Efficiency at Maximum Power of Irreversible Engines with Asymmetric

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Efficiency at Maximum Power of Irreversible Engines with Asymmetric Nonlinear Flux−Force Relations Jesper Koning and Joseph O. Indekeu* Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium ABSTRACT: The efficiency at maximum power (EMP) is investigated for classical irreversible thermal or chemical model engines. The heat or particle transport is governed by flux−force relations of the general power-law type. Special attention is given to engines that feature asymmetric transport laws, one for input (of heat or particles) and a different one for output. It is shown that in a couple of case studies, the EMP of such engines is close to the lowest of the two EMPs that would result for symmetric implementations of the transport laws. As a consequence, ideal efficiency at maximum power is only possible in the model in which both flux−force relations are of step-function type.

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PRELIMINARIES Efficiency at maximum power and especially the possibility of reaching close to ideal efficiencies at nonzero power is a subject that is (again) enjoying lively interest in recent months. Some of the studies are dealing with the nonequilibrium thermodynamics of classical and quantum heat engines,1−3 others with the stochastic thermodynamics of micro and nanomachines4,5 or heat engines for black holes at criticality.6 In this Article, we will be concerned with classical macroscopic engines, and we will consider only cyclic processes. The paradigm of these, in the context of reversible thermal engines is, of course, the Carnot cycle. In each phase i of a thermal cycle, the first law of thermodynamics relates the change of internal energy ΔUi to the heat Qi absorbed by the working fluid and the mechanical work Wi done on the working fluid through ΔUi = Qi + Wi. In view of the second law of thermodynamics, not all of the input heat Qin (defined as the sum of all the positive Qi) of a thermal engine can be converted to useful mechanical work Wmech done on the surroundings (defined as minus the sum of all Wi). In particular, the Carnot efficiency eC = 1 − Tl/Th is an upper bound for the efficiency e = Wmech/Qin of all thermal engines that operate between a highest temperature Th and a lowest one Tl. Besides thermal engines, we will be concerned with simple isothermal “chemical” engines. The term simple signifies here that these engines involve no chemical reactions but take up ΔN particles from a reservoir at high chemical potential μh and release the same amount of particles ΔN into a reservoir at low chemical potential μl. In each phase i of the cycle, the first law of thermodynamics reads ΔUi = Qi + Wi + μiΔNi. These engines convert the net chemical energy input Wchem ≡ ΔN(μh − μl) to useful work Wmech ≡ − ∑i Wi, at a single temperature T. In view of the isothermal character of the engine, the reversible heat Q is in this case a state function, since dQ = TdS, S being the entropy (which is always a state function), so that in each cycle Qin = Qout. The first law of thermodynamics then © XXXX American Chemical Society

implies that, in each cycle of the engine, only the net chemical energy is converted to useful work. Therefore, the often used chemical efficiency defined as η ≡ Wmech/Wchem equals unity for a reversible isothermal (chemical) engine, and the thermal contribution to Wmech is zero, because Qin − Qout = 0. The assumption of reversibility is essential here, since for an irreversible isothermal process dQirr < TdS (Qirr is not a state function), and in each cycle, a part of the net chemical energy is lost to heat in the amount ∮ dQirr < T∮ dS = 0. We mention in passing that there is no consensus (yet) on a good definition for the efficiency of a hybrid engine that involves particle transport and operates between two different temperatures. To avoid confusion, we will keep separate track of thermal efficiency e and chemical efficiency η. One could have hoped for a definition of chemical efficiency of the form echem ≡ Wmech/Wchem,in, with Wchem,in the chemical energy associated with the uptake of particles. This would be the closest possible logical analogue to the thermal efficiency e = Wmech/Qin. However, in classical thermodynamics, the chemical potential is only defined up to an arbitrary constant (the value of which can be determined quantum mechanically for specific systems such as the ideal gas by virtue of knowing Planck’s constant), so the form of Wchem, in is undetermined, and consequently, alternatives to η such as echem cannot be used within our classical description. Note that classically the quantity μhΔN can have any nominal value (and sign) so that it cannot be used to represent Wchem, in. A paradigm for a four-phase isothermal chemical cycle of this simple kind was proposed in refs 7 and 8. The first phase consists of isobaric expansion during particle uptake at constant Special Issue: Benjamin Widom Festschrift Received: November 4, 2017 Revised: February 7, 2018 Published: February 9, 2018 A

