Proc. R. Soc. A-2008-Sezer-3107-28

December 2008, published 8, doi: 10.1098/rspa.2008.0190464 2008 Proc. R. Soc. A

Ismet Sezer and Atilla Bilgin second laws of thermodynamicsoperation via the combination of the first and Mathematical analysis of spark ignition engine

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Mathematical analysis of spark ignition engineoperation via the combination of the first and

second laws of thermodynamics

BY ISMET SEZER1,* AND ATILLA BILGIN

2

1Besikduzu Vocational School, and 2Mechanical Engineering Department,Karadeniz Technical University, 61080 Trabzon, Turkey

This study aims at the theoretical exergetic evaluation of spark ignition engine operation. Forthis purpose, a two-zone quasi-dimensional cycle model was installed, not considering thecomplex calculation of fluid motions. The cycle simulation consists of compression, combus-tion and expansion processes. The combustion phase is simulated as a turbulent flamepropagation process. Intake and exhaust processes are also computed by a simple approxima-tion method. The results of the model were compared with experimental data to demonstratethe validation of the model. Principles of the second law are applied to the model to performthe exergy (or availability) analysis. In the exergy analysis, the effects of various operationalparameters, i.e. fuel–air equivalence ratio, engine speed and spark timing on exergetic termshave been investigated. The results of exergy analysis show that variations of operationalparameters examined have considerably affected the exergy transfers, irreversibilities andefficiencies. 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 haveminimum values for the specified engine speed and optimum spark timing, while the firstand second law efficiencies reach a maximum at the same engine speed and optimumspark timing.

Keywords: spark ignition engine; quasi-dimensional model; availability;irreversibilities; exergy analysis

*A

RecAcc

1. Introduction

The cyclemodels of spark ignition (SI) engines are one of the most effective tools forthe analysis of engine performance, parametric examinations and assistance to newdevelopments. These models are mainly classified as thermodynamics-basedmodels 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 spatialresolutions and the cylinder composition is regarded as mean values. Geometricfeatures 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 ofcomputation. On the other hand, fluid dynamics-based models are multi-dimensional (MD) models that are a solution of the full conservation differential

Proc. R. Soc. A (2008) 464, 3107–3128

doi:10.1098/rspa.2008.0190

Published online 22 July 2008

uthor for correspondence ([email protected]).

eived 7 May 2008epted 27 June 2008 3107 This journal is q 2008 The Royal Society


Table 1. Nomenclature.

A availability or exergy, JAexh exergy transfer with exhaust gases, JAf area of flame front, m2

Af,ch fuel chemical exergy, JAQ exergy transfer with heat, JAtm thermomechanical exergy, JAW exergy transfer with work, JAtot total exergy, Jbmep brake mean effective pressure, barbsfc brake-specific fuel consumption, kg (kW h)K1

Cp specific heat at constant pressure, J kgK1 Kd iv intake valve diameter, mD cylinder diameter, mE energy, Jf mass residual gas fraction, dimensionlessh specific enthalpy, J kgK1

I irreversibilities, JlT characteristic length scale of turbulent flame, mLcr connecting rod length, mLiv maximum intake valve lift, mLs stroke length, mm mass, kgMBT maximum brake torquen engine speed, rpmp pressure, barR cylinder radius, mRs radius of spark plug location from cylinder axis, mSA spark advance, CADQLVH lower heating value of fuel, kJ kgK1

Qw total heat transfer to cylinder walls, JT absolute temperature, Kr compression ratio, dimensionlessrf flame radius, mS entropy, JU internal energy, Jv specific volume, m3 kgK1

Vc volume of combustion chamber, m3

Vf enflamed volume, m3

xb mass fraction burned, dimensionlessXsZR/Rs spark plug location from cylinder axis, dimensionless

Greek lettersFed charge-up efficiency, dimensionlessf fuel–air equivalence ratio, dimensionlesshI first law efficiency, %hII second law efficiency, %hv volumetric efficiency, %q crank angle, CADqs spark timing crank angle, CADr density, kg mK3

(Continued.)

