FINITE TIME THERMODYNAMIC MODELING AND ANALYSIS … · FINITE TIME THERMODYNAMIC MODELING AND...

FINITE TIME THERMODYNAMIC MODELING AND ANALYSIS FOR AN IRREVERSIBLE ATKINSON CYCLE

by

Yanlin GE, Lingen CHEN *, and Fengrui SUN

Post grad u ate School, Na val Uni ver sity of En gi neer ing, Wuhan, P. R. China

Orig i nal sci en tific pa perUDC: 536.27:517.957

DOI: 10.2298/TSCI090128034G

Per for mance of an air-stan dard Atkinson cy cle is an a lyzed by us ing fi nite-timether mo dy nam ics. The ir re vers ible cy cle model which is more close to prac tice isfounded. In this model, the non-lin ear re la tion be tween the spe cific heats of work -ing fluid and its tem per a ture, the fric tion loss com puted ac cord ing to the mean ve -loc ity of the pis ton, the in ter nal ir re vers ibil ity de scribed by us ing the com pres sionand ex pan sion ef fi cien cies, and heat trans fer loss are con sid ered. The re la tions be -tween the power out put and the com pres sion ra tio, be tween the ther mal ef fi ciencyand the com pres sion ra tio, as well as the op ti mal re la tion be tween power out putand the ef fi ciency of the cy cle are de rived by de tailed nu mer i cal ex am ples. More -over, the ef fects of in ter nal ir re vers ibil ity, heat trans fer loss and fric tion loss on thecy cle per for mance are an a lyzed. The re sults ob tained in this pa per may pro videguide lines for the de sign of prac ti cal in ter nal com bus tion en gines.

Key words: finite-time thermodynamics, Atkinson cycles, heat resistance, friction, internal irreversibility, performance optimization

In tro duc tion

Fi nite time ther mo dy nam ics can an swer some global ques tions which clas si cal ther -mo dy nam ics does not try to an swer and con ven tional ir re vers ible ther mo dy nam ics can not an -swer be cause of its mi cro, dif fer en tial view point. Ex am ples of such ques tions are: (1) What isthe least en ergy re quired by a given ma chine to pro duce a given work in a given time? (2) Whatis the most work that can be pro duced by a given ma chine in given time, uti liz ing a given en -ergy? (3) What is the most ef fi cient way to run a given ther mo dy namic pro cess (op ti mal path) infi nite time? (4) What is the op ti mal time-de pend ent (on and off) pro cess? (5) What is the op ti mal dis tri bu tion be tween heat exchanger heat trans fer sur face ar eas or heat con duc tances cor re -spond ing to the op ti mal per for mance of the ther mo dy namic de vices for the fixed to tal heatexchanger heat trans fer sur face area or to tal heat con duc tance? (6) What are the quan ti ta tive and qual i ta tive fea tures of the ef fects of heat re sis tance, in ter nal ir re vers ibil ity and heat leak age onthe per for mance of real ther mo dy namic pro cesses and de vices? A se ries of achieve ments havebeen made since fi nite-time ther mo dy nam ics was used to an a lyze and op ti mize per for mance ofreal heat en gines [1-10]. Chen et al. [11] stud ied the ef fi ciency of an Atkinson cy cle at max i -

Ge, Y., et al.: Finite Time Thermodynamic Modeling and Analysis for an Irreversible ...

THERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp. 887-896 887

* Corresponding author; e-mails: [email protected], [email protected]

