OPTIMALLY STAGGERED FINNED CIRCULAR AND ELLIPTIC TUBES IN FORCED CONVECTION

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1 OPTIMALLY STAGGERED FINNED CIRCULAR AND ELLIPTIC TUBES IN FORCED CONVECTION R. S. Matos a, T. A. Laursen b, J. V. C. Vargas a, and A. Bejan c, a Universidade Federal do Paraná Departamento de Engenharia Mecânica Centro Politécnico Bairro Jardim das Américas CP:19011, Curitiba, Paraná, Brasil rudmar@demec.upr.br b Duke University Department o Civil and Environmental Engineering Durham, NC, , USA c Duke University Department o o Mechanical Engineering & Materials Science Durham, NC, , USA ABSTRACT This work presents a three-dimensional (-D) numerical and experimental geometric optimization study to maximize the total heat transer rate between a bundle o inned tubes in a given volume and a given external low both or circular and elliptic arrangements, or general staggered conigurations. The optimization procedure started by recognizing the design limited space availability as a ixed volume constraint. The experimental results were obtained or circular and elliptic conigurations with a ixed number o tubes (1), starting with an euilateral triangle coniguration, which itted uniormly into the ixed volume with a resulting maximum dimensionless tube-to-tube spacing S/ = 1.5, where S is the actual spacing and b is the smaller ellipse semi-axis. Several experimental conigurations were built by reducing the tube-to-tube spacings, identiying the optimal spacing or maximum heat transer. Similarly, it was possible to investigate the existence o optima with respect to other two geometric degrees o reedom, i.e., tube eccentricity and in-to-in spacing. The results are reported or air as the external luid in the laminar regime, or Re 100 and 15, where is the ellipses smaller axis length. Circular and elliptic arrangements with the same low obstruction cross-sectional area were compared on the basis o maximum total heat transer. This criterion allows one to uantiy the heat transer gain in the most isolated way possible, by studying arrangements with euivalent total pressure drops independently o the tube cross section shape. This paper reports three-dimensional (- D) numerical optimization results or inned circular and elliptic tubes arrangements, which are validated by direct comparison with experimental measurements with good agreement. Global optima with respect to tube-to-tube spacing, eccentricity and in-toin spacing ( S/ 0.5, e 0.5 and or Re 100 and 15, respectively) were ound and reported in general dimensionless variables. A relative heat transer gain o up to 19% is observed in the optimal elliptic arrangement, as compared to the optimal circular one. The heat transer gain, combined with the relative material mass reduction o up to % observed in the optimal elliptic arrangement in comparison to the circular one, show the elliptical arrangement has the potential or a considerably better overall perormance and lower cost than the traditional circular geometry. INTRODUCTION The optimization o industrial processes or maximum utilization o the available energy (exergy) has been a very active line o scientiic research in recent times. The increase in energy demand in all sectors o the human society reuires an increasingly more intelligent use o available energy. Many industrial applications reuire the use o heat exchangers with tubes arrangements, either inned or non-inned, unctioning as heat exchangers in air conditioning systems, rerigeration, heaters, radiators, etc. Such devices have to be designed according to the availability o space in the device containing them. A measure o the evolution o such euipment, thereore, is the reduction in size, or in ocuppied volume, accompanied by the maintenance or improvement o its perormance. Hence, the problem consists o identiying a coniguration that provides maximum heat transer or a given space (Bejan, 000). The main ocus o this work is on the experimental and numerical geometric optimization o staggered inned circular and elliptic tubes in a ixed volume to obtain global optima with respect to tube-to-tube spacing, eccentricity and in-to-in spacing. In this work, a three-dimensional (-D) numerical optimization procedure or inned circular and elliptic arrangements is conducted