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AERONAUTICS. AND SPACE ...... of variables, the energy equation becomes, after letting. Tf = Tm,. [ {IOS. + .... of a run in which there is no dead space between.
/x,' =3

FINAL

c,'C._

REPORT

FEASIBILITY

ANALYSIS

OF RECIPROCATING

MAGNETIC

HEAT

PUMPS

By A. V. Larson --

--J.

G. Hartley Sam V. Shelton M. M. Smith

-

--

Prepared for NATIONAL AERONAUTICS LEWIS

RESEARCH

AND

SPACE

ADMINISTRATION

CENTER A

CLEVELAND,

OHIO

44135

Under NASA

Grant

NAG-3-N)0 T

December

1989 2

GEORGIA

INSTITUTE

OF TECHNOLOGY

A Unit of the University System THE GEORGE W. WOODRUFF ATLANTA,

GEORGIA

of Georgia SCHOOL

OF MECHANICAL

ENGINEERING

30332-0405 N'_O - I S 3(,:,

-_

_ 5

(NASA-CR-186Z05) FFASI_IL ' _iY .... _.I y I'< ,l;. RECIPROCATING MAGNETIC _L.,%T PU_PS I-in_l Robert (georrzi-_ Tl_st. of Tech.) 59 p 20D G31_4 "

ii

I

Unclas u253177

J

FEASIBILITY

ANALYSIS

OF RECIPROCATING

MAGNETIC

HEAT

PUMPS

By A. V. Larson,

Co-P.I.

J. G. Hartley,

Co-P.I.

Sam V. Shelton, M. M. Smith

w

Co-P.l.

r

FINAL

REPORT

for the period

July

1985 to July

Prepared for NATIONAL AERONAUTICS LEWIS m

RESEARCH

CLEVELAND,

1986

AND

SPACE

ADMINISTRATION

CENTER

OHIO

44135

Under NASA

Grant

December

GEORGIA

NAG-3-600

1989

INSTITUTE

OF TECHNOLOGY

A Unit of the University System THE GEORGE W. WOODRUFF ATLANTA, w

w

GEORGIA

30332-0405

of Georgia SCHOOL

OF MECHANICAL

ENGINEERING

ABSTRACT

w

A reciprocating warm

Gadolinium

bore of a superconducting

refrigeration presented magnet

to

frequency

the

(1000

amount

Estimated

reciprocation

analysis

exhibits

of

in a regeneration

solenoidal

MW

of

approaches

Officer

costs

A cable

for

an

to conventional

10 Hertz. cycling

at

grant

in the

for magnetic procedure needed

ideal

in

chillers

Hertz

is Gerald

V.

is the

magnetic as the

A one-dimensional 0.027

seen in the experiments

for this

column

is considered

superconducting

comparable

a regenerator

fluid

applications.

capital

some of the features

The NASA Technical RESEARCH CENTER.

magnet

ton)

system

of this type become

of

difference

3.517

minimize

design.

refrigerator

which

in

core

finite

is presented

of G.V. Brown.

Brown,

NASA

LEWIS

ACKNOWLEDGEMENTS

The on

the

authors

wish

mechanical

manuscript; drawings.

and

to thank

system;

students,

our colleague,

Betty Steven

Holtsinger Corson

J.W. for

Brazell, typing

and Michael

for discussions and

proofing

L. Robertson

for

the the

TABLE OF CONTENTS

ABSTRACT ACKNOWLEDGEMENTS NOMENCLATURE I

II.

,

INTRODUCTION BACKGROUND Magnetic

............................. ..............................

Cooling

Reciprocating

III.

Devices

Magnetic

Pumps ...................

ESTIMATES

.........................

7

Cost Model

.........................

7

Logic

Pumps

4

ECONOMIC

.....................

6

.......................

12

........................

12

Minimum

Radius

Ratio, e

Optimum

Magnet

Dimensions .......................

14

Input Data Case

I

..........................

14

Input Data Case

II ..........................

16

Results

...............................

Operating

Costs

REGENERATOR Results

Column Column

DISCUSSION/CONCLUSION

FIGURES APPENDIX

.......

20

...................

21

...............................

B. Isothermal

REFERENCES

17

...........................

COLUMN

A. Adiabatic

VI.

2

Magnetic

Cost Minimization

Vo

.......................

Heat

Heat

2

Rotating

Capital

IV.

1

Ends, Ends,

Initial

29

Thermal

Initial

Internal

........................

............................. ............................... ..............................

Equilibrium Thermal

...... Gradients

29 30

32 34 39 51

NOMENCLATURE

r

A

Column

B

Magnetic

induction

Bo

Magnetic

induction

Bw

Magnet

C

Cost of magnet

Ca

Cost of auxiliary

Cm

Specific

cost of magnetic

metal

Cp

Specific

heat at constant

pressure

Cs

Specific

cost of superconducting

Cst

Cost of magnet

Ct

Total

CI

Cost coefficient

CV

Control

F

Form factor

f

Frequency

H

Magnetic

Hz

Hertz

H*

Enthalpy

cross-section;

induction

Ampere

at magnet

center

in central

plane

at wall

and magnetic

metal

core

of core

.

£_

equipment

structural

cable

support

cost of system defined

by eq.

(16)

E_

w

defined

by eq. (21)

of demagnetization intensity

(cycles/sec)

m

I

Volume

(cycle frequency)

E

h

Specific

enthalpy

J

Globally

averaged

Jsa

Characteristic

K

Unit

k

Thermal

L

Length

M

Magnetization

Mm

Mass

of core magnetic

Ms

Mass

of superconducting

N

Unit

of force,

Ri

Innerradius

of magnet

winding

Ro

Outer radius

of magnet

winding

P

Cost term defined

Pmi n

P at emi n

QL

Heat transfer

to column

QH

Heat transfer

from column

Refrigeration

load rate

current

current

of temperature,

density

density

in magnet

of superconducting

Kelvin

conductivity of magnet

winding

v

L_

V

w

metal cable

Newton

by Eq. (19)

_in

Thermal

input power

_out

Thermal

output

power

winding

end from

source

end to sink

alloy

Specific

refrigeration

S

Specific

entropy

T

Temperature;

t

Time

U

Internal

V

Speed of core relative

Vb

Volume

Vabs

Speed of core relative

V

Speed

qL

col

capacity

Unit of magnetic

of magnetic

field,

energy

of magnet

of column

to column

bore to magnet

relative

Speed of fluid

V

Volume

of superconducting

cable

Volume

of superconducting

alloy

Volume

of magnet

including

V

in core,

to magnet

Vf

S

Tesla

relative

to column

in cable

sa

V w

windings

V

Spec ifi c volume

W

Unit of

X

Coordinate

relative

to magnet

X

Coordinate

relative

to column

power,

Watt

spacing

metal

Greek

w

Symbol s Ratio of outer to inner radii

Ratio of length

V

to inner diameter

winding

for magnet

winding

_min

Value

of _ at emi n

6

Fraction

of winding

volume

filled

by superconducting

alloy

6s

Fraction

of winding

volume

filled

by superconducting

cable

6sa

Fraction

of cable cross-section

(

Porosity

defined

Y

Fraction

of core filled

P

Density

Pm

Density

of magnetic

Ps

Density

of superconducting

#o

Magnetic

H

high

L

low

f

fluid

m

magnetic

min

minimum

metal

filled

by superconducting

by Af/A

permeability

Subscripts

m

for magnet

by magnetic

metal

metal

(non-porous cable

of vacuum

form)

alloy

I.

