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Influence of temperature on stability of trapped flux magnets
Cryogenics 50 (2010) 215–221
Contents lists available at ScienceDirect
Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Influence of temperature on stability of trapped flux magnets J.E. Pienkos a, P.J. Masson a, B. Douine b, J. Leveque b, C.A. Luongo a,c,* a
Florida State University, Tallahassee, FL, USA Groupe de Recherche en Electronique de Nancy, Nancy, France c ITER Organization, Magnet Division, 13067 Saint Paul-lez-Durance Cedex, France b
a r t i c l e
i n f o
Article history: Received 8 November 2008 Received in revised form 25 August 2009 Accepted 15 September 2009
Keywords: A. High TC superconductors C. Heat capacity C. Critical current density
a b s t r a c t Trapped flux in YBCO plates has an inherent dependence on temperature. The electro-magnetic current density and thermal specific heat are both highly dependent upon the temperature. Modeling and experimental data investigate the nature of a YBCO sample that has a heat pulse forced upon a trapped magnetic field by measuring the change in the magnetic field before and after the energy input. The experimental data suggest interesting trends that are further explained with models. An analytical approach and FEA simulations are performed to discover the physics behind the experimental data. By comparing the data, the temperature dependence on the YBCO sample is further investigated and illustrated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Single domain YBCO material has the unique capability of trapping magnetic flux. Different techniques can be used to achieve flux trapping such as zero field cooling with pulsed magnetization and field cooling (FC). FC represents the most effective way of achieving the highest magnitude of trapped flux but requires a large flux density (typically several T) to be applied. Pulsed magnetization is more convenient as small coils can be used but the associated pulse generator is not trivial since during the pulse, the superconductor can enter in a lossy flux flow mode and degrade the critical current density. Trapped flux magnets (TFM) have enabled numerous applications such as high power density electric motors [1] or magnetic levitation for flywheels. It is therefore critical to understand the sensitivity of trapped flux to energy input. Indeed, energy inputted in a TFM can lead to a local decrease of the current density and trigger local flux jumps or a decrease of the trapped flux. If the superconductor is not in full current penetration, a small energy input should only lead to a local current redistribution and very little modification of the trapped flux is expected, however, in the case of full current penetration, any variation of current density would lead to a decrease of the trapped flux and lead to degraded performance. This case, a YBCO sample with full current penetration, will be investigated in this paper through numerical analysis and experiments [2]. The sensitivity S of the trapped flux to distur* Corresponding author. Address: ITER Organization, Magnet Division, Bldg. 507/ Room 019, Route de Vinon, CS 90 046, 13067 Saint Paul-lez-Durance Cedex, France. Tel.: +33 442 176956; fax: +33 442 256628. E-mail address: [email protected] (C.A. Luongo). 0011-2275/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2009.09.005
bances is represented by the variation of trapped flux with respect to the origin trapped flux.
S¼
DB B0 Bf ¼ B0 B0
ð1Þ
B0 and Bf, the initial and final trapped flux, depends on the critical current density JC that is a function of temperature. The disturbance considered in this study is a pulse of heat input directly in the material. The heat is brought from the outside and conducted into the material; therefore the change in temperature of the superconductor depends on Cp, the specific heat that depends on temperature. Because of the dependence of JC and Cp upon temperature, the factor S is expected to also depend on temperature. Intuitively, from the variation of JC and Cp, one can expect S to exhibit a minimum as shown in Fig. 1. Finding a temperature at which the trapped flux is less sensitive to external perturbations would enable the development of more stable TFM based systems. 2. Experimental observation of the phenomenon 2.1. Experimental procedure The YBCO sample is a high quality single crystal disk with a diameter of 21 mm and a thickness of 8 mm, seen in Fig. 2. The sample is placed inside a Quantum Design PPMS vacuum chamber with a helium gas flow temperature controllability range of 300– 4.2 K. The sample has two Hall probes and one Cernox gallium–aluminum–arsenide RTD (resistance temperature diode) thermal sensor attached to the top and a heater attached to the bottom. The
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Fig. 1. Expected variation of S with temperature.
Fig. 2. The bulk YBCO experimental sample to test the stability of trapped flux via FC.
Hall probes measure the localized magnetic field trapped in the plate. The RTD outputs a voltage that has been previously calibrated to a temperature curve. Additionally, the RTD is attached to the plate with thermal grease to increase the thermal contact to the sample and accurately record the temperature. The schematic for the sample can be seen in Fig. 3. The heater is a wound nichrome wire that acts as a heat pulse, and is discussed in depth below. The sample is supported on an acrylic base that is attached to the top of the chamber by a G10 rod. The G10 rod therefore has one end at 300 K (room temperature) and the other end at cryogenic temperatures. The G10 rod has a low thermal conductivity and reduces the conduction load on the sample. The vacuum chamber is 27 cm in diameter and 88 cm in length. The helium gas flows from the bottom of the chamber upwards and cools the sample by convective heat transfer. The applied magnetic field is a superconducting coil surrounding the vacuum chamber and has an operational range of 0–7 T. The energy into the heater is a pulse from a capacitor bank. The time constant for a heat pulse is 0.1 s. During a pulse, the heat transferred to the sample is not complete. Conductive losses in the heater leads and the heat transferred to the sample support represent a very significant amount of energy. The minimum energy to raise the sample temperature on the RTD, which is located across the sample thickness from the heater, is about 10 J. How-
Fig. 3. The sample with the sensors attached at the top and heater at the bottom.
