OPTIMAL PATHS OF PISTON MOTION OF IRREVERSIBLE DIESEL CYCLE FOR MINIMUM ENTROPY GENERATION. By Yanlin GE, Lingen CHEN *, and Fengrui SUN

Transcription

1 OPTIMAL PATHS OF PISTON MOTION OF IRREVERSIBLE DIESEL CYCLE FOR MINIMUM ENTROPY GENERATION By Yanlin GE, Lingen CHEN *, and Fengrui SUN A Diesel cycle heat engine with internal and external irreversibilities of heat transfer and friction, in which the finite rate of combustion is considered and the heat transfer between the working fluid and the environment obeys Newton s heat transfer law [ q ( T) ], is studied in this aer. Otimal iston motion trajectories for minimizing entroy generation er cycle are derived for the fixed total cycle time and fuel consumed er cycle. Otimal control theory is alied to determine the otimal iston motion trajectories for the cases of with iston acceleration constraint on each stroke and the otimal distribution of the total cycle time among the strokes. The otimal iston motion with acceleration constraint for each stroke consists of three segments, including initial maximum acceleration and final maximum deceleration boundary segments, resectively. Numerical examles for otimal configurations are rovided, and the results obtained are comared with those obtained when maximizing the work outut with Newton s heat transfer law. The results also show that otimizing the iston motion trajectories could reduce engine entroy generation by more than %. This is rimarily due to the decrease in entroy generation caused by heat transfer loss on the initial ortion of the ower stroke. Key words: finite rate of combustion, Newton s heat transfer law, irreversible Diesel cycle heat engine, minimum entroy generation, otimal iston motion trajectories, finite time thermodynamics. 1. Introduction Since the efficiency bound of a Carnot engine at maximum ower outut was derived by Curzon and Ahlborn [1], much work [-14] has been erformed in the field of finite-time thermodynamics. The study on the otimal aths of iston motion for internal combustion engines with different otimization objectives and heat transfer laws mainly includes the following two asects The otimal ath with Newton s heat transfer law [ q ( T) ] Mozurkewich and Berry [15, 16] investigated a four stroke Otto cycle engine with losses of iston friction and heat transfer, in which the heat transfer between the working fluid and the cylinder wall obeys Newton s heat transfer law [ q ( T) ]. The otimal iston trajectory for maximizing the 1

2 work outut er cycle was derived for the fixed total cycle time and fuel consumed er cycle. The results showed that otimizing the iston motion could imrove engine efficiency by nearly 1%. Hoffmann and Berry [17] further considered the effect of the finite combustion rate of the fuel on the erformance of engines, and studied the otimal iston motion of a four stroke Diesel cycle engine for maximum work outut with losses of iston friction and heat transfer, in which the heat transfer between the working fluid and the cylinder wall also obeys the Newton s heat transfer law. Blaudeck and Hoffman [18] studied the otimal ath of a four stroke Diesel cycle with the Newton s heat transfer law by using Monte Carlo simulation. Teh et al. [19-1] investigated the otimal iston motions of internal combustion engines for maximum work outut [19] and maximum efficiency [, 1] when the chemical reaction loss and heat leakage are the main losses of internal combustion engine. By excluding the entroy generation due to friction loss, heat transfer loss and ressure dro loss in ractical internal combustion engine, Teh et al. [, 3] isolated combustion as the sole source of entroy generation and investigated the otimal iston motion of an adiabatic internal combustion engine for minimum entroy generation [] as well as the otimal iston motion for minimum entroy generation with fixed comression ratio [3]. Based on the heat engine models established in Refs. [15, 16], Ge et al. [4] further considered the entroy generation which was not included in Refs. [, 3], derived the otimal iston motion trajectories of Otto cycle for minimizing entroy generation due to friction loss, heat transfer loss and ressure dro loss when the heat transfer between the working fluid and the environment obeyed Newton s heat transfer law, and comared the results obtained with those obtained for maximizing the work outut with Newton s heat transfer laws [15, 16]. Band et al [5, 6] studied otimal configuration of irreversible exansion rocess for maximum work outut obtained from an ideal gas inside a cylinder with a movable iston when the heat transfer between the gas and the bath obeyed Newton s heat transfer law, and discussed the otimal configurations of the exansion subjected to eight different constraints, including constrained rate of change of volume, unconstrained final volume, constrained final energy and final volume, constrained final energy and unconstrained final volume, consideration of iston friction, consideration of iston mass, consideration of gas mass and unconstrained total rocess time, resectively. Salamon et al [7] and Aizenbud and Band [8] used the results obtained in Refs. [5, 6] to further investigate the otimal configurations of the exansion rocess for maximizing ower outut [7] and the otimal configurations for maximizing work outut with fixed ower outut [8] with Newton s heat transfer law. Aizenbud et al [9] and Band et al [3] further alied the results obtained in Refs. [5, 6] to otimize the configurations of internal [9] and external [3] combustion engines with Newton s heat transfer law, resectively. 1.. The effect of heat transfer law on the otimal ath of cycle In general, heat transfer is not necessarily Newtonian and also obeys other laws; heat transfer law has the significant influence on the otimal configuration of heat engine cycles. Burzler and 4 Hoffman [31, 3] considered the effects of convective-radiative heat transfer law [ q ( T) + ( T )] and non-ideal working fluid, and derived the otimal iston motion for maximizing ower outut during the comression and ower strokes of a four stroke Diesel engine. Xia et al [33] studied the otimal iston trajectory of the Otto cycle engine for maximizing work outut with fixed total cycle 1

