Korean Chem. Eng. Res., Vol. 45, No. 3, une, 007, pp. 49-57 m o n m m o-nmo o n m Çmm * 608-739 e n 100 * 100-715 ne t 3 6 (006 10o p r, 007 3o 9p }ˆ Optmal Desgn of Generalzed Process-storage Networ Applcable To Polymer Processes Gyeongbeom Y and Euy-Soo Lee* Department of Chemcal Engneerng, Puyong Natonal Unversty, San 100 Yongdang-dong Nam-gu, Busan 608-739, Korea *Department of Chemcal Engneerng, Donggu Unversty, 3 Ga 6 Pl-dong ung-gu, Seoul 100-715, Korea (Receved October 006; accepted 9 March 007 Ž p e r-rqs sp r l rp rn l. s q p ep q ~. l l Ž p rn o l e r l rl v. prp l l o s p r pp v l. l l p rp m l o s p r pp r lr. p rp ro ql r rr r p. o s r pp p p rp q l Ž p lr rp rn ep r. r q l rn l p on p lž q p m ˆ l. l Š rp r p p } re m. p } p rp p vr p rp ˆ v on p. rp np o p r l q l v 6/10 o. h Abstract The perodc square wave (PSW model was successfully appled to the optmal desgn of a batch-storage networ. The networ structure can cover any type of batch producton, dstrbuton and nventory system, ncludng recycle streams. Here we extend the coverage of the PSW model to multtasng sem-contnuous processes as well as pure contnuous and batch processes. n prevous solutons obtaned usng the PSW model, the feedstoc composton and product yeld were treated as nown constants. Ths constrant s relaxed n the present wor, whch treats the feedstoc composton and product yeld as free varables to be optmzed. Ths modfcaton maes t possble to deal wth the poolng problem commonly encountered n ol refnery processes. Despte the greater complexty that arses when the feedstoc composton and product yeld are free varables, the PSW model stll gves analytc lot szng equatons. The ablty of the proposed method to determne the optmal plant desgn s demonstrated through the example of a hgh densty polyethylene (HDPE plant. Based on the analytcal optmalty results, we propose a practcal process optmalty measure that can be used for and of process. Ths measure facltates drect comparson of the performance of multple processes, and hence s a useful tool for dagnosng the status of process systems. The result that the cost of a process s proportonal to the square root of average flow rate s smlar to the well-nown sx-tenths factor rule n plant desgn. Key words: Optmalty, Lot-sze, Sem-contnuous, Poolng, Multtas, Polymer 1. Ym Relats[1]l Ž p n q p pn l e r-rqs p r ep To whom correspondence should be addressed. E-mal: gby@pnu.ac.r o m. Ym Relats[]l rq l p rp e r-rqsp r l r ep o lp, q p rp s l [3]. sp l Ž p erp qrp nr n e l rp o p p. p qrp q p l q 49
