Modeling of Coupled Non Modeling of Coupled Non linear Reactor Separator linear Reactor Separator

SystemsSystemsProf S.PushpavanamProf S.Pushpavanam

Chemical Engineering DepartmentChemical Engineering DepartmentIndian Institute of Technology Indian Institute of Technology

MadrasMadrasChennai 600036 IndiaChennai 600036 India

http://www.che.iitm.ac.inhttp://www.che.iitm.ac.in

Outline of the talkOutline of the talk

► Case study of a Case study of a reactive flashreactive flash► Singularity theorySingularity theory, principles, principles► Coupled Coupled Reactor SeparatorReactor Separator systems systems Motivation for the studyMotivation for the study Issues involvedIssues involved Different Different control strategiescontrol strategies for reactor/separator for reactor/separator Mass Mass couplingcoupling, energy , energy couplingcoupling Effect of Effect of delaydelay or transportation lag or transportation lag Effect of an Effect of an azeotropeazeotrope in VLE in VLE Operating reactor under Operating reactor under fixed pressure dropfixed pressure drop ConclusionsConclusions

Industrial Acetic acid Plant

Reactive flashReactive flash

Reactive flash continued…Reactive flash continued…►Model assumptionsModel assumptions

nnthth order irreversible exothermic reaction order irreversible exothermic reaction

Reactor is modeled as a CSTRReactor is modeled as a CSTR

CSTR is operated under boiling conditionsCSTR is operated under boiling conditions

Dynamics of condenser neglectedDynamics of condenser neglected

Ideal VLE assumedIdeal VLE assumed

Model equationsModel equations

( )

(1 )A A

AF AA

dx g xx x Da

dt x

Where Where xxAA is the mole fraction of component is the mole fraction of component AA αα is ratio of activation energy of reaction to is ratio of activation energy of reaction to latent heat of vaporization latent heat of vaporization

And And ββ is related to the difference in the is related to the difference in the boiling pointboiling pointSteady state is governed by Steady state is governed by xxAfAf,,Da, Da, αα,, ββ andand n.n.

Multiple steady states in two-phase reactors under boiling Multiple steady states in two-phase reactors under boiling conditions may occur if the order of self-inhibition conditions may occur if the order of self-inhibition αα is greater is greater

than the order than the order nn of the concentration dependency of the of the concentration dependency of the reaction rate.reaction rate.

Physical cause of multiplicityPhysical cause of multiplicity

►Here a phase equilibrium driven self Here a phase equilibrium driven self inhibition action causes steady state inhibition action causes steady state multiplicity in the systemmultiplicity in the system

When the reactant is more volatile then When the reactant is more volatile then the product, then a decrease in reactant the product, then a decrease in reactant concentration causes an increase in concentration causes an increase in temperature. This causes further increase temperature. This causes further increase in reaction rate and hence results in a in reaction rate and hence results in a decrease in reactant concentration.decrease in reactant concentration.

This autocatalytic effect mentioned just This autocatalytic effect mentioned just above causes steady state multiplicityabove causes steady state multiplicity

Singularity theorySingularity theory

►Most models are non linear. The processes Most models are non linear. The processes occurring in them are non linearoccurring in them are non linear

►Non linear equations which are well Non linear equations which are well understood are polynomialsunderstood are polynomials

►Hence we try to identify a polynomial Hence we try to identify a polynomial which is identical to the nonlinear system which is identical to the nonlinear system which models our processwhich models our process

Singularity theory can beSingularity theory can beused forused for

► To determine maximum number of To determine maximum number of solutions solutions

► and to determine the different kinds of and to determine the different kinds of bifurcation diagrams , dependency of x on bifurcation diagrams , dependency of x on DaDa

► and identify parameter values and identify parameter values αα,,ββ where where the different bifurcation diagrams occur the different bifurcation diagrams occur

► Singularity theory draws analogies Singularity theory draws analogies between polynomials and non linear between polynomials and non linear functionsfunctions

► Consider a cubic polynomial Consider a cubic polynomial 3 2( )P x ax bx cx d

► It satisfies It satisfies

2 3

2 30, 0

dP d P d PP

dx dx dx

► Consider a non linear function Consider a non linear function

( , , , )f x Da ► If the function satisfies If the function satisfies

2 3

2 30, 0

f f ff

x x x

► Then f has a maximum of three Then f has a maximum of three

solutionssolutions

Singularity theory Singularity theory continued…continued…

►x i.e. the state variable of the x i.e. the state variable of the system is dependent on Da. system is dependent on Da.

