Isothermal Reactor Design - Reactor Engineering

CH4: Isothermal Reactor Design

RE4

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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 Reactor Engineering Methodology

– In terms of Conversion– Flow Rates and Concentration

• Section 2 Batch Reactor– Batch Reactor & time of cycle

• Section 3 CSTR Design– 1 CSTR and the Dahmköhler number– Series– Parallel

• Section 4 PFR Design– Liquid-phase PFR– Gaseous-phase PFR

• Section 5 PBR Design– Pressure Drop in a PBR (one reaction)

• Section 6 Semi-Continuous Reactors– Start-up of a CSTR (Unsteady state!)– Semi-Batch Reactors (Multiple Reactions ONLY)


Section 1

Reactor Engineering Methodology


Reactor Engineering Methodology• Using Conversion in our Design Equations

– CSTR (Single Reaction)

– Batch (Single Reaction)

– PFR (Single Reaction)

– PBR (Single Reaction)

• Using Flow/Concentration for Design…– Semi-Continuous (Single and Multiple Reactions)

Due to the “Differential Equations” and many species involved…

Its easier to calculate conversion at the end


Methodology for Batch, CSTR, PFR

General Mole Balance Equation

Design Equation for Reactor Type

-ra = f(X) given?

Determine Rate Law f(CA)

Use Stoichiometry Tables

Gas-Phase with Pressure Drop

Start

Combine:• Mole Balance• Design Equation• Rate Law + Tables• Pressure DropSolve them

If no change in moles and no Pressure Drop:• Combine rate law and Stoichiometry Tables• Get –ra = f(X)

Evaluate Equations. Solve. Analyze Data.Get Final Answer

End

Yes

No

-ΔP


Methodology for PBR and SemiCont.






Start

Combine:• Mole Balance• Design Equation• Rate Law + Tables• Pressure DropSolve (Software)

Analyze Data.Get Final Answer

End

-ΔP


Relate Rates of Reaction

Section 2

Batch ReactorIsothermal Design





-ra = f(X) given?




Start




End

Yes

No

-ΔP


Revisiting the Batch

• If liquid-phase– Typical change in density may be neglected

• If gas-phase– The volume of the vessel is fixed, no change in volume

• Assumptions– Well mixed– Reactants enter at the same time– No side reactions– Filling time may be neglected (tf = 0)– Isothermal Operation


Revisiting the Batch

• For both the cases we use constant volume:

– We will use Concentrations!

• This is the form we will use for analyzing rate of reaction data in the next chapter


Example of time required for ConversionFirst Order

• Given a First Order Elementary Reaction

-ra = k·CA

• Calculate the time needed to achieve certain conversion XA=90%








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• Substitute all values

• You DON’T need initial concentration!

• Note K values!

Example of time required for ConversionSecond Order

• Given a Second Order Elementary Reaction

-ra = k·CA2

• Calculate the time needed to achieve certain conversion XA=90%









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• Substitute all Values

• It DOES depends on the Initial Concentration!

• Note the K values

Compare First vs. Second Order reaction times (Batch Reactor)

• Note on Constant Values!

• Time of reaction decreases

• K value is adapted to the rate of reaction

– Must match dimensions (time, concentration, volume, moles, etc)



• Time depends on initial concentration only for 2nd order

• Why does the 1st order does not depends on concentration!?



• Procedure

– Get the “Design Equation” for the Batch in terms of Concentration (or Conversion for 1 rxn)

– If no rate of reaction vs. Conversion is given• You need a rate law

– Substitute the rate law in the Design Equation

– Develop Mathematically• Analytical solution if possible!

– Get the answer!www. Chemical Engineering Guy .com

Example of time required for a Batch

• Imagine a third order, or even a non-elementary order…

• Try those examples to practice and compare!

• The more you practice the math behind this, the more you learn about reactions and reactors


Exercise 4-1




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Questions and Problems

• There are 33 problems in this Chapter 4.

• 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

• CH4 – Isothermal Reactor Design


http://www.ChemicalEngineeringGuy.com

Reaction Time

• We speak of the Batch and all the time required to the reactor to “react” the materials

• This “time” is actually the “reaction time”

• It is not the TOTAL time needed to perform a cycle

• Check out the Course for more problems like this!

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Cycle of a Batch


Timetf

Timeth

Cycle of a Batch


Timetr

Timete

Cycle of a Batch


Timetc

Timetk

Cycle Time

• A normal cycle goes as:– tf: time necessary to feed

– th: time necessary to heat/cool before RXN

– tr: time necessary to react that reaction

– te: time necessary to empty the reactor

– tk: time necessary to heat/cool after RXN

– tc: time necessary to clean the reactor

• The cycle starts again for a new batch:

Cycle Time = tf + th + tr + te + tk+ tc


Cycle Time

• The fraction of time required to do the actual reaction vs. the total time

– Must be near 1.0 as possible– If near 0, then the time we spend is mainly to

“prepare” the reactor for that specific reaction

• Be sure not to mix the times when given in data– Reaction time, feeding time, time needed to clean,

time required to heat, time spent in maintenance, etc.

