Unified cycle model of a class of internal combustion engines and their optimum performance characteristics

Abstract

The unified cycle model of a class of internal combustion engines is presented, in which the influence of the multi-irreversibilities mainly resulting from the adiabatic processes, finite-time processes and heat leak loss through the cylinder wall on the performance of the cycle are taken into account. Based on the thermodynamic analysis method, the mathematical expressions of the power output and efficiency of the cycle are calculated and some important characteristic curves are given. The influence of the various design parameters such as the high-low pressure ratio, the high-low temperature ratio, the compression and expansion isentropic efficiencies etc. on the performance of the cycle is analyzed. The optimum criteria of some important parameters such as the power output, efficiency and pressure ratio are derived. The results obtained from this unified cycle model are very general and useful, from which the optimal performance of the Atkinson, Otto, Diesel, Dual and Miller heat engines and some new heat engines can be directly derived.

Keywords

internal combustion engine / irreversibility / power output / efficiency / optimization

Introduction

In the last few years, many researchers have investigated the optimal performance of internal combustion engines by using finite-time thermodynamics [1-3] and many important conclusions have been obtained. Mozurkewich et al. [4] applied optimal control theory to optimize piston movement for ideal Otto cycles. Angulo-Brown et al. [5] optimized a non-endoreversible Otto cycle to improve power output and efficiency. Chen et al. [6] discussed the heat transfer effects on the net work output and efficiency characteristics for an air-standard Otto cycle. Bhattacharyya [7] investigated the effect of compression ratio and cut-off ratio on an irreversible Diesel cycle. Akash [8] presented the effect of heat transfer on the performance of an air-standard Diesel cycle. Ge et al. [9] analyzed the heat transfer loss and friction-like term loss on reciprocating heat-engine cycles. Chen et al. [10] and Zhao et al. [11-14] established an irreversible cycle and investigated the influence of multi-irreversibilities such as the non-isentropic compression and expansion processes, finite rate heat transfer processes and heat leak loss through the cylinder wall. These results are closer to the reality than the classical thermodynamic theory.

Based on Refs. [10-14], a further step taken in this paper is to build a unified cycle model of a class of internal combustion heat engines, in which the cycle of the heat engine is composed of two heat additions and two heat rejections at constant volume and constant pressure, and two adiabatic processes. The influence of multi-irreversibilities mainly resulting from the adiabatic processes, finite-time processes and heat leak loss through the cylinder wall on the performance of the cycle are taken into account. The power output and efficiency of the cycle are derived and optimized with respect to the high-low pressure ratio of the working substance, and some important characteristic curves are given. The results obtained, which include the performance characteristics of the irreversible Atkinson, Otto, Diesel, Dual and Miller heat engines, are very general and useful.

A unified cycle model

The temperature-entropy (T-S) diagram of an irreversible internal combustion engine is shown in Fig. 1, where T_{\mathrm{1}} , T_{\mathrm{2}S} , T_{\mathrm{2}} , T_{\mathrm{3}} , T_{\mathrm{4}} , T_{\mathrm{5}} , T_{\mathrm{5}S} and T_{\mathrm{6}} are, respectively, the temperatures of the working substance in state points 1, 2S, 2, 3, 4, 5, 5S and 6. Process 1-2S is a reversible adiabatic compression, while process 1-2 is an irreversible adiabatic process which takes into account the internal irreversibility in the real compression process. The heat addition occurs in two steps: processes 2-3 and 3-4 are heat additions at constant volume and constant pressure, respectively. Process 4-5S is a reversible adiabatic expansion, while process 4-5 is an irreversible adiabatic process that takes into account the internal irreversibility in the real expansion process. The heat rejection occurs in two steps: processes 5-6 and 6-1 are heat rejections at constant volume and constant pressure, respectively. In this paper, an irreversible cycle consisting of states 1-2-3-4-5-6-1 are considered, which may include a reversible cycle consisting of 1-2S-3-4-5S-6-1.

For the sake of calculation, it is often assumed that the working substance of the cycle is an ideal gas such that the heats added to and rejected by the working substance are, respectively, given by

(1)Q_{in}=Q_{\mathrm{23}}+Q_{\mathrm{34}}=C_V(T_{\mathrm{3}}-T_{\mathrm{2}})+C_p(T_{\mathrm{4}}-T_{\mathrm{3}}),

and

(2)Q_{out}=Q_{\mathrm{56}}+Q_{\mathrm{61}}=C_V(T_{\mathrm{5}}-T_{\mathrm{6}})+C_p(T_{\mathrm{6}}-T_{\mathrm{1}}),

where, C_p and C_V are the heat capacities at constant pressure and constant volume, respectively.

