Performance characteristics of an irreversible magnetic Brayton refrigeration cycle

international journal of refrigeration 31 (2008) 138–144

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Performance characteristics of an irreversible magnetic Brayton refrigeration cycle Ji-Zhou He*, Xin Wu, Xin-Fa Deng Department of Physics, Nanchang University, Nanchang 330047, PR China

article info

abstract

Article history:

Based on the thermodynamic properties of a paramagnetic salt, an irreversible model of

Received 11 February 2007

the magnetic Brayton refrigeration cycle is established, in which the working substance

Received in revised form 24 July 2007

is a special paramagnetic material. The expressions of the important performance param-

Accepted 29 August 2007

eters, such as the coefficient of performance, refrigeration load and work input, are de-

Published online 02 September 2007

rived. Moreover, the optimal performance parameters are obtained at the maximum coefficient of performance. The results obtained here may include the ones of the magnetic

Keywords:

Brayton refrigeration cycle using the magnetic material obeyed the Curie law as the work-

Magnetic refrigerator

ing substance, the magnetic Brayton refrigeration cycle without regeneration and the ever-

Thermodynamic cycle

sible magnetic Brayton refrigeration cycle. Therefore, the results obtained here have

Brayton

general significance and will be helpful to deeply understand the performance of a

Modelling

magnetic Brayton refrigeration cycle. ª 2007 Elsevier Ltd and IIR. All rights reserved.

Calculation Performance

Cycle frigorifique magne´tique irre´versible de Brayton Mots cle´s : Re´frige´rateur magne´tique ; Cycle thermodynamique ; Brayton ; Mode´lisation ; Calcul ; Performance

1.

Introduction

Magnetic refrigeration is one of the important and new refrigeration technologies in the field of cryogenic refrigeration (Tegus et al., 2002a; Yu et al., 2003; Kitanovski and Egolf, 2006). Comparing with conventional gas refrigeration it has many distinctive advantages, which not only reduces environmental pollution but also offers the possibility of much greater operating efficiency and reliability. Recently, an increasing interest has been paid to magnetic refrigeration technology (Yang et al., 2005; Lin et al., 2004; Bru¨ck et al., 2003; Tegus et al., 2002b; He et al., 2002).

Besides a Carnot cycle, magnetic refrigeration cycles may have other typical cycle models, such as a Stirling cycle, Ericsson cycle and Brayton cycle. Magnetic Brayton and Ericsson refrigeration cycles can be expected to realize a much larger temperature span than a magnetic Carnot refrigeration cycle and execute more easily than a magnetic Stirling refrigeration cycle. The previous investigations were mainly concentrated on the influence of various irreversibilities on the performance of magnetic refrigeration cycle, for example, finite-rate heat transfer, heat leak, regenerative loss and internal irreversibility. Moreover, most of literatures have only researched the optimal performance characteristics of a magnetic Stirling or Ericsson

* Corresponding author. E-mail address: [email protected] (J.-Z. He). 0140-7007/$ – see front matter ª 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.08.007

international journal of refrigeration 31 (2008) 138–144

Nomenclature A C CH H H1 H2 k M N Qc  Qcm

Qh Qr r

Helmholtz free energy (J) Curie constant heat capacity at iso-magnetic field (J K1) external magnetic field (T) high magnetic field intensity (T) low magnetic field intensity (T) Boltzmann constant (J K1) magnetization (T K1) the numbers of magnetic dipoles amount of heat exchange in the low iso-magnetic field process (J) corresponding dimensionless refrigeration load at the maximum coefficient of performance amount of heat exchange in the high iso-magnetic field process (J) amount of heat exchange between the working substance and the regenerator (J) ratio of the two magnetic fields

refrigeration cycle, in which the working substance obeys the Curie law (J. Chen and Yan, 1991, 1998; L. Chen and Yan, 1994; Hakuraku, 1987; Hashimoto and Kuzuhara, 1987; Yan and J. Chen, 1990, 1991, 1992; Chen et al., 2001; C. Wu et al., 1996, 1998; F. Wu et al., 2001). However, when the temperature of the magnetic material is low enough or external magnetic field is large enough, the property of the magnetic material will deviate from the Curie law. Thus, the physical properties of the magnetic material will play an important role in the performance analysis and parametric optimum design of a magnetic refrigeration cycle. In the present paper, we investigate the main performance characteristics of an irreversible magnetic Brayton refrigeration cycle. This paper is organized as follows: in Section 2, the main thermodynamic properties of a paramagnetic salt are analyzed. In Sections 3 and 4, an irreversible model of the magnetic Brayton refrigeration cycle with regeneration is established and the expressions of the coefficient of performance, refrigeration load and work input are derived. The characteristic curves between these three important performance parameters and the ratio of the two magnetic fields at different regeneration factor and degree of irreversibility will be plotted. Finally, in Section 5, we discuss the results of the magnetic Brayton refrigeration cycle using special magnetic material obeyed the Curie law as the working substance.

