2.22 Process Gains, Time Lags, Reaction Curves

Transcription

1 . Proc Gain, Tim Lag, Racion Curv P. S. SCHERMANN (995) W. GARCÍA-GABÍN (5) INTRODUCTION An unraning of a proc can b obain by vloping a horical proc mol uing nrgy balanc, ma balanc, an chmical an phyical law; howvr, hi complica an im-conuming ffor can b omi in many ca. A goo approximaion of proc ynamic can b obain by uing implifi calculaion mho, hu rucing h wor rquir o ign conrol ym. Th ynamic bhavior of ypical proc in inurial applicaion (pip vl combinaion, ha xchangr, ranpor piplin, furnac, boilr, pump, compror, urbin, an iillaion column) can b crib uing implifi mol compo of proc gain, a im, an proc ynamic. In hi cion h concp of proc gain an a im an hir rol in conrol ym ign ar xplain. Pracical guilin ar givn for vloping xprimnal mol uing racion curv for variou quipmn cagori ha ar frqunly u in h proc inury. Th ffc of inrumnaion on loop ynamic i alo xamin. Alhough mahmaical rivaion ar minimiz in hi cion, full cripion an applicaion of h mahmaical ool can b obain from h bibliography. Th gain fin h niiviy rlaing h oupu an inpu variabl. I can b calcula a follow: K = Oupu [Tranmir uni] Inpu [ Conrollr uni].() Th ranmir an conrollr ignal can b xpr in milliampr [ma], poun pr quar inch [pig] or prcnag [%] uni, if pnumaic or lcronic analog inrumnaion or DCS ym ar u, rpcivly. Th proc gain frqunly pn on h loa or opraing poin. For xampl, h gain chang whn h cro-cion of a vl vari (hi i h ca wih horizonal cylinrical an), or if h loa i moifi (for xampl, h gain of ha ranfr proc will rop wih riing loa). Th variaion inrouc nonlinarii, maing h proc gain highly pnn on h opraing poin. Thi man ha h proc gain i a funcion of h opraing poin, which mu b an ino accoun in igning h conrol ym. Th conrol ym nginr houl b awar of h phnomnon of proc gain variaion, an h variaion in loop gain mu b compna for. Th u of aapiv (Scion.9) or robu (Scion.6) conrol can omim compna for gain variaion. PROCESS GAINS Proc gain i rmin by a numbr of facor. Th inclu h phyical propri of h quipmn uch a h vl volum, compror characriic, an conncing piping imnion; proc variabl uch a prur, mpraur, an flui vlociy; an variou law of phyic an chmiry. Whil om of h aocia gain ar a funcion of h proc, conirabl improvmn can b ma by rviing h conrol ym, bcau h inrumnaion ha a rong ffc on h loop ynamic. A ranmir gain, for xampl, can b moifi by ajuing h maurmn rang. Th proc gain inica how much a proc propry (oupu) chang pr uni of inpu chang. Th inpu can b a flow ha i by a valv opning; i affc h oupu, which can b a proc propry, uch a mpraur. DEAD TIMES AND TIME CONSTANTS Tim lag, ranporaion lag, im lay, or a im ( ) i commonly prn in inurial proc. Da im can rul from maurmn lag, analyi an compuaion im, communicaion lag, or h ranpor im rquir for a flui o flow hrough a pip. For xampl, if a concnraion analyzr i loca ownram of a racor, i a a crain a im bfor h flui laving h racor arriv a h concnraion maurmn poin. Da im i alo a rul of analyi im an communicaion im bwn fil (inrumn an final conrol lmn: valv) an h conrol room. Th ffc of h lag in h proc rpon i hown in Figur.a. Thi figur illura h prformanc of a loop, whn a im =, a p chang wa appli o h poin of a conrollr (an inpu p chang) in h conrol 96 6 by Béla Lipá

