FEASIBILITY ANALYSIS OF RECIPROCATING 

HAGNElXC HKCLT PUNPS 

Semiannual Status Report 

for the Period 7/3/85 to 1/2/86 

A. V. Larson 

J. G. Hartley 

S. V. Shelton 

H. H. Smith 

NASA Mearch Grant NAG 3-600 

Georgia Institute of Technology 

Atlanta, Georgia qn-9- 

JV-)-)L 
Semiannual Report 

Feasibility Analysis of Reciprocating Magnetic Heat Pumps 

Obi ective: 

The goal of the project is to study the commercial feasibility of magnetic 

heat pumps and refrigerators operating near room temperature. An analytical 

Phase Zero study is being to quantify the expected performance and 

economics of commercial magnetic thermodynamic devices and to recommend 

whether a Phase One program should be started to include experimental R and D. 

conducted 

Background: 

Until recently, interest in magnetic heat pumps and refrigerators was 

limited to temperatures under 20K. In 1976, Brown [ 1,2] at NASA/Lewis 

suggested the possibility of practical magnetic devices operating near room 

temperature. Noting that all heating/cooling effects in magnetic materials 

are greatest near the Curie point, he considered available materials, e.g. 

gadolinium, with Curie points near room temperature. Several thermodynamic 

cycles were considered. From those, the magnetic Stirling cycle with 

regeneration was selected for further study. A reciprocating, porous 

gadolinium core in a column of regenerative fluid inside the bore of a 

superconducting magnet was envisioned. A proof-of-concept laboratory device 

was successfully demonstrated. 

Status : 

The conceptual design selected for detailed system analysis and 

optimization is the reciprocating gadolinium core in a regenerative fluid 

ine ir'nermo6ynamic 

properties of gadolinium are giver? by Griffel, et al. [ 31 , Brow. [4], axd 

Benford and Brown [5]. The general thermodynamic relations for magnetic 

materials are given by Hatsopoulos and Keenan [6] and Booker [7]. The 

relation between the applied fields (no magnetic material present) and the 

internal fields in the gadolinium in place is taken to be that of an ellipsoid 

of gadolinium [8,9]. Initially, it is assumed that the gadolinium does not 

perturb the sources of the applied field. The forces are calculated on the 

basis of the approximation of a small dipole in a field gradient [8]. 

m. -*l..-- -.-.LL.:- LL- .I----- -c WALlIAIL Lut: UULC VI a iiiipersOduCtitig magriei. 

A computerized literature search for relevant papers has been conducted 

and is being analyzed. Contact has been made with suppliers of 

superconducting magnets and accessories, magnetic materials, and various types 

of hardware. model for the thermal analysis of the core 

and regenerator fluids is included in the following section. 

A description of the 

Thermal Analysis of Regenerator Coluum: 

The thermal analysis of the regenerator fluid and gadolinium core is 

presently based upon a one-dimensional, transient model. This model will be 

used to identify important design performance parameters and to identify 

areas which require further investigation. 

and 

-2- 
The assumptions listed for the model in the previous status report [lo] 

have been altered: 

1. Viscous forces are neglected. 

2. Thermal equilibrium has been assumed between the gadolinium and the 

fluid at any cross section in the regenerator column. A onetemperature model follows in which the heat transfer coefficient is 

not required. 

To facilitate the computation, the coordinate systems have been changed. 

Now two coordinate system are used. The first (x,y) is fixed to one end of the 

regenerator column. The second (x , y ) is fixed at the magnet center. 

The energy equation based on the assumption of local thermal equilibrium 

is : 

where 

- 

Pf 

C constant pressure specific heat of the fluid 

porosity 

intensity of the magnetic field 

density of the Gd (fluid) 

entropy of Gd 

temperature 

time 

velocity of the plug relative to the regenerator column 

specific volume of Gd 

column fixed coordinate along the regenerator axis. 

-3- 
is the total change in the field intensity observed at the magnetic material. 

The first term appears because the field appears to be time varying relative 

to the column due to the column motion. This term can be referenced to the 

magnet fixed coordinates which removes the time varying component. Then 

] = [ 'abs] >t * [ a(vii) + v a(vii) 

ax X,Y ax" XI Y 

at 

where 'abs = 'col -k vplug/col 

If E = 1 we are outside the Gd and the equation is 

aT aLT 

at 

Pf 'pf - = kf - 2 

ax 

As expected, the energy equation reduces to conduction in the fluid. 

Boundary Conditions: 

1. Initially adiabatic at the regenerator column ends: 

At the plug ends: 

- 

plug - Tfluid 2. T 

3. aT 

[ks (1-E) + kf&1 I = kf ax Ifluid Plug 

These conditions represent the continuity of temperature and heat flux. 

