## Abstract

This study aims at the theoretical exergetic evaluation of spark ignition engine operation. For this purpose, a two-zone quasi-dimensional cycle model was installed, not considering the complex calculation of fluid motions. The cycle simulation consists of compression, combustion and expansion processes. The combustion phase is simulated as a turbulent flame propagation process. Intake and exhaust processes are also computed by a simple approximation method. The results of the model were compared with experimental data to demonstrate the validation of the model. Principles of the second law are applied to the model to perform the exergy (or availability) analysis. In the exergy analysis, the effects of various operational parameters, i.e. fuel–air equivalence ratio, engine speed and spark timing on exergetic terms have been investigated. The results of exergy analysis show that variations of operational parameters examined have considerably affected the exergy transfers, irreversibilities and efficiencies. For instance, an increase in equivalence ratio causes an increase in irreversibilities, while it decreases the first and also the second law efficiencies. The irreversibilities have minimum values for the specified engine speed and optimum spark timing, while the first and second law efficiencies reach a maximum at the same engine speed and optimum spark timing.

## 1. Introduction

The cycle models of spark ignition (SI) engines are one of the most effective tools for the analysis of engine performance, parametric examinations and assistance to new developments. These models are mainly classified as thermodynamics-based models and fluid dynamics-based models, namely computational fluid dynamics (CFD) models. Thermodynamic-based models consist of zero- and quasi-dimensional (QD) models. Zero-dimensional (ZD) models do not have any spatial resolutions and the cylinder composition is regarded as mean values. Geometric features of the fluid motion cannot be predicted in these models due to the non-existence of fluid flow modelling. However, they are fast and cheap in terms of computation. On the other hand, fluid dynamics-based models are multidimensional (MD) models that are a solution of the full conservation differential (Navier–Stokes) equations in both time and spatial dimensions. CFD models can predict chamber geometry and fluid motion in detail. Many sub-models for turbulence, combustion and chemical reactions are required in CFD models. Moreover, a computational mesh and detailed boundary and initial conditions are also needed for the solution. For these reasons, they are very slow and expensive in terms of computation. Furthermore, CFD models suffer from some uncertainties during numerical modelling due to the complexity of engine processes. Therefore, CFD models are mostly used to determine the turbulent fluid flow field and optimum combustion chamber geometry in the case of no combustion conditions (Wengang 1995; Tallio 1998; Sezer 2008). QD models, on the other hand, are fundamentally thermodynamics-based models, as cited before, and they need empirical inputs for turbulent intensities and mass burn rate. QD models have the advantages of both ZD and MD models due to some influence of chamber geometry and fluid motion, are fast in terms of computation time and are easy to use (Dai *et al*. 1996; Caton 2001; Georgios 2005; Sezer 2008). For these reasons, QD models have been widely used for predicting the effects of flow parameters, mixture composition and combustion chamber geometry in SI engine combustion and heat transfer, and also overall engine performance and emissions (Agarwal *et al*. 1998; table 1).

Both ZD and QD models have been commonly used over the years for research and design studies, but most such simulation studies have been used only for the first law of thermodynamics. Furthermore, in the few past decades, it has been clearly understood that the first law of thermodynamics is not capable of providing a suitable insight into engine operations. For this reason, the use of the second law of thermodynamics has been intensified in internal combustion engines (Caton 2000*a*; Rakopoulos & Giakoumis 2006). The analysis of a process or a system with the second law of thermodynamics is termed availability or exergy analysis. The application of exergy analysis in engineering systems is very useful because it provides quantitative information about irreversibilities and exergy losses in the system. In this way, the thermodynamic efficiency can be quantified and poor efficiency areas identified, so that systems can be designed and operated more efficiently (Cengel & Boles 1994; Moran & Shapiro 2000; Rezac & Metghalchi 2004).

