**About the Editor**

**Michel Feidt** is an emeritus professor at the University of Lorraine, France, where he has spent his entire career in education and research. His main interests are thermodynamics and energy. He is a specialist in infinite physical dimensions optimal thermodynamics (FDOT), which he considers from a fundamental point of view, illustrating the necessity of considering irreversibility to optimize systems and processes and to characterize upper bound efficiencies. He has published many articles in journals and books: more than 120 papers and more than 5 books. He actively participates in numerous international and national conferences on the same subject. He has developed 55 final contract reports and has advised 43 theses. He has been a member of more than 110 doctoral committees. He is a member of the scientific committee of more than five scientific journals and editor-in-chief of one journal.

## *Editorial* **The Carnot Cycle and Heat Engine Fundamentals and Applications II**

**Michel Feidt**

Laboratory of Energetics, Theoretical and Applied Mechanics (LEMTA), URA CNRS 7563, University of Lorraine, 54518 Vandoeuvre-lès-Nancy, France; michel.feidt@univ-lorraine.fr

This editorial introduces the second Special Issue entitled "Carnot Cycle and Heat Engine Fundamentals and Applications II" https://www.mdpi.com/si/entropy/Carnot\_ Cycle\_II (accessed on 29 January 2022).

The editorial of this Special Issue comes after the review process. Nine papers have been published between 26 February 2021 and 4 January 2022 due to the COVID-19 pandemic. These papers are listed hereafter in the inverse order of date of publication. Thanks to all the authors for the various viewpoints expressed that unveil fundamental and application aspects of the Carnot cycle and heat engines.

Authors are from Europe (four papers) and China (five papers). Each paper has been viewed by 400 to 1100 persons, except the last published one. Four papers have been presently cited 8 to 20 times.

Five papers address heat engines and Carnot configurations [1–5]. Papers by [2,4,5] concern, respectively, diesel engine, Lenoir, and Brayton cycles. The papers by [1,2] are related to Carnot engines. However, these five papers address real, irreversible cases. Three papers from Chinese authors [2,4,5] deal with finite-time thermodynamics (FTT). Papers by [2,5] use numerical methods such as genetic algorithm NASCA II (through LINMAP method, TOPSIS method, and Shannon entropy method) to optimize engines. The various objectives considered include power, power density, ecological function, and first law efficiency.

The paper that discusses the Lenoir cycle is from a more conventional point of view. It deals with the steady flow (such as Chambadal's original modeling). Objectives are power and first law efficiency. The corresponding allocation of heat transfer conductance is proposed, due to finite size constraints (i.e., the Utotal imposed).

In [2], the authors consider the optimization of an irreversible Carnot engine, comparing the FTT approach to the finite speed thermodynamics approach (FST). The direct method combined with the first law efficiency takes irreversibility into account (heat transfer gradients, pressure losses, and mechanical frictions). The main results include the following:


Paper [1] concerns the modified Chambadal model of Carnot engines. It, too, addresses irreversibility but from a global point of view. This paper completes and improves the one proposed in the preceding Special Issue. A sequential optimization corresponding to various finite physical dimensions constraints is developed with the three objectives of energy, first law efficiency, and power. Two new concepts of entropic action are proposed and used—entropic action relative to production of entropy and entropic action relative to the transfer of entropy.

**Citation:** Feidt, M. The Carnot Cycle and Heat Engine Fundamentals and Applications II. *Entropy* **2022**, *24*, 230. https://doi.org/10.3390/e24020230

Received: 27 January 2022 Accepted: 28 January 2022 Published: 2 February 2022

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

**Copyright:** © 2022 by the author. 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/).

Papers by [6,7] extend the configuration from engines to reverse cycle machines including Stirling refrigerating machine [6] and Brayton refrigerating machine [7]. The paper by [7] combines, in fact, direct and inverse Brayton cycles, constituting more of a system, with regeneration purposes (regeneration before the inverse cycle). Constraints regarding pressure losses and size are considered.

The study by [6] is, in fact, related to the paper by [7], published in the preceding Special Issue: It discusses a finite physical dimension in a Stirling refrigerating machine according to Schmidt modeling. The paper uses entropy and exergy analysis. The most important irreversibility mechanisms are thermal ones and, more precisely, those due to regeneration.

Papers of [8,9] are specific but very interesting.

In [8], the authors discuss the chemical aspects of entropy and exergy analysis, including reconsideration of concepts and definitions relating the entropy–exergy relationship, with applications in industrial engineering and biotechnologies. The main objective is to evaluate the performance associated with all interactions between the system and the external environment. This is a crucial challenge today due to environmental concerns.

Paper by [9] is related to a very important and up-to-date subject—superconducting quantum circuits. It concerns a new approach mixing finite-time and quantum thermodynamics: quantum heat engine cycle. Closely linked to these fundamental aspects are corresponding applications for quantum computers.

To conclude, this second Special Issue confirms and improves the preceding one in terms of the following aspects:


Perhaps these features could pave the way toward a third Special Issue, to expand and build upon concepts and approaches presented thus far.

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

**Acknowledgments:** We express our thanks to the authors of the above contributions, and to the journal *Entropy* and MDPI for their support during this Special Issue.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **References**


## *Article* **A New Step in the Optimization of the Chambadal Model of the Carnot Engine**

**Michel Feidt <sup>1</sup> and Monica Costea 2,\***


**\*** Correspondence: monica.costea@upb.ro; Tel.: +40-021-402-9339

**Abstract:** This paper presents a new step in the optimization of the Chambadal model of the Carnot engine. It allows a sequential optimization of a model with internal irreversibilities. The optimization is performed successively with respect to various objectives (e.g., energy, efficiency, or power when introducing the duration of the cycle). New complementary results are reported, generalizing those recently published in the literature. In addition, the new concept of entropy production action is proposed. This concept induces new optimums concerning energy and power in the presence of internal irreversibilities inversely proportional to the cycle or transformation durations. This promising approach is related to applications but also to fundamental aspects.

**Keywords:** optimization; Carnot engine; Chambadal model; entropy production action; efficiency at maximum power

#### **1. Introduction**

Sadi Carnot had a crucial contribution to thermostatics that designated him as a cofounding researcher of equilibrium thermodynamics. He has shown that the efficiency of a thermo-mechanical engine is bounded by the Carnot efficiency *η<sup>C</sup>* [1]. Assuming an isothermal source at *THS*, and an isothermal sink at *TCS* < *THS*, and in between the cycle composed by two isothermals in perfect thermal contact with the source and sink, and two isentropics, he obtained:

$$
\eta\_{\mathbb{C}} = 1 - \frac{T\_{\mathbb{C}S}}{T\_{HS}}.\tag{1}
$$

Since that time, many papers have used the keyword "Carnot engine" (1290 papers from Web of Science on 17 September 2021). That same day on Web of Science, we noted 104 papers related to the keyword "Carnot efficiency".

