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Thermal anchoring of conduction-cooled current leads for superconductivity applications near liquid nitrogen temperature
Cryogenics 50 (2010) 287–291
Contents lists available at ScienceDirect
Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Technical Note
Thermal anchoring of conduction-cooled current leads for superconductivity applications near liquid nitrogen temperature Sangkwon Jeong *, Youngkwon Kim 1 Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 June 2009 Received in revised form 18 November 2009 Accepted 17 December 2009
Keywords: Thermal anchoring Conduction-cooled Current lead High temperature superconductor (HTS)
a b s t r a c t As the YBa2Cu3O7–d, or YBCO, superconductor is commercially developed and utilized for various HTS (high temperature superconductor) applications such as motor, generator, and fault current limiter, the cryocooling for 50 or 60 K range is more demanded than ever. In this case, non-superconducting current leads instead of HTS ones need to be used for energization from room temperature all the way to the cryogenic operating temperature. This technical note describes a simple method of reducing cooling load requirement for those HTS applications. Non-superconducting current leads are to be thermally anchored at an appropriate intermediate cryogenic temperature before they are connected to the application target temperature. The optimum thermal-anchoring temperature and its configuration have been obtained to minimize the required cryocooler’s cooling capacity for practical as well as ideal cases. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Development of commercially available YBCO (YBa2Cu3O7–d) superconductor impact several application areas as an enabling technology due to its ever-elevated operating temperature range at moderately high magnetic field where the performance of 1G HTS wire (BSCCO; Bi2Sr2Ca1Cu2O8 or Bi2Sr2Ca2Cu3O10) was not acceptable [1]. The operating temperature over 60 K is now feasible for realizing not only superconducting cable but also superconducting fault current limiter, transformer, motor, and generator [2– 6]. It is also noticeable that recent development of Bi-2223 tape could elevate the operating temperature and widen its application range as a competition of YBCO [7]. These applications indispensably require electrical connection from room temperature for energization and the most significant heat load to the cryogenic system is often one due to current leads. For successful commercialization and wide spread of superconducting application, using a small and therefore inexpensive cryogenic refrigeration system is of paramount importance, which demands minimization of cooling load. It was common practice to incorporate superconducting current leads partially in previous low-temperature superconductor (LTS) or 1G HTS wire systems. Superconducting current leads were connected between the low-temperature superconducting system which was typically cooled below 30 K and the non-superconducting current leads which were connected to room temperature
region. Since the joint between the non-superconducting current leads and the superconducting ones were thermally anchored, the resultant heat leak through the current lead assembly has been greatly reduced due to low thermal conductivity of superconducting current leads. In the case of recent or future superconductor application near liquid nitrogen temperature, however, non-superconducting current leads need to be used from room temperature all the way to the cryogenic temperature where it operates. Therefore, heat leak through the non-superconducting current leads will again impose major cooling load to the cryogenic cooling system of such an application. This paper emphasizes the importance of thermal anchoring of non-superconducting current leads at intermediate cryogenic temperature and describes how to reduce the cooling load requirement in general. For simple analysis, we will consider a conduction-cooled superconducting coil which is cooled by a single or multiple cryocooler units. This thermal anchoring concept will be also applicable for cases where only one cryocooler is available for cooling superconducting coil. In such a case when the cold head of cryocooler is available at single location, the thermal anchoring position can be the middle part of recuperative heat exchanger or regenerator as suggested by others [8].
2. Conduction-cooled current leads 2.1. Cooling for thermally anchored current leads
* Corresponding author. Tel.: +82 42 350 3039; fax: +82 42 869 8207. E-mail addresses: [email protected] (S. Jeong), [email protected] (Y. Kim). 1 Tel.: +82 42 350 3079; fax: +82 42 350 8207. 0011-2275/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2009.12.006
Fig. 1 shows the schematic configuration of refrigeration cycle where the major cooling load is due to current leads. Fig. 1a is
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Nomenclature A COP I L L0 k Q Q* T W W* x
cross-sectional area (m2) coefficient of performance () current (A) length of current lead (m) Lorenz number (24.45 nW X K2) thermal conductivity (W m1 K1) cooling load or heat transfer rate (W) normalized cooling load or heat transfer (K) temperature (K) work (W) normalized work (K) position (m)
2 a avg CL0 CL1 CL2 H L m tot
Greek
a f
current lead between intermediate temperature and load temperature additional thermal load average current lead connected between room temperature and load temperature current lead connected between room temperature and intermediate temperature current lead connected between intermediate temperature and load temperature room temperature stage load temperature stage intermediate stage total
configuration parameter () non-dimensional position (=x/L)
Subscript 0 current lead not thermally anchored 1 current lead between room temperature and intermediate temperature
the schematic configuration of current lead connection without thermal anchoring while Fig. 1b is one with thermal anchoring at intermediate temperature, Tm. Assuming that an ideal Stirling refrigerator is utilized as a cryocooler, we can hypothetically construct two different cryogenic cooling systems and calculate how much overall cooling load is reduced solely by thermal anchoring. In the case of thermal anchoring, conduction heat leak through the warmer current lead from room temperature is blocked by a refrigerator operating between room temperature, TH and the intermediate temperature, Tm. When the current leads between Tm and the load temperature, TL are also optimized, the cooling load will be further diminished. It is interesting to notice that proper selection of the intermediated temperature, Tm shall minimize the total refrigeration work to maintain cryogenic load temperature, TL of the system. In the following analysis, QCL is the conduction heat load through the current leads and W0 is the required ideal work for refrigeration without thermal anchoring. W1 and W2 are the required cooling work for refrigeration operating between TH and Tm and between Tm and TL respectively when thermal anchoring is implemented. The work required to cool down the current leads is expressed as the following equations.