In order to describe the irreversibility of the two adiabatic processes, the compression and expansion isentropic efficiencies are introduced, which are defined as [2,3,15,16]

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

and

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

From Eqs. (3) and (4), and the adiabatic equations of an ideal gas, Eqs. (5) and (6) can be obtained

(5)

(6)

where,r_T=T_{\mathrm{4}}/T_{\mathrm{1}}>\mathrm{1} is the high-low temperature ratio of the working substance, r_p=p_{\mathrm{4}}/p_{\mathrm{1}}>\mathrm{1} is the high-low pressure ratio, and r=C_p/C_V is the specific heat ratio.

For a real heat engine cycle, the heat leak loss between the working substance and the ambient outside cylinder wall during combustion is not negligible. The heat leak loss through the cylinder wall is often assumed to obey the Newtonian law [17-19] and the heat added to the working substance by combustion is given in Eq. (7) [10-14]

(7)Q_{in}=Q_{\mathrm{T}}-\beta (T_{\mathrm{2}}+T_{\mathrm{4}}-\mathrm{2}{T}_{\mathrm{0}}),

where, Q_{\mathrm{T}} is the total heat released by combustion, \beta is the constant related to combustion and heat transfer, and T_{\mathrm{0}} is the average temperature of the cylinder wall.

Furthermore, it is often proposed that the time spent on a process is proportional to the temperature difference of the process [10-14,20-22]. Thus, the time spent on the heat exchange processes can be, respectively, given by

(8)t_{\mathrm{23}}=k_{\mathrm{1}}(T_{\mathrm{3}}-T_{\mathrm{2}}),

(9)t_{\mathrm{34}}=k_{\mathrm{2}}(T_{\mathrm{4}}-T_{\mathrm{3}}),

(10)t_{\mathrm{56}}=k_{\mathrm{3}}(T_{\mathrm{5}}-T_{\mathrm{6}}),

and

(11)t_{\mathrm{61}}=k_{\mathrm{4}}(T_{\mathrm{6}}-T_{\mathrm{1}}),

where, k_{\mathrm{1}}, k_{\mathrm{2}} ,k_{\mathrm{3}} and k_{\mathrm{4}} are proportional coefficients. Also it can be assumed that the time spent on the two adiabatic processes are proportional to those spent on the four heat exchange processes [10-14] and may be expressed as

(12)t_{\mathrm{12}}+t_{\mathrm{45}}=a(t_{\mathrm{23}}+t_{\mathrm{34}}+t_{\mathrm{56}}+t_{\mathrm{61}}),

where, a is a proportional coefficient. Thus, the cycle period is given by

(13)$$\begin{gathered} \tau =t_{\mathrm{12}}+t_{\mathrm{23}}+t_{\mathrm{34}}+t_{\mathrm{45}}+t_{\mathrm{56}}+t_{\mathrm{61}} \\ =(\mathrm{1}+a)(t_{\mathrm{23}}+t_{\mathrm{34}}+t_{\mathrm{56}}+t_{\mathrm{61}}) \\ =(\mathrm{1}+a)k_{\mathrm{1}}[(T_{\mathrm{3}}-T_{\mathrm{2}})+{\displaystyle \frac{k_{\mathrm{2}}}{k_{\mathrm{1}}}}(T_{\mathrm{4}}-T_{\mathrm{3}})+{\displaystyle \frac{k_{\mathrm{3}}}{k_{\mathrm{1}}}}(T_{\mathrm{5}}-T_{\mathrm{6}})+{\displaystyle \frac{k_{\mathrm{4}}}{k_{\mathrm{1}}}}(T_{\mathrm{6}}-T_{\mathrm{1}})]. \end{gathered}$$

From Eqs. (1), (2), (5)-(7) and (13), the power output and efficiency of the cycle can be derived

(14)

(15)where,

(16a)d_{\mathrm{1}}=T_{\mathrm{1}}(\mathrm{1}/{\eta }_c+r_T{\eta }_e+r-\mathrm{1})+T_{\mathrm{6}}(\mathrm{1}-r),

(16b)d_{\mathrm{2}}=(T_{\mathrm{1}}{r}_T^{\mathrm{1}-r})/{\eta }_c,

(16c)d_{\mathrm{3}}={\eta }_e{(r_T{T}_{\mathrm{1}})}^r{T}_{\mathrm{6}}^{\mathrm{1}-r},

(16d)d_{\mathrm{4}}=T_{\mathrm{1}}[r_T-\mathrm{1}+\mathrm{1}/{\eta }_c+k_{\mathrm{3}}{r}_T(\mathrm{1}-{\eta }_e)/k_{\mathrm{1}}-k_{\mathrm{4}}/k_{\mathrm{1}}]+T_{\mathrm{6}}(k_{\mathrm{4}}/k_{\mathrm{1}}-k_{\mathrm{3}}/k_{\mathrm{1}}),

(16e)d_{\mathrm{5}}=T_{\mathrm{1}}[r_T-\mathrm{1}+\mathrm{1}/{\eta }_c+\beta (\mathrm{1}-\mathrm{1}/{\eta }_c+r_T)/C_V]-(\mathrm{2}\beta /C_V)T_{\mathrm{0}},

