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Efficiency of a Miller engine

A. Al-Sarkhi a, *, J.O. Jaber a , S.D. Probert b

a Department of Mechanical Engineering, Hashemite University, Zarqa 13115, Jordanb

School of Engineering, Craneld University, Bedford MK43 0AL, UK Available online 9 June 2005

Abstract

Using nite-time thermodynamics, the relations between thermal efficiency, compressionand expansion ratios for an ideal naturally-aspirated (air-standard) Miller cycle have beenderived. The effect of the temperature-dependent specic heat of the working uid on theirreversible cycle performance is signicant. The conclusions of this investigation are of importance when considering the designs of actual Miller-engines.

2005 Elsevier Ltd. All rights reserved.

Keywords: Finite-time thermodynamics; Miller cycle; Heat resistance; Friction; Temperature-dependentspecic-heat

Introduction

The Miller cycle, named after its inventor R.H. Miller, has an expansion ratio

exceeding its compression ratio. The Miller cycle, shown in Fig. 1 , is a modern mod-ication of the Atkinson cycle (i.e., a complete expansion cycle). The compressionratios of spark-ignition, gasoline-fueled engines are limited by knock and fuel qualityto being in the range between 8 and 11, depending upon various factors, such as theengine s bore and stroke as well as engine speed.

Signicant achievements have ensued since nite-time thermodynamics was devel-oped in order to analyze and optimize the performances of real heat-engines [13].

0306-2619/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.apenergy.2005.04.003

* Corresponding author. Tel.: +962 5 3826600x4574; fax: +962 5 3826348.E-mail address: [email protected] (A. Al-Sarkhi).

Applied Energy 83 (2006) 343351

www.elsevier.com/locate/apenergy

APPLIEDENERGY
mailto:[email protected]:[email protected]


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Hoffman et al. [4] and Mozurkewich and Berry [5] used mathematical techniques,developed for optimal-control theory, to reveal the optimal motions of the pistonsin Diesel and Otto cycle engines, respectively. Aizenbud et al. [6] and Chen et al.[7] evaluated the performances of internal-combustion engine cycles using the opti-mal motion of a piston tted in a cylinder containing a gas pumped at a speciedheating-rate. Orlov and Berry [8] deduced the power and efficiency upper limitsfor internal-combustion engines. Angulo-Brown et al. [9], Chen et al. [10] and Wanget al. [11] modeled the behaviors of Otto, Diesel and Dual cycles, with friction losses,over a nite period. Klein [12] investigated the effects of heat transfer on the perfor-mances of Otto and Diesel cycles. Chen et al. [13,14] and Lin et al. [15] derived therelations between the net power and the efficiency for Diesel, Otto and Dual cycleswith due consideration of heat-transfer losses. Chen et al. [16,17] determined thecharacteristics of power and efficiency for Otto and Dual cycles with heat transferand friction losses. Chen et al. [18], Al-Sarkhi et al. [19] and Sahin et al. [20] studiedthe optimal power-density characteristics for Atkinson, Miller and Dual cycles with-out any such losses. Qin et al. [21] deduced the universal power and efficiency char-acteristics for irreversible reciprocating heat-engine cycles with heat transfer andfriction losses. Parlak et al. [22] optimized the performance of an irreversible Dualcycle: the predicted behavior was corroborated by experimental results. Fischer

and Hoffman [23] concluded that a quantitative simulation of an Otto-engine sbehavior can be accurately achieved by a simple Novikov model with heat leaks.Al-Sarkhi et al. [24] recently found that friction and the temperature-dependent spe-cic heat of the working uid of a Diesel engine had signicant inuences on itspower output and efficiency. This paper describes a corresponding analysis of thebehavior for an irreversible Miller-cycle with losses arising from heat resistanceand friction.

An air standard Miller-cycle model

As in Fig. 1 , the compression 1 ! 2 process is isentropic; the heat addition 2 ! 3,an isochoric process; the expansion 3 ! 4, an isentropic process; and the heat rejec-

3

1

2

4

5

v

P

S

T

2

1

3

4

5

q i n

q o u t

q o u t

6

Fig. 1. P V and T S diagram of a Miller cycle.

