An experimental heat pump that uses a rare earth element as the refrigerant is modeled using NASTRAN. The refrigerant is a ferromagnetic metal whose temperature rises when a magnetic field is applied and falls when the magnetic field is removed. The heat pump is used as a refrigerator to remove heat from a reservoir and discharge it through a heat exchanger. In the NASTRAN model the components modeled are represented by one-dimensional ROD elements. Heat flow in the solids and fluid are analyzed. The problem is mildly nonlinear since the heat capacity of the refrigerant is temperature-dependent. One simulation run consists of a series of transient analyses, each representing one stroke of the heat pump. An auxiliary program was written that uses the results of one NASTRAN analysis to generate data for the next NASTRAN analysis

Schroeder, E. A.

English

NASA Technical Reports Server

Erwin A. Schroeder 
Yuwrical Ilcchanics Division 
David W. Tcylor Yaval Ship R&D Center 
Bethesda, nvyland 20084 
& experimental heat pmp tnat uses a rare earth elcnnt as the retrigerant is 
W e l e d  usicg UETLUI.  The refrigerant is a ferromagnetic metal whose temperature 
rises vhta a lagnetic field is applied and falls uben the magnetic field 5s 
removed. The htzt p mp is used as a refrigerator to r m v e  heat from a reservoir 
ad Cischaree it through a heat exchanger. I C  the MASTRAM -1 the CoIponents 
&led are represented by onediaensiimal ROD elements. T!w? analysis acccunts for 
heat flow in the aoliGs. heat trrsport in the fluid, and hert transfer beween the 
solids and fluid. The problem is rildlv nonlinear 3ince the heat capscity of the 
refriberatt is te!aperature-dcpendent,. Cne simulation run conrists of a series of 
transient aalyses, each representing one strowe of the heat p p .  b. auxiliary 
program -3 written that uses the results of one IZmtW analysis to generate data 
for the next MASTRAN analysis. C a p u t t d  results are oarpara with experimental 
data. 
An experimental heat pump king developed at the David Y. Taylor Naval Ship 
R&D Center (DTNSRDC) uses me or Eore rare earth elements as the working substance. 
These elements are frrrosag;~etic retals whose temperature increases when a magnetis 
field is applied and decreases when the field is removed. nris thermodynamic pro- 
perty is used in a tuo-stroke cpclt to pump heat fram a reservoir and discharge it 
into the surrounding air. Thus. a refrigeration cycle is implied and the heat pump 
tiill be referred to as a refrigeratar. Using rare eartn elements as the rofri- 
gerant rakes possible magnetic refrigerators that discharge heat at r a m  tempera- 
ture and refrigerate to very low tcsperatures. Brown (Reference 1) gives con- 
siderations that are required tc  make magnetic refrigeration practical for such 
applications and (Reference 2) several configurations that have been proposed for 
qnetic heat pups. In the apparatus developed at DTWSRK, a regenerator serves 
as the reservoir f r a  which %eat is removed and discharged through a heat 
exchanger. Figure 1 is a s~neaatic description of this test apparatus. Heat is 
3arried by a fluid, compressed nitrogen, that moves through the refrigerator. The 
refrigerant is forred into thin bands that are wrapped aromd landrels to form 
coils. These coils are formed with gaps between the layers through which fluid can 
https://ntrs.nasa.gov/search.jsp?R=19850017580 2020-03-22T19:22:22+00:00Z
Figure 1. The Hagnetic Refr igera tor  
flow. Severa l  c o i l s  a r e  s tacked i n  a s t a i n l e s s  steel chamber. The chamber is ccn- 
nected on the cool  side to the regenera tor  and on t h e  warm s i d e  to t h e  hea t  
exchanger. The f l u i d  c i r c u i t  is coopleted by a cy l inde r  conta in ing  a p i s ton  t h a t  
m v e s  t h e  f l u i d  back and f o r t h  through t h e  chamber, regenera tor ,  and hea t  
exchanger. The l a r g e  magnetic f i e l d  rega i red  through the r e f r i g e r a n t  is provided 
by a superconduct ing magr.et . 
The cooling cycle consists of two strokes which will be calied the exhaust and 
intake strokes. During the exhaust stroke the magnetic field is applied to thc 
refrigerant, raising its temperature, and fluid drawn through the cooled regenera- 
tor passes over and cools the refrigerant. T k  heat acquired ~y the fluid during 
the exhaust stroke is discharged by a heat exchanger. During the intake stroke, 
the magnetic field is removed: the temperature of the refrigera~t falls, and fluid 
dram through the heat exchanger and passing over the refrigerant in the opposite 
direction is cooled and in turn cools the regenerator. 
