Ecological Optimization Performance of An Irreversible ... - Springer Link

Open Sys. & Information Dyn. (2006) 13: 55–66 DOI: 10.1007/s11080-006-7267-4

c Springer 2006 °

Ecological Optimization Performance of An Irreversible Quantum Otto Cycle Working with an Ideal Fermi Gas Feng Wu1,2 , Lingen Chen1 , Fengrui Sun1 1 Postgraduate School, Naval University of Engineering Wuhan 430033, P. R. China 2 School of Science, Wuhan Institute of Technology Wuhan, 430073, P. R. China

Chih Wu Department of Mechanical Engineering, U. S. Naval Academy Annapolis, MD 21402, U.S.A.

Fangzhong Guo School of Energy and Power Engineering, Huazhong University of Science and Technology Wuhan 430074, P. R. China

Qing Li Technical Institute of Physics and Chemistry, Chinese Academy of Science Beijing 100080, P. R. China

(Received: August 22, 2005) Abstract. The model of an irreversible Otto cycle using an ideal Fermi gas as the working fluid, which is called as the irreversible Fermi Otto cycle, is established in this paper. Based on the equation of state of an ideal Fermi gas, the ecological optimization performance of an irreversible Fermi Otto cycle is examined by taking an ecological optimization criterion as the objective, which consists of maximizing a function representing the best compromise between the exergy output and exergy loss (entropy production) of the cycle. The relationship between the ecological function E and the efficiency η is studied. Three special cases are discussed in detail. The results obtained herein may reveal the general performance characteristics of the irreversible Fermi Otto cycle.

1.

Introduction

The Otto cycle is one of the important thermal cycle modes [1 – 9]. According to the theory of classical thermodynamics, the performance parameters of the cycle can be derived by using the classical ideal gas equation of state or some equations based on it. However, when the gas temperature is low enough or density is high enough, the ideal gas will deviate from classical gas behaviour

56

F. Wu, L. Chen, F. Sun, C. Wu, F. Guo, Q. Li

and quantum degeneracy of the gas will become important [10, 11]. Under the conditions, the gas called quantum gas obeys Fermi statistics instead of classical statistics. One can call the cycle with quantum Fermi gas as quantum cycle or Fermi cycle. Thus, when the working fluid is a quantum gas, the Otto cycle will have some new performance characteristics, which are different from those with conventional working fluid [1 – 9], to be researched. Many authors have studied the performance of the quantum engines, refrigerators and heat pumps [10 – 37] based on the semi-group approach and the generalized quantum master equation. Recently, finite time thermodynamics [38 – 44] has made considerable progresses, and the object of study now has been extended from classical thermal systems to quantum thermal systems. Most of the previous works have concentrated on power optimization, or the minimization of fixed cost for a heat engine. Another criterion for heat engines is the thermal-efficiency optimization that can be considered the variable-cost minimization. Alternatively, Angulo-Brown in [45] proposed an ecological criterion E 0 = W − TL ∆S for finite-time Carnot heat engines, where TL is the temperature of cold heat reservoir, W is the work output and ∆S is the entropy generation. They derived a general property of non-endoreversible thermal cycles with this ecological criterion [46]. Yan [47] showed that it might be more reasonable to use E 0 = W − T0 ∆S if the cold reservoir temperature TL is not equal to the environment temperature T0 because in the definition of E 0 , two different quantity, exergy output W and non-exergy TL ∆S, were compared together. This criterion function is more reasonable than that presented by Angulo-Brown. The optimization of the ecological function represents a compromise between the work output W and the loss work T0 ∆S, which is produced by entropy generation in the system and its surroundings. Ecological optimization has been carried out for endoreversible and irreversible Carnot, Brayton, Stirling and Ericsson heat engines, refrigerators, and heat pumps [46 – 50]. The purpose of this paper is to investigate the quantum performance of an irreversible Fermi Otto cycle with an ideal Fermi gas by using ecological objective. For an irreversible Fermi Otto cycle, the entropy in two adiabatic branches is variable due to the irreversible processes. The irreversibility degree is determined by using the factors of internal irreversibility degree. In this paper, the quantum degeneracy effect on the ecological function E and the efficiency η of an irreversible Fermi Otto cycle with 3 He gases is analyzed. Some special cases are discussed in detail. 2.

