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A COMPUTATIONAL MODEL FOR
DETERMININGTHE EFFECTIVE
THERMAL CONDUCTIVITY OF A

POROUS MEDIUM

By

Daniel K. Lucas

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Master of Science

Department of Mechanical Engineering

1996

ABSTRACT
A COMPUTATIONAL MODEL FOR
DETERMINING THE EFFECTIVE
THERMAL CONDUCTIVITY OF
POROUS MEDIUM
By

Daniel K. Lucas

There are many systems of interest in engineering that involve heat
transfer through a medium consisting of two or more'distinct materials. Such a
medium is often considered to be a porous medium when one of the materials is
in solid phase and another material is in fluid phase. Some examples of a
porous medium are the packed bed of a cooling tower, building insulation, and
the core of a catalytic converter. In evaluating these systems and others of
interest in engineering thermodynamics, heat transfer, or fluid mechanics, the
effective thermal conductivity of a porous medium must first be evaluated. It is
difficult to predict the thermal conductivity of a medium consisting of two or more
separate materials.

There is no general model to predict the effective thermal conductivity of a
porous medium. This paper proposes such a model starting with the two
dimensional heat conduction equation. Finite differencing is employed to
develop the numerical model. From this numerical model, a computer code is

developed that predicts the effective thermal conductivity of a porous medium.

To my wife and family: Susan, Lisa, Kerri, and Daniel.

TABLE OF CONTENTS

LIST OF FIGURES

NOMENCLATURE

Chapter

APPENDIX

INTRODUCTION
Problem Statement
Literature Review
Proposal Statement
Summary

METHOD OF SOLUTION
Governing Equations
Boundary Conditions

RESULTS AND DISCUSSION
Porous Media Generation
Results of Numerical Validation

CONCLUSIONS AND RECOMMENDATIONS
Conclusions
Recommendations

LIST OF REFERENCES
Specific References
General References

47

48
51
52

Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.

Figure 7.
Figure 8.
Figure 9.

Figure 10.
Figure 11.

Figure 12.
Figure 13.

Figure 14.
Figure 15.
Figure 16.
Figure 17.
Figure 18.
Figure 19.
Figure 20.
Figure 21.
Figure 22.
Figure 23.

LIST OF FIGURES

Conductive Heat Transfer in Porous Medium
Parallel Resistance Model
Parallel Resistance Circuit
Series Resistance Model
Series Resistance Circuit
Parallel and Series Models of Effective
Thermal Conductivity for y = 0.1
Loeb’s Model
Thermal Circuit of Loeb’s Model
Heat Conduction Model
Array of Cylinders with Imposed Finite Element Grid
Simulated Porous Media Where
Shaded Area Represents Solid
Finite Difference Element
Normalized Porosity for Target
Porostiy of 0.1 to 0.9
Frequency vs. 0.1 Normalized Porosity
Frequency vs. 0.9 Normalized Porosity
Calculated porosity vs. N x N Elements
Calculated porosity vs. N x N Elements
Effective Thermal Conductivity vs. N x N Elements
Effective Thermal Conductivity vs. N x N Elements
Effective Thermal Conductivity vs. Log (DT)
Effective Thermal Conductivity vs. Target Porosity
km vs. Generated Porosity
km/kf vs. Target Porosity

32
33

36
37
39
4O
41
43

Nomenclature

oQO>—I.or—x

Symbols

Subscripts

h“"l'"13CD-IQO'O¢DC 3-~

Superscripts
k

Overbar

Thermal Conductivity

Length

Heat Transfer Rate
Temperature

Area

Thermal Resistance

Diameter

Average Temperature Gradient
Time Step

Porosity

Solid Thermal Conductivity I Fluid
Thermal conductivity
Geometrical Factor

Void Fraction

Fluid

Lower Boundary

Effective Value

Upper Boundary

Solid

Pore

Continuous Phase
Discontinuous Phase

Top Cell Node

Bottom Cell Node

Right Cell Node

Left Cell Node

Index Transverse to Heat Flow
Index Parallel to Heat Flow

Index Direction

Average Conditions

CHAPTER I
INTRODUCTION

1.1 Problem Statement

There are many systems of interest in engineering that involve heat
transfer through a medium consisting of two or more distinct materials. Such a
medium is often considered to be a porous medium when one of the materials is
in solid phase and the other material is in fluid phase. Some examples of a
porous medium are the packed bed of a cooling tower, building insulation, and
the core of a catalytic converter. In evaluating these systems and others of
interest in engineering thermodynamics, heat transfer, or fluid mechanics, the
effective thermal conductivity of a porous medium must first be evaluated. The
thermal conductivity of a single material is defined as the amount of heat flowing
by conduction through a unit area per unit time per unit temperature gradient.
The evaluation of thermal conductivity for a single material is very straight
forward. However, it is more difficult to predict the thermal conductivity of a
medium consisting of two or more separate materials.

There has been previous work to determine the thermal conductivity of a
porous medium. However, this work has dealt with either specific systems or
specific geometries, and there is no general model for evaluating the effective

thermal conductivity of a two component system.

2
There are three modes of heat transfer in a porous medium: conduction,

convection, and radiation. Conduction is heat transfer by molecular, atomic, and
electronic motion, and is often the prominent mode of heat transfer. Convection
is heat transfer by fluid motion and can be important as porosity increases.
Finally, radiation is heat transfer by electromagnetic wave motion and becomes
increasingly important as the temperature increases.

