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Evaluation of loss of current leads for HTS power apparatuses
Cryogenics 49 (2009) 263–266
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Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Evaluation of loss of current leads for HTS power apparatuses Mitsuho Furuse *, Koh Agatsuma, Shuichiro Fuchino National Institute of Advanced Industrial Science and Technology, Energy Technology Research Institute, Umezono 1-1-1, Tsukuba, Ibaraki 3058568, Japan
a r t i c l e
i n f o
Article history: Received 14 March 2008 Received in revised form 26 August 2008 Accepted 22 September 2008
Keywords: A. High-Tc superconductors C. Electrical conductivity C. Thermal conductivity D. Flow meters F. Current leads
a b s t r a c t This paper evaluates the losses of current leads made of various metals used in high-Tc superconducting power apparatuses operating at liquid nitrogen temperatures. The heat flow into liquid nitrogen conducting adiabatic current leads was measured by the nitrogen boil-off method. The results were used to derive average Lorenz numbers for metals between room temperature and liquid nitrogen temperatures. The average Lorenz number is a good index of performance of current leads. A theory of loss evaluation of current leads using the average Lorenz number and a method for the optimum current lead design were described as well. In addition, the loss of current leads made of Bi2223/Ag tape conductors was evaluated above the liquid nitrogen temperature, and a higher efficiency was achieved than with pure metal current leads. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Heat loads of current leads used for cryogenic apparatuses operating at liquid helium temperatures have been significantly reduced by the use of high-Tc superconductors (HTSs) [1–6]. Generally, HTSs are used below a thermal anchor at around 77 K. A conventional copper lead conducts current from a room temperature terminal to the thermal anchor stage. The Joule heat of the upper copper lead section and the heat conducted from the room temperature terminal are removed by the thermal anchor, vaporize liquid nitrogen (LN2) or become loads of refrigerators. Current leads of HTS power apparatuses operating at LN2 temperatures are also made of copper, usually operated under adiabatic conditions, just like current leads of refrigerator-cooled superconducting magnets. In a large current HTS power system, such as an electric power distribution system using several kA class HTS DC power cables [7], losses at the conventional copper leads greatly lowers the efficiency of the system. Thus, the heat flows of current leads from room temperature to LN2 temperatures (losses of current leads) should be carefully estimated in designing the superconducting systems and minimized as much as possible. There is much literature on the theoretical evaluation of losses of metal current leads [8–15]. But most of the work has been done under the assumption that the relationship between electrical resistivity q(T) and thermal conductivity k(T) of pure metals obeys the Wiedemann–Franz law, which states that the Lorenz number L = k(T)q(T)/T = 2.45 108 W X K2 is constant at any temperature T, or expanded Wiedemann–Franz laws, correcting discrepan* Corresponding author. Tel.: +81 29 861 3420; fax: +81 29 861 5822. E-mail address: [email protected] (M. Furuse). 0011-2275/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2008.09.004
cies in Lorenz numbers between the actual and theoretical values. The Wiedemann–Franz law is a good approximation when the temperature of the pure metal is above the Debye temperature, however, it is not suitable for precise analyses of losses of current leads used for cryogenic apparatuses. Fig. 1 shows the calculated Lorenz numbers for various metals determined using the typical literature values for electrical resistivity [16] and thermal conductivity [17]. But the values used in the calculation are of different specimens. Properties of metals generally depend on their purity and manufacturing process. Thus, Lorenz numbers should be determined from the measured electrical resistivity and thermal conductivity of the identical specimen. There have been some reports on the experimental evaluation of the losses of current leads made of various metals by the helium boil-off method [11,14,18,19]. In these studies conducted before the discovery of HTSs, losses of current leads from room temperature to liquid helium temperatures were evaluated for cryogenic apparatuses operating at liquid helium temperatures. As shown in Fig. 1, aluminum has the smallest Lorenz number in the 77– 300 K range. Mallon estimated the losses of aluminum current leads numerically [20], however, to the best of the present authors’ knowledge, no experimental evaluation of losses of aluminum current leads has yet been done in the range from room temperature to LN2 temperatures. We experimentally evaluated the heat flows into liquid nitrogen conducting adiabatic current leads made of various metals by the nitrogen boil-off method. We used the results to derive average Lorenz numbers for metals between room temperature and the LN2 temperature. The average Lorenz numbers of metals are quite useful in designing optimum current leads because they incorporate the electrical resistivity and thermal conductivity of
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M. Furuse et al. / Cryogenics 49 (2009) 263–266
from (1) by numerical analysis was 4.1 W, as determined by substituting I = 100 A, optimized l/S = 3.68 104 m1 and the literature data for resistivity [16] and thermal conductivity [17]. Likewise, the calculated heat flow using (3) was 4.3 W, obtained by substituting as average thermal conductivity ka = 430 W/m K and as gradient of resistivity a = 5.18 1011 X m/K; these values are taken from the literature [16,17]. Though there is a 5% error caused by the approximation of thermal conductivity and resistivity, heat flows of current leads can be obtained without numerical analyses. 2.2. Measurement of losses of current leads by the nitrogen boil-off method
Fig. 1. Temperature dependence of Lorenz numbers of various metals calculated from thermal conductivity and electrical resistivity data reported in the literature. The theoretical value of the Lorenz number according to the Wiedemann–Franz law is 2.45 108 W X K2 and this value temperature-independent. The calculated Lorenz numbers of aluminum are the lowest in the 77–300 K range. The Lorenz numbers shown here are calculated using the electrical resistivity and thermal conductivity of different specimens when in fact the properties of the same specimen should be used.
