Indi an Journal of Pure & Applied Physics

Vol. 39, October 200 I, pp. 628-635

; r ••

Optimal efficiency of an irreversible heat engine with t,herm1!1 reservoir of finite heat capacitance using method of Lagrangian multiplier

'---- - -

.S C JK aushik & P Kumar

I Ce~Hre for Energy St'udies. Indian Institute of Technology , Hauz Khas, New De lhi - IIOOI Y

Rece ived 4 Deeemher 2000; accepted 9 April 200 I

The ~nite time thermodynami c optimization or an irreversi bl e heat engi ne havi ng thermal reservoi rs of rinite heat capacitance has been presented. A general exp ression ror the optimum e lTiciency or heat engine is de ri ved at max imum work output and given heat in put Il ow rate conditi on. Ex tern al irreversi bility is due to linite temperature dirfe rence hetween system working Iluid and thermal reservoirs (source/sink) while internal irreversibility is due to non-isentrop ic ex pansion and compress ion within the system whi ch is represented hy an irreversibility parameter indi cating the dev iati on rrom endoreversible case. It is round that the d rect or internal irreversibility is more pronounced than the ex ternal irreve rsibilit y assoc iated with the heat engine. The errect or va ri ous operating parameters on the efli ciency of the heat engine is stud ied and

') c: /1 ,~/ ~ I ~ C\. L I (' . . ~ c'"

numerical res ults are presented in the ell(~. ,

1 Introduction

French engineer Sacli Carn ot has deri ved an

express ion for thermal effi ciency of a reversible heat

engine workin g between any two thermal reservoirs

of temperature TH and TL as gi ven by Refs 1-3.

T 11=I--L

Til

Thi s is the highest effici ency of any heat engine

work ing between the same temperature level and no

heat engine can achieve thi s ef fi ciency in real

practice becau se thi s formul a is derived by

considerin g reversible processes. In ac tu al case, all

processes are irreversible and occur in finit e time

hence actual effi ciency w ill be less than the Carn ot

reversible ef f iciency due i o irreversibility . M ain

obj ective functi on power (work output per cyc le

time) will be zero at thi s Carn ot effi ciency since

reversible process is infinite time executable process.

Thi s effi ciency fo rmul a i s highl y important in theory

but has no practical utili ty because o f ideal

assumptions4. There are two types of i rreversibility in

the system. Ex tern al irreversib ility is mainl y due to

r

temperature difference between the sys tem and

thermal reservoirs and internal irreversibility is due

to non-i sentropic expan sion and compress ion w ithin

the sys tem. Curzon-Ahlborn considered ex ternal

irreversibility into account and deri ved optimal

efficiency expression, -I o at max imum power output of

heat engi ne.

11 = 1- ~T, Ci\ T

H

Thi s effi c iency formul a w ill always have lower

va lue than reversible Carn ot ef f iciency formula.

Rubin9

has carri ed out finit e time analysi s of heat

engine with given input hea t fl ow rate and thermal

reservoirs of infinite heat capacit y. He did not

consider any effect of internal irreversibility and

finite heal capac itance of thermal reservoirs in the

an alys is. In thi s paper we are ex tending Rubiri's

analys is by taking thermal reservo irs of finite heat

capac itance and presenting a general analys is for heat

engine with given constant input heat fl ow rate

thereby considering ex ternal as well as in tern al

irreversibiliti es . Curzon-Ahlborn - an alys is IS not

KAUS HIK & KUMAR: EFFICIENCY OF IRREVERSIBLE HEAT ENGINE 629

appl icable due to some constraints li ke given in put heat fl ow rate"- 18

. In thi s paper we have applied the method of Lagrangian multiplier for the finite time thermodynamic analys is of an irreversible heat engine with given input heat flow rate and finite heat capacitance of the thermal reservoir (source/sink). It is fo und that the effect of internal irreversibility is more pronounced as compared to the ex ternal irreversibili ty associated with the heat engine. Numerica l results and parametric studi es are presented in the end .

