THE KINETICS OF METHYLCYCLOPENTANE REACTIONS OVER A

PLATINUM ON T^-ALUMINA CATALYST

by

FERNANDO C. VIDAURRI JR., B.S. in Ch.E., M.S. in Ch.E.

A DISSERTATION

IN

CHEMICAL ENGINEERING

Submitted to the Graduate Faculty of Texas Technological Collese

in Partial Fulfillment of the' Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

June, 19(18

(\BC-O'\OO

Ad HI

hlo. (^

Dedicated to the memory of

my father

in grateful appreciation of his

pers evering encouragement,

ii

.dm'fi'AtmoMi.mii

ACKNOWLEDGEMENT

The author expresses his sincere appreciation to Dr.

A. J. Gully, under whose direction this work was carried out,

for his valuable guidance. Acknowledgement is also aiven to

the other members of the committee for their assistance,

through helpful criticism, in the writing of the dissertation

Special acknowledgement is given to my wife, Manuela,

for her patience and help during the preparation of the

manuscript.

lii

TABLE OF COriTENTS

Page

DEDICATION ii

ACKNOWLEDGHEIIT iii

LIST OF TABLES vii

LIST OF ILLUSTRATIONS ix

I. INTRODUCTION 1

II. CATALYSIS AND CATALYTIC MECHANISMS 7

Inorganic Catalysis 7

Hydrocarbon Isomerization-Dehydroisomer-ization Mechanisms 10

III. KINETICS OF HETEROC-EriSOUSLY CATALYZED

REACTIONS 15

On Epistomology 15

Kinetic Model Fitting 16

Method of Hougen and "/atson 17

External Diffusion 19

External Heat Transfer 24

Internal Di^*'usion, Effectiveness Factor . . 25

Adsorption , • • • • • 31

Analysis of Heterogeneous Reaction Rates , . 33

Catalyst Poisoning 34

IV. MATHEMATICAL MODELING 36

Mathematical Modeling 36

Linear Least Squares 38

Nonlinear Least Squares 41

iv

V

Page

Simultaneous Least S- uares 45

Nonlinear Parameter Estimation ^6

Special Techniques and Interpretations . . . 50

Parameter Scaling 50

Interaction 52

Null Effect 54

Analysis of Variance 55

Residual Analysis 56

V. EQUIPMENT AND PROCEDURES 59

Equipment 59

Reactor Description 60

Feed Preparation and Metering 62

Product Metering and Recovery 64

Experimental Development and Catalyst

Deactivation 66

Preliminary Experimental Development . . . 66

Catalyst Deactivation 68

Sequencing of Experimental Runs 72

Operational Procedure 73

Analytical Methods and Procedures 76

Data Precision 81

VI. DATA REDUCTION AND ANALYSIS 86

Data Reduction 86 Catalyst Physical Properties and Effec­tiveness Factor of the Catalytic Reaction System 88

VI

Page

Data Analysis 90

Methylcyclopentene 91

Benzene, cyclohexane , 95

Ring Opening Products 102

Modeling of Experimental Rates 108

Methylcyclopentene 109

Benzene-cyclohexane 116

Ring Opening Products 122

Nonisothermal Rate Models, . . , . , . . , 125

Summary 130

LIST OF REFERENCES 134

NOMENCIATURE 140

APPENDIX 144

A. Estimation of Transport Properties 145

Density 146

Heat Capacity 146

'iscosity , 147

Thermal Conductivity 148

Diffusion Coefficient 149

B. Marquardt's Method 150

C. Computer Programs 155

Physical Properties and Data Reduction . . • 156

Nonlinear Parameter Estimation 166

Data Plotting Program 176

D. Tables 179

LIST OF TABLi-S

Table Pago

1. Ranges of Independent Variables 5

2. Chromatograph Operating Conditions 80

3. Estimates of Per Cent Relative Error 85

4. Parameter Estimates and Goodness of Fit Data for Methylcyclopentene Models 112

5. Individual Confidence Limits for Methyl­cyclopentene Model II Parameters 114

6. Methylcyclopentene Parameter Correlation Matrix 114

7. Goodness of Fit of M< 'thyleyelopentene Rate Model II 115

8. B^nzr^e plus Cyclohexane Models Parameter Values 119

9. Individual Confidence Limits for Each Parameter , . 120

10. Benzene plus Cyclohexane Rate Model Parameter Correlation iiatrix 121

11. Goodness of Fit of Benzene plus Cyclohexane

Rate Model ' 123

12. Goodness of Fit of Ring Opening Rate Models , . 124

13. Non-isothermal Parameter Estimates 129

14. Reactor and Catalyst Data 180 15. Hydrocarbon Retention Time in a 10 Ft.

Squalane Column at 115 °C 181

16. Chromatographic Response Factors 182

17a. Component Thermodynamic and Transport Properties 183

17b. Component Thermodynamic and Transport Properties 184

vii

« * *

Vlll

Table Page

18. Liquid Feed Molar Composition and Definition

of Computer Output Symbols 185

19. Independent Variables Settings for 850 °F Runs. 186

20. Hydrogen Inlet Feed Rate (SGFH) and Liquid

Product Molar Concentration for All Runs. . . 187

21. Component Rates for S50 °F Runs 188

22. Normalized Rates 189 23. Catalyst Surface Ter ^ratures and Surface

Partial Pressures (Atm) for All Runs 190 24. Grouped Paraffin Average Molar Bulk Concen­

trations in Reactor for 850 °F Runs 191

25. Mixture Properties for 850 °F Runs 192

26. Mass Transfer Coefficients for 850°F Runs . . . 193

27. Component Diffusivities in fixtures for

850 Op Runs 194

28. Schmidt Numbers for 850 °F Runs 195

29. Miscellaneous Properties for 850 °F Runs. , . . 196

30. Independent Variable Settings, Mixture

Properties 197

31. Mixture Properties 198

32. Component Rates, Schmidt Numbers 199

33. Miscellaneous Properties 2C0

34. Catalyst Deactivation Data, Accumulated Running

Time, Liquid Product Molar Composition, , . . 201

35. Catalyst Deactivation Rates 202

36. Catalyst Deactivation, Ave-f fl e Bulk Composition 203

LIST OF ILLUSTRATIOi\^S

F i g u r e Page

1. Average Pore in Spherical Catalyst Pellet . . . 28

2. Reactor Detail 61

3. Feed Preparation and Measurement System . . . . 63

4. Product Recovery System 65

5. Catalyst Deactivation, 2-methylpentane Rate . , 70

6. Hydrocarbon detention in Squalane Column, , , , 82

7. Diagnostic Test for External Diffusion, 850 °F, 92

8. Effect of MCP Surface Partial Pressure on HCP= Rate 93

9. Effect of MCP Surface Partial Pressure on Benzene Rate 96

10. Benzene-cyclohexanc-hydr« -'< n Thermodynamic Equilibrium and Experimental Ratios at 850°F, 98

11. An Isomerization-Dehydroisomerization

Mechanism 97

12. Cyclohexane Production Flux Ratio 101

13. Sum of Ring Opening Rates . , 103

14. n-hexane to 2-methylpentane Production Ratio. . 104

15. Temperature Dependence of n-hexane to 2-methylpentane Production Ratio 106

16. Temperature Dependence of Ring Opening Products/Six-membered Ring Products Ratio . . 107

17. Benzene Arrhenius Plot 127

ix

CHAPTER I

INTRODUCTION

A petroleum refinery is a complex system composed

of many non-simple processes. The mathematical modeling

of the individual processes and, if possible, the system,

are necessary steps in the determination of optimal

operating conditions or of optimal control systems. In

a large number of cases, the limiting factor in the

optimization is the lack of adequate process models. The

formulation of these models requires a tremendous amount

of proper data, an intimate knowledge of the process,

and a dexterity of ideas along the process modeling lines.

The modeling and subsequent optimization are desirable•

because even incremental process or system improvements

are significant when large production quantities are

invoIved.

The reforming of petroleum naphthas is one of the

more important processes in a refinery. Petroleum naphthas

are complex hydrocarbon mixtures which have usually been

characterized by standardized group characterization

factors. The fuel properties of the naphthas are

enhanced by reforming reactions which yield aromatized

and isomerized products. The industrial reforming of

petroleum naphthas is presently one of the largest

catalytic operations, the installed capacity in the

1

United States being over two million barrels per day. (22)

The chemical reactions occuring in petroleum reforming

are diverse and have been classified as follows: (35)

1. Dehydrogenation of cyclohexanes to aromatics.

2. Dehydroisomerization of alkylcyclopentanes to

aromatics.

3. Dehydrocyclization of paraffins to aromatics,

4. Isomerization of n-paraffins to isoparaffins.

5. Isomerization of alkylcyclopentanes to cyclo­

hexanes .

6. Isomerization of substituted aromatics.

7. Hydrocracking of paraffins.

An approach to the study of the reforming process

is to individually consider and model representative

reactions from each of the general classes of reactions

and then to compound all of the information into a process

model. Specific studies have been made for each of the

general classes of reactions (21, 34, 36, 61, 63) in an

effort to elucidate their role in the overall reforming

process. The scope of each of these studies, however,

can best be described as qualitative.

The dehydrogenation of cyclohexane to benzene, repre­

sentative of reaction type (1), was studied by the author

and others on a 0.5 per cent platinum on J-alumina. (32,

33) This reaction was studied not only because it was

representative of type (1), but because it was a necessary

reaction step in some of the other classes of reactions.

The general purpose of this dissertation is to

investigate the kinetic behavior of the isomerization-

dehydroisomerization reactions of methylcyclopentane

(representative of reaction types (5) and (2)) on a

commercial platinum on 'Vj-alumina catalyst. These

reactions have been qualitatively studied (21, 53, 60,

61, 63, 65, 74), but no kinetic modeling was attempted

in any of the reported work. Most of these works are

discussed under section 2,2.0, Hydrocarbon Isomerization-

Dehydroisomerization Mechanisms.

The study of the isomerization-dehydroisomerization

reactions of methylcyclopentane is of immediate interest

because the monetary gains in feeding this component

to a reformer are considered to be marginal. The

elucidation of the kinetics of the relatively slower

methylcyclopentane aromatization rates is a necessary

step for the optimization of this process or in the

study of catalyst and/or process modification.

The isomerization-dehydroisomerization reactions

are simultaneous and consecutive, making the modeling

in a flow system difficult. The difficulty can be

compounded by the presence of

1. Side reactions, e.g., hydrocracking.

2. Temperature and concentration gradients from

the bulk phase to the catalyst surface.

3. Temperature and concentration gradients within

the catalyst,

4. Catalyst instability.

5. Independent variable effects requiring nonlinear

(in the parameters) rate models.

The presence of all of these factors in a commercial

process is not unusual.

A specific purpose of this dissertation is then to

develop mathematical models which will adequately describe

the effect of the independent variables on the rates of

catalytic isomerization-dehydroisomerization and other

related catalytic reactions of methylcyclopentane.

Although the determination of the reaction mechanisms of

the reactions is not one of the prime objectives of this

study, the feasibility of determination of reaction

mechanisms by the use of mechanistic models will be

cons idered.

In pursuance of the dissertation objectives, experi­

mentation was carried out to obtain reliable data on the

effects of temperature, pressure, modified Reynolds

number, and of methylcyclopentane and hydrogen concentra­

tions on the rates of the various reactions. The reactions

were carried out under a large hydrogen partial pressure

to minimize catalyst deactivation by coking. The ranges

of the independent variables over which the experimentation

was conducted are presented in Table 1,

Table 1

Ranges of Independent Variables

Minimum

13.4

810

17.5

21.3

Maximum

33.8

870

25.0

41.7

Modified Reynolds number

Temperature, °F

Methylcyclopentane feed concen­tration, mole per cent

Total pressure, atm.

The range of the modified Reynolds number (feed throughput

rate) was determined by practical operational limitations

on the experimental system. The temperature range was

limited to the region where the reactor could be operated

as a differential reactor within the feed throughput

limits at a fixed catalyst charge. The methylcyclopen­

tane feed concentration and total pressure were reasonably

bounded to study their effects within a region of interest.

It is hoped that the investigation within these ranges

and with this feedstock will allow construction of rate

models which can later be used in conjunction with other

models to describe the industrial reforming process.

In order to study the kinetics of and to develop

mathematical models of the methylcyclopentane system, a

general knowledge of inorganic catalysis is useful. A

specific knowledge of kinetic modeling of heterogeneous

catalytic systems and of statistical mathetoatieal modeling

techniques is essential. An individual chapter will be

devoted to each of these three topics. Most of the material

found in these three chapters was originated elsewhere but

will be presented here because it is highly pertinent and

so that the material will be readily available.

CHAPTER II

CATALYSIS AND CATALYTIC MECHANISMS

2.1,0 Inorganic Catalysts

The inorganic catalysts may be broadly described as

acidic (electron pair-accepting), basic (electron pair-

donating), or mono-electronic (oxidative-reductive, hydro­

gen active). Catalysts having both acidic and electronic

or both basic and electronic properties are also known.

Furthermore, acid catalysts are not without basicity and

bases are not without acidity. Attention is usually focused

on one of the attributes of a catalyst, with the conjugate

attribute fading into the background and subsequently

forgotten. The catalytic activity is probably due to the

presence of both conjugate properties to some degree.

With basic catalysts, isomerization has been found to

be limited to shifts in the position of double bonds of

olefins and to certain alkyne-alkadiene interconversions,

(23) Examples of basic catalysts are alkali metal hydrox­

ides, amides, hydrides, and organosodium compounds. Mono-

electronic catalyst isomerization appears to be limited to

shifts in the position of double bonds, interconversion of

geometrical isomers of olefins, and inversion of configur­

ation at a saturated carbon. Examples of electronic cata­

lysts are various forms of pure or supported nickel,

7

8

palladium, or platinum, and activated charcoal.

Acid catalysts are generally the most effective for

the isomerization of hydrocarbons and may be divided into

three groups on the basis of chemical constitution: (23)

1. Acid ha1ides.

2. Hydrogen acids.

3. Acidic chalcides.

Aluminum chloride and aluminum bromide are the most active

and most widely used of the acidic halides. Hydrogen

acids (Bronsted acids) are definite molecules with readily

ionizable protons. They are essentially fully hydrated

chalcides (oxides) or can be regarded as derivatives of a

fully hydrated chalcide. Sulfuric acid and its derivative

organic sulfonic acids are typical examples.

The elements of Group VI A of the Periodic Table

have been called chalcogens, a term analogous to halogens

used for the elements of Group VII A, This group includes

oxygen, sulfur, seleniiom, and tellurium. Compounds of

these elements are sometimes called chalcides and chalcide

catalysts include a large variety of solid oxides and sul­

fides. The most widely used are alumina, silica, and

mixtures of alumina and silica in which other oxides, such

as chromia, magnesia, molybdena, and zirconia, may also be

present.

In contrast to sulfuric acid, which may be regarded as

a fully hydrated chalcide, the chalcides of this group are

seldom very highly hydrated under conditions of use. It

is probable, however, that adsorbed protons are essential

to their activity as catalysts. They are sometimes acti­

vated for use by treatment with aqueous mineral acid and

then dried at high temperatures. (46, 47)

The acidic chalcides are physically and chemically

stable and catalytically active at temperatures approaching

the threshold for thermal decomposition of hydrocarbons.

They are not sufficiently acidic to form stable complexes

with unsaturated hydrocarbons, as do the aluminum halides.

The acidic chalcides are frequently used at high temper­

atures and are preferred for isomerization of unsaturated

hydrocarbons, which are polymerized by strongly acidic

catalysts at low temperatures. Polymerization is thermo-

dynamically disfavored at high temperatures.

Chalcide catalysts are not very effective for iso­

merization of saturated hydrocarbons unless they possess

electronic as well as acidic properties. Electronic pro­

perties (hydrogen activity) may be imparted by the presence

of a transition metal or a transition metal oxide, such as

cobalt, nickel, platinum, or molybdena, tungsten oxide,

and zirconia. The use of hydrogen with these catalysts is

beneficial and may be essential in the isomerization of

saturated hydrocarbons, (23)

The isomerization of paraffins in which either the

reactant or product has no tertiary carbon is achieved only

10

at conditions under which most of the yields are cracked

products. Acidic-electronic catalysts having both cracking

and hydrogenation activity are often effective for isomer­

ization of saturated hydrocarbons at temperatures below

those required for cracking.

There is a general interest in acidic and acidic-

electronic catalysts in this study. Of particular interest

are the various aluminas and platinum impregnated aluminas.

The physical properties, surface chemistry, and catalytic

properties of eta and gamma aluminas and of nickel and

platinum impregnated eta and gamma aluminas have been

reported in a series of publications. (46, 47, 69) The

determination of surface areas of the various aluminas,

heated aliimina hydrates, and the thermal transformations

of aliiminas and alumina hydrates has also been reported.

(57, 58, 68)

2.2.0 Hydrocarbon Isomerizat ion-Dehydro isomerizat ion

Mechanisms

A study of the kinetics of a reaction has often given

much insight into its mechanism. However, this has not

been so for catalytic isomerization-dehydroisomerization

of hydrocarbons, for here the rates of reaction depend on

many hard-to-control parameters. The system is hetero­

geneous and the rate of reaction may depend on the degree

of turbulence in a fixed bed reactor, particle size and

11

surface area of the catalyst, and other factors influencing

the nature of contact between catalyst and feedstock.

Traces of impurities in the feedstock have a profound

effect on their concentration on the catalyst surface at

which they accumulate and therefore the magnitude of their

effect changes with time. Catalyst activity is sometimes

hard to duplicate in separate preparations and often declines

with use. These many hard-to-control factors must be shown

or assumed to be constant, or must be corrected for, before

an observed variation in rates can be attributed to a con-J

troliable independent variable under investigation. \

A mechanism advanced to explain the isomerization of

saturated hydrocarbons with so-called dual function catalysts

involves the formation of olefins as intermediates. (21,

53, 74) The function of the hydrogenation-dehydrogenation

component is to establish equilibrium between saturated

hydrocarbons, olefin, and hydrogen. Only a small amount

of olefin would be present at equilibriiim under the condi­

tions used. The function of the acidic component is then

to isomerize the olefin, presumably by a carbonium ion

mechanism. The rearranged olefin is then transported in

either the gas phase and/or on the catalyst surface

to a hydrogenation site where it is either saturated with

hydrogen or dehydrogenated to an aromatic. Observations

by different experimenters on several reaction systems

have suggested that a mechanism is

12

Aromat ic Meta 1 Acid ic Met a 1 .'

Saturate ^—^ Unsaturate ^~" ^ Rearranged ^^^ site site unsaturate \

site X ^ Rearranged saturate

Some evidence is available supporting a mechanism in which

isomerization of saturated hydrocarbons proceeds via gas

phase olefin intermediates that were formed on platinum

sites of the catalyst. (53, 60, 65, 74) The independence

of isomerization rate on platinum content (dehydrogenation

activity) above a certain level has been interpreted as

indicating that the formation of olefin intermediates is

not a limiting factor in the reaction. (65) A similar

type of interpretation for the dehydrocyclization was not

adequate since the rate increased with an increase in

plat inum content.

It has been suggested that the role of hydrogen with

the dual function catalysts (74) and cracking catalysts

(62,64) is primarily to keep the catalyst surface clean

of hydrocarbon residue and thus maintain its activity.

Hydrogen minimizes the concentration of unsaturated hydro­

carbons that form a complex with the catalyst and the

concentration of carbonacious residues on the catalyst

surface that reduce its activity.

In cracking studies with platinum free alumina cata­

lysts, the hydrogen pressure had an effect on the

13

paraffin-to-olefin ratio in the reaction products and was

interpreted as evidence for the hydrogenation activity of

alumina catalysts. (40, 64, 77) These works did not mention

if corrections for homogeneous hydrogenation were made.

Sinfelt (63) noted that the predominant reaction of

methylcyclopentane over alumina was ring splitting with

essentially no isomerization to cyclohexane, whereas over

platinum supported on the same alumina appreciable isomer­

ization-dehydro isomerizat ion to cyclohexane and benzene was

observed. (62) This suggested that the intermediates invol­

ved in these reactions were different in detail, although

both were presumably of the carbonium ion type.

Barron et.al. (4) and Maire et al. (49), after study­

ing the hydrogenolysis of various cyclic hydrocarbons on

metal films and on the corresponding metal-on-alumina

catalysts at very low pressures, suggest that at low temp­

eratures (less than 600 °F) where the alumina is not very

acidic, the isomerization of hydrocarbons takes place on

the platinum hydrogenation sites and not on the alumina

carrier. In a second publication, Barron et al. (5)

stated that the alumina carrier plays an important part

in the isomerization and ring enlargement on platforming

catalyst at higher temperatures, but that the platinum

by itself catalyzes a large part of the isomerization and

aromatization. Anderson and Avery (2), from molecular

orbital calculations have similarly concluded that

14

isomerization on a platinum on alumina catalyst takes place

by a dual-function mechanism involving carbonium ion rear­

rangement on the acid function of the catalyst and that

this is augmented by another isomerization process occur­

ing on the platinum alone.

In siimmary, the roles of the constituents of a plat­

inum on alumina catalyst remain unclear. The isomerization

acid function (alumina) has been reported as having hydro­

genation characteristics. The hydrogenation electronic

function (platinum) has likewise been reported as having

isomerization characteristics. The platinum isomerization

function must be relatively weak at higher temperatures

and pressures or the platinum must be specially treated,

since this experimenter found essentially no isomerization

and subsequent dehydrogenation of methylcyclopentane over

a platinum on 5-alumina catalyst, although the dehydro­

genation activity was present. (32, 33) The isomerization

reaction did occur, however, on platinum on t|-alumina

under similar thermal and mechanical conditions. The degree

of dispersion of the platinum on the alumina carriers may

determine the strength of this function. It appears more

likely, however, that since the acid sites of the 'y-alumina

are stronger than on the ^-alumina (46, 47), that the

isomerization reactions take place on the alumina sites.

CHAPTER III

KINETICS OF HETEROGENEOUSLY CATALYZED REACTIONS

3.0.0 On Epistemology

Theories of knowledge are generally classified as

either rationalistic or empirical. Rationalists hold

that some ideas are self-evidently true, independent of

experience, and known by reason only. Empiricists believe

that all ideas are derived from experience, and the truth

of any idea can be established by reference to experience

alone. The doctrine by which empiricist philosophy has

been inspired is that all human knowledge is uncertain,

inexact, and partial.

P. W. Bridgman (17) has stated that no epistemology

can be logically rigorous, but between rival epistemologies

it can only be a question of which is logically more toler­

able in a particular setting. The mode today appears to be

more toward extreme empiricism, or phenomenalism. Pheno­

menology limits knowledge to phenomena only, its object

being the scientific description of actual phenomena, with

avoidance of all interpretation, explanation, and evaluation

It is an attempt to obtain the most objective observations.

The term "scientific description" in its definition leads

to the realization that even the most objective of obser­

vations are steeped in conventions that are adopted at the

15

16

outset and by forms and habits of thought developed during

the course of investigation.

3.0.1 Kinetic Model Fitting

In dealing with complex reaction systems, it is often

expedient to initially take a phenomenological viewpoint

and use rate models that exist somewhere in the spectrum

between purely empirical interpolation polynomials and

precisely formulated theoretical models. If prevailing

environmental and system conditions allow, a rationalistic

approach may then result in insight which could be useful

in generalizations or data extrapolation.

The Hinshelwood model based upon the Langmuir theory

of adsorption has provided a pragmatic approach to the

correlation of experimental rate data for heterogeneous

catalytic reactions, Hougen and Watson (42) pioneered in

this approach and showed the significant features of the

method. There exists in the literature, however, (10, 45,

76) considerable discussion of the need for analyzing data

with the resulting relations. It has been proposed that

many heterogeneous reactions, in particular, gas-solid

catalytic reactions, can be represented by the simpler

pseudo-homogeneous (noncatalytic) rate equations instead of

the more complex heterogeneous (catalytic) forms. Lapidus

and Peterson (45) have noted that such problems as degeneracy

of the heterogeneous equations and errors in measurement

17

obscure some effects in determining the best model. They

conclude that when these difficulties are involved, it is

questionable that the relatively complex heterogeneous

rate equations are more warranted than the simpler pseudo-

homogeneous equations. Wei and Prater (73) have questioned

the Hougen and Watson approach because of its sometimes

dependence on nonmeasurable intermediate species and

because of the difficulty in obtaining models which fit

the data and in which the parameters are uncorrelated and

"realistic". They propose a different, more complex ap­

proach to first order and pseudo-first order chemical

reaction systems, but it is not known if the approach can

be extended to a general catalytic reactor,

3.1.0 Method of Hougen and Watson

The method of Hougen and Watson (42, 43, Chap. 19)

based on the Langmuir-Hinshelwood approach has been widely

accepted as a means of representing heterogeneous catalytic

reactions involving a bulk (fluid) phase composed of pro­

ducts and reactants and of a catalyst phase. In the so-

called Langmuir-Hinshelwood mechanisms, all the reactions

take place between species adsorbed on the surface of the

catalyst. Those reactions which take place at the surface

by reaction between an adsorbed species and one from the

homogeneous fluid phase are sometimes referred to as Rideal

mechanisms. The distinction between the two types is not

always clear, (7)

18

In commercial applications the fluid phase is often

gaseous and the catalyst is dispersed in a porous solid

matrix, The Langmuir-Hinshelwood approach is to build a

model based on a sequence of steps and to assxime that a

particular step or combination of steps provide the control­

ling rate, all other steps being near equilibrium. The

assumption of a controlling step is essential in the

modeling in order to prevent a large number of parameters

in the modeling system whose determination would require

an excessive number of experimental data points. Also,

computationally, multidimensional search techniques for

optimum parameter values for a large number of parameters

do not always yield a unique best set of parameter values.

The steps usually considered in the model building

are: (42, 43)

1. Diffusion of reactant to external surface of

catalyst particle.

2. Diffusion into the pores of the catalyst.

3. Adsorption of the reactants onto the catalyst

surface.

4. Actual chemical reaction or series of reactions

taking place in the adsorbed phase on the catalyst surface.

5. Desorption of product molecules.

6. Diffusion of products to exterior of catalyst

pellet.

7. Diffusion from pellet to fluid phase.

19

In practice, the presence of steps (1) and (7) can be

determined if they are influential. In a fixed bed catalytic

reactor the classical diagnostic test for the presence of

concentration and/or temperature gradients external to the

catalyst is to determine conversions at various levels of

flow rate with space velocity (F/W, gmol feed/hr/unit weight

of catalyst) held constant. The conversion X» is indepen­

dent of flow rate level in the absence of external gradients.

Chambers and Boudart (19) have pointed out that this test

may be insensitive at low values of modified Reynolds

number, NRQ = d GAU • If external gradients are found or

suspected, compensation can be made for their effect, as

shown in the next two subsections.

