THEORY OF CUMULATIVE FUEL CONSUMPTION BY LPG POWERED CARS

Transcription

1 Journal of KONES Powerrain an Transor, Vol. 22, No THEORY OF CUMULATIVE FUEL CONSUMPTION BY LPG POWERED CARS Lech Jerzy Sinik Wroclaw Universiy of Technology Faculy of Mechanical Engineering Wysianskiego Sree 27, Wroclaw, Polan el.: , fax lech.sinik@wr.eu.l Absrac Theory of cumulaive consumion is shown for he firs ime in [] an [2]. The heory of cumulaive consumion has been resene also in his work. The examle of LPG car research resuls have shown he way of geing o mahemaical moel of cumulaive consumion an he inensiy of cumulaive consumion. In his case, he suies were carrie ou 6 cars whose engines were owere by LPG. The vehicles are oerae in a small flee of vehicles, which run in ciy raffic. Daa on oucomes of exloiaion an oeraion consumion is acquire from he accouning ocumens of he comany. Very goo resuls reicion mahemaical moel of oeraional aa are obaine. The high value of rescience quoiens (in his case R-sq > 9.999) are similar o he values ha were obaine in various oher cases. Conversance of mahemaical moel of cumulaive consumion allows carrying ou comrehensive analysis of his significan exloiaive arameer. The resene heory mus no only be regare as a heory of cumulaive consumion bu also can be seen more broaly as a heory of cumulaive energy consumion. Research is being conuce on he alicaion of he heory o evaluae energy consumion by hybri vehicles. The resuls can be very ineresing. Keywors: consumion, heory, alicaions. Theory of cumulaive consumion Fuel is injece ino he combusion chamber in a quanum. Quana are he elivery. Fuel quana have a ranom size. Toal volume (or weigh) of quana of sulie o he engine is calle consumion. Fuel quanum summaion leas o eermining he cumulaive consumion. Fuel consumion cause by he ime of he engine work, can be esignae as: n () Qs () = ν i = n () ν (), () i= where: ν i i-h quanum of sen (e.g. elivery er engine revoluion) ν () he average size of he quanum of consume o he ime, n() quanum number of he consume o he ime, Qs() he cumulaive consumion a ime. To know he cumulaive consumion o ime i shoul be familiar wih he average size of he quanum-sulie an he number of quana of consume o ha ime. T is a ranom variable reresening he ime beween successive oses of. The isribuion of he ranom variable is: F () = P { T < }, (2) ISSN: e-issn: DOI: /

2 L. J. Sinik where P {T < } is he robabiliy ha T assumes smaller values of ; is any amoun of ime. Derivaive of he cumulaive F () = F () = f(), (3) is he ensiy of ranom variable T. If we assume furher ha P {, + } is he robabiliy ha in he ime inerval, an hus a erio of ime (, + ) here will be no suly, an ha he was also aminisere over a ime erio (0, ), accoring o Bayes rule where: If here is P{ T + } R( + ) P {, + } = =, (4) P { T } R () P { T } = P { T < } = F () = R (). (5) P {, + } = P {, + }, (6) a robabiliy ha in he ime inerval here will be ose, rovie ha here was no suly in he ime erio (0, ) hen: P {, + } = P { T + }, (7) R ( + ) P{, + } = P{, + } =. (8) R () Afer iviing boh sies of equaion (8) hrough, here is obaine P{, + } R( + ) R( ) R( + ) = =. (9) R() R() The limes of he exression (9) wih 0 is: P{, + } R( + ) R() R () lim = lim =. (0) R() R() 0 0 Limes (0) can be eermine by λ(). I reresens he inensiy of he osing a ime : R () λ () =. () R () The number of quana given o he engine unil can be efine as: n() = λ () = = [ln R () + C] = ln C R (). (2) R() 0 R() o o 0 A ime = 0 is no given any, an no amoun of from he efiniion of R( = 0) = = F( = 0) = 0 =, 0 = ln C ln () C C R ( = 0) = =, (3) 276

3 Theory of Cumulaive Fuel Consumion by LPG Powere Cars furher means ha C = 0 an he number of oses given o he ime is exresse as: n ( ) = ln R (). (4) Furher, i is necessary o know he isribuion of he ranom variable T an herefore he isribuion of ime inervals beween successive oses of. The size of iniviual quanumνi has a ranom size. This quaniy can be escribe as a wo-imensional saisical isribuion wih ensiy f (ν, ). The average size of he quana of fe o he engine over a erio (0, ) can be eermine generally as: ν ( ) = ν f( ν, )ν. (5) 00 The limis of inegraion, he amoun of oses no change from zero o infiniy bu a a cerain inerval o (5), i can be wrien as: ν max ν ( ) = ν f( ν, )ν. (6) ν min 0 Taking he above ino accoun, i is in accorance wih () an (4) o wrie: Qzs () = ν () n() = ν ()ln = ν ()ln. (7) R () F () In he case of he Poisson isribuion F() = e λ. In his equaion, he secific oeraion of he engine an R () are unknown. Boh of hese values can be eermine in oeraional research. I was assume ha hese inervals coul escribe he ye of saisical isribuion of he Poisson isribuion funcion. There is also a well-known form of he saisical isribuion of he ose of. Because i is a a funcion of ime, however, i can be assume aroximaely ν () = ν. Base on boh assumions an aking ino accoun () an (4) here can be wrien: a Qzs () = ν () n() = ν () ln = ν ln = λ F () ( e ) (8) a a a = ν ln = ν ln ex( λ) = ν λ. ex( λ) Because assumions, ν cons an λ cons, also: ν λ = c = cons, (9) Then: a a Q () = c = c +. (20) zs This is a relaively simle relaionshi escribing he cumulaive consumion as a funcion of ime. The cumulaive consumion, base on he engine run ime, is he inensiy of he cumulaive consumion afer a given erio of engine oeraion. Q zs = ca ( + ) a. (2) The inensiy of he cumulaive consumion akes values infiniely large when a < 0 an 277

