Radiation effect on natural convection over a vertical cylinder embedded in porous media

Vol.26, No. 2, pp. 259-267, 1999 Copyright© 1999ElsevierScienceLtd Printedin the USA.All rightsreserved 0735-1933/99IS-seefront matter

Int. Comm. HeatMass Transfer,

Pergamon PII S0735-1933(99)00012-3

RADIATION VERTICAL

EFFECT ON NATURAL CONVECTION CYLINDER EMBEDDED IN POROUS

OVER MEDIA

A

K.A. Yih Air Force Aeronautical and Technical School D e p a r t m e n t of General Course Kangshan, Kaohsiung, Taiwan 90395-2, R.O.C.

(Communicated by J.P. Hartnett and W.J. Minkowycz) ABSTRACT In this p a p e r numerical solutions are presented for the effect of r a d i a t i o n on n a t u r a l convection about an isothermal vertical cylinder e m b e d d e d in a s a t u r a t e d porous medium. These p a r t i a l differential equations are t r a n s f o r m e d into the nonsimilar b o u n d a r y layer equations which are solved by an implicit finite-difference m e t h o d (Keller box m e t h o d ) . Numerical results for the dimensionless t e m p e r a t u r e profiles and the local Nusselt n u m b e r are presented for the transverse curvature p a r a m e t e r (, conductionr a d i a t i o n p a r a m e t e r Rd and surface t e m p e r a t u r e excess ratio H. In general, the local Nusselt n u m b e r increases as the transverse curvature p a r a m e t e r increases. F u r t h e r m o r e , decreasing the c o n d u c t i o n - r a d i a t i o n p a r a m e t e r Rd and increasing surface t e m p e r a t u r e excess ratio H augments the local heat transfer rate. © 1999 ElsevierScienceLtd

Introduction S t u d y of b u o y a n c y - i n d u c e d convection flow and heat transfer in a f l u i d - s a t u r a t e d porous m e d i u m has recently a t t r a c t e d considerable interest because of a n u m b e r of imp o r t a n t energy-related engineering and geophysical a p p l i c a t i o n s such as t h e r m a l insulation of buildings, g e o t h e r m a l engineering, enhanced recovery of p e t r o l e u m resources, filtration processes, g r o u n d water p o l l u t i o n and sensible heat storage beds. Free convection a b o u t a vertical flat plate e m b e d d e d in a porous m e d i u m at high Rayleigh n u m b e r s was analyzed by Cheng and Minkowycz [1]. Na and Pop [2] studied free convection flow p a s t a vertical flat p l a t e m a i n t a i n e d at a nonuniform surface t e m p e r a t u r e e m b e d d e d in a s a t u r a t e d porous m e d i u m and p r e s e n t e d numerical results by employing a 259

260

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two-point finite difference method. G o r l a a n d Z i n a l a b e d i n i [3] s t u d i e d the free convection from a vertical plate with nonuniform surface t e m p e r a t u r e and e m b e d d e d in a s a t u r a t e d porous m e d i u m .

In the aspect of vertical cylinder, Minkowycz and Cheng [4] were the first a u t h o r s to present free convection about a vertical cylinder e m b e d d e d in a porous medium.

In

their p a p e r , the numerical results for variable wall t e m p e r a t u r e are presented and the comparisons between the local similarity and local n o n s i m i l a r i t y m e t h o d s of solution are obtained.

Yiicel [5] employed an implicit finite difference m e t h o d to examine the free

convection about a vertical cylinder in a porous m e d i u m .

K u m a r i et al.

[6] used the

finite-difference (Keller box m e t h o d ) and i m p r o v e d p e r t u b a t i o n solution (Shanks transform a t i o n ) for free convection on a vertical cylinder e m b e d d e d in a s a t u r a t e d porous medium. Merkin [7] investigated the free convection from an i s o t h e r m a l vertical cylinder e m b e d d e d in a s a t u r a t e d porous medium. Bassom a n d Rees [8] e x t e n d e d the work of Merkin [7] to investigate the variable wall t e m p e r a t u r e case. T h e governing equations are also solved numerically using the Keller box m e t h o d .