3.2.0 External Diffusion

The passing of fluid over the surface of a pellet causes

a boundary layer to be developed in which the velocity para­

llel to the surface varies rapidly over a very short distance

normal to the direction of flow. The fluid velocity ap­

proaches the bulk stream velocity at a plane not far from

the interface, and decreases to essentially zero at the

interface. In the region very near the interface, the fluid

velocity is low and there is little mixing; consequently

transport normal to the surface is by molecular diffusion

and very near the surface the rate of diffusion is propor­

tional to Dj.. , the molecular diffusion coefficient of

I I ( ;;l 1 > •

20

component i in the mixture. In the turbulent main fluid

stream, mass transfer is essentially independent of Dw,,

The overall process of transport between pellet and fluid

is therefore found experimentally to be proportional to

^Mi ^here n is an empirical constant with a value between

zero and one. (59)

Data on mass transfer have generally been correlated

by an expression of the form

% = ' Gi (Pi - Psi) (3-1)

where N^ = mass flux of component i (moles/sec cm^)

^Gi ~ J ss transfer coefficient for component

i (moles/sec cm^ atm)

i> ^Si " partial pressures of component i in the

bulk and at the surface respectively.

Chilton and Colbum, (20) from a dimensional analysis of

mechanisms involving both turbulent and laminar flow,

suggested as a basis of correlation of mass transfer data

the equation

JD = ^^^^^" yU / (3.2)

L GM J e " Mi

where JQ = mass transfer number (dimensionless)

^fi ~ P^®ssure factor for component i (atm)

Gw = molal mass fltix of gas based on total cross

section of bed (moles/sec cm^)

yU = viscosity of mixture

Q = density of mixture.

21

The methods of obtaining the physical properties required

by the correlations are presented in Appendix A. A com­

puter program utilizing these methods, similar to that of

Holmes and Baems (41) is listed in Appendix C.

For the case of flow through packed beds, the Reynolds

number, upon which JQ depends, is modified and is defined as

NRe„> = S°//« (3-3)

where G = mass fl\ix of gas based on total cross section 2

of bed (grams/sec cm )

d = effective diameter of packing (cm),

defined as the diameter of a sphere with

the same external surface area as the

catalyst particles.

DeActic and Thodos (25) presented a correlation of

simultaneous heat and mass transfer data in which J^

was found to be

Jj5 = 0.725/(NRO-^1 - 1.5) . (3-4)

They stated that this relationship was valid over a

modified Reynolds number range of ten to two thousand.

In order to evaluate the pressure factor required in

equation (3-2), the particular system under study must be

considered. In the case of one way diffusion through a

stagnant film (such as evaporation of water from a surface

into air), the pressure factor is equal to the logarithmic

22

mean value, between the interface and the main ambient

stream, of the partial pressure of the non-diffusing gas.

In the special case of equimolar counter-diffusion, the molar

flux normal to the main stream is zero, and the pressure

factor is equal to the total pressure. In most cases, how­

ever, there is a flux normal to the surface. Where a bulk

flux of fluid occurs in the direction of diffusion, then

additional transfer of mass occurs in this direction. In

order to account for the effect of bulk flux, the pressure

factor must be used rather than total pressure. The pressure

factor (43, p. 978) for component A is given by

PfA = ^T " ^A^^ t b - r - s)/a (3-5)

where a, b, r, and s are the stoichiometric coefficients

for the general reaction

aA *»• bB -^ rR • sS. (3-6)

The pressure factor is calculated by taking the logarithmic

mean of the pressure factor calculated from bulk conditions

and the pressure factor calculated from surface conditions.

This is an iterative procedure since the surface conditions

cannot be measured. The necessity of using the pressure

factor term arises because the reported correlations of

J^ were obtained from experiments using the evaporation

of water into a stagnant film.

23

The diffusivity (Dj ) required to evaluate the Schmidt

number (M/Q ^m) In equation (3-2) must be the effective

diffusivity of component i in the multicomponent mixture

under conditions of a net molar flux of each component.

Starting with the Stefan-Maxwell equations, Stewart (8,

p. 571), developed an equation for the effective diffusi­

vity in terms of the binary diffusivities:

^ =

/n y —^ ^fA

(Yi - Y^(Ni/N^))/D^i (3-7)

Where Y^^ = 1 - Y^(a + b - r - s)/a (3-8)

YA = mole fraction of component A

DA- = binary diffusivity of component i in

component A (see equation A-12)

N. = mass flux of component i.

In a simple reaction system, the flux ratios N^/N^

are equal to the ratio of the stoichiometric coefficients.

In a reaction system in which competing reactions occur, the

flux ratios are equal to the ratios of the rates. Any one

of the products or reactants may be chosen as a basis for

normalization, but there is usually one that is of more

interest or is known more accurately.

24

3.3.0 External Heat Transfer

Heat and mass are transferred between the catalyst and

the flowing fluid by analogous mechanisms provided that

heat transfer by radiation is negligible. Radiant heat

transfer within the catalyst bed was not found to be

significant at the conditions of the present investigation.

The J factor for heat transfer is analogous to the Jj

factor for mass transfer and was defined by Chilton and

Colbum (20) as

JH = ^G

Cp^M

%/i 2/3 • (3-9) i

JL"^ Jfi ilm

2 where h^ = heat transfer coefficient (cal/sec cm °K)

C = molal heat capacity (cal/mol °K)

k = thermal conductivity (cal/hour cm ®K/cm).

DeActis and Thodos (25) found the J-^ factor to be

adequately correlated over the modified Reynolds number

range of ten to two thousand by the expression

J^ = 1.10/(NR2.'^1 - 1.5) . (3-10)

In a heterogeneously catalyzed reaction system at steady

state, if the reaction rates are known, the temperature

difference between the catalyst surface and the local

fluid stream may be calculated from a heat balance,

yielding

25

Tfl . T = ( £ r [ AH^i)/hQ (3-11)

Where Tg, Tg = bulk temperature and surface

temperature respectively

r! = reaction rate of product i

(moles/sec cm^)

/\ Hj j = heat of reaction to product i

(cal/mol).

Temperature gradients are generally unaffected by ordinary

mass fluxes.

3.4.0 Internal Diffusion. Effectiveness Factor

Steps 2 and 6 of the Langmuir-Hinshelwood approach

(section 3,1,0) deal with the diffusion of reactants and

products within the pores of the catalyst. In most com­

mercial catalysts, most of the catalytic surface area is

contributed by the internal area that is exposed by the

pores. Under conditions of fast reaction and/or small pore

diameter, however, the internal surface may not be as effec­

tive as that at the exterior because of pore diffusion in­

fluence leading to concentration gradients and reduced rates.

(66) Pore diffusion may be a composite of ordinary diffu­

sion, surface diffusion, and Knudsen diffusion, and in most

cases, cannot be isolated from the surface phenomena (steps

3, 4, and 5), (66, p, 232) Conventionally, these cases have

been treated by basing the rate of reaction upon the total

26

catalytic surface (internal and external). An effectiveness

factor is then introduced to take into accotint the additional

diffusion resistance for the part of the reactants that react

on the internal surface of the pores. This approach is not

necessarily rigorous because the kinetics of the reaction

deep in the pores may not be the same as near the outer

surface.

The effectiveness factor 77 is a measure of the pore

diffusion influence. It is defined as the ratio of the

actual reaction rate to that which would occur if all the

surface throughout the inside of the catalyst particle

were exposed to reactant of the same concentration and

temperature as that existing at the outside surface of the

particle. (59) Techniques for the calculation of effec­

tiveness factors for general reactions have not been devel­

oped. It has been stated (67) that in all probability no

generalization exists.

The calculation of effectiveness factors has been

worked out for the cases A^B, A-vnB, and isothermal and

polytropic A-*-B. These are summarized by Satterfield and

Sherwood, (59) They generally require catalyst character­

ization data, approximations of effective diffusivity of the

reactants and products in the pores, and approximation of

surface reaction rates. The development of a calculational

technique for a specific case is certainly not a trivial

undertaking. It is doubtful whether any useful relation-

27

ships can be developed for complex reaction systems composed

of competitive reactions, although a rough check for pore

diffusion effects might be made assuming first order kin­

etics for each reaction. The exact solution involves the

simultaneous solution of a system of differential equations

which can possibly be nonlinear and which have both initial

and final boundary conditions. The equations are partial

differential equations for any catalyst geometry other

than spherical.

A general equation of diffusion and reaction in the

pores of a spherical pellet will be derived. The effec­

tiveness factor for the special case of irreversible A-^B

will then be presented without derivation.

Consider a spherical catalyst pellet with n average

pores. The mean pore radius may be approximated by (66)

r = 2V^/S^ (3-12) S S

where V = pore volume, cm /gram 2

and S = surface area, cm /gram, g

An estimate of the mean pore length may be obtained from

L =VpfT/S^ (3.^3)

3 where V = volume of single particle, cm

o and S = external surface area, cm^.

28

An average pore with the plane of symmetry at the center

of the catalyst particle is depicted in Figure 1.

External surface of particle

Center-line of pellet

Figure 1

Average Pore in Spherical Catalyst Pellet

The concentration gradients at the center line are zero.

The concentration of component i at the outer surface is

^io ^^^ ^^ ^^y distance x measured from the outer surface

is C.. A steady-state mass balance on an element of pore

dx is

(Rate of diffusion into element) - (Rate of diffusion

out of element) - (Rate of reaction) = 0.

Mathematical expression of this statement results in a

system of equations composed of (k - 1) differential equa­

tions where k equals the number of components present in the

reaction system:

-2 ,..,,.., ,2. . X---2 -D^(dCi/dx)nr -(-D^) (dC^/dx 4- dfc^ dx)7Tr

-r^27xr dx = 0.

29

These expressions may be reduced to

D^ r d^C^/dx^ - 2 r^ = 0 (3-14)

where D. = diffusivity of component i in the pores

r = pore radius

X = distance measured from outer surface

0^ = concentration of component i

^4 = generation rate (form depends on surface

model).

In the simplified first order case of A-rB irreversi-

bly, the expression (3-14) may be further simplified to J

2^ , ...2

r2k' V?D

d^C^/ dx' = 2k»CA/(rD) •

where k* = a first order rate constant/unit of

surface area.

The boundary conditions are jl;

°A = C^o «t ' = 0 *'-• •

dC./dx =0 at X = L

The solution of this differential equation and subsequent

manipulations may be found in Smith. (66, p. 269) The

effectiveness factor rj is defined as

^ = tanh (h) = factual rate 7] (3-15) ' h I rate if no diffusional resis./

I— J pore

where h is the Thiele modulus defined as

h = L r2k' . (3-16)

30

The Thiele modulus is concentration dependent for reaction

rates other than first order.

Experimentally, the effectiveness factor can be de­

termined most accurately by measuring reaction rates on

several catalyst particle sizes under otherwise identical

conditions. The effectiveness factor approaches unity

when no increase in rate per unit quantity of catalyst

occurs on subdivision. If data are available on finely

divided catalyst having an effectiveness factor approach­

ing unity, then the ratio of rate per unit quantity of

catalyst for the larger size to that for the fines is

directly equal to the effectiveness factor of the larger

size, A general criterion for absence of significant

diffusion effects can be taken as those conditions for

which yi exceeds about 0.95. (75)

The solution of the general system of equations (3-14)

will be nonlinear if the surface models used are nonlinear

in the concentrations. A predictor-corrector type numerical

solution will probably be necessary for this type system.

It is also possible that the numerical solution will be

unstable since the final boxmdary conditions involved are

that the changes in concentrations are equal to zero and

in reality some of the concentrations may become constant

before the final boundary is reached.

In the mechanistic analysis of complex reaction systems,

it can only be hoped that pore diffusion corrections are

31

negligible or that the parameters which are obtained in the

attempted description of the surface rates account for this

effect. Smith (66, p. 267) states that if internal diffu­

sion is present, a careful kinetic analysis, say by Langmuir-

Hougen and Watson approach, is not warranted,

3,5,0 Adsorption

Fluids can be adsorbed on solid surfaces by physical

adsorption or by chemisorption. Physical adsorption is

only of secondary interest in the study of heterogeneous w

catalysis. When it occurs, the adsorbed molecules are held J

by weak bonds. The quantity of heat evolved from the process

is of the same order of magnitude as a latent heat. The

process is readily reversible and the equilibrium between

the adsorbed and fluid molecules is rapidly attained. In

this case the adsorption process does not significantly alter

the interatomic forces, and as a result there is no redis­

tribution of energy states. There could therefore not be a

significant change in the energy of activation for the reac­

tion from that in the fluid phase. Catalytic behavior resul­

ting from the adsorption on the solid surface would be small.

Physical adsorption is of value, however, in studying phy­

sical properties of a porous catalyst, such as surface area.

With chemisorption the adsorbed material is held by

forces of the same nature as exists between atoms. In effect,

there has been a "reaction" between the fluid and solid

4,

it

32

during the adsorption process. The heat evolution is of

the same order of magnitude as in a chemical reaction and

bond energies are redistributed altering the activation

energy of the desired reaction from that existing prior

to chemisorption. This process results in catalytic beha­

vior (increased rate) if the activation energy is reduced.

A quantitative treatment of the adsorption process will

not be presented here but may be readily found in standard

references. (43, 50, 54, 66)

It is important to note, however, that if a reaction

takes place among molecules or species specifically

adsorbed on the catalyst surface, there may not be a

simple relation between the concentrations on the surface

and the pressures of reactant species in the fluid phase.

In the absence of precise adsorption isotherms, the

molecular interpretation of rate laws for catalytically

controlled reactions is subject to considerable uncertainty.

(7, p. 627) For this reason, whenever possible, the absorp­

tion isotherms of the reactants and products should be

measured independently and as close to the experimental

conditions of the kinetic runs as is practical.

It is interesting to note (7, p. 662) that since

chemisorption, which is a prerequisite for catalysis, has

an activation energy, the chemisorption of the reactants is

generally slower than the desorption of the products. This

means that products in the fluid phase are often in

33

equilibrixjm with some species in the adsorbed phase. The

resultant is that either step 3 or 4 of section 3.1.0 is

usually the controlling step.

3.6.0 Analysis of Heterogeneous Reaction Rates

It is useful to consider adsorption and desorption as

individual steps in a series of chemical reactions that

occur sequentially in the conversion of reactants to products.

If one of these steps in the series is assumed to determine

the over-all rate of reaction, the others may be assumed ^

to be occuring at near equilibrium conditions. j

In testing whether a step in a postulated mechanism

may be rate controlling, the following procedure is often

useful: (54)

1, If external concentration and/or temperature grad­

ients are present, they are compensated for by the methods

of section 3,2,0 and 3.3,0,

2, An attempt is made to determine the influence of

pore diffusion on the over-all rate.

3, In the absence of internal diffusion, an adsorp­

tion-surface reaction-desorption mechanism is postulated,

A rate expression is derived asstiming one or more of the

steps in this mechanism to be rate-controlling and the others

to occur at near equilibrium conditions,

4, The values of the parameters are determined by the

use of the experimental data and regression techniques.

34

5. The derived rate expression with "best fit" para­

meter values is tested to see if the experimental data can

be predicted with reasonable accuracy or if a detectable

bias exists.

This testing procedure cannot ascertain the true

mechanism of the reaction. The strongest statement that

can be made is that a mathematical equation was developed

which agrees reasonably well with experimental data over

the range studied.

The derivation of a postulated mechanism is demonstra­

ted in Perry, (54, p, 4-13) A table summarizing the rate

expressions resulting from several assumed mechanisms with

different rate controlling steps is also presented.

3,7.0 Catalyst Poisoning

Catalyst poisons are substances which exert an appre­

ciable inhibitive effect on catalysts even when they are

present in trace quantities. They are frequently strongly

adsorbed on the catalyst surface, blocking active centers.

This blocking may result in decreased catalyst activity, a

change in selectivity, or the catalyzation of additional

undesirable by-product reactions. Simple deposition of

inert material may also occur on the catalyst surface

(e,g,, carbon deposition on cracking catalysts). The

result of the deposition is the physical covering of the

active catalyst sites and/or in blocking off the pores.

35

thus making the interior of the catalyst inaccesible.

Some poisons cause phase changes in the catalyst (e.g.,

water-vapor poisoning of platinum-alumina catalyst).

For some catalysts, if catalyst activity is plotted

against accumulated surface poison concentration, the

activity decreases linearly with increasing poison concen­

tration over a large part of the poisoning curve. If poi­

sons are present in a catalyst system, the effectiveness

factor (section 3,4,0) is a function of time until the

poison concentration reaches an equilibrium value. Analytic

solutions have been obtained for effectiveness factors as a 3

function of time for simple cases, (54, 59)

n ' . • •

CHAPTER IV

MATHEMATICAL MODELING

4.0.0 Mathematical Modeling

Mathematical modeling is an integral part of the study

of the kinetics and mechanisms of reacting systems. Some

of the general modeling schemes have been known for over a

century, but were impractical before the advent of the

computer. The accumulation of computational experience

has resulted in the modification of the classical schemes

into practical algorithms, especially within the last five

or six years.

There are many ramifications of the general modeling

problem, and each application has to be considered in the

light of its own needs. In the mathematical modeling of

heterogeneous catalytic systems, the majority of the rate

equations are nonlinear in the parameters. This nonlinear-

ity necessitates a study of iterative techniques used to

obtain parameter estimates. Problems associated with iter­

ative techniques are those of convergence, extreme nonlin-

earity, and of obtaining good initial parameter estimates.

Confidence intervals and regions can only be rigor­

ously derived for linear models, but useful approximations

exist for nonlinear models as well. Box and Lucas (14)

have pointed out that the functional form of the hyperbolic

36

37

models (Hougen and Watson) is such that the parameter

estimates are usually highly correlated and the parameters

are poorly estimated. White and Churchill (78) have remarked

on the problem of "over-fitting" data, that is, where the

number of parameters contained in a model is too great for

the range of the experimental data. The problem of high

correlation between the parameters is attacked by repara-

meterization. Reparameterization is achieved by writing

the model in an equivalent mathematical form with differ­

ent parameters. Reparameterization also often helps the

convergence of the search for the best parameter values.

The adequacy of a model is checked by the residuals

(actual value minus predicted value). All information in

the data about the inadequacy of a given model is contained

in the residuals. An analysis of variance is an overall

test that is essentially concerned only with the length of

the residual vector. A residual analysis is used to examine

the direction of this vector. This is usfeful because it is

sometimes possible that the overall fit is adequate, but

subtle model inadequacies such as biases exist. Residual

analysis is the key to model development or tuning.

Experiment designs using the models to obtain maximum

information and to discriminate between rival tentative

models are also two facets of interest, but will not be

discussed here.

38

4,1.0 Linear Lest Squares

Consider a model which can be written in the form

Y = X ^ t £. (4-1)

where X ^s an (nxl) vector of observations

X is an (nxp) matrix of known mathematical

variables,

^ is a (pxl) vector of parameters,

and £_ is an (nxl)vector of errors (or residuals).

The underlined variables represent matrices or vectors.

Rules of matrix manipulation may be found in standard

references, (39)

The assumptions that are most often made involving

probability distributions are (3, 26)

1, ^ is a vector composed of E. , i = 1, 2, •••, n

random variables each with a mean zero; i,e, E(^) = 0,

where 0 is the null matrix.

2, Each €i has the same variance CT (unknown),

3, The vector elements €^ and €. are uncorrelated,

i j, so that

cov( e^, ^j) = 0,

The assumptions (2) and (3) are written as

V(£.) = IC^

where I is the identity matrix,

4, £ is a normally distributed random variable, with

39 2

mean zero and variance cy .

£^— N(o,cr2).

The last assumption is not as restrictive as it appears.

If the errors can be regarded as being a sum of other inde­

pendent errors, the Central Limit Theorem indicates that in

many cases the observations will be approximately normally

distributed, A more critical assumption is (3) that the

errors in the observation are independent. The dangers

of violating this assumption can frequently be reduced by

careful experiment design.

Since E(€^) = 0, an alternative way of writing equation

(4-1) is

E(I) = Xyd . (4-2)

The error sum of squares (or sum of square of the residuals)

in matrix notation (26, 39) is then

£JS^ = (I - X^)»(Y - X ^ ) |j|

= VI - 2^»X»Y • A*X*2L£. (4-3)

where the primes denote the transpose of a matrix.

The least squares estimate of S is the value b which

minimizes ^*J^, This estimate b is obtained by differenti­

ating equation (4-3) with respect to £^ and setting the

resultant matrix equation equal to zero. This yields the

normal equations

(X*X)b = X»Y (4-4)

whose solution is

%u c

40

k = (X'X)"'^^'! (4-5)

providing (X»X)"^ exists (X'X is nonsingular). If X'X is

singular, there are fewer than p independent equations in

the p unknowns, in which case additional constraints are

needed for the parameters or the model needs to be recast

in terms of fewer parameters, A

The predicted values are obtained from Y = )Cb, the A

residuals from e = Y - Y, and the variance and covariances

from V(b) = (X«X)"^cr^.

If assumptions (1) through (4) hold (i.e. £^ N(0, JLCT^),

then b is the maximum likelihood estimate of B^ where the

Fisher likelihood function (29) for the n data points is

given by

exp -ej^lKS^ (4-6)

Q-n(2-^)n/2

For a fixed value of <5", maximizing the likelihood function

is equivalent to minimizing the sum of squares of the

residuals £ ' €.

\

C 1

V

41

4,2,0 Nonlinear Least Squares

In reaction kinetics, the independent variables ^ .

can be temperature, pressure, feed flow rate, feed compo­

sition, catalyst weight, and in some cases, the age and

physical and chemical composition of the catalyst. The

parameters 0^ are usually complicated lumped combinations

of thermodynamic equilibrixim constants, chemisorption

equilibrium constants, and reaction rate constants. The

dependent variables are the exit concentration of the

reaction intermediate and final products, from which can

be deduced by data reduction, the rates of formation of

the individual species. The relationships between the

dependent and independent variables are usually nonlinear

in the par£uneters, requiring special mathematical modeling

techniques.

Consider a vector of settings of k independent variables

1 = (^1. ^ 2 ' •"' ^k>' and a vector of the p parameters present in a model

The equation for a generalized model for the dependent

variable Y can be written as

Y = f(l., fi) • € . (4-7)

If there are n observations of the form

Yu» ^ lu» S 2u» •'•• ^ ku

for u = 1, 2, •••, n, the model can be written as

42

Yu = f(£.u» i) -^ €^' (4-3)

The error sum of squares for the nonlinear model and the

given data is defined as

S(£) = ^ 6 ' = ^ (Y^ - f(l^, Q))2 (4-9)

Since Y^ and ^^ are fixed observations the sum of squares

is a function of 0. The vector 0 is the least squares esti­

mate of ©which minimizes S(£). If £,-'^N(0, iCT^), the

least squares estimate of 0 is also the maximum likelihood

estimate of £,

To obtain 0, equation (4-9) is differentiated with

respect to 0, yielding p normal equations to be solved for

9, The normal equations are of the form

y (.\ - f(£.u. £))raf(£_a. £)1 ^ - = 0 (4-10)

for r = 1, 2, • • • , p, The teirm inside the brackets is the

partial of f (^ . £) with respect to the rth parameter ©,.

When the function f(_^^^, £) is linear in the parameters,

the resulting partial derivatives are not functions of the

parameters £.

The likelihood function is given by

.5^ exp .((Y^ - f ( ^ , £))V2G-2) i(£,cr2) =

43

wo ^ 2 ^ ^^P -(£.'jL/2<r2) 1(£,C^) = (4-11)

Cjn(2;T)n/2

The maximization of the likelihood function with respect

to the parameters is the same as the minimization of the

sum of squares function equation (4-9), The Fisher infor­

mation matrix (29) can be formed with respect to all

the parameters, including all cross partials

fii = - 3^L = 3_^£A£. (4-12)

where L = ln(lC5''(27T)"'^^) .

The true second partial derivatives of the sum of

squares fxinction can be obtained from the partial deriva­

tives of the residuals by continued differentiation (27)

^ 2 S(£) = €je_ = A € (4-13)

— I ^

as(0)

a-1 i lid^j) (a^^iaOj,

(4-15)

where £ = uth residual. If the second term in the

suuimation in equation (4-15) is dropped, a second derivative

-!

i; )

44

approximation, g^^, is obtained which can be computed from

the first partial derivatives of the residuals:

^{

This has two advantages: (a) it eliminates any requirement

for calculation of actual second partial derivatives, and

(b) the resultant G matrix (Gauss-Newton) is always at least

positive semi-definite. It will be positive definite unless

the rows of partial derivatives of the residuals with res­

pect to the parameters do not possess enough degrees of

linear independence.

It is to be noted that in the vicinity of S(G) . the • — m i n

residuals (£^) go towards zero and the dropped terms in

equation (4-15) approach zero. In the process of achieving

the minimum S(£), the residuals tend to scatter about zero

producing effective cancellation of this term during the

summat ion.

In practice, the Gauss-Newton matrix is obtained from

G = D'D

where the elements d . of matrix D are composed of partials U l — * *r

of the sum of squares function with respect to the 0.

parameters evaluated at the uth set of experimental con-

ditions, (44) The variance-covariance matrix of £ is -1 2 then approximated by G (3* .

45

4.2.1 Simultaneous Least Squares

Complex reaction systems may have either a sequence

of reactions and/or parallel reactions. In establishing

reaction mechanisms, it may be necessary to model the

rates of the intermediate species in a sequence of reactions.

In a competing reaction system, it is usually desirable to

model the rates of most of the products. There are usually

some parameters which are common in some of the models

composing the complex system. After the best fit has

been obtained for each model, the model system may be con­

sidered as a unit.

The problem of simultaneous fit of a system of equa­

tions has been considered by Box and Draper (16), and in

various approaches, by others, (6, 71, 80) The function

to be minimized is

S(0) = r (Yi„ - f(i_u,£i)) *//^2u - f(lu.£2))

4 • •

if it can be assumed that the various sets of observations

are statistically independent of each other and have the

same variance.

Box and Draper (16) have pointed out the necessity of

measurement of the intermediate species postulated to be

present in a sequential mechanism. If only the final

products are followed, the intermediate specie have to be

46

lumped together in the modeling without excessive constants

and cannot be isolated. They point out the importance of

not resorting immediately to the joint analysis of responses.

