effects of heat loss as percentage of fuel’s energy, friction

8/7/2019 Effects of heat loss as percentage of fuels energy, friction

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Effects of heat loss as percentage of fuels energy, frictionand variable specic heats of working uid on performance

of air standard Otto cycle

Jiann-Chang Lin a, *, Shuhn-Shyurng Hou b

a Department of General Education, Transworld Institute of Technology, Touliu City, Yunlin County 640, Taiwan, ROC b Department of Mechanical Engineering, Kun Shan University, Yung-Kang City, Tainan County 71003, Taiwan, ROC

Received 30 June 2006; received in revised form 28 January 2007; accepted 9 September 2007Available online 22 October 2007

Abstract

The objective of this study is to analyze the effects of heat loss characterized by a percentage of the fuels energy, friction and variablespecic heats of working uid on the performance of an air standard Otto cycle with a restriction of maximum cycle temperature. A morerealistic and precise relationship between the fuels chemical energy and the heat leakage that is based on a pair of inequalities is derivedthrough the resulting temperature. The variations in power output and thermal efficiency with compression ratio, and the relationsbetween the power output and the thermal efficiency of the cycle are presented. The results show that the power output as well asthe efficiency where maximum power output occurs will increase with increase of the maximum cycle temperature. The temperaturedependent specic heats of the working uid have a signicant inuence on the performance. The power output and the working rangeof the cycle increase with the increase of specic heats of the working uid, while the efficiency decreases with the increase of specic heatsof the working uid. The friction loss has a negative effect on the performance. Therefore, the power output and efficiency of the cycledecrease with increasing friction loss. It is noteworthy that the effects of heat loss characterized by a percentage of the fuels energy, fric-tion and variable specic heats of the working uid on the performance of an Otto cycle engine are signicant and should be consideredin practical cycle analysis. The results obtained in the present study are of importance to provide good guidance for performance eval-uation and improvement of practical Otto engines. 2007 Elsevier Ltd. All rights reserved.

Keywords: Otto cycle; Heat leakage; Friction; Irreversible; Variable specic heat

1. Introduction

Since there are some differences between real airfuel

cycles and ideal air standard cycles, the results from airstandard thermodynamic analysis will deviate from actualconditions. However, it is very interesting to note thatthe errors are not great, and the property values of temper-ature and pressure are very representative of actual enginevalues, depending on the geometry and operating condi-tions of the real engine [1]. Therefore, to make the analysisof the engine cycle much more manageable, air standard

cycles are used to describe the major processes occurringin internal combustion engines. Air is assumed to behaveas an ideal gas, and all processes are considered to be

reversible [1,2]. In practice, air standard analysis is usefulfor illustrating the thermodynamic aspects of an engineoperation cycle. Meanwhile, it can provide approximateestimates of trends as major engine operating variableschange. Good approximations of power output, thermalefficiency and mep (mean effective pressure) can beexpected.

For an ideal engine cycle, heat losses do not occur, how-ever, for a real engine cycle, heat losses indeed exist andshould not be neglected. It is recognized that heat lossstrongly affects the overall performance of the internal

0196-8904/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2007.09.002

* Corresponding author. Tel.: +886 5 5370988; fax: +886 5 5370989.E-mail address: [email protected] (J.-C. Lin).

www.elsevier.com/locate/enconman

Available online at www.sciencedirect.com

Energy Conversion and Management 49 (2008) 12181227
mailto:[email protected]:[email protected]


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combustion engine. If it is neglected, the analysis will justdepend on the ideal air standard cycle. Some attentionhas been paid to analyzing the effects of heat transfer losseson the performance of internal combustion engines [37].Klein [3] examined the effect of heat transfer through a cyl-inder wall on the work output of Otto and Diesel cycles.

Chen et al. [4,5] derived the relations between net poweroutput and the efficiency of Diesel and Otto cycles withconsiderations of heat loss through the cylinder wall.Hou [6] studied the effect of heat transfer through a cylin-der wall on the performance of the dual cycle.

In addition to heat loss, friction has a signicant effecton the performance, but it is omitted in ideal engine cycles.Taking into account the friction loss of the piston, Angulo-Brown et al. [7], Chen et al. [8] and Wang et al. [9] modeledOtto, Diesel and dual cycles with friction like loss, respec-tively. Furthermore, Chen et al. [10,11] and Ge et al. [12]derived the characteristics of power and efficiency for Otto,dual and Miller cycles with considerations of heat transferand friction like term losses. The above studies [312] weredone without considering the variable specic heats of theworking uid. However, in real engine cycles, the specicheat of the working uid is not a constant and should beconsidered in practical cycle analysis [1316].

