Effect of variable heat capacities on performance of an irreversible Miller heat engine

Abstract

Based on the variable heat capacities of the working fluid, the irreversibility coming from the compression and expansion processes, and the heat leak losses through the cylinder wall, an irreversible cycle model of the Miller heat engine was established, from which expressions for the efficiency and work output of the cycle were derived. The performance characteristic curves of the Miller heat engine were generated through numerical calculation, from which the optimal regions of some main parameters such as the work output, efficiency and pressure ratio were determined. Moreover, the influence of the compression and expansion efficiencies, the variable heat capacities and the heat leak losses on the performance of the cycle was discussed in detail, and consequently, some significant results were obtained.

Keywords

Miller cycle / variable heat capacity / irreversibility / parametric optimization

Introduction

The Miller cycle engine was patented by an American engineer, Ralph Miller, in the 1940s, to which much attention have been paid because its expansion ratio is longer than its compression ratio. In recent years, some scholars have presented a thermodynamic analysis of the performance of an ideal air-standard Miller cycle, and a large number of significant results have been obtained [1–3]. In these investigations, the heat capacities of the working substance are assumed to be constant. In fact, the heat capacities of the working substance are variable in practical cycles. Some authors [4–7] have optimized the performance of the heat engine with variable heat capacities. For example, Ge et al. [4] have analyzed the influence of the working substance with variable heat capacities on a endoreversible Otto cycle; several authors have investigated, respectively, the effect of multi-irreversibilities including temperature-dependent heat capacities, the irreversibility coming from the compression and expansion processes and the heat leak losses through the cylinder wall on the performance of a Diesel heat engine [5], a Atkinson heat engine [6], a Miller heat engine [7] and a Dual heat engine [8]. The results obtained indicate that the effect of temperature-dependent heat capacities is obvious. For this reason, it is very significant and necessary to investigate the performance of the heat engine with variable heat capacities.

An irreversible Miller cycle model

In this paper, the multi-irreversibilities existing in a Miller heat engine are taken into account and an irreversible cycle model of a Miller heat engine is established. The cycle model is more complex than the cycle model in Ref. [7], which includes the Otto cycle model and the Atkinson cycle model. The influence of temperature-dependent heat capacities and other important parameters on the performance of the Miller cycle is discussed in detail by numerical examples and graphic method. The results obtained here will provide some references for the parametric design of a Miller heat engine.

Figure 1 shows the T-S diagram of an irreversible Miller heat engine, where processes 1–2 and 3–4 represent, respectively, an irreversible adiabatic compression and adiabatic expansion, processes 2–3, 4–5 and 5–1 represent, respectively, an isochoric heat addition, an isochoric heat rejection and an isobaric heat rejection, and processes 1–2S and 3–4S are two reversible adiabatic processes. In this paper, an irreversible cycle consisting of states 1–2–3–4–5–1 will be studied, which may include a reversible cycle consisting of states 1–2S–3–4S–5–1.

Usually, the compression and expansion efficiencies [9,10],

(1){\eta }_{\mathrm{c}}=\left(T_{\mathrm{2}S}-T_{\mathrm{1}}\right)/\left(T_{\mathrm{2}}-T_{\mathrm{1}}\right)

and

(2){\eta }_{\mathrm{e}}=\left(T_{\mathrm{4}}-T_{\mathrm{3}}\right)/\left(T_{\mathrm{4}S}-T_{\mathrm{3}}\right)

are introduced to describe the irreversibility of the two adiabatic processes. In Eqs. (1) and (2), T_{\mathrm{1}}, T_{\mathrm{2}}, T_{\mathrm{2}S}, T_{\mathrm{3}}, T_{\mathrm{4}} and T_{\mathrm{4}S} represent, respectively, the temperatures of the working substance in state points 1, 2, 2S, 3, 4 and 4S. From Eqs. (1) and (2), Eqs. (3) and (4) can be obtained.

(3)T_{\mathrm{2}}=\left(\mathrm{1}-\mathrm{1}/{\eta }_{\mathrm{c}}\right)T_{\mathrm{1}}+\left(\mathrm{1}/{\eta }_{\mathrm{c}}\right)T_{\mathrm{2}S},

(4)T_{\mathrm{4}}=\left(\mathrm{1}-{\eta }_{\mathrm{e}}\right)T_{\mathrm{3}}+{\eta }_{\mathrm{e}}{T}_{\mathrm{4}S}.

