ChE 505
Chapter 10A
10A. EVALUATION OF REACTION RATE FORMS IN
STIRRED TANK
REACTORS
Most of the problems associated with evaluation and determination of proper rate forms from
batch data are related to the difficulties in estimating derivatives of concentrations or in fitting
cumbersome integral forms to the data.
In contrast, if a perfectly mixed stirred tank reactor shown below is operated at steady state,
direct estimates of the rate itself are obtained.

 FAo mol A h
q L  C mol 
 h  Ao  L 

FA mol A h

q CA mol L

For a reaction A  products that proceeds without appreciable change in volume of the reaction
mixture a mass balance on A gives at steady state:
Rate in   Rate out   Rate of Re action  O
FAo  FA   RA V  0
FAo , F A  in mol / hr

 RA   mol  V L  reaction mixture in the reactor
hr lit 
 L 
FAo  q CAo
F A  q CA q
hr 
F  F A CAo  C A
 RA  Ao

V
V / q 
V
By varying   we can readily obtain a set of data  RA vs CA and plot directly
q
log  RA   log k   log CA as Y vs X .
X
Y
Let us in addition assume that every run was performed isothermally, but at a different
temperature, and that the rate can be expressed by an n-th order form. Now we have generated a
set of data - r A vs T vs C A from which we have to extract  , ko , E since it was assumed that
the rate is of an n-th order form:
 RA  ko e  E /RT CA 
log  RA   log k o 
Y
E 1
  log CA
2.3R T
X1
X2
1
ChE 505
Chapter 10A
Say we had N-measurements. Then
Yi  A  B X1i  CX 2i ; i  1,2... N
X1i 
1
Ti
;
X2i  log CAi
Yi  log  RA  predicted at i ; A  log k o , B  
E
,C 
2.3 R
X1i , X2i , Yi are known at every data point. We want to minimize the sum of the squares of the


errors E  yi  Yi where yi  log  RAi measured at point i.
N
S   wi yi  Yi 
2
i1
S  S  S


O
 A  B C
which yields the three linear equations for best estimates of A,B,C (when wi = 1)
N
N
N
 yi  AN  B  X1i  C  X2i
i1
N
i1
N
i1
N
N
 yi  A  X1i  B  X 1i  C  X1i X2i
i1
N
i1
N
i 1
2
i1
N
N
 X2i yi  A  X2i  B  X1i X2i  C  X 2i
i1
i1
i1
2
i1
Thus, multiple linear regression can be used successfully to determine all three parameters but
at some loss of statistical rigor due to the logarithmic transformation performed. Often multiple
linear regression is adequate to determine the parameters with satisfactory accuracy. When
further refinements are needed, one should use the results of linear regression analysis as the
starting estimates of the parameters and proceed to refine them by nonlinear regression.
Suppose we wanted to find now the values of parameters that minimize the sum of the squares
of errors in the original space. Now S is defined by:
N

S   yi  Ae
i1
BX 1i
C
 
2
X 2i wi
A
All of the techniques will rely on some sort of iteration scheme. Briefly if P  
B
 is the vector
C
of parameters and P k is the k-th guess for parameters, P k1 is k  1 improved guess given by
P
k1
k
 P   RG
2
ChE 505
Chapter 10A
where
T
 S S  S 
G
A  B  C P k
is the gradient vector in the direction of increasing (decreasing S), R is a matrix, the
components of which determine whether in search of the minimum of S we will move directly in
the gradient direction or at some angle to the gradient direction,  is a scalar regulating the size
of the step taken in improving the parameter value.
For example if R  I is the identity matrix
1 0 0
R  0 1 0

