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Chcmicol Engineering Scirnee, Vol. 46, No. 516, pp 1361-1383,
1991. oab9 2509/91 s3.m + 0.00 Printed in Great Britain 0 1991
Pergamon Press plc
SYNTHESIS OF ISOTHERMAL REACTOR-SEPARATOR-RECYCLE SYSTEMS
ANTONIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS+ Department of
Chemical Engineering, Princeton University, Princeton, NJ
085444263, U.S.A.
(First received 12 March 1990; accepted in revised firm 31 July
1990)
Abstract-A systematic synthesis approach is presented for
isothermal reactor-separator-recycle systems. The approach proposes
a general superstructure of different reactors and separation tasks
and features all the potential interconnections among the proposed
units. The synthesis problem based upon the proposed superstructure
results in a mixed integer nonlinear programming (MINLP)
formulation in which the objective function involves both integer
and continuous variables and is subject to a nonlinear set of
constraints. A variety of objectives was selected for the synthesis
problem such as the minimization of the total annual cost of the
plant and the maximization of its profit, as well as objectives
traditionally used for optimizing the performance of a reactor
network such as the product yield and selectivity. Discussion of
the results and comparison among the different solutions obtained
provided the ground for conclusions related to the potential
trade-offs and the performance of the isothermal chemical systems
under consideration.
1. INTRODUCTiON
In most chemical processes reactors are sequenced by systems
that separate the desired products out of their outlet reactor
streams and recycle the unconverted reactants back to the reactor
system. Despite the fact that process synthesis has been developed
into a very active research area for the last two decades, very few
systematic procedures have been proposed for the synthesis of
reactor-separator-recycle systems. The proposed evolutionary
approaches are always based upon a large number of heuristic rules
to eliminate the wide variety of choices. Many of these heuristics
are actually extensions of results obtained by separately studying
the synthesis problem of reactor networks or separator systems and,
therefore, the potential trade- OITS resulting from the coupling of
the reactors with the separators have not been investigated.
The delay in the development of a general synthesis strategy
that will set the basis for a rigorous and systematic search for
the optimal reactor-separ- ator-recycle configuration is mainly due
to the diffi- culties arising from the large number of structural
alternatives and the nonlinear design equations of the
participating units. Instead of focusing on an optimal search
procedure, the various proposed methods have restricted the
synthesis problem into a limited search around a feasible operation
point. Thus, although it is often possible to identify directions
of potential im- provements, the finally obtained solution can
never be claimed, structurally and/or operationally, to be even a
local optimal point.
In the optimization of reactor-separator-recycle systems a very
limited number of publications exist. Conti and Paterson (1985)
proposed a heuristic evolu- tionary technique to solve the
synthesis problem. First a hierarchy of heuristics is adopted that
target to: (i) minimize process complexity, (ii) maximize
process
Author to whom correspondence should be addressed.
yield, and (iii) select the distillation column which minimizes
total heat load. According to the proposed hierarchy a base case
design is quickly established where everything is specified.
Incremental changes are then introduced to the system by
successively relaxing the heuristics so that a low cost process
topology is obtained at each stage. It is evident that since there
is no unique way of relaxing any of the above heuristics, arbitrary
decisions should often be made to provide directions of potential
changes in the system. Per- forming a case study for the Van der
Vusse reaction, the authors were able to compare their results with
reported optimal solutions for the isolated reactor system. The
comparison made clear that a design based upon the maximization of
reactor yield is much inferior to the design based upon the
integrated flow sheet.
Floquet el ul. (1985a, b) proposed a tree searching algorithm (a
branch and bound method) in order to synthesize chemical processes
involving reactor- separator-recycle systems interlinked with
recycle streams. The reactor network of this approach is restricted
to a single isothermal CSTR or PFR unit and the separation units
are considered to be simple distillation columns. The conversion of
reactants into products and the temperature of the reactor, as well
as the reflux ratio of the distillation columns, were treated as
parameters. Once the values of the para- meters have been
specified, the composition of the outlet stream of the reactor can
be estimated and application of the tree searching algorithm on the
alternative separation tasks provides the less costly distillation
sequence. The problem is solved for sev- eral values of the
parameters and conclusions are drawn for different regions of
operation.
Pibouleau et al. (1988) provided a more flexible representation
for the synthesis problem by replacing the single reactor unit by a
cascade of CSTRs. They also introduced parameters for defining the
recovery rates of intermediate components into the distillate,
CES *6-5/6-K 1361
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1362 ANTONIS C. KOKOSSIS and CHRISTODOLJLOS A. FLOLJDAS
the split fractions of top and bottom components that are
recycled toward the reactor sequence, as well as parameters for the
split fractions of the reactor outlet streams. A benzene
chlorination process was studied as an example problem for this
synthesis approach. In this example, the number of CSTRs in the
cascade was treated as a parameter that ranged from one up to a
maximum of four reactors. By repeatedly solving the synthesis
problem an optimum number of CSTRs was determined.
In this paper a general approach based upon math- ematical
programming techniques is proposed for the synthesis of
reactor-separator-recycle systems. A superstructure is postulated
with all the different al- ternatives for the reactor and separator
network, as well as for their possible interconnections. Different
separation tasks, different types of reactors, different reactor
configurations and different feeding, recycling and bypassing
strategies are considered. The synthesis problem is formulated as a
mixed integer nonlinear programming problem (MINLP). The continuous
variables include the stream flow rates and composi- tions of the
reactors and separators while the integer variables describe the
existence of the reactors and the distillation columns. The
solution of the (MINLP) formulation will provide an optima1
configuration of the reactor-separator-recycle system.
2. PROBLEM STATEMENT
For a chemical process in which a reactor network with a
reaction system of known kinetics is followed by a sequence of
separation tasks that are required to extract the desired products
and recycle the uncon- verted reactants, the synthesis problem
consists of systematically determining the reactor-separator-re-
cycle system that operates so that a given performance criterion
(e.g. total cost or profit of the plant, yield or selectivity of
desired products, conversion of re- actants) is optimized. The
solution of such a problem should provide information about:
(a) the reactor network (types and sizes of reactors, feeding
strategy and interconnections among the reactors);
(b) the separator network (appropriate separation se- quence and
sizes of separators);
(c) the interconnection between the two networks via the
allocations of the outlet streams from the reactors and the
allocations of the recycles from the separators back to the
reactors.
In the proposed synthesis problem the following as- sumptions
are made.
(1) All separators are considered to be distillation columns
while the available reactor units consist of continuous stirred
tank reactors and plug flow reactors that are approximated by equal
volume CSTRs (SCs). In the separation level, distillation was
assumed to be the only method available,
although, as will become clear in the following sections.
alternative separation methods can also be handled in a way similar
to distillation.
(2) All distillation columns are simple (i.e. one feed and two
products) and operate as sharp splits of the light and heavy key
components. No dis- tribution of components is allowed in both the
distillate and bottom products and configurations that include
nonsharp separators, like those de- scribed by Aggarwal and Floudas
(19901, are not considered.
(3) The thermodynamic state of the various streams in both the
reactor and separation networks are supposed to be known. For the
examples pre- sented in this paper, the feed streams of all columns
are saturated liquids at the pressure of the column. Distillate and
bottom streams are also considered to be saturated liquids at their
bubble points.
(4) The operating conditions of the various units (pressure,
temperature, reflux ratio of the columns) are not considered as
optimization variables and are fixed at nominal values. The heating
and cooling requirements are directly provided by hot and cold
utilities and, therefore, no heat integra- tion takes place.
Assumption (1) does not impose any restrictions on the potential
of the proposed reactor network. Due to the fact that CSTRs and
PFRs represent the two extreme types of reactor performance,
superstructures of these units are capable of handling simple and
complex types of reaction mechanisms since they can approximate
reactors with various degrees of mixing and can provide for
different feeding, recycling and bypassing strategies (Kokossis and
Floudas, 1989, 1990). Furthermore, batch and semi-batch processes
can also be studied from their space equivalent plug flow
reactors.