2 System Description and Analysis

A Heat engine working between two therma l reservoirs of finite heat capac itance is shown in Fig. I . Heat engine takes heat fro m high-temperature reservoir (heat source) and converts a fr acti on of thi s amount of heat into mechanica l work and the rest of heat is rejected to the low temperature reservo ir (heat sin k). Since heat source and sin k are of finite heat capacitance so during heat absorpti onirejecti on processes, temperature of the heat source and sin k varies from Ttl I to Tm and TLJ to Tu. respec ti ve ly. Temperatures of hot and co ld side workin g fluid of the heat engine are Til and Te, respecti ve ly. Its T-S diagram is shown in Fig. 2.

The rate of heat fl ow from hi gh temperature heat source to the system is given by:

QH = Q H = U H A H (LMTD )H = l~l H C PII (T HI - Tin ) tH

. . . ( I )

Similarly, the rate of heat rejected from system to the heat sink:

.. . (2)

where

(L MTD ) = (T H I - T1J - (T II 2 - TI,) II In (T il l - T I,)

(TH 2 - T I,)

Heat Source

Tm

QII -----+------ Th(t)

Heat Engine

Tc(t)

~QI T

lL<::--~ __ -----,1 1.2

TI.I lIeat Sink

Fig. I - Schemati c diagram of a heat engine

Til l

Till

I' Warm working fluid ....

lh , , , , I , : , , I

Heat engine cycle \

~ ~ I I \ I , I , I , I , I

Tc \

I I

Cuitl working fihid j'

~TLl Tu~

\ \

J

w

Entropy (S)

Fig. 2 - T-S diagram of heat engine cycle

630 INDIAN 1 PURE & APPL PHYS VOL 39, OCTOBER 2001

have:

(LMTD)L = (Tc - TLI)-(Tc - TL2) In (Tc - TLI)

(Tc - T L2)

Using LMTD expressions in Eqs. (1-2), we

where HE = CH £H, LE = CL £L, CII = U,.,A"

mHcPH, CL = m LCPL, tH =I-e - 1~,I CPlI and ULAL

tL = 1 - e --. -. tH and tL are heat absorption and mLCPL

rejection time to and from the system respectively. We are assuming that adiabatic expansion and compression processes take negligible time as compared to heat absorption and rejection processes so total cycle time of heat engine7

-8 is taken to

be '( = tl-I + tL. The heat conductances between high­temperature reservoir and warm side working fluid , the cold side working fluid and low temperature reservoir are HE(t) and LECt), respectively. The relations between the heat conductance quantities and time t are assumed to be as follows:

HE(t) = HE tL ~ t ~ '( LE(t) = LE til ~ t ~ '(

= 0 0 ~ t ~ tL = 0 0 ~ t ~ tH

where HE and LE are gi ven above.

t

QH= fHE(t) [THI-Th(t)]dt ... (3) o

t

QL = fLEet) [Tc(t) - TLI]dt ... (4) o

Now, from the first law of thermodynamics, we

have

t

W = f [ HE( t) (T H 1- T h (t) ) - LE( t) (T c (t) - T L I)] d t o

.. . (5)

where QI-I, QL and Ware the quantities of input heat, rejected heat and total work output of the heat engine, respectively.

2.1 Internal irreversible cycle

If internal irreversibility (s ch as due to friction) is also accounted for the system then the two isentropic processes become adiabatic processes with entropy generations. This internal irreversibility of the system can be characterised by an irreversibi lity parameter representing compression in terms h Id 'f't' II 11 C anges I erences ' -.

non-isentropic expansionl of the ratio of entropy On T-S diagram, the four

processes of heat engine system are shown in Fi g, 2.

