Second law analysis of active magnetic regenerative hydrogen liquefiers

Second law analysis of active magnetic regenerative hydrogen liquefiers L. Zhang, S.A. Sherif*, T.N. Veziroglu t and J.W. Sheffield* Department of Mechanical Engineering, University of Miami, Coral Gables, FL 33124, USA * Department of Mechanical Engineering, University of Florida, Gainesville, FL 32611, USA t Clean Energy Research Institute, University of Miami, Coral Gables, FL 33124, USA :i: Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, Rolla, MO 65401, USA

Received 1 April 1992; revised 1 August 1992 Recent published work in the area of magnetic refrigeration reports on its potential for greater efficiency and high reliability. This paper presents an exergy analysis of a threestage active magnetic regenerative (AMR) hydrogen liquefier which cools a hydrogen gas stream at 77 K and 1 atm to hydrogen liquid at 20 K. O r t h o - p a r a cofiversion of hydrogen is accomplished in a heat exchanger employing a 10 atm helium fluid that cycles in the refrigerator. The performance of the system is described in terms of the cooling capacity and exergy losses as functions of the magnetic material type, magnetic bed size and temperature, helium mass flow rate, ratio of isofield and adiabatic process times, and the operating frequency.

Keywords: magnetic refrigeration; hydrogen; liquefiers

Nomenclature A

Coefficient of magnetic material entropy due to ferromagnetic transitions defined by Equation (1) (J mol 1 K-i) A c Cross-section area of material bed (m 2) B Magnetic field (T) Bj(x) Brillouin function as defined by Equation (3) C Specific heat of magnetic material (J mol -~ K l) Ex Exergy (J tool -1) g Lande factor of magnetic material H Helium enthalpy (J tool -l) J Total angular momentum number of magnetic materials K Boltzmann constant (= 1.38 x 10 -23 J K -~) L Length of material bed (m) m Mass flow rate of helium fluid (mol s -~) M Magnetic moment of magnetic material (J tool- l T -i) N Number of atoms per molar magnetic material Ntu Number of transfer units of heat exchanger Q Heat transfer rate in material bed (W) Q~ Cooling load R Gas constant (J mol -~ K -l) S Entropy (J mol l K ~) t Strip thickness for offset strip plate-fin heat exchanger T~ Helium fluid outlet temperature after heating magnetic material bed (K) Th Hot end temperature of auxiliary refrigerator (K)

Tn T,, T~c T~, T~ U V W

Nitrogen boiling point temperature (K) Curie temperature of magnetic material (K) Stage cold temperature of magnetic liquefier (K) Stage hot temperature of magnetic liquefier (K) Helium fluid outlet temperature after cooling magnetic material bed (K) Internal energy of magnetic material (J mol-~) Volume flow rate of fluid (m 3 s ~) Work done on auxiliary refrigerator (W)

Greek symbols c~ ~P T/ @D /xB r r(a) r(i)

Thermal conductivity of bed (W m ~ K ~) Pressure drop across magnetic material bed (N m 2) Strip length for offset strip plate-fin heat exchanger Debye temperature of magnetic material (K) Bohr magneton (= 9.27 × 10 24 j T ~) Total cycle time for one stage (s) Duration of adiabatic process (s) Duration of isofield process (s)

Subscripts 1,2,3 cold hot i mat

Stage Demagnetized state Magnetized state Section of material bed Magnetic material

