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Multiplicity of solutions due to combined mixed convection and radiation in a channel discreetly heated from below A. BAHLAOUI*, A. RAJI* and M. HASNAOUI** Department of Physics * Faculty of Sciences and Techniques, Unit of Formation and Research of Chemistry and Environment Sciences, B. P. 523, Beni-Mellal ** Faculty of Sciences Semlalia, Unit of Formation and Research of Thermics and Fluid Mechanics, B. P. 2390, Marrakech MOROCCO Abstract: - In this work, we present a numerical study of the phenomenon of the mixed convection coupled with radiation in a channel inclined with respect to the horizontal, discreetly heated by the bottom and subjected to an external ventilation of fresh air. We study the emissivity effect, , of the walls of the channel on the multiplicity of solutions. The dynamical and thermal flow structures is examined by presenting typical streamlines and isotherms for Ra = 105, Re = 10 and = 30°. The average convective, radiative and total Nusselt numbers, at the levels of the various walls are calculated and presented for various values of with an aim of quantifying the contribution of the radiation to the overall heat transfer. The ratio of the heat quantities leaving the channel through the cold top wall, Qtop, and through the exit, QE, is also presented in order to better know the most favorable issue to the heat transfer. Key-Words: - Mixed convection, radiation, discrete heating, numerical study, multiplicity of solutions. Nomenclature tot total B aspect ratio of the channel, B L/H T0 dimensionless reference temperature, cv convection T0 = TC / (TH TC ) Fij view factor between Si and Sj elements u0 velocity of the entering flow, m/s g acceleration due to the gravity, m/s2 u, v dimensionless horizontal and vertical H height of the channel, m velocities, (u, v) (u , v ) / u 0 Ii 4 dimensionless irradiation, I i I / TC i x, y dimensionless coordinates, dimensionless radiosity, J i J / TC (x, y) (x , y) / H 4 Ji i L length of the channel, m Nr convection-radiation interaction parameter, Greek symbols 4 N r TC H / (TH TC ) thermal diffusivity of the fluid, m2/s Nu average Nusselt number thermal expansion coefficient of the fluid, * Nu average normalised Nusselt number, 1/K * Nu = Nu/Nu( = 0) emissivity of the walls of the channel Pr Prandtl number, Pr / thermal conductivity of fluid, W/mK Qr dimensionless radiative heat flux, kinematic viscosity of fluid, m2/s Q r Q r / TC 4 dimensionless vorticity, H / u 0 Ra Rayleigh number based on H , dimensionless stream function, / u H Ra g (TH TC )H 3 / 0 Stéfan-Boltzman constant, rd radiation = 5,669 10-8 W/(m2 K4) Re Reynolds number, Re u H / 0 angle of inclination of the channel t dimensionless time, t t u / H 0 T dimensionless fluid temperature, Subscripts T (T TC ) / (TH TC ) b bottom wall TC temperature of the cold wall, K CR critical H heated surface TH temperature of the heated elements, K I inlet of the channel E exit of the channel top top wall Superscripts ' dimensional variable 1 Introduction The study of heat transfer by mixed convection in inclined channels has received a growing interest The non-dimensional governing equations, written during the last decades. This interest is dictated by in - formulation are: the role played by such configurations in the field of 1 2 2 Ra T T u v 2 2 2 cos θ sin (1) the habitat, the design of the solar heat collectors and t x y Re x y Re Pr x y more recently the cooling of the electronic cards T T T 1 2T 2T (because of the tendency to the miniaturization of u v (2) t x y RePr x 2 y 2 the components). An exhaustive review of the literature shows that the case of the mixed 2Ψ 2Ψ 2 Ω (3) convection in a rectangular channel was examined x 2 y by several authors and the references [1-4] are quoted only by way of indication. However, the The stream function and the vorticity are related to effect of the radiation was often neglected in the the velocity components by the following majority of the available studies in spite of its expressions: significant contribution to the heat transfer in such Ψ Ψ v u u , v and Ω = (4) systems. Its effect was rather taken into account in y x x y rectangular closed [5-6] or opened [7-8] geometries in the case of natural convection. In comparison with these studies, the case of opened systems with 2.1 Boundary conditions radiant walls in mixed convection hardly starts to u=v=0 on the rigid walls arouse interest . T = v = = 0, u = 1 at the inlet of the channel and = y The objective of this work consists in studying T=0 on the top cold wall the coupling between