To achieve very low temperatures. a refrigerator with several stages operating 
at progressively lower temperatures will be needed. Several rare earth elements or 
alloys of these elements will be used as refrigerants: each has a temperature ran&e 
at uhich it works m s t  effectively. Om rare earth elenent, gadoliniw, has its 
largest temperature change with a change in magnetic field at about 300 K which is 
near room temperature. Other rare earth metals that may be used are terbium and 
dysprosiue which have largest temperature c3anges near 225 K and 180 K. respec- 
tively. 
REQUIREHEWE FOR THE l4ATHEUTICAL ANALYSIS 
Different designs for the refrigerator, using various materials must be tested 
to determine which configurations and material properties hold the most promise for 
effective and efficient operation. It is lore efficient to model the refrigerator 
analytically and simulate its operation with different designs and materials than 
to rake coils cf various materials and then build and test experimental refrigera- 
tors, 
Since PTUSRDC has considerable experience with the IASTRAN program, it was 
decided to use its heat transfer capability for this problem. However, several 
aspects of the problem make it a non-ro~tine application. The material properties 
of the refrigerant are sufficiently temperature-dependent in the temperature rbnge 
used to make the problem nonlinear. Also, each test run of the refrigerator con- 
sists of many strokes. Each stroke corresponds to a t i e  segment in the solution 
with initial ccnditions different from, but determined, by the final conditions of 
the previous segrent. NASTRAN provides for neither nonlinear material prcperties 
nor changing temperature values during a transient analysis. To accoamodate these 
requirements, a sequence of WASTRAM analyses is used, one for each stroke. An aux- 
iliary program HEATSTEP was developed that uses results from the NASTRAN analysis 
of the previous stroke, computes the initial conditions for the next stroke, and 
prepares data for the next NASTRAN analysis. 
A number o f  s impl i fy ing  assumptions f o r  t h e  mathematical model keep t h e  
a n a l y s i s  manageable. Only t h e  r e f r i g e r a n t .  t h e  cha lber ,  and t h e  f l u i d  passing 
through t h e  chamber are m d e l e d ,  and a simple approximation is made o f  the hea t  
capaci ty  o f  t h e  tubing  leading  from the ch-r to t h e  regenera tor .  S ince  the 
l a y e r s  of r e f r i g e r a n t  and f l u i d  and t h e  s t a i n l e s s  steel chamber wall are t h i n ,  it 
5s  assumed t h a t  t h e r e  is no s i g n i f i c a n t  temperature v a r i a t i o n  through t h e s e  l aye r s .  
It is a l s o  assumed t h a t  t h e r e  is no s i g n i f i c a n t  temperature v a r i a t i o n  a c r o s s  t h e  
t h e  chharber. Therefore,  t h e  temperature o f  each type  9f material is assumed to 
vary only  along t h e  length  o f  t h e  r e f r i g e r a n t  chamber, t h a t  is, t h e  problem is 
one-dimensional. S ince  seve ra l  seconds a r e  requi red  to apply and r m v e  t h e  mag- 
n e t i c  f i e l d ,  it is assumed t h a t  a t  t h e  begin t ing  o f  each s t r o k e  t h e  temperature o f  
t h e  f l u i d  equa l s  the temperature o f  the r e f r i g e r a n t  a t  the  same pos i t ion .  The tep- 
pera tu re  d i s t r i b u t i o n  i n  t h e  chamber wal l  is assumed not to change whi le  t h e  mag- 
n e t i c  f i e l d  is changing. The r e f r i g e r a n t  coils are stacked one on t o p  o f  another., 
so t h e r e  is a r e l a t i v e l y  small thermal connection between t h e  c o i l s ,  and it is 
approxiamcet by a co~duc-tor wi th  a r e l a t i v e l y  small thermal conduct iv i ty .  
During t h e  i n t a k e  s t r o k e  t h e  hea t  exchanger, and during t h e  exhavst  s t r o k e  the 
regenera tor ,  a r e  assumed to provide sources  o f  f l u i d  wi th  oons tant  temperatures. 
Therefore, during t h e  in t ake  s t roke ,  t h e  temperature o f  t h e  f l u i d  flowing i n t o  t h e  
r e f r i g e r a n t  c h e r  equa l s  t h e  ambient temperature. Determining t h e  temperature o f  
t h e  f l u i d  e n t e r i n g  t h e  chamber dur ing  t h e  exhaust  s t r o k e  is not  so s t ra ight forward .  