3 1 time, the fixed fuel consumed er cycle and linear henomenological heat transfer law [ q ( T )] in the heat transfer rocess between working fluid and the environment, and the results showed that otimizing the iston motion could imrove work outut and efficiency of the engine by more than 9%. Based on the heat engine models established in Refs. [15, 16, 33], Ge et al [4, 34] derived the otimal iston motion trajectories of Otto cycle for minimizing entroy generation due to friction loss, heat transfer loss and ressure dro loss when the heat transfer between the working fluid and the environment obeys linear henomenological [4] and radiative [34] heat transfer laws, and comared the results obtained with those obtained for maximizing the work outut with Newton s [15, 16] and linear henomenological [33] heat transfer laws. On the basis of Ref. [6], Chen et al [35] determined the otimal configurations of exansion rocess of a heated working fluid in the iston cylinder with the linear henomenological heat transfer law. Song et al [36] and Chen et al [37] used the results obtained in Ref. [35] to otimize the configuration of external [36] and internal [37] combustion engines with linear henomenological heat transfer law. Song et al [38], Ma et al [39] and Chen et al [4] determined the otimal configurations of exansion rocess of a heated working fluid in the iston cylinder with the generalized radiative n [ q ( T )] [38], Dulong-Petit [ q ( T) 5/4 ][39] and convective-radiative [4] heat transfer laws, obtained the first-order aroximate analytical solutions about the Euler-Lagrange arcs by means of Taylor series exansion. Ma et al [41, 4] reeated the investigation on the otimal configurations of exansion rocess with the generalized radiative heat transfer law by means of elimination method. Ma et al [41, 43] used the results obtained in Refs. [41, 4] to otimize the configuration of external 4 combustion engine with radiative [ q ( T )] [41, 43], generalized radiative [43] and convectiveradiative [41] heat transfer laws, resectively. Chen et al [44] considered the effects of iston motion on the heat conductance, established a model closer to ractical exansion rocess of heated working fluid with the generalized radiative heat transfer law and time-deendent heat conductance, and determined otimal configuration of exansion rocess for maximum work outut. Based on Refs. [17, 4], this work studies the otimal iston trajectory of the Diesel cycle engine for minimizing entroy generation er cycle with the finite rate of combustion, the fixed total cycle time, the fixed fuel consumed er cycle and Newton s heat transfer law in the heat transfer rocess between working fluid and environment, and the results obtained are comared with those obtained for maximizing the work outut with the Newton s heat transfer law [17].. Diesel cycle engine model In order to analyze ractical Diesel cycle, some assumtions are made: (1) the fixed fuel consumed er cycle is equivalent to the given initial working fluid temerature on the ower stroke; () the working fluid in the cylinder consists of an ideal gas which is in internal equilibrium all the time; and (3) the major losses in real internal combustion engine and the iston motion of the conventional engine are simlified and described qualitatively and quantitatively below according to Refs.[15-18, 4, 33, 45, 46]..1. Finite combustion rate [17]

