IN HFATA~)MASSTRANSFER 0094-4548/79/0301-0093502.00/0 VOI. 6, pp. 93-102, 1979 ~P~Press Ltd. Printed inGreatBritain

ON THE BOUNDARY LAYER REGIME IN A

VERTICAL ENCLOSURE FILLED WITH A POROUS MEDIUM

A d r i a n BeJan Depar tmen t of M e c h a n i c a l E n g i n e e r i n g

U n i v e r s i t y of Co lo rado , B o u l d e r , Co lo rado

(Ccm~icated bv J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The floating constants in Weber's boundary layer solution for free convection in a differentially heated vertical porous slab are re- evaluated with a new approach. This approach uses Weber's solution to calculate the net vertical heat flux whlch is equated to zero near the top and bottom ends of the enclosure. It is shown that the Nusselt numbers predicted with the new constants are in excel- lent agreement with experimental and numerical results.

I n t r o d u c t i o n

The p r e s e n t emphas i s on h i g h pe r fo rman ce i n s u l a t i o n s f o r b u i l d i n g s and

c o l d s t o r a g e s p a c e s has g e n e r a t e d an i n c r e a s e d i n t e r e s t i n b a s i c f l u i d mechan-

i c s phenomena i n porous media . Of p a r t i c u l a r i n t e r e s t i s t h e f r e e c o n v e c t i o n

e f f e c t on h e a t t r a n s f e r a c r o s s v e r t i c a l r e c t a n g u l a r c a v i t i e s f i l l e d w i t h

g r a n u l a r or f i b r o u s i n s u l a t i o n , t h e v e r t i c a l w a l l s of t h e c a v i t y b e i n g main-

t a i n e d a t d i f f e r e n t t e m p e r a t u r e s w h i l e t h e top and b o t t o m w a l l s a r e i n s u l a t e d .

The s y s t e m i s p r e s e n t e d s c h e m a t i c a l l y i n F i g . 1. E x i s t i n g n u m e r i c a l and

e x p e r i m e n t a l s t u d i e s have shown c o n c l u s i v e l y t h a t when t h e t e m p e r a t u r e d i f f e r -

ence be tween t h e two v e r t i c a l w a l l s i s l a r g e t h e f r e e c o n v e c t i o n e f f e c t can

domina te c o m p l e t e l y t h e h e a t t r a n s p o r t a c r o s s t h e i n s u l a t i o n . I n such c a s e s

t h e h e a t t r a n s f e r r a t e i s c o n s i d e r a b l y l a r g e r t h a n t h e t h e o r e t i c a l e s t i m a t e

based on c o n d u c t i o n h e a t t r a n s f e r a l o n e ( s e e , f o r example t h e work of

Schneider [i], Klarsfeld [2], Chan, Ivey and Barry [3], Bankvall [4] and

Burns, Chow and Tien [5]).

93

94 A. Bejan Vol. 6, No. 2

In a series of graphic displays, Bankvall [4] demonstrated that as the

temperature difference across the cavity (T a - Tb) increases the isotherms

become increasingly more horizontal over the central (core) region of the

cavity. This observation suggests that, in this limit, heat is being trans-

ported from T a to T b not by direct conduction but by free convection, by

heating the fluid rising through the porous medium along the warm wall (Ta)

and cooling it as it descends along the cold wall (Tb). To a certain degree,

the fluid present in the core region can be viewed as stagnant and thermally

stratified, the fluid motion being concentrated near the two vertical walls.

This is the boundary layer regime studied analytically by Weber [6] who

relied entirely on the powerful method of solution developed earlier by Gill

[7] for free convection in a vertical slot filled with viscous fluid.

H / 2 ~ ' ~ ' ~ ' ~

y"

ot P O R O U S M E D I U M ' -"

- H /2

I I

I

0

~ I N S U L A T E D

V"

I u

L x"

FIG. i

Schematic of two-dimensional vertical slot filled with a porous material

~oi. 6, No. 2 ]~KX.OSL~REF~X.~DWITHAIK)ROi~M~DiTJM 95

The Weber theory for the boundary layer regime began with the assumption

that a vertically stratified core region exists far away from both vertical

walls. Boundary layer solutions were then obtained for the two vertical

layers. The process of matching the two boundary layer solutions with the

same (unknown) core solution yielded a consistent picture for the free con-

vection pattern in the cavity, in areas situated sufficiently far from the top

and bottom adiabatic and impermeable ends. The limitations of this theory are

derived from the boundary layer-type approximations involved. Consequently,

the Weber solution cannot satisfy all four boundary conditions present along

the horizontal walls

v* = 0 , ~T*/~y* = 0 at y* = ±H/2 (I)

where v* and T* represent vertical velocity and temperature, respectively.