rm rmin S T Th Tc U W  Wm

139

corresponding ratio of the two magnetic fields at maximum coefficient of performance minimum ratio of the two magnetic fields entropy (J K1) absolute temperature (K) temperature of heat reservoir (K) temperature of cold reservoir (K) internal energy (J) work input (J) dimensionless work input at the maximum coefficient of performance

Greek symbols a regeneration factor 3 coefficient of performance maximum coefficient of performance 3max hc ðhe Þ degree of irreversibility in two irreversible adiabatic processes m magnetic moment (J T1) Bohr magneton (J T1) mB s temperature ratio of two reservoirs

N    N m H ZN ¼ emB H=kT þ emB H=kT ¼ 2 cosh B ; kT

(1)

where mB is the Bohr magneton, k is the Boltzmann constant and T is the absolute temperature. Accordingly, the Helmholtz free energy is given by    m H : A ¼ kT ln ZN ¼ NkT ln 2 cosh B kT

(2)

Moreover, the entropy, internal energy, magnetization and heat capacity at iso-magnetic field may be calculated as         vA m H m H m H  B tanh B ; ¼ Nk ln 2 cosh B S¼ vT H kT kT kT

(3)

  m H U ¼ A þ TS ¼ NmB H tanh B ; kT

(4)

    vA m H ; ¼ NmB tanh B M¼ vH T kT

(5)

and CH ¼

2      vU m H 2 mB H ; ¼ Nk B sech vT H kT kT

(6)

respectively. Eqs. (2)–(6) are general expressions of main thermodynamic quantities of magnetic systems.

2. Main thermodynamic properties of magnetic systems Let us consider a system of N magnetic dipoles, each having a magnetic moment m, placed in an external magnetic field H. For simplicity, each dipole has a choice of two orientations, the corresponding energies being mB H and þmB H, respectively. The partition function of the system is given by (Pathria, 1972)

3. An irreversible model of the regenerative magnetic Brayton refrigeration cycle A magnetic Brayton refrigeration cycle consists of two adiabatic and two iso-magnetic field processes. Its temperature– entropy diagram is shown in Fig. 1. T1 , T2 , T2s , Th , T4 , T5 , T5s

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S

H2

H1

1

2s

Qr 6

The fourth stage, 5/6, is an iso-magnetic field process in which H ¼ H2 and the working substance raises its temperature from T5 to Tc by absorbing heat Qc , i.e. refrigeration load, from the cold reservoir. Using Eq. (6), one can obtain Z Tc CH2 dT ¼ NmB H2 ½tanh x5  tanh x6 ; (11) Qc ¼ Q56 ¼

3

2

Qh

T5

Qc Qr 5 5s

4

Tc

Th

T

where x5 ¼ mB H2 =kT5 and x6 ¼ mB H2 =kTc . In order to increase the temperature span of refrigeration, the regenerative magnetic refrigeration cycle is generally applied and the regenerator is often used for regeneration when the two iso-magnetic field processes have the same temperature ranges, i.e. the temperature ranges from the states 3 to 4 and from the states 6 to 1. The processes, 3/4 and 6/1, are called the regenerative processes. Using Eq. (6), one can obtain the amount of heat exchange in two regenerative processes, Z T4 CH1 dT ¼ NmB H1 ½tanh x4  tanh x3 ; (12) Q34 ¼ Th

Fig. 1 – The entropy–temperature diagram of a magnetic Brayton refrigeration cycle.

and Q61 ¼

Z

T1

Tc

CH2 dT ¼ NmB H2 ½tanh x6  tanh x1 ;

(13)

and Tc are the temperatures of the working substance in the states 1, 2, 2s, 3, 4, 5, 5s and 6, respectively. Th and Tc are the temperatures of the heat reservoir and the cold reservoir, respectively. H1 and H2 are high and low magnetic field intensity ðH1 > H2 Þ. The temperatures of the working substance have the following relation:

When the influence of finite-rate heat transfer is negligible in the regenerator, the magnetic refrigeration cycle may possess the condition of perfect regeneration. Thus, the amount of heat Q34 flowing into the regenerator in the one regenerative process, 3/4, is equal to that of Q61 flowing from the regenerator in the other regenerative process, 6/1, i.e.

