*Article* **Optimization Modeling of Irreversible Carnot Engine from the Perspective of Combining Finite Speed and Finite Time Analysis**

**Monica Costea 1,\*, Stoian Petrescu 1, Michel Feidt 2, Catalina Dobre <sup>1</sup> and Bogdan Borcila <sup>1</sup>**


**Abstract:** An irreversible Carnot cycle engine operating as a closed system is modeled using the Direct Method and the First Law of Thermodynamics for processes with Finite Speed. Several models considering the effect on the engine performance of external and internal irreversibilities expressed as a function of the piston speed are presented. External irreversibilities are due to heat transfer at temperature gradient between the cycle and heat reservoirs, while internal ones are represented by pressure losses due to the finite speed of the piston and friction. Moreover, a method for optimizing the temperature of the cycle fluid with respect to the temperature of source and sink and the piston speed is provided. The optimization results predict distinct maximums for the thermal efficiency and power output, as well as different behavior of the entropy generation per cycle and per time. The results obtained in this optimization, which is based on piston speed, and the Curzon–Ahlborn optimization, which is based on time duration, are compared and are found to differ significantly. Correction have been proposed in order to include internal irreversibility in the externally irreversible Carnot cycle from Curzon–Ahlborn optimization, which would be equivalent to a unification attempt of the two optimization analyses.

**Keywords:** irreversible Carnot engine; optimization; thermodynamics with finite speed; internal and external irreversibilities; entropy generation calculation; thermodynamics in finite time

#### **1. Introduction**

Recent work [1] has emphasized that an analysis using the *finite time of the process* rather convey to a "physical potential optimization" than to an "engineering optimization" of thermal machine [2]. What is called *physical optimization* could provide more realistic performance compared to reversible Carnot cycle one, but it is still overvalued with respect to the actual one. Thus, the results of the physical optimization can be considered as upper bounds for real machine performance [3–5].

Moreover, criticisms have been addressed [6–11] to the results of Finite Time Thermodynamics (FTT) analysis of thermal machines, claiming that it failed to keep the promises, at least from the engineer's point of view. The main reason is the fact that FTT does not consider the internal losses generated by irreversibilities on a *fundamental basis*, since they have been introduced through a constant coefficient [12], factor of non-endoreversibility [13], degree of internal irreversibility [14], entropy variation ratio [15], ratio of two entropy differences [16], or entropy generation term as a function of temperature [17,18]. Therefore, the studies based on FTT approach cannot be effectively used by engineers for a better design and optimization study, leading to the conception and build of more efficient thermal machines since to apply optimization in a thermodynamic analysis, it needs to advance to

**Citation:** Costea, M.; Petrescu, S.; Feidt, M.; Dobre, C.; Borcila, B. Optimization Modeling of Irreversible Carnot Engine from the Perspective of Combining Finite Speed and Finite Time Analysis. *Entropy* **2021**, *23*, 504. https://doi.org/10.3390/e23050504

Academic Editor: Peter Salamon

Received: 19 March 2021 Accepted: 18 April 2021 Published: 22 April 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the higher phases of the system design than the one based on endoreversibility assumption that is considered very early [10]. Furthermore, the internal irreversibilities contributed by the system components are inherently interconnected with external irreversibilities in real operation conditions, so the performance reported by FTT analysis may be even smaller compared to that of a real system [8].

These criticisms did not remain without reply [19–23]. Thus, some authors of the anti-criticism papers addressed the clarification of finite-time thermodynamics objectives and their inclusion in the efforts to approach the irreversible systems and their performance [21]. Others emphasized the meaning of time for thermodynamic processes, namely that of providing bounds by discussing nine general principles for finding bounds on the effectiveness of energy conversion [22] or bounds relative to the efficiency versus maximum power efficiency of heat engines [23].

However, regarding the usefulness of the FTT, the endoreversible model has the merit of launching nowadays the competition of finding new upper bounds of thermal machines performance, closer to the real one. Thus, progress has been made in the modeling and optimization of thermodynamic processes and cycles [24–32], with special attention to the common ones in thermal machines: Otto cycle [27], Stirling engine [28], Kalina cycle [30], and Brayton cycle [31,32]. The results obtained [30,31] have shown that besides the gains of FTT optimization with three or four objectives, the original results reported in the initial work of the FTT theory [3–5] are also revealed.

The engineering optimization is mainly concerned about internal irreversibility assessment by insight in dissipation mechanism, to approach and model the irreversible cycle performance. Both internal and external irreversibility are considered, conveying an actual optimization of thermal machine performance.

Although there is no operational Carnot machine, much has been written on the optimization of Carnot cycle, and in particular, on the heat engine cycle, endoreversible [33–39] or with internal and external irreversibilities [40–61]. One reason could be that the performance of the Carnot cycle represents upper bounds for actual operating machines. However, only in the 1990s was attention focused on analysis of the Carnot cycle that also includes internal irreversibilities [12,16–18,41,42,46–49].

The Thermodynamics with Finite Speed (TFS) has been shown to be able to provide analytical evaluation of internal irreversibilities in several machines (Stirling, Otto, Diesel, Brayton, Carnot) [60–68] and electrochemical devices [69], as a function of the speed of the piston. Actually, the finite speed of the piston (and process implicitly) is also responsible of external irreversibilities, namely the finite heat transfer rate from source to cycle fluid and then to sink. The computation scheme developed in TFS using the Direct Method is based on the *First Law of Thermodynamics for Processes with Finite Speed* that contains the main internal irreversibility causes of thermal machines expressed as a function of the average piston speed. By integration of the new expression of the First Law on each cycle process, analytical expression for performance (Power and Efficiency) is provided. It can be used to optimize theoretical cycles of actual thermal machines and most importantly, it was validated for 12 performing Stirling Engines (in 16 operational regimes) [63,64] and 4 Solar Stirling Motors [49,50].

In recent publications [54–58], it has been mentioned that only *Thermodynamics with Finite Speed* (TFS) developed the necessary tools to optimize thermal machines by considering internal losses in addition to external ones by analytical means. Based on these statements, it was concluded that using the above-mentioned achievements of TFS in combination with FTT tools could convey a more realistic and efficient approach of thermal machines.

The analytical approach relative to this combination is presented here by original models introducing irreversibilities step by step and leading to important results that are more accurate than those obtained by each irreversible thermodynamics branch separately.

Firstly, a brief presentation of the Curzon–Ahlborn modeling of an endoreversible Carnot engine is given, together with the discussion relative to the presence of the nice radical in other works.

Then, optimization models for a Carnot cycle engine in a closed system that operates with finite speed of the piston are presented. The speed is considered constant and equal to the average speed of the piston that moves with a classical rod–crankshaft mechanism; by using the First Law of Thermodynamics for Processes with Finite Speed and the Direct Method, the optimization analysis of this cycle with external and internal irreversibilities is developed. Heat losses between the two heat reservoirs temperature level through the engine are considered. External irreversibilities are due to the finite heat transfer rate at the source and sink are modeled by an irreversible coefficient added to the classical expression of heat transfer on isothermal process. Internal irreversibilities are included in the mathematical expression of the First Law of Thermodynamics for Processes with Finite Speed as non-dimensional pressure losses due to the non-uniformity of the fluid pressure in the cylinder and friction. The piston speed for maximum power and for maximum efficiency is found for a particular set of engine parameters and it is shown that the minimum entropy generation per cycle occurs at maximum power. This analysis provides lower values of Carnot cycle efficiency than predicted by the Curzon–Ahlborn approach that was considered for comparison.

A further development of the model aims to combine the analysis of the Carnot cycle engine with only external irreversibility from Finite Time Thermodynamics (FTT) with the main advantage of the Thermodynamics with Finite Speed (TFS) approach, namely the internal irreversibility quantification as a function of the speed of the process (piston). Thus, corrections of the power output, efficiency, and optimized cycle fluid temperature in FTT optimization results based on the calculated speed of processes from the duration time in FTT and average piston speed in TFS. It results that when internal ireversibilities (speeds and friction) are included, the performance predicted by a TFS analysis is better than that predicted by an FTT analysis.

The first unification attempt between TFS and FTT considers only pressure losses due to the non-uniformity of the pressure in the cylinder as a function of piston speed. The analytical development of the model provides modified Curzon–Ahlborn expression for the externally irreversible Carnot cycle to also include the internal irreversibility. Equations for the optimum cycle temperature, maximum power, and efficiency for the internally and externally irreversible cycle are presented. The corrections are shown to increase with increased piston speed and to be significant at high but realizable piston speeds. The optimum temperature corresponding to maximum power is shown to increase with increased piston speed.

Then, a further step in the unification attempt between TFS and FTT is done by considering in addition to the Finite Speed, two other causes of internal irreversibility given by friction and throttling. Thus, based on the first unification achievement, new expressions are derived for the power output and efficiency of the direct Carnot cycle with finite speed processes. The results emphasize optimum speed values generating maximum power output, as well as the effect of irreversibilities on the optimum high temperature of the cycle.

The overview on the results of these models emphasizes that a significant difference exists between the results of the two optimization analyses in the sense that FTT optimization seems to be an upper bound when compared to the engineering optimization based on TFS and the Direct Method.

#### **2. Optimization Models of Carnot Cycle Engine**

#### *2.1. Models in Thermodynamics in Finite Time Analysis Seeking for Maximum Power Output of Carnot Cycle Engine*

The Curzon–Ahlborn modeling of the Carnot-type engine [3] refers to a cycle that is internally reversible but with no thermal equilibrium between the working fluid and the thermal reservoirs during the isothermal heat input and heat rejection, respectively. Furthermore, there exists a finite time duration of heat transfer given by Newton's heat transfer law during the isothermal processes. The expression of the power output of the

Curzon and Ahlborn cycle allows a maximum for which the corresponding efficiency is given by what was called nice radical.

Actually, the efficiency of a Carnot engine is treated for the case where the power output is limited by the rates of heat transfer to and from the working substance. It is shown that the efficiency, *ηCA*, at maximum power output is given by the expression *ηCA* = 1 − (*T*2/*T*1) 1/2 where *T*<sup>1</sup> and *T*<sup>2</sup> are the respective temperatures of the heat source and heat sink. It results in an efficiency less than the one introduced by Carnot (*η* = 1 − (*T*2/*T*1)), and it is shown that the existing engines performance is well described by the above result.

Before the Curzon and Ahlborn analysis, a similar approach aiming to maximize the power output and the nice radical has appeared in Chambadal modeling of the Carnot engine [4], but its model used heat capacity rate instead of heat conductances.

Almost at the same time, Novikov [5] has also found the nice radical.

The above-mentioned models and mainly the Curzon–Ahlborn one, which remain as references for the Carnot machine optimization in the frame of what was called Thermodynamics in Finite Time.

#### *2.2. Models of Irreversible Carnot Cycle Engine in Thermodynamics with Finite Speed*

#### 2.2.1. First Law of Thermodynamics for Processes with Finite Speed in Closed System

The optimization modeling presented in this section proceeds from a basis of thermodynamic fundamentals, systematically detailed and developed, starting from a unique equation called the *First Law of Thermodynamics for Processes with Finite Speed* [59,70–79]. The advantages of using this equation instead of the one from Classical Reversible Thermodynamics consists of its capability to account for both causes and mechanisms of irreversibility generation in complex cycles or real machines such as Stirling Engines, as well as in other cycles such as Otto, Diesel, Brayton, and Carnot cycles [60,71–73]. In addition, it is capable to consider both internal and external irreversibilities.

By integrating this equation for irreversible process step by step on each transformation of the cycle, the efficiency and power output are determined *analytically*. These expressions contain the *causes of irreversibility*, namely, the *finite speed of the piston*, an important parameter that can be optimized, for *Maximum Efficiency* or *Maximum Power*.

The mathematical expression of the First Law of Thermodynamics for Processes with Finite Speed in a closed system in its differential form is [59,70–76,78]:

$$dUI = \delta Q - p\_{av,i} \left( 1 \pm \frac{aw}{c} \pm \frac{f \cdot \Delta p\_f}{p\_{av,i}} \right) dV\_\prime \tag{1}$$

and the irreversible work for these processes [59,70–76,78]:

$$
\delta \mathcal{W}\_{irrev} = p\_{av,i} \left( 1 \pm \frac{aw}{c} \pm \frac{\Delta p\_f}{p\_{av,i}} \right) dV \tag{2}
$$

where *U*—internal energy, *Q*—heat, *W*—mechanical work, *pav,i*—instantaneous average pressure of the gas, *w*—average speed of the piston, *c*—average molecular speed, Δ*pf* pressure losses due to friction, *a*—coefficient depending the gas nature, *f*—coefficient relative to the amount of heat generated by friction that remains in the cycle, and *V* volume.

In the previous equations, the plus sign corresponds to the compression processes and the minus sign corresponds to the expansion ones.

Regarding the terms appearing in the right member, the first term in the parenthesis accounts for the irreversibility generated by the Finite Speed of the piston, *w*, and due to the non-uniformity of the pressure in the cylinder. Therefore, the pressure on the piston *pp* is larger during compression and smaller during expansion than the pressure on the head of the cylinder *pc*, and this is also the case for the instantaneous average pressure in the gas *pav.i* [47,59–61,76]. The experimental verification of this term is described in references [51,59–61]. The second term in the parenthesis takes into account the irreversibility generated by the friction between moving parts of the machine (piston–cylinder, bearings, etc.) [47,60,61]. When the processes in the machine involve internal throttling, a third term is added in the First Law for Processes with Finite Speed [47,60,61], playing an important role in the optimization of Stirling machines [51,59–67,77,80]. This term is less important in the Carnot cycle modeling, so that it is neglected in this study.

Other terms from the right member of Equations (1) and (2) have the following expressions:

$$a = \sqrt{3\gamma}, \ c = \sqrt{3RT},\tag{3}$$

with *γ*—ratio of specific heat at constant pressure and constant volume, and *R*—gas specific constant.

The pressure losses due to friction expressed as function of rotation per minute and based on their experimental evaluation for classical thermal engines operating upon Otto and Diesel cycles [81] were adapted to speed [76], and their expression resulted as:

$$
\Delta p\_f = (0.97 + 0.045w) / N \tag{4}
$$

where *N*—parameter depending on structural characteristics of the engine.

Note that Equations (1) and (2) completed by Equations (3) and (4) clearly show that the finite speed of the piston is responsible for all irreversibility causes, since it appears in both terms in the parentheses.

#### 2.2.2. Model of Carnot Cycle Engine with Analytically Modeled Internal and External Irreversibility

The cyclic system of a Carnot heat engine, including irreversibilities of finite-rate heat transfer between the gas in the thermal engine and its heat reservoirs, heat leakage between the reservoirs, and internal dissipations of the working fluid, is shown schematically in Figure 1 [48,49]. The working fluid in the system is alternately connected to a hot reservoir at constant temperature *TH,S* and to a cold reservoir at constant temperature *TL,S* and its temperatures are, respectively, *TH* and *TL*.

**Figure 1.** Carnot engine cycle with finite speed of the piston illustrated in p-V diagram [48,49].

Heat losses between the two heat reservoirs temperature level through the engine are considered by the heat rate term . *Qlost*. In addition, irreversible adiabatic processes are shown by the curves 2-3 and 4 -1.

Inside the cylinder with the piston illustrated in the bottom side of Figure 1 appears several pressures that are used in a process with finite speed analysis: on the piston, *pp*, on the cylinder, *pc*, and the instantaneous average pressure in the gas, *pav,i*.

By integrating Equations (1) and (2) over the isothermal processes of the Carnot cycle, the following expressions for the energy exchanges are dependent of the average piston speed yield:

• The irreversible heat received by the cycle gas from the source:

$$Q\_H = z\_H' \cdot mRT\_H \ln \frac{V\_4}{V\_3} = z\_H' \cdot mRT\_H \cdot \ln \varepsilon\_r \tag{5}$$

with *z <sup>H</sup>*—irreversible coefficient that accounts for a limited heat input in the cycle due to the finite speed of the process:

$$z\_H' = \left(1 - \frac{aw}{\sqrt{3RT\_H}} - \frac{f \cdot \Delta p\_f}{p\_{av, 34}}\right). \tag{6}$$

This irreversible coefficient shows that regardless of the heat available at the source, the cycle gas can only receive a limited amount of heat from the source.

• The irreversible heat rejected by the cycle gas to the sink:

$$Q\_L = z\_L' \cdot mRT\_L \ln \frac{V\_2}{V\_1} = -z\_L' \cdot mRT\_L \cdot \ln \varepsilon\_r \tag{7}$$

with *z <sup>L</sup>*—irreversible coefficient that accounts for a limited heat rejected by the cycle gas to the sink due to the finite speed of the process:

$$z\_L' = \left(1 + \frac{aw}{\sqrt{\Im RT\_L}} + \frac{f \cdot \Delta p\_f}{p\_{av,12}}\right). \tag{8}$$

• The irreversible work produced/consumed during the isothermal processes of the cycle:

$$\mathcal{W}\_{H,w} = z\_H \cdot mRT\_H \cdot \ln \varepsilon\_\prime \tag{9}$$

$$|\mathcal{W}\_{L,\text{nr}}| = z\_L \cdot mRT\_L \cdot \ln \varepsilon\_\prime \tag{10}$$

with the corresponding irreversible coefficients:

$$z\_H = \left(1 - \frac{aw}{\sqrt{\Im RT\_H}} - \frac{\Delta p\_f}{p\_{av, \text{34}}}\right),\tag{11}$$

$$z\_L = \left(1 + \frac{aw}{\sqrt{\Im RT\_L}} + \frac{\Delta p\_f}{p\_{av,12}}\right). \tag{12}$$

with

$$mR = P\_{1r}V\_{1r}/T\_{1r} \tag{13}$$

and

$$T\_{1r} = T\_{L, \mathbb{S}\_r} \ \ V\_{1r} = \ V\_1. \tag{14}$$

and

$$\frac{V\_4}{V\_3} = \frac{V\_1}{V\_2} = \varepsilon.\tag{15}$$

The work per cycle results from Equations (9) and (10) as:

$$\mathcal{W}\_{\text{cycle},\text{w}} = m\mathcal{R}(z\_H T\_H - z\_L T\_L) \ln \text{\dots} \tag{16}$$

The non adiabaticity of the engine suggested in Figure <sup>1</sup> by the term . *Qlost* is better explained in Figure 2 by the insulating wall between the two semi-cylinders that form the heat conduction path between the heat source and sink.

**Figure 2.** The cylinder configuration used in heat transfer area computation [48,49].

The heat transfer rate lost through this conduction path is:

$$
\dot{Q}\_{\rm lost} = k\_{\rm ins} A\_{\rm lost} (T\_{\rm HS} - T\_{\rm LS}) / B\_{\rm ins} \tag{17}
$$

where *kins*—thermal conductivity of the insulation, and *Bins*—insulation thickness. Equation (17) expressed on the cycle becomes:

$$Q\_{\text{lost,cycle}} = \dot{Q}\_{\text{lost}} \cdot \pi\_{\text{cycle}}.\tag{18}$$

The cycle time duration can be expressed as:

$$
\pi\_{cycle} = \frac{2(V\_1 - V\_3)}{wA\_p},
\tag{19}
$$

with *Ap*—piston area.

The area associated to the heat transfer rate lost between the source and sink yields (see Figure 2):

$$A\_{last} = (D + 2L\_4)(D\_\varepsilon - D),\tag{20}$$

where *D* is the inner diameter of the cylinder.

This heat transfer rate lost per cycle will modify the heat supply from the source and the heat rejected to the sink as follows:

$$Q\_{H,tot} = Q\_H + Q\_{lost,cycle} \tag{21}$$

$$|Q\_{L,tot}| = |Q\_L| + Q\_{lost,cycle}.\tag{22}$$

In the above equations, the heat input to the cycle gas and heat rejected from the gas to the sink may be considered those already given by Equations (5) and (7), or it can be expressed in terms of heat transfer as follows:

$$Q\_H = \mathbb{U}\_H(w) \cdot A\_H \cdot (T\_{H,S} - T\_H) \cdot \tau\_{H\prime} \tag{23}$$

$$|Q\_L| = \mathcal{U}\_L(w) \cdot A\_L \cdot (T\_L - T\_{L,S}) \cdot \tau\_L. \tag{24}$$

where *UH*(*w*) and *UL*(*w*) are the overall heat transfer coefficient during the heat exchange at the source and sink, respectively, and *AH* and *AL* are the area of the heat transfer surfaces.