2 . Proc Gain, Tim Lag, Racion Curv 97 Tranmir oupu Oupu Inpu Dpning on h lngh an iamr of ubing bwn I/P convrr an conrol valv, pnumaic conrol valv can inrouc conirabl lay. Th u of valv poiionr can conirably ruc hi lay. Th roing im of convnional (pnumaic) valv vary from o min, whil high-p lcric an hyraulic acuaor can ruc hm o h millicon rang. A an xampl, a in. (.54 m) valv wihou poiionr an wih /4 in. (6.5 mm) pnumaic ubing of f (9 m) lngh bwn conrollr an valv ha a im conan of abou Tim FIG..a Th rpon of a proc having boh a im an a im conan. room. Thi inpu p chang rul in h no rpon of h conroll variabl (h oupu rpon), which ar afr a a im ( ) ha pa an rach a nw ay a afr h paing of h im conan (τ ). Th prnc of a im ignificanly complica h analyi an ign of fbac conrol loop. I gra h conrol loop prformanc bcau i inrouc an unabl bhavior, which ma i ifficul o achiv a aifacory conrol. Spcial car houl b an whn h a im o im conan raio ( /τ) xc, bcau h PID conrollr canno hanl uch proc an uch conrol rucur a h Smih pricor coul b rquir. Inrumnaion Effc on Proc Dynamic Th inrumnaion an final conrol lmn ar no par of h proc bu ar par of h conrol loop. Y hy affc h ynamic bhavior of h proc, an h locaion of h nor an/or final conrol lmn alo chang h oal a im. Th proc conrol nginr mu now boh h p of cion an h im i a o ranpor ha informaion. Th im can chang conirably a a funcion of maurmn variabl an yp of inrumnaion u. Th im conan of prur an iffrnial prur maurmn ar on h orr of.. Tmpraur maurmn ar lowr. Thir im conan ar uually bwn an. Compoiion maurmn (analyzr) ar vn lowr, varying from 5 o min. Maurmn ignal propagaion lay ar ngligibl wih lcronic inrumnaion, bu ignal procing lay can occur in igial ym bcau h loop ar proc qunially, on by on. Th can prio of inpu an oupu normally vari bwn. an, an procing lay ini DCS ym ar of h am orr of magniu. Tranporaion Lag Whn marial or nrgy i phyically mov in a proc plan, hr i a a im aocia wih ha movmn. For inanc, if a mpraur chang ravl hrough a pip wih h flui, wihou mixing, h ruling a im i a funcion of h lngh of h pip. If h pip i conir o b long (i lngh i much grar han i iamr), plug flow occur. Th ruling a im bwn wo poin on h piplin can b calcula by whr i h aim V i h volum of h pip F v i h volum flow ra L i h lngh of pip v v i h flui vlociy V L = = F v v.() Thi a im qual h rinc im of h flui in h pip. No ha h a im i invrly proporional o h flow ra. Th fac ha h a im rop a h proucion ra of h plan ri mu b an ino accoun whn igning h conrol ym. Evn whn h plug flow aumpion i no vali, ranporaion lag uually can approximaly b mol by a pur a im. For liqui flow in a pip, h plug flow aumpion i mo accura whn h axial vlociy profil i fla, a coniion ha occur whn Nwonian flui ar ranpor in urbuln flow. For non-nwonian flui an for laminar flow, h flui ranpor lag can ill b mol a a pur a im, which i calcula on h bai of h avrag flui vlociy. Suppo ha a proc ram i flowing hrough a pip in plug flow. Th ram i an incompribl flui. Th pip i prfcly inula an ha a lngh of L = f (6.96 m). Th flui flow a a conan vlociy of v = f/ (6.96 m/) an i iniially a a conan mpraur of T = F (7.778 C) hroughou h lngh of h pip. In hi ca, h ranporaion lag of a v 6 by Béla Lipá

