Ind.Eng. Chem. Res. 1993,32, 1087-1092
1087
PROCESS DESIGN AND CONTROL Optimal Synthesis for the Retrofitting of Multiproduct Batch Plants Ho-Kyung Lee,In-Beum Lee,' Dae Ryook Yang, and Kun So0 Chang Automation Recrearch Center, Pohang Institute of Science and Technology, P.O. Box 125, Pohang 790-600, Korea
For change of a market condition such as increase of demand and new selling price, an optimal retrofit method is proposed through adding new equipment to an existing multiproduct batch plant. Until now, the optimum volume and the appropriate position of adding equipment have been solved using a mixed integer nonlinear program. In this paper, a heuristic procedure is presented which first determines the positions of adding equipment. Then a nonlinear programming is formulated to obtain their optimum sizes. The effectiveness of this method is verified by solving several literature problems. 1. Introduction
The continuing shift toward the production of highervalue-added specialty products by the chemical processing industries has stimulated studies for batch processes. Batch operations are used because of the flexibility they provide in accommodating variability in feed materials, the particular nature of certain processing steps (e.g., bacterial culture), and the seasonality of the market demands. Because of the same merits as above, not only the optimum design for new plant but also the retrofitting problem for existing plants has been treated in many papers. Due to the change of production quantities and selling price, the retrofitting problem adding new equipment to the existing batch plants occurs. Thus, we can set up the optimum retrofitting problem to economically determine the size of new equipment. A batch plant consists of a sequence of batch and semicontinuous stages. Because the semicontinuous stages make negligible contribution to the total cost, no semicontinuous stages are considered for the retrofit design. This paper uses the conventions used in the batch plant design problem by Sparrow et al. (1975) and Grossmann and Sargent (1979). The heuristic sizing procedure suggested by Yeh and Reklaitis (1987)could accommodate some forms of retrofitting application of single-product plants. The optimum retrofitting problem of the multiproduct batch process was dealt with by Vaselenaket al. (1987)andFletcher et al. (1991). As their papers (Vaselenak et al., 1987;Fletcher et al., 1991)show, the optimum retrofit design problem for multiproduct batch plants may be formulated as a mixed-integer nonlinear programming (MINLP) problem. The MINLP formulation adopted by Vaselenak et al. (1987)is to add units either all in parallel or all in sequence. The MINLP formulation by Fletcher et al. (1991)shows a unit may have different operating mode for different products. Although they used binary variables in position determination in their formulations, this paper shows that binary variables in position determination are substituted for heuristic rules. Furthermore, current MINLP codes *Author to whom correspondence should be addressed. Telephone, 011-82-662-79-2274; FAX,011-82-662-79-3499.
1
bCycle
I
Tie
Time Cycle time of E
time of A
::sf%$ stage 2
Cycle time of A
Ti:q%/
~
T,mc
Cycle time of E
(b)
Figure 1. Out-of-phase mode retrofitting. (a) Existing plant. (b) Modified plant.
Table I. Data for Illustrative Example
tii (h) product A
B
stage 1 0.5 1
product
atage 1
stage 2 1 0.5 Sij
A B
V
vy
V
v/2
(Lkg') stage 2
VI2 V V
cannot solve the problem with large nonlinear and many binary variables. In this paper, a heuristic procedure is presented which first determines the positions of new equipment. Then a nonlinear programming is formulated to obtain their optimum sizes and we use these procedures to treat four examples (Vaselenak et al., 1987;Fletcher et al., 1991).
0888-5885/93/2632-1087$04.00/00 1993 American Chemical Societv
1088 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 ~
stage 1
stage 2
~
stage 1
~
stage 2
~
Rcdua B
Rcducl A
B,=80
B,=lM)
B,,=IOO
B,,=80
B,=lM)
w
B,=lW
w
U
U
U
B,,=ZO
no use
(b)
Figure 2. In-phase mode retrofitting. (a) Existing plant. (b) Modified plant.
