NonDar&n mixed convection along a vertical plate embedded in a porous medium ChienHsin
Chen and Cha’oKuang
Chen
Department of Mechanical Engineering, Taiwan, Republic of China
National
Cheng
Kung
University,
Tainan,
The analysis is performed for a mixed convective boundary layer flow in a fluidsaturated porous medium near a vertical surface. In modelling the flow in the porous medium the nonDarcian effects such as the flow inertia, noslip boundary conditions, nonuniform porosity, and thermal dispersion are taken into account. Because of the porosity variation in the nearwall region, the stagnant thermal conductivity also varies accordingly. These effects have a signzficant influence on velocity and temperature profiles and heat transfer rate from the flat plate. This work examines these effects on mixed convective transport and demonstrates the variation in heat transfer predictions based on different flow models. The present study also discusses the dependence of fluid flow and heat transfer characteristics on the problem parameters. Keywords:
porous
material, mixed convection,
nonDarcian
Introduction In recent years, heat transfer and fluid flow through porous media have received considerable attention from many researchers. This is primarily due to a broad range of applications, such as geothermal systems, grain storage, thermal insulation, packed sphere beds, heat exchangers, chemical catalytic reactors, and filtration. Most of the earlier studies pertinent to heat and fluid flow in porous media have been based on Darcy’s law,’ which states that the volumeaveraged velocity is proportional to the pressure gradient. In many practical applications, for example, packed sphere beds, the porous medium is bounded by an impermeable wall, has higher flow rates, and reveals nonhomogeneous porosity variation near the wall, making Darcy’s law inapplicable. Therefore the research interest in recent studies has been focused on the important nonDarcian phenomenon of convective heat transport in porous media. The Brinkman’s extension, which includes a viscous shear stress term in the momentum equation, has been used to account for the boundary effects. The inertia effects can be modelled through the addition of a quadratic term in velocity, which is known as Forchheimer’s extension. The boundary and inertia effects on forced convective heat transfer from a flat plate were
Address reprint requests to Professor Chen at the Department of Mechanical Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China.
Received
482
6 October
1989; accepted
Appl. Math. Modelling,
20 March
1990
1990, Vol. 14, September
flow
first examined by Vafai and Tien2 for a constantporosity medium. These effects were shown to decrease the velocity in the thermal boundary layer and reduce the heat transfer rate. A similar analysis was carried out recently by Beckermann and Viskanta.3 For packed spheres, owing to the variation in packing next to the solid wall, the measurements of Benenati and Brosilow4 show a distinct porosity variation with a highporosity region close to the external boundary. The nonhomogeneous porosity variation near the wall results in a flowchanneling phenomenon, which yields a maximum velocity near the solid surface. This nonuniform porosity effect has been reported by several investigators such as Vortmeyer and Schuster,s and Vafai.” When the inertia effects are prevalent, the studies by Cheng7 and Plumb8 suggested that the transverse thermal dispersion effect becomes important. This dispersive transport results from the mixing of local fluid streams as the fluid moves past the solid particles. The present analysis investigates the fluid flow and heat transfer characteristics of a mixed convective flow about a vertical surface in a porous medium. A similar problem was first analyzed by Cheng,9 who performed a boundary layer analysis based on Darcy’s law for mixed convection flow about inclined surfaces. Similarity solutions were found to exist in the case in which the wall temperature distribution and the free stream velocity of the inclined surface vary according to the same power function of distance. The boundary friction and flow inertia effects on mixed convective flows were taken into consideration in a recent paper.‘O The present work examines the influence of all the aforementioned nonDarcian phenomena on flow field and heat transfer in packed beds of spheres. The flow in