In the actual cycles, the heat capacities of the working substance are dependent on the temperature, which would inevitably affect the performance of the cycle. According to Refs. [4–8], it may be assumed that the heat capacities of the working substance are linear functions of temperature. Thus, the heat capacities at constant pressure and constant volume can be, respectively, expressed as

(5)C_{\mathrm{P}}=a+k_{\mathrm{1}}T,

(6)C_{\mathrm{V}}=b+k_{\mathrm{1}}T,

where a, b and k_{\mathrm{1}} are constants. It should be noticed that for ideal gases, the parameters a and b must satisfy a/b=k and a-b=nR, where k, n and R represent, respectively, a constant, a molar number and a molar gas constant of ideal gases.

According to Refs. [5,6], when the heat capacities of the working substance vary with the temperature, the adiabatic equation of ideal gases can be expressed as

(7)T{V}^{a/b-\mathrm{1}}{\mathrm{e}}^{k_{\mathrm{1}}T/b}=C,

where C is a constant. Using Eq. (7), two adiabatic processes of a Miller heat engine can be obtained and given by

(8)T_{\mathrm{2}S}{\mathrm{e}}^{k_{\mathrm{1}}{T}_{\mathrm{2}S}/b}=T_{\mathrm{1}}^k{T}_{\mathrm{3}}^{\mathrm{1}-k}{\mathrm{e}}^{k_{\mathrm{1}}{T}_{\mathrm{1}}/b}{r}_{\mathrm{P}}^{k-\mathrm{1}},

(9)

where r_{\mathrm{P}}=P_{\mathrm{2}}/P_{\mathrm{1}}>\mathrm{1} is the pressure ratio, P_{\mathrm{1}} and P_{\mathrm{2}} are, respectively, the lowest and highest pressure of the working substance.

Based on Eq. (6), the heats added to by the working fluid in process 2–3 is given by

(10)Q_{\mathrm{i}\mathrm{n}}=Q_{\mathrm{23}}=b\left(T_{\mathrm{3}}-T_{\mathrm{2}}\right)+{\displaystyle \frac{k_{\mathrm{1}}\left(T_{\mathrm{3}}^{\mathrm{2}}-T_{\mathrm{2}}^{\mathrm{2}}\right)}{\mathrm{2}}}.

Based on Eqs. (5) and (6), the total heat rejected by the working fluid in processes 4–5 and 5–1 is given by

(11)\begin{array}{l}{Q}_{\mathrm{o}\mathrm{u}\mathrm{t}}=Q_{\mathrm{45}}+Q_{\mathrm{51}}=b\left(T_{\mathrm{4}}-T_{\mathrm{5}}\right)+{\displaystyle \frac{k_{\mathrm{1}}\left(T_{\mathrm{4}}^{\mathrm{2}}-T_{\mathrm{5}}^{\mathrm{2}}\right)}{\mathrm{2}}}\hfill \\ +a\left(T_{\mathrm{5}}-T_{\mathrm{1}}\right)+{\displaystyle \frac{k_{\mathrm{1}}\left(T_{\mathrm{5}}^{\mathrm{2}}-T_{\mathrm{1}}^{\mathrm{2}}\right)}{\mathrm{2}}}.\hfill \end{array}

In the actual Miller cycle, the heat leak losses between the working substance and the cylinder wall can not be ignored. It is often assumed that the heat leak losses through the cylinder wall are proportional to the average temperature of both the working substance and the cylinder wall [11–13]. Thus, the total heat supplied by a combustion process is given by

(12)Q_{\mathrm{23}}=Q_{\mathrm{T}}-B\left(T_{\mathrm{2}}+T_{\mathrm{3}}-\mathrm{2}{T}_{\mathrm{0}}\right),

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

From Eqs. (10)–(12), the work output and efficiency of the cycle can be calculated, and respectively, given by