A series of papers was published on second law or exergy analysis applied to internal combustion engines in the last few decades. A review study was published by Caton (2000*a*) and was extended by Rakopoulos & Giakoumis (2006). It can be seen from these review papers that numerous studies have been performed on the application of exergy analysis to SI engines (Shapiro & Van Gerpen 1989; Gallo & Milanez 1992; Rakopoulos 1993; Alasfour 1997; Caton 1999*a*, 2000*b*, 2002; Kopac & Kokturk 2005) and compression ignition engines (Flynn *et al*. 1984; Alkidas 1988; Kumar *et al*. 1989; Van Gerpen & Shapiro 1990; Velasquez & Milanez 1994; Rakopoulos & Giakoumis 1997; Kanoglu *et al*. 2005; Parlak *et al*. 2005). In most SI engine studies, the combustion process has been generally simulated by using simple empirical equations (Shapiro & Van Gerpen 1989; Gallo & Milanez 1992; Caton 1999*a*, 2000*b*), and studies performed with detailed combustion models are limited (Rakopoulos 1993; Caton 2002). However, the combustion process is the most important stage during SI engine operation and modelling of combustion in a realistic way is very important for exergetic computations. Therefore, this study has been devoted to the investigation of SI engine operation from the second law perspective by using a QD two-zone cycle model, and the combustion period is simulated as a turbulent entrainment process. The details of the cycle model and exergy analysis are presented in the next sections.

## 2. Mathematical model

### (a) Governing equations of the cycle model

A QD thermodynamic cycle model, which is mainly based on the model of Ferguson (1985), has been used in this study. Equations of the original model have been rearranged for the model used here. The governing equations of the cycle model were derived from the first law of thermodynamics (the energy equation) by assuming that the cylinder content obeys the ideal gas law. The energy equation in crank angle basis is written as(2.1)where *m* is the mass of cylinder content; *u* is the specific internal energy; *Q* is the heat transfer; *p* is the pressure; *V* is the volume; and *θ* is the crank angle. Equation (2.1) shows the variations of the thermodynamic properties with respect to the crank angle. Moreover, these thermodynamic properties are also functions of temperature and pressure, which will be determined in the model.

To determine instantaneous cylinder volume, pressure, and burned and unburned gas temperatures, the following governing equations have been used:(2.2)where *r*_{cr} is half of the ratio of stroke length *L*_{s} to connecting rod length *L*_{cr}, *r*_{cr}=*L*_{s}/2*L*_{cr}.(2.3)where(2.4)and(2.5)The extra governing equations are also used to compute work output and heat loss,(2.6)and(2.7)In the above equations, heat transfer coefficients *h*_{g} and the cylinder wall temperature *T*_{w} are taken as 500 W m^{−2} K^{−1} and 400 K, respectively, as suggested by Ferguson (1985). *A*_{b} and *A*_{u} are also the wetted areas by burned and unburned gases, respectively, which are computed by a geometrical sub-model given below. Further details of the cycle model can be found in Ferguson (1985) and Sezer (2008).

### (b) Combustion model

After the initiation of combustion at a specified crank angle, two zones, namely burned and unburned, exist in the combustion chamber. Each zone is assumed to be uniform in temperature and homogeneous in composition. It is also assumed that a uniform pressure distribution exists in the cylinder. The combustion process is simulated as turbulent flame propagation and it is supposed that the flame front progresses spherically in the unburned gases. Under these assumptions, the following equations, first presented by Blizard & Keck (1974) and Keck (1982), developed by Tabaczynski *et al*. (1977, 1980) and also by Bayraktar & Durgun (2003), have been used for the determination of burnt mass fraction:(2.8)(2.9)and(2.10)where *A*_{f} is the area of the flame front and *τ*_{b} is the characteristic burning time of the eddy of size *l*_{T}.