DOI: 10.1021/acs.jpcb.7b10836 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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transport equations governing these phases correspond to Fick’s law, apart from a simple change of variables. The force that drives the particle flux is expressed in terms of a chemical potential difference instead of, as is usually done, a density (or concentration) difference. This chemical potential difference is denoted by μh − μh* > 0 for uptake and by μl* − μl > 0 for release. The cycle is assumed to be operating endoreversibly between the (starred) intermediate chemical potentials, and the chemical efficiency then becomes η = (μ*h − μ*l )/(μh − μl) < 1. The intermediate chemical potentials are now the variational parameters with respect to which the power can be maximized. For both kinds of engines, it has been shown that the efficiency at maximum power is about 50% of the ideal efficiency. To be more precise

particle number density. The second phase is an isothermal expansion at constant particle number (“isocardinal”). In the compressive third phase, particles are released at constant pressure (and constant particle number density). The cycle closes with the fourth phase of isocardinal isothermal compression. A cartoon of the cycle is shown in Figure 1.

eMP 1 = + 6((Th − Tl )/Th) eC 2

(1)

for the thermal engine, and Figure 1. Cartoon of the four-phase isothermal chemical cycle. (I) Uptake of particles. (II) Expansion at constant number of particles. (III) Release of particles. (IV) Compression at constant particle number.

ηMP =

Tl Th

(2)

for the chemical engine. The former result was obtained using the general arguments in refs 10−14 and the latter in ref 15. For the chemical engine, the correction term can be absent. This is the case for the cycle proposed in refs 7 and 8 when a simple linear flux−force law is employed. The leading term in these expressions, 1/2, is universal for all engines in the sense that it is robust to modifications of the engines and of the transport laws within a wide range of specifications. In particular, for the chemical engine, it was verified that it is robust with respect to including higher-order terms (i.e., going beyond linear response) in the flux−force laws7,8 and with respect to continuously varying the reservoir chemical potential during particle exchange.16 However, it turned out that the universality of the fraction 1/2 can be broken by considering flux−force laws that do not obey linear response at small driving.7,8 Such situations may arise when the transport coefficient behaves anomalously. Transport coefficients that vanish would lead to flux−force laws without a linear term but with a leading third-order term. On the other hand, diverging transport coefficients would also lead to flux−force laws without a linear term but with presumably a leading sublinear term, which reproduces the divergence of the derivative of the flux with respect to the force at vanishing force. Divergent transport coefficients are known to occur in the immediate vicinity of critical points in condensed matter systems. In the context of chemical engines, for example, we could consider working fluids consisting of binary liquid mixtures. We then envisage the possibility of a divergent osmotic compressibility (the derivative of the composition with respect to the chemical potential) close to the critical solution point.17 Alternatively, in the context of thermal engines, it is obvious that a diverging heat conductivity near the critical point of a one-component fluid18−21 would fundamentally alter the heat transport equation. The consequences of considering flux−force laws without a linear term were explored in refs 7 and 8 in the context of chemical engines. For power-law flux−force laws of the type

Reversible cycles (thermal or chemical) proceed infinitely slowly, and therefore, the output power is zero. In order to obtain nonzero power, irreversible processes must be included. For the Carnot cycle, this was implemented by making the isothermal phases irreversible while leaving the adiabatic phases unchanged, except for assuming that all phases now take some finite time. Note that the speeding up of an adiabatic phase does not compromise its character, because no heat must be exchanged with the surroundings. For the isothermal phases, the irreversibility is implemented more drastically. It is assumed that the heat input is driven by a temperature difference between the reservoir temperature Th and the highest cycle temperature T*h < Th. Likewise, the heat output occurs between Tl* > Tl and Tl. The heat transport is irreversible and takes a finite time. It is governed by a transport equation, in which the heat current is proportional to a temperature difference or temperature gradient (Fourier’s law). The cycle itself is now imagined to be shrunk to a (reversible) Carnot cycle between the shifted temperatures Th* and Tl*. This is called the endoreversible approximation. Therefore, the thermal efficiency of the cycle is now e = 1−T*l /T*h . The key idea now is to treat the intermediate temperatures Th* and Tl* as auxiliary variational parameters with respect to which the power of the engine is maximized. The engine now comprises the cycle plus the two irreversible processes attached to the cycle. Clearly, the power becomes zero when the auxiliary temperatures approach the reservoir temperatures, because then, the heat transport becomes extremely slow. Also, the power becomes zero when the auxiliary temperatures approach each other, because then, the cycle can do no work (as the area in the pV diagram shrinks to zero). It is thus plausible that the power is maximal somewhere in between these two limits. It was shown by several scientists, and reported in an often cited later pedagogical note,9 that this variational exercise is simple and that the efficiency at maximum power is given by eMP = 1 −

