Entropy generation and unified optimization of Carnot-like and low-dissipationrefrigerators

Julian Gonzalez-Ayala1, A. Medina1, J. M. M. Roco1,2 and A. Calvo Hernandez1,2

1Departamento de Fısica Aplicada and 2Instituto Universitario de Fısica Fundamental y Matematicas (IUFFyM)Universidad de Salamanca, 37008 Salamanca, Spain

The connection between Carnot-like and low-dissipation refrigerators is proposed by means oftheir entropy generation and the optimization of two unified, compromise-based figures of merit.Their optimization shows that only a limited set of heat transfer laws in the Carnot-like model arecompatible with the results stemming from the low-dissipation approximation, even though thereis an agreement of the related physical spaces of variables. A comparison between two operationregimes and relations among entropy generation, efficiency, cooling-power and power-input are ob-tained, with emphasis on the role of dissipation symmetries. The results extend previous findingsfor heat engines at maximum power conditions.

I. INTRODUCTION

The pursue of energy converter models to describemore accurately efficient real-life devices, and refrigera-tors in particular, has brought a variety of models and op-timization criteria according to the different nature andscale range: from quantum [1–11] to macroscopic [12–19]passing through mesoscopic [20–29] levels and includingsolar-assisted cooling techniques [30, 31].

Most models are based on finite-time thermodynamicsconsiderations [32, 33]. Among them, but very differentin nature are the Carnot-like (CL) [13–18, 34] and thelow-dissipation (LD)[35–39] models whose predicted re-sults are quite close to the region of experimental data[12]. The key peculiarities of each model can be summa-rized as follow. The CL engine makes use of thermal con-ductances and heat transfer laws to model the heat fluxesand entropy production, this gives information about theheat-exchanges nature and thermal properties of the ma-terial involved in the device. The LD model, on theother hand, allows obtaining upper and lower perfor-mance bounds under certain operation regime depend-ing only on the dissipations’ symmetries, accounted byan specific entropy generation without any informationabout the heat-fluxes nature. The optimization variablesare, in the CL case, the working substance temperaturesat the isothermal processes, while in the LD case are thecontact-times at the isothermal processes. The general-ity of the LD model reproduces behaviors of both en-doreversible and irreversible engines, depending on time-constraints [40]. For the CL engine the difference betweenboth behaviors is the presence of a heat-leak. This arisethe question of whether a correspondence between thesemodels can be established, and if so, to explain the roleof the heat-leak in time-constraints.

In recent years, the LD heat-engine (HE) perspectiveshave increased with its applicability to models that incor-porate fluctuations in microscopic and quantum systems[39, 41–43]. Their relevance in microscopic refrigerator-engines (RE’s), whose studies are mostly based in specificheat transfer mechanisms, have not been widely explored,making the unified study of HE’s and RE’s energetics an

on-going task.

In a recent paper [44] the connection between the CLand the LD models was proposed for HE’s at maximumpower, showing that the set of variables that describeeach model can be related through the entropy genera-tion. The particular case of an inverse-of-temperatureheat law has been recently addressed in [45]. The physi-cal space of parameters is equivalent in the two descrip-tions, however, maximum-power efficiencies do not matchexactly for arbitrary heat transfer laws. This could un-derlay in the nature of each approach: the LD modelbased on a specific entropy-generation law and the CLmodel over the heat fluxes. Nevertheless, for a rangeof heat transfer laws the correspondence between thesemodels is reasonably good.

Beyond models, another important point is the pro-posal of unified figures of merit for any kind of energyconverter [19, 39, 46–51]. In particular, we consider inthis paper two optimization criteria: the so called χ [36]and Ω [52] criteria, whose validity for both RE’s and HE’shas been widely acknowledged [53–58].

To highlight the resemblance between the treatment ofHE’s and RE’s we present a parallel description to thatappearing in [44] for HE’s with special emphasis on therole played by the heat-leaks and contact-times. We willpresent how-well the efficiencies from the CL RE (for awide family of heat transfer laws) fit the LD assumption.