W 0 ¼ Q CL0 =COP0 ; for not-anchored current leads W tot ¼ W 1 þ W 2 ¼ Q CL1 =COP1 þ Q CL2 =COP0 ; for thermally anchored current leads
d dT 1 I2 L2 qðT 1 Þ þ 1 kðTÞA1 ¼ 0 and B:C: : T 1 ð0Þ ¼ T m ; df1 A1 df1 T 1 ð1Þ ¼ T H for CL1 d dT 2 I2 L2 qðT 2 Þ kðTÞA2 ¼ 0 and B:C: : T 2 ð0Þ ¼ T L ; þ 2 df2 A2 df2
ð3Þ
T 2 ð1Þ ¼ T m
ð4Þ
for CL2
where f = x/L and L, A, I, k, q and T are length of current lead, crosssectional area of current lead, current, thermal conductivity, resistivity and temperature, respectively. The governing equations above
ð1Þ ð2Þ
2.2. Optimization of current leads For not-thermally-anchored current leads shown in Fig. 1a, its cooling load (QCL0) and optimized configuration can be easily determined as reported by previous researchers [9,10]. In the case of thermally anchored current leads, the separate conduction heat transfer equations including Joule-heating by the flowing current are solved to determine the optimum configuration. Two current leads are considered at different temperature ranges. They can have dissimilar configuration but they should have same temperature at the anchored position. From now on, the current leads installed between TH and Tm will be called ‘‘CL1” and another one called ‘‘CL2”.
Fig. 1. Schematic energy flow in refrigerator with current lead connection (a) without thermal anchoring and (b) with thermal anchoring at intermediate temperature.
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are simplified if the thermal conductivity is assumed constant as an average value for the temperature range we are interested in, but the resistivity is a function of temperature to satisfy the Wiedemann–Franz law. 2
d T1 df21
þ a21 T 1 ¼ 0 and B:C: : T 1 ð0Þ ¼ T m ; T 1 ð1Þ ¼ T H
for CL1 ð5Þ
2
d T2 df22
þ a22 T 2 ¼ 0 and B:C: : T 2 ð0Þ ¼ T L ; T 2 ð1Þ ¼ T m
for CL2 ð6Þ
pffiffiffiffiffi pffiffiffiffiffi where a1 ¼ L0 IL1 =ðA1 kav g Þ; a2 ¼ L0 IL2 =ðA2 kav g Þ and L0 is Lorenz number. We used the constant theoretical value of the Lorenz number neglecting its temperature dependence [11]. The variable a represents the physical configuration of the current lead at a given material and an operating current. The temperature profile of each current lead can be achieved by solving the above equations.
T 1 ðf1 Þ ¼ T m cosða1 f1 Þ þ T 2 ðf2 Þ ¼ T L cosða2 f2 Þ þ
½T H T m cosða1 Þ sinða1 f1 Þ sinða1 Þ
½T m T L cosða2 Þ sinða2 f2 Þ sinða2 Þ
for CL1
for CL2
ð7Þ
ð8Þ
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 pffiffiffiffiffi T 2H T 2m L0 I L1 A a1 ¼ ð14Þ ¼ arctan@ Tm kav g A1 pffiffiffiffiffi L I L a2 ¼ 0 2 kav g A2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h i3 2 2 ðT T Þ ðT þ T Þ T ðT þ T Þ 2T T T m L m L m L H m L H 6 7 7 ð15Þ ¼ arctan6 4 5 T m T L ð2T H þ T m þ T L Þ Fig. 2 shows the optimized configuration parameters according to the intermediate temperature when the warm end temperature is 300 K and the superconductivity load temperature is 60 K. The optimum value of configuration parameter (a) tends to decrease as the two end temperatures of the current lead approach. This means a large cross-sectional area of current lead is favorable for small temperature span of the current lead. The cooling load due to the current lead is composed of the conduction heat transfer through it and Joule-heating of electrical current. When the temperature difference between two ends is smaller, it is advantageous to have large cross-sectional area to reduce the amount of Jouleheating in the current lead. Especially, the optimum configuration of CL1 (a1) is expressed as Eq. (14) and it is the same result achieved by conventional optimization method of current lead that is situated between TH and Tm
The cooling loads at the anchored stage and the load temperature can be calculated from the achieved temperature profiles. For example, the conduction heat transfer at the end of the CL1 is expressed as Eq. (9) and other cooling loads can be expressed in the same way.
