As discussed in previous section, the CS and MS coils are operated in a high magnetic field so that the temperature limitation is only about 2 K at minimum. Furthermore
Chapter 3. Thermal Stability during the Beam Operation
Table 3.1: The current sharing temperature Tcs, critical temperature Tc and critical current estimated with the maximum magnetic fieldBmax and bath temperature of 4.5 K for each coil.
It is noted that the operation currents for TS1b, TS1d, TS1e and TS1f are 2581 A, 2619 A, 2538 A and 2916 A, respectively.
CS0 CS1 MS1 MS2 TS1(a,c) TS1b TS1d TS1e TS1f Tcs [K] 7.0 6.5 7.2 7.2 7.5 7.4 7.5 7.5 7.6
Tc [K] 7.6 7.0 7.7 7.7 7.9 7.9 8.0 8.0 8.1
Bmax [Tesla] 4.1 5.3 3.8 3.7 3.2 3.3 3.0 3.1 2.9
Ic [kA] 15.8 12 16.8 17.2 18.9 18.6 19.7 19.3 20.1 these coils are exposed under irradiation environment with very high nuclear heating as presented in chapter 2. Therefore, the coil temperature margin could be vanished as accumulated irradiation, and the temperature rise is essential to be investigated for design the superconducting magnet in a high radiation environment.
In a longer time duration, the heat generation from nuclear heating can be treated as a time independent term, and the coil temperature can be obtained by solving the following heat transfer equation numerically:
%CdT
dt =∇ ·(k∇T) +Q (3.3)
where T is the temperature, C is the specific heat as a function of temperature, % is the mass density, k is the thermal conductivity, and Q is the heat generation, respectively. In general, the thermal conductivity is a function of temperature, however, it also relies on the magnetic field and RRR for some metals.
Monte Carlo
Simulation Thermal Analysis Quench Analysis
Energy Deposition Degradation Rate of RRR
(Experimental Data) Magnetic Field Analysis
Neutron Flux
or DPA Thermal Conductivity
Beam Operation Time
Temperature
Figure 3.2: Overview of the method to take the irradiation effects into account in thermal analysis.
As for the PCS, the coil temperature is expected to be increased by not only the nuclear heating but also the degradation of thermal conductivity on thermal path during a long-term operation. To take into account the irradiation effect in the process of thermal analysis, a method to combine the Monte Carlo simulation and thermal analysis is implemented as presented in Fig. 3.2:
1). The profiles of energy deposition and neutron flux are calculated with the Monte Carlo simulation as described in chapter 2.
2). The profile of neutron flux is converted to that of effective RRR with the experimental data and a continuous beam operation time to take the irradiation effect into account.
3). The three-dimensional thermal conductivity is estimated with the profiles of effective RRR, temperature and magnetic field.
4). Passing the profiles of three-dimensional thermal conductivity and energy deposition to the terms of k and Qin equation (3.3), the temperature is obtained by solving the heat transfer equation with a numerical method described in Sec. 3.2.1.
In the quench analysis which will be described in chapter 5 later utilizes the same thermal model presented here to take the irradiation effect into account.
Simulation Parameters
In this simulation, the structure of superconducting coil is modeled in accordance with the current coil design as described in Ref. [46]. The superconducting cable is insulated with composite of polyimide film and glass cloth with a thickness of 0.2 mm, of which the thermal conductivity is assumed as polyimide (Kapton), conservatively. As illustrated in Fig. 3.3, the 1 mm thick aluminum strips are inserted in between each layer of CS and MS coils. In CS and MS coils, totally 24 sheets of aluminum strips with a width of 15 cm is inserted between the coil layer along the azimuth. In addition to the cable insulation, a 0.25 mm thick ground insulation is positioned between the coil layer and the aluminum strip, of which the material is also assumed to be polyimide (Kapton). The both ends of aluminum strips in CS1 and MS1 coils are extracted from the coil and fixed at 4.5 K with a distance of 25 cm from the coil ends, while only one side of strip is fixed at 4.5 K in CS0 and MS2 coils. The gap between the support shell and coil is filled with 3 mm thick resin whose material property is treated as the fiber glass epoxy-G10, and outer surface of the support shell is anchored at 4.5 K.