Among these papers, some are related to the connection between energy, efficiency, and power optimization. The most cited paper is probably that of Curzon and Ahlborn [2,3]. These authors proposed in 1975 an expression of the efficiency according to the first law of thermodynamics *ηI*(*MaxW*) at the maximum mechanical energy and at the maximum power . *W* for the endo-reversible configuration of the Carnot cycle (no internal irreversibility for the converter in contact with two isothermal heat reservoirs):

$$
\eta\_{I,endo}(\text{MaxW}) = 1 - \sqrt{\frac{T\_{CS}}{T\_{HS}}} \tag{2}
$$

This result is well-known as the *nice radical*, and it has been recently reconsidered in the previous Special Issue *Carnot Cycle and Heat Engine Fundamentals and Applications*

**Citation:** Feidt, M.; Costea, M. A New Step in the Optimization of the Chambadal Model of the Carnot Engine. *Entropy* **2022**, *24*, 84. https:// doi.org/10.3390/e24010084

Academic Editor: José Miguel Mateos Roco

Received: 21 November 2021 Accepted: 24 December 2021 Published: 4 January 2022

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

**Copyright:** © 2022 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/).

<sup>060042</sup> Bucharest, Romania

*I* [3] and particularly in [4]. This last paper reports on the progress in Carnot and Chambadal modeling of thermomechanical engines by considering entropy production and heat transfer entropy in the adiabatic case (without heat losses).

The proposed paper gives back the basis of the modeling and a summary of the main results obtained recently for an endo-irreversible Carnot engine. Furthermore, the performance analysis of an extended Chambadal configuration is considered by including the converter irreversibilities. Emphasis is placed on the entropy production method, which is preferred over the ratio method.

#### **2. Summary of Obtained Results for Carnot Endo-Irreversible Configuration**

The consideration of *endo-irreversible* Carnot engine modeling was recently developed [5]. The approach considering as a reference the heat transfer entropy released at the sink Δ*SS* (maximum entropy available at the source in the reversible case) [5] confirmed that the work per cycle results (see Appendix A):

$$\mathcal{W} = (T\_{HS} - T\_{CS})(\Delta S\_S - \Delta S\_I), \tag{3}$$

where Δ*SI* is the entropy production due to the internal irreversibilities of the cycle throughout the four thermodynamic transformations (two adiabatic and two isothermal ones).

For an engine without thermal losses, the following expression of the thermal efficiency was retrieved:

$$
\eta\_I = \eta\_{\mathbb{C}} (1 - d\_I),
\tag{4}
$$

where *dI* = <sup>Δ</sup>*SI* <sup>Δ</sup>*SS* is a coefficient of the converter's internal irreversibility during the cycle.

This parameter was introduced by Novikov [6] and Ibrahim et al. [7] in slightly different forms.

The reversible limit (*dI* = 0) in Equation (4) restores the Carnot cycle efficiency associated with equilibrium thermodynamics.

Since the reversibility is unattainable, it appears that the optimization (maximization) of the mechanical energy at the given parameters (Δ*SS*, *THS*, and *TCS*) is related to the minimization of the entropy production (as was proposed by Gouy [8]).

The assumption that each of the four transformations of the endo-irreversible cycle takes place with a duration *τ<sup>i</sup>* (*i* = 1–4), leading to the inverse proportionality to *τ<sup>i</sup>* of the corresponding entropy production:

$$
\Delta S\_{Ii} = \frac{C\_{Ii}}{\tau\_i} \,\,\,\,\tag{5}
$$

where *CIi* represents the irreversibility coefficients, whose unit is Js/K [5].

These coefficients are *irreversible entropic actions* by analogy to the energy (mechanical) action (Js).

By performing cycle energy optimization using the Lagrange multipliers method with the constraint of the cycle's finite time duration *τ,* one obtains the maximum work per cycle [5]:

$$\text{Max}\_1 W = W\_{endo} - \frac{\Delta T\_S}{\tau} \left(\sum\_i \sqrt{\mathbb{C}\_{Ii}}\right)^2 \tag{6}$$

where Δ*TS* = *THS* − *TCS*.

The efficiency at the maximum finite time work becomes

$$\eta\_I(\text{Max}\_1\mathcal{W}) = \eta\_\mathbb{C} \left( 1 - \frac{\left(\sum\_i \sqrt{\mathbb{C}\_{li}}\right)^2}{\pi \cdot \Delta S\_S} \right), \tag{7}$$

where *τ*Δ*SS* is the available entropic transfer action of the cycle.

The new result provided by Equation (7) gives back the Carnot efficiency limit for the reversible case (*CIi* = 0). These calculations have been pursued for the case of power optimization, where Δ*SS*, *THS*, and *TCS* remain parameters. It was shown that a value of the cycle duration *<sup>τ</sup>*<sup>∗</sup> corresponding to *Max* . *W*, the mean power output over the cycle, exists, and it is expressed as

$$
\pi^\* = 8 \frac{C\_{Ii}}{\Delta S\_S},
\tag{8}
$$

and

$$\text{Max}\overline{\dot{W}} = \frac{\Delta T\_S \cdot \Delta S\_S}{16 \text{ C}\_{Ii}} \,\text{.}\tag{9}$$

Equation (9) proves that *Max* . *W* is a decreasing function of the total entropic action of the cycle and that the associated efficiency with the maximum of the mean power corresponds to half the Carnot efficiency, as appeared repeatedly in some recent works [9–11].

#### **3. Summary of the Obtained Results for the Chambadal Configuration**

In the present paper, we intend to reconsider the approach of the Chambadal model of a Carnot engine [12]. This configuration is common for thermomechanical engines, since the cold sink mainly refers to the environment (i.e., the atmosphere or water sink). This corresponds to the Chambadal approach (Figure 1), with a temperature gradient at the hot source (*THS*, *TH*) but with perfect thermal contact at the sink (*TCS* or *T*<sup>0</sup> at ambient temperature).

**Figure 1.** Representation of the associated cycle with the Chambadal engine in a *T*-*S* diagram.

We propose here to extend the results (Equations (6)–(9)) to enhance the Chambadal configuration modeling. This extension starts from the endo-irreversible case, to which external irreversibilities due to heat transfer between the hot finite source and the converter are added. Thus, the new results obtained complete the endo-irreversible Carnot model [5] and an earlier paper on Chambadal configuration [12].

#### *3.1. The Modified Chambadal Engine*

To help understand the extension of the modeling in Section 3, we report here the case with the following hypothesis:


$$Q\_H = G\_H (T\_{HS} - T\_H)\_\prime \tag{10}$$

where *GH* is the heat transfer conductance expressed by *GH* = *KHτ* when we consider the mean value over the cycle duration *τ* or *GH* = *K <sup>H</sup>τ<sup>H</sup>* when we consider the mean value over the isothermal heat transfer at the hot source (as was performed by Curzon and Ahlborn [2]).

Equation (10) corresponds to the heat expense of the engine.

Note that other heat transfer laws, namely the Stefan–Boltzmann radiation law, the Dulong–Petit law, and another phenomenological heat transfer law can be considered in the maximum power regime search [13];

3. *Presence of irreversibility* in the converter (internal irreversibility).

Two approaches are proposed in the literature, which introduce the internal irreversibility of the engine by (1) the *irreversibility ratio IH*, [6,7], respectively (2) the *entropy production over the cycle* Δ*SI*, [5].