The rate of mass burned is proportional to the flame front area and the flame speed. The *A*_{f} has been computed instantaneously depending on the enflamed volume *V*_{f} with the help of the geometrical sub-model. Moreover, the following equations were used for determining the turbulent entrainment speed *U*_{e}, the turbulent speed *U*_{T} and the characteristic length scale of the turbulent flame *l*_{T}:(2.11)(2.12)(2.13)and(2.14)where is the speed of gases entering the cylinder during induction; *A*_{pc} is the area of the piston crown; *A*_{iv} is the maximum opening area of the intake valve; and *L*_{iv} is the maximum intake valve lift.

Laminar flame speed *S*_{L} is also calculated by the equations developed by Gülder (1984),(2.15)where *S*_{L,0} is the laminar flame speed at standard conditions; *T*_{0}=298 K; and *p*_{0}=1 bar and is calculated as(2.16)The values of *α*, *β*, *Z*, *W*, *η* and *ξ* are given in Gülder (1984) and *ψ* is selected as 2.5 for 0≤*f*≥0.3.

### (c) Geometric sub-model

In the geometric sub-model, it is supposed that the spherical flame propagates from the spark plug through the combustion chamber, as shown in figure 1. Under this assumption, the enflamed volume *V*_{f}, flame surface area *A*_{f} and heat transfer surface area *A*_{w} have been computed, depending on the instantaneous enflamed volume *V*_{f} and chamber height *h*, by using an iterative method. The values of *V*_{f} and *h* have been determined from the thermodynamic cycle model as follows:(2.17)and(2.18)

The geometric features of the flame front in figure 1 have been determined for any flame radius from the following mathematical relations (Blizard & Keck 1974; Keck 1982; Bilgin 2002):(2.19)and(2.20)where ; ; and .

The total combustion chamber surface area wetted by the burned gases is the sum of the areas of the cylinder head, cylinder wall and piston crown wetted by the burned gases as below,(2.21)

(2.22)

(2.23)and(2.24)

The area wetted by the unburned gases is determined by assuming the total chamber area is the sum of the areas in contact with the burned and unburned zones as follows:(2.25)

### (d) Computation of the cycle

As known, SI engine cycles consist of four consecutive processes, namely intake, compression, expansion and exhaust. Intake and exhaust processes are computed by using the approximation method given by Bayraktar & Durgun (2003). In this method, pressure loss during the intake process is calculated from the Bernoulli equation for one-dimensional uncompressible flow, and intake pressure and temperature are determined as(2.26)and(2.27)where *p*_{0} and *T*_{0} are the ambient pressure and temperature, respectively, and Δ*p*_{a} is the pressure loss.

The volumetric efficiency is determined as(2.28)where *Φ*_{ed} is the charge-up efficiency; *r* is the compression ratio; and *γ*_{r} is the molar residual gas fraction.

Compression, combustion and expansion processes have been computed by arranging the governing equations (2.3)–(2.7) for each process in a suitable manner.

Exhaust pressure *p*_{r} and exhaust temperature *T*_{r} are also specified in terms of ambient pressure *p*_{0} and burned gas temperature *T*_{b}, respectively (Bayraktar & Durgun 2003),(2.29)and(2.30)

Once the computation of cycle is completed, engine performance parameters, i.e. brake mean effective pressure (bmep) and brake-specific fuel consumption (bsfc), can be determined by using well-known equations (Ferguson 1985; Bayraktar & Durgun 2003; Sezer 2008).

### (e) The second law (exergy) concept

The second law is analogous to the statement of entropy balance as follows (Cengel & Boles 1994; Moran & Shapiro 2000):(2.31)where *σ* is the total entropy generated due to the internal irreversibilities. Considering the combination of the first and second laws of thermodynamics, the exergy or availability equation can be stated for a closed system (Cengel & Boles 1994; Moran & Shapiro 2000),(2.32)where *E* is the total energy, which is the sum of the internal, kinetic and potential energies (*E*_{tot}=*U*+*E*_{kin}+*E*_{pot}); *V* and *S* are the volume and entropy of the system, respectively; and *p*_{0} and *T*_{0} are the fixed pressure and temperature of the dead state, respectively.

