Optimal Design of Chemical Process by Feasible Decomposition Method Tomio Umeda,l Akio Shindo, and Eiichiro Tazaki Chiyoda Chemical Engineering & Construction Co., Yokoha?na, Japan
A feasible decomposition method for optimal design has been developed and successfully applied to the optimal design of a chemical processing system which consists of three subsystems: reaction, distillation, and heating units. The convergence behavior of this two-level approach has been compared with that of the single-level approach, and the results showed that the present method gave better convergence character than the single-level approach. As a search method of optimization, the Complex method of M. J. Box has been used in optimizing the subsystems and in coordinating the solution for subproblems.
T l i c o1)tinial design or chemical processes in practical cases has bccn carried out by applying sequential search methods, owiig to tlic complexity of required mathematical inanipulatious or to tlie discontiiiuous nature of an objective function and/or coiistraints. The sequential search methods for optimal design involve the simulation of t'he process using a set of feasible values foi the decision variables and the evaluation of tlic correspoiiding objective fuiiction. This process is coilt i m e d by choosing better sets of feasible values for the decision variables aiid comparing the calculated values of the objectivc fuiictions, until a preset stopping criterion is satisfied. Clieniical processes, however, are generally complex and large scale, and are often linked to the necessary utilities, so that the alqilication of sequential search met'liods is limited. T o eliminate computational difficulties associated with the optimal design of such large-scale systems, methods of decomposing the integrated problem into many smaller subproblems have been devised. Two methods of tlie decomposition called feasible aiid 11011feasible methods, have been present'ed (Brosilow et al., 1965; Brosilow and Lasdon, 1965). In the former method, the overall system equations are always satisfied, while the stationary conditions for optirnality are violated duriiig the iterative computations. These condit,ions are satisfied when t'he optimal solut'ion is obtained. In t,he lat,ter method, the overall system equations are not' satisfied during the computations except a t the optimal solution. In solving practical problems, the feasible method of deconipositioii is more useful since the apglicatioii of the noiifeasible inethod requires complicated mathematical manipulation for obtaining the stationary conditions, and also initial values for Lagrange multipliers are not easily found. T h e authors, iii the previous work (Tazaki et al., 1970), presented a feasible decompositioii method with the application of general search methods instead of using the st'ationary conditions. The method is generally applicable regardless of opt'imization techniques, if a iiiathernatical condition described later is satisfied. In this paper, this method of feasible decomposition is combined with the Complex method ( I ~ o s , 1965). The Complex method has been successfully applied t o several constrained optimization problems by one of the authors (Umeda, 1967, 1969) and others (Robbins and
Francis, 1969; St'evens, 1970). One of the advantages of this method is that it handles general inequality constraints, and it has been considered to be a rather generally applicable search method. The combination of an effective problem-solving approach with this search method may thus serve to extend its applicability to large-scale problems. Furthermore, this approach may give useful information o n the sensitivity characteristics with respect to coordination variables, since the subsystems are optimized a t every iteration for giveu values for the coordination variables. As a practical example, the optimal design of a typical chemical process consisting of two coiitiiiuous stirred tank reactors, two distillation columns, and a simple utility facility is solved to sliow the applicability of the proposed method. Feasible Decomposition Method
iiii integrated optimization problem inay be defined:
maximize + ( x ,u) subject' to x = F(u) and G(x, u) 0 where x E X and u E U are state aiid decision vectors, respectively. S and U are subsets of linear finite dimensiional spaces and F and G are vector-valued functions. It, is assumed that the integrated problem may be decomposed into smaller subproblems, each of which has the following objective fuiiction and subsystem equation: objective funct'ion +m(Xm, urn1,Urn2) syst'em equation X m = fm'(Urn', Urn2) inequality constraint g m ( x m , Urn', Urn*)
The overall objective function is represented by the separable M
form of these objective functions, +(x, U)
=
+m(Xm, Urn', m=l
urn2).Among these variables, the following interrelation exists: Urn2 = f m 2 ( X 1 , xz,
. . . , ~ m - 1 ,Xm+l, . . . , X M ,
ull, up1, . . . ,
where
Xm
E
X m =
To whom correspondence should be addressed.
