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Eur. Phys. J. Plus (2016) 131: 394

DOI 10.1140/epjp/i2016-16394-9

Maximum cycle work output optimization for generalized radiative law Otto cycle engines Shaojun Xia, Lingen Chen and Fengrui Sun

Eur. Phys. J. Plus (2016) 131: 394 DOI 10.1140/epjp/i2016-16394-9

THE EUROPEAN PHYSICAL JOURNAL PLUS

Regular Article

Maximum cycle work output optimization for generalized radiative law Otto cycle engines Shaojun Xia1,2,3 , Lingen Chen1,2,3,a , and Fengrui Sun1,2,3 1 2 3

Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China College of Power Engineering, Naval University of Engineering, Wuhan 430033, China Received: 12 September 2016 / Revised: 8 October 2016 c Societ` Published online: 10 November 2016 –  a Italiana di Fisica / Springer-Verlag 2016 Abstract. An Otto cycle internal combustion engine which includes thermal and friction losses is investigated by finite-time thermodynamics, and the optimization objective is the maximum cycle work output. The thermal energy transfer from the working substance to the cylinder inner wall follows the generalized radiative law (q ∝ ∆(T n )). Under the condition that all of the fuel consumption, the compression ratio and the cycle period are given, the optimal piston trajectories for both the examples with unlimited and limited accelerations on every stroke are determined, and the cycle-period distribution among all strokes is also optimized. Numerical calculation results for the case of radiative law are provided and compared with those obtained for the cases of Newtonian law and linear phenomenological law. The results indicate that the optimal piston trajectory on each stroke contains three sections, which consist of an original maximum-acceleration and a terminal maximum-deceleration parts; for the case of radiative law, optimizing the piston motion path can achieve an improvement of more than 20% in both the cycle-work output and the second-law efficiency of the Otto cycle compared with the conventional near-sinusoidal operation, and heat transfer mechanisms have both qualitative and quantitative influences on the optimal paths of piston movements.

1 Introduction Finite-time thermodynamics (FTT), or entropy generation minimization (EGM) [1–27], is one of the branches of modern thermodynamics, and has made a great progress in recent years. Recently, it has been used for analyzing the performances of various thermodynamic cycles including quantum Otto [28], Rankine [29], double-effect absorption refrigerator [30], Dual-Atkinson [31, 32], Diesel [33, 34], Brayton [35] and so on. One of the classical problems of FTT is to investigate the optimal piston trajectories for improving the performance of the practical, engineering internalcombustion engines. A model of an ideal Otto cycle with the thermodynamic irreversibility losses of heat leakage and friction was firstly established by Mozurkewich and Berry [36, 37], the maximum cycle work output was selected as the optimization objective, and the optimal piston movement path of the engine with the Newtonian law (q ∝ ∆(T )) was determined. The results indicated that both the cycle work output and second-law efficiency could achieve a 10% improvement by optimizing the piston movement. After considering the finite-rate fuel combustion, an ideal Diesel cycle internal combustion engine model was further established by Hoffmann et al. [38, 39], and the optimal piston movement was also determined. Based on refs. [36–39], the minimum entropy generation was also selected as the optimization objective to optimize the piston movements of the Diesel [40] and Otto [41] cycle engines by Ge et al. [40, 41]. An adiabatic internal-combustion engine model was established by Teh et al. [42–45], and different optimization objectives such as work output [42], entropy production [43, 44] and thermal efficiency [45] were adopted for optimizing the piston movement. The optimal architecture (i.e., the process sequence and the process lengths of the cycle) for the maximum thermal efficiency of a steady-flow combustion engine was determined by Ramakrishnan et al. [46–50], and the results showed that the combustion process should be performed in a partly adiabatic and partly isothermal manner to achieve the maximum thermal efficiency [49], while the compression process should be performed in a partly intercooled and partly non-intercooled manner [50]. The optimal piston movement path of the a

e-mail: [email protected]

