EC 20,2. Received February 2002 Revised November 2002 Accepted November 2002

Transcription

1 The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at EC 20,2 152 Received February 2002 Revised November 2002 Accepted November 2002 Mixed convection in rectangular enclosures with adiabatic fins attached on the heated wall S. Kasbioui, E.K. Lakhal and M. Hasnaoui Department of Physics, University Cadi Ayyad, Marrakech, Morocco Keywords Mixed convection, Structures, Numerical methods, Heat transfer Abstract The investigation of heat transfer and fluid flow by mixed convection in a vertical rectangular cavity containing adiabatic partitions attached to the heated wall is numerically studied. The parameters governing this problem are the Rayleigh number (10 3 # Ra # ), the Reynolds number (5 # Re # 100), the aspect ratio of the cavity (2.5 # A # 15), the partitions length (0.1 # B # 0.95), the aspect ratio of the micro cavities (0.33 # C # 0.66) and the Prandtl number (Pr ¼ 0.72). The results obtained indicate that the heat exchange between the system and the external medium, through the cold wall and the upper vent, are considerably affected by the presence of the partitions and for all the values of A and Ra considered. However, the quantity of heat released by the higher opening remains insensitive to the presence of the partitions; it depends only on the intensity of the forced flow. Moreover, it is shown that for critical values of Re and Ra, these rates of heat transfer pass by maxima of which the value is independent of A when this parameter is equal to or higher than 10. For high Reynolds numbers, the flow is dominated by forced convection for low values of Ra and high values of B. Finally, the competition between natural and forced convection occurs when Ra $ The heat transfer is correlated with the main parameters and presented for an eventual utilization in design. Engineering Computations Vol. 20 No. 2, 2003 pp q MCB UP Limited DOI / Nomenclature A ¼ aspect ratio of the cavity ð¼ H 0 =L 0 Þ B ¼ dimensionless length of the partitions ð¼ l 0 =L 0 Þ C ¼ aspect ratio of the micro cavities ð¼ h 0 =L 0 Þ C p ¼ coefficient of specific heat, J/kg/K g ¼ acceleration due to gravity, m /s 2 h 0 ¼ micro cavity height, m h ¼ local convection coefficient, W/m 2 /K H 0 ¼ height of the cavity, m L 0 ¼ width of the cavity, m l 0 ¼ length of partitions, m n ¼ number of micro cavities N ¼ number of partitions Pr ¼ Prandtl number ð¼ n=aþ Q ¼ dimensionless heat quantity Ra ¼ Rayleigh number ½¼gbDT 0 L 03 =ðnaþš Re ¼ Reynolds number ½¼u 0 0 L0 =nš Pe ¼ Peclet number ð¼pr ReÞ t ¼ dimensionless time ½¼t 0 u 0 0 =L0 Š t 0 ¼ time, s T H 0 ¼ temperature of heated wall, K T C 0 ¼ temperature of cooled wall, K T ¼ dimensionless fluid temperature ½¼ðT 0 2T 0 C Þ=DT0 Š DT 0 ¼ temperature difference ð¼t 0 H 2T0 C Þ; K (u, v) ¼ dimensionless velocity in x and y directions ½¼ðu 0 ;v 0 Þ=u 0 0 Š (u 0, v 0 ) ¼ velocities in x 0 and y 0 direction, m/s u 0 0 ¼ velocity of the imposed flow, m/s (x, y) ¼ dimensionless Cartesian coordinates ½¼ðx 0 ;y 0 Þ=L 0 Š (x 0, y 0 ) ¼ Cartesian coordinates, m

2 Greek symbols a ¼ thermal diffusivity, m 2 /s b ¼ adiabatic coefficient of thermal expansion of fluid, 1/K m ¼ dynamic viscosity, kg/(m s) l ¼ thermal conductivity, W/(mK) n ¼ kinematic viscosity, m 2 /s r ¼ fluid density, kg/m 3 C 0 ¼ stream function, s/m 2 C ¼ dimensionless stream function ð¼ C 0 =aþ V 0 ¼ vorticity, 1/s V ¼ dimensionless vorticity ð¼ V 0 : L 02 =aþ Subscripts C ¼ cooled surface H ¼ heating surface cr ¼ critical value loc ¼ local value E ¼ exit (upper vent) max ¼ maximum value min ¼ minimum value Superscript 0 ¼ dimensional variable Mixed convection in enclosures Introduction Heat transfer by convection in rectangular cavities has been the object of an impressive literature. Researchers have often considered rectangular cavities with vertical walls subjected to imposed temperatures and adiabatic horizontal walls. By adding partitions, regularly distributed on an active wall, these configurations have given rise to repetitive structures which are often used in building elements and solar collectors in order to reduce the heat losses by convection and radiation. Usually, a transparent cover is inserted in the habitat field in order to minimize the losses to the surroundings. The repetitive structures thus formed have dimensions of a rectangular cavity constituted of several small open cavities (micro cavities). Each micro cavity is limited at top and bottom by two adiabatic (Hasnaoui et al., 1990, 1992) or perfectly conducting (Lakhal et al., 1996) partitions attached to the heated wall. Previous experimental studies carried out on vertical cavities containing adiabatic partitions attached to the heated wall showed that the vertical walls (transparent cover and massive wall) are isothermal and maintained at different temperatures (Dumas, 1985). In such configurations, the temperature gradient in the vertical direction is negligible and the heat transfer is mainly carried out in the horizontal direction of the cavity. A literature review shows that the published works in this field are related to the natural convection phenomenon developed in similar configurations where the partitions are attached to one of the active walls. Hence, in the numerical study conducted by Hasnaoui et al. (1992), the partitions are adiabatic and attached to the hot wall. The researches have considered aspect ratios varying from 2.5 to infinity (by using periodic conditions) and partitions of lengths ranging between 0 (cavity without partitions) and They conclude that the length of the partitions is an important parameter to control the heat transfer and fluid flow within the cavity. Thus, the presence of the adiabatic partitions leads to the reduction of the quantity of heat released by the system to the external medium and this reduction is accentuated by increasing their length.

