Mathematical Modeling of Spark-Ignition Engine Cycles

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Mathematical Modeling of Spark-Ignition Engine Cycles a

Hakan Bayraktar a Mechanical Engineering Department, Karadeniz Technical University, Trabzon, Turkey. To link to this article: DOI: 10.1080/00908310390212345 URL: http://dx.doi.org/10.1080/00908310390212345

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ESO 25(7) #3896

Energy Sources, 25:651–666, 2003 Copyright © 2003 Taylor & Francis 0090-8312/03 $12.00 + .00 DOI: 10.1080/00908310390212345

Mathematical Modeling of Spark-Ignition Engine Cycles Downloaded By: [ANKOS 2007 ORDER Consortium] At: 14:56 21 February 2007

HAKAN BAYRAKTAR ORHAN DURGUN Mechanical Engineering Department Karadeniz Technical University Trabzon, Turkey

A quasi-dimensional spark-ignition (SI) engine cycle model has been developed. In this model, combustion is modeled as a turbulent flame propagation process. During the combustion, it is assumed that the cylinder charge consists of unburned and burned gas zones. A computer code was developed for the mathematical cycle model presented. By using this code for any engine running at specified conditions, parameters that characterize the combustion, cycle, and engine performance can be computed practically. The above-mentioned parameters for several engines with different geometries and running at different operating conditions are determined theoretically, and these predicted values are compared to experimental data given in the literature. Good agreement between predicted and experimental results is obtained. Comparisons of predicted and measured results show that the presented model is reliable for analyzing SI engine cycles. Keywords engine cycles, quasi-dimensional engine models, thermodynamic cycle models, turbulent flame propagation

For many years, researchers have spent considerable effort to improve the performance of spark-ignition (SI) engines. Research studies concerning internal combustion engines are grouped mainly as experimental and theoretical. Although experimental methods provide actual results about the engine operation, design, manufacture, and testing of new engines or new systems is more expensive and requires more time than theoretical methods. If the mathematical model of an engine cycle can be proposed with realistic assumptions and this is arranged as a computer code, the engine cycle and performance can be predicted at the beginning of design or development study. For this reason, cycle simulation studies have had great interest to date. Internal combustion engines consist of open thermodynamic systems such as cylinder, intake, and exhaust manifolds (Bayraktar, 1997; Heywood, 1988). These systems are open to the transfer of mass, enthalpy, and energy in the form of work and heat (Poulos and Heywood, 1983). In the simulation studies, determining the thermodynamic state of the charge involved by the mentioned systems is the aim. For this purpose, a set of equations Received 7 April 2002; accepted 26 May 2002. Address correspondence to H. Bayaraktar, Mechanical Engineering Department, Karadeniz Technical University, 61080 Trabzon, Turkey. E-mail: [email protected]