0011-2275/93/070667-08 @ 1993 Butterworth-Heinemann Ltd

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Active magnetic regenerative hydrogen liquefiers: L. Zhang et al. Magnetic refrigeration research has intensified in recent years because of the potential of magnetic refrigerators for greater efficiency, high reliability and more rugged construction than the present gas cycle refrigerators 1. Seyfert 2 reported on two studies, one investigating the feasibility of magnetically active regeneration in the temperature range 4 . 2 - 15 K and the other looking into the possibility of building a magnetic Carnot refrigerator that can be used in large scale superfluid helium cooling systems. Helvensteijn and Kashani 3 released the design of a magnetic refrigerator operating in the 10-2 K temperature range with a cooling capacity of 0.1 W at 2 K. Savage et al. 4 reported on test results for adiabatic demagnetization refrigerators for use in a NASA spacecraft. Serlemitsos et al. 5 reported on significant changes to an adiabatic demagnetization refrigerator built in the early 1980s for space use. Smith et al. 6 pointed out that proper matching of thermodynamic cycles to core properties requires the rate of magnetic field change and the helium flow rate to be matched at every location along the core and at every instant during the cycle. Researchers from the David Taylor Research Center designed and tested an active regenerator of a reciprocating magnetic refrigerator. The active materials they chose were gadolinium (Gd) and terbium (Tb), having a Curie temperature of 293 and 235 K, respectively. Based on their experimental results, they suggested combining several magnetic materials in a single regenerator. Numazawa et al. 7 developed a magnetic refrigerator which operated on the Carnot cycle between 10 and 1.4 K. Li et al. 8 built a test apparatus for studying the thermal performance of cryogenic regenerators between 4 and 35 K. Kral and Barclay 9 pointed out that as the properties of magnetic refrigerators became better known through actual validated models and performance, incorrect preconceptions, similar to the small cooling capacity myth, would be identified and clarified. DeGregoria et al. 1° proposed the design of an active magnetic regenerative (AMR) refrigerator operating in the 1.8-4.7 K range, using a particle bed of gadolinium gallium garnet (GGG) as the magnetic material and liquid helium as the heat transfer fluid. DeGregoria et al. 11 also reported on an AMR experimental refrigerator operating in the 6 - 2 0 K temperature range. Zimm et al. 12 provided thermodynamic properties of the ferromagnetic materials GdPd, GdNi, GdNi 2 and Er0.86Gdo.taAl: and studied these materials with respect to their use as magnetic refrigerants in the 2 0 - 8 0 K range. DeGregoria et al. 13 reported test results of an AMR refrigerator which operates within the 4 - 80 K temperature range and described a one-dimensional time-dependent model for an AMR refrigerator. Janda et al. 14 demonstrated the design of a two-stage 0.1 ton per day AMR hydrogen liquefier operating between 77 and 20 K. Carpetis ~5 presented a thermodynamic cycle simulation both for magnetic refrigerator stages with an external regenerator and for the AMR refrigerators. Other work includes that of Schroeder 16 who used NASTRAN to model an experimental refrigerator employing the finite element method 17. Their numerical results were compared with the measured performance of a prototype experimental apparatus. They also applied the model to a new design that was expected to

668

Cryogenics 1993 Vol 33, No 7

have a larger capacity than the previous designs and examined the effects of material properties, operating characteristics and heat loads on the temperatures attained by the model 1.

Analysis The system to be considered in this paper is a three-stage AMR hydrogen liquefier. This system is described in some detail by Waynert et al. 18 but is modified here to the system shown in F i g u r e 1. The three temperature ranges 2 0 - 4 0 K, 4 0 - 6 0 K and 6 0 - 7 7 K correspond to three stages of an AMR cycle. For identification purposes, the lowest temperature range stage will be labelled the first stage. Heat is rejected to a 77 K liquid nitrogen boil-off tank at the third stage. The boiled-off nitrogen is used to precool the incoming hydrogen gas from 300 K to near 77 K in a series of heat exchangers. The three stages of active magnetic regeneration considered employ the cooperative Brayton cycle. The components of an AMR cycle are shown in F i g u r e 2a. A packed bed of magnetic material is sandwiched between a hot and cold reservoir. Since the temperature span of a single material in a Brayton cycle is limited to = 10 K, a series of cascaded Brayton cycles was proposed by Tausczik 19 to substantially expand the temperature span. As can be seen in F i g u r e 2, a heat transfer fluid shuttles back and forth through the bed to cool or b~at the material bed. The operation of the AMR cycle is as follows. The bed is magnetized with no flow. Fluid is then passed from the cold to the hot reservoir with the bed in the magnetized state. The bed is then demagnetized with no flow. Fluid is then passed from LNm

Figure 1 Three-stage AMR hydrogen liquefier

r-~=

r.~,

r..~la

Active magnetic regenerative hydrogen fiquefiers: L. Zhang et al.

