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Dynamic Simulation and Optimization using EMSO

Argimiro R. Secchi

– Lecture 5 –Simulation of tubular reactors.

Solutions for Process Control and OptimizationPEQ/COPPE-UFRJ

January, 2013

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Dynamic Modeling of Tubular ReactorsDynamic Modeling of Tubular Reactors

Assumptions:

• Constant physical properties (, , cp);

• Newtonian fluid;

• Angular symmetry;

• vr = v = 0

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Model classification based on physical-chemistry principles

Microscopic Model

Multiple Gradients Model

Maximum Gradient Model

Macroscopic Model

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Microscopic Model

( )( , ) ( ) ( D )li iz i i i

C Cv r t C v C Rt z

( ) ( ) ( ) ( ) ( )( , ) ( ) ( ) ( )l l l t tp p z p v v r

T Tc c v r t c v T k T St z

( ) 2[ ] lv v v v v P v gt

( ) ( )t ti i iv c J C D

( ) ( )t tpc v T q k T

( ) ( )t tv v v

Turbulence Model (simple example):

( . ) 0v v v r tz z ( , )

Computational Fluid Dynamic (CFD)

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Recommended Numerical MethodRecommended Numerical Method

Finite Volume:Consist in carry out balances of properties in elementary volumes (finite volumes), or in an equivalent form in the integration over the elementary volume of the differential equation in the conservative form (or divergent form, where the fluxes appearing in the derivatives).

Use of CFD software (Computational Fluid Dynamics)

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Simple example using finite-volume method:Reaction-diffusion equations of a spherical catalytic particle

2 22

0

0

1 ( , )

0 ; (1, ) 1

( ,0)

r

C Cr C r tt r r rC C tr

C r C

2 2 2 2e e e

w w w

C Cr dr r r C drt r

First-order selections

24dV r dr

7

2

22 2

2

3

e

ew

p ewe w

w

C r drC C r dr

r rr dr

2 2

3 3

3 ep

pwe w

dC Cr Cdt rr r

mean value

e w

E p p W

e wr r r r

C C C CC Cr r r r

2pW W p p E E p

dCA C A C A C C

dt

2

3 33 w

We w w

rAr r r

2 2

3 33 e w

pe we w

r rAr rr r

2

3 33 e

Ee w e

rAr r r

p = 2, ..., N 1

Simple example using finite-volume method:

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0wr

Cr

1

e

e p p

f fr

C C CCr r r

dC AC bdt

Resulting systems: where A is a tridiagonal matrix

Boundary conditions

( )dC F Cdt

Or a non-linear system for reaction of order 1:

Simple example using finite-volume method:

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Multiple Gradients Model

( , ) ( D )i iz i i

C Cv r t C Rt z

( , ) ( )p p z v rT Tc c v r t k T St z

( ) ( )t l D D D( ) ( )t lk k k ( ) ( )t l

2zz

v P v gt

Boundary Conditions:

reaction zonevz, Ci0, T0

z = 0 z = Lz = 0 ¯ z = 0+ z = L¯ z = L+

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2

2

1D ( , ) D ( , ) ( , )i i i iz R z i

C C C Cr t r r t v r t Rt z r r r z

0(0, , )( , ) ( ) ( , ) (0, , ) D ( , ) i

z i z i zC r tv r t C t v r t C r t r t

z

Cz

L r ti ( , , ) 0

Cr

z ti ( , , )0 0

Cr

z R ti ( , , ) 0

_( , , 0) ( , )i i inC z r C z r

no reaction at exit

Symmetry

Initial condition

Impermeable wall

Component Mass Balance(removing time average notation)

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( , , ) 0T L r tz

( ,0, ) 0T z tr

( , , 0) ( , )inT z r T z r

No reaction at exit

Symmetry

Initial condition

Heat exchange with the wall

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2

1( , ) ( , ) ( , )p z R p z r AT T T Tc k r t r k r t c v r t H Rt z r r r z

0( , ) (0, , )( , ) ( ) ( , ) (0, , ) z

z zp

k r t T r tv r t T t v r t T r tc z

( , ) ( , , ) ( , , )R wTk R t z R t U T T z R tr

Energy Balance(removing time average notation)

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(0, ) 0zv tr

_( ,0) ( )z z inv r v r

Symmetry

Initial Condition

Fixed wall

1z zv P vrt z r r r

v R tz ( , ) 0

Momentum Balance(removing time average notation)

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2

2

DDi i i iRL z i

C C C Cr v Rt z r r r z

2

2R

p L p z r AT T k T Tc k r c v H Rt z r r r z

Additional assumptions:

- constant effective diffusive coefficients

- constant velocity

( , , ) 0T L r tz

( ,0, ) 0T z tr

( , , 0) ( , )inT z r T z r

0(0, , )( ) (0, , ) L

z zp

k T r tv T t v T r tc z

( , , ) ( , , )R wTk z R t U T T z R tr

0(0, , )( ) (0, , ) D i

z i z i LC r tv C t v C r t

z

Cz

L r ti ( , , ) 0

Cr

z ti ( , , )0 0

Cr

z R ti ( , , ) 0

_( , ,0) ( , )i i inC z r C z r

2

2

DDi i i iRL z i

C C C Cr v Rt z r r r z

2

2R

p L p z r AT T k T Tc k r c v H Rt z r r r z

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Method of LinesMethod of Lines