3 EC 20,2 154 In the same way, the numerical work conducted by Scozia and Frederick (1991) in a tall, vertical cavity, differentially heated and provided with conducting partitions attached to the cold wall, underlines the effect of the geometrical and thermophysical parameters on the dynamic and thermal behaviors of the fluid. These authors showed the strong dependence of the fluid flow on the variation of the geometrical parameters. The mean heat transfer obtained in the case of a cavity with finite aspect ratio was correlated with that obtained for a cavity with infinite aspect ratio. More recently, the numerical results obtained by Lakhal et al. (1996), while studying the natural convection developed in a rectangular cavity containing perfectly conducting partitions attached to the heated wall, showed that it is possible to combine the geometrical and thermophysical parameters to enhance the transfer of heat evacuated towards the outside (case favorable to the cooling of the electronic components) or to reduce it for the benefit of the system (case favorable to the heating of the habitat). In the habitat domain, Lakhal et al. (1995a, b) numerically analyzed the heat transfer by natural convection and conduction in a similar cavity limited by a massive wall containing conducting partitions on one of its faces (the one in contact with the outside). These authors showed that the heat transfer which leaves the system through its left wall, is dominated by conduction for the weak values of the relative conductivity of the wall (case of very important thermal resistance of the wall), while for high values of the conductivity, the heat exchange through the right wall of the system is due to convection. The heat transfer by natural convection and conduction in a partitioned nonconvective solar collector was also studied by them. They analyzed the effect of the main parameters governing the problem when the system s work conditions correspond to those of summer and winter. Recommendations concerning the design have been given in terms of useful correlations. They conclude that the operating conditions of this kind of wall are favorable to the summer working mode. In addition, these authors have shown that such systems are well performing when the temperature difference between the active walls is about 20 K. In the light of this literature review, it appears that the natural convection is able to adequately ensure the evacuation of the overheat when the powers released are weak. However, the recourse to an external ventilation becomes necessary to bring back the temperature of the system to a desired level in situations where the generated quantities of heat are important. This choice can be justified by the interest of using this kind of wall under double operation to ensure energy saving on the one hand, and the search of suitable thermal comfort on the other hand. Thus, in the present work, we study numerically a problem of mixed convection in a rectangular cavity provided with adiabatic fins attached to the heated wall. In practice, the objective is to make the energy transferred through

4 the massive wall, towards the habitation maximum, by limiting the amount of energy lost through the cover. The literature review related to this type of configuration shows an obvious deficiency of mixed convection works being attached to it. This justifies our interest to study the contribution of a forced flow in terms of quantities of heat exchanged between the system and the external medium with an aim to contribute to the enrichment of the specialized literature in this domain. Mixed convection in enclosures Problem formulation The studied configuration is shown in Figure 1. It is a vertical rectangular cavity of height H 0 and width L 0. For each given aspect ratio, the number n of micro cavities depends on the height of the cavity and the number of partitions N according to the relation n ¼ A=C ¼ ðn 2 1Þ: The partitions attached to the hot wall are considered adiabatic and having a length l 0 and a thickness e 0. The horizontal surfaces of the cavity in contact with the openings are adiabatic while its vertical left (cover) and right-hand side (receiving wall) walls are, respectively, maintained at constant temperatures T 0 C and T 0 H ðt 0 H. T 0 C Þ: Figure 1. Geometrical configuration of the studied system and computer domain