ever, the exact amount of energy to the sample support is unknown so all energy values given are calculated directly from the voltage capacitor bank. The voltage across the capacitor bank allowed control of the energy into the heater. Due to the large amount of energy, an electronic gate was used to ease the heat impulse into the heater. The sample chamber (PPMS) is convectively cooled by a controllable helium flow. This helium flow is automatically controlled by the systems controls and cannot be altered. The helium gas is always flowing and cooling the sample, even during the heat input and the redistribution of the current density. The experimental procedure begins by trapping flux in the plate. The flux is trapped in the sample by field cooling; 7 T is applied while the sample temperature is at 95 K; temperature is then ramped down to the desired level for the experiment. The applied magnetic field is then ramped down to zero and the field remaining trapped in the plate is measured. Hall probes output voltages are recorded along with the sample temperature coming from an RTD. The trapped flux at this point represents the maximumtrapped flux. After the flux is trapped at the desired temperature, an electric pulse is inputted to a heater attached to the bottom of the sample. The transient voltage from the two Hall probes and the temperature are recorded immediately after the heat pulse. The sample is then heated above the critical temperature and the process begins for the next datum point. The instabilities of the sample were quantified by the change of trapped field after the heat was inputted into the system. This provides good results, however the specific instabilities of quench or flux jumping were not studied. The instabilities were also only examined from the surface of the sample, thus finding only local data and not averaged for the sample. Although multiple Hall probes were used, the largest field is in the center and thus the only data used. 2.2. Experimental results First consider the maximum-trapped flux curve for the sample vs. the operating temperature, given in Fig. 4. The evolution of the maximum-trapped flux magnitude vs. temperature is not linear due to the non-linear variation of the critical current density of YBCO with temperature and magnetic flux. This data are before any heat impulse is applied to the sample and the current has fully penetrated the sample. The additional curve in Fig. 4 is the data supplied by the manufacturer of the YBCO sample, shown for comparison. Now consider the effect of the heat impulse when the YBCO plate is fully saturated in current. The current density decreases locally leading to energy dissipation and hence a decrease of the stored electromagnetic energy. Fig. 5 shows the change in trapped
Fig. 4. The experimental found and manufacturer supplied maximum-trapped flux in the sample.
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Fig. 5. The change in trapped flux vs. the energy input into the heater.
urated sample. The energy balance accounts for the heat impulse, the change in stored energy from the magnetic field, and the change in stored thermal energy, seen in Eq. (2). The energy balance has two states of the sample, the initial state before the heat input and the final state after the sample has achieved the maximum temperature from the energy input and magnetic-thermal energy dissipation. In the initial state, it is assumed that the current has fully saturated the plate and the current density is JC. The initial state of the plate is considered to be isothermal and constant JC. The heat impulse is added and raises the temperature of the plate, which therefore decreases the JC value of the plate and changes the stored magnetic energy from the trapped flux to thermal energy and raises the temperature of the plate further. The final temperature of the plate is solved for using the energy balance. This simple energy balance does not take into account any cooling on the plate, the process is assumed to be adiabatic. Thus, the model is a one-dimensional (radius) analysis of the energy balance that occurs during the heat impulse on a trapped flux, bulk YBCO plate.
Ein þ DEMagnetic ¼ DEThermal
ð2Þ
The DEMagnetic is the change in the stored magnetic energy in the sample. This is dependent upon the JC value, which is dependent on the temperature. The change in temperature allows for different values of JC and thus different values of stored magnetic energy. The stored magnetic energy at a given temperature is calculated in the following equations [3]. Beginning with the stored magnetic energy, W, as follows:
WðJÞ ¼
1 2
ZZZ
* *
J A dV
ð3Þ
V
Fig. 6. The normalized change in magnetic flux vs. the operating temperature with constant energy input.
magnetic flux vs. the amount of energy inputted into the heater. The change in magnetic flux was measured for three temperatures: 80 K, 60 K, and 30 K for various energy amounts input to the heater. Fig. 5 shows that the decrease of the trapped flux is proportional to the energy of the disturbance. As temperature changes, trapped flux magnitude changes and hence the electromagnetic stored energy changes. Fig. 6 shows the change in magnetic flux for a constant amount of energy inputted. Hundred volts was charged across the capacitor bank representing 22 J of energy. The YBCO plate was, for each experiment, fully saturated and very different flux variations were measured. Greater variation occurred when the stored energy was larger. The flux variation decreased continuously as temperature increases. For a given heat input, the trapped flux variation is minimum in the temperature range 55–60 K, showing a better behavior in reaction to heat disturbances. This is explained by the fact that as temperature increases so does the heat capacity of the sample. So, for a constant heat input, the temperature increase due to the heat pulse is decreasing as the experiment is performed at higher temperature. On the other hand, the JC is not decreasing as rapidly as the temperature increases, so in relative terms, there is a region for the heat capacity increases as which the sensitivity S is minimum as shown in Fig. 6. 3. Zeroth order analytical analysis 3.1. Model description The phenomenon is first modeled by applying an energy balance to the YBCO plate as the heat pulse is added to the current sat-
where J is the vector current density in A/m2, V is the volume, and A is the magnetic vector potential and is found in Eq. (4) using BiotSavart equation
A¼
l0 4p
ZZZ
*
J
* dV
V
ð4Þ
S
In Eq. (4), l0 is the permeability of vacuum and s is the unit direction vector of integration. For the sample, s is the radius and Eq. (4) becomes
A¼
J l0 RH 2
ð5Þ
where R and H are the radius and height of the sample. Now using Eqs. (3) and (5), one can find:
W¼
pl0 4
J 2 H2 R3
ð6Þ
Using Eq. (6), one can calculate the stored magnetic energy of the plate by knowing J, which is equal to JC, and is a function of the temperature. The third energy term in Eq. (2) is DEThermal and is the change in the stored thermal energy. This term, since the system can be assumed to be incompressible is found from the following:
DEThermal ¼ m
Z
Tf
cdT
ð7Þ
Ti
where m is the mass in kg, c is the specific heat in J/kg K, and is calculated by the integral with limits of the change in temperature T, in K [4]. The specific heat is a function of the temperature and is critical to determine accurate data at cryogenic temperatures. The specific heat of YBCO in the initial state is calculated from the following Eq. (8) and is valid from 20 to 150 K:
c ¼ 2:2819 T 19:902
ð8Þ
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The specific heat for the final temperature is integrated from the initial to final temperature to accurately portray the temperature change in the material for the energy balance. The specific heat is dependent on the YBCO material and can vary greatly from sample to sample. Inserting Eqs. (6) and (7) into Eq. (2) one finds Eq. (9) to calculate the temperature at the final state, f, from the initial state, i.