The energy equation was implemented on the computer using a finite difference 

scheme. Non-dimensionalizing the equation was attempted. This proves to be 

very'difficult due to non-constant coefficients and the lack of global 

geometric scales that can be for references. A dimensional approach was 

adopted for the present. 

used 

Problems in implementation: 

(1) Node Types. 

The nodes are picked to be fixed relative to the regenerator column. As 

the plug moves along the column the character of the nodes changes. Also, 

the plug-fluid boundary is in general not at a node location. The 

numerical scheme must recognize different node types and use various 

schemes to calculate new temperatures. This leads to bookkeeping problems 

with the nodes. 

-4- 
(2) Stability and Convergence. 

This is the usual problem with the ratio of step sizes. The space 

increment At is computed to keep the coefficients in the numerical scheme 

positive. The increment At is also checked so that the plug advances by 

fractions of a space step. 

aT The convective terms (those with - 

wind difference to improve stabid&. As V, the relative velocity between 

column and plug changes sign, the - terms change relative to the upwind 

directions. Higher order difference was also tried but did not improve 

stability. 

(3) Convective Terms. 

had to be treated as one-sided upax 

Results: 

Some initial predictions from the model are given in Figures 1-3. 

The regenerator column has a length of lm, the gadolinium plug, 0.2m. The 

fluid is half water, half methyl alcohol by volume. The porosity of the 

gadolinium (open area/section area) is 80%. The cylinder and ends of the 

regenerator are adiabatic. The vacuum field of the superconducting magnet is 

assumed constant in time and given by American Magnetics for their 8 Tesla 

unit operating at 6T maximum. 

At the start of the first the fluid and gadolinium are uniform in 

temperature at 295K. The end of the regenerator nearest to the magnet is lm 

from magnet center and the gadolinium is at the near end. After a half-cycle, 

the gadolinium and magnet are concentric, the gadolinium is at the other 

end of the regenerator. 

Figure 1 shows the temperature profile within the regenerator after 14.5 

and 15 cycles. The cycle period is 37 seconds with no pauses and with 

velocities given by appropriate step functions. During this computer run, the 

minimum separation between the gadolinium and regenerator end was 5cm at each 

end. The column position, 0.00, marks the near end of the column at the start 

of a cycle. The drop in temperature at the other end between the two profiles 

by repositioning the column to complete the cycle). At present the model 

includes thermal conduction in the column, but excludes mixing. The gradients 

at the column ends reveal that axial conduction has a minor effect. 

Figure 2 shows the maximum and maximum temperatures in the column after 

each half-cycle. The nearly isothermal sections again reveal the minor effect 

of axial conduction. The curvatures suggest that asymptotes will be 

approached, but the AT is already close to the value found in the experiment 

by Brown and Pappel. 

Figure 3 gives the results of a run in which there is no dead space 

between the gadolinium and the ends of the regenerator. The temperatures at 

the column ends are now changing with each cycle. 

cycle, 

and 

chnw_c C-C~~ZO 0 nffnrrt ------ cf r=~eyi~g the g&clini*z from magnet (fslloued 

current Plans: 

Next we plan to incorporate mixing into the model and also the transfer of 

energy between load and sink. - 

-5- 
1. G. V. Brown, "Magnetic Heat Pumping Near Room Temperature," J. Appl. 

Phys., 47, 3673-3680, (1976). 

2. G. V. Brown, "Basic Principles and Possible Configurations of Magnetic 

Heat Pumps," ASHRAE Transactions, 87, pt. 2, 1981. 

3. M. Griffel, et al., "The Heat Capacity of Gadolinium from 15 to 355"K, 

Phys. Rev., 93, 657-661, (1954). 

4. G. V. Brown, "Magnetic Stirling Cycles--A New Application for Magnetic 

Materials," IEEE Trans. Magnetics, MAG-13, No. S., 1146-1148, Sept. 1977. 

5. S. M. Benford and G. V. Brown, "T-S Diagram for Gadolinium near the Curie 

Temperature," J. Appl. Phys., 52, 2110-2112, March 1981. 

6. G. N. Hatsopoulos and J. H. Keenan, Principles of General Thermodynamics, 

Wiley, New York, 1965, 

7. H. G. Booker, Energy in Electromagnetism, the Institute of Electrical 

Engineers, London and New York, Peter Peregrinus Ltd., New York, 1982. 

8. P. Moon, Magneto-Solid Mechanics, Wiley, New York, 1984. 

9. C. W. Chen, Magnetism and Metallurgy of Soft Magnetic Materials, North 

Holland, Amsterdam and New York, 1977. 

10. A. V. Larson, J. G. Hartley, S. V. Shelton, Feasibility Analysis of 

Reciprocating Magnetic Heat Pumps, Semiannual Status Report, NASA Grant 

NAG 3-600, Georgia Tech, July 1985. 

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