1 + 6((μh − μl )/μh ) 2

< eC .

flux ∝ force θ

For the chemical cycle, the reasoning is similar, and the calculations are even simpler. The phases that are made irreversible are the particle uptake and release phases. The

(3)

with θ ≥ 0 (for linear response, θ = 1), it was shown that the efficiency at maximum power is nonuniversal in the manner B

DOI: 10.1021/acs.jpcb.7b10836 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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ηMP =

1 1+θ

solutions corresponding to maxima are retained. An example of this procedure is detailed in ref 7. In practice, it is profitable to start from “linearity” (θh,θl) = (1,1), for which the solutions are known analytically and from there to explore the entire θ space moving out using small steps. In Figure 2, the chemical efficiency at maximum power (EMP) is plotted versus the exponent of the particle uptake

(4)

and a concordant result was subsequently obtained for thermal engines.22 This remarkable result suggests that special attention ought to go to the sublinear regime and that even ideal efficiency (at maximum power) is possible in the limit of a stepfunction flux−force relation (θ → 0). It turns out that in this limit, ηMP → 1 and eMP → eC, and moreover, the maximum power is itself maximal for θ → 0.22 It was also shown in ref 22 that these results need not violate recently derived thermodynamical inequalities relating the fluxes and the dissipation (or entropy production).23 We speculate that step-function (discontinuous) flux−force laws might arise in classical thermal systems close to a firstorder phase transition. A latent heat current associated with a moving interface between two coexisting phases of matter is possible at a vanishing temperature difference. Near equilibrium, this transport may be extremely slow, but if the phase transition takes place in an undercooled/overheated system, finite power appears compatible with infinitesimal driving. This is an avenue worth exploring. While first-order phase transitions are not common in the working fluids of engines, they do occur familiarly in working fluids of refrigerators. Of course, for these devices, the coefficient of performance replaces the concept of efficiency. In order to discuss their efficiency as engines, one must run these cycles in reverse.

Figure 2. Chemical efficiency at maximum power (EMP) versus flux− force exponent θh, for the case θl = 0 (numerically 10−6). A fixed stepfunction flux−force law governs the release of the particles. Furthermore, Δμ/kBT = 10−3, and several transport coefficient ratios λh/λl have been used (the upper curve corresponds to the high ratio of 100). For reference, the curve 1 is shown (dashed line), which is

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RESULTS FOR ASYMMETRIC FLUX−FORCE RELATIONS We now proceed to present some new results within the framework that we have recalled. For a simple isothermal chemical engine, we consider, following up on ref 22, that the particle transport from and to the reservoirs is governed by the following generalized flux−force relation μ − μi* dN = λikBT sgn(μi − μi*) i dt kBT

1 + θh

the EMP for symmetric nonlinear transport (θl = θh).

process when particle release obeys a step-function flux−force relation (θl = 10−6) in the small driving regime (Δμ/kBT = 10−3). When θh = 1, the relative efficiency is 1/2, which equals the EMP when both particle exchange processes would be of the linear kind. For decreasing θh, the efficiency rises and reaches unity when θh → 0. For reference, 1/(1 + θh) is also plotted, and we observe that the efficiency gain is modest with respect to symmetrical transport, in which both particle exchange processes would be modeled by one and the same θ equal to θh. In the symmetric case (θl = θh), the EMP is independent of the ratio of the transport coefficients λh/λl and independent of the driving strength Δμ/kBT. This is no longer the case for unequal exponents (θh ≠ θl). For example, for a transport coefficient ratio favorable for uptake of particles (high λh/λl), the EMP is enhanced in the range of 0 ≤ θh < 0.5. We observe in Figure 2 the striking feature that the EMP of the engine is dominated by the largest of the two exponents, which is θh (since θl = 0). In contrast, in Figure 3, we depict the EMP when the particle release phase obeys the usual linear flux−force relation (θl = 1). The efficiency at maximum power remains close to 1/2 for 0 ≤ θh ≤ 1 and is only slightly enhanced when 0.4 < θh < 1. This effect is more pronounced for low λh/λl. On the other hand, for θh > 1, the EMP decreases and follows the characteristic 1/(1 + θh) well, as if we were dealing with a symmetric system. We conclude, for both cases investigated, that the EMP of the engine is dominated by the EMP that it would display if both particle transports would have the same exponent with a value equal to the largest of θh and θl.

