Darcy-Forchheimer natural, forced and mixed convection heat transfer in non-Newtonian power-law...

R. EHRLICH

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Transport in Porous Media 11: 219-241, 1993. 219 © 1993 Kluwer Academic Publishers. Printed in the Netherlands.

Darcy-Forchheimer Natural, Forced and Mixed Convection Heat Transfer in Non-Newtonian Power-law Fluid-Saturated Porous Media

A. V. SHENOY Department of Energy and Mechanical Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432 Japan

(Received: 13 June 1991; revised: 17 December 1991)

Abstract. The governing equation for Darcy-Forchheimer flow of non-Newtonian inelastic power-law fluid through porous media has been derived from first principles. Using this equation, the problem of Darcy-Forchheimer natural, forced, and mixed convection within the porous media saturated with a power-law fluid has been solved using the approximate integral method. It is observed that a similarity solution exists specifically for only the case of an isothermal vertical flat plate embedded in the porous media. The results based on the approximate method, when compared with existing exact solutions show an agreement of within a maximum error bound of 2.5%.

Key words. non-Newtonian fluids, power-law fluids, Darcy-Forchheimer flow, natural convection, forced convection, mixed convection

Nomenclature

A cross-sectional area bi coefficient in the chosen temperature

profile B1 coefficient in the profile for the

dimensionless boundary layer thickness

C coefficient in the modified Forchheimer term for power-law fluids

C 1 coefficient in the Oseen approximation which depends essentially on pore geometry

Ci coefficient depending essentially on pore geometry

C0 drag coefficient C, coefficient in the expression for K* d particle diameter (for irregular

shaped particles, it is characteristic length for average-size particle)

fp resistance or drag on a single particle FR total resistance to flow offered by N

particles in the porous media g acceleration due to gravity

gx component of the acceleration due to gravity in the x-direction

GrK. Grashof number based on permeability for power-law fluids

K intrinsic permeability of the porous media

K* modified permeability of the porous media for flow of power-law fluids

1, characteristic length m exponent in the gravity field n power-law index of the inelastic non

Newtonian fluid N total number of particles Nu,,a,F local Nusselt number for Darcy

forced convection flow Nu,,a-F,F local Nusselt number for Darcy

Forchheimer forced convection flow Nu,,0 .M local Nusselt number for Darcy

mixed convection flow Nux,D-F.M local Nusselt number for Darcy

Forchheimer mixed convection flow Nux.D,N local Nusselt number for Darcy

natural convection flow

220 A. V. SHENOY

Nux.D-F.N local Nusselt number for Darcy- U1 dimensionless velocity profile Forchheimer natural convection u, external forced convection flow

flow velocity

ji pressure u, seepage velocity (local average

p exponent in the wall temperature velocity of flow around the particle)

variation u., wall slip velocity

Pe, characteristic Peclet number U,M characteristic velocity for mixed

Pe, local Peclet number for forced convection convection flow U,N characteristic velocity for natural

Pe~ modified local Peclet number for convection mixed convection flow x,y boundary-layer coordinates

Ra, characteristic Rayleigh number X1'Y1 dimensionless boundary layer

Ra, local Rayleigh number for Darcy coordinates natural convection flow x coefficient which is a function of flow

Ra~ local Rayleigh number for Darcy- behaviour index n for power-law Forchheimer natural convection fluids

flow r,i effective thermal diffusivity of the

Re convectional Reynolds number porous medium for power-law fluids r,i' shape factor which takes a value of

ReK' Reynolds number based on per- n/4 for spheres meability for power-law fluids /3' shape factor which takes a value

T tern per at ure of n/6 for spheres

T, ambient constant temperature /3. expansion coefficient of the fluid

Tw,ref constant reference wall surface br boundary-layer thickness

temperature ()Tl dimensionless boundary layer

T.,(X) variable wall surface temperature thickness

/'i.Tw temperature difference equal to E porosity of the medium

Tw,ref-Te 1J similarity variable

Ti term in the Darcy-Forchheimer (} dimensionless temperature difference

natural convection regime for A' coefficient which is a function of the

Newtonian fluids geometry of the porous media (it

Tz term in the Darcy-Forchheimer takes a value of 3n for a single sphere natural convection regime for non- in an infinite fluid) Newtonian fluids (first approx- µo viscosity of Newtonian fluid

imation) µ* fluid consistency of the inelastic

TN term in the Darcy/Forchheimer non-Newtonian power-law fluid natural convection regime for non- ~ constant equal to X(2e2 -"A')/r.i' Newtonian fluids (second approx- p density of the fluid

imation) </> dimensionless wall temperature

u Darcian or superficial velocity difference

1. Introduction

The tremendous research work that has gone into the area of convective heat transfer in porous media can be adjudged from the many reviews that have appeared on this subject, such as those of Combarnous and Bories (1975), Cheng (1978), Bejan (1987), Kafoussias (1990), Tien and Yafai (1990), and Trevisan and Bejan (1990). The continuing interest in heat transfer through porous media is mainly due to several practical applications of the subject matter in geothermal engineering, thermal insulation systems, ceramic processing, enhanced oil recovery, filtration processes, chromatography, etc.

DARCY-FORCHHEIMER CONVECTION

A careful study of the above revi focusses on steady-state heat transfer Even the comprehensive review articl transfer to non-Newtonian fluids, sho non-Newtonian fluid-saturated porou graph by Kakac et al., (1991) also be fluid porous media heat transfer is a studies on the subject have begun to (1988a, 1988b), Wang and Tu (1989), P Wang et al. (1990), Nakayama and (1993). This area has begun to attract come into contact with porous media ior, especially in ceramic processing, e

Chen and Chen (1988a) were the firs of non-Newtonian fluids past an isothe medium. Their analysis was later exten body shapes such as horizontal cylinde transfer of non-Newtonian fluids in p (1989). They obtained expressions for t free convection of a Hershel-Bulkley isothermal semi-infinite plate in porou the nonlinear effects of non-Newtonia medium. They analyzed the two cases boundary-layer flow of Hershel-Bulkle rheological effects of some nonisothe studied by Pascal (1990a). An unsteadyfluid in the presence of a yield stress non-Newtonian fluids through a por (1990b). Buoyancy-induced flow of nonof arbitrary shape in a fluid-saturated p Koyama (1991). Combined free and fo fluid-saturated porous media was anal presented possible similarity solution cylinders, and spheres.

The one common feature saturated porous media studies is that which truly neglects the boundary and · It is well known that when the velocity · regime and then the porous inertia effe fluid flow, Forchheimer (1901) propose Darcian velocity term to account for Forchheimer term. This pioneering, w

M

N

y ,y1

T

Tl

A. V. SHENOY

dimensionless velocity profile external forced convection flow velocity seepage velocity (local average velocity of flow around the particle) wall slip velocity characteristic velocity for mixed convection characteristic velocity for natural convection boundary-layer coordinates dimensionless boundary layer coordinates coefficient which is a function of flow behaviour index n for power-law fluids effective thermal diffusivity of the porous medium shape factor which takes a value of n/4 for spheres shape factor which takes a value of rr/6 for spheres expansion coefficient of the fluid boundary-layer thickness dimensionless boundary layer thickness porosity of the medium similarity variable dimensionless temperature difference coefficient which is a function of the geometry of the porous media (it takes a value of 3rr for a single sphere in an infinite fluid) viscosity of Newtonian fluid fluid consistency of the inelastic non-Newtonian power-law fluid constant equal to X(2s2

-• A')/rl density of the fluid dimensionless wall temperature difference

one into the area of convective heat the many reviews that have appeared

and Bories (1975), Cheng (1978), Bejan 0), and Trevisan and Bejan (1990). The porous media is mainly due to several r in geothermal engineering, thermal need oil recovery, filtration processes,

_.....