3o
urn' E
=
1, 2, . , . ,31
E
X m
X
Xm-1
X
Xm+1
X
X U,X , . . X
Urn-1
X
Um+l
X . . . >< U M
Xm,
XI X X Z X
U m = Til
. , , , uitfl),m U m , Urn2
X
Urn
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
1
X , and LYm are subsets of X and C , respectively, and these are associated with the subsystem m . Furthermore, urn2iii tlie above i:, espressed by tlie following set of relation5 which shows the iiiterconnection betn-eel1 subsystems: urn2
=
[Umi2, um22,
. . , , umn2, . , . ,
~m.rr~]
umn2 = fmn2(Xn, U n ' ) , n
=
1, 2, . . . , III
?n
=
1, 2 , . , , , JI, n #
171
where umn2is the coordiiiatioli variable associated with tlie output of subsystem n and t'he input of subsystem m. This is an elenient of urn2.If the optiniization of each sub carried out taking this interrelation into accouiit, fcnsible solutiolis are obtained. Kow let us define the followiiig problenis 011 the first' and second levels.
Subproblem ?n on First Level iuasiruize qTn(xni, unhl> um2)
(1)
by choice of urn1 subject to xm = frn1(um1, urn2) gm(xm,
urn1,urn2) 3
u n m 2 = fnrn2(Xrn,
(2)
o
(3)
urn') n = 1, 2, . ,
,,M, n
where unm2are the nth elements of urn2(m and urn2is determined on the second level.
=
# m
1, 2 ,
.
,
(4)
. , M),
Coordination on Second Level 'I4
maximize
C +m(Xm,
Urn',
um2)
(5)
m=l
by choice of U m 2 . X m and Urn' are determined on the first level, and the coordination determines urn2in such a way as to maximize the objective function of the integrated system. T o apply the feasible decomposition method, it is necessary that the number of decision variables on the first level should be greater than that of the coordination variables on the second level; the following condition must be satisfied: M n=l
dim. [unm2] 6 dim. [urn1], m
=
1, 2 , . .
,M
(6)
streams as decision variables Unm2 on the second level. Some appropriate value is assigned for each of these variables and these become the iiiput aiid output conditioiis for the subsystems associated with the severed streams. The optimization of the subsystem problems is carried out with respect to the decision variables urn1for the given input' and output condit'ions. I n solving the subproblem which is defined by Equations 1 through 4, we must, solve Equations 2 and 4 for xmand u,l. for given U n m 2 (n = 1, 2, , X ) ,or Urn2. Then um1 is determined in such a way that the objective function 1 is niaxiniized, subject to tlie inequality constraint 4. The set of xnL, urn',and urn2obtained in this way always satisfies the equality and inequality constraints (Equations 2-4), and feasible solutions for the subproblems are determined by finding optimal values of urn1. To solve the problem 011 the second level is to determine urn2,for given xrn aiid Urn', in such a way that the overall objeclive function 5 is masimized. Only inequality constraints on urn2are imposed in this problem. .Iccording to tlie above statement, the followiiig cornputmation procedure for the feasible decomposition method can be derived : Step 1: ilssunie values of Urn2, m = 1, 2 , . . , , M, in Equations 1-4. Step 2 : Optimize each subsystem subject to Equatioiis 2-4. Step 3: Check whebher the assumed values of urn2should be corrected by the optimalit'y crit'erion for the integrated system. If satisfied, stop the computations. Otherwise go to Step 4. Step 4: Revise the values of urn2 and return to Step 2. For solving the problems on the first and second levels, gradient-type search methods have been widely used (13rosilow and Lasdon, 1965; Brosilow et al., 1965; 13rosilow and Suiiez, 1968). However, the gradient methods have their limited applicability in solving problems with large dimension. In this work, t'he Comples method (Uos, 1965) is applied both to optimize t'he subsystems and to coordinate these solutiolis so as to achieve the overall optimum. The new trial point8sfor urn1and urn2can be obtained in vector form by the following relations:
where dim. [ ] represents the dimension of respective variables (Tazaki et al., 1970). I n Figure 1, the above-mentioned decomposition aiid the interaction among the variables are shown. K2
Method of Solution
