Optimisation of a model external combustion engine with linear phenomenological heat transfer law H. J. Song, L. G. Chen* and F. R. Sun The optimal motion of a model external combustion engine with a piston fitted inside a cylinder containing an ideal gas is examined. The gas is heated at a given rate f(t) and coupled to a heat bath with linear phenomenological heat transfer law q˙!D(T21) for a finite amount of time. The optimal motion of the model external combustion engine with maximum power output objective is comprised of fully cyclic operation and semicyclic operation. The maximum work output per cycle and the corresponding efficiency are obtained respectively. Furthermore, the finite optimal compression ratio of such engine with special working condition is obtained. Numerical examples are provided, and the obtained results are compared with those obtained with Newton’s heat transfer law. The results presented herein can provide the basis for both determination of optimal operating conditions and design of real systems operating with the linear phenomenological heat transfer law. Keywords: Linear phenomenological heat transfer law, Maximum power output, Maximum work output, External combustion engine, Finite time thermodynamics, Generalised thermodynamic optimisation
Introduction 1–3
Finite time thermodynamics has been a powerful tool for performance analysis and optimisation of real energy systems. Band et al.,4–6 Salamon et al.7 and Aizenbud et al.8,9 used finite time thermodynamics to investigate the problem of maximising the work obtained from an ideal gas inside a cylinder with a moveable piston. The gas was coupled to an external heat bath at constant temperature Tex, and the heat transfer between the gas and bath obeyed Newton’s heat transfer law, q˙!DT. The optimal configuration for maximum work output was obtained,4,5 the power output optimisation was performed,7,8 and the applications in the model internal combustion engine9 and external combustion engine6 were discussed. Since the optimal results of these investigations are dependent on the concrete evolution law of the heat transfer process, the effects of heat transfer are important to the analysis. Chen et al.10 investigate the problem of maximising the work obtained from an ideal gas inside a cylinder with linear phenomenological heat transfer law q˙!D(T {1 ) instead of Newton’s one. Song et al.11 performed the similar work further based on generalised radiative heat transfer law q˙!D(T n ). In this paper, a further step is made based on Ref. 10, just as Ref. 6 done based on Refs 4 and 5. The optimal motion of the model external combustion engine with linear phenomenological heat transfer law Postgraduate School, Naval University of Engineering, Wuhan 430033, China *Corresponding author, email
[email protected]
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ß 2009 Energy Institute Published by Maney on behalf of the Institute Received 13 April 2008; accepted 14 October 2008 DOI 10.1179/014426009X12448168550226
and maximum power output objective is solved step by step. There exist fully cyclic operation, semicyclic operation and a finite optimal compression ratio. The maximum work output per cycle and the corresponding efficiency are obtained respectively. The limiting performance is determined by using finite time thermodynamics. Numerical examples are provided, and the obtained results are compared with those obtained with Newton’s heat transfer law. The results presented herein can provide the basis for both determination of optimal operating conditions and design of real systems operating with linear phenomenological heat transfer law.
Optimal solutions Let us consider the problem of maximising the power output obtained from an external combustion engine with a piston fitted inside a cylinder containing an ideal gas. There is just 1 mol ideal gas in the cylinder, and the gas is uniformly heated by a heat source whose rate of output, f(t), is an arbitrary given function of time. The gas is coupled to an external heat bath at constant temperature Tex. One will maximise the power output obtained from this system over a specified time interval by controlling the motion of the piston as shown in Fig. 1. The model adopted here is the same as that used in Ref. 6. However, the heat transfer between the gas and the external heat bath is assumed to obey linear phenomenological heat transfer q˙!D(T {1 ),10 where q˙ is the heat flux across the cylinder walls with heat conductance K, T is the gas temperature, Tex is the bath temperature. In general, K depends on the area of the walls in contact with the gas. However, for simplicity, K