INTRODUCTION

r_

Some

materials

become

hotter

(cooler)

when

magnetized

(demagnetized).

E

Refrigerators

and heat pumps

based

on the effect

In practice,

adiabatic

demagnetization

few

Kelvin.

This

can be imagined

has been

important

in principle.

in cooling

below

a

w

degrees

application The

report

of a magnetocaloric

motivation

flexibility

is

than

to are

find

considers

refrigerator

devices

found

in

of

the

feasibility

operating

greater

conventional

near

capacity,

of

room

commercial

temperature.

economy

technology

to

or

meet

design

particular

applications. In 1976,

Brown

magnetocaloric earths

such

magnets

__k w

magnetic concept

gadolinium

effect

is

and less

suggested

temperatures. the

advent

power

stronger

the possibility

The bulk availability of

higher

consumption

near

the

of practical

was

Curie

field

superconducting

intriguing

point

of rare-

(Gd

because

293K)

and

the with

field changes.

Brown

w

at normal

considerably

magnetocaloric larger

devices as

with

[1,2] at NASA/Lewis

discussed Stirling

laboratory

feasibility

several cycle

with

device

was

was thought

of the regenerator.

possible

thermodynamic

regeneration successfully

to rest on economics

These

are the factors -

I

-

for

cycles

further

demonstrated

and

selected

study. [3].

A

of

The commercial

and the thermodynamic addressed

proof

the

in this paper.

performance

Comparisons conventional gadolinium

are 1000

core

made

ton

here

on

chillers

and

the a

in a fluid regenerating

Progress

is

regenerator.

This

and irreversible

system

and

using

operating

a

costs

reciprocating

of

porous

column.

reported

on

modelling

is to eventually

processes,

capital

modelling

but hasn't

the

done

gadolinium

take

into

-

account

fluid rate

column

dependent

so yet.

v

II.

BACKGROUND

r_

W

Magnetic

Cooling

Devices

The magnetocaloric 1918.

Prior

refrigerators

to

this

effect Edison

and engines

Temperatures

was

based

first

[5]

and

observed Tesla

by Weiss

[6]

had

and

Piccard

patented

on the ferromagnetic-paramagnetic

down to I K can be obtained

to liquefy

He.

[4] in

designs

for

transition.

Debye

[7] in 1926

w

and Giauque

[8] in 1927

be produced

by the adiabatic

method

was

Adiabatic w

independently

successfully demagnetization

suggested

demagnetization tested has

in been

research.

-2

1933 used

that

lower

temperatures

of a paramagnetic by

Giauque

since

then

and in

substance. MacDougall

could The [9].

low-temperature

v

In

"one-shot"

contact

w

with

devices

a low

a

temperature

low

temperature.

An applied

the

paramagnetic

material

established field

the

is

paramagnetic

thermal

lowered

to

substance

reservoir

of

magnetic into

the

contact

to

zero.

The

placed

He and a material

field

causes

Once

reservoir

experimental

energy

broken

material

and

thermal

be studied

thermal

is

in

to

thermal

reservoir. the

is

to

flow

at from

equilibrium

is

and

the

magnetic

the

paramagnetic

L

substance

will

then

drop

Refrigerators

have

temperatures

below

to

a temperature

been

I

K

in

using

built

for

below

using

loads

that

of

paramagnetic

less

than

the

reservoir.

substances

i

mW [10,

to

11].

maintain There

is

w

substantial V

to

20

[13,

interest

K)

in

14,

superconducting

15].

There

similar

These

are

devices

devices

devices

would

basically

two

and need

to

to

to

maintain

cool

temperatures

instruments

handle

competing

low

loads

designs

in

greater in

(I

space

than

current

craft

I W. magnetic

r

refrigeration

w

magnetic

research material

moving

other

design

uses

fluid

acting

as

referred rotating

to

applications industrial

as

and

at

temperatures

with

a rotating the

the

link

a reciprocating wheel

between

reciprocating

reciprocating in

waste

space,

above

laser

magnetic

the

source

and

rotating

heat recovery

amplifier

K.

heat

in

material and

with

designs, pumps

cooling,

involves

a fluid

sink.

[13, 14, 15, 16, 17]. - 3

One

motion

of

magnetic

i

have helium

a

porous

column.

The

a counterflowing

The

two

designs

respectively. been

proposed

liquefication

are Both for and

Reciprocating

The

Magnetic

reciprocating

magnetic

material,

A typical In

cycle

the

Pumps

magnetic

heat

a fluid-filled

mechanical

material

cooled.

The

increase

in

core.

In the

is completed

for

Here,

process

temperature

be vertical

I.

cycle,

in the

cycle

pump

consists

regenerator

is shown in Fig.

magnetic

likely

Heat

by In

proper

fluid

the magnet I-2

part

control.

magnet.

of

the

2-3 the material 3-4

necessary

At

porous

magnetization

process

the

the

is on continuously.

demagnetization

practice,

of

and the external

includes

isofield

core

4-I.

column

in

steady

and an

isofield

translation

operating

is

would

conditions

_m

the fluid

in the regenerator

and has an overall

Fig. I the left to the magnet and off with

high-field

column

temperature

end is hot.

is not always the

magnetic

superconducting

is stratified

difference

The motion required.

with

respect

of TH - T L.

In the column

of the magnetic Instead,

material

inside

the

magnets

may favor

material

the magnet bore.

to temperature

with

could

However,

shown

respect

be turned the

the use of relative

in

nature

motion

on of

and a

v

constant The

field. details

of

the actual

ends

of the regenerator

column

and

these,

the

variation is

along

with

in the magnetic

achieved.

Fig.