I. Sezer and A. Bilgin3108

Proc. R. Soc. A (2008)



Table 1. (Continued.)

tb characteristic burning time of eddy at size of lT, su angular speed, sK1

subscripts0 reference or dead-state conditionsa intake conditionsb burnedcr connecting rodcomb combustiondest destructione entrainmentg gaskin kineticmax maximumpot potentialr exhaust conditionss sparktot totalT turbulentu unburnedw wall

3109Exergy analysis of SI engine operation


(Navier–Stokes) equations in both time and spatial dimensions. CFD models canpredict chamber geometry and fluid motion in detail. Many sub-models forturbulence, combustion and chemical reactions are required in CFD models.Moreover, a computational mesh and detailed boundary and initial conditions arealso needed for the solution. For these reasons, they are very slow and expensive interms of computation. Furthermore, CFD models suffer from some uncertaintiesduring numerical modelling due to the complexity of engine processes. Therefore,CFD models are mostly used to determine the turbulent fluid flow field andoptimum combustion chamber geometry in the case of no combustion conditions(Wengang 1995; Tallio 1998; Sezer 2008). QD models, on the other hand, arefundamentally thermodynamics-based models, as cited before, and they needempirical inputs for turbulent intensities and mass burn rate. QD models have theadvantages of both ZD andMDmodels due to some influence of chamber geometryand fluid motion, are fast in terms of computation time and are easy to use (Daiet al. 1996;Caton2001;Georgios 2005; Sezer 2008).For these reasons,QDmodels havebeen widely used for predicting the effects of flow parameters, mixture compositionand 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 researchand design studies, but most such simulation studies have been used only for thefirst law of thermodynamics. Furthermore, in the few past decades, it has beenclearly understood that the first law of thermodynamics is not capable of providinga suitable insight into engine operations. For this reason, the use of the second lawof thermodynamics has been intensified in internal combustion engines (Caton2000a; Rakopoulos & Giakoumis 2006). The analysis of a process or a system withthe second law of thermodynamics is termed availability or exergy analysis.





The application of exergy analysis in engineering systems is very useful becauseit provides quantitative information about irreversibilities and exergy losses in thesystem. In this way, the thermodynamic efficiency can be quantified and poorefficiency areas identified, so that systems can be designed and operated moreefficiently (Cengel & Boles 1994; Moran & Shapiro 2000; Rezac &Metghalchi 2004).

A series of papers was published on second law or exergy analysis appliedto internal combustion engines in the last few decades. A review study waspublished by Caton (2000a) and was extended by Rakopoulos & Giakoumis(2006). It can be seen from these review papers that numerous studies have beenperformed on the application of exergy analysis to SI engines (Shapiro & VanGerpen 1989; Gallo & Milanez 1992; Rakopoulos 1993; Alasfour 1997; Caton1999a, 2000b, 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 beengenerally simulated by using simple empirical equations (Shapiro & Van Gerpen1989; Gallo & Milanez 1992; Caton 1999a, 2000b), and studies performed withdetailed combustion models are limited (Rakopoulos 1993; Caton 2002).However, the combustion process is the most important stage during SI engineoperation and modelling of combustion in a realistic way is very important forexergetic computations. Therefore, this study has been devoted to the investiga-tion of SI engine operation from the second law perspective by using a QDtwo-zone cycle model, and the combustion period is simulated as a turbulententrainment process. The details of the cycle model and exergy analysis arepresented 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 ofFerguson (1985), has been used in this study. Equations of the original modelhave been rearranged for the model used here. The governing equations of thecycle model were derived from the first law of thermodynamics (the energyequation) by assuming that the cylinder content obeys the ideal gas law. Theenergy equation in crank angle basis is written as

DU ZQKW0mdu

dqCu

dm

dqZ

dQ

dqKp

dV

dq; ð2:1Þ

where m is the mass of cylinder content; u is the specific internal energy; Q is theheat transfer; p is the pressure; V is the volume; and q is the crank angle.Equation (2.1) shows the variations of the thermodynamic properties withrespect to the crank angle. Moreover, these thermodynamic properties are alsofunctions of temperature and pressure, which will be determined in the model.

To determine instantaneous cylinder volume, pressure, and burned andunburned gas temperatures, the following governing equations have been used:

V ðqÞZVc 1CrK1

21Kcos qC

1

rcr

h1Kð1Kr 2cr sin

2qÞ0:5i� ��

; ð2:2Þ





where rcr is half of the ratio of stroke length Ls to connecting rod length Lcr,rcrZLs/2Lcr.

dp

dqZ

ACBCC

DCE; ð2:3Þ

where

AZ1

m

dV

dq; BZ

hg

um

vbCpb

v lnvbv lnTb

1KTw

Tb

� �AbC

vuCpu

v lnvuv lnTu

1KTw

Tu

� �Au

� �;

