turbulent natural convection flow in partitioned enclosure

Pergamorl

Computers & Fluids Vol. 26. No. 6, pp. 541-563, 1997 0 1997 Elsevier Science Ltd. All rights reserved

Printed in Great Britain PII: SOW-7930(97)80010-8 0045-7930/97 $17.00 + 0.00

TURBULENT NATURAL CONVECTION FLOW IN PARTITIONED ENCLOSURE

S. A. M. SAID, M. A. HABIB and M. A. R. KHAN Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran,

31261, Saudi Arabia

(Received 22 August 1996; in revised form 5 December 1996)

Abstract-This study represents the numerical solutions of the buoyancy driven turbulent flows in an inclined two-dimensional rectangular enclosure in which one of the inclined walls is heated and the other is cooled. The low Reynolds number k-s model is used to model the turbulent flow. The effect of various parameters such as the angle of inclination, the Rayleigh number and the number of partitions on the flow field and the average Nusselt number have been investigated and presented. 0 1997 Elsevier Science Ltd.

NOMENCLATURE

A

c, g H k Kt KS K L N

8!5 NU P P, Ra Re, se t T u,v X,Y Y+ YP

aspect ratio of enclosure ( = H/L) specific h’eat acceleration due to gravity height of the enclosure kinetic energy of turbulence thermal conductivity of the fluid thermal conductivity of the solid partition conductivity ratio (K,/Kr) distance between the two isothermal walls number of partitions local Nusselt number along the vertical wall average Nusselt number pressure laminar I’randtl number (&./Kr) Rayleigh number (gB(Th - ?JH’/av) turbulence Reynolds number (k*/e x l/v) source term in the energy equation dimensionless thickness of the partition (XI: - x,/L) temperature velocities in the xg directions, respectively horizonta.l and vertical coordinate directions dimensionless wall distance y-coordinate at grid node p

Greek letters

!T

thermal diffusivity (Kr/pC,) coefficient of thermal expansion temperature difference between the two walls

E rate of dissipation of kinetic energy IJ laminar viscosity pt turbulent viscosity

: laminar kinematic viscosity fluctuating temperature

; mean temperature angle of inclination

Subscripts

ij direction x, y or z max maximum value min minimum value

Superscripts

fluctuating quantity - mean quantity

547

548 S. A. M. Said et al.

1. INTRODUCTION

Natural convection in enclosures is a topic of contemporary importance, because enclosures filled with fluid are central components in a long list of engineering applications. Hence, the study of natural convection flows in vertical enclosures has received considerable attention. The review article by Ostrach [I] in addition to the recent literature [2-61 represents an excellent summary of research activities on the subject. Natural convection in enclosures is the result of the complex interaction between the finite size fluid system in thermal communication with all the walls. The level of complexity depends on the geometry and orientation of the enclosure.

The knowledge of heat transfer characteristics of natural convection flow across inclined fluid layers is often of interest. Such fluid layers occur, for example, between the absorber and cover plates of a solar collector, insulation of buildings (air gap between inside and outside panels of the exterior walls) and in window glazing (air gap between double pane window systems). An active research in this problem led to numerous analytical and experimental studies.

Arnold et al. [7] investigated the effect of angle of inclination on the heat transfer across rectangular regions of several aspect ratios in the laminar range. The angle of inclination was varied from 0” (heated from above) to 180” (heated from below) with aspect ratio of 1, 3, 6 and 12. It was concluded that the scaling law could not be applied for the cases with 4 > 90” as the flow becomes more complex.

Extensive experiments involving high aspect ratio enclosures heated from below are reported by Hollands et al. [8]. The Rayleigh number range covered is from lo3 to IO’, and the angle of inclination is in the range 0” < $J < 70” (heated from below). A correlation was developed which gives the Nusselt number as a function of Ra x cos C#I and 4.

Another significant contribution to the study of inclined rectangular enclosures was given by El-Sherbiny et al. [9]. Measurements are reported for high aspect ratios between 5 and 110 and Ra ranging from lo* to 2 x 10’. It was concluded that the average Nusselt number depends on Ra, aspect ratio and angle of inclination. Correlations have been provided for inclined and vertical layers.

Badr and Siddiqui [lo] investigated the effect of angle of inclination on the coupling effect between natural convection inside a rectangular enclosure and forced convection outside the enclosure. It was concluded that the coupling effect resulted in a reduction in the heat transfer rate. The angles of inclination were varied between 40” and 90”.