DARCY-FORCHHEIMER CONVECTION HEAT TRANSFER 221

A careful study of the above reviews indicates that the bulk of the literature focusses on steady-state heat transfer in porous media to only Newtonian fluids. Even the comprehensive review articles of Shenoy (1986, 1988), which discuss heat transfer to non-Newtonian fluids, show that there were no studies until then, wherein non-Newtonian fluid-saturated porous media were considered. The recent monograph by Kakac et al., (1991) also bears evidence of the fact that non-Newtonian fluid porous media heat transfer is a much neglected area, although very recently studies on the subject have begun to appear including those by Chen and Chen (1988a, 1988b), Wang and Tu (1989), Pascal and Pascal (1989), Pascal (1990a, 1990b), Wang et al. (1990), Nakayama and Koyama (1991), and Nakayama and Shenoy (1993). This area has begun to attract attention because a number of fluids which come into contact with porous media certainly exhibit non-Newtonian flow behavior, especially in ceramic processing, enhanced oil recovery, and filtration.

Chen and Chen (1988a) were the first to consider the simplest free convection flow of non-Newtonian fluids past an isothermal vertical flat plate embedded in a porous medium. Their analysis was later extended (Chen and Chen, 1988b) to include other body shapes such as horizontal cylinders and spheres. Boundary-layer flow and heat transfer of non-Newtonian fluids in porous media was explored by Wang and Tu (1989). They obtained expressions for the local Nusselt numbers for the forced and free convection of a Hershel-Bulkley (1926) type non-Newtonian fluid past an isothermal semi-infinite plate in porous media. Pascal and Pascal (1989) considered the nonlinear effects of non-Newtonian fluids on natural convection in a porous medium. They analyzed the two cases of constant temperature and constant flux boundary-layer flow of Hershel-Bulkley fluid along a heated vertical cylinder. The rheological effects of some nonisothermal flows through a porous medium were studied by Pascal (1990a). An unsteady-state solution for the case of a shear thinning fluid in the presence of a yield stress was obtained. The case of two-phase flows of non-Newtonian fluids through a porous medium was also analyzed by Pascal (1990b). Buoyancy-induced flow of non-Newtonian fluids over a nonisothermal body of arbitrary shape in a fluid-saturated porous medium was treated by Nakayama and Koyama (1991). Combined free and forced convection heat transfer to power-law fluid-saturated porous media was analyzed by Nakayama and Shenoy (1993). They presented possible similarity solutions for vertical flat plates, cones, horizontal cylinders, and spheres.

The one common feature of all the above-mentioned non-Newtonian fluidsaturated porous media studies is that they only deal with the Darcy flow model which truly neglects the boundary and inertia effects on fluid flow and heat transfer. It is well known that when the velocity increases, the flow enters a nonlinear laminar regime and then the porous inertia effects are no longer negligible. For Newtonian fluid flow, Forchheimer (1901) proposed a square velocity term in addition to the Darcian velocity term to account for this effect which Muskat (1946) called the Forchheimer term. This pioneering, work was followed by other proposals (a) for

222 A. V. SHENOY

mathematically describing non-Darcy flows in such works as by Ergun (1952) and Ward (1969) and (b) for emphasizing its importance in different flow and heat transfer situations in the works of Plumb and Huenefeld (1981), Bejan and Poulikakos (1984), Poulikakos and Bejan (1985), Nield and Joseph (1985), Ingham (1986), Vasantha et al. (1986), and Fand et al. (1986). When dealing with non-Darcy flow, thermal boundarylayer effects also need to be considered by the inclusion of the Brinkman (1947) term and there are studies which take this aspect into account such as those of Tong and Subramanian (1985) and Lauriat and Prasad (1987) for Newtonian fluids.

Forchheimer effect would certainly be relevant in non-Newtonian power-law fluids, especially, in ceramic processing and enhanced oil recovery, wherein low porosity densely packed media are encountered. Practical situations involving high porosity flow of power-law fluids are rather hard to find and, therefore, the Brinkman effect would not be as important when dealing with power-law fluids as the Forchheimer effect. Hence, in the present analysis, the focus is on the important

and relevant Darcy-Forchheimer flow. In the present paper, the problem of steady state Darcy-Forchheimer natural,

forced, and mixed convection past nonisothermal bodies of arbitrary shape embedded in non-Newtonian power-law fluid - saturated porous medium has been studied. Starting from the first principles, the governing momentum equation for DarcyF orchheimer flow of power-law fluids is derived. Appropriate transformations of variables are attempted in order to obtain similarity solutions from the governing equations. It is found that similarity exists only for the case of the isothermal vertical flat plate embedded in the porous medium. The results of the analysis are compared with existing exact and approximate solutions for certain limiting cases to show an agreement within 2.5% of reported exact values.

2. Analysis

The porous medium is assumed to be composed of individual discrete particles and completely filled, i.e. saturated with a non-Newtonian power-law fluid. In order to take a force balance, a representative volume element within the porous medium has to be chosen. Any arbitrary shaped volume element can be selected; however, for convenience, an imaginary cylindrical element is chosen as shown in Figure 1, which is assumed to contain a sufficient number of particles such that the porosity of the element is representative of the entire porous medium. Summation of the forces acting in the x-direction on the fluid in the element results in a zero net force under assumed steady-state conditions and can be written in the following simplified form

(1)

where pis the pressure, p the fluid density, 9x is the component of the gravitational force in the x-direction, e is the porosity of the element, e dA dx is the fluid volume


z

Fig. 1. Schematic diagram of the forces acti element of the porous medium [taken from R

of the element, dA is the total cross-s resistance offered by all the N particl

FR=Nfp,

where fp is the resistance or drag on particles given as

N (1-e)dA dx P'd3 ,

where dis the particle diameter and for spheres.