and validated by means o direct comparison to experimental measurements to search the optimal geometric parameters in general staggered inned circular and elliptic conigurations or maximum heat transer. Circular and elliptic arrangements with the same low obstruction cross-sectional area are then compared on the basis o maximum total heat transer. Appropriate nondimensional groups are deined and the optimization results reported in dimensionless charts. THEORY A typical our-row tube and plate in heat exchanger with a general staggered coniguration is shown in Fig. 1. Fowler and Bejan, (1994) showed that in the laminar regime, the low through a large bank o cylinders can be simulated 65

2 accurately by calculating the low through a single channel, such as that illustrated by the unit cell seen in Fig. 1. Because o the geometric symmetries, there is no luid exchange or heat transer between adjacent channels, or at the top and side suraces. At the bottom o each unit cell, no heat transer is expected across the plate in midplane. In Fig. 1, L, H and W are the length, height and width (tube length) o the array, respectively. The ins are identical, where t is the thickness and, is the in-to-in spacing. W t H u u 1 u tube x z y L in unit cell (S+)/ air low u (t +)/ Figure 1. Arrangement o inned elliptic tubes, and the three dimensional computational domain. The governing euations are the mass, momentum and energy euations which were simpliied in accordance with the assumptions o three-dimensional incompressible steady-state laminar low with constant properties, or a Newtonian luid (Bejan, 1995): U U 0 X Y Z ( 1 ) U U X Y Z P 1 X Re L X Y Z ( ) U U X U U Y U Z P 1 U U U ( ) Y Re L X Y Z U U X U U Y U Z P 1 U U U ( 4 ) Z Re L X Y Z U U X Y Z 1 ( 5 ) Pe L X Y Z The symmetries present in the problem allow the solution (computational) domain to be reduced, to one unit cell, represented by the extended domain shown in Fig. 1, o height (S/ b), and width (/ + t /). In Es. (1) - (5), dimensionless variables have been deined based on appropriate physical scales as ollows: x, y,z p X, Y, Z ; P ( 6 ) L u U, U, U 1 u1, u u, u ; T Tin T T U U Re and Pe ( 7 ) where (x, y, z) are the Cartesian coordinates, m; p the pressure, N/m ; the luid density, kg/m ; u the ree stream velocity, m/s; (u, v, w) the luid velocities, m/s; T the temperature, K; T in unit cell inlet temperature, K; T w the tubes surace temperature, K; L the array length in the low direction, m; the luid kinematic viscosity, m /s and the luid thermal diusivity, m /s. The solution domain o Fig. 1 is composed by the external luid and hal o the solid in. The solid-luid interace is included in the solution domain such that mass, momentum and energy are conserved through the domain. Es. (1) - (5) model the luid part o the domain. Only the energy euation needs to be solved in the solid part o the domain, accounting or the actual properties o the solid material. The dimensionless energy euation or the solid in is written as: 1 Re s L X w Y Z where a dimensionless time is deined by in ; ( 8 ) t L/u, t is the time, and s is the solid in thermal diusivity, m /s. For steady-state solutions is assumed to be zero. The solution to Es. (1) - (8) subject to appropriate boundary conditions or the extended domain o Fig. 1 delivers the velocities (luid) and temperature (luid and solid) ields. The objective is to ind the optimal geometry, such that the volumetric heat transer densityis maximized, subject to a volume constraint. The engineering design problem starts by recognizing the inite availability o space, i.e., an available space L H W as a given volume that is to be illed with a heat exchanger. To maximize the volumetric heat transer density means that 66

