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A COMPUTATIONAL APPROACH TO SIMULTANEOUS
TWO-DIMENSIONAL HEAT AND MASS TRANSFER IN
A HEAT GENERATING POROUS MEDIA

presented by

Laura J. Genik

has been accepted towards fulfillment
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1/98 mm“

A COMPUTATIONAL APPROACH TO SIMULTANEOUS TWO-DIMENSIONAL
HEAT AND MASS TRANSFER IN A HEAT GENERATING POROUS MEDIA

By

Laura J. Genik

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

1998

ABSTRACT

A COMPUTATIONAL APPROACH TO SIMULTANEOUS TWO-DIMENSIONAL
HEAT AND MASS TRANSFER IN A HEAT GENERATING POROUS MEDIA

By

Laura J. Genik

The use of activated carbon in the recovery of evaporative fuel emissions has
become common place with the advent of stricter requirements imposed upon the
automobile industry by the Environmental Protection Agency and the California Air
Resources Board. The adsorption of the hydrocarbon based fuel vapors on the activated
carbon is an exothermic process that is a function of temperature and mass concentration
of fuel vapors. To further understand the interaction of the fuel vapors and the activated
carbon, an investigation was undertaken to analyze the complex relationship of the
coupled heat and mass transfer in this heat generating porous media. The investigation
includes the analysis of linear and non-linear flow in several different porous media
resulting in a recommended approach to determine the permeability and Forchheimer
coefficient for non-linear flows. The heat transfer is analyzed with a uniform flow model
using the classic approaches to porous media analysis: single and dual energy equations.

The importance of axial conduction is determined for this model as well as the

appropriateness of the single and dual energy equations to the specifics of the given
porous media as well as generalizations to different porous media. The geometric nature
of the given problem allows for the application of an iterative boundary condition to
balance the heat flux across the separation wall that adds a unique nature to the problem.
An experimentally determined adsorption isotherm for hydrocarbon based fuels onto
wood-based activated carbon is utilized to model the source term in the heat transfer
equations. This source term is closely coupled with the mass adsorption term in the mass
species equation. The presented heat and mass transfer equations are coupled partial
differential equations that are numerically solved utilizing a fully implicit second order

correct finite difference scheme.

Dedicated to Douglas M. Gatrell, P.E.

Acknowledgements

One can never pay in gratitude; one can only pay “in kind” somewhere else in life.
Anne Morrow Lindbergh 1935

I would like to acknowledge the American Business Women’s Association, Zonta
International Foundation, Ford Motor Company, GE Foundation, Graduate School,
College of Engineering, and Department of Mechanical Engineering for their financial
support during my graduate education. My continued interaction with the Zonta
International Foundation has been a great support for me in this ongoing educational
endeavor. Zonta International is an organization of dedicated professional women who
have done a tremendous amount to further engineering graduate studies for women in the
memory of Amelia Earhart. Dr. George Lavoie of Ford Scientific Research is owed
many thanks for sparking Michigan State University’s interest in this work and in
steering me towards doctoral work.

I wish to express my thanks to several members of the faculty for their
contributions to my graduate education as well as this work. I would like to begin with
the members of my committee: Drs. Beck, Lamm, Somerton, and Zhuang for their efforts
in bringing this dissertation to completion. Our Chairman, Dr. Ronald C. Rosenberg,
gave me the opportunity to teach several courses and has always been supportive of my
continued education. I greatly appreciate Dr. James Beck for always being helpful,

encouraging, and supportive of my professional as well as personal goals. Our Director

of Communication, Craig J. Gunn, has been instrumental in clarifying and cleaning-up
every paper, fellowship application, cover letter, and resume, not to mention this
document, that I have written, and for that I am very grateful. Dr. Craig W. Somerton has
been my major professor and mentor throughout my graduate work. From him I have
learned much about engineering, education, and life and to him I owe many debts of
gratitude too numerous to list here.

There are several fellow graduate students and friends who should be
acknowledged for their contribution to my survival of graduate school. This includes
such activities as late nights in the heat transfer lab doing convection or ‘vicious’ flows or
perhaps an afternoon excursion for ice cream or malling or just commiserating and the
ever-popular dinner and card game. Thanks to all: Brooks Byam, Nancy Chism, Gayle
Errner, Lisa Kim, Mark Minor, Heidi Relyea, Leslie Scott Schoof, Bob Vance, and the
Pinochle gang. It should be pointed out that the Pinochle gang shall not be disbanded
with the completion of this document, they will just be invoking longer recesses.

The familial contributions to my graduate education are also almost too many to
list, but I shall attempt it anyway. First and foremost are my parents, Linda and Rick
Genik, for, among other things, allowing me to stay on the family plan while I was in
college: I’m sure by now they regret the ‘as long as you’re in school...’ credo that they
live by. For getting me through difficult times and for being there with a cold beer or a
much-needed ride, Aunt Shirley and ‘Unca’ Earl are owed many thanks. My brothers,
Robert and Richard, have always been supportive during these years; from changing the
oil-pressure sensor in the ’79 Thunderbird in the dead of winter to supplying the

competition in the snails race for degree completion. My in-laws, Lyle and Jean Gatrell,

vi

are owed many thanks for their support and encouragement. Jenny, Joey, and Mark
Matten are acknowledged with thanks for supplying those much-needed study-breaks and
for reminding me why I completed this degree. Though the dedication speaks for itself, I
cannot complete these gratuitous remarks without mentioning my husband, Douglas M.
Gatrell. He has been my cohort in crime throughout this event; and without him being
who he is, I would not be here in this time and place to do what has been done. The
accomplishment is as much his as mine. Last and certainly not least are the purry-furry
clan of Kismet, the gray ghost, Craftsman, and Sinbad: the greatest experimental heat

transferists known to engineers.

vii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

NOMENCLATURE

CHAPTER 1

INTRODUCTION

1.1
1.2

CHAPTER 2

Literature Review
Objective

PHYSICAL DEVELOPMENT OF MODEL

2.1
2.2

CHAPTER 3

Background for Porous Media
Heat and Mass Transfer in Porous Media

xi

xiv

1
2
11

12
12
14

DESCRIBING EQUATIONS AND BOUNDARY AND INITIAL CONDITIONS 16

3.1
3.2

3.3

3.4

Fluid Mechanics
Heat Transfer
3.2.1 One Energy Equation Model
3.2.1.1 Order of Magnitude Analysis
3.2.1.2 Initial Condition and Boundary Conditions
3.2.1.3 Non-dimensional Analysis
3.2.2 Two Energy Equation
3.2.2.1 Fluid and Solid Regions
3.2.2.2 Interphase Transport
3.2.2.3 Initial Condition and Boundary Conditions
3.2.3 Addition of Axial Conduction
Mass Transfer Model
3.3.1 Initial Condition and Boundary Conditions
3.3.2 Non-dimensional Analysis
Development of the Source Terms
3.4.1 Development of Mass Adsorption Term
3.4.2 Development of Heat Generation Term

viii

17
17
18
18
21
23
25
25
27
28
29
31
32
33
34
35
4O

CHAPTER 4
METHOD OF SOLUTION
4.1 Heat Transfer
4.1.1 One-Energy Equation Model
4.1.2 Two-Energy Equation Model
4.1.3 ADI Method for Addition of Axial Conduction
4.1.3.1 Single-Energy Equation Model with ADI
4.1.3.2 Single-Energy Equation Model with ADI
4.2 Mass Transfer
4.3 Coupling of Heat and Mass Transfer Models

CHAPTER 5
ANALYSIS OF TRANSPORT PROPERTIES AND EXPERIMENTATION
5.1 Flow Parameters

5.1 .l Permeability-Forchheimer Development
5.1.2 Method of Solution

5.1.3 Analysis
5.1.4 Conclusions and recommendations of Permeability Study
5.2 Experimentation

5.2.1 Determination of Density, Porosity, and Specific Surface
5.2.1 Non-reacting Gas Heat Transfer Experiment

CHAPTER 6
RESULTS AND DISCUSSION
6.1 Numerical vs. Simplified Analytical Solutions
6.1.1 Heat Transfer Model
6.1.2 Mass Transfer Model
6.2 Single Energy Equation Model
6.2.1 Parametric Study
6.2.2 Boundary Condition at y = 0
6.2.3 Addition of Axial Conduction
6.3 Dual Energy Equation Model
6.4 Coupled Model
6.4.1 Parametric Study
6.4.2 Comparison to Experimental
6.5 Summary

CHAPTER 7
CONCLUSIONS AND RECOMMENDATIONS

APPENDIX
RESIDUAL SUM OF SQUARES ANALYSIS FOR AXIAL CONDUCTION

BIBLIOGRAPHY

42
42
43
46
48
50
51
54
57

6O
6O
6O
63
66
76
77
77
79

82
82
82
85
86
86
96
101
107
l 10
1 10
120
120

122

125
136

LIST OF TABLES

Table 5.1 F -Test for all Data Sets 68
Table 5.2 Vector B for All Data Sets 69
Table 5.3 Experimental Data Sets 69
Table 5.4 Permeability Calculated From Quadratic Curve Fit and Maximum
Likelihood Estimation Procedure 70

Table 5.5 B2 Calculation From Quadratic Curve Fit and Maximum Likelihood

Estimation Procedure 70
Table 5.6 [33 for All Data Sets 71
Table 5.7 Confidence Intervals for 83 71
Table 5.8 Porous Media Characteristics for all Data Sets 75
Table 5.9 Measured Data for Density Calculation 77
Table 5.10 Density Calculation and Uncertainty Analysis 78
Table 5.1] Specific Surface Area 79

Table 6.1 Energy Equation Model Selection Matrix 121

Figure 2.1
Figure 2.2
Figure 3.1
Figure 3.2

Figure 3.3

Figure 3.4
Figure 3.5
Figure 4.1
Figure 5.1
Figure 5.2
Figure 5.3

Figure 6.1

Figure 6.2

Figure 6.3
Figure 6.4
Figure 6.5

Figure 6.6

LIST OF FIGURES

Example of Porous Media
Physical Model Geometry

Model Geometry

Illustration of Balanced Heat Flux

Magnification of Balanced Heat Flux for Dual Energy Equation
Model

Mass Transfer Potential Model

Adsorption Isotherrn Based on Experimental Data
Coupling of Heat and Mass Transfer Models
Experimental Data of Ahmed and Sunada

Differential Control Volume

Comparison of Numerical Model to Experimental Data

Comparison of Analytical to Numerical Solution for Heat
Transfer Model

Comparison of Analytical to Numerical Solution for Mass
Transfer Model

Temperature Distribution at 10% of Steady State
Temperature Distribution at 50% of Steady State
Temperature Distribution at Steady State

Temperature Distribution for Various Heat Generation Terms

xi

12

15

16

23

29
35

38

59

64

73

81

83

84
87

88

89

92

Figure 6.7
Figure 6.8
Figure 6.9

Figure 6.10

Figure 6.11

Figure 6.12

Figure 6.13
Figure 6.14
Figure 6.15
Figure 6.16

Figure 6.17

Figure 6.18

Figure 6.19

Figure 6.20

Figure 6.21

Figure 6.22
Figure 6.23
Figure 6.24
Figure 6.25
Figure 6.26

Figure 6.27

Temperature Distribution for Various Aspect Ratios
Temperature Distribution for Various Heat Capacity Ratios
Temperature Distribution for Various Peclet Numbers

Effect on Exit Temperature of Media due to Various Boundary
Conditions at y = 0

Comparison of Various Boundary Conditions at y = O in
First Half of Media

Comparison of Various Boundary Conditions at y = 0 in
Second Half of Media

Balanced Heat Flux Boundary Condition

Residual Sum of Squares with respect to Peclet number
Magnification of Figure 6.14

Example of Effect of the Axial Conduction at Peclet number of 5

Example of Effect of the Axial Conduction at Peclet number of
100

Importance of Specific Surface Area in Selection of Dual
Energy Equation Model

Increase in Magnitude of Heat Generation Term in Selection of
Dual Energy Equation Model

Thermal Effect of the Heat Capacity Ratio in Coupled Model

Mass Transfer Effect of the Heat Capacity Ratio in Coupled
Model

Thermal Effect of the Peclet Number in Coupled Model

Mass Transfer Effect of the Peclet Number in Coupled Model
Thermal Effect of the Biot Number in Coupled Model

Effect of the Lewis Number in Coupled Model

Comparison of Sherwood Correlations in Coupled Model

Comparison of Experimental Data to Coupled Model

xii

93

94

95

97

98

99
100

103

104

105

106

108

109
112

113
114

115

116

117

118

119

Figure A.1 Example of Effect of Axial Conduction at Peclet number of 2 125
Figure A.2 Example of Effect of Axial Conduction at Peclet number of 10 126
Figure A.3 Example of Effect of Axial Conduction at Peclet number of 15 127
Figure A.4 Example of Effect of Axial Conduction at Peclet number of 20 128
Figure A.5 Example of Effect of Axial Conduction at Peclet number of 50 129
Figure A.6 Example of Effect of Axial Conduction at Peclet number of 75 130
Figure A.7 Example of Effect of Axial Conduction at Peclet number of 200 131
Figure A.8 Example of Effect of Axial Conduction at Peclet number of 300 132
Figure A.9 Example of Effect of Axial Conduction at Peclet number of 500 133
Figure A. 1 0 Example of Effect of Axial Conduction at Peclet number of 1000 134

Figure A.ll Example of Effect of Axial Conduction at Peclet number of 2000 135

xiii

z~2~0°
§~

ambient

:.
'1‘

gbgwxf"

ES

.D'U'U'03
”‘0

NOMENCLATURE

Diameter of circular control volume
Width in radial direction

Constant in adsorption isotherm
Biot number

Constant in coefficient matrix
Constant in coefficient matrix
Constant in coefficient matrix
Concentration of adsorbate in solid
Specific heat

Constant in source vector
Mass diffusivity of component i in to gas mixture, k

Constant in adsorption isotherm
particle diameter

gravity

Enthalpy change of adsorption
Outside heat transfer coefficient
Interphase heat transfer coefficient

Enthalpy change from gas to liquid of active gas

Permeability

Thermal conductivity
Thermal conductivity of separation wall
Length in axial direction
Lewis number
Forchheimer Coefficient
Nusselt number

F orchheimer Exponent
Pressure

Peclet number

Prandtl number

Specific Discharge

xiv

Q.
3 i

Q.
a
Q

UJN ‘5

8H°N3Hggfcbnh

“I

T(z )inltial
I

U
w
x

2

Y
y.
z

Volumetric heat generation rate (W/m3)

Scaling quantity for volumetric heat generation rate

Residual sum of squares of maximum likelihood estimation
Volumetric mass adsorption rate (g/m3)

Estimated variance of data

Sum of squares function

Schmidt number

Sherwood number

Temperature

Temperature at inlet to continuum
Ambient temperature
Maximum temperature in continuum

Initial temperature distribution

Time

Overall heat transfer coefficient

Darcian velocity component in axial direction
Specific Discharge in general model

Radial position

Experimental data point squared

Axial position

Greek Symbols

>3 wzmm,ooi>'-iue

M'D<T—'T—'
N

N

'96-'19

Thermal diffusivity
Vector of estimated parameters
Maximum temperature difference
Differential operator term
Wall thickness
Residual difference
Porosity
Model
Mass transfer coefficient
k 6

W

Thermal conductivity ratio of media to wall, ,1 = —"'k—-
a

Dynamic viscosity

Unknown multiplicative constant
Viscosity

Density

Specific surface area

Known variance of the errors

Tortuosity
Hydraulic gradient
Angle fluid element enters control volume

XV

<lD-E~€

Superscripts

1:

~

Y
z

Subscripts

mm: 23 gfiwh'mm

Covariance matrix

Angle at which the fluid element impacts particle
Non-dimensional heat capacity ratio

Lapacian Operator

Non-dimensional quantity
Half time step

y-plane in ADI method
z-plane in ADI method

Fluid property

Indices for y

Indices for 2

Indices fort

Based on length of media
Maximum value of j
Media property
Maximum value of i
Downstream position
Solid property

xvi

Chapter 1
Introduction

A man with a talent does what is expected of him, makes his way, constructs, is an engineer, a
composer, a builder of bridges. It’s the natural order of things that he construct objects outside
himself and his family. The woman who does so is aberrant. . . . We have to expiate for this cursed
talent someone handed out to us, by mistake, in the black mystery of genetics.