(16f)e_{\mathrm{1}}=T_{\mathrm{1}}[\mathrm{1}/{\eta }_c+r_T{\eta }_e+r_T(r-\mathrm{1})]+T_{\mathrm{3}}(\mathrm{1}-r),

(16g)e_{\mathrm{2}}=(T_{\mathrm{1}}^r{T}_{\mathrm{3}}^{\mathrm{1}-r})/{\eta }_c,

(16h)e_{\mathrm{3}}=r_T^r{T}_{\mathrm{1}}{\eta }_e,

(16i)e_{\mathrm{4}}=T_{\mathrm{1}}[\mathrm{1}/{\eta }_c-\mathrm{1}+k_{\mathrm{2}}{r}_T/k_{\mathrm{1}}+k_{\mathrm{3}}{r}_T(\mathrm{1}-{\eta }_e)/k_{\mathrm{1}}+T_{\mathrm{3}}(\mathrm{1}-k_{\mathrm{2}}/k_{\mathrm{1}}),

and

(16j)e_{\mathrm{5}}=T_{\mathrm{3}}(\mathrm{1}-r)+T_{\mathrm{1}}[\mathrm{1}/{\eta }_c-\mathrm{1}+r{r}_T+(\mathrm{1}-\mathrm{1}/{\eta }_c+r_T)\beta /C_V]-\mathrm{2}\beta T_{\mathrm{0}}/C_V.

It is important to note that the temperatures T_{\mathrm{3}} and T_{\mathrm{6}} of the working substance in the internal combustion engine cycle are two controllable parameters. T_{\mathrm{3}} can be theoretically varied from T_{\mathrm{2}} to T_{\mathrm{4}} while T_{\mathrm{6}} should be varied from T_{\mathrm{1}} to T_{\mathrm{5}}. For different values of T_{\mathrm{3}} and T_{\mathrm{6}}, the power output and efficiency of the cycle have different expressions. When T_{\mathrm{3}}=T_{\mathrm{2}} and T_{\mathrm{6}}=T_{\mathrm{1}}, the unified cycle becomes the Diesel cycle; when T_{\mathrm{3}}=T_{\mathrm{4}} and T_{\mathrm{6}}=T_{\mathrm{5}}, it becomes the Atkinson cycle; when T_{\mathrm{3}}=T_{\mathrm{4}} and T_{\mathrm{6}}=T_{\mathrm{1}}, it becomes the Otto cycle; when T_{\mathrm{3}}=T_{\mathrm{4}} and , it becomes the Miller cycle; and when and T_{\mathrm{6}}=T_{\mathrm{1}}, it becomes the Dual cycle. So the performance of the Diesel, Atkinson, Otto, Miller and Dual cycles can be discussed as long as the different values of T_{\mathrm{3}} and T_{\mathrm{6}} are chosen. The results in Refs. [10-14] can be derived easily from this paper. So the conclusions obtained here are very general and useful.

Performance analysis and parametric optimization

From Eqs. (14) and (15), curves of power output and efficiency varying with the high-low pressure ratio can be plotted, as illustrated in Figs. 2 and 3, where the parameters T_{\mathrm{1}}=\mathrm{350}K, T_{\mathrm{0}}=\mathrm{300}K, r_T=\mathrm{8}, r=\mathrm{1.4}, k_{\mathrm{1}}=k_{\mathrm{2}}=k_{\mathrm{3}}=k_{\mathrm{4}}, \beta /C_V=\mathrm{0.1} are chosen, P^{*}=P{k}_{\mathrm{1}}(\mathrm{1}+a)/C_V is the dimensionless power output, and r_{pP} and r_{p\eta } are, respectively, the high-low pressure ratios at the maximum dimensionless power output P_{\max }^{*} and the maximum efficiency {\eta }_{\max }. The curves in Figs. 2 and 3 demonstrate clearly that when the compression and expansion isentropic efficiencies are smaller than one \mathrm{a}\mathrm{n}\mathrm{d}\beta /C_V>\mathrm{0}, there exist a maximum power output and a maximum efficiency. From Eq. (14) and its extremal condition \partial P/\partial r_p=\mathrm{0}, the high-low pressure ratio r_{pP} at the maximum dimensionless power output can be obtained, which can be expressed by the transcendental Eq. (17).

(17)

Similarly, from Eq. (15) and its extremal condition \partial \eta /\partial r_p=\mathrm{0}, the high-low pressure ratio r_{p\eta } at the maximum efficiency can also be derived

(18)

Moreover, the curves in Figs. 2 and 3 also indicate that the pressure ratio r_{pP} at the maximum dimensionless power output is greater than the pressure ratio r_{p\eta } at the maximum efficiency, i.e., r_{pP}>r_{p\eta }.

