제5장
Chemical Reaction Engineering
Collection and Analysis of Rate Data
Yogi Berra, New York Yankees
You can observe a lot
just by watching
Objective
Determine the reaction order and specific reaction rate from experimental
data obtained from either batch or flow reactors.
Describe how to use equal-area differentiation, polynomial fitting,
numerical difference formulas and regression to analyze experimental
data to determine the rate law.
Describe how the methods of half lives, and of initial rate, are used to
analyze rate data.
Describe two or more types of laboratory reactors used to obtain rate
law data along with their advantages and disadvantages.
 Describe how to plan an experiment.
Collection and Analysis of Rate Data
 Two techniques of data acquisition
- concentration-time measurements in a batch reactor
- concentration measurements in a differential reactor
 Methods of analyzing the data
- the differential method
- the integral method
- the method of half-lives
- method of initial rates
- linear regression
- nonlinear regression
 Software packages
- POLYMATH
- MATLAB
Algorithm for Data Analysis
Algorithm for Data Analysis
Algorithm for Data Analysis
Batch Reactor Data
 Irreversible reaction
- the reaction order and the specific rate constant numerically
differentiating concentration versus time data
- essentially a function of the concentration of only one reactant
For example
-for decomposition reaction
A  products
-rA = kACA
then differential method may be used
Batch Reactor Data: Method of Excess

Method of Excess
- Consider the irreversible reaction
A + B  products
-rA = kCACB
- Excess B experiments: CB remains unchanged during the reaction (CBo)
-rA = kCA CB = kCBoCA = k'CA
- Excess A experiments: CA remains unchanged during the reaction(CAo)
-rA = k''CB
- Once  and  are determined, kA can be calculated from the measurement of
-rA at known concentration of A and B
kA= -rA /CACB [dm3/mol] --1/s
Batch Reactor Data: Differential Method

Differential Method of Analysis
A  Products
(Assumption: Constant Volume Batch Reactor)
Rate raw:
-rA = kACA = -
dCA
dt
Taking the logarithm of both sides
 dCA 
ln 
  ln k A   ln C A
 dt 
Batch Reactor Data: Differential Method
 dCA 
ln 
  ln k A   ln C A
 dt 
ln
ln
p
slope=
CA
ln
kA=
CAp
p
(CAp)
ln
How to Get –dCA/dt (Graphical Method)
Time (min)
t0
t1
t2
t3
t4
t5
Concentration (mol/dm3)
CA0
CA1
CA2
CA3
CA4
CA5
 Graphical Method (Equal-Area Graphical Differentiation)
Ti Ci
Dt
DC
DC/ Dt
t1 C1
(dC/dt)1
t2-t1
C2-C1
(DC/ Dt )2
t2 C2
(DC/ Dt )2
(dC/dt)2
t3-t2
C3-C2
(DC/ Dt )3
t3 C3
(dC/dt)2
(dC/dt)3
t4-t3
C4-C3
A
(DC/ Dt )4
t4 C4
(dC/dt)4
t5-t4
t5 C5
Draw smooth curve
that best approximates
the area under histogram
dC/dt
C5-C4
(DC/ Dt )5
1
2
3
4
5
How to Get –dCA/dt (Numerical Method)
Time (min)
t0
t1
t2
t3
t4
t5
Concentration (mol/dm3)
CA0
CA1
CA2
CA3
CA4
CA5
 Numerical Method (Independent variables are equally spaced)
 3C A0  4C A1  C A 2
 dC A 

 
2Dt
 dt  t 0
C  C A0
 dC 
Interior po int s :  A   A 2
2Dt
 dt  t1
Initial po int :
C  C A1
 dC A 

