- Email: [email protected]
Thermal stress analysis for the meander-shape YBCO fault current limiter
Physica C 411 (2004) 25–31 www.elsevier.com/locate/physc
Thermal stress analysis for the meander-shape YBCO fault current limiter Hideyoshi Takashima *, Ken-ichi Sasaki, Toshitada Onishi Hokkaido University, Kita 8 Nishi 5, Sapporo 060-0808, Japan Received 10 November 2003; received in revised form 3 February 2004; accepted 24 March 2004 Available online 8 July 2004
Abstract A thermal stress in a meander-shape YBCO fault current limiter has been calculated numerically. The FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis has been developed. From the comparison between the results of simulation and the measurement results of thermal expansion in a small piece of stainless steel, it has been found that the our code is useful for the estimation of thermal stress. The thermal stress and temperature distribution in the meander-shape YBCO fault current limiter under a normal operating condition have been calculated in two conditions; the critical current density of YBCO degrades at the inside of corner and at the middle part of straight section. In both cases, the thermal stress is concentrated around the inside of corner. In the former case, there is high possibility that YBCO bulk is cracked around the corner because the maximum thermal stress exceeds fracture strength of YBCO. On the contrary, in the latter case, there is high possibility that YBCO bulk burns out around the straight section because the maxim thermal stress in YBCO is lower than the fracture strength at the temperature enough to melt YBCO bulk. 2004 Published by Elsevier B.V. Keywords: Superconducting fault current limiter; Bulk high-Tc superconductor; Thermal stress analysis
1. Introduction A fault current limiter, which limits short-circuit current before the fault isolation, is very attractive because it reduces the stresses on all the components beyond the fault. A superconducting fault current limiter (SCFCL) is a more desirable device in power systems because it does not cause considerable voltage drop and energy loss during a normal operating condition. In addition, if the
*
Corresponding author.
0921-4534/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physc.2004.03.248
fault occurs, SCFCL automatically insert impedance into the power systems to limit the peak of the fault current. There are three different concepts of SCFCL; the resistive type [1–3], the shielded core type [4,5] and the diode bridge type [6]. Especially, many groups have vigorously studied the resistive SCFCL [7–9]. Because, the resistive SCFCL has advantages against other type; smaller size than the shielded core type, and easier system configuration than the diode-bridge type. The size of high Tc superconductor (HTS) bulk becomes large and its critical current density, Jc ,
26
H. Takashima et al. / Physica C 411 (2004) 25–31
increases by the development of technology. YBCO has a high Jc in comparison with other HTS materials, therefore, it is commonly used for resistive SCFCL. YBCO SCFCL is often formed of meander shape in order to obtain required resistance, which is cut from disciform material of melt textured YBCO. In such YBCO SCFCL, there might be a local area having degraded Jc because of the inhomogeneity in YBCO material and the effect of the shock during the cutting process. When the current flows in the SCFCL, this degraded area acts as the hot spot, and the SCFCL may be broken. A purpose of this study is to analyze the thermal stress in resistive YBCO SCFCL with the local degradation of Jc under a normal operating condition. We developed the FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis and calculated the thermal stress under the condition that Jc around the corner and the straight section was degraded. The validity of the developed code was confirmed by the experiment.
In the current analysis FEM code is used an equation as oT ¼ 0; ðr T ¼ JÞ; ð2Þ r ðqr TÞ þ l0 ot where T, q and l0 are the electric vector potential, the resistivity and the magnetic permeability in air, respectively. The boundary condition is n T ¼ 0, where n is a normal unit vector. The thermal equilibrium equation is given as oh r ðkrhÞ c þ qJ 2 ¼ 0; ð3Þ ot where h, k and c are the temperature, the thermal conductivity, specific heat, respectively. For the thermal analysis, we considered the superconductor was cooled by liquid nitrogen. In the stress analysis, the principle of virtual work is used. The principle of virtual work is described as, Z Z T T dfeg frg dV dfU g fFV g dV X X Z T dfU g fFS g dS ¼ 0; ð4Þ C
2. Analysis code The developed analysis code composes three FEM codes; the thermal analysis code, the current analysis code and the stress analysis code. The thermal analysis code and the current analysis code are combined by common variable, resistivity. One of these solutions, a temperature, is used in the stress analysis code. Usually a resistivity of a type II superconductor in a superconducting state has a finite value and it is proportional to a current density in a superconductor. The resistivity in the calculation, q, is obtained from following equation: n1 1 J qðJ Þ ¼ E0 ; ð1Þ Jc ðhÞ Jc ðhÞ where Jc is the critical current density, J is the current density in the superconductor, n is the nvalue and E0 is the voltage criterion. Generally E0 is 1.0 · 104 (V/m) at 77 K. In this study, Jc ðhÞ is supposed to depend on the temperature linearly. Eq. (1) is used to combine the two codes.