Furthermore, the curves in Figs. 2 and 3 reveal, too, that both the power output and efficiency increase quickly as the compression and expansion efficiencies increase. When the compression and expansion efficiencies are equal to one, the power output is a monotonically increasing function of the pressure ratio, while there is still a maximum value for the efficiency when r_p=r_{p\eta }.

From Eqs. (14) and (15), the characteristic curves of the dimensionless power output versus the efficiency for a class of internal combustion engines including the irreversible Diesel, Atkinson, Otto, Miller and Dual heat engines can be obtained, as displayed in Fig. 4, where P_m^{*}and{\eta }_m are the dimensionless power output at the maximum efficiency and the efficiency at the maximum power output. Figure 4 also exhibits that when the heat engine is operated in those regions of the P^*-η curve which has a positive slope, the dimensionless power output decreases with decreasing efficiency. These regions are not the optimal operating regions of the heat engine. Obviously, the optimal operating region of the heat engine should be situated in the part of the P^*-η curve which has a negative slope. When the heat engine is operated in the optimal operating region, the dimensionless power output increases with decreasing efficiency, and vice versa. Thus, the optimal ranges of the dimensionless power output and efficiency should be

(19)P_{\max }^{*}\le P^{*}\le P_{\max }^{*},

and

(20){\eta }_m\le \eta \le {\eta }_{\max }.

From Eqs. (19) and (20), the optimal region of the pressure ratio can be derived

(21)r_{p\eta }\le r_p\le r_{pP}.

This conclusion can be further explained by Figs. 2 and 3. When or r_{p\eta }, either the dimensionless power output or efficiency decreases with the decrease of the pressure ratio; when r_p>r_{pP} or r_{p\eta }, the dimensionless power output and efficiency decrease with the increase of the pressure ratio. Apparently, the regions that the pressure ratios are located in r_p>r_{pP} or are not the optimal operating ones of a class of internal combustion engines including irreversible Diesel, Atkinson, Otto, Miller and Dual heat engines.

The above results show that P_{\max }^{*} , {\eta }_{\max }, P_m^{*},{\eta }_m, r_{pP} and r_{p\eta } are six important parameters of the heat engine. P_{\max }^{*} and {\eta }_{\max } determine the upper bounds of the dimensionless power output and efficiency, P_m^{*}and{\eta }_m determine the lower bounds of the dimensionless power output and efficiency, and r_{pP} and r_{p\eta } determine, respectively, the upper and lower bounds of the high-low pressure ratio. It should be pointed out that P_{\max }^{*}, {\eta }_{\max }, P_m^{*},{\eta }_m, r_{pP} and r_{p\eta } not only depend on the compression and expansion efficiencies but also on other parameters such as T_{\mathrm{1}}, T_{\mathrm{3}}, T_{\mathrm{6}} and r_T. Thus, in the design of a class of internal combustion engines including the irreversible Diesel, Atkinson, Otto, Miller and Dual heat engines, a reasonable high-low pressure ratio should be chosen for engineers according to Eqs. (17), (18) and (21), so that the heat engine can be operated in the optimal region.

It is worthwhile to point out that the heat leak loss does not influence the power output of the heat engine cycle but it always decreases the efficiency, which can be clearly seen from Fig. 5. In particular, when the compression and expansion efficiencies are equal to one and the heat leak loss through the cylinder wall of the engine is equal to zero, both the power output and efficiency are monotonically increasing functions of the pressure ratio.

The P^*-η curve of the cycle for some given values of the high-low temperature ratio of the working substance can also be generated, as depicted in Fig. 6. It is very obviously seen from Fig. 6 that when r_T increases, the cycle of P_{\max }^{*}, {\eta }_{\max }, P_m^{*}and{\eta }_m increase, respectively.

In order to further expound the influence of temperatures T_{\mathrm{3}} and T_{\mathrm{6}} on the performance of heat engine, the P^*-η curve of the cycle for some given values of T_{\mathrm{3}} and T_{\mathrm{6}} can be plotted, as shown in Fig. 7. It is clearly seen from Fig. 7 that when T_{\mathrm{3}} varies from T_{\mathrm{2}} to T_{\mathrm{4}}, and T_{\mathrm{6}} varies from T_{\mathrm{1}} to T_{\mathrm{5}}, the cycle will vary from the Diesel to the Dual cycle, then to the Otto cycle, and then to the Miller cycle and finally to the Atkinson cycle. Furthermore, it is important to point out that the power output and efficiency increase when T_{\mathrm{3}} is close to T_{\mathrm{2}} and T_{\mathrm{6}} approaches T_{\mathrm{1}} accordingly.