1 dT 1 A1 kav g dT 1 Q m1 T T m cosða1 Þ p ffiffiffiffi ffi ffi ¼ H ¼ ¼ pffiffiffiffi sinða1 Þ a1 df1 f¼1 L0 IL1 df f¼1 L0 I
ð9Þ
For the simplicity of expression, the normalized load and the normalized work are defined respectively as the following equations and the length of the current lead is considered as unity since the configuration of the current lead can be determined as the ratio of length to cross-sectional area [9].
pffiffiffiffiffi pffiffiffiffiffi Q Q =ð L0 IÞ; W W= L0 I
ð10Þ
Normalized refrigeration work at each stage is presented with the heat load and the coefficient of performance (COP) of each refrigerator.
W 1 ¼
Q m1 Q m2 T H T m T H T m cosða1 Þ ¼ sinða1 Þ COP1 Tm
T m cosða2 Þ T L sinða2 Þ
Fig. 2. Optimized configuration parameter (a ¼ anchoring temperature.
pffiffiffiffiffi L0 IL1 =ðkav g AÞ) according to the
ð11Þ W 2
Q L T H T L T m T L cosða2 Þ ¼ ¼ sinða2 Þ COP0 TL
ð12Þ
The total work ðW tot ¼ W 1 þ W 2 Þ is a function of the configurations of two current leads and the intermediate temperature when two end temperatures are fixed. The optimum configuration of each current lead (a1 and a2) that minimizes the total refrigeration work can be determined by solving the differential equation shown in Eq. (13).
@W tot ¼ 0 and @ a1
@W tot ¼0 @ a2
ð13Þ
The solutions of Eq. (13) are expressed as the function of temperatures such as TH, Tm and TL.
Fig. 3. Temperature profile along the current lead anchored at 180 K.
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[9]. With the configuration above, the temperature gradient at the warm end (f1 = 1) is zero and it means that no conduction heat transfer at the warm end of current lead to minimize the cooling load. As shown in Fig. 2, the optimum configuration of CL2 (a2) has a different tendency since its temperature profile has a different shape from CL1. For example, the temperature profile along the current lead with the thermal-anchoring temperature of 180 K is shown in Fig. 3. The indicated region from zero to one presents the CL2 and the position from one to two presents the CL1. The configuration parameters (a1, a2) are 0.86 and 1.16 for CL1 and CL2, respectively. The normalized heat load of CL1 at the warm end is zero and that at the anchoring stage is 240 ðQ m1 Þ. The normalized heat load of CL2 at the anchoring stage is 28 ðQ m2 Þ and that at the load temperature ðQ CL2 Þ is 172. The normalized heat load at the load temperature ðQ CL2 Þ includes the heat inflow at the anchoring stage ðQ m2 Þ and the Joule-heating in CL2. The difference of heat loads at the anchoring locations, Q m1 and Q m2 , represents the cooling load that should be pumped out by the refrigerator ðQ CL1 Þ as following equation.
non-anchored one. The anchoring temperature to minimize the refrigeration work is 137 K and the minimum normalized work is 787 in this case. Although this optimum temperature may change under the influence of actual refrigeration condition, it is important to notice that there exists an optimum thermal-anchoring temperature in the case of utilizing the optimized geometry of conduction-cooled current leads. Advantage of thermal anchoring concept is not new and found in other Refs. [8,10]. Series of cascade refrigerators [12,13] or just good thermal contact with regenerator’s exterior wall [8] can realize the efficacy of thermal anchoring and the more thermal interactions with cryogenic refrigerator along temperature axis from room temperature to cryogenic temperature, the greater its effect. Considering feasible practice of electrically must-be-insulated current leads for super conducting, however, we propose only one thermal anchoring location at intermediate temperature in this paper.
Q CL1 ¼ Q m1 Q m2
Fig. 4 has revealed the optimum thermal-anchoring temperature for ideal refrigeration system. However, it is obvious that there exists no such a refrigerator. Since actual cryocooler behaves differently from ideal refrigeration system, we want to investigate how this thermal anchoring concept is influenced by the performance of real refrigeration system. For realistic estimation of cooling capability, FOM (figure of merit, COPactual/COPideal) of actual refrigerator is considered. FOM values are deduced from the Ref. [14] which contains numerous commercially-available cryocooler data. Fig. 5 shows the actual work requirement of refrigeration system incorporating thermal anchoring concept with realistic FOM value of cryocooler. The optimum thermal-anchoring temperature has been reduced to 123 K from 137 K and the amount of saving has been greatly amplified as a viewpoint of refrigeration work. This analysis clearly reflects the fact that cryocooler has lower thermodynamic efficiency as temperature goes down. It is favorable to decrease the cooling load for the coldest cryogenic stage by lowering the thermal-anchoring temperature.