The possible degradation caused by neutron irradiation is considered as the change in RRR for aluminum stabilizer and strips. The 3D profile of RRR converted from the neutron flux and profile of magnetic field are applied to calculate the thermal conductivity.
The irradiation influence on the thermal conductivity of insulation tape is investigated with a gamma-ray irradiation, and no significant degradation is observed, which will be presented in chapter 4. Hence, the thermal insulation is set to be a radiation independent variable in simulation. All of material properties are derived from the ROXIE and NIST databases besides the aluminum. For the aluminum, the electrical resistivity and thermal conductivity are treated as a function of the temperature, RRR with Hust’s semi-empirical equations, in which its magnetoresistivity is obtained by fitting a Kohler plot with the experimental data [47]. The effect of magnetic field on the thermal conductivity is converted from the magnetoresistivity by the Wiedemann-Franz law. The details of material properties applied in this simulation will be described in Sec. 3.2.3. The other parameters are listed in Table 3.2.
Chapter 3. Thermal Stability during the Beam Operation
Al stabilizer Thermal Path
Ground Insulation (0.25 mm)
Resin (3 mm) Support shell (A5083)
4.73 mm 15 mm NbTi/Cu
Turn-to-turn Insulation (0.2 mm)
4.5 K 4.5 K
4.5 K
Support shell Thermal Path
Conductor
Figure 3.3: Cross sectional view of the CS1 coil in thermal simulation. Red, green and blue domain indicate the aluminum strip, conductor and support shell, respectively (copyright:
IEEE [37]).
Table 3.2: Parameters used in the heat transfer simulation for the CS and MS coils.
Parameters Value
Coil turns 35 (CS0), 270 (CS1), 285 (MS1) and 140 (MS2)
Coil layers 9 (CS0, CS1), 5 (MS1) and 7 (MS2)
Conductor Rutherford cable stabilized with aluminum Al:Cu:NbTi = 7.3:0.9:1.0
Conductor size 15.0×4.73 mm2
Turn-to-turn insulation thickness: 0.2 mm, material: polyimide (Kapton) Ground insulation thickness: 0.25 mm, material: polyimide (Kapton) Support shell thickness: 80 mm, material: aluminum (A5083) Resin thickness: 3 mm, material: GRFP-G10
Aluminum strips thickness: 1 mm, material: aluminuma
number: 24 (along the azimuth), width: 150 mm Initial RRR 400 (Al stabilizer), 2000 (Al strip)b
a. Aluminum as a function of temperature, RRR and magnetic field.
b. The RRR is a function of beam operation time, which is decreased as the accumulated neutron flux increases.
Modeling the Irradiation Effect
To take the irradiation effect into account in the thermal analysis, a parameter, effective residual resistance ratio RRRef f, is defined to correlate the impurity of metals with a
damage rate as
RRRef f(top) = ρRT ρ0 +ζφtop
(3.4) where φ is the neutron flux (En >0.1 MeV), ρRT is the electrical resistivity at room tem-perature, ρ0 is the residual resistivity at cryogenic temperature,ζ is the degradation rate induced by radiation damage, and top is the elapsed time in a thermal cycle, respectively.
The parameter φ can also be a DPA rate, meanwhile the damage rate ζ must be in the unit of Ω·m/DPA. Figure 3.4 shows the correlation between the neutron fluence (En >
0.1 MeV) or DPA and the effective RRR. With a peak of neutron flux, 1.2×1014n/m2/sec, the RRRs of thermal path as well as the aluminum stabilizer are expected to be degraded from 2000 to 100 for about 10-day beam operation, while, by using the damage rate of DPA, the RRR of thermal path is expected to be degraded slightly. In order to give a conservative simulation for the thermal design of superconducting magnet, the neutron flux is used in the further thermal and quench analysis for the PCS.
1016 1018 1020 1022
Neutron Fluence (En>0.1 MeV) [n/m2] 10-1
100 101 102 103 104
Effective RRR
RRR0 = 400 RRR0 = 2000
10-8 10-6 10-4 10-2
Displacement Per Atom [DPA]
10-1 100 101 102 103 104
Effective RRR
RRR0 = 400 RRR0 = 2000
Figure 3.4: RRR correlated with the neutron flux (left) and DPA (right) for aluminum, where RRR0 is the initial RRR before the degradation. The degradation rate of 0.03 nΩ·m per 1020 n/m2 (En >0.1 MeV) measured by M. Yoshida et al. and 6.8×10−4 Ω·cm measured by J. Horaket al. are used here. Red and green line indicate the aluminum with the initial RRRs of 400 and 2000 used as stabilizer and thermal path in PCS, respectively.
