Quasi Steady State Modelling of an Evaporator

Transcription

1 Qasi Steady State Modelling o an Evaporator Ed. Eitelberg NOY Bsiness 58 Baines Road, Drban 400, RSA controle@pixie.dw.ac.za Ed. Boje Electrical, Electronic & Compter Eng. University o Natal, Drban 404, RSA boje@n.ac.za Abstract: Even thogh the thermal capacity o the evaporator tbes and o the water within these tbes are not very dierent, their combined dynamic model is sti. Application o the conventional singlar pertrbation simpliication to the ll model yields an inaccrate reslt. Here, a modiied qasi steady state techniqe is shown to yield a good low reqency approximation. Key-words: qasi steady state; singlar pertrbation; model redction; sti systems; strong copling.. Introdction. Most real dynamical processes are composed o components that proceed at very dierent speeds, or in very dierent reqency ranges. This is a grave problem or nmerical simlation and is known as system stiness the eigenvales o a sti system s Jacobian have very dierent magnitdes. In the context o eedback control, the ast components o the system are an nnecessary complication, i their dynamic behavior alls signiicantly beyond the loop bandwih. In many sitations, it is advantageos to eliminate the ast process components (the large-magnitde eigenvales o the Jacobian) rom a complex model withot x remaining impairing the model s accracy in the lower reqency range. It is not necessarily easy to identiy the ast components in a complex model. One o the diiclties is that the so-called slow and ast physical variables generally contain both slow and ast transients. However i they are identiied, then the corresponding (non-linear) state dierential eqations can be written as ( x x ) x &,, ( x, x, ) ( x, x, ) () where x and x are the slow and ast state vectors respectively and is the system inpt vector. Readers sed to block diagrams may appreciate the eqivalent Figre. x x 'ast' x & x, x, ( ) x Figre : A model with an identiied ast component.

2 The two sbsystems are generally in a (mltivariable) eedback loop. The slow sbsystem inpts are x and and the ast sbsystem inpts are x and. The corresponding slow and ast otpts are x and x respectively. Now it shold not be diiclt to see what slow and ast may mean generally. In particlar, the ast sbsystem otpt x (t) reacts to any changes in its inpts, x (t) and (t), mch aster than the state, x (t) and x (t), o the combined system. Eqivalently, the ast sbsystem otpt x (t) reacts to siciently slow changes in its inpts, x (t) and (t), with negligible dynamic delays x (t) is strongly copled to x (t) (Sandell et al., 978). This relationship o system s state to the inpts is called qasi steady state (QSS) and is deined by ( x (), t x (), t () t ) 0 () The same modelling approach has been characterised as singlar pertrbation (Tihonov, 95; Hoppenstead, 97 and 974; Kokotovic et al., 976; Eitelberg, 983 and 985), whereby a pertrbation parameter ε is introdced as ollows: ε ( x, x, ) ( x, x, ) (3) Nominally ε, bt letting ε go to zero will, nder certain conditions, lead to the qasi steady state condition in eq. (). The singlarly pertrbed redced-order model is then deined by 0 ( x, x, ) ( x, x, ) (4) Eitelberg (003) has shown why this singlar pertrbation o sti models cannot be expected to yield good low reqency approximations and has instead proposed a modiied qasi steady state techniqe as ollows. First, the otpt o the identiied ast sbsystem is deined exactly as x () g ( x () t, () t () t ) t, (5) Sbstittion into the slow sbsystem yields withot approximation ( x g ( x,, x ) ) x &, (6), & Then the qasi steady state derivative o the ast state is evalated, either rom the implicit eqation (), or eqivalently rom the explicit eqation (5): x & () t dg ( x () t, () t, 0 ) g x g + & (7) Sbstittion into eq. (6) yields the loworder approximation x g g x, g x,, x, & + & x () t g ( x () t, () t, 0 ) It may be o some interest that g x g ; x x x. A simple evaporator model. (8) (9) In the evaporator o a power boiler, most o the heat energy is transerred rom the rnace into the steel pipes by radiation. This heat low rate q rad is essentially imposed on the evaporator and independent o the evaporator temperatres. With this assmption, the thermal state o any section o an evaporator can be characterised by the ollowing dierential eqations. The heat energy mct m in the metal mass m (with the heat capacity C and temperatre T m ) o the tbes is increased by the imposed radiated heat low rate q rad and decreased by the heat low rate q(t m,t) into the water or steam with the temperatre T within the pipes: dmctm q rad q m, (0) The heat energy h o the water/steam with density ρ and speciic enthalpy h in the inside volme V o the tbes is increased by the heat that is transported into the evaporator section in the water and by the heat low rate q(t m,t) into the water. It is decreased by the heat that is transported ot o the evaporator section in the steam or water:

3 m& in h in m& oth + q( Tm T ), () The section mass in- and ot-low rates, m& in and m& ot respectively, are related by the mass conservation eqation d m& in m& ot () A dierential eqation or the metal temperatre is obtained rom eq. (0) simply by dividing with the (approximately) constant mc: dt m q rad q m, (3) mc A dierential eqation or the steam/water speciic enthalpy is obtained by developing eq. () as ollows: d h + m& in h in m& oth + q m, (4) Sbstittion o eq. () yields ( h h ) q min in + m &, (5) We have ond that, nder some realistic conditions, the combined thermal model ( h h ) + q( T, T ) m& in in q rad q m mc ( T, T ) m (6) is sti and very time-consming to simlate. The stiness is related to the very good thermal condctivity between the metal and lid. The dierential eqation or the metal temperatre T m will be considered here as the ast sbsystem. Singlar pertrbation. The conventional singlar pertrbation techniqe wold set the metal temperatre derivative to zero in eq. (6), hence ( T m, T ) q rad q (7) And sbstittion into eq. (6) wold yield the simpliied model ( h h ) min in + q rad & (8) However, this cannot be right generally good thermal condctivity between the metal and lid does not mean that the heat capacity o the one or the other can be atomatically ignored. The heat capacity o metal is ignored in eq. (8). Qasi steady state. According to the qasi steady state techniqe rom Eitelberg (003), we irst have to eliminate T m rom the speciic enthalpy dierential eqation. For this prpose, we can solve the metal temperatre eqation or the heat low rate q m, q rad mc (9) Sbstittion o this heat low rate into the speciic enthalpy dierential eqation yields ρ V + mc m& in ( h in h ) + q rad (0) Now we have to evalate the qasi steady state derivative o the metal temperatre (similarly to eq. (7)). This can be ond by dierentiating the implicit eqation (7) (the qasi steady state eq. (9)) with respect to time, dt m + dt m dq rad and then solving or m dq rad dt () () Mostly, bt not always, the heat transer seems to be symmetrical in the sense that m (3) This wold not be (strictly) correct, or example, in the case o radiated heat transer. However, radiation is negligible between the tbe-wall and H O in conventional evaporators. Thereore dq rad + m dt (4) The steam temperatre derivative is related to the steam speciic enthalpy h and pressre p via the thermodynamic state relationships (steam tables): 3

4 dt + (5) h p Sbstitting eq. (5) into eq. (4) yields dq rad + + m h p (6) Sbstitting eq. (6) into eq. (0) yields inally ( h h ) m& in in + q rad + mc h dq mc rad + p m + mc h (7) In systems that we have investigated recently, and mc/ h have similar magnitdes when water is below satration temperatre hence, neither heat capacity can be ignored. In twophase low, however, T is a nction o pressre alone and ( h h ) m& in in + q rad dq mc rad m + p (8) The singlarly pertrbed model in eq. (8) describes the thermal dynamics identically only nder the condition o two-phase low at constant pressre p and with constant heat ptake q rad. 3. Simlation. Figre compares the dierences o the ll order, singlarly pertrbed and qasisteady state models developed above applied to the model o a large once throgh (Benson ) boiler dring start-p. The model divides the economiser and evaporator o the boiler into ive and ten spatial sections respectively. As shown in Figre a, the start-p has irst one and then two mills broght into service to move the load demand rom 0% to 30% and then to 50% at a rate o arond 0 MW/min. In order to excite a aster transient behavior, a (temporary) mill trip is simlated, starting at 45 mintes. All simlations have constant eedwater pressre (6 MPa), economiser inlet masslow (80 kg/s) and eedwater temperatre (80 C). Figre b shows the water otlow rate o the boiler (it is not yet in Benson mode). Figre c shows the speciic enthalpy o the water or two-phase lid at the evaporator otlet. As can be seen in Figres b and c, the ll order and qasi-steady state model give almost exactly the same response. In contrast, direct application o the singlar pertrbation method reslts in a model that does not inclde the eect o energy absorbed by the boiler tbes dring transients and is thereore too ast. Figre 3 shows the eect o a step change in the eedwater temperatre rom 80 C to 0 C on the water otlet masslow rate. For this simlation, the ollowing are held constant: eedwater pressre (6 MPa); economiser inlet masslow (80 kg/s); and 50% boiler demand. The simlation highlights that the qasisteady state method is somewhat limited in that the transient heat exchange between the hotter lid and boiler tbes is not captred correctly. The singlarly pertrbed model is however mch worse. 4. Conclsion. This paper has shown that while direct application o the singlar pertrbation method yields a model with incorrect transient behavior, the qasi steady state techniqe achieves model order redction with good low reqency approximation. This techniqe in the particlar application o a power plant boiler also makes technical sense as it correctly captres the thermal capacity o both the evaporator tbes and o the water within them. 4

5 Boiler demand [MW] water otlow [kg/s] speciic enthalpy [kj/kg] ll model 400 singlar pertrbation qasi steady state MILL MILLS "MILL TRIP" 500 ll model singlar pertrbation qasi steady state Figre : Simlation o start-p o once-throgh boiler sing qasi-steady state model order redction ll model singlar pertrbaiton qasi steady state water otlow [kg/s] Figre 3: Simlation o once-throgh boiler with rate-limited step distrbance in eedwater (inlet) temperatre (enthalpy) rom 80 C to 0 C 5

6 Reerences. Eitelberg, Ed. (983): Pertrbationstechniken bei der Optimierng grosser linearer Regelngssysteme. Fortschr.-Ber. VDI-Z. VDI-Verlag, Desseldor.. Eitelberg, Ed. (985): A transormation o non-linear dynamical systems with a single singlar singlarly pertrbed dierential eqation. Int. J. Control, Vol. 4, No. 5, pp Eitelberg, Ed. (003): Qasi steady state modelling. Sbmitted. 4. Hoppenstead, F. (97): Properties o soltions o ordinary dierential eqations with small parameters. Comm. Pre Appl. Math., Vol XXIV, pp Hoppenstead, F. (974): Asymptotic stability in singlar pertrbation problems. II: Problems having matched asymptotic expansion soltions. J. Dierential Eqations. Vol 5, pp Kokotovic, P.V.; O Malley, R.E.; Sannti, P. (976): Singlar pertrbations and order redction in control theory an overview. Atomatica. Vol, pp Sandell, N.R.; Varaiya, P.; Athans, M.; Saonov, M.G. (978): Srvey o decentralized control methods or large scale systems. IEEE AC-3, No., pp Tihonov, A.N. (95): Systems o dierential eqations with small parameters at the derivatives. Mat. Sb. Vol. 3 (73), no. 3, pp (in Rssian). 6