Availability (or exergy) is defined as the maximum theoretical work that can be obtained from a combined system (combination of a system and its reference environment) when the system comes into equilibrium (as thermally, mechanically and chemically) with the environment (Cengel & Boles 1994; Caton 2000*a*; Moran & Shapiro 2000; Rezac & Metghalchi 2004; Rakopoulos & Giakoumis 2006). The maximum available work from a system emerges as the sum of two contributions: thermomechanical exergy *A*_{tm} and chemical exergy *A*_{ch}. Thermomechanical exergy is defined as the maximum extractable work from the combined system as the system comes into thermal and mechanical equilibria with the environment, and it is determined as (Van Gerpen & Shapiro 1990; Moran & Shapiro 2000; Rakopoulos & Giakoumis 2006)(2.33)where *m*_{i} and *μ*_{0,i} are the mass and chemical potential of species ‘i’, respectively, calculated at restricted dead-state conditions.

At the restricted dead-state conditions, the system is in thermal and mechanical equilibria with the environment and no work potential exists between the system and the environment due to temperature and pressure differences. But the system does not reach chemical equilibrium with the environment, as the contents of the system are not permitted to mix with the environment or enter the chemical reaction by environmental components (Van Gerpen & Shapiro 1990). In principle, the difference between the compositions of the system at the restricted dead-state conditions and the environment can be used to obtain additional work to reach chemical equilibrium. The maximum work obtained in this way is called chemical exergy and is determined as (Van Gerpen & Shapiro 1990; Moran & Shapiro 2000; Rakopoulos & Giakoumis 2006)(2.34)where is the chemical potential of species ‘*i*’ calculated at true dead-state conditions.

The availability balance for a closed system for any process can also be written as (Caton 1999*a*, 2000*a*,*b*)(2.35)where Δ*A* is the variation of the total system availability; *A*_{2} is the total availability at the end of the process; *A*_{1} is the total availability at the start of the process; *A*_{Q} is the availability transfer due to heat transfer interactions; *A*_{W} is the availability transfer due to work interactions; and *A*_{dest} is the destroyed availability by irreversible processes.

Considering fuel chemical exergy, the exergy balance equation for the engine cylinder is written as (Zhang 2002; Sezer 2008)(2.36)

The left-hand side of equation (2.36) is the rate of change in the total exergy of the cylinder contents. The first and second terms on the right-hand side represent exergy transfers with heat and work, respectively. The third term on the right-hand side corresponds to burned fuel exergy; here, *m*_{f} and *m*_{tot} are the masses of fuel and total cylinder contents and *a*_{f,ch} is the fuel chemical exergy. *a*_{f,ch} is calculated by using the following equation that is developed for liquid fuels by Kotas (1995):(2.37)where *Q*_{LHV} is the lower heating value of the fuel that is calculated by using the Mendeleyev formula(2.38)The quantities *h*′, *c*′, *o*′, *s*′ and *w*′ in equations (2.37) and (2.38) represent the mass fractions of the elements carbon, hydrogen, oxygen, sulphur and water content in the fuel, respectively.

The last term on the right-hand side of equation (2.36) illustrates exergy destruction in the cylinder due to combustion. It is calculated as(2.39)where is the rate of entropy generation due to combustion irreversibilities. It is calculated from the two-zone combustion model depending on entropy balance as (Zhang 2002; Sezer 2008)(2.40)where *m*_{b} and *m*_{u} are the masses and *s*_{b} and *s*_{u} are the specific entropy values of the burned and unburned gases of the cylinder contents, respectively.

Moreover, the total exergy destruction considered here consists of combustion and heat transfer irreversibilities as follows (Sezer 2008):(2.41)

Entropy production sourced from the heat transfer process is as in equation (2.42), and exergy destruction due to heat transfer has already been determined in equation (2.36),(2.42)where and are the rates of heat loss from the burned and unburned gas zones at temperatures *T*_{b} and *T*_{u}, respectively.

