Energy Fuels 2010, 24, 295–301 Published on Web 09/23/2009
: DOI:10.1021/ef900773n
Performance Limits for a Class of Irreversible Internal Combustion Engines Lingen Chen,* Shaojun Xia, and Fengrui Sun Postgraduate School, Naval University of Engineering, Wuhan 430033, P. R. China Received July 22, 2009. Revised Manuscript Received September 6, 2009
The performance of a class of irreversible internal combustion engines with finite rate heat exchange with the environment and nonzero entropy generation due to combustion chemical reactions in the cylinder is studied in this paper by using finite time thermodynamics. It is assumed that the heat transfer between the working fluid in the cylinder and the environment obey linear phenomenological law [q Δ(T -1)] in the irreversible thermodynamics, and the combustion chemical reactions in the cylinder obey a general rate equation of reactions. The upper bounds of power output and efficiency of the internal combustion engines are derived by applying optimal control theory. For the special examples with one chemical reaction and some linearly independent chemical reactions, the average optimal control problems are transformed into nonlinear programming problems, and the Kuhn-Tucker conditions corresponding to the optimal solutions are found. Analytical solutions of minimum entropy generation for the two cases are provided. The results obtained herein are compared with those obtained with different rate equation of reactions and Newton’s heat transfer law ([q Δ(T )]). The methods used and the results obtained in this paper can provide some theoretical guidelines for the optimal design and operation of practical internal combustion engines.
combinations) and constant or variable specific heats (including linear and nonlinear relations of temperature) of the working fluid.10-17 The second is to a utilize simplified model to predict internal combustion heat engine performance and compare with detailed numerical simulation result.18,19 The third is to study the optimal paths of piston motion for internal combustion engines. Mozurkewich and Berry20,21 investigated a four-stroke Otto cycle engine with losses of piston friction and heat leakage, in which the heat transfer between the working fluid and the cylinder wall obeys Newton’s heat transfer law [q Δ(T)]. The optimal piston trajectory for maximizing the work output per cycle was derived for the fixed total cycle time and fuel consumed per cycle. It turned out that optimizing the piston motion could improve engine efficiency by nearly 10%. Hoffmann et al.22 and Blaudeck and Hoffmann23 further considered the effect of
1. Introduction Since the mid 1970s, the research into identifying the performance limits of thermodynamic processes and optimizing thermodynamic processes has made great progresses in the fields of physics and engineering.1-9 The research line of finite time thermodynamics (FTT) typically is as follows: making some assumptions for a real process to establish a thermodynamic model; given a series of constraints, defining the probable time pathway of the process; then solving for the given path (or the optimal path) of the specified process variable to obtain optimum performance of the defined process. The heat engine is always one of the major research objects in FTT. The researches on optimal configurations and performances of various internal combustion heat engines include four aspects at least. The first is to study the power output and efficiency of various air-standard cycle models (including diesel, Otto, dual, Miller, Atkinson cycles and universal cycle model) with different loss items (including heat transfer, friction, internal irreversibility, and the different
(10) Angulo-Brown, F; Fernandez-Betanzos, J; Diaz-Pico, C. A. Eur. J. Phys. 1994, 15, 38–42. (11) Angulo-Brown, F; Rocha-Martinez, J. A.; Navarrete-Gonzalez, I. D. J. Phys. D: Appl. Phys. 1996, 29, 80–83. (12) Klein, S. A. Trans. ASME J. Eng. Gas Turbines Power 1991, 113, 511–513. (13) Abu-Nada, E.; Al-Hinti, I.; Al-Aarkhi, A.; Akash, B. Int. Comm. Heat Mass Transfer 2006, 33, 1264–1272. (14) Al-Sarkhi, A.; Al-Hinti, I.; Abu-Nada, E.; Akash, B. Int. Comm. Heat Mass Transfer 2007, 34, 897–906. (15) Parlak, A.; Yasar, H.; Soyhan, H. S.; Deniz, C. Energy Fuels 2008, 22, 1930–1935. (16) Ge, Y.; Chen, L.; Sun, F. Appl. Energy 2008, 85, 618–624. (17) Ge, Y.; Chen, L.; Sun, F. Math. Comp. Model. 2009, 50, 101–108. (18) Curto-Risso, P. L.; Medina, A.; Calvo-Hernandez, A. J. Appl. Phys. 2008, 104, 094911. (19) Curto-Risso, P. L.; Medina, A.; Calvo-Hernandez, A. J. Appl. Phys. 2009, 105, 094904. (20) Mozurkewich, M.; Berry, R. S. Proc. Natl. Acad. Sci. U. S. A. 1981, 78, 1986-1988. (21) Mozurkewich, M.; Berry, R. S. J. Appl. Phys. 1982, 53, 34–42. (22) Hoffman, K. H.; Watowich, S. J.; Berry, R. S. J. Appl. Phys. 1985, 58, 2125–2134. (23) Blaudeck, P.; Hoffman, K. H. Proc. Int. Conf. ECOS’95; Istambul, Turkey, 1995; vol. 2, pp754.