Assumption (2) can be justified for a large number of chemical
processes where product specifications require very pure products
and high recoveries only should be considered. Although assumption
(3) is a useful postulation that enables the effective formula-
tion of the synthesis problem, assumption (4) should be considered
as the most restrictive assumption of the approach. This is not
only because noniso- thermally operated units that are excluded
according to this assumption may appear more profitable choices for
the synthesis problem, but also because the potential heat
integration among the hot and cold streams of the
reactor-separator-recycle system is always expected to reduce the
energy requirements for the plant and result in a much less costly
alternative solution. The nonisothermal operation and the heat
integration issue will be addressed in a future publication.
3. THE SYNTHESIS APPROACH
Based upon the set of assumptions introduced in the previous
section, it is intended to present a sys-
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Synthesis of isothermal reactor-separator-recycle systems
1363
tematic strategy potent to handle the isothermal syn- thesis
problem of reactor-separator-recycle systems. The basic parts that
constitute the proposed approach consist of the following.
3.1. Derivation of cost models via simulation data The synthesis
problem involves cost expressions of
the various units. Since the specifications of the feed streams
of the columns are variables in this approach, a series of
simulations is performed by slightly relax- ing the split
assumption in order to provide cost data for different feed flow
rates and compositions. Sub- sequent regression analysis of these
data determines the expressions for the cost of each column as a
function of these variables.
3.2. Generation of the superstructure The basic idea behind the
generation of the super-
structure is to include all different configurations for the
reactor-separator-recycle system. In the reactor network, the
superstructure should feature aItern- atives for reactors in
series, reactors in parallel or series-parallel, reactors with
multiple feeds and re- cycles, as well as bypasses around the
reactor units. The superstructure should account for all the poten-
tial interconnections between the reactor network and the separator
network, it should include all potential allocations of the recycle
streams originated from the separators and leading to the reactor
units, and, in the separation level, it should include all the
alternative separation sequences.
3.3. Formulation of the synthesis problem Based upon the
proposed superstructure, the syn-
thesis problem is formulated as a constrained optim- ization
problem. Continuous, as well as integer, var- iables are
introduced. The integer variables represent the existence of each
particular reactor and separator unit; these variables are assigned
a nonzero value if the associated unit exists in the solution and a
zero value otherwise. The continuous variables consist of the flow
rates and compositions of all streams, the volumes of the reactors,
the operation time of the plant, and the costs associated with each
unit. The objective function is generally a mixed integer nonlin-
ear function which is subject to a nonlinear set of constraints
defined by the mass balances around the various splitters and
mixers of the superstructure, the mass balances around the reactors
and separators, and the logical constraints.
3.4. The proposed solution algorithm The above mathematical
formulation results in a
mixed integer nonlinear programming problem (MINLP), the
solution of which is obtained by de- composition and iteration
according to the gen- eralized Benders decomposition algorithm
(Geoffrion,
1972). According to the specified criterion (e.g. profit or
venture cost, yield or selectivity of the product), the solution
will provide information about an optimum structure of the
reactor-separator-recycle system, as well as information about the
flow rates and composi- tion of the various streams and the sizes
of the reactor and distillation units.
4. COST MODELS AND SIMULATION DATA
The formulation of the synthesis problem involves expressions
for the capital and operating cost of the reactors and the
distillation columns. Although for the reactor units such
expressions are readily avail- able, for the distillation columns
additional effort is required. This is due to the fact that the
feedfiow rate and composition of each column are considered as
optimization variables in the synthesis problem. The cost of a
column is clearly a function of these variables and, since the
function itself is not known, it has to be extracted from a series
of simulation data obtained for a range of these variables.
Guthries cost module (1969) summarizes the data necessary for
estimating the cost of a column to be: (i) the shell size and
material, (ii) the pressure of the column, (iii) the number,
diameter and type of trays, and (iv) the heat duties of the
reboiler and condenser. The tray sizing can be based on some
reasonable values of the flooding factor. A value of 78% is used
for columns with diameters greater than 1.2 m and 75% for columns
with smaller diameters. Valve trays were used in all cases. The
shell size and the material can generally be prespecified and the
pressure of the columns has already been assumed to be known. With
the economically optimal reflux ratio being 1.2 times the minimum
(King, 1971), shortcut distillation calcu- latious were performed
for a wide range of feed flow rates and compositions and provided
the number of trays and the heat duties for the condenser and re-
boiler. Fixing the number of trays to the number calculated by the
shortcut distillations, a second series of rigorous calculations
provided additional informa- tion about the diameter of the trays.
In this work, all the simulation data was obtained using the
PROCESS flow sheeting system (Simulation Sciences, 1985).
Once the number of trays, the size of trays and the heat duties
of the reboiler and the condenser have been calculated, the data
necessary for estimating the cost of the columns have been
obtained. Mathemat- ical expressions are next derived that provide
the cost of a column as a function of its feed flow rate and
composition. This is achieved by fitting various mo- dels to the
simulation data and using regression ana- lysis via SAS (Statistics
Analysis Software). The capi- tal cost Cosr:lp and the operating
cost Cost:: of each column were expressed in terms of the
compositions of the feed components of the column XFC,+_, (kth
degree polynomial), as well as the feed Aow rate of the columns
FC,,,. The cost expressions are in the general
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1344 ANTONIS C. KOKOSSIS and CHRISTOD~ULOS A. FLOUDAS
form:
+a clo,.mXFCm..., + . . . + &mXFC& >
(11
+b hnX~Cm,w~ + . . + bL,m XFCk,,, >
G-9
where h, d,..,, &.,,, . - , 4,,col, bLOl. b,l col, . . . ,
bk m co, are constants available from the regression ana- lysis for
each column and Msd is the set of feed components of the column.
The above expressions for the costs are nonlinear in both the feed
flow rates and the compositions of the considered components. The
cost dependence on the composition vector is ex- pressed via a
polynomial form that features positive and negative coefficients
while the dependence on the flow rate is expressed in a bilinear
form that includes the polynomial of the composition vector and the
feed flow rate. The exact type of the nonlinearities, how- ever, is
dependent on the problem under considera- tion and the fitting
expressions resulting from the regression analysis.
The capital cost Cmt,,,, Cap of the reactors is assumed to be a
linear expression of the reactor volume in the form:
COStCaP - relf - Y,,.. + La, vi (3)
where Y,,,, and L, are respectively the fixed charge and
variable charge investment cost of the reactor and V Tea= is the
reactor volume. In all cases, the operating cost of the reactors is
assumed to be negligible.
5. THE GENERATION OF THE SUPERSTRUCTURE
A reactor-separator-recycle system consists of two different
networks: the reactor network and the separ- ator network. The
reactor network consists of the reactor units and the
interconnecting streams associ- ated with these units while the
separator network consists of the different separation tasks that
consti- tute the desired separation process. A reactor-
separator-recycle superstructure should generally provide options
for all the structural alternatives for the units (reactors,
separators), as well as a complete stream network that will allow
for all the different interconnections among them. At the reactor
network level, the superstructure should feature all the possible
arrangements of the reactors and should consider cases of different
types of reactors with multiple feeds, recycles and bypasses, as in
Kokossis and Floudas (1989, 1990), as well as different allocations
for the recycled reactants that flow purified as bottom or
distillate streams out of the separators. In the separa- tion
level, the superstructure should include all differ- ent separation
sequences and all different interconnec- tions among the separators
and the reactors. It should
be noted that the specifics of a system may provide additional
alternatives that cannot be included in a general purpose
superstructure. These cases are not addressed by the present
approach which intends to handle the synthesis problem of
reactor-separ- ator-recycle systems in a general form. Once the
mechanism is known, however, the extension of the superstructure in
order to accommodate special fea- tures is always possible.
The proposed superstructure is best described by defining a
basic set of mixers and splitters associated with the various
units. These include the set of mixers MX:,, prior to each i,,
CSTR, mixers MXSCi,, sk prior to each sk SC-unit of the i,, PFR,
mix&s MXRC, associated with each recycled component rc, mixers
MXSS,, prior to each separation sequence sq, splitters SL$
associated with each feed stream of mf component and splitters SLP
in the outlet of each reactor i.