Several special cases

1) When T_{\mathrm{3}}=T_{\mathrm{2}} and T_{\mathrm{6}}=T_{\mathrm{1}}, processes 2-3 and 3-4 become isobaric processes, processes 5-6 and 6-1 become isochoric processes, so the unified cycle reduces the Diesel cycle. Equations (14) and (15) may be written as

(22)P={\displaystyle \frac{C_V}{(\mathrm{1}+a)k_{\mathrm{1}}}}·{\displaystyle \frac{e_{\mathrm{1}D}+e_{\mathrm{2}D}{r}_p^{\mathrm{1}-\mathrm{1}/r}-e_{\mathrm{3}D}{r}_p^{\mathrm{1}-r}-e_{\mathrm{4}D}{[(\mathrm{1}-\mathrm{1}/{\eta }_c)/r_p+\mathrm{1}/({\eta }_c{r}_p^{\mathrm{1}/r})]}^{\mathrm{1}-r}}{e_{\mathrm{5}D}+e_{\mathrm{6}D}{r}_p^{\mathrm{1}-\mathrm{1}/r}+k_{\mathrm{3}}{e}_{\mathrm{3}D}{r}_p^{\mathrm{1}-r}/k_{\mathrm{1}}-e_{\mathrm{4}D}{[(\mathrm{1}-\mathrm{1}/{\eta }_c)/r_p+\mathrm{1}/({\eta }_c{r}_p^{\mathrm{1}/r})]}^{\mathrm{1}-r}}}

and

(23)\eta ={\displaystyle \frac{e_{\mathrm{1}D}+e_{\mathrm{2}D}{r}_p^{\mathrm{1}-\mathrm{1}/r}-e_{\mathrm{3}D}{r}_p^{\mathrm{1}-r}-e_{\mathrm{4}D}{[(\mathrm{1}-\mathrm{1}/{\eta }_c)/r_p+\mathrm{1}/({\eta }_c{r}_p^{\mathrm{1}/r})]}^{\mathrm{1}-r}}{e_{\mathrm{7}D}+e_{\mathrm{2}D}{r}_p^{\mathrm{1}-\mathrm{1}/r}+e_{\mathrm{4}D}(\beta /C_V-\mathrm{1}){[(\mathrm{1}-\mathrm{1}/{\eta }_c)/r_p+\mathrm{1}/({\eta }_c{r}_p^{\mathrm{1}/r})]}^{\mathrm{1}-r}}},

where,

(24a)e_{\mathrm{1}D}=T_{\mathrm{1}}[r/{\eta }_c+r_T{\eta }_e+(\mathrm{1}-r_T)(\mathrm{1}-r)],

(24b)e_{\mathrm{2}D}=T_{\mathrm{1}}(\mathrm{1}-r)/{\eta }_c,

(24c)e_{\mathrm{3}D}=e_{\mathrm{3}}=r_T^r{T}_{\mathrm{1}}{\eta }_e,

(24d)e_{\mathrm{4}D}=T_{\mathrm{1}}/{\eta }_c,

(24e)e_{\mathrm{5}D}=k_{\mathrm{2}}{T}_{\mathrm{1}}(\mathrm{1}/{\eta }_c-\mathrm{1})/k_{\mathrm{1}}+T_{\mathrm{1}}\{k_{\mathrm{2}}{r}_T/k_{\mathrm{1}}+k_{\mathrm{3}}[r_T(\mathrm{1}-{\eta }_e)-\mathrm{1}]/k_{\mathrm{1}}\},

(24f)e_{\mathrm{6}D}=T_{\mathrm{1}}(\mathrm{1}-k_{\mathrm{2}}/k_{\mathrm{1}})/{\eta }_c,

and

(24g)e_{\mathrm{7}D}=T_{\mathrm{1}}[r(\mathrm{1}/{\eta }_c-\mathrm{1}+r_T)+\beta (\mathrm{1}-\mathrm{1}/{\eta }_c+r_T)/C_V]-(\mathrm{2}\beta /C_V)T_{\mathrm{0}}.

Equations (22) and (23) have been given in Ref. [11]. The characteristic curves of the Diesel cycle are presented by the solid curves in Figs. 2-4 and Figs. 6 and 7. It is necessary to note that the Diesel heat engine cycle has higher power output and efficiency than other internal combustion engines.

2) When T_{\mathrm{3}}=T_{\mathrm{4}} and T_{\mathrm{6}}=T_{\mathrm{5}}, processes 2-3 and 3-4 become isochoric processes, processes 5-6 and 6-1 become isobaric processes, so the unified cycle reduces the Atkinson cycle. Equations (14) and (15) may be written as