3 the overall heat transer rate between the luid inside the tubes and the luid side the tubes will be maximized. Next, the optimization study proceeds with the identiication o the degrees o reedom (variables) that allow the maximization o the overall heat transer rate between the tubes and the ree stream, Q. Three geometric degrees o reedom in the arrangement are identiied in this way, i.e.: (i) the spacing between rows o tubes, S; (ii) the tubes eccentricity, e, and (iii) the in-to-in spacing,. The behavior o S, e and at the extremes indicate the possibility o maximum Q in the intervals, 0 S Sm, 0 e 1 and 0 W. A comparison criterion between elliptic and circular arrangements with the same low obstruction cross-sectional area is adopted, i.e., the circular tube diameter is eual to two times the smaller ellipse semi-axis o the elliptic tube. This criterion was also adopted in previous studies Bordalo and Saboya, (1999), Saboya and Saboya (001), Rocha et al. (1997), and Matos et al. (001). To complete the problem ormulation, the ollowing boundary conditions are then speciied or the extended three-dimensional computational domain in agreement with Fig. : (F) (A) U U 0; 1; 0 ( 9 ) (B) and (C) U U 0; 0 Z Z Z ( 10 ) (D) and (E) U U 0; 0 ( 11 ) Y Y Y (F) U U 0 X X X X ( 1 ) (G) U U 0; 1 ( 1 ) (H) U U 0; 0 Z ( 14 ) In order to represent the actual low with boundary conditions (A) and (F), two extralengths need to be added to the computational domain, upstream and downstream, as shown in Fig.. The actual dimensions o these extralengths need to be determined by an iterative numerical procedure, with convergence obtained according to a speciied tolerance. Once the geometry o the extended computational domain represented by the unit cell o Fig. is speciied, Es. (1) - (14) deliver the resulting velocities, pressure and temperature ields in the domain. The dimensionless overall thermal conductance is described by Matos (00): (D) (C) (E) N Pr Re ec L H S L 1 1 ( 15 ) (D) (G) (G) (H) (G) (E) (G) (B) n t t where, is the dimensionless in W t density in direction z (0 n t W), and Pr the luid Prandtl number, The dimensionless mass o solid material was calculated through the ollowing euation: (D) (C) (E) m m L s W n t (ab (att )(b t t )) (LH n tab) (16) L x (A) air low z y where t t is the thickness o the tube wall and n t is the total number o tubes in the arrangement. u,t NUMERICAL METHOD Figure. The boundary conditions o the -D computational domain. The numerical solution o Es. (1) - (14) was obtained utilizing the inite element method (Zienkiewicz and Taylor, 1989), giving the velocities and temperature ields in the unit cell o Fig.. The implementation o the inite element method or the solution o Es. (1) - (5) and (8) starts rom obtaining the variational (weak) orm o the problem as described by Reddy and Gartling, (1994). The weak orm 67

4 is discretized with an upwind scheme proposed by Hughes, (1978), where it is possible to adeuate the discrete orm o the problem to the physical characteristics o the low. Ater developing the discrete orm o the problem, the resulting algebraic euations are arranged in matrix orm or the steady state three dimensional problem as described by Matos, (00). For the -D problem o Fig. 1, the computational domain contains both the external luid and the solid in. Thus, the solution o E. (8) is also reuired in order to obtain the complete temperature ield. Instead o solving separately or the two entities (luid and solid) and imposing the same heat lux at the interace solid-luid, as a boundary condition, the solution is sought or the entire domain, simultaneously, with the same set o conservation euations, imposing zero velocities in the solid in. EXPERIMENTS An experimental rig was built in the laboratory to produce the necessary experimental data to validate the - D numerical optimization o inned arrangements, and to perorm the experimental optimization o inned arrangements. Figure shows a schematic drawing o the experimental apparatus utilized in this study power source an p dierential pressure transducer electric resistance T8, T9, T10 T11, T1 test module T1, T T T4, T5 T6, T7 connecting hub PC computer power source T1 extended region u anemometer Figure. Experimental apparatus. low straightener air low The circular and elliptic tube arrangements were made rom copper circular tubes, all the ins were made rom aluminum plates and the electric heaters consisted o double step tubular electric resistances. Twelve high precision thermistors were placed in each test module. All the thermistors were placed in the midplane between the side walls o the wind tunnel and at the midline o the elemental channels. Three thermistors were placed at the arrangement inlet (T1 - T), ive at the let (T8 - T1), and our at the tubes suraces in one elemental channel (T4 - T7). An