(13)\begin{array}{l}W=Q_{\mathrm{i}\mathrm{n}}-Q_{\mathrm{o}\mathrm{u}\mathrm{t}}\hfill \\ =b\left(T_{\mathrm{3}}+T_{\mathrm{5}}-T_{\mathrm{2}}-T_{\mathrm{4}}\right)\hfill \\ -a\left(T_{\mathrm{5}}-T_{\mathrm{1}}\right)+{\displaystyle \frac{k_{\mathrm{1}}\left(T_{\mathrm{1}}^{\mathrm{2}}+T_{\mathrm{3}}^{\mathrm{2}}-T_{\mathrm{2}}^{\mathrm{2}}-T_{\mathrm{4}}^{\mathrm{2}}\right)}{\mathrm{2}}},\hfill \end{array}

(14)\eta ={\displaystyle \frac{W}{Q_{\mathrm{T}}}}={\displaystyle \frac{b\left(T_{\mathrm{3}}+T_{\mathrm{5}}-T_{\mathrm{2}}-T_{\mathrm{4}}\right)-a\left(T_{\mathrm{5}}-T_{\mathrm{1}}\right)+k_{\mathrm{1}}\left(T_{\mathrm{1}}^{\mathrm{2}}+T_{\mathrm{3}}^{\mathrm{2}}-T_{\mathrm{2}}^{\mathrm{2}}-T_{\mathrm{4}}^{\mathrm{2}}\right)/\mathrm{2}}{b\left(T_{\mathrm{3}}-T_{\mathrm{2}}\right)+k_{\mathrm{1}}\left(T_{\mathrm{3}}^{\mathrm{2}}-T_{\mathrm{2}}^{\mathrm{2}}\right)/\mathrm{2}+B\left(T_{\mathrm{2}}+T_{\mathrm{3}}-\mathrm{2}{T}_{\mathrm{0}}\right)}}.

From Eq. (14), the efficiency of the cycle is found to be closely dependent on the heat leakage losses, and thus the heat leakage losses should be considered in the actual analysis of heat engines.

Performance characteristics and optimal ranges

In the Miller heat engine, T_{\mathrm{5}} is an important controllable parameter, which theoretically varies between T_{\mathrm{1}} and T_{\mathrm{4}}. When T_{\mathrm{5}}=T_{\mathrm{1}}, processes 4–5 and 5–1 become an isochoric process, and the Miller cycle reduces the Otto cycle. When T_{\mathrm{5}}=T_{\mathrm{4}}, processes 4–5 and 5–1 become an isobaric process, and the Miller cycle reduces the Atkinson cycle. Therefore, when the different values of T_{\mathrm{5}} are selected, the performance of the Miller cycle, the Otto cycle and the Atkinson cycle can be discussed.

Using Eqs. (3), (4), (8), (9), (13) and (14), the characteristic curves W-r_{\mathrm{P}}, \eta -r_{\mathrm{P}} and W-\eta can be plotted, as illustrated in Figs. 2–4, where the related parameters are given by T_{\mathrm{1}}=\mathrm{350}\mathrm{}\mathrm{K}, T_{\mathrm{0}}=T_{\mathrm{1}}, T_{\mathrm{3}}=\mathrm{2000}\mathrm{}\mathrm{K}, n=\mathrm{1.57}\times {\mathrm{10}}^{-\mathrm{2}}\mathrm{}\mathrm{m}\mathrm{o}\mathrm{l}, {\eta }_{\mathrm{c}}={\eta }_{\mathrm{e}}=\mathrm{0.9}\mathrm{}\mathrm{and}\mathrm{}\mathrm{1}, k_{\mathrm{1}}=\mathrm{6.035}\times {\mathrm{10}}^{-\mathrm{5}}\mathrm{}J/K^{\mathrm{2}}, k=\mathrm{1.4}, B/b=\mathrm{0.1}, T_{\mathrm{5}}=400K, 350 K and T_{\mathrm{4}} corresponding to the Miller, the Otto and the Atkinson heat engine, respectively, and W_{\max }, {\eta }_{\mathrm{m}} and r_{\mathrm{P}\mathrm{W}} are, respectively, the maximum work output, the efficiency and the pressure ratio at the maximum work output, and {\eta }_{\max }, W_{\mathrm{m}} and r_{\mathrm{P}\mathrm{\eta }} are, respectively, the maximum efficiency, the work output and the pressure ratio at the maximum efficiency.