*To whom all correspondence should be addressed. E-mail: lgchenna@ yahoo.com and
[email protected]. Fax: 0086-27-83638709. Telephone: 0086-27-83615046. (1) Andresen, B. Finite-Time Thermodynamics; Physics Laboratory II, University of Copenhagen: Copenhagen, 1983. (2) Sieniutycz, S.; Shiner, J. S. J. Non-Equilib. Thermodyn. 1994, 19, 303–348. (3) Bejan, A. J. Appl. Phys. 1996, 79, 1191–1218. (4) Berry, R. S.; Kazakov, V. A.; Sieniutycz, S.; Szwast, Z.; Tsirlin, A. M. Thermodynamic Optimization of Finite Time Processes; Chichester: Wiley, 1999. (5) Chen, L; Wu, C; Sun, F. J. Non-Equilib. Thermodyn. 1999, 24, 327– 359. (6) Hoffman, K. H.; Burzler, J.; Fischer, A.; Schaller, M; Schubert, S. J. Non-Equilib. Thermodyn. 2003, 28, 233–268. (7) Sieniutycz, S. Prog. Energy Combust. Sci. 2003, 29, 193–246. (8) Chen, L.; Sun, F. Advances in Finite Time Thermodynamics: Analysis and Optimization; New York: Nova Science Publishers, 2004. (9) Sieniutycz, S.; Jezowski, J. Energy Optimization in Process Systems; Elsevier: Oxford, UK, 2009. r 2009 American Chemical Society
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: DOI:10.1021/ef900773n
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the finite combustion rate of the fuel on the performance of engines, and studied the optimal piston motion for a fourstroke diesel cycle engine with losses of piston friction and heat leakage, in which the heat transfer between the working fluid and the cylinder wall also obeys the Newton’s heat transfer law. Teh and Edwards24-26 and Teh et al.27 investigated the optimal piston motions of adiabatic internal combustion engines for maximum work output,24 minimum entropy generation,25,26 and maximum efficiency.27 The fourth is to use a more detailed, “realistic” model and try to find approximate upper limits for possible performance directly.28-30 Orlov and Berry28 first investigated the maximum power output of an irreversible heat engine with a nonuniform working fluid and Newton’s heat transfer law. Both the lumped-parameter model with uniform temperature and the distributed-parameter model described by partial differential equations are put forward, and the results show that the maximum power output of the heat engine in the distributed-parameter model is less than or equal to that in the lumped-parameter model. Orlov and Berry29 further investigated the efficiency limit of an irreversible heat engine with distributed working fluid and Newton’s heat transfer law. For practical internal combustion engines, Orlov and Berry30 established an open internal combustion engine model taking into account a finite rate of Newton’s law heat exchange with the environment and nonzero entropy generation due to combustion chemical reactions, and derived the upper bounds of power and efficiency of the internal combustion engine model by using optimal control theory. Besides, for combustion chemical reactions with a special type of rates of the reactions, they also derived the analytical expression of the low bound of the entropy generation. The results showed that it might be stimulus for research in the field of constructing highly efficient nonconventional internal combustion engines with heating systems instead of cooling systems. In general, heat transfer is not necessarily Newton’s heat transfer law and also obeys other laws; heat transfer laws not only have significant influences on the performance of the given thermodynamic process,31-37 but also have influences on the optimal configurations of thermodynamic process for
Burzler and Hoffthe given optimization objectives. mann51 and Burzler52 considered the effects of convectiveradiative heat transfer law [q Δ(T) þ Δ(T 4)] and nonideal working fluid and derived the optimal piston motion for maximizing power output during the compression and power strokes of a four-stroke diesel engine. Xia et al.53 considered an Otto cycle engine with internal and external irreversibilities of friction and heat leakage, in which the heat transfer between the working fluid and the cylinder wall obeys the linear phenomenological heat transfer law [q Δ(T -1)] in irreversible thermodynamics, and derived the optimal piston trajectory for maximizing the work per cycle with the fixed total cycle time and fuel consumed per cycle. Xia et al.54 investigated the maximum power output of an irreversible heat engine with a nonuniform working fluid and linear phenomenological heat transfer law. Based on ref 30, this paper studies a class of irreversible open internal combustion engines with finite