PROCESS OF SELF-IGNITION ENGINE LOADS AND ITS PROPERIES

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1 Jounal of KONES Powetain and Tanspot Vol. 3 No. 3 PROESS OF SELF-IGNITION ENGINE LOADS AND ITS PROPERIES Jezy Gitle Gdansk Univesity of Technology Faculty of Ocean Engineeing & Ship Technology Depatment of Ship Powe Plants tel.: fax: jgitl@pg.gda.p Abstact The pape pesents a pobabilistic intepetation of self-ignition combustion engine load with egads to known paametes (ates) of engine opeation. It has been shown that load of engines of this kind consideed in any moment can be egaded as a multi-dimensional andom vaiable. hanges of engine load in time of opeation ae consideed heein as a pocess of loads and pesented in a fom of multidimensional stochastic pocess of which states ae loads as andom vaiables. The pocess of loads can be egaded as a stochastic pocess with gains asymptotically independent being stationay and egodic. The eseaches on self-ignition engine loads should conside the stochastic dependence between themal and mechanical loads. Load consideed in abitay moment of wok of each engine has been accepted as a andom vaiable. Loads consideed in successive moments make the pocess of loads.. The moments howeve ae not andom vaiables but paametes of the pocess. Apat fom the hypothesis specified above also othe ones have been fomulated. Fo veification of the hypotheses thee have been poposed: the method of non-deductive (inductive) infeence called eductive infeence and the method of deductive infeence called the modus tollens ule. The poposed appoach to analysis of self-ignition engine loads is essential because the loads ae ones of the most significant causes of suface (linea) as well as volume wea of the engines what in consequence leads to engine failues. The engine failues happening duing opeation ae andom events. The failues consideed in successive moments of thei occuence make the pocess of failues which is a andom pocess. Keywods: tanspot combustion engine engine load hypothesis stochastic pocess PROES OBIĄŻEŃ SILNIKA O ZAPŁONIE SAOZYNNY I JEGO WŁASNOŚI Steszczenie Zapoponowano pobabilistyczną intepetację obciążenia silników spalinowych o zapłonie samoczynnym z uwzględnieniem znanych paametów (wskaźników) ich pacy. Wykazano że obciążenie tego odzaju silników ozpatywane w dowolnej chwili może być uważane za zmienną losową wielowymiaową. Zmiany obciążenia silnika w czasie jego pacy uznane zostały za poces obciążeń i pzedstawione w fomie wielowymiaowego pocesu stochastycznego któego stanami są obciążenia jako zmienne losowe. Wykazano że poces obciążeń może być uznany za poces stochastyczny o pzyostach asymptotycznie niezależnych stacjonany i egodyczny. Wykazano także że w badaniach obciążenia silnika o zapłonie samoczynnym należy uwzględnić zależność stochastyczną między ich obciążeniem cieplnym a mechanicznym. Obciążenie ozpatywane w dowolnej chwili pacy każdego silnika uznano za zmienna losową dlatego że może być uważana za pawdziwą następująca hipoteza: obciążenie silnika jest zmienną losową dlatego ponieważ jego watości w kolejno wykonywanych pomiaach można pzewidzieć jedynie z okeślonym pawdopodobieństwem. Obciążenia ozpatywane w kolejnych chwilach twozą poces obciążeń. Zatem istnieje potzeba opacowania koncepcji analizy i oceny tych obciążeń w aspekcie pobabilistycznym z uwzględnieniem faktu że zmiany obciążeń następujące kolejno po sobie we wspomnianych chwilach twozą poces stochastyczny. hwile te nie są jednak zmiennymi losowymi lecz paametami tego pocesu. Opócz wymienionej hipotezy sfomułowano także inne. Do weyfikacji pzedstawionych hipotez zapoponowano metodę wnioskowania niededukcyjnego (indukcyjnego) nazywaną wnioskowaniem edukcyjnym oaz metodę wnioskowania dedukcyjnego nazywaną egułą modus tollens. Zapoponowane podejście do analizy obciążeń silników o zapłonie samoczynnym jest istotne dlatego że należą one do najistotniejszych pzyczyn zużycia powiezchniowego (liniowego) jak ównież

2 J. Gitle objętościowego tych silników a w konsekwencji do ich uszkodzeń. Zaś zachodzące w czasie eksploatacji uszkodzenia silników są zdazeniami losowymi. Uszkodzenia te ozpatywane w kolejnych chwilach ich pojawiania się twozą poces uszkodzeń któy jest pocesem losowym. Słowa kluczowe: tanspot silnik spalinowy obciążenie silnika hipoteza poces stochastyczny. Intoduction Loads of self-ignition engines mechanical as well as themal ae among the most significant causes of suface (linea) and capacity wea what leads in esult to thei failues. Excessive themal loads of components making the engine woking spaces (combustion chambe) have extemely unfavouable effect on wea of the engines. uently loads of combustion engines ae analyzed geneally in deteministic aspect [ ]. Such analyses in pobabilistic aspect ae made in a