Coefficient of performance under optimized figure of merit in minimally nonlinear irreversible refrigerator

0.23 MB·English

by  Y. Izumida

epl draft
Minimally nonlinear irreversible thermodynamics for low dissipa-
tion Carnot refrigerators
2
1 Y. Izumida1,2, K. Okuda3, A. Calvo Herna´ndez4 and J. M. M. Roco4
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2 1 Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-8656, Japan
l 2 Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
u 3 Division of Physics, Hokkaido University, Sapporo 060-0810, Japan
J
4 Departamento de F´ısica Aplicada, and Instituto Universitario de F´ısica Fundamental y Matem´aticas (IUFFyM),
4
Universidad de Salamanca, 37008 Salamanca, Spain
1

]
h
c PACS 05.70.Ln–Nonequilibrium and irreversible thermodynamics
e
m Abstract – We apply the model of minimally nonlinear irreversible heat engines developed by
- IzumidaandOkuda[EPL97,10004(2012)]torefrigerators. ThemodelassumesextendedOnsager
t relations including a new nonlinear term accounting for dissipation effects. The bounds for the
a
t optimizedregimeunderanappropriatefigureofmeritareanalyzedandsuccessfullycomparedwith
s
those obtained previously for low dissipation Carnot refrigerators in the finite-time framework.
.
t Explicit derivation of theOnsager coefficients is also presented.
a
m
-
d
n
o
c Nowadays, the optimization of thermal heat devices is dissipation Carnot heat engine model. In this model the
[ receivingaspecialattentionbecauseofitsstraightforward entropy generation in each heat exchange process is as-

relation with the depletion of energy resources and the sumed to be inversely proportional to the time duration
1
v concerns of sustainable development. A number of dif- of the process, the reversible regime is approached in the
0 ferent performance regimes based on different figures of limit of infinite times and the maximum power regime al-
9 merit have been considered with special emphasis in the lows us to recover the Curzon-Ahlborn value η when
CA
3
analysisofpossibleuniversalandunifiedfeatures. Among symmetric dissipation is considered, but without the re-
3
them, the efficiency at maximum power (EMP) η for quirement of assuming any specific heat transfer law nor
. maxP
7
heat engines working along cycles or in steady-state is thelinear-responseregime,sincethederivationisindepen-
0
largely the issue most studied independently of the ther- dent of the external temperature values. Considering ex-
2
1 mal device nature (macroscopic, stochastic or quantum) tremely asymmetric dissipation limits, these authors also
: and/or the model characteristics [1–14]. Even theoretical derived that EMP is bounded from the lower and upper
v
results have been faced with an experimental realization sides as
i
X for micrometre-sizedstochastic heat engines performing a η η
C C
r Stirling cycle [15]. ηmaxP , (1)
2 ≤ ≤ 2 η
a C
−
For most of the Carnot-like heat engine models ana- where η 1 τ is the Carnot efficiency. The Curzon-
C
≡ −
lyzed in the finite-time thermodynamics (FTT) frame- Ahlborn value η (i.e., the symmetric dissipation limit)
CA
work[5,6]the EMPregimeallowsforvaluableandsimple is located between these bounds. Additional studies on
expressions of the optimized efficiency, which under en- various heat engine models [18–20] confirmed and gener-
doreversible assumptions (i.e., all considered irreversibili- alized above results.
ties coming from the couplings between the working sys- Work by de Toma´s et al. [21] extended the low dissipa-
tem and the external heat reservoirs through linear heat tionCarnotheatenginemodeltorefrigeratorsandbesides
transfer laws) recover the paradigmatic Curzon-Ahlborn proposed a unified figure of merit (χ-criterion described
value [16] η =1 √τ η using τ T /T , where below) focusing the attention in the common character-
maxP CA c h
− ≡ ≡
T andT denotethetemperatureofthecoldandhotheat istic of every heat energy converter (the working systems
c h
reservoirs, respectively. A significant conceptual advance and total cycle time) instead of any specific coupling to
was reported by Esposito et al. [17] by considering a low external heat sources which can vary according to a par-
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Y. Izumida et al.