The article is organized as follows: In Section 2 wepresent the unified criteria χ and Ω. In Section 3 themathematical correspondence among the characteristicsvariables of both models for RE’s is proposed. In Sec-tions 4 and 5, respectively, we analyze the maximum-χand Ω regimes in both models; in Section 6 we presenta comparison of the some relevant energetic magnitudes,including the coefficient of performance (COP) and en-tropy generation within the LD model framework. Fi-nally, some concluding remarks are presented in Section7.

2

C

Qh

Qc

Th

Pin

Tc

(b)

Th

Thw

Qh

Qc

C

Tc

QL

Pin

Tcw

(a)

ΔSTh ΔS +Σh

th

ΔSTc -ΔS +Σc

tc

ΔSTh =Qh -QL

Th

ΔSTc = -Qc -QL

Tc

FIG. 1. Sketchs of a LD refrigeration (a) and of an irreversible Carnot-like RE with a heat-leak QL(b).

II. UNIFIED FIGURES OF MERIT χ AND ΩFOR HE’S AND RE’S

The function χ is defined as the efficiency of the enginetimes the heat-flux entering into the working substancein a cycle period, that is,

χ =zQint

, (1)

being z = η (ε) the efficiency of the energy converter andQin = Qh (Qc) for HE’s (RE’s), then

χ(HE) =η Qh

t= P, (2)

χ(RE) =εQc

t= εR, (3)

where P = (Qh −Qc) /t, is the power-output of the HEand R = Qc/t is the cooling-power of the RE. For REthe role of the efficiency is taken by the COP, ε, and theanalogue of Carnot efficiency ηC = 1 − τ and Curzon-Ahlborn (CA) efficiency ηCA = 1 −

√1− ηC is played

by εC = τ/ (1− τ) and εCA =√

1 + εC − 1 [36], whereτ = Tc/Th being the cold-to-hot bath temperature ratio.

In the line of unified studies of HE’s and RE’s a rel-evant role is played by the so-called Ω function. Thisecological-like figure of merit takes into account the un-avoidable losses caused by the irreversible nature of thefinite-time periodic processes and is defined as follows[53]

Ω ≡ (2z − zmax)Eint, (4)

where Ein is the heat input for HE’s and the work inputWin for RE’s. Then,

ΩHE = (2η − ηmax)Qh

t, (5)

ΩRE = (2ε− εmax)Pin, (6)

where Pin = (Qh −Qc)/t.

III. CORRESPONDENCE BETWEEN THE RE’SVARIABLES OF BOTH MODELS

The LD model for RE considers a base-line Carnotrefrigerator working between the temperatures Tc andTh > Tc (see Fig. 1a). A deviation from the reversible sce-nario is modeled by additive terms in the entropy changesat the heat reservoirs, given by [53]

∆STh= ∆S +

Σh

th, (7)

∆STc= −∆S +

Σc

tc, (8)

where Σh and Σc are some dissipative coefficients thatcontain all the information of intrinsic dissipative device-properties; th and tc are the contact-times with the hotand cold reservoirs. We assume that the adiabatic pro-cesses time can be neglected. ∆S is the entropy changeof the working fluid in the cold isothermal process. Thetotal entropy change is

∆Stot =∆STh

Th+

∆STc

Tc=

Σh

th+

Σc

tc≥ 0. (9)

where the reversible scenario (∆Stot = 0) is achieved inthe limits th →∞ and tc →∞.

By means of the dimensionless variables defined in [60]that take into account the size of the system: α ≡ tc/t,

Σc ≡ Σc/ΣT and t ≡ (t∆S)/ΣT, where t = th + tc andΣT ≡ Σh + Σc; we define a characteristic total entropy

3

production per unit time

˙∆Stot ≡

∆Stot

t∆S=

∆Stot

t

ΣT

∆S2=

1

t

[1− Σc

(1− α)t+

Σc

αt

].