The problem will be studied as a two-dimensional system similar to the
system shown in Figure 1. This system consists of two or more anisotropic
materials having different thermal characteristics and has fixed temperatures
applied to the top and bottom surfaces and adiabatic sidewalls. The effective
thermal conductivity is defined as km such that the heat transfer may modeled
using the equation

_ kmAT

q: L. (1)

 

Using the above equation, a control volume may be defined, that is small enough
to have local thermal equilibrium yet large enough to retain the properties of the
porous medium, over which Fourier’s law of heat conduction is valid, i.e.,

9 = kaT. (2)

1.2 Literature Review
Previous work by Baradat and Combarnous [1] has shown that the two

simplest models for determining effective thermal conductivity also describe the

V

 

 

 

 

 

 

 

TL

Figure 1. Conductive Heat Transfer in Porous Medium

4
upper and lower boundaries for the actual thermal conductivity of a porous

medium. These are the parallel resistance model and the series model. In the
parallel resistance model, the voids and solids hypothetically move in opposite
directions transverse to the heat flow as shown in Figure 2. Working with the
parallel thermal circuit shown in Figure 3, the thermal conductivity can be shown

to be

51:841-.» (3)

where e is the porosity and y is the ratio of solid thermal conductivity to the fluid
thermal conductivity. In the series resistance, the voids and solids hypothetically
move in opposite directions in the direction of the heat flow as in Figure 4. Now,
working with the series thermal circuit shown in Figure 5, the thermal conductivity
can be shown to be

km (1 — s) 4

T(,— = [a + T] . (4)
The current models are bounded by the series and parallel models as shown in
the graph in Figure 6 as km/kf versus porosity with the solid to fluid thermal
conductivity ratio, 7, set equal to 0.1.

These two simple models represent a whole group of models that have
simple solutions. In these simple models, either heat flow lines run parallel to
the direction of the heat flux or the isotherms are perpendicular to the heat flow.
The group of models based on these simple models have one common

approach for solution. They use a thermal circuit network and allow the

 

 

/

 

 

 

 

 

Fluid Solid L
/ .IL
|-— (1-e)A + 2A 4

 

Figure 2. Parallel Resistance Model

 

 

 

 

m< <R.

 

 

 

Figure 3. Parallel Resistance Circuit

 

 

 

 

 

 

 

 

W
. /. 4

Figure 4. Series Resistance Model

Rf

Rs

Figure 5. Series Resistance Circuit

 

1.00

0.90 .

0.80 -

0.70 .

0.60 s

kin/kc 0.50 -

0.40 <

0.00

r

I

T

I

T

 

 

----Parallelkm/kf
---'Seneskmlkf

 

4 I

 

 

0.00

 

0.20 0.40 0.60 0.80

Porosity

1.00

 

Figure 6. Parallel and Series Models of Effective

Thermal Conductivity for y = 0.1

 

1 0
problems to be solved by means of reducing them to algebra and geometry.

Numerous models have been proposed to make use of the simple
solutions. Russell [2] proposes that porous media be modeled as cubical pores

in a cubic lattice arrangement and developed

k y s -1
—m = 1 - 3 + . 5
kr ( 8) ‘Y [8% + (1 - 8)%y] ( )

Another model proposed by Loeb [3] views the porous medium as a

 

square matrix with a lattice of pores situated inside the matrix. The effective
thermal conductivity as determined by Loeb is

k

 

 

8
_m = 1 - + 1 6
kf ( 81) 82ks + 1 ( )
k ( - 82)
p
where 31: perpendicular void fraction
22: parallel void fraction
kp: 4ydeoT3
and y: geometrical factor
d: largest dimension of pore in direction of heat flow.

Figure 7 shows Loeb’s model and the corresponding thermal circuit is shown in
Figure 8. Note that the effective thermal conductivity, kp, of a pore is a function
of radiative heat transfer. This is a disadvantage when the fluid is a liquid rather
than a gas due to the opaque nature of most liquids. Woodside [4] modeled a

porous medium putting a matrix of spheres in a continuum. Using this model, he

11

 

 

 

 

 

------------------------

 

 

 

 

 

 

 

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

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

 

 

 

Figure 7. Loeb's Model

 

12

 

 

 

 

\/\

 

 

AAA/W

Figure 8. Thermal Circuit of Loeb’s Model

13
determined the equation for the effective thermal conductivity of a porous

medium to be

'.:—:={FRVIFi-Inlz—fllll"

where

 

a: 1+ 4 (8)

“(k—“AIM
kc Tl:

I. _I

 

 

 

and c and d denote the continuous and discontinuous phase respectively. This
relation is valid only where

o s e, 5 0.5236.

There are many geometries that can be used to develop expressions for
the effective thermal conductivity of a porous medium. Specifying a geometry
and developing the corresponding thermal circuit can, in principle, be reasonably
straight forward. However, the geometry and algebra may make obtaining an
expression for the effective thermal conductivity very difficult. Other geometries
have been investigated by Godbee and Ziegler [5], Luikov, et. al. [6] and
Krupicka [7] and corresponding expressions for effective thermal conductivity of
these porous media have been developed.

The exact solution of Laplace’s equation forms the basis for another
group of models. Many of these are heat transfer applications that have been

developed from electromagnetic applications. Using a heterogeneous body

14
composed of an element with one them'Ial conductivity value inserted into

another larger element having another thermal conductivity, Maxwell [8]

developed an expression for the effective thermal conductivity

k = k [awn-22.8.40] .

k, +2k, +e,(k, _k,) (9)

This solution is based on an explicit assumption that the spheres have no
influence on one another due to the distance between. Therefore, it is implicit
that the expression above is good only as the porosity, ed gets small. To account
for influence between spheres, Rayleigh [9] developed models for spheres in a

cubic array and cylinders in a square array. He obtained the equation

 

 

( ~28d
1.4)
k
51: ° (10)
f 2 _d
+kc
kd +8d
—k—

for the cylindrical model.