[UrnYi)IIc- um,w2
The integrated problem can be decomposed into many smaller subproblems in various ways. The only requirement in making this decomposition is the satisfaction of the condition 6 above. The subproblems are formed by severing streams between subsystems aiid by treating the severed
I
Coordination
ISubproblern ml
Ui
UL
1
Second level
ISubprubIem MI
First level
Uh
Figure 1. Coordination and subproblems 2
Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 1 , 1972
urn2(i+1) = (1 + a )
iz=l
K2
- auin,w2(i) (8)
where the argument i denotes the ith iteration and subscript w stands for the worst point. K,' and K 2 show the number of vertices for the mth subproblem and t,he coordinat,ion, respectively. If each of these new trial points is also the worst or does not satisfy some inequality constraint, it is moved halfway toward the centroid of the remaining points. Explicit forms of inequality constraints are more easily handled by keeping the independent variables within feasible regions. The detailed computational procedures in practical design have been described in a previous paper (Umeda, 1969), and further explanation is omitted here. The comput'atioiial algorit'lim is given in Figure 2. The iterative computations, using the above relationships 7 and 8, are continued until prescribed stopping criteria for each
pidKGi-
obiective function I
algorithm (1st leve I )
,
I
( 2nd.leve I )
Figure 2. Computational procedure by feasible decomposition method
a reaction unit, a separation unit, and a n auxiliary utility facility. The reaction unit involves a two-stage continuous stirred tank reactor (CSTR) equipped with reflux condensers, chlorine recycle systems, and cooling systems to remove the heat of reaction. The separation unit consists of two distillation columns with reboilers and condensers. The auxiliary utility facility consists of a furnace and its accessories. For numerical computations, this problem is simplified to that of producing monochlorobenzene (Bodman, 1968). Benzene and chlorine are reacted in the CSl'Rs, and after removal of unreacted benzene in the first column, the monochlorobenzene is separated as a product from other chlorobenzenes in the second column. The unreacted benzene is recycled to the first CSTR. The total heat required in the reboilers is provided by a furnace. For the optimal design problem, the following venture profit is defined in terms of the symbols given a t the end of the paper: objective function
= m
where IX2*
2
@m
is defined by:
r
I
Subsystem 2
L
s
k
n
ULUA
Figure 3. Illustrative example
iteration are satisfied. For illustration, a simple example shown in Figure 3 is taken up. I n the form of scalar-valued functions, the subproblems 1 and 2 , and coordination problem are defined : Subproblem 1 maximize $1 (21, ulll, ud,u2,u212) b y choice of uIl1,ulZ1 subject to z1 = jll(ulll, ud,ulz2),x1 = u212 and gl(rl, ulll, ud,u2,w 1 2 ) 3 0 Subproblem W maximize b ( 5 2 1 , ZZZ, uzll, U Z ~u2, , UZI~) by choice of u ~ ~uZz1 l, subject to z21 = f ~ ~ ~ ( U ZuZ '2 ,uz12), ~~, zzz= u 1 2 ~ 2 2 2 = f221(?L211, uz21, u219 and g2(z21, 2 2 2 , uzll, ud,U ~ Zu212) ~ , 3 0
Coordiibation Problem maximize
h(z1, ulll,
+ +Z(Z~I,
ulzl, u2,u2l2)
2 2 2 , U Z I ~UZZ', ,
for the mth subsystem. The first term represents the difference between product sales and raw materials cost. The second and third ternis are ut'iiity costs and fixed charges, respectively. This objective function should be maximized by finding some proper values for independent design variables within admissible regions. The analysis of the optimization problem gives us 13 independent variables shown in Table I and Figure 5. Values of these variables are properly assigned so t'hat the mass and heat balance relationships within feasible regions can be set up. The mathematical models for process design calculations are given in the next section. Mathematical Models. For the present purpose shortcut methods for process design calculations are adopted. The
Table 1. Independent Variables for Application Problem Single-level
Two-level
u2,
Coordination
... . .