The superstructure of the reactor-separator- recycle system is
then generated so that:
(1) streams originated from any of the SL$-, MXRC,, and SLp lead
to any of the mixers MXiB and MXSC! z,=,=.sk; (2) streams
originated from any of the SLput lead to any of the mixers MXSS,;
(3) streams originated from each MXSS,, lead to each one of the
mixers MXRC,,; (4) outiets of the MXF,,, the MXSCi,,.,,, and the
MXSS, feed the CSTRs, the SCs and the leading separator of the sq
separation sequence respectively; (5) splitters SLP are fed by the
outlets of the reactors and mixers MXRC,, (apart from MXSS,,) by
these streams of the separators that produce pure rc.
In order to illustrate examples of superstructures generated
according to the above propositions, the examples of two different
reaction mechanisms are considered. In the first example the
reactor network consists of three CSTRs and the components to be
separated are A, B and C with their relative volatility in this
order. The consecutive reaction mechanism
kl kl n-E- C is assumed to take place with B denoting the
desired product and A the fresh feed component. The potential
separation tasks are A/EC, AS/C, B/C and A/B. The component which
should be considered For recycling is A and flows as distillate
from columns A/BC and A/B. The generated super- structure, shown in
Fig. 1, features all the possible interconnections among the
reactor units (streams 7, 8, 10, 11, 15 and 16), among the reactor
and the separation units (streams, 5, 6, 12, 13, 17 and 18), and
all the potential recycles from the separation network to the
reactors (streams 25, 26 and 27).
The different configurations for the reactor- separator-recycle
system can be obtained by elimin- ating the appropriate streams of
the proposed super- structure. Thus, elimination of all but streams
1, 7, 11, 14, 18, 22, 24 and 27 results in the configuration
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Synthesis of isothermal reactor-sepaiator-recycle systems
1365
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1366 ANT~NIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS
shown in Fig. 2(a) where the three CSTRs are con- nected in
series, the separator system includes columns A/BC and B/C and a
total recycle of A is fed into the first CSTR. Should all but
streams 1, 2,3,4, 5, 9, 12, 14, 17, 21, 23 and 26 be eliminated
from the superstructure, the configuration of Fig. 2(b) is ob-
tained where the CSTRs are connected in parallel, the separator
network consists of columns AB/C and A/B and the recycle stream
from A/B feeds the second CSTR. A different configuration,
illustrated in Fig. 2(c), is the case where streams 1, 2, 8, 9, 14,
17, 12, 21, 23, 25 and 27 only are activated. In the reactor
network, the first and third CSTRs are connected in series and the
second reactor is in parallel with the serial arrangement of the
other two CSTRs. The separator system consists of columns AB/C and
A/B, while the recycle of A is fed into the first and third CSTRs.
As a final example, the reactor-separ- ator-recycle configuration
of Fig. 2(d) is shown. The configuration results by considering
only streams 1, 7, 9, 12, 14, 17, 21,23, 25 and 27 from the
superstructure of Fig. 1 and consists of reactors CSTR- 1 and
CSTR-2 in series with CSTR-2 feeding column A/BC, the distillate of
which feeds CSTR-3 and CSTR-1.
In case the reactor network consists of two CSTRs and one PFR
and the assumed reaction mechanism is
of the form: As B 2 C the generated reactor- separator-recycle
superstructure is shown in Fig. 3. As in the previous example, B is
assumed to be the desired product while C is a byproduct, the
recycling of which may or may not be desirable. Thus, the
superstructure features the potential recycle streams 25, 26, 27,
32, 33 and 34 which lead toward CSTR-2, CSTR-1 and PFR-1. According
to the proposed features for the plug flow reactor, each potential
feed stream of the PFR (1,4, 10, 15,27 and 34) is split into a
number of substreams equal to the number of the SCSTRs in the PFR
representation. As a result, differ- ent feeding strategies can be
obtained for the PFR unit as illustrated in the configurations
shown in Fig. 4.
In Fig. 4(a), the reactor network consists of the CSTR-1 in
series with the PFR, the inlet stream of which is unevenly
distributed along the reactor.-The separator system consists of
columns A/BC and B/C and A is the only recycled component. In Fig.
4(b), the reactor network is restricted to the plug flow reactor,
columns AB/C and A/B constitute the separator sys- tem and
components A and C are both recycled toward the PFR. A total
recycle of A is fed in the front section of the PFR while a
substream of the produced C is fed in the middle and ending section
of the reactor. In the final example, shown in Fig. 4(c), the
reactor network consists of all three reactors. The leading PFR
unit is fed by fresh and recycled reactant A and is connected in
series with a parallel arrange- ment of the two CSTRs. The
separator system con- sists of columns A/BC and B/C. The recycled
com- ponent A is feeding the PFR while the recycled com- ponent C
is feeding the CSTRs.
4. VARIABLES AND PARAMETERS OF THE SYNTHESIS PROBLEM
The basic index sets used to define the variables and parameters
of the synthesis problem include the set of components M = {ml, the
set of reactors Z = (i}, the set of the columns L = (I} and the set
of intermediates N = { nJ. The latter set consists of all groups of
two or more components discharged in the same outlet stream of a
separator.
A number of subsets of these sets are also intro- duced and
include:
Subsets of M
Mfeed _ 1 -
Mbot = 1
Mdis _ I -
{m/me M is a recycled component} {mlmt M is a fresh feed
component) {m/me M is a desired product] { mlm E M participating in
rp E RP path kin- etic expression} {mlmE M is a feed component of
column IEL) {m/me M is a bottom component of column IEL) {rnlrn~ M
is a distillate component of column I E L}
Subsets of Z ZcSrR = { ili~Z is a continuous stirred
ZPFR = {iii E Z is a pIug flow reactor}
Subsets of L J = { I[! E L is a leading column}
tank reactor}
L!F:d = {l/f EL with feed the intermediate n E IV) L.tz = {ZIIE
L with bottom the intermediate n E N} I$: = (I//E L with distillate
the intermediate n E N) L1.E = (ZIIEL produces recycled
component
mEM) L;tm= {ZllsL produces desired mcMdp).
In addition to the above sets, there is also the set of the
reaction paths RP = {rp} and the set SK = (sk} of the subunits
(CSTRs) that represent each individual plug flow reactor.
The variables of the synthesis problem are defined over the
above sets and subsets and consist of con- tinuous and integer
variables. The first set of con- tinuous variables includes
variables associated with the cost or the profit of the plant,
CO&: annualized cost of the plant Pro$r: annualized profit
of the plant,
as well as variables associated with the cost of the various
units and the operation time of the plant,
CustFap: capital cost of column 1 Co@? capital cost of reactor i
CostFper: operating cost of column 1 8: total operation time of the
plant.
Another set of variables is related to the flow rates of the
streams that constitute the proposed super-
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Synthesis of isothermal reactor-separator-recycle systems
1367
c 9 f
-l!I!b! A Eel--h B L-J C
B II C (a) 4 J
B
c----
C
CSTR-2 CSTR - 3 B
(d)
Fig. 2. Reactor-separator-recycle configurations obtained from
the superstructure of Fig. 1
-
f 24
t
2
-
2
2
B 6
c T 28 d I-
t
A 6
22 5 c 26
A
:41
0 L Fi
g. 3
. Rea
ctor
-sep
arat
or-r
ecyc
le su
pers
truc
ture
cons
istin
g of o
ne C
STR
and
one
PFR
and
sep
arat
ing t
hree
com
pone
nts.