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governing the fluid mechanic and thermodynamic behavior of the engine working fluid throughout the cycle is derived and then solved numerically on a computer. In general, mathematical engine simulation models are divided into 2 main groups. The first is fluid-dynamic-based models and the second is thermodynamic-based models (Bayraktar, 1997; Heywood, 1988). Fluid-dynamic-based models are also called multidimensional models due to the fact that their formulation is based on the conservation of mass, chemical species, and energy at any location within the engine cylinder or manifolds at any time. That is, space dimensions (spatial coordinates) and time are the independent variables, hence the governing equations are partial differential equations. By using the multidimensional models, detailed information about the spatial distribution of the gas velocity, temperature, and composition within the engine cylinder can be obtained. From the above information, it is obvious that solving such equations will be difficult and a large amount of computer time will be needed. For this reason, these models are not suited to predicting engine performance parameters (Novak and Blumberg, 1978). However, multidimensional models are used for the calculation of the flow field at motored conditions within the reciprocating engines running at low speeds. Thermodynamic cycle models are based on the thermodynamic analysis of the cylinder contents during the engine operating cycle. In these models the First Law of Thermodynamics is applied to an open system composed of a fuel-air-residual gas charge within the engine manifolds and cylinders (Bayraktar, 1997; Poulos and Heywood, 1983; Novak and Blumberg, 1978). Thermodynamic models are zero dimensional because the only independent variable is time. Thus the governing equations consist of ordinary differential equations instead of partial differential equations. In such engine models, the combustion process is simulated in 2 ways for SI engines. In the first approach, the rate of burning of the charge is obtained empirically by using the Wiebe function or the cosine burn rate formula (Heywood, 1980; Heywood et al., 1979). That is, combustion chamber geometry and flame geometry are not considered. Therefore these models are called zero-dimensional models. In the second approach, the mass burning rate is determined from a mathematical model of the turbulent flame propagation (Bayraktar, 1997; Heywood, 1988; Poulos and Heywood, 1983; Heywood, 1980; Blizard and Keck, 1974; Tabaczynski et al., 1977; Beratta et al., 1983; Tabaczynski et al., 1980). These types of thermodynamic cycle models that include detailed combustion modeling are called quasi-dimensional models. From the above information, it is well understood that the dimensional models are useful for predicting the detailed spatial information of the cylinder charge during the engine cycle, but are not practical for parametric studies of the effects of changes in design and operating variables on engine performance, efficiency, and emissions. The presented theoretical study has been performed to investigate the SI engine’s combustion, cycle, and performance for various engine geometries, operating conditions, and fuels. For practical and reliable evaluation, considering the above comparisons, a quasi-dimensional SI engine cycle model has been developed. In this paper, a proposed mathematical model and its applications for SI engines are presented. Applications of this model include checking the reliability of the presented method and various comparisons of results to those of the other studies that have been presented. Finally, discussions and recommendations are given at the end of the article. In brief, a quasi-dimensional SI engine model that was developed for various theoretical investigations concerning the SI engines is presented here. Alternative applications such as using alternative fuels in engines, examining the effects of engine design and operating parameters, etc., are left for future articles.

Mathematical Modeling of SI Engine Cycles

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Mathematical Model


For theoretical investigation of a SI engine cycle, a mathematical model for the cylinder charge must be established. In this study, a quasi-dimensional cycle model was developed. Details of the model, basic assumptions, solution procedures, and sample applications will be given in the next sections. Broad descriptions of the model were given previously by Bayraktar (1997) and Bayraktar and Durgun (1997). The SI engine cycle consists of 4 consecutive processes: intake, compression, combustion-expansion, and exhaust. These are different in nature, thus each process must be computed with different approximations. In general, to investigate changes in the thermodynamic state of the cylinder contents, expressions for the time rates of the charge temperature and pressure must be established. By using the First Law of Thermodynamics and ideal gas laws, the following equations for the time rates of pressure and temperature of an open system have been derived: B m h V˙ 1 ˙ ˙ ˙ T = 1− − + m ˙ i hi − Q w , A m B V Bm p˙ =

ρ ∂ρ ∂p

−

1 ∂ρ ˙ m ˙ V˙ − T + , V ρ ∂T m

(1)

(2)

where 

 ∂ρ  1 ∂h ∂h  ∂T   + − ; A= ∂ρ  ρ ∂T ∂p ∂p

1 ∂h B = 1−ρ . ∂ρ ∂p ∂p

Here the dots denote the differentiation with respect to time; m ˙ i hi is the net rate of ˙ is the total heat transfer to the wall; W˙ = p V˙ is the work done by influx of enthalpy; Q the gases on the piston; and h is the enthalpy. Terms included in Equations (1)–(2) must be specified in order to solve the above system of ordinary, first-order differential equations. During the cycle, these equations must be arranged with reasonable assumptions for each process due to the existence of different physical and chemical events in the engine processes. Intake and exhaust processes are computed by using the approximate method developed by Durgun and given by Bayraktar (1997). Compression, combustion (including expansion) processes are computed by using the above set of differential equations. The solution procedures are given below.

