1. Introduction
Fig. 1 shows the HTS CORC cable structure consisting of multiple functional layers, including the skeleton, conductor, insulation, shielding, etc. Considering the advantages of low loss, compactness, flexibility, and environmental friendliness, CORC cables are one of the most promising superconducting applications [1], [2]. For long-distance and large-capacity power transmission, several hundred-meter scale CORC cables have been connected to the grid in the past two years [3], [4]. Since HTS cables must be cooled by liquid nitrogen (LN2), the axial shrinkage caused by thermal expansion seriously threatens their safety, which has been reported and verified by our trials [5], [6], [7]. Therefore, it is critical to estimate thermo-mechanical behaviors and provide appropriate compensation measures to prevent HTS power cables, which cost tens of millions for construction, from being damaged.
Fig. 1. Structure of tri-axial HTS CORC cable. |
With the wide applications of CORC cables, many researchers focused on their mechanical properties in the past five years, which were first investigated experimentally. Based on the electro-mechanical properties of superconducting tapes, D C van der Laan et al. systematically studied the current-carrying performance of CORC cables under mechanical behaviors, including transverse compressive stress [8], [9], axial tensile strain [10], [11], cyclic load [8], [11], and electromagnetic force [12], [13], [14]. With the introduction of nonlinear finite element (FE) software ABAQUS, the spiral winding process of CORC cables was simulated [15], [16], and then the parametric and analytical analyses were carried out [17], [18]. As the subsequent mechanical behaviors of cabling, bending and torsion models were developed [16], [19], [20], [21], which have been verified by experiments [20], [21], [22], [23]. However, further experimental and simulation results showed that the thermal stress of the cable dominated its stresses and strains [24], [25], even when subjected to electromagnetic forces exceeding 10 kiloampere and 10 Tesla levels [12], [13], [14], [26]. Besides, based on its local defects and predictions [27], [28], more attention should be paid to the mechanical behavior of HTS cables in practical applications at the cable level. Therefore, it is necessary and meaningful to develop new approaches to further focus on the thermal-stress-dominated mechanical properties of multilayer power cables.
Referring to the modeling and thoughts from FEM to analytical methods [29], [30], this paper provides an analytical solution to evaluate the effect of thermo-mechanical loads on cable shrinkage, as follows: In Section 2, the radial temperature distribution of the CORC cable is calculated based on the simplified geometric model. Then, three analytical methods are developed for stress and strain analysis in Section 3, in which the recursive method is introduced by assuming cable ends are axially fixed. Then, taking the axial strain as a variable, a more precise algebraic method is yielded based on the recursive method. Finally, concluded from the above derivation, a matrix equation with multi-variables is constructed, which is user-friendly and widely useable. In Section 4, the accuracy of analytical methods is verified by FEM results with the introduction of calculation procedures, in which the matrix method can be applied to isotropic and orthotropic materials. In Section 5, analytical methods are employed for the thermo-mechanical analysis of the tri-axial HTS CORC cable, and trial results show that the calculation error for the axial shrinkage of a 360 m cable is less than 1.63 cm. Moreover, the axial strain computation time of the 26-layer cable is around 10 s, which is thousands of times faster than the detailed three-dimensional (3D) FE model in ABAQUS. Combined with a cylindrical coordinate system with additional rotation [29], more work will be carried out from the functional layers of the cable to the constituent layers of the coated conductor.
2. Analytical solution for thermal loads
The thermal load dominates the mechanical behavior of the superconducting device, especially the thermal stress generated by the cyclic temperature. Based on reasonable assumptions, this section derives the radial temperature distribution of the CORC cable under applied conditions.
2.1. Simplified geometric model
The cable core and terminals are entirely fixed in multilayer power cables to prevent each functional layer from delamination under operating conditions. Therefore, when the HTS cable is subjected to thermo-mechanical loads, the axial displacement of each functional layer of the cable core is consistent, meaning each functional layer has the same axial strain. Compared with the thermal stress, the friction between interfaces is negligible, and all contact pairs can be considered perfectly bound. Furthermore, ignoring the air gap between superconducting tapes, the helically wound CORC cable can be simplified as an axisymmetric composite structure.
Based on the above assumptions, the schematic diagram of an N-layer CORC cable is shown in Fig. 2. From the inner layer to the outer layer, the radius and temperature at each surface are named ri and Ti (i = 0, 1, 2, …, N), respectively. The tri-axial HTS CORC cable is cooled by inflowing liquid nitrogen in the inner pipe and outflowing liquid nitrogen in the outer pipe. Pressured to p0 to maintain a subcooled state, the inflowing liquid nitrogen at temperature Ti cools the cable with a convective heat transfer coefficient hi. Considering the pressure drop along the cable, the outflowing liquid nitrogen at temperature To and pressure pN cools the cable with a convective heat transfer coefficient ho.