More general solutions were developed by assuming the particles to be

ellipsoids. Fricke [10] developed the expression

1 + ed(F-k—d — 1]
kc

° 1+ 8,,(F -1) (11)

 

 

 

15

1 3 k, "
5 J1 + IE - 1H (12)

where
F =

and

:1: = 1. (13)

The ratio of the overall average temperature gradients in the solid and the fluid
phases is represented F and the semi-principle axes of the ellipsoid are
represented by f1. As with Maxwell, Fricke also assumed no particle to particle
interactions in his model.

Laplace’s equation is solved over one fourth of a cell for spheres in a
cubic array and cylinders in a square array by Krupiczka [7]. A nearly unusable
and very complex mathematical expression is obtained for the effective thermal
conductivity by applying artificial boundary conditions.

The solution for a matrix of particles that are in or are nearly in contact
with one another is obtained by Batchelor and O’Brien [11]. They developed this

solution for systems where y is large. They developed the expression

X

A = 4.01 lny (14)
kt

for the special case where a random array of uniform spherical particles are in
contact with one another.

There are some empirical and numerical models being used to determine

the effective thermal conductivity of a porous medium. Numerical models may

16
also be used to obtain simplified solutions to the series resistance and the

parallel resistance models. Woodside and Messmer [12] noted that the
distribution for the parallel model corresponds to a weighted arithmetic mean and
the distribution for the series model corresponds to a weighted harmonic mean.
An intermediate model has been proposed by developing the weighted

geometric mean which is expressed by

x.

k
A: __.o 15
k, 8k, ( )

This expression is the foundation for the most widely used empirical relationship,

x

k
__m = C _3__ 16
k, 2k ( )

This expression was first proposed and obtained empirically by Lichtender [13].
C is an empirical constant that is adjusted to fit the experimental data. Yet
another proposed relationship is
km = k,(1 - a). (17)

This expression was proposed by Francl and Kingery [14] who conducted
experiments on a test section with a fabricated porosity of isometric spherical
pores and anisometric cylindrical pores using air as the fluid. They were able to
correlate their data for temperatures below 500°C. There have been other
empirical expressions developed, however, they are of limited in usefulness.

Another numerical model is developed by Jaguaribe and Beasley [15]. In

this model, they consider radial flow in a porous medium as an array of solid

17
cylinders in a stagnant fluid. They use a thermal circuit with complicated algebra

that is solved numerically by determining the resistance to the flow of energy.

Many of the thermal conductivities that have been obtained experimentally
are currently available. Some of the better compilations are Cheng and Vachon
[16], Luikov, et al. [6], Ofuchi and Kunii [17], and Krupiczka [7]. However, the
experimental data appear to have deficiencies in that there is no data for
systems with y < 1 and only a limited amount of date for y = 0(1) systems. Most
of the data available are for systems where y is large. For systems where y is
small or of order one, it is recommended that experiments be run to determine
the thermal conductivity.

After completing this review of models developed for predicting thermal
conductivity in a porous medium, it is apparent that there is no acceptable
general model. Some models have been shown to be acceptable in some
special cases. Cheng and Vachon [16] have shown that most all of the
commonly models used give results that differ by more than ten percent with
experimentally determined thermal conductivities. This points toward the need to
develop an acceptable general model for determining the effective thermal

conductivity of porous media.

1.3 Proposal Statement

The objective of this study is to implement a general numerical model that
can be used to determine the effective thermal conductivity of many different

porous media geometries.

18
1.4 Summag

In this thesis, a numerical model based on the heat conduction model in
Figure 9 will be developed by the superpositioning of a finite difference grid on a
simulated porous medium. This porous medium is generated using a random
number generator and will be composed of either solid or fluid elements. The
ensuing difference equations will be solved utilizing the Alternating Direction
Implicit (ADI) method. Employment of the ADI method begins by introducing a
quasi-time derivative represented by stepping first in the x-direction (k+‘/2
iteration) and then in the y-direction (k or k+1 iteration). Further manipulation will
yield a series of N equations with N unknowns that can be solved using a
Gaussian elimination method. The heat flux is then averaged in the x-direction,
thus leading to determining the effective thermal conductivity.

This thesis continues next with the presentation of the method of solution
followed by the results and discussion, and finishes with recommendations and

conclusions.

19

 

 

 

 

Figure 9. Heat Conduction Model

CHAPTER II

METHOD OF SOLUTION

2.1 Governing Eguations

A porous medium can be arranged from a series of cylinders as shown in
Figure 10. If a square grid is imposed over the porous media, then a grid that is
fine enough to have single elements that are either a solid or a fluid can be
visualized as in Figure 11. From this, a simulated porous medium can be
generated. By placing a finite difference grid over this porous medium grid, a
finite difference algorithm can be employed to solve the ensuing heat conduction
problem. From this, the effective thermal conductivity can be determined for the
entire porous medium.

The porous medium is generated by first determining a porosity to be
used for the porous medium. Once a porosity value is decided upon, a random
number between zero and one is generated and assigned to each element in the
grid. Those elements assigned a random number less than the porosity value
are fluid elements and those elements with a random number greater than the
porosity value are solid elements.

