Performance of an Endoreversible Atkinson Cycle with Variable Specific Heat Ratio of Working Fluid
Marsland Press Journal of American Science 2010 ;6(2)
Performance of an endoreversible Atkinson cycle with variable specific heat ratio of working fluid
Rahim Ebrahimi
Department of Agriculture Machine Mechanics, Shahrekord University, P.O. Box 115, Shahrekord, Iran [email protected]
Abstract : The p erformance of an air standard Atkinson cycle is analyzed using finite -time thermodynamics. In the endoreversible cycle model, the linear relation between the specific heat ratio of the working fluid and its temperature, and the heat transfer loss are consi dered. The relations between the net work output, the thermal efficiency, and the compression ratio are indicated by numerical examples. Moreover, the effects of variable specific heats of the working fluid on the endoreversible cycle performance are analy zed. The results show that the effect of the temperature dependent specific heat of the working fluid on the endoreversible cycle performance is significant. The conclusions of this investigation are of importance when considering the designs of actual Atk inson engines. [Journal of American Science 2010 ;6(2): 12 -17 ]. (ISSN: 1545 -1003).
Key words : Atkinson heat -engine; Finite -time processes; Heat loss; Performance; Thermodynamics
1. Introduction and thermal efficiency. Ge et a l. (2005a) derived the Recently, the analysis and optimization of performance characteristics of a universal generalized thermodynamic cycl es for different optimization cycle model, which included the Atkinson cycle with objectives has made tremendous progress by using heat transfer and friction -like term losses. Zhao and finite -time thermodynamics (Aizenbud and Band , 1982; Chen (2006) performed analysis and parametric Bejan , 1996; Chen et al. , 1999; Wu et al. , 1999; Chen optimum criteria of an irr eversible Atkinson heat engine and Sun 2004; Aragon -Gonzalez et al. , 2006). Leff using finite time thermodynamics. Performance analysis (1987) determined th e thermal efficiency of a reversible of an Atkinson cycle with heat transfer, friction and Atkinson -engine cycle at maximum work output. A variable specific heats of the working fluid was studied power density maximization of a reversible Atkinson by Ge et al. (2006). Their results showed that the effects cycle has been performed by Chen et al. (1998a). Their of variable specific heats of working fluid and results showed that the efficiency at maximum power friction -like term losses on the irreversible cycle density is a lways greater than that at maximum power, performance are significant. Ge et al. (2007) analyzed and the design parameters at maximum power density the effects of the heat transfer and variable specific lead to smaller and more efficient Atkinson engines with heats of working fluid on the performance of a n larger pressure ratios. Al -Sarkhi et al. (2002) compared endoreversible Atkinson cycle. Hou (2007) compared the performance characteristic curves of the Atki nson the performances of air standard Atkinson and Otto cycle with those of the Miller and Joule –Brayton cycles cycles with heat transfer loss considerations. Lin and by using numerical examples, and outlined the effect of Hou (2007) investigated the effects of heat loss, as maximizing power density on the performance of the characterized by a percentage of fuel’s ener gy, friction cycle efficiency. Qin et al. (2003) derived the and variable specific heats of the working fluid, on the performance characteristics of a univers al generalized performance of an air standard Atkinson cycle under the cycle model, which included the Atkinson cycle, with restriction of the maximum cycle -temperature. Chen et heat -transfer loss. Wang and Hou (2005) studied the al. (2007) built a class of generalized irreversible performance analysis and comparison of an Atkinson universal steady flow h eat engine cycle model cycle coupled to variable temperature heat reservoirs consisting of two heating branches, two cooling under maximum work and maximum power density branches, and two adiabatic branches with consideration conditions, assuming a constant specific heat, too. Their of the losses of heat resistance, heat leakage, and results showed an engine design based on maximum internal irreversibility. The performance characteristics power density is better than that based on maximum of Diesel, Otto, Br ayton, Atkinson, Dual and Miller work conditions, from the view points of engine size cycles were derived. Chen et al. (2008) analyzed and
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compared the performance characteristics of combustion engine, which was designed and built by endoreversible and irreversible reciprocating Diesel, James Atkinson in 1882 (Ge et al. , 2005a). The Otto, Atkinson, Brayton, Braysson, Carnot, dual, and Atkinson cycle, one of the most heat -efficient, Miller cycles with constant and variable specific heats high -expansion ratio cycles, is designed to provide of the working fluid. Thermodynamic analysis of an efficiency at the expense of power. The Atkinson cycle ideal air -standard Atkinson cycle with temperature allows the intake, compression, power, and exhaust dependant specific heat is presented by Al -Sarkhi et al. strokes of the four -stroke cycle to occur in a single turn (2008). This paper outlines the effect of maximizing of the crankshaft. By the use of clever mechanical power d ensity on the performance of the cycle linkages, the expansion ratio is greater than the efficiency. Ge