INSTITUTE OF PHYSICS PUBLISHING

SUPERCONDUCTOR SCIENCE AND TECHNOLOGYSupercond. Sci. Technol. 19 (2006) S540–S544

doi:10.1088/0953-2048/19/7/S23

Heat propagation analysis in HTSC bulks
during pulse field magnetization
Hiroyuki Fujishiro1 , Shusuke Kawaguchi1, Masahiko Kaneyama1,
Atsushi Fujiwara1 , Tatsuya Tateiwa1 and Tetsuo Oka2
1
2

Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, JapanIMRA Material R&D Co. Limited, 5-50 Hachiken-cho, Kariya 448-0021, Japan

E-mail: fujishiro@iwate-u.ac.jp

Received 16 December 2005, in final form 14 February 2006
Published 2 May 2006
Online at stacks.iop.org/SUST/19/S540
Abstract
The time evolutions of the three-dimensional temperature profiles in asuperconducting bulk disc have been calculated after applying a pulse field in
the pulse field magnetization (PFM) by use of a finite element method(FEM). The total generated heat Q , experimentally estimated using the
maximum temperature rise Tmax and specific heat C of the bulk, used in theanalysis and the distribution of Q in the periphery region of the bulk, issuitably supposed in order that the calculated time evolutions of temperatures
T (t) reproduce the measured ones on the bulk surface. From the analysis, theheat generation during PFM takes place under adiabatic conditions because
the total Q value is about one or two orders of magnitude larger than thecooling power of the cryocooler used. The enhancement of the total heat
capacity by setting a stainless steel ring onto the bulk as a heat reservoir is
one of the effective methods to reduce the temperature rise and to enhance
the trapped field.

1. Introduction
For the practical applications of high- Tc superconducting(HTSC) bulks as a strong bulk magnet system, pulse fieldmagnetization (PFM) has been intensively developed, as wellas conventional field cooled magnetization (FCM). In the PFM
process, however, a large amount of the heat generation occurs
due to the dynamical motion of the magnetic fluxes relative
to the FCM process, resultantly reducing the trapped field
BTP at temperatures below 77 K [1]. We have measured thetime dependences of the temperature T (t) on the bulk surface
during PFM at several positions and have systematicallyinvestigated the relation between BTP and T (t) [2–4]. We haverecently developed a new PFM method using the modified
multi-pulse technique with stepwise cooling (MMPSC), andattained BTP as high as 5.20 T on the surface of a GdBaCuO
bulk superconductor 45 mm in diameter, which originates fromthe precise investigation of the temperatures on the bulk [5].
BTP = 5.20 T is currently the highest value using PFM.
In this way, monitoring the temperature is a valuable toolin understanding the field trapping phenomena during PFM.
The temperature rise is, however, generally inhomogeneous0953-2048/06/070540+05$30.00

in the bulk due to the inhomogeneous distribution of thecritical current density Jc and the existence of growth sector
boundaries (GSBs). Since the number of measuring positionsin the temperature and trapped field must be finite in an
actual experimental setup, it is desirable to numericallyobtain the three-dimensional profile of temperature T (t, x) in
the bulk in order to precisely understand the field trapping
mechanism in PFM. The heat propagation analysis combinedwith the electro-magnetic phenomena using a finite element
method (FEM) is a powerful tool in analysing the PFM
procedure [6]. However, it is difficult for the numericalanalysis to take account of the inhomogeneous distribution of
Jc in an actual bulk.
In this paper, we apply the FEM analysis only to theheat propagation in the bulk during PFM. The time evolution
and spatial distribution of the three-dimensional temperatures
T  (t, x ) have been calculated to reproduce the measured
temperatures T (t, x ) on the bulk surface. We discuss theheat propagation phenomena in the bulk cooled by the helium
refrigerator after applying the pulse field and suggest thedesired direction in PFM to enhance the trapped field BTP .

© 2006 IOP Publishing Ltd Printed in the UK

S540

Heat propagation in HTSC bulks during PFM

Figure 1. (a) Experimental setup around the bulk and themagnetizing coil. (b) The positions of the temperature (T 1–T 5) andthe trapped field ( BTP ) measurements on the bulk surface areindicated. The bulk disc is equally divided into the elements for thenumerical analysis (see text).