Q in þ
pl0 4
J 2Ci H2 R
pl0 4
J 2Cf H2 R ¼ m
Z
Tf
cdT
ð9Þ
Ti
So by inputting a heat impulse at a given initial temperature, Eq. (8) allows for calculation of the final temperature of the sample. Correlations for JC have been found in many different studies [5– 7] since JC can vary greatly from one sample to the next. The linear model is used in this study and JC can be found from the following equation:
J C ¼ J C0
1 T=T C 1 T 0 =T C
ð10Þ
In Eq. (10), JC0 is the current density at the temperature T0, typically 77 K, and is characteristic of the sample, and TC is the critical temperature. From Eq. (10), the current density is found from the operational temperature and the characteristic temperature and characteristic current density. To determine the trapped field, one can utilize the Bio-Savart law. The Biot-Savart law is seen in Eq. (11) and allows for the calculation of magnetic field B, at any point. *
B¼
l0
ZZZ
4p
The next step is to solve the energy balance. The energy from the heater and the initial temperature are input into the balance and the final temperature of the sample is calculated. When the final temperature is found, the initial and final magnetic fields are then calculated. This data then gives the change in magnetic field as a function of the initial temperature and heater energy input. The methodology to solve the energy balance can be seen in the flow chart in Fig. 7. 3.3. Analytical model results
*
*
J S *
V
Fig. 8. Change in magnetic flux vs. heat input for the analytical model.
dV
ð11Þ
S
As in the case for the experiment, calculating the magnetic field at the top center of the plate can be achieved. It is important to note at this point that the vector B is entirely in the axial direction since the vector J is the current density and is completely in the theta direction. This model is used to explain the physics of the phenomenon of the stability experiment on the YBCO sample. 3.2. Model implementation The first step to implement the model is to characterize the current density of the sample using Eq. (10), where JC is a function of temperature. To characterize the sample, JC0 must be found at T0 = 77 K using the magnetic field from experimental data presented in this paper B0 is 0.35 T. The current density at T0 is JC0 = 150 A/mm2. The current density for the sample is therefore characterized by JC0 at T0 and the operating temperature of the sample.
The energy balance model outputs change in trapped flux after a heat pulse was applied following the same behavior as observed during the experiment as shown in Fig. 8. The magnitude of the loss in magnetic flux comparison between the model and experiments do not match due to several reasons. The current density is modeled as a function of temperature due to the linear model ignoring the dependence on magnetic field. The model also simplifies the heat transfer by assuming an adiabatic case as opposed to the experiment which has constant cooling from the flow of helium gas through the chamber. The model also assumes that the final state has full current penetration throughout the sample which is needed due to the simplification of the dimensions and lack of instantaneous transient effects. The use of the assumptions simplified the model but the overall trends of the data are captured quite well. The sensitivity S calculated shows the same minimum between 50 K and 60 K as observed experimentally, a comparison with experimental data as shown in Fig. 9. The physical parameters used to describe the material allowed for reproducing the behavior observed experimentally thus validating our hypothesis of the mini-
Fig. 7. The flow chart to solve the energy balance.
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J C ¼ J C0
J C0 ¼
1 T=T C 1 1 T 0 =T C 1 þ kBk=B0
1=n1 JC E EC EC
ð12Þ
ð13Þ
The different parameters used are shown in Table 1 [8–10]. The specific heat is modeled as depending on temperature. Since the thermal conductivity only varies of about 20% over the temperature range on interest, it was considered constant. During the simulation, a probe outputs the flux density above the plate as shown in Fig. 11 where the different phases are described from ramping up of the field to flux trapping and heat pulse application. Fig. 9. S as function of temperature of the analytical model, shown with the experimental data for comparison.
mum occurring due to the dependence upon temperature of the heat capacity and critical current density. In order to visualize the variation of current distribution through fully transient simulations, a finite element model was developed and is presented in following section. 4. Finite element analysis 4.1. Model description The energy balance analysis gave interesting results that we will validate using a transient simulation of the trapped flux in a YBCO plate. The model coupled electromagnetic to thermal analysis to determine the sensitivity S of trapped flux vs. thermal disturbances. The model is defined as axial–symmetrical and the YBCO is considered having uniform properties. Flux is trapped by applying a magnetic field to the sample and then cooling the sample from the cooling environment. Cooling is provided surrounding environment and is modeled using conduction through surrounding material, altering conductivity of cooling environment gives control of cooling power to sample. The energy is inputted from the bottom of the plated via a heater in which a pulse of heat is applied. The energy is transferred to the heater volumetrically and to the sample by conduction. The model implemented in Comsol multiphysics is shown in Fig. 10. The current density for the FEA analysis was based on Kim’s model [6] and the electrical model used for the superconducting material follows the n power law obtained experimentally and represented by the following equation:
Fig. 10. Model implemented in FEA software.