Fig. 2. Simplified schematic diagram of an N-layer CORC cable. |
2.2. Heat balance equations
The general form of heat conduction in a cylindrical coordinate system is:
$k_{i}\left(\frac{\partial^{2} T_{i}(r)}{\partial r^{2}}+\frac{1}{r} \frac{\partial T_{i}(r)}{\partial r}+\frac{1}{r^{2}} \frac{\partial^{2} T_{i}(\theta)}{\partial \theta^{2}}+\frac{\partial^{2} T_{i}(z)}{\partial z^{2}}\right)+f_{i}=\rho_{i} c_{i} \frac{\partial T_{i}(t)}{\partial t}$
where ki, fi, ρi, and ci are thermal conductivity, heat source, density, and specific heat capacity of each layer, respectively. Moreover, in respect, r, θ, and z are the radial, hoop, and axial coordinates. Also, t is the time operator.
Since superconducting tapes have almost no losses when carrying current, the heat source term in Eq. (1) can be ignored. Under the steady state condition, the heat balance equation of the ith layer can be simplified as follows:
$\frac{k_{i}}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T_{i}(r)}{\partial r}\right)=0$
According to Fourier’s law, heat flux at any radius is:
$q(r)=-k_{i} \frac{\partial T_{i}(r)}{\partial r}$
Without heat accumulation or dissipation, the heat transfer of each cylindrical surface is equal. Therefore, the radial temperature distribution of the cable core can be obtained by integrating Eq. (3) in r.
$T_{i}(r)=-\frac{r q(r)}{k_{i}} \ln \frac{r}{r_{i-1}}+T_{i-1}$
2.3. Boundary conditions
For working conditions shown in Fig. 2, the inner and outer surfaces of the cable core satisfy mixed boundary conditions as follows:
$\left\{\begin{array}{l} q\left(r_{0}\right)=-k_{1} \frac{\partial T_{1}(r)}{\partial r}=h_{\mathrm{i}}\left(T_{\mathrm{i}}-T_{0}\right) \\ q\left(r_{\mathrm{N}}\right)=-k_{\mathrm{N}} \frac{\partial T_{\mathrm{N}}(r)}{\partial r}=h_{\mathrm{N}}\left(T_{\mathrm{N}}-T_{\mathrm{o}}\right) \end{array}\right.$
Similarly, from the equal heat transfer of each cylindrical surface, Eq. (5) yields the following relation:
$\left\{\begin{array}{l} T_{0}-T_{\mathrm{i}}=-\frac{q\left(r_{0}\right)}{h_{\mathrm{i}}} \\ T_{\mathrm{o}}-T_{\mathrm{N}}=-\frac{q\left(r_{\mathrm{N}}\right)}{h_{\mathrm{N}}} \end{array}\right.$
To calculate heat flux, the temperature difference between inner and outer surfaces can be obtained by adding Eq. (4) for the N layers:
$T_{\mathrm{N}}-T_{0}=-r \sum_{i=1}^{\mathrm{N}} \frac{q(r)}{k_{i}} \ln \frac{r_{i}}{r_{i-1}}+T_{i-1}$
Substituting Eq. (6) into Eq. (7), the heat flux at radius r is [31]:
$q(r)=\frac{T_{1}-T_{0}}{r\left(\frac{1}{r_{0}^{k_{1}}}+\sum_{i}^{\mathrm{N}} 1 \frac{1}{k_{k}} \ln \frac{r_{i}}{r_{i-1}}+\frac{1}{r_{N} \hat{N}_{0}}\right)}$
When heat flux is determined, the radial temperature distribution can be calculated by Eq. (4). Specifically, when the temperature of cable surfaces is constant, or the convective heat transfer coefficient is very large, inner and outer surfaces can be considered to satisfy the Dirichlet boundary conditions. At this time, Eq. (8) can be simplified as follows:
$q(r)=\frac{T_{1}-T_{0}}{r \sum_{i=1}^{N} \frac{1}{k_{i}} \ln \frac{r_{i}}{r_{i-1}}}$
3. Analytical methods for stresses/strains
Combined with the calculated temperature in Section 2, this section introduces three analytical methods for stress-strain calculation of CORC cables, and shows their derivation process in detail. According to the number of variables, they are recursive method, algebraic method and matrix method.
