Mixed convection heat and mass transfer within a vertical channel

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1 Mied convection heat and mass transfer within a vertical channel K. SELLAMI a *, N. LABSI a, M. FEDDAOUI b, M. OUBELA b, Y.K. BENKAHLA a a. Laboratoire des Phénomènes de Transfert, Equipe RSNE Département de Génie chimique et de Crogénie, Facultéde Génie Mécanique et de Génie des Procédés. Université des Sciences et de Technologie Houari Boumediene, BP. 32 El Alia, Bab Ezzouar, Alger, Algérie b. Laboratoire Génie de l'énergie, Matériau et sstèmes Département de Génie des Procédés, École Nationale des Sciences Appliquées d Agadir, Université Ibn Zohr, Maroc sellami_karima@ahoo.fr, nabilalabsi@ahoo.fr, m.feddaoui@uiz.ac.ma, m.oubella@uiz.ac.ma, ubenkahla@ahoo.fr Résumé : L objectif de ce travail est d étudier numériquement le transfert de chaleur et de masse lors de l'évaporation d'un film liquide d'acétone, d'épaisseur négligeable, mouillant les parois d'un canal vertical. Les équations générales de conservation et les conditions au limites associées sont discrétisées par le bais de la méthode des volumes finis. Le couplage vitesse-pression est traité par l'algorithme SIMPLER. L'étude se focalise sur l'analse de l'effet de la vitesse et de l'humidité relative de l'air à l entrée du canal, sur le comportement hdrodnamique, thermique et massique du flu d'air humide. Abstract: The purpose of this work is to evaluate heat and mass transfer during the evaporation of a negligible thickness liquid film of acetone, wetting the walls of a vertical channel. The governing equations and the associated boundar conditions are discretized b means of the finite volume method and the pressure-velocit coupling is treated with the SIMPLER algorithm. The stud focuses on the analsis of the effect of the inlet velocit and relative humidit of the air, on the hdrodnamic, thermal and mass behavior of the flow of moist air. Kewords: Mied convection; Heat and mass transfer; Evaporation, wet wall. Nomenclature Cp specific heat of the fluid at constant pressure, J.kg -1.K -1. D h hdraulic diameter, m D mass diffusivit, m 2.s -1

2 g gravitational acceleration, m.s -2 H channel width, m Gr M solutal Grashof number Gr T thermal Grashof number. k thermal conductivit, W.m -1.K -1 L channel length, m P pressure, Pa Re Renolds number, T temperature, C U 0 gas inlet velocit, m.s -1 V velocit, m.s -1 W mass fraction, kg of vap/kg of air longitudinal coordinate, m transverse coordinate, m Greek smbols coeficient of thermal epansion, K -1 0 relative humidit of the air-vapor miture. dnamic viscosit, kg.m -1.s -1. densit, kg.m -3 Indices and eposants w wall v liquide vapor m avenage 0 inlet 1 Introduction Coupled heat and mass transfer is widel encountered in man practical applications and processes such as: distillation, film cooling, liquid film evaporator, cooling towers and cooling of microelectronic equipments. Due to such widespread applications, heat and mass transfer of air stream with liquid film evaporation associated with latent heat transfer has received considerable attention of researchers over the past decades, b considering several configurations. Yan and Lin [1] investigated laminar natural heat and mass transfer in a vertical plate channel with pure liquid film evaporation. The show that the influence of the liquid is substantial near the interface. Ait Hammou et al. [2] studied numericall the effects of the inlet conditions on a downward laminar flow of humid air in a vertical channel with isothermal wetted walls; the cases of film evaporation and vapor condensation were considered. Lin et al. [3] as well as Yan [4,5] investigated the influence of the wetted wall on laminar mied convection heat transfer in vertical ducts. In their studies, the liquid film on the wetted wall was assumed etremel thin so that it was regarded as a boundar condition for heat and mass transfer onl. Ben Jabrallah et al. [6] studied the effects of the flu densit of heating, the wall temperature and the gas mass flow, on evaporation b convection. Their results show that the heat and mass transfer can be intensified b decreasing the flow rate of the mass of feed and increasing the feed temperature due to the increase of the effective surface of the evaporation. Oulaid et al. [7,8] studied numericall the effect of buoanc forces on the heat and mass transfer b mied convection for laminar upward flow of humid air in a vertical and inclined channel constituted of two flat plates. Their results show that buoanc forces have significant effect on the thermal field and hdrodnamic mass fraction; these forces decelerate the flow near the walls and induce the flow reversal at different temperatures between the gas and the walls and thus, reduce the heat and mass transfer. Debbissi et al. [9] have numericall studied the evaporation of water b mied and free convection, resulting from thermal and solutal buoanc forces, acting in the downward direction within a vertical channel, is this stud the authors eplain the eistence of the inversion temperature b considering heat and mass transfer in the evaporation process. An analsis has been developed for studing the evaporative cooling of liquid film falling inside a vertical insulated tube in turbulent gas stream has been also reported b Feddaoui et al. [10]. A numerical stud of turbulent convection heat and mass transfer of evaporative cooling in an asmmetricall heated wet channel was presented b Fedorov et al. [11]. The results obtained suggest that the heat and mass transfer analog begins to break down after the flow transitions from laminar to turbulent.