2

energy

are not shown.

details

field,

shows

addition

of

the

and rejection

Various relative

will determine

a cycle - 4

consisting

what of

processes

methods motion

could and

at

be used,

the

type thermodynamic two

isothermal

the

spatial cycle and

two

isofield

paths

regeneration then

the

Carnot

which

is the representation

paths 2-3 and 4-I were

ideal

cycle

device.

the partial

congruent

would

have

the

(The dashed

path

could

magnetization

of

same

the cycle

considered

as suggested

coefficient

be achieved

of the core during

by the dashed

of

which

If

line,

performance

in practice

warming,

here.

as

the

by programming

is not considered

initially.) The reciprocating

magnetic

heat pump was first

proposed

by J. R. Van Geuns

m

in 1966 m

[18].

More

by G. V. Brown

recently

[i, 2, 191.

this device Brown

has been discussed

and Papell

reciprocating

magnetic

device

maximum

employed

was 7 T producing

field

80 K.

In separate

K

328

and

K.

refrigerator

fluid, issuing

but

Two

factors

depends the

from

the

on

the

gradient core

The

in

would

the

test

fluid

the

device mixing

of

a

a temperature was

region

a small

span

of

by

The

of

about

were an

241

actual

reciprocating

gradient

degraded

in the

papers

sink).

attained

performance

operation of

or

temperature

temperatures

limit

successful

and tested

(no source

a maximum

maintenance

causing

walls

and highest

which

noted.

[3] have built

adiabatic

tests the lowest

were

refrigerator

with

in several

jets

behind

in of

it.

the

fluid Also,

L_

more

surface

area was needed

Two other

reciprocating

to enhance

magnetic

heat transfer.

refrigerators

have been

tested.

Barclay

v

et

al.

[20]

temperatures noted:

built of

and

tested

2.2 K and 4.2

(i) frictional

heating

a

device

K, respectively. (mechanical 5 -

= w

which

operated These

contact),

at

source

limiting

(2) viscous

and

sink

factors

were

heating,

and

(3) mixing tested

owing

a double

device

acting

one-half

watt

Rotating

Magnetic

The

was tested

Heat

as

magnetic

rotates

through

magnetic

refrigerator The cold

and thermal

between

et al.

in

section

energy

[21]

1981.

This

is located

is rejected

from

1.8 K and 4.2 K and produced

at

each

nearly

capacity.

refrigerator in

Fig.

material.

3.

Fluid

high-field

A prototype

magnetic

C. Delpuech

Pumps

illustrated

porous

core.

and magnets.

column

of refrigeration

rotating

exchanger

cores

of the regenerator

The refrigerator

of the porous

reciprocating

has two paramagnetic

the middle end.

to the motion

is The

is

rim

pumped

and low-field

to test the rotary

arranged of

as

the

through

a

counterflow

wheel the

is

heat

composed

porous

rim

of

as

it

regions.

magnetic

heat pump

principle

was designed,

T

built was

and tested

in 1977

a forerunner

[23].

Also,

of

The

two

concentrated

heat

refrigeration

This

device

temperature

magnetic

operated

device

refrigerator

that

at room was

operating

temperature

reported between

on

about

and

in

1981

2 K and

[24].

main

problems

field

at

and controlling expected

a room

a rotating

4 K has been tested

[22].

one

with

rotary

location

on the

the flow of the fluid. transfer capacity

between of

400

a

wheel

These

the fluid W,

designs

obtaining

and a zero

problems,

and magnetic

Coefficient 6 -

are

of

field

along

a

high

elsewhere,

with

lower

material,

resulted

Performance

(COP)

than in a of

26

r

percent

of the Carnot

COP and a maximum

of 1000 W, 70 percent

and 40 K, respectively

that

might

the flow problem

would

be driven

A Different

through

Magnetic

the previous

w

T.

be alleviated

device

[26].

material

connected

operating

device

somewhat

the porous material

using

two categories,

Hashimoto

[23].

Barclay

goals

[25] has suggested

by using

by magnetic

to design

a ferrofluid

which

forces.

Heat Pump

One other current

L_

AT of 7 K as compared

is described

They to

magnetic

envision

one-way

a

heat

materials,

that does

in a 1984 patent refrigerator

pipes.

No

not fall

into

by H. Nakagome

and

composed published

of

a

accounts

magnetic of

an

have been located.

m

III.

Capital To

magnet

ECONOMIC

ESTIMATES

Cost Model obtain

economic

estimates

and a reciprocating

assumptions

were

the

magnetic

basic

porous

system

of

Fig.

core was chosen.

I with

Some

a steady

simplifying

made.

I)

The porous

2)

The

core just fills

the bore of the magnet

windings.

w

vacuum

field

of

the magnet

is

uniform

in



equal to that calculated

for the magnet -

7

-

center.

the

bore

with

a value

w

f

v

3)

Any

metallic

field

in

the

ellipsoid. 4)

The

The

6)

is

vacuum

The

same

voids

spatially

of magnetic manner

as

Demagnetization

magnet

adequate 5)

piece

averaged

the

demagnetization Eddy currents

8)

The

the

center

stacking

channels magnet

uniform

refrigeration

7)

by

for cooling at

responds

to

the

element

uniform of

a

applied

long

thin

coils

with

is negligible.

assembled

field

core

is

disk-shaped

and structural

center

current

rate

thin

members.

is calculated

using

a global

density.

proportional

to

the

frequency

of

of the core. are ignored.

coefficient

of

performance

is

the

maximum

COP:

There

are

no

irreversibilities.

The

logic that

of magnetic the

follows

material

superconducting

optimized uniform

to

provide

starts with

to satisfy cable the

is

a load

specification.

the load requirement chosen,

necessary

the

shape

field

at

of

Then

is found. the

minimum

Finally

magnet cable

the

after

solenoid

weight

is

(for

winding).

The refrigeration

rate

is (_L -- qL Mm f

where

amount

qL is the refrigeration

capacity

(1)

per unit mass

per each demagnetization,

Mm is the mass of the core magnetic -8-

metal,

and

of core magnetic

metal

a

7 V

f J

is the frequency

Therefore

of demagnetization.

the required

volume

of the magnetic

material

is

_L V

(2)

= m

where

Pm

is the density

for the porosity

qL f Pm

of the magnetic

metal

in non-porous

form.

of the core,

(3)

Vm = _ Vb where

Vb

is the volume

of the magnet

is the filling

The magnet where

Allowing

is illustrated

fraction

in more

bore,

and

of the magnetic

detail

metal

in the core.

in Fig. 4

Ro

is the outer radius

of the winding,

Ri

is the inner radius

of the winding,

L

is the length

J

is the globally

of the winding, averaged

current

density,

and

Bo = Bw = B is assumed. Defining R 0

(4)

R. l

L

(5)

# " 1

it follows

that Vb = 27r R3'] /9. -9

(6)

V

The windings

and their spacing

occupy

a volume,

given

Vw

by

(7)

Vw : (_2 _ I) V b.

