ANALYSIS AND TEMPERATURES CONTROL IN A TUBULAR CHEMICAL REACTOR

Transcription

1 PHYSCON 9 Catania Italy Septembe Septembe 9 NLYSIS ND TEMPERTURES CONTROL IN TUBULR CHEMICL RECTOR Pet Dotál Vladimí Bobál and Jiří Vojtěšek Depatment of Poe Contol Faulty of pplied Infomati Toma Bata Univeity in Zlin Czeh Republi dotalp@fai.utb.z btat The pape deal ith analyi and ontinuoutime adaptive ontol of a tubula hemial eato ith a ounteuent ooling a a non-linea ingle input ingle output poe. The output eatant tempeatue and the mean eatant tempeatue ae hoen a the ontolled output and the oolant flo ate a the ontol input. The paamete of it ontinuou-time extenal linea model ae etimated via oeponding delta model The eulting ontolle ae deived uing polynomial appoah. The ontol ytem tutue ith to feedbak ontolle i onideed. The appoah i teted on a mathematial model of the tubula hemial eato. Key od Tubula hemial eato nonlinea model extenal linea model delta model polynomial method. Intodution Tubula hemial eato ae unit fequently ued in hemial and biohemial induty. Fom the ytem theoy point of vie tubula hemial eato belong to a la of nonlinea ditibuted paamete ytem. Thei mathematial model ae deibed by et of nonlinea patial diffeential equation (PDR). The method of modelling and imulation of uh poee ae deibed eg. in [Luyben 989; Ingham et al. 99; Dotál et al. 8 and Seveane ]. It i ell knon that the ontol of hemial eato and tubula eato epeially often epeent vey omplex poblem. t all event a peviou analyi of the poe behaviou i obligatoy. One poible method to ope ith thi poblem i appliation of adaptive tategie baed on an appopiate hoie of a ontinuou-time extenal linea model (CT ELM) ith euively etimated paamete (ee e.g. [Rao and Unbehauen 5]). Thee paamete ae onequently ued fo paallel updating of ontolle paamete. The pape deal ith analyi and ontinuoutime adaptive ontol of a tubula hemial eato ith a ounteuent ooling a a non-linea ingle input ingle output poe. With epet to patial poibilitie of a meauement and ontol the output eatant tempeatue and the mean eatant tempeatue ae hoen a the ontolled output and the oolant flo ate a the ontol input. The paamete of it CT ELM ae etimated via oeponding delta model (ee e.g. [Middleton and Goodin 99; Mukhopadhyay et al. 99; Steike and Sinha 99]. The eulting ontolle ae deived uing polynomial appoah [Kučea 99]. The ontol ytem tutue ith to feedbak ontolle [Dotál et al. 7] i onideed. The appoah i teted on a mathematial model of the tubula hemial eato. Model of the Plant Conide an ideal plug-flo tubula hemial eato ith a imple exothemi oneutive k k eation B C in the liquid phae and ith the ounteuent ooling. Heat loe and heat ondution along the metal all of tube ae aumed to be negligible but dynami of the metal all of tube ae ignifiant. ll denitie heat apaitie and heat tanfe oeffiient ae aumed to be ontant. Unde above aumption the eato model an be deibed by five PDR

