GASLIQUID 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 GASLIQUID CSTR.....24
FOUR STEADY STATE RESULTS............................ .. .....51
FIVE STABILITY OF THE STEADY STATES.. ........................103
SIX CONCLUSIONS............................................... 117
APPENDICES
A STEADY STATE ANALYSIS FOR A SINGLEPHASE 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 GASLIQUID CSTR.................151
F THE APPLICATION OF POORE'S METHOD FOR DETERMINING
MULTIPLICITY TO A GASLIQUID CSTR........................160
G THE FRANKKAMENETSKII APPROXIMATION ......................173
H THE EXISTENCE OF MUSHROOMS AND ISOLAS IN A
SINGLEPHASE CSTR......................................... 179
I THE EXISTENCE OF MUSHROOMS AND ISOLAS IN A
GASLIQUID CSTR.......................................... 194
J STABILITY CONDITIONS FOR AN UNENHANCED GASLIQUID 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 GASLIQUID
CSTR WITH B IN EXCESS AND e = 0.... ..........................109
F.1 UNIQUENESS AND MULTIPLICITY CONDITIONS FOR A
NONADIABATIC, UNENHANCED GASLIQUID CSTR WITH
B IN EXCESS................................................. 170
G.1 ERROR DUE TO THE FRANKKAMENETSKII APPROXIMATION..............176
J.1 STABILITY CONDITIONS FOR AN UNENHANCED GASLIQUID
CSTR WITH ec = 0............................................. 208
LIST OF FIGURES
FIGURE PAGE
2.1 Heat generation and heat removal functions versus
temperature for a singlephase 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: sshaped 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 singlephase CSTR........................... 126
A.2 Multiplicity patterns for a singlephase 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 gasliquid 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 singlephase 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 singlephase CSTR.....198
1.2 Steady state analysis for a twophase 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, cm1
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, secI
hL, hL heat transfer coefficient with and without reaction,
cm/sec
k second order rate constant, cm3/gmolesec
k0 frequency factor for k, cm3/gmolesec
kI pseudofirst order rate constant, k1 = kCBf, secI
k10 frequency factor for kI, sec'
kL, k[ mass transfer coefficient with and without reaction,
cm/sec
Thermal conductivity, cal/cmsec.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/gmoleK
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 liquidphase reactant
b bulk
c cooling water
FK FrankKamenetskii 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
GASLIQUID 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 gasliquid reactors. In this study
a model is proposed for a specific class of gasliquid reactors and is
investigated to establish the basis for such oscillations.
We study the class of gasliquid 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: sshaped, mushroom and
isola. A method for predicting the existence of isolas and mushrooms
is given. For a singlephase 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 twophase, gasliquid 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 twophase 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 twophase systems. However, these oscillations
can not be like those found in singlephase reactors which can only
originate in the nonlinearity of the reaction rate. Here the primary
cause is the interaction of the temperaturedependent 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 singlephase continuous stirredtank 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 twophase 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
singlephase 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 temperaturedependent 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
gasphase mass balance. For this reaction scheme it should be possible
to analyze the multiplicity, stability and oscillatory behavior of a
twophase 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 singlephase CSTR's and their multiplicity and stability
analysis were developed over the years 19551976, 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 sshaped 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 singlephase 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 singlephase 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 singlephase 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 singlephase CSTR model
Schmitz and Amundson (1963ad) proposed a model for a twophase 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 singlephase 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
singlephase model and the latter is linear; hence only a unique stable
steady state exists.
There are two problems with this model. Most gasliquid 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 ndecane 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 wellmixed 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 liquidphase 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 (pseudofirst order because B is assumed to be in excess) of a gas
A through a stagnant liquid film. At the gasliquid 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 surfacerenewal. 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 timevarying models from the penetration or
surfacerenewal 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 filmbulk boundary which is assumed
equal to the bulk concentration.
Solving the diffusionreaction 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 pseudofirst 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 KrevelenHoftijzer
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 KrevelenHoftijzer 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 KrevelenHoftijzer 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 gasliquid 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 singlephase 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 singlephase 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, visavis a gasliquid 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 gasliquid 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 gasliquid
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 (1963ad)
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 ammoniawater 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 absorptionreaction 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
surfacerenewal theory and the result can be generalized for all reac
tion rates up to instantaneous.
