- Email: [email protected]
Mixed convection with heat and mass transfer in horizontal tubes
International Communications in Heat and Mass Transfer 32 (2005) 511 – 519 www.elsevier.com/locate/ichmt
Mixed convection with heat and mass transfer in horizontal tubesB Jamel Orfia,*, Nicolas Galanisb a
De´partement de Physique, Faculte´ des Sciences de Monastir, Monastir, 5019, Tunisie De´partement de Ge´nie Me´canique, Universite´ de Sherbrooke, Sherbrooke, Qc, J1K 2R1, Canada
b
Abstract This paper presents the influence of the Lewis number on laminar mixed convective heat and mass transfer in a horizontal tube with uniform heat flux and uniform concentration at the fluid–solid interface. The results of this numerical study show that, for these boundary conditions, the effect of the Lewis number on the Sherwood number is most important near the tube inlet. In the case of the Nusselt number and the wall shear stress, this effect is limited to the intermediate region between the entrance and the fully developed regions. D 2004 Elsevier Ltd. All rights reserved. Keywords: Heat and Mass transfer; Mixed convection; Lewis number effect; Horizontal tubes
1. Introduction Convection flows with simultaneous thermal and chemical species diffusion are encountered in many natural processes and practical situations including the evaporation from a body of water with or without wind, the chemical vapour deposition of solid layers and the cooling of an air stream by evaporation. Here, we are interested in confined mixed convection with double diffusion caused by temperature and concentration gradients. Many studies have dealt with such flows. Thus, Yan [1] studied numerically the developing laminar mixed convection with heat and mass transfer in inclined rectangular ducts. Typical developments of velocity, temperature and concentration profiles were shown. The local friction factor, B
Communicated by J. Taine and A. Soufiani. * Corresponding author. Tel.: +216 73 500 511; fax: +216 73 500 278. E-mail addresses: [email protected] (J. Orfi), [email protected] (N. Galanis).
0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.04.029
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J. Orfi, N. Galanis / Int. Commun. Heat and Mass Transf. 32 (2005) 511–519
Nusselt number and Sherwood number were presented for species diffusion of interest in air (Pr=0.7) over a Schmidt number range of 0.2–2. Zheng and Worek [2] investigated experimentally the combined heat and mass transfer processes at an air–water interface in a rectangular inclined channel. They used a fiber-optic holographic interferometer to measure local heat and mass transfer coefficients as well as a set of thermocouples and a dewpoint hygrometer to determine the average transfer rates. Lee [3] investigated numerically the laminar natural convection heat and mass transfer in open vertical rectangular ducts with uniform temperature and uniform concentration or uniform heat flux and uniform mass flux at the wall. Results were presented for a fluid with Pr=0.7 and Sc equal to 0.2 (hydrogen), 0.6 (water vapour) and 1.3 (ethyl alcohol). The author proposed correlations for the rates of heat and mass transfer. The literature review shows that the problem of developing mixed convection with heat and mass transfer in circular ducts has not received much attention. In particular, the effect of the Lewis number has not been studied. In a recent paper, the present authors [4] analysed the transport phenomena for laminar mixed convection with heat and mass transfer in the entrance region of horizontal and vertical tubes. These results were limited to the case when Le=1. The present study extends these results by presenting the effect of the Lewis number for flow in a horizontal tube.
