• TABLE OF CONTENTS
HIDE
 Title Page
 Table of Contents
 List of Tables
 List of Figures
 Key to symbols
 Abstract
 Introduction
 Literature review
 Model for absorption and reaction...
 Steady state results
 Stability of the steady states
 Conclusions
 Appendices
 Bibliography
 Biographical sketch














Title: Gas-liquid reactors
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Permanent Link: http://ufdc.ufl.edu/UF00099505/00001
 Material Information
Title: Gas-liquid reactors predictions of a model respecting the physics of the process
Physical Description: xii, 214 leaves : ill. ; 28 cm.
Language: English
Creator: White, Daniel D., 1950-
Publication Date: 1982
Copyright Date: 1982
 Subjects
Subject: Chemical kinetics   ( lcsh )
Chemical reactions   ( lcsh )
Gases -- Absorption and adsorption   ( lcsh )
Chemical Engineering thesis Ph. D
Dissertations, Academic -- Chemical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis (Ph. D.)--University of Florida, 1982.
Bibliography: Bibliography: leaves 211-213.
Statement of Responsibility: by Daniel White, Jr.
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00099505
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000365973
oclc - 09967335
notis - ACA4801

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Table of Contents
    Title Page
        Page i
    Table of Contents
        Page ii
        Page iii
    List of Tables
        Page iv
    List of Figures
        Page v
        Page vi
    Key to symbols
        Page vii
        Page viii
        Page ix
        Page x
    Abstract
        Page xi
        Page xii
    Introduction
        Page 1
        Page 2
        Page 3
    Literature review
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
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        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
    Model for absorption and reaction in a gas-liquid CSTR
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
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    Steady state results
        Page 51
        Page 52
        Page 53
        Page 54
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    Stability of the steady states
        Page 103
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    Conclusions
        Page 117
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    Appendices
        Page 125
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    Bibliography
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    Biographical sketch
        Page 214
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Full Text












GAS-LIQUID REACTORS: PREDICTIONS OF A MODEL RESPECTING
THE PHYSICS OF THE PROCESS










By

Daniel White, Jr.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA

1982










TABLE OF CONTENTS

PAGE

LIST OF TABLES............................. ..................... iv

LIST OF FIGURES...................................................... v

KEY TO SYMBOLS................................................... vii

ABSTRACT........................ ........... .....................xi

CHAPTER

ONE INTRODUCTION............................................... 1

TWO LITERATURE REVIEW........................................... 4

THREE MODEL FOR ABSORPTION AND REACTION IN A GAS-LIQUID CSTR.....24

FOUR STEADY STATE RESULTS............................ .. .....51

FIVE STABILITY OF THE STEADY STATES.. ........................103

SIX CONCLUSIONS............................................... 117

APPENDICES

A STEADY STATE ANALYSIS FOR A SINGLE-PHASE CSTR.............125

B MANN AND MOYES' INTERFACIAL TEMPERATURE RISE:
GENERAL RESULT............................................ 131

C PREDICTION OF THE MASS TRANSFER ENHANCEMENT CAUSED
BY THE REACTION OF THE DISSOLVED SOLUTE.................134

D INTERFACIAL TEMPERATURE RISE.............................. 143

E THE SLOPE CONDITION FOR A GAS-LIQUID CSTR.................151

F THE APPLICATION OF POORE'S METHOD FOR DETERMINING
MULTIPLICITY TO A GAS-LIQUID CSTR........................160

G THE FRANK-KAMENETSKII APPROXIMATION ......................173

H THE EXISTENCE OF MUSHROOMS AND ISOLAS IN A
SINGLE-PHASE CSTR......................................... 179

I THE EXISTENCE OF MUSHROOMS AND ISOLAS IN A
GAS-LIQUID CSTR.......................................... 194

J STABILITY CONDITIONS FOR AN UNENHANCED GAS-LIQUID CSTR....202











PAGE

BIBLIOGRAPHY.................... ............................... 211

BIOGRAPHICAL SKETCH.................. ............................ 214

















































iii










LIST OF TABLES


TABLE PAGE

3.1 CHARACTERISTIC TIMES............................... .......... 32

3.2 DIMENSIONLESS EQUATIONS ........................................ 47

3.3 EQUATIONS IN TERMS OF TIME SCALES................................49

4.1 OPERATING REGIONS... .......................................... 63

4.2 CRITICAL POINTS SEPARATING OPERATING REGIONS...................64

4.3 HEAT GENERATION FUNCTION IN EACH OPERATING REGION..............68

4.4 PARAMETERS FOR THE CHLORINATION OF DECANE AS
STUDIED BY DING ET AL. (1974)..................................70

5.1 STABILITY CONDITIONS FOR AN UNENHANCED GAS-LIQUID
CSTR WITH B IN EXCESS AND e = 0.... ..........................109

F.1 UNIQUENESS AND MULTIPLICITY CONDITIONS FOR A
NONADIABATIC, UNENHANCED GAS-LIQUID CSTR WITH
B IN EXCESS................................................. 170

G.1 ERROR DUE TO THE FRANK-KAMENETSKII APPROXIMATION..............176

J.1 STABILITY CONDITIONS FOR AN UNENHANCED GAS-LIQUID
CSTR WITH ec = 0............................................. 208










LIST OF FIGURES


FIGURE PAGE

2.1 Heat generation and heat removal functions versus
temperature for a single-phase CSTR....................... .....5

3.1 Concentration profiles near the interface for
instantaneous reaction ...................................... 36

3.2 Enhancement factor versus Ik ............................... 38

4.1 Series and parallel resistances for absorption
and reaction...................................................55

4.2 rI and r2 versus rate constant................................ 57

4.3 Relative rate of absorption versus rate constant...............59

4.4 Heat generation and heat removal functions versus
temperature .............. .............. ................. 71

4.5 Comparison of enhanced mass transfer region model
and full model................................................ 73

4.6 Effect of T on the heat generation function.................... 76

4.7 Effect of k10 on the heat generation function..................77

4.8 Effect of E on the heat generation function...................78

4.9 Effect of kL on the heat generation function.................. 79

4.10 Effect of a on the heat generation function...................80

4.11 T versus T for the data of Ding et al. (1974)................. 83

4.12 T versus T: isola behavior.................................... 85

4.13 T versus T: mushroom behavior ............................... 86

4.14 T versus r: s-shaped multiplicity............................ 87

4.15 k10 versus r: multiplicity regions and type...................89

4.16 Da = g(e) versus e............................................ 95

5.1 The relationship of the zeros of the determinant, mI and
m2, to the heat removal and heat generation curves............107

5.2 Region where uniqueness is guaranteed and the first
stability condition is satisfied............................. 110










FIGURE PAGE

5.3 Region where the second stability condition is
guaranteed: BRB versus ..................................... 111

5.4 Region where the second stability condition is
guaranteed: n/Y versus BR ................................... 112

5.5 Steady state temperature versus Damkohler number .............. 114

A.1 Heat generation and heat removal functions versus
temperature for a single-phase CSTR........................... 126

A.2 Multiplicity patterns for a single-phase CSTR................ 129

C.1 Concentration profiles near the interface for
instantaneous reaction................................. .. 139

D.1 Temperature profile near the interface for
instantaneous reaction ..................................... 145

E.1 An uniqueness condition for a gas-liquid CSTR.................158

F.1 Da = g(e) versus e ...................................... 162

F.2 The left and right sides of Equation F.8.................... 165

H.1 Multiplicity patterns for a single-phase CSTR.................180

H.2 Number of zeros of de/dT..................................... 186

H.3 Zeros of f for ec = 0 .................................... 191

I.1 Formation of multiplicity patterns in a single-phase CSTR.....198

1.2 Steady state analysis for a two-phase CSTR...................200

J.1 The left and right sides of Equations J.9 and J.10............. 205

J.2 Steady state temperature versus Damkohler number..............210










KEY TO SYMBOLS

A gaseous reactant

a interfacial surface area per unit volume, cm-1

B liquid reactant

BR dimensionless heat of reaction

BS dimensionless heat of solution

C concentration in the liquid phase, gmole/cm3

Cp heat capacity of the liquid, cal/gmole.K

D diffusivity, cm2/sec

Da Damkohler number

Dai Damkohler number for the interfacial region

Dal, Da2' value of the Damkohler number such that 0 equals ml, m2,
Da3 Da4 s1 and s2
E activation energy, cal/gmole

ER error

E* reaction factor

fb fraction of gas absorbed into the bulk liquid that reacts
fi fraction of gas absorbed that reacts near the interface

fR fraction of gas absorbed that reacts

g(e) function of the steady state temperature that equals the
Damkohler number

H dimensionless heat transfer coefficient

HR heat of reaction, -AHR, cal/gmole

HS heat of solution, -aHS, cal/gmole

H Henry's Law coefficient, atm.cm3/gmole

H, inverse of the characteristic time for heat transfer,
H = H1T, sec-I

hL, hL heat transfer coefficient with and without reaction,
cm/sec










k second order rate constant, cm3/gmole-sec

k0 frequency factor for k, cm3/gmole-sec

kI pseudo-first order rate constant, k1 = kCBf, sec-I

k10 frequency factor for kI, sec-'

kL, k[ mass transfer coefficient with and without reaction,
cm/sec

Thermal conductivity, cal/cm-sec.K

M dimensionless group, klD/(kL)2

mi, m2 dimensionless temperatures where the determinant is zero

PA partial pressure of gas A, atm

QG heat generation function

QR heat removal function
Q,, Q2 slope of QR and QG with respect to e
q volumetric flow rate of the liquid, cm3/sec

R gas constant, cal/gmole-K

Ra rate of absorption, gmole/sec

Rr relative rate of absorption

rl inverse of the resistance due to reaction

r2 inverse of the resistance due to mass transfer

S stoichiometric ratio of the concentration of A to that
of B at feed conditions

Sc surface area available for cooling, cm2

sI, s2 dimensionless temperatures where the trace is zero
T temperature, K

t time, sec

t' dimensionless time

U overall heat transfer coefficient, cal/cm2.0Ksec

V volume of the liquid phase, cm3










dimensionless concentration


xi dimensionless concentration of A at the interface

z distance from the interface, cm

GREEK SYMBOLS

a thermal diffusivity, cm2/sec

B dimensionless mass transfer coefficient

y dimensionless activation energy

6 film thickness, cm

v stoichiometric coefficient

p liquid density, gmole/cm3

n dimensionless exponent for solubility

e dimensionless temperature

X coefficient which defines the location of the reaction
front for instantaneous reaction, cm2/sec

T holding time, sec

Ti interfacial exposure time, sec

TH I M' characteristic times for heat transfer, mass transfer,
TR' TRi reaction and reaction at the interface, sec
j enhancement factor

,I enhancement factor for instantaneous reaction

SUBSCRIPTS

A gaseous reactant

B liquid-phase reactant

b bulk

c cooling water

FK Frank-Kamenetskii approximation

f feed

H heat transfer










i interfacial

A bulk liquid

M mass transfer

R reaction
















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GAS-LIQUID REACTORS: PREDICTIONS OF A MODEL RESPECTING
THE PHYSICS OF THE PROCESS

By

Daniel White, Jr.

December 1982

Chairman: L. E. Johns
Major Department: Chemical Engineering

The opposing effects of temperature on gas solubility and chemical

reaction rate imply oscillations in gas-liquid reactors. In this study

a model is proposed for a specific class of gas-liquid reactors and is

investigated to establish the basis for such oscillations.

We study the class of gas-liquid continuously fed stirred tank

reactors where excess gas is bubbled through the liquid and where the

reaction occurs in the liquid phase. The model accounts for the inter-

facial temperature rise due to fast reaction which can significantly

affect enhanced absorption and which has been neglected in previous

studies. Six characteristic times scale the individual processes occur-

ring in the reactor and these allow us to identify two controlling

regimes of reactor performance and five operating regions. A maximum

of five steady states is possible with at most one in each region.

Uniqueness and multiplicity conditions are given for the steady

states and their stability is ascertained. These conditions completely

specify the parameter space for a reactor with unenhanced absorption.










The predictions of the model agree with the experimental results for

the chlorination of decane.

Three multiplicity patterns are found: s-shaped, mushroom and

isola. A method for predicting the existence of isolas and mushrooms

is given. For a single-phase reactor the roots of a fifth degree

polynomial determine precisely where isolas exist. A previously

hypothesized result, that isolas and mushrooms are not possible for

an adiabatic reactor, is proven. For a two-phase, gas-liquid reactor

the conclusions are less sharp but yield the result that these phenom-

ena can not exist for an adiabatic reactor.

Directions in the search for oscillations are given. The opera-

ting region of most interest is the mass transfer controlled region

and general conditions for the existence of oscillations are discussed.

The results herein indicate how the model can be simplified to quantify

these oscillations.















CHAPTER ONE
INTRODUCTION

If a reactor could control its feed rate, varying it with time,

then there would appear to be a physical basis for anticipating and

explaining a natural oscillation in the reactor's performance. The

elementary fact that reaction rates increase with temperature whereas

absorption rates decrease suggests that a two-phase system where

dissolved A reacts with B in the liquid phase but where A is fed in

the gas phase should exhibit this phenomenon.

