Measurements of natural convection in partially porous rectangular enclosures of aspect ratio 5

INT. COMM. HEAT MASSTRANSFER Vol. 15, pp. 203-212, 1988 ©Pergamon Press

MEASUREMENTS OF IN P A R T I A L L Y P O R O U S

NATURAL CONVECTION RECTANGULAR ENCLOSURES

OF A S P E C T

Department

0735-1933/88 $3.00 + .00 . Printed in the United States

plc

RATIO

5

S. B. Sathe and T. W. Tong of M e c h a n i c a l and A e r o s p a c e E n g i n e e r i n g A r i z o n a State U n i v e r s i t y Tempe, AZ 8 5 2 8 7

(Conrmmnicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT This paper p r e s e n t s e x p e r i m e n t a l results for s t e a d y - s t a t e natural c o n v e c t i v e heat transfer in p a r t i a l l y porous r e c t a n g u l a r e n c l o s u r e s of aspect ratio 5. A d i a b a t i c and isothermal b o u n d a r y c o n d i t i o n s were imposed for the h o r i z o n t a l and vertical b o u n d a r i e s , respectively. The e n c l o s u r e was d i v i d e d v e r t i c a l l y into a f l u i d - s a t u r a t e d porous region and a f l u i d - f i l l e d region by an i m p e r m e a b l e partition. M e a s u r e m e n t s were c o n d u c t e d for d i f f e r e n t R a y l e i g h numbers, porous m a t e r i a l s and t h i c k n e s s e s of the porous region. The results agreed w i t h an earlier p r e d i c t i o n that heat t r a n s f e r could be m i n i m i z e d by p a r t i a l l y f i l l i n g the e n c l o s u r e with a porous m a t e r i a l rather than f i l l i n g it entirely. A c o r r e l a t i o n has been d e v e l o p e d e x p r e s s i n g the Nusselt number in terms of the g o v e r n i n g parameters.

Introduction Recently, theoretical transfer

Sathe

prediction

[I] e x p e r i m e n t a l l y

of Tong and S u b r a m a n i a n

in a t w o - d i m e n s i o n a l

minimized

as a function

experiment

was

the porous

and fluid

partition.

To date,

results

et al.

conducted

for p a r t i a l l y

partially

of the porous with

regions

porous

region

an e n c l o s u r e separated

porous

enclosures 203

the

[2] that heat enclosure

thickness.

of aspect

by an

there has been very

verified

could be The

ratio

i0 and

impermeable

limited in the

experimental

literature

despite

204

the

S.B. Sathe and T.W. Tong

importance

of s u c h

applications. conducted ratio

Since

objective

for

paper

porous

to m a k e

The m e a s u r e m e n t s and b o t h

commercial

were

Nickel

nickel)

and

distilled

materials

were

chosen

thermal

conductivity medium

(Rc)

R c values

containing

are

a fluid

In a d d i t i o n problem

under Pr,

number

Ra o and

of t h e s e

kef f .

The

arithmetic the

The

mean

experimental

DeWitt

[3].

previously

less

such

for

It is the available

porous

the

porous

one

the

of

These

ratio

conductivity

insulation

test

of the

or a p p r o x i m a t e l y

fluid of the

one.

systems

can be

using

Da,

fraction

in Ref.

S.

I.

of Pr,

Ra,

properties

of the

hot

values

reported

are:

the

Prandtl

The

Rayleigh

definitions

Ha,

Da,

R c and

since

N c and Ha o r e q u i r e

c, kf,

are

for

modified

varied

Da, ~,

A,

Among

independently

of ~ and keff

results

ratio

number

properties:

of t h e s e

parameters

aspect

Darcy

found

temperature runs

the

as the

foam)

medium.

thermal

thermal

thickness

determination

five

than

water

made

(polyurethane

to m a i n t a i n

are Ra,

can be

of t h e m

The

as

material

the d i m e n s i o n l e s s

number

The v a l u e s [i].