simplified way and in pinciple ae limited to stochastic methods. ombustion engine loads should be howeve mapped in the space-time as andom vaiables assigned to andom events in which measuement of load of pope value is taken. The fact that the taken measuement of load is a andom event means that it is the event which detemined pobabilities should be assigned to. oeove the loads should be analysed in successive moments of combustion engines opeation. Loads consideed in these moments ceate the pocess of data. That is why thee is a need to elaboate a conception of analysis and estimation of the loads in pobabilistic aspect with egads to the fact that the changes of loads following one afte one in the mentioned moments ceate the stochastic pocess. The moments howeve ae not andom vaiables but paametes of the pocess. To elaboate such conception the pobabilistic popeties of the pocess of combustion engine load should be established fist beginning fom analysis of states of the pocess which ae the loads of the consideed engines undestood as andom vaiables... Engine load as a andom vaiable Empiical eseaches show that the values of engine loads ae not pedictable to the end [ ]. Thus the following hypothesis H can be fomulated: engine load is a andom vaiable because its values in measuements taken successively can be pedicted only with detemined pobability. uality intepetation of combustion engine load in any moment t can be pesented in a fom of the following dependence: whee: ( t) = ( t) ( t) () D + ( t) = ΔU ( t) ( t) + O at: D (t) themal enegy supplied to engine in time t (t) themal enegy tansmitted by engine components duing opeation (themal load) in time t (t) mechanical enegy connected with existing teestial gavity foce and aising gas foces fiction and inetia (mechanical; load) in time t ΔU(t) incease of mateial intenal enegy of components which a pat of themal enegy (t) maked as O (t) go though in time t O (t) enegy tansmitted to the envionment in time t though walls of engine components foming ex. woking spaces (combustion chambe) and the othes. 0

3 Pocess of Self-Ignition Engine Loads and ITS Popeies The mentioned engine loads can be consideed as andom vaiables. In the case when one paamete (one andom value) is chosen to descibe the load the load (as a value of the loading pocess) is a one-dimension andom vaiable. But when moe paametes ae applied to descibe the loads the load is a multidimensional andom vaiable. Then the load will be descibed by system of many (n in geneal) andom vaiables. The load can be then egaded as a n- dimensional andom vaiable so a system of many (n) eal-valued functions assigning unequivocally many numeical values to each andom event (which is measuement of load) [ ]. In opeation pactice self-ignition engine load is expessed the most often by the following paametes: p max maximal combustion pessue t max maximal combustion tempeatue p e aveage effective pessue c ś aveage piston speed Δϕ pś aveage ate of pessue incease. The paametes can be egaded as values of andom vaiables which can be designated adequately with: P max T max P e ś and ΔΦ pś. In case when the load is detemined by such andom vaiables as: maximal pessue (P max ) and maximal tempeatue (T max ) the two andom vaiables (P max i T max ) can be consideed at the same time. The vaiables can be egaded as jump vaiables. Then abitay ealizations of the andom vaiables ae the quantities p imax and t jmax. So the pai (p imax and t jmax ) is ealization of a twodimensional andom vaiable (P max and T max ). Occuing the events P max = p imax and T max = t jmax in the same time is detemined by the pobability q(p imax t jmax ). In case of consideing values of all p imax and t jmax which may occu the following dependence is effective [ 8]: i= j = q( p i max t ) =. () Set of pobabilities p(p imax t jmax ) is two-dimensional distibution of andom vaiable (P max T max ). The pobability q(p imax ) of the andom event P max = p imax without consideation of the value of andom vaiable T max is equal to the sum of pobabilities q(p imax t jmax ) including all possible values t jmax so: q ( pi max i max j = ) = q( p t ). (3) Set of pobabilities q(p imax ) detemined accoding to the fomula (3) is a maginal distibution of andom vaiable P max. In pactice it is essential to detemine the conditional pobability q(p imax /t jmax ) of the andom event P max = p imax unde condition (at the assumption) that T max = t jmax. This follows fom that the engine is the most loaded mechanically and themally when the maximal pessues and tempeatues occu in the engine woking spaces (cylindes). The pobability can be detemined fom the fomula: q( pi max t ) q ( pi max t ) =. (4) q( t ) Set of conditional pobabilities q( p i max t ) fo the same condition (assumption ) T max = t jmax is a conditional distibution of andom vaiable P max unde the condition T max = t jmax. The sum of conditional pobabilities q( p t ) including all possible p imax values is equal to so: i max i= q ( pi max t ) =. (5) 03

