STOCHASTIC MODEL OF THE PROCESS OF STARTING PISTON COMBUSTION ENGINES AND THE PRACTICAL USE OF THE PROCESS

Joural of KONES Powertrai ad Trasort, Vol. 3, No. 2 STOCHASTIC MODEL OF THE PROCESS OF STARTING PISTON COMBUSTION ENGINES AND THE PRACTICAL USE OF THE PROCESS Jerzy Girtler Gdas Uiversity of Techology Faculty of Ocea Egieerig & Shi Techology Deartmet of Shi Power Plats tel.: +48 58 3472430, fax: +48 58 34798 e-mail: jgirtl@g.gda.l Abstract The aer resets a roosal of a model of diesel egie startig rocess as semi-marov rocess. States of the rocess are the followig: cold egie state (s ), warm egie state (s 2 ) ad hot egie state (s 3 ). Idea of usig the model for determiig a quatity of harmful substaces emitted i exhaust gases durig startig of egie, has also bee roosed herei. The aer rovides a ossibility of cosiderig i researches o the quatity of harmful substaces cotaied i exhaust gases, a mass of harmful substaces as a radom variable. It has bee acceted that statistics of the radom variable has distributio asymtotically ormal. The stochastic model of self-igitio egies startig rocess eables derivig the limitig distributio of the rocess, beig the robabilities of occurrig the metioed states (s ), (s 2 ) ad (s 3 ) oe by oe. The distributio is idisesable to determie the exected value of the mass of harmful substaces emitted i time of successive re-starts of egie erformed durig log oeratio. The secod aroach to estimatio of the metioed mass cosidered as a radom variable eables estimatig its exected value i time of log-term wor of egies till the first failure or i the eriod betwee two successive revetive maiteaces. The aer rooses the iterval estimatio because maes it ossible to assess accuracy of the estimatio. I this case the metioed value is determied i the form of iterval with radom edges (limits), which comrises uow value of the mass of harmful substaces emitted by egie, at the robability called cofidece level. Keywords: trasort, diesel egie, semi-marov rocess, toxic substace, modellig MODEL STOCHASTYCZNY PROCESU URUCHAMIANIA TOKOWYCH SILNIKÓW SPALINOWYCH I JEGO PRAKTYCZNE ZASTOSOWANIE Streszczeie Zarooowao model rocesu rozruchu siliów o zaoie samoczyym, w formie rocesu semimarowsiego, dysretego w staach i cigego w czasie esloatacji. Staami tego rocesu s: sta zimy silia (s ), sta ciey silia (s 2 ) i sta gorcy silia (s 3 ). Zarooowao rówie ocecj zastosowaia oracowaego modelu do oreleia iloci substacji szodliwych, jaie emitowae s w saliach odczas rozruchu silia. Przedstawioo tae moliwo uwzgldieia, w badaiach iloci substacji szodliwych zawartych w saliach, masy substacji zaieczyszczoej jao zmieej losowej. W rozwaaiach tych rzyjto, e statystya tej zmieej losowej ma rozad asymtotyczie ormaly. Przedstawioy model stochastyczy rocesu rozruchu silia o zaoie samoczyym umoliwia wyzaczeie rozadu graiczego tego rocesu, tórym s rawdoodobiestwa ojawiaia si olejo wsomiaych staów: (s ), (s 2 ) i (s 3 ). Rozad te jest iezbdy do oreleia wartoci oczeiwaej masy substacji szodliwych emitowaej odczas olejych rozruchów silia, dooywaych w czasie dugotrwaej ich esloatacji. Drugie odejcie do oszacowaia wsomiaej masy substacji szodliwych, rozumiaej jao zmieej losowej, umoliwia oszacowaie jej wartoci oczeiwaej w czasie dugotrwaej racy siliów do ierwszego uszodzeia bd w oresie midzy dwoma olejymi ich obsugami rofilatyczymi. Oszacowaie wartoci oczeiwaej masy substacji szodliwych emitowaej odczas dugotrwaej racy siliów moe by dooae rzez zastosowaie estymacji utowej lub rzedziaowej. W oracowaiu zarooowao estymacj rzedziaow, oiewa umoliwia

