Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities

Journal of the Energy Institute xxx (2014) 1–12

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Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities X.W. Liu a, b, c, L.G. Chen a, b, c, *, F. Wu c, d, F.R. Sun a, b, c a

Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430033, China Military Key Laboratory for Naval Ship Power Engineering, Naval University of Engineering, Wuhan 430033, China c College of Power Engineering, Naval University of Engineering, Wuhan 430033, China d School of Science, Wuhan Institute of Technology, Wuhan 430074, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 February 2011 Accepted 21 May 2013

By using quantum master equation, semi-group approach and finite time thermodynamics (FTT), this paper derives the expressions of cycle period, power and efficiency of an irreversible quantum Carnot heat engine with irreversibilities of heat resistance, internal friction and bypass heat leakage, and provides detailed numerical examples. The irreversible quantum Carnot heat engine uses working medium consisting of many non-interacting spin-1/2 systems and its cycle is composed of two isothermal processes and two irreversible adiabatic processes. The optimal performance of the quantum heat engine at high temperature limit is deduced and analyzed by numerical examples. Effects of internal friction and bypass heat leakage on the optimal performance are discussed. The endoreversible case, frictionless case and the case without bypass heat leakage are also briefly discussed. Ó 2014 Energy Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Finite time thermodynamics Spin-1/2 systems Quantum heat engine cycle Power Efficiency

1. Introduction In the past few decades, tremendous progress has been made in optimizing the performance of thermodynamic processes and cycles by using the theory of finite time thermodynamics (FTT) [1–8]. However, for some special fields and systems, such as, magnetic system, infrared techniques, laser system, superconductivity system, and so on, the working medium in these systems has quantum characteristics and obeys quantum statistical mechanics instead of classical statistical mechanics. Therefore, the classical thermodynamics based on phenomenological law and classical statistical mechanics based on equilibrium statistical mechanics are inapplicable. By considering the quantum characteristics of working medium, some researchers have applied the FTT to analyze and optimize the performance of quantum thermodynamic processes and cycles, and obtained many novel results. Kosloff [9] first established a quantum heat engine model and derived the power and efficiency of the quantum heat engine. This engine was called quantum harmonic engine and consisted of three parts: (1) the engine: the engine is similar to a quantum amplifier. It consists of two harmonic oscillators with different frequencies and produced power through a population inversion between the harmonic oscillators; (2) the power output mechanism: the power output mechanism is accomplished by a periodic coupling which manipulated the population difference between the oscillators; (3) the heat reservoirs: there are two heat reservoirs coupled the harmonic oscillators, respectively, and the reservoirs are describe by a semi-group approach. Geva and Kosloff [10] established an endoreversible quantum heat engine model and analyzed the optimal performance of the engine by using semi-group approach and FTT. This quantum heat engine is similar to the quantum harmonic engine in Ref. [9]. Differently, this quantum engine uses working medium consists of non-interacting spin-1/2 systems and is called quantum spin-1/2 heat engine. In the quantum spin heat engine, the time dependence of the external driving field is controllable, and the spin medium is carried along a Carnot cycle, which is composed of two isothermal branches and two adiabatic branches, by changing the magnitude of the external driving field over time. Geva and Kosloff [11] compared the performance of endoreversible spin quantum Carnot heat engine and endoreversible harmonic quantum Carnot heat engine, and concluded that the optimal thermodynamic pass of these two kinds of quantum heat engines are not Carnot type by using optimal control theory. Besides quantum Carnot cycle, Feldmann et al. [12] established an

* Corresponding author. Naval University of Engineering, College of Power Engineering, Wuhan 430033, China. Tel.: þ86 27 836 15046; fax: þ86 27 836 38709. E-mail addresses: [email protected], [email protected] (L.G. Chen). http://dx.doi.org/10.1016/j.joei.2014.02.008 1743-9671/Ó 2014 Energy Institute. Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

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X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

List of symbols parameter of heat reservoir (s1) heat reservoir external magnetic field (T) dimensionless factor that describes the magnitude of the bypass heat leakage c parameter of heat reservoir (s1) E internal energy of the spin-1/2 systems (J) b H Hamiltonian Z reduced Planck’s constant (J s) kB Boltzmann constant (J K1) L1, L2 Lagrangian functions b M magnetic moment operator m intermediate variable nc population of the thermal phonons of the cold reservoir P power (W) Q amount of heat exchange (J) b þ operator in the Hilbert space of the system and b a, Q Q a Hermitian conjugate Q0 amount of heat exchange between heat reservoir and working medium (J) Q_ rate of heat flow (W) q parameter of heat reservoir S expectation value of spin operator b Sz b S  spin creation and annihilation operators Sþ, b b Sy ; b S z Þ spin operator Sðb Sx ; b Seq asymptotic value of S a B ! B Ce

T 0 T t W

absolute temperature (K) absolute temperature of the working medium (K) time (s) work (J)

