Linear irreversible heat engines based on the local equilibrium assumptions

We formulate an endoreversible finite-time Carnot cycle model based on the assumptions of local equilibrium and constant energy flux, where the efficiency and the power are expressed in terms of the thermodynamic variables of the working substance. By analyzing the entropy production rate caused by the heat transfer in each isothermal process during the cycle, and using an endoreversible condition applied to the linear response regime, we identify the thermodynamic flux and force of the present system and obtain a linear relation that connects them. We calculate the efficiency at maximum power in the linear response regime by using the linear relation, which agrees with the Curzon-Ahlborn efficiency known as the upper bound in this regime. This reason is also elucidated by rewriting our model into the form of the Onsager relations, where our model turns out to satisfy the tight-coupling condition leading to the Curzon-Ahlborn efficiency.


I. INTRODUCTION
The physics of heat engines originates from the Carnot's great discovery of the fundamental upper bound of the thermodynamic efficiency η of heat engines working between two heat reservoirs with temperatures T R h and T R c (T R h > T R c ) [1][2][3]: where the equality holds for an infinitely slow process (quasistatic limit) with zero dissipation realized in, e.g., the Carnot cycle. Indeed, this discovery may be regarded as the origin of thermodynamics itself. However, the quasistatic limit is an ideal case, and the thermodynamic processes observed in daily life occur at finite rates. Remembering that we always demand power for our use of electric devices, which may originally be generated from power plants converting thermal energy into electric power, we require the physics of powerful heat engines free from the limitation of the equilibrium thermodynamics. This deep understanding of powerful heat engines is becoming more important due to the worldwide energy crisis and climate change.
The physics of heat engines maximizing the power rather than the efficiency was developed in a classical paper by Curzon and Ahlborn [4] (see also [5,6] for similar previous studies). They showed that, under the assumption of the endoreversible condition and the Fourier law of heat conduction between the working substance and the heat reservoir, the efficiency at maximum power η * of a finite-time Carnot cycle is given by the following Curzon-Ahlborn (CA) efficiency: Because this CA efficiency displays a similar simplicity to the Carnot efficiency, it led to the development of a new discipline of finite-time thermodynamics that aims to account for the efficiency of actual power plants and thermal devices [7][8][9][10][11][12][13]. The key to the derivation of the CA efficiency is the phenomenological assumption of the endoreversible condition, which means that the irreversibility occurs only by the heat transfer process between the working substance and the heat reservoir, and that the state of the working substance is internally reversible whose entropy change along the cycle is expressed by a Clausius-like equality (see Eq. (13)). Despite its importance, even until recently, there had been no argument showing whether the CA efficiency is universal as η * from the viewpoint of fundamental physics. The role of the CA efficiency has become increasingly important after Van den Broeck [14] proved that the CA efficiency is the upper bound of η * in the linear response regime by using the Onsager relations of the linear irreversible thermodynamics framework [15][16][17]: where the bound is realized under the tight-coupling (no heat-leakage) condition [14]. As reviewed in [18][19][20][21], since then, various studies on finite-time heat engines have been conducted, which include linear response [22][23][24][25][26][27][28][29], nonlinear response [30][31][32][33][34][35][36][37][38][39], stochastic [40][41][42], quantum [43][44][45][46], thermoelectric [47][48][49], photoelectric [50], molecular dynamics [51][52][53][54], and experimental [55][56][57] studies.
where X 1 is a "mechanical" thermodynamic force and X 2 is a "thermal" thermodynamic force that is proportional to the temperature difference between the heat reservoirs ∆T R ≡ T R h −T R c , J 1 and J 2 are their conjugate thermodynamic fluxes, and L ij 's are the Onsager coefficients with reciprocity L 12 = L 21 (see Sec. II C for details). While this viewpoint is familiar in steady-state heat energy conversion, such as in thermoelectric devices usually analyzed with the Onsager relations [1,21,48,49], an identical formulation has also been established even for cyclic heat engines such as a finite-time Carnot cycle [26,27].