Irreversible Fermi Otto Cycle

Consider an Otto cycle working with an ideal Fermi gas, which is called as the Fermi Otto cycle. It is well know that the Otto cycle is made of two isochoric branches connected by two adiabatic branches. Fig. 1 shows schematically the diagram for an irreversible Otto cycle. Processes 1–2 and 3–4 are non isentropic adiabatic, and processes 2–3 and 4–1 are isochoric. Processes 1–2’ and 3–4’ are

Ecological Optimization Performance of An Irreversible Quantum Otto Cycle. . .

57

Fig. 1: T–S diagram of an irreversible Otto power cycle isentropic adiabatic processes. Hence, cycle 1–2–3–4 is an irreversible one, and cycle 1–2’–3–4’ is an endoreversible one. The irreversible and endoreversible cycles are distinguished by using the factors ϕ1 and ϕ2 called the factor of internal irreversibility degree inside the system. The internal temperatures of the working fluid at state 1, 2, 2’, 3, 4, and 4’ are T1 , T2 , T20 , T3 , T4 , T40 , respectively. The highest temperature and the lowest temperature of the gas in the cycle are TH = T1 and TL = T3 , respectively. 3.

The Thermodynamic Properties of an Ideal Fermi Gas

According to the classical Maxwell-Boltzmann statistics, the equation of state for a classical ideal gas can be written as follows p = nkT ,

(1)

where k is Boltzmann constant, T is the gas temperature, n is the number density and p is the pressure of ideal gas. However, under sufficiently low temperature or high density conditions an ideal gas obeys quantum statistics instead of the Maxwell-Boltzmann one although it is still an ideal gas. Using Fermi-Dirac statistics, the equation of state for an ideal Fermi gas can be given as follows [10] p = nkT F (z) ,

(2)

where F (z) is called as the correction factor and it is given by [51] F (z) =

f5/2 (z) f3/2 (z)

(3)

58


with 1 Γ(l)

fl (z) =

Z∞ 0

xl−1 dx , +1

z −1 ex

(4)

where fl (z) is called the Fermi function, z = exp(µ/kT ) is the fugacity of the gas, Γ(l) is the Γ-Euler function, and µ is the chemical potential of the gas. Based on quantum statistics and (2), the internal energy of the system can be written as follows [51] 3 (5) U = pV . 2 ¿From (2) and (5), the heat capacity at constant volume can be expressed as

CV =

∂U

∂T

V

=

3 ∂ [T F (T, V )] , Nk ∂T 2

(6)

where the correction factor F (T, V ) is a function of temperature T and volume V . By using (6) for an isochoric process, the heat quantity Q1 supplied by the heat source in process 4–1 and the heat quantity Q2 released to the heat sink in process 2–3 are, ZT1

Q1 =

CV dT = T4

Q2 =

h i 3 N k TH F (TH , VL ) − T4 F (T4 , VL ) , 2

ZT3 h i 3 N k T2 F (T2 , VH ) − TL F (TL , VH ) , CV dT = T2

2

(7)

(8)

where TH = T1 and TL = T3 (see Fig. 1). 4.

The Ecological Function of the Fermi Otto Cycle

Since the entropy S of the system and the correction factors F (z) depend on z only, the expressions in the isentropic adiabatic processes 3–4’ and 1–2’ can be obtained F (T40 , VL ) = F (TL , VH ) , F (T20 , VH ) = F (TH , VL ) .