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maximum net work output for the four-stroke free-piston engine with irreversible Miller cycle was determined by Lin et al. [51] via Gauss pseudospectral method, and that for the Daniel cam engine with low heat rejection was determined by Badescu [52]. Besides the combustion-type engines discussed above, the maximum ecological performance of a bimolecular, light-driven engine was investigated by Chen et al. [53]. Besides the Newtonian law, other different heat transfer laws may also exist in engineering. The thermodynamic optimal performances of thermal systems were influenced by different heat transfer laws greatly [54–57], and so were the thermodynamic optimal configurations of thermal systems [58–68]. A class of endoreversible theoretical heat engines with linear phenomenological heat transfer law (q ∝ ∆(T −1 )) [69], radiative heat transfer law (q ∝ ∆(T 4 )) [70, 71] and generalized radiative heat transfer law (q ∝ ∆(T n )) [72–74] were established, and the maximum thermal efficiency [69–72] and the maximum power output [69, 73, 74] were selected as the optimization objectives to determine the corresponding optimal cycle configurations. The optimal piston trajectories of the Diesel cycle engine for the cases of q ∝ ∆(T ) + ∆(T 4 ) (i.e., convective-radiative law) [75, 76], q ∝ ∆(T −1 ) [77] and q ∝ ∆(T n ) [78] were investigated by Burzler et al. [75, 76], Xia et al. [77] and Chen et al. [78], respectively, and optimization results different from those for the Newtonian law [40, 41] were obtained. The optimal trajectories of the Otto cycle engine with q ∝ ∆(T −1 ) for maximizing the cycle work output was investigated by Xia et al. [79], and that with (q ∝ ∆(T 4 )) for minimizing the entropy generation was further studied by Ge et al. [80]. For the bimolecular, light-driven engine with q ∝ ∆(T −1 ), the optimal trajectories for the maximum cycle work output [81], the minimum entropy generation [81] and the maximum ecological performance [82] were determined by Ma et al. [81, 82]. For the reaction of [A] ⇆ [B] type, light-driven engine with q ∝ ∆(T n ), the optimal trajectories for both the maximum cycle work output and the minimum entropy generation were determined by Chen et al. [83]. More detailed reviews about this topic can be seen in refs. [6, 7, 14, 84]. Ge et al. [84] also provided both the mathematical models and numerical optimization results about the optimal trajectories of the Otto cycle engine with q ∝ ∆(T n ) for minimizing the entropy generation and maximizing the ecological performance, and more details can be seen in ref. [85]. On the basis of the research work mentioned above, this paper will further investigate the optimal trajectories of the Otto cycle engine with q ∝ ∆(T n ) for maximizing cycle work output. The research on optimal piston trajectories of the Otto cycle internal-combustion engines with both different thermal resistance models (i.e., from Newtonian [36, 37, 41] and linear phenomenological [79] to generalized radiative laws in this paper) and different optimization objectives (i.e., from the minimum entropy generation [41, 84, 85] and the maximum ecological performance [84, 85] to the maximum cycle work output in this paper) enriches FTT, and can provide some guidelines for the design of engineering highperformance Otto cycle type internal combustion engines.

2 Physical model of the Otto cycle engine The Otto cycle contains four strokes, i.e. intake, compression, power and exhaust ones. For the Otto cycle considered herein, the combustion process of the fuel is assumed to complete instantaneously, which is different from the Diesel cycle considered in refs. [38, 39, 75–78]. For the power stroke, the initial gas temperature is also given, which is equivalent to the fact that the consumed fuel per cycle is fixed. Since the sound wave speed of the ideal gas is always much larger than the speed of the piston movement, the thermodynamic process of the working fluid always keeps in internal equilibrium, and its thermodynamic state can be denoted as a set of unitized state parameters. Besides, in accordance with refs. [36–41, 75–80, 85–87], both the conventional piston motion and the main losses including heat leakage and frictional loss in real Otto cycle heat engines are simplified. 2.1 Conventional piston motion A typical model of the piston-cylinder structure is shown in fig. 1. X0 is the position of the top dead center, r is the brace length, l is the connecting rod length, and θ is the crankshaft rotating angle. From a geometrical point of view, the motion equation of the piston is given by [86] v = X˙ = (2π∆X/τ )(sin θ){1 + r cos θ[1 − (r/l)2 sin2 θ]/l}−1/2 ,

(1)

where ∆X = 2r and θ = 4πt/τ . The position of the bottom dead center is denoted as Xf , and Xf = 2r + X0 . The cycle period is denoted as τ . When t = 0, one has X = X0 . It is noticed that the piston motion of eq. (1) will turn to be pure sinusoidal when r/l = 0. 2.2 Main losses 2.2.1 Loss caused by heat transfer The heat transfer in real heat engines dissipates nearly 12% of the total engine power [87]. For the model of piston cylinder shown in fig. 1, the thermal energy transfer Q˙ between the working substance of the ideal gas in the cylinder

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Fig. 1. Conventional piston-cylinder structure.

and the inner wall of the cylinder obeys generalized radiative law q ∝ ∆(T n ), i.e. Q˙ = απb[X + (b/2)](T n − Twn ),

