optimal design and operation of batch reactors
TRANSCRIPT
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I n d . Eng. Chem. Res. 1993,32, 866-881
Optimal Design and Operation
of
Batch Reactors. 1. Theoretical
Framework
Masoud Soroush and Costas Kravaris*
Depar tment
of
Chemical Engineering, The University
of
Michigan, Ann Arbor, Michigan 48109 2136
In
this work, we propose
a
framework for integrated design and operation
of
single-stage batch
or
semibatch reactors. This includes systematic decoupling of optimization and design through
conceptual decomposition of the reactor dynamics into two subsystems with distinct characteristics.
In this framework, notions of feasibility, flexibility, controllability, and safety of the design for batch
processes are introduced for the first time and some criteria for their assessment are presented. The
proposed framework includes (a) mathematical modeling of the process dynamics, (b) dynamic
optimization tha t involves simultaneous optimization of loading conditions and operating temperature
and/or concentration profiles,
( c )
design of the heat exchange and/or feeding system(s)
and
investigation of process operability (feasibility, flexibility, controllability,
and
safety of the design),
and (d) design of
a
control scheme for automatic st artup and optimal operation of the reactor.
Introduction
Batch processes play a very important role in the
chemical process industry. Because of their great flexi-
bility, they are extensively used in the production of fine
and specialty chemicals, pharmaceuticals, polymers, and
bioproducts, as well as other products for which efficient
continuous production is not feasible. Thus, batch pro-
cesses contribute to a significant proportion of the world’s
chemical production (especially in value). The increasing
technological trends toward the manufacture of specialty
chemicals (Anderson, 1984) make the efficient design and
operation of batch processes even more important.
Batch processes are different from continuousprocesses
in the following major aspects:
1.
Mode of operation: Their mode of operation is
intrinsically dynamic (the operating conditions are time-
varying).
2.
Role of initial loading: The role of initial conditions
(initial loading of batch processes) is very important in
the operation of batch processes, while the loading
conditions of continuous processes becomes a major
operational issue when there is a possibility for existence
of multiple steady states.
3. Flexibility of operation: Batch processes possess
greater flexibility of operation and ability to cope with the
fluctuations in the market conditions.
4. Small-volume production: Batch processes are
usually used for the manufacture of low-volume high-value
products such
as
pharmaceuticals and other fine chemicals.
5.
Finite time of operation (limited batch cycle time).
6. Wide range ofoperation: This makes the control of
operating conditions difficult and necessitates the use of
measuring instruments with much wider measuring ranges.
Moreover, because a batch process model will have to
describe the behavior of the system over a wide range of
conditions, the requirement for accuracy of the batch
process model is somewhat more rigorous than that of a
continuous process model.
7. Irreversible behavior: Once an off-specification
material was produced, because of an upset in the operating
conditions, they may be no means for any correction, and
this may lead to shutdown of the process and discard of
the reacting mixture. This is in contrast to continuous
processes, where upsets in operating conditions eventually
~~ ~~ ~ ~~~
* To
whom
correspondence should be addresse d.
0888-5885/93/2632-0S66~0~.~/0
wash out of the system and the process can return to the
desired steady state.
During the past decades, there have been significant
contributions in the area of design of continuousprocesses
[see, e.g., the recent review paper by Seider et al. (1991)l.
Steady-state chemical process simulators, which were first
developed in the early 1960s, now play a very significant
role in process simulation and design work in the chemical,
petrochemical, and petroleum industries. Also, for con-
tinuous processes, on-line calculation of the optimum
steady-state control is rapidly becoming state-of-the-art
in several companies, e.g.,
IC1
and Shell (Fisher et al.,
1988b).
Because of the transient mode of operation of batch
processes, the contributions in the area of continuous
process design (e.g., Grossmann and Morari, 1983; Lang
et al., 1988; Fisher and Douglas, 1985; Palazoglu et al.,
1985; Birewar and Grossmann, 1989) are usually not
applicable
to
batch processes. On the other hand, in the
area of batch process design, a major research direction
has been the study of batch scheduling (e.g., Wellons and
Reklaitis, 1989; Karimi and Modi, 1989; Faqir and Karimi,
19891, that is, optimization of a set of batch equipment in
which each equipment is counted
as
a static system. In
today’s competitive industry, efficient design, planning,
and operation
of
batch chemical plants have become
extremely important due to competitive pressure, in-
creasing difficulties in discovering new products and
obtaining official approval for their production (Rippin,
1983).
In many batch reactors, because of the higher value of
the product compared to the value of reactants and the
cost of energy, the process economics depend much more
on the product quality and/or yield than the amount
of
energy or reactant used. Thus, significant economic
benefits can be realized from maximizing product quality
and/or yield rather than minimizing the capital and/or
operating cost(s). For example, a key factor that should
be considered in the selection of a design candidate for
polymer products is the product quality
as
reflected in
the polymer molecular weight and composition distribution
(Malone and McKenna, 1990). On the other hand, only
computation of the optimal operating conditions
[as
in
Thomas and Kiparissides (1984), Tsoukas et al., (1982),
and Wu et al. (198211 of a batch reactor does not guarantee
the feasibility of implementation of such optimal trajec-
tories. Manoquinand Luyben (1973)have addressed some
0
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American Chemical Society
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of the practical problems related
to
the implementation
of these optimal profiles. In this paper, by considering
the above process-economic reality, the general term
'optimization is used as maximization of the product
quality and/or yield of the batch process.
It has always been recognized that in deeiding on the
best operating conditions, he problem of control will have
a direct effect on whether and how the optimal conditions
will be realized. Juba and Hamer (1986) have reviewed
some of the difficulties involved in the control of hatch
processes. For example, in a hatch cycle, these is no steady
state and, therefore, no nominal condition a t which
controllers can he tuned. Moreover, the requirement for
startup and shutdown control in batch processes demands
good dynamic response over the entire operating range of
the controlled variables. This contrasts with the precise
control over a small range that is required in many
continuous processes.
The static and dynamic behavior of a process isdirectly
influenced hy the design of ita control system. Because
of this interaction between process design and control,
the synthesis of a control structure should be confronted
during the stage of the process design (Stephanopoulos,
1983). In the area of continuous processes, recently there
have been some attempts
to
remove the discontinuity
which currently exists at he interface between design and
control (Fisher et al., 1988a-c).
Considering the whole range of the previowly-men-
tionedchallengingproblemsrelatedtohatchreactors, one
can see that dynamic optimization, design, control, and
optimaloperationof a hatch reactor arenot separatehues,
and theymustheconfrontedatonestage. Inthisdirection,
we propose an integrated methodology in which the issues
of design, modeling, dynamic optimization,and control of
batch reactors are investigated in a unified framework. In
this framework, the above
tasks
are mathematically
formulated, organized,and performed interactively. More
specifically,
1.
The reactor model
is
partitioned systematically into
twosuhsystemswhich posseasdistinctcharacteristics.One
includesonlyintensivevariahles
(itwillbecalledthe inner
system ), and the other one includes both intensive and
extensive variables (it will be called the 'outer system ).
The optimization of a quality index is formulated only in
terms of the intensive variables of the inner system and
is, therefore, independent of the design. The design
parameters appear as tunable parameters of the outer
system, which acta
as
a feedback loop around the inner
system. Consequently, dynamic considerations will have
to
be accounted for in the design.
2. Notions of feasibility, flexibility, safety, and con-
trollability for batch processes are introduced for the first
time, and some criteria for their assessment are developed.
3. The proposed frameworkclarifies theroleof dynamic
modeling in optimization and design, and the interaction
between design and control becomes explicit.
There are three levels of modeling in the proposed
framework (see Figure 1). In the first level (modelo), the
model has the lowest level of information and is used as
a basis for making
somepreliminaryoperationaldecisions.
In the second level (modelI; inner system), the model
is
more complete and is used for dynamic optimization. In
the third level (model II; overall model), the model has
the highest level of information which characterizes the
overall dynamics of the reactor and is used for design
purposes.
An
overview of the proposed design and operation
framework is depicted in Figure
2;
it shows the logical
.
.
Modeling
i
Design
,--
Figure
1.
Different
levels
of mod eling and their
roles
in preliminary
decisions, optimization, design, and control
stapes.
G
Kinetic Data & Physical Parameters.