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where CV is molar heat capacity, and R is the universal gas constant. E9(0) and V9(0) satisfy the following equations ½2KCV zE’(0)F (0) ( ) ð tm ½2KCV zE’(0)F (0)2 1 exp tm { F 2 (t)dt 4KCV 0 4KCV ½E’(0)2 E’(0)KCV Vm R=CV ~ (5) E(0) V (0) h i{R=CV ’(0) 2KCV z VV(0) E(0)F (0) h
i{R=CV
V ’(0) V (0)
E(0)
2 h i{R=CV 2KCV z VV’(0) E(0)F (0) (0) exp 1 Schematic diagram of model: mole of ideal gas inside cylinder is pumped by given heating function f(t) and is coupled to heat bath by linear phenomenological heat transfer law
is taken to be a constant in this paper, as did by Refs. 4– 10. Furthermore, the inertia of the gas and the piston is assumed to be negligible and there is no friction associated with the movement of the piston. For the system, the first law of thermodynamics takes the form : : E(t)~f (t){W (t){K Tex {1 {T {1 (t) (1) : where E is the rate of change of internal energy of the : gas, and W (t) is the power of expansion against the piston. One wishes to maximise the power output, i.e. maximise the work output per cycle for given f(t), initial volume of the gas, V0, final volume of the gas, Vm, and initial internal energy of the gas, E0. The results of Ref. 10 show that the optimal control for the expansion of a heated working fluid consists of, at most, three stages: (i) an initial instantaneous adiabat (ii) an intermediate Euler–Lagrange arc (E–L arc) (iii) a final instantaneous adiabat. Stage (i) is the initial adiabat from V(0) to V9(0) at t50. The equation is E’(0)~E(0)½V ’(0)=V (0){R=CV
(2)
where E9(0) is the initial value of internal energy of such optimal process, but not the initial value of internal energy of the whole optimal expansion process E(0). Stage (ii) is the E–L arc and proceeds from the initial V9(0) and E9(0) at t50 until time t5tm E(t)~
2KCV E’(0) 2KCV {E’(0)½F (t){F (0)
(3)
CV =R E’(0) V (t)~V ’(0) 1{ ½F (t){F (0) 2KCV ( ) ð ½2KCV zE’(0)F (0)2 1 t 2 t{ exp F (t)dt 4KR 0 4KRE’2 (0) (4)
4KCV E 2 (0)
h
i{2R=CV
tm
V ’(0) V (0)
ð tm KCV Vm R=CV 1 ~ exp F (t)dt 4KC V 0 E(0) V (0)
(6)
where F (t)~f (t){KTex {1 , tm is the total time allowed for the cycle. Stage (iii) is the final adiabat from V(tm) to Vm at t5tm Em ~E(tm )½Vm =V (tm ){R=CV
(7)
where V(tm) and E(tm) can be obtained from equations (5) and (6) at time tm.
Optimal cycles In this section, one obtains the optimal operation for a fully cyclic engine. Upon completion of the cycle, the volume and the temperature of the working fluid are periodic over the compression and power strokes (no other strokes are presented). The solution to the fully cyclic problem is obtained from the equations of E–L arc with the condition that at time t5tiztm, V(t) and E(t) return to their initial values. Since F(t) is assumed to be periodic with period tm, the exponential of the righthand side equation (4) should vanish at t5tiztm, yields the condition 8" Ð 9 # < ti ztm F 2 (t)dt 1=2 = ti E(ti )~2KCV = {F (ti ) (8) : ; tm If one assumes that tm is given, one obtains directly from equation (8) the value of E(ti), and hence completely determines the motion of the piston. There are two solutions of equation (8), a positive one and a minus one, and the positive one should be held. The constraint E(ti)5E(tiztm) reduces the number of degrees of freedom available in controlling the system operation. If the Ð t zt integral tii m F 2 (t)dt is large (large pumping energy), then T(ti) (equals to E(ti)/CV) will larger then Tex, and consequently there will be a reduction of efficiency of the engine operation as compared with the semicyclic operation (just V(ti)5V(tiztm)) due to larger heat leak to the bath. At the very start of engine operation, the temperature of the working fluid equals to the external temperature,
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2 Optimal volume versus time of fully cyclic operation with linear phenomenological heat transfer law and Newton’s heat transfer law
3 Optimal volume versus time of semicyclic operation with linear phenomenological heat transfer law and Newton’s heat transfer law
Tex. It will therefore require a number of cycles (semicyclic operation) before the initial temperature reaches the initial steady state temperature as determined by equation (8). Such a steady-state fully cyclic operation of the engine is assured to occur eventually if the piston motion (equation (4)) is maintained for a number of cycles until E(ti) given by equation (8) is reached. Alternatively, the working fluid may be preheated to the temperature of equation (8). Once the initial temperature E(ti)/CV is ensured, the optimal cyclic solution contains no adiabat (regardless of the form of the heating function). The cyclic nature of the solution forced the elimination of the adiabatic arcs. In the semicyclic operation, the optimal solution is comprised of an E–L arc and adiabatic jump arcs. From the result of Ref. 10, the final point of the initial adiabatic jump satisfies the following equation