(10)In the CL-RE the entropy change of the internal re-

versible cycle is zero and the total entropy production isthat generated at the couplings with the external heatreservoirs as depicted in Fig. 1b. According to this fig-ure Qh = Thw ∆S ≥ 0 and Qc = Tcw ∆S ≥ 0, where ∆Sis the entropy generation at the heat reservoir Tcw, andQL ≥ 0 is a heat-leak between the reservoirs Th and Tc,then

∆STh=Qh

Th−QLTh

= ∆S+(−1 + a−1

h − τQL)

∆S, (11)

∆STc = −Qc

Tc+QLTc

= −∆S+(−ac + 1 + QL

)∆S, (12)

where ah = Th/Thw ≤ 1, ac = Tcw/Tc ≤ 1 and a char-

acteristic heat-leak is defined as QL ≡ QL/ (Tc∆S). Acomparison between Eqs. (7) and (8) gives the expres-sions associated to Σh and Σc,

Σh

th=(−1 + a−1

h − τQL)

∆S (13)

Σc

tc=(−ac + 1 + QL

)∆S. (14)

The heat-leak is not a feature appearing in the LDmodel, nevertheless (as will be shown later) it’s possi-ble to link it within the effects of the dissipation term

TcΣc/tc. From the above two equations it is easy to ob-tain the following relations between the variables of theLD and CL models,

Σ−1c = 1 +

(1− αα

)(−1 + a−1

h − τQL

−ac + 1 + QL

), (15)

t =1

α(−ac + 1 + QL

) [1 +

(1−αα

) (−1+a−1h −τQL

−ac+1+QL

)] ,which can be summarized as:

Σc

αt= −ac + 1 + QL. (16)

This equation allows for a consistent, thermodynamicsinterpretation of the heat-leak in the LD model. It canbe rewritten as

QL = TcΣc

tc− (Tc − Tcw) ∆S, (17)

where the first term of the right-hand-side is the heat dis-sipated to the cold reservoir in the LD model (QLDc, diss).Since ∆S is the working-substance entropy change while

in contact with the Tcw heat reservoir, then, Tc∆S cor-responds to the heat exchanged when Tcw = Tc and(Tc − Tcw) ∆S is the difference between the heat-input atthe totally reversible situation (Qc,rev) and that of non-equilibrium (Qc, neq). We name this quantity Qendoc, loss ≡Qc,rev − Qc, neq, a heat-input loss due to the endore-versibility of the CL engine. In this way, Eq. (16) is

QL = QLDc, diss −Qendoc,loss. (18)

Thus, the heat-leak is the part of the dissipated heat tothe cold reservoir that has not an endoreversible originin the CL model. The term Qendoc, loss is very similar to the

one obtained in the so-called “geometric dissipation”[61],where a dissipation-like term can be attached to a re-versible cycle if one subtracts the heat released by aCarnot cycle from the heat released by the reversiblecycle, when both cycles operate between the same heatreservoirs and with the same heat input.

Regarding the physical region for the RE’s it can bededuced from the Clausius inequality in Eq. (9). Thus we

require that for any ac,h and QL values −∆STc ≥ 0 and∆STh

≥ 0 (Eqs. (11) and (12), respectively). This yields

to a−1h ≥ τQL and ac ≥ QL and since the latter condition

is achieved first it defines the physical constraint QL ∈[0, ac]. This heat-leak constraint (from CL model-basedarguments) applied to Eq. (16) gives a restriction uponthe LD RE variables

Σc ≤ αt. (19)

Later on, we will show that this is equivalent to requiringχ ≥ 0 (with only LD model considerations), showing thatboth physical spaces of variables are in agreement.

The correspondence among the variables of both mod-els (Eq. (16)) is independent of heat transfer laws andoperation regime. Despite of this, it is possible to see theinfluence of the heat-leak on Tcw and Thw (which usu-ally requires an explicit heat transfer law) and on thetotal operation time as can be seen in Fig. 2a and 2b,respectively.