(a p o r o u s

distilled

as air.

thermophysical

first

using

to the e f f e c t i v e

to Rc,

the

three

following

in o r d e r

typical

Rayleigh

N a o = R a D a H c. the

of a s p e c t

new measurements

plastic

water

either

parameters

only

performed

consideration

number

Hao,

have

of the E x P e r i m e n t

a foam

with

Such

enclosures

materials.

these

Foametal

saturated

porous

porous

I, the a u t h o r s

literature. Description

fluid,

insulation

of Ref.

partially

two d i f f e r e n t

of this

in t h e r m a l

the p u b l i c a t i o n

measurements

5 using

in the

results

Vol. 15, No. 2

p,

@,

evaluated

and c o l d w a l l s

at

the

for e a c h

in I n c r o p e r a

have been

K and

and

measured

x : 6 . 7 6 x l O -8 m e and k e f f = O . S S 9

of

Vol. 15, No. 2

W/m 2

for

for 24

CONVECTION IN POROUS ~ C L O S U R E S

Nickel

the

foam

plastic.

oC w h e r e a s Two

aspect (lO

ratio

the

same

for

enclosure

m

the

for

For

adjusting

each the

average

are

was plate The heat

entire

the

temperature but

and

earlier

0.2540

(2 in.)

end w a l l s

had b e e n

ratio

from

lO to 5.

This

the

foam plastic.

high,

0.4064

constructed

enclosure

Ra o.

The

was

and h a v e

been

wide. [I]

A smaller

m (16

employed and

m

enclosure

in.)

for the N i c k e l

operation

°C.

investigation

and

for

25

the s a m e

was

m

at

of

with

0.0508

W/m 2

conducted

enclosures

in the

(5 in.)

same

ke~=0.599

~ were

a mean

deep

and

bottom

wide was

smaller

were

conducted

S value,

test

hot

and

temperature

approximately

total

m

for

of the

used

aspect

m2

dimensions,

One

top,

smaller

-7

deep

Foametal

to y i e l d construction

described

of

in d e t a i l s

[1].

1.00.

the

at

(16 in.)

0.1270

The

Measurements

state

m

of the

(l in.)

enclosures

before

were

of d i f f e r e n t

enclosure

was

measurements. measurements

kef f

the m e a s u r e m e n t s

that

and 0 . 0 2 5 4

x=2.369xlO

measurements

employed.

to c h a n g e

used

both

for

0.4064

the w i d t h

doubled was

high,

was

except

those

were

and The

enclosures

in.)

This

Foametal,

205

cold

reached,

the

Nu = Q o d / k f A o

to was

filled

(Th-T c)

for

0.25,

the

different

ambient steady

input

0.50,

to

the

0.75,

Ra w e r e while

hot

made

keeping

temperature. state.

and

It

Once plate

by

the took

steady

heater

and

recorded.

(Nu) the

S=O,

temperatures

reach

were

number

transfer

enclosure

to power

temperatures Nusselt

to

hours

runs

plate

close

8-10

for

was

heat with

defined transfer the

fluid

as

the

by

conduction alone.

ratio

of

the

when

the

Hence:

(1)

206

S.B. Sathe and T.W. Tong

where

Oo

corresponded

plate

heater.

to the

electrical

Results Shown

in T a b l e s

conditions heat Eq.

and

transfer (I).

percent cases

The for

where

tabulated. order

the

are

highest

S=0,

the

and

have been

literature

(shown

difference

when

computer largest with

difference

is 4.2

results

presented

in h e a t 0.25.

for

for

obtained

percent.

from

The

The

For

therefore

are

the

not

included

in

the m e a s u r e m e n t s

are

For

accurate,

some

of

in the

S=0,

of Seki

and 0 . 5 0

the

by

to + 7 . 7

available

S=0.25

the

largest

et al. are

[4]

compared

finite-difference

investigation results

of T o n g

the N i c k e l

was

amount

Foametal

in Fig.

obtained

when

of r e d u c t i o n

[2].

for S=I

The

agree

and S u b r a m a n i a n

increased

from

0.25

transfer

increased

when

S was

is c l e a r

that

heat

Foametal

transfer

thickness.

as s h o w n

1 (a). S was

in h e a t

further

Nickel

at w h i c h

in the e a r l i e r

graphically

transfer The

S was

and

correlations

solution

+2.6

temperatures

those

runs.

