INVESTIGACION´ REVISTA MEXICANA DE FISICA´ 48 (3) 228–234 JUNIO 2002 Otto and Diesel engine models with cyclic variability J. A. Rocha-Mart´ınez, T. D. Navarrete-Gonzalez,´ C. G. Pav´ıa-Millery, R. Paez-Hern´ andez´ Area de F´ısica de Procesos Irreversibles, Depto. de Ciencias Basicas,´ Universidad Autonoma´ Metropolitana Azcapotzalco. Av. San Pablo 180, Col. Reynosa, 02200, Mexico´ D. F., Mexico.´ y Also ESFM-IPN F. Angulo-Brown Depto. de F´ısica, Escuela Superior de F´ısica y Matematicas,´ Instituto Politecnico´ Nacional. edificio 9, U. P. Zacatenco, 07738, Mexico´ D. F., Mexico.´ Recibido el 23 de octubre de 2001; aceptado el 15 de febrero de 2002 Typically, in an internal combustion engine, thousands of cycles are performed in a minute. In this sequence of cycles many physical and chemical quantities change from cycle to cycle. For example, the combustion heat changes due to residual gases, imperfect combustion and other reasons. In this work, we present two finite-time thermodynamics models for both an Otto and a Diesel cycle, in which the cyclic variability is studied as occurring in the heat capacities of the working fluid. The fluctuations considered are of the uncorrelated type (uniform and gaussian) and one correlated case (logistic map distribution). We find that in the correlated fluctuations case, the power output and the efficiency of both cycles reach bigger fluctuations than in the uncorrelated cases. This result can provide insights over the performance of internal combustion engines. Keywords: internal combustion engine; cyclic variability; fluctuations En maquinas´ de combustion´ interna, t´ıpicamente, miles de ciclos son realizados en un minuto. En esta secuencia de ciclos algunas cantidades f´ısicas y qu´ımicas cambian de ciclo a ciclo. Por ejemplo, el calor de combustion´ cambia debido a gases residuales, a combustion´ imperfecta y a otras razones. En este trabajo presentamos dos modelos a tiempo finito para los ciclos Otto y Diesel, en los cuales se estudia la variabilidad c´ıclica como si estuviera ocurriendo en las capacidades calor´ıficas de la sustancia de trabajo. Consideramos tanto fluctuaciones descorrelacionadas (uniforme y gaussiana) como correlacionadas (distribucion´ tipo mapa log´ıstico). Encontramos que en el caso de las fluctuaciones correlacionadas, la potencia y la eficiencia de ambos ciclos alcanzan mayores fluctuaciones que en los casos descorrelacionados. Este resultado puede ayudar a comprender mejor el funcionamiento de maquinas´ de combustion´ interna. Descriptores: maquinas´ de combustion´ interna, variabilidad c´ıclica, fluctuaciones PACS: 44.60.+K; 44.90.+C 1. Introduction cycle model including chemical reactions [5]. In that model, we analyzed the performance of an Otto engine (see Fig. 1) In 1996, Badescu and Andresen [1] proposed that finite-time taking into account power losses due to a kind of lumped thermodynamics (FTT) can be complemented with some friction and we also consider the combustion reaction at the probabilistic concepts allowing a more accurate description end of the adiabatic compression. In other work [6], we took of the performance indicators of a power system. These au- the concept of fluctuant combustion heat proposed by Daw et thors studied a continuous flow tube reactor which supplies al. [3] as the input of our irreversible Otto cycle model, and heat to an engine from a chemical reaction with linear ki- then we analyze the behavior of performance outputs, such as netics. In general, typical FTT-models of thermal cycles are the power (P) and the efficiency (´) of the Otto cycle model. worked in steady state and only one cycle is taken as repre- In that work, we found that the size of the fluctuations in P sentative of all the other cycles pertaining to a sequence of and ´ around their mean values can be driven by the size of them. In a typical internal combustion engine, several thou- the combustion heat fluctuations and also by thermodynamic sands of cycles are performed in a minute and there exist the- properties of the states of the working fluid. In the present oretical and experimental reasons to expect important vari- work, we study two thermal cycles, namely the Otto and the ations from one cycle to the next [2, 3]. These variations Diesel cycles, under heat fluctuations, but using an alterna- can be found for example in the combustion heat of an Otto tive approach, where the fluctuations are taken as occurring cycle [3, 4]. In a recent paper, Daw et al. [3] proposed a dis- in the heat capacity