Optimization of the power, efficiency and ecological function for an air-standard irreversible Dual-Miller cycle

Abstract

This paper establishes an irreversible Dual-Miller cycle (DMC) model with the heat transfer (HT) loss, friction loss (FL) and other internal irreversible losses. To analyze the effects of the cut-off ratio (ρ) and Miller cycle ratio (r_M) on the power output (P), thermal efficiency (η) and ecological function (E), obtain the optimal ρ_opt and optimal r_Mopt, and compare the performance characteristics of DMC with its simplified cycles and with different optimization objective functions, the P, η and E of irreversible DMC are analyzed and optimized by applying the finite time thermodynamic (FTT) theory. Expressions of P, η and E are derived. The relationships among P, η, E and compression ratio (ε) are obtained by numerical examples. The effects of ρ and r_M on P, η, E, maximum power output (MP), maximum efficiency (MEF) and maximum ecological function (ME) are analyzed. Performance differences among the DMC, the Otto cycle (OC), the Dual cycle (DDC), and the Otto-Miller cycle (OMC) are compared for fixed design parameters. Performance characteristics of irreversible DMC with the choice of P, η and E as optimization objective functions are analyzed and compared. The results show that the irreversible DMC engine can reach a twice-maximum power, a twice-maximum efficiency, and a twice-maximum ecological function, respectively. Moreover, when choosing E as the optimization objective, there is a 5.2% of improvement in η while there is a drop of only 2.7% in P compared to choosing P as the optimization objective. However, there is a 5.6% of improvement in P while there is a drop of only 1.3% in η compared to choosing as the optimization objective.

Graphical abstract

Keywords

finite-time thermodynamics / Dual-Miller cycle / power output / thermal efficiency / ecological function

1 Introduction

Thermodynamic analysis of internal combustion engines (ICE) cycles can develop new technologies of ICE and perfect ICE cycles. A lot of scholars have performed the first-law and second-law analyses for ICE cycles by applying classical thermodynamic method. Now, more and more scholars have been analyzing and optimizing ICE cycles by using the finite time thermodynamic (FTT) theory [1–10]. Besides, the performance bounds [11–20] and the optimal paths [21–26] of thermal systems have also been obtained. The study objects of the FTT theory include the conventional thermodynamic devices [27–33], the unconventional systems [34–36], and the operation (on/off) of time-dependent processes.

Lots of FTT studies have been performed on ICE cycles [37], such as the Otto cycle (OC), the Dual cycle (DDC), the Otto-Miller cycle (OMC), and the Dual-Miller cycle (DMC). Many of the studies focus on optimizing the power output (P) and thermal efficiency (η) of the ICE cycle, because the P of ICE determines the working capacity per unit time, which reflects the dynamic performance of the facility, and the η of ICE determines the utilization rate of the fuel, which reflects the environment friendly and energy saving performance of the facility. For the OC, Chen et al. [38] took an endoreversible OC with the heat transfer (HT) loss as research€ object, €analyzed€ the €work€ (W) €and€ η €characte-ristics. Angulo-Brown et al. [39] analyzed and optimized an irreversible OC model by considering the finite time characteristic and friction loss (FL). Chen et al. [40] investigated the characteristic relationship between P and η for an irreversible OC with the HT loss and FL. Chen et al. [41] optimized and analyzed the P and η of an irreversible OC by considering the internal irreversibilities and the HT loss. Zhao and Chen [42] conducted an analysis and optimization for irreversible OC based on P and η by taking into account the HT loss, the FL and other internal irreversibilities. Noroozian et al. [43] compared the P and η of OC with those of the Diesel cycle and the Atkinson cycle by taking HT into account. For the DDC, Lin et al. [44] and Hou [45] optimized W and η for an endoreversible DDC. Wang et al. [46] derived the expressions of the P and η of DDC with the FL. Chen et al. [47] and Zheng et al. [48] studied the P and η of the irreversible DDC with the HT loss and FL. Ge [49] performed a study on an irreversible DDC based on P and η by taking the HT loss, the FL and other internal irreversibilities into account. For the OMC, Fukuzawa et al. [50] introduced the history of OMC heat engines. Wu et al. [51] compared the W and η between OC and OMC, and optimized W and η characteristics of OMC. Ge et al. [52] analyzed and optimized the P and η of irreversible OMC with the HT loss and FL. Ye [53] investigated the P and η of an irreversible OMC with linear variable specific heat of working fluid. Gonca and Sahin [54] analyzed the effects of turbo charging and steam injection methods on the P and η of OMC. Based on simple cycle, Gonca et al. [55–57] studied the performances of an air-standard irreversible DMC by taking the HT loss and internal irreversibilities into account, optimized the performance characteristics of P, η, maximum dimensionless power (MP), maximum thermal efficiency (MEF) and maximum dimensionless power density, analyzed and compared the performance differences among OMC, DMC and Diesel-Miller cycle. Wu et al. [58] analyzed and optimized the P and η of an air-standard irreversible DMC with nonlinear variable specific heat ratio of working fluid. Gonca [59] and Gonca & Sahin [60] performed P and η optimization for irreversible DMC [59] and gas cycle engines [60] with internal irreversibility, heat leakage and HT of finite rate.