(25)P={\displaystyle \frac{C_V}{(\mathrm{1}+a)k_{\mathrm{1}}}}·{\displaystyle \frac{d_{\mathrm{1}A}+d_{\mathrm{2}A}{r}_p^{\mathrm{1}/r-\mathrm{1}}-d_{\mathrm{3}A}{r}_p^{r-\mathrm{1}}-d_{\mathrm{4}A}{[{\eta }_e{r}_p^{\mathrm{1}/r}+r_p(\mathrm{1}-{\eta }_e)]}^{\mathrm{1}-r}}{d_{\mathrm{5}A}+d_{\mathrm{6}A}{r}_p^{\mathrm{1}/r-\mathrm{1}}-d_{\mathrm{3}A}{r}_p^{r-\mathrm{1}}+k_{\mathrm{3}}{d}_{\mathrm{4}A}{[{\eta }_e{r}_p^{\mathrm{1}/r}+r_p(\mathrm{1}-{\eta }_e)]}^{\mathrm{1}-r}/k_{\mathrm{1}}}},

and

(26)\eta ={\displaystyle \frac{d_{\mathrm{1}A}+d_{\mathrm{2}A}{r}_p^{\mathrm{1}/r-\mathrm{1}}-d_{\mathrm{3}A}{r}_p^{r-\mathrm{1}}-d_{\mathrm{4}A}{[{\eta }_e{r}_p^{\mathrm{1}/r}+r_p(\mathrm{1}-{\eta }_e)]}^{\mathrm{1}-r}}{d_{\mathrm{5}}+d_{\mathrm{3}A}(\beta /C_V-\mathrm{1})r_p^{r-\mathrm{1}}}},

where,

(27a)d_{\mathrm{1}A}=T_{\mathrm{1}}[\mathrm{1}/{\eta }_c+r_{T}r{\eta }_e+(r-\mathrm{1})(\mathrm{1}-r_T)],

(27b)d_{\mathrm{2}A}=r_T{T}_{\mathrm{1}}(\mathrm{1}-r){\eta }_e,

(27c)d_{\mathrm{3}A}=T_{\mathrm{1}}{r}_T^{\mathrm{1}-r}/{\eta }_c,

(27d)d_{\mathrm{4}A}={\eta }_e{r}_T{T}_{\mathrm{1}},

(27e)d_{\mathrm{5}A}=T_{\mathrm{1}}\{r_T-\mathrm{1}+\mathrm{1}/{\eta }_c+(k_{\mathrm{4}}/k_{\mathrm{1}})[(\mathrm{1}-{\eta }_e)r_T-\mathrm{1}]\},

and

(27f)d_{\mathrm{6}A}=r_T{T}_{\mathrm{1}}{\eta }_e(k_{\mathrm{4}}/k_{\mathrm{1}}-k_{\mathrm{3}}/k_{\mathrm{1}}).

Equations (25) and (26) have been given in Ref. [12]. The characteristic curves of the Atkinson cycle are presented by the dashed curves in Figs. 2-5 and 7.

3) When T_{\mathrm{3}}=T_{\mathrm{4}} and T_{\mathrm{6}}=T_{\mathrm{1}}, processes 2-3 , 3-4, 5-6 and 6-1 become isochoric processes, so the unified cycle reduces the Otto cycle. Equations (14) and (15) may be written as

(28)P={\displaystyle \frac{C_V}{(\mathrm{1}+a)k_{\mathrm{1}}}}·{\displaystyle \frac{d_{\mathrm{1}o}-d_{\mathrm{2}\mathrm{o}}{r}_p^{r-\mathrm{1}}-d_{\mathrm{3}o}{r}_p^{\mathrm{1}-r}}{d_{\mathrm{4}o}-d_{\mathrm{2}o}{r}_p^{r-\mathrm{1}}+k_{\mathrm{3}}{d}_{\mathrm{3}o}{r}_p^{\mathrm{1}-r}/k_{\mathrm{1}}}},

and

(29)\eta ={\displaystyle \frac{d_{\mathrm{1}o}-d_{\mathrm{2}o}{r}_p^{r-\mathrm{1}}-d_{\mathrm{3}o}{r}_p^{\mathrm{1}-r}}{d_{\mathrm{5}}+d_{\mathrm{2}o}[(\beta /C_V)-\mathrm{1}]r_p^{r-\mathrm{1}}}}.

where,

(30a)d_{\mathrm{1}o}=T_{\mathrm{1}}(\mathrm{1}/{\eta }_c+r_T{\eta }_e),

(30b)d_{\mathrm{2}o}=d_{\mathrm{2}}=(T_{\mathrm{1}}{r}_T^{\mathrm{1}-r})/{\eta }_c,

(30c)d_{\mathrm{3}o}={\eta }_e{T}_{\mathrm{1}}{r}_T^r,

and

(30d)d_{\mathrm{4}o}=T_{\mathrm{1}}[r_T-\mathrm{1}+\mathrm{1}/{\eta }_c+k_{\mathrm{3}}{r}_T(\mathrm{1}-{\eta }_e)/k_{\mathrm{1}}-k_{\mathrm{3}}/k_{\mathrm{1}}].

Equations (28) and (29) have been given in Ref. [10]. The characteristic curves of the Otto cycle are presented by the short dash-dot curves in Figs. 2-5 and 7.

4) When T_{\mathrm{3}}=T_{\mathrm{4}} and , processes 2-3 and 3-4 become isochoric processes, so the unified cycle reduces the Miller cycle. The power output and efficiency are still expressed by Eqs. (14) and (15), which have been given in Ref. [14]. The characteristic curves of the Miller cycle are presented by the long dash-dot curves in Figs. 2-4, 6 and 7.