It is assused t h a t  the t e m p e r a ~ u r e s  o f  t h e  f l u i d  and t h e  tubing  on t h e  co ld  s i d e  
are equal  and cons t an t  during t h e  exhaust  s t r o k e ,  and a t  t h e  beginning c f  t h e  
in t ake  s t r o k e  t h e  t eape ra tu re  o f  t h e  cold  s i d e  is the same a s  it was dur ing  :he 
l a s t  exhaust  s t roke .  Therefore,  t h e  temperature o f  t h e  f l u i d  e n t e r i n g  t h e  refri- 
ge ran t  chamber dur ing  t h e  present  exhaust  s t r o k e  is computed by t ak ing  a weighted 
average of  t h e  average temperature o f  f l u i d  en te r ing  t h e  regenera tor  dur ing  t h e  
prereding i n t a k e  s t r o k e  and t h e  temperature o f  f l u i d  leaving  t h e  regenera tor  dur ing  
t h e  preceding e x h a m t  s t roke .  I n  computing t h e  average, t h e s e  temperatures a r e  
weighted by t h e  va lues  of  t h e  hea t  capac i ty  o f  t he  f l u i d  moved dur ing  one sSroke 
and t h e  hea t  capaci ty  o f  the  cool side o f  t h e  apparatus,  respec t ive ly .  The tep- 
pera tu re  change dur ing  one s t r o k e  is a t  most a few aegrees ,  so t h e  temperature- 
dependent ma te r i a l  p r o p e r t i e s  a r e  determined f o r  t h e  i n i t i a l  temperatures f o r  each 
s t roke  and remain cons t an t  during t h e  s t roke .  
EiE EQUATIONS THAT DESCRIBE THE REFRIGERATOR HODEL 
The p a r t i a l  d i f f e r e n t i a l  equat ions  t h a t  desc r ibe  t h e  temperature d i s t r i b u t i o n  
i n  t h e  r e f r i g e r a n t  and t h e  chamber wal l  a r e  one-dimensional hea t  equations.  A 
loading term on t h e  r i g h t  hand s i d e  r ep resen t s  f i l m  convection o f  hea t  t.o t h e  
f l u i d .  These equat ions  a r c  o f  t h e  form 
rhere m = r, s or f for refrigerant, stainless steel or fluid 
m 
= density of material 
c = heat capacity of Inaterial 
m 
km = conductivity of material 
Am = area of the surface between the material and fluid 
"m 
= volue of material 
Tm 
= temperature of material 
h = convection coefficient 
The partial differential equation that describes the temperature distribution 
in the fluid is 
The term containing aT /ax represents transfer of heat due to the flow of the f 
fluid. The two terms on the ribht hand side represent Til& convection to the 
refrigerant and stainless steel chamber dall. Boundary conditions fcr the one- 
dimensional heat equation consist of the initial temperature distribution in the 
chamber and specified temperatures or derivatives of the temperature at the ends of 
the chamber. The boundary conditions are provided by the initial temperatures and 
constraints carrtained in the NASTRAN input data. 
THE FINITE ELEMENT W D E L  
The problem is sclved using the HASTRAN transient thermal analysis rigid 
format. The refrigerant, stainless steel, and flcid are modeled by NASTEAN one- 
dimensional ROD elements. The thermal contacts betweer; the coils of refrigerant 
are represented by YASTRAW ELAS1 elements. Transf?r of heat to aad from the fluid 
from the refrigerant material and the stainless steel chamber is modeled using HBDY 
elements of type LINE. Figure 2 Is a schematic representation of the finite 
elelnent model. 
11 n 61 81 - REFRIGERANT 
6 - - ELEMENTS 
12 22 32 u 52 sz n 82 FLUID 
a - - - - 
- - - - - - ELEMENTS 
13 P 33 43 53 63 n 83 a 93 STAINLESS STEEL 
- - - - - - 
ELEMENTS 
Figure 2. The Finite Element W e 1  
The term in the fluid differential equation containing the spatial derivative 
of the fluid temperature is obtained by a finite difference approximation using a 
NASTRAN transfer function. This term is included in the differential equation by 
applying a load to the appropriate node using the capability for applying nonlinear 
loads. For example, to compute the finite difference approximation for the fluid 
node number 32, a value, u is assigned to an extra point, number 732, by the 
equation 732 
for '732 = (uh2 - u ) /  Ax, where Ax is the distance between adjacent nodes, and 32 
u is the temperature at node number 32, so B = AX, A1 = -1, and A 32 = 1. The 
preceding approxination is for fluid flowing in the direction of increasing nodal 
identification numbers; for fluid flowing in the other direction the approximate 
value is 
'732 I (up - uZ2 I /  AX 
The term with aT/ax is added in the form of a load using a NASTRAN NOLIN card where 
'732 
t aT/ax is the load and p c vV is a scale factor. 