- spatial discretization (finite differences, finite volumes, finite elements)

- time integration

Example: finite differences in axial direction and orthogonal collocation in radial

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, , , 1, , , , 1, , 1, , 1,, , , , , ,2

0

2D 4 D

2

ni j k i j k i j k i j k i j k i j k

L R k k m k m i j m z i j km

dC C C C C Cu B A C v R

dt z z

1

, 1, , 1, 1, 1,, , , , ,2

0

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2

nj k j k j k j k j k j k

p L R k k m k m j m p z r A j km

dT T T T T Tc k k u B A T c v H R

dt z z

.( )m k

k mdl uA

du

2

, 2

( )m kk m

d l uBdu

1

0( )

np

mp m pp m

u ul u

u u

2u rwhere:

j = 1,2,...,N k = 1,2,...,n1

( , )1

0( ) ( ) ( )

n

n m mm

y u P u l u y

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,2, 00 ,1,

( )( ) D

2i k i

z i z i k L

C C tv C t v C

z

, 1, , , 0i N k i N kC C

1

0, , ,0

0n

m i j mm

A C

1

1, , ,0

0n

n m i j mm

A C

, , _ , ,i j k i in j kC C

Method of LinesMethod of Lines

Boundary conditions

1, , 0N k N kT T

1

0, ,0

0n

m j mm

A T

, , ,j k in j kT T

2, 00 1,

( )( )

2kL

z z kp

T T tkv T t v Tc z

1

1, , , 10

n

R n m j m w j nm

k A T U T T

j = 1,2,...,N k = 1,2,...,n

Results in a DAE system!

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Another Multiple Gradients Model(ignoring radial gradients) – PFR with axial dispersion

2

2Di i iL z i

C C Cv Rt z z

2

2

2p L p z w r A

T T T Uc k c v T T H Rt z z R

0(0, )( ) (0, ) D i

z i z i LC tv C t v C t

z

( , ) 0iC L tz

_( ,0) ( )i i inC z C z

( , ) 0T L tz

( ,0) ( )inT z T z

0(0, )( ) (0, ) L

z zp

k T tv T t v T tc z

0

0

( , , )( , )

R

i

i R

C r z t r drC z t

r dr

0

0

( , , )( , )

R

R

T r z t r drT z t

r dr

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, , 1 , , 1 , 1 , 1,2

2D

2i j i j i j i j i j i j

L z i j

dC C C C C Cv R

dt z z

1 1 1 1,2

2 22

j j j j j jp L p z w j r A j

dT T T T T T Uc k c v T T H Rdt z z R

j = 1,2,...,N

,2 00 ,1

( )( ) D

2i i

z i z i L

C C tv C t v C

z

, 1 , 0i N i NC C

, _ ,i j i in jC C

1 0N NT T

,j in jT T

2 00 1

( )( )2

Lz z

p

T T tkv T t v Tc z

Method of LinesMethod of Lines

- spatial discretization (F.D., F.V., orthogonal collocation on finite elements)

- time integration

Example: finite differences in axial direction

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Maximum Gradient Model(ignoring axial dispersion) – PFR without axial dispersion

i iz i

C Cv Rt z

2p p z w r A

T T Uc c v T T H Rt z R

0(0, ) ( )i iC t C t

_( ,0) ( )i i inC z C z

( ,0) ( )inT z T z

0(0, ) ( )T t T t

Results in an EDO system!

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Macroscopic Model

0 ( )ii z i z i

dCV C t v S C v S R Vdt

0 ( ) ( )p p z p z t w r AdTc V c v S T t c v S T U A T T H R Vdt

_(0)i i inC C

(0) inT T

0

0

( , )( )

L

i

i L

C z t S dzC t

S dz

0

0

( , )( )

L

L

T z t S dzT t

S dz

Results in a EDO system!

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Method of Orthogonal Collocation with EMSO

EMSOSimulator

RootsA and B Matrices

• Boundaries• Internal Points• Alfa and Beta

• Jacobi roots• A and B matrices

Plugin

DD as Plugin (Type=“OCFEM”, Boundary="BOTH”, InternalPoints=5 alfa=1, beta=1)

Plugin: ocfem_emso.dll

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Fixed-bed Reactor with Axial Dispersion(reaction of order m)

2

2

1 my y y Da yx Pe x

Boundary conditions:

0

1 1 ( ,0)x

y yPe x

1

0x

yx

( ,0) 1y or

Initial conditions:

(0, ) 0y x

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Example: add Plugin ocfem_emso.dll and execute flowsheets of files FDM_ss.mso, OCM_ss.mso and OCFEM_ss.mso, and compare results of discretizations. Repeat for the dynamic simulation in files FDM_din.mso e OCM_din.mso.

