Performance optimization of quantum Brayton refrigeration cycle working with spin systems

APPLIED ENERGY

Applied Energy 84 (2007) 176–186

www.elsevier.com/locate/apenergy

Performance optimization of quantum Brayton refrigeration cycle working with spin systems Jizhou He *, Yong Xin, Xian He Department of Physics, Nanchang University, Nanchang 330047, People’s Republic of China Received 6 February 2006; received in revised form 6 May 2006; accepted 28 May 2006 Available online 23 August 2006

Abstract The new model of a quantum refrigeration cycle composed of two adiabatic and two isomagnetic field processes is established. The working substance in the cycle consists of many non-interacting spin-1/2 systems. The performance of the cycle is investigated, based on the quantum master equation and semi-group approach. The general expressions of several important performance parameters, such as the coefficient of performance, cooling rate and power input, are given. It is found that the coefficient of performance of this cycle is a close analogue of the classical Carnot-cycle. Some performance characteristic curves relating the cooling rate, the coefficient of performance and power input are plotted. Further, for high temperatures, the optimal relations between the cooling rate and the coefficient of performance are analyzed in detail. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Spin systems; Quantum refrigeration cycle; Performance parameters; Optimization analysis

1. Introduction In recent years, the optimal analysis of the performance characteristics of thermodynamic cycles has been extended to the regime of quantum cycles. The performances of quantum Carnot, Ericsson and Stirling cycles have been intensively studied [1–14]. Many novel conclusions have been obtained. Besides, the investigations have also dealt with the performance of quantum Brayton refrigeration cycle [7,15], such as the behaviour of the *

Corresponding author. Tel./fax: +86 791 8305626. E-mail addresses: [email protected], [email protected] (J. He).

0306-2619/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2006.05.002

J. He et al. / Applied Energy 84 (2007) 176–186

177

Nomenclature B E ^ H h M P Q Qc Qh R S S1 S2 T Tc Th t tc th W b bc bh e ec lB Cc x x1 x2 xc xh Ch

magnetic field (T) internal energy (J) Hamiltonian (J) Planck’s constant (J s) magnetic moment (J T1) power input (J s1) amount of heat (J) amount of heat absorbed by the working substance from the hot reservoir (J) amount of heat released to the hot reservoir from the working substance (J) cooling rate (J s1) spin angular momentum mean value of the spin angular momentum in one adiabatic process mean value of the spin angular momentum in another adiabatic process absolute temperature (K) temperature of cold reservoir (K) temperature of hot reservoir (K) cycle period (s) time of isomagnetic field process (s) time of isomagnetic field process (s) work per cycle (J) inverse of temperature (T1) inverse of temperature of cold reservoir (T1) inverse of temperature of hot reservoir (T1) coefficient of performance coefficient of performance of the Carnot refrigeration cycle Bohr magneton (J T1) thermal conductivity of cold reservoir material (s1) magnetic-field (J) high magnetic-field (J) low magnetic-field (J) the upper bound of low magnetic-field (x2) (J) the low bound of high magnetic-field (x1) (J) thermal conductivity of hot reservoir (s1)

molecular refrigerators toward attaining ultra-low temperatures and three-level laser refrigeration [11]. In fact, the Brayton cycle is an important cycle in engineering thermodynamics. The investigation relative to Brayton cycles has continuously attracted a good deal of attention [16–18]. It has some distinctive merits, which are noteworthy in theory and practice. In classical thermodynamic cycles, there are the Stirling cycle, Ericsson cycle, Brayton cycle, etc., besides the Carnot cycle. The performance of the Carnot cycle is independent of the property of the working substance, while the performances of other cycles are, in general, dependent on the property of the working substance [19,20]. Quantum-mechanical