2 t + v z = k () B B + v = k k B () t z T T Q U + v = ( T T )() t z ( ρ ) d ( ρ ) p p T = du( T T ) + t ( d d )( ρ ) p () + d U ( T T T T ndu v = ( T T )(5) t z ( d n d )( ρ ) p ith initial and bounday ondition ( z) = ( z) B ( z) = B ( z) T ( z) = T ( z) T ( z) = T ( z) T ( z) T ( z) = ( t) = ( t) B ( t) = B ( t) T ( t) = T ( t) T ( L t) = T L ( t). Hee t i the time z i the axial pae vaiable ae onentation T ae tempeatue v ae fluid veloitie d ae diamete ρ ae denitie p ae peifi heat apaitie U ae heat tanfe oeffiient n i the numbe of tube and L i the length of tube. The ubipt ( ) tand fo the eatant mixtue ( ) fo the metal all of tube ( ) fo the oolant and the upeipt ( ) fo teady-tate value. The eation ate and heat of eation ae nonlinea funtion expeed a E j k j = k j exp j = (6) RT Q = ( Δ H ) k + ( Δ H ) k (7) B hee k ae pe-exponential fato E ae ativation enegie ( - ΔH ) ae in the negative onideed eation entalpie and R i the ga ontant. The fluid veloitie ae alulated via the eatant and oolant flo ate a q q v = v = (8) π nd ( ) π d n d The paamete value ith oepondent unit ued fo imulation ae in Tab.. Fom the ytem engineeing point of vie ( ) L t = out ( ) B L t = B out T ( ) L t = T out and T ( t ) = T ae the output vaiable and out q () t q () t () t T () t and T () t L ae the input vaiable. mong them fo the ontol pupoe motly the oolant flo ate Tab.. Ued Paamete Value L = 8 m n = d =. m ρ = 985 kg/m ρ = 78 kg/m ρ = 998 kg/m U =.8 kj/m K d = m d =. m p =.5 kj/kg K p =.7 kj/kg K p =.8 kj/kg K U =.56 kj/m K k = / k =.8 8 / E /R = 77 K (-ΔH ) = 5.8 kj/kmol q E /R = 59 K (-ΔH ) =.8 kj/kmol () t an be taken into aount a the ontol vaiable heea othe input ente into the poe a ditubane. the ontolled output ae onideed the eatant output tempeatue T and the mean eatant tempeatue given by out L Tm ( t) = T ( z t) d z. (9) L Computation Model Fo omputation of both teady-tate and dynami haateiti the finite difeene method i employed. The poedue i baed on ubtitution of the pae inteval z < L > by a et of diete node point { z i } fo i = n and ubequently by appoximation of deivative ith epet to the pae vaiable in eah node point by finite diffeene. The poedue i in detail deibed in [Dotál et al. 8]. Steady-tate Chaateiti Steady-tate haateiti ee omputed fom dietized model ith zeo time deivative uing fixed point iteation algoithm. The dependene of output onentation and tempeatue on the oolant flo ate in the teadytate fo.85 = = T = T = 9 and q =.5 ae hon in Fig.. The nonlineaity of all haateiti i evident. 5 Dynami Chaateiti The dynami haakteiti ee invetigated fo both uppoed ontolled output and eleted tep hange of the ontol input in the neighbouhood of the hoen opeating point q =.7 m / T out = 6. K and T m =. K. ll vaiable ee onideed a deviation fom thei teady B

3 value Δ q = q() t q Δ Tout () t = Tout () t Tout Δ Tm() t = Tm() t Tm. Simulated tep epone ae in Fig. and. out (kmol/m ) q (m /) out Bout Figue. Dependene of output onentation on the oolant flo ate. Tempeatue (K) q (m /) T out Figue. Dependene of eatant output and mean tempeatue on the oolant flo ate. ΔT out (K) ΔT m (K) - - Δq -6 = -. - Δq = -. - Δq =. - Δq = t () Figue. Reatant output tempeatue tep epone Δq = -. - Δq = -. - Δq =. - Δq = t () Figue. Reatant mean tempeatue tep epone. T m 6 CT and Delta Extenal Linea Model Fo the ontol pupoe the ontolled output and the ontol input ae defined a y () t =Δ T = T () t T () out out out q () t q y () t =Δ T () t = T () t T u() t =. m m m q Thee expeion enable to obtain vaiable of appoximately the ame magnitude. Fo both ontolled output the eond ode CT ELM have been hoen in the fom of the eond ode linea diffeential equation yt () + a yt () + a yt () = b ut () () o in the tanfe funtion epeentation a Y () b G () = =. () U() + a+ a Etablihing the opeato q = () T hee q i the foad hift opeato and T i the ampling peiod the delta ELM oeponding to () take the fom yt ( ) + a yt ( ) + a yt ( ) = b ut ( ) () hee t i the diete time. When the ampling peiod i hotened the delta opeato appoahe the deivative opeato and the etimated paamete a b eah the paamete a b of the CT model a hon in [Steike and Sinha 99]. 7 Delta ELM Paamete Etimation poedue of the euive CT ELM paamete etimation via a oeponding delta model an be biefly deibed in folloing tep: Subtituting t = k equation () may be eiten to the fom yk ( ) + a yk ( ) + a yk ( ) = (5) = b u( k ). In the pape the euive identifiation method ith exponential and dietional fogetting aoding to [Bobál et al. 5] a ued. Etablihing the egeion veto Φ T ( k ) = ( y( k ) y( k ) u( k ) )(6) the veto of delta model paamete Θ T ( k) = ( a a b ) (7) an be euively etimated fom the RX model ( ) T y k = Θ ( k ) Φ ( k ) +ε ( k ) (8) 8 Contol Sytem Deign The ontol ytem ith to feedbak ontolle i onideed a hon in Fig. 5. Hee i the