Ratio = Time of Reaction/Time of cycle


Cycle Time Exercise

• If– tf: 25 min

– th: 2 hr

– tr: 6.7 hr

– te: 23 min

– tk: 1.2 hr

– tc: 30 min

• What is the total Cycle Time?

• What is the fraction of time of that reaction vs. batch time?






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Cycle Time Exercise


– th: 2 hr

– tr: 6.7 hr

– te: 23 min

– tk: 1.2 hr

– tc: 30 min





a) 25+120+402+23+72+30 = 672 min

b) 402/672 = 0.598




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Cycle Time Exercise


– th: 2 hr

– tr: 6.7 hr

– te: 23 min

– tk: 1.2 hr

– tc: 30 min





a) 25+120+402+23+72+30 = 672 min

b) 402/672 = 0.598

60% of time the reactor is having a reaction

40% is dead time




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Section 3

Continuous Stirred Tank Reactor Isothermal Design





-ra = f(X) given?




Start




End

Yes

No

-ΔP


Revisiting the CSTR

• Typical liquid-phase reactions!

• We will make the next assumptions:– Well mixed

– No change in volume/density

– Reactants enter at the same time

– No side reactions

– Filling time may be neglected (tf= 0)

– Isothermal Operation


Space time + CSTR

• Lets force “Space Time” into our Design Equations in the CSTR


First-Order Single CSTR





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The Dahmköhler Number

• Dimensionless number

• “quick” estimate to know the degree of conversion

• Ratio of “Rate of reaction at entrance” vs. “Entering Flow Rate of A”

• Also ratio of “rate of reaction vs. convection rate”


Dahmköhler for CSTR 1st Order


Dahmköhler for CSTR 2nd Order


Dahmköhler for CSTR nth Order

• Verify it by yourself…

• Try zeroth, third, and higher order…


The Dahmköhler Number

• Rule of Thumb for Da

– If Da > 10 Conversion may achieve 90%

– If Da < 0.1 Conversion will me max 10%

• Conversion in terms of Da Number


CSTR and Da Number• We will be using this for further analysis

• The Dahmköhler Number help us analyze “faster” and “easier” 1st and 2nd Order reactions

• Specially for Series or Parallel CST-Reactors of the same characteristics– Size– Temperature

CSTR in Series

• Suppose we got 2 CSTR

• Same Size (Volume)

• Same Temperature of Operation

• Same “k” or constant rate

• Series Arrangement (dependent of previous)


CSTR in Series

• Now lets suppose there are “n” reactors of same characteristics


CSTR in Series





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CSTR in Series





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CSTR in Series




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CSTR in Series




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CSTR in Series

• We get this equation

• Obviously, as n increases, the conversion increases

• If Da increases, conversion also increases!


CSTR in Series


Excel Spread Sheet Download in Web-Page!

Analysis of Number of Reactors

• We actually want Da increase, not n

• Da Number depends

– Volume of tank (generally fixed)

– “k” Constant… We can Increase Temperature!

– Volumetric Flow Rate We can Adjust it


Analysis of Number of Reactors

• To increase conversion… the most normal operation technique is:

– decrease volumetric flow rate (increase time in reactor)

– increase temperature


CSTR in Parallel Arrangements• Suppose we got n CSTR

• Same Size (Volume)

• Same Temperature of Operation

• Same “k” or constant rate

• Parallel Arrangement (independent of each other)


CSTR in Parallel Arrangements

• Conversion WILL be the same (same reactors)

• Rate of Reaction WILL be the same

• Therefore, you need N tanks to get the total Volume

– V = n·Vn

– Ft = n·FA0



• Its like having 3 identical reactors

• Same Volumes, Same Volumetric Flow, Same Flow Rates, Same “k” Constant

• The Total Volume 3 times that RKT volume

• Total Flow Rate 3 times that RKT Flow Rate

• For n reactors n times that…



• We’ve proved then that the parallel arrangement would be the same if we would actually have one LARGE reactor of that Volume



Exercise





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Section 4

Plug Flow ReactorIsothermal Design





-ra = f(X) given?




Start




End

Yes

No

-ΔP


Liquid Phase PFR

• We will analyze two cases

– First Order Rate Law

– Second Order Rate Law

• Get equations in terms of Conversion & Da!


Liquid Phase PFR

• Assumptions

– Plug Flow Profile

– No dispersion or radial gradients in Temp, Vel, Conc.