rate of heat exchange with the environment and nonzero entropy generation due to combustion chemical reactions in the cylinder. It is assumed that the chemical reactions obey general rate equation of the reactions and the heat transfer between the working fluid in the cylinder and the environment obeys the linear phenomenological heat transfer law. The upper bounds of power output and efficiency of the internal combustion engines are derived by applying optimal control theory. For the special examples with one chemical reaction and some linearly independent chemical reactions, the average optimal control problems are transformed into nonlinear programming problems, and the Kuhn-Tucker conditions corresponding to the optimal solutions are found. Analytical solutions of minimum entropy generation for the two cases are provided. The results obtained herein are compared with those obtained by Orlov and Berry.30 2. Model of a Class of Internal Combustion Engines 2.1. Balances of Energy and Entropy for the Engines. For simplicity, a process in a one-cylinder internal combustion engine is considered in this paper. Such process is shown schematically in Figure 1. Under the assumptions of negligible potential and kinetic energy of flows, and of the linear phenomenological heat transfer law in the heat transfer process between the working fluid in the cylinder and the external environment, applying the first law of thermodynamics gives the energy balance for the mixture inside the cylinder: Z dE ¼ Jinl hinl - Jout hout RðT0 -1 - T -1 Þda - PðtÞ ð1Þ dt AðtÞ
(24) Teh, K. Y.; Edwards, C. F. Proc. IMECE2006,Paper No. IMECE2006-13622, 2006 ASME Int. Mech. Engng. Congress & Exposition, November 5-10, 2006, Chicago, Illinois, USA. (25) Teh, K. Y.; Edwards, C. F. Trans. ASME J. Dyn. Sys. Measure. Control 2008, 130, 041008. (26) Teh, K. Y.; Edwards, C. F. Proc. IMECE2006, Paper No. IMECE2006-13581, 2006 ASME Inte. Mech.l Engng. Congress & Exposition, November 5-10, 2006, Chicago, Illinois, USA. (27) The, K. Y.; Miller, S. L.; Edwards, C. F. Int. J. Engine Res. 2008, 9, 449–481. (28) Orlov, V. N.; Berry, R. S. Phys. Rev. A 1990, 42, 7230–7235. (29) Orlov, V. N.; Berry, R. S. Phys. Rev. A 1992, 45, 7202–7206. (30) Orlov, V. N.; Berry, R. S. J. Appl. Phys. 1993, 74, 4317–4322. (31) Gutowicz-Krusin, D.; Procaccia, J.; Ross, J. J. Chem. Phys. 1978, 69, 3898–3906. (32) de Vos, A. Am. J. Phys. 1985, 53, 570–573. (33) Chen, L.; Yan, Z. J. Chem. Phys. 1989, 90, 3740–3743. (34) Gordon, J. M. Am. J. Phys. 1990, 58, 370–375. (35) Angulo-Brown, F.; Paez-Hernandez, R. J. Appl. Phys. 1993, 74, 2216–2219. (36) Chen, L.; Sun, F.; Wu, C. J. Phys. D: Appl. Phys. 1999, 32, 99– 105. (37) Huleihil, M.; Andresen, B. J. Appl. Phys. 2006, 100, 014911. (38) Yan, Z.; Chen, J. J. Chem. Phys. 1990, 92, 1994–1998. (39) Chen, L.; Zhu, X.; Sun, F.; Wu, C. Appl. Energy 2004, 78, 305– 313. (40) Andresen, B.; Gordon, J. M. Int. J. Heat Fluid Flow 1992, 13, 294–299. (41) Badescu, V. J. Non-Equilib. Thermodyn. 2004, 29, 53–73. (42) Sieniutycz, S.; Kuran, P. Int. J. Heat Mass Transfer 2006, 49, 3264–3283.
(43) Sieniutycz, S.; Kuran, P. Int. J. Heat Mass Transfer 2005, 48, 719–730. (44) Sieniutycz, S. Int. J. Heat Mass Transfer 2007, 50, 2714–2732. (45) Sieniutycz, S. Appl. Math. Model. 2009, 33, 1457–1478. (46) Sieniutycz, S. Energy 2009, 34, 334–340. (47) Song, H.; Chen, L.; Li, J.; Sun, F.; Wu, C. J. Appl. Phys. 2006, 100, 124907. (48) Song, H.; Chen, L.; Sun, F. J. Appl. Phys. 2007, 102, 094901. (49) Chen, L.; Xia, S.; Sun, F. J. Appl. Phys. 2009, 105, 044907. (50) Xia, S.; Chen, L.; Sun, F. Braz. J. Phys. 2009, 39, 98–105. (51) Burzler, J. M.; Hoffman, K. H. In Thermodynamics of Energy Conversion and Transport; Sienuitycz, S., de Vos, A., Eds.; Springer: New York, 2000; Ch 7. (52) Burzler, J. M. Performance Optima for Endoreversible Systems; Ph. D. Thesis, University of Chemnitz: Germany, 2002. (53) Xia, S.; Chen, L.; Sun, F. Sci. in China Ser. G: Phys. Mech. Astron. 2009, 52, 708–719. (54) Xia, S.; Chen, L.; Sun, F. Sci. in China Ser. G: Phys. Mech. Astron. 2009, 52, 1081–1089.
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cycle efficiency, but also different from the second-law efficiency,20-22,51-53 that is, the ratio of the actual work output to the reversible work output.54 It is more realistic to compare efficiency of real engines to the efficiency of eq 7 than to the air-standard cycle efficiency and the second-law efficiency. Combining eqs 6 and 7 yields: η¼1þ
Figure 1. Model of the one-cylinder internal combustion engine.
