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A MATHEMATICAL MODEL FOR THE PYROLYSIS OF
OIL SHALE USING DISTRIBUTION OF

ACTIVATION ENERGY KINETICS

BY

John Arthur Marlatt

A THESIS

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

MASTER OF SCIENCE

Department of Chemical Engineering

1980

ABSTRACT

A MATHEMATICAL MODEL FOR THE PYROLYSIS OF
OIL SHALE USING DISTRIBUTION OF
ACTIVATION ENERGY KINETICS

BY

John Arthur Marlatt

Mathematical simulations of commercial oil shale
retort Operations require an understanding of the rate
processes which occur in individual oil shale particles.
Previously, the kinetics of oil shale devolatilization have
been modeled using simple kinetic schemes but have failed
to predict the production of volatile products over a wide
range of pyrolysis conditions. A prototype model was
developed in this study which incorporated the kinetics of
devolatilization, a mechanism for intraparticle mass
tranSport of volatile products, and the kinetics for intra-
particle product degradation. The complex kinetics associated
with oil shale devolatilization were interpreted using the
distribution of activation energies theory develOped pre-
viously for coal pyrolysis. A Gaussian distribution
function was used to represent the activation energies for
the reactions producing volatile products. The parameters

in the model were estimated from isothermal weight loss data

John Arthur Marlatt

for western oil shale. A favorable comparison was made
between the model prediction and data from a nonisothermal

eXperiment.

Dedicated to

Janet Lee Marlatt

ii

ACKNOWLEDGMENTS

The author gratefully acknowledges the guidance and
editorial assistance of Dr. Charles Petty.

Special appreciation is also expressed to my wife,
Jan, for her encouragement and help in the preparation of

this thesis.

iii

TABLE OF CONTENTS

Page
LIST OF TABLES vii
LIST OF FIGURES. . . . . . . . . . . viii
LIST OF NOMENCLATURE . ix
Chapter

1. INTRODUCTION . . . . l
1.1 Physical Description of Oil Shale 2
1.2 Description of Oil Shale Pyrolysis. 4
1.3 Kinetic Models for Pyrolysis. . . 7
1.4 Objectives of This Reseach . 15

2. A MATHEMATICAL MODEL FOR THE PYROLYSIS OF
OIL SHALE. . . . . . . . . . 17
2.1 Devolatilization Kinetics. 18
2.2 Coking Kinetics . 21
2.3 Material and Energy Balances. . 22

2.3.1 Material Balance for the Reactive
Volatile Species in the Gas Phase
of the Particle. . . . . 22
2.3.2 Material Balance for the Non-
reactive Volatile Species in the
Gas Phase of the Particle . 26
2.3.3 A Material Balance for the
Volatile Species Bound to the
Solid Phase . . 27
2.3.4 Material Balance for the Weight
of Volatiles Collected . . . 28
2.3.5 Energy Balance . . . . . . 28
3. SPECIAL CASES TO BE STUDIED . . . . . . 31
3.1 Chemical Kinetics Controlling Regime--
Isothermal . . . . . . . . . 31
3.1.1 Reactive Species . . 31
3.1.2 Nonreactive Species . . . 32
3 1.3 Weight of Volatiles Collected . 33

iv

Chapter

3.2 Chemical Kinetics Controlling Regime--

Nonisothermal . . .
3.3 Diffusional Limitations for Pyrolysis--
Isothermal . . . . . . . . .

3.3 l Reactive Species . . . .
3.3.2 Nonreactive Species . . .
3 3 3 Weight of Volatiles Collected

3.4 Convection Limitations for Pyrolysis--

Isothermal . . . . . . . . .
3.4.1 Nonreactive Species . .
3.4.2 Reactive Species . . . .
3.4.3 Weight of Volatiles Collected

4. THE EFFECT OF KINETIC PARAMETERS ON "OIL"

YIELD . . . . . . . . . .

4.1 The Effect of 0 on "Oil" Yield . .
4.2 The Effect of E0 on "Oil" Yield. .
4.3 The Effect of k0 on "Oil" Yield.
4.4 The Effect of the Heating Rate on

Oil Yield' . . . . . . . . .

5. PARAMETER ESTIMATES USING DATA FOR WESTERN
OIL SHALE. . . . . . . . . . .

5.1 Estimates of Parameters for the Distri-
bution of Activation Energy Model .

5.2 Estimates of Parameters for a Lumped
First Order Model.

5.3 Prediction of Nonisothermal Pyrolysis
Using Kinetic Parameters Estimated
from Isothermal Data. .

5.4 Prediction of Isothermal Pyrolysis for
Large Particles . . . . .

6. MODEL PREDICTIONS FOR EASTERN OIL SHALE

6.1 Chemical Kinetics Controlling Regime .
6.2 Diffusional Limitations for Pyrolysis.
6.3 Convection Limitations for Pyrolysis

7. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER
RESEARCH . . . . . . . . . . . .

Page

34
36
36
37
38
4O
40
41
43
48
49
51
S4

56

59

59

65

66
7O
75
77

79
79

APPENDICES
Appendix

A. Procedure for Estimating the Devolatilization
Parameters . . . . . . . . . . . .

B. Computer Program for Estimating the
Devolatilization Parameters . . . . .

C. Procedure for Estimating Y and C*(l - e).

D. Procedure for Estimating the Kinetic Para-
meters in the Lumped First Order Model. .

E. Computer Program for Calculating the Volume
of Oil Collected for Nonisothermal Conditions
for the Chemical Kinetics Controlling Regime.
F. Computer Program for Calculating the Weight of
Volatiles Collected as a Function of Time for
Isothermal Convection Limitations . . . .

LIST OF REFERENCES . . . . . . . . . . . .

vi

Page

87

91

97

101

103

107

111

LIST OF TABLES

Table Page

1. Expected Values for the Kinetic Parameters in
the Distribution Model . . . . . . . . 4S

2. Physical and Transport PrOperties for Western
Oil Shale . . . . . . . 46

3. Physical and Transport Properties for Eastern
Oil Shale . . . . . . . . . . . 47

vii

Figure

1.

10.

11.

12.

13.

14.

15.

LIST OF FIGURES

A schematic representation of an oil shale
particle .

The
The
The
The

The

effect
effect
effect
effect

effect

Results of

of
of
of
of

of

g on the "oil" yield at 648°K
0 on the "oil" yield at 775°K
E0 on the "oil" yield at 775°K.
k0 on the "oil" yield at 775°K.

the heating rate on oil yield

parameter estimation at 648°K for the
distribution and first order models . . .

Page

23
50
52
53
55

57

61

Results of parameter estimation at 673°K to obtain

Distribution model with diffusional limtations

at 703°K

Distribution model with convection limitations

at 703°K

y and C*(1 - e). . . . . . . . . .
'The distribution and first order models for a
heating rate of 0.033°K/sec. . . . .

The effect of the heating rate . . . .

Distribution and second order models for the
chemical kinetics controlling regime. . .

Distribution and second order models for
diffusional limitations . . . . .

Distribution and second order model for
convection limitations . . . . . . .

viii

63

67

69

71

73

78

80

82

iR

iBR

iBR

iRco

iNR

iBNR

iBNR

CiNRm

LIST OF NOMENCLATURE

(kg sec}

collection of constants given on page 44 [ 3

heating rate [°K/Sec]

collection of contants given on page 44 [“—§&Q-]
(m sec)

mass concentration of a reactive volatile Species
i in the gas phase of the particle [kg/m3 of gas
phase in the particle]

mass concentration of a reactive volatile Specie?
i bound in the solid phase of the particle [kg/m

of solid phase in the particle]

initial mass concentration of a reactive volatile
species i bound in the solid phase of the particle
[kg/m3 of solid phase in the particle]

mass concentration of a reactive volatile Species
i in the sweep gas far from the particle [kg/m3 of
gas phase in the particle]

mass concentration of a nonreactive volatile Species
i in the gas phase of the particle [kg/m3 of gas
phase in the particle]

mass concentration of a nonreactive volatile Species
i in the solid phase of the particle [kg/m3 of
solid phase in the particle]

initial mass concentration of a nonreactive volatile
Species i bound in the solid phase of the particle
[kg/m3 of solid phase in the particle]

mass concentration of a nonreactive volatile s ecies

i in the sweep gas far from the particle [kng of
gaS phase in the particle]

ix

NR

Roo

NR°°

C*

total mass concentration of all reactive volatile
Species in the gas phase of the particle [kg/m3
of gas phase in the particle}

total mass concentration of all nonreactive volatile
species in the gas phase of the particle [kg/m

of gas phase in the particle]

total mass concentration of all reactive volatile
Species in the sweep gas far from the particle
[kg/m3 of gas phase in the particle]

total mass concentration of all nonreactive
volatile species in the sweep gas far from the

particle [kg/m3 of gas phase in the particle]

total volatile content of the particle [kg/m3
of solid phase in the particle]

heat capacity of the solid phase of the particle
[J/(kg°K)]

diameter of the particle [m]
effective diffusion coefficient [mZ/sec]

activation energy associated with the devolatiliza-
tion reaction liberating species i [cal/g-mole]

activation energy [cal/g-mole]

mean activation energy associated with the Gaussian
distribution function [cal/g-mole]

distribution function for the activation energies
ratio defined in Eq. (4)

convective heat transfer coefficient between the
sweep gas and the particle surface [J/(m2°K sec)]

heat of reaction for each devolatilization reaction
liberating Species i [J/kg]

Arrhenius rate constant associated with the reaction
liberating Species i [sec‘l]

Arrhenius frequency factor associated with the
rate constant ki [sec‘ ]

Arrhenius frequency factor for any reaction i
[sec' ]

k"

iR

iNR

Arrhenius rate constant for the decomposition of

the reactive volatile Species

mass transfer coefficient

[sec-l]

[m/sec]

thermal conductivity of gas [J/(m sec °K)]

thermal conductivity of the solid phase of the

particle [J/(m sec °K)]

typical molecular weight for a reactive Species

[q/g-mole]

typical molecular weight for a nonreactive species

[g/g-mole]

effective mass flux of a reactive volatile Species
i in the particle [kg/(mzsec)]

effective mass flux of a nonreactive volatile

species i in the particle

total effective mass flux
Species i in the particle

total effective mass flux
volatile Species i in the

total effective mass flux
Species at the surface of

total effective mass flux

[kg/(mzsec)]

of all reactive volatile
[kg/(mzseC)]

of all nonreactive
particle [kg/(mzsec)]

of all reactive volatile
the particle [kg/(mzsec)]

of all nonreactive

volatile Species at the surface of the particle

[kg/(m sec)]

number of particles

partial pressure of reactive volatiles in the gas
phase of the particle [psia]

partial pressure of nonreactive volatiles in the

gas phase of the particle

[psia]

total pressure of volatiles in the gas phase of

the particle [psia]

heat flux at the surface of the particle [J/(mzsec)]

intrinsic rate of reaction for a devolatilization

reaction i [k g/ (m3sec)]

radius of the particle [m]

xi

w(t)

W(t)

O.

(l - Y)

ideal gas constant

intrinsic rate of devolatilization [kg/(sec m3 of
solid phase)]

intrinsic rate of coking [kg/(sec m3 of gas phase)]
surface area of a Spherical particle [m2]

time [sec]

uniform temperature of the particle [°K]

initial temperature of the particle [298°K]
temperature of the sweep gas [°K]

velocity of volatile Species in the gas phase of
the particle as they are convected out of the

particle [m/sec]

weight of reactive volatiles collected at time t
[kg]

weight of reactive and nonreactive volatiles
collected at time t [kg]

Greek Letters

 

thermal diffusivity of solid [mZ/Sec]

fraction of C* that represents the total reactive
volatiles content

fraction of C* that represents the total nonreactive
volatiles content

void Space of the particle

 

[m3 of gas phase in the particle

1

m3 of particle
Darcy permeability [m2]

characteristic viscosity of the nonreactive volatile
Species in the gas phase of the particle [kg/m sec]

3.1416....

density of the solid phase of the particle

xii

Da

Nu

Pr

Re

So

Sh

standard deviation of the Gaussian distribution

of activation en

integrating fact

ergies

or

[cal/g-mole]

Dimensionless Groups

R k"

Damkéhler number

Nusselt number

Prandtl number Cp'

Reynolds number

where Um is the superficial velocity of the sweep

gas

Schmidt number

p

Sherwood number

Thiele Modulus V

u
*0

kg

 

Pu
u

 

AB
k

mi!

xiii

D

2R

1 . INTRODUCTION

In view of dwindling petroleum reserves and the
ubiquitous nature of Oil shale deposits, oil from Oil shale
is being considered once again as a source of chemical
feedstock. Although more than one trillion barrels of oil
exist in the eastern oil shale formations in Michigan (Katz
and Goddard, 1964) and another 700 billion barrels of oil
are present in the western formations of Colorado, Wyoming,
and Utah (Lewis and Rothman, 1975), it is unclear whether
significant fractions of this oil can be recovered.

Combustion and hot gas retorting are two methods
currently being investigated as a means of recovering oil
from oil shale. The proposed methods include both above
ground and in Situ processing (Jones, 1976; Lewis and Rothman,
1976). Mathematical Simulations of these large scale com—
mercial Operations are presently being develOped by Braun
and Chin (1977) and Crowl and Piccirelli (1979) to study
alternative Operating strategies. These models necessarily
require an understanding of the rate processes which occur

in individual Oil shale particles under pyrolysis conditions.

Devolatilization of shales has been modeled using
simple kinetic schemes but, unfortunately, this strategy
fails to predict the production of volatile products under
a wide range of pyrolysis conditions. A satisfactory model
has not been developed which incorporates the kinetics of
devolatilization, a mechanism for intraparticle mass trans-
port of volatile products, and the kinetics for intraparticle
product degradation. Therefore, a prototype model will be
developed in this study which describes all these rate
processes during the pyrolysis of Oil Shale particles. The
complex kinetics characteristic of oil shale devolatiliza—
tion will be interpreted using the distribution of activation
energies theory developed previously for coal pyrolysis
(see, eSp., Anthony and Howard, 1976). The kinetic model
will be combined with material balances to describe the
pyrolysis of large particles. For these larger particles,
diffusion and bulk flow will affect the production of

volatile products.