2. Problem formulation and numerical procedure We consider a mixture of two nonreacting components (small quantity of B with large quantity of A), entering a horizontal circular tube with uniform velocity V0V, temperature T0V and concentration C0V. A uniform heat flux q and a uniform concentration CwV are applied at the fluid–solid interface. The flow is assumed to be steady and laminar. The fluid properties are considered constant except for the density in the buoyancy terms which varies linearly with both temperature and concentration. Dissipation, pressure work, axial diffusion as well as the Soret and Dufour effects have been neglected [4,5]. The following reference quantities were used to nondimensionalise the corresponding variables: the diameter 2R for the radial coordinate, the average velocity for the axial velocity component and the ratio of thermal diffusivity to the diameter for the other two velocity components. All the other nondimensional variables are defined in the nomenclature. The mass, momentum, energy and concentration equations in cylindrical coordinates can then be written in the following nondimensional form [4]: 1 Bðrvr Þ 1 Bvh Bvz þ þ ¼0 r Br r Bh Bz
ð1Þ
2 Bvr vh Bvr Bvr v2h BP1 B vr 1 Bvr 1 B2 v r vr 2 Bvh þ 2 2 vr þ þ vz ¼ þ Pr þ 2 r Bh r Br r Bh2 r Bh Br Bz r Br Br2 r 2 ð2aÞ þ Pr ðGrT T þ GrC CÞcosh 2 Bvh vh Bvh Bvh vr vh BP1 B vh 1 Bvh 1 B2 vh vh 2 Bvr þ þ vz þ ¼ þ Pr þ 2 vr þ 2 þ 2 r Br r Bh2 r Bh Br r Bh Bz r rBh Br2 r 2 ð2bÞ Pr ðGrT T þ GrC CÞsinh
J. Orfi, N. Galanis / Int. Commun. Heat and Mass Transf. 32 (2005) 511–519
2 Bvz vh Bvz Bvz dP2 B vz 1 Bvz 1 B2 v z þ þ vz ¼ þ Pr þ 2 vr þ r Br r Bh2 Br r Bh Bz dz Br2 BT vh BT BT B2 T 1 BT 1 B2 T þ þ vz ¼ 2 þ þ 2 Br Bz Br r Br r Bh2 r Bh 2 BC vh BC BC BC 1 BC 1 B2 C : þ þ þ vz ¼ Le þ 2 vr Br Bz Br2 r Br r Bh2 r Bh
vr
513
ð2cÞ
ð3Þ
ð4Þ
One constraint to be satisfied and used to deduce the axial pressure gradient in the axial momentum equation is the overall mass balance at every axial location: Z
0:5 0
Z
p
p 8
vz rdhdr ¼ 0
ð5Þ
The boundary conditions based on the assumptions of uniform entry, nonslip and impermeability on the tube wall as well as symmetry about the vertical diameter are : At z ¼ 0: At r ¼ 0:5: At h ¼ 0; p:
vz ¼ 1 and vh ¼ vr ¼ T ¼ C ¼ 0: vh ¼ vr ¼ vz ¼ 0; C ¼ 1 and ðBT =BrÞ ¼ 1 vh ¼ Bvr =Bh ¼ Bvz =Bh ¼ BT =Bh ¼ BC=Bh ¼ 0:
ð6aÞ ð6bÞ ð6cÞ
The set of coupled nonlinear equations was solved numerically using a version of the control volume method by Patankar [6] for three-dimensional axially parabolic flows. A marching technique based on the SIMPLEC algorithm [7] for handling the in-plane pressure velocity coupling, the power law scheme for the approximation of the in-plane convective–diffusive terms and a proven procedure for the calculation of the axial pressure gradient [8] were used. To obtain enhanced accuracy, grids were chosen to be uniform in the tangential direction but nonuniform in the radial and axial directions to account for the uneven variations of velocity, temperature and concentration near the tube wall and in the entrance region. Numerical experiments were made to ensure that the results are grid independent. To check the adequacy of the model and the numerical scheme, the developing flow and thermal or solutal fields were computed without buoyancy effects, and the results were compared with those in Kays and Crawford [9]. In addition, the calculated results for developing mixed convection without mass gradients in heated and in isothermal tubes were successfully compared with experimental and numerical published results. More information concerning these validation tests can be found in Orfi and Galanis [4].
3. Results and discussion In the following, results for the simultaneous development of the momentum, temperature and concentration fields inside horizontal circular tubes are presented and discussed. They were calculated
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Fig. 1. (A) Evolution of the secondary flow for Lewis number=0.5. (B) Evolution of the secondary flow for Lewis number=5. (C) Axial evolution of the radial profile of the axial velocity for Le=0.2 and 5.