Greek symbols intermediate variable “temperature” b ¼ 1/(kBT) (J1) 0 0 “temperature” of working medium b ¼ 1/(kBT ) (J1) phenomenological positive coefficients efficiency parameter of the heat reservoir Lagrangian multipliers friction coefficient Bohr magneton (J T1) interaction strength operator time (s)/cycle period (s) frequency of the thermal phonons (s1)

a b 0 b gþ , g h l l1, l2 m mB b G s uc

Subscripts B heat reservoir c cold side h hot side S working medium system SB interaction between heat reservoir and working medium system m ¼ 0, Ce ¼ 0 maximum point for endoreversible case 0 environment 1, 2, 3, 4 cycle states

endoreversible spin quantum Brayton heat engine cycle model and investigated its optimal performance, while Wu et al. [13] established endoreversible forward and reverse harmonic quantum Stirling cycle models and investigated the optimal performance of these quantum Stirling cycles. Wu et al. [14] investigated the optimal exergoeconomic performance of an endoreversible harmonic quantum Stirling engine. Lin and Chen [15] established an endoreversible harmonic quantum Brayton heat engine model and investigated the optimal performance of the quantum heat engine by using numerical solutions. Chen [16] investigated the optimal ecological performance of an endoreversible spin quantum heat engine, and the quantum heat engine cycle is composed of an adiabatic process, an isomagnetic field process and two isothermal processes. Similar to analysis and optimization of heat engine with classical working medium, various sources of irreversibility, such as the heat resistance, bypass heat leakage, dissipation processes inside the working medium, etc., were considered in the analysis and optimization of quantum heat engines. Jin et al. [17] introduced a bypass heat leakage in the investigation of the optimal exergoeconomic performance of an irreversible harmonic quantum Carnot engine. The bypass heat leakage arises from the thermal coupling action between the hot reservoir and cold reservoir. Feldmann and Kosloff [18] introduced an internal friction into the investigation of the optimal performance of an irreversible spin quantum Brayton heat engine and heat pump. The internal friction describes the effects of non-adiabatic phenomenon, which arises from the rapid change of the external magnetic field, in the adiabatic process. Since then, origin of quantum friction and effects of it on the performance of irreversible quantum thermodynamic cycles have attracted a lot of attentions [19–26]. Lin and Chen [27] investigated the optimal performance of an irreversible harmonic quantum Stirling heat engine by taking into account irreversibilities of heat resistance and inherent regeneration. Wu et al. [28,29] established generalized irreversible harmonic [28] and spin [29] quantum Brayton heat engine models with heat resistance, internal irreversibility and bypass heat leakage, and investigated the optimal performance of the two quantum heat engines. Different from the internal friction in Ref. [18], internal irreversible factors f were used to describe the irreversibility inside the irreversible adiabatic processes in the two quantum heat engine cycles. Wu et al. [30] established a generalized irreversible spin quantum Carnot heat engine model with heat resistance, internal irreversibility (described by an internal irreversible factors f) and bypass heat leakage. Different from the works mentioned above, a new function, the Helmholtz free energy of a two-level system, was used to calculate the heat exchanges between the working medium and heat reservoir in Ref. [30]. Liu et al. [31] established a generalized irreversible harmonic quantum Carnot heat engine model with heat resistance, internal friction and bypass heat leakage, and investigated the optimal ecological performance of the quantum heat engine. The irreversibility of non-adiabatic phenomenon was described by internal friction coefficient and that was different from the internal irreversible factor used in Refs. [28–30]. In the performance analysis and optimization of quantum heat engines which use harmonic or spin working medium, the thermal coupling between the working medium and the heat reservoir is described by the Hamiltonian or Liouvilles’s operator, and the motion equation of arbitrary operator the working medium system is given by the quantum master equation and semi-group approach. At weak coupling limit and high temperature limit, the thermodynamic quantities and the performance parameters of the engine cycle are obtained Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

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by solving the quantum master equation, and the optimal relationships among the performance parameters is obtained by using optimal control theory. In this paper, a model of an irreversible quantum Carnot heat engine working with working medium consisting of many non-interacting spin-1/2 systems will be proposed. The optimal performance of the model cycle with irreversibilities of heat resistance, internal friction and bypass heat leakage will be investigated. The quantum Carnot heat engine cycle is composed of two isothermal processes and two irreversible adiabatic processes. The expressions of cycle period, power and efficiency of the quantum heat engine will be derived by using quantum master equation, semi-group approach and FTT. Detailed numerical examples will be provided. At high temperature limit, the optimal performance of the quantum heat engine is deduced and analyzed by detailed numerical examples. The effects of internal friction and bypass heat leakage on the optimal performance of the quantum heat engine will be discussed. Some special cases, such as endoreversible case, frictionless case and the case without bypass heat leakage will also be shown. The results obtained are general and can enrich the FTT theory for quantum thermodynamic cycles. 2. Quantum dynamics of a spin-1/2 system ! When a single spin-1/2 particle is placed in a time-dependent magnetic field B which is along the positive z axis, the Hamiltonian of the ! b is HðtÞ b b ! interaction between the magnetic field ð B Þ and the magnetic moment ð MÞ ¼  M$ B . For a single spin-1/2 system, the magnetic b b b b moment M is proportional to the spin angular momentum S, and the directions of S and M are opposite. The Hamiltonian of the single spin1/2 system is given by [32]