Despite these successes, these theories of finite-time heat engines may still be abstract compared to the theory of the quasistatic heat engine. One reason could be that a general state of a finite-time heat engine cannot be drawn on a thermodynamic plane, thus a clear picture is lacking, unlike the quasistatic cycle with well-defined thermodynamic variables of the working substance such as the pressure, volume, and so on. Even in a finite-time cycle, however, it may still be possible to assume that the state of the working substance is specified by a unique combination of the thermodynamic variables at any instant along the cycle, and hence the working substance and the heat reservoirs are in a local equilibrium, but not in a global equilibrium similar to the quasistatic cycle. According to this local equilibrium assumption, we can draw the finite-time cycle on the thermodynamic plane as well as the quasistatic cycle. In fact, Rubin introduced this local equilibrium thermodynamic description to the endoreversible cycle [7] (see also [35]), where it is shown that the endoreversible condition automatically holds as a natural consequence of the local equilibrium assumption. Therefore we are naturally motivated to elucidate how these endoreversible heat engine models based on the local equilibrium assumption are compatible with the more recent linear irreversible thermodynamic description using the Onsager relations Eqs. (4) and (5).
In the present study, we formulate an endoreversible finite-time Carnot cycle model based on the local equilibrium assumption. In our framework, the power and the efficiency can be expressed in terms of the thermodynamic variables of the working substance. From the analysis of the entropy production rate caused by the heat transfer in each isothermal process during the cycle, we identify the thermodynamic flux and force in each isothermal process, where the flux and force are assumed to be related by the Fourier law. From the calculation of the efficiency at the maximum power by using the thermodynamic flux and force, we obtain the CA efficiency. We also elucidate that the relationship between the thermodynamic flux and force in our framework can be rewritten into the form of the Onsager relations by a variable change, from which we can directly confirm the tight-coupling condition leading to the CA efficiency.
The present paper is organized as follows. In Sec. II A, we introduce our model based on the local equilibrium assumption. In Sec. II B, we analyze the entropy production rate of our heat engine, and naturally identify the thermodynamic fluxes and forces to describe the heat engine. We then calculate the efficiency at the maximum power in the linear response regime by using them. In Sec. II C, we elucidate the relationship between the thermodynamic fluxes and forces obtained in Sec. II B and the Onsager relations Eqs. (4) and (5), explicitly showing that our model surely satisfies the tight-coupling (no heat-leakage) condition leading to the CA efficiency. In Sec. III, we discuss a few aspects related to our formulation and a possible extension of our analysis developed in the Sec. II. We summarize our study in Sec. IV.

II. MODEL AND RESULTS
A. Local equilibrium thermodynamic formulation of the endoreversible finite-time Carnot cycle Our heat engine model consists of the working substance, the hot heat reservoir with temperature T R h and the cold heat reservoir with temperature T R c . We assume that the working substance is always in a local equilibrium state specified by a unique combination of the well-defined thermodynamic variables, and the heat reservoirs are also in a local equilibrium state. Denoting the internal energy and the entropy of the heat reservoir by U R i and S R i (i = h, c), respectively, the temperature is defined by 1 We also denote the internal energy and entropy of the working substance by U and S, respectively. Hereafter, we use the suffix i to denote the thermodynamic variable of the working substance when it contacts with the heat reservoir with the temperature T R i . We then obtain the first law of thermodynamics (energy-conservation law) by using these thermodynamic variables as [7] − where we define 1 Ti ≡ ∂Ui ∂Si and Pi Ti ≡ ∂Si ∂Vi through the fundamental thermodynamic relation with T i , P i , and V i being the temperature, pressure, and volume of the working substance, respectively. Eq. (6) states that the heat flux from the heat reservoir − dU R i dt , which is the internal-energy change rate of the heat reservoir, is decomposed into the internal-energy change rate of the working substance dUi dt and the instantaneous power output P i dVi dt . Our heat engine experiences a thermodynamic cycle that consists of (I) an isothermal expansion process in contact with the hot heat reservoir with the temperature T R h , (II) an adiabatic expansion process, (III) an isothermal compression process in contact with the cold heat