Mixed convection in a horizontal tube with external longitudinal fins

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1 Mixed convection in a horizontal tube with external longitudinal fins M. Ouzzane' and N. Galanis2 I Laboratoire de diversification e'nergetique, CANMET, Canada Dkpartement de gknie mkanique,universite' de Sherbrooke, Canada Abstract The coupled non-linear PDEs modeling developing laminar mixed convection in a horizontal tube and conduction in the surrounding solid have been solved numerically for bare and finned tubes. The results, for heat losses varying linearly with the temperature of the solid, show that the corresponding thermal and hydrodynamic fields are quite different and that the heat loss from the top fins is significantly greater than that from the bottom fins. 1. Introduction Combined free and forced convection in the entrance region of tubes occurs in many applications such as solar collectors and nuclear reactors. The secondary flow induced by the buoyancy force and its effects on the hydrodynamic and thermal fields have been investigated experimentally [l, 21 and numerically 13, 4, 51 for isothermal and uniform or non-uniform flux conditions applied at either the solid-fluid interface or at the outer surface of the tube. All of these studies show that the heat transfer coefficient, or Nusselt number, is significantly higher than the corresponding value for forced convection and that it varies considerably in the entrance regionof the tube. Several other studies have investigated the hydrodynamic and thermal fields in tubes with longitudinal internal fins 161 or with external annular fins 171. They have confirmed the results for bare tubes, namely that the hydrodynamic and temperature fields are three-dimensional and that the heat transfer coefficient is strongly dependent on the local flow conditions. The performance of flat plate

2 solar collectors, which consist of tubes with two external longitudinal fins, has also been the subject of many studies. Most of these use a global energy balance applied to the fluid circulating within such tubes. Thus, Prabhakar [8] calculated the axial variation of the fluid bulk temperature and the two-dimensional temperature field in the solid for different characteristics of the system. He showed that, for high mass flow rates, axial heat conduction in the solid can be neglected. Oliva et al [9] presented a numerical model for the determination of the thermal behavior of solar collectors, which takes into account two- dimensional effects in the solid but neglects the influence of free convection within the fluid. Very recently, Ouzzane & Galanis [lo] presented a numerical analysis of the effects of mixed convection on the performance of solar collectors. A literature review shows that the study of mixed convection in tubes with external longitudinal fins is essentially limited to the case of solar collectors. Nevertheless, heat transfer between a fluid in a tube and a gas around it can be enhanced by using several external longitudinal fins. Contrary to the case of solar collectors, the thermal boundary conditions for this problem are circumferentially (and axially) uniform. The purpose of this paper is to evaluate the effectiveness of such a heat transfer system. We consider developing mixed convection within a horizontal tube with several external longitudinal fins. Heat losses from the external surface of the solid by both convection and long wavelength radiation are taken into account. The coupled non-linear three- dimensional equations for this conjugate heat transfer problem have been solved numerically. The results provide a description of the velocity and temperature distributions in the fluid and in the solid. They also show the axial and circumferential variations of the heat flux evacuated by the system. Figure 1 : Schematic representation of the system.

3 Advancc.dCompututiodMcthods in HatTrumfc.r Mathematical formulation and numerical procedure Figure 1 shows a schematic representation of the problem under study. The flow in the tube is steady and laminar. The fluid is Newtonian, incompressible with constant properties except for the density in the gravity force which varies linearly with temperature. Viscous dissipation is neglected, as in all studies cited earlier, since the corresponding rate of energy release is considerably lower than the external heat loss. Axial diffusion of momentum is neglected, as in other studies 11, 31, since flow reversal is highly unlikely for the conditions of interest here. Similarly, axial conduction in the fluid is considered to be negligible with respect to convection, as elsewhere 13, 41. Furthermore, axial conduction in the solid is neglected based on the results of Prabhakar [g]. These assumptions lead to a considerable simplification of the fundamental equations, which become parabolic in the axial direction while remaining elliptical in the other directions. Finally, following Patankar & Spalding [ll], the pressure is expressed as the sum of a cross-sectional average value p and an in plane perturbation p. The following non-dimensional variables are introduced. r z R-, E-, v - >.=Q, vz=% P1 4 q.repr -af Q 0 with p,= p (r,o)+ po.g.r.cos I and With these assumptions the non-dimensional governing equations for the cylindrical domain, which includes the fluid and the tube (R 2 A), are as follows:

