Collection & Analysis of Rate Data

CH5: Collection & Analysis of Rate Data
RE5
Chemical Engineering Guy
www. Chemical Engineering Guy .com

Chemical Reaction Engineering Methodology
CH3: Elements of Chemical Reaction Engineering
H. Scott Fogler (4th Edition)

Content
• Section 1 Batch Reactor Data Analysis
– Excess Method
– Differential Method
• Graphical Method
• Numerical Method
• Polynomial Fit
– Integral Method
• 0, 1st, 2nd orders
– Initial Rates Method
– Half-Lives Method
• Section 2 Differential Reactors Data Analysis
– Analysis for PBR

Section 1
Batch Reactor Data Analysis

Batch Reactor Data Analysis
• Typical for homogeneous (liquid, gas)
• We collect data for
– Concentration
– Pressure
– Volume
– Time

General Methodology: Batch Reactor
1. Postulate a Rate Law
– Power Law –rA = k·Ca
n
– Special Rate Laws known: Langmuir models, etc
2. Select Reactor Type  (Batch) 
3. Process Data in terms of measured data
– Moles, Concentration, Partial Pressure
4. Simplify as possible
– If reactant in excess Concentration is constant
– If yA0 of reactant is small  ε = 0

General Methodology
5. Calculate –rA as a function of Concentration of A,
CA and determine reaction order
6. Apply any of these methods
– Excess Method
– Differential Analysis: Graphical Method
– Differential Analysis: Numerical Method
– Differential Analysis: Polynomial Fit
– Integral Method
– Initial Rates Method
– Half-Lives Method

Excess Method
• Advantage  we analyze only CA and then CB
• May be treated as CA dependent ONLY
• May then be treated as CB dependent Only

Excess Method

Excess Method
• Notes:
– You will need to do two experiments
• CA in excess with changes in CB vs. time
• CB in excess with changes in CA vs. time
You get K’’ You get K’

Excess Method: Examples
• We will show later a combined example
• IF you want to see more…
– Check out the course at:
– www.ChemicalEngineeringGuy.com/Courses
• Reactor Engineering Course
• Solved Problems

Differential Method
• Force a x-y graph
• We will analyze:
– Zero Order
– First Order
– Second Order
• We get
– Y axis  Derivative of Concentration with respect of time
– Slope  Reaction Order
– X axis  Concentration of A

Differential Method

Differential Method: The Order α
• Remember that the slope, m is equal to the
order
m = α

Differential Method: The Rate Constant

Differential Method
• One detail…
• We need to find the value of dCa/dt
• We could do it as:
– Graphical Method
– Numerical Method
– Polynomial Fit

D.M. Graphical Method
• Its visual and fast to calculate
• Not so “perfect”, actually not recommended
• Plot –ΔC/Δt vs. Time
• We mean differences and not differentials
• See appendix A-2 of textbook

• Our final table must include:

• An example is shown after the next two methods
are explained
– Numerical Method
– Polynomial Fit
• Or go to the Course @
www.ChemicalEngineeringGuy.com/Courses

D.M. Numerical Method
• Remember we need a table with the
derivatives of Concentration and time!
Difference of time must be EQUAL!

• We are going to use formulas to get those
values!
• Only possible with Δx of same “size” (equally
spaced)
– Initial Point Formula
– Interior Points
– Final Point Formula

This Material is only Available at
If you are interested in getting this Slides, please send an email to:
Contact@ChemicalEngineeringGuy.com
OR visit our VIDEO-Presentation in YouTube:
www.YouTube.com/ChemEngineeringGuy

Initial
Point
Intermediate Points
Final
Point
Difference of time must be EQUAL!

• Then we apply the Differential Method
Order of reaction Constant Rate

• An example is done after the next method is
explained:
– Polynomial Fit
• Or go to the Course @

D.M. Polynomial Fit
• Fit a polynomial to the curve
CA= a0+a1·t + a2t2 + … antn
• Derivative of that function
dCA/dt= a1+ 2a2t + … n·antn-1
• Plot dCA/dt vs. t
Normally done in Excel

D.M. Polynomial Fit

D.M. Polynomial Fit
• Input x-axis data (Concentration of lnCA)
• Input y-axis data (Derivative of Ca vs t)
• Get the polynomial fit
– “By hand” (not included)
– Using a Software (excel for example)
• Get the Equation of CA(t) = a0+a1·t+a2·t2…
• Differentiate it… get values of derivatives

D.M. Polynomial Fit
• Use the equation (already differentiated) and
substitute date into it!

D.M. Polynomial Fit
• Be sure to include many points as needed
• If the order is too low
– Will not capture all the function
• If the order is to high
– Will have peaks and valleys between points

• Once again, we can now we apply the
Differential Method
Order of reaction Constant Rate

• An example is shown in the next slides…
• The example will show the 3 methods!

Differential Method: Getting the values
• Either method you use (graph, numerical or
polynomial) should get you the next data

Differential Method: Review
• Now you got all “derivatives”
• Its time to plot dCa/dt vs. t
m = order
Round to nearest number!

Differential Method: Example

• Part 1: Determine Rate of Reaction
2. Select Reactor Type
5. Calculate –rA as a function of
Concentration of A, CA and determine
reaction order
6. Apply any of these methods

• Postulate Rate Law
-rA = k·CA
αCB
β
• Select Rector Type  Batch
Its already set…
• Process Data in meassurable variable (CA)
No need to re-process/ re-calculate
• Simplify if possible
Concentration of B is in excess!
Apply Excess Method also!