►The behavior of x Vs Da depends The behavior of x Vs Da depends on the values of on the values of αα and and ββ..

►Critical surfaces are identified in Critical surfaces are identified in

αα--ββ plane across which the nature plane across which the nature of bifurcation diagram changes.of bifurcation diagram changes.

Hysteresis varietyHysteresis variety

2

20

f ff

x x

►We solve for x, Da and We solve for x, Da and αα when other when other parameters are fixedparameters are fixed

Isola varietyIsola variety

0f f

fx Da

►We solve for x, Da and We solve for x, Da and αα when other when other parameters are fixedparameters are fixed

Bifurcation diagrams across Bifurcation diagrams across hysteresis Varietyhysteresis Variety

Low density Polyethylene Low density Polyethylene PlantPlant

HDA processHDA process

Coupled Reactor SeparatorCoupled Reactor Separator

Motivation to study Coupled Reactor Motivation to study Coupled Reactor Separator systemsSeparator systems

► Individual reactors and separators have Individual reactors and separators have been analyzedbeen analyzed

► They exhibit steady-state multiplicity as well They exhibit steady-state multiplicity as well as sustained oscillations caused by a as sustained oscillations caused by a positive feedback or an autocatalytic effectpositive feedback or an autocatalytic effect

► A typical plant consists of an upstream A typical plant consists of an upstream reactor coupled to a downstream separatorreactor coupled to a downstream separator

►We want to understand how the behavior of We want to understand how the behavior of the individual units gets modified by the the individual units gets modified by the couplingcoupling

Issues involved in modeling Coupled Issues involved in modeling Coupled Reactor Separator systemsReactor Separator systems

►Degree of freedom analysis tells us how many Degree of freedom analysis tells us how many variables have to be specified independentlyvariables have to be specified independently

► The different choices give rise to different The different choices give rise to different control strategiescontrol strategies

►Our focus is on behavior of system using Our focus is on behavior of system using idealized models to capture the essential idealized models to capture the essential interactions by including important physicsinteractions by including important physics

► This helps us understand the interactions and This helps us understand the interactions and enable us to generalize the resultsenable us to generalize the results

► This approach helps us gain analytical insightThis approach helps us gain analytical insight

Mass Coupled Reactor Separator Mass Coupled Reactor Separator networknetwork

VLE of a Binary MixtureVLE of a Binary Mixture

Control strategies for ReactorControl strategies for Reactor

Control Strategies for Control Strategies for SeparatorSeparator

Flow control strategiesFlow control strategies

►Coupled Reactor separator networks Coupled Reactor separator networks can be operated with different flow can be operated with different flow control strategiescontrol strategies

FF00 is flow controlled and M is flow controlled and MRR is fixedis fixed

F is flow controlled and MF is flow controlled and MRR is fixed is fixed

FF00 and F are flow controlled. and F are flow controlled.

Coupled Reactor Separator Coupled Reactor Separator systemsystem

““FF00 is flow controlled and M is flow controlled and MRR is is

fixedfixed””1

1 1

dz

dt af e

e e

e

x y Da ze

d y xDa BDa ze

dt z x

The reactor is modeled as CSTR and The reactor is modeled as CSTR and separator as a Isothermal Isobaric flashseparator as a Isothermal Isobaric flash

The steady state behavior is described byThe steady state behavior is described by

1 11

( , , ) log ( ) 0af ee eaf e

e

x yy xf z Da p Da B x y

z x Da z

Steady state behavior of the Steady state behavior of the coupled systemcoupled system

► It can be established that the coupled It can be established that the coupled Reactor Separator network behaves as Reactor Separator network behaves as a quadratic when Fa quadratic when F00 is flow controlled is flow controlled and Mand MRR is fixed. is fixed.

►So the system either admits two So the system either admits two steady states or no steady state for steady states or no steady state for different values of bifurcation different values of bifurcation parameters.parameters.