! b ! b ¼  M$ B ¼ 2mB b S$ B =Z ¼ 2mB b S z Bz =Z H S

(1)

where mB is the Bohr magnetron and Z is the reduced Planck’s constant. Similar to Ref. 10, one can define u(t) ¼ 2mBB(t)z and refer it rather than Bz as “the magnetic field”. Therefore, the Hamiltonian becomes

b ðtÞ ¼ uðtÞb S z =Z H S

(2)

Based on the statistical mechanics, the expectation value of b S z is given by

S ¼

D

b Sz

E

  bu Z ¼  tanh 2 2

(3)

where Z=2 < S < 0 and b ¼ 1/(kBT). Here kB is the Boltzmann constant and T is the absolute temperature. For simplicity, one can refer b rather than T as the temperature. The internal energy of the spin-1/2 system is given by

ES ¼

D

b H S

E

D E ¼ u b S z =Z ¼ uS=Z

(4)

When the spin-1/2 system couples thermally to a heat reservoir (bath), it becomes an open system. The system-bath Hamiltonian is given by

b þH bB b ¼ H b þH H S SB

(5)

b ,H b and H b B stand for Hamiltonians of the spin-1/2 system, system-bath and bath, respectively. For the system operators, effects of where H S SB b and H b B are included in the Heisenberg equation as additional relaxation-type terms. In the Heisenberg picture, one can obtain the H SB motion of an operator of the spin-1/2 system by using the quantum master equation:

  b b dX i h b b i vX b ¼ þ LD X H S; X þ dt Z vt

(6)

b Þ is a dissipation term (the relaxation-type term). It originates from a thermal system-bath coupling. The system-bath coupling is where LD ð X further assumed as

b ¼ H SB

X a

b aB ba Ga Q

(7)

b a, B b a and Ga are operators of the spin-1/2 system, the bath and the interaction strength, respectively. Using semi-group approach, where Q one obtain the dissipation term [33,34]

  i  þ   X  þh b ;X b X b b þ Q b Q b ¼ b; Q ga Q LD X a a a a

(8)

a

þ

b a and Q b are operators of the system in the Hilbert space and are Hermitian conjugates; ga is the phenomenological positive where Q a coefficient. b ¼ H b ¼ ub Substituting X S=Z into Equation (6) yields S

dES d Db E ¼ HS ¼ dt dt

* + D  E b vH du dS S b ¼ S =Z þ u =Z þ LD H S dt vt dt

(9)

Comparing Equation (9) with the differential form of the first law of thermodynamics Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

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X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

dES dW dQ þ ¼ dt dt dt

(10)

one can find that the instantaneous power and the instantaneous heat flow are

P ¼

D

Q_ ¼

b =vt vH S

E

¼ u_ S=Z ¼ dW=dt

D  E _ b ¼ uS=Z ¼ dQ =dt LD H S

(11) (12)

respectively. Hence, the work inexact differential and the heat inexact differential are

dW ¼ Sdu=Z

(13)

dQ ¼ udS=Z

(14)

respectively. For a spin-1/2 system, Equation (9) gives the time derivative of the first law of thermodynamics. b þ and Q b a may be chosen to be the spin creation and annihilation operators: b In Equation (8), Q Sþ ¼ b S x þ ib S y and b S ¼ b S x  ib Sy. a 2 2 2 Substituting b S þ and b S  into Equation (6) and using ½b Sx; b S y  ¼ iZb S z , ½b Sy; b S z  ¼ iZb S x , ½b Sz; b S x  ¼ iZb S y and b Sx ¼ b Sy ¼ b S z ¼ Z2 =4 yields

S_ ¼ 2Z2 ðgþ þ g ÞS  Z3 ðg  gþ Þ

(15)

The coefficients gþ and g are constants if u is fixed. The solution of Equation (15) is

 SðtÞ ¼ Seq þ Sð0Þ  Seq e2ðgþ þg Þt

(16)

where S(0) and Seq ¼ Zðg  gþ Þ=½2ðg þ gþ Þ are the initial and asymptotic value of S, respectively. This asymptotic spin angular momentum must correspond to the value at thermal equilibrium Seq ¼ ½Z tanhðbu=2Þ=2. Comparison of these two expressions for Seq yields g/gþ ¼ ebu. It is assumed that [10]

gþ ¼ aeqbu

(17)

g ¼ aeð1þqÞbu

(18)

where a and q are both constants. gþ, g > 0 requires a > 0. If bu / N, gþ / 0 and g / N hold, it requires that 0 > q > 1. Substituting Equations (17) and (18) into Equation (15) yields

h    i S_ ¼ aZ2 eqbu 2 1 þ ebu S þ Z ebu  1

(19)