reservoir with the temperature T R c , and (IV) an adiabatic compression process (see Fig. 1 (a)). We assume that the durations of the adiabatic processes are negligible compared to the ones of the isothermal processes, and the thermodynamic states of the working substance move along the quasistatic adiabatic curves in the thermodynamic plane. From this, our heat engine can be regarded to run one cycle in the cycle time t cyc ≡ t h + t c , where we denote by t i the duration of the isothermal process in contact with the heat reservoir at T R i . Additionally, in our formulation, we also assume that the energy flux corresponding to each term in Eq. (6) is constant at any instant along each isothermal process, and the temperature T i does not change during the isothermal process, where these assumptions are also adopted in the original CA model [4]. Using Eq. (7), we can rewrite Eq. (6) as The heat from the heat reservoir during the isothermal process Q i is calculated by using Eq. (8) as where we defined the entropy change of the working substance during the isothermal process as Because we require that the cycle is closed after one cycle and we also assume that the adiabatic processes are regarded as quasistatic processes, the following relations should hold: We note that the endoreversibility condition [4] automatically holds from Eqs. (9), (10) and (12) [7]. This condition manifests that the the entropy production is occurred only by the heat transfer between the heat reservoirs and the working substance, and the internal state of the working substance is, as assumed above, in a local equilibrium state, which implies that the entropy change of the working substance along the cycle is expressed by the Clausius-like equality using the temperature of the working substance. Eq. (13) also implies that the efficiency of this type of the heat engines is given by the "endoreversible Carnot efficiency" using the temperatures of the working substance as This differs from the usual Carnot efficiency Eq. (1), which uses the temperatures of the heat reservoirs. The power outputẆ of the heat engine is also expressed by using the entropy change aṡ where ∆T ≡ T h − T c and we used the first law of thermodynamics W = Q h + Q c for one cycle. Hereafter we denote by the dot the quantity divided by the cycle period. Then the rightmost expression in Eq. (15) using the thermodynamic variables of the working substance may be regarded as "endoreversible power." The power and the efficiency, which are defined by using only the thermodynamic variables of the working substance in this way, are a characteristic of our local equilibrium description of the endoreversible heat engine model.

B. Efficiency at maximum power in the linear response regime
In this section, we consider the efficiency at the maximum power of our heat engine in the linear response regime ∆T R → 0, based on the local equilibrium assumption introduced in Sec. II A.
Our analysis using the linear irreversible thermodynamics begins from the entropy production rate. Because we assume that the heat reservoirs and the working substance are always in a local equilibrium state with the well-defined entropies, the entropy production rate ds dt of the total system (the working substance and the heat reservoirs) at any instant along the isothermal process is expressed by the sum of the entropy change rates of these partial systems: where we used Eq. (6). Then, the entropy production rate for one cycleσ is written aṡ where we defined the internal energy change of the heat reservoir during the isothermal process as From Eq. (17), we naturally define the thermodynamic force as the (inverse) temperature difference between the working substance and the heat reservoir, and we define the conjugate thermodynamic flux as the heat flux from the heat reservoir [35] (see Fig. 1 (b)): Because the energy flux and the temperature of the working substance is assumed to be constant along each isothermal process, we obtain by using Eq. (8). Then the thermodynamic flux J Qi can also be expressed by using the time derivative of the thermodynamic variable of the working substance as where we denote by a i the ratio of the duration of each isothermal process t i to the cycle time t cyc . To proceed further, we need a relation that connects J Qi and X Ti , in addition to the local equilibrium thermodynamic formulation mentioned in Sec. II A. Because we are adopting the local equilibrium thermodynamic assumption, it is also quite natural to assume that the heat flows in proportion to the temperature difference (the Fourier law) in the same way as the original CA model [4]: where we denote by κ i the thermal conductance between the heat reservoir with the temperature T R i and the working substance. Using Eq. (22), we then obtain the following relationship between J Qi and X Ti : In the following, we consider the linear response regime ∆T R → 0. Using Eq. (23), we can write the