A class of irreversible Carnot refrigeration cycles wtih a general heat transfer law

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A class of irreversible Carnot refrigeration cycles with a general heat transfer law

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1990 J. Phys. D: Appl. Phys. 23 136

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J. Phys. D: Appl. Phys. 23 (1990) 136-141.rinted in the UK

A class of irreversible Carnotrefrigeration cycles with a general heattransfer law

Zijun Yan t* and Jincan Chen*t China Cen tre of Advanced Science a nd Technology (World Laboratory), PO Box8730, Beijing, Peoples Republic of C hina* Department of Physics, Xiamen University, Xiamen, Fujian, Peoples Republic ofChinaReceived 20 February 19 89, in final form 17 July 1989

Abstract. The p erforman ce of a class of irreversible C arnot refrigeration cyclesoperating between two heat reservoirs at a low temperature T, and a hightemperature TH, or which the on iy irrev ersib ility results from the finite rate of heatconduction, is studied. The relation between the optimal rate of refrigeration andthe coefficient of p erformance of these cycles is derived, based on a general heattransfer law. Moreov er, the op timal performance of these cycles is discussed andthe effect of different heat transfer laws on the optimal performance s investigated.Consequen tly, some new and u seful results for refrigeration cycles are obtained.

1. IntroductionIt has been demonstrated by many authors that underdifferent heat transfer aws irreversible cycles have dif-ferent optimal configurations and characters [l-51. Forexample, oraCar not cycle operatingbetween woheat eservoirs at a high temp erature T H an da lowtemperature TL , and with theheatconductances K ,an d K 2between the working m aterial and the two heatreservoirs, when heat transfer obeys Newtonsaw (i .e .the heat flux q a AT), the efficiency of the cycle atmaximum power output is

v,,, = 1- T ~ / T ~ ) ~ / ~ . (1)This is the CA efficiency [6], justs theCarnotefficiency is on ly a fu nction of reservoir temperatures.Wh en heat transfer obeys another linear heat transferlaw in irreversible thermodynam ics (i.e. q m A(l/T)),the efficiency of the cycle at maximum power outp utas shown previously [ 5] sv,,, = (m1 V??*)1 - T J T H ) / ( V Z l + 2 C K 2

+ -1 TL/TH). (2 )It is not only a function of reservoir tem peratures, bu talso of theheatconductances. As anotherexample,when the heat transfer law is

4 = ~ ( I / T - / T R ) + P(I/T - 1 p R ) 9 ( 3 )(where CY > 0 and P > 0 are the heat condu ctances,an d T an d TRare the temperatu res of the system and0022-3727/90/020136 06 $03.50 1990 OP Publishing Ltd

-+Figure 1. A C arnot refrigeration cycle affected by thermalresistance.

reservoirs, respectively) th e optim al cycle of maximu mefficiency with a given amo un t of heat npu t is nolonger Carnots form , but consists of three isothermsand three adiabats [l].For refrigeration cycles, there is naturally the sameproblemaspow er cycles. How ever,up o now theeffects of heat ransfer law on he performance ofrefrigeration cycles have never bee n investigated sys-tematically. T her efo re, it is very worthwhile to studythe problem further.In this paper the relation between the optimal rateof refrigera tion and the coefficient of perform ance ofa class of Ca rnot re frige ration cycles, whose only irre-versibility results rom heat ond uction , is derivedbased on a general heat ransfer aw. Moreover, heoptimal perform ances of such a class of cycles for thr ee


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A class of irreversible Carnot refrigeration cycles

common heat transfer aws are discussed. In particular,th eoptimalperform ance of aNewton's law Carnotrefrigeration cycle is analysed in detail. It is pointedou t hat he effect of the rm al resistance is aseriousproblem in ultra-low -tem peratu re technology andbrings ab ou t twofold difficulties in ach ieving ultra-lowtempera tures .At he ame ime, he nherent ndintrinsic cha rac ter of the effect of the heat trans fer lawon he performan ce of sucha class of cycles is alsorevealed.

2. The model of a class of irreversible Carnotrefrigeration c yclesTh e model of the type of irreversible Carno t refriger-ation cycles considered in this paper is shown in figure1, and satisfies the following conditions.