(9) (10)

According to classical thermodynamics, for the isochoric processes 4–1 and 2–3 one obtains the following equations TH F (TH , VL ) T4 F (T4 , VL ) T2 F (T2 , VH ) TL F (TL , VH )

=

=

p1 = λ1 , p4 p2 = λ2 , p3

(11)

(12)


where λ1 and λ2 are called as the isochoric compression ratio. ¿From the endoreversible cycle 1–2’–3–4’, one obtains p1 p20 = λ, = p3 p40

59

(13)

where λ is the endoreversible isochoric compression ratio. Combining equations (11), (12) and (13) gives λ1 =

λ2 =

λ p1 p40 · = , p40 p4 ϕ1 p20 p2 · = λϕ2 , p3 p20

(14)

(15)

where factors ϕ1 = p4 /p40 and ϕ2 = p2 /p20 of internal irreversibility degree are assumed as constants in the paper. Substituting (9)–(15) into (7) and (8) yields Q1 =

Q2 =

3 ϕ1 N kTH F (TH , VL ) 1 − , 2 λ 3 N kTL F (TL , VH )(λϕ2 − 1) . 2

(16)

(17)

By using (16) and (17), the net work output W per cycle can be written as W = Q1 − Q2 =

h 3 ϕ1 F (TL , VH ) i N kTH F (TH , VL ) 1 − , − τ (λϕ2 − 1) 2 F (TH , VL ) λ

(18)

where τ is the temperature ratio defined as τ = TL /TH . Combining (16) and (17) gives the entropy generation of the cycle as follows ∆s =

h Q2 Q1 F (TL , VH ) ϕ1 i 3 − − 1− = N kF (TH , VL ) (λϕ2 − 1) . TL 2 F (TH , VL ) TH λ

(19)

Substituting (18) and (19) into ecological function E = W − T0 ∆s yields E =

h 3 ϕ1 F (TL , VH ) i N kTH F (TH , VL ) a1 1 − − τ a2 (λϕ2 − 1) , 2 λ F (TH , VL )

(20)

where a1 = 1 + T0 /TH and a2 = 1 + T0 /TL . ¿From (16) and (17) the efficiency η of the cycle can be obtained in the following equation: η = 1−

Q2 τ (λϕ2 − 1)F (TL , VH ) = 1− . Q1 (1 − ϕ1 /λ)F (TH , VL )

(21)

Eqs. (18)–(21) indicate that work W , entropy generation ∆s, efficiency η and ecological function E of the Fermi Otto cycle are functions of the endoreversible isochoric compression ratio λ for given TH , TL , T0 , ϕ1 , ϕ2 , VH , VL . Taking the derivatives of E with respect to λ and setting it equal to zero (∂E/∂λ = 0) yields the optimum isochoric compression ratio Ê

λ = λ0 =

a1 ϕ1 θ , a2 τ ϕ2

(22)

60


where θ = F (TH , VL )/F (TL , VH ). The corresponding optimal work, the optimal entropy generation, the optimal thermal efficiency and the optimal ecological function are as follows h 3 ϕ1 τ (λ0 ϕ2 − 1) i , − N kTH F (TH , VL ) 1 − θ λ0 2 i h 3 λ0 ϕ2 − 1 ϕ1 , ∆s = N kF (TH , VL ) − 1− λ0 2 θ τ (λ0 ϕ2 − 1) η = 1− , (1 − ϕ1 /λ0 )θ h 3 ϕ1 τ a2 (λ0 ϕ2 − 1) i E = N kTH F (TH , VL ) a1 1 − . − 2 θ λ0

W

=

5.

(23)

(24)

(25)

(26)

Discussion

In (25) and (26), the factors F (TH , VL ) and θ display the effect of gas degeneracy for the performance index of an irreversible Fermi Otto cycle. Three special cases are discussed. 5.1.