0 1990 ButterworthHeinemann
NonDarcian mixed convection in a porous medium: the packed bed is assumed to be governed by the BrinkmanDarcyErgun flow model* with nonuniform porosity taken into account. The variations of the porosity in the vicinity of the solid boundary can be approximated by exponential functions.6 The effects of transverse thermal dispersion are included in the energy equation with variable stagnant conductivity taken into account. An efficient finite difference algorithm has been employed to solve the governing equations for the present problem. The results show that the heat transfer rate is prominently increased by considering thermal dispersion effects, which is similar to the findings of previous researches.‘~” Analysis Consider the problem of mixed convection about a vertical flat plate, which is embedded in a fluidsaturated porous medium at uniform temperature. The streamwise coordinate is denoted by X, and that normal to it is denoted by y. In the formulation of the present problem the following common assumptions are made: The flow is steady, incompressible, and twodimensional; the fluid and the porous medium are everywhere in local thermodynamic equilibrium; and the boundary layer and Boussinesq approximations are valid. Under these assumptions the continuity, momentum, and energy equations for mixed convection in porous media with the nonDarcian effects included may be expressed as? (1) 2
dP p&4 + pcu* =  z + ~ay2 +
PHT

TcJ
dT (3) dy ( (yc;ly> where u and u are the components of velocity along the x and ydirections; T, P, and g are the temperature, pressure, and gravitational constants; p, p, and p are the density, viscosity, and thermal expansion coeffcient of the fluid; K, C, and 4 are the permeability, inertial force parameter, and porosity of the porous medium; and (Y, = k,l(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. The second term in the momentum equation based on the BrinkmanDarcyErgun model accounts for highflowrate inertial pressure losses. The fourth term of this equation is the viscous shear force, which represents the noslip boundary effect and is significant near the wall surface. The energy equation is based on the assumption of local thermal equilibrium. This assumption allows only a small temperature difference between the fluid and solid phases. The boundary conditions for the problem are u=v=o, T = T,, aty = 0 (4) T 
T,
asy 
m
The porosity of the porous medium depend on location to account for porosity variation. This variation is mated by a simple exponential function studies:5,6T12
is assumed to the nearwall often approxias used in the
4 = & + (&  6) exp (NY/~
(6)
where & = 0.4 is the free stream porosity; #L, = 0.9 is the porosity at wall; and the constant N = 7 is used to represent the porosity decay.4 It is noted that the oscillations of the porosity, which are considered to be secondary, are neglected in the present study. For a packed sphere bed, both the permeability K and the inertia coefficient C of the bed depend on the sphere diameter and the porosity and are determined from the wellknown correlations developed by Ergun:13 d2+3 K = 150(1  4)*
(7)
C = l75(1  4) d4’
(8)
where d is the particle diameter. It is known that the effective thermal conductivity k, of a saturated porous medium is composed of a sum of the stagnant thermal conductivity kd (due to molecular diffusion) and the thermal dispersion conductivity k, (due to mechanical dissipation), that is, k, = kd + k,
(9)
The stagnant thermal conductivity of a packed sphere bed can be given by the following semianalytical expression:‘4
(2)
dT aT u+uu2. dx ay
u = u,,
C.H. Chen and C.K. Chen
(5)
2c+ ,_hB
kd =(lC4)+ kf
(10) where B = 1.25[(1  $)/$]1°‘” and A = k,lk, is the ratio of the thermal conductivity of the fluid phase to that of the solid phase. Equation (10) shows that the stagnant thermal conductivity is a function of position for a nonuniform porosity medium. This kind of representation for kd is also supported by the previous works.12,‘5 As proposed by Hsu and Cheng,‘” it is assumed that the thermal dispersion conductivity is of the form
kDpef 4
d
!L&* 42
(11)
where Ped = Pr Red is Peclet number with Pr denoting the Prandtl number of the fluid and Red = u,d/u. In the above equation, u* is a dimensionless Darcian velocity given by u* = u/u,., and D, is an empirical contant. The governing equations (l)(3) are nondimension
Appl. Math. Modelling,