T5s < T5 < Tc < T4 < T1 < Th < T2s < T2 :

Qr ¼ Q61 ¼ Q34 ;

(7)

The first stage of the cycle, 1/2, is an irreversible adiabatic magnetizing process in which the magnetic field intensity is increased from H2 to H1 and the temperature of the working substance is raised from T1 to T2 . In this process, the entropy of the working substance is increased. The process, 1/2s, is the corresponding reversible adiabatic process in which the entropy of the working substance is unvary. Using Eq. (3), one can obtain the following equation:   coshðx1 Þ ¼ x1 tanhðx1 Þ  x2s tanhðx2s Þ; (8) ln coshðx2s Þ where x1 ¼ mB H2 =kT1 and x2s ¼ mB H1 =kT2s . The second stage, 2/3, is an iso-magnetic field process in which H ¼ H1 and the working substance decreases its temperature from T2 to Th by releasing heat Qh to the heat reservoir. Using Eq. (6), one can obtain Z T2 CH1 dT ¼ NmB H1 ½tanh x3  tanh x2 ; (9) Qh ¼ Q23 ¼

where Qr is called the regeneration. In order to describe the degree of irreversibility in two irreversible adiabatic processes, one may define two parameters he ¼

ðT4  T5 Þ ; ðT4  T5s Þ

(15)

ðT2s  T1 Þ : ðT2  T1 Þ

(16)

and hc ¼

When hc ¼ he ¼ 1, the magnetic Brayton refrigeration cycle is reversible in two adiabatic processes. According to Eqs. (9) and (11), work input and the coefficient of performance of the magnetic Brayton refrigeration cycle are given by W ¼ Qh  Qc ¼ NmB H1 ½tanh x3  tanh x2   NmB H2 ½tanh x5  tanh x6 ; (17)

Th

where x2 ¼ mB H1 =kT2 and x3 ¼ mB H1 =kTh . The third stage, 4/5, is an irreversible adiabatic demagnetizing process in which the magnetic field intensity is decreased from H1 to H2 and the temperature of the working substance drops from T4 to T5 . It is similar to the irreversible adiabatic magnetizing process. The process, 4/5s, is the corresponding reversible adiabatic process, which satisfies the following equation:   coshðx5s Þ ¼ x5s tanhðx5s Þ  x4 tanhðx4 Þ; (10) ln coshðx4 Þ where x4 ¼ mB H1 =kT4 and x5s ¼ mB H2 =kT5s .

(14)

and 3¼

Qc H2 ½tanh x5  tanh x6  ¼ ; W H1 ½tanh x3  tanh x2   H2 ½tanh x5  tanh x6 

(18)

respectively.

4.

Optimal performance parameters

For Brayton refrigeration cycle, the coefficient of performance, refrigeration load and work input are three important

international journal of refrigeration 31 (2008) 138–144

141

performance parameters. In order to obtain the optimal performance of the refrigeration cycle, we introduce three external parameters. r ¼ H1 =H2 is ratio of the two magnetic fields, s ¼ Tc =Th is the temperature ratio of two reservoirs and a ¼ T1 =Th is a regeneration factor. According to Fig. 1, when a ¼ 1, the cycle is operated with a maximum regeneration, i.e. Qr max ¼ NmB H2 ½tanh x6  tanh x01 , where x01 ¼ mB H2 =kTh ; when a ¼ s, the cycle is operated without regeneration, i.e. Qr ¼ 0. Thus the range of a is s  a  1. Eqs. (8), (10) and (14) may be rewritten as

numerical computation method and Eqs. (26)–(28), as shown in Figs. 2–7, respectively, where Qc ¼Qc =Nk is the dimensionless refrigeration load and W ¼W=Nk is the dimensionless work input. In these figures, Tc ¼ 100 K, T2 ¼1K and s¼0:45 are adopted. It is shown from Figs. 2 and 5 that the ratio of the two magnetic fields must be larger than the minimum value rmin . When r
          rT2 srT2 T T  tanh ¼ tanh 2  tanh 2 ; r tanh T4 Tc Tc T1

(19)

r>rmin ;

           cosh T2 =T1 T2 T rT2 rT2    ¼ tanh 2  tanh ; ln T1 T1 T2s T2s cosh rT2 =T2s

(20)

             cosh T2 =T5s T2 T rT2 rT2  ¼   tanh 2  tanh ; ln T5s T5s T4 T4 cosh rT2 =T4

(21)

where T2 ¼ mB H2 =k, its unit is that of absolute temperature (K). By using Eqs. (15), (16) and (19), the temperatures in state points 1, 2, 4 and 5 of the working substance may be expressed as T1 ¼

aTc ; s

(22)