May Sarton 1965

Interest in the analysis of porous media has existed for innumerable centuries. In
early Roman times, considerable attention was given to the design of fountains that
entailed an understanding of the flow through porous media. In France, circa 1800, the
design of public works led to the observation of the relationship between pressure drop
and the specific discharge of the flow (Darcy 1856). In modern day society, porous
media analysis is still undertaken by the traditional civil engineer in the application of
groundwater flow, decontamination and remediation of soil, solid waste analysis, and
design of public works. Porous media analysis has branched out beyond the traditional
analysis as engineers investigate the world in which they live as a combination of solid
and fluid interaction. Several applications exist from the analysis of chemical reactors to
the movement of fire through forests. Widening the scope of the analysis into the design
of electronic equipment and components through to the design of heat exchangers has
occurred in recent years. The addition of catalytic converters and activated carbon

canisters as standard equipment on automobiles has ignited the interest of the automobile

industry in porous media analysis. To paraphrase from the state of Michigan motto, ‘if

you seek a pleasant porous media, look about you’.

1.1 Literature Review

Considerable work has been done to further the modeling of flow in a porous
media since Darcy (1856) recorded his observations in his engineering report on the
modernization of the public water works of Dijon. On the centennial anniversary of the
publication of "Les Fontaines Publiques de la Ville de Dijon," Hubbert (1956) re-
acquainted the engineering world with the original experimentation and data of Darcy and
then went on to derive Darcy’s law from Navier-Stokes. Forchheimer (1901) dealt with
some of the non-linearity of Darcy’s law through the addition of a quadratic term that has
come to be known as the Forchheimer extension. This extension takes into account the
inertia effects of the flow through the media. Tek (1957) investigated the non-linearity of
Darcy’s law for high flow rates through various sands through the addition of a constant
term that is a function of porosity of the media. Included in non-linear flow analysis in
porous media is Ward’s (1964) treatment of turbulent flow analysis in the media that
leads to an equation quite similar to the F orchheimer extension. Masuoka and Takatsu
(1996) modeled the turbulent flow in the medium from a volume averaged form of the
momentum equation where the Forchheimer term is modeled as a drag force term due to
turbulent mixing rather than inertia effects. Masuoka and Takatsu go on to propose a 0-

equation model for the turbulent flow in a porous media that relates the Forchheimer

parameter to the turbulent eddy viscosity. Vafai and Tien (1981) derive the momentum
equation from a volume average analysis and proceed to rigorously investigate the
boundary and inertia effects of flow in porous media. The effect of variable porosity and
channeling of flow at the wall of a porous medium is analyzed by Vafai (1984) as well.
The magnitude of the Reynold's number based on permeability is determined for the
demarcation between pre-Darcy, Darcy, F orchheimer, and turbulent flow in a medium of
packed spheres by Kececioglu and Jiang (1994). Vafai and Kim (1995) utilize the
Brinkman extension to the Darcy-Forchheimer relation to analyze the flow through
porous media.

Investigating other effects found in the flow modeling in a porous media, the
validity of the infinite porous medium assumption in a bounded flow is analyzed by
Somerton and Wood (1988). The development of analytical expressions for varying
porosity based on a two-region model for the flow are presented. The governing effect on
the infinite porous medium assumptions appears to depend on the particle diameter to
container diameter ratio. This type of two-region analysis is also employed by Srinivasan
et al. (1994) to model a spirally fluted tube of a heat exchanger using a porous substrate
analysis. In this work the flow is divided into a region with no media and a region with a
porous substrate which incorporates a directionally dependent permeability and flow
which is then numerically solved utilizing the finite element method. This is a novel
application to the heat exchanger analysis to account for the swirl in the flow typically
attributed to turbulence and modeled with a k-e approach. Srinivasan et a1. confirmed

their numerical model with experimental data and found excellent results.

Closely tied to the flow in porous media is the mechanism of heat transfer within
the media. Several examinations of both natural and forced convection have been
investigated by numerous people. Analysis involving natural or free convection in a
porous media are many and varied. Dona and Stewart (1989), to elicit the understanding
of variable properties such as viscosity and density, undertook the study of heat
generation in a natural convection situation in porous media. Stewart and Burns (1992)
model the free heat transfer in heat generating porous media as it applies to grain storage.
This investigation included the interesting application of a permeable inner boundary
condition in an annulus, which lead to the conclusion that the flow was forced deeper into
the media as a function of temperature. Stewart et al. (1994) extended this work to a new
design of a grain storage bin to determine if there is an increase in the effective heat
transfer without the aid of external cooling. Chen and Wang (1993) analyzed the effect
of a horizontal conducting baffle in a porous enclosure. The specific application was to
the modeling of aluminum backed insulation in a roof with natural convection occurring.
The authors determined that the stability of the convection was a function of the
placement of the baffle through a parametric study of various geometric configurations.
Parsad and Chui (1989) examined the natural convection in a cylindrical porous enclosure
with heat generation in a steady state situation. The thermal stability of the flow was
determined depending on the applied isothermal boundary condition. Considerable data
is presented for the Rayleigh number, aspect ratio, and maximum temperature. Jang and
Tsai (1988) studied the stability of natural convection in a porous medium between
parallel plates with uniform heating from below with a conductive partition. The authors

concluded that the onset of stable thermal convection for the two horizontal porous layers

is most likely when the conducting partition bisects the channel. An increase in the
partition thickness or a decrease in the thermal conductivity of the partition will result in
the system becoming stable as well. The effect of natural convection in a chemically
reacting media is investigated by Vafai et a1. (1993) to determine the parameters that
govern the thermal runaway of the application. This would occur when the chemical
reaction rate exceeds the removal of energy from the media. An interesting application of
a boundary condition in a free convection problem was presented by Rees and Pop (1994)
who investigated the effect of a isothermally heated vertical wavy surface that has
undulations of order 1. With the proper similarity transformation, the governing
boundary layer equations reduce to those of a smooth vertical surface. The authors
concluded that this is only valid if the boundary layer thickness is of the same order as the
undulations.

Additional investigations in porous media analysis leads to several applications of
forced convection. Gupta et a1. (1974) attempted to develop a Reynold's number-Colburn
factor correlation for interphase heat transfer to cover a wide range of flows and
geometries. Vafai and Kim (1989 and 1995) determine an exact solution for fully
developed flow with steady state conduction and convection in a bounded isotropic
media. Comer and Kleinstreuer (1995) utilize porous media analysis to model liquid
droplets in gas flow by applying the Galerkin method to the continuity, momentum, and
energy equations. Jiang et a1. (1996) examined the effect of variable properties and
porosity within a vertical porous media subject to forced convection. The flow was
modeled as non-Darcian and seemed to concur with recently published experimental data.

Gunn and De Souza (1974) completed considerable work on establishing the need for

axial dispersion to be included in low Reynold's number flows. Axial dispersion is a
widely accepted modification to axial conduction to include a velocity component in
porous media analysis. Adnani et al. (1995) investigated the necessity of correcting the
axial dispersion coefficient in a media with two-dimensional conduction and axial
convection. The authors concluded that for a Peclet number less then 10, the corrected
axial dispersion should be included since as Peclet number increases, the dependency of
the thermal conductivity on flow decreases. Hadim (1994) performed a numerical study
of forced convection in a saturated porous media with discrete thermal heaters. The study
was done for two configurations: one in a completely filled porous media and one where
the media is only directly over the thermal heater. It was determined that the partially
filled media could be a useful heat transfer augmentation method for the cooling of
electronic equipment.

Several authors have also examined the existence of local thermal equilibrium
between the solid and fluid regions in a porous media. Zarubin (1991) compared and
contrasted the one- and two-energy equation model for porous media applications. The
author concludes that for low Peclet number flow the one-energy equation model is
preferred; however, both models are valid. Wakao et al. (1978) investigated numerous
published data sets to determine an appropriate model for particle to fluid heat transfer in
a porous media that is not in thermal equilibrium. The result of this investigation was a
Nusselt number correlation to model the interphase heat transfer present in the dual-
energy equation. Shen et al. (1981) undertakes an experimental investigation to measure
the interphase heat transfer coefficient and correlates it to a Nusselt number. Baker-Jarvis

(1989) develops an analytical solution for forced convection with the dual-energy

equation model in cylindrical coordinates for simplified boundary conditions. Hwang
and Chao (1994) examined the cooling of electronic components through the addition of
high conductivity porous media. The experimental data were used to determine fully
developed and local Nusselt number correlations. A numerical model using both the
single- and dual-energy equation models for steady state was investigated as well. It was
concluded that the thermal equilibrium assumption over-predicted the fully developed
Nusselt number. Spiga and Spiga (1981), who solved the dual-energy equation model for
transient convection using Green’s functions, investigated the application of porous
media analysis to thermal rock storage. Several different thermal inputs are compared
and contrasted for the rock storage media.

Combined natural and forced convection solutions have also been attempted in
this branch of engineering. Nakayama (1994) attempts to determine a unified theory for
non-Darcian free, forced, and mixed convection from a buried heated solid. For the
appropriate conditions the model appears to be accurate. Shenoy (1992) used the integral
method and similarity solution to find analytical solutions to Darcy natural, forced, and
mixed convection for porous media where the energy is balanced with the heat transfer
from the wall.

Variability in media geometry has consistently been an avenue of investigation.
Chou et al. (1994) evaluated the non-Darcian heat transfer in cylindrical packed tubes
accounting for a variable porosity as an exponentially decaying function of the distance
from the wall. The thermal conductivity and permeability are developed as functions of
the porosity. Specific conclusions are made that the fluid media has a significant effect

on the appropriate model for porosity and thermal dispersion; in particular, water and air

cannot be consistently modeled the same. Huang and Vafai (1994) analyzed an external
boundary covered with porous substrate that is described by four non-linear ordinary
differential equations. Utilizing a Runge-Kutta method as opposed to a finite difference,
finite volume, or finite element solution method numerically accelerated the solution.
Daszkowski and Eigenberger (1992) determined the appropriate values for the wall heat,
transfer coefficient in the model of a cylindrical tube filled with a reacting catalyst where
the porosity varies in two dimensions. The authors found that the type of wall boundary
condition (isothermal or isoflux) greatly affected the value of the heat transfer coefficient.

Several applications in porous media are not only closely coupled to heat transfer
but mass transfer as well. Goyeau et al. (1996) performed a parametric study of several
dimensionless parameters in analyzing a porous media where coupled heat and mass
transfer are occurring. The mass transfer is modeled as double-diffiasive, and the heat
transfer is primarily natural convection. Irudayaraj and Wu (1994) examined coupled
heat and mass transfer in porous biomaterials with a specific application to the drying
process emphasizing the moisture content of the materials. Kececioglu and Rubinsky
(1989) perform a theoretical development of the movement of trace salts in the freezing
of water in a rock bed. These equations are model as coupled heat, mass, and momentum
with interface conditions to account for sudden changes. Dekker et a1. (1995) determined
the mass transfer from the bulk phase to the surface of a catalyst in a isothermal case.
Considerable attention is paid to extra- and intra-particle mass transfer. The coupling of
the mass transfer to heat transfer model is considered to be unimportant in comparison to
the modeling of the external and internal mass transfer. Somerton (1985) determined an

analytical solution of coupled heat and mass transfer in a stacked-tray desiccant bed and

developed a Sherwood number correlation for the mass transfer from the porous layer to
the fluid layer. Wakao and Funazkri employed a steady state potential model to
determine the particle to fluid mass transfer coefficient for a packed bed of spheres. A
Sherwood number correlation is based on their experimental work as well as
experimental data spanning the prior thirty years. Jackson (1974) utilized the dusty-gas
model to develop a set of integro-differential equations to describe heat and mass transfer
in a highly permeable porous catalyst. In an application of the mass transfer to heat
transfer analogy, Carley et al. (1995) measured the mass transfer associated with
naphthalene particles in a medium of crushed oil shale in an attempt to experimentally
quantify the heat transfer between irregularly shaped particles. The authors investigate
various published correlation’s and then determine that a logarithmic relation best suits
their data. Baladi et al. (1981) undertook an experimental investigation on wet and dry
soil to determine the proper modeling of the heat and mass transfer in the coupled model.
The authors find that simplistic heat transfer models do not effectively model
experimental data in the wet soil. Resnick and Golt (1981) estimate the mass transfer
coefficient for an active particle in a active-inert packed bed specifically analyzing the
geometric arrangement of active and inert particles in the media. Ogniewicz and Tien
(1981) analyze the condensation of water vapor in porous insulation. There are three
zones defined (two wet and one dry) with various interface boundary conditions. The
authors conclude that the condensation rate greatly affects the thermal performance of the
insulation. Banks and Ali (1964) analyze the mass transfer from a dilute mixture to a

fluid and solid matrix.

10

Understanding the chemical process that occurs in the coupled heat and mass
transfer in porous media analysis leads to several investigations into the process of
adsorption. Grant and Manes (1966) utilize experimental data to confirm the Polanyi
theory of adsorption by illustrating that the ideal assumption is valid for non-ideal gas
mixtures. The authors present their development of isotherm data for binary hydrocarbon
gas mixtures. Lee and Weber (1969) characterize the isothermal adsorption process in
the mass transfer equations with a mass transfer coefficient similar to a film coefficient.
Lee and Weber interpret the experimental breakthrough data for adsorption of methane on
to activated carbon using isothermal and non-isothermal models. The authors find that the
interaction of heat and mass transfer can be very pronounced. Aranovich and Donohue
(1995) expand upon the Type I adsorption isotherms to determine a model that collapses
to Henry’s law at low pressures.

One of the applications of porous media analysis that has come into vogue in
recent years is the use of activated carbon canisters in the automobile industry. Johnson
and Williams (1990) compare and contrast different based activated carbons for
performance in evaporative emissions systems. A study by Fuher et al. (1994)
experimentally investigated the effect of high humidity ratios on the performance of
activated carbon canisters in vehicle evaporative emissions systems. Pakko et al. (1994)
investigated the use of activated carbon canisters during cold start to contain post
combustion emissions when the catalytic converter is not yet at it’s optimal operating
temperature. Lavoie et al. (1996) analyzes a coupled heat and mass transfer model in a
one-dimensional case for an activated carbon canister used in evaporative emissions

containment.

11

1.2 Objective

The objective of this study is to completely characterize the two-dimensional
transient nature of the coupled heat and mass transfer in a heat generating porous media
due to a reactive fluid in a U-shaped geometry. The analysis entails both single- and
dual-energy equation approaches to the heat transfer modeling. The determination of the
importance of axial conduction in the thermal model is included. The geometric nature of
the problem allows for the application of an iterative boundary condition to balance the
heat flux across the separation wall that adds a unique nature to the problem. Due to the
presence of a reactive fluid, an adsorption process is occurring that closely couples the
heat and mass transfers. The adsorption process is appropriately modeled for these
coupled equations with a simple potential model that utilizes a widely accepted
isothermal approach. These equations are numerically solved utilizing a fully implicit

second order correct finite difference scheme.

Chapter 2
Physical Development of Model

Work: Something made greater by ourselves and in turn that makes us greater.
Maya Angelou 1977

2.1 Background for Porous Media

The continued analysis of porous media is an endeavor that is undertaken by an
assortment of individuals in various realms of engineering. In the most general sense a
porous material is a solid with holes. This configuration may be constructed as a
collection of packed solids with holes produced by the void spaces in the packing. Figure

2.1 illustrates an example of a uniformly packed porous media.

 

 

 

 

 

Figure 2.1 Example of Porous Media

13

However, to be more specific, a porous medium could be defined as a two-phase mixture
composed of a solid matrix fixed in space and a fluid that fills the void spaces in this
matrix. Initial interest in porous media analysis led to a continuum model for the
treatment of the two-phase mixture where the solid and fluid are treated as one
homogenous substance. Further modeling efforts led to the porous media being described
by coupled equations for each phase.

In any analysis of porous media, understanding of the geometry and its effect on
the mechanisms of transport is necessary. Transport properties associated with porous
media clearly must depend on the transport properties of the individual components as
well as the geometric configuration of the media. There are three basic geometric
properties that are used on the macroscopic level to describe porous media: porosity,
tortuosity, and specific surface area (Lee 1987). The porosity, e, is defined by the ratio of

the volume of the voids to the bulk volume of the media as follows:

V .
8 = ——VO’d (2.1)
Vbulk

Clearly from this definition, the parameter 8 would maintain values between zero and 1.
At zero the medium would be strictly a solid, and at 1 the medium would be strictly a
fluid. The tortuosity, t, is defined by Scheidegger (1960) as the ratio of the path that a

fluid element follows through the media to the length of the media as follows:

L
T: __I£'Lh_ (2.2)

L
sample

14

From this definition the reciprocal of r is clearly bounded between zero, when the path is
extremely large, and 1, when the path is equal to the length of the sample. The specific
surface area, 2, is defined as the total interstitial surface area of the porous media per unit
bulk volume of the porous media as follows:

Asurface

Vbulk

2 = (2.3)

The value of 2‘. will vary over a great range of numbers based on the packing of the media

and the surface contact between particles.

