Embed Size (px)
Citation preview
This article was downloaded by: [Massachusetts Institute of Technology]On: 25 November 2014, At: 18:35Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Numerical Heat Transfer, Part A: Applications: AnInternational Journal of Computation and MethodologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/unht20
NATURAL CONVECTION HEAT TRANSFER IN ANENCLOSURE WITH A PARTITION-A FINITE-ELEMENTANALYSISP. K. Ghosh a , A. Sarkar a & V.M.K. Sastri ba Department of Mechanical Engineering , Jadavpur University , Calcutta, 700 032, Indiab Department of Mechanical Engineering , Indian Institute of Technology , Madras, 600 036,IndiaPublished online: 27 Apr 2007.
To cite this article: P. K. Ghosh , A. Sarkar & V.M.K. Sastri (1992) NATURAL CONVECTION HEAT TRANSFER IN AN ENCLOSUREWITH A PARTITION-A FINITE-ELEMENT ANALYSIS, Numerical Heat Transfer, Part A: Applications: An International Journal ofComputation and Methodology, 21:2, 231-248, DOI: 10.1080/10407789108944874
To link to this article: http://dx.doi.org/10.1080/10407789108944874
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Numerical Heat W f e r , Part A, vol. 21, pp. 231-248, 1992
NATURAL CONVECTION HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION- A FINITE-ELEMENT ANALYSIS
I I! K. Ghosh and A. Sarkar
I Department of Mechanical Engineering, Jahvpur University. Calcutta 700 032, India
K M. K. Sastri Department of Mechanical Engineering, Indian Institute of Technology, Madras 600 036, India
h i ~ r nahlml convection in rectangular enclosures divided by o single thin pa&n is studied numericdy by (he finite-element method. Special emphasis is given to enclosures with aspect ratios less than 1, dhough the resuks for aspect ratios of 1 have also been reporred for comparison. llre effect on flow structure and heat transfer of shifting the portiiion in the enclosure has also been investigated. While the Pmndrl number is held at unily, the hkyleigh number is considered over o wide mnge, thor is, from ld to 1 6 and the aspect ratio assumed the values of 1, 0.8, 0.5, and 0.4.
INTRODUCTION
Natural convection in enclosures is an area of keen interest to research. Applica- tions include thermal design of buildings, cryogenic storage, and solar collector design. The literature in this area has been critically reviewed by Ostrach [I] and Catton [2]. The configurations that are most widely studied in the literature involve a square cavity with differentially heated sidewalls and a cavity heated from below. Of late, however, the effects of different boundary conditions as well as varied aspect ratios and inclina- tions have also been investigated. The motivation stems from the thermal design of buildings, and efforts are being made to ascertain the amount of leakage of heat through the building walls. Considerable understanding of the flow and heat transfer phenomena are also necessary, since the buoyancy effects arise due to temperature differences exist- ing between various pans of the building.
One of the situations that occurs frequently in practice involves the analysis of enclosures separated by a partition. The problem is inherently complex in the sense that the thermal conditions on the partition are not known a priori. The thermal conditions on the partition are strongly dependent on the interaction of the natural convection boundary layers on either side of the partition. In order to reduce the associated computational complexity, several simplified versions have been proposed. For example, some works employed constant heat flux through the wall [3, 41, while others reported results with power law wall temperature distributions [5]. Conjugate problems seem to have been taken up first by Sparrow and Prakash [6] , who considered natural convection in a