4 J. Gitle In pactice it may happen that andom vaiables P max and T max will be independent vaiables ex. in the esult of incoect pefomance of engine egulation. Then the following dependences ae effective: q p t ) = q( p ) (6) Taking obtained: ( i max i max q t p ) = q( t ). (7) ( i max into account the dependences (6) and (7) in the equation (4) the following fomula is q p t ) = q( p ) q( t ). (8) ( i max i max Random vaiables P max and T max satisfying the condition (8) ae stochastically independent. In case when the condition is not satisfied the andom vaiables P max and T max ae andom vaiables stochastically dependent (so coelated). Fom the dependence (8) esults that eseaches should conside the fact that in case of independent andom vaiables P max and T max conditional distibutions of the vaiables do not diffe fom thei maginal distibutions. In a simila way it can be consideed the load which is descibed by thee paametes ex. maximal pessue P max maximal tempeatue T max and aveage effective pessue p e. The pesented poposal of desciption of combustion engine load is significant because the load effects the wea of engine components. Fom this eason the wea in a chosen moment of engine opeation time is also a andom vaiable and being analyzed in successive moments of the time it should be egaded as a andom pocess. 3. Engine load as a stochastic pocess Engine loads (mechanical and themal) can be consideed in dynamic (shot) time measued in [ms] (this is the time in which thee exists one o a few themodynamic cycles of engine) and in quasi-statistic (long) time being the time of coect wok of engine measued the most often in w [hous]. ompaing the loads existing in successive themodynamic cycles so in shot time (t d ) it can be said that they ae diffeent fom diffeent easons [ ]. With incease of the value of time t d of engine opeation the diffeences gow. Thus at the fixed value of dynamic time and investigated load in paticula engine cycles in moments t d t d the diffeent values of load can be obtained. Obtaining a specific value of the load is a andom event. That means that to each moment of the time t d it can be assigned a andom vaiable indicating the load assigned just to this moment. Similaly to each moment of quasi-statistic time (t q ) it can be assigned a andom vaiable indicating the engine load just in this moment. Engine load as a andom vaiable in any moment t is a value of the pocess of loading. This pocess foms the un of loads occuing one afte one and being connected casually duing changes of load. The load can be descibed by diffeent andom quantities of which values can be pedicted only with detemined pobability. Thus the pocess of loading is a stochastic pocess so function of which values ae andom vaiables assigned to defined moments of engine opeation time t [ 4 5]. Theefoe the following hypothesis H can be fomulated: pocess of loading an engine is a stochastic pocess because the values of engine loads assigned to abitay moment ae andom vaiables. Fom the theoy of stochastic pocesses follows that a set of such moments is a set of paametes of the pocess [4 5]. Engine load can be chaacteized by diffeent quantities (paametes ates). Geneally the load can be expessed in a fom of the following dependence (): 04

5 Pocess of Self-Ignition Engine Loads and ITS Popeies whee: ( t) = f [ ( t) ( t)] (9) engine load mechanical load of engine themal load of engine t engine opeation time. Thus in empiical eseaches on load thee can be consideed at least two stochastic pocesses { (t): t 0} and { (t): t 0}. The pocesses ae components of the vecto pocess {(t: t 0} [4 5]. As fo the loads i it can be said that they ae vectos of the following components in abitay moment t: : [ pmax p p c ϕ ϕ n P P K] (0) : [ & & q T ρ p c Tmax T T K] e ś z sw whee: p max maximal combustion pessue p z combustion pessue p e mean effective pessue (p e = η m p i η m mechanical efficiency p i aveage indicated pessue) c ś aveage piston speed ϕ - (isochoic) pessue incease ate ϕ p moment ate of pessue incease n (engine) shaft speed ate P g foce coming fom gases pessue Pb foce of inetia q& themal enegy flux density T tempeatue gadient ρ ate of peliminay (isobaic) expansion T max maximal combustion tempeatue T z combustion tempeatue T sw exhaust gases tempeatue & themal flux T ol tempeatue of oil tempeatue of cooling wate. T w z e ś Fom the dependences (9) - () follows that the engine load depends on many quantities (paametes indexes) so the load can be consideed as a stochastic pocess including supeposition (composition) of paticula individual pocesses ( and in the most simple case). This undestanding of load is convenient fo geneal consideations but may be of a little usability fo pactical needs. Geneally load can be undestood as a pocess in which its states can be consideed in the fom of andom vaiables. The pocess as it is seen fom the dependences (0) and () is a multidimensional pocess. Thus the following n andom vaiables: t d t d t dn can be consideed in any moment t dl (l = n) of the dynamic time t d. The vaiables fo futhe consideations ae designated as n ). The vaiables can be n diffeent quantities (paametes ates) of the examined load (ex. = p max = T max 3 = p e 4 = q& etc.). p g b () 05