2.2 Heat and Mass Transfer in Porous Media

The major focus of this work is the analysis of coupled simultaneous heat and
mass transfer in a heat generating porous media. Modeling of this problem includes two
of the three basic modes of heat transfer: convection and conduction in a transient
process. In any forced convection situation, the first concern must be the fluid mechanics
of the model. The flow within a porous media is generally accepted to be uniform or slug
flow in the strearnwise direction. This is a reasonable assumption in the physical model
presented here. The complex condition of energy transport arising from simultaneous
heat and mass transfer in a heat generating porous media may be described by one of two
methods: a one-energy equation model or a two-energy equation model. The first
method recognizes the fluid and solid regions as a single continuum and utilizes effective

thermal properties which are combinations of the fluid and solid phase properties based

15

on the characteristics of the porous media. This model assumes local thermal equilibrium
between the solid and fluid phases. The second approach allows each region to be
analyzed independently with an energy equation specified for both the fluid and solid
phases along with a specified interphase transport coefficient which couples the
equations. Each of these methods will be employed to analyze the following physical
situation: consider volumetric heating in a well-packed porous media in a U-shaped
container with an interior separation wall as illustrated in Figure 2.2, where the arrows

indicate the direction of the flow.

raj/1‘61

V —-—>

 

 

 

 

 

 

Figure 2.2 Physical Model Geometry

In conjunction with the energy transport, simultaneous mass transfer is modeled
as well. The combined heat and mass transfer is the result of an exothermic reaction with
the media that releases energy into the system. The temperature is therefore dependent on
the equilibrium condition of the media and the enthalpy of adsorption thus coupling the

energy transfer to the simultaneous mass transfer occurring in the media.

Chapter 3
Describing Equations and Boundary and Initial Conditions

We love to overlook the boundaries which we do not wish to pass.
Samuel Johnson
For simplicity, the given physical geometry is analyzed as a straight channel of
length L with convection from the outer surface to the ambient conditions and balanced
heat flux across the interior wall. The outer surface and the interior wall are assumed to be
impermeable to mass transfer. The following will develop the describing equations for the

fluid mechanics, heat transfer, and mass transfer through this given porous media.

 

 

 

 

 

 

 

 

 

 

 

 

Interior
Separation
Wall L
z < fl)
’l /i / ’
a/ / \|/ Too, hambiem
w

Figure 3.1 Model Geometry

16

17

3.1 Fluid Mechanics

The fluid mechanics within a porous media are generally accepted to follow

Darcy‘s Law, which is expressed for one-dimensional flow as follows:

i?- ” (3.1)

=—w

de

with w being the area averaged Darcian velocity and u being the dynamic viscosity of the
fluid. The proportionality constant, K, is known as the permeability, which has units of
square length, and is based on the geometric properties of the medium (Scheidegger 1960).
For a constant cross section channel and a constant pressure gradient, Equation (3.1) will

result in a uniform flow through the media.

3.2 Heat Transfer

The heat transfer within the porous media will be analyzed with respect to two
fundamentally different procedures: the single- and the dual-energy equation models.
Each model also includes computations with and without axial conduction. The boundary

and initial conditions are essentially the same for both models.

18

3.2.1 One-Energy Equation Model

The energy equation for an ideal gas or incompressible liquid in the Cartesian

coordinate system ignoring viscous dissipation effects is as follows:

6T é’T 8T é‘T 032T 527’ ézT .,
(pep) —+ cp) u—+v—+w—— =k,,, 2 + 2 + 2 +q (3.2)
'" at I fix é’y a2 fix é’y é’z

 

where the subscript m refers to effective thermal properties of the porous medium or fluid-
solid mixture and the f refers to the thermal properties of only the fluid phase. The
temperature distribution described would be of the fluid-solid continuum where u, v and w
are the x, y and 2 components of the Darcian velocity, and q'" is an internal volumetric
heat generation term. The heat generation term may be a function of time and position as

well as being coupled to the mass transfer within the porous medium.

3.2.1.1 Order of Magnitude Analysis

Figure 3.1 clearly illustrates that one may assume that the fluid flow is one-
dimensional with only the z-component present and that conduction and convection in the
x-direction may be ignored. However, the assumption that the heat transfer is two-
dimensional with conduction in the y-direction and convection in the z-direction thus
ignoring axial or streamwise conduction may not be as clearly obvious. To clarify this
issue, an order of magnitude analysis will be performed. Equation (3.2) may be reduced to

the following, with streamwise conduction remaining as follows:

 

19

 

 

 

2 2 - n
Qfl+wfl=am _§_7§‘_+0" { + q (3.3)
5t 52 fly é’z (pep) /
where
Q =____(pcp )m (3.4)
is the non-dimensional heat capacity and
m : km (3.5)

is the effective thermal diffusivity. An order of magnitude analysis of Equation (3.2) will
show that the axial or streamwise conduction is negligible for large Peclet numbers and
further investigation would determine a specific value for the Peclet number to be
nominally 20 for this assumption to be valid. Recall that the Peclet number is the non-
dimensional ratio of fluid momentum to thermal diffusion. To begin this analysis,

Equation (3.3) will be sealed with the following parameters:

2
z =— 3.6
L ( )
where L is the length of the continuum in the z-direction
y‘ =X (3.7)
a
where a is the width of the continuum in the y-direction
T‘__I_-IL_ (3.8)

20

where Tm is defined by determining the temperature distribution within a porous media

with only conduction and heat generation. Thus, the following second order differential

equation is solved easily:

5’27" .,
- km [a—yz]:q (39)

Applying constant temperature boundary conditions to the solution results in the following

being defined as Tm:

-n2

q a
T = +T 3.10

m

 

From this definition the term F may be defined as

an 2
firm—T, 45%“— (3.11)

with km being the effective thermal conductivity of the continuum. Finally, time is non-

dimensionalized by using the following:
I =_ (3.12)

where t, is the appropriate scaled time to be determined, it will either be based on fluid
motion or thermal diffusivity. Applying these relations to Equation (3.2) results in the

following:

 

 

 

(3.13)

a

may" aT‘_[L]’ 1 air‘ 1 aZT’ [L]2 1 q"
a

+ + + — —-—
O 0 .2 .2 .
Wt: at a2 Pel..m ay Pel.,m a2 P el.,m qiiiax

where Pew" is defined as the Peclet number

Pew, =IE (3.14)
a

"I

21

From Equations (3.6) and (3.7), the differential of the distances are clearly of order 1.
From Equation (3.8) the temperature differential will be of order 1 since it has been sealed

with the maximum temperature. The first term on the right hand side of Equation (3.13)

2
will go like [11] since L is greater than a and the differential is of order 1. This also
0

applies to the last term of Equation (3.13). The axial or streamwise conduction will go

like

 

and is therefore negligible for large Peclet numbers (Bejan 1984). Equation

Pew

(3.13) also clearly illustrates that time should be sealed with the thermal diffusivity and
the length of the continuum to ensure that the temperature is a function of time in the zero

flow case.

3.2.1.2 Initial Condition and Boundary Conditions

The physical geometry of the continuum dictates the boundary conditions and the
initial condition to solve the governing differential equation; hence, to satisfy the

differential equation the following initial condition is utilized:
Tm, 12,2) =T(z)..-.,-., (3.15)

which assigns an initial temperature distribution, T(z) to the continuum. Three

initial ’

boundary conditions are needed to analyze this physical situation. The first boundary

condition

T(t, y,O) :7" (3.16)

0

22

maintains a uniform temperature, 7; , at the inlet of the continuum. The second boundary

condition

—k—— =U(T' —T) (3.17)

I,_v=u,z
represents the heat transfer from the outer surface to the ambient conditions at y = a where
U is an overall heat transfer coefficient accounting for the conduction through the wall and
the convection away from the wall. The overall heat transfer coefficient is derived from a
thermal circuit analysis for one-dimensional heat transfer. This analysis determines an
overall resistance to heat flow for a system. Essentially

1 5
+
k

 

(3.18)

 

l
U humble"! wall

where 5 is the outer wall thickness, h is the outer convection coefficient, and kw” is

ambient

the thermal conductivity of the wall. The final boundary condition is

0"_T
m ay

6T

"gy- =—"-(T.....o.., 4..-...) (3.19)

',_V=O.:2 W

—k =k

 

 

I.y=0.:,
which represents the continuity of the heat flux across the separation wall at y = 0 between
the locations zI and z, and the thermal capacity of the separation wall. A physical

representation is found in Figure 3.2

 

 

‘1"(21) >‘ > q"(zz)

NI!“

 

 

 

 

 

Figure 3.2 Illustration of Balanced Heat Flux

where the two locations are related by the following:

2, = L-z, (3.20)

3.2.11.3 Non-dimensional Analysis

To ease the solution of the differential equation it will be non-dimensionalized.
The same parameters will be used here as in the scaling with the exception of temperature.
In the order of magnitude analysis it was imperative that the derivatives be of order 1;
however, that is not a necessary condition for the solution of the differential equation. The

scaling of temperature will be

 

T = 0 (3.21)

Recall that the time is scaled by the thermal diffusivity as follows:

 

z =——'~— (3.22)

Applying these non-dimensional quantities results in the following differential equation:

o o 2 2 o . n,
(23?.— +Pe, mil-=15] iii—+L (3.23)
é’t “ 52 a 0" y qfl,
where
‘ M 7:) km
qs'cl = L2 (3.24)

The non-dimensionalized boundary conditions are as follows:

 

 

T’(0,y'.z‘1= W)?“ ' T" =T'(z'1.. (3.25)
T‘(t‘, y‘,0)=0 (3.26)

aT' . . .
ay' =81]; ‘ T ww) (3.27)

In Equation (3.27), the Biot number is defined as

Bi =— (3.28)

ak

_ w
.. k 6 ( I‘,y‘=0.z; _ ,o‘y030.:|')
"I W

a;
fiy'

-22
03y.

 

(3.29)

 

 

the non-dimensionalized relationship between the two points on the separation wall is as

follows:

22 Zn
—=l-—- 3.30
L L ( )

With this definition Equation (3.29) may be rewritten as

25

 

 

 

 

5T. é’T’ 1
_ . = . : — T . _ T . 3.31
6y l.y=0.:: fly l.y=0,I-gl' l( 1.y=0.l-z. l.y=0,:l ) ( )
where
’1 = k"? (3.31a)
a

3.2.2 Two-Energy Equation Model

The second manner in which a porous media may be analyzed is with the two-
energy or dual-energy equation model. In this case, the solid and fluid temperatures are
not in thermal equilibrium and are related by an interphase transport coefficient. This
leads to an energy equation being expressed for both the solid phase and the fluid phase

that are coupled by the interphase transport.

3.2.2.1 Fluid and Solid Regions

The transport of energy between the two phases is represented by a potential

temperature difference. The following is the energy equation for the fluid phase:

é’T 5T air
(peg/7!!— + op)! wa—zf = gk,;;{—+h,sz[r, —T,] (3.32)

where e is the porosity of the media, 2 is the specific surface area, and hf, is the interphase

transport coefficient. In the fluid energy equation there is no volumetric heat generation

26

term since the adsorption reaction is strictly associated with the solid. The solid energy

equation is as follows:

2
(pep 1% = (1—£)ks%;7—;i+hfiZ[Tf —T_,]+q"' (3.33)

Both the fluid and solid equations are solved in a non-dimensional form with the
temperature and geometric variables being sealed in the same fashion as the single-energy
equation; however, the time scale for each will be the fluid thermal diffusivity. The non-

dimensional fluid equation is as follows:

a

5T; + Pe 3T; ( L12 aZT,’
t l.,/ ’2
5t ‘ 0’72 é’y

2
+1., 5—2[T,' — 77] (3.34)
The Peclet number is based on the fluid thermal diffusivity and the length of the porous

media in this case.

Pew :11: (3.35)

01,
Taking the interphase transport ' coefficient, one can develop an interphase transport
Nusselt number based on the bed length and the fluid thermal conductivity. The
remaining portion of the coefficient is non-dimensional since 2 has units of LZ/L3. Finally,
the fluid energy equation may be re-written in terms of the interphase transport Nusselt

number as follows:

6T, 5T;
~ +

L 252T . .
3;,— e”??? = g(—] #+Nub(L2)[T, —T,] (3.36)

Cl

The solid energy equation is presented in non-dimensional form as follows:

27

 

 

—= (1— g)—'— [:6 J— a T; +Nu,,(L2)[T; -T. H (3.37)
3.2)) sci,
where
11‘
(lit/f = L21 Qf = (pcl’)s (3.38)

3.2.2.2 Interphase Transport

The interphase transport coefficient can be determined from a plethora of Nusselt
number correlations. Wakao et a1 (1978) developed, from a mass transfer analogy, the

following for interphase heat transport for spheres in a porous media:

N11,,” = 2 +1.1 Pry3 ReDPO'G (3.39)

where Pr is the Prandtl number which is a ratio of the thermal diffusivity, 01,—, to the fluid
viscosity, vf. In this correlation, both the Reynold’s number and Nusselt number are based
on particle diameter. For this to be consistent with the derivation of the two-energy
equation model presented here, the Reynold's number is defined in terms of the Peclet

number as follows:

 

D
Re =P 3.40
I) 9L Pr L ( )
then Equation (3.3 9) is modified in the following manner:
Nu -Nu 1‘— (3 41)
fr Dp DP ‘

28

3.2.2.3 Initial Condition and Boundary Conditions

The boundary conditions for the solid and fluid energy equations are similar in
nature to those presented in the single-energy equation model. The solid and fluid are
assumed to be in thermal equilibrium initially; therefore, the temperature distribution is
the same throughout the media as in the single-energy equation model. The incoming gas
temperature is assumed to be the instantaneous temperature at the media entrance for both
the solid and fluid. The non-dimensionalized boundary conditions for the fluid energy

equation are as follows:

 

 

 

 

 

 

 

 

 

 

 

TI. (0’ y. ’ Z. ): T]. (2. )initial (3 .42)
71.113)”. 901=0 (3.43)
5T ' . .
fly]. "Jul.” =Bi(T°° _ TI I‘,y‘=l,:‘) (344)
_ 3T]. _o”Tf' _ “kw _
5y. I.y=0,:.° — fly. i,y=0,I-zl° _ kfaw ( jJ‘.y‘,I-z,° T/.I'.y',zl’) (345)
for the solid energy equation
T.‘ (0.y'.z‘)=T.'(z‘),-..... (3.46)
8.110301% (3.47)
Z". W =Bi1T; — Emu.) (3.48)
03717 _ 6T; _ akw _

 

 

29

These boundary and initial conditions are relatively straight forward with the exception of
Equation (3.45) and Equation (3.49), the interior separation wall condition. In this case
the solid particles are assumed to be arranged across the interior wall such that solid
conducts to solid and fluid conducts to fluid, refer to the magnification of the wall in

Figure 3.3.

q 5%le
q f"(21)

 

V

 

Figure 3.3 Magnification of Balanced Heat Flux for Dual-energy Equation Model

3.2.3 Addition of Axial Conduction

In the given physical model, the axial conduction is only negligible for large
Peclet number; however, if the Peclet number is not sufficiently large, the axial conduction
will be included. The appropriate forms of the energy equations for both models are as

follows:

 

 

(3.50)

&T é’T_ é’ZT é’ZT + q"
0"! £2 m

Q—+w— a —7+ 2
5y 52 (pep),

30

 

é’T 6T 032T dzT
(pcp)f——f—+ cp)fw—f Sk/[5y2f+ azzf]+hfi2[Ts—Tl]

 

 

5T. 52T fiZT ,,,,
(pop )3 0"; = (1—£)k_‘.[ 5y; + azz’]+hfl.2[Tf -7;]+q

In so doing, an additional boundary condition is needed in the axial plane

 

 

 

£2“. :0
32 [M
6T
___f _0
a2 I,y,l.
3T, =0
52

l,y,l.