ð16Þ
The amount of heat pumped by the refrigerator at the thermally anchored location is only 212 ðQ CL1 Þ and the rest of heat flows to the CL2. The total normalized refrigeration work with an ideal refrigerator is 829. It may sound better to design the CL2 so that all the heat flowing from the CL1 can be pumped out at the anchoring location for minimizing refrigeration work because the COP of refrigerator at the anchoring position is higher than that at the load temperature. However, the total refrigeration work is 839 when the configuration of CL2 is selected to have a zero temperature gradient at the anchoring position, which means that Q m2 is zero and Q CL1 is the same with Q m1 . The geometric configuration parameter (a2) of CL2 to achieve zero temperature gradient at the anchoring location is 1.52, which implies a thinner current lead at the same length that that of the optimized one. In this case, the thinner current lead configuration causes much more Joule-heating load due to electrical resistance. It is concluded that all the heat flowing from the CL1 need not to be lifted at the anchoring stage to minimize the total cooling work. Fig. 4 shows the total cooling loads cooled by the refrigerators and the total refrigeration work to keep the current lead at proper cryogenic temperature for various anchoring temperatures. As shown in Fig. 4, the total refrigeration work with thermally anchored current lead is always less than that of not-anchored current lead while the total heat load is always more than that of
Fig. 4. Ideal refrigeration work of refrigeration system and total heat load vs. thermal-anchoring temperature of optimally designed conduction-cooled current leads.
2.3. Cooling work requirement with an actual refrigerators
2.4. Additional thermal load Additional cryogenic thermal loads besides those of current leads surely impose more cooling loads for refrigeration system. They can be practically AC loss, HTS index loss, radiation heat leak and auxiliary conduction heat leak. The following simple equations
Fig. 5. Work requirement of refrigeration system and total cooling load for optimally designed conduction-cooled current leads incorporating thermal anchoring at intermediate temperature, Tm.
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where X ¼ Q a =Q CL0 . As shown in Fig. 6b, the additional cooling load at Tm increases the optimum thermal-anchoring temperature. The additional cooling load at Tm in most cases may be due to radiation heat leak from room temperature environment and it is a good common practice to install radiation shield at this location. 3. Summary This technical note describes simple thermal anchoring concept of conduction-cooled non-superconducting current leads for application using HTS superconductors. With properly selected optimum thermal-anchoring temperature and configuration, the whole refrigeration work to maintain cryogenic cooling can be greatly reduced. The optimum thermal-anchoring temperature of the conduction-cooled current leads between 300 K and 60 K is approximately 123 K with actual cryocoolers while it is 137 K with ideal refrigerators Acknowledgments This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Education, Science and Technology (No. R0A-2007-000-20062-0). References
Fig. 6. Ideal refrigeration work of refrigeration system for various additional cooling loads at (a) load temperature and (b) anchoring temperature.
calculate ideal refrigeration work for various additional cooling loads with thermal anchoring. (1) With additional thermal load, Qa at TL
Q Q Q a W tot ¼ W 1 þ W 2 þ W a ¼ CL1 þ CL2 þ COP1 COP0 COP0 Q CL1 Q CL2 ¼ þ W 0 þX COP1 Q CL0
ð17Þ
where X ¼ Q a =Q CL0 . The anchoring temperature for minimum refrigeration work does not change due to additional thermal load at TL because W 0 X is not a function of Tm as shown in Fig. 6a. (2) With additional thermal load, Qa at Tm
W tot
Q Q Q a ¼ W 1 þ W 2 þ W a ¼ CL1 þ CL2 þ COP1 COP0 COP1 Q CL0 Q CL1 Q ¼ þ X þ CL2 COP1 Q CL0 COP0
ð18Þ
[1] Young MA, Demko JA, Gough MJ. Measurements of the performance of BSCCO HTS tape under magnetic fields with a cryocooled test rig. IEEE Trans Appl Supercon 2003;13:2964–7. [2] Iwakuma M, Kajikawa K. Development of a 22 kV/6.9 kV single-phase model for a 3 MVA HTS power transformer. IEEE Trans Appl Supercon 2001;11:1578–81. [3] Chang HM, Choi YS, Van Sciver SW, Choi KD. Cryogenic cooling system of HTS transformers by natural convection of subcooled liquid nitrogen. Cryogenics 2003;43:589–96. [4] Lee C, Seok BY. Design of the 3 phase 60 MVA HTS transformer with YBCO coated conductor windings. IEEE Trans Appl Supercon 2005;15:1867–70. [5] Yazawa T, Koyanagi K, Takahashi M. Superconducting fault current limiter using high-resistive YBCO tapes. Physica C 2008;468:2046–9. [6] Barns PN, Sumption MD. Review of high power density superconducting generators: present state and prospects for incorporating YBCO windings. Cryogenics 2005;45:670–86. [7] Otabe ES, Kikuchi M, Matsushita T, Fujino K, Ohmatsu K, Ni B. Fabrication of a working Bi-2223 superconducting magnet cooled by liquid nitrogen. Cryogenics 2009;49:267–70. [8] Radebaugh R, Marquardt ED, Gary J, O’Gallagher A. Regenerator behavior with heat input or removal at intermediate temperatures. Cryocoolers 2001;11:409–18. [9] Chang HM, Van Sciver SW. Thermodynamic optimization of conduction-cooled HTS current leads. Cryogenics 1998;8:729–36. [10] Hust JH. Thermal anchoring of wires in cryogenic apparatus. Rev Sci Instrum 1970;41:622–4. [11] Furuse M, Agatuma K, Fuchino S. Evaluation of loss of current leads for HTS power apparatuses. Cryogenics 2009;49:263–6. [12] Chuang TC, Yoshida S, Frederking THK. Optimum intermediate temperatures of two-stage Gifford-Mchahon type coolers. Cryocoolers 2002;11:401–8. [13] Jeong S, Smith Jr JL. Optimum temperature staging of cryogenic refrigeration system. Cryogenics 1994;34:929–33. [14] ter Brake HJM, Wiegerinck GFM. Low-power cryocooler survey. Cryogenics 2002;42:705–18.