But when P (or η) reaches the maximum, η (or P) is lower. To find a condition that the P and η of ICE can reach relatively higher values simultaneously, it is necessary to propose a new optimization objective function. In 1991, Angulo-Brown et al. [61] first proposed the ecological function E = P−T_Lσ for the heat engine cycle, and proved that T_Lσ reflected the power dissipation of the heat engine, where T_L is the cold-source temperature and σ is the entropy generation rate of the cycle. This definition ignored the difference between exergy and energy. Yan [62] revised the definition as E = P−T₀σ later, where T₀ is the environment temperature and T₀σ means the exergetic loss. Because the optimization of E is claimed to represent a compromise between P and the loss power which is produced by entropy generation in the system and its surroundings and also between P and η, lots of studies have been performed. Moscato and Oliveira [63] conducted a study on an irreversible OC based on E criterion by considering the heat leakage, finite rate of HT, and internal irreversibilities. Ge et al. [64] performed E optimization for an irreversible OC with linear variable specific heat of working fluid. Gonca [59] and Gonca & Sahin [60] analyzed and optimized E of irreversible DMC [59] and gas cycle engines [60] with the heat leakage, finite rate of HT, and internal irreversibilities. Wu et al. [58] performed an analysis and optimization for an air-standard irreversible DMC with a nonlinear variable specific heat ratio of working fluid based on E criteria. You et al. [65] studied E of an air-standard irreversible DMC with polytropic processes.

It can be seen that in relevant literatures, the E of an irreversible DMC with constant specific heat, HT loss, FL and other internal irreversibilities does not appear, the effects of ρ and r_M on P, η and E are rarely analyzed, and the optimal design parameters are rarely obtained. This paper will adopt irreversible DMC model with HT loss, FL and other internal irreversibilities, derive expressions of P, η and E, analyze effects of the design parameters on P,η, E, MP, MEF and maximum ecological function (ME), investigate performance differences among OC, DDC, OMC and DMC, and analyze and compare performance characteristics of DMC under different optimization objective functions. The presented results could be assessed by engine designer to optimize the performance of DMC engines.

2 Cycle model and performance analyses

The P-V and T-s diagrams of an irreversible DMC are shown in Fig. 1, in which there are two adiabatic, two constant volume, and two constant pressure processes. Processes 1→2s and 4→5s are isentropic compression and expansion processes, respectively, while processes 1→2 and 4→5 are irreversible adiabatic compression and expansion processes that take internal irreversibilities into account. The real heat addition occurs in constant volume process 2→3 and constant pressure process 3→4, and real heat rejection occurs in constant volume process 5→6 and constant pressure process 6→1. At each state point, temperatures are T_i/K (i=1, 2, 2S, 3, 4, 5, 5S, 6) and volumes are V_i/m³ ((i=1, 2, 2S, 3, 4, 5, 5S, 6)).

The working fluid of the DMC engine is the air under standard conditions, and its specific heat can be considered as a constant. The total heat addition rate ({\dot{Q}}_{\mathrm{in}}) and heat rejection rate ({\dot{Q}}_{\mathrm{out}}) are given as

(1){\dot{Q}}_{\mathrm{in}}={\dot{Q}}_{\mathrm{23}}+{\dot{Q}}_{\mathrm{34}}=\dot{m}{C}_{\mathrm{v}}(T_{\mathrm{3}}-T_{\mathrm{2}})+\dot{m}{C}_{\mathrm{p}}(T_{\mathrm{4}}-T_{\mathrm{3}}),

(2){\dot{Q}}_{\mathrm{out}}={\dot{Q}}_{\mathrm{56}}+{\dot{Q}}_{\mathrm{61}}=\dot{m}{C}_{\mathrm{v}}(T_{\mathrm{5}}-T_{\mathrm{6}})+\dot{m}{C}_{\mathrm{p}}(T_{\mathrm{6}}-T_{\mathrm{1}}),

where \dot{m} is the working-fluid mole flow rate (mol/s) while C_v and C_p are the specific heat at constant volume and constant pressure, respectively, J/(mol•K).

The design parameters, compression ratio (ε), cut-off ratio (ρ) and Miller cycle ratio (r_M), for a DMC are defined as

(3)\epsilon ={\displaystyle \frac{V_{\mathrm{1}}}{V_{\mathrm{2}}}},

(4)\rho ={\displaystyle \frac{V_{\mathrm{4}}}{V_{\mathrm{3}}}}={\displaystyle \frac{T_{\mathrm{4}}}{T_{\mathrm{3}}}},

(5)r_M={\displaystyle \frac{V_{\mathrm{6}}}{V_{\mathrm{1}}}}={\displaystyle \frac{T_{\mathrm{6}}}{T_{\mathrm{1}}}}.