4 In the modern Diesel cycle engine, fuel is injected into the cylinder at the end of the comression stroke and evaorates in the hot comressed air. After a short delay, art of the injected fuel is ignited and burned raidly. The remaining fuel burns relative slowly as it evaorates and diffuses into oxygen-rich regions where combustion can be sustained. In moderately and heavily loaded engines, the combustion rocess will continue until the end of the ower stroke. The finite combustion rate is one of the main features of a Diesel engine and can be aroximated by the following time-deendent function which describes the extent of the reaction [17] Gt ( ) = F+ (1 F)[1 ex( t/ t b )] (1) where F is the fraction of the fuel charge consumed in the initial instantaneous burn, and t b is the burn time during which most of the combustion occurs. For the corresonding heating function ht (), one has ht () = NQGt &() () c where Q is the heat of combustion er molar fuel-air mixture charge. Q c c is assumed to be temerature indeendent. The mole number N and heat caacity C are assumed to be influenced by the extent of the combustion reaction within the iston chamber. This deendence can be exressed as [17] N = Nt () = N+ ( N N) Gt () (3) i f i C = C() t = C + ( C C ) G() t (4) i f i The subscrits i and f refer to conditions at G = and G = 1, resectively. Furthermore, the heat caacities of the reactants and roducts are temerature indeendent... Loss terms [15-18, 4, 33, 45] For the Diesel cycle, the major losses are as follows: (1) friction, () ressure dro, (3) heat leakage, (4) fuel injection, (5) incomlete combustion, and (6) exhaust blowdown. The non-negligible losses from this list are cast into simlified functional forms and incororated into this model. This aroach reroduces the Diesel engine s salient features in a flexible, easily deciherable model...1. Friction loss Friction force is assumed to be roortional to the iston velocity v [15-18, 4, 33], the frictional work W in a stroke taking time t is exressed as f W t = µ v dt f (5) o Due to greater ressure on the iston, the value of friction coefficient is usually about twice as large on the ower stroke as on the other strokes. If the friction coefficient on the nonower stroke is µ, the friction coefficient on the ower stroke will be µ [15-18, 4, 33].... Pressure dro 3

5 There is an additional friction-like loss term on the intake stroke. This is due to the ressure differential that develos, due to viscosity, as the gas flows through the inlet valve. The ressure differential is roortional to velocity, so it may be included in the friction term for the intake stroke, and then the friction coefficient on the intake stroke is assumed to be 3µ [15-18, 4, 33]...3. Heat leakage Loss due to heat transfer from the working fluid to cylinder walls tyically cost about 1% of the total ower [45]. The same as Refs. [15-18, 4], the heat transfer between the working fluid and the environment obeys Newton s heat transfer law. The heat transfer exression used here assumes that the rate is linear in the inside surface area of the cylinder and in the difference between the temerature of working fluid T and that of the cylinder wall T. T is assumed to be a constant. w w For heat transfer coefficient k and cylinder diameter b, the rate of heat leakage at osition X (see fig. 1) is Q & = kπb( b + X)( T T w ) (6) Figure 1. Conventional iston linkage. The effect of heat transfer is only imortant on the ower stroke. Since the average value of ( T T w ) is much smaller on the nonower stroke, the rate is negligible on them...4. Incomlete combustion If the exhaust valve oens before the burning fuel-air mixture reaches chemical equilibrium, there can be minor efficiency losses even in well-adjusted engines oerating at normal loads. These losses have been included in the model by reresenting the combustion reaction as the exonential function give exlicitly in eq. (1) [17]...5. Other losses 4