In simulation, the degradation of copper is no taken into account and its RRR is fixed at 50 because its damage rate is lower than that of aluminum about factor 3 as well as the cross-sectional area of copper is small enough. Also, as reported in Ref. [48], the magnetoresistance of copper is not affected by neutron irradiation as the effective RRR degrades.
3.2.1 Simulation Method
A three-dimensional thermal analysis is essential since the spatial distribution of the neutron flux and energy deposition in the coil are not symmetric. The calculation of heat transfer can be boosted by solving implicitly with the three-dimensional finite differential
Chapter 3. Thermal Stability during the Beam Operation
method (FDM) [49], however, it is difficult to handle the computation memory for the sparse matrix, thereby, the temperature is calculated with a explicit method in this work1.
To solve the heat transfer equation, first of all, the geometry of coil is meshed and discretized into a cylindrical element. Assuming each cylindrical element has the identity number of (i, j, k) with a size of dz ×rdθ × dr along z,θ and r direction as plotted in Fig. 3.5, the heat transfer equation can be rewritten as
%(i,j,k)C(i,j,k)V(i,j,k)T(i,j,k)t −T(i,j,k)t−1
∆t = ∆q(i,j,k)z + ∆qθ(i,j,k)+ ∆qr(i,j,k)+q(i,j,k) (3.5) where V(i,j,k) is the volume of each element, ∆qz, ∆qθ and ∆qr are the potential heat for each element along the z, θ and r direction, q(i,j,k) is the heat load for each element, T(i,j,k)t is the temperature of the (i, j, k) element in current time step, T(i,j,k)t−1 is that of temperature in the previous time step and ∆t is the time step, respectively.
d𝜃
dr dz
d𝜃(r+dr/2) d𝜃(r-dr/2)
qj-1
qj+1
qk+1
qk-1
qi+1
qi-1 Z
R 𝜃
Ar𝜃
A’z𝜃
Az𝜃
Azr O
(i,j,k) (i+1,j,k) (i-1,j,k)
(i,j,k-1) (i,j,k+1)
Z R
dz dr
Figure 3.5: The meshed element of (i, j, k) at position of (z, θ, r) with the dimension ofdz × rdθ ×dr (Left) and the connected thermal network from the view of θaxis (Right).
The potential heat for each element is presented as
∆qz = kavgz (T(i+1,j,k)t−1 −T(i,j,k)t−1 )Arθ
lz −kavgz0 (T(i,j,k)t−1 −T(i−1,j,k)t−1 )Arθ lz0
∆qθ = kavgθ (T(i,j+1,k)t−1 −T(i,j,k)t−1 )Azr
lθ −kavgθ0 (T(i,j,k)t−1 −T(i,j−1,k)t−1 )Azr
l0θ
∆qr = kavgr (T(i,j,k+1)t−1 −T(i,j,k)t−1 )Azθ lr
−kravg0 (T(i,j,k)t−1 −T(i,j,k−1)t−1 )A0zθ
l0r (3.6) where λ and λ0 are the effective thermal conductivity between the adjacent elements, l andl0 are the distance of the adjacent elements, andA andA0 are the contacted area with
1Since the three-dimensional neutron flux resulting in the thermal conductivity is difficult to be implemented into the FEM software, the thermal analysis is performed with a standalone code in this work, which is written in C++ and managed withCmake.