Yield
h
Product Quality Objectives
*
.
f
Mathematical Model
0
1
Reactor Sizing
.
Mathematical Model
I1
I
Formulation of Control Proble
& Synthesis o Control Law
Figure
2. Flow
diagram
of
the design and opsrationmethodology.
sequence of steps, based on the intuitive considerations,
which one should follow.
As
we proceed through the
discussion of the steps, one should realiie the theoretical
and practical issues in each step, the logic of sequencing
of the steps, and the interaction between the steps.
In this paper, the steps of the methodology will
be
discussed in detail, and the main focus will be the
theoretical formulation. In part
2
of this study, the
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integrated design and operation methodology will be
illustrated through application to a batch polymerization
reactor.
Design and Operation Methodology
In order to use the proposed integrated design and
operation methodology, one either needs to know the
kinetic rate laws of the reactions which take place in the
reactor and the related thermodynamic and physical
properties,or should have enough experimental data from
lab or pilot plant experiments to obtain the necessary rate
laws and properties. Moreover, it is assumed that (a) the
production rate of the batch reactor (i.e., the volume/
mass of the desired product per unit time) and (b) a set
of objectives related to product quality and/or yield are
specified.
In what follows, the proposed steps of the design
framework are discussed one by one.
1. Development of Model
0.
The purpose of the
development of model
0
is to (i) understand the interactions
among the different variables of the system and how these
variables affect the yield and/or product quality objectives
(which will be precisely formulated in the next step) and
(ii) define the operation variables which characterize the
operation of the reactor.
These considerations will lead to the selection of the
mode of operation of the reactor (batch or semibatch) in
the next step. Throughout this paper, mathematical
models appear as sets of ordinary differential equations.
Model 0 consists of a set of differential equations that
describe the system under closed (no feeding into the
reactor) and adiabatic conditions. As a typical situation,
consider a batch reactor in which liquid-phase reactions
with no significant density change take place.
(As
will be
seen in part 2, when density changes, the design procedure
and the related theoretical analysis remain the same.)
Under the assumptions of s independent concentrations
and constant density and heat capacity of the reacting
mixture, model 0 of this typical reactor is of the general
form:
. . .
. .
. .
--Cs - Rs(Cl, ..,C,,n
dt
dt PC
In other words, model 0 essentially consists of all the rate
expressions that describe the physical and chemical
phenomena in the reactor. The variables of model 0
(dependent variables of the ordinary differential equa-
tions) will be called the operation variables, since they
characterize the operation of the reactor. It is of course
understood that, for a complete model, one must be able
to precisely specify all performance indices (which will be
defined in the next step) n terms of the operation variables;
this will be done in the next step.
2.
Formulation of Performance Indices and Se-
lection
of
Modeof Operation. The purpose in this step
is to do the following:
Table I. Typical Polymer Product Quality Ind ices and
Performance Indices (Nunes et al..
1982)
product quality index
(end use polymer properties’)
performance index
flow properties average molecular weights
tensile stress average molecular weights
melting point average molecular weights
stress crack resistance copolymer composition
corrosion resistance copolymer composition
(i) Translqte the given product quality objectives into
a set of performance indices which can be formulated in
terms of the operation variables.
(ii) Select a subset of the operation variables (that will
be
called the optimizing variables), which have a reasonably
strong and direct effect on the performance indices and
can be easily manipulated. By continuous adjustment of
these variables, the optimal operation
of
the batch reactor
will be realized.
(iii) Select the mode of operation (batch or semibatch)
on the basis of the optimizing variables selected in (ii).
In every optimal design, there is a t least one objective
which should be maximized or minimized. The purpose
of this step is first to find a relationship between the given
objectives and a set of performance indices which can be
defined in terms
of
the operation variables of process.
The necessity for this translation arises whenever the
objectives are defined in an “abstract” form and not in
terms of operation variables.
For
product quality objec-
tives, in most cases, there is an empirical relationship
between the quality objective (actual customer specifi-
cations) and a performance index which can be defined in
terms of the reactor operation variables. Typical product
quality indices and the corresponding performance indices
for the case of polymerization are given in Table I. Because
of the complexity of polymerization processes (Ray, 1983;
Nunes et al., 1982), for a given polymer/copolymer, a
product quality index cannot usually be characterized
completely by a single performance index. Rippin (1983)
has
studied the optimization of bioreactors, polymerization
reactors, and other reactors, and tabulated some typical
performance indices. A list of typical performance indices
is given in Table
11.
Once the performance indices are formulated, one needs
to find a subset of the operation variables, which have
reasonably strong and direct effect on the performance
indices and can be easily manipulated. This is done on
the basis of our knowledge
of
the process and/or model
0.
This subset of the operation variables will be called the
optimizing variables (Rippin, 1983). The vector of the
optimizing variables is denoted by
W.
Typical optimizing
variables corresponding to certain performance indices
are also given in Table 11. The proper selection of the
performance indices and the optimizing variables W is a
crucial step in the design methodology.
A
necessary
condition
for an
operation variable to be
an
optimizing
variable “ui is that there should exist an external input
(e.g., inlet flow or heat input) by which any profile of the
optimizing variable Wi(t) can be enforced during the batch
period. For instance, if an optimizing variable candidate
is the concentration of a reactant in the reactor, practically
one should be able to enforce the concentration profile
Ui(t) to the reactor, independent of the reactor operating
conditions, through addition of the reactant to he reactor.
If reactor temperature is an optimizing variable, i t can be
manipulated by building a heat exchange. Nonisothermal
temperature trajectories are common means for optimizing
the throughput of many batch reactors (Thomas and
Kiparissides, 1984; Horak and Jiracek, 1983; Kiparissides
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T a b l e 11. Typical Pe r fo rm a nc e Ind ic e s a nd O p t imiz ing V a r ia b le s .
reaction typ e performance index optimizing variables
polymerization PD I temp erature
polymerization PD I temperature and initiator concn
polymerization end time temperature
polymerization end time temp erature and initiator concn
polymerization copolymer comp drift temp erature
polymerization copolymer comp drift monom er concentration
polymerization AMW drift temp erature and initiator concn
polymerization AMW drift and conversion temperature and initiator concn
polymerization AMW and PD I drifts temperature
bioreaction final enzyme activity temp erature
bioreaction final conversion temp erature
bioreaction yield of product temp erature and pH
bioreaction biomass growth sub stra te concentration
bioreaction yield of produc t sub stra te concentration
classical reactions
A + P ,E 1 Ez yield of product (P) temperature
A
, E1
=El
yield of pro duct (P) concen trations of A and/or P
A + P + W , E i
#E2
yield of product (P) temperature
A + P + W , E l = E z y ie ld of p ro du ct
(P)
concentrations of A and/or
P
A
selectivity concentration of A
selectivity temperature
El and
Ez
= activation energies; AMW
=
average molecular weight; PDI
=
polydispersity index.
and Shah,
1983)
and could be enforced by a heat exchanger
system. A good understanding of the physics and chem-
istry of the process is very important in the selection of
the proper operation variables as optimizing variables.
Once the optimizing variables are selected, this imme-
diately specifies the mode of operation of the reactor. For
example, if one of the selected optimizing variables is a
concentration, then feeding will be needed and the mode
of operation will be semibatch.
Remark 1:
From the point of view of control, we would
like an optimizing variable to be measurable or to be
accurately inferred from some measurements. In this case,
we will be able to use closed-loop control
to
enforce a
desirable profile of the optimizing variable to the process.
Otherwise we have
to
use open-loop control, which provides
poor tracking performance in the presence of process
disturbances. The superiority of closed-loopcontrol over
open-loop control implies that an optimizing variable,
which can be measured or accurately inferred from some
measurements, is preferred
to
an optimizing variable which
does not have this property.
Remark
2: In case closed-loop control is used to enforce
desirable optimizing variable profiles Cul(t),...,Cu,(t)
to
the process, there is a tradeoff between number of
optimizing variables
Cui’s
and the simplicity of the control
law which will later by synthesizedat he controller design
step. Although a higher number of optimizing variables
Wi)s
provides a better product (in terms of the defined
performance indices) and probably smoother optimal
profiles of optimizing variables
‘ U i ’ s ,
when it comes
to
tracking the optimal profiles of Cui’s more measurements
and control loops (a multivariable controller with higher
dimensionality) will be needed.