4 where (7) In the fluid re= r R=Tz= Pr,rT=1 while in the solid Ts = r R=Tz=M, TT=at/af. On the other hand, the temperature distribution in the fin (05X 5 LIZ) is given by the following analytical solution to the one dimensional conduction equation: T': - Tzmb - (I/Bi.kp) = cosh(m.x) + Y.sinh(m.X) Tzi:(X= 0) - Tzmb - (l/bi.kp) where m2= Bia/S* and Y = [Bia.tgh(m.Ll *)+m] [m.tgh(m.ll*)+bia]-' (Sb) The heat flux transferred by the fin to the tube is used as a boundary condition for the cylindrical domain. Thus at the external surface of the tube (R=A or X$), thermal conditions are expressed by the following form of the heat flux: * 'L --I- -& (9) where y = 6 /(L *.F) in the part without fins (10a) and y= 2 in the part with fins The fin efficiency is F = (2L14')-'[P+(2/m).tgh(Ll*.m)][l+(Biah).tgh(Ll~'.m)]~' (lob) (10c) At the entry of the tube (Z=O), the temperature and the axial velocity of the fluid are assumed to be uniform: Vz= l and T*=0 The vertical diameter is an axis of symmetry; therefore the calculations were performed on one half of the circular domain. At the interface between the tube wall and the fluid, the continuity conditions for temperature and heat flux are applied so that:

5 Advancc.dCompututiodMcthods in HatTrumfc.r 49 The above formulation indicates that the solution to this complex conjugate heat transfer problem depends on the values of the following eleven nondimensional parameters: athf, Bi, Bia, Gr, kp, Ll':', Pr, Re, Tamb:':,8"' and A. Their definitions are specified in the nomenclature. The coupled non-linear partial differential equations were integrated and discretised using the staggered grid approach. The SIMPLEC procedure [ 121 was used for the linkage of velocities and pressure while the iterative method by Raithby &L Schneider [13] was used to calculate the axial pressure gradient. The set of linearized difference equations was solved with the tridiagonal matrix algorithm. The three diffusion coefficients for the solid were set equal to l@). The grid distribution is non-uniform in all directions, providing a closely spaced mesh in regions of pronounced gradients. The results were checked to ensure mesh size independence. The adopted grid has 38 nodes in the circumferential direction. In the radial direction it has 34 non-equidistant nodes in the fluid and 6 equidistant nodes in the solid. In the axial direction 1100 cross- sections are considered over a tube length of 28.3 diameters; the distance between these cross-sections is 1.6 ~ 10 diameters ~ ~ at the inlet and increases auradually as the flow moves downstream. After the 400fhstep the distance between successive cross-sections is diameters. The numerical code has been validated by comparing its predictions with analytical, numerical and experimental results from the literature [5, lo]. In particular, the calculated velocity and temperature profiles for mixed convection in a uniformly heated vertical tube are in good agreement with corresponding measured distributions by Zeldin & Schmidt [l]. The axial evolution of the Nusselt number is in good agreement with the experimental results by Petukhov &L Polyakov 121. Finally, the fully developed values of Nu for mixed convection in vertical and horizontal tubes agree quite well with corresponding experimental and numerical values for lo4<gri lo6. 3. Results and discussion The results were calculated for water entering the tube at 45 "C. The tube and fins are made of steel. The ambient temperature is S "C, the external convection coefficient is h, = 60 W/m2K and the radiative losses from the external surface are %Id= -10 W/m2.The values of the non-dimensional parameters are: daf= 80, Bi=Bia= 391x10m4,Gr= -1.5x10s, kp= 70, Ll+= 2/3, Pr= 4. Re= 400, T:kan,h= P=0.093 and A= 16/30.