With this background a model for gasliquid 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 surfacerenewal
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 GASLIQUID 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 twophase 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 indepth analysis and physically interesting predictions.
We investigate this simple class of twophase, 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 liquidphase 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 singlephase (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 gasliquid 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 singlephase 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 singlephase 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 singlephase
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 surfacerenewal 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 wellmixed liquid this
contact time might be random, suggesting the surfacerenewal 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 gasliquid 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 3.53.7, 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 gasliquid 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 absorptionreaction 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. 3.53.7 are compatible. We therefore add
the assumption:
A10. The thermal diffusivity a is much greater than the molecular dif
fusivity D; therefore the absorptionreaction 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 gasliquid 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. 3.23.4, 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 gasliquid 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
gasliquid 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 pseudofirst 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 et/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 singlephase 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 KrevelenHoftijzer 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 KrevelenHoftijzer 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 Sshaped
functions of temperature. In fact fi is the same function that deter
mines the multiplicity in a singlephase 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 gasliquid 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
surfacerenewal 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) [ enel/(+1i) XA
S= (I + H) +H + BR Da xA xB e/(+)
+ [ BS + BR fi ] p a (en6i/(l+Oi) XA)
9i = e + [ BS + BR fi ] /
if S2 Dai e(Y2n)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(Y2n)ei/(+
then
" > XB
S= 1 + (xg/S) enei/(1+ei)
fi = 1
erai/(I+0i)
48
TABLE 3.2continued
where
CA / CAif
CB / CBf
(PA / Ho) en
HS / (R Tf)
= (T Tf) / Tf
=(TTf)/Tf
= (Tc Tf) / Tf
=t/T
= o C Bf eY
= Tk a
= (Hs CAif) / (p Cp Tf)
= (v CAif) / CBf
ei = (Ti Tf) / Tf
y = E / (R Tf)
Dai = k CBf eY
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 = en i/(l+9i)
where
x = CAi / CAif
T=V/q
T = V / q
R 1 / (k CB)
TM = / (kL a)
k = k eE/(R T)
o
50
TABLE 3.3continued
(sec) Ti = D / (kL)2
(sec) TRi = 1 / (k(Ti)CB)
(sec) TH = T/H = (p Cp V)/(U Sc)
(cm3 /gmolesec)
(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
Sx (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 singlephase 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 singlephase 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 pseudofirst 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 visavis 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 flowbulk
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 flowbulkreaction 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
12
23
34
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
sshaped 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 righthand 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 sshaped 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 sshaped 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 sshaped 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 103 gmole/cm3
Cp = 85. cal/gmole,0K
j = 3.35 X 10" cal/cm.secK
D = 6. X 105 cm2/sec
V = 400. cm3
Dimensionless Parameters:
Da = 9.28 X 108 T
Dai = 3.48 X 109
BR = 3.57 X 102
BS = 6.43 X 103
B = 0.12 T
= 2970K
= 2980K
= 0.03 cal/Ksec
= 1 atm
= 1.12 X 107 atm.cm3/gmole
= 4500. cal/gmole
= 0.04 cm/sec
= 3.0 cm1
= 50.8
= 3.61 X 102
= 7.63
= 1.73 X 104 T
= 3.37 X 103
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 sshaped
regions for the heat generation function. That is, for multiplicity
the existence of at least one of the sshaped 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
sshaped 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 sshaped 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 gasliquid 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
sshaped region and compresses the second sshaped 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 sec1
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. cm1
.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 sshaped 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 sshaped 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 singlephase CSTR. In Appendix A we review the types of patterns
possible for this reactor and how they are formed. For a singlephase
CSTR the heat generation function has one sshaped 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 sshaped 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 cm1. 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 gasliquid CSTR
exhibits two sshaped 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 sshaped 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 sshaped 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 sshaped 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 sec1
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: sshaped multiplicity.
k10 = 1018 sec1
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 sshaped (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