0 0 1

P k 1  P k   G
and we have the steepest descent method.
If, however, R is related to the Hessian matrix
1
 2 S  2 S  2 S 
A 2  A B  A C 
  2 S  2 S  2 S 
R
A B  B2  B C 
2
2
2
  S  S  S 
 A C B C C 2 
one has the Newton-Raphson Method, and with approximations to R the Gaus-Newton
Method.
Some available computer programs would require explicit expressions for the partial
S  2 S
,
, etc. to evaluate G and R . Others estimate them by finite differences.
derivatives
 A  A B
Further information on parameter estimation in rate equations can be found in:
1. Hanns Hofmann, "Industrial Process Kinetics and Parameter Estimation", in 1st Int. Symp.
Chem. React. Eng. (H. Hulburt, editor), Advances in Chemistry 109, pp 519-534 (1972).
2. John H. Seinfeld and Leon Lapidus, "Mathematical Methods in Chemical Engineering, Vol. 3
Process Modeling, Estimation and Identification", Chapter 7, pp. 339-418, Prentice-Hall, N.J.
(1974).
3
ChE 505
Chapter 10A
10A.1
Review of Kinetic Forms
The following few pages summarize the integrated most commonly encountered kinetic
forms in batch reactors. Only the frequently encountered reaction rates for single reactions are
integrated and presented. Expressions for some simpler multiple reactions can also be readily
found and are given in almost all textbooks (Carberry, Levenspiel, etc.).
We have seen that reaction rates are functions of temperature and composition and usually are
much more sensitive to changes in temperature than in composition (i.e the rate changes much
more dramatically with temperature than with composition).
Consider an n-th order irreversible reaction
 RA  k oe
E /RT
n
CA
with E = 20,000 cal; n = 2.
At constant temperature doubling the concentration would increase the rate by a factor of 4.
However, doubling the temperature from 300K to 600K would increase the rate at constant
concentration by a factor of 17 x 106, i.e, seventeen million times! An increase of 27o i.e of
1.09 times the original temperature (less than 10%) would increase the rate by a factor of 4.
Thus, for single irreversible reactions we should choose the highest allowable temperatures in
order to get the highest rates.
For a reversible reaction the problem is slightly more complicated. For example, for a first
order reversible reaction we have
RA  k10e
E1 / RT
CA  k20e
E2 /RT
CP
for a reaction A = P. At equilibrium - RA = 0 .
k10
e
k 20
( E 1  E 2 )
RT
CP  CAo x A
xA
e
1  x Ae
 Kc
C 
  P   Kc
CA eq
CA  CAo 1  x A 
x Ae 
Kc
1  Kc
Now, if we raise the concentration of A, the net rate forward will increase while the
equilibrium remains unaffected.
4
ChE 505
Chapter 10A
1.0
locus of maximu m rates
direction of
increasing rates
xA
-RA =0 (X A )
e
T
Figure 1a. Conversion vs Temperature for an Exothermic Reaction
-RA =0 (X A)
e
xA
T
Figure 1b. Conversion vs Temperature for an Endothermic Reaction
If we raise the temperature, the rate will increase both in forward and reverse direction, the
portion of the rate with the higher activation energy will increase more and the equilibrium will
be affected. In the above example H  E  E . K  K RT   j . For this example
1
2
c
P
  j  O and thus
d n Kc  H

dT
RT 2
dKc
dK
1  Kc   c
dX Ae
Kc
dK c
dT 
 dT
2
2
dT
1  Kc 
1  Kc  dT
d n Kc
K 2c
H

2
2
1  Kc  dT
1  Kc  RT 2
dx A e
The sign of the above derivative syn
depends exclusively on the sign of H since all
dT
dx A e
other terms are positive. Thus
 0 for  H  0 i.e endothermic reactions, and
dT

Kc2
xAe goes up as temperature rises
5
ChE 505
Chapter 10A
X xAe  as T  . See Figure 1b.
dx A e
 0 for  H  0 i.e exothermic reactions, and the equilibrium conversion drops as the
dT
temperature rises x A e  as T  . See Figure 1a.
This conclusion can be generalized and it holds for all types of reversible reactions. Thus, in
principle for every reversible reaction we could plot on a conversion versus temperature plot a
locus for the equilibrium conversion as function of temperature and loci of constant rates. The
locus of constant rates is obtained by connecting the points in the x A vs T plane which have the
same rates.
For endothermic reactions no peculiar behavior is observed (Figure 1b). The higher the
temperature at fixed conversion the higher the rate, i.e if we travel along a line parallel to T-axis
from left to right we will cross the lines of constant rates moving from lower to higher rates. At
constant temperature the lower the conversion (the higher the reactant concentration) the higher
the rate i.e, when moving from the top to the bottom along a line parallel to x A - axis we will
constantly get into regions of higher rates.
For exothermic reactions, we can quickly observe (Figure 1a) that at constant temperature T
the lower the conversion the higher the rate. However, when we look at lines of constant
conversion x A we see that if we move from left to right we go through a region of increasing
rates, hit a maximum rate and go then through a region of decreasing rates. Thus, we can plot on
the x A vs T diagram a locus of maximum rates. For the given example of first order reversible
exothermic reaction this locus would be obtained as follows:
RA  k10eE1 / RT CAo 1 xA  k20eE2 / RT CAo xA
E
E
d  RA    12 k10 e  E1 / RT CA o 1 x A   22 k20 e E 2 / RT CA o x A dT
RT