SUPERCIII~I~I~ MAGNEr

10 HEAT EXCHANGER CQ4.D RESERVER

PACKEDl I D []1" ~ I I C

MATERIAL

MAGNETIZED STALE " Tv

40 K

DE)~GNET]ZEO STATE rcl

b

lED P(~H[0N

Figure 2 (a) Schematic of first stage of AMR liquefier and (b) its temperature profile for the bed and fluid

the hot to the cold reservoir with the bed in the demagnetized state. The magnetic material bed is a blend from the ferromagnetic material series Gd~_~Er~AI2 whose Curie temperature can be adjusted from 165 to 15 K as the concentration x is varied from zero to one. So the material bed is made of a variety of magnetic materials whose temperature profiles in the magnetization and demagnetization states are approximately similar to the ones shown in Figure 2b. The three stages of active magnetic regeneration work together to cool the incoming 77 K hydrogen gas to a 20 K hydrogen liquid. The temperature of the 1 arm 77 K hydrogen gas is reduced in three g a s - g a s heat exchangers while the o r t h o - p a r a conversion is accomplished in three other heat exchangers employing a 10 atm helium fluid that is cooled in the three stages of the magnetic beds. The modifications employed in the system in question are as follows: 1, an auxiliary classical refrigerator box having two classical gas-cycle refrigerators that operate between the first and second stages and between the second and third stages, respectively, is added; 2, two additional pipes with separate valves for the two lower stages are also added. The aim of the first modification is to reduce the helium temperature from T~,~(i = 1, 2) to T~ (i = 2, 3), so that the reduced temperature helium in the lower stage may enter the next higher stage and be cooled by the magnetic bed. For this purpose, auxiliary classical refrigerators are chosen. The aim of the second modification is to ensure that helium can be cooled from T~hi (i = 1, 2) to T~i (i = 1, 2) under the specified cooling load of each stage. As mentioned earlier, the overall system is divided into three stages, each operating on a cycle comprising four processes. The magnetic material bed is magnetized isentropically by applying a magnetic field, followed by releasing heat to the cycled cool helium fluid. The hot helium fluid then releases its heat to an auxiliary gascycle refrigerator. The magnetic material bed is then demagnetized and the cycled helium is cooled. Finally,

the cool helium cools the hydrogen gas. Also, an offset strip plate-fin heat exchanger is employed in six heat exchanger positions in the overall system, as suggested by Zukauskas 2°. In the calculation procedure, several assumptions are employed. These are described below. 1. Although the superconductor magnet is charged by electricity before operating the magnetic refrigeration system, the magnet is assumed to have a continuous ability to provide the required field strength throughout the entire refrigeration process. In other words, the energy consumed by charging the superconductor magnet is negligible. 2. Each magnetic material bed is a blend from 10 materials of different Curie temperatures (series of the compound Gdl ~Er,AI2). The Curie temperatures chosen for each bed are based on the stage temperature range (T~,-T~,) and the need to increase the temperature difference between the magnetization and demagnetization states. Knowledge of the Curie temperature enables calculation of the concentration x by assuming that the Curie temperature changes linearly with x. Additional information is based on the experimental data of Zimm et al. 2~ who reported that T,, = 15 w h e n x = 1 and To = 165 K w h e n x = 0 . 3. Because the available thermal properties of the magnetic material compound Gdl JSr~AI2 do not cover the whole computation operating range, the entropy-temperature relationship which is used to calculate entropy in the magnetization and demagnetization states is computed from the mean field theory according to the following equation

S = R ln(2J + 1.0) - A

when B = 0

(1)

where A(To/T) 2 is the component of entropy due to ferromagnetic transitions 22. Also, for a general value of magnetic field strength S

[smh[~J + ° : 5 ) x ] ] - xBjIx) In[ sinh(0.5x) J

(2)

where the Brillouin function Bj(x) is given by

Bj(x) = (J + 0.5) coth [(J + 0.5)x] - 0.5 coth(0.5x)(3) with x for ferromagnetic materials being given by

g>a [(~,,,) 3KBj(x))] x = KiT~To) + gtx~JCJ + i

(4)

The magnetic moment is computed using

M = NgJlxBBj(x)

(5)

The thermal conductivity of the material compound Gd~ ,Er,A12 is taken from the experimental results of Zimm et al. 21. These results have been correlated to c~ = [0. 1875T + 8.0] for purposes of computation.