In the case of n a t u r a l convection, the existence of the large t e m p e r a t u r e difference between the surface and the ambient causes the r a d i a t i o n effect m a y become i m p o r t a n t . Hossain and Pop [9] investigated the effect of r a d i a t i o n on D a r c y ' s buoyancy induced flow along an inclined surface placed in porous m e d i a e m p l o y i n g the implicit finite difference m e t h o d together with Keller box elimination technique. S t e a d y two-dimensional n a t u r a l convection flow through a porous m e d i u m b o u n d e d by a vertical infinite porous plate in the presence of r a d i a t i o n is considered by R a p t i s [10]. In the present analysis, we are concerned with the effect of r a d i a t i o n on the heat transfer characteristics in n a t u r a l convection over an isothermal vertical cylinder e m b e d d e d in porous media.

Analysis Let us consider the problem of the r a d i a t i o n effect on n a t u r a l convection b o u n d a r y layer flow of optically dense viscous incompressible fluids over an isothermal vertical cylinder e m b e d d e d in a s a t u r a t e d porous m e d i u m . T h e wall t e m p e r a t u r e of the vertical cylinder T~ is higher than the ambient t e m p e r a t u r e T ~ . Thus, as a result of the buoyancy force, an upward convective fluid movement is induced. T h e variations of fluid properties are limited to density variations which affect the b u o y a n c y force t e r m only. The governing

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VERTICAL CYLINDER EMBEDDED 1N POROUS MEDIA

261

equations for the p r o b l e m under consideration with the b o u n d a r y layer a n d Boussinesq a p p r o x i m a t i o n s a n d the D a r c y law can be w r i t t e n as

o(r~) O(rv) o ~ + or - o, 04

g 3 K OT

Or

OT

OT

(1)

u

0 (rOT5

u ~-~z + V Or -- -r c% \

(2)

Or'

1 1 0 (3)

pCp -r ~-r(rq~).

Or]

T h e b o u n d a r y conditions are defined as follows: r=ro:

v=O,

r--~oc: u = 0 ,

T=Tw,

(4)

T=

(5)

To.

where x a n d r are coordinates m e a s u r e d along a n d n o r m a l to the surface, respectively, u a n d v are the D a r c i a n velocities in the x and r directions, respectively, g is the g r a v i t a t i o n a l acceleration. /3 is the t h e r m a l expansion coefficient of the fluid. K is the p e r m e a b i l i t y of the porous medium. • is the kinematic viscosity of the fluid. T is the t e m p e r a t u r e of the fluid a n d the porous m e d i u m which are in local t h e r m a l equilibrium, a is the equivalent t h e r m a l diffusivity, p is the density of the fluid. Cp is the specific heat at constant pressure. qT is the r a d i a t i v e heat flux in the r-direction, ro is the radius of the vertical cylinder. By using the Rosseland a p p r o x i m a t i o n [11], we take 4a ¢9T4 qr-

16aT 30T -

33R Or

33R

Or

(6)

with a the S t e f a n - B o l t z m a n n constant, ~R the Rosseland m e a n extinction coefficient. T h e t e r m 16~Ta/(3flR) can be considered as the "radiative conductivity". We now define a s t r e a m function ¢ such t h a t

ru = o--7-

and

rv = - b ~ ; "

(7)

The continuity equation (1) is then a u t o m a t i c a l l y satisfied. Invoking the following dimensionless variables: ~-

2x , , 1/2,

ro.l~ax

(8a)

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K.A. Yih

Vol. 26, No. 2

~, 1/2 q --

r2

r ° z2txa ~ ( ~ o 2

f(~, r~) -

-

I)

,

(Sb)

, , 1/2' aro~ax

(8c)

T-To 0(~,~)

-

T~

-

Too

(8d)

a n d s u b s t i t u t i n g e q u a t i o n (8) into equations (1)-(6), we o b t a i n the following t r a n s f o r m e d governing equations

f ' = 0,

(9)

4

3

T h e b o u n d a r y conditions are defined as follows:

r/=O:

f=0,

0=1,

rl--*oc : 0 = 0.

(11) (12)

In the above equations, the primes denote the differentiation with respect to r/. E q u a t i o n (9) is o b t a i n e d by integrating equation (2) once with the help of equation (5).

R a , = 9J3K(Tw - r ~ ) x / ( u ~ )

is defined as the modified local Rayleigh n u m b e r for the flow

t h r o u g h the porous m e d i u m . ~ is the transverse curvature p a r a m e t e r . Re = k 3 R / ( 4 e T ~ ) is the c o n d u c t i o n - r a d i a t i o n p a r a m e t e r .