4 efeene ignal v denote the load ditubane e i the taking eo u i the output of the ontolle y i the ontolled output and u i the ontol input. v e R - - u Figue 5. Contol ytem ith to feedbak ontolle. In geneal them G epeent an ELM ith the tanfe funtion b () G () = (9) a () and Q and R ae feedbak ontolle ith tanfe funtion q () Q () = () R () = () p () p () hee q and p ae polynomial in. Both and v ae onideed to be tep funtion ith tanfom v W ( ) = V ( ) =. () The ontolle deign appea fom the polynomial appoah and the pole aignment method. The eulting ontolle obtained by a olution of polynomial equation enue the ontol ytem intenal popene and tability a ell a aymptoti taking of tep efeene and tep load ditubane attenuation. The poedue to obtain admiible ontolle an be biefly deibed a follo: Etablihing the polynomial t a t ( ) = ( ) + q ~ ( ) () the ontol ytem tability i enued hen polynomial ~ p and t ae given by a olution of the polynomial Diophantine equation a ( ) ~ p( ) + b( ) t( ) = d( ) () ith a table polynomial d on the ight ide. Evidently the oot of d detemine pole of the loed-loop. Taking into aount the tanfom of the taking eo E () = [( ap + bqw ) () bpv ()] () d and both tanfom () the aymptoti taking and load ditubane attenuation ae povided by polynomial ~ p and q ~ having the fom p () = p () q () = q (). (5) Then the tanfe funtion of ontolle () take fom u G Q y q( ) ( ) Q ( ) = R ( ) =. (6) p( ) p( ) table polynomial p() in denominato of (6) (exepting the integating pat) enue the tability of ontolle. No the polynomial t an be eitten into the fom t ( ) = ( ) + q( ). (7) Taking into aount olvability of () and ondition of intenal popene of the ontol ytem the degee of polynomial in () and (7) an be eaily deived a deg t = deg = deg a deg q = deg a deg p deg a deg d deg a. (8) Denoting deg a = n polynomial t and q have fom t n i ( ) = t i ) = i= n n i ( i q ) = i= i= i ( q i (9) hee thei oeffiient fulfill equalitie = t i + qi = ti fo i =... n () Then unknon oeffiient i and q i an be obtained by a hoie of eletable oeffiient β i uh that i = βi ti q i = ( βi ) ti fo i =... n. () The oeffiient β i plit a eight beteen numeato of tanfe funtion Q and R. The ontolle paamete then follo fom olution of the polynomial equation () and depend upon oeffiient of the polynomial d. Fo the eond ode model () ith deg a = the ontolle' tanfe funtion take peifi fom q + q + + Q () = R () = () + p ( + p ) hee = βt = β t q = ( β) t q = ( β ) t. () In thi pape the polynomial d ith oot detemining the loed-loop pole i hoen a d () = n ()( +α ) () hee n i a table polynomial obtained by petal fatoization a () a() = n () n() (5) and α i the eletable double pole. Note that a hoie of d in the fom () povide the ontol of a good quality fo apeiodi ontolled poee. No it follo fom the above intodued poedue that tuning of the ontolle an be pefomed by a uitable hoie of eletable paamete β and α.