J. Girtler oa orelei doadoci tego oszacowaia. W tym rzyadu, wsomiaa warto jest orelaa w formie rzedziau o losowych ocach (graicach), w tóry zawarta jest iezaa warto masy emitowaych rzez sili substacji szodliwych, z rawdoodobiestwem azywaym oziomem ufoci. Sowa luczowe: sili o zaoie samoczyym, roces semi-marova, substacja tosycza. Itroductio Cosideratios i the hases of combustio egies desigig ad oeratig iclude ot oly arameters characterizig the eergy coversio, lie [2, 4, 5, 7, 8]: geeral ability of egie, heat emissio rate, heat usage rate, heat evolutio rate, ressure escalatio rate, etc. Emissio cotet should also be submitted to aalysis for such toxic comouds as carbo mooxide (CO), hydrocarbos (C H m ), itric oxide (NO x ), articulate solid (PM- Particulate Matte) ad sulfur comouds (SiO 2, SiO 3, H 2 SO 3, H 2 SO 4 ), aldehydes ad others [, ]. From researches o fuel combustio rocesses results that hysical (ot chemical) rocesses have great ifluece o the quatity of emissio of articular toxic comouds icluded i exhaust gases. The other issue from the researches is that their emissio is closely coected with each other ad ossibility of reductio of oe of them ca cause emissio icrease of aother oe. Therefore, ay actio to reduce the cotet of toxic comouds i exhaust gases must be followed by a comromise esurig miimizatio of harmful imact of toxic exhaust comouds o atural eviromet. Emissio of the toxic comouds deeds, amog others, o egie thermal state, esecially while startig egie. Thus, this aer rovides a roosal of a origial method to determie the mass of toxic comouds emitted by self-igitio or sar-igitio egie durig startig. The method cosiders the stochastic model of startig egies of this id. It has bee show that the model ca be reseted i the form of semi-marov rocess. Professioal literature roves that the most toxic comouds are formed i exhaust gases durig so called cold egie startig, so whe the egie is started after ot sufficiet rewarmig hase [, 9,, 9]. From this reaso there is a eed to create a model of egie startig rocess cosiderig at least three states of egie: cold, warm (warmed u) ad hot. 2. Model of self-igitio egie startig rocess Cosiderig the rocess of startig a self-igitio egie, at least three thermal states of it ca be distiguished, that exist directly before startig, i.e. cold state (s ), warm state (s 2 ), ad hot state (s 3 ). The states may be values of the rocess {U(t): t T} beig a real model of the egie startig rocess. The model ca be reseted i the form of semi-marov rocess havig the set of states [5]: S = {s, s 2, s 3 } () with the followig iterretatio of the states: s cold state which eables startig egie i ambiet coditios (coditios of shi ower lat) with tyical low temeratures (T) ot higher tha 290 K; s 2 warm state which eables startig egie i coditios of re-warmig (T > 300 K) erformed o egie i state s before startig; s 3 hot state which eables startig egie i its oeratio (wor) coditios followig from the eed of stoig the egie wor uder eve large load ad the re-startig. Thus, the rocess is a three-state rocess with cotiuous realizatios (rocess cotiuous i time). This is a semi-marov rocess [4, 5, 6, 7] ad, as a model of chages of the states of selfigitio egie startig, is the simlest model which may be of ractical sigificace. 74