(2)

where α is the generalized heat transfer coefficient, b is the inner diameter of the cylinder, X is the transient piston position, and T and Tw are the temperatures of the ideal gas and the wall, respectively, n is the power exponent related to different heat transfer mechanisms, i.e. n = 1, n = −1 and n = 4 are chosen for Newtonian, linear phenomenological and radiative laws, respectively. Besides, n may also be set to other values, such as n = 2 and n = 3 given in ref. [54]. However, only n = 4 will be considered in this paper in detail due to the fact that the radiation is the dominant energy transfer behavior inside the cylinder when the temperature of the combustion gas reaches about 2800 K in the real internal combustion engine [86], and the optimization results will be compared with those obtained for the cases of both n = 1 in refs. [36, 37] and n = −1 in ref. [79]. The thermal energy transfer is considered on the power-stroke alone, while it is neglected on other strokes. 2.2.2 Loss caused by friction [36–41, 75–80] The friction dissipates nearly 20% of the total engine power. Of this, the friction between the piston rings and the cylinder wall is about 75%, while that in the crankshaft bearings is about 25% [87]. The former depends on the piston movement, while the latter does not. In this case, if the reversible cycle work is denoted as WR , and the total power loss due to the friction is denoted as Wf,τ , one further obtains Wf,τ = 0.15WR . The friction force f is a linear function of the velocity v and the gas viscosity is neglected herein. Let the friction coefficients of both the compression and the exhaust strokes be µ, and the corresponding friction forces are given by f = µv. For the power stroke, the gas pressure is greater than other strokes, so the corresponding friction coefficient is set to be 2µ. For the intake stroke, the corresponding friction coefficient is set to be 3µ due to the finite pressure differential flow of the gas through the inlet valve [87]. Besides, all other losses are neglected since they are so smaller compared to those caused by the heat transfer and the friction.

3 Optimal trajectories The problem is to maximize the cycle work output and determine the corresponding optimal piston movement path. This complex problem can be simplified and separated into two aspects. Firstly, the optimal piston motion for each stroke to be optimized is determined. Secondly, the total cycle period to be distributed on the four strokes is optimized. As the influence of thermal energy transfer on the three non-power strokes is negligible, optimization for the power stroke is distinct from those for other non-power ones. The former is relatively much more complicated than the latter. The three non-power strokes can be treated together in order to make the optimization problem simple, and their

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Eur. Phys. J. Plus (2016) 131: 394

optimization objective is chosen to be the minimum frictional loss. The total time tnp to be distributed among all the non-power strokes is further optimized for the minimum total frictional losses, which is denoted as Wf,tnp . The optimization objective of the power stroke is chosen to be the work output Wp per stroke for the given time tp spent on the stroke. Finally, the optimal relation between tnp and tp for the given cycle period τ is determined for maximizing the cycle work output, which is denoted as Wτ . 3.1 Optimal trajectories for non-power strokes Firstly, for the non-power stroke chosen to be optimized, the aim is to minimize the frictional loss on that stroke. Let the stroke duration be t1 , and then the corresponding friction loss is given by  t1 Wf,t1 = µv 2 dt. (3) 0

From refs. [36–41, 75–80], the optimal piston movement path for the case with no acceleration limit is that where the velocity keeps constant along the stroke, and the value of the velocity is equal to ∆X/t1 strictly; while for the case with limited acceleration am contains three segments, including an original acceleration segment with acceleration am and duration ta , a middle segment with uniform velocity v = am ta and duration t1 − 2ta , and a final deceleration segment with acceleration am and duration ta , where t1 and ∆X denote the total duration and distance on the non-power stroke, respectively. One further obtains ∆X = am t2a + am ta (t1 − 2ta ).

(4)

ta = t1 (1 − y1 )/2,

(5)

From eq. (4), one gets ta as follows: where y1 = (1 − 4∆X/am t1 2 )1/2 . Integrating eq. (3) yields the frictional loss Wf,t1    ta  t1 −ta 2 2 (am ta ) dt . (am t) dt + Wf,t1 = µ 2

(6)

ta

0

Equation (5) is further substituted into eq. (6), and one obtains Wf,t1 = µam 2 t1 3 (1 + 2y1 )(1 − y1 )2 /12.

(7)

Secondly, for the three non-power strokes, the distribution of tnp on these strokes is optimized for minimizing the total frictional loss Wf,tnp . For the compression and exhaust strokes, µ is equal in each, and their stroke durations are also equal. The friction coefficient of the intake stroke is 3µ, which is distinct from the former two non-strokes. If the duration for the intake stroke is given by t2 , the total duration for all of the non-power strokes is tnp = t2 + 2t1 . From eq. (7), one obtains the total frictional loss for the three non-power strokes as follows: Wf,tnp = µa2m [2t1 3 (1 − y1 )2 (1 + 2y1 ) + 3t2 3 (1 − y2 )2 (1 + 2y2 )]/12.