3 98 Conrol Thory Tmpraur ( F) Inpu T() Oupu T ( ) = Paé Approximaion Paé approximaion rprn h rm by h raio of low-orr polynomial in h Laplac omain, M () = N ().(4) whr i h orr approximaion an M () an N () ar polynomial: M( ) = m + m+ m( ) + L+ m( ) N( ) = n + n+ n( ) + L+ n( ).(5).(6) Tim () 5 FIG..b Tranporaion lag ruling from plug flow in a piplin. mpraur chang bwn om poin of inpu an oupu i givn by Thy ar calcula by M N i i i () = ( )!! i!!( i)! ( ) ( ) i= i i () = ( )!! i!!( i)! ( ) i=.(7).(8).() Figur.b illura hi inpu/oupu ranporaion lag. Da Tim Rprnaion Da im rprnaion vari accoring o h mahmaical mol. Tabl.c how iffrn way o rprn h a im. Da Tim Approximaion L f = = = v f/ v Th frquncy (Laplac) omain i frqunly u o uy conrol ym. In hi ca h a im i rprn by. Bcau h mho u o analyz an ign conrol ym canno incorpora xponnial funcion, polynomial funcion ar u o rprn uch xponnial rm. Th mo common rprnaion ar Paé approximaion, ri xpanion, an fir-orr ym ri approximaion. TABLE.c Da Tim Rprnaion Domain Inpu Oupu Sri Expanion Th ri xpanion of h xponnial rm i fin a = + + ( ) ( ) ( ) + L!!!.(9) Howvr, imulaion rquir h polynomial numraor gr o b qual or mallr han ha of h nominaor; for hi raon, ina of Equaion.(9), h following approximaion i u in pracical applicaion: = = ( ) ( ) ( ) L +L!!!.() Fir-Orr Sym Sri (FOSS) An xponnial rm i quivaln o a ym compo of an infini qunc of fir-orr ym ri (FOSS): = Lim +.() Tabl. ummariz h ranfr funcion of h Paé,, an FOSS approximaion. I prn only fir-, con-, an hir-orr approximaion bcau hy ar h mo common in pracical conrol applicaion. + L Tim T() T( ) Laplac T() T() Dicr T(z) m m : ampl im T(z) z Uing h following a im ym: G ()=.() 6 by Béla Lipá

4 . Proc Gain, Tim Lag, Racion Curv 99 TABLE. Tranfr Funcion Approximaion of h Exponnial Trm ( ) of Da Tim Orr Paé FOSS Diffrn approximaion hav bn compar o how how clo h variou approach ar o h xponnial rm. Figur. an.f how Bo iagram for fir- an con-orr approximaion. Paé approximaion i oo clo o h xponnial rm in h magniu iagram, bu h pha lag i inaqualy approxima a highr frqunci. an FOSS prouc a imilar approximaion; hy mach raonably h xponnial rm in magniu an pha a low frqunci, bu hy ar unuccful a high frqunci. Th Paé approach provi br rul for h am approximaion orr han an FOSS. In orr o illura how h Paé,, an FOSS approximaion moify h im rpon, a p i appli o a pur a im ym (Equaion.[]). In hi ca, h ial oupu woul b h am inpu wih a a im ( ). Thi ial oupu ha bn compar wih fir- an conorr a im approximaion, rpcivly, in Figur.g an.h. an FOSS approach prouc br approximaion in h im omain for a uni p rpon han o h Paé approximaion. Th Paé approximaion ha righ half plan zro (RHPZ); alo, h numbr of RHPZ incra wih h orr of approximaion. Th prnc of RHPZ in h ranfr funcion i rponibl for h invr rpon. I i alo h ourc of a conirabl amoun of ifficuly in conrollr ign. Som conrol algorihm hav abiliy problm whn conrolling a proc wih RHPZ. Thi occur 5 Pha (g) Magniu (B) w Paé Paé Pha (g) Magniu (B) Paé Paé w FIG.. Bo iagram of a a im ym (Equaion.[]) uing fir-orr approximaion. FIG..f Bo iagram of a a im ym (Equaion.[]) uing con-orr approximaion. 6 by Béla Lipá