1.1. Physical Description of Oil Shale

Oil Shale is a low porosity material which yields
gaseous volatile products upon heating. Volatiles, other
than water, which condense at temperatures above -78°C are
classified as "oil." The noncondensable products are

typically CO, CO CH4, and C2H6' The nominal void

2! H2!
fraction of eastern oil shale particles is 0.05 (Crowl and

Piccirelli, 1979); the western oil shales typically have

higher porosities on the order of 10% (Tisot and Murphy,
1965).

The solid phase of the particle is composed of
inorganic and organic material. The solid organic phase
contains kerogen, a complex polymeric material. The
structure of kerogen is highly napthenic with aromatic,
nitrOgen, and sulphur heterocylic ring systems distributed
throughout the kerogen molecule (Jones and Dickert, 1965).
When an oil shale particle is heated, the volatile products
are produced by devolatilization reactions which occur in
the solid organic phase at temperatures between 475°K and
775°K.

The inorganic solid phase of western oil shales
contains dolomite, CaMg(CO3)2; calcite, CaCO3; and magnesium
carbonate, MgCO3. At temperatures of 840°K or higher,

Jukkola, et a1. (1953) determined that these inorganic

carbonates decomposed by the following endothermic reactions:

+

CaMg(CO3)2 + MgO + CO2 + CaCO3
CaCO3 + Ca0 + CO2

Mg’CO3 + MgO + C02.

Allred's (1967) eXperiments also Show that the inorganic
carbonates do not decompose if the temperature is below
880°K. These endothermic reactions act as heat sinks in
the retorting process and their presence could potentially

limit the efficiency of a particular recovery strategy.

However, for the temperature range of the pyrolysis of
kerogen (i.e., 473°K to 773°K) the decomposition reactions
of the carbonates will not occur, so the heat sinks associated

with these reactions should not limit the extent of pyrolysis.

1.2. Description of Oil Shale Pyrolysis

AS an oil shale particle is heated the bonds of the
kerogen molecule begin to break. Parts of the molecule are
released as volatile Species into the gas phase of the
particle. These bond breaking processes are the devolatili-
zation reactions which occur in the solid organic phase
uniformly throughout the particle. Reactions producing
certain classes of volatile species may be identified with
a characteristic activation energy associated with the
strength of the chemical bond being broken by thermal energy.
Moreover, the volatiles released into the gas phase may be
classified as either reactive or nonreactive, depending on
their tendency to coke.

The reactive volatiles, associated with "oil" forma-
tion, are tarry, higher molecular weight Species possibly
existing as free radicals (Russell, et a1., 1979). These
reactive volatiles are subject to gas phase coking and
cracking reactions and may deposit as inert char inside the
particle instead of being released as "oil." The nonreactive
volatiles are typically lower molecular weight species such
as C0, C02, CH4, C2H6’ hydrogen, and water. Although these

Species are not subject to gas phase decomposition for

T < 775°K, they do react at higher temperatures. The activa-
tion energies associated with the gas phase decomposition of
the nonreactive species are of the order of 80-104 kcal/g-
mole (Benson, 1968; Murphy, et al., 1958; Palmer, 1963);
whereas, the activation energies for the gas phase decomposi-
tion of the reactive volatiles are of the order of 30-60
kcal/g-mole (Benson, 1968; Murphy, et al., 1958; Palmer,
1963). In typical cracking and coking reactions, additional
nonreactive volatiles are produced when reactive volatiles
decompose (Hirt and Palmer, 1963; Palmer and Cross, 1966;
Hudson and Heicklen, 1968; Heicklen, et al., 1969); however,
in this study it is assumed that the volatile species pro-
duced by the decomposition reactions are negligible.

Upon release from the solid, the volatiles are
transported to the particle surface by convection and
diffusion, but the reactive volatiles are subject to decom-
position reactions. Therefore, the yield is limited by
internal as well as external resistances to mass transfer.
These limitations cause the buildup of reactive volatiles
in the gas phase of the particle which increases the rate
Of gas phase decomposition.

For large particles the rate of volatiles production
may be affected by mass transport processes. Anthony (1974)
points out that very little eXperimental work has been con-
ducted to find this limiting particle size for coal pyrolysis.
It is noteworthy that no influence on the rate of coal

devolatilization has been observed for particle diameters

up to 200 microns. Anthony (1974) states that the onset of
mass transport limitations arises somewhere in the particle
size range of 200-2000 microns. It will be assumed that the
limiting particle Size for oil shale devolatilization will
also fall in this range.

Shih and Sohn (1978) have demonstrated that for
large particles internal temperature gradients within the
particle affect the retOrting process; however, if the
particle sizes are sufficiently small, the particle can be
considered spatially isothermal. This assumption can be
justified by estimating the time needed for a solid Sphere
of radius R and thermal diffusivity a(EkS/pSCp) to approach
a uniform temperature when subjected to a step change at
its boundary. According to Bird, et a1. (1960), this occurs
when oat/R2 a 0.4. Granoff and Nuttall (1977) and Campbell,
et a1. (1977) have measured the following physical properties

for Colorado oil shale:

cp = 1.13 x 103 J/kg °c
p8 = 2.25 x 103 kg/m3
kS = 1.25 J/m sec °C.

Thus, the thermal diffusivity, a, corresponding to these

prOperties is ~ 5 x 10-7 m2/sec. For particle radii ranging

from hundreds of microns (i.e., 10-4

-3

m) to tenths of a
centimeter (i.e., 10 m), the real times corresponding to

the dimensionless time 0.4 are approximately 10—2 sec and

1 sec, respectively. The time scale for devolatilization
in most experiments is on the order of 105 sec; therefore,
regardless of whether the environment of the particle is
isothermal or nonisothermal, the particle will behave as
though it is Spatially isothermal. This will be true for
particle diameters up to approximately one centimeter.

As the solid organic phase of the Oil shale pyrolyzes,
the void Space within the particle will increase as volatiles
leave the solid. For non—coking lignite coal, the particles
maintain their porous structure throught out pyrolysis and
leave behind a very porous structure of ash and char (see,
Russell, et al., 1979). However, coking bituminous coal
particles swell and soften during pyrolysis and volatiles
escape as bubbles. Although the physical structure of oil
shale is comparable to that of coking, bituminous coals, the
prototype model developed hereinafter assumes that the shale
particle maintains its geometric configuration and integrity

throughout pyrolysis.

1.3. Kinetic Models for Pyrolysis

Hubbard and Robinson (1950) conducted isothermal
experiments on samples of oil shale in the form of cylinders
one centimeter in diameter and six centimeters in length.
They determined the amount of kerogen reacted by measuring
the amounts of oil, gas, and bitumen produced as a function
of time and interpreted their data with a simple mechanism

involving two consecutive first order reactions, viz.,

k1. k2.
kerOgen + bitumen + Oil + gas.

Their experiments showed that a carbonaceous residue was
formed as a pyrolytic product. However, no measurements
were made on the amounts of residue formed, and the residue
does not appear in the reaction mechanism.

Allred (1967) was able to calculate the amount of
residue directly from the data of Hubbard and Robinson (1950).
He also conducted nonisothermal experiments in which
the weight of volatiles collected were measured as a function
of time and found that the production of oil and gas occurred
over temperatures ranging from 477°K to 744°K. No further
appreciable weight loss was observed until 880°K where
inorganic carbonates began to decompose giving off additional

volatile Species.

Allred tried to explain ..is data and that of Hubbard
and Robinson using the kinetic mechanism prOposed by Hubbard
and Robinson. However, his measurements of the overall rate
constant showed three distinct regions in which k vs. l/T was

linear. This observation led him to the following

mechanism

k

kerogen +1 gas + bitumen + carbon residue
k

bitumen +2 oill + gas

At temperatures below 744°K, the decomposition of bitumen

was observed to be rate limiting with an Arrhenius activation

energy of 40 kcal/g-mole. Above 744°K the vaporization of
oily liquids became rate limiting with an activation energy
of 15 kcal/g-mole. Jones and Dickert (1965) state that the
activation energies of 45 kcal/g-mole reported for kerOgen
pyrolysis is to be expected for reactions involving carbon-
carbon bond breaking.

Braun and Rothman (1975) observed that the experi-
ments of Hubbard and Robinson were not true isothermal
experiments because of the large particle size. They con-
sidered a kinetic scheme similar to that of Allred's but
introduced a heating up period before retorting started.

No devolatilization occurred until a certain thermal
induction time was reached. Braun and Rothman Observed that
the duration of this initial stage decreases linearly with
increasing temperature. The reaction scheme employed by
Braun and Rothman is

k

kerogen +1 fl bitumen + f3 gas + carbon
k

bitumen +2 f2 oil + f4 gas + carbon

where the stoichiometric coefficients f f f f and

ll 2' 3’ 4,

the initial concentration of kerogen are functions of the

 

pyrolysis temperature. The rate constants kl and k2 are

 

_ -lO,650 cal/ -mole -1
k1 — 14.4 exp ( R T ) , sec
G
k = 2.025 x 1010 -421443 cal/g-mole) e1.

 

exp ( R T , sec

G

10

Although these results are in general agreement with Allred's
eXperimentS in which the decomposition of bitumen is the rate
limiting step at temperatures below 760°K, above 760°K
kerogen decomposition is rate determining. Apparently, the
decomposition of kerogen involves the breaking of relatively
weak chemical bonds, while the decomposition of bitumen
involves breaking much stronger bonds.

The data of Hubbard and Robinson (1950) and Cummins
and Robinson (1972) Show that the weight of volatiles col-
lected as a function of time asymptotes to a value character-
istic of a particular temperature. If the temperature is
increased this asymptote also increases. The upper limit
of this asymptote occurs around 773°K which Allred measured
as the upper limit for oil and gas production. Braun and
Rothman show that the limiting value of the fraction of
kerogen converted to oil is 0.62 and occurs at temperatures
over 748°K.

Because the molecular structure of oil shale is not
well defined, Johnson, et a1. (1975) concluded that it was
unrealistic to describe the mechanism for each chemical
Species undergoing pyrolysis. By lumping groups of similar
products, they were able to arrive at a more complicated

kinetic scheme, viz.,

kl k2 k5 k7
kerogen + rubberoid + bitumen + semicoke + coke
i
* k4 .¥k3 y k6 k8 k9
gas Oil oil heavy oil + oil.
3‘ klO

gas

11

With this assumption they were able to interpret a wide range
of isothermal and nonisothermal experiments conducted with
various particle size shales. Their model contains no tem-
perature dependent coefficients but contains twenty adjustable
kinetic parameters. Such agreement between their model and
experimental data is expected given the number of adjustable
parameters. Furthermore, such a complex kinetic scheme may
not be convenient for describing large scale retorts.

Campbell, et al. (1978) assumed that kinetic experi-
ments could be interpreted as an average of many processes
occurring simultaneously. They postulated that devolatili-
zation of shale could be modeled as a single first order
reaction with an effective activation energy of 52 kcal/g-
mole. As with the two previous simple kinetic models, the
initial concentration of reactive kerogen must be interpreted
as a function of temperature to describe experiments at
different temperatures. In another study Campbell, et al.
(1977) Observed that the heating rate in nonisothermal
experiments also affected the final yield of volatiles.

They postulated that the heating rate affected some intra-
particle mechanism degrading the oil once it had been
liberated from the solid phase.

The bond breakages that occur during pyrolysis
include carbon-carbon, carbon-oxygen, and carbon-hydrogen
bonds. The bond strengths of these three groups depend on
neighboring functional groups but range from 80 kcal/g-mole

to 104 kcal/g-mole. Anthony and Howard (1976) point out a

12

study by Juntgen and Van Heek (1970) where it was demonstrated
theoretically that a set of parallel first order reactions
can be represented by a single first order expression having
a lower activation energy. This phenomena can explain the
lower activation energies, 40-60 kcal/g—mole, Observed in
kinetic experiments. This might also explain the low value
of 19 kcal/g-mole obtained by Cummins and Robinson (1972) in
their experimental study. Many of the qualitative features
of oil shale kinetics have also been observed by researchers
investigating coal devolatilization. Anthony and Howard
(1976) give an extensive review of this work.

Pitt (1962) adapted Vand's (1943) treatment of a
large number of simultaneous parallel rate processes with
differing activation energies to interpret coal devolatili-
zation experiments. In a refinement of Pitt's theory,
Anthony and Howard (1975, 1976a, 1976b) described the
devolatilization of the solid organic phase of coal as an
infinite set of simultaneous, parallel, irreversible, first
order reactions. The intrinsic rate of reaction for one of

the reactions can be described by

ri = -kiCiB'

These reactions are all first order in the mass concentration
of unreacted material and have rate constants of the

Arrhenius form

1“
J-.J

v-n
_"‘

ki = floi exp(§gf).

The Arrhenius frequency factors, koi’ for each reaction i

can be estimated from transition state theory to have a
value of 1013 sec—l, provided the entropy of activation is
close to zero (Anthony and Howard, 1976; Benson, 1968).
The activation energies for all the reactions
represent a continuous distribution characterized by some
distribution function. Anthony and Howard assumed a
Gaussian distribution characterized by the parameters E0,
the mean activation energy, and f, the variance of the
distribution. If the frequency factor is the same for
all the reactions, then only three, intrinsic, adjustable,

kinetic parameters are introduced: E 0, and k0. If a

o’
more complex kinetic scheme is used, as in the Johnson,

et al., model, two more adjustable parameters are required
for each new reaction proposed. By using the distribution
of activation energies approach and assuming a Gaussian
distribution, only one additional parameter, c, is intro-
duced, although the number of reactions in the kinetic
scheme is infinite. The distribution of activation energies
model retains the complexity of many simultaneous first
order reactions yet keeps the number of adjustable para-
meters to a minimum.