3. An irreversible spin quantum Carnot heat engine The irreversible quantum heat engine model has the following constraints: (1) The working medium of the quantum heat engine consists of many non-interacting spin-1/2 systems. The spin-1/2 system in the cycle is not only coupled thermally to the heat reservoirs but also coupled mechanically to an external “magnetic field” and is a two-level system. The external magnetic field is along the positive z axis and is time dependent. The magnitude of the external magnetic is not allowed to be zero when the two energy levels are degenerate. (2) The engine operates between a hot reservoir B at constant temperature Th and a cold reservoir Bc at constant temperature Tc. The two reservoirs are thermal phonon systems. The two heat reservoirs are infinitely large and their internal relaxations are strong. Therefore, the two heat reservoirs are assumed to be in thermal equilibrium. (3) The quantum heat engine cycle is composed of two isothermal branches connected by two irreversible adiabatic branches. The S–u diagram of the cycle is shown in Fig. 1. In the two isothermal processes, the spin systems are coupled thermally to the heat reservoirs. The temperatures of the working medium 0 0 0 0 in processes 1 / 2 and 3 / 4 are b h and b c , respectively. For heat engine, the second law of thermodynamics requires bc > b c > b h > bh . The amounts of heat exchanges between the heat reservoir and the working medium can be calculated from Equation (14)

Qh0 ¼

1 Z

Qc0 ¼ 

Z

2 1

1 Z

Z

udS ¼ 4

3

0 

 1

 cosh bh u2 =2 1 1 u1 tanh b0h u1 =2  u2 tanh b0h u2 =2 þ 0 ln

0  2 2 bh cosh bh u1 =2

udS ¼



 1

 1 cosh b0c u4 =2 1 u4 tanh b0c u4 =2  u3 tanh b0c u3 =2  0 ln

0  2 2 bc cosh bc u3 =2

(20)

(21)

where Q 0 h and Q 0 c are amounts of heat exchanges in processes 1 / 2 and 3 / 4, respectively. The working medium releases heat in process Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

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Fig. 1. The S–u diagram of a spin quantum Carnot heat engine with multi-irreversibilities.

3 / 4 indicated by a minus sign before the integral in Equation (21). The work done by the system along these two processes can be calculated from Equation (13)

W12 ¼ 

W34

1 Z

1 ¼  Z

Z

2

1

Z

4

3

Sdu ¼

Sdu ¼

Z

1

b0h

0  cosh bh u2 =2

0  cosh bh u1 =2

(22)

0  cosh bc u4 =2 tanh xdx ¼ 0 ln

 bc cosh b0c u3 =2

(23)

2

tanh xdx ¼

1

Z

1

b0c

4 3

1

b0h

ln

1

There are no thermal coupling action between working medium and heat reservoir in adiabatic processes 2 / 3 and 4 / 1. According to quantum adiabatic theorem [35], rapid change in the external magnetic field causes quantum non-adiabatic phenomenon and that will cause heat generation in the adiabatic processes. The effect of the quantum non-adiabatic process on the performance of the quantum heat engine is similar to that of internally dissipative friction in the classical analysis. Therefore, one can use a friction coefficient m to describe the non-adiabatic process. It is assumed that the external magnetic field changes linearly with time

uðtÞ ¼ uð0Þ þ u_ t

(24)

The friction coefficient m satisfies [18]:

 m 2 S_ ¼ Z 0 t

(25)

where t0 is the time spent on the corresponding adiabatic process. The spin angular momentum is [18]

SðtÞ ¼ Sð0Þ þ Z

 m 2 t0

t

(26)

where 0  t  t0 . Substituting t ¼ sa and t ¼ sb into Equation (26) yields

S2 ¼ S3  Zm2 =sa

(27)

S4 ¼ S1  Zm2 =sb

(28) 0 ðZ=2Þtanhðbh u1 =2), S2

0 ðZ=2Þtanhðbh u2 =2),

where sa and sb are the required times of the processes 2 / 3 and 4 / 1, respectively. S1 ¼ ¼ 0 0 S3 ¼ ðZ=2Þtanhðbc u3 =2) and S4 ¼ ðZ=2Þtanhðbc u4 =2) are the spin angular momentums at states 1, 2, 3 and 4, respectively. Using Equations (27) and (28) yields

u2 ¼

u4 ¼

2

b0h 2

b0c

1

tanh

tanh

1

tanh

b0c u3

tanh

b0h u1

2

2

þ

þ

2m2

! (29)

sa 2m2

! (30)

sb

There is no heat exchange between the working medium and heat reservoir along the adiabatic processes. Therefore, the work done by the system along the adiabatic processes can be calculated from Equations (9), (24) and (26), respectively

W23 ¼ 

Z sa 0

dES ¼ 

1 Z

Z sa 0

Sdu 

1 Z

Z sa 0

udS ¼ ðu2  u3 Þ



m2 S3  Z 2sa

 

m2 ðu3 þ u2 Þ 2sa

(31)

Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

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X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