In particular, if a gas containing A is bubbled through a liquid

containing B then the feed rate of A into the reacting liquid depends

on the saturation concentration of A, a strong function of temperature.

Thus it is anticipated that if a system cools, the solubility of A in-

creases, resulting in an increase in the rate of absorption, in the con-

centration of A in the liquid, in the generation of heat via reaction,

and hence in the temperature. So cold systems show a tendency to heat

up. Conversely, in hot systems the decreased solubility of A reduces

the rate of absorption, the rate of reaction and hence the generation of

heat, cooling the system. This seems to suggest that oscillations are

characteristic of such two-phase systems. However, these oscillations

can not be like those found in single-phase reactors which can only

originate in the nonlinearity of the reaction rate. Here the primary

cause is the interaction of the temperature-dependent solubility and

the rate of reaction. The ultimate objective of this study then is to

find conditions which will guarantee the existence of such oscillations

1










and to show that they are relatively commonplace. But before this can

be done a model of the system must be constructed and its multiplicity

and stability implications investigated. This then is the primary

objective of this study.

In a single-phase continuous stirred-tank reactor (CSTR)1 for a

simple but important class of reactions, the model equations are simple

and the multiplicity and stability problems are essentially resolved.

But in two-phase systems the model equations are more complex and

results have been obtained only in simple limiting cases. Most recent

investigations have focused on the special case where one reactant

enters in the gas phase and another nonvolatile reactant enters in the

liquid phase with reaction occurring in the liquid phase only. The

models proposed for this system are simplified so that they resemble the

single-phase model for which the method of analysis is well established.

But this simplification confuses the physics and may alter the interac-

tion of the mass transfer and the reaction processes. And it is just

this interaction between gas solubility and reaction that must exist

to produce the above scenario for oscillations. So a model must be

constructed which is capable of describing the requisite physics for

natural oscillations. That is, the opposing effects of temperature on

the solubility of the gaseous reactant and on the reaction rate must be

adequately described so that the competition between reaction and mass

transfer is evident. This requires a temperature-dependent solubility

(Henry's law constant) and separate descriptions of the mass transfer

and the reaction processes.



1The name CSTR identifies an idealization satisfying the following con-
dition: on some scale the contents of the reactor are spatially uniform.










In full generality such a model would be too complex to give useful

conditions for multiplicity and stability so a less general case but one

that still contains the above physics is necessary. Many of the gas-

liquid reactions carried out in a CSTR involve a pure gas (and/or a gas

fed in excess) and a liquid reactant, e.g. chlorination, hydrogenation,

nitration, oxidation and sulfonation. The gaseous reactant dissolves in

the liquid and reacts forming a liquid product. If a gaseous product is

formed, it is often so dilute that there is no significant effect on the

gas phase. Under these conditions the only influence the gas phase

exerts on the reactor is through the pressure dependence of the solubil-

ity (partial pressure if not pure but only fed in excess), obviating the

gas-phase mass balance. For this reaction scheme it should be possible

to analyze the multiplicity, stability and oscillatory behavior of a

two-phase CSTR without neglecting the requisite physics for natural

oscillations. Thus with this description of the system such phenomena

can be sought.
















CHAPTER TWO
LITERATURE REVIEW

Models for single-phase CSTR's and their multiplicity and stability

analysis were developed over the years 1955-1976, culminating in Poore's

method of analysis, see Poore (1973). Before this the classical method

was graphical. The steady state mass balance was solved and substituted

into the energy balance. This was written in the form: rate of genera-

tion of heat equals the rate of removal. When these two functions are

plotted versus temperature their intersections are the steady states.1

The heat removal function is a straight line hence the shape of the heat

generation function determines the number of possible steady states.

For a first order reaction this is

rk
1 + rk

where r is the holding time and k is the rate constant. This function

is s-shaped with respect to temperature and thus the maximum number of

steady states is three (Figure 2.1). In addition to providing multi-

plicity information these graphs give the first of two conditions for

stability, the slope condition. This is the determinant condition for

the corresponding linearized system and is a sufficient condition for

instability and a necessary one for stability:

If the slope of the heat removal function at a steady state is less

than the slope of the heat generation function then the steady



IThe analysis for a single-phase CSTR is summarized in Appendix A.



















3 solutions


QR: 1 solution


QR: 1 solution


Figure 2.1. Heat generation and heat removal functions versus
temperature for a single-phase CSTR.










state is unstable.

For example, in Figure 2.1 the above is true for steady state B; hence it

is unstable. That is, if the reactor is operating at B and is perturbed

to a slightly higher temperature, then the graph shows that the rate of

heat generation is now greater than the rate of removal; thus the tem-

perature continues to rise, ultimately reaching steady state C. Simi-

larly if a perturbation lowers the temperature the steady state shifts

to A. For steady states A and C the above does not hold so they may be

stable.

This graphical technique gives a necessary, but not sufficient,

condition for stability and limited information on regions in phase

space of uniqueness and multiplicity. Poore's technique completes the

former and generalizes the latter but it is model bound and works only

because the single-phase system is so nice. For a CSTR the dynamic

equations are ordinary differential equations in only two state vari-

ables and hence the linearized system can be characterized by the deter-

minant and the trace of the Jacobian. Also because the system has at

most three steady states, the determinant and trace conditions lead to

quadratic equations in the temperature which can be solved explicitly

to obtain the critical points for uniqueness and stability. From this,

regions in phase space for uniqueness and stability can be determined

for simple kinetics. Also oscillations in temperature and concentration

due to the nonlinearity of the reaction rate constant can be found. If

the system were more complex this would be much more difficult, and might

be impossible.

In conjunction with the development of the single-phase CSTR model

Schmitz and Amundson (1963a-d) proposed a model for a two-phase CSTR. The










model is quite general with reaction possible in both phases and heat

and mass transfer between phases. The analysis is only manageable be-

cause they assume dilute systems and represent the transport processes

in their simplest possible forms. With these simplifications the model

becomes very similar to two single-phase reactors with linear coupling.

Since the multiplicity and stability analysis was not well developed at

the time, only partial answers were possible for these questions.2 For

the limiting cases of physical equilibrium and chemical equilibrium the

model takes particularly simple forms: the first is very similar to the

single-phase model and the latter is linear; hence only a unique stable

steady state exists.

There are two problems with this model. Most gas-liquid reactors

are not this complex and the transport processes are not this simple.

In the preceding model mass transfer and reaction are treated as inde-

pendent phenomena but in fact the two interact. In particular, chemical

reaction enhances mass transfer by sharpening the concentration gradient

at the interface. This enhancement can be significant and needs to be

taken into account. Similarly the interfacial temperature may be

greater than that in the bulk due to reaction near the interface. This

rise needs to be investigated and included because it is the interfacial

temperature that determines the solubility of gas.

On the other hand the reactor model is too general, e.g. reaction

in both phases, heat and mass transfer resistances in both phases, etc.,

thus making analysis difficult. Fortunately many systems of industrial



20ne result which is of interest in our work is that for a system in
thermal equilibrium with reaction in one phase only up to three steady
states are possible (Henry's law is also independent of temperature).








importance are simpler. Most involve reactions in the liquid phase only

with one reactant fed in each phase. Also heat transfer to the gas

phase is often negligible due to its low heat capacity and the fact that

the most important temperature of the gas is its interfacial temperature

which is usually equal to that of the liquid. For example Ding et al.

(1974) studied the chlorination of n-decane in an adiabatic CSTR. Pure

chlorine was bubbled through liquid decane where it dissolved and re-

acted. Multiple steady states were observed over a range of holding

times. Two stable steady states were found and one unstable state was

located via a feedback control scheme.

Hoffman et al. (1975) in an effort to understand Ding et al. (1974)

and account for the enhancement of mass transfer via reaction modelled

this system using a reaction factor. When a reaction factor is used the

rate of mass transfer is defined as

E* k' a CAi

where E* is the reaction factor which accounts for the enhancement and

is a function of the rate of reaction near the interface, kt is the

mass transfer coefficient with no reaction, a is the specific surface

area, and CAi is the interfacial concentration of the gas A in the

liquid. Then the reaction factor is found via a theory for the dif-

fusion and reaction processes near the interface.3


3Alternatively one could have used the more common enhancement factor
* which is the ratio of the mass transfer rate with and without reac-
tion. With this the rate of mass transfer is

k a (CAi CA)

where CAZ is the dissolved gas concentration in the bulk liquid. Thus
the reaction factor is smaller than the enhancement factor by
E* = (1 CA/CAi)








The Hoffman model is the basis for all the recent studies of two-

phase reactors but it pertains only to the class of well-mixed systems

(CSTR) where the reaction

A + v B Products

occurs in the liquid phase and where the reactant A is fed in the gas

and the nonvolatile reactant B in the liquid. For this system it is the

most complete description available and includes most of the important

phenomena. For this reason it will be reviewed more fully.

The model consists of two overall mass balances, one for each reac-

tant, and an overall energy balance. This differs from previous models

in that the balances are not written for each phase and the reaction

term is replaced by the reaction factor term RAV = E* k a CAi V q CA

so it is unnecessary to describe the reaction in the bulk. Thus the

rate of reaction is the rate of mass transfer (or absorption) to the

liquid minus the rate at which unreacted A flows out of the reactor with

the liquid. This leaves five unknowns (the concentration of A in the

gas, in the liquid, and at the interface, the concentration of B, and

the temperature) with only four equations (three overall balances and the

Henry's law equilibrium relationship for the interface). Thus another

equation is necessary, relating the bulk concentration CAZ to the other

variables. In general such a relationship would be derived from a mass

balance on either of the phases, leading to separate terms for mass

transfer and reaction, but the liquid-phase mass balance has already

been used to define the reaction rate. Hoffman et al. (1975) circum-

vent the need for another equation via the use of a clever boundary

condition which determines CAP from the interfacial model. This is

unusual because the interfacial model is normally used only to determine










the reaction factor. With the bulk concentration determined from an

interfacial model the choice for this model assumes undue importance.

The reaction factor, like a mass transfer coefficient, is found by

modelling the interfacial region, calculating the flux at the interface

and equating it with the mass transfer rate in the reaction factor form:

E* kt a CAi V


Several interfacial models can be used but the results, at least for the

enhancement factor (the ratio of the mass transfer coefficients with and

without reaction, see previous footnote), are insensitive to the one

chosen. Hoffman et al. chose the simplest, the film theory.4 To deter-

mine the reaction factor Hatta's film theory is followed, including

corrections to his final result, except that a new boundary condition at

the bulk liquid is used. This condition accounts for reaction in the

bulk and effectively defines the reaction rate in a mass balance on the

liquid phase. Hatta's model is for the steady state diffusion and reac-

tion (pseudo-first order because B is assumed to be in excess) of a gas

A through a stagnant liquid film. At the gas-liquid interface, the gas

concentration is CAi whereas at the bulk liquid boundary it is zero.



4Although the main features of results derived from the film theory are
usually correct, e.g. the reaction factor, occasionally this theory
gives incorrect parameter dependencies. Thus using this model to deter-
mine the bulk concentration CAZ may be suspect. This result needs to be
verified and compared with the other more physically plausible models,
e.g. penetration and surface-renewal. Also these models were con-
structed to estimate the rate of mass transfer at the interface and
extrapolating their results to the bulk to obtain a quantitative value
seems unjustified. There is some question as to whether the film model
is the proper choice since this is a steady state model and the mass
transfer process it describes is a dynamic one. A more appropriate
choice might have been the time-varying models from the penetration or
surface-renewal theories.










But in Hoffman's model this zero boundary condition is replaced by a

flux condition. The flux from the film to the bulk is equated with the

rate of reaction of A in the bulk plus the loss of A due to the unre-

acted A flowing out with the liquid:
dCA
-V a D A- = V k CA CB (E a 6) + q CA

Here CA is the concentration at the film-bulk boundary which is assumed

equal to the bulk concentration.

Solving the diffusion-reaction equation with this boundary con-

dition yields a reaction factor similar to Hatta's and an expression for

CAt. But Hatta's result was for the liquid reactant B in excess and its

concentration equal to the bulk concentration. Historically this was

corrected by van Krevelen and Hoftijzer (1948) who estimated the inter-

facial concentration of B and used this in the film theory equations.

Thus the model is still pseudo-first order but with a more accurate

estimate of the B concentration. Hoffman et al. (1975) also use this

correction to account for the lower value of the B concentration due to

reaction. But originally this correction was only valid for CA equal

to zero, so an additional correction due to Teramoto et al. (1969) is

also used. The resulting equations for E* and CA must then be solved

iteratively with the original reactor equations to determine the steady

state.

