Collection & Analysis of Rate Data
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Transcript of Collection & Analysis of Rate Data
CH5: Collection & Analysis of Rate Data
RE5Chemical Engineering Guy
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Chemical Reaction Engineering Methodology
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CH3: Elements of Chemical Reaction EngineeringH. Scott Fogler (4th Edition)
Chemical Reaction Engineering Methodology
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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
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Section 1
Batch Reactor Data Analysis
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Batch Reactor Data Analysis
• Typical for homogeneous (liquid, gas)• We collect data for– Concentration– Pressure– Volume– Time
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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
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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
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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
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Excess Method
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Excess Method
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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’www. Chemical Engineering Guy .com
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
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Differential Method
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Differential Method: The Order α
• Remember that the slope, m is equal to the order
m = α
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Differential Method: The Rate Constant
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Differential Method
• One detail…• We need to find the value of dCa/dt• We could do it as:– Graphical Method– Numerical Method– Polynomial Fit
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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
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D.M. Graphical Method
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D.M. Graphical Method
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D.M. Graphical Method
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D.M. Graphical Method
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D.M. Graphical Method
• Our final table must include:
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D.M. Graphical Method
• An example is shown after the next two methods are explained– Numerical Method– Polynomial Fit
• Or go to the Course @
www.ChemicalEngineeringGuy.com/Courses
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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!
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D.M. Numerical Method
• 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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D.M. Numerical Method
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D.M. Numerical Method
• Our final table must include:Initial Point Intermediate Points
FinalPoint
Difference of time must be EQUAL!
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D.M. Numerical Method
• Then we apply the Differential Method
Order of reaction Constant Rate
D.M. Numerical Method
• An example is done after the next method is explained:– Polynomial Fit
• Or go to the Course @www.ChemicalEngineeringGuy.com/Courses
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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
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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!
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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
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D.M. Numerical Method
• Once again, we can now we apply the Differential Method
Order of reaction Constant Rate
D.M. Numerical Method
• An example is shown in the next slides…• The example will show the 3 methods!
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Differential Method: Getting the values
• Either method you use (graph, numerical or polynomial) should get you the next data
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Differential Method: Review
• Now you got all “derivatives”• Its time to plot dCa/dt vs. t
m = orderRound to nearest number!
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Differential Method: Example
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Differential Method: Example
• 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
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Differential Method: Example
• 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!
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Differential Method: Example
• Now we got alfa, the order of CA…• Remember we applied Excess Method• Where we actually got k’ not k
Differential Method: Example
• 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:
Differential Method: Example
• 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
Integral Method: First Order
• This Equation must be integrated:
Integral Method: First Order
• And the plot you get is:
Slope = k
Integral Method: Second Order• This Equation must be integrated:
Integral Method: Second Order
• And the plot you get is:
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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If you are interested in getting this Slides, please send an email to:
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End of Topic
Half-Lives Method
Section 2
Differential Reactors Data Analysis
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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
Differential Reactor: PBR
• 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
Differential Reactor: PBR
Differential Reactor: PBR
• 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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Differential Reactor: PBR Example
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Differential Reactor: PBR Example
• 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
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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
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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:• [email protected]
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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)
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Bibliography
Elements of Chemical Reaction EngineeringH. Scott Fogler (4th Edition)
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We’ve seen CH5