Results concerning turbulent natural convection in inclined enclosures have so far been very limited. Kuyper et al. [l l] studied numerically both the laminar and turbulent natural convection in inclined enclosures utilizing the standard k--E model to account for turbulence. They reported that the Nusselt number shows strong dependence on the orientation of the cavity and power law dependence on the Rayleigh number of the flow.

Yedder and Bilgen [12] studied turbulent natural convection in an enclosure bounded by a massive wall and concluded that maximum heat transfer occurs for an inclination angle of W-90”, and also stated that the heat transfer is an increasing function of the Rayleigh number and of the wall conductivity.

One of the least studied cases is that of an inclined enclosure in which partitions are inserted in order to reduce the heat transfer rate. Most of the studies in a partitioned enclosure have been concerned with vertical air-filled enclosures. Tong and Gerner [13] studied numerically the effect of partition position on the heat transfer rate. They compared the results of the bisected air-filled enclosure (partitioned enclosure) with that of an enclosure fully filled with a porous insulation, and concluded that bisecting the enclosure with a partition is an effective method of reducing heat transfer. Maximum reduction in heat transfer occurs when the partition is placed midway between the vertical walls.

Anderson and Bejan [14] studied enclosures with a single partition analytically based on the Oseen linearization method. The study was in the boundary layer regime and the effect of the conductance through the partition was neglected. They confirmed their results experimentally using an enclosure with a double partition. The experimental results were correlated to obtain a relation of heat transfer between the two ends. It was proportional to (1 + N)0.6’, where N is the number of partitions.

Turbulent natural convection in partitioned enclosure 549

Nishimura et ar’. [15] developed a boundary layer solution for natural convection in enclosures with a partition and the solution validity was confirmed by experiments. It has been concluded that the heat transfer rate is independent of the position of the partition if the boundary layer thickness is less than the half width of each cell constructed by the partition.

Nishimura et al. [16] studied the effect of multiple partitions on the heat transfer rates in horizontal enclosures. It was also found from the engineering standpoint that the horizontal and vertical enclosures are equivalent in the thermal insulation capability of partitions under the same conditions, in spite of different flow patterns.

Kangni et al. [ 171 studied laminar natural convection and conduction in enclosures having multiple partitions. The effect of Rayleigh number (Ra), aspect ratio (A), thickness of partition (t) and conductivity ratio (K,, solid to fluid conductivity) has been studied for air as the fluid medium. It was concluded that at high Ra the heat transfer decreases with increasing N, increasing partition thickness and increasing conductivity ratio K,.

Mamou et al. [ 181 studied extensively the effect of different parameters like Rayleigh number, angle of inclination, solid to fluid conductivity ratio, thickness of fluid layer, thickness of solid partitions, and number of partitions on the overall Nusselt number for laminar natural convection in inclined enclosures.

Analytical and1 numerical studies of natural convection in inclined enclosures were also performed by Va.sseur er al. [19]. The studies were limited to Rayleigh numbers up to lo’, i.e. laminar flow.

As can be seen from the literature review, the fluid flow and heat transfer calculations for the natural convection in inclined partitioned enclosures are limited to laminar flow, i.e. Ra < 10’. No such calculations exist in the literature for turbulent natural convection flow in inclined partitioned rectangular enclosures. Hence, turbulent natural convection flow in partitioned rectangular enclosures will be investigated numerically.

2. PROBLEM FORMULATION

The problem considered is depicted schematically in Fig. 1. It is a two-dimensional flow in a rectangular enclosure with the upper and lower walls being insulated while the two vertical walls are heated and cooled uniformly. The enclosure is inclined at an angle $J measured from the heated

Fig. I. A rectangular enclosure.


side (Le. # = 90” corresponds to a vertical enclosure). A solid partition of thickness t, (x, - x,/L) and conductivity KS is placed at a distance x1 from the origin.

The flow is assumed to be steady, two-dimensional with negligible viscous dissipation. The particular form of the general transport equations which govern the process of natural convection in an enclosure is given as follows.

Continuity equation:

Momentum equation:

Energy equation:

&(pzG)=S,+~ Ji ae I axi[ Pr( axi)-(~q. (3)

In order to investigate turbulent flows numerically, turbulence modelling is required. The low Reynolds number extension of the two equation k-s model [20] is used in this study. The equations for the kinetic energy of turbulence and its dissipation rate are given as follows. k Equation:

pui ax, 3 = &[(p+ $g+P,+G,-PC. E Equation:

where p, is called the turbulent viscosity and is given by

(4)

The last term in equation (4), ps, is the destruction rate, and Pk is the rate of generation of turbulent kinetic energy and Gk is the buoyancy production term, which are given as:

G,=gi@t g . [ il

The constants are given in Table 1 and in equations (8) - (10).

