Optimized efficiency at maximum $\dot {{\Omega}}$ figure of merit and efficient power of power law dissipative Carnot-like heat engines

The worldwide problem concerning the energy and dwindling fossil fuels has created a renewed interest in optimizing the efficiency of heat engines. Heat engines, the crucial components of the industrial revolution, convert heat energy to mechanical energy. Thus, finding more realistic upper bounds on the efficiency would pave the way for reducing the energy consumption in heat engines and hence provide a suitable solution to the concerns related to existing energy problems. For a Carnot heat engine, an ideal one has the maximum efficiency ${\eta }_{\mathrm{C}}=1-\left(\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}\right)$, where T_c and T_h are the temperatures of the cold and hot reservoirs, respectively. The Carnot cycle consumes infinite time to complete the process and hence its power output is zero.

The real heat engines operate in a finite time duration with non-zero power output whose efficiency is always bounded below the ideal Carnot efficiency. In order to obtain a non-zero power, the thermodynamic processes for a heat engine should take place in a finite time duration with optimized efficiency. Finite time thermodynamics (FTT) is one such wider field of thermodynamic optimization, providing more realistic bounds on the efficiency of real systems by considering finite time irreversible processes [1, 2]. The theoretical bounds determined from the FTT provide optimal conditions for designing real systems. Yvon [3], Novikov [4], Chambadal [5] and later Curzon and Ahlborn [6] were the pioneers in obtaining the efficiency optimized at maximum power by using an endo-reversible concept in the reversible Carnot cycle. The so-called Curzon–Ahlborn expression for the efficiency at maximum power obtained from the above model is given by ${\eta }_{\mathrm{C}\mathrm{A}}=1-{\left(\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}\right)}^{1/2}$ [6].

Recently, there has been tremendous progress in identifying the performance limits of thermodynamic processes through various optimizations of thermodynamic cycles of heat engines using finite-time thermodynamics [7, 8, 9, 10]. In particular, irrespective of any heat transfer laws, Esposito et al obtained the extreme bounds on the efficiency at maximum power for low dissipation Carnot-like heat engines [11]. The main assumption of their model is that the irreversible entropy production in each isothermal process is inversely proportional to the time required for completing that process. On the other hand, Ma used the parameter called per-unit time efficiency, optimized at maximum power, and obtained the extreme bounds on the efficiency [12]. This criterion was found to be a compromise between the efficiency and speed of the thermodynamic cycle.

Few heat engine studies based on the low dissipation reported the extreme bounds on the efficiency of Carnot-like heat engines with the consideration of non-adiabatic dissipation in finite time adiabatic processes [13, 14]. The dissipation that occurs due to the effects of inner friction during the finite time adiabatic process is known as non-adiabatic dissipation [13, 15, 16]. These investigations showed that the additionally incorporated non-adiabatic dissipation does not influence the extreme bounds on the efficiency at maximum power. Followed by the investigation of Yang and Tu [17], one of the present authors obtained the generalized bounds on the efficiency at maximum power by incorporating the power law dissipation in a finite time Carnot-like heat engine model [18] and also showed that the generalized extreme bounds are not influenced by the additionally incorporated non-adiabatic dissipation [19]. In all of these studies, the optimization of efficiency at maximum power is used to investigate the performance of heat engines.

Even though the efficiency at maximum power is a desirable operational regime, several other optimization parameters are also used to enhance the heat engine performance. In particular, $\dot {{\Omega}}$ figure of merit and efficient power, χ_ep are two such criteria that have attracted much attention nowadays to study the heat engine performance. The former, $\dot {{\Omega}}$ figure of merit, is defined as [20] $\dot {{\Omega}}=\left(2\eta -{\eta }_{\mathrm{max}}\right)\dot {{Q}_{\mathrm{h}}}$, where η is the efficiency, η_max is the maximum efficiency and $\dot {{Q}_{\mathrm{h}}}$ is the rate of heat flow or heat exchanged with the hot reservoir per cycle time [21]. The $\dot {{\Omega}}$ figure of merit unifies the trade-off between the useful energy delivered and energy lost of heat engines [20], whereas the latter, efficient power, provides a compromise between efficiency and power P, which is defined as χ_ep = ηP [22–24].