Stochastic Model of the Process of Startig Pisto Combustio Egies ad the Practical use of the Process I this relatio the set of states of startig self-igitio egies S = {s, s 2, s 3 } ca be cosidered as the set of values of the stochastic rocess {U(t): t T} with costat itervals ad right-side cotiuous realizatios. Iitial distributio of the rocess {U(t): t T} of ay self-igitio or sar-igitio egie is defied by the formula [5]: dla i Pi PU (0) si. (2) 0 dla i 2,3 Fuctioal matrix of the rocess is as follows: 0 Q( t) Q3( t) Q(t) Q ( ) 0 ( ) 2 t Q23 t. (3) Q3( t) Q( t) 0 Thus, for the reseted rocess {U(t): t T} with fuctioal matrix exressed by the formula (3), the followig limitig distributio ca be determied: at: 2 3 2 3 23 P 23 3 3 E( T ) 2 E( T2 ) 3E( T3 ), P2, P3 H H H (4) 2 3 3 2 2, 2, 2 23 3 3 3 2, H E T ) E( T ) E( ), ( 2 2 3 T3 P, P 2, P 3 robability that combustio egie (self-igitio or sar-igitio) is started from the states accordigly: s, s2, s3, j limitig robability of the Marov chai iserted i the rocess {U(t): t T}, describig ossibility of occurrig the state s j, j =, 2, 3, ij robability of the rocess {U(t): t T} trasitio from the state s i to the state s j ; E(T j ) exected value of duratio of the state s j, (j =, 2, 3). The reseted states of startig ay self-igitio or sar-igitio egie are coected with articular thermal states of the egie. They reflect states existig i ractice. Therefore, the roosed model ca be emloyed for determiig the mass of toxic comouds. 3. Determiatio of toxic comouds mass durig hase of egie startig. Aalysis of research results o the combustio rocess ru, shows that durig combustio there are differet coditios for toxic comouds to form i emissio.. The most sigificat of them are: carbo mooxide (CO), hydrocarbos (C H m ), itric oxide (NO x ), articulate solid ad substaces i very little amouts (formig low cocetratios) lie sulfur comouds (SiO 2, SiO 3, 75

J. Girtler H 2 SO 3 ad H 2 SO 4 ), aldehydes ad others [,, 4, 8]. I ractice there is a eed to cotrol emissio of the metioed substaces. Emissio of toxic substaces ca be determied from the followig formula []: e v K s at =, 2,..., tye (umber) of the give toxic substace, e c mix H 0 6 (5) mass of toxic substace [g/m], v mix volume of dilute emissio i ormal coditios [dm 3 /test], desity of toxic substace i ormal coditios [g/dm 3 ], K M correctio rate of humidity of itric oxide mass, c cocetratio of the substace toxic for surroudigs [m], s rout traveled by a trasort mea (car, sea-goig shi) durig testig [m]. Emissio e ( =, 2,..., ) is differet deedig o whether the startig of egie rus from the cold state (s ), warm state (s 2 ) or hot state (s 3 ). Durig test erformace, o the rout (s) traveled by the give trasort mea (car, sea-goig shi), egie ca be started several times from its differet states metioed above. The emissio will be also differet for the egie deedig o techical state of the fuel system ad fuel quality. Ambiet coditios i which the egie is started are imortat, as well. Therefore, the value of toxic substaces emissio determied from the formula (5) may be acceted as emissio realizatio cosidered as a radom variable. Hece, taig ito accout the deedece (4), the mass of toxic substaces ca be exressed by the followig formula: at: ( 3 ) E( T ) ( 3E( T2 ) ( 2) E( T3 ) E ( E ) e e2 e3 (6) M M M M E T ) E( T ) ( ) E( ), ( 2 23 T3 e mass of the substace toxic for the atural eviromet ( =, 2,..., ), ij robability of trasitio of the egie startig rocess {U(t): t T} from the state s i to the state s j, i j; i, j =, 2, 3, E(T j ) exected value of state s j (j =, 2, 3) duratio. The formula (6) follows from cosideratio of all deedeces rovided i the formula (4) which determies the robabilities P, P 2 i P 3. The metioed toxic substaces cotaied i emissio lie: carbo mooxide (CO), itric oxides (NO x ), the most of all NO, hydrocarbos (C H m ), articulate solid (PM) ad sulfur comouds, as sulfur dioxide (SO 2 ), sulfur trioxide (SO 3 ), sulfurous acid (H 2 SO 3 ), sulfuric acid H 2 SO 4 ), ca be ad should be cosidered as radom variables. That is because these characteristics are such variables which, i the result of successive measures, tae differet umerical values with determied robability. So, the mass e of the metioed toxic substaces i emissio, calculated from the formula (5), ca be cosidered as realizatio of radom variable E, where = CO, NO, 76