3. Development of Model I (Inner System). The
objective of the development of model I is to obtain a
quantitative mathematical description of the impact of
the optimizing variables
41
on the performance indices for
the selected mode of operation. The description must be
exact and at the same time of minimal order to facilitate
computation of the solution of the dynamic optimization
problem. This can be performed in two substeps: (i)
modification and extension of model
0,
on the basis of the
mode of operation selected in step
2,
to
obtain the
nonadiabatic and/or open dynamic model of the reactor;
(ii) reduction of order of the model; the balances for the
optimizing variables
w i l l
not be included in model I,
because the optimizing variables
w i l l
have tobe viewed as
inputs to model I. Also, there may be a need for a variable
transformation in the model.
The development of model I is illustrated by considering
the typical situation examined in the development
of
model
0.
In order
to
modify the model for nonadiabatic and/or
open conditions, one must include the rate of heat addition
to the reaction
8)
n the energy balance equation and the
feeding rate of species
“j”
(Fj) n the species mass balance
equation of the model of eq
1.
For simplicity,we consider
the case where only species1 s added tothe reactor. Then,
the reactor model takes the form
dC2
Fl
R2(Cl,&,T)
-
T C 2
dt
. . .
... .
Note that when there is no feeding
to
the reactor
(F1=
0 ) , he volume of the reacting mixture V does not appear
as
a variable in the model of eq
2.
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On the other hand, when F1 #
0,
it is convenient to
perform change of variable for some of the variables in the
model of eq 2. Defining the new variables, relative
concentrations, as
. . .
. .
I
and assuming Cl,, C1 for all t
1 0
for semibatch reactors
usually CI,,
>
Cl), the model of eq 2 becomes
. . .
. .
. .
In order to illustrate the development of model I, it is
instructive to consider the following three cases:
(i) Batch operation [F1=01and the optimizing variable
is the reactor temperature T: The first equation of eq 2
immediately leads to
V
= constant and therefore V is no
longer a state variable. Furthermore, with T being viewed
as input, the last equation in eq 2 is no longer relevant.
This leads to
. . .
. .
. .
-=
Cs R,(C
,..., c,,n
dt
(4)
(ii) Semibatch, isothermal operation
[F1 0,
T =
constant] and the optimizing variable is C1: Since T
=
constant, the last equation in eq
3
and the dependence of
theRj's on
T
rop out. Furthermore, with C1 being viewed
as input, the first two equations of eq 3 are no longer
relevant. These lead to
. . .
. .. .
(iii) Semibatch
[FI
01, nonisothermal operation and
the optimizing variables are T and C1: From eq
3,
by
setting aside the first two and last equations we obtain
. . .
. .. .
In all the cases (i, i, iii), the reduced-order model has the
optimizing Variables
as
inputs. In general, a reduced-
order model of the form
= 3,(z, u) (7)
is obtained. Here u is the vector of optimizing variables,
and z is the vector of remaining operation variables (not
included in u). 3 s a vector function. The dynamic
system of eq 7 is called the inner system for reasons that
will become clear later. At this point one must note that
the inner system depends
on
the physics and chemistry
only and not on the design parameters of the system.
Also, all the variables of the inner system are intensive
variables.
Remark 3: One must emphasize once again the
advantage of reduction of order of the model: I t helps
encounter fewer numerical difficulties in the computation
of optimal operating profilesor possibly finds an analytical
solution for the problem. It is also a basis for the
decoupling of optimization from design
as
will be seen
later. One must also point out that this model reduction
is standard in the theoretical optimal control literature.
The above change of variables isa special case of Kelley's
transformation (Kelley,
1964).
Remark 4:
The development of the reduced-order
model (eq 7) and then the calculation of the optimal
operating conditions in terms of a subset of operating
conditions ( u), which are independent of design, is in
complete analogy with the calculation of optimal operating
conditions for continuous systems in a stage prior to the
design stage. For example, operating conditions of a
continuous tank reactor are usually defined in terms of
reactor temperature, concentrations, etc. (intensive vari-
ables), rather than the steam pressure and flow rate in t he
jacket and flow rate of the reactants and products. These
desirable operating conditions (intensive variables) are
usually fixed prior
to
the design stage.
4.
Dynamic Optimization (Computation
of
Optimal
Loading Concentrations and Operating Conditions).
In this step, we formulate an optimization index
J to
be
minimized as well as the pertinent constraints. The
optimization index will be one of the performance indices
or a weighted sum of some performance indices. The
constraints will include the inner system (eq 7), the
remaining performance indices which were not included
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and time-varying, constant, piecewise constant (bang-bang
type), or a combination of these.
The optimality of the batch time ( t f * ) nd the operating
conditions
[U*(t )
and
2*(0)1
for a given reactor model
depends on how accurate the model of the inner system
is. An analysis of the sensitivity of the optimal operating
conditions to modeling errors and disturbances in the inner
system is very important and should be taken into
consideration in any optimization study. The general
treatment of the parametric sensitivity of optimization
results is an open issue and beyond the scope of this paper.
The existence of varying and uncertain parameters in
adynamic process model and the sensitivity of the dynamic
optimization results to these parameter variations have
motivated the development of an on-line dynamic opti-
mization method (Palanki et al., 1992). This mathemat-
ically rigorous method involves on-line dynamic optimi-
zation of aclass of processes using nonlinear state feedback
laws.
5.
Reactor Sizing.
After the optimization problem is
solved, we know (a)tf*,he optimal batch time, (b)
Cp*(tf*),
the optimal product concentration at he end of the batch
cycle, and (c)
cv*(tf*),
he volume-increase factor during
the optimal operation of the batch reactor (V( t f* )/ O ) .
In the case of semibatch operation, where only com-
ponent
1
s fed, this is given by
t v*( t f* )=
in the optimization index
J
(that must be within given
limits), and safety, operational and other constraints,
expressed as bounds on z and U.
In a typical situation, the dynamicoptimization problem
can be mathematically formulated as
minimize J =
K(z( t , ) , t , , z (O)) JotfL(z(t),U(t))
t
(8)
subject to
8 0 ) = S , ( z ( t ) , W ( t ) )
(inner system)
Uj, j ( t ) jh, j
=
1
...,
q
(optimizing variable constraints)
h ( z ( t ) , t )
O
(state inequality constraints)
g,(z(O),O)
= 0 ,
go E Rjo
(initial constraints)
(terminal constraints)
f(z(tf),tf)
= 0 , g f E
RJf
The optimization problem can involve calculating the
optimal optimizing variable profiles
U * ( t ) ,
he optimal
loading conditions z * O ) , and the optimal final time
tf*
so
thatJisminimized and the remainingperformance indices
(which were considered
as
constraints in eq 8) are within
the wanted bounds. This is a standard dynamic optimi-
zation problem [e.g., Bryson and Ho (197511.
For
the
purpose of completeness, a brief review of necessary
conditions for optimality is given in the Appendix.
Remark 5: In some cases it may be meaningful to
mathematically formulate a multiobjective dynamic
op-
timization problem, i.e., have more than one J to be
minimized. An example of this type is given in Tsoukas
et al. (1982). For the sake of simplicity, the treatment
here will be limited
to
a single optimizing index.
Remark
6:
In the cases of existence of some infeasible
optimal profiles (because of the sharp changes of the
optimal profiles), one may have to introduce constraints
(lower and/or upper bounds) on the time derivative of
some of the Ut 's . This can be done by defining an
appropriate extension of the system (the time derivative
of some UCi(sbecome new inputs and the corresponding
Uj s
become states) and considering the optimization of
Jsubjec t to the extended dynamic system with additional
constraints on the new state and input variables.
Remark 7: The selection of the initial and terminal
constraints should be performed with enough care. There
may be cases for which there are no optimizing variable
profiles within given constraints which take the batch
reactor from the given initial conditions to the requested
terminal conditions. In other words, the operating con-
ditions of the reactor a t the end of batch time should be
reachable from the operating conditions of the reactor at
the beginning of batch cycle by use of profiles of the
optimizing variables within the given limits. In the optimal
control theory [e.g., Leitmann (196711,this issue is referred
to
as reachability of the terminal conditions.
The dynamic optimization of eq
8
can calculate the
optimal batch time tf*, the optimal profiles of the
optimizing variables U , * ( t ) , ..,U , * ( t ) , and the optimal
loading conditions
z * O ) .
