NELINEARNI MATEMATIČKI MODELI POMOĆNIH UREĐAJA KOTLA. Slobodan Bogdanović, Elektrotehnički institut "Nikola Tesla" u Beogradu
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1 Pretdn apštenje UDK: : =861 BIBLID: (2000,13,p NELINEARNI MAEMAIČKI MODELI POMOĆNIH UREĐAJA KOLA Slbdan Bgdanvć, Elektrtenčk nttut "Nkla ela" u Begradu Sadržaj: U radu u frmran nelnearn matematčk mdel pmćn uređaja ktla u termelektran. Defnan u vektr ulazn zlazn velčna mdela. Pvezvanjem mdela pjednačn uređaja prek njv vektra ulazn zlazn velčna frmra e mdel kupa pmćn uređaja ktla. Ključne reč: matematčk mdel / parn kta / pmćn uređaj ktla 1 UVOD U lteratur je bjavljen vše matematčk mdela namenjen prkazvanju lžen prcea kj e dvjaju u termblkvma [1,2,3,4]. Publkvan u nelnearn mdel vrl vkg reda kj detaljn puju prcee u parnm ktlu parnj turbn. Pr matematčkm mdelvanju pjedn vrl značajn delva parng ktla (parvač, pregrejač prce u njma e tretraju, trg egzaktn, ka tem a rapdeljenm parametrma, pa e dlaz d tema nelnearn parcjaln dferencjaln nelnearn algebark jednačna. Međutm, mng u brjnj mdel prcea u parnm ktlu turbn kj ka nvnu pretptavku uvde da e rad temma a kncentranm parametrma. Eventualna nedvljna tačnt takv mdela, za defnanu namenu, ppravlja e frmranjem mdela manj kntruktvn celna parng ktla u kjma e dvjaju pmatran prce. Stem nelnearn dferencjaln algebark jednačna takv mdela atje e d vše deetna dferencjaln jednačna bar dvtruk vše algebark jednačna. Al, čak u vrl detaljnm glmaznm matematčkm mdelma [1,2,3,4] prkazuju e prce d uvđenja napjne vde u kta, grva vazdua u lžšte pa d zlaza z tepena turbne nkg prtka. Ogrmna većna mdela upn z lterature ne buvata prcee kj e dvjaju u celm parnm ckluu termblka. Za praktčnu prmenu mdela prcea u termblku ptrebn je buvatt prcee u delu zatvreng parng cklua d zlaza z tepena turbne nkg prtka pa d ulaza u predgrejač parng ktla. Energetka prema u kjj e dvjaju prce z tg dela parng cklua u termelektranama nazva e pmćn uređaj ktla. U vm radu prkazuju e nelnearn matematčk mdel prcea kj e dvjaju u pmćnm uređajma ktla. Pr tme u buvaćen kndenzatr, parn zagrejač vde a parm nkg prtka, rezervar napjne vde a degazatrm, parn zagrejač vde a parm vkg l rednjeg prtka, pumpa za napjnu vdu pgnk anrn mtr. Namena v mdela je da e pvežu a detaljnm nelnearnm mdelm parng ktla turbne za analzu dnamčkg pnašanja termblka. 2 O PRISUPU PRI FORMIRANJU MAEMAIČKOG MODELA U radu e razmatraju nelnearn matematčk mdel pmćn uređaja ktla čja je namena da budu de mdela kmpletng parng cklua kj e dvja u termblku. Onvn zatev kj e ptavljaju pr frmranju takv mdela pdudaraju e, dakle, 21