Once the generated porous media grid is in place, the solution for the two
dimensional heat conduction equation for the grid may be considered. The

following equation may be written for each element as shown in Figure 12:

azT
V

E

k3” + kg) W, = o. (18)

20

21

 

Figure 10. Array of Cylinders with Imposed Finite Element Grid

22

 

Figure 11. Simulated Porous Media Where
Shaded Area Represents Solid

23

(k + 1’2)

TB,i + 1,]

 

 

1

TT,ij

“RM-1? TL,ij 9 TRj+ TL,I,j+1 ——* (k or k + 1)
II

TB,ij

O

 

 

 

 

TT,I - 1,;

 

 

Ln

Figure 12. Finite Difference Element

23

(k + 1/2)

T31 + 1,]

 

TT,ij

 

 

 

TR,i,j-1. TL,ij 9 TRJ. TLIJH —> (k or k + 1)
il

TB,ij

 

 

‘

TT,i - 1,]

 

 

ka

Figure 12. Finite Difference Element

24
The superscript i on the thermal conductivity indicates whether an element is to

be either a solid (5) or a fluid (f) and the subscript indicates thermal conductivity
as directionally dependent. The upper boundary is set at some temperature, Tu,

and the lower boundary at some temperature, TL, where Tu > TL. Further, the

sidewalls in the porous system will be considered to be adiabatic (Q = 0).

Applying finite difference to Equation (18), the terms in Equation 18 become

@21- (Tu; " 2Tij '1' TR,ij)

 

 

ax2 = (sz) (19)
and
62T (TB.ij — 2Tij + Tm.)
_ = , 2o
63/2 (Ayz) ( )

Substituting Equations (19) and (20) into Equation (18), we can write Equation

(18) in the finite difference form

[TU] _ :Ti; + T”) + R91
x .

k (i)

x,ij

 

 

Now, the Alternating Direction Implicit (ADI) method will be utilized to
solve the set of equations represented in Equation (21). To employ the ADI
method, we must first set a quasi-time derivative that corresponds to first
stepping in the y-direction (k + 1I2 iterate) and then in the x-direction (k + 1 or k

iterate).

25
The following equation is thus obtained:

k

(“)9 (k) _ X-l' (k) (k) (kl
T, — Tij _ :(TL,j — 2Tij + TM)
k ” + + +
+ AWE?“ V2) — 2T)" ’2’ + TI}, V9) (22)
GT

where

2 2 2
5T = __(Ax) = __(Ay) = i . (23)
AtT AtT AtT

Before the set of equations represented by Equation (22) can be solved,

adjacent finite difference elements must be coupled. This is done by observing

that
TTJI' = Tam.) (24)
T3."- = TU.” (25)
TU]. = 1',"qu (26)
TR."- = TLJJH (27)

which represents the temperature continuity at the boundaries between adjacent
elements. There is also a continuity of the heat flux at the boundaries between

adjacent elements. Therefore, we can also write the following equation:

 

 

T .. - T.. T. . - T . _
ky‘ij 1ij I] = ky'H M 1+ 1,) 9/ B,I+1,) . (28)
2 2
Substituting Equation (24) into Equation (28) yields the equation
T .. - T. T. . - T ..
ky,ij TJl ll = ky,i+1,j 1+1.J T,IJ ’ (29)
96 96

26
and solving for T1),- gives

T - ky,i+1,j 12%;?) + ky "iTIK+}§I 30
_ . ( )

k' ‘+ k:.:1

YJ " 1-l
Similarly, the terms T3,“, TL], and TR), can all be expressed in terms ofTi-1_,-, Ti,- -
1, and T1,). 1 such that

11M“ + ky" "TiIKTyI

 

 

 

k .
T8, = y,I-1j I-1.j (31)
J ky,i-1,j + kyiiu
k .. TI“) + k MT”)
TLi. = XM" H 1 (32)
J kx,i,j-1 T kxjju
k .. TIKI+ k .. TI")
TR, = X,l,j*1 i.j+1 XJI 'I (33)
J kxjj + 1 + k X,lj

respectively. Substituting Equations (30), (31), (32), and (33) into Equation (22)

yields the equation

TIIMM) _ TiIk I

 

 

 

 

k k k
= kx,iI kx,i-1.ITift.)I+ kx.iITiI ) _2T(k) + kx.i+1.j TigIIT- kX'lT'I )
5 T kx,i-1.j + kxII 1) kx.i+1.1 + kxii
(“)6 k+)6) (“)6 (“)9
+kyii [km- 1_T,,_ 1 I +k,,,,T,,( _2Tu(k+ )4) + ky.ij+1Ti.j+1 ) +kinTii ] . (34)
GT ky,,,,-_, + kal J ky,i,j+1 T ky.ij

By rewriting Equation (34) and collecting the coefficients on each element
temperature, T, the coefficients can be written for

TIE-W: B =- Mk” ‘ (35)

ij cT(kY'I+kYJI 1)

27
k2.. k2 k

 

 

 

 

 

 

M: c, :1. - +24% (36’
‘ ‘ O’T(kY.iI +ky.i.I-1) °T(ky.ii +kaI+ 1) OT
+ k i' R 1'1'
T515216): 00:" y" H 1 I (37)
o-T(ky,ij+ky.i-I*1)
k k
1,0,)” E,,- = x,ij X.i-1.I (38)
' O’T(kxij+kx,I- 1j)
Ti(jk)2 Fij =1+ kiij + kid _zffl’ (39)
OT(kx.ij+kx,i-1.I) OT(kx.iI+kx.i+1J) OT
k k .
11(911' Gil: x“ H U (40)
61(kxij+kx,i+1,j)

and the following equation can be written

B, TOW) + c, T,"”Y2’ + D, To”)

'1 1 i,j+1
= 5,119,, + F, T,"" + G, ,T,‘f,, (41)

Further, Equation (41) can be written as a set of equations

=0" )6) —(k+ )3) -(k)

M) T) = S (42)

=k+

2)
where M,- is represented by a tridiagonal matrix. By employing a Gaussian

elimination routine called the Thomas algorithm , Equation (42) can be solved

for TE“ ”

. This routine has the advantage of being able to store the required
coefficients in a compact 3 x N2 matrix. This is done as j is stepped form 1 to N.