344 A. Al-Sarkhi et al. / Applied Energy 83 (2006) 343351


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tion 4 ! 5, an isochoric process, while the rejection of heat 5 ! 1, an isobaric process.Finally, the exhaust from 1 ! 6 is also an isobaric process. As is usual in nite-timethermodynamic heat-engine cycle models, there are two instantaneous adiabatic-pro-cesses namely 1 ! 2 and 3 ! 4 . For the heat addition (2 ! 3) and heat rejection(4 ! 5 ! 1) stages, respectively, it is assumed that heating occurs from state 2 to state3 and cooling ensues from state 4 to state 1: these processes proceed according to

dT d t

1C 1

for 2 ! 3; dT

d t

1C 2

for 4 ! 5; and dT

d t

1C 3

for 5 ! 1 1

where T is the absolute temperature and t is the time, C 1, C 2 and C 3 are constants.Integrating Eq. (1) yields

t 1 C 1T 3 T 2; t 2 C 2T 4 T 5; and t 3 C 2T 5 T 1 2

where t1 is the heating period and t2 and t3 the cooling periods. Then, the cycle per-iod is

s t 1 t 2 t 3 C 1T 3 T 2 C 2T 4 T 5 C 2T 5 T 1 3

In a real cycle, the specic heat of the working uid depends upon its temperatureand this will inuence the performance of the cycle. Over the temperature range gen-erally encountered for gases in heat engines (i.e., 3002200 K), the specic-heat curveis nearly linear, i.e., to close approximations

C p a k 1T 4

C v b k 1T 5where a, b and k 1 are constants: C p and C v are the molar specic heats with respect toconstant pressure and volume, respectively. Accordingly, the constant, R, of theworking uid is

R C p C v a b 6

The heat added to the working uid, during the process 2 ! 3, is

Q in M

Z

T 3

T 2

C v dT M

Z

T 3

T 2

b k 1T dT M bT 3 T 2 0.5k 1T 23 T 22

7

where M is the molar number of the working uid.The heat rejected by the working uid, during the process 4 ! 5, is

Qout 1 M Z T 4

T 51

C v dT M Z T 4

T 51

b k 1T dT M bT 4 T 5 0.5k 1T 24 T 25

8The heat rejected by the working uid, during the process 5 ! 1, is

Qout 2 M Z T 1

T 5C p dT M Z

T 1

T 5 a k 1T dT M aT 5 T 1 0.5k 1T

25 T

21

9

A. Al-Sarkhi et al. / Applied Energy 83 (2006) 343351 345


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Because C p and C v are dependent on temperature, the adiabatic exponent k = C p /C v will also vary with temperature. Therefore, the equation often used for a revers-ible adiabatic process, with constant k , cannot be used for a reversible adiabatic pro-cess with a variable k . However, a suitable engineering approximation for thereversible adiabatic process with a variable k can be assumed; i.e., the process canbe considered to occur in many innitesimally-small steps and for each of these,the adiabatic exponent k can be regarded as a constant. For example, any revers-ible-adiabatic process between states i and j can be regarded as consisting of a seriesof numerous innitesimally-small processes, for each of which a slightly different va-lue of k applies. For any of these processes, when innitesimally-small changes intemperature d T , and volume d V of the working uid ensue, the equation for thereversible adiabatic process with variable k can be written as follows:

TV k 1

T dT V dV k 1

10From Eq. (10), we get

k 1T j T i b lnT j=T i R lnV j=V i 11

The compression, rc, and expansion, re, ratios are dened as

r c V 1=V 2 and r e V 4=V 3 V 5=V 2 12

Therefore, equations describing processes 1 ! 2 and 3 ! 4 are, respectively, asfollows:

k 1T 2 T 1 b lnT 2=T 1 R ln r c 13k 1T 3 T 4 b lnT 3=T 4 R ln r e 14

For an ideal Miller-cycle model, there are no heat-transfer losses. However, for areal Miller-cycle, heat-transfer irreversibility between the working uid and the cyl-inder wall is not negligible. It is assumed that the heat loss through the cylinder wallis proportional to the average temperature of the working uid and the cylinder wall,and that, during the operation, the wall temperature remains approximately invari-ant. The heat added to the working uid by combustion is given by the followinglinear-relation [8,12,14,18,22] :

Q in M A BT 2 T 3 15where A and B are constants related to the combustion and heat-transfer processes.