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Synthesis of isothermal reactor-separa(or-recycle systems
s 0
1369
(b)
Fig. 4. Reactor-separator-recycle configurations obtained from
the superstructure of Fig. 3.
structure and consist of the fresh feed stream flowrates,
FD,: total fresh feed stream of component m E M mJ FRi,m: fresh
feed of reactant m E M w in reactor i FRSw>,: sk substream of
FR,., (i E IPFR, sk E SK),
streams leading to or originated from the reactor units (CSTRs
and SCs),
IN,: inlet stream of reactor i OUT,: outlet stream of reactor i
TNSi,,k: inlet stream of sk SCSTR of reactor i (i.zlPFR, skESK)
OUTSi,sk: outlet stream of sk SCSTR of reactor i (i E IPFR, sk E
SK) BP+: bypass around sk SCSTR of reactor i (i E IPFR, sk E
SK),
streams leading to or originated from the separator units (feed,
bottom and distillate streams of the distill- ation columns),
FC,: total feed of column I B,: bottom product of column I D,:
distillate product of column 1,
as well as streams that interconnect different units of the
superstructure such as the streams among the reactors,
RR,.,: interconnecting stream from reactor i to reactor k; i,
ke1 RRSi.r,,r: sk substream of RR,_, (itz I, k E IPFR, Sk E SK)
or streams connecting the reactor network to the separator
system and vice versa,
MM,.,: reactor network recycle in front of column 1 merging with
REC, (1 E J, m E M) RMi,I: stream from reactor i to leading column
1 E J MRSi,,.Sk 1 sk substream of MRi,,, (i E I PER, m E M 111, sk
E SK) MRi,m: recycle stream of component rnE M to reactor i
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1370 ANTONIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS
REC,: total recycle of pure m (me M) a: payout time PG,: total
purge stream of m (m~A4~). p: income tax rate
Figure 5 shows the reactor-separator-recycle superstructure (one
PFR, one CSTR and a three-
SP,: sales price of component m PR,: minimum production of
desired m E Mdp,
component separation process) along with the stream as well as
the chemical process (reaction mechanism, variables that have been
introduced above. A different reaction mixture) and the plug flow
reactor repres- set of variables is associated with the reaction
rates entation (maximum allowable SC units), and the volumes of the
reaction units,
R,,.i: reaction rate of reaction rp in reactor i R&i,&
reaction rate of reaction path rp in sk SCSTR of reactor c (iEIPFR)
V,: volume of reactor i vSi.J* ; volume of sk SCSTR of reactor i
(if IPFR, Sk E SK),
Y,~,,: stoichiometric coefficient of component m in reaction
path rp N,,: number of SK elements k,: reaction constant for
reaction path rp MVm: molar volume of m.
as well as the concentrations of the species that appear in the
reaction rate expressions, 7. FORMULATION OF THE SYNTHESIS
PROBLEM
CNi,,: concentration of component m in the output 7.1. Objective
function(s): the difSerent cases
of reactor i The different objectives that have been
considered
CNSi,sk,m: concentration of component m in in this approach
include those related to the eco- nomics of the nlant. like the
profit and the annualized
0 U TSi,,r. venture cost df the system, *and those related to
the
A final set of continuous variables considers the efficient
utilization of the reactants like the overail
molar fractions of all the streams of the superstructure yield
and selectivity. The exDression for the annualized venture cost of
the and consists of:
XFC,,,: molar fraction of component m in FC, XB,,,: molar
fraction of component m in BI XD,,,: molar fraction of component m
in D, XMKwsw: molar fraction of component m MRi,,r XIN,,,: molar
fraction of component m in INi XOuTi,,k3,: molar fraction of
component m OUT, XINSi,sk,nt: molar fraction of component m INS+
XOUTS,,,: molar fraction of component m OUTSi,,*.
plant is of the general form:
in
costann = - :.{z# CostfP + c COStfaP &I. 1 (4)
in where
in Costa = the annual venture cost of the plant Cosp = the total
capital cost of reactor unit i
in Costpap = the total capital cost of column I Cosf;lper = the
annual operating cost of column 1.
Apart from the continuous variables, the proposed formulation
also includes integer variables that de- scribe the structural
alternatives of the reactor- separator-recycle system. The set of
integer variables includes:
Zi: binary variables associated with reactor i Y,: binary
variable associated with column 1
zsi,sli : binary variable associated with sk SCSTR of reactor i
(i E IPPR).
The above binary variables represent the existence or not of
each unit. Thus, if the reactor i (column r) participates in the
superstructure, then Zi ( Yr) takes the value of 1. Otherwise it
takes the value of 0. The introduction of the binary variables
ZSiVslrr in addition to Zi, reflects the alternative of having
activated only a subset of the available SCSTRs given by N,,.
Besides the continuous and integer variables of the problem, a
set of parameters is introduced which is related to the costing and
the specifications for the plant,
In Section 4 it was shown how to derive the capital and
operating costs of the units. In the case where a particular unit
does not participate in the solution the cost associated with the
unit should be eliminated. Logical constraints that are introduced
later in this section force the feed flow rates and the volumes of
nonexisting reactors and coIumns to take zero values, thus
eliminating the operating costs of the units and their variable
charge investment costs. For the fixed charge investment costs also
to be eliminated in the case where the unit does not exist, the
expressions for the capital costs are written in the equivalent
form:
Costpa~ = yiz, + 6, v,. (6) For the total profit of the plant
the expression used
-
Synthesis of isothermal reactorseparator-recycle systems
A
1371
-
1372 ANTONIS C.Ko~ossrs and CHRISTODOULOS A. FLOUDA~
is:
Pro& = c c SP, 13. FC,,, . XFC,,, [E If,-; ?nehfLp
- ,s; /SP, 0. FD, - Cost (7)
where an upper bound is usually set for the total operation time
0 of the plant.
The efficient utilization of the reactants can be evaluated in
terms of the overall yield defined as the molar rate of desired
products over the molar rate of fresh feed reactants, the overall
selectivity defined by the molar rate of desired products over the
molar consumption rate of reactants and/or the overall con- version
defined by the molar consumption rate of reactants over the molar
fresh feed rate of reactants. Since detailed expressions are not
generally possible unless the particular reaction mechanism is
available, objective functions based upon these criteria are re-
presented by the general symbol R.
7.2. Constraints of the synthesis problem The desired objective
is subject to a set of con-
straints constituted by the mass balances and the logical
constraints. The mass balances include:
(i) molar balances for the feed stream splitters, and PFR stream
splitters
FD, - CFR~,~ = 0 mEMmf (8) i.zr
FL - c FRSis,k,, = 0 iE IPFR, mEMm/ (9) skcSK
RR,,, - c RRSi,;p,,t = 0 iE I, ipe ZPFR (10) sk&K
MRi,m - 1 MRSi,,,,k = 0 iEZPFR, rnE Mm (11) sk&K
(ii) molar balances for the mixers in the inlet of each CSTR
unit
+ F%rq -ZNi.XZNi.,=Om~M,
i E ZCsTR, mf e Mm- (121
(iii) molar balances for the mixers prior to each SC unit
Bpip,sk - 1 XNSip,s* - I
+ OUTSip,sk- I _ XOuTSip,rk- 1.m
- INS+ XINSip,s~.rn = 0
(13) ipEIPFA, 1 < sk < N,,, mEM
INi, . XINips, + c RRSi,ip.sk . XOUT,., iEI
+ FRSip.ti,mf
+ m,&_~MRSip,m,.,k _ XMRm,.m
- BPip.sr+ I . XINSip.sk+ 1.m
- INSip,sk+ I . XINSip,sk+ 1.m = 0
ip E ZPFR, skESK, sk = 1, rnEM (14)
(iv) moIar balances for the splitters in the outlet of each
reactor
c RR,,, . XOUT,,, + c RM,,j. XOUTiv, ksl Id
- OUT,- XOUT,,, = 0 iEI, meM (15)
BPip.slr . XINS+,,, + OUTSip.slr . XOWTSip.sk,m
- OUT,; XOiJTip.m = 0 QxIPFR, skESK,
sk = NsK, me M (16)
(v) molar balances for the mixers prior to each leading
column
c RM,,I. XOUT,,, - 1 MM,,, . XFC,,, rn.z&frc
- FC, . XFCI,, = 0 IEJ, mEM (17)
(vi) molar balances for recycle mixers of the super-
structure
REG, - 1 MM,,,, . XFC,sm Id
-~MR,,,;XMR,=,=O mEM,mrEM (18)
REC, + PG, - c XFC,,, . FCI = 0 mr EMC ,ELiZ
(19)
(vii) summation of the volumes of the SC units
Vi - C VSi,sk = 0 ifz ZPFR skfSK
(20)
