Effects of a random porosity model on heat transfer performance of porous media

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\

PERGAMON International Journal of Heat and Mass Transfer 31 "0888# 02Ð14

E}ects of a random porosity model on heat transfer

performance of porous media

Wu!Shung Fu\ Hsin!Chien Huang

Department of Mechanical Engineering\ National Chiao Tung University\ Hsinchu\ 29949\ Taiwan\ Republic of China Received 7 January 0886^ in _nal form 06 April 0887

Abstract

Due to the non!uniform distribution and fractural structure of the beads inside a porous medium\ the porosity distributed in the porous medium is random for most realistic situations[ Therefore\ the e}ects of the porosity distributed casually inside a porous block mounted on a heated region with a laminar slot impinging jet on ~ow and thermal _elds are investigated numerically[ A numerical method of SIMPLEC is adopted to solve governing equations\ as for the energy equation\ a one!equation thermal model with Van Driest|s wall function is adopted[ All the non!Darcian e}ects including the solid boundary and inertial e}ects are considered and three di}erent porosity models of constant\ variable

and random are examined[ The results indicate that the relationship between the local Nusselt number Nuxand the near

wall local porosity oxis a negative correlation[ Consequently\ in order to enhance the thermal performance of the porous

Nomenclature

b width of the slot jet ðmŁ

B9 coe.cient of stagnant conductivity

Cf speci_c heat of ~uid ðkJ kg

−0

>C−0

Ł

dp mean bead diameter ðmŁ

Da Darcy number "K:b1

#

DT empirical constant in thermal dispersion con!

ductivity

F inertial factor

hx local heat transfer coe.cient ðW m

−1_>C−0_Ł

Hj dimensional distance from the jet inlet to the top

surface of the block ðmŁ

Hp dimensional height of the block ðmŁ

Hz dimensional distance from the jet inlet to the solid

wall ðmŁ

HJ dimensionless distance from the jet inlet to the top

surface of the block "Hj:b#

HP dimensionless height of the block "Hp:b#

HZ dimensionless distance from the jet inlet to the solid

wall "Hz:b#

Corresponding author[ Tel[] 775 2 4601010^ fax] 775 2 4619523

kd stagnant conductivity ðW m

−0_>C−0_Ł

ke e}ective thermal conductivity of the porous block

ðW m−0_>C−0_Ł

kf thermal conductivity of the ~uid ðW m

−0

>C−0

Ł

ks thermal conductivity of solid phase in porous block

ðW m−0

>C−0

Ł

kt thermal dispersion conductivity ðW m

−0_>C−0_Ł

K permeability ðm1

Ł

l Van Driest|s wall function

Lp dimensional length of the block ðmŁ

LP dimensionless length of the block "Lp:b#

m¾ dimensionless ~ow rate of ~uid

N"a\ b1_# _{normal distribution with mean a and standard}

deviation b

Nux local Nusselt number along the heated wall of the

block "hxb:kf#

Nu mean Nusselt number

p dimensional pressure ðN m−1_Ł

P dimensionless pressure "p:rv1

9#

Prf Prandtl number of ~uid "rfCfnf:kf#

Prp Prandtl number of porous medium "rfCfnf:ke#

r0\ r1 coe.cient in equation "1#

Re Reynolds number "v9b:nf#

Rep bead diameter based Reynolds number "=up=dp:nf#

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u dimensional velocity in the x direction ðm s−0_Ł

U dimensionless velocity in the X direction "u:v9#

v dimensional velocity in the y direction ðm s−0_Ł

v9 jet inlet velocity ðm s

−0

Ł

V dimensional velocity in the Y direction "v:v9#

x\ y dimensionless Cartesian coordinates ðmŁ

X\ Y dimensionless Cartesian coordinates "x:b\

y:b#[ Greek symbols

a thermal di}usivity ðm1_s−0_Ł

Ds shortest distance from the calculated point to the

boundaries of the porous block

o porosity

o¹ mean porosity

oe e}ective porosity

ox near wall local porosity at Y 9[9904

u dimensionless temperature ""T−T9#:"Tw−T9##

L ratio of thermal conductivity of solid phase to ~uid

phase in porous block

m viscosity ðkg m−0 s−0 Ł n kinematic viscosity ðm1_s−0_Ł j random variable r ~uid density ðkg m−2_Ł

so standard deviation of porosity

F computational variable

c dimensionless stream function

v empirical constant in Van Driest|s wall function[

Superscripts

n the nth iteration index

* mean value[

Subscripts

c[v[ control volume

e e}ective value

f external ~ow _eld

i index

in ~owing into porous block

out leaving from porous block

p porous medium

s solid block

w solid wall

x along the X direction

9 inlet condition[

Other

= = magnitude of velocity vector[

0[ Introduction

It is a well!known method that a metal porous block is adopted to disturb ~uid ~ow and enlarge heat transfer surface which enhances heat transfer rate[ The issue is studied widely and deeply in the last decade[

