YANG Xiao, LIU Xuemei

Thermally developing forced convection and the corresponding thermal stresses in a porous plate channel

© Higher Education Press and Springer-Verlag 2007

manufactured with resin flow with high temperature in the fibrous materials [3].

One of the most important characteristics of the porous medium is its larger contact surface between the solid and pore fluid, which increases the macroscopic heat exchange coefficients and enhances heat transfer efficiency. In order to precisely investigate the thermo-hydro-mechanical couplings in porous media, behavior of the fluid flow and heat exchange should be investigated in the pore scale, and the mechanisms of the macroscopic phenomena are revealed from the microscale, such as improvement of the heat transfer effici-ency. However, it is very difficult to precisely investigate behavior of the porous media in such a manner because of the complexities of the pore structures. Therefore, with homo-genization method, macroscopic mathematical model are employed on a representative elementary volume (REV) which comprised particles of the solid skeleton and the pore fluid, and the macroscopic behavior is revealed with the macroscopic constitutive equations, in which the effect of the pore structures is considered. At present it has been proved that such a homogenization method is very useful for solving many problems [4].

In the field of heat and mass transfer, if the heat exchange between the solid skeleton and pore fluid is completely achieved, the temperatures of the solid skeleton and pore fluid are the same in an REV, and then, the analysis can be conducted on the model of the local thermal equilibrium (i.e., one-energy equation model). Based on this model, the ther-mal conduction effect along the direction of fluid flow can be omitted for sufficiently large Péclet number (Pe), i.e., the case of the fully developed heat transfer, and vast achievements have been made [4], for instance, Sheikh [5] investigated the heat transfer for Brinkman fluid in porous media imbedded inside various shaped ducts, and the comparison of the results between the modified Graetz method and the method of weighted residuals are presented. For the heat transfer that takes the conduction along the direction of the fluid flowing into consideration, Simacek [6] analyzed the heat transfer of the Darcy resin flow in a porous medium for the manufac-turing processes of the composite materials. The series solutions of the steady temperature in the porous medium are

Front. Mech. Eng. China 2007, 2(1): 57–61DOI 10.1007/s11465-007-0009-7

RESEARCH ARTICLE

Abstract Based on the Darcy fluid model, by considering the effects of viscous dissipation due to the interaction between solid skeleton and pore fluid flow and thermal conduction in the direction of the fluid flow, the thermally developing forced convection of the local thermal equili-brium and the corresponding thermal stresses in a semi-infinite saturated porous plate channel are investigated in this paper. The expressions of temperature, local Nusselt number and corresponding thermal stresses are obtained by means of the Fourier series, and the distributions of the same are also shown. Furthermore, influences of the Péclet number (Pe) and Brinkman number (Br) on temperature, Nusselt number (Nu) and thermal stress are revealed numerically.

Keywords Brinkman number, local thermal equilibrium, Nusselt number, Péclet number, saturated porous medium, thermal stress, thermally developing forced convection

1 Introduction

The porous media theory was first developed in the fields of mining, soil mechanics, and hydrogeology [1], in which the main focuses were on the behavior of the isothermal fluid flow and deformation of solid skeleton. Recently, behavior of nonisothermal fluid flowing in porous media has attracted the attention of many researchers. It is mainly because the hyperporous media have been used extensively in energy storage units in power systems, solar heat exchange equipments and in cooling system of electron devices [2]. In addition, reinforced fibrous polymer materials can be

Translated from Journal of Lanzhou University (Natural Science Edition), 2006, 42(2): 114–119 [译自: 兰州大学学报(自然科学版)]

YANG Xiao ( )Department of Mechanics, Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200444, ChinaE-mail: xyang@staff.shu.edu.cn

LIU XuemeiBox 339, Chang’an University, Xi’an 710064, China

58

obtained, and the temperature profiles were discussed in detail. It was shown that the low-order approximation solution of the temperature can give good results for the area away from the resin flow inlet. Furthermore, the transient temperature profile was obtained when the thermal conduc-tion along the direction of the fluid was neglected. Based on the Brinkman fluid model and the modified Graetz method, Kuznetsov [7] examined the thermal developing forced convection in a circular porous medium duct by considering the viscous dissipation of the fluid. The Nusselt number (Nu) was expressed and a parameter study was conducted. However, to the best of authors’ knowledge, thermal stresses in porous medium have not been investigated widely in the field of heat and mass transfer.

