Optimization of power and efficiency for an irreversible Diesel heat engine

Abstract

A cyclic model of an irreversible Diesel heat engine is presented, in which the heat loss between the working fluid and the ambient during combustion, the irreversibility inside the cyclic working fluid resulting from friction, eddies flow, and other irreversible effects are taken into account. By using the thermodynamic analysis and optimal control theory methods, the analytical expressions of power output and efficiency of the Diesel heat engine are derived. Variations of the main performance parameters with the pressure ratio of the cycle are analyzed and calculated. The optimum operating region of the heat engine is determined. Moreover, the optimum criterion of some important parameters, such as the power output, efficiency, pressure ratio, and temperatures of the working fluid at the related state points are illustrated and discussed. The conclusions obtained in the present paper may provide some theoretical guidance for the optimal parameter design of a class of internal-combustion engines.

Keywords

Diesel heat engine / irreversibility / power output / efficiency / parameter optimization

Introduction

The Diesel cycle is one of the important cycle models of internal combustion engines, which has a higher pressure ratio than other internal combustion engines, such as Otto heat engines, and a higher thermal efficiency, and it has attracted much attention from investigators and engineers. Some investigators have provided stylized descriptions of the air-standard Diesel cycle, and the conclusions reached have some significant guidance for real heat engines. In recent decades, some scholars have applied the finite time thermodynamics or entropy generation minimization theory to analyze and evaluate the performance of heat engine cycles [1-5], while some investigators have used the optimal control theory to optimize piston movement for the internal combustion engines [6-8]. The effect of heat-transfer on the performance of the air-standard Diesel heat engine has been studied [9], the effect of piston friction on the performance of the internal combustion engines with finite time constraint have been explored [10-13], and the Diesel heat engine cycle has been optimized by using the ecological function [14-18]. These investigations are helpful to the optimal design and performance evaluation of internal combustion engines. It is of practical value to study further the influences of the irreversibilities in the expansion and compression processes, and the heat loss between the cycle working fluid and the ambient during combustion on the cyclic performance of the Diesel heat engine.

In the present paper, an irreversible cycle model of Diesel heat engines is established, in which finite-rate heat transfer, the heat loss between the cyclic working fluid and the ambient during combustion, the internal irreversibility resulting from friction, eddies flow, and other irreversible effects inside the cyclic working fluid are taken into account. On the basis of finite time thermodynamics and optimal control theory approaches, the power output and efficiency of the Diesel heat engine are, respectively, optimized with respect to the pressure ratio. By using the numerical value calculation technology, the optimal operating regions of the related design parameters and the performance bounds of the heat engine cycle are analyzed and evaluated. Moreover, several interesting special cases are deduced and discussed from the results obtained.

An irreversible Diesel heat engine cycle

The p-V diagram of an irreversible Diesel heat engine cycle is illustrated in Fig. 1, which consists of two adiabatic (compression or expansion) processes, an isobaric heat addition process, and an isochoric heat rejection process, where the processes 1–2 and 3–4 are isentropic or reversible adiabatic processes; 1–2i and 3–4i are two irreversible adiabatic ones; 2–3 is an isobaric heat addition process; and 4i–1 is an isochoric heat rejection process.

For convenience of calculation, it is assumed that the cyclic working fluid is an ideal gas such that the heat added to the cyclic working fluid during the isobaric process and the heat rejected to the ambient during the isochoric process are expressed as, respectively,

(1)Q_{\mathrm{2}i-\mathrm{3}}=k{c}_v(T_{\mathrm{3}}-T_{\mathrm{2}i}),

(2)Q_{\mathrm{4}i-\mathrm{1}}=c_v(T_{\mathrm{4}i}-T_{\mathrm{1}}),

where k=c_p/c_v, c_p, and c_v are, respectively, the specific heats at constant pressure and constant volume, and T_{\mathrm{1}}, T_{\mathrm{2}i}, T_{\mathrm{3}}and T_{\mathrm{4}i} are the temperatures at the state points of 1, 2i, 3, and 4i, respectively.