5 EC 20,2 156 The two openings are placed at bottom and top, between the horizontal adiabatic surfaces and the heated wall, to allow the admission of outside fresh air and the passage of the heated air towards the adjacent room, respectively. The forced flow of fresh air, imposed at the level of the lower opening, has a temperature T 0 C and a horizontal velocity u 0 0. The study is limited to the space located between the partitioned face of the receiving wall and the cover. Moreover, the cavity is supposed to be sufficiently deep in the third direction so that the hypothesis of two-dimensional study is valid. By using the stream function C and vorticity V formulation, the dimensionless equations governing this problem, in the case of incompressible fluid obeying the Boussinesq approximation, are written in the following form: V t þ ðuvþ x þ ðvvþ y ¼ 1 Re 2 V x 2 þ 2 V y 2 2 Ra T Pe Re x ð1þ T t þ ðutþ x þ ðvtþ y ¼ 1 2 T Pe x 2 þ 2 T y 2 ð2þ 7 2 C ¼ 2V ð3þ with u ¼ 2 C y and v ¼ C x Equations (1)-(4) have been written in dimensionless form by using as characteristic scales the width L 0 of the cavity, the temperature difference ðt 0 H 2 T0 C Þ between the two vertical active walls, the imposed velocity u0 0, and the time (u 0 0/L 0 ). The hydrodynamic boundary conditions are characterized by the no-slip of the particles on the rigid boundaries and the impermeability of the latter leading to u ¼ v ¼ 0 on all these boundaries. The other boundary conditions relative to this problem are defined as:. at the level of the lower opening, ð4þ u ¼ 21; T ¼ V ¼ v ¼ 0; and C ¼ y ð5aþ. on the left vertical wall, T ¼ 0 and C ¼ 0 ð5bþ. on the active faces of the right vertical wall, T ¼ 1 ð5cþ

6 . on the horizontal adiabatic surfaces, T y ¼ 0 and C ¼ 0 ð5dþ Mixed convection in enclosures. on the heated wall and the partitions surface, C ¼ C ¼ h 0 =L 0 ð5eþ 157. at the exit of the system (upper opening), the quantities T, C and V are extrapolated by considering their second derivatives equal to zero (Yücel et al., 1993). 3. Heat transfer In terms of heat transfer, the emphasis lies essentially on the quantities of energy released by the system towards the external surroundings (through the cover) and the adjacent room (through the higher opening). The evaluation of these losses is done according to the evaluation of the rates of heat transfer through the cover (conduction and natural convection) and through the higher opening (conduction and forced convection). They are calculated, respectively, by the following relations: Q C ¼ Z A 0 Q loc dy ¼ Z A 0 T x dy x¼0 ð6þ where Q E ¼ Z A A2C T þ ut Pe x Q loc ¼ hl0 l ¼ T x x¼0 dy x¼1 represents the quantity of heat calculated along the cover. ð7þ 4. Numerical method Equations (1) and (2) have been discretized by using a finite difference method. Hence, the central differences were used to discretize the stream function equation and the diffusive terms. While a second-order upwind scheme was used for the discretization of the convective terms (Roache, 1982), to avoid eventual numerical instabilities frequently encountered in such problems. The integration of these two equations is ensured by using the alternative direction

7 EC 20,2 158 implicit (ADI) method. At each time step, equation (3) is integrated by the method of successive over relaxation (SOR) with an optimum relaxation factor equal to 1.85 for the considered grids (Frankel, 1950). The velocity field is then deduced from equation (4) by using, for the derivatives of C, second order precise centered differences. In order to satisfy the continuity equation at each time step, the convergence criterion, X X C kþ1 i; j 2 C k i; j k10 24 C kþ1 i; j ; i; j i; j relative to the stream function was adopted (k indicates the number of iterations). The time steps dt were varied between and depending on the values of the governing parameters of the study. Thus, at low values of Re and high values of Ra, the smallest values of dt were used to ensure the convergence of the numerical code. Uniform grid in x and y directions is used with a size proportional to the aspect ratio in the vertical direction. Hence, it is 21 79; and ; respectively, for A ¼ 2:5; 10, and 15. The various preliminary tests carried out led us to consider these grids as sufficient to correctly model the fluid flow and heat transfer in the cavity. The computer code was validated in the absence of ventilation (pure natural convection; Re ¼ 0) with the results of Hasnaoui et al. (1992). Maximum differences of about 3 and 4 percent were found, respectively, in terms of rate of heat transfer and maximum steam function. Moreover, in the case of mixed convection, a systematic test of checking the energy balance was carried out for each studied case. In fact, for all the cases considered, the energy released to the fluid by the hot wall should leave the system through its cover and its higher opening. This balance was checked within 4 percent difference for the most unfavorable case. 5. Results and discussion For this study, the temperature difference between the active walls is considered lower than 708C so that the assumption of Boussinesq remains valid. This restriction implies that the Rayleigh number will not exceed for each studied configuration. The fluid flow and temperature distribution within the system as well as the heat transfer rates will be studied for 10 3 # Ra # ; 5 # Re # 100; 2:5 # A # 15; 0:1 # B # 0:95 and C ¼ 0:5 and 0.33 by considering air ðpr ¼ 0:72Þ as the working fluid. 5.1 Streamlines and isotherms For A ¼ 2:5; Figure 2(a)-(f) shows streamlines (on the left) and isotherms (on the right), obtained for Re ¼ 10; Ra ¼ 10 4 and 10 5 and different values of B (B ¼ 1=4; 1/2 and 3/4). The figure indicates an important effect of the governing parameters (Ra, Re and B) on the flow structure and the temperature