The extreme bounds on the efficiency by optimizing both these target functions were studied for low dissipation Carnot-like heat engines without incorporating non-adiabatic dissipation [25, 26]. This raises a question whether the inclusion of non-adiabatic dissipation will influence the extreme bounds on the efficiency in the low dissipation model by optimizing both these target functions. Although the low-dissipation model is a well-founded model for many heat engines [24, 27], it has been observed that this model might not be suitable for real heat engines operating at different dissipation levels [17, 18, 24, 28]. Hence, the present work investigates the generalized minimum and maximum bounds on the efficiency at maximum $\dot {{\Omega}}$ figure of merit and efficient power, χ_ep of power law dissipative Carnot-like heat engines [18], which incorporates the generalized dissipation and also addresses the influence of non-adiabatic dissipation on these extreme bounds in the low dissipation model as a special case.

This paper is organized as follows: in section 2, the model of the power law dissipative Carnot-like heat engine is explained. In sections 3 and 4, the optimization of efficiency at maximum $\dot {{\Omega}}$ figure of merit and at maximum efficient power are derived and its extreme bounds are discussed. The paper concludes with the conclusion in section 5.

Figure 1 represents the temperature (T) and entropy (S) plane of an irreversible Carnot-like cycle (A → B → C → D → A) [14]. The system is in contact with the hot (cold) reservoir at the constant temperature T_h (T_c) in the finite time interval t_h (t_c) during A → B (C → D) for an isothermal expansion (compression) process. During B → C (D → A), the system is decoupled from the hot (cold) reservoir and undergoes a finite time adiabatic expansion (compression) process in a finite time duration t_a (t_b). The value of entropy at the time the system completes a particular process is denoted by S_i (i: a, b, c, h). Here, (A → B → C' → D' → A) represents the (reversible) Carnot cycle in which S_a = S_h and S_b = S_c.

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In this work, a power law dissipative Carnot-like heat engine cycle of two irreversible isothermal and two irreversible adiabatic processes with finite time non-adiabatic dissipation is considered [18, 19]. The four processes involved in the present model are discussed below:

• Isothermal expansion: during this process, the working substance is in contact with the hot reservoir at a higher temperature T_h for a time interval t_h. In this process there is an exchange of Q_h amount of heat between the working substance and the hot reservoir and the change in entropy is given as
  $$\begin{equation}{\Delta}S={\Delta}{S}_{\mathrm{h}}={Q}_{\mathrm{h}}/{T}_{\mathrm{h}}+{\Delta}{S}_{\mathrm{h}}^{\mathrm{i}\mathrm{r}}\end{equation} \tag{ 1 }$$
  where ${\Delta}{S}_{\mathrm{h}}^{\mathrm{i}\mathrm{r}}$ being the irreversible entropy production during the isothermal expansion process.
• Adiabatic expansion: the non-adiabatic dissipation increases the entropy during this adiabatic expansion process and the irreversible entropy production during the time interval t_h < t < t_h + t_a is denoted by [13],
  $$\begin{equation}{\Delta}{S}_{\mathrm{a}}^{\mathrm{i}\mathrm{r}}={S}_{\mathrm{a}}-{S}_{\mathrm{h}}\end{equation} \tag{ 2 }$$
  where S_a and S_h denotes the entropy at the instant t_a and t_h, respectively.
• Isothermal compression: now the working substance is in contact with the low temperature (T_c) cold reservoir for the time period t_h + t_a < t < t_h + t_a + t_c with the exchange of Q_c amount of heat between the working substance and the cold reservoir. The change in entropy is given as
  $$\begin{equation}{\Delta}{S}_{\mathrm{c}}={Q}_{\mathrm{c}}/{T}_{\mathrm{c}}+{\Delta}{S}_{\mathrm{c}}^{\mathrm{i}\mathrm{r}}\end{equation} \tag{ 3 }$$
  where ${\Delta}{S}_{\mathrm{c}}^{\mathrm{i}\mathrm{r}}$ is the irreversible entropy production during the isothermal compression process.
• Adiabatic compression: in the process of adiabatic compression during the time interval t_h + t_a + t_c < t < t_h + t_a + t_c + t_b, the working substance is removed from the cold reservoir and now the entropy production due to non-adiabatic dissipation is given by [13],
  $$\begin{equation}{\Delta}{S}_{\mathrm{b}}^{\mathrm{i}\mathrm{r}}={S}_{\mathrm{b}}-{S}_{\mathrm{c}}\end{equation} \tag{ 4 }$$
  where S_b and S_c denote the entropy at the instances t_b and t_c, respectively.