The efficiency is defined to be able to compare different engine size applications or evaluate various improvement effects from the perspective of either the first or second laws (Rakopoulos & Giakoumis 2006). The first law (or energy-based) efficiency is defined as(2.43)where *W* is the indicated work output.

Various second law efficiencies (exergetic or availability efficiency, or effectiveness) have been defined in the literature (Rakopoulos & Giakoumis 2006). In this study, the following definition is used for the second law efficiency:(2.44)

## 3. Numerical applications

### (a) Computer program and solution procedure

A computer code has been written for the presented SI engine cycle model. Input parameters in the software were *r*, *n*, *ϕ*, *X*_{s}, *θ*_{s}, properties of the fuel and ambient pressure and temperature. Once determining the intake conditions, the thermodynamic state of the cylinder charge is predicted by solving the governing differential equations. To integrate these differential equations, the DVERK subroutine is used. The composition and thermodynamic properties of the cylinder content in the simulation are computed using the Fortran subroutines fuel–air–residual gas and equilibrium-combustion-products, which were originally developed by Ferguson (1985). Exergetic calculations are performed simultaneously, depending on the thermodynamic state of the cylinder content.

Finally, the results obtained have been corrected using error analysis as follows (Ferguson 1985; Sezer 2008):(3.1)and(3.2)Selecting the values of *ϵ*_{1} and *ϵ*_{2} at the 10^{−4} level, the confidence of the analysis and the computer program is fulfilled.

### (b) Validation of the model

To demonstrate the reliability of the presented cycle model, the predicted values are compared with the experimental data in figure 2 for the conditions specified in the figure and the engine specifications given in table 2. Cylinder pressure is selected as the comparison parameter, and predictions for pressure are in good agreement with the experimental data, as shown in the figure. Therefore, it can be said that the presented model has a sufficient level of confidence for the analysis of engine performance and parametric investigation.

## 4. Results and discussion

Figure 3 shows typical variations of the exergetic terms during the investigated part of the cycle for the conditions given in the figure. As shown in the figure, the thermomechanical exergy (*A*_{tm}) increases gradually during the compression stroke up to the beginning of combustion, while fuel chemical exergy (*A*_{f,ch}) remains constant. During compression, increase in the thermomechanical exergy directly depends on the exergy transfer with work (*A*_{W}) and *A*_{W} shows a symmetrical variation by thermomechanical exergy with a negative sign. There is no remarkable variation in irreversibilities (*I*) due to a negligible exergy transfer with heat (*A*_{Q}) in this period. The variation in the total exergy (*A*_{tot}) reflects the behaviour of the thermomechanical exergy variation. With the start of combustion, at the crank angle of –30 CAD before top dead centre, the chemical exergy of the fuel decreases rapidly due to conversion to heat. As a result of this conversion, i.e. combustion of fuel, the temperature and pressure increase in the cylinder, which results in a steep increase in thermomechanical exergy and an increase in exergy transfer from the cylinder contents to the walls by heat transfer. Both heat transfer and, especially, combustion give a rapid increase in the irreversibilities due to entropy generation. Combustion ends at the crank angle of 34 CAD after top dead centre and the expansion process continues until the piston reaches the bottom dead centre. During this part of the cycle, decreases in *A*_{tm} and *A*_{tot} continue due to the exergy transfers (both work and heat) from the system, while irreversibilities stand at almost a constant level. The remaining exergy in the cylinder at the end of the expansion emits with exhaust gases, which is called exergy transfer with exhaust (*A*_{exh}). The distributions of the exergetic terms in the fuel exergy, i.e. *A*_{Q}, *A*_{W}, *A*_{exh} and *I* are approximately 8, 36.5, 36.9 and 18 per cent, respectively, for the conditions given in figure 3.