To reflect the irreversibilities that contain the FL in the irreversible adiabatic processes 1→2 and 4→5, according to Ref. [66], the isentropic compression efficiency (η_c) and isentropic expansion efficiency (η_e) are defined as

(6){\eta }_c={\displaystyle \frac{T_{\mathrm{2}s}-T_{\mathrm{1}}}{T_{\mathrm{2}}-T_{\mathrm{1}}}},

(7){\eta }_e={\displaystyle \frac{T_{\mathrm{4}}-T_{\mathrm{5}}}{T_{\mathrm{4}}-T_{\mathrm{5}s}}}.

Combining Eqs. (3)–(7) yields

(8)T_{\mathrm{2}s}=T_{\mathrm{1}}{\epsilon }^{k-\mathrm{1}},

(9)T_{\mathrm{2}}=T_{\mathrm{1}}\left({\displaystyle \frac{{\epsilon }^{k-\mathrm{1}}-\mathrm{1}}{{\eta }_{\mathrm{c}}}}+\mathrm{1}\right),

(10)T_{\mathrm{3}}={\displaystyle \frac{T_{\mathrm{4}}}{\rho }},

(11)T_{\mathrm{5}s}=T_{\mathrm{4}}{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}},

(12)T_{\mathrm{5}}=T_{\mathrm{4}}\left\{\mathrm{1}-{\eta }_e\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]\right\},

(13)T_{\mathrm{6}}=T_{\mathrm{1}}{r}_{\mathrm{M}}.

The fuel combustion produces heat in the cylinder for the ICEs. There is the HT loss from the working fluid to the outside through the cylinder wall because of the large temperature difference. Assuming the ambient temperature is equal to T₀ and the HT loss obeys linear relation, according to Ref. [67], the HT loss is a linear relationship with the temperature difference of the working-fluid average temperature and the ambient temperature, that is

(14){\dot{Q}}_{\mathrm{in}}=A_{\mathrm{1}}-B_{\mathrm{1}}\left({\displaystyle \frac{T_{\mathrm{2}}+T_{\mathrm{4}}}{\mathrm{2}}}-T_{\mathrm{0}}\right),

where A₁ is the heat releasing rate of fuel combustion, J/s, and B₁ is the heat leakage coefficient of cylinder wall, J/(K•s).

It can be known from Eq. (14) that the total heat addition rate of the working fluid is equal to the heat releasing rate of fuel combustion minus the HT loss rate. Thus, the HT loss rate can be expressed as

(15){\dot{Q}}_{\mathrm{leak}}=B[(T_{\mathrm{2}}+T_{\mathrm{4}})-\mathrm{2}{T}_{\mathrm{0}}],

where B=B₁/2, J/(K•s).

The working process of the four strokes ICEs consist of the intake, the compression, the expansion, and the exhaust strokes. There are FLs between the piston and the cylinder wall in the four strokes. For actual irreversible DMC (1→2→3→4→5→6→1) shown in Fig. 1, processes 1→2 and 4→5 are with the internal irreversibilities, that is FL, and are reflected by η_c and η_e, respectively. Thus, the power loss (P_μ) caused by FL merely refers to the FL of piston in intake and exhaust strokes. According to Refs. [68–70], the friction coefficients in the exhaust stroke and intake stroke are μ and 3μ, respectively, where μ is friction coefficient, kg/s. The friction force (f_μ) is linear with the velocity (u), f_μ = –μdX/dt, where X is the displacement of piston, m; and t is the time, s. The power loss caused by the FL of piston in intake and exhaust strokes is

(16)P_{\mu }=-\mathrm{4}{f}_{\mu }{\displaystyle \frac{\mathrm{d}X}{\mathrm{d}t}}=\mathrm{4}\mu {\displaystyle \frac{\mathrm{d}X}{\mathrm{d}t}}\cdot {\displaystyle \frac{\mathrm{d}X}{\mathrm{d}t}}=\mathrm{4}\mu {\overline{u}}^{\mathrm{2}},

where \bar{u} is piston average velocity, m/s.

For the four-stroke heat engine, the piston average velocity \bar{u} is

(17)\bar{u}=\mathrm{4}Ln,

where L is stroke length, m; and n is cycle index, s^–1.

Thus, the power loss caused by the FL of piston in intake and exhaust strokes is

(18)P_{\mu }=4\mu {(4Ln)}^2=\mathrm{64}\mu {(Ln)}^2.

The power output of the cycle is

(19)\begin{array}{l}P={\dot{Q}}_{\mathrm{in}}-{\dot{Q}}_{\mathrm{out}}-P_{\mu }\\ =\dot{m}{C}_{\mathrm{v}}[(T_{\mathrm{3}}+T_{\mathrm{6}}-T_{\mathrm{2}}-T_{\mathrm{5}})+k(T_{\mathrm{1}}+T_{\mathrm{4}}-T_{\mathrm{3}}-T_{\mathrm{6}})]-\mathrm{64}\mu {(Ln)}^{\mathrm{2}}\\ =\dot{m}{C}_v{T}_{\mathrm{1}}\left\{\left(r_{\mathrm{M}}-{\displaystyle \frac{{\epsilon }^{k-\mathrm{1}}-\mathrm{1}}{{\eta }_{\mathrm{c}}}}-\mathrm{1}\right)+\tau \left\{{\displaystyle \frac{\mathrm{1}}{\rho }}+{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]-\mathrm{1}\right\}+k\left[(\mathrm{1}-r_{\mathrm{M}})+\tau \left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\rho }}\right)\right]\right\}-\mathrm{64}\mu {(Ln)}^{\mathrm{2}},\end{array}

where τ = T₄/T₁ is cycle temperature ratio, and k = C_p/C_v is specific heat ratio.

