Forced convection in a parallel plate channel partially filled with a high porosity medium

0735-1933/92 $5.00 + .00 Copyrighte1992 Pergamon Press plc

INT. COMM. HEAT MASS TRANSFER VoL 19, pp. 263-273, 1992 Printed in the USA

FORCEDCONVECTIONIN A PARAI.IFI. PLATE CHANNELPARTIALLYFIIJ .El) WITH A HIGHPOROSITYMEDIUM

J.Y. Jang and J.L Chen Dept. of Mechanical Engineering National Cheng-Kung University Tainan, Taiwan 70101 R. O. C. (Communicated by I. Tanasawa and R. Echigo) ABSTRACT This paper presents a numerical study of non-Darcy effects on the fully developed forced convection parallel plate channel flow partially filled with a porous medium. The Navier-Stokes equations govern the fluid motion in the fluid region, while Darcy-Brinkman-Forchheimer model is assumed to hold within the porous media. The effect of thermal dispersion in the porous matrix is also considered. Inclusion of these effects significantly alters the velocity and temperature profile for those predicted by Darcy and DarcyBrinkman models. It is shown that the Nusselt number strongly depends on the open space thickness ratio S, and a critical value of the porous layer thickness exists at which the Nusselt number reaches a minimum. The critical value of the porous layer thickness shifts to lower S as the Darcy number decreases. In addition, it is found that thermal dispersion effect is only significant for small values of open space thickness ratio S. Introduction The existence of a fluid layer adjacent to a layer of fluid saturated porous medium is a common occurance in both geophysical and engineering enviroments. There are numerous industrial situations that require analyses of the interaction between a fluid layer and fluid saturated porous medium, such as porous bearing, porous heat pipe, blood flow in lungs or in arteries. Modeling of such systems requires an understanding of the convective interaction between the fluid layer and the adjacent permeable systems. At present there appears to be very limited research on convection in composite fluid and porous layers. A recent review of this subject was presented by Prasad [1]. Most of the existing studies focus on the problem of natural convection. Somerton and Catton [2] studied the stability condition for a fluid-superposed p o r o u s layer isothermally heated from below. Poulikakos et al. [3] obtained numerical solutions for a 263