  A3
2Dt
 dt  t 2
C  C A2
 dC A 

  A4
2Dt
 dt  t 3
Last po int :
C  C A3
 dC A 

  A5
2Dt
 dt  t 4
C  4C A 4  3C A5
 dC A 

  A3
2Dt
 dt  t 5
How to Get –dCA/dt (Polynomial Fit)
 Polynomial
Fit
Time (min)
t0
t1
t2
t3
t4
t5
Concentration (mol/dm3)
CA0
CA1
CA2
CA3
CA4
CA5
Polynomial fit with software program to get best value of ai
n-th order polynomial
C A  a0  a1t  a 2 t 2  ...  a n t n
Differential equation
dCA
 a1  2at  3a 2 t 2  ...  nan t n 1
dt
How to Get –dCA/dt (Polynomial Fit)
3rd-order
polynomial
5th-order
polynomial
Finding the Rate Law Parameter
Example 5-1: Determining the Rate Law
The reaction of triphenyl methyl chloride (trityl) (A) and methanol (B) was carried out
in a solution of benzene and pyridine at 25oC. Pyridine reacts with HCl that then precipitates
as pyridine hydrochloride thereby making the reaction irreversible. The concentration-time
data was obtained in a batch reactor. The initial concentration of methanol was 0.5 mol/dm3.
(C6H5)3CCl (A) + CH3OH (B)  (C6H5)3COCH3 (C) + HCl (D)
Time (min)
CA (mol/dm3) x 103
0
50
50
38
100
30.6
150
25.6
200
22.2
250
19.5
300
17.4
CAo=0.05 M
1. Determine the reaction order with respect to trityl (A)
2. In a separate set of experiments, the reaction order with respect to methanol was found
to be first order. Determine the specific reaction rate constant.
Solution: See the text (p.261-266)
Integral Method
 The integral method uses a trial-and-error procedure to find rxn order
- if the order we assume is correct, the appropriate plot of the
concentration-time data should be linear.
 It is important to know how to generate linear plots of functions of CA
versus t for zero-, first-, and second-order reactions.
 The integral method is used most often when the reaction order is
known and it is desired to evaluate the specific reaction rate constants
at different temperatures to determine the activation energy.
 Finally we should also use the formula to plot reaction rate data in
terms of concentration vs. time for 0, 1st, and 2nd order reactions.
zero order
first order
second order
dCA
 k
dt
dCA
 kC A
dt
dCA
2
 kC A
dt
@ t  0, C A  C A0
@ t  0, C A  C A0
@ t  0, C A  C A0
C A  C A0  kt
ln
CA
 kt
C A0
1
1

 kt
C A C A0
C A0
slope = -k
CA
ln
t
CA
C A0
slope = k
t
1
CA
slope = k
t
The rxns are zero, first, and second order respectively since the plots are linear.
Example 5-2
Use the integral method to confirm that the reaction is second order with respect
to trityl(A) as described in example 5-1 and to calculate the specific reaction rate k'.
Trityl(A) + Methanol (B)  Products
dCA
' A2
 kC
dt
@ t  0, C A  C A0
1
1

 kt'
C A C A0
Time (min)
CA (mol/dm3)
1/CA (dm3/mol)
0
50
0.05 0.038
20
26.3
100
0.0306
32.7
150
0.0256
39.1
200
0.0222
45
250
300
0.0195 0.0174
51.3
57.5
Example 5-2
Nonlinear Regression (Nonlinear Least-Squares Analysis)
-Minimize the sum of squares of the differences
between the measured values and the calculated values
• N experiments
N
(rim  ric )
s2
 

N  K i 1 N  K
2
N = number of runs
K = number of parameters to be determined
rim = measured reaction rate for run i (i.e., -rAim)
ric = calculated reaction rate for run i (i.e., -rAic)
Nonlinear Regression (Nonlinear Least-Squares Analysis)
Nonlinear Least-Squares Analysis
Constant-volume Batch reactor
Searching
technology
dCA
 kC A
dt
C 1A0  C 1A  (1   )kt