At the instance of completing the single cycle, the system recovers to its initial state and the total change in entropy of the system is zero [11, 13, 14], i.e. ${\Delta}S+{\Delta}{S}_{\mathrm{a}}^{\mathrm{i}\mathrm{r}}+{\Delta}{S}_{\mathrm{b}}^{\mathrm{i}\mathrm{r}}+{\Delta}{S}_{\mathrm{c}}=0$. Therefore, ${\Delta}{S}_{\mathrm{c}}=-\left({\Delta}S+{\Delta}{S}_{\mathrm{a}}^{\mathrm{i}\mathrm{r}}+{\Delta}{S}_{\mathrm{b}}^{\mathrm{i}\mathrm{r}}\right)$. Since the present model also considers the finite time non-adiabatic process, there will be an additional irreversible entropy production ${\Delta}{S}_{\mathrm{a}}^{\mathrm{i}\mathrm{r}}$ and ${\Delta}{S}_{\mathrm{b}}^{\mathrm{i}\mathrm{r}}$ during the adiabatic processes [13, 14]. From equations (1) and (3), the amounts of heat Q_h and Q_c exchanged between the hot and cold reservoirs and the working substance are obtained as [13, 14]

$$\begin{equation}{Q}_{\mathrm{h}}={T}_{\mathrm{h}}\left\{{\Delta}S-{\Delta}{S}_{\mathrm{h}}^{\mathrm{i}\mathrm{r}}\right\}\end{equation} \tag{ 5 }$$

$$\begin{equation}{Q}_{\mathrm{c}}={T}_{\mathrm{c}}\left\{-{\Delta}S-{\Delta}{S}_{\mathrm{c}}^{\mathrm{i}\mathrm{r}}-{\Delta}{S}_{\mathrm{a}}^{\mathrm{i}\mathrm{r}}-{\Delta}{S}_{\mathrm{b}}^{\mathrm{i}\mathrm{r}}\right\}.\end{equation} \tag{ 6 }$$

Even though many studies incorporated the 1/τ scaling of the irreversible entropy production both in a finite-time isothermal process [11, 29] and a finite-time adiabatic process [13, 14, 16], a recent theoretical study on quantum Otto engine showed (in terms of extra adiabatic work) 1/τ² scaling of the irreversible entropy production in a finite-time adiabatic process [30]. Here, τ is the controlling or contact time in which each process take place. This raises a possibility that the irreversible entropy production may scale with τ with other values of exponents for various real heat engines. Considering the above facts, the more generalized power law dissipative Carnot-like heat engines have been proposed earlier and studied in detail for efficiency at maximum power [17–19].

The irreversible entropy production associated with the isothermal processes and the adiabatic processes can be written in a generalized power law dissipative form as [19],

$$\begin{equation}{\Delta}{S}_{i}^{\mathrm{i}\mathrm{r}}={\alpha }_{i}{\left(\frac{{\sigma }_{i}}{{t}_{i}}\right)}^{1/\delta },\end{equation} \tag{ 7 }$$

where i: h, a, b, c and σ_i = λ_i Σ_i , in which α_i and λ_i are the tuning parameters and Σ_i is the isothermal and adiabatic dissipation coefficients [13, 14]. The level of dissipation present in the system is signified by the value of δ [18], in which δ = 1 represents the normal or low-dissipation regime, 0 < δ < 1: sub dissipation regime and δ > 1: super dissipation regime [17]. It should be noted that the employed model contains the parameter α_i , which might be related to the control scheme that tunes the system energy levels during the isothermal and adiabatic processes [31] and the parameter λ_i is related to some external controlled parameters that drive the system during the isothermal and adiabatic processes in a given time interval [17, 32]. A suitable combination of control schemes [31, 32] can be employed to control the irreversible entropy generation by using these tuning parameters.

