SELECTION OF MODEL PARAMETERS OF BIOGAS IC ENGINE. Karol Cupiał, Grzegorz Katolik

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1 TEKA Km. Mt. Energ. Rln., 2006, 6A, SELECTION OF MODEL PARAMETERS OF BIOGAS IC ENGINE Karl Cupał, Grzegrz Katlk Insttute f Internal Cmbustn Engnes and Cntrl Engneerng Techncal Unversty f Częstchwa Arm Krajwej Ave 21, Częstchwa, Pland e-mal: cupal@mc.pcz.czest.pl Summary. The statstcal methds f selectn f fur chsen engne mdel s value parameters are presented n ths paper. The am f ths paper s t create a mdel takng nt cnsderatn nnrepeatablty f cnsecutve engne wrk cycles. Mdellng f cnsecutve engne wrk cycles was carred ut wth usng zer-dmensn, tw-zne engne mdel. In ths mdel heat release was descrbed wth Vbe functn and heat exchange was descrbed wth Wschn equatn. Values f fur characterstc parameters btaned by applyng the presented methds allws fr a cmparatvely precse mdellng f ndvdual cycles as well as seres f engne wrk cycles. Key wrds: value parameters, engne wrk cycles, methds f selectn NON-STATIONARY MODEL OF ENGINE WORK The am f nn-repeatablty cnsecutve engne wrk cycles mdellng s takng ths nn-repeatablty nt cnsderatn, fr nstance durng an analyss f an assembly f crankshaft, pstns and cnnectng-rd r durng an analyss f therm-chemcal prcesses n a cylnder durng ndvdual engne wrk cycles. These phenmena are very mprtant fr an applcatn t generatng sets drven by bgas engne, whch have a hgher level f nn-repeatablty f engne wrk cycles than generatng sets drven by a CI engne fuelled by lqud fuels. The mdel utlzed n ths paper permts t reach a satsfactry agreement f value f elementary ndcated wrk, maxmum pressure and maxmum pressure rse n ndvdual cycles. Mdellng was carred ut usng a zer-dmensn, tw-zne engne mdel [Cupał 2003], whch takes nt accunt varable mle fractns f ten half-deal gases: N 2, O 2, CO 2, CO, H 2, CH 4, C 2 H 6, C 3 H 8, C 4 H 10 and H 2 O. In ths mdel heat release was descrbed wth Vbe functn [Vbe 1957] and heat exchange was descrbed wth Wschn [2001] equatn. The grups as fllws were dstngushed n ths mdel (Fg. 1): parameters wth cnstant values determned by engne settngs; parameters wth values defnte by well-knwn functnal dependences; parameters wth quas-randm varable values.

2 SELECTION OF MODEL PARAMETERS OF BIOGAS IC ENGINE 33 Selectn f mdel parameters fr each ndvdual cycle was realzed as lng as btans the best cnsstent tw pressure traces: real and mdelled. Values f these parameters are changed n each engne wrk cycle. These parameters are: dspersn f temprary values f elementary energy E j (caused by varable values f a fllng factr n cnsecutve cycles); energy suppled nt cylnder expressed as a prduct f tw factrs: calrfc value f flammable mxture (at 1 bar, 273 K) and a fllng factr; flame develpment angle ϕ rp the crank angle nterval between the spark and the tme at whch 2% f heat s released; burnng duratn angle ϕ spal the crank angle nterval between the end f the flame develpment stage and the tme at whch the 99.89% ( 100%) f heat s released; relatve angle peak ϕ pku - the angle f maxmum heat flux and burnng duratn angle rat. MODEL PARAMETERS Knwn values, cnstant n whle seres Values defnte by well - knwn functnal dependences Quas - randm varable values ϕ zap Ej ϕ rp ϕ spal ϕ pku Ej Qchł Kąt wy przedzena zapłnu Mean elementary energy Heat exchange Dspersn f elementary energy F lame develpment angle Burnng Relatve angle duratn angle peak Parameters wth cnstant values determned by engne settngs Wschn equatn The am f the paper Fg. 1. Dvsn f mdel parameters Durng mdellng were cnsdered, n ndrect way, nn-repeatablty f heat exchange prcess by ntrducng nt Wschn equatn curse f pressure and temperature n ndvdual engne wrk cycles. Mdellng was carred ut fr mean values f elementary ndcated wrk, whch was calculated fr mean fuel cnsumptn and fuel s calrfc value. Mdellng f nn-repeatablty f cnsecutve cycles was based n the results f supercharged bgas engne ndcatns. The ndcatns f an engne (1100 cntnuus cycles f engne wrk and pressure curses n the nlet and n the utlet channels) were carred ut fr every 0.5 CA fr mean effectve pwer N ef = 630 kw, whch crrespnds t mean ndcated wrk 1.7 MJ/m 3. ITERATIVE SELECTION OF MODEL PARAMETERS VALUES FOR INDIVIDUAL CYCLES The am f the wrk was checkng a pssblty t acheve the best cnsstence f the hgh-pressure part f engne wrk cycle (wthut charge exchange) f tw pressure