Contents lists available at ScienceDirect
Cryogenics journal homepage: www.elsevier.com/locate/cryogenics
Technical Note
Thermal anchoring of conduction-cooled current leads for superconductivity applications near liquid nitrogen temperature Sangkwon Jeong *, Youngkwon Kim 1 Cryogenic Engineering Laboratory, Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea
a r t i c l e
i n f o
Article history: Received 9 June 2009 Received in revised form 18 November 2009 Accepted 17 December 2009
Keywords: Thermal anchoring Conduction-cooled Current lead High temperature superconductor (HTS)
a b s t r a c t As the YBa2Cu3O7–d, or YBCO, superconductor is commercially developed and utilized for various HTS (high temperature superconductor) applications such as motor, generator, and fault current limiter, the cryocooling for 50 or 60 K range is more demanded than ever. In this case, non-superconducting current leads instead of HTS ones need to be used for energization from room temperature all the way to the cryogenic operating temperature. This technical note describes a simple method of reducing cooling load requirement for those HTS applications. Non-superconducting current leads are to be thermally anchored at an appropriate intermediate cryogenic temperature before they are connected to the application target temperature. The optimum thermal-anchoring temperature and its configuration have been obtained to minimize the required cryocooler’s cooling capacity for practical as well as ideal cases. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Development of commercially available YBCO (YBa2Cu3O7–d) superconductor impact several application areas as an enabling technology due to its ever-elevated operating temperature range at moderately high magnetic field where the performance of 1G HTS wire (BSCCO; Bi2Sr2Ca1Cu2O8 or Bi2Sr2Ca2Cu3O10) was not acceptable [1]. The operating temperature over 60 K is now feasible for realizing not only superconducting cable but also superconducting fault current limiter, transformer, motor, and generator [2– 6]. It is also noticeable that recent development of Bi-2223 tape could elevate the operating temperature and widen its application range as a competition of YBCO [7]. These applications indispensably require electrical connection from room temperature for energization and the most significant heat load to the cryogenic system is often one due to current leads. For successful commercialization and wide spread of superconducting application, using a small and therefore inexpensive cryogenic refrigeration system is of paramount importance, which demands minimization of cooling load. It was common practice to incorporate superconducting current leads partially in previous low-temperature superconductor (LTS) or 1G HTS wire systems. Superconducting current leads were connected between the low-temperature superconducting system which was typically cooled below 30 K and the non-superconducting current leads which were connected to room temperature
region. Since the joint between the non-superconducting current leads and the superconducting ones were thermally anchored, the resultant heat leak through the current lead assembly has been greatly reduced due to low thermal conductivity of superconducting current leads. In the case of recent or future superconductor application near liquid nitrogen temperature, however, non-superconducting current leads need to be used from room temperature all the way to the cryogenic temperature where it operates. Therefore, heat leak through the non-superconducting current leads will again impose major cooling load to the cryogenic cooling system of such an application. This paper emphasizes the importance of thermal anchoring of non-superconducting current leads at intermediate cryogenic temperature and describes how to reduce the cooling load requirement in general. For simple analysis, we will consider a conduction-cooled superconducting coil which is cooled by a single or multiple cryocooler units. This thermal anchoring concept will be also applicable for cases where only one cryocooler is available for cooling superconducting coil. In such a case when the cold head of cryocooler is available at single location, the thermal anchoring position can be the middle part of recuperative heat exchanger or regenerator as suggested by others [8].