This technique appears to be somewhat backwards. Here the liquid

phase mass balance is placed in the interfacial model in the form of a

boundary condition. From this both the reaction factor and the liquid

phase concentration of A are calculated. Normally an interfacial model

is used just to estimate the enhancement of the mass transfer rate which

is then substituted into the mass balance. As shown in Sherwood,








Pigford and Wilke (1975) the interfacial model used for this is not

very important since the results are all similar. But in the analysis

of Hoffman et al. (1975) not only is the enhancement found but also the

liquid concentration. This would appear to make the choice of inter-

facial model quite important. This technique does have an obvious

drawback, confusion. Combining the liquid balance with the interfacial

model and using the two corrections discussed in the last paragraph

yields a system where the physics have been scrambled. That is because

all the reaction has been obscured in the function E*, any structure the

system once had is lost. Thus none of the recent authors can describe

their results so that it is clear physically which processes are impor-

tant and when they are important. As an example of some unnecessary

confusion and complexity, consider the two corrections used in the film

theory. These are not needed together. The van Krevelen-Hoftijzer

correction accounts for the lower concentration of the liquid reactant

at the interface which is necessary only for fast reactions, whereas

the Teramoto correction is used when the gas concentration in the bulk

liquid is nonzero which happens only for slow reaction.5 Thus adding the

Teramoto correction to that of van Krevelen-Hoftijzer is an unnecessary

complication which, even though it does not create any error, obfuscates

the results with extraneous complexities. This can lead to errors in

the calculation scheme if not handled properly.



5This can be shown via a time scale argument which will be explained in
the text. Astarita (1967) also discusses when the bulk concentration
goes to zero and finds this to be true for fast reactions. Some con-
ditions are given in the literature for the bulk concentration to go to
zero and they are in terms of the Hatta number, see van Dierendonck and
Nelemans (1972).










Solving the model equations numerically Hoffman et al. (1975) find

up to five steady states with at least two unstable. The slope condi-

tion which is a necessary condition for stability and also a sufficient

condition for uniqueness can not be used to find regions of guaranteed

stability or uniqueness due to the iterative nature of the solution.

Thus the utility of the model is reduced because of its inherent com-

plexity. Yet its complexity does not seem necessary or even appropriate

considering the interfacial model and corrections which are used.

Sharma et al. (1976) extend this analysis to two consecutive reac-

tions in the liquid phase of a nonadiabatic CSTR and find up to seven

steady states. They are also able to fit their model to the data of

Ding et al. (1974) and predict the ignition and extinction points quite

closely. A parametric study revealed the existence of an isola. An

isola is a multiplicity pattern where a shift from the low temperature

to the high temperature steady state can not occur by increasing or

decreasing the independent parameter, e.g. the residence time.

Raghuram and Shah (1977) simplify Hoffman's model in order to

obtain explicit formulae for uniqueness and multiplicity. The most

important assumption is that the liquid reactant B is in excess and the

concentration of B in the film is the same as that in the bulk. Thus in

Hoffman's film theory the van Krevelen-Hoftijzer and the Teramoto cor-

rections are not necessary, and analytical expressions for the reaction

factor and the concentration of the gas in the bulk liquid can be writ-

ten explicitly. This obviates Hoffman's iteration thereby allowing

conditions for uniqueness and multiplicity to be deduced. In addition

they assume that the volumetric flow rates of the gas in and out of the

reactor are the same, whereas Hoffman et al. (1975) correct the flow










rates for the gas that is transferred to the liquid. This is true if

the reactant gas is dilute in the gas phase, or if very little gas is

transferred, or if the gas flow rate is very large and in excess.

Raghuram and Shah (1977) also analyze the case of instantaneous reaction

where the reaction factor is very simple and the concentration of gas in

the bulk is zero.

An observation on the expression of Henry's law used by Raghuram

and Shah (1977) seems pertinent. Henry's law is a proportionality be-

tween the partial pressure of a gas and its mole fraction in the liquid:

PA = H(T) XA

where H(T), the Henry's law constant, can be represented via

-HS/RT
H(T) = Ho e

They write it in terms of the molar concentration in both phases

CAg = H'(T) CAi

and presume
-Hs/RT
H'(T) = Ho e-

But if CAg = PA/RT and CAi = p XA then

H'(T) = H(T)/pRT

It appears that Raghuram and Shah have missed a factor of 1/T.

The result of this simplified model is a seemingly explicit

inequality (slope condition) for uniqueness and stability. This

inequality is a sufficient condition for uniqueness and a necessary

condition for stability. Unfortunately the inequality is not fully

explicit requiring a solution of the steady state equations for its

evaluation. This makes it very difficult to determine regions of

uniqueness, multiplicity, and stability in parameter space. But for a










specific system (Ding et al., 1974) where the values of the parameters

are taken from Hoffman et al. (1975) they are able to show numerically

regions of one, three or five steady states. A result more conservative

than the slope condition but explicit is also deduced from the inequal-

ity and is a sufficient condition for uniqueness. This condition is

that the activation energy of the reaction be less than twice the heat

of solution (where the heat of solution is the exponent in Henry's law).

It is a very weak condition in that it is almost never true except for

very slow reactions.

More explicit results are found for two special cases, fast reac-

tion and instantaneous reaction. For both, the reaction occurs near the

interface and the bulk concentration of the gas is zero. The corre-

sponding reaction factors are simple. For fast reaction Raghuram and

Shah (1977) find up to three steady states whereas for instantaneous

reaction the steady state is always unique. They conclude that "the

possibility of five steady states is a direct consequence of the liquid

phase concentration of the gaseous reactant being nonzero." Again the

formulation of their model, like Hoffman's, confuses the physics so an

interpretation of the results becomes difficult. Accounting for the

finite value of the bulk concentration in the fast reaction model can

not give five steady states, in fact we will show that the bulk concen-

tration is vanishingly small here. The reason only three are possible

is that the fast reaction model holds only in the region where the reac-

tion is fast. Thus any solutions in the regions of slow reaction or

physical absorption without reaction are lost. That is, simplifying a

general model to a specific region can only give useful results perti-

nent to that region and can not be expected to give results pertaining








to the solution of the general model outside this region. Thus if

any solutions are found outside the assumed region they are probably

meaningless. This will be shown in the text. Raghuram et al. (1979)

extend this analysis to a nonadiabatic CSTR and to a cascade of n CSTRs.

They obtain five steady states but only for an extremely narrow range of

parameters, thus concluding that five steady states are probably not

physically observable for a gas-liquid reactor.

Huang and Varma (1981a) use a simplified version of Raghuram and

Shah's (1977) model for fast reaction to determine explicit necessary and

sufficient conditions for uniqueness and multiplicity, and for stabil-

ity. They also predict the direction and stability of limit cycles.

Their model is the same as Raghuram and Shah's except that they assume

that the concentration of B is constant at the feed composition and

their reactor is nonadiabatic. The fast reaction assumption implies

that the reaction factor has the simple form

E* = = /kD/k

and that the bulk liquid concentration of the gas is zero hence no

iteration is necessary. The model then consists of two equations, one

for the gas concentration and one for the temperature. But due to the

form of the reaction factor and Henry's law these equations are equiva-

lent to those for a single-phase CSTR. In establishing the equivalence

it is necessary only to identify the apparent activation energy with one

half the activation energy minus the heat of solution:

EApp 12 E HS

Thus the entire theory of single-phase reactors (Uppal et al., 1974

and 1976) can be applied to this problem although Huang and Varma

(1981a) do not avail themselves of this.








Sufficient conditions for uniqueness are obtained in terms of the

parameters, free of the steady state temperature. Thus these conditions

can be used without solving the steady state equations and as such are

a great simplification over the results of Raghuram and Shah (1977).

Also, simplifying the sufficient condition for uniqueness gives a sim-

ilar condition to that of Raghuram and Shah (1977):

E < 2HS + 8RTf Huang and Varma (1981a) (2.1)

E < 2HS Raghuram and Shah (1977). (2.2)

The two models give the same maximum number of steady states, three as

expected. Comparing a prediction of Huang and Varma's (1981a) model

with the data of Ding et al. (1974) shows a reasonable prediction of

the ignition and extinction points and a qualitative prediction of the

multiplicity pattern, even though the reaction of Ding et al. was not

fast. But it should be noted that not all the parameters necessary

for this model were found or reported by Ding et al. so there is some

discrepancy between the values used by Huang and Varma (1981a) and those

used by the previous authors. Apparently some of these parameters have

been used to fit the authors' model to the data of Ding et al. (1974).

Finally Huang and Varma (1981a) find conditions implying the existence

of limit cycles but the corresponding values of the parameters necessary

are unrealistic, vis-a-vis a gas-liquid reactor.

Both Raghuram and Shah's (1977) and Huang and Varma's (1981a)

models are fast reaction models and make sense only when the reaction is

fast. That is, if the model predicts a solution outside the fast reac-

tion region the solution is meaningless. Limiting cases are instructive

here. If the rate constant k1 goes to zero the reaction factor E* goes

to zero and the mass transfer from the gas to the liquid vanishes. But








of course there are nonreactive conditions under which the amount

absorbed can be quite large. So any solution found that is in the slow

reaction region is incorrect or at the least is suspect. Conversely, in

the opposite limit of instantaneous reaction (k1 + -), the reaction fac-

tor overpredicts the rate of mass transfer, viz.,

limit E* +
kl-

But in reality the reaction factor is bounded, viz.,

E* < 1 + CB / u CAi

where CBZ is the bulk concentration of the liquid phase reactant and

v is the stoichiometric coefficient. Again any result here is meaning-

less. Finally it will be shown in the text that of the three steady

states found from this model, two are outside its region of applicabil-

ity and the one remaining solution, which is the only real steady state

for a gas-liquid reactor, is the middle unstable steady state. Hence

the multiplicity and stability results of this model do not apply to any

real system.

Huang and Varma (1981b) remove the requirement of fast reaction

allowing all reaction regimes, the concomitant price being a return to

the complex model of Hoffman et al. (1975) and numerical solutions. The

film theory used is identical to that of Hoffman et al. but the overall

balances for the reactor differ as they did in the previous paper. Pre-

dicted results for the experiment of Ding et al. (1974) using values of

the parameters as given by Sharma et al. (1976) show reasonable agree-

ment with the data for short residence times and for the prediction of

the extinction point. The numerical results also show the existence of

an isola, a multiplicity pattern where no ignition point from the low










temperature steady state exists and two extinction points exist for the

high temperature branch. For the adiabatic case an isola with five

steady states was found to be possible. But even a small heat loss

reduced the system to three steady states. Explicit conditions for

uniqueness and stability are not possible due to the complexity of the

model. As in the previous papers little insight into the interaction

of the mass transfer and reaction processes is obtained.

All the models incorporating the reaction factor suffer from con-

fusion due to their combination of mass transfer and reaction into one

complex function E*. This complexity essentially eliminates the possi-

bility of a multiplicity analysis. Also the results from the inter-

facial model are extrapolated to deduce results in the bulk liquid,

whereas originally these models were hypothesized to estimate the mass

transfer coefficient only. This has not been justified and at the least

makes one uncomfortable. Thus if an understanding of a gas-liquid

reactor is to be found, the model should handle the reaction and mass

transfer processes separately except where the mass transfer process is

influenced by interfacial reaction. Here an interfacial model should be

used only to determine the factors describing this interaction.

None of the previous studies in this area is based upon a model

that captures the full effect of the temperature interaction between

gas solubility and liquid phase reaction. Schmitz and Amundson (1963a-d)

neglect the effect of temperature on the distribution coefficient

(Henry's law constant). Hoffman et al. (1975) and the studies based on

this work assume the temperature is everywhere uniform even though all

the reaction may take place in the neighborhood of the interface, cf.

the fast reaction studies. Whenever mass transfer is enhanced, i.e.








whenever appreciable reaction takes place in the interfacial region,

the interfacial temperature will be greater than the bulk temperature.

Because heats of reaction and solution can be large, this effect could

be significant. And it is just the highly exothermic systems that

would be of interest when searching for unusual phenomena such as

oscillations, where the surface temperature rise is especially likely

to be important. Because temperature sensitivity seems to be required

for natural oscillations and this sensitivity appears in the gas solu-

bility which is determined by the interfacial temperature, the over-

looked phenomenon of an interfacial temperature rise would seem to be

of importance.

Interfacial temperature rises in gas absorption with and without

reaction have been observed. Recently Verma and Delancy (1975) found

temperature rises in the nonreacting systems ammonia-water and propane-

decane. They predicted this increase to be as large as 180C. In

reacting systems larger values have been observed. Beenackers (1978)

estimated from flux measurements a rise up to 50C. Mann and Clegg

(1975) and Mann and toyes (1977) indirectly measured temperature rises

up to 530C for chlorination and 58C for sulfonation using a laminar jet

technique. Temperature increases of this magnitude will significantly

affect reaction rates and solubilities, hence they should be accounted

for in absorption-reaction systems.

Recent theoretical analyses have been able to predict large tem-

perature rises similar to those observed experimentally. Shah (1972)

constructs numerical "solutions" to the partial differential equations

from the penetration theory model for mass and energy diffusion with

reaction and shows that large temperature rises are possible. Clegg










and Mann (1969) obtain an expression for the rise but it is quite com-

plicated even after some strong restrictions are made. The analysis is

similar to the early work of Danckwerts (1951), who noted that since the

thermal diffusivity is much greater than the mass diffusivity, the dis-

tance over which the thermal gradient exists is much greater than that

over which the concentration gradient exists. Thus the mass transfer

and reaction process appears to take place at a constant temperature,

whereas in the heat transfer process, the heat of reaction appears as an

interfacial flux. But Danckwerts assumed the interfacial concentration

of the gas, the solubility, to be constant, i.e. independent of tem-

perature. Clegg and Mann (1969) correct this but approximate the solu-

bility with a linear relationship. This restricts their results to a

small region about the initial temperature. Also they use a temperature

independent rate constant. This weakens their results because as solu-

bility decreases with temperature, resulting in a lower prediction for

the temperature rise, the reaction rate increases thereby enhancing the

rise. But even with the latter assumption, the predictions demonstrate

the possibility of a large interfacial temperature rise for the chlori-

nation of toluene.