the same specimen. In this paper, we describe a theory of loss evaluation of current leads using the average Lorenz number and a method for optimum current lead design. In addition, we attempted further loss reduction of current leads using Bi2223/Ag tape conductors and obtained promising results. 2. Analysis and experimental evaluation of losses of current leads 2.1. Analysis of heat flow of current leads The one-dimensional thermal equilibrium equation for a metal current lead of cross-sectional area S and current flow I under adiabatic conditions can be expressed as
d dT qðTÞI2 þ kðTÞS ¼ 0: dx dx S
We evaluated losses of current leads using the measurement vacuum cryostat shown in Fig. 2. Heat flow into LN2 in a measurement reservoir, which is connected to the cold-ends of test leads with a thick copper block, is measured from the mass flow rate of nitrogen boil-off gas. The measurement LN2 reservoir is surrounded by a thermal shield at LN2 temperature. The radiation from the thermal shield to the test leads is estimated to be less than 10 mW, which is negligible. The flow meter was calibrated with a heater attached to the copper block. Fig. 3 plots the measured nitrogen gas flow rate vs. the heater input. Since the latent heat of liquid nitrogen at the boiling temperature is 160 kJ/l, the gas flow rate at 273 K for 1 W heater input can be estimated to be 60 [J/min]/160 [kJ/l] 650 [l@273 K/l@77 K] = 0.244 [l/min]. The measured data are well consistent with the calculated value. The materials of test leads are listed in Table 1. The dimensions of all test leads were length l = 0.5 m and diameter d = 5 mm. The aluminum test leads were soaked in a 5% sodium hydroxide solution in order to remove the surface layers of aluminum oxide. First, we measured the average thermal conductivity, ka, of the test leads between room temperature and the LN2 temperature. A single rod of the test leads, rather than a pair, was attached to the measurement cryostat. The temperature of the warm-end of the rod, Th, was fixed at room temperature (300 K) using the upper heater q during the measurement in order to eliminate heat flow from the atmosphere to the test specimen. From the measured heat flow into LN2, Q, which was consistent with q, and the temperatures at the warm-end (Th) and cold-end (Tl), ka is given by
ð1Þ
We denote the length of the lead by l, the temperature at the warm-end x = 0 by Th, the temperature at the cold-end x = l by Tl, and the average thermal conductivity of the material between Th and Tl by ka. We note that ka is temperature-independent. By integrating Eq. (1) assuming electrical resistivity of the material is proportional to temperature (q(T) = aT), the temperature distribution along the current lead is obtained
T¼
T l sin bx þ T h sin bðl xÞ ; sin bl
b¼
I S
rffiffiffiffiffi a : ka
ð2Þ
This approximation is reasonable for most metals in the range from room temperature to LN2 temperatures. The heat flow at the cold-end, Q, can be calculated via
Q ¼ ka S
pffiffiffiffiffiffiffiffi T h T l cos d dT ¼ I ka a ; dx x¼l sin d
rffiffiffiffiffi d¼I
al : ka S
ð3Þ
Current I is allowable in the range d < p/2 for a current lead with dimensions l/S. We validate (3), which is derived under the assumptions of constant thermal conductivity and proportional resistivity. The heat flow at the cold-end Q of an oxygen-free copper lead calculated
Fig. 2. Schematic diagram of the measurement vacuum cryostat.
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M. Furuse et al. / Cryogenics 49 (2009) 263–266 Table 2 Measurement results
(1) (2) (3) (4) (5)
CuOF CuETP AlLP AlHP Ag
Heat flow into LN2, Qc (W)
Average thermal conductivity, ka (W m1 K1)
Average Lorenz number, La (W X K2)
5.11 4.87 2.78 2.82 4.80
436 436 242 261 481
2.17 108 2.30 108 2.40 108 2.21 108 2.10 108
Table 3 Optimized designs losses of 100 A current leads
Fig. 3. Gas nitrogen flow rate vs. heater input. (1) (2) (3) (4) (5)
Table 1 Materials of test leads
(1) (2) (3) (4) (5)
CuOF CuETP AlLP AlHP Ag
Material
Purity (%)
Copper, oxygen free Copper, electrolytic tough pitch Aluminum, low purity Aluminum, high-purity Silver
99.99 99.9 99.7 99.99 99.99
a
CuOF CuETP AlLP AlHP Ag
Minimum heat flux, Qmin (W)
Optimized, l/S (m1)
Diameter, dopt (mm)a
4.3 4.4 4.5 4.3 4.2
3.9 104 3.8 104 2.0 104 2.3 104 4.4 104
4.0 4.1 5.6 5.3 3.8
Calculated values for l = 0.5 m.