2 a zatevma kj e ptavljaju pr frmranju dgvarajuć matematčk mdela parng ktla turbne. u [4,5]: dvljna tačnt za željenu prmenu, št jedntavnj mdel za željenu prmenu dn zmeđu prmenljv parametara mdela fzčk velčna karaktertka realng tema treba da u št neprednj, jedntavnj janj. Pr ptavljanju mdela pmćn uređaja ktla uvde e dređene pretptavke a cljem dbjanja jedntavnj preglednj mdela. Među mngbrjnm čet kršćenm pretptavkama, nvne u: a v element mdelvang fzčkg tema mgu e mdelvat ka tem a kncentranm parametrma, b kada e mdeluje kmpnenta atavljena d vše paraleln ptavljen elemenata tg tpa (npr. cev, pumpe, td. nda e paraleln ptavljen element zamenjuju a jednm ekvvalentnm elementm, c bl kja karaktertka fluda (vda, para, grejn ga je unfrmna na učenm pprečnm preeku št mgućava kršćenje jedndmenznaln umet prtrn mdela, d pr ptavljanju mdela a kncentranm parametrma reprezentatvn pprečn preek za razmatran pmćn uređaj je zlazn pprečn preek (pmatran u meru trujanja radng fluda, a ne centraln l ulazn preek (vakav zbr ulvljen je prblemma kj e javljaju pr numerčkm rešavanju mdela mgućntma upršćavanja mdela, e nercja gavt prdukata agrevanja je mng manja neg nercja radng fluda, dnn dnamčk prce u prtru grejng gaa dvjaju e mng brže neg dgvarajuć prce u prtru radng fluda, f ne mže e zanemart prmena gutne radng fluda uled prmene prtka u dnu na prmenu gutne uled prmene temperature radng fluda. Nelnearn matematčk mdel pmćn ktlvk uređaja frmraju e kršćenjem analtčk zraza za pvanje zakna držanju mae, energje klčne kretanja, analtčk zraza za zakne prenšenju tplte za prkazvanje knttutvn dna zmeđu termdnamčk velčna malzvan pljn karaktertka pump anrn mtra. Onvne karaktertke dbjen mdela u: Mdel u dbjen uz nvnu pretptavku da e mdelvan tem mže razmatrat ka tem a kncentranm parametrma. Prkladnm prmenm tg prtupa dlaz e d matematčkg mdela kd kga ptj nepredna jana veza zmeđu prmenljv parametara mdela fzčk velčna karaktertka realng tema. Ova bna mdela značajn lakšava računavanje parametara mdela. Mdel uključuje nelnearnt vred za šrke pege prmene pterećenja termblka za razlčte prmene ulazn velčna. Knttutvn dn zmeđu termdnamčk velčna (dređen tablcama pare uvršten u u mdel prek aprkmacn plnma. Pmću nj e p želj tačn računavaju termdnamčke velčne u pecfcranm pezma prtka temperature. Frmran mdel uvažavaju mgućnt velk prmena napna učetant u temu št značajn utče na rad pmćn uređaja ktla. Parametr mdela računavaju e na nvu kntruktvn pdataka, pdataka načnu prenšenja tplte nva pterećenja termblka. 22
3 3 NELINEARNI MAEMAIČKI MODELI POMOĆNIH UREĐAJA KOLA Funkcnaln blk djagram pmćn ktlvk uređaja u E "Nkla ela" - A prkazan je na l.1. On e dn na termblk a prtčnm ptkrtčnm parnm ktlm. Izrađena para z parne turbne e uvd u kndenzatr pare. Iz kndenzatra e vda pumpama dvd u parne zagrejače vde a parm nkg prtka pa u rezervar za napjnu vdu. U njemu e vda meša zagreva a parm lađenm na parnm zagrejačma vde rednjeg vkg prtka pa e zatm dvd krz zagrejače vde a parm rednjeg nkg prtka u predgrejač ktla. Rapred međubna pvezant pmćn ktlvk uređaja u traktu napjne vde je praktčn jednaka za termblkve već naga. Male razlke ptje u brju uređaja (npr. zagrejača vde u metma a kj e duzma para za zagrevanja radng fluda. Sl.1 Blk djagram pmćn uređaja ktla 3.1. Matematčk mdel kndenzatra pare Kada e uvaže čnjence da e prtak radng fluda u kndenzatru mal menja, dnn da je prblžn talan, da je radn flud u kndenzatru u zaćenm tanju, nda je = Cc (3.1.1 c c c = f ( p (