CZKdxbdq

ðvbKvuÞCvbv lnvbv lnTb

hbKhu

CpbTb

� �; DZx b

v2bCpbTb

v lnvbv lnTb

� �2

Cvbp

v lnvbv lnp

� �

and

EZð1KxbÞv2u

CpuTu

v lnvuv lnTu

� �2

Cvup

v lnvuv lnp

� �:

dTb

dqZ

KhgAbðTbKTwÞumCpbxb

CvbCpb

v ln vbv ln Tb

dp

dqC

huK hbxbCpb

dxbdq

ð2:4Þ

and

dTu

dqZ

KhgAuðTuKTwÞumCpuð1K xbÞ

dp

dqC

vuCpu

v ln vuv ln Tu

: ð2:5Þ

The extra governing equations are also used to compute work output and heat loss,

dW

dqZ p

dV

dqð2:6Þ

and

dQw

dqZ

hg

u½AbðTbKTwÞCAuðTuKTwÞ�: ð2:7Þ

In the above equations, heat transfer coefficients hg and the cylinder walltemperature Tw are taken as 500 W mK2 KK1 and 400 K, respectively, assuggested by Ferguson (1985). Ab and Au are also the wetted areas by burnedand unburned gases, respectively, which are computed by a geometrical sub-modelgiven 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, namelyburned and unburned, exist in the combustion chamber. Each zone is assumed tobe uniform in temperature and homogeneous in composition. It is also assumedthat a uniform pressure distribution exists in the cylinder. The combustion processis simulated as turbulent flame propagation and it is supposed that the flame





front progresses spherically in the unburned gases. Under these assumptions, thefollowing 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:

dme

dqZ ruAfUe; ð2:8Þ

dmb

dqZ ruAfSLCmeKmb=tb ð2:9Þ

and

tb Z lT=SL; ð2:10Þwhere Af is the area of the flame front and tb is the characteristic burning time ofthe eddy of size lT.

The rate of mass burned is proportional to the flame front area and the flamespeed. The Af has been computed instantaneously depending on the enflamedvolume Vf with the help of the geometrical sub-model. Moreover, the followingequations were used for determining the turbulent entrainment speed Ue, theturbulent speed UT and the characteristic length scale of the turbulent flame lT:

Ue ZUTCSL; ð2:11Þ

UT Z 0:08 �U iðri=reÞ1=2; ð2:12Þ

�U i ZhvðApc=AivÞLsn

30ð2:13Þ

and

lT Z 0:8L ivðre=riÞ3=4; ð2:14Þ

where �U i is the speed of gases entering the cylinder during induction; Apc is thearea of the piston crown; Aiv is the maximum opening area of the intake valve;and L iv is the maximum intake valve lift.

Laminar flame speed SL is also calculated by the equations developed byGulder (1984),

SLðf;T ; pÞZSL;0½Tu=T0�a½p=p0�bð1Kjf Þ; ð2:15Þ

where SL,0 is the laminar flame speed at standard conditions; T0Z298 K; andp0Z1 bar and is calculated as

SL;0ðfÞZZWfh exp½KxðfK1:075Þ2�: ð2:16Þ

The values of a, b, Z, W, h and x are given in Gulder (1984) and j is selected as2.5 for 0%fR0.3.

(c ) Geometric sub-model

In the geometric sub-model, it is supposed that the spherical flame propagatesfrom the spark plug through the combustion chamber, as shown in figure 1.Under this assumption, the enflamed volume Vf, flame surface area Af and heat



spark plug cylinder axis

Figure 1. Flame front geometry.



transfer surface area Aw have been computed, depending on the instantaneousenflamed volume Vf and chamber height h, by using an iterative method. Thevalues of Vf and h have been determined from the thermodynamic cycle modelas follows:

VfðqÞZVbðqÞC ½meðqÞKmbðqÞ�=ru ð2:17Þ

and

hðqÞZLs½1KðrK1Þ�C0:5½1KcosðqÞ�: ð2:18Þ

The geometric features of the flame front in figure 1 have been determined forany flame radius from the following mathematical relations (Blizard & Keck1974; Keck 1982; Bilgin 2002):

AfðrfÞZ 2rf

ðzZy

zZ0aðzÞdz ð2:19Þ

and

VfðrfÞZ 2rf

ðzZy

zZ0½aðzÞr2ðzÞCbðzÞR2KXsR sin bðzÞ�dz; ð2:20Þ

where cos aZðX2s Cr2ðzÞKR2Þ=½2XsrðzÞ�; cos bZðX2

s Cr2ðzÞKR2Þ=½2XsrðzÞ�;and r2ðzÞZr 2f Kz2.

