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Physics Procedia 36 (2012) 687 – 692

Superconductivity Centennial Conference

Three-dimensional simulation of magnetic flux dynamics andtemperature rise in HTSC bulk during pulsed fieldmagnetization
H. Fujishiro, T. Naito and M. Oyama
Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka 020-8551, JapanAbstract
We have performed a three-dimensional (3D) numerical simulation of the dynamical motion of the magnetic flux and the heatpropagation in the superconducting bulk after applying a pulsed magnetic field. An inhomogeneous Jc distribution was supposed in
the bulk; the Jc in the growth sector boundary (GSB) is four times higher than that in the growth sector region (GSR). For lower
applied pulsed field, magnetic flux was penetrated and trapped in the GSR, and for higher applied pulsed field, the magnetic flux
was trapped more preferentially in the GSB. These results of the simulation reproduce the experimental ones and are valuable forthe understanding the flux dynamics in the bulk during pulsed field magnetization.
© 2011by ElsevierLtd. Selectionand/or peer-review
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Rogalla and PeterKes.
©
2012PublishedPublishedby Elsevier
B.V. Selection
and/or underpeer-review
underresponsibilityof theGuest Editors.
Keywords; pulsed field magnetization; inhomogeneous Jc distribution; flux dynamics; heat propagation; numerical simulation

1. Introduction
RE-Ba-Cu-O superconducting bulk (RE: rare earth element or Y) can trap higher magnetic field, Bz,
using a pulsed field magnetization (PFM). The Bz value by PFM is, however, pretty smaller than that byfield-cooled magnetization (FCM) because of a large heat generation due to the rapid magnetic flux
motion. We have investigated experimentally the enhancement of Bz by PFM based on the results oftemperature rise and flux movement [1, 2]. To explore the desired research direction to the enhancementof Bz, it is necessary to analyze both the flux dynamics and the heat propagation numerically. We

Corresponding author
E-mail address: fujishiro@iwate-u.ac.jp

1875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer-review under responsibility of the Guest Editors.
doi:10.1016/j.phpro.2012.06.268

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constructed a framework of the two-dimensional (2D) numerical simulation in the bulk withhomogeneous critical current density Jc distribution during PFM, and simulated the time and spatial
dependence of local field Bz(t, r) and temperature T(t, r) for a single magnetic pulse application [3, 4] andfor a multi-pulse field application [5], all of which reproduced the experimental results qualitatively.
However, an inhomogeneous Jc distribution exists in the actual bulk because of the characteristic crystal
growth mechanism, and the inhomogeneous flux motion and heat propagation were measured
experimentally. In this study, we constructed and performed a three-dimensional (3D) numericalsimulation of the dynamical motion of the magnetic flux and the temperature rise in the superconducting
bulk with an inhomogeneous Jc distribution after applying a pulsed magnetic field.
2. Numerical model and simulation
Based on the procedures and results of the axi-symmetric simulation [6], a framework of the 3D
numerical simulation was constructed using the finite element method (FEM). A superconducting bulk
disk 46 mm in diameter and 15 mm in thickness was cooled from the bottom surface. The spacing platewith the thermal conductivity, cont (=0.5 Wm-1K-1), between the bulk and the cold stage (Ts=40 K) with 1
mm in thickness was set, which imaginarily represented both the cooling power of the refrigerator and thethermal contact of the bulk to the cold stage. The upper and side surfaces of the bulk were supposed to beadiabatic. For the numerical analysis, one forth of the superconducting bulk disk was considered, which
was equally divided into 9 elements along the circumferential direction, into 6 elements along the radialdirection and into 6 elements along the thickness direction, respectively, as shown in Fig. 1. The pulsed
field Bex(t) with a rise time of =0.01 s and with a pulse duration of 0.1 s was applied to the bulk using asolenoid coil (82 mm I.D., 116 mm O.D., and 50 mm height). Commercial software, Photo-EddyTM,
combined with Photo-ThermoTM (Photon Ltd, Japan) was adopted for the analysis using a personalcomputer.
The power-n model (n=8) was used to describe the nonlinear E-J characteristic in superconducting
bulk. The temperature and magnetic field dependence of the critical current density Jc(T, B) wasdescribed as,3  2 2TB0J c (T,B) =  1  TB+ B0 ,
 c