In those studies [316], the heat addition process for anair standard cycle has been widely described as subtractionof an arbitrary heat loss parameter times the average tem-perature of the heat addition period from the fuels chem-ical energy. That is, the heat transfer to the cylinder walls isassumed to be a linear function of the difference betweenthe average gas and cylinder wall temperatures during the

energy release process. However, the heat leakage parame-

ter and the fuels energy depend on each other. Their validranges given in the literature affect the feasibility of airstandard cycles. If they are selected arbitrarily, they willpresent unrealistic results and make the air standard cyclesunfeasible [17]. For this reason, a more realistic and preciserelationship between the fuels chemical energy and the

heat leakage needs to be derived through the resulting tem-perature [17]. Thereby, the performance analysis of anyinternal combustion engine can be covered by a more real-istic and valid range of the heat loss parameter and thefuels energy.

However, Ozsoysals study [17] was only focused on thetemperature limitations and no performance analysis waspresented. Moreover, his study was done without consider-ing the effects of variable specic heats of the working uidand friction. In particular, no performance analysis isavailable in the literature with emphasis on the Otto cyclewith considerations of variable specic heats of the work-ing uid, friction and heat leakage characterized by a per-centage of the fuels energy. This study is aimed atanalyzing these effects (i.e. variable specic heats of work-ing uid, friction and heat loss characterized by a percent-age of the fuels energy) on the net work output and theindicated thermal efficiency of an air standard Otto cycle.In the present study, we relax the assumptions that thereare no heat losses during combustion, that there are no fric-tion losses of the piston for the cycle, and that specic heatsof the working uid are constant. In other words, heattransfer between the working uid and the environmentthrough the cylinder wall is considered and characterizedby a percentage of the fuels energy; friction loss of the pis-

ton in all the processes of the cycle on the performance is

Nomenclature

ap constant, dened in Eq. (4)bv constant, dened in Eq. (5)C pm molar specic heat at constant pressure

C vm molar specic heat at constant volumef l friction force, dened in Eq. (21)k specic heat ratio, k = C pm /C vmk 1 constant, dened in Eqs. (4) and (5)L stroke of Otto enginema mass of air per cyclemf mass of fuel per cycleN cycles per secondP net actual power output of cycle, dened in Eq.

(25)P R power output without friction losses, dened in

Eq. (20)P l lost power due to friction, dened in Eq. (22)Q fuel total energy of fuel per second input into engineQLHV lower heating value of fuelQ leak heat leakage per secondQ in heat input

Qout heat rejectedR gas constant of working uidT temperature

T 1, T 2, T 3, T 4 temperatures at state points 1, 2, 3, 4V volumev piston velocityv piston mean velocityx piston displacementx1 piston position corresponding to maximum vol-

ume of trapped gasesx2 piston position corresponding to minimum vol-

ume of trapped gases

Greek lettersa heat leakage percentagecc compression ratio, cc = V 1/V 2g efficiency of cyclek excess air coefficientl coefficient of friction

J.-C. Lin, S.-S. Hou / Energy Conversion and Management 49 (2008) 12181227 1219


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taken into account. Furthermore, we consider the variablespecic heats of the working uid that is signicant in prac-tical cycle analysis. The results obtained in the study mayoffer good guidance for design and operation of the Ottocycle engine.

2. Thermodynamic analysis

Fig. 1 shows the limitation of the maximum cycle tem-perature due to heat leakage in the temperature-entropydiagram of an air standard Otto cycle model. Thermody-namic cycle 1230401 denotes the air standard Ottocycle without heat leakage, while cycle 12341 desig-nates the air standard Otto cycle with heat leakage. Process12 is an isentropic compression from BDC (bottom deadcenter) to TDC (top dead center). The heat addition takesplace in process 23, which is isochoric. The isentropicexpansion process, 34, is the power or expansion stroke.The cycle is completed by an isochoric heat rejection pro-cess, 41. The heat added to the working uid per unitmass is due to combustion. The temperature at the comple-tion of the constant volume combustion ( T 3) depends onthe heat input due to combustion and the heat leakagethrough the cylinder wall. In this study, the amount of heatleakage is considered to be a percentage of the deliveredfuels energy [17]. The fuels energy then is the sum of theactual fuel energy transferred to the working uid andthe heat leakage through the cylinder walls. If any heatleakage occurs, the maximum cycle temperature ( T 3)remains less than that of the no heat leakage case ( T 3 0).When the total energy of the fuel is utilized, the maximum

cycle temperature reaches undesirably high levels withregard to structural integrity. Hence, engine designersintend to restrict the maximum cycle temperature. Assum-

ing that the heat engine is operated at the rate of N cyclesper second, the total energy of the fuel per second inputinto the engine can be given by