θi

(5)

where the left-hand side constitutes the particle current, λi (i = l or h) is a transport coefficient, μi is a reservoir chemical potential, μi* is an auxiliary chemical potential of the working fluid in the engine, kB is the Boltzmann constant, and T is the absolute temperature. The exponent θi determines the degree to which the transport is intrinsically nonlinear (θ = 1 for the usual linear law). In previous work on nonlinear transport, it was assumed that the exponent θ is the same in both uptake and release branches of the thermodynamical cycle.7,8,22 Here we give this a natural generalization and consider the situation in which the two transport processes do not follow the same flux−force relations. For example, when one transport process occurs near a phase transition or a critical point, the other might be farther away. Two such situations are examined further (numerically). First, particle uptake occurs at variable θh, but particle release occurs in the step-function limit, θl → 0 (θl = 10−6). Second, particle uptake still occurs at variable θh, but particle release is now governed by the usual linear transport, θl = 1. The computations proceed as follows. The partial derivatives of the power with respect to the auxiliary variables (chemical potentials or temperatures) are set equal to zero. The solutions of these coupled equations are obtained numerically (in general) or analytically (in special cases). The extremal C

DOI: 10.1021/acs.jpcb.7b10836 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 3. Chemical EMP versus θh for θl = 1, Δμ = 10−3/kBT, and several values of the ratio λh/λl. It is conspicuous that the EMP closely follows, and only slightly exceeds, the minimum value of the two EMPs 1 and 1 , associated with symmetric engines in which 1 + θh

Figure 5. Relative thermal EMP versus θh for θl = 1, 1−Tl/Th = 10−3, and several values of κh/κl. For reference, 1 , which is the EMP for 1 + θh

symmetric nonlinear transport, is shown for θh > 1, and

1 + θl

both flux−force relations carry the same exponent θ equal to θh and θl, respectively.

dt

= κiT̅ sgn(Ti − Ti*)

Ti − Ti* T̅

is shown

for 0 < θh ≤ 1. Note that, in accord with the result for the chemical engine, the EMP closely follows the EMP curve for a symmetric engine (dotted and dashed lines) with the largest of the two exponents θh and θl = 1.

The same investigation can be made for the thermal engine. The force−flux relations when the working fluid is in contact with the reservoirs are dQ i

1 1 + θl

irreversible processes of, respectively, heat and particle transport that were built around an endoreversible cycle. Following recent works, we have studied intrinsically nonlinear transport laws in the form of flux−force relations that can carry an exponent θ ≠ 1. We have generalized these to the case of asymmetric engines, in which the input and output transport feature different exponents. Studying these engines with asymmetric flux−force relations in a couple of interesting limits (one with a linear law combined with a general nonlinear one and another with a step-function law combined with a general nonlinear one) has indicated two properties. The first of these we formulate as a conjecture. The EMP of an engine with asymmetric transport laws is close to the lowest of the two EMPs of engines, in which the transport laws are applied symmetrically. That is

θi

(6)

where κi (i = l or h) is a coefficient related to the heat conductivity, Ti is the temperature of the reservoir in contact with the working fluid, which is operated at auxiliary temperature T*i , and T ≡ (Th + Tl)/2. Again, θi determines the extent of the nonlinearity of the flux−force relation. The EMP relative to the Carnot efficiency as a function of the exponent θh is shown in Figures 4 and 5. Again, we observe the same qualitative features as for the chemical engine.

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CONCLUSION AND OUTLOOK We have considered classical (macroscopic) thermal and chemical engines that operate at nonzero power by virtue of

EMP(θh , θl) ≈ min{EMP(θh , θh), EMP(θl , θl)} ⎧ 1 1 ⎫ ⎬ = min⎨ , ⎩ 1 + θh 1 + θl ⎭

(7)

The second property that can be observed upon inspection of the figures is that the EMP never appears to fall below the righthand side of eq 7. Moreover, by manipulating the transport coefficient ratio κh/κl (thermal engine) or λh/λl (chemical engine) the EMP can be improved to reach values somewhat above this lower bound. We therefore conclude, based on the few cases studied, that the following inequality is likely to hold ⎧ 1 1 ⎫ ⎬ EMP(θh , θl) ≥ min⎨ , ⎩ 1 + θh 1 + θl ⎭

(8)

A further comment concerns the possibility of ideal efficiency at nonzero power. Apparently, based on our explorations so far, the ideal efficiency at maximum power that was found in previous work only occurs in the case of symmetrically applied step-function flux−force relations. A heat engine which might have some features in common with this model system is one in which the working fluid is close to gas−liquid coexistence and therefore can undergo one (or two) first-order phase