The general expression for resistan

f P = C v<l d2 (pu; /2),

where C v is the drag coefficient and rx; spheres. Chhabra (1986) has detaile Newtonian power-law fluid past asp of the same expression applicable to

Cv=X(2e2 -n Jc')/r:l Re,

where X is a function of the flow b coefficient dependent upon particle minimum value of 3n for a single s number for a power-law fluid given a

pu2-ndn Re

µ*

A. V. SHENOY

such works as by Ergun (1952) and ance in different flow and heat transfer d (1981), Bejan and Poulikakos (1984), (1985), Ingham (1986), Vasantha et al.

th non-Darcy flow, thermal boundaryinclusion of the Brinkman (194 7) term to account such as those of Tong and (1987) for Newtonian fluids. levant in non-Newtonian power-law enhanced oil recovery, wherein low

ted. Practical situations involving high er hard to find and, therefore, the hen dealing with power-law fluids as

analysis, the focus is on the important

dy state Darcy-Forchheimer natural, al bodies of arbitrary shape embed

ated porous medium has been studied. ing momentum equation for Darcyived. Appropriate transformations of milarity solutions from the governing y for the case of the isothermal vertical e results of the analysis are compared

s for certain limiting cases to show an es.

sed of individual discrete particles and ewtonian power-law fluid. In order to element within the porous medium has element can be selected; however, for is chosen as shown in Figure 1, which particles such that the porosity of the us medium. Summation of the forces ement results in a zero net force under ritten in the following simplified form

(1)

x is the component of the gravitational he element, e dA dx is the fluid volume

..


z

/' - ap

-~~:~J''d' t'9E dA dx

Fig. 1. Schematic diagram of the forces acting on the power-law fluid within an imaginary cylindrical element of the porous medium [taken from Rumer (1969)].

of the element, dA is the total cross-sectional area of the element, and FR is the total resistance offered by all the N particles in the element. Thus,

FR=Nfp, (2)

where f P is the resistance or drag on a single particle and N is the total number of particles given as

N = _(1_-_e)_d_A_d_x /3' d3 , (3)

where d is the particle diameter and /3' is the shape factor which takes a value of n/6 for spheres.

The general expression for resistance of a single particle can be written as

fp= Cvr:t.'d2 (pu;/2), (4)

where C v is the drag coefficient and r:t.' is a shape factor which takes a value of n/4 for spheres. Chhabra (1986) has detailed the expression for Cv for the flow of nonNewtonian power-law fluid past a sphere in the Stokes regime. A more general form of the same expression applicable to any shaped particle can be written as

Cv=X(2e2 -n A1 )/r:t.1 Re, (5)

where X is a function of the flow behavior index n for power-law fluids, ),.' is a coefficient dependent upon particle shape and size and which reaches a limiting minimum value of 3n for a single sphere, and Re is the conventional Reynolds number for a power-law fluid given as

pu2-ndn Re=-

µ* (6)

224 A. V. SHENOY

where µ* is the fluid consistency and u is the Darcian or superficial velocity defined

as

U=BU5 • (7)

It is to be noted that using Equation (7) and the values of J,' = 3n, rx' = n/4 in Equation (5) gives the exact form presented by Chhabra (1986) for CD past a sphere in the Stokes region. There is no general solution for a drag coefficient beyond the Stokes fl.ow range for non-Newtonian fluids. However, an approximate solution along the lines of Goldstein (1938) for Newtonian fluids is worth extending for laminar flow of power-law fluids past a sphere at a Reynolds number of up to 2, taking into account, at least partially, the inertial forces that are neglected in Stokes' solution. It is assumed that the general form for the drag coefficient proposed by Goldstein (1938)

holds good for power-law fluids as well. Thus,

Cv=(~/Re{ 1 + t(- l)i+1C;ReJ

(8)

where ~ is a constant which is taken as equal to X(2e2 - "A')/rx' to maintain

consistency with Equation (5). C; are coefficients that are dependent essentially on the pore geometry. If only the first term of the above series is retained in line with the

approximation of Oseen (1927), then

CD=(X2e2 -"A'/rx' Re)[l+C 1 Re]. (9)

Combining Equations (1), (2), (3), (4), and (7) gives

_ ap _ _ rx'(l - e)pu2CD _ O ax pgx 2{3'e3 d - .

(10)

Substituting the expression for CD from Equation (9) and using the definition of Re

from Equation (6) gives

(11)

The quantity {3's"+ 1/A'(l - e)X is dependent only on the pore geometry and the fluid power-law index which would be constant for each system and can be replaced by a single coefficient C0 • The average particle diameter dis related to the property of the solid matrix, namely, intrinsic permeability K of the porous medium as

follows:

e3d2 K=----150(1 - s)2 .

Combining Equations (11) and (12) gives

ap µ*u" Cpu2

- ax - pgx = K* + K 112 '

(12)

(13)


where K* is introduced as the m non-Newtonian power-law fluids can

K* = _1_ (~)" (50K)<" + 01

2C, 3n + 1 3e

where

{

25/12 (Christopher an (2.5)" 20 - nl/ 2 (Kemblo

C, = 2( 8n )"(lOn - 3)( 3 9n + 3 6n + 1

For n = 1, the above Equation (13) gi often used for solving Darcy-Forchh and Huenefeld, 1981; Bejan and Po 1990; Nakayama and Pop, 1991).

2.1. DARCY-FORCHHEIMER NATURAL

It is assumed that the geometrical c shown in Figure 2. The wall surface than the ambient temperature T. w considered to be occurring under nat

u

Fig. 2. Schematic diagram of the physical m

A. V. SHENOY

arcian or superficial velocity defined

(7)

e values of).'= 3n, r:/ = n/4 in Equation

I

ra (1986) for C0 past a sphere in the r a drag coefficient beyond the Stokes r, an approximate solution along the is worth extending for laminar flow of umber of up to 2, taking into account,

neglected in Stokes' solution. It is efficient proposed by Goldstein (1938)

(8)

equal to X(2e 2 -n ).')/a' to maintain

nts that are dependent essentially on above series is retained in line with the

(9)

ives

(10)

tion (9) and using the definition of Re

- e)Xpu2

n+lJ (11)

t only on the pore geometry and the t for each system and can be replaced

e diameter d is related to the property ability K of the porous medium as

(12)

(13)


where K* is introduced as the modified permeability. Expression for K* for non-Newtonian power-law fluids can be written as

K* = _l_(~)n(50K)(n+1)/2 2Ci 3n + 1 3e ' (14)

where

{

25/12 (Christopher and Middleman, 1965)

(2.5)nzO -nJ/l (Kemblowski and Michniewicz, 1979)

Ci= ~(~)n(lOn -3)(75)[3(1on-3ll/<1on+11i (Dharmadhikari 3 9n + 3 6n + 1 16 and Kale, 1985).

(15)

For n = 1, the above Equation (13) gives the familiar Forchheimer equation which is often used for solving Darcy-Forchheimer flow problems (see, for example, Plumb and Huenefeld, 1981; Bejan and Poulikakos, 1984; Nakayama et al., 1988, 1989, 1990; Nakayama and Pop, 1991).

2.1. DARCY-FORCHHEIMER NATURAL CONVECTION

It is assumed that the geometrical configuration has an arbitrary curved shape, as shown in Figure 2. The wall surface temperature Tw(x) is considered to be higher than the ambient temperature Te which is assumed to be constant. The flow is considered to be occurring under natural convection conditions.

u •

L· .. . . . . . . . . . . .

• . . v. . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . , . . .. . . . . . . . . . . . . . . "' . . . . . . ' . . . . . . . . . . . . . . . . .. . ~y • • • • • , • •

> ..... > <O "-0 -0

"' u "-0

.......

Fig. 2. Schematic diagram of the physical model and its coordinate system.