Q fuel Nmf QLHV ; 1

and then the heat leakage per second is

Q leak aQfuel aNmf QLHV ; 2where mf is the delivered fuel mass into the cylinder, QLHVis the lower heating value of the fuel and a is an unknownpercentage parameter having a value between 0 and 1.

Since the total energy of the delivered fuel Q fuel isassumed to be the sum of the heat added to the workinguid Q in and the heat leakage Q leak ,

Q in Qfuel Q leak 1 aNmf QLHV : 3

In practical internal combustion engine cycles, constantpressure and constant volume specic heats of the workinguid are variable, and these variations will greatly affect the

performance of the cycle. According to Ref. [13], it can beassumed that the specic heats of the working uid arefunctions of temperature alone and have the following lin-ear forms:

C pm ap k 1T 4

and

C vm bv k 1T 5

where C pm and C vm are, respectively, the specic heats withrespect to constant pressure and volume. ap , bv and k 1 areconstants. Accordingly, the gas constant ( R) of the working

uid can be expressed as

R C pm C vm ap bv 6

The temperature is restricted as the maximum temperaturein the cycle is T 3, and the available energy Q in during theheat addition per second can be written as

Q in Nma Z T 3

T 2

C vmdT Nma Z T 3

T 2bv k 1T dT

Nma bvT 3 T 2 0:5k 1 T 23 T 22 : 7

Combining Eqs. (3) and (7) yields

Nma bvT 3 T 2 0:5k 1 T 23 T 22 1 aNmf QLHV 8Dividing Eq. (8) by the amount of air mass ma , we have

a 1 kma=mf s

QLHVbvT 3 T 2 0:5k 1 T 23 T

22 9

or

T 2 bv ffiffiffiffiffiffiffiffiffib

2v 2k 1 0:5k 1T

23 bvT 3 1 a

QLHVkma =mf sh ir

k 1

10

3

4

3

4

Q in Q leak

Q leak

Q out

Q fuel

s

1

2

V=C

V=C

T

Fig. 1. T s diagram of an air standard Otto cycle model.

1220 J.-C. Lin, S.-S. Hou / Energy Conversion and Management 49 (2008) 12181227


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where k is the excess air coefficient dened as k = ( ma /mf )/(ma /mf )s, (ma /mf ) is the airfuel ratio and the subscripts a, f,and s, respectively, denote air, fuel and the stoichiometriccondition.

The rst condition for realizing a feasible cycle isT 2 6 T 3 (= T max ), so that

a 6 1 11The upper limit for the percentage of heat leakage is thenfound as amax = 1. The second condition, T 2 P T 1(= T min ),is utilized to determine the lower limit as follows

a P 1 kma=mf s

2QLHVk 1 T 23 T

21 2bvT 3 T 1 12

Hence, the minimum value of a is expressed as

amin 1 kma=mf s

2QLHVk 1 T 23 T

21 2bvT 3 T 1 13

The heat rejected per second by the working uid ( Qout )during process 4 ! 1 is

Qout Nma Z T 4

T 1

C vmdT Nma Z T 4

T 1bv k 1T dT

Nma bvT 4 T 1 0:5k 1 T 24 T 21 14

The adiabatic exponent k = C pm /C vm will vary with tem-perature since both C pm and C vm are dependent on temper-ature. Accordingly, the equation often used in reversibleadiabatic processes with constant k cannot be used inreversible adiabatic processes with variable k . Accordingto Ref. [13], however, a suitable engineering approximationfor reversible adiabatic processes with variable k can bemade, i.e. this process can be divided into innitesimallysmall processes, and for each of these processes, the adia-batic exponent k can be regarded as constant. For instance,for any reversible adiabatic process between states i and j ,we can regard the process as consist of numerous innites-imally small processes with constant k . For any of theseprocesses, when small changes in temperature d T and vol-ume dV of the working uid take place, the equation for areversible adiabatic process with variable k can be writtenas follows:

TV k 1 T dT V dV k 1: 15

Re-arranging Eqs. (4)(6) and (15), we get the followingequation

dT =T R=bv k 1T dV =V 0: 16

Integrating Eq. (16) from state i to state j , we obtain

k 1T j T i bv lnT j =T i R lnV j =V i: 17

The compression ratio ( cc) is dened as cc = V 1/V 2. There-fore, the equations for processes 1 ! 2 and 3 ! 4 areshown, respectively, by the following equations:

k 1T 2 T 1 bv lnT 2=T 1 R ln cc

18

and

k 1T 3 T 4 bv lnT 3=T 4 R ln cc 19

From Eqs. (7) and (14), the power output without frictionlosses is given by:

P R Qin Qout Nma bvT 3 T 1 T 2 T 4 0:5k 1 T 23 T 21 T 22 T 24 20

Every time the piston moves, friction acts to retard the mo-tion. Considering the friction effects on the piston in all theprocesses of the cycle, we assume a dissipation term repre-sented by a friction force ( f l ) that is linearly proportionalto the velocity of the piston [79], which can be writtenas follows:

f l l v ldxdt

21

where l is the coefficient of friction, which takes into ac-count the global losses on the power output, x is the pis-tons displacement and v is the pistons velocity.Therefore, the power lost due to friction is

P l f l v ldxdt

2

l v2 22

For a four stroke cycle engine, the total distance the pistontravels per cycle is

4L 4x1 x2 4x2x1=x2 1 4x2cc 1 23

where x1 and x2 are the pistons position corresponding tothe maximum and minimum volume, respectively, and L is

the stroke of the piston.Running at N cycles per second, the mean velocity of thepiston is

v 4LN 24

Therefore, the net actual power output of the Otto cycleengine can be written as

P P R jP l j

Nma bvT 3 T 1 T 2 T 4 0:5k 1 T 23 T 21 T

22 T

24

16l Nx2cc 12;

25

The efficiency of the Otto cycle engine is expressed by

g P

Q in nma bvT 3 T 1 T 2 T 4 0:5k 1

T 23 T 21 T

22 T

24 16l N x2cc 1

2o ma bvT 3 T 2 0:5k 1 T 23 T

22

1: 26

When T 1, T 3 and cc are given, T 2 can be obtained from Eq.(18) and T 4 can be found from Eq. (19). Finally, by substi-tuting T 1, T 2, T 3 and T 4 into Eqs. (25) and (26), respec-tively, the power output and the efficiency of the Otto

cycle engine can be obtained. Therefore, the relations



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between the power output, the efficiency and the compres-sion ratio can be derived.

3. Results and discussion

According to Refs. [13,16,18], the following constants

and ranges of parameters are used in the calculations:bv = 0.68580.8239 kJ/kg K, ma = 1.26 10 3 kg, T 1 =

300400 K, k 1 = 0.0001330.00034 kJ/kg K 2, x2 = 0.01 m,N = 30, QLHV = 44,000 kJ/kg and l = 0.01290.0169kN s/m. This study focuses on the limitation of the maxi-mum cycle temperature T 3 instead of T 3 0 due to the varyingheat leakage conditions. Numerical examples are shown asfollows.

Fig. 2 shows the variation of the heat leakage percentage(a) with respect to the maximum cycle temperature ( T 3)and the volumetric compression ratio ( cc). It is foundthat the maximum cycle temperature plays a dominant roleon the quantity of heat leakage. For a xed compressionratio, the larger maximum cycle temperature can beobtained as the heat leakage percentage parameter is smal-ler. For a xed cycle maximum temperature, the heatleakage percentage parameter increases with the increaseof compression ratio. The most important point here is thatsome values of the heat leakage percentage might beinsufficient for a feasible Otto cycle. Therefore, acceptablevalues for a could only be achieved from the denition of amin given by Eq. (13).

Fig. 3 illustrates the variation of the heat leakage per-centage (a) with respect to the airfuel ratio ( ma /mf ) orthe excess air coefficient (k) and the volumetric compres-

sion ratio ( cc). The results show that the excess air coeffi-

cient also plays an important role on the quantity of heatleakage. For a xed maximum cycle temperature(T 3 = 1900 K), the heat leakage percentage parameterincreases with the increase of compression ratio. For xedmaximum cycle temperature and compression ratio, theheat leakage percentage parameter decreases with increas-

ing excess air coefficient. Similar to Fig. 2, suitable valuesfor a could only be achieved from the denition of amingiven by Eq. (13).