Copyright @ 1992 by Hemisphere Publishing Corporation 231
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
232 P. K. GHOSH ET AL.
I NOMENCLATURE I acceleration due to gravity
GrH Grashof number [-&(TH - T~)H ' IV~] H height of the cavity
NuH Nusselt number [j: (a€Ilax),_, dl') p, P dimensional and nondimensional pres-
sure Pr Prandtl number ( - vlu) LaH Rayleigh number ( - Gr,, Pr) TH. Tc hot nnd cold wall temperatures, respec-
tively u, u velocities in x and y directions, respec-
tively
nondimensional velocities width of the cavity dimensional coordinates nondimensional coordinates position of partition, Fig. I thermal diffusivity coefficient of thermal expansion of fluid nondimensional temperalure kinematic viscosity density streamfunction
square enclosure where one vertical wall was cooled by external natural convection of the boundary layer flow. They concluded that the Nusselt numbers under such circum- stances are 60% of those for the standard enclosure. Results have also been reported for cavities separated by single or multiple partitions [7-91. Acharya and Tsang [8] have considered a rectangular cavity with a centrally located partition. They critically evalu- ated the flow structure and heat transfer characteristics in these cavities when the aspect ratio assumed values of 1 and 2. The effect of inclination has also been studied by Acharya and Tsang. Tong and Gerner [lo] considered natural convection in partitioned air-filled rectangular enclosures for which the aspect ratios were 5, 10, and 15. They [lo'] employed the finite-difference method to determine the effect of the position of the partition for Rayleigh numbers up to loS. They concluded that placing a partition mid- way between the vertical walls of an enclosure produced the greatest reduction in heat transfer. Results of heat transfer by natural convection in enclosures with multiple verti- cal partitions have also been reported [9]. The effect of an off-center partition in a high aspect ratio cavity has been experimentally investigated by Nishimura et al. [I I].
It appears that literature is scarce insofar as low aspect ratios are concerned. The main application is thermal design of buildings, in which low aspect ratios are often encountered. Also, in many of the above cases numerical or experimental investigations have been carried out with high Prandtl number fluids. The present work aims at a thorough numerical investigation of a low aspect ratio cavity, and the effect of position of the partition is reported in some detail. Results for cavities without partitions have also been presented for effective comparison. The Prandtl number has been chosen to be unity, and the Rayleigh number ranges from 10' to lo7.
PROBLEM STATEMENT
A fully partitioned enclosure, shown schematically in Fig. 1, may be viewed as two separated nonpartitioned enclosures designated CI and C2. A thermally active parti- tion that separates the enclosures is assumed to be thin. It may be noted that for a partition to be thermally thin, it is necessary that the partition thickness be smaller than 0.1 of the partition height [12]. Insofar as the higher Rayleigh numbers are concerned, Bajorek and Lloyd [I31 observed an unsteady fringe structure at Rayleigh numbers near
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION 233
lo6. However, Nansteel and Grief [14] used water as the working fluid and observed only slight and localized unsteadiness even at a Rayleigh number of 1 0 ' ~ . With the introduction of a partition, the unsteady behavior is expected to be further suppressed. In light of these observations, reports of steady state analysis have been presented in this work, even at high Rayleigh numbers.
GOVERNING EQUATIONS AND SOLUTION PROCEDURE
Assuming steady, two-dimensional, incompressible flow, the conservation equa- tions may be written as follows:
The fluid is assumed to be Newtonian, and the Boussinesq approximation is valid. The same fluid is assumed to occupy both containers.
On introduction of the following nondimensional quantities, u - duo , V - u/u,, 8 - (T - Tc)I(T, - T,), X - x/H, Y - y/H, and P - P / ~ U ; , the governing equations take the following form:
Fig. 1 Computational domain and boundary conditions for a cavity with a partition.
"
k = o
T = T c / U = V = O
ae ,o ,av m
- .-
H
I
- ' u 3 v = o Q
t I T-T, Partition u : v = o (U=V=O)
1 g
u:v=o - .. 1/ - 4
XP L - W
/
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
P. K. GHOSH ET AL.
where A - (Gr,)In and u, - [OgH(T, - T,)I'~. The boundary conditions are shown in Fig. 1. As noted earlier, the main problem is that the thermal condition of the partition is not known a priori. Hence, an iterative scheme is to be devised. The salient feature then is how to pass the thermal information from one cavity to the other across the wall. Sparrow and Prakash [6] observed that more rapid convergence of an iterative solution is achieved if, whenever possible, thermal information is transferred via the heat transfer coefficient rather than via the temperature flux or heat flux. Their reasoning was that, at any stage of an iterative solution, the heat transfer coefficient is usually closer to its converged value than are other quantities.