endoreversibility condition Eq. (13) as for the zeroth and first orders of ∆T and ∆T R of Eq. (13), where T ≡ T h +Tc 2 and T R ≡ are the averaged temperatures. Then, order by order, we obtain the following relations in the linear response regime as the consequence of the endoreversibility condition Eq. (13). Using Eq. (26), we find that X Ti 's are expressed in terms of ∆T and ∆T R as follows: where we defined the "reduced thermodynamic force" X T as From Eqs. (28) and (29), we find that X T h and X Tc have opposite signs in the linear response regime. We can simplify the entropy production rate Eq. (17) by using X T up to the quadratic order of X T aṡ where J Q is the averaged heat flux [38,58,59] defined as Therefore the description of the present heat engine model is reduced to this linear relation Eq. (32). In the linear response regime, the endoreversible Carnot efficiency Eq. (14) and the endoreversible power Eq. (15) are also approximated by using X T as respectively, whereẆ is a quadratic function of the thermodynamic force X T . The quasistatic limitẆ = 0 is realized at X qs T = 0 (∆T qs = ∆T R ), while the maximum power ∂Ẇ ∂XT = 0 is realized at In the quasistatic limit X qs T = 0, we attain the Carnot efficiency η C ≃ ∆T R T as the maximum efficiency from Eq. (33). At the maximum power, from Eqs. (33) and (35), the efficiency η * is given by which is the CA efficiency corresponding to the equality in Eq. (3). The maximum powerẆ * iṡ which depends on the thermal conductivity [4]. We note that the endoreversible power as given in Eq. (34) should satisfy the first law of the thermodynamicṡ W = J Q h + J Qc for one cycle. To this end, we need to consider J Qi with higher-order correction of ∆T and ∆T R , while we have considered only the lowest order so far. As a simplest choice, we can assume the following form that includes a term in proportion to the power itself, which is a quadratic term of ∆T and ∆T R as given in Eq. (34): where b h + b c = 1. We note that these higher-order terms do not affect the entropy production rate up to the quadratic order of ∆T and ∆T R in Eq. (31), and Eqs. (38) and (39) including these nonlinear terms still satisfy the endoreversibility condition Eq. (13). We here choose b i as b h = b c = 1 2 for simplicity (see Sec. III A for this reason). Then, by using Eq. (34), we can approximate the heat fluxes in Eq. (23) in the linear response regime as Similar expressions to Eqs. (38) and (39) have been previously obtained based on the weighted reciprocal of the temperatures and thermodynamic forces in [38].
C. Formulation of the endoreversible finite-time Carnot cycle model using Onsager relations As we have shown in Sec. II B, the efficiency at the maximum power η * in Eq. (36), which is based on the local equilibrium thermodynamic formulation using the linear relation Eq. (32), is the upper bound in Eq. (3), while the inequality in Eq. (3) comes from the formulation based on the Onsager relations [14]. Therefore, we elucidate the relationship between these formulations in this section.
First, we briefly review the derivation of the inequality for the efficiency at maximum power in Eq. (3) [14]. Denoting an external force and its conjugate variable by F and x, respectively, we can generally express the power of the heat engineẆ asẆ = −Fẋ. Then the entropy production rate of the total systemσ is decomposed into the sum of the entropy increase rate of each heat reservoir because the state of the working substance should return to the original state after one cycle: Here, the thermodynamic fluxes J 1 ≡ẋ and J 2 ≡Q h , and their conjugate thermodynamic forces X 1 ≡ F T R and X 2 ≡ ∆T R T R2 are related through the Onsager relations Eqs. (4) and (5). We note that, in this case, we do not necessarily assume that the working substance along the cycle is expressed in terms of the well-defined thermodynamic variables, in contrast to our formulation in Sec II B. Using these thermodynamic fluxes and forces, the power and the efficiency are given asẆ With these expressions as well as the Onsager relations Eqs. (4) and (5), we find that the maximum power is realized at X * 1 = − L12X2 2L11 from ∂Ẇ ∂X1 = 0. Its efficiency η * is given as which is a monotonically increasing function of |q|, where the coupling strength q is defined by From the non-negativity of the entropy production rateσ = J 1 X 1 + J 2 X 2 , the Onsager coefficients L ij 's should satisfy L 11 ≥ 0, L 22 ≥ 0, and L 11 L 22 − L 2 12 ≥ 0, and they impose the following constraint on q: where the equality is known as the tight-coupling (no heat-leakage) condition [14,17]. Under this tight-coupling condition, η * in Eq. (45) attains the upper bound given by the CA efficiency η CA as in Eq. (3). An essential point of the derivation of the formula Eq. (45) is that the non-zero cross-coefficient L 12 plays an important role in η * in Eq. (45), which is clear from the definition Eq. (46). Returning to our original