50 p Ëpp l [4]. l l Ž p n p rp q s e rl p qlp l e rp ˆ q p. l rp HDPE, PP, LDPE, LLDPE, PVC, PS, ABS, CB, PBm v p n q rl rn tn rp. o o r, nr r ~ ll lt r p l. o s r pp n rom o rl n. l l p rp m l o s r pp vr l. l rp r redš l tn l trp. p r p re l p [5, 6]. l l l p rp nmp r l p r p q. n Ž p pn l p r p l rp r r e l r o p. rp r psrp r n, q ov n q p q Œ q np p. o ˆp l rl rn p r p } m tr r re p. r p o l v r p o p. p rp r r p } p ep r, v nr r p qlp r np. r r l rp r qop r q rl r Œl p q p. l l re p rs r p qp pp rp d rl q n. p rp vp p l e rp p p. n l l re p r-rqs s e, l e r r p p q l rn p. l p ˆp q l r r r re. Ym Relats[4]l l q p rp l l p v v d p q p.. n Kuhn-Tucer l l n p rpm n Ym Relats[3]m p. l p rp Fg. 1l ˆ } e l r p. p r p p q lp rp, Fg. l ˆ } p rl v r qlp v n p ˆ. q lp,,, N p v, t rp p. rl p ql l l l p Ym Relats [3]p m p. rp o rqs, p r p l rqs v. rqs p v rq r vp m p. qlp o n r p. r l p ql np o p o p F n p rp, r l p ql np r p o p G n r p. o s r pp p. r l p ql nl v vep p. o45 o3 007 6 Fg. 1. General structure of batch-storage networ. Fg.. Multtasng batch process. F n 1 G n 1 B n ------ l B n p r l ql np n nre p ˆ. rl Fg. l ˆ m p t ω m eq e `t tn nr p. ω l N p t p ˆ. v ω ω, y p, 0 1p. t p rp Kuhn-Tucer s p p. e p qlp l p v r ol qlp n l. r l ql nl o tp eqe p `t n p ˆ, r eqe p t n ' p ˆ, p ep. t n ' `t n t n ( n 1 `t n `t ω (3 n 1 l t n o tp r eqe p o tpe } e p p. t n p ppp p, m, Ym Relats[3]l t n (1 x n ' p vr l. l (`x n, x n ' (o tp r rqs nr e p 1 q. p ep rl (1
rr v d l rl rr v. rl t n p p rr. rqs l v vep p. q rl rn p p r-rqs s r 51 1 G n K( 1 D 1 F n l D o q K( v p o p, D m p q m M( l Ž v p o p. rp rep p. (, n l θ o v p (, n p, θ r v ( n, p p. o vp p o D, t ω, eq e t rqs nr e p x l p rp. p B D ω p. sr p Ž p rp o D m, t ω m, eqe, rqs nr e p p Ž B m D m ω m p v p. r l t o p qlp Ž l p l. q l p qlp e np v p Fg. 3l ˆ m p t rp qlp p v p p. qlp v l Ž p p pn l p p p qll lp p. p rp Fg. 4l ˆ } l rl rn. Ym Relats[3]m p p rqs l q V(tp p. rqs Fg. (cl ˆ m p o q m rp lm qm rp l l l p. q V(t p. M( θ F ( n θ G n, n (, ( n,, n 1, n 1 D m (4 (5 Fg. 4. Bloc operatons of sem-contnuous process. V ( t V ( 0 K( B nt t t 1 ω Ì (6 l V(0 q p. q, q q p r p vl p l v []. q m q V p. V V ( 0 ( 1 x D ω t ---------- mn 1 ----res 1 t, ---------- x ω ( G n nt t t n 1 n 1 -------------- mn 1 -------------- 1 r e s t t n, ------------ ω x n ω M( B m nt t ----------- mn 1 -----res 1 t, ----------- ω m ω m V 1 V F n ( K( 1 nt n ( 1 x n G n 1 F n t 1 n t m t t --------------- 1 mn 1, ----------------res ω x n y n M( D m t n K( 1 D t 1 G n t n n ω t t --------------- Ì (7 (8 q V V V ( 0 1 1 K( 1 ( 1 x ------------------ D ω ( 1 x n --------------------------G n ( 1 x n ------------------------------F n K( 1 1 1 D t G n t n F n t n Fg. 3. Decomposton of multple tass nto multple flows of sngle tas. M ( M( ( 1 ------------------- D mω m D m Ì (9 Korean Chem. Eng. Res., Vol. 45, No. 3, une, 007
