Modeling of Coupled Non linear Reactor Separator SystemsProf S.PushpavanamChemical Engineering DepartmentIndian Institute of Technology MadrasChennai 600036 Indiahttp://www.che.iitm.ac.in

Outline of the talkCase study of a reactive flashSingularity theory, principlesCoupled Reactor Separator systemsMotivation for the studyIssues involvedDifferent control strategies for reactor/separatorMass coupling, energy couplingEffect of delay or transportation lagEffect of an azeotrope in VLEOperating reactor under fixed pressure dropConclusions

Industrial Acetic acid Plant

Reactive flash

Reactive flash continuedModel assumptions

nth order irreversible exothermic reaction

Reactor is modeled as a CSTR

CSTR is operated under boiling conditions

Dynamics of condenser neglected

Ideal VLE assumed

Model equationsWhere xA is the mole fraction of component A is ratio of activation energy of reaction to latent heat of vaporization

And is related to the difference in the boiling pointSteady state is governed by xAf,Da, , and n.

Multiple steady states in two-phase reactors under boiling conditions may occur if the order of self-inhibition is greater than the order n of the concentration dependency of the reaction rate.

Physical cause of multiplicityHere a phase equilibrium driven self inhibition action causes steady state multiplicity in the systemWhen the reactant is more volatile then the product, then a decrease in reactant concentration causes an increase in temperature. This causes further increase in reaction rate and hence results in a decrease in reactant concentration.This autocatalytic effect mentioned just above causes steady state multiplicity

Singularity theoryMost models are non linear. The processes occurring in them are non linear

Non linear equations which are well understood are polynomials

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

Singularity theory can beused forTo determine maximum number of solutions

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

and identify parameter values , where the different bifurcation diagrams occur

Singularity theory draws analogies between polynomials and non linear functions

Consider a cubic polynomial It satisfies

Consider a non linear function If the function satisfies Then f has a maximum of three solutions

Singularity theory continuedx i.e. the state variable of the system is dependent on Da. The behavior of x Vs Da depends on the values of and .Critical surfaces are identified in - plane across which the nature of bifurcation diagram changes.

Hysteresis varietyWe solve for x, Da and when other parameters are fixed

Isola varietyWe solve for x, Da and when other parameters are fixed

Bifurcation diagrams across hysteresis Variety

Low density Polyethylene Plant

HDA process

Coupled Reactor Separator

Motivation to study Coupled Reactor Separator systemsIndividual reactors and separators have been analyzedThey exhibit steady-state multiplicity as well as sustained oscillations caused by a positive feedback or an autocatalytic effectA typical plant consists of an upstream reactor coupled to a downstream separatorWe want to understand how the behavior of the individual units gets modified by the coupling

Issues involved in modeling Coupled Reactor Separator systems

Degree of freedom analysis tells us how many variables have to be specified independentlyThe different choices give rise to different control strategiesOur focus is on behavior of system using idealized models to capture the essential interactions by including important physicsThis helps us understand the interactions and enable us to generalize the resultsThis approach helps us gain analytical insight

Mass Coupled Reactor Separator network

VLE of a Binary Mixture

Control strategies for Reactor

Control Strategies for Separator

Flow control strategiesCoupled Reactor separator networks can be operated with different flow control strategies F0 is flow controlled and MR is fixed

F is flow controlled and MR is fixed

F0 and F are flow controlled.

Coupled Reactor Separator systemF0 is flow controlled and MR is fixedThe reactor is modeled as CSTR and separator as a Isothermal Isobaric flashThe steady state behavior is described by

Steady state behavior of the coupled systemIt can be established that the coupled Reactor Separator network behaves as a quadratic when F0 is flow controlled and MR is fixed.So the system either admits two steady states or no steady state for different values of bifurcation parameters.

Bifurcation diagrams corresponding to different regions

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

Coupled Reactor Separator networkF is flow controlled and MR is fixedThe coupled system is described by the following equationsThe steady state behavior is described by

Steady state behavior of the coupled systemIt can be established that the coupled system behaves as a cubicQualitative behavior of the coupled system is similar to that of a stand-alone CSTRThis implies that the two units are essentially decoupledHysteresis variety and Isola variety can be calculated to divide the auxiliary parameter space

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

F0 and F are flow controlledIn this case coupled system is described by the following equations

Steady state behavior

It can be established that the system always possesses unique steady state when MR is allowed to vary and F0 ,F are flow controlled

Mass and Energy coupled Reactor Separator network

Mass and Energy Coupled Reactor Separator NetworkThe coupled system in this case is described by

Steady state behavior of the system is described byIt can be established analytically that system posses hysteresis variety at =0.5 when =0 i.e. for adiabatic reactor

Bifurcation diagram for =2,B=0.7

Delay in coupled reactor separator networksDelays can arise in the coupled reactor separator networks as a result of transportation lag from the reactor to separator