W41 ¼ 

Z sb 0

1 dES ¼  Z

Z sb 0

Sdu 

1 Z

Z sb 0



udS ¼ ðu4  u1 Þ

m2 S1  Z 2sb

 

m2 ðu4 þ u1 Þ 2sb

(32)

(4) . There exists bypass heat leakage between hot and cold reservoirs. It comes from the coupling action between the hot and cold reservoirs by the working medium of the heat engine. The calculation of bypass heat leakage is similar to that of the heat flow between the working medium and heat reservoirs. The hot and cold reservoirs are thermal phonon systems. The population of the thermal phonons of the cold reservoir is _ the time derivative of nc can be derived at the condition of small thermal disturbance nc ¼ 1=ðeZuc bc  1Þ. Similar to the derivation of S, [11,17]

n_ c ¼ 2celZbh uc

h  i eZbh uc  1 nc  1

(33)

where uc is the frequency of the thermal phonons, c and l are two constants. The rate of heat flow from the hot reservoir to the cold reservoir (i.e. rate of bypass heat leakage) is

h   i Q_ e ¼ Ce Zuc n_ c ¼ 2Ce cZuc elZbh uc 1  eZbh uc  1 nc

(34)

where Ce is a dimensionless factor introduced to describe the magnitude of the bypass heat leakage. The rate of bypass heat leakage is assumed to be a constant. Therefore, the bypass heat leakage quantity Qe per cycle is given by

h   i Qe ¼ Q_ e s ¼ 2Ce cZuc elZbh uc 1  eZbh uc  1 nc s

(35)

The model established in this paper is similar to the model of classical generalized irreversible Carnot heat engines by considering multiirreversibilities, such as heat resistance, bypass heat leakage and internal irreversibility [36–40].

4. Cycle period Substituting S_ ¼ u_ ðdS=duÞ into Equation (19) yields the time of isothermal process

s0 ¼

Z

Sf

Si

dS ¼ S_

Z Z uf dS=du 1 uf ðdS=duÞdu du ¼  

 

a ui Z2 eqbu 2 ebu þ 1 S þ Z ebu  1 S_ ui

(36)

0

Substituting S ¼ ðZ=2Þtanhðbh u=2Þ and b ¼ bh into Equation (36) yields the time of isothermal process 4 / 1

Z b0h u1 1 dmh

 2aZ2 b0h u4 eqah mh ðeah mh  emh Þ 1 þ emh

sh ¼

0

(37)

0

where mh ¼ bh u and ah ¼ bh =b h . 0 Substituting S ¼ ðZ=2Þtanhðb c u=2Þ and b ¼ bc into Equation (36) yields the time of isothermal process 2 / 3

sc ¼

1

Z b0c u3

dmc

 2aZ2 b0c u2 eqac mc ðeac mc  emc Þ 1 þ emc 0

(38)

0

where mc ¼ bc u and ac ¼ bc =bc . Consequently, the cycle period is:

s ¼ sh þ sc þ sa þ sb bZ0h u1

¼

1 2aZ2

b0h u4

eqah mh ðeah mh

dmh 1

þ  emh Þ 1 þ emh 2aZ2

bZ0c u3 b0c u2

dmc

 þ sa þ sb eqac mc ðeac mc  emc Þ 1 þ emc

(39)

5. Power and efficiency Using Equations (22), (23), (31) and (32) yields the total work per cycle of the quantum heat engine

W ¼ ¼

H

dW ¼ W12 þ W23 þ W34 þ W41

1 ln coshðbh u2 =2Þ 0 b0h coshðbh u1 =2Þ 0

  0 coshðb u4 =2Þ u u u1 u4 ÞS1 þ b10 ln cosh bc0 u =2 þ ð 2  3 ÞS3 ð  m2 usa2 þ usb4 Z ð c 3 Þ c

(40)

Combining Equation (39) with (40) yields the power of the heat engine Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

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0

0

Fig. 2. The dimensionless power P=Pmax;m¼0;Ce ¼0 versus “temperatures” bh and bc .

P ¼ W s1 "

0 

0   # cosh bh u2 =2 u4 1 1 cosh bc u4 =2 ðu2  u3 ÞS3  ðu1  u4 ÞS1 2 u2 s1



 þ 0 ln þ m ¼ ln þ sa sb Z b0h cosh b0h u1 =2 bc cosh b0c u3 =2

(41)

Using Equations (20) and (40) yields the efficiency of the heat engine

h ¼ W=Qh 1

¼

b0h

   m2 usa2 þ usb4 c 0  

1 ln coshðbh u2 =2Þ þ u2 S2 u1 S1  2C cZu elZbh uc eZbh uc  1 n  1 s 0 e c c Z b0h coshðbh u1 =2Þ 0

ln

coshðbh u2 =2Þ 0 coshðbh u1 =2Þ

þ b10 ln

0

coshðbc u4 =2Þ 0 coshðbc u3 =2Þ

þð

u2 u3 ÞS3 ðu1 u4 ÞS1 Z

(42)