The superconducting

cable has a volume

Vs

(B)

V s = 6 s Vw where

6s is the fraction

of the winding

volume

filled

with

superconducting

cable. A

typical

superconductor

superconducting necessary cable

alloy

to specify

cross-section

embedded

the cable which

cable

in

consists For

copper.

the

tiny

filaments

purposes

here,

Let 6sa be the fraction

in more detail.

is superconducting

of

alloy,

Vsa is the volume

of the superconducting

Defining

6

then

Vsa = 6Vw. Now

the

global

superconducting

average

alloy characteristic J

A complication

arises

density current

(10) (11) J

can

density,

be

related

to

sa (B)

qL = qL (B' TL) 10 -

the

known

Jsa-

= 6 Jsa

. J

of the

alloy.

in that qL and Jsa each depend

Jsa

is

(9)

_ 6sa 6s

current

it

then

Vsa = 6sa Vs where

of

(12)

on the field.

(13)

(14)

Thus

one must

equation

choose

(2),

equations

then

B, then

proceed

find the volume

to

find

the

of magnetic

optimum

material,

winding

shape

Vm, through

starting

with

(13) and (12).

Since Vm and Vsa (expected may seek to estimate

to be high cost variables)

the system cost

as a function

are coupled

by B one

of B.

Let C t be the total cost of the system

(15)

C t : Cm Mm + CsM s + Cst + Ca

.y

where

Cm

is the specific

cost of the magnetic

Cs

is the specific

cost of the superconducting

Cst

is the cost of the magnet

Ca

is the cost of the auxiliary

structure,

metal, cable,

and

equipment.

Assume Cs t + CsMs L

C1CsM s.

(16)

Define C _ CmPmVm + ClCs#sVs where

Pm

is

the

density

Ps

is the density

of

the

(17)

magnetic

material

in

non-porous

form, and

Rewrite

equation

(17) using

equations

of the superconducting

cable.

(3), (7) and (8):

C - CmPmVm

(18)

[I + P]

where IC1CsPsSsl p:[

Cm--_m?)(2_

-

L

11 -

(19) i).

L

L_

Cost Minimization

Recall

Logic

equation

(2).

_L V

=

--

m

The

application

Choose

the

Choose

B which

the

cable:

Choose

the

filling

B also

fixes

field

can

equations Cs,

B by minimizing

CI

to

Minimum

minimize

Radius

Following

CI

is

e for

Ratio,

#s,

" Jsa(B) the

that

magnetic

#m.

and

?,

6 s.

(18).

6sa-



cost

C of

(:2 not

metal,

TL).

(2)

fractions:

minimize

Assuming seeks

Jsa

and the

qL = qL (B,

in

Choose

Now one

f,

fixes

fixed

qL#m

QL, TL.

frequency,

Now Vm is

r

fixes

f

1)

the in

a sensitive

a given

magnet

equations

and magnetic (19)

function

and

of

material

for

given

(18).

the

design

variables,

one

B.

e

Reference

[28] for this type magnet B = J Ri F 12 -

we have

(20)

where

F is a form factor F = F (e,#)

1/2 -

°

(21)

where

#o is the magnetic

In equations relate

only

to

(20) and (21), the

shape

shape which minimizes The

latter

can

permeability

be

of

B and J have

the magnet:

e subject

recast

of vacuum. been

Ro,

R i, L.

to the constraints

in an

informative

fixed.

The parameters

One

seeks

of equations

manner.

Combine

to

e, #

find

the

(20) and

(21).

equations

(3),

(6), (12), (20) and (21) to get:

(22) F3 or, using equation

(2)

(23)

Since the parameters

on the right #-- =

have been fixed, constant.

(24)

F3

From equation One

typical

(21), #/F 3 is a function

can easily

curve

find

emi n subject

for e vs # is shown

of # associated

with

to

only

the constraint

in Fig. 5.

emin13 -

of e and #. in equation

Now let #min

stand

(24).

A

for the value

Optimum Magnet

Combine

Dimensions

equations

(3) and (6) after

inserting

the values

for emi n, #min to

get

-_

R

IvllJ 3

=

m 2_ _/ #min

(25)

R° = emi n Ri.

(26)

i

From equation

The "build"

(4)

of the winding

defined

as Ro - Ri, is

Ro - Ri " Ri (emin

From equation

E--

the inner and outer

Input Data Case

diameters

Ri.

(28)

are

Di = 2 Ri

(2g)

DO - 2 Ro

(3O)

I

For the parameters

of equation

(23),

Load Requirements:

QL " 3517 kw (1000 Tons of Refrigeration) f

(27)

(5), L = 2 #min

Of course

I).

= IHz

- 14

Geometric

Design: - .8

6 Magnetic

- .1125 (6 s = .g, 6sa - .125) Metal:

Gadolinium

Pm

= 7.9 x 103 Kg/m 3

Cm

= $200/Kg

qL

=

[.589B

- .0817 B3/2] KJ/Kg

B in Tesla, Superconducting

Ref.

m

[29].

Cable:

I part superconducting

alloy,

Ps

= 8.6 x 103 Kg/m 3

Cs

= $66/Kg,

Superconducting

Ref.

as given

= Jsa(B)

Jsa

= (5-90) 1010

- Equations

below.

(.773) B Amp/m2, 2.5T _ B _ 12T.

(18) and (19):

CmP m = 1.58 x 106 $/m 3 CsP s -

copper

Alloy:

Jsa

Formula

7 parts

[30].

for B in the range Cost

at T L = 280K

.568 x 106 $/m 3.

_ 15 _

Ref

[31].

Equation (18) becomes C = $(I.58E6) [I + Pmin] VM,

Where

Pmin is given by setting

Equation

e = emi n in equation

(31)

(19).

(19) becomes P - (.404)CI

In the above,

the cost

(emin 2

I).

(32)

of the Gd is: (33)

$(I.58E6)VM The cost of the superconducting

cable

is:

P (34) $(I.58E6)VM

The cost of the superconducting

magnet

---_1 •

is:

(35)

$(I.58E6)V M P.

Input Data Case

II

The only change

frequency

made for the second

by a factor

of 10.

- 16

evaluation

was to increase

the

Results The

cable

results

of

are given

magnet

in Tables

and the auxiliary

The magnet appropriate

cost

data.

