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Non-darcian effects on conjugate mixed convection about a vertical circular pin in a porous medium
Compurers & Smcrures Vol. 38, No. S/6, pp. 529-535. 1991 Printed in Great Britain.
0
0045.7949/91 $3.00 + 0.00 1991 Pergamon Press plc
NON-DARCIAN EFFECTS ON CONJUGATE MIXED CONVECTION ABOUT A VERTICAL CIRCULAR PIN IN A POROUS MEDIUM CHA’O-KUANG CHEN and CHIEN-HSIN CHEN Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China (Received 8 January 1990)
Abstract-The problem of mixed convection heat transfer along a vertical circular pin embedded in a fluid-saturated porous medium is analyzed based on the conjugate convection-conduction theory. The effects of non-Darcian flow phenomena, which are neglected in Darcy’s law, on conjugate heat transfer are examined. These effects include the solid-boundary shear, inertial forces, variable porosity distribution, and transverse thermal dispersion. The numerical results show that the non-Darcian effects significantly alter the heat transfer predictions from those using the Darcy flow model. The boundary and inertia effects tend to decrease the heat transfer rate, while the near-wall porosity variation and thermal dispersion increase it. The effects of the conjugate convection-conduction parameter and the surface curvature on heat transfer characteristics of the pin are also presented.
NOTATION f; A B c ;a 4 ;
Gr g h h* K k
k, kf k, k, L N NC P
Pe Pe, Pr :* 4 4* Re r r0 T u 4 v X
a 5 P ;
empirical constant in eqn (6) constant defined in eqn (23) inertia coefficient specific heat of the fluid Darcy number, Km/L2 empirical constant defined in eqn (24) particle diameter dimensionless stream function Grashof number, gpK, L(T,, - Tm)/v2 gravitational constant local heat transfer coefficient dimensionless local heat transfer coefficient permeability effective thermal conductivity stagnant thermal conductivity thermal conductivity of fluid thermal conductivity of particles thermal dispersion conductivity pin length empirical constant in eqn (6) conjugate convection-conduction parameter pressure Peclet number, u,L/u, Peclet number based on particle diameter Prandtl number of the fluid total heat transfer rate dimensionless total heat transfer rate local heat flux dimensionless local heat flux Reynolds number, u, L/v radial coordinate radius of pin temperature x-component velocity convective velocity, -(K, /p)(dP/dx) r-component velocity axial coordinate effective thermal diffusivity thermal diffusivitv of fluid thermal expansion coefficient pseudo-similarity variable dimensionless temperature
lr ; P $ 0
parameter in eqn (14) thermal conductivity ratio of the solid phase to fluid phase viscosity of the fluid kinematic viscosity of fluid d’lmensionless streamwise coordinate density of the fluid 0, porosity stream function surface curvature parameter
Subscripts
b” P
quantities away from the wall quantities at pin base quantities associated with the pin
INTRODUCITON Convective heat transfer and fluid flow in porous media have attracted considerable attention in recent years due to applications in a number of science and engineering disciplines. These include geothermal operations, filtration, chemical catalytic reactors, packed-sphere beds, grain storage, thermal insulation systems, etc. Most of the previous studies pertinent to flow through porous media have been based on Darcy’s law which neglects the non-Darcian flow effects. The non-Darcian flow situation, which may prevail in some of the above applications, makes Darcy’s law inapplicable. Therefore, it is necessary to include the non-Darcian effects in the analysis. These effects have been investigated for heat transfer problems induced by various convection mechanisms (forced, free, mixed convection) in porous media [l-6]. Boundary effects can be modeled by adding a viscous term to the momentum equation. A velocity
529
530
CHA’O-KUANGCHEN and CHEN-HSIN CHEN
squared term is incorporated into the momentum equation to account for the inertia effects. When the effect of porosity variation is considered, the flow channeling will occur adjacent to the surface [4,7]. This effect is demonstrated to be very important on heat transfer. In studying the nonuniform porosity effects a simple exponential function is usually used to approximate the porosity variation in the vicinity of the solid boundary. As pointed out by Cheng [8], transverse thermal dispersion effects may become significant when inertia effects are prevalent. This dispersion effect is also examined in the present analysis. The majority of existing studies for heat transfer problems in porous media were based on some assumed temperature distributions along the impermeable surface. However, in many practical applications, heat conduction at the solid boundary interacts closely with convection in the boundary layer. Instead of being prescribed as in conventional fin studies, the heat transfer coefficient is unspecified and a part of the solution to the problem. This consideration is necessary and more closely approximates the physical situation. The important conclusions drawn from the studies by Sparrow and his co-workers [9, lo] were that the conventional fin model based on a uniform input of heat transfer coefficient gives a good estimate for the overall rate of fin heat transfer, but substantial errors could arise in the local predictions. The conjugate mixed convectionconduction heat transfer problem for a cylindrical fin embedded vertically in a