Heat input to the reversible heac engine Q'II = Th (S2 - S' I) while heat input to the irreversible heat

engine is QI-I = Th (S2 - SI) since SI > S; and so QII <

Q'H (primes are added to quantities associated with reversible heat engine). Thus we can define irreversible heat absorption parameter CI such that

QI-I = CI Q'H with CI = (S2 - SI)/(S2 - S;) < I ,

similarly irreversible heat rejection parameter C2 can be defined as:

Using second law of thermodyn amics,

f dQ = Q H _ Q L < 0 T Th Tc

this inequality can be written

as:

QH_R

QL_O 6S - with

Th Tc

R 6S is internal irreversibility parameter of the heat engine cycle. R6S = 1.0 is for endo-reversible heat engine cycle and R6S < 1.0 is fo r irrevesible heat engine cycle. Now including this internal irreversibility parameter and using the second law of thermodynamics net entropy generation for irreversible heat engine is given by:

~S= f[HE(t) (~- IJ+R 6sLE(t/TLI( t)-I J~cl l =0 o l T,,(I) . l1', (I) IJ

... (6)

KAUSHIK & KUMAR: EFFICIENCY OF IRREVERSIBLE HEAT ENGINE 631

2.2 Power output maximization under given constraints i.e. given heat input flow rate

For seeking maximum power output for given

input heat flow rate, we have introduced a modified

Lagrangian 18 at any time t defined as:

L=[HE(t) (TH I-Th(t») -LE(t) (Te(t)-TLI )]­

A [H E(t) (THI-Th (t))]

- J.1 [HE(t) (~-Il+ LE(t) (~- 1 )11 Th(t)) Te (t) ~

... (7)

where A and J..l are Lagrangian undetermined

multipliers. We see that L is a function of Th(t) and

TeCt). Now USIng Euler-Lagrangian equations, we

have:

and aL ") = 0 give optimal values

aTc(t)

... (8)

.. . (9)

2.3 Relation between optimal efficiency and rate of input heat flow rate

Substituting the optimal values of TI,(t) and

T eCt) from Eqs. (8-9) into Eqs. (3-4 and 6) , we have:

... ( 10)

.. . (I I)

6S= HEIH(~'-I)+R " LE I, (i: -1)=0

.. . (12)

Now using Eqs. (10-12), we have:

. .. (13)

. .. ( 14)

... ( 15)

Efficiency of heat engine system is given by :

2.4 Maximization of efficiency wrt heat absorption/rejection time tH and k

and

't ~E gives -= 1 + tH R6S LE

~= 1 +JR6S LE gIves t2 HE

. .. (16)

... ( 17)

... (18)

... ( 19)

putting the optimal values of tH and tL from Eqs.

(18-19) into Eqs. (13-14) and (17), we have opt imal

value of working fluid temperatures and optimal efficiency of heat engine is given by :

... (20)

632 INDIAN J PURE & APPL PHYS VOL 39. OCTOBER 200 I

Table I (a-b) - Effect of source/sink inlet temperature, on working tluid temperatures, efficiency and power

output of the heat engine system.

TLI Th Tc 11 P

K K K kW

297 411.78 303.38 0. 18 0.54

300 41 1.78 306.45 0.17 0.52

303 411.78 309.5 I 0. 16 0.49

306 411.78 3 12.57 0. 16 0.47

309 411.78 3 15.64 0. 15 n.44

3 12 4 11.78 3 18.70 0. 14 0.42

3 15 411.78 32 1.77 0.13 0.40

3 18 411.78 324.83 n. 12 0.37

32 1 41 1.78 327.90 n.12 0.35

324 411.78 330.96 n. I I 0. 32

T c = . .. (2 1)

and

. .. (22)

where PH = QH • If we introd uce the equiva lent

'[

* PH [ ~E J2 temperature T III = TH I - - 1+ HE LE R 6 S

then the effi ciency of heat engine system becomes, T

11 = I - _ L_I This equation shows that heat engine TI~ I .

cyc le operatin g between the reservoir temperatures Till and TLI at a given input heat flow rate, is eq uival ent to a single reversib le Carnot heat engine

Till Th T(, P

K K K kW

400 391.78 306.78 0.1 3 0.39

405 396.78 306.69 n.14 0.42

410 40 1.78 306 .61 O. IS 0.46

415 406.78 306.5 3 0.16 0.49

420 411.78 306.45 0. 17 0.S2

425 416.78 306.37 n.1 g n.ss

430 421.78 306.29 0.19 O . S~

435 426.78 306.2 1 0.20 (l .6 1

440 431.78 306.14 o.n n.M

445 436.78 306.07 0.22 0.66

cyc le operating between reservoir at temperatures

TI~ I and TLI . Thus TI~I is the effec tive sou rce

temperature of the heat engine cyc le caused by the existence of therma l resistance.