j ¼1
kfi
j ¼1
kri
k
R
k
β
Wi ¼ kfi ðTÞ P nj ij - kri ðTÞ P nj ij j ¼1
j ¼1
ð10Þ
where nj(t, ξ) is the molar density of the jth component. It satisfies the following equation r X ðvij Wi Þ ð11Þ dnj =dt ¼ i ¼1
where vij = βij - Rij. The total energy balance equation of chemical reactions is given by 0 1 k X nj hj A=dt ¼ 0 ð12Þ d@
P ¼ -f h - Q0 ð3Þ Rτ where f =R(1/τ) 0f(t)dt, fh(t) = Jout(t)hout - Jinl(t)hinl, and R Q0 = (1/τ) τ0 A(t)R(T0-1 - T -1)da dt is the average heat flow out of the engine into the environment. For heat engines with cooling systems, Q0 > 0. From eq 3, one can obtain
j ¼1
where hj is the molar enthalpy of the jth component. The total entropy balance equation of chemical reactions is given by 0 1 k X nj sj A=dt - σch ¼ 0 ð13Þ d@
ð4Þ
j ¼1
From an entropy balance of eq 2, one further obtains
where hj is the molar entropy of the jth component. According to refs 55 and 56, the chemical potential of the jth component in a mixture can be written as μj = μj0(T, P) þ RT ln(nj), where μj0(T, P) is the standard chemical potential of the jth component. The chemical potential μj can be expressed as the difference of the molar enthalpy hj and the product Tsj, that is, μj = hj - Tsj. Combining eqs 12, 13, and the equation μj = hj - Tsj yields 0 1 k d @X nj μj A þ Tσch ¼ 0 ð14Þ dt j ¼1
ð5Þ Rτ
out - Jinl(t)sinl and δ = -1/τ 0[σ(t) þ Rwhere fs(t) = Jout(t)s 2 A(t)R(1/T0 - 1/T) da]dt. In terms of the second law of thermodynamics, it is easy to see that δ e 0 with equality holding for a reversible process. Combining eqs 3 and 5 yields P ¼ -f h þ T0 f s þ T0 δ ð6Þ
According to ref 30, the efficiency of an internal combustion engine is defined as follows η ¼ P=ð -f h Þ
ki
where and are the forward and backward reaction rate constants of the ith chemical reaction, respectively; Rij and βij are the stoichiometric coefficients. The reaction rate of the ith chemical reaction Wi is determined by the law of mass actions
where S(t) is total entropy of the fuel-air mixture in the cylinder, and σ(t) g 0 is the total entropy generation inside the system. The process in the cylinder is periodic. From periodicity it follows that E(0) = E(τ), S(0) = S(τ), and M(0) = M(τ). Here M(τ) is the total mass of the working fluid and τ is the period. These conditions were called weak periodicity in refs 28-30. 2.2. Power Output and Efficiency for the Engines.30 Assume that T0 is a constant, then from eq 1 it follows that
f s þ δ ¼ -Q0 =T0
ð8Þ
2.3. Entropy Generation for Combustion Chemical Reactions. It is assumed that reactions are reversible and the law of mass action holds. Effects of the inert gas and other materials that excluded in the chemical reactions on the chemical reactions are neglected. For better understanding, the description of chemical reactions occurring uniformly in the space in an isolated constant-volume vessel is first considered here. For r types of chemical reactions including k species, the stoichiometric equations for the reactions have the form k k kfi X X ð9Þ Rij Nij ar βij Nij , i ¼ 1, :::, r
30
where E(t) is a total energy of the system; Jinl and Jout are the mass flows of fuel-air mixture and exhaust gases, respectively; hinl and hout are specific enthalpies of the flows correspondingly; R = R(ξ) is a heat transfer coefficient between the inner surface of the cylinder or piston and the environment; T0(t, ξ) is the temperature of the thermal boundary layer of the environment; P(t) is the indicated power, that is, the power that is delivered as the work done on a piston; T(t, ξ) is the temperature of the fuel-air mixture; and ξ = (ξ1, ξ2, ξ3) is a vector of coordinates of a point in some Cartesian system. Integration in eq 1 is carried out over the area (A(t)) of the surface, which is the inner boundary of the cylinder and the piston. Applying the second law of thermodynamics gives the entropy balance of the system Z dS RðT0 -1 - T -1 Þ ¼ Jinl sinl - Jout sout da þ σðtÞ ð2Þ dt T AðtÞ
P e -f h
T0 f s T0 δ þ -f h -f h
ð7Þ (55) Andresen, B.; Rubin, M. H.; Berry, R. S. J. Chem. Phys. 1983, 87, 2704–2713. (56) Tsirlin, A. M.; Kazakov, V.; Kan, N. M.; Trushkov, V. V. J. Phys. Chem. B 2006, 110, 2338–2342.
