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Buoyancy-induced heat transfer in a vertical enclosure with offset partial vertical dividers
Buoyancy-induced heat transfer in a vertical enclosure with offset partial vertical dividers R. Jetli and S. Acharya Mechanical Engineering Department, Louisiana State University, Baton Rouge, LA 70803, USA (Received May 1987; revised November 1987)
A numerical study has been made of buoyancy-induced flow in a square, externally heated vertical enclosure with two offset baffles or vertical dividers, in order to investigate the effect of baffle height on the flow and heat transfer characteristics. Air is taken to be the working fluid, and the horizontal end walls are assumed to be perfectly conducting. Two baffle positions are considered: in one, the upper baffle is offset toward the hot wall, and the lower baffle toward the cold wall (position A): in the other, the upper baffle is offset toward the cold wall, and the lower baffle toward the hot wall (position B). The effect of baffle conductivity and height on the thermal stratification and heat transfer is investigated. Both factors are found to play an important role, with lower heat transfer at higher baffle conductivities and baffle heights. The nature of thermal stratification between the baffles and in the baffle-near wall region controls the corresponding flow behavior. In position B, strong vertical stratification between the baffles inhibits the crossflow, whereas in position A stratification and correspondingly reduced flow penetration into the baffle-near wall region is noted. These differing flow patterns have important ramifications on the heat transfer. Keywords: buoyancy-induced flow, vertical enclosure, baffle conductivity, thermal stratification, heat transfer
Buoyancy-induced heat transfer in an externally heated, undivided vertical enclosure has been extensively studied in the literature. Ostrach, ~ Catton, 2 de Vahl Davis and Jones, 3 and Bejan 4 have reviewed the reported studies in this area. More recently, studies have been made on natural convection in partially divided enclosures. These studies include investigations with a single vertical baffle mounted on the enclosure floor or roof 5-~ and investigations with two vertically mounted baffles, one on the floor and the other on the roof. t~-~7 However, with the exception of Refs. 15-17, none of the reported studies have considered the effect of offset baffles on the flow and heat transfer behavior. Work
reported in this paper deals with buoyancy-induced heat transfer in a vertical enclosure with offset baffles. Janikowski, Ward, and Probert '5 performed an experimental study for air-filled cavities of high aspect ratio ( = 5) with offset baffles. The baffle configuration with the top baffle offset toward the cold wall and the bottom baffle offset toward the hot wall was found to have the lowest heat transfer rates. Probert and Ward j6 extended the study in Ref. 15 to consider very high aspect ratio ( = 16 to 22) enclosures. In neither of these studies were detailed flow patterns or isotherm distributions presented. A numerical study was undertaken by Jetli, Acharya, and Zimmerman ~7 for studying the effect of baffle location on free convection in a vertical square enclosure with offset vertical baffles. It was
© 1988 Butterworth Publishers
Appl. Math. Modelling, 1988, Vol. 12, August
Introduction
411
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya found that the enclosure heat transfer decreases as the upper baffle is moved toward the cold wall and the lower baffle moved toward the hot wall. Thermal stratification effects in the baffle-near wall region were found to have an important influence on the flow pattern and heat transfer. However, only baffle heights (h) of one-third the enclosure height (H) were studied; therefore, the enclosure opening (i.e., the vertical distance between baffle tips) was always one-third the enclosure height. Even for this configuration, some degree of thermal stratification was noted along the vertical midplane of the enclosure when the upper and lower baffles were offset toward the cold and hot walls, respectively. Clearly, as the baffle heights are increased, stratification effects in the baffle-near wall region and along the vertical midplane will become increasingly important and can have a significant effect on the flow pattern and heat transfer behavior. Since the effect of baffle height on stratification, and the corresponding effect on the flow pattern and enclosure heat transfer has not yet been reported, the calculations to delineate this effect have been performed and presented in this paper for an externally heated, vertical square enclosure with offset baffles and air as the working fluid. Two offset baffle configurations were considered in the present study and are shown in Figure 1. In the first configuration, called position A, the upper baffle is offset toward the hot wall and the lower baffle is offset toward the cold wall. In the second arrangement, called position B, the direction of offset is reversed, with the upper baffle placed closer to the cold wall and the lower baffle moved toward the hot wall. In either case, the horizontal distance between the baffles, and between the baffles and the nearer wall is one-third the enclosure width (on height). Three baffle heights are considered. In one, h = L/3, so the vertical distance between baffle tips is also L/3. In the second case, h = L/2, and the vertical distance between baffle tips is zero. The third baffle height considered is h = 2L/3, so the baffles overlap each other. Since the baffles are offset with respect to each other, for all three h values, the enclosure is partially, and not fully, divided. In all cases, the baffle thickness is taken to be L/20, to simulate a thin baffle. The horizontal end walls are assumed to be perfectly conducting in this study. This assumption is based on earlier calculations by Zimmerman and Acharya, '4.'8 where it is shown that with air as the working fluid, realistic simulations are obtained with the perfectly conducting end wall assumption. Results with adiabatic conditions assumed along the end walls are shown not to compare well with experimental data even if the end walls are made of a poorly conducting material such as Plexiglas.
been previously verified by comparing with experimental data in both nonpartitioned t8 and partially partitioned enclosures. 14 With the aforesaid assumptions, the governing differential equations, in nondimensional terms, can be written as Continuity:
V . V =O
(1)
X-momentum:
U.VU
)'-momentum:
-aP U. V V = - + V 2 V + Ra. O/Pr aY
Energy:
U. VO = - Pr
-aP
=-~--
(2)
+ V2U
V20
(4)
where the following dimensionless variables have been used: X = x/L
u
U-
v -
tx/ p L p* e - _ _
wherep* = p + pgy
A+T=
T-To --
-
(5b)
tz/pL
p(p./pL) 2
0
(5a)
Y = y/L
u
where To - - 2
-
T . - Tc
UT
COLD WALL
HOT WALL
~L/3-,'~ Post%ion A
TLJ
COLD WALL
UXLL H
d - , i pP
Governing equations The flow is assumed to be steady, laminar, and twodimensional, and the Boussinesq approximation is used to model the density variation. The validity of these assumptions, for moderate temperature differences, has
412 Appl. Math. Modelling, 1988, Vol. 12, August
(3)
:
2[-,'3 [. Po6].~,lon B
Figure I
Schematic of the square partitioned enclosure
(5c) (5d)
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya In the baffle, the governing differential equation is the dimensionless Laplace's equation, expressed as
kJk V20h = 0
(6)
Pr where kb/k = kr is the dimensionless baffle conductivity and 0b denotes the temperature in the baffle region. The energy balance at the baffle-air interface can be stated as Prkan];
- N\-~-n ],
(7)
where n denotes the normal to the baffle interface and i refers to the interface. The boundary conditions used are U = V = 0on all walls
(8a)
0 = 0.5 on hot walls
(8b)
0 = - 0.5 on cold walls 0 = 0.5 - Xalong horizontal end walls
(8c)
Solution procedure To arrive at the numerical solution to the governing differential equations, we obtain finite difference solutions by using the control-volume-based procedure described by Patankar. ~9 In this procedure, the calculation domain is subdivided into control volumes with a grid point located at the center of each control volume. The control volume faces are chosen such that the baffle-air interfaces fall along certain control volume faces. The governing differential equations (equations (1)-(4)) are discretized by first integrating them over each control volume and then using Green's theorem to replace the volume integral by the surface integral. With suitable profile approximations in each coordinate direction (the power-law approximation ~9is used in this study), a system of algebraic equation results that can be solved iteratively. To resolve the pressure-velocity interlinkage, we use the SIMPLER (semiimplicit method for pressure-linked equations--revised) algorithm, ~9 in which the continuity equation is used to derive an equation for pressure and an equation for pressure correction (P'), and a predictor-corrector procedure is invoked to update pressures and velocities until the velocity field satisfies both momentum and continuity equations. To avoid checkerboard pressure and velocity fields, pressure is stored at the main grid points while U is stored at the X control volume faces and V at the Y control volume faces. To account for the presence of baffles, we use the strategy recommended by Patankar. 2° By this strategy, a very high viscosity ( - - 1 0 3°) is specified in the baffle, together with kr/Pr as the dimensionless baffle conductivity. The specification of the high viscosity in the baffle and the no-slip boundary condition suppresses the velocities in the baffle to near zero values and reduces equation (4) to the appropriate Laplace's equation in the baffle region (equation (6)). Further, since the numerical scheme is conservative, flux leaving a
control volume through one face is exactly equal to the flux entering the adjacent control volume through the common face. Thus, the interface energy balance (equation (7)) is exactly satisfied at the baffle-air interfaces. In the baffle region the coefficients of the velocity equation have very high values (-1030 ) in view of the very high viscosity values (-103°). The coefficient in the pressure or pressure correction equation that links the P or P' at a point immediately outside the baffle to a point inside it is inversely proportional to the coefficient of velocity at the baffle-air interface. Since this velocity coefficient has an extremely high value, the corresponding P or P' coefficient is nearly zero, and therefore the pressure or pressure correction field in the fluid is independent of the P and P' values in the baffle, as it should be. Thus spurious or incorrect pressure or pressure correction fields are avoided. All calculations have been done on a 48 x 42 nonuniform grid with the grid points clustered more closely along all solid boundaries. The choice of the grid point distribution is based on a number of preliminary calculations with successively finer grids. The accuracy of the calculations is verified by comparing representative results with those obtained on a 78 x 72 grid. The maximum difference between the two solutions is 2.0% in peak local Nusselt number and 1.6% in the peak mid-height horizontal velocity values. As further verification of the calculation method and the assumptions, reference is made to earlier calculations reported by the authors in both nonpartitioned ~8 and centrally partitioned TM enclosures. In both cases, calculations are compared with measurements and found to compare well. Computed results in a nonpartitioned enclosure have also been compared with the benchmark numerical solution of de Vahl Davis and J o n e s ) and the two results compare to within 1% at Ra = 10 6. In view of the above-described comparisons and tests, the numerical accuracy of the solution is considered to be adequate.