2. Conduction-cooled current leads 2.1. Cooling for thermally anchored current leads
* Corresponding author. Tel.: +82 42 350 3039; fax: +82 42 869 8207. E-mail addresses: [email protected] (S. Jeong), [email protected] (Y. Kim). 1 Tel.: +82 42 350 3079; fax: +82 42 350 8207. 0011-2275/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cryogenics.2009.12.006
Fig. 1 shows the schematic configuration of refrigeration cycle where the major cooling load is due to current leads. Fig. 1a is
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Nomenclature A COP I L L0 k Q Q* T W W* x
cross-sectional area (m2) coefficient of performance () current (A) length of current lead (m) Lorenz number (24.45 nW X K2) thermal conductivity (W m1 K1) cooling load or heat transfer rate (W) normalized cooling load or heat transfer (K) temperature (K) work (W) normalized work (K) position (m)
2 a avg CL0 CL1 CL2 H L m tot
Greek
a f
current lead between intermediate temperature and load temperature additional thermal load average current lead connected between room temperature and load temperature current lead connected between room temperature and intermediate temperature current lead connected between intermediate temperature and load temperature room temperature stage load temperature stage intermediate stage total
configuration parameter () non-dimensional position (=x/L)
Subscript 0 current lead not thermally anchored 1 current lead between room temperature and intermediate temperature
the schematic configuration of current lead connection without thermal anchoring while Fig. 1b is one with thermal anchoring at intermediate temperature, Tm. Assuming that an ideal Stirling refrigerator is utilized as a cryocooler, we can hypothetically construct two different cryogenic cooling systems and calculate how much overall cooling load is reduced solely by thermal anchoring. In the case of thermal anchoring, conduction heat leak through the warmer current lead from room temperature is blocked by a refrigerator operating between room temperature, TH and the intermediate temperature, Tm. When the current leads between Tm and the load temperature, TL are also optimized, the cooling load will be further diminished. It is interesting to notice that proper selection of the intermediated temperature, Tm shall minimize the total refrigeration work to maintain cryogenic load temperature, TL of the system. In the following analysis, QCL is the conduction heat load through the current leads and W0 is the required ideal work for refrigeration without thermal anchoring. W1 and W2 are the required cooling work for refrigeration operating between TH and Tm and between Tm and TL respectively when thermal anchoring is implemented. The work required to cool down the current leads is expressed as the following equations.
W 0 ¼ Q CL0 =COP0 ; for not-anchored current leads W tot ¼ W 1 þ W 2 ¼ Q CL1 =COP1 þ Q CL2 =COP0 ; for thermally anchored current leads
d dT 1 I2 L2 qðT 1 Þ þ 1 kðTÞA1 ¼ 0 and B:C: : T 1 ð0Þ ¼ T m ; df1 A1 df1 T 1 ð1Þ ¼ T H for CL1 d dT 2 I2 L2 qðT 2 Þ kðTÞA2 ¼ 0 and B:C: : T 2 ð0Þ ¼ T L ; þ 2 df2 A2 df2
ð3Þ
T 2 ð1Þ ¼ T m
ð4Þ
for CL2
where f = x/L and L, A, I, k, q and T are length of current lead, crosssectional area of current lead, current, thermal conductivity, resistivity and temperature, respectively. The governing equations above
ð1Þ ð2Þ
2.2. Optimization of current leads For not-thermally-anchored current leads shown in Fig. 1a, its cooling load (QCL0) and optimized configuration can be easily determined as reported by previous researchers [9,10]. In the case of thermally anchored current leads, the separate conduction heat transfer equations including Joule-heating by the flowing current are solved to determine the optimum configuration. Two current leads are considered at different temperature ranges. They can have dissimilar configuration but they should have same temperature at the anchored position. From now on, the current leads installed between TH and Tm will be called ‘‘CL1” and another one called ‘‘CL2”.
Fig. 1. Schematic energy flow in refrigerator with current lead connection (a) without thermal anchoring and (b) with thermal anchoring at intermediate temperature.
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are simplified if the thermal conductivity is assumed constant as an average value for the temperature range we are interested in, but the resistivity is a function of temperature to satisfy the Wiedemann–Franz law. 2
d T1 df21
þ a21 T 1 ¼ 0 and B:C: : T 1 ð0Þ ¼ T m ; T 1 ð1Þ ¼ T H
for CL1 ð5Þ
2
d T2 df22
þ a22 T 2 ¼ 0 and B:C: : T 2 ð0Þ ¼ T L ; T 2 ð1Þ ¼ T m
for CL2 ð6Þ
pffiffiffiffiffi pffiffiffiffiffi where a1 ¼ L0 IL1 =ðA1 kav g Þ; a2 ¼ L0 IL2 =ðA2 kav g Þ and L0 is Lorenz number. We used the constant theoretical value of the Lorenz number neglecting its temperature dependence [11]. The variable a represents the physical configuration of the current lead at a given material and an operating current. The temperature profile of each current lead can be achieved by solving the above equations.