In Fig. 2a the temperatures Tc, Tcw, Thw and Th are

displayed as function of ac and QL. For the CL RE itis well-known that thermal equilibrium between internaland external reservoirs (reversible situation) is achieved

only if QL = 0. As the heat-leak increases the temper-ature Thw departs from Th, producing a larger thermalgradients, and preventing thermal equilibrium. On theother hand, for the LD RE, t as a function of ac and

QL (see Eq. (16)) is depicted in Fig. 2b, showing that asthe heat-leak increases the total-time decreases and only

when ac → 1 and QL → 0 the reversible limit t → ∞can be achieved (if additionally ah → 1). Fig. 2, wherephysical constraints are considered, displays a similar be-havior of that reported for HE’s (see Fig. 2 of ref. [44]).

4

FIG. 2. a) Thw and Tcw from Eq. (15). Note that as heat-leak appears both internal temperatures cannot be in thermal equi-

librium with the external reservoirs. b) Total operation time t(QL, ac) according to Eq. (16). In both figures the representative

values α = 15, τ = 0.4 and Σc = 1

2are considered, the qualitative behavior for other values is the same.

IV. MAXIMUM-χ REGIME

A. Low dissipation refrigerator engine

The input and output heats, (see Fig. 1a), are givenby [60]

˙Qh ≡

Qh

t=

Qh

Tc∆S

ΣT

t∆S=

(−1− 1− Σc

(1− α)t

)1

t,(20)

˙Qc ≡

Qc

t=

Qc

Tc∆S

ΣT

t∆S=

(1− Σc

α t

)τ

t= R, (21)

the corresponding dimensionless power input Pin ≡Pin/

(Tc∆S t

)=

˙Qh −

˙Qc is

Pin =

(1− τ +

1− Σc

(1− α)t+τ Σc

αt

)1

t, (22)

thus, ε and χ (see Eq. 3) are given by

ε ≡ R

Pin=Qc

W= ε =

(1− Σc

αt

)τ

1− τ + 1−Σc

(1−α)t+ τΣc

αt

. (23)

χ ≡ εQc

t=

εQc

Tc∆S

ΣT

t∆S=

(1− Σc

αt

)2τ2

t

1− τ + 1−Σc

(1−α)t+ τΣc

αt

,(24)

Note that from Eqs. (23) and (24) ε and χ are posi-

tive if Σc ≤ αt, which is exactly the constraint appearing

in Eq. (19) from heat-leak arguments. The maximiza-

tion is achieved through α and t by solving the system(∂χ

∂t

)α

= 0 and(∂χ∂α

)t

= 0. The first condition leads to

the optimum value t∗

t∗ =Σc

2α

3 +

√√√√9− τ + 8(

α1−α

)(1−Σc

Σc

)1− τ

(25)

and the further maximization of χ∗(α; Σc, τ) =

χ(α, t∗; Σc, τ) with respect to α is made numerically. Thelower and upper bounds for εχmax are

ε−χmax = 0 ≤ εχmax ≤√

9 + 8εC − 3

2= ε+χmax (26)

corresponding to the Σc = 0 and Σc = 1, respectively.

The symmetric dissipation (Σc = 1/2) optimized COPis,

εsymχmax=√

1 + εC − 1 ≡ εCA, (27)

a result which could be considered as the counterpartfor refrigerators of the CA efficiency η = 1 −

√τ = 1 −√

1− ηC [19].

B. CL refrigerator model without heat-leak(endoreversible model).

Consider a reversible Carnot RE operating between theabsolute temperatures Thw and Tcw (see Fig. 1b) with Qh

5

and Qc given by the heat transfer laws

Qh = σh

(T khw − T kh

)th = T kh σh

(a−kh − 1

)th ≥ 0,(28)

Qc = σc

(T kc − T kcw

)tc = T kc σc

(1− akc

)tc ≥ 0. (29)

where the exponent k 6= 0 is a real number, σh and σc

are the conductances in each process and th and tc arethe times at which isothermal processes are completed.Adiabatic processes’ times are neglected. According toEq. (3) χ is a function depending on the variables ac,ah, tc/th, k, τ and σhc. The endoreversible hypothesis∆STcw