3 Discussion of Results

In order to have numerical apprec iation of the theoretical analys is of the heat engine system. we have studi ed the effect of various input parameters on the effi ciency of the system and res ults are show n in

Tables I A. During the va ri ati on of anyone parameter, all other parameters are ass umed to be constant as given below:

Till = 420 K, T LI = 300 K, ell = CL = 1.00 kW/K, Ell = EJ. = n.75 , R 6S = O.7S and input heat flow rate is assumed to be fixed as PI I = Qlh: = 3.0 kW during all parameters va ri ati on.

It is seen from the Table I-(a) that by inne,\s in g the inlet temperature of the external Sl)UrCe side fluid , power outpu t of the sys tem IIlcreases and consequentl y eff iciency of the heat engine inneases since input heat fl ow rate is fixed. Tell1perature or the working fluid 011 both sou rce and sin ' side increases by inneas ing sou rce in let temperature.

KAUSHIK & KUMAR: EFFICIENCY OF IRREVERSIBLE HEAT ENGINE 633

Table 2(a-b) - Effect of heat capacitance of source/sink s ide external fluid , on working fluid tempe ratures.

effic iency and power output of the heat engine system.

kW/K

0.5

0 .6

0 .7

0 .8

0 .9

1.0

1.1

1.2

1.3

1.4

K

410.04

410.56

410.96

411.29

411.56

411 .78

411.98

412.15

412.30

41 2.44

Tc

K

3 11.27

309.69

308.55

307 .68

307 .00

306.45

305 .99

305 .60

305 .27

304.98

0 .16

0 .16

0 .17

0 .17

0 . 17

0 .17

0 .17

0 .18

0 . 18

0 .18

P

kW

0.47

0.49

0 .50

0.51

0 .51

0 .52

0 .52

0 .53

0 .53

0 .54

Table I-(b) shows that by inc reas ing s ink inle t

temperature, power output of the heat engIne

decreases he nce efficiency of the heat eng ll1 e ~

decreases . Temperature of intern al working fluid on

hi gh temperature side re ma in s constant whe reas on

si nk side it Increases with increas ing sink inlet temperature.

T ables 2(a-b) shows that as heat capac itance

rates o f ex tern a l source/s ink s ide fluid inc rease on

e ither side, the power output and consequentl y

effic iency of the heat engine system increase since

input heat fl ow rate is fi xed. These T abl es 2(a-b) a lso

show as we increase the heat capac itance rate of

external fluid on e ither s ide whil e keeping othe r as

constant , tempe rature of the working fluid on hot s ide

increases and cold side decreases.

Tables 3(a-b) show the effect of effecti veness £ 11

or £1. of source or s ink s ide heat exchanger in the heat

eng ine syste m. By increas in g anyone of the m whi le

other be in g kept as constant, power output o f the

system increases and consequentl y e ffic ie ncy of' the

sys te m increases because input heat flow rate is

fixed. Te mperature of the interna l working flui d

changes in the same way as in case of heat

capac itance va ri ation. As we inc rease effecti veness

kW/K

0.5

0 .6

0 .7

0.8

0 .9

1.0

1.1

1.2

1.3

1.4

K

406 .04

407 .89

409 .25

410.29

411.11

411.78

41 2. 34

412 .82

41 3.23

41 3.58

T c

K

307 .89

307.45

307.1 2

306.85

306 .63

306.45

306. 29

306.15

306.03

305 .92

0.16

0 . 16

0 . 17

0 . 17

0 .17

0 .17

0 .17

0 .18

0 .18

0.1 8

P

kW

0 .47

0.49

0 .50

0 .51

0 .5 1

0 .52

0 .52

0 .53

0 .53

0 .53

of the heat exchanger on e ither s ide (whil e keep in g

other as con stan t), tempe rature of the working flui d

on hot side increases and co ld s ide decreases.