Assuming the local thermal equilibrium and based on the Darcy fluid model, the thermally developing forced convec-tion and the corresponding thermal stresses in a semi-infinite porous plate channel are investigated in this paper. Firstly, by neglecting the effect of the porous medium deformation on the temperature, taking into account the viscous dissipation due to the interaction between solid and pore fluid and effect of the thermal conduction in the direction of the fluid flow, the expressions of temperature and local Nu are obtained by means of the Fourier series and the superposition principle. Secondly, by neglecting the longitudinal displacement, the corresponding thermal stresses are derived with thermoela-sticity. Finally, the local Nu, temperature profile, and thermal stresses are examined numerically, and the effects of various parameters on them are discussed.

2 Mathematical model

Consider an ideal Darcy fluid in a homogeneous porous medium parallel-plate channel with thickness H and length L (L � H) (as shown in Fig. 1). According to the porous media theory [8, 9], if the pore pressures are p0 and p1 at the two channel ends x = 0 and x = L, respectively, and the upper and lower surfaces are impermeable, in the steady state, the velocity of the pore fluid flow is in the direction of ox-axis and equals to a constant w = w(p1-p0) / (Sn L), where w is the porosity; Sv is the coupling coefficient between the solid and fluid phases, related with the Darcy permeability KF by Sv = w2cFR/KF, where cFR is the real density of the fluid.

fluid-saturated porous medium and the equilibrium equation of the solid skeleton in oy direction are

kx y

c wx

S wyy

yy

yy

2

2

2

22 0

h hr

hn+ + =-

⎛⎝⎜

⎞⎠⎟

F F (1)

( )l m m ahS S

2 S

2S

2 S

2S S+ + - =2 3 0

yy

yy

yy

u

y

u

xK

y (2)

where h(x, y) is the temperature of porous medium mixture; uS(x, y) is the displacement of the solid skeleton in the direc-tion of oy-axis; lS and mS are the macroscopic Lamé constants of the solid skeleton, and ΚS = lS + 2mS/3, and aS is the thermal expansion coefficient of the solid skeleton; rF and cF are the macroscopic mass density and the specific heat of the fluid flow, respectively. If real thermal conduction co-efficients of the solid skeleton and fluid flow are kS and kF, respectively, the effective thermal conduction coefficient of the mixture is k = wkS + (1 – w)kF.

Assume that the temperatures of the upper and lower unmovable surfaces are h0, the temperature of the mixture at the fluid inlet x = 0 is hin and the boundary x = 0 is traction-free. Then, the boundary conditions for Eqs. (1) and (2) are as follows

h h

h h s

h

= = =

= = =

+

0

in

S

, , ,

, ,

,

u y H

x

u x

xy

S

SE

finite,

0 0

0 0

m m ∞

⎧

⎨⎪

⎩⎪

(3)

Introducing following dimensionless quantities

y

y

Hx

x

Lu

u

Hc Pe

S= = = =

= = =

, , ,

, ,

max

max max max

SS

F

in

in

a hl

hh

hh

h

hh

h

h0

0 ,, l=H

L

⎧

⎨⎪⎪

⎩⎪⎪

(4)

where hmax = max{h0,hin}, Péclet number Pe = rFcFwH/k, and Brinkman number Br = Snw2H2 / (hmaxk).

For simplification, the bar ‘–’ over the dimensionless quantities are omitted in the following equations. Using the constitutive equation s mxy u xSE S S= y y , the following dimensionless boundary value is obtained

lh h h2

2

2

2yy

yy

yyx y

cx

Br+ - + =2 0F

(5)

yy

yy

yy

2 S

2

2 2 S

2

u

y

u

x y+

--

-+-

( )

( )

1 2

2 1

1

10

n l

n

n

n

h=

(6)

h h

h h

h

= = =

= = =

+

0 , , ,

, ,

,

u y

u x x

u x

S

inS

S finite,

0 0 1

0 0y y

m → ∞

⎧

⎨⎪

⎩⎪

(7)

where, n is the Poisson constant of the solid skeleton.

