Lecture 11

Chemical Reaction Engineering (CRE) is the

field that studies the rates and mechanisms of
chemical reactions and the design of the
reactors in which they take place.
 Block 1: Mole Balances
Block 2: Rate Laws
Block 3: Stoichiometry
Block 4: Combine

 Determining the Rate Law from Experimental Data

Integral Method
Differential (Graphical) Method
Nonlinear Least Regression

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 Integral Method
Consider the following reaction that occurs in a constant
volume Batch Reactor: (We will withdraw samples and
record the concentration of A as a function of time.)
A  Products
dN A
Mole Balances:  rAV
dt

Rate Laws: rA  kCA


Stoichiometry: V  V0
dCA
Combine:    kCA
3
dt

 Finally we should also use the formula to plot reaction
rate data in terms of conversion vs. time for 0, 1st and
2nd order reactions.
Derivation equations used to plot 0th, 1st and 2nd order
reactions.
These types of plots are usually used to determine the
values k for runs at various temperatures and then used
to determine the activation energy.

Zeroth order First Order Second Order

dCA dCA dCA
 rA  k  rA  kC A  rA  kC A2
dt dt dt

at   t  0,  CA  CA 0 at   t  0,  CA  CA 0 at   t  0,  CA  CA 0

CA 0  1 1
 CA  CA 0 
 kt  ln  kt    kt
4 CA  CA C A 0
 
 Integral Method
Guess and check for α = 0, 1, 2 and check against
experimental plot.
 0  1  2
 C A0  1 1
rA  C A0  kt ln    kt   kt
 CA  C A C A0
ln(CA0/CA)
CA

1/CA
t t t
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 Differential Method
 dC A 
Taking the natural log of   kC A 
 dt 
 dC A 
ln     ln k   ln C A
 dt 
 dCA 
The reaction order can be found from a ln-ln plot of:    vs   CA
 dt 
dC A ln

dt

  dC A 
dC A  

dt Slope = α  dt  p
P
k
C Ap
ln
6 C AP CA
 Methods for finding the slope of log-log and semi-log
graph papers may be found at

http://www.physics.uoguelph.ca/tutorials/GLP/

However, we are usually given concentration as a

function of time from batch reactor experiments:
time (s) 0 t1 t2 t3
concentration CA0 CA1 CA2 CA3
(moles/dm3)

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Three ways to determine (-dCA/dt) from concentration-time data
 Graphical differentiation
 Numerical differentiation formulas
 Differentiation of a polynomial fit to the data
1. Graphical

CA

t

8
t
 CA

t

dC A 
 
 dt  0
dC A 
 
dt t1
dC A 
 
dt  t2

t
0 t1 t2
9 The method accentuates measurement error!
 Example – Finding the Rate
Law
t(min) 0 1 2 3

CA(mol/L) 1 0.7 0.5 0.35

C A
 0.3 0.2 0.15
t

C A Areas equal for both


t .3 sides of the histogram
.2
.1

t
1 2 3
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 Example – Finding the Rate
Law
C A
Find f(t) of  using equal area differentiation
t
CA 1 0.7 0.5 0.35
-dCA/dt 0.35 0.25 0.175 0.12

Plot (–dCA/dt) as a function of CA

ln

dCA/dt
Slope = α
ln
11 CA
 Example – Finding the Rate
Law
Choose a point, p, and find the concentration and
derivative at that point to determine k.

ln dCA/dt
 dC A 
 
 dt  p  dC A 
  Slope = α
k  dt  p
C Ap
ln CA
CA p

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 Non-Linear Least-Square
Analysis
We want to find the parameter values (α, k, E) for
which the sum of the squares of the differences, the
measured rate (rm), and the calculated rate (rc) is a
minimum.