The efficiency for the engine of eq 7 should be paid attention, which is not only different from the conventional first-law efficiency,29,31-33,35 that is, the air-standard 297
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: DOI:10.1021/ef900773n
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Substituting μj = μj0(T,P) þ RT ln(nj) into eq 14 yields the entropy generation due to chemical reactions σch:30,56,57 r X Ai Wi ð15Þ σch ¼ T i ¼1
3. Minimum Entropy Generation of Combustion Chemical Reactions The change of the volume of practical internal combustion engine is finite, which is assumed to satisfy the constraint of R Vmin eR V(t) e Vmax. With the notation F = (1/τ) t0[(dt)/ (V(t))] Ω(t)F(t,ξ)dξ, eqs 16 and 18 further give r X Ai W i σ ch ¼ -VðtÞ ð21Þ T i ¼1
Pk where Ai ¼ j ¼1 ðμj vij Þ is the chemical affinity of the ith reaction. Averaged over the cycle time τ, the entropy generation σhch due to chemical reaction in the cylinder is given by # Z Z "X r 1 τ Ai Wi dξ dt ð16Þ σ ch ¼ τ 0 ΩðtÞ i ¼1 T
r X VðtÞ ðvij Wi Þ ¼ Φj , j ¼ 1, :::, k
R respectively. In eq 22,Φj = (1/τ) τ0[Jout(t)(cj,out/Mj) - Jinl(t)(cj,inl/Mj]dt. Our problem now is to determine the minimum entropy generation of eq 21 subject to the volume constraint of Vmin e V(t) e Vmax and the mass balance of eq 22. This is a typical average optimal control The ifollowh problem. Pk f f ing variables are defined: Bi ¼ Ai - j¼1 ðRij μj0 Þ =ðRTÞ h i P Pk Bri ¼ Ari - kj¼1 ðβij μj0 Þ =ðRTÞ. The constant Di = j=1-
For the open system in Figure 1, the material balances for the cylinder are given by # Z "X r cj , inl cj , out dNj ¼ Jinl - Jout þ ðvij Wi Þ dξ, dt Mj Mj ΩðtÞ i ¼1 R
j ¼ 1, ::::, k
ð22Þ
i ¼1
ð17Þ
where Nj(t) = Ω(t)nj(t, ξ) dξ is the total molar amount of P component j in the cylinder, cj = (Mjnj)/( kj=1Mjnj) is the mass fraction of jth component, and Mj is the molar mass of jth component. From the periodicity condition Nj(0) = Nj(τ) and eq 17, one can obtain # Z Z "X Z " r cj, out 1 τ 1 τ Jout ðtÞ ðvij Wi Þ dξ dt ¼ τ 0 ΩðtÞ i ¼1 τ 0 Mj # cj , inl dt, j ¼ 1, ::::, k ð18Þ - Jinl ðtÞ Mj
(vijμj0)/T is also defined. Additional physical restrictions on the these variables are Afi e 0, Ari e 0, i = 1, ..., r, which follow f ..., k,30 so one obtains Bfi e BP from T > 0, μj e 0, j = 1,P i0 and Bri e Bri0, where Bfi0 = -[ kj=1(Rijμj0)]/(RT), Bri0 = -[ kj=1(βijμj0)]/(RT), Rij g 0, and βij g 0. Equations 21 and 22 further give σ ch ¼ VðtÞ
r X f½RðBfi - Bri Þ þ Di ½kfi ðTÞexpðBfi Þ - kri ðTÞexpðBri Þg i ¼1
From μj = μj0(T, P) þ RT ln(nj), one obtains nj = exp[(μj-μj0(T, P))/(RT)]. Substituting it into eq 10 yields 9 8 k P > > f > > A ðR μ Þ > ij j0 > = < i j ¼1 f Wi ¼ ki ðTÞexp > > RT > > > > ; :
- kri ðTÞexp
9 8 k P > > r > > A ðβ μ Þ > i ij j0 > = < j ¼1
> > > :
RT
> > > ;
ð23Þ r X VðtÞ fvij ½kfi ðTÞexpðBfi Þ - kri ðTÞexpðBri Þg ¼ Φj , j ¼ 1, :::, k i ¼1
ð24Þ respectively. According to the categories of chemical reactions in the cylinder and the relationship between them, there are three different cases should be analyzed, respectively. 3.1. One Chemical Reaction Including k Species. Pk In * = [ this case, r = 1. With the notation Φ 1 j=1P (v1jΦj)]/( kj=1v21j), the optimization problem of eqs 23 and 24 is equivalent to the minimization of V½RðBf1 -Br1 ÞþD1 ½kf1 ðTÞexpðBr1 Þ -kr1 ðTÞexpðBr1 Þ with the constraint of VðtÞ½kf1 ðTÞexpðBf1 Þ -kr1 ÞexpðBr1 Þ ¼ Φ1 , that is,
ð19Þ
It is noted that the minimum entropy generation due to chemical reactions is obtained in ref 30, which was based on the following rate equation of the reactions: 1 0 k P 3 0 ! μ ðTÞRij C2 r B f C B j ¼1 j A A 0 f i i C B Wi ¼ ki ðTÞexpB C4exp RT - exp RT 5 RT A @
σ ch g
min
Bf1 eBf10 , Br1 eBr10
fVmax ½kf1 ðTÞexpðBf1 Þ
- kr1 ðTÞexpðBr1 Þ½RðBf1 - Br1 Þ þ D1 g
ð20Þ
¼
P P where Afi = kj=1(μjRij), Ari = kj=1(μjβij), and Ai = Ari - Afi . Comparing eq 19 with eq 20, one can see that the rates of reactions are significantly different from each other. This paper will derive the minimum entropy generation due to chemical reactions based on the rate equation of reactions of eq 19.
min fΦ ½RðBf1 - Br1 Þ þ D1 g , Br1 eBr10 1
Bf1 eBf10
ð25Þ
The problem is further equivalent to minimize the function f r f f Φ*[R(B 1 1 - B1) þ D1] with the constraints of [k1(T) exp(B1) r r f f r r k1(T) exp(B1)] = Φ1* /Vmax, B1 e B10, and B1 e B10. The minimum entropy generation due to chemical reactions always exists, and this minimum value could not be derived from the method of general Lagrange multipliers. Then, the convex objective function achieves its minimum value at the
(57) Tsirlin, A. M.; Kan, N. M.; Trushkov, V. V. Theor. Found. Chem. Eng. 2006, 40, 32–37.