Stochastic Model of the Process of Startig Pisto Combustio Egies ad the Practical use of the Process C H m, PM, SO 2, SO 3, H 2 SO 3, H 2 SO 4. That meas that determied robability is assiged to each ossible value of ay radom variable E,. This fact ca be described i the followig form [3, 3]: P(E = e i ) = i (i =, 2,..., ). (7) The formula shows that robability i is a fuctio of the values which ca be tae by a radom variable E. So, for ay radom variable E the followig descritio ca be made: i = f(e i ). (8) The characteristic of the fuctio is that the sum of robabilities determied by the deedece (8) is equal to, what ca be described as follows: i ( ). (9) Aalyzig toxic substaces cotaied i the emissio as radom variables E ad maig the successive measures of their mass value, the average mass value (arithmetic average) ca be determied from the formula [3, 8, 0, 6]: e i e e i i, (0) e i value of the aalyzed characteristic (radom variable) E. Average value e, determied from the formula (0) is the observed value of statistics E. The statistics is a radom variable [3] E E i, () where E i radom variable with the same (arbitrary) distributio of a commo exected value E(E i ) = m ad variatio D 2 ( E i ) 2 0. Thus, radom variable E is a arithmetic average with ideedet radom variables E i of idetical distributio. Exected value ad variatio of radom variable E are defied i the form of deedeces [3, 8]: i E (, E) E( Ei) m 2 2 2 D ( E) D ( Ei). () From the Lideberg-Levy theorem follows [3], that the radom variable E (statistics) has got a distributio asymtotically ormal N(m, ) regardless of the character of the radom variable E. That meas that the arithmetic average with ideedet radom variables E i, 77

J. Girtler arbitrary but idetical distributio, commo exected value E(E i ) m ad variatio 2 2 D ( Ei), has got distributio asymtotically ormal N(m, ). Aalysis of toxic substaces cotaied i emissio accordig to the reseted roosal is iterestig i the asect that covergece of the statistics E distributio to the ormal distributio N(m, ) is so quic that it ca be used for all 4, so almost always. Whe the value is ow ad the distributio N(m, ) of the statistics E is cosidered, it is ossible to determie a cofidece iterval for uow exected value m =E( E ) from the formula [3, 8, 0]: Pe y E ( E e y, (3) ) cofidece level, y stadardized variable of ormal distributio, meetig the cofidece level = ( sigificace level). I research ractice the value is, as a rule, uow ad for its evaluatio the followig formula [3, 6] is alied: s ei e i 2. Because the statistics E has always distributio asymtotically ormal N(m, ), ad its covergece to ormal distributio N(m, ) is very quic, so ractically it ca be acceted that aalyzed characteristic E of emissio, beig a cotet of a toxic substace, has ormal distributio N(m, ). Hereby, it has to be oited out that accetace of assumtio that the metioed characteristic E has ormal distributio N(m, ) is ot a limitatio i research ractice because the statistics E has always distributio asymtotically ormal N(m, ). Additioally, covergece of this distributio to ormal distributio is very quic \[3]. Thus, the E E( E ) radom variable has distributio t-studet with = degrees of freedom. s That meas that cofidece iterval for uow exected value E(E ) of the radom variable E ca be determied from the formula [3, 8]: 4. Fial coclusios s s Pe t, E ( E ) e t,. 78

