Abstract

Cable-in-conduit conductor (CICC) has wide applications, and this structure is often served to undergo heat force-electromagnetic coupled field in practical utilization, especially in the magnetic confinement fusion (e.g., Tokamak). The mechanical behavior in CICC is of relevance to understanding the mechanical response and cannot be ignored for assessing the safety of these superconducting structures. In this chapter, several mechanical models were established to analyze the mechanical behavior of the CICC in Tokamak device, and the key mechanical problems such as the equivalent mechanical parameters of the superconducting cable, the untwisting behavior in the process of insertion, the buckling behavior of the superconducting wire under the action of the thermo-electromagnetic static load, and the Tcs (current sharing temperature) degradation under the thermo-electromagnetic cyclic loads are studied. Finally, we summarize the existing problems and the future research points on the basis of the previous research results, which will help the related researchers to figure out the mechanical behavior of CICC more easily.

Keywords: Nb3Sn, cable in conduit conductor (CICC), cable stiffness, coefficient of thermal expansion, untwisting, current sharing temperature

### 1. Introduction

The ITER program is one of the largest and the most influential international energy technology cooperation projects, to verify the engineering feasibility of the magnetic confinement fusion. The core device of the magnetic confinement fusion reactor is the cable-in-conduit conductor (CICC). CICCs were used to build up the superconducting coil for generating strong magnetic fields to confine the hightemperature plasma in a confined space and maintaining the fusion reaction [1]. The ITER superconducting magnet systems mainly consist of four kinds of coils: 6 central solenoids (CS), 18 toroidal field (TF) coils, 6 sets of poloidal field (PF) coils, and 9 pairs of correction fields coils (CC) [2].

As early as the 1960s, the low-temperature superconducting material NbZr was processed into round wire and cables [3]. Subsequently, the superconducting magnets were wound with the structure of an internally cooled conductor (ICS) [4]. The superconducting strand is cooled to the superconducting state by the heat transfer copper tube with liquid helium in it [5]. But, the contact cooling method of ICS is inefficient, and the superconducting material is inclined to have a magnetic

flux jump which will make the magnetic system to be quenched. In 1975, Hoenig et al. suggested subdividing the superconductor into strands to suppress the flux jump and twisting them into a cable to reduce the AC losses [6]. In 1980, Lue et al. proposed a cable-in-conduit design, and the innermost part was a perforated copper tube or a high-hardness stainless steel spring to form a liquid helium fast-flowing channel [7]. These two designs are the prototypes of modern CICC conductors. Nowadays, the CS and TF conductors with higher magnetic fields in the ITER project were fabricated by more than 1000 Nb3Sn wires. Disadvantageously, the superconducting properties of Nb3Sn are sensitive to mechanical deformation, which means that the tensile, compressive, and torsional deformations all lead to the reduction of the critical current [8]. Therefore, the strain state of the Nb3Sn strand cannot be ignored. Therefore, during its design, manufacture, and operation stage of the CICC, the mechanical analysis is needed.

Many studies have been published on the equivalent mechanical parameters of the twisted cable with two dimensions, such as the compression modulus of the cross section [9, 10], rather than built a complete three-dimensional model of the twisted cable. Feng et al. have applied the thin rod model to CICC conductor analysis and established the spatial geometry of each superconducting strand in the CICC conductor [11]. Qin et al. have applied the thin rod model to the mechanical analysis of superconducting cables and derived the axial stress–strain curves of primary cables and high-order strands [12]. The influence of pitch on the elastic modulus of the stranded cable and the curvature of the strand has been discussed. The introduction of the copper strand has been found to greatly reduce the axial stiffness of the strand, and the contact deformation between the strands has been found to have a great influence on the stiffness of the strand. The theoretical calculation results have appeared to be in good agreement with the experiments [13, 14]. Yue et al. have conducted a systematic mechanical analysis of the CICC in the design, preparation, and operation stage [15–18].