Fig. 4 depicts the inuence of maximum cycle tempera-ture (T 3) on the cycle performance. The power output givenby Eq. (25) is a convex function with a single maximum forthe optimum compression ratio, as shown in Fig. 4a. Anincrease in compression ratio rst leads to an increase inpower output, and after reaching a peak, the net poweroutput decreases dramatically with further increase in com-pression ratio. As shown in Fig. 4b, the behavior of the effi-ciency versus compression ratio plot is similar to that forthe power output. Additionally, Fig. 4 illustrates thatincreasing T 3 corresponds to increasing the amount of heatadded to the engine due to combustion, and therefore, T 3has a positive effect on the P cc and g cc characteristiccurves. In other words, for a given cc , the power outputand efficiency increase with the increase of T 3, and the max-imum power output and its corresponding efficiencyincrease with increasing T 3. Furthermore, it is found thatthe values of cc at maximum power output or at maximumefficiency increase with increasing T 3.

It has been reported that for a real heat engine, the max-imum power and maximum efficiency operating points areusually relatively close [18]. This is reected through loop

shaped power versus efficiency plots. As is shown in

0 10 20 30 c

40

60

80

100

Heat Leakage Percentage (%)

(1700 K)

min (1700 K)

T3

(1800 K)

(1900 K)

min (1800 K)

min (1900 K)

T3

T1=350 Kbv=0.6858 kJ/kg-Kk 1=0.000202 kJ/kg-K 2

x2=0.01 m=0.0129 kN-s/m =1.2

Fig. 2. The variation of the heat leakage percentage ( a) with respect to themaximum cycle temperature ( T 3) and the volumetric compression ratio

(cc).

0 10 20 30

c

20

40

60

80

100

Heat Leakage Percentage (%)

(1.1)

min (1.1)

(1.2) (1.3)

min (1.2) min (1.3)

T1=350 KT3=1900 Kbv=0.6858 kJ/kg-Kk 1=0.000202 kJ/kg-K 2

x2=0.01 m=0.0129 kN-s/m

Fig. 3. The variation of the heat leakage percentage ( a) with respect to theairfuel ratio ( ma /mf ) or the excess air coefficient (k) and the volumetriccompression ratio ( cc).



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Fig. 5, we also obtain the loop shaped power output versusefficiency curves, which reect the performance characteris-tics of a real irreversible Otto cycle engine. It is depictedthat the maximum power output, the maximum efficiency,the power at maximum efficiency and the efficiency at max-imum power will increase with the increase of T 3.

Figs. 6 and 7 show the inuence of the parameter bvrelated to the variable specic heats of the working uidon the performance of the Otto cycle. For a xed k 1, a lar-ger bv corresponds to a greater value of the specic heatwith constant volume ( C vm ) or the specic heat with con-stant pressure ( C p m ). Fig. 6a demonstrates that for a givencc in a feasible range, the maximum power output of thecycle increase with the increase of bv, nevertheless,Fig. 6b shows that the maximum efficiency decreases withthe increase of bv. It is noted that the parameter bv hasan important inuence on the compression ratio wherethe maximum power or efficiency occurs. The values of ccat the maximum power output or at the maximum effi-ciency increase with the increase of bv, as shown inFig. 6. As can be found in Fig. 7, the curves of power out-

put versus efficiency are also loop shaped. It shows that,

0

10

20

30

P (kW)

0 10 20 30 40 c

0.0

0.2

0.4

0.6

a

b

1700

1800

1900

1700

1800

1900

T3 (K)

T3 (K)


x2=0.01 m=0.0129 kN-s/m =1.2

Fig. 4. (a) The inuence of maximum cycle temperature ( T 3) on thevariation of power output ( P ) with compression ratio ( c

c); (b) The

inuence of maximum cycle temperature ( T 3) on the variation of efficiency(g) with compression ratio ( cc).

0 0.2 0.4 0.6

0

8

16

24

P (kW)

1700

T3 (K)

1800

1900


x2=0.01 m =0.0129 kN-s/m =1.2

Fig. 5. The inuence of maximum cycle temperature ( T 3) on the poweroutput ( P ) versus efficiency (g) characteristic curves.