3 34 Karl Cupał, Grzegrz Katlk traces: a real and a mdelled ne. It was carred ut by usng teratve selectn f values fr fur characterstc mdel parameters fr ndvdual cycles wthn a seres f 100 cntnuus engne wrk cycles. Iteratve selectn f values fr fur chsen, characterstc parameters f an engne wrk cycle depended n change values f ndvdual parameters as well as n bservatn f an nfluence f ths change n the value f the chsen crtern f mdellng qualty. An agreement f results f mdellng wth real pressure curse was estmated by usng the abslute values f elementary dfferences f ndcated wrks cunted fr real and mdelled cycles dvded by value f real ndcated wrk. It was expressed by numercal crtern ADIW (Equatn 1). Ths crtern s mst senstve n lcal nnadjustment f pressure curses amng the ther analysed crterns [Katlk 2005]. The value f COV f elementary ndcated wrk was used t estmate the precsn f mdellng, t (Equatn 2). ADIW crtern was used t estmate the precsn f mdellng f ndvdual cycles. Values f COV p were used t estmate the precsn f mdellng f a cycle s seres wthut takng nt cnsderatn features f ndvdual cycles enterng nt the seres. ( ϕ= 540 CA ) lreal lmd ( ϕ= 180 CA) ADIW = (1) ( ϕ= 540 CA) lreal ( ϕ= 180 CA ) where: l real real elementary ndcated wrk; l md mdelled elementary ndcated wrk. Mdellng wth teratve selectn values f fur characterstc mdel parameters (dspersn f elementary energy E j, early flame develpment angle ϕ rp, burnng duratn angle ϕ spal and relatve angle peak ϕ pku ) permts t exact mdellng f hgh-pressure part f engne wrk cycle. Maxmal errr f mdellng f ndvdual cycles, expressed by ADIW crtern s lwer than 4.46%, mean values f ADIW crtern reach 3.21% (Fg. 2). 35% 30% 25% ADIW 20% 15% 10% 5% Max value 0% Mean value Iteratve selectn Samplng 1 Samplng 2 Samplng 3 Samplng 4 Range 1 Range 2 Range 3 Range 4 Mn value Fg. 2. Cmparsn f accuracy f engne wrk cycle mdellng

4 SELECTION OF MODEL PARAMETERS OF BIOGAS IC ENGINE 35 Obtaned values f COV p reach 2.55% and are very apprxmate t COV p values btaned fr seres f real engne wk cycles reach 2.34% (Fg. 4). Cmparsns f real and mdelled pressure curses btaned durng teratve mdellng (ADIW = 3.21%) are shwn n Fg. 3. Fg. 3. Example f SI engne wrk cycle mdellng (ADIW 3.21%) [Cupał 2003] The teratve methd requres multple repettns f calculatns f fr each cycle and allws t btan the enugh large precsn f mdellng. Hwever ths methd requres cnsderable amunts f jb, whch makes dffcult applyng f ths methd n practce. The randm selectn f values f fur parameters frm ther range f changeablty des nt allw btan the relable mdellng f ndvdual cycle, because the errr f mdellng expressed by the value ADIW that can reach very large values. SELECTION OF MODEL PARAMETERS VALUES FOR CYCLE SERIES Gdness-f-fts f real and mdelled parameters wthut takng cnsderatn cnfrmty f ndvdual cycles s analysed n ths part f paper. Analyss f cnfrmty f real and mdelled seres f engne wrk cycles was based n COV p value (Equatn 2), whch s statstcal measure f devatn f elementary ndcated wrk n seres: were: COV p σ = p p σ P standard devatn f p ; p mean elementary ndcated wrk; n number f measurements. = n = 1 ( p p ) n 1 p (2)

5 36 Karl Cupał, Grzegrz Katlk Values f fur selected, characterstcs mdels parameters were selected wth usng the pseudrandm numbers generatr (dstrbutn can be apprxmated wth usng nrmal dstrbutn). These values were generated frm emprcal determned ranges f parameters values, whch are descrbed by standard devatns. The ranges were defned n bass f real cycles analyss and teratve mdellng results. Samplng values f parameters were dne fur tmes fr cmplete seres f engne wrk cycles. Obtaned values weren t cnnected by emprcal equatns. Precsn f mdellng was estmated wth usng the COVp values. COVp values btaned durng randm mdellng (Fg. 4), n all cases, are hgher than COVp value calculated fr seres f real engne wrk cycles reach 2.34%. It shws, that usng randm factr n prcess f mdel parameters values selectn can cause the larger dspersn lmts f p n cmparsn wth seres f real cycles. Ths enlargement was expressed by hgher values f COVp abut % n reference t the analgus value cunted fr real cycles seres. 5,0% 4,5% 4,60% 4,0% COV p [%] 3,5% 3,0% 2,5% 2,0% 1,5% 2,34% 2,55% 3,32% 3,20% 3,01% 2,14% 1,92% 1,38% 2,05% 1,0% 0,5% 0,0% Real cycles Iteratve selectn Samplng 1 Samplng 2 Samplng 3 Samplng 4 Range 1 Range 2 Range 3 Range 4 Fg. 4. Cmparsn f COV p values f engne wrk cycle mdellng fr smplfed methds f values parameters mdel selectn Usng randm selectn f mdel parameters values requre defntn f changeablty parameters lmts. Determnatn f these lmts requred a statstcal analyss f fur characterstc mdel parameters. Calculatn f standard devatns f parameters s requred t. One f the methds f determnatn f lmts s smplfed range methd (knwn as rzstęp ) [Vlk 1973]. The calculatn f R value (Equatn 3) depends n calculatn f mnmal and maxmal value n data seres. R = x max x mn (3) There are requres nly tw values, maxmum and mnmum n seres, t make a rzstęp calculatn. In case f engne wrk cycle, values f elementary ndcated wrk