(viii) summation of the mole fractions of the streams
c XMR,,,, - 1 =O mrEMrC (21) m
C XIZVi_- 1=0 iEI (22) msA4
c XOUTiv, - 1 = 0 i~l (23) tn.&f
1 XZNSi,wa - 1 =O iEIPER, skeSK (24) n.
-
Synthesis of isothermal reactor-separator-recycle systems
1373
c Xi?,,,- 1 =o 1EL (27) mEM
c XL,., - l=O ZEL (28) ma&f
(ix) molar balances around each reactor unit
IN, . XINi,,, - 0 U Ti . X0 U T,,,
- vi . Crp.m - Rr,,i = 0 ieIcsrn, mEA (29) rl,
INSi,.pk . XINSi,mt - OUTsi. XOUTSi.,k_,
- VSi,,~Cv,p.m.RS,p,i,,* = 0 iEIPFR, mEM. VP
(30)
The reaction rates I&,,,, and RS,,,.,ti.i can be ex- pressed
as a general functlonf,, of the outlet stream concentrations times
the reaction constant k as:
R,p_i - k,, . f,,(CNi,,) = 0 rp E RP, i E ICSTR,
rnEMP (31)
RS,p,i& - k,p S,p (CNSi.sk,m) = 0
rp E RP, i E IPPR, s~ESK, rn~M~ (32)
where the outlet concentrations are given in terms of molar
fractions as:
CNi,, . (
c XOUTi,m. MV,,, - XOUT,,, = 0 me&f >
~EI, rn~M (33)
CNSi.s*,m (
C X0UTSi.sk.m . MVm rnEM !
- XOUTSi,sL.m = 0 iEiPFR, mEM, skESK (34)
(x) molar balances around each separator of the
superstructure
FC, - B, - D, = 0 EEL (35)
D, - FC, c
c XFC,
= 0 IEL (36)
XDL, (
c XFC,,, nlE A#:~ >
- XFCI,, = 0
IEL, rnEMP6 (37)
X%n . (
c XFC,., - XFC,,, = 0 meMy >
IcL, mEMy (38)
(xi) molar balances for the intermediate components
c Bi. X6,, - c Di.XQ,, I6LE 1 E Ld. I.1
+ ,e$d FC, . XFC,,, = 0 n E N (39) r..
as well as the specifications for the desired products,
c XFCL, -FC, 2 PR, mEMdp rsp
(xii) logical constraints
(40)
The logical constraints are used in order to force a continuous
variable of a nonactive unit of the super- structure to take the
value of zero. They are of the general form:
var; - u . varpt 6 0
where VULVA denotes the continuous variable associ- ated with
unit k, vary denotes the integer variable that describes the
existence of this particular unit and U is a large number. If the
unit participates in the superstructure, varp takes the value of 1
and the above inequality yields:
varconL < U
while if the unit does not exist, varp becomes zero and the
inequality yields:
rrar;O < 0
which, since the continuous variable is always positive, can be
true only if uarrn is zero.
For the present formulation the following set of these
constraints was introduced:
lNi - U.Zi
-
1374 ANTONIS C. KOKOSSIS and CHRWCODOULOS A. FLOUDAS
A final set of constraints imposes that all variables are
non-negative (non-negativity constraints) or binary variables
(integrality constraints) and is defined as
FD,, FRi,,, MRi,,, INi, OUT, 3 0
D,, XFC,.,, -X-B,,,, XD,.,, XMR,,,, 2 0
v,, I%.&, &,,ir R.%~,i,,~, CNi,,t, CJJSi,,r,, 3 0
(51)
zi, r,, ZSi,Sk E (0, 11. (521
The complete mathematical formulation of the MINLP problem is
given in Appendix A.
8. THE PROPOSED SOLUTION ALGORITHM
The optimization problem that is formulated in the previous
section consists of two types of variables: continuous and integer
variables. The integer vari- ables participate linearly in both the
objective func- tion and the constraints while the continuous vari-
ables participate linearly and nonlinearly. The nonlin- earities
include the polynomial expressions obtained from the simulation
data, the expressions of the reac- tion rates and the bilinearities
associated with the mass balances. Such constraints define a
nonconvex feasible region and therefore there is no guarantee for
the global optimum.
The proposed solution algorithm for the MINLP problem is based
upon a decomposition of the ori- ginal problem and iteration
according to the Gen- eralized Benders decomposition algorithm. The
al- gorithm suggests the solution of a sequence of nonlin- ear
programming (NLP) primal problems and mixed integer linear
programming (MILP) master problems that are based upon the dual
representation of the primal problem. The solution of each NLP
problem provides an upper bound (lower bound) for a minim- ization
(maximization) direction while the solution of each MILP provides a
lower (upper) bound. The MILP solution also provides the integer
variables to project on the next NLP problem and convergence of the
algorithm is achieved whenever the MILP master problem fails to
provide a new feasible integer combi- nation.
The NLP subproblems of the solution algorithm are large scale
nonlinear programming problems and initialization is possible by
solving a so-called pseudo- primaI NLP subproblem. The pseudoprimal
problem is the projection of the full primal problem in the
subspace of the continuous variables that are associ- ated only
with the active units at the current iteration. Around a small
region of the solution vector of this subproblem a relaxed problem
is next solved that relaxes the equality constraints H(X) of the
primal in the form - a d n(x) d 0: and minimizes the infeasi-
bilities CL. The solution vector of the relaxed problem is used to
initialize the primal problem. In the case where
an infeasible primal problem is found, the Lagrangian
multipliers that are associated with the next MILP master problem
are obtained by the solution of the relaxed problem.
The suggested solution procedure which is included in OASIS
(Floudas, 1990) was automated in the com- puter program ORESS
(Optimization of REactor Separator recycle Systems) that uses the
high level modeling language GAMS (General Algebraic Modeling
System) and the algorithmic development methodology APROS (Paules
and Floudas, 1989). At each iteration the program calls the
appropriate solvers for the various subproblems, updates relevant
parameters and checks for stopping criteria. If the criteria are
met it stops, otherwise it continues to the next iteration.
Due to the nonconvex type of nonlinearities in- volved in the
mathematical formulation, infeasibilities are very likely to occur
in the NLP subproblems in the case where improper initialization
has been used or inappropriate bounds have been allowed for the
continuous variables. In order to avoid unfortunate solution
trials, ORESS dynamically sets and updates lower and upper bounds
for all variables, generates a number of different starting points
at each iteration and makes use of all of these points before it
declares as infeasible a nonlinear programming (NLP) problem.
The proposed approach is general and, under the assumptions
imposed in Section 2, accounts for all possible structural and
operational alternatives of the reactor-separator-recycle system.
As an offset to the advantages of the approach, however, one should
consider that the solution algorithm is restricted to provide a
local solution. The solution, which evidently depends on the
initial starting point, cannot be ciaimed to represent the global
optimum, the search of which is not addressed by this paper. The
global optimum search for nonlinear programming and mixed integer
programming problems is itself an area of research and newly
developed techniques that have provided promising results might
also be proved ap- plicable for the reactor-separator-recycle
synthesis problem as in the proposed formulation.
In the following section, the potential of the pre- sented
algorithm is illustrated in two different examples of chemical
processes by considering a variety of specifications for the
reactor network and the desired products, as well as by optimizing
in terms of different objective functions.
9. EXAMPLES
9.1. Benzene chlorination The design of a benzene chlorination
process is
considered as a first example. The chemical reactions of this
liquid-phase process are given as:
kl C,H, + Cl, - C,H,Cl + HCl
LZ C,H,Cl + Cl, - C,H,Cl, + HCl.
-
Synthesis of isothermal reactor-separator-recycle systems
137.5
Further chlorination reactions can also take place but since
they involve insignificant amounts of re- actants they have been
considered to be negligible. The kinetics of the process were
studied by McMullin (1948), who showed that the chlorination of
benzene (A), monochlorobenzene (B) and dichlorobenzene (C) is in
all cases first-order and irreversible.