The total combustion chamber surface area wetted by the burned gases is thesum of the areas of the cylinder head, cylinder wall and piston crown wetted by





the burned gases as below,

AwðrfÞZAw;chðrfÞCAw;cwðrfÞCAw;pcðrfÞ; ð2:21Þ

Aw;chðrfÞZað0Þr2ð0ÞCbð0ÞR2KXsR sin bð0Þ; ð2:22Þ

Aw;cwðrfÞZ 2

ðzZy

zZ0RbðzÞdz ð2:23Þ

and

Aw;pcðrfÞZaðyÞr2ðyÞCbðyÞR2KXsR sin bðyÞ: ð2:24ÞThe 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 unburnedzones as follows:

Atot ZAb CAu: ð2:25Þ

(d ) Computation of the cycle

As known, SI engine cycles consist of four consecutive processes, namelyintake, compression, expansion and exhaust. Intake and exhaust processes arecomputed by using the approximation method given by Bayraktar & Durgun(2003). In this method, pressure loss during the intake process is calculated fromthe Bernoulli equation for one-dimensional uncompressible flow, and intakepressure and temperature are determined as

pa Z p0KDpa ð2:26Þ

and

Ta Z ðT0 CDT CgrTrÞ=ð1CgrTrÞ; ð2:27Þ

where p0 and T0 are the ambient pressure and temperature, respectively, and Dpais the pressure loss.

The volumetric efficiency is determined as

hv ZFed

r

rK1

pap0

T0

T0CDT CgrTr

; ð2:28Þ

where Fed is the charge-up efficiency; r is the compression ratio; and gr is themolar residual gas fraction.

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

Exhaust pressure pr and exhaust temperature Tr are also specified in terms ofambient pressure p0 and burned gas temperature Tb, respectively (Bayraktar &Durgun 2003),

pr Z ð1:05K1:25Þp0 ð2:29Þand

Tr ZTb=ðpb=prÞ1=3: ð2:30Þ





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):

DS Z

ððQ=TÞboundaryCs; ð2:31Þ

where s is the total entropy generated due to the internal irreversibilities.Considering the combination of the first and second laws of thermodynamics, theexergy or availability equation can be stated for a closed system (Cengel & Boles1994; Moran & Shapiro 2000),

AZECp0VKT0S; ð2:32Þwhere E is the total energy, which is the sum of the internal, kinetic and potentialenergies (EtotZUCEkinCEpot); V and S are the volume and entropy of thesystem, respectively; and p0 and T0 are the fixed pressure and temperature ofthe dead state, respectively.

Availability (or exergy) is defined as the maximum theoretical work that canbe obtained from a combined system (combination of a system and its referenceenvironment) when the system comes into equilibrium (as thermally,mechanically and chemically) with the environment (Cengel & Boles 1994;Caton 2000a; Moran & Shapiro 2000; Rezac & Metghalchi 2004; Rakopoulos &Giakoumis 2006). The maximum available work from a system emerges as thesum of two contributions: thermomechanical exergy Atm and chemical exergyAch. Thermomechanical exergy is defined as the maximum extractable work fromthe combined system as the system comes into thermal and mechanical equilibriawith the environment, and it is determined as (Van Gerpen & Shapiro 1990;Moran & Shapiro 2000; Rakopoulos & Giakoumis 2006)

Atm ZECp0VKT0SKX

m0;im i; ð2:33Þ

where m i and m0,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 andmechanical equilibria with the environment and no work potential exists betweenthe system and the environment due to temperature and pressure differences. Butthe system does not reach chemical equilibrium with the environment, as thecontents of the system are not permitted to mix with the environment or enterthe chemical reaction by environmental components (Van Gerpen & Shapiro1990). In principle, the difference between the compositions of the system at therestricted dead-state conditions and the environment can be used to obtainadditional work to reach chemical equilibrium. The maximum work obtained inthis way is called chemical exergy and is determined as (Van Gerpen & Shapiro1990; Moran & Shapiro 2000; Rakopoulos & Giakoumis 2006)

Ach ZX

m i m0;iKm0i

� �; ð2:34Þ





where m0i 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 1999a, 2000a,b)

DAZA2KA1 ZAQKAWKAdest; ð2:35Þwhere DA is the variation of the total system availability; A2 is the totalavailability at the end of the process; A1 is the total availability at the start ofthe process; AQ is the availability transfer due to heat transfer interactions; AW

is the availability transfer due to work interactions; and Adest is the destroyedavailability by irreversible processes.

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

dA

dqZ 1K

T0

T

� �dQ

dqK

dW

dqK p0

dV

dq

� �C

m f

m

dxbdq

a f;chKdIcomb

dq: ð2:36Þ

The left-hand side of equation (2.36) is the rate of change in the total exergy ofthe cylinder contents. The first and second terms on the right-hand siderepresent exergy transfers with heat and work, respectively. The third term onthe right-hand side corresponds to burned fuel exergy; here, m f and mtot are themasses of fuel and total cylinder contents and af,ch is the fuel chemical exergy.a f,ch is calculated by using the following equation that is developed for liquidfuels by Kotas (1995):

Af;ch Z a f;chm f ZQLHV 1:0401C0:01728h 0

c 0C0:0432

o0

c 0

�

C0:2196s 0

c 0 1K2:0628h 0

c 0

� ��; ð2:37Þ

where QLHV is the lower heating value of the fuel that is calculated by using theMendeleyev formula

QLHV Z ½33:91c0 C125:6h 0K10:89ðo0Ks 0ÞK2:51ð9h 0Kw 0Þ�: ð2:38Þ

The quantities h0, c0, o0, s0 and w0 in equations (2.37) and (2.38) represent themass fractions of the elements carbon, hydrogen, oxygen, sulphur and watercontent in the fuel, respectively.