For a real Diesel heat engine, the heat loss between the cyclic working fluid and the ambient outside cylinder wall during combustion is not negligible. The heat loss through the cylinder wall may be assumed to obey the Newtonian heat transfer law and is given by

(3)Q_L={\displaystyle {\int }_{\mathrm{0}}^{t_p}\mathit{\gamma }(T-T_{\mathrm{0}})}\mathrm{d}t,

where T and T_{\mathrm{0}} are the absolute temperatures of the working fluid (assuming the temperature of cylinder wall is the same as that of the cyclic working fluid) and the ambient, respectively; t_p is the time of the isobaric process, and \gamma is the heat loss coefficient.

Since processes 1-2i and 3-4i have no heat-exchange, they may be treated as instantaneous [11,16]. Furthermore, the heating from state 2i to 3 and cooling from state 4i to 1 are considered to proceed according to constant temperature variation rates [4,9-13], that is

(4)\begin{array}{ll}\mathrm{d}T/\mathrm{d}t=k_{\mathrm{1}}^{-\mathrm{1}}& (\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{2}i-\mathrm{3}),\end{array}

(5)\begin{array}{ll}\mathrm{d}T/\mathrm{d}t=-k_{\mathrm{2}}^{-\mathrm{1}}& (\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{4}i-\mathrm{1}).\end{array}

From Eqs. (4) and (5), the cycle period can be obtained as follows:

(6)\tau =t_p+t_v=k_{\mathrm{1}}(T_{\mathrm{3}}-T_{\mathrm{2}i})+k_{\mathrm{2}}(T_{\mathrm{4}i}-T_{\mathrm{1}}),

where k_{\mathrm{1}} and k_{\mathrm{2}} are two positive constants, and t_v is the time of the isochoric process. Combining Eqs. (3) and (4), the heat loss through the cylinder wall during combustion is obtained and is given by

(7)Q_{\mathrm{L}}=\mathit{\gamma }{k}_{\mathrm{1}}(T_{\mathrm{3}}-T_{\mathrm{2}i})[(T_{\mathrm{3}}+T_{\mathrm{2}i})/\mathrm{2}-T_{\mathrm{0}}].

Besides the heat loss during combustion, there also exists the internal dissipation for the cyclic working fluid. Thus, the two adiabatic processes are considered as irreversible. In order to describe the irreversibility of the two adiabatic processes, the expansion and compression isentropic efficiencies can be introduced and defined as [19-22]

(8){\eta }_{\mathrm{1}}=(T_{\mathrm{3}}-T_{\mathrm{4}i})/(T_{\mathrm{3}}-T_{\mathrm{4}}),

(9){\eta }_{\mathrm{2}}=(T_{\mathrm{2}}-T_{\mathrm{1}})/(T_{\mathrm{2}i}-T_{\mathrm{1}}),

where T_{\mathrm{2}} and T_{\mathrm{4}} are the absolute temperatures at states 2 and 4, respectively. It is clear that, in general, , unless the irreversibility of the adiabatic processes may be ignored that {\eta }_{\mathrm{1}}, {\eta }_{\mathrm{2}}=\mathrm{1}.

For an air-standard Diesel cycle, it yields

(10)T_{\mathrm{2}}/T_{\mathrm{1}}=r_p^{(k-\mathrm{1})/k},

(11)T_{\mathrm{3}}/T_{\mathrm{4}}={(V_{\mathrm{1}}/V_{\mathrm{3}})}^{k-\mathrm{1}},

(12)(p_{\mathrm{1}}{V}_{\mathrm{1}})/T_{\mathrm{1}}=(p_{\mathrm{2}}{V}_{\mathrm{3}})/T_{\mathrm{3}},

where r_p=p_{\mathrm{2}}/p_{\mathrm{1}} is the pressure ratio, p_{\mathrm{1}} and p_{\mathrm{2}} are the pressures at state 1 and the process 2-3, V_{\mathrm{1}} and V_{\mathrm{3}} are the volumes at the process 4i-1 and state 3. From Eqs. (8)-(12), the following equations can be obtained:

(13)T_{\mathrm{2}i}=(\mathrm{1}+a)T_{\mathrm{1}},

(14)T_{\mathrm{4}i}=b{T}_{\mathrm{3}},

where a=[r_p^{(k-\mathrm{1})/k}-\mathrm{1}]/{\eta }_{\mathrm{2}} and b=\mathrm{1}-{\eta }_{\mathrm{1}}+{\eta }_{\mathrm{1}}{[T_{\mathrm{3}}/(r_p{T}_{\mathrm{1}})]}^{k-\mathrm{1}}.