The theory is able to predict the weight loss at low

temperatures since the reactions which produce volatiles at

14

these temperatures have characteristically low activation
energies. As the temperature is increased, the additional
yield of volatiles is again predicted. Volatiles released
at higher temperatures are produced by reactions which have
much higher activation energies than the mean, so these
reactions occur extremely Slow at low temperatures.

Hill, et a1. (1967) observed that the density of the
oil produced in pyrolysis eXperiments increased with tempera-
ture due to increasing amounts of higher molecular weight
Species produced at higher temperatures. This experimental
evidence provides additional motivation to develop a theory
of Oil shale pyrolysis similar to the one already developed
for coal. Basically, the idea is that higher molecular
weight volatiles are bound more tightly to the solid and are
liberated by reactions having high activation energies;
whereas, the lower molecular weight volatiles are produced
by reactions which have low activation energies. Because of
the complex nature of kerogen, a continuous distribution of
reactions characterized by different activation energies
occurs.

The value for the initial concentration of volatiles
in the distribution of activation energies theory represents
the ultimate volatile content of the oil shale. This is in
contrast to the initial concentrations in the Simpler models
which represent apparent volatile concentrations for a

particular experiment. Anthony and Howard (1976a)

15

successfully used this approach to predict the weight of
volatiles collected as a function of time for non-coking

lignite coals.

1.4. Objectives of This Research

The kinetic model to be developed in this study uses
the distribution of activation energies theory develOped by
Anthony and Howard and others. By comparing the kinetic
model to experimental data for powdered Oil shale samples
where no mass transfer limitations exist, the parameters of
the model can be determined for a particular shale. The
kinetic model can then be used to predict weight loss as a
function of time in cases where intraparticle and inter-
particle mass transfer limitations are important.

Simple kinetic approaches are not able to predict
experimental data for a wide range of eXperimental conditions
without the use of temperature dependent initial concentra-
tions and stoichiometric coefficients. Anthony (1974) was
able to overcome these problems in his coal devolatilization
study by using the distribution of activation energies
theory. Because the pyrolysis behavior of oil shale is
similar to that of coal, it is natural to assume that this
approach will also apply to oil shale.

Anthony (1974) did not extend his theory to situa-
tions where mass transfer limitations are important.
Although Russell, et a1. (1979) developed a model for coal

pyrolysis which included intraparticle mass transfer as well

16

as gas phase decomposition, they did not employ the distri-
bution of activation energy kinetics as developed herein-
after.

In the approach develOped here, the heating rate
enters the problem eXplicitly through the boundary condition
for the temperature. This is in contrast to the approach of
Campbell, et a1. (1977) where the heating rate appears
implicitly in the kinetic rate parameters. It is anticipated
that the appearance of the heating rate in the boundary
condition will allow for more accurate model predictions

over a wider range of experimental conditions.

2. A MATHEMATICAL MODEL FOR THE PYROLYSIS

OF OIL SHALE

The porous oil shale particle will be treated as an
effective homogeneous mixture of organic and inorganic
compounds. Each devolatilization reaction which liberates
a volatile Species i occurs homogeneously throughout the
particle. Because the porosity of the particle is very
small and the rate of devolatilization is slow compared to
the intraparticle mass transport, the pseudo-steady state
assumption will be applied to the gas phase material balances.

The mechanism for intraparticle mass transfer
includes both diffusion and convection. The diffusive flux
will be described by Fick's law of diffusion characterized
by an effective diffusion coefficient, and the convective
flux will be described by Darcy's law. A resistance to mass
transfer will also occur between the gas surrounding the
particle and the surface of the particle. This resistance
occurs across an imaginary film surrounding the particle
and is characterized by a convective mass transfer coeffi-
cient. For very small particles, intraparticle mass

tranSport occurs extremely fast, so the concentration of

17

18

Species i will be the same everywhere in the particle.
Therefore, the only resistance to mass transfer will occur
across the imaginary film surrounding the particle.

A finite resistance to heat transfer will also occur
across this imaginary film. This resistance to characterized
by a convective heat transfer coefficient, which will be
estimated from correlations found in Bennett and Myers
(1962) for a sphere suspended in a gas stream. An estimate of
the mass transfer coefficient will be made using the Colburn
analogy (i.e., jm = jh).

The temperature of the particle is approximately
uniform (see, Section 1.2) for particle diameters less than
one centimeter. However, the endothermic heats of reaction
for the devolatilization reactions will act to lower the
temperature of the particle relative to the surroundings.
Hence, the temperature of the particle will be determined
by the amount of heat transferred between the particle and
its surroundings and the heat removed by the pyrolysis

reactions.

2.1. Devolatilization Kinetics
The intrinsic rate of reaction for Species 1 under-

going devolatilization can be described by
r. = ki C. (1)

where

19

-E.
_ 1
ki — kO exp(§—T). (2)
G
C. is the mass concentration of species i bound to the solid

1B

(i.e., mass of Species i per unit volume of solid phase).

Eqs. (1) and (2) can be combined to give

.3.
_ ___1 o
ri — kO eXp(RGT)g(Ei,t) CiB (3)
where
c.
g(Ei,t) s 7:31 . (4)
C13

The ratio in Eq. (4) will be determined later by a material
balance over the solid phase of the particle.

The constant CIB represents the initial mass con-
centration of species i which decomposes by a reaction
having an activation energy between Bi and Bi + AEi. The
probability that a reaction, which produces Species i, has

a certain activation energy is represented by f(Ei)AEi.

Thus,

0 _
CiB — C* f(Ei) AEi (5)

where C* is the total initial mass concentration of volatile

species that decompose. Because ZCIB = C*, the density
1
function must satisfy
I f(Ei) AEi = 1. (6)

l

20

The total intrinsic rate of reaction is obtained by
summing the contributions from all species i. Eqs. (3) and
(5) imply that

-Ei
‘2‘ - * .____.
a r. - k C E exp(R T)g(Ei,t)f(Ei)AEi. (7)
J. 1 G
If the number of different activation energies in the

reaction set is very large, then the sum appearing in

Eq. (7) can be calculated as an integral with the result

that
z: r = R = k c* Jbexp’i)g(E t)f(E)dE (8)
. i ' o a ‘R T '
l G

where the limits of the integral in Eq. (8) define the

interval over which f(E) is defined.

Eq. (8) is the total intrinsic rate of devolatiliza-
tion for all volatile species. To describe the intrinsic
rate of devolatilization for the reactive volatile Species,
Eq. (8) is multiplied by y, the fraction of C* which yields
reactive volatiles. Similarly, the intrinsic rate of
reaction for the nonreactive species is given by Eq. (8)
multiplied by (l-Y).

The density function f(E) for the activation energies
is unknown. In this study it is assumed that EO>>0 and that

o<<EO so a Gaussian density can be used:

2
'(E-EO)

exp 2

f(E)= 20
/2? c

(9)

21

Anthony (1974) was successful in using Eq. (9) to interpret
coal devolatilization experiments. It is noteworthy that
this model requires only one more parameter than the simple
lumped first order kinetic model, yet the number of reactions
being considered is infinite. The devolatilization para-
meters, which do pp; depend on the temperature, include a
mean activation energy E0, the variance 0, and a frequency
factor k0. The parameter‘rmay be viewed as a stoichiometric
coefficient. One of the objectives of this thesis will be

to estimate these parameters from pyrolysis data on western

oil shale.

2.2. Coking Kinetics
The intrinsic rate of decomposition of a reactive
Species i in the interstices of the particle can be described

by a single first order irreversible reaction

r . = k" C.

Ci iR' (10)

CiR is the mass concentration of reactive volatile species

(i.e., mass per unit volume of gas phase). This type of
expression is consistent with the studies of Murphy, et a1.
(1958); Palmer and Cross (1966); and Hirt and Palmer (1963).
The coking constant k" is assumed to be the same for each
species i. This rate constant was determined by Campbell,

et a1. (1977) as

1

g-mole) , sec- . (11)

n _ 7 -35 kcal/
k - 3.1 x 10 exp( RGT

22

The summation of all reactive species i in Eq. (10) gives

the total decomposition rate,

RC = k CR (12)

where CR is the total mass concentration of reactive species.

2.3. Material and Energy Balances

2.3.1. Material Balance for the Reactive Volatile Species
in the Gas Phase of the Particle
A mass balance for a reactive volatile Species i in

the gas phase of a shale particle is

.9..-

at (CiRE) + g (v-Ilir) = Yri(]_-E) - r .8. (13)

Cl

air is the intraparticle mass flux of reactive volatile
Species i and s is the void fraction of the particle.
Figure 1 gives a schematic of the physical situation and
helps to define some of the notation.

If Eq. (13) is summed over all species i, then for

pseudo-steady state

V°§R = (iii) Y R - RC. (14)

Using Spherical coordinates with p varying in the radial

R
direction only, Eq. (14) becomes

) = ‘ Y R - R (15)

 

where n denotes the radial component of p

R R'

24

The mass flux of reactive volatiles with respect to
stationary coordinates is defined by Bird, et a1. (1960) as

dx

nR = -QDe_dE + xR (nR + nNR). (16)

w

nNR is the radial component of the flux of nonreactive

C
Species. With XR = —%, Eq. (16) becomes for p: constant
dC C
_ .. .33 .3.
DR — Dedr + o (nR + nNR) (17)
Substituting pvr = nR + nNR’ Eq. (17) becomes
dCR
nR = “pea?- + C RVr. (18)

De is an effective diffusion coefficient and vr can be

related to the pressure by Darcy's law

(19)

'1

CIA
(LID:
H. "U

Eq. (19) is a constitutive equation which describes
the velocity of a fluid flowing through a porous media. K
is the permeability of the porous media, u is the viscosity
of the fluid, and the gradient of the total pressure is the
driving force for flow. The total pressure is equal to the

sum of the partial pressures of the nonreactive and reactive

Species, that is, P = P + P . Assuming ideal gas behavior,

NR R

the total pressure is related to the concentrations by

c c
p=RGT(—3+—153). (20)

MNR

25

Direct substitution of Eqs. (20), (19), and (18) into Eq.

(15) results in a nonlinear differential equation for CR.

However, if a nonreactive species is typically of lower

C C
molecular weight than the reactive Species, then —E§ >> MB’
R R

and the total pressure is independent of CR’ Therefore,

RTC
p~—§——§3 (21)

_ MNR

This simplification uncouples the material balances for C

R
and CNR'
The boundary conditions for Eq. (15) are
dC
R -
dE_ r=o _ 0 (22a)
anr=Rp = kg (CR‘r=Rp ’ CROO) ' (22b)

CR is the mass concentration of reactive species far from

the particle and k9 is a convective mass transfer coeffi-
cient. For situations where a sweep gas flows past the
R;*0.

The first boundary condition specifies symmetry for

particle, C

the concentration profile at the center of the particle.
The second boundary condition requires continuity of mass
flux at the surface of the particle. The mass transfer
coefficient will be taken to be a constant independent of
concentration and temperature. kg will be calculated by
using the Colburn analogy (see, p. 646 in Bird, et al.,

1960)

26

j = Nu Re'lpil/3 = j = Sh Re'15c‘l/3.
H In
An estimate of Nu can be made using the following correla-

tion (see, p. 386 in Bennett and Myers, 1962)

Nu = 2 + 0.6 (Pr)l/3(Re)l/2.

This correlation was developed for a sphere suspended in
a gas stream. The sweep gas will be assumed to be nitrOgen.
As an approximation, the diffusivity of methane in nitrogen
is used to calculate Sh and So. The transport prOperties
of the nonreactive volatile Species appearing in Eqs. (19)
and (21) will also be assumed to be those of methane. For
powdered shale particles, Nu== 2 will be employed.
2.3.2. Material Balance for the Nonreactive Volatile Species
in the Gas Phase of the Particle
A mass balance for a nonreactive volatile species i

in the interstices of the particle is

8(C a)

 

iNR - .
at + e(v EiNR) - (1 - {)(l e)ri (23)
where EiNR is the intraparticle mass flux of nonreactive

volatiles. Once again, if Eq. (23) is summed over all

nonreactive species, then

V'BNR = il—g—El (1 - v) R . (24)

provided pseudo-steady state applies. In spherical coordi-

nates, Eq. (24) is

27

1.,

r

2 _ - _ I
(r nNR) - ———:—- (l x) R. (25)

(LID;
H

The intraparticle mass flux of nonreactive Species includes
both diffusion and convection

dC

__ NR
r1NR De dr + CNRVr' (26)

 

v is given by Eqs. (19) and (21) as before.

r
The boundary conditions for Eq. (25) are
dC
NR _
dr Ir=o - 0 (27a)
nNer=Rp _ kg (CNer=Rp - CNRw)' (27b)

The mass transfer coefficient for the nonreactive species
is estimated in the same way as for the reactive Species.
CNRmis approximately zero, if a sweep gas flows past the

particle.
2.3.3. A Material Balance for the Volatile Species Bound
to the Solid Phase
A material balance for species i bound to the solid
phase of the particle is

d(CiBVp(l - s)) -E

= - __£ - r
dt koexp(RGT)CiBVp(l -). (28)

 

If the volume of the particle and the void fraction are

constant, then Eq. (28) is

28

 

dCiB 'Ei
dt = ‘ko eXp(RGT) Cis‘ (29)
Thus,
c. -E.
—£§ = eXp[-k ft exp(-—$)dt] (30)
o o R T
ciB 0 G

which defines g(Ei,t) appearing in Eq. (4).

2.3.4. Material Balance for the Weight of Volatiles Collected
The weight of volatiles collected as a function of

time from an arbitrary sample of oil shale particles can be

described by performing an overall material balance around

all the particles. This yields

dW _
5E ‘ Sp€(nRs + nNRS)’ (31)
The surface area of a particle is S (i.e., 4nRé), p is the

number of particles, n and nNRS are the mass fluxes of

RS
reactive and nonreactive Species at the surface of the

particle, and W is the total weight of volatiles collected.

