Carnot-Like Heat Engines Versus Low-Dissipation Models

Abstract

In this paper, a comparison between two well-known finite time heat engine models is presented: the Carnot-like heat engine based on specific heat transfer laws between the cyclic system and the external heat baths and the Low-Dissipation model where irreversibilities are taken into account by explicit entropy generation laws. We analyze the mathematical relation between the natural variables of both models and from this the resulting thermodynamic implications. Among them, particular emphasis has been placed on the physical consistency between the heat leak and time evolution on the one side, and between parabolic and loop-like behaviors of the parametric power-efficiency plots. A detailed analysis for different heat transfer laws in the Carnot-like model in terms of the maximum power efficiencies given by the Low-Dissipation model is also presented.

1. Introduction

A cornerstone in thermodynamics is the analysis of the performance of heat devices. Since the Carnot’s result about the maximum possible efficiency that any heat converter operating between two heat reservoirs might reach, the work in this field is mainly focused on how to fit real-life devices as close as possible to the main requirement behind the Carnot efficiency value, i.e., the existence of infinite-time, quasi-static processes. However, real-life devices work under finite-time and finite-size constraints, thus giving finite power output. Over the last several decades, one of the most popular models in the physics literature to analyze finite-time and finite-size heat devices has been the so-called Carnot-like model. Inspired by the work reported by Curzon–Ahlborn (CA) [1], this model provides a first valuable approach to the behavior of real heat engines.

In this model, it is assumed an internally reversible Carnot cycle coupled irreversibly with two external thermal reservoirs (endoreversible hypothesis) through some heat transfer laws and some phenomenological conductances related with the nature of the heat fluxes and the properties of the materials and devices involved in the transport phenomena. Without any doubt, the main result was the so-called CA-efficiency $\eta =1-\sqrt{\tau }$ (where $\tau =T_{\mathrm{c}}/T_{\mathrm{h}}$ is the ratio of the external cold and hot heat reservoirs). It accounts for the efficiency at maximum–power (MP) conditions when the heat transfer laws are considered linear with the temperature difference between the external heat baths and the internal temperatures of the isothermal processes at which the heat absorption and rejection occurs. Later extensions of this model included the existence of a heat leak between the two external baths and the addition of irreversibilities inside of the internal cycle. With only three main ingredients (heat leak, external coupling, and internal irreversibilities) the Carnot-like model has been used as a paradigmatic model to confront many research results coming from macroscopic, mesoscopic and microscopic fields [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. Particularly relevant have been those results concerned with the optimization not only of the power output but also of different thermodynamic and/or thermo-economic figures of merit and additionally the analysis on the universality of the efficiency at MP (or on other figures of merit as ecological type [29,30,31,32,33,34]).

Complementary to the CA-model, the Low-Dissipation (LD) model, proposed by Esposito et al. in 2009 [35], consists of a Carnot engine with small deviations from the reversible cycle through dissipations located at the isothermal branches which occur at finite-time. The nature of the dissipations (entropy generation) are encompassed in some generic dissipative coefficients, so that the optimization of power output (or any other figure of merit) is made easily through the contact times of the engine with the hot and cold reservoirs [36,37,38,39]. In this way, depending on the symmetry of the dissipative coefficients, it is possible to recover several results of the CA-model. In particular, the CA-efficiency is recovered in the LD-model under the assumption of symmetric dissipation. Recently, a description of the LD model in terms of characteristic dimensionless variables was proposed in [40,41,42]. From this treatment, it is possible to separate efficiency-power behaviors typical of CA-endoreversible engines as well as irreversible engines according to the imposed time constraints. If partial contact times are constrained, then one obtains open parabolic (endoreversible) curves; otherwise, if total time is fixed, one obtains closed loop-like curves.

The objective of this paper is to analyze in which way the Carnot-like heat engines (dependent on heat transfer laws) and the LD models (dependent on a specific entropy generation law) are related and how the variables of each one are connected. This allows for an interpretation of the heat transfer laws, including the heat leak, in terms of the bounds for the efficiency at MP provided by the LD-model, which, in turn, are dependent on the relative symmetries of the dissipations constants and the partial contact times.

The article is organized as follows: in Section 2, a correspondence among the variables of the two models for heat engines (HE) is made. In Section 3, we study the region of physically acceptable values for the Carnot-like HE variables depending on the heat leak. In Section 4, the study of the MP regime is made, showing that a variety of results between both descriptions can be recovered only in a certain range of heat transfer laws; in particular, we analyze the efficiency vs. power curves behaviors. Finally, some concluding remarks are presented in Section 5.

2. Correspondence between the HE’s Variables of Both Models

A key point to establish the linkage between both models is the entropy production. By equaling the entropy change stemming from both frameworks it is possible to give the relations among the variables that describe each model (see Figure 1).