The methodology used in the present work is different from that of Ref. [6] insofar as thermal communication between the two chambers C1 and C2 is concerned. When the solution is sought from scratch, that is, initial values of all variables are zeros, a constant temperature other than zero is assigned at the partition, and the governing equations are solved in C2. Heat fluxes at the nodes along the partition are calculated. These serve as boundary conditions for C 1, and corresponding temperatures are determined from the solution of governing equations in C1. Then the average of the two temperatures is used for C2. However, when solutions for higher Rayleigh numbers are sought, the solutions already obtained for the low Rayleigh number cases are used as starting solutions. The iterative scheme thus alternates from one chamber to the other until the solution con- verges. Before the criterion of convergence is taken up for discussion, a few words are in order regarding the implementation of the Newton-Raphson method in the finite- element formulation of the constitutive equations, Eqs. (&)-(2d).
According to Galerkin's formulation, the governing equations, Eqs. (&)-(2d), are written in the form
where w - ui + Vj and Z is a unit vector in the direction of gravity. The weight
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION 235
functions N~ and MT also denote the interpolating polynomials for (V, U, 8) and P, respectively. [F,], [F,], and [F,] indicate the residue matrices. The element-based area integrals constitute the element stiffness matrix. However, for the Newton-Raphson method, the element stiffness matrix is based on the gradient of the right-hand side of Eqs. (3a)-(3c) with respect to the variables. It is, therefore, often called the gradient or Jacobian matrix. Details of the method have been described in Ref. [ I S ] . In brief, in the Newton-Raphson method the magnitude of the residues indicates the closeness of the variables to the actual solution. It may be noted that although the objective is to keep the residues as close to zero as possible, in practice the solution is assumed to converge when the largest of the residues reaches a preassigned value of 5 x lo-'. It has been observed that at convergence, the absolute value of the difference of two successively iterated values of any variables does not exceed 0.0001.
An eight-node quadratic element has been used throughout the present work for the purpose of discretization. The X extent of the thermally active walls was below 0.003, while for regions around the comer, the Y extent was not allowed to exceed 0.002. However, in the regions away from the comer, as well as at the active walls, the
(b)
Fig. 2 (a) Isotherms and (b) streamlines for AR - 0.4, XP - I . RaH - lo7, and Pr - 1
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
236 R K. GHOSH ET AL.
Fig. 3 (a) Isotherms and (b) streamlines for AR - 0.4, XP - 1.25, Ra,, - lo7, and Pr - I
X and Y extents were considerably increased. A 15 x 12 mesh size is used in the present work.
Insofar as a convergence criterion in the present problem is concerned, the authors note that in Ref. [8] the iterative scheme is terminated when the changes in the heat fluxes and the temperature of the partition wall do not exceed 1 %. The present work goes a step further in the sense that even after meeting the aforementioned criterion, the iteration is continued until the largest of the residues in each of the cavities, that is, in C, and C,, becomes less than 5 x lo-'. This addition criterion is necessary for the accurate prediction of gradient-based quantities such as hot wall Nusselt number and partition wall heat flux.