mum power den sity with out any loss. Qin et al. [12] and Ge et al. [13] de rived the per for mancechar ac ter is tics of Atkinson cy cle with heat trans fer loss [12] and with heat trans fer and fric -tion-like term losses [13], re spec tively. Ge et al. [14, 15] con sid ered the ef fect of vari able spe -cific heats on the cy cle pro cess and stud ied the per for mance char ac ter is tic of endoreversible and ir re vers ible Atkinson cy cles when vari able spe cific heats of work ing fluid are lin ear func tions of its tem per a ture and the max i mum tem per a ture of the cy cle is not fixed. Wang et al. [16] an a -lyzed and com pared the per for mance of an Atkinson cy cle cou pled to vari able-tem per a ture heatres er voirs un der max i mum power and max i mum power den sity con di tions. Zhao et al. [17] an a -lyzed the per for mance and op ti mized the para met ric cri te ria of an ir re vers ible Atkinson heat en -gine. Hou et al. [18] com pared the per for mance of air stan dard Atkinson and Otto cy cles withheat trans fer con sid er ations. Lin et al. [19] an a lyzed the in flu ence of heat loss, as char ac ter izedby a per cent age of fuel’s en ergy, fric tion, and vari able spe cific heats of work ing fluid on the per -for mance of an air-stan dard Atkinson cy cle when vari able spe cific heats of work ing fluid arelin ear func tions of its tem per a ture and the max i mum tem per a ture of the cy cle is not fixed.Al-Sarkhi et al. [20] out lined the ef fect of max i mum power den sity on the per for mance of theAtkinson cy cle ef fi ciency when the vari able spe cific heats of work ing fluid are lin ear func tionsof its tem per a ture. Abu-Nada et al. [21] and Al-Sarkhi et al. [22] ad vanced a non-lin ear re la tionbe tween the spe cific heats of work ing fluid and its tem per a ture and com pared the per for manceof the cy cle with con stant and vari able spe cific heats. Parlak et al. [23] de fined the in ter nal ir re -vers ibil ity by us ing en tropy pro duc tion, and an a lyzed the ef fect of the in ter nal ir re vers ibil ity onthe per for mance of ir re vers ible re cip ro cat ing heat en gine cy cles. Zhao et al. [24-26] de fined thein ter nal ir re vers ibil ity by us ing com pres sion and ex pan sion ef fi cien cies and an a lyzed the per -for mance of die sel, Otto, dual, and Miller cy cles when the max i mum tem per a ture of the cy cle isfixed and the ef fi ciency has a new def i ni tion. Zhao et al. [27, 28] used the model of spe cificheats ad vanced in refs. [14, 15], the in ter nal ir re vers ibil ity de fined in [24-26], and stud ied theop ti mum per for mance of Otto and die sel cy cles when the max i mum tem per a ture of the cy cles isfixed. Ge et al. [29-30] adopted the spe cific heat model ad vanced in refs. [21, 22], the in ter nal ir -re vers ibil ity de fined in refs. [24-28], and the fric tion loss de fined in ref. [31], and stud ied theper for mance of an ir re vers ible Otto, die sel, and dual cy cles when heat trans fer, fric tion, and in -ter nal ir re vers ibil ity losses are con sid ered. This pa per will adopt the spe cific heats model ad -vanced in refs. [21, 22, 29, 30], the in ter nal ir re vers ibil ity and ef fi ciency de fined in refs. [24-30]and the fric tion loss de fined in refs. [29-31], and study the per for mance of an ir re vers ibleAtkinson cy cle when heat trans fer, fric tion, and in ter nal ir re vers ibil ity losses and non-lin ear

vari able spe cific heats of the work ing fluid are con sid ered.

Cy cle model and anal y sis

An air stan dard Atkinson cy cle model is shown in fig. 1.Pro cess 1 ® 2S is a re vers ible adi a batic com pres sion, whilepro cess 1 ® 2 is an ir re vers ible adi a batic pro cess that takesinto ac count the in ter nal ir re vers ibil ity in the real com pres sion pro cess. The heat ad di tion is an isochoric pro cess 2 ® 3. Pro -cess 3 ® 4S is a re vers ible adi a batic ex pan sion, while 3 ® 4 is an ir re vers ible adi a batic pro cess that takes into ac count the in -ter nal ir re vers ibil ity in the real ex pan sion pro cess. The heatre jec tion is an iso baric pro cess 4 ® 1.


888 THERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp. 887-896

Fig ure 1. T-S di a gram for the cy cle model

In most cy cle model, the work ing fluid is as sumed to be have as an ideal gas with con -stant spe cific heats. But this as sump tion can be valid only for small tem per a ture dif fer ence. Forthe large tem per a ture dif fer ence en coun tered in prac ti cal cy cle, this as sump tion can not be ap -plied. Ac cord ing to ref. [21], for the tem per a ture range of 200-1000 K, the spe cific heat ca pac ity with con stant pres sure can be writ ten as:

Cp = (3.56839 – 6.788729·10–4T + 1.5537·10–6T2 – 3.29937·10–12T3 – 466.395·10–15T4)Rg (1)

For the tem per a ture range of 1000-6000 K, the equation is writ ten as:

C T Tp = + × - × +

+ ×

- -( . . .

.