Mann and Moyes (1977) derive a relatively simple transcendental

equation for the interfacial temperature using the film theory. As

above they assume the thermal diffusivity to be greater than the molec-

ular diffusivity and hence the film thickness for mass transfer to be

much less than the film thickness for heat transfer. The mass trans-

fer problem can then be solved at a constant temperature, that of the

interface. The temperature rise is obtained via an overall energy

balance over the heat transfer film. The rate of energy transport to








the bulk can be equated to the heat of solution times the mass flux at

the interface plus the heat of reaction times the excess flux into the

film, i.e. the difference between the mass flux in and out of the film.

From this it is possible to obtain for a general reaction a transcenden-

tal equation for the interfacial temperature, but Mann and Moyes did so

only for the two limiting cases: no reaction and fast reaction. If a

linear solubility relationship is used, the no reaction case gives an

explicit expression for the interfacial temperature. This is not pos-

sible for the fast reaction limit unless the rate constant is indepen-

dent of temperature. The general expression that can be obtained from

this technique, see Appendix B, is applicable over the range of no

reaction to fast reaction. However, it is not applicable when the reac-

tion is not first order, i.e. when the liquid reactant is not in excess,

and when the reaction becomes instantaneous. But the correction is

straightforward and involves only correcting the enhancement factor for

these cases. This technique presents the same problem that is encoun-

tered whenever the film theory is used. Even though the result seems

reasonable the physical basis for the calculation is suspect, especially

in this particular case. The existence of a stagnant film is difficult

to believe in a turbulent system but it is even more improbable for two

films of different thicknesses to exist. The technique used by Mann and

Moyes assumes a different film thickness for mass and heat transfer,

with the latter being much greater than the former.6 This does not

agree with the film theory view that a stagnant film separates the



6Note: if the film thicknesses are the same the temperature rise is
less than the above by factor of Jo/ D or about ten!










gas from the bulk liquid. Here the bulk liquid is characterized by a

uniform concentration and temperature whereas in the above analysis the

mass transfer film does not contact liquid with the bulk temperature.

It will be shown that this inconsistency can be avoided by using the

surface-renewal theory and the result can be generalized for all reac-

tion rates up to instantaneous.

With this background a model for gas-liquid reactors is constructed

which contains the processes necessary for understanding the natural

oscillations expected. It contains separate terms for the chemical

reaction and the mass transfer processes. The role of the reaction in

enhancing the mass transfer coefficient is deduced via the surface-

renewal theory. The enhancement factor is then approximated piecewise

by very simple explicit expressions of reasonable accuracy in the slow,

fast and instantaneous reaction regimes. Simultaneously the reaction-

induced surface temperature rise is deduced via the surface-renewal

theory. These are used together with a temperature dependent form of

Henry's law to represent the gas solubility.















CHAPTER THREE
MODEL FOR ABSORPTION AND REACTION IN A GAS-LIQUID CSTR

A simple method for contacting a gaseous reactant with a liquid

reactant is to bubble the gas through the agitated liquid. In general

each component of each phase may transfer to the other phase and react.

The resulting system may exhibit multiple reactions in each phase and

mass and energy transport between them. The concentrations and tem-

perature of each phase and their dependence on the reactor parameters

must be taken into account. In this generality the complexity of the

process precludes all but the most superficial results.

For a large class of important reactions, viz., for chlorinations,

hydrogenations, oxidations, etc., the reaction takes place in the liquid

phase only. Here a pure gas A is bubbled through a nonvolatile liquid

containing a reactant B into which the gas absorbs and reacts, via

A + v B + Products. The gas is often supplied in excess, obviating a

mass balance for this phase and implying that the only way the gas phase

can affect the two-phase reactor is through the pressure dependence of

the solubility. For this type of reaction the physical phenomena imply

a reactor model that strikes a reasonable balance between the possibili-

ties of in-depth analysis and physically interesting predictions.

We investigate this simple class of two-phase, continuous stirred-

tank reactors. The assumptions that define this class are:

Al. The gas flow rate is greatly in excess of that required for absorp-

tion so that the interfacial area, a, is constant in space and time.

A2. The partial pressure of gas A, PA' is constant in space and time.










A3. The reactant A enters the reactor only in the gas phase.

A4. The liquid-phase solvent, the reactant B and the products are

nonvolatile.

A5. The reaction occurs only in the liquid phase.

A6. Thermal equilibrium exists between the two phases and the thermal

capacity of the gas phase is vanishingly small.

A7. Physical equilibrium exists at the interface between the phases and

can be described by a temperature dependent Henry's law constant:

PA = H(T) CAi(T) (3.1)
-Hs/RT
H(T) = H e

A8. Mass transfer is controlled by the liquid phase.

A9. The volume of the interfacial "film"1 is much less than that of the

bulk liquid.

The first five assumptions obviate balances on the gas phase. Now

the system is like a single-phase (liquid) reactor with separate feeds

for each reactant, component A via absorption and component B in the

liquid feed. Therefore the feed rate of A is controlled by the reactor

temperature and this is an essential part of the natural oscillation

scheme proposed. Assumptions six and seven describe the temperature

dependence of the solubility. The interfacial concentration of A is a

decreasing function of temperature and an increasing function of the

partial pressure of gas A. This pressure dependence is the only influ-

ence the gas phase will have on the gas-liquid reactor. The last



By film we do not mean to suggest the film theory of intraphase mass
transfer. Here film refers to regions where concentration variations
are significant; bulk refers to regions where they are not.










assumption makes it unnecessary to distinguish between the bulk liquid

volume and the total liquid volume because in most systems this differ-

ence is vanishingly small.

The model consists of two parts, the overall mass and energy bal-

ances describing the performance of the system and the interfacial mass

and energy balances describing the absorption process. The second

establishes the feed rate of A in the first. The overall mass balances

for A and B in the bulk liquid are

dCA
V --= q CA V k Cg CA + V k' a(CAi CA)( fi) (3.2)


dCB
V---= q(CBf CB) v V k Cg CA v V k' a(CAi CA)fi. (3.3)


The similarity to the single-phase reactor equations is evident, see

Appendix A. The only difference in the first balance is that the feed

rate of A is the rate of mass transfer to the bulk liquid. This is the

product of the overall rate of mass transfer times the fraction that

does not react in the interfacial region, (1 f.). The feed rate is a

function of temperature hence the conditions inside the reactor control

the feed as described in the introduction. That is, if a system becomes

cold the solubility of A, CAi, increases resulting in a larger feed rate

or rate of absorption, V i k' a (CAi CA), which increases the con-

centration of A in the liquid, the generation of heat via reaction and

hence the temperature. So cold systems show a tendency to heat up.

Conversely in hot systems the decreased solubility of A reduces the

rate of absorption hence the rate of reaction, the generation of heat

and the temperature. We note that this description holds when there

is no enhancement of mass transfer, i.e. when 4 = 1 and f. = 0.










Otherwise, inasmuch as 4 and fi increase with temperature, the rate

of absorption and heat generation may not follow the above, depending

on the interaction of the solubility and the interfacial reaction.

Consequently we anticipate that oscillations will be found even when

there is not enhancement. The balance on B is also similar to the

corresponding balance for the single-phase reactor except that there are

two reaction terms, one for reaction in the bulk and the other for reac-

tion near the interface. The reaction near the interface is the product

of the rate of mass transfer of A and the fraction of A absorbed that

reacts, f .

The energy balance is also very similar to that of the single-phase

reactor, viz.,

Vp Cp dT= q p Cp(Tf T) U Sc(T Tc) + HS V k[ a(CAi CA)

+ HR [V k CB CA + V k[ a(CAi CA)f] (3.4)

The only difference is that the source term is more complex. Here there

are three sources, the heat of solution, HS, the heat of reaction, HR,

in the bulk and the heat of reaction in the interfacial region.2 The

generation terms also follow the scenario for oscillations, being large

for high mass transfer rates corresponding to high interfacial gas con-

centrations and low temperatures and vice versa.

Before this system of equations becomes a complete representation

of the reactor performance a model for the interfacial processes is

needed so that the enhancement factor, *, the fraction that reacts in



The heat of solution, HS, is normally positive and the heat of reac-
tion, HR, is positive for exothermic reactions.










the interfacial region, f and the interfacial temperature, T can be

estimated. Several physical descriptions of the contacting of a gas and

liquid are available. The most common models are from the film, pene-

tration and surface-renewal theories. But because all the models con-

tain an unknown constant which sets the mass transfer coefficient, k,

and because our purpose is not so much to get the right numbers for a

specific system but to get a general picture of how the performance of

the mass transfer process changes from system to system over a wide

range of systems, kt is to be viewed as a free parameter, defining the

contacting. Thus what we seek from these theories is a way of pre-

dicting the enhancement of k[ via reaction. Sherwood, Pigford and Wilke

(1975) say that these standard models are interchangeable for this pur-

pose. So we will adopt the model that is most physically plausible for

this system and yields the quantities sought most easily.

The film theory, which is a steady state description of the mass

transfer process, is less appealing physically than the others because

the mass transfer process between a system of rising bubbles and an agi-

tated liquid is a dynamic one. For a rising gas bubble the simplest

physical picture suggests the penetration model insofar as the liquid

elements attach themselves at the leading edge of the bubble, flow

around the surface and detach at the trailing edge, thereby contacting

the gas for the length of time necessary for the bubble to pass. But

for a system of variously sized bubbles and for a well-mixed liquid this

contact time might be random, suggesting the surface-renewal theory

which averages the penetration theory results over a distribution of

contact times. We adopt this model for these reasons and because cer-

tain quantities of interest can most easily be found.










At the gas-liquid interface the concentration of A is established

by the condition that the liquid phase be in equilibrium with the gas

phase whose state we are free to set. The A that dissolves reacts with

B while diffusing away from the interface; the B that reacts is replaced

by diffusion from the bulk, thus obtaining
3CA 32CA
t- DA 2 k CB CA (3.5)
at A az2

CB a=2C
DB 2 v k CB CA (3.6)
at 3z
Because the absorption and the reaction processes generate heat, a tem-

perature gradient or at least a temperature rise can be expected. This

is not considered in the standard models mentioned but can be handled in

a similar fashion. The heat of absorption is released at the interface

whereas the heat of reaction is generated in the liquid phase bordering

the interface. Whence an energy balance on this region gives

P CP a3 HR k C A (3.7)
at az
We can not write the solution to Equations 3.5-3.7, as simple as they

are. However, because the thermal diffusivity, a = A /p Cp, is much

greater than the molecular diffusivity, D, (this is almost always true

for a gas-liquid system) we may construct a useful approximation to the

solution. This condition, a >> D, implies that the characteristic

distance over which the temperature variation is significant is much

greater than the corresponding distance for the concentration vari-

ation. Thus we may assume that the absorption-reaction process takes

place at a uniform temperature and that the heat generation in the

energy balance can be taken to be released at the interface. This

allows us to investigate Eqs. 3.5 and 3.6 independently of the energy










balance equation, Eq. 3.7, and we may do this assuming a uniform tem-

perature Ti. This temperature is not necessarily the bulk temperature T

and must be set so that Eqs. 3.5-3.7 are compatible. We therefore add

the assumption:

A10. The thermal diffusivity a is much greater than the molecular dif-

fusivity D; therefore the absorption-reaction process takes place

isothermally at the interfacial temperature Ti.

The interfacial equations, Eqs. 3.5 and 3.6, remain dependent on

the reactor equations through the initial and the boundary conditions:

at t = 0 CA = CA CB = CBZ

at z = 0 (interface) CA = CAi(Ti) dCB/dz = 0 (3.8)

as z + CA = CA C = CB

where the subscript X denotes the value in the bulk liquid. What makes

it possible for us to get the information that we seek is that the phys-

ics of gas-liquid reactions enables us to handle the problem for reac-

tant B via a useful approximation (to be discussed) and to uncouple the

interfacial and bulk equations for reactant A. The latter follows on

comparing the time scales of the several processes necessary in gas-

liquid absorption and reaction. In particular, we can show that when-

ever mass transfer is enhanced by reaction the bulk liquid concentration

of A is vanishingly small, i.e. whenever 4 is other than one and fi is

other than zero their correct values are found by assuming that the bulk

concentration of A is zero, CAt = 0. This means either that i = 1 and

f. = 0 (and the interfacial equation for A becomes very simple) or that

we can investigate the interfacial equation for A independently of the

overall reactor equations. Thus the equations are independent of the

bulk. This will now be shown.










To each process in the reactor we associate a characteristic time.