mixed convection and T=1 on the heated elements radiation in a rectangular channel by examining the T NrQr 0 on the adiabatic portions effect of the emissivity, , of the walls on the y temperature distribution and the flow structure within the channel. A detailed attention will be given The vorticity on the rigid walls is calculated by to the contribution of the radiation to overall heat using the Woods formula : transfer for different combinations of the governing parameters. ΩW 1 Ω W 1 3 Ψ W 1 Ψ W (5) 2 Δη 2 2 Analysis and modelling where, w, denotes the wall and the space step of The geometry considered is depicted in Fig. 1. It space in the normal direction to the latter. consists of a channel of finite length, with an aspect ratio B L/H = 10, inclined with respect to the The boundary conditions at the exit of the channel horizontal and discretely heated from below. The are obtained by mean of extrapolation technique top wall of the channel is maintained at a cold similar to that used in references [1-4]. uniform temperature. The system is submitted to an imposed flow of ambient air, parallel to the plates. 2.2 Radiation equations The flow is considered two-dimensional and The calculation of the radiative heat exchange laminar. The fluid properties are considered between the channel and its surrounding (through constant and the assumption of an incompressible the inlet and the exit) is based on the radiosity fluid obeying the Boussinesq approximation is method. In addition, the radiative heat transfer valid. between the surfaces is expressed by the following y’, v’ Exit set of equations: Cold wall x’, u’ Heated section Inlet Adiabatic portion T 4 Equations (1) and (2) were discretized by using a J i (1 i ) Fij J j i i 1 (6) finite difference technique. Hence, all the diffusive sj To terms were approached by centred differences while a second order upwind scheme was used for the The radiative heat flux leaving a surface Si is convective terms in order to avoid possible evaluated by: instabilities frequently encountered in mixed convection problems. The integration of equations T Q r J i I i i ( i 1) 4 To F ij Jj (7) (1) and (2) was ensured by the Alternate Direction S j Implicit method (ADI). At each time step, the Poisson equation was treated by using the point Since the working fluid (air) is transparent to the successive over-relaxation method (PSOR) with an radiation, the contribution of the latter appears only optimum over-relaxation coefficient equal to 1.74 in the thermal boundary conditions. Thus, the heat for the considered grid. The set of equations received by radiation on the adiabatic portions is described by Eq. (6), representing the radiative restored to the fluid by conduction according to the transfer between the different elementary surfaces of relation: the channel is solved by using the Gauss-Seidel T method. The numerical code performed was Nr Qr 0 (8) validated by comparing the results obtained with y those of Yücel et al.  in the absence of the radiation effect. The relative difference observed in 2.3 Heat transfer terms of the Nusselt numbers is lower than 1.6% for The average Nusselt numbers characterising the various combinations of the governing parameters contributions of the mixed convection and radiation (Ra, Re and ). through the heated wall (bottom), the cooled one (top), the inlet and the exit of the channel are respectively defined as: 3 Results and discussion For this problem, multiple steady solutions have been obtained. Hence, in Figs. 2a-2c and 3a-3c, 1 T B B 1 Nu b (cv) = - B 0 y dx ; Nu b (rd) = B0 N rQr dx (9) streamlines and isotherms illustrating two different y0 y0 solutions are presented for various values of , Ra = 105, Re = 10 and = 30°. The solution 1 T B 1 B corresponding to Figs. 2 was obtained using a pure Nu top (cv ) = - B y dx ; Nu top ( rd ) = B NrQr dx (10) mixed convection solution ( = 0) corresponding to 0 y 1 0 y 1 lower value of Ra as initial condition. Hereafter, this kind of solution will be called as S1 type. The T 1 1 Nu I (cv) = - dy ; Nu I (rd) = N r Q r dy (11) solution presented in Figs. 3, known as S2 and S3 0 x x 0 0 x 0 type, was obtained by using the pseudo-conductive regime. Other kinds of initial conditions were also 1 1 tried, but the final state was either one of those T Nu E (cv ) = x + Re Pr u T dy ; Nu E (rd ) N r Q r x B dy (12) presented in Figs. 2 and 3. Fig. 2a, obtained for = 0 0 x B 0, shows a complex multicellular structure characterised by the presence of a large cell For each of the precedent equations, the total occupying, from the inlet, a good part of