Similarly, an equation can be developed for the k + 1 iteration and solving for

28

k+1)

T: as i is stepped from 1 to N. The solutions are obtained alternately in

each direction iteratively until the solutions converge.

2.2 Boundam Conditions

It is observed that at the left boundary, where the adiabatic walls exist, the

heat flux equation becomes

fl

8x ”0 = 0 (43)
which becomes
L—T— = o.
Ax (44)
Therefore,
TL = TR. (45)

Equation (45) is then substituted into Equation (22) to obtain

Til-(Mg) _ Tillk) = &(2Téz) _ 213(k))
oT '

k i' + + +
+ Gil—(T3; 72) — 2T;k V2) + T3; 72)) (46)
Similarly, the boundary condition at the right wall is

E
ax

=0

x=N

 

and the finite difference equation becomes

29

k ..
(kW?) ('0 _ X-' (k) (k)

T

k .. + + +
+ firs: V2) - 2T9 y» + Tl}, Y»). (47)
T

The B,C,D,E,F, and G coefficients are then obtained as before. In addition, the
initial condition at the bottom of the porous medium is

T3,, = TL (48)
and at the top, the initial condition is

Tm = TU. (49)
The B, C, D, E, F, and G coefficients are then again obtained as before.

The effective thermal conductivity can be obtained by determining the
average heat flux in the y-direction while x = O

_ 1 " 6T

and then applying the definition from Equation (1) and assuming L = 1,

k =

q
, 48

CHAPTER III
RESULTS AND DISCUSSION

3.1 Porous M Generation
The porous media generation is accomplished by the use of the pseudo-random
number generator function call available in Microsoft FORTRAN. Figure 13
shows the results of running the random number generator one hundred times
for each porosity from 0.1 to 0.9. The values are then normalized by dividing the
generated porosity by the target porosity. For example, with a target porosity of
0.1, the calculated porosities range from 0.04 to 0.18 with a population standard
deviation of 0.027. This distribution is shown in Figure 14. This results in a
normalized porosity of 0.4 to 1.8. Further, running the random number generator
for a target porosity of 0.9 results in a range from 0.83 to 0.96 with a population
standard deviation of 0.026 giving a normalized porosity of 0.92 to 1.07. This
distribution is shown in Figure 15. As will be seen in the discussion on numerical
testing, the randomness of the generated porous media will result in slight
deviations as the different data are put into graphical form.

Figure 16 shows calculated porosity as a function of grid size. A thermal
conductivity of 0.3 was used. As before, the actual porosities tend toward the

target porosities of 0.3, 0.5, and 0.8 as the grid size is increased.

30

31

Distribution of Cells for Porosity of 0.1 to 0.9

 

 

7 10.?
no.2
no.3
no.4
.05
[0.6
. no.7

I08

L201 l i

Solid Gels in
Porous Nbdia

 

 

 

 

Target
Fbrosity

 

 

Target Porosity g

Figure 13. Normalized Porosity for Target
Porostiy of 0.1 to 0.9

 

32

 

 

 

 

 

16

14 i»
12 ..
10 ~»

 

 

88.?
88..
$8..
89;
«83
5:3 .
8o:

nomad

0.1 Target Pbrosity

hemmd
oomhd
named
um 56

ooovd

 

 

Figure 14. Frequency vs. 0.1 Normalized Porosity

 

 

33

 

 

 

 

 

 

 

20
i 184»
16<~
141»
12»
l
~FREIBCY10«~
3..
l
6
4T
2.
o.
N N 1") t") V to to IO
N O N no
s§§§sa§§s§s§§
O O O O O O O 1- !- v- v- ‘-
W
0.9 Target Porosity

Figure 15. Frequency vs. 0.9 Normalized Porosity

 

 

0.900

0.800 4

0.700 4»

0.600 1

0.500 . ~

Calculated
Porosity

0.400

0.300 «»

0.200

0.100

 

0.000

l
he.-.)

 

 

9 12
N x N Bernents

15 18

f-- L - . — Target W031

....... Target Porosity = 0.5

Figure 16. Calculated porosity vs. N x N Elements

- — — - Target Porosity = 0.8

35
Further, it is expected that as the grid size increases, the calculated porosity will

converge around a single value as determined by the set target porosity. This is
confirmed by the graph in Figure 17 which represents the calculated porosity as
a function of grid size. Figure 18 shows that for a kl/ks = 0.5 (0.3/0.6) and 3 OT =
0.1, the effective thermal conductivity will approach a single value as the grid

size is increased.