In the LD case (see Figure 1a), the base-line Carnot cycle works between the temperatures $T_{\mathrm{c}}$ and $T_{\mathrm{h}}>T_{\mathrm{c}}$, the entropy change along the hot (cold) isothermal path is $\Delta S=-\frac{Q_{\mathrm{h}}}{T_{\mathrm{h}}}$ ($\Delta S=+\frac{Q_{\mathrm{h}}}{T_{\mathrm{c}}}$) and the times to complete each isotherm processes are $t_{\mathrm{h}}$ and $t_{\mathrm{c}}$, respectively. The adiabatic processes, as usual, are considered as instantaneous, though the influence of finite adiabatic times has been reported in the LD-model in [43]. The deviation from the reversible scenario in the LD approximation is considered by an additional contribution to the entropy change at the hot and cold reservoirs given by

$$\begin{array}{ccc}\hfill \Delta S_{T_{\mathrm{h}}}& =& -\Delta S+\frac{{\Sigma }_{\mathrm{h}}}{t_{\mathrm{h}}},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \Delta S_{T_{\mathrm{c}}}& =& \Delta S+\frac{{\Sigma }_{\mathrm{c}}}{t_{\mathrm{c}}},\hfill \end{array}$$

(2)

where ${\Sigma }_{\mathrm{h}}$ and ${\Sigma }_{\mathrm{c}}$ are the so-called dissipative coefficients. The signs $-(+)$ take into account the direction of the heat fluxes from (toward) the hot (cold) reservoir in such a way that $Q_{\mathrm{c}}$ and $Q_{\mathrm{h}}$ are positive quantities. Then, the total entropy generation is given by

$$\Delta S_{\mathrm{tot}}=\frac{{\Sigma }_{\mathrm{h}}}{t_{\mathrm{h}}}+\frac{{\Sigma }_{\mathrm{c}}}{t_{\mathrm{c}}}.$$

(3)

At this point, it is helpful to use the dimensionless variables defined in [40]: $\alpha \equiv t_{\mathrm{c}}/t$, ${\tilde{\Sigma }}_{\mathrm{c}}\equiv {\Sigma }_{\mathrm{c}}/{\Sigma }_{\mathrm{T}}$ and $\tilde{t}\equiv (t\Delta S)/{\Sigma }_{\mathrm{T}}$, where $t=t_{\mathrm{h}}+t_{\mathrm{c}}$ and ${\Sigma }_{\mathrm{T}}\equiv {\Sigma }_{\mathrm{h}}+{\Sigma }_{\mathrm{c}}$. In this way, it is possible to define a characteristic total entropy production per unit time for the LD-model as

$${\dot{\widetilde{\Delta S}}}_{\mathrm{tot}}\equiv \frac{\Delta S_{tot}}{\tilde{t}\Delta S}=\frac{{\Delta S}_{\mathrm{tot}}}{t}\frac{{\Sigma }_{\mathrm{T}}}{\Delta S^2}=\frac{1}{\tilde{t}}\left[\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{(1-\alpha )\tilde{t}}+\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}}\right].$$

(4)

In the irreversible Carnot-like HE, the entropy generation of the internal reversible cycle is zero and the total entropy production is that generated at the external heat reservoirs (see Figure 1b). By considering the same sign convention as in the LD model $Q_{\mathrm{h}}=T_{\mathrm{hw}}\Delta S\ge 0$ and $Q_{\mathrm{c}}=T_{\mathrm{cw}}\Delta S\ge 0$, where $\Delta S$ is the entropy change in the hot isothermal branch of the reversible Carnot cycle, and a heat leak $Q_{\mathrm{L}}\ge 0$ between the reservoirs $T_{\mathrm{h}}$ and $T_{\mathrm{c}}$, then

$$\begin{array}{ccc}\hfill \Delta S_{T_{\mathrm{h}}}& =& -\frac{Q_{\mathrm{h}}}{T_{\mathrm{h}}}-\frac{Q_{\mathrm{L}}}{T_{\mathrm{h}}}=-\Delta S+\left(1-a_{\mathrm{h}}^{-1}-\tau \frac{Q_{\mathrm{L}}}{T_{\mathrm{c}}\Delta S}\right)\Delta S,\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill \Delta S_{T_{\mathrm{c}}}& =& \frac{Q_{\mathrm{c}}}{T_{\mathrm{c}}}+\frac{Q_{\mathrm{L}}}{T_{\mathrm{c}}}=\Delta S+\left(a_{\mathrm{c}}-1+\frac{Q_{\mathrm{L}}}{T_{\mathrm{c}}\Delta S}\right)\Delta S,\hfill \end{array}$$

(6)

where $a_{\mathrm{h}}=T_{\mathrm{h}}/T_{\mathrm{hw}}\ge 1$ and $a_{\mathrm{c}}=T_{\mathrm{cw}}/T_{\mathrm{c}}\ge 1$. By introducing a characteristic heat leak ${\tilde{Q}}_{\mathrm{L}}\equiv Q_{\mathrm{L}}/\left(T_{\mathrm{c}}\Delta S\right)$ and a comparison with Equations (1) and (2) gives the expressions associated with the dissipations

$$\begin{array}{ccc}\hfill \frac{{\Sigma }_{\mathrm{h}}}{t_{\mathrm{h}}}& =& \left(1-a_{\mathrm{h}}^{-1}-\tau {\tilde{Q}}_{\mathrm{L}}\right)\Delta S,\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill \frac{{\Sigma }_{\mathrm{c}}}{t_{\mathrm{c}}}& =& \left(a_{\mathrm{c}}-1+{\tilde{Q}}_{\mathrm{L}}\right)\Delta S.\hfill \end{array}$$