1990, Vol. 14, September
483
NonDarcian
mixed convection
alized by introducing ables:
in a porous medium:
the following dimensionless
vari
n=vs
5=;,
(12) T  T, ~ ’ = T,  T,
f(& 77)= ccl(~,YwGz
C.H. Chen and C.K. Chen
where the stream function I+!I is introduced
such that
arCI u= a* and (13) u=ex ay The continuity equation is automatically satisfied by the above relations. In terms of the new variables the momentum and energy equations are
(14) (15)
where DaL = K,IL2, PeL = u,LIaf, r = K,C,u,lv, (T = c~Jq, and Gr/Re = gfiK,(T,  TJu,.v; K, and C, are the permeability and inertia coefficient in the bulk region. The dimensionless boundary conditions are f’ = 0,
f + 252
e=
= 0,
Results and discussion 1
d
at n = 0 f’ = 
1+v?FZ 21_
9
(16)
o=o
as 71 
00 (17)
where the primes indicate differentiation with respect to n. The free stream boundary condition on the velocity is obtained from the momentum equation, equation (14), by neglecting the viscous and buoyancy terms. A main objective of this work is to determine the heat transfer rate from the vertical plate. Consider first the local heat flux along the surface of the flat plate, which can be computed from
(18) The results for the local heat transfer rate from the vertical surface can be represented in terms of the local Nusselt number Nu, which is defined as Nu = hxlk,, where h is the local heat transfer coefficient. Combining equation (18) with the definition of h, that is, 4 = h(T,  T,) gives the heat transfer parameter Nu/(RePr)O.’ =  /?‘(& 0)
(19)
where Re = u,x/u. The numerical solutions for equations (14) and (15) with the boundary conditions (16) and (17) have been obtained on the basis of an implicit finite difference scheme, namely, the Keller Box method.“~‘* This numerical scheme has several very desirable features that make it appropriate for the solution of partial differential equations. The main features of this method involve secondorder accuracy with arbitrary 5 and 77 spacings, allowing very rapid 5 variations and allowing easy programming of the solution of large numbers of coupled equations. The details of the solution procedure by this method are described in the literature (see,
484
Appl. Math. Modelling,
for example, Ref. 18). Hence in the interest of brevity they are not repeated here.
1990, Vol. 14, September
In this section the effects of the nonDarcian flow phenomena on mixed convective flow and heat transfer in porous media are examined and discussed. In obtaining the numerical results, the following values of physical quantities were employed: Pr = 5.4, L = 0.5 m, u, = 0.01 m/s, and k, = 1.05 Wm‘Ki for glass spheres. The empirical constant D, in equation (11) should be determined from experiment. According to the recent reports,i6 the value of D, = 0.02 is adequate for the numerical computations. Since various nonDarcian flow effects are taken into consideration, the following legends are used: BIV, which indicates boundary, inertia, and variable porosity effects; nBnIU, which denotes no boundary, no inertia, and uniform porosity effects, and so on. It is noted that nBnIU, which is the widely known Darcy’s law, is the case reported previously by Cheng.9 In this study the results are illustrated by using particles that are 3 mm and 5 mm in diameter. Figures l8 present the results of numerical simulations with a focus on the variable porosity, boundary, and inertia effects by neglecting the thermal dispersion effect. The heat transfer predictions accounting for the thermal dispersion effects are reported and discussed in Figures 911. Figures
Z(a) and l(b) depict the typical velocity protiles resulting from the inclusion of various nonDarcian effects. The velocity profiles at the wall are steeper at downstream positions as shown in these figures. This behavior agrees qualitatively with the results reported in a recent study for constant porosity media.‘O When the effects of nonuniform porosity are taken into consideration, the channeling profiles are created by the highporosity regions near the solid surface. The inertia effects with the noslip boundary condition cause an additional pressure loss and reduce the magnitude of the velocity. The free stream velocity for the case in which inertia effects are considered is given by equation (17). Figures 2(a) and 2(6) show the dimensionless tem
NonDarcian 5.0
mixed convection
in a porous medium:
C.H. Chen and C.K. Chen
r I
4.0
3.0
! t
f 2.0
?.O
0.0
1 i I
0
2
I
4
3
6
5
7
?