4.2. FEA simulation results FEA simulation allows for representation of the different physical values of the problem, Fig. 12 shows the current distribution before and after the heat pulse. Since the plate was completely saturated in current before the pulse, the decrease of JC due to the elevation of temperature inevitably leads to a decrease of trapped flux. The bottom portion of the plate no longer contributes to the trapped flux and all the current is located on the top portion of the plate. The simulation is able to reproduce the experiment described earlier and the sensitivity S can be found for different temperatures. Fig. 13 shows the maximum-trapped flux vs. the operating temperature for the FEA simulations, as well as the experimental and analytical data for comparison. Fig. 14 shows the variation of trapped flux for different temperatures and different energy inputs for the FEA analysis. The results differ slightly from the experimental data as linear variation with energy inputs were expected. The linearity of the data given in the previous model uses current density as a function of temperature. It is thought that the simple variation of JC(B, T) modeled after Kim’s work [6] may not be accurate enough to model a thick YBCO plate thus generating a different behavior. From the data of Fig. 14, the parameter S can be plotted as function of temperature. As for the energy balance model, the FEA model is able to simulate the minimum of S at 57 K. The magnitude is
Table 1 Parameters used to represent the YBCO for FEA. n k T0 TC B0 JC0
E–J power law constant Thermal conductivity Char. temp Crit. temp Magnetic field at T0 Current density at T0
20 0.7 W/m K 77 K 92 K 0.5 T 850 A/mm2
Fig. 11. Different phases of the simulation: 1 – ramp up current in coil to apply magnetic field, 2 – cool sample to target operational temperature, 3 – ramp down current in coil to zero, 4 – reduce the heat transfer from the cooling environment, 5 – apply heat pulse, 6 – measure change in magnetic field, 7 – investigate the transient current density profile.
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Fig. 12. Current density distribution before (a) and after (b) the heat pulse.
Fig. 13. Trapped flux for the Comsol (FEA) analysis vs. the operating temperature. Shown for comparison is the experimental and analytical data.
Fig. 15. Variation of S vs. operating temperature for the FEA analysis with the experimental data shown for comparison.
ple and obviously, a large cooling power would lead to lower values of S. 5. Conclusion
Fig. 14. Change in trapped flux vs. energy input.
very close to that of the experimental data and the behavior is successfully obtained. A comparison of the variation of S vs. temperature is shown in Fig. 15. The results obtained with the FEA show similar trends to the experimental and analytical model data, the trapped flux variation with temperature is similar to the one generated with the analytical energy balance model. It is to be noted that no complete loss in magnetic flux occurred unless an appropriate energy was deposited from the heater. A minimum for the sensitivity S was found at 57 K a heat pulse. We have validated that the phenomenon comes from the dependence upon temperature of the specific heat and of the critical current density. The FEA simulations showed a complete loss of current density near the heater, the current density recedes to the coldest spot of sample. It is also to be noted that the cooling greatly affects the total loss of trapped flux in the sam-
The results presented show that there is an operating temperature for the trapped flux magnets for which the sensitivity to heat inputs is minimum; the preferred temperature depends on the value of the critical current density and its dependence upon temperature. We have shown through the use of an analytical model based on energy balance that the phenomenal comes from the competing dependence of the specific heat and the current density upon temperature. Finite element simulations were also able to reproduce the phenomenon and showed how the current redistributes in the material after a heat pulse. The stability of operation of TFMs can be improved by choosing the operating temperature properly and the material itself could be engineered to present the minimum of sensitivity to disturbances at a convenient temperature. Acknowledgements This research was done with support from GREEN Laboratory and from NASA Vehicle Systems Program and the Department of Defense Research and Engineering (DDR&E) division under the URETI on Aeropropulsion and Power. References [1] Masson PJ, Soban DS, Upton E, Pienkos JE, Luongo CA. HTS motors in aircraft propulsion: design considerations. IEEE Trans Appl Supercond 2005;15(2):2218–21. [2] Pienkos JE, Douine B, Leveque J, Masson PJ, Luongo CA. Experimental investigation of trapped flux stability in bulk YBCO. Adv Cryogen Eng Mater: Trans ICMC 2008;54:535–42.
J.E. Pienkos et al. / Cryogenics 50 (2010) 215–221 [3] Durand E. Magnetostatique. Paris: Saint-Germain; 1968. [4] Mills AF. Heat and mass transfer. 3rd ed. Chicago: Irwin; 1995. [5] Bean CP. Magnetization of high-field superconductors. Rev Mod Phys 1964;36:31–9. [6] Kim YB, Hempstead CF, Strnad AR. Critical persistent currents in hard superconductors. Phys Rev Lett 1962;9:306–9. [7] Schonborg N, Hornfeldt S. Model of the temperature dependence of the hysteresis losses in a high-temperature superconductor. Physica C 2002;372:1734–8.
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[8] Douine B, Berger K, Netter D, Leveque J, Rezzoug A. Influence of JC(B) on the full penetration current of superconducting tube. Physica C 2006;443:23–8. [9] Hong Z, Campbell AM, Coombs TA. Numerical solution of critical state in superconductivity by finite element software. Supercond Sci Technol 2006;19:1246–52. [10] Berger K, Leveque J, Netter D, Douine B, Rezzoug A. Influence of temperature and/or field dependences of the E–J power law on trapped magnetic field in bulk YbaCuO. IEEE Trans Appl Supercond 2007;17(2):3028–31.
