used fo developin he sepoin ackin calculao while he exenal is applied fo he conolle synhesis. he as empeaue is applied wih I/O lineaizaion o develop h

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1 empeaue Conol of hemal Cackin Funace wih a Coupled ODE and D-PDEs Model Chawin aweeojkulsi and Chanin Panjaponpon Absac his pape pesens a new conol echnique fo he hemal cackin funace modeled by ses of odinay diffeenial equaion (ODE) and D-paial diffeenial equaions (PDEs). he dynamics of coupled D- PDEs-ODE model have been divided ino subsysems, se of sae vaiables of he inenal and exenal cackin coil. Wih he concep of inpu-oupu (I/O) lineaizaion, hese inne and oue dynamics ae applied o desin he sepoin ackin calculao and he appoximae I/O feedback conolle eecively. he fis-ode eo dynamics and he finie-based, open-loop obseve ae ineaed wih he poposed conolle sysem o compensae he model mismach and o pedic he unmeasued sae infomaion. he pefomances of he poposed mehod ae evaluaed houh he sevo es. he esuls showed ha he conol mehod effecively foces he oupu o he desied sepoin. I. INRODUCION Vinyl chloide monome (VCM) is a aw maeial fo Poly Vinyl Chloide (PVC) poducion. I is ypically obained fom he cackin of,-dichlooehane () unde C, of which hydoen chloide (HCl) is a bypoduc. he eacion can poceed by followin: CH4Cl( ) CH3 Cl ( ) + HCl ( ) () VCM Hydoen Chloide he cackin ae sonly depends on he eacion empeaue; incease on he eacion empeaue esuls in he hih cackin ae. he vapo is eaced alon he lenhy empy coil suended in he chambe of he as-fied cackin funace. Funace dynamics ae hihly nonlinea due o he aial disibued empeaue and concenaion of he as inside he cackin coil, as well as he effec of he empeaue of he funace wall. hese complex behavios lead o deeioae he pefomance of he as empeaue conol by a popoional ineal deivaive (PID) conolle. hey may cause off ec of he poducs, hemal unaway, plan shu down o, in he wos case, explosion. heefoe, he conol mehod ha can handle he empeaue of he cackin funace effecively is needed o achieve a hih qualiy poduc. Reseach woks eadin he empeaue conol of he funace wee mosly focused on he dynamics of he ubula Chanin Panjaponpon (coeondin auho: ex. 30; fencnp@ku.ac.h) and Chawin aweeojkulsi ae wih Cene of Excellence on Peochemical and Maeials echnoloy, Depamen of Chemical Enineein, Faculy of Enineein, Kasesa Univesiy, Bankok 0900, hailand. eaco. Some woks applied he model educion echnique o lump he eaco model befoe pefomin he conolle synhesis. Fo example, he PDE was lumped by Galekin mehod and hen applied wih infinie dimensional sae feedback [] lumped by mehod of chaaceisic and applied wih obus conol [] and lumped by infinie dimensional mehod and applied wih he linea quadaic eulao (QR) [3]. Some woks use he pocess daa o develop an empiical model by he neual newok mehod befoe applied wih he obus conol [4] o eneic model conol (GMC) [5]. Besides, hee ae few woks considein o he ineacion of wall adiaion in he conol of funace. Masoumi and colleaues [6] sudied he empeaue conol of he naphha hemal cackin wih muli cackin coils by usin he PI conolle. he desied sepoins wee obained fom he opimizaion of he empeaue pofile. In Zeybek [7], he oule as empeaue is conolled by manipulain he fuel mass flow ae by usin he adapive heuisic conolle based on hee layes of feed fowad aificial newok (ANN). Panjaponpon e al. [8] poposed he conol of coupled PDE-ODEs fo cackin funace by usin