the adjacent element. The effective thermal conductivity between the adjacent elements is averaged in a serial thermal circuit. For a cylindrical coordinate, each effective thermal conductivity is calculated as
kavgz = (dz(i,j,k) k(i,j,k)z
+ dz(i−1,j,k)
kz(i−1,j,k)
)−1·(dz(i,j,k)+dz(i−1,j,k)) kavgz0 = (dz(i,j,k)
k(i,j,k)z
+ dz(i+1,j,k) kz(i+1,j,k)
)−1·(dz(i,j,k)+dz(i+1,j,k)) (3.7)
kθavg = (rdθ(i,j,k)
k(i,j,k)θ + rdθ(i,j−1,k)
k(i,j−1,k)θ )−1·(dθ(i,j,k)+dθ(i,j−1,k))r kθavg0 = (rdθ(i,j,k)
k(i,j,k)θ + rdθ(i,j+1,k)
k(i,j+1,k)θ )−1·(dθ(i,j,k)+dθ(i,j+1,k))r (3.8)
kravg = (dr(i,j,k) kr(i,j,k)
+ dr(i,j,k−1)
k(i,j,k−1)r
)−1·(dr(i,j,k)+dr(i,j,k−1)) kravg0 = (dr(i,j,k)
kr(i,j,k)
+ dr(i,j,k+1) k(i,j,k+1)r
)−1·(dr(i,j,k)+dr(i,j,k+1)) (3.9) where dz,dθ anddr are the size of each element, kz,kθ andkr is the thermal conductivity in the given element, andr is the radius of the element. Each contacted area between the element and the adjacent element is calculated with the size of element as
Arθ = dθ
2 ·[(r+dr/2)2 −(r−dr/2)2] Azr = dr·dz
Azθ = (r−dr/2)dθ·dz
A0zθ = (r+dr/2)dθ·dz (3.10)
where Arθ and Azr are the contacted area in the r−θ and z−r plane, and Azθ and A0zθ are the inner and outer surface in z−θ plane due to the change of the radius.
The boundary condition must be set at the element of boundary to link the element with the external condition. Two kinds of boundary, the Dirichlet and Neumann condition [49]
are implemented to produce the boundary condition. Since treatment of the heat transfer equation is complicated at boundary, a virtual element Tb is added next to the boundary element, and linked with the boundary element to transfer the heat by updating the virtual element in each time step. For the different boundary, the temperature of the virtual element is updated as
Tb =
(T0, Dirichlet boundary
T(i,j,k)+ qhA, Neumann boundary (3.11)
whereqA is the heat flux through the boundary area,his the thermal contact conductance at boundary and T0 is the temperature fixed at boundary, respectively. In this analysis,
Chapter 3. Thermal Stability during the Beam Operation
the boundaries which are not fixed to 4.5 K are assumed to be adiabatic boundary, thus the temperature of the virtual element is set to be Tb =T(i,j,k) for adiabatic boundary.
The time step is difficult to be defined because the large time step could cause the oscillation in the simulation. To suppress the oscillation, a method to determine the minimum time step calculated from the thermal diffusivity and element size is implemented as [50]
∆t≤min{2[αz
dz2 + αθ
(rdθ)2 + αr
dr2]−1(i,j,k)} (3.12)
whereα =k/%C is the thermal diffusivity of each element. Because the thermal diffusivity relies on the temperature and the other parameter, the time step is calculated and optimized in each time loop. For an instance, due to a 1 mm thick thermal path made of aluminum with very high purity, the time step at cryogenic temperature is estimated automatically to be around 5 µsec for the calculation in each time loop.
3.2.2 Simulation Model
Treatment of Conductors
In PCS, the aluminum-stabilized superconducting cable is wound with 2 layers of insulation tapes, and the conductor consists the strands and stabilizer, which is difficult to reproduce the details in the simulation. Therefore, the conductor with two layers of insulation tapes as well as the aluminum strip with insulation sheets are treated as one element, then, their material properties are averaged with the composition of materials.
The treatment of the material properties of conductor is illustrated as follows:
- Mass density:
%mean =ηCu%Cu+ηAl%Al+ηsc%sc (3.13) where
ηCu, ηAl,ηsc: cross-sectional ratio of copper, aluminum and NbTi in conductor,
%Cu,%Al,%sc: mass density of each component.
- Specific heat capacity:
Cmean= (ηAl%AlCAl+ηCu%CuCCu+ηsc%scCsc)
%mean (3.14)
where
CCu, CAl,Csc: specific heat of each component.