As expected, in general,
U * ( t )
and
z * ( t )
[ z t ) orresponding
to
z (0 )
= z*(O)
and
U ( t )=
U * ( t ) ] re not constant and vary with time. A computed
optimaloptimizingvariable profile [Ui*( t ) l ,an be smooth
Then given the desired production rate P , the initial
volume
( V O )
s calculated from
and the reactor vessel size V , is obtained from
(1
+
+,)q *VO I
v,
where av
1
is the vessel size over-design margin and
aEV*s the maximum value of
ev*(t)
during the optimal
operation, Le.,
8,,*
= supt cv*(t) for 0 I t I tf*. tb and
t,,
are, respectively, the loading/startup time and the
shutdown/cleaning ime for each batch cycle. Because tb
and
t , ,
depend on the initial loading volume VO, ne may
need to perform some iterations to obtain the value of VO.
The loading/startup time
( t k )
s s u m of two time periods,
the time needed to load the reactor and the time required
for startup (usually heatup). Both depend on the reactor
initial loading volume (VO);he latter also depends on the
maximum available heating rate through the heat ex-
changer. Here, the startup part of
tk
is guessed and later
a better estimate can be obtained when the design is
completed (step 6).
6. Designof Heating/Coolingand Feeding Systems
and Selectionof Actual Manipulated Inputs. In this
step, one must find ways of forcing the reactor to follow
the optimal rajectories of the optimizing variables through
the use of external inputs. This includes the design of
heating/cooling (H/C) and/or feeding systems, and the
selection of the
actual
manipulated inputs.
Instep 2, the optimizing variablesUwere selected among
the operation variables of the process. Associated with
these optimizing variables, there must be a set of ma-
nipulated inputs
denoted by
u ( t )
which
directly
affect
them and can
actually
enforce the optimal operation to
the process. Therefore, a necessary condition for the actual
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1993
Table 111. Typical O ptimizing Variables and
Corresponding Actual Manipulated Inputs
u U
reactor temper ature coolant flow ratea
reactor temper ature steam flow ratea
reactor temperature
reactor temperature
reactor temper ature jacket tempera turea
concentration of species y
coolant and steam
flow
ratesa
current input to heatera
rate of addition of component j
a
The choice
of
u also depends on kind of the H/C ystem which
is designed in step
6.
manipulated inputsui's is that each u i should be accessible
(Morari and Stephanopoulos, 1980) at least from one
ui.
The effect of
u ( t )
nd uwill be through the heat exchange
and/or feeding systemswhich must be designed . Onemust
assure that the chosen set of manipulated inputs u t ) ffect
V ( t ) t rongly enough
to force
W ( t )
o track the precal-
culated U * ( t ) . In other words,U ( t )must be controllable
(in a strong enough sense) by u ( t ) . One can usually find
more than one ui which affect Vi. However, the actual
manipulated variable which has the strongest effect on
ui(t)
is chosen as
ui.
The selection of the
actual
manipulated inputsui)sdepends on the H/C and/or feeding
schemes that will be used. For instance, when the reactor
temperature is an optimizing variable, we need a heat
exchange system, Depending on the heat exchangescheme
we select, the actual manipulated input can be steam
pressure, heating fluid flow rate, current input to an
electrical heater, etc.
In general, when the reactor
temperature is an optimizing variable, a heat exchange
system is needed, and when the concentration of the species
j
is an optimizing variable, a feeding system is required.
Typical optimizing variables and the corresponding actual
manipulated inputs are given in Table
111.
For example,
a good actual manipulated input candidate for the
optimizing variable
Cj
is the rate of addition of the chemical
species
j
tothe reactor
(Fj).
However, there is an inherent
disadvantage associated with these addition rates (Fis)
as manipulated inputs, that is,
Fj( t )2 0,
for all
t
(the
species
j
cannot be removed arbitrarily from the reactor).
This lower limit of the
Fj(t)'s
may result in lack of
controllability in some situations. An advantage associated
with the feedings is the higher effective heat-removal
capacity of the reactor when the feed is cold enough.
Remark 8: There may exist cases in which for tracking
of a ui(t)more than one ui( t )are required. This usually
happens when the optimal
U i * ( t )
profile is not smooth
enough. A tradeoff like the one mentioned in remark 2
exists between the dimensionality of u and the simplicity
of the control law which will be synthesized later. Higher
dimensionality of
u
provides more controllability , which
results in better tracking of the optimal profile ui*(t).
The substeps, which one must follow here, are the
following:
(i) Decide on the method of cooling and/or heating,
coolant and/or heating fluid, heater, feeding, etc. The
resulting decisions, naturally, depend on the optimal
operating conditions and the size of the reactor vessel.
There are numerous standard and nonstandard H/C and
feeding schemes available in the literature [e.g., Liptak
(1986)1, that one can choose from.
(ii) Decide on the actual manipulated inputs which the
H/C and feeding systems offer, once the H/C and feeding
systems are specified.
(iii) Specify the design parameters (those that are
inherently constant because of the design and those which
are adjustable), based on the selection of the method of
H/C and/or feeding. Those design parameters which are
U i O ) , t l O i
=FoTo(z,U,~,u:Pd)
Outer
System
Figure 3. Block diagram of the o verall dynamic model.
adjustable are denoted by
P d .
The design parameters
P d
may be determined through some iterations. Initially some
reasonable values are chosen for them, and if the design
does not satisfy some of the operability requirements
(which will be given later), they have
to
be readjusted.
Remark
9:
An increase in the heat exchange surface
area may be accompanied by a rise in the heat exchanger
holdup. This increases the sluggishness of the exchanger
dynamics which is undesirable. Also,despite the increase
in the heat exchanger area, the expected rise in the rate
of heat exchange will not always be achieved. Marroquin
and Luyben (1973) have shown tha t there is an optimum
(in the sense of maximum rate of heat removal capacity)
heat exchanger size for a jacketed reactor.
After
designing the H/C and the feeding systems, we
have enough rough knowledge of the overall reactor system.
This knowledge will be used in the next step to develop
the overall process model.
7.
Development
of Model
I1
(Overall Model). In
step
3,
the model of the inner system, 6 = 3,(z,Y) was
developed. Now, after designing the H/C and/or feeding
systems, one can develop a dynamic model which describes
the relationship between ui's and uj's. This model will
have the general form of
where
90
s a vector function. Here
7
is the vector of the
states of the H/C and/or feeding systems (e.g., jacket
temperature) as well as the dynamics which were not
included in modelI (e.g., jacket dynamics). u s the vector
of actual manipulated inputs. The dynamic system of eq
9
is called the outer system . Figure 3 shows how the inner
and outer systems are interconnected. The outer and inner
syst em have interesting characteristics. The inner system
is independent of the design. However, the outer system
directly and strongly depends on the design. The objective
is to design the outer system and later the controller to
force the system
to
track the optimal profiles
u,*(t) ,
..,
u,*(t) during the time period 0 ,< t
S
tf.
Remark 10:
As
can be seen from Figure 3, the outer
system acta
as
a
feedback loop around
the inner system.
It
wll
therefore
affe ct the stability characteristics and
the speed of response of the overall system. The design
parameters of the H/C and feeding systems can be viewed
as tunable parameters of this feedback mechanism.
Because of the effect of design parameters of the outer
system on the overall dynamics of the batch reactor,
dynamic criteria (stability and speed of response) must be
incorporated in the design of the outer system.
As
will be
seen, the design scheme and values of the corresponding
design parameters
(Pd)
are finalized when some operability
requirements are met.
Combining the inner system (eq 7) and the outer system
(eq91,we obtain the complete dynamic model of the reactor
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. . .
. .
. .
Figure 4. Schematic diagram of the cooling and feeding systems.
which will be called model
11.
In order
to
illustrate the notion of outer system, let us
consider the typical case described by eq 2 in which both
the concentration of species
1
and the reactor temper-
ature are chosen
as
optimizing variables and the reactions
are exothermic (only cooling is needed during the oper-
ation). The inner system of this case is given by eq 6.
Suppose the scheme shown in Figure 4 is chosen as the
cooling and feeding systemsfor this typical reactor. Here
the actual manipulated inputs (u1 and UZ re the flow
rate of cooling water
(FW)
nd the inlet flow rate of solution
of species 1 (Fl), espectively.