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Active magnetic regenerative hydrogen liquefiers: L. Zhang et al.

4. The exergies of the magnetic material and the helium fluid are defined as follows Ex = U -

T, S - B M

for the magnetic material

(6)

at the beginning of the process, i.e. EXin

,(U i - TnS, - BMi)ho,

~-

]

mat

and + [m(TnS Ex = TnS - H

for the helium fluid

(7)

The internal energy U of the magnetic material is computed using U = ~T C dT

(8)

J0 where C is the specific heat of the magnetic material. Two different equations for calculating the specific heat under the magnetization and demagnetization states are employed. For magnetization, the following is employed

/"/)T~h ]helium

(ll)

For the exergy output, we should account not only for the exergy that the material bed and the helium fluid have at the end of the process, but also for the exergy loss due to the bed geometry 23. For the bed geometry employed in this analysis, the exergy loss is TnAS. The entropy generation AS is considered to be the sum of three entropy generation terms AS = AS~ + AS2 + AS3

(12)

where

AS~

dS C = T dT

-

-

Ntu + ~

~

Tw

(13)

(9)

while for demagnetization, Debye's theory is employed

AS2 -

VAP

(14)

Tw

3

C = __127r2R 5

(10)

5. The material bed plays the role of regenerator, thus being responsible for heat transfer with the cycled helium fluid. Four different geometries of magnetic material beds have been investigated by Barclay and Sarangi 23. In this analysis a geometry with a stack of perforated plates separated by small spacers normal to the fluid flow direction has been chosen, in part because of its relatively higher FOM. The heat transfer performance of the bed is calculated from the equations given by Barclay and Sarangi 23. 6. The operating efficiency of the auxiliary classical refrigerator is taken to be 80% of that of Carnot. Also, the heat transfer efficiency of the material bed and that of the offset strip plate-fin heat exchangers are assumed to be 80%. 7. Since simulation of continuous operation of the cycle should not be affected by the initial conditions, iteration should continue until the influence of the initial conditions is eliminated. 8. All three stages of the magnetic refrigerator are assumed to operate identically but covering different temperature ranges.

Based on the aforementioned assumptions, the following calculation procedure for a single stage of an AMR system is described. 1. The magnetic material bed is magnetized and is allowed to transfer heat to the cycled cool helium fluid to reduce the bed temperature. The exergy input is the exergy that the material bed and the helium fluid have

670

Cryogenics 1993 Vol 33, No 7

c~.4c (Tw AS3 - - - -

L

Lc) 2

TwTsc

(15)

These entropy generation terms are due to the finite nature of the heat transfer coefficient, viscous dissipation of energy and axial conduction along the bed, respectively. Therefore, the exergy output becomes

+ [m(TnS -/-/)r~ ] helium-- TnAS

(16)

2. Helium at Tw transforms into helium at T,c through the auxiliary gas-cycle refrigerator. This process deals only with the auxiliary refrigerator, so the exergy input can be expressed as Exi, = W + [ m ( T . S - H)T, ] he,urn

(17)

while the exergy output is written as Exout = [ m ( T , S - H)T,~]helium

(18)

3. The magnetic material bed is demagnetized and is allowed to release the cooling load to the cycled helium fluid. The exergy input of the material bed and the helium fluid is

Ex~.=[E

(U~- TnSi)e°ld]mat+[m(TnS--H)T'~]helium (19)