The last term on the right-hand side of equation (2.36) illustrates exergydestruction in the cylinder due to combustion. It is calculated as

_Icomb ZT0_S comb; ð2:39Þ

where _S comb is the rate of entropy generation due to combustion irreversibilities.It is calculated from the two-zone combustion model depending on entropybalance as (Zhang 2002; Sezer 2008)

_S comb ZdðmbsbÞ

dqC

dðmusuÞdq

; ð2:40Þ

where mb and mu are the masses and sb and su are the specific entropy values ofthe burned and unburned gases of the cylinder contents, respectively.





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

_Itot Z _IcombC _IQ: ð2:41ÞEntropy production sourced from the heat transfer process is as in equation

(2.42), and exergy destruction due to heat transfer has already been determinedin equation (2.36),

_S Q Z_Q b

Tb

C_Q u

Tu

; ð2:42Þ

where _Q b and _Q u are the rates of heat loss from the burned and unburned gaszones at temperatures Tb and Tu, respectively.

The efficiency is defined to be able to compare different engine sizeapplications or evaluate various improvement effects from the perspective ofeither the first or second laws (Rakopoulos & Giakoumis 2006). The first law (orenergy-based) efficiency is defined as

hI Zenergy out ðas workÞ

energy inZ

W

m fQLHV

; ð2:43Þ

where W is the indicated work output.Various second law efficiencies (exergetic or availability efficiency, or

effectiveness) have been defined in the literature (Rakopoulos & Giakoumis2006). In this study, the following definition is used for the second law efficiency:

hII Zexergy out ðas workÞ

exergy inZ

AW

m fa f;ch

: ð2:44Þ

3. Numerical applications

(a ) Computer program and solution procedure

A computer code has been written for the presented SI engine cycle model. Inputparameters in the software were r, n, f, Xs, qs, properties of the fuel and ambientpressure and temperature. Once determining the intake conditions, thethermodynamic state of the cylinder charge is predicted by solving the governingdifferential equations. To integrate these differential equations, the DVERKsubroutine is used. The composition and thermodynamic properties of thecylinder content in the simulation are computed using the FORTRAN subroutinesfuel–air–residual gas and equilibrium-combustion-products, which wereoriginally developed by Ferguson (1985). Exergetic calculations are performedsimultaneously, depending on the thermodynamic state of the cylinder content.

Finally, the results obtained have been corrected using error analysis asfollows (Ferguson 1985; Sezer 2008):

31 Z 1K½vm=V � ð3:1Þand

32 Z 1C ½W=DðmuÞCQw�: ð3:2Þ



Table 2. Specifications of the engines.

specification r Xs D (mm) Ls (mm) Lcr (mm) d iv (mm) Liv (mm)

experimental;Rakopoulos (1993)

7.5 0.1 76.2 110 241.3 35 4.8

parametric;Ferguson (1985)

10 0 100 80 160 40 5

–120 –80 –40 0 40 80 120

crank angle (CAD)

0

10

20

30

40

50

pres

sure

(ba

r)

Rakopoulos (1993)

presented model

Figure 2. Comparisons of computed and experimental cylinder pressures. rcZ7; fZ1.0; fZ0.1;XsZ0.1; qsZK30 CAD; nZ2500 rpm; fuel, gasoline.



Selecting the values of 31 and 32 at the 10K4 level, the confidence of the analysisand the computer program is fulfilled.

(b ) Validation of the model

To demonstrate the reliability of the presented cycle model, the predictedvalues are compared with the experimental data in figure 2 for the conditionsspecified in the figure and the engine specifications given in table 2. Cylinderpressure is selected as the comparison parameter, and predictions for pressure arein 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 theanalysis of engine performance and parametric investigation.