(1)

where Tc (=92 K) is the critical temperature at B=0, and B0 (=1.3 T) is constant. An inhomogeneous Jc
distribution was supposed in the bulk; one segment of the bulk shown in Fig. 1 was regarded as the GSB
and the other 8 segments were regarded as the GSRs. In Eq. (1), the  values of 1.83 x 109 for GSB and
0.46 x 109 for GSR were adopted, which corresponded to the Jc(40 K, 0 T) values of 1.33 x 109 A/m2 and0.33 x 109 A/m2, respectively. The anisotropic thermal conductivities ab=20 Wm-1K-1 in the ab-plane andc=4 Wm-1K-1 along the c-axis are adopted. For comparison, the numerical simulations for homogeneous
Jc distribution with Jc(40 K, 0 T)=0.33 x 109 A/m2 and 1.33 x 109 A/m2 were performed.
3. Results and discussion
3.1. Time dependence of the flux penetration and temperature rise
Examples of the time and position dependence of the local field Bz and temperature T are shown in Fig.
1 for Bex=6 T. At the beginning stage of the magnetic pulse application, i.e., at t=0.002 and 0.006 s shownH. Fujishiro et al. / Physics Procedia 36 (2012) 687 – 692

Fig. 1. The time and spatial dependences of the local field Bz and temperature T for the bulk after applying the pulsed field of 6 T at(a) t=0.002 s, (b) 0.006 s, (c) 0.01 s, (d) 0.1 s and (e) 7 s. The model of the simulation and the time dependence of the pulsed fieldBex(t) are also shown.

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70(a)on GSB (Jc=1.33e9)
on GSR (Jc=0.33e9)
uniform (Jc=0.33e9)
Bex pulsez

B (r=0) (T)54

B =6T, T =40K

3ex

(b)
on GSB (Jc=1.33e9)
on GSR (Jc=0.33e9)
uniform (Jc=0.33e9)60

T (r=0) (K)6s2

B =6T, T =40Kexs50GSBGSR

1010-3r4010-210-1100
Time (s)

101

102

10-3

10-2

10-1
100
Time (s)101102

Fig. 2. The time evolution of the local field Bz and temperature T at the center of the bulk surface after applying the magnetic pulseof Bex=6 T. Thein Figs. 1(a) and 1(b), the magnetic flux mainly intrudes into the bulk along the GSR. Because of the flux
movement (viscous loss, Qv) and the flux trap (pinning loss, Qp), the temperature rise took place mainly at
the GSRs. At the peak of t=0.01 s shown in Fig. 1(c), the magnetic flux intrudes also into the GSB, but
the temperature rise in the GSB was delayed because of the lower thermal conductivity ab. Thereafter, alarge amount of the intruded magnetic flux was escaped from the bulk periphery as shown in Fig. 1(d),and the final trapped field in the GSB became larger than that in the GSR as shown in Fig. 1(e). The
generated heat around the bulk periphery diffused in the bulk with increasing time and was exhausted
from the cold stage.
Figures 2(a) and 2(b) show the time evolution of the local field and temperature at the center (r=0) ofthe bulk surface, respectively, after applying the magnetic pulse of Bex=6 T. The results for the uniform Jc
distribution are also shown. Bz(r=0) takes a maximum at t=0.02 s and T(r=0) takes a maximum at t=7 s,and then decreasing with time. Both results are almost independent of the Jc distribution in the bulk.
3.2. Trapped field in a steady state
Figure 3(a) shows the applied field dependence of the trapped field Bz(r=0) around the center of thebulk surface on the GSB and GSR. The results for the uniform Jc distribution were also shown. For lower
applied field (Bex3 T), the magnetic flux was not trapped around the bulk center, but at the bulk
periphery. For higher applied field (Bex4 T), the magnetic flux was intruded into the center and theBz(r=0) takes a maximum at Bex=5 T and then decreases with further increase in Bex because of the heatgeneration. The difference of Bz between on the GSB and on the GSR was small. However, Bz on theGSR was larger for lower Bex, but that on the GSB was larger for higher Bex. For the uniform Jcdistribution of Jc=0.33 x 109 A/m2, the Bz vs. Bex curve is an intermediate character of that on GSB and on
GSR. However, for the higher and uniform Jc distribution of Jc=1.33 x 109 A/m2, the magnetic flux starts
to intrude and trap at Bex=7 T and Bz takes a maximum at 9 T.
Figures 3(b) and 3(c) show the angular dependence of the trapped field Bz for each radius of the bulk
surface for Bex=3 T and 6 T, respectively. Figures 4(a) and 4(b) show respectively, the trapped field Bz onthe GSB and GSR of the bulk surface, as a function of radius r at Bex=3 T and 6 T. For lower applied fieldof Bex=3 T, the magnetic flux is preferably trapped on GSR. However, for the higher Bex of 6 T, thetrapped field on the GSB is higher than that on the GSR. These results well reproduced the experimental
ones reported by Yanagi [7] and Fujishiro [8].