226 A. V. SHENOY

Equation (13) when combined with the Boussinesq approximation, for the present case of natural convection in a power-law fluid-saturated porous medium, can be

written as

[CpK*u1-n ]1/n [K* ]1/n

U µ* jK + 1 = Ji* [pgxf3o(T - Te)] ·

The energy integral equation is given by

j__f h u(T- Te)dy = - a oT\ . dx o oy y=O

The boundary conditions on the velocity and temperature are

u(x, 0) = uw; u(x, br) = 0;

T(x, 0) = T w(x); T(x, br) = Te.

(16)

(17)

(18)

A scale analysis can now be performed using u,....,Q(UcN), x,....,Q(fc), Y"'O(br), 9x,....,Q(g) and T- Te,....,Q(LiTw = Tw.rer - Te). The energy equation based on the

above scales can be written as

(19a)

and the modified Forchheimer equation (16) scales as

CpK*U1 K* ft.Nor U~N,....,* [pgf30 LiTwJ. µ* µ

(19b)

Depending upon whether the fl.ow is slow enough to be in the pure Darcy regime or is fast enough to be in the pure Forchheimer regime, one of the scales in Equation (19b) will dominate. For the intermediate Darcy-Forchheimer regime, U cN is so

chosen as to satisfy the two limiting cases. Thus

(20)

where

(21a)

(21 b)

(21c)


- __5;_(K*2p2[gJ1oLiTw]2 GrK* - IV 2

...;K µ*

The term T1 is chosen based on Nakayama and Pop (1991) for Dar fluids. Nakayama and Pop (1991) ar of Equation (16) for Newtonian flui non-Newtonian fluids; but because o the Darcy term, one would have reso as attempted in Equation (21). Equa mation in the iteration using a form si the non-Newtonian character of the given by Equation (22) for a non-Ne Equations (19a) and (20) leads to

br ["' Ra112'

c c

where

Ra=~ o w T l (K*pg/3 LiT )lfn c ('/. µ* N·

Since there is no characteristic lengt method of Hellums and Churchill ( Rae= 1. Thus,

The nondimensional variables can n

X1 = x//c;

U1 = u/UcN;

Further, it is assumed that the gravit

9x = gx'I'

and the wall temperature is given as

<P = xf.

Thus, Equations (16) and (17) can be

A. V. SHENOY

ssinesq approximation, for the present id-saturated porous medium, can be

]

1/n

- Te)] . (16)

(17)

temperature are

(18)

sing u~O(U,N), x~O(lc), y~O(br), . The energy equation based on the

(19a)

scales as

(19b)

ugh to be in the pure Darcy regime or regime, one of the scales in Equation arcy-Forchheimer regime, UcN is so us

(20)

(21a)

(21 b)

(21c)


_ C (K*2p2[g/Jol1Tw]2-n)l/n Grx• - ft µ*2 . (22)

The term T1 is chosen based on the similarity transformation suggested by Nakayama and Pop (1991) for Darcy-Forchheimer-free convection of Newtonian fluids. Nakayama and Pop (1991) arrived at this term by solving the quadratic form of Equation (16) for Newtonian fluids. The same procedure could be applied for non-Newtonian fluids; but because of the presence of exponent n on the velocity in the Darcy term, one would have resort to the method of successive approximations as attempted in Equation (21). Equation (21a) thus represents the second approximation in the iteration using a form similar to Equation (21c) but modified to include the non-Newtonian character of the fluid. The appropriate definition for Grx• as given by Equation (22) for a non-Newtonian fluid is used. Solving the scales given by Equations (19a) and (20) leads to

(jT

T~Ra~i2 '

where

Rae= ~(K*pg/JoATw)l/n IY. µ* TN.

(23)

(24)

Since there is no characteristic length for the external flow being considered, the method of Hellums and Churchill (1964) is followed and le is chosen such that Rae= 1. Thus,

l -ry, -( µ* )1/n 1

e - K* pg/30

1'1.Tw TN.

The nondimensional variables can now be written as

X1 = x/lc;

U1 = u/UcN;

Y1 = y/lc;

<P = (Tw - T.)/(Tw,ref - T.);

br1 = br/le; r/ = Y1/C5r1;

() = (T - T.)/(Tw - T.)

Further, it is assumed that the gravity field 9x is given as

9x = gxT

and the wall temperature is given as

<P = xf.

(25)

(26)

(27)

(28)

Thus, Equations (16) and (17) can be written in the nondimensional forms as follows:

Grx• nuf + TNu~ = xT+P(), (29a)

228 A. V. SHENOY

(29b)

Equations (29a,b) are now to be solved subject to the following boundary conditions:

U1(Xi,0)=1; U1(X1,bT 1)=0;

B(xi.0)= 1; 8(xi,bT1)=0.

Combining Equations (29a) and (29b) gives

d f 1 ' {G T2 3 + T" n + 1} - m d xf (}(,II - UT, rK. Nul NU1 Xi '1 = - - -dx 1 0 bT, 011 q=o·

(30)

(31)

The velocity and temperature profiles are now to be specified. It should be noted that besides the above boundary conditions, the profiles must satisfy the following conditions for smoothness at the edges of the boundary layer:

OU1 I = 0 811 q=1 ,

081 -0 01} q= 1 - .

The temperature profile is assumed to be of the form

i

e = :L b;11i. i=O

(32a)

(32b)

(33)

It can be readily established that for satisfying the conditions given by Equations (30, 32b), the least that is needed is i = 2. The following profile is thus chosen

(34)

Choosing the velocity profile is a rather tricky problem for the Darcy-Forchheimer regime. It can be seen from Equation (29a) that the velocity profile becomes automatically defined for each of the limiting cases of Darcy and Forchheimer, respectively, once the temperature profile is selected. Thus,

for pure Darcy convection

U1 = Xlm+p)/n(l - 11)2/n,

for pure Forchheimer convection

U1 = x\m+p)/2(1 - IJ).

(35a)

(35b)

In order that the solution be valid for the two limiting cases, Equations (35a) and (35b) are both used in respective dominant terms of Equation (31). Thus

_!15 x-m{GrK*T2x3<m+p)/2 + n T" x<n+l)(m+p)/n} = 2xf dx Ti 1 4 N 1 (3n + 2) N

1 bT,. (36)


For solving the above Equation it is

bT, = B 1 xi.

Thus

B {[(s - m) + 3(m + p)/2]GrK• 1 4

+ n[(s - m) + (n + l)(m + (3n + 2)

For a similarity solution to exist, the

s - m - 1 + 3(m + p) = s - m 2

Solving Equation (39) gives

n = 2 or m = - p and s

Since the aim is to get a similarity discarded, thus leaving Equation (40b can be seen to be rather unrealistic Hence, the only way similarity would to zero. This is the well known sim Thus, for this case, m = p = O and s =

{GrK* n }112 2 -4- T~ + (3n + 2) TN = B

The local Nusselt number is defined a

Nux,D-F,N = - (08

) ~ OIJ q=O bT,

{GrK. 2 n

= -4~TN + (3n + 2)

where

Ra = ::(K*pgpojj_Tw)l/n x IX µ* .