3.1. Recursive method without variables
The target variable for the cable shrinkage under thermo-mechanical loads is the axial strain εzz. However, the recursive method for stresses and strains requires no variables in the calculation procedure [32]. Therefore, the developed recursive method without variables shills axial strain to zero. According to the cylindrical coordinate system shown in Fig. 3, the stress-strain relationship of the ith layer with isotropic materials is given by:
$\left\{\begin{array}{l} \sigma_{r r}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \varepsilon_{r r}^{i}+v_{i} \varepsilon_{\theta \theta}^{i}-\left(1+v_{i}\right) \alpha_{i} \Delta T_{i}(r)\right] \\ \sigma_{\theta \theta}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \varepsilon_{\theta \theta}^{i}+v_{i} \varepsilon_{r r}^{i}-\left(1+v_{i}\right) \alpha_{i} \Delta T_{i}(r)\right] \end{array}\right.$
where Ei, vi, and αi are Young's modulus, Poisson's ratio, and thermal expansion coefficient of the ith layer, respectively. Moreover, △Ti indicates the temperature difference, equal to temperature Ti(r) minus reference temperature Tref.
Fig. 3. Stress components in a cylindrical coordinate system. |
Under axisymmetric loads, stresses and strains are independent of θ, and radial and axial displacements only depend on their corresponding coordinates. Thus, the simplified strain-displacement relationship is as follows:
$\left\{\begin{array}{l} \varepsilon_{r \gamma}^{i}=\frac{\partial u_{r}^{i}}{\partial r} \\ \varepsilon_{\theta \theta}^{i}=\frac{u_{\tau}^{i}}{r} \end{array}\right.$
Similarly, the stresses of the model are independent of z. Furthermore, the equilibrium equation for radial and hoop stresses reduces to:
$\frac{\partial \sigma_{r r}^{i}}{\partial r}+\frac{\sigma_{\pi r}^{i}-\sigma_{\theta \theta}^{i}}{r}=0$
Substituting Eq. (10) with Eq. (11) into Eq. (12), the equilibrium equation can be expressed by radial displacement:
$\frac{\partial}{\partial r}\left[\frac{1}{r} \frac{\partial\left(u_{r}^{i}(r) \cdot r\right)}{\partial r}\right]=\frac{1+v_{i}}{1-v_{i}} \alpha_{i} \frac{\partial \Delta T_{i}(r)}{\partial r}$
Integrating Eq. (13) in r, the radial displacement of cable is as follows:
$u_{r}^{i}(r)=\frac{1+v_{i}}{E_{i}} \times\left[\left(1-2 v_{i}\right) A_{i} r-\frac{B_{i}-\frac{B_{i} a_{i}}{1-v_{i}} \int_{r_{i-1}}^{r} \Delta T_{i}(r) r d r}{r}\right]$
where Ai and Bi are constants to be determined.
Since all layers are assumed to be perfectly bound, radial displacements and stresses between interfaces are continuous. Substituting Eq. (14) into Eq. (10), radial continuity relations can be expressed by constants Ai and Bi.
$\left\{\begin{array}{l} \frac{1+v_{i}}{E_{i}} \times\left[\left(1-2 v_{i}\right) A_{i} r_{i}-\frac{B_{i}-\frac{E_{i} a_{i}}{1-v_{i}} \int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}}\right]=\frac{1+v_{i}+1}{E_{i}+1} \\ \times\left[\left(1-2 v_{i}+1\right) A_{i+1} r_{i}-\frac{B_{i}+1}{r_{i}}\right] \\ A_{i}+\frac{B_{i}-\frac{B_{i} a_{i}}{1-\tau_{i}} \int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}^{2}}=A_{i+1}+\frac{B_{i+1}}{r_{i}^{2}} \end{array}\right.$
To determine constants in Eq. (15), radial stresses of interfaces are given by corresponding radial pressures:
$\left\{\begin{array}{l} A_{i}+\frac{B_{i}}{r_{i-1}^{2}}=-p_{i-1} \\ A_{i+1}+\frac{B_{i+1}-\frac{B_{i}+1 \alpha_{i}+1}{1-v_{i}+1} \int_{r_{i}}^{r_{i}}+1 \Delta T_{i}+1(r) r d r}{r_{i+1}^{2}}=-p_{i+1} \end{array}\right.$
Substituting Eq. (16) into Eq. (15), constants Ai and Bi can be expressed by radial pressures:
$\left\{\begin{array}{l} A_{i}=\frac{1}{C_{i}^{1}-C_{i}^{2}} \\ \times\left[-p_{i-1} C_{i}^{1}+p_{i}+1 C_{i}^{3}-\frac{\int_{t_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{\left(1-v_{i}\right) r_{i-1}^{2}} E_{i} \alpha_{i} C_{i}^{1}-\frac{\int_{T_{i}}^{r_{i}}+1{ }_{1} \Delta T_{i}+1(r) r d r}{\left(1-v_{i}+1\right) r_{i}^{2}+1} E_{i}+{ }_{1} \alpha_{i}+{ }_{1} C_{i}^{3}\right] \\ B_{i}=\frac{r_{i-1}^{2}}{C_{i}^{1}-C_{i}^{2}} \times\left[p_{i-1} C_{i}^{2}-p_{i}+{ }_{1} C_{i}^{3}+\frac{\int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{\left(1-v_{i}\right) r_{i-1}^{2}} E_{i} \alpha_{i} C_{i}^{1}-\frac{\int_{r_{i}}^{r_{i}}+1{ }_{1} \Delta T_{i}+1(r) r d r}{\left(1-v_{i}+1\right) r_{i}^{2}+1} E_{i}+1 \alpha_{i}+{ }_{1} C_{i}^{3}\right] \end{array}\right.$
where
$\left\{\begin{array}{l} C_{i}^{1}=\frac{\frac{1+v_{i}}{E_{i}} r_{i-1}^{2}\left(r_{i+1}^{2}-r_{i}^{2}\right)-\frac{1+v_{i+1}}{E_{i+1}} r_{i-1}^{2} r_{i+1}^{2}-\frac{\left(1+v_{i+1}\right)\left(1-v_{i+1}\right)}{E_{i+1}} r_{i}^{2} r_{i+1}^{2}}{r_{i}^{2} r_{i+1}^{2}} \\ C_{i}^{2}=-\frac{\frac{1+v_{i+1}}{E_{i+1}} r_{i+1}^{2}+\frac{\left(1+v_{i}+1\right)\left(1-v_{i}+1\right)}{E_{i}+1} r_{i}^{3}+\frac{\left(1+v_{i}\right)\left(1-v_{v_{i}}\right)}{E_{i}}\left(r_{i+1}^{2}-r_{i}^{2}\right)}{r_{i+1}^{2}} \\ C_{i}^{3}=-\frac{2\left(1+v_{i+1}\right)\left(1-v_{i+1}\right)}{E_{i+1}} \end{array}\right.$
According to Eq. (16), radial pressure on the outer surface of the ith layer is:
$A_{i}+\frac{B_{i}-\frac{E_{i} \omega_{i}}{1-\theta_{i}} \int_{r_{i}-1}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}^{2}}=-p_{i}$
Substituting Eq. (19) into Eq. (17), the radial pressure at the interface of each layer is:
$\begin{array}{l} p_{i+1} \\ =\frac{1}{r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ \times\left[\begin{array}{c} p_{i-1} r_{i-1}^{2}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-p_{i} r_{i-1}^{2} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right) \\ +\frac{\int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{1-v_{i}} E_{i} \alpha_{i}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)+\frac{\int_{r_{i}}^{r_{i+1}} \Delta T_{i+1}(r) r d r}{1-v_{i+1}} \times \frac{E_{i+1} \alpha_{i+1} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}{r_{i+1}^{2}} \end{array}\right] \\ \end{array}$
Since pressures on the inner and outer surfaces, p0 and pN, can be measured by the refrigeration system of superconducting cables, the radial pressure of each interface can be determined by the recursive relation described in Eq. (20).
$p_{i}=D_{i}^{2} p_{0}+\frac{D_{i}^{1}}{D_{N}^{1}}\left(p_{N}-D_{N}^{2} p_{0}\right)$
where
$\left\{\begin{array}{l} D_{i+1}^{1}=\frac{D_{i-1}^{1}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-D_{i}^{1} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right)}{\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ D_{i+1}^{2}=\frac{D_{i-1}^{2}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-D_{i}^{2} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right)}{\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}+\frac{1}{p_{0} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ \times\left[\frac{\int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{1-v_{i}} E_{i} \alpha_{i}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)+\frac{\int_{i+1}^{1} \Delta T_{i+1}^{1}(r) r d r}{1-v_{i+1}} \times \frac{E_{i+1} \alpha_{i+1} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}{r_{i+1}^{2}}\right] \\ D_{0}^{1}=0, D_{1}^{1}=1, D_{0}^{2}=1, D_{1}^{2}=0 \end{array}\right.$
As the radial pressure of each interface is known, constants Ai, Bi, and radial displacement distribution can be determined. Furthermore, radial and hoop strains can be calculated by Eq. (11), while axial stress under the assumption of no strain is expressed as:
$\sigma_{z Z}^{i}=v_{i}\left(\sigma_{r r}^{i}+\sigma_{\theta \theta}^{i}\right)-E_{i} \alpha_{i} \Delta T_{i}(r)$
At this time, cable shrinkage is considered. According to the force balance condition, the resultant axial force of the cable end is equal to the external load. Therefore, the actual axial strain is:
$\varepsilon_{z z}=\frac{\sum_{i=1}^{N} \int_{r_{i-1}}^{\tau_{i}} 2 \pi r \sigma_{x z}^{i} d s-F_{x}}{\sum_{i=1}^{\mathrm{N}} \int_{r_{i-1}}^{1_{i}} 2 \pi r E_{i} d r}$
where Fz is the axial external force.