2.3.5. Energy Balance

The energy balance for the particle is

d(vppstT)
dt = qS - R]AHIVp(l - e) (32)

 

where Vp is the volume of the particle, Cp is the heat

capacity of the solid phase, ps is the density of the solid

29

phase, T is the uniform temperature of the particle, and
[AH{ is the endothermic heat of reaction. The heat flux at

the surface of the spherical particle is q and equals

q = h(Tg - T) (33)

where h is a convective heat transfer coefficient and Tg
is the temperature in the gas far from the particle. The
heat transfer coefficient in Eq. (33) is assumed to be
independent of concentration and temperature.

For pseudo-steady state, Eqs. (32) and (33) imply

that
h(T ~ T)S = RIAHIV (1 - e), (34)
g p

which yields

RIAHXR (1 - s)
T = T - P
9 3h

 

(35)

where Rp is the radius of the particle. Eq. (35) shows
that the temperature of the particle will differ from that
of the gas depending upon Rp and h. The magnitude of this
difference can be investigated by calculating the second
term on the right hand side of Eq. (35) for large and small
particles.

The heat of pyrolysis for Colorado Oil shale is 335
kj/kg (see, Campbell, et al., 1977), and the porosity is 10%.
With Nu = 2 and kga§=0.03 J/(m sec °K), h=65 J/(m2 sec °K)

for 0 = 8 x 10‘4m; and, h = 4 J/(m sec °K) for 0 = 0.0127 m.

30

These particle diameters were used in experiments to

R
be investigated later, so the values of giH| are 0.7

3 3

(m sec °K)/kg and 17 (m sec cI)/kg.

For isothermal experiments, R is a maximum at t = 0
and decreases as the reaction progresses. In nonisothermal
experiments, R increases to a maximum and then decreases to

zero. It will be assumed as a first approximation that the
R|AH|R (l-E)
P
3h

to Tg' If this is the case, then the temperatures of the

particle and the sweep gas will be approximately the same.

 

values of in Eq. (35) are small compared

This assumption will be verified a posteriori when the
values of E0, 0, kc, and C* are determined and values for
R are calculated.

For nonisothermal experiments the heating rate of
the particle enters the problem through T9. The heating
rate does not affect the intrinsic kinetic rate parameters
as postulated by Campbell, et a1. (1977) but enters the rate
expression through the eXplicit temperature dependence of

k..
1

3. SPECIAL CASES TO BE STUDIED

3.1. Chemical Kinetics Controlling Regime--Isothermal

The equations develOped in the previous section can
be solved for the situation where intraparticle mass
tranSport occurs much faster than chemical kinetics. These
cases are typical of small particles, and the concentration
of volatile Species in the gas phase of the particle will

be the same everywhere in the particle.

3.1.1. Reactive Species
Integrating Eq. (15) and applying the boundary con-

ditions (22a) and (22b) yields

 

 

2 (l ' F) YRRP: 12.939:
Rp kg(CR-CRm) = s 3 - 3 . (36)
Solving Eq. (36) for CR gives
R
_p (1 - e)
c = 3 Y __-E__- R + kgCRm (37)
R k + R °
9 32 k"

Substituting Eq. (37) into Eq. (22b) gives the flux of

reactive species out of the particle

31

32

Rp -—-———Y(l - E) R — R k"CR°°

= p
nRs 3 + Da (38)

 

where Da is the Dakahler number

Da = —E—— . (39)

The Dakahler number is the ratio of the rate of gas phase

decomposition to the rate of interphase mass transfer.

3.1.2. Nonreactive Species
Integrating Eq. (25) and applying the boundary

conditions (27a) and (27b), results in

2 (l-€)(l-‘{)RR3

- = p
Rp kg(CNR CNRm) 3s (40)

 

Thus,

 

(1 - a) (1 - Y)
NR ' 38 k RRp + CNRm (41)
Substituting Eq. (41) into Eq. (27b) gives the flux of

nonreactive Species out of the particle

1
[11 - a) (l - Y)RRP;

NRS =| 3e J'

 

n (42)

Note that the flux does not depend on the external resistance
to mass transfer. This is a consequence of the pseudo-steady

state assumption and the absence of coking reactions. As

the external resistance to mass transfer becomes very small

(i.e., kg+m, Da+0), CR-rCRoo and CNRICNRw' As the external
resistance increases (i.e., kaeo, Da+m), CR+0 since R goes
to zero at long times. This indicates that all reactive
volatiles will eventually go to coke if not transported out

of the particle. If k"=0, then Eqs. (37) and (38) reduce

to Eqs. (41) and (42), respectively.

3.1.3. Weight of Volatiles Collected

Substituting Eqs. (38) and (42) into Eq. (31) gives

P

 

dw i ‘1’ Vppe k "CR“:
3? ‘ Vp‘”l ‘ ”’)(i‘¥‘5a) + (l ‘ I) R + Da ° (43’
L 3— (l + 3“)

Eq. (43) describes the rate of volatiles collected as a
function of time and is applicable in both isothermal and
nonisothermal situations. The only difference will be the
form of g(E,t) as given by Eqs. (4) and (30).

Substituting Eqs. (30), (9), and (4) into Eq. (8)

gives for the isothermal case

k c* +w
O

 

 

-E ~15:
R = f exp(——-) exp[-k t exp(—-)) x
/2Ro -m RGT O RGT
'(E - EO)2
exp[ 2 ] dB. (44)
20

When CR3? 0, Eqs. (44) and (43) yield

34

 

 

 

 

gig L._ 5 Da + (1 - y) x
v/Z’T 7 '(l + T)
f exp(-——) exp[-k t exp(-——)] exp[ ]dE.(45)
_w RGT o RGT 202
Integration of Eq. (45) over time gives
W(t) = v p(1 - e)c*[———Y——— + (1 - «M x
p Da
(1 + 3—)
i 1 +m -E -(E - Ho)2
1 - I__ f eXp[-k t exp(-——)] exp[ ]dE (46)
o R T 2
/2w O -w G 20

Eq. (46) describes the total weight of volatiles collected
as a function of time for isothermal situations where
chemical kinetics controls the production of volatiles.
The weight of "oil" collected is given by Eq. (46) without
the term (1 - y) and will be represented by the variable

w(t).

3.2. Chemical Kinetics Controlling Regime-~Nonisothermal
Many experiments are designed so the temperature

far from the particle increases linearly with time, i.e.,

T = TO + bt. The parameter b is a constant heating rate.

Substituting Eqs. (30), (9), and (4) into Eq. (8) gives

35

 

 

 

 

 

 

 

 

 

k0c* +m E t _ -(E-EO)2
R = f expf—-¥:<———] exp(-k f exp[-—fi———-dt) exp[————————jdE.
R m
VERO -m G(Toot) O RG(igbti 20(2
(47)
IfR———:§—— = x, then
C(To+bt)
-E
kOC* +co _E -k E RG(TO+bt) ex (x) -(E-EO)2
R = f “MFR—67:7] exp[ R b x 412—31") exp[“z—ldfi-
/2F0 -m G o G -E x 20
RGTO (48)
Recoqnizing that TO + bt = T
-.:32
R T _,_ 2
koc* +m -E -koE G exp(x) (L E )
R = f exp(§f¥)eXp[ R b .f 2 dx] exp[ 2 ] dE.
/§;J -m G G -E x 20
RGTO (49)
Integrating over x gives
kOC* +w _E -(E-EO)2
R = _ £00 exp(E—f) exp[ 2 ] X
/2fl0 G 20
- exp(:§*)
-koE -E -E RGT
EXP R—'b_‘ [E 1(R—Tf)‘ ELI-(W) "E +
G G G o (-—)
R T
- G
-E
eXP‘R—GT"
-E ] dB. (50)
( )
RGTO _

where Ei(') is the exponential integral function and T0 is
the initial temperature of the particle. Substitution of

Eq. (50) into Eq. (43) with CRm=O yields

36

 

dw __ L. Y _ V 3
5E; - Vpp(l -) [(l+2§)+ (1 g)] b (5].)
3
where R is defined by Eq. (50). Eq. (51) can be integrated

numerically to obtain the weight of volatiles collected as
a function of temperature for nonisothermal situations
where chemical kinetics controls the rate of volatiles
production. The weight of "oil" collected (Ew(t)) is Eq.

(51) without the term (1 - v).

3.3. Diffusional Limitations for Pyrolysis--Isothermal

For very short times or for very low pyrolysis tem-
peratures, the concentration of reactive and nonreactive
volatiles in the interstices of the particle will be rela-
tively small. If the mass concentration of nonreactive
volatiles is small, the total pressure given by Eq. (21)
will be small. If this pressure is low enough so that Cer

and CNRV in Eqs. (l8) and (26) are small relative to
dC r chR
De-EE and De d , then the dominant mechanism for intra-

r r
particle mass transport of volatiles will be diffusion.

 

3.3.1. Reactive Species
For intraparticle diffusional limitations, Eqs. (15)

and (18) can be combined to give

dCR

e dr

(1 - a)

(I'D )=-——€——YR-R. (52)

~13.
(LID:
H

H

the general solution of Eq. (52) is

37
C ‘? C ‘7 _ - v

c =--]-'-Coshx/}5-r+-—25inh /-;—-r+-(—];———EZ--‘-—R. (53)
r e r

R
e

Applying the boundary conditions Eqs. (22a) and (22b) results

 

 

 

 

in
:{k R 2(1 — C)R _ _
S;E - _ R acc ’3:;r
C = )0“; P QR“ Sinh / DE + “/(l _ E) R
R D [Sinh¢-¢cosh¢]-k R sinh¢ r *———;:g
e g p
‘ ~J (54)
where
.111
$ = v De Rp. (55)

Substitution of Eq. (54) into Eq. (22b) gives the flux of
reactive volatile species out of the particle.

- q

(ILLEZ_EL R - k" CRm)R
— 53 (56)
RS $2

Da - [l - ¢ coth @]

- J

 

 

 

 

For ¢+0 (i.e., De+w), Eq. (56) reduces to Eq. (38) for the
case of no intraparticle mass transfer limitations. By

applying L'Hospital's rule, it is simple to show that

lim o2
¢+0 (1 - ¢coth¢)

 

=-3

3.3.2. Nonreactive Species
For intraparticle diffusional limitations, Eqs. (25)

and (26) can be combined to give

38

 

 

(r21) NR) = (l " ”fl " S) R. (57)

8‘)“
(D
D.)
H

:l
2
r

The solution of Eq. (54) subject to the boundary conditions,

Eqs. (27a) and (27b),is

F 2 2

(l - ‘rHRp -r ) (l - Y)R
NR ‘ 60 3k
e g

 

 

 

 

 

_ J (58)

Substitution of Eq. (58) into Eq. (27b) gives the flux of

nonreactive volatile species

- . - a R
(l ()(l ) P

nNRs = 35 R' (59)

 

Eq. (58) shows that the concentration profile is affected
by diffusional limitations; however, Eq. (59) is identical
to Eq. (42) owing to the pseudo-steady state assumption.

As k"+0 (i.e., ¢+0 and Da+0), Eq. (56) and Eq. (59) are the

same except for the parameter y. Thus,

n=n +n

RS + (l - €)RpR/3€.

NRS

3.3.3. Weight of Volatiles Collected
Eqs. (56) and (59) can be substituted into Eq. (31)

with the result that

 

 

. 3V pek"C m
3% ‘ VP? (1 - E) 3‘2 + (l - Y) R + -p 2R
Da-(-——$———-) Da—(——¥g———.)
_ l-¢coth¢ l-¢coth¢

(60)

39

Eq. (60) describes the rate of volatiles collected as a
function of time for both isothermal and nonisothermal
situations where diffusional mass transfer limitations
exist. For isothermal cases the total intrinsic rate of
devolatilization is given by Eq. (44). Substitution of Eq.

(44) into Eq. (60) and assuming CRufO yields

 

 

 

 

 

 

‘ 1
V l - €)k C* V
w - pp( _ O 3! + (l __ Y) X
“it (‘2'?) v Dal—(J1..-)
E l-ocotho
L -
r+w -E -E '(E-EO)2
exp(———) exp[-k t exp(~—~)] exp[--—-~—]dE.
RGT o RGT 202
(61)
Integrating Eq. (61) over time gives
_ - * 3Y .
W(t) - Vpp(l - c)C 2 + (1 - f) X
_ Da-(l-¢coth$)
l +m -E -(E"Eo)2
l - I__ f exp[-kot exp(E—T)] exp( 2 )dE
/2n 0 -m G 20
L (62)

 

 

Eq. (62) describes the total weight of volatiles collected
as a function of time for isothermal situations where
diffusional limitations affect the production of volatiles.
Note that the only difference in this result and Eq. (46),
which describes the situation of no intraparticle mass
transfer limitations, is the constant multiplicative factor

containing o. The weight of "oil" collected is Eq. (62)

v.

92

Ct
\ufi

40

without the term (1 - Y) and will be represented once again

as w(t).

3.4. Convection Limitations for Pyrolysis--Isothermal

For high pyrolysis temperatures, the concentration of
reactive and nonreactive volatiles in the interstices of the
particle will be large. Therefore, the total pressure
within the shale particle relative to the surroundings may
be high enough to cause the convective terms in Eqs. (18)
and (26) to dominate the diffusive terms. Thus, intra-
particle mass transport of volatiles will be controlled by

convective processes.

3.4.1. Nonreactive Species

For intraparticle convection limitations, Eqs. (25)

and (26) can be combined to give

(1 - a)

(T) A

1 ‘ “R. (63)

Substituting Eqs. (19) and (21) into Eq. (63) and inte—

grating the result yields

(' .-

2
u (l - Y)(l - €)R A

NR
1 3K€RGT

- 1

where Al and A2 are arbitrary constants of integration.

(64)

 

 

 

Applying the boundary conditions to Eq. (64) gives

38k + cNRoo +
g .