A. V. SHENOY

(29b)

to the following boundary conditions:

(30)

= - x1 i)(JI . (31) fJT, 017 q=O

to be specified. It should be noted that e profiles must satisfy the following

'boundary layer:

(32a)

(32b)

the form

(33)

lg the conditions given by Equations (30, Bowing profile is thus chosen

(34)

y problem for the Darcy-Forchheimer 9a) that the velocity profile becomes ting cases of Darcy and Forchheimer,

selected. Thus,

(35a)

(35b)

two limiting cases, Equations (35a) and terms of Equation (31). Thus

2) T~Xln+l)(m+p)/•} = 2x1 b .

Ti

(36)

J

J

DARCY-FORCHHEIMER CONVECTION HEAT TRANSFER

For solving the above Equation it is assumed that

bT, = B1x~.

Thus

{[(s - m) + 3(m + P)/2]Grx• y2 (s-m-1)+3(m+p)/2 B NX1 1 4

229

(37)

+ n[(s - m) + (n + l)(m + p)/n] yn x<s-m- l)+(n+ l)(m+ p)/n} = 2xl-s (38) (3n + 2) N

1 B1

For a similarity solution to exist, the following equations must hold good

s - m - 1 + 3(m _+ p) = s - m - 1 + (n + I)(m + p) = p - s. (39)

Solving Equation (39) gives

1 (m + p) n = 2 or m = - p and s = 2 - ~· (40a,b,c)

Since the aim is to get a similarity solution for all values of n, Equation (40a) is discarded, thus leaving Equation (40b) as the necessary condition. But this condition can be seen to be rather unrealistic which is not achievable in practical situations. Hence, the only way similarity would exist is when m and p are both identically equal to zero. This is the well known simplest case of the isothermal vertical flat plate. Thus, for this case, m = p = 0 and s = 1/2 are substituted in Equation (38) to obtain

{Gr n } 1

12 2

4K*y~ + (3n + 2) r::i = B1° (4 l)

The local Nusselt number is defined as

where

Nux,D-F,N = (of)) X1

011 q=O fJT,

2 1/2 =~x1

B1

= {Grx• y2 + n yn }1/2 (Ra' )112 4 N (3n + 2) N x '

Ra~= Rax TN,

Rax= ~(K*pg{30 ATw)l/n a µ* .

(42a)

(42b)

(42c)

(43a)

(43b)

230 A. V. SHENOY

It is to be noted that the Nusselt number equation for pure Darcy natural convection can be easily derived from Equations (42b), (43a), and (43b) by putting Grx• = 0.

Thus,

{ n }1/2

Nux,D,N = (3n + 2) Ra;12

• (44)

2.2. DARCY-FORCHHEIMER FORCED CONVECTION

In the present case of Darcy-Forchheimer forced convection, it is a priori assumed that the geometrical configuration is a vertical flat plate with a constant wall surface temperature T w which is higher than the ambient constant temperature Te. The flow is considered to be occurring by forced convection with a uniform parallel velocity of ue. Equation (13) is the momentum equation for this case and is rewritten as

[K* ( ap )]1;n Ue[Rex• + 1]1/n = µ* - ax - pg '

where

CpK*u;-n Rex•= µ*JK .

The energy integral equation for the flow under consideration is given by

~f h Ue(T - Te)dy = - rt ar\ . dX O ay y=O

(45)

(46)

(47)

Since Equations (45), (47) are decoupled unlike in the earlier natural convection case, the solution of the above equations is rather simple. Hence, the details are not provided here. Following the same procedure as in Section 2.1, the expression for the local Nusselt number for the Darcy-Forchheimer forced convection flow can be

obtained as

Nux,D-F.F = 0.5774 Pe;12, (48)

where

(49)

2.3. DARCY-FORCHHEIMER MIXED CONVECTION

It is well known that in any heat transfer situation, density differences are bound to arise and a forced field is likely to be superimposed by natural convection effects. In forced convection if the momentum transport rates are significantly strong, then the effects of natural convection can be neglected. Similarly, if the bouyancy forces are of


relatively greater magnitude, then t ignored. But in many practical heat convection effects are of comparable o mixed convection that actually deter problems are also as difficult to solv coupling of the momentum and ener

From Section 2.1, it is clear that a natural convection exists only for th present case of Darcy-Forchheimer m geometrical configuration is a verti temperature T w which is higher than t is considered to be occurring under convection conditions. The external sidered to be uniform and parallel.

Equation (13), when combined with case of mixed convection in a powe written as

CpK*u2

n K* (ap ) µ*ft + u = Ji* ax - pg

At the edge of the boundary layer, u = be written as

CpK*u; n _ K*( ap µ*ft + Ue - µ* - ax - pg

Using Equation (51), Equation (50) is

CpK*u2 CpK*u 2

--~+un= e+un+ µ*ft µ*ft e

Solving the Darcy-Forchheimer mix an extra assumption is needed for sim is assumed that although the forced c flow in the Darcy-Forchheimer regi strong enough to include the Forchhei not unreasonable because for non-N would be very likely that Darcy r convection. Under this circumstance, t (52) is neglected and the equation is w

CpK*u 2 K* u" = µ*fie + u: + µ* [pgpo(

A. V. SHENOY

''on for pure Darcy natural convection 43a), and (43b) by putting GrK* = 0.

(44)

ION

ced convection, it is a priori assumed fiat plate with a constant wall surface nt constant temperature Te. The flow ion with a uniform parallel velocity of 'or this case and is rewritten as

(45)

(46)

er consideration is given by

(47)

in the earlier natural convection case, er simple. Hence, the details are not as in Section 2.1, the expression for the eimer forced convection flow can be

(48)

(49)

N

ation, density differences are bound to posed by natural convection effects. In

t rates are significantly strong, then the . Similarly, if the bouyancy forces are of

i

J


relatively greater magnitude, then the effects of forced convection flow may be ignored. But in many practical heat transfer situations, the forced and the natural convection effects are of comparable order and, hence, it is the combined effect of this mixed convection that actually determines the heat transfer rate. Mixed convection problems are also as difficult to solve as the natural convection, again due to the coupling of the momentum and energy equations.

From Section 2.1, it is clear that a similarity solution for the Darcy-Forchheimer natural convection exists only for the isothermal vertical fiat plate. Hence, for the present case of Darcy-Forchheimer mixed convection, it is a priori assumed that the geometrical configuration is a vertical flat plate with a constant wall surface temperature T w which is higher than the ambient constant temperature Te. The flow is considered to be occurring under the combined effects of forced and natural convection conditions. The external velocity ue for the forced convection is considered to be uniform and parallel.

Equation (13), when combined with the Boussinesq approximation, for the present case of mixed convection in a power-law fluid-saturated porous medium, can be written as

CpK* u2 " K* (op ) K* µ*ft + U = µ* OX - pg + Ji*[pgf3o(T- Te)J. (50)

At the edge of the boundary layer, u = ue and T = Te. Thus, the above equation can be written as

CpK*u; "_ K*( op ) IV + Ue - - - - - pg µ* v ft µ* OX .