36

 

 

0.40
0.35 11

0.30

Calculated
Porosity

0.25
0.20 -

0.15 1

 

 

 

 

95
105
115 .,
125
135

N x N Bements

 

Figure 17. Calculated porosity vs. N x N Elements

36

0.40

 

035 ~

0.30 1-

Calculated
Porosity

0.25 .,

0.20 1»

0.15 ,_

 

 

 

 

0.10:1A.L.e.+1

IO ID ID

5
15
25
35 1
45
55
6
7
8
95

105
115

N x N Bements

Figure 17. Calculated porosity vs. N x N Elements

125 »

135

0.50

0.45 1

0.40

0.35

0.30

Km 0.25

0.20

0.15 ,_

0.101

0.05 <

0.00

37

 

 

 

 

l
1
ID ID In In In to to ID ID ID In In to I!)
v- N ('0 1‘ In (D N G) O) o v- N co
v- ‘- v- 1-
N x N Bements

 

Figure 18. Effective Thermal Conductivity vs. N x N Elements

 

38
3.2 Results of Numerical Validation

For the following numerical testing, a target porosity of 0.3456 was used
unless specifically stated otherwise. Thermal conductivity values of either 0.3 or
0.03 were used for the fluid and solid materials in the following numerical tests.
Figure 19, showing effective thermal conductivity as a function of the grid size,
indicates the relative range of thermal conductivities for kf/ks of 0.1, 1.0, and
10.0. As expected, these curves trend closer to one value as the grid size is
increased.

In Figure 20, the effective thermal conductivity is plotted as a function of
the log1o of the time step. As before, thermal conductivities of either 0.3 and
0.03 were used for the fluid and solid materials. These graphs were expected to
converge around one value as the time step is increased. The fluctuation that is
seen is attributed to the randomness of the grid generation. However, these
curves do tend toward a central value. Figure 21 shows the effective thermal
conductivity as a function of the target porosity for thermal conductivity ratios of
0.5, 1.0, and 2.0 using thermal conductivities of 0.3 and 0.6. These curves also
behave as expected. When kf/k,3 = 2.0 at a target porosity of 0.0 to 1.0, the curve
is equal to 0.3 and 0.6 respectively, and when kf/ks = 0.5 with the target porosity
once again running from 0.0 to 1.0, the curve runs from 0.6 to 0.3 respectively.
This is also the expected result. Finally, with kf/ks =1.0 and the thermal
conductivities set equal to 0.3, the curve is equal to 0.3 from a target porosity of

0.0 to 1.0.

 

39

 

 

 

 

 

 

 

 

 

0.35 .
0.30 “ /—\ d‘ ......................
.I \v”
.I
V
0.25
l
0201 .
—.-—- kf/ks=0.1 I
-.—--l<f/ks=1.0 I .
....... l<flks=10.01 Q
0.15 . . .
-— '.~.‘\. ’- ....... ~\
0.10 .
0.051 """"""""""""""" - """"""""" 1 I
0.00 f # 5 + L .
3 6 9 12 15 1a 21 24
NxNBerrents

 

Figure 19. Effective Thermal Conductivity vs. N x N Elements

40

 

 

 

0.4 AVE— — , , - .. ____,

 

kf k - 1.0
0.3 —~,.-_._--__’. 3 -__ _- ._-- ¥ %-

 

 

Effective
Thermal .
Conducfivny

 

 

 

0.2 —1I I
a ____k_[I_Igs - 0.1 1
1 -
L ki/ks_,;,_1_g.02-_---wr9---;ng ______ ‘ I
0'0 1‘ ‘7 77 7 T 7 V ‘7 ” T T " ", i f 5 .
-3.000 -2.000 -1.000 0.000 1.000 1

Log(DT) |

 

Figure 20. Effective Thermal Conductivity vs. Log (DT)

 

0.4

0.3

Effective
Thermal
Conductivity

0.2

0.1

40

 

 

 

 

 

kflks - 1.0 -

_ _.-2 7—0- 2

—4

._ kings - 0.1

~2.000
Log(DT)

1 ki/ks 7/100 _,—~\

-1.000

0.000

 

Figure 20. Effective Thermal Conductivity vs. Log (DT)

1.000

 

 

 

 

 

 

 

 

0.6000 ,
\I . ,
\\ o
\ ,-
\\
‘. .
\
0.5000 ‘\ .‘
\. ,0
\O
‘. .0
\. 0
\V.
o...\\
0.4000 ." \.
. ' x
. \.
.0... \‘\I
.". \
.o'. .\._. "— fl
\.'\ 1. ...... kf/ks=2.0‘
19.0.3000 1:1 ..................................... .1. 1_,___,k,,k3=1Io‘
—-—--kf/ks=0.5
0.2000 -
0.1000 -
0.0000 1 * r; 4 a c f f
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Target Porosity

Figure 21. Effective Thermal Conductivity vs. Target Porosity

42
The three curves in Figure 22 show the effective thermal conductivity for

kfx/ks" = 0.1, kfx/k.’ = 1.0, ks" = constant and ks’ being varied such that k,‘/ k..y =
0.5, 1.0, and 2.0.

Finally, Figure 23 is the same graph as Figure 6 with an additional curve.
The two curves in Figure 6 show the theoretical boundaries for porous media.
The one curve being the graphical representation of the parallel model and the
other curve being the graphical representation of the series model. The third
curve in Figure 22 represents the model as proposed in this paper. The shape
to the curve is due to the random generation of the fluid and solid cells in the
porous medium. As stated previously, all current models for determining the
effective thermal conductivity of a porous medium fall between the boundaries as
established by the parallel and series models. As expected, Figure 22 shows

that the curve for this model also lies between these two limits.