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cycle differs from the classical cycle in two respects. On the one respect, the working substances may be the spin systems, harmonic oscillator systems, ideal quantum gases, and micro-particle systems confined to a potential well [21,22]. On the other respect, the relaxation dynamics is modelled by the semi-group approach. The feature of quantum cycles is that the use of phenomenological heat-transfer laws can be avoided. In the present paper, a new model of the quantum Brayton refrigeration cycle using the spin-1/2 systems as the working substance is established, it is composed of two adiabatic and two isomagnetic field processes, which is a microscopic analog of the magnetic Brayton refrigeration cycle [16]. First, the thermodynamic property of a spin-1/2 system is given, based on the quantum master equation and semi-group approach. The performance characteristics of the quantum Brayton refrigeration cycle are analyzed. Secondly, the important performance parameters, such as the coefficient of performance, cooling rate, and power input are optimized. Especially, at high temperatures, the optimal relations between the cooling rate and the maximum cooling rate are derived in detail. Finally, the results obtained here may be generalized for the spin-J systems. The optimum performance of the quantum Ericsson or Carnot refrigeration cycles may be derived similarly. 2. Quantum refrigeration cycle Let us consider a quantum spin-1/2 system with a magnetic moment M placed in a magnetic field B. The magnitude of the magnetic field can change over time, but is not allowed to reach zero. The Hamiltonian of the interaction between the magnetic moment M in the quantum system and the magnetic field B is given by [23–25] ^  B ¼ 2l Bz ðtÞS^z ; ^  B ¼ 2lB S ^ ðtÞ ¼ M H B

ð1Þ

where lB is the Bohr magneton, S is a spin angular momentum, h = h/(2p), and h is the Planck’s constant. Throughout this paper we adopt h = 1 and define x(t) = 2lBBz(t) for simplicity: x is positive since the spin angular-momentum and magnetic moment are in opposite directions. One can refer to x rather than Bz as ‘‘the field’’. Thus, the Hamiltonian of an isolated single spin-1/2 system in the presence of the field x(t) may be expressed as ^ ðtÞ ¼ xðtÞS^z : H

ð2Þ

The internal energy of the spin-1/2 system is the expectation value of the Hamiltonian, i.e., ^z i ¼ xS: ^ i ¼ xðtÞhS E ¼ hH

ð3Þ

Based on statistical mechanics, the expectation value of the spin angular momentum Sz is expressed by the following relation [26–28] 1 ð4Þ S ¼ hS^z i ¼  tanhðbx=2Þ; 2 where 1/2 < S < 0. In quantum refrigeration cycles, the spin-1/2 system is not only coupled mechanically to the given ‘‘magnetic field’’ x(t), but also coupled thermally to a heat reservoir at temperature T. Based on the semi-group formalism [24], the equation of motion of an operator in the Heisenberg picture is given by the quantum master equation, i.e., ^ ^ dX ^ þ oX þ LD ðXÞ; ^ ^ ; X ¼ i½H dt ot

ð5Þ

J. He et al. / Applied Energy 84 (2007) 176–186

where ^ ¼ LD ðXÞ

X

179

^ þ ½X; ^ V ^ a  þ ½V ^ þ ; X ^V ^ aÞ ca ð V a a

ð6Þ

a

is a dissipation term and originates from a thermal coupling of the spin to a heat reservoir: ^ a and V ^ þ are operators in the Hilbert space of the system and are Hermitian conjugates, V a ^ in Eq. (5) by H ^ and using and ca are phenomenological positive coefficients. Substituting X Eq. (3), one can obtain the rate of change of the internal energy as   ^ dE d ^ oH ^ ¼ dx S þ x dS : ¼ hH i ¼ ð7Þ þ hLD ðHÞi dt dt ot dt dt Comparing Eq. (7) with the time derivative of the first law of thermodynamics dE dW dQ ¼ þ ; dt dt dt

ð8Þ

one can easily find that the instantaneous power is   ^ dW oH dx P¼ ¼ ¼ S dt ot dt