(8)

By assuming that the ratio $t_{\mathrm{c}}/(t_{\mathrm{c}}+t_{\mathrm{h}})$ is the same in both descriptions, then we introduce $\alpha =1/(1+t_{\mathrm{h}}/t_{\mathrm{c}})$ into the Carnot-like model, and ${\tilde{\Sigma }}_{\mathrm{c}}={\Sigma }_{\mathrm{c}}/{\Sigma }_{\mathrm{T}}$ and $\tilde{t}=t\Delta S/{\Sigma }_{\mathrm{T}}$ are

$$\begin{array}{ccc}\hfill {\tilde{\Sigma }}_{\mathrm{c}}^{-1}& =& 1+\left(\frac{1-\alpha }{\alpha }\right)\left(\frac{1-a_{\mathrm{h}}^{-1}-\tau {\tilde{Q}}_{\mathrm{L}}}{a_{\mathrm{c}}-1+{\tilde{Q}}_{\mathrm{L}}}\right),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill \tilde{t}& =& \frac{1}{\alpha \left(a_{\mathrm{c}}-1+{\tilde{Q}}_{\mathrm{L}}\right)\left[1+\left(\frac{1-\alpha }{\alpha }\right)\left(\frac{1-a_{\mathrm{h}}^{-1}-\tau {\tilde{Q}}_{\mathrm{L}}}{a_{\mathrm{c}}-1+{\tilde{Q}}_{\mathrm{L}}}\right)\right]},\hfill \end{array}$$

(10)

which are the relations between the characteristic variables of the LD model and the variables of the Carnot-like HE. This is summarized in the following expression:

$$\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}}=a_{\mathrm{c}}-1+{\tilde{Q}}_{\mathrm{L}}.$$

(11)

3. Physical Space of the HE Variables

We stress that all the above results between variables hold for arbitrary heat transfer laws in the Carnot-like model. As a consequence, above equations provide the generic linkage between both descriptions, and, from them, useful thermodynamic information can be extracted.

In Figure 2a, the internal temperatures for the irreversible Carnot-like HE, contained in $a_{\mathrm{h}}$ and $a_{\mathrm{c}}$, are depicted with fixed values $\tau =0.2$, $\alpha =0.2$ and ${\tilde{\Sigma }}_{\mathrm{c}}=0.5$. Notice that, in order to obtain thermal equilibrium between the auxiliary reservoirs and the external baths (i.e., to achieve the reversible limit), it is necessary that $Q_{\mathrm{L}}=0$. As soon as a heat leak appears, $T_{\mathrm{hw}}<T_{\mathrm{h}}$, meanwhile $T_{\mathrm{cw}}=T_{\mathrm{c}}$ is always a possible configuration. As the heat leak increases in the HE, the internal temperatures get closer to each other until the limiting situation where $T_{\mathrm{hw}}=T_{\mathrm{cw}}$ (see contact edge in Figure 2a).

As a heat leak appears, the reversible limit $\tilde{t}\to \infty$ is no longer achievable. This is better reflected in Figure 2b, where we plot the total operation time $\tilde{t}$ depending on $a_{\mathrm{c}}$ and ${\tilde{Q}}_{\mathrm{L}}$ (see Equation (11)). Only when $a_{\mathrm{c}}\to 1$ and ${\tilde{Q}}_{\mathrm{L}}\to 0$ can large operation times be allowed. We can see in this figure that the existence of a heat leak imposes a maximum operational characteristic time to the HE. The total time is noticeably shorter as the heat leak increases, in agreement with the fact that, for $\tilde{t}\le 1,$ the working regimes are dominated by dissipations. It could be said that the heat leak behaves as a causality effect in the arrow of time of the heat engine.

Notice that, in Figure 2, there is a region of prohibited combinations of ${\tilde{Q}}_{\mathrm{L}}$ and $a_{\mathrm{c}}$. This has to do with the physical reality of the engine (negative power output and efficiency). In [41], the region of physical interest in the LD model under maximum power conditions was analyzed. In the Carnot-like engine, some similar considerations can be addressed as follows: in a valid endoreversible HE, the internal temperatures may vary in the range $a_{\mathrm{h}}\in (1,{\tau }^{-1})$ and $a_{\mathrm{c}}\in (1,{\tau }^{-1}{a}_{\mathrm{h}}^{-1})$ in order to have $T_{\mathrm{c}}\le T_{\mathrm{cw}}\le T_{\mathrm{hw}}\le T_{\mathrm{h}}$, $a_{\mathrm{c}}={\tau }^{-1}{a}_{\mathrm{h}}^{-1}$ being the condition for $T_{\mathrm{cw}}=T_{\mathrm{hw}}$ implying null work output and efficiency. From Equation (9), it is possible to obtain two conditions on ${\tilde{Q}}_{\mathrm{L}}(\alpha ,{\tilde{\Sigma }}_{\mathrm{c}},a_{\mathrm{c}},a_{\mathrm{h}},\tau )$ (initially assumed to be $\ge 0$) according to the values $a_{\mathrm{h}}=1$ and $a_{\mathrm{h}}={\tau }^{-1}$. For $a_{\mathrm{h}}=1,$ we obtain that

$$\begin{array}{c}\hfill {\tilde{Q}}_{\mathrm{L}}=-\frac{\left(a_{\mathrm{c}}-1\right)\left(\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{{\tilde{\Sigma }}_{\mathrm{c}}}\right)}{\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{{\tilde{\Sigma }}_{\mathrm{c}}}+\tau \left(\frac{1-\alpha }{\alpha }\right)}\le 0,\end{array}$$