Figure l(a).
The velocity distributions
at 5 = 0.05
Figure 2(a).
The temperature
distributions
at 5 = 0.05
7.0
B
B
f WIU
2wTLru 3 wrv 4wnlv
1
0.8
0.6
8
0.4
0.2
0
I
I
I
I
2
3
The velocity distributions
1
0
17
Figure lib).
J
0.0
4
3
z
4
6
8
7
11
at 5 = 0.70
perature profiles corresponding to the velocity profiles in Figures l(a) and Z(b). Notice that the profiles plotted in Figures 1 and 2 are for the limiting case of Gr/Re = 0, which corresponds to the case of pure forced convection. The inertia effects are found to thicken the thermal boundary layers and reduce the temperature gradients at the wall. It can also be seen that with porosity variations included, the temperature gradient is increased owing to the strong flowchanneling effect. Figures 3 and 4 aim to identify the contribution of each of the various effects on the heat transfer parameter for the heated plate. Including only the Brinkman friction term (BnIU) yields a decrease in the heat trans
Figure 2(b).
The temperature
distributions
at 5 = 0.70
fer parameter especially near the leading edge when compared with the case in which the Darcy’s flow model is assumed (nBnIU). The value of Nu is reduced further when the Forchheimer inertia term is also included in the momentum equation (BIU). The effect of the flowchanneling phenomenon, caused by the nearwall porosity variations, is shown to augment the heat transfer from the wall. The facts that the boundary and inertia effects decrease the heat transfer rate and nonuniform porosity effect increases it have been reported in the literature (see, for example, Ref. 6). Whether the heat transfer is enhanced or reduced as compared to the Darcy flow model depend on the relative mag
Appl.
Math.
Modelling,
1990,
Vol.
14, September
485
NonDarcian
mixed convection
in a porous medium:
C.H. Chen and C.K. Chen 6.0
f WIU
d=3mm
d=5mm
2wnlu 3 WIV 4wnJv
( = 0.4
7.0

BN

Bfl
I 6.0
5.0
f' 4.0
3.0 2.0
1.0
Figure 3.
Local heat transfer parameters for different flow models I
a Figure 6.
r.4
Influences of GriRe on velocity distributions
d=5mm Gr/Re = 0
1.2 f.L
~ 
d=fimm
f.0 4 .!! 0.8 <
WIU wfl
0.6
2 0.6 0.6
0.4 0 I
0.2 A
0.2
0.0
Figure 4.
I
I
I
0.4
0.6
0.8
7.0
0.4
Local heat transfer parameters for different flow models 0.2
2.0
f WIU
0.0
2wnl7J 3 WIV 4wnrv 5?t.EnIu
1.6
0”
0.0
I
1 0.0
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
Figure 5. Local heat transfer parameters with Gr/Fie = 1
486
Appl.
Math.
Modelling,
1990,
for mixed convection
Vol.
2
3
4
5
6
D
Figure 7.
0.4
I
14, September
Influence of Gr/Re on temperature
distributions
nitude of these nonDarcian mechanisms. This is clearly seen from Figures 3 and 4. Although the above predictions are shown for Gr/Re = 0, it is reasonable to expect that the nonDarcian flow phenomena should exhibit a similar influence on mixed convection heat transfer. Figure 5 provides an example that meets our expectations. Figures 68 illustrate the influence of buoyancy force on the velocity and temperature distributions and on the heat transfer rate. The quantity Gr/Re is known as a measure of relative importance of natural to forced
NonDarcian f.8
I
1
0.0
0.0
in a porous medium:
C.H. Chen and C.K. Chen
BIU _____Bp
d=5mm
1.6
mixed convection
I
I
I
I
0.2
0.4
0.6
0.8
I.0
t Figure 8.
Influence of Gr/Re on local heat transfer parameters
3.0
Figure 11. Influence transfer parameters
2.5
_
g"."