In such a case, the temperature T_{\mathrm{6}} varies from T_{\mathrm{1}} to T_{\mathrm{5}} and the power output and efficiency of the cycle are higher when T_{\mathrm{6}}=T_{\mathrm{5}} than others.

5) When and T_{\mathrm{6}}=T_{\mathrm{1}}, processes 5-6 and 6-1 become isochoric processes, so the unified cycle reduces the Dual cycle. The power output and efficiency are still expressed by Eqs. (14) and (15), which have been given in Ref. [13]. The characteristic curves of the Dual cycle are presented by the dot curves in Figs. 2-5 and 7.

In this case, the values of the power output and efficiency will vary from great to little when the temperature T_{\mathrm{3}} varies from T_{\mathrm{2}} to T_{\mathrm{4}}.

6) When the irreversibility in the adiabatic processes is negligible, i.e., {\eta }_c={\eta }_e\to \mathrm{1}, the power output and efficiency of the cycle may be, respectively, written as

(31)

and

(32)

where,

(33a)d_{\mathrm{1}}^{\prime }=T_{\mathrm{1}}(r_T+r)+T_{\mathrm{6}}((\mathrm{1}-r),

(33b)d_{\mathrm{2}}^{\prime }=T_{\mathrm{1}}{r}_T^{\mathrm{1}-r},

(33c)d_{\mathrm{3}}^{\prime }={(r_T{T}_{\mathrm{1}})}^r{T}_{\mathrm{6}}^{\mathrm{1}-r},

(33d)d_{\mathrm{4}}^{\prime }=T_{\mathrm{1}}(r_T-k_{\mathrm{4}}/k_{\mathrm{1}})+T_{\mathrm{6}}(k_{\mathrm{4}}/k_{\mathrm{1}}-k_{\mathrm{3}}/k_{\mathrm{1}}),

(33e)d_{\mathrm{5}}^{\prime }=T_{\mathrm{1}}[r_T+(\beta /C_V)r_T]-(\mathrm{2}\beta /C_V)T_{\mathrm{0}},

(33f)e_{\mathrm{1}}^{\prime }=T_{\mathrm{1}}(\mathrm{1}+r_{T}r)+T_{\mathrm{3}}(\mathrm{1}-r),

(33g)e_{\mathrm{2}}^{\prime }=T_{\mathrm{1}}^r{T}_{\mathrm{3}}^{\mathrm{1}-r},

(33h)e_{\mathrm{3}}^{\prime }=r_T^r{T}_{\mathrm{1}},

(33i)e_{\mathrm{4}}^{\prime }=T_{\mathrm{1}}(k_{\mathrm{2}}{r}_T/k_{\mathrm{1}}-k_{\mathrm{3}}/k_{\mathrm{1}})+T_{\mathrm{3}}(\mathrm{1}-k_{\mathrm{2}}/k_{\mathrm{1}}),

and

(33j)e_{\mathrm{5}}^{\prime }=T_{\mathrm{3}}(\mathrm{1}-r)+r_T{T}_{\mathrm{1}}(r+\beta /C_V)-(\mathrm{2}\beta /C_V)T_{\mathrm{0}}.

In such a case, the power output is a monotonically increasing function of the pressure ratio, while there is still a maximum value for the efficiency when r_p=r_{p\eta }, as shown by the curves c in Figs. 2 and 3 and the curves b in Fig. 4.

7) When the heat leak loss through the cylinder wall of the engine is negligible, i.e., \beta /C_V\to \mathrm{0}, the power output is still expressed by Eq. (14), while the efficiency may be simplified as

(34)

where,

(35a)d_{\mathrm{5}}^{″}=T_{\mathrm{1}}(r_T-\mathrm{1}+\mathrm{1}/{\eta }_c)

and

(35b)e_{\mathrm{5}}^{″}=T_{\mathrm{3}}(\mathrm{1}-r)+T_{\mathrm{1}}(\mathrm{1}/{\eta }_c-\mathrm{1}+r{r}_T).

A comparison of the curves b (\beta /C_V\to \mathrm{0}) with the curves a (\beta /C_V=\mathrm{0.1}) in Fig. 5 reveals that the influence of the heat leak loss on the efficiency of the cycle is obvious and has to be considered in the performance analysis of the engine.

8) When both the irreversibility in the adiabatic processes and the heat leak loss through the cylinder wall of the engine are negligible, i.e., {\eta }_c={\eta }_e\to \mathrm{1} and \beta /C_V\to \mathrm{0}, the expression of the power output is still given by Eq. (31), while the efficiency may be further simplified as

(36)

where

(37)e_{\mathrm{5}}^{‴}=T_{\mathrm{3}}(\mathrm{1}-r)+r{r}_T{T}_{\mathrm{1}}.