The heat transfer expressed using the Finite Speed analysis (Equations (5) and (7)) should be the same as the heat transfer corresponding to the above Equations (23) and (24). Therefore, the two equalities allow expressing the *temperature of the gas at the hot end and at the cold end* respectively, in connection with the source and sink temperature:

$$T\_H = T\_{H,S} \cdot \left[1 + \frac{z\_H' \cdot m\mathbb{R} \cdot \ln\varepsilon}{\mathcal{U}\_H(w) \cdot A\_H \cdot \tau\_H}\right]^{-1} \tag{25}$$

$$T\_L = T\_{L,S} \cdot \left[1 - \frac{z\_L' \cdot mR \cdot \ln \varepsilon}{\mathcal{U}\_L(w) \cdot A\_L \cdot \tau\_L} \right]^{-1}. \tag{26}$$

The overall heat transfer coefficients of the heat exchanger at source and sink, *UL*, *UH* are calculated based on average bulk fluid temperatures by using well-known equations [82]:

$$Nu\_D = \begin{cases} 1.86 (Re\_D Pr)^{\frac{1}{3}} \left( \frac{D}{L} \right)^{\frac{1}{3}} \left( \frac{\mu}{\mu\_{wall}} \right)^{0.14}, & for \ Re\_D \le 2300 \\\ 0.023 \ R e\_D^{0.8} Pr^{\mu}, & for \ Re\_D \ge 3000 \end{cases} \tag{27}$$

with *n* = 0.4 for heating, respectively, *n* = 0.3 for cooling.

Similarly, the dynamic viscosity and the thermal conductivity of the gas are calculated using polynomial functions [64], based on the bulk gas temperature.

The contact time per cycle for the heat transfer from the heat source to the engine corresponding to the isothermal process is:

$$\tau\_H = (L\_4 - L\_3) / w = \frac{L\_1 \left(1 - \frac{1}{\varepsilon}\right) \left(\frac{T\_L}{T\_H}\right)^{\frac{1}{\gamma} - 1}}{w},\tag{28}$$

while the contact time per cycle for heat transfer from the gas engine to the sink is:

$$
\pi\_{\mathcal{L}} = (L\_1 - L\_2) / w = \frac{L\_1 \left(1 - \frac{1}{\mathcal{\tilde{\varepsilon}}}\right)}{w}. \tag{29}
$$

The area for the heat transfer between the source and the hot gas during the isothermal heat addition process (see Figure 2) is:

$$A\_H = 0.5D\left(\frac{\pi D}{4} - B\_{\rm ins}\right) + 0.5L\_1\left(1 + \frac{1}{\varepsilon}\right)\left(\frac{\pi D}{2} - B\_{\rm ins}\right) \cdot \left(\frac{T\_L}{T\_H}\right)^{\frac{1}{\gamma} - 1}.\tag{30}$$

Similarly, the area for heat transfer between the cold gas and the sink during the isothermal heat rejection process is expressed as:

$$A\_L = 0.5D\left(\frac{\pi D}{4} - B\_{\rm ins}\right) + 0.5L\_1\left(1 + \frac{1}{\varepsilon}\right)\left(\frac{\pi D}{2} - B\_{\rm ins}\right),\tag{31}$$

with

$$\frac{L\_1}{\varepsilon} = L\_2.\tag{32}$$

The power output of the irreversible Carnot engine is given by:

$$P\_{\Delta T, w, Q\_{last}} = \frac{W\_{cycle, w}}{\pi\_{cycle}}.\tag{33}$$

The efficiency of the Carnot cycle with internal and external irreversibility is:

$$\eta\_{\Delta T, w, Q\_{\rm lost}} = 1 - \frac{|Q\_{L, \rm w}|}{Q\_{H, \rm w}} = 1 - \frac{T\_L}{T\_H} \cdot \frac{z\_L'}{z\_H'}.\tag{34}$$

Then, the entropy generation per cycle can be expressed as:

$$
\Delta S\_{\text{cycle}\text{e}} = \frac{Q\_{H,w}}{T\_H} + \frac{Q\_{L,w}}{T\_L} = mRln\varepsilon \cdot (z\_H' - z\_L'),\tag{35}
$$

and its corresponding expression per unit time is:

$$
\dot{S}\_{\text{gcell}} = \frac{\Delta S\_{\text{cycle}}}{\pi\_{\text{cycle}}}.\tag{36}
$$

The results of this optimization model will be given in Section 3.

#### *2.3. The Curzon–Ahlborn Model of the Carnot Cycle Engine Combined with the Analysis Based on Thermodynamics with Finite Speed (TFS)*

The model aims to combine the analysis of the Carnot cycle engine with only external irreversibility in Thermodynamics in Finite Time (FTT) with the main advantage of the Thermodynamics with Finite Speed (TFS) approach, namely the internal irreversibility quantification as a function of the speed of the process.

The main differences of this model compared to the previous one are represented by:


$$\mathcal{W}\_{\text{lost, and, int}} = \left(\frac{aw}{c\_{22'}} + \frac{\Delta p\_f}{p\_{22'}}\right) (V\_{3'} - V\_2)\_{23'} - \left(\frac{aw}{c\_{4'1}} + \frac{\Delta p\_f}{p\_{4'1}}\right) (V\_1 - V\_{4'})\_{4'1}.\tag{37}$$

where *p*23 and *p*4 <sup>1</sup> are the average gas pressure on the irreversible adiabatic compression and expansion, respectively.

This lost work term is then subtracted from the work per cycle given by Equation (16), since it does not include the effect of internal irreversibilities of the adiabatic processes.

By including this lost work term in the analysis, an expression for the efficiency of the Carnot cycle, considering all internal and external irreversibilities yields as:

$$\eta\_{\Delta T, w, f} = \left(\frac{z\_H}{z\_H'} - \frac{z\_L \cdot T\_L}{z\_H' \cdot T\_H}\right) - I\_{\rm ad} \frac{1 - T\_L/T\_H}{z\_H'(\gamma - 1)\ln\varepsilon'}\tag{38}$$

where the irreversible adiabatic process contribution of the internal irreversibility of the cycle, due to the finite piston speed and friction, *Iad*, results as:

$$I\_{ad} = aw\left(\frac{1}{c\_{23'}} + \frac{1}{c\_{4'1}}\right) + \Delta p\_f \left(\frac{1}{p\_{23'}} + \frac{1}{p\_{4'1}}\right). \tag{39}$$

Note that the second term in Equation (38) is obtained by integration of the First Law for Processes with Finite Speed (TFS) for the adiabatic processes 23 and 4 1 (see Figure 1), Equations (1) and (2).

The combination of the two analyses based on FTT and TFS models will include a similar term to that given by Equation (39) in the Curzon–Ahlborn approach. As previously mentioned, this approach included the time duration of the cycle processes, with the assumption that the adiabatic processes occur rapidly and accordingly consume far less time than the isothermal processes. Based on this assumption, the FTT and TFS analyses can be rationally compared only if the Carnot cycle engine dimensions and number of cycles per unit time are made equal in both cases. In a TFS analysis, the speed of the piston, *w*, is assumed constant in each of the four processes and equals the average speed based on the number of cycles per unit time. However, in a Curzon–Ahlborn type analysis (FTT optimization), the speed of isothermal compression *wL*, the speed of isothermal expansion *wH*, and the speed of the adiabatic processes *wad* (assumed equal for both adiabatic processes), are calculated. The result must be consistent with the total cycle time optimized for maximum power.

When this comparison is performed, the following process speeds, in terms of the average speed, are obtained (see Figure 2) [49]:

$$w\_L = \frac{a'(L\_1 - L\_2)(1 + Z^\*)}{2L\_1/w},\tag{40}$$

$$w\_H = \frac{a'(L\_4 - L\_3)(1/Z^\* + 1)}{2L\_1/w},\tag{41}$$

$$w\_{ad} = \frac{a'w[(L\_2 - L\_3) + (L\_1 - L\_4)]}{2L\_1(a'-1)},\tag{42}$$

where *Z*\*—ratio of the optimized duration of the isothermal processes in the Curzon– Ahlborn treatment (FTT), *a*'—coefficient depending on time to speed transfer.

The optimized temperatures in the Curzon–Ahlborn analysis [3] are expressed based on corresponding optimized times for each process, as follows:

$$T\_{L,FTT} = T\_L \frac{1 + \sqrt{\frac{T\_H}{T\_L}} \cdot \frac{1}{Z^t}}{1 + \frac{1}{Z^t}},\tag{43}$$

$$T\_{H,TTT} = T\_H \frac{1 + \sqrt{\frac{T\_L}{T\_H}} \cdot Z^\*}{1 + Z^\*}. \tag{44}$$

By using the above expressions of temperatures and including the effect of internal irreversibility, the corresponding power of Carnot cycle in FTT analysis is:

$$Power\_{FTT} = \frac{\frac{A\_L \text{LI}\_L}{d} \left(\sqrt{T\_H} - \sqrt{T\_L}\right)^2}{\left(Z^\* + 1\right)^2} - \left(\mathcal{W}\_{loss, \text{ad}, \text{int}} + \mathcal{W}\_{loss, \text{isat}, \text{int}}\right) \frac{1}{\tau\_{\text{cycle}}}.\tag{45}$$

Equation (45) appears as a combination of the two analyses as the first term is the original Curzon–Ahlborn term [3] taking account of only external irreversibilities generated by the temperature difference, and the second term accounts for internal irreversibilities generated by the finite speed and friction from the TFS approach.

Nevertheless, a simpler expression of the power output can be also given as:

$$Power\_{\Lambda T, w, f, FTT} = Q\_H \cdot \eta'\_{\Lambda T, w, f, FTT} \cdot \frac{1}{\pi\_{cycle}},\tag{46}$$

where the efficiency term contains all irreversibility causes of the Carnot cycle engine.

The passage from the efficiency of the Carnot cycle including only external irreversibilities and corresponding to maximum power output in the original Curzon–Ahlborn analysis [3]:

$$\eta\_{\Delta T, FTT} = 1 - \frac{T\_{L, FTT}}{T\_{H, FTT}} = 1 - \sqrt{\frac{T\_{LS}}{T\_{HS}}},\tag{47}$$

will be performed here by including the effects of internal irreversibilities. Similarly, Equations (5)–(12) are expressed by evaluating ZFTT and Z FTT irreversible coefficients at the appropriate speeds (*wL* and *wH*) on the isothermal processes at *TL* and *TH* respectively, and on the adiabatic processes (*wad*) conveying to the following corrected efficiency:

$$\eta\_{\text{irr,int,FTT}} = \frac{Z\_{H,\text{FTT}}}{Z\_{H,\text{FTT}}^{!}} - \frac{Z\_{L,\text{FTT}} \cdot T\_{L,\text{FTT}}}{Z\_{L,\text{FTT}}^{!} \cdot T\_{H,\text{FTT}}} - I\_{\text{ad}}^{\prime} \frac{1 - T\_{L,\text{FTT}} / \cdot T\_{H,\text{FTT}}}{Z\_{H,\text{FTT}}^{!} (\gamma - 1) \ln \varepsilon} \tag{48}$$

where the equivalent term *Iad* to that from Equation (39) is similar, but it is based on *wad* (Equation (42)) instead of *w* and also on the resulting temperatures and pressures from the Curzon–Alhborn. Ref. [3] analysis of the Carnot cycle completed by TFS tools (Equations (43) and (44)).

#### *2.4. Unification Attempts of Thermodynamics in Finite Time and Thermodynamics with Finite Speed Analyses*

The first unification attempt is based on [47] that had a very important role in the development of Thermodynamics with Finite Speed (TFS) and the Direct Method, for analytical evaluation of the performances of irreversible cycles with internal and external irreversibilities. Later, it was completed by [31,34].

Specific issues addressed in this model are illustrated on cycle Carnot engine represented in *T-S* coordinates in Figure 3. There are shown to have external irreversibility due to heat transfer from the source (with fixed temperature *TH,S*) to the cycle temperature at the hot end, *TX*, during the isothermal heat addition process 2–3. Then, internal irreversibilities due to the finite piston speed are considered during only the adiabatic compression and expansion processes. The sink temperature and the cycle temperature at the cold end are the same. The sink temperature, *T*0, is fixed, while the cycle temperature at the hot end, *TX*, is a variable.

**Figure 3.** Carnot engine cycle with internal irreversibilities illustrated in T-S diagram [47,52].

Another novelty compared to previous model consists of the use of entropy variation calculation on the irreversible cycle processes that will provide a term in the cycle efficiency expression that could unify the two analyses.

The first unification attempt is based on the First Law of Thermodynamics for Processes with Finite Speed [70–73] in its reduced form that considers only the internal irreversibility due to the finite speed of the piston:

$$d\mathcal{U} = \delta \mathcal{Q} - p\_{av,i} \left( 1 \pm \frac{aw}{c} \right) dV. \tag{49}$$

From the equation for adiabatic irreversible processes of ideal gases with constant specific heats that is derived from Equation (49) by integration [72,73,75,76], one can express the temperature *T*<sup>2</sup> at the end of an irreversible adiabatic process as

$$T\_2 = \frac{\left(1 \pm \frac{aw}{c\_1}\right)^2}{\left(1 \pm \frac{aw}{c\_2}\right)^2} T\_1 \left(\frac{V\_1}{V\_2}\right)^{\gamma - 1} = \delta\_{irr} T\_1 \left(\frac{V\_1}{V\_2}\right)^{\gamma - 1} \tag{50}$$

where *γ* is the ratio of the specific heat at constant pressure and at constant volume.

For a compression process with finite speed *w* << *c*, one could express *δirr*.*cpr* as follows:

$$\delta\_{irr,cpr} = \frac{\left(1 + \frac{aw}{c\_1}\right)^2}{\left(1 + \frac{aw}{c\_2}\right)^2} \stackrel{\approx}{=} \left[\left(1 + \frac{aw}{c\_1}\right)\left(1 - \frac{aw}{c\_2}\right)\right]^2 = \left[1 + \frac{aw}{c\_1} - \frac{aw}{c\_2}\right]^2,\tag{51}$$

if a2*w<sup>2</sup>* << *<sup>c</sup>*1·*c*<sup>2</sup> and the corresponding term is neglected.

Note that for compression, the plus sign is used in parenthesis.

Note that the average molecular speed *c*<sup>2</sup> depends on temperature *T*<sup>2</sup> that contains *δirr*.*cpr*. Thus, the calculation should be done by using approximations.

The first approximation considers the temperature at the end of the reversible adiabatic compression for which one gets (see Equation (3)):

$$T\_2 = T\_1 \left(\frac{V\_1}{V\_2}\right)^{\gamma - 1} \Rightarrow c\_2 = c\_1 \left(\frac{V\_1}{V\_2}\right)^{\frac{\gamma - 1}{2}}.\tag{52}$$

By substituting Equation (52) in Equation (51), a first evaluation of *δirr*.*cpr* is done:

$$\delta\_{irr.cpr} = \left[1 + \frac{aw}{c\_1} - \frac{aw}{c\_1} \left(\frac{V\_2}{V\_1}\right)^{\frac{\gamma - 1}{2}}\right]^2. \tag{53}$$

Note that a more precise approximation is possible by combining Equations (50) and (53) that yields:

$$T\_2 = \delta\_{irr.cpr} T\_1 \left(\frac{V\_1}{V\_2}\right)^{\gamma - 1} \,\, \, \, \, \, \tag{54}$$

and a better approximation for the adiabatic irreversible coefficient is given by:

$$\delta'\_{irr.cpr} = \left[1 + \frac{aw}{c\_1} - \frac{aw}{c\_1} \left(\frac{V\_2}{V\_1}\right)^{\frac{\gamma - 1}{2}} \left(\delta\_{irr.cpr}\right)^{-\frac{1}{2}}\right]^2. \tag{55}$$

For simplicity, the first approximation expression of the adiabatic irreversible coefficient (Equation (53)) is used hereafter.

The entropy variation computation in the case of an adiabatic irreversible process of compression with finite speed when the results from Equations (50) and (53) are introduced in the classical formula of Δ*S*:

$$
\Delta S = S\_f - S\_i = mc\_v \ln \frac{T\_f}{T\_i} + mR \ln \frac{V\_f}{V\_i} \tag{56}
$$

which provides:

$$
\Delta S\_{irr, \varepsilon pr} = mc\_v l \ln \left[ 1 + \frac{aw}{c\_1} - \frac{aw}{c\_1} \left( \frac{V\_2}{V\_1} \right)^{\frac{\gamma - 1}{2}} \right]^2. \tag{57}
$$

Similarly, the entropy variation expression on the adiabatic irreversible expansion can be derived showing that the only difference consists in the change of signs in the parentheses, so that one can give a general form of both compression and expansion processes, as:

$$
\Delta S\_{ad,irr}^{w} = mc\_v \ln \left[ 1 \pm \frac{aw}{c\_1} \mp \frac{aw}{c\_1} \left( \frac{V\_2}{V\_1} \right)^{\frac{\gamma - 1}{2}} \right]^2. \tag{58}
$$

By using Equations (56) and (58) in the present analysis on the two irreversible adiabatic processes and on the isothermal expansion, the following expressions result:

$$
\Delta S\_{\text{ad.irr.}\underline{r}\underline{r}\underline{r}}^{\underline{w}} = \Delta S\_{12} = m c\_v \ln(a\_1),
\text{ with } a\_1 = \left[1 + \frac{aw\_{c\underline{r}\underline{r}}}{c\_1} - \frac{aw\_{c\underline{r}\underline{r}}}{c\_1} \left(\frac{V\_2}{V\_1}\right)^{\frac{\gamma - 1}{2}}\right]^2,\tag{59}
$$

$$
\Delta S\_{ad.irr.exp}^{w} = \Delta S\_{34} = mc\_v \ln(a\_2),
\text{ with } a\_2 = \left[1 - \frac{aw\_{exp}}{c\_3} + \frac{aw\_{exp}}{c\_3} \left(\frac{V\_4}{V\_3}\right)^{\frac{\gamma - 1}{2}}\right]^2,\tag{60}
$$

$$
\Delta S\_{23} = S\_3 - S\_2 = mR \ln \frac{p\_2}{p\_3}.\tag{61}
$$

with *cv*—specific heat at constant volume, *R*—specific constant of the cycle fluid.

Then, the actual thermal efficiency of the Carnot cycle engine with irreversibilities can be expressed based on previous calculation (see Figure 3) as:

$$\eta\_{act} = 1 - \frac{Q\_{41}}{Q\_{23}} = 1 - \frac{T\_{\mathbb{C}} \Delta S\_{14}}{T\_X \Delta S\_{23}} = 1 - \frac{T\_0 (\Delta S\_{23} + \Delta S\_{12} + \Delta S\_{34})}{T\_X \Delta S\_{23}},\tag{62}$$

and together with Equations (59)–(61), the following expression results:

$$\eta\_{act} = 1 - \frac{T\_0}{T\_X} \left[ 1 + \frac{2 \ln(a\_1 a\_2)}{(\gamma - 1) \ln \frac{P\_2}{P\_3}} \right]. \tag{63}$$

When the piston speed is much less than the average molecular speed, namely *awcpr* << *c1*, and *aexp* << *c3*, one gets a simplified form of Equation (63):

$$\eta\_{act} = 1 - \frac{T\_0}{T\_X} \left[ 1 + \frac{2(\beta\_1 + \beta\_2)}{(\gamma - 1) \ln \frac{p\_s}{p\_S}} \right] \tag{64}$$

where

$$\beta\_1 = \frac{aw\_{cpr}}{c\_1} \left(1 - \sqrt{\frac{T\_0}{T\_X}}\right) \tag{65}$$

$$\beta\_2 = \frac{aw\_{\exp}}{c\_3} \left( \sqrt{\frac{T\_X}{T\_0}} - 1 \right). \tag{66}$$

For the same speed of the piston on the two adiabatic processes of the cycle, Equation (64) becomes:

$$\eta\_{act} = 1 - \frac{T\_0}{T\_X} \left\{ 1 + \frac{4aw}{c\_1} \frac{\left(1 - \sqrt{\frac{T\_0}{T\_X}}\right)}{(\gamma - 1) \ln \frac{p\_2}{p\_3}} \right\}. \tag{67}$$

Once having the actual efficiency of the cycle, the power output of the engine can be easily derived as: .

$$
\dot{W}\_{\rm act} = \dot{Q}\_{H} \eta\_{\rm act} = \mathcal{U}\_{H} A\_{H} (T\_{H,S} - T\_{X}) \eta\_{\rm act}.\tag{68}
$$

To render the model more general, a non-dimensional form of the power output of the Carnot engine will be optimized, namely:

$$P\_{\rm ND} = \frac{\dot{W}\_{\rm act}}{\underline{U}\_{\rm H} A\_{\rm H} T\_{\rm H,S}}.\tag{69}$$

Moreover, the actual efficiency is expressed as a product of the Carnot reversible efficiency:

$$
\eta\_{\rm CC} = \left(1 - \frac{T\_0}{T\_X}\right)\_{\prime} \tag{70}
$$

and the second law efficiency accounting for irreversibilities:

$$\eta\_{IIad.irr}^{w} = \left[1 - \frac{\mathbb{C}\left(\frac{T\_0}{T\_X}\right)}{\left(1 + \sqrt{\frac{T\_0}{T\_X}}\right)}\right],\tag{71}$$

with the internal irreversible coefficient *C* given by:

$$C = \frac{4aw}{c\_1(\gamma - 1)\ln\frac{p\_2}{p\_3}}.\tag{72}$$

By combining Equation (69) with Equations (68), (70)–(72) and term rearrangement, one gets:

$$P\_{\rm ND} = \left(1 - \frac{T\_X}{T\_{H,S}}\right) \left(1 - \frac{T\_0}{T\_X \Phi}\right) \tag{73}$$

With

$$\Phi = \frac{1}{1 + \mathcal{C}\left(1 - \sqrt{\frac{T\_0}{T\_X}}\right)}.\tag{74}$$

Note that for a given cycle fluid, coefficient *Φ* depends only on the fluid temperature at the hot end, *TX*, and the piston speed, *w*. Thus, the non-dimensional power (Equation (73)) is seen to be a complex function of *TX* and the piston speed by the term *C*. Searching for an analytic expression of the optimum temperature to maximize the non-dimensional power can be done in the first approximation, for *Φ* = constant in Equation (73). This is in good agreement with Ibrahim's approach [16], where for *Φ* constant, the expression of the optimal temperature of the cycle fluid at the hot end that maximizes the power output of the engine was established as:

$$T\_X^{\max P\_{ND}} \to T\_{opt} = \sqrt{\frac{T\_{H,S} \cdot T\_0}{\Phi}}.\tag{75}$$

Although this is a simple expression, the value of *Φ* is not known. It is indicated as a parameter with a given (not computed) value.