RESULTS AND DISCUSSION
As mentioned earlier, the main objective of the present work is to highlight the effect of shifting the partition from its central position within the cavity. For a cavity of given aspect ratio, the results are presented for situations in which the partition has been
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION 237
placed at 40%, 50%, and 60% of the cavity width. For this purpose, an aspect ratio of 0.4 has been chosen. In the second phase, the emphasis is on the results of cavities with other low aspect ratios. Figures 2, 3, and 4 represent the streamline maps and isotherm plots for a cavity of aspect ratio 0.4. Its nondimensional height has been assumed to be 1, while the width is given by the reciprocal of the aspect ratio. In the present case, the nondimensional width is then 2.5. Cases have been considered when the partition is placed at distances of XP - 1, 1.25, and 1.5 from the heated wall. In all cases, the Rayleigh number is lo7, while the Prandtl number is held at unity. Such high Rayleigh number flows are characterized by fast-moving, thin, shear layers along the thermally active walls. As expected, the streamlines near the comer exhibit a wavy nature, and the core is characterized by the presence of small secondary cells. The effect of the partition is to lower the effective temperature difference across each cavity. In effect, then, this phenomenon delays the transition to turbulence. When the cavity is provided with a centrally located partition, the flow pattern is symmetric (Fig. 3). Cavities with other aspect ratios show the same pattern (Fig. 5). However, shifting of the partition to either side of the center results in a loss of symmetry of the flow field. In fact, the average
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
238 P. K. GHOSH ET AL.
Fig. 5 (a) Temperature distributions and (b) streamline maps in the cavity for RaH - lo7, AR - 0.5. and XP - 0.5.
temperature of the partition depends on its position and determines which side of the cavity will have stronger convection. The table below indicates the magnitude of 4," for different positions of the partition in chambers C1 and C2 (Fig. 1).
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION
0 0.2 0.4 0.6 0 .8 1.0
V-
Fig. 6 Temperature distribution of the partition wall for XP - 0.5 and AR - 1.
Fig. 7 Heat flux distribution on the hot wall for XP - 0.5 and AR - I
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
P. K. GHOSH ET AL.
Fig. 8 Heat flux distribution on the panition wall for XP - 0.5 and AR - I
Fig. 9 Temperature distributions on the partition wall for aspect ratio of 0.5 and XP - I .
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION
Fig. 10 Heat flux distribution on the partial wall for aspect ratio of 0.5 and XP - I.
Fig. 11 Heat flux distribution along the hot wall for aspect ratio of 0.5 and XP - 1.
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
242 P. K. GHOSH ET AL.
Now, if it is assumed that the magnitude of $,, provides an insight about the strength of convection in C1 or C2, then it is obvious, from a close look at the table, that as the partition is shifted progressively away from the hot wall, the temperature differential across C1 increases and a stronger convection seems evident. In the limit, when the partition is shifted very close to the cold wall, conduction will have an increasingly dominant role in C2. Insofar as the distribution of the isotherms is concerned, it is observed that thin thermal boundary layers are formed along the thermally active walls. The core, as expected, is thermallystr&ied, with fluid of higher temperature at the top. The result of this thermal stratification is that the buoyancy force increases as one moves up the core from the bottom. This means that the largest buoyancy force is experienced by the layers near the top. The presence of the horizontal boundary there then forces the fluid to move horizontally. This explains the near-horizontal shape of the streamlines within the core.
0.25
- 0.25 0 1 .O 2.5
(6)
Fig. 12 (a) Temperature distribution and (b) vertical velocity distribution along the cavity at midheight for AR - 0.4, XP - I , RaH - lo7, and PI - 1 .
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION 243
Fig. 13 (a) Temperature distribution and (b) vertical velocity distribution along the cavity at midheight for AR - 0.4, XP - 1.25, Ra, - 10'. and Pr - 1.