Taking into account the friction loss of the piston, as deduced by Angulo-Brownet al. [9] and Chen et al. [16], and assuming a dissipation term resulting from the fric-tion force as being a linear function of the velocity, then

f l l v ld xd t

16

where l is the coefficient of friction and x is the piston displacement. Then, the lostpower is

P l dW l

d t l

d xd t

d xd t

lm 2 17



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The piston s mean-velocity is

v x5 x2

Dt 5! 2

x2r e 1

Dt 5! 2

18

where x2 is the piston s position corresponding to the minimum volume of thetrapped gases and D t5 ! 2 is the time spent in the power stroke. Thus, the power out-put P output W s P l can be written as

P output M bT 3 T 2 T 4 T 5 aT 5 T 1 0.5k 1T 23 T

21 T

22 T

24

C 1T 3 T 2 C 2T 4 T 5 C 3T 5 T 1

b1r e 12 19

where b1 l x22

D t 5! 22, and the efficiency of the cycle g th P output

Q in =s . Thus

g th

M bT 3 T 2 T 4 T 5 aT 5 T 1 0.5k 1T 23 T 21 T

22 T

24

b1r e 12C 1T 3 T 2 C 2T 4 T 5 C 3T 5 T 1( )

M bT 3 T 2 0.5k 1T 23 T 22

20When the values of rc, re and T 1 are given, T 2 can be obtained from Eq. (13); then,

substituting from Eq. (7) into Eq. (15) yields T 3, and T 4 can be found using Eq. (14).The last unknown is T 5, which can be deduced from the entropy change assuming anideal-gas: rst; the entropy change D S

3!

2 between states 2 and 3, is equal to the en-

tropy change D S 4 ! 1 between states 4 and 1. Thus

D S 3! 2 D S 4! 1 D S 4! 5 D S 5! 1 21

dS C vdT T

RdV V

or d S C pdT T

Rd P P

22

Processes 2 ! 3 and 4 ! 5 occur at constant volume and 5 ! 1 is a constant-pres-sure process. By substituting the specic heat from Eqs. (4) and (5) and integratingfrom the initial to the nal state of the process, then:

b ln T 3

T 2 ln T 4

T 5 a ln T 5

T 1 k 1 T 3 T 2 T 4 T 1 0 23

Substituting T 1, T 2, T 3 and T 4 into Eq. (23), we get T 5 and substituting T 1, T 2, T 3, T 4and T 5 into Eqs. (19) and (20) permits the efficiency and power to be estimated.Then, the relations between the power output and the compression ratio, as wellas between the thermal efficiency and the expansion ratio, of the Miller cycle, canbe derived.

Numerical example and discussion

The following constants and parameter values have been used in this exercise:A = 60,000 J/mol, B = 25 J/molK, b1 = 33 kW, M = 1.57 10

5 kmol, T 1 = 300 K,

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k 1 = 0.004 ! 0.008 J/mol K 2, b = 20 ! 23 J/mol K, a = 27.5 ! 30 J/mol K, re =6 ! 13, C 1 = 8.128 10 6 s/K and C 2 = 18.67 10 6 s/K, and C 3 = 10 10 6 s/K[4]. Cases were studied numerically for values of the expansion ratio ( r e) from6 to 13, for k 1 = 0.004, 0.006 or 0.008, for b = 20, 21 or 22, and for a = 27.5, 28,29 or 30.

Figs. 27 show the effects of the temperature-dependent specic-heat of the work-ing uid on the thermal efficiency of the cycle with heat resistance and irreversiblefriction-losses. The thermal efficiency versus compression-ratio characteristics areapproximately exponential-like curves. The efficiency versus expansion-ratio charac-teristics approximate to parabolic-like curves. They reect the performance charac-teristics of a real irreversible Miller-cycle engine.