 

Presenting these equations in non-dimensionalized form

 

o o 2 2 o 2 o .m
9.22: 5.)... H L r. .L 2.; .__(
31‘ ‘ 32 a é’y é’z q“,
ar‘ 37“ 2 527" 527’ . .
—{ + Pe, ,—-’ = a [5) ,g 4 ,2’ +Nu,,Lz[T, 4“,]
fit é’z a é’y 072 ' '

 

 

 

aT,‘ k, L 2527‘; 827‘: . . q"
917=(1—£);[[;) ‘ :3 ' ]+NufiLZ[Tj—71]+

 

 

 

 

3y” 2’2 '32,,
é’T' __0
$2 I,“
ar,‘ —o
52' I l —
ar.‘ _0

 

l,y,l

(3.51)

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

(3.57)

(3.58)

(3.59)

(3.60)

(3.61)

31

3.3 Mass Transfer Model

Within the given physical geometry, the incoming gas may be a mixture of several
ideal gases. The mass concentration equation for a component i of an ideal gas mixture
with a mass source/sink term in the Cartesian coordinate system is as follows (Edwards,

Denny, and Mills 1978):

%+VL0,V}+V[/,]=I§" (3.62)

where pi is the density of the component i and j: is the mass flux component i. The rate of
volumetric mass adsorption, ff", is dependent on the equilibrium condition within the
media and the enthalpy of adsorption. To simplify Equation (3.62) the following is
defined:

p, =m,pf (3.63)
where ml is the mass fraction of the component and of is the total density of the ideal gas

mixture. This allows Equation (3.62) to be rewritten as follows:
0" - , -
mi(7+V(/0/ V)]+pf%+p/ vai+vji=fim (3-64)

The bracketed term on the left-hand side of Equation (3.64) is continuity and is zero by

definition. Recalling Fick's Law

ji : _pj pikvmi (3-65)

32

where 2,, is the mass diffusivity of component i into the mixture k with units of length
squared per unit time. Applying Equation (3.65) to Equation (3.64) and simplifying

results in the following:
flmi ~ .,,,
p,—at—+p,va,=v(p,fl,ka,)+r, (3.66)
Assuming that the density and mass diffilsivity are independent of direction and that the

flow is one-dimensional in the axial direction, Equation (3.66) may be rewritten as

follows:

 

+r'," (3. 67)

8m. 5m, fizm, 032m,
p,E-+p,w3z—=pf '1‘ é’yz + 322

The axial diffusion term is retained to allow the migration of vapor in a low or zero flow

condition.

3.3.1 Initial Condition and Boundary Conditions

Similar to the heat transfer model, the initial mass fraction distribution is known
for the porous media

mi (0, y, z) =m(z), (3.68)

illilittl
The walls of the media are assumed to be impermeable; therefore, the following may be

defined:

a
fly

_0"m,

__ :0 3.69
5y ( )

 

 

Ly=0x Ly=u;

33

This boundary condition implies no radial dependency for the mass transfer; however,
with the coupling of the mass transfer and the heat transfer, the temperature distribution in
the radial direction will lead to a mass distribution in the y-plane. Similar to the heat
transfer boundary condition, the mass fraction at the inlet of the continuum is known and
is constant

m. (t, y,O) =m,o (3.70)
At the end of the media the gas is assumed to continue to flow; however, the axial
diffusion is no longer present due to the absence of the media. No mass adsorption is

occurring either; therefore, the final boundary condition is as follows:

+ wflm, —p p 032m,
pf 82 I.v.:=l. l m ayl

5m,

 

 

I.y.:=l. I r :=L

3.3.2 Non-dimensional Analysis

The mass transfer model, Equation (3.67), is scaled in the same manner as the heat
transfer model. The scaled parameters described in Equations (3.6), (3.7), and (3.22) are

applied to Equation (3.67) resulting in the following:

 

 

2 2 2 -m 2
a’premi’lgL [5) a "’2 +& ”1 +341“— (3.72)
at “ 62 Le a fly é’z plan,
where the Lewis number is defined as
Le = (3.73)

21
Z.

34

One should note that the mass fraction, mi, is a non-dimensional quantity and is bounded

between 0 and 1. The non-dimensionalized boundary conditions are as follows:

 

 

 

 

 

 

mi (0’ y. ’ z. ) = "1(2. )’ mm! (3 '74)
L"? :07"; :0 (3.75)
fly I..y.=0‘:. é’y ME”.
m,(t',y°,0)=m,0 (3.76)
2 2
giswfi Pew-€12} =-—‘— [5) L’f’; (3.77)
t 52 "yum Le a fly 1
, .. (“yflz‘a

3.4 Development of the Source Terms

The source terms in both the energy and mass transfer equations are closely
coupled. The mass source term development is similar in nature to the interphase
transport term that was developed in the dual-energy equation model. It is continually
pointed out in the literature that the mass transfer or heat transfer analogy (Wakao and
Kaguei 1982) allows for the development of a corresponding Sherwood or Nusselt number

correlation. The heat generation term is developed based on an energy balance.

35

3.4.1 Development of Mass Adsorption Term

Consistent with the development of mass potential by Edwards, Denny, and Mills

(1979) the mass source/sink term may be defined by the following potential:

in = pr2(mi,ss - mi) (3'78)
where K is the particle to fluid mass transfer coefficient with units of length per time, 2 is

the specific surface area, and mLss is the mass fraction of the adsorbate at the surface of a

given solid particle. Figure 3.4 illustrates this mass transfer potential.

mi, Ta __—9

 

Figure 3.4 Mass Transfer Potential Model

Taking Equation (3.78) and scaling it with the appropriate term from Equation (3.72)

W} _ LZKZ

plan: am

 

(m

— ml.) (3.79)

i ,55‘

Introducing the mass diffusion coefficient leads to a Lewis number in this representation

4'2 2
rL =LKzfl(mm—m,)=LZ&(m. —m,) (3.80)
plan, a 2k ' Le

I .u
m

 

where the Sherwood number is defined as follows:

36

ShL = LK (3.81)
”.1.

Several correlations exist for this Sherwood number. Similar to the Nusselt number
development in the dual-energy equation model, Wakao and Funazkri (1978) present a
Sherwood number correlation for the fluid phase mass transfer in a porous media of

spherical particles

8th = 2 +1.1Sc3 Reg-j (3.82)
where Sc is the Schmidt number which is a ratio of the fluid viscosity, vf, to the mass
diffusivity, 2,, . Once again the Reynold's number as well as the Sherwood number are

based on particle diameter and, to be consistent with the development here, will be re-

written appropriately
D s l 2 i
ShL= 2+1.1(T”] Lw3Pef Pr" Di (3.83)

Incropera and DeWitt (1990) present a different correlation for the heat transfer in a

porous media in terms of the Colburn factor, j and the porosity of the media, a

e j = 2.06 C‘ Refif“

l 1 spheres
Nun . L, (3.84)
j = ——"—l—- C = l 0.79 cylinders 54— = 1
Rev Pr3 ”
r L 0.71 cubes

 

This correlation, through a mass transfer analogy similar to the one used by Wakao and

Kaguei (1982), may be re-defined as a Sherwood number correlation

. 0575 I _”
Sh, = fic— Pef‘zf—L—J 1.81 Pr '23 (3.85)
‘ ‘ D

P
The other term in Equation (3.79) that needs to be defined is the mass fraction at the

surface, m. The mass fraction at the surface may be represented as

mm = mm (Tycm) (3.86)
a function of the temperature of the particle, Tp, and the concentration of adsorbate, cm, in
the solid. The adsorption process occurs when the reacting fluid is in equilibrium with the
adsorption pressure. This pressure is determined from the isotherm for the solid and the
reacting fluid. The isotherm is determined from the theory originally developed by

Polanyi in 1914 and adapted by Dubinin and Radushkevich who proposed this form for

the relation

where cm, is the concentration of adsorbed vapor in the solid per unit mass of solid;

therefore, it is related to the mass fraction as follows:

6c”): pIZShLL (m,— ‘ )
a: Lea—8);),r ' '”

 

(3.88)

In Equation (3.87), Do and a0 are constants determined experimentally (Lavoie et al. 1996)
at a given temperature for the given reacting fluid and solid. A representation of the
isotherm for butane on to carbon based on the experimental data of Lavoie et al (1996) is

found in Figure 3.5. The constant a0 may be sealed with temperature as follows:

38

3.5 Econ—toaxm .8 woman H.353.— aoqumu< m.n 953..—

 

 

 

 

 

Emmi
H _.o 5.0 Sod Good
Sod
m1 oofio
n _ a J weaq+__ . . wjaufile 2 Ea...q~l q COO.H
8mm _S:o§Bme

Bob onEBoQ 8:582 cosmomue. BoEaEm £35033an 98 £585

pnos JO ssem/JodeA poqlospe

39

I'

a..(T,.> =a..('1“. 807]
(3.89)
T,=310.9 K

The constant Do is assumed to be ten percent of a0. The saturation pressure is determined
from the Antoine equation which is a well-known equation used to determine the pressure,
at a given temperature at which a hydrocarbon will vaporize (Rossini et al. 1953)

B
T—C

P

 

10310 (P30!) = A "’ (3-90)

where A, B, and C are empirical constants. The temperature at which the Antoine
equation and the adSorbed vapor are calculated is the equilibrium temperature of the
particle. Once the equilibrium pressure is determined from Equation (3.87), the mole
fraction of at the surface can be determined

1?.
yi,.s's =—I—)Lq_ (3.91)

Consistent with Edwards, Denny, and Mills (1979) the mass fraction at the surface, mm,
may be determined from the following:
yi,.s'sMI4/i

mi,” = n (3.92)
2% MW}
j=|

where MW is the molecular weight of the of the active gas.

40

3.4.2 Development of Heat Generation Term

The heat generation term is closely coupled with the mass adsorption term in the

mass transfer modeling. The volumetric heat generation term may be written as

q"=q"(h....,h.,,c.,.) (3.93)
a function of the enthalpy of adsorption, hadss the enthalpy of phase change of the gas to
liquid, hi“, and the concentration of the adsorbate in the solid, cm. Taken these parameters
into account the following expression may be derived:

.. ac...
q = (1—6)px 7(Ahm + Ahg,) (3.94)

where Ben/6t is the rate change of concentration of adsorbate in the solid per unit of solid
as defined by Equation (3.88). The adsorbate in this case would be the reactive fluid
flowing in the channel. Determination of the appropriate enthalpy change is done through
the application of equilibrium thermodynamics and the second law of thermodynamics

resulting in the following:

a_P
8T

-55

6v

-29.- Ah

— — (3.95)
r T6v TAv

 

 

V

This equation is known as the Clapeyron relation originally derived in 1834. Since the
substance is in equilibrium and therefore is undergoing a reversible process, Ah is known
as the ordinary heat of vaporization. The Ahgl is determined from the Clapeyron relation

for an ideal gas at the saturation pressure (Lewis and Randall 1961)

Ah, = E T2 W (3.96)
1‘ dT

41

The saturation pressure is determined from Equation (3.90). The Clapeyron relation is
used to determine the enthalpy of adsorption as well (Lewis and Randall 1961); however,

enthalpy of adsorption is evaluated at the equilibrium adsorption pressure. The heat

generation term, Equation (3.94), is then scaled with 4;, from Equation (3.24).

Chapter 4
Method of Solution

The scientific mind does not so much provide the right answers as ask the right questions.
Claude Lévl-Strauss 1964

It is desired to utilize a fully implicit finite difference scheme to solve the
previously defined partial differential equations for energy and mass transport. The
momentum transport solution results in a constant flow situation for this analysis.
Different numerical situations arise based on the order of the spatial terms in the partial

differential equations; therefore, more than one method is applied.

4.1 Heat Transfer

The energy equations in both the single-energy equation model and the dual-
energy equation model are to be solved numerically both with and without axial
conduction. The dual- or two-energy equation model presents a unique challenge since
the two equations are coupled. Adding axial conduction to both models requires the
diffusion or second-order terms to be solved numerically in both spatial directions. This
leads to the utilization of two different numerical procedures to allow all the solutions to

be fully implicit.

42

43

4.1.1 One-Energy Equation Model

Equation (3.23) is classified as a non-homogenous constant coefficient, linear
partial differential equation that is first-order in t and z and second-order in y. This
equation may be solved numerically by employing a finite difference approach. The
difference equation form of Equation (3.23) can be obtained by applying first-order
correct backward differencing to the first-order derivatives and second-order correct
central differencing to the second-order derivative resulting in a fully implicit solution.
This will lead to a tri-diagonal matrix that is rather efficiently solved using a method
based on the Thomas algorithm (Anderson et al. 1984). This method was adapted from
Anderson et al. by McDonough (1981). For convenience, the following notation is

introduced:
T.(t.9y.az‘)=Tk,i,j (4'1)
where the maximum values of i and j are N and M respectively and k is the time index.

Executing the appropriate finite difference scheme for each derivative results in the

 

 

 

 

following:
2 on
QTkJJ _Tk-l.i.j + Pe Tu} " TkJJ-l =[£] Tk.i-l,j -2714}; + Tun.) + q I (4 2)
l..m up '
At A2 a (Ay)2 q,,, M
Rearranging the terms from Equation (4.2) results in
2 2 2
P At
-A. 2. Tm, . (uh—+223: L T. -T, L T =
aAy ' ' A2 aAy " aAy ' ‘
T m (4.3)
k,i, '-1 q
0TH 1.1 +Pem At _A; +At,—

 

In
qscl

kJ'J

44

This form of the Equation (4.2) allows for the tri-diagonal matrix of unknown
temperatures to be formed with respect to 7,7 or the y-plane. For simplicity, Equation

(4.3) is rewritten as follows:

CIT .+C2T . +C3T D. (4.4)

“-1., lug, Tm.)- = T

where C is a set of coefficients and D is referred to as the source term. Equation (4.4) is
applicable for the interior nodes; however, for the exterior nodes, the boundary and initial
conditions need to be applied. For the first time step, Equation (3.25) thoroughly defines

the initial temperature distribution as

T, T

initial

(4.5)

1.}:
Therefore, the k=1 time step is known. The temperature distribution is also known at
the inlet to the continuum, j=1 , from Equation (3.26), rewritten here as follows:

7‘, 1.: =0 (4.6)
Numerically proceeding through the finite difference scheme with respect to the y-

component will solve Equation (4.3); therefore, Equation (4.3) must be defined at i = N,

recalling that N is the maximum value of i

613...-..- +028...) +6371...” =0, (4.7)

The node temperature TkMU is referred to as an image point since it is out of the realm

of the prescribed nodal scheme (Anderson et al. 1984). This image point may be solved
for by determining the second-order correct forward differencing equation for the

appropriate boundary condition, Equation (3.27) in this case, written here as follows:

Tk,N+l.j " TIT
2Ay

 

”"4 = Bib; — 71ml (4.8)

45

Solving this equation for the image point and making the appropriate substitution in

Equation (4.7), the following is obtained:
(C1+ C3)rr,‘~_,,j +[C2— 2C3A yBi]T,,.N’ 1:19,, + 2A yBiTw (4.9)

Application of the final boundary condition, Equation (3.29), is done in two parts and

results in the following condition applying second-order correct forward differencing:

-3T .+4T .—T .
_/1 u.) 2A;.2.J k'3‘j=Tk,1.p_Tk.l./' (4.10)

 

 

 

= 4.11
My ( )

_ "' 377.1,} + 477.2,} - Tk.3.j — 3Tk.l.p + 4Tk.2.p — Tin-3.11
2Ay

Recall that A is defined by Equation (3.31a). The position p is a point downstream such
that

p=M+1—j (4.12)
recalling that M is the maximum value of j. The point at i = 3 in Equation (4.10) is found
from evaluating Equation (4.4) at i = 2 for the appropriate j location. Equations (4.10)
and (4.11) may then be written in terms of the i = l and i = 2 nodes only for each j
location, thus maintaining the tri-diagonal nature of the matrix. Equation (4.10) is solved

for Tm, and is then substituted into the Equation (4.11), which results in the following:

“—9509 ”fl Tm+ 31-4—92 1+"—5’i T,,j=4T,,,,—T,,,,—Qi ”LE (4-13)
2Ay C3 Ay " Ay C3 Ay " " " C3 Ay

Equation (4.13) is applied at i = 1 node in place of Equation (4.4). One should note that
for p less then or equal to j/2, the temperature at p is taken from the previous time step.

For p greater then j/2 the point p is taken from the current time step since the previous j

46

locations have been solved for in the march downstream. The solution is then iterated on

within the current time step. The tri-diagonal matrix may now be formed as follows:

'IP 1 I- -

 

 

 

 

 

"0203005520007, Dl
0102030052000r2 D,
001020303350 T, D,
=00102030555 ‘ (4.14)
001020303: _
"001020303 "
.. “00102030
00 '00102030. .
00 '0C'1C‘2C37‘,,_,D,,,_l
_0 0 o o 01 C'2_LT~_ .~.,

where [C] is an N by N matrix and [T] and [D] are 1 by N vectors. The prime notation
indicates the application of the boundary conditions and therefore the coefficients are

different then the rest of the matrix.

