Efficiency improvements of a double acting reciprocating magnetic refrigerator

Continuing our work with gadolinium gallium garnet (GGG) after the previously obtained results of 1.2 W at 1.8 K and 45% of the Carnot cycle, we now have accomplished new steps using half of the volume of GGG, a more suitable field profile, improved heat transfer and we have obtained 0.9 W at 2 1 K with efficiency of 64%. We have also computed the competitive threshold values for a magnetic stage refrigerator.

Efficiency improvements of a double acting reciprocating magnetic refrigerator A.F. Lacaze, R. Beranger, G. Bon Mardion, G. Claudet and A.A. Lacaze Key words: cryogenics,adiabatic demagnetization, superfluid helium, magnetic refrigeration, gadolinium gallium garnet, heat pump

Magnetic cooling and refrigeration efficiency Classical refrigeration by gas cycle becomes less efficient as the refrigerating power and the temperature decrease, a Magnetic cycle systems could a priori be more reliable than classical systems. These magnetic systems which have a high specific power within a certain temperature range should be able to replace the classical systems, especially if they can also be more efficient. We have therefore attempted to define the minimal value for the efficiency of magnetic cooling above which it could compete with cooling by helium gas.

Threshold of efficiency The coefficient of performance, COP, is usually defined as:

COP = Qc/W where Qe is the refrigerating power, and W is the power consumed by the refrigerator. Similarly, the figure of merit, FOM, relates real COP of the refrigerator to the ideal COP, ie to the coefficient of performance of a refrigerator functioning according to the Carnot cycle. For a classical refrigerator which works between 300 K and a cold temperature T¢, lg represents the total mechanical power and the FOM is:

the study carried out by Strobridge. 1 The author of this study collected a large number of efficiency values compared to the Camot cycle (FOMw), for machines of varying size, operating at varying temperatures, and showed that a single curve FOMw (Qc, To) could account for all these values. In this study, we only retained the points which correspond to recent machines, which means that our curve is slightly different from Strobridge's one as Fig. 1 shows: curve 1 from Strobridge and curve 2 proposed by the present authors. From curve 2 we plotted (Fig. 2) the specific work in terms of Qc in Watts, for some cold source temperature To, versus the capacity of the refrigerator, that is to say the ratio X(Tc) = w/a c = l/COP, WW-1 . Let us now imagine two refrigeration systems: one, a classical refrigerator operating between 300 K and Tc, absorbing power Qe at T e and consuming power WI ; and two, a two stage refrigerator, with a classic stage operating between 300 K and an intermediate temperature Tw, absorbing Qw power at Tw and consuming I¢2 power, plus a magnetic stage absorbing Qe power at T c and giving out Qw at Tw while consuming W m power. The magnetic refrigeration stage will be competitive in terms of efficiency if the power consumed (at 300 K) by the second system, W2 + Wm, is less than the power W~ 10 2

FOMw = (300/T e - 1) (Qc/W) For a refrigerator operating between Tw and T e we will be more concerned with the thermal power Qw released to the hot source at temperature Tw below ambient temperature, and the figure of merit will then be different:

FOMQ = (Qe/Qw) (Tw/Tc) The subscripts W and Q are introduced to distinguish clearly between the different definitions for the two FOMs, one being concerned with the total power consumed, the other with the thermal power. Tiffs difference will be justified later. In order to give the degree of efficiency for gas machines operating at different cold temperatures, we used

I

~io~

I-

IO'

!

o LL I0 c

,o-l,o_ ' , ,,, too

, ,,, , ,,t , ,,, , ,,, , ,,, , ,, Io' ,o2 ,o3 ~o4 io5 tos Capocity Oc, W

Fig. 1 Efficienclesfor heliumrefrigerators from Strobridge 1 - reference 1 ; 2 - only recent machines

study. 1

0011-2275/83/080427-06 $03.00 © 1983 Butterworth & Co (Publishers) Ltd. CRYOGENICS . AUGUST 1983

427

The condition then becomes

105

FOMQ >

Tw)/FOMw(Q,) + Tw)/

( ( 3 0 0 - Tc)/FOMw(Q¢) + To)

104

-

T

((300 -

~

T

c

By successive iteration, Qw and the threshold value of the FOMQ are calculated. Subsequently we will write FOM instead of FOMQ. Fig. 3 shows some threshold value curves for different temperature ranges in terms of Qc.