2 
n
 Cim  Cic  2 
S2
i 1 N K N K
2
That is, we want  to be a minimum.
N= number of runs
K= number of parameters to be determined
Cim= 
measured conc for run i
13 Cic= calculated conc for run i
 Non-Linear Least-Square
Analysis
For concentration-time data, we can combine the
r
mole balance equation for A  kC 
toA obtain:
dC A
  kC A
dt
t  0 C A  C A0

C1A0  C1A  (1   )kt

Rearranging to obtain the calculated concentration

as a function of time, we obtain:
1 1/(1 )
C Ac  C A  [C A0  (1   )kt ]
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 Non-Linear Least-Square
Analysis
Now we could use Polymath or MATLAB to find the values of α and k
that would minimize the sum of squares of differences between the
measured (CAm) and calculated (CAc) concentrations.
That is, for N data points,

Similarly one can calculate the time at a specified concentration, t c

and compare it with the measured time, tm, at that same concentration.
That is, we find the values of k and α that minimize:

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 Non-Linear Least Squares
Analysis
Guess values for α and k and solve for measured
data points then sum squared differences:
CAm 1 0.7 0.5 0.35
CAc 1 0.5 0.33 0.25
(CAc-CAm) 0 -0.2 -0.17 -0.10
(CAc-CAm)2 0 0.04 0.029 0.01 0.07

for α= 2, k = 1 → s2 = 0.07
for α = 2, k = 2 → s2 = 0.27
16 etc. until s2 is a minimum
 Non-Linear Least Squares
Analysis

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 Non-Linear Least Squares
Analysis

 
N N
1 1 2
s2   CAmi  CAci   CAmi  C1
A 0  1   kt i 
2 

i1 i1

We find the values of alpha and k which minimize s2

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Minimum Sum of Squares
20
21
 Residuals

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23
 Method of initial rates
Differential method of data analysis is
one of the easiest method
However, other effects (reverse reaction),
could render differential method
ineffective
In such cases, methods of initial rates can
be used to determine reaction order and
specificofrate
Series constants.are carried out at
experiments
different initial conc (CA0) and initial rate
of the reaction (-rA0) is determined for
each
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run. -rA0=
 Example 5-4 Method of initial rates in
solid-liquid dissolution kinetics

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 Example 5-4 Method of initial rates in
solid-liquid dissolution kinetics

Concentration-time data
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 Example 5-4 Method of initial rates in
solid-liquid dissolution kinetics

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 Example 5-4 Method of initial rates in
solid-liquid dissolution kinetics

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 Method of half-lives (t1/2)
t1/2 is the time it takes for the
concentration of the reactant to fall to
half
The of its initial
method of value.
half lives requires many
experiments
When two reactants are involved, half life
method is used in conjunction with
method of excess to get rate law of the
form
-rA= kCA

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 Method of half-lives (t1/2)
For irreversible reaction
A Products
Mole balance on species A in constant
volume batch reactor
-dCA/dt=-rA=
kCA

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 Method of half-lives (t1/2)
For irreversible reaction
A Products
Mole balance on species A in constant
volume batch reactor
-dCA/dt=-rA=
kCA

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 Method of half-lives (t1/2)
For irreversible reaction
A Products
Taking natural log of both sides

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 Differential reactors
Differential reactor is normally used to
determine rate of reaction as a function
of
It either conc
consists of or partial
tube pressurevery small
containing
amount catalyst in the form of thin wafer
or disk
The criterion for reactor being differential
are
- Conversion of reactant in the bed is
extremely small
- Reactor is considered to be
It gradientless
is easy to construct at low cost
33- Reactor is considered to be isothermal
 Differential reactors
AP

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 Differential reactors
AP

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 Differential reactors
AP
Mole balance in terms of concentration

Differential reactor design

equation

In terms of conversion or
product flowrate, Fp

In terms of concentration
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product
 Differential reactors
AP
Mole balance in terms of concentration

Rate as a function of conc of

A in the bed
Arithmatic mean of inlet &
outlet conc
Bed conc is essentially the
inlet conc
Rate as a function of CA0
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 Criteria for evaluation of
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

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 Types of reactors

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 Summary of reactor rating

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