It is suggested instead, that the individual fit of each

response be studied by analyzing the residuals and that the

consistency of the information from the various responses

be considered. The consistency may be studied by looking

at the posterior distributions of the common parameters to

see if they are an estimate of the same thing, A joint

analysis is in orxier only if these two conditions are

satisfactorily met,

4.3.0 Nonlinear Parameter Estimation

When the model is nonlinear in the 0»s, the system of

normal equations (4-10) is also nonlinear. The p equations

thus obtained are usually quite complicated, almost never

possessing an explicit solution, so the usual procedure is

to resort to iterative nximerical methods. In most cases

multiple solutions exist which correspond to multiple

stationary values of the function S(Q). W. R, Carradine

and the author conducted a computer study along these

lines. (18)

There are three iterative techniques currently in wide

use industrially and academically. (27) These algorithms

are: (a) Gauss-Newton method, (b) steepest descent, and

(c) Marquardt's compromise of the first two methods. Each

47

of these algorithms is based on the repeated use of two

steps:

1, At a base point evaluate partial derivatives of

the quantity to be minimized, and from this information

determine a search trajectory in the independent parameter

space.

2, Find a stopping point along this trajectory having

a smaller value than the base point of the quantity to be

minimized. This becomes a new base for the repetition of

step (1).

Although there may be a considerable variation on the

method for choosing the stopping point along the search

trajectory, the essential requirement for reliable perfor­

mance of an algorithm is that the search trajectory display

"truncation convergence". This requires that whatever the

behavior of the functions involved at long distances from

the base point, it must always be possible to obtain a re­

duction in the quantity to be minimized by truncating the

search to a small distance. The method of steepest descent

provides a search trajectory which possesses truncation

convergence, while the Gauss-Newton method provides a tra­

jectory having this convergence only if the Gauss-Newton

matrix G is positive definite.

Gauss (31) originally pointed out that fitting of non­

linear fxinctions by least-squares can often be achieved by

an iterative method involving a series of linear approx-

48

imations. At each iteration, linear least-squares theory

is used to obtain the next approximation.

One such method of rather rapidly obtaining the para­

meter values which lead to a least-squares fit of the data

is that of performing operations on the nonlinear model

such that parameter corrections are obtained. That is, the

linear method at each iteration produces parameter correc­

tions rather than correct parameters as in truly linear

models. The process is repeated until the corrections be­

come neglible. Marquardt (51) has discussed this method.

It has the advantage of converging rapidly if the initial jj

trial parameter values are good, but the disadvantage of

diverging if they are poor. Another method which does the

same thing in a slightly different manner is presented by

Box, (15) Again, the method is quite efficient if the

initial trial values are good.

The steepest descent method is based on methodology

developed primarily by Box. (11, 12, 13) It is an iter­

ative process, the object being to proceed from iteration

to iteration in such a manner that the siim-of-squares always

decreases. This is accomplished by examinig the slope of

the contour surface at the current trial point in the

parameter space, Andersen (1), Boas (9), and Hoerl (38)

have discussed fine points and limitations of the method,

and other closely related methods.

Several attempts have been made to accelerate the

p:-

49

convergence of the steepest descent method. The technique

of Forsythe and Motskin (30) is quite successful if only

two parameters appear in the model, and Finkel (28) reports

that it is fairly useful with more than two.

The steepest descent method has been modified by Mar­

quardt (51) to give the modified steepest descent method.

Marquardt*s method represents a compromise between the

Gauss-Newton (linearization) method and the steepest des­

cent method and appears to combine the best features of

each while avoiding their more serious limitations. The

need for such an algorithm is based on two observations.

The method of steepest descent often works well on the

initial iterations, but grows progressively slower as the

minimum is approached. The Gauss-Newton method works well,

however, when the minimum of S(£) is near, but often gives

trouble on the initial iterations.

The idea of Marquardt's method can be explained briefly

as follows. Suppose we start from a certain point in the

parameter space £, If the method of steepest descent is

applied, a certain vector direction, 6 gj where g stands

for gradient, is obtained for movement away from the initial

point. Because of attenuation in the S(£) contours this may

be the best local direction in which to move to attain

smaller values of S(£) but may not be the best overall

direction. However the best direction must be within 90°

of 6fv or else S(£) will become larger locally. The

50

Gauss-Newton method leads to another correction vector 6 .

Marquardt found that for a number of practical problems he

studied, the angle, say ^, between 5^ and 6 fell in the

range 80°< <90 , In other words, the two directions were

almost at right angles. The Marquardt algorithm provides

a method for interpolating beteen the vectors 8^ and b^

and for obtaining a suitable step size to proceed towards

the minimum.

The method of Marquardt is presented in Appendix B,

A computer program utilizing Marquardt*s technique is

presented in Appendix C, Some of the features of a program

developed by D, A, Meeter (52) are incorporated into the

program,

4.4.0 Special Techniques and Interpretat ions

Maddison (48) has noted that a number of difficulties

may arise in practical applications of nonlinear regression.

The difficulties that interfere with efficient operation of

the minimization algorithm can generally be traced to impro­

per scaling, excessive parameter interaction, and/or null

effect,

4.4.1 Parameter Scaling

Parameter scaling is the division of parameter values

by scalars characteristic of the individual parameters.

This device can be used external to the minimization

51

algorithm so that the algorithm can operate on a scaled

parameter set of which the members are of similar magni­

tudes in effect.

Ideally, scaling should be designed to cause the change

in the partial derivative of the quantity to be minimized

with respect to any scaled parameter due to movement of

that parameter to be the same for all parameters:

A(..)s(&)/3 0i'\ = AOs,^a)/?) h) = ••• ( -17) A ei A 62

Geometrically, this means producing a scaled space in which

the function to be minimized is a symmetrical hyperpara-

baloid, with hypersphere contours:

S(£) = S(£)^i^ * c;^(Oi - G?)^ (4-18)

(c;:-0) ^

This is poss ible only if the minimization function S(£) is

a pos i t i ve de f in i t e quadratic, which w i l l not in general be

the case. However, it should be possible to approximate a

reasonably well-behaved S(£) by a quadratic function over a

small space around a given point. The approximation to

the second order terms in the quadratic obtained from the

Gause-Newton matrix has the advantage of always being at

l eas t pos i t i ve semi-def ini te . (27)

52

4.4.2 Interaction

Parameter interaction occurs when the partial deriva­

tive of the quantity to be minimized with respect to one

parameter is dependent on the value of another parameter.

When the Gauss-Newton matrix is used as an approximation

to the second derivative for a sum of squares function,

the degree of interaction between two parameters is iden­

tical to the correlation coefficient between the two

vectors representing the partial derivatives of the resi­

duals with respect to the two parameters. A zero correla- 3

tion coefficient indicates no interaction, and maximimi {y\

interaction corresponds to a coefficient of unity.

It is not possible to scale parameters to satisfy

equation (4-17) in the presence of interaction. Interaction

can be noted as present if the off-diagjnal elements of the

matrix of partial second derivatives are non-zero. When

interaction exists, each ratio in equation (4-17) cannot

be uniquely defined, since the change in a partial deri­

vative with respect to a parameter will depend not only on

the movement of that parameter but also on the movement of

any other parameters that interact with it. It is there­

fore important to eliminate or at least minimize parameter

interaction before scaling.

There are two general approaches for the minimization

of parameter interaction. The first possibility is to

53

perform a specific parameter transformation designed to

eliminate or at least reduce the cross-partials throughout

the parameter space. The second is to perform a normalized

linear transformation on the parameters, i.e., a rotation

of the co-ordinates of the parameter space. This approach

can locally completely eliminate interaction. It is effec­

tive over the space over which the minimization function

can be adequately approximated by a quadratic.

Parameter scaling is possible after interaction has

been minimized by some parameter transformation. Geometri­

cally, the combined process of transformation and scaling

consists of first rotating so that the co-ordinates will be

parallel with the principal axes of the contours, and then

expanding or contracting along these axes to obtain a

hypersphere. In terms of the transformed parameters, ideal

scaling is accomplished when

aisipi = 3^s(Q) = ,,. = dhcQ) . ^ , g s 3 0*2 -^^ "J^ ^ ^

and the scaling factors can be obtained as ratios from

the unsealed derivatives:

Sj/Sj = O2s(e)/ao|)/o2s(o)/3e5) (4-20)

where 0* = O^/S^ (4-21)

54

4.4.3 Null Effect

Parameter null effect occurs when perturbation of a

parameter or of some combination of parameters has no sig­

nificant effect on any of the residuals involved in the

summation of squares. Geometrically, this situation makes

it impossible to achieve the objective of converting the

minimization function to a symmetrical hyperparabaloid by

parameter transformation and scaling, since at least one

aspect of the Gauss-Newton approximation to the function

surface has no curvature. Analytically, null effect for

the ith parameter will be indicated by a zero as the ith

element on the diagonal of the Gauss-Newton matrix, since 2

for all k, 9€j^/3 0^ is zero, and g^^ =21(^^/«^^i) •

K

In general cases when a combination of parameters is invol­

ved, the zero diagonal will not be apparent until after

a parameter transformation has been used to eliminate

interaction

Since perturbation of null effect parameters has no

effect on the sum of squares function, there is no rationale

for seeking better values for these parameters. This is

similar to the case when the search for better values for

other parameters is terminated when their perturbations

have no effect on the sum of squares function.

55

4,4,4 Analysis of, Variance

The sxim of squares due to regression is given by

A 2 Z Y , = (Xb)»(Xb)

and the residual sum of squares by

SSj = 1&1 = 6.'^ = (Y - Xb)'(Y - Xb) .

The residual sum of squares has (n - p) degrees of freedom,

where n is the number of data points and p is the number

^' 2

;n - p) - (m. - 1 and has (n - p) - ^ (m. - 1) degrees of freedom, A check 3 ^

it'}

of parameters in the model, J^\

The sum of squares due to pure error comes from repli--1

cations and is given by §, \ J €

SSg = r (Y. - Y)^ (4-22) ^I

for replications at a single point and by Ijfr

SS„ = Z ^ ( Y . . - Y.)^ (4-22a)

for replications of various points. The total degrees of

freedom for the error sum of squares is ^ ( m . - 1), where J J

m. is the number of replications at point j and k is the

number of points that were replicated. The lack of fit

sum of squares is given by

SSLP = SSj - SSg (4-23)

56

of whether a model is grossly inadequate can be made by

considering the ratio of the lack of fit mean square to

the pure error mean square; if this ratio is very large

then there is evidence that the model fits the data in­

adequately. In particular, when

SS LF/"\)

ssE/^).

V i I

- ^ F.05<^'i. S)e) (*-24) e

^ k where ' V i = ( n - p ) - 2 r ( m . - l ) = degrees of

^ J 3 freedom for lack of fit

-. K and \ ) _ = 2 ( m . - 1 ) = degrees of freedom for

pure error.

4,4,5 Residual Analysis

A plot of the residual versus the predicted value, j^i"

Y, of a model can indicate whether the model truly repre­

sents the rate data. For example, residuals which are

generally positive at low rates and negative at high rates

can indicate a model inadequacy even though the overall

test of an analysis of variance indicates that the model

is acceptable.

These plots can also provide information about the

assvimption of constant error variance made in unweighted

linear or nonlinear least squares analyses. If the residuals

continually increase or continually decrease in such plots,

a nonconstant error variance could be present. Here,

57

either a weighted least squares analysis or a transfor­

mation should be found to stabilize the error variance. A

weighted least squares involves the minimization of

'V

S(£) = Zweight^(Y^ - f(£^, £))^ ,

One useful transformation is that which is used when the

errors in the observations are proportional to the size

of the observation. This might occur in experimental

situations where the data range through several orders

of magnitude and where rough error bounds on the data are

typically expressed in percentages rather than in some

absolute system of units. In these cases, it is appro­

priate to minimize

^Y\

S(£) = t . (In Y^ - In f(£^, Q)r ,

Geometrically, failure to scale residuals properly

results in elongation of the sum of squares contours,

creating unnecessary burden on the algorithm used for

minimization. In extreme cases, where residual scaling

is in error by two or more orders of magnitude, a situa­

tion similar to that caused by null effect parameters is

created. This occurs because the siim of squares function

is insensitive to changes in that subset of residuals for

which the weighting fails to produce adequate response to

parameter pertubation.

58

If the residuals are plotted against each of the

independent variables, non-randomness may be detected

with respect to one or more of these variables indicating

that they are not properly taken into account in the model.

The plot of residuals against some measure of the time

at which experiments were run can also be informative.

If the number of hours on stream or the cumulative volume

of feed passed through the reactor is used, non-random

residuals could indicate improper treatment of catalyst

activity decay. Variables not taken into account that

affect activity change might be ascertained in this

manner. ^

CHAPTER V

EQUIPMENT AND PROCEDURES

5,1,0 Equipment

An isothermal, fixed catalyst bed, flow reactor was

chosen to study the effects of the independent variables on

the rates of the various reactions of methylcyclopentane

over a platin\im on 'Vi-aliimina catalyst. The catalyst charge

to the reactor was fixed at a level that yielded low methyl­

cyclopentane conversions (less than 14 per cent) in the

practical ranges of the independent variables. The reactor

was operated isothermally and as a differential reactor (low

conversions) so that the reaction rates did not change

throughout the length of the reactor. The mathematical rate

modeling from this type of data is considerably easier than

from data obtained from nonisothermal and/or integral reac­

tors. The tentative rate models that are being entertained

in these second cases must be integrated throughout the

length of the reactor and must satisfy the final boundary

conditions. This makes the estimation of the parameters in

the rate models more difficult. Integral reactor data are

useful, however, in checking out the rate models developed

in a differential reactor. Data from commercial reactors

are of the integral type and a large portion of the commer­

cial modeling of complex reactions is done on the total

59

60

conversion instead of specific rates which must be

integrated.

The experimental system can be divided into three

parts for purposes of description: the catalytic reactor,

feed preparation and metering system, and the product

metering and recovery system,

5.1,1 Reactor Description

The catalytic reactor (Figure 2) had a volxime of about

250 cubic centimeters. It was made from 15/16 inch inside

diameter stainless steel (ss.) tube. The reactor had a

SS. wire screen for catalytic support and was provided with

a concentric 1/4 inch outside diameter ss. thermowell. The ^

reactor was mounted within a lead bath which was maintained o\ % '•'

in a molten state by electrical resistance heaters. Temp- %',

erature measurements were made using calibrated iron-

constantan thermocouples and a Honeywell indicating poten­

tiometer. In order to prevent excessive temperature grad­

ients within the catalyst bed, the catalyst was mixed with

sufficient stainless steel pellets to fill the reaction

chamber. After temperature stabilization, the reactor could

easily be maintained to within 1.2 ^F by manual manipulation

of the auto-transformers controlling the voltages to the

resistance bead-heaters. This is further discussed in the

operational procedure section 5,3,1, The reactor and

catalyst specifications are presented in Table 14 in

Appendix D,

61

Scale: approximately actual size

Reactant In

Thenmowell (may be removed to charge catalyst)

Catalyst Bed

Thermowell (1/4 inch 0,D. stain­less steel tubing)

1 inch Schedule 80-Type 347 stainless steel pipe

Product Out

Figure 2

Reactor Detail

62

'• •2 Feed Preparation and Metering

The feed preparation and measurement system is shown

in Figure 3, The system will be described by following the

feed flow from its inlet to the system to its exit,

A 4 foot section of Linde 4-A molecular sieve was

provided for drying the liquid feed. The feed system was

designed for precision metering of gases and liquids since

small errors in the flow quantities could cause relatively

large errors in the experimental results. A positive dis­

placement, metal diaphragm type metering piipip was selected

for pumping the liquid feed at highly accurate though

adjustable rates. Adjustment of liquid feed rates could

be made without shutting the piimp down.

Gaseous hydrogen was supplied from commercial cylinders,

A standard pressure regulator was used to reduce the cylin­

der pressure, after which the hydrogen passed through a 4-A

molecular sieve to remove traces of water vapor. The hydro­

gen then passed through a deoxygenating unit used to remove

small traces of oxygen by combination with hydrogen on a

palladium catalyst. The water formed here was absorbed in

a dryer packed with calcium sulfate. The purified and

filtered gas passed first through a mass flow transducer

used for flow measurement and then through a precision

needle valve which was used for controlling the flow

rate.

63

M

H,

Oxygen Removal Unit

Preheater

To Reactor

Thermocouples

Insulation

Lead Bath

Liquid Charge Graduated Cylinders

Kg X Filter

Transducer

i

J' >

Figure 3

Feed Preparation and Measurement

System

64

The two reagent streams were mixed and then conducted

into a combination vaporizer-preheater section. The vapor­

izer-preheat er consisted of a section of coiled 3/8 inch

outside diameter stainless steel tubing 25 feet long im­

mersed in a molten lead bath, A glass-packed mixing chamber

mounted downstream from the vaporizer-preheater was used to

insure a homogeneous feed to the reactor. The vaporized

and heated reactants then passed through a second coil of

tubing which was submerged within the lead bath surrounding

the reactor. This second preheating section was needed to

offset any heat loss by the gas stream after leaving the

previous vessel. The reactants entered near the top and

passed vertically down through the reaction chamber, dis- g \

cussed in section 5.1.1, | '>•

5.1.3 Product Metering and Recovery ;!,v[

?:-After leaving the reactor, the product gases passed '"

through a water-cooled condenser and into a vapor-liquid

separator (shown in Figure 4), The phase separation was

made at the pressure of the system (20 to 40 atm) and at

a temperature of about 70 ^F, After separation, the gaseous

portion of the product stream passed through a filter and

a pneumatically operated pressure control valve. This valve

was used to maintain the desired system pressure. Following

pressure reduction the product gas could be by-passed

through a gas sampling system. The product gas was metered

65

From Reactor

" ^ 1

Sight Glass

Condenser

Pressure Controller

Gas Sanpling Gas to Vent

Wet Test Meter

Condenser

Liquid Product

Figure 4

Product Recovery System

V*'

66

using a calibrated wet test meter and vented.

The liquid product from the high pressure separator

passed through a manual pressure reduction valve and into

a collection reservoir from which samples were periodically

metered and removed for analysis,

5.2.0 Experimental Development and Catalyst Deactivation

The requirements and limitations of any experimental

system should be established early in the experimental

program. A knowledge of the system limitations can some­

times lead to design modifications which make the system

more flexible. The flexibility of a research system is of

such importance that it can control the scope and scheduling

of a research program,

5.2.1 Preliminary Experimental Development

The checking of the newly constructed experimental

system for adequacy of the system and parameter design is

reported in previous work by the author and others, (32, 33)

Preliminary work must be done on each catalyst-feedstock

system to be studied to determine

1, Product composition,

2, Size of catalyst charge (determined by desired

conversion level),

3, Independent variable levels (fixed by conversion

level and system parameters fixed by the apparatus).

67

4. Catalyst stability.

In this reaction system, a 98 mole per cent methyl­

cyclopentane liquid feed along with hydrogen was fed over

an Englehardt RD-150-C reforming catalyst having a 0.35

per cent platinum on 'vy-alumina (1/16 inch diameter extru­

ded pellet, 405 square meters/ gram nitrogen surface area),

A charge of 3,5745 grams of this catalyst mixed with 230

milliliters of stainless pellets was reduced jn situ at

980 ^F for three hours under a hydrogen pressure of 500

psig, and a hydrogen flow rate of 2,4 ft^/hr. Two runs

were made to obtain product samples to use in the devel­

opment of the analytical techniques. This sequence of

development was necessary because it was not known a

priori what products would be present and in what concen­

tration ranges. The chromatographic column that was devel­

oped for the initial study (32) was found to be inadequate

in this case. The development of the analytical technique

is presented in the following analytical subsection 5.3,0.

During the interim six weeks required to develop the ana­

lytical procedure, the reactor was kept under hydrogen

pressure and at ambient temperature. The catalyst reduc­

tion procedure was repeated prior to the start of the next

series of runs of this catalyst charge.

During this next series of runs, some mechanical diffi­

culties with the chromatograph developed. These had to be

reconciled before the study could continue. This also

68

resulted in the recalibration of the analytical apparatus.

At this same time, an instability of the catalyst due to

possible poisons in the feedstocks was detected (non-

reproducible results, even after regeneration),

5,2.2 Catalyst Deactivation

The preliminary runs indicated that the catalyst acti­

vity was not constant, especially for the ring opening

products. It was deduced that the change in catalyst acti­

vity was due to the presence of traces of either nitrogen

bases or water in the feed, or to a self-poisoning effect

brought about by the carbonization and plugging of some

sites by the reacting hydrocarbons.

It is known that nitrogen bases present in a hydro­

carbon feed to a platinum-containing or silica-alumina

catalyst poison it for isomerization of cyclohexane, pre­

sumably by tying up protons. (23) Analytical methods for

the detection of and feed preparatory techniques for the

elimination of trace amounts of nitrogen bases in the feed­

stock were not readily available and the development of

such techniques was considered as infeasible for the objec­

tives of the project.

Alumina, the platinum support, is sometimes used as a

dessicant, and its physical structure and chemical proper­

ties are sensitive to moisture content. After the variable

catalyst activity was detected, Linde 4-A molecular sieves

69

were installed in the two reagent feed streams in an at­

tempt to minimize the moisture being carried to the ca'ta-

lyst bed.

It was decided to operate the reactor until the cata­

lyst activity leveled out. This was the only alternative

left if some reactions on the catalyst surface were deposi­

ting carbonaceous material. Subsequent investigations re­

vealed a 29 per cent reduction in the surface area and a

39 per cent reduction in the pore volume of the catalyst.

This made the calculated mean pore radius essentially the

same for the fresh and the used catalyst. These facts,

in addition to the change in the catalyst color from white

to medium grey, strongly indicated that pore plugging by

carbonaceous material occured. There is also a possibility

of an alteration of the alumina phase by the presence of

traces of water in the feedstocks,

A 2,1125 gram catalyst charge was activated by main­

taining it at 980 °F for 19 hours with a small hydrogen

flow rate at 500 psig, A series of runs was then made to

monitor the change in the catalyst activity. Figure 5 is a

plot of 2-methylpentane reaction rates against the total

accumulated running time. The standard runs made every

day to check the catalyst stability were at the same condi­

tions as the catalyst deactivation study and are also

plotted on this graph.

70

F igu r6 5 Goto ly$t Deact i vat ion,

2-methylpentane Rate

o X

>s ir 2 0 0

u E 0)

0 £

0)

Le0 t n d

Rates, Stdhdby conditions dftor rocKtor shutdown

A rirst day, 600 pstg hydrogen B Second day, 600 psIg hydrofl^n C Third day , 300 Psig nitrogen 0 Fourth day, 100 t»sig hydrogen K Standord runs, 100 pilg hydrogen

o

11

^ ' ; '

— i

To 2(5 3(5 45 5b 6 " Ti me , Hours

75 80 90

wi.

71

Curve A of Figure 5 shows the catalyst deactivation

for the formation of 2-methylpentane during the first day.

After shutdown, the reactor was maintained close to the

operating temperature of 850 °F and 610 psig of hydrogen

pressure were left on the system overnight. After startup

the following day, it was foxind that the catalyst activity

for this particular reaction had been increased to close

to its initial value (start of curve B), The reactivation

was attributed to the hydrogen reacting with the deposited

residues, forming light hydrocarbons which could diffuse

into the gas phase. The deactivation was postulated as

a mechanical blocking of some of the pores. Curve B shows

the catalyst re-deactivation on the second day of operation.

The reactor was maintained on the same stand-by conditions

overnight, and curve C is a record of the operations on the

third day. It can be seen that the catalyst was only par­

tially reactivated. The reactor system was then left

overnight with 300 psig nitrogen pressure in an attempt to

prevent change in the catalyst ring opening activity.

Curve D indicates that this was achieved. For subsequent

operations, the reactor system was left overnight with

100 psig hydrogen pressure. Curve E was obtained from the

daily standard runs and shows that relative catalyst stabi­

lity was achieved.

In the deactivation study, the 3-methylpentane and

n-hexane rates were found to exhibit behavior similar to

72

that of 2-methylpentane, The ring opening products were

initially present in equilibrium ratios, but the ratios

changed as a stable catalyst activity was achieved, indi­

cating a change in selectivity as well as activity. The

methylcyclopentene rates were essentially unaffected by

the changes in catalyst activity and the benzene rates

were only slightly affected. Barron (5) has stated that

the deactivation of a catalyst is faster for hydrogenolysis

than for aromat izat ion, and this appears to have been the

case here.

The eight standard runs that were made after the

catalyst activity had stabilized were used to estimate

the pure error mean squares for the reaction rates. These

were used in determining the adequacy of fit of the models.

The relative error of the rates of the individual compon­

ents are reported in section 5,4,1.

5,3,0 Sequencing of Experimental Runs

In setting up a sequence of runs to be performed to

obtain the desired experimental data, there may be an opti­

mum plan that will conserve time, feedstock, and minimize

experimental errors. In kinetic studies, it is well to

remember that mechanical equilibrium can be attained faster

than thermal equilibritim. For this reason, on a given day,

the operations should be kept in an isothermal sequence as

much as possible. Pressure and feed throughput changes

I I

73

equilibrate rapidly, and all that is required is to allow

a sweepout time to insure that the voliome from the reactor

inlet to the point of sampling has been adequately displaced.

The variables whose measurements are the most critical

should be changed the least often, A longer continuous

measurement at a fixed condition allows a better estimate

of that variable. At a fixed temperature and pressure level,

the hydrogen to hydrocarbon ratio was adjusted by changing

the hydrogen flow rate using the mass flow meter and main­

taining the more critical liquid feed rate constant. At a

fixed pump setting, the feed rate varied with a change in

pressure because of imperfect check valves. This meant

that the liquid feed rate had to be adjusted to a desired

level after each pressure change, A near-optimum experi­

mental sequence was found to be

I, Temperature

A, Descending pressure

1, Liquid feed rate

a. Increasing hydrogen feed rate,

where the changes to be made first are the innermost parts

of the scheme,

5.3.1 Operational Procedure

In setting up to make a daily series of runs, the mass

flow meter calibration curve for hydrogen feed rate was

checked using a calibrated wet test meter with readings

74

corrected for changes in the barometric pressure. In ini­

tiating an experimental run, the temperatures of the lead

baths in both the preheater and the reactor sections were

brought to the desired levels and maintained there by man­

ual adjustment of the voltage supplied to the resistance

heaters. When a temperature change was made, two to three

hours were required for temperature stabilization. The

fastest overall stabilization was achieved by setting the

liquid and hydrogen flow rates at their required levels

while the temperature was still changing. The hydrogen

rates were set before the liquid feed pump was turned on.

This was to insure that there was sufficient hydrogen in

the system to keep the feed from coking up the catalyst.

After system startup, the only manual adjustments necessary

during an experimental run were occasional temperature

adjustment and product sampling.

Temperatures at five points within the reactor assembly

and at two points in the preheater were monitored using a

multipoint indicating potentiometer. The reaction itself

was monitored with two thermocouples. One could be moved

within a thermowell to any vertical position, and the

second was inserted into the top of the reactor near the

point of reactant introduction. The movable thermocouple

was used to obtain a temperature profile of the catalyst

bed. In this particular study, no discemable temperature

variation was found because the catalyst was well dispersed

75

throughout the reactor and the net heat of reaction (endo-

thermic) for the multireactions was about one kilocalorie

per hour per gram of catalyst.