(i) nsucha mod el, a reversible Carno t refriger-ation cycle is carried out inside the working material.(ii) Th e only rreversibility in themodel resultsfrom he finite rate of heatconductionbetween heworkingmaterialand he woheatreservoirs.Whenth ewosothermalrocessesnsideheworkingmaterial are carried out , their temperatures T 1 an d T 2are different from the reservoir temperatu res TH an dTL , and there exists the relation T 1> TH > TL > T2 .Thus, the heat condu ction may be carried out in finitetime, and the cycle may h ave a certain rate of refriger-at ion.(iii) The sentropic processesoccu r in negligibletime. This mean s that they must occur on a timescalethat is fast compa red with the slow rates for h eat leaksto the environm ent , but slow comp ared with the rapidinternal relaxation of pressure gradients in the workingmaterial [7]. If this is the case, cycle time is given byt = t 1 + t , (4 )

where t l an d t 2 are the imes of releasing and absorbingheat processes respectively.This model is similar to the Curzon-Ahlborn cyclemodel which has been extensively adop ted by manyauthors [&lo], buthemod el dopted ere is arefrigeration cycle and is somew hat different. The tem-peratures T 1 and T , of the wo sothermalprocessesinside the work ing material are not between TH an dTL. The y satisfy the elation T 1> T H> TL > T 2 .Therefore, ora given TH and T L , th epa th of suchrefrigeration cycles cannot coincide with that of a Cur-zon-Ahlbo rn cycle. This is different from a reversibleCa rn ot cycle in which the pa th of the reverse cycle iscoincident with the original one.Th is just ndicatesthat he Carno t refrigeration cycles mentioned aboveare irreversible.In order to make anified desc ription of the variousoptimum performances of the above Carnot refriger-ation cycles under several comm on heat transfer laws,we use ageneralheat ransfer law which hasbeenadopted by Vos [ 2 , 41 a nd Ch en [ 5 ] , i .e . , the heat Q,

released to the high -temperature reservoir and the heatQ 2 absorbed rom he ow-temperature eservoir bythe w orking material per cycle are assum ed to satisfythe following relations:

an d

K 1 an d K 2 are heheatconductancesbetween heworking m aterial and the two heat reservoirs at tem-peratures TH and TL. n is anon-zero nteger.Whenn < 0, K 1 and K 2 ar enegative.Thegeneralityan dsignificance of such a heat transfer law lies in the factthat , when a different value f n is cho sen, it represe ntsa different heat transfer law. In particular, n = 1 rep-resents Newton's law, n = -1 stands for another linearheat ransfer law in irreversible hermodynamics andn = 4 stands for he hermal radiation aw. They areoftenused in practiceand or his reaso n, we takeequations ( 5 ) an d (6) as general heat transfer laws.According to the model in which a reversible Car -not refrigeration cycle is carried o ut inside theworkingmaterial, one has

Q J Q I = T d T 1 (7 )and the coefficient of performance is

E = Q 2 / ( Q 1 - Q 2 ) = T2/(T1 - T2). (8)Using eq uatio ns (4)-(7), on e can obtain he averagerate of refrigeration of the cycle asR = Q,/t KZ(% - T 2 ) / ( 1 + t l / t , )- K2- 1 (9 )

3. The relation between optimal rate of refriger-ation and coefficient of performanceIn orde r to find the relation between the optimal rateof refrigerationand the coefficient of perfo rma nce,equat ion (8) is written as

T 1 / T 2= 1 + E - ' , ( 1 0 )and ubstitution of equa tion (10 ) intoequation (9)gives

Using the ex trema1 condition( a R / d T , ) , = 0 ( 1 2 )

we can find( T ? - T I ) - , = ( K 2 / K l ) ( l E - ' ) ~ + ~

X [ ( l+ E - ' ) ~ T $ T L] - 21 3 )from equation ( 1 1 ) . Solving equation (13), we get137


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Zijun Yan and Jincan Chen

T k + [ ( K 2 / K l ) ( l E - ' ) ~ + ' ] ~ / ~t( 1 + + [ ( K 2 / K 1 ) ( 1 E - ' ) ~ + ' ] ~ / ~ 'Tq =

Then , we obtainT t - T" - T r - T $ / ( l +- 1 + [ K 2 / K l ( l+ E - ' ) ' " ~ ] ~ / * ( 1 5 )

an d( 1 + E - ' ) ~ T " HT"- K 2 ( 1+ E - ~ ) " + ' / K ~ ] ~ / ~ [ T ET",(l + E - ' ) " ]- 1 + [ K ~ / K ~ ( I~ - l y - l ] l / ~