RT

 k10 e
 E1 /RT
CAo  k 20 e
 E2 /RT

CAo d x A
Now for the loci of constant rates - RA = const, d ( -RA ) = 0 and
E1
E2
k 2x A
2 k1 1  x A  
dx A
RT
RT 2

dT RA  cons t
k1  k2
If we want to identify the points at which the loci of constant rates go through maxima in the
x A vs T plot then at those points we must have:
6
ChE 505
dx A
dT
Chapter 10A
 R A  const
0
which leads to:
E1k1 1  x A   E2 k2 x A  0
E k 1 x   E1  E 2 
A
E1k10 e
x A  1 10
e RT  1  0
E2 k20


E1 k20 1  x A
E  E2
e 1
E2 k20 x A
RT
 E1 / RT
Tm 
E1  E2
E2  E1

E k 1 x A 
 E k x

R n  1 10
 R n  2 20 A 
 E2 k20 x A 
E1k10 1  x A  
This is the locus of maximum rates (the dashed line on our graph) in Figure 1a. In other words,
pick a set of values for 0  x A  x Ae T min  and from the above formula calculate the
corresponding T . Every pair x A - T represents a point on the locus of maximum rates.
By solving for conversion one can express the locus of maximum rates as
1
xA 
E2 k20
E  E2
exp 1
1
E1k10
RT
Thus, one can pick a set of values for temperature, T , in the region of interest, and from the
above formula calculate the corresponding x A. Every pair x A - T is a point on the locus of
maximum rates on our graph (dashed line).
The above expressions were developed for a first order reversible reaction only. Nevertheless
the procedure can be readily generalized for any rate form
r
r
dr 
dT 
dx A  0
T
 xA
dx A
dT
r
  T  0
r
r const
x A
The locus of maximum rates can thus always be obtained by setting
r
0
T
The above discussion leads to two basic conclusions. In the case of reversible endothermic
reactions, the higher the temperature the higher the rate and the more favorable the equilibrium.
7
ChE 505
Chapter 10A
For reversible exothermic reactions, the higher the temperature the less favorable the
equilibrium. Thus, in a batch system one should move along the locus of maximum rates by
starting at high temperature and then as conversion increases, lower the temperature accordingly
in such a manner that at every temperature T one stays at the maximum of the rate loci.
Towards the end of reaction low temperature will allow favorable equilibrium while the rates
will still stay as high as possible at that temperature.
Besides being affected by temperature, the equilibrium composition is also affected by initial
composition. For example for a reaction aA = pP
CA xA 
Kc  a
C A 1 x A 
p
o
e
a
o
 C Ao
pa

x A ep
1 x Ae
e
 const at fixed T

a
Differentiating the above with respect to CAo we get
0  p  a CAo
x Ap
p a1
e
1  x A 
a
a1
p x A p1 1  x A   a 1  x A  x A p dx A
2a
d CA
1  x A 
a
e
 CAo p a
e
e
e
e
o
e
Thus, the sign of
sgn
d x Ae
d CA o
dx Ae
d CAo
e
depends only on p-a
  p  a
This can readily be generalized to:
S
dx Ae
sgn
   j
dCAo
j1
If the reaction proceeds with an increase in the number of moles then  j  0 and the
equilibrium conversion decreases with increased initial concentration i.e CAo  x A e  . If the
reaction proceeds with the decrease in the number of moles
conversion increases with increasing CAo
  j  O and equilibrium
i.e CAo  x Ae  .
This of course assumes ideal mixtures in which the activity or fugacity coefficients will not
change with composition.
mol 
For example, consider A = 2P, a = 1, p = 2 at the temperature at which Kc  1 
. If we
lit 
mol 
start with CAo  1 
the attainable equilibrium conversion is obtainable from the expression
lit 
8
ChE 505
Chapter 10A
for the equilibrium constant:
CAo
x Ae