T able 4 shows the effect of intern al

irreve rs ibility paramete r Rns. As thi s parameter

increases, powe r output of the heat e ngine in creases

and consequentl y the effi c iency II1creases. T he

te mperature of hot s ide working fluid increases

whe reas co ld side working fluid tempe rature

decreases with increas ing interna l irrevers ibility

parameter. The case Rns = 1.0 correspond s to the

endoreversible case. Thu s the effec t of int e rn al

irrevers ibility paramete r is more pronoun ced as

compared to the e ffect o f effec ti veness and heat

capac itance rate of the source/s in k s ide thermal

reservoi rs.

4 Conclusions

Finite time thermodynami c anal ys is of' an

irrreve rsibl e heat e ng ine syste m with finite heat

capac itance of heat source/ s in k reservo irs, has been

carri ed out using the method of Lagrangian mu lt ip li er

by maximi zing power output for a g iven in put heat

fl ow rate. It has been shown that in tern al

irrevers ibility in the sys tem can be c haracteri sed by a

634 INDIAN J PURE & APPL PHYS VOL 39, OCTOBER 200 I

Table3(a-b) - Effect of effectiveness of source/sink side heat exchanger, on working fluid temperatures , efficiency and power output of the heat engine system.

£L ThTc T\ P

K K kW

0.20 407.84 319.45 0.13 0.39

0.30 409.33 313.62 0. 15 0.45

0.40 410.23 310.68 0.16 0.48

0.50 410.84 308.90 0 .16 0.49

0.60 411.29 307.68 0 .17 0.51

0 .70 411 .64 306.80 0 .17 0 .52

0.75 411.78 306.45 0 .17 0.52

0.80 411.92 306.13 0 .17 0 .52

0 .90 412.15 305.60 0 .18 0.53

1.00 412.35 305.17 0 . 18 0.53

single irreversible parameter representing the ratio of

two entropy differences. This parameter appears in

both the equations for maximum output power and

efficiency of the system. The equations clearly show that a heat engine system with internal irreversibility

produce less power and lower efficiency than

endoreversible heat engine system. The effect of

various parameters on thermal efficiency has been

investigated and numerical results show that the

effect of internal irreversibility is more pronou nced

than external irreversibi lity.

Nomenclature

All = Surface area of heat exchanger between high­

temperature reservoir and the hot working fluid

(m\

AL = Surface area of heat exchanger between the cold

working fluid and 10w- temperature reservoir

(m\

C II = Heat capacitance rates of external fluid in high­

temperature reservoir (heat source) (kW/K).

£1-1 ThTc T\ P

K K kW

0.20 396.84 309.85 0 . 13 0.40

0.30 403 .33 308.50 0.15 0.45

0.40 406.73 307.73 0.1 6 0.48

0.50 408.84 307.22 0.1 7 0.50

0.60 410.29 306.85 0.17 0.51

0.70 411.35 306.56 0 .17 0 .52

0.75 411.78 306.45 0.1 7 0.52

0.80 412. 17 306.34 0 . 17 0.52

0.90 412.82 306.15 O.IH 0.53

1.00 4 13.35 305 .99 0.18 0.53

CL = Heat capacitance rates of external fluid in low­

temperature reservoir (heat sink) (kW/K).

CPH = Specific heat of external fluid in hi gh­

temperature reservoir of the system (kJ/kg-K ).

CPL = Specific heat of external fl uid in low­

temperature reservoir of the system (kJ/kg-K ).

III H = Mass flow rates of ex ternal fluid in hi gh-

temperature reservoir of the system (kg/s).

IllL = Mass flow rates of external fluid in low-

temperature reservoir of the system (kg/s).

P = Power output of the heat engine system (kW).