Therefore, each wire in the cable can be simplified into a thin rod which is elongated in the axial direction under the axial tensile load, and the wires can be contacted tightly or rotate in the lateral direction. The deformation and the force analysis of

; <sup>τ</sup> <sup>¼</sup> <sup>v</sup>d<sup>t</sup> cos <sup>α</sup> sin <sup>α</sup>

τ þ Hk<sup>0</sup>

<sup>d</sup><sup>s</sup> � Hk <sup>þ</sup> <sup>G</sup><sup>τ</sup> <sup>þ</sup> <sup>N</sup> <sup>þ</sup> <sup>K</sup><sup>0</sup> <sup>¼</sup> <sup>0</sup>;

k þ Θ ¼ 0:

<sup>r</sup>d<sup>s</sup> <sup>¼</sup> cos <sup>α</sup> sin <sup>α</sup>

� N<sup>0</sup> þ K ¼ 0;

; H ¼ GIzΔτ: (3)

sin <sup>2</sup><sup>α</sup> : (4)

<sup>r</sup> : (1)

(2)

the wires are shown in Figure 1(a) and (b), respectively. The curvature and torsion of the spiral are as follows:

The position of a spiral rod (a) and loads acting on the wire (b) [19].

DOI: http://dx.doi.org/10.5772/intechopen.82349

<sup>r</sup>d<sup>s</sup> <sup>¼</sup> cos <sup>2</sup><sup>α</sup>

r

The Mechanical Behavior of the Cable-in-Conduit Conductor in the ITER Project

τ þ X ¼ 0;

k þ Z ¼ 0;

G ¼ EIxΔκ; G<sup>0</sup> ¼ EIyΔκ<sup>0</sup>

The equilibrium equation of forces and moments can be expressed as

dG <sup>d</sup><sup>s</sup> � <sup>G</sup><sup>0</sup>

dG<sup>0</sup>

dH

Assuming that the thin rod is isotropic and elastic, the moments in any cross section with respect to the axis x, y, z can be written as Ix, Iy, Iz, and the constitutive

In the result of solving Eq. (3) with the account for the temperature terms, the expression of the equivalent coefficient of thermal expansion in the axial direction

<sup>α</sup>eff <sup>¼</sup> <sup>α</sup><sup>L</sup> � <sup>α</sup><sup>T</sup> cos <sup>2</sup><sup>α</sup>

In Eq. (4), αeff is the equivalent coefficient of thermal expansion of the strand, α<sup>L</sup> is the coefficient of thermal expansion of the strand in the longitudinal direction, α<sup>T</sup> is the transverse coefficient of thermal expansion of the strand, and α denotes the

<sup>d</sup><sup>s</sup> � Gk<sup>0</sup> <sup>þ</sup> <sup>G</sup><sup>0</sup>

<sup>κ</sup> <sup>¼</sup> <sup>0</sup>; <sup>κ</sup><sup>0</sup> <sup>¼</sup> <sup>v</sup>d<sup>t</sup> cos <sup>2</sup><sup>α</sup>

þ Tκ<sup>0</sup> � N<sup>0</sup>

<sup>d</sup><sup>s</sup> � Nk<sup>0</sup> <sup>þ</sup> <sup>N</sup><sup>0</sup>

of the triplet can also be given by [15]

<sup>d</sup><sup>s</sup> � Tk <sup>þ</sup> <sup>N</sup><sup>τ</sup> <sup>þ</sup> <sup>Y</sup> <sup>¼</sup> <sup>0</sup>;

dN ds

Figure 1.

dN<sup>0</sup>

dT

equations are given by

helix angle.

83

In this chapter, first of all, we focus on the equivalent mechanical parameters of the superconducting cable; second, we concentrate on the untwisting behavior in the process of insertion; third, we want to explain the buckling behavior of the superconducting wire under the action of the thermo-electromagnetic static load; and finally, the Tcs degradation under the thermo-electromagnetic cyclic loads is studied. Our goal is to relate the cable stresses and buckling behavior to the thermal and electromagnetic loads so that relations between cable stress and current transport characteristics are built completely.