308793 12 4597 10 0 42372 10

67 4775

4 6 2

10 397077 1012 3 15 4- -- ×T T R. ) g (2)

Equa tions (1) and (2) can be ap plied to a tem per a ture range of 200-6000 K which is too wide for the tem per a ture range (300-3500 K) of prac ti cal en gine. So a sin gle equa tion was usedto de scribe the spe cific heat model which is based on the as sump tion that air is an ideal gas mix -ture con tain ing 78.1% ni tro gen, 20.95% ox y gen, 0.92% ar gon, and 0.03% car bon di ox ide.

C T T Tp = × + × - × + ×- - -2506 10 1454 10 4246 10 316211 2 7 1 5 7. . . .. 10

13303 1512 10 3063 10 2212

5 0 5

4 1 5 5 2

-

- -

+

+ - × + × -

T

T T

.

.. . . . × -107 3T (3)

Ac cord ing to the re la tion be tween spe cific heat with con stant pres sure and spe cificheat with con stant vol ume:

Cv = Cp – Rg (4)

the spe cific heat with con stant vol ume can be writ ten as:

C C R T T Tv p g= - = × + × - × +- - -2506 10 1454 10 4246 1011 2 7 1 5 7. . .. 3162 10

10433 1512 10 3063 10

5 0 5

4 1 5 5 2

.

. . .

.

.

× +

+ - × + ×

-

- -

T

T T - × -2212 107 3. T (5)

where Rg = 0.287 kJ/kgK is the gas con stant of the work ing fluid. The unit of Cv and Cp is[kJkg–1K–1].

The heat added to the work ing fluid dur ing pro cess 2 ® 3 is:

Q M C T M T TT

T

in vd == ò × + × -- -

2

3

2506 10 1454 10 42411 2 7 1 5( . . .. 6 10

3162 10 10433 1512 10

7

5 0 5 4 1 52

3

× +ò

+ × + - ×

-

- -

T

T TT

T

. . .. . + × - × =

= × +

- -

-

3063 10 2212 10

8353 10 58

5 2 7 3

12 3

. . )

[ . .

T T T

M T

d

16 10 2123 10 2108 10

10433 3

8 2 5 7 2 5 1 5× - × + × +

+ +

- - -T T T

T

. .. .

. . . . ].024 10 3063 10 1106 104 0 5 5 1 7 22

3× - × + ×- - -T T TTT

(6)

The heat re jected by the work ing fluid dur ing pro cess is:

Q M C T M T TT

T

out pd == ò × + × -- -

1

4

2506 10 1454 10 4211 2 7 1 5( . . .. 46 10 4246 10

31162 10 13303 1

7 7

5 0 51

4

× - × +ò

+ × + -

- -

-

.

. ..

T

TT

T

. . . )

[ .

.512 10 3063 10 2212 10

835

4 1 5 5 2 7 3× + × - × =

=

- - -T T T T

M

d

3 10 5816 10 2123 10 2108 1012 3 8 2 5 7 2 5 1× + × - × + ×- - - -T T T T. . .. .

.

.

. . .

5

4 0 5 5 1 7

13303

3024 10 3063 10 1106 10

+ +

+ × - × + ×- -

T

T T T -21

4]TT (7)



where M is the mass flow rate of the work ing fluid, T1, T2, T3, and T4 [K] are the tem per a tures atstates 1, 2, 3, and 4.

For the two adi a batic pro cesses 1 ® 2 and 3 ® 4, the com pres sion and ex pan sion ef fi -cien cies can be de fined as [24-30]:

hcS=

-

-

T T

T T2 1

2 1

(8)

hcS

=-

-

T T

T T4 3

4 3

(9)

These two ef fi cien cies can be used to de scribe the in ter nal ir re vers ibil ity of the pro -cesses.

Since Cp and Cv are de pend ent on tem per a ture, adi a batic ex po nent k = Cp/Cv will varywith tem per a ture as well. There fore, the equa tion of ten used in re vers ible adi a batic pro cess withcon stant k can not be used in re vers ible adi a batic pro cess with vari able k. How ever, ac cord ing to refs. [14, 15, 29-40], a suit able en gi neer ing ap prox i ma tion for re vers ible adi a batic pro cess withvari able k can be made, i. e. this pro cess can be bro ken up into a large num ber of in fin i tes i mallysmall pro cesses and for each of these pro cesses, adi a batic ex po nent k can be re garded as a con -stant. For ex am ple, for any re vers ible adi a batic pro cess be tween states i and j can be re garded ascon sist ing of nu mer ous in fin i tes i mally small pro cesses with con stant k. For any of these pro -cesses, when an in fin i tes i mally small change in tem per a ture dT, and vol ume dV of the work ingfluid takes place, the equa tion for re vers ible adi a batic pro cess with vari able k can be writ ten asfol lows.