4. Results and discussion

Figure 3 shows typical variations of the exergetic terms during the investigatedpart of the cycle for the conditions given in the figure. As shown in the figure, thethermomechanical exergy (Atm) increases gradually during the compression



–180 –150 –120 –90 –60 –30 0 30 60 90 120 150 180crank angle (CAD)

–500

0

500

1000

1500

2000

2500

3000

exer

getic

term

s (J

)

Figure 3. Typical variations of exergetic terms as a function of crank angle. fZ1.05; fZ0.1; rZ10;XsZ0; qsZK30 CAD; nZ3000 rpm; DqbZ64 CAD.



stroke up to the beginning of combustion, while fuel chemical exergy (Af,ch)remains constant. During compression, increase in the thermomechanical exergydirectly depends on the exergy transfer with work (AW) and AW shows asymmetrical variation by thermomechanical exergy with a negative sign. There isno remarkable variation in irreversibilities (I ) due to a negligible exergy transferwith heat (AQ) in this period. The variation in the total exergy (Atot) reflects thebehaviour of the thermomechanical exergy variation. With the start ofcombustion, at the crank angle of –30 CAD before top dead centre, the chemicalexergy of the fuel decreases rapidly due to conversion to heat. As a result of thisconversion, i.e. combustion of fuel, the temperature and pressure increase in thecylinder, which results in a steep increase in thermomechanical exergy and anincrease in exergy transfer from the cylinder contents to the walls by heat transfer.Both heat transfer and, especially, combustion give a rapid increase in theirreversibilities due to entropy generation. Combustion ends at the crank angle of34 CAD after top dead centre and the expansion process continues until the pistonreaches the bottom dead centre. During this part of the cycle, decreases in Atm andAtot 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 thecylinder at the end of the expansion emits with exhaust gases, which is calledexergy transfer with exhaust (Aexh). The distributions of the exergetic terms in thefuel exergy, i.e. AQ, AW, Aexh and I are approximately 8, 36.5, 36.9 and 18 percent, respectively, for the conditions given in figure 3.



–250

–200

–150

–100

–50

0

50(a) (b)

(c) (d )

(e) ( f )

exer

gy tr

ansf

er w

ith h

eat (

J)

1.0

1.1

–180 –120 –60 0 60 120 180crank angle (CAD)

–180 –120 –60 0 60 120 180crank angle (CAD)

–400

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Figure 4. Effects of equivalence ratio on exergetic terms. (a–f ) fZ0; rZ10; XsZ0; qsZMBT;nZ3000 rpm.



Figure 4 shows the effects of equivalence ratio (f) on exergetic terms duringthe periods of compression, combustion and expansion. As shown in the figure,the effects of f appear clearly after the start of combustion in AQ and AW, theirreversibilities and also the thermomechanical exergy. Af,ch and Atot variations,



Table 3. The values obtained from a parametric study for different conditions.

parameter m f (g) Qf ( J) Af,ch ( J) Tb,max (K) bmep (bar) Dqb (CAD)

f

0.9 4.36!10K2 1957.72 2112.32 2796.91 12.79 701.0 4.86!10K2 2180.55 2352.84 2875.05 13.86 651.1 5.38!10K2 2194.49 2604.59 2877.81 13.98 66

n (rpm)1500 4.86!10K2 2180.82 2353.06 2892.88 13.33 363000 4.86!10K2 2180.55 2352.84 2875.05 13.86 654500 4.87!10K2 2187.43 2359.99 2858.49 13.85 110

SA (CAD)15 4.85!10K2 2179.22 2351.41 2790.36 13.50 7730 4.86!10K2 2180.55 2352.84 2875.05 13.86 6545 4.80!10K2 2155.90 2331.53 2939.42 13.06 67



however, are affected from the beginning of the compression. The values of AQ infigure 4a have the maximum values for a stoichiometric mixture (fZ1.0), whilethey decrease to lower values for both the lean mixture (fZ0.9) and rich mixture(fZ1.1). This can be attributed to the temperature values given in table 3; thestoichiometric and rich mixtures give higher combustion temperatures incomparison with the lean mixture due to more fuel (m f) and fuel energy (Qf).The values of AQ for rich and lean mixtures are less than the value for astoichiometric mixture, approximately 6.9 and 9.2 per cent, respectively.However, the AW values in figure 4b increase with increasing equivalence ratio;thus the maximum values have been obtained with a rich mixture, but it is alsonoted that the rich mixture gives much closer AW values to those of astoichiometric mixture. These variations can be attributed to the bmep valuesgiven in table 3, which are also a direct result of the fixed maximum brake torquetiming. The values of AW for fZ0.9 and 1.0 are less than the value of fZ1.1,approximately 7.6 and 0.8 per cent, respectively. The irreversibilities in figure 4chave their maximum values for fZ1.1 and decrease to lower values withdecreasing equivalence ratios. The variations in the irreversibilities can beexplained by combustion efficiency. Decreases in the equivalence ratio beyondthat of a stoichiometric mixture result in the leaning of the mixture, andcombustion becomes more efficient due to the abundant oxygen in the fuel–airmixture, which results in a decrease in the combustion irreversibilities.Furthermore, increases in the irreversibilities with increasing equivalence ratiocan be explained by the composition and the amounts of combustion products, ascited by Caton (1999b). He stated that the destroyed value of availability, on aper mass of mixture basis, increased with increasing equivalence ratio. Thevalues of the irreversibilities for fZ0.9 and 1.0 are less than the value of fZ1.1,approximately 6.2 and 5.7 per cent, respectively. There are closer variations infigure 4d between the curves of Atm for fZ1.0 and 1.1, while a lean mixture givesconsiderably lower values, especially during expansion. These variations can beexplained by the fact that the increased fuel exergy contribution for the richmixture by excess fuel cannot be converted to work due to insufficient oxygen,