appoximae I/O lineaizaion; he ube empeaue was conolled by manipulain he fuel as flow while he mass poducion ae of VCM was handled by he PI conolle by manipulain he feed. he funace model was developed by assumin a plu-flow velociy pofile and nelecin he effec of he adius hea ansfe. Howeve, hee ae some woks menioned abou sinifican diffeence of pediced pocess dynamics when he edial effec and velociy pofile has been aken ino accouned [9-0]. his bins abou he quesion of he impovemen of conol pefomance when he D model has been applied. In fac, he as empeaue epesened he eacion empeaue is measued by a hemocouple insalled a he cene of he exi ube. he convesion calculaed by he D model will be lowe han he acual pocess value. he pefomance of D-based PDE-ODE conolle in pacice may deeioae due o a sinifican pocess-model mismach. his wok pesens a new sucue of he couplin D PDEs-ODE model fo he cackin funace by usin he I/O lineaizaion. he dynamics of cackin funace consis of he concenaion and as empeaue consideed as he inenal saes and he ube empeaue and funace wall consideed as exenal saes. All he dynamics ae descibed by PDEs excep he funace wall dynamics descibed by ODE. he pupose of his wok is o conol he as empeaue a he exi ube by manipulain he fuel as flow. Insead of applyin he I/O conolle o he objecive diecly, he inenal subsysem is 543

2 used fo developin he sepoin ackin calculao while he exenal is applied fo he conolle synhesis. he as empeaue is applied wih I/O lineaizaion o develop he mappin funcion of he equivalen ube empeaue sepoin fo he I/O feedback conolle. he fis-ode eo dynamics and he finie-based, open-loop obseve ae ineaed ino he conol sysem o eliminae he offse and pedic he unmeasued sae infomaion. An advanae of poposed conol mehod wih he paiionin sae dynamics is o educe he complexiy of he conolle equaion wih a bee pedicin qualiy by usin he D pocess model. II. MAHEMAIC MODE OF CRACKING FURNACE A simple pocess scheme of an cackin funace is shown in Fi.. In he opeaion, vapo is fed o he cackin coil and conveed o be VCM and HCl. he naual as is used as a combusion fuel o supply he eney o he funace o ise up he funace wall empeaue ( w ). he funace wall adiaes and ansfes he eney o he ube inside leadin o he chane of he ube empeaue ( ), he as empeaue ( ) and concenaion (C ) consequenly. In his wok, he poposed conol saey is applied wih D PDEs-ODE model of cackin funace. he followin model assumpions ae applied: ) All ases in he sysem ae ideal. ) Only he eacion in () occuin in he ube is concened. 3) Nelec effecs of all elbows and fiins; saih ube is assumed. 4) he popeies of ases in he ube ae consan. 5) he ube empeaue is vaied alon he z-diecion only because of he pipe hickness << he coil disance. 6) he as empeaue and concenaion ae vaied in boh he adius and disance of he coil. Fi.. Coninuous sied ank eaco wih coolin sysem C C k C = v + + Cp k AFσ C V C i = v + + ( ) p p ''' ( H) + 0 C Ea R p = ke C wih he followin boundaies and iniial condiions: fo he concenaion, BC: C (0, z, ) = 0 BC : C ( Ri, z, ) = 0 BC3 : C (,0, ) = C IC: C (, z,0) = C,0,0 and fo he as empeaue, BC4 : (0, z, ) = 0 BC5: ( Ri, z,) = (,) z BC6 : (,0, ) =,0 IC : (, z, = 0) =,0 he velociy pofile of he as flowin in he coil is efeed o an empiical/analyical soluion of k ε ubulence model in []: dp dz Ri v ( ) =.85 + dp dz Ri Ri ln 0.3 µ 6 dp dz R i whee he oal pessue adien and fannin ficion faco ae appoximaed by usin analyical/empiical equaions poved fom he Moody ficion []: dp P dz f vav P = (4) 4 R i () (3) he dynamic models of he cackin funace ae epesened by followin equaions - he dynamics of concenaion and eaco empeaue in he cackin coil: f 0.84 = 0., Re 0,000 Re 544