Stochastic Model of the Process of Startig Pisto Combustio Egies ad the Practical use of the Process Emissio of toxic substaces, occurrig durig oeratio of combustio egies, deeds, amog others, o their thermal; states. The highest emissio of toxic substaces is recorded i time of startig the egies. Startig a egie from its cold state, so the state havig a ambiet temerature, is articularly disadvatageous. Egies i warm state ad esecially hot state geerate smaller amout of toxic substaces. Therefore, it is sigificat to ow the robabilities of startig egies from their articular states. For determiig the robabilities it has bee roosed herei the semi-marov model of the egie startig three-state rocess with the followig iterretatio of the states: cold state (s ), warm state (s 2 ), hot state (s 3 ). It has bee show usability of the model to determie the mass of toxic substaces emitted to the surroudigs durig startig egies. There has also bee reseted aother aroach to testig the cotet of toxic substaces i exhaust gases. It cosists i cosiderig the mass of ay toxic substace as a characteristic of emissio, beig a radom variable. It has also bee show usability of the Lideberg-Levy s theorem, from which follows that the statistics occurred from results of testig a determied mass of emitted toxic substace is a radom variable with distributio asymtotically ormal regardless of the character of the radom variable which is the tested mass of the metioed substace. This eables alyig the iterval estimatio for assessig uow exected value of radom variable which ca be a mass of ay toxic substace existig i exhaust gases. Refereces [] Bielaczyc, P., Merisz, J., Pielecha, J., Sta ciely silia saliowego a emisja zwizów szodliwych, Wydawictwo Politechii Pozasiej, Poza 200. [2] Bru, R., Szybobiee silii wysoore. WKi, Warszawa 973. Dae o orygiale: Sciece et Techique du Moteur Diesel Idustriel et de Trasort, Coyright by Societe des Editios Techi et Istitut Fracais du Petrole, Paris 967. [3] Firowicz, S., Statystycza ocea jaoci i iezawodoci lam eletroowych, WNT, Warszawa 963. [4] Girtler, J., Probabilistycza metoda oreleia masy substacji szodliwych emitowaych rzez sili o zaoie samoczyym odczas jego rozruchu, Zeszyty Nauowe AM w Szczeciie Nr EXPLO-SHIP 2004. Am, Szczeci 2004. [5] Girtler, J., Semi-Marovia models of the rocess of techical state chages of techical objects, Polish Maritime Research No 4/2004. Gdas. [6] Girtler, J., Physical asects of alicatio ad usefuless of semi-marovia rocesses for modelig the rocesses occurrig i oeratioal hase of techical objects. Polish Maritime Research No 3/2004. Gdas. [7] Grabsi, F., Teoria semi-marowsich rocesów esloatacji obietów techiczych, Zeszyty Nauowe AMW, r 75 A, Gdyia 982. [8] Gre, J., Modele i zadaia statystyi matematyczej, PWN, Warszawa 968. [9] Höglud, P.G., Ydstedt, A., Reduced Air Pollutio ad Fuel Cosumtio with Preheated Car Egies, Urba Trasort ad the Eviromet for the 2 st Cetury, Iteratioal Coferece, Lisbo 998. [0] Krzysztofia, M., Urbae, D., Metody statystycze, PWN, Warszawa 979. [] Larso, R.E., Vehicle Emissio Characteristics uder Cold Ambiet Coditios, SAE Techical Paer 89002. [] Loto, W., Zasilaie siliów saliowych aliwami alteratywym, ITE, Radom 995. [3] Pawow, B., W., Badaia diagostycze w techice, WNT, Warszawa 967. [4] Piotrowsi, I., Ortowe silii saliowe, WM, Gdas 97. [5] Piotrowsi, I., Witowsi, K., Esloatacja ortowych siliów saliowych, AM, Gdyia 2002. 79

J. Girtler [6] Vol, W., Statystya stosowaa dla iyierów, WNT, Warszawa 965. [7] Wajad, J.A., Silii o zaoie samoczyym, WNT, Warszawa 988. [8] Zaboci, M., Wtrys i salaie w siliach wysoorych, WKi, Warszawa 976. [9] Swedish Evirometal Protectio Agecy ad Motortestceter of the Swedish Motor Vehicle Isectio Comay. Cold Climate Emissios 993. 80