- Thermal conductivity kmean = (kz, kθ, kr):
kz = (dzins+dzcdt)·(dzins
kins +dzcdt
kAl )−1 kθ = Ainskins+AcdtkAl
Ains+Acdt
kr = (drins+drcdt)·(drins
k + drcdt
k )−1 (3.15)
where
kAl, kins: thermal conductivity of aluminum and insulation tape, Acdt, Ains: cross section of conductor and insulation tape,
dz,dr: thickness of each component along the z and r-axis.
It is noted that the mean of thermal conductivity is treated as a equivalent parallel circuit along the winding direction, while the others are calculated as a serial circuit. Vice versa, the element of support shell is averaged with the support shell and half thickness of epoxy resin, and the one of aluminum strip is averaged with the aluminum and ground insulation respectively.
Heat Transfer along the Winding Direction
In practice, the coil is wound with one superconducting cable from the inner layer to outer layer. The element along the winding direction is thermally connected to each other. To implement the heat transfer along the winding direction, a virtual element next to the element at boundary is added to contain the boundary condition. The Dirichlet boundary and Neumann boundary can be implemented by reseting the temperature at the temperature in virtual element (e.g. Adiabatic condition can be implemented by copying the temperature at last element to the virtual element that the temperature flux at boundary is calculated to be zero) as plotted in Fig. 3.6. Thereby, the heat transfer along the helical direction is treated with a procedure as follows:
1). The temperature at the last element in thei-th turn is copied to the virtual element next to the first element in the (i+ 1)-th turn.
2). The temperature at the first element in the (i+ 1)-th turn is copied to the virtual element next to the last element in the i-th turn.
3). The first element in the i-th turn and last element in the n-th turn is set to be adiabatic.
Alignment of Thermal Path
Totally 24 pure aluminum strips with a width of 150 mm and thickness of 1 mm are inserted between coil layer every 15 degree along the azimuth, in which the space between the strips are filled with the epoxy resin. The aluminum strips are extracted from the coil end, and connected to the cooling pipe to cool down the superconducting coil as shown in Fig. 3.7.
To implement the alignment of thermal path in simulation, the algorithm is developed as follows:
1). The filling ratio of total aluminum strips along the azimuth is calculated, in which the filling ratio at the inner and outer of conductor layer are defined as fi and fo.
Chapter 3. Thermal Stability during the Beam Operation
0 2π
π 𝜃
T(0,2π,k)
T(1,1,k) T(0,2π+1,k)
T(1,0,k) virtual element
T(i,2π,k)
T(i+1,1,k) T(i,2π+1,k)
T(i+1,0,k)
Figure 3.6: Extended each turn at kth layer. At the boundary of θ axis, each element is connected with the virtual element thermally. Exchanging the temperature at next turn with virtual element, the heat transfer along the winding direction has been implemented.