Doubtless\ porosity is a remarkable factor during ana! lyzing the ~uid ~ow and heat transfer of porous medium[ For facilitating analyses\ in the beginning the porosity was usually assumed as a constant\ which is conveniently called a constant porosity model in this study[ However\ Roblee et al[ ð0Ł and Benenati and Brosilow ð1Ł based on their experimental results observed that porosity varied signi_cantly in the near!wall region[ Schwartz and co! workers ð2\ 3Ł conducted experimental studies and mea! sured the maximum velocity in the near wall region which is normally called the channelling e}ect[ These phenom! ena directly validated that the porosity which was regarded as a variable was more realistic[ Furthermore\ Cheng et al[ ð4Ł pointed out that in much of the literature the porosity was simulated as a damped oscillatory func! tion of the distance from the wall and the damped oscil! latory phenomenon was insigni_cant as the distance was larger than _ve!bead diameters for packed beds[ There! fore\ in concerning both of the practical use and con! venient theoretic model\ the variation of the porosity is assumed as an exponential function of the distance from the solid wall and is called a variable porosity model for comparing with the constant porosity model[ Based upon the above experience\ two di}erent models have been adopted to derive the individual equations of ~uid ~ow and heat transfer for the porous medium[

For the constant porosity model\ Vafai and Tien ð5Ł according to the concept of local volume averaging analy! sis derived the governing equations of ~uid ~ow and heat transfer for the porous medium[ As for the variable porosity model\ Hsu and Cheng ð6Ł utilized the volume averaging technology to derive the governing equations of the ~uid ~ow and heat transfer for the porous medium[ Both the above two di}erent types of the governing equa! tions were widely employed to study the e}ect of the inertia term\ solid boundary and variable porosity on the ~uid ~ow and heat transfer of the porous medium in both forced and natural convections\ such as Vafai and co! workers ð7\ 8Ł\ Kaviany ð09Ł\ Cheng and Zhu ð00Ł\ Hadim ð01Ł\ Hunt and Tien ð02Ł and Fu and co!workers ð03Ð05Ł\ etc[

However\ Georgiadis and co!workers ð06Ð08Ł studied the unidirectional transport phenomena of ~ow and heat transfer in the random porous medium with stochastic models and obtained the results that for the same pressure gradient along the channel the mean ~ow rate U"o# based on random porosity was larger than U"o¹# based on mean porosity as the Forchheimer model of ~ow was held[ Saito et al[ ð19Ł studied the e}ects of the porosity and void distributions on the permeability by using Direct Simulation Monte Carlo method and found that the per! meability depended not only on the porosity but also on the void distribution strongly[ These facts indicated that except for special screen process the sizes of the beads are extremely di.cult to be uniform[ Besides\ the geometry of the broad de_nition of beads is not always spherical

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and sometimes is fragmental[ Therefore\ the charac! teristics of the porosity distribution being disordered and random which results from the non!uniform size and fractural structure of the beads should be considered for a more realistic model of the porous medium[ This model is abbreviated to the random porosity model[ Although the e}ect of the random porosity distribution on the ~ow _eld had been discussed in the past\ however\ most of them were only concentrated on simple phenomena[ As the knowledge of the authors\ the results and discussion of the heat transfer and ~uid ~ow mechanisms of the porous medium with the random porosity model were seldom presented in detail[

Consequently\ the aim of this study is to investigate e}ects of a random porosity model on heat transfer per! formance of a porous block under an impinging jet numerically[ The distribution of the porosity follows the normal "Gaussian# distribution rule and is generated by KindermanÐRamage procedure ð10Ł[ The diameters of 090 beads of a porous block are measured to obtain a reasonable mean porosity and standard deviation of the porosity[ The constant and variable porosity models are also considered[ A numerical method of SIMPLEC ð11Ł is adopted to solve the governing equations\ as for the energy equation\ a one!equation thermal model with Van Driest|s wall function is adopted[ All the non!Darcian e}ects including the solid boundary e}ects and inertial e}ects are considered[ Other important parameters such as Reynolds number\ geometry size ratios of the porous block and Prandtl number\ etc[\ are selected based on the author|s previous studies ð03Ð05Ł[ The results indicate that the relationship between the local Nusselt number

Nux and the near wall local porosity ox is a negative

correlation[ Consequently\ in order to enhance e}ectively the thermal performance of the porous medium\ the porosity near the solid plate should be smaller to make the conductive heat transfer dominant[

1[ Physical model

The physical model is shown symmetrically in Fig[ 0[ There is a two!dimensional laminar slot jet impinging on a partially heated plate[ The width of the jet inlet is b[ The uniform inlet velocity and temperature of the jet are

v9 and T9\ respectively[ A portion of the impingement

plate is heated and the other region is insulated[ The

length of the heated region is Lp\ and the temperature of

the heated region is Twwhich is higher than T9[ A porous

block of which the porosity distribution is corresponding to the random porosity model is mounted on the heated region[ Based on the results of Fu and Huang ð04\ 05Ł\

the height Hpand the length Lpof the block are chosen

to 9[4b and 1b\ respectively[ The distances from the jet inlet to the top surface of the block and the impingement

plate are Hj "2[4b# and Hz"3[9b#\ respectively[ The

Fig[ 0[ Physical model[

whole computation domain is large enough for fully developed distributions of the velocity and temperature to be formed[ Under this con_guration\ the ~ow _eld can be decomposed into two conjugate regions] one stands for the internal ~ow _eld where it is bounded by the porous block\ and the other is called the external ~ow _eld which excludes the porous media[