For the exergy output, the exergy loss due to the bed

Active magnetic regenerative hydrogen fiquefiers: L. Zhang et al. 80

geometry is taken into account. This yields

---

EXout=[E(Ui-- TnSi)h°tlmatq-[m(TnS-H)T~]taelium

I

stage 3 '

70-

stage 2 60-

g

stage 1

~, 5o-

- T, AS

(20)

4. The helium fluid leaving after heating the magnetic material bed drives the heat transfer mechanism with the hydrogen gas to be liquefied in the offset strip plate-fin heat exchanger. For the two-fluid heat exchanger, the exergy equations are as follows

O

Tsc

30-

=

{Exin]hcli ..... q- [EXin]hs, drogen

3K

c~_ = 1 Hz 1

2O 100

Exi,,

!~a~

40-

(21)

0

1'0 1'5 2'0 Volume of Material Bed (10-ax m 3)

2'5

30

(a)

and 1400 ¸ EX,,ut =

[El:,,m]heli

.... ~-

[Ex,,u,]h~a~og~.

(22) 1200

5. The correlation used in predicting the friction factors for the offset strip plate-fin heat exchanger is given by Webb 24

-i--

Tc - Tsc = 3 K

stage 3

/~/

frequency = 1 Hz r(i)/r(a) = 1

, 1000

/ / ~

stage 2 __..__ stage 1

/

800 tlJ

f =

0.44t

.

+ 1.328Ref °5

O

6. There are two driving systems for this liquefier, one is for driving the helium fluid shuttled between the heat exchanger and material bed, while the other is for driving the superconductor magnet back and forth relative to the magnetic material bed. For the power of the driving magnet, only practical experimental data are available. Regarding the power needed to drive the helium fluid, the pressure drop through the magnetic material bed (for both cooling and heating) and through the external offset strip plate-fin heat exchanger is employed. The friction factor correlations for the bed and heat exchangers were given by Barclay and Sarangi 23 and Webb 24, respectively. 7. The helium pressure is taken as 10 atm, while the hydrogen pressure is taken as 1 atm. Both of these values were suggested by Waynert et al. 18. The properties of helium and hydrogen are calculated by employing the code developed by McCarty 25.

Results

00

(24)

and discussion

There are several ways to increase liquid hydrogen production using magnetic refrigeration: 1, employ a large magnetic material bed size: 2, increase the mass flow of the cycled helium; 3, improve heat exchanger performance; and 4, apply a large magnetic field. The following are some observations about how the aforementioned factors affect both the cooling capacity and exergy losses. I. Since the heat released due to o r t h o - p a r a conversion at 60, 40 and 20 K are 229.9, 312.6 and 352.0 J g m respectively, the size of the magnetic material bed of the three stages should be different to satisfy the respective stage cooling load requirements. It is obvious that the larger the bed the larger the cooling capacity produced. Figures 3a and b show the magnetic material bed volume

400 2OO 0

1'0

1'5

;/_'0

2'5

~o

Volume of Material Bed (1Uex m ]

(b) Fi0ure 3 Material bed volume dependence o f (a) c o o l i n g c a p a c i t y o f t h e t h r e e s t a g e s a n d (b) o n e c y c l e e ~ e r g y l o s s o f t h e t h r e e stages

versus both the cooling capacity and exergy losses for the three stages. From Figure 3a we can see that the cooling capacity increases almost linearly with the bed volume for all three stages. However, the rates of increase of the cooling capacity are. different, with the smallest increase being associated with the lower temperature stage. Since increasing the material bed volume requires increasing the size of the superconductor magnet, the system's performance cannot be improved solely by increasing the bed size. Also, since the materials used in magnetic refrigerators are usually expensive rare metal compounds, increasing the material bed size is an expensive way to improve the performance. Since for the same material bed volumes the cooling capacities are different for all three stages, different volumes for the material bed should be used for proper operation. For the case of a 1 ton per day liquid hydrogen production rate, a volume ratio of 15:3:1 is computed. Figure 3b shows the effect of the bed volume on the exergy loss for one cycle. As can be seen, there is a point at which exergy loss is a maximum. When the bed volume is smaller than that corresponding to the maximum exergy loss, exergy losses are large. When the bed volume is large, exergy losses are reduced. 2. Thermal properties of the magnetic material can greatly affect the performance. This effect has been taken into account by the bed Curie temperature