( 1 6 )Finally, ubstitution of equations ( 1 5 ) an d ( 1 6 ) intoequation ( 1 1 ) gives the relation between he optimalrate of refrigeration R and he coefficient of per-formance E :

K2[TT - T",(1 + E - ' ) " ]R = { l + [ K 2 / K 1 ( 1 E - ' ) ~ - ~ ] ~ / ~ } ~ ' 1 7 )Since

(aE/aT2)R = - ( ~ R / d T 2 ) , / ( W W r z ( 1 8 )the condition ( d e /d T 2 ) R= 0 corresponds to equat ion( 1 2 ) under the circumstances (d R/d E ) T 2 # 0. There-fore,quat ion ( 1 7 ) alsoeterminesheelationbetween heoptim al coefficient of perform ancean dthe ate of refrigeration. In act, equa t ion ( 1 7 ) is ageneral a nd fu ndam ental relation for the mod el ycles.Th e various optimu m perform ances of such a modelcycle an d the effect of the heat trans fer law on theseperformances can all be deduced from it.

4. Optimal performance of the model cycles forthree common heat-transfer laws4.1. Case n = 1The case n = 1 is an mportantheat ransfermodelwhich is extensivelyused in practice and finite-timethermodynamics [6-141. Whe n n = 1 , itaneobtained, from equation 1 7 ) , hat therelation betweenth eoptim al ate of refrigerationand he coefficientof perform ance, or he relation between he optimalcoefficient of perf orm ance and the ratef refrigeration,of the model cycles is

R = K[TL- & T H / ( l E ) ] ( 1 9 )o r

E = ( T ~R / K ) / ( T , - TL + R / K ) (20)where K = KIK2/(


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coefficient of performance of a reversible Carnotrefrigeration cycle, whereas equation (23) correspondsto the rate of entropy production of a heat conductionprocess between two reservoirs at temperatures TL andT t in which the heat flux is R. It is thus obvious thatunder the circumstances of a given R and K, the per-formance of a Carnot refrigeration cycle affected bythermal resistance corresponds to that of a reversibleCarnot refrigeration cycle whose low-temperature res-ervoir temperature is lower than T L by R/K, whereasthe irreversibility of heat conduction may be treated asa simple heat conduction process between the res-ervoirs at temperatures TL and T t . Therefore, T t istheconcentrated expression of the irreversibility ofheat conduction in theCarnot refrigeration cycle.When TL is small, T t is much smaller than TL . Then,it s seen from equations (22) and (21) that E and Rbecome very small, and tend to zero as T L approacheszero. It is thus clear that T t also indicates the twofolddifficulties n lowering temperature when TL isverylow.

As mentioned above, owing to the effect of thermalresistance, the coefficient of performance and the rateof refrigeration of a refrigerator must be consideredsimultaneously. Equation (19) or (20) just providessome theoretical bases for the question ofhow toreasonably choose the two parameters. For example,when we pay equal attention to both the coefficient ofperformance E and he ate of refrigeration R , themultiplication ER may be taken as an objectivefunction. We can then find from equation (19) or (20)that he coefficient of performanceand he ate ofrefrigeration of a efrigerator at maximum ER con-ditions are respectively given by

0 = [TH/(TH- TL)]1/2-1 (24)and

R0 = K(TH - T L ) EO . (25)Thus, we get

(ER),, = EORO= K(TH - TL)E$. (26)These arehe optimal operation conditions of arefrigerator when the coefficient of performance andthe rate of refrigeration are paid equal attention.

We can find easily from equations (24) and (25)E ~ / E ~Ro/R,, = 1/[1 + (1 + E C ) < 1/2.27)Equation (27) shows clearly that when the coefficientof performance nd he ate of refrigeration of arefrigerator are paid equal attention, the coefficient ofperformance cannot attain the value of eC/2.Thus, forpractical refrigerators, in order to get a certain rate ofrefrigeration, their coefficients of performance have tobe smaller thanHowever, its unsuitable if therate of refrigeration is chosen too large, because thecorresponding coefficient of performance would be toosmall to be beneficial to the reasonable use of energy.In general, R should be smaller than R@ Only in some

special cases, e.g., when a very ow temperature isrequired for the process, can the rate of refrigerationbe chosen to be larger than Ro.Because in such a case,R,,, is very small, heat leakage is quite serious sothata larger rate of refrigeration is needed for therefrigeration. But, this will result in the coefficient ofperformance becoming reduced, so obviously highercosts must be met. It is thus clear that equation (19)or (20) may provide some theoretical bases which aremore significant than he heory of classical thermo-dynamics for choosing the optimal operation conditionsof refrigerators.