8 distribution inside the system. The natural convection effect is characterized by the presence of closed cells, with negative sign, indicating a counterclockwise fluid circulation. For Re ¼ 10 and B ¼ 0:75 (Figure 2(a) and (d)), the flow regime is pseudo-conducting for Ra ¼ 10 4 as indicated by the low distortions of the isotherms. Moreover, the results show a periodicity of the solution in all the micro cavities, except those at the two ends, where the openings were introduced. By decreasing the value of B to 0.5 (Figure 2(b) and (e)), a natural convection cell of weak intensity appears ðc min ¼ 20:109Þ for Ra ¼ 10 4 : It is located at the upper part of the cavity, in contact with the cold wall. For this value of Ra, the competition between this cell and the forced flow generates a limited effect on the distortion of the isotherms. However, this effect is accentuated with Ra since its increase generates a tightening of the isotherms in the higher part of the system (in the vicinity of the cold surface) and in the lower part of each micro cavity (in the vicinity of the heated walls). It should be noted that any additional reduction in the length of the partitions supports the formation of bulkier and more intense cells, which leads to a better stratification of the temperature within the system when Ra is increased (compare Figure 2(c) and (f) with others). The effect of the length of the partitions, on the flow structure and the temperature distribution within the system, also appears significant. In fact, for a given couple (Re, Ra), the natural convection cells are more and more intensified by decreasing B; they invade the space released by this reduction. For Re ¼ 100 and Ra ¼ 10 4 and 10 5, the obtained results (not presented) indicated, in general, the disappearance of the natural convection cells following the increase of this parameter (Re) and this, for the various values Mixed convection in enclosures 159 Figure 2. Steady-state streamlines (on the left) and isotherms (on the right) for A ¼ 2.5, Re ¼ 10, C ¼ 0.5 and different values of B. (a-c) Ra ¼ 10 4, (a) C min ¼ , (b) C min ¼ , (c) C min ¼ , (d-f) Ra ¼ 10 5, (d) C min ¼ , (e) C min ¼ , (f) C min ¼

9 EC 20,2 160 considered for B and Ra. For the lower value of Ra, the fluid flow was mainly outside the micro cavities; it was between the latter and the cold surface. The absence of the isotherms was observed between the end of the partitions and the cold wall due to the absence of thermal interaction between the ventilated fluid and the heated wall. This interaction remained negligible in spite of the increase in the Rayleigh number, showing that the forced convection was dominant for this value of Re. For A ¼ 10; the results presented in Figures 3 and 4 show streamlines and isotherms for Re ¼10 and 100, respectively. The other parameters are Ra (Ra ¼ 10 4 and 10 5 ) and B (B ¼ 0:75; 0.5 and 0.25). We should remark that for B ¼ 0.25 and relatively large Ra, instabilities phenomena have led to the divergence of the numerical code. In fact, the conditions imposed at the level of the lower opening become incompatible with the physical nature of the problem since the formation of a large natural convection cell prevents the forced flow penetration inside the cavity. This explains the absence of Figure 3(f) which would correspond to B ¼ 0:25; Re ¼ 10 and Ra ¼ 10 5 : Contrary to the case of small aspect ratios (case of A ¼ 2:5), the results obtained for A ¼ 10 show that the periodicity of the flow structure is practically recovered for B $ 0:5: In addition, the reduction of the relative height of the partitions precipitates the establishment of the mixed convection mode by supporting the formation and intensification of the closed cells. As an indication, for Ra ¼ 10 4 ; the value of C min passes from to when B passes from 0.75 to 0.25; whereas for Ra ¼ 10 5 ; it passes from to when B passes from 0.75 to 0.5. Moreover, the increase of the aspect ratio of the cavity generates the formation of closed cells for Ra ¼ 10 5 and B # 0:5; supporting the establishment of the mixed convection flow to the detriment of a dominant forced convection observed for A ¼ 2:5: 5.2 Effect of B on the heat transfer Figures 5(a) and (b) and 6(a) and (b), show, respectively, for Re ¼ 10 and Re ¼ 100; the effect of the length of the partitions on the quantities of heat Q C, released by the system through the cover and Q E, released by the system through the higher opening. The results are presented for different aspect ratios A (A ¼ 2:5; 10 and 15) and different Ra (Ra ¼ 10 4 and 10 5 ). Figure 5(a) shows that the heat loss by natural convection through the cover depends slightly on B for Ra ¼ 10 4 because of the weak natural convection cells which have just been formed. For Ra ¼ 10 5 ; the numerical solutions have been obtained only for B $ 0:5: Indeed, for values of B lower than this value, the computation code diverges for the reasons explained previously. Globally, the results indicate that the quantity of heat evacuated through the cover increases quickly with B and this increase is as important as the aspect ratio A is higher. As an indication, when B passes from 0.5 to 0.95, an increase in Q C of about 200 percent is obtained for A ¼ 10 and about 300 percent for A ¼ 15:

10 Mixed convection in enclosures 161 Figure 3. Steady-state streamlines (on the left) and isotherms (on the right) for A ¼ 10, Re ¼ 10, C ¼ 0.5 and different values of B. (a-c) Ra ¼ 10 4, (a) C min ¼ , (b) C min ¼ , (c) C min ¼ , (d, e) Ra ¼ 10 5, (d) C min ¼ , (e) C min ¼