Resonance

Imaging

length

those

the

the calculations

as

need

for

homogeneity hand,

the

withstand

cost

equipment

There

in Tables

of

large

a large

commercial

over

not so critical will

forces

uncertainty

Gadolinium

and the

for the costs

a

the

of the

same

of the lack of

for bore

are strongly

relatively

in magnetic

the

because

magnets

MRI costs

escalate as

bore

of about

The

homogeneity

the latter

estimates

warm

applications

I and 2.

internal

with

of the

[30].

have

are

and other

is probably

the costs

I and 2, along

estimates

field

for

large

heat

due to the Gadolinium

is

Magnetic

diameter

and

escalated

volume.

pumps. need

the

by

Field

On the other

of the magnet

withdrawn

from

to the

magnet. For the cost of the magnet,

Cs + Cst, we have

Cs + Cst - $(14B

This

m_

cost

shape

and

costs

of

system, quantity from

formula

size a rack

both

fits

similar of

used

production

high quantity

fairly

to those

electronics for the and

do

well

,

of the

include

manufacturing.

- 17

non-MRI

! and 2.

and a closed-loop

the

used the expression:

B in Tesla.

to marketed

of Tables

operation not

+ 19)k

simply

The

liquid

magnet.

The

significant

magnets

$19k helium costs

with

bore

represents

the

refrigeration reflect

reductions

small

expected

o

.

Table

1.

Capital

Cost

Estimates

Vs.

Applied

Field,

f

= 1 Hz.

3_TT

6T

9T

.3315

.1908

.1438

_/F 3, A6/N 3

7.036E25

4.970E23

1.093E22

emi n

1.00265

1.01384

1.04972

_min

1.4

1.4

1.45

Vm, m3

R i, cm

36.12

Ro-R i, cm L, cm Cost,

Gd $

Cost,

Cable,

.0957 101.1 524k $

Cost of Magnet,

s

Cost of Auxiliaries Cost of System,

Table

$

2.

Capital

12__!T

3.819E20 1.15458 1.5 25.24

30.04

27.02

.416 84.1

1.112 78.4

301k

.1212

3.90 75.7 192k

227k

1.1k

3.4k

9.4k

25.8k

61k

I03k

145k

187k

30k

30k

30k

30k

615k

434k

402k

409k

Cost

Estimates

vs. Applied

Field,

f - 10 Hz

3T

6T

9T

12T

.03315

.01908

.01438

.01212

_/F 3, A6/N 3

7.036E24

4.970E22

1.093E21

3.819E19

emi n

1.00572

1.02991

1.10808

_min

1.4

1.45

1.5

Vm, m3

R i, cm Ro-Ri, L, cm Cost,

cm Gd, $

Cost, Cable,

$

Cost of Magnet,

Cost of Auxiliaries, Cost of System,

$

$

1.65 ]1.34

16.76

13.77

12.39

.0959 46.9

.419" 39.9

1.339 37.2

37.4

52.4k

30.Ik

22.7k

19.2k

2.1k

6.2k

.2k $

1.3425

.7k

3.89

61k

103k

145k

187k

30k

30k

30k

30k

143k

163k

198k

236k

- 18-

L

Separately

listed

to drive

the motion

magnetic

refrigerator

and the work

operates

of 377 kN with level

vertical,

the

estimated

level

Even

the

length a peak

would

be

about

weight

of

the

37.7

with

the

steel

5 cm

(an

effect

internal 3-4

rod of

say

end

of the

each each

would

neglected

latching

(Fig.

I),

needed

to

mechanical drivers.

must

it may

be

diameter

use only

between

forces

core). a small

I and

the core

the

be

COP is 9.33,

force

would

be on

At f = 10 Hz, the

actual

added,

ideal

be 377 kJ of work

high.

the

power

possible

motion

which

would

be

increases

the

2).

would

If

f -

to

have

i) can

actuators. suffice

the rods

percentage

by

with

non-magnetic

as the

major

had

traverse

to

of the volume

However,

the major

A

be met

using

driver

the

available

some

walls

in paths

drivers

attached

kind

of

1-2 and to

the

column.

column

costs

valves

relative

requirements

The initial

(B = 3T,

and the regenerator

set of piston/cylinders, move

Since

If the

ideal

the average

twice as

electromagnetic

metal

in Tables

ends of the regenerator Another

about kN.

greatest

stainless

column,

I m, then

Gadolinium

and

fluid

the

required

about 8%.

case

on

and 310K,

system

core.

MW load at f - I Hz would

force

piston/cylinders

rod

280K

were

hydraulic

(one

is t_e mechanical

and the magnetic

between

for a 35.17

If the stroke

the order

equipment

of the regenerator

required

per cycle.

average

as auxiliary

to

here

estimated

the

are

and rods core much

(the minor

in paths smaller

for the mechanical

- 19 -

2-3 than

system

drivers) and

for

4-I. the

are [39]:

are The

major

$ 6k

two cylinders

4k

two pumps

6k

valves

4k

controller

10k TOTAL

Operating

An of

ideal

thermodynamic

irreversibilities

increase

cost

with

A serious

power

efficiency much

better

motors,

cycle

due

and other

advantage

shaft

$ 30k

Cost

Gadolinium

do

miscellaneous

to

causes

cycle

has been

core/fluid

to cylinder is about

80%

efficiencies

and valves

assessed

and will

of higher frequency

offset

observed

power.

[39].

system

higher

look_ very encouraging

- 20 -

the

[39].

The

some

in

the

irreversibilites

extent

the

in the conversion

systems

existing

pressures

currents

effects

capital

I and 2.

In conventional

However,

by using

to

quantitative eddy

yet.

in Tables

in the mechanical

rod

The

interactions,

have not been

frequency

loss occurs

assumed.

trend

and very

this

of motor

conversion

in hydraulics efficient

to

pumps,

IV.

The

conceptual

reciprocating

a dissertation treat

the

transient

design

core

superconducting

selected

for

in a regenerative

magnet

regenerator

detailed

fluid

as in Fig. I.

in progress

column

The analysis

by one of the authors

fluid

and

core

using

system

within

the bore

is the subject, [32].

a

analysis

The first

simplified

is

the

of a steady in part,

of

task was to

one-dimensional,

model.

The assumptions I.

REGENERATOR COLUMN

The core

in the model is assumed

are:

to have a porous

structure

composed

of gadolinium

w

having 2.

a uniform

Temperature of motion

gradients

equilibrium

3.

Viscous

4.

Fluid properties

5.

The magnetic

6.

in the

are negligible.

in thermal

steady

porosity.

forces

magnetic

forces

are independent intensity,

of the gadolinium

field

normal

to the direction

and gadolinium

in the

The gadolinium

H, is a known

function

of position

to be the vacuum

field

of the magnet.

is a known

and the fluid - 21 -

m

and

of temperature.

function

of temperature

intensity. is rigid

are

are ignored.