The thermal efficiency of the cycle is

(20)\begin{array}{l}\eta ={\displaystyle \frac{{\dot{Q}}_{\mathrm{in}}-{\dot{Q}}_{\mathrm{out}}-P_{\mu }}{{\dot{Q}}_{\mathrm{in}}+{\dot{Q}}_{\mathrm{leak}}}}\\ ={\displaystyle \frac{\dot{m}{C}_{\mathrm{v}}[(T_{\mathrm{3}}+T_{\mathrm{6}}-T_{\mathrm{2}}-T_{\mathrm{5}})+k(T_{\mathrm{1}}+T_{\mathrm{4}}-T_{\mathrm{3}}-T_{\mathrm{6}})]-\mathrm{64}\mu {(Ln)}^{\mathrm{2}}}{\dot{m}{C}_{\mathrm{v}}[(T_{\mathrm{3}}-T_{\mathrm{2}})+k(T_{\mathrm{4}}-T_{\mathrm{3}})]+B(T_{\mathrm{2}}+T_{\mathrm{4}}-\mathrm{2}{T}_{\mathrm{0}})}}\\ ={\displaystyle \frac{\dot{m}{C}_{\mathrm{v}}{T}_{\mathrm{1}}\left\{\left(r_{\mathrm{M}}-{\displaystyle \frac{{\epsilon }^{k-\mathrm{1}}-\mathrm{1}}{{\eta }_{\mathrm{c}}}}-\mathrm{1}\right)+\tau \left\{{\displaystyle \frac{\mathrm{1}}{\rho }}+{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]-\mathrm{1}\right\}+k\left[(\mathrm{1}-r_{\mathrm{M}})+\tau \left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\rho }}\right)\right]\right\}-\mathrm{64}\mu {(Ln)}^{\mathrm{2}}}{\dot{m}{C}_{\mathrm{v}}{T}_{\mathrm{1}}\left\{\left[{\displaystyle \frac{\tau }{\rho }}-{\displaystyle \frac{({\epsilon }^{k-\mathrm{1}}-\mathrm{1})}{{\eta }_{\mathrm{c}}}}-\mathrm{1}\right]+k\tau \left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\rho }}\right)\right\}+B{T}_{\mathrm{1}}\left[{\displaystyle \frac{({\epsilon }^{k-\mathrm{1}}-\mathrm{1})}{{\eta }_{\mathrm{c}}}}+\mathrm{1}+\tau -\mathrm{2}{\chi }^{-\mathrm{1}}\right]}},\end{array}

where \chi =T_{\mathrm{1}}/T_{\mathrm{0}}.

The entropy generation of the air-standard irreversible DMC comes from the HT loss, the FL in intake and exhaust strokes, the irreversible compression and expansion, which include all irreversibilities in processes 1→2 and 4→5, and the heat loss of the working fluid with energy exhausting to the environment.

According to Ref. [71], the entropy generation rate coming from the HT loss is

(21)\begin{array}{l}{\sigma }_{\mathrm{q}}={\dot{Q}}_{\mathrm{leak}}\left[\mathrm{1}/T_{\mathrm{0}}-\mathrm{2}/\left(T_{\mathrm{2}}+T_{\mathrm{4}}\right)\right]\hfill \\ =B{\left(T_{\mathrm{2}}+T_{\mathrm{4}}-\mathrm{2}{T}_{\mathrm{0}}\right)}^{\mathrm{2}}/\left[T_{\mathrm{0}}\left(T_{\mathrm{2}}+T_{\mathrm{4}}\right)\right]\hfill \\ =B\chi {\left[\left({ϵ}^{k-\mathrm{1}}-\mathrm{1}\right)/{\eta }_{\mathrm{c}}+\mathrm{1}+\tau -\mathrm{2}{\chi }^{�\mathrm{1}}\right]}^{\mathrm{2}}\hfill \\ /\left[\left({ϵ}^{k-\mathrm{1}}-\mathrm{1}\right)/{\eta }_{\mathrm{c}}+\mathrm{1}+\tau \right]\hfill \end{array}

The FL in intake and exhaust strokes comes from the relative motion between the piston and the cylinder. Because the piston motion is driven by the working fluid with a high energy, the power loss caused by the FL of piston is a part of the working-fluid heat addition rate. Assuming this part of loss heat addition rate can be totally transformed into the power loss caused by FL, thus, the entropy generation rate caused by the loss heat addition rate can be calculated by dividing the environment temperature by the power loss caused by FL. Therefore,

(22){\sigma }_{\mu }={\displaystyle \frac{P_{\mu }}{T_{\mathrm{0}}}}={\displaystyle \frac{\mathrm{64}\mu {(Ln)}^{\mathrm{2}}}{T_{\mathrm{0}}}}.