In the reaction level, pure A reacts to the desired product B,
waste product C and hydrochloric acid. The kinetic constants are k,
= 0.412 h- and k, = 0.055 h - . The hydrochloric acid produced
is
eliminated at the reaction level output by a stripping operation
whose cost is not taken into account. Although all the reactions
are exothermic internal coil are used in the reactors to remove the
evolved heat and therefore keep the temperature in the reactors
constant.
In the separation level, unreacted A is separated and recycled
toward the reactor network, valuable product B, of which the demand
is assumed to be 50 kmol/h, and product C. The volatility ranking
of these components is aA > ug > cc,. Thus, the possible
separation tasks are: A/BC (column l), AB/C (column 2), B/C (column
3) and A/B (column 4).
Simulation results reported by Auzerais (198X) pro- vided
estimates for the capital and operating cost of all the sharp
distillation columns of the chlorination process (purity
specification: 99%) and are given along with the cost expressions
for the reactors in Appendix B.
The synthesis problem was studied in four different cases. In
the first three cases, the objective was to minimize the annualized
total cost of a plant that produces a minimum of 50 kmol/h
chlorobenzene by considering a variety of specifications for the
reactor network. In the fourth case, different objective func-
tions were used and conclusions were drawn by com- paring the
different solutions. The postulated super- structure always
consisted of four CSTRs and four PFRs and each PFR unit has been
approximated by a cascade of seven CSTRs (SCs). The large-scale
optim- ization problem included 947 continuous variables, 36
51.35 kmolfir
integer variables and 1063 constraints. In all cases, a MIPS RC
2030 workstation computer was used.
Case 1: utilization of a least number of CSTRs. In this case a
minimum of two CSTRs is required for the reactor network. The
starting point of the algorithm was one PFR and four CSTRs along
with columns 2 and 4 and the optimal reactor-separator-recycle sys-
tem was found to consist of two CSTRs and 1 PFR and columns 1 and
3. In the optima1 solution, shown in Fig. 6, a fresh benzene stream
(53.14 kmol/h) along with a total recycle stream of purified
benzene (51.35 kmol/h) originating from the distillate of col- umn
1, are feeding the first CSTR of the reactor network. The reactors
are connected in a CSTR-PFR-CSTR series combination and feature
volumes V,,,, = 3.427 m3, V,,, = 10.565 m3 and V ,-sTR = 3.482 m3
respectively. The production of chlorobenzene is the minimum
required (50 kmol/h) and the total annual cost is $314,361. The
reported solution was obtained in nine iterations with an aver- age
CPU time per iteration of 5.88 s per prima1 and 2.00 s per master
problem.
Case 2: utilization of CSTRs and/or PFRs. No requirements were
imposed for the reactor network in this case. Providing an initial
guess of one PFR and columns 2 and 4, the minimum annual cost is
found to be 5304,911 and the optimal structure consists of a single
PFR and coIumns 1 and 3. Convergence of the solution algorithm
required five iterations with an average consumption of 5.50 CPU
seconds per primal and 1.78 seconds per master problem. In the
optimal solution, shown in Fig. 7, both the volume require- ments
(V,,, = 17.478 m3) and the total recycle from the separation
network (52.3 kmol/h) are greater than in case 1. Fresh feed is
required at a rate of 53.147 kmol/h and no splitting for any of the
feeding streams of the PFR was necessary.
Case 3: utilization of a least number of PFRs. A minimum number
of three PFRs is required in this
II A 50 kmoLbr
[
B d C c - Fig. 6. Solution for the benzene chlorination
process--case 1.
-
1376 ANTONIS C. KOKOSSIS and CHRISTODOULOS A.FLOUDAS
52.31 kmobhr
t
Fig. 7. Solution for the benzene chlorination processdase 2.
case. The initial guess for the reactor-separator- recycle
system consists of four PFRs along with columns 2 and 4. The
minimum annual cost is found to be $309,840 and the optimal
structure, shown in Fig. 8, is an arrangement of three PFRs in
series and columns 1 and 3. The first PFR features a recycle (10.10
kmol/h) and has volume Y - 5.562 m3 PFR-L - while the other two
PFRs have volumes VPFR_a = 5.734 m3 and V PFR-3 = 5.873 m3
respectively. Al-
though a larger total cost is found for the plant, the total
volume requirements, the fresh feed require- ments (53.02 kmol/h)
and the tota recycle from the separation network (47.98 kmol/h) are
less than in the previous case. The solution algorithm converged in
two iterations consuming 6.24 CPU seconds per pri- mal and 1.98 CPU
seconds per master problem.
Case 4: multiple objectives. The importance of ap- plying
different performance criteria for the reactor- separator-recycle
system is next studied by solving the synthesis problem using
different objective func- tions. The different objectives include
the profit of the plant, the annualized cost of the plant and the
overall yield of the product. In order to facilitate comparisons
among the solutions obtained, a minimum production of 50 kmol/h
chlorobenzene is required for all but the last of the presented
examples.
(a) Profit: providing an initial structure of one CSTR and one
PFR with columns 2 and 4, the
maximum profit is found to be $1,224,038. The solu- tion is
obtained in two iterations and the average CPU time is 5.56 seconds
per primal and 1.85 seconds per master problem. The optimal
structure consists of three CSTRs and columns 1 and 3 is shown in
Fig. 9. A fresh feed stream of 51.97 kmol/h is required and the
recycled benzene stream is 159.43 kmol/h. Both the fresh feed and
the recycle stream lead to the first CSTR. The CSTRs are connected
in series and are of almost the same size: V,-STR_L = 4.88 m3,
Vcsm_2 = 4.97 m3 and V,,,,_, = 5.04 m3. The annual cost
of this plant is $321,158, the yield 0.314 and the benzene
conversion 0.326.
(b) Annualized cost: with the same initial structure as in the
previous example, the minimum cost is found to be $293,324. The
optimal structure, shown in Fig. 10, consists of two CSTRs with
columns 1 and 3. The CSTRs are connected in series and their
volumes are VCSI.R_I = 10.12 In3 and V,,,,_, = 10.47 m3. A larger
feed stream (54.45 kmoI/h) and a much smaller recycle stream (54.23
kmol/h) are required than in the previous case. The profit
associated with the plant is $1,166,420, the yield 0.460 and the
benzene conversion 0.501. The solution algorithm converged in two
iterations consuming 5.275 s per primal and 1.86 s per master
problem.
(c) Overall yield (a): the maximum overall yield is found to be
0.5653 and the optimal structure, shown in Fig. I 1, consists of a
single PFR and columns 1 and 3. The initial structure was one CSTR
and columns 1
47 98 km&l-c 50 krnoLhr
A B
10.10 kmol/hr
Fig. 8. Solution for the benzene chlorination process--case
3.
-
Synthesis of isothermal reactor-separator-recycle systems
1377
r
L59.43 kmolfhr l 50kmol/hr
I3 E C Fig. 9. Solution for the benzene chlorination process-se
4(a).
-d 54.23 kmoyhr
54.45kmolhm j wm
V CSTB=10.12 In3 V csTR=10.47m3
Fig. 10. Solution for the benzene chlorination process-ase
4(b).
4 50 kmolfhr
B t
54.57kmoh V
PFR = 20.73~~~ * C
Fig. 11. Solution for the benzene chlorination process+ase
4(c).
and 3, and the solution algorithm converged in two iterations
with an average consumption of 5.48 CPU seconds per primal and 1.85
seconds per master prob- lem. In the final solution, the PFR has
volume I,,,_, = 20.73 m3 and does not feature any side streams.
The required fresh feed stream is 54.57 kmol/h and much less
recycle (33.87 kmol/h) is needed than in the previous cases. The
annual cost of the plant is $785,509, the profit $667,109 and the
benzene conver- sion 0.617.
(d) Overall yield (b): instead of a minimum pro- duction of
chlorobenzene, a minimum consumption of 50 kmol/h of benzene is
required. Starting from one CSTR and columns 1 and 3, the optimal
reactor-
CES 46-5/6-L
separator-recycle system consisted, as in the previous case, of
a single PFR unit with columns 1 and 3. However, in the final
solution, shown in Fig 12, a PFR was at each upper bound (V,,,., =
30.0 m3), a much larger feed stream is required (78.96 kmol/h) and
a larger recycle stream (49.00 kmol/h). Although a larger annual
cost ($840,214) and less profit (5615,405) are found, the overall
yield and the benzene conver- sion are, as in case (a), 0.5653 and
0.617 respectively. The solution algorithm converged in two
iterations consuming 5.62 CPU seconds per primal and 1.75 seconds
per master.