This is a measure of the relative speed of the process. A long time

signifies a slow process, whereas a short time signifies a fast one.

These times can be used to scale the processes. Six characteristic

times are apparent (Table 3.1); four originate in the reactor model and

two in the absorption model. In the reactor equations, Eqs. 3.2-3.4, we

identify a holding time T which characterizes the feed rate of B, a mass

transfer time rM which characterizes the feed rate of A, a reaction time

TR and a heat transfer time TH. In the absorption model we find an
average interfacial exposure or contact time ri and an interfacial reac-

tion time TRi. The interfacial reaction time differs from the bulk

reaction time because the temperature and composition of the bulk and

interface may differ.

Mass transfer is enhanced via reaction near the interface. This is

significant if the interfacial reaction rate is fast compared to the

exposure time of the liquid to the gas, that is, if the interfacial

reaction time is less than or equal to the interfacial exposure time,

viz., if3

1 i Ri

Suppose we have enhanced absorption and consider the corresponding con-

ditions in the bulk liquid. The bulk reaction time is always greater

than or equal to the interfacial reaction time because the bulk tem-

perature is always less than or equal to the interfacial temperature.



31n the following we are speaking qualitatively. Hence the symbol
~ will denote an approximate equality or an order of magnitude rela-
tionship between two quantities.
















TABLE 3.1
CHARACTERISTIC TIMES


Reactor


Holding Time


Reaction Time


Mass Transfer Time


Heat Transfer Time



Absorption Process


Exposure Time


Reaction Time


T = V / q
-r=V/q


R= 1 / k CB

M = 1 / kL a


TH = Cp V / U Sc






Ti = D / (k)2


TRi = 1 / k(Ti) CB










At the first sign of enhancement the two temperatures are nearly the

same (because little reaction has occurred at the interface) and we have

Ti ~ TRi ~ 'R
It is the nature of gas-liquid systems that the mass transfer time is

much greater than the interfacial exposure time, see Astarita (1967),

viz.,

TM >> Ti

whence, at the onset of enhancement

T >> R

It follows that the reaction rate in the bulk is very fast as compared

with the feed rate of A (mass transfer rate) and hence that mass trans-

fer is the controlling process when enhancement begins. This implies

that the bulk concentration of A must be vanishingly small and that we

may take the initial and boundary conditions appropriate to the absorp-

tion model to be

at t = 0 CA = 0

at z = 0 CA = CAi(Ti) (3.9)

as z + CA = 0

thereby uncoupling the interfacial and bulk problems. This continues to

hold throughout the enhanced mass transfer region because

TM >> T ~ TR (onset) > TR (enhanced)

We can construct the solution of the absorption model equations,

Eqs. 3.5, 3.6 and 3.9, in two limiting cases: reactant B in excess and

instantaneous reaction. The first is almost always true for a practical

gas-liquid reaction. That is, the concentration of the liquid phase

reactant is usually much greater than the concentration of the dissolved








gaseous reactant. Thus to a reasonable approximation the concentration

of B is constant throughout the reactor (albeit not necessarily at the

feed value). The interfacial model is then reduced to a single dif-

fusion equation with pseudo-first order reaction. Solving this linear

equation for an exposure time t and averaging the result over all the

elements of liquid at the interface assuming a random distribution of

exposure times

1 e-t/i
e
Ti

obtains the mass transfer coefficient, kL, and enhancement factor, 4,

see Appendix C:

= kL/kL = I + i/TR (3.10)

where kL is the mass transfer coefficient without reaction. The frac-

tion of A that reacts near the interface in the process of absorption is

the ratio of the rate of reaction to the rate of absorption, viz.,

(I/-r) I e [ f k CB CA dz (Va)] dt

1 V kL a CAi


Using results in Appendix C obtains

f (3.11)
1 + i hRi
This is the same as the fraction that reacts in a single-phase CSTR with

holding time Ti and reaction time TRi, see Appendix A.

The time scale argument given previously now can be understood

better. When the ratio ri/ Ri is much less than one, practically no

reaction and hence no enhancement occurs, i.e. -~ 1, fi ~ 0. Only when

the ratio is of the order of one does the enhancement become important.










It follows that if the bulk concentration of A becomes significant so

that the above boundary conditions, Eq. 3.9, become inappropriate, the

above result nonetheless exhibits the correct limit.

The second limiting case which admits simple results is that in

which the interfacial reaction becomes instantaneous. This means that

the reactants A and B can not coexist together in the liquid, that is

they react immediately upon contact. Thus when an element of liquid

first contacts the gas all of the liquid reactant B at the interface is

instantly consumed. As the absorption of A progresses a reaction plane

moves into the liquid. At this plane and only there the reaction takes

place and the concentrations of A and B vanish, see Figure 3.1. This

uncouples the diffusion and reaction equations for A and B by removing

the reaction. To the left of the plane, A diffuses from the interface

to the reaction plane; to the right B diffuses from the bulk to the

plane. Solving the two diffusion equations and equating the fluxes at

the reaction plane, adjusted for the stoichiometry, gives the concen-

tration profiles and the position of the reaction plane, see Appendix C.

Averaging the flux at the interface over all exposure times obtains the

enhancement factor, viz.,

C
S= 1 + (3.12)
S uCAi

This is the upper bound for the enhancement factor. On comparing this

with the previous expression for i we can estimate the condition for the

onset of instantaneous reaction, viz.,4



'This is an estimate because in the first expression CB = CB9 everywhere
and in the second CB vanishes throughout part of the interfacial region.
Evidently there is a transition region and its description is the aim of
the van Krevelen-Hoftijzer approximation.



























Liquid


Interface Reaction
Front


Figure 3.1. Concentration profiles near the interface for instantaneous
reaction.










1 + i / TRi > 1 + CB / UCAi

That is, whenever the noninstantaneous enhancement factor exceeds the

instantaneous value then the latter must be used. Whenever enhancement

is significant, Ti/Ri is much greater than one and because CBA/VCAi is

almost always much greater than one, a simpler form of the above con-

dition is possible, viz.,

>i CAi (3.13)
TRi \vAi

All of this is consistent with the intuitive idea that the fraction that

reacts for instantaneous reaction must be approximately one.

There appears to be three limiting cases for the enhancement factor

corresponding to slow, fast and instantaneous interfacial reaction:

1 = 1 slow

= 1 + i/T fast (3.14)

p = 1 + CB,/vCAi instantaneous.

This can be seen in Figure 3.2 where the enhancement factor is plotted

against the rate constant (similar to a plot against temperature). Now

that the enhancement factor is known for the limiting cases we must

establish what happens in the transition region between fast and instan-

taneous reaction. The fast reaction expression, because it is based on

B in excess, breaks down when the concentration of B becomes small near

the interface. But the reaction does not become instantaneous until B

goes to zero there. So in the transition region the variation in the

concentration of B must be considered. This can be done using the

van Krevelen-Hoftijzer approximation, see Sherwood, Pigford and Wilke

(1975), which corrects for the variation in the concentration of B by

making an estimate of its interfacial value and using that in place of
























Instantaneous Reaction -


I

/


Fast Reaction

S= TikI


Reaction


Figure 3.2. Enhancement factor versus k .
1*









the bulk concentration in the excess B result. The interfacial con-

centration of B is estimated by equating the rate of diffusion of B

toward the interface, stoichiometrically corrected, to the excess rate

of absorption of A, i.e. the fraction that reacts. This concentration

can be used in place of CBn in the original calculation of the enhance-

ment factor to obtain


= i + -- ~--- (3.15)
V Ri QI 1

where 1, is the instantaneous enhancement factor, Eq. 3.12.

This result is shown in Sherwood, Pigford and Wilke (1975, p 328)

to correctly bridge the limiting conditions of excess B and instanta-

neous reaction. But this result is more complicated than the previous

ones and using it makes the analyses to follow quite difficult. How

much error would be introduced by replacing this intermediate regime

result by an extrapolation of its limits? The transition region is

small as illustrated in Figure 2.3 and the enhancement factor does not

differ greatly from the extrapolation of its limits. This suggests that

without the loss of any important phenomena, we can represent the tran-

sition region by this approximation. Thus in keeping with the desire

for a simple model, the enhancement factor will be approximated by

extrapolating these two limiting cases to their intersection in the

transition region.

The results of the interfacial model are:
/ CBZ 2
for '< then = /I + T
Ri v Ai

f i/Ri
1 + Ti/TRi











and for > then I = 1 + CB/CAi
Ri A = 1 + C/vCAi

fi = 1 (3.16)


Both the enhancement factor and the fraction that reacts are S-shaped

functions of temperature. In fact fi is the same function that deter-

mines the multiplicity in a single-phase CSTR if Ti is the holding time.

The function f. varies from zero to one. The enhancement factor ranges

from one to ,I where I depends on temperature through the gas solubil-

ity, i.e. CAi. Both functions increase monotonically with temperature.

But the temperature they depend on is the interfacial temperature which

has yet to be determined.

Just as reaction near the interface increases the concentration

gradient, and hence the mass transfer, it increases the local temper-

ature through the heat of reaction. The interfacial temperature can

also rise due to the release of the heat of solution of the gas upon

absorption. We can construct the solution of the energy balance

equation, Eq. 3.7, in two special cases: that of no reaction where

the temperature rise is due solely to the heat of solution and that of

instantaneous reaction where the heat of solution is vanishingly small.

The temperature rise due to gas absorption without reaction will be

found first.

For no reaction the concentration and temperature near the inter-

face are described by the diffusion equations:

aCA a2CA
-= D -
at az2

aT 2 T
at az2









with the initial and boundary conditions:

at t = 0 CA = 0 T T

at z = 0 CA = CAi(Ti) T = T

as z + CA = T T

The relationship between CAi and Ti is given by Henry's law, viz.,


PA PA HS/RTi
Ai H(Ti) Ho

where PA is the gas partial pressure and H(T ) is the Henry's law coef-

ficient which depends on the heat of solution HS. The solution of these

equations is given in Appendix D. The interfacial temperature Ti can be

found by equating the heat released on absorption to the flux of energy

into the liquid at the interface:

A IT
D Hz = -A
zi z=0 p az z=0

Solving obtains an implicit equation for the temperature rise:


T TZ =- CAi(T ) (3.17)


Since HS and the ratio D/a are usually small the temperature rise due

to the heat of solution will usually be small. Using parameters as

reported for the chlorination of decane studied by Ding et al. (1974),

yields a temperature rise of three degrees Celsius.

When the reaction is instantaneous, all the reaction takes place at

a plane which moves into the liquid. The solution of the mass balance

equation is known and so therefore is the motion of the reaction plane,

see Appendix C. For the energy balance equation the heat generation is









confined to the reaction plane and is known. The problem then is simply

to find the temperature field that disposes of this heat. If the gas-

liquid interface is assumed to be an adiabatic surface, which is a

reasonable approximation, then on the interfacial side of the reaction

plane the energy balance equation is satisfied if the temperature is a

constant, T.. That is, the heat generated at the moving reaction front

heats the liquid behind it to a constant temperature. On the bulk side

of the front the heat of reaction is conducted from the moving plane at

a temperature Ti to a reservoir at infinity at temperature T The

corresponding temperature profile is known, see Appendix D. The inter-

facial temperature T. can be obtained by equating the heat generated by

reaction at the front to the heat conducted away from the front:

aCA T T
-HR D at z =
R 3z 3z

The result is complicated but if the thermal diffusivity is much greater

than the molecular diffusivity, a >> D, and CBZ/(vCAi) >> 1 (which are

both true for most gas-liquid systems) it can be simplified to, see

Appendix D,


T. -T -Z - i CAi (3.18)
1 P pCp a

This equation is still implicit in Ti because the instantaneous enhance-

ment factor and the solubility are functions of Ti. We expect the tem-

perature rise for instantaneous reaction to be much larger than that for

absorption not only because heats of reaction are usually larger than

those of solution but also because the enhancement factor can be much

greater than one, that is, the rate of absorption is much greater with

reaction.










Both temperature rise results have the same form (i.e. the product

of the heat released and the rate of absorption) which suggests that the

instantaneous reaction result can be corrected for the effect of the

heat of solution by replacing the heat of reaction with the sum of the

heat of reaction and the heat of absorption. Using the data of Ding et

al. (1974), this prediction yields a temperature rise of 92 degrees

Celsius. Even though this is an upper limit, any rise of this order of

magnitude must be important.

In the intermediate regime where the reaction near the interface is

fast enough to be important but not instantaneous, we can not construct

the solution to the problem. Nonetheless we may deduce a reasonable

approximation as follows. Because a >> D, the mass transfer process

appears to take place at a constant temperature Ti with known result.

This implies a known release of energy due to absorption and reaction.