## 2. The equivalent mechanical parameters of the CICC

The mechanical behaviors of CICC have two main problems of structure and operation. On the one hand, the equivalent modulus of the cable is dependent on the manufacture parameters such as pitch, porosity, and radius. On the other hand, the electromagnetic load and the extremely low temperature make the internal stress and strain state of the cable difficult to analyze. Therefore, the thin rod model is applied to calculate the equivalent mechanical parameters of CICC conductors.

#### 2.1 The tensile stiffness of the triplet

From the geometry characteristics of the cable, we know that the CICC superconducting cables have a complex structure with five stages of spirals. The Mechanical Behavior of the Cable-in-Conduit Conductor in the ITER Project DOI: http://dx.doi.org/10.5772/intechopen.82349

Figure 1.

flux jump which will make the magnetic system to be quenched. In 1975, Hoenig et al. suggested subdividing the superconductor into strands to suppress the flux jump and twisting them into a cable to reduce the AC losses [6]. In 1980, Lue et al. proposed a cable-in-conduit design, and the innermost part was a perforated copper tube or a high-hardness stainless steel spring to form a liquid helium fast-flowing channel [7]. These two designs are the prototypes of modern CICC conductors. Nowadays, the CS and TF conductors with higher magnetic fields in the ITER project were fabricated by more than 1000 Nb3Sn wires. Disadvantageously, the superconducting properties of Nb3Sn are sensitive to mechanical deformation, which means that the tensile, compressive, and torsional deformations all lead to the reduction of the critical current [8]. Therefore, the strain state of the Nb3Sn strand cannot be ignored. Therefore, during its design, manufacture, and operation

Nuclear Fusion - One Noble Goal and a Variety of Scientific and Technological Challenges

Many studies have been published on the equivalent mechanical parameters of the twisted cable with two dimensions, such as the compression modulus of the cross section [9, 10], rather than built a complete three-dimensional model of the twisted cable. Feng et al. have applied the thin rod model to CICC conductor analysis and established the spatial geometry of each superconducting strand in the CICC conductor [11]. Qin et al. have applied the thin rod model to the mechanical analysis of superconducting cables and derived the axial stress–strain curves of primary cables and high-order strands [12]. The influence of pitch on the elastic modulus of the stranded cable and the curvature of the strand has been discussed. The introduction of the copper strand has been found to greatly reduce the axial stiffness of the strand, and the contact deformation between the strands has been found to have a great influence on the stiffness of the strand. The theoretical calculation results have appeared to be in good agreement with the experiments [13, 14]. Yue et al. have conducted a systematic mechanical analysis of the CICC in

In this chapter, first of all, we focus on the equivalent mechanical parameters of the superconducting cable; second, we concentrate on the untwisting behavior in the process of insertion; third, we want to explain the buckling behavior of the superconducting wire under the action of the thermo-electromagnetic static load; and finally, the Tcs degradation under the thermo-electromagnetic cyclic loads is studied. Our goal is to relate the cable stresses and buckling behavior to the thermal and electromagnetic loads so that relations between cable stress and current trans-

The mechanical behaviors of CICC have two main problems of structure and operation. On the one hand, the equivalent modulus of the cable is dependent on the manufacture parameters such as pitch, porosity, and radius. On the other hand, the electromagnetic load and the extremely low temperature make the internal stress and strain state of the cable difficult to analyze. Therefore, the thin rod model is applied to calculate the equivalent mechanical parameters of CICC conductors.

From the geometry characteristics of the cable, we know that the CICC superconducting cables have a complex structure with five stages of spirals.

stage of the CICC, the mechanical analysis is needed.

the design, preparation, and operation stage [15–18].

2. The equivalent mechanical parameters of the CICC

port characteristics are built completely.

2.1 The tensile stiffness of the triplet

82

The position of a spiral rod (a) and loads acting on the wire (b) [19].

Therefore, each wire in the cable can be simplified into a thin rod which is elongated in the axial direction under the axial tensile load, and the wires can be contacted tightly or rotate in the lateral direction. The deformation and the force analysis of the wires are shown in Figure 1(a) and (b), respectively.