Since axial shrinkage is a uniaxial mechanical behavior, radial and hoop strains remain almost unchanged. Based on the stress-strain relationship, cable stresses under thermo-mechanical loads are:
$\left\{\begin{array}{l} \sigma_{r r}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \varepsilon_{r r}^{i}+v_{i} \varepsilon_{\theta \theta}^{i}+v_{i} \varepsilon_{z z}\right]-\frac{E_{i} \alpha_{i} \Delta T_{i}(r)}{1-2 v_{i}} \\ \sigma_{\theta \theta}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \varepsilon_{\theta \theta}^{i}+v_{i} \varepsilon_{r r}^{i}+v_{i} \varepsilon_{z z}\right]-\frac{E_{i} \alpha_{i} \Delta T_{i}(r)}{1-2 v_{i}} \\ \sigma_{z z}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \varepsilon_{z z}^{i}+v_{i} \varepsilon_{r r}^{i}+v_{i} \varepsilon_{\theta \theta}\right]-\frac{E_{i} \alpha_{i} \Delta T_{i}(r)}{1-2 v_{i}} \end{array}\right.$
3.2. Algebraic method with a variable
According to the analysis in Section 3.1, the axial strain εzz is the target variable for this mathematical problem. On the other hand, the initial assumption and uniaxial tension may bring errors to the calculation results. Thus, an algebraic method is developed based on the recursive method by introducing the axial strain as a variable. Considering the effect of axial strain, radial and hoop stresses of the ith layer are given by:
$\left\{\begin{array}{l} \sigma_{r r}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \frac{\partial u_{F}^{i}}{\partial r}+v_{i}\left(\frac{u_{r}^{i}}{r}+\varepsilon_{z z}\right)-\left(1+v_{i}\right) \alpha_{i} \Delta T_{i}(r)\right] \\ \sigma_{\theta \theta}^{i}=\frac{E_{i}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}\left[\left(1-v_{i}\right) \frac{u_{r}^{i}}{r}+v_{i}\left(\frac{\partial u_{r}^{i}}{\partial r}+\varepsilon_{z z}\right)-\left(1+v_{i}\right) \alpha_{i} \Delta T_{i}(r)\right] \end{array}\right.$
Substituting Eq. (25) into the equilibrium Eq. (12), the radial displacement of the cable core can still be expressed by Eq. (14). Based on the assumption of perfectly bound, radial continuity relations of radial displacements and stresses are presented in the following equation, in which case, axial strain must be considered.
$\left\{\begin{array}{l} \frac{1+v_{i}}{E_{i}} \times\left[\left(1-2 v_{i}\right) F_{i} r_{i}-\frac{G_{i}-\frac{E_{i} a_{i}}{1-v_{i}} \int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}}\right]=\frac{1+v_{i}+1}{E_{i}+1} \\ \times\left[\left(1-2 v_{i}+1\right) F_{i}+1 r_{i}-\frac{G_{i}+1}{r_{i}}\right] \\ F_{i}+\frac{G_{i}-\frac{E_{i} a_{i}}{1-v_{i}} \int_{r_{i-1}}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}^{2}}+f_{i}\left(\varepsilon_{z z}\right)=F_{i+1}+\frac{G_{i+1}}{r_{i}^{2}}+f_{i+1}\left(\varepsilon_{z z}\right) \end{array}\right.$
where Fi and Gi are constants as Ai and Bi. For the convenience of comparison with the recursive method, the term with axial strain is defined as follows:
$f_{i}\left(\varepsilon_{z z}\right)=\frac{E_{i} v_{i} \varepsilon_{z z}}{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}$
To determine constants in Eq. (27), radial stresses of interfaces are given by corresponding radial pressures:
$\left\{\begin{array}{l} F_{i}+\frac{G_{i}}{r_{i-1}^{2}}+f_{i}\left(\varepsilon_{z z}\right)=-p_{i-1} \\ F_{i+1}+\frac{G_{i+1}-\frac{\varepsilon_{i}+1 \alpha_{i}+1}{1-v_{i}+1} \int_{r_{i}}^{r_{i}+1} \Delta T_{i}+1(r) r d r}{r_{i+1}^{2}}+f_{i+1}\left(\varepsilon_{z z}\right)=-p_{i+1} \end{array}\right.$
where radial pressure on the outer surface of the ith layer is:
$F_{i}+\frac{G_{i}-\frac{E_{i} \alpha_{i}}{1-v_{i}} \int_{r_{i}-1}^{r_{i}} \Delta T_{i}(r) r d r}{r_{i}^{2}}+f_{i}\left(\varepsilon_{z z}\right)=-p_{i}$
Substituting Eq. (30) into Eq. (27) with Eq. (29), the radial pressure at the interface of each layer is:
$\begin{array}{l} p_{i+1} \\ =\frac{1}{r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ \times\left\{\begin{array}{c} p_{i-1} r_{i-1}^{2}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-p_{i} r_{i-1}^{2} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right) \\ +\frac{\int_{r_{i-1}-1}^{r_{i}} \Delta T_{i}(r) r d r}{1-v_{i}} E_{i} \alpha_{i}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)+\frac{\int_{\tau_{i}}^{t_{i+1}} \Delta T_{i+1}(r) r d r}{1-v_{i+1}} \\ \times \frac{E_{i+1} \alpha_{i+1} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}{r_{i}^{2}+1} \\ +\frac{r_{i-1}^{2} r_{i+1}^{2}\left(r_{i+1}^{2}-r_{i}^{2}\right)\left(r_{i}^{2}-r_{i-1}^{2}\right)}{r_{i+1}^{2}} \times\left[\frac{\left(1+v_{i+1}\right)\left(1-2 v_{i+1}\right)}{E_{i+1}} f_{i+1}\left(\varepsilon_{z z}\right)-\frac{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}{E_{i}} f_{i}\left(\varepsilon_{z z}\right)\right] \end{array}\right\} \\ \end{array}$
Similarly, the radial pressure of each interface can be determined by the recursive relation:
$p_{i}=H_{i}^{2} p_{0}+\frac{H_{i}^{1}}{H_{\mathrm{N}}^{1}}\left(p_{\mathrm{N}}-H_{\mathrm{N}}^{2} p_{0}\right)$
Where
$\left\{\begin{array}{l} H_{i+1}^{1}=\frac{H_{i-1}^{1}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-H_{i}^{1} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right)}{\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ H_{i+1}^{2}=\frac{H_{i-1}^{2}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)-H_{i}^{2} r_{i}^{2}\left(C_{i}^{1}-C_{i}^{2}\right)}{\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}+\frac{1}{p_{0} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}} \\ \times\left\{\begin{array}{l} \frac{\int_{i-1}^{r_{i}} \Delta T_{i}(r) r d r}{1-v_{i}} E_{i} \alpha_{i}\left(r_{i}^{2} C_{i}^{1}-r_{i-1}^{2} C_{i}^{2}\right)+\frac{\int_{r_{i}}^{r_{i+1}} \Delta T_{i+1}(r) r d r}{1-v_{i+1}} \times \frac{E_{i+1} \alpha_{i+1} r_{i-1}^{2}\left(r_{i}^{2}-r_{i-1}^{2}\right) C_{i}^{3}}{r_{i+1}^{2}} \\ +\frac{r_{i-1}^{2} r_{i+1}^{2}\left(r_{i+1}^{2}-r_{i}^{2}\right)\left(r_{i}^{2}-r_{i-1}^{2}\right)}{r_{i+1}^{2}} \times\left[\frac{\left(1+v_{i+1}\right)\left(1-2 v_{i+1}\right)}{E_{i+1}} f_{i+1}\left(\varepsilon_{z z}\right)-\frac{\left(1+v_{i}\right)\left(1-2 v_{i}\right)}{E_{i}} f_{i}\left(\varepsilon_{z z}\right)\right] \end{array}\right\} \\ H_{0}^{1}=0, H_{1}^{1}=1, H_{0}^{2}=1, H_{1}^{2}=0 \end{array}\right.$
As the radial pressure of each interface is known, constants Fi and Gi of radial displacement expressions can be determined by:
$\left\{\begin{array}{l} F_{i}=\frac{\left(p_{i-1}+f_{i}\left(\varepsilon_{k z}\right)\right) r_{i-1}^{2}-\left(p_{i}+f_{i}\left(\varepsilon_{z z}\right)\right) r_{i}^{2}+\frac{E_{i} a_{i}}{1-v_{i}} \int_{r_{i-1}}^{r} \Delta T_{i}(r) r d r}{r_{i}^{2}-r_{i-1}^{2}} \\ G_{i}=\frac{\left[\left(p_{i}-p_{i-1}\right) r_{i}^{2}-\frac{E_{i} \alpha_{i}}{1-i_{i}} \int_{i-1}^{r} \Delta T_{i}(r) r d r\right] r_{i-1}^{2}}{r_{i}^{2}-r_{i-1}^{2}} \end{array}\right.$
According to Eqs. (11) and (25), radial strain, hoop strain, radial stress, hoop stress, and axial stress can be calculated in turn, all of which include variable εzz. Similarly, the cable core satisfies the condition that the resultant axial force equals the external load. Using Eq. (24), the symbolic function of MATLAB and Python can calculate the axial strain and other physical quantities.