In order to facilitate the problems\ the following assumptions are made

"0# The porous block is made of copper beads which have di}erent sizes of diameters[ The beads do not chemically react with the ~uids[

"1# The ~ow _eld is steady state\ two!dimensional\ single phase\ laminar and incompressible[ The symmetrical assumption shown in Fig[ 0 exists in the random porosity model[

"2# The ~uid properties are constant and the e}ect of gravity is neglected[

"3# The transverse thermal dispersion is modeled by Van Driest|s wall function ð12Ł\ hence\ a one!equation model of the energy equation is used for the porous medium[

"4# The e}ective viscosity of the porous medium is equal to the viscosity of the external ~uid[

As mentioned above\ the values of the porosity dis! tributed in the porous medium are random[ Then the following process is used to obtain the data of the mean

porosity o¹ and standard deviation so[ The diameters of

090 beads are ensampled and measured from decompo! sition of a sintered brass porous block[ The diameters are

classi_ed and shown in Fig[ 1[ The mean diameter dpof

the beads is about 0[24 mm and its standard deviation is 9[05 mm which is about 01) of the mean diameter of

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Fig[ 1[ The size distribution of 090 beads in a porous block[

the beads[ Shown in Fig[ 1\ the distribution of the bead sizes is like a normal distribution[ As a result\ the space occupied by the solid phase in the porous medium is corresponding to the normal distribution[ Then that the residual space "void phase# in the porous medium also follows the normal distribution rule can be drawn[ Based on the data of the density of the beads provided by the bead maker\ the maximum mean porosity of the porous block is approximately estimated as 9[36[ Therefore\ in this study\ the mean porosity o¹ is conveniently regarded

as 9[4 and the standard deviation sois 9[94 which is 09)

of the mean porosity[ The mean diameter dpis selected

as a characteristic bead diameter and equal to 0[14 mm for easily presenting the results[

In order to compare the di}erence among the models of constant porosity\ variable porosity and random porosity\ three models of porosity distribution are taken into consideration[ They are

"0# constant porosity model] o o¹ "0#

"1# variable porosity model] o oeð0¦r0e−r1Ds:dpŁ "1#

where Ds is the shortest distance from the calculated point

to the boundary of the porous block\ and r0and r1are

both empirical constants[ The oeis an e}ective porosity[

The mean porosity o¹ can be obtained from integrating the local porosity o in the full domain of the porous block as follows o¹  0 Lp×Hp

g

Lp 9

g

Hp 9 oeð0¦r0e −r₁Ds:d_p_{Ł dy dx} _"2#

where the r1 is obtained from Vafai ð7Ł and the r0 is

selected to make the local porosity in the near wall region to be equal to one[ As a result\ for the cases of o¹ 9[4\ the

oe\ r0and r1are equal to 9[296\ 1[145 and 1\ respectively[

"2# Random porosity model] according to the results of the measuring process\ the porosity distribution of the porous medium approximately follows the form of the

normal distribution with mean porosity o¹ and standard

deviation soshown in Fig[ 1[ For necessity of computing

process\ the theoretic form of the porosity distribution of the random porosity model is obtained from the fol! lowing method[ The KindermanÐRamage procedure ð10Ł "Appendix# is used to generate a random variable j of the standard normal distribution\ N"9\ 0#\ _rst[ And the random variable j is transformed to gain a general ran! dom variable o corresponding to a general normal dis! tribution\ N"o¹\ s1

o#\ of which the mean o¹ and standard

deviation soare equal to designed constants\ respectively[

Therefore\ the distribution of the general random vari! able o is regarded as the porosity distribution of the random porosity model in this study[ Shown in Fig[ 2\ the solid line is the distribution of the general random

variable o of the normal distribution N"9[4\ 9[941

# obtained from the KindermanÐRamage procedure and the dashed line is the result of the theoretical normal

distribution N"9[4\ 9[941_{#[ The deviation between both}

lines are small[

The permeability K\ and inertia factor F are de_ned as ð7Ł K o 2_d1 p 049"0−o#1 "3# F 0[64 z049o0[4[ "4#

The e}ective thermal conductivity of a porous medium

keis a combination of the stagnant conductivity kdand

the thermal dispersion conductivity ktð12Ł\ which simu!

lates the transverse thermal dissipation[ The relationship

between ke\ kdand ktis then

ke kd¦kt "5#

Fig[ 2[ The distributions of probability density of random vari! able o of normal distribution N"9[4\ 9[941_{# generated by}

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and kdis de_ned as kd kf 0−z0−o¦1Lz0−o L−B9 x

$

B9L"L−0# "L−B9# 1 ln

0

L B9

1

−B9¦0 1 − L"B9−0# L−B9

%

"6# where L ks kf "7# B90[14

0

0−o o

1

09:8 "8#

and ktis de_ned by Van Driest|s wall function as

kt

kf

DTPrfRep=up=l "09#

where DTis an empirical constant\ Repis the bead diam!