UZl')
b y choice of u1z2,uz12 F o r assumed values of u1Z2and U Z ~ ~the , subproblems 1and 2 are solved for ulll, ulz1 and uZl1,uzzl,respectively. I n the subproblem 1, ud is obtained by solving the subsystem equation for given X I ( = u212)),uy22,and for an assumed value of ul2I. Similarly, U Z Zis~ found for given zl( = u , ~ ~ xZ2( ) ) , = u122) and for an assumed value of uPp1. The value for z21is obtained from one of the subsystem equations. Inequality constraints g1 and g2 must be satisfied, respectively. The revision for uIQ2and u212 and the solution of the above-mentioned subproblems are continued until the quantity $*) is maximized.
21,4 23.4
SI,1
S1,l
s2.1
S2,l
Th,l
Th,i
Th,2
Th,2
Subsystem 1 Tr,1 T?,z Subsystem 2
T?,I T,,2
s2.2
+
Application to Optimal Design of Chemical Process
Statement of Optimization Problem. T o apply t h e method presented in the preceding section, a n optimization problem of chemical process design (Figure 4) is chosen. It is a typical one and can be decomposed into three subsystems:
Subsystem 3 7
t 71 72 ~
~~
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 ,
NO. 1 , 1972 3
The heat of reaction is removed in the external cooler and in the jacket of the reactor. The holding time of the nth reactor, 712, and the heat transfer area of the cooler, should be optimally determined. Separation Unit. The product, monochlorobenzene, is separated from other materials in the separation unit. The mass balance relation for the n t h distillation unit is represented as follows:
Distillation columns
h
1
+ bt,n for the ith component
fi,n = d z , n Furnace
m -
the heat balance relation around the nth column is given b y FnCHt,n(Tf,n) a
-
DnCHt,n(Td,n)
- BnCHt,n(Tb,n) z
%
Figure 4. Typical processing system with utility facility
&b,n
1
1-s~l.n
log
Sm,n
~
ui’:1 Figure 5. Coordination and subproblems for application problem
=
log
- ~ 2 , n=-
S Z , ~
0 (12)
steady-state mathematical models for each subsystem are described as follows: Reaction Unit. The chemical reactions considered here are represented by
+ HCl C&Cl + C12 CsH4C12 + HC1 Although the reaction, C6H4C1z + C12 CsH3C13 + HC1,
1- ~ 3 , n
log a23 ,n
01z,n
-
(dt,n/ft,n),
(13) and
for Underwood’s equation where Tn is a root of a certain value between c ~ ~and i , a~i h , n (suffixes 1 and h denote the light and heavy key components, respectively) of the following equation: (ft,nlFn)
+
=
~2,n 1- ~ 2 , n ~ 3 , n - log ___ ___
1
S1,n
for Fenske’s equation where S i , n = 1 c ~ 1 2 , nand 0 2 3 , denote ~ relative volatilities.
+ Clz
- &c,n
+
T h e number of trays can be calculated by the short-cut method based on Fenske’s (1932), Cnderwood’s equations (1948) , and Gilliland’s correlation (1940). These relations are given by
I,I,SU
CBHB
(11)
? T-
alt,n
- 1 - qn - 0 rn
and Sm,, and R m , n are used to obtain S n for a given R n by the Gilliland correlation:
C&&1
+
The diameter of the n t h column is calculated by:
+
occurs simultaneously, it is neglected for the sake of simplicity. The validity for this simple treatment may be confirmed by the result of optimization with the upper constraint for a holding time. The mass balance for the nth reactor under perfect mixing is expressed by: C1,n
Cz,n
-
-
C1,n-1
= -k1,nCl,nTn
C2,n-1 = kl,nCl,nTn
C3,n
- C3,n-1
-k~,nC~,n~n
= kz,nCZ,nTn
(9b)
where us,n is a superficial velocity obtained as a function of a tray spacing and a height of liquid seal on a tray. The heat transfer areas of the reboiler and the condenser are calculated by Ab,n = &b,n/Ub,n(ATLM)b,n (174
(9c)
for the reboiler
(94
where the reaction rate constants k,,n are given in the Arrhenius type of expression (Bodman, 1968) k1,n =
kio exp
(-El/RTr,n)
kz,n =
IC20
exp
(-Ez/RTr,n)
The heat balance relation associated with the n t h reactor is given by
C Httn-1(Tr,n-d -
4
Hi,n(T’T,n)
z
+
(-AHl,n)k1,nCj,nTn
Ind. Eng. Chem. Process Der. Develop., Vol. 1 1 , No. 1 , 1972
+
A c ,n
=
Qc
, n / U c ,n (ATLM) c,n
(17b)
for the condenser where ( A T L M ) b , n and ( A T L M ) c , n are calculated after obtaining the bubbling point T b , n and the dew point T d , n respectively. Utility Facility. The generally known utilities required for the operation of chemical plants are cooling and heating media. However, the heating facility considered in this application consists of a furnace and its accessories only. Since the emphasis is placed on the decomposition of a total system into subsystems, it is considered sufficient t o include a simple facility in the total system. The furnace design equation for obtaining heat transfer areas is obtained by
Uh
where
7,)
Th,3,and Th,4are obtained by empirical functions of heat of combustion of fuel.