5 p Ëpp o v p t A $/orderp, r l ql n p t A n $/batchp. rqs p l q ov H $/L/yearp. rp r np p, l np o, o t, r t, q ov m q p q n p p l p. TC 1 K( 1 A ----- a B P D ω [ H V b V ] 1 (10 l a, a n b r, r rqsp l q np, P o p. q e (8p 0 ƒ. v s V 0p rp rep. K( V ( 0 D t G n 1 1 N( ( 1 x F n n F n 1 M( n ( 1 D mω m Ì (11 rqs q V v, e (7l p r. r p t (ω m ω, eqe (t m Ít o (D, F n m G n p. ql eqe t n ' mít n e (m e (3l p Ít p l. D, F n m G n, r e (10 p m p, re p D, F n m G n l p. ω, ω m t l m p e v. r D, F n m G n t, ω, ω, t mít n l Kuhn-Tucer s p p D, F n m G n l r p. p r r lv rep ep n o rm p r p Kuhn-Tucer s p p [3]. p rp ~w r m r m e rep v p, w r m p r m e rep v p. w r r o rrl o r r o rrl. ~ w rp Kuhn-Tucer p p ( p A. A *ω ------------ D Ψ l Ψ m Ψ p p rp. M( An n 1 *ω -------------------------------------- (,, H Ψ F n G n 1 D m 1 A n ------- a n B n (1 (13 Ψ ---- b ( 1 x (14 a Ψ F n, G n, ( ω n 1 t ω 0 n 1 t n n 1 t ω n 1 e (13p s rrp qll t p n p p p p p. v A n n 1 ----------------- Ψ ( ( ω ( F n, G n, ( * ω Ψ ( F n, G n, A n * p ep l r rrl lv r ˆ. p rp r r r ˆ p p } re. r p r r r A n ω (16 l ny op r p ˆ. e (16p r m r n p p. e (16p r p nr ˆp r r p o p. rrp v p p l v l. rnr e qe p rrl p p p p s. N( D t 1 1 M( G n F n ( t ( 1 x n F n Ì (17 1 o ep q sp p t m t n p sr p. p p o l ep seˆv l q n eqe p vleˆ p. p vlp np lt n Ž oe np. p np r p p r ep r n t rp l n pp. e (17p ep eˆ p p. m, q V (0 p vp sp. p, p, s r p r 3q l l p. l v tl l p q rr pv r dv. q l e (17p l e ˆ v p v. r r p p. ----------------- Ψ ( F n, G n, ω ----------------------------------------------------------------------------- Ψ ( * F n, G, y A n n V ( 0 ( 1 D ω m D m M( n 1 ( Fn G n ω n 1 1 N( * TCD, F n, G n K( * n * n 1 ( A Ψ D P ( D 1 1 t n G n [ a n F n ( 0.5H b Ψ ( F n, G n A n 1 1 F n { ( 1 x n G n ( 1 x n }] o45 o3 007 6 (15 1 H M( ---- b D mω m ( 1 (18
e (18p p r l. r p np o p r l ep r l rp v 6/10 o [7]. rqsp r q p e (17. * V [( 1 x n F n ( 1 x n G n ]ω 1 q rl rn p p r-rqs s r 53 K( ( 1 x D ω ( 1 x m m D mω m 1 M( (19 r e (18p re e (4, e (5m o D, F n m G n l l l. p w r m r r r o rrp p q p l. qlp n p r r n l. o ql p v nl l p p rp e (1 ~ e (19 rn p rp l p p. rp n o p p r p, LDPEp n qlp p r r p er o p ql p p p n p l p mrl re tr p n n lr on. 3. m m n op r n q q l r n p r-rqs sp r r. (1 r rqs v p r o t. p p n p r l m p l. ( Fg. 5l lv } r l p ql nl l rqs p p. r p o rqs q. o s, r p, rqs nr e p, t, q ov, q n o p p q t. Fg. 5. Optmal sze of multtasng batch process connected by storage unts. Fg. 6. Optmal sze of storage unt connected by multple supply and consumpton processes. r l p ql np r p e (13l o pel p. B n ω F n F n 1 1 (0 o r p e (1l p. (3 Fg. 6l ˆ m p rqs l l r p p. v rp v p, ` rp v p q. p p q ` p. d p o sr v l. p eqe p e (17l o p ep pn l sr e rp. D t D m G n F n t r rqs e (19. A n ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- a n F n 0.5H b [ ( { F n ( 1 x n G n ( 1 x n }] 1 K( 1 M( M( 1 4. HDPE m o ω Ì (1 4-1. m m 11 p HDPE v p p p l rp q. Table 1p ll p v l p v p v np lt. p np v r p l. Table r n o (D m, t ( m q ov (H ˆ. p p l r n p t d p. r rq} m q n, o m V ( 0 ( 1 D mω m ( 1 x n F n Korean Chem. Eng. Res., Vol. 45, No. 3, une, 007