4 gde je a f c (p c značena funkcja (plnm kjm e prkazuje zavnt entalpje zaćene vde na zlazu kndenzatra d prtka zaćene vde u blat prmene prtka d nterea. Parametar mdela kndenzatra je C c. Matematčk mdel kndenzatra pare u vektrkm blku je 0 = f (x (3.1.3 c c c xc = (p, c (3.1.4 yc = xc ( Matematčk mdel parng zagrejača vde a parm nkg prtka Analzran parn zagrejač vde a parm nkg prtka je zagrejač zatvreng tpa. Para nkg prtka za zagrevanje vde duzma e z tepena turbne nkg prtka. Prmenm zraza za zakne državanju mae, energje mpula na velčne na ulazu zlazu parng zagrejača dbja e: d d( 1 = V u (W 1 = V -W (W -W +Q pz (3.2.1 ( (W p - p = k (3.2.3 Jednačne tanja vde na ulazu zlazu parng zagrejača vde u = f ( p, (3.2.4 = f (, (3.2.5 p p = f (, (3.2.6 prkazane plnmma kjma e aprkmraju pdac z tablca pare u dređenm pegu. Matematčk prkaz zakna držanju mae, energje mpula za zaćenu paru nkg prtka kjm e zagreva vda u zagrejaču vde u W =W (3.2.7 Q pz =W ( - (
5 p = p (3.2.9 a jednačna tanja zaćene pare nkg prtka je = f ( p ( Parametar mdela parng zagrejača vde je V. Nelnearn matematčk mdel parng zagrejača vde a parm nkg prtka mže e ple prevđenja u prtr tanja prkazat u vektrkm blku dx 1pz = f 1pz (x 1pz, x 2pz, u pz ( = f (x, x, u ( pz 1pz 2pz pz ypz = Cpz xpz ( x1pz = (, u ( x 2pz = (W,p,,Q,, W,p, ( pz xpz = (x, x ( pz 2pz u pz = (W,, p, W, p, ( C pz = dag(0,0,1,1,1,0,0,1,1,1 ( Matematčk mdel parng zagrejača vde a parm duzetm z tepena turbne rednjeg l vkg prtka maju t blk ( razlkuju e am prema vredntma parametara mdela ulazn zlazn velčna mdela Matematčk mdel pumpe za vdu z kndenzatra Spljne karaktertke pump bčn u zadate pmću grafčk djagrama kj pkazuju zavnt vne pumpanja ΔH tepena krng dejtva η d prtka Q za razlčte brzne ω. Ova dva kupa djagrama mgu e prkazat u blku dve krve kje dređuju prmene malzvane vne ΔH/ω 2 malzvang tepena krng dejtva η/ω 2 pr prmenama malzvang prtka Q/ω. Oblk malzvan pljn karaktertka pump prkazan je na l.2. Nrmalzvane karaktertke pumpe e za ptrebe frmranja matematčkg mdela aprkmatvn prkazuju plnmma. Matematčk mdel pumpe za vdu z kndenzatra ma ledeć blk d 1 = J ( M - M (3.3.1 M = gq H /( (
6 Q Q 2 2 = [ K 1 + K 2 + K 3( ] (3.3.3 Sl. 2. Nrmalzvane pljne karaktertke pumpe = f ( p (3.3.4 Q =W / (3.3.5 Q Q 2 2 H = [ K 4 + K 5 + K 6 ( ] (3.3.6 p - p = g H (3.3.7 W = N W (3.3.8 = f ( p, (3.3.9 Prve dve jednačne prkazuju zakn državanju mpula rtacng kretanja zlazn mmenat pumpe. Kntante K 1,...,K 6 u ( e dređuju pr aprkmacj malzvan pljn karaktertka pumpe pr čemu e ka krterjum krt mnmzacja kvadratng upanja. Jednačna ( uvažava čnjencu da paraleln rade N pump. Parametr mdela u J, K 1,...,K 6, g N. Pr tme a J u uzet mment nercje pumpe njeng pgnkg mtra. Matematčk mdel pumpe e u vektrkm blku prkazuje pmću ledeć jednačna dx 1 = f 1 ( x, u ( = f ( x, u ( y = C x (