ð9Þ

and the instantaneous heat-flow is dQ ^ Þi ¼ x dS : ¼ hLD ðH dt dt

ð10Þ

It is thus clear that, for a spin-1/2 system, Eq. (7) gives the time derivative of the first law of thermodynamics. Fig. 1 shows a schematic diagram of a quantum refrigeration cycle, which is composed of two adiabatic and two isomagnetic field processes. This cycle is a microscopic analogue of the magnetic Brayton refrigeration cycle [29], where the working substance consists of magnetic salts. For convenience of writing, ‘‘temperature’’ will refer to b rather than T, where b = 1/T and T is the absolute temperature. In adiabatic process 1 ! 2, no heat exchange is involved. Increasing x corresponds to the performance of work by the work0.0

S2

eq

S2

1

2 Qc

S S1 S1

Qh

4

3

eq

h

c

-0.5 2

c

h

1

Fig. 1. The spin angular momentum (S)-magnetic field versus (x) diagram of a spin-1/2 Brayton refrigeration cycle, where the unit of x is the Joule.

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J. He et al. / Applied Energy 84 (2007) 176–186

ing substance on the surroundings. The reverse process 3 ! 4 of adiabatically decreasing x corresponds to the performance of work by the surroundings on the working substance. In the isomagnetic field process 2 ! 3, the working substance is coupled to the hot reservoir at constant ‘‘temperature’’ bh. The amount of heat Qh is released to the hot reservoir from the working substance. In the isomagnetic field process 4 ! 1, the working substance is coupled to the cold reservoir at constant ‘‘temperature’’ bc. The amount of heat Qc is absorbed by the working substance from the cold reservoir. The ‘‘temperatures’’ of the working substance are different from those of the heat reservoirs. They are, respectively, given by b1, b2, b3, and b4, and there is a relation, b4 > b1 P bc > bh P b3 > b2; x1 and x2 represent the high and low ‘‘magnetic fields’’, respectively. 3. Performance characteristics Using Eqs. (4) and (10), we can calculate the amounts of heat exchange in two isomagnetic field processes as   Z S1 1 1 Qh ¼ x dS ¼ x1  tanh ðb3 x1 =2Þ þ tanh ðb2 x1 =2Þ ð11Þ 2 2 S2 and Qc ¼

Z

S2

S1

  1 1 x dS ¼ x2  tanh ðb1 x2 =2Þ þ tanh ðb4 x2 =2Þ ; 2 2

ð12Þ

where S1 and S2 are the mean values of the spin angular momentum in two adiabatic processes, respectively, and S1 < S2. During the adiabatic process, S remains constant. Hence 1 1 1 S 1 ¼  tanhðb3 x1 =2Þ ¼  tanhðb4 x2 =2Þ ¼  tanhðbh xh =2Þ 2 2 2

ð13Þ

1 1 1 S 2 ¼  tanhðb1 x2 =2Þ ¼  tanhðb2 x1 =2Þ ¼  tanhðbc xc =2Þ; 2 2 2

ð14Þ

and

where xh is the low bound of x1 in the high-isomagnetic field process and xc is the upper bound of x2 in the low-isomagnetic field process. Using Eqs. (11) and (12), we obtain the work input per cycle and the coefficient of performance as W ¼ jQh þ Qc j   1 1 ¼ x1 tanhðb3 x1 =2Þ  tanhðb2 x1 =2Þ 2 2   1 1 þ x2 tanhðb1 x2 =2Þ  tanhðb4 x2 =2Þ 2 2

ð15Þ

and e¼

Qc x2 xc ¼ < ¼ emax ; W x1  x2 xh  xc

ð16Þ

where emax is the maximum value of the coefficient of performance. It is found that the coefficient of performance only depends on the high and low ‘‘magnetic fields’’. It is well