Figs. 24 show the effects of k 1, a and b on the performance of the cycle at anexpansion ratio of 10. The thermal efficiency increases with increasing compressionratio, and increasing value of a, but decreases with increases of k 1 and b. The effect of changing k 1 is less than for b and even less than for a. This is due to the increase of (i)the heat rejected by the working uid and (ii) the heat added by the working uid.The magnitude of the thermal efficiency becomes much smaller when the parameterb increases (see Eqs. (7), (8) and (20 )). Eq. (20) shows that the parameter b is multi-plied by the highest temperature-difference in the cycle: this indicates that the effectof parameter b will be greater than the effects of the other parameters. Figs. 57 indi-cate the effects of the parameters a, b and k 1 on the efficiency of the cycle for differentvalues of the expansion ratio re. The efficiency increases with increasing expansionratio, reaches a maximum value and then decreases. The effect of the parameter bon the efficiency is the largest among all parameters for the same reason as men-tioned earlier. The efficiency decreases sharply even with only a slight increase of b: when b increases by about 1%, the maximum efficiency will decrease by about33%. This is due to the increase in the heat rejected by the working uid as a resultof increasing b (see Eq. (8)).

According to the above analysis, it can be concluded that the effects of the tem-perature-dependent specic heat of the working uid on the cycle performance aresignicant, and should be considered carefully in practical-cycle analysis and design.

a=28, b=20, r e =10

0

5

10

15

20

25

30

35

40

45

50

0 4 8r c

(

% )

10

=0.004

=0.006

=0.008

k 1

2 6

Fig. 2. Effect of k 1 on the variation of the efficiency with compression ratio.



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re =10, k 1=0.008, b=20

0

5

10

15

20

25

30

35

40

45

50

55

0 4 8

r c

( % )

10

a=28

a=29

a=30

2 6

Fig. 3. Effect of a on the variation of the efficiency with compression ratio.

k 1=0.008, r e = 10, a=28

0

5

1015

20

25

30

35

40

45

0 2 4 6 8

r c

( % )

10

b=20b=21

b=22

Fig. 4. Effect of b on the variation of the efficiency with compression ratio.

rc =8, a=28, b=20

20

25

30

35

40

45

50

2 6 10 12 14

r e

( % )

0.004

0.006

0.008

k 1

4 8

Fig. 5. Effect of k 1 on the variation of the efficiency with expansion ratio.



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Conclusion

An air-standard Miller-cycle model, assuming a temperature-dependent specicheat of the working uid as well as heat resistance and irreversible friction losses,has been investigated numerically. The performance characteristics of the cycle show

that there are signicant effects of the temperature-dependent specic heat of theworking uid. A slight increase in some parameters will have a signicant impacton the thermal efficiency of the studied cycle. The results obtained from this researchare compatible with those in the open literature, for other cycles, and may be used withassurance to provide guidance for the analysis of the behavior and design of practicalMiller-engines. Future studies should discuss the possible effects of fuel additives inorder to achieve a less temperature-dependent specic heat of the working uid.

References

[1] Chen L, Wu C, Sun F. Finite-time thermodynamic optimization or entropy-generation minimizationof energy systems. J Non-Equil Thermodynamics 1999;24(4):32759.

r c =8, b=20, k 1=0.00420

25

30

35

40

45

50

2 10 12 1

re

(

% )

4

a=27.5

a=28

a=28.5

4 6 8

Fig. 6. Effect of a on the variation of the efficiency with expansion ratio.

rc =8, k 1=0.004, a= 28

20

25

30

35

40

45

50

2 4 6 8 10 12 14re

(

% )

b=20

b=21

b=22

Fig. 7. Effect of b on the variation of the efficiency with expansion ratio.



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[2] Bejan A. Entropy-generation minimization: the new thermodynamics of nite-size devices and nite-time processes. J Appl Phys 1996;79(3):1191218.

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[12] Klein SA. An explanation for the observed compression-ratios in internal-combustion engines. Trans.ASME J Eng Gas Turbine Power 1991;113(4):5113.

[13] Chen L, Wu C, Sun F, Cao S. Heat-transfer effects on the net work-output and efficiencycharacteristics for an air standard Otto cycle. Energ Conversion Management 1998;39(7):6438.

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