Weber's solution contains two arbitrary constants which could conceiv-

ably be determined by taking into account only two of conditions (i). Like

Gill earlier, Weber chose to invoke impermeability at y* = ±H/2. However,

there remains a certain arbitrariness in the manner in which the two con-

stants were estimated and, as discussed by Quon [8], it may be that a totally

different procedure must be used to determine these constants. In a recent

paper Bejan [9] altered Gill's theory by calculating the net flow of energy

(conduction and convection) in the vertical direction and setting it equal to

zero at the top and bottom ends. In doing so, the four boundary conditions

(I) were all taken into account in an average way. Bejan showed that the

Nusselt number predicted by the Gill theory in combination with the average

zero energy flux boundary conditions agrees very well with available experi-

mental and numerlcal heat transfer correlations for convection in vertical

enclosures filled with viscous fluids.

The objective of this note is to modify the Weber theory for convection

in a vertical porous layer by combining Weber's boundary layer solution with

zero energy flux boundary conditions at y* = ±H/2 . We show that the Nusselt

number predicted by the modified boundary layer theory is in excellent agree-

ment with published experimental results. The ability to reliably predict the

heat transfer rate across the system of Fig. I is the motivation behind the

presen~ extension of Reference [9] to convection in vertical porous layers.

The presentation which follows below has been abbreviated intentionally.

The presentation is tied closely to the analyses of Weber [6] and Bejan [9]

and, for further details, the reader is encouraged to consult their papers.

96 A. Bejan Vol. 6, No. 2

Modified Boundary Layer Analysis

The Weber boundary layer analysis yielded the following expressions for

the flow and temperature fields near the two vertical walls:

near the left wall (x = 0)

X

2 2C(l+q)] ~L = C(I - q )[i - e (2)

a)

X

i T L = -~ [q + (1 - q ) e 2C(1+q)] (3)

near the r i g h t w a l l (~ = 0 , where ~ = ~ - x) b)

2C(l-q)] ~R = C(1 - q2 ) [ 1 - e (4)

{ 1 T R = -~ [q - (I + q) e 2C(l-q)] (5)

The streamfunction ~* was defined in the usual manner, u* = - ~*/~y* , and

v* = ~*/~x* , u* and v* being the fluid velocity components shown in Fig. i.

Solution (2) - (5) is in nondimenslonal form already, the result of having

defined

x = x*/6 , y = y*/H , L/6 = (Ra L/H) I/2 , (6,7,8)

1 ~'6 T* - ~ (T a- Tb)

= ~H ' T = Ta_ Tb (9,10)

where the asterisks indicate the dimensional variables of the problem. In

equation (8), 6 is the boundary layer thickness and Ra the Rayleigh number

based on cavity spacing L and medium permeability K,

g8 KL (T a - r b ) Ra , (ii) v~

g, 8, u and ~ standing for gravitational acceleration, coefficient of thermal

expansion, kinematic viscosity and thermal dlffusivlty, respectively. In (2~

(5), the dependence of ~ and T on altitude y is contained in parameter q

for which, in the case of temperature-independent viscosity, Weber found

3 y = C2(q- 3 ~) + C.. (12)

Constants C and C. represent the backbone of the boundary layer sol ion.

As mentioned in ~ I, Weber determined these constants by using the imp~ eable

Vol. 6, No. 2 ENCLO6%RE ~-~7.TY~ WITH A POROL~ ~EDIUM 97

condition along the two horizontal walls, T - 0 at y = • 1/2 . Thus, he found

I

C - 32/2 and C, ~ O, or q = ~I at y - ±1/2 . (13)

The second constant, C. , is zero based on symmetry, the result of having ap-

plied the same condition to both ends, y - • 1/2.

Alternately, let us consider the upward flow of energy between the two

vertical walls of the cavity. The vertical energy transport per unit width

measured perpendicular to the cross-sectlon shown in Fig. i, Qy (not to be

confused with ~Q/~y), is

3 L/6 1 L/6 E3 QY~_ -- Ra 2(L ) ! ~ kL(T a Tb ) ~-~ Tdx- (R~'~) .~ ~J~T ~,,dx. (14)

0

Here, k is the thermal conductivity of the medium. The first term repre-

sents the net flow of energy convected upward by the counterflow pattern

formed by warm fluid rising near x = 0 and cold fluid descending near

x = L/6 . The second term of equation (14) accounts for the downward energy

flow due to heat conduction in the vertically stratified porous medium.