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for Re=400, Pr=0.7, Gr T =Gr C =1.01105 and values of the Lewis number ranging from 0.2 to 5. Since the Prandtl number is constant, they also show the effect of the relative importance of mass to momentum diffusion (effect of the Schmidt number). The buoyancy forces due to the combined thermal and solutal gradients generate a secondary flow consisting of two symmetrical vortices in a plane normal to the tube axis. Fig. 1A (Le=0.5) and B (Le=5) show such secondary flow patterns at six different cross-sections, B, C, D, E, F and G, of the tube situated at z=0.002, 0.0076, 0.0281, 0.078, 0.116 and 0.156, respectively. For both values of Le, the secondary flow is fairly weak near the tube inlet (section B). It becomes stronger quickly reaching a maximum intensity close to section C and eventually settles back into a fully developed state (section F and beyond).
Fig. 2. (A) Concentration distribution for different Lewis numbers (Le=0.5, 1.5 and 5). (B) Axial evolution of the radial concentration profiles for different Lewis numbers.
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In both cases, the centre of circulation moves away from the wall as z increases. A comparison of the patterns in Fig. 1A and B shows that the ascending part of the vortices is thinner when Le=0.5. Near the tube inlet (section B), the secondary flow for Le=5 is stronger, while in section D, the maximum ascending velocity is greater for Le=0.5. This is due to the fact that mass diffusion occurs faster for large values of Le. Fig 1C presents the axial evolution of the radial profile of the axial velocity for two values of Le. It is shown that, for the region far from the inlet, the axial velocity is reduced with increasing the Lewis number. Fig. 2A shows the iso-concentration contours. At the first cross-section, the region of nonuniform concentration (corresponding to the thickness of the ascending part of the vortices in Fig. 1A,B) is much thicker for Le=5. Further downstream, at section D, the concentration for Le=5 is almost uniform. Therefore, the corresponding secondary flow in Fig. 1B is essentially generated by thermal gradients only. On the other hand, for Le=0.5 at section D, the concentration is far from uniform and the corresponding secondary flow is due to both thermal and concentration gradients. The effect of Lewis number on the development of the concentration field can also be understood by looking to Fig. 2B, which presents the axial evolution of the radial concentration profiles for different values of Le. The tendency of the concentration to become uniform is a direct consequence of the imposed interfacial boundary condition. This state of uniform concentration is approached earlier as Le increases. On the other hand, thermal effects persist throughout the length of the tube since heating does not stop. Therefore, the fully developed flow depends on Gr T only. The axial evolutions of the circumferentially averaged Nusselt, wall shear stress and Sherwood numbers are presented in Figs. 3, 4 and 5, respectively. It is obvious that Nu z and s z exhibit a similar behaviour. In the entrance region, up to z=3103, they decrease in close agreement with the corresponding evolution for forced convection. Further downstream, the vigorous secondary flow induced by the combined thermal and solutal gradients causes an enhancement of the heat transfer and an increase in the flow resistance over the corresponding forced convection values. For a short distance and a fixed value of Le, their values remain essentially constant. Finally, they decrease due to the gradual disappearance of free convection effects generated by concentration gradients and approach their fully
Fig. 3. Axial variation of the circumferentially averaged Nusselt number.
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517
Fig. 4. Axial variation of the circumferentially averaged wall shear stress.
developed values. The Lewis number effects are nonexistent very near the entrance, where the buoyancy induced flow is extremely weak, and in the fully developed region where the concentration field is uniform. It should be noted that the length of the intermediate region, where Lewis number effects are important, decreases as Le increases since the latter increase promotes an earlier uniformization of the concentration field. It should also be noted that the fully developed values of Nu z and s z are considerably higher than the corresponding ones for forced convection. The axial evolution of Sh z (Fig. 5) is quite
Fig. 5. Axial variation of the circumferentially averaged Sherwood number.