(iv) It is interesting to compare the above resultswith the corresponding results of aCarnotengine.These results are listed in table 1, where P is powerand the subscript m denotes the parameter values atthe maximum power conditions. It can be seen clearlyfrom table 1 thathe relation between E,, and issimilar in form to the relation between v, and v,-; therelation between (ER),and c0 is similar in form to therelation between P, and v, and so on. Thus it can beseen that hereare some common characteristics inusing the objective functions P and ER to discuss theoptimal performance of Carnot engines and refriger-ators, respectively. Consequently, theparametersand R. for practical refrigerators play the same instruc-tive role as theparameters r],nd P , for practicalengines. They will be helpful to he urtherunder-standing of the performance limits of practical appar-atus.

4.2. Case n = 4The case n = 4 is a model which is valid for heat trans-fer by thermal radiation [4]. When n = 4, equation (17)may be expresed as

K2[Ti- T&/(1+R = {l+ [K2/K1(1+ E - ~ ) ~ ] ~ / ~ } * ' (28)It is shown from equation (28) that dR/dE < 0. Thisshows that for the case using the thermal radiation law,the optimal rate of refrigeration R of a refrigerationcycle s also a monotonically decreasing function ofthe coefficient of performance E . When R = 0, E =when E = 0, R = K 2 T : = R,,,. This also shows thatonly if the rate of refrigeration is equal o zero canthe coefficient ,of performance attain he value ofreversible thermodynamics foraCarnot efrigeratoraffected by thermal resistance. In fact, this is a generalconclusion which s independent of the heat transferlaw. Its physical meaning is obvious, because it willalways be restricted by the second law of thermo-dynamics whichever heat transfer law is used. Accord-ing to the second law of thermodynamics, heat canonly flowspontaneously from a hotter to aooler body.Consequently, only if the relation T , > TH > TL > T 2is satisfied can the heat conduction be carried out infinite time. On theotherhand, he value of R,,, isobviously dependent on theheat transfer law. Asshown above, R,,, is independent of K1 in the case of

139


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Zijun Yan and Jincan ChenTable 1. The corresponding relations between the Carno t engine and therefrigerator.

0 Em CC EFigure 3. The curve R varying with E for n = -1.n = 4, but is dependent on K1 for the case n = 1. Forsomeheatransferaws, venheoptima l ate ofrefrigeration R is not a monotonic function of E . Forexample, when n = - , he relat ion between R an d Eis just th e circumstances (see figure 3) .4.3. Case n = - 1The case n = -1 is another linear mod el in irreversiblethermo dyn amics wh ich. is also used in practicean dfinite time hermodyn amics [l-3, 51. When n = - ,equation (17) may be expressed as

&(l E )R = - K Z [ E + ( K ~ / K I ) ~ ( I&>]*THTL

where -K1 and -K2 are he so-calledkineticcoef-ficients [15]. It is seen romequation (29) that E =when R = 0. This shows again that it is a genera lconclusion.O n the other hand, R expressed by equation (29)is not a monotonic function of E , and there exists anextremevalue R,,,, asshown in figure 3. From heextrema1 condition dR/d&= 0, we obtain that when& = T I [ ~ T H(K1/K2)12TL- T,] G E , (30)

R at tains the maximum , i .e .R,,, = -K,TL/[4T& -t4(K 1/K 2)12 THTL]. (31)

When R < R,,,, under a given R ,we can get two valuesof E from equation (29), where one is larger han E,and the other is smaller than E,. Evidently, only thelarger one is theoptim al coefficient of perform anceunder the given R. Therefo re, the optimal coefficient140

of performancehould lie betwee n E, a nd heimportant significance of E, lies in the fact that itdeterm ines a owe r limit of the value of the optim alcoefficient of perf orm ance orCarno t efrigerationcycles in this case.In addition, it can be seen from equation (30) thatE , is not only a function of reservoir temp eratures butis also dependent on the atio of hea tconductancesKl/K2. This matter is worthnoting in the tudy ofcycles affected by therm al esista nce,otherwise heconclusions would lose their universality [ 5 ] .The par-ticuiar importance of the identification of heat transferlaws in real herm odyn amic devices canbe nferredfrom the possibility of the existence of d iffere nt heattransfer laws.