(Th,4
Coordination
ip, C Q h , n , and 7'
Decomposition of Problem. T h e study for decomposing the integrated problem determines the decision and coordination variables. T h e number of independent variables of the integrated problems is 13, as shown in Table I. If the system is decomposed into three subsystems b y severing the streams 4,6,9, and 10 in Figure 4, the corresponding subproblems can be separately solved. Although there are three degrees of freedom in streams with numbers 4 and 6, respectively, mole fract,ions for two components in these streams are chosen as the coordination variables, so that t,here remnin two and four decision variables in the reaction arid separation units, respectively. For the utility facility, two of three variables associated with the corresponding subprobleni are chosen as the coordination variables. When we take computational convenience into account , the decision and coordiiiat,ioii variables are chosen as presented in Table I. 011 the basis of this table, the following subproblems are defined :
Subsystem 1 (Reaction Unit) maximize
$1
= (1
-
(3) X
by choice of T,,1,T,,2 subject to the subsystem Equations 9a, b, c, and 10, and the inequality constraints, 40 5 T,,,, T,,z 5 SO("C), and 0.1 5 T ~ 7 ,2 5 1.0 (hr).
- T h , 2 ) - ( T h , 3 - Th.1) 3
maximize4
$ ,
= m=l
by choice of (Fl,J/F4)r (F3,4/F4), &,I, Th,l,Th,2 subject t o 0 5 (Flt4/F4)t (F8,4/F4), s I j lS, Z ,I ~ 1, T h , l and 10 5 ( T . b , 1 - Th,l) 5 100 ("C)
5 Th,?
The interrelation of these variables is shown in Figure 5 where u l " , UZ", and u31'denote the subset chosen from ull, uzl,and u3l, respectively. Computational Results
The optimization was carried out using the procedure shown in Figure 2 . The number of vertices for the second-level problem is seven and those for the subproblems are four, six, and four for subsystems 1 , 2 , and 3, respectively. With the Complex method, undesirable contraction is t.0 be avoided by adding two or three vertices to the number of independent variables in each subsystem problem. The optimal results under the given conditions (Table 11) are presented in Table 111. These results are considered to be global optimum since several trials using different initial values for the decision variables gave almost t,he same results. The stopping criterion was defined by the relative error of the greatest and least function values a t the vertices of a simplex during the iterative search: thus calculation was stopped when - $min)/$max becomes less than 0.015. T o show the convergence behavior of this method, the value of the objective function for the integrated system is depicted in Figure 6 with respect to computing time. The computational result obtained by the single-level approach with the Complex method is also shown for comparison purpose.
Subsystem 2 (Separation Cnit) maximize4z = (1 - P ) X
by choice of s ~ , s ~3 , 3,, (R/R& ( R / R m h subject to the subsystem Equations 11 through 17b, and the inequality constraints, 1 5 ( R / R m ) l ,(R/R,)* 5 4,and 0 I S t , ] , sz,z I 1, (i = 1, 2, 3).