Under the condition of strong gas degeneracy

In the first approximation the correction factor for Fermi gas can be simplified as [26] 2TF (VF ) π2T F (T, V ) = + , (27) 5T 6TF (VF ) where 3N 2/3 h2 TF (V ) = πV 8mk is the Fermi temperature. Substituting (27) into (25) and (26) gives

τ (λ0 ϕ2 − 1) , (1 − ϕ1 /λ0 )θ1 h ϕ1 τ a2 (λ0 ϕ2 − 1) i 3 N kTH F1 a1 1 − − , 2 λ0 θ1

η = 1−

(28)

E =

(29)

where

Ê

λ0 =

F1 =

θ1 =

a1 ϕ1 θ1 , a2 τ ϕ2 3rN 2/3 h2 4π 2 TH mk πVH 2/3 + , 20TH mk πVH 3h2 3rN 3rN 2/3 h2 4π 2 TH mk πVH 2/3 + 20TH mk πVH 3h2 3rN , 2 2 h 3N 2/3 4π TH τ mk πVH 2/3 + 20TH τ mk πVH 3h2 3N

(30)

(31)

(32)


61

Fig. 2: Variations of E versus to η under the condition of strong gas degeneracy and in the first approximation. where r = VH /VL . The ecological function E versus the efficiency η for 3 He gas with VH /N = 1.1 · 10−28 m3 , τ = 0.2, a1 = 1.9, a2 = 1.5 and ϕ1 = ϕ2 = 1.1 is shown in Fig. 2. It is seen from Fig. 2 that the ecological function E increases and the efficiency η decreases with the increase in r. It is clear that both the ecological function E and the efficiency η decrease with increases in the factors ϕ1 and ϕ2 of internal irreversibility degree. 5.2.

Under the condition of weak gas degeneracy

To the first approximation the correction factor for Fermi gas can be simplified as [26] GN F (T, V ) = 1 + 3/2 (33) T V with h3 G = . (34) 32(πmk)3/2 Substituting (33) and (34) into (25) and (26) yields

τ (λ0 ϕ2 − 1) , (1 − ϕ1 /λ0 )θ2 h 3 ϕ1 τ a2 (λ0 ϕ2 − 1) i N kTH F2 a1 1 − − , 2 λ0 θ2

η = 1−

(35)

E =

(36)

where

Ê

λ0 =

a1 ϕ1 θ2 , a2 τ ϕ2

(37)

62


Fig. 3: Variations of E versus η under the condition of weak gas degeneracy and in the first approximation. F2 = 1 +

θ2 =

GN r

, 2/3 TH VH GN r 1 + 2/3 TH VH . GN 1+ 2/3 τ TH VH

(38)

(39)

The ecological function E versus the efficiency η for 3 He gas with VH /N = 1.1 · 10−28 m3 , τ = 0.2, a1 = 1.9, a2 = 1.5 and ϕ1 = ϕ2 = 1.1 is shown in Fig. 3. Fig. 3 shows that both the ecological function E and the efficiency η increase with the increase in r and that the ecological function E varies strongly with η for large r. The efficiency η is a constant when r ≥ 5. 5.3.

Under the condition of TH À TF (VL ) and TL À TF (VH )

In the first approximation the correction factor for Fermi gas can be simplified as F (TH , VL ) = 1 +

GN r 3/2

TH VH

(40)

and

3N 2/3 h2 4π 2 TH τ mk πVH 2/3 + . 20TH τ mk πVH 3h2 3N Substituting (40) and (41) into (25) and (26) yields

F (TL , VH ) =

η = 1−

τ (λ0 ϕ2 − 1) , (1 − ϕ1 /λ0 )θ3

(41)

(42)


63

Fig. 4: Variations of E versus η under the condition of TH À TF (VL ), and TL À TF (VH ) and in the first approximation. E =

h 3 ϕ1 τ a2 (λ0 ϕ2 − 1) i N kTH F (TH , VL ) a1 1 − − , 2 λ0 θ3

(43)

with Ê

λ0 =

θ3 =

a1 ϕ1 θ3 , a2 τ ϕ2 F (TL , VH ) . F (TH , VL )