An
energy balance for
the cooling system gives (assuming heat transfer only to
the reactor)
d Tj P w UA
m0 cwmo
t
F,.,-(Tov
- Tj) + -( T - Tj)
By including the above heat balance to the set of
ODES
which were not included in development of the inner
system (eq 6), we obtain the outer system, i.e.,
dTj
PW UA
,-(TCw -
Tj)
+ -(T - Tj)
dt m, cwmo
dV
dt = Fl
or in the general form of eq 9, i.e.,
where p d = [A TWIT,u = [Few FJT, u = [T CIIT, and T
= [TjQ T . Combining the inner system (eq 6) and outer
system (eq l l ) , we obtain the overall dynamic model of
the reactor, i.e.,
Tj- (
U A T -
Tj) +
Few-(w
T,,
-
Tj)
dt c,m, m,
% = F l
dt
which is in the general form of
8. Assessment of Feasibility, Flexibility, Safety,
and Controllability of the Design.
In this step, one
must check whether the H/C and feeding systems were
properly designed or not. The basis for this analysis is
that the designed H/C and feeding systems should be
flexible, controllable, and safe and must, of course, also
guarantee feasibility of the computed optimal trajectories.
In what follows, we will introduce notions of feasibility,
flexibility, controllability, and safety for batch processes.
These will be defined in mathematical terms in the context
of the proposed design framework and will account for the
dynamic nature of batch processes. It will be seen that
feasibility, flexibility, safety, and controllability are tech-
nically different concepts and constitute important op-
erability conditions in batch processes.
Here, the general term feasibility is used to describe
the ability of the plant to perform satisfactorily under
nominal design and optimal operating conditions, Le., no
modeling error and no disturbances.
Definition of Feasibility:
Consider the outer system
with the operation variables z(t) fixed at the optimal
conditions z*(t),
u
as input and u
as
output:
Y
= u
(13)
The optimal conditions z*(t) and U*(t) will be feasible,
if the input/output system described by eq 13 s invertible
at Y(t)
=
U*(t) [Le., there exists au(t ) that produces Y(t)
=
CU*(t) under appropriate initialization] and the corre-
sponding u(t) lies within the bounds imposed by the
design.
Remark
11: The invertibility property can be guar-
anteed under relatively mild assumptions, and standard
techniques are available in the literature (Hirschorn, 1979)
for calculating the corresponding inputs.
In order t o illustrate this feasibility criterion, consider
the typical outer system described by eq
11.
Here, the
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874
Ind. Eng. Chem. Res. Vol. 32,No. ,1993
optimal profiles z*( t ) and W*(t) will be feasible, if the
corresponding cooling water and reactant feed profiles
F,*W
andF~*(t)ever exceed their lowerandupper
l i t s
for0 5 t 5
tt.
F,*(t) andF~*(t)reobtained by computing
first
and
V*(t)
=
v,
x
then
PCV*(t)
Tj* t)
P t )
Q*(t)-----
UA
finally
F,*(t)
=
and
dV*(t)
dt
,*(t)
=
uestions tha t may arise here are what should be done
if the optimal operation is not feasible, and what can cause
this infeasibility. If the optimal operation is not feasible,
then this may be due
to
any of the following reasons:
(a) The unacceptable shape of the optimal profile(s) of
temperature and/or concentration(s), e.g., sharp slopes,
toohigh value, or too low value of the profile(s) at some
times. 1nthiaease.onehastoimpsesomenewcomtraints
on the optimizing variables and/or their derivatives (see
remark 6 ) . and redo step 4.
(b) The improper design of the H/C and/or feeding
systems. In this case the design parameters (pd) should
be adjusted (e.g., larger surface area, lower coolant
temperature), or maybe the method of H/C should be
replaced hy one which is capable of providing such
optimizing variable profiles (return to step 6).
Remark 12
In
certain cases, in order
to
produce a
product with very high quality, one
has
to
enforce very
special profile(s)
to
the reactor optimizing variable(s).
In
such cases, one should either use more expensive equip-
ment in the design (e.g., a coolant with a freezing
temperature instead of cooling water), obe able
to
enforce
that profile to the reactor, or add constraints on the
optimization and then use cheaper equipment. The
decision on what
to
do depends
on
the equipment cost
and the profit from the high-quality product.
In every design, there are uncertain parameters, e.g.,
feed or ambient conditions, which may vary widely during
the plant operation. It is always a major design objective
to ensure that the chemical plant has the required
flexibilitytooperateoveragivenrangef parametervalues.
The study of operational flexibility or static resiliency for
FiyreS. Typicaleffect o fth e parameter variationson the operation
variables r t ) .
continuous processes
has
been an active research area for
many yeara (e.g., Grossmann et al. 1983;Grossmann and
Morari, 1983; Fisher and Douglas, 1985; Floudas and
Grmmann,1987;Malik and Hughes,1979;Palazoglu and
Arkun, 1987;Pistikopulos and Grossmann,1988). Flex-
ibility or static resiliency (for continuous processes) is
mainly concerned with the problem of ensuring feasible
steady-state operation of the plant not for only a single
set of nominal conditions, but for a whole range of
conditions that may be encountered in the operation. In
the area of batch scheduling, some studies on the oper-
ational flexibility of parallel batch units have also been
reported (e.g., Oi et al., 1979).
Here flexibility is the problem of ensuring optimal
operationofabatchprocesaforarangeofparametervalues,
i.e., feasibility n the presence of variations of parameters.
Therefore, once the optimal operating conditions are
feasible, one should investigate the ability of the design
inenforcingthe optimaloperatingconditiomto thereaetor
in the presence of the parameter variations.
Because of the dynamic mode of operation of batch
processes, the flexibility issue in these is much more
involved than in continuous processes. In a mathematical
context, he difference stems from the fact that continuous
processes are characterized at steady state by a set of
algebraic equations, while batch processes are character-
ized by a set of differential equations.
For
the purpose of formulating a notion of flexibility
for batch processes, t is assumed
that
the controller (which
will be synthesized later) is always able
to
force the
optimizing variables W ( t ) o track their optimal profiles
W * ( t )
satisfactorily, i.e., under control
W ( t ) W * ( t ) .
When there are uncertainties in the loading concen-
trations and temperature and the kinetic and physical
parameters, the solution z ( t ) of the inner system will not
matchz*(t). Letz*(t)+ 6z*(t;yln+6yr) denote thesolution
of the inner system in the presence of uncertainties when
W ( t )
=
W * ( t ) ,where 71 s the vector of nominal values of
the uncertain parameters in the inner system (e.g., loading
concentrations and temperature, the kinetic and physical
parameters). 671 is the vector of the deviations of the
uncertain parameters from their nominal values. Note
that under nominal conditions (nouncertainties; 671= 0 )
6z*(t;y1 ) =
0 .
As 671 aries within a certain range, the set
of the profiles
z*( t )
+ 8z*(t;yln+6y1) corresponding to all
possible values of 671s a tube which contains z*( t ) . This
is depicted in Figure 5 , for the case where the vector
~ ( t )
has two components.
A t
t = 0, the uncertainties in the
initial conditions z (0 ) are due
to
the errors in the loading
conditions. Because of the existence of uncertainties in
the model parameters,
as
ime proceeds the uncertainties
in
z ( t )
propagate.
Uncertainties must also be considered in the outer
system which can be written as
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875
A
where yonis the vector of nominal values of the uncertain
parameters in the
outer
ystem (e.g., ambient temperature,
coolant temperature, heat transfer coefficient) and 6y0
is
the vector of the deviations of the uncertain parameters
from their nominal values.
Definition of Flexibility: Consider the outer system
described by eq
14
with the operation variables
z ( t )
at
z*(t)
+
Gz*(t;yf
+
by^),
u
as
input, and
u
as
output:
[
Et : ] =
3,(z*(t)+6z*(t;r, +br,), u(t),S(t),u(t)rP,,yon+Gy,)
Y(t) =
U( t )
(15)
The designed outer system will be
flexible,
f the input/
output system described by eq 15 is invertible at Y(t) =
u*(t)
and the correspondingu t )
ies within the bounds
imposed by the design for ally
E
,. Here, y is the vector
of all uncertain parameters, i.e.,
y
=
171 + Gy~Iyo~
6yolT,
and Q, is the set of all possible values of y.