– No Pressure Drop and Isothermal Operation

– Steady State

– Constant Volume/Density


Liquid Phase PFR: First Order


Liquid Phase PFR: First Order





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Liquid Phase PFR: Second Order


Liquid Phase PFR: Second Order


Liquid Phase PFR: Conclusion

• Its easy because Volume is Constant

• Main “problem” the integral

• Check out for zero and third order!











Gas Phase PFR

• Typical Gas-phase operation

• Assumptions– Turbulent Flow

– Plug Flow Profile

– No dispersion

– No radial gradients in Temp, Vel, Conc.

– No Pressure Drop

– Isothermal Operation

– Steady State


Gas Phase PFR

• We will analyze:

– First Order

– Second Order

• New Model for Concentration of A


Effect of Change in moles δ: First Order

• Express this Equations in terms of Concentration

• We analyze the effect of ε which is f(δ)


Gas Phase PFR: First Order





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Integral from Appendix A-5



Conversion of A

Effect of Change in moles δ: Second Order

• Express this Equations in terms of Concentration

• We analyze the effect of ε which is f(δ)


Gas Phase PFR: Second Order





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Integral from Appendix A-7




Conversion of A

Conclusion of PFR with change in Volume


• For negative changes in volume…– You need less volume for the same conversion

• Economics favor this type of reactions

• If you are producing moles… you will need to invest in a larger reactor

• Volume of Reactor Changes dramatically when Second order

– Due to the exponent (square) in Concentration!

Conclusion of PFR with change in Volume


• For higher conversion, the Volume Required goes exponential

Change in Volume “punished” by factor of 1

Change in Volume “punished” by factor of 2 twice!

Exercise 4-3 for PFR

• 4-3





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Section 5

Packed Bed ReactorIsothermal Design








Start



End

-ΔP



PBR Revisited

• Typical gas-solid phase reactions

• Packed Bed (catalyst on it)

• Liquid-solid may also be used… there is no Pressure Drop


PBR Revisited

• Gas-Solid interaction Drop of Pressure

• Drop Pressure due to the friction of solid-gas

• The higher the velocity, the higher the -ΔP


PBR Mole Balance + First Order

• Let’s Suppose we have a 1st order rate law.

• Get the Design Equation of a PBR in terms of Conversion/Mass of Catalyst






PBR Mole Balance + Second Order

• Let’s Suppose we have a 2nd order rate law.

• Get the Design Equation of a PBR in terms of Conversion/Mass of Catalyst





PBR Revisited

• One small detail…

• P changes as Conversion advances

• This conversion is dependent of the mass of catalyst


PBR Revisited

• Pressure is a key factor now!

• We will need to model the Pressure Drop so we can accurately use this equation

• This means Simultaneous Solving!


Accounting for Pressure Drop

• As you may remember…

We have a change in Pressure

• If pressure changes Concentration changes

• If concentration changes rate of reaction changes

• If rate of reaction changes concentration changes


First Order PBR Second Order PBR

Ergun Equation

• This is used in a packed bed or fluidized bed reactors/towers

• Models the Pressure change vs. Length of reactor/tower

• Laminar (term 1)

• Turbulent (term 2)

• Only gas density changes with Pressure Drop


Ergun Equation• Definition of variables

• dP: Differential (change) of Pressure• Dz: differential (change) of bed length• G: mass flux (mass flow per unit area)• gc: 1 for SI units (Force-weight ratio)• DP: Particle/Pellet Diameter• µ: viscosity of gas• Ρ: gas density• ϕ: Free space / Bed volume• 1-ϕ: Volume of solids / Bed volume• 150 Laminar Correction Factor• 1.75 Turbulent Correction Factor


Ergun Equation for PBR

• From Steady State Mass balance



Ergun Equation for PBR• Make a single constant!


Ergun Equation for PBR• Change “length” of catalyst vs. “mass” of

catalyst


Bulk Density vs. Solid Density of Catalyst

• Solid Density is the “normal” density you are used to

• Bulk Density includes “volume spaces”

• Bulk Density is ALWAYS less than Solid Density

• Porosity is taken into consideration in Bulk Density




• Once again, use a constant

*We will revisit this model for Multiple Reactions!www. Chemical Engineering Guy .com


• Let “y” be the P/P0



• Changing Flow Rates to Conversion



• We get this equation for One Reaction!

• For Isothermal Design


Ergun Equation for PBR• We got this equation for Pressure drop vs.

mass of catalyst

• It is a Differential Equation!

• And this equation is dP/dW=F2(X,P)– Depends on Conversion


Our PBR model

• We got a Two Coupled Differential Equation System

dX/dW = F1 (X,P)

dP/dW = F2 (X,P)

• Two Equations, Two Variables Can be solved!