In the present analysis, one can approximate the value of *Topt* by iterations. Thus:

• For *w =* 0, which means an internally reversible cycle, Equations (72) and (74) lead to *Φ* = 1, so that Equation (75) becomes:

$$T\_{opt}^{(w=0)} = \sqrt{T\_{H,S} \cdot T\_0}.\tag{76}$$

• For *w* = 0, by combining Equations (74) and (76), a first approximation of the term responsible for cycle irreversibilities is expressed as:

$$\Phi\_w = \left[ 1 + \mathcal{C} \left( 1 - \sqrt[4]{\frac{T\_0}{T\_{H,S}}} \right) \right]^{-1} \text{.} \tag{77}$$

and the corresponding optimum temperature yields from Equation (75) as:

$$T\_{opt}^{(w \neq 0)} = \sqrt{T\_{H,S} \cdot T\_0 \left[1 + \mathcal{C} \left(1 - \sqrt[4]{\frac{T\_0}{T\_{H,S}}}\right)\right]}.\tag{78}$$

Equation (78) is the first approximation of the optimum temperature to maximize the non-dimensional power when the piston speed is not zero and when therefore both internal and external irreversibilities are accounted for.

Furthermore, the next step in the approximation procedure is to replace *Tx* in Equation (74) by Equation (78), that allows obtaining a more accurate expression of *Φ* term:

$$\Phi\_w' = \left[1 + \mathbb{C}\left(1 - \sqrt[4]{\frac{T\_0 \Phi\_w}{T\_{H,S}}}\right)\right]^{-1}.\tag{79}$$

One could continue the iteration, but the gain in accuracy would become insignificant. Thus, the optimized temperature of the cycle fluid at the hot end of the engine coming out of TFS analysis is:

$$T\_{opt}^{'(w \neq 0)} = \sqrt{\frac{T\_{H,S} \cdot T\_0}{\Phi\_w'}},\tag{80}$$

and the maximum non dimensional power output of the internally and externally irreversible Carnot cycle becomes:

$$P\_{\rm ND,maxZ} = \left(1 - \frac{T\_{opt}^{'(w \neq 0)}}{T\_{H,S}}\right) \left(1 - \frac{T\_0}{\Phi\_w' T\_{opt}^{'(w \neq 0)}}\right) = \left(1 - \sqrt{\frac{T\_0}{T\_{H,S} \Phi\_w'}}\right)^2. \tag{81}$$

Then, the efficiency of the irreversible Carnot cycle is calculated by substituting *T* (*w* =0) *opt* into Equation (67) that leads to:

$$\eta\_{act} = 1 - \sqrt{\frac{T\_0}{T\_{H,S}} \Phi\_{\text{iv}}'} \cdot \left[1 + \mathbb{C} \left(1 - \sqrt[4]{\frac{T\_0}{T\_{H,S}} \Phi\_{\text{iv}}'}\right)\right]. \tag{82}$$

One can see now that Equation (82) unifies the FTT and TFS analyses by the same expression of the actual efficiency of an irreversible Carnot cycle engine. Thus:

• For internally reversible, externally irreversible Carnot cycle engine for which *w* = 0 and consequently, *Φ <sup>w</sup>* = 1, one gets the Curzon–Ahlborn "nice radical" [3]:

$$
\eta\_{CA} = 1 - \sqrt{\frac{T\_0}{T\_{H,S}}}.\tag{83}
$$

• For an internally and externally irreversible Carnot cycle engine for which *w* = 0 and consequently, *Φ <sup>w</sup>* > 1, one gets:

$$\eta\_{\rm act} = 1 - \sqrt{\frac{T\_0}{T\_{H,S}}} \mathcal{J}\_{\rm uv} \tag{84}$$

with

$$\zeta\_{\rm w} = \sqrt{\Phi\_{\rm w}^{\prime}} \left[ 1 + \mathbb{C} \left( 1 - \sqrt[4]{\frac{T\_0}{T\_{H,S}} \Phi\_{\rm w}^{\prime}} \right) \right]. \tag{85}$$

Note that *ζ<sup>w</sup>* ≥ 1 and it accounts for internal irreversibilities of the cycle when depending on the piston speed. Equations (83)–(85) clearly show that the nice radical of FTT analysis overestimates the actual efficiency of the engine evaluated by TFS analysis.

A second unification attempt is under development. It aims to extend the modeling by considering, in addition to the finite speed, two other causes of internal irreversibility: friction and throttling.

Based on previous equations of the first unification attempt, a new expression was derived for the actual efficiency of the Carnot cycle engine:

$$\eta\_{act}^{irr} = 1 - \frac{T\_0}{T\_X} \left\{ 1 + 4 \left( \frac{aw}{c\_1} + \frac{\Delta p\_f}{p\_{av,34}} + \frac{\Delta p\_{thr}}{p\_{av,34}} \right) \frac{\left(1 - \sqrt{\frac{T\_0}{T\_X}}\right)}{(\gamma - 1)\ln\frac{p\_2}{p\_3}} \right\},\tag{86}$$

where Δ*pthr* is estimated as [62–64,83]:

<sup>Δ</sup>*pthr* <sup>=</sup> *Cthr* · *<sup>w</sup>*2, (87)

with *Cthr* = 0.005.

Then, the irreversibility coefficient yields:

$$\mathcal{C}\_{irr} = 4\left(\frac{aw}{c\_1} + \frac{\Delta p\_f}{p\_{av,34}} + \frac{\Delta p\_{thr}}{p\_{av,34}}\right)\frac{1}{(\gamma - 1)\ln\frac{p\_2}{p\_3}}.\tag{88}$$

The power output and efficiency of the Carnot cycle engine with finite speed processes considering all internal irreversibility causes are smaller compared to those determined from Equations (81) and (82), since the new correction is more substantial by its three terms (Equation (88)).

The results of this modeling emphasize optimum speed values generating maximum power output, as well as the effect of irreversibilities on the optimum cycle high temperature.

#### **3. Results**

The results of TFS analysis presented in Section 2.2 relative to a Carnot cycle engine with internal and external irreversibilities generated by losses due to (1) heat transfer between the cycle and the heat source and sink, (2) the effect of variation in the area for heat transfer and in the dwell time for heat transfer due to the movement of the piston during the isothermal expansion and compression processes, and (3) non adiabaticity of the engine are presented in Figures 4–6. The following fixed parameters entering in the equations of the model were used: *D* = 0.015 m; *L*<sup>1</sup> = 2 m; *ε* = 3; *f* = 0; *p*1r = 0.05 bar (pressure of the gas in state 1r); Δ*pf* = (0.97 + 0.045 w)/80; *TH,S* = 1200 K; *TL,S* = 300 K; *γ* = 1.4; *Bins* = 0.002 m; *kins* = 0.01 W/mK; *De* = 0.019 m. The cycle fluid is air that is considered as an ideal gas with specific heat, conductivity, and viscosity varying as a function of temperature.

Figure 4 illustrates the effect of irreversibilities introduced gradually on the power output showing the important difference between the cycle power output for the reversible Carnot cycle and for the Carnot cycle with irreversiblities due to the finite speed of the piston. Then, the cycle efficiency including internal and external irreversibilities, *η*Δ*T*,*w*,*Qlost* , is represented as a function of piston speed showing optimum values for maximum performance. In addition, the time rate of entropy generation is added in order to compare the optimization results in terms of optimal speed.

One can see that the piston speed for maximum efficiency is only 4 m/s, for which the rate of entropy generation (per unit of time) is very low. Moreover, the piston speed for maximum power is near 17 m/s, and the rate of entropy generation (per unit of time) at this speed is significantly higher. As expected, the power output decreases, as additional irreversibilities are included in the analysis.

**Figure 4.** Power, efficiency and entropy generation per time as a function of the piston speed.

**Figure 5.** Power, efficiencies and entropy generation per cycle as a function of average piston speed.

**Figure 6.** Power, efficiencies and temperatures as a function of average piston speed.

Figure 5 brings together the efficiency of the Carnot cycle determined by the TFS analysis when it is gradually affected by irreversibility, the one based on Curzon–Ahlborn analysis, the power output, and the entropy variation per cycle as functions of piston speed. The efficiency of the Carnot cycle as determined by TFS analysis is at all piston speeds less than the efficiency based on the Curzon–Ahlborn analysis. In addition, for piston speeds greater than *wopt*, the efficiency of the Carnot cycle at maximum power as determined by TFS is less than the efficiency based on the Curzon–Ahlborn analysis, even if only the external irreversibility is included. For example, the TFS efficiency, at the speed corresponding to maximum power, is 0.29 when only external irreversibilities are included and is 0.15 when both internal and external irreversibilities are included in the analysis.

An important aspect is related to the entropy generation per cycle and per time as functions of piston speed from Figures 4 and 5. Their evolution with the piston speed is completely different, in that only Δ*Scycle* shows a minimum for the speed as the maximum power output.

The hot and cold heat reservoir temperatures, the hot and cold end gas temperatures, and the Curzon–Ahlborn optimized temperature are shown in Figure 6 as a function of the piston speed. The hot-end gas temperature optimized for maximum power is shown to be nearly the same over a large variation range of piston speeds (5 to 10 m/s), as the Curzon– Ahlborn optimized temperature. In addition, the predicted temperature difference between the high and low gas temperature is shown to increase as the piston speed decreases and to be especially great at piston speeds less than the speed for maximum efficiency.

Some results of the second model (Section 2.3) are shown in Figures 7–9.

Figure 7 illustrates the relative speed of the adiabatic processes and of each of the isothermal processes in FTT optimization compared to the average speed of the piston considered in TFS optimization. The curves show that the optimization results in lower speed than the average speed of the piston *wTFS*, for the two isothermal processes in FTT optimization. In addition, the high temperature isothermal process has the lowest speed; then, it follows the low temperature isothermal process with a higher speed, while the adiabatic processes occur at a much higher speed. However, the internal irreversibilities were not included in the original Curzon–Ahlborn analysis [3], so the high piston speed during the adiabatic process had no negative effect on the cycle efficiency and power. In fact, the resulting slower piston speed during the isothermal processes significantly enhanced the cycle efficiency and power in FTT optimization.

**Figure 7.** Piston speeds for process in TFS and FTT analyses as function of average piston speed.

**Figure 8.** Power output of the Carnot engine for processes in TFS and FTT optimizations as function of average piston.

**Figure 9.** Carnot cycle efficiency based on TFS and FTT optimizations as function of average piston speed.

The effect of the piston speed on the power output and efficiency for a Carnot engine with external irreversibilities and internal ones gradually introduced in both TFS and FTT analyses is shown in Figures 8 and 9, respectively. These results are based on the following fixed parameters: *D* = 0.015 m; *L*<sup>1</sup> = 0.5 m; *ε* = 2; *f* = 0.3; *a* = 1.1; *p1r* = 0.01 bar (pressure of the gas in state 1r); Δ*pf* = (0.97 + 0.045 w)/60 bar; *THS* = 800 K; *TLS* = 300 K; *γ* = 1.4.

The FTT optimization predicts greater power output from the Carnot engine at almost all piston speeds than the TFS optimization when only external irreversibilities (Δ*T*) are considered. It is due to the little cycle time that was allocated to the adiabatic processes in the FTT optimization. This allowed more time for the isothermal processes without any penalty associated with the more rapid adiabatic processes, since the internal irreversibilities of these processes are not considered. In the TFS optimization for example, at 9 m/s the power is 0.33 W, and the efficiency is 25%. In the FTT optimization at the same speed, by comparison, the power is 0.6 W, and the efficiency is 39%. However, when the internal irreversibilities are included in the analyses, the TFS optimization results in greater power and efficiency than FTT, even though both are less than when the internal irreversibilities were neglected.

It is also important to keep in mind that a cycle that operates with three different piston speeds for the four processes presents a huge mechanical complication in the design of the actual engine. While it may be possible to design such an engine (for example, using cams with different profiles for each process), there is no need to do so, since the TFS optimization predicts superior operating performance.

The non-dimensional power as determined from Equation (77) as a function of the cycle high temperature and the piston speed is shown in Figure 10. In addition, the power output of the reversible Carnot cycle is added for comparison purposes. The nondimensional power reveals the maximum value for any fixed piston speed or internal irreversibility consequence, and this maximum is moving toward growing temperature *Tx* as the piston speed increases.

**Figure 10.** The non-dimensional power of the Carnot cycle engine as a function of the cycle high temperature and the piston speed *w*, as determined from TFS analysis.

Figure 11 presents the second law efficiency variation versus the cycle high temperature for different values of the piston speed. The curves show that this irreversibility coefficient decreases as piston speed increases, as expected, and the decrease is more important at lower values of the cycle high temperature.

**Figure 11.** The effect of the piston speed, *w*, on the second law efficiency variation with the cycle high temperature.

Regarding the irreversible term *Φ* determined from Equation (74), its variation with the cycle high temperature and piston speed becomes important mainly at high speeds, as illustrated in Figure 12. However, there is little change of *Φ* in the region of optimal temperatures (from 800 to 1000 K).

The comparison of the results before (Figure 10) and after (Figure 13) using approximations in search of optimal temperature expression that optimizes the power output of the engine shows good agreement and lends confidence that a first iteration provides sufficiently accurate results for most purposes. However, it is possible to improve the accuracy of the results by making a new iteration.

**Figure 12.** Parameter *Φ* variation with the cycle high temperature *TX* for different piston speed values.

**Figure 13.** Graphical determination of optimal temperature.

#### **4. Conclusions**

Important performance parameters of an irreversible Carnot cycle engine based on optimization models developed in Thermodynamics with Finite Speed and by using the Direct Method have been presented. This analysis predicts lower values of Carnot cycle efficiency than is predicted by the Thermodynamics in Finite Time (FTT), as originated by Chambadal and Curzon–Ahlborn. The piston speed for maximum power and for maximum efficiency has been found for two sets of engine parameters, and it has been shown that entropy generation per time clearly differs from entropy generation per cycle. Moreover, a minimum occurs for the entropy generation per cycle at optimum piston speed corresponding to maximum power.

This study produces a more realistic model for the design of Carnot cycle engines since it includes many of the various internal and external irreversible processes that occur in the actual operation of these engines and correlates them with the finite speed of the piston.

The present analysis has shown that the first unification attempt of TFS and FTT optimization involves analytical correction of the Curzon–Ahlborn efficiency, which is well known as a nice radical, by a term accounting for internal irreversibilities of the Carnot cycle engine. They were evaluated based on the Fundamental Equation of TFS, the First Law for Processes with Finite Speed, where the main irreversibility causes are accounted for, namely, finite speed of the piston, friction, and throttling. This correction appears not only in the Carnot cycle efficiency but also in the optimum temperature of the gas at the hot end of the engine for maximum power, and in the non-dimensional power output of the engine. Thus, the engine performances were derived analytically for a Carnot engine with external and internal irreversibilities generated by finite speed w.

A step further in this first unification approach did a comparison between TFS and FTT optimization results for a Carnot cycle emphasizing that TFS analysis can account for both kind of irrevesibilities, and it can also provide improvement of FTT results.

Thermodynamic analysis based on the Direct Method and Finite Speed of the processes is shown to be especially effective for engineering optimizations since the efficiency and power can each be optimized based on gas temperatures and process speed. The fact that it is already used by other researchers [54–58,84–87] proves its capability to become a useful tool in thermal machine analysis and optimization.

We do hope that this work marks an important step toward the development of a more powerful Engineering Irreversible Thermodynamics, which could be a synthesis unifying the three important branches, namely Thermodynamics with Finite Speed, Thermodynamics with Finite Dimensions, and Thermodynamics in Finite Time.

**Author Contributions:** M.C. contributed to the development of the model, synthesis, and preparation of the manuscript; S.P. contributed substantially to the development of the model and interpretation of the results; M.F. contributed to the analysis and interpretation of the results; C.D. contributed to the development of the model and analysis of the results; B.B. contributed to the development of the model and results illustration. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Power and Thermal Efficiency Optimization of an Irreversible Steady-Flow Lenoir Cycle**

**Ruibo Wang 1,2, Yanlin Ge 1,2, Lingen Chen 1,2,\*, Huijun Feng 1,2 and Zhixiang Wu 1,2**


**\*** Correspondence: lgchenna@yahoo.com

**Abstract:** Using finite time thermodynamic theory, an irreversible steady-flow Lenoir cycle model is established, and expressions of power output and thermal efficiency for the model are derived. Through numerical calculations, with the different fixed total heat conductances (*UT*) of two heat exchangers, the maximum powers (*P*max), the maximum thermal efficiencies (*η*max), and the corresponding optimal heat conductance distribution ratios (*uLP*(*opt*)) and (*uL<sup>η</sup>* (*opt*)) are obtained. The effects of the internal irreversibility are analyzed. The results show that, when the heat conductances of the hot- and cold-side heat exchangers are constants, the corresponding power output and thermal efficiency are constant values. When the heat source temperature ratio (*τ*) and the effectivenesses of the heat exchangers increase, the corresponding power output and thermal efficiency increase. When the heat conductance distributions are the optimal values, the characteristic relationships of *P* − *uL* and *η* − *uL* are parabolic-like ones. When *UT* is given, with the increase in *τ*, the *P*max, *η*max, *uLP*(*opt*), and *uL<sup>η</sup>* (*opt*) increase. When *τ* is given, with the increase in *UT*, *P*max and *η*max increase, while *uLP*(*opt*) and *uL<sup>η</sup>* (*opt*) decrease.

**Keywords:** finite time thermodynamics; irreversible Lenoir cycle; cycle power; thermal efficiency; heat conductance distribution; performance optimization

#### **1. Introduction**

Finite time thermodynamic (FTT) theory [1–4] has been applied to the performance analysis and optimization of heat engine (HEG) cycles, and fruitful results have been achieved for both reciprocating and steady-flow cycle models. For the steady-flow models, FTT was also termed as finite physical dimensions thermodynamics by Feidt [5–10]. The famous thermal efficiency formula *<sup>η</sup>* <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>√</sup>*TL*/*TH*, where *TH* and *TL* are the temperatures of the heat source and heat sink of a HEG, was derived by Moutier [11] in 1872, Cotterill [12] in 1890, and Novikov [13] and Chambadel [14] in 1957 for steady-flow power plants, while the systematical analysis combining thermodynamics with heat transfer for Carnot cycle was performed by Curzon and Ahlborn [15] in 1975 for reciprocating model, and FTT development was promoted by Berry's group [4].

A large number of works have been performed for reciprocating (finite time) models [16–25] by applying FTT. While finite size is the major feature for steady-flow devices, such as closed gas rubine (Brayton cycle) power plants and steam (Rankine cycle) and organic Rankine cycle power plants, many scholars have performed FTT studies for various steady-flow cycles with the power output (POW), thermal efficiency (TEF), exergy efficiency, profit rate, and ecological function as the optimization goals, under the conditions of different losses and heat transfer laws [26–51].

Lenoir [52] first proposed the Lenoir cycle (LC) model in 1860. The simple LC consists of only three processes of constant-volume endothermic, adiabatic expansion, and constantpressure exothermic; the LC is also called the triangular cycle. According to the cycle

**Citation:** Wang, R.; Ge, Y.; Chen, L.; Feng, H.; Wu, Z. Power and Thermal Efficiency Optimization of an Irreversible Steady-Flow Lenoir Cycle. *Entropy* **2021**, *23*, 425. https://doi.org/10.3390/e23040425

Academic Editor: Michel Feidt

Received: 15 March 2021 Accepted: 31 March 2021 Published: 2 April 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

form, LC can be divided into steady-flow and reciprocating. Georgiou [53] first used classical thermodynamics to study the performances of simple, regenerated, and modified regenerated steady-flow Lenoir cycles (SFLCs).