5 Conrol Thory Tranmir oupu (%) Inpu Paé 6 8 FIG..g Sp rpon of a a im ym (Equaion.[]) uing fir-orr approximaion. whn h conrollr conain an invr mol of h proc an high prformanc i ir. In ummary, Paé i a br approximaion han an FOSS, bu Paé approximaion prn RHPZ, which can cau h conrollr o bcom unabl. In uch ca h or FOSS approximaion can rul in a br choic o approxima h xponnial rm. REACTION CURVES Ial oupu Tim/ A compl mahmaical cripion of h proc can b conruc uing ma an nrgy balanc, nginring rlaion, valv quaion, c. I i a ifficul an im-conuming Tranmir oupu (%) Inpu Ial oupu 4 Paé Tim/ FIG..h Sp rpon of a a im ym (Equaion.[]) uing con-orr approximaion. job o vlop a ynamic proc mol. An approximaion of h ynamic proc mol can b obain by uing h racion curv mho. Thi provi a impl an fa procur o rmin an approxima linar proc mol. Thi xprimnal chniqu i ba on applying a chang in h manipula variabl (inpu of h proc) wih h loop opn (conrollr in manual) an rcoring h rpon of h conroll variabl (oupu of h proc). Th paramr of h linar mol ar calcula on h bai of h locaion of om pcific poin in h oupu rpon. Th linar mol rprn an approxima mol ha i aqua for many nginring purpo; howvr, i i only vali in an opraing poin an i o no a ino accoun h high-orr bhavior an nonlinariy of h proc. Mo of h inurial proc can b rprn by a fir-orr plu a im mol. Ohr proc can b approxima by a con-orr unramp plu a im ym or by an ingraing plu a im ym. Th following procur can b u o approxima linar mol for h in of proc. Fir-Orr Plu Da Tim Proc A high prcnag of all chmical proc can b mol by uing fir-orr plu a im ym. K G ()= τ + whr K i h ay-a gain τ i h im conan i h a im.() Th paramr of h proc mol, K,, τ ar obain by uing h following procur:. Th conrol loop i opn by wiching i o manual.. Thi i on whn h conroll variabl (ym oupu) i a a conan valu an no iurbanc or ohr up ar allow o occur, whil h racion curv i vlop.. A p chang i appli o h manipula variabl (conrollr oupu, which i an inpu o h proc). Th p chang ar uually 5 o %. Th p im houl b long nough for h manipula variabl (ym inpu) o rach a nw ay a. 4. Th manipula variabl (ym inpu) rpon i rcor o provi goo viibiliy on boh h ampliu an im cal. Two lighly iffrn graphical chniqu ar uiliz in h procur. Th mho ar xplain in h nx paragraph. 6 by Béla Lipá

6 . Proc Gain, Tim Lag, Racion Curv Tmpraur ( F) Manipula variabl (%) u() FIG..i Racion curv valuaion uing h fir mho o rmin a im an im conan of a fir-orr plu a im proc. y() u() y() Tmpraur ( F) Manipula variabl (%) u() u() y().6 y() y().8 y() FIG..j Racion curv valuaion uing h con mho o rmin a im an im conan of a fir-orr plu a im proc. Fir Mho Th fir mho u a angn o h iniial proc rpon curv, rawn a h poin of maximum lop; h poin whr i inrcp h final ay-a valu i no. In hi ca, h proc im conan i h im bwn h angn inrcping h original an h nw ay-a lin. Th a im i maur a h im i a bwn applying h p chang an h bginning of h im conan. In Figur.i, a 5% p chang o h manipula variabl (conrollr oupu) ha bn appli, an proc rpon in rm of h conroll mpraur i rcor. If h fir mho i uiliz, h mol paramr ar y () 5 F F K = = = u () 5% % τ = 4., =..