PDEMethod of Lines: D.F. and Orthogonal Collocation

PDEMethod of Lines: D.F. and Orthogonal Collocation

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Comparing Results

OCM by EMSONumber of internal points: 5

y(x=1) = 0.151475 (error of 0.038%)

1 1

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

0.0000 0.2000 0.4000 0.6000 0.8000 1.00000.0000

0.2000

0.4000

0.6000

0.8000

1.0000

1.2000

0.0000 0.2000 0.4000 0.6000 0.8000 1.0000

Method of Finite DifferencesNumber of internal points: 6000

y(x=1) = 0.15155 (error of 0.087%)

y(x=1) = 0.151418 (exact)

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Case Study

• Production of acetic anhydride in adiabatic PFR– Acetic anhydride is often produced by reacting acetic acid with ketene,

obtained by heating acetone at 700-770oC.– A important step is the vapor phase cracking of acetone to ketene and

methane:

– The second step is the reaction of ketene with acetic acid.

Ref: G. V. Jeffreys, A Problem in Chemical Engineering Design: The Manufacture of Acetic Anhydride, 2nd ed. (London: Institution of Chemical Engineers, 1964)

3 3 2 4C H C O C H C H C O C H

2 3 3CH CO CH COOH CH CO O

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34222exp 34.34kT

Problem Definition – The first production step is carried out in a vapor phase reaction

of acetone in an adiabatic PFR.

where A = acetone; B = ketene and C = methane

– The reaction is of 1a order in relation to acetone in the cracking reaction, with Arrhenius constant given by:

• k – seconds-1

• T – Kelvin

CBA

Case Study

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Process Description

– Reactor geometry• adiabatic continuous tubular reactor;• bank of 1000 tubes of 1 in sch. 40 with cross section of 0.557 m2;• total length of 2.28 m;

– Operating conditions• feed temperature 762oC (1035 K);• operating pressure: 1.6 atm• feed flow rate of 8000 kg/h (137.9 kmol/h);

– Composition• acetone, ketene and methane• feed of pure acetone

– Kinetics• first order reaction, • pre-exponential factor (k0): 8.2 x 1014 s-1

• activation energy (E/R): 34222 K• heat of reaction: -80.77 kJ/mol

Case Study

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Example: run FlowSheet in file PFR_Adiabatico.mso and plot steady-state temperature and composition profiles. Show also the evolution of the temperature profile. Discuss the type and quality of discretization.

Case study– Production of acetic anhydride –

Case study– Production of acetic anhydride –

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ExerciseExerciseSolve the reaction-diffusion problem in a spherical catalytic particle, given by:

2 2 1/ 22

1y yr yt r r r

0

0r

yr

1( , ) 1

ry t r

0( , ) 0

ty t r

2 (Thiele modulus)

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ReferencesReferences• Himmelblau, D. M. & Bischoff, K. B., "Process Analysis and Simulation - Deterministic Systems", John Wiley & Sons, 1968.• Finlayson, B. A., "The Method of Weighted Residuals and Variational Principles with Application in Fluid Mechanics, Heat and Mass Transfer", Academic Press, 1972.• Villadsen, J. & Michelsen, M. L., "Solution of Differential Equation Models by Polynomial Approximation", Prentice-Hall, 1978.• Davis, M. E., "Numerical Methods and Modeling for Chemical Engineers", John Wiley & Sons, 1984.• Denn, M., "Process Modeling", Longman, New York, 1986.• Luyben, W. L., "Process Modeling, Simulation, and Control for Chemical Engineers", McGraw-Hill, 1990.• Silebi, C.A. & Schiesser, W.E., “Dynamic Modeling of Transport Process Systems”, Academic Press, Inc., 1992.• Biscaia Jr., E.C. “Método de Resíduos Ponderados com Aplicação em Simulação de Processos”, XV CNMAC, 1992• Ogunnaike, B.A. & Ray, W.H., “Process Dynamics, Modeling, and Control”, Oxford Univ. Press, New York, 1994.• Rice, R.G. & Do, D.D., “Applied Mathematics and Modeling for Chemical Engineers”, John Wiley & Sons, 1995.• Maliska, C.R. “Transferência de Calor e Mecânica dos Fluidos Computacional”, 1995.• Bequette, B.W., “Process Dynamics: Modeling, Analysis, and Simulation”, Prentice Hall, 1998.• Fogler, H.S., “Elementos de Engenharia de Reações Químicas”, Prentice Hall, 1999.

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For helping in the preparation of this material

Special thanks to

For supporting the ALSOC Project.

Prof. Rafael de Pelegrini Soares, D.Sc.Eng. Gerson Balbueno Bicca, M.Sc.Eng. Euclides Almeida Neto, D.Sc.Eng. Eduardo Moreira de Lemos, D.Sc.Eng. Marco Antônio Müller

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... thank you for your attention!

Process Modeling, Simulation and Control Lab• Prof. Argimiro Resende Secchi, D.Sc.• Phone: +55-21-2562-8307• E-mail: arge@peq.coppe.ufrj.br• http://www.peq.coppe.ufrj.br/Areas/Modelagem_e_simulacao.html

http://www.enq.ufrgs.br/alsoc

EP 2013

Solutions for Process Control and Optimization