Results and discussion The main parameters of interest in this study are the two different baffle positions (Figure 1), the three different baffle heights (h = L/3, L/2, 2L/3), the Rayleigh number (Ra), and the conductivity ratio (kr). Three different Rayleigh numbers are considered in the range of 104-3.55 × 10-~,and two different conductivity ratios are used (kr = 2,500) to represent a poorly conducting and highly conducting baffle, respectively. The results are presented in the form of representative streamlines and isotherms, mid-height temperature and mid-height vertical velocity profiles, and local and average Nusselt number values.
Streamlines and isotherms Streamline and isotherm plots presented are for Ra = 105 and kr = 2 (Figures 2, 3), and for Ra = 3.55 × 105 and both baffle conductivities (Figures 47). The notation used to describe the streamline and
Appl. Math. M o d e l l i n g , 1988, Vol. 12, A u g u s t
413
Buoyancy-induced heat transfer in a vertical enclosure." R. Jetli and S. Acharya ISOTHERMS RANGE - 0.50 (0.1) 0.50
STREAMLINES RANGE 14.46(1.92) 1,00 ~mox= 15.22
COLD WALL
COLD WALL
STREAMLINES RANGE 9.35 ( 1.26 ) 0.50 ~mox = 9.84
t
HOT WALL
HOT WALL ISOTHERMS RANGE - 0.50 (0.1) 0.50
COLD ~ WALL
COLD WALL HOT WALL
III Y/11 I1{1//~~\/111 HOT I ~ j
~ _ _ ~
WALL
STREAMLINES RANGE 4.70 ( 0.66 ) O. I 0 ~mox = 4.95
ISOTHERMS RANGE -0.50(0.1)0.50
COLD WALL HOT WALL
i
HOT WALL
Figure 2 Streamline and isotherm plots for position A (Ra = 10S, k, = 2)
isotherm values is RANGE ~,(A~)~2, where dh and ~z are the limiting values of the contours and A~ is the uniform contour increment used. The maximum streamfunction value (gtmax) is also shown in the figures. For both baffle locations, the strength of the flow (indicated by ~bm,x)expectedly decreases with increasing baffle height..This reduction in flow strength, at all Rayleigh numbers, is more noticeable for baffles in position A. At Ra = l0 5, and particularly for baffles in position A (Figure 2), a relatively weaker flow is noted in the region between the baffles and the nearer vertical side walls. This is due to thermal stratification in the baffle-near wall region, which inhibits flow penetration into these regions. Thus, for h = L/3, much of the flow moving up the hot wall is deflected off it near the upper baffle tip toward the cold wall. However, for h = L/2 and 2L/3 the flow deflected near the baffle tip is heated by only a small portion of the hot wall and is not sufficiently buoyant to move up and
414 Appl. Math. Modelling, 1988, Vol. 12, August
around the upper baffle. As a result, this deflected flow, on meeting the cooler lower baffle, moves down it. Consequently, for these baffle heights, an eddy is formed between the base of the hot wall and the lower baffle (Figure 2). This feature is not observed for a baffle height of L/3 and has an important effect on the heat transfer behavior. For position B, the warm fluid stream rising along the hot wall is cooled by both the conducting horizontal end wall and the top baffle. Consequently, it has a lower tendency to rise behind the baffle, causing a weak flow in that region. For all baffle heights and Rayleigh numbers studied, the flow in the confined baffle-near side wall region is weaker for baffles in position A compared with position B (Figures
2-7). The flow pattern in the region between the baffles, in position B, is dictated by the thermal stratification in this region (see isotherms in Figure 3), which inhibits the crossflow between the hot wall side and the cold
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya STREAMLINES 10.54(I.36)
~rnox
RANGE 1.00
ISOTHERMS RANGE -0.50(0.1)0.50
= 11.09
COLD WALL
COLE WALL
HOT WALL
HOT WALL
S T R E A M L I N E S RANGE 8.54 (I.15)0.50
~rnox
ISOTHERMS RANGE -0.50 (0.,)0.50
= 8.99
-oL %
STREAMLINES RANGE 5.07 (0.71)0.10
.o, WALL
ISOTHERMS RANGE -0.50(0.1 )0.50
~rnox = 5.:54
COLD WALL
COLD WALL
HOT
HOT WAL L
WALL
Figure 3 Streamline and isotherm plots for position B (Ra = 105, kr = 2)
wall side, particularly for baffles with h = L/2 and 2L/3. Thus, for h = L/2 and 2L/3, separate eddies in the hot wall-upper baffle region and the cold walllower baffle regions are formed, with a relatively weak crossflow (compared to position A) between the two eddies. For Ra = 3.55 x 105 (Figures 4-7) the stronger convective motion along the vertical walls ensures greater penetration of the fluid into the baffle-side wall regions. In position A, as mentioned earlier, part of the fluid stream rising along the hot wall detaches near the baffle tip and tends to skip over the top baffle. For baffle height of L/3, due to the relatively higher velocity of the detached fluid at the higher Rayleigh number, flow separation behind the baffle can be observed (Figure 5). As the baffle height increases to L/2, flow separation disappears, since the flow velocity at the baffle tip is lower due to the lower heating of the fluid
detaching off the hot wall near the baffle tip. For h = L/2 and 2L/3, as at Ra = 105, separate eddies are observed between the side wall and the farther baffle, but as the overall flow is stronger at the higher Rayleigh number the corresponding eddy sizes are smaller (Figures 4-5). In position B, as mentioned earlier, the predominant factor affecting the flow for all the three baffle heights is the thermal stratification in the cavity core, which is clearly reflected in the near horizontal isotherms in the center of the cavity (Figures 6-7). This strong thermal stratification at the higher Rayleigh numbers greatly inhibits the crossflow motion. Consequently, the @max values for position B are always lower compared with the corresponding values for position A. Because the flow velocities across the baffle tip are lower in this position (compared to velocities in position A), no flow separation is seen behind the baffles, even at the baffle height of L/3.
Appl. Math. Modelling, 1988, Vol. 12, August 415
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya STREAMLINES RANGE 23.66 ( 3.24 ) 1.O0 ~max = 24.91
COLD
~
ISOTHERMS RANGE -0.50(0.1 )0.50
~
"' Htkk , lS?l II\W/IF1 \\\\\\~J/IHI
I ~
HOT
HOT
~ I W A L L
WALL
STREAMLINES RANGE 18.30 (2.47) 1.00 ~max = 19.26
ISOTHERMS RANGE -0.50 (O.t) 0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STREAMLINES RANGE 13.94 ( I . 8 6 ) 0 . 9 4 ~mox = 14.67
ISOTHERMS RANGE - 0 . 5 0 (0.1)0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
Figure 4 Streamline and isotherm plots for position A (Ra = 3.55 × 10s, kr = 2)
The baffle conductivity has a noticeably different effect on the flow with baffles in position A compared with the corresponding effect with baffles m position B. Since the higher-conductivity baffle readily attains the temperature of the perfectly conducting end wall at the base of the baffle, the upper baffle is cooler and the lower baffle warmer for the greater-conducting baffle cases (Figures 4, 7). In position A, the higher-conductivity baffle has the effect of reducing the thermal stratification in the baffle-nearer vertical wall region, which in turn, induces a stronger flow in that region (Figures 4-5). Since the fluid moving down the moreconducting baffle is cooler and has a lower tendency to negotiate around the tip of the top baffle, a larger separation eddy is observed at kr = 500, for the baffle
416
Appl. Math. Modelling, 1988, Vol. 12, August
height of L/3 (Figure 5). For the baffle height of L/2, the cooler fluid descending down the upper baffle reduces the crossflow in the cavity and promotes separate eddies between the baffles and the farther side walls. In position B, with increasing baffle conductivity, the fluid coming off the cooler top baffle has a lower tendency to rise into the baffle-cold wall region, leading to a weaker flow in the baffle-nearer vertical wall regions (Figures 6, 7). Since at the higher baffle conductivity the upper baffle is colder and the lower baffle warmer, the crossflow in the core, with flow down the upper baffle and up the lower baffle, is strengthened. For the baffle height of 2L/3 this leads to higher q~maxvalues for kr = 500 compared with kr = 2 case (Figures 6-7).
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 2 2 , 8 9 ( 3 . 1 3 ) 1.00 ~mox= 24.10
ISOTHERMS RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STR EAMLINES RANGE 16.62(2.23)1.00 ~rnax = 17.49
I S O T H E R M S RANGE -0.50 (0.1)0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
S T R E A M L I N E S RANGE 12,14 ( I . 5 9 ) 1.00
ISOTHERMS RANGE -0.50(0,1)0.50
~/rnax = 12.78
COLD WALL
COLD WAL L
HOT WALL
HOT WALL
Figure 5 Streamline and isotherm plots f o r position A (Ra = 3.55 × 10s, k, = 500)
In general, the greater-conducting baffles lead to a slower flow in the enclosure since the cooler upper baffle and warmer lower baffle tend to reduce the temperature gradients along the hot and cold walls. The reduction in these temperature gradients is greater in position B compared with position A.