U (g)
Figure 4. Alternative retrofittinge for illustrative example. (a) Case 1(existing plant); (b) case 2; (c) case 3; (d) case 4; (e) case 6; (0case 6; (9) case 7.
stage 1
which consists of M batch stages. The indices i and j identify products and batch stages, respectively. Unit batch size Bij is defined as follows (Grossmann and Sargent, 1979).
stage 2 (in sequence mode)
Figure 3. Two-stage batch plant. Table 11. Process Analysis of Illustractive ExamDle product LCT LBS stagel stage2 A max(0.6,l) = 1 min(V/V,V/(V/2)) = 1 LBS LCT (stage 2) (stage 1) B max(l,0.5) = 1 min(Vl(V/B),VIV)= 1 LCT LBS (stage 1) (stage 2)
B, = Vj/Sij i = 1,...,N j = 1,...,M
where Vj is the volume of the j t h equipment and Sij is the sizingfactor of the jth equipment which produces product i. Therefore LBS (limiting batch size) Bi is expressed as
Bi = min B, j = l ,...,hi
2. Optimal Retrofit
Generally, the optimum retrofitting problem is divided into two branches: to which position equipment is added and how large equipment is added in order to maximize profit. 2.1. Position Determination of Added Equipment. We assume that N producta are to be produced in a process
(1)
i = 1, ...,N
(2)
Unit cycle time Tij is usually expressed as (Grossmann and Sargent, 1979)
T V. .= t V. .+ ci$y
i = 1, ...,N j = 1, ...,M
(3)
where C i j and dij are correlating coefficients between Tij
Table 111. Retrofit Strategies of Illustrative Example operationa case 1
product A B A B A B A B A B A B A B
atage 1
cycle times (h) 2
stage 1
0.5 1
2 1
batch sizes (kg)
limit (LCT)
stage 1
2 2
1 2 1 2 P 0.5 1 2 2 S 0.5 0.5 2 1 3 S 0.5 213 213 1 2 P 1 0.5 1 2 312 4 S 0.5 0.5 0.5 1 2 P 1 0.5 1 2 2 5 P S 0.5 213 213 2 2 S P 0.5 0.5 0.6 2 312 6 P S 0.5 0.5 0.5 2 2 S P 0.5 0.5 0.5 2 2 7 P S 0.5 113 0.5 3 2 S P 113 0.5 0.5 2 2 4 Operating strategy: P, operate in phase with existing units, S, operate out of phase with existing units.
0.5
1 1 1 0.5
liiit (LW 1 1
2 1 1
312 1 2 2 312 2
2 2
2
r a t e (kg/h) (=LBS/LCT) 1
1 2 2 312 312 2 2 3 3 4 4
4 4
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1089 Raw Matcnals
b
c
4000L
3000L
b Roduct A,
B
Figure 6. Process network of Example 2.
~~
Droduct A B
~
vy
9
v
Kj ($1 cj ($ L-')
staae 1 2.0 1.5
staae 2 1.0 2.25
4000 0 4000 30560 32.64
3000 0
3000 30560 32.24
1.0 2.0
A B
heuristic methods are proposed in order to determine the optimum retrofitting position as follows. Rule 1. First analyze the existing batch plant by eq 5, and then add new out-of-phase mode equipment to the stage where LCT occurs. Rule 2. By analyzing the existing process by eq 2, add new in-phase mode equipment to the stage where LBS occurs. We explain this algorithm,which determinesthe position of added equipment with reference to Figure 3 (Fletcher et al., 1991). First, when each product is produced, we analyze using eqs 2 and 5 where LCT and LBS occur. As the result of analysis, we make a process analysis table the same as Table 11. Three suggestions based on rules 1and 2 with reference to Table I1 are made. Suggestion 1: Out-of-phase mode equipment is added in the case of producing product A at stage 1,and inphase mode equipment is added in the case of producing product B at stage 1. Suggestion 2: Out-of-phase mode equipment is added in the case of producing product A at stage 2, and in-phase mode equipment is added in the case of producing product B at stage 2. Suggestion 3: We add both equipment of rule 1at stage 1 and equipment of rule 2 at stage 2. We consider the synthesis of seven cases from the three suggestions (refer to Figure 4). Table I11summarizes the effects between these seven cases and production rate (this means production quantities per unit time; LBS divided by LCT gives production rate). Case 1 is an existing process (production rate = 1). In case 6, a four times increase in production rate means adding three equipmenta. Unit cycle time out-of-phase mode and unit batch size in-phase mode at the jth stage are calculated by the following equations.