(61

(7)

c,

0.09

c,,

1.44

Table I. Constants for k--E model

G G 4

1.92 1.0 1.0


( > 3

fi=l.O+ F ) Ir

(9)

fi = 1.0 - [ee-(Rd)], (10)

where Re, = & x YnlV , Yn = normal distance to the nearest wall, and Re, = k2/s x I/v . Boundary conditions are: k = 0; @s)/(Z?y) = 0 at the wall.

2.1. Boundary co,rzditions

To solve the governing equations comprising the mathematical model, boundary conditions are needed at each part of the domain boundary. The problem has been solved in the x-;v plane. At

x=0, u=O, v=O, T=T,;

x=L, u=O, v=O, T= X;

y = 0, u = 0, v = 0,

y = H, u = 0, v = 0,

The energy equation across the solid partition takes

@T a2T ax’+ayZ=

The boundary conditions at the interface where the

c.r =: 0.0; aY

g = 0.0.

the form

0.0. (11)

partition is introduced (i.e. at x = x,) are:

3. NUMERICAL SCHEME

The coupled conservation equations (l)-(5) along with their boundary conditions are solved numerically using the general purpose computer code PHOENICS (Parabolic Hyperbolic Or Elliptic Numerical Integration Code Series) 121,221. PHOENICS is based on a finite volume method with fully implicit hybrid scheme. Convergence is obtained when the sum of the residuals for the field variables (P,U,V&T) fall below 0.001. The relaxation parameters had to be adjusted for each simulation in order to accelerate the convergence. All computations were carried out on a 486 PC machine.

4. RESULTS

Computations of the heat transfer and fluid flow characteristics of natural convection in an inclined enclosurfe have been performed for both laminar and turbulent flows.

The first set of results pertains to laminar flow. Figs 2 and 3 show the isotherms and streamlines for different angles of inclination. The flow field and temperature distributions for the case where the cavity is heated from the top (# = 180”) confirm the situation that can be fully determined by conduction heat transfer. As soon as the cavity is rotated, the fluid is set in motion. The isotherms shown in Fig. 2(a.) (4 = 135") indicate that the temperature gradient in the core region gets larger, Diffusion is still the dominating process in the enclosure since the hot fluid is positioned in the upper corner and the cold fluid in the lower comer. Figure 2(b) shows that the essential part of the flow is the thermal boundary layer structure aiong the hot and cold surfaces. The figure shows a zero tem~rature gradient along the x-direction in the core region and higher gradients near the walls. The thickness of the thermal boundary layer increases with the height along the hot wall while it decreases along the cold wall. Figure 2(c) shows that the stratified temperature is broken up as the flow in the core region pushes the flow along the adiabatic walls to the side. Figure 2(d) shows that the counterclockwise rotation of the cell in the core region is accelerated, bending the isotherms


in the core region such that they are no longer orthogonal to the gravitational field. This distortion of the temperature field is due to the increase in speed of the counterclockwise rotating cell.

The streamlines in Fig. 3(a) show that the fluid leaving the heated and cooled walls is being decelerated back to the wall due to the effect of gravity forming a stretched cell along both walls. The velocity along the hot and cold walls increases with rotation. As can be seen from Fig. 3(b), a small portion of the fluid is entering the boundary layer upstream again, forming a narrow cell. These cells get stretched along the adiabatic walls as the angle is reduced to C#J = 60”, as shown in Fig. 3(c). As the angle of inclination is further reduced to 4 = 40”, the two cells begin to interact, forming a counterclockwise rotating closed cell in the core region of the enclosure as shown in Fig. 3(d).

The calculated average Nusselt number as a function of the enclosure angle of inclination is shown in Fig. 4. In this figure the numerical results of Kuyper et al. [ 1 l] and the experimental results

283 (5) 333 min (interval) max

(a) t$ = 135”

min (interval) max

(c) 4 = 60”


(b) 0 = 90”

min (interval) max

(d) @ - 40’

Fig. 2. Isotherms for laminar natural convection in an enclosure without partition, Ra = 1 x 106, A = 1 .O, P, = 0.71.