(4).(5).(6) Scon Mho Th con mho limina h n o raw a angn. Thi approach propo ha h valu of τ an ar o lc ha h mol will coinci wih h proc rpon a wo poin. On h im cal, h wo poin ar a im whn h conroll variabl (ym oupu) rach 8. an 6.% of i final ay-a valu. Onc h poin mach, h paramr of h proc mol ar calcula uing Equaion.(7) o.(9). A hown in Figur.j, a p chang in h manipula variabl i appli o h proc, an h rpon hu, of h conroll variabl i valua. Th mol paramr can b calcula uing h following quaion: y K = () u () τ = ( 6 8) = 6 τ 5 F F K = = 5% % τ = 49 (.. ). = = = 6 τ ( 4.. )..(7).(8).(9).().().() Th fir an con mho ar compar wih h proc rpon in Figur.. Th con mho rul in a br approximaion han h fir on. On limiaion of h fir mho i ha rawing h angn lin a h poin of maximum lop i no an ay a. Th ohr raon why h con mho i uprior i bcau i mach h proc rpon curv a wo poin in h rgion of maximum lop, ina of on. Figur.l how how h con mho can b u o obain a ru approximaion of a nonminimum pha ym by a fir-orr plu a im approximaion. Th principal ia i o approxima h invr rpon by a a im. 6 by Béla Lipá

7 Conrol Thory Tmpraur ( F) Scon mho Proc rpon 9 Fir mho FIG.. Comparion bwn h fir an con mho of approximaing h proc rpon. Unramp Proc Chmical proc rarly ar unramp; howvr, lcrical mchanical ym wih im lag (u o communicaion lag or ignal analyi lag) ypically ar, whn an inpu p chang i appli. A racion curv can b u o obain h paramr for a con-orr unramp wih a im proc in h am way a wa on wih a firorr plu a im ym. Clo-loop chmical proc ar ofn un o hav an unramp rpon. In ha ca, a proc mol i calcula uing h onlin p rpon. 4 A proc ha i unramp wih a im can b approxima by h quaion blow: 5 G ()= τ.() an h paramr of hi mol can b ima from h following xprion: whr K i h ay-a gain i h a im OR i h ovrhoo raio OR i h im o rach h maximum ovrhoo y OR i h maximum pa ovrhoo τ i h im conan ξ i h amping facor K + τξ+ OR = y y.(4).(5).(6).(7) A p ignal, u = %, i appli a = min o h manipula variabl (proc inpu). Th oupu rpon (vlociy) i hown in Figur.m. Th figur alo inifi h raing ha hav o b ma o calcula h con-orr unramp mol paramr. OR π θ = arc an ln OR in( θ) τ = OR π θ ξ = co θ Tranmir oupu (ma) Proc rpon Fir orr plu a im approximaion 5 5 FIG..l Approximaion of a nonminimum pha proc by a fir-orr plu a im mol uing h con mho. yo 5 rpm OR = = = y rpm 5. π θ = arc an = 6. ln 5. ra in( 6. ) τ = 4. =. π 6. ξ = co 6. = 4. y K = u = rpm = rpm % % = Finally, h approxima mol i a follow: G () = (8).(9).().().().().(4) 6 by Béla Lipá