(51)

Using Equation (51), Equation (50) is rewritten as

CpK*u2 n CpK*u2 K* IV+u- e+n µ*..._;ft - µ*ft Ue + J:*[pgf3o(T- Te)J. (52)

Solving the Darcy-Forchheimer mixed convection case is a nontrivial task. Hence, an extra assumption is needed for simplification before a solution is sought. Thus, it is assumed that although the forced convection velocity is high enough to have the flow in the Darcy-Forchheimer regime, the natural convection currents are not strong enough to include the Forchheimer term in Equation (52). This assumption is not unreasonable because for non-Newtonian fluids with higher consistencies, it would be very likely that Darcy regime may prevail most often for natural convection. Under this circumstance, the first term of the left-hand side of Equation (52) is neglected and the equation is written in the form

Un = CpK*u; n K* µ*ft + Ue +Ji* [pgf3o(T - Te)J. (53)

232

The energy integral equation is given by

~f<lT u(T- T.)dy = - tt.~TI . dx o vy y=O

The boundary conditions on the velocity and temperature are

u(x, 0) = uw;

T(x,O) = Tw;

u(x, bT) = u.,

T(x,bT) = T •.

A. V. SHENOY

(54)

(55)

A scale analysis can now be performed using u-O(UeM), x"'O(lc), y-O(bT), and T- Te-O(ATw = Tw - Te). The energy equation based on the above scales can be

written as

ueMATwbT ATw le "'ti.Tr.

From Equation (53), UeM is chosen as

UcM"'uZ[ReK* + 1] + [:: [pgf3oATw] J1

n.

Solving the scales given by Equation (56a) and (56b) leads to

JT ["' Pe112•

c c,M

where

and

(56a)

(56b)

(57)

(58)

(59)

(60)

Since there is no characteristic length for the external flow being considered, the method of Hellums and Churchill (1964) is followed and le is chosen such that

PeeM = 1. Thus,

le= tt./UeM· (61)

The nondimensional variables can now be written as

X1 = x/le

U1 = u/UcM; (62)

DARCY ~FORCHHEIMER CONVECTION

Thus Equations (53) and (54) can be

u1 = [1 + (Rax/Pe~t8]/[1 + (

Combining Equations (63a) and (63b

__i_ (1 b [1 + (Rax/Pe~rBJ 11" dx1 Jo T, [1 + (Rax/Pe~tJ 1 i•

The integration in Equation (64) cann most important contributing terms i terms which actually account for t convection. Thus,

The temperature profile used earlier by Equation (34) can be used in the p Equation (64) simplifies to

d L1 (1 - l'/)2 di} + (Rax/P

~d JT,~~~~~~~~---1 X1 [1 +(Rax/

Integration of Equation (66) is straigh expansion was used, the result of the assuming that the following holds go

[1/3] + [n/(3n + 2)](Rax/Pe~)-

Using Equation (67) and following the Darcy-Forchheimer natural convec'ti derived as

Nux.D~F,M = {[1/3]" + [n/(3n +

or

N 2n N 2n N 2n Ux,D-F.M = Ux,D-F,F + Ux,D,N

It is worth noting that Equation (68b) equations for combined laminar force by Churchill (1977) and Ruckenstein (1980a,b) for non-Newtonian fluids in interpolate the two extremes of forced

A. V. SHENOY

(54)

tetnperature are

(55)

g u~O(UcM), x,.._,O(U, Y"'O(br), and tion based on the above scales can be

(56a)

l/n (56b)

d (56b) leads to

(57)


Thus Equations (53) and (54) can be written in the nondimensional forms as follows:

u~ = [1 + (Rax/Pe~)"()]/[1 +(Rax/Pe~)"]

d fl 1 8()1 - Dr,u1()d17 = - -- . dx1 o br, 017 ~=o

Combining Equations (63a) and (63b) gives

d fl [1 +(Rax/Pe~)"()] 11" 1 OfJI - Dr, ' 1/ () d17 = - -- . dx1 0 [1 + (Rax/Pexn n Dr, 017 ~=O

(63a)

(63b)

(64)

The integration in Equation (64) cannot be easily done. Hence, it is assumed that the most important contributing terms in the series expansion are the first and the last terms which actually account for the two limiting cases of forced and natural convection. Thus,

[1 + (Rax/Pe~)"()] 11""' 0(1 + (Rax/Pe~)() 1 i"). (65)

The temperature profile used earlier for the pure natural convection case and given by Equation (34) can be used in the present case of mixed convection as well. Thus, Equation (64) simplifies to

d L1 (1 - 17)2 d17 + (Rax/Pe~) f

0

1 (1 - 17)<2n+ 2 >/n d17 l o()

-bTt I 1/ = ---1 • (66) dx1 [1 + (Rax/Pex)"J " br, 017 ~=o

Integration of Equation (66) is straightforward. Since an approximation of the series (58) I expansion was used, the result of the integration is reverted back into the series by

assuming that the following holds good.

(59) [1/3] + [ n/(3n + 2)](Rax/Pe~)"' 0([1/3]" + [n/(3n + 2)]n(Rax/Pe~)") 1 1•. (67)

Using Equation (67) and following the procedure identical to that used earlier for the Darcy-Forchheimer natural convection case, the Nusselt number can be easily

(60) I derived as

the external flow being considered, the s followed and le is chosen such that

(61)

ritten as

17 = yif Dr,

(62)

Nux,D-F,M = {[1/3]" + [n/(3n + 2W(Rax/Pe~)n)} 1 i2"(Pe~) 1 1 2 (68a)

or

Nu~:'o-F,M = Nu~~o-F,F + Nu~'.b.N· (68b)

It is worth noting that Equation (68b) is of a form similar to those of the correlating equations for combined laminar forced and free convection heat transfer suggested by Churchill (1977) and Ruckenstein (1978) for Newtonian fluids and by Shenoy (1980a,b) for non-Newtonian fluids in homogeneous media. Such equations which interpolate the two extremes of forced and free convection have been shown to give

234 A. V. SHENOY

reasonably accurate results by these authors. In the derivation of Equation (68b), it was assumed that the forced convection flow through the porous media was in the Darcy-Forchheimer regime, while natural convection was still in the Darcy region. In order to get the equation which is valid for mixed convection when both forced and natural convection are in the Darcy-Forchheimer region, it is reasonable to expect that the above correlating equation would hold good when the appropriate Nusselt number for this case, i.e. Nux,D-F,N is used in place of Nux,D,N in Equation

(68b). Thus,

Nu;~D-F,M = Nu;~D-F,F + Nu;~o F.N (68c)

or

~;:·~~~~;M = {[1/3]" + [(Gr* /4)T~ + { n/(3n + 2)} TN]"(Ra~/Pe~)")}1 12 ". (68d)

It is to be noted that the Nusselt number equation for pure Darcy mixed convection can be easily derived from Equations (68a) by putting Rex• = GrK* = 0. Thus,

Nux,D,M = {[1/3]" + [n/(3n + 2)]"(Rax/Pex)")}1i2"(Pex)1i2 (69a)

or

(69b)

3. Results and Discussion

3.1.1. Darcy-Forchheimer Natural Convection

The first step is to check the propriety of Equation (42c) for Newtonian fluids when n = 1 and it simplifies to the following form

(70)

where the Newtonian fluids

(71a)

(7lb)

(71c)

(71d)


Table I. Nux,0-F.N/(Ra~) 1 i 2 for Darcy-Forchh

GrK. Present work Plumb-Huenefeld (1 Equation (70) Exact solution

0 0.4472 0.4439 I0-2 0.4456 0.4423 10-1 0.4325 0.4297 1 0.3680 0.3662

IO 0.2528 0.2513 I02 0.1527 0.1519

The predictions of Equation (70) literature, as shown in Table I, give t thus instilling confidence in the s analysis.