tabulated in Table 2. The La can be considered to be the average Lorenz number between room temperature and the LN2 temperature. 2.3. Optimization of dimensions of current leads
dT Th Tl ¼ ka S ; dx l l Q : ) ka ¼ S Th Tl
Q ¼ ka S
ð4Þ
The measured Tl was higher than the LN2 boiling temperature (77.3 K) because of the temperature gradient caused by the copper block connecting the cold-end of the test lead and the LN2 reservoir. The temperature gradient of the copper block was 1.6 K for a heat flow of 1 W. We estimated discrepancies in the average thermal conductivity by Eq. (5) using the literature data on k(T)
R 300
kðTÞdT Err: ¼ 77 300 77
,R T h Tl
kðTÞdT
Th Tl
ð5Þ
:
The maximum error was negligible (1.5%) in the measurement of (3) AlLP. Thus, we take the measured ka to be the average thermal conductivity between room temperature and the LN2 temperature. Next, we evaluated the average Lorenz number La of test leads between room temperature and the LN2 temperature by the following procedure. A pair of test leads was attached to the measurement cryostat and conducted a direct current Ir, which was the rated current that Th reached room temperature. We measured the heat flow into LN2, 2Qc, since two leads were attached to the measurement cryostat. Here, we define the average Lorenz number La as
La ¼
ka qðTÞ ka aT ¼ ¼ ka a: T T
ð6Þ
When the material obeys the Wiedemann–Franz law, La equals L = 2.45 108 W X K2. But since the test leads contain impurities and the temperature is below the Debye temperature, La does not necessarily equal L. Eq. (3) reduces to
Q c ¼ Ir
pffiffiffiffiffi T h T l cos d La ; sin d
d ¼ Ir
pffiffiffiffiffi La l : ka S
ð7Þ
By numerical analyses substituting measured Qc, Th, Tl, k a and Ir into Eq. (7), we obtained the average Lorenz number La. The results are
From Eq. (7), loss of a current lead, carrying current I between room temperature and the LN2 temperature, attains a minimum when the dimensions of the lead satisfy
l 1 ka 77 ¼ pffiffiffiffiffi cos1 : Sopt I La 300
ð8Þ
The minimum loss Qmin is given by
Q min ¼ I
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi La ð3002 772 Þ:
ð9Þ
That is to say, Qmin is proportional to current I and the square root of the average Lorenz number of the material, La1/2. Table 3 tabulates the calculated Qmin of current leads made of different materials for a rated current I = 100 A, l/Sopt, diameter dopt for a lead length l of 0.5 m. 3. Results and discussion We concluded that among the materials tested, oxygen-free copper is the most suitable material for current leads. As shown in Table 2, pure silver has the smallest average Lorenz number, however, the advantage over oxygen-free copper is only slight. La of low purity electrolytic tough pitch copper is 6% higher than that of oxygen-free copper. Electrolytic tough pitch copper is used for electric wires and is readily available. It may be used for current leads when efficiency lowering is allowable. Experimentally evaluated La of high-purity aluminum was larger than that of oxygen-free copper, though the value calculated from data reported in the literature was the smallest, as shown in Fig. 1. The measured average thermal conductivity ka of highpurity aluminum was in accord with the data reported in the literature, indicating that the electrical resistivity was high. The most likely cause of high electrical resistivity is the insufficient removal of surface layers of aluminum oxide. Aluminum is readily oxidized in the atmosphere, and thus, complete removal of the oxide layer is difficult.
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M. Furuse et al. / Cryogenics 49 (2009) 263–266
Table 4 Test results of Bi2223/Ag current lead
(6) Bi2223/Aga a
Heat flow Rated into LN2, Qc current, (W) Ir (A)
Average thermal conductivity, ka (W m1 K1)
Average Lorenz number, La (W X K2)
1.62
254
2.07 108
40
Ten tapes are bound.