All that remains then is to find the temperature field which accommo-

dates this heat generation. In particular, the interfacial temperature

is maintained by the heats of solution and reaction which appear to be

released at the interface. That is, because the heat transfer process

takes place over a much greater distance than the mass transfer process,

the generation of heat via absorption and reaction appears as a surface

source for heat transfer rather than a bulk source. On this basis the

temperature profile can be found from the diffusion equation with

constant interfacial and bulk temperatures, viz.,

aT 32T
at 3Z

at t = 0 T = T


and as z+ T = T .


at z = 0 T = T i









Then the interfacial temperature is obtained by an energy balance at the
interface. The heat generated by absorption and reaction is equated to
the energy flux at the interface where both have been averaged for the
surface-renewal theory, see Appendix D, viz.,

ICA IaT
(HS + HR fi) (- D a - I

The resulting temperature rise is

Hs HR
Ti T ( P + f. ) CA (3.19)
1 9 pCp pCp i a Ai

This result is similar in form to the two previous limiting results. In
each the temperature rise is the product of three terms: the rate of
absorption, the heat released, and the inverse of the heat transfer
coefficient. The heat released for each mole of gas absorbed is the sum
of the heat of solution and the product of the heat of reaction and the
fraction that reacts. The rate of absorption is t k[ CAi. The heat
transfer coefficient is \l/D p Cp k where 7j/D has to do with con-
verting a mass transfer coefficient into a heat transfer coefficient.
Eq. 3.19 also has the proper limits:


limit (Ti T ) =.p CAi




k pCp I Ai

The first is the result for absorption without reaction. The second is
the result for instantaneous reaction, where f. = 1 and $ = I,, with the
effect of the heat of solution added. Thus the general result for the
interfacial temperature rise, Eq. 3.19, seems to be reasonable, it is










easily interpreted physically and gives the proper limits for slow and

fast reaction.

This result also agrees with what one might estimate from the

instantaneous reaction temperature rise by equating the ratios of the

reaction rates and the temperature rises for instantaneous and non-

instantaneous reaction. That is the ratio of the temperature rise for

noninstantaneous reaction to that for instantaneous should be equal to

the corresponding ratio of the interfacial reaction rates. The reaction

rates are the mass transfer rates times the fraction that reacts. Thus

we have

(Ti T,) fi k' CAi

(Ti T)Inst. I k CAiI

The instantaneous rise is known, i.e. Eq. 3.18, so

HR D c
(T T ) f CAi
PCp *1 Ai


This is the result obtained above if the heat of solution is neglected.

Thus the predicted temperature rise agrees with physical intuition.

The mass and energy balances for the reactor complemented by the

interfacial results for the enhancement factor, the fraction absorbed

that reacts, and the temperature rise, constitute a complete descrip-

tion of the reactor dynamics. The use of the averaged interfacial

results in a dynamic reactor model is justified because the charac-

teristic time for variations near the interface is much less than that

for the reactor. In Table 3.2 these equations are shown in their dimen-

sionless form. But as previously seen, a better understanding of the

physics of the processes may be obtained by writing the equations in









terms of the time scales, Table 3.3. This will become more evident when

the steady state results are examined.

The scenario for oscillations, outlined originally, can be made

more precise. In the temperature equation, the generation terms,

containing the heats of solution and reaction, operate to decrease the

temperature at high temperatures and increase it at low ones. At high

temperatures the solubility xi is small hence the heat generated via

absorption, BS (T / T~) p (xi XA), and interfacial reaction,

BR (T / TM) i fi (Xi XA)' can be small. Also the bulk concentration
of A, xA, must be small for small xi, whence the bulk reaction term,

BR (T / R) xA, must be small. Thus as the heat generation decreases
and the heat removal increases for high temperatures, we anticipate

dT/dt < 0. Conversely, for decreasing temperatures the rate of absorp-

tion increases hence the rate of reaction may also increase because xA

is large. Thus we anticipate that for low temperatures, dT/dt > 0. But

we observe that the effect of the enhancement factor and of the fraction

that reacts on absorption is contrary to the above, viz., increasing

temperature increases absorption and interfacial reaction, hence the

generation of heat. So it may be that oscillations do not appear in

systems with enhanced mass transfer and that the most likely regime for

oscillations is the one that precedes enhancement. This discussion is

premature because before the dynamics can be adequately understood, the

steady state behavior must be known. The steady state results will be

investigated first to provide a basis for dynamic studies.








TABLE 3.2
DIMENSIONLESS EQUATIONS



dxA
- = (1 + Da B eY/(1+)) xA + Dx (1 fi) [ e-nel/(+1i) XA







S= (I + H) +H + BR Da xA xB e/(+)


+ [ BS + BR fi ] p a (e-n6i/(l+Oi) XA)


9i = e + [ BS + BR fi ] /


if S2 Dai e(Y-2n)8i/(1+i) < XB

then = /I + Dai xB eYi/(1+i)


Dai xB eYi/(l+6i)
i + Dai B ei(
1 + Dai XB ei/ i)


or if S Dai e(Y-2n)ei/(+

then


" > XB

S= 1 + (xg/S) enei/(1+ei)


fi = 1


e-rai/(I+0i)






48


TABLE 3.2--continued


where


CA / CAif

CB / CBf


(PA / Ho) en

HS / (R Tf)


= (T Tf) / Tf
=(T-Tf)/Tf

= (Tc Tf) / Tf



=t/T

= o C Bf e-Y


= Tk a


= (Hs CAif) / (p Cp Tf)

= (v CAif) / CBf


ei = (Ti Tf) / Tf






y = E / (R Tf)

Dai = k CBf e-Y


H = (U Sc) / (p Cp q)


BR = (HR CAif) / (p Cp Tf)






49


TABLE 3.3
EQUATIONS IN TERMS OF TIME SCALES



= (1 + T / TR) A + ( / ) (xi ) (1 fi)



= (1 XB) S (T / TR) XA S < (r /TM) (xi XA) fi


= (1 + T / TH) 6 +


+ BR [ (T / TR) XA


(T / TH) Oc + BS (T / TM) (Xi XA)


+ ( (1 / TM)


(xi XA) fi ]


i = + BR fi ] 7R fi xi


then = + i / TRi


S i / RRi
f
S+ / TRi


then + = 1 + x / (S xi)


= Dai XB e'y/(1+6i)


< B 2
S x.


2


Tr
if
TRi


1.
if
'Ri


- Da x e/(l+6)



i e /(
xi = e-n i/(l+9i)















where


x = CAi / CAif


T=V/q
T = V / q


R 1 / (k CB)



TM = / (kL a)



k = k e-E/(R T)
o


50

TABLE 3.3--continued












(sec) Ti = D / (kL)2


(sec) TRi = 1 / (k(Ti)CB)


(sec) TH = T/H = (p Cp V)/(U Sc)



(cm3 /gmole-sec)


(sec)


(sec)


(sec)















CHAPTER FOUR
STEADY STATE RESULTS

The steady state equations are obtained by setting the time deriva-

tives of the concentrations and temperature to zero in the model. In

terms of the time scales, see Table 3.3, these are

0 = -(1 + T/TR) xA + (T/TM) 4 (xi xA) (1 fi)
(4.1)
0 = (1 xB) S (T/TR) xA S (T/TM) (xi A) fi

0 = -(1 + /TH) + H) BS (T/TM) ( (Xi XA)

+ BR { (/TR) XA + (T/TM) 4 (xi A) f }

The concentration of B and the temperature also appear implicitly; xB

appears in TR, R and fi whereas e appears in TR and in 4, fi and xi

through their ei dependence, see Table 3.3. Only the concentration of

A can be found explicitly, viz.,

x ( fi)
XA M (4.2)
{ 1 + + (1 fi)
TR TM

Eliminating the concentration of A using Eq. 4.2 obtains two implicit

equations for xB and e, viz.,

S x. (1 + ) + f.
xB = 1 S ( ) (4.3)
{ + + (i f) } i + T
R TM R

and









4 x (1 + )
TM TR
(1+ T/TH) e (T/TH) ec = T R
( 1 -+ + (1 fi)
TR TM

_+ fi
{ BS + BR ( ) } (4.4)
1 +-
'R

Although complicated, these balances can be easily interpreted.
The amount of A absorbed in time T is
T T
S--x (1 +- )
T ( TR
4 (x x)= (4.5)
TM {I 1 + + (1 f)
'R M 1

and the total fraction of absorbed A that reacts is1
T/TR + f.
fR = ( ) (4.6)
1 + /rR

Equation 4.3 then says that the concentration of B is its initial con-

centration minus the product of the amount of A absorbed, the fraction



1Because fR is the ratio of the rate of reaction in the bulk and in the
interfacial region to the rate of absorption, viz.,

fb V kL a (CAi CA) (1 fi) + fi V kL a (CAi CA)
fR = V k a (CAi CA)

= fb (1 fi) + f
and fb is the fraction of A absorbed into the bulk that reacts in the bulk
V k CB CA r/R
b V < k a (CAi CA) (1 f) 1 + T/TR
we obtain

/TR + fi
R 1 + T/TR










that reacts and the stoichiometric constant S. The energy balance,

Eq. 4.4, is written in the classical form: the heat removed from the

reactor in time T equals the heat generated. The heat removed is the

sum of the energy lost to the effluent and to the cooling medium, viz.,

e + (T/rH) (e eO)

and is linear in temperature. The heat generated is the product of the

amount of A absorbed and the total heat released via absorption and

reaction per mole absorbed. The latter is the sum of the heat of solu-

tion and the product of the heat of reaction times the fraction that

reacts. Inasmuch as the heat removal function is a straight line with

respect to e, identical to that for the single-phase reactor, it is the

shape of the heat generation function with respect to e which determines

the steady state multiplicity, as will be seen (see also Appendix A

where the same is true for the single-phase CSTR).

Classically the steady states are found by plotting the left and

right sides of the energy balance, Eq. 4.4, versus the temperature. But

Eq. 4.4 is complicated, so before doing this we seek a better under-

standing of the system and of the interaction between the physical

processes by considering a special case: an isothermal reactor with

the concentration of reactant B equal to its feed value (therefore the

reaction is pseudo-first order), see Sherwood, Pigford and Wilke (1975).

By varying the rate constant for this system, we can consider an ensem-

ble of cases without the complicated temperature equation and determine

how the reactor is affected. Inasmuch as the greatest temperature

dependence is through the rate constant, the results should give us an

indication of how the system will respond to variations in the temper-

ature.









Because the heat generation is proportional to the rate of absorp-

tion, one should expect that a knowledge of this function will greatly

help in the understanding of the heat generation function and hence

the multiplicity of solutions to the steady state reactor equations.

The rate of absorption of A is

S~- (qCAi) (1 + T/R)
R = (V/TM) (CAi CA = i (4.7)
a Ai { 1 + + (1 fi
T T 1
R M

The effect of the solubility can be removed by introducing a relative

rate of absorption, viz.,

TM (1 + T/T)
Rr= Ra/ (qCAi) = T (- (4.8)
{ 1 + + (1 fi)
ER M

where qCAi is the rate of absorption if the liquid is in equilibrium

with the gas. Observing that 1/Rr can be written as a sum of resist-

ances gives

1/Rr = I/r1 + I/r2

(1 + T/TR)
where r (1 fi) =(1 + T/TR) (1 + Ti/TRi) (4.9)

and r2 = M (T/TM) (4.10)

Here we have scaled the system by T which is a measure of the flow rate

or, more specifically, a measure of the rate of flow of dissolved A out
of the reactor. Thus the overall resistance to absorption is the sum

of the resistance to the removal of A from the reactor 1/rl, and the

resistance to the mass transfer of A into the reacting liquid, 1/r2,

cf. Figure 4.1. The removal of A is accomplished in two ways, via flow

and via reaction, where reaction consists of two parts: bulk reaction




















Liquid
Phase



Reaction


Mass
Transfer

Flow






Solute In Solute Out


Figure 4.1. Series and parallel resistances for absorption and reaction.









and interfacial reaction. Because T >> ri always and TR = TRi at the

onset of enhanced mass transfer, initially for low temperatures or small

rate constants r, will go as (1 + T/TR). The resistance to mass trans-

fer depends on the mass transfer coefficient and the specific surface

area, r2 = T:koa. This is a constant until the temperature or the rate

constant is large enough so that the mass transfer becomes enhanced.

The only temperature dependence in these resistances is in the rate

constant k1 which appears in TR' TRi and q. If r, and r2 are plotted

versus the rate constant kI then insight into the behavior of the rate

of absorption as a function of temperature can be gained because the

relative rate of absorption will approximately equal the lesser of rl

and r2. For small values of k1, r1 is approximately one, cf. Figure

4.2. Only when T/TR = 1 does it begin to increase and when T/TR >> 1,

it goes as I/rR = Ikl. This continues until enhancement begins,

Ti/TRi = 1, and thereafter rl increases even faster, eventually going

to the limit


r T i = T kl(e) kl(i)
1 [R TRi

For the mass transfer process r2 is t4kLa and has no k1 dependence (or

temperature dependence) except through the enhancement factor, viz.,

4 = /1 + Ti/Ri = 1 + i k (ei)

So r2 is a constant, TkLa, with kI until i/TRi = 1, cf. Figure 4.2;

then r2 goes as Tkoa Frik until 4 reaches its maximum for instantaneous

reaction viz.,

xB
S= 1 +Sx
~I Sx







57









r = (1 + 1) (1
R
1000
B
CAi TM


100 2





10




1





-4 -2 0 2
log k1

T = 200 sec Ti = 0.0375 sec

CB
TM 12.5 sec VAi 50



Figure 4.2. rl and r2 versus rate constant.


