PT326 PROCESS TRAINER

Transcription

1 PT326 PROCESS TRAINER

2 1. Descrptn f the Apparatus PT 326 Prcess Traner The PT 326 Prcess Traner mdels cmmn ndustral stuatns n whch temperature cntrl s requred n the presence f transprt delays and transfer lags. The prcess cntaned n the PT 326 nvlves ar that s drawn frm the atmsphere by a centrfugal blwer, and s heated as t passes ver a heater grd befre beng released nt the atmsphere thrugh a duct. The cntrl bjectve s t mantan the temperature f the ar at a desred level. Temperature cntrl s acheved by varyng the electrcal pwer suppled t the heater grd. The ar temperature may be sensed by usng a bead thermstr placed n the flw at any f three pstns alng the duct. The spatal separatn between the thermstr and the heater cl ntrduces a transprt delay nt the system. Fgure 1 shws the frnt panel f the PT 326 apparatus. The mass flw f ar thrugh the duct can be adjusted by settng the penng f the thrttle. The temperature sensr can be placed at any f the three lcatns marked P1, P2 and P3. The leads f the temperature sensr are cnnected t the termnals marked detectr. The varatn n the resstance f the temperature sensr s cnverted t a vltage varatn by a brdge crcut, and s avalable at termnal Y. The utput f the temperature sensr can be used fr feedback by cnnectng termnals X and Y. The desred temperature may be ndcated by adjustng the knb labeled set value. An addtnal step change n the set value can be cmmanded by thrwng the swtch marked nternal. An external reference nput can be prvded at termnal D labeled external. The dfference between the reference nput + set value and the feedback sgnal s avalable at termnal B, and can be appled t the heater pwer supply thrugh a gan that can be adjusted by usng the knb marked prprtnal band. A prprtnal band f PB % crrespnds t a gan factr f 100/PB. The sgnal at B s avalable at termnal A after amplfcatn by the gan factr. The sgnal at A may be appled drectly t the heater pwer supply by usng the swtch S t select cntnuus cntrl. Alternatvely, the sgnal may be used n an n-ff cntrller by usng the swtch S t select tw-step cntrl. The n-ff cntrller cmmands maxmum heater pwer when n, and zer heater pwer when ff. The hysteress n the n-ff cntrller can be adjusted by usng the knb marked verlap whle the maxmum heater pwer t be suppled can be adjusted by usng the knb marked max heater pwer. The sgnal drvng the heater pwer supply can be mntred at termnal C. 2. System Mdelng. The physcal prncple whch gverns the behavr f the thermal prcess n the PT326 apparatus s the balance f heat energy. The rate at whch heat accumulates n a fxed vlume V enclsng the heater s, qa = q + q q qt (1)

3 Thrttle Cntnuus Cntrl Detectr P1 P1 P2 P3 Bead Thermstr X Y S A B D C Tw-Step Cntrl Set Value Max Heater Pwer Overlap Prprtnal Band External Internal Fgure 1: Frnt panel f the PT326 apparatus

4 where q s the rate at whch heat s suppled by the heater, q s the rate at whch heat s carred nt the vlume V by the cmng ar, q s the rate at whch heat s carred ut f the vlume V by the utgng ar and q t s the heat lst frm the vlume V t the surrundngs by radatn and cnductn. Fgure 2 belw depcts the vlume V. q q V q q t Fgure 2: Heat transfer frm the vlume V The accumulatn f heat n the vlume V causes the temperature T f ar n V t rse. Assumng a unfrm temperature n the vlume V, the rate f heat accumulatn s gven as, q a dt = C, (2) dt where C s the heat capacty f the ar ccupyng the vlume V. The rates at whch the ar flw carres heat n and ut f the vlume V are gven respectvely by the mass flw rate and q. = C mt and a a q. = C mt, where C s the specfc heat capacty f ar,. m s T s the ambent ar temperature. Assumng that the rse n ar temperature T = T Ta s small, the rate q t at whch heat s lst frm the vlume V s prprtnal t the temperature rse T. Thus, where yeld, 1 q q qt = T, (3) R 1 s a prprtnalty cnstant called the thermal resstance. Equatns (1), (2) and (3) R Takng Laplace transfrm yelds d T 1 C + T = q. (4) dt R