In order to determine the actual set of parameter values
y, for which the optimal conditions
z * ( t )
and U*( t ) re
feasible (for a f i e d design), he following parametric region
of feasibility is defined
Q,, A ( y l z * ( t )+ 6z*(t;yI)
and
U*( t ) re feasible for all yo]
This region provides the basic information on the flexibility
of operation of a given design. In general the actual shape
of this region could be rather complex. A particular
example of this region is depicted in Figure 6a for the case
where the region Q,, is a convex set. The above definition
of the feasible region Q,, provides a conceptual tool for
analyzing the feasibility of operation for a specified set of
bounded parameters
Q,. A
designer is interested in
knowing whether the optimal operation
is
feasible
for
all
y E Q,
or
not. Figure 6b illustrates the case when the set
Q, is totally contained within the region
Q,,
which shows
that the design possesses enough flexibility. On the other
hand, Figure 6c shows an example where the rectangle
Q,
is infeasible sincea subset of it lies outaide from the feasible
region
fly,
This case is undesirable, since the design does
not possess Yenough lexibility.
Remark 13:
The set of bounded parameters
Q,
is
typically a hypercube. If the parametric region of feasi-
bility
a
is a convex set, then a necessary and sufficient
condition for
Q,
to be contained in a is that all corner
points of
Q,
are inside
Q,,
This suggests
a
simple method
for checking flexibility.
The above flexibility analysis checks whether the design
is feasible over the set of uncertain parameters y or not,
but it does not provide a measure of flexibility in the design.
Also,
it does not determine the maximum feasible pa-
rameter set that a given design can handle. Further
research should be done to define
a
quantitative measure
for flexibility in a given design and to develop algorithms
for ita assessment. Finally, it should be mentioned that ,
ifthe uncertain parameters are time-varying, the definition
and theoretical framework for flexibility analysis is still
valid. However, in this case,y,
Q,
nd Qyrare time-varying
and this creates further complications.
A traditional and heuristic way to increase the flexibility
of a design is to overdesign the equipment. In particular,
Y1
Figure 6.
(a, top ) Param etric feasible region of ope ration
a
for a
fixed design. (b, middle) Feasible parameter set. (c, bottom)
Infeasible parameter set.
if
ui*(t)
is the ith actual manipulated variable profile
corresponding
to
the optimal operation, and aut
>
0 is a
flexibility margin for the ith actual manipulated variable,
one may request (1
+
au,)ui*(t)to be within the bounds.
One question that arises here is what should be done if
the optimal operation is not flexible enough. In such a
case, one has
to
adjust
Pd
(e.g., larger surface area, lower
coolant temperature) and/or redesign the H/C and/or
feeding systems.
In every design, it is essential to assure that hazardous
conditions cannot be created during the plant operation.
Here, the general term safety
is
used
to
describe the
ability of a design to prevent hazardous conditions as a
consequence of mechanical and/or electrical failures and/
or human errors.
A
rigorous and general definition of the notion of safety
in batch reactors and the developmentof rigorous, general,
and easy-to-use safety
tests
is an
open
issue. However,
one can provide a basis for the analysis of safety through
defining appropriate saf ety indices. The safety indices
(denoted by
&' i (z, u ,~) ,
= 1, ...,
n,)
are functions of the
states of the overall system and must remain within certain
safety bounds, even when the reactor operates under the
most severe possible conditions. By the most severe
conditions, we mean operating at he highest possible initial
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B
Figure 7.
Alternative
operation profiles
under
different loading
conditions.
temperature, the loadingconcentrations atwhich the rates
of reactions
are
maximum, the highest possible temper-
ature of the inlet stream and the most adverse concen-
trations and flow rate of the inlet stream, the most
dangerous settings of uncertain parameters, and
so
on.
Under these conditions (which will be denoted by super-
script
+),
and failure of any actuator (e.&, control valves
fully closed and/or open), he resultingprofdesof thestates
of the overall system are denoted by
z + ( t ) , ' u + ( t ) ,
and
I+(t).
Definition of Safety: A design will be safe, if the
inequalities
s , (Z+( t ) ,P l+ ( t )s+( t ) )
5 L8? =
1,
...,np
hold for all t
2 0.
Here L re the safety limits
corresponding
to
the indices
si(z+(t),'ll+(t).I+(t))
hich
must not be exceeded in any situation.
The above notion of safety is illustrated in Figure 7,
where 'u and z are s d a r and must lie within the safety
limits
2 5 z+ ( t ) -< P d 91,s 91+(t)5 Pl ,
The
f m e epicta alternative pathsthat the system follows
under the most severe conditions for different initial
loading conditions. Because the trajectories A and B are
not within the
safety
limits
all
the time, this situation
represents an unsafe design.
Certainly, under the severe conditions which are non-
optimal, the main concern of an operator is
to
prevent
creation of any dagerous situation and then drain and
clean the reactor to
s t a r t
a new batch cycle.
Safety concerns are usually handled by adjusting the
design parametersPd and/or loosening the bounds on the
actual manipulated inputs of the H/C system or replacing
the whole or part of the design by a system with higher
ability
to
cope with dangerous situations (return
to
step
6).
For
nstance, if the reactor is a semibatch one and the
concern is thermal containment, the addition of a colder
stream
to
the reactor will increase the heat removal
capacity of the reactor.
Controllability of the designed system must also he
checked
to
ensure the quality and stability of its dynamic
response
as
well
as
ita ability in forcing the optimizing
variables tovary arbitrarily in thevicinity of their optimal
profiles.
Definition of Controllability: Consider the system
The overall system will be
controllab le, if
the system of
eq 16 is controllable (in the sense of nonlinear systems
theory) for all
y E fly
A
necessary condition for controllability of the system
of eq 16 is that each
state
variable of the above system
should be accessible (Morari and Stephanopoulos, 1980)
from an input variable
ui.
If one follows the straightfor-
ward intuitive procedure for the selection of
uj s
(given in
step 6), the above accessibility requirement can be met.
9.
Formulat ion of Control Problem
and
Synthesis
of Contro l
Law.
Once
'u*(t)
as been computed and the
outer system has been designed, the issue becomes how
to
force
' u ( t ) o
follow
'u* ( t )
now that we know that it is
feasible and controllable). Although the outer system is
a feedback mechanism, it does not guarantee tracking of
a desirable
% ( t )
unless we incorporate a controller. The
importance of the controller is further supported by the
presence of possible disturbances and modeling errors in
the outer system.
More generally, one can define a vector of controlled
outputs
where h(z,Pl) is
a
vect ~r unction, which is strongly
correlated
to
the performance indices, can be measured
or
accurately estimated on-line, and
is
controllable by the
actualmanipulatedinputsu(t)
nasystem theoreticsense,
the objective being to track the trajectories y*(t)
=
h(z*(t),'u*(t)).
Here,
it
is assumed that the vectorsy,
u,
and
91
have the same number of components, e.&,
m
components. The control problem is characterized by its
multiinput/multioutput (MIMO) nature (in general), its
nonstationary behavior (no steady state), the nonlinear
dynamic model
(a
linear time-invariant approximation
is
very inaccurate because of lack of steady state and the
wide range of operation), the need for high servo and
regulatory performance in the presence of modeling errors
and disturbances, and the possibility of reactor destabi-
lization during the operation. Traditionally, the main
concern in the operation of batch reactors has been the
possibility of reactor runaway in the case of exothermic
reactions. Temperature control (possibly isothermal
operation)has beenusedto try toovercome thisdifficulty.
Many strategies for temperature control have been de-
veloped, and some of them have been implemented
industrially [e.& standard PID contro1,self-tuningcontrol
(Hodgson and Clarke, 1984). the dual-mode control of
Shinskey and Weinstein (1965), predictivecontrol(Merkle
and Lee, 1989), and nonlinear control (Kravaris et al.,
1989)l.
In practice, many control problems of this nature
in
batch reactors have been attacked by the use of open-loop
control, that is, (a) the computation of the manipulated
inputprofilesul*(t),
...,u,* t) thatconespondsto'ul*(t),
...,
Urn*@)
nd then (b) open-loop implementation of the
trajectories u,* t ) ,...,
u,*(t).
A major problem with this
open-loop strategy is that it only provides satisfactory
tracking performance in the absence of process distur-
bances and modeling errors.