Velocity and temperature distributions Figure 8 shows the vertical velocity at the cavity mid-height for both conductivities at Ra = 3.55 × 105. The boundary layer velocity along the cold wall decreases significantly as the baffle height is increased. The decrease is most significant for position A, reflecting the influence of the weaker flow in the cold wall-bottom baffle thermally stratified region. For the baffles in position A and height of 2L/3, higher velocities are observed in the region between the baffles
than along the walls. For the other two baffle heights, the peak velocities occur along the wall boundary layers. In position B, the stabilizing influence of the thermal stratification results in lower velocities in the cavity core. As the baffle conductivity is raised, the fluid velocities along the wall are reduced while the corresponding values in the core increase in magnitude. This behavior is directly attributable to the fact that at higher conductivities, the upper baffle being colder results in a stronger downflow along the baffle into the core and a weaker upflow around the baffle into the baffle-cold wall region. The mid-height temperature profiles are shown for Ra = 3.55 x 105 in Figure 9. Since the profiles are antisymmetric about the diagonal, temperature profiles for kr = 2 are plotted only for x/L <=0.5, and profiles for kr = 500 are plotted only for x/L >=0.5. In position
Appl. Math. Modelling, 1988, Vol. 12, August 417
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 1 7 . 7 6 ( 2 . 5 4 ) 1.00 ~/mox = 1 8 . 6 9
ISOTHERMS
RANGE
- 0 . 5 0 (0.! ) 0 , 5 0
COLD WAL L
COLD WALL
HOT WALL
HOT WALL
STREAMLINES 16.85(2.26)
RANGE 1.00
I S O T H E R M S RANGE -0.50 (0.1)0.50
~mox = I Z 7 4
COLD WAL L
COLD WALL!
HOT WAL L
HOT WALL
STREAMLINES RANGE 12.82 ( 1.71 ) 0 . 6 2
I S O T H E R M S RANGE - 0.50 (0.1 )0.50
t//mo x = 13.49
J
COLD WALL
COLD WALL
HOT WALL
Figure 6 Streamline and isotherm plots for position B (Ra = 3.55 x 10s, k, = 2)
A, the near wall region is characterized by a thermal boundary layer. Hence, close to the wall the temperature exhibits a sharp gradient. With increasing x / L , the temperature and the gradients diminish due to the cold fluid moving up along the lower baffle. Close to the midplane, the mixing of the hot and cold streams results in sharper temperature gradients. In position B, due to the lower velocities and the importance of thermal stratification, the temperature gradients, particulady in the core, are significantly lower compared with the corresponding values in position A. In the baffle region, a slight distortion of the temperature profile is noted for the less-conducting baffles (near x / L = t), and distinct plateaus in the temperature profile for the greater-conducting case (near x / L = ~). The more dramatic effect on the temperature profile, for kr =
418 Appl. Math. Modelling, 1988, Vol. 12, August
500, is due to the stronger end wall effect when the baffle is more conducting. In particular, the baffle conductivity appears to have a significant influence for the h = L/2 case, with sharp temperature gradients on either side of the more-conducting baffle. With increasing Rayleigh numbers, the stratification in position B and the temperature gradients in position A increase in the cavity core. Increasing the baffle conductivity, however, decreases the core stratification in position B and the core temperature gradients in position A. N u s s e l t n u m b e r distributions
The local Nusselt number distributions along the cold wall are presented for both baffle conductivities
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 17.57 ( 2 . 3 4 ) 1.00 ~rnax = 1 8 . 2 8
I S O T H E R M S RANGE - 0.50 (0.! ) 0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STREAMLINES RANGE 16.44 ( 2 . 2 1 ) 1.00
I S O T H E R M S RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
~rnox = 17.31
COLD WALL
COLD WALL
;L
HOT WALL
STREAMLINES RANGE 13.14 ( 1 . 7 3 ) 1.00 ~bmax = 1 3 . 8 3 "'
I S O T H E R M S RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
COLD WAll
COLD WALL
HOT WALL
HOT WALL
Figure 7 Streamline and isotherm plots f o r position B (Ra = 3.55 x 10s, kr = 5001
at Ra = 105 in Figures 10 and 11, respectively. At both Rayleigh numbers, in position A a single prominent peak exists in the upper region of the cold wall, whereas in position B a dual peak is observed, with the first peak in the upper region of the cold wall followed by a second peak situated somewhat below the location corresponding to the tip of the upper baffle. For position A, at the baffle height of L/3, the fluid stream is warmed as it rises along the hot wall and tends to skip over the top baffle (Figures 2, 4) and is deflected directly toward the cold wall. This causes large temperature gradients and Nusselt number values to occur at y/L -~ 0.75, for Rayleigh numbers of 105 and 3.55 x 105. As the baffle height is increased from L/3 to L/2 at Ra = 105, about 33% reduction in the peak Nusselt number value is. observed. This sharp decrease is a direct consequence of the formation of separate eddies
between the side wall and the farther baffle (Figures 2, 4), which inhibits direct crossflow of the heated fluid toward the cold walls. Figures 10 and 11 show the near constant Nusselt number between 0 =< y/L <- 0.5 reflecting the influence of the thermal stratification in this region. At Ra = 3.55 x 105, the reduction in the peak Nusselt number caused by increasing the baffle height from L/3 to L/2 in position A is only 23%. This is partially due to the increased crossflow in the cavity and reduction in the size of the eddies between the walls and farther baffles at the higher Rayleigh number. However, at the higher Rayleigh number, the increase in the stratification in the baffle-near side wall region causes the Nusselt number value in that region (0 y/L <=0.2) to fall to values in the vicinity of the conduction value of 1.0. Similar effects are noted when the baffle height is increased to 2L/3. The peak Nusselt
Appl. Math. Modelling, 1988, Vol. 12, August 419
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya 250
,.o i
kr:2 , POSITION B h = 2L/3 h= L/2 = l.J'3
200
0.9 t
~ /i
150.
'°I
0.80.70.6-
)SITION A h=L/3 h =L/2 h = 2L/3
_J "~ 0.50.4-
/
-,O011 ,/"OS'T'ON" t",.J
0.3
0.2 0.1 • kr =
- 25d . o.o
oJ,
o:2
500
I I 0:4 0.5 o'.o o'.,
o;3
0.0
o.'2 o.~
o14
o .....
o15
; .......
~
"3
Figure 8 Mid-neight vertical velocity profiles at Ra = 3.55 x 105
.....
,: ....... ; .....
;
Nu
x/L
Figure 10 Local N u s s e l t n u m b e r d i s t r i b u t i o n a l o n g t h e c o l d w a l l (Ra = 10 s,k, = 2)
I.o0.5
' kr= 500
".\
0.9-
/:
o.s-
0.3
POSITION A
/
h= L / 3 ~ h = L/2 - - ~ h =2 L/3~
0.2
to
I /""•
',
',, /
0.7"
'// ./~ ~'=~
0.1
"'-...
I
/°
0.6.
/
4%
POSITION A
,
/.
0.5
w
i
,/; /o
/
d
J
"'-°% "\
0.4"
/°/
h =L/2 h =2L/3
0.0
q~
/ -0.1.
" ~ ...... / / I
P\\\
"
~ -~-.= - .= -3LI2
/ i
/ /
,/ "
r
" 0"2"1
JJ
/
,i
~-.
/"
O.~
. o.,
~
0.1
k =2 , o.o
P/~.~L)SITIO
I:l~
J
f
-o
/
0.3
=2 L / 3
--/1""
- 03-
~'
0.4
)SITION S
o.z
0.3
s 0.4
N B
h= L 1 3 h = L/2 h=2L/3
0.O
a5
~ 0.6
0 ....
0.7
o.s
o.s
i
,.o
i
......
i
....
4 . . . .
; .......