8 . Proc Gain, Tim Lag, Racion Curv Vlociy (rpm) y OR OR y FIG..m Th approximaion of an unramp, con-orr plu a im proc rquir h maurmn of h no four valu from h proc rpon curv. Ingraing Plu Da Tim Proc Somim inurial proc conain pur ingraion lmn. Thy ar no lf-rgulaing proc an o no hav a ay a. Thrfor, any p chang will cau h conroll variabl (proc oupu) o incra or cra linarly wih im. Thi i h ca wih conrolling lvl in a an, whn h manipula variabl i f flow ra ino h an. Th following mol in Laplac omain can approxima an ingraing ym wih a im: whr K i h proc gain i h a im K G ()=.(5) Manipula variabl (%) r u Tranmir oupu (%) 5 5 y FIG..n A pul i u o ablih h ynamic characriic of an ingraing plu a im proc. 6 by Béla Lipá

9 4 Conrol Thory Th proc mol of an ingraing proc i obain by applying a pul an no a p bcau h lar woul prouc an oupu curv ha will chang linarly, wihou raching a ay-a valu. Th gain of h ingraion lmn can b calcula by 5.(6) Whn a pul p i appli, Equaion.(6) can b rwrin a.(7) Th raing ha n o b obain ar hown in Figur.n, conqunly CONCLUSION y K = r u () y K = u r y K = u = % MV = % MV 5. r 5% TO 8 TO = 5..(8).(9) Th ynamic bhavior of mo chmical proc can b approxima by h u of fir-orr plu a im mol. Ohr mho, alo crib in hi cion, inclu h mho of moling con-orr unramp an ingraing plu a im proc. Th racion curv mho wr u o mol h ynamic characriic of h proc. Th ar inpu/oupu (blac-box) linar mol ha o no a ino accoun nonlinarii or high-orr ynamic. Th mol ar uually vali only in a rgion nar h opraion poin of h proc, bu hy ar powrful ool in h ign an uning of conrol loop. Rfrnc. Sborg, D., Thoma, E., an Mllichamp, D., Proc Dynamic an Conrol, Nw Yor: John Wily & Son, Ogunnai, B. A., an Ray, W. H., Proc Dynamic, Moling, an Conrol, Nw Yor: Oxfor Univriy Pr, Smih, C., an Corripio, A., Principl an Pracic of Auomaic Proc Conrol, Nw Yor: John Wily & Son, Yuwana, M., an Sborg, D., A nw mho for on-lin conrollr, AIChE Journal, May Luybn, W., Proc Moling, Simulaion, an Conrol for Chmical Enginr, Nw Yor: McGraw-Hill, 99. Bibliography Ari, R., an Amunon, N. R., Mahmaical Mol in Chmical Enginring, Englwoo Cliff, NJ: Prnic Hall, 97. Bichoff, K. B., an Himmlblau, D. M., Proc Analyi an Simulaion of Drminiic Sym, Nw Yor: John Wily & Son, 968. Frily, J. C., Dynamic Bhavior of Proc, Englwoo Cliff, NJ: Prnic Hall, 97. Goul, L. A., Chmical Proc Conrol: Thory an Applicaion, Raing, MA: Aion-Wly, 969. Kcman, V., Sa-Spac Mol of Lump an Diribu Sym, Hilbrg: Springr-Vrlag, 988. Koppl, L. B., Inroucion o Conrol Thory wih Applicaion o Proc Conrol, Englwoo Cliff, NJ: Prnic Hall, 968. Luybn, W. L., Proc Moling, Simulaion an Conrol for Chmical Enginr, Nw Yor: McGraw-Hill, 97. Marlin, T., Proc Conrol: Digning Proc an Conrol Sym for Dynamic Prformanc, Nw Yor. McGraw-Hill,. Pczowi, J. L., an Sain, M. K., Nonlinar Conrol by Coorina Fbac Synhi wih Ga Turbin Applicaion, Procing of h Amrican Conrol Confrnc, Jun 985. Phillip, C. L., an Harbor, R. D., Fbac Conrol Sym, Englwoo Cliff, NJ: Prnic Hall, 99. Ramar, O., Rijnorp, J. E., an Maarlvl, A., Dynamic an Conrol of Coninuou Diillaion Uni, Amram: Elvir, 975. Rijnorp, J. E., MacGrgor, J. F., Tyru, B. D., an Taamau, T., Dynamic an Conrol of Chmical Racor, Diillaion Column an Bach Proc, Oxfor: Prgamon Pr, 99. Sphanopoulo, G., Chmical Proc Conrol: An Inroucion o Thory an Pracic, Englwoo Cliff, NJ: Prnic Hall, by Béla Lipá