43

 

 

 

O
03
l
1
l

1 V ‘5 0 V 1:. ks(x)lks(y) - 2.0

 

2 < ks(x)lks(y) - 1.0 I
, ‘_ I * ks(x_)lks(y) - 0.5...
0.5 "1
l
o. _
Effective
Thermal :
Conductivity ~
0.3 .. 1
1 ‘7 1
0.2 + l
l
I :
0.1 *I 3 I? I
- -,, 1... “55*

l ' i , ’ I I . 1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Generated Porosity

Figure 22. km vs. Generated Porosity

0.6

0.5

0.4

Effective
Thermal
Conducfivny

0.3

0.2

0.1

0.0

43

 

.L *5 T , T 1 k8(X)Ik8(y) - 2-0 1
  = ~ k8(X)IkS(y) - 1.0 1
1-; k8(X)/ks(y) - 0-5 r

J

1

 

 

'1 1
1
1 I
I f}... ._, \ H 4P
3‘ -1 .2. ..
T Tl‘e-fi
T 1 T . 1 ' f 'T 1TTTT‘TTT'TT1T 1

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Generated Porosity

 

 

Figure 22. km vs. Generated Porosity

k../Ki

1.200

1.000 <»

0.800 «

0.600 ..

0.400 ..

0.200 ~

0.000

44

 

knlk, vs. Target Porosity

 

 

 

 

,1
If;
r .'
I.’ I
I I
I, .' ’
I
I . l
/
I l
I, ' O
l/ ' I,
1’ I
I
’I/ . l
I ‘ I
[I .f“. .' .I
/” \T I
’ .' I
I ’1 .’
/ . .
I, ,o'. .r’T’
. / .o-"TTT .’ _l’
’ . ".’
I. ’ -. ' .I .
e:.---"'
§ § ‘2’ .3. § § § § § § §
0' o' o" o' :5 d o' o d o .-

Target Pbrosity

Figure 23. kmlkf vs. Target Porosity

 

T; — — — Paralel Nbdel
—--—- SeriesNbdel 1
....... noposedModer

L

 

CHAPTER IV

CONCLUSIONS AND RECOMMENDATIONS

4.1 Conclusions
1. As matrix size is increased the effective thermal conductivity will trend
toward a close set of values that are dependent upon the chosen target porosity.
2. The effective thermal conductivity is well behaved between the set
solid and fluid thermal conductivities for 0.0 target porosity to 1.0 target porosity.
3. The validity of this model is supported by the ratio of effective thermal
conductivity to fluid thermal conductivity curve falling between the two theoretical
boundaries of the series and the parallel model.
4. The calculated porosity does not necessarily match a specified target

porosity due to the random generation of the porous medium grid.

4.2 Recommend_ations

1. The current program requires more substantial testing using materials
with anisotropic properties.

2. The next extension of this numerical model would be to expand the
capability to model three, four, five, or more components. This would allow for
the determination of the effective thermal conductivity of systems consisting of
multi-component materials.

3. Another refinement would be to utilize the Monte Carlo method for

generating the material matrix. Each cell in the matrix would have an equal

45

46
opportunity of being assigned a value from one to N that would be used in

determining whether that cell would be a fluid cell or a solid cell.

APPENDIX

APPENDIX

Sample output for kf = 0.3 and ks = 0.3, DT = 10 seconds, and
convergence criteria set to 0.01 degrees.

RESULTS OF EFFECTIVE THERMAL CONDUCTIVITY CALCULATION

TARGET POROSITY: 0.346
GENERATED POROSITY: 0.361
GRID SIZE: 6
EFFECTIVE THERMAL CONDUCTIVITY: 2.999E-01
ITERATION COUNT: 2203

MATERIAL MATRIX:

_bN—l—t—LN
ANA-KNN
A—bN-L—L—t
ANA—t—t—t
A-ANNN-A
ANN-84M

TEMPERATURE FIELD:

345.8
337.5
329.2
320.8
312.5
304.2

345.8
337.5
329.2
320.8
312.5
304.2

345.8
337.5
329.2
320.8
312.5
304.2

345.8
337.5
329.2
320.8
312.5
304.2

345.8
337.5
329.2
320.8
312.5
304.2

47

345.8
337.5
329.2
320.8
312.5
304.2

48

Sample output for kf = 0.6 and ks = 0.3, DT = 10 seconds, and
convergence criteria set to 0.01 degrees.

RESULTS OF EFFECTIVE THERMAL CONDUCTIVITY CALCULATION

TARGET POROSITY: 0.346
GENERATED POROSITY: 0.333
GRID SIZE: 6
EFFECTIVE THERMAL CONDUCTIVITY: 3.819E-01
ITERATION COUNT: 741

MATERIAL MATRIX:

AANAAN
A—b—SNNN
”JANA—3
_sANAAN
N-L—t—tN—L
AAAANA

TEMPERATURE FIELD:

345.9
337.5
329.0
319.8
311.2
304.0

346.1
338.5
329.4
319.5
311.9
304.6

345.8
337.2
327.8
319.2
312.6
304.6

345.5
336.3
328.2
320.4
311.5
303.3

345.4
336.7
328.5
320.4
312.2
303.9

345.0
336.8
328.6
320.6
312.9
304.2

49

Sample output for kf = 0.3 and ks = 0.6, DT = 10 seconds, and
convergence criteria set to 0.01 degrees.

RESULTS OF EFFECTIVE THERMAL CONDUCTIVITY CALCULATION

TARGET POROSITY: 0.346
GENERATED POROSITY: 0.333
GRID SIZE: 6
EFFECTIVE THERMAL CONDUCTIVITY: 4.678E-01
ITERATION COUNT: 1123

MATERIAL MATRIX:

_LNANAN
JN—L-A—t—L
.AN—tN-L—t
ANN-INA
ANN-3AA
N—F—h—fi—l—h

TEMPERATURE FIELD:

346.3
338.7
329.9
321.0
313.6
305.0

346.4
339.3
330.0
320.3
312.7
304.4

346.4
338.8
330.6
320.8
312.5
304.7

347.0
339.0
328.9
319.6
312.1
304.1

346.9
338.5
329.1
320.3
312.2
304.1

345.7
336.9
328.8
320.4
312.2
304.1

50

Sample output for kx(f) = 0.6, ky(f) = 0.3 and kx(s) = 0.2,
ky(s) = 0.8, DT = 10 seconds, and convergence criteria set to
0.01 degrees.