Liquid product was removed and measured volumetrically

at 15 to 20 minute intervals. Due to the high pressure at

which the phase separation was made, the hydrocarbon recov­

ery in the liquid phase averaged approximately 97 per cent

by weight. For each data point, samples were collected

over a time interval of from 1 to Ih hours. Chromatographic

time checks on product composition were made in the deter­

mination of the system stabilization time.

Gas evolution was metered through the wet test meter

and some samples were taken to run on the chromatograph.

It was found that the compositions of the gas samples did

not vary sufficiently from those of the liquid samples to

cause significant changes in the rate calculations. Small

corrections were made by the use of experimentally deter­

mined phase equilibriiim, k = y/x, data.

The analyses of the reactor product composition were

made at the same time that the runs were being made. This

served as a check for gross calculational errors made in

setting up the runs, as an immediate check on the system

stability, and as a basis to decide whether to continue

the original daily experimental design or to experiment

with changes in different variables or at different levels

than originally anticipated.

76

5.4.0 Analytical Methods and Procedures

The analytical procedure to be used in conjunction

with an experimental system is determined by

1. The power of the procedure to resolve the compo­

nents at the concentration levels of interest (qualitative).

2. The precision of the procedure.

3, The sensitivity of the prt>cedure to small changes

in concentration (quantitative).

4, The ease of experimental operation and interpre­

tation of the data.

The gas chromatograph, developed since the mid-1950's,

has taken a prominent place in the analyses of liquid and

gaseous mixtures. The basic apparatus requires four units:

a carrier gas supply, a sample port, a column, and a detec­

tor for determining the composition of the column effluent.

The detector and column can generally be independently

thermostated over a wide temperature range. In this rela­

tively new technique a packed column is used to separate

gas mixtures in a reproducible fashion. The passage of

a component through the detector is recorded as a peak

whose normalized height varies with the concentration.

Gas-Liquid Chromatographv by Dal Nogare and Juvet (24)

is one of the standard references in this field.

The gas chromatograph can detect the presence and

time of passage of each component, but does not establish

its identity. Auxiliary data are needed for component

77

identification. These can be obtained from the passage

of the unknown mixture through several different columns

at standard conditions. The peak locations can be compared

to those of known samples that have also been passed through

these columns at the standard conditions.

The chromatogram can be calibrated for each expected

component to obtain precise quantitative results (1% pre­

cision or better). The detector sensitivity can be changed

by the changes in the current to its resistor filaments.

With the selection of the proper column and chromatograph

operating conditions, most analysis for which the method

can be used can be performed in less than 20 minutes.

The key to the analysis is the proper selection of

the chromatographic column and the determination of the

optimum settings for the column length, column tempera­

ture, detector block temperature, carrier gas rate, and

detector current. The following procedure has proved

useful in the selection of a chromatographic column and

operating conditions,

1. An attempt is made to identify the components

in the product samples by passing then through several

long columns at low temperatures. This is to insure the

minimization of peak overlap (maximization of component

separation). Known standards containing components

suspected of being present in the product samples are

also passed through these columns for comparison of

78

retention times (time from injection until peak height is

recorded by the detector). The total elution time in this

step may be 40 minutes, which would probably be prohibitive

as a control indicator in many cases. Maximum final elu­

tion times in the 10 to 15 minute range are preferable,

2, Using literature information as a basis for column

formulation, several 10 to 15 foot chromatographic columns

are prepared which could be capable of making the desired

separation in a reasonable length of time with sharp peak

heights. The columns usually need to be cured for 10 hours

with the carrier gas flowing through them and at tempera­

tures somewhat above that to be used in the separation,

3, Chromatograms are made for each colvimn at differ­

ent temperatures (a five degree centigrade change can have

considerable effect). Higher temperatures reduce the elu­

tion time, but decrease peak separation and column life.

Each column may have a different maximum temperature above

which the column material develops a substantial partial

pressure or decomposes and is deposited on the detector.

It is desirable to run at the lowest temperature at which

the total separation can be made in a reasonable length

of time.

4, The carrier gas rate is usually somewhere between

30 and 90 ml/min for i inch columns. The effect of this

variable can be studied within the permissible temperature

range.

79

5, The thermal conductivity detector sensitivity

can be studied by checking the response to changes in the

concentration. For a resistor type detector this is done

at several current levels, at a detector temperature at

least 50 °C above that of the column. The detector response

is more linear to changes in concentration at the lower

currents, but is less sensitive to these changes at the

lower currents.

In the screening of chromatographic columns for this

system, the following columns were tried and found to be

inadequate, either because of failure to make the separa­

tion and/or because the retention time was too long.

1. Di-isodecylphthalate 10 gm/100 gm on 60/80 mesh

firebrick (§ 100 °C,

2. Benzyl cyanide-silver nitrate (acetone soluble)

30 gm/120 gm on 60/80 acid washed (A/W) Chromasorb P

(ASTM D1717) (§ 75 °C,

3. Tricresylphosphate (acetone soluble) 30 gm/70 gm

on 30/60 A/W Chromasorb W (§ 70 °C.

4. Bis-(2-ethylhexyl) adipate (ether soluble) 30 gm/

100 gm on 30/60 A/W Chromasorb W (? 25 °C and 40 °C.

A squalane column which had been observed to separate

hydrocarbons according to their boiling points (70) was

constructed. The 10 foot column was packed with a mixture

having a ratio of 35 grams of squalane (ether soluble) de­

posited on 100 grams of 30/60 mesh acid washed firebrick.

80

The column was cured at 170 °C for 15 hours.

The chromatograph used in this study was a Varian

Aerograph Model 1520 B with a Hone3rwell recorder. Using

the squalane column, operating conditions were found that

gave a good separation for all the components that were

detected. The established operating conditions are pre­

sented in Table 2,

Table 2

Chromatograph Operating Conditions

Column temperature 115 C o

Detector block temperature 203 C

Injection port temperature 196 C

Carrier gas, rate Helium 40 ml/min

Carrier gas pressure 60 psig

Detector current 150 ma

High purity benzene and cyclohexane were available for

peak identification by retention time and for the making

of known standards for the determination of response factors,

Methylcyclopentane, the feedstock, was available in 95 mole

per cent purity and was further purified by distillation

at a 30/1 reflux ratio to approximately 99 mole per cent.

High purity (greater than 99 mole per cent) 2-methylpentane,

81

3-methylpentane, and n-hexane were purchased from Phillips

Petrolexim Company. The retention times (in minutes) were

determined for the compounds suspected to be present in the

reactor effluent and are listed in Table 15 of Appendix D.

These results are presented in Figure 6, which shows the

techniques used in determining the unknown peaks.

After peak identification, the response factors were

determined for the components found to be present in the

product mixture. The actual determination is presented in

reference 72 Volume 3, The chromatographic coliamn was a

15 foot squalane colximn at 135 °C, The response factors

(mol%/height%) are presented in Table 16 in Appendix D,

5,3,1 Data Precision

The response factors were used to find an estimate of

the standard deviations of the compositions in mole fractions

for a single run (run 5-19-A, Ref, 72, Volume 3), Since

each of the components stayed in essentially the same con­

centration range, the error in measurement associated with

the analytical procedure was constant as required by the

mathematical section 4.2.0. It is probable that after

catalyst stabilization the errors between runs due to small

changes in catalyst activity were random and did not intro­

duce a bias into the data.

The rates of the individual components in gram moles

90

82

80 •<5lO

u o

70 8

O

•H

O

o

60

50

40

30

o 4

3

^ 5

1 2 3 4 5 6 7 8 9 10

Legend

n-pentane 2-2 dimethylbutane (est.) 2-3 dimethylbutane (est,) 2 methylpentane 3 methylpentane n-hexane methylcyclopentene (est.) methylcyclopentane benzene cyclohexane

5 6 7 8 9 10 11 12 13

Retention time in 10 ft, squalane column, minutes

Figure 6 Hydrocarbon Retention in Squalane Column

14

83

per gram of c a t a l y s t were ca lcu la ted from the eqtiation

Ratej^ = (0,97 m^ • COSm^ kj - f^)F/W (5-1)

where 0.97 a fract ion of l iquid feed recovered as

l iquid

ift s mole fraction of component i in

recovered liquid

kj = experimental vapor-liquid equilibritim

ratio

f. = mole fraction of component i in liquid

feed

F = molar liquid feed rate

W s grams of catalyst in the reactor.

The variance of a general function of independent variables

involves the square of the partial, derivatives of the

function with respect to each variable. If a function X

is calcualted from some fxinction of x., x , x , •••

X = f(xj, x^, x^, •••)

then C^CX) = < /3X \ CS^iXi)

The variance of equation (5-1) is then

Cr^(Rate^) = t^r (2(0,97 • 0,03kj ) ^^(mj^) •

Ca;03mj^)V^(k^) • (y^(fj]/(0.97m^ • 0,03m kj - f^)^

• Cr2(F)/F2 • (r2(W)/w2 j . (5-2)

\^

!i ,

84

Using equation (5-2) and estimates of the required varian­

ces, the relative errors ((T (Ratej^)/Rate^) in the measure­

ments were calculated and are listed in column 1 of Table

3. Also listed in column 2 are the actual per cent rela­

tive errors obtained from the standard runs. In coltimn 3

are listed the relative errors for the standard runs after

normalization to make the methylcyclopentane rates equal.

For the experimental data, only 11 out of 44 data points

were normalized, with the normalization of the standard

run for a particular day serving as a basis for normalizing

the other rurs. Column 3 is a lower bound on the per cent

relative error, and column 2 is an upper bound. The actual

per cent relative errors are probably somewhere near the

column 2 values. The column 3 values place more stringent

requirements on mathematical models describing the rates

and were therefore used in the model testing.

The methylcyclopentene rates were not normalized in

the reduction of the data because they appeared to be

unaffected by the changes in catalyst activity. The high

relative error in cyclohexane was due to the fact that

cyclohexane was an intermediate product and also because

it was present in relatively small (sensitive to change)

quantities.

85

Table 3

Estimates of Per Cent Relative Error

Component

2-methylpentane

3-methylpentane

n-hexane

methylcyclopentene

methylcyclopentane

benzene 2.75 5.97 1.27 ^

cyclohexane

1

8.83

2.67

3.22

2.21

1,87

2.75

7.21

2

8,99

7,85

8,35

1.74

7,00

5,97

16.05

3

4,95

4,35

2,46

1.27

11.50

Colximn 1 e Estimates from s i n g l e run

Column 2 = Estimates from standard runs

Colximn 3 = Estimates from normalized standard runs

11

CHAPTER VI

DATA REDUCTION AND ANALYSIS

6.1.0 Data Reduction

The independent variables in this study were feed

component partial pressures, feed rates, and temperature.

The partial pressures of the components in the feed were

changed by altering the total pressure, the hydrogen to

hydrocarbon ratio, and the composition of the liquid feed.

Table 19 in Appendix D is a listing of the independent

variable settings for the isothermal runs made at 850 °F.

In Table 30 are found the values of the independent variables

for the runs that were made at other temperature levels.

The dependent variable? were the product compositions.

These are recorded in reference 72, Volume 3, and in Table

20.

Assuming that the total number of moles of hydrocarbons

was constant throughout the reactor (no cracking) and that

each reaction rate was constant throughout the differential

reactor, the rates of production of the individual compon­

ents were calculated from equation (5-1). The component

rates are presented in Table 21 for the runs at 850 °F and

in Table 32 for the runs at other temperature levels.

The bulk phase partial pressure of each component in

the differential reactor was taken as the average between

86

87

its inlet and exit partial pressure. These partial pres­

sures, along with estimates of transport properties (Appen­

dix A) and a mass transfer correlation (Section 3,2,0)

were used to determine the average component partial pres­

sures at the catalyst surface. The surface temperatures

were calculated using the conversion rate data, heats of

reaction data (Tables 17A and 17B), and a heat transfer

correlation (Section 3.3.0), The catalyst surface temper­

atures and surface partial pressures are listed in Table 23

for all of the runs. In Tables 24 through 29 are listed the

various transport data for the 850 °F mixtures, miscellane­

ous mixture properties necessary for the calculations,

and heat and mass transfer coefficients. Tables 30 through

33 are similar presentations of the data obtained for the

other stabilized catalyst runs.

The deactivation of the catalyst to a stable activity

level was followed by making a series of long runs at

850 °F with 203 ml of methylcyclopentane input at 400 psig

and a 3/1 hydrogen to hydrocarbon ratio. The pertinent

data are recorded in Tables 34 through 36, After the cata­

lyst activity was relatively stabilized, an experimental

design was constructed similar to that in section 5.2.2.

Standard runs at the above conditions were made each day,

and these served as a basis for normalizing the data.

These standard runs are also listed in the catalyst deacti­

vation tables.

88

The product distributions did not change appreciably

in the standard runs, but the methylcyclopentane conversion

varied slightly. The methylcyclopentane rates in these

standard runs were used as a basis for normalizing the data

obtained on a given day to the same catalyst activity level.

Run number 28 in Table 21 served as the normalization basis.

The runs made at different temperature levels or with dif­

ferent liquid feedstocks were not normalized. In Table 22

are listed the settings of the independent variables and

the normalized rates for the adjusted data,

6,1,1 Catalyst Physical Properties and Effectiveness

Factor of the Catalytic React ion System

The physical properties of the catalyst used in this

study were determined by the author before loading the

reactor and after the completion of the runs. A 0.5 per

cent platinum on ^-alumina in 1/8 inch diameter pellets

and having a 90 square meters per gram nitrogen surface

area was initially tried as a catalyst for the conversion

of methylcyclopentane to benzene. This reaction was not

catalyzed on this Englehardt catalyst, but the reaction of

cyclohexane to benzene was and subsequent work on this

reaction has been reported. (32, 33)

The multistep reaction of methylcyclopentane to ben­

zene was found to be catalyzed by a 0.35 per cent platinum

on 'M-alximina. This catalyst (Englehardt RD-150-C) was

89

a 1/16 inch diameter extrudate having an external surface

area of 26 square centimeters per gram of catalyst and an

initial B. E, T. nitrogen surface area of 405 square meters

per gram. The used catalyst had a nitrogen surface area of

286 square meters per gram.

The pore volume of the catalyst was determined by boil­

ing samples of weighted catalyst in cyclohexane for 45

minutes. The cyclohexane was decanted from the catalyst

samples and the samples were placed on filter paper until

the external surface was visibly dried. The gain in weight

of each sample was determined, and along with the specific

gravity of the cyclohexane, the weight gains were used to

calculate the catalyst pore volume. The pore volxime was

initially 0,313 cubic centimeters per gram of catalyst and

was 0,192 cubic centimeters per gram on the used catalyst.

The average pore radius r was calculated (equation

(3-12)) to be 15,5 A for the fresh and 13,5 A for the used

catalyst, and the mean pore length L to be 0,358 cm (equa­

tion (3-13)), Smith (66) states that at 20 atmoshperes o

and pore radius below 20 A, Knudsen diffusion predominates,

given by the dimensional equation

D„ = 9.7xlo\(T/M)^''^ , cm^/sec

Where T = temperature, °K

M = molecular weight

r = pore radius, cm.

90

A pseudo-first order rate constant for the total con­

version of methylcyclopentane was determined for several

points. The Thiele modulus (equation (3-16)) was calcul­

ated for these points and used to determine the effective­

ness factor (equation (3-15)). The minimum effectiveness

factor that was calculated was 0.90, strongly indicating

the absence of internal diffusion effects. Since effective­

ness factors are larger for lower rates and methylcyclopen­

tane had the largest rate, this also established the

absence of internal diffusion effects for the other compon­

ents.

6.2.0 Data Analysis

The reaction of methylcyclopentane over the 0.35

per cent platinum on yi -alumina was found to yield a nxim-

ber of products. The products that were present in sig­

nificant amounts were 2-methylpentane, 3-methylpentane,

n-hexane, methylcyclopentene, benzene, and cyclohexane.

The dehydrogenation activity of the catalyst was

higher than the isomerization activity. The conversion

of methylcyclopentane to products was about 12 per cent,

and of this, 50 to 60 per cent was benzene. With a 5

per cent cyclohexane in the methylcyclopentane feed, 50

per cent of the benzene that was formed was from the

conversion of cyclohexane in the feed, A standard run

using pure cyclohexane as a feedstock yielded a product

91

containing 52% benzene and only 1,2% methylcyclopentane.

Diagnostic plots of component rates versus throughputs

to check for the presence of external diffusion effects,

such as Figure 7, showed that diffusion effects were pre­

sent in the production of methylcyclopentene and benzene,

indeterminate in the production of cyclohexane, and insig­

nificant in the ro4ttcti0n of the ring opening products.

Subsequent calculations demonstrated the absence of temper­

ature gradieats between the bulk phase and the catalyst

surface,

6,2.1 Methylcyclopentene

Figure 8 is a plot of methylcyclopentene rates versus

methylcyclopentane surface partial pressures. Curves A, B,

and C are at 3/1 hydrogen to hydrocarbon ratios, and at

different throughputs. It can be seen that the methyl­

cyclopentene rates increase with increasing throughputs,

again indicating the presence of diffusion effects. This

phenomenon is demonstrated again in curves D, E, and F,

which are at 4,7/1 hydrogen to hydrocarbon ratio. The

rates within each individual curve correlate? fairly well

with the methylcyclopentene mass transfer coefficients,

but the rates that are on different curves do not.

All of the curves show a decrease in rate with increas­

ing methylcyclopentane surface partial pressures. This

indicates that at higher pressures, the reverse reaction

92

300 psi to

o

o rt u o g

I-H

o &

I-H k

is

0

100 200 MCP Liquid feed rate,-ml/hr

Figure 7 Diagnostic Test for External Diffusion, 850 *F

300

I

93

8

rt O

X

o u

E O)

o E 4 4-

E

O)

0).

o

1

F igure 8 Effect of MCP Surface Partial

Pressure on MCP" Rate

Legend 3/1 H^ / HC 4 . 7 / 1 H^/HC

G, H 5.1 \ cyclohexane or 6.1 \ benzene In MCP

All liquid feed rotes 203 m l / hr

except -A & D at 125 m l / h r C & F at 300 m l / h r

"-S

1 i i—i 1 1 7 8 MCP Surface Part ia l Pressure , Atm

;H

94

of methylcyclopentene to methylcyclopentane becomes more

significant. A comparison of curves A with D, B with E, and

C with F shows that for the same methylcyclopentane partial

pressures, the methylcyclopentene rate is decreased with

an increase in hydrogen pressure. This again demonstrates

the importance of the reverse reaction.

Curve H, which is composed of runs containing about

5.2 mole per cent cyclohexane and also of runs containing

6.1 mole per cent benzene in the feed, is not significantly

different from its corresponding curve B containing only

methylcyclopentane in the feed. Similarly, curve G is not

significantly different from curve E. This indicates that

at these relatively low levels of cyclohexane and benzene

(which roughly correspond to their conversion levels in the

differential reactor), the reaction from cyclohexane or

benzene to methylcyclopentene is relatively minor.

The observations that the methylcyclopentene reverse

reaction to methylcyclopentane is significant and that

the formation of methylcyclopentene from cyclohexane or

benzene at their low concentrations is insignificant

indicate that if methylcyclopentene is a precursor to

cyclohexane, the isomerization of methylcyclopentene to

cyclohexene is the slow step in the reaction sequence and

is essentially irreversible at these concentration levels.

95

6,2.2 Benzene. Cyclohexane

Figure 9 is a plot of benzene rates versus methylcyclo­

pentane surface partial pressures. Curves A, B, and C are

at 3/1 hydrogen to hydrocarbon ratios and curves D, E, and

F are at 4.7/1 ratios. The benzene rate increases with

increasing throughput, indicating diffusional effects.

Again, the rates within an individual curve correlate fairly

well with the benzene mass transfer coefficients, but the

rates that are on different curves do not.

All of the curves show a decrease in the benzene rate

with increasing methylcyclopentane partial pressure. There

may be two reasons for this s (1) cyclohexane is thermodyham-

ically favored at higher pressures, and therefore the con­

version of benzene to cyclohexane can be significant, and

(2) the methylcyclopentene precursor is present in smaller

concentrations at the higher pressures.

Curves I and J are at 4.7/1 and 3/1 hydrogen to hydro­

carbon ratios respectively and are composed of runs made

with approximately 5,2 mole per cent cyclohexane in the

feed. The significant difference between curves E and I

and between B and J indicates that the dehydrogenation of

cyclohexane to benzene is rapid.

Curves G and H are at 4,7/1 and 3/1 hydrogen to hydro­

carbon ratios, and are composed of runs made with a 6.1

mole per cent benzene in the feed. Curve G is signifi-

8

Figure 9 Effect of MCP Surface Par t ia l

Pressure on Beniene Rate

— —

• • • •

1. J 0, H

Legend 3/1 H^ / HC

4 . 7 / 1 H ^ / H C

S.I % cyc lohexane in MCP 6.1 % benzene in MCP

96

x6

0 u

E o> \»

X 4 4)

"o E

CO

4)

I2

1 -

Ali liquid feed rates 203 ml /hr except A A D at 125 m l /h r C & F at 300 m l / h r

1 2 MCP Surface Par t ia l Pressure, A t m .

97

cantly different from curve E, as is curve H from curve B.

Curves G and H indicate that the hydrogenation of benzene

is also rapid. The reverse reaction, benzene to cyclo­

hexane, is significant and/or the benzene is preferentially

adsorbed on the reaction sites, blocking its production

by methylcyclopentene precursors. Figure 10 is a plot of

the ratios of benzene to cyclohexane in ternary equilibriiim

with hydrogen at 850 ^F. Also plotted on this figure are

the experimental benzene to cyclohexane partial pressure

ratios at the catalyst surface. Curve A depicts the equil­

ibrium ratio values of benzene to cyclohexane, curve B the

experimental values of the feed containing 6,1 mole per

cent benzene, curve C the regular methylcyclopentane feed,

and curve D the experimental values of the runs containing

a 5.2 mole per cent cyclohexane in the feed. Po-int E

represents the result of a pure cyclohexane feedstock run.

The methylcyclopentane curve coincides with the

equilibrium curve at hydrogen partial pressures greater

than about 23 atmospheres. If the reaction sequence

Q ^ J^^r^ A = adsorbed < .- k^ D = desorbed

Figure 11

An Isomerizat ion-Dehydro isomerizat ion Mechanism

98

o •H

iS s o g o 9)

o o

§

10

0

Legend

A Thermodynamic equilibrium ratio B 6,1% benzene in MCP feed C MCP feed D 5,1% cyclohexane in MCP feed E Cyclohexane feed

t . I L 1 L J 1 1 L

10 .4_.- I -_ I I - J. » < I I I I V I )

20 25 30 35 15

Hydrogen pressure, atm.

Figure 10 Benzene-cyclohexane-hydrogen Thermodynamic Equilibrium

and Experimental Ratios at 850**F

\ <;i>^

USUI "

99

is considered, curve C indicates that at the higher pres­

sures the reaction rates 8 and 9 are equal. At lower

pressures, curve A indicates that benzene is heavily

favored by thermodynamics. Curve C lies below curve A

at the lower pressures, indicating that there is too much

cyclohexane or not enough benzene to maintain the equil­

ibrium ratio. At these conditions, rate 9 is greater than

rate 8,

If the only source of a specie is the catalyst surface

and the specie does not react in the desorbed phase, it is

possible that the adsorbed and desorbed species are in equil«

ibrium. For the methylcyclopentane feed this would result

in the adsorbed and desorbed benzene being in equilibrium,

as well as the adsorbed and desorbed cyclohexane. Curve B

lies below the equilibrium curve below 19 atmospheres, but

lies above curve C. This indicates that even with a 6.1

mole per cent benzene in the feed, rate 8 is still not fast

enough to convert the cyclohexane to the equilibrium level.

Curve B lies above the equilibrium curve A above 21 atmos­

pheres of hydrogen partial pressure. With an external

benzene source, this points out the possibility of lack of

equilibrium between the desorbed benzene and that adsorbed

on the proper catalyst site to be converted to cyclohexane,

or of a slow surface reaction. Similarly, curve C lies

below the equilibrium curve for the entire pressure range.

Again, with an external cyclohexane source, this indicates

100

the possibility of lack of equilibrium between the desorbed

cyclohexane and that adsorbed on the proper catalyst sites,

or of a slow surface reaction.

Point E represents the ratio of benzene to cyclohexane

obtained from a pure cyclohexane feed and strongly indicates

that the cyclohexane-benzene hydrogenation- dehydrogenation

surface reactions are not fast enough to achieve the thermo­

dynamic equilibrium ratios, A 53 per cent conversion of

pure cyclohexane to benzene was obtained under the same

conditions that yielded 6 to 8 per cent conversions of

methylcyclopentane to these two products. This suggests

that the hydrogenation-dehydrogenation reactions are from

7 to 9 times faster than the isomerization reactions. The

failure of conversion of the relatively small concentration

(approximately 5 per cenjt) benzene and cyclohexane spiked

methylcyclopentane feedstock to equilibrium indicates that

they were competing with methylcyclopentane for the platinum

hydro-dehydrogenation sites.

Figure 12 is a plot of the cyclohexane production flxix

ratios based on benzene production. A negative production

flux ratio indicates a net production of cyclohexane while

a positive flux ratio denotes a consximption of cyclohexane.

It can be seen from Figure 12 that the cyclohexane consump­

tion rate (net rate of 13 minus 12 of Figure 11) was approx­

imately equal to the methylcyclopentane isomerization rate

(rate 3) or to one-half the benzene production rate for the

101

+1

g • H 4-» O

S

P.

0)

0

-1

« -2

o •

Legend

+ cyclohexane converted to benzene - benzene converted to cyclohexane A 5ol mol % cyclohexane in MCP feed B 0,2 mol % cyclohexane in MCP feed C loO mol % cyclohexane, 6,1 mol %

benzene in MCP feed

A.

-3 —

10 J J_.^.i„._L .J_„.L.. I I II

25 15 20 25 30

Hydrogen partial pressure, atm.