(12)

whose only physical solution is ${\tilde{Q}}_{\mathrm{L}}=0$. Then, as long as there is a heat leak in the device, the internal hot reservoir cannot reach equilibrium with the external hot reservoir and the reversible configuration is not achievable. On the other hand (as can be seen in Figure 2a), the largest possible heat leak (i.e., the largest dissipation in the system) has as an outcome that $T_{\mathrm{hw}}\to T_{\mathrm{cw}}\to T_{\mathrm{c}}$, that is, $a_{\mathrm{c}}\to 1$ and $a_{\mathrm{h}}\to {\tau }^{-1}$. In that limit, Equations (9) and (11) give

$$\begin{array}{c}\hfill {\tilde{Q}}_{\mathrm{L},\max }=\frac{\left(1-\tau \right)\left(\frac{1-\alpha }{\alpha }\right)}{\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{{\tilde{\Sigma }}_{\mathrm{c}}}+\tau \left(\frac{1-\alpha }{\alpha }\right)}=\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}},\end{array}$$

(13)

and, since in this case all the input heat is dissipated to the cold external thermal reservoir, the HE has a null power output. In Figure 3, we depict the range of possible values that ${\tilde{Q}}_{\mathrm{L}}$ can take (from 0 up to ${\tilde{Q}}_{\mathrm{L},\max }$) in terms of $\alpha$ and $\tilde{t}$. By means of Equation (13), it is established a boundary condition for physically acceptable values of the irreversible Carnot-like HE in terms of the LD variables, which is

$$\begin{array}{c}\hfill \tilde{t}=\frac{\alpha \left(1-{\tilde{\Sigma }}_{\mathrm{c}}\right)+{\tilde{\Sigma }}_{\mathrm{c}}\tau \left(1-\alpha \right)}{\alpha \left(1-\alpha \right)\left(1-\tau \right)}.\end{array}$$

(14)

Up to this point, we have proposed a generic correspondence between the variables of both schemes: the LD treatment, based on a specific entropy generation law, and the irreversible Carnot-like engine based on heat transfer laws. In the following, we will further analyze the connection given by Equation (11) with the focus on different heat transfer laws and the maximum power efficiencies given by the low-dissipation model.

4. Maximum-Power Regime

As is usual, the power output is given by

$$\begin{array}{c}\hfill P=\frac{\eta Q_{\mathrm{h}}}{t_{\mathrm{c}}+t_{\mathrm{h}}}.\end{array}$$

(15)

In [41], it was shown that, in the MP regime of an LD engine display, an open, parabolic behavior for the parametric $P-\eta$ curves when $\alpha ={\alpha }_{{\tilde{P}}_{max}}$ is fixed and for ${\Sigma }_{\mathrm{c}}\in \left[0,1\right]$. On the other hand, by fixing the value of $\tilde{t}={\tilde{t}}_{{\tilde{P}}_{max}}$, one obtains for the behavior of $\eta$ vs. P loop-like curves (see Figure 4 in [41]). In the irreversible Carnot-like framework, open $\eta$ vs. P curves are characteristic of endoreversible CA-type engines, and, when a heat leak is introduced, one obtains loop-like curves. The apparent connection between the behavior displayed by fixing $\tilde{t}$ or $\alpha$ in the low dissipation context with the presence of a heat leak, or the lack of it, is by no means obvious. A simple analysis of the MP regime in an irreversible Carnot-like engine in terms of the LD variables will shed some light on this issue and will also provide us a better understanding of how good the correspondence is between both schemes.

4.1. Low Dissipation Heat Engine

In terms of the characteristic variables, the input and output heat are

$$\begin{array}{ccc}\hfill {\dot{\tilde{Q}}}_{\mathrm{h}}& \equiv & \frac{{\tilde{Q}}_{\mathrm{h}}}{\tilde{t}}=\frac{Q_{\mathrm{h}}}{T_{\mathrm{c}}\Delta S}\frac{{\Sigma }_{\mathrm{T}}}{t\Delta S}=\left(1-\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{(1-\alpha )\tilde{t}}\right)\frac{1}{\tau \tilde{t}},\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill {\dot{\tilde{Q}}}_{\mathrm{c}}& \equiv & \frac{{\tilde{Q}}_{\mathrm{c}}}{\tilde{t}}=\frac{Q_{\mathrm{c}}}{T_{\mathrm{c}}\Delta S}\frac{{\Sigma }_{\mathrm{T}}}{t\Delta S}=\left(1-\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}}\right)\frac{1}{\tilde{t}},\hfill \end{array}$$