f p  
The temperature of the fluid entzring the chamber is maintained by a large 
source specified by a TLOADl card. The magnitude of this force multiplied by a 
large scale factor M is given on a DAREA card and applied to a scalar point. A 
large conductor from the scalar point to ground is specified by an ELAS4 element 
t h a t  has  conduc t iv i t y  equal  t o  the Scale f a c t o r  H. Thus, t h e  va lue  assigned t o  t h e  
s c a l a r  po in t  is cons t ra ined  t o  equal  t h e  i npu t  temperature,  and then  t h e  appropri- 
a t e  f l u i d  g r i d  po in t  temperature is cons t ra ined  by a mu l t i po in t  c o n s t r a i n t  t o  equal  
t h i s  value.  The temperature of  t h e  f l u i d  e n t e r i n g  t h e  r e f r i g e r a n t  chamber is main- 
t a i n e d  by a t t a c h i n g  t o  t h e  f l u i d  node loca t ed  where t h e  f l u i d  enters t h e  chamber, a 
l a r k e  l i n e a r  conductor represented  by a NASTRAN ELAS4 element, and applying t o  t h i s  
node a source  o f  hea t  wi th  a magnitude t h a t  w i l l  enforce  t h e  r equ i r ed  temperature. 
THE PROGRAM HEATSTEP 
The program HEATSTEP, developed a t  DTNSRDC, is run a f t e r  each ;!ASTRAN a n a l y s i s  t o  
prepare d a t a  f o r  t h e  fo l lowing  NASTRAN run.  To determine t h e  temperatures  a t  t h e  
end o f  the s t r o k e  and t h e  average temperature of  the f l u i d  l eav ing  t h e  r e f r i g e r a n t  
chamber du r ing  an i n t a k e  s t r o k e ,  HEATSTEP reads  t h e  f i l e  of  nods1 temperatures  pro- 
duced by NASTRAN. Using t h i s  information it computes t h e  temperature drop o r  rise 
a t  each r e f r i g e r a n t  node depending on whether t h e  preceding s t r o k e  was an in t ake  o r  
exhaust  s t roke .  The program then  determines t h e  temperature o f  each r e f r i g e r a n t  
and s t a i n l e s s  steel node f o r  t h e  beginning o f  t h e  fol lowing s t roke .  The program 
a l s o  computes hea t  c a p a c i t i e s  f o r  each r e f r i g e r a n t  element. 
HEATSTEP determines :he temperature of  t h e  f l u i d  t h a t  w i l l  be pass ing  i n t o  t h e  
chamber on t h e  fol lowing s t r o k e ,  and selects t h e  f l u i d  g r i d  p o i n t  t o  which t h e  
incoming f l u i d  temperature const . raint  is t o  be appl ied .  This  is accomplished by 
s e l e c t i n g  t h e  one o f  two MPC3 which connects  t h e  app rop r i a t e  f l u i d  g r i d  po in t  t o  
t b e  s c a l a r  po in t  whose t e e p e r a t u r e  has  been spec i f i ed .  For an i n t a k e  s t r o k e ,  t h e  
temperature of  t h e  f l u i d  g r i d  po in t  n e a r e s t  t h e  h e a t  exchanger is cons t ra ined  t o  
equal  t h e  ambiefit temperature. The r a t i o r i a l e  f o r  t h i s  average is given i n  t h e  sec- 
t i o n  on Assumptions. The inpu t  temperature T is given by t h e  equat ion  I N  
where C is t h e  h e a t  capac i ty  o f  t h e  volume of  f l u i d  moved dur ing  one s t r o k e  and C F T 
is che hea t  capac i ty  o f  t n e  tub ing  on t h e  co ld  s i d e ;  T~~~ is t h e  average tempera- 
t u r e  of f l u i d  leaving  t h e  chamber dur ing  t h e  preceding in t ake  s t r o k e ,  and T. LAST is 
t h e  temperature T f o r  t h e  l a s t  exhaust  s t roke .  I N  
F i n a l l y ,  HEATSTEP prepare3  an inpu t  f i l e  f o r  t h e  next  NASTRAN run t h a t  incor-  
po ra t e s  t h e  new d a t a  it has  computed. 