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J.Y. Jang and J.L. Chen

Vol. 19, No. 2

t w o - l a y e r c o m p o s i t e system h e a t e d from below. Taslim a n d Narusawa {4] r e p o r t e d the stability analyses of a fluid (porous) layer sandwiched between two p o r o u s (fluid) layers h e a t e d f r o m below. B e c k e r m a n n et al. [5] i n v e s t i g a t e d the n a t u r a l c o n v e c t i o n heat t r a n s f e r in a v e r t i c a l e n c l o s u r e that is p a r t i a l l y filled with a vertical l a y e r of a fluids a t u r a t e d p o r o u s m e d i u m . Campos et al. [6] s t u d i e d the n a t u r a l c o n v e c t i o n in a vertical a n n u l a r e n c l o s u r e d i v i d e d into a fluid region and p o r o u s region. For t h e e x t e r n a l f o r c e d c o n v e c t i o n in c o m p o s i t e f l u i d a n d p o r o u s l a y e r s , a n u m e r i c a l s t u d y of f o r c e d c o n v e c t i o n o v e r an e x t e r n a l b o u n d a r y lined with a p o r o u s s u b s t r a t e b a s e d on the Darcy-Brinkman-Forchheimer (DBF) flow m o d e l was i n v e s t i g a t e d b y Vafai a n d Kim [7]. Later, Vafai a n d Kim [8] also p r e s e n t e d an exact solution of a fully d e v e l o p e d flow o v e r a flat plate, where a fluid l a y e r is s a n d w i c h e d b e t w e e n a p o r o u s m e d i u m from a b o v e a n d an external b o u n d a r y from below. As for the i n t e r n a l f o r c e d c o n v e c t i o n flow, R u d r a i a h [9] s t u d i e d a p a r a l l e l p l a t e c h a n n e l flow b o u n d e d below b y a p o r o u s l a y e r o f finite thickness a n d a b o v e b y a rigid m o v i n g p l a t e with a u n i f o r m velocity. Poulikakos a n d K a z m i e r c z a k [10] p r e s e n t e d an exact solution of the f o r c e d convection in a d u c t p a r t i a l l y filled with a p o r o u s m e d i u m . Both o f t h e s e works in [9] a n d [10] are b a s e d on the D a r c y - B r i n k m a n (DB) m o d e l . However, at h i g h e r flow rates o r in high p e r m e a b i l i t y p o r o u s m e d i a , t h e i n e r t i a a n d t h e r m a l d i s p e r s i o n effects not i n c l u d e d in DB m o d e l m a y b e c o m e significant. The nonDarcy b e h a v i o r on the forced convection in a parallel plate channel p a r t i a l l y filled with a p o r o u s m e d i u m s e e m s n o t to have b e e n i n v e s t i g a t e d . This m o t i v a t e d t h e p r e s e n t investigation. Unlike the work in [10], the p r e s e n t f o r m u l a t i o n d o e s n o t a d m i t an exact solution. Numerical solutions a r e p r e s e n t e d for Darcy n u m b e r f r o m 0.001 to 0.05 a n d for o p e n space thickness ratio S from S=0 ( p u r e p o r o u s layer) to S=1 ( p u r e fluid layer), a n d c o m p a r e d with those o b t a i n e d from the Darcy a n d Darcy-Brinkman models [10]. Mathematical formulation The g e o m e t r y of the p r o b l e m u n d e r c o n s i d e r a t i o n is shown in Fig. 1. A p a r a l l e l p l a t e c h a n n e l with a d i s t a n c e of s e p a r a t i o n 2H is p a r t i a l l y filled with a p o r o u s m e d i u m a n d b o u n d e d b y two h o r i z o n t a l walls at c o n s t a n t t e m p e r a t u r e Tw. The following a s s u m p t i o n s a r e m a d e for this s t u d y (1) fluid a n d solid p h a s e s a r e in local t h e r m a l e q u i l i b r i u m (2) b o t h t h e t e m p e r a t u r e a n d the flow fields in the c h a n n e l a r e fully d e v e l o p e d (3) viscous d i s s i p a t i o n a n d axial c o n d u c t i o n t e r m s in the e n e r g y e q u a t i o n h a v e b e e n n e g l e c t e d (4) l a m i n a r flow is a s s u m e d in the fluid r e g i o n (5) t h e wall c h a n n e l i n g effect in the p o r o u s layer is neglected. (6) the effective viscosity of the p o r o u s l a y e r is equal to the d y n a m i c viscosity of the fluid layer.

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CONVECTION IN A PARA!.I.F.L PLATE CHANNEL

265

For the fluid region, the m o m e n t u m and energy equations are:

aT u

aZr

a-x" = c t f -ay - 2

(2)

For the porous region, the governing equations which include the boundary, inertia and thermal dispersion effects can be written as dp ~_O2u ~tf pfc u2 O - - ~ - x + ttf aY2 ~ - u - . ~ aT --

u Ox

(3)

a2'I` =

- -

(4)

ct eft aY2

where K is the permeability and c is the flow inertia p a r a m e t e r [11]; the effective thermal diffusivity o~ffof a saturated porous medium can be expressed as [12]: o~rf=So+ o~, where ~ is the stagnant diffusivity and oh is the diffusivity due to thermal dispersion. Here we adopt the following thermal dispersion model proposed by Cheng [13], that is Ctd= 0.04

1-E

ud

(5)

The corresponding b o u n d a r y conditions for the above equations are as follows:

y=0,

0u=0 0y '

0T=0 ay

y=s,

T ] y..~= Tly.s÷

(k,fr~-y) y.÷ ay ' y.,-" ~

(6)

u ] y.,~" uly.~"

y-H.

u=0,

T-Tw

These conditions express no slip of velocity, temperature and the continuity of shear stress and heat flux~ Due to the s y m m e t r y of the geometry, the temperature and velocity gradients vanish at the center line of the channel. The conservation equations can be recast into dimensionless forms by introducing the following dimensionless variables.

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J.Y. Jang and J.L. Chen

* X

x

*

Y

= "~" , y = "~ s

S

u*=

,

Nu = h(4H) -'~

H

T -- Z w

u

Uc

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O= -

,

Tm-Tw

4UmH2 dTm ctf(Tw- Tin) dx

(~.) -4k*

y*.i

(7)

where Uc = HX(-~x)/l~fis the characteristic velocity, Um is the mean axial velocity, and S is the o p e n space thickness ratio. The other symbols are defined in the Nomenclature. The dimensionless m o m e n t u m and energy equations are Huid: = - 1

dy.2

-T

(8)

e - ay,2

(9)