Thus, the expression for the amount of heat exchanged Q_h and Q_c can be rewritten as [13, 19],

$$\begin{equation}{Q}_{\mathrm{h}}={T}_{\mathrm{h}}\left\{{\Delta}S-{\alpha }_{\mathrm{h}}{\left(\frac{{\sigma }_{\mathrm{h}}}{{t}_{\mathrm{h}}}\right)}^{1/\delta }\right\}\end{equation} \tag{ 8 }$$

and

$$\begin{equation}{Q}_{\mathrm{c}}={T}_{\mathrm{c}}\left\{-{\Delta}S-\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\alpha }_{i}{\left(\frac{{\sigma }_{i}}{{t}_{i}}\right)}^{1/\delta }\right\}.\end{equation} \tag{ 9 }$$

During the total time period t = t_h + t_c + t_a + t_b, the work performed by the engine is given by −W = Q_h + Q_c. Throughout this paper, the convention used is that the work done and heat absorbed by the system are positive. The power generated during the Carnot cycle is $P=\frac{-W}{t}$. On substituting the values of Q_h and Q_c, the expression for power can be written as

$$\begin{equation}P=\frac{1}{t}\left\{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S-{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\left(\frac{{\sigma }_{\mathrm{h}}}{{t}_{\mathrm{h}}}\right)}^{1/\delta }-{T}_{\mathrm{c}}\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\alpha }_{i}{\left(\frac{{\sigma }_{i}}{{t}_{i}}\right)}^{1/\delta }\right\}.\end{equation} \tag{ 10 }$$

The efficiency of the heat engine is then given by,

$$\begin{equation}\eta =\frac{{Q}_{\mathrm{h}}+{Q}_{\mathrm{c}}}{{Q}_{\mathrm{h}}}\end{equation} \tag{ 11 }$$

which, on substituting the values of Q_h and Q_c,, becomes [19],

$$\begin{equation}\eta ={\eta }_{\mathrm{C}}-\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}\left[\frac{{\Delta}S}{{\alpha }_{\mathrm{h}}}{\left(\frac{{t}_{\mathrm{h}}}{{\sigma }_{\mathrm{h}}}\right)}^{1/\delta }-1\right]}\left[1+\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}\left(\frac{{\alpha }_{i}}{{\alpha }_{\mathrm{h}}}\right){\left(\frac{{\sigma }_{i}{t}_{\mathrm{h}}}{{\sigma }_{\mathrm{h}}{t}_{i}}\right)}^{1/\delta }\right].\end{equation} \tag{ 12 }$$

In the following sections, the $\dot {{\Omega}}$ figure of merit and the efficient power χ_ep are used as target functions and are optimized for analyzing the performance of heat engines with (isothermal and non-adiabatic) power law dissipation.

The $\dot {{\Omega}}$ figure of merit is a trade-off function that provides a compromise between the useful energy and the lost energy. It can be defined as $\dot {{\Omega}}=\left(2\eta -{\eta }_{\mathrm{max}}\right)P/\eta$, where η_max is the maximum efficiency of a heat engine, which is nothing but the Carnot efficiency η_C [21]. With the inclusion of $P=\frac{{Q}_{\mathrm{h}}+{Q}_{\mathrm{c}}}{t}$ and $\eta =\frac{{Q}_{\mathrm{h}}+{Q}_{\mathrm{c}}}{{Q}_{\mathrm{h}}}$, one can obtain

$$\begin{equation}\dot {{\Omega}}=\left(2\eta -{\eta }_{\mathrm{C}}\right)\frac{{Q}_{\mathrm{h}}}{t}.\end{equation} \tag{ 13 }$$

On substituting the values of η, η_C and Q_h in equation (13), the expression for $\dot {{\Omega}}$ figure of merit can be rewritten as

$$\begin{equation}\dot {{\Omega}}=\frac{1}{t}\left\{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S-\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\left(\frac{{\sigma }_{\mathrm{h}}}{{t}_{\mathrm{h}}}\right)}^{1/\delta }-2{T}_{\mathrm{c}}\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\alpha }_{i}{\left(\frac{{\sigma }_{i}}{{t}_{i}}\right)}^{1/\delta }\right\}.\end{equation} \tag{ 14 }$$

Optimizing the $\dot {{\Omega}}$ figure of merit with respect to time t_i (i: h, c, a, b) gives the values of ${\tilde {t}}_{i}$ at which the $\dot {{\Omega}}$ figure of merit is maximum. The values for ${\tilde {t}}_{i}\left(i:\mathrm{h},\mathrm{c},\mathrm{a},\mathrm{b}\right)$ by considering $\frac{\partial \dot {{\Omega}}}{\partial {t}_{i}}=0$ are given below:

$$\begin{equation}{\tilde {t}}_{\mathrm{h}}={\left[\frac{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{1}{\delta }\right)\left(1+\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\left(\frac{2{T}_{\mathrm{c}}{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right)\right\}\right]}^{\delta },\end{equation} \tag{ 15 }$$