(17)

giving a power output and efficiency as follows:

$$\begin{array}{ccc}\hfill \tilde{P}& \equiv & -\frac{\tilde{W}}{\tilde{t}}=-\frac{W}{T_{\mathrm{c}}\Delta S}\frac{{\Sigma }_{\mathrm{T}}}{t\Delta S}=\left[\frac{1}{\tau }-1-\frac{1}{\tau }\left(\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{(1-\alpha )\tilde{t}}\right)-\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}}\right]\frac{1}{\tilde{t}},\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill \tilde{\eta }& \equiv & \frac{\tilde{P}}{{\dot{\tilde{Q}}}_{\mathrm{h}}}=-\frac{W}{Q_{\mathrm{h}}}=\eta =\frac{1-\tau -\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{(1-\alpha )\tilde{t}}-\frac{\tau {\tilde{\Sigma }}_{\mathrm{c}}}{\alpha \tilde{t}}}{1-\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{(1-\alpha )\tilde{t}}}.\hfill \end{array}$$

(19)

The optimization of $\tilde{P}(\tilde{t},\alpha ;{\tilde{\Sigma }}_{\mathrm{c}},\tau )$ is accomplished through the partial contact time $\alpha$ and the total time $\tilde{t}$, whose values are

$$\begin{array}{ccc}\hfill {\alpha }_{{\tilde{P}}_{max}}\left({\tilde{\Sigma }}_{\mathrm{c}},\tau \right)& =& \frac{1}{1+\sqrt{\frac{1-{\tilde{\Sigma }}_{\mathrm{c}}}{\tau {\tilde{\Sigma }}_{\mathrm{c}}}}},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\tilde{t}}_{{\tilde{P}}_{max}}\left({\tilde{\Sigma }}_{\mathrm{c}},\tau \right)& =& \frac{2}{1-\tau }{\left(\sqrt{\tau {\tilde{\Sigma }}_{\mathrm{c}}}+\sqrt{1-{\tilde{\Sigma }}_{\mathrm{c}}}\right)}^2,\hfill \end{array}$$

(21)

with an MP efficiency given by

$${\eta }_{{\tilde{P}}_{max}}\left({\tilde{\Sigma }}_{\mathrm{c}},\tau \right)=\frac{\left(1-\tau \right)\left[1+\sqrt{\frac{\tau {\tilde{\Sigma }}_{\mathrm{c}}}{1-{\tilde{\Sigma }}_{\mathrm{c}}}}\right]}{{\left[1+\sqrt{\frac{\tau {\tilde{\Sigma }}_{\mathrm{c}}}{1-{\tilde{\Sigma }}_{\mathrm{c}}}}\right]}^2+\tau \left(1-\frac{{\tilde{\Sigma }}_{\mathrm{c}}}{1-{\tilde{\Sigma }}_{\mathrm{c}}}\right)}.$$

(22)

One of the most relevant features of this model is the capability of obtaining upper and lower bounds of the MP efficiencies without any information regarding the heat fluxes nature. These limits are

$${\eta }_{{\tilde{P}}_{max}}^{-}=\frac{{\eta }_C}{2}\le {\eta }_{{\tilde{P}}_{max}}(\tau ,{\tilde{\Sigma }}_{\mathrm{c}})\le \frac{2-{\eta }_C}{{\eta }_C}={\eta }_{{\tilde{P}}_{max}}^{+},$$

(23)

corresponding to ${\tilde{\Sigma }}_{\mathrm{c}}=1$ and ${\tilde{\Sigma }}_{\mathrm{c}}=0$ for the lower and upper bounds, respectively. For the symmetric dissipation case, ${\tilde{\Sigma }}_{\mathrm{c}}=1/2(={\tilde{\Sigma }}_{\mathrm{h}})$, the well known CA-efficiency ${\eta }_{{\tilde{P}}_{max}}^{sym}=1-\sqrt{\tau }={\eta }_{CA}$ is recovered.

4.2. Carnot-Like Model without Heat Leak (Endoreversible Model)

Now, let us consider a family of heat transfer laws depending on the power of the temperature to model the heat fluxes $Q_{\mathrm{h}}$ and $Q_{\mathrm{c}}$ (see Figure 1b) as follows:

$$\begin{array}{ccc}\hfill Q_{\mathrm{h}}& =& T_{\mathrm{h}}^k{\sigma }_{\mathrm{h}}\left(1-a_{\mathrm{h}}^{-k}\right)t_{\mathrm{h}}\ge 0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill Q_{\mathrm{c}}& =& T_{\mathrm{c}}^k{\sigma }_{\mathrm{c}}\left(a_{\mathrm{c}}^k-1\right)t_{\mathrm{c}}\ge 0,\hfill \end{array}$$

(25)

where $k\ne 0$ is a real number, ${\sigma }_{\mathrm{h}}$ and ${\sigma }_{\mathrm{c}}$ are the conductances in each process and $t_{\mathrm{h}}$ and $t_{\mathrm{c}}$ are the times at which the isothermal processes are completed. The adiabatic processes are considered as instantaneous, a common assumption in the two models. According to Equation (15), power output is a function depending on the variables $a_{\mathrm{c}}$, $a_{\mathrm{h}}$ and the ratio of contact times; k, $\tau$ and ${\sigma }_{\mathrm{hc}}$ are not optimization variables for this model. The endoreversible hypothesis $\Delta S_{T_{\mathrm{hw}}}=-\Delta S_{T_{\mathrm{cw}}}$ gives the following constriction upon the contact times ratio

$$\frac{t_{\mathrm{c}}}{t_{\mathrm{h}}}={\sigma }_{\mathrm{hc}}{a}_{\mathrm{c}}{a}_{\mathrm{h}}{\tau }^{1-k}\left(\frac{1-a_{\mathrm{h}}^{-k}}{a_{\mathrm{c}}^k-1}\right),$$