5. Relation between E and other parameters(i) Owing to the relation between the average pow erinput P and the rate f refrigeration R of a refrigerationcycle being P = R/E ,we get

from quation (17). Equation (32) determinesherelation etween theoptim al coefficient of per for -mance and he power nput of a Ca rno t refr igq atio ncycle with th e only irreversibility of he at conduction .(ii) Owing to he ate of average ntropypro-duction in a cycle being0 = AS/ t = ( Q I / T H Q $ L > / ~= R[(1+ & - ~ ) / T H ~ / T L ] (33)we getfromequation (17) that he relation betweenthe optim al coefficient of performan ce and the rate ofaverage entro py is

TT-T&/(l+E-l)n l + & 1IT=K2 j 1 + [K , / K l ( I+ & - l )n -1 ] i ) i l , -T , ) .

(34)Equation (34) indicates that no matter how the valueof n is chosen, i .e., no matte r which heat transfer lawis used, we can have no irreversible loss (i.e., U = 0)if, and only if, E = Therefo re he rreversible lossis inevitable in a practical refrigerator.


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Th e relations etween E and ther arametersbesides thosementioned bov e, .g. ,heelat ionbetween heoptim al coefficient of perform ancean dthe rate of loss of availability, etc , a re also determ inedfrom equation (17). In other words, equat ion (17) canplay a more instructive role than the theory of classicalthermodynam ics in the optimal design of refrigerators.

heat condu ctances. Therefo re, for a practical refriger-ator, he uitableheat ransfer law, the easonab lecoefficient of performanceand ate of refrigerationshouldbechosen in accordance with theparticularcircumstances such that the refrigerator can op erat e inthe opt imum condit ions. Theconclusions of finite-timethermodynamics can just provide somenew theoreticalbases for such an optimal design.

6. Conclusion ReferencesIt is shown from the above discussion that even thoughth eperformances of Carno t efrigeration cycles aredifferent romeachother ordifferentheat ransferlaws (fo r different value of n ) , the optimal coefficientof perform ance always decreases as the rate f refriger-ation ncreases, tca nattain heboun d of classicalthermodynamics if, and onlyf, the rateof refrigerationis equ al to zero, and the i rreversible loss of the finiterate of heat condu ction is inevitable. These all resultfrom the secondaw of thermodynam ics. Different heattransfer laws can only change the value of the lowerlimit of theoptimal coefficient of perfo rma nce, hemaximum rate of refrigeration R,,, and other relativequantities. Moreover, for a concrete heat transfer law,in general , these quanti t ies are also depen dent on th e

[ l ] Orlov V N 1985 Sou. Phys.-Dokl. 30 506[2] Vos A D 1985 Am . J. Phys. 53 570[3] Orlov V N and Rudenko A V 1985 Autom. Remote[4] Vos A D 1987 J. Phys. D: Appl . Phys. 20 232[5] Chen L and Yan Z 1989 J. Chem. Phys. 90 3740[6] Curzon F L and Ahlborn B 1975Am. J. Phys. 43 22[7] Rubin M H and Andresen B 1982 J. Appl . Phys. 53 1[S] Rubin M H 1979 Phys. Reu. A 19 1272[9] Salamon P and Nitzan A 1981 J. Chem. Phys. 74 3546

Control 46 549

[lo] Rebhan E and Ahlborn B 1987 Am . J. Phys. 55 423[l11 Ondrechen M J , Rubin M H and Band Y B 1983[l21 Chen J and Yan Z 1988 J. Appl . Phys. 63 4795[l31 Yan Z and Chen J 1989 J. Appl . Phys. 65 1[l41 Yan Z 1984 Cryogenics (in Chinese) 3 17[l51 Callen H 1960 Thermodynamics (New York: Wiley)

J. Chem. Phys. 78 4721

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A class of irreversible Carnot refrigeration cycles wtih a general heat transfer law