Conjugate free convection from long vertical plate fins in a non-Newtonian fluid-saturated porous medium

Conjugate free convection from long vertical plate fins in a non-Newtonian fluid-saturated porous medium

Pergamon International Communicationsin Heat and Mass Transfer, Vol.21, No. 2, pp, 297-305, 1994 Copyright© 1994 ElsevierScieaw.eLtd Printed in the U...

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International Communicationsin Heat and Mass Transfer, Vol.21, No. 2, pp, 297-305, 1994 Copyright© 1994 ElsevierScieaw.eLtd Printed in the USA.All fights re,served 0735-1933/94 $6.00 + .00

CONJUGATE FREE CONVECTION IN A NON-NEWTONIAN

FROM LONG VERTICAL PLATE FINS

FLUID-SATURATED

I. Faculty

POROUS MEDIUM

Pop

of M a t h e m a t i c s

University

of Cluj

CP 235, Romania

A. Nakayama Dept.

of Energy and M e c h a n i c a l

ShizuQka U n i v e r s i t y ,

Hamamatsu,

Engineering 432 J a p a n

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

ABSTRACT The p r o b le m of c o n j u g a t e f r e e c o n v e c t i o n from long v e r t i c a l f i n s in a n o n - N e w t o n i a n f l u i d - s a t u r a t e d p o r o u s medium has been i n v e s t i g a t e d e x p l o i t i n g t h e boundary l a y e r a p p r o x i m a t i o n . The g o v e r n i n g e q u a t i o n s based on t h e p o w e r - law model a p p r o p r i a t e f o r t h e Darcy flow a r e s o l v e d e x p l o i t i n g an i n t e g r a l method. The e f f e c t s of t h e f i n shape e x p o n e n t and p o w e r - l a w i n d ex p a r a m e t e r s on t h e flow and heat transfer characteristics ar e d i s c u s s e d .

INTRODUCTION

The s t u d i e s vertical source

plate

on c o n j u g a t e

have been r e s t r i c t e d

importance

c o n v e c t i o n from an i n f i n i t e l y

f i n embedded in a p o r o u s medium and h e a t e d

However, t h e p r e d i c t i o n Newtonian f l u i d

free

o n l y to t h e D a r c i a n

of f l o w and h e a t

transfer

in a p o r o u s medium a d j a c e n t

in a number of g e o p h y s i c a l

fluid

by a p l a n e

in t h e p a s t

characteristics

to a h e a t e d

surface

and c h e m i c a l e n g i n e e r i n g

297

long heat

[ 1,

2 ].

of a noni s of applications

298

I. Pop and A. Nakayama

Vol. 21, No. 2

such as petroleum d r i l l i n g s and polymer solutions. The aim of the present paper is to study the free convection of a non Newtonian fluid from a long, v e r t i c a l palte fin embedded in a porous medium using the power-law mode] proposed by Christopher and Middleman [ 3 ], and Dharmadhikari and Kale [ 4 ]. T h i s model was successfully used in several recent studies on convective flow in a porous medium [ 5-11 ]. The thin fin approximation for the fin is invoked and the boundary layer approximation for the convective flo~ outside the fin is applied. To attack the set of highly nonlinear governing equations, an integral treatment is proposed along the lines of Pop et al.

i I ] and Nakayama and Koyama [ 2 ]. Closed

form expressions as functions of the fin shape exponent m and power-law index n are obtained

for the isotherms and local heat t r a n s f e r rate from

a fin. These r e s u l t s extend the findings in Refs.

[ I, 2 ] by accounting

for the non-Newtonian nature of the fluid.

ANALYSIS

The p h y s i c a l the

vertical

the

horizontal

chosen will

problem

coordinate coordinate

such that

the

be d e t e r m i n e d

i s T~,, w h i c h model t h e

under

is

buoyancy

later,

flo~

Darcy momentum e q u a t i o n

along

normal

distance

higher

consideration

located

the

to t h e

fin.

from t h e o r i g i n

?he p r e s c r i b e d

than

that

of t h e

shown in F i g .

to the

surrounding

non-Newtonian so t h a t

it

of t h e

The o r i g i n

temperature

of t h e

modified

is

center-line

fluid

of t h e

fin at

i s x, the

porous in t h e

is pertinent

1, w h e r e

fin

coordinate

is

whose v a l u e

base

of the

medium at porous

for

x is

and y i s

fiH

T~. To

medium,

the

power-law

fluids [ 3, 4 ] will be used. In addition, we assume that the porous medium and the fluid are in thermodynamic equilibrium and that the usual Boussinesq approximation holds. Based on these assumptions,

the boundary layer equat-

ions for the momentum and energy can be written as

o~

u

o~

v

+

a

x

= ~

y

0

(1)