Porous: 02u*

~ - u s2

1 u*

ay*2- ' ~ Nu ( ~ ) O 4 k* where Da = K/H z, Rec =

-1

(10)

0_..~I ( l + Ds * O0 Oy

7")7]

(1,)

CUc~fK ^ ~. 1-~ ucd vf and Ds = u.u,~-7"---@- represent the Darcy number, inertial

Reynolds n u m b e r and dispersion parameter, respectively; k* (=ko/kf) is the ratio o f s t a g n a n t t h e r m a l c o n d u c t i v i t y of the p o r o u s m e d i u m to that of the fluid, which is assumed unity in this study. The inertial Reynolds n u m b e r appears due to the inclusion of the inertia term. Thus, omitting the inertia term is equivalent to assuming that the inertial Reynolds n u m b e r is equal to zero. The corresponding dimensionless b o u n d a r y conditions are as follows y*= 0 ,

y*-S

,

0

ay

0__0.. 0 Oy

Oly*=S- -Oly*.S ÷

(12) Oy

Oy

u* [ y'-S- =u* ly*-S÷

Oy y*= I

I y*..S- =

U*= 0 0

=0

oy

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CONVECTION IN A PARALLEL PLATE CHANNEL

267

Method of solution The equations derived above are solved numerically by using a sixth order RungeKutta variable step size integration routine and the shooting technique. For the fluid region, the velocity profile could be obtained theoretically by integrating eq. (8), that is u* = - 0.Sy.2 + m, where m is an integration constant to be determined. Then the velocity gradient at the fluid/porous interface is du*/dy*= - y*. The momentum equation for the porous layer can be solved as follows. By guessing a value of velocity gradient at the wall, it is possible to begin the Runge-Kutta integration starting from the wall and proceeding toward the fluid/porous interface. After the velocity is obtained, the continuity of velocity gradient at the interface is checked. The above procedure is repeated until the guessed value of velocity gradient yields a velocity field that satisfies the boundary condition at the porous/fluid interface (error less than 10-4). After the velocity field in the porous layer is obtained, the integration constant m can be determined. For the system of energy equation and its boundary conditions, it constitutes a differential eigenvalue problem with Nu being the eigenvalue. Similar to the a b o v e procedures, first Nu is guessed. Then, a vanishing temperature gradient at the center line of the channel (y*= 0) is checked (error less than 10"4). In the limiting case of Rec=Ds=O, the present results are in good agreement with the exact solutions obtained by using the DB model [10]. Results and discussion Numerical results for the velocity profiles and the Nusselt numbers are presented for different values of Darcy number Da (Da=0.001, 0.01 and 0.05) and for open space thickness ratio S from S=0 (porous layer alone) to S=I (fluid layer alone). It should be noted that using the present realistic flow model (DBF with thermal dispersion effect) removes some of the arbitrariness in varying the parameters affecting the velocity and t e m p e r a t u r e fields. Once the nature of the porous matrix is decided, the relative contributions of the Darcy number (Da), inertia Reynolds number (Rec) and thermal dispersion parameter (Ds) are practically fixed. Water is taken as the fluid m e d i u m , and the data and the various parameters used in the present study are listed in Table 1. Graphical representations of the velocity profiles across the channel for open space thickness ratio S=0.4 are presented in Figs. 2-4 for Darcy number Da=0.05, 0.01 and 0.001, respectively. Since the velocity profiles are symmetric with respect to the center line of the channel, only half of the profiles are shown in the figures. Also shown in these figures are the results obtained by using the Darcy flow model and Darcy-Brinkman (DB) model [10]. It is seen that, as would be expected, both the Darcy and Darcy-Brinkman