=

~ Io3

First experimental device

11

~10 2

I0

I

J

t tl I0

I

t

I

,I

I

I

I I]

I0 2

I

I

I0 B

I l

104

Capacity Oc, W Fig. 2 Specific work versus capacity and cold temperature of the refrigerator

consumed (at 300 K) by the first system, for the same useful power Qc absorbed at the cold source. From the curves in Fig. 2, we can determine the values x(Te)= Wl/Qc and x(Tw)= W2/Qw. We note that Qw may be much greater than Qc if the ratio Tw[Te is high enough. According to Strobridge, the higher the refrigerating power, the better the efficiency. It follows that x(Te) may be considerably greater than x(Tw). The mechanical power Wm absorbed by the magnetic refrigerator may be written Wm = Qw - Qc in the case of a perfect thermal cycle.

The machine used has already been described elsewhere. :'3 Its main part is a bar comprising two single crystals of gadolinium gallium garnet Gda Gas O12 or GGG (Fig. 4). The bar is moved vertically with a hydraulic jack controlled by a function generator. Two sintered-alumina bearings direct the bar and isolate the central chamber from the 4.2 K bath. Both magnetic elements are moved periodically in an unvarying magnetic field to perform a cycle consisting of magnetization up to a magnetic induction B at 4.2 K, and subsequent demagnetization down to 0 Tesla at 1.8 K. Inside the central He II bath, an electrical heater and an auxiliary liquid helium refrigerator allow us to determine the useful power at 1.8 K and the different losses as a function of the frequency. We can also determine the heat released to the warm source at 4.2 K by measuring the liquid helium consumption. The drive system can be used to transmit well-defined movements to the bar. A high speed movement ensuring quasi adiabatic process is followed by a low speed movement which allows good heat transfer. So the cycle performed in theory by the GGG would be a Carnot cycle (two isotropics and two isotherms). I00

90

The condition I#2 + Wm < IgI may be written: x(Tw) Qw + Qw

-

Qe

<

80

x(Te)Oc

7O

that is to say

Qc/Qw >

(1

+x(Tw))/(1 +x(T¢))

6O

For the magnetic stage we must then have:

1.8-4.5

50 -

FOMQ > (1 +x(Tw))/(1 +x(Tc) ) (Tw/Tc)

o

4O

For refrigeration at very low temperatures, 1 is negligible with regard to x(T), so we have FOMQ > x(Tw)/ x(Te) (Tw/Te), which amounts to neglecting the power Wm compared to W2 and Wx . In this case we have FOMw = FOMQ, this justifies the choice of FOMQ for magnetic refrigeration. In order to calculate the FOM thresholds, the curves in Fig. 2 can be used, or the values in Fig. 1 directly by using the following equation: x(Tw)

='

(300

-

3O

2O

I0

0

Tw)/(TwFOMw(Qw)) = W=/Qw

428

=

(300- Tc)/(TcFOMw(Qc))

=

WI/Qc

I I t [

I

t I II

IO

I

t I I

IOO

IOOO

Capacity Qc, W

Fig. 3

x(Te)

I

Competitive thresholdvaluefor a magneticstagerefrigerator

for different temperature ranges as function of its capacity

C RYOGEN ICS . AUGUST 1983

According to a simple analysis, thermal losses warm up the cold source and cool down the warm source, so: Qcu = Qcr - losses