Following on from [53], Shen et al. [54] applied FTT theory to optimize the POW and TEF characteristics of the endoreversible SFLC with only the loss of heat resistance, and they studied the influences of heat source temperature ratio and total heat conductance (HC) on cycle performance. Ahmadi et al. [55] used a genetic algorithm to carry out multiobjective optimization for endoreversible SFLC, and they obtained the optimal values of ecological performance coefficient and thermal economy under different temperature ratios.

In this paper, an irreversible SFLC model will be established on the basis of [54], while the cycle performance will be analyzed and optimized with the POW and TEF as objective functions, the optimal HC distributions of hot- and cold-side heat exchangers (HACHEX) of the cycle will be studied under different fixed total HCs, and the characteristic relationships between POW and TEF versus HC distribution are obtained. The effect of the internal irreversibility will be analyzed.

#### **2. Cycle Model**

Figures 1 and 2 show the *T* − *s* and *p* − *v* diagrams of the irreversible SFLC. As can be seen, 1 → 2 is the constant-volume endothermic process, 2 → 3 is the irreversible adiabatic expansion process (2 → 3*S* is the corresponding isentropic process), and 3 → 1 is the constant-pressure exothermic process. Assuming the cycle WF is an ideal gas, the entire cycle needs to be completed between the heat source (*TH*) and heat sink (*TL*).

**Figure 1.** *T* − *s* diagram for the irreversible steady-flow Lenoir cycle (SFLC).

**Figure 2.** *p* − *v* diagram for the irreversible SFLC.

In the actual work of the HEG, there are irreversible losses during compression and expansion processes; thus, the irreversible expansion efficiency *η<sup>E</sup>* is defined to describe the irreversible loss during the expansion process.

$$
\eta\_E = \frac{T\_2 - T\_3}{T\_2 - T\_{3S}} \,\mathrm{}\tag{1}
$$

where *Ti* (*i* = 2, 3, 3*S*) is the corresponding state point temperature.

Assuming that the heat transfer between the WF and heat reservoir obeys the law of Newton heat transfer, according to the theory of the heat exchanger (HEX) and the ideal gas properties, the cycle heat absorbing and heat releasing rates are, respectively,

$$
\dot{Q}\_{1 \to 2} = \dot{m} \mathbb{C}\_{\text{v}} E\_H (T\_H - T\_1) = \dot{m} \mathbb{C}\_{\text{v}} (T\_2 - T\_1) \, \tag{2}
$$

$$
\dot{Q}\_{3 \to 1} = \dot{m} \mathcal{C}\_P E\_L (T\_3 - T\_L) = \dot{m} \mathcal{C}\_P (T\_3 - T\_1) \, . \tag{3}
$$

where . *m* is the mass flow rate of the WF, *Cv*(*CP*) is the constant-volume (constant-pressure) SH (*CP* = *kCv*, *k* is the cycle SH ratio), and *EH*(*EL*) is the effectiveness of hot-side (cold-side) HEX.

The relationships among the effectivenesses with the corresponding heat transfer unit numbers (*NH*, *NL*) and HCs (*UH*, *UL*) are as follows:

$$N\_H = \mathcal{U}\_H / (\dot{m} \mathcal{C}\_v)\_\prime \tag{4}$$

$$N\_L = \mathcal{U}\_L / (\dot{m}k\mathcal{C}\_v)\_\prime \tag{5}$$

$$E\_H = 1 - \exp(-N\_H),\tag{6}$$

$$E\_L = 1 - \exp\left(-N\_L\right). \tag{7}$$

#### **3. Analysis and Discussion**

*3.1. Power and Thermal Efficiency Expressions*

According to the second law of thermodynamics, after a cycle process, the total entropy change of the WF is equal to zero; thus, one finds

$$\mathbb{C}\_{\mathcal{V}} \ln \left( T\_2 / T\_1 \right) - \mathbb{C}\_P \ln \left( T\_{3S} / T\_1 \right) = 0. \tag{8}$$

From Equation (8), one obtains

$$\frac{T\_2}{T\_1} = (\frac{T\_{3S}}{T\_1})^k. \tag{9}$$

From Equations (2) and (3), one has

$$T\_2 = E\_H(T\_H - T\_1) + T\_{1\prime} \tag{10}$$

$$T\_{\mathfrak{B}} = (E\_L T\_L - T\_1) / (E\_L - 1). \tag{11}$$

Combining Equations (1), (9), and (10) with Equation (11) yields

$$T\_1 = \frac{E\_H T\_H (\eta\_E - 1) + (T\_1 - E\_L T\_L)/(1 - E\_L)}{\left\{(1 - E\_H)(1 - \eta\_E) + \left\{[E\_H T\_H + (1 - E\_H)T\_1]/T\_1\right\}^{\frac{1}{T}} \eta\_E\right\}}. \tag{12}$$

From Equations (2), (3) and (9)–(11), the POW and TEF expressions of the irreversible SFLC can be obtained as

$$P = \dot{Q}\_{1 \to 2} - \dot{Q}\_{3 \to 1} = \dot{m} \mathbb{C}\_{\mathbb{P}}[E\_H(T\_H - T\_1) - \frac{kE\_L(T\_1 - T\_L)}{1 - E\_L}],\tag{13}$$

$$\eta = P/\dot{Q}\_{1\to 2} = 1 - \frac{kE\_L(T\_1 - T\_L)}{E\_H(1 - E\_L)(T\_H - T\_1)}.\tag{14}$$

When *η<sup>E</sup>* = 1, Equation (12) simplifies to

$$T\_1 - E\_L T\_L = (1 - E\_L) \left[ E\_H T\_H + (1 - E\_H) T\_1 \right]^{\frac{1}{k}} T\_1^{1 - \frac{1}{k}}.\tag{15}$$

Equation (15) in this paper is consistent with Equation (15) in [54], where *T*<sup>1</sup> was obtained for the endoreversible SFLC. Combining Equations (13)–(15) and using the numerical solution method, the POW and TEF characteristics of the endoreversible SFLC in [54] can be obtained.

#### *3.2. Case with Given Hot- and Cold-Side HCs*

The working cycles of common four-branch HEGs, such as Carnot, Brayton, and Otto engines, can be roughly divided into four processes: compression, endothermic, expansion, and exothermic. Compared with these common four-stroke cycles, the biggest feature of the SFLC is the lack of a gas compression process, presenting a relatively rare three-branch cycle model.

When the hot- and cold-side HCs are constant, it can be seen from Equations (4)–(7) that the effectivenesses of the HACHEX which are directly related to each cycle state point temperature will be fixed values; as a result, the POW and TEF will also be fixed values.

#### *3.3. Case with Variable Hot- and Cold-Side HCs When Total HC Is Given*

When the HC changes, the POW and TEF of the cycle will also change; therefore, the HC can be optimized and the optimal POW and TEF can be obtained. Assuming the total HC is a constant,

$$
\mathcal{U}\_L + \mathcal{U}\_H = \mathcal{U}\_T.\tag{16}
$$

Defining the HC distribution ratio as *uL* = *UL UT* (0 < *uL* < 1), from Equations (4)–(7), the effectivenesses of the HACHEX can be represented as

$$E\_H = 1 - \exp[-(1 - \mu\_L) \mathcal{U}\_T / (\dot{m} \mathcal{C}\_v)],\tag{17}$$

$$E\_L = 1 - \exp[-\mu\_L \mathcal{U}\_T / (\dot{m} k \mathcal{C}\_{\upsilon})].\tag{18}$$

Combining Equations (12)–(14) and (17) with Equation (18) and using a numerical solution method, the characteristic relationships between POW and the hot- and cold-side HC distribution ratio, as well as between TEF and the hot- and cold-side HC distribution ratio, can be obtained.

#### **4. Numerical Examples**

It is assumed that the working fluid is air. Therefore, its constant-volume specific heat and specific heat ratio are *Cv* = 0.7165 kJ/(kg·K) and *k* = 1.4. The turbine efficiency of the gas turbine is about *<sup>η</sup><sup>E</sup>* <sup>=</sup> 0.92 in general. According to the [51–55], . *m* = 1.1165 kg/s and *TL* = 320 K were set.

Figure 3 shows the POW and TEF characteristics when the HCs of the HACHEX and temperature ratio are different values. When the HCs and temperature ratio are fixed values, the effectivenesses of the HEX are fixed values, and the corresponding POW and TEF are also fixed values. The POW and TEF characteristics are reflected in the graph as a point. As can be seen, when *τ*(*τ* = *TH*/*TL*) and the HCs of the HEXs increase, the corresponding POW and TEF increase. Figure 4 shows the influence of *η<sup>E</sup>* on *P* − *η* characteristics when the HCs of HACHEX and temperature ratio are given. As can be seen, with the increase in *η<sup>E</sup>* (the decrease of irreversible loss), the corresponding *P* and *η* increase.

**Figure 3.** The power output (POW) and thermal efficiency (TEF) characteristics when the HCs of HACHEX are given.

**Figure 4.** Effect of *η<sup>E</sup>* on *P* − *η* characteristics when the HCs of HACHEX are given.

Figures 5–8 show the influences of *UT* on the *P* − *uL* and *η* − *uL* characteristics when *τ* = 3.25 and *τ* = 3.75. The relationship curves of *P* − *uL* and *η* − *uL* are parabolic-like changes. With the increase in *uL*, the corresponding POW and TEF first increase and then decrease, and there are optimal HC distribution values *uLP*(*opt*) and *uL<sup>η</sup>* (*opt*), which lead to POW and TEF reaching their maximum values *P*max and *η*max.

Figures 5 and 6 show the influence of *UT* on *P* − *uL* characteristics when *τ* = 3.25 and *τ* = 3.75. As can be seen, with the increase in *UT*, *P*max increases and *uLP*(*opt*) decreases. When *UT* is 2.5, 5, 7.5, and 10 kW/K and *τ* = 3.25, the corresponding *P*max is 23.04, 56.58, 70.25, and 74.39 W, while *uLP*(*opt*) is 0.58, 0.575, 0.574, and 0.573, respectively. When *UT* changes from 2.5 to 10 kW/K, the corresponding *P*max increases by about 222.9%, while the *uLP*(*opt*) decreases by about 1.21%. When *UT* is 2.5, 5, 7.5, and 10 kW/K and *τ* = 3.75, the corresponding *P*max is 33.06, 80.06, 90.24, and 105.06 W, while *uLP*(*opt*) is 0.586, 0.579, 0.5785, and 0.5782, respectively. When *UT* changes from 2.5 to 10 kW/K, the corresponding *P*max increases by about 217.8%, while the *uLP*(*opt*) decreases by about 1.33%.

Figures 7 and 8 show the influence of *UT* on *η* − *uL* characteristics when *τ* = 3.25 and *τ* = 3.75. As can be seen, with the increase in *UT*, *η*max increases and *uL<sup>η</sup>* (*opt*) decreases. When *UT* is 2.5, 5, 7.5, and 10 kW/K and *τ* = 3.25, the corresponding *η*max is 0.066, 0.111, 0.126, and 0.1303, while *uL<sup>η</sup>* (*opt*) is 0.629, 0.614, 0.605, and 0.6, respectively. When *UT* changes from 2.5 to 10 kW/K, the corresponding *η*max increases by about 97.4%, while *uLP*(*opt*) decreases by about 4.61%. When *UT* is 2.5, 5, 7.5, and 10 kW/K and *τ* = 3.75, the corresponding *η*max is 0.0774, 0.129, 0.1458, and 0.1506, while *uL<sup>η</sup>* (*opt*) is 0.644, 0.624, 0.608, and 0.606, respectively. When *UT* changes from 2.5 to 10 kW/K, the corresponding *η*max increases by about 94.6%, while *uLP*(*opt*) decreases by about 5.9%.

From Figures 5–8 and Equations (12)–(14), (17), and (18), one can see that, when *τ* is given, the POW and TEF are mainly affected by the total HC; with the increase in *UT*, the *P*max and *η*max increase. When the total HC is small, the corresponding *P*max and *η*max change more significantly. When the total HC is large, the corresponding *P*max and *η*max change little. When *UT* is given, with the increase in *τ*, the *uLP*(*opt*) and *uL<sup>η</sup>* (*opt*) increase. When *τ* and *UT* are given, the corresponding *uL<sup>η</sup>* (*opt*) > *uLP*(*opt*).

**Figure 5.** Effect of *UT* on *P* − *uL* characteristics when *τ* = 3.25.

**Figure 6.** Effect of *UT* on *P* − *uL* characteristics when *τ* = 3.75.

**Figure 7.** Effect of *UT* on *η* − *uL* characteristics when *τ* = 3.25.

**Figure 8.** Effect of *UT* on *η* − *uL* characteristics when *τ* = 3.75.

Figures 9 and 10 show the influences of *η<sup>E</sup>* on *P* − *uL* and *η* − *uL* characteristics when *τ* = 3.75 and *UT* = 7.5 kW/K. As can be seen, when *τ* = 3.75 and *UT* = 7.5 kW/K, with the increase in *η<sup>E</sup>* (the decrease in irreversible loss), the *P*max and *η*max increase, while the corresponding *uLP*(*opt*) and *uL<sup>η</sup>* (*opt*) decrease. When *η<sup>E</sup>* is 0.75, 0.8, 0.85, 0.9, 0.95, and 1.0, the corresponding *P*max is 30.2431, 50.4808, 70.7674, 91.0982, 111.4719, and 131.8876, *η*max is 0.0445, 0.0743, 0.1041, 0.1339, 0.1637, and 0.1935, *uLP*(*opt*) is 0.601, 0. 593, 0.586, 0.581, 0.576, and 0.572, and *uL<sup>η</sup>* (*opt*) is 0.619, 0.617, 0.615, 0.613, 0.611, and 0.609, respectively. When *η<sup>E</sup>* changes from 0.75 to 1.0, the corresponding *P*max increases by about 336.1%, *η*max increases by about 334.8%, *uLP*(*opt*), and *uL<sup>η</sup>* (*opt*) decreases by about 4.83% and 1.62%, respectively.

**Figure 9.** Effect of *η<sup>E</sup>* on *P* − *uL* characteristics.

**Figure 10.** Effect of *η<sup>E</sup>* on *η* − *uL* characteristics.

#### **5. Conclusions**

In this paper, an irreversible SFLC model is established on the basis of [54], while the POW and TEF characteristics of the irreversible SFLC were studied using FTT theory, and the influences of *τ*, *UT* and *η<sup>E</sup>* on *P*max, *η*max, *uLP*(*opt*), and *uL<sup>η</sup>* (*opt*) were analyzed. The main conclusions are as follows:


(5) When *τ* = 3.75 and *UT* = 7.5kW/K, with the increase in *ηE*, *P*max and *η*max increase, while the corresponding *uLP*(*opt*) and *uL<sup>η</sup>* (*opt*) decrease.

**Author Contributions:** Conceptualization, R.W. and L.C.; data curation, Y.G.; funding acquisition, L.C.; methodology, R.W., Y.G., L.C. and H.F.; software, R.W., Y.G., H.F. and Z.W.; supervision, L.C.; validation, R.W., H.F. and Z.W.; writing—original draft preparation, R.W. and Y.G.; writing—reviewing and editing, L.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This paper is supported by the National Natural Science Foundation of China (Project No. 51779262).

**Acknowledgments:** The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**


#### **Abbreviations**


#### **References**

1. Andresen, B.; Berry, R.S.; Ondrechen, M.J. Thermodynamics for processes in finite time. *Acc. Chem. Res.* **1984**, *17*, 266–271. [CrossRef]


## *Article* **Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle**

**Chenqi Tang 1,2,3, Lingen Chen 1,2,\*, Huijun Feng 1,2,\* and Yanlin Ge 1,2**


**Abstract:** An improved irreversible closed modified simple Brayton cycle model with one isothermal heating process is established in this paper by using finite time thermodynamics. The heat reservoirs are variable-temperature ones. The irreversible losses in the compressor, turbine, and heat exchangers are considered. Firstly, the cycle performance is optimized by taking four performance indicators, including the dimensionless power output, thermal efficiency, dimensionless power density, and dimensionless ecological function, as the optimization objectives. The impacts of the irreversible losses on the optimization results are analyzed. The results indicate that four objective functions increase as the compressor and turbine efficiencies increase. The influences of the latter efficiency on the cycle performances are more significant than those of the former efficiency. Then, the NSGA-II algorithm is applied for multi-objective optimization, and three different decision methods are used to select the optimal solution from the Pareto frontier. The results show that the dimensionless power density and dimensionless ecological function compromise dimensionless power output and thermal efficiency. The corresponding deviation index of the Shannon Entropy method is equal to the corresponding deviation index of the maximum ecological function.

**Keywords:** closed simple Brayton cycle; power output; thermal efficiency; power density; ecological function; multi-objective optimization

#### **1. Introduction**

Some scholars have studied performances of gas turbine plants (Brayton cycle (BCY)) [1–4] all over the world for their small size and comprehensive energy sources. The gas-steam combined, cogeneration, and other complex cycles have appeared for the requirements of energy conservation and environmental protection. The thermal efficiency (*η*) of a simple BCY is low, and the NOx content in combustion product is high. To further improve the cycle performance, it has become a key research direction to improve the initial temperature of the gas or to adopt the advanced cycles (such as regenerative, intercooled, intercooled and regenerative, isothermal heating, and other complex combined cycles).

In the case of simple heating, when the compressible subsonic gas flows through the smooth heating pipe with the fixed cross-sectional area, the gas temperature increases along the pipe direction; in the case of simple region change, when the compressible subsonic gas flows through the smooth adiabatic reductive pipe, the gas temperature decreases along the pipe direction. Based on these two gas properties, the isothermal heating process (IHP) can be realized when the compressible subsonic gas flows through the smooth heating reductive pipe. The combustion chamber, which can recognize the IHP, is called the convergent combustion chamber (CCC). The pipe of the CCC is assumed to be smooth. During the heating process, the temperature of the gas is always constant. According to the energy conservation law, the kinetic energy of the gas increases, that is, the pushing work of

**Citation:** Tang, C.; Chen, L.; Feng, H.; Ge, Y. Four-Objective Optimizations for an Improved Irreversible Closed Modified Simple Brayton Cycle. *Entropy* **2021**, *23*, 282. https:// doi.org/10.3390/e23030282

Academic Editor: Michel Feidt

Received: 17 January 2021 Accepted: 22 February 2021 Published: 26 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the gas increases. From the definition of enthalpy, it can be seen that enthalpy includes two parts: the thermodynamic energy and the pushing work. Therefore, the enthalpy increases. Based on this, Vecchiarelli et al. [5] proposed the CCC to perform the IHP of the working fluid. The power output (*W*) and *η* of the BCY could be improved, and the emission of harmful gases such as NOx could be reduced by adding this combustion chamber model. The regenerative BCYs [6–8] and binary BCY [9] with IHPs were also studied by applying the classical thermodynamics.

Finite time thermodynamics (FTT) is a useful thermodynamic analysis theory and method [10–19]. In general, it is known that Curzon and Ahlborn [12] initialized FTT in 1975. In fact, the classical efficiency bound at the maximum power was also derived by Moutier [10] in 1872 and Novikov [11] in 1957. The applications of FTT include majorly two fields: optimal configurations [20–36] and optimal performances [37–61] studies for thermodynamic cycles and processes. The *W* and *η* have been often considered as the optimization objectives (OPOs) of the heat engines [62–72]. When the power density (*P*) [73–81] was taken as the OPO, the operating unit had a smaller size and higher *η*. Aditionally, the ecological function (*E*) [82–88] is also an OPO that balances the conflict between *W* and *η*.

Kaushik et al. [89] first applied the FTT to studying the regenerative BCY with an IHP. The regenerative, intercooled and regenerative complex BCYs with isothermal heating combustor were further investigated [90–96]. Based on this, Chen et al. [97–99] studied the endoreversible simple isothermal heating BCY with the *W*, *η* and *E* as OPOs. Arora et al. [100,101] adopted NSGA-II and evolutionary algorithms to optimize the irreversible isothermal heating regenerative BCY with the *W* and *η* as the OPOs. Chen et al. [102] considered the variable isothermal pressure drop ratio (*πt*), established an improved isothermal heating regenerative BCY model, and studied the regenerator's role on cycle performance. Qi et al. [103] demonstrated a closed endoreversible modified binary BCY with IHPs and found the *W* and *η* raised as the heat reservoirs' temperature ratios. Tang et al. [104] considered the variable *π<sup>t</sup>* and established an improved irreversible binary BCY model modified by isothermal heating. The heat exchanger's heat conductance distributions (HCDs) and the top and bottom cycles' pressure ratios were taken as optimization variables to optimize the cycle performance.

In the process of the thermodynamic system optimization, single-objective optimization often led to unacceptable objectives for other objectives when there were conflicts among the considered goals. Multi-objective optimization would consider the trade-offs among the goals, and the optimized results were more reasonable [99,100,102,105–125].