$$\begin{align}\hfill {\tilde {t}}_{\mathrm{a}}& =\left[\frac{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{1}{\delta }\right)\left(1+{\left(\frac{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right.\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left.\sum\limits _{i=\mathrm{b},\mathrm{c}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right\}\right)\right]}^{\delta },\hfill \end{align} \tag{ 16 }$$

$$\begin{equation}\begin{aligned}\hfill {\tilde {t}}_{\mathrm{b}}& =\left[\frac{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{1}{\delta }\right)\left(1+{\left(\frac{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right.\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left.\sum\limits _{i=\mathrm{a},\mathrm{c}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right\}\right)\right]}^{\delta },\hfill \end{aligned}\end{equation} \tag{ 17 }$$

and

$$\begin{align}\hfill {\tilde {t}}_{\mathrm{c}}& =\left[\frac{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{1}{\delta }\right)\left(1+{\left(\frac{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right.\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left.\sum\limits _{i=\mathrm{a},\mathrm{b}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right\}\right)\right]}^{\delta }.\hfill \end{align} \tag{ 18 }$$

The ratios of ${\tilde {t}}_{\mathrm{h}}$ and ${\tilde {t}}_{i}\left(i:\mathrm{a},\mathrm{b},\mathrm{c}\right)$ can also be obtained from the optimized $\dot {{\Omega}}$ figure of merit and are given below:

$$\begin{equation}{\left(\frac{{\tilde {t}}_{\mathrm{h}}}{{\tilde {t}}_{i}}\right)}^{\frac{1}{\delta }+1}=\frac{\left({T}_{\mathrm{h}}+{T}_{\mathrm{c}}\right){\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{i}{\sigma }_{i}^{1/\delta }}.\end{equation} \tag{ 19 }$$

Similarly the ratios for $\frac{{\tilde {t}}_{j}}{{\tilde {t}}_{i}}$ with i, j = a, b, h are also given by

$$\begin{equation}{\left(\frac{{\tilde {t}}_{i}}{{\tilde {t}}_{j}}\right)}^{\frac{1}{\delta }+1}=\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{j}{\sigma }_{j}^{1/\delta }}.\end{equation} \tag{ 20 }$$

Using equations (15) and (19) on equation (12) provides the efficiency at maximum $\dot {{\Omega}}$ figure of merit, ${\eta }_{\dot {{\Omega}}}$ and is given by

$$\begin{equation}{\eta }_{\dot {{\Omega}}}={\eta }_{\mathrm{C}}-\frac{\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}\left(1+\frac{{T}_{\mathrm{h}}+{T}_{\mathrm{c}}}{2{T}_{\mathrm{c}}}\zeta \right)}{\left(\frac{{T}_{\mathrm{h}}+{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}-{T}_{\mathrm{c}}}\right)\left(\frac{1}{\delta }+1\right)\left(1+\zeta \right)-1}\end{equation} \tag{ 21 }$$

where $\zeta ={\sum }_{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\left\{\left(\frac{2{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}+{T}_{\mathrm{c}}}\right)\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}\right\}}^{\frac{\delta }{\delta +1}}$. It is observed that when neglecting the adiabatic dissipation coefficients, σ_a = 0 and σ_b = 0, the efficiency of the heat engine (equation (21)) reduces to the one derived for Carnot-like heat engines without adiabatic dissipation [21].

It can be observed from equation (21) that the value of efficiency at the maximum $\dot {{\Omega}}$ figure of merit depends on the ratio between values of σ_i 's and σ_h. The generalized extreme bounds of the efficiency at maximum '$\dot {{\Omega}}$' figure of merit are obtained from equation (21) as

$$\begin{equation}{\eta }_{\mathrm{C}}\left(1-\frac{1}{2\left(1+\frac{1}{\delta }\right)}\right)\equiv {\eta }_{\dot {{\Omega}}}^{-}{\leqslant}{\eta }_{\dot {{\Omega}}}{\leqslant}{\eta }_{\dot {{\Omega}}}^{+}\equiv {\eta }_{\mathrm{C}}\left[\frac{\left(1-{\eta }_{\mathrm{C}}\right)\left(1+\frac{1}{\delta }\right)+\frac{1}{\delta }}{\left(1-{\eta }_{\mathrm{C}}\right)\left(2+\frac{1}{\delta }\right)+\frac{1}{\delta }}\right].\end{equation} \tag{ 22 }$$