Subsystem S (Ltility Facility) maximize 43
=
(1
- p)
X
by choice of 7 subject t o the subsystem Equation 18, and the inequality constraints, 5 I ( T h , l (Th,l - T b , 2 ) I 100 ("C). The coordination of the solutions of these subproblems is defined as follows:
-55.001 50
Computation time
100
Second
Figure 6. Convergence behavior of total objective function Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
5
Table 111. Optimal Results
Table II. Specified Conditions for Optimal Design
Single-level
Unit
Flow rate Benzene Product specification Monochlorobenzene, min Conditions for reaction system Ai A2
Kg-mol/hr
97
%
5 . 1 X 10l2 l/hr 2 . 9 X lozo l / h r 19.6 Kcal/mol 32.6 Kcal/mol 31.26 Kcal/mol 29.46 Kcal/mol 150.0 Kcal/m2hr "C 200.0 Kcal/mzhr "C
E1 E2 (- AH') (- AHz) U (jacket) U (cooler) Conditions for distillation system Tray efficiency Tray spacing U (condenser) U (reboiler) Conditions for utility facility
c,
20
M
Kcal/m2hr "C Kcal/mzhr "C
Th,4 Uh4
Economic data Cooling water
Kcal/kg "C Kcal/kg "C Kcaly'mzhr "C
5.0
Heating medium
Yeny'ton or centy'3.6 ton Yeny'kg or cent/3.6 kg Yen/103 kcal or cent/3.6 X lo3kcal Yen/kg or cent/3.6 kg Yen/kg or centy'3.6 kg
158.0
Fuel
0.65
Benzene
30.0
Chlorine
35.0
Minimum acceptable rate of return Tax rate
0.15 0.50
Each point of the two-level approach involves several iterations for finding the optima of each subsystem a t each stage of coordination. Therefore, comparison has been made by computing time rather than the number of interations. Twolevel approach gives better convergence characteristic than the single-level approach. The required computing time by the two-level approach is about 1 min by I B M 360/75I and it is about half of that by the single-level approach. Discussion
The first step necessary in the application of the feasible decomposition method is to examine whether Equation 6 is satisfied in the decomposed problems. In the example of the preceding section, Equation 4 reduces to the following relationships:
and 6
57.6 49.5 0.0009 0.9418 3.1 2.3
sz,2
83.2
(R/Rm)l (R/Rm)z
0.341 9600.0 1660.0 30.0
(- A H ) fuel
T,,i T,,2
%
60.0 0.6 250.0 300.0
uz12
=
f212(Xl, u 1 1 )
Ul22
=
f12YX2,
uz')
uZ32
=
f232(X3,
u3')
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1, 1972
Two-level
Objective function 26.12 26.35 Net sales 873.59 869.06 Raw material cost 765.19 762.29 16.47 14.20 Operating cost Fixed cost 20.32 20.15 [Unit of above figures is 106 yen/year or (106/360) $/year] Independent variables Coordination variables 21,4 ... 0.7113 x3,4 ... 0.0057 Sl ,1 0.0024 0.0095 SZJ 0.9537 0.9825 Th ,I 204.4 194.4 Th,2 272.9 285.3
0.701 0.117 0.263
17 71
72
Subsystem 1 45.1 43.3 Subsystem 2 0.0009 0.9005 1.93 1.57 Subsystem 3 0.694
... ...
where uz12 = [(F1,4/F4), (F3,4/F4), F41, u12 (F2
u232 =
and
(Th,lr
=
[(F1,6/F6),
,dFd F6 1 i
T ~ , z~) ,1 =' (TT,,,T,,z,7 1 ,
72)
SI,^, ~ 2 , 1 ,(R/Rm)l,SZ,Z,&,2, (R/Rm)21
UZ'
=
u3l
= (Th,l,
Th,d
State vectors xl,x2, and x3 are determined by these decision and coordination variables using Equations Sa through 18. The decomposition into three subproblems satisfies Equation 6 as follows: dim.
[uz12] =
dim.