3.3. Matrix method with multi-variables
From Section 3.1 and Section 3.2, the key to solving stresses and strains under thermo-mechanical loads lies in the radial displacement of the cable. Substituting Eqs. (11) and (25) into the equilibrium Eq. (12), the radial displacement of the cable can be expressed in the following form:
$u_{r}^{i}(r)=H_{i} r+\frac{I_{i}}{r}-\frac{\alpha_{i} r^{2} q(r)(2 \ln r-1)\left(1+v_{i}\right)}{4 k_{i}\left(1-v_{i}\right)}$
where Hi and Ii are constants, which can be solved by a system of algebraic equations.
The interface of each layer satisfies continuity conditions of radial displacement and stress
$\left\{\begin{array}{l} u_{r}^{i}\left(r_{i}\right)=u_{r}^{i+1}\left(r_{i}\right) \\ \sigma_{r r}^{i}\left(r_{i}\right)=\sigma_{r r}^{i+1}\left(r_{i}\right) \end{array}\right.$
Besides, boundary load conditions satisfied by the inner and outer surfaces of the cable core are:
$\left\{\begin{array}{l} \sigma_{r r}^{i}\left(r_{0}\right)=-p_{0} \\ \sigma_{r r}^{i}\left(r_{\mathrm{N}}\right)=-p_{\mathrm{N}} \end{array}\right.$
Last but not least, the axial stress of the cable core should satisfy the following condition:
$\sum_{i=1}^{N} \int_{r_{i-1}}^{r_{i}} 2 \pi r \sigma_{z z}^{i} d r=F_{z}$
Therefore, for an N-layer CORC cable, combining the continuity conditions of Eq. (36), boundary conditions of Eq. (37), and integral formula of Eq. (38), 2 N + 1 parametric equations can be obtained to solve a total of 2 N + 1 variables for Hi and Ii in Eq. (35) and εzz in Eq. (25). In this case, the solve function of MATLAB or the NSolve function of Mathematica can solve the 2 N + 1-dimensional matrix equation.
4. Calculation procedures and validation
According to the derivation in Section 3, calculation procedures for estimating the thermo-mechanical behavior of multi-layer hollow CORC cable are proposed and verified in this section. To be noticed the steady-state radial temperature distribution is the common basis of three analytical methods.
4.1. Introduction of calculation procedures
4.1.1. Radial temperature
1) Based on geometric parameters ri, input thermal parameters ki for each material layer.
2) For Dirichlet boundary conditions, calculate heat flux at radius r by Eq. (9) with surface temperatures Ti and To.
3) For mixed boundary conditions, calculate heat flux at radius r by Eq. (8) with convective heat transfer coefficients hi and ho.
4) Lastly, calculating heat flux q(r), steady-state radial temperature distribution can be obtained by Eq. (4).
4.1.2. Recursive method
1) Determine the sequences of $C_{i}^{1}$, $C_{i}^{2}$, and $C_{i}^{3}$ in Eq. (18).
2) Based on $D_{0}^{1}$, $D_{1}^{1}$, $D_{0}^{2}$, and $D_{1}^{1}$, determine the sequences of Di1, and Di2 in Eq. (22).
3) Calculate the radial pressure of each interface using Eq. (21) with surface pressures p0 and pN.
4) Determine the sequences of Ai and Bi in Eq. (17).
5) With the calculated constants Ai and Bi, radial displacement at radius r of each layer can be obtained by Eq. (14).
6) Furthermore, radial and hoop strains can be calculated by the strain-displacement relationship of Eq. (11).
7) Assuming cable ends are fixed, axial stress can be calculated by Eq. (23).
8) Considering cable shrinkage, calculate actual axial strain using Eq. (24) with axial external force Fz.
9) Lastly, calculating axial strain, radial, hoop, and axial stresses can be determined by the stress-strain relationship of Eq. (25).
4.1.3. Algebraic method
1) Like the recursive method, determine the sequences of $C_{i}^{1}$, $C_{i}^{2}$, and $C_{i}^{3}$ in Eq. (18).
2) Based on $H_{0}^{1}$, $H_{1}^{1}$, $H_{0}^{2}$, and $H_{1}^{2}$, determine the sequences of $H_{I}^{1}$ and $H_{I}^{2}$ in Eq. (33).
