VIETNAM NATIONAL UNIVERSITY – HO CHI MINH CITY
UNIVERSITY OF TECHNOLOGY
FACULTY OF CHEMICAL ENGINEERING
Chemical Reaction Engineering
(Homogeneous Reactions in Ideal Reactors)
Mai Thanh Phong, Ph.D.
FCE – HCMC University of Technology
Chemical Reaction Engineering
Chapter 2. Interpretation of Batch Reactor Data
1. Rates of reaction
1.1. Description of reaction rates
Reaction rates depend usually in a complex manner on the concentrations, the
temperature and often on the effect introduced by catalysts:
r = f ( ci ,T, catalyst)
The order of a reaction is related to the concentration dependence. Typical
examples are:
Irreversible Reactions:
• First order:
A → Products:
r = k(T)CA
k in [s-1]
• Second order:
A + B → Products:
2A → Products:
r = k(T)CACB
r = k(T)CA2
k in [m3 mol-1s-1]
k in [m3 mol-1s-1]
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Chapter 2. Interpretation of Batch Reactor Data
Reversible Reactions:
• First order:
A ' R:
r = k1CA – k2CR
k in [s-1]
• Second order:
A + B ' R + S:
2A ' 2R:
2A ' R + S:
r = k1CACB – k2CRCS
r = k1CA2 – k2CR2
r = k1CA2 – k2CRCS
k in [m3 mol-1s-1]
k in [m3 mol-1s-1]
k in [m3 mol-1s-1]
r = k1CACB – k2CR2
k in [m3 mol-1s-1]
A + B ' 2R:
The dependency of the reaction rate on the temperature can be described using
Arrhenius law:
⎛ EA ⎞
k (T ) = k0 exp⎜ −
⎟
⎝ RT ⎠
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• EA is the activation energy of the reaction.
• ko is the pre-exponential factor (not dependent on
the reaction temperature
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Chapter 2. Interpretation of Batch Reactor Data
1.2. Rate laws of simple reactions
In this section, rate equations of simple reactions and the corresponding
temporal change of concentration are analyzed. A closed system (isothermic,
batch reactor) and aconstant volume are assumed.
1.2.1. Irreversible first-order reactions (Decomposition reactions)
A → Products
The rate equation is:
r=
1 dCi
= kCi
ν i dt
(2.1)
dC A
= kC A
With i is A and vi = -1, then: r = −
dt
The integration form:
−
CA
∫
C Ao
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(2.2)
t
CA
dC A
= kt
= k ∫ dt or − ln
C A0
CA
0
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(2.3)
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Chapter 2. Interpretation of Batch Reactor Data
The eq. (2.3) leads to the temporal course of concentration cA.
C A = C A0 e − kt
(2.4)
C A0 − C A
, the eq. (2.3) leads to:
XA =
C A0
(2.5)
X A = 1 − e − kt
(2.6)
With
− ln(1 − X A ) = kt
or
1.2.2. Irreversible bimolecular-type second-order reactions
Consider the reaction (A + B → Products) with corresponding rate equation
(2.7)
r=
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Chapter 2. Interpretation of Batch Reactor Data
Noting that the amounts of A and B that have reacted at any time t are equal and
given by CA0XA, eq. (2.7) can be written in terms of XA as
(2.8)
r=
(2.9)
(2.10)
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Chapter 2. Interpretation of Batch Reactor Data
After breakdown into partial fractions, integration, and rearrangement, the final
result in a number of different forms is
(2.11)
For the reaction: 2A → Products, the defining second-order differential
equation becomes
(2.12)
r=
which on integration yields
(2.13)
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Chapter 2. Interpretation of Batch Reactor Data
1.2.3. Empirical rate equations of nth order
When the mechanism of reaction is not known, we often attempt to fit the data
with an nth-order rate equation of the form
(2.14)
which on separation and integration yields
(2.15)
1.2.4. Zero-order reactions
Integrating and noting that CA
can never become negative,
we obtain directly:
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Chemical Reaction Engineering
(2.16)
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Chapter 2. Interpretation of Batch Reactor Data
1.2.5. Irreversible Reactions in Parallel
Consider the simplest case, A decomposing by two competing paths
The rates of change of the three components are given by
(2.17)
(2.18)
(2.19)
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Chapter 2. Interpretation of Batch Reactor Data
The k values are found using all three differential rate equations. First of all,
eq. (2.17), which is of simple first order, is integrated to give
(2.20)
Then dividing eq. (2.18) by eq. (2.19) we obtain the following
(2.21)
which integrated gives simply
(2.22)
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Chapter 2. Interpretation of Batch Reactor Data
1.2.6. Irreversible Reactions in Series
Consider the reaction
Rate eqautions for the three components are
(2.23)
(2.24)
(2.25)
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Chapter 2. Interpretation of Batch Reactor Data
Assuming that at t = 0, concentration of A is CA0, and no R or S present,
integration of eq. (2.23) gives
(2.26)
To find the changing concentration of R, substitute the concentration of A from
eq. (2.26) into the differential equation governing the rate of change of R, eq.