= −∆SThwgives a constraint upon the time ratio

tc/th:

tcth

= σhcacahτ1−k

(a−kh − 1

1− akc

), (30)

where σhc ≡ σh/σc. Whenever a heat-leak is not presentTcw/Thw = acahτ = ε/(1 + ε) and the dependence on ah

can be replaced by ε. In terms of α = (1 + th/tc)−1,Eq. (30) can be written as

α

1− α= σhcτ

k

(1 + ε

ε

)(ετac1+ε

)k− 1

1− akc

. (31)

The optimization of χ = χ (ac, ε;σhc, τ, Th, k) is achieved

through ac and ε by solving(∂χ∂ac

)ε

= 0 for ac and(∂χ∂ε

)ac

= 0 for ε. From the first condition (a∗c) we obtain

χ∗ as a function of ε

χ∗ (ε;σhc, τ, Th, k) = χ (a∗c , ε;σhc, τ, Th, k)

= σhTkh ε

τk−( ε1+ε )

k(√σhc+( ε

1+ε )k−12

)2 , (32)

which shows a unique maximum, obtained by solving

numerically(∂χ∗

∂ε

)a∗c

= 0 for ε. Only for the cases

σhc → 0,∞ the solutions are analytical, which are dis-played in Fig. 3a (dot-dashed curves, green online). InFig. 3, for σhc →∞ beyond k ≈ 2 there are no mathemat-ical solutions for χmax, meanwhile, in the case σhc → 0below k = −1 the solutions are non-physical (ε < 0). Forany σhc ∈ (0,∞) all εχmax ’s are located between thosecurves. It can be seen that the LD bounds provided bythe LD-model (Eqs. (26) and (27)) are fulfilled by the en-doreversible RE only for k = −1 which also occurred forHE’s. Outside the region bounded by the two curves noσhc value can reproduce the LD-model COP’s. The New-tonian heat transfer law (k = 1) is the only case whereall possible values of σhc give the same COP (εCA). Onthe other hand, in the LD model εCA is attached only to

Σc = 1/2. To reconcile these situations it will be shownthat there is only one σhc corresponding to a symmetricdissipation.

From the LD optimization α and t for the case Σc =1/2 are

αsymχmax=

1 +√

1− τ2 +√

1− τ, (33)

tsymχmax=

2 +√

1− τ√1− τ

. (34)

then, ac can be computed according to Eq. (16) and ah isobtained from the condition Tcw/Thw = acahτ = ε

1+ε =

1−√

1− τ by using ε = εCA, then,

asymc,χmax=

2 +√

1− τ2(1 +√

1− τ) (35)

asymh,χmax=

2

2 +√

1− τ(36)

that is, the above ac and ah are obtained from the LDmodel optimization and the endoreversible hypothesis.If the optimization of both models were equivalent it isexpected that by means of Eq. (31) the correspondingσhc, given by the following expression

σsymhc = τk(

1 +√

1− τ1−√

1− τ

) 2k −(

2+√

1−τ1+√

1−τ

)k(2 +√

1− τ)k − 2k

,

(37)would reproduce εCA. In Fig. 3b it can be seen thatthe εCA value is not perfectly reproduced but only in thecases k = −1, 1. However, there is a good agreement inthe region between these two values, a common featurewith the HE case. For k = 1, σhc = 1 +

√1− τ and for

k = −1, σhc = (1−√

1− τ)−2.