0

8

16

24

32

P (kW)

0 10 20 30 40 c

0.0

0.2

0.4

0.6

a

b

0.6858

0.75480.8239

bv (kJ/kg-K)

0.6858

0.75480.8239

bv (kJ/kg-K)

T1=350 KT3=1900 Kk 1=0.000202 kJ/kg-K 2

x2=0.01 m =0.0129 kN-s/m =1.2

Fig. 6. (a) The inuence of bv on the variation of power output ( P ) withcompression ratio ( cc); (b) The inuence of bv on the variation of efficiency

(g) with compression ratio ( cc).



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with the increase of bv, the maximum power output and thepower at maximum efficiency increase, while the maximumefficiency and the efficiency at maximum power outputdecrease.

Figs. 8 and 9 represent the inuence of the parameter k 1related to the variable specic heats of the working uid onthe performance of the Otto cycle. For a given bv, a largerk 1 corresponds to a greater value of the specic heats with

constant volume ( C vm ) or the specic heat with constantpressure (C pm ). Fig. 8 shows that k 1 has the same inuenceas bv (shown in Fig. 6) on the performance of the cycle.That is, for a given cc in a feasible range, the power outputof the cycle increase with increasing k 1, as shown in Fig. 8a,while the efficiency decreases with the increase of k 1, asdepicted in Fig. 8b. It is also found that the parameter k 1has a signicant inuence on the loop shaped curves forthe power output versus efficiency plots. With the increaseof k 1, the maximum power output and the power at maxi-mum efficiency increase, while the maximum efficiency andthe efficiency at maximum power output decrease, asshown in Fig. 9.

Figs. 10 and 11 show the inuence of the friction liketerm loss (l ) on the performance of the Otto cycle. It isclear that the parameter l has a negative effect on the per-formance. As is seen in Fig. 10, the maximum power out-put, the maximum efficiency and the value of thecompression ratio at maximum power output or at maxi-mum efficiency will decrease with increasing l . Moreover,Fig. 11 shows that the maximum power output, the maxi-mum efficiency, the power at maximum efficiency and theefficiency at maximum power will decrease with theincrease of l .

Figs. 12 and 13 depict the inuence of intake tempera-

ture (T 1) on the performance of the Otto cycle. As is seen

0 0.2 0.4 0.6

0

10

20

30

P (kW)

bv (kJ/kg-K)

0.8239

0.7548

0.6858

T1=350 KT3=1900 Kk 1=0.000202 kJ/kg-K 2

x2=0.01 m =0.0129 kN-s/m =1.2

Fig. 7. The inuence of bv on the power output ( P ) versus efficiency (g)characteristic curves.

0

12

24

36

P (kW

)

0 10 20 30 40 c

0.0

0.2

0.4

0.6

0.000133

0.0002020.000340

k 1 (kJ/kg-K 2)

0.000133

0.000202

0.000340k 1 (kJ/kg-K 2)

T1=350 KT3=1900 Kbv=0.6858 kJ/kg-Kx2=0.01 m =0.0129 kN-s/m =1.2

a

b

Fig. 8. (a) The inuence of k 1 on the variation of power output ( P ) withcompression ratio ( cc); (b) The inuence of k 1 on the variation of efficiency


0 0.2 0.4 0.6

0

8

16

24

P (kW)

k 1 (kJ/kg-K 2)

0.000340

0.000202

0.000133

T1=350 KT3=1900 Kbv=0.6858 kJ/kg-Kx2=0.01 m =0.0129 kN-s/m =1.2

Fig. 9. The inuence of k 1 on the power output ( P ) versus efficiency (g)characteristic curves.



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in Fig. 12, for a restricted maximum cycle temperatureT 3 = 1900 K, the maximum power output, the maximumefficiency, the compression ratio at maximum power outputand the compression ratio at maximum efficiency of theOtto cycle decrease with the increase of T 1. Loop shapedcurves of power versus efficiency plots are also shown inFig. 13. It is also found that as T 1 increases, the maximumpower output, the maximum efficiency, the efficiency atmaximum power output and the power output at maxi-mum efficiency decrease.

4. Conclusions

The effects of heat loss as a percentage of the fuelsenergy, friction and variable specic heats of the workinguid on the performance of an Otto engine under therestriction of maximum cycle temperature are presentedin this study. The results are summarized as follows.