T .
TRi


TM
= TM










Ti xB 2
TRi Sxi
TRi 1

Figure 4.2 shows rl and r2 and Figure 4.3 shows the relative rate of

absorption Rr versus the rate constant.

Figure 4.2 is drawn for a fixed value of T. Because r is ordinar-

ily an order of magnitude greater than TM, for small k1 mass transport

easily supplies what flow can dispatch. Thus rl and r2 stand in the

small kI or low temperature relation shown in Figure 4.2 where it is

fair to say that flow controls, viz.,

CA CAi -

When the rate constant increases we have the following:

i) bulk reaction increases, opening a parallel pathway for the

removal of the solute which reduces the resistance, 1/rl, without limit

and ii) interfacial reaction increases which reduces the mass transfer

resistance, 1/r2, but it opens no new pathway and its influence is lim-

ited. This sets the general shapes of the curves in Figure 4.2. The

details follow.

Reaction can affect absorption in two ways, through the overall

concentration difference and the mass transfer coefficient, i.e. the

local concentration gradient. Reaction in the bulk liquid increases

the overall concentration difference from the interface to the bulk

thereby increasing the rate of absorption. Exceptionally fast reaction

steepens the local gradient at the interface, increasing the value of

the mass transfer coefficient as reflected by the enhancement factor.

In Figure 4.2 r1 illustrates how reaction increases the overall differ-

ence and hence the rate of absorption. For slow reaction (small k1) we

have Rr = r, = 1, hence Ra is (qCAi). That is, mass transfer easily

















Instantaneous
Reaction

1000







10
R II
100 I





SEnhanced
I Mass
I MaTransfer
Mass
I Transfer
Control
Fast
Bulk
Physical Reaction
Absorption


-4 -2 0 2 4
log k1


Figure 4.3. Relative rate of absorption versus rate constant.









supplies all that flow can remove whence the rate of absorption equals

the rate of A flowing out of the reactor at the equilibrium concentra-

tion. This condition persists until 1/rR increases to the order of

1/T. Thereafter the loss of A via bulk reaction becomes significant

and the rate of absorption increases due to the larger concentration

difference, viz.,

r1 = T/R = Tk1 '

so that rl becomes proportional to k1. Thus rl increases rapidly,

driving CA to zero and turning control over to r2. Of course, ri

increases even more rapidly when 1/TRi increases to the order of I/Ti,

but as shown in Chapter Three, this occurs only after rl is already so

large vis-a-vis r2 that it is no longer of interest, i.e. after absorp-

tion becomes mass transfer controlled. In Chapter Three we concluded

that when enhancement begins, ri/TRi = 1 and TR = [Ri (i.e. the inter-

facial temperature equals the bulk temperature because before the onset

of enhancement there can be no significant reaction near the interface).

Because TM >> Ti this implies TM >> TR and T/TM << /TR. Thus r2 << r,

when enhancement begins and r2 will be controlling Rr. Therefore the

rate of absorption has two important subregions when it is under rl

control, one for slow bulk reaction and the other for fast bulk reaction.

The first is the physical absorption region where CA = CAi and is de-

fined by

T/TR << 1 or TkI << 1.

The second is the fast bulk reaction region, defined by

1 << T/TR T/TM









1 << Ik << Tka .

The critical point separating these two subregions of rl control is

defined by

T/TR = 1

We say that control of absorption shifts from the rl or flow-bulk-

reaction regime to the r2 or mass transfer regime when

T/TM = T/TR or TR = M

Here Rr = r2 and hence Rr is a constant with respect to k, because the

reaction in the bulk has gone essentially to completion. It remains

constant until enhancement begins and the mass transfer coefficient

increases. This unenhanced mass transfer controlled subregion is the

second plateau in Rr, cf. Figure 4.3. Here the relative rate of absorp-

tion is T/TM and the region is bounded by TR = TM and Ti/TRi = 1, i.e.

TM > TR I Ti where TR = TRi here. So this subregion of the mass trans-

fer controlled regime lies between the fast bulk reaction subregion of

the flow-bulk-reaction controlled regime and the enhanced mass transfer

subregion of the mass transfer controlled regime. For larger values of

the rate constant the reaction near the interface becomes significant,

enhancing the mass transfer coefficient. Hence the enhancement factor

becomes greater than one, increasing r2 and Rr. This region of enhanced

mass transfer is defined by

Ti xB 2
1 << << ( )
TRi Sxi

where TRi is now the variable and the relative rate of absorption is

approximately

Rr =2 = (T/TM) /RI = (T/M) Ti kl(ei)









In this region the rate increases as the square root of k1 as seen in

Figures 4.2 and 4.3. The enhancement factor o has a maximum pI, the

instantaneous enhancement factor, which establishes the upper bound

for Rr and establishes the final subregion. Here we have

xB
R = r= -- ( 1 + )
r 2 M M Sxi

which holds for

i xB )2
>> ( )
Ri Sxi

This is the third plateau in the figures.

The rate of absorption graph can now be divided into five operating

regions: physical absorption, fast bulk reaction, mass transfer con-

trolled, enhanced mass transfer, and instantaneous reaction, cf. Figure

4.3 and Table 4.1. In the second and fourth regions, fast bulk reaction

and enhanced mass transfer, the reaction increases the rate of absorp-

tion and Rr is an increasing function of k1, proportional to k, and k1

respectively. Otherwise Rr is independent of k1. In the first region,

physical absorption, the reaction is vanishingly small and the rate of

absorption is controlled by the gas solubility. In the third region,

mass transfer control, the reaction is so fast that it has essentially

gone to completion in the bulk, resulting in the maximum overall concen-

tration gradient. In the fifth region, instantaneous reaction, the

interfacial reaction is instantaneous, resulting in the maximum mass

transfer coefficient. The rate of absorption for each region and an

order of magnitude estimate for the critical points are shown in Tables

4.1 and 4.2, both in terms of the time scales and the reactor parameters.















TABLE 4.1
OPERATING REGIONS


Region


1. Physical
Absorption



2. Fast Bulk
Reaction


Rate of
Absorption



(qCAi)




rkl(qCAi)


Bounds
(Time Scales)


T/TR << 1




1 << T/TR << T/TM


3. Mass Transfer
Control


4. Enhanced Mass
Transfer



5. Instantaneous
Reaction


rkoa(qCA )
L Ad


Tkoa(qCAi) Tikl(ei)




TkLa(qCAi)( + )
Sx+ i


1 1 -=- _1
TM TR TRi i



Ti XB 2
1 << << (
Ri Sx



XB 2 T
( s ) << -
Sxi Ri















TABLE 4.2
CRITICAL POINTS SEPARATING



Order of Magnitude
(Time Scales)

T/TR = 1



T/TR = T/TM



*i/TRi = 1


Ti xB 2
Si)
Ri 1


OPERATING REGIONS



Order of Magnitude
(Parameters)

Tk1 = 1


k1 = kL a


D k1(e) = ( kL )2


Dk1 ) = xB )2 ( 2
Dkl(ei) = ( ( )i


Transition
Region


1-2


2-3


3-4



4- 5










Using this picture we can anticipate the structure of the results.

Because the rate of heat generation is proportional to the rate of

absorption, Eq. 4.4 may have as many as five solutions. This is evident

in Figure 4.3, where the relative rate of absorption is composed of two

s-shaped curves, suggesting that it may intersect the straight line for

the rate of heat removal as many as five times, once in each operating

region. Of course Figure 4.3 shows only the relative rate of absorption

which needs to be multiplied by (qCAi) and to be corrected for varying CB.

If the reactor is not isothermal, the relative rate of absorption

behaves with temperature almost exactly as it does with the rate con-

stant k1. The only difference will be in the area of enhanced mass

transfer. Here the rates may increase because of a higher interfacial

temperature which increases kI and hence 4. Also (I increases because

of its dependence on the temperature through CAi. More importantly the

actual rate of absorption is the product of the relative rate and qCAi.

Inasmuch as the solubility decreases with temperature, the plateau

regions seen in Figure 4.3 will not be constant with temperature but

will decrease. The two regions affected by reaction will still increase

with temperature as long as the products klCAi and [i CAi increase

with temperature. But if the solubility decreases fast enough then the

rate of absorption will decrease with temperature everywhere. This

implies that only one intersection with the heat removal line is possi-

ble and hence only one steady state is possible, which leads us to

anticipate an uniqueness condition, viz. that (klCAi) decreases with

temperature. Because

PA -(E HS)
klCAi k e10 p H RT
o









this condition is

E < HS (4.11)

that is, if the activation energy is less than the heat of solution then

only one steady state solution exists. This is similar to but stronger

than Huang and Varma's (1981a) and Raghuram and Shah's (1977) results,

Eqs. 2.1 and 2.2. If the solubility is not so rapidly decreasing then

it will not significantly affect the shape of the rate of absorption

function. Hence five steady state solutions should be possible.

The final assumption used in the above is that the concentration

of the liquid reactant B is constant, i.e. CB = CBf. If this is not

true, the major effect is on TR which now depends not only on the rate

constant but also on CB, viz.,

1/TR = k1 = kCGB
As seen in Table 4.2 this changes the critical temperatures separating

the regions. In general because CB should decrease with increasing

temperature, this should shift all the critical points toward higher

temperatures except the critical point for instantaneous reaction which

should shift to lower temperatures due to the CB dependence of I. The

effect of CB on the control regions is seen in Table 4.1. Of the three

regions independent of the rate constant, only that for instantaneous

reaction has any CB dependence and this is through 1I. The two remain-

ing regions which depend on reaction, the fast bulk reaction and the

enhanced mass transfer regions, will not increase as rapidly with the

temperature as they did with the rate constant k1 because the rates of

absorption in these regions are proportional to k1 = kCB and k = J k

and here CB decreases as temperature increases.










The heat generation function (the right-hand side of Eq. 4.4) is

the product of the rate of absorption, now well understood, and the heat

released. But the heat released also depends on temperature through the

total fraction of absorbed A that reacts, viz.,

T/TR + fi
f = ( ) .
fR =( 1 + T/TR

Now fi is the fraction that reacts near the interface and is essentially

zero until enhancement begins. But it has been shown previously that

when enhancement begins, T/TR >> 1. That is, bulk reaction is already

fast. Therefore fR goes as the s-shaped function

T/TR
R
( __--)-
1 + T/TR

and is essentially one when enhancement begins. In the physical absorp-

tion region fR is approximately zero hence the heat generation function

is a constant multiple of the rate of absorption there. In passing

through the fast bulk reaction region fR is an s-shaped function going

from zero to one. Thus in this region the heat generation curve looks

like the rate of absorption curve except that the rate of increase is

greater due to fR' so the curves should qualitatively be the same. For

the mass transfer controlled regime, viz. the last three subregions, fR

equals one so the rate of heat generation is another, higher, constant

multiple of the rate of absorption. In summary, the heat generation

function has the same critical points as the rate of absorption, see

Table 4.2, and is directly proportional to the rate of absorption in

all regions except the transition to the fast bulk reaction region, see

Table 4.3. Thus the time scale picture for the absorption process can

also be applied to the reactor itself.















TABLE 4.3
HEAT GENERATION FUNCTION IN EACH OPERATING REGION



Heat Generation Function
Region (right side of Eq. 4.4)


1. Physical Absorption




2. Fast Bulk Reaction




3. Mass Transfer Controlled



4. Enhanced Mass Transfer




5. Instantaneous Reaction


BS xi


(BS + BR)




(BS + BR)



(BS + BR)




(BS + BR)


-x




- x.
TM




T xB
M ( Ri


S(xi + XB
T M S









Plotting the heat generation function, i.e. the right side of

Eq. 4.4, versus temperature gives a graph similar to Figure 4.3. For

the system studied by Ding et al. (1974)2 where the parameters are

given in Table 4.4, this function is graphed in Figure 4.4. The heat

generation curve (labelled QG) has two s-shaped regions like the rate

of absorption and one can identify the five regions described. The

heat removal function (labelled QR), i.e. the left side of Eq. 4.4, is

a straight line of slope (1 + r/TH) and intercept (-(T/TH)eB). Although

Ding et al. found a maximum of three steady states for the conditions in

Table 4.4, it is evident from the figure that TH can be chosen such that

QR intersects QG five times. Thus a maximum of five steady state solu-
tions is possible. Each of these steady state solutions lies in a

different region, that is, only one steady state is possible in each

region.

The first stability condition can also be seen in this figure.

This is the slope condition which says that if the slope of the heat

generation function is greater than the slope of the heat removal func-

tion at a steady state solution then that solution is unstable. Thus

for multiple solutions every other one must be unstable. Moreover if

multiple solutions exist then any one in either the fast bulk reaction

or enhanced mass transfer regions must be unstable.

At this point we can draw an important conclusion about approximate

models. If we assume that a reactor operates in a specific region, we



2The values of the parameters for this system have variously been
reported by several investigators (Hoffman et al. (1975), Sharma et al.
(1976), Raghuram and Shah (1977), and Huang and Varma (1981a))with no
two in complete agreement. The values used here are the most commonly
quoted ones from the above collection.
