In applying the FTT, the heat transfer was introduced into the thermodynamic analysis of the thermodynamic process, and finite temperature difference was considered in Refs. [11,12]. In this paper, the same method in Refs. [11,12] will be used, and the finite temperature difference will be considered when establishing the model, which is the key relation among this paper and the Refs. [11,12]. On this basis, the cycle's irreversibility will be further considered, and the corresponding conclusion will be more in line with the actual situation. The compression and expansion losses in the model in Refs. [97–99] were not considered, and they will be further considered in this paper alongside the losses in the heat exchangers. Meanwhile, the thermal resistance loss and the optimal HCD will be considered. With the *W*, *η*, *P* and *E*, respectively, as the OPOs, an improved irreversible closed modified simple BCY with one IHP and coupled to variable-temperature heat reservoirs (VTHRs) will be optimized, and the optimization results will be compared. The effects of the compressor and turbine efficiencies on optimization results will be analyzed. The NSGA-II algorithm will be applied for multi-objective optimization to obtain the Pareto frontier further. The results obtained in this paper will reveal the original results in Refs. [10–12], which were the initial work of the FTT theory.

#### **2. Cycle Model and Performance Analytical Indicators**

The schematic diagram of an improved irreversible closed modified simple BCY with one IHP and coupled to VTHRs is shown in Figure 1. A compressor (C), a regular combustion chamber (RCC), a CCC, a turbine (T), and a precooler are the main parts of the cycle. The corresponding *T* − *s* diagram of the cycle is shown in Figure 2. The cycle consists of five processes in total:


**Figure 1.** Schematic diagram of the cycle.

**Figure 2.** Diagram of the cycle.

The working fluid is the ideal gas. The pressures and temperatures of the working fluid are *pi*(*i* = 1, 2, 3, 4, 5, 2*s*, 5*s*) and *Ti*, and the ratio of specific heat is *k*. The outside fluids' temperatures are *Tj*(*j* = *H*1, *H*2, *H*3, *H*4, *L*1, *L*2). The specific heat at constant pressure and the working fluid's mass flow rate are *Cp* and . *m*. The working fluid's thermal capacity rate is *Cw f* where *Cw f* = *Cp* . *m*. The outer fluids' thermal capacity rates at the RCC, CCC, and precooler are *CH*, *CH*<sup>1</sup> and *CL*, respectively; then, one has:

$$\mathbb{C}\_{\text{Hmax}} = \max \left\{ \mathbb{C}\_{H}, \mathbb{C}\_{wf} \right\}, \\ \mathbb{C}\_{\text{Lmax}} = \max \left\{ \mathbb{C}\_{L}, \mathbb{C}\_{wf} \right\}, \\ \mathbb{C}\_{\text{Hmin}} = \min \left\{ \mathbb{C}\_{H}, \mathbb{C}\_{wf} \right\}, \\ \mathbb{C}\_{\text{Lmin}} = \min \left\{ \mathbb{C}\_{L}, \mathbb{C}\_{wf} \right\} \tag{1}$$

The heat exchangers' heat conductance is the product of the heat transfer coefficient and the heat transfer area. The heat exchangers' heat conductance values in the RCC, CCC, and precooler are *UH*, *UH*<sup>1</sup> and *UL*, the heat transfer units' numbers are *NH*, *NH*<sup>1</sup> and *NL*, and the effectiveness values are *EH*, *EH*<sup>1</sup> and *EL*, respectively:

$$N\_H = \mathcal{U}\_H / \mathcal{C}\_{H\text{min}} \; N\_{H1} = \mathcal{U}\_{H1} / \mathcal{C}\_{H1'} \; N\_L = \mathcal{U}\_L / \mathcal{C}\_{L\text{min}} \tag{2}$$

$$E\_H = \frac{1 - e^{-N\_H(1 - \mathbb{C}\_{H \text{min}}/\mathbb{C}\_{H \text{max}})}}{1 - (\mathbb{C}\_{H \text{min}}/\mathbb{C}\_{H \text{max}})e^{-N\_H(1 - \mathbb{C}\_{H \text{min}}/\mathbb{C}\_{H \text{max}})}} \tag{3}$$

$$E\_{H1} = 1 - \varepsilon^{-N\_{H1}} \tag{4}$$

$$E\_L = \frac{1 - e^{-N\_L(1 - \mathbb{C}\_{L\text{min}}/\mathbb{C}\_{L\text{max}})}}{1 - (\mathbb{C}\_{L\text{min}}/\mathbb{C}\_{L\text{max}})e^{-N\_L(1 - \mathbb{C}\_{L\text{min}}/\mathbb{C}\_{L\text{max}})}} \tag{5}$$

When *CH*max = *CH*min and *CL*max = *CL*min, Equations (3) and (5) are, respectively, simplified as:

$$E\_H = N\_H / \left(N\_H + 1\right) \tag{6}$$

$$E\_L = N\_L / (N\_L + 1) \tag{7}$$

The outside fluids' temperature ratios at the RCC and CCC are:

$$
\pi\_{H1} = T\_{H1}/T\_0 \tag{8}
$$

$$
\pi\_{H3} = T\_{H3}/T\_0 \tag{9}
$$

where *T*<sup>0</sup> is the ambient temperature.

The process 1 → 2*s* is the isentropic one, namely:

$$T\_{2s}/T\_1 = \pi^{\text{""}} = \text{x} \tag{10}$$

where *m* = (*k* − 1)/*k* and *π* is the pressure ratio of the compressor.

The process 4 → 5*s* is the isentropic one, namely:

$$T\_4/T\_{5s} = \pi^m \pi\_t^m = xy \tag{11}$$

The process 3 → 4 is the isothermal one, namely:

$$T\_3 = T\_4 \tag{12}$$

$$\dot{Q}\_{3-4} = \dot{m}(h\_4 - h\_3) - \dot{m} \int\_3^4 vdp = -\dot{m}R\_\circ T\_3 \ln \pi\_\text{f} \tag{13}$$

where *πt*, *M*<sup>3</sup> and *M*<sup>4</sup> must satisfy the following relation:

.

$$\ln \pi\_t = -c\_p (k-1) (M\_4^2 - M\_3^2) / (2R\_\chi) \tag{14}$$

where the working fluid's flow velocity must be subsonic, namely, *M*3, *M*<sup>4</sup> < 1 Because the working fluid has an initial speed, (*M*<sup>2</sup> <sup>4</sup> − *<sup>M</sup>*<sup>2</sup> <sup>3</sup>) < 0.96 and *π<sup>t</sup>* > 0.5107 when *M*<sup>3</sup> = 0.2. Because of *M*<sup>4</sup> > *M*3, *π<sup>t</sup>* < 1. When *π<sup>t</sup>* = 1, the cycle model in this paper can be simplified to a simple Brayton cycle.

According to the definition of *πt*, it can be obtained that:

$$
\pi\_1 = \frac{p\_4}{p\_3} = \frac{p\_4}{p\_3} \cdot \frac{p\_1}{p\_1} = \frac{p\_4}{p\_1} \cdot \pi^{-1} \ge \pi^{-1} \tag{15}
$$

Considering the irreversibilities in the compressor and the turbine, the efficiencies of them are:

$$
\eta\_{\mathbb{C}} = (T\_1 - T\_{2\mathbb{S}}) / (T\_1 - T\_{2}) \tag{16}
$$

$$
\eta\_t = (T\_5 - T\_4) / (T\_{5\mathfrak{s}} - T\_4) \tag{17}
$$

The pressure drop is not considered in this paper. It will be considered in future, as it was by Ref. [126]. The study in Ref. [126] showed that the pressure drop loss has a little influence on the cycle performance quantitatively, and has no influence qualitatively.

The working fluid's heat absorption rates at RCC and CCC are . *<sup>Q</sup>*2−<sup>3</sup> and . *Q*3−4, respectively: .

$$Q\_{2-3} = \mathbb{C}\_{H}(T\_{H1} - T\_{H2}) = \mathbb{C}\_{wf}(T\_3 - T\_2) = \mathbb{C}\_{H \text{min}} E\_H (T\_{H1} - T\_2) \tag{18}$$

$$\dot{Q}\_{3-4} = \mathbb{C}\_{H1}(T\_{H3} - T\_{H4}) = \mathbb{C}\_{H1}E\_{H1}(T\_{H3} - T\_3) = \dot{m}(V\_4^2 - V\_3^2)/2\tag{19}$$

The heat releasing rate at the precooler is . *Q*5−1, namely:

$$\dot{Q}\_{5-1} = \mathcal{C}\_{L}(T\_{L2} - T\_{L1}) = \mathcal{C}\_{wf}(T\_5 - T\_1) = \mathcal{C}\_{L \text{min}} E\_L(T\_5 - T\_{L1}) \tag{20}$$

The heat leakages between the heat source and the environment [127,128] are neglected. Therefore, the *W* and *η* are:

$$\mathcal{W} = \dot{Q}\_{2-3} + \dot{Q}\_{3-4} - \dot{Q}\_{5-1} \tag{21}$$

$$
\eta = W / (\dot{Q}\_{2-3} + \dot{Q}\_{3-4}) \tag{22}
$$

The dimensionless power output (*W*) is:

as:

$$\mathcal{W} = \mathcal{W} / (\mathbb{C}\_{wf} T\_0) \tag{23}$$

The maximum specific volume corresponding to state point 5 is *v*5. The *P* is calculated

$$P = \mathcal{W}/\upsilon\_5 \tag{24}$$

The specific volume corresponding to state point 1 is *v*1. The dimensionless power density (*P*) and dimensionless maximum specific volume (*v*5/*v*1) are obtained as:

$$\overline{P} = \frac{P}{\mathbb{C}\_{wf} T\_0 / v\_1} = \frac{W / v\_5}{\mathbb{C}\_{wf} T\_0 / v\_1} = \frac{W}{\mathbb{C}\_{wf} T\_0} \times \frac{T\_1}{T\_5} = \overline{W} \times \frac{T\_1}{T\_5} \tag{25}$$

$$
v\_5/v\_1 = T\_5/T\_1\tag{26}$$

There are two different methods for calculating the entropy production rate. One was suggested by Bejan [129,130], and the another was suggested by Salamon et al. [131]. In this article, the method used is the one suggested by the latter.

The entropy production rate (*sg*) and *E* are, respectively, calculated as:

$$s\_{\mathcal{g}} = \mathbb{C}\_{H} \ln(T\_{H2}/T\_{H1}) + \mathbb{C}\_{H1} \ln(T\_{H4}/T\_{H3}) + \mathbb{C}\_{L} \ln(T\_{L2}/T\_{L1})\tag{27}$$

$$E = \mathcal{W} - T\_0 \mathbf{s}\_{\mathcal{S}} \tag{28}$$

The dimensionless ecological function (*E*) is obtained as:

$$E = E / (\mathbb{C}\_{wf} T\_0) \tag{29}$$

Equations (10)–(12) and (16)–(29) are combined, and the four dimensionless performance indicators of the cycle are obtained as follows:

$$\begin{array}{c} \mathbf{C}\_{wf} \mathbf{x} \mathbf{y} (\mathbf{C}\_{H1} \mathbf{E}\_{H1} T\_{H3} + \mathbf{C}\_{L\text{min}} \mathbf{E}\_{L} T\_{L1}) + \mathbf{C}\_{H\text{min}} \mathbf{E}\_{H} T\_{H1} \begin{cases} \mathbf{x} \mathbf{y} [\mathbf{C}\_{wf} \\ -\mathbf{C}\_{H1} \mathbf{E}\_{H1} + \mathbf{C}\_{L\text{min}} \mathbf{E}\_{L} (\eta\_{t} - 1)] - \mathbf{C}\_{L\text{min}} \mathbf{E}\_{L} \eta\_{t} \end{cases} + a\_{1} \{\mathbf{C}\_{L\text{min}} \mathbf{E}\_{L} \\\ \times [(\eta\_{t} - 1) \mathbf{x} \mathbf{y} - \eta\_{t}] (\mathbf{C}\_{wf} - \mathbf{E}\_{H} \mathbf{C}\_{H\text{min}}) - \mathbf{x} \mathbf{y} [\mathbf{C}\_{wf} \mathbf{C}\_{H\text{min}} \mathbf{E}\_{H} \\\ \overline{\mathbf{W}} = \frac{+ \mathbf{C}\_{H1} \mathbf{E}\_{H1} (\mathbf{C}\_{wf} - \mathbf{C}\_{H\text{min}} \mathbf{E}\_{H}) \big| \biglyeq \end{array} \tag{30}$$

$$\begin{aligned} &\mathbf{C\_{H\min}}\mathbf{C\_{L\min}}\mathbf{E\_{H}}\mathbf{E\_{H}}\boldsymbol{\eta}\_{1}\mathbf{T\_{H}} - \left\{\mathbf{C\_{H\min}}\mathbf{E\_{H}}\mathbf{T\_{H1}}\mathbf{T\_{H1}}\right\}\mathbf{C\_{wf}} - \mathbf{C\_{H1}}\mathbf{E\_{H1}} + \mathbf{C\_{L\min}}\mathbf{E\_{L}}(\eta\_{t} - 1)\right\} \\ &+ \mathbf{C\_{wf}}\mathbf{x}\mathbf{y}\left(\mathbf{C\_{H1}}\mathbf{E\_{H1}}\mathbf{T\_{H3}} + \mathbf{C\_{L\min}}\mathbf{E\_{L}}\mathbf{T\_{L1}}\right)\right) + a\_{1}\left\{\left\{\mathbf{C\_{H\min}}\mathbf{C\_{wf}}\mathbf{E\_{H}} + \mathbf{C\_{H1}}\mathbf{E\_{H1}}\right\}\mathbf{C\_{wf}} \\ &\eta = \frac{-\mathbf{E\_{H}}\mathbf{C\_{H\min}}\left[\left(\mathbf{x}\mathbf{y} - \mathbf{C\_{L\min}}\mathbf{E\_{L}}\left(\mathbf{C\_{wf}} - \mathbf{C\_{H\min}}\mathbf{E\_{H}}\right)\right)\left[\left(\eta\_{t} - 1\right)\mathbf{x}\mathbf{y} - \eta\_{t}\right]\right]}{\mathbf{x}\eta\left\{a\_{1}\left[\mathbf{C\_{H1}}\mathbf{C\_{wf}}\mathbf{E\_{H1}} + \mathbf{C\_{H1}}\mathbf{m}\mathbf{E\_{H}}\right]\mathbf{C\_{wf}} - \mathbf{C\_{H1}}\mathbf{E\_{H1}}\right\} + \mathbf{C\_{H\min}}\mathbf{E\_{H1}}\mathbf{E\_{H1}}} \\ &- \mathbf{C\_{wf}}\right)T\_{H1} - \mathbf{C\_{H1}}\mathbf{C\_{wf}}\mathbf{E\_{H1}}\mathbf{T\_{H3}}\end{aligned} (31)$$

*P* = ( *a*1(*Cw f* − *CH*min*EH*)(*Cw f* − *CL*min*EL*)[*xy*(*η<sup>t</sup>* − 1) − *ηt*] − *CL*min*Cw f ELTL*1*x* ×*y* + *EHCH*min*TH*1(*Cw f* − *CL*min*EL*)[(*η<sup>t</sup>* − 1)*xy* − *ηt*] )(*Cw f xy*(*CH*1*EH*1*TH*<sup>3</sup> <sup>+</sup>*CL*min*ELTL*1) + ( *xy*[*Cw f* − *CH*1*EH*<sup>1</sup> + *CL*min*EL*(*η<sup>t</sup>* − 1)] − *CL*min*ELη<sup>t</sup>* ) *CH*min ×*EHTH*<sup>1</sup> + *a*<sup>1</sup> ( *CL*min(*Cw f* − *EHCH*min)*EL* ( (*η<sup>t</sup>* − 1)*xy* − *η<sup>t</sup>* − *xy*[*CH*min*Cw f EH* <sup>+</sup>*CH*1*EH*1(*Cw f* <sup>−</sup> *CH*min*EH*)]) }} *C*3 *w f <sup>T</sup>*0*xy*[*a*1(*Cw f* <sup>−</sup> *CH*min*EH*) + *CH*min*EHTH*1][(*η<sup>t</sup>* <sup>−</sup> <sup>1</sup>)*xy* <sup>−</sup> *<sup>η</sup>t*] (32) *E* = ( *Cw f xy*(*CH*1*EH*1*TH*<sup>3</sup> + *CL*min*ELTL*1) + *CH*min*EHTH*<sup>1</sup> ( *xy*[*Cw f* − *CH*1*EH*<sup>1</sup> +*CL*min*EL*(*η<sup>t</sup>* − 1)] − *CL*min*ELηt*} + *a*<sup>1</sup> ( *CL*min*EL*(*Cw f* − *CH*min*EH*)[(*η<sup>t</sup>* <sup>−</sup>1)*xy* <sup>−</sup> *<sup>η</sup>t*] <sup>−</sup> *xy*[*Cw f CH*min*EH* <sup>+</sup> *CH*1*EH*1(*Cw f* <sup>−</sup> *CH*min*EH*)]) }/(*T*<sup>0</sup> <sup>×</sup>*xy*) <sup>−</sup> *Cw f* ( *CL* ln( 1 + ( *CL*min*EL* ( *a*1*Cw f η<sup>t</sup>* − *Cw f xy*[*a*1(*η<sup>t</sup>* − 1) + *TL*1] +*CH*min*EH*(*a*<sup>1</sup> − *TH*1)[(*η<sup>t</sup>* − 1)*xy* − *ηt*]}}/(*CLCw f TL*1*xy*) ) + *CH* ln{[*a*<sup>1</sup> <sup>×</sup>*CH*min*EH* + (*CH* <sup>−</sup> *CH*min*EH*)*TH*1]/(*CHTH*1)} <sup>+</sup> *CH*<sup>1</sup> ln( 1 + ( *EH*1[*Cw f* ×(*a*<sup>1</sup> − *TH*3) + *EHCH*min(*TH*<sup>1</sup> − *a*1)]}/(*Cw f TH*3) ) } *C*2 *w f* (33)

$$\begin{aligned} \text{where} \\ a\_1 &= \frac{(\eta\_c + \mathbf{x} - 1) \left\{ \mathbf{C}\_{L \text{min}} \mathbf{C}\_{wf} E\_L T\_{L1} \mathbf{x} \mathbf{y} - \mathbf{C}\_{H \text{min}} \mathbf{E}\_H T\_{H1} (\mathbf{C}\_{wf} - \mathbf{C}\_{L \text{min}} \mathbf{E}\_L) [(\eta\_t - 1) \mathbf{x} \mathbf{y} - \eta\_t] \right\}}{\mathbf{C}\_{H \text{min}} \mathbf{C}\_{L \text{min}} E\_H \mathbf{E}\_L (\eta\_c + \mathbf{x} - 1) (\eta\_t \mathbf{x} \mathbf{y} - \mathbf{x} \mathbf{y} - \eta\_t) + \mathbf{C}\_{wf}^2 [\mathbf{x} \mathbf{y} - \mathbf{x}^2 \mathbf{y} + \eta\_t (\eta\_c + \mathbf{x} \mathbf{y} - \mathbf{x} \mathbf{y})]} \\ &- 1) (\mathbf{x} \mathbf{y} - 1) \left[ -\mathbf{C}\_{wf} (\eta\_c + \mathbf{x} - 1) (\mathbf{E}\_H \mathbf{C}\_{H \text{min}} + \mathbf{E}\_L \mathbf{C}\_{L \text{min}}) [(\eta\_t - 1) \mathbf{x} \mathbf{y} - \eta\_t] \right] \end{aligned} \tag{34}$$

Parameters *x* and *y* in Equations (30)–(34) can be obtained by Equations (13) and (19), and then the arithmetic solution of *W*, *η*, *P* and *E* can be gained. When *CH*, *CH*1, *CL*, *EH*, *EH*1, *EL*, *η<sup>c</sup>* and *η<sup>t</sup>* are specific values, the cycle could be transformed into different cycle models. Equations (30)–(34) could be simplified into the performance indicators of the various cycle models, which have certain universality.