Cryogenics 1993 Vol 33, No 7

671

Active magnetic regenerative hydrogen fiquefiers." L. Zhang et al. distribution of each stage. Three different Curie temperature distributions for each stage bed were assumed. Results pertaining to the cooling capacity and exergy loss for each cycle are listed in Table 1. As can be seen, the lower the Curie temperature, the larger the cooling capacity and the smaller the exergy loss. The Curie temperature should be higher than the magnetized temperature, thus limiting the minimum Curie temperature of any section by the section magnetized temperature and the magnetic field strength. The authors suggest that the Curie temperature should equal the magnetized temperature. For example, the first stage Curie temperature distribution is chosen as follows. From Figure 2 we know that the first section Curie temperature should be 20 + AT, where AT is the temperature difference necessary for transferring heat between the material bed and the helium fluid. For the remaining sections, the Curie temperatures may be obtained by linearly interpolating between the first and last sections. 3. The magnetic field is another important factor affecting the performance. Usually, the larger the magnetic field~ the larger the cooling capacity. However, in this case, the magnetic field has little effect on the cooling capacity primarily because the internal energy at a high field and low temperature is smaller than that at a low field and high temperature. 4. The cycled helium outlet temperatures Tw and Tc (see Figure 1) have a direct influence on the performance. Tw and T~ are determined by the material bed temperature and the helium mass flow rate. Generally, the lower the temperature To, the larger the cooling capacities produced. Figure 4 shows the difference between the helium output temperature and the stage cold temperature ( T o - T,c) versus both the cooling capacities and exergy losses for each cycle of the three stages. As seen in Figure 4a, the cooling capacity monotonically increases with the temperature difference. In Figure 4b the exergy losses do not change much with increasing temperature difference. Therefore, higher Tw and lower Tc values are normally recommended. However, Tw and T~ are dependent upon the bed heat transfer performance. Under certain circumstances, the needed cooling capacity cannot be achieved due to certain limitations on the heat transfer performance of the material bed (see discussion items 5

Table 1

4.5 4 stage 1

.

3.5 stage 2 3 s

t

a

g

~

_-

~" 2.5. o

2.

1.5 1'

Bed Volume : 5000, 980, 375 cm3 frequency = 1 Hz r(i)/r(a) = 1

0.5 i

115

:~

215

;3

Temperature Difference Between Helium Output Temperature and Stage Temperature

(a)

120stage 1

100

stage 2 -.-)K.--

80- stage

3

~, ~ 60. ~, 40.

Bed Volume : 5000, 980, 375 cm3

20.

frequency = 1 Hz r(i)/r(a) = 1

0

015

i

1'.5

2

2'.5

3

(b) Figure 4

Dependence on the difference between the helium output temperature and stage temperature of (a) cooling capacity of the three stages and (b) one cycle exergy loss of the three stages

Cooling capacities for different Curie temperature ranges Curie temperature range (K) First section

Last section

Cooling capacities, Qc (W)

One cycle exergy losses, Ex (J mol - 1 K - 1)