1 1.. J L 35

Figure 12

Cyclohexane Production Flux Ratio

102

entire pressure range when cyclohexane was present in the

feed. This indicates that the isomerization step 3 and the

cyclohexane dehydrogenation path have the same pressure

dependence. With benzene in the feed, the benzene and

cyclohexane production rates were equal at a hydrogen

partial pressure of about 31,5 atmospheres, with the cyclo­

hexane production rate rapidly increasing with increasing

hydrogen partial pressures beyond this point,

6.2,3 Ring Opening Products

The conversion of methylcyclopentane to the ring open­

ing products 2-methylpentane, 3-methylpentane, and n-hexane

was essentially constant, as is shown in Figure 13, The

presence of small concentrations (less than 6 mole per

cent) of cyclohexane or benzene in the feedstock did reduce

the ring opening rates. This suggests that the cyclohexane

and benzene may have been preferentially adsorbed on the

catalyst sites, and that the source of some of the ring

opening products was some precursor from the methylcyclo­

pentane reaction sequence. This hypothesis is strengthened

by the fact that no ring opening products were observed

with a pure cyclohexane feedstock run. Figure 14 is a plot

of the ratios of n-hexane to 2-methylpentane. The ratio is seen

to increase with increasing total pressures and increasing

hydrogen to hydrocarbon ratio. This might be expected

if the n-hexane comes from two sources. A methylcyclo-

8

o 6

I'

103

F i g u r e 13

Sum of Ring Opening Rates

Legend

A All MCP feed runs

B Al l runs containing cyclohexane or

benzene wi th the MCP

0 u

E 0)

4-

"5 £ E 2

4)

0 t^ 1

A

B

h . | S - i

-»4-s s

.t. i l l

•t5-

t ^ 4 5 6 /

MCP Surface Partial Pressure. A t m .

t

2.8 104

o

o 00

I

I

o

2o6

Legend

MCP feed 5.1% cyclohexane in MCP feed^ 6.1% benzene in MCP feed

2o4

2c2

2o0

lo8

1,6

lo4 _

/

I 600 200 400

Total pressure, psig

Figure 14

n-hexane to 2=methylpentane Production Ratio

105

pentane or methylcyclopentene ring opening precursor yields

the three ring opening products. A hydrocarbon^catalyst

complex which is an immediate precursor to cyclohexane yields

only n-hexane, Methylcyclopentene is disfavored at the higher

pressures and the higher hydrogen to hydrocarbon ratios, con­

ditions which favor cyclohexane. If n-hexane comes from these

two sources, the n-hexane to 2-methylpentane ratio should be

very sensitive to changes in the total pressure and hydrogen

to hydrocarbon ratio, which was found to be the case.

The n-hexane to 2-methylpentane ratios were found to

decrease with the presence of benzene and cyclohexane in

the feed. Plots of this ratio against both benzene and

cyclohexane external surface partial pressures (not shown)

did not show a uniform correlation for all the data, indi°

eating that the concentrations inside the catalyst control

the relative rates of the ring opening products. The reduc­

tion in the ratios with the presence of benzene and cyclo­

hexane indicates that the formation of the complex cyclohex-

ane precursor is essentially irreversible.

Figures 15 and 16 show the effect of temperature on the

experimental product distributions. In Figure 15 is shown the

temperature dependence of the n-hexane to 2«methylpentane ratio.

This ratio appears to go through a maximum near 830 ^ F. This

maximum might be expected if the formation of n-hexane was from

two sources. The temperature dependence of the n-hexanre forma=

tion rate could be different for each source or the concen­

tration of each source could have a different temperature

2.6

2,4

106

o

§ rH

4->

i I

g

I

C

o

2,2

2;o

loS

lo6

psig

psig

1.4

800 820 860 840

Tenperature, °F

Figure 15

Temperature Dependence of n-hexane to

2-methylpentane Production Ratio

880

> » ^

107

dependence. Catalyst ring opening selectivity might

also be affected by changes in temperature.

Figure 16 shows the temperature dependence of the

ratio of the ring opening products to the six-membered

ring products. It can be seen that the formation of the

six-membered ring products was favored at all of the

temperatures, but that the relative production of the

ring opening products was slightly increasing as the

temperature was increased.

6.3,0 Modeling of Experimental Rates

The initial modeling strategy was to write several

mechanistic Hougen-Watson rate models for each of the

components. The experimental rates were used in conjunct­

ion with the nonlinear parameter estimation program to

obtain parameter estimates and goodness of fit data.

Since the nonlinear parameter estimation program required

estimates of the parameter values to initiate the search,

each model had assigned to it several initial estimates

for its parameters. It was hoped that this strategy would

yield information that could prove useful in screening the

models. It was found, however, that the lack of fit of the

models to the data was so large for all the models that

no model discrimination could be made.

A resort was made to simpler empirical rate models

which had the same general form as the mechanistic models.

y^

1.2

1.0

108

o •H

•H U

T3 0 ?-< a> •i B I

t/5

g p. o

Oo8

0.6

0.4 300^ psig

0,2

0.0 800 820 , 840

TCTiperature, °F

860 880

Figure 16

Temperature Dependence of Ring Opening

Products/Six-membered Ring Products Ratio

109

The trends in the lack of fit of each of the empirical

models were followed closely. Plots were made of the

residuals against the independent variables to see if

definite trends existed, e.g., positive residuals at high­

er pressures and negative residuals at lower pressures. A

definite trend in a plot was interpreted as indicating

that the effects of that independent variable had not been

properly compensated for by the model. The model was then

adjusted to allow for a better compensation of the indepen­

dent variable. These residual plots were the key to the

tuning of the rate models. The models were tuned to levels

where the error in prediction was approximately equal to

the experimental error,

6.3.1. Methylcyclopentene

The methylcyclopentene concentrations in the product

stream were small in comparison with some of the other

reaction products, e.g., the benzene concentrations.were

about ten times larger than that of methylcycloperlten^. It

might be suspected that there might be relatively large

error in the measurement of methylcyclopentene concentrations

As it turned out, this was not the case. The methylcyclo­

pentene rates appeared to be independent of the changes

in catalyst activity and did not have to be normalized.

Several semi-empirical rate models were tested to see

if their functional forms fitted the general trends of the

• - ^

no data. These forms were usually of a net rate type, composed

of a forward rate term minus a reverse rate term. In most

of the forms tested, the residuals were highly correlated

with the feed throughput rates. This indicated that the

calculation of some of the mass transfer coefficients in­

volved in the production of methylcyclopentene might have

been in error and that there was a possibility of adjust­

ment of the mass transfer correlation. The final forms of

the best two tuned models contained a feed rate correction

term and are presented below.

The methylcyclopentene isothermal rate model I, which

gives the methylcyclopentene net production rate in

gram moles/hour/gram of catalyst is given by

A = 10

B = 10

(-5488./Tg • 6.6207)

(-6152.16/Ts • 6.8828)

(.6475.84/Tg t 6.7957)

AA = (A*01 & B*02 ^ C*03)*(Pj cp /PH2B^

05 Z = 04*(No /21.9)

R®m

Rate MCP^ = k *(AA - P^^^a) •Z C^-l) g "Crg'

where A,B,C = thermodynamic equilibrium constants

for 1-methylpentene, 3-methylpentene,

and 4-methylpentene, based on

methylcyclopentane

• ^

Ill

Tg = catalyst surface temperature, °K

^MCPg> PH2 * ^MCPg = bulk partial pressures, atm.

kg = methylcyclopentene mass transfer coefficient,

calculated from experimental flux and exper­

imental conditions, units in Appendix D.

^Re ^ modified Reynolds number

21 o9 SS modified Reynolds niimber at the conditions

of a standard run

91, 02, 03, 04, 05 = parameters to be estimated by

nonlinear least squares

and for convenience, the Fortran symbol (*) denotes multi=

plication.

The methylcyclopentene rate model II which calculates

the rates in the same units as model I is given by

2 - e2*(NR^/21.9)®'^

Q3 Rate MCP^ - ^S*^*^^'^*^m?B^^^n2 ^ '^ PMCP|) (6-2)

where all of the variables have the same definitions as

under model I and the O's are parameters.

In Table 4 are listed the parameter values for the

two models9 the residual sum of squares, and the residual

mean squares«

112

Table 4

Parameter Estimates and Goodness of Fit

Data for Methylcyclopentene Models

Model 01 02 Q3 04 Q5 A B

1 18,28 0.3302 0.4369 0.7692 0.6332 2.43 6.25

2 0.0549 8.965 0.8900 0.6728 - 0.95 2.38

Column A = Sum of squares x 10 Q

Colximn B = Residual mean square x 10

The residual mean squares can be used as an estimate

of the variance if the lack of fit mean square is not

significantly different from the pure error mean square,

which will be shown to be the case here in a later para­

graph. A hypothesis that the variance of model I is equal

to the variance of model II can be tested by taking the

ratio of their residual mean squares. The comparison of

this ratio to the F statistic can only be approximate here

due to the lack of independence of the mean squares.

2 2 Test H ; s (model I) = s (model II)

o RMS model I F ^^(^1,^0) F ^.(41,40) RMS model II '^^ "^^

2.63 1,69 2.09

Reject H at 90% and 98% level on linear

hypothesis.

P!.9H

113

The rejection of the hypothesis indicates that model II

is probably significantly better than model I and only the

goodness of fit data for model II will be presented.

The minimum experimental methylcyclopentene rate was

0.001014 gm moles/hr/gm catalyst, while the maximum was

0.005096, a range of about a five-fold increase. In Table

5 are listed the individual confidence limits for each

parameter of model II and Table 6 is a correlation matrix

showing the dependence of the parameters.

Model II was found to fit the experimental data extreme­

ly well, with an average relative error of 4„14 per cent and

a maximum relative error of 13.3 per cent. Table 7 lists

the mean squares used in checking the goodness of fit of

the model. The pure error sum of squares is calculated

from equation (4-22) using unnormalized data obtained from

the standard runs. The lack of fit mean square is obtained

from equation (4-23). The ratio of the lack of fit mean

square to the pure error mean square (equation (4-24)) is

not found to be significant. This indicates that there

appears to be no reason to doubt the adequacy of the model

and that the pure error and the lack of fit mean squares

can be pooled to get an estimate of the variance.

"x

114

Table J

Individual Confidence Limits for

Methylcyclopentene llodel II Parameters

(Un Linear Hypothesis)

01

O.U793

U.0304

02

16 .85

1.0b

03

0.931

0 .b^9

0^

0.751

0 .59^

Table 6

Methylcyclopentene Parameters

Correlation Matrix

01

02

03

Qi+

01

1.0000

- 0 . 9 6 1 8

0.3076

0 . 1 ^ 2 ^

02

1.0000

- 0 . 0 3 6 ^

»Ool699

03

1.0000

-0 .0362

0if

.0000

^ -Tt^iMs

""mp':, • J pv 'l

115

Table 7

Goodness of Fit of Methylcyclopentene

Rate Madel II

Residuals

Pure firror

Lacic of Fit

Sum of Squares

x10

2.38

1.5^

0.8k

Degrees Freedom

ko

7

33

oT Mean Square

x10^

2.2

0.25

F.05^30,7) = 3.38

\

(Lacl?: of f i t meam square ) / (Er ror mean square) = 0.116 •1

• ^

116

6.3.2 Benzene-cvclohexane

The cyclohexane was usually present in smaller con­

centrations than benzene under the conditions that were

studied. When cyclohexane was present in the feed, it

was readily converted to benzene, resulting in negative

cyclohexane rates. Changes in pressure and hydrogen to

hydrocarbon ratios affected the distribution ratio of these

two products. Since the rates of formation of benzene and

cyclohexane were so highly coupled, the modeling of the

individual rates proved to be very difficult. The rates

were therefore grouped and a conversion rate to these two

products was modeled.

The isothermal rate models predicting the sum of the

cyclohexane and benzene rates were more empirical than the

methylcyclopentene models. The benzene-cyclohexane models

included a term which was analogous to the Hougen-Watson

catalyst active center concentration term. In this term, the

components \<rtiich had proved to be affected by diffusion were

asstimed to be preferentially adsorbed. The two term net rate

equations were found to be inadequate, containing pressure

biases. The final form of the equation was that of a

summation of three terms, with the third term being an

empirical pressure tuning term. Essentially, a single

model was found to give the best fit, A second form of the

model included corrections to the mass transfer coefficients

and was found to improve the fit slightly.

117

The benzene and cyclohexane model I which gives the

six membered ring production rates in gm moles/hr/gm catalyst

is given by

A = 2. - 04*(Pwp«= • p • p ^ p ) . Q5*p MCPj^ MCP^ BZjL CY^^ H2

EQKCB = lo(20.679 - 11490./T3)

03. Rate (BZ • CY) = ©l*PMrD *A " (©2*?^^ *P„ )/EQKCB MCP^"^ " "" "BZ %

• 06/(P„ ®^) (6-3) "2

where A = catalyst concentration t erm

^MCPJ' ^MCP^' ^BZj^' ^CY^ "^ interfacial partial

pressures, atm.

Pjj = hydrogen partial pressure, essentially the

same in either bulk or interface

EQKCP as thermodynamic equilibrium constant for

cyclohexane and benzene

01, 02, •••,07 = parameters to be estimated by

nonlinear least squares.

In the second form of the model, the interfacial

partial pressures of the methylcyclopentane and benzene

were calculated from 09

Z = 08*(N^^ /21.9) ^®m

^MCPi = ^ M C P B • ^^^'^^("CPJ/C^gMcp*^)

~->«iW

BZi = ^ Z B ^ Rate(BZ)/(k *Z) (6-4)

118

'fZ)

'BZ

where j Qp > ^^Zn " ^^^^ partial pressures, atm. D B

Rate(MCP), Rate(BZ) = experimentally measured

fluxes

^g > Jg = mass transfer coefficients calcul­

ated from the experimental fluxes

and experimental conditions, units

in Appendix D.

In Table 8 are listed the parameter values for the

two models, the residual sum of squares, and the residual

mean squares. Listed under Table 8 is the test to see if

the two models are significantly different. The test indi­

cates that the second model is significantly better than

the first at the 90% level, and only the goodness of fit

data for the second model will be presented.

The minimum experimental six membered ring net forma­

tion rate was 0.0159 gm mole/hr/gm catalyst and the maximum

was 0.0468, The range consists of a threefold increase in

the rate. The individual confidence limits for each para­

meter are listed in Table 9 and Table 10 is the parameter

correlation matrix.

The experimental data were again fit extremely we'll.

The average relative error for the second model was 3.42

per cent and the maximum relative error was 12.0 per cent.

i.i.t, • m w u ,

119

2

<5r o o

00 I-H

rH

u <D

(D

o V)

3

o

C/5 i^ X

00

in 00 to

00

0)

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rt

iH O

I-H

0 0

in CD

o

V c o N c J-i

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to CD

<x>

CM CM 'St

r>. .

00 vO 00 1-H •

oo LO •^ to o o CM r-i (N| I-H O e

00 I-H "«* O O •

• (NJ t^ 00 I-H

r o to o o

to p^ 'SJ-t*>. «

VO to Oi I-H o

t^ t^ ">* to o « r 00 00

o o • Oi

o% "t (SJ

o e

o

r--LO r-rH to CM to o o

CM X

II

X

•H

O

o

u d » (i> c S '. I-H •»->

V)

5 i2

to a> r>- fH

o a

UO r-

o c

l O LO 00

0)

rt cr

i

(1)

U

'd

O

u •I-t

0

00

o

0) CM

120

I-H

(D

rt

.8 Vi •P •H

0) O

§ •H

c o u

•H >

•H

•a

• H

is •H

O^ OD

00 CD

r--cr

\o 05

LO CD

'tf CD

to a>

CM cx>

f-i

a>

VO to to

CVJ

to rH «

CM r>j

o r r-i

• I-H

C7i l^ Oi to .

** r--^ o o

-St vO '!t o o

00 • ^

^ I-H e

• ^ Oi 'rt CM

•«* O 'St

o

"Sj-

to

o

o

t- VO o

I-H

vO rj-CM to « o

00 o I-H o 1

iH CM CM O o

vO 00 CM o 1

cr> '^ en o e 1

o iH

CM O I-H

LO •«* CM O

121

o tH 0) iH

-9 a H

X •H u p s c o nH •i <« rH

rre

6 u o •p

^1

rt Oi

r-i

•%

s o

C7i CD

00 CD

t^ CD

vO CD t

LO CD

P4

CD

to CD

CM CD

CD o o

o o o rH

LO to o

o o e r-\

to to Oi

t 1

r> 00 o

o o I-H

to "^ CM « 1

o ->!t 1-H •

O

00 ^ r-

o o I-H

rH to cr» 1

r--to (SI o

o

t^ LO I-H « 1

OO lO in

o o «

rH

vO t^ 00

1

i^ Oi vO

o

'^ to to « 1

C7 vO CM o

O

LO •«t to

o o o rH

c: 00 c a o

LO rH o\ 1

t^ r>-r o

"t iH to a 1

00 to CM o

o

cy> vO -"st

o o a iH

vO r o • o

1^ r o a o

"«:t CT> O

1

00 00 o o

T-i to r-{ a 1

r--"^ rH a

o

^ •^ CM

O o iH

'^ «* 00 o

o

CM CT> O a

o

CM o rH a

o

LO cn o 1

rH 00 O

O

00 CT> r-i a 1

LO LO r-{ a

o

en to CM

CD CM CD

to CD

" J-CD

LO CD

vO CD

r» CD

00 CD

C7k CD

" " ^

122

The first model had yielded an average relative error

of 4.35 per cent and a maximum relative error of 16.5

per cent. The mean squares used in checking the goodness

of fit of the second model are presented in Table 11. A

statistical F ratio test indicated that the model fitted the

data within the bounds of the experimental error.

6.3.3 Ring Opening Products

The summation of the rates of formation of the three

ring opening products, 2-methylpentane, 3-methylpentane,and

n-hexane, remained constant for all the isothermal runs

made with the methylcyclopentane feed. The presence of

less than 6 per cent benzene or cyclohexane in the feed

reduced the total rates slightly. Although the overall

rates were constant, there were changes in the 2- and 3-

methylpentane to n-hexane ratios, with the n-hexane rates

being favored at the same conditions that favored cyclo­

hexane, i.e., higher pressures and higher hydrogen to

hydrocarbon ratios.

For the methylcyclopentane feedstock, the average total

ring opening rate was 0.02046 gm moles/hr/gm catalyst and

for the feeds containing benzene and cyclohexane was 0.01534,

The average relative error was 6 per cent in each case and

the maximum relative error was 13 per cent for the methyl­

cyclopentane feed and 14.5 for the other feeds. In Table

12 are listed the mean squares used to test the adequacy

123

.e 11

Goodness or Fit of Benzene plus

Cyclohexane Rate Model

Residuals

Pure Error

Lack of Fit

Sum of Squares

xlO^

7.6

3.03

•.61

Degrees of Freedom

35

7

28

Mean Square

x10^

.33

1.65

^.05C30j7) = 3.38

(Lack of fit mean square)'/(Error mean square) = 0.382

124

Table 12

Goodness of Fit of Ring Opening Rate Models

Residual

Pure Error

Lack of Fit

MCP feed Sum of Squares

X 10^

.59

0.81

3.78

F^^^(2U,7) = 6.16

Degrees of Freedom

30

7

23

Mean Square

X 10^

(Lack of fit mean square)/(Error mean square) = 1. 3

0.116

0.16^

Residual

Pure Error

Lack of Fit

Other feeds Sum of Squares

X 10^

9.91

8.11

1.80

Degrees of Freedom

Mean Square

X 10^

11

7

^-

(Lack of fit mean square)/(Error mean square)

= 0.388

1.16

0. 5

125

of fit of the single valued models. In each case, the

model fitted the data within the limits of experimental error.

6.3.4 Nonisothermal Rate Models

All of the previous models were for isothermal runs

at 850 F, Several isothermal runs were made at three

other levels: 810°, 830°, and 870 °F. In these runs, the

feed throughput rates were the same as the standard runs

used to check the catalyst activity. The feedstock for

most of these runs was the standard methylcyclopentane

feedstock containing 0.2977 per cent cyclohexane. This

feedstock was used up during the course of experimentation,

and a different methylcyclopentane feedstock containing

1.0339 per cent cyclohexane was used at the 870 °F level.

In developing nonisothermal models from the isothermal

data taken at different temperature levels, the only data

points that were considered were those having the same feed

rate and the same approximate feed compositions. Data

points at different pressure levels were used.

A simple first order rate constant was assumed for

each component, the constant calculated from

k, = Rate./P^^„ i i MCP

where k as pseudo-first order rate constant for

component i

Rate^ = experimental rate for component i

|i!IU

• = ^

126

P = methylcyclopentane surface partial pressure MGJr

Arrhenius plots of In (pseudo k Q,-, H ' versus 1/T were

made to see if this simple form could be used to correlate

the data. Figure 15 is the benzene Arrhenius plot. The

trends in this plot are the same as the trends exhibited

by the other components. An examination of Figure 15 shows

that the data in this form plot out as straight lines having

the same slopes but different intercepts. The intercepts

are inversely proportional to the total pressure. This

suggests an equation of the form

In (k^) = A - B*f(P^) - C/Tg

as a basis for correlation. The form of f(P,p) was found

to be In (P-,), yielding the equation

In (k.) = A - B^ln (P^) - C/Tg . (6-5)

An estimate of the slope C was obtained at each pressure

level and then an average of the estimates was taken. This

average was used to calculate a value for A. The values

of A and C were then used to obtain estimates of B, which

were then averaged. These parameter estimates were then

used as initial values for the nonlinear parameter estima­

tion program. Equation (6-5) can be transformed to an

equivalent form which can be more useful

k = In"^ (A)*P^"^ exp (-C/Tg) . (6-5a)

X" 'i"jnmi

' ^

- l o O

2o0

127

3o0

0

O

g t/J a.

=4o0

- 5 , 0 -^

6o0 K

-7o0

1.3

o

o 00

o

o LO 00

o

o to oo

o

O r-t 00

1.4

300 psig

400 psig

500 psig

600 psig >< L_

1.5

Reciprocal temperature (1/T°K) x 10"

Figure 17

Benzene Arrhenius Plot

128

The rates of each of the components were fitted to

equation (6-5) except for cyclohexane. The cyclohexane

rates were negative at the higher temperatures with the

new feedstock which contained 1 per cent cyclohexane.

The rates were positive at the other conditions, making

a fit difficult. The values of the parameter estimates

are listed in Table 13. Also listed in this table are the

goodness of fit data. The models did not fit any of the

experimental data within the isothermal experimental

error established at 850 °F, however, the degree of lack

of fit of the models is normal in kinetic modeling. The

lack of fit can be attributed to several factors, A

slightly different feedstock was used at the higher temper­

atures. In addition, the temperature dependence models

are more sensitive and require more and better data. Fi­

nally, in a competing reaction system, the form of the

temperature dependent reaction rates is probably more

complex than equation (6-5), . In the case of methylcyclo­

pentene and benzene, the pseudo-first order rate constant

is a grouped forward rate and thermodynamic constant. The

pressure dependence of equation (6-5) occurs because the

concentration terms in the equilibrixim constant are pres­

sure dependent.

The factors found to be significant in this study are

likely to be significant in any process where the reactions

of methylcyclopentane on >7-alumina take place. The

"X

129

to

o I-H

Xi cd

4)

O i/i

I

£4^ *4-l C/D O »

IS ^ vO

o <-> o

8-w to

0) o

o C

< w

O-^o

vO 0 0

o t ^

o LO

« o r-i

LO OC

o t ^

o o to r^

r g 0 0

o (VI I—1

0 0 o

r^ to

Ln

o

VO

e (NJ

00

X

vO

ec

LO o

CM

Ln

O CO

00

o o o

00

o o e

o

o

to

o CM (M (M

LO

rH

o (NI

LO

vo 'St

O

o CM

00

to o

LO CM

00 to

o LO en

vO to o

00

LO 00

to 00 CO

to to

LO o

rH (M

ro 0) o ^ ri TJ X

^

cr> VO

e 0 0

o vO

o LO

LO r-

o LO i H

C> CM

o

*!*•

o CvJ

o

o CTi

to T t

o

i H vO

1—1 <M

o

to

0 0 o

o

r H

r H O

CM ( M

CM LO

o O J

cn CO o

(M CM

00

o to

o O t o

C7> O

o to to

CM

to o

0 0 CNJ

«?!•

•^ •

O

o to CM

LO

o LO CM

o vO

o

to

II

' t t LO

o vO

II

LO

o Ii.

o

CM

0)

Pi s t^)

s X o 1

C

II Cl. W -^

a. w ^

a> ta rt <D pq

130

use of the forms of the mathematical models that were

developed and the values of the empirical constants is^

limited to the ranges of the variables that were studied.

6.4.0 Summary

The kinetics of the reactions of methylcyclopentane

over a platinum on n<l -alumina catalyst were successfully

studied. The major reaction products were found to be

benzene, cyclohexane, and the ring opening products:

2-methylpentane, 3-methylpentane, and n-hexane, Methyl­

cyclopentene was also detected at a smaller but stable

concentration level.

The complexity of an initially difficult problem of

describing the kinetics of a catalytic reaction system

composed of simultaneous and consecutive reactions was

compounded by a variable catalyst activity. The cata­

lyst was treated and operating conditions were developed

to minimize the activity changes. The data obtained under

these conditions were then treated in an attempt to com­

pensate for the remaining catalyst variability, making

a mathematical kinetic description of the system possible.

The kinetic descriptions of the effects of the inde­

pendent variables on the rates of formation of the several

reaction products were successfully achieved by the use of

empirical rather than theoretical mechanistic models. The

average relative error in the estimation of the rates was

i»«-N

131

about 4% for the isothermal models and about 18% for the

non-isothermal models.

In the course of the investigation, the following

observations were made:

1. The changes in the catalyst activity affected the

rates of formation of all the products except methylcyclo­

pentene. The rates of formation of the ring opening pro­

ducts were affected the most, dropping to about % of their

original values before catalyst stabilization was attained.

These rates were probably affected by the 29% decrease in

B. E. T. surface area and the 39% decrease in the pore vol­

ume of the used catalyst as compared to the fresh catalyst.

2. External diffusion effects were present in the

formation of methylcyclopentene and benzene. In the

mathematical rate modeling of these components, the flow

rate was found to be a significant variable whose effect

could not be removed by the usual mass transfer correla­

tions.

3. The following observations from which a possible

mechanism could be construed were made:

a. The benzene-cyclohexane hydrogenation-dehydrogen­

ation reactions were from 7 to 9 times faster than the

methylcyclopentane isomerization reactions. The hydro-

dehydrogenation reactions were still not fast enough to

maintain thermodynamic equilibrium between cyclohexane

and benzene. The benzene and cyclohexane competed with

\N

132

the methylcyclopentane for the platinum hydro-dehydrogen-

ation sites.

b. In the formation of methylcyclopentene, the

reverse reaction to methylcyclopentane was found to be

significant. The formation of methylcyclopentene from

cyclohexane or benzene was found to be insignificant.

These observations, in addition to observation (3a),

indicate that if methylcyclopentene is a precursor to

cyclohexene, the isomerization of methylcyclopentene to

cyclohexene is the slow step in the reaction and is

essentially irreversibleat the conditions studied.

c. The sources of the ring opening products were

some intermediates in the methylcyclopentane to cyclo­

hexane and benzene reaction sequence. The cyclohexane

and benzene were relatively stable toward ring opening

reactions. The isothermal total conversion to the ring

opening products was constant, although changes in pressure

and hydrogen to hydrocarbon ratio altered their distribution,

The following mechanistic scheme is consistent with

thes e obs ervat ions t C c

6 ^6—->0^=^@ \l ^ 0

2-methylpentane 3-methylpentane li-hexane

133

4o The distribution of ring opening products was

affected by changes in temperature. The n-hexane to

2-methylpentane ratio passed through a maximum near o

830 F. The presence of a maximum suggests the presence

of at least two sources for some of the ring opening

products.