6 Besides, there are effects of the time loss due to beginning the exansion stroke while combustion is still taking lace and the exhaust blowdown due to oening the exhaust valve before comletion of the exansion stroke. They are so small comared with the heat leakage and friction losses that they are all negligible [4]..3. Piston motion of a conventional engine [46] To determine imrovements resulting from the otimized iston motion, work outut and engine efficiencies are calculated for conventional iston trajectories. Piston motion within a conventional Diesel engine can be described by 1 π X sinθ rcosθ r v = X& = 1 + [1 ( ) sin θ ] (8) τ l l As shown in fig. 1, X denotes iston osition, X = r, and θ = 4πt τ. X = X when t =. The four-stroke cycle eriod is τ. Pure sinusoidal iston motion occurs when rl=. Tyical r/ l is between.16 and.4. Varying the value of r/ l has little effect on the results. 3. Otimization rocedure The otimization roblem is minimizing the entroy generation er cycle for fixed fuel consumtion and total cycle time. Thus, the only difference between the otimized engine and the conventional one is in the iston motion. The otimization rocedure consists of two arts. The first is to determine the otimal trajectory on each stroke. The second is to otimize the distribution of the total cycle time among the strokes. A real engine includes ower and nonower strokes. Comared with the ower stroke, the heat leakage is negligible on the nonower strokes which include intake, comression and exhaust strokes. Therefore, the otimization of these strokes is relatively simle, and the three strokes can be treated together to simlify the otimization roblem. For the three nonower strokes, at first, the iston ath on each stroke is determined with the objective of minimum entroy generation, resectively, and then the time allocation among three non-ower strokes for a fixed total time t is determined with the objective of minimum total entroy generation S n t. While for the n ower stroke, the iston ath is determined with the objective of minimum entroy generation S t when heat transfer loss is considered. For the total cycle, the distribution of the total cycle time τ between the total time t of nonower strokes and the ower stroke time t is determined with the n objective of minimum entroy generation S er cycle Otimization for nonower strokes The otimal rocess of nonower strokes is the same as Ref. [4]. After otimization, the total entroy generation on nonower strokes is S = µ a [ t (1 + y )(1 y ) + 3 t (1 + y )(1 y ) ] (1 T ) (9) tn 3 3 m For a given value of nonower stroke time n t, the otimal allocation of the nonower stroke time is determined by following two equations 5

7 t = t + t (1) n 1 t (1 y ) = 3 t (1 y ) (11) 1 1 where 1/ 1 y = (1 4 X / a t ), y = (1 4 X a t ) and a 1 m 1 m m is the value of constrained acceleration. For the case without constraint on acceleration, i.e. a, eq. (11) becomes m t = 3t (1) 1 And the total entroy generation, from eq. (9), is S = µ + X t T (13) tn ( 3) ( ) ( ) n 3.. Otimization for ower stroke Different from the otimization of the nonower strokes, in which only the friction loss is considered, the otimization of the ower stroke will consider the effects of heat transfer on the otimal iston trajectory. So, the entroy generation in ower stroke is due to friction loss and heat transfer loss, and which can be exressed as S f, t and S qt,, resectively. t µ vdt S = (14) f, t T t kπ b( b + X)( T T ) dt w S = (15) qt, T Power stroke with unconstrained acceleration In order to obtain the minimum entroy generation during the ower stroke, the corresonding otimization roblem becomes t [ µ + π ( + )( )] w v k b b X T T dt min S = S + S = (16) t f, t q, t T In terms of the first law of thermodynamics, one has 1 NRTv b T & = [ + kπb ( + X )( T Tw ) h ( t )] NC X (17) where R is the gas constant. The rate of heat roduced during the combustion is described by the heating function NQ (1 F) c t ht () = ex( ) (18) t t b b Furthermore, there is X& = v (19) The Hamiltonian for this roblem is µ v + kπb( b/ + X)( T T ) w λ1 NRTv b H = [ + kπb( + X)( T T ) h( t)] + λ v () w T NC X 6

8 The canonical equations are H b λ 1 λ Rv & 1 1 = = kπb( + X)( ) + 1 T NC T CX (1) λ & H λ 1 λ RTv λ = = k π b ( T T )( ) () 1 1 w X NC T CX The extremum condition is H / a =, one has T ( λ RT λ CX) 1 v = (3) 4µ CX In solving these equations, there are four boundary conditions to be satisfied. They are T () T P =, X() = X, X( t ) = X, λ ( ) = (4) 1 where T is the initial temerature of working fluid on the ower stroke. Eqs. (17), (19), (1) and () determine the otimal solution of this roblem, which could be solved for the minimum value of S t as a function of time and the otimal ath of iston motion, i.e. otimal relationshi between iston velocity v and time t when the ower stroke time t is given. Furthermore, the initial osition of the iston must be constrained, and the iston osition during the whole ower stroke should satisfy the following equation X X (5) Without the above constraint the iston will move above the to dead center. After constrained the iston osition, the otimal ath of the iston with unconstrained acceleration on the ower stroke is a two branch ath. For t t, the velocity of iston is d vt () = (6) where t is the motion delay time. During the motion delay time t d d, the iston osition will not satisfy the eq. (5), and the iston must kee still. While for t t t, the iston osition will satisfy d the eq. (5), and the velocity of iston will be eq.(3). f t 3... Power stroke with constrained acceleration For the case with constrained acceleration, the iston velocity is required to be zero at both start and end oints and the acceleration is constrained to lie within finite limits. The otimization objective is still to minimize the function given by eq. (16), the differential constraints on the state variables T and X remain as given in eqs. (17) and (19), resectively. Besides, the deendence of the state variable v on the control variable a and the inequality constraints on variable a are given by v& = a (7) a a a (8) m 7 m The Hamiltonian for this roblem is µ v + kπb( b/ + X)( T T ) w λ1 NRTv H = [ + kπb( b/ + X)( T T ) h( t)] + λ v+ λ a (9) w 3 T NC X The canonical equations conjugate to eqs. (17), (19) and (7) are eqs. (1), () and & H 4µ v λ RT 1 λ = = + λ 3 v T CX (3) The extremum condition is H / a =, one has λ = (31) 3