5 9 Simulation Reult The ontol imulation fo the eatant output tempeatue ae hon in Fig Simulation ee pefomed in the neighbouhood of the opeating point q =.7 m / T = 6. K. out Fo the tat (the adaptation phae) a P ontolle ith a mall gain a ued in all imulation. The effet of the pole α on the ontol epone i tanpaent fom Fig. 6 and 7. Hee to value of α ee eleted. The ontol eult ho enitivity of the ontolled output and ontol input to α. Obviouly aele eletion of thi paamete an lead to oillatoy ontolled output o even to untability. Futhe a ineaing α lead to highe value and hange of the ontol input. The effet of β on the ontolled output i hon in Fig. 8. The ontol imulation fo the eatant mean tempeatue hon in Fig. 9 ee pefomed in the neighbouhood of the opeating point m q =.7 m / and T =. K unde the ame efeene and ontaint on the ontol input a in the peviou ae. Peent eult again demontate an poibility to detemine a fom and peed of the ontolled output a ell a an amplitude of the ontol input and it hange. Fo intane the ontol input in Fig. and ho thei high enitivity to a hoie of the paamete β. Thi fat may be vey impotant in a patial ontol. y (K) α =.5 - α = t () Figue 6. Contolled output y dependene on α.(β = β = ). u () t () - α =.5 - α =.75 Figue 7. Contol input dependene on α (β = β = ). Conluion In thi pape one appoah to the ontinuou-time y (K) y (K) - β = - β =.5 - β = t () Figue 8. Contolled output y effet of β (β = ) α =.5 - α =. - α = t () Figue 9. Contolled output y dependene on α (β = β = ). y (K) y (K) β = - β =.5 - β = t () Figue. Contolled output y effet of β (β = ) β = - β =.5 - β = t () Figue. Contolled output y effet of β (β = ). adaptive ontol of the output and mean eatant tempeatue in a tubula hemial eato a popoed. The ontol tategy i baed on a peliminay teady-tate and dynami analyi of the poe behaviou and on the aumption of the tempeatue meauement at the output a ell a along the eato. The popoed algoithm employ an altenative ontinuou-time extenal linea

6 model ith paamete obtained though euive paamete etimation of a oeponding delta model. The eulting ontinuou-time ontolle ae deived uing the polynomial appoah and given by a olution of a polynomial Diophantine equation. Tuning of thei paamete i poible eithe via the paamete affeting the loed-loop pole o by a hoie of eletable oeffiient plitting a eight beteen numeato of ontolle' tanfe funtion. The peented method ha been teted by ompute imulation on the nonlinea model of the tubula hemial eato ith a oneutive exothemi eation. The eult demontate the appliability of the peented ontol tategy. u () y (K) u () - β = - β =.5 - β = - 5 t () Figue. Contol input effet of β (β = ) t () - β = β = - β = β =.5 - β = β = Figue. Contolled output y effet of β β. - β = β = - β = β =.5 - β = β = t () knoledgment Thi ok a uppoted by the Minity of Eduation of the Czeh Republi unde the gant MSM Refeene Bobál V. Böhm J. Fel J. and Maháček J. (5). Digital Self-tuning Contolle. Spinge Velag. Belin. Dotál P. Gazdoš F. Bobál V. and Vojtěšek J. (7). daptive ontol of a ontinuou tied tank eato by to feedbak ontolle. In Po. 9th IFC Wokhop daptation and Leaning in Contol and Signal Poeing LCOSP'7. Saint Petebug Ruia. P5- P5-6 Dotál P. Bobál V. and Vojtěšek J. (8). Simulation of teady-tate and dynami behaviou of a tubula hemial eato. In Po. nd Euopean Confeene on Modelling and Simulation. Nioia Cypu. pp Ingham J. Dunn I.J. Heinzle E. and Přenoil J.E. (99). Chemial Engineeing Dynami. Modelling ith PC Simulation. VCH Velaggeellhaft. Weinheim. Kučea V. (99). Diophantine equation in ontol uvey. utomatia vol. 9 pp Luyben W. (989). Poe modelling imulation and ontol fo hemial enginee. MGa- Hill. Ne Yok. Middleton R.H. and Goodin G.C. (99). Digital Contol and Etimation - Unified ppoah. Pentie Hall Engleood Cliff. Mukhopadhyay S. Pata.G. and Rao G.P. (99). Ne la of diete-time model fo ontinuo-time ytem. Intenational Jounal of Contol vol. 55 pp Rao G.P. and Unbehauen H. (5). Identifiation of ontinuou-time ytem. IEE Po.-Contol Theoy ppl. vol. 5 pp Seveane F.L. (). Sytem Modeling and Simulation. Wiley Chihete. Steike D.L. and Sinha N.K. (99). Identifiation of ontinuou-time ytem fom ample of input-output data uing the - opeato. Contol-Theoy and dvaned Tehnology vol. 9 pp. -5. Figue. Contol input effet of β β.