The formation of six-membered ring products was

favored over the formation of the ring opening products.

An increase in temperature appeared to slightly favor

the ring opening products.

5. The total pressure had an abnormal effect on

the formation of benzene. An adequate unbiased mathe­

matical description of the isothermal rate data was

obtained only after the form of the model was changed to

one containing an empirical pressure term,

6. In a complex system, the probability of deter­

mining the governing mechanisms by theoretical model

fitting is so low as to make the search impractical. For

commercial reactors which are characterized by complex

parallel-series reactions, the empirical modeling approach

is highly preferred

^

134

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•w

139 73. V/ei, J., and Prater, C. D. American Instute of

Chemical Enr^ineerinp^ Journal. 9: i:o. 1., p. 11. 1963.

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140

NOMENCLATURE

A = catalyst concentration term

A = constant for viscosity estimation equation

A,B,C = thermodynamic equilibrium constants for 1-methyl­

pentene, 3-methylpentene, and 4-methylpentene,

based on methylcyclopentane

A,B,C = constants for nonisothermal rate models

a = activity

a,b,c = coefficients for molal heat capacity equation

B subscript on partial pressure = bulk term

C. = molal concentration of component i (moles/cm )

C = molal heat capacity at constant pressure (cal/mol °K)

C^ = molal heat capacity at constant volume (cal/mol °K)

D = scaling matrix for method of Marquardt

D.. = binary diffusivity of component j in i (cm /sec)

Dj = Knudsen diffusion coefficient (cm^/sec)

Dj . = effective diffusivity of component i in the 2

mixture (cm /sec)

d = effective diameter of packing P

F = molar liquid feed rate

f. = mole fraction of component i in the liquid feed

f ^ = element of Fisher information matrix

f( ) = function of ( )

G = mass flux of gas based on total cross section of bed

(gm/sec cm^)

/ ' ^

141

^m = molal mass flux of gas based on total cross section

of bed (moles/sec cm )

^ij " ® ®" ® t of Gauss-Nexton matrix

h = Thiele modulus

hg = heat transfer coefficient (cal/sec cm^ °K)

^^ri ~ ^ eat of reaction to product i (cal/mole)

i subscript on partial pressure = interface

Jjj = mass transfer number (dimensionless)

Jj = heat transfer number (dimensionless)

O r\

K = temperature, "Kelvin

k = Boltzman's constant (cal/sec cm^ Oj )

k = experimentally determined phase equilibrium ratio

k = thermal conductivity (cal/cm/°K/sec)

k^ = pseudo-forward rate constant

k-. = mass transfer coefficient for component i

(moles/sec cm^ atm)

L = catalyst mean pore length, cm

M = molecular weight

m. = mole fraction of component i in recovered liquid No« = modified Reynolds number ^®m

N. = mass flux of component i (moles/sec cm^)

P = critical pressure, atm

P-. = pressure factor for component i, atm

Pi > ^QA = partial pressure of component i in the bulk and i Si

at the surface respectively, atm

iMWi '

142

P„ = reduced pressure

P^ = system total pressure, atm

R = gas constant

r = mean pore radius, cm

r[ = generation rate of product i (moles/sec cm^)

r^ = generation rate of product i (gmole/hr/gm cat)

S = catalyst surface area (cm /gm)

^x ~ catalyst external surface area (cm^/gm)

S( ) = sum of squares function

•B> ^S ~ bulk temperature and surface temperature

respectively, °K

T = critical temperature, °K

T = reduced temperature

V = volume (cm^/mole)

V = critical volume (cm /mole)

V = catalyst pore volume (cm- /gm) o

3 Vp = volume of single catalyst particle, cm W = weight of catalyst, gm

X* = fraction of feedstock converted

X = matrix of known mathematical variables

X = distance in pore measured from outer surface of catalyst

YA = mole fraction of component i

V = vector of obs ervat J.ons A.

V = value of independent variable calculated from model

A SS vector of parameters for linear model 8 = direction vector

143

£. = force constant

€. = vector of errors

77 = effectiveness factor; type of alumina

6" = vector of parameters for nonlinear models

yi/ = viscosity, centipoises

\) , \) = degrees of freedom associated with lack of fit L ®

mean square and error mean square

S = vector of settings of independent variables

$ = density of mixture

QT = standard deviation; force constant

<P = function of ( )

-O- = collision integral

Abbreviations and S5nnbols

BZ = Benzene = \0\

CY = Cyclohexane

Cyclohexene =

nHex = n-hexane

MCP = Methylcyclopentane

MCP" = Methylcyclopentene

2-MP = 2-methylpentane

3-MP = 3-methylpentane

144

APPENDIX

A. Estimation of Transport Properties

B. Marquardt's Method

C. Computer Programs

D. Tables

145

APPENDIX A

ESTIMATION OF TRANSPORT PROPERTIES

The successful application of the mathematical rela­

tionships describing heat and mass transfer is dependent

upon accurate physical data. The direct measurem.ent of

transport properties at all experimental conditions is

not feasible, therefore accurate estimation and extra­

polation procedures are required. The book of Reid and

Sherwood i'p^) provides an excellent summary of estima­

tion techniques, estimation parameters, and comments as

to the goodness of the techniques.

The transport properties required to obtain heat

and mass transfer coefficients are gas density, heat capa­

city, viscosity, thermxal conductivity, and diffusion coef­

ficients. Data in the literature for these properties

are usually for a particular pressure and temperature and

for the pure substance. Experimentally determined physical

properties of gas mixtures are seldom available. Multi-

component physical properties must therefore be estimated

from data commonly available in the literature.

Since the calculation of physical properties at

various temperatures and pressures for several complex

multicomponent mixtures is very tedious, a computer pro­

gram similar to that of Holmes and Baerns (1 ) was adapted

for these calculations. The techniques that were used

146

in the calculation of the required transport properties

are presented in this Appendix.

A.1.0 Density

The density of the pure gases was* calculated from the

familiar ideal gas law using the compressibility factor Z.

e -- (^Mv^RT (A-1)

The density of the gas mixture was calculated assuming

Amagat's law. This assui.'iption is usually valid at low

densities. The density of the gas mixture is given by

where X ^ is the mole fraction of component i.

A.2.0 Heat Capacity

Molar heat capacity at constant pressure for the

pure components in this system have been measured and

correlated as a function of temperature by the following

equation: (56)

C^= a+ b( io"^)T +c(l0"^)Tf (A_3)

The molar heat capacity of the mixture was calculated

by again assuming Amagat's lav/ of additive volume. Before

the heat capacity of the mixture is calculated, the effect

_«^.

147

of temperature and pressure on the heat capacity of the

pure component is calculated using the equation (A-3).

The correction for pressure effect was neglible at the

conditions of interest since the pseudo-reduced temper­

ature and pressure of the mixture would be used for this

correction. The expressions for mixture heat capacity,

reduced temperature, and reduced pressure veve as follows

( >Dx* ~ -^—— ^or '

6-1

i = ;

A.3.0 Viscosity

The equation used for determining the effect of temp­

erature on the viscosities of the pure components was

Where <PU.Z5T^\ ^ /. 05 S f'' ^ - ^ f ^

T-D = reduced temperature.

The viscosity of the gas mixture was determined using

the semi-empirical equation of Wilke: (79)

148

)

No pressure correction was necessary since the correction

would be made using the pseudo-reduced temperature and

pressure of the mixture (see equations A-5, A-6). Using

these temperatures and pressures the correction was negli-

ble at the conditions of this study because of the large

amount of hydrogen in the gas mixtures.

A.^,0 Thermal Conductivity

The thermal conductivity of the pure gases was extra­

polated from the reported literature values at T>| to the

desired temperature, Tp, by the follo\\ring expression:

V^-.U^ (t)(l.3 Tft ^ (Cv2+4.47) (A-10)

^(/•33rRj) ^Cvl 4 4-.47J

where Ct)(1.33Tj ) is defined the same as for the viscosity

calculation.

The thermal conductivity of the gas mixtures was

calculated from the equation of Bird et. a^. (37)

149

r2k.

k IrvMX

XLVL (A-1T)

where Yii is defined by equation (A-9).

A.5.0 Diffusion Coefficient

The binary diffusion coefficients were calculated

from the following dimensional equation:

v/here

(A-12)

The collision integral H^A is given in a tabulated form

(55). 'as a function of / K'T.l, which was calculated from

the following expression:

1^] = l-30(Tp,^TRi^ 1-6; i

^

(A-13)

^Ij These binary diffusion coefficients were then used in

equation (3-7) to obtain the effective mixture diffusion

coefficients.

150

APPEPDIX B

MARQUARDT'S METHOD

In the notation of section ^.0.0, the computational

problem of interest is the minimiization of

Si9) -^ L^^- -7^] . 2 C y^- Cl^ 0)] (B-1

as a function of e. If 0^^^ is an initial guess, the

first order Taylor Series expansion about ^ is:

)

u = 1 , ..., n.

or in more compact form

ro) 2^(9) = 2J + S 5

9-(0)

(B-2)

(B-3)

where X is the nxp matrix

X yy\ V^ ^

(0) The vector £ is defined as W - £^ ? lj(£) is the nxl

ctor [f(Si,©), •.., fCf-n'® ' ^^^'^ (0) ,

ve

vector'77(£^^^).

is the nxl

The approximation on the right side of equation

(B-3; is now linear in the parameters . If- ^ is replaced

by 1 (o; + X_5 , an approximation for S(Gj is:

151

(£>J

and corresponding to the formula {h-k),. the value of

_5 which minimizes S(£) is:

where 1. - ^ - 2Z

By the definition of Q_, the new estimate is then (1) ^ (0)

£ ~ S •'• £ 5 SJ" "the next iteration can be started

by expanding about £^''). This method is originally due

to Gauss around 1821. The approximation (B-2) is frequently not sufficiently good, making S(£) a poor approximation

(1 ) for S(£). It may even be possible that S(£ ) is greater

than S(£^^), contrary to the objective. Some way is

therefore needed to systematically control the size of the

region over which the linear approximation to f ( ,£) Is

allowed to hold, limiting the size of the correction vector

g. Marquardt's algorithirx (51, 1959) is very good in this

respect and, v/ith slight modifications, is used in the

nonlinear least squares program.

In Marquardt's algorithm, a correction vector is com­

puted by using, instead of (B-5), the formula

v/here is a non-negative number. In using this correction

152

vector, it is observed that

(1) The solution ^^ of equation (B-6) minimizes

S(£), given by equation (B-^), on the boundary of a

(0) sphere centered at 0^ ^ and whose radius is NTS ' S =

— ' -m -m

^2) II 5^11 is a continuous decreasing function of

such that 115 11 -^0 as A^o^.

(3) If ^ is the angle between the correction vector

^ and 6 , the vector of "steepest descent", then S-^O

monotonically as -^-OD , and 6 rotates toward 6„. ' —m —g

As indicated by condition (3), the correction vector 5^

is effectively an interpolation between the vector produced

by the Gauss method and that produced by the method of

steepest descent. Since it is well known that the method

of steepest descent is not scale invariant, it is necessary

to scale the £ - space. Marquardt chose to scale in units

of the standard deviations of the df/<3© that make up the

matrix X. If D is a pxp diagonal matrix whose i-th

diagonal element is the same as that of X'X, then, after

scaling, the equation which gives the correction vector is

S -D'^(D'^xVD"^+;^ir'D ^xV. (B-7)

The idea behind the algorithm is based on the following

observations. The method of steepest descent often works

well on the initial iterations, but the approach to the

minimum grows progressively slower. The method of Gauss,

«=\

153

however, works well v/hen the minimum of S(£) is near, but

often gives trouble on the initial iterations. From

equations (B-5), (B-6), and condition (3), it can be seen

that these two extremes are represented by/|-ya> and^-^'O,

respectively. On the first few iterations when the minimum

is far away, there is also a danger of having the correction

vector become so large that the approximation breaks down.

From conditions (1) and (2) and considering the possibility

of an excessively large correction vector, it is seen that

a relatively large value of ^ should be used initially and

then decreased steadily as the Iterations progress. As

is decreased, the region of linear approximation is enlarged.

The decrease in /i should be made only if the sum of squares

S(£; at the new estimate is smaller than at the old. The

basic strategy at the i-th iteration is as follows:

Let S(^) be the value of S(£j obtained by using /^ in equa­

tion (B-7) to get £ ' from £ ' ~ ^ . Let ;)(i-'') be the

value of ^ from the previous iteration. Let V7I • Then

compute Si?!'^''^^) and S(;\ i-1 )/9 ).

(i) If s ( ; i ( - iV'9)^s(£^^-^)) , let;^^'^ =^^'-'V^ .

(ii) If S(^^^~^ V-i? ) 7 S(£^^-"''^), and S(;\^^"''^)^

SC£^i-^)), let;^^^^ =^^^•^^K

(iii) Otherwise, increase ^ by successive multiplication

by v> until for some smallest w, S(; ^ "'' V ) - S(£^^"'^^).

Let: ^ ^ =a^'-^V.

154

This algorithm should share with the gradient or

steepest descent method the ability to converge from a

region far from the minimum, and like the method of Gauss,

should converge rapidly once the vicinity of the minimum

is reached.

155

APPENDIX C

COMPUTER PROGRAMS

Uuk.

C PHYSICAL PROPERTIES AND DATA REDUCTION PROGRAM ^^^

C SIBJOB NODECK SIBFTC C C PROSRAM FOR ESTIMATION OF PHYSICAL PROPERTIES OF MIXTURES. C DETERMINATION OF KINECTIC RATE DATA FOR THE METHYLCYCL3PENTANE C DIFFERENTIAL REACTOR, HEAT AND MASS TRANSFER COEFFICIENTS, C AND AVERAGE CONDITIONS AT THE EXTERNAL SURFACE OF THE CATALYST. C EXECUTION TIME ON IBM 7040 FOR 60 DATA POINTS AND C 9 COMPONENTS WAS 2 MINUTES C

DIMENSION TCt6), PC(6), VC{6), EM(6), ZI6), A{6), B{6), C{6), I D(6), E(6), X(6,45), TK(5), PKI5), TR(6,5), PRt6,5), SIG(6), ?ENPUT(6), CP(6,5), RH0{6,5), HMU(6,5),CAY(6,5), TE(2), FI(2), 3 C11I6), C12(6), C13(6), C14(6),C15(6), CAYT(79), HMD(79), ^ TSR(5), DEE(6,6,4), R0W(<f5,<^), :PM(<t5,<f), EMU(<f5,4), PHn5,&,4) DIMENSION C<(<»5,4), DE(6,45,4), C0MP(6), TRED(45,4), PRED(<^5,4), IPRANDT(45), SCHMIT(6,45), REYN0L<45), CJD(45), CJH(45), CH(45), 2CKGI6t45), G(45), GM(45), AVM0L{45), RATEt8,45), 3FMC(5,B), WTM3(8), WTMDF(5), V0M0F(5), SPGR(5), 0UMM(45), 4V0LR(45), EMC(8,45), VLE(8), HYM0L(45), V(45), HCIPR(8), OHCIB) DIMENSION HCAVCB), XX{9,45), ST3IN{6,45), RR2(3,45), RR3(3,45) DIMENSION HH(45) ,DDUM(5) , SUMNJ(45) ,DELHR(45) DIMENSION TS(45) Cl=1.0E-03 C2=l.0E-06 C3=1.0E^05 C4=I.0E-09 C10»3.33E-03 CNT * (68B.»28.32»10.»»3.)/(760.»82.D6«296.)

400 FORMAT!72H I )

401 F0RMAT(5I3) 402 F0RMAT(10F7.0,F2.0,A6) 403 F0RMAT(6F12.0) 404 FORMAT(2F12.0) 405 F0RMAT(12E6.0) 406 FORMAT!/////20H TEMPERATURECDEG K)=F7.1,35X, 19HTEMPERATURE(DEG K)

1=F7.1/15H PRESSURE!ATM)sF9.4,38X,19HPRESSURE(ATM)*l.000//) 407 F0RMAT(1X,A6,3X,F8.3,4XF9.5,4XEI1.4,15XE11.4,5XEI1.4,4XE11.4) 408 F0RMAT(////23H DIFFUSIVITY(SQ CM/SEC)/9H C0MP0UND,7X,A6,8(5XA6)I 409 F0RMAT(1X,A6,4X,9(E12.4)) 413 F0RMAT{4H MIX,5X, 9HT REDUCED,5X, 9HP REDUCED,6X,7HDENSITY,

117X, 13HHEAT CAPACITY,4X, 9HVISC0SITY,4X, ISHTH.CONDUCTIVITY,/ 238X, 8HGRAM/CCM,16X, l3HCAL/M0Lft3EG K,4X, lOHCENTIPOISE,3X, 3 16HCAL/CM»DEG K»SEC)

411 F0RMAT(1X,I2,5XF8.3,6XF9.5,5XE11.4,15XH11.4,5XE11.4,4XE11.4) 412 FORMAT! 23H DIFFUSIVITY!SQ CM/SEC) /4H MIX,9X,A6,8(5X,A6)) 413 FORMAT !IX,I2,4X,9!F12.5)) 414 FORMATI7F10.0) 415 FORMAT! 19H MIXTURE PROPERTIES,//14H MOLE FRACTION,/4H MIX,

19X,A6,B!5X,A6)) 415 FORMAT!//9H COMPOUND,7X,2HTC,9X,2HPC,9X,2HVC,9X,1HM,lOX, IHZ, 9X,

llHA,10XvlHB,10X,lHC,10X,lH0,10X,lHE) 417 F0RMAT!1X,A6,5!F11.2),5!F11.4)) 418 F0RMAT!lXtI2,4X,9!E12.4)) 419 F0RMAT!/////28H ADDITIONAL INPUT PROPERTIES,/9H COMPOUNDtTX,

1 7HDENSITY,6X, 10HT!DENSITY),4X,10HP!DENSITY),5X. 9HVISC0SITY,3X, 2 12HT!VISC0SITY),3X, 15HTH.C0N0UCTIVITY.3X. 1IHTITH.COND.)/l5X i?y JJurAp/jI'^Acr"?^^!^'''^* 3HATM,10X, 1OHCENTIPOI SE .5X, 5H0EG K. 47X. 16HCAL/CM»DEG K»SEC.5X, 5HDEG K)

423 F0^MAT!1X,A6,4X,E14.5,4XF8.1.6XF8.3.2XE14.5,4XF8.1,2XE16.5.6XF8.1) 421 F0RMAT!9H COMPOUND.3X. 9HT REDUCED.3X. 9HP REDUCED,5X.7HDENSITY.

1 17X, 13HHEAT CAPACITY,4X, 9HVISCOSITY,4X. ISHFH.CONDUCTIVITY,/ 238X, 8HGRAM/CCM,I6X, 13HCAL/M0L»DEG K,4X, lOHCENTIPOISE,3X, 3 16HCAL/CM»DEG K»SEC) u.toc,3A,

450 FORMAT! 15H PRANOTL NUMBER,/ 4H MIX) 451 FORMAT!13,E15.7) 452 FORMAT! 15H SCHMIDT NUMBER,/IX, 3HMIX,6X,A6,5!12X,AS)) 453 F0RMAT!I3,4X,6!3X,E15.7)) 454 FORMAT! 4H MIX,2X,IHG,16X,2HRE,15X,2HJD,15X,2HJH,15X,IHH) 455 FORMAT! 13, 7!2X,E15.7)) 460 FORMAT! 20H MASS TRANSFER COEF.) 461 F0RMATI/4H MIX,2X,2HGM.15X.6!A6,1IX)) 464 F0RMAT!////36H AVERAGE MOLECULAR WEIGHT OF MIXTURE/4H MIX) 465 F0RMATI////28H HEAT CAPACITY, CAL/GM»DEG K,/4H MIX) 467 F0RMAT!E15.7) 470 F0RMAT!4H.MIX,2X,8HTEMP !K),6X,lOHPRES !ATM), 6X,5HHYD/HC,9X,

112HF0 RATE !ML),3X,9HFEED TYPE) 471 F0RMAT!I3,5E15.7) 501 FORMAT !4E16.8) 502 FORMAT I3E16.8) 503 F0RMATI/6E17.B) 543 F0RMAT!21H DID NOT CONVERGE, N*,I3) 3149 F0RMAT!//23H RATE, !GMOL/HR/GM CAT)/4H MIX,2X,5H22DMB,10X,3H2MP,

112X,3H3MP,12X,4HNHEX.llX.9HMCPENTENE.6X,9HMCPENTANE,6X,7HBENZENE, 28X,11HCYCL0HEXANE)

3150 F0RMAT!I3,8E15.7) 3151 FORMAT! 18H PARTIAL PRESSURES,/79H HEAT OF REACTION, KCAL/HR.GM

ICAT. GROUPED PARIFFINS STOICHIOMETRIC NUMBERS,/25H PARIFFIN RAT 210 2MP BASIS/ 25H PARIFFIN HATIO 3MP BASIS)

3152 FORMAT!4H MIX,2Xf10H220MBUTANE,5X,9H2MPEMTANEt6X,9H3MPENTANE,6Xt 17HNHEXANEt8X,9HMCPENTENE,6X,9HMCPENTANE,6X,7HBENZENE,8X, 211HCYCL0HEXANE, 4X, 8HHY0R0GEN)

3153 FORMAT! I3f9F13.B) 3154 F0RMAT!4X,E13.5,25X,6P13.8) 3155 F0RMAT!16X,3F13.8) 3156 FORMAT!13H1TABLE NUMBER //) 3160 FORMAT!/4H MIX,3X,5H22DMBf8X,9H2MPENTANEf4Xt9H3MPENTANE,4X«7HNHEXA

lNEf6X,9HMCPENTENE,4X,9HMCPENTANE,4X,7HBENZENEf4XfUHCYCL0HEXANE, 22X,8HHY0R0GEN/)

3161 F0RMAT!20H1SJRFACE TEMPERATURE/ 26H SURFACE PARTIAL PRESSURES) 4000 FORMAT I6E12.5)

C CAVr • QUANTUM MECHANICS REDUCED TEMPERATURE READ 405,!CAyT!n,I«l,79)

C HMO " COLLISION INTEGRAL FOR MASS DIFFUSIVITY !LOW PRESSURE) READ 405t!HM0(I),l-l,79)

C OBJECT OF READ 400 IS DATA IDENTIFICATION CARD HAVING A C 1 PUNCHED IN COLUMN 1.

5 READ 400 C NC - NUMBER OF COMPONENTS FOR WHICH PHYSICAL PROPERTIES ARE TO C BE ESTIMATED C NFM > NUMBER OF DIFFERENT PEED MIXTURES C NCF • NUMBER OF COMPONENTS IN REACTION SYSTEM C NM • NUMBER OF DATA POINTS

7 READ 401, NC, NFM, NCF, NT, NM

r'SB^s

IX = 0 ^^8 PRINT 400 PRINT 416

C TC = CRITICAL TEMPERATURE. DEGREES K C PC = CRITICAL PRESSURE. ATM. C VC = CRITICAL VOLUME, CUBIC CM./MOLE C EM = MOLECULAR WEIGHT. GRAMS/MOLE C Z = COMPRESSIBILITY FACTOR C A.B.C.3.E, = HEAT CAPACITY CONSTANTS C ENPUT = INDICATOR TO SHOW IF ADDITIONAL PHYSICAL PROPERTY DATA C IS PRESENT C COMP = 6 LETTER COMPONENT IDENTIFICATION

,?^J°.??^*^^^'^^'''^'^*'^^*^^'^^<'>'Z<J>»AJI''B«I>.C(I).D(I).E!I). lENPUTd ).COMP(I),I = I.NC) DO 10 1=1.NC PRINT <^17,C0MP(I),TC(I),PC(I).v:(I).EM(I},Z!I).A!I).B!I).C!I),

1D!I),E(I) IF!IX)10.8,10

8 IF!ENPUT!I))9,I0.9 9 IX =1

10 CONTINUE C C DATA REDUCTION C C DUMM = FEED IDENTIFICATION C VOLR = LIQUID FEED RATE. ML. C V = HYDROGEN FEED RATE, UNCORRECTED CUBIC FEET/HOUR

READ 403. DUMM!l). VOLR(I). V(l) C WTMO = M3LECJLAR WEIGHT

READ 403, (WTMO(J), J=1,NCF) C VLE = EXPERIMENTAL VAPOR-LIQUID EQUILIBRIUM K

READ 403.(VLE!J).J=l.NCF) DO 3070 1=1.NFM

C SPGR = SPECIFIC GRAVITY OF LIQUID FEED READ 403, SPGR(I)

C FMC = MOLAR CONCENTRATION IN LIQUID FEED. PERCENT 3070 READ 403.!FMC!I.J).J=l,NCF)

C DDUM = DUMMY VARIABLE. NUMBER OF DATA POINTS AT A GIVEN C TEMPERATURE AND PRESSURE

READ 403, !DDUM(J), J=1,NT) INN = 0 DO 2060 K=1,NT

C TK. PK = TEMPERATURE AND PRESSURE OF A GROUP OF DATA POINTS READ 434, TKIK) , PK(K) IF!INN)2050,2050,2051

2050 INN = 1 NO = 1 NOT = DDUMIK) GO TO 2053

2051 NO = NOT • 1 NNN = DDUM!X) NOT = NOT • NNN

2053 DO 2060 J=NO,NOT C EMC = PRODUCT LIQUID MOLAR CONCENTRATION 2063 READ 433, IEMC!II,J), 11=1,NCF)

DO 2061 J=2,NM DUMMIJ) = DUMM!J-1) VOLRIJ) = VOLRIJ-1)

2061 V!J) = V!J-l)

FEED

DO 3002 I = 1,NFM SSUM = 0.0 00 3001 J = 1,NCF FMC!I,J) = FMC!I,J)/100.

3001 SSUM = FMC!I,J)»WTMO!J) • SSUM WTMOF(I) = SSUM

C V0M3F = MOLAR VOLUME OF LIQUID 3002 VOM0F!I) = WTMOF!I)/SPGRCI)

INN = 0 DO 3008 K = 1,NT TK!<) = ITK!<) • 460.)/1.8 PK(K) = PK!K)/14.7 • 682./760. IF!INN)3020,3020,3021

3020 INN = 1 NO = 1 NOT = DDUMIK) GO TO 3023

3021 NO = N3T • 1 NNN = DDUMIK) NOT = NOT • NNN

3023 DO 3008 J=?NO,NOT 1 = DUMMIJ)

C FLMOL = MOLAR FEED RATE FLMOL = VOLR!J)/VOMOF!I) DO 3003 II = l.NCF EMC!II,J) = EMC!II,J)/I00.