C. CL refrigerator with heat-leak.

Now, let us consider a heat-leak of the same kind thanQc and Qh,

QL = σL

(T kh − T kc

)(th + tc)

= T kh σL

(1− τk

)(th + tc) ≥ 0, (38)

where σL is the heat-leak conductance. Then

QL =QL

Qc= ac

T kh σL

(1− τk

)(th + tc)

T kc σc (akc − 1) tc

=ac σLc

(1− τk

)τk (akc − 1)

(thtc

+ 1

)(39)

where σLc ≡ σL/σL, in this case R takes into account theheat transferred by QL, thus, ε and χ read as

ε ≡ R

Pin=

(Qc −QL)

(Qh −Qc)=

(1− QL

)(Qh

Qc− 1) , (40)

χ ≡ εR =R2

Pin=

(1− QL

)2

Qc(Qh

Qc− 1)t, (41)

6

0-1-2-3 1 2 3 4

CO

P (ϵ)

ϵχmax-

ϵχmaxsym

ϵχmax+

Heat transfer law exponent (k)

(a)

ϵχmaxsym

ϵχmaxsym +0.004

ϵχmaxsym -0.004

(b)

0-1 1

σhc →∞

σhc → 0

σhcsym

FIG. 3. In a) the upper and lower bounds of the COP for the CL RE at maximum-χ regime versus the heat transfer lawexponent k. The σhc → 0,∞ along with the σsymhc case (Eq. (37)) are plotted. The bounds provided by Eqs. (26) and (27)are labeled. In b) a close view of εχmax using σsymhc from Eq. (37). In all cases the representative value of τ = 0.7 is used.

where R = (Qc −QL) /t. The heat-leak diminishes theCOP and is responsible for the loop-like COP versus χcurves.

The maximization of Eq. (41) is obtained through ah

and ac by solving numerically(∂χ∂ac

)ah

= 0 for ac and(∂χ∂ah

)ac

= 0 for ah. In the limit situations σc, σh →∞ the solutions can be obtained analytically but in thegeneral case their obtaining requires numerical methods.In Fig. 4 they are depicted for three cases. The effect ofthe heat-leak is more noticeable for higher COP values,meanwhile for lower values it is scarce.

The similarity between the above results for RE andthose reported for HE [44] strengthen the road towardunified studies of heat devices. In this sense, the Ω func-tion will provide useful insights due to the capability ofobtaining analytical closed expressions.

V. MAXIMUM-Ω REGIME

From Eq. (6) and since Pin = W/t = (Qh −Qc) /t,R = Qc/t and ε = Qc/W = R/Pin, if εmax = εC , then

ΩRE = Ω = 2R− τ

1− τPin. (42)

A. Maximum-Ω regime for a LD RE

From Eqs. (21) and (22) Ω ≡ Ω t/(Tc ∆S t

)is given

by

Ω =τ

t

[1− Σc

α t

(2− τ1− τ

)− 1− Σc

(1− α) (1− τ) t

]. (43)

Its optimization is achieved through α and t as inthe χmax case by solving simultaneously the conditions(∂Ω∂α

)t

= 0 and(∂Ω∂t

)α

= 0, leading to

αΩmax=

1

1 +

√1−Σc

Σc(2−τ)

, (44)

tΩmax =2

1− τ

(√1− Σc +

√Σc (2− τ)

)2

, (45)

Ωmax and εΩmax are functions of Σc and τ , the latter

being an increasing function of Σc bounded by

ε−Ωmax

=2

3εC ≤ εΩmax ≤

3 + 2εC4 + 3εC

εC = ε+Ωmax

, (46)

where the lower and upper bounds are obtained in the

limits Σc → 0 and Σc → 1, respectively. In the symmet-

ric case Σc = 1/2,

εsymΩmax

=εC√

(1 + εC)(2 + εC)− εC. (47)

These limits are in agreement with those reported in[53], but obtained in a quite different manner.

B. Maximum-Ω regime for a CL endoreversible RE

The Ω function is given by

Ω = 2R− τ

1− τPin =

Qc

(1− τ) t

(2− τ − τQh

Qc

). (48)

7

0-1-2-3 1 2 3 4Heat transfer law exponent (k)

CO

P (ϵ)

ϵ χ max-

ϵ χ max

sym

ϵ χ max+

Q˜L = 0.005

Q˜L = 0.0

Q˜L = 0.015 Q

˜L = 0.005

Q˜L = 0.0

Q˜L = 0.015

σhc → 0

σhc →∞

FIG. 4. Influence of the heat-leak on the upper and lower bounds of the COP at the χmax regime. The cases σL =0, 0.005, 0.015 are displayed. The representative value τ = 0.7 is used.