Contents lists available at ScienceDirect
Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Influence of temperature on stability of trapped flux magnets J.E. Pienkos a, P.J. Masson a, B. Douine b, J. Leveque b, C.A. Luongo a,c,* a
Florida State University, Tallahassee, FL, USA Groupe de Recherche en Electronique de Nancy, Nancy, France c ITER Organization, Magnet Division, 13067 Saint Paul-lez-Durance Cedex, France b
a r t i c l e
i n f o
Article history: Received 8 November 2008 Received in revised form 25 August 2009 Accepted 15 September 2009
Keywords: A. High TC superconductors C. Heat capacity C. Critical current density
a b s t r a c t Trapped flux in YBCO plates has an inherent dependence on temperature. The electro-magnetic current density and thermal specific heat are both highly dependent upon the temperature. Modeling and experimental data investigate the nature of a YBCO sample that has a heat pulse forced upon a trapped magnetic field by measuring the change in the magnetic field before and after the energy input. The experimental data suggest interesting trends that are further explained with models. An analytical approach and FEA simulations are performed to discover the physics behind the experimental data. By comparing the data, the temperature dependence on the YBCO sample is further investigated and illustrated. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Single domain YBCO material has the unique capability of trapping magnetic flux. Different techniques can be used to achieve flux trapping such as zero field cooling with pulsed magnetization and field cooling (FC). FC represents the most effective way of achieving the highest magnitude of trapped flux but requires a large flux density (typically several T) to be applied. Pulsed magnetization is more convenient as small coils can be used but the associated pulse generator is not trivial since during the pulse, the superconductor can enter in a lossy flux flow mode and degrade the critical current density. Trapped flux magnets (TFM) have enabled numerous applications such as high power density electric motors [1] or magnetic levitation for flywheels. It is therefore critical to understand the sensitivity of trapped flux to energy input. Indeed, energy inputted in a TFM can lead to a local decrease of the current density and trigger local flux jumps or a decrease of the trapped flux. If the superconductor is not in full current penetration, a small energy input should only lead to a local current redistribution and very little modification of the trapped flux is expected, however, in the case of full current penetration, any variation of current density would lead to a decrease of the trapped flux and lead to degraded performance. This case, a YBCO sample with full current penetration, will be investigated in this paper through numerical analysis and experiments [2]. The sensitivity S of the trapped flux to distur* Corresponding author. Address: ITER Organization, Magnet Division, Bldg. 507/ Room 019, Route de Vinon, CS 90 046, 13067 Saint Paul-lez-Durance Cedex, France. Tel.: +33 442 176956; fax: +33 442 256628. E-mail address: [email protected] (C.A. Luongo). 0011-2275/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2009.09.005
bances is represented by the variation of trapped flux with respect to the origin trapped flux.
S¼
DB B0 Bf ¼ B0 B0
ð1Þ
B0 and Bf, the initial and final trapped flux, depends on the critical current density JC that is a function of temperature. The disturbance considered in this study is a pulse of heat input directly in the material. The heat is brought from the outside and conducted into the material; therefore the change in temperature of the superconductor depends on Cp, the specific heat that depends on temperature. Because of the dependence of JC and Cp upon temperature, the factor S is expected to also depend on temperature. Intuitively, from the variation of JC and Cp, one can expect S to exhibit a minimum as shown in Fig. 1. Finding a temperature at which the trapped flux is less sensitive to external perturbations would enable the development of more stable TFM based systems. 2. Experimental observation of the phenomenon 2.1. Experimental procedure The YBCO sample is a high quality single crystal disk with a diameter of 21 mm and a thickness of 8 mm, seen in Fig. 2. The sample is placed inside a Quantum Design PPMS vacuum chamber with a helium gas flow temperature controllability range of 300– 4.2 K. The sample has two Hall probes and one Cernox gallium–aluminum–arsenide RTD (resistance temperature diode) thermal sensor attached to the top and a heater attached to the bottom. The
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Fig. 1. Expected variation of S with temperature.
Fig. 2. The bulk YBCO experimental sample to test the stability of trapped flux via FC.
Hall probes measure the localized magnetic field trapped in the plate. The RTD outputs a voltage that has been previously calibrated to a temperature curve. Additionally, the RTD is attached to the plate with thermal grease to increase the thermal contact to the sample and accurately record the temperature. The schematic for the sample can be seen in Fig. 3. The heater is a wound nichrome wire that acts as a heat pulse, and is discussed in depth below. The sample is supported on an acrylic base that is attached to the top of the chamber by a G10 rod. The G10 rod therefore has one end at 300 K (room temperature) and the other end at cryogenic temperatures. The G10 rod has a low thermal conductivity and reduces the conduction load on the sample. The vacuum chamber is 27 cm in diameter and 88 cm in length. The helium gas flows from the bottom of the chamber upwards and cools the sample by convective heat transfer. The applied magnetic field is a superconducting coil surrounding the vacuum chamber and has an operational range of 0–7 T. The energy into the heater is a pulse from a capacitor bank. The time constant for a heat pulse is 0.1 s. During a pulse, the heat transferred to the sample is not complete. Conductive losses in the heater leads and the heat transferred to the sample support represent a very significant amount of energy. The minimum energy to raise the sample temperature on the RTD, which is located across the sample thickness from the heater, is about 10 J. How-
Fig. 3. The sample with the sensors attached at the top and heater at the bottom.