Intake Process Intake pressure and temperature are specified by a simple method given by Bayraktar (1997). In this method, pressure loss p is computed from the Bernoulli equation for the one-dimensional noncompressible flows and then intake pressure is specified as pa = p0 − pa ,

(3)

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where p0 is the ambient pressure. Then intake temperature Ta is computed from the following formula given by Bayraktar (1997): Ta =

T0 + T + γr Tr , 1 + γr

(4)


where T0 is the ambient temperature, T is the increase in temperature throughout the intake system, Tr is the exhaust temperature at the previous cycle, and γr is the coefficient of the residual gases. Volumetric efficiency is given by ηv = ed

ε pa T0 , ε − 1 p0 T0 + T + γr Tr

(5)

where ed is the charge-up efficiency and ε is the compression ratio. Compression Process During the compression stroke, Equations (1)–(2) must be arranged by making reasonable assumptions. For this process, basic assumptions are as follows. The system that is considered consists of instantaneous contents of a single cylinder. The cylinder is a variable volume plenum and is spatially uniform in pressure. These assumptions are also valid for other strokes. Throughout the compression, the cylinder charge consists of a mixture of air, fuel vapor, and residual gases. These gases are nonreacting ideal gases and are characterized by a single mean temperature. The total mass of the system is taken as constant during the cycle. By considering the above assumptions, Equations (1)–(2) were arranged as follows: ˙ ˙ wu K V Q Bu ˙ Tu = − − , (6) Au V Bu mu rad where

∂ρ 1 ∂T + cpu ; Au = u ∂ρ ρu ∂p u

Bu =

1 . ∂ρ ∂p u

Here u denotes the unburned gas, V is the instantaneous total cylinder volume, Q is the total heat transfer to the wall, ρ is the density, and cp = (∂h/∂T )p is the specific heat at a constant pressure. The rate of change of the cylinder pressure is ˙ V bar 1 ∂ρ −5 ρu ˙ − − Tu . (7) p˙ = 10 ∂ρ V ρu ∂T u rad ∂p u Combustion Period The compressed fuel-air-residual gas mixture within the combustion chamber is ignited by a spark discharge and then combustion begins. From experimental observations (Beratta et al., 1983; Gatowski et al., 1984; Namazian et al., 1980; Tagalian and Heywood, 1986) it is known that 3 distinct zones exist during combustion. These are identified as an


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unburned fuel-air mixture zone ahead of the flame front, a flame zone, and a fully burned zone behind the flame zone. For these reasons, governing equations must be arranged separately for each zone. In this study, it was assumed that 2 distinct zones separated by a thin spherical flame that has a negligible volume occurs during the combustion. Thus for the unburned and burned zones, Equations (1)–(2) can be written as ˙ m ˙ hi V B 1 ˙ wi + m 1− − + (−Q ˙ i hu ) , A i m i Bi V i (Bm)i       ρ ˙ ∂ρ V m ˙  p˙ = 10−5  ∂ρ  − , −  ∂T  T˙i + V i m i ∂p ρ i i

T˙i =

(8)


(9)

where ∂ρ 1 ∂T Au = cpu + u ; ∂ρ ρu ∂p u

Bu =

1 ; ∂ρ ∂p u

∂ρ ∂h ∂T b 1 −5 , − 10 Ab = cpb + ∂ρ ∂p b ρb ∂p b ∂h 1 −5 ρ . Bb = 1 − 10 ∂ρ ∂p b ∂p b

Here for the unburned mixture i = u and for the burned mixture i = b. From the conservation of mass it is obvious that m ˙ b = −m ˙ u. Expansion Process After burning of the entire charge within the cylinder, the expansion process starts. During the expansion the cylinder charge consists of fully burned gases and the total mass is constant. By making the above assumptions for the expansion process, Equations (1)–(2) can be written as ˙ ˙ wb V B −Q ˙ Tb = − + , (10) A b V (Bm)b       ρ ∂ρ ˙ V p˙ = 10−5  ∂ρ  − −  ∂T  T˙b  . (11) V ∂p ρ b b In this process, Ab and Bb are the same form as that of the combustion process. Exhaust Process Exhaust pressure pr and temperature Tr are calculated by a simple method developed by Durgun and given by Bayraktar (1997). Exhaust pressure is specified depending on the ambient pressure p0 as pr = (1.05 ÷ 1.25)p0 ,

(12)

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and exhaust temperature Tr is specified using the burned gas temperature Tb at the end of the expansion: Tr =

Tb 1/3 .

pb pr

(13)


If the difference between the predicted and chosen values of Tr is reasonable or is within the given limits, the last value of Tr is accepted. Otherwise the cycle must be computed again for last value of Tr .