6 It should be noted that, since q& is negative and heat is lost from the system (T<T,), the non-dimensional temperature T* is everywhere positive. Therefore an increase of T* corresponds to a decrease of the temperature. All the results for the finned tube are compared with those for a bare tube under identical operating conditions. 3.1 Hydrodynamicand thermal fields Figure 2 shows the axial evolution of the secondary flow caused by the buoyancy force. The heat losses through the solid create a temperature gradient in the fluid with colder fluid situated near the fluid-wall interface. This heavier fluid moves downwards while the lighter fluid near the center of the tube moves upwards. Very close to the tube inlet this movement is extremely weak. Further downstream, in the case of the bare tube it consists of two symmetrical counter- rotating vortices. In the finned tube this movement is more vigorous and more complex. Thus, for z > it consists of two large and two small vortices. At z=o.o0064 the small vortex is near the bottom of the tube, while further downstream it is near the top. It should be noted that for both bare and finned tubes this secondary flow reaches a maximum intensity at about z=o.o0064. Figure 3 shows the corresponding isotherms. Very close to the tube inlet, z= 6xlU6, the isotherms are circular and the temperature gradients are extremely weak. Therefore, as mentioned before, the secondary buoyancy induced motion is negligible in this short region. At z=o the isotherms are quite distorted, especially in the finned tube. Cold fluid is near the bottom of the tube while hot fluid is near the top. The temperature difference between top and bottom is considerably greater in the case of the finned tube. For the latter, the temperature of the tube and of the fins decreases from top to bottom. Further downstream, the isotherms in the lower part of the tube become horizontal indicating that stratification is taking place. This explains the fact that the secondary motion for ~ is weaker in the lower part of the tube. It should also be noted that in the case of the bare tube the maximum temperature is always on the vertical diameter. This is also true for the finned tube at z=o.o0064 but not at ~ Indeed, at the last two cross sections, the maximum fluid temperature occurs near the interface between the top two fins. Thus, in this case, there are two symmetrically situated maxima. This is obviously due to the cooling effect of the top fins and explains the presence of the small corresponding vortex in figure 2 which does not exist in the bare tube. Finally, the bulk temperature of the fluid, which decreases with increasing distance from the inlet, is always considerably lower in the finned tube. Axial velocity distributions have been calculated but are not shown here for lack of space. Near the tube inlet (z=6xlo") this distribution is axisymmetrical and identical for both finned and bare tubes. This is due to the fact that, in this short entry region, natural convection is essentially non-existent and the axial velocity is only influenced by boundary layer growth. Further downstream

7 Advancc.d Compututiod Mcthods in HatTrumfc.r 5 1 Figure 2 : Secondary flow streamlines =9 Figure 3 : Isotherms U

8 however, the velocity profile along the vertical diameter is more irregular for the finned tube than for the bare tube. The maximum velocity at any cross section is greater in the bare tube. Its value increases with z for both bare and finned tubes. 3.2 Distribution of heat losses Figure 4 shows the circumferential distribution of the interfacial heat flux at different axial positions. As expected, the heat flux for the finned tube is in general much higher than that for the bare tube. Very close to the tube inlet, the latter is uniform while the former exhibits peaks at the position of the fins. Further downstream, the heat flux from the bottom part decreases for both bare and finned tubes reflecting the fact that the fluid in this region is colder (cf. figure 3). As a result of the stratification, the bottom fins become ineffective as z increases. The maximum interfacial heat flux occurs between the top two fins and corresponds to the position of the maximum fluid temperature (cf. figure 3). Figure 5 shows the axial evolution of the heat loss fraction evacuated by each fin. The progressive deterioration of the cooling contribution by the lower fins and the corresponding improvement of the contribution by the upper fins is quite obvious. It is also consistent with the previously presented temperature distributions. The contribution of the circular part of the external tube surface is independent of the axial position. The axial integration of the heat fluxes over the entire length of the tube shows that the circular part contributes 21.26% of the total. The corresponding contributions of the fins are: l 1.94% for fin #l, 11.66% for fins #2 and #8, 10.43% for fins #3 and #7, % for fins #4 and #G and 6.6% for fin #5. Finally, it should be mentioned that, for the conditions of this study, a bare tube should have a length of m to evacuate the same total heat as the finned tube m long oo Figure 4 : Circumferential distribution of interfacial heat flux