eter based Reynolds number\ de_ned as

Rep

=up=dp

nf

"00# where l is the Van Driest|s wall function de_ned as l 0−e−Ds:vd

p "01#

and v is an empirical constant[

Based on the above assumptions and with the following

characteristic scales of b\ Tw−T9\ rv

1

9 and v9\ the gov!

erning equations\ boundary conditions and geometry dimensions are normalized as follows]

Uf9\ 1Vf 1X9\ 1uf 1X9 Up9\ 1Vp 1X9\ 1up 1X9 "10#

on surface BC "heated region#

Up9\ Vp9\ up0 "11#

on surface CD "insulated region#

Uf9\ Vf9\ 1uf 1Y9 "12# on surface ED "X : # 1Uf 1X9\ 1Vf 1X9\ 1uf 1X9 "13# on surface EF "Y :# 1Uf 1Y9\ 1Vf 1Y9\ 1uf 1Y9 "14# on surface FG "wall# Uf9\ Vf9\ 1uf 1X9 "15#

on surface GA "jet inlet#

Uf9\ Vf −0\ uf9[ "16#

There are some interfacial conditions at the interfaces between the porous block and external ~ow _eld[ These are the matching conditions of the horizontal and vertical velocities\ normal and shear stresses\ temperature\ heat ~ux and pressure[ However\ these conditions will make the problem more complex[ A simpli_ed method suggested to solve these interfacial problems was dis! cussed in the study of Hadim ð01Ł[ The interfacial conditions at the ~uid:porous medium interface are

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automatically satis_ed ð01Ł due to the Brinkman exten! sion in the momentum equations for the porous medium[

2[ Numerical method

The SIMPLEC algorithm ð11Ł with TDMA solver ð13Ł is used to solve the governing equations "02#Ð"19# for the ~ow and thermal _elds[ Equations "02#Ð"19# are _rst discretized into algebraic equations by using the control volume method ð13Ł with a power!law scheme[ The underrelaxation factor is 9[1 for both the _elds of velocity and temperature[ The conservation residues ð11Ł of the equations of the momentum\ energy and continuity and the relative errors of each variable are used to examine the convergence criteria which are de_ned as follows]

"S=Residue of F equation=1

c[v[#0:1¾ 09−3\

F U\ V\ u\ and mass flow rate "17#

max =Fn¦0 −Fn = max =Fn¦0 = ¾ 09 −4_\ _{F U\ V\ P\ u[} _"18#

In order to reduce the computation time\ a non!stag! gered mesh is used[ The _ner meshes are placed in both the interfacial region of the block and near the solid wall region[ The meshes are then expanded outwards from the interfacial boundary and the solid wall with a scale ratio of 0[94[ Also on the basis of the suggestions of Patankar ð13Ł\ the harmonic mean formulation of thermophysical properties is used to avoid the e}ects of abrupt change of these properties across the interfacial region of the block and the external ~ow _eld on the computation accuracy[

The numerical method and accuracy are validated by Fu et al[ ð04\ 05Ł[ The comparison of the results of Miyar! zaki and Silberman ð14Ł\ which were derived by an ana! lytical method for a case of a laminar slot jet impinging on a smooth wall\ and the results of this study "solid line# are indicated in Fig[ 3[ The deviation between these two results is small[

The parameters which include the Reynolds number Re\ block height HP\ block length LP and mean porosity

o¹\ adopted in this study are tabulated in Table 0[ The

Darcy number Da listed in Table 0 is based on the mean porosity o¹[ Since the porosity o is not a constant in both the variable and random porosity models\ hence the Da in each control volume is also a variable during the com! putation[ For the Re 349 cases\ the whole dimen! sionless domain X×Y is 04[9×01[9 and the fully developed conditions at the outlet sections can be satis! _ed[

Table 1 shows the empirical constants used in the de_! nitions of the porosity o for the variable porosity model and the Van Driest|s wall function l ðequation "01#Ł[

Where the DTand v are provided by Cheng and Hsu

ð12Ł[

Fig[ 3[ The results of local Nusselt number distributions of jet impinging normally on smooth wall\ compared with Miyazaki and Silberman ð14Ł[

Table 0

The main parameters

b"m# Re HP HJ LP o¹ so dp"m# Da Pr

9[90 349 9[4 2[4 1 9[4 9[94 0[14E−2 4[197E−4 9[6

Table 1

The empirical constants for o¹ 9[4 "oe 9[296#

r0 r1 DT v

1[145 1 9[2 2[4

The results of grid tests are listed in Table 2\ in order to gain more accurate results of the random porosity model\ the 216×071 meshes are chosen in this study\ and there are 65×57 meshes inside the porous block[

Table 2

Grid tests of porous block for Re 349\ HP 9[4\ LP 1\ HJ 2[4\ o¹ 9[4 and Pr 9[6

Meshes of X Meshes of Y Nup Iterations

216 071 09[697 10 001 092 071 09[628 17 799 75 071 09[642 04 871 51 071 09[739 06 041 75 119 09[640 33 989 75 83 09[760 2078