11 EC 20,2 162 Figure 4. Steady-state streamlines (on the left) and isotherms (on the right) for A ¼ 10, Re ¼ 100, C ¼ 0.5 and different values of B. (a-c) Ra ¼ 10 4, (a) C min ¼ , (b) C min ¼ , (c) C min ¼ , (d-f) Ra ¼ 10 5, (d) C min ¼ , (e) C min ¼ 20.01, (f) C min ¼

12 Mixed convection in enclosures 163 Figure 5. Variations of Q C and Q E versus B for Re ¼ 10, Ra ¼ 10 4 and 10 5 and different values of A: (a) Q C variations and, (b) Q E variations The role of the parameter B is thus significant in the control of the dissipation towards the external medium. Also, to support the heat transfer towards the interior room, it is enough to reduce the length of the partitions. However, to support the losses of heat through the cover, we should give an impulse to the role of the natural convection by increasing the length of the partitions. Similar conclusions, about the role played by the length of the partitions on the heat dissipation through the cover, were formulated in pure natural convection case (Hasnaoui et al., 1992). Finally, to allow an operation of the system in mixed convection mode, it becomes necessary to proceed to a reduction of the fin

13 EC 20,2 164 Figure 6. Variations of Q C and Q E versus B for Re ¼ 100, Ra ¼ 10 4 and 10 5 and different values of A: (a) Q C variations and, (b) Q E variations lengths and the imposed velocity to the fluid at the lower opening of the system. The evolution of the quantity of heat, Q E, which leaves the system through its higher opening is marked by a strong reduction when the length of the partitions increases (Figure 5(b)). This tendency is explained by the fact that the long partitions slow down the evolution of the jet towards the top of the cavity supporting its interaction with the heated wall and this, to the detriment of the forced convection since the effect of the latter is reduced. Let us also note that the quantity of heat released by the system through the higher opening is

14 slightly dependent on Ra and A when the length of the partitions is lower than 0.6 ðb # 0:6Þ: Beyond this value of B, Q E becomes practically independent of these two parameters (Ra and A). As an indication, the relative reductions of Q E are situated between 80 and 95 percent (depending on A and Ra) when the length of the partitions passes from 0.1 to Finally, it should be noted that, in the conduction regime (weak Ra and Re), the ratio Q C =Q E is always higher than 1 and this, independently of the aspect ratio of the cavity and the length of the partitions. By increasing the intensity of the forced flow, the results of Figure 6(a) and (b), obtained for Re ¼ 100; show a change in the behavior of Q C and a continuous diminution of Q E when B increases. In fact, by increasing B from 0.1, the quantity of heat released by the system through the cover, Figure 6(a), undergoes variations, more important is that the values of A and Ra are high. For A $ 10; the tendencies are similar and characterized by a decrease of Q C due to the absence of natural convection cells. This decrease of Q C is stopped once its minimum is reached for a critical value, B cr,ofb which depends on Ra and A. Beyond B cr, the heat transfer increases with B and reaches a maximum. This increase of Q C is supported by the apparition and the development of natural convection cells which compete with the forced flow (Figure 4(e) and (f)). The more the deviations registered between the maximum and the minimum of Q C are important, the more high are A and Ra. The tendencies of Q C are rather different for A ¼ 2:5: In fact, for Ra ¼ 10 4 ; Q C remains practically constant until B ¼ 0:7 and undergoes a slight increase beyond this limit value of B. The curve obtained for Ra ¼ 10 5 undergoes a decrease until B ¼ 0:7 and then increases monotonously without presenting any sign of change. In Table I, the extreme values of Q C and the corresponding critical lengths are presented for different A and Ra. The variations of the curves previously described can be attributed to the absence or to the presence of natural convection cells which are favorable to the heat transfer through the cold wall. It is important to point Mixed convection in enclosures 165 Aspect ratio Critical values of A ¼ 15 Q C,min B cr Q C,max B cr A ¼ 10 Q C,min B cr Q C,max 5 10 B cr A ¼ 2.5 Q C,min B cr Q C,max 2 2 B cr Table I. Critical values of B and extremum points of Q C

15 EC 20,2 166 out that the tendencies of Q E, presented in Figure 6(b), remain qualitatively similar to those of Figure 5(b). 5.3 Conjugate effect of the parameters Ra and Re Figure 7(a) and (b) shows the effect of Ra, respectively, on the quantities of heat Q C and Q E for A ¼ 10 and different Re. The results obtained indicate that, for a given Ra, Q C decreases by increasing Re. Moreover, for a given Re, Q C is practically insensitive to the variations of Ra as long as the effect of the natural convection remains negligible. However, in mixed convection regime (existence Figure 7. Variations of Q C and Q S versus Ra for A¼10, B ¼ 0.75, C ¼ 0.5 and different values of Re: (a) Q C variations and, (b) Q E variations