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m o d e l s o v e r e s t i m a t e the velocities (or mass flow rates) in the fluid a n d p o r o u s layers for a given p r e s s u r e g r a d i e n t (dp/dx=O.02 N/m3). As the inertia effect is c o n s i d e r e d (DBF model), the m a g n i t u d e of velocity decreases. This is because the form d r a g of the p o r o u s m e d i u m is i n c r e a s e d when the inertia effect is included, it is also o b s e r v e d that, as the Da d e c r e a s e s , b o t h the velocities in the fluid a n d p o r o u s layers decrease. This is d u e to t h e fact t h a t f o r small Da t h e D a r c y r e s i s t a n c e to t h e flow in t h e p o r o u s m e d i u m increases, so t h e velocity decreases. Fig. 4 indicates t h a t the effects of t h e B r i n k m a n friction a n d flow i n e r t i a can be neglected when the Da is small (Da=O.O01). The Nusselt n u m b e r s Nu versus open space thickness ratio S a r e i l l u s t r a t e d in Figs. 5-7 for Da=O.05, 0.01 a n d 0.001, respectively. T h e r e are f o u r r e s u l t s shown in e a c h figure : t h e D a r c y model, DB model, DBF m o d e l w i t h o u t t h e r m a l d i s p e r s i o n , a n d DBF m o d e l with t h e r m a l dispersion. It is seen that the Nusselt n u m b e r s t r o n g l y d e p e n d s on S, a n d a critical v a l u e o f the p o r o u s l a y e r thickness exists at which t h e Nusselt n u m b e r r e a c h e s a m i n i m u m . Also, it is shown that neglecting the i n e r t i a effect [10] c a n l e a d to serious e r r o r s for t h e Nusselt n u m b e r calculations. The n u m e r i c a l results i n d i c a t e t h a t t h e t h e r m a l d i s p e r s i o n effect t e n d s to e n h a n c e the h e a t t r a n s f e r d u e to t h e m o r e v i g o r o u s mixing of the fluid, a n d its effect is only significant for small values of S (large p o r o u s l a y e r thickness). Finally, a close look at Figs. 5-7 indicates that the critical value of the p o r o u s l a y e r thickness shifts to lower S as the Darcy n u m b e r decreases. Conclusions The p r o b l e m of f o r c e d c o n v e c t i o n in a c h a n n e l p a r t i a l l y filled with a p o r o u s m e d i u m a n d b o u n d e d b y two p a r a l l e l plates is a n a l y z e d n u m e r i c a l l y . It is shown t h a t b o t h t h e Darcy a n d Darcy-Brinkman models [10] overestimate the velocities (or mass flow rates) in t h e fluid a n d p o r o u s l a y e r s for a given p r e s s u r e gradient. The n u m e r i c a l results d e m o n s t r a t e t h a t n e g l e c t i n g the i n e r t i a effect [10] can l e a d to s e r i o u s e r r o r s for t h e Nusselt n u m b e r calculations. The t h e r m a l d i s p e r s i o n effect i n c r e a s e s t h e t e m p e r a t u r e g r a d i e n t a d j a c e n t to t h e wall resulting in the e n h a n c e m e n t o f c o n v e c t i v e h e a t transfer, a n d its effect is o n l y significant for small values of o p e n space t h i c k n e s s r a t i o S. T h e p r e s e n t realistic m o d e l also indicates t h a t the Nusselt n u m b e r curve exhibits a m i n i m u m a r o u n d S=0.2 to 0.4 d e p e n d i n g o n the value of the Darcy n u m b e r . This critical value of t h e p o r o u s l a y e r thickness shifts to the lower S as the Darcy n u m b e r decreases. Nomenclature c

inertial coefficient

d

particle diameter

Da

Darcy number, K/H 2

VoL 19, No. 2

CONVECrION IN A PARAI J.~L PLATE CHANNEL 1-~ ucd ctf

Ds

dispersion parameter, 0.04

h

heat transfer coefficient

H

half of parallel plate separation distance

K

permeability of the porous medium effective thermal conductivity of the porous medium

~ff kf

thermal conductivity of the fluid stagnant conductivity

k*

thermal conductivity ratio,

Nu

Nusselt number, h.(4H)/kf

P Re Rec

pressure fluid Reynolds number, uc I-I/vf inertial Reynolds number, cuc'fK/vf

s

thickness of the fluid region

S

open space thickness ratio, s/H

T

temperature mixed mean fluid temperature

Tm Tw u

ko,/k f

wall temperature

Urn

axial velocity mean axial velocity

Uc

characteristic velocity, tt2(-~xx)/~f

x

axial coordinate

Y otf

coordinate perpendicular to x axis thermal diffusivity of fluid

oteff

effective thermal diffusivity

Oto

stagnant diffusivity

Otd

diffusivity due to thermal dispersion

¢

porosity

0

dimensionless temperature, (T - Tw)/(Tm - Tw) dynamic viscosity of the fluid

i,tf vf

of

kinematic viscosity of the fluid fluid density Superscripts dimensionless quantity References

1. V. Prasad, NATO Advanced Study Institute on Convective Heat and Mass Transfer in Porous Medium, Izmir, Turkey (1990). 2. C.W. Somerton and I. Catton, J. Heat Transfer, 104, 160 (1982).