;
Nu
x/L
Figure 11 Local Nusselt number distribution along the cold wall Figure 9 Mid-height temperature profiles at Ra = 3.55 x 105
420
Appl. Math. Modelling, 1988, Vol. 12, August
(Ra = 105, kr = 500)
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya number still occurs in the upper region of the cold wall, but the peak value is significantly lower due to the overall flow becoming weaker for this baffle position. The location of the peak Nusselt number value gradually increases from y/L = 0.75 for baffle height of L/3 to y/L = 0.85 for baffle height of 2L/3. In position B, for the baffle height of L/3, the warm air rising along the hot wall is cooled by the horizontal end wall and the top baffle. Part of the air negotiating the top baffle rises into the baffle-cold wall region, causing a Nusselt number peak in that region, while part of it impinges directly on the cold wall, causing a second Nusselt number peak below the baffle (Figures 10-11). As Ra increases, due to the stronger flow, larger peaks are observed. As the baffle height is increased to L/2, dual Nu peaks are again observed. For this baffle height, however, the second peak below the baffle (y/L ~- 0.35) is due to the effect of the eddy that exists between the cold wall and bottom baffle, which carries the fluid warmed along the bottom baffle toward the cold wall and causes the second Nu peak. At the baffle height of 2L/3, the thermal stratification in the cavity core effectively divides the flow into two separate zones, thus reducing the strength of the flow and the associated temperature gradients. Increasing the baffle conductivity significantly reduces the heat transfer occurring in the upper bafflecold wall region in position B (Figure 11). Since the upper baffle is cooler for the higher-conductivity case, the air coming off that baffle is less amenable to rise behind the baffle. Consequently, the first Nu peak behind the baffles in position B is considerably reduced. Furthermore, this cooler air coming off the top baffle (for the baffle height of L/3) impinges directly on the cold wall, causing the second Nu peak to be more pronounced (Figure 11). In position A, the peak values of Nu are reduced in all cases, due to the reduced temperature gradients (Figure I 1), although the overall profile retains the same shape as for a low-conductivity
Table I
baffle. For h = 2L/3, at Ra = 105 the local increase in the vigor of the downward flow along the top baffle causes the peak Nu value to increase with baffle conductivity. The average Nu values for the cavity are presented in Tables I and 2 for baffles in positions A and B, respectively. For h = L/3, the heat transfer is generally lower for position B compared with position A. This is partly due to the stabilizing thermal stratification in the cavity core and partly due to the effect of the baffles on the flow pattern. As the baffle height is increased to L/2, at the lower Rayleigh numbers of l04 and l05, position A is more effective in reducing the overall heat transfer compared with position B. This reversal in the trend is the effect of the near-stagnant regions between the side walls and the nearer baffles in position A. Even though the peak Nusselt number is still higher in this position, the near constant values in the baffleside wall regions cause a lower average value. At Ra = 3.55 x l05, position B has a lower average Nu because as more flow penetrates the baffle-side wall region in position A. At h -- 2L/3, position A has lower heat transfer rates at all Rayleigh numbers, which is again a consequence of the weak flow in the bafflenear side wall region. This is somewhat unexpected and contrary to the results in Ref. 17, since intuitively, the configuration with the stabilizing core (position B) would be expected to be associated with lower heat transfer rates. In general, raising the baffle conductivity reduces the average Nusselt number values, except in some cases of 2L/3 baffle height, where the increase in conductivity causes increased local flow along the baffles.
Concluding remarks Natural convection in a square vertical enclosure with two offset baffles has been studied numerically. The study shows that increasing the baffle height causes a
A v e r a g e Nusselt n u m b e r v a l u e s f o r p o s i t i o n A Ra = 104
L/3 L/2 2L/3 No baffles
Table 2
Ra = 10 s
Ra = 3.55 x 10 s
k, = 2
k, = 500
k, = 2
kr = 500
k, = 2
k, = 500
1.17 1.00 1.01
1.18 1.03 1.04
2.71 1.59 1.16
2.61 1.59 1.23
4.25 2.81 2.07
4.11 2.67 2.10
1.76
3.37
4.91
A v e r a g e Nusselt n u m b e r v a l u e s f o r p o s i t i o n B Ra = 10 a
L/3 L/2 2L/3 No baffles
Ra = 10 s
Ra = 3.55 x 10 s
kr = 2
k, = 500
k, = 2
kr = 500
k, = 2
k, = 500
1.22 1.11 1.05
1.18 1.12 1.08
2.39 1.75 1.38
2.10 1.64 1.36
4.00 2.74 2.19
3.22 2.41 2.12
1.75
3.37
4.91
Appl. Math. Modelling, 1988, Vol. 12, August
421
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya significant reduction in the flow strength and heat transfer for both of the baffle locations investigated. As the baffle height is increased, separate eddies tend to form between the side wall and the farther baffles. In the baffle position with the top baffle offset toward the hot wall and the bottom toward the cold wall, these eddies are an effect of the thermal stratification in the top baffle-hot wall and bottom baffle-cold wall regions. Conversely, in the position with the top baffle offset toward the cold wall and the bottom toward the hot wall, the eddies are a consequence of the stabilizing stratification in the cavity core that inhibits crossflow in the cavity. At lower Rayleigh numbers and longer baffleheights, lower overall heat transfer is observed in the position with the top baffle offset toward the hot wall, compared with the position where it is offset toward the cold wall, with the results being reversed at higher Rayleigh numbers and smaller baffle heights. In general, the greater-conductivity baffle leads to lower average heat transfer compared with the less-conducting baffle, with the reduction being greater for the position with top baffle offset toward the cold wall.
Nomenclature
k~ L Nu Nu P P Pr Ra
Rayleigh number,
c, d g H h k kb
T
T~ Th To I1
U o
V x,X
y,Y
References I 2 3
4 5
6 7 8
aperture ratio, (H-h)/L specific heat of fluid thickness of the baffle acceleration due to gravity enclosure height baffle or divider height thermal conductivity of air thermal conductivity of baffle conductivity ratio, kb/k enclosure width Nusselt number, hL/k average Nusselt number thermodynamic pressure dimensionless pressure Prandtl number, tzCp/k
mp
dimensionless temperature in baffle stream function I/Jrflax maximum stream function 0b
9
10
II
12 13
g/3(Th -- T~)L3
va dimensional temperature cold wall temperature hot wall temperature reference temperature, (T~ + Th)/2 dimensional velocity in x direction dimensionless velocity in x direction dimensional velocity in y direction dimensionless velocity in y direction dimensional and dimensionless coordinate along the horizontal direction dimensional and dimensionless coordinate along the vertical direction
14 15
16
17 18
Greek symbols a /3 z, p 0
thermal diffusivity volume expansion coefficient at constant pressure kinematic viscosity density dimensionless temperature
422
Appl. Math. Modelling, 1988, Vol. 12, August
19 20
Ostrach, S. Natural convection in enclosures. In Advances in Heat Transfer, Vol. 8, eds. J. P. Hartnett and T. F. Irvine, Jr. Academic Press, New York. 1972, pp. 161-227 Catton, I. Natural convection in enclosures. Proceedings o f the Sixth International Heat Transfer Conference, Vol. 2, pp. 13-31, 1978 de Vahl Davis, G. and Jones. J. Natural convection in a square cavity--a comparison exercise. In Numerical Methods in Thermal Problems, Vol. II, eds. R. W. Lewis, K. Morgan, and B. A. Schnefler. 1981, pp. 552-572 Bejan, A. A synthesis of analytical results for natural convection heat transfer across rectangular enclosures. Int. J. Heat and Mass Tran~fer 1980, 23, 723-726 Nansteel, M. W. and Greif, R. Natural convection in undivided and partially divided enclosures. J. Heat Tran.~fer 1981, 103, 623-629 Lin, N. N. and Bejan, A. Natural convection in a partially divided enclosure. Int. J. Heat and Mass Transfer 1983, 26, 1867-1878 Winters, K. H. The effect of conducting divisions on the natural convection of air in a rectangular cavity with heated side walls. ASME Paper No. 82-HT-69 Winters, K. H. Laminar natural convection in a partially divided rectangular cavity at high Rayleigh number. AERE, Harwell Laboratory, Report No. HL85/1527, 1985 Chao, P. K. B., Ozoe, H., Lior, N., and Churchill, S. W. The effect of partial baffles on natural convection in an inclined rectangular enclosure. Chemical Engineering Fandamentals 1986, 2, 23-49 White, M.D., Kirkpatrick, A. T., and Winn, C. B. Numerical study of high Rayleigh number natural convection flows through openings. Second ASMEIJSME Thermal Engineering Joint Conference. Hawaii, 1987 Duxbury, D. An interferometric study of natural convection in enclosed plane air layers with complete and partial central vertical divisions. Ph.D. Thesis, University of Salford, UK, 1979 Bajorek, S. M. and Lloyd, J. R. Experimental investigation of natural convection in partitioned enclosures. J. Heat Transfer 1982, 104, 527-532 Chang, L. C. Finite difference analysis of radiation--convection interactions in two-dimensional enclosures. Ph.D. Thesis, Dept. of Aerospace and Mechanical Engineering, University of Notre Dame, 1981 Zimmerman, E. and Acharya, S. Free convection heat transfer in a partially divided vertical enclosure with conducting end walls. Int. J. Heat and Mass Tran.~fer 1987, 30, 319-331 Janikowski, H. E., Ward, J., and Probert, S. D. Free convection in vertical air filled rectangular cavities fitted with baffles. Proceedings o f the Sixth International Heat Transfer Conference, Vol. 6, 1978, pp. 257-263 Probert, S. D. and Ward, J. Improvements in the thermal resistance of vertical, air filled enclosed cavities. Proceedings o f the Fifth International Heat Tran~sferConference, Tokyo, NC3.9, 1974, pp. 124-128 Jetli, R., Acharya, S., and Zimmerman, E. Influence of baffle location on natural convection in a partially divided enclosure. Numerical Heat Transfer 1986, 10, 521-536 Zimmerman, E. and Acharya, S. Natural convection in a vertical square enclosure with perfectly conducting end walls. Proceedings o f A S M E Solar Energy Conference, Anaheim, California, 1986, pp. 57-65 Patankar, S. V. Numerical Heat Transfer and Fhdd Flow. Hemisphere, Washington, D.C., 1980 Patankar, S. V. A numerical method for conduction in composite materials, flow in irregular geometries and conjugate heat transfer. Proceedings o f the Sixth International Heat Transfer Conference, Toronto, Vol. 3, 1978, pp. 297-302