Vol. 21, No. 2

FINS IN A N O N - N E W T O N I A N P O R O U S M E D I U M

~z ~

--

299

n

( - U )

pg,8(

:

T - T~ )

( 2 )

K'

and 0 U

T

0

~

modified

the

:

O

power-law

permeability

Koyama [ 9 ] b a s e d The b o u n d a r y

y

-->

y

=

on t h e

the

OY-

~'

is the

power-law

studies

conditions

(3)

Ot

Y

index,

for

~

a~T

V

÷

O x

where n i s

T

for

consistency

fluids

in R e f s . the

index,

as g i v e n

[ 3,

and

K'

is the

by Nakayama and

4 ].

f l o w and t e m p e r a t u r e

at

infinity

are

:

u

=

O,

T

:

T~

(

4a,

b

)

:

v

:

0,

T

=

TF( x )

( 4c,

d )

fin

which

while

0

at

the

fin-fluid

interface

is

t o be d e t e r m i n e d .

( 4d ) s u c h t h a t Under the across

the

follows

fin

that

variable

the

Note t h a t

the thin-fin

thickness

of t h e

thin-fin

approximation,

and h e a t

conduction

the

governing

conductivity

d

along

approximation

the

is used

in Eq.

can be n e g l e c t e d . is

fin

for heat is given

is

no t e m p e r a t u r e

variation

quasi-one-dimensional.

conduction

It

in a t h i n - f i n

of

by

O T

d x

O Y

tF and kF a r e t h e conditions

temperature

there

) + kL - - I

dx

boundary

equation

d TF

fin

in t h e

or t h i c k n e s s

( tFkF

where

w h e r e TF i s t h e

for

half-thickness Eq.

=

0

(5)

y=0

and c o n d u c t i v i t y

of t h e

fin.

The

( 5 ) are

X

=

X~

:

TF

=

T~,

(

6a

)

x

~

oo

:

TF

=

T~

(

6b

)

300

I. P o p a n d A. N a k a y a m a

The p r e s e n t equation

integral

( 3 ) along

d

treatment

with

the

start

with

continuity

S

dx

0 u ( T - T~ ) dy

=

integrating

equation

(~

--

Vol. 21, No. 2

-a

the

energy

1 ) as

T

( 7

~

0

0

y

y=O

I t can e a s i l y be shown that a s i m i l a r i t y s o l u t i o n Is p o s s i b l e when the wall-ambient temperature d i f f e r e n c e v a r i e s according to

T,

(TF

)

x =

( T

where

A

is

satisfied.

a negative Moreover,

be a p p r o x i m a t e d the

thermal

x~

exponent the

to

layer,

iT

be d e t e r m i n e d

temperature

by an e x p o n e n t i a l

boundary

)

)

T~

(8

k

(

profile function

such

within with

that

the

an a r b i t r a r y

( 6b

) is

medium scale

6

may for

as

T~. )

y :

exp

(

)

(T;-T,)

( 9

6

such that Eq. ( 4a ) tc ( 4d ) are a u t o m a t i c a l l y s a t i s f i e d . gqs. ( 2 ),

Eq.

porous

Substitutlng

( 8 ) and ( 9 ) into ( 7 ), and carrying out i n t e g r a t i o n and

d i f f e r e n c i a t i o n , the following expression can be obtained:

2( ( d/

x )

Rax

l + n

: (1+2n) A

( 10

)

( 11

)

n

where pK'gB( Ra,:

1/n

T~:-T~ ) x ]

= [ /.,/~

(z ~

Vol. 21, No. 2

is the

FINS IN A N O N - N E W T O N I A N P O R O U S M E D I U M

local

Rayleigh

To s o l v e product

the heat

number f o r a p o w e r - l a w f l u i d . conduction

=

where m i s t h e p r e s c r i b e d ( 8 ),

tbk~ ( x/xb

fin

( 5 ),

the conductivity-thickness

-k(

m

+k-

+

(

1/2

n

similarity

( 12 ) a l o n g

1/2

k~Ra:~

:

=

0

( 13 )

x

solution:

1/2 ) Rax~

(

kF t ~

Eq.

yields

]

k~ x~ :

Substituting

2( 1 + n )

t h e following

) Ra×

( 12 )

[

X

k~ x

( 5 ),

-(l+2n)k-

1 ) -

reveals

)m

shape e x p o n e n t .