where Qh ¼ Qh0 þ Qe is the total heat released by the hot reservoir. 0 0 From Equations (41) and (42), one can see clearly that the power and efficiency are dependent on bh and bc for given bh, bc, b0, q, a, c, l, u1, u3, uh, m and Ce. For the general case, the integral in the denominator of the expression of cycle period in Equation (39) is unable to evaluate in close form. Therefore, it is unable to obtain the analytical fundamental optimal relation between the power and efficiency. Using 0 0 0 0 Equations (41) and (42), one can plot three-dimensional diagrams of the power (P=Pmax;m¼0;Ce ¼0 , bh , bc ) and efficiency (h, bh , bc ) as shown in Figs. 2 and 3, where Pmax;m¼0;Ce ¼0 is the maximum power for endoreversible case. According to Refs. [10,18], Z ¼ 1 and kB ¼ 1 are set in the numerical calculations for simplicity, and the other used parameters are a ¼ c ¼ 2, q ¼ l ¼ 0.5, bh ¼ 0.2, bc ¼ 1, b0 ¼ 1.25, sa ¼ sb ¼ 0.01, u1 ¼ 5, u3 ¼ 1, uc ¼ 0.05, m ¼ 0.01 and Ce ¼ 0.05. Fig. 2 shows that there exist two optimal “temperatures” b0h and b0c for given heat reservoir temperatures and other parameters, which correspond to the maximum power of the irreversible spin quantum Carnot heat engine. As a result of the internal friction and bypass heat leakage, the maximum dimensionless power ðP=Pmax;m¼0;Ce ¼0 Þmax < 1. Fig. 3 shows that there 0 0 also exist two optimal “temperatures” bh and bc for given heat reservoir temperatures and other parameters, which correspond to the 0 0 maximum efficiency when there exits a bypass heat leakage. The optimal “temperature” bh (or bc ) is close to the heat reservoir “temperature” bh(or bc). 6. Optimal performance at high temperature limit The obtained results above can be simplified when the temperatures of the heat reservoir and working medium are high enough, i.e.

bu  1. At the first order approximation, Equations (29), (30), (37) and (38) can be, respectively, simplified to

0 0 Fig. 3. The efficiency h versus “temperatures” bh and bc .

Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

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X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

u2 ¼

b0c u3 sa þ 4m2 b0h sa

(43)

u4 ¼

b0h u1 sb þ 4m2 b0c sb

(44)

sh ¼ sc ¼

1 4aZ2 ðah  1Þ 1 4aZ ðac  1Þ 2

u2 u1

(45)

u4 u3

(46)

ln

ln

Using Equations (43)–(46), Equations (34), (39), (41) and (42) can be, respectively, simplified to

Q_ e zCe ½2cZuc ð1 þ lZbh uc Þ=bc ðbc  bh Þ ¼ Ce aðbc  bh Þ

s¼



 

b0h bc  b0c ln b0c u3 sa þ 4m2



b0h u1 sa



  0 



 0

0   0 bc u3 sb þ 4aZ2 bh  b0h bc  b0c ðsa þ sb Þ þ bc bh  bh ln bh u1 sb þ 4m2



 0 0 4aZ2 bh  bh bc  bc

(47)

(48)

h

0 

2

0 2 i 0 

0 0 0 0 0 0 0 aZ2 bh  bh bc  bc bh bc s2a s2b bh u21 þ bc u23  bc s2b u3 bc sa þ 4m2  bh s2a bh sb u1 þ 4m2 P¼  0

 0 

 







 0 0 0   0 bh u1 sa þ b0c bh  b0h ln b0h u1 sb þ 4m2 b0c u3 sb þ 4aZ2 bh  b0h bc  b0c ðsa þ sb Þ 2bh bc s2a s2b bh bc  bc ln bc u3 sa þ 4m2 (49)

h¼





b0h b0c s2a s2b b0h u21 þ b0c u23  b0c s2b u3 b0c sa þ 4m2

0 2 2 2 0 2 0 2 b02 h bc sa sb u1  bc sb bc sa u3 þ 4m

2

2

0 2 0  bh s2a bh sb u1 þ 4m2

0

0

þ 8bh bc s2a s2b Ce aðbc  bh Þs

(50)

where a ¼ 2cZuc ð1 þ lZbh uc Þ=bc . 0 0 0 0 Using Equations (49) and (50), one can plot three-dimensional diagrams of the power (P=Pmax;m¼0;Ce ¼0 , bh , bc ) and efficiency (h, bh , bc ), as shown in Figs. 4 and 5, where Pmax;m¼0;Ce ¼0 is the maximum power for endoreversible case at high at high temperature limit. The parameters used the numerical calculations are a ¼ c ¼ 2, l ¼ 0.5, bh ¼ 0.001, bc ¼ 1/320, b0 ¼ 1/300, sa ¼ sb ¼ 0.01, u1 ¼ 10, u3 ¼ 2, uc ¼ 6, m ¼ 0.001 and 0 0 Ce ¼ 0.0001. Fig .4 shows that the relationship among dimensionless power P=Pmax;m¼0;Ce ¼0, bh , and bc at high temperature limit is similar to that in general case, and there also exist two optimal “temperatures” which correspond to a maximum dimensionless power and the 0 0 maximum dimensionless power ðP=Pmax;m¼0;Ce ¼0 Þmax < 1. Fig. 5 shows that the relationship among efficiency h, bh , and bc at high temperature limit is also similar to that in general case, and there also exist two optimal “temperatures” which correspond to a maximum efficiency. To determine the maximum power of the quantum Carnot heat engine for a fixed efficiency or the maximum efficiency for a fixed power, one can introduce Lagrangian functions L1 ¼ P þ l1h or L2 ¼ h þ l2P, where l1 and l2 are two Lagrangian multipliers. Theoretically, combining Equations (49) and (50) the extremal conditions 0