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3[ Results and discussion

Without notice\ Re 349\ o¹ 9[4\ so 9[94\ Pr 9[6\

HP 9[4\ LP 1 and HJ 2[4 are _xed in the following

situations[

Theoretically\ there are in_nite patterns generated by a random porosity model with a given mean porosity o¹

and standard deviation so[ It is di.cult to solve all of the

patterns\ therefore\ only ten patterns with the same mean porosity o¹ "9[4# are presented to investigate the e}ects of the random porosity model on the ~ow and thermal _elds[ The _rst three patterns "runs 0Ð2# have the same random variable j of the standard normal distribution

but three di}erent standard deviation soof 9[90\ 9[94 and

9[0\ respectively[ The latter seven patterns "Runs 3Ð09#

have the same standard deviation so "9[94# with the

di}erent random porosity patterns[

The global porosity distribution maps and the near

wall local porosity oxdistributions along the X direction

at Y 9[9904 where it is the central position of the _rst control volume in Y direction during the computation\ of the variable porosity model and two selected cases "Runs 1\ 3# of the random porosity models are shown in Figs 4"a#Ð"f#[ In the global porosity distribution map\ the total area is approximately divided into three main di}erent porosity regions with di}erent colors where the darker color represents the large porosity[ For the vari! able porosity model\ each color region has the same porosity interval of 9[11[ However\ for the random porosity models\ the central region of the porosity index

means the variation of the porosity to change from o¹−so

to o¹¦so[ In general\ for the porous medium made of

monosized and nonconsolidated beads\ the pack between the solid beads and the solid wall is sparser than that between the beads and beads in the core region[ There! fore\ as shown in Figs 4"a# and "b# for the variable

porosity model\ the near wall local porosity oxis almost

equal to unity\ and the porosity in most regions varies from 9[21 to 9[43[ Oppositely\ for the random porosity model\ as mentioned above the porosity distributions are not in order\ then the pack between the beads and the solid wall is no longer sparer than that of the other positions[ Hence\ for the random porosity models shown in Figs 4"c#Ð" f#\ the variation of porosity in the most

region is from o¹−so to o¹¦so and the variations of the

near wall local porosity ox are drastic and disorder[

Although the Runs 1 and 3 have the same mean porosity and standard deviation\ the two cases have the di}erent random variables\ then the global porosity distributions

and near wall local porosity oxdistributions are di}erent[

Shown in Figs 5"a#Ð"f#\ there are streamlines for the cases of the constant and variable porosity models and four selected cases "Runs 0Ð3# of the random porosity models\ respectively[ In order to illustrate the ~ow and thermal _elds more clearly\ the phenomena near the porous block are presented only[ The dimensionless stream function c is de_ned as]

Fig[ 4[ Global porosity distribution map and near wall local porosity oxdistribution at Y 9[9904 along the X] "a# and "b#

variable porosity model\ o¹ 9[4 "oe 9[296#^ "c# and "d# random

porosity model\ Run 1 "o¹ 9[4\ so 9[94#^ and "e# and " f#

random porosity model\ Run 3 "o¹ 9[4\ so 9[94#[

U 1c

1Y and V −

1c

1X[ "29#

The ~uids are issued from the jet inlet and impinge on the top surface of the porous block _rst[ Due to the existence of the ~ow resistance inside the porous block\ only a portion of the ~uids can penetrate into the porous block[ A circulation region neighboring to the right side of the block occurs[ This ~ow pattern is disadvantageous to the heat transfer performance of the heated region[ The ~ow pattern outside the porous block are similar for the six di}erent cases[ However\ that the streamline of

c 9[990 of the variable porous model is close to the

solid wall\ which means that more ~uids ~ow through the near wall region[

m¾p\in and m¾p\out are the ~ow rates of the ~uids pen!

etrating into and leaving from the porous block\ respec!

tively\ and indicated in Table 3[ Where m¾9is the ~ow rate

of the ~uid issued from the jet inlet[ The ~uids penetrate into and leave from the porous block through the top and right side surfaces\ respectively[ The ~ow rate of the variable porosity model penetrating into the porous

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Fig[ 5[ Streamlines of di}erent porosity models] "a# constant\ "b# variable\ "c# random\ Run 0 "so 9[90#\ "d# random\ Run 1 "so 9[94#\

"e# random\ Run 2 "so 9[0# and "f# random\ Run 3 "so 9[94#[

Table 3

Ratio of ~ow rate "m¾p# penetrating into the porous block to that

of jet inlet "m¾9#

Ratio of ~ow rate

Top surface Right side surface Porosity

model m¾p\in:m¾9 m¾p\out:m¾ m¾p\in:m¾9 m¾p\out:m¾

Constant 9[9349 9 9 9[9349 Variable 9[9824 9 9 9[9824 Random Run 0 9[9337 9 9 9[9337 Run 1 9[9305 9 9 9[9305 Run 2 9[9227 9 9 9[9227 Run 3 9[9310 9 9 9[9310 Run 4 9[9306 9 9 9[9306 Run 5 9[9316 9 9 9[9316 Run 6 9[9323 9 9 9[9323 Run 7 9[9304 9 9 9[9304 Run 8 9[9312 9 9 9[9312 Run 09 9[9306 9 9 9[9306

block is almost two times of those of the other two models[ In addition\ the ~ow rate of the constant porosity model is larger than those of the random porosity models[ As for the ~ow rates of the _rst three cases of the random porosity models "Runs 0\ 1\ and 2#\ the smaller the stan!