• They need “initial conditions” each

• How to solve:

– Analytical Methods “By hand” (not common)

– With Software (common and easier)


Analytical Methods: PBR

• If εX may be approximated to 0…

• We get this…


Analytical Methods: PBR





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Analytical Methods: PBR• Take in mind the constant values, alpha and beta





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Numerical Methods: PBR• If εX may NOT be approximated to 0…

• Use Euler Method


Numerical Methods: PBR

• EULER Method small review

• You should know by now that method!

• If you don’t know it… check your numerical method course

• Check out Topic:

– Typical Numerical Methods for Solving Differential Equations (1st Order)


Numerical Methods: PBR

• Euler methods…

• Runge-Kutta

Software Solving: PBR• This is just an overview

• I see this type of problems in other course

• Computer Solving in Chemical Engineering


Software Solving: PBR

• Essentially:– Set all constants with values

• R = 8.314• Mass Flow = 4.5

– Set all variables to equations• Volumetric Flow = Ideal Gas law • Mass Flux = Mass Flow / Area

– Set a First Order Differential equation• F1 (Rate Law + Design Equation + Stoichiometry)• Set initial Point (e.g. X=0, W = W0)

– Set a Second Order Differential Equation• F2 (Ergun Equation for PBR)• Set Initial Point (e.g. P = P0, W = W0)

– Click “Run” to Solve in the Software


Exercise 4-4 of TextBook





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Exercise

• 4-4, 4-5, and 4-6

• Are in the course are of the web-page

– www.ChemicalEngineeringGuy.com/Courses

• Check the Reactor Engineering Course– Solved Problems Section












Section 6

Semi-Continuous ReactorIsothermal Design








Start



End

-ΔP



Quick Notes on Semi-Continuous Reactors

• Start-up of a CSTR

– Seen in this Chapter

– Helps to see the “basics”

• Semi-batch Operation for Multiple Reactions

– Not shown in this Chapter

– You need to know the fundamentals of Multiple Reactions!

– Multiple Reactions CH6


Start-up of a CSTR

• CSTR are always continuous operation

• To get to this “steady state” you need to “star-up”

• This process means

– Start from some initial conditions to final conditions

– The final conditions are the “steady state” conditions


Start-up of a CSTR

• This is done because

– New Process

– New Equipment Installed

– Quality/Maintenance shut down

– Electrical Failure Shut Down

– Scale-Up or Scale-Down


Start-up of a CSTR

• Time necessary to achieve Steady state

• Concentration and Conversion function of time!

• Analytical Solutions Zeroth and First Order Rates

• ODE Superior Orders (2nd and up)


Start-up of a CSTR

• The Mole Balance Equation “Modified”

• Conversion we cannot account it because of the accumulation!

• Use concentration (Methodology 2)

• We will suppose 99% of Steady State Concentration is when we achieve S-S


Start-up of a CSTR: 1st Order• For a First Order Reaction


Start-up of a CSTR: 1st Order

• For First Order Reactions

Start-up of a CSTR: 1st Order

Time needed for Steady State

• We will suppose 99% of Steady State Concentration is when we achieve S-S




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• k·τ>>1 then model as ts= 4.6/τ

• k·τ<<1 then model as ts = 4.6τ

Exercise: First Order

• Reaction: AB

– Elementary Rate of Reaction

– k = 2.2 dm3/ s·mol

– V rate = 0.05 dm3/s

– V0 = 2.5 dm3

– CA0 = 0.05 mol/dm3

• Calculate the time needed to achieve Steady State


Exercise: First Order

• Reaction: AB

– Elementary Rate of Reaction

– k = 2.2 dm3/ s·mol

– V rate = 0.05 dm3/s

– V0 = 2.5 dm3

– CA0 = 0.05 mol/dm3

• Calculate the time needed to achieve Steady State





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Semi-batch Reactor

• Will be analyzed after CH6: Multiple Reactions


End of Block RE4

• You are now prepared to model Isothermal Reactors with one reaction!

• You understand now when to apply the Design Equations

• You know why is it important to study rates of rection laws

• You know the methodology and why we need a Design Equation + Reaction Rate Data


End of Block RE4

• You know when it is convinient to work with conversion, mole flow, concentration

• You can model Batch in 1st, 2nd and you could model higher rates!

• Now you know Dahmlköhler Number and its importance in Reactor Engineering

• You can model CSTR with Da Number

• You can Model Multiple CSTR in Series and Parallel Arrangements


End of Block RE4

• You can model PFR in liquid and gas phase!

• You know that in PBR there is a drop in pressure

• You can model that drop of pressure with Ergun Equation!

• You know how to solve a ODE for a PBR in terms of conversion and pressure drop


End of Block RE4

• You know now how to calculate stability times for CSTR Starting up!

• For next chapter, we will analyze the Data for Rate of Reactions!











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Text Book & Reference

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

Chemical Reactor Analysis and Design FundamentalsJ.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)


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