3 he aveae velociy is calculaed by R i νav = νz () d R (5) i 0 - he dynamics of ube and funace wall empeaue: AF = + c d w k p, cp V, h, σ ( ) w w π AF σ ( ) ( ) i w ln( R / R ) o i m H σ FA = ( m c ) ( ) fuel comb w w w p w wih he bounday and iniial condiions: BC7 : (0, ) =,0 BC8: (, ) = 0 IC3 : ( z,0) =,0 IC4 : ( z,0) = w w,0 k + Rh All pocess paamees defined in he noaion secion ae iven in able I. he model of he fied-funace in ()-(6) descibed by paial diffeenial equaions in and z coodinaes and odinay diffeenial equaion can be ouped ino wo subsysem. he subsysem of (7.a) expesses he ineacion of he sae vaiables inside he cackin coil and he subsysem in (7.b) expesses he ineacion of he sae vaiables ouside he cackin coil and he adiain wall. xp (, z,) xp xp = a + b + M( xp, xp) (7.a) xp(,) z xp = c + Nx ( p, xo( )) + Ox ( p, xo) dxo () = f( xo, xp, u ( )) y= hx ( ) p wih he iniial and bounday condiions of (7.a): xp (0, z, ) = 0, xp ( Rz,,) BC.. xp ( R, z,) = xp(,) z o = 0, xp (,0,) = xp (,) IC.. x (, z,0) = x (, z) p p,0 and he iniial and bounday condiions of (7.b): o f ] (6) (7.b) able I. PARAMEER VAUES FOR HE CRACKING FURNACE Symbol Quaniy Value A w Aea of he funace wall 8 m C p BC.. IC.. xp(0, ) = xp( ), xp = 0 z= xp( z,0) = xp,0( z), x ( = 0) = x o Aveae hea capaciy of cacked ases o, m 3 C p Hea capaciy of he ube 444 J/k K C pw Hea capaciy of he 000 /k K funace wall D i Inenal ube diamee 0.9 m E a Acivaion eney J/mol F Shape faco H Hea of eacion J/mol H comb Hea of combusion J/mol k 0 Kineic consan k k hemal conduciviy of ases in ube hemal conduciviy of he W/m K 0.5 W/m K ube ube lenh 300 m m ube weih k m w Mass of funace wall k m f Mass flow ae of he fuel k/s Mw molecula weih /mol Pe hemal Pecle numbe P Pandl numbe 0.7 R Gas consan 8.34 J/mol K R i Inenal ube adius m R o Exenal ube adius 0. m Re Reynolds numbe V Pipe volume m 3 Cacked as densiy k/m 3 ube densiy 8470 k/m 3 σ Sefan-Bolzman consan W/m K µ Viscosiy of cacked ases k/m s ν Feed velociy 5 m/s whee xp (, z,) denoes he veco of he sae vaiables dependin on and z coodinaes, x (,) p z denoes he sae vaiable of he exenal ube dynamics which is diecly 545

4 affeced by x o, xo () denoes he sae vaiable which depend on ime, y denoe he oupu vaiable, z [0, ] and [0, R o ] ae aial coodinaes, [0, ] is he ime, and u () is he manipulaed vaiable. III. CONRO SYSEM DESIGN In ou case, he pocess model is hihly complex due o coupled PDEs and ODE. he conol objecive is o eulae he oupu a he exi of he ube (y=), he sae in he subsysem (7.a), by adjusin he inpu (u) in he subsysem (7.b). o educe complexiy of he conolle desin, in his wok, he se of PDE in (7.a) descibed he inenal ube dynamics will be used o ceae a ackin coelaion beween he oupu ( y ) and he disibued sae vaiable elaed o he lumped dynamics ( x ). he se of couplin p PDEs-ODE in (7.b) descibed he exenal ube dynamics will be used o develop he I/O feedback conolle ha he conol acion ( u ) is obained by solvin closed-loop eonse of x. A schemaic diaam of he conol sysem p shown in Fi. is poposed. he conol sysem consiss of a sepoin ackin calculao, I/O lineaizin conolle, and a finie-based, open-loop obseve. Moe deails of he conol sysem desin ae iven as