There is an FL in each stroke of the four-stroke heat engine. For the irreversible DMC in Fig. 1, the irreversible adiabatic processes 1→2 and 4→5 have taken the internal irreversibilities into account compared to reversible adiabatic processes 1→2s and 4→5s. All the irreversibilities in processes 1→2 and 4→5 have contained the FL of the piston in compression and expansion strokes, and they are reflected by η_c and η_e, respectively.

According to Ref. [71], the entropy generation rate coming from the irreversible compression and expansion are

(23){\sigma }_{\mathrm{2}s\to \mathrm{2}}=\dot{m}{\displaystyle {\int }_{T_{\mathrm{2}s}}^{T_{\mathrm{2}}}{C}_{\mathrm{v}}{\displaystyle \frac{\mathrm{d}T}{T}}=\dot{m}{C}_{\mathrm{v}}\ln {\displaystyle \frac{T_{\mathrm{2}}}{T_{\mathrm{2}s}}}=}\dot{m}{C}_{\mathrm{v}}\ln \left[{\displaystyle \frac{(\mathrm{1}-{\epsilon }^{\mathrm{1}-k})}{{\eta }_c}}+{\epsilon }^{\mathrm{1}-k}\right],

(24){\sigma }_{\mathrm{5}s\to \mathrm{5}}=\dot{m}{\displaystyle {\int }_{T_{\mathrm{5}s}}^{T_{\mathrm{5}}}{C}_{\mathrm{v}}{\displaystyle \frac{dT}{T}}=\dot{m}{C}_{\mathrm{v}}\ln {\displaystyle \frac{T_{\mathrm{5}}}{T_{\mathrm{5}s}}}=}\dot{m}{C}_{\mathrm{v}}\ln \left\{{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{\mathrm{1}-k}-{\eta }_{\mathrm{e}}\left[{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{\mathrm{1}-k}-\mathrm{1}\right]\right\}.

According to Ref. [71], the entropy generation rate coming from the heat loss of the working fluid with energy exhausting to the environment is

(25)\begin{array}{l}{\sigma }_{\mathrm{pq}}=\dot{m}\left[{\displaystyle {\int }_{T_{\mathrm{1}}}^{T_6}{C}_{\mathrm{p}}\mathrm{d}T\left({\displaystyle \frac{\mathrm{1}}{T_{\mathrm{0}}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right)}+{\displaystyle {\int }_{T_{\mathrm{6}}}^{T_{\mathrm{5}}}{C}_{\mathrm{v}}\mathrm{d}T\left({\displaystyle \frac{\mathrm{1}}{T_{\mathrm{0}}}}-{\displaystyle \frac{\mathrm{1}}{T}}\right)}\right]\\ =\dot{m}{C}_{\mathrm{v}}\left\{k\left({\displaystyle \frac{T_{\mathrm{6}}-T_{\mathrm{1}}}{T_{\mathrm{0}}}}-\ln {\displaystyle \frac{T_{\mathrm{6}}}{T_{\mathrm{1}}}}\right)+\left({\displaystyle \frac{T_{\mathrm{5}}-T_{\mathrm{6}}}{T_{\mathrm{0}}}}-\ln {\displaystyle \frac{T_{\mathrm{5}}}{T_{\mathrm{6}}}}\right)\right\}\\ =\dot{m}{C}_{\mathrm{v}}\left\{\begin{array}{l}k[\chi (r_{\mathrm{M}}-\mathrm{1})-\ln (r_{\mathrm{M}})]+\chi \left\{\tau \left\{\mathrm{1}-{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]\right\}-r_{\mathrm{M}}\right\}\\ -\ln \left\{{\displaystyle \frac{\tau \left\{\mathrm{1}-{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]\right\}}{r_{\mathrm{M}}}}\right\}\end{array}\right\}.\end{array}

The total entropy generation rate is the sum of that in five aspects for the irreversible DMC. Therefore,

(26){\sigma }_{\mathrm{sum}}={\sigma }_{\mathrm{q}}+{\sigma }_{\mu }+{\sigma }_{\mathrm{2}\to \mathrm{2}s}+{\sigma }_{\mathrm{5}\to \mathrm{5}s}+{\sigma }_{\mathrm{pq}}.