4.1.2 Two-Energy Equation Model

The solution method for the dual-energy equation model is similar in nature to the
single-energy equation model; however, the coupling of the equations makes the solution
more challenging. The fluid energy equation, Equation (3.36), is classified as a coupled,
non-homogenous constant coefficient, linear partial differential equation that is first-order
in t and z and second-order in y. The solid energy equation, Equation (3.37), is classified
as a coupled non-homogenous constant coefficient, linear partial differential equation that
is first-order in t and second-order in y. A similar numerical procedure is employed to

develop the finite difference form of these equations: first-order correct backward

47

differencing to the first-order derivatives and second-order correct central differencing to
the second-order derivative. For convenience, the following notation is introduced:
Tf(t ,y ,z )=Tf (4.15)

:k,i,j
T,‘(T‘, y‘,z‘)=T,.,,J (4.16)
Applying this notation and the finite difference scheme to Equation (3.36) and re-

arranging terms results in the following for the fluid energy equation:

 

L 2 Pe/At L 2
—eAt —— TNT--1)- + 1+ +2eAt — +Nuva2At THU. —
aAy ' ' ' Az aAy ' ' "

(4.17)

T/Ik ,I'.j-l

2
L
gAt[—] an,“ —Nu,.L>:AtT. = T,.,,_,,,J. +Pe/At

.:k.'.'
aAy s I]

and applying the same to the solid energy equation, Equation (3.3 7) becomes

2 2
—(l—£)Atk" -L— Tm,l .+ o,+2(1—T)Atl‘L i- +NufiLZAt 7}“.
k f aAy " ‘J k aAy ‘ ' "’

 

2 ’ (4.18)
+ “(1 _ 3)At£ [—11—] Ts:k,i+1_j - NujstzAtTerJJ = Q.Ts~.k—l,i,j +At£ifq77
kl aAy kl q“! k,i. j
For simplicity, Equation (4.17) and Equation (4.18) are rewritten as follows:
AfT/:k,i-l.j +B/T/2k.i.j +C1Tf:k.i+l.j + D/TmJ = G/zi (4-19)
gsTCkJ—IJ + flsTrthJ +€STVIkJ+lJ + 7737‘]:ij =Zszi (4'20)

where AT, Bf, Cf, Dr, and Cs, [3,, £5, 11, are coefficients for the fluid and solid equations and
Gf and x, are the source vectors. The numerical scheme follows the previously described
solution procedure: the equations are solved with respect to unknown temperatures in the
y-plane, then numerically march in z-plane and finally step forward in time. The coupled

nature of these equations does not allow for a tri-diagonal matrix to be formed; however,

48

if the equations are properly ordered, a penta-diagonal matrix may be formed that allows
the equations to be solved simultaneously. This matrix will have dimensions of 2N by N
and can be readily solved using a modification of the Thomas algorithm (Anderson et al.
1984) resulting in a fully implicit solution to the coupled equations. Hence, the equations
are ordered in the matrix alternating between fluid and solid energy equations with

appropriately placed zeros as follows:

’8, U, C, o a 5 5 0 0 0 ‘ 7“,, ”0,,
T7; 5; 0 g; 0 5 5 0 0 0 T.. 1,.
A, o B, D, 0, 0 3 5 0 0 7,, 0,,
0 4.. '7. l3. 0 5. 0 5 5 5 7:2 1:2 (4-2 1)
= 0 A, 0 B, D, 0, 0 5 5 _ s
I O 4: ”1 fl: 0 61 0 E
. I o A, 0 B, D, C, 0
O 0 : O C: ”I fix 0 :3 '
0 0 0 : 0 A, 0 B} U] Tm GIN
_0 0 0 - 0 4; '7: 131-17... 1.”-

 

 

 

 

 

 

Once again, the prime notation indicates that the boundary conditions have been applied
in the finite difference analysis. The application of the boundary conditions follows the
procedure outlined in Equations (4.5)-(4.13) with respect to Equations (3.42)-(3.49). It
should be noted that the initial temperature distribution is assumed to be in thermal

equilibrium; therefore, the solid and fluid temperature distributions are the same initially.

4.1.3 ADI Method for Addition of Axial Conduction

The inclusion of axial conduction alters the classification of both models of the
energy equation by adding a second-order term in the z-plane. The two-dimensional

nature of the diffusion limits the number of numerical methods that can be utilized

49

efficiently. Explicit methods begin to have strong stability restrictions and the previously
described implicit method no longer forms a tri-diagonal or a penta-diagonal system of
algebraic equations (Anderson et al. 1984) making the solution to the matrix
computationally intense. The recommended solution method for equations that contain
two-dimensional diffusion is the alternating direction implicit method (ADI). ADI is
unique in that there is a splitting in the time step that occurs. First, the equation is solved
implicitly in the z or j direction at a half time step; and then the equation is solved
implicitly in the y or i direction a half time step further, based on the prior calculated half
time step. This also implicitly makes the time derivative second-order correct. For the
single-energy equation model, at each half time step a tri—diagonal matrix is formed
which is then solved using a modification of the Thomas algorithm. For the dual-energy
equation model, at each half time step a penta-diagonal matrix is formed that is then
solved using a similar algorithm. In both cases, the addition of the solution of a matrix in
the z-plane allows for second-order correct, central differencing to be utilized with
regards to the convection terms as opposed to the first-order correct, backward
differencing that is utilized in the non-axial conduction models. Therefore, the finite
difference scheme for the energy equations that include axial conduction will be second-
order correct, backward differencing in time and second-order correct, central

differencing in space.

50

4.1.3.1 Single-Energy Equation Model with ADI

Utilizing the notation of Equation (4.1) and executing the appropriate finite
difference scheme for each derivative, the finite difference form of the single-energy

equation model in the z-direction results in the following:

 

 

 

Q TEL} _ Th1} P TEN” - Tim-1 _ L 2 Tk.i—l j -2Tk ij + Tk 7+1 j
—'Z— + em, — —' 2 +
V2 2A2 a (Ay)
. (4.22)
Ting-I _ZTitt' j+TkJi+l qnl
(AZ)2 4;: iEJJ

 

where I? is the half time step, re-arranging terms

— AtPeL At At AtPeL At
'"'— T._ Q —T~.. ,,._ T... =
[ 4Az 2(Az)2] “J" :l + (Az)2] """ +[ 4Az 2(Az)2] “'1”
L 2 At L 2 At [L]2___ At A_t__q"’
Q- — — T . . + ———T._ + T +—

Now the y-direction a half time step further at k + 1

 

 

(4.23)

 

kJJ

 

 

 

+ Fe”, +

At ' 2A2 a Ay 2
/2 ( ) (4.24)
T —2T +T .. WI

ki k ..ij k,t,j+l

(Az)2 4.1:,

2
Q Tk+l.i.j Tit-,1 TEJJH TIM—1 _ [L] Tk+l.i—I.j _ 2Tk+l,i,j + Tk+l,i+l.j

 

 

k+l,i.j

re-arranging terms

_ AI L 2 L 2 A! _ A1 _[L]2
" T 0+ T +—— T . .=
2(Ay)2 [a] k+l, i-l .j +|: [a] (:L);] It+l.i.J 2(Ay)2 k+l,l+l,j

AtPe m - AtPe At
L, + 2 Tit ,i,j- -l + Q— 77: t',j + 1"" + 2 TEJJ‘H (425)
4A2 2(Az) 4A2 2(AZ)

An)"
24;,

 

 

 

 

It+l.i,j

51

With the inclusion of axial conduction, a fourth boundary condition is necessary,

Equation (3.59), presented here in second-order correct finite difference form:

T.

k,i,M+l _

2Az

T~ .
“M" = 0 (4.26)

 

Application of this boundary condition results in the convection term being zero at the
end of the media in Equation (4.23) and (4.25). The remaining initial and boundary
conditions are applied at the appropriate half time step consistent with the procedure
outlined in Equations (4.5)-(4.13) with respect to Equations (3.25)-(3.29). Now Equation
(4.23) and Equation (4.25) in conjunction with the boundary conditions are used to form
the two tri-diagonal matrices and two source vectors that are prescribed by the ADI

method.

4.1.3.2 Dual-Energy Equation Model with ADI

The numerical representation of the fluid energy equation with axial conduction,
Equation (3.57), is determined consistent with the outlined differencing scheme for the z-

direction as follows:

 

 

 

 

2
T/JEJJ — TIM.) +Pe Thin/41 _ Trim-1 = i Tf:k,i-l.j " 2TH.” + Tf:k,i+l,j +
Ay 1“] 2A2 a (Ay)2
2 (4.27)
T ~ . —2T ~ +T ~. .
[Jr J, -l f:k,i,' f2]: ,, +1
5 1 (AZ); H + NufiLz TWIJJ — T111131]

re-arranging for the terms z-direction results in the following:

52

 

 

— AtPe 1.. , 5A1 5A1 Nu ,SELAI
- 2 [JET-.74 + 1+ 2 + T121?“
4A2 2(Az) (A2) 2

AtPe , 8A! - Nu ,ZLAI L 2 8A!
4Asz — 2( 2 TIIITJJH + f 71:117.}: 1_]:_] —__2 Tf:k,i.j (428)
A2) 2 a (Ay)

{LT 8A! T {LT 8A1 T
a 2(Ay)2 .lzk.I-|./ a 2(Ay)2 /.k.t+l.,/

 

 

Now for the y sweep

 

 

 

Tf$+l.i.j _ TfiJJ +Pe T/JTJJH — Tf;l:,i,_,'_| =g[£:]2 T/rhlj—LJ‘ -' 2szk+l,i,j + Tf:k+l,i+l,j
% I..f 2Az a (Ay)2
(4.29)
T,~.. —2T,~. .+T,~..
g 1.1.1.1—1 1.1. ,IJ 1.1. T._/+1 + NuISLZ [lbw-J _ Tth+l,i.j]

 

(AzY

re-arranging terms for the y-sweep results in the following:

 

 

 

-14” - 5N L 2 6A! Nu ,,2LAI
_ Tum-1.1 + 1 _ 2 + um.)
- .. 2(A)’)2 a (Ay) 2
PL- 2 - 6A! - Nu flXLAt AtPe, , aAt
— - ' +_T . .= “ + ~, _ 4.30
_ a _ 2(Ay)2 fl:+l.l+l.j 2 .s.k+l.l.j [ 4A2 2(AZ)2 [1; $1.] ( )

 

 

8A! - AIPEU eAt
+[1- (Az)2 :lT/JFJJ +] 4A2 + 2(Az)2 ]T/:I7.i,_j+l

The fourth necessary boundary condition to solve this equation numerically from

Equation (3.60) shown here for j = M node

szit‘uwn _ TffJM—l = 0 (4.31)

2Az
As in the prior two-energy equation development, a simultaneous solution is sought;
therefore, the penta-diagonal matrix to be formed will be a 2N by N for each sweep in the

solution method. To complete this formulation, the solid energy equation for each

53

direction will be presented. From Equation (3.58), implementing the second-order

correct differencing for the z-sweep results in the following:

Q] t—ijA— yTh':,lti'j':(1m £)k__‘[aL :lz skll/“(A$.k)lzj+]~\kl+lj+

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T/2 —2T + T -~ (4'32)
(1 _E); .vtk,i,j—l .1".k,i2,j .\:k Hij+l + NuI‘Lz[T TIIJ 1,]. _T‘k [j]+m L
kl (AZ) q“, k,ij
rearranging terms in Equation (4.32)
F —(1—e)At£-T *1+ Q +(___1-- e)At k _+ AtNujJaz-T ~ +
2(AZ)2 kl] s'jr,i,j (AZ )2 k] 2 ‘ .1".k,i,j
r-1—.TAtk.l —AtNu,.L>: 1—TAtk,’L 2
( 2) 4 Tr:k,i,j+l + 2 f Tf:k,i.j =( )2 k. _] T1tk,i-l,j + (433)
_ 2(Az) 1‘1] 2(AJ’) L.
— 1— Atk. L2 l—sAtk. L2 At‘”
Q] — ( 8)2 A [—] T1:k,i,j + ( )2 A [—] 7If1'.lr,i+l,j +-—.q,;—
_ (Ay) k, a 2(Ay) k f a 2g“, 1:1,;
Now for the y-sweep
Q] T1“,,k+ll'j/T1':k ,,ij __(1_8)_[_a:|2T.1:k+li-lj _ 2::J;$,2i.j + T1‘:k+l,i+l,j +
2T + T .., (4.34)
1__£T1".k,,ij—l .1tk.i.j 1:ki,j+l +Nu‘.L2T —T~_ +q_
( )% (AZ)2 1 [T f:k,i i,j 1.k.ij] q-zld [”le
rearranging terms in Equation (4.34)
' _ _ ‘2 i" _ 2 A N .L:
M k, [é T1“k+li-1 '+ Q] + (1 82A! [‘11:] £+l Tnk+lij 'l'
L 2(Ay) k, a_ " ' J ] (Ay) a k, 2 "
P— 1-6‘ At k, L“2 —AtNu..L2 1-6' Atk,
-—2((A_y)—2)_k—][; T1:k+l,i+l,j + ———_2__£-_ f:k+l.i./ = (2(Azg2 kl .1':I:.i,j—l + (435)
j _
Q] _ (1_£ZAtks:]T1-£U + (1—6)ZAtk.. 7.7:.- 1+1 + Art?"
_ (AZ) H ' 2(Az) kl H H 26]“, k+l.i.l

 

54

Alternating between fluid and solid equations in each matrix, the penta-diagonal matrices
for each sweep can now be formed. The z sweep for each phase, Equation (4.28) and
Equation (4.33), is re-written here where the superscript notation on the coefficients

indicates for which direction the equation is being solved in:

z z z z _ z

A/sziIJJ—I +Bij;I?,i,j +C/T1:E,i,j+1 + D/Tsziu - Girl (4'36)
2 z z z __ z

Cs TyJEJJ-l + fix Tm?” +53 Tr.l?,i,j+l + ”S T]:E,i,j _ Z-vrj (4°37)

Now for the solution in the y-direction, Equation (4.30) and Equation (4.34) are re-

written as follows:

y .V y y
A] T/:k+l.i-l.j +Bf Tf:k+l.i.j +CfT/:k+l,i+l.j + Dst'JHlJJ

= G}, (4.38)
CsyTstkHJ-Lj + flsyTslelJJ +6.:Tszk+l.i+l,j + ”snyzkhiJ = Iii (439)

Now matrices similar to Equation (4.21) may be formed for each direction of the solution

with the appropriate application of initial and boundary conditions.