et al. (2008a, 2008b) analyzed the compression ratio, resulting in greater efficiency than performance of an air standard Otto and Diesel cycles. with engines using th e alternative Otto cycle. The cycle In the irreversible cycle model, the non -linear relation for this engine is depicted in figure 1. The cycle is also between the specific heat of the working fluid and its called the Sargent cycle by several physics oriented temperature, the friction loss computed according to the thermodynamic books (Ge et al. 2007). mean velocity of the piston, the internal irreversibility Figure 1 presents pressure -volume (P− V ) and described by using the compression and expansion temperature -entropy (T− S ) diagrams for the efficiencies, and the heat transfer loss are considered. thermodynamic processes performed by an air standard Ust (2009) made a comparative perfo rmance analysis Atkinson cycle. Process ( 1→ 2 ) is an adiabatic and optimization of irreversible Atkinson cycle under (isentropic) compression; process ( 2→ 3 ) is a heat maximum power density and maximum power addition at a constant volume; process ( 3→ 4 ) conditions. adiabatic (isentropic) expansion; process ( 4→ 1 ) is heat All of the above mentioned research, the specific rejection at a constant pressure. heats at constant pressure and volume of working fluid As already mentioned in the previous section, it are assumed to be constants o r functions of temperature can be supposed that th e specific heat ratio of the alone and have the linear and or the non -linear forms. working fluid is function of temperature alone and has But when calculating the chemical heat released in the linear forms: combustion at each instant of time for internal γ= γ − k T (1) o 1 combustion engine, the specific heat ratio is generally where γ is the specific heat ratio and T is the modeled as a linear function of mean charge absolute temperature. γ and k are constants. o 1 temperature (Gatowski et al., 1984; Ebrahimi, 2006). The heat added to the working fluid, during The model has been widely used and the phenomena processes ( 2→ 3 ) is that it takes into account are well knows (Klein, 2004). T3 T 3 R QMin= cdTM v = dT = However, since the specific heat ratio has a great ∫T ∫ T ( 2) 2 γ −k1 T − 1 influence on the heat release peak and on the shape of o (2) the heat release curve (Brunt, 1998), many researchers MR γ −k1 T 2 − 1 ln o kγ − k T − 1 have elaborated different mathematical equations to 1o 1 3 describe the dependence of specific heat ratio from where M is the molar number o f the working fluid. temperature (Gatowski et al., 1984; Brunt, 1998; Egnell, R and cp are molar gas constant and molar specific 1998; Klein, 2004; Klein and Erikson, 2004; Ceviz and heat at constant volume for the working fluid, Kaymaz, 2005). It should be mentioned here that the respectively. most important thermodynamic property used in the The heat rejected in the isobaric heat rejection process heat release calculations for engines is the specific heat ( 4→ 1 ) may be written as ratio (Ceviz and Kaymaz, 2005). Therefore, the T4 T 4 γ − k T R ( o 1 ) Qout= McdTM p = dT = objective of this study is to examine the effect of ∫T ∫ T 1 1 γ −k1 T − 1 variable specific heat ratio on the net work output and o (3) 1 γ −k T − 1 the thermal efficiency of air standard Atkinson cycle. o 1 1 MR T4− T 1 + ln kγ − k T − 1 1o 1 4 2. Cycle model where cp is the molar specific heat at constant The Atkinson cycle engine is a type of internal
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P T 3 3
2 2 Pressure 4 Temperature
1 1 4 (a) Volume V (b) Entropy S Figure 1. (a) P− V diagram;(b) T− S diagram for the air standard Atkinson cycle
pressure for the working fluid. released heat by combustion per second, and the second According to references (Ge et al., 2006; Ebrahimi part is the heat leak loss per second,
2009), the equation for a reversible adiabatic process Qleak = MB( T2 + T 3 ) . with variable specific heat ratio can be writing as From equations (2) and (3), the net work output of follows: the Atkinson cycle engine is given by: γ −1 γ −1 W= Q − Q = TV=()() T +d TV + d V (4) out in out Re -arranging equations (1) and (4), we get the following MR ()()γ−kT −1 γ − kT − 1 (11) ln o12 o 15 +MR T − T equation ()4 3 k1()()γ− kT 14 −1 γ − kT 11 − 1 γ −1 o o Tγ− kT −=1 T γ − kT − 1 VV / o (5) i( o1 j) j() o 1 i( ji ) The thermal efficiency of the Atkinson cycle ∗ engine is expressed by: The specific compression, rc , and compression, rc , ratios are defined as 1 (γ−kT −1)( γ − kT − 1 ) ln o12 o 15 +T − T 4 3 rc = V1 V 2 (6) kγ− kT −1 γ − kT − 1 1()()o 14 o 11 ηth = (12) and 1 γ −k T − 1 ln o 1 2 +T − T ∗ V4 T 4 4 3 r= = r (7) k1γ − k 1 T 4 − 1 cV T c o 2 1 When the values of r and T are given, T Therefore, the equations for processes ( 1→ 2 ) and c 1 2 can be obtained from equation (8), then, substituting ( 3→ 4 ) are shown, respectively, by the following: equation (2) into equation (10) yields T3 . The last unknown is T , which can be deduced from equation γ −1 4 Tγ−− kT1 ro = T γ −− kT 1 (8) 1()()()o 12c 2 o 11 (9). Finally, by substituting T1 , T2 , T3 and T4 into γ −1 o equations (1 1) and (1 2), respectively, the net work T4 T3()()γ− kT 14 −=1 T 4 γ − kT 13 − 1 r c (9) output and thermal efficiency of the Atkinson cycle o o T 1 engine can be obtained. Therefore, the relations between The energy transferred to the working fluid during the net work output, the thermal efficiency and the combustion is given by the following linear relation compression ratio can be derived. (Chen et al. 1998b; Ge et al., 2008a).