porous medium has recently been considered by Liu et al. [l l] using Darcy’s law. The objective of the present investigation is to analyze the effects of non-Darcian terms on conjugate mixed convectionconduction heat transfer about a vertical circular pin immersed in a porous medium. The differential equations for the boundary layer flow and the heat conduction equation in the pin are solved simultaneously through the interfacial conditions. Numerical solutions of the governing differential equations are generated by an efficient finite difference method. The various non-Darcian effects are demonstrated by comparing the results with those of the Darcy flow model. Results of interest, such as local heat transfer coefficient, local heat flux, pin temperature distribution, and total heat transfer rate, are presented as functions of controlling parameters. ANALYSIS
Consider a vertical circular pin with length L and radius r, whose upper end is maintained at a constant temperature Tb. The pin is embedded vertically downward in a fluid-saturated porous medium at an ambient temperature of T, which is assumed to be smaller than Tb. The coordinate system with the origin at the tip of the pin is oriented so that the gravitational acceleration is in the opposite direction
to the positive x-axis and the r-axis is perpendicular to the pin surface. It is assumed that the convective fluid and the porous matrix are in local thermodynamic equilibrium. The Boussinesq approximation and the boundary-layer approximation are invoked. The governing equations for the flow field in a cylindrical coordinate system are given as [2]
& (rz4)+ f
(ru) = 0
ap
gu+pcu2=
-ax
+fig
r: (
aT
aT
u~+u~=--~
+pg/I(T-T,)
(2)
>
i a
az-
ma,,
(
)
(3)
where u and v are the components of velocity in the x- and r-directions; T, P, and g are the temperature, pressure, and gravitational constant; p, p, and p are the density, viscosity, and the thermal expansion coefficient of the fluid; K, C, and I$ are the permeability, inertia coefficient, and porosity of the porous medium; and CI= k/pc is the effective thermal diffusivity of the porous medium with k denoting the effective thermal conductivity of the saturated porous medium and pc the product of the density and specific heat of the fluid. Equations (lH3) are subject to the boundary conditions u=u=O,
T=T(x,r,,)
u=u,,
T=T,
atr=r, asr+co.
(4) (5)
In order to study the variable porosity effects, an exponential decrease is usually assumed to approximate the near-wall porosity variation such as in packed-sphere beds [12] 4 = 4, + { 1 + A exp[-N(r -
ro)14),
(6)
where 4, = 0.4 is the free-stream porosity, d the particle diameter, and A and N are the empirical constants which depend on the packing of particles near the solid wall. The value for A is determined so that the porosity at the wall is 0.9. The value of N = 6 is used to represent the decay of porosity from the solid wall [12]. Both the permeability K and the inertia coefficient C of the porous matrix can be expressed in terms of the particle diameter and porosity from the correlations developed by Ergun [13] K=
d24’
150(1 - 4)’
(7)
Non-Darcian effects on conjugate mixed convection
The continuity equation (1) is satisfied automatically by introducing the stream function $ as
,=2
N=-‘~
and
ar
ax’
To facilitate the analysis, eqns (2) and (3) along with the boundary conditions (4) and (5) are nondimensionalized according to the following transformations: r/ =
( = x/L g&,a,x)O.5, 0
T-T,
f(C
7)
=
+(x9
~Yko(y4x)o~51, 0 = -.
T,-
Substituting the transformations yields
T,
(10)
$‘=
-1+&l
2r
+4r)
531
,
8= 0
as V+OO,
(14)
where 0, = (T, - Tm)/(Tb - T,) is the dimensionless temperature of the pin and the primes indicate differentiation with respect to q. The free-stream boundary condition on the velocity is obtained from the momentum equation, eqn (1 1), by neglecting the viscous and buoyancy terms. If the pin is considered to be relatively long compared with its radius r,, heat conduction in the pin can be considered to be one-dimensional. In addition, the heat loss from the tip of the pin is assumed to be negligible. Under these assumptions, the thin pin conservation equation and boundary conditions can be expressed, in dimensionless form, as
into eqns (2) and (3)
(15)
DaPe a -6+ art
de d6
P=O
(11)
(12) where
e,=l
atr=O
(16)
at{=l,
(17)
where NC = (k/k,)(2L/r,),/Re is the conjugate convection-conduction parameter with k, denoting the thermal conductivity of the pin, and h*(r) = hL/k,/R e is the dimensionless local heat transfer coefficient. Equations (11)-(14) are coupled with eqns (15)-(17) through the following interface conditions: T(x, ro) = T,(x)
at r = r.
(18)
Da=3 -kz=h(x)(T,-
Pe = u,L tll r=
T,)
at r =r,.
These two conditions state the physical requirements that the temperatures and heat fluxes of the pin and the porous medium are continuous at the pin-fluid interface. When the transformations given by eqn (10) are applied to eqns (18) and (19) there results
K, C, u, V
e,(i;) = e(r, 0) h*(t) = -JPre’(5,O)it&,(t;)1, and w is the transverse curvature parameter, defined as
w= Boundary conditions f’=O,
f+25$=0,
s
(cr,u,L)O.~.
(4) and (5) reduce to e=e,(r)
(19)
at q
=o
(13)
(20) (21)
where Pr is the Prandtl number of the fluid. It is well known that the effective thermal conductivity k of a saturated porous medium is composed of a sum of the stagnant thermal conductivity k, (due to molecular diffusion) and the thermal dispersion conductivity k, (due to mechanical dissipation), i.e. k=k,+k,.