problem, from Eq. (23), we formally obtain the following "Onsager coefficients" under our choice of the thermodynamic fluxes J Qi and forces X Ti : where there are no nondiagonal elements. This contrasts to the formulation using Eqs. (4) and (5) where the crossterms play an important role in the heat-energy conversion into work [14]. Eq. (48) is natural if the entropy production originating from the heat transfer between the working substance and the heat reservoir in each isothermal process is independent of each other. However, as seen from Eqs. (28) and (29), X Ti 's are not independent of each other, unlike X i 's in Eq. (42), but X Ti 's and X i 's should be related with each other by a variable change. To elucidate this point, we restate our expression of the entropy production rate Eq. (17) with X Ti 's using independent thermodynamic forces as (see Fig. 1 We can make this restatement by using the endoreversibility condition Eq. (13), the first law of the thermodynamics W = Q h + Q c for one cycle, and Eq. (15), where we defined the heat flux from the hot heat reservoir as a new thermodynamic flux and its conjugate new thermodynamic force as In addition, we defined the entropy flux as another new thermodynamic flux and its conjugate new thermodynamic force as In this way, all thermodynamic fluxes and forces are expressed in terms of the combination of the thermodynamic variables of the working substance and the heat reservoirs owing to the local equilibrium assumption. From Eq. (51), in particular, the new thermodynamic force Y T is proportional to the temperature difference of the working substance between the isothermal processes. In the linear response regime, these new thermodynamic fluxes and forces are approximated as Here, we approximated J Q h as the averaged heat flux J Q defined by Eq. (32), and use J Q instead of J Q h as the thermodynamic flux in the following. We also note that J Q h and J S are proportional as J Q h = T h J S ; hence, J Q is also expressed as In fact, the proportionality between the two thermodynamic fluxes in Eq. (53) indirectly implies the tight-coupling condition of this system |q| = 1 [35], because we can easily show from Eqs. (4) and (5) the relation J 2 = L21 L11 J 1 + L 22 (1 − q 2 )X 2 between the two thermodynamic fluxes. However, to understand this relationship more directly and precisely, we express the present system by the following Onsager relations using the new thermodynamic fluxes and forces: To obtain the new Onsager coefficients from the previous coefficients in Eq. (48), we relate the thermodynamic forces X Ti (i = h, c) and Y m (m = T, T R ) by using Eqs. (28), (29), and (52) as follows: Rewriting Eq. (56) as X Ti ≡ F im Y m in Einstein notation, we obtain the new Onsager matrix from the relationL mn = F T mi L ij F jn that conserves the entropy production rate asσ = L ij X Ti X Tj =L mn Y m Y n . Alternatively, we can directly obtain the Onsager coefficientsL T R T andL T R T R from the expression of which is obtained using Eqs. (32) and (56), and we can also obtainL T T andL T T R from the relation J S = JQ T in Eq. (53). From Eq. (57), it is straightforward to confirm that the Onsager reciprocity and the tight-coupling (no heat-leakage) condition are fulfilled:L Therefore, for our endoreversible heat engine model based on the local equilibrium assumption, we conclude that the efficiency at the maximum power attains the upper bound in Eq. (3) as from a viewpoint of the linear irreversible thermodynamics framework using the Onsager relations [14]. We note that the above derivation is quite general because it does not rely on any particular working substance or thermal conductance.

III. DISCUSSION
As we have shown in Sec. II C, our formulation of the efficiency at maximum power based on the local equilibrium assumption can be related to the linear irreversible thermodynamics framework using the Onsager relations. In the present section, we here remark a few aspects related to our formulation in Sec. II and also discuss a possible extension of our formulation. such that the work output during the adiabatic expansion process of each small cycle exactly cancels with the work output during the adiabatic compression process of its neighboring cycle. The number of the small cycles N is taken as sufficiently large such that the temperature variation during the isothermal processes in each small cycle is negligible.
a cycle based on the local equilibrium thermodynamic formulation. To this end, at first, we virtually decompose the endoreversible finite-time Carnot cycle into N small endoreversible finite-time Carnot cycles labeled by index l (l = 1, · · · , N ) such that the work output during the adiabatic expansion process of each small cycle exactly cancels with the work output during the adiabatic compression process of its neighboring cycle (see Fig. 2). Then we can essentially apply the same arguments in Sec. II A to each small cycle.