Qin = MABT −( 2 + T 3 ) (10) 3. Numerical examples and discussion where A and B are two constants related to According to references (Ebrahimi , 2009, Ghatak combustion and heat transfer which are function of and Chakraborty, 2007; Ge et al. , 2007, Chen et al. , engine speed. From equation (10), it can be seen that 2006; Ge et al. , 2005b) the following constants and Q contained two parts: the first part is MA , the ranges of parameters are us ed in the calculations: in T= 360 K , γ =1.31 − 1.41 , A= 60000 J . mol −1 , 1 o
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−5 −1 M=1.57 × 10 kmol , k1 =0.00003 − 0.00009 K and They reflect the performance characteristics of an B= 28 J . mol−1 K − 1 . Numerical examples are shown as endoreversible Atkinson cycle engine. It should be follows. noted that the heat added and the heat rejected by the Figures 2 -5 sh ow the effect of the parameters γ and working fluid decrease with increases of γ , while o o k1 related to the variable specific heat ratio of the increase with increasing k1 . (see Eqs. (2) and (3)). It working fluid on the Atkinson cycle performance with can be seen that the effect of γ is more than that of o considerations of heat transfer. From these figures, it k1 on the net work output and thermal efficiency. It can be found that γ and k play a key role on the should be mentioned here that for a fixed k , a larger o 1 1 work output and the thermal efficiency. It is clearly seen γ corresponds to a greater value of the specific heat o that the effects of γ and k on the work output a nd ratio and for a given γ , a larger k corresponds to a o 1 o 1 thermal efficiency are related to compression ratio. lower value of the specific heat ratio.
0.1 0.1 k= 0.00003 K −1 0.09 γ = 1.31 0.09 1 o k= 0.00006 K −1 γ = 1.36 1 o k= 0.00009 K −1 0.08 γ = 1.41 0.08 1 o 0.07 0.07 )
0.06 0.06
0.05 0.05
0.04 0.04 Net work output (kJ output Net work
0.03 (kJ) output Net work 0.03
0.02 0.02
0.01 0.01
0 0 0 20 40 60 0 10 20 30 40 Compression ratio Compression ratio Figure 2. Effect of γ on the variation of the net work Figure 3. Effect of k on the variation of the net o 1 output with compression ratio k= 0.00006 K −1 work output with compression ratio γ = 1.36 ( 1 ) ( o )
The effects of γ and k on the net work output are to the heat rejected. One can see that the maximum net o 1 shown in Fig ures 2 and 3. It can be found from these work output, the working range of the cycle and the figures that the net work output versus compression optimal compression ratio corresponding to maximum ratio characteristic is approximately parabolic like net work output decrease (increase) about 11% (4.8%) curves. In other words, the net work output increases and 66.5% (54.7%), 43.6% (25%) when γ ( k ) o 1 with increasing compression ratio, reach their ma ximum increases 7.6% (200%). This is due to the fact that the values and then decreases with further increase in ratio of heat added to heat rejected increases (decreases) compression ratio. It can also be found from the figures wit h increasing γ ( k ) in this case. It should be noted o 1 2 and 3 that if compression ratio is less than certain here that both the heat added and the heat rejected by value, the increase (decrease) of γ ( k ) will make the the working fluid decrease with increasing γ (see Eq. o 1 o net work output bigger, due to the increase in the ratio (4)), and increas e with increase of k1 (see Eq. (5)). of the heat added to the heat rejected. In contrast, if The effects of γ and k on the thermal efficiency are o 1 compression ratio exceeds certain value, the increase shown in Fig ure s 4 and 5. It can be found that the (decrease) of γ ( k ) will make the net work output thermal efficiency increases with the increase of γ o 1 o less, because of decrease in the ratio of the heat added and the decrease of k1 throughout the compression
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70 70
60 60
50 50
40 40
30 30
−1 k1 = 0.00003 K
γ = 1.31 (%) efficiency Thermal −1 Thermal efficiency (%) efficiency Thermal 20 o 20 k1 = 0.00006 K γ =1.36 −1 o k1 = 0.00009 K γ = 1.41 10 o 10
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