9 If eq. (31) holds for more than isolated oints between a and a m m, one also has & λ = (3) 3 Eliminating λ by using eqs. (3) and (3), the exression of the velocity is the same as eq. 3 (3). On this basis one can conclude that the otimal trajectory with acceleration constraint on the ower stroke has two cases. The first case is that when the iston motion exists motion delay time t, d the otimal trajectory is a three-branch ath: (1) From the initial time t = to the motion delay time t = t d, the iston kees still, and the iston osition is the initial osition. () From the motion delay time t = t to the switch time d t = t, the iston trajectory satisfies the system of eqs. (17), (19), (1) and (). (3) From the switch time t = t to the ower stroke time t = t, the iston trajectory is the maximum deceleration segment. The second case is that when the iston motion does not exist motion delay time t d, i.e. the iston osition is X X during the whole ower stroke time, the otimal trajectory is also a three-branch ath, i.e. two boundary segments (maximum acceleration and maximum deceleration) connected by a segment which satisfies the system of eqs. (17), (19), (1) and (). For these equations, only numerical results could be obtained. 4. Numerical examles and discussions 4.1. Determination of the related constants and arameters Table 1. Constants and arameters used in the calculations [14] Mechanical arameters initial osition X =.5cm final osition X f = 8cm cylinder bore b= 7.98cm cycle time τ = 33.3ms corresonding to 36 rm Thermodynamic arameters number of moles of gas comression stroke N =.144, ower stroke N i f =.157 initial temerature comression stroke T = 39K, ower stroke T = 36K C P constant volume heat caacity comression stroke C =.5R, ower stroke C i f = 3.35R cylinder wall temerature T = 6K w Loss term coefficients friction coefficient µ = 1.9 kg / s heat transfer coefficient 1 k 135W K = m Heat function arameters exlosion fraction F =.5 burn time t =.5ms b 4 heat of combustion Q = J / mol er molar fuel-air mixture charge c Gas constant 1 1 R 8.314J mol = K In order to calculate the entroy generation, the environment temerature T = 3K is set. 8

10 Other constants and arameters are listed in tab. 1 according to Ref. [17]. In the following calculations, v is the maximum velocity of the iston on the ower stroke max and T is temerature of working fluid at the end of ower stroke. f 4.. Numerical examles for the case with constrained acceleration The otimal trajectory with constrained acceleration on the ower stroke is a three-branch ath. The system of differential equations for the case with constrained acceleration is solved backwards, i.e. taking the final osition of the iston as the initiate oint of calculation. For the iston motion maybe exists motion delay time, the detail calculation method has two cases. (1) There is exist the motion delay time of iston motion When the time t sent on the ower stroke is fixed, the first is to calculate the maximum deceleration segment. The values of the final temerature T f and the time sent on the maximum deceleration segment t 3 are guessed, then solving eqs. (17), (19) and (5) for the initial various arameters on this segment. The second is to calculate the interior segment. Using the calculation results of the former ste as the initial values of this ste, one can solve the differential equations (17), (19), (1) and () backwards with the iterative method. X X < is used as the terminate condition of the calculating rocess. The time sent on this segment t can be obtained, and the motion delay time can be written as t = t t t. The third is to calculate the first segment During the motion d 3 delay time, the iston osition is initial osition and velocity kee still, i.e. X = X and v =. The initial temerature T P can be solved by eq. (17). The resultant value of T P is comared with the desired one. The guessed initial value of T f and the time sent on the maximum deceleration segment t are then modified to minimize the square of the deviation between the resultant and 3 desired values, and the entroy generation on the ower stroke S is solved. The fourth is to t calculate the entroy generation and the time distribution of nonower strokes. Using the calculation result of the time sent on the ower stroke t, one can solve eqs. t = τ t = t + t, (9) and (11) for n 1 the entroy generation S t and time distribution of nonower strokes, and then the total entroy n generation of the cycle S can be obtained. The fifth is to modify the value of the time sent on the ower stroke t, and to reeat the former four stes until all of the values of the time sent on the ower stroke are calculated. The sixth is to comare the entroy generation er cycle S with different values of time sent on the ower stroke and to select the minimum S. () There is not exist the motion delay time of iston motion When the time t sent on the ower stroke is fixed, the first is to calculate the maximum deceleration segment. The values of the final temerature T f and the time sent on the maximum deceleration segment t 3 are guessed, then solving eqs. (17), (19) and (5) for the initial various arameters on this segment. The second is to calculate the interior segment. Using the calculation results of the former ste as the initial values of this ste, one can solve the differential equations (17), (19), (1) and () backwards with the iterative method. The iston velocity on the initial osition of the interior segment is related to the iston osition, which could be a switching oint from the interior segment to the maximum acceleration segment. The initial arameters and the time sent on this segment t can be obtained. The third is to calculate the maximum acceleration segment. Using the calculation results of the former ste as the initial values of this ste, one can solve eqs. (17), (19) and (5) backwards for the initial temerature T P and the time sent on the maximum acceleration 9