7 x 1 = ( x 2 = (M 1,,Q, H,, W, W, ( x = (x, x ( u = (p, p, M ( C = dag[1,0,1,0,0,0,0,1,1] ( Jednačne matematčkg mdela pumpe za napjnu vdu z rezervara napjne vde maju ptpun jednak blk uz pretptavku da je vda na zlazu z napjng rezervara, dnn na ulazu u pumpu za napjnu vdu, u zaćenm tanju Matematčk mdel anrng mtra za pumpu za vdu z kndenzatra (zajedn a regulacnm draulčkm pjncm Zavnt mmenta M d klzanja anrng mtra analtčk e prkazuje M C = M max M M U + 2 max M M (3.4.1 M - M M = (3.4.2 M M el = (3.4.3 C n = (3.4.4 M K Parametr mdela u velčne klzanja M max, makmaln mment mtra, kntanta C M mtra, napn U na krajevma mtra, elektrčna brzna mtra ω el kja je dređena učetanšću mreže brj par plva C n mtra. Matematčk mdel ( mže e prkazat u vektrkm blku 0 = f ( x, u (3.4.5 x M M = (M M M, M M M,, (3.4.6 M ym = CM xm (3.4.7 um = (, K (3.4.8 C M = dag[1,0,0,0] (
8 Matematčk mdel mtra kj pgn pumpu za vdu z rezervara napjne vde zajedn a njegvm regulacnm draulčkm pjncm je dentčan p blku a mdelm ( Matematčk mdel rezervara za napjnu vdu a degazatrm Rezervar za napjnu vdu je zagrejač tvreng tpa. U njega e dvd vda zagrevana u zagrejačma vde a parm nkg prtka. Vda e u njemu dalje zagreva mešanjem a zaćenm parm natalm d pare rednjeg vkg prtka ple njeng lađenja na zagrejačma vde a parm rednjeg vkg prtka. Matematčke frmulacje zakna držanju mae energje za prtr rezervara za napjnu vdu u: d( V + V =W +W -W d( u V + u V = W + W - W (3.5.1 (3.5.2 Zapremna pare u napjnm rezervaru V zapremna vde V u vremenk prmenljve velčne al je njv zbr kntantan V +V =V (3.5.3 Zapremna vna vde H u rezervaru u pštem lučaju pvezan u aprkmacnm plnmm drugg tepena 2 H = K 1 + K 2 V + K 3 V (3.5.4 Pad prtka napjne vde u rezervaru za napjnu vdu je jednak nul p = p (3.5.5 Vda para u rezervaru za napjnu vdu e nalaze u zaćenm tanju pa e ve termdnamčke velčne prkazuju pmću am jedne nezavne termdnamčke velčne = f ( p (3.5.6 = f ( p (3.5.7 u u = f ( p (3.5.8 n u = f ( p (3.5.9 = f ( p ( Parametr matematčkg mdela u zapremna rezervara V kntante aprkmacng plnma K 1, K 2, K 3. 28
9 Matematčk mdel rezervara za napjnu vdu ktla u prtru tanja prkazan je u vektrkm blku ledećm jednačnama: 29
10 29 Sl. 3. Pvezan matematčk mdel kupa pmćn uređaja ktla
11 dx 1 = f 1 (x 1, x 2, u ( = f (x, x, u ( x = (x, x ( y = C x ( x1 = (p, V ( x 2 = (W,,, u, u,, V, H ( u = (W, W,,, p ( C = dag[1,0,1,0,0,0,0,1,0,0] ( Matematčk mdel kupa pmćn uređaja ktla U radu u frmran matematčk mdel za vak d pmćn uređaja ktla a l. 1 m za ventle. Mdel ventla ne navde e u vm radu pšt e bčn frmranju u kvru mdela parng ktla (npr. ventl za ubrzgavanje raadne vde u pregrejanu paru u pregrejače međupregrejače ktla. Om tga t u vrl jedntavn mdel kjma e algebarkm jednačnm prkazuje pad prtka radng fluda. Mdel pmćn ktlvk uređaja prkazan u u vektrkm blku a defnanm ulaznm zlaznm velčnama mdela. Matematčk mdel dela prcea u zatvrenm parnm ckluu kj e dvja d zlaza