[ U I Z ~ ]=
dim. [u2s2] =
< dim. 3 < dim. 2 < dim. 3
=
4
[UZ'] =
6
[ull]
[ugl]
= 3
If these ulzZand uZl2 were chosen as the coordination variables, there would remain no degree of freedom for the optimal design of the first distillation column, and only one degree for the reaction unit. Thus, two of these three variables in ulZ1and uZ12 have been chosen as the coordination variables, so that two- and four-decision variables have been left for the optimization of the reaction and separation units, respectively. I n addition, the form of design equations should be taken into account since the proper choice of decision and coordination variables strongly affects the overall computational efficiency. In particular, explicit form of relations requires less computing time. In the example, the choice of sl,l and ~ 2 instead , ~ of (F1,6/F6) and (F2,6/F6) makes it possible to eliminate the iterative computation with respect to the recycled stream. This leads to incomplete decomposition stream, but since the design equations employed here are widely used, the incompletely severed streams will often be encountered in the optimal design of chemical processing systems. These con-
Acknowledgment
Sn,I rl.00
The authors are grateful to Chiyoda Chemical Engineering and Construction Co. for permission to publish this paper. Nomenclature
Ab,n
,
50.0
400
30.0
I
I
1
0.5
0.6
1
0.7
Fs
0.8 F1,4/F4
Tr,
I
Figure 7. Interrelationship among variables along optimal condition of subsystem Between subsystems 1 and 2 0 : optimal point
Th.2-Th.1
1
230
220
210
203
Th,l
070
A
075
heat transfer area for the reboiler associated with the column n Ac,n = heat transfer area for the condenser associated with the column n Ah = heat transfer area for the furnace Ar,n = heat transfer area for the cooler associated with the reactor n Bn = total molar flow rate of the bottoms of the column n bi,n = molar flow rate of component i in the bottoms of the column n bi,n = molar flow rate of component i in the bottonis of the column n Ci,n = molar concentration of component i in the reactor outlet stream n Dn = total molar flow rate of the distillate of the column n D T ,n = diameter of the column n dt,n = molar flow rate of component i in the distillate of the column n Ei = activation energy of the i t h reaction Ek,n = consumption of the kth utility in t'he unit n ex = unit cost of the kth utility Fn = flow rate of stream n, or total molar flow rate of the feed of the column Fi.n = flow rate of component i in F n fa,n = molar flow rate of component i in the feed of the column n H j , n ( T j , n=) molar enthalpy of component i a t T j , nof the unit n = latent heat of component i a t T b , ,of the unit n T = investment cost of the unit n = number of vertices of a simplex associated with solving the subproblem m = number of vertices of a simplex associated v i t h coordination = the ith, reaction rate constant (i = 1, 2) = frequency factor of the it'h reaction rate (i = =
0 80
'I Figure 8. Interrelationship among variables along optimal condition of subsystem Between subsystems 2 and 3 0 : optimal point
siderations have given us the final choice of variables shown in Table I and Figure 5 . In addition to having better convergence characteristics, this method has practically useful information contained in its results. Figures 7 and 8 are obtained by using the optimal designs of the subsystems for iteratively given values of the coordination variables during the computations. Each point in the figures is the optimum point for the associated subsystem; thus, the lines show approximate optimal trajectories. For example, Figure 6 shows that the decrease in T,,, requires the increase of s ~ and , ~ Fg to maintain a suboptimal set of design variables. Knowledge of such variations is most useful in practical process design. I t is also possible to make sensitivity analysis when several optimal results for subsystems are obtained by varying the values for parameters with fixed coordination variables (Unieda e t al., 1969).
