L), fitted with an electric resistance water heater, was circulated through the inner tube by means of a positive displacement pump and then returned to the storage tank which exhibited sufficient thermal buffer characteristics. The hot-water flow rates were controlled with a hand-operated bypass system. Similarly, cold water was supplied from a cold-water storage tank (1,000 L) connected to a chiller and pumped through the annulus by means of two series-connected centrifugal pumps to ensure high flow rates through the test sections before being returned to the storage tank.

Table 2. Diameters of tubes used for the different heat exchangers Heat exchanger

D1 (mm)

D2 (mm)

a

1 2 3 4 5 6 7 8

6.35 6.35 6.35 6.35 6.35 6.35 9.45 19.05

11.15 14.10 17.30 20.30 26.50 32.00 32.00 32.00

1.76 2.22 2.72 3.20 4.17 5.04 3.39 1.68

CONVECTIVE HEAT TRANSFER IN CONCENTRIC ANNULI

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EXPERIMENTAL PROCEDURE Experimental tests [12] were performed at a wide range of inner tube and annular flow rate combinations. A wide annular flow rate range was covered in the turbulent flow regime in order to facilitate the development of an accurate annular heat transfer correlation. At first the inner tube flow rate was fixed at an arbitrary level while changing the annular flow rate. For each annular flow rate, adequate time was given for steady-state conditions to be established before inlet and outlet temperatures were captured by means of a data logger and flow rates were measured with both displacement flow meters and variable-area flow meters. The process was repeated for an array of inner tube flow rates, which produced a wide range of inner tube and annular flow rates. Hot- and cold-water inlet temperatures were maintained at in the vicinity of 50 and 10◦ C, respectively. A high level of accuracy in the experimental data was maintained. More than 90% of all data points represented conditions with energy balance errors of less than 1% between the inner tube and annular heat transfer rates. A Reynolds number range, based on the hydraulic diameter, of 2,600 to 35,000 was covered in experiments performed on the eight heat exchangers setups. PROCESSING OF DATA To facilitate the Wilson plot process the internal and annular Nusselt numbers can be written by means of Eqs. (1) and (2), respectively: Nui =

hi Di 1/3 = Ci Re0.8 i Pr i ki

µ µw

ho Dh 1/3 = Co RePo,Dh Pr o Nuo = ko

0.14 (1) i

µ µw

0.14 (2) o

P , Ci , and Co are added to account for geometry influences. For the inner tube the exponent of the Reynolds number was kept at 0.8 as proposed in the literature [1]. With the modified Wilson plot method (Briggs and Young [1]), the values of P , Ci , and Co were obtained as functions of the annular diameter ratio. The value of Ci was found to be constant, namely, in the region of 0.027, as can be expected. The values of P and Co are given in Figures 2 and 3. By using these values, more than 95% of all data points were predicted within 3% accuracy by the Wilson plot-obtained correlations for the different heat exchangers. All Wilson plot correlations exhibited a median error of less than or in close proximity to 1%. Standard deviations for error values were less than 2%. DERIVATION OF CORRELATION P and Co showed a dependence on the annular diameter ratio. The value of P exhibited a downward trend when the annular diameter ratio was increased. (See Figure 2.) On the other hand, the value of Co had an upward trend for an increasing annular diameter ratio. (See Figure 3.)

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Figure 2. P values obtained from Wilson plot analyses.

Results obtained for annular diameter ratios of 4.17 and 3.39, which are encircled in Figures 2 and 3, did not agree with the general trend of the rest of the heat exchangers. These annular cases were rebuilt and the experimental tests repeated. The values of P and Co were reaffirmed. A great possibility exists that even though these cases were rebuilt, concentricity could not be maintained due to the great distance between the inner tube and outer tube of these annuli. From the available experimental results, the behavior of P and Co can be described relatively precisely for annular diameter ratios below 3.2. For ratios greater than 3.2, this is unfortunately not the case, and more experimental data are needed. Unfortunately, commercial-sized tubes, which could produce more annular ratios ranging from 3 to 5, are not readily available, making the investigation process difficult.

Figure 3. Co values obtained from Wilson plot analyses.


