Modeling of Coupled Non linear Reactor Separator Systems Prof S.Pushpavanam Chemical Engineering Department Indian Institute of Technology Madras Chennai India

Outline of the talk ► Case study of a reactive flash ► Singularity theory, principles ► Coupled Reactor Separator systems  Motivation for the study  Issues involved  Different control strategies for reactor/separator  Mass coupling, energy coupling  Effect of delay or transportation lag  Effect of an azeotrope in VLE  Operating reactor under fixed pressure drop  Conclusions

Industrial Acetic acid Plant

Reactive flash

Reactive flash continued… ► Model assumptions  n th order irreversible exothermic reaction  Reactor is modeled as a CSTR  CSTR is operated under boiling conditions  Dynamics of condenser neglected  Ideal VLE assumed

Model equations Where x A is the mole fraction of component A α is ratio of activation energy of reaction to latent heat of vaporization And β is related to the difference in the boiling point Steady state is governed by x Af, Da, α, β and n.

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

Physical cause of multiplicity ► Here a phase equilibrium driven self inhibition action causes steady state multiplicity in the system  When 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 theory ► Most 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 be used for ► To determine maximum number of solutions ► and to determine the different kinds of bifurcation diagrams, dependency of x on Da ► and identify parameter values α,β 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 continued… ► x i.e. the state variable of the system is dependent on Da. ► The behavior of x Vs Da depends on the values of α and β. ► Critical surfaces are identified in α-β plane across which the nature of bifurcation diagram changes. α-β plane across which the nature of bifurcation diagram changes.

Hysteresis variety ► We solve for x, Da and α when other parameters are fixed

Isola variety ► We solve for x, Da and α 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 systems ► Individual reactors and separators have been analyzed ► They exhibit steady-state multiplicity as well as sustained oscillations caused by a positive feedback or an autocatalytic effect ► A typical plant consists of an upstream reactor coupled to a downstream separator ► We 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 independently ► The different choices give rise to different control strategies ► Our focus is on behavior of system using idealized models to capture the essential interactions by including important physics ► This helps us understand the interactions and enable us to generalize the results ► This 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 strategies ► Coupled Reactor separator networks can be operated with different flow control strategies  F 0 is flow controlled and M R is fixed  F is flow controlled and M R is fixed  F 0 and F are flow controlled.

Coupled Reactor Separator system “ F 0 is flow controlled and M R is fixed ” The reactor is modeled as CSTR and separator as a Isothermal Isobaric flash The steady state behavior is described by

Steady state behavior of the coupled system ► It can be established that the coupled Reactor Separator network behaves as a quadratic when F 0 is flow controlled and M R 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 x e =0.9, y e =0.5, B=1.2

Coupled Reactor Separator network “F is flow controlled and M R is fixed” ► The coupled system is described by the following equations ► The steady state behavior is described by

Steady state behavior of the coupled system ► It can be established that the coupled system behaves as a cubic ► Qualitative behavior of the coupled system is similar to that of a stand-alone CSTR ► This implies that the two units are essentially decoupled ► Hysteresis variety and Isola variety can be calculated to divide the auxiliary parameter space

Bifurcation Diagram for x e =0.9, y e =0.2 and B=4

F 0 and F are flow controlled ► In 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 M R is allowed to vary and F 0,F are flow controlled

Mass and Energy coupled Reactor Separator network SEP HX V y e F,z,T F 0 x af L,x e

Mass and Energy Coupled Reactor Separator Network ► The coupled system in this case is described by

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

Bifurcation diagram for γ=2,B=0.7

Delay in coupled reactor separator networks ► Delays 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 flash F is flow controlled and M R is fixed F 0 is flow controlled and M R is fixed

Linear stability analysis ► when F is flow controlled and M R is fixed, delay can induce dynamic instability ► when F 0 is flow controlled and M R 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

Critical Delay contours for ‘F’ fixed Unstable Stable Unstable

VLE of a Binary System with an Azeotrope

Influence of azeotrope on the behavior of the coupled system ► When 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 reactor  The coupled system admits multiple steady states even for endothermic reactions

Bifurcation Diagram for B=-3

Autocatalytic effect  Consider a perturbation where z increases  This causes L to decrease  This results in an increase in τ  The temperature decreases, lowering the reaction rate  This causes an accumulation of reactant amplifying the original perturbation in z

Dynamic behavior of coupled system ► The 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 fixed ► The control strategy of fixing pressure drop across the reactor is useful when pressure drops across the reactor are large like manufacture of low density polyethylene ► An 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 reactor ► The coupled system admits multiple steady states even when the reactor is operated isothermally ► The coupled system behaves in a similar fashion as the stand-alone reactor because of decoupling between the two units

Bifurcation diagrams across Hysteresis Variety

Conclusions ► We 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 methods ► This approach has helped us identify different sources of instabilities which can possibly arise in Coupled Reactor Separator systems

Thank You