Modeling of Coupled Non linear Reactor Separator Systems

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Modeling of Coupled Non linear Reactor Separator
Systems

• Prof S.Pushpavanam
• Chemical Engineering Department
• Indian Institute of Technology Madras
• Chennai 600036 India
• http//www.che.iitm.ac.in

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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

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Industrial Acetic acid Plant
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Reactive flash
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Reactive flash continued

• Model 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

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Model equations
Where xA is the mole fraction of component A a
is ratio of activation energy of reaction to
latent heat of vaporization And ß is related to
the difference in the boiling pointSteady
state is governed by xAf,Da, a, ß and n.
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Multiple steady states in two-phase reactors
under boiling conditions may occur if the order
of self-inhibition a is greater than the order n
of the concentration dependency of the reaction
rate.
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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

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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

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Singularity theory can beused 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 a,ß where the
  different bifurcation diagrams occur

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• Singularity theory draws analogies between
  polynomials and non linear functions
• Consider a cubic polynomial

• It satisfies

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• Consider a non linear function

• If the function satisfies

• Then f has a maximum of three solutions

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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
  a and ß.
• Critical surfaces are identified in
• a-ß plane across which the nature of
  bifurcation diagram changes.

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Hysteresis variety

• We solve for x, Da and a when other parameters
  are fixed

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Isola variety

• We solve for x, Da and a when other parameters
  are fixed

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Bifurcation diagrams across hysteresis Variety
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Low density Polyethylene Plant
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HDA process
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Coupled Reactor Separator
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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

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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

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Mass Coupled Reactor Separator network
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VLE of a Binary Mixture
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Control strategies for Reactor
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Control Strategies for Separator
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Flow control strategies

• Coupled 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.

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Coupled Reactor Separator systemF0 is flow
controlled and MR is fixed
The reactor is modeled as CSTR and separator as a
Isothermal Isobaric flash
The steady state behavior is described by
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Steady state behavior of the coupled system

• It 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.

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Bifurcation diagrams corresponding to different
regions
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Bifurcation Diagram at xe0.9, ye0.5, B1.2
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Coupled Reactor Separator networkF is flow
controlled and MR is fixed

• The coupled system is described by the following
  equations

• The steady state behavior is described by

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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

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Bifurcation Diagram for xe0.9, ye0.2 and B4
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F0 and F are flow controlled

• In this case coupled system is described by the
  following equations

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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

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Mass and Energy coupled Reactor Separator network
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Mass and Energy Coupled Reactor Separator Network

• The coupled system in this case is described by

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Steady state behavior of the system is described
by
It can be established analytically that system
posses hysteresis variety at ?0.5 when ß0 i.e.
for adiabatic reactor
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Bifurcation diagram for ?2,B0.7
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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

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Model equations for Isothermal CSTR coupled with
a Isothermal Isobaric flash
F0 is flow controlled and MR is fixed
F is flow controlled and MR is fixed
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Linear stability analysis

• when 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

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Dependence of dimensionless critical delay on Da
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Critical Delay contours for F fixed
Stable
Unstable
Unstable
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VLE of a Binary System with an Azeotrope
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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

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Bifurcation Diagram for B-3
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Autocatalytic effect

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

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Dynamic behavior of coupled system

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

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Oscillatory branch of solutions
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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

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Stand-alone CSTR
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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

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Bifurcation diagrams across Hysteresis Variety
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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

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• Thank You