*E* =

1. When *CH*<sup>1</sup> = *CL* → ∞, Equations (30)–(34) can be simplified into the performance indicators of the irreversible simple BCY with an IHP and coupled to constant-temperature heat reservoirs (CTHRs) whose *T* − *s* diagram is shown in Figure 3a:

$$\begin{array}{c} \mathbf{C}\_{wf} \mathbf{x} \mathbf{y} (\mathbf{C}\_{H1} \mathbf{E}\_{H1} T\_{H3} + \mathbf{C}\_{L \text{min}} \mathbf{E}\_{L} T\_{L1}) + \mathbf{C}\_{H \text{min}} \mathbf{E}\_{H} T\_{H1} \mathbf{I} \left\{ \mathbf{x} \mathbf{y} [\mathbf{C}\_{wf} \\\quad - \mathbf{C}\_{H1} \mathbf{E}\_{H1} + \mathbf{C}\_{L \text{min}} \mathbf{E}\_{L} (\eta\_{t} - 1)] - \mathbf{C}\_{L \text{min}} \mathbf{E}\_{L} \eta\_{t} \right\} + a\_{1} \{\mathbf{C}\_{L \text{min}} \mathbf{E}\_{L} \\\quad \times [(\eta\_{t} - 1) \mathbf{x} \mathbf{y} - \eta\_{t}] (\mathbf{C}\_{wf} - \mathbf{E}\_{H} \mathbf{C}\_{H \text{min}}) - \mathbf{x} \mathbf{y} [\mathbf{C}\_{wf} \mathbf{C}\_{H \text{min}} \mathbf{E}\_{H} \\\hline \overline{W} = \frac{+ \mathbf{C}\_{H1} \mathbf{E}\_{H1} (\mathbf{C}\_{wf} - \mathbf{C}\_{H \text{min}} \mathbf{E}\_{H}) \big[ \\\end{array} \tag{35}$$

$$\begin{aligned} &C\_{\text{wf}}E\_{\text{f}}E\_{\text{f}}v\_{\text{f}}H\_{\text{f}} - \left\{ \begin{aligned} &C\_{\text{f}}E\_{\text{f}}E\_{\text{f}}v\_{\text{f}}H\_{\text{f}} - \left\{ \text{E}\_{\text{f}}E\_{\text{f}}E\_{\text{f}} + \text{C}\_{\text{f}}E\_{\text{f}}(\eta\_{\text{f}}-1) \right\} + \left\{ \text{C}\_{\text{f}}E\_{\text{f}}E\_{\text{f}}E\_{\text{f}} \right\} \\ &+ \text{C}\_{\text{wf}}E\_{\text{f}}E\_{\text{f}} \right\} \left\{ \text{x}\eta\_{\text{f}} + 2\left\{ \text{C}\_{\text{wf}}E\_{\text{f}} + \text{C}\_{\text{f}}E\_{\text{f}}E\_{\text{f}}(1-E\_{\text{f}}) \right\} \mathbf{x} - \text{C}\_{\text{wf}}E\_{\text{f}}(1-E\_{\text{f}}) \right\} \\ &\quad \times \left\{ \text{x}\eta\_{\text{f}} + (1-\eta\_{\text{f}})\{\text{x}\} \right\} \\ &\quad \times \left\{ \text{x}\eta\_{\text{f}} \left[ 2\left(\text{C}\_{\text{f}}E\_{\text{f}} + \text{E}\_{\text{f}}\right) \left(\text{x}\eta\_{\text{f}} - \text{C}\_{\text{f}}E\_{\text{f}}\right) \right] + \text{E}\_{\text{f}}\Pi\_{\text{f}} \left(\text{C}\_{\text{f}}E\_{\text{f}} - \text{C}\_{\text{f}}\right) \right] \eta\_{\text{f}} \\ &\quad \times \left\{ \text{x}\} \left\{ \text{x}\} \left$$

*Cw f*

where

$$a\_{2} = \frac{(\eta\_{\text{c}} + \mathbf{x} - 1)\{-E\_{L}T\_{L1}\mathbf{x}\mathbf{y} - E\_{H}T\_{H1}(1 - E\_{L})[(\eta\_{\text{f}} - 1)\mathbf{x}\mathbf{y} - \eta\_{\text{l}}]\}}{E\_{H}E\_{L}(\eta\_{\text{c}} + \mathbf{x} - 1)[(\mathbf{x}\mathbf{y} - 1)\eta\_{\text{f}} - \mathbf{x}\mathbf{y}] + [\mathbf{x}\mathbf{y} - \mathbf{x}^{2}\mathbf{y} + \eta\_{\text{f}}(\eta\_{\text{c}} + \mathbf{x} - 1)}\tag{39}$$

2. When *ηc*<sup>1</sup> = *ηt*<sup>1</sup> = 1, Equations (30)–(34) can be respectively simplified into the performance indicators of the endoreversible simple BCY with an IHP and coupled to VTHRs [99], whose *T* − *s* diagram is shown in Figure 3b:

$$\begin{aligned} &\mathbf{C\_{wf}x}\{\mathbf{C\_{L\min}}\mathbf{C\_{wf}}E\_{L}T\_{L1}(y-1) + \mathbf{C\_{H1}}E\_{H1}[\mathbf{C\_{wf}}T\_{H3}(y-1) + \mathbf{C\_{L\min}}E\_{L}(T\_{H3} \\ &- T\_{L1}\mathbf{x}y)] \} + E\_{H}\mathbf{C\_{H\min}}\{\mathbf{C\_{L\min}}E\_{L}[\mathbf{C\_{wf}}T\_{H1}(\mathbf{x}-1) + \mathbf{C\_{wf}}T\_{L1}\mathbf{x}(1-\mathbf{x}y) + \mathbf{C\_{H1}} \\ &\mathbf{W} = \frac{\times E\_{H1}\mathbf{x}(T\_{L1}\mathbf{x}y - T\_{H3})) + \mathbf{xC\_{wf}}[(y-1)\mathbf{C\_{wf}}T\_{H1} + \mathbf{C\_{H1}}E\_{H1}(T\_{H3} - T\_{H1}\mathbf{y})] \} \\ &\mathbf{C\_{wf}T\_{0}x[\mathbf{C\_{wf}}y - (\mathbf{C\_{wf}} - \mathbf{C\_{H\min}}E\_{H1})(\mathbf{C\_{wf}} - \mathbf{C\_{L\min}}E\_{L})]} \end{aligned} \tag{40}$$

$$\begin{aligned} &\mathbb{C}\_{wf}T\_0\mathbf{x}\left\{\mathbb{C}\_{L\text{min}}\mathbb{C}\_{wf}E\_{L1}T\_{L1}(y-1) + \mathbb{C}\_{H1}\mathbb{E}\_{H1}[\mathbb{C}\_{wf}T\_{H3}(y-1) + \mathbb{C}\_{L\text{min}}\mathbb{E}\_{L}(T\_{H3} - T\_{L1}\mathbf{x}\mathbf{y})]\right\} \\ &+ \mathbb{C}\_{H\text{min}}\mathbb{E}\_{H}\left\{\mathbb{C}\_{L\text{min}}\mathbb{E}\_{L}[\mathbb{C}\_{wf}T\_{H1}(\mathbf{x}-1) + \mathbb{C}\_{wf}T\_{L1}\mathbf{x}(1-\mathbf{x}\mathbf{y}) + \mathbb{C}\_{H1}\mathbb{E}\_{H1}\mathbf{x}(T\_{L1}\mathbf{x}\mathbf{y} - T\_{H3})]\right\} \\ &= \frac{\mathbb{C}\_{wf}\mathbb{E}\_{wf}[\mathbb{C}\_{wf}T\_{H1}(y-1) + \mathbb{C}\_{H1}\mathbb{E}\_{H1}(T\_{H3} - T\_{H1}\mathbf{y})]}{\mathbb{C}\_{wf}\mathbb{E}\_{wf}[\mathbb{C}\_{wf}T\_{L1}(\mathbf{x}-1) + \mathbb{C}\_{L\text{min}}\mathbb{E}\_{L}(T\_{L1}\mathbf{x}\mathbf{y}) + \mathbb{C}\_{L\text{min}}\mathbb{E}\_{L}(T\_{L1}\mathbf{x}\mathbf{y})]} \end{aligned} \tag{41}$$

*η* = *Cw f <sup>T</sup>*0*x*{*CH*min*EH*[*C*<sup>2</sup> *w f TH*1(*y* − 1) + *CH*1*Cw f EH*1(*TH*<sup>3</sup> − *TH*1*y*) + *CL*min*Cw f EL*(*TH*<sup>1</sup> − *TL*1*xy*) +*CH*1*CL*min*EH*1*EL*(*TL*1*xy* − *TH*3)] + *CH*1*Cw f EH*1[*Cw f TH*3(*y* − 1) + *CL*min*EL*(*TH*<sup>3</sup> − *TL*1*xy*)]}

$$\begin{aligned} & \quad \left\langle \mathcal{C}\_{\text{lim}} \text{E}\_{H} \text{H}\_{11} (\text{C}\_{\text{cV}} - \mathcal{C}\_{\text{cV}} \text{E}\_{L1}) + \text{C}\_{\text{cLm}} \text{E}\_{C} \text{E}\_{L} \text{T}\_{1} \text{T}\_{1} \right\rangle \left\langle \mathcal{C}\_{\text{cV}} \text{C}\_{\text{cLm}} \text{C}\_{\text{cT}} \text{T}\_{1} \text{T}\_{1} (y - 1) + \text{C}\_{\text{cH}} \text{E}\_{L1} \\ & \times \left\langle \mathcal{C}\_{\text{cV}} \text{T}\_{1} (y - 1) + \text{C}\_{\text{cm}} \text{E}\_{L1} (\text{C}\_{\text{cV}} - \text{T}\_{1} \text{x}) \text{y} \right\rangle + \text{C}\_{\text{cH}} \text{E}\_{C} \left\langle \mathcal{C}\_{\text{cV}} \text{C}\_{\text{cV}} \text{T}\_{1} (y - 1) + \text{T}\_{1} \text{y} - \text{T}\_{1} \text{y} \right\rangle \\ & \times \left\langle \mathcal{C}\_{\text{cV}} \text{T}\_{1} \text{E}\_{L1} \right\rangle + \text{C}\_{\text{cm}} \text{E}\_{L1} \left\langle \mathcal{C}\_{\text{cV}} \text{T}\_{1} (y - 1) + \text{C}\_{\text{cV}} \text{T}\_{1} \text{x} (1 - \text{x}) \right\rangle + \text{C}\_{\text{cH}} \text{E}\_{L1} \text{E}\_{L1} \text{y} \\ & \times \left\langle \mathcal{C}\_{\text{cV}} \text{T}\_{1} \text{C}\_{\$$

3. When *ηc*<sup>1</sup> = *ηt*<sup>1</sup> = 1 and *CH*<sup>1</sup> = *CH*<sup>2</sup> = *CL* → ∞, Equations (30)–(34) can be simplified into the performance indicators of the endoreversible simple BCY with an IHP and coupled to CTHRs, whose *T* − *s* diagram is shown in Figure 3c:

$$\begin{aligned} \mathbb{C}\_{wf} & \mathbf{x} \left\{ \mathbb{C}\_{wf} E\_L T\_{L1} (y - 1) + \mathbb{C}\_{H1} E\_{H1} [E\_L T\_{H3} - E\_L T\_{L1} \mathbf{x} y + T\_{H3} (y - 1)] \right\} \\ & + \mathbb{C}\_{wf} E\_H \left\{ \mathbb{E}\_L [T\_{H1} \mathbb{C}\_{wf} (\mathbf{x} - 1) + \mathbb{C}\_{wf} T\_{L1} \mathbf{x} (1 - \mathbf{x} y) + \mathbb{C}\_{H1} E\_{H1} \mathbf{x} (T\_{L1} \mathbf{x} y \\ \overline{W} &= \frac{-T\_{H3} \left[ \right] + \mathbf{x} [\mathbb{C}\_{wf} T\_{H1} (y - 1) + \mathbb{C}\_{H1} E\_{H1} (T\_{H3} - T\_{H1} y)] \right\}}{\mathbb{C}\_{wf}^2 T\_0 \mathbf{x} (E\_H + E\_L + y - E\_H E\_L - 1)} \end{aligned} \tag{44}$$

*η* = *T*0*x* ( *Cw f ELTL*1*y* − *Cw f ELTL*<sup>1</sup> + *CH*1*EH*1[*TH*3*y* − *TH*<sup>3</sup> + *TH*3*EL* − *ELTL*1*xy*] ) + ( *EH* ( *x*[*Cw f TH*1*y* − *Cw f TH*<sup>1</sup> + *CH*1*EH*1(*TH*<sup>3</sup> − *TH*1*y*)] + *EL*[*Cw f TH*1*x* − *Cw f* <sup>×</sup>*TH*<sup>1</sup> <sup>+</sup> *Cw f TL*1*x*(<sup>1</sup> <sup>−</sup> *xy*) + *CH*1*EH*1*x*(−*TH*<sup>3</sup> <sup>+</sup> *TL*1*xy*)]) *Cw f T*0*x* ( [*Cw f TH*1*y* − *Cw f TH*<sup>1</sup> + *CH*1*EH*1(*TH*<sup>3</sup> − *TH*1*y*) + *Cw f EL*(*TH*<sup>1</sup> − *TL*1*xy*) +*CH*1*EH*1*EL*(*TL*1*xy* − *TH*3)]*EH* + *CH*1*EH*1[*TH*3(*y* − 1) + *EL*(*TH*<sup>3</sup> − *TL*1*xy*)]} (45) *P* = [*EHTH*1(1 − *EL*) + *ELTL*1*xy*] ( *Cw f x* ( *Cw f ELTL*1(*y* − 1) + *CH*1*EH*1[*TH*3(*y* − 1) <sup>+</sup>*EL*(*TH*<sup>3</sup> <sup>−</sup> *TL*1*xy*)]} <sup>+</sup> *EHCw f* ( *x*[*Cw f TH*1(*y* − 1) + *CH*1*EH*1(*TH*<sup>3</sup> − *TH*1*y*)] <sup>+</sup>*EL*[*Cw f TH*1(*<sup>x</sup>* <sup>−</sup> <sup>1</sup>) + *Cw f TL*1*x*(<sup>1</sup> <sup>−</sup> *xy*) + *CH*1*xEH*1(*TL*1*xy* <sup>−</sup> *TH*3)]) } *C*2 *w f <sup>T</sup>*0*x*(*EH* <sup>+</sup> *EL* <sup>+</sup> *<sup>y</sup>* <sup>−</sup> *EHEL* <sup>−</sup> <sup>1</sup>)(*EHTH*<sup>1</sup> <sup>+</sup> *ELTL*1*<sup>x</sup>* <sup>−</sup> *ELTL*1*xEH*) (46)

*E* = ( *Cw f ELTL*1(*y* − 1) + *CH*1*EH*1[*TH*3(*y* − 1) + *ELTH*3*TL*1*xy*] ) *x* +*EH* ( *EL*[*Cw f TH*1(*x* − 1)+(1 − *xy*)*Cw f TL*1*x* + *CH*1*EH*1*x*(*TL*<sup>1</sup> <sup>×</sup>*xy* <sup>−</sup> *TH*3)] + *<sup>x</sup>*[*Cw f TH*1(*<sup>y</sup>* <sup>−</sup> <sup>1</sup>) + *CH*1*EH*1(*TH*<sup>3</sup> <sup>−</sup> *TH*1*y*)]) *Cw f T*0*xy* − *Cw f T*0*x*(1 − *EH* − *EL* + *EHEL*) <sup>−</sup> *CH*<sup>1</sup> *Cw f <sup>T</sup>*<sup>0</sup> ln *Cw f TH*3(1 − *EH* − *EL* + *EHEL*)(*EH*<sup>1</sup> − 1) + *Cw f* ×*y*[*EHEH*1*TH*<sup>1</sup> − *TH*3(*EH*<sup>1</sup> − 1)] + *EH*1*ELTL*1*xy* ×(1 − *EH*) *Cw f* [*TH*3*y*−*TH*3(1−*EH*)(1−*EL*)] <sup>−</sup> *CH Cw f <sup>T</sup>*<sup>0</sup> ln{<sup>1</sup> <sup>+</sup> *EHCw f*(*TH*1−*EL*×*TH*1−*TH*1*y*+*ELTL*1*xy*) *TH*1*CH*[*y*−(1−*EH*)(1−*EL*)] } <sup>−</sup> *CL Cw f <sup>T</sup>*<sup>0</sup> ln{<sup>1</sup> <sup>+</sup> *ELCw f* [*EH*(*TH*1−*TL*1*x*)−*TL*1*x*(*y*−1)] *CLTL*1[(*EH*−1)(1−*EL*)*x*+*xy*] } (47)

4. When *EH*<sup>1</sup> = 0, Equations (30)–(34) can be simplified into the performance indicators of the simple irreversible BCY coupled to VTHRs [79], whose *T* − *s* diagram is shown in Figure 3d:

$$\begin{aligned} \mathbf{C}\_{L\min} \mathbf{C}\_{wf} E\_L T\_{L1} \mathbf{x} &+ \mathbf{C}\_{H\min} E\_H T\_{H1} \left\{ \mathbf{C}\_{L\min} E\_L [\eta\_t(\mathbf{x} - 1) - \mathbf{x}] + \mathbf{C}\_{wf} \mathbf{x} \right\} \\ \overline{\mathcal{W}} &= \frac{+a\_3 \left\{ \mathbf{C}\_{L\min} (\mathbf{C}\_{wf} - \mathbf{C}\_{H\min} E\_H) E\_L [\eta\_t(\mathbf{x} - 1) - \mathbf{x}] - \mathbf{C}\_{H\min} \mathbf{C}\_{wf} E\_H \mathbf{x} \right\}}{\mathbf{C}\_{wf}^2 T\_0 \mathbf{x}} \end{aligned} \tag{48}$$

$$\begin{aligned} \eta &= \frac{a\_3 \left\{ \mathbb{C}\_{H \text{min}} \mathbb{C}\_{wf} E\_H \mathbf{x} - \mathbb{C}\_{L \text{min}} E\_L (\mathbb{C}\_{wf} - \mathbb{C}\_{H \text{min}} E\_H) [\eta\_t(\mathbf{x} - \mathbf{1}) - \mathbf{x}] \right\} - \mathbb{C}\_{L \text{min}} \\ \eta &= \frac{\times \mathbb{C}\_{wf} E\_L T\_{L1} \mathbf{x} + \mathbb{C}\_{H \text{min}} E\_H T\_{H1} [\mathbb{C}\_{L \text{min}} E\_L (\eta\_t + \mathbf{x} - \eta\_I \mathbf{x}) - \mathbb{C}\_{wf} \mathbf{x}]}{\text{xC}\_{H \text{min}} E\_H \mathbb{C}\_{wf} (a\_3 - T\_{H1})} \end{aligned} \tag{49}$$

$$\begin{aligned} & \left\{ -a\_3[\eta\_l(\mathbf{x} - 1) - \mathbf{x}](\mathbf{C}\_{wf} - \mathbf{C}\_{H\text{min}}E\_H)(\mathbf{C}\_{wf} - \mathbf{C}\_{L\text{min}}E\_L) - \mathbf{C}\_{H\text{min}}E\_HT\_{H1}[\eta\_l(\mathbf{x} - 1) - \mathbf{x}] \right. \\ & \left. \times (\mathbf{C}\_{wf} - \mathbf{C}\_{L\text{min}}E\_L) + \mathbf{C}\_{L\text{min}}\mathbf{C}\_{wf}E\_LT\_{L1}\mathbf{x} \right\} \left\{ a\_3 \left\{ \mathbf{C}\_{L\text{min}}E\_L[\eta\_l(\mathbf{x} - 1) - \mathbf{x}](\mathbf{C}\_{wf} - \mathbf{C}\_{H\text{min}}E\_H) \right. \\ & \left. \mp \mathbf{ - C\_{H\text{min}}}\mathbf{C}\_{wf}E\_H\mathbf{x} \right\} + \mathbf{C}\_{H\text{min}}E\_HT\_{H1}\left\{ \mathbf{C}\_{L\text{min}}E\_L[\eta\_l(\mathbf{x} - 1) - \mathbf{x}] + \mathbf{C}\_{wf}\mathbf{x} \right\} + \mathbf{C}\_{L\text{min}}\mathbf{C}\_{wf}E\_LT\_{L1}\mathbf{x} \right\} \\ & \left. - \mathbf{C}\_{wf}^3T\_0 \mathbf{x} [\eta\_l(\mathbf{x} - 1) - \mathbf{x}][a\_3(\mathbf{C}\_{wf} - \mathbf{C}\_{H\text{min}}E\_H) + \mathbf{C}\_{H\text{min}}E\_HT\_{H1}] \right. \\ & \left. \times \mathbf{1} \end{aligned} \tag{50}$$

$$\begin{cases} \mathbb{C}\_{\mathcal{L}\min} \mathbb{C}\_{wf} E\_{L} \mathbb{T}\_{11} \mathbf{x} + \mathbb{C}\_{H\min} E\_{H} T\_{\mathcal{H}1} \left\{ \mathbb{C}\_{\mathcal{L}\min} \mathbb{E}\_{L} [\eta\_{t}(\mathbf{x} - 1) - \mathbf{x}] + \mathbb{C}\_{wf} \mathbf{x} \right\} + a\_{3} \Big\{ \mathbb{C}\_{\mathcal{L}\min} (\mathbb{C}\_{wf} \\ \quad - \mathbb{C}\_{H\min} E\_{H}) E\_{L} [\eta\_{t}(\mathbf{x} - 1) - \mathbf{x}] - \mathbb{C}\_{H\min} \mathbb{C}\_{wf} E\_{H} \mathbf{x} \Big\} / (T\_{0} \mathbf{x}) - \mathbb{C}\_{wf} \{ \mathbb{C}\_{\mathcal{L}} \ln \Big| 1 + \mathbb{C}\_{H\min} E\_{H} \\ \quad \times (a\_{3} - T\_{H1}) / (\mathbb{C}\_{\mathcal{L}} T\_{H1}) ] + \mathbb{C}\_{L} \ln \Big\{ 1 + \mathbb{C}\_{L\min} E\_{L} \Big\} a\_{3} \mathbb{C}\_{wf} \eta\_{t} + \mathbb{C}\_{H\min} E\_{H} (a\_{3} - T\_{H1}) [\eta\_{t}(\mathbf{x} - 1) - \mathbf{x}] \\ \quad \times [-\mathbf{x}] - \mathbb{C}\_{wf} [a\_{3} (\eta\_{t} - 1) + T\_{L1} ] \mathbf{x} \Big\} / (\mathbb{C}\_{L} \mathbb{C}\_{wf} T\_{L1} \mathbf{x}) \Big] \Big{} \\ \end{cases}$$

where

$$\begin{split} \eta\_{3} &= \frac{(\eta\_{\rm c} + \mathbf{x} - 1) \left\{ \mathbb{C}\_{L \text{min}} \mathbb{C}\_{wf} \mathbb{E}\_{L} T\_{\text{L1}} \mathbf{x} - \mathbb{C}\_{H \text{min}} \mathbb{E}\_{H} T\_{\text{H1}} (\mathbb{C}\_{wf} - \mathbb{C}\_{L \text{min}} \mathbb{E}\_{L}) [(\eta\_{\rm t} - 1) \mathbf{x} - \eta\_{\rm t}] \right\}}{\mathbb{C}\_{H \text{min}} \mathbb{C}\_{L \text{min}} \mathbb{E}\_{H} E\_{L} (\eta\_{\rm c} + \mathbf{x} - 1)(\eta\_{\rm t} \mathbf{x} - \mathbf{x} - \eta\_{\rm t}) + \mathbb{C}\_{wf}^{2} [\mathbf{x} - \mathbf{x}^{2} + \eta\_{\rm t}(\eta\_{\rm c} + \mathbf{x} - 1)(\mathbf{x} - \mathbf{x} - 1)]}} \\ &\quad - 1) \left[ -\mathbb{C}\_{wf} (\eta\_{\rm c} + \mathbf{x} - 1)(\mathbb{E}\_{H} \mathbb{C}\_{H \text{min}} + \mathbb{E}\_{L} \mathbb{C}\_{L \text{min}}) \times [(\eta\_{\rm t} - 1)\mathbf{x} - \eta\_{\rm t}] \right] \end{split} \tag{52}$$

