The characteristics of polytropic magnetic refrigeration cyclesZijun Yan and Jincan Chen Citation: Journal of Applied Physics 70, 1911 (1991); doi: 10.1063/1.349472 View online: http://dx.doi.org/10.1063/1.349472 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/70/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Composite magnetic polytropes AIP Conf. Proc. 895, 234 (2007); 10.1063/1.2720428 Two Quantum Polytropic Cycles AIP Conf. Proc. 643, 302 (2002); 10.1063/1.1523821 The effect of thermal resistances and regenerative losses on the performance characteristics of amagnetic Ericsson refrigeration cycle J. Appl. Phys. 84, 1791 (1998); 10.1063/1.368349 Main characteristics of a Brayton refrigeration cycle of paramagnetic salt J. Appl. Phys. 75, 1249 (1994); 10.1063/1.356427 Regenerative balance in magnetic Ericsson refrigeration cycles J. Appl. Phys. 71, 5272 (1992); 10.1063/1.350591

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The characteristics of polytropic magnetic refrigeration cycles Zijun Yan and Jincan Chen China Center of Advanced Science and Technology (World Laboratory), P.O. Box 8730, Beijing 100080, People’s Republic of China and Department of Physics, Xiamen University, Xiamen, Fujian, People’s Republic of Chinti’

(Received 7 February 1991; accepted for publication 22 May 1991)

The concepts of generalized polytropic processes and polytropic refrigeration cycles, which consist of two isothermal processes and two generalized polytropic processes of paramagnetic salt, are introduced. It is shown that such a class of general magnetic refrigeration cycles, which includes the Camot, Stirling, and other useful magnetic refrigeration cycles, possesses the conditions of perfect regeneration and can have the same coefficient of performance as the Carnot cycle for the same temperature range. Thus, they have many applications in the research and manufacture of magnetic refrigerators.

I. INTRODUCTION

In the research and manufacture of magnetic refiiger- ators, Ericsson and Stirling magnetic refrigeration cycles are two important cycle models. An Ericsson refrigeration cycle of paramagnetic salt consists of two isothermal pro- cesses and two isofield processes and cannot possess the conditions of perfect regeneration, ’ so its coefficient of per- formance cannot attain that of a Carnot refrigerator, and it has some notable characteristics2 which are different from a perfect regeneration cycle. A Stirling refrigeration cycle of paramagnetic salt consists of two isothermal processes and two isomagnetization processes and possesses the con- ditions of perfect regeneration,is3 so its coefficient of per- formance can attain that of a Carnot refrigerator. Al- though, in practice, actually executing an isomagnetization process will not be easy because of the required field change with temperature, the Stirling magnetic refrigera- tion cycles with perfect regeneration are expected to realize even larger temperature spans than that of the Camot cy- cle and still come into notice. Of course, except for the Stirling magnetic refrigeration cycle, there are many other magnetic refrigeration cycles with the conditions of perfect regeneration. Therefore, the question of how to find such a class of cycles and its main characteristics is worthy to be studied.

In this paper we discuss the main characteristics and some applications of a class of magnetic refrigeration cy- cles which are referred to as polytropic or quasi-Carnot magnetic refrigeration cycles. It is of special interest that such a class of cycles possesses the conditions of perfect regeneration and their coefficient of performance can attain that of a Carnot refrigerator.

II. THERMODYNAMIC PROPERTIES OF PARAMAGNETIC SALT

To begin with, let us consider a unit volume paramag- netic salt and assume that its volume change is negligible. Then the fundamental equation of thermodynamics of paramagnetic salt may be expressed as

“Mailing address.

du=Tds +p&dM, (1)

where u and s are, respectively, the internal energy and entropy of paramagnetic salt per unit volume. M and H are, respectively, the magnetization and magnetic intensity of paramagnetic salt. T is the absolute temperature, and p. is the permeability of the vacuum.

Next, it may be shown by using a statistical theory that the relation between M and H for a simple paramagnetic salt is given by4,5

M=wmJ~.hw.J~~W) 1, where

(2)

B&l = Fcoth(Tx) --fcoth(&-)- (3)

is the Brilloum function, PU, = &/(2mc) iS referred to as the Bohr magneton, g is Land& g factor, J is the quantum number of angular momentum, and n is the number of magnetic moments per unit volume.

Using Eqs. ( 1 )-( 3), one can provelt2

u=u( T),

C,u=CdT),

and

s=s,,( T) - ,u&hi/T

(4)

(5)

+eank[lnsinh(yx)-lnsinh$], (6)

where x = gpg JH/(kT), C,(T) is the heat capacity at isomagnetization, and sc( T) is the entropy of paramag- netic salt when M = 0. Both Ch( T) and se(T) are only a function of the temperature, but do not depend on the magnetization. Their values are dependent on the specific properties of paramagnetic salt.