(8)

The optimal t2 for the frictional loss Wf,tnp to achieve its extreme value should satisfy the following equation, which is obtained from ∂Wf,tnp /∂t2 = 0, i.e. t1 2 (1 − y1 )2 = 3t2 2 (1 − y2 )2 . (9)

If tnp is given, t1 and t2 will be further obtained by combing eq. (9) with tnp = 2t1 + t2 and then solving them numerically. If there is no maximum acceleration limit, eqs. (8) and (9) further, respectively, give √ Wf,tnp = µ(2 + 3)2 (∆X)2 /tnp (10) and t2 =

√ 3t1 .

(11)

3.2 Optimal trajectory for power stroke Distinct from non-power strokes, the optimization objective for the power stroke is to maximize the work output Wp per stroke for the given duration tp spent on the stroke, and the influence of thermal energy transfer on the optimal trajectory should be also considered.

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3.2.1 Unconstrained time In order to obtain a set of reasonable initial guess parameters for the calculation of the case with constrained time, the simplest case with unconstrained time is firstly considered herein. If N and R are the mole number and universal gas constant, the ideal gas follows the state equation pV = N RT . From the law of energy conservation and through some mathematical derivations (see appendix A for details), one has     N RT 1 απb b dT =− + + X (T n − Twn ) , (12) dX N CV X v 2 where CV is the molar constant-volume heat capacity. The optimization objective is maximizing the work output per stroke, i.e.   Xf  N RT max Wp = − 2µv dX. (13) X X0 From eqs. (12) and (13), the Hamiltonian function for this functional optimization problem is given by     N RT N RT λ απb b n n H= − 2µv − + + X (T − Tw ) . X N CV X v 2

(14)

The co-state equation corresponding to eq. (12) is dλ ∂H NR λR λnαπb(b/2 + X)T n−1 =− =− + + . dX ∂T X CV X N CV v

(15)

From eq. (14) and ∂H/∂v = 0, the optimal control variable for the optimization problem satisfies the following equation:  λαπb(b/2 + X)(T n − Twn ) v= . (16) 2µN CV The boundary conditions for the optimization problem are T (X0 ) = T0p ,

λ(Xf ) = 0.

(17)

There is no analytical solution for the above optimization problem, and it will be solved numerically.

3.2.2 Constrained time and no acceleration constraint If the duration tp of the power stroke is further constrained, the mathematical expression of the optimization objective turns to be  tp ˙ max Wp = (N RT X/X − 2µX˙ 2 )dt. (18) 0

Equation (12) further gives −1 T˙ = N CV



  b N RT X˙ n n + απb + X (T − Tw ) , X 2

(19)

where T˙ = dT /dt and X˙ = dX/dt denote the time derivatives of the temperature and the piston position, respectively. For the functional optimization problem considered herein, the Lagrangian is

   ˙ RT b X N RT X˙ απb 2 n n L= − 2µX˙ + λ T˙ + + + X (T − Tw ) , (20) X CV X N CV 2 where λ is a Lagrange multiplier depending on the time t. The optimality conditions corresponding to eq. (20) are given by     ∂L d ∂L d ∂L ∂L − − = 0, = 0. (21) ∂X dt ∂ X˙ ∂T dt ∂ T˙

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Eur. Phys. J. Plus (2016) 131: 394

Equation (20) is substituted into eq. (21), and one further obtains ˙ λαπb n λRT λRT˙ N RT˙ + 4µv˙ − − = 0, (T − Twn ) − N CV X CV X CV X   N RX˙ λR X˙ λnαπbT n−1 b + + + X − λ˙ = 0. X CV X N CV 2

(22) (23)

From eqs. (19), (22) and (23), one obtains X˙ = v,

(24)







RTwn

NR b απb + X (Twn − T n ) + λ 4µN CV X 2 CV X   n−1 b N Rv λRv λnαπbT λ˙ = + + +X . X CV X N CV 2 v˙ =



b +X 2



 n n − (T − Tw ) ,

(25) (26)

The corresponding boundary conditions are X(0) = X0 ,

X(tp ) = Xf ,

T (0) = T0p ,

λ(tp ) = ∂L/∂T |t=tp = 0,

(27)

where T0p is the initial gas temperature. Equations (19) and (24)–(26) determine the optimal trajectory of the power stroke with unlimited acceleration. If n = 1, they become the optimization results for the Newtonian law [36, 37]; if n = −1, they become the optimization results for the linear phenomenological law [79]; if n = 4, they become the optimization results for the radiative law. 3.2.3 Constrained time and constrained acceleration Besides the stroke time, if the acceleration is further constrained, both the optimization objective and the corresponding differential constraints on the temperature and the piston position are the same as those for the case with constrained time and no acceleration constraint discussed above, i.e. eqs. (18), (19) and (25). For the additional constraint of the maximum acceleration limit am , the corresponding mathematical expressions are given as follows: v˙ = a, −am ≤ a ≤ am . For the functional optimization problem considered herein, the Hamiltonian function turns to be     b N RT N RT v λ1 H= v − 2µv 2 − + απb + X (T n − Twn ) + λ2 v + λ3 a. X N CV X 2