4.2 Mass Transfer

Equation (3.36) is classified as a non-homogenous constant coefficient, linear
partial differential equation that is first-order in t and second-order in y and z. This
equation may be solved numerically by employing a finite difference approach. For

convenience, the following notation is introduced:

mi(t.9y.az.)=mk,i,j (440)

55

where k is the time index and the maximum values of i and j are N and M respectively.
One should note that the component subscript has been dropped; however, the equation
still refers to a single component of an ideal gas mixture. The ADI method will be
utilized to develop the tri-diagonal systems of algebraic equations (Anderson et al. 1984).
Once again, the time derivative will be second-order correct, backward differencing and
the space derivatives will be second-order correct, central differencing. Executing the
appropriate finite difference scheme for each derivative for the implicit solution in the z-

direction results in the following:

 

 

 

 

 

 

 

 

P 2
mm,- - mun} +Pe mk,i,j+l _ mil-.14 = 5 i ka-IJ "' 2mm.)- + mk,i+l.j
Ayz 2Az _a Le (Ay)2
(4.41)
_1__ mic'JJ-i _ 2mm]. + m; n+1 (“"142
2
Le (A2) ap, £11.,
where k is the half time step. Now the y-direction
'- 2
mk+lJJ _ mFJJ +Pem17,i,j+l _ mk.i.j-l = £ _1__ mk+1,i—l,j - 2mk+l.i,j + mk+l,i+l,j
13% 2A2 _a Le (Ay)2
(4.42)
_l_ mi,i.j-I — 2mi’,i,j + min“ ’WLZ
2
Le (AZ) apf k+l,i.j

 

Now both equations will be simplified into the implicit form to set-up the tri-diagonal

systems to be solved in each direction. First in the z-direction

 

 

PeAt At m +1+ At m~ + —PeAt_ At m~
4Az 2Le(Az)2 "*"J’*‘ (Az)2Le w 4A2 2(Az)2Le w-u

L 2 At L 2 At L 2 At
=_ “”-—--————..4.43
[L] 2(Ay)2Le]m*'”"’+[1 [a] (Ay)zie]”’w+[[,] 2(Ay)2Le:lm""“" < >

At r""I.2
+ .—
2 ap/

 

 

17.131

56

Rearranging terms in the y-direction

AH m + ”Hi; m + _-_At_[£] m
2Le(Ay)2 a k+l.i+l.j a Le(Ay)2 k+l.i,j 2L6(Ay)2 a k+l,i—l.j

 

 

 

At PeAt At PeAt At
= - mkij+l+ 1—— Eif" + 2 mi; -—| (4.44)
2Le(Az)2 4Az ~ Le(Az)2 ' 4A2 2Le(Az) J
Atr""L2
+
2 (1p! k+l,i.j

 

For simplicity, Equation (4.43) and (4.44) are rewritten as follows:

Azm~.. +Bzm~. .+C’m~
k.l.j—l k,i,j

G: (4.45)

”J“ = J

AymMHj +B"'m,‘+,.,‘j +C"’mk+l_,+l.j = G3 (4.46)
The initial and boundary conditions outlined in Equations (3.74)-(3.76) are a relatively
straightforward application; however, Equation (3.75) requires some detail. Equation

(3.75) may be written for first half time step, maintaining second-order correct finite

differencing by utilizing a backward scheme for the convection term is as follows:

 

 

 

 

 

 

kaM _ kaM 4P 3mi,i,M — 4miZJM-1 + kaJtl-Z _
I e -
Ay 2Az
2 2 (4.47)
[£] _1_ ka—IJW - 2mm»! + mk,i+l,M
0 Le (Ay)2
and for a half time step further:
mk'JM — kaM i e 3mm” _ 4ka.M-l + mk.i.M—2 =
At r 2Az
/2 2 (4.48)
[£:| _1_ mk+l,i—l,M - 2mk+IJM + mk+l,i+l,M
a Le (Ay)2

Equation (4.47) is a direct replacement for Equation (4.45) since, once the terms are

rearranged, only the source vector is different. Therefore, by simply changing the j = M

57

term in the source vector the boundary condition is applied. To apply Equation (4.48),
Equation (4.45) is written for the j = m-l node and Equation (4.48) is solved explicitly

for the j = m node as follows:

A’mhw2 +B’mE'W_l +sziuw = GL4 (4.49)

where m i . is replaced with the following:
,I

.M

F _
PeAt 4mi,i,M—l mE,iM-2
kaM +

2 2Az
AHZ "Lu—1M4 - 2mi,.,M-. + m.,,...,,-, (4.50)
2L8 (Ay)2 ‘

 

_ 3PeAt _l
mE’W— 1+ 4A2

 

 

 

a

The tri-diagonal matrix for the z-sweep is only solved for an M-l by M-l matrix and then

the mass fraction at j = M is assigned explicitly with Equation (4.50).

4.3 Coupling of Heat and Mass Transfer Models

The coupling of the models focuses on the interdependency of the source terms.
First, the mass transfer equation will be written with the appropriate temperature
dependency. Equation (4.41) and Equation (4.42) need to be expressed with respect to

Equation (3.80), the definition of the mass source term. The z-direction will be written as

 

 

 

follows:
mm ' mu} +Pe mm” ‘ min--1 =[_L_:|2 _1_ mix—1,,- - 2mm]. + mka
A/ 2Az at Le (Ay)2
2 (4.51)
1 mEJJ—I -2mk,i,j +mif,i,j+1 + LE&(msC' . -m~. )

 

fe- (AZ)2 Le ..".k,l.j k,i,j

58

where k is the half time step. Now the y-direction

 

 

 

mk+lJJ -mi?.i.j *Pe min” _mk.i.j-I _ £1 2 1 mk+l,i—l.j - 2mk+l.l,j + mk+|,i+l.j
A! 1 2Az — a Le (A )2
/2 y (4.52)

1 mivipj‘l - 2mg~loj Evin/‘24 L2 Sh]. ( )
—‘ + m.1‘.1‘:k+l,i,j — mk+|JJ
Le (A2)2 Le

 

Rearranging terms in Equation (4.51) and Equation (4.52) to be consistent with Equation

(4.45) and Equation (4.46) results in the following for the z-sweep:

-

PeAt At At AtSh,
L 4A2 2Le(Az)2 J‘” (A2) Le 2Le "’

' 2
_PeAt _ At mi, --. = F] __4’_2_ mm.)- + (4.53)
_ 4A2 2(Az)2Le N a 2(Ay) Le

- L 2 A’ L 2 At AtSh
1‘ — — - ' - . . L2 L - . j
b [a] (Ay)2 Le ]mk'l'j + [[ a ] 2(Ay)2 Le ]mk'l—l’j + 2L3 (msszk ,I._] )

for the y-sweep

—At L‘2 L 2 At At Sh,
——T " mk+li+lj+1+ _ ——"2'+LZ——‘ mk+lij+
2Le(Ay) a_ ’ ‘ a Le(Ay) 2 Le "

—At [_Lfm :1 At _PeAtm~ +1- At W + (454)
2Le(Ay)2 a_ M'H'j 2Le(Az)2 4A2 “J“ Le(Az)2 ”J '

PeAt At At Sh
[ 4A2 + 2Le(Az)Z ]m‘“" + LEY—Lfimmw )

 

 

 

 

 

 

 

 

The temperature dependency of mass fraction at the solid surface, In“, is determined from
Equation (3.87), Equation (3.91), and Equation (3.92). The concentration of adsorbate in
the solid is determined from the integration of Equation (3.88) from zero to the current

time, t, as follows:

<[wmacip — prShl.L (mi " hub! = I EShLL (

" ~ - - . t 4.55
’ Lea-6);). Le(l—£) CW “(Mb ( )

59

evaluating this by using a simple trapezoidal rule,

+ ZShLLAt [c,.,,, (t) — c,,,,...(t>]+ [c.,,,(t — 4r) — c..,,...(t — 40]
Le(1 — a) 2

 

cw, (t + At) = c,,,(t) (4.56)

Now that the concentration of adsorbate is determined the volumetric heat generation
term may be evaluated directly from Equation (3.94) written here in finite difference form

as follows:

 

c,‘,,(t) —c,‘,,(t — At) (Ah

At ads. + Ahw) (4.57)

4'" = (l — am.

where the enthalpy’s are determined directly from Equation (3.96). Figure 4.1 shows a

schematic of the interdependency of the heat transfer with the mass transfer.

Heat Transfer Mass Transfer
model model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1 Coupling of Heat and Mass Transfer Models

Chapter 5
Analysis of Transport Properties and Experimentation

Data is what distinguishes the dilettante from the artist.
George V. Higgins 1988

5.1 Flow Parameters

5.1.1 Permeability-Forchheimer Development

Considerable interest exists in engineering problems that require an understanding
of flow through porous media. Among these problems are the chemical contamination of
aquifers, design of super insulation, the decontamination of soil, use of vapor recovery
systems in automobile fueling, and use of catalytic converters in automobile emissions
control. The understanding of the flow through porous media dates back to early Roman
times as illustrated by the fountains of Rome, in particular the Trevi Fountain in Rome.
These fountains utilize the pressurized water from underground rivers to drive the flow
through the fountain, thus illustrating to the world that the ancient Romans had a basic
understanding of the engineering principles linked to porous media. In the mid-eighteen
hundreds, a French civil servant Henri Darcy (1856), while designing the public water
works for the village of Dijon, concluded that the specific discharge through a porous

medium is directly proportional to the pressure drop across the medium. He became the

60

61

first person to mathematically express the engineering principles associated with flow in

porous media. Darcy's Law is expressed for one-dimensional flow as follows:
d .0 fl
_ = 501
K 61 ( )

with q being the specific discharge and [.1 being the dynamic viscosity of the fluid. The
proportionality constant, K, is known as the permeability and is based on the geometric
properties of the medium. Darcy's Law is a reliable engineering approximation for a
porous medium; however, it does falter in some physical situations. For instance, non-
homogenous media, compressible fluid flow, solutions and absorptive media, and high
flow rates all deviate from Darcy's Law. In a high flow rate situation the data show
marked departure from Darcy’s linear model leading to the Forchheimer extension of

Darcy's Law, which takes the following quadratic form:

dp

2 5.2
dx q ( )

# p
=— +—
Kq M

with p defined as the density of the fluid. The parameter M is known as the Forchheimer
coefficient and, similar to K, is also a function of the geometric properties of the porous
medium. Commonly Equation (5.2) is rewritten in a non-dimensional form by defining

the hydraulic gradient as a normalized pressure gradient where g is gravity

(1): _Lfl’. (5.3)

pg dx

Applying this to Equation (5.2) and simplifying

(5.4)

62

Though Equation (5.2) is a reliable engineering approximation for flow in porous media,
it also falters for some experimental data. The imperfections that the Forchheimer
extension shows may be traced to the manner in which the constants K and M are
determined. The accepted method for determining the permeability is to take the slope of
the linear region of a plot of pressure drop versus specific discharge, disregarding any
non-linear effects (API 1952). The Forchheimer coefficient is usually taken as the
appropriate result of a quadratic curve fit to experimental data (Ahmed and Sunada 1969).
Based upon experimentally derived values for K and M, appropriate predictive models
have been developed by Firoozabadi and Katz (1979) and Geertsma (1974). In an
attempt to numerically determine the values of the permeability and the F orchheimer
coefficient using various estimation methods, the possibility has arisen that the exponent,
n, associated with the Forchheimer extension is not exactly 2 and may also be a function
of the geometric properties of the media. It is the goal of this investigation to show that
by using maximum likelihood estimation procedures a better approximation may be made
for the permeability and F orchheimer coefficient and that the actual form of the

Forchheimer extension should be

-22

61" (5-5)

=aq+£
dx K M

where n is a function of the geometric properties of the porous media.

63

5.1.2 Method of Solution

Before continuing with an estimation procedure, some standard assumptions need
to be made concerning the errors involved with the data. The experimental data that are
utilized are taken from Ahmed and Sunada (1969) and a representation of these data is
shown in Figure 5.1. The errors are assumed to be multiplicative in nature, a distinctive
aspect of this work. The errors are also assumed to have a normal distribution, and the
covariance matrix for the errors is assumed to be known to within a multiplicative
constant. There are no errors in the independent variables, and the parameters that are to
be estimated are assumed to be constant with no prior information. The least squares
estimation procedure utilized, based on these assumptions, is the maximum likelihood
approach (Beck and Arnold 1975). In this approach the sum of the squares of the
residuals will be minimized to determine the vector [3, which consists of the parameters
(Beck 1991). This approach differs from other least squares methods by the definition of

the covariance matrix for the errors, which takes the form

s1: #2 diag[of oi oi mafi] (5.6)
where u2 is an unknown multiplicative constant and the of are known but not constant.
Since the errors are assumed to be multiplicative, the of are assumed to be yiz, where Yi

are the known experimental values. The sum of squares function to be minimized will

have the following form:

S = [y-n]"‘¥"[y-n] (5.7)

64

 

«.356 6.5 3.52 me an: _auuofitomnm fim 0.5»:—

 

 

 

 

 

 

 

 

 

owhaauma 959on
2: A: _ _d 56 Sod
a Illa. _ . . . A . 7- . . . 414.4} .ilIJIIE. . "000.0
_ z , M.
i + .il .\ . ; n 53
~_+n i |ll i i J l I l . ,, 1 n .
2+ \0 H. ~O 0
all. i t - - w- - - n 3
n
0+” iii i [I H 1 # w
_n
filo...” ill: . . - Iill - »1 -ll ll- A. i i # S
N w
...... i i .. !. ?_ |--! Fill ’ , H o3

 

 

 

 

 

‘1 -.| - l ill . . o2:

3on gm Ea Bag
«:5: 380m 3093 8m b628, ism msmuo> ~5ng £3333

(ssaluogsuamgq)
warping) annmpAH

65

To begin the model building process, successive terms will be added to a
polynomial expansion model, 11, that is linear in its parameters. A statistical F-test will
then be performed on the sum of the squares of the residuals for the expansion model
(Beck and Arnold 1976). This test will indicate the number of terms that the model needs
to accurately represent the experimental data within a 95% confidence region. Statistical
reviews on the F-test are given by Lapin (1983), Bhattacharyya and Johnson (1977), and
Gibra (1973). The F-statistic is calculated for the model as terms are added and
compared to the critical F -statistic for the 95% confidence region. If the calculated value
exceeds the critical value, then that particular term is needed in the model. This model

will take the following form:

n=fi,x + ,6,x2 + + flnx" (5.8)

with the sensitivity coefficients appearing as

fl_ 1'
3,6. —x (5.9)

I

At this point one should note that the permeability and the Forchheimer coefficient have
been combined with the appropriate fluid property to ease the estimation procedure. In
Equation (5.8) the x0 term is absent since physically, if there is no flow through the
porous medium, the pressure drop will be zero within the medium. To confirm the
Forchheimer extension the F-test should result in a quadratic model for all data sets.
Table 5.1 illustrates the results of the F-test. In this Table, Rml is the residual sum of

squares, ARml is the change in the residual sum of squares as successive terms are added,

66

the 32 value is the estimated variance, F is the calculated F -statistic, and F95 is the critical
F-statistic for the 95% confidence region. The F-test suggests that to properly model
some of the experimental data a cubic model is needed instead of a quadratic model. This
implies that the actual value of the power I: in Equation (5.5) may not be equal to 2;
therefore, the model used to best represent the experimental data and estimate the vector

[3 should be of the following form:

n=fllx+fl2xfl3 (5.10)
where the sensitivity coefficients are

27-7- = x (5.1 1)
0%

fl = x'63 (5.12)
0752

0”?) :83

— = l 5.13
0763 flzx “05) ( )

Using the maximum likelihood estimation procedure with the above model the vector [3

was estimated for all the data sets. The results are found in Table 5.2.

5.1.3 Analysis

Ahmed and Sunada estimated the permeability for the data sets analyzed here;
however, they did not determine a Forchheimer coefficient but a number similar to [32. It

appears that their analysis consisted of a quadratic curve fit to the data. It should be noted

67

that the difference in the various data sets is the particle diameter of the porous material,
which leads to different porosity, tortuosity, and specific surface area for the material.
Table 5.3 identifies these differences.

Table 5.4 shows the calculated permeability from the data sets and the
permeability determined by using the maximum likelihood estimation procedure along
with the percent difference between the two procedures. Note that the units on the
permeability are square centimeters. The different procedure for determining the
permeability accounted for up to a 32% difference in the estimated values. Table 5 .5
shows similar data for the [32 evaluation. In the B2 evaluation the percent difference
ranges from 0.76% to 28.52%. When looking at this difference, one must keep in mind
that Ahmed and Sunada were determining values for the Forchheimer extension; whereas
this investigation has allowed the value of n to vary according to the media. The change
in the value of n has considerable impact on the value of the Forchheimer coefficient.
From Table 5.6 one might conclude that an average value of 2 is a good approximation;
however, recalling that this value is an exponent, the percent difference of up to 24.73%
will have considerable impact on the model. This was illustrated by the percent
differences observed in the [52 value. A confidence interval was determined for the [33
parameter. This interval allows one to determine the range that the data may take based
on the variance of the errors in the data. Table 5.7 shows the 95% confidence interval for

the B3 parameter.