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Ill. GENERALIZED POLYTROPIC PROCESSES OF PARAMAGNETIC SALT

The polytropic process is a useful process whose main character is that the heat capacity of the process remains constant. The generalized polytropic process is a direct extension of the polytropic process so that it is more gen-

1 era1 than the polytropic process. Its main character is that the heat capacity C, of the process is only a function of the temperature, i.e.,

C,,=C,(T). (7) Such processes are more appropriate to describe many practical processes, because the heat capacities of many useful processes in practical systems such as the isomagne- tization processes in paramagnetic salt are only dependent on the temperature.

When a generalized polytropic process is carried out in the system of paramagnetic salt, one can obtain

C,(T)dT=C,(T)dT-p&dM (8)

from Eqs. ( 1) and (4). Using Eqs. (2) and (3) again, we have

C,(T)=C,(T) -ponW/W’)lk,w02 x (dH/dT-H/T)dB.,(x)/dx. (9)

When gpdH/(kT) 4 1, Eq. (2) becomes the Curie law and Eq. (9) may be written as

C,,(T) = C& T) - ,uoCH d(H/T)/dT. (10)

The equation of magnetic generalized polytropic process,

T2 s

T GdT) - C,(T) dlL& H2 To T 2

or

s T C,(T) - C,(T) dT= PO &fz

TO T 2c ’

(11)

(12)

can be derived from Eq. ( 10). Here C = ng$$ J x (J+ 1)/(3k) is the Curie constant, and To is the tem- perature of paramagnetic salt at H = 0. In general cases, the equation of magnetic generalized polytropic process must be derived from Eq. (9). Equations (9) and ( 10) show clearly that an isomagnetic intensity process is not a generalized polytropic process while H must change with T in generalized polytropic processes. For example, in the isomagnetization generalized polytropic process, the equa- tion of H changing with T is

H/T=const. (13) This shows clearly that the key of executing a generalized polytropic process is to control the magnetic field change with temperature.

It should be pointed out that C,(T) has many choices. When C,,(T) = 0, a generalized polytropic process be- comes an adiabatic process; when C,(T) = C&T), a generalized polytropic process becomes an isomagnetiza- tion process, and so on.

0 s

FIG. 1. Temperature-entropy schematic diagram of a quasi-Camot mag- netic refrigeration cycle.

IV. POLYTROPIC MAGNETIC REFRIGERATION CYCLES

For the sake of convenience, we give a schematic dia- gram of the polytropic or quasi-Carnot magnetic refriger- ation cycle which consists of two isothermal processes and two generalized polytropic processes, as shown in Fig. 1. In Fig. 1, Qh and Q, are the heats released to the hot reservoir at temperature T, and absorbed from the cold reservoir at temperature T, per cycle. Qb, and Qd, are the heat trans- ferred into and out of the regenerator in the two general- ized polytropic processes. All the heats Qh, Q,, QbP and 12d, are positive.

According to Fig. 1 and the properties of generalized polytropic processes, we can find

Q,i= - Th(S/, - Sa>,

Qbc= - j-; C, 0,

SC -sb= s =’ (C,/T)dT, Th

Qc= T&d - 4,

Qcja= j-; G dT c

and

(17)

(18)

s, - sd= I

Th (C,,/T)dT, (19) TC

where s,, sb, s, and sd are the entropies of the states A, B, C, and D in Fig. 1, respectively. It is easily seen from Eqs. (15) and (18) that

C&c= t&z- (20)

1912 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991 Z. Yan and J. Chen 1912

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Equation (20) shows clearly that the polytropic magnetic refrigeration cycle possesses the conditions of perfect re- generation. Using Eqs. (14), (16), (17), and (19), we obtain the coefficient of performance of the cycle:

QC Tc Tc ‘=m= T,,(s, - sb)/(sd - s,) - T,=m.

(21) It is seen from Eq. (21) that the polytropic magnetic re- frigeration cycles can have the same coefficient of perfor- mance as the Carnot refrigerator for the same temperature range. This is one of the most important characteristics for polytropic magnetic refrigeration cycles.