CA  C
1
A0
 (1   )kt
N
s   (C Ami C Aci )
2
i 1

1 /(1 )
N
2
 
i 1
Find  and k that will make
s2 a minimum
C Ami
C
1
A0
 (1   )kt 
1 /(1 )
i
2 (5-19)
Nonlinear Regression (Nonlinear Least-Squares Analysis)
Method of Initial Rates
 The presence of a significant reverse reaction
 Initial concentration CA0
- a series of experiments is carried out at different initial concentrations.
 Initial rate of reaction –rA0
- can be found by differentiating the data and extrapolating to zero time
 Form of the rate law
 rA0  kC A0
- the slope of the plot of ln(-rA0) versus lnCA0 is the reaction order a
Example 5-4
Example 5-4
Example 5-4
Example 5-4
Method of Half-Lives
A  products (irreversible)
 The half-life of a reaction
= the time it takes for the
concentration of the reactant
to fall to half of its initial value

dCA
 rA  kCA
dt
 C  1 
1  1
1 
1
  1   1    1
 A0   1
t
k (  1)  C A
C A0  kCA0 (  1)  C A 


If two reactants are involved in the
rxn, the experimenter will use the
method of excess.
ln t1/ 2
Slope=1-
t  t1/ 2
t1/ 2
2 1  1  1 
  1 

k (  1)  C A0 
ln t1/ 2
ln CA0
1
when C A  C A0
2
2 1  1
 ln
 (1   ) ln C A0
k (  1)
Differential Reactors
• Most commonly used catalytic reactor to obtain experimental data
- use to determine the rate of reaction as a function of either
concentration or partial pressure
- the conversion of the reactants in the bed is very small
- gradientless
- spatially uniform
FA0
catalyst
FAe
DL
Inert filling
Differential Reactors
Steady State Mole Balance on Reactant A
 flow  flow
rate   rate  

 

in  out 
DL
FA0
FAe
CA0
FP
CP
rate of

 generation


rate of



accum uation 
 rate of reaction

(m ass of cat.)  0
[ FA0 ]  [ FAe ]  
 m ass of cat. 

FA 0
 FAe
 (rA' )((DW)
W)  0
DW
 rA' 
FA0  FAe
DW
r 
'
A
Reactor Design Equation
 0 C A0  C Ae
DW
FA0 X FP
r 

DW
DW
'
A
(5-27)
(5-28)
Differential Reactors
For constant volumetric flow
Can be determined
r 
'
A
 0C A0 C Ae
DW
known

 0 (C A0  C Ae )  0CP
DW

Measuring
the product
concentration
DW
known
- using very little catalyst and large volumetric flow rates
(CA0  CAe ) ~ 0
 rA'  rA' (CAb )
where CAb the concentration of A within the catalyst bed
- the arithmetic mean of the inlet and outlet concentration:
- very little reaction takes place within bed:
CAb ~ CA0
 rA'  rA' (CA0 )
C Ab
C A0  C Ae

2
Evaluation of Laboratory Reactor
The successful design of industrial reactors lies primarily with the reliability of the
experimentally determined parameters used in the scale-up.
CRITERIA USED TO EVALUATE LABORATORY REACTORS
1. Ease of sampling and product analysis
2. Degree of isothermality
3. Effectiveness of contact between catalyst and reactant
4. Handling of catalyst decay
5. Reactor cost and ease of construction
Evaluation of Laboratory Reactor (Types of Reactors)
Integral reactor
(Fixed bed)
Stirred-Batch Reactor
catalyst slurry
Stirred Contained-Solids Reactor
(SCSR)
Continuous-Stirred Tank Reactor
(CSTR)
Recirculating transport reactor
Straight-through transport reactor
Evaluation of Laboratory Reactor (Reactor Ratings)
Reactor type
Sampling
Isothermality F-S contact Decaying Catalyst Ease of construction
Differential
P-F
F-G
F
P
G
Fixed bed
G
P-F
F
P
G
Stirred batch
F
G
G
P
G
Stirred-contained solids
G
G
F-G
P
F-G
Continuous-stirred tank
F
G
F-G
F-G
P-F
Straight-through transport
F-G
P-F
F-G
G
F-G
Recirculating transport
F-G
G
G
F-G
P-F
Pulse
G
F-G
P
F-G
G
G=Good; F=Fair; P=Poor
Homework
P5-9B
P5-12A
P5-13B
Due Date: Next Week