C RATE = GM MOLES/HR/GM CATALYST RATE!II,J) = !0.97»EMC( 11,J)+0.03»EMC(11,J)»VLE!11)-FMC! I, II) )

3003 RATECII.J) = RATE{11.J)#FLM0L/2.1125 C DELHR = HEAT OF REACTION

DELHR!J)=-RATE!1.J)»18.72-RATE!2.J)»15.94-RATE!3,J)»15.55 DELHR!J)=DELHR!J)-RATE!4,J)»14.48*RATE!5.J)»27.47^RATE!7.J)»48.92 DELHR!J) = DELHRiJ) - RATE!8,J)•3.71 HYM3L!J) = CNT»V!J) HH!J) = HYMOL!J)/FLMOL

C TOTMO = TOTAL MOLES TOTMO = HYMOLIJ) • FLMOL HPRES = PK!K)»HYMOL!J)/TOTMO DO 3004 II = 1,NCF

3004 HCIPR!II)=P<!K)»FLMOL»FMC!I,II)/TOTMO C G = MASS FLOW RATE GM/SEC/CM»«2

G!J) ^ !VOLR!J)»SPGR!I) • HYMOL!J)»2.016)/!3600.»4.294) OHYMO = HYMOLIJ) • 3.•FLMOL^EMC(7,J) • FLM3L»EMC!5,J) 0HYM3 = OHYMO -FLMOL»!EMC!I,J) • EMC!2,J) • EMC!3,J) • EMC!4,J)) OHYMO = OHYMO - 3.»FLM0L»FMC!I,7) OTOMO = OHYMO • FLMOL DHYC = 0HYMO«PK!K)/OTOMO DO 3005 II = 1,NCF

3005 OHCIII) = FLMOL»EMC!II,J)*PK!K)/OTOMO HAV = IHPRES • 0HYC)/2.0. SSUM = HAV 00 3006 II - 1,NCF HCAVIII) » !HCIPR!II) • 0HC!II))/2.0

3006 SSUM - SSUM *• HCAVIII) DO 3007 1 1 = 1 , NCF XX = AVERAGE MOLE FRACTION BETWEEN INLET AND OUTLET CONDITIONS

3007 XXIIItJ) = HCAV!II)/SSUM XX!9,J) = HAV/SSUM

159

«s

C ESTIMATION OF PHYSICAL PROPERTIES C

X!l,J) = XXII,J) • XX!2.J) • XX(3.J) • XX(4,J) X(2,J) = XX!5,J) X!3,J) = XX!6,J) X!4,J) = XX!7,J) X!5,J) = XX!8,J) X!6,J) = XX!9,J) NCFl = NCF • 1 DO 3010 IP = 1,NCF1

3010 XX(IP,J) = XX!IP,J)»PK(K) C STOIN = STOICHIOMETRIC NUMBER

ST3IN!I,J) = -(RATE(I,J)*RATE{2.J)4-RATE(3,J)+RATE!4.J)) ST0IN(2.J) = -RATE!5.J) ST0IN(3.J) =-RATE(6,J) ST3IN|(4.J) = -RATE!7.J) ST0INI5.J) = -RATE(8.J) ST3IN(6.J) = -STOINll.J) + ST0IN{2,J) • 3.0»ST0IN!4.J) DO 3011 II = 1,6

3011 STOIN(II.J) = STOIN(II,J)/<RATE{7,J)) DO 3008 II = 2.4 RR2(II-l.J) = RATE! II.J)/RATEI2,J)

3008 RR3(II-1.J) = RATE( II.J)/RATE{3.J) IF!IX)13,15.13

13 PRINT 419 15 DO 24 1=1,NC

C13!I)=0.0 C14!I)=0.0 C15(I)=0.0 CUBER=!ALOGIVC!I)))/3. CUBER = EXPICUBER) SIG!I)=0.833»CUBER Cll!I)=C10»S0RT(EM(I)»TC(I))/(CUBER'CUBER) C12!I) = EM!I)/Z! I) IFIENPUT!I))24.24,16

C ADDITIONAL INPUT PROPERTIES. DENSITY !GM/CC) AT GIVEN T AND P. C VISCOSITY !CENTIPOISES) AT T. AND THERMAL CONDUCTIVITY C CAL/CM/SEC/DEGREE K) AT T.

16 READ 4l4.RH3I.TEE.PEA.HMUI.TE!l).CAYI.TE!2) PRINT 420.COMP!I).RHOI.TEE,PEA,HMUI,TE!I),CAYI,TE!2) DO 1811=1.2 IF!TE!Il))17,18,17

17 TRI=TE!I1)/TC!I) EX=0.390865»ALOG(1.9»TRI) FI!I1)=1.05B»!TRI»».645)-.261/(!1.9»TRI)»*EX)

18 CONTINUE IF(RH0I)19,20,19

19 C13!I)=TEE»RHOI/PEA 20 IF!HMUI)21,22,21 21 C14!I)=HMUI/FI!1) 22 IF!CAYI)23,24,23 23 TES=TE!2)»TE!2)

CPI=A!I)*^B!I)»Cl»TE(2)*C!n»C2»TES«^D!I)»C3/TES*E!I)»C4»TE!2)»TES IF!1-3)5000,5000,5002

5003 IF!I-1)5002,5002.5001 5001 CPI = CPI - 2.35 5002 CVI = CPI - 1.987

C15!I)=CAYI/!FI!2)»!CVI*4.47)) 24 CONTINUE

1000 CONTINUE ^^^ DO 30 K=1,NT TSR!K)=TX!<)»»1.5/PK!K) TKS=TK!<)»TX!K) C5=C1»TK!K) C6=C2»TKS C7=C3/TKS C8=C4»T<!<)»TKS C9=P<!<)/TK!<) 00 30 1=1,NC TR!I,K)=TK!K)/TC! I) PR!I,K)=PK!K)/PC!I) EX=0.390865*ALOG!1.9»TR! I.K)) FIK=1.058»!TR!I,K)»».645)-.261/!!1.9»TR!I,K))»*EX) CP!I,K)=A!I)*B!I)»C5*C!I)«C6+D!I)»C7*E(I)»C8 IF!1-3)5010,5010,5012

5010 IF!I-1)5012,5012,5011 5011 CP!I,K) = CP!I,K) - 2.35 5012 CVIK = CP!I,X) - 1.987

IFIC13!I))725,25.725 25 RH0!I,K)=C12!I)»C9/82.057

GO TO 726 725 RH0!I,K)=C9»C13!I) 726 IF!C14!I))727,27,727 27 HMU!I,K)=C11!I)»FIK

GO TO 728 727 HMU!I,<)=FI<«C14! I) 728 IF!C15!I))729,28,729 28 CAY!I,K)=HMU!I,K)»!CVI<*4.47)/!EM!I)•100.0)

GO TO 30 729 CAY!I,K»=FIK»!CVIK+4.47)»C15!I) 30 CONTINUE

DO 40 IQ=1,NC DO 40 1=1,NC ILL = 1 EMR=SQRT!!EM!I)+EM!IQ))/!EM!I)»EM!IQ))) EMCC = EMR«!.00214 - .000492»EMR) PA=EM!I)/EM!IQ) PBB= SQRT!8.0»(1. • PA)) PQ=!EM!IQ)/EM!I))»».250 IF!SIG!I))31,32,31

31 IF!SIG!IQ))33,32,33 32 ILL=0

GO TO 733 33 SIGPS=!SIG!l)»SIG!I)+2.0»SIG!I)»SIG!IQ)*SIG!IQ)»SIG!IQ))/4.0

DC = EMCC/SIGPS 733 00 40 K=1,NT

IF!ILL)735,734,735 734 DEE!I,IO,K)^0.0

GO TO 39 LINEAR INTERPOLATION FOR COLLISION INTEGRAL

735 GAMP«1.30*SQRT!TR!I,K)«rR!IQ,K)) J«2

34 IF!GAMP-CAYT!J))38,38,36 36 J=J*1

GO TO 34 38 CONTINUE

CD*&AMP-CAYT!J-1) CY«HMD!J-1)*IIHMD!J)-HMD!J-1))/!CAYTCJ)-CAYT!J-l)))«CO

HMIP = CY 162

DEE!I,IQ,K)=TSR(K)»DC/HMIP

39 PHI!I,IQ,K)=!!l.0*SQRT(HMU!I,K)/HMU!IQ,K))«PQ)»»2.)/PBB 40 CONTINUE

DO 41 K=l,NM SUMNJ(<) = 0.0 DO 41 J=1,NC

41 SUMNJtO = SUMNJ(K) • STOIN(J,K) INN = 0 DO 50 K=l,NT IF!INN)2000.2000,2001

2000 INN = I NO = 1 NOT = DDUM!K) GO TO 2003

2001 NO = NOT • I NNN = DDUMIK) NOT = NOT • NNN

2003 DO 50 J=NO.NOT CK!J.K)=0.0 EMU(J.K)=0.0 CPM(J.<)=6.0 ROW!J.K)=0.0 TRED!J,K)=0.0 PRED1J,IC)=0.0 DO 45 1=1.NC

C ROW = DENSITY ROW( J.K) = RDW( J.K)4^RH0( I.K)»X!I .J)

C CPM = CONSTANT PRESSURE MOLAR HEAT CAPACITY CPM( J.K)=CPM! J.K)4-CP( I.K)»X(I,J) TREDlJ,K)=TRE0(J,K)+TR!I,K)»X(I,J) PRED( J,K)=PRE0( J,K)-»-PR( I,K)»X( I, J) SUM1=0.0 SUM2=0.0 DO 42 IQ=1»NC SUMl = SUMl*X(IO,J)»PHn I,IQ,K) SUM2=SJM2^(X(IQ.J)»ST0IN(I.J)-X!I.J)»ST0IN!IQ.J))/DEE!I.IQ.K)

42 CONTINUE C DE = DIFFUSION COEFFICIENT OF COMPONENT IN MIXTURE CORRECTED FOR C M3LAR FLUX

DEII.J.K) = !STOIN!I.J) - X!I.J)»SUMNJ!J))/SUM2 C EMU = VISCOSITY IN CENTIPOISE

EMU!J,K)=EMU!J.K)+!X(I,J)«HMU!I,K))/SUMl C CK = THERMAL CONDUCTIVITY

CK!J,K)=CK!J,K)+!X(I,J)•CAY!I,K))/SUMl 45 CONTINJE 50 CONTINUE

C C DETERMINATION OF EXTERNAL TEMPERATURE AND PRESSURE GRADIENTS C

DO 98 J=1,NM AVMOLIJ) = 0.0 DO 98 1=1,NC

98 AVMOLIJ) = AVMOLIJ) • EM!I)*X!I,J) PRINT 464 DO 800 J=1,NM

800 PRINT 451, J, AVMOLIJ) DO 54 J=1,NM

54 GM!J) = G!J)/AVMOL!J)

: GM = MOLAR FEED RATE PRINT 3156 PRINT 470 INN a 0 DO 105 K=1,NT IF!INN)2010,2010,2011

2010 INN = 1 NO = 1 NOT - DDUMIX) GO TO 2012

2011 NO = NOT • 1 NNN = ODUMIX) NOT = NOT *• NNN

2012 DO 105 J=NO.NOT HH = HYDROGEN/HYDROCARBON

t-l

105

75

PRINT PRINT PRINT DO 75 PRINT PRINT PRINT INN = DO 80

RATIO J. TK!K) f PK!K), HH!J),

1=1,NC)

2015

2016

2017 83

2020

2021

2022 90

< 71. 3156 415.!C0MP{I), J=1.NM 413.J.!X!I,J),I=1,NC) 3156 410 0 K=1,NT

IF!INN)2015,2015,2016 INN = 1 NO = 1 NOT = ODUMIO GO TO 2017 NO = NOT * 1 NNN = ODUMIK) NOT = NOT • NNN DO 80 J=NO.NOT PRINT 411.J.TRED(J.K).PRED!J,K),^0W!J.K), PRINT 3156 PRINT 412,!C0MP!I),I=1,NC) INN = 0 DO 90 <=1,NT IF!INN)2020,2020,2021 INN = 1 NO = 1 NOT = DDUM!K) GO TO 2022 NO = NOT 4- 1 NNN = DDUMIK)

NOT • NNN J=NO,NOT 418,J,!DE!I,J,K),I=1,NC) 3156

VOLR!J), DUMM!J)

;PM!J,K),EMU!J,K),CK!J,K)

NOT = 00 90 PRINT PRINT PRINT 3149

NM !RATE(II,J), 11=1,NCF)

00 1053 J=l 1050 PUNCH 4000,

PRINT 3156 PRINT 3151 PRINT 3160 DO 1070 J*1,NM PRINT 3153, J, !XX!II,J),II=1,9) PRINT 3154, DELHRIJ), !STOIN!11,J),I 1*1,6) PRINT 3155, !RR2! II,J),II = 1,3)

1070 PRINT 3155, !RR3!11,J),I 1 = 1,3) 55 INN = 0

DO 100 K=1,NT IF!INN)2005,2005,2006

2005 INN = 1 NO = 1 NOT = DDUMIK) GO TO 2007

2006 NO = NOT • 1 NNN = DDUMIX) NOT = NOT * NNN

2007 DO 100 J=NO,NOT AVCP = CPM!J,K)/AVMOL!J)

Z PRANDT = PRANOTL NUMBER 93 PRANDT!J)=CPM(J,K)/AVM0L!J)^EMU!J,K)*0.01/CK!J,<)

DO 94 1=1,NC C SCHMIT = SCHMIDT NUMBER

94 SCHMIT!I,J)=EMU(J.K)•O.01/!ROW!J,K)•DE!I,J .K) ) C DIAPAR = AVERAGE PARTICLE DIAMETER, CM

DIAPAR = 0.40 REYNOL(J) = DIAPAR^G!J)/!EMU!J,K)^0.01) REY = REYNOL!J)^*0.41 - 1.5 CJH!J) = 1.1/REY CJOIJ) ' CJHIJ)/1.5

: CH = HEAT TRANSFER COEFFICIENT, CAL/CM^^2/SEC/0EGREE K 95 CH!J)=:JH(J)^CPM{J,K)/AVMOL!J)^G!J)/!PRANDT!J)^^0.6667)

DO 96 11=1,NC CKG!II,J)=CJD!J)^GM!J)/!PK!K)^(1.-SUMNJ!J)/ST0IN!II,J)*X!II,J))) SCHMIT!II,J) = ABS!SCHMIT!II,J))

C CKG = ^ASS TRANSFER COEFFICIENT, M0LE/CM**2/SEC/ATM 96 CKG!I I,J)=CKG!II,J)/SCHMIT!II,J)^^0.6667

100 CONTINJE PRINT 3156 PRINT 450 DO 300 J=1,NM

300 PRINT 451, J, PRANDT!J) PRINT 3156 PRINT 452, !COMP!I), 1=1,NC) DO 301 J=1,NM

301 PRINT 453, J, !SCHMIT!I,J), 1=1,NC) PRINT 3156 PRINT 454 DO 302 J=1,NM

302 PRINT 455, J, PRINT 3156 PRINT 460 PRINT 461, !COMP!I) , DO 303 J=1,NM

303 PRINT 455, J, GM!J), PRINT 3161 PRINT 3160 CATALYST SURFACE AREA/ CCT = l./!26.38*3600.) INN « 0 DO 3310 K=1,NT IF1INN)3300,3300.3301

3300 INN « 1 NO * I NOT = DDUMIK)

164

G!J). REYNOL!J), CJDIJ). CJHIJ), CH!J)

1*1,NC)

!CKG!I.J). I«1»NC)

GM » 26.38 CM»*2

GO TO 3302 3301 NO = NOT • 1

NNN = DDUMIK) NOT = NOT • NNN

3302 DO 3310 J=NO,NOT C TS = CATALYST SURFACE TEMPERATURE

TS!J) = T<!K)-DELHR{J)/!26.38^3.6^CH(J)) DO 3309 11=1,NCF IF!11-4)3307,3307,3308

3307 XX!II,J)=RATE!II,J)^CCT/CKG!1,J) + XX!II,J) GO TO 3309

3308 XX(II.J)=RATE!II.J)^CCT/CKG!II-3,J) + XX!II,J) 3309 CONTINJE

Z XX = SURFACE PARTIAL PRESSURE 3306 XXI9,J)=RATE!7,J)^(-ST0IN(6,J))^CCT/CKS!6,J) • XX!9,J)

PRINT 404, TSCJ) 3310 PUNCH 4000, TS!J). !XX!II.J). 11=1.9)

CALL EXIT END

SENTRY

165

166 C C NONLINEAR PARAMETER ESIMATION PROGRAM C SIBJOB NODECK SIBFTC C C MAIN PROGRAM SETS PROGRAM OPERATION PARAMETERS AND READS VALUES C OF THE DEPENDENT AND INDEPENDENT VARIABLES. INDEPENDENT C VARIABLES ARE PLACED IN COMMON FOR USE IN SUBROUTINE MODEL. C SUBROUTINE M3DEL CALCULATES MODEL VALUE OF DEPENDENT VARIABLE. C EXECUTION TIME ON IBM 7040 FOR 6 PROBLEMS OF 3 PARAMETERS WAS C ABOUT 5 MINUTES, FOR 3 PROBLEMS OF 8 PARAMETERS ABOUT C 7 MINUTES. C C NPROB = PROBLEM NUMBER C MODEL = NAME OF SUBROUTINE WHICH COMPUTES VALUE OF MODEL C NOB = NUMBER OF OBSERVATIONS C Y = ONE-DIMENSIONAL ARRAY CONTAINING THE VECTOR OF THE C OBSERVED FUNCTION VALUES C NP = NUMBER OF UNKNOWN PARAMETERS C TH = ONE-DIMENSIONAL ARRAY CONTAINING A VECTOR OF INITIAL C PARAMETER VALUES C DIFF = ONE-DIMENSIONAL ARRAY CONTAINING A VECTOR OF PROPORTIONS C FOR TAKING PARTIAL DERIVATIVES NUMERICALLY C SIGNS = ONE-DIMENSIONAL ARRAY A PRIORI RESTRICTING THE SIGNS ON C EACH PARAMETER TO THAT AFFIXED BY THE SIGN OF THE INITIAL C ESTIMATE. A POSITIVE NUMBER FOR SIGN(I) RETAINS THE FEATURE C WHILE A ZERO DISABLES IT. C EPSl = SUM OF SQUARES CONVERGENCE CRITERION. l.OE-7, ITERATIONS C AGREE TO 7 DECIMAL PLACES C EPS2 = PARAMETER CONVERGENCE CRITERION. ALL PARAMETERS AGREE C TO WITHIN THIS TOLERANCE AFTER SUCESSIVE ITERATIONS. C MIT = MAXIMUM NUMBER OF ITERATIONS (70 HERE) C F = VECTOR OF PREDICTED VALUES CALCULATED IN SUBROUTINE MODEL C FNU = ALGORITHM INITIATING CONSTANT, .01 RECOMMENDED. C FLAM = ALGORITHM INITIATING CONSTANT, 10. RECOMMENDED. C OBJECT 3F READ 400. PROGRAM IDENTIFICATION CARD HAVING A 1 C PUNCHED IN COLUMN I C TRAPS(-l) = BINARY SUBROUTINE PERMITTING FLOATING POINT C UNDERFLOWS C SCRATCH DIMENSIONS IN SUBROUTINE GAUSHS TO BE GREATER THAN C 5^NP • 2^{NP^^2) * 2^N0B + NP»NOB C

DIMENSION RATE(9.58). XX(9.58),TSI 60). E(90) DIMENSION TH(IO). SIGNSIIO). DIFFIIO) DIMENSION CKG(6.58). REMI60), M(21) DIMENSION A!8).B(8).C(8),F(25),R(25) COMMON XX, TS, JJJ. LLL. CKG. RATE. REM EXTERNAL MODEL

400 F0RMAT(72H 1 )

401 FORMAT !5I3) 402 F0RMAT(6E12.5) 403 F0RMATI6F12.5) 404 FORMAT !I3, 10E12.5)

READ 400 READ 401, NC, NFM, NCF, NT, NM PRINT 400 CALL TRAPS !-l)

1050

3310 C

c

3320

C

c

DO 1050 J=l,NM READ 432,(RATE!I I,J), 11 = 1,NCF) RATE UNITS = GMOL/HR/GM CATALYST DO 3310 J=l,NM READ 402, TS!J), {XX(II,J), 11=1,9) TS JNITS = DEGREES KELVIN XX UNITS = SURFACE PARTIAL PRESSURES IN ATM. DO 3320 J=1,NM READ 402,!C<G(II,J), II=l,NC) CKG UNITS = M0LE/CM*^2/SEC/ATM READ 403,!REM(I), 1=1,44) REM = »^ODIFIED REYNOLDS NUMBER EPSl = l.OE-7 EPS2 = l.OE-7 DATA SIGNS(1),SIGNS(2),SIGNS(3),SIGNS(4),SIGNS(5)/I.,1. ,1. , 1. ,1. / DATA SIGNS! 6).SIGNS(7).SIGNS(8).SIGNS(9).SIGNS(10)/I..1..1..1.,1./ DATA DIFF!l),DIFF!2),DIFF!3),DIFF(4),DIFF(5)/.01,.01,.Ol..01..01/ DATA DIFF(6).DIFF(7),DIFF(8).DIFF(9),DIFF(I0)/.0I..01..0l..0l..01/ CCT = 1-/(26.38*3600.) THE ARRAY M LOCATES THE NONISOTHERMAL, CONSTANT THROUGHPUT

AND FEED COMPOSITION DATA POINTS DATA M(l).M!2).M{3).M(4),M(5),M(6).M(7)/l,7.10.14.16.17.21/ DATA M(8).M(9).M(10).M(I1),M(12).M(13)/24.27.28.33,41,<»5/ DATA M(14).M(15).M(16),M(17).M(18).M(19)/46,47,48,49.50.51/ DATA M(20),M(21)/53,55/ DATA A(2).A(3).A(4),A(5)/33.4818,36.5519.39.2988,18. 198/ DATA A!6),A!7)/29.lttb5,29.648/ DATA B(2),B!3),B(4),B(5)/2.4331,2.583.2.2799,2.5039/ DATA B!6).B(7)/1.6759,1.849/ DATA:(2).C(3),C!4),C(5)/2.3622,2.5788.2.7726.1.2839/ DATA C!6),C!7)/2.0593,2.0917/ NOB = 21 NP = 3 NPROB = I DO 3 1=1.NOB J = M(I)

167

10

TS( I ) K = 1 K = K TH(1) TH(2) TH!3) NPROB DO 11

= TS!J)

• I

11

• 1 = A(K) = B I O = C!K) = NPROB 1=1.NOB

J = M(I ) E d ) = RATEK.J) DO 4 1=1.NOB

4 E!I) = ABSIEII)) ^. ^„^, ,^ CALL GAUSHS(NPROB.MODEL.NOB.E.NP.TH,DIFF.SIGNS.EPSl.EPS2.70. 1.01,10.) IF{K .LT. 7) GO TO 10 CONTINUE CALL EXIT END

SIBFTC MODEL SUBROUTINE MODEL(NPROB,TH,F.NOB.NP) DIMENSION TH!1). F(l). XX(9.58), TS(60) DIMENSION JJJ! I). LLL(l) DIMENSION CKG(6.58), RATE(9.58)

20

DIMENSION REM(60) COMMON XX, TS, JJJ. LLL. CC = l.OE+4 DO 50 1=1.NOB SUM = 0.0 DO 20 J=l,9 SUM = SUM • XX!J,I) ALNK = TH!1) - TH(2)•ALOG(SUM) -

168 CKG, RATE, REM

50 EXP(ALNK) ALNK^XX(6,I)

TH(3)^CC/TS(I) ALNK = F(I) = RETURN END

SIBFTC GAUSHS

SUBROUTINE GAUSHS(NPROB,FOF,NOB,Y.NP.TH,DIFF.SIGNS.EPSl.EPS2. 1 MIT,FLAM,FNU) DIMENSION Y!NOB), TH(NP). DIFF(NP). SIGNS(NP) DIMENSION SCRATC(750) IA=1 IB=IA4-NP IC=IB*NP ID=IC*NP IE=ID*NP IF=IE+NP IG=IF*-NOB IH=IG+NOB II = lH-»-NP^NP IJ=II+NP^NP CALL GSHS59(NPR0B.F0F,N0B,Y,NP.TH,DIFF,SIGNS.EPS1,EPS2.MIT.

1 FLAM,FNU,SCRATC!IA),SCRATC( IB),SCRATC(IC),SCRATC(ID), 2 S C R A T C ( I E ) , S C R A T C ! I F ) , S C R A T C ! 1 3 ) . S C R A T C ! I H ) , S C R A T C ( 1 1 ) , 3 SCRATC(IJ)) RETURN END

SIBFTC GSHS59 SUBROUTINE GSHS59(NPRRO,FOF.NBO,Y,NQ.TH.DIFZ.SIGNS.EPIS.EP2S.