Moreover, using Eq. (6), Eqs. ( 14) and ( 17) may be respectively written as

QF=PO~AI [HbBh) - H3.k) 1 + ~onkT$‘, (22)

and

Qc=~odl tKB.r(xc) - H$.r(-d I + ponkT,Y,, (23)

where

Yi=ln sinhC[(W+ 1)/W]x,)sinh[(1/2J)xb] sinh[(1/2J)x,,]sinh{[(W+ l)/W]xb)’

Y,=ln sinh[(l/W)x,] sinh{[(W+ 1)/2J]xd} sinh{[(W-t l)/W]x,}sinh[ ( l/W)xd] ’

A,=wzJ, -%=AtHd(kTh), xb=A,H/(kTh),

x,=A,W(kTc),

and

xd=A,HJ(kT,).

H,, Hb, Ho and Hd are the magnetic intensities in states A, B, C, and D in Fig. 1, respectively. From Eqs. (22) and (23), we find that the coefficient of performance of the polytropic magnetic refrigeration cycle may be written as

TC E=TdCkYl +Al[HbB.r(%) - H$.r(x,)]/Th)/{kY, + A,[HaJ(xc) - HdBJ(xd)]/Tc}) - T,’

(24)

Comparing Eq. (24) with Eq. (21), we have

kY1 + Al[HbBJ(%) - HnB.kz) ]/Th ky, + A1[fM&,> - HdBAxd l/T,= ‘* 05)

This shows clearly that for polytropic magnetic refrigera- tion cycles, the magnetic intensities of states A, B, C, and D in Fig. 1 must be restricted by Eq. (25). When gpBJH/(kT) ( 1, Eq. (25) may be written as

(H; - Hi) Tf=== (Hz - H;) c. (26)

Equation (25) or (26) provides the theoretical basis for the reasonable choice of the applied magnetic field in the design of a magnetic refrigeration cycle with perfect regen- eration.

V. TWO SIMPLE APPLICATIONS

( 1) Polytropic magnetic refrigeration cycles are a class of general cycles. They include many important practical cycles such as the Carnot magnetic refrigeration cycles in which the heat capacity C,(T) of generalized polytropic processes is chosen to be equal to zero,47 the Stirling mag- netic refrigeration cycles in which C,(T) is chosen to be equal to C,(T), and so on. It is thus obvious that poly- tropic magnetic refrigeration cycles are a class of useful cycles. One can expect that some new magnetic refrigera- tion cycles which are suitable for the various uses in the research and manufacture of magnetic refrigerators will be found through the reasonable choice of C,(T). For exam-

I

ple, in some new practical two-heat-source magnetic refrig- eration cycles, the heat capacities of the two nonisothermal processes are negative values’ so that these cycles are nei- ther the Carnot’s nor Stirling’s form.g However, their per- formances may be described approximately by selecting suitable polytropic magnetic refrigeration cycles in which C,(T) is taken to be negative.

(2) Using the concept of polytropic cycles, it is easily judged that the Ericsson magnetic refrigeration cycles can- not possess the conditions of perfect regeneration, and are not polytropic magnetic refrigeration cycles, because the isomagnetic intensity (or isomagnetic induction) process is not a generalized polytropic process. Thus, if one wishes to have a magnetic refrigeration cycle with perfect regenera- tion and retain some advantages of the Ericsson magnetic refrigeration cycle, through the reasonable choice of C,(T) or the use of the working substance not to be with a single magnetic material, one can seek out polytropic magnetic refrigeration cycle that comes close to the Eric- sson cycle.

Obviously, for polytropic magnetic refrigeration cy- cles, there are also other applications in the new technol- ogy of magnetic refrigeration. Therefore, the characteris- tics and applications of polytropic magnetic refrigeration cycles are worth investigating further.

‘Z. Yan and J. Chen, J. Appl. Phys. 66, 2228 (1989). ‘J. Chen and Z. Yan, J. Appl. Phys. 69, 6245 ( 1991). 3X. Jin, R. Ouyang, and W. Chen, Cryog. Supercond. 15, 15 (1987) (in

1913 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991 Z. Yan and J. Chen.. 1913 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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Chinese). ‘Y. Hakuraku and H. Ogata, Jpn. J. Aepl. Phys. 24, 1538 (1985); 25, 144 ‘R Kubo, Statistical Mechanics (North-Holland, Amsterdam, 1965), p. (1986).

149. ‘Y. Hakuraku and H. Ogata, J. Appl. Phys. 60, 3266 (1986). _ ‘M. W. Zemansky, Heat and Thermodynamics, 5th ed. (McGraw-Hill, ‘Y. Hakuraku and H. Ogata, Cryogenics 26, 171 (1986). New York, 1968), p. 452. 9Z. Yan, Cryog. Supercond. 17, 8 (1989) (in Chinese).

1914 J. Appl. Phys., Vol. 70, No. 4, 15 August 1991 Z. Yan and J. Chen 1914

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