2. Experimental details
Highly c-axis oriented YBaCuO and SmBaCuO bulks (DowaMining Co, Ltd, 45 mm in diameter and 15 mm in thickness)were used for the measurements and the analyses. The
experimental setup around the bulk disc is presented infigure 1(a). The bulk was stacked on the cold stage of the
Gifford McMahon (GM) cycle helium refrigerator using an
indium foil and was cooled down and fixed at Ts = 40 K
in a vacuum chamber. Five sets of fine chromel–constantan
thermocouples were adhered at the bulk centre (P1) and eachgrowth sector region (GSR) (P2–P5, 12 mm distant from P1).
The time dependences of temperature (T 1(t)–T 5(t)) were
measured about seven times per second after applying themagnetic pulse. The bulk was magnetized using a solenoidtype pulse coil dipped in liquid N2 , where pulse fields of
Bex = 3.83 and 5.53 T were applied with the rise time of 12 ms.
The trapped field BTP was measured using a Hall sensor (Bell,model BHA 921) adhered on the centre of the bulk surface(position H).
The heat propagation analysis was performed by solving athree-dimensional heat diffusion equation using a commercial
code (PHOTO-THERMOTM by PHOTON Ltd, Japan). For
the numerical analysis, the bulk disc was equally divided into
20 elements along the circumferential direction, 10 elements
along the radial direction and 6 elements along the thickness
direction, respectively, as shown in figure 1(b). The upperand side surfaces of the bulk were supposed to be adiabaticand the cold stage to be a constant temperature ( Ts =40 K) as boundary conditions. The thermal contact resistance
Rc between the lower bulk surface and the cold stage wasconsidered as a fitting parameter for the calculation. In theanalysis, the magnitude and the spatial distribution of thegenerated heat after applying a magnetic pulse must be decided
in order to reproduce the observed T (t, x ) at each position on
the bulk surface. The total generated heat Q was estimatedusing the maximum temperature rise Tmax and the specificheat C of the bulk [7]. C(T ) for each bulk was calculated using
the relation C = κ/α , i.e., the thermal conductivity κ divided
by the thermal diffusivity α , both of which were measured
simultaneously [8]. Since the magnetic fluxes always intrude
into the bulk from the periphery region during PFM, theheat generation should be the largest at the periphery, and it

Figure 2. The time evolutions of temperatures T (t) for respective
positions on the YBaCuO bulk after applying a pulse field of
Bex = 3.83 and 5.53 T at Ts = 40 K.

decreases on approaching the bulk centre. For simplicity inthe analysis, we suppose that the heat generation occurs onlyin the outermost periphery element layer. The time evolutionsof the three-dimensional temperature T  (t, x ) were calculated
to minimize the residual differences between the measured andcalculated temperatures for each bulk and for various appliedfields Bex . In the calculation, the ab-plane and c-axis thermal
conductivities κab , κc and the density d of each bulk werereferred to in the database [9].

3. Results and discussion
3.1. Determination of thermal contact resistance Rc
Figure 2 presents the time evolutions of temperatures T (t)for respective positions on the YBaCuO bulk after applying
a pulse field of Bex = 3.83 and 5.53 T [2]. For each Bex ,the nearly homogeneous temperature changes occur at T 1–T 5
with a small difference in the rise time. These results suggestthat the pinning force Fp is nearly identical in each GSR andthat the magnetic fluxes homogeneously intrude into the bulk
from the periphery. The temperature rise for Bex = 5.53 T islarger than that for Bex = 3.83 T. The Q value estimated using
the Tmax for T 2 is 160 J and 450 J, and the trapped field BTP
at position H is 0.17 T and 1.62 T for Bex = 3.83 and 5.53 T,
respectively.
Figure 3 shows the calculated temperature T 2 (t) on the
YBaCuO bulk for several boundary conditions after applying
a pulse field of Bex = 3.83 T. In this analysis, we supposedthat Q = 160 J was equally divided by 20 and each heatelement was arranged in the outermost surface layer, taking
account of the results shown in figure 2. In the ideal boundary
condition with Rc = 0, the calculated T 2 (t) shows a clear
peak at t = 1 s, suddenly decreases and then returns to theinitial temperature for t = 20 s. This result does not reproduce
the measured T 2 (t ). The T 2 (t) profile depends on the Rc
value and can fairly reproduce the measured one, when the Rc
value is supposed to be 5000 times the thermal resistance of thebulk along the c-axis (=5000/κc ). The heat generation after
applying a pulse field is terminated within the pulse duration(∼0.1 s) [10] and the Q value is about one or two ordersof magnitude larger than the cooling power of the GM cycle
helium refrigerator used, which is as small as 1 W (=1 J s−1 )S541

H Fujishiro et al

Figure 3. The calculated temperature T 2 (t) on the YBaCuO bulk
for several thermal contact resistances Rc after applying a field of
Bex = 3.83 T at Ts = 40 K.