J. He et al. / Applied Energy 84 (2007) 176–186

181

known that the coefficient of performance of the Carnot refrigeration cycle is ec = T2/(T1  T2). Comparing these two results, we find the analogue of the classical thermodynamic result of Carnot cycle as long as the temperature T replaced by ‘‘magnetic field’’ x. 4. Time evolution of the spin angular-momentum and cycle period In order to calculate the time of the heat exchange processes, one must solve the equation of motion that determines the time evolution of the spin angular-momentum. For a ^ a are chosen to be the spin creation and annihilation operators: spin system, V ^ ¼ xS^z . Substituting S^þ , S^ , H ^ , and X^ ¼ S^z into S^þ ¼ S^x þ iS^y and S^ ¼ S^x  iS^y , and H Eq. (5), one can show that [23] dS ¼ 2ðcþ þ c ÞS  ðc  cþ Þ: dt

ð17Þ

If x is constant, c+ and c are also constants, and the solution of Eq. (17) is given by SðtÞ ¼ S eq þ ½Sð0Þ  S eq  exp ½Ct;

ð18Þ

where Seq = (c  c+)/2(c + c+) is the asymptotic value of S and C = 2(c + c+) is the thermal conductivity. This asymptotic spin angular momentum must correspond to the value at thermal equilibrium, S eq ¼  12 tanhðbx=2Þ. Eq. (18) is a general expression of time evolution for a spin-1/2 system coupling with the heat reservoir and the external magnetic field. In the isomagnetic field process x1, the ‘‘temperature’’ of the working substance changes from b2 to b3. Substituting SðtÞ ¼ S 1 ¼  12 tanhðb3 x1 =2Þ, Sð0Þ ¼ S 2 ¼  12 tanhðb2 x1 =2Þ, 1 and S eq ¼ S eq 1 ¼  2 tanhðbh x1 =2Þ into Eq. (18), one can obtain the time spent on the isomagnetic field process x1 t1 ¼

1 tanhðbh x1 =2Þ  tanhðb2 x1 =2Þ ; ln Ch tanhðbh x1 =2Þ  tanhðb3 x1 =2Þ

ð19Þ

where Ch is thermal conductivity of the hot reservoir. Similarly, substituting S ðtÞ ¼ S 2 ¼ 1  12 tanh ðb1 x2 =2Þ, S ð0Þ ¼ S 1 ¼  12 tanh ðb4 x2 =2Þ, and S eq ¼ S eq 2 ¼  2 tanh ðbc x2 =2Þ into Eq. (18), one can obtain the time spent on the isomagnetic field process x2, t2 ¼

1 tanhðbc x2 =2Þ  tanhðb4 x2 =2Þ ; ln Cc tanhðbc x2 =2Þ  tanhðb1 x2 =2Þ

ð20Þ

where Cc is thermal conductivity of cold reservoir. In the two adiabatic processes, since S is a constant of the motion, irrespective of the time dependence of x, the times spent for the adiabatic processes are negligible. Consequently, the cycle period is given by t ¼ t1 þ t2 : ð21Þ 5. Optimization of the performance parameters The coefficient of performance, cooling rate, and power input are three important performance parameters, which are often considered in the optimal design and theoretical analysis of refrigerators. Using Eqs. (12), (15) and (21), one can find that the cooling rate and power input may be, respectively, expressed as

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  Qc 1 1 ¼ x2  tanhðb1 x2 =2Þ þ tanhðb4 x2 =2Þ R¼ ðt1 þ t2 Þ 2 2 t and P¼

W ¼ t



 1 1 tanhðb3 x1 =2Þ  tanhðb2 x1 =2Þ 2 2   1 1 þx2 tanhðb1 x2 =2Þ  tanhðb4 x2 =2Þ ðt1 þ t2 Þ: 2 2

ð22Þ



x1

ð23Þ

Using Eqs. (22) and (23), one can optimize these important performance parameters of the quantum refrigeration cycle. According to Eqs. (13) and (14), Eqs. (22) and (23) are rewritten as R¼