4 L. J. Sinik where 0, an so almos immeiaely afer he sar of he engine (bu raily ecreases wih increasing lengh of he isance ravelle). 2. A meho of eermining he cumulaive consumion on he basis of exerimenal aa Running a mahemaical moel of consumion, which is he equaion (8), an a he assumions, equaion (20), is known when hese facors are known as c an a. Their values are eermine, e.g. he mahemaical suy of exerimenal resuls. Equaion (20) can be convere o a convenien form in he firs-egree olynomial because afer aking he logarihm on boh sies hereof is relace wih: a+ ln Q ( ) = ln ( c ) = ln c+ ( a+ ) ln. (22) zs Afer subsiuing: (22) is convere ino: ln Qzs( ) = y, ln c= b0, ( a+ ) = b, ln = x, (23) y= b0 + bx. (24) This is he equaion of a sraigh line. The cumulaive consumion is consiere in he ime omain. To mee his requiremen i shoul be given ime insance moo hours. So given he working ime machine for examle agriculural racors. In he ransor vehicle uime aoe aminisere as heir course. However, he course is no he same as working ime vehicle engines. When he vehicle is saionary an he engine is, running here is consumion wihou increase mileage. For hese reasons, i is imossible o irecly aly he moel ye (8), namely (20), because no aa is collece relaing o he consumion in he ime omain, unersoo an as is aaren from he nee o moel he cumulaive consumion. In furher exloiaion, he ime of he vehicle is exresse in kilomeres. Wih his assumion: where: km mileage. km, (25) The aoion of his assumion makes ha he moel (20) is in he form Qzs ( km) = c( km ) a+. (26) The assumion (25) can lea o errors because consumion is recore, an he life of he engine is no because i is recore mileage. Deails of he meho of eermining he mahemaical moel of he cumulaive consumion an saisical verificaion of his reicion, an a aricular examle of he resuls of invesigaions resuls LPG owere cars, are given below. I was invesigae he consumion of foureen LPG owere cars. The resuls of hese ess were use o eermine he mahemaical moel of he cumulaive consumion. An examle of he resuls is given in Tab.. Table 2 shows he coefficiens of a mahemaical moel. Table 3 shows he resuls of he analysis of he qualiy of he mahemaical moel. Achieve arameers are exceionally goo. This furher confirms he char in Fig.. Is course is isincive an is reeae in he case of research of many cars. Table 4 conains he moel coefficiens of cumulaive consumion LPG of all ese cars. Preicion arameers of moels are very high. R-square for cumulaive consumion moels of all vehicles is no less as

5 3. Conclusions Theory of Cumulaive Fuel Consumion by LPG Powere Cars ) The heory of cumulaive consumion has been resene. Tab.. The measuremen an calculaion aa of LPG consumion of he engine of he car (Renaul Traffic 3) Mileage Cumulaive consumion Mileage (logarihmically) Cumulaive consumion (logarihmically) Cumulaive consumion (moel) Cumulaive consumion (ifference) Inensiy of cumulaive consumion km m 3 ln(km) ln(m 3 ) m 3 % m 3 /km Tab. 2. Moel coefficiens of cumulaive LPG consumion (Renaul Traffic 3) Coefficiens b 0 b c a Values Tab. 3. Parameers of he moel reicion of he cumulaive consumion of LPG owere Renaul Traffic 3 car Regression saisics Mulile of R R-square Ajuse R-square Sanar error SA Observaions Fig.. Cumulaive consumion of by LPG owere Renaul Traffic 3 car 2) The examle of LPG car research resuls have shown he way of geing o mahemaical moel 279

6 L. J. Sinik of cumulaive consumion an he inensiy of cumulaive consumion. 3) Daa on oucomes of exloiaion an oeraion consumion acquire from he accouning ocumens of he comany. Tab. 4. The moel coefficiens of cumulaive consumion LPG of cars Tese cars b 0 b R-sq. c a Fia Ducao Renaul Traffic Renaul Kang Renaul Kang Renaul Kang Renaul Traffic Fia Doblo Fia Doblo Oel Asra Renaul Traffic Renaul Kang Skoa Fabia Skoa Fabia Skoa Fabia Fig. 2. Cumulaive consumion all invesigae cars Fig. 3. Inensiy of cumulaive consumion all invesigae cars 4) Very goo reicion resuls in mahemaical moel of oeraional aa are obaine. The high value of rescience quoiens (in his case R-sq > 9.999) are similar o values which were obaine in various oher cases. 5) Conversance of mahemaical moel of cumulaive consumion allows o carry ou comrehensive analysis of his significan exloiaive arameer. 6) Presene heory mus no only be regare as a heory of cumulaive consumion bu also can be seen more broaly as a heory of cumulaive energy consumion. Research is being conuce on he alicaion of heory o assess he energy consumion of hybri vehicles he resuls can be very ineresing. References [] Sinik, L. J., Teoria skumulowanego zużycia aliwa, Raor P/00/2004 BiONT I-6 Poliechniki Wrocławskiej, Wroclaw [2] Sinik, L., J., Skumulowane zużycie aliwa, Cumulae consumion. Archiwum Mooryzacji, Vol. 7, Nr 3, 2004 [3] Kacrzyński, B., Planowanie ekserymenu. Posawy maemayczne, WNT Warszawa