Turbulent natural convection in partitioned enclosure

min (interGal) max

(a) I$ - 13W

vO.0 (0.542E-04) 0.542603 min (interval) max

‘h

0.0 10.335E-04) 0.335E-03 min (interval) max

(b) i$ = 90”

0.0 (0.695E-051 0.695503 min (interval) max

Fig. 3. Streamlines for laminar natural convection in an enclosure without partition, Ru = 1 x 1O6, A= 1.0, P,=O.71.

of Hamady et al. [23] are compared with the results of the present study. As can be seen, the results of the present study show a very good agreement with the experimental results of Hamady et al. [23] in comparison with the numerical results of Kuyper et al. [l 11. The difference with the experimental results is due to the conduction through the connecting walls in their experiments. As can be seen from Fig. 4, a decrease in the angle of inclination from 180” (heated from above) causes an increase in the average Nusselt number which is due to the increase in driving potential for natural convection. A decrease in the angle of inclination below 90” will reduce the gravity component along the heated %wall, although the average Nusselt number continues to increase until a maximum value is reached at an angle of 75”. Beyond this angle a decrease in angle of inclination will result in a decrease in the average Nusselt number until a local minimum is obtained at an angle of 30”. This reduction in % is due to the interaction of the two cells forming a counterclockwise rotating closed cell in the center of the cavity, as seen in Fig. 3(d). Because of the good mixing obtained

554 S. A. M. Said ef al.

in the core region, temperature gradients are small as seen in Fig. 2(d). Further reduction in the angle of inclination to 20” shows an upward trend in the average Nusselt number. The experimental results of Hamady et al. [23] show an increase in the averageNusselt number as the angle of inclination is further reduced towards 0”. This increase in NU is due to the transition of the unicellular flow structure to a highly unstable flow. Because of the unstable and probably three-dimensional nature of the flow at angles close to o”, the present numerical results were obtained up to an angle of inclination of 20” only.

The second set of results pertains to turbulent flow. At high Rayleigh numbers (Ra > 109) the flow becomes turbulent with thin boundary layers along the heated walls of the enclosure. The velocity and temperature gradients within this thin boundary layer are very large and require the use of many computational grid points. In most turbulence calculations the velocity, temperature and other dependent variables in this part of the boundary layer are approximated by logarithmic wall functions. However, it is reported in the literature that using the standard k--E model with the wall functions does not give an accurate solution. Hence the k--E model was modified and named as the low Reynolds number k--E model [3,4]. In the present study both models were used to model turbulent natural convection in a rectangular enclosure, and the results are compared with the experimental results of Giel et al. [24] in order to choose the one that will exhibit good agreement. Figs 5 and 6 show a comparison of the temperature and vertical velocity distributions at the mid-height of the enclosure at Ra = 8 x 10” and A = 10. As can be seen from the figures, the low Reynolds number k-e model results exhibit a very good agreement with the experimental results as compared to the standard k--E model results. This is due to the fact that the standard k---E model uses logarithmic wall functions which were originally derived for forced convection viscous sublayers, and hence does not predict well the natural convection behavior, while the low Reynolds number k-c model equations can be integrated right up to the wall. Based on the above findings, it has been decided to use the low Reynolds number k--E model to account for turbulence when performing further computations.

10.0

8.0

XExpt Hamadyetal

I 1 1 I ,

30.0 60.0 Angle

o~ihtion 120.0 166.0 1

Fig. 4. Effect of angle inclination on the average Nusselt no. for laminar natural convection in an enclosure without partition, Ra = 1 x 106, A = 1.0, P, = 0.71.

0.6

0.4

0.2

0.0


I I I

Low Reywldsnumbermodel

------- Standardk-ernodalwllhwallhuwtbns

xxxxx Expellmmltalresldtsof Glel&.schddl

y x x n 6 6 x

I _____ __--.------

0.0 0.1

Fig. 5. Temperature distribution

0.2 0.3 0.4 0.5

Mstance along the x- ckaeUon , X ! L

at the mid-height of the enclosure without partition, Ra = 8 x IO”, A = lO:l, I$ = 90”.

The third set of results pertains to the effect of partitions on the heat transfer characteristics of a vertical (4 = 90’) partitioned enclosure. Computations were carried out for different numbers of partitions ranging from 0 to 4 at Ra = 8 x lOlo, A = 10, P, = 0.71, K, = 1.0, t = 0.1 and q5 = 90”. Figure 7 shows the effect of number of partitions on the streamlines for a rectangular enclosure.