The two integrals of equation (14) are next evaluated by splitting each

integral into two parts corresponding to the two wall solutions, equations

(2)- (5). The procedure is laborious but straightforward, therefore it will

be omitted here. The final form of (14) is

i 3 QyH - H ] = Ra 2 (~) C(l- q2). dqdy (IS)

kL (T a - T b)

in which we have already dropped a number of terms multiplied by exp [- (L/6)/

[4C(I + q)]}, a small number in the boundary layer limit discussed here.

Before proceeding further, it is worth commenting on the physical sig-

nificance of result (15). The vertical energy flowrate Q -~ is an even func- ~" jl y

tion of altitude reaching its maximum across the ~Id-section of the cavity,

1 _3 QyH - H 2

[ kL(Ta-Tb) ]max = CRa 2 (~-) - C "2 , at y = 0 . (16)

Using the terminology of Eckert and Carlson [I0] for free convection in air- i

filled enclosures, between y = -~ and y = 0 Qy increases as more heat is 1 being pumped into the cavity near the "starting end" (x = 0, y = -~) rela-

tive to the heat being extracted near the facing "departure end" (~ = 0,

98 A. Bejan Vol. 6, No. 2

i y = - ~)" In the upper half of the cavity Qy steadily decreasesldue to in-

tense cooling present near the other starting end (~ = 0 , y = +~). Another

interesting fact is that the convective contribution to Qy, the first term

on the right hand side of (15), dominates the vertical flux of energy as

H Ral/3 either Ra or H/L increases, In fact, if the group ~ is infinite, i

the zero energy flux condition Qy = 0 at y = ±~ - yields q = ± i which is i

exactly the result obtained by Weber claiming impermeability at Y = ±3 '

equation (13). Therefore the impermeability condition is equivalent to the

zero energy flux condition in the limit where the vertical energy flux is all

by free convection.

1 . . . . . . . l 1

0.5

0 I I I I I I I I I I I I

10 1 ~LRag

v~ 2

I

I I I I I

100

FIG. 2

Variation of boundary layer solution parameters

and Nu 6/L with the new group (H/L) Ra I/3"

C, qe

Vol. 6, No. 2 ~CSOSURE FTT,T,k'D WITH A PCROUS MEDIUM 99

When Ra and H/L are moderate as in all applications, the condition of i

zero net energy transfer across the y = ±3 planes yields the more general

result 2 i

O 6 C = (i - qT) ma (¢) , C, = 0 , (17)

i where qe = q( ~ ) " To obtain expression (17) we used equation (12) to eval-

uate dq/dy appearing in equation (15), Equation (17) and equation (12) with i

y - ~ , constitute a parametric result for the unknown C as a function of

Ra and H/L. This result is shown plotted in Fig. 2 showing that C asymp-

totically approaches the Weber value ~r~/2 as the new group Ra I/3 H/L tends

to infinity.

i Having not invoked impermeability at y - ~ , the modified boundary

layer theory based on the zero energy fhu~ condition is characterized by non-

i particularly near all four corners, zero vertical velocities at y = ±~ ,

away from the core region. For example, the vertical velocity at x = 0, i i

y = ~ is v = ~ (i - qe ) where qe' plotted in Fig. 2, is less than one.

The solution worked out in this section suggests that the warm fluid rising

along the left wall on Fig. i tends to reach the top level with finite velo- I

city. Since in the real system the fluid cannot advance above y = ~, its

only chance is to hit the wall and move horizontally to the right in a curving

fashion. Now we also have a theoretical explanation fo= the opposite effect I

occurring at the two departure corners, (x = 0 , y = -3) and (~ = 0 and i

y = ~) , where the modified boundary layer solutien predicts a finite verti-

cal velocity into the cavity. Since this is made impossible by the physical

presence of the top and bottom walls, a local vacuum is created near the two

corners. Consequently, fluid leaving one boundary layer tends to rush in

from many angles to enter the opposite boundary layer. These effects are all

illustrated by Bankvall's displays. The present theory indicates that the

corner effects should die down rapidly if, for constant Ra, the aspect ratio

H/L increases (see Fig. 2). Conversely, the corner effects should be more

visible in relatively short cavities, particularly in shallow cavities as

analyzed by Bejan and Tien [ii] and Walker and Homay [12].