According to the definition of the ecological function in Refs. [61,62], the ecological function of the irreversible DMC is

(27)\begin{array}{l}E=P-T_{\mathrm{0}}{\sigma }_{\mathrm{sun}}\hfill \\ =\dot{m}{C}_{\mathrm{v}}\left\{\left(T_{\mathrm{3}}+\mathrm{2}{T}_{\mathrm{6}}-T_{\mathrm{2}}-\mathrm{2}{T}_{\mathrm{5}}\right)+k\left(\mathrm{2}{T}_{\mathrm{1}}+T_{\mathrm{4}}-T_{\mathrm{3}}-\mathrm{2}{T}_{\mathrm{6}}\right)+T_{\mathrm{0}}\ln \left({\displaystyle \frac{T_{\mathrm{2}s}{T}_{\mathrm{5}s}{T}_{\mathrm{6}}^{k-\mathrm{1}}}{T_{\mathrm{1}}^k{T}_{\mathrm{2}}}}\right)\right\}\hfill \\ \hfill \\ \hspace{0.17em}\hspace{0.17em}-B\left(T_{\mathrm{2}}+T_{\mathrm{4}}-\mathrm{2}{T}_{\mathrm{0}}\right)-\mathrm{128}\mu {\left(Ln\right)}^{\mathrm{2}}\hfill \\ =\dot{m}{C}_{\mathrm{v}}{T}_{\mathrm{1}}\left\{\begin{array}{l}{\displaystyle \frac{\tau }{\rho }}+\mathrm{2}{r}_{\mathrm{M}}-\left({\displaystyle \frac{{ϵ}^{k-\mathrm{1}}-\mathrm{1}}{{\eta }_{\mathrm{c}}}}+\mathrm{1}\right)-\mathrm{2}\tau \left\{\mathrm{1}-{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{ϵr_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]\right\}\hfill \\ \hfill \\ +k\left(\mathrm{2}+\tau -{\displaystyle \frac{\tau }{\rho }}-\mathrm{2}{r}_{\mathrm{M}}\right)+{\chi }^{-\mathrm{1}}\ln \left({\displaystyle \frac{\tau {\rho }^{k-\mathrm{1}}}{{\displaystyle \frac{{ϵ}^{k-\mathrm{1}}-\mathrm{1}}{{\eta }_{\mathrm{c}}}}+\mathrm{1}}}\right)\hfill \end{array}\right\}\hfill \\ \hfill \\ \hspace{0.17em}\hspace{0.17em}-B{T}_{\mathrm{1}}{\left[{\displaystyle {ϵ}^{k-\mathrm{1}}-\mathrm{1}/{\eta }_{\mathrm{c}}}+\mathrm{1}+\tau -\mathrm{2}{\chi }^{-\mathrm{1}}\right]}^{\mathrm{2}}/\left[{\displaystyle {ϵ}^{k-\mathrm{1}}-\mathrm{1}/{\eta }_c}+\mathrm{1}+\tau \right]-\mathrm{128}\mu {\left(Ln\right)}^{\mathrm{2}}.\hfill \end{array}

3 Special example analyses

Parameters ρ and r_M must satisfy the following conditions in order that the DMC engines can operate normally:

(28)\mathrm{1}\le \rho \le {\displaystyle \frac{\tau {\eta }_{\mathrm{c}}}{{\epsilon }^{k-\mathrm{1}}+{\eta }_{\mathrm{c}}-\mathrm{1}}},

(29)\mathrm{1}\le r_{\mathrm{M}}\le \tau \left\{\mathrm{1}-{\eta }_{\mathrm{e}}\left[\mathrm{1}-{\left({\displaystyle \frac{\rho }{\epsilon r_{\mathrm{M}}}}\right)}^{k-\mathrm{1}}\right]\right\}.

Equations (19), (20) and (27) are expressions of P, η and E of an air-standard irreversible DMC, which can be simplified to those of other cycles when design parameters ρ and r_M take certain values.

(1) When r_M = 1, DMC is simplified to DDC, and Eqs. (19), (20) and (27) are simplified to those of the P, η and E of DDC [49].

(2) When ρ = 1, DMC is simplified to OMC, and Eqs. (19), (20) and (27) are simplified to those of the P, η and E of OMC [49].

(3) When ρ = 1 and r_M = 1, DMC is simplified to OC, and Eqs. (19), (20) and (27) are simplified to those of the P, η and E of OC [49].

4 Results and discussion

When the values of T₁, T₄, ε, ρ, r_M, η_c and η_e are set, T_2s, T₂, T₃, T_5s, T₅ and T₆ can be calculated according to Eqs. (8)€–€(13). Substituting those temperatures into Eqs. (19), (20) and (27) yields the P, η and E of irreversible DMC.

According to Refs. [72,73], the parameters are set as \dot{m}=\mathrm{1}\hspace{0.17em}\mathrm{mol/s}, C_v = 20.78 J/(mol·K), k = 1.4, B = 2.2 J/(K·s), T₀ = 300 K, \chi =\mathrm{1.2}, τ = 5.83, n = 30 s^–1, L = 0.07 m, μ =1.2 kg/s, η_c = η_e= 0.97,ρ = 1, 2, 3, 4, r_M = 1, 2, 3, 4, and ε = 1€–€180 in the numerical computations. According to numerical simulations, the performance differences of OC, OMC, DDC and DMC can be obtained, the effects of ρ and r_M on P, η and E can be analyzed, and the optimal cut-off ratio ρ_opt and optimal Miller cycle ratio r_{{\mathrm{M}}_{\mathrm{opt}}}can also be obtained.