Power output and efficiency

From Eqs. (1), (2), (6), (7), (13) and (14), the power output and efficiency of the irreversible Diesel heat engine cycle can be obtained and given by

(15)P={\displaystyle \frac{Q_{\mathrm{2}i-\mathrm{3}}-Q_{\mathrm{4}i-\mathrm{1}}}{\tau }}={\displaystyle \frac{c_v}{k_{\mathrm{1}}}}{\displaystyle \frac{k-A}{\mathrm{1}+(k_{\mathrm{2}}/k_{\mathrm{1}})A}},

(16)\eta ={\displaystyle \frac{Q_{\mathrm{2}i-\mathrm{3}}-Q_{\mathrm{4}i-\mathrm{1}}}{Q_{\mathrm{2}i-\mathrm{3}}+Q_L}}={\displaystyle \frac{k-A}{k+[\left(\gamma k_{\mathrm{1}}{T}_0\right)/(\mathrm{2}{c}_v)]\{[T_{\mathrm{3}}+T_{\mathrm{1}}(\mathrm{1}+a)]/T_{\mathrm{0}}-\mathrm{2}\}}},

where A=(b{T}_{\mathrm{3}}-T_{\mathrm{1}})/[T_{\mathrm{3}}-T_{\mathrm{1}}(\mathrm{1}+a)]. Equations (15) and (16) are two important equations of the irreversible Diesel heat engine, it is seen that the power output and the efficiency are related to the functions and the pressure ratio, the temperatures of the cyclic working fluid in states 1 and 3, the isentropic efficiencies, and so on. In other words, the pressure ratio, the operating temperatures of the working fluid, and the irreversibility of the two adiabatic processes have important effects on the power output and efficiency.

Performance analysis and parametric optimization

By analyzing Eqs. (15) and (16) , it can be found that as the pressure ratio r_p increases, there exist, respectively, a maximum power output and a maximum efficiency. The following discussions will be an evaluation, in detail, of the optimal performance of the heat engine at two different kinds of objective functions.

Optimum power

According to Eq.(15) and the extremal condition \partial P/\partial r_p=\mathrm{0}, it is found that when the pressure ratio r_p equals r_{p^p}, the power output attains its maximum P_max. Moreover, the pressure ratio r_{p^p} at maximum power output may be solved from the following equation:

(17)c-A=\mathrm{0},

where c=k{\eta }_{\mathrm{1}}{\eta }_{\mathrm{2}}{r}_p^{\mathrm{1}/k}{[T_{\mathrm{3}}/(r_p{T}_{\mathrm{1}})]}^k. In principle, r_{p^p} could be solved from Eq. (17), such that P_max could be obtained by combining Eqs. (15) and (17). However, Eqs. (15) and (17) are all transcendental equations of the pressure ratio r_p, which shall be solved by a numerical value and graphical methods.

Based on Eq. (15), one can generate the P^{*}-r_p characteristic curve of the Diesel heat engine, as shown in Fig. 2, where P^{*}=P{k}_{\mathrm{1}}/c_v is the dimensionless power output. In Fig. 2, the relative parameters value k=\mathrm{1.4}, k_{\mathrm{2}}/k_{\mathrm{1}}=\mathrm{1.1}, \gamma k_{\mathrm{1}}{T}_{\mathrm{0}}/(\mathrm{2}{c}_v)=\mathrm{0.3}, T_{\mathrm{0}}=\mathrm{300}\mathrm{K}, T_{\mathrm{1}}=\mathrm{350}\mathrm{K}, and T_{\mathrm{3}}=\mathrm{1500}\mathrm{K} are chosen, and curves I, II, III, and IV correspond to the cases of {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, 0.99, 0.98, and 0.97, respectively. In Fig. 2, it can be seen clearly that when the pressure ratio attains r_{p^P}, there exists a maximum dimensionless power output P_{\max }^{*}. In addition, as the isentropic expansion and compression efficiencies η₁, η₂ increase, both P_{\max }^{*}and r_{pP} increase remarkably. However, when both η₁ and η₂ are equal to 1, that is, when the irreversibilities of the two adiabatic processes may be ignored, there does not exist a maximum power output. Similarly, by using Eqs.(13)-(15), the curves P^{*} versus T_2i and P^{*} versus T_4i that are presented in Figs. 3 and 4 can be obtained. In Figs. 3 and 4, T_{\mathrm{2}iP} and T_{\mathrm{4}iP} are the values of T_2i and T_4i at the maximum dimensionless power output.