Qwu = Qwr - losses In order to estimate these losses, we carried out measurements and analysis by computation. There are static losses occuring either by conduction in all the elements linking the cold source to the warm source in solid elements or in channels ftlled with superfluid helium, or by thermosiphon effect due to the difference between the densities of helium at 4.2 K and 1.8 K. Dynamic losses occur due to friction of the piston on the bearings and to the movement of warm helium towards the cold source and vice versa during piston movement. The lower part of Fig. 5 shows the curve of the total losses measured at 1.8 K and for a field of 2.5 Tesla. The total losses vary according to the magnetic field, since applying a magnetic field increases the losses by friction. Dynamic losses were measured using a bar with iron cores which replace the GGG elements and gives the same stresses. Fig. 5 also shows the curve of useful power in terms of frequency at 1.8 K and 2.5 Tesla as a positive value. Thus, we can work out the relative importance of the losses and of the useful power, and have direct value for the total power exchanged.

12

Fig. 4 Cross sectional view of experimental device. 1 - expansion valve, 2 - level gauge, 3 - copper well, 4 - superfluid helium bath, 5 - saturated helium bath, 6 - insulator, 7 - bearing, 8 - magnetic bar, 9 - superconducting coil, 10 - magnetic elements, 11 - corrective superconducting coil

1.2

1.0

C O P - FOM 0.8

For the ideal refrigerator or Camot refrigerator: Qci/Qwi

=

Tc/Tw ,~ O.6

m

and COP i = 0.426

o.4

The real refrigerator has a COP r equal to Qer/Qwr, Qcr and Qwr being the thermal power exchanged by the magnetic elements with the two sources.

Q2

Similarly:

I

0

F O M = C O P / C O P i = COP/0.426

We will note F O M r for the real cycle. From the experimental values we define the useful coefficient of performance COP u and the useful figure of merit F O M u :

m

-0.2

-/

/

/

/0.2

0.4

0.6

0.8

1.0

Frequency, Hz

~ -O.4

3 COPu

=

Qcu/Qwu

-0.6

F O M u = C O P u / C O Pi

The useful power Qeu is the electrical power released to the cold source. Similarly, Qwu is the thermal power computed from liquid helium consumption.

CRYOGEN ICS . A U G U S T 1983

Fig. 5 Useful power and losses at 2.5 Tesla and 1.8 K. For 0.9 Hz losses are 0.72 W, useful power is 1.18 W and total cooling power is 1.9 W

429

Heat transfer at the heat sources The thermal conductivity of GGG is very high - about 0.3 W cm-1 IC 1 at 1.8 K4 - and the temperature homogeneity is always quite good or acceptable inside the block of GGG (2.4 cm in diameter) for all the frequencies involved here. The thermal power absorbed by the magnetic elements from the cold source remains proportional to the frequency until the limitations imposed by heat transfer at the sources are reached. At the cold source, heat transfer is limited by the Kapitza resistance. If we assume for heat transfer the Value computed with the theoretical phonon radiation limit, that is to say a thermal power proportional to the 4th power of the temperature, the maximum cooling power would be about 2 W with our two magnetic elements at 0 K in the helium bath at 1.8 K. At the warm source, the limitation comes mainly from the transition between nucleate and Film boiling convection. In the condition of nucleate boiling peak flux the magnetic element would transfer a maximal thermal power of 30 W to the warm source. This occurs when the temperature of the GGG, at the end of the magnetization with high speed movement (ie when the element enters the bath at 4.2 K), is slightly lower than 5.2 K. In this case any power limitation of the refrigerator would come only from heat transfer at the cold source. On the other hand, if the temperature of GGG is slightly greater than 5.2 K, heat is transferred with film boiling convection; the maximum thermal power decreases sharply from 30 W to about 3 W. In this condition, the limitation would come mainly from the warm source. The experimental curves of useful power versus frequency (Fig. 6) show the different stages of limitation