Submodels Basic differential equations for temperature and pressure for each process were given in the previous sections. To solve these equations numerically requires determining the numeric values of T˙ and p. ˙ For this reason, the terms included in these equations must be determined with reasonable approximations. In the next sections, the submodels included in the cycle models will be given. Thermodynamic Properties The thermodynamic properties of the unburned and burned gases must be determined continuously during the entire cycle analysis. The unburned gas is assumed to be composed of 3 nonreacting ideal gases. These are air-fuel vapor and residual gases. At any instant of the cycle, for calculation of this mixture composition and thermodynamic properties and their partial derivatives with respect to pressure and temperature, a submodel was proposed by using the method developed by Komiyama and Heywood (1973). For burned gases during combustion and expansion, the model developed by Olikara and Borman (1975) was used. In this method, it is assumed that the burned gases are reacting gases that are in chemical equilibrium. The composition of the mixture and thermodynamic properties and their thermodynamic derivatives are computed simultaneously at any instant via the developed submodel. Heat Transfer Heat transfer between the gases and cylinder walls must be specified. Heat transfer in SI engines is due to the convection and radiation. Several empirical correlations have been developed for determination of the instantaneous heat transfer coefficient. These correlations were given by Borman and Nishiwaki (1987). The empirical formula developed by Annand (1963), which is widely used, is used in this study. By using this formula, the ˙ w is arranged as term Q ki b 4 4 ˙ (14) Qwi = Awi a Rei (Ti − Tw ) + c(Ti − Twi ) (Watt), D where a, b, and c are constants and are chosen as a = 0.35–0.8, b = 0.7, and c = 4.3 × 10−9 W/m2 K4 (for combustion and expansion); D is the cylinder bore; subscript i can be u (unburned gas) or b (burned gas); and Tw is the wall temperature. Reynolds number Re is calculated empirically (Heywood, 1988).


657

Geometric Model


During the cycle analysis, instantaneous cylinder volume V and its differentiation with respect to time t or crank angle θ (V˙ ), flame front area, and the area of the combustion chamber in contact with unburned and burned gases are computed from the geometric submodel. The total cylinder volume and its derivative V˙ at any crank angle can be computed easily. Geometric features of the flame at any instant of combustion require a specific calculation method. The experimental observations have shown that the geometry of the flame in SI engines is approximately spherical (Beratta et al., 1983; Gatowski et al., 1984; Namazian et al., 1980). For this reason, to determine the instantaneous flame geometry, a mathematical method developed by Annand (1970), which is widely used in combustion modeling studies, was used. At any piston position, enflamed volume Vf and surface area Af have been calculated depending on the instantaneous flame radius Rf and chamber height h by use of Simpson’s integration scheme. During combustion, the volume enveloped with the flame front, Vf , is expressed as follows: V f = Vb +

me − mb , ρu

(15)

where me is the entrained mass by the flame front and mb is the burned mass. The wall contact area of the gases corresponding to the instantaneous burned gas volume and the flame front surface area corresponding to the volume enclosed by flame Vf have been computed by the Newton–Raphson iteration method. Thus the wall contact areas of the gases required for heat transfer calculation and the flame geometry required for the turbulent flame propagation model are determined.

