Collection & Analysis of Rate Data

CH5: Collection & Analysis of Rate Data

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Chemical Reaction Engineering Methodology


CH3: Elements of Chemical Reaction EngineeringH. Scott Fogler (4th Edition)

Chemical Reaction Engineering Methodology


CH3: Elements of Chemical Reaction EngineeringH. 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


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

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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


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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 @

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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




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• Our final table must include:Initial Point Intermediate Points

FinalPoint

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 @www.ChemicalEngineeringGuy.com/Courses


D.M. Polynomial Fit

• Fit a polynomial to the curveCA= a0+a1·t + a2t2 + … antn

• Derivative of that functiondCA/dt= a1+ 2a2t + … n·antn-1

• Plot dCA/dt vs. t

Normally done in Excel


D.M. Polynomial Fit





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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

• Our final table must include:

• 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 = orderRound to nearest number!


Differential Method: Example





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• Part 1: Determine Rate of Reaction

1. Postulate a Rate Law2. Select Reactor Type3. Process Data in terms of measured data4. Simplify as possible5. 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

αCBβ

• Select Rector Type BatchIts already set…

• Process Data in meassurable variable (CA)No need to re-process/ re-calculate

• Simplify if possibleConcentration 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 = -kb or y-intercept = CA0


• This Equation must be integrated:



Slope = k

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

Integral Method: Second Order


Slope = +kb or y-intercept = 1/CA0

Integral Method: Unknown Order

• Failing to model…

• Probably a First order? Even a Zeroth Order…

Integral Method: Example

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End of Topic

Integral Method

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

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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

Half-Live Methods

Half-Live Methods

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

www.ChemicalEngineeringGuy.com–Go to the Courses!• Select Reactor Engineering Course– Solved Problems Section!» Chapter 5 – Collection & Analysis of Rate Data




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End of Topic

Half-Lives Method

Section 2

Differential Reactors Data Analysis


General Methodology: Differential Reactor

1. Postulate a Rate Law– Power Law –rA = k·Ca

n

– Special Rate Laws known: Langmuir models, etc

2. Select Reactor Type (Differential) 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: 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




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• This and more solutions can be seen in the course of Reactor Engineering

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–Go to courses• Reactor Engineering– Solved Problems Section» CH5 – Collection and Anlysis of Rate Data


End of Topic

Differential ReactorPBR: 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…

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Text Book & Reference

Essentials of Chemical Reaction EngineeringH. Scott Fogler (1st Edition)

Chemical Reactor Analysis and Design Fundamentals J.B. Rawlings and J.G. Ekerdt (1st Edition)

Elements of Chemical Reaction EngineeringH. Scott Fogler (4th Edition)


Bibliography

Elements of Chemical Reaction EngineeringH. Scott Fogler (4th Edition)


We’ve seen CH5

Collection & Analysis of Rate Data

Engineering

differential method

excess method www

excess method advantage

excess method notes

reactor engineering

com ch3

pbr www

problems www