Figure 4. The calculated temperature profile along the thickness
direction (z -direction) as a function of time t for (a) Rc = 0 and(b) Rc = 5000/κc .

Figure 5. The calculated temperature profiles T  (t) on the bulk
surface along the radius direction (r -direction) after applying a pulse
field of Bex = 5.53 T to the YBaCuO bulk.

Figure 5 presents the time evolution of the calculated
temperature profile T  (t, r ) on the surface of the YBaCuO
bulk along the radius direction (r -direction) through the bulk
centre and the position P2, after applying a higher pulse fieldof Bex = 5.53 T. It should be noticed that the calculated
temperature T  (t, r ) at the edge of the bulk (r = 22.5 mm)reaches 113 K at t = 0.2 s, which is far higher than the
superconducting transition temperature Tc (=90 K). T  (t, r )at the edge quickly decreases to 78 K at t = 1 s and thengradually decreases. T  (t, r ) at the centre (r = 0) increases
relatively more slowly than that at the outer surface and thendecreases. These results at P1 (r = 0 mm) and P2 (r = 12 mm)
reproduce the experimental ones shown in figure 3 well.
3.3. Inhomogeneous heat generation in SmBaCuO bulk

at 40 K. These results suggest that the heat generation during
PFM in the bulk must take place under adiabatic conditions. As
a result, the Rc value determined by the analysis mainly comesfrom the excess heat generation, which cannot be absorbed by
the cooling power. In the subsequent analysis, we use the Rc
value (=5000/κc ) as a thermal contact resistance between thelower bulk surface and the cold stage.
3.2. Spatial dependence of temperature change in YBaCuO
bulk
Figure 4 shows the calculated temperature profile T  (t, z) in
the YBaCuO bulk along the thickness direction (z -direction)
as a function of time t in the case of (a) Rc = 0 and (b) Rc =5000/κc after applying a pulse field Bex = 3.83 T. In thisfigure, the position z = 15 mm coincides with the position
P2 on the bulk surface. For Rc = 0, which is an unrealistic
condition, the temperature T  (t, 0) at the lower bulk surface(z = 0) is fixed at 40 K and is independent of t . The calculated
temperature T  (t, z) in the bulk increases after applying a pulse
field, has a maximum at t = 1 s and then decreases. On theother hand, for the actual situation with Rc = 5000/κc shownin figure 4(b), T  (t, z) rises up uniformly and sharply in thewhole bulk due to the adiabatic character, has a maximum at
t = 5 s and then decreases at t = 60 s. It is found that atemperature gradient hardly exists along the z -direction.
S542

Figure 6(a) indicates the experimental results of temperaturechanges T (t) of the SmBaCuO bulk after applying a pulse
field of Bex = 3.83 T [3]. T 4 and T 5 sharply rise up witha clear peak but T 2 and T 3 increase moderately, in contrast to
the YBaCuO bulk shown in figure 2(a). These results suggestthat the magnetic fluxes preferably intrude into the bulk fromthese two positions (P4 and P5) due to the weaker pinning force
Fp . The trapped field BTP was as small as 0.24 T at position H
because the Bex value to magnetize the bulk sufficiently was
fairly small. Figure 6(b) indicates the calculated temperature
profiles T  (t) at each position. In this case, the generated heat
Q was 240 J, which was estimated using the average of the
maximum temperature rises at P2–P5. The distribution of theheat generation was decided most suitably at the most outersurface as shown in the inset of figure 6(b). The calculatedT  (t) can reproduce the measured T (t) at each position.
Figure 7 shows the measured T (t) and the calculatedT  (t) for the SmBaCuO bulk after applying a pulse field of
Bex = 5.53 T [3]. In figure 7(a), all the maximum temperatures
Tmax increase compared with those for Bex = 3.83 T shown infigure 6(a) and the spatial dependences of T (t) in four GSRs
become small. These results suggest that the heat-generating
region expands due to the large amount of flux intrusions. For
the T  (t) calculation, the estimated Q value of 760 J was used.
As a result of the optimum distribution of Q as shown in theinset of figure 7(b), the calculated T  (t) can reproduce theHeat propagation in HTSC bulks during PFM

Figure 6. The time evolutions of (a) the measured temperatures T (t)and (b) the calculated ones T  (t) on the SmBaCuO bulk surface after
applying a pulse field of Bex = 3.83 T. The spatial distribution of thegenerated heat is also shown in the inset of (b).