[5]

to

9 percent.

The

Heat

reported

hot

test

respectively.

meaningless

with

to the

results

the b o u n d a r y - l a y e r

within

also

program

Ra,

the m e a s u r e m e n t s

The m e a s u r e m e n t s

numerical

to the

of Nu as d e f i n e d

from

the p a r e n t h e s e s ) .

compared

of the

experimental

in t e r m s

ranges

compared

within

the

lowest

cold wall

the

the

the

temperature

supplied

summaries

in Nu r a n g e d

and

To d e m o n s t r a t e

with

all

presented

w e r e made.

is 5 p e r c e n t .

the

R c and Ra o w e r e

The hot

results

for

uncertainties

power

and D i s c u s s i o n

2 are

results

results

the

to s h o w

1 and

Vol. 15, No. 2

Significant increased

transfer

to 0.50, changed

and

in Fig.

became from

zero

are

the

as

to 0.75. 1.00.

as a f u n c t i o n

1 (b)

to

smaller

0.50 to

I are

reduction

from

from 0.75

can be m i n i m i z e d Shown

in T a b l e

It

of the

results

VOI. 15, NO. 2

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N38NITN 1 7 3 8 $ N N

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208

S.B. Sathe and T.W. Tong

Summary

Run

of e x p e r i m e n t a l Nickel Foametal,

Ra

S

Rc

Vol. 15, No. 2

TABLE ] conditions and results A=5, Da=l. O 4 8 x l O -4.

Ra o

Pr

0 0 0 0 0 0 0

O0 O0 O0 O0 O0 O0 O0

0 0 0 1 1 2 2

61 80 95 25 66 30 34

Nu

Th

Tc

(oC)

(°C)

6.79 6.96 6.79 6.68 6.86 6.75 6.70

22.52 22.13 23.33 24.65 24.78 26.90 27.24

19.48 18.05 18.62 18.53 16.44 15.56 15.76

5.98(6.23) a 6.75(6.68) a 6.87(6.94) a 7.67(7.45) a 8.23(8.01) a 8.79(8.68) a 8.99(8.72) a

xlO - s

1 2 3 4 5 6 7

for

8 9 lO ll 12

0.25 0.25 0.25 0.25 0.25

0.79 1.09 1.48 1.91 2.22

0.696 0.695 0.694 0.693 0.693

57.57 79.40 107.62 138.67 161.17

6.78 6.85 7.01 7.li 7.03

23.01 23.40 23.64 24.21 25.41

19.09 17.93 16.06 14.25 13.99

2.72(2.84) 2.95(2.97) 3.15(3.10) 3.25 3.38

13 14 15 16 17

0.50 0.50 0.50 0.50 0.50

0.53 0.97 1.48 1.72 2.03

0.694 0.694 0.694 0.694 0.693

38.32 70.62 107.62 125.10 147.38

6.94 7.00 6.96 6.96 7.13

21.54 22.38 23.86 24.38 24.45

18.86 17.41 16.33 15.62 13.88

1.82(1.83) b 2.08(2.03) 0 2.19(2.25) b 2.31 2.44

18 19 20 21 22

0.75 0.75 0.75 0.75 0.75

0.99 1.36 1.65 2.10 2.38

0.695 0.694 0.693 0.695 0.696

71.89 98.90 I19.83 152.99 173.68

6.86 7.00 7.05 7.11 6.98

23.12 23.35 23.85 24.75 25.97

18.17 16.38 15.33 13.82 13.83

1.85 2.01 2.20 2.38 2.55

23 24 25 26 27 28

1.00 l . O0 l . O0 1.O0 1.O0 1.00

0 I 1 l 1 2

0 0 0 0 0 0

57.33 79.53 102.12 i15.98 117.43 160.97

6.79 6.79 6.71 6.71 6.71 6.85

22.95 23.69 24.82 25.26 25.36 26.17

19.04 18.25 17.95 17.42 17.43 15.11

2.57(2.76) 3.02(3.25) 3.45(3.66) 3.76(3.91) 3.88(3.94) 4.39(4.58)

79 09 40 59 61 21

a Results

from

b Results

obtained

c Results

from

696 696 696 696 696 695

correlation from

the

boundary-layer

for

laminar

computer solution

flow

program [5]

0 b b

c c c c c c

[4] reported

in

Ref.