where feg, frg, fU g, fFV g and fFS g are the strain vector, the stress vector, the displacement vector, the body force vector and the surface force vector, respectively. For the stress analysis, we considered the free boundary condition. Eqs. (2)–(4) are formularized with the finiteelement method using Galerkin’s method and Crank–Nicolson’s scheme. The algorithm for the numerical calculation is as follows. (a) (b) (c) (d) (e) (f) (g) (h)
Initialize. Calculate T by Eq. (2). Calculate q by Eq. (1). Calculate h by Eq. (3). Calculate q by Eq. (1). Iterate from (b) to (e) q until converge. Calculate r by Eq. (4). Iterate from (b) to (g).
3. Measurement of the displacement by the thermal expansion In order to evaluate the validity of our code, the displacement by the thermal expansion was mea-
H. Takashima et al. / Physica C 411 (2004) 25–31
sured and those results were compared with the calculation results. Usually, the displacement is measured by the strain gauge. However, the strain gauge is difficult to set on the surface of the bulk superconductor. Therefore, we employed a laser displacement sensor, which can measure a thermal expansion without contact with the sample. Fig. 1 shows the experimental set-up for the measurement of the displacement by the thermal expansion, which contains the acrylic airtight container, the XYZ-stage, the laser displacement sensor and the sample stand. The sample is set on the sample stand, which holds liquid nitrogen. The sample is not in contact with liquid nitrogen directly, and it cooled by the conductive cooling through the electrodes. The platform of equipments and XYZ-stage were covered with thermal insulation for preventing thermal contraction caused by the chill. Since the gas in the airtight container of the acrylic cube is replaced by nitrogen gas, the surface of the sample, that is the laser spot, is almost not frosted. The shape of sample is also shown in Fig. 1. The material of sample is
27
stainless steel. Electrical currents flowed in the sample, and the thermal expansion due to Joule’s heat was measured. The thin part around the center expands larger than around the ends of sample due to the higher current density. We measured the difference of thermal expansions between at the center and the end of the sample so that there was no influence of the thermal contraction of experimental equipments on the measurement. Fig. 2 shows experimental results and numerical results. Fig. 2(a) and (b) are the temperature change and the displacement change, respectively. The excitation pattern of the current is as follows: (1) current increases to 100 A at the rate of 1 A/s, (2) current is held in 100 A during 100 s and (3) current decreases to 0 A at the rate of 1 A/s. Two bold lines in Fig. 2(b) represent the displacements at the thin part and the end of the sample, respectively. It can be found from Fig. 2(b) that both results are good agreement. The average thermal displacement during the excitation was about 4 lm, and the numerical result was 3.37 lm.
Fig. 1. The experimental equipment.
28
H. Takashima et al. / Physica C 411 (2004) 25–31 200
25
experimental result
180
center of the sample
20
end of the sample 160
15
140
10
120
5
100
numerical result
0
80 0
50
100
150
(a)
200
= 4 µm
250
300
-5
350
0
50
100
150
200
250
300
350
Time[sec]
(b)
Time[s]
Fig. 2. The experimental results: (a) the temperature change and (b) the displacement change.
non-linear characteristic. However, because the cooling on the surface of the FCL does not affect largely in this calculation, the constant value for the film boiling is adopted. The calculation was performed with three cases by two conditions.
The numerical result almost corresponds with the measurement result, indicating that the result calculated by our analysis code is reliable.
4. Calculation of the thermal stress in meandershaped current limiting device
Case (a): the critical current density Jc of YBCO degrades to 75% of the normal critical current density Jc0 of YBCO at the inside of corner ( in Fig. 3). Case (b): Jc degrades to 25% of Jc0 at the same place as Case (a). Case (c): Jc degrades to 75% of Jc0 at the middle part of straight section ( in Fig. 3).