T 1 ðf1 Þ ¼ T m cosða1 f1 Þ þ T 2 ðf2 Þ ¼ T L cosða2 f2 Þ þ
½T H T m cosða1 Þ sinða1 f1 Þ sinða1 Þ
½T m T L cosða2 Þ sinða2 f2 Þ sinða2 Þ
for CL1
for CL2
ð7Þ
ð8Þ
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12 pffiffiffiffiffi T 2H T 2m L0 I L1 A a1 ¼ ð14Þ ¼ arctan@ Tm kav g A1 pffiffiffiffiffi L I L a2 ¼ 0 2 kav g A2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 h i3 2 2 ðT T Þ ðT þ T Þ T ðT þ T Þ 2T T T m L m L m L H m L H 6 7 7 ð15Þ ¼ arctan6 4 5 T m T L ð2T H þ T m þ T L Þ Fig. 2 shows the optimized configuration parameters according to the intermediate temperature when the warm end temperature is 300 K and the superconductivity load temperature is 60 K. The optimum value of configuration parameter (a) tends to decrease as the two end temperatures of the current lead approach. This means a large cross-sectional area of current lead is favorable for small temperature span of the current lead. The cooling load due to the current lead is composed of the conduction heat transfer through it and Joule-heating of electrical current. When the temperature difference between two ends is smaller, it is advantageous to have large cross-sectional area to reduce the amount of Jouleheating in the current lead. Especially, the optimum configuration of CL1 (a1) is expressed as Eq. (14) and it is the same result achieved by conventional optimization method of current lead that is situated between TH and Tm
The cooling loads at the anchored stage and the load temperature can be calculated from the achieved temperature profiles. For example, the conduction heat transfer at the end of the CL1 is expressed as Eq. (9) and other cooling loads can be expressed in the same way.
1 dT 1 A1 kav g dT 1 Q m1 T T m cosða1 Þ p ffiffiffiffi ffi ffi ¼ H ¼ ¼ pffiffiffiffi sinða1 Þ a1 df1 f¼1 L0 IL1 df f¼1 L0 I
ð9Þ
For the simplicity of expression, the normalized load and the normalized work are defined respectively as the following equations and the length of the current lead is considered as unity since the configuration of the current lead can be determined as the ratio of length to cross-sectional area [9].
pffiffiffiffiffi pffiffiffiffiffi Q Q =ð L0 IÞ; W W= L0 I
ð10Þ
Normalized refrigeration work at each stage is presented with the heat load and the coefficient of performance (COP) of each refrigerator.
W 1 ¼
Q m1 Q m2 T H T m T H T m cosða1 Þ ¼ sinða1 Þ COP1 Tm
T m cosða2 Þ T L sinða2 Þ
Fig. 2. Optimized configuration parameter (a ¼ anchoring temperature.
pffiffiffiffiffi L0 IL1 =ðkav g AÞ) according to the
ð11Þ W 2
Q L T H T L T m T L cosða2 Þ ¼ ¼ sinða2 Þ COP0 TL
ð12Þ
The total work ðW tot ¼ W 1 þ W 2 Þ is a function of the configurations of two current leads and the intermediate temperature when two end temperatures are fixed. The optimum configuration of each current lead (a1 and a2) that minimizes the total refrigeration work can be determined by solving the differential equation shown in Eq. (13).
@W tot ¼ 0 and @ a1
@W tot ¼0 @ a2
ð13Þ
The solutions of Eq. (13) are expressed as the function of temperatures such as TH, Tm and TL.
Fig. 3. Temperature profile along the current lead anchored at 180 K.
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[9]. With the configuration above, the temperature gradient at the warm end (f1 = 1) is zero and it means that no conduction heat transfer at the warm end of current lead to minimize the cooling load. As shown in Fig. 2, the optimum configuration of CL2 (a2) has a different tendency since its temperature profile has a different shape from CL1. For example, the temperature profile along the current lead with the thermal-anchoring temperature of 180 K is shown in Fig. 3. The indicated region from zero to one presents the CL2 and the position from one to two presents the CL1. The configuration parameters (a1, a2) are 0.86 and 1.16 for CL1 and CL2, respectively. The normalized heat load of CL1 at the warm end is zero and that at the anchoring stage is 240 ðQ m1 Þ. The normalized heat load of CL2 at the anchoring stage is 28 ðQ m2 Þ and that at the load temperature ðQ CL2 Þ is 172. The normalized heat load at the load temperature ðQ CL2 Þ includes the heat inflow at the anchoring stage ðQ m2 Þ and the Joule-heating in CL2. The difference of heat loads at the anchoring locations, Q m1 and Q m2 , represents the cooling load that should be pumped out by the refrigerator ðQ CL1 Þ as following equation.
non-anchored one. The anchoring temperature to minimize the refrigeration work is 137 K and the minimum normalized work is 787 in this case. Although this optimum temperature may change under the influence of actual refrigeration condition, it is important to notice that there exists an optimum thermal-anchoring temperature in the case of utilizing the optimized geometry of conduction-cooled current leads. Advantage of thermal anchoring concept is not new and found in other Refs. [8,10]. Series of cascade refrigerators [12,13] or just good thermal contact with regenerator’s exterior wall [8] can realize the efficacy of thermal anchoring and the more thermal interactions with cryogenic refrigerator along temperature axis from room temperature to cryogenic temperature, the greater its effect. Considering feasible practice of electrically must-be-insulated current leads for super conducting, however, we propose only one thermal anchoring location at intermediate temperature in this paper.