ever, the exact amount of energy to the sample support is unknown so all energy values given are calculated directly from the voltage capacitor bank. The voltage across the capacitor bank allowed control of the energy into the heater. Due to the large amount of energy, an electronic gate was used to ease the heat impulse into the heater. The sample chamber (PPMS) is convectively cooled by a controllable helium flow. This helium flow is automatically controlled by the systems controls and cannot be altered. The helium gas is always flowing and cooling the sample, even during the heat input and the redistribution of the current density. The experimental procedure begins by trapping flux in the plate. The flux is trapped in the sample by field cooling; 7 T is applied while the sample temperature is at 95 K; temperature is then ramped down to the desired level for the experiment. The applied magnetic field is then ramped down to zero and the field remaining trapped in the plate is measured. Hall probes output voltages are recorded along with the sample temperature coming from an RTD. The trapped flux at this point represents the maximumtrapped flux. After the flux is trapped at the desired temperature, an electric pulse is inputted to a heater attached to the bottom of the sample. The transient voltage from the two Hall probes and the temperature are recorded immediately after the heat pulse. The sample is then heated above the critical temperature and the process begins for the next datum point. The instabilities of the sample were quantified by the change of trapped field after the heat was inputted into the system. This provides good results, however the specific instabilities of quench or flux jumping were not studied. The instabilities were also only examined from the surface of the sample, thus finding only local data and not averaged for the sample. Although multiple Hall probes were used, the largest field is in the center and thus the only data used. 2.2. Experimental results First consider the maximum-trapped flux curve for the sample vs. the operating temperature, given in Fig. 4. The evolution of the maximum-trapped flux magnitude vs. temperature is not linear due to the non-linear variation of the critical current density of YBCO with temperature and magnetic flux. This data are before any heat impulse is applied to the sample and the current has fully penetrated the sample. The additional curve in Fig. 4 is the data supplied by the manufacturer of the YBCO sample, shown for comparison. Now consider the effect of the heat impulse when the YBCO plate is fully saturated in current. The current density decreases locally leading to energy dissipation and hence a decrease of the stored electromagnetic energy. Fig. 5 shows the change in trapped
Fig. 4. The experimental found and manufacturer supplied maximum-trapped flux in the sample.
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Fig. 5. The change in trapped flux vs. the energy input into the heater.
urated sample. The energy balance accounts for the heat impulse, the change in stored energy from the magnetic field, and the change in stored thermal energy, seen in Eq. (2). The energy balance has two states of the sample, the initial state before the heat input and the final state after the sample has achieved the maximum temperature from the energy input and magnetic-thermal energy dissipation. In the initial state, it is assumed that the current has fully saturated the plate and the current density is JC. The initial state of the plate is considered to be isothermal and constant JC. The heat impulse is added and raises the temperature of the plate, which therefore decreases the JC value of the plate and changes the stored magnetic energy from the trapped flux to thermal energy and raises the temperature of the plate further. The final temperature of the plate is solved for using the energy balance. This simple energy balance does not take into account any cooling on the plate, the process is assumed to be adiabatic. Thus, the model is a one-dimensional (radius) analysis of the energy balance that occurs during the heat impulse on a trapped flux, bulk YBCO plate.
Ein þ DEMagnetic ¼ DEThermal
ð2Þ
The DEMagnetic is the change in the stored magnetic energy in the sample. This is dependent upon the JC value, which is dependent on the temperature. The change in temperature allows for different values of JC and thus different values of stored magnetic energy. The stored magnetic energy at a given temperature is calculated in the following equations [3]. Beginning with the stored magnetic energy, W, as follows:
WðJÞ ¼
1 2
ZZZ
* *
J A dV
ð3Þ
V
Fig. 6. The normalized change in magnetic flux vs. the operating temperature with constant energy input.
magnetic flux vs. the amount of energy inputted into the heater. The change in magnetic flux was measured for three temperatures: 80 K, 60 K, and 30 K for various energy amounts input to the heater. Fig. 5 shows that the decrease of the trapped flux is proportional to the energy of the disturbance. As temperature changes, trapped flux magnitude changes and hence the electromagnetic stored energy changes. Fig. 6 shows the change in magnetic flux for a constant amount of energy inputted. Hundred volts was charged across the capacitor bank representing 22 J of energy. The YBCO plate was, for each experiment, fully saturated and very different flux variations were measured. Greater variation occurred when the stored energy was larger. The flux variation decreased continuously as temperature increases. For a given heat input, the trapped flux variation is minimum in the temperature range 55–60 K, showing a better behavior in reaction to heat disturbances. This is explained by the fact that as temperature increases so does the heat capacity of the sample. So, for a constant heat input, the temperature increase due to the heat pulse is decreasing as the experiment is performed at higher temperature. On the other hand, the JC is not decreasing as rapidly as the temperature increases, so in relative terms, there is a region for the heat capacity increases as which the sensitivity S is minimum as shown in Fig. 6. 3. Zeroth order analytical analysis 3.1. Model description The phenomenon is first modeled by applying an energy balance to the YBCO plate as the heat pulse is added to the current sat-
where J is the vector current density in A/m2, V is the volume, and A is the magnetic vector potential and is found in Eq. (4) using BiotSavart equation
A¼
l0 4p
ZZZ
*
J
* dV
V
ð4Þ
S
In Eq. (4), l0 is the permeability of vacuum and s is the unit direction vector of integration. For the sample, s is the radius and Eq. (4) becomes
A¼
J l0 RH 2
ð5Þ
where R and H are the radius and height of the sample. Now using Eqs. (3) and (5), one can find:
W¼
pl0 4
J 2 H2 R3
ð6Þ
Using Eq. (6), one can calculate the stored magnetic energy of the plate by knowing J, which is equal to JC, and is a function of the temperature. The third energy term in Eq. (2) is DEThermal and is the change in the stored thermal energy. This term, since the system can be assumed to be incompressible is found from the following:
DEThermal ¼ m
Z
Tf
cdT
ð7Þ
Ti
where m is the mass in kg, c is the specific heat in J/kg K, and is calculated by the integral with limits of the change in temperature T, in K [4]. The specific heat is a function of the temperature and is critical to determine accurate data at cryogenic temperatures. The specific heat of YBCO in the initial state is calculated from the following Eq. (8) and is valid from 20 to 150 K:
c ¼ 2:2819 T 19:902
ð8Þ
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The specific heat for the final temperature is integrated from the initial to final temperature to accurately portray the temperature change in the material for the energy balance. The specific heat is dependent on the YBCO material and can vary greatly from sample to sample. Inserting Eqs. (6) and (7) into Eq. (2) one finds Eq. (9) to calculate the temperature at the final state, f, from the initial state, i.