5 T( s) k1 = q ( s) τ s + 1, (5) 1 where k1 = and τ = RC s the tme cnstant. On assumng the heater heat supply rate t be R prprtnal t the heater nput vltage V, (5) yelds the transfer functn between the heater nput vltage and the temperature rse as, T( s) k1k2 = V ( s) τ s + 1, (6) where k 2 s the prprtnalty cnstant between q and V. In equatn (6), T represents the ncrease n temperature f the ar cntaned n the vlume V cnsdered abve. The temperature sensr prduces a vltage V that s prprtnal t T, that s V = k3 T. Hwever, snce the sensr s physcally lcated at a certan dstance frm the heat surce, the sensr utput respnds t a temperature change wth a pure tme delayτ d, whch s the tme taken by the flwng heated ar t cver the dstance between the heater and the sensr. Thus the transfer functn between the heater nput vltage and the sensr utput vltage s, τ ds V ( s) ke = V ( s) τ s + 1, (7) d where k = k1k2k3 s the DC gan f the system. The term s n (6) arsng due t flud transprt 1 s called a transprt delay, whle the term ( τ s + 1) arsng due t the heat transfer dynamcs s called transfer delay. Fgure 3 belw llustrates equatns (5)-(7). e τ Temperature at Sensr V Heater k 2 Heat Transfer k1 τ s + 1 Ar Flw Sensr T V e τ d s k 3 τ ds ke τ s + 1 Fgure 3: Blck dagram f the PT326 prcess

6 Questn: Hw are the parameters k, τ d and τ affected by the sensr lcatn and thrttle penng? 3. System Step Respnse The utput f the temperature sensr V and the heater nput vltage are related by the frst-rder transfer functn (7) fr small temperature changes frm the ambent. The transfer functn n (7) s characterzed by tw parameters, namely, the DC gan K and the tme cnstant τ. Bth f these parameters can be determned frm the respnse f the temperature t a step ncrease n the heater nput vltage frm a state f thermal equlbrum. T fnd an expressn fr the unt step respnse f the temperature varatn, recall that the Laplace transfrm f a unt step nput s 1. Hence the Laplace transfrm f the respnse f s the temperature varatn V t an ncrease f 1V n the heater nput vltage s τ ( ) ds Ke V s = s( τ s + 1), (8) Takng nverse Laplace transfrm yelds V t = K e, (9) ( t τ d )/ τ ( ) (1 ) We ntce that V, the change n temperature frm the ntal equlbrum value, cnverges t the value K. Thus the DC gan K s smply the rat f the steady state change n the utput t the steady state value f the nput. We als bserve that, at t = τ + τ, the value f 1 the utput s V ( τ ) K (1 e = ) = K. Thus the tme cnstant f the system s the tme that t takes fr the utput t change by % f the steady state change. Bth the parameters K and τ can thus be determned expermentally frm the step respnse. d Experment 1: In ths experment, yu wll gve step nputs t the pen-lp system t determne the nature f the transfer functn between the set value and the utput f the temperature sensr. Set the prprtnal band t 100% and the set value t 2. Use the apprprate swtches t select ``cntnuus cntrl'' and ``prprtnal band''. Perfrm ths experment by placng the sensr at each f the three lcatns P1, P2 and P3. Fr each sensr lcatn, repeat the experment at the three values 40, 65 and 90 f thrttle settng. Fr every cmbnatn f the sensr pstn and thrttle settng, gve a set-value dsturbance by thrwng the swtch marked ``nternal''. Ths gves a step nput vltage f a certan fxed magntude, whch can be seen n