Bifurcation diagrams Bifurcation diagrams corresponding to different corresponding to different

regionsregions

Bifurcation Diagram at xe=0.9, ye=0.5, B=1.2

1.8 1.82 1.84 1.86 1.88 1.9 1.920.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1/Da

z

Coupled Reactor Separator networkCoupled Reactor Separator network“F is flow controlled and M“F is flow controlled and MRR is fixed” is fixed”

► The coupled system is described by the The coupled system is described by the following equationsfollowing equations

2

2 2

( )( )

( )

(1 )

e af e

e e

x z x ydzDa ze

dt x y

dDa BDa ze

dt

► The steady state behavior is described byThe steady state behavior is described by

22 2

( )( ) ( )( )( , , ) log 0

( ) ( )(1 )e af e e af e

e e e e

x z x y B x z x yf z P Da

x y Da z x y Da

Steady state behavior of the Steady state behavior of the coupled systemcoupled system

► It can be established that the coupled It can be established that the coupled system behaves as a cubicsystem behaves as a cubic

►Qualitative behavior of the coupled Qualitative behavior of the coupled system is similar to that of a stand-system is similar to that of a stand-alone CSTRalone CSTR

►This implies that the two units are This implies that the two units are essentially decoupledessentially decoupled

►Hysteresis variety and Isola variety can Hysteresis variety and Isola variety can be calculated to divide the auxiliary be calculated to divide the auxiliary parameter spaceparameter space

Bifurcation Diagram for xe=0.9, ye=0.2 and B=4

0.15 0.152 0.154 0.156 0.158 0.16 0.162 0.164 0.166 0.168 0.170.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

Da

z

FF00 and F are flow controlled and F are flow controlled

► In this case coupled system is described In this case coupled system is described by the following equationsby the following equations

*

0

2* *0

2* *0

( )1

( )

( ) ( )( )

( )

( )1

( )

e

e e

af e e

e e

e

e e

z x FdM

d x y F

x z x z z y FdzDa ze

d M M x y F

z y FdBDa ze

d M x y F M

Steady state behaviorSteady state behavior

► It can be established that the system It can be established that the system always possesses unique steady state always possesses unique steady state when Mwhen MRR is allowed to vary and F is allowed to vary and F00 ,F ,F are flow controlledare flow controlled

Mass and Energy coupled Reactor Mass and Energy coupled Reactor Separator networkSeparator network

SEPHX

V ye

F,z,T

F0 xaf

L,xe

Mass and Energy Coupled Reactor Mass and Energy Coupled Reactor Separator NetworkSeparator Network

► The coupled system in this case is The coupled system in this case is described by described by

1dz

Dazedt

*1( )

1

dBDaze Da

dt z

*

* *1( )

1

dDa

dt z

Steady state behavior of the Steady state behavior of the system is described bysystem is described by

1

(1 )(1 (1 ))( , , , , ) 1 0

B

Daz Da zf z Da B Daze

It can be established analytically that It can be established analytically that system posses hysteresis variety at system posses hysteresis variety at γγ=0.5 when =0.5 when ββ=0 i.e. for adiabatic =0 i.e. for adiabatic

reactorreactor

Bifurcation diagram for Bifurcation diagram for γγ=2,B=0.7=2,B=0.7

Delay in coupled reactor Delay in coupled reactor separator networksseparator networks

►Delays can arise in the coupled reactor Delays can arise in the coupled reactor separator networks as a result of separator networks as a result of transportation lag from the reactor to transportation lag from the reactor to separatorseparator

►Delay can induce new dynamic Delay can induce new dynamic instabilities in the coupled system and instabilities in the coupled system and introduce regions of stability in introduce regions of stability in unstable regionsunstable regions

Model equations for Isothermal CSTR coupled Model equations for Isothermal CSTR coupled with a Isothermal Isobaric flashwith a Isothermal Isobaric flash

1e**e

e**

af*Da)z(fzx

)t(zx

y)t(zzx

dt

dz

2afeee

e**

af*Da)z(fxx

yx

y)t(zzx

dt

dz

F is flow controlled and MF is flow controlled and MRR is fixed is fixed

FF00 is flow controlled and M is flow controlled and MRR is fixed is fixed

Linear stability analysisLinear stability analysis

► when Fwhen F is flow controlled and Mis flow controlled and MRR is fixed, delay can is fixed, delay can induce dynamic instabilityinduce dynamic instability