TV T T V Vk k- -= + +1 1( )( )d d (10)

For an isochoric heat ad di tion pro cess i ® j, the heat added is Qin = Cv(Tj – Ti) = TDSi®j = = TCvln(Tj/Ti). So one has T = (Tj – Ti)/ln(Tj/Ti), where T is the equiv a lent tem per a ture of heat ab -sorp tion pro cess. When Cv is the func tion of tem per a ture, the Cv(T) can be re garded as mean spe -cific heat with con stant vol ume.

From eq. (10), one gets

CT

TR

V

Vv gln ln

j

i

i

j

(11)

where the tem per a ture in the equa tion of Cv is T = (Tj – Ti)/ln(Tj/Ti).The com pres sion ra tio is de fined as:

g =V

V1

2

(12)

There fore, equa tions for re vers ible adi a batic pro cesses 1 ® 2S and 3 ® 4S are:

CT

TRS

v gln ln2

1

= g (13)

CT

TR

T

TRgv

S

Sgln ln ln4

3

1

4

- = - g (14)

For an ideal Atkinson cy cle model, there are no heat trans fer losses. How ever, for areal Atkinson cy cle, heat trans fer ir re vers ibil ity be tween work ing fluid and the cyl in der wall is



not neg li gi ble. One can as sume that the heat trans fer loss through the cyl in der wall (i. e. the heatleak age loss) is pro por tional to av er age tem per a ture of both the work ing fluid and the cyl in derwall and that the wall tem per a ture is con stant, T0 [K]. If the re leased heat by com bus tion per sec -ond is A1 [kW] the heat leak age co ef fi cient of the cyl in der wall is B1 [kJkg–1K–1] which has con -sid ered the heat trans fer co ef fi cient and the heat ex change sur face, one has the heat added to thework ing fluid per sec ond by com bus tion in the fol low ing lin ear re la tion [12-15]:

Q A MBT T

Tin = -

+

-1 1

2 3

02(15)

From eq. (15), one can see that Qin con tained two parts, the first part is A1, the re leasedheat by com bus tion per sec ond, and the sec ond part is the heat leak loss per sec ond, it can bewrit ten as:

Qleak = MB(T2 + T3 – 2T0) (16)

where B = B1/2.Tak ing into ac count the fric tion loss of the pis ton as rec om mended by Chen et al. [31]

for the Dual cy cle and as sum ing a dis si pa tion term rep re sented by a fric tion force which in aliner func tion of the ve loc ity gives

f vx

tm m m= =

d

d(17)

where m [Nsm–1] is a co ef fi cient of fric tion which takes into ac count the global losses and x is the pis ton dis place ment. Then, the lost power is:

PW

t

x

t

x

tvm

mm m= = =

d

d

d

d

d

d2 (18)

If one spec i fies the en gine is a four stroke cy cle en gine, the to tal dis tance the pis tontrav els per cy cle is:

4L = 4(x1 – x2) (19)

For a four stroke cy cle en gine, run ning at N cy cles per sec ond, the mean ve loc ity of the pis ton is:

v LN= 4 (20)

where x1 and x2 [m] are the pis ton po si tion at max i mum and min i mum vol ume and L [m] is thedis tance that the pis ton trav els per stroke, re spec tively.

Thus, the power out put is:

P Q Q P

M T T T T

at in out= - =

= × + - - +-

–

[ . ( ) .

m

8353 10 51233

13

23

43 816 10

2123 10

832 5

12 5

22 5

42 5

732

× + - - -

- × +

-

-

( )

. (

. . . .T T T T

T T T T T T T T12

22

42 5

31 5

11 5

21 5

41 52108 10- - + × + - - +

+

-) . ( ). . . .

10433 13303 3024 103 2 4 14

30 5

10. ( ) . ( ) . ( . .T T T T T T- - - + × +- - 5

20 5

40 5

53

11

12

14

13063 10 1

- - -

- × + - - +

- -

- - - -

T T

T T T T

. . )

. ( ) . ( )]106 1073

21

22

24

2 2× + - - -- - - -T T T T vm (21)

The ef fi ciency of the cy cle is:



hatat

in leak

=+

=

=

× + - - +-

P

Q QM T T T T[ . ( )8353 10 512

33

13

23

43 . ( )

. (

. . . .816 10

2123 10

832 5

12 5

22 5

42 5

732

× + - - -

- ×

-

-

T T T T

T + - - + × + - - +-T T T T T T T12

22

42 5

31 5

11 5

21 5

41 52108 10) . ( ). . . .