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different. The influence of the Lewis number is very important in the entrance region where values of Sh z are larger or smaller than the corresponding values for forced convection for Leb1 and LeN1, respectively. This influence vanishes far downstream where the concentration field becomes fully developed. There, all the curves tend towards the well known value of 3.66 corresponding to pure forced flow with uniform wall concentration. It can also be observed that, for a fixed z, Sh z decreases as the Lewis number increases. This effect is analogous to that of the Prandtl number on the Nusselt one in the case of forced convection [9]. It should finally be noted that when the Lewis number is large (DNa), the concentration field develops very quickly compared to the temperature field. And, since Pr=0.7, this condition also implies that the Schmidt number is small (DNm). Therefore, the concentration field develops much quicker than the hydrodynamic field as well. In the limit Le=l and Sc=0 (or equivalently DJa and DJm), the concentration field can be calculated by considering that the velocity and temperature are uniform while the hydrodynamic and thermal fields can be calculated by considering that the concentration is uniform (C=1). Analogously, when Le is small and Sc is large, the concentration field develops much slower than both the hydrodynamic and thermal fields. In the limit Le=0 and Sc=l, the hydrodynamic and thermal fields can be calculated by considering C=0 and the concentration field can be calculated using the developed velocity and temperature profiles. Thus, for both these limiting cases, the concentration field is independent of the hydrodynamic and thermal fields, and vice versa. This situation is analogous to that for forced convection with heat transfer only in the limiting cases of Pr=l and Pr=0 [9].
4. Conclusion The numerical solution of the equations modelling the developing velocity, temperature and concentration fields in a horizontal tube with uniform heat flux and concentration at the fluid–solid interface has provided a description of this heat and mass transfer problem. For the conditions under consideration, the effects of mass diffusion disappear entirely after an axial position which is closer to the tube entrance for high values of Le. The effects of thermal diffusion persist throughout the tube length. The results show that the axial evolution and the effects of Le on Nu z and s z are similar and quite different from those of Le on Sh z . Nomenclature a Thermal diffusivity, m2 s1 C Dimensionless species concentration, (CVC 0V)/(C wV C 0V) D Mass diffusivity, m2 s1 g Gravitational acceleration, m s2 Gr C Solutal Grashof number, gb*(C wV C 0V) (2R)3/m 2 Gr T Thermal Grashof number, gbq (2R)4/km 2 k Thermal conductivity, Wm1K1 Le Lewis number, D/a Rp R 0:5 R p Nu z Average Nusselt number, p=ð 0 Tw dh 8 0 0 vz TrdhdrÞ p m Mean cross-sectional pressure, Pa pV In-plane perturbation about the mean pressure p m, Pa P 1 Dimensionless modified in plane pressure variation, ( pV+q 0rVgcosacosh)/(q 0 (a/2R)2)
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519
P 2 Dimensionless modified mean cross-sectional pressure, ( p m+q 0zVgsina)/(q 0 (V 0V)2) Pr Prandtl number, m/a q Heat flux imposed at the wall, W m2 r Dimensionless radial coordinate, rV/2R R Tube radius Re Reynolds number, 2RV 0V/m Sc Schmidt number, m/DR R 0:5 R p p Sh z Sherwood number, ½ 0 ð BC 0 vz Crdhdr Br Þr¼0:5 dh=½p 8 0 T Dimensionless temperature, (T VT V0)/(2Rq/k) v r, v h, v z Dimensionless velocity components in r V, h and zV directions, respectively z Dimensionless axial coordinate=zV/(2RRePr) Greek letters a Tube inclination b Coefficient of thermal expansion, K1 b* Coefficient of concentration expansion h Circumferential coordinate m Kinematic viscosity, m2 s1 q Density, kg m3 Rp s w Dimensionless average wall shear stress,½ 0 ðBvz =BrÞr¼0:5 dh=p Subscripts 0 Refers to entry r, h, z Refer to the radial, circumferential and axial directions, respectively V Refers to dimensional quantity
Acknowledgements The authors thank the bFaculte des Sciences de MonastirQ, the bLaboratoire d’Etudes des Syste`mes Thermiques et Energe´tiques de MonastirQ and the Natural Sciences and Engineering Research Council of Canada for their financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M.W. Yan, Int. J. Heat Mass Transfer 38 (1995) 2905. G.S. Zheng, W.M. Worek, Int. J. Heat Mass Transfer 39 (1996) 97. K.T. Lee, Int. J. Heat Mass Transfer 42 (1999) 4523. J. Orfi, N. Galanis, Int. J. Therm. Sci. 41 (2002) 319. B. Gebhart, Y. Jaluria, R.L. Mahajan, B. Sammakia, Buoyancy Induced Flows And Transport, Hemisphere Publishing, New York, 1988, p. 971. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing, New York, 1980, p. 197. J.P. Van Doormal, G.D. Raithby, Numer. Heat Transf. 7 (1984) 147. G.D. Raithby, G.E. Schneider, Numer. Heat Transf. 2 (1979) 417. M.W. Kays, M.E. Crawford, McGraw-Hill Ser. Mech. Eng. (1993) 601.