3 Wet and isotherm 23 ème Congrès Français de Mécanique Lille, 28 Août au 1 er Septembre 2017 The objective of the present stud is to analze the coupled heat and mass transfer processes in the gas stream in the evaporation of negligibl thin film of acetone along a vertical channel where the liquid film can be replaced b the approimate boundar conditions for the gas stream 2 Phsical and mathematical model 2.1 Phsical model Let s consider a negligible thickness liquid film of acetone wetting the walls of a vertical channel of length L and a channel width H, shown schematicall in figure 1. The walls of the channel are maintained at a constant temperature T w. The channel is traversed b a downward flow of miture gas with a constant temperature T 0, a uniform relative humidit 0 and a uniform velocit profile U 0 At the inlet. The flow is assumed to be laminar. Radiation, viscous dissipation, and other secondar of Soret and Dufour effects are assumed negligible. T 0, W 0,U 0 o V Wet and isotherm L H Figure 1: Schematization of the phsical problem. 2.2 Governing equations Stead laminar mied convection in the gas flow can be epressed b the boundar laer conservation equations of mass, aial-momentum, energ and concentration. This sstem can be written as follows: m V mv 0 (1)

4 mvv V V m mvv mvv P V m mg P V m W C V C V k D C C m p m p m m pv pa (4) mv W mv W md mv W (2) (3) (5) 2.3 Boundar conditions At the inlet of the channel: = 0 0 < H V = U0 V = 0 T = T0 W = W0 (6) At the walls: 0 L = 0 V = 0 T = Tw W = Ww (7) Dmv W V = Ve 1 Ww (8) w At the outlet: = L 0 H V V T W 0 (10) 3 Numerical method The set of equations governing the flow, the heat and mass transfer as well as the associated boundar conditions are discretized b means of the finite volume method proposed b Patankar [12]. The velocit-pressure coupling is treated using the SIMPLER algorithm [12]. The mesh is non-uniform in both and directions with a higher node densit near the entrance and close to the walls where the temperature and the concentration gradients are the highest. To validate the computer code, the obtained results are compared to those of Desraaud et al. [13] for the case of a thermal mied convection problem within a vertical channel, in the presence of dr air with a uniform velocit profile (Re = 300) and temperature T 0 = 10 C at the entrance. The vertical walls are maintained at a constant temperature T w = 60 C. This comparison is illustrated in figure 2, which represents the aial evolution of the sensible Nusselt number according to the non-dimensional aial length (/D h Pe), for different values of the channel width and the thermal Grashof number. The figure shows a ver good agreement between both results, since the relative difference is less than 2%.

5 Nu Present stud Desraaud et al [13] d = 0,020 m d = 0,025 m d = 0,030 m d = 0,060 m Forced convection Results and Discussion /(D h Pe) Figure 2: Aial evolution of the sensible Nusselt number. The following results concern the case of the evaporation in the presence of a laminar downstream air flow within a vertical channel of an aspect ratio of = 1/100, a length L = 1.5 m and a width H = 0.02 m. The channel walls are wetted b a thin film of acetone and maintained at a constant and uniform wall temperature (T w = 20 C). 4.1 Velocit and temperature profiles Figure.3 presents the longitudinal velocit profiles at three different positions = 0.01 m, = 0.20 m and = 1.50 m. Closest to the entrance region, the velocit profile is characterized b an acceleration of the flow near the walls and smmetrical concavit in channel center, resulting in distortion to satisf the mass conservation, while we notes a deceleration at the center channel = 1.5 m = 0.2 m = 0.01 m V (m/s) (m) Figure 3: Longitudinal velocit profiles. 0 = 10% ; T 0 = 30 C ; T w = 20 C ; Re = 300 ; = 1/100.