+ - - - + × +- -10433 13303 3024 103 2 4 14

30 5

10. ( ) . ( ) . ( .T T T T T T . . . )

. ( )

52

0 54

0 5

53

11

12

14

13063 10

- - -

- × + + - +

- -

- - - -

T T

T T T T 1106 10

8353 10

73

21

22

24

2 2

12

. ( )]

[ . (

× + - - -

×

- - - -

-

T T T T

M T

mn

33

23 8

32 5

22 5 7

32

225816 10 2123 10- + × - - × -- -T T T T T) . ( ) . (. . )

. ( ) . ( ) .. .

+

+ × - + - + ×-2108 10 10433 3024 10531 5

21 5

3 24T T T T ( )

. ( ) . (

. .T T

T T T3

0 52

0 5

53

12

1 73

3063 10 1106 10

- -

- -

- -

- × - + × - -- + + -22

22 3 02T MB T T T)] ( )

(22)

When g, T1, T3, hc, and he are given, T2S can be ob tained from eq. (13), then, sub sti tut -ing T2S into eq. (8) yields T2, T4S can be ob tained from eq. (14), and the last, T4 can be solved outby sub sti tut ing T4S into eq. (9). Sub sti tut ing T2 and T4 into eqs. (21) and (22) yields the powerand ef fi ciency. Then, the re la tions be tween the power out put and the com pres sion ra tio, be -tween the ther mal ef fi ciency and the com pres sion ra tio, as well as the op ti mal re la tion be tweenpower out put and the ef fi ciency of the cy cle can be ob tained.

Nu mer i cal ex am ples and dis cus sion

Ac cord ing to ref. [29-31], the fol low ing pa ram e ters are used: T1 = 350 K, T3 = 2200 K,x1 = 8·10–2, x2 = 1·10–2 m, N = 30, and M = 4.553·10–3 kg/s. Fig ures 2-4 show the ef fects of thein ter nal ir re vers ibil ity, heat trans fer loss, and fric tion loss on the per for mance of the cy cle. Onecan see that when the above three irreversibilities are not in cluded, the power out put vs. com -pres sion ra tio char ac ter is tic and the power out put vs. ef fi ciency char ac ter is tic are par a bolic-likecurves, while the ef fi ciency will in crease with the in crease of the com pres sion ra tio. When morethan one irreversibilities are in cluded, the power out put vs. com pres sion ra tio char ac ter is tic andthe ef fi ciency vs. com pres sion ra tio char ac ter is tic are par a bolic like curves and the power out put vs. ef fi ciency curve is loop-shaped one.

Ac cord ing to eq. (21), the def i ni tions of thepower out put, the heat trans fer loss has no ef fecton the power out put of the cy cle. So fig. 2 onlyshows the ef fects of the in ter nal ir re vers ibil ityand fric tion loss on the power out put of the cy cle.Com par ing curves 1 with 1' and 2 with 2', one can see that the power out put in creases with the de -crease of in ter nal ir re vers ibil ity. Com par ingcurves 1 with 2 and 1' with 2', one can see that thepower out put de creases with the in crease of fric -tion loss.

Fig ure 3 shows the ef fects of the in ter nal ir re -vers ibil ity, heat trans fer loss and fric tion loss onthe ef fi ciency of the cy cle. Curve 1 is the ef fi -ciency vs. com pres sion ra tio char ac ter is tic with out ir re vers ibil ity. Un der this cir cum stance, the ef fi -



Figure 2. The influences of the internalirreversibility and friction loss on the poweroutput

ciency in creases with the in crease ofcom pres sion ra tio. Other curves are ef fi -ciency vs. com pres sion ra tio char ac ter is -tic with one or more irreversibilities andthese curves are par a bolic-like ones.Com par ing curves 1 with 1', 2 with 2', 3with 3', and 4 with 4', one can see that theef fi ciency in creases with the de crease ofin ter nal ir re vers ibil ity. Com par ing curves 1 with 3, 2 with 4, 1' with 3', and 2' with 4', one can see that the ef fi ciency de creaseswith the in crease of heat trans fer loss.Com par ing curves 1 with 2, 3 with 4, 1'with 2', and 3' with 4', one can see that theef fi ciency de creases with the in crease offric tion loss.