These generalized lower and upper bounds of the efficiency at maximum $\dot {{\Omega}}$ figure of merit are achieved when σ_h → 0 and σ_h → ∞. When δ = 1, the lower bound becomes 3η_C/4 for σ_h → 0 and the upper bound becomes ${\eta }_{\mathrm{C}}\left(\frac{3-2{\eta }_{\mathrm{C}}}{4-3{\eta }_{\mathrm{C}}}\right)$ for σ_h → ∞, which is the bound of the efficiency of the Carnot-like heat engine at the maximum $\dot {{\Omega}}$ figure of merit obtained for the low dissipation case [25]. This shows that the inclusion of finite time non-adiabatic dissipation on low-dissipation model does not influence the lower and upper bound on the efficiency optimized at maximum $\dot {{\Omega}}$ figure of merit. It is to be noted that when δ → 0 (no dissipation), ${\eta }_{\dot {{\Omega}}}^{-}$ and ${\eta }_{\dot {{\Omega}}}^{+}\to {\eta }_{\mathrm{C}}$ and when δ → ∞ (high super dissipation limit), ${\eta }_{\dot {{\Omega}}}^{-}$ and ${\eta }_{\dot {{\Omega}}}^{+}\to {\eta }_{\mathrm{C}}/2$. This shows that the $\dot {{\Omega}}$ figure of merit provides half the Carnot efficiency, even at a very high level (super) of power law dissipation, which further confirms that the $\dot {{\Omega}}$ figure of merit provides a compromise between the useful energy and the energy lost. Thus, more generalized upper and lower bounds on the efficiency of a Carnot-like heat engine can be obtained under the combined adiabatic and isothermal power law dissipation in the asymmetric limits.

This section discusses the optimization of efficient power χ_ep = ηP and its significance in detail. The efficient power can be expressed using equation (11) and the fact that $P=\frac{{Q}_{\mathrm{h}}+{Q}_{\mathrm{c}}}{t}$ as

$$\begin{equation}{\chi }_{\mathrm{e}\mathrm{p}}=\eta P=\frac{{\left({Q}_{\mathrm{h}}+{Q}_{\mathrm{c}}\right)}^{2}}{{Q}_{\mathrm{h}}t}.\end{equation} \tag{ 23 }$$

Using equations (8) and (9), the following relation for χ_ep is obtained:

$$\begin{equation}{\chi }_{\mathrm{e}\mathrm{p}}=\frac{{\left[\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S-{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\left(\frac{{\sigma }_{\mathrm{h}}}{{t}_{\mathrm{h}}}\right)}^{1/\delta }-{T}_{\mathrm{c}}{\sum }_{i=\mathrm{a},\mathrm{b},\mathrm{c}}{\alpha }_{i}{\left(\frac{{\sigma }_{i}}{{t}_{i}}\right)}^{1/\delta }\right]}^{2}}{t{T}_{\mathrm{h}}\left({\Delta}S-{\alpha }_{\mathrm{h}}{\left(\frac{{\sigma }_{\mathrm{h}}}{{t}_{\mathrm{h}}}\right)}^{1/\delta }\right)}.\end{equation} \tag{ 24 }$$

Similar to the $\dot {{\Omega}}$ figure of merit, optimizing the efficient power χ_ep with respect to the time t_i (i: h, c, a, b) gives the values of ${\tilde {t}}_{i}$ at which χ_ep is maximum. The values for ${\tilde {t}}_{i}\left(i:\mathrm{h},\mathrm{c},\mathrm{a},\mathrm{b}\right)$ by considering $\frac{\partial {\chi }_{\mathrm{e}\mathrm{p}}}{\partial {t}_{i}}=0$ are given below:

$$\begin{equation}\begin{aligned}\hfill {\tilde {t}}_{\mathrm{h}}& =\left[\frac{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{2-\eta }{\delta }\right)+\left(2-\eta \right)\sum\limits _{i=\mathrm{a},\mathrm{b},\mathrm{c}}\left(\frac{1}{2}+\frac{1}{\delta }\right)\right.\right.\hfill \\ \hfill & \quad {\left.\left.{\times}{\left(\frac{2{T}_{\mathrm{c}}{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}\left(2-\eta \right)\right)}^{\frac{\delta }{\delta +1}}\right\}\right]}^{\delta },\hfill \end{aligned}\end{equation} \tag{ 25 }$$