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: DOI:10.1021/ef900773n
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boundary of feasible zone of variables (Bf1, Br1). This problem
eq 25 gives
could be transformed into a nonlinear programming problem, which is given by
σ ch g σf
8 > min σ f ðBf1 , Br1 Þ ¼ Φ1 ½RðBf1 - Br1 Þ þ D1 > > < g1 ðBf1 , Br1 Þ ¼ ½kf1 ðTÞexpðBf1 Þ - kr1 ðTÞexpðBr1 Þ - Φ1 =Vmax ¼ 0 > g2 ðBf1 , Br1 Þ ¼ -Bf1 þ Bf10 g 0 > > : g3 ðBf1 , Br1 Þ ¼ -Br1 þ Br10 g 0
3.2. Some (r) Linearly Independent Chemical Reactions Pk = v v , i,l = 1, Including k Species. The notations C il j=1 ij lj Pr -1 Pr ..., r and j=1(vljΦj), i = 1, ..., r are used, PrΦi* = -1l=1Cli where l=1CilClj = δij, δij = 1 when Pi = j, and δij = 0 when i 6¼ j. Consider the constraints of ri=1{vij[kfi (T)exp(Bfi ) kri (T)exp(Bri )]} = Φj*/Vmax, j = 1, ..., k, Bfi e Bfi0, and Bri e Bri0, then one follows the same procedure as previously and yields r X σ ch g σfi ð31Þ
ð26Þ The gradient of the objective function has the form Δσf(Bf1, = (Φ1*, -Φ1*), and the gradients of three constraints g1(Bf1, Br1), g2(Bf1, Br1), and g3(Bf1, Br1) are given by rg1 (Bf1, Br1) = (kf1 (T) exp(Bf1), -kr1(T) exp(Br1)), rg2(Bf1, Br1) = (-1,0), and rg3(Bf1, Br1) = (0, -1), respectively. For the above three constraints, one introduces three generalized Lagrange multipliers λnu, nu = 1, 2, 3, correspondingly. Let r* (Bf* 1 , B1 ) be the point corresponding to the minimum objective function, then it should satisfy the following famous Kuhn-Tucker condition 8 3 X > r r > < rσf ðBf , B Þ ½λnu rgnu ðBf 1 , B1 Þ ¼ 0 1 1 ð27Þ nu ¼1 f r > > : λnu gnu ðB1 , B1 Þ ¼ 0, nu ¼ 1, 2, 3 λnu g 0, nu ¼ 1, 2, 3
Br1)
i ¼1
where σfi* is the minimum entropy generation corresponding to the ith chemical reaction. 3.3. Some (r) Linearly Dependent Chemical Reactions Including k Species. The optimization problem could also be analyzed by using methods of nonlinear optimization. Introducing Lagrange multipliers λj, j = 1, ..., k, one has ( r h X σ ld ¼ min Vmax RðBfi - Bri Þ Bfi eBfi0 , Bri eBri0 i ¼1 þ Di þ
k X
ðλj Φj Þ
ð32Þ
* is the minimum entropy generation due to r linearly where σld * dependent chemical reactions. As soon as one computes σld for given dependent chemical reactions, an inequality estimate of the entropy generation σhch g σld* could be obtained. 4. Maximum Power Output and Maximum Efficiency Our problem now is to find an upper limit of average power of eq 5 with the constraint of weak periodicity S(0) = S(τ). This is an average optimal control problem, correspondingly, the modified Lagrange function is given by Z τZ P ¼ -f h þ 1=τ RðT -1 - T0 -1 Þð1 þ λ=TÞda dt 0
AðtÞ
þ λðσ - f s Þ
ð33Þ
where λ < 0 is a Lagrange multiplier. Using the temperature of fuel-air mixture T(t, ξ) as a control variable and minimizing the right side of eq 33 over T(t, ξ) > 0 with fixed λ < 0, one obtains T^ = -2λT0/(T0 - λ). Substituting it into eq 33 yields Z τZ P e -f h þ λðσ - f s Þ - 1=τ RðT0 þ λÞ2 =ð4λT0 2 Þda dt 0
ð28Þ
AðtÞ
ð34Þ
r If Bf* */Vmax þ kr1(T) exp(Br1)]/kr1(T)} and Br* 1 = ln{[Φ1 1 = B10,
Inequality 34 is true for all λ < 0. Minimizing the right-hand side of eq 34 with respect to λ gives the optimum λˆ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ λ ¼ - β=½γ þ 4ðf s - σÞ ð35Þ R R R R where β = 1/τ τ0 A(t)R(ξ)da dt and γ = 1/τ τ0 A(t)R(ξ)/ 2 [T0(t, ξ)] da dt. From eq 35 and β > 0, the value of λˆ is reasonable only when the inequality σh < γ/4 þ f s holds. When σh = γ/4 þ f s, one obtains λˆ = -¥. This implies that
the minimum entropy generation σ* f is given by σf
-
j ¼1
) i
- kri ðTÞexpðBri Þ
σf ¼ Φ1 fRfBf10 - lnf½kf1 ðTÞexpðBf10 Þ - Φ1 =Vmax =kr1 ðTÞgg þ D1 g
k ih X ðλj vij Þ kfi ðTÞexpðBfi Þ j¼1
When the objective function is a convex function, the Kuhn-Tucker condition is not only the necessary condition that the objective function achieves its minimum value, but also the sufficient condition. Form eq 27, one can see that if (Bf10, Br10) satisfies the equation [kf1(T) exp(Bf1) - kr1(T) exp(Br1)] = Φ1*/Vmax, the point (Bf10, Br10) is an extreme point. Besides, the function σf(Bf1, Br1) is a convex function, so the point (Bf10, Br10) is also the point corresponding to the minimum objective function. If (Bf10, Br10) does not satisfy the equation [kf1(T) exp(Bf1) - kr1(T) exp(Br1)] = Φ1*/Vmax, f there are other three different cases. First, let Bf* 1 = B10, then f f r = ln {[k (T) exp(B ) Φ */V ]/k one obtains Br* 1 1 1 1 max 1(T)}, if r f* r* the inequality Br* 1 < B10 further holds, the point (B1 , B1 ) is one of the extreme points of the objective function, else it is r f* */ not. Second, let Br* 1 = B10, then one obtains B1 = ln{[Φ1 r r r f Vmax þ k1(T) exp(B1)]/k1(T)}, if the inequality Bf* 1 < B10 f* r* further holds, the point (B1 , B1 ) is one of the extreme points of the objective function, else it is not. Third, if one extreme point is omitted from the above two steps, the remain extreme point is the desired one. If two extreme points are obtained from the above two steps, one could compare the values of objective function at the two extreme points and f* r* f* f choose the smaller one. Let σ* f = σf(B1 , B1 ), if B1 = B10 and r* f f r B1 = ln{[k1(T) exp(B1) - Φ*/V 1 max]/k1(T)}, one obtains the minimum entropy generation σf*, as follows:
ð30Þ
¼ Φ1 fRflnf½Φ1 =Vmax þ kr1 ðTÞexpðBr10 Þ=kf1 ðTÞg - Br10 g þ D1 g ð29Þ
For given chemical reactions, the values of parameters kf1(T), kr1(T), Bf10, and Br10 are known, then one can obtain r* the point (Bf* 1 , B1 ) corresponding to the minimum value r* of the function σf(Bf1, Br1) from eq 27. Let σf* = σf(Bf* 1 , B1 ), 299
Energy Fuels 2010, 24, 295–301
: DOI:10.1021/ef900773n
Chen et al.
the inequality σh < γ/4 þ f s is true for any weakly periodic process. Substituting eq 35 into eq 34 yields P e P^max ðσÞ ¼ Z τZ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi -f h þ β½γ þ 4ðf s - σÞ=2 -1=ð2τÞ ðR=T0 Þda dt 0
5. Comparisons of Results with Different Rates of the Reactions and Heat Resistance Models 5.1. Comparison for Minimum Entropy Generations. Based on rates of the reactions of eq 20, Orlov and Berry30 investigated the minimum entropy generation due to combustion chemical reactions. For one chemical reaction including k species and r linearly independent chemical reactions including k species, the results were given by
AðtÞ
ð36Þ Combining eq 7 with 36 yields ^ max ðσÞ ¼ -P^max ðσÞ=f h ηeη
ð43Þ
ð44Þ
σ ch g -RjΦ1 jln½1 - RjΦ1 j=ðVmax kmax 1 Þ
ð37Þ
When T0(t, ξ) = const, one further obtains β = γT02. Equation 36 gives qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P^ max ðσÞ ¼ -f h þ β½γ þ 4ðf s - σÞ=2 - γT0 =2 (qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) ¼ -f h þ 2T0 ðf s - σÞ= 1 þ 4ðf s - σÞ=γ þ 1 ð38Þ
σ ch g
r X
f -RjΦi jln½1 - RjΦi j=ðVmax kmax Þg i
i¼1
¼ maxT>0 fRkfi ðTÞexp½ respectively, where kmax i
k P
μj0 -
j¼1
ðTÞRij =ðRTÞgFor some (r) linearly dependent chemical reactions including k species, there is no analytical solution, which could also only be solved by using methods of nonlinear optimization. Based on rates of the reactions of eq 19, this paper investigates the minimum entropy generation due to combustion chemical reactions. For one chemical reaction including k species and r linearly independent chemical reactions including k species, the Kuhn-Tucker conditions corresponding to the optimum solutions are also given. Comparing eq 28 or eq 29 with eq 43, one can see that results obtained with different rates of the reactions are significantly different. The minimum entropy generation due to one chemical reaction obtained in ref 30 relates to parameters kfi (T), μj0, Rij (i = 1, ..., r, j = 1, ...,k) and so on, while that obtained in this paper not only relate to kfi (T), μj0, Rij, but also relate to kr1(T), βij, and so on. The methods applied in this paper can be used in other researches related to chemical reactions. From eqs 31 and 34, the minimum entropy generation due to r linearly independent chemical reactions including k species is the sum of that due to each chemical reaction. 5.2. Comparisons for Power Output and Efficiency Limits. Orlov and Berry30 investigated the power output and efficiency performance limits of an internal combustion engine with Newton’s heat transfer law in the heat transfer process between the working fluid in the cylinder and the external environment. If the entropy generation due to finite rate heat transfer is considered and that due to chemical reactions is neglected, the inequality estimates for the power and efficiency are given by30 P e P^max ð0Þ ¼ -f h þ T0 f s γ0 =ðf s þ γ0 Þ ð45Þ
Function P^max from eq 36 is monotonically decreasing in the argument σh, because qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð39Þ dP^max =dσ ¼ -β= β½γ þ 4ðf s - σÞ < 0 The entropy generation σh g 0, so one could substitute σh = 0 into eq 38 in order to get an upper limit for power output. A corresponding estimate for eq 38 is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ P e Pmax ð0Þ ¼ -f h þ 2T0 f s = 1 þ 4f s =γ þ 1 ð40Þ Combining eqs 8 with 40, one obtains the inequality for the efficiency of the internal combustion engine, as follows sout - sinl 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ max ð0Þ ¼ 1 þ T0 ð41Þ ηeη hinl - hout 1 þ 4f s =γ þ 1 Substituting the minimum entropy generation due to chemical reactions σh = σ*ld into eq 39 yields qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dP^max =dσ ¼ -T0 = ½1 þ 4ðf s - σld Þ=γ ð42Þ Equation 42 shows that it is hard (but theoretically not possible) to construct heat engines with efficiency ηˆ max(0) in eq 41. This is mainly