RESULTS AND DISCUSSION 
Severa l  ana lyses  were made us ing  t h i s  method. Some of  t he se  ana lyses  were 
made t o  g e t  r e l a t i v e  i n d i c a t i o n s  of  t h e  performance o f  proposed con f igu ra t ions .  
One a n a l y s i s  modeled t h e  experimental  device ;  r e s u l t s  from t h i s  a n a l y s i s  a r e   lot- 
t e d  i n  F igure  3. 
MAGNETIC REFRIGERATOR 
240 1 1 I ,  I I 1 1 1 I 1 I I I I 1 I I I 1 I 
0 2 4 6 8 10 12 14 16 18 20 
CYCLES 
300 
290- 9 
. . o  
- 280- 
Figure 3. Compsrison o f  Computed and Experimental Temperatures 
Y 
- 
W 
a 
3 
no 
a 
W 
0 
5 
'-260- 
250- 
Severa l  f a c t o r s  may c o n t r i b u t e  t o  improving agreement between t h e  computed and 
experimental  r e s u l t s .  To avoid pene t r a t i ng  t h e  pressur ized  chamber, thermocouples 
were mounted on t h e  o u t s i d e  su r f ace  of  t h e  s t a i n l e s s  s t e e l  r e f r i g e r a n t  chamber. 
S t eps  a r e  being taken t o  move t h e  tinermocouples i n s i d e  t h e  chamber t o  more accu- 
r a t e l y  record  t h e  temperature of  t h e  r e f r i g e r a n t  and f l u i d .  The f i n i t e  element 
model uses  a simple r ep re sen ta t i on  o f  t h e  regenera tor  and probably jts hea t  capa- 
c i t y  is under represented ;  a l a r g e r  hea t  capac i ty  would l i k e l y  slow t h e  temperature 
- 
- 
EXPERIMENTAL 
- COMPUTED - 
drop, prodacing b e t t e r  agreement i n  t h e  r e s u l t s .  I t  is a l s o  l i k e l y  t h a t  b e t t e r  
values  may be obtained f o r  t h e  f i l m  convect ion c o e f f i c i e n t s  and thermal conduc- 
t i v i t y  of t h e  c o n t a c t  between r e f r i g e r a n t  c o i l s .  These imprcvements w i l l  be made 
during future  work. 
REFERENCES 
l.Brown, G .  V . ,  "Hagnetic Heat Pumping Near Room Temperatures," J .  Appl. Phys., 
Vol. 47, pp. 3673-3680 ( 1976). 
2.Brown, G .  V., "Basic Pr inc ip les  and Poss ib le  Configurations o f  Magnetic Heat 
Pumps," ASHRAE Transactions, Vol. 87, Part 2 ,  pp. 783-793 (1981 1. 
1. RW No. 2. tovnmVnt kcusion Yo. 3. Rec~pmt's C.U@ ;M 
NASA CP-2373 
- 
4. T i e  md Suhtitla 5. Rwort Date 
Thirteenth FASTRAN' Users * rolloquium 
10. Work Unit No. 
8. Rrforw:ngOr imtion NUN a d  Mdrm 
COSMIC, GSA* s Computer Software Management 
and Informatlon Cer.ter 11. Contract or Gnnt NO. 
The Universi ty of Georgia 
Athens, GA 30602 , 1.7. T V ~  of ~ e w t  and b:od cowred 
2 . S a c r a i t q ~ N n u m d M . l m n  
National Aeronautics and Space Administration Publication 14 Sponsuritq Agmcy Cudr 
Washington, D.C. 20546 
Available from COSMIC, Athens, GA 30602 
This publ ica t ion is the proceedings of a colloquium, and it contains 
technical  papers contributed during the  Thirteenth NASTRAN~ Users* 
Colloquium held i n  Boston, Massachusetts on Maj- 6-10, 1985. The authors 
review general  appl ica t ion of f i n i t e  element methodology and the  s p e c i f i c  
appl ica t ion of t h e  NASA St ruc tu ra l  Analysis System, NASTRAN, t o  a va r i e ty  
of s t a t i c  and dynamic s t r u c t u r a l  problems. 
- 
7. Kw Wonk I ~ n J  bv Authrls: 18. Distribution Sutmrnt 
NASTRAN Unclassif ied - Unlimited 
S t ruc tu ra l  Analysis 
St ructures  
F i n i t e  Element Analysis 
Colloquium Subject Category 39 
8. Socuity Clrrif. (of thls report) 20. h r i t v  W f .  (of this prgl) 11. NO. of P l p n  22. Rice 
Unclassif ied Unclassif ied 462 A2 0 
- 


Transient Analysis of a Magnetic Heat Pump

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