$$\begin{align}\hfill {\tilde {t}}_{\mathrm{a}}& =\left[\frac{{T}_{\mathrm{c}}{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{2}{\delta }\right)\left(1+\sum\limits _{i=\mathrm{b},\mathrm{c}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right)\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left(\frac{2}{2-\eta }+\frac{2}{\delta }\right){\left(\frac{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{a}}{\sigma }_{\mathrm{a}}^{1/\delta }}\left(2-\eta \right)\right)}^{\frac{\delta }{\delta +1}}\right\}\right]}^{\delta },\hfill \end{align} \tag{ 26 }$$

$$\begin{align}\hfill {\tilde {t}}_{\mathrm{b}}& =\left[\frac{{T}_{\mathrm{c}}{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{2}{\delta }\right)\left(1+\sum\limits _{i=\mathrm{a},\mathrm{c}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right)\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left(\frac{2}{2-\eta }+\frac{2}{\delta }\right){\left(\frac{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{b}}{\sigma }_{\mathrm{b}}^{1/\delta }}\left(2-\eta \right)\right)}^{\frac{\delta }{\delta +1}}\right\}\right]}^{\delta },\hfill \end{align} \tag{ 27 }$$

and

$$\begin{align}\hfill {\tilde {t}}_{\mathrm{c}}& =\left[\frac{{T}_{\mathrm{c}}{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}{\left({T}_{\mathrm{h}}-{T}_{\mathrm{c}}\right){\Delta}S}\left\{\left(1+\frac{2}{\delta }\right)\left(1+\sum\limits _{i=\mathrm{a},\mathrm{b}}{\left(\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}\right)\right.\right.\hfill \\ \hfill & \quad +{\left.\left.\left(\frac{2}{2-\eta }+\frac{2}{\delta }\right){\left(\frac{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{\mathrm{c}}{\sigma }_{\mathrm{c}}^{1/\delta }}\left(2-\eta \right)\right)}^{\frac{\delta }{\delta +1}}\right\}\right]}^{\delta }.\hfill \end{align} \tag{ 28 }$$

The ratios of $\frac{{\tilde {t}}_{\mathrm{h}}}{{\tilde {t}}_{i}}\left(i:\mathrm{a},\mathrm{b},\mathrm{c}\right)$ can also be obtained from the optimized efficient power χ_ep and are given below:

$$\begin{equation}{\left(\frac{{\tilde {t}}_{\mathrm{h}}}{{\tilde {t}}_{i}}\right)}^{\left(\frac{1}{\delta }+1\right)}=\frac{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}{2{T}_{\mathrm{c}}{\alpha }_{i}{\sigma }_{i}^{1/\delta }}\left(2-\eta \right).\end{equation} \tag{ 29 }$$

Similarly the ratios for $\frac{{\tilde {t}}_{i}}{{\tilde {t}}_{j}}$ with i, j = a, b, c are also given by

$$\begin{equation}{\left(\frac{{\tilde {t}}_{i}}{{\tilde {t}}_{j}}\right)}^{\left(\frac{1}{\delta }+1\right)}=\frac{{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{\alpha }_{j}{\sigma }_{j}^{1/\delta }}.\end{equation} \tag{ 30 }$$

Similar to that done for the $\dot {{\Omega}}$ figure of merit, equation (12) on further substitution of equations (25) and (29) with $\eta ={\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}$ yields the efficiency at maximum efficient power, χ_ep and is given by

$$\begin{equation}{\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}=\frac{{\eta }_{\mathrm{C}}}{\delta }\left\{\frac{\phi +{\phi }^{\frac{1}{\delta +1}}\xi }{1+\frac{\phi }{\delta }+{\phi }^{\frac{1}{\delta +1}}\xi \left(\frac{1}{2}+\frac{1}{\delta }\right)-{\eta }_{\mathrm{C}}}\right\}\end{equation} \tag{ 31 }$$