It can be also found in Figs. 3 and 4 that the dimensionless power output first increases and then decreases as T_{\mathrm{2}i} and T_{\mathrm{4}i} increases when {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}\ne \mathrm{1}. This shows clearly that there also exists a maximum dimensionless power output with respect to T_{\mathrm{2}i} or T_{\mathrm{4}i}. Moreover, the dimensionless power output decreases with decreasing {\eta }_{\mathrm{1}} and {\eta }_{\mathrm{2}}. This is natural because the decrease of {\eta }_{\mathrm{1}} and {\eta }_{\mathrm{2}} implies the increase of the irreversibility of the two adiabatic processes, such that the power output of the heat engine goes down. If the two adiabatic processes are reversible, namely, {\eta }_{\mathrm{1}}, {\eta }_{\mathrm{2}}=\mathrm{1}, the dimensionless power output of the heat engine increases with increasing T_{\mathrm{2}i} or decreasing T_{\mathrm{4}i}.

Optimum efficiency

By using Eq.(16) and the extremal condition \partial \eta /\partial r_p=\mathrm{0}, it is found that when the pressure ratio r_p satisfies the following equation,

(18){\displaystyle \frac{(k-A)(\mathit{\gamma }{k}_{\mathrm{1}}/\mathrm{2})}{k{c}_v+\gamma k_{\mathrm{1}}[T_{\mathrm{3}}+T_{\mathrm{1}}(\mathrm{1}+a)-\mathrm{2}{T}_{\mathrm{0}}]/\mathrm{2}}}={\displaystyle \frac{c-A}{T_{\mathrm{3}}-T_{\mathrm{1}}(\mathrm{1}+a)}},

the efficiency of the heat engine attains its maximum {\eta }_{\max }. Using Eqs. (13), (14), and (16), the performance characteristic curves of the heat engine including the \eta -r_p, \eta -T_{\mathrm{2}i}, and \eta -T_{\mathrm{4}i} ones can be generated, as shown in Figs. 5–7. In Figs. 5–7, the related parameter values are the same as those used in Fig. 2. In addition, {\eta }_{\max }, r_{p\eta }, T_{\mathrm{2}i\eta }and T_{\mathrm{4}i\eta } are, respectively, the maximum efficiency, the pressure ratio at the maximum efficiency, and the values of T_{\mathrm{2}i} and T_{\mathrm{4}i} at the maximum efficiency. Curves I, II, III and IV correspond to the cases of {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, 0.99, 0.98, and 0.97, respectively. It can be seen in Figs. 5–7 that there exists a maximum efficiency {\eta }_{\max } with respect to r_p, T_{\mathrm{2}i}or T_{\mathrm{4}i}, respectively. Even though {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, the maximum efficiency can be still found in Figs. 5– 7. This is different from that at the optimum power output. Moreover, the effect of {\eta }_{\mathrm{1}} or {\eta }_{\mathrm{2}} on the efficiency of the heat engine is also evident. The efficiency decreases with decreasing {\eta }_{\mathrm{1}} and {\eta }_{\mathrm{2}}. It can be also seen in Figs. 5–7 that with decreasing {\eta }_{\mathrm{1}} and {\eta }_{\mathrm{2}}, r_{p\eta } and T_{\mathrm{2}i\eta } decrease, while T_{\mathrm{4}i\eta } increases.