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Figure 5. Effects of engine speed on exergetic terms. (a–f ) fZ1.0; fZ0; rZ10; XsZ0; qsZMBT.



which presents incomplete combustion. Furthermore, the reduced cylinderpressures and temperatures come out for the lean mixture due to fuel deficiency.Thus, the stoichiometric mixture gives the maximum Atm values due to the lowerirreversibilities than the rich mixture. Enriching of the fuel–air mixture increasesAf,ch in figure 4e, but, as cited above, excess fuel, more than required, cannot be





converted efficiently to useful work and the remaining fuel is wasted withexhaust, as seen in the figure. The main contribution to variations in Atot givenin figure 4f is supplied by the fuel exergy during compression, while boththermomechanical and fuel exergies donate to it in the rest of the cycle. Aexh

increases by increasing the equivalence ratio so rich mixtures give the maximumexhausted exergy. The values of Aexh for fZ1.0 and 1.1 are higher than that offZ0.9, approximately 24.5 and 42.7 per cent, respectively.

Figure 5 shows the effects of engine speed (n) on exergetic terms during theexamined part of the cycle. As shown in figure 5a, AQ values decrease withincreasing 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 theengine runs slowly due to the enlarging of the cycle duration. The combustiontemperatures in table 3 have also affected the AQ values. The values of AQ for3000 rpm and 4500 rpm are less than the value of 1500 rpm, approximately 42.5and 58.7 per cent, respectively. However, AW in figure 5b has the maximumvalues for 3000 rpm during expansion, while 4500 rpm has the maximum AW

during compression. The variations in AW can be explained by the bmep valuesin table 3; the useful work and AW also increase as the bmep increases. Thevalues of AW for 1500 rpm and 4500 rpm are less than that of 3000 rpm,approximately 3 and 1 per cent, respectively. Irreversibilities in figure 5c alsoreach to the minimum for 3000 rpm. The values of the irreversibilities for1500 rpm and 4500 rpm are greater than the value for 3000 rpm, approximately3.5 and 0.3 per cent, respectively. The results for AW and the irreversibilitiesshow that the engine speed of 3000 rpm is an optimum speed among examinedspeeds. Atm in figure 5d is significantly affected by the variation of engine speed,especially during expansion. The increase in engine speed has not varied the peakvalues of Atm, but it causes an increase in Aexh at the end of expansion. The Af,ch

values in figure 5e do not vary with engine speed for the same amount of fuelsupplied to the cylinder, as shown in table 3. The steep variation takes place inAf,ch during combustion and the slopes of the lines decrease with increasingengine speed due to the variation of combustion duration. The variations in Atot

in figure 5f reveal the combination of thermomechanical and fuel exergies. TheAexh increases by increasing the engine speed due to the enlarging of thecombustion duration. The values of Aexh for 1500 rpm and 4500 rpm are greaterthan the value for 3000 rpm, approximately 35 and 47.3 per cent, respectively.

Figure 6 shows the effects of the spark timing (qs) on exergetic terms duringthe compression, combustion and expansion periods. AQ values in figure 6adecrease to lower values with the retarding of the spark timing, thus theminimum AQ values are obtained by K15 CAD. The advances in the beginningof combustion increase AQ due to the extension of the contact duration of hotgases with the cylinder walls. The combustion temperatures in table 3 are theother factor in the variations in AQ; increasing combustion temperature increasesheat transfer naturally. The values of AQ for K15 and K30 CAD are less thanthat of K45 CAD, approximately 3.2 and 6.96 per cent, respectively. In figure 6b,AW has the maximum values for K30 CAD during expansion, while increasingspark timing causes increases in the compression work. Thus, additionalcompression work is required during compression for the advanced spark timing,but the advanced spark timing cannot contribute to the exergy output as usefulwork during expansion. The values of AW for K15 and K45 CAD are less than



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Figure 6. Effects of spark timing on exergetic terms. (a–f ) fZ1.0; fZ0; rZ10; XsZ0; nZ3000 rpm.