(28) (29)

(30)

The co-state equations related to the Hamiltonian function of eq. (30) are   N Rv λ1 Rv λ1 nαπbT n−1 b ∂H =− + + +X , λ˙ 1 = − ∂T X CV X N CV 2 N RT v RT v απb ∂H λ λ 1 1 λ˙ 2 = − = − + (T n − Twn ), ∂X X2 CV X 2 N CV N RT λ1 RT ∂H =− + 4µv + − λ2 . λ˙ 3 = − ∂v X CV X

(31) (32) (33)

As stated by the maximum value principle, one obtains ∂H/∂a = 0 along the optimal path, i.e. λ3 = 0.

(34)

If other points except isolated points during [−am , am ] satisfy eq. (34), one further obtains λ˙ 3 = 0;

(35)

λ2 is eliminated from eqs. (32), (33) and (35), and the same differential equations as those for the case of unlimited acceleration are further obtained. It can be found that the optimal trajectory of the power stroke for maximizing the work output with the constraint of the maximum acceleration limit contains three segments, which includes an original maximum acceleration segment, a terminal maximum deceleration segment, and a middle maximum work arc related to eqs. (19) and (24)–(26).

Eur. Phys. J. Plus (2016) 131: 394

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Table 1. Parameters for different cases with unlimited acceleration and the radiative law. µ

α (×10−7 )

τ

n

(kg/s)

4

W/(K · m )

(ms)

(rpm)

(1)

12.9

1.51

33.33

3600

(2)

7.5

3.00

33.33

3600

(3)

17.2

0.75

33.33

3600

(4)

12.9

1.51

25.00

4800

(5)

12.9

1.51

50.00

2400

Case

2

4 Numerical calculations and discussions 4.1 Parameters for calculations According to ref. [87], the following parameters are set: the dead-center positions X0 = 1 cm and Xf = 8 cm, the stroke distance ∆X = 7 cm, the inner diameter of the piston cylinder b = 7.98 cm, the cylinder displacement V = 400 cm3 , the cycle period τ = 33.3 ms for the case with rotating speed n = 3600 r/min, the universal gas constant R = 8.314 J/(mol · K); for a power stroke, the original temperature T0p = 2795 K, the molar number Np = 0.0157 mol, and the constant-volume molar heat capacity CV,p = 3.35R; for compression stroke, T0c = 333 K, Nc = 0.0144 mol and CV,c = 2.5R; the cylinder wall temperature Tw = 600 K; the total frictional loss per cycle WB = 50 J; and r/l = 0.25. The reversible cycle work output for the Otto cycle is WR = Np CV,p T0p [1 − (X0 /Xf )R/CV,p ] + Nc CV,c T0c [1 − (Xf /X0 )R/CV,c ].

(36)

The result of eq. (36) is WR = 435.9 J. The friction coefficient is set to be µ = 12.9 kg/s [36–41, 77–82]. In order to compare the optimization results with each other, the heat transfer coefficients α are set to be 1305 W/(K · m2 ) [36–41], −1.41 × 109 W · K/m2 [79] and 1.5106 × 10−7 W/(m2 · K4 ) for Newtonian (n = 1), linear phenomenological (n = −1) and radiative (n = 4) law, respectively. For the power stroke, vmax denotes the peak velocity, Tf denotes the terminal gas temperature, and WQ denotes the work loss related to thermal energy transfer; WQ is obtained from WQ = Np CV,p T0p [1 − (X0 /Xf )R/CV,p ] − Wp − Wf,tp ,

(37)

where Wf,tp is the frictional loss on the power stroke. ε is the thermodynamic second-law efficiency [88], i.e. the ratio of Wτ to WR . The cycle work output Wτ is calculated by Wτ = Wp − Wf,tnp + Nc CV,c T0,c [1 − (Xf /X0 )R/CV,c ] − WB .