The thermal stress and temperature distribution in the 1-kA class meander-shaped YBCO fault current limiter having degraded area under a normal operating condition was calculated using our code. Fig. 3 shows the calculation model and Table 1 shows the parameters for the calculation. In the calculation, the electrical conductivity, the thermal conductivity, the critical current density and the specific heat have the dependence of temperature, and the FCL is cooled by liquid nitrogen. The heat transfer coefficient of liquid nitrogen has
Because of the symmetry of the model, the calculation is performed in the area enclosed by dashed line. The transport current is 1000Apeak
31.6
Unit: [mm]
6.4 1 2
2.0 2
42.0
2.0 2.2
1.1
1.0
1.0 2.0 4.0
1
7.0
Temperature data Points
Thickness: 0.8
Fig. 3. The calculation model for meander-shaped SCFCL.
2.0
H. Takashima et al. / Physica C 411 (2004) 25–31 Table 1 The parameters for the FEM calculation Temperature [K]
1000
90.0 K 1.0 · 109 A/m2 5.0 · 106 Xm 15.0 120.0 GPa 0.3 10.0 · 106 50 Hz, 1000 A 77.0 K 1.0 · 103 W/m2 K
800 600 400 200 0 1.2336
1.2342
1.2344
Fig. 6. The stress distributions in Case (b).
1200 1000 800 600 400 200 0 3.6345
Temperature [K]
1.2340
Fig. 5. The maximum temperature in Case (b).
with 50 Hz, and we assume that the current is constant during the calculation. The sample is immersed in liquid nitrogen at 77 K. Fig. 4 shows the maximum temperature in the device with a function of time in the Case (a), that point is shown in Fig. 3. In this case, the temperature became a steady state. In other words, even if Jc degrades to 75% in the corner, the sample does not quench. Fig. 5 shows the maximum temperature in the Case (b). In this case, the sample quenches at the time of about 1.2337 s, and after that, the temperature rise at the ratio of 180 K per 0.1 ms. Fig. 6 shows the stress distribution when the maximum temperature in the sample reaches to 1000 K. The maximum stress is about 85 MPa at the corner, and this value exceeds the fracture strength of YBCO adding Ag [10]. Namely, if Jc degrades to 25% in the corner, a sample breaks by the thermal stress. Fig. 7 shows the maximum temperature change of Case (c). In this case, the sample quenches at the
85 84 83 82 81 80 79 78 77 76 0.0
1.2338
Time [sec]
Temperature [K]
Tc : critical temperature Jc : critical current density at 77 K qTc : normal resistivity at Tc n: n-value Y : Young’s modulus v: Poisson’s ratio a: thermal expansion I: current h0 : initial temperature ac : Heat transfer coefficient
29
3.6350
3.6355 Time [sec]
3.6360
3.6365
Fig. 7. The maximum temperature in Case (c).
2.0
4.0 Time [sec]
6.0
Fig. 4. The maximum temperature in Case (a).
8.0
time of about 3.635 s, and the temperature rise at the ratio of 160 K per 0.1 ms. The stress distribution is shown in Fig. 8 when the maximum temperature in the sample reaches to 1000 K. In this case, the maximum stress is about 40 MPa at the corner. This value does not exceed the fracture strength of YBCO with Ag, indicating that the
30
H. Takashima et al. / Physica C 411 (2004) 25–31
Fig. 8. The stress distributions in Case (c).