A numerical study has been made of buoyancy-induced flow in a square, externally heated vertical enclosure with two offset baffles or vertical dividers, in order to investigate the effect of baffle height on the flow and heat transfer characteristics. Air is taken to be the working fluid, and the horizontal end walls are assumed to be perfectly conducting. Two baffle positions are considered: in one, the upper baffle is offset toward the hot wall, and the lower baffle toward the cold wall (position A): in the other, the upper baffle is offset toward the cold wall, and the lower baffle toward the hot wall (position B). The effect of baffle conductivity and height on the thermal stratification and heat transfer is investigated. Both factors are found to play an important role, with lower heat transfer at higher baffle conductivities and baffle heights. The nature of thermal stratification between the baffles and in the baffle-near wall region controls the corresponding flow behavior. In position B, strong vertical stratification between the baffles inhibits the crossflow, whereas in position A stratification and correspondingly reduced flow penetration into the baffle-near wall region is noted. These differing flow patterns have important ramifications on the heat transfer. Keywords: buoyancy-induced flow, vertical enclosure, baffle conductivity, thermal stratification, heat transfer
Buoyancy-induced heat transfer in an externally heated, undivided vertical enclosure has been extensively studied in the literature. Ostrach, ~ Catton, 2 de Vahl Davis and Jones, 3 and Bejan 4 have reviewed the reported studies in this area. More recently, studies have been made on natural convection in partially divided enclosures. These studies include investigations with a single vertical baffle mounted on the enclosure floor or roof 5-~ and investigations with two vertically mounted baffles, one on the floor and the other on the roof. t~-~7 However, with the exception of Refs. 15-17, none of the reported studies have considered the effect of offset baffles on the flow and heat transfer behavior. Work
reported in this paper deals with buoyancy-induced heat transfer in a vertical enclosure with offset baffles. Janikowski, Ward, and Probert '5 performed an experimental study for air-filled cavities of high aspect ratio ( = 5) with offset baffles. The baffle configuration with the top baffle offset toward the cold wall and the bottom baffle offset toward the hot wall was found to have the lowest heat transfer rates. Probert and Ward j6 extended the study in Ref. 15 to consider very high aspect ratio ( = 16 to 22) enclosures. In neither of these studies were detailed flow patterns or isotherm distributions presented. A numerical study was undertaken by Jetli, Acharya, and Zimmerman ~7 for studying the effect of baffle location on free convection in a vertical square enclosure with offset vertical baffles. It was
© 1988 Butterworth Publishers
Appl. Math. Modelling, 1988, Vol. 12, August
Introduction
411
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya found that the enclosure heat transfer decreases as the upper baffle is moved toward the cold wall and the lower baffle moved toward the hot wall. Thermal stratification effects in the baffle-near wall region were found to have an important influence on the flow pattern and heat transfer. However, only baffle heights (h) of one-third the enclosure height (H) were studied; therefore, the enclosure opening (i.e., the vertical distance between baffle tips) was always one-third the enclosure height. Even for this configuration, some degree of thermal stratification was noted along the vertical midplane of the enclosure when the upper and lower baffles were offset toward the cold and hot walls, respectively. Clearly, as the baffle heights are increased, stratification effects in the baffle-near wall region and along the vertical midplane will become increasingly important and can have a significant effect on the flow pattern and heat transfer behavior. Since the effect of baffle height on stratification, and the corresponding effect on the flow pattern and enclosure heat transfer has not yet been reported, the calculations to delineate this effect have been performed and presented in this paper for an externally heated, vertical square enclosure with offset baffles and air as the working fluid. Two offset baffle configurations were considered in the present study and are shown in Figure 1. In the first configuration, called position A, the upper baffle is offset toward the hot wall and the lower baffle is offset toward the cold wall. In the second arrangement, called position B, the direction of offset is reversed, with the upper baffle placed closer to the cold wall and the lower baffle moved toward the hot wall. In either case, the horizontal distance between the baffles, and between the baffles and the nearer wall is one-third the enclosure width (on height). Three baffle heights are considered. In one, h = L/3, so the vertical distance between baffle tips is also L/3. In the second case, h = L/2, and the vertical distance between baffle tips is zero. The third baffle height considered is h = 2L/3, so the baffles overlap each other. Since the baffles are offset with respect to each other, for all three h values, the enclosure is partially, and not fully, divided. In all cases, the baffle thickness is taken to be L/20, to simulate a thin baffle. The horizontal end walls are assumed to be perfectly conducting in this study. This assumption is based on earlier calculations by Zimmerman and Acharya, '4.'8 where it is shown that with air as the working fluid, realistic simulations are obtained with the perfectly conducting end wall assumption. Results with adiabatic conditions assumed along the end walls are shown not to compare well with experimental data even if the end walls are made of a poorly conducting material such as Plexiglas.
been previously verified by comparing with experimental data in both nonpartitioned t8 and partially partitioned enclosures. 14 With the aforesaid assumptions, the governing differential equations, in nondimensional terms, can be written as Continuity:
V . V =O
(1)
X-momentum:
U.VU
)'-momentum:
-aP U. V V = - + V 2 V + Ra. O/Pr aY
Energy:
U. VO = - Pr
-aP
=-~--
(2)
+ V2U
V20
(4)
where the following dimensionless variables have been used: X = x/L
u
U-
v -
tx/ p L p* e - _ _
wherep* = p + pgy
A+T=
T-To --
-
(5b)
tz/pL
p(p./pL) 2
0
(5a)
Y = y/L
u
where To - - 2
-
T . - Tc
UT
COLD WALL
HOT WALL
~L/3-,'~ Post%ion A
TLJ
COLD WALL
UXLL H
d - , i pP
Governing equations The flow is assumed to be steady, laminar, and twodimensional, and the Boussinesq approximation is used to model the density variation. The validity of these assumptions, for moderate temperature differences, has
412 Appl. Math. Modelling, 1988, Vol. 12, August
(3)
:
2[-,'3 [. Po6].~,lon B
Figure I
Schematic of the square partitioned enclosure
(5c) (5d)
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya In the baffle, the governing differential equation is the dimensionless Laplace's equation, expressed as
kJk V20h = 0
(6)
Pr where kb/k = kr is the dimensionless baffle conductivity and 0b denotes the temperature in the baffle region. The energy balance at the baffle-air interface can be stated as Prkan];
- N\-~-n ],
(7)
where n denotes the normal to the baffle interface and i refers to the interface. The boundary conditions used are U = V = 0on all walls
(8a)
0 = 0.5 on hot walls
(8b)
0 = - 0.5 on cold walls 0 = 0.5 - Xalong horizontal end walls
(8c)
Solution procedure To arrive at the numerical solution to the governing differential equations, we obtain finite difference solutions by using the control-volume-based procedure described by Patankar. ~9 In this procedure, the calculation domain is subdivided into control volumes with a grid point located at the center of each control volume. The control volume faces are chosen such that the baffle-air interfaces fall along certain control volume faces. The governing differential equations (equations (1)-(4)) are discretized by first integrating them over each control volume and then using Green's theorem to replace the volume integral by the surface integral. With suitable profile approximations in each coordinate direction (the power-law approximation ~9is used in this study), a system of algebraic equation results that can be solved iteratively. To resolve the pressure-velocity interlinkage, we use the SIMPLER (semiimplicit method for pressure-linked equations--revised) algorithm, ~9 in which the continuity equation is used to derive an equation for pressure and an equation for pressure correction (P'), and a predictor-corrector procedure is invoked to update pressures and velocities until the velocity field satisfies both momentum and continuity equations. To avoid checkerboard pressure and velocity fields, pressure is stored at the main grid points while U is stored at the X control volume faces and V at the Y control volume faces. To account for the presence of baffles, we use the strategy recommended by Patankar. 2° By this strategy, a very high viscosity ( - - 1 0 3°) is specified in the baffle, together with kr/Pr as the dimensionless baffle conductivity. The specification of the high viscosity in the baffle and the no-slip boundary condition suppresses the velocities in the baffle to near zero values and reduces equation (4) to the appropriate Laplace's equation in the baffle region (equation (6)). Further, since the numerical scheme is conservative, flux leaving a
control volume through one face is exactly equal to the flux entering the adjacent control volume through the common face. Thus, the interface energy balance (equation (7)) is exactly satisfied at the baffle-air interfaces. In the baffle region the coefficients of the velocity equation have very high values (-1030 ) in view of the very high viscosity values (-103°). The coefficient in the pressure or pressure correction equation that links the P or P' at a point immediately outside the baffle to a point inside it is inversely proportional to the coefficient of velocity at the baffle-air interface. Since this velocity coefficient has an extremely high value, the corresponding P or P' coefficient is nearly zero, and therefore the pressure or pressure correction field in the fluid is independent of the P and P' values in the baffle, as it should be. Thus spurious or incorrect pressure or pressure correction fields are avoided. All calculations have been done on a 48 x 42 nonuniform grid with the grid points clustered more closely along all solid boundaries. The choice of the grid point distribution is based on a number of preliminary calculations with successively finer grids. The accuracy of the calculations is verified by comparing representative results with those obtained on a 78 x 72 grid. The maximum difference between the two solutions is 2.0% in peak local Nusselt number and 1.6% in the peak mid-height horizontal velocity values. As further verification of the calculation method and the assumptions, reference is made to earlier calculations reported by the authors in both nonpartitioned ~8 and centrally partitioned TM enclosures. In both cases, calculations are compared with measurements and found to compare well. Computed results in a nonpartitioned enclosure have also been compared with the benchmark numerical solution of de Vahl Davis and J o n e s ) and the two results compare to within 1% at Ra = 10 6. In view of the above-described comparisons and tests, the numerical accuracy of the solution is considered to be adequate.