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H. Fujishiro et al. / Physics Procedia 36 (2012) 687 – 692(a)on GSB (Jc=1.33e9)
on GSR (Jc=0.33e9)
uniform (Jc=0.33e9)
uniform (Jc=1.33e9)0(b) Bex=3T-902zz

1r=0
r=3.1
r=6.2
r=9.3
r=12.3r=15.4
r=18.5(c) Bex=6TrGSRr=0

B (T)

B (r=0) (T)23GSB90GSB

GSB

100GSB

0246810-20 0

20 40 60 80
 (deg.)B (T)ex-20 0GSB

20 40 60 80 100
 (deg.)Fig. 3. (a) The applied field dependence of the trapped field Bz(r=0) on the GSB and GSR. Those for the uniform Jc distribution arealso shown. Angular and radius dependences of the trapped field Bz on the bulk surface for (b) Bex=3 T and (c) Bex=6 T.

(a) Bex=3T2(b) Bex=6T

on GSB (Jc=1.33e9)
on GSR (Jc=0.33e9)
uniform (Jc=0.33e9)

2on GSB (Jc=1.33e9)
on GSR (Jc=0.33e9)
uniform (Jc=0.33e9)

z

B (T)

z

B (T)

GSB

1GSB

GSR

1r

GSR

r

0051015r (mm)20250051015r (mm)2025Fig. 4. The trapped field Bz on the GSB and GSR of the bulk surface, as a function of radius r at (a) Bex=3 T and (b) 6 T. The results
for the uniform Jc distribution are also shown.

4. Summary
We have performed a three-dimensional (3D) numerical simulation of the dynamical motion of themagnetic flux and the temperature rise for the superconducting bulk after applying a pulsed magneticfield. An inhomogeneous Jc distribution was supposed in the bulk; the Jc of the growth sector boundaries(GSBs) is four times higher than that of the growth sector regions (GSRs). For lower applied pulsed field,
magnetic flux was penetrated and trapped on the GSRs, and for higher applied field, the magnetic flux
was trapped more preferentially on the GSBs. These results of the simulation reproduce the experimental
ones and are valuable for the understanding the flux dynamics in the bulk.
This work is supported in part by a Grant-in-Aid for Scientific Research from the Ministry of theEducation, Culture, Sports, Science and Technology, Japan (No. 23560002).

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References[1] Fujishiro H, Hiyama T, Naito T, Tateiwa T, Yanagi Y. Mater. Sci. Eng. B 2008; 151: 95–100.
[2] Fujishiro H, Tateiwa T, Fujiwara A, Oka T, Hayashi H. Physica C 2006; 445-448: 334–8.
[3] Fujishiro H, Naito T. Supercond. Sci. Technol. 2010; 23: 105021(1)-(8).
[4] Fujishiro H, Naito T, Oyama M. Supercond. Sci. Technol. 2011; 24: 075015(1)-(7).
[5] Fujishiro H, Naito T, Furuta D. IEEE Trans. Appl. Supercond. 2011; 21: 2723–6.
[6] Kimi Y, Sekino M, Ohsaki H. Physica C 2009; 469: 1262–5.
[7] Yanagi Y, Itoh Y, Oka T, Ikuta H, Mizutani U. Supercond. Sci. Technol. 2005; 18: 839-49.
[8] Fujishiro H, Naito T, Furuta D, Arayashiki T, Yanagi Y, Itoh Y. Supercond. Sci. Technol. 2011; 24: 105003(1)-(6).