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
Outline of the talk
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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
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 equations
dxA
g ( xA )
 xAF  xA  Da

dt
(1   xA )
Where xA 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 xAf,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.
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
P( x)  ax  bx  cx  d
3
► It
2
satisfies
2
3
dP
d P
d P
P

 0,
0
2
3
dx
dx
dx
► Consider
a non linear function
f ( x, Da,  , )
► If
the function satisfies
f  f
 f
f   2  0, 3  0
x x
x
2
► Then
3
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.
Hysteresis variety
f  f
f 
 2 0
x x
2
► We
solve for x, Da and α when other
parameters are fixed
Isola variety
f
f
f 

0
x Da
► 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
 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 system
“F0 is flow controlled and MR is fixed”
dz
 xaf  ye  Da1 ze
dt
 ye  xe

d
  
  Da1   BDa1 ze
dt
 z  xe

The reactor is modeled as CSTR and separator as
a Isothermal Isobaric flash
The steady state behavior is described by
 ye  xe
  xaf  ye
f ( z, Da1 , p)  
  Da1  log 
 z  xe
  Da1 z

  B( xaf  ye )  0

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.
Bifurcation diagrams corresponding
to different regions
0.9
0.85
0.8
0.75
z
0.7
0.65
0.6
0.55
0.5
0.45
1.8
1.82
1.84
1.86
1.88
1.9
1.92
1/Da
Bifurcation Diagram at xe=0.9, ye=0.5, B=1.2
Coupled Reactor Separator network
“F is flow controlled and MR is fixed”
► The
coupled system is described by the
following equations
( xe  z )( xaf  ye )
dz


 Da2 ze
dt
( xe  ye )
d
  (1   Da2 )  BDa2 ze
dt
► The
steady state behavior is described by
 ( xe  z)( xaf  ye )  B( xe  z)( xaf  ye )
f ( z, P, Da2 )  log 
0

 ( xe  ye ) Da2 z  ( xe  ye )(1   Da2 )
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
0.65
0.6
0.55
0.5
z
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.15
0.152
0.154
0.156
0.158
0.16
0.162
0.164
0.166
0.168
0.17
Da
Bifurcation Diagram for xe=0.9, ye=0.2 and B=4
F0 and F are flow controlled
► In
this case coupled system is described by
the following equations
( z  xe ) F
dM *
1
d
( xe  ye ) F0
( xaf  z )
( xe  z )( z  ye ) F
dz




Da
ze
2
d
M*
M * ( xe  ye ) F0
d


d
M*

( z  ye ) F 


1



BDa
ze


2
*
(
x

y
)
F
M
e
e
0 

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
F0 xaf
V ye
HX
F,z,T
SEP
L,xe
Mass and Energy Coupled Reactor Separator
Network
► The
coupled system in this case is
described by
dz

 1  Daze
dt
d
1
*


(   )  BDaze   Da
dt 1  z
d
1 *

*
   Da (   ) 
 
dt
1 z 

*
Steady state behavior of the system
is described by
B
f ( z , Da, B,  ,  )  1  Daze
 Da 
1
(1 z )(1 Da (1 z ))
0
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
F0 is flow controlled and MR is fixed
*
*

z
(
t


)  ye 
dz


x e  z   f (z)Da 1

x

z

af
*
*
* 

dt
 x e  z( t   ) 
F is flow controlled and MR is fixed
 z( t   )  y e 
dz





x

z

x

x

f
(
z
)
Da
af
e
af
2
*


dt
 x e  ye 
*
*
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, isothermalisobaric 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
16

*
12
8
4
0
0
0.5
1
Da
1.5
Critical Delay contours for ‘F’ fixed
Stable
Unstable
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