Sliding Mode Control: An Approach To Regulate Nonlinear Chemical Processes

Transcription

1 Sliding Mde Cnrl: An Apprach T Regulae Nnlinear Chemical rcesses Oscar Camach Deparamen de Circuis y Medidas Universidad de Ls Andes Mérida 5. Venezuela Carls A. Smih Chemical Engineering Deparmen Universiy f Suh Flrida Tampa Flrida. USA Absrac A new apprach fr he design f Sliding Mde Cnrllers based n a firs-rder-plus-deadime mdel f he prcess is develped. This apprach resuls in a fixed srucure cnrller wih a se f uning equains as a funcin f he characerisic parameers f he mdel. The cnrller perfrmance is judged by simulains n w nnlinear chemical prcesses. Keywrds: Sliding Mde Cnrl, Variable Srucure Cnrl, Nnlinear Chemical rcesses.. Inrducin Sliding Mde Cnrl SMC is a rbus and simple prcedure synhesize cnrllers fr linear and nnlinear prcesses. T develp a Sliding Mde Cnrller, SMCr, knwledge f he prcess mdel relaing he cnrlled variable, XBCB, he manipulaed variable, U, is necessary. Hwever, here are w prblems wih he use f a mdel as far as chemical prcesses are cncerned. Firs, he develpmen f a cmplee mdel is difficul due mainly he cmplexiy f he prcess iself, and he lack f knwledge f sme prcess parameers. Secnd, ms prcess mdels relaing he cnrlled and he manipulaed variables are f higher-rder. Generally, he SMC prcedure prduces a cmplex cnrller, which culd cnain fur r mre parameers resuling in a difficul uning jb. Therefre, he use f he radiinal prcedures f SMC presens disadvanages in heir applicain chemical prcesses. An efficien alernaive mdeling mehd fr prcess cnrl is he use f empirical mdels, which use lw rder linear mdels wih deadime. Ms imes, firs-rder-plus deadime FODT mdels are adequae fr prcess cnrl analysis and design. Bu, hese reduced rder mdels presen uncerainies arising frm imperfec knwledge f he mdel, and he prcess nnlinear effecs cnribue perfrmance degradain f he cnrllers. Cnveninal cnrllers, such as ID, Lead-Lag r Smih redicrs, are smeimes n sufficienly versaile cmpensae fr hese effecs. Thus, a SMCr culd be designed cnrl nnlinear sysems wih he assumpin ha he rbusness f he cnrller will cmpensae fr mdeling errrs arising frm he linearizain f he nnlinear mdel f he prcess. The aim f his paper is design a SMCr based n a firs-rder-plus-deadime FODT mdel f he acual prcess. The verall idea is develp a general SMCr, which can be used fr self-regulaing chemical prcesses. The parameers f he mdel, prcess gain, K, prcess ime cnsan,, and prcess deadime, BB, are used bain he iniial esimaes f he uning erms in he SMCr.B This aricle is rganized as fllws. Secin briefly presens he prcess mdel. Secin, presens sme basic cnceps f he SMC mehd. Secin 4 shws he prcedure design a SMCr using he FODT mdel. Tuning equains fr he cnrller are als given in his secin. In Secin 5 he simulain f he SMCr fr w nnlinear chemical prcesses is presened. Secin 6 cncludes he paper.

2 and. rcess Mdel The prcess reacin curve, Figure, is an fen-used mehd fr idenifying dynamic mdels []. I is simple perfrm, and prvides adequae mdels fr many applicains. The curve is bained by inrducing a sep change in he upu frm he cnrller and recrding he ransmier upu. [Figure ] Frm he prcess curve shwn in he figure, and he prcedure presened in he reference, he numerical values f he erms in he FODT mdel given in Eq. are bained s X s Ke = U s s where Xs is he Laplace ransfrm f he cnrlled variable, he ransmier upu, and Us is he Laplace ransfrm f he manipulaed variable, he cnrller upu. Bh Xs and Us are deviain variables. In his paper we use he uni f Xs as fracin f he ransmier upu, fracin TO; he uni f Us is fracin f he cnrller upu, fracin CO. K, BB were previusly defined.. Basic Cnceps abu Sliding Mde Cnrl Sliding Mde Cnrl is a echnique derived frm Variable Srucure Cnrl VSC which was riginally sudied by []. The cnrller designed using he SMC mehd is paricularly appealing due is abiliy deal wih nnlinear sysems and ime-varying sysems [-5]. The rbusness he uncerainies becmes an impran aspec in designing any cnrl sysem. The idea behind SMC is define a surface alng which he prcess can slide is desired final value; Figure depics he SMC bjecive. The srucure f he cnrller is ineninally alered as is sae crsses he surface in accrdance wih a prescribed cnrl law. Thus, he firs sep in SMC is define he sliding surface S. S is chsen represen a desired glbal behavir, fr insance sabiliy and racking perfrmance; The S seleced in his wrk, presened by [4], is an inegral-differenial equain acing n he racking-errr expressin n d S = λ e d d where e is he racking errr, ha is, he difference beween he reference value r se pin, R, and he upu measuremen, X, r e = R - X. λ is a uning parameer, which helps define S ; This erm is seleced by he designer, and deermines he perfrmance f he sysem n he sliding surface, n is he sysem rder [Figure ] The bjecive f cnrl is ensure ha he cnrlled variable be equal is reference value a all imes, meaning ha e and is derivaives mus be zer. Once he reference value is reached, Eq. indicaes ha S