16 of closed cells), the quantity Q C increases quickly with Ra. The variations of Q E remain, on the contrary, insensitive to Ra for all values of Re considered if we disregard the case Re ¼ 50 and Ra $ ; for which a sudden increase of this parameter is noted, followed by a decrease after reaching a maximum. This phenomenon is attributed to the complete disappearance of the natural convection cells which are more favorable to the heat transfer through the cover and the apparition of the return flow at the level of the higher opening of the system as illustrated by the streamlines and isotherms as is appropriate for Re ¼ 50; Ra ¼ and B ¼ 0:75: The variations observed can also be explained by the analysis, versus Ra, of the intensity (C min ) of the natural convection cells developed for different Reynolds numbers (Figure 8). The results obtained show a progressive reduction of the size of these cells when Re is progressively increased. For Re ¼ 100; the closed cells (natural convection cells) disappear completely, which leads to a net reduction of the heat losses through the cover. The results also show that, without depending on A and Re, the closed cells disappear once the value of B exceeds 0.5 and this, for Ra ¼ 10 4 : However, for moderate Rayleigh numbers, they tend to occupy more and more of space as Ra increases and Re decreases. This occupation becomes well marked for the great aspect ratios as indicated in Figure 9. Note that the results of Figure 8 indicate that for Ra # 10 4 and B ¼ 0:75; the convective cells are absent for all the considered values of Re. In fact, the plots of C min versus Ra show the intensification of the natural convection cells once Ra increases and Re decreases. As an indication, for Ra ¼ 10 5 and A ¼ 10; the intensity of the cells increases by about 300 percent when the value of Re passes from 10 to 5. Mixed convection in enclosures 167 Figure 8. Variation of C min versus Ra for A ¼ 10, B ¼ 0.75, C ¼ 0.5 and different values of Re

17 EC 20,2 168 Figure 9. Variation of C min versus B for Ra ¼ 10 4, B ¼ 0.75, C ¼ 0.5 and different values of A and Re 5.4 Influence of Re The effect of the Reynolds number, Re, on the quantities of heat Q C and Q E has been studied for 5 # Re # 100; B ¼ 0:75 and different values of Ra and A ¼ 2:5 (Figure 10(a) and (b)) and 10 (Figure 11(a) and (b)). A global outline of the results obtained shows that the effect of Re on the evolution of Q C and Q E is important and depends on Ra. The decrease of Q C, resulting from an increase of Re, can be explained by the fact that the increase of the velocity imposed on the inlet of the cavity, engenders an intensification of the forced flow to the detriment of the natural convection cells which are at the origin of the increase of heat losses towards the outside, through the cover. We should note that for A ¼ 2:5 and Re sufficiently high ðre $ 85Þ; the quantity of heat Q C becomes independent of Re because of the total dominance of the forced flow and this, for all the values considered for Ra. However, for A ¼ 10; the critical value of Re passes to 150 by reason of the inertia imposed by this high value of the aspect ratio. The variations of Q E are marked by a linear increase with Re and they are independent of Ra as this parameter is lower than or equal to Ra cr ¼ : Beyond this limit value of Ra, three tendencies can be noted in the evolution of Q C when Re increases: for Re # 45; the heat losses through the cover decrease to reach their first minima which are, for Ra ¼ 10 5 ; of the order of 1.8 and 16.5, respectively, for A ¼ 2:5 and 10. From these critical values of Q C, the heat losses towards the ambience increase until reaching their maxima, leading to the resonance phenomena analogous to those observed in natural convection by Lage and Bejan (1993) and Lakhal et al. (1999) following the temporal variations of the parameters characterizing the thermal boundary conditions. It should be remarked that these critical values of Ra and Re, and also the extremum values of the heat transfer depend strongly on the relative length, B, of the partitions. We can thus say that the mixed convection

18 Mixed convection in enclosures 169 Figure 10. Variations of Q C and Q E versus Re for A ¼ 2.5, B ¼ 0.75, C ¼ 0.5 and different values of Ra: (a) Q C variations and (b) Q E variations enters in resonance with the forced flow imposed on the inlet of the cavity. The fluctuations in terms of Q C and Q E are about 300 percent for A ¼ 2:5 when compared with the rate of heat released for Re ¼ 5 (weak velocity of the imposed flow). The results show that the limit values, Re m, for which these important energetic fluctuations are observed, depend on the aspect ratio and the Rayleigh number as shown in Figure 12(a). In fact, when Ra reaches a certain limiting value, the value of Re m engendering these important energetic transformations depend only on A. The limiting values of Ra are 1: ;

19 EC 20,2 170 Figure 11. Variations of Q C and Q E versus Re for A ¼ 10, B ¼ 0.75, C ¼ 0.5 and different values of Ra: (a) Q C variations and (b) Q E variations 1: ; and 1: ; respectively, for A ¼ 2:5; 10 and 15 and this, for B ¼ 0:75: Beyond these limiting values of Ra, the critical values of Re for which the heat losses become maximum are of the order of 80, 90 and 100 for the same respective values of A. Note that for a couple (Ra, Re) of critical values, the mixed convection regime dominates, thus leading to a better contribution of the system to the thermal exchanges. Note also that, when the value of Ra is lower than the limiting value, the linear tendency of the curves of Figure 12(a) can be correlated with the simple form Re m ¼ a Ra b : The regression parameters a and b are given in Table II. When the value of Re exceeds this limit, a drastic diminution of Q C towards a value close to that of the other cases is observed.