Table 5 Comparison of losses of current leads optimized for 40 A
Qcmin Normalized value
(6) Bi2223/Ag
(1) CuOF
(5) Ag
1.62 W 1.00
1.71 W 1.02
1.68 W 1.01
In an attempt to reduce the losses of current leads, we studied the use of HTSs for current leads. Since the critical temperature Tc of Bi2223 is approximately 110 K, the electrical resistivity of current leads can be reduced if current flows through Bi2223 between Tc and the LN2 temperature. It is desirable for superconductors to be embedded within the matrix so that current transfers from metals to HTSs with small resistance. Thus, we fabricated a current lead binding 10 commercial Bi2223/Ag tape superconductors (manufactured by Sumitomo Electric Industries Ltd.; critical current of a tape conductor was 120 A). The transport current flows through the silver alloy matrix above Tc, gradually transfers to Bi2223 between Tc and the current-sharing temperature. Iwasa et al. have examined Bi2223/Ag current leads operating in the current-sharing mode [21], however, only below 80 K. We have obtained promising results, as summarized in Table 4. Although the matrix of Bi2223/Ag tape conductors is a silver alloy, the average Lorenz number obtained is smaller than that of pure silver. This is thought to be because the Joule heat is eliminated by current flowing through Bi2223. Eqs. (7) and (8) are not applicable to HTS current leads because of the assumption of electrical resistivity linearity. Thus, we designed 40 A current leads made of oxygen-free copper and pure silver. The evaluated Qcmin are tabulated in Table 5. The loss of the Bi2223/Ag current lead is 2% smaller than that of the oxygen-free copper lead, and 1% smaller than that of the pure silver lead. The amount of Bi2223 is clearly superfluous for the transport current and Bi2223 is useless above Tc. Optimization of the crosssectional area and the ratio of Bi2223 to metal are crucial for the practical use of HTS tape current leads. Numerical analyses of temperature distribution along the HTS tape current leads would be required to estimate the upper limit of transport current because the excess current causes burnout. These are topics for future study. 4. Conclusion We have experimentally estimated the average thermal conductivity ka and the average Lorenz numbers La of metals between room temperature and the LN2 temperature through measure-
ments of losses of current leads. ka and La are good indices of performance for current leads. While it is difficult to simultaneously measure the data of temperature dependence of thermal conductivity and electrical resistivity of the same specimen, La is the arithmetic mean of a function having both factors. By substituting ka and La into Eqs. (8) and (9), we can readily design optimized current leads without knowing the accurate temperature dependence of material properties. The measured losses and La of aluminum leads were unexpectedly larger than the values estimated on the basis of data reported in the literature. The fabrication of low-loss aluminum current leads requires the development of oxide removal technique. We have attempted to use commercial Bi2223/Ag tape conductors as current leads between room temperature and the LN2 temperature. The measured losses of the HTS current leads were lower than those of pure metals. We believe that the efficiency of the HTS current leads could be improved by optimizing the amount of HTS. References [1] Matrone A, Rosatelli G, Vaccarone R. Current leads with high Tc superconductors bus bars. IEEE Trans Magn 1989;25:1742. [2] Mumford FJ. Superconducting current-leads made from high Tc superconductor and normal metal conductor. Cryogenics 1989;29:206. [3] Hull JR. High temperature superconducting current leads for cryogenic apparatus. Cryogenics 1989;29:1116. [4] Wu JL, Dederer JT, Eckels PW, Singh SK, Hull JR, Poeppel RB, et al. Design and testing of a high temperature superconducting current lead. IEEE Trans Magn 1991;27:1861. [5] Grivon F, Leriche A, Cottevieille C, Kermarrec JC, Petitbon A, Fevrier A, et al. YBaCuO current lead for liquid helium temperature applications. IEEE Trans Magn 1991;27:1866. [6] Niemann RC, Cha YS, Hull JR. Performance measurements of superconducting current leads with low helium boil-off rates. IEEE Trans Appl Supercond 1993;3:392. [7] Furuse M, Fuchino S, Higuchi N, Ishii I. Feasibility study of low voltage DC superconducting distribution system. IEEE Trans Appl Supercond 2005;15: 1759. [8] McFee R. Optimum input leads for cryogenic apparatus. Rev Sci Instrum 1959;30:98. [9] Mercouroff W. Minimization of thermal losses due to electrical connections in cryostats. Cryogenics 1963;3:171. [10] Donadieu L, Dammann C. Theory of gas-cooled current-carrying leads for cryogenic apparatus. In: Proceedings of ICEC2; 1968. p. 200–2. [11] Scott JP. Current leads for use in liquid-helium cryostats. In: Proceedings of ICEC3; 1970. p. 176–81. [12] Rauh M. Optimum dimensions of current-carrying leads to cryogenic apparatus. In: Proceedings of ICEC3; 1970. p. 182–6. [13] Kohler JWL, Prast G, DE Jonge AK. Calculation of losses induced by currentcarrying leads in cryogenic installations. In: Proceedings of ICEC3; 1970. p. 192–6. [14] Buyanov YL, Fradkov AK, Shebalin IY. A review of current leads for cryogenic devices. Cryogenics 1975;15:193. [15] Wilson MN. Superconducting magnets. Oxford: Clarendon Press; 1983. [16] For example Schwartz FR et al. Cryogenic materials data handbook. Air Force Materials Laboratory; 1970. [17] For example Touloukian YS et al. Thermal conductivity – metallic elements and alloys. Plenum; 1970. [18] Deiness S. The production and optimization of high current leads. Cryogenics 1965;5:269. [19] Oberhauser CJ, Sukhatme SP. Evaluation of optimum current-carrying leads for cryogenic apparatus. Adv Cryo Eng 1967;3:322. [20] Mallon RG. Optimum electrical leads of aluminum and sodium for cryogenic apparatus. Rev Sci Instrum 1962;33:564. [21] Iwasa Y, Lee H. High-temperature superconducting current lead incorporating operation in the current-sharing mode. Cryogenics 2000;40:209.