• Now we got alfa, the order of CA…
• Remember we applied Excess Method
• Where we actually got k’ not k

• Part 2  If the rate of reaction is said to be
also first order with respect to B!
• Its time to find k…
• Remember we used Excess Method!
• Lets find k’’ and then:

• Given β=1 get k’’
• Then apply:

End of Topic
Differential Method

Integral Method
• We propose the order of reaction
– Zero
– First
– Second
• If any of these is correct, the plot will be a
straight line when applied
• We integrate every equation (0, 1st, 2nd, etc)

Integral Method: First Order
• This Equation must be integrated:

Integral Method: Zero Order
• And the plot you get is:
Slope = -k
b or y-intercept = CA0

Integral Method: First Order
Slope = k

Integral Method: Second Order
• This Equation must be integrated:

Integral Method: Second Order
Slope = +k
b or y-intercept = 1/CA0

Integral Method: Unknown Order
• Failing to model…
• Probably a First order? Even a Zeroth Order…

Integral Method: Example
Check out this material in my webpage!
www.ChemicalEngineeringGuy.com

Initial Rates
• Series of experiments (not only one)
• Good for REVERSE reactions (equilibrium)
• Determines the initial rate of reaction of each
run/experiment
– Differentiate and extrapolate to 0
• Y-axis: ln(-ra0)
• X-axis: lnCA0
• Slope: order of reaction

Initial Rates: Example
Check out this material in my webpage!

End of Topic
Initial Rates Method

Half-Live Methods
• By definition: half-life of a reaction t/2 is
defined as the time it takes for the
concentration of the reactant to fall to half of
its initial value
• We will use the Method of Excess parallel to it

Half-Live Methods
• Doesn’t need to be half life… you may use any concept:
– 1/8
– 1/3
• Be sure to set the time to that fraction!
• Just substitute “n” in the “2” of the half-life formula

Half-Live Methods
• For graphing the method…
• Take out the natural logarithm in both sides
α = 1-m

Half-Live Methods: Example
• There are no examples in the book
• If you want to check out more problems go to my
web-page here
–Go to the Courses!
• Select Reactor Engineering Course
– Solved Problems Section!
» Chapter 5 – Collection & Analysis of Rate Data

End of Topic
Half-Lives Method

Section 2
Differential Reactors Data Analysis

General Methodology: Differential Reactor
– Power Law –rA = k·Ca
n
– Special Rate Laws known: Langmuir models, etc
2. Select Reactor Type  (Differential) 
– Moles, Concentration, Partial Pressure
– If reactant in excess Concentration is constant
– If yA0 of reactant is small  ε = 0

General Methodology: Differential Reactor
5. Calculate –r’A as a function of
– Concentration of A CA
– Partial Pressure of A PA
6. Solve and apply linear regression
– Given –r’A as f(CA)
– Choose a Model
– Use non-linear regression for best model fit
7. Use GOF for a correlation coefficient

Differential Reactor: PBR
• Typical for solid-fluid phase reactors
• Packed Bed Reactors suit this description
– Solid  Catalyst Bed
– Fluid  Gas inlet (reactants and products)
• Concentration or Partial Pressure are used

• Conversion is usually very small  CA is
constant
• Gradientless, spatially uniform!
• The analysis is done on the product rather
than the reactants
• Apply Excess Method if given more than 1
reactant

• These are the equations we are going to use
• After having different values of rate of reaction…
• Use Linear Regression
• Use God of Fitness to

Differential Reactor: PBR Example

Differential Reactor: PBR Example
• This and more solutions can be seen in the
course of Reactor Engineering
–Go to courses
• Reactor Engineering
– Solved Problems Section
» CH5 – Collection and Anlysis of Rate Data

End of Topic
Differential Reactor
PBR: Method

End of Block RE5
• You’re done with the chapter 5!
• You know now more information on the rate
law topics!
• You know why the Excess Method is important
• After this Chapter, you are able to propose
zero, first, second orders of reactions to
different reactions

End of Block RE5
• You know that a Batch Reactor Rate Data is useful
for CSTR, PFR as well!
• You know now many methods to find out the
reactor order
– Differential Method (graphical, numerical, polynomial
fit)
– Half-lives
– Initial Rate
– Integral Method
• You know you need a differential reactor for a
PBR

End of Block RE5
• Congratulations! You are now ready to study
chapter 6  Multiple Reactions!

Questions and Problems
• There are 16 problems in this Chapter 5.
• I also included some extra problems and
exercises
• All problems are solved in the next webpage
– www.ChemicalEngineeringGuy.com
• Courses
–Reactor Engineering
»Solved Problems Section
• CH5 – Collection & Analysis of Rate Data

More Information…
• Get extra information here!
– Directly on the WebPage:
• www.ChemicalEngineeringGuy.com/courses
– FB page:
• www.facebook.com/Chemical.Engineering.Guy
– Contact me by e-mail:
• Chemical.Engineering.Guy@gmail.com

Text Book & Reference
Essentials of Chemical
Reaction Engineering
H. Scott Fogler (1st Edition)
Chemical Reactor
Analysis and Design
Fundamentals
J.B. Rawlings and J.G.
Ekerdt (1st Edition)
Elements of Chemical
Reaction Engineering

Bibliography
Elements of Chemical Reaction Engineering
We’ve seen  CH5

Collection & Analysis of Rate Data

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