11 segment t. The resultant values of T and the total time sent on the three segments 1 P t + t + t 1 3 are comared with the desired ones. The guessed initial values of T f and the time sent on the maximum deceleration segment t 3 are then modified to minimize the square of the deviation between the resultants and desired values, and the entroy generation on the ower stroke S is t solved. The fourth is to calculate the entroy generation and the time distribution of nonower strokes. Using the calculation result of the time sent on the ower stroke t, one can solve eqs. t = τ t = t n 1 + t, (9) and (11) for the entroy generation S t and time distribution of nonower n strokes, and then the total entroy generation of the cycle S can be obtained. The fifth is to modify the value of the time sent on the ower stroke t, and to reeat the former four stes until all of the values of the time sent on the ower stroke are calculated. The sixth is to comare the entroy generation er cycle S with different values of time sent on the ower stroke and to select the minimum S. Table. Parameters for different cases Case Variation from Table 1 1 none t =.1ms b 3 t = 1.ms b 4 t = 5.ms b 5 τ = 66.66ms,corresonding to18rm 6 k = 61 W K / m 7 µ = 5.8 kg / s Tab. lists some arameters (other arameters unchanged) varied from tab. 1. Tab. 3 lists the calculation results for the corresonding cases, where the modified sinusoidal motion ( r/ l =.5) is chosen as the conventional motion. From tab. 3, one can see that the influences of different choices of burn time, friction coefficient, heat leakage coefficient and cycle time on the otimal configurations of iston movement. For different cases, the results in tab. 3 show that the eak velocity v is much max larger than that of with the conventional motion, while the time sent on ower stroke t with the otimal iston motion is smaller than that of with conventional motion, resectively. After otimization, the decrease of time sent on the ower stroke t has two influences. On the one hand, with the decrease of t, the time sent on the nonower strokes t will increase and the mean iston n velocity during the nonower strokes will decrease, so the entroy generation due to the friction loss on the nonower strokes will decrease which can be seen from the change of S t in tab. 3. On the n other hand, with the decrease of t, the heat leakage loss will decrease for the decrease in the time sent on the contact between the high-temerature working fluid and the environment outside the cylinder, so the entroy generation due to heat leakage loss on the ower stroke will decrease which can be seen from the change of S qt, in tab. 3. The entroy generation due to friction loss on the ower stroke will increase with the increase of the eak velocity, this can be seen from the change of S f, t. The amount of increase in the entroy generation due to friction losses is always smaller than the amount of the decrease in the entroy generation due to heat leakage loss on the ower stroke, so the total entroy generation on the ower stroke is decreased, which can be seen from the change of 1