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

Model equations for Isothermal CSTR coupled with a Isothermal Isobaric flashF is flow controlled and MR is fixedF0 is flow controlled and MR is fixed

Linear stability analysiswhen F is flow controlled and MR is fixed, delay can induce dynamic instability

when F0 is flow controlled and MR is fixed, delay cannot induce dynamic instability

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

Dependence of dimensionless critical delay on Da

Chart3

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7.7343944

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13.663166

Da

t*

fig1

1.00E-022.00E-023.00E-024.00E-025.00E-026.00E-027.00E-028.00E-029.00E-021.00E-011.10E-011.20E-011.30E-011.40E-011.50E-011.60E-011.70E-011.80E-011.90E-012.00E-012.10E-012.20E-012.30E-012.40E-012.50E-012.60E-012.70E-012.80E-012.90E-013.00E-013.10E-013.20E-013.30E-013.40E-013.50E-013.60E-013.70E-013.80E-013.90E-014.00E-014.10E-014.20E-014.30E-014.40E-014.50E-014.60E-014.70E-014.80E-014.90E-015.00E-015.10E-015.20E-015.30E-015.40E-015.50E-015.60E-015.70E-015.80E-015.90E-016.00E-016.10E-016.20E-016.30E-016.40E-016.50E-016.60E-016.70E-016.80E-016.90E-017.00E-017.10E-017.20E-017.30E-017.40E-017.50E-017.60E-017.70E-017.80E-017.90E-018.00E-018.10E-018.20E-018.30E-018.40E-018.50E-018.60E-018.70E-018.80E-018.90E-019.00E-019.10E-019.20E-019.30E-019.40E-019.50E-019.60E-019.70E-019.80E-019.90E-011.00E+001.01E+001.02E+001.03E+001.04E+001.05E+001.06E+001.07E+001.08E+001.09E+001.10E+001.11E+001.12E+001.13E+001.14E+001.15E+001.16E+001.17E+001.18E+001.19E+001.20E+001.21E+001.22E+001.23E+001.24E+001.25E+001.26E+001.27E+001.28E+001.29E+001.30E+001.31E+001.32E+001.33E+001.34E+001.35E+001.36E+001.37E+001.38E+001.39E+001.40E+001.41E+001.42E+001.43E+001.44E+001.45E+001.46E+001.47E+001.48E+001.49E+00

8.69E-018.72E-018.76E-018.80E-018.83E-018.87E-018.91E-018.95E-018.99E-019.03E-019.07E-019.11E-019.15E-019.19E-019.23E-019.27E-019.31E-019.36E-019.40E-019.45E-019.49E-019.53E-019.58E-019.63E-019.67E-019.72E-019.77E-019.82E-019.87E-019.92E-019.97E-011.00E+001.01E+001.01E+001.02E+001.02E+001.03E+001.03E+001.04E+001.05E+001.05E+001.06E+001.06E+001.07E+001.08E+001.08E+001.09E+001.09E+001.10E+001.11E+001.11E+001.12E+001.13E+001.13E+001.14E+001.15E+001.16E+001.16E+001.17E+001.18E+001.19E+001.20E+001.20E+001.21E+001.22E+001.23E+001.24E+001.25E+001.26E+001.26E+001.27E+001.28E+001.29E+001.30E+001.31E+001.32E+001.34E+001.35E+001.36E+001.37E+001.38E+001.39E+001.40E+001.42E+001.43E+001.44E+001.46E+001.47E+001.48E+001.50E+001.51E+001.53E+001.54E+001.56E+001.58E+001.59E+001.61E+001.63E+001.65E+001.67E+001.68E+001.71E+001.73E+001.75E+001.77E+001.79E+001.82E+001.84E+001.87E+001.89E+001.92E+001.95E+001.98E+002.01E+002.04E+002.08E+002.11E+002.15E+002.19E+002.23E+002.27E+002.32E+002.36E+002.42E+002.47E+002.53E+002.59E+002.65E+002.72E+002.80E+002.88E+002.96E+003.06E+003.16E+003.28E+003.40E+003.54E+003.70E+003.88E+004.08E+004.32E+004.60E+004.94E+005.37E+005.91E+006.65E+007.73E+009.55E+001.37E+01