First stage

34.O 35.O 36.O

70.O 80.O 90.0

2.67 2.71 2.28

88.10 96.87 246.3

Second stage

59.0 60.0 61.0

95.0 105.O 11 5.0

3.78 3.53 3.28

58.1 2 91.70 76.22

Third stage

83.0 84.0 85.0

114.0 122.0 131.0

3.93 3.79 3.57

30.20 44.57 37.85

672

315

Temperature Difference Between Helium Output Temperature and Stage Temperature

Cryogenics 1993 Vol 33, No 7

Active magnetic regenerative hydrogen fiquefiers: L. Zhang and 6 below). Trying to remedy this problem by increasing the temperature difference leads to increasing the helium mass flow rate, which requires an increase in the helium inlet pressure and a consequent increase in the pumping power. 5. Since the magnetic material bed goes through four processes during one cycle (magnetization without flow, isofield cooling by the helium fluid, demagnetization without flow and isofield heating by the helium fluid), the duration of these processes should be selected such that the duration of the first two processes and the last two processes is the same. The ratio of the time duration for magnetization to that of the isofield cooling is treated as an independent variable. If the time for any of the two adiabatic processes is assumed to be ~'(a), while that for the two isofield processes is assumed to be ~-(i), these time functions can be adjusted such that the total cycle time T = 2[~-(a)+ T(i)]. Figure 5 demonstrates that longer isofield times help achieve higher cooling capacities. Unfortunately, the associated exergy losses are also larger. In other words, increasing the cooling capacity requires increasing the isofield process time, while increasing the cooling capacity and reducing exergy losses requires parametric optimization for every stage. 6. The effect of the operating frequency on the cooling capacity and exergy losses for each cycle can be seen in

2018-

4.5

f//" /

/J

/

16-

1412CL

0= 10L) ~

8-

0

6stage 2 stage 3

Operating Frequency (Hz) (a)

300-

Bed Volume : 5000, 980, 375 cm~ 250-

Te-Tsc = 3 K r(i)/~(a) = 1

/ 1 =

~,, 200-

'"

150

©

100

5

5-

/2_

Bed Volume : 5000, 980, 375 cm3

T0 T,c=3K

et al.

50

4 3.5 Operating Frequency (l-lz) ._~

3.

(b) Bed Volume 5000 980 375 cm s

~ 2.5. 0

Tc - Tsc = 3 K

g 2. 8 0

Figure 6 O p e r a t i n g f r e q u e n c y d e p e n d e n c e c a p a c i t y and (b) o n e c y c l e e x e r g y loss

frequency = 1 Hz

(a)

cooling

1.5 1

stage 2

0.5

stage 3

0

Ratio of Isofield Duration to Adiabatic Process Duration

(a) 500.

450400-

_9 °

of

35o300-

,

uJ 250og

2oo-

0

150-

/m

5 cm3

Tc-Tsc= 3 K . "

100-

tage 2

Figure 6. Obviously, higher operating frequencies result in larger cooling capacities and higher exergy losses. However, higher frequencies may result in higher frictional heating as well. Also, the frequency is restricted by the performance of the moving parts in the system. As stated by Sherif and Zhang 26 regarding the frequency effect, there is always an optimal operating frequency that takes into account all the heat losses. 7. The exergy losses per cycle are the sum of the exergy losses from the four processes. Table 2 lists the exergy loss distribution of these four processes for the three stages. As can be seen, the maximum exergy losses occur during the bed demagnetization and helium heating processes. This is primarily because the bed specific heat under zero magnetic field is relatively large. A material with a relatively small zero magnetic field specific heat could have given a better performance from the exergy loss standpoint.

--i--

500

o

+//

i

stage 3

~

3

~

~

Ratio of Isofield Process Duration to Adiabatic Process Duration

(b) Figure 5 D e p e n d e n c e on t h e r a t i o o f i s o f i e l d p r o c e s s t i m e to a d i a b a t i c p r o c e s s t i m e o f (a) c o o l i n g c a p a c i t y and (b) o n e c y c l e e x e r g y loss

The model developed here is also capable of providing the performance of auxiliary equipment. Obviously, better efficiencies and larger sizes of heat exchangers should increase the cooling capacity. The total power needed to drive the helium fluid in the system according to item 6 above is negligible compared to the cooling capacity.