(22)
532
CHA’~-KUANG CHENand CHIEN-HSIN CHEN
The stagnant thermal conductivity for packed-sphere beds can be given by the following semi-analytical expression [ 141: 2&l-
-&l-4))+
+1
4)
l-*1B
where B = 1.25[( 1 - 6)/4 ]10i9 and I = k,//c# is the ratio of the thermal conductivity of fluid to that of particles. Equation (23) reveals that the stagnant thermal conductivity is a function of position for variable porosity media. As proposed by Hsu and Cheng [ 151, the thermal dispersion conductivity can be expressed in terms of the new variables as
k, -= kr
l-4
D
’
2
hf’,
4
where D, is an empirical constant and Ped = u,d/a,. In the present study a fixed value of D, = 0.02 1151 is used to examine qualitatively the thermal dispersion effect. Quantities of practical interest are local heat transfer coefficient, local heat flux, total heat transfer rate, and pin temperature distribution. The dimensionless form of local heat transfer coefficient is given by eqn (21). The local heat flux can be computed from q=-k;
0’
(25)
, - ro
which in a dimensionless
q*(5)=
qL
k(T, - T,)Re’.’
form becomes = -fi@Y&
0)/J’?.
RESULTSAND DISCUSSION Equation (15) is coupled with eqns (11) and (12) through the interface conditions, eqns (20) and (21). To obtain a solution to the problem, eqns (11)-(14) must be solved simultaneously with eqns (15)-(17). The numerical solution of the complete set of equations, eqns (11)-(17), was accomplished by the Keller Box method described in [16]. Computations were carried out for the following values of physical quantities: Pr = 5.4, L = 1 m, u, = 0.01 m/set and k, = 1.05 W m-’ K-’ for glass beads of particle size d = 3 mm. Since various non-Darcian effects are taken into consideration, the following legends are used in the figures: nBnIU, which indicates no Boundary, no Inertia, and Uniform porosity effects; and BIV, which refers to Boundary, Inertia, and Variable porosity effects, etc. It may be remarked that nBnIU is the Darcy flow case reported in [l 11. In order to confirm the accuracy of the present numerical scheme, results using the Darcy flow mode1 have been compared and found in good agreement with those of [ll]. The comparison is not presented here for brevity. The distributions of dimensionless local heat transfer coefficients along the pin surface are plotted in Figs 1 and 2 for w = 0.2 and 1.O. Each figure contains value of Gr/Re = 1.0 and values of NC = 0.1, 1.O. When the boundary and inertia effects are considered (BIU), the heat transfer coefficients are decreased as compared with those of a Darcy flow case (nBnIU). The basic cause of this behaviour is the decrease of velocity in the boundary layer, which yields a reduction in heat transfer. For a variable porosity medium the coefficients are increased due to the flow-channeling effect [4,7]. It is also observed that for larger values of NC the coefficients decrease at first
(26)
The value of total heat transfer rate can be obtained either from integrating the local heat flux along the pin surface or from the heat conducted from the base to the pin at r = 1.0. Thus Q = 2nro
q(x) dx
(27)
or Q = k,nrid
dT
(28)
ax x=L’
In terms of the new variables defined in eqn (lo), these become
Q ‘*
= &k(T,
_ T,)L
1 =q
I
’ o q*(e)dT
(2g)
or
Q Q*=4nk(Tb-Tm)L=~d~
1
NC = 0.1
. 000 de, c_,
(30)
#*
0.8
7.0
Fig. 1. Local heat transfer coefficients for o = 0.2.
Non-Da&an effects on conjugate mixed convection
533
10
8
6 l
26
01 4
2
NC = 0.1
0.0
I
I
0.4
0.4
I
I
0.6
1
0.8
1.0
0 c 0 .O
I
I
OS?
0.4
I
0.6
I
0.8
#
[
Fig. 2. Local heat transfer coefficients for o = 1.
Fig. 4. Local heat fluxes for o = 1.
to a minimum value, and then increase with the increasing streamwise direction. The nonmonotonic distributions of heat transfer coefficient can be attributed to an enhanced buoyancy encountered by the fluid as it passes from the pin tip to the pin base. Comparing Fig. 1 with Fig. 2, for given values of NC and Gr/Re, one can see that local heat transfer coefficients for larger values of o are higher than those for smaller values of o. Figures 3 and 4 present the results of dimensionless local heat fluxes. The boundary and inertia effect tend to reduce the heat transfer and these effects are more pronounced for a higher NC, especially downstream.