PII = QII/'t = Input heat power or inpu t heat fl ow rate

(kW)

PL = QJ't = Output heat power (kW)

QH = Heat transfer rates from the high-temperature

reservoir to sink (kW)

KAUSHIK & KUMAR: EFFICIENCY OF IRREVERSIBLE HEAT ENGINE 635

Table 4 - Effect of internal irreversibility parameter, on working fluid temperatures, efficiency and power

output of the heat engine system. R 6S Til Tc 11 P

K K kW

0.80 411.53 307.07 0.07 0.20

0.82 411.58 306.93 0.09 0.27

0.85 411.66 306.74 0.12 0.37

0.90 411.78 306.45 0.17 0.52

0.95 411.90 306. 18 0.22 0.65

0.96 411.92 306.13 0.23 0.68

0.97 411.94 306.08 0.23 0.70

0.98 411.96 306.03 0.24 0.73

0.99 4 1 1.98 305.99 0.25 0.75

1.00 4 12.00 305 .94 0.26 0.77

Q L = Heat transfer rates from cold working fluid to

the low temperature reservo ir (kW )

RfiS = Intern al irreversibility parameter

'( = Total cycle time (s)

Till = Inl et temperature of the heat source fluid (K).

TLI = In let temperature of the heat sink fluid (K).

Til = Temperature of working substance In hi gh-

temperature reservoir side(K).

T, = Temperatu re of working substance In low

temperature reservoir side(K).

til = Time of heat absorpti on from heat source to heat

engine (s)

II. = Time of heat rej ecti on from heat engi ne to heat

sink (s)

UII = Overall heat transfer coeffi cient on high­

temperature reservoir side (k W /m2 -K).

UL = Overa ll heat transfer coefficient on low­

temperature reservoir side (kW/m2-K).

W = Work ou tput of the heat engine (kJ)

£11 = Effectiveness of source side heat exchanger.

EL = Effectiveness of sink side heat exchanger.

A = Lagrangian mUltiplier.

Acknowledgement

The authors gratefully acknowledge the financial support from CSIR, India .

References Single 0 P. Engineering therl/lOdvnall/ics. Macmill an India Press. ( 1(90).

2 Kau shik S C. Solar refrigeratioll ({lid .Ilwce conditioning. Divya Jyoli Prakashan, Geo-environ Acadmia press. Jodhpur (India), ( 1989).

3 Arora C P. Refrigeration and Airconditiol1ing. Tata McGraw-Hili Publi shing Company. ( 1995).

4 Chen W Z. Su n FR. Cheng S M & Chen L G. Int . .I. EI/{'rgr r(,.\"(,llrch. 19 ( 1995) 75 1-759.

5 Curzon F L & Ahlborn B. AII/erican JOllmol (~l PII\ ·sics. 43 ( 1975) 22.

6 Lee W Y. Kim S S & Won S H. Lnerg.\'. 15 ( 1990) lJ7lJ-') '5. 7 Lee W Y & Kim S S. Energ.\' Conl'l'.I·.& Ivlgllll .. 3:1 ( l lJ()2)

5')-67.

X WuC.Energv, 13 ( l lJ8S)6SI -687. I) Rubin M H. Phrsical r(' vielllA. 19 ( 197R) 1272- 127(1. I I I Wli C. En'ergr Conl'en. & MglI/t .. 30 ( I lJ(0) 261-26(1. II Wu C& Ki angR L, Energ.\', 17 ( l lJ92) 1173-11 7R. 12 Chen J & Yan Z . .I. Apil/. IJhrs .. 6:1 ( 19SR) 4795-4708. U Sahin B &. Kodal A. !:n ergr , 20 ( 1905) 1285 -1 289. 14 Kodal A. Sahin B & Yilmaz T. En{'/'gr CO /II '('/'. & MglI/t .. 41

(2000) 607-619. 15 Wu C. Int('motional JourI/{{l (~r AII/hienl ('n(' rg-" . I,,) ( 11)').\ )

25-28.

16 Wli C. International Joum({l (~r AII/himt ('nerg.\' . I ,,) ( 1 lJ'J:I ) 77-82.

17 LelT H S &. Teeters W D. AIII('rican Jourl/ol (~f PIl\'siD. ,,)6 ( 1978) 19-22.

I S Blanchard C H. Joum{{l of 1I1'1,lil'd PhrsiCJ. 5 I ( I lJX(J ) 2471-2472.