where, in the above equation, $\phi =2-{\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}$ and $\xi ={\left(\frac{2{T}_{\mathrm{c}}{\alpha }_{i}{\sigma }_{i}^{1/\delta }}{{T}_{\mathrm{h}}{\alpha }_{\mathrm{h}}{\sigma }_{\mathrm{h}}^{1/\delta }}\right)}^{\frac{\delta }{\delta +1}}$. It is to be noted that the above expression contains ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}$ on both sides, which is too complicated to solve in the generalized fashion. However, the solution with δ = 1 (low dissipation) is found when neglecting the adiabatic dissipation coefficients, σ_a = 0 and σ_b = 0, which is the same as the efficiency derived for Carnot-like heat engines without adiabatic dissipation [26]. Similar to the efficiency at maximum $\dot {{\Omega}}$ figure of merit, the value of efficiency at maximum efficient power ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}$ also depends on the ratio between values of σ_i and σ_h. The generalized extreme bounds on the efficiency at maximum 'χ_ep' are obtained when σ_h → 0 and σ_h → ∞. That is, when σ_h → 0, ξ → ∞, which gives ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}=\frac{2{\eta }_{\mathrm{C}}}{\delta +2}$ and when σ_h → ∞, ξ → 0, which provides ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}=\frac{{\Upsilon}+\sqrt{{{\Upsilon}}^{2}-8{\eta }_{\mathrm{C}}}}{2}$, where ϒ = (δ + 2) − η_C(δ − 1). This shows that the efficiency at the maximum efficient power lies between these two extreme bounds, which is given by

$$\begin{equation}\frac{2{\eta }_{\mathrm{C}}}{\delta +2}\equiv {\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{-}{\leqslant}{\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}{\leqslant}{\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{+}\equiv \frac{\left(\delta +2\right)-{\eta }_{\mathrm{C}}\left(\delta -1\right)+\sqrt{{\left(\left(\delta +2\right)-{\eta }_{\mathrm{C}}\left(\delta -1\right)\right)}^{2}-8{\eta }_{\mathrm{C}}}}{2}.\end{equation} \tag{ 32 }$$

Thus, the generalized lower and upper bounds of efficiency at maximum efficient power are obtained for the asymmetric dissipation limits of σ_h → 0 and σ_h → ∞ respectively, for any finite values of σ_i (i: a, b, c). When δ = 1, the values of optimized efficiency at the maximum efficient power at low dissipation regime are obtained, which are $\frac{2{\eta }_{\mathrm{C}}}{3}$ and $\frac{3+\sqrt{9-8{\eta }_{\mathrm{C}}}}{2}$, the lower and upper bound, respectively [24, 26]. Similar to the $\dot {{\Omega}}$ figure of merit, this result also shows that the inclusion of finite time non-adiabatic dissipation on the low-dissipation model does not influence the lower and upper bound on the efficiency optimized at maximum efficient power. It is to be noted from equation (31) that when δ → 0 (no dissipation), ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{-}$ and ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{+}\to {\eta }_{\mathrm{C}}$, and when δ → ∞ (high super dissipation limit), ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{-}$ and ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}^{+}\to 0$. Thus, the generalized universal nature of the lower and upper bounds on the efficiency of Carnot-like heat engines at maximum efficient power ${\eta }_{{\chi }_{\mathrm{e}\mathrm{p}}}$ (equation (32)), under the combinations of isothermal and adiabatic asymmetric dissipation limits, is obtained.

In this paper, the generalized extreme bounds of the efficiency for the power law dissipative Carnot-like heat engines under $\dot {{\Omega}}$ figure of merit and efficient power χ_ep optimization criteria were investigated. Since the $\dot {{\Omega}}$ figure of merit provides the trade-off between the useful energy delivered and energy lost of heat engine and the efficient power governs the compromise between power and efficiency of heat engine, finding generalized bounds of the efficiency with these target functions is very relevant in direct correlation to the actual needs of energy consumption, availability of resources and environmental impact. Too much mathematical complexity is observed while obtaining the generalized expression for optimized efficiency at maximum χ_ep as compared to $\dot {{\Omega}}$ figure of merit. When δ = 1, the bounds of the efficiency with the $\dot {{\Omega}}$ figure of merit and efficient power χ_ep in the asymmetric dissipation converge to the same bounds as the corresponding ones obtained from the previous low dissipation model. In corroboration with the efficiency at maximum power, these results also showed that the presence of non-adiabatic dissipation does not influence the minimum and maximum bounds on the efficiency optimized at maximum $\dot {{\Omega}}$ figure of merit and also the maximum efficient power obtained in the low dissipation model, which does not take into account the non-adiabatic dissipation. Future work will focus on comparison of these figure of merit predictions with different heat engine models and with the observed efficiency of real heat engines [28].