Data Set 1

Data Set 2

Data Set 3

Data Set 4

Data Set 5

Data Set 6

Data Set 7

Data Set 8

Data Set 9

Data Set 10

Data Set 11

Data Set 12

Data Set 13

68

Table 5.1 F-Test for all Data Sets

 

 

 

 

 

 

 

 

 

 

 

 

 

Model p n-p Rm] S2 AR”. F F,5
1 1 17 1.95 0.1 14705882
2 2 16 0.0245 0.00153125 1.9255 1257.469388 4.49
3 3 15 0.0238 0.001586667 0.0007 0.441176471 4.54
4 4 14 0.0235 0.001678571 0.0003 0.178723404 4.6
1 1 15 2.3 0.153333333
2 2 14 0.0078 0.000557143 2.2922 4114.205128 4.6
3 3 13 0.00715 0.00055 0.00065 1.181818182 4.67
4 4 12 0.00606 0.000505 0.00109 2.158415842 4.75
1 1 20 3.83 0.1915
2 2 19 0.0128 0.000673684 3.8172 5666.15625 4.38
3 3 18 0.00532 0.000295556 0.00748 25.30827068 4.41
4 4 17 0.00527 0.00031 5E-05 0.161290323 4.45
1 1 21 0.545 0.025952381
2 2 20 0.0135 0.000675 0.5315 787.4074074 4.35
3 3 19 0.0131 0.000689474 0.0004 0.580152672 4.38
4 4 18 0.0123 0.000683333 0.0008 1.170731707 4.41
1 1 21 1.18 0.056190476
2 2 20 0.0112 0.00056 1.1688 2087.142857 4.35
3 3 19 0.0111 0.000584211 1E-04 0.171171171 4.38
4 4 18 0.00815 0.000452778 0.00295 6.515337423 4.41
1 1 18 1.47 0.081666667
2 2 17 0.00894 0.000525882 1.46106 2778.302013 4.45
3 3 16 0.00861 0.000538125 0.00033 0.613240418 4.49
4 4 15 0.00782 0.000521333 0.00079 1.515345269 4.54
1 1 15 5.62716 0.375144
2 2 14 0.0615451 0.004396079 5.5656149 1266.040816 4.6
3 3 13 0.020652 0.001588615 0.0408931 25.74134709 4.67
4 4 12 0.0196303 0.001635858 0.0010217 0.624565086 4.75
1 1 1 1 0.116646 0.010604182
2 2 10 0.00343253 0.000343253 0.1 1321347 329.8251436 4.96
3 3 9 0.00337976 0.000375529 5.277E-05 0.140521812 5.12
4 4 8 0.00209354 0.000261693 0.00128622 4.915005206 5.32
1 1 9 0.447455 0.049717222
2 2 8 0.00486292 0.000607865 0.44259208 728.109169 5.32
3 3 7 0.00472156 0.000674509 0.00014136 0.209574802 5.59
4 4 6 0.0030687] 0.000511452 0.00165285 3.231683672 5.99
1 1 9 1.16095 0.128994444
2 2 8 0.00682882 0.000853603 1.154121 18 1352.059278 5.32
3 3 7 0.006483 58 0.000926226 0.00034524 0.372738518 5.59
4 4 6 0.00501316 0.000835527 0.00147042 1.759872017 5.99
1 1 6 0.0722215 0.012036917
2 2 5 0.0194795 0.0038959 0.052742 13.53782181 6.61
3 3 4 0.0173099 0.004327475 0.0021696 0.501354716 7.71
4 4 3 0.0159886 0.005329533 0.0013213 0.247920393 10.13
1 1 13 2.24121 0.172400769
2 2 12 0.0502295 0.004185792 2.1909805 523.4327636 4.75
3 3 11 0.0502295 0.004566318 0 0 4.84
4 4 10 0.0225439 0.00225439 0.0276856 12.28075 4.96
1 1 21 4.85632 0.231253333
2 2 20 0.0492643 0.002463215 4.8070557 1951.537198 4.35
3 3 19 0.0492643 0.002592858 0 0 4.38
4 4 18 0.0343495 0.001908306 0.0149148 7.815729487 4.41

69

Table 5.2 Vector B for All Data Sets

 

Data Set 8, [3, B3
1 1.45 0.250 2.043
2 0.941 0.186 2.025
3 0.690 0.143 2.066
4 7.408 0.927 1.935
5 3.826 0.580 1.973
6 2.195 0.366 2.002
7 0.00603 0.016 1.884
8 0.0122 0.000927 2.021
9 0.00421 0.000391 2.065
10 0.070 0.038 2.061
11 18.592 8.443 2.657
12 0.150 0.192 1.842
13 0.127 0.083 1.931

Table 5.3 Experimental Data Sets

 

Data Set Medium Particle Diameter (cm)
1 Sand 0.140
2 Sand 0.199
3 Sand 0.258
4 Sand 0.0764
5 Sand 0.054
6 Sand 0.107
7 Marble 1.6
8 Sand 0.3
9 Sand 0.5
10 Glass Beads 0.53
11 Ottawa Sand 0.07
12 Sand 0.5
13 Glass Spheres 0.3

70

Table 5.4 Permeability Calculated From Quadratic Curve Fit
and Maximum Likelihood Estimation Procedure

Data Set Ahmed and Sunada ML Estimation Procedure °/o Difference

 

1 1.00E-05 1.03E-05 2.57
2 1.69E-05 1.68E-05 0.35
3 2.21505 2.221305 0.55
4 2.10E-06 2.09E-06 0.24
5 3.96E-06 3.93E-06 0.70
6 6.91E-06 7.24E-06 4.53
7 1.191303 1.771303 32.68
8 7.60E-04 7.65E-04 0.65
9 2.301303 22315-03 3.21
10 1.38E-04 1321304 4.22
11 8.20E-07 7.321307 12.00
12 4.941305 62113-05 20.44
13 6.45E-05 7.36E-05 12.40

Table 5.5 BzCalculated From Quadratic Curve Fit and
Maximum Likelihood Estimation Procedure

 

Data Set Ahmed and Sunada ML Estimation % Difference
1 0.24 0.25 4.00
2 0.179 0.186 3.76
3 0.165 0.143 15.38
4 0.745 0.927 19.63
5 0.454 0.58 21.72
6 0.308 0.366 15.85
7 0.0117 0.0159 26.58
8 0.00092 0.000927 0.76
9 0.0005 0.000391 27.88
10 0.0368 0.0379 2.83
11 7.96 8.44 5.72
12 0.137 0.192 28.52
13 0.0648 0.0833 22.21

71

Table 5.6 B, for All Data Sets

 

Data Set [33 % Difference
1 2.044 2.15
2 2.025 1.23
3 2.066 3.19
4 1.935 3.36
5 1.973 1.37
6 2.002 0.10
7 1.884 6.16
8 2.021 1.04
9 2.065 3.15
10 2.061 2.96
11 2.657 24.73
12 1.842 8.58
13 1.931 3.57

Table 5.7 Confidence Intervals for [53

Data Set Low Value High Value

 

fl

\OOOQQUI-bbJN

1.967
1.987
2.041
1.83
1.675
1.946
1.847
1.806
1.913
1.932
0.303
1.722
1.884

2.121
2.063
2.091
2.04
2.271
2.058
1.921
2.236
2.217
2.19
5.011
1.962
1.978

72

Since the permeability and Forchheimer coefficients have been determined for
these data sets, one may now investigate the governing geometric parameters of the
porous mediums. The Carman permeability model may be utilized to generate an

iterative calculation of the porosity for each data set

__ 63613.
-180(1- g)2

(5.14)
where (1,, is the particle diameter. Once the porosity is determined, the specific surface
area may be calculated by assuming that the particles are spherical. From Equation (2.3)
one may determine the following:

d 2
4 /
Asurface ”i 2 j

z = V = V (5-15)
bulk bulk

 

Recalling Equation (2.1), the bulk volume may be defined in terms of the volume of
solids and porosity

Vbulk ‘ Vsolid = 1 _ 13042 (5.16)

Vbulk Vb ulk

8:

Solving Equation (5.16) for the bulk volume results in

V
_ solid
Vbulk— 1-8 (5'17)

The specific surface area may be determined by making the appropriate substitutions into
Equation (5.15) and simplifying to obtain

_6(1-6)
:— d (5.18)

P

73

From Equation (2.2), the tortuosity is a difficult parameter to determine. The tortuosity
will be determined as an integrated average over a control volume. Figure 5.2 illustrates
a simple model of a spherical particle in a spherical differential control volume where 4) is
the angle at which a fluid element enters the control volume and w is the angle at which
the fluid element will impact the particle. The fluid element is assumed to attach itself to
the particle until it detaches at the same angle 11! and then exits the control volume at the

same angle 0 about the plane of symmetry or the y-axis.

Y /\Plane of Symmetry

 

 

 

 

Figure 5.2 Differential Control Volume

The following averaging integral will be evaluated:

% L
1:2 L {Md}, (5.19)

% 0 Loommple

From the geometry of Figure 5.2 the angles will be defined as functions of y as follows:

¢=ArcSin(ZZ-)
A

. 2y
= ArcSm —
w [d

P

] (5.20)

74

The particle diameter has already been specified for each data set. The diameter of the

control volume is determined from the definition of the porosity in the following form:

dP
A = 1_6 (5.21)

 

w

The length of the path, L(y)p,,h, and the length of the sample, L(y)s,mp.e, need to be
determined as functions of y. The length of the sample will be the path that a fluid

element would take if the particle did not exist in the control volume, defined here as

LU)sampIe =ACOS(ArcSin(—2AZD 0 S y S (5.22)

The length of the path will be the distance that the fluid element travels prior to impact.
the arc length which the element travels attached to the particle and the exit path of the

element. This combination of distances is represented mathematically as follows:

Liy)pa t h=ACos(A rcSir{—2fD—d ”C 0.1[ArcSin 1%)]
P

+£[n-2ArcSin[E¥—J] OSySfl'i (5.23)
2 d 2

I)

__ . 2y 61,, A
My)path—ACOS[ArcSm[-;D ? <yS—2-

Combining Equation (5.23) and Equation (5.22) with Equation (5.19) results in the

following:

7

f /

\\ \\
,3 d , _
«1,,1'ACos ArcSin£2JJ —d Cos ArcSm 2y + ” 7r—2Arc51‘n‘ 2) I
I2 A P d], 2 (d, //
0 AC as (ArcSin(2y J]
l A
11,,

ll .

 

 

(5.24)

% y) ACos [ArcSin[
+ l

1 ”/2 ACos [ArcSin[

 

32’ 2.1::

 

 

 

 

3* l

75

Simplifying Equation (5.24) and evaluating where possible

(

d

 

_P_
2

w
11
halb

—.:- 2 M
Cos [ArcSin [{D ”

 

1

,/ Cos [ArcSin [ 23) D .17.,

1 d (1,, 2 7rd,,
1 d

0

W 2 dy
C os[ArcSin (7': D "

l

"'/2

 

-l

[ArcSin[
in.

.V

:11

P

 

A Cos[ArcSin (2’)]

dy

A
+——
2

5.».
2

\

 

)

(5.25)

Solving the first integral in Equation (5.25) results in a hypergeometric function that is

readily evaluated. The second integral is relatively simple, and the third is evaluated

through numerical integration. Table 5.8 shows the results of the evaluation of all the

geometric parameters.

Table 5.8 Porous Media Characteristics

 

for all Data Sets
Data Set A e 23 'c
1 0.161 0.344 28.127 5.001
2 0.227 0.326 20.312 5.090
3 0.292 0.307 16.121 5.215
4 0.087 0.313 53.988 5.296
5 0.065 0.429 63.418 4.748
6 0.124 0.360 35.893 5.015
7 1.864 0.368 2.371 3.754
8 0.411 0.612 7.750 3.766
9 0.689 0.617 4.594 3.669
10 0.607 0.335 7.530 4.789
11 0.077 0.248 64.474 5.648
12 0.559 0.284 8.592 5.220
13 0.352 0.383 12.344 4.911

76

5.1.4 Conclusions and recommendations of Permeability Study

From this investigation the following conclusions may be made:

1. A statistical F -test suggests that the form of the
F orchheimer extension could be redefined with a linear and
non-linear part of variable power where that power is based
on the geometry of the media.

2. Utilizing the maximum likelihood estimation procedure
results in up to a 28% difference in the value of the
permeability for a given media.

3. The Forchheimer coefficient may be redefined to account
for the variable power associated with the non-linear term

in the Forchheimer extension.

4. Once the porosity is determined, the remaining geometric
properties may be determined.

The results of this study give rise to several interesting future investigations.
First, predictive models need to be developed for the permeability and the Forchheimer
coefficient based on the governing geometric parameters. Second, these models need to
be confirmed with experimental data. Finally, a predictive model needs to be developed
for the power It in the Forchheimer extension to determine the precise nature of the

geometric relation that the power has to the media.

77

5.2 Experimentation

5.2.1 Determination of Density, Porosity, and Specific Surface

In a given specific example the solid matrix of interest can be made up of non-
uniform cylindrical particles of wood—based carbon. The density of these solid particles is
relatively simple to determine by measuring the diameter and length of a random
sampling of particles with calipers accurate to within +/- 0.0005 in. and measuring the
mass of the individual particles to within +/- 0.00005 g. The measured data are found in
Table 5.9.

Table 5.9 Measured Data For Density Calculation

Length (inches) Diameter (inches) Mass (grams)

 

0.223 0.084 0.0110
0.167 0.088 0.0096
0.254 0.083 0.0138
0.239 0.084 0.0100
0.247 0.086 0.0110
0.262 0.083 0.0099
0.208 0.084 0.0091
0.160 0.084 0.0078
0.172 0.081 0.0089
0.219 0.082 0.0100

Once the measurements are made the density may be determined from the standard

definition of density

 

= (5.26)

78

The uncertainty in the density is found by an error analysis based on the uncertainty in the
measured data which in this example would be the uncertainty in the measurement of

mass, diameter, and length.

6_p (23
6D 6L

Applying this to (5.27) the specific form of the uncertainty is as follows:

apam +
6m

6D + 5L (5.27)

5p=

 

 

 

 

 

6p=—1-6m +—2—"—’aD+—6L (5.28)
V DV VL

Simplifying Equation (5.28) results in the following:

5p: p[1 6m + gar) +2613] (5.29)

Table 5.10 shows the density calculation and the uncertainty analysis.

Table 5.10 Density Calculation and Uncertainty Analysis

Volume (cm’) Density (g/cm’) Uncertainty

 

0.02025 0.54 +/- 0.01
0.01664 0.58 +/- 0.01
0.02252 0.61 +/- 0.01
0.0217 0.46 +/- 0.01
0.02351 0.47 +/- 0.01
0.02323 0.43 +/- 0.01
0.01889 0.48 +/- 0.01
0.01453 0.54 +/- 0.01
0.01452 0.61 +/- 0.01
0.01895 0.53 +/— 0.01

By using Equation (2.3) and Equation (5.17) and defining the volume based on a
cylinder, the specific surface area can be determined once the porosity is known. For the

given porous media, with knowledge of the pellet density, the porosity may be

79

determined by measuring the weight of a 1-liter graduated cylinder filled with the given
cylindrical pellets and applying the definition of the porosity, Equation (5.16). The
average porosity determined in this manner over several measurements is 8 = 0.333.
Table 5.11 illustrates the results of the specific surface area calculation along with the

uncertainty calculation that is consistent with Equation (5.27).

Table 5.11 Specific Surface Area

 

2 (m2/m3) Uncertainty

1250.33 +/-0.044
1193.5 +/-0.040
1265.39 +/-0.045
1250.33 +/-0.044
1221.25 +/-0.042
1265.39 +/-0.045
1250.33 +/-0.044
1250.33 +/-0.044
1296.64 +/-0.047
1280.83 +/-0.046

5.2.2 Non-reacting Gas Heat Transfer Experiment

An experiment was conducted to substantiate the numerical solution of the heat
transfer processes within the porous media. A well-packed U-shaped container with a
porosity of 0.33 is insulated and equipped with several thermocouples to record the
heating of the porous media with a non-reacting pre-heated gas. Figure 5.3 illustrates the
experimental data along with numerical results for various Biot numbers. The thermal

conductivity of the media is determined to be 0.095 W/[m-K] and the thermal diffusivity

80

of the media is 8.2x10'5 m2/s. These properties are taken to be a combination of the
properties of the non-reacting gas and the particles based on the porosity of the media. In
comparing the numerical results to the experimental results, it appears that the difference
is due to a changing Biot number in the experiment. Natural convection is occurring on
the outside surface of the container; and, hence, the outside convective heat transfer
coefficient and the Biot number will increase as the temperature of the bed rises. This
explains the variability in the experimental data, but the numerical results still

demonstrate an ability to accurately represent the data at these different Biot numbers.