9 , I 2002 WIT Press, Ashurst Lodge, Southampton, SO40 7AA, UK. All rights reserved. khncmi CompututiodMethods in HatT,-~~tlsfpr r 0.20: ~, I I, I I I I I I,, I I ~ I I,,,,,, I I,,,, I,,,,,,,,, TUBE 0.18: 0.16: {ai, ~ 0.14: %t Fly %N.2 \ FIN.3 FIN.4 FlN : I I I, # I,,,, l,,, I I,,, ~ Z,, I, I,, I ~ I,,,, I,,,, Figure 5 : Axial evolution of the heat flux fractions 4. Conclusion This numerical study of mixed convection in a horizontal tube and conduction in the surrounding solid has shown that the hydrodynamic and thermal fields for finned and bare tubes are very different. Both are also very different from the axisymmetrical fields for forced convection. The results also show that, due to stratification, the heat loss from the top fins is significantly greater than that from the bottom fms. Furthermore, the relative contribution of the top fins increases with distance from the inlet while the opposite is true for the bottom fins. Nomenclature at, af Bi, Bia Di, De L1 L2 r, x, z R,X Z To, v0 P 6 ht, hf V thermal diffusivity of tube, fluid Biot number for tube, fins internal, external tube diameters fin width (LI" = LUDi) tube length (L2:':= LYDi) dimensional coordinates non-dimensional coordinates temperature, velocity at tube inlet volumetric expansion coefficient fin thickness (6" = 8Di) thermal conductivity of tube, fluid kinematic viscosity Gr Grashof number Pr Prandtl number Re Reynolds number A = De/2.Di kp = htaf

54 Advmced Compututiod Methods it1 HwtTrmsf?r References Schmidt F. W. & Zeldin B., Laminar Heat Transfer in the Entrance Region of Ducts. App. Sc. Res.,23, pp. 73-94, 1970. Petukhov B. S. 8~ Polyakov A.

10 54 Advmced Compututiod Methods it1 HwtTrmsf?r References Schmidt F. W. & Zeldin B., Laminar Heat Transfer in the Entrance Region of Ducts. App. Sc. Res.,23, pp , Petukhov B. S. 8~ Polyakov A. F., Experimental Investigation of Viscogravitational Fluid Flow in a Horizontal Tube. Scientific Research Institute of High Temperatures, Translated,from Teplofizika Vywkikh Ternperatur,ql),pp , Nesreddine H., Galanis N. & Nguyen C.T., Effects of Axial Diffusion on Laminar Heat Transfer With Low Peclet Numbers in the Entrance Region of Thin Vertical Tubes. Num. Heat Transfer Pt. A,33,pp , Choudhury D. & Patankar S. V., Combined Forced and Free Laminar Convection in the Entrance Region of an Inclined Isothermal Tube. ASME.l. of Heat Trunsfer, 110,pp , Ouzzane M. & Galanis N., Effets de la Conduction PariCtale et de la RCpartition du Flux Thermique sue la Convection Mixte Pres de I'Entree d'meconduite InclinCe.Int. J. ofthermal Sciences, 38, pp , Shome I. B., Mixed Convection Laminar Flow and Heat Transfer of Liquids in Horizontal Internally Finned Tubes. Num. Heut Transfer Pt. A, 33, pp , Moukalled F., Kasamani J. & Acharya S., Turbulent Convection Heat Transfer in Longitudinally Conducting Externally Finned Pipes. Num. Heat Transfer Pt. A, 21, pp , Prabhakar P. R., Two Dimensional Configuration Analysis of a Flat Plat Solar Collector, Ph. D thesis, Mech. Eng.,Univ. of Oklahoma, 92 p,, Oliva A., Costa M. & Perez Segara C. D., Numerical Simulation of Solar Collectors: The Effect of Non uniform and Non Steady State of the Boundary Conditions. Solar Energy,47(5), pp , [IO] Ouzzane M. & Galanis N., Numerical Analysis of Mixed Convection in Inclined Tubes with External Longitudinal Fins. Solar Energy, 71(3), pp , l] Patankar S.V. & Spalding D.B., A Calculation Procedure for Heatt:Mass and Momentum Transfer in Three Dimensional Parabolic Flows. Int. J. Heat Mass Transfer, 15, pp , [12] Van Doormaal J.P. & Raithby G.D., Enhancements of the SIMPLE Method for Predicting Incompressible Fluid Flows. Num Heat Transfer, 7,pp , [13] Raithby G.D. & Schneider G.E., Numerical Solution of Problems in Incompressible Fluid Flow: Treatment of the Velocity-Pressure Coupling. Num. Heut Transjer, 2,pp , 1979.

Laminar Mixed Convection in the Entrance Region of Horizontal Quarter Circle Ducts

Proceedings of the 5th IASME/WSEAS Int. Conference on Heat Transfer Thermal Engineering and Environment Athens Greece August 5-7 007 49 Laminar Mixed Convection in the Entrance Region of Horizontal Quarter