_=

7.

core

in any cross-section.

and inertial

field

and core

The fluid

in time, and is taken

The entropy

fluid

is incompressible.

and

8.

The velocity

of the core with respect to the regenerator column is

constant during core traversals 9.

There is no dwell time. motion relative

10.

Any effects

Either the core or column (or both) is in

to the magnet at all

of magnetically

The thermodynamic properties Brown [2],

The relation

and the internal ellipsoid

times.

induced eddy currents are negligible. of gadolinium are given by Griffel

and Benford and Brown [34].

for magnetic materials [36].

in the column.

The general thermodynamic relations

are given by Hatsopoulos and Keenan [35] and Booker

between the applied fields

fields

(no magnetic material

present)

in the gadolinium in place is taken to be that of an

of gadolinium [37,38] with no demagnetization effect.

To write the energy balance, fix

a reference frame (x) to the left

the column in Fig. I and assumethe Gd core is moving to the right V.

[33],

For a differential

auI

_-_

end of

with speed

control volume located at x in this frame

: Q + W + Net Enthalpy

Input

(36)

Rate

CV

au

I cv

*j aH = _

ah I m cv " PmAm Ax _ + #fAfax

ahf (37)

at

I'Tml 'Ti]m ,A,f I'T

aTf

(_=kA m

m

ax

x + Ax

ax

x + Ax

x

dM : Am r% Ax H--dt

- 22 -

ax

(38)

(39)

Net Enthalpy where

Input Rate - -

U is the internal

hx + Ax

hxlm PmAmV"

lhx +Ax"

hxlf

energy,

t is the time,

is the thermal

power,

is the rate of magnetic

work,

H* is the enthalpy,

h is the specific

enthalpy,

m, f are subscripts

for magnetic

metal,

fluid,

# is the density,

Am, f is the cross-section k is the thermal

area of metal

or fluid,

conductivity,

T is the temperature,

Po is the magnetic H is the magnetic

permeability

of free space,

intensity,

M is the magnetization, V is the metal speed,

relative

to the column,

Vf is the fluid speed

in the core,

u

23 -

relative

and

to the column.

#fAfVf

(40)

In this model the two magnetic vectors are colinear. Combining the equations, dividing

by AAx, and taking the limit,

one

obtains

Pm (I-()

E°hm 1 dt

/_oVm

dT

+ pfCpf

E°Tf l

a2Tm k (I-() _ m ax 2

where

is the porosity

Cp is the constant and use has been made

volume defined

(1-()V

a2Tf (41)

+ kf E

A is the column cross-section

vm is the specific

E a-t-

ax 2

area,

of the solid, by E _ Af/A,

pressure

specific

heat,

of the relations: dh = C dT P

d dt

a = a_

- 24 -

a + V a--_

(42)

It is convenient to change variables of Gd are available

because the thermodynamic properties

[33,34] in the form of the entropy function s - s(T,P,H)

at atmospheric pressure.

From Ref. [35], Tds = dh - _ovHdM-vdP

(43)

so that for the metal in a constant pressure process,

ds P = I dt dhm T _-_

_oVmHdM

(44)

L

which

is the factor to be transformed.

From calculus,

at constant

pressure,

_si.T[a.] dT [as I d_vH_ _4s_

T_-_

P

_

By this transformation

p,vH

dE

of variables,

+ T

_

the energy

p,T

equation

dt

becomes,

after

letting

Tf = Tm,

[

OS

Pm (l-E)

+

#mT

{I

T

+

a-T p, vH

_

#fE Cpf

#fCpf

Ot

8--_

p,vH

[

#m (I-E)

T

{osl a(vH)

8tS(vH) + V

8--x---a(vH) ]] = [kf E + km(1-E)]

p,T

The factor

a(vH) St

+V

a(vH) ax

- 25

a2Tax 2

is the total The first

change in the field

intensity

term appears because the field

the column due to the column motion.

observed at the magnetic material.

appears to be time varying relative

to

This term can be referenced to the magnet

fixed coordinate (X) which removes the time varying component. Then

a--t-_

a--x--

--

aX

(47)

Vbs

=

where

Vab s = Vco I + V.

If E = I, there

Vco I is the speed

of the column

is no Gd and the equation

relative

to the magnet.

is

aT

a2T (48)

#f as expected,

then the energy

The boundary

conditions

I.

At the regenerator

Initially

2.

Cpf_-_

change

-

kf--ax 2

is due only to conduction

in the fluid.

are: column

adibatic,

ends:

aT --:0; 8x

then

later

isothermal,

T = fixed.

At the Gd core ends:

Tcore " Tflui d

w

o

aT I

[km(1-( ) + kfE] _-_ ore -

These conditions

represent

kf

(49)

aT Ifluid _

the continuity

- 26 -

of temperature

and heat flux.

Initially,

the core and regenerator column are in thermal equilibrium

the temperature of the environment. mechanical cycle of Fig. I will

The column ends are adiabatic

cause a temperature gradient

regenerator column, the left

end being the hotter.

not a cycle

each mechanical cycle,

differs. thermal

because after

However, eventually conduction

limits

so that the

to develop in the

The thermodynamic path is the temperature

a thermodynamic cycle should result

in the fluid

at

profile

as the axial

the maximumtemperature difference

between the column ends. After the column ends have reached temperatures suitable

for refrigeration,

the column ends are to be put into appropriate thermal contact with the source and sink of the refrigeration calculations walls

included thermal reservoirs

at the column ends.

conduction in the fluid Solution finite

scheme. As a step in that direction, in contact with perfectly

Heat transfer

with

diathermal

the reservoirs

occurs via

and Gd.

of the energy equation was implemented on a computer using a

difference

scheme.

Non-dimensionalizing

This proved to be very difficult of global

the first

geometric

scales that

the equation was attempted.

due to non-constant coefficients can be used for

references.

and the lack Therefore a

dimensional approach was used initially. Someimplementation difficulties,

discussed more completely later 27 -

[32], are

categorized

(I)

as:

Node Types. The

finite

difference

regenerator the nodes node

column.

are

As the core

changes.

location.

nodes

Also,

picked

moves

to

along

the core-fluid

The numerical

scheme

be

relative

the column

boundary

must

fixed

is

recognize

to

the

the character

in general different

of

not at a node

types

w

and

use

various

bookkeeping (2)

Stability This

schemes

problems

to

with

calculate

new

temperatures.

This

leads

to

the nodes.

and Convergence.

is

the

usual

problem

with

the

ratio

of

step

sizes.