follows. A. Sepoin ackin calculao he inpu/oupu lineaizaion is a mehod ha ceaes a linea elaionship beween inpu and oupu based on he coodinae ansfomaion. I is adiionally applied fo he ODE sysem. Fo he applicaion of PDEs-ODEs sysem, le conside he sysem in (8). dx = f( x, x, x, xz, xzz, u) y = hx ( ) = = 0, z whee x is he veco of sae vaiables, x z = x z and x = x ae he fis-ode aial deivaives of x (8) eec o z-diecion and -diecions, xzz = x z and x = x ae he second-ode aial deivaives of x eec o z-diecion and -diecion, u is he manipulaed inpu and h is he veco of nonlinea funcions. he elaive ode of he conolled oupu y,, can be defined by followin noaion: y = h ( x) = 0, z= dy h x h ( x, x, x, xz, xzz ) x = z= h x h x h x d = 0, z= + + y x x x h xz h xzz xz xzz = h ( xzz ) = 0, z= x x x, x, x, xz, xzz,,, ( x ),,, xz xzz ( x ),,, ( ),,, ( ) x z x zz h x h x h x + + x x x h xz h xzz ( x ) zz xz xzz = 0, z= d y x x x, x, x, xz, xzz,,, ( x ),,, = h xz xzz ( x ),,, ( ),,, ( ), x z x zz u = 0, z= = 0, z= he sepoin ackin calculao is applied o develop a coelaion beween y x p. Fom he subsysem (7.a), he closed-loop eonse of he oupu a he cene of he exi ube is in linea fom as follows: (9) Fi.. Schemaic diaam of he poposed conol sysem. 546

5 ( ε + ) y = y, (0) whee is he diffeenial opeao, y is he oupu a he posiion =0 and z=, y, is he desied sepoin, ε is he unin paamee used o adjus he eed of he oupu eonse and is he elaive ode of y wih eec o x. p By subsiuin he ime deivaives of (9) ino (0) and sein all ime deivaives of he sae adiens o be zeo, he closed-loop eonses of he oupu can be pesened in a compac fom φ ( x, x, x, x, x ) = y () p p, p, p, z p, he ackin sepoin funcion (ν ) of can be obained by solvin () fo x p, in followin fom: ν = ψ ( x, x, x, x, x, y ) () p p, p, p, z p, B. Feedback I/O lineaizin conolle Fom he subsysem (7.b), he closed-loop eonses of he sae xp a he posiion z= ae equesed in linea fom as follows: ( β ) xp + = ν (3) whee ν is he ackin sepoin funcion, β is he unin paamee and is he elaive ode of xp wih eec o u. We subsiue he ime deivaive in (9) ino (3) and se all ime deivaives of he sae adiens o be zeo. he closed-loop eonses of he sae xp can be pesened in a compac fom φ ( x, x, x, u) = ν (4) p p zz hus, he feedback conolle (u) is obained by solvin (4). he compac fom of he conolle equaion is denoed by (5) u = Ψ ( x, x, x, ν ) (5) p p zz C. Finie-based sae obseve he CFD echnique is a useful ool o pedic behavio of he sysem of he complex PDE poblem by usin he numeical calculaion. hus, in his wok, a CFD-based, open-loop sae obseve is developed o povide he esimaion of he unmeasued pocess concenaion, C, and he sae deivaives. D. Ineao o compensae he pocess-model mismach and he eo fom he esimae saes, he fis-ode eo dynamics in (6) is applied: ε ν = λ ( y y) = y ε = 0, z= (6) ε whee is he oupu eo, ν is he coeced sepoin. λ is a posiive paamee, and IV. RESU AND DISCUSSION he velociy wih plu-flow paen is pimaily assumed in many lieaues fo a conol of he ubula eaco. Howeve, his assumpion is pope fo a hih viscosiy fluid. o achieve a ealisic pedicion, he k ε ubulence model is applied wih he developed D model, which he compaed velociy pofiles ae shown in Fi.3. Fo he sevo es, he as empeaue a exi ube is conolled a he desied sepoin y = 700 K. he iniial condiions of he dynamics ae C (,0,) = mol/l, (,0,)=644 K, (0,)= 76 K, and w =808 K. he unin paamees of he poposed conol sysem ae ε =8, β =8 and λ =0.00. he closed-loop eonses of he cackin funace ae illusaed in Fis he esuls show ha Velociy (m/s) k ε Plu flow Radius (m) Fi.3. he flow paen of cacked as inside he ube. Gas empeaue (K) Poposed conolle Sepoin ime (min) Fi.4. he closed-loop eonse of he as empeaue a he cene exi ube. 547