From the endoreversible hypothesis Qc

Qh= Tcw

Thw= acahτ

and since ε =(Thw

Tcw− 1)−1

it follows that

Qh

Qc=Thw

Tcw=

1 + ε

ε, (49)

and by means of Eqs. (28) and (29) it leads to

thtc

=τkφ (1− x)

σhc (x τkφk − 1)(50)

where φ ≡ 1+εε and x ≡ akc . Then, Ω is given by

Ω =T kc σhτ

k

1− τ

(1− x) (2− τ − τφ)

1 + τkφ(1−x)σhc(x τkφk−1)

. (51)

Its maximization is achieved by solving(∂Ω∂x

)φ

= 0 for

x and(∂Ω∂φ

)x

= 0 for φ. From the first condition one

obtains x∗, given by

x∗ =

√φ1−k+τ−kφ−k√σhc√

φ1−k+√σhc

, k > 0,

=

√φ1−k−τ−kφ−k√σhc√

φ1−k−√σhc

, k < 0, (52)

both cases lead to the same Ω∗ function, given by

Ω∗ (φ;Th, σc, σhc, τ, k) =

T kh σcσhc

1− τ

(2− τ − τφ)(τk − φ−k

)(√φ1−k −√σhc

)2

. (53)

Finally, εΩmax (σhc, τ, k) is obtained from the condition(∂Ω∗

∂φ

)x∗

= 0. In Fig. 5 the limiting cases σhc → 0,∞of εΩmax are depicted (continuous lines, purple online),these two curves bound any other possible values of theCOP; for k < −1 the lower bound is given by ε = 0.Once more, the region of heat transfer laws where theendoreversible and the LD RE results overlap is limited.Only at k = −1 the upper and lower LD bounds arerecovered.

C. Maximum-Ω for a CL RE with heat-leak

From Eq. (48), with the use of Eqs. (28), (29), (30)and (38),

Ω = 2(Qc−QL)t − τ Qc

(1−τ)t

(Qh

Qc− 1)

=

σcTkc

σhc(2−τ)acah−1

(1−τk)

(σhcacah1−akc

+ τk−1

a−kh

−1

) − 2σLc1−τkτk

(54)

8

-3 -2 -1 0 1 2 3 4

Heat transfer law exponent (k)

CO

P (ϵ)

ϵΩmax-

ϵΩmax

sym

ϵΩmax

+

QL = 0.005; σhc → 0

QL = 0.0; σhc →∞

QL = 0.015; σhc → 0

QL = 0.005; σhc →∞

QL = 0.0; σhc → 0

QL = 0.015; σhc →∞

FIG. 5. COP of the endoreversible (QL = 0) RE operating at Ωmax and two cases with QL = 0.005, 0.015. The representativevalue of τ = 0.7 is used.

that is, the heat-leak contributes with an additive termwhich does not alter the optimization in terms of ac andah. In Fig. 5 it is depicted the influence of the heat-leakon ε for σhc → 0,∞. A noticeable difference with re-spect to the χmax (Fig. 4) is that here the entire boundsare lowered by the heat-leak.

Below, we present a comparison between the tworegimes focused on the role of dissipation symmetries andthe entropy generation within the LD-model framework.

VI. χmax VERSUS Ωmax FOR A LD RE.

A comparison between energetic properties involvingentropy production and COP in the two regimes in theLD model is quite illustrative to get insights about theinfluence of the dissipation symmetries beyond the par-ticularities on the heat transfer mechanisms, taking ad-vantage that the LD approximation is good for certainset of them.

In Fig. 6(a–f) we show the influence of the dissipation

symmetries on α, t, Pin, R, ε and˙

∆S under the χmaxand Ωmax regimes (dashed lines and continuous lines,respectively).