0

0

0

vL1 =vbh ¼ 0; vL1 =vbc ¼ 0

(51)

or

vL2 =vbh ¼ 0; vL2 =vbc ¼ 0

(52)

0 0 Fig. 4. The dimensionless power P=Pmax;m¼0;Ce ¼0 versus “temperatures” bh and bc at high temperature limit.

Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

9

0 0 Fig. 5. The efficiency h versus “temperatures” bh and bc at high temperature limit.

0

0

yields the optimal relation between bh and bc . However, it is hard to solve these equations analytically as the result of strong complexity and nonlinearity of these equations. Therefore, it is also hard to obtain the fundamental optimal relations between power and efficiency analytically at high temperature limit. Solving Equation (51) or (52) numerically, one can plot characteristic curves of the dimensionless power P=Pmax;m¼0;Ce ¼0 versus efficiency h, as shown in Figs. 6 and 7. Except m and Ce, the parameter values used in the numerical calculations are the same as those used in the calculations for Fig. 4. Figs. 6 and 7 show that the P=Pmax;m¼0;Ce ¼0  h curves are parabolic-like ones when there is no bypass heat leakage (Qe ¼ 0) and the dimensionless power have a maximum. The P=Pmax;m¼0;Ce ¼0  h curves are loop-shaped ones when there exists bypass heat leakage (Qe s 0), and both the dimensionless power and efficiency have maxima. For a fixed bypass heat leakage Qe, both the available maximum dimensionless power and available maximum efficiency decrease with the increase in internal friction m. For a fixed internal friction m, the available maximum efficiency decreases with the increase in the bypass heat leakage Qe. The effects of bypass heat leakage Qe on dimensionless power is small due to the fact that the work of the cycle is independent of bypass heat leakage Qe. For a given dimensionless power, there are two different efficiencies. Obviously, the heat engine should work at the point where the efficiency is higher.

7. Three special cases Case 1 Endoreversible case (i.e. m ¼ 0 and Ce ¼ 0). There exits only irreversibility of heat resistance in the cycle. Compared to the time spent on the two isothermal processes, the time spent on the two adiabatic processes is negligible (i.e. sa ¼ sb ¼ 0). The cycle period, power and efficiency of the quantum heat engine can be, respectively, expressed simply as

2 1 6 6 s ¼ 2aZ2 4

Z b0 u2 h 0 h

b u1

Z b0 u4 h

eqah mh ðeah mh

3

7 dmh dmc



þ 7 5 a a m m q m m m m 0 c c c c c c h h e Þ 1þe ðe e Þ 1þe bh u3 e

(53)

Fig. 6. Effects of m and Ce on dimensionless power P=Pmax;m¼0;Ce ¼0 versus efficiency h

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10

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

Fig. 7. Effects of m and Ce on dimensionless power P=Pmax;m¼0;Ce ¼0 versus efficiency h.

"

0 #

0 cosh bc u3 =2 S1 u1 1 0  S3 u3 P ¼ bh  bc  þ ln

 s1 0 0 b0c b0h cosh b0h u1 =2 Zbh Zbc

h ¼ 1

b0h b0c

(54)

(55)

At high temperature limit, the cycle period and power can be, respectively, simplified to



s¼



 

b0h bc  b0c bh ln b0c u3 = b0h u1

0 

0 4aZ2 bh  bh bc  bc



h i

 0 

0 0 0 0 0 03 03 aZ2 bh  bh bc  bc bh bc bh u21 þ bc u23  u23 bc  bh u21 P ¼

0    0 0 0 0 0 2bh bc bh bc  bc bh ln bc u3 = bh u1

(56)

(57)

From Equations (55) and (57), one can obtain the analytical fundamental optimal relation of the endoreversible quantum Carnot heat engine

P ¼

h i aZ2 ð1  hÞ2 hu21 þ ð1  2hÞu23 ½bh  ð1  hÞbc  8ð1  hÞ2 ln½u3 =ðu1  hu1 Þ

(58)

One can derive the maximum power and corresponding efficiency in the endoreversible case, and these are the results obtained in Ref. 10. Case 2 Frictionless case (i.e. m ¼ 0 and Ce s 0). There exist irreversibilities of heat resistance and bypass heat leakage in the cycle. Similar to the endoreversible case, the time spent on the adiabatic processes is negligible (i.e. sa ¼ sb ¼ 0). The efficiency can be expressed simply as