P-\eta characteristics and the optimum operating region

On the basis of Eqs. (15)-(18), the P^{*}-\eta curves of the irreversible Diesel heat engine can be generated, as shown in Fig. 8, where {\eta }_P and P_{\eta }^{*} are, respectively, the efficiency at the maximum power output and the dimensionless power output at the maximum efficiency. In Fig. 8, the values of the related parameters k, k_{\mathrm{2}}/k_{\mathrm{1}}, \mathit{\gamma }{k}_{\mathrm{1}}{T}_{\mathrm{0}}/(\mathrm{2}{c}_v), T_{\mathrm{0}}, T_{\mathrm{1}}and T_{\mathrm{3}} are the same as those used in Fig. 2. In Fig. 8, it is seen that when the irreversibilities of the two adiabatic processes are taken into account, the P^{*}-\eta curve is a loop-type one. In such a case, there exists a maximum dimensionless power output and a maximum efficiency. On the other hand, it is also seen in Fig. 8 that when or , the dimensionless power output decreases as the efficiency decreases, and thus, these regions are not the optimal operating ones of the heat engine. The optimal operating region should be located in the part of the P^{*}-\eta curve with negative slope, namely, the region with P_{\max }^{*}\ge P^{*}\ge P_{\eta }^{*} and {\eta }_P\le \eta \le {\eta }_{\max }. The above results and the corresponding values can provide some reference for the optimum design of the Diesel heat engine.

Discussion

1) The optimal performance characteristics of the irreversible Diesel heat engine at the maximum power output and at the maximum efficiency are, respectively, analyzed and evaluated by using the numerical value calculation technology. The choice of the optimum criterion should be analyzed concretely according to the desiderative optimal objective for applications in engineering. For example, it is suitable to choose an optimal power output criterion when the heat engine is to be operated at a state of power output as large as possible; while the heat engine is to be operated at a state of the efficiency as large as possible, it should be considered to choose an optimal efficiency criterion. Furthermore, if the heat engine is to be operated at a compromise state in which there is no particular emphasis on efficiency or power output, the other optimal criterions, such as the ecological criterion [14-18], are worthy to be taken into account.

2) When {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, i.e., when the irreversibilities in the two adiabatic processes is negligible, the dimensionless power output is a monotonically increasing function with regard to the pressure ratio; while as long as the related parameters are large or small enough, when {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, there exist still the maximum values of the efficiency, as shown in Figs. 5–7. Moreover, when {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1}, Eqs.(13)–(16) may, respectively, be simplified into

(19)T_{\mathrm{2}i}=T_{\mathrm{1}}{r}_p^{(k-\mathrm{1})/k},

(20)T_{\mathrm{4}i}=T_{\mathrm{3}}^k/{(r_p{T}_{\mathrm{1}})}^{k-\mathrm{1}},

(21)P={\displaystyle \frac{c_v}{k_{\mathrm{1}}}}{\displaystyle \frac{k-A_{\mathrm{1}}}{\mathrm{1}+(k_{\mathrm{2}}/k_{\mathrm{1}})A_{\mathrm{1}}}},

and

(22)\eta ={\displaystyle \frac{k-A_{\mathrm{1}}}{k+(\gamma k_{\mathrm{1}}{T}_{\mathrm{0}}/\mathrm{2}{c}_v)[(T_{\mathrm{3}}+T_{\mathrm{1}}{r}_p^{(k-\mathrm{1})/k})/T_{\mathrm{0}}-\mathrm{2}]}},

where A_{\mathrm{1}}=(T_{\mathrm{3}}^k/r_p^{k-\mathrm{1}}{T}_{\mathrm{1}}^{k-\mathrm{1}}-T_{\mathrm{1}})/(T_{\mathrm{3}}-T_{\mathrm{1}}{r}_p^{(k-\mathrm{1})/k}).

3) If \mathit{\gamma }=\mathrm{0}, it implies that the heat loss between the working fluid and the ambient during combustion is ignored. In such a case, the expressions of the temperatures at the state points 2i and 4i and the power output are still given by Eqs.(13)-(15), respectively, while the efficiency equation is simplified as

(23)\eta =\mathrm{1}-A/k.