Mixed convection with heat and mass transfer in horizontal tubesB Jamel Orfia,*, Nicolas Galanisb a
De´partement de Physique, Faculte´ des Sciences de Monastir, Monastir, 5019, Tunisie De´partement de Ge´nie Me´canique, Universite´ de Sherbrooke, Sherbrooke, Qc, J1K 2R1, Canada
b
Abstract This paper presents the influence of the Lewis number on laminar mixed convective heat and mass transfer in a horizontal tube with uniform heat flux and uniform concentration at the fluid–solid interface. The results of this numerical study show that, for these boundary conditions, the effect of the Lewis number on the Sherwood number is most important near the tube inlet. In the case of the Nusselt number and the wall shear stress, this effect is limited to the intermediate region between the entrance and the fully developed regions. D 2004 Elsevier Ltd. All rights reserved. Keywords: Heat and Mass transfer; Mixed convection; Lewis number effect; Horizontal tubes
1. Introduction Convection flows with simultaneous thermal and chemical species diffusion are encountered in many natural processes and practical situations including the evaporation from a body of water with or without wind, the chemical vapour deposition of solid layers and the cooling of an air stream by evaporation. Here, we are interested in confined mixed convection with double diffusion caused by temperature and concentration gradients. Many studies have dealt with such flows. Thus, Yan [1] studied numerically the developing laminar mixed convection with heat and mass transfer in inclined rectangular ducts. Typical developments of velocity, temperature and concentration profiles were shown. The local friction factor, B
Communicated by J. Taine and A. Soufiani. * Corresponding author. Tel.: +216 73 500 511; fax: +216 73 500 278. E-mail addresses: [email protected] (J. Orfi), [email protected] (N. Galanis).
0735-1933/$ - see front matter D 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2004.04.029
512
J. Orfi, N. Galanis / Int. Commun. Heat and Mass Transf. 32 (2005) 511–519
Nusselt number and Sherwood number were presented for species diffusion of interest in air (Pr=0.7) over a Schmidt number range of 0.2–2. Zheng and Worek [2] investigated experimentally the combined heat and mass transfer processes at an air–water interface in a rectangular inclined channel. They used a fiber-optic holographic interferometer to measure local heat and mass transfer coefficients as well as a set of thermocouples and a dewpoint hygrometer to determine the average transfer rates. Lee [3] investigated numerically the laminar natural convection heat and mass transfer in open vertical rectangular ducts with uniform temperature and uniform concentration or uniform heat flux and uniform mass flux at the wall. Results were presented for a fluid with Pr=0.7 and Sc equal to 0.2 (hydrogen), 0.6 (water vapour) and 1.3 (ethyl alcohol). The author proposed correlations for the rates of heat and mass transfer. The literature review shows that the problem of developing mixed convection with heat and mass transfer in circular ducts has not received much attention. In particular, the effect of the Lewis number has not been studied. In a recent paper, the present authors [4] analysed the transport phenomena for laminar mixed convection with heat and mass transfer in the entrance region of horizontal and vertical tubes. These results were limited to the case when Le=1. The present study extends these results by presenting the effect of the Lewis number for flow in a horizontal tube.