2

Vol. 15, NO. 2

CONVECgION IN PORfTJS ~ L O S U R E S

Summary

Run

S

209

TABLE 2 of e x p e r i m e n t a l c o n d i t i o n s and r e s u l t s the f o a m p l a s t i c , A=5, D a = 9 . 1 7 5 x l O -s.

Ra

Rc

Ra o

Pr

Th

Tc

(oc)

(oC)

6.15 6.34 6.21 6.15

26.67 27.18 28.58 29.14

23.20 20.24 20.68 20.29

xl0-V

for

Nu

11.41(12.02) 13.29(13.07) 13.89(13.86) 14.13(14.47)

1 2 3 4

0.00 0.00 0.00 0.00

0.87 1.62 1.98 2.29

5 6 7 8

0.25 0.25 0.25 0.25

0.87 1.46 2.00 2.61

1.018 1.018 1.018 1.018

812.9 1364.2 1868.8 2438.7

6.16 6.16 6.21 6.18

26.52 27.72 28.60 29.97

23.05 21.88 20.44 19.45

4.37 5.15 5.95 6.88

9 10 11 12 13

0.50 0.50 0.50 0.50 0.50

0.83 1.39 1.46 1.91 2.62

1.018 1.018 1.018 1.018 1.018

775.5 1298.8 1364.2 1784.7 2448.1

6.16 6.14 6.18 6.18 6.19

26.52 27.69 27.66 28.54 29.90

23.05 22.18 21.74 20.80 19.26

3.69 4.70 4.90 5.43 6.33

14 15 16 17 18 19 20

0.75 0.75 0.75 0.75 0.75 0.75 0.75

0.84 0.85 1.37 1.86 1.95 2.47 2.58

1.018 1.018 1.018 1.018 1.018 1.018 1.018

784.9 794.2 1280.1 1738.0 1822.0 2307.9 2410.7

6.15 6.16 6.13 6.24 6.19 6.21 6.16

26.60 26.51 27.66 28.17 28.59 29.54 29.98

23.23 23.11 22.23 20.51 20.67 19.45 19.62

4.09 4.08 4.71 5.48 5.47 5.95 6.29

21 22 23 24 25 26

1.00 1.00 1.00 1.00 1.00 1.00

0.83 1.38 1.90 1.97 2.61 2.65

1.018 1.018 1.018 1.018 1.018 1.018

775.5 1289.4 1775.3 1840.7 2438.7 2476.1

6.15 6.15 6.19 6.23 6.21 6.22

26.53 27.66 28.45 28.44 29.85 29.90

23.24 22.16 20.72 20.32 19.20 19.02

laminar

flow

a Result

from

correlation correlation

for

b Results

from

for

c Results

from boundary-layer

transition solution

[5]

[4] flow

[4]

a b b b

6.32(6.77) c 7.93(8.68) c 9.37(10.14) c 9.52(10.31) c 10.92(11.79) c 10.84(11.90) c

210

for

S.B. Sathe and T.W. Tong

the

foam plastic.

The

existence

Vol. 15, No. 2

of a m i n i m u m

is e v i d e n t

again. The given

expl

in d e t a i l

Foametal, combined

increase

transfer

in the

because

decrease trends

increased

was

heat

in c o n v e c t i v e

change

transfer

1 (a)

and

the

the

region

(b)

transfer

foam

was

fluid

are

in

heat

plastic,

increased

increase

in the

the

in c o n v e c t i v e

of

been

of N i c k e l

when

heat

as S was

when

in the p o r o u s

in Figs.

for A = I O

In the c a s e

Nu has

case

Nu o c 9 ~ r e d

by the d e c r e a s e

occured

heat

in the

and c o n d u c t i v e

did not

transfer

in

offset

by

region.

consistent

the The

with

those

[I].