1.2

Experimenl'al Computed

~

a

Bearing i

k\\\\\\\~ I~'\\\\\\\~1 Bearing

I0

r~ \ \ \ \ \ \ \ ~ I k\\\\\\\\~l

i

Z, cm

-2 o

g 0

4

/ b

Fig. 7

,

~\\\\\\\~ k\\\\\\\\l

Z, cm

Magneticfield profile

of heat exchange. At low frequencies no limitation occurs and the graphs are straight lines. At higher frequencies the graphs become curved; then the limitation comes from the cold source due to the Kapitza resistance and possibly due to internal exchanges. If the maximum magnetic field is sufficiently high, a break appears in the curve: this is the transition from nucleate to f'flm boiling, and thus the limitation comes from the hot source. Real cycle

2.6T

The magnetic induction was produced by two main coils facing each other and two small compensating coils also facing each other. The magnetic induction prof'fle was such that at 1.8 K the whole bath was in zero field (Fig. 7a). At the end of the high speed movement during the demagnetization process, the paramagnetic element was therefore in a zero field region. So in the lower temperature part of the cycle, the GGG was then subjected to an isofield increase of temperature. Fig. 8 is an entropy diagram for GGG for various magnetic fields: a Camot cycle ABCD is superimposed on four refrigeration cycles a, b, c and d. The four refrigeration cycles are computed for two maximum magnetic inductions and two frequencies. These computations take into account heat transfer in all parts of the cycle.

1.0 2.7 T

0.8

o3 (3. D

2 0.6

3T 0.4

Useful power versus frequency curves 3.5T 0.2

%

I IIIII

I

I

I

0.2

0.4

0.6

Frequency, Hz Fig. 6

430

Useful power at 1.8 K

0.8

1.0

The computation which enabled the cycles to be drawn up also allowed the calculation of the total power exchanged at the sources. The loss values obtained experimentally were included in the computation and the calculated useful power was deduced. Fig. 6 shows: the measured curves of useful power (continuous line); and the power curves calculated for 2.5 and 3 Tesla fields (dotted line). There is a good agreement between computed and experimental curves concerning the powers exchanged with

CRYOGENICS . AUGUST 1983

B=OT f

6-

-

/

t

D/

/

c

COPu = 0.19

F O M u = 45%

COPr -- 0.27

F O M r = 64%

IT

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A

Influence

o

[/ C/I

g

/

/

d

I /

//'B /

/

w o-3T

0.9Hz

b-2.ST

0.9Hz

c - 2.5T d-

5T

I I

0.15 Hz O.15Hz

I 2

I 5

I 4

I 5

I 6

I 7

Temperature, K Fig. 8

profile

This study shows that whatever the movement, heat exchange occurred in a constant field, zero in this case, at the cold source. Efficiency can thus be increased by carrying out demagnetization during the exchange at the cold source. In order to do this, the field profile must be modified. Ideally the exchanges at the sources should be isothermal. Knowing the entropy diagram, the desired cycle and the heat exchangers at the sources, we can accurately calculate the exact magnetic field to be created in every respect. Indeed let us consider any point in the cycle at temperature T and field H. We wish to move from field H to field H + dH by isothermal transformation. In order to do this, a quantity of heat dQ must be exchanged. From the known temperatures of the GGG and of the helium, and the laws governing the exchange, we deduce the time necessary for the elementary transformation. By repeating this computation throughout the chosen cycle, we obtain a function H(t) relating magnetic field to time. Given the piston movement z(t), we can then deduce the static profile of the field H(z).

j //-~ ~-4

of field

Computed cycles on entropy diagram for Gd3Gas012

the sources and in particular for the break in the curve at 3 Tesla due to the transition to f'tim boiling convection. Using the cycles traced under different conditions we can check the qualitative interpretation of the shapes of the power curves, and can give a few characteristic working points: At 2.5 Tesla (cycles b and c of Fig. 8), at high frequency the GGG no longer has sufficient time to balance the temperature with the sources. The temperature at the end of magnetization is above 4.2 K and at the end of demagnetization is below 1.8 K. At 3 Tesla (cycles a and d of Fig. 8) even at low frequency the temperature of the GGG rises above 6 K during magnetization, which greatly increases the irreversibility. At low frequencies we clearly observe the increase in the entropy variation corresponding to the increase in magnetic field. But as soon as the frequency rises, film boiling causes a spectacular flattening of the cycle. The entropy variation at 0.9 I-Iz is only one third of its 0.15 I-Iz value.