"

h¼

1

b0h

ln

#1

0 

0 

0   #" cosh bh u2 =2 cosh bh u2 =2 u2 u4 u2 S2  u1 S1 1 cosh bc u4 =2 ðu  u3 ÞS3  ðu1  u4 ÞS1 1

0

0

 m2  þ 0 ln þ 2  þ þ ln sa sb Z Z bc cosh bc u3 =2 b0h cosh b0h u1 =2 cosh bh u1 =2 (59)

The cycle period and power are independent of bypass heat leakage and thus the expressions of cycle period and power are still Equations (53) and (54), respectively. At high temperature limit, Equation (59) can be simplified to



h ¼ 1

b0h s2a b0h sb u1 þ 4m2 0 2 2 2 b02 h bc sa sb u1



2

0 02  bh bc s2a s2b u23 2 0 2 0 bc sb bc sa u3 þ 4m2

(60)

From Equations (56), (57) and (60), one can derive the maximum power and corresponding efficiency of the irreversible quantum Carnot heat engine in the frictionless case analytically for given S1 and S3. Case 3 The case without bypass heat leakage (i.e. m s 0 and Ce ¼ 0). There exist irreversibilities of heat resistance and internal friction in the cycle. In this case, the efficiency can be expressed simply as Please cite this article in press as: X.W. Liu, et al., Optimal performance of a spin quantum Carnot heat engine with multi-irreversibilities, Journal of the Energy Institute (2014), http://dx.doi.org/10.1016/j.joei.2014.02.008

X.W. Liu et al. / Journal of the Energy Institute xxx (2014) 1–12

  0  Su coshðbc u3 =2Þ S1 u 1 1 3 3 0  0 þ 0 0 ln 0 bc bh Zbh Zbc coshðbh u1 =2Þ  h¼  0  

b u cosh =2 ð Þ 0 S3 u3 3 bh Zb0  SZ1bu0 1 þ b01b0 ln coshðb0c u =2Þ þ 2Ce cZuc elZbh uc 1  eZbh uc  1 nc s h c c h h 1

11



b0h  b0c

(61)

The cycle period and power are independent of bypass heat leakage and thus the expressions of cycle period and power are still Equations (39) and (41), respectively. At high temperature limit, Equation (61) can be simplified to



h¼





b0h b0c s2a s2b b0h u21 þ b0c u23  b0c s2b u3 b0c sa þ 4m2

2

0 2 2 2 0 0 2 b02 h bc sa sb u1  bc sb bc sa u3 þ 4m

2

2

0 2 0  bh s2a bh sb u1 þ 4m2

0

0

þ 8bh bc s2a s2b Ce aðbc  bh Þs

(62)

Based on Equations (49) and (62), it is unable to obtain the fundamental optimal relation between the power and efficiency analytically. Fig. 6 (lines 1, 2 and 3) and Fig. 7 (lines 1 and 2) give the characteristic curves of dimensionless power P=Pmax;m¼0;Ce ¼0 versus efficiency h of the quantum heat engine without bypass heat leakage. The P=Pmax;m¼0;Ce ¼0  h curves are parabolic-like and the dimensionless power has a maximum. 8. Conclusions The optimal performance of an irreversible spin quantum Carnot heat engine with irreversibilities of heat resistance, internal friction and bypass heat leakage is investigated in this paper. The irreversible quantum Carnot heat engine uses many non-interacting spin-1/2 systems as working medium and the cycle is composed of two isothermal processes and two irreversible adiabatic processes. By using quantum master equation, semi-group approach and FTT, the expressions of cycle period, power and efficiency of the quantum heat engine are derived, and detailed numerical examples are provided. The numerical examples show that both the power and efficiency have maxima in general case. At high temperature limit, the optimal performance of the quantum heat engine is deduced and analyzed by using detailed numerical examples. The effects of internal friction and bypass heat leakage on the optimal performance of the quantum heat engine are discussed. Three special cases (endoreversible case, frictionless case and the case without bypass heat leakage) are also briefly discussed. Both the power and efficiency have maxima at high temperature limit. The internal friction does decrease the power and efficiency but does not change the shape of power versus efficiency characteristic curves. The bypass heat leakage changes the power versus efficiency characteristic curves from parabolic-like ones to loop-shaped ones and decreases the efficiency, but has no effects on the power. The obtained results are general, which include the fundamental optimal power and efficiency characteristics of endoreversible (the sole irreversibility of heat resistance) and various irreversible (the heat resistance and bypass heat leakage; the heat resistance and internal friction; the heat resistance, internal friction and bypass heat leakage) conditions at general and high temperature limit. They can enrich the FTT theory for quantum thermodynamic cycles. Acknowledgments This paper is supported by the National Natural Science Foundation of P. R. China (Project No. 10905093), the Natural Science Fund of China (Project No. 50846040) and the Innovation Research Foundation for Ph. D Candidates of Naval University of Engineering (HGBSJJ2012002). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

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