4) When {\eta }_{\mathrm{1}}={\eta }_{\mathrm{2}}=\mathrm{1} and \gamma =\mathrm{0}, the expressions of the temperatures at the state points 2i and 4i and the power output are still the same as Eqs. (13)–(15), respectively, while the efficiency may further be simplified as

(24)\eta =\mathrm{1}-A_{\mathrm{1}}/k,

which is just the efficiency equation of the reversible Diesel heat engine, and it has been presented in the textbooks.

Conclusions

An irreversible Diesel heat engine cycle is established by considering the irreversibilities coming from the compression and expansion processes, finite-rate heat transfer, and heat loss through the cylinder wall. The power output, and the efficiency of the heat engine cycle are, respectively, optimized with respect to the pressure ratio for other parameters given. The optimal regions and bounds of some important performance parameters, such as the power output, efficiency, and so on, are determined, and several interesting special cases may be derived. The results obtained in the present paper are general and can provide some significant guidance for the optimal parameter design of a class of internal-combustion engines.

References

[1] Rubin M H. Optimal configuration of a class of irreversible heat engines. I. Physical Review A, 1979, 19(3): 1272–1276

[2] Salamon P, Nitzan A, Andresen B, Berry R. Minimum entropy production and the optimization of heat engines. Physical Review A, 1980, 21(6): 2115–2129

[3] Angulo-Brown F, Rocha-Martinez J, Navarrete-Gonzalez I. A non-endoreversible Otto cycle model: improving power output and efficiency. Journal of Physics. D, Applied Physics, 1996; 29(1): 80–83

[4] Chen Lingen, Wu Chih, Sun Fengrui, Cao Shui. 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

[5] Chen Lingen, Wu Chih, Chen Jincan. Recent Advances in Finite-Time Thermodynamics. New York: Nova Sci Publishers, Inc., 1999

[6] Mozurkewich M, Berry R. Finite-time thermodynamics: Engine performance improved by optimized piston motion. Proc. Natl. Acad. SciUSA, 1981, 78(4): 1986–1988

[7] Mozurkewich M, Berry R. Optimal paths for thermodynamic systems: The ideal Otto cycle. Journal of Applied Physics, 1982, 53(1): 34–42

[8] Hoffman K, Watowich S, Berry R. Optimal paths for thermodynamic systems: The ideal diesel cycle. Journal of Applied Physics, 1985, 58(6): 2125–2134

[9] 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

[10] Calvo A, Medina A, Roco J, Velasco S. On an irreversible air standard Otto-cycle model. European Journal of Physics, 1995, 16(1): 73–75

[11] Bhattacharyya S. Optimizing an irreversible diesel cycle—Fine tuning of compression ratio and cut-off ratio. Energy Convers Mgmt2000, 41(8): 847–854

[12] Chen Lingen, Lin Junxing, Luo Jun, Sun Fengrui, Wu Chih. Friction effect on the characteristic performance of diesel engines. International Journal of Energy Research, 2002, 26(11): 965–971

[13] Chen L, Zheng T, Sun F, Wu C. The power and efficiency characteristics for an irreversible Otto cycle. International Journal of Ambient Energy, 2003, 24(4): 195–200

[14] Angulo-Brown F. An ecological optimization criterion for finite-time heat engines. Journal of Applied Physics, 1991; 69(11): 7465–7469

[15] Yan Zijun. Comment on “An ecological optimization criterion for finite-time heat engines” [J. Appl. Phys. 69, 7465(1991)]. Journal of Applied Physics, 1993, 73(7): 3583

[16] Cheng Ching-Yang, Chen Cha’o-Kung, The ecological optimization of an irreversible Carnot heat engine. J. Phys. D: Appl Phys, 1997, 30(1): 1602–1609

[17] Yan Zijun, Lin Guoxing. Ecological optimization criterion for an irreversible three-heat source refrigerator. Applied Energy, 2000, 66(3): 213–224

[18] Cheng Ching-Yang. The optimum allocation of heat transfer equipment for an irreversible combined heat engine with ecological criteria. Int. Comm. Heat Mass Transfer, 2004, 31(4): 573–584

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