Burn Rate Model In a SI engine, combustion is initiated by a spark discharge. At the beginning of combustion, the flame is a smooth-surfaced, roughly spherical kernel about 1 mm in diameter and grows quite spherically for the next few degrees (Gatowski et al., 1984; Namazian et al., 1980; Tagalian and Heywood, 1986). During this period a negligible fraction of the mass is burned and the measured initial burning speeds are close to laminar flame speeds (Tagalian and Heywood, 1986). This stage is called the initial burning phase. After only a few degrees, the interaction of the flame and the turbulent flow field produces a highly wrinkled and convoluted outer surface of the flame (Gatowski et al., 1984; Tagalian and Heywood, 1986). During this period burning speeds are equal to turbulent flame speed. This phase is called the faster burning phase. Experimental studies also show that a specified amount of gas is still unburned after the termination of the flame propagation process. This last stage is called the final burning phase. The experimental observations are summarized above. For realistic predictions, the combustion process must be modeled considering the details of the turbulent flame propagation process. For this purpose, several modeling studies have been performed (Heywood, 1980; Blizard and Keck, 1974; Tabaczynski et al., 1977; Beratta et al., 1983; Tabaczynski et al., 1980). These models were originally based on the method developed by Blizard and Keck (1974). In the presented study, the method that was postulated originally by Blizard and Keck (1974) and extended later by Keck and coworkers (1982) is used. According to this theory, the turbulent eddies having characteristic radius lt are

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entrained into the flame zone at the entrainment velocity Ue and burned in a characteristic time τb . The combustion is described by the following equations: dme = ρu Af (Ut + Sl ), dθ

(16)

dmb (me − mb ) , = ρu Af Sl + dθ τb

(17)


where τb = lt /Sl is the characteristic reaction time to burn the mass of an eddy of size lt , me is the mass entrained by the flame front, Ut is the characteristic speed, and Sl is the laminar flame speed. In this study, flame surface area Af corresponding to enflamed volume Vf is determined from a geometric model. Sl is computed using the empirical correlations given by Bayraktar (1997) and Heywood (1988). Ut is determined empirically depending on the mean inlet gas speed Ui and the ratio of unburned gas density ρu to inlet gas density ρi (Keck, 1982). lt is calculated empirically depending on the maximum intake valve lift Liv and the density ratio ρi /ρu (Keck, 1982) as follows: (18) Ut = 0.08Ui ρu /ρi , lt = 0.8Liv (ρi /ρu )3/4 .

(19)

During the initial burning, it is apparent that Ut = 0. During the fully-developed flame propagation (faster burning), Equations (16) and (17) are used, and during the final burning phase (Af = 0) the burning rate is computed from m ˙b = e−(θ−θF )/τb , m ˙ bF

(20)

where F denotes the conditions at the end of the flame propagation.

Simulation Program A computer code has been developed for the presented mathematical cycle model. The inputs to begin the calculation are as follows: engine speed n; equivalence ratio ; distance of ignition point from edge of chamber (spark plug location) a; spark advance angle θs ; properties of fuel; intake valve geometry; ambient pressure; and temperature. After determining the intake conditions, the thermodynamic state of the cylinder charge is predicted by solving the arranged first-order differential equations for each process with appropriate crank angle increments. The Euler predictor-corrector technique is used to integrate the equations. At each step the variables in the equations are computed from submodels. At the start of combustion (at θs before Top Dead Center–TDC), the initial value of the burned gas temperature is determined as adiabatic flame temperature. The initial value for the mass fraction burned is computed from the cosine burn rate formula. During the ignition delay period, laminar burning is assumed. After burning of an eddy, combustion is computed as a fully-developed turbulent flame process. Throughout the combustion, the thermodynamic state of unburned and burned gas is determined separately. Once the total mass is burned the expansion process starts. During this process, the cylinder charge

105 90 95.25 63.5 82.6 82.6 83 78

Abraham et al. (1985) Bayraktar (1991) Benson and Baruah (1977) Blizard and Keck (1974) Heywood et al. (1979) McCuiston et al. (1977) Tabaczynski et al. (1980) Tomita and Hamamoto (1988)

95.25 120 69.24 76.2 114.3 114.3 74 85

H (mm) 8.56 variable 8.5 5 7 8 9.9 4.8

ε 158 240 136.5 127 254 254 122.1 130

Lc (mm) 7.7 6 4.5 4.83 6.5 6.1 5.3 6.3

Liv (mm)

2.5 11 52.5 21 30 0 41 39

a (mm)

−27 −10 −25 −30 −32 −15 −27 −20

θs (CA)

D = diameter, H = stroke, ε = compression ratio, Lc = connecting rode length, Liv = intake valve lift, a = spark plug location, θs = spark angle.