fig1

0.86872356

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Da

t*

da

1.00E-022.00E-023.00E-024.00E-025.00E-026.00E-027.00E-028.00E-029.00E-021.00E-011.10E-011.20E-011.30E-011.40E-011.50E-011.60E-011.70E-011.80E-011.90E-012.00E-012.10E-012.20E-012.30E-012.40E-012.50E-012.60E-012.70E-012.80E-012.90E-013.00E-013.10E-013.20E-013.30E-013.40E-013.50E-013.60E-013.70E-013.80E-013.90E-014.00E-014.10E-014.20E-014.30E-014.40E-014.50E-014.60E-014.70E-014.80E-014.90E-015.00E-015.10E-015.20E-015.30E-015.40E-015.50E-015.60E-015.70E-015.80E-015.90E-016.00E-016.10E-016.20E-016.30E-016.40E-016.50E-016.60E-016.70E-016.80E-016.90E-017.00E-017.10E-017.20E-017.30E-017.40E-017.50E-017.60E-017.70E-017.80E-017.90E-018.00E-018.10E-018.20E-018.30E-018.40E-018.50E-018.60E-018.70E-018.80E-018.90E-019.00E-019.10E-019.20E-019.30E-019.40E-019.50E-019.60E-019.70E-019.80E-019.90E-011.00E+001.01E+001.02E+001.03E+001.04E+001.05E+001.06E+001.07E+001.08E+001.09E+001.10E+001.11E+001.12E+001.13E+001.14E+001.15E+001.16E+001.17E+001.18E+001.19E+001.20E+001.21E+001.22E+001.23E+001.24E+001.25E+001.26E+001.27E+001.28E+001.29E+001.30E+001.31E+001.32E+001.33E+001.34E+001.35E+001.36E+001.37E+001.38E+001.39E+001.40E+001.41E+001.42E+001.43E+001.44E+001.45E+001.46E+001.47E+001.48E+001.49E+00

8.69E-018.72E-018.76E-018.80E-018.83E-018.87E-018.91E-018.95E-018.99E-019.03E-019.07E-019.11E-019.15E-019.19E-019.23E-019.27E-019.31E-019.36E-019.40E-019.45E-019.49E-019.53E-019.58E-019.63E-019.67E-019.72E-019.77E-019.82E-019.87E-019.92E-019.97E-011.00E+001.01E+001.01E+001.02E+001.02E+001.03E+001.03E+001.04E+001.05E+001.05E+001.06E+001.06E+001.07E+001.08E+001.08E+001.09E+001.09E+001.10E+001.11E+001.11E+001.12E+001.13E+001.13E+001.14E+001.15E+001.16E+001.16E+001.17E+001.18E+001.19E+001.20E+001.20E+001.21E+001.22E+001.23E+001.24E+001.25E+001.26E+001.26E+001.27E+001.28E+001.29E+001.30E+001.31E+001.32E+001.34E+001.35E+001.36E+001.37E+001.38E+001.39E+001.40E+001.42E+001.43E+001.44E+001.46E+001.47E+001.48E+001.50E+001.51E+001.53E+001.54E+001.56E+001.58E+001.59E+001.61E+001.63E+001.65E+001.67E+001.68E+001.71E+001.73E+001.75E+001.77E+001.79E+001.82E+001.84E+001.87E+001.89E+001.92E+001.95E+001.98E+002.01E+002.04E+002.08E+002.11E+002.15E+002.19E+002.23E+002.27E+002.32E+002.36E+002.42E+002.47E+002.53E+002.59E+002.65E+002.72E+002.80E+002.88E+002.96E+003.06E+003.16E+003.28E+003.40E+003.54E+003.70E+003.88E+004.08E+004.32E+004.60E+004.94E+005.37E+005.91E+006.65E+007.73E+009.55E+001.37E+01

da

Da

t*

Critical Delay contours for F fixedUnstableStableUnstable

VLE of a Binary System with an Azeotrope

Influence of azeotrope on the behavior of the coupled systemWhen the feed to the flash has an azeotrope in the VLE at the operating pressure of the flash then the system admits two branches of solutions Recycle of reactant lean stream can take place from the separator to the reactorThe coupled system admits multiple steady states even for endothermic reactions

Bifurcation Diagram for B=-3

Autocatalytic effectConsider a perturbation where z increasesThis causes L to decreaseThis results in an increase in The temperature decreases, lowering the reaction rateThis causes an accumulation of reactant amplifying the original perturbation in z

Dynamic behavior of coupled systemThe coupled system shows autonomous oscillations even when the reactor coupled with the separator is operated adiabatically

Oscillatory branch of solutions

Operating a reactor with pressure drop fixedThe control strategy of fixing pressure drop across the reactor is useful when pressure drops across the reactor are large like manufacture of low density polyethyleneAn important issue in modeling polymerization reactors is incorporation of concentration, temperature dependent viscosity

Stand-alone CSTR

Operating a coupled reactor separator system with pressure drop fixed across the reactorThe coupled system admits multiple steady states even when the reactor is operated isothermallyThe coupled system behaves in a similar fashion as the stand-alone reactor because of decoupling between the two units

Bifurcation diagrams across Hysteresis Variety

ConclusionsWe have seen how a comprehensive understanding can be obtained using simple models which incorporates the essential physical features of a process.The simplicity of the models enables us to use analytical or semi-analytical methodsThis approach has helped us identify different sources of instabilities which can possibly arise in Coupled Reactor Separator systems

Thank You