L. -J

 

2
_ { (l y)(l €)RPR
NR

 

41

 

 

uMN (l - Y)(l - €)(R 2 - r2)R) %
R p (65)
3K€RGT
Substituting Eq. (65) into Eq. (27b) gives the flux of
nonreactive species out of the particle
(l - Y)(l - €)R R
WNRS = 36 p ° (66)

Eq. (66) is the same as Eqs. (42) and (59), because coking
is ignored and pseudo-steady state is employed.
Substitution of Eq. (66) into Eqs. (19) and (21) gives

the velocity profile within the shale particle

 

 

 

 

(1 - y)(l - €)r R (1 - ()(1 - s)RER 2
Vr = 36 38k + CNR +
L g
2 2 )‘%
MkRU.-€)U.-y)mp-r)Ri
.- (67)
3<cRGT :
Note that at the surface of the particle, if CNROO is zero,
v = k .
r 9

3.4.2. Reactive Species
For intraparticle convection limitation, Eqs. (15) and
(18) can be combined to give

= Y(l ‘55)R - RC. (68)

(r2C

 

er)

0110-:
'H

L
r2

By introducing the variable, f(r)5r2vrCR, Eq. (68) becomes

42

 

 

n . _ A 2
dr v E
r
Eq. (69) can be solved by using the integrating factor
‘ : Fr E:
8 _ exp[a) Vr dr]. (70)

The lower limit on the integral in Eq. (70

and does not have to be specified.

) is arbitrary

Substituting Eq. (67) into Eq. (70) results in

 

 

 

 

 

 

 

 

)(r) -.= exp[H(r) - H(a)] (71)
where
7—7 ——'2'
_ 3k"€/z+yl: 3 Rue/2 »/z+yr - fi
H(r) : (l.- )(1 - €)R + i (].-‘\)(1 - t)R ln I / 2 I
Y { vz+yr + /E
- (72)
(l - 7) (l - €)R R )2 uMNRu - Y) (1 - €)R 2R
z 2 l - p + C i + AP '
. 3tk NRm| 3K€R T
a g - G
L j
(73)
and
-M%RU-7Hl-'UR
y E 3K€R T ' (74)
G
If CNRm=O and kg+w, Eq. (73) becomes
MN (1 - “()(1 - e)R 2R
z E R p (75)
3K€RGT

)

Now, if Eq. (69) is multiplied by W and integrated, then

I

I

43

-H(r)
R = )(1 ‘ EéRe f e 3 df. (76)

I‘VE O
r

 

The boundary condition at the surface of the particle is

l _
nR'r=R - VrCR!r=R (77)
P P

Substituting Eq. (76) into Eq. (77) gives the flux of

volatile species out of the particle.

 

R \
Y(l - €)R f p eH(1:‘) 2

nRS = 2 r dr. (78)
R e o
P
Note that at r = Rp, eH(Rp)=l, and
2 %
H(r) 3k"€(z+yr )

 

(1 - Y)(l - €)R] X

.3- kn€(z);5
2 (l ' ‘1")(1 - €)R

 

 

(79)

3.4.3. Weight of Volatiles Collected

Eqs. (66) and (78) can be substituted into Eq. (31) to

give

dW 2 (l " N{)(1 ' €)RPR + ‘1’(l -€)R

38 R 28

P

 

X

R A
)9 eH(r)f2df]
O

(80)

44

Substituting Eqs. (74), (7S), and (79) into Eg. (80) and

 

 

 

 

making the change of variable fir/RP, Eq. (80) becomes
dw ‘ .1 [ Be
8? = Vpp(l - e) (1 - HR + 3-,'Vpp(l - :WOJ eXp L'fi—J‘E’fi X
Fr 1.2 1.7 21 1 B: L5
; . _ - _ ___« ‘,
(1-E2)]2 x ' (AR[l-r2]) - (ARLj) ) 4[(1 - 5:2]
' _ ’5 u 1' - _
- ’ (AR[l-r2]) + (AR) g ' rzdr (8l)
.- ~ 1
where
MN (1 - wR 2
A s R P (82)
3KR T
G
"2‘ 2
B : 3k LMNRRp (83)

 

KRGT(l - y)°

For isothermal cases, the total intrinsic rate of devolatili—
zation is given by Eq. (44).

Eq. (81) can be integrated numerically to obtain the
total weight of volatiles collected as a function of time
for isothermal situations where convection limitations
affect the production of volatiles. Eq. (81) gives the rate
at which oil is collected provided the term containing
(1 - y) is deleted.

Table 1 lists the expected range of kinetic para-
meters for Colorado Oil Shale and Table 2 gives the values
of certain physical and transport prOperties used in subse-
quent model calculations. Table 3 lists typical physical

and tranSport prOperties for eastern oil shales.

45

Table l.-—Expected Values for the Kinetic Parameters in the
Distribution Model.

 

Parameter Expected Range

 

13 14

Arrhenius frequency factor, k0 lO - lO , sec-l

Mean activation energy, E 40 — 6O kcal/g-mole

O

Variance of the mean, 0 O - 20 kcal/g-mole

 

 

46

01!! " .‘I .‘II I n- tli‘-‘ u -III'UII-I.III"-A1II .IO' I 1.0.. It- II. ‘nln u'ntl'l"‘lll ill}! -1“... -Q“!!'l"|-l.l.

m

owm\E m. - mo. x .ucoMOmemoo ummmcmuu mmmz

Amamsvxn ow . v z .ucmfloflmmmoo ummmcmuu “mm:

Aonmav chHmuwuumm owm\me oHnoa x a ma .ucmfiowwmwoo cofimDMMHc w>euommwm

Annmav Hanuusz cam mwocmuo AUmmnmEV\o mN.H mx .wmmza cflaom mo >ua>wuozccoo Hmsumne

mmfluquOHm uuommcmue

Annmav Hamuusz com wwocmuo ms\@x mmm *0 .ucmucoo waflumao> mumEHuH:
Amhmav cmenuom can csmum No.0 >.Hao ou Umuum>coo cmmouox mo cofluomum
Anwmav .Hm um .Hawnmemo mx\hx mmm :4 .:0Huommu mo pom:
Ammmav Haaouwooflm can Hzouu we macoa x vmm.a Amamcm cumummmv y .>uHHHnmmEHmm
Amomav snaps: can acmfle oH.o o .suwmouom
Ashmao .Hm um .Hamnmsmo ms\mx moH x mm.m ma .mmanm oflHOm mo suflmcma
Annmav Hamuusz can mmocmuu Ava mxv\hmoa x MH.H mo .ummn owwflowmm

mofluummoum Havamwnm

 

umumfimumm
oozoumuwm 05Hm> HmcflEoz

 

 

I! II .-'..l|. ‘ll I‘l‘ .iv "li‘ a- .I.yulll. . I .‘I‘I .0 II .51- .-| I’I. .III

.mamnm HHO cuvummz “Om mmfluuomoum uuommcmue can HmOflmxnmtl.N mange

47

 

m

oom\E mo. 1 mo. x .ucmwOfiwwmoo ummmcmuu mmmz

0mm NE\n ow : v S .ucmflOflwmwoo umwmcmuu umwm

Aonmav pamflwumuumm 0mm\mE OHIOH x H mQ .ucmonwmmoo :OHmDMMflc m>fluommwm

Amhmav Haaoufiooam cam Hzouo Axoum-e\h mmo.o mx .ommcm pflHOm mo >uw>wuoswcoo Hmeumse

mofluuomoum uuommcmue

 

 

 

: : mE\@x sum *0 .ucoucoo mafiumao> mumeuH:
83: 33.383 new 386 3. s .30 3 nouuogoo ammouox «0 coflomum
:8: .E S .2838 3 \E mmm :< .8383 mo 83:
= : me mH-oH x va.H y .s»a~wnmmsumm
: : mo.o w .wufimouom
: : mE\mx MOM x vv.m mo .mmmzm Ufiaom mo muflmcmo
Ammmav Haamuflooflm ocm Hzouu Axolaxv\n moH x mflo.H mo .ummn Dawwommm
mmfiuuomoum amOflm>nm
mocvuowom msam> Hmcfleoz umuwfimumm

 

 

.-..- It'll;
(.Y(- i) U. I: Q I‘l'- 'l'lli -.‘l| A- II -.!I.|I| luuiiy) I I I Q. I -llbllo! (1"-‘ll' .'l' ~10 I.-t'.l It :l’u|tvl, 'Jl:|.". ‘A'It’ill‘l:

.moamzm Hfio cumummm MOM mofluuomoum uuommcmnfi Ucm Hmofim>£mnu.m magma

4. THE EFFECT OF KINETIC PARAMETERS ON

"OIL" YIELD

For very small particles, the weight of "oil" pro—
duced will be limited by chemical kinetics. The effect of
the kinetic parameters on the weight of "oil" collected for
small particles can be calculated using Eq. (46) with the
term containing (1 - Y) deleted. In what follows, the
fraction of the total possible oil yield will be calculated
for different values of k0, Eo’ and o. The base case for
these calculations will be k0 = 6.95 x 1013sec-l, E0 = 55,
333 cal/g-mole, and o = 2000 cal/g-mole. For this set of
calculations the Damkohler number will be set equal to zero.
The effect of the heating rate on the weight of oil collected
for small particles can be calculated using Eqs. (50) and
(51).

The integration of E over the interval (-m, +m) was
approximated by integrating over the finite interval
[BO-20, Eo+201. Even with this truncation, 95.45% of the
reactions are still included in the reaction set. The
error introduced by the truncation will be small since the

probability that a reaction has an activation energy in the

truncated region is very small. The integral was calculated

48

49

using Simpson's rule (see, p. 79 of Carnahan, et al., 1969).
It was determined that thirty applications of Simpson's
rule were necessary to obtain an error of less than 10-6

in the evaluation of the integral.

4.1. The Effect of 0 on "Oil" Yield

Figure 2 is a plot of the dimensionless oil weight
collected versus time at 648°K for three values of O. The
sigma values of 2 and 10 kcal/g-mole represent narrow and
broad distributions. At 648°K, the curve representing
0 = 10 kcal/g-mole displays the greatest "oil" yield during
the time represented. This behavior is a consequence of
the higher probability that a reaction has an activation
energy much lower and much higher than E0. For these
reactions, the rate constants will be relatively large
despite the low temperature, and a significant production
of volatiles will occur.

For large and small values of O, the ultimate "oil"
yield is the same. However, 0 influences the time required
for pyrolysis significantly. Note that for 0 = 2 kcal/g-mole
the probability that a reaction has an activation energy
less than or greater than E0 is much smaller than the case
with 0 = 10 kcal/g-mole. Thus, with ZRGT<EO, the rate of
devolatilization is extremely slow--esp. if ZRGT<EO-20. For
T = 648 K, o = chal/g-mole, and E0 = 55,333 cal/g—mole,
2R T = 2.6 kcal/g-mole<<EO-20 = 51,333 kcal/g-mole. Thus,

G

for the time scale shown on Figure 2, the effect of 0 on

50

.mHoe-m\Hmox ooowusomm . owm OH x mm.muox .mflos-m\amox mmm.mmuom

HI ma

.x.mqm um cams» =Hflo= may go o «0 powwow are

Amazoumw. mth

omo coo ovm omv owv com com ovN om" ow“ ow
_ _ _ _ _ _ _ _ P b

_

\llll\\|n
/

 

   

macs-m\fimox ca
mHOEIm\Hmox mno .mvm
mHOEIm\Hmox ouo

OHOElm\Hmox Nuo

II
‘I-

 

.N wusmflm

031331703 lM 110 SSBWNUISNEHIU

51

the "oil" yield is dramatic. Note, however, that o =
10 kcal/g-mole gives EO-20==35 kcal/g-mole.
Figure 3 shows the effect of d on the oil yield
for a higher temperature, vif., 775°K. For this case, ZRGT
is closer to EO-Zo than previously, so the temperature is
high enough to cause reactions with activation energies
around E0 to contribute significantly. This is a direct
result of the dependence of the Arrhenius rate constant on
the temperature. Note that the behavior for very short
times (i.e., t<30 sec) is similar to the results shown
in Figure 2, but that the long time behavior is opposite.
Once again, for t+w, all of these curves will approach unity.
It is interesting that the case where 0 = 10 kcal/g-
mole shows the smallest "oil" yield for t>60 sec. This
occurs because ZRGT<<Eo + 20, so the rate constants for the
reactions with activation energies much higher than EO are
very small. Therefore, these reactions cannot contribute
significantly to the weight of "oil" collected during the
time scale displayed by Figure 3. This type of behavior

may explain differences between oil shale resources having

different organic compositions. A simple lumping strategy

equivalent to setting 0 0 for all shales would not have

the flexibility exhibited by Figures 2 and 3.

4.2. The Effect of E0 on "Oil" Yield
Figure 4 is a plot of the dimensionless weight of

"oil" collected versus time at 775°K for three values of

52

.mfloe-m\amo ooamuaomm . omm

H- maoaxmm.ouox .maesum\fimo mmm.mmuom

.xomss um camws =Hwo= man go o no nommmm was

Amazoumm. msz

oww omm cum om¢ omv 0mm com ocm cod emu we

_ b _ L

o
m

 

maosum\amox oHuo
GHOEum\Hmox Nuo

mHOEIm\Hmox onoll

 

 

.mvw
V

mHOEIm\Hwox cano

I.
‘
‘
\
‘
I.
I.
II
‘I
‘0‘
‘E
‘I‘
“‘
“‘|"'
‘|"
II' '
II-"II
"'|“

\\\wHOEIm\Hmox mno

lllllllllllllllll «AQEhmmmmox ouo

 

      

2'0

t’O
031333303 lM 110 SS33NOISN3N10

 

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.xomss gm camfls gage: an» no om to 008000 are .q musmflm

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com com ovm omv omv omm com 03.. of own cm 0
_ . _ r _ _ _ h u b

WA

 

      

mHOEIm\Hmox ownomlu //
mHOEIm\Hmox mmnom
mHOEIm\Hmox omuom

3
5

 

 

9'0

8'0

 

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llllllllllllllfiil

 

031331103 1M 110 SS31NUISN3H10

0'!

54

E0. As expected, the rate of "oil" production is greatest
for the short time behavior for the case of E0 = 50,000
cal/g-mole. This is a direct consequence of the dependence
of the Arrhenius rate constant on E. As E0 is lowered,
reactions with lower activation energies are included in
the reaction set. Therefore, the rate constants for all
the reactions are increased and consequently the production
of "oil" increases.