Figure 7. The time evolutions of (a) the measured temperature T (t)and (b) the calculated ones T  (t) on the surface of the SmBaCuO
bulk after applying a pulse field of Bex = 5.53 T. The spatial
distribution of the generated heat is also shown in the inset of (b).

measured T (t) at each position well. The slight difference
between the calculated T  (t) and the measured T (t) may result
from our assumption for the Q distribution; the heat source
exists only in the outermost periphery element layer.

T  (t) on the SmBaCuO bulk with SUS ring after applying
a pulse field of Bex = 5.53 T. In the calculation of T  (t),the thermal conductivity κ (=5.5 W m−1 K−1 ), the density d(=7.8 g cm−3 ) and the specific heat C (=48.7 J kg−1 K−1 ) of
the stainless steel were used. The thermal contact resistanceRc (ring) between the bulk and the SUS ring was decided tobe 1000/κab . The calculated T  (t) at each position shown infigure 8(b) can roughly reproduce the measured T (t) shownin figure 8(a). In this experimental setup, the SUS ring acts
as a heat reservoir with a large heat capacity and a part of thegenerated heat in the bulk quickly transfers to the ring. Since
the heat generation takes place under an adiabatic condition
during PFM, the enhancement of the total heat capacity byattaching a heat reservoir with a large mass to the bulk is oneof the possible techniques to reduce the temperature rise andthen to enhance the trapped field BT in the bulk.

3.4. Effect of stainless steel ring setting
In our previous paper, we investigated the effect of setting a
metal ring on the SmBaCuO bulk disc on the trapped field andthe temperature rise during PFM [10, 11]. It was found that the
total trapped flux TP and the trapped field BTP were enhanced
about 10–20% by setting a stainless steel (SUS) ring. A part of
the heat generated in the peripheral region promptly transfers to
the SUS ring. Figure 8(a) shows the time dependences of themeasured temperature T (t) of the SmBaCuO bulk with SUS
ring after applying a pulse field of Bex = 5.53 T at Ts = 40 K.
An SUS ring 4 mm in thickness and 15 mm in height was
tightly fixed onto the bulk disc using APIEZON-N grease. The
temperature Tring (t) on the SUS ring was also monitored. It
should be noticed that the T (t) of T 1–T 5 increase, have amaximum similar to those for the same bulk without SUS ring,as shown in figure 7(a), and then decrease. Tring (t) gradually
increases by about a half of T 1–T 5 and has a maximum at
15 s. The heat generation in the SUS ring due to the eddy
current loss was confirmed to hardly take place because of
the higher electrical resistivity. These results mean that apart of the generated heat Q in the bulk promptly transfers to
the SUS ring. Figure 8(b) shows the calculated temperatures

4. Summary
The three-dimensional temperature profiles T  (t, x) in the
superconducting bulk disc after applying the pulse field havebeen calculated by the use of a finite element method (FEM).
The calculated T  (t, x) well reproduced the experimentallyobtained T (t, x) by distributing the generated heat Q
appropriately in the periphery region of the bulk disc. From
the analysis, the heat generation takes place in the bulk underadiabatic conditions during PFM and the Q values are about
one or two orders of magnitude larger than the cooling power
S543

H Fujishiro et al

and that it cannot be absorbed by the helium refrigerator
promptly. In order to reduce the temperature rise during PFM,
the dispersion of the generated heat to a metal ring set onto thebulk is one of the possible methods from the viewpoint of theheat propagation analysis.

Acknowledgments
This work is supported in part by a Grant-in-Aid for Scientific
Research from the Ministry of the Education, Culture, Sports,
Science and Technology, Japan (No. 17560001) and from Iwate
Prefecture, Japan.

References

Figure 8. The time evolutions of (a) the measured temperature T (t)and (b) the calculated ones T  (t) of the SmBaCuO bulk with astainless steel (SUS) ring after applying a pulse field of
Bex = 5.53 T.

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