1200000 1000000
Table VI. Process Analysis of Example 1 product LCT LBS stage1 400012 LBS A 6 300012.5 LCT B 5
stage2 LCT LBS
and Bi. TOcompare with Fletcher et al. (1991), Tij is defined as t i j ( C i j = 0). Tij
= tij
(4)
(unit cycle time)j = unit cycle time of existing units (no. of existing unita + no. of adding units j t h ntrge (6)
)
LCT (limiting cycle time) T Lis~ expressed as
Tu = max (Tij/Nj) i = 1,...,N j = l , ...,M
(5)
(unit batch size)i = vol of existing units + vol of adding units size factor of existing units )itbB*
where Nj is the number of equipment which is set up in the jth stage in parallel. The total production rate of a batch plant is specified by two parameters, the LBS and LCT. When a new equipment is added to existing plant, the operating policy of added equipment has two modes. One is out-of-phase mode retrofitting which reduces limiting cycle time; the other is in-phase mode retrofitting which increases limiting batch size. These policies are explained in Figures 1and 2. Figure 1 shows effects which reduce LCT through adding equipment in one stage in parallel. Figure 2 shows effects which increase LBS from 80 to 100through adding equipment in one stage in parallel. Therefore, two Table VII. Result of Examde 1 new unita NLP sue 1 (existing) 2 1 3 2 4 1 2
(
(7)
When the selling quantities are infinite, continuous addition of new equipment is good to increase the production rate. Fixed cost is considered whenever new equipment is added. The optimum retrofit is determined in the following objective function F
F = nR - xC
(8) where n is the increase of production rate, x is the number of added equipment, R is an annual selling profit by the increase of production rate, and C is the retrofitting cost. operation
size (L)
A
B
1358 1395 1545 1554
P S P S
S P S P
~
Profit (8 275 160 3 125 200 3 124 600
CPU time (e) 0.6 0.7 0.8
3 041 600
1.1
1090 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993 Table XI. Data for Example 3
id
3 w ~ h product A B D E
stage 1 4.7 4.1 2.9 5.3
stage 2 10.1 3.8 2.7 11.8
product A B
stage 1 4.8 3.6 3.9 8.3
stage 2 2.0 4.9 5.6 3.6
12800 13.63 4000 1000 4000
11400 12.14 3000 1000 3000
Figure 7. Process network of Example 3.
Sij
Table VIII. Data for Example 2 stage 1 (h) 4.0 5.0
product
tjj
A B
stage 2
(h) 6.0 3.0
tjj
Cij
dij
0.1723-1 0.3643-1
0.865 0.823
Cij
dij
0.812E-2 0.5743-1
0.967 0.623
D E
(8
K j c j ($ L-')
q M (L)
Droduct
stage 1
A
2.0 1.5
stage 2 1.0 2.25
4000 0 4000 30560 32.54 0.6
3000 0 3000 30560 32.24 0.54
B
vy
e T
(8
K j c j ($ L-')
Ti product
pi ($ kg')
A
1.0 2.0
B
Qi
(L)
y (L)
~
Table X. Result of Example 2 new units operation NLP stage size(L) A B 1 (existing) 2 1 2000 P S 3 2 3000 s P
E
(W
stage2 LCT LBS
(ke;)
stage2 LCT LBS LBS LCT
selling profit is expressed as (Vaselenak et al., 1987) N
CpiniBi
(12)
Pl
profit($) CPUtime (8) 861 090 1.0 1688500 1.1 1069600 1.1
i = 1, ...,N
where pi is the selling price of product i. Therefore, a objective function is defined maximizing the profit expressed as eq 12 minus eq 11. The objective function and constraints are summarized as follows.
subject to
niBi I Qi
g n i T u IH
TI y"I
i = 1,...,N
Bi, ni 2 0 The linearized annual cost of equipment is expressed as (Vaselenak et al., 1987) j = 1, ...,M
i = 1,...,N
(9)
N
Kj+ c j y
Qi
290 000 300 000 350 000 140 000
Table XII. Process Analysis of Example 3 product LCT(h) LBS(kg) stage1 LBS A 10.1 400014.8 300014.9 LCT B 4.1 300015.6 LCT D 2.9 400018.3 LBS E 11.8
where nJ3i is the production quantity of product i, Qi is the upper bound on the production of product i, and ni is the number of cycles. The operating time H is greater than the total production time:
&niT, I H
0.63 0.53 0.44 1.09
D
Table IV shows the F value of each case. If C > R, the optimum retrofit is case 2. If C < R, the optimum retrofit is case 6. 2.2. Size Determination of Added Equipment. Generally, production quantities are limited to less than the predicted demand; therefore the following constraint becomes
niBi I Qi
Pi ($ kg')
A B
1200000 1000000
Table IX. Process Analysis of Example 2 Droduct LCT LBS stage1 LBS 400012 A 16.3 LCT 300012.25 B 8.1
product
(L kg')
(11)
where Kj is the fixed cost and cj is the cost coefficient. The
j = 1, ...,M
i = 1, ...,N
where Bj and T u are obtained from eqs 2 and 5. Since ni is the number of cycles,ni is an integer variable. However, the number of predicted batches will not in general take integer values. To make them integer, the number of batches can simply be rounded down to the nearest integer.