H = T w / T ~ is the surface t e m p e r a t u r e excess

ratio. In t e r m s of the new variables, the velocity components are given by u --

aRa~

f',

(13)

X

and

v --

,-, 1/2 ar~a~

-x

ro(~ 1 Of ~ ) rf -t- -~
(14)

The r a t e of heat transfer qw at the surface of the cylinder is

16aT 3 "~OT

(i~)

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VERTICAL CYLINDER EMBEDDED IN POROUS MEDIA

263

For practical applications, it is usually the local Nusselt number that is of interest. This can be expressed as

N~

-

hx k

qwX -

(16)

k(:r~ - ro~)

where h denotes the local heat transfer coefficient and k represents the thermal conductivity. Substituting Eqs. (8b) and (15) into Eq. (16), we obtain AT..ux = - / { 1 + 4 H 3

. 1/2

-/~ax

.

5-~)

x0 t

(~,0).

(17)

W h e n Rd---~c there is no radiation interaction, and Eqs. (9)-(10) reduce to those for pure natural convection flow over an isothermal vertical cylinder embedded in a saturated porous medium [4-8].

Numerical

Method

The above governing equations (9)-(10) and the boundary conditions

(11)-(12)

are

nonlinear partial differential equations depending on the transverse curvature parameter ~, conduction-radiation parameter Rd and surface temperature excess ratio H. The present analysis integrates the system of equations (9)-(10) by the implicit finite difference approximation together with the modified Keller box m e t h o d of Cebeci and Bradshaw [12]. The computations were carried out on an AcerPower 590hd computer with A~ = 0.1 (uniform grid), the first step size zX~l = 0.01, the variable grid parameter is chosen 1.01. The value of r?~ = 1000 was found to be sufficiently accurate for 10~l < 10 -3. The requirement that the variation of the temperature distribution is less than 10 -5 between any two successive iterations is employed as the criterion of convergence.

Results and Discussion In order to verify the accuracy of the present method, we have compared our results with those of Minkowycz and Cheng [4], Yficel [5], Kumari et al. [6] and Bassom and Rees [8]. The comparisons between our results and data of Bassom and Rees [8] are found to be in very good agreement, as shown in Table 1.

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K.A. Yih

Vol. 26, No. 2

Numerical results are presented graphically for the transverse curvature parameter r a n g i n g from 0 to 10, the conduction-radiation p a r a m e t e r Rd ranging from 0.1 to 10 l° and the surface t e m p e r a t u r e excess ratio H ranging from 1.1 to 3.

TABLE 1 Values of - 0 ' ( ~ , 0) for Rd = 101° Minkowycz and Cheng [4] 0

1 2 3 4 5 6 7 8 9 10 20

- -

0.6149 0.7668 0.9085 1.044 1.176 1.305 1.435 1.565 1.696 1.830 --

Y~cel [5] - -

0.6192 0.7753 0.9201 1.0571 1.1884 ---------

K u m a r i e t al. [6] 0.4438 0.6479 0.8538 1.0576 1.2571 1.4519 1.6424 1.8290 2.0120 2.1918 2.3688 4.0268

Bassomand Rees [8] 0.4438 0.6191 0.7750 0.9191 1.055 1.185 1.310 1.431 1.549 1.665 1.777 2.8249

Present method 0.4437 0.6192 0.7750 0.9192 1.0554 1.1855 1.3110 1.4325 1.5509 1.6665 1.7797 2.8245

Figure 1 shows the dimensionless t e m p e r a t u r e profiles for various values of { with H = 1.1, Rd = 1. It is seen that the dimensionless t e m p e r a t u r e gradient at the wall increases with increasing the value of transverse curvature parameter {. The dimensionless t e m p e r a t u r e profiles for various values of Rd with ~ = 4, H = 1.5, is displayed in Fig. 2. Increasing the conduction-radiation parameter Rd enhances the dimensionless temperature gradient at the wall. Figure 3 illustrates the dimensionless t e m p e r a t u r e profiles for various values of H with { = 2, Rd = 0.5. From this figure we observe that the dimensionless t e m p e r a t u r e profiles increase owing to increase in the surface t e m p e r a t u r e excess ratio H. The local Nusselt n u m b e r for the various values of the conduction-radiation parameter Rd with H = 2.0. is presented in Fig. 4. It is found that the value of the local Nusselt n u m b e r increases due to decrease in the value of Ra.

Figure 5 shows the local Nusselt

n u m b e r for the various values of the surface t e m p e r a t u r e excess ratio H with Ra = 1. As the surface temperature excess ratio H increases, the local Nusselt n u m b e r increases. Moreover, the local Nusselt n u m b e r increases with increasing the transverse curvature parameter ~, as depicted in Figs. 4 and 5.