Results and discussion The main parameters of interest in this study are the two different baffle positions (Figure 1), the three different baffle heights (h = L/3, L/2, 2L/3), the Rayleigh number (Ra), and the conductivity ratio (kr). Three different Rayleigh numbers are considered in the range of 104-3.55 × 10-~,and two different conductivity ratios are used (kr = 2,500) to represent a poorly conducting and highly conducting baffle, respectively. The results are presented in the form of representative streamlines and isotherms, mid-height temperature and mid-height vertical velocity profiles, and local and average Nusselt number values.
Streamlines and isotherms Streamline and isotherm plots presented are for Ra = 105 and kr = 2 (Figures 2, 3), and for Ra = 3.55 × 105 and both baffle conductivities (Figures 47). The notation used to describe the streamline and
Appl. Math. M o d e l l i n g , 1988, Vol. 12, A u g u s t
413
Buoyancy-induced heat transfer in a vertical enclosure." R. Jetli and S. Acharya ISOTHERMS RANGE - 0.50 (0.1) 0.50
STREAMLINES RANGE 14.46(1.92) 1,00 ~mox= 15.22
COLD WALL
COLD WALL
STREAMLINES RANGE 9.35 ( 1.26 ) 0.50 ~mox = 9.84
t
HOT WALL
HOT WALL ISOTHERMS RANGE - 0.50 (0.1) 0.50
COLD ~ WALL
COLD WALL HOT WALL
III Y/11 I1{1//~~\/111 HOT I ~ j
~ _ _ ~
WALL
STREAMLINES RANGE 4.70 ( 0.66 ) O. I 0 ~mox = 4.95
ISOTHERMS RANGE -0.50(0.1)0.50
COLD WALL HOT WALL
i
HOT WALL
Figure 2 Streamline and isotherm plots for position A (Ra = 10S, k, = 2)
isotherm values is RANGE ~,(A~)~2, where dh and ~z are the limiting values of the contours and A~ is the uniform contour increment used. The maximum streamfunction value (gtmax) is also shown in the figures. For both baffle locations, the strength of the flow (indicated by ~bm,x)expectedly decreases with increasing baffle height..This reduction in flow strength, at all Rayleigh numbers, is more noticeable for baffles in position A. At Ra = l0 5, and particularly for baffles in position A (Figure 2), a relatively weaker flow is noted in the region between the baffles and the nearer vertical side walls. This is due to thermal stratification in the baffle-near wall region, which inhibits flow penetration into these regions. Thus, for h = L/3, much of the flow moving up the hot wall is deflected off it near the upper baffle tip toward the cold wall. However, for h = L/2 and 2L/3 the flow deflected near the baffle tip is heated by only a small portion of the hot wall and is not sufficiently buoyant to move up and
414 Appl. Math. Modelling, 1988, Vol. 12, August
around the upper baffle. As a result, this deflected flow, on meeting the cooler lower baffle, moves down it. Consequently, for these baffle heights, an eddy is formed between the base of the hot wall and the lower baffle (Figure 2). This feature is not observed for a baffle height of L/3 and has an important effect on the heat transfer behavior. For position B, the warm fluid stream rising along the hot wall is cooled by both the conducting horizontal end wall and the top baffle. Consequently, it has a lower tendency to rise behind the baffle, causing a weak flow in that region. For all baffle heights and Rayleigh numbers studied, the flow in the confined baffle-near side wall region is weaker for baffles in position A compared with position B (Figures
2-7). The flow pattern in the region between the baffles, in position B, is dictated by the thermal stratification in this region (see isotherms in Figure 3), which inhibits the crossflow between the hot wall side and the cold
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya STREAMLINES 10.54(I.36)
~rnox
RANGE 1.00
ISOTHERMS RANGE -0.50(0.1)0.50
= 11.09
COLD WALL
COLE WALL
HOT WALL
HOT WALL
S T R E A M L I N E S RANGE 8.54 (I.15)0.50
~rnox
ISOTHERMS RANGE -0.50 (0.,)0.50
= 8.99
-oL %
STREAMLINES RANGE 5.07 (0.71)0.10
.o, WALL
ISOTHERMS RANGE -0.50(0.1 )0.50
~rnox = 5.:54
COLD WALL
COLD WALL
HOT
HOT WAL L
WALL
Figure 3 Streamline and isotherm plots for position B (Ra = 105, kr = 2)
wall side, particularly for baffles with h = L/2 and 2L/3. Thus, for h = L/2 and 2L/3, separate eddies in the hot wall-upper baffle region and the cold walllower baffle regions are formed, with a relatively weak crossflow (compared to position A) between the two eddies. For Ra = 3.55 x 105 (Figures 4-7) the stronger convective motion along the vertical walls ensures greater penetration of the fluid into the baffle-side wall regions. In position A, as mentioned earlier, part of the fluid stream rising along the hot wall detaches near the baffle tip and tends to skip over the top baffle. For baffle height of L/3, due to the relatively higher velocity of the detached fluid at the higher Rayleigh number, flow separation behind the baffle can be observed (Figure 5). As the baffle height increases to L/2, flow separation disappears, since the flow velocity at the baffle tip is lower due to the lower heating of the fluid
detaching off the hot wall near the baffle tip. For h = L/2 and 2L/3, as at Ra = 105, separate eddies are observed between the side wall and the farther baffle, but as the overall flow is stronger at the higher Rayleigh number the corresponding eddy sizes are smaller (Figures 4-5). In position B, as mentioned earlier, the predominant factor affecting the flow for all the three baffle heights is the thermal stratification in the cavity core, which is clearly reflected in the near horizontal isotherms in the center of the cavity (Figures 6-7). This strong thermal stratification at the higher Rayleigh numbers greatly inhibits the crossflow motion. Consequently, the @max values for position B are always lower compared with the corresponding values for position A. Because the flow velocities across the baffle tip are lower in this position (compared to velocities in position A), no flow separation is seen behind the baffles, even at the baffle height of L/3.
Appl. Math. Modelling, 1988, Vol. 12, August 415
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya STREAMLINES RANGE 23.66 ( 3.24 ) 1.O0 ~max = 24.91
COLD
~
ISOTHERMS RANGE -0.50(0.1 )0.50
~
"' Htkk , lS?l II\W/IF1 \\\\\\~J/IHI
I ~
HOT
HOT
~ I W A L L
WALL
STREAMLINES RANGE 18.30 (2.47) 1.00 ~max = 19.26
ISOTHERMS RANGE -0.50 (O.t) 0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STREAMLINES RANGE 13.94 ( I . 8 6 ) 0 . 9 4 ~mox = 14.67
ISOTHERMS RANGE - 0 . 5 0 (0.1)0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
Figure 4 Streamline and isotherm plots for position A (Ra = 3.55 × 10s, kr = 2)
The baffle conductivity has a noticeably different effect on the flow with baffles in position A compared with the corresponding effect with baffles m position B. Since the higher-conductivity baffle readily attains the temperature of the perfectly conducting end wall at the base of the baffle, the upper baffle is cooler and the lower baffle warmer for the greater-conducting baffle cases (Figures 4, 7). In position A, the higher-conductivity baffle has the effect of reducing the thermal stratification in the baffle-nearer vertical wall region, which in turn, induces a stronger flow in that region (Figures 4-5). Since the fluid moving down the moreconducting baffle is cooler and has a lower tendency to negotiate around the tip of the top baffle, a larger separation eddy is observed at kr = 500, for the baffle
416
Appl. Math. Modelling, 1988, Vol. 12, August
height of L/3 (Figure 5). For the baffle height of L/2, the cooler fluid descending down the upper baffle reduces the crossflow in the cavity and promotes separate eddies between the baffles and the farther side walls. In position B, with increasing baffle conductivity, the fluid coming off the cooler top baffle has a lower tendency to rise into the baffle-cold wall region, leading to a weaker flow in the baffle-nearer vertical wall regions (Figures 6, 7). Since at the higher baffle conductivity the upper baffle is colder and the lower baffle warmer, the crossflow in the core, with flow down the upper baffle and up the lower baffle, is strengthened. For the baffle height of 2L/3 this leads to higher q~maxvalues for kr = 500 compared with kr = 2 case (Figures 6-7).
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 2 2 , 8 9 ( 3 . 1 3 ) 1.00 ~mox= 24.10
ISOTHERMS RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STR EAMLINES RANGE 16.62(2.23)1.00 ~rnax = 17.49
I S O T H E R M S RANGE -0.50 (0.1)0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
S T R E A M L I N E S RANGE 12,14 ( I . 5 9 ) 1.00
ISOTHERMS RANGE -0.50(0,1)0.50
~/rnax = 12.78
COLD WALL
COLD WAL L
HOT WALL
HOT WALL
Figure 5 Streamline and isotherm plots f o r position A (Ra = 3.55 × 10s, k, = 500)
In general, the greater-conducting baffles lead to a slower flow in the enclosure since the cooler upper baffle and warmer lower baffle tend to reduce the temperature gradients along the hot and cold walls. The reduction in these temperature gradients is greater in position B compared with position A.