54 p Ëpp Table 1. Grade transton costs of HDPE plant($/batch F550 F607LD F6060P F50100 TR158 TR570 TR144 TR130 TR147 F5811 HX100 F550 0,040 3,570 4,080 5,610 6,10 4,080 4,590 4,080 4,590 8,160 F607LD,889 0,408 4,815 4,334 6,741 5,97 5,97 4,815 3,371 9,630 F6060P,895,413 0 4,85 4,85 6,755 5,308 5,790 4,85,895 9,650 F50100 3,06 5,954 6,870 0 3,06 4,580 4,1 4,1 4,580 5,954 6,41 TR158 3,916 4,895 5,385 3,47 0 4,895 3,916 4,406 4,406 4,895 7,83 TR570 5,63 5,63 7,168 4,096 4,608 0 4,096 4,608 4,608 5,63 7,168 TR144 4,433 5,910 5,910,463 5,418 7,388 0 3,448 4,95 6,895 6,403 TR130 5,841 7,434 7,434 3,717 6,37 7,965 3,717 0 5,310 9,558 8,496 TR147 3,51 3,951 3,951 3,51 4,390 5,68 3,51 4,89 0 3,073 7,04 F5811 3,7 3,7 3,7 3,7 4,090 4,908 4,090 4,908 3,7 0 6,544 HX100 5,750 8,050 8,050 4,600 6,900 8,65 4,600 6,35 5,750 8,050 0 Table. nput and output data of sngle reactor desgn Sequence Product Code A ' ($/Batch D (ton/day m H ($/ton/day 0.5H (1- ($/ton/day B n (ton 1 F550,040 30,587 0.110993 84.315 4.159843 1,79.499 F607LD,408,90 0.010596 85.41 0.447709 190.4467 3 F6060P,895 5,913 0.01457 86.14 0.9043 381.41 4 F5811 6,544 7,665 0.07815 87.35 1.179457 491.5 5 HX100 4,600,847 0.010331 96.75 0.494477 185.7353 6 TR144 3,448 90,885 0.39801 86.14 9.519866 4,015.5 7 TR130 3,717 33,580 0.11854 89.79 4.80407 1,943.857 8 F50100 3,06 16,571 0.060133 85.775.43853 1,06.674 9 TR158 4,895 14,454 0.0545 84.68.10468 90.83 10 TR570 4,608 11,753 0.04649 85.045 1.736197 741.7148 11 TR147 3,51 58,400 0.1191 86.14 7.19313 3,033.89 sum 4,1873 75,575 1 34.96715 vp q ov n o r p p. v Table l ˆ m p p t l v np p r l. p r 11 p e nžo rm. p t l v p. p n e (15 Ψ 0.5 H. e ( 1 1 1 (13p t l p p t. ω ( 41, 873 ---------------------------------------------------- 4 days ( 75, 575 ( 34.96715 ( p t p pr r l rqs e (18l p. sr p q l } l rq r p rqsp p. 1,79.499*(1-0.110993190.4467*(1-0.010596381.41*(1-0.01457 491.5*(1-0.07815185.7353*(1-0.0103314,015.5*(1-0.39801 1,943.857*(1-0.118541,06.674*(1-0.06013390.83*(1-0.0545 741.7148 * (1-0.04649 3,033.89 * (1-0.1191 1,136.04 tons 4-. m p p nl p r e l q. p p p n p p rq n l nr p l. r r p n o45 o3 007 6 l n n tr r pn p. r n p q TR144 p rn v p l yp p l. p p l v pv v np tp o l Table 3l ˆ } p p v TR144 n m l p TR144 p. TR144 n p q p r d y p l p. Table 3l n l p } v p p l l p p nm p v. p p r t p. (, 386 ω 1 --------------------------------------------------------- 6 days (137, 787.5 ( 31.5484 (, 90 ω 1 --------------------------------------------------------- 31 days (137, 787.5 ( 3.15767 (3 (4 t p nl p p l pp p p p t v np v p. Table 1l p n p q p o rl v l. p np p p n. l e p pr r o p p p np l ƒv. np o l r p p p er t *4 34.