maximum temperature is more important in this case. Because the temperature of about 1000 K is the nucleation temperature of the 123 phase [11], and YBCO seems to begin to melt above 1000 K. Namely, if Jc degrades to 75% in the straight section, it has possible the melting by the temperature rise. Fig. 9 shows the relation between the maximum stress and the maximum temperature for the Case (b) and (c). The maximum stress in Fig. 9 is the xcomponent stress. The dashed lines parallel to the stress axis, represent the fracture strength of the non-Ag added YBCO and the 10 wt% Ag added YBCO, 40 MPa and 85 MPa. Furthermore, the dashed line parallel to the temperature axis is the melting temperature of YBCO, 1000 K. It is found that in the Case (b), the thermal stress causes a
100
The fracture strenght of 10wt% Ag YBCO
X direction Stress [MPa]
90
fracture, and, in the Case (c), the sample is not broken by thermal stress but melted by the temperature rise. The straight section has not enough areas in order that the current escapes from degraded section. The non-degraded area quenches as soon as the current escapes from the normal to superconducting area, and temperature rises uniformly in the cross-section. Therefore, in the straight section, the gradient of temperature is not large, and the thermal stress around there is not also large. On the other hand, in the Case (c), there is an enough area for the escape of current around the corner. Therefore, the non-degraded area stays superconducting state even if the current redistribution occurs. So, the area of temperature rising does not spread quickly in comparison with the Case (b), the gradient of temperature is large, and the thermal stress also is large around the corner. These results show that the straight section is sensitive to the degradation of Jc in comparison to the corner section, the thermal stress is concentrated around the inside of corner, and if Jc degrades in the straight section, it has a possibility of the sample melts. Probably, the same phenomenon happens when the fault occurs even if the unevenness of the critical current density is small in YBCO and there are not any problems in the normal operation. When the current increases, the region with lower Jc quenches first, and the temperature skyrockets locally. These are indicating that the meander-shape YBCO FCL is not only required to have large fracture strength around the corner section, but also to have homogeneous Jc especially in the straight section.
80 70 60 50
5. Conclusion
The fracture strength of Non-Ag YBCO
40 30 20
corner straight
10
The melting temperature
0 0
200
400
600
800
1000
1200
Temperature [K]
Fig. 9. The relation between the maximum stress and the maximum temperature for the Case (b) and (c).
A thermal stress in a meander-shape YBCO fault current limiter has been calculated numerically. We developed the FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis, and confirmed the validity of our code from the comparison between the results of simulation and the measurement
H. Takashima et al. / Physica C 411 (2004) 25–31
results of thermal expansion in a small piece of stainless steel. The thermal stress and temperature distribution in the meander-shaped YBCO fault current limiter having degraded area under a normal operating condition was calculated. From the calculation results, we found that the straight section is sensitive to Jc degraded in comparison to the corner section and the thermal stress is concentrated around the inside of corner. And if Jc degrades in the corner section, there is high possibility that YBCO bulk is cracked around the corner because the maximum thermal stress exceeds fracture strength of YBCO. On the contrary, if Jc degrades in the straight section, there is high possibility that YBCO bulk burns out around the straight section because the maxim thermal stress in YBCO is lower than the fracture strength at the temperature enough to melt YBCO bulk. Therefore, it is required to improve the fracture strength around the corner and the homogeneity of Jc in the straight section in order to improve the performance of the meander-shape YBCO fault current limiter.
31
References [1] M. Noe, K.-P. Juengst, F. Werfel, L. Cowey, A. Wolf, S. Elschner, IEEE Trans. Appl. Supercond. 11 (1) (2001) 1960. [2] P. Tixador, L. Porcar, E. Floch, D. Buzon, D. Isfort, D. Bourgault, X. Chaud, R. Tournier, IEEE Trans. Appl. Supercond. 11 (1) (2001) 2034. [3] B. Gromoll et al., IEEE Trans. Appl. Supercond. 7 (2) (1997) 828. [4] W. Paul et al., Test of 1.2 MVA high-Tc superconducting fault current limiter, Supercond. Sci. Technol. 10 (1997) 914. [5] T. Onishi, K.-i. Sasaki, R. Akimoto, Cryogenics 41 (2001) 239. [6] E. Leung, B. Burley, N. Chiwood, H. Gurol, G. Miyata, D. Morris, L. Ngyuen, B. O’Hea, et al., IEEE Trans. Appl. Supercond. 10 (1) (2000) 832. [7] K. Tekletsadik, M.P. Saravolac, A. Rowley, IEEE Trans. Appl. Supercond. 9 (9) (1999) 672. [8] P. Tixador et al., Inst. Phys. Conf. Ser. 167 (1999) 949. [9] W. Paul, et al., Superconducting fault current limiter applications, technical and economical benefits, simulations and test results, CIGRE Session, 2000, 13. [10] T. Miyamoto, K. Nagashima, N. Sakai, M. Murakami, Supercond. Sci. Technol. 13 (2000) 816. [11] M. Murakami, M. Morita, K. Doi, K. Miyamoto, Jpn. J. Appl. Phys. 28 (1989) 1189.