the value for K30 CAD in magnitude, approximately 2.5 and 5.4 per cent,respectively. The irreversibilities in figure 6c also attain the minimum values forK30 CAD. The values of the irreversibilities for both K15 and K45 CAD aregreater by 0.6 per cent than that of K30 CAD. The variations indicate that thespark timing of K30 CAD is an optimal condition in the tested spark timings. Itis also noted that there is no significant effect of varying spark timing on the



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Figure 7. Effects of (a) equivalence ratio, (b) engine speed and (c) spark timing on the engineperformance parameters. Right-hatched bars, hI; left-hatched bars, hII; horizontal bars, bsfc.



irreversibilities. Therefore, ignition timing has to be optimized to give the leastAQ and to extract maximum exergy as useful work. When the spark timing isfixed earlier, Atm rises in advance, as shown in figure 6d. Conversely, if ignition isdelayed, Atm diminishes due to the decrease in pressure and temperature, asshown in table 3. There is almost no effect on the fuel exergy in figure 6e, butAf,ch varied during combustion due to variations in the combustion durationshown in table 3. Atot variations in figure 6f reflect the combination of Atm andAf,ch, and Aexh reach the minimum for K30 CAD at the end of the expansion.The values of Aexh for K15 and K45 CAD are greater than the value ofK30 CAD, approximately 4.5 and 4.7 per cent, respectively.

Figure 7 shows the variations of hI, hII and bsfc with equivalence ratio, enginespeed and spark advance, respectively. hI and hII decrease, while bsfc increaseswith increasing equivalence ratio, as shown in figure 7a. These variations aresuitable for the definitions of the efficiencies. The leaning of the fuel–air mixtureresults in increases in hI and hII and decreases in bsfc due to the improvement ofcombustion efficiency. However, the enriching of the charge has negative effectson the efficiencies and bsfc due to the increasing fuel supplied to the cylinder, asshown in table 3. The variations in the efficiencies are as cited by Ferguson(1985). The increments obtained with fZ0.9 and 1.0 are approximately 4.9and 3.6 per cent in hI and 4.6 and 3.4 per cent in hII, respectively, when comparedwith fZ1.1. The increments in the bsfc are approximately 4.7 per cent for





fZ1.0 and 21.6 per cent for fZ1.1 in comparison with fZ0.9. The variations ofperformance parameter with engine speed are given in figure 7b. hI and hIIhave the maximum values for 3000 rpm and the minimum bsfc obtained at thisengine speed. The decrements for 1500 rpm and 4500 rpm are approximately 1.7and 1.3 per cent in hI and 1.5 and 0.5 per cent in hII, when compared with3000 rpm. The increments in the bsfc are also approximately 1.4 per centfor 1500 rpm and 10.4 per cent for 4500 rpm, in comparison with 3000 rpm. Theeffects of spark timing on the engine performance parameters are shown infigure 7c. hI and hII have the maximum values for K30 CAD, and the minimumbsfc values are obtained at that spark timing. The decrements for K15 andK45 CAD are approximately 1 and 1.9 per cent in hI and 0.9 and 1.7 per cent in hII,respectively, when compared with K30 CAD. The increments in the bsfc are alsoapproximately 5 per cent forK15 CAD and 9 per cent forK45 CAD, in comparisonwithK30 CAD.

5. Conclusions

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

(i) A parametric exergetic analysis provides a better understanding ofinteractions between operating conditions, energy conversions andtransfer processes, which permits the revelation of the magnitudeof work potential lost during the cycle in a more realistic way than thefirst law analysis, and points to several possible ways for improvingengine performance.

(ii) The effects of equivalence ratio, engine speed and spark timing showdifferent trends in manner and magnitude. Exergy transfer with heat is amaximum for the stoichiometric mixture (fZ1.0), while irreversibilities ona per mass of mixture basis increase as the equivalence ratio increases. Theexergy transfer with exhaust and work also increases by increasingthe equivalence ratio due to increasing fuel exergy in the cylinder.However, the increment in exergy transfer with work for a rich mixture(fZ1.1) is very slight. The peak values of the first and second lawefficiencies were obtained with the lean mixture (fZ0.9) and thisequivalence ratio gives the minimum bsfc.

(iii) Exergy transfer with heat and exhaust increases by increasing the enginespeed. However, exergy transfer with work reaches a maximum at theengine speed of 3000 rpm and irreversibilities reach a minimum at thisengine speed. The maximum first and second law efficiencies and theminimum bsfc were obtained for 3000 rpm.

(iv) Exergy transfer with heat increases as the spark timing increases. However,exergy transfer with work reaches a maximum, while the irreversibilitiesand exhausted exergy have minimum values at an optimum spark timing ofK30 CAD. The best first and second law efficiencies and the minimum bsfcvalue were obtained at that spark timing.





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