We preconized this second approach for a long time. We also note that the original model of Chambadal is endo-reversible [14]. Hence, we prefer to name the present model the "modified Chambadal model" due to some other differences that will be specified hereafter.

Note that only the second approach regarding the presence of irreversibilities in the converter will be considered in the following section.

*3.2. Optimization of the Work per Cycle of the Modified Chambadal Engine with the Entropy Production Method*

The first law of thermodynamics applied to the cycle implies conservation of energy, written as

$$\mathcal{W} = \mathcal{Q}\_{\rm conv} - Q\_{\rm S} \tag{11}$$

where *Qconv* and *QS* are defined in Appendix A.

One supposes here that Δ*SI* is a parameter representing the total production of entropy over the cycle composed by four irreversible transformations. Thus, the entropy balance corresponds to

$$\frac{Q\_{conv}}{T\_H} + \Delta S\_I = \frac{Q\_S}{T\_0}.\tag{12}$$

By combining Equations (11) and (12), we easily obtained

$$\mathcal{W} = Q\_{\rm conv} \left( 1 - \frac{T\_0}{T\_H} \right) - T\_0 \Delta S\_I. \tag{13}$$

If *Qconv* (Δ*Sconv*) is a given parameter, we retrieve the Gouy-Stodola theorem stating that *Max W* corresponds to min Δ*SI* with the known consequences reported in Section 4.1.

#### *3.3. Optimization of the Work per Cycle of the Modified Chambadal Engine with the Heat Transfer Constraint*

In this case, the energy balance between the source and isothermal transformation implies the combination of Equations (13) and (A1):

$$\mathcal{W} = \left(Q\_H - T\_H \Delta S\_{IH}\right) \left(1 - \frac{T\_0}{T\_H}\right) - T\_0 \Delta S\_I. \tag{14}$$

Knowing *QH* from Equation (10), one obtains

$$\mathcal{W} = G\_H (T\_{HS} - T\_H) \left( 1 - \frac{T\_0}{T\_H} \right) - T\_H \Delta S\_{IH} - T\_0 \Delta S\_{I\prime}^{\prime} \tag{15}$$

where Δ*S <sup>I</sup>* = Δ*SIEx* + Δ*SIC* + Δ*SICo*.

The maximum of *W* with respect to *TH* is obtained for

$$T\_H^\* = \sqrt{\frac{T\_{HS} T\_0}{1 + s\_I}},\tag{16}$$

where *sI* = <sup>Δ</sup>*SIH GH* , a specific ratio relative to the irreversible isothermal transformation *TH*. Finally, the expression of *Max*1*W* yields

$$\text{Max}\_1 \mathcal{W} = G\_H \left( \sqrt{T\_{HS}} - \sqrt{(1 + s\_I)T\_0} \right)^2 - T\_0 \Delta S\_I. \tag{17}$$

#### **4. Complement to the Previous Results**

Now, we will consider the time variable related to entropy production for each thermodynamic transformation, defined as <sup>Δ</sup>*SIi* = *CIi <sup>τ</sup><sup>i</sup>* . This form of the entropy production satisfying the second law induces that the entropy production method is well adapted to subsequent optimizations of energy and power as well.

#### *4.1. Work Optimization Relative to the Time Variables*

The expression of *Max*1*W* with *GH* as an extensive parameter (Equation (17)) shows that *Max*1*W* is always the optimum in the endo-reversible case. Nevertheless, if there are separate irreversibilities for each cycle transformation (as is the case with finite entropic actions), the irreversibility on the high temperature isotherm possesses a specific role (see Equation (17) and the *sI* ratio).

The constraint on the transformation duration or preferably frequencies *fi* (finite cycle duration) allows one to seek for the optimal transformation duration allocation (see Appendix B for the derivation).

We obtained *Max*2*W* for the following optimal durations:

$$
\pi\_H ^\ast = \sqrt{\sqrt{T\_0 T\_{HS}} \frac{\mathbb{C}\_{IH}}{\lambda}} \,\,\,\,\tag{18}
$$

and

$$
\pi\_i^\* = \sqrt{\frac{T\_0 C\_{Ii}}{\lambda}},
\tag{19}
$$

where *λ* is given in Appendix B and *i* = *Ex*, *C*, *Co.*

Thus, the second optimization of *W* (see Appendix B) leads to

$$
\hbar \mathbf{A} \mathbf{x}\_2 \mathbf{W} \approx \mathbf{W}\_{\rm enddo} - \frac{T\_0}{\tau} \mathbf{N}^2 \,. \tag{20}
$$

Furthermore, a third sequential optimization could be performed by considering the finite entropic action as a new constraint. This case is not developed here for brevity reasons.

#### *4.2. Power Optimization in the Case of a Finite Heat Source (When GH Is the Parameter)*

The mean power of the modified Chambadal cycle for the condition of maximum work *Max*2*W* is defined by

$$
\overline{\dot{\mathcal{W}}}(\text{Max}\_2\mathcal{W}) = \frac{\mathcal{W}\_{\text{cudo}}}{\tau} \text{--} \frac{T\_0}{\tau^2} \mathcal{N}^2 \,\text{.}\tag{21}
$$

where *Wendo* = *GH* √*THS* – <sup>√</sup>*T*<sup>0</sup> <sup>2</sup> is the mechanical work output of the endo-reversible engine.

The power is maximized with respect to the cycle period *τ*. Thus, the expression of the optimum period is

$$
\pi^\* = \frac{2T\_0 N^2}{W\_{cudo}}.\tag{22}
$$

This expression is analogous to the similar results obtained in [5], leading to

$$\text{Max}\overleftarrow{\dot{\mathcal{W}}} = \frac{\left|\mathcal{W}\_{\text{endo}}\right|^{2}}{4T\_{0}\mathcal{N}^{2}}\,. \tag{23}$$

The action of entropy production appearing in *N* diminishes the mean power of the engine. At the given endo-reversible work, the maximum power corresponds to the minimum of the *N* function, depending on the four entropy actions of the cycle, such that

$$N = \sqrt{\frac{T\_0}{T\_{HS}} \mathcal{C}\_{IH}} + \sqrt{\mathcal{C}\_{IEx}} + \sqrt{\mathcal{C}\_{IC}} + \sqrt{\mathcal{C}\_{ICo}} \, . \tag{24}$$

The main difference between Equation (23) and the previous results [5] comes from the imperfect heat transfer between the source and the converter in the Chambadal model.

#### **5. Discussion**

This paper proposed that the Special Issue *Carnot Cycle and Heat Engine Fundamentals and Applications II* completes the previous paper [12] published in Special Issue 1 and adds new results to a recently published paper [5].

Whatever variable is chosen for the modified Chambadal model work optimization (*TH* or Δ*S*), the same optimum for the work per cycle is obtained with parameters *GH*, *THS*, and *T*0.

It appears that by introducing the duration of each transformation *τ<sup>i</sup>* and the period of the cycle *τ*, the modified Chambadal model satisfies the Gouy-Stodola theorem. At the minimum of entropy production, the optimal durations are dependent on the transformation entropy actions. This result is new to our knowledge.