The curvature and torsion of the spiral are as follows:

$$\kappa = 0; \kappa' = \frac{v \text{dt} \cos^2 a}{r \text{ds}} = \frac{\cos^2 a}{r}; \tau = \frac{v \text{dt} \cos a \sin a}{r \text{ds}} = \frac{\cos a \sin a}{r}. \tag{1}$$

The equilibrium equation of forces and moments can be expressed as

$$\frac{\mathrm{d}N}{\mathrm{d}\sigma} + T\mathrm{k}' - N'\tau + X = 0; \quad \frac{\mathrm{d}G}{\mathrm{d}\sigma} - G'\tau + H\mathrm{k}' - N' + K = 0;$$

$$\frac{\mathrm{d}N'}{\mathrm{d}\sigma} - T\mathrm{k} + N\tau + Y = 0; \quad \frac{\mathrm{d}G}{\mathrm{d}\tau} - H\mathrm{k} + G\tau + N + K' = 0; \tag{2}$$

$$\frac{\mathrm{d}T}{\mathrm{d}\sigma} - N\mathrm{k}' + N'\mathrm{k} + Z = 0; \quad \frac{\mathrm{d}H}{\mathrm{d}\sigma} - G\mathrm{k}' + G'\mathrm{k} + \Theta = 0.$$

Assuming that the thin rod is isotropic and elastic, the moments in any cross section with respect to the axis x, y, z can be written as Ix, Iy, Iz, and the constitutive equations are given by

$$G = EI\_\mathbf{x} \Delta \kappa; G' = EI\_\mathbf{y} \Delta \kappa'; H = G I\_\mathbf{z} \Delta \tau. \tag{3}$$

In the result of solving Eq. (3) with the account for the temperature terms, the expression of the equivalent coefficient of thermal expansion in the axial direction of the triplet can also be given by [15]

$$a\_{\rm eff} = \frac{a\_{\rm L} - a\_{\rm T} \cos^2 a}{\sin^2 a}. \tag{4}$$

In Eq. (4), αeff is the equivalent coefficient of thermal expansion of the strand, α<sup>L</sup> is the coefficient of thermal expansion of the strand in the longitudinal direction, α<sup>T</sup> is the transverse coefficient of thermal expansion of the strand, and α denotes the helix angle.

### 2.2 The tensile stiffness of the higher stage strand

Based on the equivalent modulus and thermal expansion of the triplet, the space and the 2D view of the triplet and single wire are shown in Figure 2(a) and (b), respectively.

The conversion relationship between the local coordinates of the triplet and the higher-level strand can be expressed as

$$T\_{\mathbf{k}} = \begin{bmatrix} -\cos\theta\_{\mathbf{k}} & -\sin\theta\_{\mathbf{k}} & \mathbf{0} \\\\ \sin\theta\_{\mathbf{k}}\sin a\_{\mathbf{k}} & -\cos\theta\_{\mathbf{k}}\sin a\_{\mathbf{k}} & \cos a\_{\mathbf{k}} \\\\ -\sin\theta\_{\mathbf{k}}\cos a\_{\mathbf{k}} & \cos\theta\_{\mathbf{k}}\cos a\_{\mathbf{k}} & \sin a\_{\mathbf{k}} \end{bmatrix}. \tag{5}$$

Mp1 Mb1 Mt1

3. Rotation analysis of the CICC

behavior of the cable must be controlled [17].

cable under the insertion force FInsert.

Figure 3.

85

Mp2 Mb2 Mt2

The Mechanical Behavior of the Cable-in-Conduit Conductor in the ITER Project

transformations as the CICC conductor has a five-stage twist structure.

r2ð Þ Ft2 sin α<sup>2</sup> þ Fb2 cos α<sup>2</sup> sin θ<sup>2</sup> �r2ð Þ Ft2 sin α<sup>2</sup> þ Fb2 cos α<sup>2</sup> cos θ<sup>2</sup> r2ð Þ Ft2 cos α<sup>2</sup> � Fb2 sin α<sup>2</sup>

Theoretically, the tensile stiffness of the conductor can be deduced by four times

