Optimal Design and Operation of Batch Reactors

8/15/2019 Optimal Design and Operation of Batch Reactors

1/16

866

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

1993

American Chemical Society


2/16

Ind. Eng. Chem. Res., Vol. 32, No.6,1993 887

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


3/16

868 Ind. Eng. Chem. Res., Vol. 32,

No. 5,

1993

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


4/16

Ind. Eng. Chem. Res., Vol. 32,

No.

6,

1993

869

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


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.


5/16

870 Ind. Eng. Chem. Res., Vol. 32,No.

5,

1993

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


6/16

Ind. Eng. Chem. Res.,

Vol.

32, No.5,1993 871

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


7/16

872 Ind. Eng. Chem. Res., Vol. 32, No.

5,

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


8/16

. . .

. .

. .

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


9/16

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


10/16

Ind. Eng. Chem. Res., Vol. 32, No. 5, 1993

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


11/16

876 Ind. Eng. Chem. Res.,

Vol. 32, No.

5,1993

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,


12/16

Ind. Eng. Chem. Res., Vol. 32,

No.

5, 1993 877

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.


13/16

878 Ind. Eng. Chem. Res., Vol. 32, No.

5,

1993

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


14/16

Ind. Eng. Chem. Res., Vol.

32, No.

5,

1993

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


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


15/16

880 Ind. Eng. Chem. Res., Vol. 32,

No.

5,

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,

Optimal Design and Operation of Batch Reactors

Documents

Transcript of Optimal Design and Operation of Batch Reactors