T2 ¼

1 ½T2s  ð1  hc ÞaTc =s; hc

(23)

T4 ¼

1

tanh

rT2         ; 1=r tanh T2 =Tc tanh sT2 =aTc þtanh srT2 =Tc (24)

and T5 ¼ð1he ÞT4 þhe T5s ;

(25)

where T2s and T5s may be solved from Eqs. (20) and (21) by means of the bisection method. Substituting Eqs. (22)–(25) into Eqs. (11), (17) and (18), one may obtain the coefficient of performance, refrigeration load and work input,     tanh T2 =T5 tanh T2 =Tc           ;   3¼  r tanh rsT2 =Tc tanh rT2 =T2  tanh T2 =T5 tanh T2 =Tc (26)      Qc ¼NkT2 tanh T2 =T5 tanh T2 =Tc ;

(27)

and

        W¼NkT2 r tanh rsT2 =Tc tanh rT2 =T2  tanh T2 =T5    tanh T2 =Tc :

(29)

where rmin may be obtained by numerical computation in the condition of 3>0 or Qc >0. rmin is one important parameter of the magnetic Brayton refrigeration cycle and determines the minimum ratio of the two magnetic fields. For given a, s, he and hc the coefficient of performance 3 is not a monotonic function of r while the refrigeration load and work input are monotonically increasing function of r, as shown in Figs. 2–7. According to Eq. (30) and the extremal condition v3=vr¼0, one can calculate the maximum coefficient of performance 3max and the corresponding ratio of the two magnetic fields rm . But it is too complicate to yield a simple analytical solution. One may only obtain 3max and rm from Figs. 2 and 5. When rrm , the coefficient of performance will increase as the refrigeration load decreases, and vice versa. One always hopes to obtain both the coefficient of performance and the refrigeration load as large as possible. Consequently, the optimal region of the ratio of two magnetic fields should be (30)

r>rm :

It is found from Figs. 2–4 that when the refrigeration cycle is operated in the optimal region, the coefficient of performance and the refrigeration load decrease while the work input increases as the degree of irreversibility h decreases. That is to say, the smaller the irreversible loss in two adiabatic processes, the larger the coefficient of performance and the refrigeration load while the smaller the work input. However, it is also seen from Figs. 5–7 that when the refrigeration cycle is operated in the optimal region, the coefficient of performance increases while the refrigeration load and work input decrease as the regeneration factor a decreases, i.e. the smaller the amount of the regeneration in two regeneration processes, the larger the coefficient of performance while the smaller the refrigeration load and the work input. Finally, based on the numerical computation one may cal culate the corresponding dimensionless refrigeration load Qcm  and dimensionless work input Wm at the maximum coefficient of performance, as indicated in Table 1.

(28)

For given the temperature of cold reservoir Tc , low magnetic field H2 and temperature ratio s the characteristic curves between the important performance parameters ð3;Qc ;W Þ, and the ratio of the two magnetic fields (r) at different regeneration factor (a) and degree of irreversibility ðhc ;he Þ can be plotted by

5.

Discussion

When the temperature of the magnetic material is high enough or external magnetic field is small enough, i.e. mB H=kT << 1, magnetization of the working substance is

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international journal of refrigeration 31 (2008) 138–144

0.30

0.030

Tc = 100K τ = 0.45 α = 0.7 ηc = ηe = 1.0 ηc = ηe = 0.98 ηc = ηe = 0.96

0.25 0.20

0.025

0.020

W*

εmax 0.15

0.015 0.10

Tc = 100K τ = 0.45 α = 0.7

0.00

η = η = 1.0 c e η = η = 0.98 c e ηc = ηe = 0.96

0.010

0.05

rmin 2

1

rm

3

4

5

r

6

7

0.005

8

Fig. 2 – The coefficient of performance 3 versus ratio of the two magnetic fields r at different degree of irreversibility.