As expected, the operation time and COP are largerfor the latter. When the dissipation is mostly located in

the contact with the cold reservoir (Σc → 1) the entropygeneration is the lesser (Fig. 6f), but not necessarily at-

tached to the smallest cooling power or power-input (see

Fig. 6(c–d)) nor the largest total-time t (Fig. 6b). Noticethat ε increases and the entropy production diminishesas the contact-time α increases.

On the right side of Fig. 6 the comparison of bothregimes is complemented. Fig. 6 (g–h) offers a measureof the distance that each time variable has to be shiftedin order to change the operation regime. Note the exis-tence of well defined maxima, while the ratios depictedin Fig. 6(i–l) are monotonous functions bounded by

the limiting asymmetrical dissipation cases Σc = 0, 1.Depending on the dissipation symmetry the transition ofone regime to another can offer certain advantages. For

example, for small values of Σc (when the major part ofthe dissipation occur at the hot reservoir) the transitiongives the largest gain in terms of the COP, however, itrequires also the largest changes in the cooling power,power input and entropy production. Nevertheless thechanges of α as well as the total operation times aresmall and a passing from one regime into the otherdemands small changes of these control parameters.

In the opposite situation, when Σc → 1 the switchingof regime demands also small variations of α and t,with a small gaining in the COP, but also with smallincrements of the entropy production, cooling power andpower input, resulting also in a suitable case since ineach regime the COP is the largest (see Fig. 6e).

9

FIG. 6. Left: Ωmax (dashed lines, blue online) and χmax (continuous lines, red online). Right: Comparison between Ωmax andχmax. In all cases we use τ = 0.7.

By incorporating the study of the entropy production,relevant information regarding the benefits of having cer-tain dissipation symmetries arise. In Fig. 7 a 3D-plot ofthe total entropy production is shown under the time-constraints t = tχmax and t = tΩmax . Over the surfaces,curves of constant COP are displayed. As can be seen,the entropy can significantly increase or decrease despiteof having the same COP. This is more noticeable for theχmax regime. The curve ε = 0 bounds the physical regionof interest. As the COP value increases the set of possi-

ble α–Σc combinations that produce such COP bound anarrower region, until the limiting situation where only

α → Σc → 1 produce the upper bounds ε+Ωmax,χmax

. As

can be seen, the maximum achievable COP does not cor-respond to the minimum entropy production.

VII. CONCLUDING REMARKS

A relation between the variables that describe the low-dissipation RE model and those that describe the Carnot-like RE has been presented from considerations on theentropy generation. The physical space of the variablesis consistent for both models. However, from the opti-mization process the correspondence between both set ofvariables is only valid for the heat transfer laws with ex-ponent k = −1, 1 and very close in the region (−1, 1).Outside this range of exponents there are COP valuesstemming from the Carnot-like RE that cannot be recov-ered from the low-dissipation model. These results areconsistent with previous studies on HE’s, reinforcing theconcept of the unified criteria of merit by means of the χ

10

ΔStot

Σ˜c

α

ϵΩmax-

ϵχmax

sym

ϵΩmax

sym

ϵ = 0

ϵχmax+

ϵΩmax

+

FIG. 7.˙

∆Stot under the time-constraints t = tχmax (upper surface) and t = tΩmax (lower surface). Over each surface the level

curves denote the configurations of Σc and α that produce the same COP. We use the representative value τ = 0.7.

and the Ω functions.

The role of the entropy generation has been addressedtogether with the criteria of merit, allowing for a com-plementary vision of the role of dissipation symmetriesand the nature of the irreversibilities in the energeticsof these kind of energy converters. In this way a globalstudy of dissipation symmetries’ effects on the COP, en-tropy generation, power input and cooling power can beobtained. As a result, upper and lower bounds of therelative gaining or loosing of such quantities in a change

of operation regime can be evaluated.

ACKNOWLEDGEMENT

J. G.-A. acknowledge CONACYT-MEXICO for par-tial financial support. We acknowledge financial supportfrom Junta de Castilla y Leon under project SA017P17.

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