(26)

where ${\sigma }_{\mathrm{hc}}\equiv {\sigma }_{\mathrm{h}}/{\sigma }_{\mathrm{c}}$. Since there is no heat leak, the efficiency of the internal Carnot cycle is the same as the efficiency of the engine, then $a_{\mathrm{c}}{a}_{\mathrm{h}}\tau =1-\eta$, and the dependence of $a_{\mathrm{h}}$ is substituted by $\eta$. Then, in terms of $\alpha$, Equation (26) is

$$\frac{\alpha }{1-\alpha }={\sigma }_{\mathrm{hc}}{\tau }^k\left(1-\eta \right)\left(\frac{1-\frac{a_{\mathrm{c}}^k{\tau }^k}{{\left(1-\eta \right)}^k}}{a_{\mathrm{c}}^k-1}\right).$$

(27)

The optimization of power output $P\left(a_{\mathrm{c}},\eta ;{\sigma }_{\mathrm{hc}},\tau ,T_{\mathrm{h}},k\right)$ in this case is achieved through $a_{\mathrm{c}}$ and $\eta$. The maximum power is obtained by solving ${\left(\frac{\partial P}{\partial a_{\mathrm{c}}}\right)}_{\eta }=0$ for $a_{\mathrm{c}}$ and ${\left(\frac{\partial P}{\partial \eta }\right)}_{a_{\mathrm{c}}}=0$ for $\eta$. From the first condition, we obtain $P^{*}$, which is

$$P^{*}\left(\eta ;{\sigma }_{\mathrm{hc}},\tau ,T_{\mathrm{h}},k\right)={\sigma }_{\mathrm{h}}{T}_{\mathrm{h}}^k\left(\frac{\eta }{1-\eta }\right)\frac{{\left(1-\eta \right)}^k-{\tau }^k}{{\left(\sqrt{{\sigma }_{\mathrm{hc}}}+{\left(1-\eta \right)}^{\frac{k-1}{2}}\right)}^2}.$$

(28)

This function has a unique maximum corresponding to ${\eta }_{P_{max}}$, which is the solution to the following equation

$$\sqrt{{\sigma }_{\mathrm{hc}}}\left(1-\eta \right)\left[{\tau }^k-{\left(1-\eta \right)}^k\left(1-k\eta \right)\right]+{\left(1-\eta \right)}^{\frac{k+1}{2}}\left[\left(1-\left(1-k\right)\eta \right){\tau }^k-{\left(1-\eta \right)}^{k+1}\right]=0,$$

(29)

and depends on the values ${\sigma }_{\mathrm{hc}}$, $\tau$ and the exponent of the heat transfer law k as showed in [9]. In Figure 4a, ${\eta }_{P_{max}}$ is depicted for the limiting cases ${\sigma }_{\mathrm{hc}}\to \{0,\infty \}$. All of the possible values of ${\eta }_{P_{max}}$ for different ${\sigma }_{\mathrm{hc}}$s are located between these two curves. It is well-known that, for the Newtonian heat transfer law ($k=1$), ${\eta }_{P_{max}}={\eta }_{CA}$ is independent of the ${\sigma }_{\mathrm{hc}}$ value. As the heat transfer law departs from the Newtonian case, the upper and lower bounds cover a wider range of efficiencies. Then, the limits appearing in Equation (23) are fulfilled for a limited region of k values in the Carnot-like model. From Figure 4a, it is possible to see that the results stemming from an endoreversible engine are accessible from an LD landmark only in the region $k\in (-1,2.5]$ (for other values of k, there are efficiencies outside the range given by Equation (23)). By equaling these efficiencies with the LD one (see Equation (22)) and solving for ${\tilde{\Sigma }}_{\mathrm{c}}$, we obtain those values that reproduce the endoreversible efficiencies. This is depicted in Figure 4b. Notice also that not all ${\tilde{\Sigma }}_{\mathrm{c}}$ symmetries are allowed for every $k\in (-1,2.5]$. For example, with a heat transfer law with exponent $k=-1$, all the possible values of the efficiency can be obtained if the parameter ${\tilde{\Sigma }}_{\mathrm{c}}$ varies from 0 to 1, that is, all symmetries are allowed. Meanwhile, for $k=1$, only the symmetric case ${\tilde{\Sigma }}_{\mathrm{c}}=1/2$ is allowed, reproducing the CA efficiency. For k outside $(-1,2.5]$, there are efficiencies above and below the limits in Equation (23) with no ${\Sigma }_{\mathrm{c}}$ values that might reproduce those efficiencies, thus limiting the heat transfer laws physically consistent with predictions of the LD model.

Inside the region where the LD model is able to reproduce the asymmetric limiting cases (${\sigma }_{\mathrm{hc}}\to \{0,\infty \}$), the correspondence between the two formalisms has not an exact fitting. In order to show this, we will address the symmetric dissipation case.