3 is,,, reaches a cnsan value. T mainain S a his cnsan value, meaning ha e is zer a all imes; i is desired make ds = d Once he sliding surface has been seleced, aenin mus be urned design f he cnrl law ha drives he cnrlled variable is reference value and saisfies Eq.. The SMC cnrl law, U, cnsiss f w addiive pars; a cninuus par, UBCB and a discninuus par, UBDB [6]. Tha is U = U U 4 C The cninuus par is given by D X, R U C = f 5 where f X, R is a funcin f he cnrlled variable, and he reference value. The discninuus par, UBDB incrpraes a nnlinear elemen ha includes he swiching elemen f he cnrl law. This par f he cnrller is discninuus acrss he sliding surface. U = K S S δ D D 6 where KBDB he uning parameer respnsible fr he reaching mde. δ is a uning parameer used reduce he chaering prblem. Chaering is a high-frequency scillain arund he desired equilibrium pin. I is undesirable in pracice, because i invlves high cnrl aciviy and als can excie high-frequency dynamics ignred in he mdeling f he sysem [, 4, 6]. In summary, he cnrl law usually resuls in a fas min bring he sae n he sliding surface, and a slwer min prceed unil a desired sae is reached. 4. SMCr Synhesis frm an FODT Mdel f he rcess This secin presens he develpmen f a general SMCr, fr self-regulaing prcesses, using a firs-rderplus-deadime FODT prcess mdel. The FODT mdel is an apprximain he acual higher-rder mdel. The develpmen f his cnrller significanly simplifies he applicain f sliding mde cnrl hery chemical prcesses. The lieraure reviewed des n reveal a simple and pracical mehd apply SMC prcess wih dead ime [7-9]. In his chaper, a SMCr srucure based n he FODT mdel f he acual prcess is designed. Thus, he firs sep is prpse a way handle he deadime erm The deadime can be apprximaed in w differen ways. A firs-rder Taylr series apprximain he deadime erm prduces e s s 7 The abve apprximain can als be wrien as a firs-rder adé Apprximain

4 e s. 5 s. 5 s 8 Figure shws a cmparisn amng he deadime erm and he firs-rder Taylr series and adé Apprximains. The figure shws ha he adé Apprximain wrks very well beween and bu beynd he apprximain brakes dwn. On her hand, he Taylr series apprximain imprves as BB increases. [Figure ] In [] is shwn ha he firs-rder Taylr apprximain r he adé apprximain can be cnsidered as gd apprximains fr he deadime erm fr chemical prcesses. The nex secin shws he develpmen f a SMCr using bh apprximains. 4. SMCr Develpmen Based n a Firs-Order Taylr Series Apprximain In his secin a SMCr is develped based n he firs-rder Taylr series expansin. Addiinally, a rule chse he uning parameers will als be presened. Subsiuing Eq. 7 in Eq. prduces Xs K 9 Us s s In differenial equain frm d X dx X= KU d d and since his is a secnd-rder differenial equain, n =, frm Eq. S becmes de S = e λ e d d Where λ= λ and λ = λ Frm Eq. λ ds d d e de = λ λe = d d Subsiuing he definiin f he errr, e = R - X, in he firs w erms f he abve equain gives d R d X d d dr dx λe = d d λ 4