due to the negative influence of internal entropy generation on power output that may grow with the increase of the external environment temperature T0. If the minimum entropy generation of the working fluid in the cylinder is known and let it be a constant σ* ld > 0, an estimate of average internal entropy generation σh g σ* ld is known. From *). From eq 42, one can also see eq 39, one obtains P e P^max(σld that besides the entropy generation due to finite rate heat transfer, the entropy generation of the working fluid in the cylinder is also very important for the optimal design of * < f s, the practical internal combustion engines. When σld heating system may have a positive effect on the power and efficiency. When σ* ld g f s, it is impossible to improve the engine by heating, because it is forbidden by the entropy balance of eq 2 and the periodicity condition S(0) = S(τ), which may contradict the second law of thermodynamics. Besides the entropy generation due to finite rate heat transfer, effects of the entropy generation inside the working fluid on the power and efficiency of the internal combustion engines can not be neglected, and the entropy generation due to combustion chemical reactions as the main composition of the entropy generation inside the cylinder should be paid great attention.
^ max ð0Þ ¼ 1 þ T0 ηeη
sout - sinl γ0 hinl - hout f s þ γ0
ð46Þ
R R respectively, where γ0 = 1/τ τ0 A(t)R0 da dt and R0 is the heat transfer coefficient corresponding to Newton’s heat transfer law. Comparing eqs 40 and 41 with eqs 45 and 46, respectively, one can see that the power output and efficiency limits with different heat transfer laws are different significantly, so the researches on the power and performance limits of practical internal combustion engines with different heat resistance models are very necessary. 6. Conclusions On the basis of ref 30, this paper studies the optimal performance of a class of irreversible internal combustion 300
Energy Fuels 2010, 24, 295–301
: DOI:10.1021/ef900773n
Chen et al.
engines by taking into account the finite rate of heat exchange with the environment and nonzero entropy generation due to combustion chemical reactions in the cylinder. It is assumed that the chemical reactions obey generalized rates of the reactions and the heat transfer between the working fluid in the cylinder and the environment obeys the linear phenomenological heat transfer law. The upper bounds of power output and efficiency of the internal combustion engines are derived by applying optimal control theory. Based on general rate equations of the reactions, the minimum entropy generation due to chemical reactions is investigated. For one chemical reaction and some linearly independent chemical reactions, the average optimal control problems are transformed into nonlinear programming problems, and the Kuhn-Tucker conditions corresponding to the optimal solutions are obtained. The objective functions are convex functions, so Kuhn-Tucker conditions are not only the necessary conditions that the objective functions achieve their minimum values, but also the sufficient conditions. Analytical solutions of minimum entropy generation for the two cases are also obtained. The methods applied in this paper can be used in other researches related to chemical reactions. Analytical solutions of the maximum power output and efficiency of the internal combustion engines are obtained by using optimal
control theory, and the results also show that it may be a stimulus for research in the field of constructing highly efficient nonconventional internal combustion engines with heating systems instead of cooling systems.30 The obtained results herein are compared with those obtained in ref 30, it shows that the minimum entropy generations due to chemical reactions with different rate equations of the reactions are different and heat resistance models have significantly effects on the power output and efficiency performance limits of irreversible internal combustion engines. The researches on the performance limits of internal combustion engines with different rate equations of the reactions and heat resistance models enrich the finite time thermodynamic theory. Besides, the methods used and the results obtained in this paper could provide some theoretical guidelines for the optimal design and operation of practical internal combustion engines. Acknowledgment. This paper is supported by the Program for New Century Excellent Talents in University of P. R. China (Project 20041006) and The Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project 200136). The authors wish to thank the reviewers for their careful, unbiased, and constructive suggestions, which led to this revised manuscript.
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