Cx2 1 ½tanhðbh xh =2Þ  tanhðbc xc =2Þ½th þ tc  2

ð24Þ

P¼

C 1 ðx1  x2 Þ½tanhðbh xh =2Þ  tanhðbc xc =2Þ½th þ tc  ; 2

ð25Þ

and

tanhðbh xh =2Þtanhðbc x2 =2Þ½tanhðbc xc =2Þtanhðbh x1 =2Þ , and C = Ch = Cc. Using Eq. where th þ tc ¼ ln ½½tanhðb h xh =2Þtanhðbh x1 =2Þ½tanhðbc xc =2Þtanhðbc x2 =2Þ (24) and the extremal condition ðoR=ox2 Þx1 ¼ 0, one can obtain the following equation

ðth þ tc Þ 

bc x2 sech2 ðbc x2 =2Þ½tanhðbh xh =2Þ  tanhðbc xc =2Þ ¼ 0: 2 ½tanhðbh xh =2Þ  tanhðbc x2 =2Þ½tanhðbc xc =2Þ  tanhðbc x2 =2Þ ð26Þ

It gives an optimal relation between x1 and x2, but it is too complicated to yield an analytical solution. Based on a numerical computational method and Eqs. (16), (24) and (26), one can plot the optimal characteristic curves R*  e, P*  e and R*  P*, as shown in Figs. 2–4, where R* = 2R/(Cxc) and P* = 2P/(Cxc) are the dimensionless cooling rate and power input, bh = 0.1, bc = 0.2, xc = 6 J, xh = 12 J or xc = 4 J, xh = 8 J. It is found

ω c=6, ω h=12 ω c=4, ω h=8

0.08 * R max

0.06 *

R

0.04

0.02

0.00 0.0 ε m

0.2

0.4

0.6

0.8

1.0

ε Fig. 2. The dimensionless cooling rate R* = 2R/(Cxc) versus coefficient of the performance for different high and low ‘‘magnetic fields’’ at a given reciprocal temperature of the cold reservoir bc = 0.2 and reciprocal temperature of the hot reservoir bh = 0.1.

J. He et al. / Applied Energy 84 (2007) 176–186

183

3.0 ω c=6, ω h=12 ω c=4, ω h=8

2.5 2.0 *

P

1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

ε Fig. 3. The dimensionless power input P* = 2P/(Cxc) versus the coefficient of the performance. The values of all parameters are the same as those used in Fig. 2.

0.08 ωc =6, ω =12

0.06

h

ω =4, ω =8 c

*

R

h

0.04

0.02

0.00 0.0

0.5

1.0

1.5

2.0

2.5

3.0

*

P

Fig. 4. The dimensionless cooling rate R* versus the dimensionless power input P*. The values of all the parameters are the same as those used in Fig. 2.

from these figures that there exists a maximum cooling rate Rmax and the corresponding coefficient of the performance em for a set of given parameters bc, bh, xc, and xh. For different given parameters, the maximum cooling rate Rmax and the corresponding coefficient of performance em are different. At given xc, bc and bh, the larger xh is, the larger the maximum cooling rate Rmax; But the corresponding coefficient of the performance em is unvarying, as shown in Fig. 5. It is shown from Eq. (16) that at a given xc the larger xh is, the less the maximum value of the coefficient of the performance emax. Based on Eqs. (20)–(22), when x1 ! xh and x2 ! xc, the times spent on the two isomagnetic field processes approach infinity and the cooling rate almost to zero. Therefore, it is shown that the curves in Fig. 5 cross each other in some region of e. At given bc, and bh, the larger xc is, the larger the maximum cooling rate Rmax and the less the corresponding coefficient of the performance em, as shown in Fig. 2. At high temperatures, i.e.bx  1, the results obtained above may be simplified. For example, Eqs. (11)–(16), Eqs. (19), (20), and Eqs. (24),(25) may be, respectively, simplified to become

184

J. He et al. / Applied Energy 84 (2007) 176–186 0.10 =4, =4, =4, c

0.08

=12 =10 =8

c

h

c

h h

0.06 *

R

0.04

0.02

0.00 0.0

0.2

0.4

0.6

0.8

1.0

Fig. 5. The dimensionless cooling rate R* versus the coefficient of the performance for different high and low ‘‘magnetic fields’’ at a given ‘‘temperature’’ of cold reservoir bc = 0.2 and ‘‘temperature’’ of hot reservoir bh = 0.1 under the high-temperature condition.