the space Nusselt number is evaluated as being the sum of the offered inside the channel. The effect of radiation corresponding convective and radiative Nusselt leads to the split of this cell to three cells of numbers. different sizes when the emissivity reaches a critical value CR1 = 0.32. This behaviour is illustrated in The heat quantities leaving the channel through the Figs. 2b and 2c for two values of beyond this cold wall (top), Qtop, and through the exit, QE, are threshold. The corresponding isotherms are well respectively defined as: marked by the multicellular behaviour of the flow. B 1 T T The effect of the emissivity is also important in the Q top y dx ; Q E x Re Pr u T N Q r r dy (13) case of S2 and S3 solutions presented in Fig.3. In 0 y 1 0 x B fact, significant changes are observed in the flow structure when is increased. Hence, in Fig. 3a, 2.4 Method of solution obtained for = 0, the flow structure is characterised by the presence of a closed cell occupying almost the totality of the space offered in the channel (S3 solution). The corresponding isotherms indicate an important heat exchange between the cell and the a) end part of the cold plate, at the vicinity of the exit. However, this behaviour changes when the effect of the radiation is considered (Figs. 3b and 3c). The closed cell (S3 solution) is replaced by a reversal flow (S2 solution) more favourable to the evacuation of heat through the exit of the channel (for >0.1). The isotherms examination shows a very limited heat transfer through the cold wall. In fact, the flow b) admitted through the exit, being at the ambient temperature, enters by the upper part of the channel and skirts the cold wall of the latter. It is clear to deduce from the flow structure that the S1 solution is more favourable than the S2 and S3 solutions to the heat evacuation through the cold surface while the opposite is true concerning the heat evacuation through the exit. c) Fig. 3: Streamlines and isotherms obtained for Re = 10, Ra = 105, = 30° and various values of the emissivity of the walls (using the pseudo-conductive regime): a) = 0, b) = 0.5 and c) = 0. 8. a) Variations, with , of the normalised Nusselt numbers (with respect to pure mixed convection) evaluated on the bottom wall are presented in Fig. 4 for the two flow structures previously defined. A close examination of this curve shows that the emissivity effect on the mixed convection Nusselt b) number is very limited while it is significant on the radiation heat transfer rate which increases almost linearly with . Consequently, the total Nusselt number undergoes a linear increase with this parameter. A comparison of the mixed convection and radiation contributions to the overall heat transfer shows that the contribution of the mixed c) convection remains more important as long as is lower than a critical value which about 0.53/(0.43) in Fig. 2: Streamlines and isotherms obtained for Re = the case of S1/(S2 and S3). It is useful to note that, in general, the reversal flow (S2) is more favourable to 10, Ra = 105, = 30° and various values of the heat transfer in comparison with the multicellular the emissivity of the walls (using a pure flow structure (S1). The variations of the average mixed convection solution): a) = 0, b) = Nusselt numbers with , evaluated at the inlet, are 0.5 and c) = 1. presented in Fig. 5. It is clear that the contribution of the mixed convection to the overall heat transfer is negligible and essentially due to the cells which extend until the inlet of the channel. However, the contribution of the radiation is important and its importance increases by increasing the value of the emissivity. Finally, it is to note that the overall heat transfer at the inlet is independent of the type of increases quickly when is increased towards its solution which is expected since the latter depends critical value for which a maximum is reached mainly on the radiation effect. followed by a linear decreasee, resulting from the big change undergone by the flow structure (passage 3.5 from monocellular (S3) to reversal flow (S2)). The 3.0 S1 Solution radiative transfer varies in a monotonous way with S2 and S3 Solutions the emissivity. A comparison between the 2.5 * Nub(tot) convective and radiative heat transfer quantities, * 2.0 * Nub Nub(rd) evaluated at the level of the cold wall (Fig. 6) shows 1.5 that the contribution of radiation becomes greater 1.0 than that of the mixed convection when the 0.5 * emissivity value is higher than 0.1/(0.58) in the case Nub(cv) 0.0 of a monocellular and reversal/(multicellular) flow. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 However, the contribution of the mixed convection to the overall heat transfer is always higher than that of the radiation at the exit of the channel (Fig. 7) for Fig. 4 : Variation of the average normalized Nusselt the three flow structures. number on the bottom wall with the 3.5 emissivity of the walls for Ra = 105, Re = S1 Solution total 3.0 10 and = 30°. S2 and S3 Solutions 2.5 radiation 2.0 * 2.0 S1 Solution NuI(tot) Nutop S2 and S3 Solutions 1.5 radiation 1.5 total convection 1.0 S3 S2 NuI 1.0 0.5 convection NuI(rd) 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.5 NuI(cv) 0.0 Fig. 6: Variation of the average normalized Nusselt 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 number on the top wall with the emissivity of the walls for Ra = 105, Re = 10 and = 30°. Fig. 5: Variation of the average Nusselt number at 3.5 the inlet of the channel with the emissivity S3 S2 S1 Solution total of the walls for Ra = 105, Re = 10 and = 3.0 S2 and S3 2.5 convection Solutions 30°. * NuE 2.0 total Variations, with , of the normalised Nusselt 1.5 numbers on the top wall and the exit of the channel 1.0 convection are presented in Figs. 6 and 7. In the case of S1 solution (solid line), Fig. 6 shows that the amount of 0.5 radiation radiation heat evacuated by convection through the cold 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 surface is insensible to the variations of . However, in the case of the dashed line, the increase of the emissivity of the walls leads first to the decrease of Fig. 7: Variation of the average normalized Nusselt the convection effect due to the change of the flow number at the outlet of the channel with the structure which passes progressively from a emissivity of the walls for Ra = 105, Re = monocellular flow (S3 solution) to a reversal one (S2 10 and = 30°. solution) when reaches the value with CR2 0.1. Above this critical value of , the convection effect In such problems, it is useful to attach a special becomes constant and practically negligible (Nut(cv) importance to the quantities of heat leaving the 0.25). On the other hand, the examination of Fig. system through the cold surface and the exit in order 7, shows that, for the same solution, NuE(cv) to quantify how the energy provided by the heating elements is distributed between these two issues. The quantification of the relative amount of heat and structure fields. The radiative and total heat evacuated by each one of these issues to the total quantities evacuated through the cold wall and the heat transfer is presented in Fig. 8 in terms of the exit of the channel increase by considering the effect variations, with the emissivity, of the ratio Qtop/QE. of radiation, while the latter contributes to weaken In the absence of the radiation effect ( = 0), the the mixed convection at these issues. flow structure is characterised by the presence of a big cell. The corresponding flow structure is References: denoted as a solution of S3 type (dashed line). An  T. Tomimura, and M. Fujii, Laminar Mixed increase of the emissivity supports the appearance Convection Heat Transfer between Parallel of the return flow (S2 solution) which generates a Plates with Localized Heat Sources, Proc. Int. reduction in the quantity of the heat received by the Symp. on Cooling Technology for Electronic cold wall in favour of the exit (Qtop/QE decreases). Equipement, Honolulu, Aung, W. ed, 1988, pp. For > 0.1, the flow structure remains qualitatively 233-247. unchanged and the small increase of Qtop/QE results  C. Yücel, M. Hasnaoui, L. Robillard and E. only from the increase of the radiative heat transfer Bilgen, Mixed Convection Heat Transfer in at the level of the top wall. In the case of S1 solution Open Ended Inclined Channels with Discrete (solid line), the tendency of variation of the ratio Isothermal Heating, Num. Heat Transfer, Vol. Qtop/QE is very different. It shows that the heat 24, 1993, pp. 109-126. generated by the heated elements is mainly  A. Raji, M. 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The Vol. 18, No. 7, 2001, pp. 922-949. taking into account of the effect of radiation  L. C. Woods, A Note on the Numerical contributes to the multiplicity of the solutions. The Solution of Fourth Order Differential Equations, obtained results show the existence of three Aero. Quart, Vol. 5, 1954, pp. 176-184. solutions (multicellular S1, reversal flow S2 and monocellular S3). The radiation heat exchange was found to have a significant influence on the thermal
"Multiplicity of solutions due to combined mixed convection and"