The variation in porosity increases the momentum transport in the boundary layer, and hence results in an enhancement in heat transfer. It is also seen that for a given o, most of the heat transfer takes place in the neighborhood of the pin root as NC increase. This phenomenon is caused by the fact that a larger NC, representing a lower thermal conductivity of the pin, gives rise to a higher temperature near the root. The figures also demonstrate that higher values of o result in larger local heat fluxes. However, one should note that the radius of the pin could be different for each w. Thus, smaller values of local heat fluxes for smaller ws do not necessarily imply that the total heat transfer rates over the pin surface are less than those for larger ws. Representative pin temperature distributions for selected values of controlling parameters are illustrated in Figs 5 and 6. The reduction in heat transfer due to the boundary and inertia effects results in smaller pin temperature variations. The variable porosity effect tends to promote the nonuniformities of the pin temperature distributions. It is also shown that larger values of NC give rise to larger pin temperature variations. This is readily explained by the fact that higher values of NC correspond to higher convective coefficients and lower pin conductances, thus yielding increases in pin temperature variations. In addition, higher values of w also amplify the pin temperature variations. The quantity of major interest in the present investigation is the total heat transfer rate over the pin surface. Numerical results of this quantity in dimensionless form are given in Figs 7 and 8 as a function of NC. The corresponding total heat transfer rates solved by the two methods described in eqns (29) and (30) are found to be in excellent agreement. This confirms the previous statements that boundary
10
6
6 + (r
0.8
I
0.4
0.8
0.8
#
Fig. 3. Local heat fluxes for o = 0.2. CAS 38/1/6-D
f.0
CHA’O-KUANG CHENand CHIEN-HSINCHEN
534
0.6 0 0.4
0.X
o.a_
0.0
I
0.3
I
0.4
Fig. 5. Fin tem~rature
I
0.6 # dist~butions
L
0.6
1 1.0
0.0
0.4
0.6
1R
?.U
6.0
NC for w = 0.2.
and inertia effects decrease the heat transfer rate and the porosity variation increases it. If the thermal dispersion effect is included in the analysis, the total heat transfer rates are increased tremendously. The great heat transfer enhancement caused by the dispersive transport can be attributed to the better mixing of convective fluid within the pores. It also shows that for fixed values of w, the total heat transfer rates decrease with an increasing NC. This behavior is evident from the fact that increase of NC yields a highly nonisothermal pin, and therefore decreases the total heat transfer rate of the pin. The influence of the surface curvature on the total heat transfer rate can
Fig. 7. Total heat transfer rates for w =0.2.
be observed by comparing the results shown in Fig. 7 with those shown in Fig. 6. It is seen that an increase of w results in a decrease of total heat transfer rate. A higher value of w represents a smaller value of r,, hence the total heat transfer rate is decreased due to a smaller convection surface. CONCLUSION
A study has been conducted to investigate the importance of non-Darcian effects on conjugate mixed convection from a vertical circular pin embedded in a ~uid-saturated porous medium. The effects
1.0
0.6
6 k\
0.0-
0.6 0.6 1.0 F Fig. 6. Fin temperature distributions for w = 1. 0.0
0.0
0.4
0.0
0.4
0,s
1.6
l.6
NC Fig. 8. Total heat transfer rates for w = 1.
a.0
Non-Darcian effects on conjugate mixed convection
of the conjugate convection-conduction parameter and surface curvature on heat transfer characteristics are also illustrated. It is seen that the heat transfer predictions with the non-Darcian effects included differ significantly from those using the Darcy flow model. The boundary and inertia effects tend to reduce the heat transfer rate, while the nonuniform porosity and thermal dispersion effects increase it. The results also demonstrate that the total heat transfer rates increase with decreasing values of NC and w. REFERENCES 1. 0. A. Plumb and J. C. Huenefeld, Non-Darcy natural convection from heated surfaces in saturated porous media. Int. J. Heat Mass Transf. 24, 165-168 (1981). 2. K. Vafai and C. L. Tien, Boundary and inertia effects on flow and heat transfer in porous media. Inr. J. Heat Mass Transf. 24, 195-203 (1981).
3. C. T. Hsu and P. Cheng, The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium. Int. J. Heat Mass Tram-f. 28, 683497
(1985).
4. B. C. Chandrasekhara and P. M. Namboodiri, Influence of variable permeability on combined free and forced convection about inclined surfaces in porous media. Ini. J. Hear Mass Transf. 28, 199-206 (1985).
5. D. Poulikakos and K. Renken, Forced convection in a channel filled with porous medium, including the effects of flow inertia, variable porosity, and Brinkman friction. J. Heat Transf 109, 880-888 (1987).
535
6. M. Kumari and G. Nath, Non-Darcy mixed convection
boundary layer flow on a vertical cylinder in a saturated porous medium. Int. J. Heat Mass Transf 32, 183-187 (1989).
7. D. Vortmeyer and J. Schuster, Evaluation of steady flow profiles in rectangular and circular packed beds. Chek. Engng Sci. 38,1691-1699
(1983). _
8. P. Chena, Thermal disuersion effects in non-Darcian convective flows in a saturated porous medium. Lerr. Heat Mass Tranf.
8, 267-270
(1981).
9. E. M. Sparrow and S. Acharya, A natural convection fin with a solution-determined nonmonotonically varying heat transfer coefficient. J. Heat Transf 103, 218-225 (1981).
10. E. M. Sparrow and M. K. Chyu, Conjugate forced convectionconduction analysis of heat transfer in a plate fin. J. Heat Transf: 104, 204206 (1982). Il. J. Y. Liu, W. J. Minkowycz and P. Cheng, Conjugate mixed convection-conduction heat transfer along a cylindrical fin in a porous medium. Inf. J. Heat Mass Transf 29, 769-775 (1986).