Figure 4 shows the effects of equivalence ratio (*ϕ*) on exergetic terms during the periods of compression, combustion and expansion. As shown in the figure, the effects of *ϕ* appear clearly after the start of combustion in *A*_{Q} and *A*_{W}, the irreversibilities and also the thermomechanical exergy. *A*_{f,ch} and *A*_{tot} variations, however, are affected from the beginning of the compression. The values of *A*_{Q} in figure 4*a* have the maximum values for a stoichiometric mixture (*ϕ*=1.0), while they decrease to lower values for both the lean mixture (*ϕ*=0.9) and rich mixture (*ϕ*=1.1). This can be attributed to the temperature values given in table 3; the stoichiometric and rich mixtures give higher combustion temperatures in comparison with the lean mixture due to more fuel (*m*_{f}) and fuel energy (*Q*_{f}). The values of *A*_{Q} for rich and lean mixtures are less than the value for a stoichiometric mixture, approximately 6.9 and 9.2 per cent, respectively. However, the *A*_{W} values in figure 4*b* increase with increasing equivalence ratio; thus the maximum values have been obtained with a rich mixture, but it is also noted that the rich mixture gives much closer *A*_{W} values to those of a stoichiometric mixture. These variations can be attributed to the bmep values given in table 3, which are also a direct result of the fixed maximum brake torque timing. The values of *A*_{W} for *ϕ*=0.9 and 1.0 are less than the value of *ϕ*=1.1, approximately 7.6 and 0.8 per cent, respectively. The irreversibilities in figure 4*c* have their maximum values for *ϕ*=1.1 and decrease to lower values with decreasing equivalence ratios. The variations in the irreversibilities can be explained by combustion efficiency. Decreases in the equivalence ratio beyond that of a stoichiometric mixture result in the leaning of the mixture, and combustion becomes more efficient due to the abundant oxygen in the fuel–air mixture, which results in a decrease in the combustion irreversibilities. Furthermore, increases in the irreversibilities with increasing equivalence ratio can be explained by the composition and the amounts of combustion products, as cited by Caton (1999*b*). He stated that the destroyed value of availability, on a per mass of mixture basis, increased with increasing equivalence ratio. The values of the irreversibilities for *ϕ*=0.9 and 1.0 are less than the value of *ϕ*=1.1, approximately 6.2 and 5.7 per cent, respectively. There are closer variations in figure 4*d* between the curves of *A*_{tm} for *ϕ*=1.0 and 1.1, while a lean mixture gives considerably lower values, especially during expansion. These variations can be explained by the fact that the increased fuel exergy contribution for the rich mixture by excess fuel cannot be converted to work due to insufficient oxygen, which presents incomplete combustion. Furthermore, the reduced cylinder pressures and temperatures come out for the lean mixture due to fuel deficiency. Thus, the stoichiometric mixture gives the maximum *A*_{tm} values due to the lower irreversibilities than the rich mixture. Enriching of the fuel–air mixture increases *A*_{f,ch} in figure 4*e*, but, as cited above, excess fuel, more than required, cannot be converted efficiently to useful work and the remaining fuel is wasted with exhaust, as seen in the figure. The main contribution to variations in *A*_{tot} given in figure 4*f* is supplied by the fuel exergy during compression, while both thermomechanical and fuel exergies donate to it in the rest of the cycle. *A*_{exh} increases by increasing the equivalence ratio so rich mixtures give the maximum exhausted exergy. The values of *A*_{exh} for *ϕ*=1.0 and 1.1 are higher than that of *ϕ*=0.9, approximately 24.5 and 42.7 per cent, respectively.