Contents lists available at ScienceDirect
Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Evaluation of loss of current leads for HTS power apparatuses Mitsuho Furuse *, Koh Agatsuma, Shuichiro Fuchino National Institute of Advanced Industrial Science and Technology, Energy Technology Research Institute, Umezono 1-1-1, Tsukuba, Ibaraki 3058568, Japan
a r t i c l e
i n f o
Article history: Received 14 March 2008 Received in revised form 26 August 2008 Accepted 22 September 2008
Keywords: A. High-Tc superconductors C. Electrical conductivity C. Thermal conductivity D. Flow meters F. Current leads
a b s t r a c t This paper evaluates the losses of current leads made of various metals used in high-Tc superconducting power apparatuses operating at liquid nitrogen temperatures. The heat flow into liquid nitrogen conducting adiabatic current leads was measured by the nitrogen boil-off method. The results were used to derive average Lorenz numbers for metals between room temperature and liquid nitrogen temperatures. The average Lorenz number is a good index of performance of current leads. A theory of loss evaluation of current leads using the average Lorenz number and a method for the optimum current lead design were described as well. In addition, the loss of current leads made of Bi2223/Ag tape conductors was evaluated above the liquid nitrogen temperature, and a higher efficiency was achieved than with pure metal current leads. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Heat loads of current leads used for cryogenic apparatuses operating at liquid helium temperatures have been significantly reduced by the use of high-Tc superconductors (HTSs) [1–6]. Generally, HTSs are used below a thermal anchor at around 77 K. A conventional copper lead conducts current from a room temperature terminal to the thermal anchor stage. The Joule heat of the upper copper lead section and the heat conducted from the room temperature terminal are removed by the thermal anchor, vaporize liquid nitrogen (LN2) or become loads of refrigerators. Current leads of HTS power apparatuses operating at LN2 temperatures are also made of copper, usually operated under adiabatic conditions, just like current leads of refrigerator-cooled superconducting magnets. In a large current HTS power system, such as an electric power distribution system using several kA class HTS DC power cables [7], losses at the conventional copper leads greatly lowers the efficiency of the system. Thus, the heat flows of current leads from room temperature to LN2 temperatures (losses of current leads) should be carefully estimated in designing the superconducting systems and minimized as much as possible. There is much literature on the theoretical evaluation of losses of metal current leads [8–15]. But most of the work has been done under the assumption that the relationship between electrical resistivity q(T) and thermal conductivity k(T) of pure metals obeys the Wiedemann–Franz law, which states that the Lorenz number L = k(T)q(T)/T = 2.45 108 W X K2 is constant at any temperature T, or expanded Wiedemann–Franz laws, correcting discrepan* Corresponding author. Tel.: +81 29 861 3420; fax: +81 29 861 5822. E-mail address: [email protected] (M. Furuse). 0011-2275/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2008.09.004
cies in Lorenz numbers between the actual and theoretical values. The Wiedemann–Franz law is a good approximation when the temperature of the pure metal is above the Debye temperature, however, it is not suitable for precise analyses of losses of current leads used for cryogenic apparatuses. Fig. 1 shows the calculated Lorenz numbers for various metals determined using the typical literature values for electrical resistivity [16] and thermal conductivity [17]. But the values used in the calculation are of different specimens. Properties of metals generally depend on their purity and manufacturing process. Thus, Lorenz numbers should be determined from the measured electrical resistivity and thermal conductivity of the identical specimen. There have been some reports on the experimental evaluation of the losses of current leads made of various metals by the helium boil-off method [11,14,18,19]. In these studies conducted before the discovery of HTSs, losses of current leads from room temperature to liquid helium temperatures were evaluated for cryogenic apparatuses operating at liquid helium temperatures. As shown in Fig. 1, aluminum has the smallest Lorenz number in the 77– 300 K range. Mallon estimated the losses of aluminum current leads numerically [20], however, to the best of the present authors’ knowledge, no experimental evaluation of losses of aluminum current leads has yet been done in the range from room temperature to LN2 temperatures. We experimentally evaluated the heat flows into liquid nitrogen conducting adiabatic current leads made of various metals by the nitrogen boil-off method. We used the results to derive average Lorenz numbers for metals between room temperature and the LN2 temperature. The average Lorenz numbers of metals are quite useful in designing optimum current leads because they incorporate the electrical resistivity and thermal conductivity of
264
M. Furuse et al. / Cryogenics 49 (2009) 263–266
from (1) by numerical analysis was 4.1 W, as determined by substituting I = 100 A, optimized l/S = 3.68 104 m1 and the literature data for resistivity [16] and thermal conductivity [17]. Likewise, the calculated heat flow using (3) was 4.3 W, obtained by substituting as average thermal conductivity ka = 430 W/m K and as gradient of resistivity a = 5.18 1011 X m/K; these values are taken from the literature [16,17]. Though there is a 5% error caused by the approximation of thermal conductivity and resistivity, heat flows of current leads can be obtained without numerical analyses. 2.2. Measurement of losses of current leads by the nitrogen boil-off method
Fig. 1. Temperature dependence of Lorenz numbers of various metals calculated from thermal conductivity and electrical resistivity data reported in the literature. The theoretical value of the Lorenz number according to the Wiedemann–Franz law is 2.45 108 W X K2 and this value temperature-independent. The calculated Lorenz numbers of aluminum are the lowest in the 77–300 K range. The Lorenz numbers shown here are calculated using the electrical resistivity and thermal conductivity of different specimens when in fact the properties of the same specimen should be used.