2. Problem formulation and numerical procedure We consider a mixture of two nonreacting components (small quantity of B with large quantity of A), entering a horizontal circular tube with uniform velocity V0V, temperature T0V and concentration C0V. A uniform heat flux q and a uniform concentration CwV are applied at the fluid–solid interface. The flow is assumed to be steady and laminar. The fluid properties are considered constant except for the density in the buoyancy terms which varies linearly with both temperature and concentration. Dissipation, pressure work, axial diffusion as well as the Soret and Dufour effects have been neglected [4,5]. The following reference quantities were used to nondimensionalise the corresponding variables: the diameter 2R for the radial coordinate, the average velocity for the axial velocity component and the ratio of thermal diffusivity to the diameter for the other two velocity components. All the other nondimensional variables are defined in the nomenclature. The mass, momentum, energy and concentration equations in cylindrical coordinates can then be written in the following nondimensional form [4]: 1 Bðrvr Þ 1 Bvh Bvz þ þ ¼0 r Br r Bh Bz
ð1Þ
2 Bvr vh Bvr Bvr v2h BP1 B vr 1 Bvr 1 B2 v r vr 2 Bvh þ 2 2 vr þ þ vz ¼ þ Pr þ 2 r Bh r Br r Bh2 r Bh Br Bz r Br Br2 r 2 ð2aÞ þ Pr ðGrT T þ GrC CÞcosh 2 Bvh vh Bvh Bvh vr vh BP1 B vh 1 Bvh 1 B2 vh vh 2 Bvr þ þ vz þ ¼ þ Pr þ 2 vr þ 2 þ 2 r Br r Bh2 r Bh Br r Bh Bz r rBh Br2 r 2 ð2bÞ Pr ðGrT T þ GrC CÞsinh
J. Orfi, N. Galanis / Int. Commun. Heat and Mass Transf. 32 (2005) 511–519
2 Bvz vh Bvz Bvz dP2 B vz 1 Bvz 1 B2 v z þ þ vz ¼ þ Pr þ 2 vr þ r Br r Bh2 Br r Bh Bz dz Br2 BT vh BT BT B2 T 1 BT 1 B2 T þ þ vz ¼ 2 þ þ 2 Br Bz Br r Br r Bh2 r Bh 2 BC vh BC BC BC 1 BC 1 B2 C : þ þ þ vz ¼ Le þ 2 vr Br Bz Br2 r Br r Bh2 r Bh
vr
513
ð2cÞ
ð3Þ
ð4Þ
One constraint to be satisfied and used to deduce the axial pressure gradient in the axial momentum equation is the overall mass balance at every axial location: Z
0:5 0
Z
p
p 8
vz rdhdr ¼ 0
ð5Þ
The boundary conditions based on the assumptions of uniform entry, nonslip and impermeability on the tube wall as well as symmetry about the vertical diameter are : At z ¼ 0: At r ¼ 0:5: At h ¼ 0; p:
vz ¼ 1 and vh ¼ vr ¼ T ¼ C ¼ 0: vh ¼ vr ¼ vz ¼ 0; C ¼ 1 and ðBT =BrÞ ¼ 1 vh ¼ Bvr =Bh ¼ Bvz =Bh ¼ BT =Bh ¼ BC=Bh ¼ 0:
ð6aÞ ð6bÞ ð6cÞ
The set of coupled nonlinear equations was solved numerically using a version of the control volume method by Patankar [6] for three-dimensional axially parabolic flows. A marching technique based on the SIMPLEC algorithm [7] for handling the in-plane pressure velocity coupling, the power law scheme for the approximation of the in-plane convective–diffusive terms and a proven procedure for the calculation of the axial pressure gradient [8] were used. To obtain enhanced accuracy, grids were chosen to be uniform in the tangential direction but nonuniform in the radial and axial directions to account for the uneven variations of velocity, temperature and concentration near the tube wall and in the entrance region. Numerical experiments were made to ensure that the results are grid independent. To check the adequacy of the model and the numerical scheme, the developing flow and thermal or solutal fields were computed without buoyancy effects, and the results were compared with those in Kays and Crawford [9]. In addition, the calculated results for developing mixed convection without mass gradients in heated and in isothermal tubes were successfully compared with experimental and numerical published results. More information concerning these validation tests can be found in Orfi and Galanis [4].
3. Results and discussion In the following, results for the simultaneous development of the momentum, temperature and concentration fields inside horizontal circular tubes are presented and discussed. They were calculated
514
J. Orfi, N. Galanis / Int. Commun. Heat and Mass Transf. 32 (2005) 511–519
Fig. 1. (A) Evolution of the secondary flow for Lewis number=0.5. (B) Evolution of the secondary flow for Lewis number=5. (C) Axial evolution of the radial profile of the axial velocity for Le=0.2 and 5.