6 J. Gitle Taking into account simultaneously the paticula popeties of load (the load may be chaacteized by excessive combustion pessue excessive combustion tempeatue excessive themal flux etc.) it is obtained a system of n andom vaiables which makes n-dimensional andom vaiable (designated late on as ). In geneal (in ode to simplify the futhe consideations) consideations can be limited by accepting the andom vaiable as a two-dimensional andom vaiable ( ). Because measuements ae taken peiodically it can be assumed that andom vaiables and ae jump (discete) andom vaiables. Realizations of the vaiables ae quantities of q i and q i accodingly. Thus the pai ( q i qi ) is ealization of a andom vaiable ( ). The vaiable takes the values ( q i q i ) with detemined pobability ( q i q i ) which is the pobability of occuing the events = q i and = q i at the same time. Set of the mentioned pobabilities p( qi q i ) makes a two-dimensional distibution of the andom vaiable ( ). Just like in the pevious consideations which enabled fomulation of dependences () and (3) the distibution p ( q i qi ) satisfies the following condition: i= j= p( ) =. () q q i j aginal distibution of the andom vaiable is as follows p( q i ) = p( q q ) j = i j (3) and distibution of the andom vaiable is of the following fom: j p( q ) p( q q ). (4) = i= The above investigations shows that some quantities ex. p e c ś [0 3] chaacteize mechanical as well as themal load. Thus it is obvious that between mechanical load and themal load thee exist dependences. Because they ae andom pocesses the stochastic connection should also be expected between them. In ode to explain this connection the following hypothesis H 3 can be fomulated: thee is a stochastic dependence between the mechanical load (t) and themal load (t) because the detemined vaiants of one of the vaiables ae accompanied by diffeent vaiants of the second vaiable. Hence thee is a conclusion that the dependence between the loads ( (t) and (t)) cannot be descibed in the esult of applying a common method of algebaic equations. It seems to be tue because the load depends on big numbe of factos including these which cannot be measued [ ]: In case of combustion engine in the main pocess of enegy tansition themal enegy undegoes tansition into mechanical enegy and not the othe way ound. That is why the themal load ( ) can be accepted conventionally as independent vaiable and mechanical load ( ) as dependent vaiable. By analogy be epe can ind ndent vaiables and n. dependent vaiables n Degee in which the load vaiables can be vey diffeent. In pactice it may happen that one independent vaiable ( ) almost totally detemines dependent vaiable ( ). It also can happen that a few independent i j is detemined by the load o j (j= n) as independent 06

7 Pocess of Self-Ignition Engine Loads and ITS Popeies vaiables ( ) only in a small degee influence the dependent vaiable ( j ). Fom this it follows that thee is a need of taking into account the intensity (powe) of stochastic connection between and. Intensity (powe) of the stochastic connection between (t) and (t) can be established by applying fo empiical eseaches the following dependences [8]: χ T = T = (5) N ( k )( l ) whee: k numbe of vaiants of the vaiable l numbe of vaiants of the vaiable N maginal numbe of vaiants of the vaiable o χ value calculated fom the chi-squae fomula T ( ) convegence ate of zupow. It can be poved [8] that T takes the values fom the inteval [0]. The ate equals zeo (T = 0) when thee is no connection between values of the pocess ( i ) and equals one (T = ) when functional dependence exists. Geneal statement whethe the andom vaiables: and ae dependent is possible afte examining the pobabilities defined by genealized fomulas (4) (6) o detailed fomulas (6) (8). ompaing the loads occuing in successive cycles of engine so in time t d (shot) a conclusion can follow that they ae diffeent. So at a fixed value of time t d and by testing the load in paticula cycles in the moment t d t d diffeent values of load ae obtained. Thus the fact of obtaining a detemined (expected) value of load is a andom event. This is such event because in the