We equally decompose the cycle period of the whole cycle into the sum of the cycle period of the small cycles as t cyc ≡ N l=1 δt cyc = N δt cyc by tuning the duration of the isothermal process of l-th small-cycle such that it satisfies δt cyc = δt l h + δt l c , where t h = N l=1 δt l h and t c = N l=1 δt l c . We hereafter denote by δ the quantity related to the small cycle. Then we can define the heat during the isothermal process of the l-th small cycle δQ i,l as where δU R i,l denotes the internal-energy change of the working substance during the isothermal process of the l-th small cycle. If we take the limit of N → ∞ such the temperature variation during the isothermal process of each small cycle is negligible, we can express Eq. (67) by using the temperature of the working substance T i,l during the isothermal process of the l-th small cycle as where we defined the entropy change during the isothermal process of the l-th small cycle as We here assume δS h,l = −δS c,l as a condition for the l-th small cycle to be closed. Then, from the relation δQ i,l = T i,l δS i,l , the endoreversible condition for each small cycle holds as Then, in the same way as Eq. (49) in Sec. II C, we consider the total entropy production rateσ along the cycle in the linear response regime, which can be decomposed into the infinite sum of the entropy production rate of each small cycleσ l aṡ Or, as is equivalent, we may define J cyc 1 and X cyc 1 in Eq. (77) by inserting ∆S as in a similar way to Eq. (52), where X cyc 1 = Y T holds when the temperature of the working substance along the isothermal process is constant. The definition Eq. (77) was firstly introduced in our previous study on a finite-time Carnot cycle of a brownian particle trapped in a harmonic potential [27]. Essentially the same definition was also used in a finite-time Carnot cycle model of an ideal gas [26]. In [26,27], the Onsager coefficients under these definitions were calculated by assuming that these thermodynamic fluxes and forces are connected by the Onsager relations Eqs. (4) and (5). Because the heat flux in these systems depends on the time-dependent thermal conductance (see Eq. (7) in [26] and Eq. (34) in [40]) and hence these system must correspond to the non-constant energy flux case, we naturally have a question of how the thermodynamic fluxes J cyc i , forces X cyc j and the Onsager coefficients L ij of the whole cycle are related to the thermodynamic fluxes J l m , forces Y l n and the Onsager coefficientsL l mn of the small cycles when we compare Eq. (71) and Eq. (76).
Although we do not have a complete answer to this question at this point, we here consider the special case of the constant force condition Y l T = Y T in Eq. (72). In this case, the thermodynamic flux J cyc i and the Onsager coefficients L ij of the whole cycle are obtained by integrating the counterparts of the small cycles: where X cyc 1 defined in Eq. (78) agrees with Y T under the constant force condition Y l T = Y T . As is clear from Eqs. (79) and (80), the effect of the time-dependent thermal conductance [26,27] as embedded in the l-dependence of the Onsager coefficientsL l mn of the small cycles may be renormalized into the Onsager coefficients L ij of the whole cycle. By expecting that the above example of non-constant energy flux case but under the constant force condition may be applied to [26,27], we may understand how the infinite pairs of the Onsager relations are reduced to just one pair of the Onsager relations and why the efficiency at the maximum power of these systems agrees with the CA efficiency, as in Eq. (60) for the constant energy flux case. However, in general cases where the constant force condition is not fulfilled, we may need to describe the heat engine by using the infinite pairs of the Onsager relations as in Eqs. (73) and (74). Analyzing the efficiency at maximum power in this general case would be our future important task.

IV. SUMMARY
In the present study, we formulated an endoreversible finite-time Carnot cycle model based on the local equilibrium assumption. In our framework, the power and the efficiency are expressed in terms of the thermodynamic variables of the working substance. From the analysis of the entropy production rate caused by the heat transfer in each isothermal process, we identified the thermodynamic flux and force in each isothermal process, which are related by the Fourier law. We calculated the efficiency at the maximum power by using these thermodynamic fluxes and forces, and obtained the Curzon-Ahlborn efficiency, which is the upper bound as proved by the linear irreversible thermodynamics framework using the Onsager relations. We also elucidated that the linear relationship between the thermodynamic flux and force in our framework could be rewritten into the form of the Onsager relations by a variable change, from which we can directly confirm that our model satisfies the tight-coupling condition that ensures the Curzon-Ahlborn efficiency. We stress that our framework is quite universal because it only assumes that the working substance is in a local equilibrium state specified by a unique combination of thermodynamic variables at any instant along the cycle. We expect that our study unifies recent development of the theories of heat engines based on the universal nonequilibrium thermodynamics framework and based on the more phenomenological finite-time thermodynamics approach, which was originally designed for application to real power plants and heat devices.