This new concept [5] allows a new subsequent sequential optimization. The optimal allocation of the entropy action coefficients is slightly different from the equipartition (a new form of the equipartition theorem [15,16]).

Thus, the fundamental aspect related to irreversibilities through the *new concept of entropy production action* seems promising. Furthermore, this new concept could contribute to the improvement of the global system analysis by conducting optimal dimension allocation. In this respect, finite physical dimensions analysis could be a complementary way to correlate with exergy analysis.

Further extensions of this work are foreseen in the near future.

#### **6. Conclusions**

Similarities and differences present in the literature regarding the optimization of energy, first law efficiency, and power of the modified Chambadal engine have been resituated and summarized since the publication of [12].

This approach allows for highlighting the evolution of the obtained results from the reversible Carnot engine case (thermostatics) to the endo-irreversible models related to the approaches of Novikov [6] and Ibrahim et al. [7] or to the entropy production method that we promote.

By generalizing a proposal from Esposito et al. [9] and defining the new concept of entropic action through a coefficient *CI* (Js/K) for the entropy production of transformations all along the cycle, we achieved a new form of power optimization different from the one of Curzon and Ahlborn, since the internal converter irreversibilities and the heat transfer irreversibility between the heat source and converter were accounted for.

The maximum work per cycle was obtained for the irreversible cycle case. It depended on the entropic action coefficient of the four transformations of the cycle *CIi*, after which the power of the engine was sequentially optimized.

An optimal period of the cycle *τ*\* appeared, corresponding to the maximum mean power of the cycle. It generalized the recent published results [5] for a modified Chambadal engine.

This research continues to be developed by our team to obtain more general results.

**Author Contributions:** Conceptualization, M.F.; methodology, M.F.; software, M.C.; validation, M.F. and M.C.; formal analysis, M.F. and M.C.; writing—original draft preparation, M.F.; writing—review and editing, M.C.; visualization, M.F. and M.C.; supervision, M.F.; project administration, M.C. 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.

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

**Appendix A. Work per Cycle of the Modified Chambadal Engine with the Entropy Production Method**

**Figure A1.** Carnot engine cycle with internal irreversibilities along the four transformations of the cycle, illustrated in a *T*-*S* diagram.

It results from Figure A1 that the various heats exchanged over the irreversible cycle (1–2–3–4) are expressed as follows:


$$Q\_{conv} = T\_H(\Delta S\_H - \Delta S\_{IH}).\tag{A1}$$

• *QC* = *T*0Δ*SC*, where Δ*SC* = Δ*SS* − Δ*SIC*.

Note that Δ*SIC* is the entropy production during the irreversible isotherm at *T*<sup>0</sup> and Δ*SS* is the entropy rejected to the sink such that *QS* = *T*0Δ*SS*.

Thus, the entropy balance over the cycle is

$$
\Delta S\_{conv} + \Delta S\_I = \Delta S\_S \tag{A2}
$$

The total entropy production over the cycle Δ*SI* is represented by

$$
\Delta S\_I = \Delta S\_{IH} + \Delta S\_{IEx} + \Delta S\_{IC} + \Delta S\_{ICo\prime} \tag{A3}
$$

where

Δ*SIH* is the entropy production during the isothermal transformation at *TH*, Δ*SIE* is the entropy production during the adiabatic expansion from *TH* to *T*0, Δ*SIC* is the entropy production during the isothermal transformation at *T*0, and Δ*SICo,* is the entropy production during the adiabatic compression from *T*<sup>0</sup> to *TH*.

The energy balance over the cycle for the system comprising the converter, the heat source, and the sink (with the source and sink as perfect thermostats) provides

$$\mathcal{W} = \mathcal{Q}\_{conv} - \mathcal{Q}\_{\mathcal{S}}.\tag{A4}$$

Various forms of mechanical energy are obtainable from this point by combining the preceding relations. Thus, one may express *W* as follows:

1. With Δ*Sconv* as the reference entropy:

$$\mathcal{W} = T\_H \Delta S\_{conv} - T\_0 \Delta S\_{S\prime} \tag{A5}$$

$$\mathcal{W} = (T\_H - T\_0)\Delta S\_{conv} - T\_0 \Delta S\_I. \tag{A6}$$

2. With Δ*SS* as the reference entropy:

$$\mathcal{W} = (T\_H - T\_0)\Delta S\_S - T\_H \Delta S\_I. \tag{A7}$$

3. With Δ*SS* or Δ*SS* as the reference entropy:

$$\mathcal{W} = T\_H(\Delta S\_H - \Delta S\_{IH}) - T\_\mathbb{C}(\Delta S\_\mathbb{C} + \Delta S\_{IC}).\tag{A8}$$

We prefer to choose between Equations (A6) and (A7). Note that Equation (A7) was the one used by Esposito et al. [9].

We use Equation (A6) here because it gave back known results, particularly the Gouy-Stodola theorem, with Δ*Sconv* being a parameter. Thus, the maximum energy occurs when Δ*SI* = 0 such that

$$\mathcal{W}\_{\rm endo} = (T\_H - T\_0) \Delta S\_{\rm conv} \,. \tag{A9}$$

This corresponds to the endo-reversible model of Chambadal.

In Section 3, we proposed a complete Chambadal model taking account entropy production all along the cycle.

#### **Appendix B. Work Optimization Relative to Time (Frequency)**

Using the Lagrange multipliers method with the frequencies *fi* = <sup>1</sup> *<sup>τ</sup><sup>i</sup>* as variables, we get the following function:

$$\begin{split} L(f\_{i}) &= \left(\sqrt{\mathbf{G}\_{H}T\_{HS}} - \sqrt{(\mathbf{G}\_{H} + \mathbf{C}\_{IH}f\_{H})T\_{0}}\right)^{2} \\ &\quad - T\_{0}(\mathbf{C}\_{IH}f\_{H} + \mathbf{C}\_{I\to x}f\_{\to x} + \mathbf{C}\_{IC}f\_{\gets} + \mathbf{C}\_{I\gets o}f\_{\gets o}) \\ &\quad - \lambda \left(\frac{1}{f\_{H}} + \frac{1}{f\_{\to x}} + \frac{1}{f\_{\to}} + \frac{1}{f\_{\to s}} - \tau\right). \end{split} \tag{A10}$$

The vector of optimal values is

$$f\_{\rm Ex}^{\*} = \sqrt{\frac{\lambda}{T\_0 \mathbb{C}\_{I \to x}}} \qquad ; \qquad f\_{\rm C}^{\*} = \sqrt{\frac{\lambda}{T\_0 \mathbb{C}\_{I \gets}}} \qquad ; \qquad f\_{\rm Co}^{\*} = \sqrt{\frac{\lambda}{T\_0 \mathbb{C}\_{I \gets}}} , \tag{A11}$$

Additionally, the following is a non-linear equation to solve numerically for *f* ∗ *H*:

$$f\_H^2 = \lambda \sqrt{\frac{\mathcal{G}\_H + \mathcal{C}\_{IH} f\_H}{\mathcal{G}\_H}} \frac{1}{\sqrt{T\_{HS} T\_0 \mathcal{C}\_{IH}}}.\tag{A12}$$