Remark
14
In the ease tha t an optimizing variable
9 j
cannot be measured or inferred accurately on-line, the
optimizing variable cannot be a controlled output wj .In
this case, one can use open-loop control
to
enforce
W j W )
to
the process under consideration. This open-loop
strategy involves (a) the computation of the manipulated
input profilesul* t),
..,u,* t)
that correspondto
W P W
...,u,*(t)
and then (b) open-loop implementation of the
trajectory uj* t) . As mentioned above, in many cases,
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mainly because of the lack of reliable on-line information
on
optimizing variables, open-loop control has been used
despite the fact that it provides poor tracking performance
in the presence of process disturbances and modeling
errors.
We propose the globally linearizing control (GLC)
methodology (Kravaris and Soroush, 1990)
to
be used for
the trajectory tracking problem (see Figure8). The reason
for this choice is the ability of this control method tohandle
theMIMO nd the nonlinear nature of the control problem,
and
to
provide satisfactory servo and regulatory perfor-
mance in the presence of modeling errors and disturbances.
Here for brevity, we avoid the details of the control
methodology, and give only the steps that one should follow
to
derive the nonlinear control law. These steps are
I. Recast the model described by eqs
following form:
10and 17 in the
where
x
=
[ z
PC 7ITE
R ,
( n ) ,
g l ( x ) ,
...,
grn(x),
h ( x ) are
vector functions,
m
is number of the actual manipulated
inputs. This will always be possible because u ( t )appears
linearly in all cases.
11. Calculate the relative orders rl, ...,rrn. The relative
order ri is the smallest integer for which
[Lg1L7-lhi(x)
..
LgmL7-'hi(x)1 [O ... 01
where
n
a(L;-lhi(x))
L:hi(x) fj(x),
) = I axj
111. Calculate the
state
feedback (for an input/output
decoupled response)
rl
J
where bij are scalar tunable parameters.
of the GLC:
IV. Use m SISOPI controllersas he external controller
I I
Figure
8.
Block diagram of the controller.
% l ( y i * ( t )
-
yi(t))dt,
=
1,
...
m
(18)
TIi
where the PI biases Ub,(t)are calculated from (Soroush
and Kravaris, 1992a)
V.
Tune the parameters
Bik, K,,,
and
TI,
[see e.g., the
tuning guidelines given in Soroush and Kravaris (1992a)l.
In the case that a
state
variable is not measured on-line
a state observer should be utilized (Daoutidis et al., 1991;
Soroush and Kravaris, 1992b).
10.
Checking Servo and Regulatory Performance
and Robustness of the Controller.
In this step, the
performance of the controller is examined through sim-
ulations, to ensure tha t the controller is able to (a) force
the system to track the optimal output yi*(t), (b) reject
the effect of disturbances, and (c) be insensitive
to
modeling errors and unwanted changes in feed quality
and environmental conditions, accidental error in loading,
etc. Substep c is very crucial since batch reactors do not
have long periods of steady-state operation with the luxury
of time for on-line controller tuning, and exothermic
reactions have the potential for dangerous runaway. The
design engineer will often have to assess control system
robustness before actual implementation (Hugo, 1980).
This robustness evaluation can be done through simula-
tions. Moreover, optimal operation of the reactor also
depends on how well the controller can perform in the
above sense. Unsatisfactory performanceof the controller
may result in serious deterioration of the product quality.
The satisfactory performance of the controller in the
senses (a) and (b) can be attained by proper tuning of the
controller parameters and then examination
of
its per-
formance through simulations. For satisfactory robustness
of the controller, the controller should be sufficiently
detuned.
Conclusions
We proposed an integrated methodology in which the
issues of design, modeling, dynamic optimization, and
control of batch reactors are investigated in a unified
framework. In this framework, the above
tasks
are
mathematically formulated, organized, and performed
interactively. More specifically,
1.
A
batch reactor model is systematically partitioned
into two systems, an inner system and an outer system,
which possessdistinct characteristics. The former includes
only intensive variables, whereas the latter includes both
intensive
and
extensive
variables.
A
quality index is
formulated only in terms of intensive variables; therefore,
the outer system has no direct effect on the dynamic
optimization. The use of the inner system in the dynamic
optimization, instead of the overall model, facilitates the
dynamic optimization through the reaction of the model
order.
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2. The interaction between the inner system and outer
system clearly illustrates the idea that, in every design,
dynamic effects should be accounted for and the issues of
design, optimization, and control should be confronted in
a single stage.
3. In the framework, notions of feasibility, flexibility,
safety, and controllability for batch processes were de-
veloped for the first time, and some criteria for their
assessment were presented.
4. The development of the reduced-order model (inner
system) and then the calculation of the optimal operating
conditions in terms of a subset of operating conditions
(optimizing variables), which are independent of equip-
ment design, is in complete analogy with the calculation
of optimal operating conditions for continuous systems in
a step prior t o the equipment design step.
Because of the systematic consideration of the entire
process dynamics in the operability analysis, the final
design of a reactor within the proposed framework should
guarantee feasibility, flexibility, safety, controllability, and
optimality.
This work addressed some problems involved in batch
design and operation.
A
number of open and unsolved
problems related to batch design and operation were
highlighted throughout this paper.
An Alternative Single-Step Approach
Alternatively, the dynamic optimization and design can
be done in a single step. This approach involves the
following.
(I)
Solve the following dynamic optimization problem:
minimize J = K’(z(
f),W(t,),T( tr)
t f , z (0) ,W(O) ,~(O))
S,”L’(z(t) ,W(t),B(t) ,u(t))t
subject to
(overall system)
uj,5
uj ( t ) jh , j
= 1,
...,
q’
(manipulated input constraints)
(state inequality Constraints)
’(z(t),”u(t),q(t),t)
5
0
g,,’ z O),~ O),v O),O) =
0, goE R’”
(initial constraints)
g;(z(t ,) ,U(tf) ,v( t ,) , t , ) 0,
gf
E
R”
(terminal constraints)
to calculate the optimal manipulated input profiles
ul*(t),..., um*(t),
he optimal loading conditions
z * O ) ,
W* O),
arid ~*(0),nd the optimal terminal time tf*.
(11) In general, use open-loop control: varying the
manipulated inputs
ul,
..,
urn
according to ul*(t),
..,
u,*(t),
respectively.
The Multistep Approach versus the Single-Step
Approach
The final results of the above single-step optimization
and design approach, in principle, will be the same
as
he
results obtained from our multistep method, provided that
there are no process disturbances and modeling errors.
The reason is the systematic consideration of the entire
process dynamics in the multistep approach, which uses
the overall process model in the form of two subsystems.
The advantages and disadvantages of the above tradi-
tional single-step optimization and open-loop control
approach compared to our multistep design and operation
method can be distinguished in the following aspects:
1. In the single-step optimization and
design
approach,
optimization and design are performed in one step;
therefore, no iteration is necessary.
2. The optimal input profiles calculated by the single-
step optimization and design approach havetobe enforced
to
the process under consideration, in general, by using
open-loop control. Therefore, in this case, process mea-
surements are not used to safeguard the operation against
the process disturbances.
3. The multistep method systematically considers all
the process measurements toachieve an efficient operation;
it relies
as
much
as
possible on the process measurements
to enforce optimal operating conditions to the reactor.
4. In the single-step optimization and design approach,
the performance index
is
minimized subject to the overall
process model, whereas in the multistep method it is
minimized subject to the inner system (a subset
of
the
overall process model). Therefore, in the multistep
method, the order of the dynamic model, which is used in
dynamic optimization, is lower than in the single-step
approach. Through the reduction of the order of the
model, which is used in dynamic optimization, one
encounters fewer numerical difficulties in the computation
of optimal operating profiles
or
possibly finds an analytical
solution to the optimization problem.
5. In the multistep method, important operation
variables (optimizing variables), which characterize the
process dynamics, are systematically distinguished. This
also provides insight into the nature of the process
dynamics.
6. The theoretical developments in the multistep
method are in complete analogy with the computation of
optimal operating conditions (which are usually indepen-
dent of design) in a step prior
to
the equipment design
step.
7. In the single-step method, the feasibility of an optimal
operation can be guaranteed by imposing upper and/or
lower limits on the manipulated inputs in the dynamic
optimization.