The

space

w

increment

is chosen

arbitrarily

keep the coefficients is

also

fraction (3)

checked

difference

and column

tions.

that

the

core

increment

scheme

positive.

advances

through

At

is computed

The

the

increment

column

by

a

to

At set

Terms.

The convective

core

in the numerical

time

of a space step.

Convective

upwind

so

and the

A higher

terms

[those with _-_ aT]

to improve

changes

sign,

stability.

had to be replaced As V, the relative

aT the _-_ terms

order difference

change

was also tried

28-

relative

with

a one-sided

velocity

between

to the upwind

but did not improve

direc-

stability.

Results

Adiabatic

Ao

Some

Column

initial

component

water,

The

time

and

operating

at

At

is

length

by

of by

0.80.

vacuum

given

a

the

model

Im,

are

of

American

of the

cylinder

the

given that

core,

volume.

The

field

Equilibrium

independently

alcohol

area)

adiabatic.

from

chosen

has

methyl

area/section

in

was

column half

Initial Thermal

predictions

sizing

regenerator

Ends,

in

The

porosity

and

ends

(see

Figs.

Section

O.2m.

superconducting

Magnetics

in

The of

of

Appendix)

the

III.

The is

half

core

(open

regenerator

is

for

The

fluid

the

magnet

6-8.

assumed

their

are

constant

8 Tesla

unit

6T maximum.

the

start

of

temperature

at

from

magnet

center

and

magnet

are

the

295K.

first

The

end

and the

core

cycle, of

concentric,

the

the

fluid

and

regenerator

is

near

and

the

that

nearest

end.

core

gadolinium

After

is

are

to

the

uniform

magnet

a half-cycle,

near

the

other

end

the

magnet

is

is

O.Im/s

orzero.

in

is

Im

the

core

of

the

w

regenerator. Also, period step

the

The speed

is

37

the

core

seconds

6 shows

15 cycles.

During

and regenerator near

of

with

the

column

relative

relative no

to

pauses

the

and

to magnet

with

velocities

given

O.Im/s

by

or

zero.

The

cycle

appropriate

functions. Fig.

in

of

speed

end of

the

temperature

this

computer

end was 5cm at the

column

profile

at

the

run,

each start

end. of - 29

within the

the

minimum The column

a cycle.

regenerator separation position,

The drop

in

after

14.5

and

between

the

0.00,

marks

the

at

the

temperature

core

other

end

between

gadolinium

from

the mechanical column,

w

axial

but

the

7 shows The

At present mixing.

effect

by repositioning includes

gradients

and minimum

isothermal

close

cooling

at

the

removing

column

thermal

the

of

to

complete

conduction

column

ends

the

in the

reveal

that

effect. temperatures

sections

The curvatures

but TH-T L is already

the

the model

The

the maximum nearly

shows

(followed

has a minor

conduction.

Pappel

profiles

magnet

excludes

half-cycle. axial

two

cycle).

conduction Fig.

the

suggest

to the value

again

that

in the column

reveal

the minor

asymptotes

found

will

be

in the experiment

at

each

effect

of

approached,

by

Brown

and

[3].

Fig. 8 gives the gadolinium

the results

of a run in which

and the ends of the regenerator.

ends are now changing

by a larger

B.

Ends,

Isothermal

The

Column

column

is

amount with

Initial

operated

Internal

there

is no dead

space

The temperatures

between

at the column

each cycle.

Thermal

adiabatically

Gradients

as

before

until

the

highest

w

temperature column

in the

column

below

280K.

goes

diathermally

n

to

the

by holding

core

to travel

transfers

thermal

energy

above

Then

hot

computationally is assumed

goes

and

and

reservoirs cold

the end the entire

directly

ends,

nodes

to

the at

lowest

310K

the

temperature

and

280K

respectively.

of the column

column

- 30 W

310K

length

at 310K

so at various

reservoirs.

The

are

in

coupled

This

is

and

280K.

times

heat

the

done The

the core

transfer

is

calculated increment

from

the temperature

of time.

This

gradient

is summed

into the cold end and the energy

cycle

is achieved,

and conductivity

over

a mechanical

rejected

the difference

at each

cycle

at the hot end.

in these

two quantities

end for each

to get

Once

the energy

a thermodynamic

equals

the work

into

the system. Figure cycle

versus

0.50. the

9

QH

shows

the

the number

is the heat

source.

exchange

of successive

transfer

It appears

achievement

energy

that

of thermodynamic

with

the

mechanical

at the sink asymptotic

reservoirs

cycles.

The

per mechanical

values

may

per

porosity

cycle,

occur,

mechanical

thus

is now

and QL

is at

signalling

the

cycling.

w

In Fig.

10, the values of

QL QH - QL

are plotted does

versus

not represent

been obtained.

asymptotic

the number the work

However

approach

of successive involved

mechanical

because

cycles.

a thermodynamic

The denominator cycle

has not yet

the data of Figs. 9 and 10 are not inconsistent

to

a COP

in the vicinity

of

the

ideal

COP

of

with

9.33

an

for

a

M

Carnot

refrigerator

with

reservoirs

at 310K and 280K.

L

Such of

a limit will

the Gd execute

shown

in Fig.

mechanical

not be reached

different

11 for three

in this model

thermodynamic

sections

(left

cycle. - 31 -

i

paths.

because

different

The thermodynamic

end, middle,

right

end)

sections paths

are

for the 30th

The differ same

program

by less criteria

different

core

is stopped

arbitrarily

than

I% from their

for

program

respective

cut-off

porosities.

when

was

The results

both

QH and

values

used

in

are given

QL

in the

in the

(N-1)th

calculations

Nth cycle

cycle.

starting

The with

in Fig. 12, where

QL QH - QL

is plotted

versus

core

porosity.

Any

since Figs. 9 and 10 suggest

a closer

The

may

result

Carnot

at

low

porosity

limit

of

9.33,

Clarification

of

this

because

not a

conclusions approach

be

in

be

drawn

to the asymptote

error

thermodynamic

should

even

cycle

though has

cautiously

may be needed. it

exceeds

not been

the

achieved.

w

two-temperature

was

not

Large conventional an

electric

Performance

of

because

research

has

been

started

on

a

model.

V.