x A 2e
 K c and x Ae 
1  x Ae
1

1  1  4  0.618
2

Kc
K 2c
Kc

2 4
CAo
C Ao
CAo
2

If at the same temperature we started with
1  1
1
1 
 mol 
CAo  2 
;
x




4
x

 0.5
A
e
lit 
2  2
4
2 
Since
  j  2  1  1  O asC A
o
 x Ae  as stated above.
The effect of total pressure on the equilibrium constant is
d n KP
V

dP
RT
dx A e
Thus, sgn
  V
dP
For a reaction that is accompanied by an increase in volume  V  O an increase in pressure
on the system will reduce the equilibrium conversion, i.e P  x Ae  .
For a reaction that is accompanied by a decrease in volume  V  O an increase in pressure
on the system will increase the equilibrium conversion i.e P  x Ae  .
For multiple reactions similar conclusions can be drawn. If the activation energy for the
reaction of desired product formation is the highest, the optimum yields will be obtained at the
highest temperature. If the activation energy for formation of the desired product is the lowest,
the lowest temperature will maximize the yield but the rates may be too slow then, so a
compromise has to be found. If the activation energy for desired product formation is in between
then an optimum temperature profile exists which would maximize the yield.
Various integrated rate forms are in the appended tables.
10A.2
Rate Forms Other Than n-th Order
So far we have mainly discussed the n-th order type rate forms. For an irreversible reaction at
constant temperature these rate expressions can be plotted against concentration of the limiting
reactant or conversion. (See Figure 2). All these rates have one thing in common, i.e they never
decrease with increasing concentration of the limiting reactant (or other reactants), or
mathematically
9
ChE 505
d  RA 
d CA
Chapter 10A
 0 for all 0  CA  CA o
These are "well behaved" rate expressions.
We should watch for possible peculiar behavior when we encounter a rate which can decrease
at least over some reactant concentration range with increasing reactant concentration i.e, for
rates where
d  RA 
 0 for CAc "  CA  CA c '
d CA
n=1
n<1
n>1
-RA
n=0
CA or (1 - x A)
Figure 2. n-th Order Rate Behavior
-RA
1
2
CA
Figure 3. Non n-th Order Rate Behavior
Two of such rates are presented in Figure 3. The first one (1) results from autocatalytic reactions
A + P= P + P - RA = k CACP , the other (2) is observed in a number of catalytic processes with
10
ChE 505
self inhibition  RA 
k CA
K  CA 2
Chapter 10A
. In these particular situations the rate is low at high reactant
concentrations, reaches the maximum at some C A and then decays again as the reactant
concentration is further depleted. It is worth considering autocatalytic reactions in a little more
detail.
For example
A PP  P
with a rate of RA  k CACP
From stoichiometry
Co
CAo  CA  CP CPo
; CP  CPo  CAo  CA
In a batch system then:
dC
 RA   A  k CA Co  CA 
dt
The rate reaches a maximum value when
d  RA 
C
 0  Co  2CA  CA  o or
d CA
2
CAo 1  x A  
Co
C
 xA  1  o
2
2CAo
Then the rate takes its maximum value of
C 
C
C2
 RAmax  k o Co  o   k o
2 
2 
4
If CAo  f Co where 0 < f < 1 the ratio of the maximum rate and initial rate is
2
 RAmax
 RAi nit ial
Co
1
4


k f Co Co  f Co  4 f 1 f 
k
Thus, if the initial solution contains 99% A and 1% P (which is reasonable since P serves only to
start the reaction) f = 0.99 and the maximum rate is over 25 times larger than the initial rate.
The integrated form of the rate expression is:
CA Co  CA 
n o
 k Co t
CA Co  CAo
C
The maximum rate at CA  o would be reached after time of
2