(38)

The calculation is performed on the software Matlab 2012. The solving function bvp4c for the two-point boundary value problem is used and reasonable initial values are obtained by solving the optimization problem with unconstrained time. The solving function fsolve for nonlinear equations is used to solve eq. (9). For the problem with limited acceleration, one can refer to refs. [36, 37, 79] for detailed calculation methods. 4.2 Unconstrained acceleration and q ∝ ∆(T4 ) The parameters chosen for the calculation are listed in table 1, while the other ones are not changed. The corresponding calculation results are listed in table 2. As shown in table 2, the peak velocity for the conventional piston motion is lower than that for the optimized piston motion, and then the frictional loss on this stroke for the former is smaller than that for the latter. The stroke length is unchanged, and the stroke duration will decrease with the improvement in the average velocity. As a result, the heat transfer loss decreases with the decrease in the contact time between the high-temperature gas and the cylinder wall. At the mean time, the total duration on three non-power strokes increases, which results in the decrease in the corresponding total frictional loss (i.e., the average velocities on the strokes decrease). The decrement in the frictional loss for non-power strokes is larger than the increment in that for power stroke, so Wf,τ for the whole cycle decreases. Thus, the optimal trajectory can reduce both Q and Wf,τ compared with the convectional case, and both the cycle work Wτ and the second-law efficiency ε are improved finally.

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Eur. Phys. J. Plus (2016) 131: 394 Table 2. Numerical results for different cases with unlimited acceleration and the radiative law. vmax

tp

Wp

Wτ

Wf,τ

WQ

Q

(m/s)

(ms)

(J)

(J)

(J)

(J)

(J)

conv.

13.3

8.33

468.8

242.0

66.6

77.4

234.2

0.331

1143.9

0.555

opt.

28.9

6.03

513.5

301.9

56.3

27.7

142.5

0.195

1240.0

0.693

conv.

13.3

8.33

431.9

225.0

38.7

122.2

345.1

0.354

992.8

0.516

opt.

51.5

7.26

514.4

315.4

39.2

31.3

215.1

0.145

1082.1

0.724

conv.

13.3

8.33

495.4

252.7

88.8

44.4

144.0

0.309

1274.9

0.580

opt.

18.9

6.89

520.0

296.3

70.2

19.4

97.2

0.200

1324.5

0.679

conv.

17.7

6.25

477.6

235.0

88.7

62.3

193.9

0.321

1201.4

0.539

opt.

29.9

4.91

511.7

288.6

71.7

25.7

126.4

0.203

1272.1

0.662

conv.

8.86

12.5

450.0

239.0

44.3

102.6

297.9

0.344

1055.8

0.548

opt.

28.1

8.97

515.7

314.9

41.1

29.9

174.2

0.172

1172.4

0.723

Case

(1)

(2)

(3)

(4)

(5)

WQ /Q

Tf

ε

(K)

Table 3. Numerical results for different cases with limited acceleration and the radiative law. Case

conventional

vmax

tp

Wp

Wτ

Wf,τ

WQ

Q

(m/s)

(ms)

(J)

(J)

(J)

(J)

(J)

WQ /Q

Tf

ε

(K)

13.3

8.33

468.8

242.0

66.6

77.4

234.2

0.331

1143.9

0.555

3

15.8

7.46

475.5

256.4

60.6

68.8

214.3

0.321

1173.4

0.588

4

1 × 10

16.5

7.23

486.2

270.3

56.7

58.9

198.4

0.297

1187.8

0.620

4

19.3

6.28

495.4

282.4

56.2

47.3

172.3

0.274

1220.0

0.648

4

22.3

6.10

503.7

291.6

55.8

38.5

157.6

0.244

1233.2

0.669

unconstrained acceleration

28.9

6.03

513.5

301.9

56.3

27.7

142.5

0.195

1240.0

0.693

symmetric

15.7

8.33

486.6

268.8

56.1

61.0

209.2

0.292

1164.4

0.617

constrained acceleration

6 × 10

2 × 10

am 2

(m/s )

5 × 10

4.3 Constrained acceleration and q ∝ ∆(T4 ) For the cases of limited acceleration, am changes from 6 × 103 m/s2 to 5 × 104 m/s2 . The heat transfer coefficient, the friction coefficient and the cycle period are given by α = 1.5106 × 10−7 W/(m2 · K4 ), µ = 12.9 kg/s and τ = 33.3 ms, respectively. Table 3 lists the corresponding calculation results. Figure 2 shows the optimal piston movement configuration along the power stroke for the cases with different maximum acceleration constraints. Figure 2 shows that the middle part of the optimal path for the case of am = 5 × 104 m/s2 and the optimal trajectory for the case with unlimited acceleration are very similar to each other, which is due to the fact that both of these two different cases satisfy the same mathematical forms of differential equations. The different piston motion paths along the power stroke are shown in fig. 3, which include the sinusoidal (convention) case, the symmetric case and the optimal piston motion for the case of am = 6 × 103 m/s2 . The symmetric case indicates that each stroke has the same piston motion profile, and am = 1 × 104 m/s2 is set. As shown in fig. 3, the maximum velocity for the symmetric case is larger than that for the sinusoidal case, and is much closer to the top dead-center position. This makes for reducing the heat transfer loss. Besides, both Wτ and ε for the symmetric case are larger than for the conventional sinusoidal case. Figure 4 shows changes of the piston velocity over the whole cycle, and it includes the pure (i.e., r/l = 0) and the modified sinusoidal (i.e., r/l = 0.25) cases, and the optimal piston motion with unlimited acceleration and the limited acceleration of am = 2 × 104 m/s2 .