In the reasonable case of low irreversibility on the *TH* isotherm (*CIH fH GH*), a good approximation of *f* ∗ *<sup>H</sup>* is

$$f\_H^\* = \sqrt{\frac{\lambda}{\sqrt{T\_{HS} T\_0} \mathbb{C}\_{IH}}}.\tag{A13}$$

The finitude constraint on *<sup>τ</sup><sup>i</sup>* allows for determining the <sup>√</sup>*<sup>λ</sup>* expression as

$$
\sqrt{\lambda} = \frac{N\sqrt{T\_0}}{\pi},
\tag{A14}
$$

where

$$N = \sqrt{\frac{T\_0}{T\_{HS}} \mathcal{C}\_{IH}} + \sqrt{\mathcal{C}\_{IEx}} + \sqrt{\mathcal{C}\_{IC}} + \sqrt{\mathcal{C}\_{ICo}} \,. \tag{A15}$$

Finally, we get

$$
\hbar \mathbf{A} \mathbf{x}\_2 \mathbf{W} \approx \mathbf{W}\_{\rm enddo} - \frac{T\_0}{\tau} \mathbf{N}^2 \,. \tag{A16}
$$

#### **References**


## *Article* **Performance Optimizations with Single-, Bi-, Tri-, and Quadru-Objective for Irreversible Diesel Cycle**

**Shuangshuang Shi 1,2, Lingen Chen 1,2,\*, Yanlin Ge 1,2,\* and Huijun Feng 1,2**


**\*** Correspondence: lgchenna@yahoo.com (L.C.); geyali9@hotmail.com (Y.G.)

**Abstract:** Applying finite time thermodynamics theory and the non-dominated sorting genetic algorithm-II (NSGA-II), thermodynamic analysis and multi-objective optimization of an irreversible Diesel cycle are performed. Through numerical calculations, the impact of the cycle temperature ratio on the power density of the cycle is analyzed. The characteristic relationships among the cycle power density versus the compression ratio and thermal efficiency are obtained with three different loss issues. The thermal efficiency, the maximum specific volume (the size of the total volume of the cylinder), and the maximum pressure ratio are compared under the maximum power output and the maximum power density criteria. Using NSGA-II, single-, bi-, tri-, and quadru-objective optimizations are performed for an irreversible Diesel cycle by introducing dimensionless power output, thermal efficiency, dimensionless ecological function, and dimensionless power density as objectives, respectively. The optimal design plan is obtained by using three solution methods, that is, the linear programming technique for multidimensional analysis of preference (LINMAP), the technique for order preferences by similarity to ideal solution (TOPSIS), and Shannon entropy, to compare the results under different objective function combinations. The comparison results indicate that the deviation index of multi-objective optimization is small. When taking the dimensionless power output, dimensionless ecological function, and dimensionless power density as the objective function to perform tri-objective optimization, the LINMAP solution is used to obtain the minimum deviation index. The deviation index at this time is 0.1333, and the design scheme is closer to the ideal scheme.

**Keywords:** irreversible Diesel cycle; power output; thermal efficiency; ecological function; power density; finite time thermodynamics

#### **1. Introduction**

As a further extension of traditional irreversible process thermodynamics, finite time thermodynamics [1–13] have been applied to analyze and optimize performances of actual thermodynamic cycles, and great progress has been made. The application of finite time thermodynamics to study the optimal performance of Diesel cycles represents a new technology for improving and optimizing Diesel heat engines, and a new method for studying Diesel cycles has been developed. Assuming the working fluid's specific heats are constants [14–24] and vary with its temperature [25–32], many scholars have studied the performance of irreversible Diesel cycles with various objective functions, such as power output (*P*), thermal efficiency (*η*), and ecological functions (*E*, which was defined as the difference between the exergy flow rate and the exergy loss).

In addition to the above objective functions, Sahin et al. [33,34] took power density (*Pd*, defined as the ratio of the cycle *P* to the maximum specific volume) as a new optimization criterion to optimize Joule–Brayton engines and found that the heat engine designed under the *Pd* criterion has higher *η* and a smaller size when no loss is considered. Chen et al. [35]

**Citation:** Shi, S.; Chen, L.; Ge, Y.; Feng, H. Performance Optimizations with Single-, Bi-, Tri-, and Quadru-Objective for Irreversible Diesel Cycle. *Entropy* **2021**, *23*, 826. https://doi.org/10.3390/e23070826

Academic Editor: Michel Feidt

Received: 25 May 2021 Accepted: 23 June 2021 Published: 28 June 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/).

introduced the objective function *Pd* into the thermodynamic analysis and optimization of the Atkinson cycle. Atmaca and Gumus [36] compared and analyzed the optimal performance of a reversible Diesel cycle based on the *P*, *Pd*, and effective *P* (which was defined as the product of power output and thermal efficiency) criteria. Raman and Kumar [37] conducted thermodynamic analysis and optimization of a reversible Diesel cycle under the criteria of *P*, *Pd*, and effective *P* when the working fluid's specific heats were linearly functioning with temperature. Rai and Sahoo [38] analyzed the influences of different losses on the effective *P*, effective *Pd*, and total heat loss of an irreversible Diesel cycle when the working fluid's specific heats were non-linearly functioning with temperature. Gonca and Palaci [39] analyzed and compared design parameters under the effective *P* and effective *Pd* criteria of an irreversible Diesel cycle.

The research mentioned above only optimized a single-objective function and did not optimize multiple objective functions at the same time. Therefore, NSGA-II can be used to solve a multi-objective optimization (MOO) problem, and MOO can be performed for the combination of different objective functions.

Ahmadi et al. [40–43] carried out MOO for an irreversible radiant heat engine [40], fuel cell combined cycle [41,42], and Lenoir heat engine [43] with different objective functions. Shi et al. [44] and Ahmadi et al. [45] performed MOO of the Atkinson cycle when the working fluid's specific heats were constants [44] and varied with temperature non-linearly [45]. Gonzalez et al. [46] performed MOO on *P*, *η*, and entropy generation of an endoreversible Carnot engine and analyzed the stability of the Pareto frontier. Ata et al. [47] performed parameter optimization and sensitivity analysis for an organic Rankine cycle with a variable temperature heat source. Herrera et al. [48] and Li et al. [49] performed MOO of *η* and emissions of a regenerative organic Rankine cycle. Garmejani et al. [50] performed MOO of *P*, exergy efficiency, and investment cost for a thermoelectric power generation system. Tang et al. [51] and Nemogne et al. [52] performed MOO of an irreversible Brayton cycle [51] and an absorption heat pump cycle [52]. MOO has been applied for performance optimization of various processes and cycles [53–56].