As might be expected, the annualized cost and the
-
1378 ANTONISC. KOKOSSIS and CHRISTODOULOS A. FLOUDAS
49.00 kmol/hr
t
78.96 kmob'hr
Fig. 12. Solution for the benzene chlorination process--case
4(d).
profit provide similar solutions that favour the CSTR structure.
The overall yield, however, results in a different optimal
structure, namely a PFR, an optimal operation that accounts for
54.5% of the maximum obtainable profit and an annual cost which is
267.8% greater than the minimum possible. Furthermore, by comparing
the results from the last two examples, it becomes clear that
taking the yield as the performance criterion of the
reactor-separator-recycle system, no distinction can be made among
solutions associated with favourable annual costs and/or profits.
Since a synthesis problem based upon the maximization of the
overall yield does not actually take into account the separation
network but maximizes instead only the performance of the reactor
system, these results should be considered indicative of the
importance of the coupling between the reactor and separator net-
work and should lead to the conclusion that unless the objective
does not account for this coupling poor design results are
obtained.
9.2. Production of ethylbenzene In this example the alkylation
of benzene with
ethylene for the production of ethylbenzene is studied. The
process is an intermediate stage of the production of styrene using
the direct hydrogenation method and is carried out in the liquid
phase. The following first order reversible reactions describe this
proces:
C,H, +CH,=CH,, % C,H,CH,CH,
C,H,CH,CH, + CH, = CH 2 2 C,H,(CH,CH,),
Higher order alkylation products as well as other high boiling
materials that are obtained as by- products of the process were not
considered here. In the alkylators liquid benzene (A) reacts with a
gaseous stream of pure ethylene to produce the desired ethyl-
benzene (B) and the coproduct diethylbenzene (C). In the separation
level, B is obtained at a minimum rate of 10 kmol/h while both A
and C have potential recycles to the reactor network.
The ranking volatility is aA > air > uc and, thus, the
possible separation tasks are: A/BC (column l), AB/C (column 2) B/C
(column 3) and A/B (column 4). For
each column separate simulations were necessary for different
feed flow rates and compositions. Details for the simulations along
with the expressions for the venture costs of the distillation
columns and the capital cost of the reactors are given in Appendix
B.
In the following examples, a superstructure of four CSTRs and
four PFRs is postulated for the reactorseparator-recycle system.
With a required minimum production of 10 kmol/h ethylbenzene, the
synthesis problem is solved by considering two differ- ent cases
for the reaction kinetic constants of the alkylation process. The
objective is to minimize the annual cost of the plant and the large
scale optimiza- tion problem consists of 1050 continuous variables,
36 integer variables and 1133 constraints. An MIPS RC 2030
workstation was used for the computational part of the solution
algorithm.
Case 1. In this case the reaction constants used are: k, = k; =
k, = k; = 0.4 h-i. Starting from a structure of one CSTR and
columns 2 and 4, the solution algorithm converged in six iterations
and consumed 6.12 CPU seconds per primal and 1.6 CPU seconds per
master problem. The minimum annual cost of the plant is $79,272 and
the optimal structure, shown in Fig. 13, consists of a single PFR
and columns 1 and 3. The required volume for the PFR is V pFR =
3.68 m3 and the fresh feed benzene stream is 11.78 kmol/h. Although
a total recycle of 36.28 kmol/h is found for the benzene, a
relatively small portion (21.6%) of the produced diethylbenzene
(2.27 kmol/h) is recycled.
Case 2. In this case the reaction constants used are: k, = k; =
0.4h- and k, = k, = 4.0 h- . With an initial structure of one PFR
and columns 2 and 4, the minimum annual cost is found to be
$157,253. The optimal structure, shown in Fig. 14, is a single PFR
and columns 1 and 3. The fresh benzene stream is 16.84 kmol/h and
the volume of the PFR is V,,, = 5.7X m3. In this case, a much
larger benzene
recycle (223.57 kmol/h) is required and a larger portion (40.5%)
of the produced diethylbenzene (11.5 1 kmol/h) is led toward the
reactor network than in case 1. Such a result should be expected
since the
-
1380
CNLc.,
costann costpap Cost-P c*st+ D, FL,,,
F&y,
FRSi,sk,,
FG I ICSTR
p-FR
IN,
INSi.sk
M iced
Mbt
MY
MM,,,
MRSi,,.sk
MK,m
MV,
MXRC,,
MXSS,,
iv N SK
ANTONIS C. KOKOSSIS and CHRJSTODOULOS A. FLOUDAS
concentration of component m~ M of OuTSi*sc annualized cost of
the plant capital cost of column I E L capital cost of reactor i E
Z operating cost of column 1 E L distillate product of column 1 EL
total fresh feed stream of component mEM fresh feed of reactant m E
MI compon- ent into reactor iE1 substream of FR+,, leading to
subunit sk E SK of reactor i E I PFR total feed of column IE L set
of the reactors set of the continuous stirred tank re- actors set
of the plug flow reactors inlet stream of reactor i E Z inlet
stream into subunit skESK of re- actor i e IPFR set of the leading
columns reaction constant for reaction path rpERP set of the
columns set of 1 EL columns with feed n E N inter- mediate set of I
EL columns with bottom n E N intermediate set of I EL columns with
distillate no N intermediate set of EEL columns producing recycled
m E M} component set of Ic L columns producing recycled M E Mdp)
component set of components set of recycled components set of fresh
feed components set of desired products set of components
participating in reac- tion path rp E RP set of components feeding
column I E L set of components at the bottom of col- umn IEL set of
components at the distillate of column IE L reactor network recycle
in front of col- umn 1EL substream of MRi,, leading to subunit sk E
SK of reactor i E lPFR recycle stream of component m E M to reactor
i molar volume of rnE M mixers prior to reactor i E FSTR mixers
prior to subunit sk E SK of reactor i E IPFR mixers associated with
each recycled component rc E M mixers prior to each leading column
sq i
REC, RMi,,
RP RR,,,
RRsi~,r
RSrp,i,s:
SK
SL$ SLOU
SP:, u vi vsi,sk
XFG,,
X&t,, XDm,, XM%mr
XZNm,i
outlet stream out of reactor iEZ outlet stream out of subunit
skE SK of reactor i E IPFR total purge stream of component WI EM
minimum production of desired compon- ent m r Mdp annualized profit
of the plant reaction rate of reaction rpE RF in re- actor i4zl
total recycle of pure m E M stream from reactor i E I towards
leading column I E J set of the reaction paths interconnecting
stream from reactor i E 1 to reactor k E I substream of RR,,*
leading to subunit skESK reaction rate of reaction path rp E RP in
subunit skE SK of reactor ieZPFR set of subunits (SCs) that
represent each reactor i E ZPFR splitters of each FD,, splitters of
each OUT, sale price of component WEE M large number volume of
reactor ieZ volume of subunit sk ESK of reactor iEIPFR molar
fraction of component rnE M in FC, molar fraction of component m EM
in B, molar fraction of component m E M in D1 molar fraction of
component me M in ML,, molar fraction of component rnE M in ZNi
xuu Ti,sk,m molar fraction of component rnE M in OUT,
XINSi,sk,nt molar fraction of component me M in
ZNSi,,, XOUTS,,, molar fraction of component me M in
OuTSi,sr, Y, binary variable associated with column
1EL
zi binary variable associated with reactor icl
ZSi.sk binary variable associated with subunit sk E SK of
reactor i E IPFR
II
letters payout time income tax rate fixed charge cost of reactor
i E 2 variable charge cost of i E I reactor total operation time of
the plant fixed charge cost of column 1 EL stoichiometric
coefficient of component m E M in reaction path rp E RP functional
expression for the yield or sel- ectivity
-
Synthesis of isothermal reactor-separator-recycle systems
1381
REFF.RENCF.S
Aggarwal, A. and Floudas, C. A., 1990, Synthesis of general
distillation sequences for nonsharp separations. Currrput. Chem.