of the working fluid. In our approach, crete engine model that explains how both stochastic and de- we take two previous FTT- models for both the Otto [7] and terministic features can be observed in spark-ignited internal the Diesel [8] cycles. The paper is organized as follows: in combustion engines. These authors present a model which Sec. 2, we present a brief resume of our previous thermal cy- reproduces the experimental observations of the cyclic vari- cle models; in Sec. 3 we discuss our fluctuant models of both ability of the combustion heat in a four-stroke, spark-ignition cycles and finally in Sec. 4 we present the conclusions. Otto cycle. Recently, we have reported an irreversible Otto OTTOANDDIESELENGINEMODELSWITHCYCLICVARIABILITY229 2. Otto and Diesel FTT-models ∆t12 = ¿=2 the time spent in the power stroke. Thus, the Otto model effective power output is given by In Fig. 1 we depict the Otto cycle pressure-volume diagram of the processes followed by the working fluid consisting of 1¡γ CV ¡ CV r n moles of air. In Ref. 7 it was considered that both the 1 2 2 P = PR ¡ P¹ = 1¡γ ¡ b (r ¡ 1) ; (5) K1 ¡ K2 r “absorbed” heat Qeff and the rejected heat Qout occur at finite times given by and the cycle’s efficiency by t = K (T ¡ T ) ; 1V 1 3 2 2 P CV b (r¡1) ¡ ¢ ´ = =1¡ 2 r1¡γ ¡ K ¡K r1¡γ : (6) and (1) Q =¿ C C 1 2 eff V1 V1 t2V = K2 (T4 ¡ T1 ) ; 1¡γ Equation (6) reduces to ´0 = 1 ¡ r (the ideal Otto effi- ciency) for CV = CV and ¹ = b = 0 and Eq. (5) reduces respectively, where K and K are constants linked to the 1 2 1 2 = 0 mean variation rate of the temperatures. In this approach to P for the ideal reversible (infinite-time) case. the adiabatic processes were taken as approximately instan- taneous, as it is common in FTT-models [9]. In this way, the cycle’s period is given by, ¿ = t1V + t2V = K1 (T3 ¡ T2 ) + K2 (T4 ¡ T1 ) : (2) FIGURE 2 Pressure-volume diagram of an ideal Diesel cycle. In Fig. 2, we depict a pressure-volume diagram of a Diesel cycle. In the FTT-approach of this cycle followed in Ref. 8 mainly based in the ideas of Ref. 7, the following expressions were obtained, for the cycle’s period, ¿ = K1 (T3 ¡ T2 ) + K2 (T4 ¡ T1 ) ; (7) and for the lumped friction losses FIGURE 1 Pressure-volume diagram of an ideal Otto cycle. 2 P¹ = ¡b (rC ¡ 1) ; (8) In Ref. 7 the cycle’s power output without losses was r r = V =V taken as where C is the compression ratio C 1 2 and 1¡γ 2 2 W CV ¡ CV r b = ¹ x =∆t . The effective power output and the cycle’s P = TOT = 1 2 ; (3) 2 12 R 1¡γ efficiency are given by [8] ¿ K1 ¡ K2 r where C and C are the constant-volume heat capacities V1 V2 ¡ ¢ γ¡1 γ γ of air in both isochoric processes 2 ! 3 and 4 ! 1, and CP (rC ¡ rE )(rE rC ) ¡ CV rC ¡ rE P = γ¡1 γ γ γ = CP1 =CV1 = CP2 =CV2 , being CP and CP the constant- 1 2 K1 (rC ¡ rE )(rE rC ) ¡ K2 (rC ¡ rE ) pressure heat capacities of the working fluid during the pro- 2 cesses 2 ! 3 and 4 ! 1 respectively, r = V1=V2 is the ¡b (rC ¡ 1) ; (9) so-called compression ratio. When in the model, losses due and to friction, turbulence in the working fluid, heat leaks etc, are added, all of them lumped in only a friction-like term, we rγ ¡ rγ have [7] ´ = 1 ¡ E C 2 2 γ¡1 P¹ = ¡¹º = ¡b (r ¡ 1) ; (4) γ (rE ¡ rC )(rE rC ) h i where ¹ is a lumped friction coefficient that embraces all 2 γ¡1 γ γ b (rC¡1) K1 (rE¡rC)(rE rC) +K2 (rE¡rC ) the global losses, º is the piston speed and b = ¹x2=∆t2 , 2 12 ¡ γ¡1 ; (10) CP (r ¡ r )(r r ) being x2 the piston position at minimum volume V2 and E C E C Rev. Mex. F´ıs. 48 (3) (2002) 228–234 230J.A.ROCHA-MARTINEZ´ ,T.D.NAVARRETE-GONZALEZ´ ,C.G.PAVIA-MILLER´ ,R.PAEZ-HER´ NANDE´ ZANDF.ANGULO-BROWN where CP is the constant-pressure heat capacity and puts such as the power P and the efficiency ´. In the present rE = V1 =V3 is the expansion ratio. Eq. (10) immediately paper we extend the study to a Diesel engine model and we reduces to the ideal Diesel efficiency [10] when ¹ = b = 0 assume that the heat fluctuations can be represented by fluc- and Eq. (9) reduces to P = 0 for the ideal reversible case. tuations in the heat capacities of the working fluid along the As it was remarked in Refs. 7 and 8, Eqs. (5), (6), (9) and branches of the thermal cycles. (10) have a reasonable behavior when they are compared with real power and efficiency curves for actual Otto and Diesel 3.1. The Otto cycle case engines. In fact, the obtained values for Pmax and ´max in In this case, we assume that the intake mixture (gas r) is com- both cases are close to the real Pmax and ´max values.