the same specimen. In this paper, we describe a theory of loss evaluation of current leads using the average Lorenz number and a method for optimum current lead design. In addition, we attempted further loss reduction of current leads using Bi2223/Ag tape conductors and obtained promising results. 2. Analysis and experimental evaluation of losses of current leads 2.1. Analysis of heat flow of current leads The one-dimensional thermal equilibrium equation for a metal current lead of cross-sectional area S and current flow I under adiabatic conditions can be expressed as
d dT qðTÞI2 þ kðTÞS ¼ 0: dx dx S
We evaluated losses of current leads using the measurement vacuum cryostat shown in Fig. 2. Heat flow into LN2 in a measurement reservoir, which is connected to the cold-ends of test leads with a thick copper block, is measured from the mass flow rate of nitrogen boil-off gas. The measurement LN2 reservoir is surrounded by a thermal shield at LN2 temperature. The radiation from the thermal shield to the test leads is estimated to be less than 10 mW, which is negligible. The flow meter was calibrated with a heater attached to the copper block. Fig. 3 plots the measured nitrogen gas flow rate vs. the heater input. Since the latent heat of liquid nitrogen at the boiling temperature is 160 kJ/l, the gas flow rate at 273 K for 1 W heater input can be estimated to be 60 [J/min]/160 [kJ/l] 650 [l@273 K/l@77 K] = 0.244 [l/min]. The measured data are well consistent with the calculated value. The materials of test leads are listed in Table 1. The dimensions of all test leads were length l = 0.5 m and diameter d = 5 mm. The aluminum test leads were soaked in a 5% sodium hydroxide solution in order to remove the surface layers of aluminum oxide. First, we measured the average thermal conductivity, ka, of the test leads between room temperature and the LN2 temperature. A single rod of the test leads, rather than a pair, was attached to the measurement cryostat. The temperature of the warm-end of the rod, Th, was fixed at room temperature (300 K) using the upper heater q during the measurement in order to eliminate heat flow from the atmosphere to the test specimen. From the measured heat flow into LN2, Q, which was consistent with q, and the temperatures at the warm-end (Th) and cold-end (Tl), ka is given by
ð1Þ
We denote the length of the lead by l, the temperature at the warm-end x = 0 by Th, the temperature at the cold-end x = l by Tl, and the average thermal conductivity of the material between Th and Tl by ka. We note that ka is temperature-independent. By integrating Eq. (1) assuming electrical resistivity of the material is proportional to temperature (q(T) = aT), the temperature distribution along the current lead is obtained
T¼
T l sin bx þ T h sin bðl xÞ ; sin bl
b¼
I S
rffiffiffiffiffi a : ka
ð2Þ
This approximation is reasonable for most metals in the range from room temperature to LN2 temperatures. The heat flow at the cold-end, Q, can be calculated via
Q ¼ ka S
pffiffiffiffiffiffiffiffi T h T l cos d dT ¼ I ka a ; dx x¼l sin d
rffiffiffiffiffi d¼I
al : ka S
ð3Þ
Current I is allowable in the range d < p/2 for a current lead with dimensions l/S. We validate (3), which is derived under the assumptions of constant thermal conductivity and proportional resistivity. The heat flow at the cold-end Q of an oxygen-free copper lead calculated
Fig. 2. Schematic diagram of the measurement vacuum cryostat.
265
M. Furuse et al. / Cryogenics 49 (2009) 263–266 Table 2 Measurement results
(1) (2) (3) (4) (5)
CuOF CuETP AlLP AlHP Ag
Heat flow into LN2, Qc (W)
Average thermal conductivity, ka (W m1 K1)
Average Lorenz number, La (W X K2)
5.11 4.87 2.78 2.82 4.80
436 436 242 261 481
2.17 108 2.30 108 2.40 108 2.21 108 2.10 108
Table 3 Optimized designs losses of 100 A current leads
Fig. 3. Gas nitrogen flow rate vs. heater input. (1) (2) (3) (4) (5)
Table 1 Materials of test leads
(1) (2) (3) (4) (5)
CuOF CuETP AlLP AlHP Ag
Material
Purity (%)
Copper, oxygen free Copper, electrolytic tough pitch Aluminum, low purity Aluminum, high-purity Silver
99.99 99.9 99.7 99.99 99.99
a
CuOF CuETP AlLP AlHP Ag
Minimum heat flux, Qmin (W)
Optimized, l/S (m1)
Diameter, dopt (mm)a
4.3 4.4 4.5 4.3 4.2
3.9 104 3.8 104 2.0 104 2.3 104 4.4 104
4.0 4.1 5.6 5.3 3.8
Calculated values for l = 0.5 m.