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Aris, R., 1969, Discrete Dynamic Programming. Blaisdell, New
York.
Auzerais, F. M., 1988, Automatic synthesis of reactor/separ-
ator network configurations. Submitted as a second pro- position at
the Department of Chemical Engineering, Princeton University.
Barona, N. and Prengle, H. W., March 1973, Reactor design for
liquid-phase processes. Hydrocarbon Processes 63-79.
Benders, J. F., 1962, Partitioning procedures for solving
mixed-variable programming problems. Number. Math. 4, 238-252
B&hop, J. and Meerhaus, A., 1982, On the development of a
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Bodman, S. W., 1968, The Industrial Practice of Chemical Process
Engineering. M.I.T. Press, Cambridge, MA.
Brooke, A., Kendrick, D. and Meerhaus, A., 1989, GAMS: A Users
Guide. Scientific Press, Redwood City, CA.
Chitra, S. P. and Govind, R., 1981, Yield optimization for
complex reactor systems. Chem. Engng Sci. 36, 1219-1225.
Conti, G. A. P. and Paterson, W. R., 1985, Chemical reactors in
process synthesis. Process Syst. Engng, Symp. Ser., No. 92,
391-397.
Floquet, P., Pibouleau, L. and Domenech, S., 1985a, Reactor
separator sequenws synthesis by a tree searching al- gorithm.
Process Syst. Engng, Symp. Ser., No. 92, 415426.
Floquet, P., Pibouleau, L. and Domenech, S., 1985b, Pro&d-
ures doptimisation de cascades de rOacteurs avec ou sans recyclage.
Chem. Engng J. 30, 1 l-2 1.
Floudas, C. A., 1990, OASIS: Discrete/Contintious Optimiza- tion
Approaches in Process Systems. Computer Aided Sys- tems Laboratory,
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GeolTrion, A. M., 1972, Generalized Benders decomposition. J.
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Guthrie, K. M., March 1969, Capital cost estimating. Chem. Engng
114142.
Hartmann, K., 1979, Experience in the synthesis of optimal
chemical process systems. Proc. i2th Symp. on Computer Applications
in Chemical Engineering, Montreux, Switzerland.
Hornibrook, J. N., 1962, Manufacture of styrene. Chem. Ind.
872477.
Kokossis, A. C. and Floudas, C. A., 1989, Optimal synthesis of
reactor networks. I. Chem. E. Symp. Ser., No. 114, 261-272.
Kokossis, A. C. and Floudas, C. A., 1990, Optimization of
complex reactor networks-l. Isothermal operation. Chem. Engng Sci.
45, 595-614.
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Management and Information Systems, University of Arizona.
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9-19.
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APPENDIX A. THE COMPLETE MINLP FORMULATION
Max Profit or min Cost or max a.
Subject to:
+ a;+,XFC:., >
= 0
+ bf, XFC&
>
= 0
CostfP - yizi + ai vi = 0
Cost= - 1 a z i
cosp + c Cost,P - p c cost;pe = 0 1.s 1 ZEl
Profit - x x SP;t.FC,.;XFC,,, IELdP rnEMdP 1. m
- me-&SP,. 0. FD, - Cost- = 0
FD,-_cFRi.,=O mcMmf iEl
c RR,,, XOUT,,, + 1 M&m, XM&,, ILE, ?hrsh#-=
+ FRi.,f - INi. XINi+, = 0 mgM, iEIs, mffM
BPip.sk- I XIN.%p.sr,- I + OuTSip.slr- I XOUTSip,st- 1.m
+ 1 RRSi,ip.,k X0 u T,.m + FRSi,wt.ml id
rt _Es)fRLr.slr. XMR,.., - BP,,.,, + 1 . XIN%.,, + 1.m - fN%,sh+
I XJW,.,I,+ ,.,.a = 0
ip 6 lpFR, skESK, sk = 1, mrM
FRi,, - c FRzY~,,~.,,, = 0 i E IPFR, m E Mm r&SK
RR,,+ - x RRS,,,,,., = 0 iE I, ipEIPFR Sk&K
MR,,, - x MRSi,,..r = 0 iEIPFR, mcM .TkSK
-
1382 ANTUNIS C. KOKOSSIS and CHRISTODOULOS A. FLOUDAS
c RR,,, . XOUT;,, + c RM,,, . XOUT(., !sEI 1d
- OUT{, XOUT;,, = 0 isl, meM
BPip,sk XINSip.,k + OU TSic,sk XUU TS,,s,k m
~ OUT,; XOUTi,,, = 0 ipsIPPR, skESK, sk = N,,
~RM,.I. XOUTi,, - 1 MM,,, XFC,,, i mIE.aP
- FC,-XFC,.,=O ieJ,m~M
c XFC,,,, . FG + z MM,., . XfG., IE.q I
lE~
REC, - c MM,.,, . XFC,,, le.4
- CMR~*,,- XMR,,, = 0 rnEM, mreM 1e,
REC, + PG, - x XFC,,; FC, = 0 mr~ Mrc f E r;;;
xXMR,V., - 1 =0 mrEM m
x XIN,,, - 1 = 0 if i me%4
c XOUT,,, - 1 = 0 ieI _M
c XINSi,,r,, ~ 1 = 0 ieiPFR, sk ESK nrEM
c XOUTSi,srs, ~ 1 = 0 iEiPFR, skeSK IncM
1 XFC,_,-- 1 =0 1sL nlEM
c X&., - l=O ICL REM
c XD,,, - l=O IEL WM
IN,. XINi,, - OUT,. XOUT+,,
- Vi .I ~.p,m R,,.i = 0 in IcsT, msM P
INSi,ti X~NS
-
Synthesis of isothermal reactor-separator-recycle systems
1383
Table 1. Parameters for example 1
Cost of utilities Steam Cold water
Sale and purchase prices Chlorine Benzene Monochlorobenzcne
Payout time Income tax rate
$21.67/10 kl yr $4.65./10 kl yr
$19.88/kmol $27.98/kmol $92.67/kmol
2.5 yr 0.52
In the fourth case, however, the capital costs of the reactors
(CSTRs and PFRs) is type dependent and given by the following
expressions:
cos@$ = 25,794.255. Z,,,, + 8178.003. V,,,, @IO) Costy$p =
3894.938-Z,,, + 49.332.715. VP,,. (B11)
The cost of cold and hot utilities, the prices of the reactants
and products, and other parameters of the synthesis prob- lem, are
given in Table 1.
Example 2 For columns 1 and 2 the simulations were performed
at
50, 100, 200 and 300 kmol/h and covered molar composi- tions
40.0-53.5% of A, 3-Z% of B and 4.5-24% of C. For column 3 the flow
rates were 20, 50, 70 and 90 kmol/h and
the molar composition of B ranged from 60 to 79.5%. Finally, for
column 4 the feed flow rates were 75, 85,90 and 95 kmol/h and the
molar composition of A ranged from 45 to 64.5%. The purity
specification of the simulations was set to 99%.
The annualized venture cost of each column was calcu- lated from
the simulation results and the values af the cost coefficients
(payout time, income tax rate, unit price of utilities) used in the
previous example. Within relative error less than lo%, regression
analysis on the results provided the following expressions:
Vencosr, = 41,357.320. Y, + FC, (432.709 + 418.420. XFC,.,
- 152.456. XFC,.,)
Vencosty = 622,272.549, Yz - FC,
(562.283 + 633.467. XFC,.,
+ 389.655. XFC,,,)
Vencost\ = 54,966.389. Y, + FC,
(748.609. XFC,., - 588.673 XFC:,,)
Vencosty = 40,526.895. Yd + FC, (382.765 + 432.315 XFC,,,).
0312)
@I31
(B14)
0315)
The capital cost of the reactors was given by the
expression:
Cust;Bpita = 12,760.43 - Zi + 14.059.78. Vi (B16) while the
operating cost was assumed to be negligible.