( 9 ) and ( 10 ) i n t o

tFk~

which

equation

i s assumed to ~e of t h e form

t~kF

with

301

:

(3-2m)[n(l+n)(l+3n-(l+2n)m}]:

k~ t~

(14)

where Raxb : R a x ( x b ) ,

and t h u s ,

t h e unknown s i m i l a r i t y

length

parameter

i s g i v e n by

(3-2m)[n(l+n)(l+3n-(l+2n)m}]l:~kbt~/k~ x~/

: [

] {pK'gB(

Moreover,

2/3

t h e unknown e x p o n e n t

k

=

provided

T~-T~ ) / ~ e * a n }

k

i s g i v e n by

n (3-

2m)

l+3n m

<

(16

3 (

l+2n

( 15

~ 2''

<

)

2

(17

x~

302

I. Pop and A. Nakayama

Vol. 21, No. 2

RESULTS AND DISCUSSION

Relation ( 17 ) shows t h a t s o l u t i o n s of t h i s problem e x i s t for the f i n shape exponent m l e s s than ( l + 3 n ) / ( l + 2 n ) . However, for a tapered f i n with c o n s t a n t thermal c o n d u c t i v i t y , the shape exponent has to s a t i s f y the more r e s t r i c t e d condition m ~

O.

For a s e t of parameters m and n, the local Nusselt number along the f i n , of our primary concern, i s given by

x Nux

0

T

=

x [

(TF-T~)

0

y

n{1+3n-(1+2n)m}I/2

: -y=O

= [

]

~

i/2 Ra~

I 18 )

i + n

Figure 2 shows the heat t r a n s f e r f u n c t i o n Nu./Ra× ~ - as a function of n and m. I t

can be seen t h a t Nux/Rax ; ~ i n c r e a s e s with the power-law index n, as

in the case of non-conjugate f r e e convection in Ref. [ 9 ]. As e a s i l y can be expected from Eq. ( 18 ), Nu×/Ra~ ~ -,

for fixed n,

is higher for n e g a t i v e m

than for p o s i t i v e m, and vanishes at m = ( l + 3 n ) / ( l + 2 n ) . S u b s t i t u t i o n of Eqs.

( 8 ),

( 14 ) and ( 15 ) into ( 9 ) y i e l d s the

following closed-form e x p r e s s i o n for the isotherms:

y

i/2 Rax~

x~

x 2-m : - (--) x,

I + n [

i/2 ]

(T -T~,) In{ ~

n{1+3n-(1+2n)m}

x n(3-2m)

(

(Tr-T,.)

)

}(19)

x~

In order to i l l u s t r a t e the temperature f i e l d within the porous medium, the isotherms are obtained for the case of an i n f i n i t e l y long f i n with c o n s t a n t thermal c o n d u c t i v i t y and t h i c k n e s s ( i . e . m = 0 ). The isotherms for (T-T.)~ (T~-T~) = 0.01 and 0.05 are generated for n = 0.5,

I and 2 from Eq. ( 19 )

and p l o t t e d in Fig. 3 ~here the e f f e c t of the power index on the temperature f i e l d can be seen. Since (T~-T,)

~

x :~ ~ .....

the temperature d e c r e a s e s

more d r a s t i c a l l y away from the base for higher n, and thus a hot region diminishes away from the base and the thermal boundary layer i s kept comparatively t h i n , whereas the isotherms for the p s u e d o p l a s t i c f l u i d ( i . e . n = 0.5 ) extend deeper in the porous medium.

Vol. 21, No. 2

FINS IN A NON-NEWTONIAN POROUS M E D I U M

303

[--

0

Y

X





e



.~:



oo •

4

• •



.-.

o/ •

,'=









o

e



• •

Iu

o

o

~ o

~

0

!

g

!

Vertical fin in a non-Newtonian f l u i d -



saturated porous medium





o •

• •

Fig. 1



• •

e !

o

°

1.0

0

1/2 (Y/X b ) ~ax b I0 20

,,

]

30

l

J

J I l

m=-O, 1.5

=0

x

; 2.0

x

x/x b

2.5

]~ -4

~\~

n

sil \,\

sil,

\~0.