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Using results for annular ratios of below 3.2, it was possible to describe the trend mathematically by evaluating different curve-fits. Equations (3) and (4) exhibited the best accuracies and are indicated in Figures 2 and 3 as dotted lines. Linear-type curve approximations were also considered, but due to a loss in accuracy, Eqs. (3) and (4) were decided upon. P = 1.013e−0.067a Co =

0.003a 1.86 0.063a 3 − 0.674a 2 + 2.225a − 1.157

(3) (4)

By substituting (3) and (4) into (2), a correlation for the prediction of the Nusselt number is produced. The validity of the resulting correlation for the prediction of Nusselt numbers was tested with experimental data from all heat exchangers having an annular diameter ratio of less than 3.2. All predictions were within 3% of experimentally obtained values. Correlations from the literature (Table 1) were also compared with measured Nusselt numbers in the annular diameter range 1.7–3.2. This is demonstrated in Figure 4.

Figure 4. Predictions of correlations cited in the literature in terms of measured Nusselt numbers.

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Figure 5. Comparison between the deduced correlation and correlations from the literature for a wide range of annular diameter ratios.

Only a small number of the predictions were within 10% of the measured Nusselt numbers, except the current correlation, which predicts all values within ±3%. A large prediction scatter is also exhibited by most of the correlations. Two main prediction bands are present. The first band, being in proximity to the measured values, consists of predictions by a correlation recommended by McAdams [4], and the Dittus-Boelter correlation [9]. The second band is located outside the 25% range from the measured values and consists of predictions of various authors. Predictions of Foust and Christian [2] had the highest deviance from the measured Nusselt numbers. The deduced correlation in this article was also compared to correlations in the literature for an arbitrary thermal condition over a wide range of annular diameter ratios and Reynolds numbers. For a case where the Reynolds number is 15,000 and the Prandtl number is 3.36, the result is shown in Figure 5. For small annular diameter ratios, up to about 2.5, the predictions correspond well with the correlation by Dittus and Boelter [9], and an equation by McAdams [4]. In the region of an annular ratio of 3.5, close agreement exists with the correlation of Stein and Begell [6]. These trends were found to be true for a wide range of Reynolds numbers and Prandtl numbers. COMPUTATIONAL FLUID DYNAMICS VERIFICATION OF THE CORRELATION Figure 6 shows the grid of the tube and tube model that was used. The simulation package makes use of the finite-volume method. The equations used were the mass and momentum conservation equations (Navier-Stokes equations) and the heat transfer equations for conduction and convection. The inner tube inside diameter is 8 mm, with


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Figure 6. Grid used for the CFD simulations.

an outer diameter of 10 mm. The inside diameter of the outside tube is 16 mm. The annular diameter ratio was 1.6, which is slightly lower than the recommended values. The length of the heat exchanger is 50 mm. Boundaries were added to the surfaces of the model to simulate the boundary conditions. An adiabatic wall boundary was attached to the outer fluid’s surface. No heat transfer or fluid flow can take place over this boundary. Inlet boundaries were placed at the hot- and cold-water inlets. At the outlets, standard outlet boundaries were placed, which specify the direction of flow. The surfaces without boundaries were made wall boundaries that allowed for heat transfer. The k–ε turbulent model (k = turbulent kinetic energy, ε = dissipation rate) for high Reynolds number was selected. The values can be entered as either k–ε values or as an equivalent turbulent intensity and entrance mixing length. The latter two parameters were selected. The outer tube is not modeled; its presence is accounted for by introducing an adiabatic boundary on the outside of the outer fluid. The amount of cells used was 237,000; 120,000 cells for the modeling of the inner fluid, 27,000 cells for the copper tube, and 90,000 cells for the annulus fluid. The inner fluid temperature was taken as 355 K (82◦ C), with a density of 970.2 kg/m3 . The inlet turbulence intensity was assumed to be 0.05 and the length was 0.0002 m. Thus the flow is not fully developed at the inlet, but becomes fully developed very early in the flow. The same condition applies to the annulus flow with the same turbulence intensity and a slightly higher length of 0.00045. The increase in entrance length was done to keep the ratio of entrance length to cross-sectional area the same for the inner and annulus flow. The inlet temperature was 283 K (10◦ C) with a density of 999.2 kg/m3 . The simulations were done for inner Reynolds numbers of 11,184, 17,894, 31,315, and 44,735 (corresponding to inlet velocities of 0.5, 0.8, 1.4, and 2 m/s, respectively). The annulus Reynolds numbers used were 4,576, 12,814, 22,425, and 32,035 (corresponding to inlet velocities of 1, 2.8, 4.9, and 7 m/s, respectively). The inlet temperature of the inner tube was taken as 82◦ C and for the annulus, the inlet temperature was 10◦ C. For each inner Reynolds number, four simulations were done for every annulus Reynolds number. There were thus 16 simulations in total.