As can be seen from Figure 4, in the endoreversible CA-type HE, for every k, there is one ${\sigma }_{\mathrm{hc}}$ that reproduces the CA efficiency. On the other hand, in the LD model, the symmetric dissipation is attached to ${\eta }_{CA}$. If we use the $\alpha$ and $\tilde{t}$ values of MP of the LD model and calculate the values of $a_{\mathrm{c}}$ and $a_{\mathrm{h}}$ associated with them (instead of calculating them according to Equations (28) and (29)), we can see whether they allow us to recover the correct value of ${\sigma }_{\mathrm{hc}}$ that in the endoreversible model gives the CA efficiency or does not. That is, for ${\tilde{\Sigma }}_{\mathrm{c}}=1/2$, Equations (20) and (21) reduce to

$$\begin{array}{ccc}\hfill {\alpha }_{{\tilde{P}}_{max}}^{sym}& =& \frac{\sqrt{\tau }}{1+\sqrt{\tau }},\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\tilde{t}}_{{\tilde{P}}_{max}}^{sym}& =& \frac{1+\sqrt{\tau }}{1-\sqrt{\tau }}.\hfill \end{array}$$

(31)

From Equation (11), we obtain $a_{\mathrm{c}}$ and with the condition $\eta ={\eta }_{CA}$, with $T_{\mathrm{cw}}/T_{\mathrm{hw}}=a_{\mathrm{c}}{a}_{\mathrm{h}}\tau =1-\eta =\sqrt{\tau }$ we calculate $a_{\mathrm{h}}$, thus

$$\begin{array}{c}\hfill a_{\mathrm{c},{\tilde{\mathrm{P}}}_{\max }}^{sym}=\frac{1+\sqrt{\tau }}{2\sqrt{\tau }},\end{array}$$

(32)

$$\begin{array}{c}\hfill a_{\mathrm{h},{\tilde{\mathrm{P}}}_{\max }}^{sym}=\frac{2}{1+\sqrt{\tau }}.\end{array}$$

(33)

By using Equation (30), the ratio of contact times results in $\frac{t_{\mathrm{c}}}{t_{\mathrm{h}}}=\frac{\alpha }{1-\alpha }=\sqrt{\tau }$, and, by using the endoreversible hypothesis (Equation (26)), it is possible to obtain the value of ${\sigma }_{\mathrm{hc}}$ that would produce the CA efficiency, being

$${\sigma }_{\mathrm{hc},{\tilde{\mathrm{P}}}_{\max }}^{sym}=\frac{{\left(1+\sqrt{\tau }\right)}^k{\tau }^{\frac{k}{2}}-2^k{\tau }^k}{2^k-{\left(1+\sqrt{\tau }\right)}^k},$$

(34)

which for $k=1$ gives ${\sigma }_{\mathrm{hc},{\tilde{\mathrm{P}}}_{\max }}^{sym}=\sqrt{\tau }$ and for $k=-1$ gives ${\sigma }_{\mathrm{hc},{\tilde{\mathrm{P}}}_{\max }}^{sym}=1/\tau$. Nevertheless, by substituting Equation (34) into Equation (29), the MP efficiency is not exactly the CA one, as can be seen in Figure 5. Showing that the correspondence between both models is a good approximation only in the range $k\in \left[-1,1\right]$, and is exact only for $k=-1$ and $k=1$.

Another incompatibility of the two approaches comes up in the Newtonian heat exchange ($k=1$): meanwhile, the Carnot-like scheme ${\eta }_{CA}$ is independent of any value of ${\sigma }_{\mathrm{hc}}$, and, in terms of the LD model, ${\eta }_{CA}$ is strictly attached to a symmetric dissipation $\widetilde{{\Sigma }_{\mathrm{c}}}=1/2(=\widetilde{{\Sigma }_{\mathrm{h}}})$. Then, the only law that has an exact correspondence for all values of ${\tilde{\Sigma }}_{\mathrm{c}}$ and ${\sigma }_{\mathrm{hc}}$ is the law $k=-1$.

4.3. Carnot-Like Model with Heat Leak

Now, let us consider a heat leak of the same kind of the heat fluxes $Q_{\mathrm{c}}$ and $Q_{\mathrm{h}}$, that is,

$$Q_{\mathrm{L}}=T_{\mathrm{h}}^k{\sigma }_{\mathrm{L}}\left(1-{\tau }^k\right)\left(t_{\mathrm{h}}+t_{\mathrm{c}}\right)\ge 0,$$

(35)

then, the characteristic heat leak is

$${\tilde{Q}}_{\mathrm{L}}=\frac{Q_{\mathrm{L}}}{Q_{\mathrm{c}}}=a_{\mathrm{c}}\frac{T_{\mathrm{h}}^k{\sigma }_{\mathrm{L}}\left(1-{\tau }^k\right)\left(t_{\mathrm{h}}+t_{\mathrm{c}}\right)}{T_{\mathrm{c}}^k{\sigma }_{\mathrm{c}}\left(a_{\mathrm{c}}^k-1\right)t_{\mathrm{c}}}=\frac{a_{\mathrm{c}}{\sigma }_{\mathrm{Lc}}\left(1-{\tau }^k\right)}{\alpha {\tau }^k\left(a_{\mathrm{c}}^k-1\right)}.$$