Q in þ
pl0 4
J 2Ci H2 R
pl0 4
J 2Cf H2 R ¼ m
Z
Tf
cdT
ð9Þ
Ti
So by inputting a heat impulse at a given initial temperature, Eq. (8) allows for calculation of the final temperature of the sample. Correlations for JC have been found in many different studies [5– 7] since JC can vary greatly from one sample to the next. The linear model is used in this study and JC can be found from the following equation:
J C ¼ J C0
1 T=T C 1 T 0 =T C
ð10Þ
In Eq. (10), JC0 is the current density at the temperature T0, typically 77 K, and is characteristic of the sample, and TC is the critical temperature. From Eq. (10), the current density is found from the operational temperature and the characteristic temperature and characteristic current density. To determine the trapped field, one can utilize the Bio-Savart law. The Biot-Savart law is seen in Eq. (11) and allows for the calculation of magnetic field B, at any point. *
B¼
l0
ZZZ
4p
The next step is to solve the energy balance. The energy from the heater and the initial temperature are input into the balance and the final temperature of the sample is calculated. When the final temperature is found, the initial and final magnetic fields are then calculated. This data then gives the change in magnetic field as a function of the initial temperature and heater energy input. The methodology to solve the energy balance can be seen in the flow chart in Fig. 7. 3.3. Analytical model results
*
*
J S *
V
Fig. 8. Change in magnetic flux vs. heat input for the analytical model.
dV
ð11Þ
S
As in the case for the experiment, calculating the magnetic field at the top center of the plate can be achieved. It is important to note at this point that the vector B is entirely in the axial direction since the vector J is the current density and is completely in the theta direction. This model is used to explain the physics of the phenomenon of the stability experiment on the YBCO sample. 3.2. Model implementation The first step to implement the model is to characterize the current density of the sample using Eq. (10), where JC is a function of temperature. To characterize the sample, JC0 must be found at T0 = 77 K using the magnetic field from experimental data presented in this paper B0 is 0.35 T. The current density at T0 is JC0 = 150 A/mm2. The current density for the sample is therefore characterized by JC0 at T0 and the operating temperature of the sample.
The energy balance model outputs change in trapped flux after a heat pulse was applied following the same behavior as observed during the experiment as shown in Fig. 8. The magnitude of the loss in magnetic flux comparison between the model and experiments do not match due to several reasons. The current density is modeled as a function of temperature due to the linear model ignoring the dependence on magnetic field. The model also simplifies the heat transfer by assuming an adiabatic case as opposed to the experiment which has constant cooling from the flow of helium gas through the chamber. The model also assumes that the final state has full current penetration throughout the sample which is needed due to the simplification of the dimensions and lack of instantaneous transient effects. The use of the assumptions simplified the model but the overall trends of the data are captured quite well. The sensitivity S calculated shows the same minimum between 50 K and 60 K as observed experimentally, a comparison with experimental data as shown in Fig. 9. The physical parameters used to describe the material allowed for reproducing the behavior observed experimentally thus validating our hypothesis of the mini-
Fig. 7. The flow chart to solve the energy balance.
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J C ¼ J C0
J C0 ¼
1 T=T C 1 1 T 0 =T C 1 þ kBk=B0
1=n1 JC E EC EC
ð12Þ
ð13Þ
The different parameters used are shown in Table 1 [8–10]. The specific heat is modeled as depending on temperature. Since the thermal conductivity only varies of about 20% over the temperature range on interest, it was considered constant. During the simulation, a probe outputs the flux density above the plate as shown in Fig. 11 where the different phases are described from ramping up of the field to flux trapping and heat pulse application. Fig. 9. S as function of temperature of the analytical model, shown with the experimental data for comparison.
mum occurring due to the dependence upon temperature of the heat capacity and critical current density. In order to visualize the variation of current distribution through fully transient simulations, a finite element model was developed and is presented in following section. 4. Finite element analysis 4.1. Model description The energy balance analysis gave interesting results that we will validate using a transient simulation of the trapped flux in a YBCO plate. The model coupled electromagnetic to thermal analysis to determine the sensitivity S of trapped flux vs. thermal disturbances. The model is defined as axial–symmetrical and the YBCO is considered having uniform properties. Flux is trapped by applying a magnetic field to the sample and then cooling the sample from the cooling environment. Cooling is provided surrounding environment and is modeled using conduction through surrounding material, altering conductivity of cooling environment gives control of cooling power to sample. The energy is inputted from the bottom of the plated via a heater in which a pulse of heat is applied. The energy is transferred to the heater volumetrically and to the sample by conduction. The model implemented in Comsol multiphysics is shown in Fig. 10. The current density for the FEA analysis was based on Kim’s model [6] and the electrical model used for the superconducting material follows the n power law obtained experimentally and represented by the following equation:
Fig. 10. Model implemented in FEA software.