additional thermistor (T1) was placed at the midpoint o the extended region to measure the non-disturbed ree stream temperature. The velocity measurements were taken with a vane-type digital anemometer and the pressure drop measurements were u taken with a pressure transducer, which was connected to a digital pressure meter. The dierential pressure measurements had the inality o measuring the pressure drop across each arrangement in all experiments, as shown in Fig.. The experimental work involved the acuisition o temperature data in real time. The objective o the experimental work was to evaluate the volumetric heat transer density (or overall thermal conductance) o each tested arrangement by computing with E. (15) through direct measurements o u (Re ), and T, w T. Five runs were conducted or each experiment. Steady-state conditions were reached ater hours in all the experiments. The precision limit or each temperature point was computed as two times the standard deviation o the 5 runs (Editorial, 199). It was veriied that the precision limits o all variables involved in the calculation o were negligible in presence o the precision limit o, thereore P P. The thermistors, anemometer, T and properties, and lengths bias limits were ound negligible in comparison with the precision limit o. As a result the uncertainty o U was calculated by: 1 / P B P ( 0 ) where P is the precision limit o. The tested arrangements had a total o twelve tubes placed inside the ixed volume LHW, with our tubes in each unit cell (our rows). For a particular tube and plate in geometry, the tests started with an euilateral triangle coniguration, which illed uniormly the ixed volume, with a resulting maximum dimensionless tube-to-tube spacing S/ = 1.5. The spacing between tubes was then progressively reduced, i.e., S/ = 1.5, 0.5, 0.5 and 0.1, and in this interval an optimal spacing was ound such that was maximum. All the tested arrangements had the aspect ratio L/ = 8.5. Two ree stream velocities set points were tested, u = 0.1 and 0.1 m/s, corresponding to Re = 100 and 15, respectively. The largest uncertainty calculated according to E. (0) in all tests was U /

5 RESULTS AND DISCUSSION The results obtained in this study are divided in two parts: (i) experimental validation o -D numerical results or inned arrangements, and (ii) global optimization results with respect to tube-to-tube spacing, eccentricity and in density. To obtain accurate numerical results, several mesh-reinement tests were conducted. The monitored uantity was the dimensionless overall thermal conductance, computed with E. (15), according to the ollowing criterion: / 0.0 ( 1 ), j, j 1, j where j is the mesh iteration index, i.e., as j increases the mesh is more reined. When the criterion is satisied, the j 1 mesh is selected as the converged mesh. For all cases the mesh was established to consist o nodes and 100 elements. The numerical results obtained with the inite element code are validated by direct comparison to experimental results obtained in the laboratory or circular and elliptic arrangements. According to Fig. 1 the dimensions o the ixed volume or the experimental optimization procedure were L = 15. mm, H = mm, W = 15 mm, and D = = mm. All the arrangements had N ec = 6 and N = 4, where N is the number o tubes in one unit cell. The numerical and experimental optimization procedures ollowed the same steps. First, or a given eccentricity, the dimensionless overall thermal conductance,, was computed with E. (15), or the range 0.1 S/ 1.5. The same procedure was repeated or e = 0.45, 0.5, 0.6 and 1. The numerical and experimental double optimization results or inned tubes 0.006) with respect to tube-to-tube spacings and eccentricities are shown in Fig. (4), together with the corresponding experimental results, or Re = 100 and 15. The direct comparison o, m obtained numerically and experimentally shows that the results are in good agreement. The agreement is remarkable i we consider that in the experiments the tested arrays had uniorm heat lux, and were not large banks o cylinders. In the numerical simulations the domain was ininitely wider (i.e., no inluence rom the wind tunnel walls) and with isothermal tubes. An optimal eccentricity was not obtained experimentally, since an arrangement