ticulararrangement. Forrefrigerators,wedenotetheheat
absorbedby the working systemfrom the coldheat reser-
voir by Q , the heat evacuated from the working system
c
to the hotheat reservoirby Q and the workinput to the
h
working system by W Q Q . For heat engines, we
h c
≡ −
may inversely choose natural directions of heat transfers
and work. That is, we denote the heat absorbed by the
working system from the hot heat reservoir by Q , the
h
heat evacuated from the working system to the cold heat
reservoir by Q and the work output by W Q Q .
c h c
≡ −
Then we can define this χ-criterion as the product of the
converter efficiency z times the heat Q exchanged be-
in
tween the working system and the heat reservoir,divided Fig. 1: Schematic illustration of the minimally nonlinear irre-
by the time duration of cycle t : versible refrigerator described by eqs. (12) and (13).
cycle
zQ
in
χ= , (2) comparingtheresultsforEMPanditsboundswitheq.(1).
t
cycle Now, the main goal of the present paper is to extend
the application of this nonlinear irreversible theory [33]
where Q = Q , z = ε Q /W for refrigerators and
in c c
≡ to refrigerators and analyze their COP under maximum
Q = Q , z = η W/Q for heat engines. It becomes
in h h
≡ χ-condition. To get this we first present the model and
power as χ = W/t when applied to heat engines and
cycle
analyze the results when the tight-coupling condition is
also allows us to obtain the optimized coefficient of per-
met. Then, we show that the low dissipation Carnot re-
formance (COP) ε under symmetric dissipation con-
maxχ
frigeratormodel[26]isaparticularcaseandwederivethe
ditions when applied to refrigerators:
corresponding Onsager coefficients.
ε =√1+ε 1 ε . (3) Refrigeratorsaregenerallyclassifiedintotwotypes,that
maxχ C CA
− ≡ is, steady-state refrigerators and cyclic ones. Our theory
Thisresultcouldbeconsideredasthegenuinecounterpart canbe appliedtobothcases. Since the refrigeratorscome
of the Curzon-Ahlborn efficiency for refrigerators. It was backtothe originalstateafter one-cycle,theentropypro-
firstobtainedin FTT for Carnot-likerefrigeratorsby Yan ductionrateσ˙ ofthe refrigeratorsagreeswiththe entropy
and Chen [22] taking as target function εQ˙ , where Q˙ is increase rate of the heat reservoirsas
c c
the cooling power of the refrigerator, later and indepen-
Q˙ +W˙ Q˙ W˙ 1 1
dentlybyVelascoetal.[23,24]usingamaximumper-unit- σ˙ = c c = +Q˙ , (5)
c
time COP and by Allahverdyanet al. [25] in the classical Th − Tc Th (cid:18)Th − Tc(cid:19)
limit of a quantum model with two n-level systems inter-
where the dot denotes the quantity per unit time for
acting via a pulsed external field. Very recent results by
steady-state refrigeratorsand the quantity divided by the
Wanget al.[26]generalizedthepreviousresultsforrefrig-
time duration of cycle t for cyclic refrigerators. From
erators[21]andtheyobtainedthefollowingboundsofthe cycle
thedecompositionofσ˙ intothesumoftheproductofther-
COPconsideringextremelyasymmetricdissipationlimits:
modynamicfluxJ anditsconjugatethermodynamicforce
i
√9+8εC 3 Xi as σ˙ ≡ i=1,2JiXi, we can define J1 ≡x˙, X1 ≡F/Th
0≤εmaxχ ≤ 2 − , (4) earnadtoJr2s.≡TQPh˙ce,nXW˙2 ≡=1F/Tx˙hw−h1e/rTecFforansdteaxdya-rsetattheerteifmrige--
where εC τ/(1 τ) is the Carnot COP. These bounds independent external generalized force and its conjugate
≡ −
for refrigerators play the same role that eq. (1) for heat variable, respectively. Likewise we can also define
engines.