5 5 Slving fr he highes derivaive frm Eq., subsiuing i in he Eq., and slving fr U prvides he cninuus par f he cnrller = d dr d R d e X d dx K U C λ λ λ 4 This prcedure, invlving Eqs., and, bain he expressin fr he cninuus par f he cnrller is knwn in he SMC hery as he equivalen cnrl prcedure []. In [] is shwn ha he derivaives f he reference value can be discarded, wihu any effec n he cnrl perfrmance, resuling in a simpler cnrller. Thus, = e X d dx K U C λ λ 5 UBCB can be simplified by leing, λ = 6 I has been shwn ha his chice fr λbb is he bes fr he cninuus par f he cnrller []. T assure ha he sliding surfaces behaves as a criical r verdamped sysem, λbb shuld be 4 λ λ 7 Then, he cmplee SMCr can be represened as fllws δ λ = S S K e X K U D 8a wih = d e e d dx K sign S λ λ 8b Equains 8a and 8b cnsiue he cnrller equains be used. These equains presen advanages frm prcess cnrl pin f view, firs hey have a fixed srucure depending n he λ s parameers and he characerisic parameers f he FODT mdel, and secnd he acin f he cnrller is cnsidered in he sliding surface equain, by including he erm signk, in Eq. 8b. Ne, ha signk nly depends n he saic gain, herefre i never swiches. Frm an indusrial applicain perspecive, Eq. 8b represens a ID algrihm []. T cmplee he SMCr, i is necessary have a se f uning equains. Fr he uning equains as firs esimaes, using he Nelder-Mead searching algrihm [], he fllwing equains were bained []. Fr he cninuus par f he cnrller and he sliding surface

6 λ λ = = [=] [ime] [=] [ime] 9a 9b Fr he discninuus par f he cnrller.76.5 K D = [=] [fracin CO] 9c K δ =.68. K K λ [=] [fracin TO/ime] 9d D Eqs. 9c and 9d are used when he signals frm he ransmier and cnrller are in fracins. Smeimes, he cnrl sysems wrk in percenages ha is, he signals are in % f range. In hese cases he values f K D andδ are muliplied by. 4. SMCr develpmen based n he adé Apprximain This secin cnains he develpmen f he cnrl law when he deadime erm f he FODT prcess mdel is apprximaed by he adé apprximain, Eq. 8. The prcedure fllwed in his secin is similar ha ne presened in he previus par. Subsiuing Eq. 8 in Eq., gives X C s K - s U s s s Using a similar prcedure as shwn abve, he cninuus par f he cnrller, Ucs, is U C s = s - K λ s Rs - λ s X s - C s λ e s Eq. has a ple /BB n he righ side f he cmplex plane. Thus, he cninuus par f he cnrller cnains an unsable erm. Eq. represens a nnminimun phase sysem. Hence, he equivalen cnrl prcedure applied direcly ver his kind f sysems prduce unsable cnrllers. An apprach slve he previus prblem, and ha permi he use f SMC nnminimun phase prcesses is presened in []. In summary, up nw, he synhesis f a SMCr has been shwn frm he linearizain f a nnlinear chemical prcess. The linear mdel represening he nnlinear chemical prcess is an FODT mdel. The characerisic parameers f he FODT mdel als are used in he uning equains Thus, frm he previus resuls, he cnrller equain be used is ha bained frm he Taylr Series Apprximain. The nex par illusraes he cnrller perfrmance. 6

7 5. Simulain Resuls This secin simulaes he cnrl perfrmance f he SMCr designed and given in Eqs. 8a and 8b. The firs prcess, a mixing ank, cmpares he perfrmance f he SMCr wih respec a ID cnrller. The secnd prcess, a chemical reacr, presens furher perfrmance characerisics. 5. Mixing Tank Cnsider he mixing ank shwn in Figure 4. The ank receives w sreams, a h sream, WBB, and a cld sream, WB B. The ule emperaure is measured a a pin 5 f dwnsream frm he ank. The fllwing assumpins are acceped The liquid vlume in he ank is cnsidered cnsan The ank cnens are well mixed The ank and he pipe are well insulaed. The emperaure ransmier is calibraed fr a range f F F. Table shws he seady-sae cndiins and her peraing infrmain. The fllwing equains cnsiue he prcess mdel [Figure 4] Energy balance arund mixing ank dt W Cp T W Cp T W W Cp T = VρCv d ipe delay beween he ank and he sensr lcain T 4 = T - Transprain lag r delay ime LAρ = 4 W W Temperaure Transmier dto T4 = TO d T 5 Valve psiin dv p = m -V d p [ ] 6 Vp Valve equain 5 W = CVL V p G f v 7 6 Sliding Mde Cnrller SMCr 7