Velocity and temperature distributions Figure 8 shows the vertical velocity at the cavity mid-height for both conductivities at Ra = 3.55 × 105. The boundary layer velocity along the cold wall decreases significantly as the baffle height is increased. The decrease is most significant for position A, reflecting the influence of the weaker flow in the cold wall-bottom baffle thermally stratified region. For the baffles in position A and height of 2L/3, higher velocities are observed in the region between the baffles
than along the walls. For the other two baffle heights, the peak velocities occur along the wall boundary layers. In position B, the stabilizing influence of the thermal stratification results in lower velocities in the cavity core. As the baffle conductivity is raised, the fluid velocities along the wall are reduced while the corresponding values in the core increase in magnitude. This behavior is directly attributable to the fact that at higher conductivities, the upper baffle being colder results in a stronger downflow along the baffle into the core and a weaker upflow around the baffle into the baffle-cold wall region. The mid-height temperature profiles are shown for Ra = 3.55 x 105 in Figure 9. Since the profiles are antisymmetric about the diagonal, temperature profiles for kr = 2 are plotted only for x/L <=0.5, and profiles for kr = 500 are plotted only for x/L >=0.5. In position
Appl. Math. Modelling, 1988, Vol. 12, August 417
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 1 7 . 7 6 ( 2 . 5 4 ) 1.00 ~/mox = 1 8 . 6 9
ISOTHERMS
RANGE
- 0 . 5 0 (0.! ) 0 , 5 0
COLD WAL L
COLD WALL
HOT WALL
HOT WALL
STREAMLINES 16.85(2.26)
RANGE 1.00
I S O T H E R M S RANGE -0.50 (0.1)0.50
~mox = I Z 7 4
COLD WAL L
COLD WALL!
HOT WAL L
HOT WALL
STREAMLINES RANGE 12.82 ( 1.71 ) 0 . 6 2
I S O T H E R M S RANGE - 0.50 (0.1 )0.50
t//mo x = 13.49
J
COLD WALL
COLD WALL
HOT WALL
Figure 6 Streamline and isotherm plots for position B (Ra = 3.55 x 10s, k, = 2)
A, the near wall region is characterized by a thermal boundary layer. Hence, close to the wall the temperature exhibits a sharp gradient. With increasing x / L , the temperature and the gradients diminish due to the cold fluid moving up along the lower baffle. Close to the midplane, the mixing of the hot and cold streams results in sharper temperature gradients. In position B, due to the lower velocities and the importance of thermal stratification, the temperature gradients, particulady in the core, are significantly lower compared with the corresponding values in position A. In the baffle region, a slight distortion of the temperature profile is noted for the less-conducting baffles (near x / L = t), and distinct plateaus in the temperature profile for the greater-conducting case (near x / L = ~). The more dramatic effect on the temperature profile, for kr =
418 Appl. Math. Modelling, 1988, Vol. 12, August
500, is due to the stronger end wall effect when the baffle is more conducting. In particular, the baffle conductivity appears to have a significant influence for the h = L/2 case, with sharp temperature gradients on either side of the more-conducting baffle. With increasing Rayleigh numbers, the stratification in position B and the temperature gradients in position A increase in the cavity core. Increasing the baffle conductivity, however, decreases the core stratification in position B and the core temperature gradients in position A. N u s s e l t n u m b e r distributions
The local Nusselt number distributions along the cold wall are presented for both baffle conductivities
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya S T R E A M L I N E S RANGE 17.57 ( 2 . 3 4 ) 1.00 ~rnax = 1 8 . 2 8
I S O T H E R M S RANGE - 0.50 (0.! ) 0.50
COLD WALL
COLD WALL
HOT WALL
HOT WALL
STREAMLINES RANGE 16.44 ( 2 . 2 1 ) 1.00
I S O T H E R M S RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
~rnox = 17.31
COLD WALL
COLD WALL
;L
HOT WALL
STREAMLINES RANGE 13.14 ( 1 . 7 3 ) 1.00 ~bmax = 1 3 . 8 3 "'
I S O T H E R M S RANGE - 0 . 5 0 (0.1 ) 0 . 5 0
COLD WAll
COLD WALL
HOT WALL
HOT WALL
Figure 7 Streamline and isotherm plots f o r position B (Ra = 3.55 x 10s, kr = 5001
at Ra = 105 in Figures 10 and 11, respectively. At both Rayleigh numbers, in position A a single prominent peak exists in the upper region of the cold wall, whereas in position B a dual peak is observed, with the first peak in the upper region of the cold wall followed by a second peak situated somewhat below the location corresponding to the tip of the upper baffle. For position A, at the baffle height of L/3, the fluid stream is warmed as it rises along the hot wall and tends to skip over the top baffle (Figures 2, 4) and is deflected directly toward the cold wall. This causes large temperature gradients and Nusselt number values to occur at y/L -~ 0.75, for Rayleigh numbers of 105 and 3.55 x 105. As the baffle height is increased from L/3 to L/2 at Ra = 105, about 33% reduction in the peak Nusselt number value is. observed. This sharp decrease is a direct consequence of the formation of separate eddies
between the side wall and the farther baffle (Figures 2, 4), which inhibits direct crossflow of the heated fluid toward the cold walls. Figures 10 and 11 show the near constant Nusselt number between 0 =< y/L <- 0.5 reflecting the influence of the thermal stratification in this region. At Ra = 3.55 x 105, the reduction in the peak Nusselt number caused by increasing the baffle height from L/3 to L/2 in position A is only 23%. This is partially due to the increased crossflow in the cavity and reduction in the size of the eddies between the walls and farther baffles at the higher Rayleigh number. However, at the higher Rayleigh number, the increase in the stratification in the baffle-near side wall region causes the Nusselt number value in that region (0 y/L <=0.2) to fall to values in the vicinity of the conduction value of 1.0. Similar effects are noted when the baffle height is increased to 2L/3. The peak Nusselt
Appl. Math. Modelling, 1988, Vol. 12, August 419
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya 250
,.o i
kr:2 , POSITION B h = 2L/3 h= L/2 = l.J'3
200
0.9 t
~ /i
150.
'°I
0.80.70.6-
)SITION A h=L/3 h =L/2 h = 2L/3
_J "~ 0.50.4-
/
-,O011 ,/"OS'T'ON" t",.J
0.3
0.2 0.1 • kr =
- 25d . o.o
oJ,
o:2
500
I I 0:4 0.5 o'.o o'.,
o;3
0.0
o.'2 o.~
o14
o .....
o15
; .......
~
"3
Figure 8 Mid-neight vertical velocity profiles at Ra = 3.55 x 105
.....
,: ....... ; .....
;
Nu
x/L
Figure 10 Local N u s s e l t n u m b e r d i s t r i b u t i o n a l o n g t h e c o l d w a l l (Ra = 10 s,k, = 2)
I.o0.5
' kr= 500
".\
0.9-
/:
o.s-
0.3
POSITION A
/
h= L / 3 ~ h = L/2 - - ~ h =2 L/3~
0.2
to
I /""•
',
',, /
0.7"
'// ./~ ~'=~
0.1
"'-...
I
/°
0.6.
/
4%
POSITION A
,
/.
0.5
w
i
,/; /o
/
d
J
"'-°% "\
0.4"
/°/
h =L/2 h =2L/3
0.0
q~
/ -0.1.
" ~ ...... / / I
P\\\
"
~ -~-.= - .= -3LI2
/ i
/ /
,/ "
r
" 0"2"1
JJ
/
,i
~-.
/"
O.~
. o.,
~
0.1
k =2 , o.o
P/~.~L)SITIO
I:l~
J
f
-o
/
0.3
=2 L / 3
--/1""
- 03-
~'
0.4
)SITION S
o.z
0.3
s 0.4
N B
h= L 1 3 h = L/2 h=2L/3
0.O
a5
~ 0.6
0 ....
0.7
o.s
o.s
i
,.o
i
......
i
....
4 . . . .
; .......