12. R. F. Benenati and C. B. Brosilow, Void fraction distribution in beds of spheres. A. I. Ch. E. Jnl 8, 359-361 (1962). 13. S. Ergun, Fluid flow through packed columns. Chem. Engng Prog. 48, 89-94 (1952).
14. P. Zehner and E. U. Schluender, Waermeleitfahigkeit von schuettungen bei massigen temperature”. ChemieIngr-Tech 42,933-941
(1976).
15. C. T. Hsu and P. Chenn. Closure schemes of the macroscopic energy equation for convective heat transfer in porous media. Int. Commun. Heat Mass Transf. “.
15, 689-703 (1988).
16. T. Cebeci and P. Bradshaw, Momentum Transfer in Boundary Layers. Hemisphere, Washington, D.C. (1977).
0
0045.7949/91 $3.00 + 0.00 1991 Pergamon Press plc
NON-DARCIAN EFFECTS ON CONJUGATE MIXED CONVECTION ABOUT A VERTICAL CIRCULAR PIN IN A POROUS MEDIUM CHA’O-KUANG CHEN and CHIEN-HSIN CHEN Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China (Received 8 January 1990)
Abstract-The problem of mixed convection heat transfer along a vertical circular pin embedded in a fluid-saturated porous medium is analyzed based on the conjugate convection-conduction theory. The effects of non-Darcian flow phenomena, which are neglected in Darcy’s law, on conjugate heat transfer are examined. These effects include the solid-boundary shear, inertial forces, variable porosity distribution, and transverse thermal dispersion. The numerical results show that the non-Darcian effects significantly alter the heat transfer predictions from those using the Darcy flow model. The boundary and inertia effects tend to decrease the heat transfer rate, while the near-wall porosity variation and thermal dispersion increase it. The effects of the conjugate convection-conduction parameter and the surface curvature on heat transfer characteristics of the pin are also presented.
NOTATION f; A B c ;a 4 ;
Gr g h h* K k
k, kf k, k, L N NC P
Pe Pe, Pr :* 4 4* Re r r0 T u 4 v X
a 5 P ;
empirical constant in eqn (6) constant defined in eqn (23) inertia coefficient specific heat of the fluid Darcy number, Km/L2 empirical constant defined in eqn (24) particle diameter dimensionless stream function Grashof number, gpK, L(T,, - Tm)/v2 gravitational constant local heat transfer coefficient dimensionless local heat transfer coefficient permeability effective thermal conductivity stagnant thermal conductivity thermal conductivity of fluid thermal conductivity of particles thermal dispersion conductivity pin length empirical constant in eqn (6) conjugate convection-conduction parameter pressure Peclet number, u,L/u, Peclet number based on particle diameter Prandtl number of the fluid total heat transfer rate dimensionless total heat transfer rate local heat flux dimensionless local heat flux Reynolds number, u, L/v radial coordinate radius of pin temperature x-component velocity convective velocity, -(K, /p)(dP/dx) r-component velocity axial coordinate effective thermal diffusivity thermal diffusivitv of fluid thermal expansion coefficient pseudo-similarity variable dimensionless temperature
lr ; P $ 0
parameter in eqn (14) thermal conductivity ratio of the solid phase to fluid phase viscosity of the fluid kinematic viscosity of fluid d’lmensionless streamwise coordinate density of the fluid 0, porosity stream function surface curvature parameter
Subscripts
b” P
quantities away from the wall quantities at pin base quantities associated with the pin
INTRODUCITON Convective heat transfer and fluid flow in porous media have attracted considerable attention in recent years due to applications in a number of science and engineering disciplines. These include geothermal operations, filtration, chemical catalytic reactors, packed-sphere beds, grain storage, thermal insulation systems, etc. Most of the previous studies pertinent to flow through porous media have been based on Darcy’s law which neglects the non-Darcian flow effects. The non-Darcian flow situation, which may prevail in some of the above applications, makes Darcy’s law inapplicable. Therefore, it is necessary to include the non-Darcian effects in the analysis. These effects have been investigated for heat transfer problems induced by various convection mechanisms (forced, free, mixed convection) in porous media [l-6]. Boundary effects can be modeled by adding a viscous term to the momentum equation. A velocity
529
530
CHA’O-KUANGCHEN and CHEN-HSIN CHEN
squared term is incorporated into the momentum equation to account for the inertia effects. When the effect of porosity variation is considered, the flow channeling will occur adjacent to the surface [4,7]. This effect is demonstrated to be very important on heat transfer. In studying the nonuniform porosity effects a simple exponential function is usually used to approximate the porosity variation in the vicinity of the solid boundary. As pointed out by Cheng [8], transverse thermal dispersion effects may become significant when inertia effects are prevalent. This dispersion effect is also examined in the present analysis. The majority of existing studies for heat transfer problems in porous media were based on some assumed temperature distributions along the impermeable surface. However, in many practical applications, heat conduction at the solid boundary interacts closely with convection in the boundary layer. Instead of being prescribed as in conventional fin