The effect of the non-Newtonian ch (42c). The predictions of this equatio index n are shown in Table II. For val n, the heat transfer rates deviate fr however, at GrK* beyond 0.1, there is porous inertia effect. With increasing heat transfer rate is lower at all value rate increases for all values of GrK* fo

3.1.2. Darcy Natural Conection

The predictions of Equation (44) are c in Table III and are found to give the similarity solution. Decreasing n re enhances it.

Table II. Nux.D-F,N/(Ra~) 1 i2 for Darcy-Forchh

GrK, Present work, Equation (42c

n = 0.25 n=

0 0.3014 0.37 I0-2 0.3012 0.37 10-1 0.2952 0.36 1 0.2485 0.3

IO 0.1728 0.21 I02 0.1I09 0.13

A. V. SHENOY

n the derivation of Equation (68b), it hrough the porous media was in the vection was still in the Darcy region. mixed convection when both forced

rchheimer region, it is reasonable to uld hold good when the appropriate used in place of Nux,o,N in Equation

(68c)

(3n + 2)} TNT(Ra~/Pe~)")}1 12 ". (68d)

tion for pure Darcy mixed convection

putting ReK* = GrK* = 0. Thus,

ex)")} 1;2n(Pex)1/2 (69a)

(69b)

ation (42c) for Newtonian fluids when

(70)

(71a)

(71 b)

(71c)

(71d)

I

I


Table I. Nux,D-F.N/(Ra~) 1 i 2 for Darcy-Forchheimer natural convection flow of Newtonian fluids

GrK. Present work Plumb-Huenefeld (1981) Bejan-Poulikakos (1984) Nakayama et al. (1988) Equation (70) Exact solution Approximate solution Integral solution

0 0.4472 0.4439 0.5000 0.4205 10-2 0.4456 0.4423 0.4992 0.4191 10-1 0.4325 0.4297 0.4912 0.4085

l 0.3680 0.3662 0.4317 0.3528 10 0.2528 0.2513 0.2973 0.2435 102 0.1527 0.1519 0.1779 0.1466

The predictions of Equation (70) when compared with those available in the literature, as shown in Table I, give the closest match to the exact solution values, thus instilling confidence in the solution procedure employed in the present analysis.

The effect of the non-Newtonian character of the fluid is brought out by Equation (42c). The predictions of this equation for certain typical values of the power-law index n are shown in Table II. For values of GrK. up to 0.1, it can be seen that for all n, the heat transfer rates deviate from the pure Darcy value by less than 5%; however, at GrK. beyond 0.1, there is a rapid drop in the heat transfer rate due to the porous inertia effect. With increasing pseudoplasticity, i.e. decreasing values of n, the heat transfer rate is lower at all values of GrK*· On the other hand, the heat transfer rate increases for all values of GrK. for dilatant fluids, i.e. for increasing n.

3.1.2. Darcy Natural Conection

The predictions of Equation (44) are compared with those available in the literature in Table III and are found to give the closest agreement to the results from the exact similarity solution. Decreasing n reduces the heat transfer while increasing n

enhances it.

Table II. Nux.D-F.N/(Ra~) 1 i 2 for Darcy-Forchheimer natural convection flow of power-law fluids

GrK• Present work, Equation (42c)

n = 0.25 n = 0.5 n = 1.5 n = 2.0

0 0.3014 0.3779 0.4802 0.4993 10-2 0.3012 0.3763 0.4785 0.4953 10-1 0.2952 0.3634 0.4648 0.4683

1 0.2485 0.3048 0.3992 0.4115 10 0.1728 0.2142 0.2678 0.2745 102 0.1109 0.1360 0.1567 0.1577

236 A. V. SHENOY

Table III. Nux.o.N/Ra!12 for Darcy natural convection flow of power-law fluids

n Present work Chen-Chen (1988a) Want-Tu (1989) Nakayama-Koyama (1991) Equation (44) Exact solution Exact solution Approximate solution

0.4 0.3536 0.353 0.3333 0.5 0.3780 0.3768 0.3535 0.8 0.4264 0.4238 0.424 0.3922 1.0 0.4472 0.4437 0.444 0.4082 1.2 0.4629 0.459 0.4201 1.5 0.4804 0.4752 0.475 0.4330 2.0 0.5000 0.4938 0.4472

3.2. FORCED CONVECTION

Equation (48) shows that the heat transfer correlation for forced convection boundary-layer flow of non-Newtonian fluids in porous media is the same as that for the Newtonian case in Darcy-Forchheimer as well as pure Darcy flows, when the Peclet number is defined on the basis of the external flow velocity. For the isothermal flat vertical plate, it can be seen from Nakayama and Pop (1991) that this coefficient is equal to 0.5641 if an exact similarity solution is obtained. The present coefficient of 0.5774 in Equation (48) differs by only about 2.3%, although it is obtained by using the approximate integral method.

3.3.1. Darcy-Forchheimer Mixed Convection

No information exists in the literature for Darcy-Forchheimer mixed convection for Newtonian or non-Newtonian fluids and, hence, the propriety of Equations (68aH68d) cannot be checked. The variations in the Nusselt number for changes in the extent of the superposition of the natural convection on the forced convection are given in Table IV for certain typical values of the power-law index n. For pseudoplastic fluids, i.e.

I I


Table IV. Nux.D-F.M/(Pe~) 112 for Darcy-Forchhe

Ra~ Pres

Pe~ n=

0 0 0.57 0.5 0.8 1.0 0.95

15 2.04 50 3.25

100 4.35 200 5.92 400 8.13

0.01 0 0.57 0.5 0.84 1.0 0.95

15 2.04 50 3.26

100 4.37 200 5.94 400 8.16

0.1 0 0.57 0.5 0.85 1.0 0.96

15 2.08 50 3.33

100 4.47 200 6.09 400 8.37

0 0.57 0.5 0.87 1.0 0.99

15 2.19 50 3.53

100 4.75 200 6.49 400 8.93

decreasing values of n, the rate of increase in heat transfer slows down for larger and 10 0 0.57

larger values of the ratio Ra~/Pe~. The exact opposite trend is seen to prevail for dilatant fluids, i.e. for increasing n. With increasing GrK., the heat transfer coefficient given in Table IV shows an increasing trend for all values of n.

3.3.2. Darcy Mixed Convection

The predictions of Equation (69b) are compared with those available in the literature in Table V and are found to give very close agreement to the results from the exact solution. For values ofRax/Pex greater than zero, it can be seen that the heat transfer rates increase for all values of n. For pseudoplastic fluids, i.e. decreasing values of n, the rate of increase in heat transfer slows down for larger and larger values of the ratio Rax/ Pex. The exact opposite trend is seen to prevail for dilatant fluids, i.e. for increasing n.