Ol _

~,oo~\,\

.os

~o.!~ '\ %

oL 0

0°5

n

1.0

1.5

Fig. 2 Local Nusselt number

3.0

I/

i

Fig. 3 Isotherms

\

304

I. Pop and A. Nakayama

Vol. 21, No. 2

NOMENCLATURE g

acceleration

k

thermal

due to g r a v i t y

conductivity

K'

modified permeability

m

fin

n

power-law index

Nu~

local Nusselt numLer

Ra×

local Rayleigh number

of t h e porous media f o r p o w e r - l a w f l u i d s

shape e x p o n e n t

Rax~ local Ralyeigh number evaluated at x~ t

half-thickness of the fin

T

temperature

u,

v Darcian

x,

y boundary

x~

velocities

similarity

layer

coordinates

length

parameter

Greek symbols

a

equivalent thermal d i f f u s i v i t y c o e f f i c i e n t of thermal expansion

6

length scale for thermal boundary layer

L

exponent ~

p

fluid

of th e f i n

consistency

temperature

of t h e n o n - N e w t o n i a n p o w e r - l a w f l u i d

density of the convective fluid

Subscripts

b

base of the fin

e

ambient condition

F

fin condition

p

porous medium

REFERENCES

i.

P o p , I . , Sunada, J. K., Cheng, P. and Minkowycz, W. J . , Conjugate free c o n v e c t i o n from long v e r t i c a l f i n s embedded in a p o r o u s meaium a t h i g h R a y l e i g h number,__i~£. J . Heat Mass T r a n s f e r 28, pp. 1629-1636 ( 1985 ).

2.

Nakayama, A. and Koyama, H., E f f e c t of t h e r m a l s t r a t i f i c a t i o n on f r e e c o n v e c t i o n w i t h i n a p o r o u s medium, .l_ ThprmnDhvs. 1, pp. 282-285 (1987)

Vol. 21, No. 2

FINS IN A NON-NEWTONIAN P O R O U S M E D I U M

305

3.

Christopher. R. V. and Middleman S Power-law flow t~rough a packed tube, Ind. En~nz. Chem. Fundls. 4, pp. 422-426 ( 1965 .

4.

Dharmadhikari, R. V. and Kale, D. D., Flow of non-Newtonian fluids through porous media, Chem. EnCn¢. Sci. 40, pp. 527-529 ( 1985 ).

5.

Chert, H. T. and Chen, C. K., Natural convection of non-Newtonian fluids a b o u t a h o r i z o n t a l s u r f a c e in a p o r o u s medium, J. T e c h n . 109, pp. 119-123 ( 1987 ).

Energy R e s o u r c e s

6.

Chen, H. T. and Chert, C. K., F r e e c o n v e c t i o n flow of n o n - N e w t o n i a n f l u i d s a l o n g a v e r t i c a l p l a t e embedded in a porous medium, J . Heat T r a n s f e r 110, p p . 257-260 ( 1988 ).

7.

Chen, H. T. and Chen, C. K., N a t u r a l c o n v e c t i o n of a n o n - N e w t o n i a n f l u i d ab o u t a h o r i z o n t a l c y l i n d e r and a s p h e r e in a p o r o u s medium, I n t . Comm. Heat. Mass T r a n s f e r 15, pp. 605-614 ( 1988 ).

8.

P o u l i k a k o s , D. and S p a t z , T. L . , Non-Newtonian n a t u r a l c o n v e c t i o n at a m e l t i n g f r o n t in a p e r m e a b l e s o l i d m a t r i x , I n t . Comm. Heat Mass T r a n s f e r 15, pp. 593-603 ( 1988 ).

9.

Nakayama, A. and Koyama, H., B u o y a n c y - i n d u c e d flow of n o n - N e w t o n i a n f l u i d s o v e r a n o n - i s o t h e r m a l body of a r b i t r a r y shape in a f l u i d s a t u r a t e d p o r o u s medium, ADD1. S c i . Hes. 48, pp. 55-70 ( 1991 ).

10. Nakayama, A . , A s i m i l a r i t y s o l u t i o n f o r f r e e c o n v e c t i o n from a p o i n t h e a t s o u r c e embedded in a n o n - N e w t o n i a n f l u i d - s a t u r a t e d p o r o u s medium, .I. H~at T r a n s f e r ( in p r e s s ). 11.

Nakayama, A . , F r e e c o n v e c t i o n from a h o r i z o n t a l l i n e non-Newtonian fluid-saturated porous medium, I n t . . l . ( in p r e s s ).

h e a t s o u r c e in a Heat F l u i d Flow

Received November 22, 1993