Figure 4. Relative thermal EMP versus θh for θl = 0 (θl = 10−3), 1 − Tl/Th = 10−3, and various values of the ratio κh/κl. For reference, 1 1 + θh

is shown (dashed line), which is the EMP for symmetric nonlinear transport. D

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(10) Van den Broeck, C. Thermodynamic efficiency at maximum power. Phys. Rev. Lett. 2005, 95, 190602. (11) Esposito, M.; Lindenberg, K.; Van den Broeck, C. Universality of efficiency at maximum power. Phys. Rev. Lett. 2009, 102, 130602. (12) Van den Broeck, C. Efficiency at maximum power in the lowdissipation limit. Europhys. Lett. 2013, 101, 10006. (13) For a historical perspective, see Moreau, M.; Pomeau, Y. Carnot principle and its generalizations: A very short story of a long journey, Eur. Phys. J.: Spec. Top. 2015, 224, 769−780.10.1140/epjst/e201502426-7 (14) Calvo Hernandez, A.; Roco, J. M. M.; Medina, A.; Velasco, S.; Guzman-Vargas, L. The maximum power efficiency 1 − √τ: Research, education, and bibliometric relevance. Eur. Phys. J.: Spec. Top. 2015, 224, 809−823. (15) Van den Broeck, C.; Kumar, N.; Lindenberg, K. Efficiency of isothermal molecular machines at maximum power. Phys. Rev. Lett. 2012, 108, 210602. (16) Koning, J.; Koga, K.; Indekeu, J. O. Efficiency at maximum power for an isothermal chemical engine with particle exchange at varying chemical potential. Eur. Phys. J.: Spec. Top. 2017, 226, 427− 431. (17) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press, Oxford, 1982. (18) Anisimov, M. A. Fifty years of breakthrough discoveries in fluid criticality. Int. J. Thermophys. 2011, 32, 2001−2009. (19) Sengers, J. V In Critical Phenomena, N.B.S. Misc. Publ. No 273, Green, M. S., Sengers, J. V., Eds.; U.S. National Bureau of Standards: Washington, D. C., 1966; p 165. (20) Sengers, J. V. Transport properties of fluids near critical points. Int. J. Thermophys. 1985, 6, 203−232. (21) Sengers, J. V.; Perkins, R. A. In Experimental Thermodynamics Vol IX: Advances in Transport Properties of Fluids; Assael, M. J., Goodwin, A. R. H., Vesovic, V., Wakeham, W. A., Eds.; Royal Society of Chemistry, 2014; pp 337−361. (22) Koning, J.; Indekeu, J. O. Engines with ideal efficiency and nonzero power for sublinear transport laws. Eur. Phys. J. B 2016, 89, 248. (23) Shiraishi, N.; Saito, K.; Tasaki, H. Universal trade-off relation between power and efficiency for heat engines. Phys. Rev. Lett. 2016, 117, 190601.

transitions, as we already suggested in our consideration of preliminaries. Making the speculations more concrete, consider a working fluid that is entirely liquid at temperature Th, but thermodynamically at coexistence. Taking up (latent) heat from the reservoir, it expands isobarically and isothermally due to evaporation of the liquid and eventually enters the single-phase region (vapor) where it heats up somewhat. Next, there is adiabatic expansion of the working fluid, so that it reaches Tl, which is again at two-phase coexistence but now on the vapor side. Now, (latent) heat is rejected and some but not all of the vapor condenses, again isothermally and isobarically. Finally, adiabatic compression of the working fluid brings it back to its original state at Th. This example suggests that the step-function limit of the flux−force relation, θ = 0, could be realized in the laboratory in connection with first-order phase transitions, whereas the more subtle experimental circumstances of critical points should rather be associated with values of θ less than, but closer to, unity. Possibly, the value of θ could bear a relationship to the value of critical exponent involved in the divergence of the relevant transport coefficient. In conclusion, we presented a numerical study on the EMP involving two transport processes that are not described by the same type of flux−force relations, a condition which we called asymmetric. The lower bound of the EMP is that of the EMP with symmetric transport laws with the highest value of the exponent θ in the flux−force relation. Slight enhancements on this EMP are found when we choose more favorable ratios of the transport coefficients, but these enhancements are only noticeable over a limited range of values of the variable exponent θ.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Joseph O. Indekeu: 0000-0002-6160-6538 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS There has been no funding source for this research. REFERENCES

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DOI: 10.1021/acs.jpcb.7b10836 J. Phys. Chem. B XXXX, XXX, XXX−XXX