dard deviation so\ the larger the ~ow rate becomes[ The

reason may be suggested as that the more uniform porosity distribution is\ the more ~uids penetrate through the porous medium under o¹ 9[4 situation[

The local velocity U distribution in the near wall region along the Y direction at four di}erent X positions\ X 9[14\ 9[4\ 9[64 and 9[84 are illustrated in Figs 6"a#Ð "d#\ respectively[ Only the data of the Run 1 are selected from the cases of all random porosity models and indi! cated in the _gures[ Since the porosity varies casually\ the local velocity U of the random porosity model presents a jagged pro_le and the channeling e}ect does not appear[ In general\ the distribution of local U velocities of the random porosity model are similar to that of the constant porosity model\ the reason is that most of the values

of the local porosity vary from o¹−1soto o¹¦1so\ which

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Fig[ 6[ The distributions of velocity U along the Y direction at "a# X 9[14\ "b# X 9[4\ "c# X 9[64 and "d# X 9[84[

When the value of sois small\ the value of o¹−1s is close

to that of o¹¦1so[

For the variable porosity model\ the maximum value of the local velocity U increases _rst as X increases and decreases after X 9\64[ However\ a di}erent trend exists for the constant and random porosity models\ the maximum "or bulk# velocity U gradually increases with the increase of the X[ The velocity gradient "dU:dY# in the near wall region of the variable porosity model is larger than those of the other models[

The isotherms of the cases of the constant and variable porosity models and four selected cases "Runs 0Ð3# of the random porosity models are illustrated in Figs 7"a#Ð "f#\ respectively[ Like the results of the streamlines shown in Figs 5"a#Ð"f#\ the isotherm distributions of the four selected cases of the random porosity models are similar to that of the constant porosity model[ That the isotherms are concentrated near the heated plate in the variable

porosity model is di}erent from those of the other two models[

Shown in Figs 8"a#Ð"d# are the distributions of the

local Nusselt number Nuxof the variable and constant

porosity models and four selected cases "Runs 0Ð3# of the random porosity models[ Where the local Nusselt

number Nuxis de_ned as Nux hxb kf −ke kf 1u 1Y

b

Y9 [ "20#

The higher the standard deviation sois\ the larger the

~uctuation of the porosity distribution becomes[ Then

the ~uctuation of the local Nusselt number Nux dis!

tribution along the heated plate for the random porosity

model increases as the standard deviation soincreases as

shown in Figs 8"a#Ð"c#[ Inversely\ the variation of the

distribution of the Nuxof the smaller value of standard

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Fig[ 7[ Isotherms of di}erent porosity models] "a# constant\ "b# variable\ "c# random\ Run 0 "so 9[90#\ "d# random\ Run 1 "so 9[94#\

"e# random\ Run 2 "so 9[0# and " f# random\ Run 3 "so 9[94#[

constant porosity model with slight ~uctuations[ For the cases of Runs 1 and 3\ due to the di}erent random vari! ables\ the results of Runs 1 and 3 are di}erent[ Conse! quently\ it is possible to obtain di}erent heat transfer performances of the porous blocks with the same mean porosity and standard deviation[ For the variable porosity model\ the local Nusselt number near the right side wall of the porous block decreases with the increase

of the X\ however\ the Nuxincreases in this region for

both the constant and random porosity models[ The phenomenon is induced by the development of the local

Uvelocity along the X mentioned in Figs 6"a#Ð"d#[

The e}ect of the near wall local porosity oxde_ned

earlier at Y 9[9904 on the local Nusselt number Nux

for all eight cases of o¹ 9[4\ so 9[94 "runs 1\ 3Ð09# are

indicated in Fig[ 09[ In essence\ the larger the ox\ the

e}ective thermal conductivity keis smaller\ therefore\ the

relationship between the Nuxand oxis a negative cor!

relation as shown in Fig[ 09[ The heat transfer mechanism between the porous medium and the heated plate is the combination of conductive and convective heat transfer[ Since the smaller porosity means that more solid phase

exists which contributes the conductive path to the heat transfer and the inference that the convective heat trans! fer herein is not a role may be drawn[ Consequently\ in order to enhance the thermal performance of the porous medium\ the porosity near the heated plate should be smaller to cause the conductive heat transfer to be domi! nant[

In order to validate the above inference\ the heat trans! fer rate of an arti_cial random porosity case "arti_cial random porosity model\ Run 00# is examined[ The values of the porosity o of the arti_cial random porosity model are sorted out from the random porosity model of the

Run 1 "o¹ 9[4 and so 9[94#[ The porosity distribution

of this model is reordered and arranged by the following rules shown in Fig[ 00[ The most dense porosity is arranged at the control volume 0 "the left most of the _rst row#\ as the value of the X increases the porosity becomes sparse[ The porosity of the control volume 66 "the left most of the second row# is right behind the porosity of the control volume 65 "the right most of the _rst row# in the order of sparsity\ and as the value of the