7 the scllscpe at the termnal marked ``trgger CRO''. The utput f the temperature sensr can be seen at the termnal marked ``Y''. By bservng bth sgnals n the scllscpe, determne the DC gan, transprt lag, and tme cnstant f the pen-lp transfer functn between the set value and the sensr utput. 4. On-Off Cntrl. A typcal cntrl bjectve n thermal systems s t mantan the temperature f sme cmpnent at a user specfed value called the set pnt. Fgure 4 belw depcts a clsed-lp system desgned t mantan the utput temperature f the PT326 apparatus at a desred set pnt. (Set Pnt) + (Desred Temperature) - Errr Cntrller Heater Input PT326 Thermal Prcess Temperature Rse T Actual Temperature Fgure 4: Clsed-lp cntrl f a thermal system Lsely speakng, the cntrller uses the errr sgnal between the desred temperature and actual temperature t manpulate the heater nput vltage n such a way that the actual temperature cnverges t the set pnt value. A cst effectve means f mplementng a cntrller n thermal systems s a relay, whch gves full heater nput vltage when the errr sgnal s pstve and gves zer heater nput vltage when the errr sgnal s negatve. Such a cntrller s called an n-ff cntrller. Fgure 5 belw shws the nput-utput characterstc f an n-ff cntrller. Practcal relays suffer frm hysteress, where, fr decreasng nput, the relay utput swtches ff at a lwer value f the nput than the value at whch the relay utput swtches t ts maxmum when the nput s ncreasng. Fgure 6 belw shws the nput-utput characterstcs f a relay wth hysteress.

8 Fgure 5: Input-utput characterstcs f an n-ff cntrller Fgure 6: Input-utput characterstcs f a relay wth hysteress Fgure 7 belw shws a typcal clsed-lp respnse f a frst-rder system under n-ff cntrl wth hysteress.

9 Cntrller Output System Output Maxmum Temperature Upper Threshld Set Pnt Lwer Threshld Mnmum Temperature τ d Tme Fgure 7: Clsed-lp respnse under n-ff cntrl Experment 2: In ths experment, yu wll nvestgate hw the pen-lp behavr nfluences the behavr f the clsed-lp system under n-ff cntrl. Place the sensr at the mddle lcatn. Set the prprtnal band t 100%, the set value t 2, thrttle settng t 40, and verlap t 0. Select tw-step cntrl and swtch ff the nternal set-value dsturbance. Clse the lp by cnnectng the termnals X and Y. Use the scllscpe t bserve the heater nput vltage at termnal C, and the temperature sensr utput at termnal Y. Frm the scllscpe, measure the threshld value f sensr utput belw whch the heater turns n, threshld value f sensr utput abve whch the heater turns ff, maxmum and mnmum values f the sensr utput vltage, maxmum and mnmum values f the heater nput vltage, cycle tme, heater-n tme and heater-ff tme n each cycle. Repeat the abve steps by settng the verlap t 0.5 and 1.

10 5. Prprtnal Cntrl. On-ff cntrllers, thugh nexpensve t mplement, suffer frm the dsadvantage that they lead t cyclng r huntng arund the desred set pnt, and thus prvde less exact cntrl. An alternatve t an n-ff cntrller s a prprtnal cntrller, whse utput s prprtnal t the errr between the set pnt and the actual measured utput. Thus, when a prprtnal cntrller s used n the clsed-lp system n Fgure 5, the heater nput vltage s related t the errr e( s) = r( s) V ( s) by where K p s the prprtnal gan f the cntrller. V ( s) = Kp( r( s) V ( s)), (10) Questn: Use (10) alng wth (7) t shw that n the absence f transprt delay, the clsed-lp system s als a frst-rder system wth DC gan and tme cnstant K CL KpG =, (11) 1 + KpG τ τ CL =. (12) 1 + KpK Equatn (12) mples that the clsed-lp respnse becmes faster as the prprtnal gan ncreases. Equatn (11) mples that the DC gan appraches 1 as the prprtnal gan Kp ncreases. In ther wrds, the steady-state value f the temperature appraches the set pnt as K p ncreases. Thus, the clsed-lp system respnds faster and mre accurately t cnstant set pnt cmmands as K p ncreases. Hwever, fr n value f the gan K p s the respnse cmpletely accurate. Ths can be physcally explaned by bservng that, f the errr s zer, then the heater vltage cmmanded by the cntrller s zer, causng the ar t start clng mmedately. In ther wrds a prprtnal cntrller cannt mantan a state f zer errr (unless the set pnt equals the rm temperature!). 6: Frequency Respnse The clsed lp system shwn n Fgure 4 cntans a pure delay. Such a system typcally exhbts nstablty fr suffcently large values f the prprtnal gan. Ths can be understd by the fllwng argument.