HEAT TRANSFER
In a partitioned cavity the temperature and the heat fluxes at the partition are considered to be outcomes of the problem and are deemed to be the main part of the solution. Before the effect of shifting the partition is considered, a closer look must be taken at cavities with a centrally located partition with aspect ratios other than that considered. Figures 6-11 represent the solutions when the aspect ratio (AR) is equal to 0.5 and 1. For the partition wall temperature distribution, the trends are similar at both AR values. The average partition temperature at Ra, - lo3 is close to 0.5 in both cases. Confirming the dominant role of conduction with increase of Ra,, a drop in temperature along the height of the partition is observed. The temperature drops from the top to the bottom, and this drop increases with increase of Ra,. It is also interesting to note that for a significant length of the partition, the wall temperature varies linearly. This length actually increases with Ra,. Also, in these zones of linear temperature variation, the heat fluxes along the partition are more or less constant and assume maximum values at the
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
244 F! K. GHOSH ET AL.
Fig. 14 (a) Temperature distribution and (b) vertical velocity distribution along the cavity at midheight for AR - 0.4. XP - 1.5. Ra, - lo7, and Pr - 1 .
same time. Heat transfer seems to be minimum at the top and bonom of the partition due to the local low temperature gradients between the fluids in C1 and C2.
Heat transfer at the hot wall, after experiencing an initial increase, decreases monotonically along the height. This initial hike in heat flux is reflected in the slight compression of the isotherms in the lower corner. Figures 12, 13, and 14 represent the temperature and vertical velocity distributions along the midheight of the cavity, when XP assumes values of 1, 1.25, and 1.5. The trends are similar in all cases. A visual examination of the temperature distributions indicates only marginal change in the tem- perature gradient at the cavity midheight. Similar small changes are also noted in heat flux and temperature distributions at the interface (Figs. 15 and 16). Figure 17, however, shows what effect the position of the partition has on the distribution of heat flux on the hot wall. Higher heat fluxes are observed for the case in which XP - 1; however, for most of the rest of the hot wall, the corresponding heat flux values fall marginally below the other two cases (XP - 1.25 and 1.5).
Tables 1-4 show the values of hot wall Nusselt number for the range of aspect ratios considered, as well as for the cases in which the partition is located at 40%, 50%,
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
Fig. 16 Temperature distribution at the partition wall for AR - 0.4, Ra, - lo7, and Pr - 1.
0
1.50 1.25
"OO \u I 0 o -15-
-25.
- ---- --
I I 0 0.4 0.8 1.0
v- Fig. 15 Heat flux distribution at the interface for AR - 0.4. Ra, - lo7, and Pr - 1.
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
P. K. GHOSH ET AL.
Fig. 17 Heat flux distribution at the hot wall for AR - 0.4, Ra, - lo7, and Pr - 1 .
and 60% of the nondimensional width from the hot wall. For some of these cases, computations have also been performed for a cavity without any partition, to provide a quantitative assessment regarding the effect of the partition. In general, at higher Ray- leigh numbers, heat transfer is reduced by more than 50% (in some cases) when a partition is introduced. A closer look at these tables also indicates that, for a given value of Ra, and AR, the minimum Nusselt number does not necessarily correspond to the case with a centrally located partition. In fact, in some cases, slightly higher Nu, values than the minimum are obtained when the partition is placed at cavity midwidth.
a b l e 1 Aspect Ratio - 0.4, H - I , and W - 2.5
Numbers in parentheses are iterations.
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
HEAT TRANSFER IN AN ENCLOSURE WITH A PARTITION
lsble 2 Aspect Ratio - 0.5. H - 1, and W - 2
XP
0.8 1 1.2 0
R ~ H (40%) (50%) (60%) (No partition)
Asterisk indicates nonconvergence
CONCLUSION
From a finite-element analysis of a cavity of low aspect ratio with a partition, the position of the partition effects a lower driving force across each chamber and thereby reduces the heat transfer, compared with the case of an unpartitioned cavity. However, it seems that the effect of partition position is not of much consequence as long as it is not placed in close proximity to either a hot or cold wall where it would disturb their boundary layers.
lsble 3 Aspect Ratio - 0.8, H - I, and W - 1.25
0.5 0.625 0.75 0 Ran (40%) (50%) (60%) (No partition)
lsble 4 Aspect Ratio - 1, H - 1, and W - I
0
R ~ H 0.4 0.5 0.6 (No partition)
l d 1.0044 1.0026 1 .0045 1.117 1 o4 1.1934 1.14 1.1932 2.2558 l d 2.257 2.2782 2.255 4.599 I o6 4.3079 4.292 4.2949 8.9858 lo7 8.0782 8.0531 8.0045
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
P. K. GHOSH ET AL.
REFERENCES
I . S. Ostrach, Natural Convection in Enclosures, J . Heat Transfer, Trans. ASME, vol. 110, pp. 1175-1 190, 1988.