Figure 5 shows the effects of engine speed (*n*) on exergetic terms during the examined part of the cycle. As shown in figure 5*a*, *A*_{Q} values decrease with increasing engine speed. This can be explained by the variation of cycle duration, such that the quantity of heat transferred to the cylinder wall increases as the engine runs slowly due to the enlarging of the cycle duration. The combustion temperatures in table 3 have also affected the *A*_{Q} values. The values of *A*_{Q} for 3000 rpm and 4500 rpm are less than the value of 1500 rpm, approximately 42.5 and 58.7 per cent, respectively. However, *A*_{W} in figure 5*b* has the maximum values for 3000 rpm during expansion, while 4500 rpm has the maximum *A*_{W} during compression. The variations in *A*_{W} can be explained by the bmep values in table 3; the useful work and *A*_{W} also increase as the bmep increases. The values of *A*_{W} for 1500 rpm and 4500 rpm are less than that of 3000 rpm, approximately 3 and 1 per cent, respectively. Irreversibilities in figure 5*c* also reach to the minimum for 3000 rpm. The values of the irreversibilities for 1500 rpm and 4500 rpm are greater than the value for 3000 rpm, approximately 3.5 and 0.3 per cent, respectively. The results for *A*_{W} and the irreversibilities show that the engine speed of 3000 rpm is an optimum speed among examined speeds. *A*_{tm} in figure 5*d* is significantly affected by the variation of engine speed, especially during expansion. The increase in engine speed has not varied the peak values of *A*_{tm}, but it causes an increase in *A*_{exh} at the end of expansion. The *A*_{f,ch} values in figure 5*e* do not vary with engine speed for the same amount of fuel supplied to the cylinder, as shown in table 3. The steep variation takes place in *A*_{f,ch} during combustion and the slopes of the lines decrease with increasing engine speed due to the variation of combustion duration. The variations in *A*_{tot} in figure 5*f* reveal the combination of thermomechanical and fuel exergies. The *A*_{exh} increases by increasing the engine speed due to the enlarging of the combustion duration. The values of *A*_{exh} for 1500 rpm and 4500 rpm are greater than the value for 3000 rpm, approximately 35 and 47.3 per cent, respectively.

Figure 6 shows the effects of the spark timing (*θ*_{s}) on exergetic terms during the compression, combustion and expansion periods. *A*_{Q} values in figure 6*a* decrease to lower values with the retarding of the spark timing, thus the minimum *A*_{Q} values are obtained by −15 CAD. The advances in the beginning of combustion increase *A*_{Q} due to the extension of the contact duration of hot gases with the cylinder walls. The combustion temperatures in table 3 are the other factor in the variations in *A*_{Q}; increasing combustion temperature increases heat transfer naturally. The values of *A*_{Q} for −15 and −30 CAD are less than that of −45 CAD, approximately 3.2 and 6.96 per cent, respectively. In figure 6*b*, *A*_{W} has the maximum values for −30 CAD during expansion, while increasing spark timing causes increases in the compression work. Thus, additional compression work is required during compression for the advanced spark timing, but the advanced spark timing cannot contribute to the exergy output as useful work during expansion. The values of *A*_{W} for −15 and −45 CAD are less than the value for −30 CAD in magnitude, approximately 2.5 and 5.4 per cent, respectively. The irreversibilities in figure 6*c* also attain the minimum values for −30 CAD. The values of the irreversibilities for both −15 and −45 CAD are greater by 0.6 per cent than that of −30 CAD. The variations indicate that the spark timing of −30 CAD is an optimal condition in the tested spark timings. It is also noted that there is no significant effect of varying spark timing on the irreversibilities. Therefore, ignition timing has to be optimized to give the least *A*_{Q} and to extract maximum exergy as useful work. When the spark timing is fixed earlier, *A*_{tm} rises in advance, as shown in figure 6*d*. Conversely, if ignition is delayed, *A*_{tm} diminishes due to the decrease in pressure and temperature, as shown in table 3. There is almost no effect on the fuel exergy in figure 6*e*, but *A*_{f,ch} varied during combustion due to variations in the combustion duration shown in table 3. *A*_{tot} variations in figure 6*f* reflect the combination of *A*_{tm} and *A*_{f,ch}, and *A*_{exh} reach the minimum for −30 CAD at the end of the expansion. The values of *A*_{exh} for −15 and −45 CAD are greater than the value of −30 CAD, approximately 4.5 and 4.7 per cent, respectively.