with e 0.5 was not built. However, the numerical results were validated by the good agreement with the experimental results or e = 0.5, 0.6 and 1. Hence, the numerical results obtained or e = 0.45 are also expected to be accurate. At e = 0.45, drops considerably with respect to e = 0.5,,m determining an optimal pair (S/, e) opt = (0.5, 0.5) or the twice maximized overall heat transer., mm Figure 5 shows an intermediate step in the optimization procedure in order to allow the comparison between the optimal elliptic coniguration with the optimal circular one. It is observed that or the elliptic arrangement (e = 0.5) optimized with respect to tubeto-tube spacing is higher than or the circular arrangement (e = 1) or all in densities,. Furthermore, the elliptic coniguration reuires less ins than the circular one at optimal conditions, i.e., at the optimal pair (S/, ) opt. It is possible to determine the total mass o material in dimensionless terms, through E. (16), at (S/, ) opt or both arrangements. The result o this analysis shows that the total dimensionless mass o the optimal elliptic arrangement is % smaller than the optimal circular one. Figure 6 shows the results o global optimization with respect to the three degrees o reedom, S/, e and. An optimal set o geometric parameters was numerically determined such that was three times maximized, i.e., (S/, e, ) opt (0.5, 0.5, 0.06). Figure 7 illustrates the temperature distribution o plate ins or our-row heat exchangers or circular and elliptic (e = 0.5) tubes, S/ = 0.5 and with Re = 100. The eect o the variation o eccentricity is observed comparing cases (a) and (b) in Fig. 7. It is also shown that the elliptical arrangement is more eicient than circular one because the in temperature distribution is more uniorm in the elliptic coniguration than in the circular one, and closer to tube wall temperature, characterizing a better thermal contact between the tubes and the luid in the elliptic arrangement. As stated in section, the governing euations are or the laminar regime. Thereore, the results o Figs. 4-7 were obtained or low Reynolds numbers, i.e., Re = 100 and 15. For higher Reynolds numbers, convergence to numerical solutions becomes increasingly more diicult, indicating the low is reaching a regime o transition to turbulence. From all numerical and experimental results obtained in this study, it is important to stress that a heat transer gain o up to 19% was observed in the optimal elliptic arrangement with e = 0.5, as compared to the optimal circular one.,m Re = 15 Re = 100, m, m L/ = 8.5 Pr = numerical experimen tal e Figure 4. Numerical and experimental optimization results or inned arrangements. 69

6 100 L/ = 8.5 Pr = 0.7,m S/ = e = 0.5 opt Re = Re = Figure 5. Comparison o the numerical optimization results or inned circular and elliptic arrangements., mm Re = 15 Re = 100 Pr = 0.7 S/ = 0.5 opt e opt = Figure 6. Numerical global optimization results or inned arrangements. (a) (b) Figure 7. Fin temperature distribution or our-row tubes and plate in heat exchangers: (a) S/ = 0.5, e = 1, 0.006) and Re = 100; (b) S/ = 0.5, e = 0.5, 0.006) and Re = 100. CONCLUSIONS In this work, a theoretical, numerical and experimental study was conducted to demonstrate that inned circular and elliptic tubes heat exchangers can be optimized or maximum heat transer, under a ixed volume constraint. The internal geometric structure o the arrangements was optimized or maximum heat transer. Better global perormance is achieved when low and heat transer resistances are minimized together, i.e., when the imperection is distributed in space optimally (Bejan, 000). Optimal distribution o imperection represents low architecture, or constructal design. The results were presented nondimensionally to allow or general application to heat exchangers o the type treated in this study. A suitable euivalent pressure drop criterion permitted the comparison between circular and elliptic arrangements on a heat transer basis in the most isolated way possible. A heat transer gain o up to 19% was observed in the optimal elliptic arrangement, as compared to the optimal circular one. The heat transer gain, combined with the relative total dimensionless 70