studies, the heat transfer coefficient is unspecified and a part of the solution to the problem. This consideration is necessary and more closely approximates the physical situation. The important conclusions drawn from the studies by Sparrow and his co-workers [9, lo] were that the conventional fin model based on a uniform input of heat transfer coefficient gives a good estimate for the overall rate of fin heat transfer, but substantial errors could arise in the local predictions. The conjugate mixed convectionconduction heat transfer problem for a cylindrical fin embedded vertically in a porous medium has recently been considered by Liu et al. [l l] using Darcy’s law. The objective of the present investigation is to analyze the effects of non-Darcian terms on conjugate mixed convectionconduction heat transfer about a vertical circular pin immersed in a porous medium. The differential equations for the boundary layer flow and the heat conduction equation in the pin are solved simultaneously through the interfacial conditions. Numerical solutions of the governing differential equations are generated by an efficient finite difference method. The various non-Darcian effects are demonstrated by comparing the results with those of the Darcy flow model. Results of interest, such as local heat transfer coefficient, local heat flux, pin temperature distribution, and total heat transfer rate, are presented as functions of controlling parameters. ANALYSIS
Consider a vertical circular pin with length L and radius r, whose upper end is maintained at a constant temperature Tb. The pin is embedded vertically downward in a fluid-saturated porous medium at an ambient temperature of T, which is assumed to be smaller than Tb. The coordinate system with the origin at the tip of the pin is oriented so that the gravitational acceleration is in the opposite direction
to the positive x-axis and the r-axis is perpendicular to the pin surface. It is assumed that the convective fluid and the porous matrix are in local thermodynamic equilibrium. The Boussinesq approximation and the boundary-layer approximation are invoked. The governing equations for the flow field in a cylindrical coordinate system are given as [2]
& (rz4)+ f
(ru) = 0
ap
gu+pcu2=
-ax
+fig
r: (
aT
aT
u~+u~=--~
+pg/I(T-T,)
(2)
>
i a
az-
ma,,
(
)
(3)
where u and v are the components of velocity in the x- and r-directions; T, P, and g are the temperature, pressure, and gravitational constant; p, p, and p are the density, viscosity, and the thermal expansion coefficient of the fluid; K, C, and I$ are the permeability, inertia coefficient, and porosity of the porous medium; and CI= k/pc is the effective thermal diffusivity of the porous medium with k denoting the effective thermal conductivity of the saturated porous medium and pc the product of the density and specific heat of the fluid. Equations (lH3) are subject to the boundary conditions u=u=O,
T=T(x,r,,)
u=u,,
T=T,
atr=r, asr+co.
(4) (5)
In order to study the variable porosity effects, an exponential decrease is usually assumed to approximate the near-wall porosity variation such as in packed-sphere beds [12] 4 = 4, + { 1 + A exp[-N(r -
ro)14),
(6)
where 4, = 0.4 is the free-stream porosity, d the particle diameter, and A and N are the empirical constants which depend on the packing of particles near the solid wall. The value for A is determined so that the porosity at the wall is 0.9. The value of N = 6 is used to represent the decay of porosity from the solid wall [12]. Both the permeability K and the inertia coefficient C of the porous matrix can be expressed in terms of the particle diameter and porosity from the correlations developed by Ergun [13] K=
d24’
150(1 - 4)’
(7)
Non-Darcian effects on conjugate mixed convection
The continuity equation (1) is satisfied automatically by introducing the stream function $ as
,=2
N=-‘~
and
ar
ax’
To facilitate the analysis, eqns (2) and (3) along with the boundary conditions (4) and (5) are nondimensionalized according to the following transformations: r/ =
( = x/L g&,a,x)O.5, 0
T-T,
f(C
7)
=
+(x9
~Yko(y4x)o~51, 0 = -.
T,-
Substituting the transformations yields
T,
(10)
$‘=
-1+&l
2r
+4r)
531
,
8= 0
as V+OO,
(14)
where 0, = (T, - Tm)/(Tb - T,) is the dimensionless temperature of the pin and the primes indicate differentiation with respect to q. The free-stream boundary condition on the velocity is obtained from the momentum equation, eqn (1 1), by neglecting the viscous and buoyancy terms. If the pin is considered to be relatively long compared with its radius r,, heat conduction in the pin can be considered to be one-dimensional. In addition, the heat loss from the tip of the pin is assumed to be negligible. Under these assumptions, the thin pin conservation equation and boundary conditions can be expressed, in dimensionless form, as
into eqns (2) and (3)
(15)
DaPe a -6+ art
de d6
P=O
(11)
(12) where
e,=l
atr=O
(16)
at{=l,
(17)
where NC = (k/k,)(2L/r,),/Re is the conjugate convection-conduction parameter with k, denoting the thermal conductivity of the pin, and h*(r) = hL/k,/R e is the dimensionless local heat transfer coefficient. Equations (11)-(14) are coupled with eqns (15)-(17) through the following interface conditions: T(x, ro) = T,(x)
at r = r.
(18)
Da=3 -kz=h(x)(T,-
Pe = u,L tll r=
T,)
at r =r,.