4.2. FEA simulation results FEA simulation allows for representation of the different physical values of the problem, Fig. 12 shows the current distribution before and after the heat pulse. Since the plate was completely saturated in current before the pulse, the decrease of JC due to the elevation of temperature inevitably leads to a decrease of trapped flux. The bottom portion of the plate no longer contributes to the trapped flux and all the current is located on the top portion of the plate. The simulation is able to reproduce the experiment described earlier and the sensitivity S can be found for different temperatures. Fig. 13 shows the maximum-trapped flux vs. the operating temperature for the FEA simulations, as well as the experimental and analytical data for comparison. Fig. 14 shows the variation of trapped flux for different temperatures and different energy inputs for the FEA analysis. The results differ slightly from the experimental data as linear variation with energy inputs were expected. The linearity of the data given in the previous model uses current density as a function of temperature. It is thought that the simple variation of JC(B, T) modeled after Kim’s work [6] may not be accurate enough to model a thick YBCO plate thus generating a different behavior. From the data of Fig. 14, the parameter S can be plotted as function of temperature. As for the energy balance model, the FEA model is able to simulate the minimum of S at 57 K. The magnitude is
Table 1 Parameters used to represent the YBCO for FEA. n k T0 TC B0 JC0
E–J power law constant Thermal conductivity Char. temp Crit. temp Magnetic field at T0 Current density at T0
20 0.7 W/m K 77 K 92 K 0.5 T 850 A/mm2
Fig. 11. Different phases of the simulation: 1 – ramp up current in coil to apply magnetic field, 2 – cool sample to target operational temperature, 3 – ramp down current in coil to zero, 4 – reduce the heat transfer from the cooling environment, 5 – apply heat pulse, 6 – measure change in magnetic field, 7 – investigate the transient current density profile.
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Fig. 12. Current density distribution before (a) and after (b) the heat pulse.
Fig. 13. Trapped flux for the Comsol (FEA) analysis vs. the operating temperature. Shown for comparison is the experimental and analytical data.
Fig. 15. Variation of S vs. operating temperature for the FEA analysis with the experimental data shown for comparison.
ple and obviously, a large cooling power would lead to lower values of S. 5. Conclusion
Fig. 14. Change in trapped flux vs. energy input.
very close to that of the experimental data and the behavior is successfully obtained. A comparison of the variation of S vs. temperature is shown in Fig. 15. The results obtained with the FEA show similar trends to the experimental and analytical model data, the trapped flux variation with temperature is similar to the one generated with the analytical energy balance model. It is to be noted that no complete loss in magnetic flux occurred unless an appropriate energy was deposited from the heater. A minimum for the sensitivity S was found at 57 K a heat pulse. We have validated that the phenomenon comes from the dependence upon temperature of the specific heat and of the critical current density. The FEA simulations showed a complete loss of current density near the heater, the current density recedes to the coldest spot of sample. It is also to be noted that the cooling greatly affects the total loss of trapped flux in the sam-
The results presented show that there is an operating temperature for the trapped flux magnets for which the sensitivity to heat inputs is minimum; the preferred temperature depends on the value of the critical current density and its dependence upon temperature. We have shown through the use of an analytical model based on energy balance that the phenomenal comes from the competing dependence of the specific heat and the current density upon temperature. Finite element simulations were also able to reproduce the phenomenon and showed how the current redistributes in the material after a heat pulse. The stability of operation of TFMs can be improved by choosing the operating temperature properly and the material itself could be engineered to present the minimum of sensitivity to disturbances at a convenient temperature. Acknowledgements This research was done with support from GREEN Laboratory and from NASA Vehicle Systems Program and the Department of Defense Research and Engineering (DDR&E) division under the URETI on Aeropropulsion and Power. References [1] Masson PJ, Soban DS, Upton E, Pienkos JE, Luongo CA. HTS motors in aircraft propulsion: design considerations. IEEE Trans Appl Supercond 2005;15(2):2218–21. [2] Pienkos JE, Douine B, Leveque J, Masson PJ, Luongo CA. Experimental investigation of trapped flux stability in bulk YBCO. Adv Cryogen Eng Mater: Trans ICMC 2008;54:535–42.
J.E. Pienkos et al. / Cryogenics 50 (2010) 215–221 [3] Durand E. Magnetostatique. Paris: Saint-Germain; 1968. [4] Mills AF. Heat and mass transfer. 3rd ed. Chicago: Irwin; 1995. [5] Bean CP. Magnetization of high-field superconductors. Rev Mod Phys 1964;36:31–9. [6] Kim YB, Hempstead CF, Strnad AR. Critical persistent currents in hard superconductors. Phys Rev Lett 1962;9:306–9. [7] Schonborg N, Hornfeldt S. Model of the temperature dependence of the hysteresis losses in a high-temperature superconductor. Physica C 2002;372:1734–8.
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[8] Douine B, Berger K, Netter D, Leveque J, Rezzoug A. Influence of JC(B) on the full penetration current of superconducting tube. Physica C 2006;443:23–8. [9] Hong Z, Campbell AM, Coombs TA. Numerical solution of critical state in superconductivity by finite element software. Supercond Sci Technol 2006;19:1246–52. [10] Berger K, Leveque J, Netter D, Douine B, Rezzoug A. Influence of temperature and/or field dependences of the E–J power law on trapped magnetic field in bulk YbaCuO. IEEE Trans Appl Supercond 2007;17(2):3028–31.