(1) The maximum power output, the maximum effi-ciency, the power at maximum efficiency, the effi-

ciency at maximum power and the value of the

0

10

20

30

P (kW)

0 10 20 30 40 c

0.0

0.2

0.4

0.6

0.0169

0.0149

0.0129

0.0169

0.0149

0.0129

(kN-s/m)

(kN-s/m)


x2=0.01 m =1.2

a

b

Fig. 10. (a) The inuence of l on the variation of power output ( P ) withcompression ratio ( c

c); (b) The inuence of l on the variation of efficiency


0 0.2 0.4 0.6

0

8

16

24

P (kW)

(kN-s/m)

0.0169

0.0149

0.0129


x2=0.01 m =1.2

Fig. 11. The inuence of l on the power output ( P ) versus efficiency (g)

characteristic curves.

0

10

20

30

P (kW

)

0 10 20 30 40 c

0.0

0.2

0.4

0.6

400

350

300

400

350

300

T1 (K)

T1 (K)


x2=0.01 m=0.0129 kN-s/m =1.2

a

b

Fig. 12. (a) The inuence of intake temperature ( T 1) on the variation of power output ( P ) with compression ratio ( cc); (b) The inuence of intaketemperature ( T 1) on the variation of efficiency (g) with compression ratio(cc).



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compression ratio when the power output or the effi-ciency is maximum increase with the increase of max-imum cycle temperature T 3.

(2) The parameters bv and k 1 related to the variable spe-cic heats of the working uid have a signicantinuence on the performance of the Otto cycle. Fora xed k 1 (or bv), a larger bv (or k 1) corresponds to

a greater value of the specic heats with constant vol-ume (C vm ). For a given compression ratio cc in a fea-sible range, the power output of the cycle increasewith the increase of the parameter bv or k 1, neverthe-less, the efficiency decreases with the increase of bv ork 1. Furthermore, with the increase of bv, the maxi-mum power output and the power at maximum effi-ciency increase, while the maximum efficiency andthe efficiency at maximum power output decrease.

(3) The inuence of the friction like term loss l has anegative effect on the performance. Therefore, themaximum power output, the maximum efficiency,the power at maximum efficiency and the efficiencyat maximum power will decrease with the increaseof l .

(4) The maximum power output, the maximum effi-ciency, the compression ratio at maximum poweroutput and the compression ratio at maximum effi-ciency of the Otto cycle decrease with the increaseof intake temperature T 1. The efficiency at maximumpower output and the power output at maximum effi-ciency decrease with increasing T 1.

(5) It is noteworthy that the effects of heat loss as a per-centage of the fuels energy and friction loss on theperformance of an Otto cycle engine with consider-

ations of variable specic heats of working uid are

signicant and should be considered in practical cycleanalysis. The results obtained in the present study areof importance to provide good guidance for perfor-mance evaluation and improvement of practical Ottoengines.

In view of the analytical results from this work, werealize that the understanding and development of engines and engine cycles should be further explored byconsidering a more realistic model with advanced theo-retical and numerical techniques. For instance, in airstandard analysis, the constant volume heat input processreplaces the combustion of the real engine cycle, whichtakes place at close to constant volume conditions, andexhaust blowdown in a real engine is almost, but notquite, constant volume. As expected, the maximum tem-perature in the cycle will depend on the crank angle atwhich the exhaust valve opens. Hence, a new type of cycle analysis is needed. In other words, conceiving anew model as a function of crank angle to help under-stand, correlate, and analyze the relation between themaximum temperature and the crank angle at whichthe exhaust valve opens in the cycle. Additionally, con-sidering the combined effects of heat loss and frictionon the performance of engine cycles, detailed compari-sons between this work and numerical analysis (or exper-iments) are worthy of further study.

Acknowledgements

This work was supported by the National Science Coun-cil, Taiwan, ROC, under contract NSC95-2221-E-265-002.The authors would like to thank the reviewers and Dr.Denton for their valuable comments and helpfulsuggestions.

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0 0.2 0.4 0.6

0

10

20

30

P (kW) 400

T1 (K)

350

300


x2=0.01 m =0.0129 kN-s/m =1.2

Fig. 13. The inuence of intake temperature ( T 1) on the power output ( P )versus efficiency (g) characteristic curves.



10/10

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effects of heat loss as percentage of fuel’s energy, friction

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