Eur. Phys. J. Plus (2016) 131: 394

Fig. 2. Optimal piston motion paths on the power stroke for various maximum acceleration limits.

Fig. 3. Comparison among conventional, symmetric and optimal cases.

Fig. 4. The piston motions over the whole cycle for the case with the radiative law.

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Page 10 of 14

Eur. Phys. J. Plus (2016) 131: 394 Table 4. Comparison between optimal and conventional cases for the radiative law. am

Case

constrained acceleration

unconstrained acceleration

2

(m/s ) 3

increase in ε

decrease in Q

decrease in WQ

decrease in Wf,τ

(%)

(%)

(%)

(%)

–

6 × 10

5.95

8.50

11.17

9.01

–

1 × 104

11.71

15.29

23.90

14.86

symmetric

1 × 104

11.17

10.67

21.19

15.77

4

16.76

26.43

38.89

15.62

4

–

2 × 10

–

5 × 10

20.54

32.71

50.26

16.22

(1)

∞

24.86

39.15

64.55

15.47

(2)

∞

40.31

37.67

74.39

−1.30

(3)

∞

17.07

32.50

56.31

17.25

(4)

∞

22.82

34.81

58.75

19.17

(5)

∞

31.93

41.52

70.86

7.22

Fig. 5. The velocity versus the position on the power stroke for different cases.

4.4 Comparison between the conventional and optimal cases The results for the optimal trajectory are compared with those for the sinusoidal case, and table 4 lists the comparison results. It can be found that whether the maximum acceleration is limited or unlimited, both Q and WQ are reduced through the optimization of the piston motion path. Besides, WQ is much more sensitive to the optimization than Q. The reason is twofold. Firstly, for the power stroke, WQ decreases with the decrease in the stroke duration when the piston motion is optimized. Secondly, for the high-temperature segment of the power stroke, the peak velocity also appears in the same segment, which makes for reducing Q. As the heat leakage decreases, the final gas temperature Tf for the power stroke increases, which is listed in tables 2 and 3. For all the cases with unlimited acceleration, after optimizing the piston trajectory, ε increases by 17.07%–40.31%, the work loss WQ decreases by 56.31%–74.39%, while the frictional loss never surpasses 19.17%. Table 4 shows that there exists a little reduction of frictional loss for the optimal cases with the case am > 1 × 104 m/s2 , and the reduced WQ results in the increase in the second-law efficiency ε. For the optimal cases with the limited accelerations am = 1 × 104 m/s2 , am = 2 × 104 m/s2 and am = 5 × 104 m/s2 , the cycle second-law efficiency ε increases by 11.71%, 16.76% and 20.54% compared to the corresponding conventional cases, respectively. 4.5 Effects of thermal resistance models Figure 5 shows changes of the piston velocity with its position for three special heat transfer laws. Figure 6 shows the piston trajectory over the whole cycle period, in which the limited acceleration is set to be am = 2 × 104 m/s2 .

Eur. Phys. J. Plus (2016) 131: 394

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Fig. 6. The piston motions over the whole cycle for different cases.

The calculation results are listed in table 5. As shown in fig. 5, the velocity v changes approximately linearly with the position X for the case of q ∝ ∆(T ), changes convex-upward for the case of q ∝ ∆(T −1 ), and changes concavedownward for the case of q ∝ ∆(T 4 ). It is evident that the optimal path of the piston movement with unlimited acceleration is very sensitive to the heat transfer mechanism changes. In addition, as listed in table 5, the optimal stroke times tp with various thermal resistances are also different from each other, i.e. the optimal cycle period distributions among all the four strokes are different from each other. Both of the above two differences indicate that thermal resistances have great effects on optimal trajectories. The reason consists of two aspects. The first aspect is that the thermal resistances related to different cylinder cooling systems are distinct. The second aspect is that for different thermal resistances, the heat transfer coefficients change largely in order to make the optimization results be compared with each other. The differences and similarities for the optimal cases with various thermal resistances are summarized: 1) all of them include three parts, which include an original maximum acceleration, a middle maximum work arc and a terminal maximum deceleration segment; 2) both the optimal trajectory with unlimited acceleration and the middle maximum work arc for the case of limited acceleration follow the same differential equations, so the optimal path differences for the cases of unlimited acceleration and different thermal-energy transfer mechanism lead to the differences for the cases of limited acceleration and different thermal resistances as shown in fig. 6.