Thermal stress analysis for the meander-shape YBCO fault current limiter Hideyoshi Takashima *, Ken-ichi Sasaki, Toshitada Onishi Hokkaido University, Kita 8 Nishi 5, Sapporo 060-0808, Japan Received 10 November 2003; received in revised form 3 February 2004; accepted 24 March 2004 Available online 8 July 2004
Abstract A thermal stress in a meander-shape YBCO fault current limiter has been calculated numerically. The FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis has been developed. From the comparison between the results of simulation and the measurement results of thermal expansion in a small piece of stainless steel, it has been found that the our code is useful for the estimation of thermal stress. The thermal stress and temperature distribution in the meander-shape YBCO fault current limiter under a normal operating condition have been calculated in two conditions; the critical current density of YBCO degrades at the inside of corner and at the middle part of straight section. In both cases, the thermal stress is concentrated around the inside of corner. In the former case, there is high possibility that YBCO bulk is cracked around the corner because the maximum thermal stress exceeds fracture strength of YBCO. On the contrary, in the latter case, there is high possibility that YBCO bulk burns out around the straight section because the maxim thermal stress in YBCO is lower than the fracture strength at the temperature enough to melt YBCO bulk. 2004 Published by Elsevier B.V. Keywords: Superconducting fault current limiter; Bulk high-Tc superconductor; Thermal stress analysis
1. Introduction A fault current limiter, which limits short-circuit current before the fault isolation, is very attractive because it reduces the stresses on all the components beyond the fault. A superconducting fault current limiter (SCFCL) is a more desirable device in power systems because it does not cause considerable voltage drop and energy loss during a normal operating condition. In addition, if the
*
Corresponding author.
0921-4534/$ - see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physc.2004.03.248
fault occurs, SCFCL automatically insert impedance into the power systems to limit the peak of the fault current. There are three different concepts of SCFCL; the resistive type [1–3], the shielded core type [4,5] and the diode bridge type [6]. Especially, many groups have vigorously studied the resistive SCFCL [7–9]. Because, the resistive SCFCL has advantages against other type; smaller size than the shielded core type, and easier system configuration than the diode-bridge type. The size of high Tc superconductor (HTS) bulk becomes large and its critical current density, Jc ,
26
H. Takashima et al. / Physica C 411 (2004) 25–31
increases by the development of technology. YBCO has a high Jc in comparison with other HTS materials, therefore, it is commonly used for resistive SCFCL. YBCO SCFCL is often formed of meander shape in order to obtain required resistance, which is cut from disciform material of melt textured YBCO. In such YBCO SCFCL, there might be a local area having degraded Jc because of the inhomogeneity in YBCO material and the effect of the shock during the cutting process. When the current flows in the SCFCL, this degraded area acts as the hot spot, and the SCFCL may be broken. A purpose of this study is to analyze the thermal stress in resistive YBCO SCFCL with the local degradation of Jc under a normal operating condition. We developed the FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis and calculated the thermal stress under the condition that Jc around the corner and the straight section was degraded. The validity of the developed code was confirmed by the experiment.
In the current analysis FEM code is used an equation as oT ¼ 0; ðr T ¼ JÞ; ð2Þ r ðqr TÞ þ l0 ot where T, q and l0 are the electric vector potential, the resistivity and the magnetic permeability in air, respectively. The boundary condition is n T ¼ 0, where n is a normal unit vector. The thermal equilibrium equation is given as oh r ðkrhÞ c þ qJ 2 ¼ 0; ð3Þ ot where h, k and c are the temperature, the thermal conductivity, specific heat, respectively. For the thermal analysis, we considered the superconductor was cooled by liquid nitrogen. In the stress analysis, the principle of virtual work is used. The principle of virtual work is described as, Z Z T T dfeg frg dV dfU g fFV g dV X X Z T dfU g fFS g dS ¼ 0; ð4Þ C
2. Analysis code The developed analysis code composes three FEM codes; the thermal analysis code, the current analysis code and the stress analysis code. The thermal analysis code and the current analysis code are combined by common variable, resistivity. One of these solutions, a temperature, is used in the stress analysis code. Usually a resistivity of a type II superconductor in a superconducting state has a finite value and it is proportional to a current density in a superconductor. The resistivity in the calculation, q, is obtained from following equation: n1 1 J qðJ Þ ¼ E0 ; ð1Þ Jc ðhÞ Jc ðhÞ where Jc is the critical current density, J is the current density in the superconductor, n is the nvalue and E0 is the voltage criterion. Generally E0 is 1.0 · 104 (V/m) at 77 K. In this study, Jc ðhÞ is supposed to depend on the temperature linearly. Eq. (1) is used to combine the two codes.