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Fig[ 8[ The distributions of local Nusselt number Nuxfor four selected cases of the random porosity models] "a# Run 0 "so 9[90#\ "b#

Run 1 "so 9[94#\ "c# Run 2 "so 9[0# and "d# run 3 "so 9[94#[

Fig[ 09[ The relationship between the near wall local porosity ox

and local Nusselt number Nux[

Fig[ 00[ The array of the porosity in the arti_cial random porosity model[

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0[ Follow the same rule until the last row\ and the most sparse porosity is arranged at the right upper corner[

The local Nusselt number Nuxdistributions along the X

direction are shown in Fig[ 01[ The results of the random porosity model are the average values of the total eight cases "Runs 1\ 3Ð09# at each X position[ The local Nusselt

number Nuxof the arti_cial random porosity model is

larger than those of the other three models[ The results are in agreement with the above inference[

4[ Conclusions

A ~ow and thermal _eld of a porous block with random porosity distribution under a laminar slot impinging jet are investigated numerically[ The e}ects of the random porosity model on ~ow and thermal _elds are examined and compared with those of the constant and variable porosity models[ The results can be summarized as fol! lows]

"0# The local Nusselt number distribution of the random porosity model are more similar to those of the con! stant porosity model than those of the variable porosity model with the smaller standard deviation[ However\ as the value of the standard deviation is larger\ the ~uctuation of the local Nusselt number is drastic and apparently di}erent from those of the other two models[

"1# The relationship between the local Nusselt number

Nuxand the near wall local porosity oxis a negative

correlation[

"2# In order to enhance the thermal performance of the porous medium\ the porosity near the heated plate should be smaller to make the conductive heat trans! fer dominant[

Fig[ 01[ The local Nusselt number Nuxdistributions of four

di}erent porosity models[

Acknowledgement

The support of this work by the National Science Council\ Taiwan\ R[O[C[ under Contract NSC75!1101! E998!931 is gratefully acknowledged[

Appendix] the KindermanÐRamage procedure

The KindermanÐRamage procedure uses a mixture of

distributions\ l0\ l1\ and l2\ to generate a standard ran!

dom variable j\ where the l0\ l1\ and l2are three computer

generated pseudo!random variables with double

precision\ such that 9[9 ³ l0\ l1\ l2³ 0[9[ The algorithm

of the KindermanÐRamage procedure is as follows ð10Ł]

Step 0[ Generate l0[ If l0³ 9[773969391187647\ gen!

erate l1and deliver

j a"0[02002052433079l0¦l1#−0

where a 1[105924756055360[ Then go to Step 09[

Step 1[ If l0³ 9[862209843062787\ go to Step 3[

Step 2[ Generate l1and l2until

l1 1³ a 1 "a1 −1 ln"l2## −0 \

then if l0³ 9[8755443669758378 then deliver

j "a1_{−1 ln"l} 2##

0:1

else deliver j −"a1_{−1 ln"l}

2##0:1[ Then go to Step 09[

Step 4[ Generate l1and l2[ Set

t a−ð9[529723790810859 min "l1\ l2#Ł[

If max "l1\ l2# ¾ 9[644480420556590\ go to Step 8[

If 9[923239492649000 "=l1−l2=# ¾ `"t#\ go to Step 8^

otherwise repeat Step 4[ where

`"t# f"t#−9[07991408095452¦"a−=t=# for =t= ³ a

and

f"t# 0:z1p e−t1_:1

"normal density function#

t 9[368616393111330¦ð0[094362550911969 min "l1\ l2#Ł

If max "l1\ l2# ¾ 9[761723865560689\ go to Step 8[

If 9[938153385262017 "=l1−l2=# ¾ `"t#\ go to Step 8^

otherwise repeat Step 6[

t 9[368616393111330−ð9[484496027904839 min "l1\ l2#Ł

If max "l1\ l2# ¾ 9[79446681332706\ go to Step 8^ other!

wise repeat Step 7[

Step 8[ If l1³ l2\ deliver j t^ otherwise deliver

j −t[ Then go to Step 09[

Step 09[ Transform the standard random variable j into

the random variable o with given o¹ and so[ The relation!

ship between j and o is

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References

ð0Ł L[H[S[ Roblee\ R[M[ Baird\ J[W[ Tiern\ Radial porosity variations in packed beds\ American Institute of Chemical Engineers Journal 3 "0847# 359Ð353[

ð1Ł R[F[ Benenati\ C[B[ Brosilow\ Void fraction distribution in packed beds\ American Institute of Chemical Engineers Journal 7 "0851# 248Ð250[

ð2Ł C[E[ Schwartz\ J[M[ Smith\ Flow distribution in packed beds\ Industrial and Engineering Chemistry 34 "0841# 0198Ð0107[

ð3Ł C[E[ Schwartz\ K[B[ Bischo}\ Thermal and material trans! port in nonisothermal packed beds\ American Institute of Chemical Engineers Journal 04 "0858# 486Ð593[