6 empeaue (K) Fi.5. he closed-loop eonses of he ube empeaue a he exi and wall empeaue. Mass fuel flow (k/s) ime (min) ime (min) Fi.6. he conol acion of he manipulaed inpu. w he conolle successfully foces he as empeaue a he desied sepoin. he chanes of as, ube and wall empeaue a he iniial peiod have a linea end due o he influence fom he consan of fuel as ae a he uppe limi. he conolle is hen adjused he fuel as flow wih a lile fuzzy o pu he as empeaue a he desied sepoin. ACKNOWEDGMEN his wok was financially suppoed by he Kasesa Univesiy Reseach and Developmen Insiue (KURDI), he pojec fo Hihe Educaion Reseach Pomoion and Naional Reseach Univesiy Developmen, Office of he Hihe Educaion Commission, and he Cene of Excellence on Peochemicals and Maeials echnoloy. hese suppos ae aefully acknowleded. REFERENCES [] Shan, H., Fase Fobes, J., Guay, M., 005. Feedback conol of hypebolic disibued paamee sysems. Chemical Enineein Science 60, [] Hoo, K.A., Zhen, D., 00. ow-ode conol-elevan models fo a class of disibued paamee sysems. Chemical Enineein Science 56, [3] Mohadam, A.A., Aksikas, I., Dubljevic, S., Fobes, J.F., 00. Q conol of coupled hypebolic PDEs and ODEs: Applicaion o a CSR PFR sysem, in Poceedins of he Ninh Inenaional Symposium on Dynamics and Conol of Pocess Sysems. pp [4] Yamuna Rani, K., Pawadhan, S.C., 007. Daa-Diven Model Based Conol of a Muli-Poduc Semi-Bach Polymeizaion Reaco. Chemical Enineein Reseach and Desin 85, [5] Aeloiannaki, E., Saimveis, H., 009. Robus nonlinea H conol of hypebolic disibued paamee sysems. Conol Enineein Pacice 7, [6] Masoumi, M.E., Sadameli, S.M., owfihi, J., Niaei, A., 006. Simulaion, opimizaion and conol of a hemal cackin funace. Eney 3, [7] Zeybek, Z., 006. Role of adapive heuisic ciicism in cascade empeaue conol of an indusial ubula funace. Applied hemal Enineein 6, [8] Panjaponpon, C., impanachaiponkul, P., Chainpanikul,., 0. Conol of coupled PDEs ODEs usin inpu oupu lineaizaion: Applicaion o a cackin funace. Chemical Enineein Science 75, [9] Van Geem, K.M., Heyndeickx, G.J., Main, G.B., 004. Effec of adial empeaue pofiles on yields in seam cackin. AIChE jounal 50, [0] Han, Y.., Xiao, R., Zhan, M.Y., 007. Combusion and Pyolysis Reacions in a Naphha Cackin Funace. Chemical Enineein & echnoloy 30, 0. doi:0.00/cea [] Mecado, E.R.., Nunhez, J.R., 000. Modelaem do aquecimeno de fluidos com escoameno em ubos [WWW Documen]. UR hp:// 788 (accessed.0.4). [] Incopea, F., Dewi, D., 00. Fundamenals of Hea and Mass ansfe, 5 h ed. John Wiley & Sons, New Yok V. CONCUSION A new conolle sucue wih I/O lineaizaion echnique is developed fo he cackin funace, of which he advanaes ae a few unin paamees and decease on he complexiy of he conolle equaion. Wih he impoance of he disibuion in -diecion of fluid flow in he ube, he k ε ubulen model is applied o he velociy. he conolle is fomulaed wih he D-PDEs and ODE ino he sepoin ackin calculao and I/O feedback conolle, and ineaed wih he fis-ode eo dynamics and finie-based, open-loop obseve. he simulaion esuls show ha he conolle can foce he conol oupu a he desied sepoin effecively. 548

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