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515
for Re=400, Pr=0.7, Gr T =Gr C =1.01105 and values of the Lewis number ranging from 0.2 to 5. Since the Prandtl number is constant, they also show the effect of the relative importance of mass to momentum diffusion (effect of the Schmidt number). The buoyancy forces due to the combined thermal and solutal gradients generate a secondary flow consisting of two symmetrical vortices in a plane normal to the tube axis. Fig. 1A (Le=0.5) and B (Le=5) show such secondary flow patterns at six different cross-sections, B, C, D, E, F and G, of the tube situated at z=0.002, 0.0076, 0.0281, 0.078, 0.116 and 0.156, respectively. For both values of Le, the secondary flow is fairly weak near the tube inlet (section B). It becomes stronger quickly reaching a maximum intensity close to section C and eventually settles back into a fully developed state (section F and beyond).
Fig. 2. (A) Concentration distribution for different Lewis numbers (Le=0.5, 1.5 and 5). (B) Axial evolution of the radial concentration profiles for different Lewis numbers.
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In both cases, the centre of circulation moves away from the wall as z increases. A comparison of the patterns in Fig. 1A and B shows that the ascending part of the vortices is thinner when Le=0.5. Near the tube inlet (section B), the secondary flow for Le=5 is stronger, while in section D, the maximum ascending velocity is greater for Le=0.5. This is due to the fact that mass diffusion occurs faster for large values of Le. Fig 1C presents the axial evolution of the radial profile of the axial velocity for two values of Le. It is shown that, for the region far from the inlet, the axial velocity is reduced with increasing the Lewis number. Fig. 2A shows the iso-concentration contours. At the first cross-section, the region of nonuniform concentration (corresponding to the thickness of the ascending part of the vortices in Fig. 1A,B) is much thicker for Le=5. Further downstream, at section D, the concentration for Le=5 is almost uniform. Therefore, the corresponding secondary flow in Fig. 1B is essentially generated by thermal gradients only. On the other hand, for Le=0.5 at section D, the concentration is far from uniform and the corresponding secondary flow is due to both thermal and concentration gradients. The effect of Lewis number on the development of the concentration field can also be understood by looking to Fig. 2B, which presents the axial evolution of the radial concentration profiles for different values of Le. The tendency of the concentration to become uniform is a direct consequence of the imposed interfacial boundary condition. This state of uniform concentration is approached earlier as Le increases. On the other hand, thermal effects persist throughout the length of the tube since heating does not stop. Therefore, the fully developed flow depends on Gr T only. The axial evolutions of the circumferentially averaged Nusselt, wall shear stress and Sherwood numbers are presented in Figs. 3, 4 and 5, respectively. It is obvious that Nu z and s z exhibit a similar behaviour. In the entrance region, up to z=3103, they decrease in close agreement with the corresponding evolution for forced convection. Further downstream, the vigorous secondary flow induced by the combined thermal and solutal gradients causes an enhancement of the heat transfer and an increase in the flow resistance over the corresponding forced convection values. For a short distance and a fixed value of Le, their values remain essentially constant. Finally, they decrease due to the gradual disappearance of free convection effects generated by concentration gradients and approach their fully
Fig. 3. Axial variation of the circumferentially averaged Nusselt number.
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517
Fig. 4. Axial variation of the circumferentially averaged wall shear stress.
developed values. The Lewis number effects are nonexistent very near the entrance, where the buoyancy induced flow is extremely weak, and in the fully developed region where the concentration field is uniform. It should be noted that the length of the intermediate region, where Lewis number effects are important, decreases as Le increases since the latter increase promotes an earlier uniformization of the concentration field. It should also be noted that the fully developed values of Nu z and s z are considerably higher than the corresponding ones for forced convection. The axial evolution of Sh z (Fig. 5) is quite
Fig. 5. Axial variation of the circumferentially averaged Sherwood number.