20 Mixed convection in enclosures 171 Figure 12. (a) Couples (Ra, Re) engendering maximum heat transfer for B ¼ 0.75, C ¼ 0.5 and different values of A. (b) Variations of Q E,max versus Ra for B ¼ 0.75, C ¼ 0.5 and different values of Ra Aspect ratio a ( ) b A ¼ A ¼ A ¼ Table II. Regression parameters of Re m This behavior can be explained by the disappearance of the return flow and also by the dominance of the forced flow (Figures 10(a) and 11(a)). Concerning the effect of Re on Q E, the analysis of Figures 10(b) and 11(b) indicate that Q E increases linearly with Re for Ra, and this without depending on the aspect ratio. In addition, this quantity of heat remains insensible to the variations of the Rayleigh number by reason of the forced flow regime domination. For Ra $ ; Q E increases quickly with Re to reach its

21 EC 20,2 172 maximum value for Re ¼ Re m : Once this critical value of Re is slightly exceeded, an abrupt decrease of Q E occurs. Subsequently Q E increases linearly with Re, independent of Ra. Note that the critical values, Re m, of Re engendering the maxima of Q E and Q C are identical. In Figure 12(b), the maximum rates of the thermal exchanges, Q E,max are presented as a function of Ra. The examination of this figure shows the existence of a certain critical value of Ra for which the thermal interactions are maximum. In addition, it has been found that when A $ 10; Q E,max becomes practically insensible to the increase of A. As an indication, the values of Q E,max are of the order of 6.7, 7.1 and 7.2 obtained for the values of the critical Rayleigh numbers ; 1: and 1: and this, respectively, for A ¼ 2:5; 10 and Conjugate effect of Re and C The variations of the quantities Q C and Q E with Re are presented, respectively, in Figure 13(a) and (b) for A ¼ 10; B ¼ 0:75; C ¼ 0:33 and 0.5 and Ra ¼ 10 5 : By comparing the general tendencies obtained for the two values of C, it can be concluded that the diminution of the parameter C (increase of the number of the partitions for the same aspect ratio A) leads to the disappearance of the reverse tendencies in the evolutions of the heat transfer rates through the cover and also through the exit opening to engender a slight reduction of Q C with Re and a sensible increase of Q E with Re. As an indication, for C ¼ 0:33; reductions of about 42 percent are observed in terms of Q C when Re passes from 5 to 100 while the value of Q E is multiplied by 20 following the same variations of Re. The regular variations of these curves can be explained by the fact that the addition of the partitions is not favorable to the natural convection; it engenders a diminution of the intensity of cells (when the latter exists) and consequently supports the disappearance of the convective regime which is at the origin of the change in the tendencies of the curves for C ¼ 0:5 and Ra ¼ 10 5 : Hence, when the value of C passes from 0.5 to 0.33, that of C min passes from to for Re ¼ 5 and from to 0 for Re ¼ 70: Note that the heat evacuated through the upper opening becomes more and more important when Re is increased, engendering a favorable situation when it is a question of heating the adjacent room. The control of the heating operation passes through a judicious choice of the height of the opening, the velocity of the imposed flow and the partitions length (case not studied here). For a given Ra, the obtained results show that the establishment of the convective regime is retarded when the opening width and the velocity of the imposed flow are weak. In fact, for Re ¼ 10; this flow regime starts at Ra ¼ 10 4 and ; respectively, for C ¼ 0:5 and Recommendations for the design To allow a wide use of the obtained results, the quantities Q C and Q E have been correlated as a function of the governing parameters, namely the Rayleigh

22 Mixed convection in enclosures 173 Figure 13. Variations of Q C and Q E versus Re for A ¼ 10, B ¼ 0.75, Ra ¼ 10 5 and different values of C number ð10 3 # Ra # Þ and the Reynolds number ð5 # Re # 100Þ: The other parameters are B ¼ 0:75; C ¼ 0:5; Pr ¼ 0:72 and different aspect ratios. The proposed equation of the correlation has the form Q C=E ¼ a Ra b Re c and the used method of correlation is that of the mean squares. The results obtained are presented in Figures 14 and 15. They show that the equation of correlation, adjusted by a cloud of 300 points, has allowed us to obtain a good smoothing. Also, the correlation coefficient of the smoothing line is close to 1. In Table III, we present the smoothing parameters and the regression coefficients a for Q C and Q E.

23 EC 20,2 174 Figure 14. Regression curves of (a) Q C and (b) Q E for A ¼ 2.5, B ¼ 0.75 and C ¼ 0.5 In order to determine the forced convection predominance as compared to the natural convection, the couples (Ra, Re) for which Q E becomes higher than Q C are presented in Figure 16. The obtained results show that the critical values of Re (Re F ) favorable to the dominance of the forced flow increase with Ra and this, for the different aspect ratios considered. The tendency of the curves in Figure 16 leads to correlations of the form Re F ¼ a Ra b ð10 3 # Ra # Þ: The parameters of the correlations relevant to A ¼ 2:5; 10 and 15 are given in Table IV.