Fig. 1 Flow pattern in a porous medium

Assume that the heat exchange between solid skeleton and fluid flow in an RVE is fully developed to a local thermal equilibrium. Neglecting displacement in the direction of the ox-axis due to L � H, the energy equation of the

59

It is easy to get a particular solution of Eq. (5) by

h h* = - +Br

y y2

1( ) 0 (8)

Set h(x, y) = T(x, y) + h*(y), then the boundary value problem for T(x, y) is

l

h h

22

2

2

2F

0

f

yy

yy

yy

m

T

x

T

yc

T

x

T y

T Br y y x

T

+ - =

= =

= - - - =

0

0 0 1

21 0

, ,

( ),in

iinite xm+∞

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

(9)

With separation variable method, the solution of boundary value problem is

T C e k ykxk= +-

=

b sin( )2 1 pK 0

∞

∑ (10)

where

bl

lk =

+ + -( ) ( )c k cF 2 2 2 2 F

2

4 2 1

2

p (11)

Ck

Br

kk =-+

-+

4

2 1

4

2 1

( )

( ) ( )

h hin 03 3p p

(12)

Therefore

h hb= + +C k ykxke-

k

y=

p0

* ( )∞

∑ sin( )2 1

(13)

Using Eq. (13), the local Nu can be calculated as

Nu = q / (h0 - have) (14)

where

q Br k C

C

k

kx

kx

=- - +

= + ++

1

22 1

2

2 1

( )

( )

p

p

e

e

-

=

-

=

b

b

h h

k

k

k

k

Br

0

ave 0012

∞

∑∞∞

∑

⎧

⎨⎪⎪

⎩⎪⎪

(15)

Similarly, using Eq. (6) and corresponding boundary condi-tions, the displacement of the solid skeleton is calculated as

u u x n ynS S

1

==

( )sin( )2 pn

∞

∑ (16)

where

u x BBr

nDn n

xnk

xS e e( )( )

( )= -

+-

+=

- -a bn l

nn k

k

1

4 1

2

3 30p

∞

∑

(17)

Dn k C

n n knkk=

+ +- - - - +

16 1 2 1

1 2 8 1

1

4 2 1

( ) ( )

( ) ( ) ( )

n

n l b n2 2 2 2 2 2k p

⋅ (18)

al

n

nn =

--

2 2 1

1 2

np ( )

(19)

Bn k C

n

n

nk k

k

=+ +

- - -16 1 2 1

8 1 1 2

1

4

0

( ) ( )

( ) ( )

n b

n n l b

a

p2 2 2 2

2

k

n

⎡

⎣⎢ ⋅

∞

∑=

-- +( )2 1k 2⎡⎣ ⎤⎦

⎤

⎦⎥⎥

(20)

With the expression uS(x,y), effective thermal stresses of the solid skeleton are obtained

Tu

y

T

yy

xx

=+

= -+-

=+

s

l m a h

n

nh

s

l m a h

yy

xx

SE

S S Smax

S

SE

m

( )

( )

2

1

1

2

yy

S S Saax

SE

max

=-

-+-

= =

n

n

n

nh

s

m a hl

1

1

1

yy

yy

u

y

Tu

xxy

S

S S

Sxy

(21)

3 Numerical results and discussions

Usually, the length L of the channel is much higher than its thickness H. Here it takes l = H/L = 0.01 [6], and also assume h0 = 0, hin = 1, and n = 0.3. For different values of Pe, Br, the temperature h, local Nu and stresses, Eq. (21) can be obtained numerically. First, the effect of viscous dissipa-tion is examined. For the fluid without viscous dissipation (Br = 0) and with viscous dissipation (Br = 100), influences of Pe on Nu are shown in Fig. 2. It can be seen that Pe has a great influence on Nu for developing forced convection. Nu increases with increase in Pe, and even attains the values of fully developed forced convection. At fluid inlet, Br has little influence on Nu. However, for the fluid without viscous dis-sipation (Br = 0), Nu reaches a limit of 4.935 as x increases, which is equal to the accurate value of p2/2 of Darcy flow. For the fluid with viscous dissipation, value limit of Nu is 6, which is in good agreement with 5.953 obtained by means of Graetz method [10] and it does not depend on Br.