8. In the single-step method, one may be able to use the
on-line dynamic optimization method of Palanki et
al.
(1992).
If there is no process variable that can be measured, one
has to use open-loop control
to
enforce optimal operating
profiies (calculated by the multistep method) to the procesa
under consideration. In this case, by using the one-step
approach one can avoid the iterations of the multistep
method at the expense of solving a higher dimensional
optimization problem. Another case, in which one may
prefer to use the single-step approach, isthe case in which
we have a low-order outer system.
Acknowledgment
Financial support from the National Science Foundation
through Grant CTS-8912836 is gratefully acknowledged.
Nomenclature
A
=
species A (reactant)
A =
heat-transfer area
of
H/C equipment, m2
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879
c
=
heat capacity of reacting mixture, kJ kgl K-l
Cj =
concentration of species j kgmol m-3
Cj,,
=
concentration of speciesy in its inlet stream, kgmol
Cp*(tf)= concentration of product under optimal operation
cw
= heat capacity of water,
kJ
kgl
K-l
El,
= activation energies for classical reactions in Table 11,
91(z,'U)
=
vector field of inner system (eq
7)
~ O ( Z , ' U , ? ) ~ d )
=
vector field of outer system (eq
9
F,
= inlet flow rate of cooling water, m3
s l
F,,,,,
= maximum inlet flow rate of cooling water, m3
s-1
Fj = inlet flow rate of species 7,m3 s-1
g (O),O) = vector of initial constraints (in optimization
gf(z(tf),tf) vector of terminal constraints (in optimization
f i ( z ( t ) , t )
= vector of state inequality Constraints (in defining
h(z,'U)
=
h(x)
= output map in control problem
H =
Hamiltonian in dynamic optimization
J
= optimization index
K,,
gain of the ith external controller
K(z(tf),tf,z(O))= function of terminal conditions to be
L(z( t ) , 'U( t ) )
function whose integral should be minimized
m,
= overall effective mass of H/C system, kg
M p = molecular weight of product, kg kgmol-l
P = desired product
P,
=
vector of tuning parameters of the controller
Pd
=vector of adjustable design parameters (e.g., heat-transfer
P = power input to heater, kJ s-1
P =
maximum power of heater, kJ
s-1
Q ( t ) = overall rate of heat transfer into reactor by H/C
ri
= relative order of ith output
R
=
universal gas constant, kJ kgmol-l K-l
R, = overall rate of production of species j kgmol m-3 s-l
RH
=
overall rate
of
heat generation by reactions, kJ m-3
s-1
T = reactor temperature, K
t = time, s
tf = batch time,
s
tl, = time required for loading and startup,
s
t,, =
time required for shutdown and cleaning, s
T,,= temperature of inlet cooling water,
K
Ti,=
temperature of inlet stream, K
Tj
jacket temperature,
K
u = vector of actual manipulated inputs
u, = jth actual manipulated input
U
=
overall heat-transfer coefficient, kJ m-2
s-1
K-l
W ( t )
=
vector of optimizing variables
Vi($) jth component of vector of optimizing variables
Uj ,
=
lower bound of jth optimizing variable
U,h
=
upper bound of jth optimizing variable
'Uj*(t)= optimal profile of jth optimizing variable
u =
external input vector of the linear closed-loop system
V = volume of the reacting mixture, m3
Vo
=
initial volume of the reacting mixture, m3
V,= reactor volume (size), m3
W
= undesired product
x = vector of state variables of control problem
y = vector of output variables of control problem
yi =
ith output variable of control problem
yi* = optimal profile of the ith output variable
z
=
vector of remaining operation variables which are not
z * ( t )
=
optimal profile of z
Greek Letters
m-3
at time t
=
tf , kgmol m-3
kJ kgmol-1
problem (eq 8))
problem (eq
8))
optimization problem 8)
minimized (in defining4
(in defining
J
area)
equipment, kJ 9-1
optimizing variables
a , flexibility margin for manipulated input
ui(t)
av
=
vessel size overdesign margin
&= tunable parameters of the input/output linearized system
P k
=
constant parameters in dynamic optimization (eq A.3)
q =
vector of the states of the H/C and feeding systems (e.g.,
jacket temperature) aswellasdynamics neglected in model
I
(e.g., volume)
y =vector of the uncertain parameters in the inner and outer
systems
71
=
vector of the uncertain parameters in the inner system
yo =
vector of the uncertain parameters in the outer system
y ~ n nominal value of vector of the uncertain parameters in
the inner system
yo: = nominal value of vector of the uncertain parameters
in the outer system
6, =
vector of the deviations of the uncertain parameters in
the inner and outer systems from their nominal values
a =
vector of the deviations of the uncertain parameters in
the inner system from their nominal values
6 = vector of the deviations of the uncertain parameters in
the outer system from their nominal values
p
=
overall deneity of reacting mixture, kg m-3
pw = density of the cooling water, kg m-3
uk = constant parameters in dynamic optimization (eq A.2)
\k
=
static-state feedback in the GLC structure
TI,
=
integral time constant of the ith external controller
M a t h
Symbols
A =
is defined by
E = belongs to
R = real time
Lfhi(x)=
Lie derivative of the scalar field with respect
Lf,hi(x)
=
Izth-order Lie derivative of the scalar field hi(x)
L&fk-'hi(x) = Lie derivative of the scalar field
Lfk-'hi(x)
with
Acronyms
BIB0
=
bounded input/bounded output
GLC = globally linearizing control
H/C = heating/cooling system
MIMO
=
multiinput/multioutput
PDI = polydispersity index
PI =
proportional integral
SISO = single input/single output
to the vector field
f
with respect to the vector field f
respect to the vector field
g
Appendix: Necessary Conditions for Optimality
The minimum principle of Pontryagin (Pontryagin,
1962)
states th at the admissible profiles of the optimizing
variables
e*@),
hich minimize
J,
s the global (absolute)
minimizer of Hamiltonian H which is defined by (for the
problem of eq
8):
H ( z ( t ) , h ( t ) , V ( t ) L ( z ( t ) , W ( t ) ) [X(t)lT3,(z(t),e(t)))
where X t ) E
R p ,
which is called the vector of costates or
adjoint variables, and is solution of the ordinary differential
equation
with the initial and final conditions (so-called transver-
sality conditions) of
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1993
and
where uk's and o h ' s are constant parameters, and the initial
and terminal constraints are
These initial and terminal constraints and the above
transversality conditions eqs A.2-A.4 provide 2p conditions
for the solution of the dynamic optimization problem of
eq 8. Based on the minimum principle of Pontryagin
(Pontryagin, 1962)' a
first-order necessary condition
for
optimality is that the derivative of the Hamiltonian H
with respect to the optimizing variables?lust be zero
along the optimal trajectory,
for the cases when Ui*(t) iCi1 and Ui*(t)
Uih
for all
t E
O,tfl. Generally, he optimal profiles of the optimizing
variables Vi*@) must satisfy the inequality:
H(z*
( t )
A*
( t )
u*
t ) )
H(z*W , A (t),W (O) ,
for all admissible profiles
U( t )
nd all t f O t I (A.6)
where
z * ( t ) ,
A*@ , and U*(t) represent the optimal
trajectories. Equation A.6 is the basic result of the
minimum principle. Thus,
W * ( t )
s the global minimizer
ofH=H(z( t),h(t),U(t)) . ThesetofeqsA.1-A.5, theinner
system, and the state and control inequality constraints
are called the necessary conditions for optimality.
Numerical Methods for Solution of Necessary
Conditions.
In order to compute the solution of the
optimal control problem of eq 8, one may need to use a
numerical technique. Gradient and shooting methods
[e.g., Bryson and
Ho,
1975;Keller, 19681 are two numerical
methods which are commonly used in obtaining the
numerical solution of dynamic optimization problems.
These two methods will be used in the case study of part
2.
Literature Cited
Anderson, E. Specialty Chemicals are Mixed Bag for Growth. Chem.
Eng. News 1984, une 4, 20-25.
Birewar, D. B.; Grossmann, I. E. Incorporating Scheduling in the
Optimal Design of Mu ltiproduct Batch Plants. Comput. Chem.
Eng. 1989,13, 41-161.
Bryson,
A.
E.; Ho, Yu-Chi. Applied Optim al Control; Hemisphere:
New York,
1975;
p
216-232.
Daoutidis,