Interpretation of batch rate equations

References:
 Levenspiel O. (2002) ……Chap 3
 Fogler, HS (1992) …………Chap.5
12/31/2017 By Brhane amha (chemical engineering ) 1
Chem. Reaction Eng. I
Methods of Analysis of Rate Data
By Brhane Amha
Chemical engineering

Introduction
• A rate equation characterizes the rate of reaction, and
• its form may either be suggested by
theoretical considerations or
empirical curve-fitting procedure.
• In any case, the value of the constants of the equation
can only be found by experiment; predictive methods are
inadequate at present.
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Cont’d
•The determination of the rate equation is usually a
two-step procedure;
• First the concentration dependency is found at fixed
temperature and
• then the temperature dependence of the rate
constants is found, yielding the complete rate equation.
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Reactors for obtaining rate data
• Equipment by which empirical information(rate data) is obtained can
be divided into two types, the batch and flow reactors.
• The differential(flow) reactor,
 Used in the study of the kinetics of heterogeneous
reactions (solid-fluid reactions).
The product concentration is usually monitored for different feed conditions.
Measurements are made during steady-state operation.
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Cont’d
• The batch reactor
 is used primarily to determine rate law parameters for
homogeneous reactions,
By measuring concentration, pressure, and/or volume are at
different times during the course of the reaction.
then using different methods of data analysis to determine
the reaction order and specific reaction rate (rate constant).
 Data are collected from the batch reactor during unsteady-
state operation
Usually operated isothermally and at constant volume
because it is easy to interpret the results of such runs.
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Methods of analyzing the data collected
• Six different methods of analyzing the data collected are
• The Differential method,
• the integral method,
• the method of half-lives,
• The method of initial rates, and
• linear and nonlinear regression (least-squares analysis).
• The differential and integral methods are used primarily in
analyzing batch reactor data.
• Batch reactor data will be discussed in this chapter
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CONSTANT-VOLUME BATCH REACTOR
• In a constant-volume system the measure of reaction
rate of component i becomes:
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Integral Method of Analysis of Data
General Procedure
Guess a particular form of rate equation and,
after appropriate integration and mathematical manipulation,
predict that the plot of a certain concentration function versus
time should yield a straight line.
The data are plotted, and if a reasonably good straight line is
obtained, then the rate equation is said to satisfactorily fit the
data. If not, guess another form of rate equation.
It should be noted that the integral method is especially useful
for fitting simple reaction types corresponding to elementary
reactions
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Irreversible Unimolecular-Type First-Order
Reactions.
• Consider the following reaction: AProducts
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In terms of conversion:
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A plot of In(1 - XA) or In (CA/CAo) vs. t, as shown in Fig. 3.1, gives a
straight line through the origin for this form of rate of equation.
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Cont’d

Irreversible Bimolecular-Type Second-Order
Reactions.
• Consider the following reaction: A+B Products
with corresponding rate equation
In terms of conversion (CAOXA=CB0XB)
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CONT’D
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Let M=CB0/CAO, we get
Doing Separation and formal integration

I. Consider the following reaction:
A+B+D Products
Irreversible Trimolecular-Type Third-Order
Reactions
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Cont’d
Consider the following reaction: A+2BProducts
• Let the reaction rate be:
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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
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Zero-Order Reactions
• A reaction is of zero order when the rate of reaction is independent
of the concentration of materials; thus
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Overall Order of Irreversible Reactions from the
Half-Life t1/2.
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Fractional Life Method
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First-Order Reversible Reactions
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Second-Order Reversible Reactions.
• For the bimolecular-type second-order reactions
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Cont’d
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Derivation: Part of exercises 3

Reversible Reactions in General
• For orders other than one or two, integration of the
rate equation becomes cumbersome.
• So if Eq. 54 or 56 is not able to fit the data, then the
search for an adequate rate equation is best done by
the differential method.
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EXAMPLE 1- FIND A RATE EQUATION USING THE INTEGRAL METHOD
Reactant A decomposes in a batch reactor :A Products
The composition of A in the reactor is measured at various times with
results shown in the following columns 1 and 2. Find a rate equation to
represent the data.
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EXAMPLE 1- SOLUTION

Differential Method of Analysis of Data
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Differential Method of Analysis of Data
To outline the procedure used in the differential method of analysis, we
consider a reaction carried out isothermally in a constant-volume batch
reactor and the concentration recorded as a function of time.

GRAPHICAL
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. This curve most likely will not pass through all the
experimental points.
2. Determine the slope of this curve at suitably selected concentration
values. These slopes dCAldt = r, are the rates of reaction at these
compositions.
3. Now 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. 17,or
(b) testing an nth-order form -rA = k CAn: by taking logarithms of the
rate equation (see Fig. 3.18).

• Methods for finding -dcA/dt from concentration-time data
• i. Graphical Method
• Ii. Numerical differentiation formulas
iii. Differentiation of a polynomial fit to the data

• Numerical Method. Numerical differentiation formulas can be used
when the data points in the independent variable are equally spaced,
such as t1 -to = t2 - t 1 = delta t:

Example 3.2. FIND A RATE EQUATION TO FIT A SET OF DATA
USING THE DIFFERENTIAL METHOD
• Try to fit an nth-order rate equation to the concentration vs. time
data of Example 3.1.

SOLUTION
• The data are tabulated in the
following columns 1 and 2
and are plotted in Fig. E3.2a.

.
• Now carefully draw a smooth curve to represent the data and at CA=
10, 8, 6, 5, 3, 2, 1,draw tangents to the curve, and evaluate them (see
column 3).

5.2 Method of Initial Rates
• The use of the differential method of data analysis to determine
reaction orders and specific reaction rates is clearly one of the
easiest, since it requires only one experiment. However, other effects,
such as the presence of a significant reverse reaction, could render
the differential method ineffective. In these cases, the method of
initial rates could be used to determine the reaction order and the
specific rate constant.

Least-Squares Analysis :Linear’zation of the Rate
Law
• If a rate law depends on the concentration of more than one species
and it is not possibIe to use the method of excess, we may choose to
use a linearized least-squares method.

Differential method of analysis
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TEMPERATURE AND REACTION RATE
• So far we have examined the effect of concentration of reactants and
products
on the rate of reaction, all at a given temperature level. To obtain the
complete
rate equation, we also need to know the role of temperature on
reaction rate.
Now in a typical rate equation we have
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Interpretation of batch rate equations

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