8 WBB WBB TBB TBB TB4B BB = VBpB WBB 5. WBB 9.7 TBB 5 TBB 5 TBB 5 B.8 m.5.4 U= U cu D 8 where = mass flw f h sream, lb/min = mass flw f cld sream, lb/min Cp = liquid hea capaciy a cnsan pressure, Bu/lb- F Cv = liquid hea capaciy a cnsan vlume, Bu/lb- F = h flw emperaure, F TB B = cld flw emperaure, F = liquid emperaure in he mixing ank, F = equal TB B delayed by BB, F deadime r ransprain lag, min ρ = densiy f he mixing ank cnens, lbm/f V = liquid vlume, f TO = ransmier upu signal n a scale frm = valve psiin, frm valve clsed valve pen m = fracin f cnrller upu, frm / BC VL = Bvalve flw cefficien, gpm/psi GBf = Bspecific graviy, dimensinless v = pressure drp acrss he valve, psi BT = Bime cnsan f he emperaure sensr, min BVp = Bime cnsan f he acuar, min A = pipe crss secin, f L = pipe lengh, f B B Table. Design parameers and seady-sae values B B B Variable Value Variable Value lb/min V 5 f lb/min TO.5 CpBB Bu/lb- F Vp.478 CpBB. Bu/lb- F BC VL gpm / psi CpBB, CvBB.9 Bu/lb- F v 6 psi Se pin 5 F BTB min F Bvp min F A.6 f F L 5 f ρ 6.4 lb / f.478 CO Fllwing he prcedure, presened in Secin, bain he parameers f he FODT mdel yields: K = -.78 fracin TO/fracin CO, =. min., and B B=.97 min. Using hese values he uning parameers fr he SMCr are λ =. 767min ; K D =. 54 fracinco λ =. 47min ; δ =. 79 fracinto / min The uning parameers fr he I cnrller are K C =. 5 and I =. min, using he uning frmulas fr Dahlin synhesis, which prduce smher respnses han Ziegler-Nichls uning equains, wrking beer fr prcess wih deadime []. Ne ha he cmparisn is dne using he iniial uning parameers fr bh cnrllers, shw he gd perfrmance bained fr he SMCr iniial uning equains, bu hey can be adjused, fine uning, unil accepable cnrl perfrmance be bained. / 8

9 lease ne ha he cnrller equains, Eqs. 8a and 8b, were develped using deviain variables. The fllwing changes he deviain variables in he cnrller acual variables U = m - m and X = TO - TO e = R - TO where m is he cnrller upu, in fracin CO, TO is he ransmier upu, in fracin, and R is he reference value, r se pin, fracin TO. The verbars indicae seady-sae values. Since he prcess gain is negaive, sign K is negaive, he cnrller equain be used is K D S m= m - TO - TO λ e K S δ 8a wih dto S = λ e λ e d d 8b Figure 5 shws he respnse f he emperaure, T 4, when he flw f h waer changes frm 5 lb/min lb/min, hen 75 lb/min, 5 lb/min, and finally 5 lb/min. The curves clearly shw ha as he peraing cndiins change, he perfrmance f he ID cnrller degrades, while ha f he SMCr mainains is perfrmance and sabiliy. In his case, as he flw f h waer decreases, wih a crrespnding decrease in cld waer, he deadime beween he ank and he sensr increases. This increase in deadime cerainly adversely affecs he perfrmance f he ID cnrller. T recver sabiliy, new unings are required fr he I cnrller while nne are required fr he SMCr. [Figure 5] In spie f he cnrller was synhesized using a Taylr apprximain and he uning equains, Eqs. 9a 9d, are empirical, he prpsed mehd can be successfully used in prcesses wih a deadime ime cnsan rai larger han ne. In ur experience, hey can be applied fr BB/ arund f. 5. Chemical Reacr The reacr shwn in Fig. 6 is a cninuus sirred ank where he exhermic reacin A B akes place. T remve he hea f reacin he reacr is surrunded by a jacke hrugh which a cling liquid flws. [Figure 6] The fllwing assumpins are acceped hea lsses frm he jacke he surrundings are negligible densiies and hea capaciies f he reacans and prducs are bh equal and cnsan he hea f reacin is cnsan. level f liquid in he reacr ank is cnsan; ha is, he flw u is equal he flw in. he reacr and he jacke are perfecly mixed. The emperaure cnrller is calibraed fr a range f 8 C C. Table shws he seady-sae and her peraing infrmain. 9