New experimental device The practical construction presents a certain number of supplementary problems, and the tuning of magnetic field and movement can only be imperfectly achieved. However we were able to calculate and then to produce for our experimental prototype a series of coils which gave a satisfactory field. However we were only able to achieve this profile by reducing the height of the magnetic elements by half. This magnetic field profile is shown in Fig. 7b. It should enable the realization of the computed cycle shown in Fig. 9. It may be noted that this cycle is very close to a rectangle in spite of the field imperfection.

Q:

~4 :>,

Experimental performance

LU

3

This has already been published elsewhere, s We recall that for the best operating point achieved at 0.9 Hz and 2.5 Tesla, the following values were obtained: Qcu = 1.18w Losses = 0.72 W 0

Qwu = 6.25 W

I

2

3

Temperature, K

So:

C R Y O G E N I C S . A U G U S T 1983

Fig. 9

New computed cycle

431

With half the amount of GGG, we reduce by half the exchange surface. Moreover it appeared necessary to double the entropy variation so as to maintain the same useful power, and this was carried out. The gain on the quantity of heat exchanged at the cold source compared to the cycle in Fig. 8 corresponds to a factor greater than two, since the temperature variation is smaller during the exchange. This enables one to work at lower frequency and thus decreases losses.

Experimental results after modification The results as a whole have been published elsewhere. 6 The essential elements are summarized in Table 1

Table 1. Essentialelements of experimental results after modification Round Frequency, Hz T¢, K Peu, mW Losses, mW Pwu, mW COPu FOMu, % FOMr, %

1 0.244 2.1 562 230 1735 0.324 64.8 78.0

2 0.303 2.087 642 260 2055 0.312 62.9 78.5

3 0.363 2.1 925 280 2885 0.321 64.1 76.1

The useful power available at 1.8 K is less than anticipated. This is probably due to a misappreciation of the real value of Kapitza resistance. In order to determine experimental efficiency values, we could only use a cold temperature of 2.1 K. Three characteristic points are noted in the Table above: Point of best useful figure of merit: F O M u = 64.8%

FOMr = 78.5%

Point of best power: Qcu = 925 mW with F O M u = 64%

Conclusion We have shown that a high magnetic refrigerator efficiency can be achieved using GGG. We then showed that it is possible to obtain very high efficiency levels by adequation between the magnetic field profile and the piston movement. Figures of merit of 64% were obtained with a useful power of 0.9 W at 2.1 K with a quantity of GGG half time that in the first device. This figure, 64% is much greater than the value corresponding to the threshold beyond which magnetic refrigeration is competitive compared to gas cycle refrigeration. We hope to further improve these performances by increasing exchanges at the cold source, and to measure the real heat transfer condition between GGG and superfluid helium.

Authors A.A. Lacaze is at CRTBT-CNRS - BP 166 - Grenoble, all other authors are at the CEA-DRFC/SBT-CENG 85 X 38041 Cedex - Grenoble, France. Paper received 10 March 1983.

References 1 2 3

4 5 6

Point of best real figure of merit:

432

Strobtidge,T.R. NBS Tech Note 655 (1974) Delpuech, C., Beranger, R., Bon Mardion, G., Claudet, G., Lacaze, A.A. Cryogenics 21 (1981) 579 Lacaze, A.F., Beranger, R., Bon Mardion, G., Claudet, G., Delpuech, C., Laeaze,A.A., Verdier, J. Proc CEC81 San Diego, paper HB6 Daudin,B., Sake, B. C.R.Ac.Sc. Paris 293 (1981) 885 Lacaze, A.F., Beranger, R., Bon Mardion, G., Laeaze, A.A. Proe ICEC 9 Kobe (1982) Lacaze,A.F. Th~se 1982, Institut National Polytechnique de Grenoble

CRYOGENICS. AUGUST 1983