81

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Chapter 6
Results and Discussion

If they don’t depend on true evidence, scientists are no better than gossips.
Penelope Fitzgerald 1990

6.1 Numerical vs. Simplified Analytical Solution

6.1.1 Heat Transfer Model

To verify the accuracy of the axial conduction model, the numerical solution was

compared to an analytical solution of a simplified form of Equation (3.56) as follows:

0: (6.1)

 

[L]2 52T‘ 32T‘
— *2 '1' '2
a ay 52

This equation was solved with an adiabatic condition at y. = 0 and a convective condition
at y. = 1. In the axial direction a known temperature is assumed at z. = 0 and an
adiabatic condition is assumed at z. = 1. Applying separation of variables to solve
Equation (6.1)

T’(y',z’ ) = ZE,Sin(2,,z‘)Cosh(%/1,,y')

n=0

E = . 2Bsz f (6.2)

2,, 3 2,,Sinh(3 2,) + Bi Cosh(3 2,9]
L L L

 

2,, =(2n+1)%

82

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Figure 6.1 illustrates the comparison of the numerical model to the analytical solution for
this simplified case. The numerical model appears to accurately represent the simplified

model.
6.1.2 Mass Transfer Model

To verify the accuracy of the axial conduction model, the numerical solution was

compared to an analytical solution of a simplified form of Equation (3.56) as follows:

0771 _[L] §2m (6 3)
3t. Le 07212 '

 

This equation was solved in the axial direction with a known mass fraction assumed at z.
= 0 and a zero gradient is assumed at z. = 1 along with the known mass distribution at t

= 0. Applying separation of variables to solve Equation (6.3)

 

 

a a 1 ——"— 2.276
mt,z =- Ene "" Sin "
( ) ”Z < 2 >
Enz—1+Cos(7r2") ( 6. 4)
All
x1”=(2n+1)

Figure 6.2 illustrates the comparison of the numerical model to the analytical solution for
this simplified case. The ntunerical model accurately represents the mass transfer

equation in this simplified form.

86

6.2 Single-Energy Equation Model

6.2.1 Parametric Study

The various effects of the parameters within Equation (3.23) may be analyzed in
several manners. To begin, one should consider how the temperature is distributed
within the porous medium at various times. In this particular case the porous media is
assumed to be a uniform channel with convection occurring from both outer walls. A
surface plot of the temperature distribution throughout the porous medium at 10% of
steady state is shown in Figure 6.3. One can observe the manner in which the thermal
wave moves through the porous media and how the temperature distribution is affected
by the boundary conditions. Recall that the inlet is maintained at a given constant
temperature. Figure 6.4 illustrates the same test case at approximately 50% of steady
state. One should note the smoothing of the thermal wave due to the diffusion and
convection affects. Finally, Figure 6.5 shows the temperature distribution at steady state
that is a linear relationship with axial position. In all three figures the radial heat loss is

evident from the parabolic nature of the temperature profile in the y-direction.

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One way in which the complex relationships between the governing parameters
may be scrutinized is by observing the changes in the average exit temperature of the
porous medium. This allows for the effects of the parameters to be detected from a
stationary position over time. Considerable attention is given here to the variation of
parameters within Equation (3.23). Figure 6.6 illustrates the change in the average exit
temperature of the media with respect to the rate of heat generation as a function of non-
dimensional time. As expected the temperature increases with the increasing rate of heat
generation. It is obvious that as more energy is added to the system the, temperature
should rise. The dependency on the internal heat generation is important since in most
applications the heat generation term will not be constant but vary throughout the porous
medium as a function of temperature, mass concentration, and rate of adsorption. For
some applications the length to width ratio or the aspect ratio of the porous medium will
be of interest. Figure 6.7 shows that for small aspect ratios (less than one) the average
exit temperature of the porous medium is approximately the same as a function of time;
however, as the aspect ratio increases, the average exit temperature decreases. In
Equation (3.23) the aspect ratio scales the conduction term; therefore, the higher the
aspect ratio is, the more heat will leave the medium due to radial conduction. This will
result in a lower exit temperature, thus confirming the trends observed in Figure 6.7.
Little difference in the approach to steady state is noticed since the storage and
convection terms are not affected by the aspect ratio. Figure 6.8 illustrates the variation
of the average exit temperature with time at various heat capacity ratios, (2. Recall that Q

is the ratio of the heat capacity of the media to the heat capacity of the fluid. From

91

Equation (3.23) the larger the Q is the more energy will be absorbed by the medium.
Hence, lower exit temperatures are expected at higher Q values because the medium is
retaining more of the energy. Following the same logic, the time to steady state is also
increased with increasing Q. Figure 6.9 illustrates that as the Peclet number is varied, the
average exit temperature will decrease with increasing values as a function of time.
Referring to Equation (3.23), as the Peclet number increases; more energy is convected
out of the medium, maintaining the final temperature at smaller values. This swifier fluid
flow augments the energy that is extracted from the medium, hence increasing the

approach to steady state.

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6.2.2 Boundary Condition at y = 0

The special nature of the boundary condition described by Equation (3.19)
requires some investigation. One avenue to investigate is whether the balanced heat flux
affects the heat transfer out of the porous media. One manner in which to determine this
is to compare the balance heat flux condition to an adiabatic condition and a convective
condition at the same location and then determine the effect on the exit temperature of the
media. From Figure 6.10 the insulated or adiabatic condition gives the highest average
exit temperature, followed by the conducting wall boundary condition and finally the
convective boundary condition results in the lowest average exit temperature with respect
to time; however, the average exit temperatures only differ by a few degrees. Effectively
no dependency is observed with regards to the various boundary conditions on the
average exit temperature. Figures 6.11 and 6.12 illustrate the temperature distributions at
the same time step in the y-plane at corresponding z-locations for all types of boundary
conditions. These figures illustrate the influence of the various boundary conditions on
the radial heat transfer processes occurring within the medium. One should note that the
conducting wall boundary condition allows for a heat loss or a corresponding heat gain
on either side of the partition. Recalling the physical geometry illustrated in Figure 3.2,
the appropriate temperature distribution for one time step at corresponding z-locations is
illustrated in Figure 6.13, where the arrows indicate the direction of flow. The
temperature peak in the downstream region being off-center depicts the influence of the

upstream heat transfer process.

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6.2.3 Addition of Axial Conduction

The determination of when to include axial conduction is determined from an
order of magnitude analysis similar to the one performed in section 3.2.2.1 where the
determining factor is the Peclet number. The goal here is to determine what constitutes a
large Peclet number in order to ignore axial conduction. To investigate this analysis
further, Equation (3.23) and Equation (3.56) were numerically solved at various values of

the Peclet number and then the residual sum of squares was determined as follows:

5 = i[T(y,z,t)— Tm (y, z,t)] 2 (6.5)

y.z,!

This is essentially the difference between the two solutions over all time and space.
Figure 6.14 illustrates how dramatically the residual sum of squares is affected by the
value of the Peclet number. As the Peclet number approaches zero, 5 approaches
infinity and as the Peclet number approaches infinity, 5 tends toward zero. Figure 6.15
shows a magnification of the Peclet number range between 0 and 1000. One may
conclude that in the Peclet number range of 20 to 500 there is some error involved in
ignoring axial conduction; however, the impact of that error is small and is certainly
insignificant compared to when the Peclet number is less than 20. Common practice has
been to consider the axial conduction negligible below 10 as pointed out by Adnani
(1995); however, this analysis shows that axial conduction occurring at Peclet numbers
between 20-500 may still play a role. Certainly at a Peclet number of 5 axial conduction

is important. Figure 6.16 illustrates the magnitude of the difference between the two

solutions at a given time for a Peclet number of 5. It is clear that considerable error exists

102

if axial conduction is ignored. Figure 6.17 illustrates a similar situation for a Peclet
number of 100; however, the order of magnitude of the error is decreased by 2 orders of
magnitude. This supports ignoring axial conduction in the 20 to 500 Peclet number
range. Figures similar to 6.16 and 6.17 are in Appendix A to further support this

observation.

103

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6.3 Dual-Energy Equation Model

To determine if the dual-energy equation model is the appropriate thermal model,
consideration must be given to several parameters. These parameters are primarily the
Peclet number, the specific surface area, and the heat generation term. The selection of
these parameters is obvious from Equation (3.37) and Equation (3.41). As the Peclet
number increases, so should the interphase transport. The lower Peclet number will
decrease the convection that is occurring between the particles and therefore the
temperature difference is smaller. Peclet numbers over the ranging from 5-200 were
investigated and it was determined that the temperature differences were small enough to
make the selection of the dual-energy equation model unnecessary for a large range of the
Peclet numbers for the given porous media. The second parameter that will affect the
interphase transport is the specific surface area. Table 5.11 gives values of 2 to be on
average 1250 mZ/m3. Clearly, a decrease in the specific surface area will result in an
increase in the temperature potential between the particle and fluid. This is illustrated by
Figure 6.18. It should be noted that this decrease must be significant, on the order of
1000 in this case, to show an appreciable difference. it is readily apparent that if more
energy is added to the system, the temperature should increase. Figure 6.19 illustrates
that if the constant heat generation term is increased, an appreciable difference between

the solid and fluid temperature will occur.

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6.4 Coupled Model

6.4.1 Parametric Study

The complex relationships between the governing parameters of Equation (3.23)
may be inspected through the changes in the exit temperature of the porous media taken
at the centerline. This allows for the effects of the parameters to be detected from a
stationary position over time. Considerable attention is given here to the variation of
parameters within Equation (3.23) where the heat generation is not constant, as in the
coupled model. Figure 6.20 illustrates the variation of the centerline exit temperature
with time at various heat capacity ratios, 9. From Equation (3.23) the larger the Q is the
more energy will be absorbed by the medium. Hence, the thermal wave will take longer
to travel through the media at higher Q values because the medium is retaining more
energy. At the lowest (2 value, 10, several thermal waves travel through the media since
the media is retaining less energy. At the highest value of Q, 300, the medium is not
allowing the energy to leave the media; therefore, the exit temperature is the lowest. In
looking at the mass fraction in bulk, Figure 6.21, the lower the Q value the longer the
time to breakthrough for the mass wave since at the resulting lower temperature the
media is able to adsorb more of the active gas; this is not an appreciable difference here
however. Breakthrough occurs when the flow of the active gas is released from the
media.

Figure 6.22 illustrates that as the Peclet number is varied the centerline exit
temperature will decrease with increasing values as a function of time. Referring to
Equation (3.23), as the Peclet number increases, more energy is convected out of the

medium. This swifter fluid flow augments the energy that is extracted from the medium,

111

hence increasing the approach to steady state. The peak of the thermal wave will
therefore travel faster through the media. In the mass transfer equation, Equation (3.72),
the Peclet nmnber increases the convection through the media as well; therefore the faster
the flow, the more quickly breakthrough is achieved as Figure 6.23 indicates. Figure 6.24
illustrates the effect of a changing Biot number on the exit temperature of the media.
Since an increase in Biot number indicates an increase in the loss of energy in the radial
direction, it is expected that the exit temperature would decrease with an increasing Biot
number, as Figure 6.24 indicates. Figure 6.25 shows the effect that changing the Lewis
number has on the mass fraction in bulk at the exit. Since the inverse of the Lewis
number scales the diffusion terms, it is expected that as the Lewis number increases, the
diffusion would decrease. Figure 6.25 illustrates this; however, the effect is minimal as
compared to the more dominant convection as Figure 6.23 illustrates.

The Sherwood number correlations given by Wakao et al., Equation (3.83), and
Incropera and DeWitt, Equation (3.85), are both utilized in the mass transfer potential
model in the mass transfer equation analyzed. The first correlation is specifically for
spheres and the second has a modification for various geometries. Both corrections are
utilized to determine the appropriate Sherwood number for Equation (3.80). Figure 6.26
illustrates the average relative difference as a fimction of non-dimensional time. One
may ascertain that in this particular instance the difference is negligible as to which

correlation is used.

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6.4.2 Comparison to Experimental

Considerable experimental data for the coupled heat and mass transfer data exists
for the described physical situation. Figure 6.27 shows that the model currently under
investigation predicts the peak of the thermal wave as it moves through the porous media.
The movement of the wave, however, is beginning to form in the appropriate staging as
the media is heated. Further investigation into the experimental data utilizing an

appropriate parameter estimation method could result in a more consistent representation

of the data.

6.5 Summary

The analysis of the single- and dual-energy equation models has led to a decision
matrix based on certain criteria of the flow and the media. Table 6.1 presents the
findings. A Peclet number of approximately 20 is a point of demarcation where the
importance of axial conduction may be determined based on the minimization of the
residuals in the single-energy equation model. In the dual-energy equation model, if
there is an appreciable difference in the fluid and solid temperatures, axial conduction
should be included in the lower flow rate regime; however, this is only necessary if there
exists a need for the dual-energy equation model. For instance, if the specific surface
area is on the order of 1 then the need for the dual-energy equation is apparent. If the
solid in the porous media matrix is able to generate a volumetric heating on the order of

100,000 W/m3, then the dual-energy equation would be an appropriate model for the heat

121

transfer. Of course, this criterion is based on the specific surface area being on average

1250 m2/m3.

Table 6.1 Energy Equation Model Selection Matrix

Single-Energy Single-Energy Equation Dual-Energy Dual-Energy Equation

 

Equation Model Model with Axial Equation Model Model with Axial
Conduction Conduction
Pe > 20 < 20 Various Various
2 >O(10) >O(10) ~00) ~00)
q'" z 0(1000) z 0(1000) Z 0( 100000) 2 0(100000)

 

The coupling of the heat and mass transfer models is strictly a function of the
physics of the process that is occurring. The criterion developed is for the coupling of the
heat and mass transfer in the specified physical problem: the adsorption of hydrocarbon
based fuel vapors onto activated carbon as a function of temperature and mass
concentration of adsorbate. The implementation of the iterative boundary condition
across the separation wall appears to have little effect in the coupled model on the mass
transfer. The anticipated radial effect of the temperature gradients on the radial mass
transfer gradients did not materialize in the analysis. The mass transfer is governed by
the impermeable wall more so then the increase in mass transfer due to an increase in

temperature.

Chapter 7
Conclusions and Recommendations

We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started

And know the place for the first time.
T. S. Eliot

The substance of this investigation was to characterize the coupled heat and mass
transfer in a heat generating porous media through the implementation of a fully implicit
second order correct numerical model. This was accomplished through the staging of the
development of the model by first analyzing the heat transfer in a constant volumetric
heat generating porous media with a uniform flow and an iterative boundary condition
that represented the balance of heat flux across a separation wall. This was modeled with
both the single- and dual-energy equation models. The importance of axial conduction in
the heat transfer model arose through this modeling. Several conclusions can be made
with specific regards to the heat transfer modeling:

1. The balanced heat flux boundary condition brought a

unique aspect to the numerical modeling of this work. The
implementation of this condition allowed for the influence
of upstream heat transfer on the downstream temperature
distributions.

2. The appropriateness of the thermal model is confirmed with

experimental data for a pre-heated non-reacting gas flowing
through an activated carbon canister.

122

123

3. The addition of axial conduction to the heat transfer
modeling is determined to be valid for Peclet number flows
less then 20 through an extensive investigation of the
residual difference between the two numerical solutions,
with and without axial conduction.

4. The advantage of modeling the non-thermal equilibrium
energy transport in a porous media is only apparent for
media with a small specific surface area or a high rate of
heat generation from the solid. For constant specific
surface area, the Peclet number does not have an
appreciable effect on the interphase heat transfer.

5. The fully implicit fmite difference solution to the dual
energy equation model utilized a noteworthy ordering of
matrix coefficients such that the solid and fluid equations
could be solved simultaneously.

In the modeling of the fluid mechanics, the general approach throughout this work
has been to utilize a uniform flow model. However, considerable attention is given to the
determination of the flow parameters known as the permeability and the F orchheimer
coefficient for non—linear flow situations in various porous media. The use of
experimental data along with well established techniques of parameter estimation and
model building led to a proposed method for the determination of the permeability,
F orchheimer coefficient and exponent in the Forchheimer extension. Once the
permeability was determined, a mathematical model was developed for the tortuosity as a
function of porosity and particle diameter for a porous media of packed spheres.

The solution of the simultaneous heat and mass transfer entails considerable
numerical computation to obtain the fully implicit second order solution to the finite
difference equations. There are several conclusions to be drawn from this investigation:

1. Independent of the geometric configuration, the Sherwood

correlations resulted in approximately the same
representation of mass transfer potential.

124

2. The breakthrough of the reacting species is a function of
the flow rate of the reacting gas more so then the diffusion
as exemplified by the Peclet number dependency being
stronger then the Lewis number dependency.

3. The iterative heat transfer boundary condition had little
effect on the radial mass transfer profile.

Recommendations for further investigation into porous media analysis have arisen

from this study and include:

1. In the mass transfer analysis further quantification of the
effect of particle geometry on the Sherwood number
correlations is desired. Current representations show little
effect due to the geometry of the media particles.

2. Utilize a least squares approach to the experimental data to
estimate the media parameters such as the effective thermal
conductivity and the thermal diffusivity.

3. Expanding the Permeability-Forchheimer analysis to
include media particles other then spheres.

4. Implement a design study to optimize the physical
geometry based on the solution to the coupled heat and
mass transfer equations.

APPENDIX

APPENDIX

RESIDUAL SUM OF SQUARES ANALYSIS FOR AXIAL

CONDUCTION

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BIBLIOGRAPHY

 

 

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1

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