10 TBiB TBcB CBpB = ρbcb = BTB = = / The fllwing equains cnsiue he prcess mdel. Mle Balance n reacan A dc d F V C C kc = 9 A Ai A A Energy balance n reacr cnens dt d F = Ti - T V H ρ C p UA V C p R - T - T c kc A ρ Energy Balance n jacke dt c UA = d V c ρcc pc Reacin rae cefficien E R T 7 Fc V c T - T c T c - T ci k= k e Temperaure ransmier dto T - 8 = - TO d T Sliding Mde Cnrller SMCr U =U U C D 4 Equal percenage cnrl valve Air clse = F C Cmax α -m F 5 where CBA B = cncenrain f he reacan in he reacr, kgmle / m CBAiB = cncenrain f he reacan in he feed, kgmle / m T = emperaure in he reacr, C = emperaure f he feed, C = jacke emperaure, C TBci B = clan inle emperaure, C TO = ransmier signal n a scale frm fracin TO F = prcess feed rae, m /sec V = reacr vlume, m k = reacin rae cefficien, m kgmle-sec B HBR hea f reacin, assumed cnsan, J/kgmle ρ = densiy f he reacr cnens, kgmle/m hea capaciy f he reacans and prducs, J/ kgmle- C U = verall hea-ransfer cefficien, J /sec-m - C A = hea ransfer area, m Vc = he jacke vlume, m densiy f he clan, kg/m CBpcB = specific hea f he clan, J/kg- C FBc B =B rae, m /sec clanb ime cnsan f he emperaure sensr, sec.

11 kbb = = CBAB TBiB B CBB ρbcb J/kgmle V m - F VBcB J/kgmle U = SMCr upu signal n a scale frm fracin CO B FBC max maximum flw hrugh he cnrl valve, m /sec α = valve rangeabiliy parameer Arrhenius frequency parameer, m /sec-kgmle E = acivain energy f he reacin, J/kgmle R = ideal gas law cnsan, 84.9 J/kgmle-K m = valve psiin n a scale frm Figure 7 shws he pen lp respnse f he reacr; frm his figure prcess parameers, are: K =.6 fracin TO/fracin CO; =. min.; B B=. min. Fr his prcess, because he prcess gain is psiive, he SMCr is m= m λ K D S TO - TO e K S δ 8a wih dto S = λ e λ e d 8b d [Figure 7] Wih he values f K,, and B, Bhe cninuus par f he SMCr can be uned using he λ expressins, Eqs. 9a and 9b, =.4 min - λ.4 min - λ = And, frm Eqs. 9c and 9d KBDB =.96 fracin CO δ =.76 fracin TO/min Table. Design parameers and seady-sae values B BTB KBB TBcB Variable Value Variable Value. kgmle/m m CBaiB.88 kgmle/m.45 m /min T 88 C FBC. m /sec maxb 66 C CBcB 484 J/kg- C TBciB 7 C α 5 Se pin 88 C min HBRB 7-9.6e m /sec-kgmle 5.85e J/kgmle- C E 7.8e U 55. J/sec-m C C kg/m.54 fracin CO A ρ 5.4 m 9. kgmles/m 7.8 m Figure 8 shws he sysem respnse when a % change in inle flw ccurs. The figure shws ha, because he emperaure f he inle flw is cler han he emperaure in he reacr, he reacr emperaure firs decreases smewha. Hwever, afer a shr while he emperaure in he reacr increases since mre