It can be observed from Figs. 2 and 3 that as the pressure ratio increases, either the work output or the efficiency first increases and then decreases. It is evident that there exists an optimal value of r_{\mathrm{P}} at which the work output or the efficiency attains its maximum value. This implies that the states points at the maximum work output and the maximum efficiency are not the same, as depicted clearly in Fig. 4. W_{\max }, W_{\mathrm{m}}, {\eta }_{\max } and {\eta }_{\mathrm{m}}are four important parameters of the Miller heat engine. W_{\max } and {\eta }_{\max } determine the upper bounds of the work output and efficiency, while W_{\mathrm{m}} and {\eta }_{\mathrm{m}} determine the lower bounds of the work output and efficiency. Obviously, the optimally operating region of the Miller heat engine should be located in the part of the W-\eta curse which has a negative slope. Thus, the optimal ranges of the work output and efficiency are obtained as

(15)W_{\mathrm{m}}\le W\le W_{\max }

and

(16){\eta }_{\mathrm{m}}\le \eta \le {\eta }_{\max }.

According to Eqs. (15) and (16), the optimal region of the pressure ratio for the Miller heat engine can be further determined as

(17)r_{\mathrm{P}\mathrm{W}}\le r_{\mathrm{P}}\le r_{\mathrm{P}\mathrm{\eta }}.

When the pressure ratio r_{\mathrm{P}} is situated in this region, the work output will increase as the efficiency decreases, and vice versa.

It is also seen from Figs. 2–4 that there are the following interesting relations:

(18)W_{\max }^{\mathrm{O}}\le W_{\max }\le W_{\max }^{\mathrm{A}},

(19){\eta }_{\mathrm{m}}^{\mathrm{O}}\le {\eta }_{\mathrm{m}}\le {\eta }_{\mathrm{m}}^{\mathrm{A}},

(20)r_{\mathrm{P}\mathrm{W}}^{\mathrm{A}}\le r_{\mathrm{P}\mathrm{W}}\le r_{\mathrm{P}\mathrm{W}}^{\mathrm{O}},

(21){\eta }_{\max }^{\mathrm{O}}\le {\eta }_{\max }\le {\eta }_{\max }^{\mathrm{A}},

(22)W_{\mathrm{m}}^{\mathrm{O}}\le W_{\mathrm{m}}\le W_{\mathrm{m}}^{\mathrm{A}},

and

(23)r_{\mathrm{P}\mathrm{\eta }}^{\mathrm{A}}\le r_{\mathrm{P}\mathrm{\eta }}\le r_{\mathrm{P}\mathrm{\eta }}^{\mathrm{O}},

where {\eta }_{\mathrm{m}}^{\mathrm{O}} and r_{\mathrm{P}\mathrm{W}}^{\mathrm{O}} are the efficiency and pressure ratio of the Otto heat engine at the maximum work output W_{\max }^{\mathrm{O}}, W_{\mathrm{m}}^{\mathrm{O}} and r_{\mathrm{P}\mathrm{\eta }}^{\mathrm{O}} are the work output and pressure ratio of the Otto heat engine at the maximum efficiency {\eta }_{\max }^{\mathrm{O}}, {\eta }_{\mathrm{m}}^{\mathrm{A}} and r_{\mathrm{P}\mathrm{W}}^{\mathrm{A}} are the efficiency and pressure ratio of the Atkinson heat engine at the maximum work output W_{\max }^{\mathrm{A}}, and W_{\mathrm{m}}^{\mathrm{A}} and r_{\mathrm{P}\mathrm{\eta }}^{\mathrm{A}} are the work output and pressure ratio of the Atkinson heat engine at the maximum efficiency {\eta }_{\max }^{\mathrm{A}}. It can be seen from the above relationships, that the optimal performance of the Otto and Atkinson cycles are included in two special cases of the Miller cycle.