where feg, frg, fU g, fFV g and fFS g are the strain vector, the stress vector, the displacement vector, the body force vector and the surface force vector, respectively. For the stress analysis, we considered the free boundary condition. Eqs. (2)–(4) are formularized with the finiteelement method using Galerkin’s method and Crank–Nicolson’s scheme. The algorithm for the numerical calculation is as follows. (a) (b) (c) (d) (e) (f) (g) (h)
Initialize. Calculate T by Eq. (2). Calculate q by Eq. (1). Calculate h by Eq. (3). Calculate q by Eq. (1). Iterate from (b) to (e) q until converge. Calculate r by Eq. (4). Iterate from (b) to (g).
3. Measurement of the displacement by the thermal expansion In order to evaluate the validity of our code, the displacement by the thermal expansion was mea-
H. Takashima et al. / Physica C 411 (2004) 25–31
sured and those results were compared with the calculation results. Usually, the displacement is measured by the strain gauge. However, the strain gauge is difficult to set on the surface of the bulk superconductor. Therefore, we employed a laser displacement sensor, which can measure a thermal expansion without contact with the sample. Fig. 1 shows the experimental set-up for the measurement of the displacement by the thermal expansion, which contains the acrylic airtight container, the XYZ-stage, the laser displacement sensor and the sample stand. The sample is set on the sample stand, which holds liquid nitrogen. The sample is not in contact with liquid nitrogen directly, and it cooled by the conductive cooling through the electrodes. The platform of equipments and XYZ-stage were covered with thermal insulation for preventing thermal contraction caused by the chill. Since the gas in the airtight container of the acrylic cube is replaced by nitrogen gas, the surface of the sample, that is the laser spot, is almost not frosted. The shape of sample is also shown in Fig. 1. The material of sample is
27
stainless steel. Electrical currents flowed in the sample, and the thermal expansion due to Joule’s heat was measured. The thin part around the center expands larger than around the ends of sample due to the higher current density. We measured the difference of thermal expansions between at the center and the end of the sample so that there was no influence of the thermal contraction of experimental equipments on the measurement. Fig. 2 shows experimental results and numerical results. Fig. 2(a) and (b) are the temperature change and the displacement change, respectively. The excitation pattern of the current is as follows: (1) current increases to 100 A at the rate of 1 A/s, (2) current is held in 100 A during 100 s and (3) current decreases to 0 A at the rate of 1 A/s. Two bold lines in Fig. 2(b) represent the displacements at the thin part and the end of the sample, respectively. It can be found from Fig. 2(b) that both results are good agreement. The average thermal displacement during the excitation was about 4 lm, and the numerical result was 3.37 lm.
Fig. 1. The experimental equipment.
28
H. Takashima et al. / Physica C 411 (2004) 25–31 200
25
experimental result
180
center of the sample
20
end of the sample 160
15
140
10
120
5
100
numerical result
0
80 0
50
100
150
(a)
200
= 4 µm
250
300
-5
350
0
50
100
150
200
250
300
350
Time[sec]
(b)
Time[s]
Fig. 2. The experimental results: (a) the temperature change and (b) the displacement change.
non-linear characteristic. However, because the cooling on the surface of the FCL does not affect largely in this calculation, the constant value for the film boiling is adopted. The calculation was performed with three cases by two conditions.
The numerical result almost corresponds with the measurement result, indicating that the result calculated by our analysis code is reliable.
4. Calculation of the thermal stress in meandershaped current limiting device
Case (a): the critical current density Jc of YBCO degrades to 75% of the normal critical current density Jc0 of YBCO at the inside of corner ( in Fig. 3). Case (b): Jc degrades to 25% of Jc0 at the same place as Case (a). Case (c): Jc degrades to 75% of Jc0 at the middle part of straight section ( in Fig. 3).