2. I. Catton, Natural Convection in Enclosures, Proc. 6th Int. Hear Transfer Con$ , Paper KS-3, vol. 6, pp. 13-43, Toronto, Canada, 1978.
3. R. Anderson and A. Bejan, Heat Transfer through Single and Double Vertical Walls in Natu- ral Convection: Theory and Experiment, Int. J . Hear Mars Transfer, vol. 24, no. 10, pp. 1611-1620, 1981.
4. E. M. Sparrow and J. L. Gregg, Laminar Free Convection from a Vertical Plate with Uni- form Surface Heat Flux, Trans. ASME, vol. 79, pp. 435-440, 1956.
5. B. Gebbart, Heat Transfer, 2nd ed., chap. 8, McGraw-Hill, New York, 1971. 6. E. M. Sparrow and C. Prakash, Interaction between Internal Natural Convection in an Enclo-
sure and an External Natural Convection Boundary Layer Flow, Int. J . Heat Mass Transfer, vol. 24, no. 5, pp. 895-907, 198 1.
7. T. Nishimura, M. Shiraishi, and Y. Kawamura, Analysis of Natural Convection Heat Transfer in Enclosures Divided by a Vertical Partition Plate, Proc. Int. Symp. Heat Transfer, Paper 85- ISHT-1-6, Beijing, 1985.
8. S. Acharya and C. H. Tsang, Natural Convection in a Fully Partitioned Inclined Enclosure, Numer. Heat Transfer, vol. 8, pp. 407-428, 1985.
9. T. Nishimura, M. Shiraishi, F. Nagasawa, and Y. Kawamura, Natural Convection Heat Transfer in Enclosures with Multiple Vertical Partitions, Int. J . Hear Mass Transfer, vol. 31, no. 8, pp. 1679-1686, 1988.
10. T. W. Tong and F. M. Gerner, Natural Convection in Partitioned Air-Filled Rectangular Enclosures, Int. Comm. Heat Mass Transfer, vol. 13, no. 1, pp. 99-108, 1986.
I I . T. Nishimura, M. Shiraishi, and Y. Kawamura, Natural Convection Heat Transfer in Enclo- sures with an Off-Center Partition, Inr. J. Hear Mass Transfer, vol. 30, no. 8, pp. 1756-1758, 1987.
12. B. A. Meyer, 3. W. Mitchell, and M. M. El-Wakil, The Effect of Thermal Wall Properties on Natural Convection in Inclined Rectangular Cells, J . Heat Transfer, vol. 104, pp. I 1 1-1 17, 1982.
13. S. M. Bajorek and J. R. Lloyd, Experimental Investigation of Natural Convection in Parti- tioned Enclosures, J . Heat Transfer, vol. 104, pp. 527-532, 1982.
14. M. W. Nansteel and R. Grief, Natural Convection in Undivided and Partially Divided Rectan- gular Enclosures, J . Hear Transfer, vol. 103, pp. 623-629, 1981.
15. A. Sarkar and V. M. K. Sastri, Finite-Element Analysis of Natural Convection in the Active Corners of a Square Cavity, Numer. Heat Transfer, Pari A, vol. 19, pp. 445-470, 1991.
Received 10 January 1991 Accepted 30 April 1991
Dow
nloa
ded
by [
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y] a
t 18:
35 2
5 N
ovem
ber
2014