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J. DIRKER ET AL.

Figure 7. Comparison between the deduced correlation and CFD data for an annular ratio of 1.6.

A heat balance between the inner and annulus flow was done on the data, and the average error in heat balance was 5.1%. The Nusselt numbers of the computational fluid dynamics (CFD) results were compared with the results predicted by the Dittus-Boelter, Sieder, and Tate and Petukhov correlations [13], and the average error was 13.6% for the annulus with respect to the CFD results were found. When compared with the correlation found in this study, the average error was only 9%. These results are displayed in Figure 7. CONCLUSION As was expected, it was found that the convective heat transfer correlation for an annulus is dependent on the annular diameter ratios. A correlation was deduced from experimental results that predicts Nusselt numbers accurately within 3% from the measured values for diameter ratios between 1.7 and 3.2 and a Reynolds number range of 4,000 to 30,000. For small annular diameter ratios of below 2.5, the correlation agreed closely with those by Dittus and Boelter [9], and McAdams [3]. For higher annular diameter ratios it approaches a correlation by Stein and Begell [6]. The correlation was further validated with CFD results. REFERENCES 1. D. E. Briggs and E. H. Young, Modified Wilson Plot Technique for Obtaining Heat Transfer Correlations for Shell and Tube Heat Exchangers, Chem. Eng. Prog. Symp., vol. 65, pp. 35–45, 1969. 2. A. S. Foust and G. A. Christian, Non-boiling Heat Transfer Coefficients in Annuli, AIChE J., vol. 36, pp. 541–554, 1940.


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3. E. S. Davis, Heat Transfer and Pressure Drop in Annuli, Trans. ASME, pp. 755–760, October 1943. 4. W. H. McAdams, Heat Transmission, McGraw-Hill, New York, 1954. 5. C. C. Monrad and J. F. Pelton, Heat Transfer by Convection in Annular Spaces, AIChE J., vol. 38, pp. 593–611, 1942. 6. J. H. Wiegand, E. L. McMillen, and R. E. Larson, Discussion on: Annular Heat Transfer Coefficients for Turbulent Flow, AIChE J., vol. 41, pp. 147–153, 1945. 7. W. M. Kays and E. Y. Leung, Heat Transfer in Annular Passages—Hydrodynamically Developed Turbulent Flow with Arbitrarily Prescribed Heat Flux, Int. J. Heat Mass Transfer, vol. 6, pp. 537–557, 1963. 8. B. S. Petukhov and L. I. Roizen, Generalized Relationships for Heat Transfer in Turbulent Flow of Gas in Tubes of Annular Section, High Temp., vol. 2, pp. 65–68, 1964. 9. F. W. Dittus and L. M. K. Boelter, University of California, Berkeley, Publications on Engineering, vol. 2, p. 443, 1930. 10. R. P. Stein and W. Begell, Heat Transfer to Water in Turbulent Flow in Internally Heated Annuli, AIChE J., vol. 4, pp. 127–131, 1958. 11. R. B. Crookston, R. R. Rothfus, and R. I. Kermode, Turbulent Heat Transfer with Annuli with Small Cores, Int. J. Heat Mass Transfer, vol. 11, pp. 415–426, 1968. 12. J. Dirker, Heat Transfer Coefficient in Concentric Annuli, M.Ing. thesis, Rand Afrikaans University, Johannesburg, South Africa, 2002. 13. J. P. Holman, Heat Transfer, McGraw-Hill, London, 1992.