z tepena turbne nkg prtka d ulaza u parn kta, dnn u pmćnm uređajma ktla, dbja e pvezvanjem mdela pjedn pmćn uređaja pmću vektra ulazn zlazn velčna. Pvezan mdel kupa pmćn uređaja ktla prkazan je na l. 3. Mdel e atj d k dvadeet dferencjaln vše deetna algebark jednačna. Parametr mdela zračunavaju e z kntruktvn pdataka, pdataka z tenčke dkumentacje termblka z rezultata dređvanja aprkmacn plnma. Analzm kmpletng mdela kupa pmćn ktlvk uređaja lak e učava laba pvezant prmenljv mdela čak u kvru pjednačn mdela uređaja (u matematčkj termnlgj laba ulvljent jednačna mdela, a pgtvu u kmpletnm mdelu kupa pmćn uređaja ktla. a čnjenca mgućava dekmpzcju mdela jedntavnje brže rešavanje jednačna mdela. Matematčk mdel kupa pmćn ktlvk uređaja rešava e prmenm tg numerčkg algrtma kjm e rešava mdel parng ktla [5,6]. m algrtmm e numerčk tabln, brz tačn multan rešavaju dferencjalne algebarke jednačne mdela. 4 ZAKLJUČAK U radu u frmran nelnearn matematčk mdel pmćn uređaja ktla. Mdel u prkazan u vektrkm blku a defnanm vektrma ulazn zlazn velčna št mgućava njv lak pvezvanje u celnu. U kvru matematčkg mdela kupa pmćn uređaja ktla ptj jana pregledna veza zmeđu parametara 30
12 prmenljv mdela kntruktvn karaktertka fzčk velčna mdelvang tema. me je mgućen lak računavanje parametara mdela kršćenje rezultata merenja prračuna mdelvang tema za dređvanje prmenljv velčna u raznm radnm tanjma. LIERAURA [1] H. W. Kwan, J. H. Andern: "A matematcal mdel fr a 200MW bler", Int.J.Cntrl, Vl.12, N 6,1970, pp [2] G. Maada: "Matematcal Mdellng f Once-rug Supercrtcal Unt; Sytem Engnerng fr Pwer State Cntrl, Dav, [3] A. Bunzemeer, J. Papenfrt: "Matematce Mdellbldung Dampferzeuger mt zrkulerender Wrbelctfeurung", Autmaternugtecnk at 38, 1990, [4] "Upravljanje autmatzacja u termenergetkm ptrjenjma", Studja IN, Begrad, 1988.g. [5] S. Bdanvć: "Matematcal Mdellng f Steam Generatr and Degn f emperature Regulatr", IFAC Wrld Cngre, Bejng, 5-9 July, [6] S. Bgdanvć: "Degn f Subptmal Supereated Steam emperature Regulatr", IFAC Cnference Cntrl f Indutral Sytem, Belfrt, May, SPISAK OZNAKA C - matrce zlaza, kntanta J - mmenat nercje M - mmenat Q - klčna tplte, prtk pumpe U - elektrčn napn V - zapremna W - (maen prtk g - ubrzanje le gravtacje - entalpja k - kefcjent trenja p - prtak - klzanje u - unutrašnja energja x - vektr tanja u - vektr ulaza y - vektr zlaza η ρ ω - tepen kršćenja - gutna - ugana brzna Indek c - kndenzatr - ulaz - rezervar za napjnu vdu - zlaz - pumpa za vdu z kndenzatra M - mtr pumpe za vdu z kndenzatra pz - zagrejač pare - para w - vda Abtract: In t paper nnlnear matematcal mdel f bler auxlary devce are derved. Vectr f nput and utput varable f te mdel are defned. Matematcal mdel f all team bler auxlary devce tgeter derved by cnnectng te ndvdual mdel ung te vectr f nput and utput varable. NONLINEAR MAHEMAICAL MODELS OF SEAM BOILER AUXILIARY DEVICES Slbdan Bgdanvć 31
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