7 material mass reduction o up to % observed in this study, show the elliptical arrangement has the potential or a considerably better overall perormance and lower cost than the traditional circular one. Three degrees o reedom were investigated in the heat exchanger geometry, i.e., tube-to-tube spacing, eccentricity and in-to-in spacing. Global optima were ound with respect to tube-to-tube spacing, eccentricity and in-to-in spacing, i.e., (S/, e, ) (0.5, 0.5, 0.06) or Re = 100 and 15. Such globally optimized conigurations are expected to be o great importance or actual heat exchangers engineering design, and or the generation o optimal low structures in general. NOMENCLATURE opt a larger ellipses semi-axis, m b smaller ellipses semi-axis, m B a bias limit o uantity a c p luid speciic heat at constant pressure, J/(kg.K) D tube diameter, m e ellipses eccentricity, b/a H array height, m k luid thermal conductivity, W/(m.K) L array length, m L/ array length to smaller ellipses axis aspect ratio m total mass o the arrangement, kg m dimensionless total mass o the arrangement n number o ins n t total number o tubes N number o tubes in one unit cell N ec number o elemental channels p pressure, N/m P dimensionless pressure Pe Peclet number based on smaller ellipses axis Pr luid Prandtl number, P a precision limit o uantity a dimensionless overall thermal conductance, E.(15) Q overall heat transer rate, W Re Reynolds number based on smaller ellipses axis, u / S spacing between rows o tubes, m, Fig. 1 S/D dimensionless spacing between rows o tubes (circular arrangement) S/ dimensionless spacing between rows o tubes (elliptic arrangement) t time, s t in thickness, m t t tube thickness, m T temperature, K T average luid temperature, K u 1,u,u velocity components, m/s U 1,U,U dimensionless velocity components U a uncertainty o uantity a W x,y,z X,Y,Z Greek symbols array width, m cartesian coordinates, m dimensionless cartesian coordinates thermal diusivity, m /s mesh convergence criterion, E. (1) in-to-in spacing, m dimensionless temperature dimensionless average luid temperature luid kinematic viscosity, m /s density, kg/m dimensionless in density in direction z dimensionless time Subscripts in m opt s w unit cell inlet maximum optimal unit cell let solid tube wall and in material tube surace ree stream ACKNOWLEDGEMENTS The authors acknowledge with gratitude the support o the Program o Human Resources or the Oil Sector and Natural Gas, o the Brazilian Oil National Agency - PRH-ANP/MCT. 71

8 REFERENCES Bejan, A., 1995, Convection Heat Transer, nd Edition, Wiley, New York, (Chapters -). Bejan, A., 000, Shape and Structure, rom Engineering to Nature, Cambridge University Press, Cambridge, UK. Bordalo, S. N., and Saboya, F. E. M., 1999, Pressure drop coeicients or elliptic and circular sections in one, two and three-row arrangements o plate in and tube heat exchangers, J. Braz. Soc. Mech. Sci. XXI (4), pp Editorial, 199, Journal o heat transer policy on reporting uncertainties in experimental measurements and results, ASME Journal o Heat Transer 115, pp Fowler, A. J., and Bejan, A., 1994, Forced convection in banks o inclined cylinders at low Reynolds numbers, Int. J. Heat Fluid Flow, 15, pp Hughes, T. J. R., 1978, A simple scheme or developing upwind inite elements, Int. J. Numer. Meth. in Eng. 1, pp Matos, R. S., Vargas, J. V. C., Laursen, T. A., and Saboya, F. E. M., 001, Optimization study and heat transer comparison o staggered circular and elliptic tubes in orced convection, Int. J. Heat Mass Transer 0, pp Matos, R. S., 00, Otimização e Comparação de Desempenho de Trocadores de Calor de Tubos Circulares e Elípticos Aletados, Tese de Dorado, PIPE-UFPR, Curitiba. Reddy, J. N., and Gartling, D. K., 1994, The Finite Element Method in Heat Transer and Fluid Dynamics, CRC Press, Boca Raton, FL, (Chapters 4-5). Rocha, L. A. O., Saboya, F. E. M. Vargas and J.V. C., 1997, A comparative study o elliptical and circular sections in one and two-row tubes and plate in heat exchangers, Int. J. Heat Fluid Flow 18, pp Saboya, S. M., and Saboya, F. E.M., 001, Experiments on elliptic sections in one and two-row arrangements o plate in and tube heat exchangers, Experimental Thermal and Fluid Science 4, pp Zienkiewicz, O. C., and Taylor, R. L., 1989, The Finite Element Method, vol. 1, McGraw-Hill, London, (Chapter 15). 7