6 Figure 4 illustrates the aial development of the temperature (a) and the concentration (b) profiles in the gas stream, at different positions = 0.01 m, = 0.20 m and = 1.50 m. Figure 4(a) shows that the gas temperature decreases monotonicall from the inlet temperature T 0 = 30 C to attain the wall temperature T w = 20 C at the outlet. This indicates that the direction of the sensible heat transfer in the gas side is from the gas flow to the walls. We can see from figure 4(b) that the acetone mass fraction vapor increases graduall as the gas moves downstream (a) = 1,5 m = 0,2 m = 0,01 m (b) T ( C) (m) W (kg kg -1 ) (m) = 0,01 m = 0,2 m = 1,5 m Figure 4: Temperature (a) and concentration (b) Profiles. 0 = 10%, T 0 = 30 C, T w = 20 C, Re = 300, = 1/ Combined effects of humidit and temperature In this section, we investigate the influence of humidit and air temperature at the entrance of the channel, on heat and mass transfer characteristics as well as on the hdrodnamic behavior. Figure 5 shows the evolution of the means sensible (a) and latent (b) Nusselt numbers depending on the inlet temperature for different air humidit values at the entrance. B analzing figure 5(a), we note that the average sensible Nusselt number remains positive. These positive values mean that the sensible heat echange alwas occurs from the air to the channel walls (air-cooling) since these walls are kept at a temperature less than the air at the inlet. From this figure we also find that the mean sensible Nusselt number increases with the increase of the air-acetone miture temperature, for a humidit less than 50%. This is due to the fact that the increase of the temperature increases the wall temperature gradients which enhances subsequentl the heat echanges. However, the heat transfer decreases for a humidit more than 50%. Figure 5(b) reveals that the average latent Nusselt number has positive or negative values. Positive values correspond to a latent heat flu in the same direction as the sensible flow resulting thus, to the condensation of the steam contained in the air on the channel walls. The negative values correspond to a latent flu from walls of the channel to the gas flow; so there is evaporation of the film covering the walls. It is to note that the average latent Nusselt number decreases with both the temperature and humidit at the inlet. We note also that the condensation occurs for an input temperature higher than 80 C and a relative humidit above 50%.

7 (a) = 70 % = 50 % = 10 % (b) = 70 % = 50 % = 10 % Nu s mo 11.0 Nu l mo T 0 ( C) T 0 ( C) Figure 5: Longitudinal evolution of the average sensible (a) and latent (b) Nusselt numbers. Re = 300, = 1/100, T w = 20 C. 5 Conclusion This numerical work deals with the hdrodnamic, thermal and solutal behavior of an evaporating thin liquid film of acetone flowing over a vertical channel. The walls of the latter are wetted b a thin acetone film and maintenated isothermal. The channel is traversed b a laminar flow of air at constant temperature and humidit at the entrance. The main results obtained from this stud are as follows: Condensation occurs when the vapor mass fraction at the inlet is higher than the corresponding saturation value at the wall temperature (for a temperature higher than 80 C and humidit higher than 50%). In the other case, evaporation takes place. The increase of the inlet air temperature results in a small increase of the sensible Nusselt number for an humidit less than 50% and a significant decrease in latent Nusselt number. References [1] W. M.Yan, TF. Lin, Evaporative cooling of liquid film through interfacial heat and mass transfer in a vertical channel. Numerical stud, Int J. Heat Mass Transfer.34 (1991), pp [2] Z. Ait Hammou, B. Benhamou, N. Galanis, J. Orfi, Laminar mied convection of humid air in a vertical channel with evaporation or condensation at the wall, Int. J. Thermal Sciences. 43 (2004), pp [3] T. F. Lin, C.J. Chang, W.M. Yan, Analsis of combined buoanc effects of thermal and mass diffusion on laminar forced convection heat transfer in a vertical tube, J. Heat Transfer. 110 (1988), pp [4] W. M. Yan and T. F. Lin, Effect of wetted wall on laminar mied convection in a vertical channel, J. Thermo. Heat Transfer 3, (1988), pp [5] W. M. Yan, Y. L. Tsa, and T. F. Lin, Simultaneous heat and mass transfer inlaminar mied convection flows between vertical parallel plates with asmmetric heating.vol. 10, No, 3, (1989 ), pp [6] S. Ben Jabrallah, A. Belghith, J.P. Corriou, Convective heat and mass transfer with evaporation of a falling film in a cavit, Int. J. Thermal Sciences. 45 (2006), pp [7] O. Oulaid, B. Benhamou, N. Galanis, Simultaneous Heat and Mass Transfer in inclined channel with asmmetrical conditions, J. Applied fluid Mechanics, Vol.l5, N.3 (2012), pp [8] O. Oulaid, B. Benhamou, N. Galanis, Flow reversal in combined laminar mied convection heat and mass transfer with phase change in a vertical channel, Int. J. Heat and Fluid Flow, 31 (2010), pp

8 [9] C. Debbissi, J. Orfi, S. Ben Nasrallah, Evaporation of water b free or mied convection into humid air and superheated steam, Int. J. Heat and Mass Transfer, Vol. 46 (2003), pp [10] M. Feddaoui, H Meftah, A. Mir, The numerical computation of the evaporative cooling of falling water film in turbulent mied convection inside a vertical tube, Int. J. Heat Mass Transfer, 33 (2006), pp [11] A.G. Fedorov. R. Viskanta, A.A. Mohamed., Turbulent heat and mass transfer in asmmetricall heated, vertical parallel-plate channel, Int. J. Heat Fluid Flow,Vol. 18, (1997), pp [12] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere, [13] G. Desraaud, G. Lauriat, Heat and mass transfer analog for condensation of humid air in a vertical channel. Heat and mass transfer, Vol. 37, No. 1, (2001), pp