Ind. Eng. Chem. Res., Vol. 32, No. 6,1993 1091 Table XIII. Result of ExPmple 3 new unite NLP 1 2 3 4
stage existing 1 2 1 2
operation
size (L)
A
B
C
D
3200 1699 2204 2585
P S P S
S P S P
S P S P
P S P
Table XIV. Data for Example 4
T;w
tij (h)
product
A B
D E
product
A B
D E
stagel 6.3822 6.7938 1.0135 3.1977
stage2 4.7393 6.4175 6.2699 3.0415
stage3 8.3353 6.4750 5.3712 3.4609
stagel 7.9130 0.7891 0.7122 4.6730
Sij (L kg') stage2 stage3 2.0815 5.2268 0.2871 0.2744 2.5889 1.6425 2.3586 1.6087
stage4 4.9523 3.3951 3.5903 2.7879
15280 16.27 4000 0 4000
38200 40.68 4000 0 4000
10180 10.84 3000 0 3000
Pi (8 kg') 1.114 0.535 0.774 0.224
rn
rn
v+vy Sij
(for j = j )
(22)
Bi 5 min(yld/Sij) (for j = 1,...,M and j # j ) (23) The problem defined by eqs 13-23 corresponds to a nonlinear program (NLP). The optimum volume is determined from the above nonlinear programming using generalized reduced gradient algorithms (Lasdon and Warren, 1982). Solutions for = 0. existing processes are obtained by setting For a batch plant with M stages, the number of NLPs to solve is 2M in the worst case, but knowing stages that have bottlenecks, the number of NLPs is smaller than 2M.
(W
Because the number of batches is large, this approximation is reasonable. In case that new equipment is added in the j stage and the new equipment is operated in the out-of-phase mode when product i is produced, the following equations are added to the above constraints.
3. Applications 3.1. Problem 1 (Vaselenak et al., 1987; Fletcher et al., 1991). Figure 5 shows the process structure of problem 1. The demand for products A and B changes from 1 X 106kg of product A/year and 8 X lo6kg of product B/year to 1.2 X lo6 kg of product A/year and 1.0 X 106 kg of product B/year. The existing equipmentis used to produce a previous demand. The retrofitting problem occurs to meet a new demand. By the algorithm suggested in this paper, first we analyze given data of Table V and then
(18)
B, Imin( yid/Sij) (for j = 1,...,M and j # j ) (19) Tu of eq 15 is replaced with Tu of eq 21.
Table XVI. Result of Example 4 new unite NLP stage size (L) 1 existing 2 1 4000 3 1 697 4 4 2547 5 4 3000 6 1 1000 4 2582 2925 7 1 4 2897 8 1 1000 4 1971 3141 9 1 4 2338
1.4
1 ij =(for j = j ) Nj + 1
Bi I
Qi
Table XV. Process Analysis of E m ~ l 4e product LCT (h) LBS (kg) A 6.3822 4000/7.9130 B 6.7938 3000f3.3951 D 11.9213 3000f3.5903 E 3.3047 4000f4.6730
524 740
1.2 1.3 1.3
In case that new equipment is added in j stage and the new equipment is operated in the in-phase mode when product i is produced, the following equations are added to the above constraints.
268 200 156 000 189 700 166 100
( y w / S i j )2 Bi (forj = j )
CPU time (8)
(forj = 1,...,M a n d j # j ) (20)
stage4 3.9443 4.4382 11.9213 3.3047
45840 48.81 3000 0 3000
S
profit ($1 406 300 474 170 618 730
stage 1 LBS/LCT LCT
stage 2
stage 3
stage 4 LBS LBSfLCT LCT
LBS operation
A
B
P S
S S P P S P S P S P S P
P S P
S
C
P P S
S P P
D P P S S P S P S P S P S
profit ($) 459 000 466 120 467 OOO 521 340 521 780
CPU time (8)
496 550
1.4
461 810
1.4
503 170
1.2
464 470
1.3
1.1 1.3 1.3 1.2 1.2
1092 Ind. Eng. Chem. Res., Vol. 32, No. 6, 1993
make the analysis result as shown in Table VI. We solve three NLPs adding either new equipment in stage 1 or new equipment in stage 2 and adding new equipment both to stage 1and to stage 2. The NLP optimal solutions are obtained by using a generalizedreduced gradient algorithm (GRG2) to solve the NLP problems. The CPU times given are for a IRIS 25G operating under UNIX 4.3. The results of three NLPs are summarized in Table VII. Table VI1 shows that optimum retrofit adds equipment (1358,l) in the out-of-phase mode for product A and the in-phase mode for product B in stage 1. 3.2. Problem 2. The process structure of problem 2 is same that of problem 1. We considered C i j and dij terms in unit cycle time Ti> This problem deals with an application of this heuristics to the cases when processing times are not constant. We also implemented annual cost of equipment with rj.