Figure 2 shows optimal volume versus time of the fully cyclic operation with linear phenomenological heat transfer law and Newton’s heat transfer law, and Fig. 3 shows optimal volume versus time of the semicyclic operation with linear phenomenological heat transfer law and Newton’s heat transfer law. Figure 4 shows optimal cycle of the fully cyclic operation with linear phenomenological heat transfer law, and Fig. 5 shows optimal cycle of the semicyclic operation with linear phenomenological heat transfer law In this example, the optimal compression ratio of the fully cyclic operation with linear phenomenological heat transfer law is 65?425 : 1, which is larger than that with Newton’s heat transfer law (48 : 1),9 and the initial steady state temperature of the fully cyclic operation with linear phenomenological heat transfer law is 310?7742 K, the maximum work output per cycle is 7135?2 J, and the corresponding efficiency is 16?99%. Because of the larger heat conductance K, the maximum work output per cycle of the fully cyclic operation with linear phenomenological heat transfer law and the corresponding efficiency are all smaller than those with
½2KCV zE’(0)F (0) ( ) ð tm ½2KCV zE’(0)F (0)2 1 2 exp tm { F (t)dt 4KCV 0 4KCV ½E’(0)2 E’(0)KCV Vm R=CV ~ (9) E(0) V (0) Integrating equation (1) gives ð 1 ti ztm W (ti ztm )~ F (t)dtzE(ti ){E(ti ztm )z 2 ti KCV F (ti ) tm z 2 E(ti )
(10)
which is the maximum work output per cycle of the fully cyclic operation.
Numerical examples Now, numerical examples for the optimal motion of a model external combustion engine with linear phenomenological heat transfer law and maximum power output objective are provided. In the calculations, V(0)5Vm51 L, E(0)53780 J, Tex5300 K, CV53R/2, K566107 J K s21 and f(t)5A[sin (vt)]60, where A5204 720 J s21 and v52p/4 rad s21 are set.
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4 Optimal cycle of fully cyclic operation with linear phenomenological heat transfer law
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5 Optimal cycle of semicyclic operation with linear phenomenological heat transfer law
Newton’s heat transfer law. From Fig. 4, one can see clearly that the temperature of working fluid in the power stroke is much larger than that in the compression stroke, so the heat transfer between the gas and bath mainly takes place in the power stroke. The optimal compression ratio of the semicyclic operation with linear phenomenological heat transfer law is 66?6097 : 1, the maximum work output per cycle is 7142?3 J, and the corresponding efficiency is 17?01%. From Fig. 5, one can see that in the semicyclic operation, only V(ti)5V(tiztm) is satisfied, and the temperature of the working fluid are not periodic. Once the initial steady state temperature E(ti)/CV is ensured, the optimal motion is changed into the fully cyclic operation from the semicyclic operation.
Conclusions The problem of the optimal motion of the model external combustion engine with linear phenomenological heat transfer law and maximum power output objective is investigated in this paper. There exist fully cyclic operation, semicyclic operation and a finite optimal compression ratio. The maximum work output per cycle and the corresponding efficiency are obtained respectively. The limiting performance is determined by using finite time thermodynamics. Numerical examples
Optimisation of a model external combustion engine
are provided, and the obtained results are compared with those obtained with Newton’s heat transfer law.9 The similarities and differences of optimal motions between two heat transfer laws are given below: both the optimal motions with two heat transfer laws contain fully cyclic operation and semicyclic operation, and once the initial steady state temperature is ensured, the optimal motion is changed into the fully cyclic operation from the semicyclic operation, but for two different heat transfer laws, the maximum work output per cycle, the corresponding efficiency and the optimal compression ratio are different, so the heat transfer law has significant influence on the optimal motion of the external combustion engine. The results presented herein can provide the basis for both determination of optimal operating conditions and design of real systems operating with the linear phenomenological heat transfer law.
Acknowledgements This paper is supported by the Programme for New Century Excellent Talents in University of China (Project No. 20041006) and The Foundation for the Author of National Excellent Doctoral Dissertation of China (Project No. 200136).
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