12 reacan is added he reacr. [Figure 8] Figure 8 shws he cnrl perfrmance when he mdeling errr beween he real prcess and he FODT mdel is small. Hwever, he mdel is never perfec. [4] cnsiders ha mdeling errr f 5% in is parameers as "reasnable errr. Le us cnsider w cases. The firs case is fr -% mdel errr and he secnd ne is fr % in mdel errr. The secnd case culd be cnsidered as "unreasnable errr, bu ur inen is judge he cnrller. The errr used is he same in every parameer, ha is, he same -% errr in K, and BB. Figure 9 shws he pen lp respnses fr he acual prcess and fr he mdel wih a -% and % errr. [Figure 9] Figures and shw he prcess respnse when he inle flw changes by % and he mdeling errr used is -% and % respecively. A cmparisn f Figs. 8 and 6, when n errr in he mdel is presen shws lile difference in he prcess respnse. Fig. 9 shws ha wih % errr in he mdel, he cnrl perfrmance degrades smewha. The ms significan difference is ha i akes lnger reurn he prcess he se pin. Hwever, even wih such a large errr in he mdel, he cnrl is sill sable. 6. Cnclusins [Figure ] [Figure ] This paper has shwn he synhesis f a sliding mde cnrller based n an FODT mdel f he acual prcess. The cnrller bained is f fixed srucure. A se f equains bains he firs esimaes fr he uning parameers. The examples presened indicae ha he SMCr perfrmance is sable and quie saisfacry in spie f nnlineariies ver a wide range f peraing cndiins. The relains given in Eqs.9 prvided a gd saring se f unings. The cnrller law, Eqs. 8a and 8b shuld be raher easy implemen in any cmpuer sysem DCS[]. References [] Smih, C. A., and A. B. Crripi, 997. rinciples and racice f Aumaic rcess Cnrl, Jhn Wiley & Sns, Inc., New Yrk. [] Ukin, V. I., 977, Variable Srucure Sysems Wih Sliding Mdes, Transacins f IEEE n Aumaic Cnrl, AC, pp.. [] Sira-Ramirez, H., and O. Llanes-Saniag, 994, Dynamical Discninuus Feedback Sraegies In The Regulain f Nnlinear Chemical rcesses, IEEE Transacins n Cnrl Sysems Technlgy,, #, pp.. [4] Sline, J.J., and W. Li, 99, Applied Nnlinear Cnrl, renice-hall, New Jersey. [5] Clanin, Maria C., Alfred C. Desages, Jse A. Rmagnli, and Ahme alazglu, 995. Nnlinear Cnrl f a CSTR: Disurbance rejecin using sliding mde cnrl, Indusrial & Engineering Chemisry Research, 4, pp. 8-9 [6] Zinber, A. S. I., 994. Variable Srucure And Liapunv Cnrl, Spring Verlag, Lndn.

13 [7] Yung, K. D., V.I. Ukin and Ü. Özgumer, 999. A Cnrl Engineer s Guide Sliding Mde Cnrl. IEEE Transacins n Cnrl Sysems Technlgy, 7, #, pp [8] Hung, J.Y., W. Ga, and J.C. Hung, 99. Variable Srucure Cnrl: A Survey. IEEE Transacins n Indusrial Elecrnics, 4, #, pp. -. [9] Yung, G.E and S. Ra, 987, Rbus Sliding-Mde f a Nnlinear rcess wih Uncerainy and Delay. Jurnal f Dynamical Sysems, Measuremen, and Cnrl, 9, pp. -8 [] Camach, O., R. Rjas and W. Garcia 999, Variable Srucure Cnrl Applied Chemical rcesses wih Inverse Respnse. ISA Transacins, 8, pp [] Camach, O. E A New Apprach T Design And Tune Sliding Mde Cnrllers Fr Chemical rcesses, h.d. Disserain, 996, Universiy f Suh Flrida, Tampa, Flrida [] Camach, O., C. Smih and E. Chacón, 997. Tward an Implemenain f Sliding Mde Cnrl Chemical rcesses. rceedings f ISIE 97, Guimaraes-rugal, pp. -5. [] Himmelblau, D.M 97, Applied Nnlinear rgramming, Mc Graw-Hill, New Yrk [4] Marlin, T. E., 995, rcess Cnrl, Mc Graw-Hill, New Yrk

14 Figure. rcess Reacin Curve 4

15 Figure. Graphical inerpreain f SMC 5

16 Figure. Cmparisn amng e -x, Taylr and adé apprximains 6

17 Figure 4. Mixing ank 7

18 F l w, l b/m i n i m e, m i n 6 58 I D T, d e g F i m e, m i n 6 58 S M Cr T, d e g F i m e, m i n Figure 5. Temperaure respnse under SMCr and ID cnrller 8

19 Figure 6. Scheme f Cninuus Sirred Tank Reacr 9

20 Figure 7. rcess reacin curve fr reacr

21 Figure 8. Sysem respnses fr % in inle flw

22 Figure 9. Effec f mdeling errr.

23 Figure. Sysem respnses fr % change in inle flw fr -% errr in mdeling

24 Figure. Sysem respnses fr % change in inle flw fr % errr in mdeling 4