The thermal stress and temperature distribution in the 1-kA class meander-shaped YBCO fault current limiter having degraded area under a normal operating condition was calculated using our code. Fig. 3 shows the calculation model and Table 1 shows the parameters for the calculation. In the calculation, the electrical conductivity, the thermal conductivity, the critical current density and the specific heat have the dependence of temperature, and the FCL is cooled by liquid nitrogen. The heat transfer coefficient of liquid nitrogen has
Because of the symmetry of the model, the calculation is performed in the area enclosed by dashed line. The transport current is 1000Apeak
31.6
Unit: [mm]
6.4 1 2
2.0 2
42.0
2.0 2.2
1.1
1.0
1.0 2.0 4.0
1
7.0
Temperature data Points
Thickness: 0.8
Fig. 3. The calculation model for meander-shaped SCFCL.
2.0
H. Takashima et al. / Physica C 411 (2004) 25–31 Table 1 The parameters for the FEM calculation Temperature [K]
1000
90.0 K 1.0 · 109 A/m2 5.0 · 106 Xm 15.0 120.0 GPa 0.3 10.0 · 106 50 Hz, 1000 A 77.0 K 1.0 · 103 W/m2 K
800 600 400 200 0 1.2336
1.2342
1.2344
Fig. 6. The stress distributions in Case (b).
1200 1000 800 600 400 200 0 3.6345
Temperature [K]
1.2340
Fig. 5. The maximum temperature in Case (b).
with 50 Hz, and we assume that the current is constant during the calculation. The sample is immersed in liquid nitrogen at 77 K. Fig. 4 shows the maximum temperature in the device with a function of time in the Case (a), that point is shown in Fig. 3. In this case, the temperature became a steady state. In other words, even if Jc degrades to 75% in the corner, the sample does not quench. Fig. 5 shows the maximum temperature in the Case (b). In this case, the sample quenches at the time of about 1.2337 s, and after that, the temperature rise at the ratio of 180 K per 0.1 ms. Fig. 6 shows the stress distribution when the maximum temperature in the sample reaches to 1000 K. The maximum stress is about 85 MPa at the corner, and this value exceeds the fracture strength of YBCO adding Ag [10]. Namely, if Jc degrades to 25% in the corner, a sample breaks by the thermal stress. Fig. 7 shows the maximum temperature change of Case (c). In this case, the sample quenches at the
85 84 83 82 81 80 79 78 77 76 0.0
1.2338
Time [sec]
Temperature [K]
Tc : critical temperature Jc : critical current density at 77 K qTc : normal resistivity at Tc n: n-value Y : Young’s modulus v: Poisson’s ratio a: thermal expansion I: current h0 : initial temperature ac : Heat transfer coefficient
29
3.6350
3.6355 Time [sec]
3.6360
3.6365
Fig. 7. The maximum temperature in Case (c).
2.0
4.0 Time [sec]
6.0
Fig. 4. The maximum temperature in Case (a).
8.0
time of about 3.635 s, and the temperature rise at the ratio of 160 K per 0.1 ms. The stress distribution is shown in Fig. 8 when the maximum temperature in the sample reaches to 1000 K. In this case, the maximum stress is about 40 MPa at the corner. This value does not exceed the fracture strength of YBCO with Ag, indicating that the
30
H. Takashima et al. / Physica C 411 (2004) 25–31
Fig. 8. The stress distributions in Case (c).