5. When *EH*<sup>1</sup> = 0 and *CH* = *CL* → ∞, Equations (30)–(34) can be simplified into the performance indicators of the simple irreversible BCY coupled to CTHRs [76], whose *T* − *s* diagram is shown in Figure 3e:

$$\overline{W} = \frac{E\_L T\_{L1} \mathbf{x} - a\_4 \{ (E\_H - 1) E\_L [\eta\_l(\mathbf{x} - 1) - \mathbf{x}] + E\_H \mathbf{x} \} + E\_H T\_{H1} [E\_L \eta\_l(\mathbf{x} - 1) + \mathbf{x} - E\_L \mathbf{x}]}{T\_0 \mathbf{x}} \tag{53}$$

$$\eta = \frac{a\_4(E\_H - 1)E\_L[\eta\_t(\mathbf{x} - 1) - \mathbf{x}] + a\_4E\_H\mathbf{x} - E\_HT\_{H1}\mathbf{x} - E\_LT\_{L1}\mathbf{x}}{1 + E\_HE\_LT\_{H1}(\eta\_t + \mathbf{x} - \eta\_t\mathbf{x})} \tag{54}$$

$$\{a\_4(E\_H - 1)(E\_L - 1)[\eta\_t(\mathbf{x} - 1) - \mathbf{x}] - E\_HT\_{H1}(E\_L - 1)[\eta\_t(\mathbf{x} - 1) - \mathbf{x}]\} = 0$$

$$\overline{P} = \frac{-1(-\mathbf{x}) - E\_L T\_{L1} \mathbf{x} \left\{ a\_4 (E\_H - 1) E\_L [\eta\_t(\mathbf{x} - 1) - \mathbf{x}] + a\_4 E\_H \mathbf{x} \right\}}{-E\_H T\_{H1} \mathbf{x} - E\_L T\_{L1} \mathbf{x} + E\_H E\_L T\_{H1} (\eta\_t + \mathbf{x} - \eta\_t \mathbf{x}) \}} \tag{55}$$

$$\frac{T\_0 [a\_4 (E\_H - 1) - E\_H T\_{H1}] [\eta\_t(\mathbf{x} - 1) - \mathbf{x}] \mathbf{x}}{T\_0 [a\_4 (E\_H - 1) - E\_H T\_{H1}] [\eta\_t(\mathbf{x} - 1) - \mathbf{x}] \mathbf{x}} \tag{55}$$

$$\begin{cases} \mathbf{E} = \left\{ E\_{L} T\_{L1} \mathbf{x} - a\_{4} \{ E\_{L} (E\_{H} - 1) [\eta\_{l} (\mathbf{x} - \mathbf{1}) - \mathbf{x}] + E\_{H} \mathbf{x} \right\} + E\_{H} T\_{H1} [\mathbf{E}\_{L} \eta\_{l} (\mathbf{x} - \mathbf{1}) + \mathbf{x} - E\_{L} \mathbf{x}] \\\ - \mathbf{1} \right\} / \left( T\_{0} \mathbf{x} \right) - \mathbf{C}\_{H} \ln \left[ 1 + \mathbf{C}\_{wf} E\_{H} (a\_{4} - T\_{H1}) / \left( \mathbf{C}\_{H} T\_{H1} \right) \right] / \mathbf{C}\_{wf} \\\ - \mathbf{C}\_{L} \ln \left\{ 1 + \mathbf{C}\_{wf} E\_{L} \left\{ a\_{4} (E\_{H} - 1) [\eta\_{l} (\mathbf{x} - \mathbf{1}) - \mathbf{x}] - T\_{L1} \mathbf{x} + E\_{H} T\_{H1} (\eta\_{l} + \mathbf{x} - \eta\_{l} \mathbf{x}) \right\} \right\} / \mathbf{C}\_{wf} \\\ - \eta\_{l} \mathbf{x} \rangle \right\} / \left( \mathbf{C}\_{L} T\_{L1} \mathbf{x} \right) \Big/ \mathbf{C}\_{wf} \end{cases} \tag{56}$$

where

$$n\_{4} = \frac{(\eta\_{c} + \mathbf{x} - 1)E\_{H}T\_{H1}(E\_{L} - 1)[\eta\_{l}(\mathbf{x} - 1) - \mathbf{x}] + E\_{L}T\_{L1}\mathbf{x}\}}{(E\_{H} - 1)(E\_{L} - 1)(\mathbf{x} - 1)(\eta\_{c} + \mathbf{x} - 1)\eta\_{l} - \mathbf{x}[\mathbf{x} - 1 + E\_{H}(E\_{L} - 1)(\eta\_{c} + \mathbf{x} - 1) - E\_{L}(\eta\_{c} + \mathbf{x} - 1)]} \tag{57}$$

6. When *EH*<sup>1</sup> = 0 and *η<sup>c</sup>* = *η<sup>t</sup>* = 1, Equations (30)–(34) can be simplified into the performance indicators of the simple endoreversible BCY coupled to VTHRs [78], whose *T* − *s* diagram is shown in Figure 3f:

$$\overline{\mathcal{W}} = \frac{\mathbb{C}\_{H \text{min}} \mathbb{C}\_{L \text{min}} E\_H E\_L (-1 + \mathbf{x}) (T\_{H1} - T\_{L1} \mathbf{x})}{T\_0 \text{x} [\mathbb{C}\_{L \text{min}} \mathbb{C}\_{wf} E\_L + \mathbb{C}\_{H \text{min}} E\_H (\mathbb{C}\_{wf} - \mathbb{C}\_{L \text{min}} E\_L)]} \tag{58}$$

$$
\eta = (\mathbf{x} - \mathbf{1}) / \mathbf{x} \tag{59}
$$

$$\overline{P} = \frac{\mathbf{C}\_{H\text{min}}\mathbf{C}\_{L\text{min}}E\_{H}E\_{L}(-1+\mathbf{x})(T\_{H1}-T\_{L1}\mathbf{x})[\mathbf{C}\_{H\text{min}}E\_{H}(\mathbf{C}\_{wf} \\ \overline{P})}{-\mathbf{C}\_{L\text{min}}E\_{L})T\_{H1} + \mathbf{C}\_{L\text{min}}\mathbf{C}\_{wf}E\_{L}T\_{L1}\mathbf{x}} \\ \begin{array}{l} \left[\mathbf{C}\_{wf} \\ \mathbf{C}\_{L\text{min}}\mathbf{C}\_{wf}E\_{L} + \mathbf{C}\_{H\text{min}}E\_{H}(\mathbf{C}\_{wf} - \mathbf{C}\_{L\text{min}}E\_{L})\right][\mathbf{C}\_{L\text{min}}\mathbf{C}\_{wf} \\ \times E\_{L}T\_{L1}\mathbf{x} + \mathbf{C}\_{H\text{min}}E\_{H}(\mathbf{C}\_{wf}T\_{H1} - \mathbf{C}\_{L\text{min}}E\_{L}T\_{L1}\mathbf{x})] \end{array} \tag{60}$$

$$\begin{array}{c} \frac{\mathsf{C}\_{Him}\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{E}\_{H}\mathsf{E}\_{L}(\mathbf{x}-1)(T\_{H1}-T\_{11}\mathbf{x})}{\left[\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{C}\_{L}\mathsf{E}\_{L}+\mathsf{C}\_{Him}\mathsf{E}\_{H}\{\mathsf{C}\_{wf}\}\right]}-\mathsf{C}\_{H}\ln[1+\frac{\mathsf{C}\_{Him}\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{E}\_{L}(\mathsf{E}\_{1}\|\mathbf{x}-\mathbf{T}\_{H1})}{\mathsf{C}\_{H}\|\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{E}\_{L}+\mathsf{C}\_{Him}\mathsf{E}\_{H}\|}\right]}{-\mathsf{C}\_{Lim}\mathsf{E}\,\mathsf{E}\,\|}\,\mathsf{T}\_{0}\mathbbm{x} & \quad \left(\mathsf{C}\_{wf}-\mathsf{C}\_{Lim}\mathsf{E}\_{L}\right)\|\mathsf{T}\_{H1}\|\\\mathsf{E}=-\frac{\mathsf{C}\_{L}\ln\left\{\frac{\mathsf{C}\_{L}\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{E}\_{L}\mathsf{E}\_{L1}\mathbf{x}+\mathsf{C}\_{Him}\mathsf{E}\_{H}[\mathsf{C}\_{L}\mathsf{C}\_{wf}\mathsf{T}\_{1\cdot\mathsf{X}}\mathsf{x}+\mathsf{C}\_{Lim}\mathsf{E}\_{L}(\mathsf{C}\_{wf}\mathsf{T}\_{H1\cdots\mathsf{C}\_{L}\mathsf{T}\_{L1\cdot\mathsf{X}}\mathsf{x}-\mathsf{C}\_{L\operatorname{T}L\operatorname{X}}\mathsf{x})]}{\mathsf{C}\_{L}\|\mathsf{C}\_{Lim}\mathsf{C}\_{wf}\mathsf{E}\_{L}+\mathsf{C}\_{Him}\mathsf{E}\_{H}(\mathsf{C}\_{wf}-\mathsf{C}\_{Lim}\mathsf{E}\_{L})\$$

7. When *EH*<sup>1</sup> = 0, *η<sup>c</sup>* = *η<sup>t</sup>* = 1 and *CH* = *CL* → ∞, Equations (30)–(34) can be simplified into the performance indicators of the simple endoreversible BCY coupled to CTHRs [77], whose *T* − *s* diagram is shown in Figure 3g:

$$\overline{W} = \frac{E\_H E\_L (-1 + \mathbf{x}) (T\_{L1}\mathbf{x} - T\_{H1})}{[E\_H (E\_L - 1) - E\_L] T\_0 \mathbf{x}} \tag{62}$$

$$
\eta = (\mathbf{x} - \mathbf{1}) / \mathbf{x} \tag{63}
$$

$$\overline{P} = \frac{E\_H E\_L (\mathbf{x} - \mathbf{1}) (T\_{L1} \mathbf{x} - T\_{H1}) [E\_H (E\_L - \mathbf{1}) T\_{H1} - E\_L T\_{L1} \mathbf{x}]}{T\_0 \mathbf{x} (E\_H E\_L T\_{L1} \mathbf{x} - E\_H T\_{H1} - E\_L T\_{L1} \mathbf{x}) [E\_H (E\_L - \mathbf{1}) - E\_L]} \tag{64}$$

$$\begin{array}{c} \mathbf{C}\_{wf}E\_{H}E\_{L}(\mathbf{x}-1)(T\_{L1}\mathbf{x}-T\_{H1}) + \mathbf{C}\_{H}T\_{0}\mathbf{x}(E\_{H}+E\_{L}-E\_{H}E\_{L})\ln\left\{1-\mathbf{C}\_{wf}E\_{H}\right\} \\ \times E\_{L}(T\_{H1}-T\_{L1}\mathbf{x})/\left[\mathbf{C}\_{H}T\_{H1}(E\_{H}+E\_{L}-E\_{H}E\_{L})\right]\right\} + \mathbf{C}\_{L}T\_{0}\mathbf{x}(E\_{H}+E\_{L}-E\_{H}) \\ \mathbf{E} = \frac{\times E\_{L}\left[\ln\left(1+\mathbf{C}\_{wf}E\_{H}E\_{L}(T\_{H1}-T\_{L1}\mathbf{x})/\left(\mathbf{C}\_{L}(E\_{H}+E\_{L}-E\_{H}E\_{L})T\_{L1}\mathbf{x}\right)\right]\right]}{\mathbf{C}\_{wf}\left[E\_{H}(E\_{L}-1)-E\_{L}\right]T\_{0}\mathbf{x}} \end{array} \tag{65}$$

8. When *EH* = *EL* = 0, *η<sup>c</sup>* = *η<sup>t</sup>* = 1 and *Cw f* → ∞, the cycle in this paper can become the endoreversible Carnot cycle coupled to VTHRs [14], whose *T* − *s* diagram is shown in Figure 3h. However, Equations (30), (33), and (34) need to be de-dimensionalized to simplify to *W*, *P* and *E* of the endoreversible Carnot cycle coupled to VTHRs. The performance indicators of the cycle are:

$$\mathcal{W} = \frac{\mathbb{C}\_{H}\mathbb{C}\_{L}E\_{H}E\_{L}(\mathbf{x} - \mathbf{1})(T\_{H1} - T\_{L1}\mathbf{x})}{\mathbf{x}(\mathbb{C}\_{H}E\_{H} + \mathbb{C}\_{L}E\_{L})} \tag{66}$$

$$
\eta = (\mathbf{x} - \mathbf{1})/\mathbf{x} \tag{67}
$$

$$P = \frac{\mathbb{C}\_{H}\mathbb{C}\_{L}E\_{H}E\_{L}(\mathbf{x} - \mathbf{1})(T\_{H1} - T\_{L1}\mathbf{x})}{\mathbf{x}(\mathbb{C}\_{H}E\_{H} + \mathbb{C}\_{L}E\_{L})} \tag{68}$$

$$E = \frac{\frac{C\_H C\_L E\_H E\_L (x - 1)(T\_{H1} - T\_{L1} x)}{\left(C\_H E\_H + C\_L E\_L\right) x} - C\_H T\_0 \ln\left[1 + \frac{C\_L E\_H E\_L (T\_{L1} x - T\_{H1})}{\left(C\_H E\_H + C\_L E\_L\right) T\_{H1}}\right]}{-C\_L T\_0 \ln\left[\frac{C\_H E\_H E\_L T\_{H1} + C\_H E\_H T\_{L1} x + C\_L E\_L T\_{L1} x - C\_H E\_H E\_L T\_{L1} x}{C\_H E\_H T\_{L1} x + C\_L E\_L T\_{L1} x}\right]} \tag{69}$$

9. When *EH* = *EL* = 0, *η<sup>c</sup>* = *η<sup>t</sup>* = 1 and *CH*<sup>1</sup> = *CL* = *Cw f* → ∞, the cycle in this paper can become the endoreversible Carnot cycle coupled to CTHRs [12], whose *T* − *s* diagram is shown in Figure 3i. However, Equations (30), (33), and (34) also need to be de-dimensionalized to simplify to *W*, *P* and *E* of the cycle [12,74,82]. The performance indicators of the cycle are:

$$\mathcal{W} = \frac{\mathcal{U}\_H \mathcal{U}\_L (-1 + \mathbf{x}) (T\_{H1} - T\_{L1}\mathbf{x})}{(\mathcal{U}\_H + \mathcal{U}\_L)\mathbf{x}} \tag{70}$$

$$
\eta = (\mathbf{x} - \mathbf{1})/\mathbf{x} \tag{71}
$$

$$P = \frac{\mathbb{U}I\_H \mathbb{U}\_L(-1+\mathbf{x})(T\_{H1} - T\_{L1}\mathbf{x})}{(\mathbb{U}\_H + \mathbb{U}\_L)\mathbf{x}}\tag{72}$$

$$E = \frac{\mathcal{U}\_H \mathcal{U}\_L (T\_{H1} - T\_{L1} \mathbf{x}) [(T\_0 + T\_{H1}) T\_{L1} \mathbf{x} - T\_{H1} (T\_0 + T\_{L1})]}{T\_{H1} T\_{L1} (\mathcal{U}\_H + \mathcal{U}\_L) \mathbf{x}} \tag{73}$$

10. When *EH* = *EL* = 0, *η<sup>c</sup>* = *η<sup>t</sup>* = 1, *CH*<sup>1</sup> = *CL* = *Cw f* → ∞, and *UL* → ∞, the cycle in this paper can become the endoreversible Novikov cycle coupled to CTHRs [11], whose *T* − *s* diagram is shown in Figure 3j. However, Equations (30), (33), and (34) also need to be de-dimensionalized to simplify to *W*, *P* and *E* of the cycle [11]. The performance indicators of the cycle are:

$$\mathcal{W} = \frac{\mathcal{U}\_H(\mathbf{x} - \mathbf{1})(T\_{H1} - T\_{L1}\mathbf{x})}{\mathbf{x}} \tag{74}$$

$$
\eta = (\mathbf{x} - \mathbf{1})/\mathbf{x} \tag{75}
$$

$$P = \frac{\mathcal{U}\_H(\mathbf{x} - \mathbf{1})(T\_{H1} - T\_{L1}\mathbf{x})}{\mathbf{x}} \tag{76}$$

$$E = \frac{\mathcal{U}\_H (T\_{H1} - T\_{L1} \mathbf{x}) [T\_{H1} T\_{L1} (\mathbf{x} - \mathbf{1}) + T\_0 (T\_{L1} \mathbf{x} - T\_{H1})]}{T\_{H1} T\_{L1} \mathbf{x}} \tag{77}$$


**Figure 3.** Diagrams of (**a**) irreversible simple BCY with an IHP and coupled to CTHRs; (**b**) endoreversible simple BCY with an IHP and coupled to VTHRs; (**c**) endoreversible simple BCY with an IHP and coupled to CTHRs; (**d**) simple irreversi-ble BCY coupled to VTHRs; (**e**) simple irreversible BCY coupled to CTHRs; (**f**) simple endoreversible BCY coupled to VTHRs; (**g**) simple endoreversible BCY coupled to CTHRs; (**h**) endoreversible Carnot cycle coupled to VTHRs; (**i**) endoreversible Carnot cycle coupled to CTHRs; (**j**) endoreversible Novikov cycle coupled to CTHRs.

#### **3. Analyses and Optimizations with Each Single Objective**

*3.1. Analyses of Each Single Objective*

The impacts of the irreversibility on cycle performance indicators (*W*, *η*, *P* and *E*) are analyzed below. In numerical calculations, it is set that *CL* = *CH* = 1.2 kW/K, *Cw f* = 1 kW/K, *T*<sup>0</sup> = 300 K, *CH*<sup>1</sup> = 0.6 kW/K, *k* = 1.4, *Rg* = 0.287 kJ/(kg · K), *EH* = *EH*<sup>1</sup> = *EL* = 0.9, *Cp* = 1.005 kW/K, *τH*<sup>1</sup> = 4.33, *τH*3= 5 and *τ<sup>L</sup>* = 1.

Figures 4–6 present the relationships of *W*, *η*, *P*, *E*, *π<sup>t</sup>* and *v*5/*v*<sup>1</sup> versus *π* with different *ηt*. As shown in Figures 4 and 5, *W*, *η*, *P* and *E* increase and then decrease as *π* increases. In the same situation, *W*, *E*, *P* and *η* reach the maximum value successively. When *η<sup>t</sup>* = 0.7 and *π* = 32.3, *W* = *P* = 0. If *π* keeps going up, *W* and *P* are going to go negative. *W*, *η*, *P* and *E* increase as *η<sup>t</sup>* increases. As *π* increases, *W*, *η*, *P* and *E* are affected more significantly by *ηt*. As shown in Figure 6, *π<sup>t</sup>* goes up but *v*5/*v*<sup>1</sup> goes down as *π* goes up. *π<sup>t</sup>* and *v*5/*v*<sup>1</sup> decrease as *η<sup>t</sup>* rises. It illustrates that the degree of the IHP is improved and the device's volume is reduced as *η<sup>t</sup>* increases.