(36)

The power output of the engine is the same than in the endoreversible case; however, a difference with the previous subsection arises, and now the efficiency is given by the following expression:

$$\eta =\frac{Q_{\mathrm{h}}+Q_{\mathrm{L}}-Q_{\mathrm{c}}-Q_{\mathrm{L}}}{Q_{\mathrm{h}}+Q_{\mathrm{L}}}=1-\frac{Q_{\mathrm{c}}+Q_{\mathrm{L}}}{Q_{\mathrm{h}}+Q_{\mathrm{L}}}=1-\frac{1+\frac{Q_{\mathrm{L}}}{Q_{\mathrm{c}}}}{\frac{Q_{\mathrm{h}}}{Q_{\mathrm{c}}}+\frac{Q_{\mathrm{L}}}{Q_{\mathrm{c}}}}=1-\frac{1+{\tilde{Q}}_{\mathrm{L}}}{{\tilde{Q}}_{\mathrm{h}}+{\tilde{Q}}_{\mathrm{L}}},$$

(37)

where we have used the fact that $T_{\mathrm{c}}\Delta S=Q_{\mathrm{c}}$ to introduce the characteristic heats in the last expression.

From Equation (37), it can be derived that, if the heat leak increases, the efficiency diminishes. In Figure 6, we can observe how the upper and lower bounds of the efficiency appearing in Figure 4 are affected by the introduction of a constant heat leak. Now, by using Equation (37) and the fact that in the endoreversible case $\eta =1-{\tilde{Q}}_{\mathrm{h}}^{-1}$, we have plotted the ${\tilde{Q}}_{\mathrm{L}}$ values that leads the efficiency ${\eta }_{CA}$ for $k=1$ (of the endoreversible case) down to the value ${\eta }_C/2$ (see point A in Figure 6), which is ${\tilde{Q}}_{\mathrm{L}}=\frac{1-\sqrt{\tau }}{\sqrt{\tau }\left(1+\sqrt{\tau }\right)}$; and the value ${\tilde{Q}}_{\mathrm{L}}=\frac{1-\tau }{2\tau }$ that lead to an efficiency ${\eta }_C/\left(2-{\eta }_C\right)$ for $k=-1$ (of the endoreversible case) to ${\eta }_C/2$ (see point B in Figure 6).

Notice in Figure 6 that, for ${\tilde{Q}}_{\mathrm{L}}=0.1,$ there is a region around $k=1$ where ${\eta }_{CA}$ is outside the shaded region of maximum power efficiencies. Then, for these values of k and ${\tilde{Q}}_{\mathrm{L}}$, the symmetric dissipation case, always attached to the ${\eta }_{CA}$ efficiency, is out of reach. Additionally, there is not a heat transfer law that fulfills both upper and lower bounds for maximum power efficiency given by the LD model as occurred for $k=-1$ in the endoreversible case. The additional degree of freedom caused by the appearance of the heat leak makes more complex the analysis of the validity of the correspondence between both models, which, in general, should be handled numerically.

5. Conclusions

We analyzed Carnot-like heat engines (dependent on heat transfer laws) and the LD models (dependent on a specific entropy generation law) and studied how the variables of each one are connected. We were able to provide an interpretation of the heat transfer laws, including the heat leak, in terms of the bounds for the efficiency at MP provided by the LD-model, which, in turn, are dependent on the relative symmetries of the dissipations constants and the partial contact times.

By comparing the entropy production of the low dissipation model and the Carnot-like model, we proposed a connection between the variables that describe each model. We show that, for an HE, the region of physical interest is independent from the operation regime, being equivalent to that for a maximum power LD-HE. That is, $\eta \left({\tilde{t}}_{{\tilde{P}}_{max}},\alpha ,{\tilde{\Sigma }}_{\mathrm{c}},\tau \right)\ge 0$ defines the acceptable $\alpha$ values, and $\eta \left(\tilde{t},{\alpha }_{{\tilde{P}}_{max}},{\tilde{\Sigma }}_{\mathrm{c}},\tau \right)\ge 0$ those of $\tilde{t}$ (gray shaded areas in Figure 7a,b, respectively). These considerations on the LD model are recovered from physical considerations on the Carnot-like HE. Thus, the difference in the performance of the HEs is not due to the physical configurations of the system, but it comes from the approximations that these models rely on: one over the entropy and the other over the heat fluxes.

We show that the heat leak disappears when fixing the partial contact time in the LD engine, leading to typical endoreversible open parabolic power vs. efficiency curves. However, the connection between the endoreversible case and the LD model is exact only for heat transfer laws with exponents $k=1$ and $k=-1$, and a good approximation in the region $k\in (-1,1)$. On the other hand, the presence of a heat leak fixes the total operation time and the partial contact time is not constrained, thus, allowing the heat leak to act as an additional degree of freedom (the same efficiency is achieved with different combinations of partial contact time ratios and heat leaks). The reversible limit is not accessible in this case, a maximum operation time is established, and when the heat leak dissipation effects are important, the efficiency may be zero, which is the origin of the loop behavior of the irreversible $\tilde{P}$ vs. $\eta$ curves. The connection in the case with heat leak is more complex and its validity depends on the value of ${\tilde{Q}}_{\mathrm{L}}$.