2.5

3.0

3.5

4.0

r Fig. 4 – The dimensionless work input W  versus ratio of the two magnetic fields r at different degree of irreversibility.

described by Curie law. Eq. (5) may be simplified as M¼

CH ; T

(31)

where C ¼ Nm2B =k is the Curie constant. By using Eqs. (23)–(25) the temperatures of the working substance in states 2, 4, and 5 may be simplified as T2 ¼

aTc ðr þ hc  1Þ ; hc s

(32)

   2 a þ ðar2  1Þs  arðhe  rhe þ rÞ Qc ¼ Nk T2 ; arðhe  rhe þ rÞTc and   2 sr2 ½aðr þ hc  1Þ  hc  W ¼ Nk T2 aðr þ hc  1ÞTc 

2

T4 ¼

ar Tc ; a þ ðar2  1Þs

(33)

arTc ðhe þ r  rhe Þ : a þ ðar2  1Þs

(34)

and T5 ¼

Eqs. (26)–(28) may also be simplified as  3¼

sr3 ½aðr þ hc  1Þ  hc  ðhe  rhe þ rÞ 1 ðr þ hc  1Þ ½a þ ðar2  1Þs  arðhe  rhe þ rÞ

a þ ðar2  1Þs  arðhe  rhe þ rÞ : arðhe  rhe þ rÞTc

;

(35)

Conclusions

The irreversible regenerative Brayton refrigeration cycle working with the special paramagnetic material has been 0.30

0.007

Tc = 100K τ = 0.45 ηc = ηe = 0.98

0.006

0.25

0.005

0.20

0.004

0.15

ε

Qc*

εmax

0.003

Tc = 100K τ = 0.45 α = 0.7 ηc = ηe = 1.0 ηc = ηe = 0.98 ηc = ηe = 0.96

0.002 0.001 2.5

(37)

Eqs. (35)–(37) obtained here are the same as those in literature (Yang et al., 2005). Similarly, we can plot the characteristic curves between the important performance parameters ð3; Qc ; W Þ and the ratio of the two magnetic fields.

6.

1

(36)

3.0

3.5

0.10 0.05

4.0

r

α = 0.6 α = 0.7 α = 0.9

0.00 1

Qc

Fig. 3 – The dimensionless refrigeration load versus ratio of the two magnetic fields r at different degree of irreversibility.

2

rm 3

4

5

6

7

8

r Fig. 5 – The coefficient of performance 3 versus ratio of the two magnetic fields r at different regeneration factor a.

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international journal of refrigeration 31 (2008) 138–144

0.007

Table 1 – The corresponding dimensionless refrigeration   load Qcm , the dimensionless work input Wm , the corresponding ratio of the two magnetic fields rm and the minimum ratio of the two magnetic fields rmin are presented at the maximum coefficient of performance 3max

0.006

Q*c

0.005

hc ¼ he

3max

 Qcm

 Wm

rmin

rm

0.6

1.00 0.98 0.96

0.32497 0.26273 0.21357

0.00312 0.00322 0.00303

0.0096 0.01224 0.01421

2.26 2.26 2.07

2.71 2.85 2.93

0.7

1.00 0.98 0.96

0.29030 0.24062 0.19938

0.00344 0.00332 0.00308

0.01183 0.01381 0.01544

1.93 1.93 1.89

2.69 2.78 2.84

0.9

1.00 0.98 0.96

0.26306 0.22275 0.18823

0.00344 0.00344 0.00308

0.01307 0.01499 0.01637

1.50 1.51 1.57

2.55 2.64 2.69

a

0.004 0.003

Tc = 100K τ = 0.45 ηc = ηe = 0.98

0.002

α = 0.6 α = 0.7 α = 0.9

0.001 2.5

3.0

3.5

r

4.0

Fig. 6 – The dimensionless refrigeration load Qc versus ratio of the two magnetic fields r at different regeneration factor a.

Acknowledgments

established based on the statistics of paramagnetism, in which the regeneration in the two iso-magnetic field processes and the internal irreversibility in the two adiabatic processes are considered. It can be clearly found from the results obtained in this paper that the main performance parameters strongly depend on the regeneration and internal irreversibility. The larger the regeneration, the larger the refrigeration load and the work input, and the smaller the coefficient of performance. The internal irreversibility always reduces the coefficient of performance and refrigeration load, and increases the work input. The two important parameters, minimum ratio of the two magnetic fields and the lower bound of the optimal ratio of the two magnetic fields, are obtained. These results may not only apply for the roomtemperature magnetic Brayton refrigerators but also the low-temperature ones. Therefore these results have general significance and provide some theoretical basis for the optimal design of the magnetic Brayton refrigerators.

0.08

Tc = 100K τ = 0.45 ηc = ηe = 0.98 α = 0.6 α = 0.7 α = 0.9

W*

0.06

0.04

0.02

0.00 2.5

3.0

3.5

4.0

4.5

5.0

r Fig. 7 – The dimensionless work input W  versus ratio of the two magnetic fields r at different regeneration factor a.

This work was supported by National Natural Science Foundation (No. 10465003), Natural Science Foundation of Jiangxi, People’s Republic of China (No. 0412011) and The Science and Technology Foundation of Jiangxi Education Bureau.

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