(2.24); thus
(2.27)
Solving the above differential equation gives
(2.28)
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Chapter 2. Interpretation of Batch Reactor Data
Noting that
which eqs (2.26) and (2.28) gives
(2.29)
If k2 is much larger than k1, Eq. (2.29) reduces to
(2.30)
If k1 is much larger than k2, then
(2.31)
Thus, in general, for any number of reactions in series it is the slowest step that has
the greatest influence on the overall reaction rate.
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Chapter 2. Interpretation of Batch Reactor Data
By differentiating Eq. (2.28) and setting dCRldt = 0, the maximum concentration of
R and the time at which it occours can be found:
(2.32)
(2.33)
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Chapter 2. Interpretation of Batch Reactor Data
Figure 2.1 shows the general
characteristics of the
concentration-time curves
for the three components:
• A decreases exponentially,
• R rises to a maximum and
then falls,
• and S rises continuously, the
greatest rate of increase of S
occurring where R is a
maximum.
Eq. (2.26)
Eq. (2.29)
Eq. (2.28)
Eq. (2.33)
Eq. (2.30)
Figure 2.1. Typical concentration-time
curves for consecutive first-order reactions.
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Chapter 2. Interpretation of Batch Reactor Data
1.2.7. First-Order Reversible Reactions
The simplest case is the opposed unimolecular-type reaction
(2.34a)
Starting with a concentration ratio M = CR0/CA0 th e rate equation is
(2.34b)
At equilibrium dCA/dt = 0. Hence from Eq. (2.34) we find the fractional
conversion of A at equilibrium conditions to be
(2.35)
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Chapter 2. Interpretation of Batch Reactor Data
and the equilibrium constant to be
(2.36)
Combining the above three equations we obtain, in terms of the equilibrium
conversion gives
(2.37)
Taking integration of the above equation, the following relation can be found:
(2.38)
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Chapter 2. Interpretation of Batch Reactor Data
1.2.8. Second-Order Reversible Reactions
For the bimolecular-type second-order reactions
(2.39a)
(2.39b)
(2.39c)
(2.39d)
with the restrictions that CA0 = CB0, and CR0 = CS0 = 0, the integrated rate
equations for A and B are all identical, as follows
(2.40)
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Chapter 2. Interpretation of Batch Reactor Data
2. Determination of kinetic parameters
2.1. Integral method
For the integral method, the concentration-time-course of one commponent is
measured.
A tipycal measurement is depicted in
CA
Figure 2.2.
CA0 *
After this measured data are compared
with theoretical integrated reaction rate
*
equation, for example:
*
C A = C A0 e
*
− kt
the unknown parameters can be calculated
by non-linear regression or using the
linearised form of the integrated rate
equation:
CA
− ln
= kt
C A0
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*
*
*
*
* *
* * *
t
Figure 2.2. Typical experimental
concentration-time-course
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Chapter 2. Interpretation of Batch Reactor Data
The measured data can be inllustrated in a ln(CA/CA0) vs. T diagram. The slope
of the straight line leads to the reaction rate constant (Figure 2.3).
2.6 of 2.3
Figure 2.3. Determination of the reaction rate constant k using the
integrated method in a linearised formulation
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Chapter 2. Interpretation of Batch Reactor Data
2.2. Differential method
For the differential method, also measurements of concentration-time courses are
necessary.
By means numerical and/or graphic differentiation of all measuring data of
concentration CA(t) the derivative dCA/dt can be determined. The measured data
can be approximated as a straight line in the reaction rate-concentration-diagram.
The procedure is as follows:
1. Plot the CA vs. t data, and then by eye carefully draw a smooth curve to
represent the data.
2. Determine the slope of this curve at suitably selected concentration values.
These slopes dCA/dt = r, are the rates of reaction at these compositions.
3. Search for a rate expression to represent this rA vs. CA data, either by
(a) picking and testing a particular rate form, rA = kf (CA), see Fig. 2.4, or
(b) testing an nth-order form rA = kCAn by taking logarithms of the rate equation
(see Fig. 2.5).
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log(rA)
Chapter 2. Interpretation of Batch Reactor Data
Figure 2.4. Test for the particular rate
form rA = kf(CA) by the differential
form.
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Figure 2.5. Test for an nth-order rate
method by the differential method.
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Chapter 2. Interpretation of Batch Reactor Data
Problems
1. Aqueous A at a concentration CA0 = 1 mol/liter is introduced into a batch reactor
where it reacts away to form product R according to stoichiometry A Æ R. The
concentration of A in the reactor is monitored at various times, as shown below:
a) For CA0 = 500 mol/m3 find the conversion of reactant after 5 hours in the batch
reactor.
b) Find the rate for the reaction.
2. For the elementary reactions in series
find the maximum concentration of R and when it is reached.
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Chapter 2. Interpretation of Batch Reactor Data
3. In the presence of a homogeneous catalyst of given concentration, aqueous
reactant A is converted to product at the following rates, and CA alone determines
this rate:
We plan to run this reaction in a batch reactor at the same catalyst concentration
as used in getting the above data. Find the time needed to lower the concentration
of A from CA0 = 10 mollliter to CAf = 2 mol/liter.
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