tabulated in Table 2. The La can be considered to be the average Lorenz number between room temperature and the LN2 temperature. 2.3. Optimization of dimensions of current leads
dT Th Tl ¼ ka S ; dx l l Q : ) ka ¼ S Th Tl
Q ¼ ka S
ð4Þ
The measured Tl was higher than the LN2 boiling temperature (77.3 K) because of the temperature gradient caused by the copper block connecting the cold-end of the test lead and the LN2 reservoir. The temperature gradient of the copper block was 1.6 K for a heat flow of 1 W. We estimated discrepancies in the average thermal conductivity by Eq. (5) using the literature data on k(T)
R 300
kðTÞdT Err: ¼ 77 300 77
,R T h Tl
kðTÞdT
Th Tl
ð5Þ
:
The maximum error was negligible (1.5%) in the measurement of (3) AlLP. Thus, we take the measured ka to be the average thermal conductivity between room temperature and the LN2 temperature. Next, we evaluated the average Lorenz number La of test leads between room temperature and the LN2 temperature by the following procedure. A pair of test leads was attached to the measurement cryostat and conducted a direct current Ir, which was the rated current that Th reached room temperature. We measured the heat flow into LN2, 2Qc, since two leads were attached to the measurement cryostat. Here, we define the average Lorenz number La as
La ¼
ka qðTÞ ka aT ¼ ¼ ka a: T T
ð6Þ
When the material obeys the Wiedemann–Franz law, La equals L = 2.45 108 W X K2. But since the test leads contain impurities and the temperature is below the Debye temperature, La does not necessarily equal L. Eq. (3) reduces to
Q c ¼ Ir
pffiffiffiffiffi T h T l cos d La ; sin d
d ¼ Ir
pffiffiffiffiffi La l : ka S
ð7Þ
By numerical analyses substituting measured Qc, Th, Tl, k a and Ir into Eq. (7), we obtained the average Lorenz number La. The results are
From Eq. (7), loss of a current lead, carrying current I between room temperature and the LN2 temperature, attains a minimum when the dimensions of the lead satisfy
l 1 ka 77 ¼ pffiffiffiffiffi cos1 : Sopt I La 300
ð8Þ
The minimum loss Qmin is given by
Q min ¼ I
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi La ð3002 772 Þ:
ð9Þ
That is to say, Qmin is proportional to current I and the square root of the average Lorenz number of the material, La1/2. Table 3 tabulates the calculated Qmin of current leads made of different materials for a rated current I = 100 A, l/Sopt, diameter dopt for a lead length l of 0.5 m. 3. Results and discussion We concluded that among the materials tested, oxygen-free copper is the most suitable material for current leads. As shown in Table 2, pure silver has the smallest average Lorenz number, however, the advantage over oxygen-free copper is only slight. La of low purity electrolytic tough pitch copper is 6% higher than that of oxygen-free copper. Electrolytic tough pitch copper is used for electric wires and is readily available. It may be used for current leads when efficiency lowering is allowable. Experimentally evaluated La of high-purity aluminum was larger than that of oxygen-free copper, though the value calculated from data reported in the literature was the smallest, as shown in Fig. 1. The measured average thermal conductivity ka of highpurity aluminum was in accord with the data reported in the literature, indicating that the electrical resistivity was high. The most likely cause of high electrical resistivity is the insufficient removal of surface layers of aluminum oxide. Aluminum is readily oxidized in the atmosphere, and thus, complete removal of the oxide layer is difficult.
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Table 4 Test results of Bi2223/Ag current lead
(6) Bi2223/Aga a
Heat flow Rated into LN2, Qc current, (W) Ir (A)
Average thermal conductivity, ka (W m1 K1)
Average Lorenz number, La (W X K2)
1.62
254
2.07 108
40
Ten tapes are bound.