RESULTS OF EFFECTIVE THERMAL CONDUCTIVITY CALCULATION

TARGET POROSITY: 0.346
GENERATED POROSITY: 0.278
GRID SIZE: 6
EFFECTIVE THERMAL CONDUCTIVITY: 5.991 E-01
ITERATION COUNT: 3152

MATERIAL MATRIX:

4N4N4N
444N44
444N44
44444N

4NN444
N444N4

TEMPERATURE FIELD:

345.9
338.7
328.9
319.7
314.2
305.4

344.5
335.1
328.4
320.7
312.7
304.4

345.9
338.0
331.2
322.2
312.9
304.3

347.0
336.7
326.5
319.9
312.4
304.3

346.7
336.0
325.9
320.0
314.3
305.4

344.5
333.9
326.6
319.6
312.5
304.4

LIST OF REFERENCES

51

Specific References

1.

10.

11.

12.

Baradat, y. and Combarnous, M., “Conductivite Therrnique des Milieux
Poreux,” lnstitut Francais du Petrole, Bordeaus, France, n° 106,
September, 1967.

Russel, H. W., “Principle of Heat Flow in Porous Insulators,” Journal of the
American Ceramic Sociey, vol. 18, no. 1, pp. 1-5, 1935.

Loeb, A. L., “Thermal Conductivity: VIII, A Theory of Thermal Conductivity
of Porous Materials,” Journal of the American Ceramic Sociaty, vol. 37,
no. 2, pp. 96-99, 1954.

Woodside, W., “Calculation of the Thermal Conductivity of Porous Media,"
Canadian Journal of Physics, vol. 36, pp. 815-823, 1958.

Godbee, H. W. and Ziegler, W. T., “Thermal Conductivities of MgO, Al203,

and ZrOz Powders to 850°C. II. Theoretical, “Journal of Applied Physics,
vol. 37, no. 1, pp. 56-65, January, 1966.

Luikou, A. V., Shashkou, A. G., Basilieu, L. L., and Fraiman, Yu. E.,
“Thermal Conductivity of Porous Systems,” International Journal of Heat
and Mass Transfer, vol. 11, pp. 117-140, 1968.

Krupiczka, R., “Analysis of Thermal Conductivity in Granular Materials,”
InternationiCLemical Engineering, vol. 7, no. 1, pp. 122-144, January,
1967.

Maxwell, J. C., A Trea_tise on Electricity and Magnetism, 3rd ed., vol. I,
Chapter 9, Article 314, Dover, New York, 1954.

Rayleigh, “On the Influence on Obstacles in Rectangular Order Upon a
Medium,” Philosophical Magazine, vol. 34, pp. 481-502, 1982.

Fricke, H., “A Mathematical Treatment of the Electric Conductivity and
Capacity of Disperse Systems,” 1h_e Physical Review, vol. 24, pp. 575-
587, November, 1924.

Batchelor, G. K. and O’Brien, R. W. , “Thermal or Electrical Conduction
Through a Granular Material,” Proceedings of the Royal societv of
London, Series A, vol. 355, no. 1682, pp. 313-333, July, 1977.

Woodside, W. and Messmer, J. H., “Thermal Conductivity of Porous

Media. l. Unconsolidated Sands,” Journal of Applied Physics, vol. 27, no.
32, no. 9, pp. 1688-1699, September, 1961.

13.

14.

15.

16.

17.

52

Lichteneker, V. K., “Die Dielektrizitatskonstante nattirlicher and
KiJnstlicher Mischkoorper,” Physikalische Zeitschrift, vol. 27, no. 415, pp.
115-158, March, 1926.

Francl, J. and Kingtery, W. 0., “Thermal Conductivity IX, Experimental
Investigation of Effect of Porosity on Thermal Conductivity,” Journal of the
American Ceramic Society, vol. 37, no. 2, pp. 99-107, 1954.

Jaguaribe, E. F. and Beasley, D. E., “Modeling of the Effective Thermal
Conductivity and Diffusivity of a Packed Bed with Stagnant Fluid,”
International Journal of Heat and Mass Transfer, vol. 27, no. 3, pp. 399-
407, 1984.

Cheng, S. C. and Vachon, R. I., “A Technique for Predicting the Thermal
Conductivity of Suspensions, Emulsions, and Porous Materials,”
lntemational Journal of Heat and Mass Transfer, vol. 13, pp. 537-546,
1970.

Ofuchi, K. and Kunii, D., “Heat Transfer Characteristics of Packed Beds
with Stagnant Fluids,” International Jgimal of Heat and Mass Transfer,
vol. 8. pp. 749-757, 1965.

Gene_r_al References

1.

Chapra, Steven C. and Canale, Raymond P., Numerical Methogs for
Engineers. 2nd ed. McGraw-Hill, 1988.

Niederreiter, Harold, Random Number Generation and Quasi-Monte Carlo
Methods. Philadelphia: Society for Industrial and Applied Mathematics,
1992.

Somerton, Craig W., “A Numerical Model for the Effective Thermal
Conductivity of a Porous Medium.” (Unpublished paper.)

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