**Figure 4.** Relationships of *W* and *η* versus *π* with different *ηt*.

**Figure 5.** Relationships of *P* and *E* versus *π* with different *ηt*.

**Figure 6.** Relationships of *π<sup>t</sup>* and *v*5/*v*<sup>1</sup> versus *π* with different *ηt*.

By numerical calculations, the influences of *η<sup>c</sup>* on *W*, *η*, *P*, *E* and *π<sup>t</sup>* are the same as those of *η<sup>t</sup>* on *W*, *η*, *P*, *E* and *πt*. When *η<sup>t</sup>* = 0.7 and *π* = 32.8, *W* = *P* = 0. However, the impacts of *η<sup>c</sup>* on *W*, *η*, *P* and *E* are less than those of *η<sup>t</sup>* on *W*,

*η*, *P*, *E*. The effect of *η<sup>c</sup>* on *π<sup>t</sup>* is more significant than that of *η<sup>t</sup>* on *πt*. *η<sup>c</sup>* has little effect on *v*5/*v*1. In the actual design process, it is suggested that *η<sup>t</sup>* should be given priority.

To further explain the difference between the models in this paper and Ref. [101], the comparison of *W* under the variable and constant *π* is shown in Figure 7. As shown in Figure 7, *W* increases and then decreases as *π* increases in both cases; that is, the qualitative law is the same. However, there is an apparent quantitative difference between the two points. Under the constant *π*, *W* corresponding to the constant *π* is always greater than *W* conforming to the variable *π*. Similarly, there are quantitative differences in *η*, *P* and *E* under the variable and constant *π*. The model whose *π* is variable is more realistic.

**Figure 7.** Comparison of *W* under the variable and constant *π*.

*3.2. Performance Optimizations for Each Single Objective*

With four performance indicators as the OPOs, respectively, the HCDs are optimized under the condition of given total heat conductance (*UT*). The optimal results under different OPOs are compared. The HCDs among the RCC, CCC, and precooler are:

$$
\mu\_H = \mathcal{U}\_H / \mathcal{U}\_T \; \; \mu\_{H1} = \mathcal{U}\_{H1} / \mathcal{U}\_{T} \; \; \mu\_L = \mathcal{U}\_L / \mathcal{U}\_T \tag{78}
$$

The HCDs are must larger than 0, the sum of them is 1, and 2 ≤ *π* ≤ 50.

Figure 8 shows the flowchart of HCD optimization. The steps are as follows:


**Figure 8.** Flowchart of HCD optimization.

#### 3.2.1. Optimizations of Each Single Objective

The optimization results of four performance indicators are similar. The optimization results with *η* as the performance indicator will be mainly discussed herein, while the results with *W*, *P* and *E* as the performance indicators are briefly discussed. The relationships of the optimal thermal efficiency (*η*opt) and the corresponding dimensionless power output (*Wη*opt) versus *π* are shown in Figure 9. The relationships of the corresponding dimensionless power density (*Pη*opt) and the corresponding dimensionless ecological function (*Eη*opt) versus *π* are demonstrated in Figure 10. As shown in Figures 9 and 10, *Wη*opt , *η*opt, *Pη*opt and *Eη*opt first rise and then drops as *π* rises, which indicates a parabolic relationship with the downward opening. The corresponding isothermal pressure drop ratio ((*πt*)*η*opt) and dimensionless maximum specific volume ((*v*5/*v*1)*η*opt) versus *π* are shown in Figure 11. (*πt*)*η*opt decreases and then increases as *π* increases. It indicates that there is a *π<sup>t</sup>* that maximizes the degree of isothermal heating in the cycle. (*v*5/*v*1)*η*opt decreases as *π* increases. The relationships of the HCDs ((*uH*)*η*opt , (*uH*1)*η*opt and (*uL*)*η*opt) versus *π* are shown in Figure 12. As *π* increases, (*uH*)*η*opt decreases, (*uH*1)*η*opt increases rapidly and then slowly, and (*uL*)*η*opt decreases first and then increases gradually.

**Figure 9.** Relationships of W*η*opt and *η*opt versus *π*.

**Figure 10.** Relationships of *Pη*opt and *Eη*opt versus *π*.

**Figure 11.** Relationships of (*πt*)*η*opt and (*v*5/*v*1)*η*opt versus *π*.

**Figure 12.** Relationships of (*uH*)*η*opt , (*uH*1)*η*opt and (*uL*)*η*opt versus *π*.

By numerical calculations, *W*opt, *ηW*opt , *PW*opt , *EW*opt ,*WP*opt ,.*ηP*opt ., *P*opt, *EP*opt , *WE*opt , *ηE*opt , *PE*opt and *E*opt increase first and then decrease as *π* increases. As *π* increases, (*πt*)*W*opt , (*πt*)*P*opt and (*πt*)*E*opt reduce first and then increase, and (*πt*)*W*opt , (*πt*)*E*opt , (*πt*)*η*opt and (*πt*)*P*opt reached the minimum successively. As *π* increases, (*v*5/*v*1)*W*opt , and (*v*5/*v*1)*E*opt decline, and their values have little difference. (*uH*)*W*opt , (*uH*)*η*opt , (*uH*)*P*opt and (*uH*)*E*opt decrease as *π* increases, and (*uH*)*η*opt is always the smallest. (*uH*1)*W*opt and (*uH*1)*E*opt rise firstly and then tend to keep constant as *π* rises. (*uH*1)*P*opt first increases then decreases and finally tends to stay stable as *π* rises. (*uL*)*W*opt , (*uL*)*P*opt and (*uL*)*E*opt first increase rapidly and then slowly as *π* increases.

#### 3.2.2. Influences of Temperature Ratios on Optimization Results

With *η* as the performance indicator, the influences of the temperature ratios on the optimization results are discussed. The relationship of the maximum thermal efficiency (*η*max) versus *τH*<sup>1</sup> and *τH*<sup>3</sup> is shown in Figure 13. According to Figure 12, the surface is divided into three parts by line *τH*<sup>3</sup> = *τH*<sup>1</sup> + 0.27 (the correlation coefficient is *r*<sup>1</sup> = 0.9969) and *τH*<sup>3</sup> = 1.2*τH*<sup>1</sup> + 0.1 (the correlation coefficient is *r*<sup>2</sup> = 1.0000). *τH*<sup>1</sup> has little influence on *η*max. When *τH*<sup>3</sup> < 1.2*τH*<sup>1</sup> + 0.1, *η*max increases as *τH*<sup>3</sup> increases; when *τH*<sup>3</sup> > 1.2*τH*<sup>1</sup> + 0.1, *τH*<sup>3</sup> has little impact on *η*max. It is recommended to magnify *τH*1.

**Figure 13.** Relationships of *η*max versus *τH*<sup>1</sup> and *τH*3.

By numerical calculations, the surface is divided into three parts by line *τH*<sup>3</sup> = 0.84*τH*<sup>1</sup> + 0.41 (the correlation coefficient is *r*<sup>1</sup> = 0.9973) and *τH*<sup>3</sup> = 1.2*τH*<sup>1</sup> + 0.23 (the correlation coefficient is *r*<sup>2</sup> = 0.9988) with *W* as the performance indicator. The surface is divided into three parts by line *τH*<sup>3</sup> = 0.78*τH*<sup>1</sup> + 0.6 (the correlation coefficient is *r*<sup>1</sup> = 0.9574) and *τH*<sup>3</sup> = 1.2*τH*<sup>1</sup> + 0.33 (the correlation coefficient is *r*<sup>2</sup> = 0.9991) with *P* as the performance indicator. The surface is divided into three parts by line *τH*<sup>3</sup> = 0.93*τH*<sup>1</sup> + 0.058 (the correlation coefficient is *r*<sup>1</sup> = 0.9978) and *τH*<sup>3</sup> = 1.1*τH*<sup>1</sup> + 0.41 (the correlation coefficient is *r*<sup>2</sup> = 0.9990) with *E* as the performance indicator. In practice, the difference between *τH*<sup>1</sup> and *τH*<sup>3</sup> should be controlled and should not be too large.

3.2.3. Influences of the Compressor and the Turbine's Irreversibilities on Optimization Results

With the four performance indicators as OPOs, respectively, the influences of *η<sup>c</sup>* and *η<sup>t</sup>* on optimization results are considered, and the thermodynamic parameters under various optimal performance indicators are compared. Figures 14 and 15 show relationships of *W* and *π* under various optimal performance indicators versus *η<sup>c</sup>* and *ηt*, respectively *W*max, *P*max, and *E*max are the maximum dimensionless power output, maximum dimensionless power density, and maximum dimensionless ecological function, respectively. When *W*max, *η*max, *P*max, and *E*max are used as subscripts, they indicate the corresponding values at *W*max, *η*max, *P*max, and *E*max points.

**Figure 14.** Relationships of *W* under various optimal performance indexes versus *ηc* and *ηt*.

**Figure 15.** Relationships of *π* under various optimal performance indexes versus *ηc* and *ηt*.

As shown in Figure 14, *W* under various optimal performance indicators increases as *η<sup>c</sup>* or *η<sup>t</sup>* increases. When *η<sup>c</sup>* and *η<sup>t</sup>* both approach 1, *Wη*max first increases and then decreases as *η<sup>c</sup>* or *η<sup>t</sup>* increases. When *η<sup>c</sup>* = *η<sup>t</sup>* = 1, *η* rises monotonically as *π* gains, and there is no maximum value. In the case of the same *η<sup>c</sup>* and *ηt*, there is *W*max > *WE*max > *WP*max > *Wη*max . As shown in Figure 15, *π* under various optimal performance indicators all increase as *η<sup>c</sup>* or *η<sup>t</sup>* increases. But the influence of *η<sup>t</sup>* on *π* is more significant than that of *η<sup>c</sup>* on *π*. When *η<sup>c</sup>* and *η<sup>t</sup>* both approach 1, *πη*max is always 50. Because the upper limit of *π* is 50. In the case of the same *η<sup>c</sup>* and *ηt*, there is *πη*max > *πP*max > *πE*max > *πW*max . The given range of *π* is 2 ≤ *π* ≤ 50, so when *π*= 50, the trends of *Wη*max and *πη*max change significantly.

By numerical calculations, *η*, *P*, and *E* under various optimal performance indicators increases as *η<sup>c</sup>* or *η<sup>t</sup>* increases. When *η<sup>c</sup>* and *η<sup>t</sup>* both approach 1, *Pη*max and *Eη*max first rises and then drops as *η<sup>c</sup>* or *η<sup>t</sup>* rises. In the same *η<sup>c</sup>* and *ηt*, there are *η*max > *ηP*max > *ηE*max > *ηW*max , *P*max > *PE*max > *PW*max > *Pη*max , (when *η<sup>c</sup>* and *η<sup>t</sup>* both tend to 1, the relationship does not work) and *E*max > *EP*max > *EW*max > *Eη*max (the difference between *EP*max and *EW*max is very small).

The calculations also show that the thermal capacitance rate matchings among the VTHRs and working fluid have influences on the cycle performance. *W*max, *η*max, *P*max, and *E*max increase first and then keep constants as *CH*/*Cw f* or *CH*1/*Cw f* increases, and the effects of *CH*/*Cw f* on *W*max, *η*max, *P*max, and *E*max are more significant than that of *CH*1/*Cw f* .

#### **4. Multi-Objective Optimization**

*4.1. Optimization Algorithm and Decision-Making Methods*

It is impossible to achieve the maximums of *W*, *η*, *P*, and *E* under the same *π*. It shows that there is a contradiction among the four performance indicators. The multi-objective optimization problem is solved by applying the NSGA-II algorithm [99,100,102,105–125]. The detailed optimization process is shown in Figure 16. The Pareto frontier of the cycle performance is obtained by taking *W*, *η*, *P*, and *E* as OPOs, using the NSGA-II algorithm. The optimal scheme is selected by using the LINMAP, TOPSIS, and Shannon Entropy methods [99,102], and the algorithm of "gamultiobj" in MATLAB is based on the NSGA-II algorithm. The calculations are assisted by applying the "gamultiobj", and the corresponding Pareto frontier could be obtained. The parameter settings of "gamultiobj" are listed in Table 1.

**Figure 16.** Flowchart of NSGA-II algorithm.

**Table 1.** Parameter settings of "gamultiobj".


The positive and negative ideal points are the optimal and inferior schemes of each performance indicator. The LINMAP method is the Euclidian distance between each scheme and the positive ideal point, among which the one with the smallest distance is the best scheme. Suppose that the Pareto front contains *n* feasible solutions, and each viable solution contains *m* objective values *Fij*(1 ≤ *i* ≤ *m* and 1 ≤ *j* ≤ *n*). After normalizing *Fij*, the value *Bij* is:

$$B\_{ij} = F\_{ij} / \sqrt{\sum\_{i=1}^{n} F\_{ij}^2} \tag{79}$$

The weight of the *j*-th OPO is *w*LINMAP *<sup>j</sup>* , and the weighted value of *Bij* is *Gij*:

$$G\_{i\rangle} = w\_j^{\text{LINMAP}} \cdot B\_{i\bar{j}} \tag{80}$$

The *j*-th objective of the positive ideal point is normalized and weighted, and the corresponding value is *G*positive *<sup>j</sup>* . The Euclidean distance between the *i*-th feasible solution and the positive ideal point is *ED*<sup>+</sup> *i* :

$$ED\_i^+ = \sqrt{\sum\_{j=1}^m \left(G\_{ij} - G\_j^{\text{Positive}}\right)^2} \tag{81}$$

The best viable solution to the LINMAP method is *i*opt:

$$d\_{\rm opt} \in \min\{ED\_i^+\} \tag{82}$$

The TOPSIS method considers the Euclidean distance among each scheme and the positive and negative ideal points comprehensively, to further obtain the best scheme. The weight of the *j*-th OPO is *w*TOPSIS *<sup>j</sup>* , and the weighted value of *Bij* is *Gij*:

$$G\_{i\circ} = w\_j^{\text{TOPSIS}} \cdot B\_{i\circ} \tag{83}$$

The *j*-th objective of the negative ideal point is normalized and weighted, and the corresponding value is *G*negative *<sup>j</sup>* . The Euclidean distance between the *i*-th feasible solution and the negative ideal point is *ED*− *i* :

$$ED\_i^- = \sqrt{\sum\_{j=1}^m \left(G\_{ij} - G\_j^{\text{ncgutive}}\right)^2} \tag{84}$$

The best feasible solution of the TOPSIS method is *i*opt:

$$i\_{\rm opt} \in \min \{ \frac{ED\_i^-}{ED\_i^+ + ED\_i^-} \} \tag{85}$$

The Shannon Entropy method is a method to get the weight of multi-attribute decisionmaking.

After normalization of *Fij*, *Pij* is obtained:

$$P\_{ij} = F\_{ij} / \sum\_{i=1}^{n} F\_{ij} \tag{86}$$

The Shannon Entropy and weight of the *j*-th OPO are:

$$SE\_j = -\frac{1}{\ln n} \sum\_{i=1}^{n} P\_{ij} \ln P\_{ij} \tag{87}$$

$$w\_j^{\text{Shannon Entropy}} = (1 - SE\_j) / \sum\_{j=1}^{n} (1 - SE\_j) \tag{88}$$

The best feasible solution of the TOPSIS method is *i*opt:

$$\dot{u}\_{\text{opt}} \in \min \left\{ P\_{ij} \cdot w\_j^{\text{Shannon Entropy}} \right\} \tag{89}$$

The deviation index *D* is defined as:

$$D = \frac{\sqrt{\sum\_{j=1}^{m} \left(G\_{i\_{\rm opt}} - G\_j^{\rm positive}\right)^2}}{\sqrt{\sum\_{j=1}^{m} \left(G\_{i\_{\rm opt}} - G\_j^{\rm positive}\right)^2} + \sqrt{\sum\_{j=1}^{m} \left(G\_{i\_{\rm opt}} - G\_j^{\rm positive}\right)^2}}\tag{90}$$

In this paper, *w*LINMAP *<sup>j</sup>* = *<sup>w</sup>*TOPSIS *<sup>j</sup>* = 1 is chosen for the convenience of calculation.

#### *4.2. Multi-Objective Optimization Results*

Figure 17 shows the Pareto frontier and optimal schemes corresponding to the four objectives (*W*, *η*, *P* and *E*) optimization. The color on the Pareto frontier denotes the size of *E*. To facilitate the observation of the changing relationships among the objectives, the pure red projection indicates the changing relationship between *W* and *η*. The pure green projection shows the changing relationship between *W* and *P*, and the pure blue projection indicates the changing relationship between *η* and *P*. It is easy to know that *W* and *η*, *W* and *P*, *η* and *P* are all parabolic-like relationships with the opening downward. To analyze the influence of the corresponding optimization variables ((*uH*)opt, (*uH*1)opt, (*uL*)opt and *π*opt) on cycle performance, the distributions of (*uH*)opt, (*uH*1)opt, (*uL*)opt and *π*opt within the Pareto frontier's value range are shown in Figures 18–21. As shown in Figure 18, the value range of (*uH*)opt is 0–1, but its distribution is between 0.167 and 0.272. As (*uH*)opt increases, *W*, *P*, and *E* gradually increase, but *η* gradually decreases. As shown in Figure 19, the value range of (*uH*1)opt is 0–1, but its distribution is between 0.151 and 0.181. As (*uH*1)opt increases, *W*, *P*, and *E* gradually decrease, but the changing trend of *η* is not apparent. As shown in Figure 20, the value range of (*uL*)opt is 0–1, but its distribution is between 0.568 and 0.662. As (*uL*)opt increases, *W*, *P*, and *E* gradually decrease, but the changing trend of *η* is not apparent. As shown in Figure 21, the value range of *π*opt is 2–50, but its distribution is between 9.692 and 24.426. As *π*opt increases, *W* gradually decreases, *η* gradually increases, and *P* and *E* rise and then reduce.

**Figure 17.** Pareto frontier and optimal schemes corresponding to the four objectives (*W*, *η*, *P* and *E* ) optimization.

**Figure 18.** Distribution of (*uH*)opt within the value range in the Pareto frontier.

**Figure 19.** Distribution of (*uH*1)opt within the value range in the Pareto frontier.

**Figure 20.** Distribution of (*uL*)opt within the value range in the Pareto frontier.

**Figure 21.** Distribution of *π*opt within the value range in the Pareto frontier.

The Pareto frontier includes a series of non-inferior solutions, so the appropriate solution must be chosen according to the actual situation. The results of the triple- and double-objective optimizations are further discussed to compare the results of multiobjective optimizations more comprehensively. The comparison of the optimal schemes gotten by single- and double-, triple-, and quadruple-objective optimizations are listed in Table 2. The deviation index (*D*) is applied to represent the proximity between the optimal scheme and the positive ideal point. The appropriate optimal schemes are chosen by using the three methods. For the quadruple-objective optimization, *W*, *η*, *P*, and *E* corresponding to the positive ideal point are the maximum of the single-objective optimization. It indicates that the Pareto frontier includes all single-objective optimization results. The *D* obtained by the Shannon Entropy method is significantly smaller than that obtained by the LINMAP and TOPSIS methods. Simultaneously, it can be found that the *D* obtained by the Shannon Entropy method is the same as that with *E* as the OPO. For the triple-objective optimization, the triple-objective (*W*, *η* and *E*) optimization *D* obtained by the LINMAP or TOPSIS method is the smallest. For the double-objective optimization, the double-objective (*W* and *P*) optimization *D* obtained by the LINMAP method is the smallest. For the single-objective optimization, the *D* corresponding to *E*max is the smallest. For single- and double-, triple-, and quadruple-objective optimizations, the double-objective (*W* and *P*) optimization *D* obtained by the LINMAP method is the smallest.


*Entropy* **2021** , *23*, 282


**Table 2.** *Cont.*

#### **5. Conclusions**

Based on FTT, an improved irreversible closed modified simple BCY model with one IHP and coupled to VTHRs is established and optimized with four performance indicators as OPOs, respectively. The optimization results are compared, and the influences of compressor and turbine efficiencies on optimization results are analyzed. Finally, the cycle is optimized, and the corresponding Pareto frontier is gained by adopting the NSGA-II algorithm. Based on three different methods, the optimal scheme is gotten from the Pareto frontier. The results obtained in this paper reveal the original results in Refs. [10–12], which were the initial work of the FTT theory. The main results are summarized:


**Author Contributions:** Conceptualization: L.C. and H.F.; funding acquisition: L.C.; methodology: C.T.; software: C.T. and Y.G.; validation: L.C. and Y.G; writing—original draft: C.T. and H.F.; writing—review and editing: L.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work is supported by the National Natural Science Foundation of China (Grant No. 51779262).

**Acknowledgments:** The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Nomenclature**

*a*, *x*, *y* Intermediate variables



#### **Abbreviations**


#### **References**