A correlation relevant

Basically,

region.

The m i n i m u m

of the m i n i m u m

the m i n i n u m

offset

fluid

displayed

observed

[i,2].

component

Rc=l.

convective

the e x i s t e n c e

in c o n v e c t i v e

region

conductive

for

before

as S was

the p o r o u s

the

~tion

has

governing

been

obtained

parameters.

The

for Nu form

in t e r m s

of the

of the

correlation

is

(2)

Nu = i / C + a o N a o a x f 1 ( S ) + b o R a b l f 2 ( S ) where C = l+S(Nc-l), The

constants

separately software

for

the

components increases

NLIN

They

1/C t e r m

component.

a I,

Nickel

package

respectively. that

a o,

fl(S)

to fit

b o,

RHS

and

the p o r o u s

and

of Eq.

the

and

foam

fluid

the

plastic

in T a b l e s

(2)

terms

= ( l - S b2 ) b3

b 2 and b 3 w e r e

in T a b l e

third

to one,

b I,

the d a t a

given

on the

from zero

a 3,

Foametal

are

The s e c o n d for

a 2,

f2 (S)

: l - ( l - S a 2 ) a3'

3.

It can be

the

regions,

convective

using

I and

represents are

evaluated the

2, verified

the

[2]

conductive

convective

respectively.

component

in the

As S

Vol. 15, NO. 2

C~ION

TABLE 3 in Equation

Constants ao

aI

IN POROUS ~I~CLOSU~

az

211

(2).

aa

bo

b z

b z ........ b.3

Nickel Fosmetal

0.0288

0.9190

0.3000

0.1129

0.~26

0.2884

0.3000

1.2759

foam plastic

0.1314

0.5545

0.2000

0.2294

0.1632

0.2598

0.2000

1.1308

porous fluid and data

region region

(b).

increases

while

decreases.

For Nickel

is 7.2 percent.

the data points

the c o n v e c t i v e

Equation

Foametal,

the

(2)

is p l o t t e d

largest

Eq.

in the

in Figs.

difference

For the foam plastic,

to within

component

(2)

1 (a)

from the

agrees w i t h

7.7 percent. Conclusions

Experimental with

a porous

was 5.

Nickel

porous

medium.

from

zero

minimized

m e d i u m were Foametal

were

of the porous showed

of the porous

partially

The e n c l o s u r e

and a foam p l a s t i c

The results

as a function

for e n c l o s u r e s

performed.

The t h i c k n e s s

to one.

correlation governing

measurements

that heat region

aspect

employed

region

filled ratio

as the

was v a r i e d

transfer

thickness.

has been

developed

expressing

Nu in terms

parameters

including

the porous

region

could be A

of the

thickness.

Acknowledgement The authors National

Science

a@e grateful Foundation

for the support

un~der grant

of this work by the

MEA 86-96062.

References i.

S. B. Sathe, T. W. Tong and M. A. Faruque, AIAA J o u r n a l T h e r m o p h y s i c s and Heat Transfer, i, 260 (1987).

2.

T. W. Tong and E. Subramanian, and Fluid Flow, 7, 3~(1986).

International

Journal

of

of Heat

212

S.B. Sathe and T.W. Tong

Vol. 15, No. 2

Introduction

3.

F. P. I n c r o p e r a and D. P. DeWitt, Transfer, Wiley, New York (1985).

4.

N. Seki, S. F u k u s a k o and H. Mechanics, 84, 695 (1978).

5.

T. W. Tong and E. Subramanian, I n t e r n a t i o n a l and Mass Transfer, 28, 563 (1985).

Inaba,

Journal

to Heat

of Fluid

Journal

of Heat