maximum temperature is more important in this case. Because the temperature of about 1000 K is the nucleation temperature of the 123 phase [11], and YBCO seems to begin to melt above 1000 K. Namely, if Jc degrades to 75% in the straight section, it has possible the melting by the temperature rise. Fig. 9 shows the relation between the maximum stress and the maximum temperature for the Case (b) and (c). The maximum stress in Fig. 9 is the xcomponent stress. The dashed lines parallel to the stress axis, represent the fracture strength of the non-Ag added YBCO and the 10 wt% Ag added YBCO, 40 MPa and 85 MPa. Furthermore, the dashed line parallel to the temperature axis is the melting temperature of YBCO, 1000 K. It is found that in the Case (b), the thermal stress causes a
100
The fracture strenght of 10wt% Ag YBCO
X direction Stress [MPa]
90
fracture, and, in the Case (c), the sample is not broken by thermal stress but melted by the temperature rise. The straight section has not enough areas in order that the current escapes from degraded section. The non-degraded area quenches as soon as the current escapes from the normal to superconducting area, and temperature rises uniformly in the cross-section. Therefore, in the straight section, the gradient of temperature is not large, and the thermal stress around there is not also large. On the other hand, in the Case (c), there is an enough area for the escape of current around the corner. Therefore, the non-degraded area stays superconducting state even if the current redistribution occurs. So, the area of temperature rising does not spread quickly in comparison with the Case (b), the gradient of temperature is large, and the thermal stress also is large around the corner. These results show that the straight section is sensitive to the degradation of Jc in comparison to the corner section, the thermal stress is concentrated around the inside of corner, and if Jc degrades in the straight section, it has a possibility of the sample melts. Probably, the same phenomenon happens when the fault occurs even if the unevenness of the critical current density is small in YBCO and there are not any problems in the normal operation. When the current increases, the region with lower Jc quenches first, and the temperature skyrockets locally. These are indicating that the meander-shape YBCO FCL is not only required to have large fracture strength around the corner section, but also to have homogeneous Jc especially in the straight section.
80 70 60 50
5. Conclusion
The fracture strength of Non-Ag YBCO
40 30 20
corner straight
10
The melting temperature
0 0
200
400
600
800
1000
1200
Temperature [K]
Fig. 9. The relation between the maximum stress and the maximum temperature for the Case (b) and (c).
A thermal stress in a meander-shape YBCO fault current limiter has been calculated numerically. We developed the FEM code combining the thermal stress analysis, temperature analysis and current distribution analysis, and confirmed the validity of our code from the comparison between the results of simulation and the measurement
H. Takashima et al. / Physica C 411 (2004) 25–31
results of thermal expansion in a small piece of stainless steel. The thermal stress and temperature distribution in the meander-shaped YBCO fault current limiter having degraded area under a normal operating condition was calculated. From the calculation results, we found that the straight section is sensitive to Jc degraded in comparison to the corner section and the thermal stress is concentrated around the inside of corner. And if Jc degrades in the corner section, there is high possibility that YBCO bulk is cracked around the corner because the maximum thermal stress exceeds fracture strength of YBCO. On the contrary, if Jc degrades in the straight section, there is high possibility that YBCO bulk burns out around the straight section because the maxim thermal stress in YBCO is lower than the fracture strength at the temperature enough to melt YBCO bulk. Therefore, it is required to improve the fracture strength around the corner and the homogeneity of Jc in the straight section in order to improve the performance of the meander-shape YBCO fault current limiter.
31
References [1] M. Noe, K.-P. Juengst, F. Werfel, L. Cowey, A. Wolf, S. Elschner, IEEE Trans. Appl. Supercond. 11 (1) (2001) 1960. [2] P. Tixador, L. Porcar, E. Floch, D. Buzon, D. Isfort, D. Bourgault, X. Chaud, R. Tournier, IEEE Trans. Appl. Supercond. 11 (1) (2001) 2034. [3] B. Gromoll et al., IEEE Trans. Appl. Supercond. 7 (2) (1997) 828. [4] W. Paul et al., Test of 1.2 MVA high-Tc superconducting fault current limiter, Supercond. Sci. Technol. 10 (1997) 914. [5] T. Onishi, K.-i. Sasaki, R. Akimoto, Cryogenics 41 (2001) 239. [6] E. Leung, B. Burley, N. Chiwood, H. Gurol, G. Miyata, D. Morris, L. Ngyuen, B. O’Hea, et al., IEEE Trans. Appl. Supercond. 10 (1) (2000) 832. [7] K. Tekletsadik, M.P. Saravolac, A. Rowley, IEEE Trans. Appl. Supercond. 9 (9) (1999) 672. [8] P. Tixador et al., Inst. Phys. Conf. Ser. 167 (1999) 949. [9] W. Paul, et al., Superconducting fault current limiter applications, technical and economical benefits, simulations and test results, CIGRE Session, 2000, 13. [10] T. Miyamoto, K. Nagashima, N. Sakai, M. Murakami, Supercond. Sci. Technol. 13 (2000) 816. [11] M. Murakami, M. Morita, K. Doi, K. Miyamoto, Jpn. J. Appl. Phys. 28 (1989) 1189.