As E0 is increased, reactions with higher activation
energies are included in the reaction set. Consequently,
the rate constants are much smaller and the rate of "oil"
production is lower. Once again, for t+w, all of these
curves will approach unity. It is also apparent that the
weight of oil collected is extremely sensitive to E0, since

a 10% increase in E0 drastically reduces the weight of "oil"

collected during the short time behavior.

4.3. The Effect of k0 on Oil Yield

Figure 5 is a plot of the dimensionless weight of
oil collected as a function of time at 775°K for three
values of k0. In Eq. (46) k0 appears in the argument of
an exponential term. As kO increases, the value of this
exponential term decreases, resulting in an increase in the
weight of oil collected. This behavior is shown in Figure 5.
In Section 4.2. it was shown that if E0 is increased or
decreased by only 10%, the weight of "oil" collected was

affected dramatically. A similar effect is seen in Figure 5,

55

.mnos-m\amo ooamuaomm .mnosum\nmo ooomuo .mnosnm\flmo mmm.mmuom

.xomss um camfls =Hfio= map so ox mo powwow was .m «muons

 

‘
E
t‘
‘l
Ell

Roam. mzup
com com o3 omv one can can 3a of on“ oo o

r r» _ . t _ u _ _ _ _ rub nu
OI
nu
33
um
n.nS
ha nu
um
. nu
n:
\ IO 3
my 9
nu
HI T...
n:

0
Inca M“
l
m
Imu n.
.u n:
n3
n3
I].
an
nu

 

 

 

 

0'1

56

if k0 is decreased by an order of magnitude. Therefore,
the distribution model is much more insensitive to kO than

E0. As before, for t+m, all the curves approach unity.

4.4. The Effect of the Heating Rate on Oil Yield

Eq. (51) was integrated using a fourth order Rungeo
Kutta technique (see, p 363 Carnahan, et al., 1969). A
step size of 1°C was used in order to minimize the truncation
error. For these calculations a value for C*(l - s)Vpp
corresponding to 10 ml of potential oil was used. The
program used in this calculation can be found in Appendix E.
Figure 6 is a plot of the volume of oil collected as a
function of temperature for five values of the heating rate.
The volume of oil collected at the lower heating rates
appears to approach an asymptote, but because Da = o, the

volume of oil collected will ultimately approach 10 m1 as

t+oo

Q

It is evident in Figure 6 that as the heating rate
is lowered, the volume of oil collected is decreased. This
model prediction is consistent with the experimental obser-
vations of Campbell, et a1. (1977). They explained this
observation in terms of greater self-generated gas sweep
rates in the particle at the higher heating rates. However,
this mechanism is not present in the model discussed here,
so an alternative eXplanation is desirable. But, at this
time it is not clear what the mechanism is that causes

this behavior. The fact that the distribution model

57

O O
mnosum\nmo eoomuo . omm oaxmm.eu x .mnosnm\emo mmm.mmu m

HI ma

.UH0H> HMO :0 mumu mcwummn mnu mo uommwm one .m mudmflm

AwDHung. umapcmmmZMH

 

 

oom owe 8a com 3m
r . c . _ . (u . nu
. \\\ .\.
\ \
\
omm\xommo.one \\ \\mw \\ a
/ \\\\ Z
omm\xomeao.oun \.\ \\.\ x \\
ommEOmmooeun \ it
\\ \\\\ \\
omm\xoaeoo.un \ .
\ .AH a.
\ 1
\ \\\ omm\xomooo.oun
\\ \\\ xx
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\
\\ (\AH.\
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8H3 031331103 1I0 30 3H010A

58

qualitatively and nearly quantitatively follows observed
experimental behavior even with Da = 0 lends support for
this approach. It is anticipated that the model will also

give satisfactory predictions for cases where the heating

rate is nonlinear.

S. PARAMETER ESTIMATES USING DATA FOR

WESTERN OIL SHALE

Campbell, et a1. (1978) conducted several isothermal
and nonisothermal experiments on powdered samples of Colorado
oil shale. The diameter of the particles used in the experi-
ments was 800 microns, and the oil content of the shale was
22 gal/ton as measured by the Fisher Assay. In one isother-
mal experiment, the weight of oil collected as a function
of time at 648°K was measured. From this experiment it was

, o, and C*(1 - 5)

possible to determine the parameters k0, EO

for the distribution of activation energies model. For this

experiment Da/3=3 x 10-8, which implies that very little

degradation of the liberated "oil" is expected to occur.

This is a consequence of the small particle size and the low

temperature of pyrolysis.

5.1. Estimates of Parameters for the Distribution of
Activation Energy Model

Values for E0, 0, k and (l - c)C* were obtained

0!
by curvefitting w(t) given by Eq. (46) to the experimental
data using a direct search Optimization scheme (see, Hooke

and Jeeves, 1961). Details of the procedure can be found

59

60

in Appendix A. The results of the Optimization gave the

following values for the four parameters:
13 -1

k0 = 6.95 x 10 sec

E0 = 55,333 cal/g-mole
= 1740 cal/g-mole

yC*(l - 5)==223 kg/m3

The results of the estimating procedure can be seen
in Figure 7. Because the first nine data points were used
in the Optimization scheme, the model accurately predicts
the data for the first 40 ks. However, the long time
response of the model deviates from the data.

In a second isothermal experiment, Campbell, et al.
(1978) measured the total weight loss as a function of time
at 673°K. For this experiment, Da/3 = 8 x 10-8, indicating
once again that "oil" degradation will be negligible. The
values of y and (1 - s)C* were obtained by fitting Eq. (46)
for the total weight collected to the experimental data
E

using the values for k and 0 obtained previously.

0’ 0’

Details of this procedure can be found in Appendix C. The
values for y and (l - s)C* are:
y'=O.65

(1 - €)C* = 344 kg/(m3 of particle)

The total initial mass concentration of volatiles
as indicated by (l - s)C* represents 15 wt% of the initial
weight of the particle. This corresponds favorably with
the 15 wt% of organic material as measured by Jukkola, et a1.

(1953) for 28 gal/ton Colorado oil shale. It also compares

61

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.mHmOOE HOOHO umunm UGO

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_ . _ r _ r! _ p _ . _ . p . _ . . . mu
0

0

0

w

z

0

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x

 

\

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031331103 1M 110 SS31NDISN3N10

 

62

well with the initial concentration of kerogen (i.e., 377.3
kg/m3) for Michigan oil shale (see, Crowl and Piccirelli,
1979). The experiments of Granoff and Nuttall (1977) also
showed that the total weight loss at 793°K for 22 gal/ton
Colorado oil shale was 15 wt% of the initial weight of the
shale. Braun and Rothman (1975) determined that the fraction
of kerOgen converted to oil was 0.62. The value for y is
consistent with this value. The value for the mean activa—
tion energy E0 is typical of what has been observed pre-
viously for oil shale pyrolysis (see, Section 1.3). The
variance of the distribution is much smaller than 9,380
cal/g-mole obtained by Anthony (1974) for lignite coal.
This smaller value of 0 apparently indicates that the bonds
which make up the kerOgen molecule may be more uniform than
those of a coal molecule.

The results of the last estimating procedure are
shown in Figure 8. The total weight of volatiles collected
is made dimensionless with the initial weight of the parti-
cles. It is apparent from Figure 8 that the distribution
model does not predict a sharp plateau at long times. The
absence of a distinct plateau could be eXplained by the
shape of the distribution function. For instance, the
Gaussian distribution of activation energies requires that
the activation energies for all the reactions producing
volatile Species occur in the interval [EC-23, EO+20]. The
experiments of Allred (1967) have shown that the reactions

producing gaseous nonreactive Species nominally occur at

63

6
ms\ x eem u 16 . H..O .me.o u >

.meos-m\3mo seen u e .Huommmaoaxmm.e n ox .mnos-m\nmo mmm.mm u on
.Aw I HV«U
can > :flmuno ou Momhm um conumfinumm umuOEmqu mo muasmmm .m musmnm
Amzu mzuh
cu m w c N o

. _ . _ . _ . _ . _ . F0

I

Y

 

Amend. .Hm um .Hamnusmo no memo ..

031331103 1M SS31N013N3H10

 

64

lower temperatures than those reactions producing reactive
Species (i.e., "oil"). Given the low value of O, the lower
limit of the distribution function [EC-20] might not be
expected to include the activation energies for those
reactions producing nonreactive species. Thus, the distri-
bution function might better be described by a bimodal dis-
tribution. One peak of this bimodal distribution would be
characterized by 01 and E01 and would be centered arOund
the mean activation energy of the reactions producing non—
reactive species. The second peak of this bimodal distri-
bution would be characterized by E02 and 02 and would be
centered around the mean activation energy of the reactions
producing reactive Species. A bimodal distribution would
improve the model prediction in Figure 8 if E01 and E02
differed significantly and if ol<<EOl and 02<<E02. First,
the short time response will be more rapid since reactions
with lower activation energies are included in the reaction
set (see, Section 4.2). Secondly, the probabililty that a
reaction has an activation energy in the interval [EO +

l

201, E02 - 202] will be small. Therefore, the long time

response will be improved since reactions having activation
energies in the interval mentioned above will not contribute

to the weight loss unless 2R T=EO

G 2‘

For these two isothermal experiments the maximum
value of R occurs at t=0. At 648°K and 673°K, the values of
R at t=0 are 0.01 and 0.06 kg/m3sec, respectively. Corre-

R [Aulu - s)R
sponding values of 9* 3h in Eq. (35) are 0.01°K

 

65

and 0.04°K reSpectively. This indicates that the temperature
of the particle is constant for the duration of the experi-

ments.

5.2. Estimates of Parameters for a Lumped First Order Model
A comparison study was made by fitting the "oil"

weight loss data of Campbell, et a1. (1978) to a single,

first order, irreversible reaction (i.e., CEO). The method

for determining the parameters k0, E and C*(1 - s) was

0’
similar to the one used previously (see, Section 5.1 and

Appendix D). The parameters for this model are:
13 C-1

k0 = 4.2063 x 10 se
E0 = 52,667 cal/g-mole
yc*(1 - e) = 111.09 kg/m3

Figure 7 shows a comparison of the first order model
and the distribution model. Because the first nine data
points were used in the Optimization scheme, both models
accurately predict the data for the first 40 ks. However,
the first order model does not accurately predict the data
at long times because of the low value of C*(1 - 8).
Although the distribution model is more accurate in this
reSpect, it also deviates from the data at long times
because of the higher value of C*(1 - s). The previous
discussion suggested a means to correct this for the distri-
bution model, but for the simple lumped approach the
strategy employed by Campbell, et al. (1978) as discussed

in the Introduction, may be the only alternative available.

66
5.3. Prediction of Nonisothermal Pyrolysis Using Kinetic
Parameters Estimated from Isothermal Data

In a nonisothermal experiment, Campbell, et al.
(1978) measured the volume of oil collected as a function
of time as the temperature of the sample was heated at
0.033°K/sec. Eq. (51) was integrated using a fourth order
Runge-Kutta technique (see, p 363 Carnahan, et al., 1969)
to obtain the volume of oil collected versus temperature
for the distribution model. A step size of l°K was used in
order to minimize the truncation error. The prOgram that
was used in making this calculation is discussed in Appendix
E. Eq. (D.2) in Appendix D was used to calculate the
volume of oil collected versus temperature for the first
order model.

The data of Campbell, et a1. (1978) is compared in
Figure 9 with predictions made by the distribution model

and the first order model. With no further adjustable

 

parameters the distribution model is more representative of

 

the data than the first order model. The initial concentra-
tion of kerOgen must be adjusted in the first order model

if it is to predict the data at long times. Such a require-
ment is typical of approaches using simple kinetic schemes.
In an attempt to explain differences between the data and
the distribution model, the sensitivity of Eq. (51) to the
parameters EO and the heating rate was investigated (see
Chapter 4). It was pointed out in Chapter 4 that the two

most sensitive parameters in the distribution model were

67

ms\mx mmm u .0 I H..o>

.meos-m\nmo eeea u o .H-ommmnoaxmm.e u ox .maosnm\amo mmm.mm u om
.omm\xommo.o mo

oumu mcnumm: a ROM mHmOOE umcuo umnnw can GOHDDQHMHmHO one .m musmflm

.mDHOOMO. manpmmumzm~
owe ow? emu own

Dom
— p r0
\IIII‘)

IP b

0
00m x . u xx .6
H- neon m e x \\\ I
OHOEIO\HOO www.mm n om \\
\\ .
HOOOE HOOHO umuHh/V\ .7
\\
I I III\ 1
\ 9
HOOOE coflusnnuumflJV\ Ammmav .Hm um .HHOOQEOU mo mama

\ -9

 

(€N3J 031331103 110 30 3Nfl10A

68

E0 and the heating rate. The effect of varying a will be
negligible since such a small value of o was found by the
Optimization scheme. However, in general, the effect of o
in the distribution model is not negligible as discussed in
Section 4.1.

The effect of E0 on the "oil" yield is shown in
Figure 4. By reducing Eo by 5% and 10%, the response is
altered significantly. Even an error of 1% in the deter-
mination of EO would affect the response. Such an error is
possible since the Optimization procedure identified several
local minima very close to each other. The least of all the
local minima was chosen as the global minimum. If the
optimization process had continued, possibly one of the
other local minima could have been identified as the global
minimum. This procedure would have required excessive
computer time, and the global minimum that was arrived at
was considered satisfactory.

Figure 10 shows the experimental data along with
the predictions given by the distribution model for four
different heating rates. The data can be represented quite
well if the heating rate of the experiment were 0.0165°K/sec.
However, this would have required that the heating rate be
lower than reported by a factor of two.

For this nonisothermal experiment the maximum value

of R occurs at 723°K and equals 0.16 kg/m3sec. The corre-
RPIAHI (1 - €)R

3h

 

Sponding value of in Eq. (35) is 0.10°K.