ð4Ł P[ Cheng\ A[ Chowdhury\ C[T[ Hsu\ Forced convection in packed tubes and channels with variable porosity and thermal dispersion e}ects\ in] S[ Kakacž et al[ "Eds[#\ Con! vective Heat Mass Transfer in Porous Media\ 0880\ pp[ 514Ð542[

ð5Ł K[ Vafai\ C[L[ Tien\ Boundary and inertia e}ects on ~ow and heat transfer in porous media\ International Journal of Heat and Mass Transfer 13 "0870# 084Ð192[

ð6Ł C[T[ Hsu\ P[ Cheng\ Thermal dissipation in a porous medium\ International Journal of Heat and Mass Transfer 22 "0889# 0476Ð0486[

ð7Ł K[ Vafai\ Convective ~ow and heat transfer in variable! porosity media\ Journal of Fluid Mechanics 036 "0873# 122Ð148[

ð8Ł A[ Amiri\ K[ Vafai\ Analysis of dispersion e}ects and non! thermal equilibrium\ non!Darcian\ variable porosity incompressible ~ow through porous media\ International Journal of Heat and Mass Transfer 26 "0883# 828Ð843[ ð09Ł M[ Kaviany\ Laminar ~ow through a porous channel

bounded by isothermal parallel plates\ International Journal of Heat and Mass Transfer 17 "0874# 740Ð747[ ð00Ł P[ Cheng\ H[ Zhu\ E}ects of radial thermal dissipation

of fully!developed forced convection in cylindrical packed tubes\ International Journal of Heat and Mass Transfer 29 "0876# 1262Ð1272[

ð01Ł A[ Hadim\ Forced convection in a porous channel with localized heat sources\ Journal of Heat Transfer 005 "0883# 354Ð361[

ð02Ł M[L[ Hunt\ C[L[ Tien\ E}ects of thermal dissipation on

forced convection in _brous media\ International Journal of Heat and Mass Transfer 20 "0877# 290Ð298[

ð03Ł W[S[ Fu\ H[C[ Huang\ W[Y[ Liou\ Thermal enhancement in laminar channel ~ow with a porous block\ International Journal of Heat and Mass Transfer 28 "0885# 1054Ð1064[ ð04Ł W[S[ Fu\ H[C[ Huang\ Thermal performances of di}erent

shape porous blocks under an impinging jet\ International Journal of Heat and Mass Transfer 39 "0886# 1150Ð1161[ ð05Ł W[S[ Fu\ H[C[ Huang\ Appropriate sizes of porous and

solid blocks for heat transfer under an impinging jet\ Jour! nal of the Chinese Institute of Engineers 19 "0886# 048Ð 058[

ð06Ł J[G[ Georgiads\ I[ Catton\ Stochastic modeling of unidi! rectional ~uid transport in uniform and random packed beds\ The Physics of Fluids 29 "0876# 0906Ð0911[ ð07Ł I[ Catton\ J[G[ Georgiads\ P[ Adnani\ The impact of non!

linear convective processes on transport phenomena in porous media\ Proceedings of 0877 National Heat Transfer Conference\ Vol[ 0\ ASME HTD!85\ Houston\ 0877\ pp[ 656Ð666[

ð08Ł J[G[ Georgiads\ E}ect of randomness on heat and mass transfer in porous media\ in] S[ Kakacž et al[ "Eds[#\ Con! vective Heat Mass Transfer in Porous Media\ 0880\ pp[ 388Ð413[

ð19Ł A[ Saito\ S[ Okawa\ T[ Suzuki\ H[ Maeda\ Calculation of permeability of porous media using direct simulation Monte Carlo method "e}ect of porosity and void dis! tribution on permeability#\ Proceedings of ASME:JSME Thermal Engineering Joint Conference\ Vol[ 2\ Maui\ 0884\ pp[ 186Ð293[

ð10Ł W[ J[ Kennedy Jr\ J[E[ Gentle\ Statistical Computing\ Marcel Dekker\ New York\ 0879[

ð11Ł J[P[ Van Doormaal\ G[D[ Raithby\ Enhancements of the SIMPLE method for predicting incompressible ~uid ~ows\ Numerical Heat Transfer 6 "0873# 036Ð052[

ð12Ł P[ Cheng\ C[T[ Hsu\ Applications of Van Driest|s mixing length theory to transverse thermal dissipation in a packed! bed with boundary walls\ International Communication in Heat and Mass Transfer 02 "0875# 502Ð514[

ð13Ł S[V[ Pantankar\ Numerical Heat Transfer and Fluid Flows\ Hemisphere[ Washington D[C[\ 0879[

ð14Ł H[ Miyazaki\ E[ Silberman\ Flow and heat transfer on a ~at plate normal to a two!dimensional laminar jet issuing from a nozzle of _nite height\ International Journal of Heat and Mass Transfer 04 "0861# 1986Ð1096[

Effects of a random porosity model on heat transfer performance of porous media

\

E}ects of a random porosity model on heat transfer

performance of porous media

Wu!Shung Fu\ Hsin!Chien Huang

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Wu!Shung Fu\ Hsin!Chien Huang