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VERTICAL CYLINDER EMBEDDED IN POROUS MEDIA

1.0

=

1.1

,

Rd

=

1

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0

,.

r

I

2

4 , H

265

1.5

=

0.0 3 r]

4

5

6

0

FIG. 1 Dimensionless temperature profiles for various values of (

4

8

12

16

20

FIG. 2 D i m e n s i o n l e s s t e m p e r a t u r e p r o f i l e s for v a r i o u s values o f R d

1.0 2

, Rd

=

0.5

0.8

0.4 0.2 0.0 0

4

8

12

16

20

FIG. 3 Dimensionless temperature profiles for various values of H I00

I00 Rd

=-

1

10 7

z 1

H = 2.0 0.1 0

2

4

6

8

10

FIG. 4 Local Nusselt number for various values of Rd

0

2

4

6

8

10

FIG. 5 L o c a l N u s s e l t n u m b e r for v a r i o u s values of H

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K.A. Yih

Vol. 26, No. 2

Conclusions The radiation effect on the natural convection flow of an optically dense viscous fluid adjacent to an isothermal vertical cylinder embedded in a saturated porous medium with Rosseland diffusion approximation is numerically investigated. The transformed nonsimilar conservation equations are obtained and solved by the Keller box method. Numerical results are given for the dimensionless temperature profiles and local Nusselt number for various values of the transverse curvature parameter ~, the conduction-radiation parameter Rd, and the surface temperature excess ratio H. It is apparent that decreasing the conduction-radiation parameter Rd enhances the local Nusselt number. Furthermore, as transverse curvature parameter ~ and the surface temperature excess ratio H increase, the local Nusselt number increases.

Acknowledgment This work reported in this paper was supported by Air Force General Headquarters.

Nomenclature Cp

specific heat at constant pressure

f

dimensionless stream function, defined in Eq. (8c)

g

gravitational acceleration

H

surface temperature excess ratio, T w / T ~

h

local heat transfer coefficient

K

permeability coefficient of the porous medium

k

thermal conductivity

Nuz

local Nusselt number, h z / k

q~

radiative heat flux in the r-direction, defined in Eq. (6)

qw

surface heat transfer rate, defined in Eq. (15)

Re

conduction-radiation parameter, k 3 R / ( 4 a T ~ )

r

radial coordinate

Fo

radius of cylinder

Rax

modified local Rayleigh number, gflI(.(T~ - To~)x/(vo~)

T

temperature

7Z

Darcian velocity in the x-direction

Vol. 26, No. 2

VERTICAL CYLINDER EMBEDDED IN POROUS MEDIA

?3

Darcian velocity in the r-direction

X

streamwise coordinate

(R

thermal diffusivity

9 ~R

coefficient of thermal expansion

r~

pseudosimilarity variable, defined in Eq. (Sb)

267

Rosseland mean extinction coefficient transverse curvature parameter, defined in Eq. (8a)

o

dimensionless temperature, defined in Eq. (8d)

p

density of fluid kinematic viscosity of fluid Stefan-Boltzmann constant

cr

stream function Subscripts w

condition at the wall

cc

condition at infinity

References

1.

P. Cheng and W. J. Minkowycz, 3". Geophys. Res. 82, 2040 (1977).

2.

T. Y. Na and I. Pop, Int. Y. Engng. Sci., 21,517 (1983).

3.

R. S. R. Gorla and A. H. Zinalabedini, A S M E Y. Energy Resources Technology, 109, 26 (1987).

4.

W. J. Minkowycz and P. Cheng, Int. Y. Heat Mas~ Transfer, 19, 805 (1976).

5.

A. Yficel, Numerical Heat Transfer, 7, 483 (1984).

6.

M. Kumari, I. Pop and G. Nath, Int. J. Heat Mass Transfer, 28, 2171 (1985).

7.

J. H. Merkin, Acta Mechanica, 62, 19 (1986).

8.

A. P. Bassom and D. A. S. Rees, Acta Mechanica, 116, 139 (1996).

9.

M. A. Hossain and I. Pop, Heat and Mass Transfer, 32~ 223 (1997).

10.

A. Raptis, Int. Comm. Heat Mass Transfer, 25,289 (1998).

11.

M. A. Hossain and M. A. Alim, Heat and Mass Transfer, 32,515 (1997).

12.

T. Cebeci and P. Bradshaw, Physical and Computational A~pects of Convective Heat Transfer, 1st ed., p. 385. Springer-Verlag, New York (1984).

ReceivedAugust 12, 1998