1 2
3 4
5 6
7 8
9 10
11 12
B B
C C
D D
E E
F F
G G
H H
材質 検査 承認
名称 尺度 改正２
改正
改正1
数量 備考
番号 製図 設計 普通公差 (JI B 0405)
角度寸法の許容差 長さ寸法の許容差 基準寸法の区分 中級 (m) 中級 (m)
対象とする角度の 短辺の長さの区分
0.5～ 3 ±0.1 2017/05/11
COMET CS-MS
コイル C 0 1
0°
90°
180°
240°巻始め位置 270°
30° 層間渡り
0°
90°
180°
270°
巻終り 100 素線絶縁 20000 以上
口出し長さ20000mm以上のこと 口出しはフラットワイズのこと 巻始め
100 素線絶縁 6000 以上
口出し長さ6000mm以上のこと 口出しはフラットワイズのこと
30° 層間渡り A
A
A-A 断面
7.5° 15°
188.95+1.00-3.50 5.15 (樹脂層0.1含む)
400 アルミストリップ 35ターン＋層間接続
1344 コイル内径15.35
1644.3 コイル外径
層間詳細
(S=2:1) 1.5 層間隙間 1 アルミストリップ 0.25 ポリイミドフィルム2層+樹脂 0.25 ポリイミドフィルム2層+樹脂
内外径
内外径
内外径
内外径
全長 0°
90°
180°
270°
全長測定位置
全長測定位置
寸法検査
・内外径各4断面（下図参照）
・全長90°ピッチ
・各層毎に測定
導体詳細
(S=2:1)
・素線絶縁 ： t=0.075 B プリプレグテープ ・絶縁巻き方法 ： ダブルスパイラル ・テ―パ角 ： 0.61°
1 2
15.05 導体 15.35
0.15 素線絶縁
5.05 4.75 導体 0.15 素線絶縁
d 詳細
(S=1:2)
b 詳細
(S=1:2)
c 詳細
(S=1:2)
a 詳細
(S=1:2)
イメージ図
3 3
3 3
9層
部品表
番号 名称 材質 数量 備考
1 アルミ安定化導体 NbTi / Al / Cu 2 B プリプレグテープ U-BT
3 コイルスペーサ BT-GFRP
4 アルミストリップ Al
4
導体
c
a
d b
1 最内層、導体渡り部のアルミストリップを削除、寸法公差修正
1
1
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9 10
10 11
11 12
12
A A
B B
C C
D D
E E
F F
G G
H H
材質
図名
高エネルギー加速器研究機構 投影法 J-PARC 低温セクション
検査 承認 名称
尺度 日付
改正２ 改正
改正1
数量 備考
図 サイズ 番 番号
製図 設計 普通公差 (JI B 0405)
角度寸法の許容差 長さ寸法の許容差 基準寸法の区分 中級 (m) 中級 (m)
対象とする角度の
短辺の長さの区分 0.5～
3～6～
120～30～
1000～400～
2000～
36 30 120400 10002000 4000
±0.1±0.1
±0.2
±0.3±0.5
±0.8±1.2
±2
～ 10～50～
120～
400～
10 12050 400
±1°
±30'
±20'±10'
±5' 単位：mm
2017/05/11 COMET CS-MS
コイル C 0
1 : 8 真家
（三立）
2017/04/21
PCS02A034 コイル C 0 巻線図
A1 1
吉田 飯尾 0°
90°
180°
240°巻始め位置 270°
30° 層間渡り
0°
90°
180°
270°
巻終り 100 素線絶縁 20000 以上
口出し長さ20000mm以上のこと 口出しはフラットワイズのこと 巻始め
100 素線絶縁 6000 以上
口出し長さ6000mm以上のこと 口出しはフラットワイズのこと
30° 層間渡り A
A A-A 断面
7.5° 15°
188.95+1.00-3.50 5.15 (樹脂層0.1含む)
400 アルミストリップ 35ターン＋層間接続
1344 コイル内径15.35
1644.3 コイル外径
層間詳細 (S=2:1) 1.5 層間隙間 1 アルミストリップ 0.25 ポリイミドフィルム2層+樹脂 0.25 ポリイミドフィルム2層+樹脂
内外径
内外径
内外径
内外径
全長 0°
90°
180°
270°
全長測定位置
全長測定位置 寸法検査
・内外径各4断面（下図参照）
・全長90°ピッチ
・各層毎に測定
導体詳細 (S=2:1)
・素線絶縁 ： t=0.075 B プリプレグテープ ・絶縁巻き方法 ： ダブルスパイラル ・テ―パ角 ： 0.61°
1 2
15.05 導体 15.35
0.15 素線絶縁
5.05 4.75 導体 0.15 素線絶縁
d 詳細
(S=1:2) b 詳細
(S=1:2) c 詳細
(S=1:2) a 詳細
(S=1:2)
イメージ図
3 3
3 3
9層
部品表
番号 名称 材質 数量 備考
1 アルミ安定化導体 NbTi / Al / Cu 2 B プリプレグテープ U-BT
3 コイルスペーサ BT-GFRP
4 アルミストリップ Al
4
導体
c
a
d b
1 最内層、導体渡り部のアルミストリップを削除、寸法公差修正
1
Superconducting coil
Aluminum strips
Figure 3.7: The alignment of the aluminum strips between the coil layer with a cross-sectional view (left) and three-dimensional view (right).
2). The contact area, Azθ and A0zθ in the z-θ plane, in each element is normalized with the filling ratio, fi and fo.
3). The contacted area Arθ inr-θ plane is normalized with the filling ratio of f.
4). The thermal conductivity along the azimuth is assumed to be the one of polyimide.
5). In the innermost layer of aluminum strip, the azimuthal heat transfer is not allowed.
The filling ratio of aluminum strips to the azimuthal length at the center, inner and outer
37