Reference [24] established a relatively complete irreversible Diesel cycle model and studied the optimal performance of *E*. Firstly, based on the model established in the reference [24], this paper studies the optimal *Pd* performance of an irreversible Diesel cycle while considering the impacts of the cycle temperature ratio and three loss issues. Secondly, the maximum specific volume, maximum pressure ratio, and *η* are compared under the maximum *P* and maximum *Pd* criteria. Thirdly, applying NSGA-II with a compression ratio as the decision variable and cycle dimensionless *P* (*P*, which is defined as *P* divided by maximum *P*), *η*, dimensionless *Pd* (*Pd*, which is defined as *Pd* divided by maximum *Pd*), and dimensionless *E* (*E*, which is defined as *E* divided by maximum *E*) as objective functions, the single-, bi-, tri-, and quadru-objective optimizations of an irreversible Diesel cycle are performed. Through three different solutions, that is, LINMAP, TOPSIS, and Shannon entropy, the deviation indexes obtained under different solutions are compared, and the optimized design scheme with the smallest deviation index is finally obtained.

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

The working fluid is assumed to be an ideal gas. Figures 1 and 2 show the *T* − *s* and *P* − *v* diagrams of an irreversible Diesel cycle. It can be seen that 1 − 2 is an adiabatic process, 2 − 3 is a constant-pressure process, 3 − 4 is an adiabatic process, and 4 − 1 is a constant-volume process. The processes 1 − 2*s* and 3 − 4*s* are the isentropic and adiabatic processes, respectively.

**Figure 1.** *T* − *s* representation of the Diesel cycle.

**Figure 2.** *P* − *v* representation of the Diesel cycle.

The heat absorption and release rates are, respectively,

$$
\dot{Q}\_{\rm in} = \dot{m} \mathbb{C}\_p (T\_3 - T\_2) \tag{1}
$$

$$
\dot{Q}\_{\text{out}} = \dot{m} C\_v (T\_4 - T\_1) \tag{2}
$$

where . *m* is the mass flow rate, and *Cv* and *Cp* are the specific heats under constant volume and pressure, respectively.

Some internal irreversibility loss (IIL) is caused by friction, turbulence, and viscous stress. The irreversible compression and expansion internal efficiencies are expressed as [16,19,20,30]

$$
\eta\_{\mathbb{C}} = (T\_{2s} - T\_1) / (T\_2 - T\_1) \tag{3}
$$

$$
\eta\_{\varepsilon} = (T\_3 - T\_4) / (T\_3 - T\_{4s}) \tag{4}
$$

The cycle compression ratio *γ* and temperature ratio *τ* are

$$
\gamma = V\_1 / V\_2 \tag{5}
$$

$$
\pi = T\_3 / T\_1 \tag{6}
$$

According to the property of isentropic process, one has

$$T\_{2s} = T\_1 \gamma^{k-1} \tag{7}$$

$$\left(T\_3/T\_{2\circ}\right)^k = T\_{4s}/T\_1\tag{8}$$

According to Equations (3)–(8), one has

$$T\_2 = T\_1[(\gamma^{k-1} - 1)/\eta\_c + 1] \tag{9}$$

$$T\_{4s} = \tau^k T\_1 / \gamma^{k(k-1)}\tag{10}$$

$$T\_4 = T\_1[\tau^k \eta\_\varepsilon / \gamma^{k(k-1)} - \tau \eta\_\varepsilon + \tau] \tag{11}$$

For the actual heat engine, there is heat transfer loss (HTL) between the working fluid and the cylinder. According to Refs. [14,24,27], it is known that the fuel exothermic rate is equal to the sum of the total endothermic rate and the HTL rate; one has

$$
\dot{Q}\_{\text{lank}} = A - \dot{Q}\_{\text{in}} = B(T\_3 + T\_2 - 2T\_0) \tag{12}
$$

where *A* is the fuel exothermic rate and *B* is the HTL coefficient.

Similarly, as the piston generates friction with the cylinder wall when running at high speed, the friction loss (FL) of the cycle cannot be ignored. As a four-stroke heat engine, a Diesel heat engine has four strokes of intake, compression, expansion, and exhaust, and all of them produce FL. According to Refs. [24,32], for the treatment of FL in each stroke, the FL during compression and expansion is included in internal irreversible losses. According to Refs. [57–59], the piston motion resistance in the intake process is greater than that in the exhaust process. If the friction coefficient in the exhaust process is *μ*, the equivalent friction coefficient, which includes the pressure drop loss in the intake process, is 3*μ*. The friction coefficients on the exhaust and intake stroke are *μ* and 3*μ*, respectively. There is a linear relationship between friction force and speed: *fμ* = −*μv* = −*μdx/dt*, where *x* is the piston displacement and *μ* is the FL coefficient. The power consumed due to FL during the exhaust and intake strokes can be derived as

$$P\_{\mu} = d\mathcal{W}\_{\mu}/dt = 4\mu \left(d\mathbf{x}/dt\right)^{2} = 4\mu v^{2} \tag{13}$$

For a Diesel cycle, the average speed of the piston in four reciprocating motions is

$$
\overline{v} = 4Ln\tag{14}
$$

where *n* is the rotating speed and *L* is the stroke length.

Therefore, the power consumed by cycle FL is

$$P\_{\mu} = 4\mu (4Ln)^2 = 64\mu (Ln)^2 \tag{15}$$

The cycle *P* and *η* are, respectively,

$$P = \dot{Q}\_{in} - \dot{Q}\_{out} - P\_{\mu} = \dot{m} [\mathbb{C}\_p (T\_3 - T\_2) - \mathbb{C}\_v (T\_4 - T\_1)] - 64\mu (Ln)^2 \tag{16}$$

$$\eta = \frac{P}{\dot{Q}\_{in} + \dot{Q}\_{leak}} = \frac{\dot{m}[\mathbb{C}\_p(T\_3 - T\_2) - \mathbb{C}\_v(T\_4 - T\_1)] - 64\mu(Ln)^2}{\dot{m}\mathbb{C}\_p(T\_3 - T\_2) + B(T\_2 + T\_3 - 2T\_0)}\tag{17}$$

According to the definition of *Pd* in Refs. [33–35], the *Pd* is expressed as

$$P\_d = P / \upsilon\_4 \tag{18}$$

According to Refs. [38,39], the total volume *vt*, stroke volume *vs*, and gap volume *vc* of the cycle are defined as

$$
v\_t = v\_s + v\_c \tag{19}$$

$$w\_s = \pi d^2 L/4\tag{20}$$

$$v\_{\mathfrak{c}} = \pi d^2 L / 4(\gamma - 1) \tag{21}$$

In the Diesel cycle, *vt* = *vmax* = *v*1, *vc* = *v*2. According to Equations (5) and (17)–(19), one has

$$P\_d = P/v\_{\max} = P/v\_t = 4(\gamma - 1)P/\pi d^2 L\gamma \tag{22}$$

According to Ref. [24], an irreversible Diesel cycle has four kinds of entropy generation due to FL, HTL, IIL, and exhaust stroke to the environment. The four entropy generation rates are expressed as

$$
\sigma\_q = B \left[ 1/T\_0 - 2/\left(T\_2 + T\_3\right) \right] \left( T\_3 + T\_2 - 2T\_0 \right) \tag{23}
$$

$$
\sigma\_{\mu} = P\_{\mu}/T\_0 = 64\mu (Ln)^2/T\_0 \tag{24}
$$

$$
\sigma\_{2s \to 2} = \dot{m} \int\_{T\_{2s}}^{T\_2} \mathbb{C}\_p dT / T = \dot{m} \mathbb{C}\_p \ln(T\_2 / T\_{2s}) \tag{25}
$$

$$
\sigma\_{4s \to 4} = \dot{m} \int\_{T\_{4s}}^{T\_4} \mathbb{C}\_v dT / T = \dot{m} \mathbb{C}\_v \ln(T\_4 / T\_{4s}) \tag{26}
$$

$$\sigma\_{pq} = \dot{m} \int\_{T\_1}^{T\_4} \mathbb{C}\_v dT (1/T\_0 - 1/T) = \dot{m} \mathbb{C}\_v [(T\_4 - T\_1)/T\_0 + \ln(T\_1/T\_4)] \tag{27}$$