24 Mixed convection in enclosures 175 Figure 15. Regression curves of (a) Q C and (b) Q E for A ¼ 10, B ¼ 0.75 and C ¼ 0.5 Aspect ratio a b c a A ¼ 2.5 Q C A ¼ 2.5 Q E A ¼ 10 Q C A ¼ 10 Q E Table III. Regression coefficients 6. Conclusion A numerical study of the mixed convection in a rectangular cavity containing adiabatic partitions attached to the heated surface has been carried out. The main results of the present study are the following.

25 EC 20, The quantity of local heat along the cold surface increases by increasing the length B of the partitions and decreasing Re. Moreover, the sinuous variations of the local heat transfer, observed in the case of natural convection, are maintained in mixed convection for Reynolds numbers lower than limiting values which depend on A, B and Ra.. The mixed convection mode is established for the moderate values of Re and Ra and for B # 0:5: Lastly, the mode of forced convection dominates for the high velocities imposed on the inlet of the cavity especially for the small aspect ratios.. The heat transfer released through the cover reaches extreme values for critical values of B due to the absence or the presence of the natural convection cells. While the quantity of heat released through the higher opening decreases by increasing B for all the values of Ra, Re and A considered.. The inertia of the system and the low velocities of the forced flow support the release of heat towards the outside (situation favorable to the airconditioning in the habitat domain). However, the opposite effect interests the heating of the adjacent dwelling connected to the system by the higher opening.. The present study has allowed us to show the existence of couples of critical numbers of Ra and Re for which the rates of heat transfer released by the system towards the outside reach maximum values. Moreover, critical Reynolds numbers become insensitive with the Ra variations once Figure 16. Couples (Ra, Re) leading to Q E.Q C for B ¼ 0.75, C ¼ 0.5 and various A Aspect ratio a b Table IV. Regression coefficients of Re F A ¼ A ¼ A ¼

26 the latter exceeds a limiting value depending on A, B and C. The results showed that Q E,max passes by a maximum for a certain critical value of Ra and becomes insensitive to A when A $ 10. The presence of the partitions supports more the establishment of conduction to the detriment of the forced and mixed convection and this, without depending on the other parameters of the study. Thus, the user of this type of system may control the heat exchange through the cover and the higher opening only by a suitable choice of C. Lastly, the various results obtained are grouped in the form of correlations to facilitate a possible use in design. Mixed convection in enclosures 177 References Dumas, L. (1985), Réalisation et étude expérimentale sur un capteur solaire passif sans et avec alvéoles, Projet de Fin d Etudes, Ecole Polytechnique de Montréal. Frankel, S.P. (1950), Convergence rates of iterative treatments of partial differential equations, Math. Tables Aids Compt, Vol. 4, pp Hasnaoui, M., Vasseur, P. and Bilgen, E. (1992), Natural convection and conduction in rectangular enclosures with adiabatic fins attached on the heated wall, Waërme-und Stoffue-bertragung, Vol. 27, pp Hasnaoui, M., Zrikem, Z., Bilgen, E. and Vasseur, P. (1990), Solar radiation induced natural convection in enclosures with conducting walls, Solar and Wind Technology, Vol. 7 No. 5, pp Lage, J.I. and Bejan, A. (1993), The resonance of natural convection in an enclosure heated periodically from the side, Int. J. Heat Transfer, Vol. 36, pp Lakhal, E.K., Bilgen, E. and Vasseur, P. (1995a), Natural convection and conduction in inclined enclosures bounded by a wall honeycomb structure, Int. J. Heat Mass Transfer, Vol. 38 No. 8, pp Lakhal, E.K., Bilgen, E. and Vasseur, P. (1995b), Natural convection and conduction in rectangular enclosures bounded by a wall honeycomb structure without vent, Journal of Solar Energy Engineering, Vol. 117, pp Lakhal, E.K., Hasnaoui, M. and Vasseur, P. (1999), Etude numérique de la convection naturelle transitoire au sein d une cavité chauffée périodiquement avec différents types d excitations, Int. J. Heat Mass Transfer, Vol. 42, pp Lakhal, E.K., Hasnaoui, M., Vasseur, P. and Bilgen, E. (1996), Natural convection in inclined rectangular enclosures with perfectly conducting fins attached on the heated wall, Heat and Mass Transfer, Vol. 32, pp Roache, P.J. (1982), Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New Mexico. Scozia, R. and Frederick, R.L. (1991), Natural convection in slender cavities with multiple fins attached to an active wall, Num. Heat Transfer, Part A, Vol. 20, pp Yücel, C., Hasnaoui, M., Robillard, L. and Bilgen, E. (1993), Mixed convection heat transfer in open ended inclined channel with discrete isothermal heating, Num. Heat Transfer, Vol. 4, Part A, pp