12 S t in tab. 3. Both the entroy generations on the nonower strokes and ower stroke are decreased, so the entroy generations er cycle S with the otimal iston motion are smaller than those with the conventional iston motion. Table 3. Numerical results for different cases with constrained acceleration am = 3 1 m s Case v max m/ s t ms S tn J K 1 S f, t J K 1 S qt, J K 1 S t J K 1 S J K 1 conv ot conv ot conv ot conv ot conv ot conv ot conv ot T f K 4.3. Comarison between the otimal and conventional iston motions Table 4. Result comarison between otimal trajectories with constrained acceleration and conventional engines Case Decrease in S tn Decrease in S f, t Decrease in S qt, Decrease in S t Decrease in S Tab. 4 lists the comarison results between the otimal trajectory and the conventional motion for different cases with constrained acceleration. From the amounts of the decreases of S t and n S f, t listed in tab. 4, one can see that the amount of the decrease of the entroy generation S t due n to the friction loss on the nonower strokes is smaller than that of the increase of the entroy generation S f, t due to the friction loss on the ower stroke after otimizing the iston motion, the entroy generation of the cycle due to friction loss increases after the otimization. Although the entroy generation of the cycle due to friction loss increases, the total entroy generation of the cycle 11

13 decreases after the otimization. From the amount of the decrease of S qt, listed in tab. 4, one can see that, for the whole cycle, the amount of the decrease of the entroy generation due to the heat leakage loss is much larger than that of the increase of the entroy generation due to the friction loss, the total otimization rocess is realized by decreasing the entroy generation due to heat leakage loss on the initial ortion of the ower stroke. Comaring the otimization results of cases 1-4, one can see that the otimization effect is better when the burn time is longer. Comaring the otimization results of cases 1 and 5, one can see that the otimization effect is better when the cycle eriod is longer. Furthermore, figs. -4 show the comarison of the otimal and conventional iston motions on the ower stroke for case 1. Figure. Comarison of iston velocity of otimal and conventional motions on the ower stroke for case 1 Figure 3. Comarison of working fluid temerature of otimal and conventional motions on the ower stroke for case 1 1

14 Figure 4. Comarison of iston osition of otimal and conventional motions on the ower stroke for case Comarison between the otimal iston motions with different otimization objectives. Figs. 5-7 show the otimal iston trajectories on the ower stroke with two different otimization objectives, which include the otimal trajectory with maximum work outut (ower oerating regime) [15] and minimum entroy generation (economical oerating regime) (this aer). Tab. 5 lists the corresonding results of numerical calculations. In Tab. 5, ε = W / W is the τ R effectiveness, i.e. the second-law efficiency [47]. When heat transfer is considered, the reversible rocess which corresonds to the ractical exansion rocess of Diesel cycle after otimizing the iston motion is not a reversible adiabatic exansion rocess, but a reversible olytroic rocess, so the reversible work outut er cycle which corresonds to the ractical cycle after otimization should be W = N R T T n + N C T X X (31) RC / ( )/( 1) [1 ( / ) vc ] R P f C vc C f where n is the olytroic exonent. The first art of eq. (31) NRT ( T )/( n 1) is used to P f calculate the reversible exansion work of reversible olytroic rocess. When otimizing the iston motion, the heat leakage along the nonower strokes is negligible, so the reversible rocess which corresonds to the ractical comression rocess after otimization is still a reversible adiabatic RC / comression rocess. The second art of eq. (31) NC T [1 ( X / X) vc ] is used to calculate the C vc C f reversible comression work of reversible adiabatic comression rocess. According to Ref. [48], the olytroic exonent changes during the exansion rocess of Diesel engine, and usually the mean olytroic exonent of the exansion is n = 1. ~ 1.8. In this aer, n = 1.5 is used. W is the work outut on the ower stroke, Wf, t n is the work lost due to the friction loss on the nonower strokes, and W is the required comression work inut along the nonower strokes, so com the net work outut er cycle is W = W W W. τ f, t n com 13

15 Figure 5. Comarison of iston velocity on the ower stroke for two otimization objectives Figure 6. Comarison of working fluid temerature on the ower stroke for two otimization objectives Comaring the results of numerical calculations with two otimization objectives listed in tab. 5, one can see that the entroy generation and work outut with the minimum entroy generation objective decrease by 47% and 45%, resectively, while the efficiency increases by 5%. From figs. 5-7, as is also shown by aer [49], one can see that otimization results are essentially different for two otimization objectives. The similarities for the otimal iston motions with different otimization objectives are as follows: They both consist of three segments, including two boundary segments and a middle movement segment; both the last segments are maximum 14