69

ms\mx man ”10 u aceo»
.H-ommmneflxmm.e u ox .mnosnm\nmo mmm.mm u on
.muou manummn may mo pomwmm one .oa muamwm

AmsHunuui washcmmmEMH
om. om. omm oem

nu

.OHOEum\Hmo owha u 6

com
_ b .

\
\

\.
\
\
\
I
Z

UQW\MOMMO.O H D \
//V\ \ \

\\ x. \ Tb.
663310385 u n \ x \
\ .
_\ \\ omm\mommeo.o u n
\ 9
\ x \A I
\ omm\xoneoe.e u a
.\ \\\ T
\ E: A... em .3338 no 33 8
\ 4. I
\
. \ .I
Ol“!

 

(9N3) 031331103 110 30 3N010A

70

Therefore, for the duration of the experiment the tempera-
tures of the particles and the sweep gas will be approxi-

mately the same.

5.4. Prediction of Isothermal Pyrolysis for Large Particles

Granoff and Nuttall (1977) conducted an isothermal
eXperiment at 703°K with a 12.7 mm diameter sphere and
measured the total weight loss as a function of time.

The oil shale that was used in the experiment contained the
same oil content as that used by Campbell, et a1. (1978).
Therefore, the parameters that were determined from the data
of Campbell, et al. (1978) will be used in this set of
calculations. For this large particle size, the mechanisms
for intraparticle mass transfer are expected to be diffusion
for the short time data and convection for the long time
data. However, Da=8 x 10-5, which indicates that degradation
of the liberated "oil" will be negligible. Eq. (62) is used
to calculate the total weight of volatiles collected in the
presence of diffusion limitations.

Figure 11 is a comparison of the data with the
distribution model for the total weight loss using three
values of ¢. Up to 300 seconds, the cases where ¢=20 and
¢=40 represent the data quite well. The values of the
effective diffusion coefficients corresponding to ¢=20 and

-11 -ll

¢=4O are 4 x 10 mz/sec and 10 mZ/sec, respectively.

The case where ¢=0 represents no intraparticle mass transfer

71

.OHOE(m\HmO ownau O .Hno

om.~ no.u v0.0.fim.~ on
r _ _ _

 

hhmHv Hamuusz Cam wmocmuw m0 mama

ms\mx 44m u 13 - H016

O 0

mm onxmm.e u x .mnosum\nmo mmm.mm u m

ma

.somos um mcoHumueEflH Hmeonsuuec can; Hence cenusneuumno .HH ousmfim

“0:0 mzuh

.~ mg.“ N0.0 00.0 0v.0 m~.0 00.0
b _ b _ _ p

0

000'

osc~o szo'o
0313311103 1M SSB1NOISNBNIO

SLO'O

001'0

 

SZI'O

72

limitations. Apparently, at short times there are diffu-
sional limitations present as hypothesized in Section 3.3.

To calculate the total weight of volatiles collected
versus time for convection limitations, Eq. (81) was inte-
grated. A fourth order Runge—Kutta technique (see, p 363
of Carnahan, et al., 1969) was used with a step size of ten
seconds. The program that was used in making this calcula-
tion is presented in Appendix F. Figure 12 is a comparison
of the data with the convection case and the case where 6:0.
Both cases lag behind the data after 360 seconds but begin
to approach the data at longer times. This lag in the
weight loss at short times for both the case where ¢=O and
the convection case gives further support for a bimodal
distribution. As was pointed out in Section 4.2, if reactions
with lower activation energies are included in the reaction
set, the rate of volatiles production increases. An increase
in the production of volatiles would increase the initial
time response of the model. Depending on the magnitude of
this increase, the convection case may be expected to repre-
sent the data more accurately. It is noteworthy that pre-
vious attempts to interpret this data with other models
(cf., Granoff and Nuttall, 1977; Shih and Sohn, 1978) were
also relatively unsuccessful and gave similar predictions
as those given by the distribution model.

For this isothermal experiment the maximum value

of R occurs at t=0 and equals 0.35 kg/mBSec. The corre-
RPIAHI(1 - €)R

3h

 

Sponding value of in Eq. (35) is 5.50°K.

73

ms\mx eem n 10 n 3016

O

.mHos-m\Hmo came u e . omm oaxmm.e u x .maosum\emo mmm.mm n

H: ma

Amv_. m:10h
om.~ po.~ 4e.1 18.1 om.1 m1.1 mm.o mo.o m¢.o m~.o so.
_ _ _ _ _ L. _ _ b _

 

\
x.
\

_\

.\\

_\\.\
\\. \\
\\x .\
COHUUWN/COO \ \ \ \
\¢\ \ \
\ \ N \
\ \
\ \
\

AbhmHv Hamuusz UGO mchOHO mo mama

 

0

SLO'O 030-0 920-0 ooo~o
031331103 1M 8831N013N3010

001'0

SZI'O

o
m

.Momon um mcoflumuflafla 20nuom>coo can: HOOOE :oHuanHumno .NH whomnm

74

O
This term is approximately equal to one after 23 minutes

into the experiment. Because of the large particle size,
the temperature of the particle will be cooler relative to
the surroundings. This small difference is not eXpected
to affect the diffusion, convection, and kinetic responses

significantly.

6. MODEL PREDICTIONS FOR EASTERN OIL SHALE

The kinetics of devolatilization of eastern oil
shales has been modeled as a single, second-order, irre-
versible reaction (Crowl and Piccerelli, 1979). The rate

expression for this reaction is

R = 2.4868 x 1013 exp (

cp ~30,337 K) D2 kg (84)

T K m3sec

where 9K is the concentration of unreacted kerOgen. The
initial concentration of kerogen was measured to be 377.3
kg/m3 or 15.26 wt % of the initial weight of the shale.

Crowl and Piccirelli (1979) used this second order expression
in an intraparticle mass transport model to predict the
pyrolytic behavior of a semi-infinite slab of oil shale.

In order to make comparisons between the intrapar-
ticle mass transfer models using second order kinetics
versus using the distribution of activation energy kinetics,
it was necessary to obtain values of E0, 0, and k0 typical
of eastern oil shale. The determination of these para—
meters requires weight loss versus time data for the devola-

tilization of eastern oil shale. Since no experimental data

are presently available in the literature for eastern oil

75

76

shales, hypothetical isothermal oil loss data at 775°K for
powdered oil shale were generated using the second-order
kinetic model. Eq. (46) for the weight of oil collected
was curvefitted to the initial time data as before with
yC*(l - a) = 377.3 kg/m3. The results of this parameter

determination are

k0 = 5.88 x 1013 m3/kg sec
E0 = 53,333 cal/g-mole
o = 1563 cal/z—mole

With these parameters Eqs. (46), (62), and (81),
can be used to calculate the weight of oil collected as a
function of time at different temperatures. These equations
are made dimensionless with the maximum possible weight of
oil which can be produced (i.e., VppyC*(l - 6)). To obtain
predictions for the second order model, Eqs. (46), (62),
and (81) will differ only in the form of R. For the second

order model

 

i‘ “O 2
-E V ’K
R = k exp(-——) (85)
cp o R T -E o I
G 51 + k0 exp(§—§)pk
; G 1
where 00K = 377.3 kg/m3 and k0 and E are the kinetic para-

meters appearing in Eq. (84).

The isothermal, dimensionless oil weight collected
versus time was calculated for each model for the chemical
kinetics controlling regime, diffusional limitations, and
convection limitations using a hypothetical 5 cm in diameter

Sphere of eastern oil shale. The calculations were made

77

at temperatures of 648°K, 703°K, and 775°K. Internal tem-
perature gradients were ignored since the main purpose of
this study was to compare the differences between second

order and distribution of activation energy kinetics. The
values for Da at 648°K, 703°K, and 775°K corresponding to

a particle size of 5 cm are 4 x 10-4, 3 x 10-3, and 2 x

10-2, respectively.

6.1. Chemical Kinetics Controlling Regime
For the chemical kinetics controlling regime the
dimensionless weight of oil collected under isothermal

conditions according to the second order model is

- l l
w(t) = -———-——— [l - ]. (86)
(l + Da) -E 0
§—- (1 + k0 eXp(§;T)DK t

 

Figure 13 shows the reSponse of the distribution model and
the second order model. The close agreement between the
models is expected at 775°K since this was the temperature

at which the parameter determination was made. It is note-
worthy that a single first order expression (i.e., o = 0
kcal/g-mole) will not represent "data" from the second order
model. However, the distribution model represents the "data"
quite well because of the flexibility introduced by the
parameter 0. The model predictions at the other temperatures

are also very close.

78

.mEnmmH manaaouncoo mOAHOOAx

HmoHEmzo mzu H00 mHmOOE umwuo Ocoomm 0cm coflusnnuumno .ma mudmflm

Amazouum. 020p
owe can own owe om. own can ouw ems ow" so

lllll dmnda.anuQ.mwmwmww.

 

 

\ \\

I). Momvw um AOOOE coausnnuumflo .\

\
\

xomon um H0008 HOOHO Ocoomm/JY .\

Memos um HOOOE :oflusnnuumna

xomvw um H0008 HOOHO ocoomm

/////JV \
omnn um HOOOE :ofiasnnuumno \

      

 

Do

0

2'0

031331103 1M 110 SS31N013N3H10

 

79

6.2. Diffusional Limitations for Pyrolysis
With diffusional limitations, the dimensionless
weight of oil collected for the second order model under

isothermal conditions is

 

 

0(t) = 31 2 [1 - l _E O 1 (87)
¢ (1 + k exp(-—-)p t)
Da‘ (m) 0 RGT k

Figure 14 shows the response of the distribution model and
the second order model for ¢=40. Close agreement is again
seen at 775°K. The weight loss occurs over a much longer

time scale but all the curves will eventually attain the

 

final value of 3Y2 . Once again, close agreement

_A_)
l-Ocoth6

is seen at all temperatures.

Da-(

6.3. Convection Limitations for Pyrolysis
For convection limitations, the isothermal, dimen-

sionless weight of oil collected is

 

 

%
- l
d 3:) = 235 f eXp [[2 (l - r 2)} X
p O . 3
K
_ i B g
(AR[l - r ] - (AR) 3 _2 _
_2 8 g g r dr (88)
_lAR[l - r ] + (AR)

 

where A and B are defined by Eqs. (82) and (83). Eqs. (81)
and (88) were integrated using a fourth order Runge-Kutta

technique. It was determined that a step size of 10 seconds

80

.oeue .mconumnesna

HmconSMMHO MOM mHmOOE HOOHO OOOOOm cam :oHuOQHHumHQ .va muzvflm

 

\

 

70'0 20’0 00

90'0

.wx0 mzuh
00.0 h0.~ v0.— «0.0 mm." m_._ «0.0 00.0 0v.0 m~.0 00.0
_ _ _ u . . p u .1... nu
XMmmwleImwOMEIwmvmw OGOOOmN.
\
M6060 um H0008 OOAOOOHHumHO \
\.
\
.\
MoMOh #6 H0008 MOOHO OOOOOm \.\.
\/
\\\\\\ /...mon
\\\\\\ um HOOOE coflusnnuumwo
\
\

Momhn um HOOOE uwcuo Ocoommx/JY

1...

somhn 06 Hence coHusaeuumeQ

0‘

 

 

80'0

01'0

031331103 1M 110 SS31NOISN3NIO

81

was necessary to avoid truncation error. Figure 15 shows
the response of the distribution model and the second order
model. Once again close agreement is seen at 775°K. The
weight loss occurs over the same time scale as the diffusion
case. The final value for the dimensionless weight collected
are 0.83 for the second order model and 0.82 for the distri-
bution model showing that a yield loss has occurred. This
yield loss was caused by the buildup of reactive species in
the gas phase of the particle because of intraparticle con-
vection limitations. This buildup increased the rate of
coking, and consequently, a yield loss occurred, character-
ized by the asymptote at 0.83. It appears that the curves
for the other two temperatures will eventually approach

this asymptote. Close agreement at all three temperatures

is seen again.

 

82

.mcoflumuAEnH
:0nuom>coo How HOOOE HOOHO Ocoomm 0cm COHuOQHHumHQ .mH musmfim

Amxv mzmh

om.~ he.~ em." 18.1 om.1 m1.1 «6.9 mo.o oe.e m~.o oo.o
_ b _ 110 . _ . b u nu
momvm um HOOOE HOOHO ocoomm

V.1.11.1.|.l.l.1.11

111111111 \

||

“|

00

031331103 1M 110 SS31NOISN3N10

\
somee pm demos codesnfluumfla \ mu
7..
_\
\_
.\
\ 0
\ .Ilo
\ .7
Momon um HOOOE HOUHO Ocoomm 1111
M1
\|\\\\ 0
1111.1 9
..1 1.1. zones 66 36668 863683060636

   

 

Roman pm 36608 menusnnuumna//

1111/ 11111111111111

momhn um H0005 MOOHO Ocoomm

 

 

0'1

7. CONCLUSIONS AND RECOMMENDATIONS FOR

FURTHER RESEARCH

The low value of o that was determined for western
oil shale indicated that the bonds of the kerogen molecule
are very uniform. The predictions given by the second order
and distribution models for eastern oil shale were very
similar. This implies that perhaps these simpler kinetic
expressions are adequate in describing the kinetics of
devolatilization. However, the failure of the first order
model and the success of the distribution model in non-
isothermal situations indicates that the distribution model
is required. The need for the more complicated distribution
model to describe the kinetics of devolatilization remains
unknown until additional data from isothermal experiments
conducted with powered oil shale particles over a wide
range of temperatures are available.

The determination of the parameters k , E , O, y,

o
and C* requires a set of data from an isothermal experiment
measuring both the weight of reactive and nonreactive

volatiles collected versus time for powdered oil shale

83

84

particles. Ideally, the temperature of this experiment should
be 775°K, the data from such an experiment would not be
expected to show an apparent weight loss plateau. In the
isothermal experiment which was used to determine the para-
meters in this study, only the weight of oil collected as a
function of time at 648°K was measured. The data from this
experiment show that the weight of oil collected as a
function of time approaches an asymptote characteristic of
648°K. Steps were taken in the parameter determination
scheme so that this apparent weight loss plateau would not
be imposed on the model. However, it is anticipated that
the parameters could be determined more accurately if data
from isothermal experiments at the highest temperature of
pyrolysis were used.