Bl7 __. BILi nB?l.ru
= 0
_________________
Eth Dispersion Effect ____________________________
qt.5
__________
s 1.0 Without Dispersion Effect _~~_~_~~~~__~~~~_______________________ __
0.5
_
,
I
I
0.4
0.6
0.8
I
0.2
“.”
f.0
L
f Figure 9. Influence of thermal dispersion on the local heat transfer parameters
3.0 BIV _____Bm ?lBnrlJ
d=5mm G+Re = 0
2.5
________
t 4 2o ;‘/
dispersion
on the local heat
convection. This dimensionless parameter has been shown to be the controlling parameter for mixed convection about inclined surfaces in a porous medium.9 From Figure 6 it can be seen that with increasing Gr/Re the velocities inside the boundary layer become larger. The temperature profiles presented in Figure 7 indicate that the temperature gradients at the wall are increased by increasing the Gr/Re. Figure 8 shows that, as expected, the heat transfer at the wall is augmented as Gr/Re is increased. These behaviors are evident from the fact that the buoyancy force assists the flow and thus enhances the heat transfer rate. The last nonDarcian effect included in the analysis is the dispersion term, which is expected to be significant as the flow inertia is prevalent. Figures 9 and 10 demonstrate the influence of thermal dispersion on pure forced convection heat transfer. It can be summarized that the heat transfer rate is drastically increased by taking into consideration the transport due to thermal dispersion. Figure I1 predicts similar heat transfer results for mixed convection. The great heat transfer enhancement caused by the dispersive transport can be attributed to the better mixing of convective fluid within the pores. Figures 9 and 10 also show that the thermal dispersion effects are more pronounced in a bed with larger particles. This is realized by the fact that increasing sphere diameter d is equivalent to increasing the Peclet number Ped in equation (1 l), which in turn gives rise to a stronger dispersive transport.
1
d=3mm Gr/Re
of thermal
    kh DkpenGn Effect __________________________
_9. t.5 ts 1.0 c_ 0.5 
0.0
’ 0.0
_
Wit?wut _
Di8p emion
Nomenclature Effect
B
C
.______________________ I
I
I
I
0.2
0.4
0.6
0.8
;a, I.0
D,
c Figure 10. Influence transfer parameters
of thermal
dispersion
on the local heat
Gr
Appl.
constant defined in equation (10) inertia coefficient specific heat of the fluid Darcy number, K,lL2 empirical constant defined in equation (11) particle diameter dimensionless stream function Grashof number, gpK,x(T,,,  Tm)/v2
Math. Modelling,
1990, Vol. 14, September
487
NonDarcian
f K kd k, kf k k,
L N NU P Ped b_
Pr ze Red T u 4 .!I X
Y
mixed convection
in a porous medium:
gravitational constant heat transfer coefficient permeability stagnant thermal conductivity effective thermal conductivity thermal conductivity of fluid thermal conductivity of particles thermal dispersion conductivity characteristic length empirical constant in equation (6) Nusselt number pressure Peclet number based on particle diameter Peclet number, u,LIaf Prandtl number of the fluid local heat flux Reynolds number, U,X/V Reynolds number based on particle diameter temperature xcomponent velocity convective velocity,  (K,I~)(dPIdx) ycomponent velocity streamwise coordinate crossstream coordinate
C.H. Chen and C.K. Chen
these effects significantly alters the flow field and heat transfer characteristics from those predicted by the traditional Darcy’s model. The effects of flow inertia and viscous shear force with the noslip boundary condition were found to reduce the velocity and heat transfer. The flowchanneling phenomenon in the nearwall region, caused by the nonuniform porosity distribution, dramatically augments thermal communication between the porous matrix and the solid boundary. The thermal dispersion effect increases the heat transfer rate considerably, especially for large particle diameter. The enhancement in heat and fluid flow from buoyancy force is demonstrated by increasing the value of Gr/Re . References 1 2
3
4
Greek symbols
effective thermal diffusivity thermal diffusivity of fluid thermal expansion coefficient dimensionless crossstream coordinate dimensionless temperature parameter in equation (14), K,C,u,.Iu thermal conductivity ratio of the solid phase to fluid phase viscosity of the fluid kinematic viscosity of fluid dimensionless streamwise coordinate density of the fluid %& porosity stream function Subscripts
quantities away from the wall quantities at the wall
5
6 7
8
9
10
II
12
13
Conclusion
14
The foregoing analysis demonstrates the importance of the nonDarcian flow effects in modelling mixed convection flows along a vertical flat plate embedded in a porous medium. The effects of flow inertia, boundary friction, and nonuniform porosity are taken into account in the momentum equation. Owing to the porosity variations, it is expected that the thermal conductivity will vary across the packed bed. The effects of variable stagnant thermal conductivity and transverse thermal dispersion are taken into consideration in the energy equation. It was found that inclusion of
488
Appl.