;
Nu
x/L
Figure 11 Local Nusselt number distribution along the cold wall Figure 9 Mid-height temperature profiles at Ra = 3.55 x 105
420
Appl. Math. Modelling, 1988, Vol. 12, August
(Ra = 105, kr = 500)
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya number still occurs in the upper region of the cold wall, but the peak value is significantly lower due to the overall flow becoming weaker for this baffle position. The location of the peak Nusselt number value gradually increases from y/L = 0.75 for baffle height of L/3 to y/L = 0.85 for baffle height of 2L/3. In position B, for the baffle height of L/3, the warm air rising along the hot wall is cooled by the horizontal end wall and the top baffle. Part of the air negotiating the top baffle rises into the baffle-cold wall region, causing a Nusselt number peak in that region, while part of it impinges directly on the cold wall, causing a second Nusselt number peak below the baffle (Figures 10-11). As Ra increases, due to the stronger flow, larger peaks are observed. As the baffle height is increased to L/2, dual Nu peaks are again observed. For this baffle height, however, the second peak below the baffle (y/L ~- 0.35) is due to the effect of the eddy that exists between the cold wall and bottom baffle, which carries the fluid warmed along the bottom baffle toward the cold wall and causes the second Nu peak. At the baffle height of 2L/3, the thermal stratification in the cavity core effectively divides the flow into two separate zones, thus reducing the strength of the flow and the associated temperature gradients. Increasing the baffle conductivity significantly reduces the heat transfer occurring in the upper bafflecold wall region in position B (Figure 11). Since the upper baffle is cooler for the higher-conductivity case, the air coming off that baffle is less amenable to rise behind the baffle. Consequently, the first Nu peak behind the baffles in position B is considerably reduced. Furthermore, this cooler air coming off the top baffle (for the baffle height of L/3) impinges directly on the cold wall, causing the second Nu peak to be more pronounced (Figure 11). In position A, the peak values of Nu are reduced in all cases, due to the reduced temperature gradients (Figure I 1), although the overall profile retains the same shape as for a low-conductivity
Table I
baffle. For h = 2L/3, at Ra = 105 the local increase in the vigor of the downward flow along the top baffle causes the peak Nu value to increase with baffle conductivity. The average Nu values for the cavity are presented in Tables I and 2 for baffles in positions A and B, respectively. For h = L/3, the heat transfer is generally lower for position B compared with position A. This is partly due to the stabilizing thermal stratification in the cavity core and partly due to the effect of the baffles on the flow pattern. As the baffle height is increased to L/2, at the lower Rayleigh numbers of l04 and l05, position A is more effective in reducing the overall heat transfer compared with position B. This reversal in the trend is the effect of the near-stagnant regions between the side walls and the nearer baffles in position A. Even though the peak Nusselt number is still higher in this position, the near constant values in the baffleside wall regions cause a lower average value. At Ra = 3.55 x l05, position B has a lower average Nu because as more flow penetrates the baffle-side wall region in position A. At h -- 2L/3, position A has lower heat transfer rates at all Rayleigh numbers, which is again a consequence of the weak flow in the bafflenear side wall region. This is somewhat unexpected and contrary to the results in Ref. 17, since intuitively, the configuration with the stabilizing core (position B) would be expected to be associated with lower heat transfer rates. In general, raising the baffle conductivity reduces the average Nusselt number values, except in some cases of 2L/3 baffle height, where the increase in conductivity causes increased local flow along the baffles.
Concluding remarks Natural convection in a square vertical enclosure with two offset baffles has been studied numerically. The study shows that increasing the baffle height causes a
A v e r a g e Nusselt n u m b e r v a l u e s f o r p o s i t i o n A Ra = 104
L/3 L/2 2L/3 No baffles
Table 2
Ra = 10 s
Ra = 3.55 x 10 s
k, = 2
k, = 500
k, = 2
kr = 500
k, = 2
k, = 500
1.17 1.00 1.01
1.18 1.03 1.04
2.71 1.59 1.16
2.61 1.59 1.23
4.25 2.81 2.07
4.11 2.67 2.10
1.76
3.37
4.91
A v e r a g e Nusselt n u m b e r v a l u e s f o r p o s i t i o n B Ra = 10 a
L/3 L/2 2L/3 No baffles
Ra = 10 s
Ra = 3.55 x 10 s
kr = 2
k, = 500
k, = 2
kr = 500
k, = 2
k, = 500
1.22 1.11 1.05
1.18 1.12 1.08
2.39 1.75 1.38
2.10 1.64 1.36
4.00 2.74 2.19
3.22 2.41 2.12
1.75
3.37
4.91
Appl. Math. Modelling, 1988, Vol. 12, August
421
Buoyancy-induced heat transfer in a vertical enclosure: R. Jetli and S. Acharya significant reduction in the flow strength and heat transfer for both of the baffle locations investigated. As the baffle height is increased, separate eddies tend to form between the side wall and the farther baffles. In the baffle position with the top baffle offset toward the hot wall and the bottom toward the cold wall, these eddies are an effect of the thermal stratification in the top baffle-hot wall and bottom baffle-cold wall regions. Conversely, in the position with the top baffle offset toward the cold wall and the bottom toward the hot wall, the eddies are a consequence of the stabilizing stratification in the cavity core that inhibits crossflow in the cavity. At lower Rayleigh numbers and longer baffleheights, lower overall heat transfer is observed in the position with the top baffle offset toward the hot wall, compared with the position where it is offset toward the cold wall, with the results being reversed at higher Rayleigh numbers and smaller baffle heights. In general, the greater-conductivity baffle leads to lower average heat transfer compared with the less-conducting baffle, with the reduction being greater for the position with top baffle offset toward the cold wall.
Nomenclature
k~ L Nu Nu P P Pr Ra
Rayleigh number,
c, d g H h k kb
T
T~ Th To I1
U o
V x,X
y,Y
References I 2 3
4 5
6 7 8
aperture ratio, (H-h)/L specific heat of fluid thickness of the baffle acceleration due to gravity enclosure height baffle or divider height thermal conductivity of air thermal conductivity of baffle conductivity ratio, kb/k enclosure width Nusselt number, hL/k average Nusselt number thermodynamic pressure dimensionless pressure Prandtl number, tzCp/k
mp
dimensionless temperature in baffle stream function I/Jrflax maximum stream function 0b
9
10
II
12 13
g/3(Th -- T~)L3
va dimensional temperature cold wall temperature hot wall temperature reference temperature, (T~ + Th)/2 dimensional velocity in x direction dimensionless velocity in x direction dimensional velocity in y direction dimensionless velocity in y direction dimensional and dimensionless coordinate along the horizontal direction dimensional and dimensionless coordinate along the vertical direction
14 15
16
17 18
Greek symbols a /3 z, p 0
thermal diffusivity volume expansion coefficient at constant pressure kinematic viscosity density dimensionless temperature
422
Appl. Math. Modelling, 1988, Vol. 12, August
19 20
Ostrach, S. Natural convection in enclosures. In Advances in Heat Transfer, Vol. 8, eds. J. P. Hartnett and T. F. Irvine, Jr. Academic Press, New York. 1972, pp. 161-227 Catton, I. Natural convection in enclosures. Proceedings o f the Sixth International Heat Transfer Conference, Vol. 2, pp. 13-31, 1978 de Vahl Davis, G. and Jones. J. Natural convection in a square cavity--a comparison exercise. In Numerical Methods in Thermal Problems, Vol. II, eds. R. W. Lewis, K. Morgan, and B. A. Schnefler. 1981, pp. 552-572 Bejan, A. A synthesis of analytical results for natural convection heat transfer across rectangular enclosures. Int. J. Heat and Mass Tran~fer 1980, 23, 723-726 Nansteel, M. W. and Greif, R. Natural convection in undivided and partially divided enclosures. J. Heat Tran.~fer 1981, 103, 623-629 Lin, N. N. and Bejan, A. Natural convection in a partially divided enclosure. Int. J. Heat and Mass Transfer 1983, 26, 1867-1878 Winters, K. H. The effect of conducting divisions on the natural convection of air in a rectangular cavity with heated side walls. ASME Paper No. 82-HT-69 Winters, K. H. Laminar natural convection in a partially divided rectangular cavity at high Rayleigh number. AERE, Harwell Laboratory, Report No. HL85/1527, 1985 Chao, P. K. B., Ozoe, H., Lior, N., and Churchill, S. W. The effect of partial baffles on natural convection in an inclined rectangular enclosure. Chemical Engineering Fandamentals 1986, 2, 23-49 White, M.D., Kirkpatrick, A. T., and Winn, C. B. Numerical study of high Rayleigh number natural convection flows through openings. Second ASMEIJSME Thermal Engineering Joint Conference. Hawaii, 1987 Duxbury, D. An interferometric study of natural convection in enclosed plane air layers with complete and partial central vertical divisions. Ph.D. Thesis, University of Salford, UK, 1979 Bajorek, S. M. and Lloyd, J. R. Experimental investigation of natural convection in partitioned enclosures. J. Heat Transfer 1982, 104, 527-532 Chang, L. C. Finite difference analysis of radiation--convection interactions in two-dimensional enclosures. Ph.D. Thesis, Dept. of Aerospace and Mechanical Engineering, University of Notre Dame, 1981 Zimmerman, E. and Acharya, S. Free convection heat transfer in a partially divided vertical enclosure with conducting end walls. Int. J. Heat and Mass Tran.~fer 1987, 30, 319-331 Janikowski, H. E., Ward, J., and Probert, S. D. Free convection in vertical air filled rectangular cavities fitted with baffles. Proceedings o f the Sixth International Heat Transfer Conference, Vol. 6, 1978, pp. 257-263 Probert, S. D. and Ward, J. Improvements in the thermal resistance of vertical, air filled enclosed cavities. Proceedings o f the Fifth International Heat Tran~sferConference, Tokyo, NC3.9, 1974, pp. 124-128 Jetli, R., Acharya, S., and Zimmerman, E. Influence of baffle location on natural convection in a partially divided enclosure. Numerical Heat Transfer 1986, 10, 521-536 Zimmerman, E. and Acharya, S. Natural convection in a vertical square enclosure with perfectly conducting end walls. Proceedings o f A S M E Solar Energy Conference, Anaheim, California, 1986, pp. 57-65 Patankar, S. V. Numerical Heat Transfer and Fhdd Flow. Hemisphere, Washington, D.C., 1980 Patankar, S. V. A numerical method for conduction in composite materials, flow in irregular geometries and conjugate heat transfer. Proceedings o f the Sixth International Heat Transfer Conference, Toronto, Vol. 3, 1978, pp. 297-302