$150k,

pursued

DISCUSSION/CONCLUSION

chillers operating

5.86

and

an

(1000 ton : 3.517 use

ideal

near

COP

600

of

kW,

10 when

MW) have an

a capital

actual

operating

cost near

Coefficient between

280K

of and

308K. The

estimated

system

capital

costs

for

ideal

magnetic

refrigeration

w

systems -approaches

of

the

same

capacity

10Hz, as shown

in Table

become 2. - 32

comparable

as

the

cycle

frequency

A

significant

regenerator increase

due

with

to

driven material w

Neither

the

cost

been

and considerably been

costs

(Gd in this have

made

relative

to

is

that

motion

Such

of

the the

irreversibilities fluid

irreversibilites

and

the

in

porous

have not yet been

the metal

treated

of the regenerator.

capital by

the

trend

cycle frequency.

in our modelling The

counter

for

and

magnetic

refrigeration

temperature-entropy

report)

cost

Magnetic

materials

are available

incorporate

cost

system

characteristics

and by the structure/assembly

optimized. less

the

of

costs

of slightly

which

the

to

magnetic

performance

No attempt

accompany

be

of the magnet.

less

and may be suitable.

reductions

appear

high

has

quantity

manufacturing. Future estimate

effort

toward

may not be warranted

reducing until

have been completed.

The appropriate

case,

program

an experimental

would

the

uncertainties

estimates

the

of the COP of actual

literature be necessary.

33-

in

to do so seems

magnet

cost

regenerators

scarce.

In any

Vl.

i. Brown,

G. V.,

Applied

2. Brown,

"Magnetic

Physics,

G. V.,

Materials,"

Heat

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3. Brown,

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The

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W

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D

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H.,

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and

Steyert,

W.

A.,

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16, 1976, pp. 245-246.

13. Barclay,

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Space

Enqineer_ng

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_nvironmenta!

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LA-8991,

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16. Johnson, D. L., "Magnetic Refrigeration Te]ecQmmunications Propulsion

17. Mills,

Laboratory,

J.

Cycles

for

18. Van

19. Brown, Heat

J.

R.,

Research

Pumps,"

20. Barclay,

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Haaften,

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a

Supplements,

Principles

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19th San

N82

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Th____e

20119,

Jet

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Heat

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Pump

Enerqy

California,

0.,

and

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Refrigerating

No. 6, 1966,

pp. 1-105.

and Possible

Transactions,

Moze,

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Cycle,"

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Vol. 87, Pt. 2, 1981, pp. 783-793.

Paterson,

for 2-4 K Operation:

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21. Delpuech,

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ASHRAE

J.

Report

Cooling,"

1984, pp. 1369-1374.

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Physics,

Waste

Enqineerinq

Geuns,

Acquisition

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Industrial

19-24,

Phillips

Data

Pasadena,

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L.,

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"A

Reciprocating

Results,"

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R., Bon Mardion,

G., Claudet,

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A.

i

A.,

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LA-6581, - 36 -

u

w

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7867-I,

1981.

24. Pratt,

W. P., Jr.,

Rosenblum, S. S., Steyert,

"A Continuous Demagnetization Refrigerator of Magnetic Refrig'erants,"

25. Barclay,

J.

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W. A., and Barclay, J. A.,

Operating near 2 K and a Study

Vol. 17, 1977, pp. 689-693.

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4,464,903,

Aug.

14, 1984.

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2887-2984.

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26. Nakogome,

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27. Larbalestier,

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Fisk,

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28. Wilson,

M.,

T., U.S.

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New

York,

1983.

29. An approximation calculations

using As data

for Fig. 5 of Ref.

30. Conversations

with

31. Wilson,

ibid.,

densities

M.,

(qL = TAs)

various

Fig.

computer

[34].

commercial

12-17,

for niobium-tin." - 37

w

from Brown's

suppliers.

"The

best

recorded

critical-current

32.Smith,

M. M. Ph.D., dissertation

33. Griffel, Physical

M.,

et al.,

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"The Heat Capacity

Revie.______w, Vol. 93, 1954,

34. Benford,

S.M.

and

Brown,

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G.

Journal

Georgia

Tech.

of Gadolinium

from

15 to 355OK, "

pp. 657-661.

V.,

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of Applied

Diagram

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Gadolinium

Physics,

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52,

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1981,

the

pp. 2110-

2112.

35. Hatsopoulos,

G.

ThermodYnamics,

36. Booker,

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N.,

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in

Peter

37. Moon,

F., Maqneto-Solid

Mechanics,

38. Chen,

C. W., Maqnetism

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W.

Of

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1965.

Electromaqnetism,

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Peregrinus

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of

Electrical

New York,

1982.

1984.

of Soft

MaQnetic

Instructor

in

Materials,

North

1977.

Brazell,

of Technology.

m

38-

Mechanical

Design,

Regenerator THigh_

Magnet

Column I TLow

_Core

Fluid-J_

2

w

3

4

u

Fig.

1.

Cycle Magnetic

Motion Heat

of

a Reciprocating Pump,

Tc

F

_

u

E

Zero

Total

Fig.

2.

A Theoretical of the Magnetic

Field

Entropy

Reversible Material

Thermodynamic in the Core

- 4C

-

Cycle

High Field Region

f

() ---

_h

KTL J

Low

Region

m

L--

Fluid

Field

Fig.

3.

Schematic Magnetic

of Heat

a

w

__m

-41 w

-

Rotating Pump

Pump

L

"i

\ \

\

I !

Ro

\ \

Ri

I Bo l

I

=

I

!

\ \

i !

/ 7

=

Bw

/ /

/

Fig. 4. w

,wT...-

Schematic

of

the

Magnet

Winding

2.5

q

2.0

m

;9 F3

(_min



°_,,_

[]

0

0 C_

0 _:_ 0 o

ck_

d

[]

I o

I 0

I

I

I

t

I

I

I

0 o

I

o

o

o

o

o

o

0

o

o

0

o_

o6

e-

_5

u6

#

o"J.

od

,_

o

I-....4

cto_s

_.I_.IFOad

1_

w

I

50

_b -

o_

--

APPENDIX The magnetic below.

The

arbitrarily

field

profile

_

--

--

--

--

--

--

for

multiplied Distance

profile an

by 3/4.

for the modelling

8T

magnet

The values

from center,m

supplied

of the regenerator by

American

Tesla

0.0000

6.000

0.0127

5.966

0.0254

5.850

0.0381

5.700

0.0508

5.275

0.0635

4.669

0.0762

3.776

0.0889

2.729

0.1016

1.829

0.1143

1.210

0.1270

0.818

0.1397

0.572

0.1524

0.413

0.1651

0.308

0.1778

0.236

0.1905

0.184

0.2032

0.147

0.2159

0.119

0.2286

0.098

0.2413

0.082

0.2540

0.0691

0.2667

0.059

0.2794

0.051

0.2921

0.045

0.3048

0.038 - 51 -

Magnetics

are for the axis locations. Field,

is given was