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J.Y. Jang and J.L. Chen

Vol. 19, No. 2

3. D. Poulikakos, A. Bejan, B. Selimos and K.R. Blake, Int. J. Heat Fluid Flow, L 109 (1986). 4. M.E. Taslim and U. Narusawa, J. Heat Transfer, 111, 357 (1989). 5. C. Beckermann, S. Ramadhyani and R. Viskanta, J. Heat Transfer, 1 0 9 , 3 6 3 (1987). 6. H. Campos, J.C. Morales and U. Lacoa, Int. Comm. Heat Mass Transfer, ~ (1990).

343

7. K. Vafai and S.J. Kim, J. Heat Transfer, 112,701 (1990). 8. K, Vafai and S.J. Kim, Int. J. Heat and Fluid Flow, 11, 254 (1990). 9. N. Rudraiah, J. Heat Transfer, ~

322 (1985).

10. D. Poulikakos and M. Kazmierczak, J. Heat Transfer, 1 0 9 , 6 5 3 (1987). 11. S. Ergun, Chem. Engng. Progress, 48, 89 (1952). 12. J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York (1972). 13. C.T. Hsu and P. Cheng, Int. J. Heat Mass Transfer, 33, 1587 (1990).

Table 1. The data and the various parameters used in the present study 10-2 0.02 1.12 x 10 4 1.42 x 10 "7 0.5 2250 90.93 31.25

5 x 10- 2 0.02 1.12 x 10-3 1.42 x 10-7 0.6 2250 154.68 16.67

Da dp/dx (N/m 3) ~f (Kg/m's)

ar (reX/s) Re Rec Ds'

10- :~

0.02 1.12 x 10- 3 1.42 x 10-7 0.4 2250 40.19 56.25

Y Tw

porous layer fluid layer U

Tw Fig. 1 Schematic of the physical model and coordinate system

Vol.19,No.2

C O N V E C T I O N IN 1.00

~'

~'ql %

APARALLEL PLATE CHANNEL

I

I

..... Darcy model

%

o-o-o DB model[10] ~

%

o,o

DBF model

% b

%

\ %

0.60

W,%%

y*

~'~, %,,,.

0.40 %% M~

0.20

0.00 0 u*

Fig.2 Velocity profile for S=0.4,Da=5x10 -2

1.00

i I

0.80

..... Darcy model

~|: ~

o-o-o DB model [10 DBF model

!

Y*

0.80

0.4O 0.20

1q~l

"~

l ~

b % %

°'°°o.~

......

'o'.~ ......

'o'.6~ ......

'o'.4~ ..... •

0:~,

u* Fig.3 Velocity profile

for S=0.4,Da=10 -2

271

272

J.Y. Jang and J.L. Chen

Vol. 19, No. 2

..... Darcy model o-o-o DB model

0.M

[10]

DBF model

0.60 y*

0.40

0.20

0.%.

0.2

O.

Fig.4 V e l o c i t y

O.

u*

profile

O.

O. 0

for S = 0 . 4 , D a = 1 0

-3

12.00

..... Darcy model

11,00

o-o-o DB model

[10]

Q-D-D DBF w i t h o u t

10.00

dispersion

DBF with d i s p e r s i o n

Nu

7.00

O-.O~Q

~) . . . . . .

0~0 ......

"4~

0:40 -

O,IJO

j

0,80

1.00

S

Fig.5

Nusselt

number

versus

S for D a = 5 x 1 0 -2

VoL 19, No. 2

CONVECrION IN A PAR.A!J.F.L PLATE CHANNEL

.....

Darcy model

o - o - o DB m o d e l [ 1 0 ]

10.00

0-0-0

DBF w i t h o u t

dispersion

DBF w i t h d i s p e r s i o n

Nu

7.00 s %.7

. . . . . b 3 b . . . . . . b:,i~ . . . . . . b : ~ . . . . . . b : ~ . . . . . . S Fig.

6 Nusselt

number

versus

S for

i:Je,

D a = 1 0 -2

11.00

10.00

.....

Darcy model

o-o-o

DB m o d e l

O-O-D

DBF w i t h o u t

[10] dispersion

DBF w i t h d i s p e r s i o n

Nu

7.00

,00 . . . . . . 0 ~ o

0.40

Fig.7

number

Nusselt

:S

o.eo versus

o.ao S for

1 .~ Da=10

-3

273