'J
1'
= production rate of stream s = product sales of stream s = =
reboiling duty for the reboiler of the columii I I condensing duty for the condenser of the coluniii
n = = = =
= = = = = = =
= =
= =
q-value of the feed to the colunin n minimum reflux ratio for the columii n reflux ratio for the column n a parameter used in Equation 14 unit cost of raw material stream s minimum number of stages of the columii 71 number of stages of the column n fraction separated of component i a t the bottom of the column n bubble point of the column n dew point of the column n temperature of heating medium (i = 1 for inlet to the furnace and i = 2 for outlet from it) reaction temperature of reactor n admissible set of decision vectors heat transfer coefficient of the reboiler n+ociated with the column n heat, transfer coefficient of the coildenser aesociated with the column n
Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 1 , 1 9 7 2
7
heat transfer coefficient of the convection zone in the furnace = subset of U associated with the subsystem m Urn = heat transfer coefficient of the cooler associated Uv,n with the reactor n U = decision vector Urn’ = decision vector in the subsystem m Uml’ = decision vector, subset of u ,’ urn2 = coordination vector (umI2,u,2, , umn2, . , u,h?), m # n umn2 = coordination vector among the subsystem m and n Us,n = superficial velocity in the column n V ( T b , , ) = specific volume at T b , ,in the column n X = admissible set of state vector x m = subset of X associated with the subsystem m X = state vector Xrn = ihate vector associated with the subsystem m =
Uh
GREEKLETTERS CY = expansion factor of the Complex method = relative volatility of component i with respect “tj,n to component j in the column n P = tax rate (- A H ) I = heat of the i t h reaction mean temperature difference of the ( L ~ Z ’ ~= ~ ~logarithmic ) ~ unit n = thermal efficiency of the furnace 1 = thermal efficiency in the radiation zone of the Br furnace x = annual expeiise rate of investment = minimum acceptable rate of return P 7 = holding time of a reactor 9 = objective function = objective function for the subproblem m +7n SUBSCRIPTS = bottom or bubbling poiiit I, C = condenser
d
distillate or dew point feed heating medium or heavy key component component number light key component = subsystem number = unit number = reactor = stream number = = = = =
f
h i 1 m
n T
R
SUPERSCRIPTS 1 2
= the first level = the second level
literature Cited
Bodman, S. W., “The Industrial Practice of Chemical Process En ineering,” pp 18, The MIT Press, Cambridge, Mass., 1968. Box, J., Computer J.,8 , 42 (1965). Brosilow, C. B., Lasdon, L. S., -4.I.Ch.E-1, Chem. Engrg. Sym. Ser., No. 4, 75 (1965). Brosilow, C. B., Nunez, E. J., Can. J. Chem. Eng., 46,205 (1968). Brosilow, C. B., Lasdon, L. S., Pearson, L. M., SRC Rept. 70-A-63-23, Case Inst. Tech. (1965). Fenske, hl. R., I n d . Eng. Chem., 24, 482 (1932). Gilliland, E. R., Ind. Eng. Chem., 32, 1220 (1940). Robbins, T., Francis, N. W., AIChE 64th National Meeting, New Orleans, La., Preprint No. 3413 (1969). Stevens, W. F., AIChE 67th National Meeting, Atlanta, Ga., Preprint No. 31e (1970). Tazaki, E., Shindo, A,, LTrneda,T., IFAC Syrn., Kyoto, Preprint No. 25.1 (1970). Umeda, T., in “Saikin no Kagaku Kogaku,” (Advances in Chem. Eng.) hlaruzen Pub., Tokyo, Japan, 1967. Umeda, T., Ind. Eng. Chem. Process Des. Develop., 8 (3), 308 (1969). Underwood, A. J. V., Chem. Eng. Progr., 44, 603 (1948).
b,
RECEIVED for review May 6, 1970 ACCEPTEDAugust 2, 1971 Support for this study came from Chiyoda Chemical Engineering and Construction Co.
Experimental Determination of Catalyst Fouling Parameters Carbon Profiles James T. Richardson‘ Synthetic Fuels Research Laboratory, Esso Research and Engineering Co., P.O. Box 4255, Baytown, Tex. 77520
Fouling and poisoning in catalyst beds is an important practical problem that has generated considerable theoretical treatmcnt in the past few years. Equations describing the reaction kinetics and diffusional characteristics of the catalyst pellet have been combined in ternis of the catalyst properties, feed characteristics, and process variables which are usually transPresent, address, Department of Chemical Engineering, Universit,y uf Houston, 3801 Cullen Boulevard, Houston, Tex.
77004. 8
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
lated into a description of bed performance from which the reactor behavior may be explained or predicted. Unfortunately, the numerous parameters and assumptions make the models too cumbersome or restrictive. Applications are limited to simple or well-characterized systems. Process designers still prefer to base their performance and scale-up calculations on pilot unit deactivation data for complex feeds and reactions. Greater mechanistic insight may be obtained and some of the empirical correlation avoided if certain of the parameters