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different. The influence of the Lewis number is very important in the entrance region where values of Sh z are larger or smaller than the corresponding values for forced convection for Leb1 and LeN1, respectively. This influence vanishes far downstream where the concentration field becomes fully developed. There, all the curves tend towards the well known value of 3.66 corresponding to pure forced flow with uniform wall concentration. It can also be observed that, for a fixed z, Sh z decreases as the Lewis number increases. This effect is analogous to that of the Prandtl number on the Nusselt one in the case of forced convection [9]. It should finally be noted that when the Lewis number is large (DNa), the concentration field develops very quickly compared to the temperature field. And, since Pr=0.7, this condition also implies that the Schmidt number is small (DNm). Therefore, the concentration field develops much quicker than the hydrodynamic field as well. In the limit Le=l and Sc=0 (or equivalently DJa and DJm), the concentration field can be calculated by considering that the velocity and temperature are uniform while the hydrodynamic and thermal fields can be calculated by considering that the concentration is uniform (C=1). Analogously, when Le is small and Sc is large, the concentration field develops much slower than both the hydrodynamic and thermal fields. In the limit Le=0 and Sc=l, the hydrodynamic and thermal fields can be calculated by considering C=0 and the concentration field can be calculated using the developed velocity and temperature profiles. Thus, for both these limiting cases, the concentration field is independent of the hydrodynamic and thermal fields, and vice versa. This situation is analogous to that for forced convection with heat transfer only in the limiting cases of Pr=l and Pr=0 [9].
4. Conclusion The numerical solution of the equations modelling the developing velocity, temperature and concentration fields in a horizontal tube with uniform heat flux and concentration at the fluid–solid interface has provided a description of this heat and mass transfer problem. For the conditions under consideration, the effects of mass diffusion disappear entirely after an axial position which is closer to the tube entrance for high values of Le. The effects of thermal diffusion persist throughout the tube length. The results show that the axial evolution and the effects of Le on Nu z and s z are similar and quite different from those of Le on Sh z . Nomenclature a Thermal diffusivity, m2 s1 C Dimensionless species concentration, (CVC 0V)/(C wV C 0V) D Mass diffusivity, m2 s1 g Gravitational acceleration, m s2 Gr C Solutal Grashof number, gb*(C wV C 0V) (2R)3/m 2 Gr T Thermal Grashof number, gbq (2R)4/km 2 k Thermal conductivity, Wm1K1 Le Lewis number, D/a Rp R 0:5 R p Nu z Average Nusselt number, p=ð 0 Tw dh 8 0 0 vz TrdhdrÞ p m Mean cross-sectional pressure, Pa pV In-plane perturbation about the mean pressure p m, Pa P 1 Dimensionless modified in plane pressure variation, ( pV+q 0rVgcosacosh)/(q 0 (a/2R)2)
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P 2 Dimensionless modified mean cross-sectional pressure, ( p m+q 0zVgsina)/(q 0 (V 0V)2) Pr Prandtl number, m/a q Heat flux imposed at the wall, W m2 r Dimensionless radial coordinate, rV/2R R Tube radius Re Reynolds number, 2RV 0V/m Sc Schmidt number, m/DR R 0:5 R p p Sh z Sherwood number, ½ 0 ð BC 0 vz Crdhdr Br Þr¼0:5 dh=½p 8 0 T Dimensionless temperature, (T VT V0)/(2Rq/k) v r, v h, v z Dimensionless velocity components in r V, h and zV directions, respectively z Dimensionless axial coordinate=zV/(2RRePr) Greek letters a Tube inclination b Coefficient of thermal expansion, K1 b* Coefficient of concentration expansion h Circumferential coordinate m Kinematic viscosity, m2 s1 q Density, kg m3 Rp s w Dimensionless average wall shear stress,½ 0 ðBvz =BrÞr¼0:5 dh=p Subscripts 0 Refers to entry r, h, z Refer to the radial, circumferential and axial directions, respectively V Refers to dimensional quantity
Acknowledgements The authors thank the bFaculte des Sciences de MonastirQ, the bLaboratoire d’Etudes des Syste`mes Thermiques et Energe´tiques de MonastirQ and the Natural Sciences and Engineering Research Council of Canada for their financial support. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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