In this case, the efficiency is a monotonically increasing function of the pressure ratio, as shown by the curve c in Fig. 5. This is the result of a reversible internal combustion engine.

Conclusions

A unified cycle model of a class of internal combustion engines has been proposed, in which the Diesel, Atkinson, Otto, Miller and Dual cycles may be included. In the cycle model, the multi-irreversibilities, which come from the adiabatic compression and expansion processes, finite-time processes and heat leak loss through the cylinder wall, have been used to analyze the performance characteristics of the cycle. The power output and efficiency of the cycle are optimized with respect to the high-low pressure ratio of the working substance. The optimum criteria of some important parameters, such as the power output, efficiency and high-low pressure ratio are given. Several special cases of the cycle are discussed. The results obtained may be helpful to understand the inherent relation and the essential distinction of the optimal performance of a class of internal combustion engines including the Diesel, Atkinson, Otto, Miller and Dual heat engines.

References

[1] Wu C, Chen L, Chen J. Recent Advances in Finite-Time Thermodynamics. New York: Nova Science Publishers, Inc., 1999

[2] Bejan A. Advanced Engineering Thermodynamics. 2nd ed. New York: Wiley, 1996

[3] Moran M J, Shapiro H N. Fundamentals of Engineering Thermodynamics. 2nd ed. New York: Wiley, 1992

[4] Mozurkewich M, Stephen B. Optimal paths for thermodynamic systems: The ideal Otto cycle. Journal of Applied Physics, 1982, 53(1): 34-42

[5] Angulo-Brown F, Rocha-Martínez J, Navarrete-González T D, Navarrete- González T. A non-endoreversible Otto cycle model: improving power output and efficiency. Journal of Physics D, Applied Physics, 1996, 29(1): 80-83

[6] Chen L, Wu C, Sun F R, Cao S. Heat transfer effects on the net work output and efficiency characteristics for an air-standard Otto cycle. Energy Conversion and Management, 1998, 39(7): 643-648

[7] Bhattacharyya S. Optimizing an irreversible Diesel cycle—fine tuning of compression ratio and cut-off ratio. Energy Conversion and Management, 2000, 41(8): 847-854

[8] Akash B. Effect of heat transfer on the performance of an air-standard Diesel cycle. International Communications in Heat and Mass Transfer, 2001, 28(1): 87-95

[9] Ge Y L, Chen L G, Sun F R, Wu C. Reciprocating heat-engine cycles. Applied Energy, 2005, 81(4): 397-408

[10] Chen J C, Zhao Y R, He J Z. Optimization criteria for the important parameters of an irreversible Otto heat-engine. Applied Energy, 2006, 83(3): 228-238

[11] Zhao Y R, Lin B H, Zhang Y, Chen J C.Performance analysis and parametric optimum design of an irreversible Diesel heat engine. Energy Conversion and Management, 2006, 47(18,19): 3383-3392

[12] Zhao Y R, Chen J C. Performance analysis and parametric optimum criteria of an irreversible Atkinson heat-engine. Applied Energy, 2006, 83(8): 789-800

[13] Zhao Y R, Chen J C. An irreversible heat engine model including three typical thermodynamic cycles and their optimum performance analysis. International Journal of Thermal Sciences, 2007, 46(6): 605-613

[14] Zhao Y R, Chen J C. Performance analysis of an irreversible Miller heat engine and its optimum criteria. Applied Thermal Engineering, 2007, 27(11,12): 2051-2058

[15] Chen L G, Wu C, Sun F R.Cooling load versus cop characteristics for an irreversible air refrigeration cycle. Energy Conversion and Management, 1998, 39(1,2): 117-125

[16] Aragón-González G, Canales-Palma A, León-Galicia A. Maximum irreversible work and efficiency in power cycles. Journal of Physics. D, Applied Physics, 2000, 33(11): 1403-1409

[17] Curzon F, Ahlborn B. Efficiency of a Carnot engine at maximum power output. American Journal of Physics, 1975, 43(1): 22-24

[18] Salamon P, Nitzan A. Finite time optimizations of a Newton’s law Carnot cycle. Journal of Chemical Physics, 1981, 74(6): 3546-3560

[19] Chen J C, Yan Z J. Unified description of endoreversible cycles. Physical Review A: General Physics, 1989, 39(8): 4140-4147

[20] Angulo-Brown F, Fernández-Betanzos J, Díaz-Pico C A. Compression ratio of an optimized air standard Otto-cycle model. European Journal of Physics, 1994, 15(1): 38-42

[21] Chen L G, Ge Y L, Sun F R. Unified thermodynamic description and optimization for a class of irreversible reciprocating heat engine cycles. Proceedings of the Institution of Mechanical Engineers. Part D, Journal of Automobile Engineering, 2008, 222(8): 1489-1500

[22] Ebrahimi R. Performance optimization of a Diesel cycle with specific heat ratio. Journal of American Science, 2010, 6(1): 157-161

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag Berlin Heidelberg