Discussion

1) The two sets of curves in Figs. 2 and 3 demonstrate the performance characteristics of the Miller heat engine at different values of compression and expansion efficiencies ({\eta }_{\mathrm{c}} and {\displaystyle {\eta }_{\mathrm{e}}}). It is seen from these curves that both the work output and efficiency of the cycle decrease with an decrease of the compression and expansion efficiencies ({\eta }_{\mathrm{c}} and {\eta }_{\mathrm{e}}). As an example, letting T_{\mathrm{5}}=\mathrm{400}\mathrm{}\mathrm{K}, k_{\mathrm{1}}=\mathrm{6.035}\times {\mathrm{10}}^{-\mathrm{5}}\mathrm{}\mathrm{J}/{\mathrm{K}}^2, B/b=\mathrm{0.1} and {\eta }_{\mathrm{c}}={\eta }_{\mathrm{e}}=\mathrm{1} and 0.9, it can be obtained that W_{\max } is equal to 276.4 and 214.8 J; and {\eta }_{\max } is equal to 0.570 and 0.416. These quantitatively indicate that the irreversibilities of the adiabatic processes are an important factor to impact the performance of the Miller heat engine cycle. When the irreversibilities of the adiabatic processes increase, the performance of the Miller heat engine decreases.

2) For the Miller heat engine, the work output and efficiency depend closely on the parameters related to the variable heat capacities. Figures 5 and 6 embody clearly the effects of the parameters related to the variable heat capacities on W-r_{\mathrm{P}} and \eta -r_{\mathrm{P}} characteristics curves for a set of given conditions. It can be seen from these curves that both the work output and efficiency increase remarkably with the increase of the parameter k_{\mathrm{1}}. It can also be observed that the pressure ratios (r_{\mathrm{P}\mathrm{W}} and r_{\mathrm{P}\mathrm{\eta }}) increase with the increase of k_{\mathrm{1}}, but r_{\mathrm{P}\mathrm{\eta }} increases more quickly than r_{\mathrm{P}\mathrm{W}} as the parameter k_{\mathrm{1}} is increased, such that the optimal region of the pressure ratio widens. For example, when k_{\mathrm{1}} is increased from \mathrm{6.035}\times {\mathrm{10}}^{-\mathrm{5}} to \mathrm{7.035}\times {\mathrm{10}}^{-\mathrm{5}}\mathrm{J}/{\mathrm{K}}^{\mathrm{2}}, W_{\max } increases from 214.8 to 220.9 J, and {\eta }_{\max } increases from 0.416 to 0.418. From the above analysis, it can be concluded that the parameter k_{\mathrm{1}} has a significant impact on the performance of the cycle. When k_{\mathrm{1}}=\mathrm{0}, both the work output and efficiency decrease significantly. Therefore, it is necessary, when using the Miller heat engine, to consider the effect of the variation of the heat capacities with temperature on the cyclic performance.

3) Based on Eqs. (13) and (14), it can be known that heat leak losses have a greater influence on the efficiency of the Miller heat engine. For a set of given conditions, when the heat leak losses increase, the efficiency decreases, and the work output does not change. The curves in Fig. 7 quantitatively reflect these changes. For example, when T_5=400\mathrm{}\mathrm{K}, {\eta }_{\mathrm{c}}={\eta }_{\mathrm{e}}=\mathrm{0.9}, k_{\mathrm{1}}=\mathrm{6.035}\times {\mathrm{10}}^{-\mathrm{5}}\mathrm{}\mathrm{J}/{\mathrm{K}}^{\mathrm{2}} and B/b increases from 0 to 0.1, it can be obtained numerically that {\eta }_{\max } decreases from 0.506 to 0.416. This means that the effect of heat leak losses on the efficiency of the Miller heat engine is significant. Therefore, in order to improve the performance of the Miller heat engine, it is important to decrease the heat leak losses as effectively as possible in the parametric design of the practical engine. When the heat leak losses are ignored, i.e., B/b\to \mathrm{0}, the performance characteristic curves of the Miller heat engine are shown by dashed curves in Fig. 7.

Conclusions

In this paper, an irreversible model of the Miller heat engine was established, in which variable heat capacities of the working substance, the irreversibility coming from the compression and expansion processes and the heat leak losses through the cylinder wall were taken into account. The optimal curves between the work output and the efficiency were obtained through numerical method. Moreover, the optimal regions of some important parameters in the cycle were determined. Furthermore, the influence of the variable heat capacities of the working substance, the irreversibility coming from the compression and expansion processes and the heat leak losses through the cylinder wall on the performance of the cycle was analyzed. The studies may be directly extended to the Otto and Atkinson cycles as long as some parameters are specially chosen. The results obtained may provide some theoretical guidance for the design of practical Miller heat engines.

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