C r y o g e n i c s 1 9 9 3 Vol 33, No 7

673

Active magnetic regenerative hydrogen fiquefiers: L. Zhang et al. Table 2

Exergy loss distribution for one cycle First stage Second stage Third stage

Bed magnetization and cooling by helium Hot helium heat release to refrigerator

10.25

0.0012

Bed demagnetization 70.13 and helium heating Heat removal from hydrogen by cold helium

8.73

11.39

0.0004

12.25

0.0002

40.81

14.89

4.55

3.07

Conclusions Based on the aforementioned discussions, one may conclude the following. 1. The cooling capacity linearly increases with the material bed volume. The shape of the material bed has no effect on the cooling capacity. However, the shape of the material bed has some effect on the exergy losses for a given cycle. For a given volume of the magnetic material, the shape should be one that maximizes heat transfer between the material bed and the helium fluid. 2. The material Curie temperature distribution of each stage should be similar to that of the stage bed magnetization. Also, materials having small specific heats in a zero magnetic field are more likely to improve the performance of the system. 3. For magnetic materials whose properties are similar to those used in these analyses, applying a high magnetic field is not necessarily advantageous. 4. The lower stage material bed heat transfer performance is very important in achieving higher cooling capacities. 5. To achieve lower helium outlet temperatures and larger cooling capacities, one may increase the helium mass flow, which requires higher helium inlet pressures. Another way is to increase the operating frequency. 6. Since increasing the duration of the isofield process results in increasing both the cooling capacity and exergy losses unevenly, optimized variables should be used for each stage with the objective of maximizing the cooling capacity and minimizing the exergy losses. Acknowledgements This work was supported by a grant from the US Department of Energy under Contract No. XL-9-18168-1. Support from the Clean Energy Research Institute and the Department of Mechanical Engineering at the University of Miami is also gratefully acknowledged. References 1 Schroeder, E., Green, G. and Chafe, J. Performance prediction of a magnetocaloric refrigerator using a finite element model Adv Cryog Eng (1990) 35 1149- 1155 2 Seyfert, P. Research on magnetic refrigeration at CEA Grenoble Adv Cryog Eng (1990) 35 1087-1096 3 Helvensteijn, B.P.M. and Kasbani, A. Conceptual design of a 0.1 W magnetic refrigerator for operation between 10 K and 2 K Adv Cryog Eng (1990) 35 1105 1123

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4 Savage, M.L., Kittel, P. and Roellig, T. Salt materials testing for a spacecraft adiabatic demagnetization refrigerator Adv Cryog Eng (1990) 35 1439-1446 5 Serlemitsos, A.T., Warner, B.A., Castles, S., Breon, S.R. et al. Adiabatic demagnetization refrigerator for space use Adv Cryog Eng (1990) 35 1431 - 1438 6 Smith, J.L., Jr, Iwasa, Y. and Cngswell, F.J. Material and cycle considerations for regenerative magnetic refrigeration Adv Cryog Eng (1990) 35 1157-1164 7 Nomazawa, T., Kimura, H., Sato, M., Maeda, H. et al. Analysis of a magnetic refrigerator operating temperature between 10 K and 1.4 K Proe Sixth Int Cryocoolers Conf Vol 2 (Eds Green, J. and Knox, M.) David Taylor Research Center, Bethesda, MD, USA (1990) 199 213 8 Li, R., Yoshida, O., Hashimoto, T., Kuriyama, T. et al. Measurement of ineffectiveness on regenerators packed with magnetic regenerator materials between 4 and 35 K Adv Cryog Eng (1990) 35 1183 1190 9 Kral, S.F. and Barclay, J.A. Magnetic refrigeration: a large cooling power cryogenic refrigeration technology Proc Cryo '90 Conf." Applications of Cryogenic Technology (Ed Kelley, J.P.) Plenum Press, New York, USA (1991) 27-41 10 DeGregoria, A.J., Barclay, J.A., Claybaker, P.J., Jaeger, S.R. et al. Preliminary design of a 100 W 1.8 K to 4.7 K regenerative magnetic refrigerator Adv Cryog Eng (1990) 35 1125 ll31 11 DeGregoria, A.J., Claybaker, P.J., Trueblood, J.R., Pax, R.A. et aL Initial test results of an active magnetic regenerative refrigerator Proc 4th Interagency Meeting on Cryocoolers (Eds Green, J. and Knox, M.) David Taylor Research Center, Bethesda, MD USA (1990) 277-290

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