4.1 Performance comparison of irreversible OC, DDC, OMC and DMC

Figure 2 shows P versus η characteristic of irreversible OC, DDC, OMC and DMC. The curves are loop-shaped, each having two zero power points, two zero efficiency points, one MP point and one MEF point. It can be seen that P_max,DMC >P_max,DDC >P_max,OMC >P_max,OC, and the MP characteristic of DMC is superior to those of DDC, OMC and OC. Moreover, P_max,DMC/P_max,OC = 1.42, the efficiency corresponding to MP is η_P,DMC/η_P,OC = 1.11. Thus, DMC can improve MP and the efficiency corresponding to MP compared to OC. η_max,DMC>η_max,OMC>η_max,DDC>η_max,OC, the MEF characteristic of DMC is superior to those of OMC, DDC and OC. Moreover, η_max,DMC/η_max,OC = 1.10, the power corresponding to MEF is P_η,DMC/P_η,OC = 1.44. Thus, DMC can improve MEF and the power corresponding to MEF compared to OC. of the reasons for those are that adding a constant-pressure heat addition process in OC can increase the heat addition quantity and heat rejection quantity, but the increasing quantity of heat addition is more obvious than that of heat rejection, and P and η will improve compared to OC. Adding a constant-pressure heat rejection process to the Otto cycle can decrease the heat rejection quantity, but has no effect on the heat addition quantity, and P and η will improve compared to OC.

Figure 3 shows E versus ε characteristic of irreversible OC, DDC, OMC and DMC. The curves are parabolic-like shaped, each having a maximum ecological function (ME) point, and ε corresponding to ME is {ϵ}_{E,\hspace{0.17em}\mathrm{DDC}}>{ϵ}_{E,\hspace{0.17em}\mathrm{DMC}}={ϵ}_{E,\hspace{0.17em}\mathrm{OC}}>{ϵ}_{E,\hspace{0.17em}\mathrm{OMC}}. It can be found that E_max,DMC>E_max,OMC> E_max,DDC>E_max,OC, and the ME characteristic of DMC is superior to those of OMC, DDC and OC. Moreover, E_max,DMC/E_max,OC = 2.72, hence, DMC can improve ME compared to OC. Adding a constant-pressure heat addition process to OC may increase the heat losses which lead to entropy generation. Adding a constant-pressure heat rejection process to OC may decrease the heat losses. According to the definition of E = P– T₀σ, although the P of DDC is higher than that of OMC, the entropy generation of OMC is lower than that of DDC. Thus, the relationship of ME among DMC, DDC, OMC and OC is E_max,DMC>E_max,OMC> E_max,DDC>E_max,OC.

4.2 Effects of ρ on the P, η, and E of irreversible DMC

Figure 4 shows P versus η characteristic of irrebersible DMC for different ρ when r_M = 2. It can be seen that ρ has a great effect on P versus η characteristic of the irreversible DMC. The relationships between MP and ρ, as well as between MEF and ρ are non-monotonic, but MP and MEF reach their maxima when ρ takes certain values, respectively. The reason for this is that adding a constant-pressure heat addition process can increase the heat addition quantity and heat rejection quantity, and the increasing quantity of heat addition is more obvious than that of heat rejection when the value of ρ is small, but the increasing quantity of heat rejection will be more obvious than that of heat addition after ρ reaches a certain value. The influences of ρ on P and η are different, and MP and MEF reach their maxima when ρ = 2; when ρ = 1, MP is smaller than those with ρ = 3 and ρ = 4, but MEF is between those with ρ = 3 and ρ = 4.

Figure 5 shows E versus ε characteristic of irrebersible DMC for different ρ when r_M = 2. It can be seen that ρ has a great effect on E of the irreversible DMC. The relationship between ME and ρ is non-monotonic, but ME reaches its maximum when ρ takes certain values. The reason for this is that P first increases and then starts to decrease with increasing ρ, and the entropy generation continuously increases with increasing ρ. The influences of ρ on E, P and η are different, and ME, MP and MEF reach their maxima when ρ = 2; when ρ = 1, MP is smaller than those with ρ = 3 and ρ = 4, but MEF is between those with ρ = 3 and ρ = 4, and ME is between those with ρ = 3 and ρ = 4. The reason for this is that ε is different when P, η and E reach their maxima.

4.3 Effects of r_M on the P, η and E of irreversible DMC

Figure 6 shows P versus η characteristic of the irrebersible DMC for different r_M when ρ = 2. It can be seen that r_M has a great effect on P versus η characteristic of the irreversible DMC. The relationships between MP and r_M, as well as between MEF and r_M are non-monotonic, but MP and MEF reach their maxima when r_M takes certain values, respectively. The reason for this is that the heat rejection quantity first decreases and then starts to increase with increasing r_M when adding a constant-pressure heat rejection process to the Otto cycle, but the heat addition quantity is constant. The influences of r_M on P and η are different, and MP and MEF reach their maxima when r_M = 2; when r_M = 1, MP is smaller than those with r_M = 3 and r_M = 4, but MEF is between those with r_M = 3 and r_M = 4.