esult of establishing the same conditions of empiical eseaches the expected value of load may occu but also may not occu. That means that gains in load ae less and less dependent on each othe with incease of time distance between the mentioned moments t d t d [ ]. On basis of the pesented consideations the following hypothesis H 4 : can be fomulated: load is a pocess of gains asymptotically independent because with incease of the time distance between the time intevals in which the load undegoes testing (measuements of load ae taken) its values become less and less dependent on each othe. Anothe popety of changes of the load consists in that obseved quantities of load in dynamic time (t d ) as well as in quasi-statistic time (t q ) do not show any oiented (monotonic) vaiations. That is why it can be accepted that the peak quantities chaacteizing the load occu in a andom way. Lack of monotony of changes of engine load enables fomulating the hypothesis H 5 saying that: load is a stationay pocess because in longe time thee is no monotony of engine load changes []. Fom up-to-now eseaches on self-ignition engines follows that thei load undegoes continuous changes in a way that its paticula values measued afte small time intevals ae stongly coelated one with anothe. Howeve when the time distance between measuements of the loads gows the coelation between the loads deceases. Thus values of the load measued in time intevals (o moments) consideably distant fom each othe can be consideed as independent. This popety is called an asymptotic independence of the load value measued in the moment ex. τ i+ fom the value taken in the moment τ i when the distance Δτ = τ i+ τ i is sufficiently big [3]. Such undestood asymptotic independence between the load values measued (o calculated) in the moments τ i and τ i+ eflects the fact that along with the gowth of Δτ the dependence between the values deceases. 07

8 J. Gitle The pesented view on engine load popeties can lead to aising new possibilities of getting dependences of wea fom load via empiical eseaches. 4. Final conclusions The pesented analysis and synthesis of events show that load of each self-ignition engine consideed in any moment of its opeation (wok) time can be egaded as a multi-dimensional andom vaiable. Loads analyzed in successive moments of opeation time of this kind of engines can be consideed as ealizations of the pocess of loading. Thus the pocess of loading of each engine needs to be investigated as a multi-dimensional stochastic pocess. Heein thee have been poposed some hypotheses explaining why the pocess of loading of any self-ignition engine can be egaded as a stochastic pocess with gains asymptotically independent stationay and egodic and that thee is a stochastic dependence between its mechanical and themal loads. Intensity of stochastic connection between the loads can be fixed duing eseaches by applying the convegence test of zupow [8]. Fo veification of the pesented hypotheses thee have been poposed the method of nondeductive (inductive) infeence called eduction infeence and the method of deductive infeence called the ule of modus tollens. Leaning about the popeties of the pocesses equies constuction of applicable mathematical models on the way of the system modeling and pefomance of adequate empiical tests. Refeences. Bun R. Szybkobieżne silniki wysokopężne WKiŁ Waszawa 973. Dane o oyginale: Science et Technique du oteu Diesel Industiel et de Tanspot. opyight by Societe des Editions Technip et Institut Fancais du Petole Pais Fikowicz S. Statystyczna ocena jakości i niezawodności lamp elektonowych WNT Waszawa Gecbach I.B. Kodonski h.b. odele niezawodnościowe obiektów technicznych WNT Waszawa Gichman I.I Skoochod A.W. Wstęp do teoii pocesów stochastycznych PWN Waszawa Gitle J. Stochastyczny model widma obciążeń silnika o zapłonie samoczynnym Zagadnienia Eksploatacji aszyn. Kwatalnik PAN z. / Gitle J. Physical aspects of application and usefulness of semi-akov pocesses fo modeling the pocesses occuing in opeational phase of technical objects Polish aitime Reseach Vol. No 3 Septembe Kozłowiecki H. Łożyska tłokowych silników spalinowych Waszawa WKiŁ Kzysztofiak. Ubanek D. etody statystyczne PWN Waszawa Niewczas A. Podstawy stochastycznego modelu zużywania popzez tacie w zagadnienia twałości elementów maszyn Zeszyty Naukowe echanika n 9 Politechnika Radomska Piotowski I. Witkowski K. Eksploatacja okętowych silników spalinowych A Gdynia 00.. Rozanov Ju.A. Stacionanye slucajnye pocessy Fizmatgiz oskva Wajand J.A. Silniki z zapłonem samoczynnym WNT Waszawa Włodaski J.K. Tłokowe silniki spalinowe Pocesy Tybologiczne. WKiŁ Waszawa Voinov A.N. Sgoanie v bystochodnych posnevych dvigateliach asinostoenie oskva Wentzell A.D. Wykłady z teoii pocesów Stochastycznych PWN Waszawa