Q CL1 ¼ Q m1 Q m2
Fig. 4 has revealed the optimum thermal-anchoring temperature for ideal refrigeration system. However, it is obvious that there exists no such a refrigerator. Since actual cryocooler behaves differently from ideal refrigeration system, we want to investigate how this thermal anchoring concept is influenced by the performance of real refrigeration system. For realistic estimation of cooling capability, FOM (figure of merit, COPactual/COPideal) of actual refrigerator is considered. FOM values are deduced from the Ref. [14] which contains numerous commercially-available cryocooler data. Fig. 5 shows the actual work requirement of refrigeration system incorporating thermal anchoring concept with realistic FOM value of cryocooler. The optimum thermal-anchoring temperature has been reduced to 123 K from 137 K and the amount of saving has been greatly amplified as a viewpoint of refrigeration work. This analysis clearly reflects the fact that cryocooler has lower thermodynamic efficiency as temperature goes down. It is favorable to decrease the cooling load for the coldest cryogenic stage by lowering the thermal-anchoring temperature.
ð16Þ
The amount of heat pumped by the refrigerator at the thermally anchored location is only 212 ðQ CL1 Þ and the rest of heat flows to the CL2. The total normalized refrigeration work with an ideal refrigerator is 829. It may sound better to design the CL2 so that all the heat flowing from the CL1 can be pumped out at the anchoring location for minimizing refrigeration work because the COP of refrigerator at the anchoring position is higher than that at the load temperature. However, the total refrigeration work is 839 when the configuration of CL2 is selected to have a zero temperature gradient at the anchoring position, which means that Q m2 is zero and Q CL1 is the same with Q m1 . The geometric configuration parameter (a2) of CL2 to achieve zero temperature gradient at the anchoring location is 1.52, which implies a thinner current lead at the same length that that of the optimized one. In this case, the thinner current lead configuration causes much more Joule-heating load due to electrical resistance. It is concluded that all the heat flowing from the CL1 need not to be lifted at the anchoring stage to minimize the total cooling work. Fig. 4 shows the total cooling loads cooled by the refrigerators and the total refrigeration work to keep the current lead at proper cryogenic temperature for various anchoring temperatures. As shown in Fig. 4, the total refrigeration work with thermally anchored current lead is always less than that of not-anchored current lead while the total heat load is always more than that of
Fig. 4. Ideal refrigeration work of refrigeration system and total heat load vs. thermal-anchoring temperature of optimally designed conduction-cooled current leads.
2.3. Cooling work requirement with an actual refrigerators
2.4. Additional thermal load Additional cryogenic thermal loads besides those of current leads surely impose more cooling loads for refrigeration system. They can be practically AC loss, HTS index loss, radiation heat leak and auxiliary conduction heat leak. The following simple equations
Fig. 5. Work requirement of refrigeration system and total cooling load for optimally designed conduction-cooled current leads incorporating thermal anchoring at intermediate temperature, Tm.
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where X ¼ Q a =Q CL0 . As shown in Fig. 6b, the additional cooling load at Tm increases the optimum thermal-anchoring temperature. The additional cooling load at Tm in most cases may be due to radiation heat leak from room temperature environment and it is a good common practice to install radiation shield at this location. 3. Summary This technical note describes simple thermal anchoring concept of conduction-cooled non-superconducting current leads for application using HTS superconductors. With properly selected optimum thermal-anchoring temperature and configuration, the whole refrigeration work to maintain cryogenic cooling can be greatly reduced. The optimum thermal-anchoring temperature of the conduction-cooled current leads between 300 K and 60 K is approximately 123 K with actual cryocoolers while it is 137 K with ideal refrigerators Acknowledgments This work was supported by the Korea Science and Engineering Foundation (KOSEF) through the National Research Lab. Program funded by the Ministry of Education, Science and Technology (No. R0A-2007-000-20062-0). References
Fig. 6. Ideal refrigeration work of refrigeration system for various additional cooling loads at (a) load temperature and (b) anchoring temperature.
calculate ideal refrigeration work for various additional cooling loads with thermal anchoring. (1) With additional thermal load, Qa at TL
Q Q Q a W tot ¼ W 1 þ W 2 þ W a ¼ CL1 þ CL2 þ COP1 COP0 COP0 Q CL1 Q CL2 ¼ þ W 0 þX COP1 Q CL0
ð17Þ
where X ¼ Q a =Q CL0 . The anchoring temperature for minimum refrigeration work does not change due to additional thermal load at TL because W 0 X is not a function of Tm as shown in Fig. 6a. (2) With additional thermal load, Qa at Tm
W tot
Q Q Q a ¼ W 1 þ W 2 þ W a ¼ CL1 þ CL2 þ COP1 COP0 COP1 Q CL0 Q CL1 Q ¼ þ X þ CL2 COP1 Q CL0 COP0
ð18Þ
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