Figure 7 shows E versus ε characteristic of the irrebersible DMC for different r_M when ρ = 2. It can be seen that r_M has a great effect on E of irreversible DMC. The relationship between ME and r_M is non-monotonic, but ME reaches its maxima when r_M takes certain values. The reason for this is that P first increases and then starts to decrease with increasing r_M, and the entropy generation continuously decreases with increasing r_M. The influences of r_M on E, P and η are different, and ME, MP and MEF reach their maxima when r_M = 2; when r_M = 1, MP is smaller than those with r_M = 3 and r_M = 4, but MEF is between those with r_M = 3 and r_M = 4, and ME is between those with r_M = 3 and r_M = 4.

4.4 Effects of ρ and r_M on MP, MEF and ME of irreversible DMC

It can be seen from Figs. 4€–€7 that each curve of P-\eta and E-ϵ has corresponding MP, MEF and ME, respectively, which vary with ρ and r_M. To analyze the effects of ρ and r_M on MP, MEF and ME, and to obtain the twice-maximum power {P^{\prime }}_{\max }, twice-maximum efficiency {{\eta }^{\prime }}_{\max } and twice-maximum ecological function {E^{\prime }}_{\max }, as well as optimal cut-off ratio ρ_opt and optimal Miller cycle ratio r_{{\mathrm{M}}_{\mathrm{opt}}} corresponding to {P^{\prime }}_{\max }, {{\eta }^{\prime }}_{\max } and {E^{\prime }}_{\max }, three-dimensional diagrams of MP, MEF and ME versus ρ and r_M are shown in Figs. 8€–10, respectively. Figures 8–10 suggest that as ρ increases, MP, MEF and ME increase at first and then decrease; there exist three different optimal ρ to make MP, MEF and ME reach their maxima, respectively. MP, MEF and ME have similar variations with the increase of r_M, and there exist three different optimal r_M to make MP, MEF and ME reach their maxima, respectively. Hence, the parameters of ρ and r_M are closely related to the performance characteristics of the irreversible DMC engine, which can reach {P^{\prime }}_{\max }, {{\eta }^{\prime }}_{\max } and {E^{\prime }}_{\max } when ρ = 2.4 and r_M = 2.4, ρ = 1.6 and r_M = 1.75, as well as ρ = 1.85 and r_M = 2, respectively.

4.5 Comparison of optimization objective functions

To analyze performance characteristics of DMC heat engines under different optimization objective functions, the power output (P_η) at \mathit{MEF}, the power output (P_E) at \mathit{ME}, the thermal efficiency ({\eta }_P) at MP, and the thermal efficiency ({\eta }_E) at ME are obtained by further numerical computations. Figures 11 and 12 show three powers (P_max, P_E, P_η) and three efficiencies (η_max, η_E, η_P) versus r_M characteristics when ρ = 2. It can be found that the powers (P_max, P_E, P_η) and efficiencies (η_max, η_E, η_P) increase at first and then decrease as r_M increases. When r_M takes the same value, there are

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(31)

When P is selected as the optimization objective, P_max = 19401.4 W, {\eta }_P = 0.486, and {\eta }_P/η_max = 0.935; when η is selected as the optimization objective, η_max = 0.520, P_η = 17799.4 W, and P_η/P_max =0.917; and when E is selected as the optimization objective, P_E = 18871.6 W, P_E/P_max = 0.973, η_E = 0.513, and η_E/η_max = 0.987. Although choosing E as the optimization objective may sacrifice parts of P and η, optimizing E is the best compromise between optimizing P and optimizing η. According to the definition of E = P– T₀σ, it is as far as possible to improve P and decrease the entropy generation when optimizing E, and η will also improve because of the reducing of irreversibilities. The optimization of E represents a compromise between P and the loss power which is produced by the entropy generation in the system and its surroundings.

5 Conclusions

Based on the FTT theory and the numerical computation method, the performances of P, η and E of an air-standard irreversible Dual-Miller cycle with HT loss, FL, and other internal irreversible losses are studied. It can be concluded that the DMC engine has great advantages on P, η and E than the OC, DDC and OMC engines. MP, MEF and ME of DMC are 1.42, 1.10 and 2.72 times of those of OC under the same condition, respectively. Parameters ρ and r_M have great influences on the P, η and E of the irreversible DMC, but the effects are different. The relationships among MP, MEF, ME and ρ, and the relationships among MP, MEF, ME and r_M are non-monotonic. The irreversible DMC engine can reach twice-maximum power, twice-maximum efficiency, and twice-maximum ecological function when ρ = 2.4 and r_M = 2.4, ρ = 1.6 and r_M = 1.75, as well as ρ = 1.85 and r_M = 2, respectively. Choosing E as the optimization objective, although there is a 27% drop in MP and a 1.3% drop in MEF, it is the best compromise between P and η.

The work presented in this paper is only a theoretical study, and it is possible to design experiments to examine the result in the future researches.

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