► when Fwhen F0 0 is flow controlled and Mis flow controlled and MRR is fixed, delay is fixed, delay cannot induce dynamic instabilitycannot induce dynamic instability

► Analysis with coupled non isothermal reactor, Analysis with coupled non isothermal reactor, isothermal-isobaric flash indicates that small delays isothermal-isobaric flash indicates that small delays can stabilize regions of dynamic instability and can stabilize regions of dynamic instability and large delays can destabilize the coupled system large delays can destabilize the coupled system furtherfurther

Dependence of dimensionless Dependence of dimensionless critical delay on Dacritical delay on Da

0

4

8

12

16

0 0.5 1 1.5

Da

*

Critical Delay contours for ‘F’ Critical Delay contours for ‘F’ fixedfixed

Unstable

Stable

Unstable

VLE of a Binary System with an AzeotropeVLE of a Binary System with an Azeotrope

Influence of azeotrope on the Influence of azeotrope on the behavior of the coupled systembehavior of the coupled system►When the feed to the flash has an When the feed to the flash has an

azeotrope in the VLE at the operating azeotrope in the VLE at the operating pressure of the flash thenpressure of the flash then

the system admits two branches of the system admits two branches of solutionssolutions

Recycle of reactant lean stream can take Recycle of reactant lean stream can take place from the separator to the reactorplace from the separator to the reactor

The coupled system admits multiple The coupled system admits multiple steady states even for endothermic steady states even for endothermic reactionsreactions

Bifurcation Diagram for B=-3Bifurcation Diagram for B=-3

Autocatalytic effectAutocatalytic effect

Consider a perturbation where z Consider a perturbation where z increasesincreases

This causes L to decreaseThis causes L to decrease This results in an increase in This results in an increase in ττ The temperature decreases, lowering The temperature decreases, lowering

the reaction ratethe reaction rate This causes an accumulation of reactant This causes an accumulation of reactant

amplifying the original perturbation in zamplifying the original perturbation in z

Dynamic behavior of coupled Dynamic behavior of coupled systemsystem

►The coupled system shows The coupled system shows autonomous oscillations even when autonomous oscillations even when the reactor coupled with the separator the reactor coupled with the separator is operated adiabaticallyis operated adiabatically

Oscillatory branch of Oscillatory branch of solutionssolutions

Operating a reactor with Operating a reactor with pressure drop fixedpressure drop fixed

►The control strategy of fixing pressure The control strategy of fixing pressure drop across the reactor is useful when drop across the reactor is useful when pressure drops across the reactor are pressure drops across the reactor are large like manufacture of low density large like manufacture of low density polyethylenepolyethylene

►An important issue in modeling An important issue in modeling polymerization reactors is polymerization reactors is incorporation of concentration, incorporation of concentration, temperature dependent viscositytemperature dependent viscosity

Stand-alone CSTRStand-alone CSTR

Operating a coupled reactor Operating a coupled reactor separator system with pressure separator system with pressure

drop fixed across the reactordrop fixed across the reactor►The coupled system admits multiple The coupled system admits multiple

steady states even when the reactor is steady states even when the reactor is operated isothermallyoperated isothermally

►The coupled system behaves in a The coupled system behaves in a similar fashion as the stand-alone similar fashion as the stand-alone reactor because of decoupling reactor because of decoupling between the two unitsbetween the two units

Bifurcation diagrams across Bifurcation diagrams across Hysteresis VarietyHysteresis Variety

ConclusionsConclusions

►We have seen how a comprehensive We have seen how a comprehensive understanding can be obtained using simple understanding can be obtained using simple models which incorporates the essential models which incorporates the essential physical features of a process.physical features of a process.

►The simplicity of the models enables us to The simplicity of the models enables us to use analytical or semi-analytical methodsuse analytical or semi-analytical methods

►This approach has helped us identify This approach has helped us identify different sources of instabilities which can different sources of instabilities which can possibly arise in Coupled Reactor Separator possibly arise in Coupled Reactor Separator systemssystems

Thank YouThank You