These two conditions state the physical requirements that the temperatures and heat fluxes of the pin and the porous medium are continuous at the pin-fluid interface. When the transformations given by eqn (10) are applied to eqns (18) and (19) there results
K, C, u, V
e,(i;) = e(r, 0) h*(t) = -JPre’(5,O)it&,(t;)1, and w is the transverse curvature parameter, defined as
w= Boundary conditions f’=O,
f+25$=0,
s
(cr,u,L)O.~.
(4) and (5) reduce to e=e,(r)
(19)
at q
=o
(13)
(20) (21)
where Pr is the Prandtl number of the fluid. It is well known that the effective thermal conductivity k of a saturated porous medium is composed of a sum of the stagnant thermal conductivity k, (due to molecular diffusion) and the thermal dispersion conductivity k, (due to mechanical dissipation), i.e. k=k,+k,.
(22)
532
CHA’~-KUANG CHENand CHIEN-HSIN CHEN
The stagnant thermal conductivity for packed-sphere beds can be given by the following semi-analytical expression [ 141: 2&l-
-&l-4))+
+1
4)
l-*1B
where B = 1.25[( 1 - 6)/4 ]10i9 and I = k,//c# is the ratio of the thermal conductivity of fluid to that of particles. Equation (23) reveals that the stagnant thermal conductivity is a function of position for variable porosity media. As proposed by Hsu and Cheng [ 151, the thermal dispersion conductivity can be expressed in terms of the new variables as
k, -= kr
l-4
D
’
2
hf’,
4
where D, is an empirical constant and Ped = u,d/a,. In the present study a fixed value of D, = 0.02 1151 is used to examine qualitatively the thermal dispersion effect. Quantities of practical interest are local heat transfer coefficient, local heat flux, total heat transfer rate, and pin temperature distribution. The dimensionless form of local heat transfer coefficient is given by eqn (21). The local heat flux can be computed from q=-k;
0’
(25)
, - ro
which in a dimensionless
q*(5)=
qL
k(T, - T,)Re’.’
form becomes = -fi@Y&
0)/J’?.
RESULTSAND DISCUSSION Equation (15) is coupled with eqns (11) and (12) through the interface conditions, eqns (20) and (21). To obtain a solution to the problem, eqns (11)-(14) must be solved simultaneously with eqns (15)-(17). The numerical solution of the complete set of equations, eqns (11)-(17), was accomplished by the Keller Box method described in [16]. Computations were carried out for the following values of physical quantities: Pr = 5.4, L = 1 m, u, = 0.01 m/set and k, = 1.05 W m-’ K-’ for glass beads of particle size d = 3 mm. Since various non-Darcian effects are taken into consideration, the following legends are used in the figures: nBnIU, which indicates no Boundary, no Inertia, and Uniform porosity effects; and BIV, which refers to Boundary, Inertia, and Variable porosity effects, etc. It may be remarked that nBnIU is the Darcy flow case reported in [l 11. In order to confirm the accuracy of the present numerical scheme, results using the Darcy flow mode1 have been compared and found in good agreement with those of [ll]. The comparison is not presented here for brevity. The distributions of dimensionless local heat transfer coefficients along the pin surface are plotted in Figs 1 and 2 for w = 0.2 and 1.O. Each figure contains value of Gr/Re = 1.0 and values of NC = 0.1, 1.O. When the boundary and inertia effects are considered (BIU), the heat transfer coefficients are decreased as compared with those of a Darcy flow case (nBnIU). The basic cause of this behaviour is the decrease of velocity in the boundary layer, which yields a reduction in heat transfer. For a variable porosity medium the coefficients are increased due to the flow-channeling effect [4,7]. It is also observed that for larger values of NC the coefficients decrease at first
(26)
The value of total heat transfer rate can be obtained either from integrating the local heat flux along the pin surface or from the heat conducted from the base to the pin at r = 1.0. Thus Q = 2nro
q(x) dx
(27)
or Q = k,nrid
dT
(28)
ax x=L’
In terms of the new variables defined in eqn (lo), these become
Q ‘*
= &k(T,
_ T,)L
1 =q
I
’ o q*(e)dT
(2g)
or
Q Q*=4nk(Tb-Tm)L=~d~
1
NC = 0.1
. 000 de, c_,
(30)
#*
0.8
7.0
Fig. 1. Local heat transfer coefficients for o = 0.2.
Non-Da&an effects on conjugate mixed convection
533
10
8
6 l
26
01 4
2
NC = 0.1
0.0
I
I
0.4
0.4
I
I
0.6
1
0.8
1.0
0 c 0 .O
I
I
OS?
0.4
I
0.6
I
0.8
#
[
Fig. 2. Local heat transfer coefficients for o = 1.
Fig. 4. Local heat fluxes for o = 1.
to a minimum value, and then increase with the increasing streamwise direction. The nonmonotonic distributions of heat transfer coefficient can be attributed to an enhanced buoyancy encountered by the fluid as it passes from the pin tip to the pin base. Comparing Fig. 1 with Fig. 2, for given values of NC and Gr/Re, one can see that local heat transfer coefficients for larger values of o are higher than those for smaller values of o. Figures 3 and 4 present the results of dimensionless local heat fluxes. The boundary and inertia effect tend to reduce the heat transfer and these effects are more pronounced for a higher NC, especially downstream.