I .-

I

I I I I

_______

xxxxx

,

Lcw Reynclds nunber model

Standard k-e model with wall functbns

Expetimental re.dts of GM & schrnldl

0.2 0.3

Distance along the x- direotion , XI L

Fig. 6. Vertical velocity distribution at the mid-height of the enclosure without partition, Ra = 8 x lo”‘, A = lO:l, 4 = 90”.


0.0 f0.00047) 0.0047 min (intervalf msx

(a) N = 0

Streamlines: 0.00 (0.000243) 0.00243 min (interval) max

ib) N = 1

Streamlines: 0.00 (0.~001~~ O-000188 min (interval) mex

(c) N = 4

Fig. 7. Streamlines for turbulent natural convection in an enclosure, Ra = 8 x lOlo, A = 10, Kf = 1.0, t=O.l, 4=90°.

As can be seen from Fig. 7(a) (N = 0), most of the streamlines are concentrated near the walls indicating a steep rise in the velocity gradients near the walls. Due to the insertion of a partition, Fig. 7(b), two fluid layers are created and the effective AT in each fluid layer is considerably less when compared to enclosure without partition. The decrease in AT reduces the temperature

Turbulent natural convection in partitioned enclosure

0.0 0.2 0.4 0.6 0.8 1.0

Dtmmstenlws Distance along the xdlmctton , XI L

Fig. 8. Dimensionless temperature at the mid-height of the enclosure at different number of partitions.

1.0

0.8

s 5 0.6 n

e ‘s

t ?j 0.4

B

0.2

0.0 L

I

1.0 1

2.0 Number of Partitions, N

Fig. 9. Effect of number of partitions on the efficiency of the partition for turbulent natural convection in an enclosure with partition, C$ = W, Ra = 8.0 x IO”, A = IO, K, = 1.0, t = 0.1.

S. A. M. Said et al. 558

50.0

45.0

30.0

25.0 - 30.0

. I 1

60.0 90.0

Angle of Inclination

1.0

Fig. 10. Effect of angle inclination on the average Nusselt no. for laminar natural convection in an enclosure with partition, Ra = 8.0 x IO”, A = 1.0, IV= 1, K, = 1.0.

gradient near the walls, which in turn reduces the heat transfer rate considerably. Addition of four partitions, Fig. 7(c), resulted in five fluid layers, which in turn reduces the AT in each fluid medium resulting in a very low temperature gradient along the surfaces. Figure 8 shows a plot of the dimensionless temperature at the mid-height of the enclosure at various values of N. The temperature profile at N = 0 is of the boundary layer type with steep gradients near the walls and zero gradient in the core region, indicating the presence of convective currents near the walls. At iV = 1 the gradient near the wall reduces to a great extent, and at N = 4 the temperature profile is almost linear, indicating conduction as the main mechanism of heat transfer. The influence of number of partitions is shown in terms of the partition efficiency in Fig. 9. The partition efficiency is defined by reduction in heat transfer rate due to the insertion of partitions and is given as:

As can be seen, the introduction of a single partition increases the partition efficiency significantly. In other words, the introduction of a single partition results in a significant reduction in the enclosure heat transfer rate. The partition efficiency increases with increase in number of partitions. The plot of the average Nusselt number vs the angle of inclination is shown in Fig. 10. As can be seen from the figure, as the inclination angle q5 increases above zero, convection becomes more significant, the average Nusselt number increases and then decreases again. The peak in the average Nusselt number occurs at an angle of inclination around 60”. This result agrees with that of an isothermally heated inclined cavity with a single diathermal partition that has been studied numerically by Acharya and Tsang [25]. According to Acharya and Tsang [25], the maximum average Nusselt number was observed to occur at about 60”.

The fourth set of results pertains to the effect of the Rayleigh number on the heat transfer characteristics of a vertical enclosure with a single centrally placed partition. The Rayleigh number can be varied by either varying the temperature difference or varying the height of the enclosure. The enclosure dimensions are varied keeping the aspect ratio constant at 10.