100

0.5 1.0

15 50

100 200 400

0 0.5 1.0

15 50

100 200 400

0.89 1.02 2.3 3.73 5.04 6.90 9.52 0.57 0.9 1.04 2.39 3.88 5.26 7.19 9.94

A. V. SHENOY DARCY-FORCHHEIMER CONVECTION HEAT TRANSFER 237

w of power-law fluids Table IV. Nux,D-F,M/(Pe~) 112 for Darcy-Forchheimer Mixed Convection flow of power-law fluids

ant-Tu (1989) Nakayama-Koyama (1991) GrK. Ra~ Present work, Equation (68d)

act solution Approximate solution Pe~ n = 0.5 n = 1.0 n = 1.5

53 0.3333 I 0.3535 I 0 0 0.5774 0.5774 0.5774

0.3922 0.5 0.8446 0.6583 0.6142

0.4082

j 1.0 0.9553 0.7303 0.6719

0.4201 15 2.041 1.826 1.879

0.4330 50 3.250 3.215 3.402

0.4472 I JOO 4.357 4.509 4.807

200 5.923 6.351 6.795

400 8.137 8.963 9.608

0.01 0 0.5774 0.5774 0.5774

0.5 0.8457 0.6585 0.6142

1.0 0.9569 0.7306 0.6719

r correlation for forced convection 15 2.047 1.828 1.879

in porous media is the same as that for 50 3.261 3.2J8 3.402

100 4.373 4.515 4.807

s well as pure Darcy flows, when the 200 5.945 6.359 6.795

ternal flow velocity. For the isothermal

I 400 8.168 8.974 9.608

a and Pop (1991) that this coefficient is 0.1 0 0.5774 0.5774 0.5774

0.5 0.8530 0.6599 0.6142

is obtained. The present coefficient of I

I 1.0 0.9672 0.7332 0.6719

3%, although it is obtained by using the I 15 2.087 1.843 1.879

50 3.334 3.247 3.403

100 4.476 4.556 4.807

200 6.091 6.4J7 6.796

400 8.375 9.056 9.609

0 0.5774 0.5774 0.5774

0.5 0.8730 0.6655 0.6158 rcy-Forchheimer mixed convection for 1.0 0.9955 0.7433 0.6759

e, the propriety of Equations (68a)-{68d) 15 2.197 1.903 1.908

selt number for changes in the extent of 50 3.534 3.360 3.457

the forced convection are given in Table 100 4.758 4.716 4.884

200 6.490 6.645 6.904

index n. For pseudoplastic fluids, i.e. 400 8.939 9.379 9.762

heat transfer slows down for larger and 10 0 0.5774 0.5774 0.5774

t opposite trend is seen to prevail for 0.5 0.8936 0.6720 0.6J 78

1.0 1.025 0.7549 0.6805

asing GrK., the heat transfer coefficient J5 2.309 1.970 1.942

for all values of n. 50 3.739 3.487 3.519

100 5.049 4.897 4.972

200 6.90J 6.902 7.029

400 9.52J 9.743 9.939

100 0 0.5774 0.5774 0.5774

d with those available in the literature in l 0.5 0.9084 0.6753 0.6184

greement to the results from the exact 1.0 1.046 0.7606 0.6819

15 2.391 2.003 1.951

zero, it can be seen that the heat transfer l 50 3.888 3.549 3.537

astic fluids, i.e. decreasing values of n, the JOO 5.260 4.986 4.997

larger and larger values of the ratio Rax/ ~ 200 7.199 7.027 7.064

400 9.942 9.92J 9.989

il for dilatant fluids, i.e. for increasing n.

238

V)

d II

"

A. V. SHENOY

I I DARCY-FORCHHEIMER CONVECTION

4. Concluding Remarks

The present work derives the govern inelastic non-Newtonian power-law t provides a comprehensive treatment heat transfer during Darcy-Forchhe porous media. A search for a similari for the simplest case of the isothermal

The approximate integral method often skepticism about the reliability o work, a comparison of the results at solutions and found to give very ace simple forms of equations such as Eq (70) for various cases.

In natural convection, the porous in rate for all n at values of GrK• beyon heat transfer rate decreases with decre

In forced convection, the heat transfi is, therefore, the same for Newtonian

In mixed convection, the effect of i the forced convection, leads to an incr Ra~/Pe~ < 10, the heat transfer rate is values of n, the rate of increase in heat effects become stronger and stronger transfer rate for the dilatant fluid bee fluid for very large values of Ra~/Pe~.

References

Bejan, A., 1987, Convective heat transfer in por Handbook of Single-Phase Convective Heat Ti

Bejan, A. and Poulikakos, D., 1984, The non-Da in a porous medium, Int. J. Heat Mass Trans

Brinkman, H. C., 1947, A calculation of the vise particles, Appl. Sci. Res. Al, 27-34.

Chhabra, R. P., 1986, Steady non-Newtonian fl Vol I, Gulf Publishing Co., Ch. 30, pp. 983-1

Chen, H-T. and Chen, C-K., 1988a, Free convec embedded in a porous medium, Trans ASME

Chen, H-T. and Chen, C-K., 1988b, Natural c cylinder and a sphere in a porous medium, In

Cheng, P., 1978, Heat transfer in geothermal sys Christopher, R. H. and Middleman, S., 196

Fundamentals 4, 422-426. Churchill, S. W., 1977, A comprehensive corr

convection, AIChE J. 23, 16.

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DARCY-FORCHHEIMER CONVECTION HEAT TRANSFER

4. Concluding Remarks

239

The present work derives the governing equation for Darcy-Forchheimer flow of inelastic non-Newtonian power-law type fluids through porous media. The study provides a comprehensive treatment of the natural, forced, and mixed convection heat transfer during Darcy-Forchheimer flow of power-law fluids in saturated porous media. A search for a similarity solution revealed that similarity exists only for the simplest case of the isothermal vertical flat plate.

The approximate integral method was followed throughout the study. There is often skepticism about the reliability of the integral method. However, in the present work, a comparison of the results at every stage has been done with existing exact solutions and found to give very accurate results. The elegant solution has given simple forms of equations such as Equations (42c), (44), (48), (68a), (68d), (69a) and (70) for various cases.

In natural convection, the porous inertia effect results in a drop in the heat transfer rate for all n at values of GrK* beyond 0.1. Further, for any fixed value of GrK., the heat transfer rate decreases with decreasing value of the power-law index n.

In forced convection, the heat transfer correlation remains unchanged for all n and is, therefore, the same for Newtonian and non-Newtonian power-law fluids.

In mixed convection, the effect of increasing the extent of natural convection on the forced convection, leads to an increase in heat transfer for all n. At low values of Ra~/Pe~ < 10, the heat transfer rate is higher for lower n. However, with decreasing values of n, the rate of increase in heat transfer slows down when natural convection effects become stronger and stronger. Hence, there are situations when the heat transfer rate for the dilatant fluid becomes greater than that for the pseudoplastic fluid for very large values of Ra~/Pe~.

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DARCY-FORCHHEIMER CONVECTION H

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A. V. SHENOY

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