Therefore, the total entropy generation rate is

$$
\sigma = \sigma\_{\emptyset} + \sigma\_{\mu} + \sigma\_{2s \to 2} + \sigma\_{4s \to 4} + \sigma\_{\not p \, \eta} \tag{28}
$$

According to the definition of *E* in Ref. [24], the *E* is expressed as

$$E = P - T\_0 \sigma \tag{29}$$

According to the processing method of Refs. [35,44], *P*, *Pd*, and *E* are respectively defined as

$$
\overline{P} = P / P\_{\text{max}} \tag{30}
$$

$$P\_d = P\_d / \left(P\_d\right)\_{\text{max}} \tag{31}$$

$$E = E / E\_{\text{max}}\tag{32}$$

According to Equations (4), (9) and (11) and given the compression ratio *γ*, the initial cycle temperature *T*1, and the cycle temperature ratio *τ*, by solving the temperatures at the 2, 3, and 4 state points, the corresponding numerical solutions of *P*, *η*, *Pd*, and *E* can be obtained.

#### **3. Maximum Power Density Optimization**

The working fluid is assumed to be an ideal gas. According to the nature of the air, *<sup>T</sup>*<sup>0</sup> <sup>=</sup> <sup>300</sup> K, *<sup>T</sup>*<sup>1</sup> <sup>=</sup> <sup>350</sup> K, . *m* = 1 mol/s, *k* = 1.4, *Cv* = 20.78 J/(mol · K), and *τ* = 5.78 − 6.78. According to Refs. [24,44], the cycle parameters are determined: *γ* = 1 − 100, *B* = 2.2 W/K, *μ* = 1.2 kg/s, *L* = 0.07 m and *n* = 30 s<sup>−</sup>1.

The relationships between the objective functions (*Pd* and *η*) of an irreversible Diesel cycle and the cycle design parameters (the cycle temperature ratio, HTL, FL, and IIL) are shown in Figures 3–6. It can be noticed that the relationship between *Pd* and *γ* (*Pd* − *γ*) is a parabolic-like one. When no loss is considered, the relationship between *Pd* and *η* (*Pd* − *η*) is a parabolic-like one, and when there is loss, the relationship curve of *Pd* − *η* is a loop-shaped one.

**Figure 3.** The effect of *τ* on *Pd* − *γ*.

**Figure 4.** The effect of *τ* on *Pd* − *η*.

**Figure 5.** The effects of *ηc*, *ηe*, *B*, and *b* on *Pd* − *γ*.

**Figure 6.** The effects of *ηc*, *ηe*, and *b* on *Pd* − *η*.

Figures 3 and 4 show the effects of *τ* on the performances of *Pd* − *γ* and *Pd* − *η*. According to Figure 3, it can be seen that there is an optimal compression ratio (*γPd* ), which makes *Pd* reach the maximum. As *τ* increases, *γPd* increases; when *τ* increases from 5.78 to 6.78, *γPd* increases from 12.7 to 16 (an increase of 25.98%). According to Figure 4, there is thermal efficiency (*ηPd* ) corresponding to the maximum *Pd*. As *τ* increases, *ηPd* increases; when *τ* increases from 5.78 to 6.78, *ηPd* increases from 45.82% to 49.29% (an increase of 7.40%). It can be seen that with the increase in *τ*, *γPd* , and *ηPd* corresponding to the maximum *Pd* also increases.

Figures 5 and 6 show the *Pd* − *γ* and *Pd* − *η* curves of the cycle when there are three different losses. Table 1 lists *ηPd* when considering different losses and the percentage of the decrease in *ηPd* compared with when no loss is considered. It can be seen that, with the increase in the losses considered, *ηPd* decreases. When the three losses are considered at the same time, *ηPd* decreases by 22.55% compared to that without any losses. According to Figure 5, it can be seen that as the compression ratio increases, *Pd* first increases and then decreases. According to Figure 6, it can be seen that when there are increases in HFL, FL, and IIL, *ηPd* corresponding to the maximum *Pd* decreases.


**Table 1.** Comparison of the *ηPd* in 8 cases.

Figures 7–9 show the change trends of the corresponding maximum specific volume, maximum pressure ratio, and *η* with the *τ* under the maximum *P* and maximum *Pd* criteria of an irreversible Diesel cycle. According to Figures 7 and 8, compared with the corresponding results under the maximum *P* criterion, the maximum specific volume is smaller and the maximum pressure ratio is larger under the maximum *Pd* criterion. It is observed that the Diesel heat engine designed under the maximum *Pd* criterion has a smaller size.

**Figure 7.** Variations of various *v*1/*vs* with *τ*.

**Figure 8.** Variations of various *p*3/*p*<sup>1</sup> with *τ*.

**Figure 9.** Variations of various *η* with *τ*.

According to Figure 9, the *η* of the cycle under the maximum *Pd* criterion is higher. When *τ* = 6.28, the *η* obtained under the maximum *P* and maximum *Pd* criterion are 46.04% and 47.64%, respectively. The latter is an increase of 3.54% over the former. Therefore, compared with the maximum *P* criterion, the engine designed under the maximum *Pd* criterion has a smaller size and a higher *η*.

#### **4. Multi-Objective Optimization with Power Output, Thermal Efficiency, Ecological Function, and Power Density**

MOO cannot make multiple objective functions reach the optimal value at the same time. The best compromise is achieved by comparing the pros and cons of each objective function. Therefore, the MOO solution set is not unique, and a series of feasible alternatives can be obtained, which are called Pareto frontiers. In this section, *P*, *η*, *E*, and *Pd* are used as objective functions; the compression ratio (*γ*) is used as an optimization variable; and NSGA-II [44–52] is used to perform bi-, tri-, and quadru-objective optimizations for an irreversible Diesel cycle. Through three different solutions, that is, LINMAP, TOPSIS, and Shannon entropy, the optimization results under different objective function combinations are obtained.

In the LINMAP solution, a minimum spatial distance from the ideal point is selected as the desired final optimal solution. In the TOPSIS solution, a maximum distance from the non-ideal point and a minimum distance from the ideal point are selected as the desired final optimal solution. In the Shannon entropy solution, a maximum value corresponding to a certain objective function is selected as the desired final optimal solution.