The experimental observation that nonreactive and
reactive volatiles are produced at different temperatures
led to the suggestion that a bimodal distribution might be
more apprOpriate than a Gaussian in characterizing the
activation energies of the devolatilization reactions. This
suggestion was given further support since model predictions
for the total weight loss lagged behind the data at short
times (see, Section 5.4). This lag suggested that reactions
with lower activation energies should be included in the
reaction set in order to Speed up the short time response
(see, Section 4.2). If a bimodal distribution is indeed
the case, the parameters E01 and G which characterize one

1
peak of the bimodel distribution could be determined by the

 

85

weight of nonreactive volatiles collected versus time in

an isothermal experiment. The parameters E02 and 02 which
characterize the second mode could be determined by the
weight of reactive Species (i.e., "oil") collected versus
time. If the distribution is actually bimodal, then the
parameters E0, 0, y, and C* which were estimated in this
thesis characterized only the peak representing the activa—
tion energies of the reactions producing reactive Species.

The value of C*(1 - s), the ultimate volatile con-
tent of the oil shale, was determined to be 15.26 wt % of
the initial weight of the particle and correSponded to values
from other experimental investigations. ED was determined to
be 55,333 cal/g-mole and is typical of activation energies
represented in other studies. The fraction of kerogen con-
verted to oil, y, was determined to be 0.65. This value
was nearly identical with the value of 0.62 obtained by Braun
and Rothman (1975).

The model predictions for nonisothermal experiments
where the temperature of the sample increased linearly with
time were consistent with experimental data. These model
predictions were made using no further adjustable parameters
and the heating rate entered the calculation explicitly
through the boundary condition for the temperature. Although
favorable predictions were obtained using linear heating
rates, it is anticipated that the distribution model will
also predict situations where nonlinear heating rates are

used.

 

86

For experiments with 12.7 mm particles it was not
clear if the production of volatiles was affected by intra-
particle convection limitation (see, Figure 12). However,
Figure 11 suggests that the production of volatiles at short
times may be affected by diffusion with an effective diffu-

ll m2/sec. The calculations comparing

sion coefficient of 10-
the convection response for second order and distribution
of activation energy kinetics for a 5 cm in diameter sphere

(see, Section 6.3) show that the production of volatiles is

 

limited by intraparticle convection. The limitations cause

a buildup of reactive volatiles in the particle and conse-
quently an increase in the rate of gas phase decomposition.
Further eXperiments need to be conducted to define the
limiting particle size where the rate of production of
volatile species is no longer controlled by chemical kinetics

but by intraparticle mass transfer.

APPENDICES

APPENDIX A

PROCEDURE FOR ESTIMATING THE DEVOLATILIZATION PARAMETERS

APPENDIX A

PROCEDURE FOR ESTIMATING THE DEVOLATILIZATION PARAMETERS

The parameters in the distribution of activation
energies kinetic model can be determined using the weight
of oil collected versus time isothermal data at 648°K. The
weight of oil collected given by Eq. (46) can be made
dimensionless by dividing by the initial weight of the

particle. Thus,

 

‘ * — oo -
V_rl£1 = (C ‘1 E) [l - 7:17— f+ 8Xp[-kot exp (FET’I
ppcs ps V20 0 -m G
-(E-EO)2
x exp[—TIdE]. (21.1)
20

The estimating procedure seeks to find a minimum
in the sum of the squares of the differences between the
data and the model. For this strategy, the objective

function is

 

n
- " l +00 -E

J(A k ,E ,O) = Z [y.-A [l- f exp[-k t. exp(‘——)] X
o o 0 i=1 1 o /20 O _m o 1 RGT

87

88

2

 

-(E-EO)2
eXp[-—-—,3--—]dE] (A.2)
20“
: 7C*(l - a) . o
where AO _ O and yi are the data p01nts at 648 K.
s

In order to begin the Optimization it was necessary

, o, and

to choose starting values for the parameters k0, EO

A0. The starting points were determined using the Michigan
State University IMSL subroutine ZSRCH. ZSRCH generates a
specified number of points in an n-dimensional space for
use as starting points in nonlinear Optimization routines.

For each set of starting values, a direct search
Optimization scheme was used to find the best values for
the parameters that minimized the objective function Eq.
(A.2) (see, Hooke and Jeeves, 1961; Beveridge and Schechter,
1970; and Walsh, 1975). The Objective function is highly
nonlinear and has many local minima. Because many sets of
starting values will converge to different local minima,
several sets of starting values were tried. It was found
that twenty starting sets of parameters were sufficient to
identify the various local minima.

The Hooke and Jeeves method and ZSRCH require that

constraints be placed on the parameters. The following

constraints were imposed:

l3 l4 -1

1.0 x 10 :koil.0 x 10 sec
40,000:Eo:60,000 cal/g-mole
0:0510,000 cal/g-mole

05A030.15 ,

89

The global minimum of the objective function was used to

define the parameter estimates. For the data of Campbell,

et a1. (1978), the devolatilization parameters were found
to be

k0 = 6.95 x 1013 sec“1

E0 = 55,333 cal/g-mole

o = 1740 cal/g—mole

A0 = 0.09906 ,

The step sizes and the number of times the step
sizes were halved were chosen to economize the computing
time without sacrificing accuracy. Several combinations
of step sizes and number of halvings were tried. It was

determined that the starting step sizes should be

Ako = 0.1 x 1013 sec"1
AEO = 2000 cal/g-mole
AG = 1000 cal/g-mole
AAO = 0.01.

It was also determined that halving five times gave suffi-
cient results. Further halving did not decrease the least
squares error appreciably and only led to excessive computing
costs.

Campbell, et a1. (1978) show twelve data points in
their experiment measuring the weight of oil collected as a
function of time. The data show the weight of oil collected

varying nearly linearly with time and then leveling off to

90

a plateau characteristic of 648°K. If all data points were
used in the estimating routine, it was felt that the value
for yC*(l - s) characteristic of 648°K would be imposed upon
the model and not the true yC*(l - 6). Therefore, only the
first nine data points were used since none of these were in
the plateau region. Hence, in Eq. (A.2), n=9. In terms of
experimental accuracy, these first nine data points are
probably the most accurate. Ideally, an isothermal experiment
should be conducted at 775°K where the total weight loss
would be realized. The data from this type of experiment
would not impose an apparent yC*(1 - s) on the model provided
no degradation of the liberated oil takes place. Appendix B
contains the program as well as sample outputs of the curve-

fitting strategy.

 

I"

APPENDIX B

COMPUTER PROGRAM FOR ESTIMATING THE

DEVOLATILIZATION PARAMETERS

APPENDIX B

COMPUTER PROGRAM FOR ESTIMATING THE

DEVOLATILIZATION PARAMETERS

A*91¢Xc‘CSTAR*99X0*FUNCTION

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APPENDIX C

PROCEDURE FOR ESTIMATING Y AND C*(l - E)

APPENDIX C

PROCEDURE FOR ESTIMATING Y AND C*(l - E)

The values of Y and C*(l - a) can be obtained from
the total weight collected versus time isothermal data at
673°K. The total weight of volatiles collected, given by
Eq. (46), can be made dimensionless by dividing by the

initial weight of the particle. Thus,

 

 

W(t) C*(1 - a) l +00 -E
_____ = [1 - f exp[-k t exp(——-)] X
Vppps ps /7F 0 -w 0 RGT
-(E-EO)2
exp[—————7f——]dE]. (c.1)
20

As in Appendix A, the following objective function

is defined

 

n ' 1 +oo -E
J(B ) = X y. — B [l - f exp[-k t. exp(-——)]
0 i=1 . l O m 0' _oo 0 1. RGT
2
-(E-EO)2
x exp[ 2 ]dE] (c.2)
20
l - a

 

where B0 = C*( ) and y1 are the data points at 673°K.
s

In the estimating strategy ten data points were used; thus,

n=lO in Eq. (C.2).
97

98

Since the values of k0, E0, and 0 were determined
previously the only undetermined parameter is 80' The minimum
in J(Bo) can be found by setting the first derivative with

respect to B0 equal to zero and solving for 80‘ The result

 

 

 

 

is,
2
10 -(E-E )
2 y, [l - l f+00 exp[-koti exp(§:%)]exp[-———E——-]dE]
B = i=1 3- /2—Tr—o -oo o 20 (c 3)
o 10 1 +00 -2 -(E-E )2 2 '
E [ - f exp[-k t. eXpC--)] expD—-———-——]ds]
i=1 /§? 0 -w 0 l RGT 202
C*(1 -8 )
Thus, the value of B0 or O A can be calculated

3
directly from the data. Y can be calculated by dividing AC

from the estimating procedure in Appendix A by 80' The
program which calculates Y and C*(l -6 ) is included on the

pages which follow.

 

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APPENDIX D

PROCEDURE FOR ESTIMATING THE KINETIC PARAMETERS

IN THE LUMPED FIRST ORDER MODEL

APPENDIX D

PROCEDURE FOR ESTIMATING THE KINETIC PARAMETERS

IN THE LUMPED FIRST ORDER MODEL

The weight of reactive volatiles collected as a
function of time can be described by an overall material

balance around all the particles.

0
= ' —Y .
w(t) YCB Vpp Cvap (D l)
O .t -E .
where C = C exp[-k J exp(———)dt] for first order
B B O O RGT
kinetics. This balance assumes no intraparticle mass trans-

fer limitations and no gas phase decomposition reactions.

Thus,

t

O 1 - exp[-kO f (—:%)dt]l. (D.2)
O .'

w(t) = VppycB RG

The weight of reactive volatiles collected can be made
dimensionless by dividing by the initial weight of the

particle. Therefore,
0 ~ .

i

.. t -E l
w(t) = l - exp[-k f (——-)dt]!
5 o o RGT i

 

(0.3)

For isothermal experiments

lOl

102

O 9

w(t) = l - exp[-kot exp(§:%)] . (D.4)
s G

 

As in Appendix A the following objective function

can be defined

 

n { -E _ (0.5)
J(CO'kO'E) = .L. yi - Co[l - exp(-koti eXp(§——T)]l
1:1 I G ‘;
YCBO
where CO = p and yi are the data points at 648°K.
s

The three parameters Co’ kc, and E which minimized J
were determined using the Hooke and Jeeves direct search
scheme described in Appendix A. Starting values for the para-
meters were again obtained from ZSRCH. Once again, nine data
points were used in this estimating procedure; therefore,
n=9 in Eq. (D.5).

The starting steps sizes were

AC0 = 0.01

AB
0

Ak
o

2000 cal/g-mole

O.l x 1013 sec-1

and the parameter constraints were

0§CO§0.15

40,0005E0560,000 cal/g-mole

l3 l4 -1
1.0 x 10 ikoil.0 X 10 sec .

The step sizes were halved five times as before.

APPENDIX E

COMPUTER PROGRAM FOR CALCULATING THE VOLUME OF OIL

COLLECTED FOR NONISOTHERMAL CONDITIONS FOR THE

CHEMICAL KINETICS CONTROLLING REGIME

APPENDIX E

COMPUTER PROGRAM FOR CALCULATING THE VOLUME OF OIL
COLLECTED FOR NONISOTHERMAL CONDITIONS FOR THE

CHEMICAL KINETICS CONTROLLING REGIME

Eq. (51) was integrated using a fourth order Runge-
Kutta technique (see, p. 363 Carnahan, et al., 1969). A
step size of l°K was used to minimize the truncation error.
To calculate the volume of oil, Eq. (51) was divided by the

3

average density of the oil produced p0i = 0.9l x 10- kg/cm3

1
(see, Campbell, et al., 1978). The initial condition which

3 at T=628°K.

was used is: volume of oil collected = 0.15 cm
Eq. (50) is the intrinsic rate of reaction for a linear
increase in the temperature with time and involves the
exponential integral function. The Michigan State Univer-
sity IMSL library subroutine MMDEI was used to calculate the
exponential integral. The computer program which was used
to integrate Eq. (51) is given on the pages that follow.
Subroutine CALOK calculates the Runge-Kutta parameters.

Function DIST calculates the integrand of the integral over

E appearing in Eq. (50).

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APPENDIX F

COMPUTER PROGRAM FOR CALCULATING THE WEIGHT OF
VOLATILES COLLECTED AS A FUNCTION OF TIME FOR

ISOTHERMAL CONVECTION LIMITATIONS

APPENDIX F

COMPUTER PROGRAM FOR CALCULATING THE WEIGHT OF
VOLATILES COLLECTED AS A FUNCTION OF TIME FOR

ISOTHERMAL CONVECTION LIMITATIONS

Eq. (81) was integrated using a fourth order Runge-
Kutta technique (see, p. 363 Carnahan, et al., 1969). A
step size of 10 seconds was used to minimize the truncation
error. The intrinsic rate of reaction for isothermal cases
is given by Eq. (44). The computer program which was used
to integrate Eq. (81) is given on the pages that follow.
Subroutine CALOK calculates the Runge-Kutta parameters.
Function DIST calculates the integrand of the integral over
E appearing in Eq. (44). Function DISTZ calculates the

integrand of the integral over E in Eq. (81).

107

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LIPICIST UDV ..rawFflonr)CleD ..1

LI ST OF REFERENCES

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Anthony, D. B. 1974. "Rapid Devolatilization and Hydro-
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Anthony, D. B. and Howard, J. B. 1976. "Coal Devolatiliza-
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Beveridge, G. S. and Schechter, R. S. 1970. Optimization:
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Bird, R. B., Stewart, W. B., and Lightfoot, E. N. 1960.
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Campbell, J. H., Koskinas, G. H., Coburn, T. T., and Stout,
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Campbell, J. H., Koskinas, G. H., and Stout, N. D. 1978.
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Carnahan, B., Luther, H. A., and Wilkes, J. O. 1969.
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Crowl, D. A. and Piccirelli, R. A. 1979. Transport Pro-
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Hudson, J. L. and Heichlen, J. 1968. "Theory of Carbon
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