Fig. 2 Distribution of local Nusselt number Nu versus ox axial coordinate

The influence of the Pe on temperature profile h for invis-cid fluid is shown in Fig. 3. It is shown that the heat exchange region becomes larger with increase in Pe. When Pe = 1, the heat exchange region locates in the vicinity of fluid inlet x = 0. When x = 0.01, temperature h is very low, and when x = 0.04, h reaches nearly zero, i.e., there is no heat exchange

60

Furthermore, due to the viscous dissipation, the highest temperature may be more than 1. With increasing Br, the region of h>1 extends and the value of the temperature h increases. Fig. 6 shows the effect of Br on the effective thermal stress Tyy when Pe = 100. It is revealed that two extreme points of thermal stress in the vicinity of the fluid inlet disappear as Br increases. When Br>10, the change of the temperature with x is very small in the porous plate channel, and this phenomena is more obvious when x>0.6. Meanwhile, when Br reaches a certain value, the thermal stress Tyy becomes tensile near both sides of the porous plate channel in the vicinity of fluid inlet.

Similarly, stresses Txy and Txx also can be analyzed numeri-cally, but they are not presented here due to space limitation.

4 Conclusions

Considering the viscous dissipation of the pore fluid, ther-mally developing forced convection and the corresponding

Fig. 3 Temperature profiles for neglecting viscous dissipation of the fluid

Fig. 5 Temperature profiles for viscous dissipation of the fluid when Pe = 100

Fig. 4 Stress profiles for neglecting viscous dissipation of the fluidFig. 6 Stress profiles for viscous dissipation of the fluid when Pe = 100

in the region of x>0.04. The heat exchange region extends with increase in Pe. Numerical results show that, when Pe = 100, the heat exchange region is 0<x<0.5, and when Pe = 200, the heat exchange region is 0<x<0.8, but when Pe = 500, the heat exchange exists in the entire region of 0<x<1.

The effect of Pe on the effective thermal stress profile Tyy for inviscid fluid, i.e., neglecting the viscous dissipation of the fluid, is shown in Fig. 4. It can be seen that thermal stress changes sharply along oy direction in the vicinity of fluid inlet x = 0, and there exist two extreme points, which are almost independent of Pe. Furthermore, the thermal stress exists only in the heat exchange region, and it increases with increase in Pe. However, thermal stress distributes uniformly along oy direction in the region away from the fluid inlet and decreases in the direction of ox-axis.

The effect of Br on temperature h for Pe = 100 is given in Fig. 5. It is shown that, for higher values of x, the temperature h is nonzero. Numerical results show that the changes of the temperature with x are very small in the region of x>0.4.

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thermal stresses in a porous parallel-plate channel are discussed in this paper. Numerical results show that Pe has a remarkable influence on the local Nu, and it increases with increase in Pe. When the viscous dissipation is neglected, the limit of Nu is 4.935, and when viscous dissipation is consid-ered, it is 6. The effect of the viscous dissipation on the temperature profile is remarkable. Since viscous dissipation is equivalent to a heat source, the temperature h may be more than 1. It is proved that the temperature increases and the region extends with increase in Br. In addition, the viscous dissipation has also a remarkable influence on the thermal stresses. With increasing Br, two extreme points of thermal stress in the vicinity of the fluid inlet x = 0 disappear, the distributions of the stress along the ox-axis approach almost uniform, and the Tyy may become tensile.

Acknowledgements This study was supported by the National Natural Science Foundation of China (Grant No. 10272070) and the Development Foundation of the Education Commission of Shanghai, China.

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