The optimization problems are solved with different optimization objective combinations, which form different MOO problems.

The six bi-objective optimization problems are as follows:

$$\max \left\{ \begin{array}{c} \mathsf{P}(\gamma) \\ \eta(\gamma) \end{array}, \max \left\{ \begin{array}{c} \mathsf{P}(\gamma) \\ \mathsf{E}(\gamma) \end{array}, \max \left\{ \begin{array}{c} \mathsf{P}(\gamma) \\ \mathsf{P}\_{d}(\gamma) \end{array}, \max \left\{ \begin{array}{c} \eta(\gamma) \\ \mathsf{E}(\gamma) \end{array}, \max \left\{ \begin{array}{c} \eta(\gamma) \\ \mathsf{P}\_{d}(\gamma) \end{array}, \max \left\{ \begin{array}{c} \mathsf{P}(\gamma) \\ \mathsf{P}\_{d}(\gamma) \end{array} \right. \end{array}, \max \left\{ \begin{array}{c} \mathsf{E}(\gamma) \\ \mathsf{P}\_{d}(\gamma) \end{array} \right. \end{array} \right. \right\}$$

The four tri-objective optimization problems are as follows:

$$\max \left\{ \begin{array}{ll} \overline{\mathbb{P}}(\gamma) \\ \eta(\gamma) \\ \overline{\mathbb{E}}(\gamma) \end{array} , \max \left\{ \begin{array}{ll} \overline{\mathbb{P}}(\gamma) \\ \eta(\gamma) \\ \overline{\mathbb{P}}\_{d}(\gamma) \end{array} , \max \left\{ \begin{array}{ll} \overline{\mathbb{P}}(\gamma) \\ \overline{\mathbb{E}}(\gamma) \\ \overline{\mathbb{P}}\_{d}(\gamma) \end{array} , \max \left\{ \begin{array}{ll} \eta(\gamma) \\ \overline{\mathbb{E}}(\gamma) \\ \overline{\mathbb{P}}\_{d}(\gamma) \end{array} \right. \end{array} \right\} \right. \tag{34}$$

The one quadru-objective optimization problem is as follows:

$$\max \begin{cases} \begin{array}{l} \overline{P}\_d(\gamma) \\ \eta(\gamma) \\ \overline{E}(\gamma) \\ \overline{P}\_d(\gamma) \end{array} \end{cases} \tag{35}$$

The evolution flow chart of NSGA-II is shown in Figure 10. The optimization results obtained by the combination of different objective functions in the three solutions are listed in Table 2. It can be seen that when single-objective optimization is performed under the criterions of maximum *P*,*η*, *E*, and *Pd*, the deviation indexes (0.5828, 0.5210, 0.2086, and 0.4122, respectively) obtained are much larger than the result obtained by MOO. This indicates that the design scheme of MOO is more ideal. When taking *P*, *E*, and *Pd* as the optimization objectives to perform tri-objective optimization, the deviation index obtained by the LINMAP solution is smaller, and the design scheme is closer to the ideal scheme.

**Figure 10.** Flow chart of NSGA-II.

Figures 11–16 show the Pareto frontiers of bi-objective optimization (*P* − *η*, *P* − *E* . , *P* − *Pd*, *η* − *E* . , *η* − *Pd*, and *E* − *Pd*). When *P* increases, *η*, *E*, and *Pd* all decrease; when *η* increases, *E* and *Pd* both decrease; when *E* increases, *Pd* decreases. According to Table 1, when *P* and *η* or *P* and *E* are the objective functions, the deviation index obtained by the LINMAP solution is smaller. When *P* and *Pd* or *η* and *E* are the optimization objectives, the deviation index obtained by the Shannon entropy solution is smaller. When *E* and *Pd* are the optimization objectives, the deviation indexes obtained by the LINMAP and TOPSIS solutions are smaller than those obtained by the Shannon entropy solution. When *η* and *Pd* are the objective functions, the deviation index obtained by the TOPSIS solution is smaller.

**Figure 11.** Bi-objective optimization on *P* − *η*.


**Table 2.** Optimization results obtained by combining different objective functions.

**Figure 12.** Bi-objective optimization on *P* − *E* . .

**Figure 13.** Bi-objective optimization on *P* − *Pd*.

**Figure 14.** Bi-objective optimization on *η* − *E* . .

**Figure 15.** Bi-objective optimization on *η* − *Pd*.

**Figure 16.** Bi-objective optimization on *E* − *Pd*.

Figures 17–20 show the Pareto frontiers of the tri-objective optimization (*P* − *η* − *Pd*, *P* − *η* − *E*, *η* − *E* − *Pd*, and *P* − *E* − *Pd*). When *P* increases, *η* decreases, and *E* and *Pd* first increase and then decrease. When *η* increases, *Pd* decreases, and *E* first increases and then decreases. When *η*, *E*, and *Pd* are the optimization objectives, the deviation indexes obtained by the LINMAP and TOPSIS solutions are smaller than those obtained by the Shannon entropy solution. When the combination of the other three objective functions are the optimization objectives, the deviation index obtained by the LINMAP solution is smaller, and the result is better.

Figure 21 shows the Pareto frontier of the quadru-objective optimization (*P* − *η* − *E* − *Pd*). With the increase in *P*, *η* increases, *Pd* decreases, and *E* first increases and then decreases. When *P*, *η*, *E*, and *Pd* are the optimization objectives, the deviation index obtained by the LINMAP solution is the smallest, and the result is the best.

**Figure 17.** Tri-objective optimization on *P* − *η* − *Pd*.

**Figure 18.** Tri-objective optimization on *P* − *η* − *E*.

**Figure 19.** Tri-objective optimization on *η* − *E* − *Pd*.

**Figure 20.** Tri-objective optimization on *P* − *E* − *Pd*.

**Figure 21.** Quadru-objective optimization on *P* − *η* − *E* − *Pd*.

#### **5. Conclusions**

The expression of the *Pd* of an irreversible Diesel cycle was derived in this paper, and the impacts of *τ* and three loss issues on the cycle of *Pd* versus *γ* and *η* characteristics were analyzed. The performance parameters (maximum specific volume, maximum pressure ratio, and *η*) of an irreversible Diesel cycle based on the criteria of maximum *P* and *Pd* were compared. Using three different solutions, including LINMAP, TOPSIS, and Shannon entropy, the results of single-, bi-, tri-, and quadru-objective optimization for an irreversible Diesel cycle were analyzed and compared. Comparing the deviation indexes obtained under different objective function combinations, the optimal design scheme was selected. The results showed the following:


**Author Contributions:** Conceptualization, Y.G. and L.C.; funding acquisition, L.C.; methodology, S.S., L.C., Y.G. and H.F.; software, S.S., Y.G. and H.F.; supervision, L.C.; validation, S.S. and H.F.; writing—original draft, S.S. and Y.G.; writing—review 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) and Graduate Innovative Fund of Wuhan Institute of Technology (Project No. CX2020038).

**Data Availability Statement:** Data sharing not applicable.

**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**


#### **References**