Fig ure 4 shows the ef fects of the in -ter nal ir re vers ibil ity, heat trans fer loss,and fric tion loss on the power out put vs.the ef fi ciency char ac ter is tic. Curve 1which is a par a bolic like curve is thepower out put vs. ef fi ciency char ac ter is -tic of the cy cle with out ir re vers ibil ity,while else curves are loop-shaped oneswith one or more irreversibilities. Com -par ing curves 1 with 1', 2 with 2', 3 with3', and 4 with 4', one can see that themax i mum power out put and the ef fi -ciency at the max i mum power out put de -crease with the in crease of in ter nal ir re -vers ibil ity. Com par ing curves 1 with 3, 2 with 4, 1' with 3', and 2' with 4', one can see that the max i mum power out put is not in flu enced byheat trans fer loss, while the ef fi ciency at the max i mum power out put de creases with the in crease of heat trans fer loss. Com par ing curves 1 with 2, 3 with 4, 1' with 2', and 3' with 4', one can seethat both the max i mum power out put and the cor re spond ing ef fi ciency de crease with the in -crease of fric tion loss.

Conclusions

In this pa per, an ir re vers ible air stan dard Atkinson cy cle model which is more close toprac tice is founded. In this model, the non-lin ear re la tion be tween the spe cific heats of work ingfluid and its tem per a ture, the fric tion loss com puted ac cord ing to the mean ve loc ity of the pis ton, the in ter nal ir re vers ibil ity de scribed by us ing the com pres sion and ex pan sion ef fi ciency, andheat trans fer loss are pre sented. The per for mance char ac ter is tics of the cy cle were ob tained byde tailed nu mer i cal ex am ples. The ef fects of in ter nal ir re vers ibil ity, heat trans fer loss and fric -



Figure 3. The influences of internal irreversibility, heattransfer loss, and friction loss on the efficiency

Figure 4. The influences of internal irreversibility, heattransfer loss, and friction loss on the power output vs.efficiency characteristic

tion loss on the per for mance of the cy cle were an a lyzed. The re sults ob tained herein may pro -vide guide lines for the de sign of prac ti cal in ter nal com bus tion en gines.

Ac knowl edg ments

This pa per is sup ported by Pro gram for New Cen tury Ex cel lent Tal ents in Uni ver sityof P. R. China (Pro ject No. NCET-04-1006) and The Foun da tion for the Au thor of Na tional Ex -cel lent Doc toral Dis ser ta tion of P. R. China (Pro ject No. 200136). The au thors wish to thank there view ers for their care ful, un bi ased, and con struc tive sug ges tions, which led to this re visedmanu script.

Ref er ences

[1] Andresen, B., Salamon, P., Berry, R. S., Ther mo dy nam ics in Fi nite Time, Phys. To day, 37 (1984), 9, pp.62-70

[2] Sieniutycz, S., Salamon, P., Ad vances in Ther mo dy nam ics, Vol. 4, 1990, Fi nite Time Ther mo dy nam icsand Thermoeconomics, Tay lor & Fran cis, New York, USA

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No men cla ture

A1 – heat released by combustion per second,– [kW]

B – constant related to heat transfer,– [kJkg–1K–1]

Cp – specific heat with constant pressure,– [kJkg–1K–1]

Cv – specific heat with constant volume,– [kJkg–1K–1]

k – ratio of specific heats, [–]L – total distance of the piston traveling per

– cycle, [m]M – mass flow rate of the working fluid,

– [kgs–1]N – number of the cycle operating in a

– second, [–]P2,3 – pressure at different states 2 and 3, [Pa]Pm – power output of the cycle, [kW]

Qin – heat added to the working fluid in a– second, [kW]

Qout – heat rejected by the working fluid in a– second, [kW]

Rg – air constant of the working fluid, [kJkg–1K–1]T1-5,2S,5S– temperature at different states, [K]V1,2 – volume at different states 1 and 2, [m3]v – velocity of the piston, [ms–1]x1 – the piston position at maximum volume, [m]x2 – the piston position at minimum volume, [m]

Greek let ters

hc – efficiency of the cycle, [–]he – expansion efficiency, [–]m – coefficient of friction, [Nsm–1]g – compression ratio, [–]

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Paper submitted: January 28, 2009Paper revised: April 14, 2009Paper accepted: April 30, 2009



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