IMIT.FLAM,FNU, Q,P,E.PHI,TB,F.R.A.D,DELZ) DIMENSION TH(NO), OIFZ(NO). SIGNS(NQ). Y(NBO) DIMENSION Q(NQ). P(NQ). E(NQ). PHI(NQ). TB(NQ) DIMENSION F(NBO). R(NBO) DIMENSION A(NO.NQ). D(NO,NQ), DELZ(NBO,NQ) NP = NQ NPROB = NPRBO NOB = NBO EPSl = EPIS EPS2 = EP2S PRINT 1000, NPROB, NOB, NP PRINT 1001 CALL GSHS60!1,NP,TH,TEMP,TEMP) PRINT 1002 CALL GSHS60!1,NP,DIFZ,TEMP,TEMP) IFINP .LT. 1 .OR. NP .GT. 50 GO TO 15

15 IF! MIT .LT. 1 .OR. MIT .GT. 999 .OR. FNU .LT. l.)GO TO 99 GO TO 16

16 DO 19 1=1,NP TEMP=DIFZ!I) IF!TEMP)17,99,18

17 TEMP=-TEMP 18 IFITEMP .GE. 1. .OR. TH(I) .EQ. 0.)G0 TO 99

GAUSH007 GAUSH008

GAUSHOll

.OR. NOB .LT. NP ) GO TO 99

169 GO TO 19

19 CONTINJE GA=FLAM Nil - I GAUSH017 ASSIGN 225 TO IRAN GAUSH018 ASSIGN 265 TO JORDAN 6AUSH019 ASSIGN 180 TO KUWAIT GAUSH020 IF( EPSl .LT. 0. ) GO TO 5 GO TO 10

5 EPSl = 0 10 IF( EPS2 .GT. 0.)G0 TO 30

GO TO 40 40 IF! EPSl .GT. 0.)G0 TO 50

GO TO 60 60 ASSIGN 270 TO IRAN GAUSH027

GO TO 70 GAUSH028 50 ASSIGN 265 TO IRAN GAUSH029

GO TO 70 GAUSH030 30 IF( EPSl .GT. 0.)G0 TO 70 80 ASSIGN 270 TO JORDAN GAUSH032 70 SSQ = 0 GAUSH033

CALL F0F!NPROB,TH,F.NOB.NP) GAUSH034 DO 90 I = 1. NOB GAUSH035 R!I) = Y!I ) - F( I) GAUSH036

90 SSQ = SS3*R!I)^R( I ) GAUSH037 PRINT 1003. SSQ GAUSH038 GO TO 105

C C BEGIN ITERATION C 100 CONTINUE 105 GA=GA/FNU

INTCNT = 0 DO 130 J=1.NP TEMP = TH(J) GAUSH042 P(J)=DIFZ(J)^TH(J) TH(J)= TH!J)*P(J) Q(J)=0 GAUSH044 CALL FOF(NPROB,TH,DELZ(1,J),NOB.NP) DO 120 I = 1. NOB GAUSH046 DELZ(I,J)= DELZ(I.J)-F(I) oA..cun£.Q

120 Q( J)=Q!J)*-DELZ( I,J)^R( I ) GAUSH048 Q(J)= Q(J)/P!J)

C Q=XT^R (STEEPEST DESCENT) 130 TH(J) = TEMP

GAUSH050

DO 150 I = 1, NP flulnoll DO 151 J=l.I GAUSH052 GAUSH053

GAUSH054 SUM = 0 DO 160 K = 1, NOB rAii<HnR«i

160 SUM = SUM * OELZ(K. D^DELZIK. J) GAUSH05b TEMP= SUM/(P!I)^P(J)) GAUSH057

ici ^Ir'^i^ycMo GAUSH058 151 D(I.J)=TEMP ^_^^^^ ^^^^^^^ ^^^^^^^ 150 E(I) = SQRT!D!I,I))

GO TO < U W A I T , ( 1 8 0 . 6 6 6 ) - ITERATION 1 ONLY-

180 CONTINUE rAiKiHO^Q ASSIGN 666 TO KUWAIT GAUSH069

i70 C -END ITERATION 1 ONLY-666 DO 153 1=1,NP

DO 153 J=l,I GAUSH070 GAUSH071

A(I,J)=0(I,J)/(E(I)^E(J)) GAUSH072 153 A(J,I)=A!I,J) GAUSH073 C A= SCALED M3MENT MATRIX

DO 155 1=1,NP GAUSH074 P(I)=0(I)/E!I) GAUSH075 PHI(I)=P(I) GAUSH076

155 A(I,I)=A(I,I)+GA GAUSH077 1 = 1 CALL MATINV!A,NP,P,I,DET,NP)

C P/E = CORRECTION VECTOR STFP=1.0 GAUSH080 SUM1=0. GAUSH090 SUM2=0. GAUSH091 SUM3=0. GAUSH092 DO 231 1=1.NP GAUSH093 SUM1 = P( n ^ P H K I )-»-SUMl GAUSH094 SUM2=P(I)^P(I)+SUM2 GAUSH095

231 SUM3 = PHI( D ^ P H K I)-i-SUM3 GAUSH096 TEMP = SUM1/S0RT(SUM2^SUM3) IF(TEMP .GT. 1.)Gn TO 232 GO TO 233

232 TEMP=1.0 233 TEMP = 57.295^ARCOS(TEMP)

170 on 220 1=1,NP GAUSH081 220 TB( I)=P( I)^STEP/E( I ) -»-TH(I) GAUSH082 7000 F0RMAT(30H0TEST POINT PARAMETER VALUES )

DO 2401 1=1,NP GAUSH099 IF( SIGNS(I) .GT. 0. .AND. TH(I)^TB(I) .LE. 0.)G0 TO 663

2401 CONTINUE GAUSH102 SUMB=0 GAUSH084 CALL FOF!NPROB,TB,F,NOB,NP) GAUSH085 00 230 1=1,NOB GAUSH086 R(I)=Y(I)-F(I) GAUSH087

230 SUMB = SJMB<-R( I )^R( I) GAUSHOjBB IF(SJMB/SSQ - 1. .LE. EPSDGO TO 662 GO TO 663

663 IFI TEMP .LE. 30.)G0 TO 665 GO TO 664

665 STEP=STEP/2.0 GAUSH105 INTCNT = INTCNT ^ 1 IFdNTCNT .GE. 36)G0 TO 2700 GO TO 170

664 GA=GA^FNU GAUSH107 INTCNT = INTCNT + 1 IF!INTCNT .GE. 36)00 TO 2700 GO TO 666

662 CONTINUE DO 669 1=1,NP GAUSHllO

669 TH!I)=TB!I) GAUSHlll GO TO IRAN,1225,265,270)

225 DO 240 I = 1. NP GAUSH116 IFIABS !P!I)^STEP/E(I))/(1.0E-20^ABS (TH!I)))-EPS2) 240,240,250 GAUSH117

240 CONTINJE ^^^c^Jl® PRINT 1009, EPS2 GAUSH119 GO TO 280 GAUSH120

250 GO TO JORDAN,1265.270)

A ^

i71 265 IF!ABS!SUMB-SSQ) .LE. EPSDGO TO 260

GO TO 270 260 PRINT 1010, EPSl GAUSH123

GO TO 280 GAUSH124 270 SSQ=SUMB GAUSH125

NIT=NIT*l GAUSH126 IF(NIT .LE. MIT)GO TO 100 GO TO 280

2700 PRINT 2710 5/16/66 2710 F0RMAT(//ll5H0^^^^ THE SUM OF SQUARES CANNOT BE REDUCED TO THE SUM5/16/66

lOF SQUARES AT THE END OF THE LAST ITERATION - ITERATING STOPS /)5/l6/66 C C END ITERATION C 280 PRINT 1004, NIT

PRINT 1007 CALL GSHS60!l,NP,TH,TFMP,TEMP) PRINT 1040,GA,SUMB GAUSH114 PRINT 1011 PRINT 2001, (F(I), I = I, NOB) GAUSH129 PRINT 1012 GAUSH130 PRINT 2001, !R(I), I = 1, NOB) GAUSH131 PRINT 1013 QQ = 0. DO 281 1=1,NOR R(I) = R(I)/Y(I)^100.

281 QQ = QQ ••• ABS(R( I ) ) TNOB = NOB PRINT 2001,(R(I)t 1=1,MOB) QQ = QQ/TNOB PRINT 1018 PRINT 2001, QQ YMAX = Y d ) DO 283 1=2,NOB IFIYMAX - Y!I))282,283,283

282 YMAX = Y!I) 283 CONTINUE

YMIN = Y(I) DO 285 1=2,NOB IFIYMIN - Y!I))285,284,284

284 YMIN = Y! I ) 285 CONTINJE

PRINT 1019 PRINT 2001, YMAX, YMIN GAUSH132 SSQ=SUMB GAUSH133 IDF=NOB-NP GAUSH134 PRINT 1015 1=0 CALL MATINVID.NP.P.I.DET.NP) GAUSH137 DO 7692 1=1.NP YY = 0(1,1) YY = ABS(YY)

7692 E d ) = SQRTIYY) DO 340 1=1,NP GAUSH141 DO 340 J = I. NP GAUSH142 A(J.I)=D!J,I)/(E(I)^E(J)) ^^,,,, GAUSH143 D(J.I)=0(J,I)/!DIFZ(I)»THd)^DIFZ(J)^TH!J)) siuSH144 D(I,J)=0!J,I) GAUSH145

340 A d , J)=A! J, I)

J)

CALL GSHS60(3,NP,TEMP,TEMP,A) 7357 PRINT 1016

CALL GSHS60!I,NP,E,TEMP,TEMP) IFdOF .GT. 0) GO TO 7058 GO TO 410

7058 AABB = IDF SDEV = SSQ/AABB PRINT 1014,SDEV,IDF SDEV = SQRTISDEV) DO 391 1=1,NP P( I ) = T H ( I )4 -2 .0^E( I ) •SDEV

391 T B ( I ) = T H ( I ) - 2 . 0 ^ E { D^SDEV PRINT 1039 CALL GSHS60!2,NP,TB,P.TEMP) DO 415 K=1.N0B TEMP=0.0 DO 420 1=1.NP DO 420 J=1,NP

420 TEMP= TEMP <• DELZ ( K , I ) •DELZ ( K , J ) •D( I TEMP = ABSITEMP) TEMP = 2.^SQRT(TEMP)^SDEV R(<)=F(K)4.TEMP

415 F(K)=F(<)-TEMP PRINT 1008 IE = 0 DO 425 1=1,NOB,10 IE=IE^-10 IF(NOB-IE) 430,435,435

430 IE=NOB 435 PRINT 2001,!R(J),J=I,IE) 425 PRINT 2006,(F(J),J=I, IE) 410 PRINT 1033, NPROB

RETURN 99 PRINT 1034

GO TO 410 I0000F0RMAT(3BH1N0N-LINEAR ESTIMATION, PROBLEM NUMBER

114H OBSERVATIONS, 15, IIH PARAMETERS ) 1001 F0RMAT(/25H0INITIAL PARAMETER VALUES ) 1002 F0RMAT(/54H0PR0P0RTI0NS USED IN CALCULATING DIFFERENCE 1003 F0RMAT(/25H0INITIAL SUM OF SQUARES = E12.4) 1004 FORMAT( //45X,13HITERATION NO. 14) 1005 F0RMATd4H0DETERMINANT = E12.4) 1006 F0RMAT(/52H0EIGENVALUES OF MOMENT MATRIX - PRELIMINARY ANALYSIS ) 1007 FORMAT(/32H0PARAMETER VALUES VIA REGRESSION ) 1008 F0RMAT(////54H0APPROXIMATE CONFIDENCE LIMITS FOR

lUE ) 10090FORMAT(/62HOITERATION STOPS

ISS THAN E12.4) 10100FORMAT(/62H0ITERATION STOPS

ISS THAN E12.4) 1011 F0RMAT(//22H FINAL FUNCTION 1012 FORMAT(////10H0RESIDUALS ) 1013 F0RMAT(////23H0PERCENT RELATIVE ERROR ) 1014 F0RMAT(//24H0VARIANCE OF RESIDUALS =

120H DEGREES OF FREEDOM ) 1015 FORMAT!////19H0C0RRELATION MATRIX ) 1016 F0RMAT(////21H0N0RMALIZING ELEMENTS ) 1018 F0RMAT(///23H0AVERAGE RELATIVE ERROR ) 1019 F0RMAT(///39H0MAXIMUM MINIMUM EXPERIMENTAL VALUE )

1, 7")

GAUSH147

13,// 15,

GAUSH152

GAUSH153

GAUSH155 GAUSH156

GAUSH160

GAUSH162 GAUSH163

GAUSH168 GAUSH169 GAUSH170 GAUSH171 GAUSH172 GAUSH173 GAUSH174 GAUSH175 GAUSH176

GAUSH178 GAUSH179

QUOTIENTS )

EACH FUNCTION VAL

- RELATIVE CHANGE IN EACH PARAMETER LE

- RELATIVE CHANGE IN SUM OF SQUARES LE

VALUES )

,E12.4,1H,I4,

17J 1033 F0RMAT(//19H0EN0 OF PROBLEM NO. 13) 1034 F0RMAT(/16H0PARAMETER ERROR ) 10390FORMAT(/71HOINDIVIDUAL CONFIDENCE LIMITS FOR EACH PARAMETER (ON LI

INEAR HYPOTHESIS) ) 10400FORMAT(/9HOLAMBDA =E10.3,40X,33HSUM OF SQUARES AFTER REGRESSION =

1E15.7) 1041 FORMAT(25H0ANGLE IN SCALED COORD. = F5.2, 8H DEGREES ) 1043 F0RMAT(28H0TEST POINT SUM OF SQUARES = E12.4) 2001 F0RMAT(/10E12.4) 2006 FORMAT(10E12.4)

END SIBFTC GSHS60

SUBROUTINE GSHS60(I TYPE,NO,A,B, C ) DIMENSION A!NO),B{NQ),C(NQ,NQ) NP = NQ NR = NP/IO LOW = I LUP = 10

10 IF( NR )15,20,30 15 RETURN 20 LUP=NP 30 PRINT 500, (J,J=LOW,LUP)

GO TO (40,60,80),ITYPE 40 PRINT 600,(A(J),J=LOW,LUP)

GO TO 100 60 PRINT 600, (B!J),J=LOW.LUP)

GO TO 40 80 DO 90 I=LOW,LUP 90 PRINT 720,I,(C(J,I),J=LOW,I) 100 LOW = LOW ^ 10

LUP = LUP • 10 NR = NR - I GO TO 10

500 F 0 R M A T ( / I 8 , 9 I 1 2 ) 600 F 0 R M A T d 0 E 1 2 . 4 ) 720 F 0 R M A T ( I H 0 , I 3 , 1 X , F 7 . 4 , 9 F 1 2 . 4 )

END SIBFTC MATINV

SUBROUTINE MAT INV(A,NVAR,B,NB,DETERM,MA) DIMENSION A(MA,MA),B(MA,NB),INDEX(50,2) EQUIVALENCE (T,SWAP,PIVOT),!K,LI)

C C INITIALIZATION C

DETERM=1.0 DO 20 J=l,50

c c c

40

20

60

80

INDEX(J,1) = 0

SEARCH FOR PIVOT ELEMENT

I = 0 IRANK = 0 AMAX=-l. DO 105 J=1,NVAR IF!INDEX!J,1))105,60,105 DO 100 K=1,NVAR IF!INDEX!K,l))100,80,100 AABB = A(J,K) T = ABSIAABB)

GAUS GAUS GAUS GAUS 4 GAUS GAUS GAUS GAUS

GAUS GAUS GAUS

GAUS GAUS

465 466 467 468 469 470 471 472

474 475 476

478 479

GAUS 481

7^

c c c

c c c

85

100 105

110 c c c

140

200

250

310

350

370

400

450

500 550

C c c 630

705

720

IF(T.LE.AMAX) GO TO 100 IROW=J ICOLJM=K AMAX=T CONTINJE CONTINUE IF(AMAX) 720,720,110 INDEXdCOLUM ,1) = IRQW

INTERCHANGE ROWS TO PUT PIVOT ELEMENT ON DIAGONAL

IFdROW.EQ.ICOLUM )G0 TO 310 DETERM=-OETERM DO 200 L=l,NVAR SWAP=A(IROW,L) A(IROW,L)=A(ICOLUM,L) A(ICOLJM,L)=SWAP DO 250 L=l, NB SWAP=B(IROW,L) B(IROW,L)=B(ICOLUM,L) B(ICOLJM,L)=SWAP 1 = 1*1 INDEX!I,2)=IC0LUM PIVOT=A(ICOLUM,ICOLUM) DETERM=PIVOT^DETERM IRANK=IRANK«-1

DIVIDE PIVOT ROW BY PIVOT ELEMENT

AdCOLUM, IC0LUM) = 1.0 PIVOT=A(ICOLUM,ICOLUM)/PIVOT DO 350 L=1,NVAR A( ICOLUM,L)=AdCOLUM,L)^PIVOT DO 370 L=l,NB

B(ICOLUM,L)=B(ICOLUM,L)^PIVOT

REDUCE NON-PIVOT ROWS

DO 550 L1=1,NVAR IF(L1.EQ.IC0LUM)G0 TO 550 T=A(L1,IC0LUM) A(L1,IC0LUM)=0.0 DO 450 L=1,NVAR A(Ll,L)=A(Ll,L)-AdCOLUM,L)^T DO 500 L=1,NB B(Ll,L)=B(Ll,L)-BdCOLUM,L)^T CONTINUE GO TO 40 INTERCHANGE COLUMNS

ICOLUM=INDEX!1,2) IROW = INDEXdCOLUM , 1 ) DO 705 K=1,NVAR SWAP=A!<,IROW) A!< , IR3W)=A!< , IC0LUM) A(K,ICOLUM)=SWAP 1 = 1-1 I F d .GT. 0) GO TO 630

GAUS GAUS GAUS GAUS GAUS GAUS

GAUS GAUS GAUS

GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS

GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS GAUS

GAUS GAUS GAUS GAUS GAUS

485 486 487 488 489 490

492 493 494

496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522

524 525 526 527 528 529 530 531 532 533 534 535

537 538 539 540 541

fJ^V

740 NBsNVAR-IRANK RETURN

END SENTRY C DATA SIBSYS

175 GAUS 543

GAUS 544

176 C c c c SEXECU

DATA PLOTTING PROGRAM

C c c c c c c c c

TE WATFOR DIMENSION RAT£(9,60), XX(9,60), TS(60)

LINE(130,130), FEEDTY(60), HYDHC{60) KK(20,60), JJ(20,60), A(20), B(20) C0MP(12), TIME(60) MOLE/HR/GM CATALYST

PARTIAL PRESSURE TEMPERATURE

DIMENSION DIMENSION DIMENSION RATE = GM XX = SURFACE TS = SURFACE

401 402 403 404 405 406 407

408

1050

3310

FEEOTY = FEED TYPE HYDHC = HYDROGEN/HYDROCARBON RATIO TIME = ACCUMULATED RUNNING TIME ON THE CATALYST COMP = 6 LETTER COMPONENT IDENTIFICATION A = SCALE FACTOR FOR ORDINATE B = SCALE FACTOR FOR ABSCISSA INTEGER BLANK, AST, DOT INTEGER ONE, TWO, THREE, FOUR, FIVE, SIX. SEVEN. EIGHT INTEGER FEEOTY, HYDHC F0RMAT(513) F0RMAT(6E12.5) F0RMAT(6F12.b) F0RMATd3, 10E12.5) FORMAT (2413) FORMATdX, 130AI) FORMAT (IH1,A6,IX,20HRATE, SCALE FACTOR =E12.5,2X,6HVERSUS,2X,A6,2 LX,32HPARTIAL PRESSURE, SCALE FACTOR = E12.5,///) FORMATI12A6) READ 401, NC, NFM, NCF, NT, NM DO 1050 J=l,NM READ 402, (RATEdl.J), II = l,NCF) DO 3310 J=l,NM READ 402, TS!J), (XXdI.J), 11=1,9)

BLANK, AST, DOT/IH , IH^, IH./ ONE, TWO, THREE, FOUR/lHl, 1H2, 1H3, IH4/

SIX, SEVEN, EIGHT/1H5, 1H6, 1H7, 1H8/ (FEEDTYd), 1 = 1,NM) (HYDHCd). 1 = 1.NM) (COMPd). 1 = 1.12) ( A d ) . 1 = 1,12)

DATA DATA DATA READ READ READ READ READ B!9) JP =

!Bd) , I = l»I2)

55

54 5

FIVE, 405, 405, 408, 403, 403, = l.O 44

NCOMIX = 7 B(5) = 5000. B(8) = 300. NDUMMY = 0 NN = 0 NRATE = 2 CONTINUE NDUMMY = NDUMMY • 1 DO 1 J=l» 130 DO 1 K=l, 130 LINE(J»K) = BLANK DO 2 J=9,109 LINE(J.IOI) = DOT LINE(J,1) = DOT

77

9) GO TO 50 = ABS(RATE(NRATE,J)) A(NRATE)^RATE(NRATE, J)

10) GO TO 51 * RATE(8,J) A(NRATE)^Q

* 1.

• RATE(2,J) A(NRATE)^0

• 1.

* RATE(3»J) + 1.

= B(NCOMIX)^XX(NCOMIX,J) • 9.

DO 3 J=9,109,10 LINE(J,lOl) = AST

3 LINE(J,1) = AST DO 15 K=l,10l LINE(9,K) = DOT

15 LINEd09,K) = DOT DO 4 K=l,10l,10 LINE(9,K) = AST

4 LINE!109,<) = AST DO 6 J=1,JP IF(NRATE .GT. RATE(NRATF, J) KK(NRATE,J) = GO TO 6

50 IFINRATE .GT. Q = RATE(7,J) KK(NRATE,J) = GO TO 6

51 Q = RATEd, J) KK(NRATE,J) =

6 CONTINJE DO 7 J=1,JP JJINCOMIX,J)

7 CONTINJE 9 DO 8 J=1,JP

KKK = JJ(NCOMIX,J) JJJ = KK(NRATE,J) LFDTYP = FEEDTY(J) LHRATO = HYDHC(J) GO TO (10,20.30,40),

10 IFILHRATO .3T.4) GO LINE(KKK,JJJ) = ONE GO TO 8

11 LINE(KKK,JJJ) = FIVE GO TO 8

20 IFILHRATO .GT. 4) GO LINE(KKK,JJJ) = TWO GO TO 8

21 LINE(KKK,JJJ) = SIX GO TO 8

30 IF(LHRATO .GT. 4) GO LINE(KKK,JJJ) = THREE GO TO 8

31 LINE(KKK,JJJ) = SEVEN GO TO 8

40 IFILHRATO .GT. 4) GO LINE(KKK,JJJ) = FOUR GO TO 8

41 LINE(KKK,JJJ) = EIGHT 8 CONTINUE

PRINT 407, COMP(NRATE), DO 100 J=l,130 K = 131 - J

100 PRINT 406,(LINE!I,K), 1=1,130) 106 NN = NN • 1

GO TO(20I,202,203,204,205,206,207.208),NN 201 NRATE = 3

GO TO 54 202 NRATE = 4

• RATE(4,J)

LFDTYP TO 11

TO 21

TO 31

TO 41

A(NRATE), COMPINCOMIX), BINCOMIX)

178

203

204

205

206

207

208 209

SENTRY

GO TO 54 NRATE = 5 GO TO 54 NRATE = 7 GO TO 54 NRATE = 6 GO TO 54 NRATE = 10 GO TO 54 NRATE = 11 GO TO 54 CONTINJE CONTINUE CALL EXIT END

r

DATA SIBSYS

I <-

179

APPENDIX D

TABLES

fe

\ \

Table 1^

Reactor and Catalyst Data

180

Reactor length

Reactor volume

Reactor cross-sectional area

Weight of catalyst

Vol. of inert catalyst diluent

Bulk density of catalyst

Platinum c>ontent by weight

Catalyst support

Catalyst N2 surface area

J$2 - surffecre ar ea af t er - run s

Catalyst pralletv-slzei

Cttt^yst manufacturer

61 cm

250 cm^ 2

^.29^ cm

2.1125 gm

230 cm-

1,15 gm/cm-

0.35%

-w-alumina

-00 m /gm

280 m /gm

1/16 in. extrudate Englehard Industries RD-150-C

Catalyst pellet external area (after removad from reactor) 2^.39 cm^/gm

181

Tkble 15

Hydrocarbon Retention Time in a

10 F t , Squalane Column at 115 °C

Component Normal boi l ing Retention time point °C (minutes)

n-pentane 36.2 ^-,^8

unknovm 5.62

2-2 dimethylbutane ^9*7

2-3 dimethylbutane 58,1

2~methylpentane 60 6,88

j-methylpentane 6- 7.59

n-hexane 69 8,15

methylcyclopentene "^ 70

methylcyclopentane 72 9 •-02

benzene 80.1 11.37

-r^yclohexane 8I 13i®9

=N

Table 16

Chromatographic Response Factors

182

Component Peak height, basis 1 attenuation, full scale = 10,

Response factor

2-me thy1pen tan e

3-methylpentane

n-hexane

Bfetbylcyclopen t en e

Methylcyclopentane

Benzene

10

15

20

30

^5

2

3

'••5

10

20

35

Cyclohexane

0.5^5

0,529

0.532

0.535

0,t?^3

0.573

0.59H

1,000

1 .000

1.015

1.085

1,12

1,1^

1,15

1 .16

1. 07

183

Table 17A ^

Component Thermodynamic and Transport Properties

Heat of reaction from MCP at 700^K, Kcal/gmol

Coefficients fbr molal heat capacity equation

a 3 b x 10

c X 10^

Crit, Temp, (°K)

Crit, Press, (atm)

Thermal conductivity at T °K. ^ cal/cm/^K/sec x 10"

Viscosity at T OKj centipaise

2MP 3MP n-Hex MCP

-15.9^ -15.55 -i^o^8 +27.4-7

Assumed same as n-hexane

^98.1 50^09

29«95 J0-8i

6,011

106.75

"33.36

507.9

29.9^

7.58 @ ^33

Assumed as C cyclohex ane minus 2.35

536^

39»9^

From reference 56

^Estimated from methods given in reference 55

\ \

Table 17B ^

Component Thermodynamic and Transport Properties

184

Heat of r e a c t i o n from MCP at 700^K, Kcal/gmol

C o e f f i c i e n t s fo r molal hea t c a p a c i t y equa t ion

b X 10

c X 10

3

Cri to Temp, (°K)

C r i t , . P r e s s (atm)

Thermal c o n d u c t i v i t y a t T °K, ca-l /em/°K/sec x 10-^

V. iscosl ty a t T °K, c e n t i p o i s e

MCP Benzene Cyclohex Hydrogen

+^8.92' -3.71

Assumed as Cp c y c l o ­hexane minus 2 .35

532.8

37.36

6.20 @ »+33

- . ^ 0 9

77.629

-26 .^29

562

h8.6'

7.28 § 5-1-85

0.0172 @ 700

-7.701

125.675

-^1 o 58h-

553

J+0,0

3.929 0 375.5

0.015 § 700

6,9^

- . 2 0 0

0,^81

^ 1 . 3 ^

20 ,8^

73.63 @ 573.5

0.015 @ 700

a From reference 56

^True critical temperature = 33.3 °K True critical pressure =12.8 atm

^••BW—r-

r^

185

Feed No,

3

Table 18

Liquid Feed Molar Composition and

Definition of Computer Output Symbols

:MP 3MP

1 0.0000 0,0000

0,0000 0,0362

0,0127 0,U^38

0,0000 0,0374

nHex

1,6770

2.7573

2,7316

2,9289

MCP

98,0253

92,0048

90,1096

95.9998

Benz Cyclohex

0.0000

0,0000

6,119 -

0,0000

0,2977

5.2016

0,9829

1.0339

2-'-2DMB - 2-2 dimethylbutane

2MP5 3MP = 2 methylpentane, 3 methylpentane

nHex - n-Hexane j grouped paraffins

BenzT, Cyclohex = Benzene, Cyclohexane

MCPentene = Methylcyclopentene

MCP and MCPentane = Methylcyclopentane

Hyd/'HC - hydrogen/hydrocarbon feed molar ratio

T,, - Catalyst surface temperature, °K

G ~ mass feed rate, gm/sec/cm'

GM - molar feed rate, gmoles/see/cm'

Mas- transfer coefficients units = mole/cm'^/sec/atm

H -• heat transfer coefficient, cal/cm /see/^K

Re -'• modified Reynolds number

Jj) -- mass transfer correlation j factor

'H heat transfer correlation 3 factor

/.,

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