Figure 7 shows the variations of *η*_{I}, *η*_{II} and bsfc with equivalence ratio, engine speed and spark advance, respectively. *η*_{I} and *η*_{II} decrease, while bsfc increases with increasing equivalence ratio, as shown in figure 7*a*. These variations are suitable for the definitions of the efficiencies. The leaning of the fuel–air mixture results in increases in *η*_{I} and *η*_{II} and decreases in bsfc due to the improvement of combustion efficiency. However, the enriching of the charge has negative effects on the efficiencies and bsfc due to the increasing fuel supplied to the cylinder, as shown in table 3. The variations in the efficiencies are as cited by Ferguson (1985). The increments obtained with *ϕ*=0.9 and 1.0 are approximately 4.9 and 3.6 per cent in *η*_{I} and 4.6 and 3.4 per cent in *η*_{II}, respectively, when compared with *ϕ*=1.1. The increments in the bsfc are approximately 4.7 per cent for *ϕ*=1.0 and 21.6 per cent for *ϕ*=1.1 in comparison with *ϕ*=0.9. The variations of performance parameter with engine speed are given in figure 7*b*. *η*_{I} and *η*_{II} have the maximum values for 3000 rpm and the minimum bsfc obtained at this engine speed. The decrements for 1500 rpm and 4500 rpm are approximately 1.7 and 1.3 per cent in *η*_{I} and 1.5 and 0.5 per cent in *η*_{II}, when compared with 3000 rpm. The increments in the bsfc are also approximately 1.4 per cent for 1500 rpm and 10.4 per cent for 4500 rpm, in comparison with 3000 rpm. The effects of spark timing on the engine performance parameters are shown in figure 7*c*. *η*_{I} and *η*_{II} have the maximum values for −30 CAD, and the minimum bsfc values are obtained at that spark timing. The decrements for −15 and −45 CAD are approximately 1 and 1.9 per cent in *η*_{I} and 0.9 and 1.7 per cent in *η*_{II}, respectively, when compared with −30 CAD. The increments in the bsfc are also approximately 5 per cent for −15 CAD and 9 per cent for −45 CAD, in comparison with −30 CAD.

## 5. Conclusions

To evaluate SI engine operation from a second law perspective, a QD cycle model was used in this study. The effects of fuel–air equivalence ratio, engine speed and spark timing on the exergy transfers, irreversibilities and efficiencies have been investigated theoretically. The following general conclusions can be drawn.

A parametric exergetic analysis provides a better understanding of interactions between operating conditions, energy conversions and transfer processes, which permits the revelation of the magnitude of work potential lost during the cycle in a more realistic way than the first law analysis, and points to several possible ways for improving engine performance.

The effects of equivalence ratio, engine speed and spark timing show different trends in manner and magnitude. Exergy transfer with heat is a maximum for the stoichiometric mixture (

*ϕ*=1.0), while irreversibilities on a per mass of mixture basis increase as the equivalence ratio increases. The exergy transfer with exhaust and work also increases by increasing the equivalence ratio due to increasing fuel exergy in the cylinder. However, the increment in exergy transfer with work for a rich mixture (*ϕ*=1.1) is very slight. The peak values of the first and second law efficiencies were obtained with the lean mixture (*ϕ*=0.9) and this equivalence ratio gives the minimum bsfc.Exergy transfer with heat and exhaust increases by increasing the engine speed. However, exergy transfer with work reaches a maximum at the engine speed of 3000 rpm and irreversibilities reach a minimum at this engine speed. The maximum first and second law efficiencies and the minimum bsfc were obtained for 3000 rpm.

Exergy transfer with heat increases as the spark timing increases. However, exergy transfer with work reaches a maximum, while the irreversibilities and exhausted exergy have minimum values at an optimum spark timing of −30 CAD. The best first and second law efficiencies and the minimum bsfc value were obtained at that spark timing.

## Footnotes

- Received May 7, 2008.
- Accepted June 27, 2008.

- © 2008 The Royal Society