5 Conclusions An Otto cycle internal combustion engine with the irreversibility losses of heat leakage and friction is investigated, and the thermal energy transfer mechanism follows the generalized radiative law q ∝ ∆(T n ). The optimization objective is chosen to be the maximum cycle work output, and the optimal piston movement is determined for the given cycle period and consumed fuel. Both the cases with unlimited and limited accelerations are discussed in details. The optimal trajectory on the power stroke with limited acceleration includes three parts, an original maximum acceleration, a middle maximum work arc, and a terminal maximum deceleration segment. Numerical calculation results for the case of q ∝ ∆(T 4 ) are provided, in which both the cases with unlimited and limited acceleration are discussed. As n = 1, n = −1 and n = 4 represent Newtonian [36, 37], linear phenomenological [79] and radiative heat transfer laws, respectively, the optimization results for these three special thermal-resistance models are compared to each other. It is concluded that for the case of q ∝ ∆(T 4 ) adopted in this paper, both the engine power and its second-law efficiency can be increased by more than 20% through the optimization of the piston motion. The main contribution comes from the reduction of the heat leakage during the original part of the power stroke. The optimal trajectory is very sensitive to the change of the heat transfer mechanism. This shows that heat resistance models have significant influences on the maximum cycle work output optimization of Otto cycle internal combustion engines, which should be clarified during the optimal designs and operations of real heat engines. There exist some methods to realize the optimal paths, such as using a contoured plate or an electrical coupling [2] and so on. The optimal trajectory not only enlarges both the cycle work and the second-law efficiency of internal combustions engines, but also reduces the frictional loss and the thermal energy transfer loss.

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Eur. Phys. J. Plus (2016) 131: 394

Table 5. Optimization results for various special heat transfer laws. For the case with constrained acceleration, am = 2 × 104 m/s2 . Except α, the other parameters are the same for the three heat transfer laws in the calculations. For the Newtonian heat transfer law, α = 1305 W/(m2 · K); for the linear phenomenological heat transfer law, α = −1.41 × 109 W · K/m2 ; for the radiative heat transfer law, α = 1.51 × 10−7 W/(m2 · K4 ). vmax

Case

(m/s) (ms) conventional

Newtonian

linear phenom-enological

radiative

tp

13.3

Wp

Wτ

(J)

(J)

Wf,τ WQ (J)

(J)

Q (J)

WQ /Q

Tf (K)

ε

8.33 497.3 270.4 66.6 48.9 224.8 0.218 1100.5 0.620

constrained acceleration

20.5

6.08 511.1 298.4 57.5 29.9 171.6 0.175 1179.4 0.685

unconstrained acceleration

25.2

5.72 519.1 307.9 57.7 20.4 160.1 0.127 1183.0 0.706

conventional

13.3

8.33 504.2 277.4 66.6 42.0 242.4 0.173 1044.2 0.636

constrained acceleration

20.7

5.82 514.0 301.6 57.5 26.7 180.8 0.148 1150.8 0.692

unconstrained acceleration

18.1

5.30 518.9 308.2 56.8 21.0 168.0 0.125 1166.4 0.707

conventional

13.3

8.33 468.8 242.0 66.6 77.4 234.2 0.331 1143.9 0.555

constrained acceleration

19.3

6.28 495.4 282.4 56.2 47.3 172.3 0.274 1220.0 0.648

unconstrained acceleration

28.9

6.03 513.5 301.9 56.3 27.7 142.5 0.195 1240.0 0.693

This paper is supported by the National Natural Science Foundation of P. R. China (Project No. 51576207). The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.

Appendix A. In terms of the law of energy conservation, one obtains dQ dU dW = + , dt dt dt

(A.1)

where U is the internal energy. Substituting dU = N CV dT and dW = pdV into eq. (A.1) yields N CV

dQ dV dT = −p . dt dt dt

(A.2)

Since the relation pV = N RT holds, one has p = N RT /V . Substituting p = N RT /V into eq. (A.2) yields N CV

dT dQ N RT dV = − dt dt V dt

(A.3)

Since dT /dt = (dT /dX) · (dX/dt) and (dV )/V = (dX)/X, eq. (A.3) further gives N CV

dT dX dQ N RT dX = − . dX dt dt X dt

(A.4)

Since dX/dt = v, eq. (A.4) becomes N CV

1 dT = dX N CV



1 dQ N RT − v dt X



.

(A.5)

Q in eq. (A.5) is the heat released, so it is negative in eq. (A.5). Finally, eq. (12) is obtained by substituting eq. (1) into eq. (A.5).

Eur. Phys. J. Plus (2016) 131: 394

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