A shortcoming of the all above results is the model de- 1 W
J , X , (6)
1 1
pendence. Thus additional research work has been de- ≡ t ≡ T
cycle h
voted [27–29] to analyze its validity and generality using 1 1
J Q˙ , X , (7)
the well founded formalism of linear irreversible thermo- 2 c 2
≡ ≡ T − T
h c
dynamics (LIT) for both cyclic and steady-state models
including explicit calculations of the Onsager coefficients for cyclic refrigerators, where X1 and X2 play the role
using molecular kinetic theory [30–32]. Beyond the linear of driver and driven forces, respectively. Anyway, we as-
regime a further improvement was reported by Izumida sume that the refrigeratorsare described by the following
and Okuda [33] by proposing a model of minimally non- extended Onsager relations [33]:
linearirreversibleheatenginesdescribedbyextendedOn-
sagerrelations with nonlinear terms accounting for power J1 =L11X1+L12X2, (8)
dissipation. The validity of the theory was checked by J =L X +L X γ J 2, (9)
2 21 1 22 2 c 1
−
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Minimally nonlinear irreversible thermodynamics for low dissipation Carnot refrigerators
where L ’s are the Onsager coefficients with reciprocity Inthefollowingwerestrictouranalysistothecase q =
ij
| |
L = L . The nonlinear term γ J2 means the power 1calledthe tight-couplingcondition. Below,wewillshow
12 21 − c 1
dissipationintothecoldheatreservoirandγ >0denotes thatthisconditionholdsforthelowsymmetricdissipation
c
its strength. We also disregard other possible nonlinear Carnot refrigeratorby calculating its Onsager coefficients
termsbyassumingthattheyaretoosmallwhencompared L ’s explicitly. Generalization of the result for q = 1 is
ij
| | 6
with the power dissipation term. straightforwardalthough cumbersome.
TheOnsagerrelationsareusuallywrittenbyusingonly Forthe refrigerator,the converterefficiency z in eq.(2)
the linear terms with respect to X and X and then the is the COP ε defined as
1 2
non-negativityoftheentropyproductionrateσ˙ =J X +
J2X2 leads to restriction to the Onsager coefficient1s [130, ε Q˙c = J2 = LL1211J1−γcJ12 , (17)
34,35] ≡ P P LL2111(τ1 −1)J1+ LT1h1J12
L11 0,L22 0,L11L22 L12L21 0. (10) whereweusedeqs.(12)and(15). Wealsointroducetheχ-
≥ ≥ − ≥
criterionasaproductoftheCOPεandthecoolingpower
Althoughour model includes the power dissipationas the
J , identifying the cooling power Q /t in eq. (2) as
2 in cycle
nonlinear term, we also assume eq. (10) holds. This idea
Q˙ =J :
that the dissipation is included in the scope of the linear c 2
irreversible thermodynamics is also found in [36–38]. 2
TQ˙hTenheJthpeoiwshegeriavtiennflpuuaxtsiPntoistghievehnobtyhePat≡reWs˙er=voJir1XQ˙h1T≡h >P +0. χ= JP22 = LL2111(cid:16)LLτ12111−J11−Jγ1c+J1L2T(cid:17)1h1J12, (18)
c 3
≡ where we used eqs. (12(cid:0)) and(cid:1)(15). Though eq. (2) us-
J =P +Q˙ =J X T +J . (11)
3 c 1 1 h 2 ing t appears to be valid only for cyclic heat energy
cycle
converters,eq. (18) using J is valid even for steady-state
By solving eq. (8) with respect to X and substituting it 2
1
refrigerators. We consider the optimization of this crite-
into eqs. (9) and (11), we can rewrite J and J by using
2 3
rion with respect to J . From eq. (12), we see that J
J instead of X as (see fig. 1): 1 1
1 1
should be located between 0 and L /(γ L ), in order to
21 c 11
J = L21J +L (1 q2)X γ J 2, (12) getapositivecoolingpowerasitmustbeinarefrigerator.