TABLE 4.4
PARAMETERS FOR THE CHLORINATION OF DECANE
AS STUDIED BY DING ET AL. (1974)


= 2.10 X


= 30000.


= 25000.


1017 cm3/gmole.sec


cal/gmole


cal/gmole


o = CBf = 5.10 X 10-3 gmole/cm3


Cp = 85. cal/gmole,0K


j = 3.35 X 10"- cal/cm.sec-K


D = 6. X 10-5 cm2/sec


V = 400. cm3



Dimensionless Parameters:

Da = 9.28 X 10-8 T

Dai = 3.48 X 10-9

BR = 3.57 X 10-2

BS = 6.43 X 10-3

B = 0.12 T


= 2970K


= 2980K


= 0.03 cal/K-sec


= 1 atm


= 1.12 X 107 atm.cm3/gmole


= 4500. cal/gmole


= 0.04 cm/sec


= 3.0 cm-1








= 50.8

= 3.61 X 10-2

= 7.63

= 1.73 X 10-4 T

= 3.37 X 10-3



















1.2








.8








.4








.4 .8 1.2
e







Figure 4.4. Heat generation and heat removal functions versus
temperature.

QG for system studied by Ding et al. (1974).
QR chosen to give five steady states.









can not use the corresponding balance equations to find the number of

steady state solutions. The full equations predict at most one solution

in each region. Thus if the approximate equations predict more than one

solution then all but one are surely meaningless. For example Huang and

Varma (1981a) propose an enhanced mass transfer model where 4= /rTR/R

Jrik1(ei) always. From this they determine that three steady state

solutions are possible and they deduce conditions for uniqueness and

stability. But we know that at most one solution may exist in this

region. The two meaningless solutions arise from the slow and fast

reaction limits of this enhanced mass transfer model, cf. Figure 4.5:

limit 4 0
kYO

limit -
kl1-

But these limits for the enhancement factor should be

S+ xB
1 < 4 < = ( 1 + ) .
- I Sx

As seen in the figure, the resulting steady state solutions are not the

slow reaction and instantaneous reaction solutions. Also the only mean-

ingful solution that is found is unstable (i.e. the slope condition is

not satisfied in this region if multiple solutions exist). So the sta-

bility conditions are meaningless as are the uniqueness conditions.

Thus models which are restricted to one region have limited capa-

bilities. These pertain to answering questions only about the one

steady state found there, for example, the stability of this state

could be examined. But uniqueness and multiplicity questions can not

be answered. If this information is desired and an approximate model

must be used to do the calculations then the approximate model must be











































/!














Figure 4.5. Comparison of enhanced mass transfer region model and full
model.

-- - Approximate Model
0 Approximate Solutions










general enough to faithfully represent at least one of the two s-shaped

regions for the heat generation function. That is, for multiplicity

the existence of at least one of the s-shaped regions is necessary.

Because two such regions exist in the full model it could be divided

into two simpler ones of limited but overlapping validity. From our

description of the process, one would consist of the physical absorp-

tion, fast bulk reaction and mass transfer control operating regions.

Thus the approximations here would be that o = 1 and fi = 0 everywhere.

This unenhanced mass transfer submodel would correspond to the first

s-shaped region of the heat generation function. The second approximate

model would consist of the mass transfer control, enhanced mass transfer

and instantaneous reaction operating regions. This enhanced mass trans-

fer submodel would be for T/TR > T/TM and fR = 1 and would correspond

to the second s-shaped region of the heat generation function. The

uniqueness and multiplicity conditions obtained from each will apply

only when the model conditions hold. But it may be possible to combine

the results of the simplified models to get conditions which apply every-

where. This will be investigated when we look for uniqueness conditions.

The effect of the parameters on the steady state behavior of the

reactor is most easily understood by examining their effect on the heat

removal and heat generation functions via graphs like Figure 4.4.

Examining the critical points, see Table 4.2, and the values of func-

tions on the operating regions, see Table 4.3, for their parameter

dependence will show how these functions change. The results in these

tables depend implicitly on the concentration of B. But inasmuch as

this concentration should be relatively constant for a gas-liquid reac-

tor, we should be able to determine the important parametric effects










assuming CB to be constant (except for very large holding times, T,

where most of the B may be consumed). The heat removal function is

linear in temperature with slope (1 + T/TH) and intercept -(T/TH)ec.

Increasing the holding time increases the slope but only significantly

when the ratio T/TH is greater than one. Thus for small holding times

relative to the heat transfer time, the slope of the heat removal

function is independent of this parameter. But at large values of T

the slope becomes proportional to T. So the slope of the heat removal

function tends to increase with T, starting from a region of independ-

ence. This is the most important variation in this function. We also

note that it only varies with the three parameters: T, TH and e .
For the heat generation function we need to consider both the

variation in the magnitude of the function in the various control

regions, see Table 4.3, and the variation in the critical temperatures

separating these regions, see Table 4.2. The dependence of the heat

generation curve on the holding time, the reaction parameters k10 and

E, and the mass transfer parameters kL and a, is shown in Figures 4.6-

4.10. In summary we can see that increasing r stretches the first

s-shaped region and compresses the second s-shaped region. In the limit

of large T the second s will ultimately disappear due to the exhaustion

of the reactant B in the mass transfer controlled region; hence there

can be no enhancement, cf. Figure 4.4. The influence of the reaction

parameters, cf. Figures 4.7 and 4.8, is through the critical tempera-

tures separating each region, see Table 4.2. These parameters do not

affect the height of the plateau regions which are set by the solu-

bility and the mass transfer parameters, see Table 4.3. Thus increasing

the Arrhenius frequency factor k10 or decreasing the activation energy E






























r = 2400 sec


Figure 4.6. Effect of T on the heat generation function.


1.2






.8
Heat
Gen.
Func.


.4






0




























k1 = 1017 sec-1


0. .4 .8 1.2


Figure 4.7. Effect of k1o on the heat generation function.


Heat
Gen.
Func.


























E = 25000 cal/mole


0. .4


.8 1.


Figure 4.8. Effect of E on the heat generation function.


Heat
Gen.
Func.






























ko = 0.08 cm/sec
L


.4 .8 1.2


Figure 4.9. Effect of kL on the heat generation function.


Heat
Gen.
Func.


.4






0


































Heat
Gen.
Func.


a = 3. cm-1


.4 .8 1.2


Figure 4.10. Effect of a on the heat generation function.









shifts all the critical temperatures to the left while the plateaus

remain the same, except for the larger solubility due to the lower

temperatures. The only effect of the heat of reaction, BR, is to

increase the heat generated in all the regimes but the first, physical

absorption, see Table 4.3. It has no effect on the critical tempera-

tures. The behavior of the mass transfer parameters, kL and a, is

interesting because only their product affects the first s-shaped region

whereas they individually influence the second. These parameters have

no effect on the first critical temperature or on the first two opera-

ting regions, cf. Figures 4.9 and 4.10, and they appear as a product

through TM when determining the second critical temperature and the

height of the mass transfer control region, see Tables 4.2 and 4.3.

For the second s-shaped region of the heat generation function, the

critical temperature for the onset of enhanced mass transfer depends

only on k and the magnitude of the heat generation function in the

enhanced mass transfer region depends only on a. This is due to the

fact that the interfacial exposure time depends on kL and not on a,

viz., Ti = D/(k )2. Therefore two systems which are identical except

for ko and a but have the same product, ka, will behave exactly the

same until enhancement begins, which will be at a lower temperature

for the system with the smaller value of k.

Inasmuch as the heat removal line and the heat generation curve

are complicated functions of the parameters and because their inter-

sections determine the steady state temperatures, we expect the depend-

ence of the solutions on the parameters to be unusual. Using the data

of Ding et al. (1974) as reported in Table 4.4 we can solve Eqs. 4.3 and

4.4 numerically for various values of the holding time. The results are










shown in Figure 4.11 where we see that at most three steady states exist

and that this system exhibits an isola. The steady state temperatures

found experimentally by Ding et al. are also shown and our model predic-

tions agree closely with these.3 Ding et al. did not run their experi-

ment at large enough holding times to find the end of the multiplicity

region so we do not know if the experiments confirm the isola.

Unusual multiplicity patterns such as an isola are also found in

a single-phase CSTR. In Appendix A we review the types of patterns

possible for this reactor and how they are formed. For a single-phase

CSTR the heat generation function has one s-shaped region and hence a

maximum of three steady states is possible. This function leads to

three basic types of multiplicity with respect to holding time, cf.

Figure A.2. The first is called s-shaped multiplicity and here the

locus of steady states turns back on itself over an interval in T. Thus

three steady states exist for this interval. This multiplicity pattern

is characterized by one ignition point and one extinction point where an

ignition point is the point on the locus where a change in T in one

direction causes a jump up to a higher temperature steady state and an

extinction point is where a change in r causes a jump down to a lower

temperature steady state. In some instances the locus can turn back on

itself a second time, cf. Figure A.2b, giving two intervals of multi-

plicity. This pattern is called a mushroom and is characterized by two

ignition and two extinction points. If these two intervals intersect



30ur predictions agree most closely with the experimental results if we
replace the value of the specific surface area, a=3, given in Table 4.4
with a=2 cm-1. Both of these values have been reported in the litera-
ture for this experiment. The predictions shown in Figure 4.11 are
actually based on this latter value.




































U


3U 3 EU U U U


20 40 60
T (min)


Figure 4.11. T versus T for the data of Ding et al. (1974).


200







T(OC)


100


100 120










then an isola is formed, cf. Figure A.2c. Here two extinction points

exist but no ignition points exist, thus it is not possible to observe

the high temperature steady states when starting from the low tempera-

ture branch and varying T. More will be said about these patterns in

Appendix H.

Inasmuch as the heat generation function for the gas-liquid CSTR

exhibits two s-shaped regions, we expect all of the above multiplicity

patterns and possibly some combinations of these. We do, in fact,

observe all of the three basic types of multiplicity and some combina-

tions. This will be discussed next. Later and in Appendix I we will

show that certain combinations of multiplicity patterns are not possible.

If we vary the frequency factor, k10, and repeat the above calcula-

tions for solving Eqs. 4.3 and 4.4 using all the values of the other

parameters as given in Table 4.4, we find several interesting multiplic-

ity patterns. Increasing k10 by a factor of ten produces an isola with

an s-shaped region and hence an interval with five steady states, cf.

Figure 4.12. This is the maximum number we anticipated from the graphs

of the heat removal and the heat generation functions. If k10 is

increased further, a mushroom is formed with an s-shaped region and five

steady states, cf. Figure 4.13. This mushroom has two ignition and two

extinction points, and two "jump" points which are indeterminant. That

is, the jump points which bound the region of five steady states could

be either ignition or extinction points. The same is true for the pre-

vious isola. If k10 is increased further, the mushroom disappears and

we get double s-shaped multiplicity with five steady states, cf. Figure

4.14. The multiplicity regions can be summarized by plotting the loca-

tion of the ignition and extinction points on a graph of k10 versus T in













400






E
300




T (OC)



200








100


1 10 100 1000
S(min)


Figure 4.12. T versus T: isola behavior.

k10= 1017 sec1

I = Ignition Point, E = Extinction Point














400







E
300


T (oC)





200 -





I or E


100 -
I or E








1 10
T (min)

Figure 4.13. T versus T: mushroom behavior.

k10 = 6. x 1017 sec-1

I = Ignition Point, E = Extinct


100 1000


ion Point













400








300 E



T (C)



200





I or E


100
I or E






). I I i0 I
0.1 1 10 100
S(min)

Figure 4.14. T versus T: s-shaped multiplicity.

k10 = 1018 sec-1

I = Ignition Point, E = Extinction Point










Figure 4.15. The locus of ignition points and the locus of extinction

points enclose the region of multiplicity. Outside this region the

steady states are unique; inside there are regions of three and five

steady states. The loci of the indeterminant jump points divide this

multiplicity region into regions of three and five steady states, as

shown. This figure also gives the type of multiplicity. For a given

value of k10 the number of ignition and extinction points determines

the type; one of each implies s-shaped (region a) and two of each imply

a mushroom (region b), whereas two extinction points imply an isola

(region c). From this figure we can find the critical values of k10

and T, above which and below which we have uniqueness. For k10 these

are 1020 sec-' and 3 X 1015 sec'1; and for t these are 0.2 min and

800 min.

Any number of figures similar to Figure 4.15 can be generated but

they only apply to a subset of the parameters with specific values, viz.

all the parameters except k10 and T are taken from Table 4.4. We would

like conditions that would answer the pertinent questions that the last

figure did for all values of the parameters without having to solve the

full equations, Eqs. 4.3 and 4.4. For example conditions guaranteeing

uniqueness and three and five steady states would be very useful. Also

we would like conditions under which mushrooms and isolas exist. We

will find such conditions next, beginning with the uniqueness problem.

The simplest uniqueness condition is that derived from the slope

condition: if the slope of the heat removal function with respect to

o is always greater than the slope of the heat generation function then

only one steady state exists. This is true because, as the graph of

these functions illustrates, cf. Figure 4.4, the slope condition must




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