q rl rn p p r-rqs s r 55 Table 3. nput and output data of two reactors desgn Process Sequence Product A ' ($/Batch H ($/ton/day 0.5H (1- ($/ton/day D (ton/day m B (ton n R1 1 TR130 3,717 0.43709 89.79 8.7481 33,580,409.768 R1 F50100 3,06 0.1065 85.755 4.53649 16,571 1,189.168 R1 3 TR158 4,895 0.104901 84.68 3.975578 14,454 1,037.48 R1 4 TR570 4,608 0.08598 85.045 3.31770 11,753 11,339.94 R1 5 TR147 3,51 0.43841 86.14 10.51769 58,400 4,190.901 R1 6 TR144 3,448 0.01987 86.14 0.96149 3,09.5 17.403 sum,386 1 31.5484 137,787.5 R 1 TR144 4,433 0.637616 86.14 9.951835 87,855.5 7,446.016 R F550,040 0.1987 84.315 7.80964 30,587,59.34 R 3 F607LD,408 0.0119 85.41 0.88588,90 47.4787 R 4 F6060P,895 0.04914 86.14 1.768984 5,913 501.1444 R 5 F5811 6,544 0.05569 87.35.9145 7,665 649.6316 R 6 HX100 4,600 0.0066 96.75 0.978631,847 41.917 sum,90 1 3.15767 137,787.5 Fg. 7. An example product dstrbuton branch. 4-3. Fg. 7p HDPE qp r m lt p. q } l p r p l. p p p n r l l Table l p np 30% r. p ttl t p. q pl l p p r 5 p p q ov rn np. rp p qlp ( N ( 1, p rqs nr e pp 0.9 r. m e p rn p e (19m e (0l p. Ψ RTC * 0.5 * 0.3 * (1-0.9(30587 * 84.315 90 * 85.41 5913 * 86.14 7665 * 87.35 847 * 96.75 90885 * 86.14 33580 * 89.79 16571 * 85.775 14454 * 84.68 11753 * 85.04558400 * 86.14 380115.51 (5 Β RTC (0.3 * 75575 50, ----------------------------------- 000, 380,115.51 11,98.49 tons (6 V DC 11,98.49 * (1 0.9 (0.3 * 75,575*7*(1-5/7 166,543.49 tons (7 5. o s r pp rp p qlp e l rp r l rp rrqs sp l o l. l p r r l o, o r, q, nr, oo m p n r p rl rn p. qlp r l r l Ž p pn o e rl p rl rnp l. l r p ql t v qlp p r rp n on. t p v r l p rp rp p qlp l l p rl. p rp t e (18p e p r r. ˆ r r n l n r p l n. l l e r rn n tr p r l HDPE qp rl rn m m. rp l rl rn p evrp r r } re m. p } rp ep r-rqs s r p o ep. p } q n n r rp r o s nqop prp l l p. p r 006 r ( oprqo p qop v q p vop l p(krf-006-311- D0038. a a b A A n A ' : annualzed captal cost of raw materal purchasng faclty, dollars per unt of tem per year : annualzed captal cost of process, dollars per unt of tem per year : annualzed captal cost of storage faclty, dollars per unt of tem per year : orderng cost of feedstoc materals, dollars per order : orderng cost of noncontnuous process, tas n, dollars per order : product change-over cost from product to product ', dollars per order Korean Chem. Eng. Res., Vol. 45, No. 3, une, 007
56 p Ëpp B B n B m D D m D f n F n g n G n : raw materal order sze, unt of tems per lot : noncontnuous process sze, unt of tems per lot : fnal product delvery sze, unt of tems per lot : average materal flow of raw materal supply, unt of tems per year : average materal flow of customer demand, unt of tems per year : average materal flow through noncontnuous processes, unt of tems per year : feedstoc composton of process, tas n : average flow rate of feedstoc materal to process, tas n, unt of tems per year : product yeld of process, tas n : average flow rate of producateral from process, tas n, unt of tems per year H : annual nventory holdng costs, dollars per unt of tem per year : noncontnuous process set : storage set K( : raw materal suppler set for storage M( : consumer set for storage N( : Tas set for process P Ít Ít n t n ' : prce of raw materal from suppler, $/unts