6 SELECTION OF MODEL PARAMETERS OF BIOGAS IC ENGINE 37 were chsen. Next step was standard devatn calculatn wth usng mean value f elementary ndcated wrk n whle seres. Standard devatn values btaned wth rzstęp methd are 40% lwer than standard devatn values calculated n bass f results f teratve mdellng. Randm selectn f mdel parameters values was dne fur tmes. Cmparsn f mdellng results btaned wth usng three methds f mdel parameters values selectn are shwn n Fgure 2 and 4. The last methd appears t be the fastest and the easest methd f values selectn. Values f COV p btaned n ths case were abut % lwer than value btaned fr real engne wrk cycles seres reach 2.34%. Methds presented abve cncerns mdellng f engne wrk cycles fr ndvdual cylnders. Values f elementary ndcated wrk btaned fr all cylnders f test engne are shwn n Table 1. Table 1. Elementary ndcated wrk f cylnders f gas engne Mean value p Standard devatn σ p COV p λ Cylnder n. MJ/m 3 MJ/m 3 MJ/m Statstcal analyss f ndcatn results shws, that mean values f elementary ndcated wrk n ndvdual cylnders are sgnfcantly dversfed: they are changng n the range between MJ/m 3, values f standard devatns are changng n the range between MJ/m 3. Values λ [PN-83/N ], btaned fr ndvdual cylnders, are presented n the Table 3. In all cases btaned λ values are lwer than crtcal value. The crtcal values λ k are dependent n level f sgnfcance α and fr α = 0.05 reach [PN-83/N ]. Numercal analyss f cycles shws that dstrbutn f values f elementary ndcated wrk n ndvdual cylnders can be apprxmated by takng advantage f nrmal dstrbutn. CONCLUSIONS Mdellng wth teratve selectn f mdel parameters values fr ndvdual cycles permts t btan the maxmal errr f mdellng, expressed as a ADIW value, reach 4.46%; mean values f ADIW crtern reach 3.12%. These results were calculated fr engne, whch wrked wthut knck. Value f COV p calculated fr the same seres f cycle s reach 2.55% and s nly abut 0.21% hgher than value btaned n case f real cycles analyss.

7 38 Karl Cupał, Grzegrz Katlk Methd f randm selectn f mdel parameters, ntrduced n ths paper, requres real cycles seres analyss. The am f ths analyss was t fx the range f varatn f selected mdel parameters. COV p values btaned fr ths methd are hgher than values btaned fr real cycles seres and they are changng n the range between %. Estmatn f standard devatn values fr fur selected, characterstc mdel parameters carred ut wth usng range methd allws t reduce the tme f calculatns and allws t btan COV p values lwer abut % than COV p value calculated fr seres f real cycles. Randm mdellng, based n range methd f standard devatn calculatn, s mst sutable fr practse use. Ths mdel requres cmparatvely small number f data and can by appled t quck mdellng f the seres f cycles wth takng nt cnsderatn ther nn-repeatablty. An analyss f results f smultaneus regstratn f pressure curses n eght cylnders f engne shws a lack f pssblty t state an pnn abut qualty f whle engne wrk n bass f analyss f measurement results frm ne cylnder. REFERENCES Cupał K Slnk wersja ; prgram d analzy begu slnka spalnweg. Instytut Maszyn Tłkwych Technk Sterwana Pltechnk Częstchwskej. Katlk G Zerwymarwy, statystyczny mdel begu slnka. Rzprawa dktrska. Częstchwa. Plska Nrma PN-83/N Statystyka matematyczna. Badana statystyczne. Badane zgdnśc rzkładu właścwśc w ppulacj z rzkładem teretycznym. Vlk W Statystyka stswana dla nŝynerów. WNT, Warszawa. Wschn G Betrag zum Prblem des Wärmeüberganges m Verbrennungsmtr, MTZ 26/4, Vbe I. I Raschet rabcheh cykla dvgatela s uchetm skrst sgranya ugla peredgenya vspamenya. Avtmblnaya traktrnaya prmyshlennst 8,