Math. Modelling,
1990, Vol. 14, September
15
16
17
18
Cheng, P. Heat transfer in geothermal systems. Adv. Heat Transfer 1978, 14, I105 Vafai, K. and Tien, C. L. Boundary and inertia effects on flow and heat transfer in porous media. Internat. J. Heat Mass Transfer 1981, 24, 195203 Beckermann, C. and Viskanta, R. Forced convection boundary layer flow and heat transfer along a flat plate embedded in a porous medium. Internat. J. Heat Mass Transfer 1987, 30, 15471551 Benenati, R. F. and Brosilow, C. B. Void fraction distribution in beds of spheres. AIChE J., 1962, 8, 359361 Vortmeyer, D. and Schuster, J. Evaluation of steady flow profiles in rectangular and circular packed beds. Chem. Eng. Sci. 1983, 38, 16911699 Vafai, K. Convective flow and heat transfer in variableporosity media. J. Fluid Mech. 1984, 147, 233259 Cheng, P. Thermal dispersion effects in nonDarcian convective flows in a saturated porous medium. Left. Heat Mass Transfer 1981, 8, 267270 Plumb, 0. A. The effect of thermal dispersion on heat transfer in packed bed boundary layers. ASMEJSME Joint Thermal Conference Proceedings, Vol. 2, 1983, pp. 1721 Cheng, P. Combined free and forced convection flow about inclined surfaces in porous media. Internat. J. Heat Muss Transfer, 1977, 20, 807814 Ranganathan, P. and Viskanta, R. Mixed convection boundarylayer flow along a vertical surface in a porous medium. Numer. Heat Transfer1984, 7, 305317 Hunt. M. L. and Tien. C. L. Effects of thermal disoersion on forced convection in fibrous media. Internat. J. Neat Mass Transfer 1988, 31(2), 301309 Cheng, P., Hsu, C. T. and Chowdhury, A. Forced convection in the entrance region of a packed channel with asymmetric heating. ASME J. Heat Transfer 1988, 110, 946954 Ergun, S. Fluid flow through packed columns. Chem. Engrg. Progr. 1952,48,8994 Zehner, P. and Schluender, E. U. Waermeleitfahigkeit von schuettungen bei massigen temperaturen. ChemieIngrTech.. 1970, 42, 933941 Cheng, P. and Hsu, C. T. Fullydeveloped, forced convective flow through an annular packedsphere bed with wall effects. Internat. J. Heat Mass Transfer 1986, 29(12), 18431853 Hsu, C. T. and Cheng, P. Closure schemes of the macroscopic energy equation for convective heat transfer in porous media. Internat. Comm. Heat Mass Transfer 1988, 15, 689703 Keller, H. B. and Cebeci, T. Accurate numerical methods for boundarylayer flows. 2: Twodimensional turbulent flows. AfAA J. 1972, 10, 11931199 Cebeci, T. and Bradshaw, P. Physical and Computational Aspects of Convective Heat Transfer. SpringerVerlag, New York, 1984