11
ChE 505
t max 
 CA 
 f 
1
1
f
1
o
n 

n

n 
1 f 
k Co
1  f k Co
Co  CAo  k Co
Chapter 10A
The larger the f is the larger the time t max necessary to reach maximum rates.
This implies that if the product is present as an impurity at 0.1% level (f = 0.999) the rate of
reaction may go unnoticed initially but would peak to 250 times larger value after k Co t = 6.9.
Autocatalytic reactions should be viewed in a broader sense than presented here. Specifically,
autocatalytic reactions become especially dangerous when they proceed in the gas phase with an
increase in the total number of moles of the reaction mixture (say A + P = 3P) and are
exothermic. In that case a slow initial reaction is accelerated on one hand due to the autocatalytic
effect of the product and on the other hand due to the increased temperature caused by the heat of
reaction. The rapidly rising pressure of the system due to the expanding reaction mixture volume
may, and occasionally does, cause explosions. Even for reactions in the liquid phase rapidly
accelerating reaction due to a combined autocatalysis-temperature (caused by exothermicity)
effect may lead to rapid build-up of the vapor phase and to explosions. Thus, autocatalytic
reactions and all reactions that exhibit rate forms which allow for increases in rates with
diminishing reactant concentrations should be watched for and dealt with cautiously.
Autocatalytic reactions can be viewed even in a broader sense than that. For example heat
released by a slow reaction causing the same or a similar reaction to accelerate rapidly is also
viewed by some as an autocatalytic effect in that reaction system.
As an example of autocatalysis let us consider the hydrolysis of an ester RCOOR' in dilute
water solution:
RCOOR' 
E
ester

H2 O

RCOOH

R' OH
W

A

B
 water 
acid 
 booze
The reaction proceeds without the catalyst at a very slow rate. However, the acid, product of
reaction, catalyzes the same reaction at much faster rates.
RCOOR'

E

RCOOH
H2 O 


 RCOOH  R' OH
A
W
A B



The total rate of disappearance of the ester can be given as the sum of the rate of the spontaneous
12
ChE 505
Chapter 10A
("residual") reaction and of the catalytic reaction
dCE

 ks CE  kc CE CA
dt
From stoichiometry
CEo CE  CA CAo
also CE  CE o 1 x E 
CA  CAo  CEo x E
If we started initially with pure reactants CAo  0 in a batch system, the governing equations
dCE
dx

 CE o E  ks CE o 1 x E   kc C2Eo 1  xE x E
dt
dt
at t  0
CE  CEo or x E  0
Note: The reaction rates for both the residual and catalytic reaction were presumed independent
of water concentration (zeroth order with respect to water) since water is present in large excess.
dx E
 ks 1  x E   kcCE o x E 1  xE 
dt
at t  0
xE  0
Upon integration we get
 1  kc CEo t
1  k cC Eo t
e
1
1e
xE  


 1  k C t
1  k cC Eo t
1 e
  e   c Eo
where

ks
kcCE o
1
kc CE o

 1
1
ks
1
has units of (time) and is a measure of the characteristic reaction time for the residual
ks
reaction, i.e gives the time scale over which that reaction takes place.
1
has units of time and is a measure of the characteristic reaction time for the catalytic
kc CEo
reaction.
Clearly the time scale for the much faster catalytic reaction is much smaller than the time
scale for the residual reaction and thus  1 .
  1  kc CE t becomes a proper dimensionless time for the process overall.
1  e 
xE  
  e 
o
The maximum rate is achieved when
13
ChE 505
xE 
1
1
;    n ; t 
2
1 kcCE
o
1 
n  
 
Chapter 10A
CE2o
CEo 
C
k s  kc E o  kc
2 
2 
4
 RE max 
The ratio of the maximum rate to the initial rate of the residual reaction is:
CEo 
ks  kcCE o 
 RE max
2 
2  1 
1 
 1  1