16 deceleration segment; both the middle movement segments and corresonding otimal solutions with unconstrained acceleration satisfy the same differential equations. While the differences for the otimal iston motions with different otimization objectives are: The first segment with the minimum entroy generation objective is a maximum acceleration segment, while that with maximum work outut objective is a stock-still segment. The reason for the differences is: With the maximum work outut objective, the iston motion has a delay time on the ower stroke, during the delay time the iston is stock-still and the temerature of the working fluid increases raidly with the fuel combustion ( which can be seen from the variation of the temerature in fig. 6), so the increase of the temerature of the working fluid at the beginning of the ower stroke increases the iston exansion work outut ( which can be seen from the variation of W in tab. 5). In addition, from the results listed in tab. 5, one can see that the otimal times t sent on the ower stroke with different otimization objectives are different, i.e. the otimal distributions of the total cycle time τ among the strokes are distinct. Both the two above kinds of differences show that otimization objective has imortant effects on the otimal iston trajectory. Figure 7. Comarison of iston osition on the ower stroke for two otimization objectives Otimal objective Minimum entroy generation Maximum work outut Table 5. Numerical results for the otimal iston trajectories of ower stroke with two otimization objectives W ( J) v f, τ max t S t W W R W τ Q m/ s ms J K 1 J J J W f, t n W f, t J T f K ε 5. Conclusion On the basis of Refs. [17, 4], this aer studies a Diesel cycle engine with internal and 15

17 external irreversibilities of friction and heat leakage, in which the finite rate of combustion is considered and the heat transfer between the working fluid and the cylinder wall obeys the Newton s heat transfer law. The otimal iston trajectories for minimum the entroy generation er cycle are derived for the fixed total cycle time and fuel consumed er cycle. Otimal control theory is alied to determine the otimal iston trajectories for the cases of unconstrained and constrained iston accelerations on each stroke and the otimal distribution of the total cycle time among the strokes. The main conclusions include: (1) The otimal iston motions with minimum entroy generation and maximum work outut otimization objectives consist of three segments, including two boundary segments and a middle movement segment, the first segment of the otimal iston motion with the minimum entroy generation objective is a maximum acceleration segment, while that with maximum work outut objective is a stoke-still segment. There are in fact several ways of achieving those athways of which one oints out just two: one mechanical solution is using a contoured late to guide the iston on the desired ath as shown in fig. 8 (i.e. an eccentric shaft of roerly designed shae turns the otimal motion of the iston into a constant angular rotation) [see age 4 of Ref. [5] in detail], and another comletely different way to transform the otimized aths is the use of an electrical couling, see age 4 of Ref. [5] in detail. () After minimum entroy generation otimization, the amount of the decrease of the entroy generation due to the heat leakage loss is much larger than that of the increase of the entroy generation due to the friction loss, so the total otimization rocess is realized by decreasing the entroy generation due to heat leakage loss on the initial ortion of the ower stroke. (3) The otimization objective has influence on the otimal trajectory of the heat engine. The results obtained in this aer can rovide some guidelines for the otimal design and oeration of ractical internal combustion engines. Figure 8. The mechanical transformer. An eccentric shaft of roerly designed shae turns the otimal motion of the iston into a constant angular rotation Acknowledgments This aer is suorted by The National Natural Science Foundation of P. R. China (Project No ) and the Natural Science Foundation of Naval University of Engineering (HGDYDJJ111). The 16

18 authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscrit. Nomenclature a -acceleration, [ m/ s ] b -cylinder bore, [ m ] 1 1 C -constant volume heat caacity, [ kj kg K ] F -exlosion fraction, [ ] f -friction force, [ kg m / s ] G -finite combustion rate, [ m ] H -Hamiltonian, [ ] h -heating function, [ kj ] k -heat transfer coefficient, [ W K / m ] L -stroke length, [ m ] l -connecting rod length, [ m ] N -number of moles of gas, [ ] n -olytroic exonent, [ ] Q -heat leakage, [ J ] 1 1 R -gas constant, [ J mol K ] r -crankshaft length, [ m ] S -entroy generation, [ T -temerature, [ K ] t -time, [ ms ] v -velocity, [ m/ s] W -work outut, [ J ] X -dislacement, [ m ] Greek symbols J K - 1 ] ε -the second-law efficiency, [ ] θ -angle of crankshaft rotating, [ ] µ -friction coefficient, [ kg / s ] τ -cycle time, [ ms ] Subscrits f -the state of combustion end f, t n -the effect of friction loss on nonower strokes f, t -the effect of friction loss on ower stroke i -the state of combustion start max min -maximum -minimum q, t -the effect of heat transfer on ower stroke t n -nonower stroke t -ower stroke -the state of ower stroke start 17

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