In the CICC conductor manufacture process, they twist a superconducting cable and penetrate it into the stainless steel tube. However, due to the friction between the superconducting cable and the stainless steel armor, the drag force of the cable is as high as several tons during the cable penetration. The friction force of the pipe leads to the axial elongation of the superconducting cable, accompanied by the untwisting of the cable, which causes the cable pitch to increase. This makes that the pitch is much larger than the ITER requirement [20]. Therefore, the untwist

In this section, the untwist model is described. The large-scale cable is considered, e.g., ITER TF, CS, and CFETR CSMC. The components of the final cable include petals, central cooling spiral, and wrap, as shown in Figure 3. The model ignores the friction between the jacket and the cable, only modeling de-twists of the

The cable is divided into three parts in the model: central cooling spiral, six petals, and the wrap. The twist direction of wrap and cooling spiral is left and with the reverse direction for the petal. The torsion constraint is free for the cable when there is undering the uniaxial tension. Therefore, the boundary conditions can be set as F ¼ F0; M ¼ 0. The force of the whole cable is from those acts on wrap, sub-

F ¼ Fin þ Fp þ Fst ¼ F0,

In Eq. (9), F is the insertion force in the axial direction for the cable. Fst, Fp, Fin are the forces loading on the stainless steel wrap, petals, and inner cooling spiral,

<sup>M</sup> <sup>¼</sup> Min <sup>þ</sup> Mp <sup>þ</sup> Mst <sup>¼</sup> <sup>0</sup>: (9)

cables, and central cooling spiral, which can be described as follows:

The dimensions and parameters of a large-scale cable (e.g., CFETR CSMC).

3 7 7

5: (8)

DOI: http://dx.doi.org/10.5772/intechopen.82349

The curvature and torsion of the secondary stage strand can be given by

$$
\begin{bmatrix} \kappa\_{\rm p2} \\\\ \kappa\_{\rm b2} \\\\ \kappa\_{\rm t2} \end{bmatrix} = T\_2 T\_1 \left\{ T\_1^\mathrm{T} \begin{bmatrix} \mathbf{0} \\\\ \mathbf{0} \\\\ \frac{\cos \alpha\_2}{r\_2} \end{bmatrix} + \begin{bmatrix} \mathbf{0} \\\\ \mathbf{0} \\\\ \frac{\cos \alpha\_1}{r\_1} \end{bmatrix} \sin \alpha\_2 \right\}.\tag{6}
$$

According to the geometric compatibility of the secondary-stage strand, the deformation of the triplet is equal to the tangential strain of the secondary-stage strand, and the torsion of the triplet is equal to the twist angle of the secondary cable. The axial loads and torque of the secondary-stage strand can be obtained. The equilibrium equations can be expressed as

$$
\begin{bmatrix} 0\\ 0\\ 0\\ F\_{\rm t0} \end{bmatrix} = \mathbf{3}^\* \mathbf{3}^\* T\_1 T\_2 \begin{bmatrix} F\_{\rm p2} \\ F\_{\rm b2} \\ F\_{\rm t2} \end{bmatrix} \tag{7}
$$

Figure 2. Space line of the triplet and single wire (a) and 2D view of the triplet and single wire (b).

The Mechanical Behavior of the Cable-in-Conduit Conductor in the ITER Project DOI: http://dx.doi.org/10.5772/intechopen.82349

$$\begin{bmatrix} M\_{\rm p1} \\ M\_{\rm b1} \\ M\_{\rm t1} \end{bmatrix} = 3T\_2 \begin{bmatrix} M\_{\rm p2} \\ M\_{\rm b2} \\ M\_{\rm t2} \end{bmatrix} + 3 \begin{bmatrix} r\_2 (F\_{t2} \sin a\_2 + F\_{\rm b2} \cos a\_2) \sin \theta\_2 \\ -r\_2 (F\_{t2} \sin a\_2 + F\_{\rm b2} \cos a\_2) \cos \theta\_2 \\\ r\_2 (F\_{t2} \cos a\_2 - F\_{\rm b2} \sin a\_2) \end{bmatrix}. \tag{8}$$

Theoretically, the tensile stiffness of the conductor can be deduced by four times transformations as the CICC conductor has a five-stage twist structure.