2 1 22 2 c 1
L − − ∂χ/∂J =0 reduces to the cubic equation:
11 1
L T
J = 21 hJ +L (1 q2)X +γ J 2, (13) 3L L γ 2L ε T
3 L11 Tc 1 22 − 2 h 1 J13+ 21 11 c− 21 C hJ12
2γ L ε T
c 11 C h
whereq L12 is the usualcouplingstrengthparame- 2L2 L3
≡ √L11L22 21 J + 21 =0. (19)
ter(|q|≤1fromeq.(10))andγh >0denotesthestrength −L11εCγcTh 1 2L211γc2εCTh
of the power dissipation into the hot heat reservoir as
Then the sole physically acceptable solution [∂2χ/∂J 2 <
1
T
γ h γ . (14) 0] is given by:
h c
≡ L −
11
Eqs. (12) and (13) allow a clearer description of the re- 2L
Jmaxχ = 21 . (20)
frigerator instead of eqs. (8) and (9) by considering J1 as 1
the control parameter instead of X1 at constant X2. So, 3γcL11+γcL11 8εC 1+ γγhc +9
eachtermineqs.(12)and(13)hasexplicitphysicalmean- r
(cid:16) (cid:17)
ings(seealso[33]): thetwofirsttermsmeanthereversible Substituting eq. (20) into eq. (17), we obtain general
heat transport between the working system and the heat expression of the COP under maximum χ-condition for
reservoirs; the second ones account for coupling effects themodelofminimallynonlinearirreversiblerefrigerators
between the heat reservoirs; and the third ones account with the tight-coupling condition:
for the power dissipation ( J2) into the heat reservoirs,
∼ 1
whUicshinigneevqista.b(l1y2)ocacnudrs(i1n3a) fiinnsitteea-tdimoef meqost.io(n8)(Ja1nd6=(09))., 9+8εC 1+ γγhc −3
ε = r . (21)
maxχ (cid:16) (cid:17)
the power input P = J3 J2 and the entropy generation 2 1+ γh
σ˙ = J3 J2 are written−as γc
Th − Tc (cid:16) (cid:17)
Eq.(21)isthemainresultofthispaper. Itisamonotonic
L 1 T
P = 21 1 J + h J 2 >0, (15) decreasing function of γ /γ and we can obtain the lower
1 1 h c
L τ − L
11 (cid:18) (cid:19) 11 andthe upper bounds by consideringasymmetricdissipa-
and tion limits γh/γc and γh/γc 0, respectively, as
→∞ →
σ˙ =L22(1−q2)X22+J12(cid:18)Tγcc + Tγhh(cid:19)>0. (16) 0≤εmaxχ ≤ √9+82εC−3, (22)
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Y. Izumida et al.
in full agreement with results reported in [26] and ex- In particular, under symmetric conditions (Σ = Σ ) the
h c
pressed in above eq. (4). furtheroptimizationbyαgivesα =√1 τ+1,from
maxχ
−
Nowweprovethattheabovemodelofminimallynonlin- which we can straightforwardly obtain the symmetric op-
ear irreversible refrigerators with the tight-coupling con- timization COP in eq. (3) [21].
ditionincludesthelowdissipationCarnotrefrigerator[26] In summary, we introduced the extended Onsager re-
byexplicitlycalculatingtheOnsagercoefficients. Thislow lations as the model of minimally nonlinear irreversible
dissipationCarnotrefrigeratoris an extensionof the qua- refrigerators under the tight-coupling condition. We de-
sistatic Carnot refrigerator by assuming that heat trans- rived the general expression for the coefficient of perfor-
fer accompanyingfinite-time operationineachisothermal mance (COP) under maximum χ-condition. Calculating
process is inversely proportional to the duration of the theOnsagercoefficientsandthestrengthofthepowerdis-
process t : sipation explicitly, we provedthat our model includes the
i
low dissipation Carnot refrigerator model as a particular
T Σ
Q =T ∆S+ h h, (23) case. Then, the validity of eqs. (3) and (4) derived in the
h h
t
h low dissipation framework, is confirmed by the minimally
T Σ
c c nonlinear irreversible formalism presented here.
Q =T ∆S , (24)
c c
− t
c
where we denote by ∆S > 0 the quasistatic entropy ∗∗∗
changeofthe workingsystemduring eachisothermalpro-
Y. Izumida acknowledges the financial support from
cess, and by Σ /t the corresponding entropy production
i i
a Grant-in-Aid for JSPS Fellows (Grant No. 22-2109).
with a constant strength Σ > 0 [17,21,26]. Since it
i
JMM Roco and A. Calvo Hern´andez thank financial sup-
worksasacyclicrefrigeratorandaccordingtoeqs.(6)and
portfrom Ministerio de Educaci´ony Ciencia of Spainun-
(7), we can define its thermodynamic fluxes and forces as
J 1/t = 1/((α+1)t ), J Q˙ , X W/T and der Grant No. FIS2010-17147FEDER.
1 cycle h 2 c 1 h
≡ ≡ ≡
X 1/T 1/T , respectively, with α t /t . By using
2 h c c h
≡ − ≡
these definitions and eqs. (23) and (24), we can calculate
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