The plots of lthe isotherms and streamlines are shown in Figs 11 and 12. As can be seen from Fig. 1 l(a) and (b), as Ru increases the temperature gradient near the walls also increases. Figure 12(a) shows the primary flow rotation from the hot wall to the cold wall with a curvature near the adiabatic walls. As the Rayleigh number increases (Ra = lOu), the rotating cells are


(a) Ra = lo9

(b) Ra = 1013

Fig. Il. Effect of Ru on the isotherms for turbulent natural convection in an enclosure, A = 10, N = I, K,= 1.0, t=O.l, 4=90”.


Streamlines: 0.00 (0.00111) 0.0111 min (interval) max

(a) Ra = 109

(b) Ra = 1013

Fig. 12. Effect of Ra on the streamlines for turbulent natural convection in an enclosure, A = 10, N = I, Kr = 1 .O, t = 0.1, 4 = 90”. (I) Magnified view of the bottom; (II) magnified view of the top.

stretched back to the upper side of the hot wall and to the lower side of the cold wall as shown in Fig. 12(b). This behavior is clearly visible in the plot of velocity vectors shown in Fig. 13. The velocity vectors moving away from the top of the hot wall and the bottom of the cold wall are dragged back towards the vertical walls forming a narrow cell. This is due to the increase in buoyancy forces, forming a secondary flow region.


The effect of Rayleigh number on the average Nusselt number is shown in Fig. 14. As can be seen from the figure, as the Rayleigh number increases the Nusselt number also increases. This is expected since an increase in the Rayleigh number is associated with an increase in the temperature gradients as shown in Fig. 1 l(a) and (b).

5. CONCLUSIONS

The standard k--E model with wall functions gave a higher prediction of the heat transfer coefficient for natural convection in enclosures, whereas the low Reynolds extension of the k--E model results were in good agreement with the experimental results of Giel et al. [24].

The insertion of a single partition having a finite thickness and thermal conductivity reduces significantly the average Nusselt number. The rate of reduction decreases as more partitions are added.

There is an optimum angle of inclination at which the highest average Nusselt number occurs for natural convection in a partitioned enclosure.

_, _ _ _ . * 1 . ..a. . . . . - . . . . .5., Y ,, . . . - - - - . - -,$

,,, f . . W““‘,

, l I . * 1 I . . “,

f

a

.

L

.

_

_

.

.

l

.

l

ft rs.. . . .

s t I... . - .

It t a**. . . .

1.s.. - I I

........ I! I .......

........ 1.. ......

Fig. 13. Velocity vectors for turbulent natural convection in an enclosure with partition Ra = 1 x lo’“, A = 10, N q = 1, K, = 1.0, t = 0.1, C$ = 90”. (I) Magnified view of the bottom; (II) magnified view of the

top. CAF 26/&E


100

80

20

0 1

1 1 1 1

1o’O 10” 1o12 1

Rayleigh Number

Fig. 14. Effect of Rayleigh number on the average Nusselt no. for turbulent natural convection in an enclosure with partiton, 4 = 90”, A = 10, t = 0.1, K, = 1.0, P, = 0.71.

The average Nusselt number increases as the Rayleigh number increases. Up to a Rayleigh number of 10” the rate of increase is slow, but as the Rayleigh number increases further a steep rise in the % is observed.

Acknowledgement-The support of King Fahd University of Petroleum and Minerals in conducting this study is acknowledged.

REFERENCES

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17. Kangni, A., Yedder, R. B. and Bilgen, E., Natural convection and conduction in enclosure with multiple vertical partition. International Journal of Heat and Mass Transfer, 1991, 34, 2819-2825.

18. Mamou, M., Hazsnaoui, M., Vasseur, P. and Bilgen, E., Natural convection heat transfer in inclined enclosures with multiple conducting solid partitions. Numerical Heat Transfer, 1994, A25, 295-315.

19. Vasseur, P., Hasnaoui, M. and Bilgen, E., Analytical and numerical study of natural convection heat transfer in an inclined composite enclosure. Applied Scientific Research, 1994, 52, 187-207.

20. Betts, P. L. and Dfa’alla, A. A., Turbulent buoyant air flow in a tall rectangular enclosure. In Significant Questions in Buoyancy Affected Enclosure or Cavity Flows, ASME Winter Annual Meeting, Anahein, 1986, pp. 377-388.

21. The PHOENICS Reference Manual, TR 200 a, CHAM, 1991. 22. The PHOENICS Reference Manual, TR 200 b, CHAM, 1991. 23. Hamady, F. J., Lloyd, J. R., Yang, H. Q. and Yang, K. T., Study of local natural convection heat transfer in an inclined

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turbulent natural convection flow in partitioned enclosure

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