of tem : start-up tme of customer demand, year : start-up tme of the frst tas feedstoc feedng to noncontnuous process, year : start-up tme of feedstoc feedng of tas n to noncontnuous process, year : start-up tme of product dschargng of tas n from noncontnuous process, year t : start-up tme of raw materal purchasng, year V : upper bound of nventory hold-up, unts of tem V : lower bound of nventory hold-up, unts of tem V (t : nventory hold-up, unts of tem V (0 : ntal nventory hold-up, unts of tem V : tme averaged nventory hold-up, unts of tem x Íx n x n ' Z n ' : storage operaton tme fracton of purchasng raw materals : storage operaton tme fracton of feedng to noncontnuous process, tas n : storage operaton tme fracton of dschargng from noncontnuous process, tas n : storage operaton tme fracton of customer demand : cycle tme rato, /ω ' : Z n 1 f the n-th tas on process produces a product and ' the (n1-th tas produces a product ' and otherwse, Z n 0 m m λ φ ' ω m ω ω : Lagrangan multplers : the composton of feedstoc to produce product ' : cycle tme of customer demand, year : cycle tme of raw materal purchasng, year : cycle tme of noncontnuous process, year : duraton of noncontnuous process, tas n, year o45 o3 007 6 Ψ : aggregated cost defned by Eq. 15 Ψ : aggregated cost defned by Eq. 14 lzm : storage ndex g zm : noncontnuous process ndex : raw materal vendors m : fnshed product customers n : tas ndex Specal Functons nt[.] : truncaton functon to mae nteger res[.] : postve resdual functon to be truncated X : Number of elements n set X y 1. Y, G. and Relats, G. V., Optmal Desgn of Multple Batch Unts Wth Feedstoc/product Storages, Chem. Eng. Comm., 181(1, 79-106(000.. Y, G. and Relats, G. V., Optmal Desgn of Batch-Storage Networ Usng Perodc Square Model, AChE., 48(8, 1737-1753(00. 3. Y, G. and Relats, G. V., Optmal Desgn of Batch-Storage Networ wth Recycle Streams, AChE., 49(1, 3084-3094 (003. 4. Y, G. and Relats, G. V. Optmal Desgn of Batch-Storage Networ wth Fnancal Transactons and Cash Flows, AChE., 50(11, 849-865(004. 5. Floudas, C. A. and Ln, X., Contnuous-tme Versus Dscrete-tme Approaches for Schedulng of Chemcal Processes: A Revew, Computers & Chemcal Engneerng, 8(11, 109-19(004. 6. Karm,. A. and McDonald, C. M., Plannng and Schedulng of Parallel Semcontnuous Processes.. Short-Term Schedulng, &EC Res., 36(7, 701-714(1997. 7. Peters, M. S., Tmmerhaus, K. D. and West, R. E., Plant Desgn and Economcs for Chemcal Engneers, McGraw-Hll, New Yor, NY(003. A: zs n om Kuhn-Tucer e (m e (3l p eqe Ít n m t n ' Ít. e (7 e (9l p e (10p l pep. TC ----- n 1 ----------------- ( H b D t ω A ω H ω A n H ---- b ( 1 x a D ω a n F n ---- b 1 x ( n G n ω
q rl rn p p r-rqs s r 57 ( t n G n H b L ------ H b t ( ( F n G n λ 1 1 H ---- ( 1 x n F n ω ( F n G 0 n (A6 n 1 ( H b y n ( F n G n ω n 1 ( ( F n G n Ít n H b constants (A1 constants ( V (0 H b H ---- ( 1 x m D mω m ( H b D m (A ω, ω, t m Ít n l l e (11p r l e (10p r rp Lagrange LTC 1 (A7 (A8 (A3 l λ Lagrange d p. Kuhn-Tucer s p e (A4m e (A6p ; λ H b (A9 e (A9m e (A5m e (A7p l p e (1 m e (13p o. e (A8p e (17p. L ------ ( H b D λ lb D 0 t (A4 -------- L ω A H ------------ ---- b ( 1 x ( ω a D 0 (A5 Korean Chem. Eng. Res., Vol. 45, No. 3, une, 007