(44)

(45)

When VH /N = 1.1 · 10−28 m3 , τ = 0.04, a1 = 1.9, a2 = 1.5 and ϕ1 = ϕ2 = 1.1 are fixed, the ecological function E versus the efficiency η for 3 He gas is shown in Fig. 4. Both E and η have maximum values, as shown in Fig. 4. For given parameters, E = Emax , when η = η0 and η = ηmax when E = E0 . The optimization criteria of the Otto cycle under the condition can been obtained from parameters Emax , η0 , ηm and E0 , as follows: E0 ≤ E ≤ Emax , 6.

η0 ≤ η ≤ ηm .

(46)

Conclusion

A thermodynamic model of the irreversible Fermi Otto cycle with an ideal Fermi gas has been established in this paper. The results equations (25) and (26) show that quantum degeneracy of the gases effects the behaviours of a Fermi Otto cycle. The relationship between the ecological function E and the efficiency η is studied in this paper. In general, the ecological function E and the efficiency η are not only

64


function of temperature, but also are dependent on other parameters for example the volume ratio r. The ecological function E increases with the increase in r under all conditions. The efficiency η decreases with the increase in r under the condition of strong gas degeneracy and it increases with the increase in r under the condition of weak gas degeneracy. The irreversibility of adiabatic processes is denoted by using the factors ϕ1 and ϕ2 called the factor of internal irreversibility degree inside the system. It is understood that both the ecological function E and the efficiency η decrease with increases in the factors ϕ1 and ϕ2 of internal irreversibility degree. Finally, it should be pointed out that this paper analyzes the performance of an irreversible Fermi Otto cycle with an ideal Fermi gas in the scope of thermodynamics and statistical physics. It provides the application foundation of the quantum statistical mechanics to the practical Otto cycles. For a real Otto cycle using an ideal Fermi gas as the working fluid, one has to consider the influence of not only the irreversibility existing in the cycle but also the quantum degeneracy of the working fluid gas. Acknowledgements This paper is supported by the Program for New Century Excellent Talents in University of P. R. China (Project No. NCET–04–1006), the Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project No. 200136) and the Natural Science Fund of Hubei, P. R. China (Project No. 2004ABA016). Bibliography [1] M. Mozurkewich, R. S. Berry, Optimal paths for thermodynamics systems: The ideal Otto cycle, J. Appl. Phys. 53, 34 (1982). [2] S. A. Klein, An explanation for observed compression ratios in internal combustion engines, Trans. ASME J. Engng. Gas Turbine Pow. 113, 511 (1991). [3] C. Wu, D. A. Blank, The effects of combustion on a work optimized endoreversible Otto cycle, J. Instit. Energy 65, 86 (1992). [4] F. Angulo-Brown, J. Fernandez-Betanzos, C. A. Diaz-Pico, Compression ratio of an optimized Otto-cycle model, Eur. J. Phys. 15, 38 (1994). [5] L. Chen, C. Wu, F. Sun, S. Cao, Heat transfer effects on the net work output and efficiency characteristics for an air standard Otto cycle, Energy Convers. Mgnt. 39, 643 (1998). [6] W. Wang, L. Chen, F. Sun, C. Wu, The effects of friction on the performance of an air standard Dual cycle, Exergy, An Int. J. 2, 340 (2002). [7] A. Fischer, Hoffman, Can a quantitative simulation of an Otto engine be accurately rendered by a simple Novikov model with heat leak?, J. Non-Equilib. Thermodyn. 29, 9 (2004). [8] Y. Ge, L. Chen, F. Sun, C. Wu, Thermodynamic simulation of performance of an Otto cycle with heat transfer and variable specific heats of working fluid, Int. J. Thermal Science 44, 506 (2005). [9] Y. Ge, L. Chen, F. Sun, C. Wu, The effects of variable specific heats of working fluid on the performance of an irreversible Otto cycle, Int. J. Exergy, 2, 274 (2005).


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