It is important to keep in mind that by having applied the zero energy

flux condition we have not replaced Weber's impermeable wall argument with 1

adiabatic wall conditions ~T/Sy = 0 at y = ±~ . As emphasized earlier,

the zero energy flux condition is a device by which, in an average way, the

i00 A. Bejan Vol. 6, No. 2

impermeable and adiabatic properties of the top and bottom walls are taken

into account at the same time. Not recoEnlzlng these features simultaneously

is the failing point of the manner in which Weber [6] selected the boundary

layer solution parameter C.

Nu

10

WEBER

PRESENT SOLUTION

f / /

11 \

~ - D 4 . 5 K~RSF~LO

I X 4.5 BANKVALL

L I I ~ I I I I 30 100 200

Re

FIG. 3

Comparison between the Nusselt number predicted by the modified boundary

layer theory and the experimental results of Klarsfeld [2] and Bankvall [4]

Vol. 6, No. 2 nqCLOSURE Fw.T.~ WITH A POROUS MEDIlI~ 101

The Nusselt Number

Of particular interest in the field of thermal insulation ensineerin g is

the value and behavior of the Nusselt number for cavities filled with aporous

medium. In this sectlonwe derive analytically the heat transfer rate between

the two vertlcal walls and compare the analytical result with published ex-

perimental and numerlcal data.

The N u s s e l t number i s d e f i n e d as

1/2 QL '- ~ - (~x) dy (18)

Nu - kH(Ta " Tb) x-0 -i/2

where Q is the net heat transfer rate (per unit width) from T a Combining definition (18) with results (3), (8) and (12) we find

t o T b .

I t .

WEBER

H ~ m s L

10 100

FIG, 4

N u s s e l t number c h a r t f o r b o u n d a r y l a y e r

f r e e c o n v e c t i o n i n v e r t i c a l p o r o u s medium

102 A. Bejan Vol. 6, No. 2

8 c Nu ~ = ~ qe (3 + q ) (19)

qe and C being known functions of Ra I/3 H/L developed in the preceding sec-

tion and shown in Fig. 2. The ratio 6/L is given by equation (8). The group

Nu 6/L was plotted also on Fig. 2, reaching its Weber value i/~ only in the

limit Ra I/3 H/L - ~ . Unlike parameter C , the deviation of Nu 6/L from its

asymptotic value is sizeable over a considerably wider range.

In Fig. 3 we plotted the theoretical Nusselt number developed in this

note against experimental data available in the literature. Although the ex-

perimental Nusselt number information for vertical porous cavities is scarce,

configurations H/L = 2.25 and H/L = 4.5 are well documented by experimental

measurements due to Klarsfeld [2] and numerical experiments reported by

Bankvall [4]. What is extremely important, the experimental data are in ex-

cellent mutual agreement. We see that the theoretical Nu predicted by the

modified boundary layer analysis reproduces the experimental data extremely

well, by contrast with Weber's original theory.

We conclude this note with a Nusselt number chart for free convection

in the high Rayleigh number limit (Fig. 4). This chart is actually a replot-

ring of curve Nu 6/L of Fig. 3, in a manner which is more appropriate for

practical calculations. Weber's original theory consistently overestimates

the net heat transfer rate across the porous cavity. As pointed out in §2,

the accuracy of Weber's result improves gradually as either Ra or

very large.

References

H/L become

i. K. J. Schneider, Proc. Int. Inst. Refrigeration, 247 (1963).

2. S. Klarsfeld, R~v. Gen. Thermique , 108, 1403 (1970).

3. B.K.C. Chan, C. M. Ivey and J. M. Barry, J. Heat Transfer, 92, 21 (1970).

4. C. G. Bankvall, W~rme-und Stoff~bertragung, !, 22 (1974).

5. P. J. Burns, L. C. Chow and C. L. Tien, Int. J. Heat Mass Transfer, 20, 919 (1977).

6. J. E. Weber, Int. J. Heat Mess Transfer, 18, 569 (1975).

7. A. E. Gill, J. Fluid Mech., 26, 515 (1966).

8. C. Quon, J. Heat Transfer, 99, 340 (1977).

9. A. Bejan, J. Fluid Mech., 90, 561 (1979).

i0. E.R.G. Eckert and W. O. Carlson, Int. J. Heat Mass Transfer, 2, 106 (1961).

IL A. Bejan and C. L. Tien, J. Heat Transfer, I00, 191 (1978).

12. K. L. Walker and G. M. Homsy, J. Fluid Mech., 87, 449 (1978).