D (mm)

Author

Table 1 SI engine parameters


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consists of fully-burned combustion products. After the piston reaches the Bottom Dead Center (BDC), the exhaust stroke starts. Then exhaust temperature and pressure are computed. If the predicted value of the temperature is close to the previous value, the cycle analysis is terminated, otherwise cycle computation is repeated again.

Numerical Applications Downloaded By: [ANKOS 2007 ORDER Consortium] At: 14:56 21 February 2007

The computer program has been run for different engine geometries at various operating conditions to compare the predicted results to other experimental results presented by several researchers. Thus the reliability of the presented model has been checked. The basic geometric parameters of the engines used in calculations are given in Table 1. Various comparisons have been performed and some of them are given in the following section.

Comparison of the Predicted and Experimental Results In order to determine the validity of the proposed SI engine cycle model, predicted parameters for different engines at various operating conditions have been compared to experimental data given in the literature (Abraham et al., 1985; Bayraktar, 1991; Benson and Baruah, 1977; Blizard and Keck, 1974; Heywood et al., 1979; McCuiston et al., 1977; Tabaczynski et al., 1980; Tomita and Hamamoto, 1988). In this study, burned mass fraction Xb , burn duration or burning interval θb , cylinder pressure p, and performance parameters (effective power, specific fuel consumption, and effective efficiency) were chosen as comparison parameters. For the combustion process, experimental data for mass fraction burned given by Blizard and Keck (1974), Tabaczynski et al. (1980), and Tomita and Hamamoto (1988) are compared to the predicted results in Figures 1–3. As seen from the figures, the calculated

Figure 1. Comparison of predicted and experimental burned mass fractions.


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Figure 2. Comparison of predicted and measured burned mass fractions.

values of mass fraction burned are in good agreement with the measured results. Another comparison for combustion is performed for experimental and theoretical burn duration values in Figure 4. For different equivalence ratios, computed and measured burning intervals are in well agreement. Most general insights about the cycle can be obtained from the cylinder pressure variation. For this reason, predicted pressure values have been compared to the measured pressure data given by Heywood et al. (1979), Benson and Baruah (1977), and Abraham

Figure 3. Comparison of predicted and experimental burned mass fractions.

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Figure 4. Comparison of predicted and experimental burning intervals.

et al. (1985). It can be clearly seen that the model predictions reasonably agree with the experimental values (Figures 5–7). The last comparisons have been performed for performance parameters. In Figures 8– 10, calculated performance parameters from the measured data for engine geometry given by Bayraktar (1991) have been compared to predicted values. It can be seen that the proposed model has a reasonable accuracy for prediction of engine performance parameters.

Figure 5. Comparison of predicted and experimental cylinder pressures.


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Conclusions A quasi-dimensional mathematical SI engine cycle model has been developed. In this model, the First Law of Thermodynamics has been used to determine the first-order differential equations for pressure and temperature. The combustion process has been modeled as a turbulent flame propagation process. In this paper, basic details of the simulation are presented. To determine the validity of the presented model, computed


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Figure 8. Comparison of predicted and experimental effective powers at various compression ratios.

results were compared to the measured results for the different engine geometries given by several authors. In general, the presented model predicts the combustion parameters, cycle parameters, and performance parameters adequately. The predicted results are in good agreement with the experimental results. In the research studies on SI engines, this model can be used for examining the effects of fuel type, engine geometry, and operating parameters on combustion, cycle, and performance.

Figure 9. Comparison of calculated and experimental specific fuel consumptions at various compression ratios.


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Figure 10. Comparison of calculated and experimental effective efficiency at various compression ratios.

In the presented model, intake and exhaust processes are computed by using a simple method. Also, mechanical losses have been calculated empirically as a function of mean piston speed. Detailed modeling of these strokes and losses would improve the validity of the model. In the combustion process, the characteristic speed Ut and the characteristic length lt are determined empirically. If these scales are computed via detailed turbulence modeling, the combustion process can be computed more reliably. In the presented study, details of the model are presented and the practical utility and validity of this model for SI engine cycle analysis are proved. In future studies, the effects of various parameters on SI engine cycle and performance could be examined by using the presented mathematical model.

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