11 A phase lag f 180 degrees s added t the sensr utput vltage as t s fed back t the heater nput vltage wth a negatve sgn. The transprt lag and the transfer lag add a further phase lag t the heater nput vltage. Because the phase lag due t the transprt delay ncreases ndefntely wth frequency, there exsts a frequency at whch the phase lag added by the transfer and transprt lags s 180 degrees. Ths frequency s called the phase crssver frequency. Thus, at the phase crssver frequency, the ttal phase lag accumulated as a sgnal traverses the clsed lp n Fgure 4 s 360 degrees. If the prprtnal gan s such that the prduct f the gans arund the lp s als unty, then a snusd at the phase crssver frequency can pass arund the lp ntact. In ther wrds, the clsed-lp system can exhbt sustaned scllatns at the phase crssver frequency, ndcatng the nset f clsed-lp nstablty. It s clear frm the argument abve that the clsed-lp system becmes unstable when the prprtnal gan equals the recprcal f the gan f the pen-lp transfer functn (7) at the phase crssver frequency. Ths recprcal s the gan margn f the system. In decbels, the gan margn s the negatve f the gan f the transfer functn (7) n decbels at the phase crssver frequency. The gan margn s the smallest value f the prprtnal gan abve whch the clsed-lp system s unstable. The gan margn f the thermal prcess n PT326 can be determned by expermentally generatng the frequency respnse f the prcess. Ths s dne by applyng snusdal sgnals f dfferent cnstant frequences at the heater nput vltage and bservng the sensr utput vltage. At each nput frequency, the sensr utput wll be snusdal at the same nput frequency. Hwever, the utput wll be attenuated (r amplfed) and suffer a phase lag when cmpared t the nput. The rats f the ampltudes f the utput t the nput s the gan f the system at the nput frequency, whle the phase lag between the utput and the nput s called the phase f the system at the nput frequency. The gan and the phase f the system are measured expermentally at dfferent nput frequences and pltted as a functn f the frequency. The phase crssver frequency and s determned frm the phase plt. The gan margn s then the recprcal f the gan f the system at the phase crssver frequency. Experment 3: In ths experment, yu wll expermentally btan the pen-lp frequency respnse data fr a thermal system and use t t predct the nset f clsed-lp nstablty. Part I: Open-lp frequency respnse In ths part, yu wll btan frequency respnse plts fr the pen-lp system. Adjust Set Value t 35 degrees and blwer nlet (thrttle) t 40 degrees. Place the sensr n pstn P3. Select Cntnuus Cntrl and External Cntrl. Use a functn generatr t gve a 2V peak-t-peak snusdal nput at termnal A. Use an scllscpe t cmpare the nput sgnal and the sensr utput (at Y). Vary the nput frequency frm 0.1 Hz t 3Hz, and measure the amplfcatn and phase shft between the utput and nput at each frequency. Use yur readngs t draw Bde plts f the system. Fnd the phase crssver frequency and the crrespndng magntude frm yur Bde plts. Calculate the gan margn f the system.

12 Part II: Clsed-lp stablty Use the same settngs fr Set Value and blwer nlet as abve. Make cnnectns such that the lp s clsed wth the prprtnal cntrller n the lp. By bservng the utput (n the absence f any external nput), fnd the value f gan at whch the clsed-lp system becmes unstable. Cmpare ths value wth the crtcal value f gan calculated frm the gan at phase crssver frequency. Questn: Fnd the value f gan at whch the clsed-lp system becmes unstable when the sensr s placed n pstn 2. Are the values fr pstns 2 and 3 same? Why? Thery predcts that a negatve feedback clsed-lp cnsstng f a stable frst-rder system and a prprtnal gan s always stable. In ths experment, hwever, yu bserved that, fr a suffcently large gan, the clsed lp s unstable even thugh the basc heat transfer prcess n the lp has frst rder dynamcs. The nstablty f the clsed lp s caused by the presence f the pure delay n the frm f transprt lag. Ths experment thus hghlghts the danger that excessve delays and phase lags n the lp pse fr stablty. Delays and phase lags can als be cmpunded by hgher-rder dynamcs, samplng, r cmputatns.