Qh ¼ x21 ðb2  b3 Þ=4; Qc ¼

x22 ðb4

ð27Þ

 b1 Þ=4;

ð28Þ

b1 x2 ¼ b2 x1 ¼ bc xc ;

ð29Þ

b3 x1 ¼ b4 x2 ¼ bh xh ;

W ¼ x21 ðb3  b2 Þ þ x22 ðb1  b4 Þ =4; 1 b  b2 ln h ; th ¼ Ch bh  b3 1 b  b4 tc ¼ ln c ; Cc bc  b1  1 Cx2 ðb xh  bc x2 Þðbc xc  bh x1 Þ ðbh xh  bc xc Þ ln h ; R¼ ðbh xh  bh x1 Þðbc xc  bc x2 Þ 2

ð30Þ ð31Þ ð32Þ ð33Þ ð34Þ

and ðx2  xc Þðbc x2  bh xh Þ ln

ðbh xh  bc x2 Þðbc xc  bh x1 Þ  x2 ðbh xh  bc xc Þ ¼ 0: ðbh xh  bh x1 Þðbc xc  bc x2 Þ ð35Þ

It is thus clear that the performance of a Brayton refrigeration-cycle depends closely on the properties of the working substance, and so is different from that of the Carnot refrigeration cycle. Using Eqs. (25) and (26), one can plot the optimal characteristic curves R*  e, as shown in Fig. 6. It is seen from these figures that the cooling rate monotonically decreases as the coefficient of performance increases. These results obtained are different from those in the general case. When e ! 0, the cooling rate attains its maximum value Rmax. The maximum cooling rate Rmax ¼

 1 Cx2m ðb xh  bc x2m Þ ðbh xh  bc xc Þ ln h ; ðbc xc  bc x2m Þ 2

ð36Þ

J. He et al. / Applied Energy 84 (2007) 176–186

185

0.30 ωc=6, ωh=12 ω c=4, ω h=8

0.25 0.20 *

R

0.15 0.10 0.05 0.00 0.0

0.2

0.4

0.6

0.8

1.0

ε Fig. 6. The dimensionless cooling rate R* versus the coefficient of the performance. The values of all the parameters are the same as those used in Fig. 2.

can be calculated from Eqs. (34) and (35). The corresponding low ‘‘magnetic field’’ x2m is given by ðx2m  xc Þðbc x2m  bh xh Þ ln

ðbh xh  bc x2m Þ  x2m ðbh xh  bc xc Þ ¼ 0: ðbc xc  bc x2m Þ

ð37Þ

In such a case, the power input is infinite. This shows that the cooling rate of the cycle cannot reach the maximum value Rmax. 6. Conclusions We have established the cycle model of a typical quantum Brayton refrigerator consisting of two adiabatic and two isomagnetic field processes and using non-interacting spin1/2 systems as the working substance. Based on the spin theory, the motion equation of an operator, and semi-group formalism, we have analyzed the optimal performance characteristics of the quantum refrigeration cycle and derived concrete expressions for the important parameters, (coefficient of the performance, cooling rate and power input). The characteristics curves among these performance parameters are generated. Further, the optimal performance of the quantum refrigerator in the high temperature limit is discussed in detail. The maximum cooling rate has been calculated. Acknowledgements This study was supported by the National Natural Science Foundation (No. 10465003) and Natural Science Foundation of Jiangxi, People’s Republic of China (No. 0412011). References [1] Sisman A, Saygin H. The improvement effect of quantum degeneracy on the work from a Carnot cycle. Appl Energ 2001;68:367–76. [2] Sisman A, Saygin H. On the power cycles working with ideal quantum gases: The Ericsson cycle. J Phys D: Appl Phys 1999;32:664–78.

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