Table 5 Comparison of losses of current leads optimized for 40 A
Qcmin Normalized value
(6) Bi2223/Ag
(1) CuOF
(5) Ag
1.62 W 1.00
1.71 W 1.02
1.68 W 1.01
In an attempt to reduce the losses of current leads, we studied the use of HTSs for current leads. Since the critical temperature Tc of Bi2223 is approximately 110 K, the electrical resistivity of current leads can be reduced if current flows through Bi2223 between Tc and the LN2 temperature. It is desirable for superconductors to be embedded within the matrix so that current transfers from metals to HTSs with small resistance. Thus, we fabricated a current lead binding 10 commercial Bi2223/Ag tape superconductors (manufactured by Sumitomo Electric Industries Ltd.; critical current of a tape conductor was 120 A). The transport current flows through the silver alloy matrix above Tc, gradually transfers to Bi2223 between Tc and the current-sharing temperature. Iwasa et al. have examined Bi2223/Ag current leads operating in the current-sharing mode [21], however, only below 80 K. We have obtained promising results, as summarized in Table 4. Although the matrix of Bi2223/Ag tape conductors is a silver alloy, the average Lorenz number obtained is smaller than that of pure silver. This is thought to be because the Joule heat is eliminated by current flowing through Bi2223. Eqs. (7) and (8) are not applicable to HTS current leads because of the assumption of electrical resistivity linearity. Thus, we designed 40 A current leads made of oxygen-free copper and pure silver. The evaluated Qcmin are tabulated in Table 5. The loss of the Bi2223/Ag current lead is 2% smaller than that of the oxygen-free copper lead, and 1% smaller than that of the pure silver lead. The amount of Bi2223 is clearly superfluous for the transport current and Bi2223 is useless above Tc. Optimization of the crosssectional area and the ratio of Bi2223 to metal are crucial for the practical use of HTS tape current leads. Numerical analyses of temperature distribution along the HTS tape current leads would be required to estimate the upper limit of transport current because the excess current causes burnout. These are topics for future study. 4. Conclusion We have experimentally estimated the average thermal conductivity ka and the average Lorenz numbers La of metals between room temperature and the LN2 temperature through measure-
ments of losses of current leads. ka and La are good indices of performance for current leads. While it is difficult to simultaneously measure the data of temperature dependence of thermal conductivity and electrical resistivity of the same specimen, La is the arithmetic mean of a function having both factors. By substituting ka and La into Eqs. (8) and (9), we can readily design optimized current leads without knowing the accurate temperature dependence of material properties. The measured losses and La of aluminum leads were unexpectedly larger than the values estimated on the basis of data reported in the literature. The fabrication of low-loss aluminum current leads requires the development of oxide removal technique. We have attempted to use commercial Bi2223/Ag tape conductors as current leads between room temperature and the LN2 temperature. The measured losses of the HTS current leads were lower than those of pure metals. We believe that the efficiency of the HTS current leads could be improved by optimizing the amount of HTS. References [1] Matrone A, Rosatelli G, Vaccarone R. Current leads with high Tc superconductors bus bars. IEEE Trans Magn 1989;25:1742. [2] Mumford FJ. Superconducting current-leads made from high Tc superconductor and normal metal conductor. Cryogenics 1989;29:206. [3] Hull JR. High temperature superconducting current leads for cryogenic apparatus. Cryogenics 1989;29:1116. [4] Wu JL, Dederer JT, Eckels PW, Singh SK, Hull JR, Poeppel RB, et al. Design and testing of a high temperature superconducting current lead. IEEE Trans Magn 1991;27:1861. [5] Grivon F, Leriche A, Cottevieille C, Kermarrec JC, Petitbon A, Fevrier A, et al. YBaCuO current lead for liquid helium temperature applications. IEEE Trans Magn 1991;27:1866. [6] Niemann RC, Cha YS, Hull JR. Performance measurements of superconducting current leads with low helium boil-off rates. IEEE Trans Appl Supercond 1993;3:392. [7] Furuse M, Fuchino S, Higuchi N, Ishii I. Feasibility study of low voltage DC superconducting distribution system. IEEE Trans Appl Supercond 2005;15: 1759. [8] McFee R. Optimum input leads for cryogenic apparatus. Rev Sci Instrum 1959;30:98. [9] Mercouroff W. Minimization of thermal losses due to electrical connections in cryostats. Cryogenics 1963;3:171. [10] Donadieu L, Dammann C. Theory of gas-cooled current-carrying leads for cryogenic apparatus. In: Proceedings of ICEC2; 1968. p. 200–2. [11] Scott JP. Current leads for use in liquid-helium cryostats. In: Proceedings of ICEC3; 1970. p. 176–81. [12] Rauh M. Optimum dimensions of current-carrying leads to cryogenic apparatus. In: Proceedings of ICEC3; 1970. p. 182–6. [13] Kohler JWL, Prast G, DE Jonge AK. Calculation of losses induced by currentcarrying leads in cryogenic installations. In: Proceedings of ICEC3; 1970. p. 192–6. [14] Buyanov YL, Fradkov AK, Shebalin IY. A review of current leads for cryogenic devices. Cryogenics 1975;15:193. [15] Wilson MN. Superconducting magnets. Oxford: Clarendon Press; 1983. [16] For example Schwartz FR et al. Cryogenic materials data handbook. Air Force Materials Laboratory; 1970. [17] For example Touloukian YS et al. Thermal conductivity – metallic elements and alloys. Plenum; 1970. [18] Deiness S. The production and optimization of high current leads. Cryogenics 1965;5:269. [19] Oberhauser CJ, Sukhatme SP. Evaluation of optimum current-carrying leads for cryogenic apparatus. Adv Cryo Eng 1967;3:322. [20] Mallon RG. Optimum electrical leads of aluminum and sodium for cryogenic apparatus. Rev Sci Instrum 1962;33:564. [21] Iwasa Y, Lee H. High-temperature superconducting current lead incorporating operation in the current-sharing mode. Cryogenics 2000;40:209.