The variation in porosity increases the momentum transport in the boundary layer, and hence results in an enhancement in heat transfer. It is also seen that for a given o, most of the heat transfer takes place in the neighborhood of the pin root as NC increase. This phenomenon is caused by the fact that a larger NC, representing a lower thermal conductivity of the pin, gives rise to a higher temperature near the root. The figures also demonstrate that higher values of o result in larger local heat fluxes. However, one should note that the radius of the pin could be different for each w. Thus, smaller values of local heat fluxes for smaller ws do not necessarily imply that the total heat transfer rates over the pin surface are less than those for larger ws. Representative pin temperature distributions for selected values of controlling parameters are illustrated in Figs 5 and 6. The reduction in heat transfer due to the boundary and inertia effects results in smaller pin temperature variations. The variable porosity effect tends to promote the nonuniformities of the pin temperature distributions. It is also shown that larger values of NC give rise to larger pin temperature variations. This is readily explained by the fact that higher values of NC correspond to higher convective coefficients and lower pin conductances, thus yielding increases in pin temperature variations. In addition, higher values of w also amplify the pin temperature variations. The quantity of major interest in the present investigation is the total heat transfer rate over the pin surface. Numerical results of this quantity in dimensionless form are given in Figs 7 and 8 as a function of NC. The corresponding total heat transfer rates solved by the two methods described in eqns (29) and (30) are found to be in excellent agreement. This confirms the previous statements that boundary
10
6
6 + (r
0.8
I
0.4
0.8
0.8
#
Fig. 3. Local heat fluxes for o = 0.2. CAS 38/1/6-D
f.0
CHA’O-KUANG CHENand CHIEN-HSINCHEN
534
0.6 0 0.4
0.X
o.a_
0.0
I
0.3
I
0.4
Fig. 5. Fin tem~rature
I
0.6 # dist~butions
L
0.6
1 1.0
0.0
0.4
0.6
1R
?.U
6.0
NC for w = 0.2.
and inertia effects decrease the heat transfer rate and the porosity variation increases it. If the thermal dispersion effect is included in the analysis, the total heat transfer rates are increased tremendously. The great heat transfer enhancement caused by the dispersive transport can be attributed to the better mixing of convective fluid within the pores. It also shows that for fixed values of w, the total heat transfer rates decrease with an increasing NC. This behavior is evident from the fact that increase of NC yields a highly nonisothermal pin, and therefore decreases the total heat transfer rate of the pin. The influence of the surface curvature on the total heat transfer rate can
Fig. 7. Total heat transfer rates for w =0.2.
be observed by comparing the results shown in Fig. 7 with those shown in Fig. 6. It is seen that an increase of w results in a decrease of total heat transfer rate. A higher value of w represents a smaller value of r,, hence the total heat transfer rate is decreased due to a smaller convection surface. CONCLUSION
A study has been conducted to investigate the importance of non-Darcian effects on conjugate mixed convection from a vertical circular pin embedded in a ~uid-saturated porous medium. The effects
1.0
0.6
6 k\
0.0-
0.6 0.6 1.0 F Fig. 6. Fin temperature distributions for w = 1. 0.0
0.0
0.4
0.0
0.4
0,s
1.6
l.6
NC Fig. 8. Total heat transfer rates for w = 1.
a.0
Non-Darcian effects on conjugate mixed convection
of the conjugate convection-conduction parameter and surface curvature on heat transfer characteristics are also illustrated. It is seen that the heat transfer predictions with the non-Darcian effects included differ significantly from those using the Darcy flow model. The boundary and inertia effects tend to reduce the heat transfer rate, while the nonuniform porosity and thermal dispersion effects increase it. The results also demonstrate that the total heat transfer rates increase with decreasing values of NC and w. REFERENCES 1. 0. A. Plumb and J. C. Huenefeld, Non-Darcy natural convection from heated surfaces in saturated porous media. Int. J. Heat Mass Transf. 24, 165-168 (1981). 2. K. Vafai and C. L. Tien, Boundary and inertia effects on flow and heat transfer in porous media. Inr. J. Heat Mass Transf. 24, 195-203 (1981).
3. C. T. Hsu and P. Cheng, The Brinkman model for natural convection about a semi-infinite vertical flat plate in a porous medium. Int. J. Heat Mass Tram-f. 28, 683497
(1985).
4. B. C. Chandrasekhara and P. M. Namboodiri, Influence of variable permeability on combined free and forced convection about inclined surfaces in porous media. Ini. J. Hear Mass Transf. 28, 199-206 (1985).
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535
6. M. Kumari and G. Nath, Non-Darcy mixed convection
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(1983). _
8. P. Chena, Thermal disuersion effects in non-Darcian convective flows in a saturated porous medium. Lerr. Heat Mass Tranf.
8, 267-270
(1981).
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12. R. F. Benenati and C. B. Brosilow, Void fraction distribution in beds of spheres. A. I. Ch. E. Jnl 8, 359-361 (1962). 13. S. Ergun, Fluid flow through packed columns. Chem. Engng Prog. 48, 89-94 (1952).
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15, 689-703 (1988).
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