K j + cj( y l r j (24) Table X shows an optimal retrofit method of problem 2. 3.2. Problem 3 (Fletcheret al., 1991). Figure 7 shows the process structure of problem 3. This problem is about a plant that produces four products in two stages. Using the given data of Table XI, we make the analysis result shown in Table XI1 and then determine the optimum volume of adding equipment with the same procedure as for problem 1. Adding new equipment (1699,l) in stage 2 in parallel brings the maximum profit. 3.3. Problem 4 (Vaselenak et al., 1987;Fletcher et al., 1991). Problem 4 is about a plant producing four products in four stages. Analyzing the given data of Table XIV, we make the basic data of Table XV. Using the process analysis table, we know that stages 1 and 4 are limiting. Since mixed integer programming investigations in stages 2 and 3 of whether new equipment is added or not are not necessary, we do not investigate in stages 2 and 3 by just using the process analysis table. However, we may have to solve 16 NLPs in the worst case. We solved 9 NLPs in this problem. In conclusion,the optimum retrofit is to add new equipment (3000,l) in stage 4.If the upper bound on the volume of new units is raised, it is found that the optimum new unit volumes are changed. 4. Conclusion
We suggest a retrofit method that optimally adds new equipment to a multiproduct batch plant. The method proposed in this paper is nonlinear programming which is based on heuristic rules. The advantages of this method are in simple and stable computation. These advantages result from using rules 1and 2 which are based on heuristic rule in the synthesisstep of equipment addition for process retrofit. The results of examples 1, 3, and 4 are same as those suggested by Fletcher et al. (1991). Current MINLP codes cannot treat the problem with large nonlinear and many binary variables. Though the
number of NLPs is large, our method can be applied to a retrofitting problem with many stages and products.
Acknowledgment We are grateful to acknowledge financial support from the Automation Research Center at Pohang Institute of Science and Technology. Nomenclature Bij = unit batch size of product i in stage j Ei = limiting batch size of product i ET = new batch size of product i in stage j cj = annualized cost coefficients of a new unit in stage j dij = correlation coefficient of batch size and unit cycle time in stage j H = operating time period K, = annualized fixed charge of a new unit in stage j ni = number of batches of product i pi = expected net profit per unit of product i Qi = upper bound on production of product i rj = annualized cost coefficients of a new unit in stage j Sij = unit size factor of product i in stage j t i j = processing time of product i in stage j Tij = unit cycle time of product i in stage j qT = new unit cycle time of product i in stage j TL = limiting cycle time of product i = volume of new unit in stage j Pd= volume of existing unit in stage j = upper and lower bound of adding unit in stage j
vf.,
Abbreviations LBS = limiting batch size
LCT = limiting cycle time NLP = nonlinear programming
Literature Cited Fletcher, R.;Hall,J. A. J.; Johns, W. R. Flexible Rstrofit Design of Multiproduct Batch Plants. Comput. Chem. Eng. 1991,15,843852. Grosemann,I. E.;Sargent,R.W. H. OptimumDesignofMultipurpoee Chemical Plants. Znd. Eng. Chem. Process Des. Dev. 1979,18, 343-348. Lasdon, L. S.; Warren, A. D. GRG2 User’s Guide; The Scientific Press: New York, 1982. Sparrow, R. E.; Forder, G. J.; Rippin, D. W. T. The Choice of Equipment Sizes for Multiproduct Batch Plank. Heuristics vs Branch and Bound. Znd. Eng. Chem. Process Des. Dev. 1987,14, 197-203. Vaeelenak, J.A.;Groeemann,I.E.; Weaterberg,A. W.OptimalRetrofit Design of Multiproduct Batch Plants. Znd. Eng. Chem.Res. 1987, 26,718-726. Yeh, N. C.; Reklaitis, G. V. Synthesis and Sizing of Batch/ Semicontinuoua Processes: Single Product Plants. Comput. Chem. Eng. 1987,11,639-654. Received for review Auguet 4, 1992 Revised munwcript received February 3, 1993 Accepted February 23,1993