 1 
 RE si nit
ks CEo
2  2   4 
10A.3
Some Comments on the Variable Volume Batch System
We have derived some time ago how the volume of the reaction system varies with reaction
extent for an ideal mixture of ideal gases:
S




j


j1
V  Vo 1 
X 
NT o




Molar extent can be expressed in terms of fractional conversion since
N  N Ao NP  N Po
X A

 etc
A
xA 
X
P
NA o  NA
N Ao
N Ao
A
xA 
N Ao
A
xA
Then V  Vo 1   A xA where the coefficient of expression  A is defined by:
 S

  j yAo
V x A  1  V xA  O
j1 
A 

A
V x O
A
y Ao - mole fraction of A in the starting mixture.
Let us suppose that we are considering the following irreversible n-th order reaction
aA  pP p  a in a constant pressure batch reactor starting with pure A .
The material balance on A yields
dN A
n
  kC AV
dt
14
ChE 505
Chapter 10A
We know that
N A  CAV but the volume now varies with reaction.
d CA V 
d CA
dV
n
V
 CA
  k CA V
dt
dt
dt
In order to integrate the above expression we need to know how V and
dV
vary either with
dt
concentration C A or with time t . We could start from:
N
CA  A
V
In any system (constant P , or constant V ) by definition of convesion moles of A at time t are
given by the initial moles of A times fraction unconverted, i.e: N A  N Ao 1  x A 
and we have seen above:
V  Vo 1   A xA
CAo  C A
1 x A
CA  CAo
or x A 
1  A xA
CAo   ACA
We could substitute x A in terms of CAo ,CA in the formula for the volume:
1   A  CA o
V  Vo
CAo   ACA
 A 1   A  CAo d CA
dV
  Vo
dt
CAo   A CA 2 dt


d CA V 
After substitution in the first equation for
dt
CAo
and some algebra we get
d CA
n
  k C A ; t  0 CA  CAo
CAo   ACA dt
p  a x 1 p  a
For our example  A 

a
a
It is of interest to note that if we did not remember the variation of reaction volume with
extent or with conversion, and starting from
d CA V 
dt
V
dCA
dV
n
 CA
  k CA V
dt
dt
dV
. It is instructive to go through that process.
dt
dV  L 
The change in volume of the reaction mixture per unit time  
is proportional clearly
dt min 
moles A 
to the total rate of disappearance of A,  RAV 
multiplied by the change in volume of
min 
we would have had to derive an expression for
15
ChE 505
 L 
the reaction mixture per each mole of A reacted 
.
mole A 
dV

pa
n
n
 A k C A V  (in our example)
kC A V
dt CAo
aCA o
Chapter 10A
Now we would have two coupled differential equations to solve simultaneously.
Since differential equations for concentrations lead to relatively complex forms it is advisable
to use conversion whenever possible in variable volume systems.
d CA V 
n
  kC A V
dt
1 x A
V  V o 1   A x A 
but CA  CA o
1  A xA
CA V  CAo Vo 1  x A 
d CA V 
dx
1 x A 
  C Ao A   k C nAo
V o 1   A x A 
dt
dt
1   A x A n
n
dx A
1 x A 
 k C n1
Ao
dt
1   A x A n1
n
t0
xA  0
The equation for conversion can be more readily integrated for most orders and some of the
integrated forms were given earlier.
For example for first order reactions n
dx A
 k 1 x A 
dt
=1
xA  1  e  k t
This expression is exactly the same as in a constant volume system, since in the case of first
order reactions it does not make any difference to conversion whether the process is performed
at constant volume or constant pressure.
The differential equation for concentration is for n = 1:
CAo
dCA
  k CA
CAo   A CA dt
CA

 C
A

CA   A CA CA
n
CAo
o
o
t
CA o d CA
  AC A CA
 1   C C
A  Ao A
 
o
 k  dt
o

 k t

Solved for concentration this yields:
16
ChE 505
CA 
CAo e

Chapter 10A
kt
1   A 1e kt

in V  const
We remember that in a constant volume system for first order reaction:
CA  CAo e k t
V  const
Thus, although at given reaction time t fractional conversion for a first order reaction is the
same in a V = const and P T = const system the reactant (or product) concentration is not. This
is to be expected since in a V = const system the change in concentration is due exclusively to
reaction. In P T = const system the change in concentration reflects the combined effect of
depletion or formation of the number of moles by reaction and of volume expansion or
contraction due to reaction. Naturally for  A  0 (no expansion or contraction) the two
expressions are identical.
17