3) Calculate the radial pressure of each interface using Eq. (32) with surface pressures p0 and pN.
4) Calculate sequences of Gi and Fi in Eq. (27).
5) With the undetermined constants Gi and Fi, each layer’s radial displacement at radius r can be expressed by Eq. (14).
6) Moreover, radial and hoop strains can be expressed as axial strain εzz by Eq. (11).
7) With the axial strain εzz as a variable, radial, hoop, and axial stresses can be expressed by Eq. (25).
8) Utilizing Eq. (24), axial strain can be determined, as well as radial displacement, radial and hoop strains, and radial, hoop, and axial stresses.
4.1.4. Matrix method
1) Express radial displacement at radius r of each layer in the form of variables Hi and Ii using Eq. (35).
2) Express radial and hoop strains of each interface in the form of variables Hi and Ii employing Eq. (11).
3) Express radial and axial stresses in the form of variables Hi, Ii, and εzz using Eq. (25).
4) Construct 2 N-2 continuity equations for an N-layer CORC cable using Eq. (36), including Hi and Ii, for 2 N variables.
5) Construct two boundary equations using Eq. (37), including H1, I1, HN, IN, and εzz for five variables.
6) Construct an integral equation using Eq. (38), including Hi, Ii, and εzz for 2 N + 1 variables.
7) Based on the system of 2 N + 1 equations, 2 N + 1 variables can be determined by the matrix solver.
4.2. Validation of analytical models
4.2.1. Case-1: Isotropic materials
To verify analytical models for estimating the thermo-mechanical behavior, a 6-layer composite model is built as described in [33]. Geometric and material parameters are presented in Table 1. For thermal loads, temperatures of the inner and outer surfaces are 200 ℃ and 150 ℃, respectively. Moreover, pressures of the inner and outer surfaces for mechanical loads are 22 MPa and 1.5 MPa, respectively.
Table 1. Geometric and materials parameters of the 6-layer model [33]. |
| Layer | ri-1/mm | ri/mm | Ei/GPa | vi/1 | αi/K−1 | ki/Wm-1K−1 |
|---|---|---|---|---|---|---|
| 1 | 500 | 510 | 190 | 0.28 | 18.7 × 10-6 | 18.4 |
| 2 | 510 | 520 | 194 | 0.28 | 17.4 × 10-6 | 25.4 |
| 3 | 520 | 530 | 198 | 0.28 | 16.1 × 10-6 | 32.4 |
| 4 | 530 | 540 | 202 | 0.28 | 14.8 × 10-6 | 39.4 |
| 5 | 540 | 550 | 206 | 0.28 | 13.5 × 10-6 | 46.4 |
| 6 | 550 | 580 | 210 | 0.3 | 12.2 × 10-6 | 53.4 |
For comparison, a two-dimensional (2D) axisymmetric numerical model is built based on the FE analysis software COMSOL Multiphysics. Since the length of the model does not affect the axisymmetric mechanical behavior, the axial height of the model can be considered quite small for better computational efficiency. To ensure computational convergence, the model should be treated as a half model in the axial direction, which can constrain the degrees of freedom in the corresponding direction. At this time, the internal symmetrical cross-section is subjected to a symmetric boundary condition, and the external free cross-section is subjected to a normal-free symmetric boundary condition. According to above mentioned the thermo-mechanical loads, the temperature and boundary load on the inner surface are 200 ℃ and 22 MPa, respectively. These values on the outer surface are 150 ℃ and 1.5 MPa, respectively. Most importantly, through the thermal expansion properties of the solid mechanical module, the thermal and mechanical behavior is fully coupled.
Based on the analytical and 2D axisymmetric numerical models, the radial temperature distribution of the 6-layer model is shown in Fig. 4. According to the thermal-hydraulic properties of HTS power cables [34], [35], [36], the radial temperature distribution is usually regarded as continuous, ignoring the possible temperature that jumps between each interface. In Fig. 4 (b), the radial temperature distribution of the analytical solution is smooth and agrees well with the numerical model, and results in [32].
Fig. 4. Radial temperature distribution of the 6-layer model. (a) Temperature contour. (b) Temperature curves. |
For comparison, presentation, and consistency, the normalized radius is used for the horizontal coordinates in Fig. 4 and subsequent figures. The transformation between the true radius and the normalized radius is as follows:
$R(r)=\frac{r-r_{i}}{r_{0}-r_{i}}$
where ri and ro are the inner and outer radii of the composite structure, respectively.
Moreover, displacement and strains are shown in Fig. 5. For thermo-mechanical loads, three analytical methods and the numerical model are in good agreement. Among them, the error of the recursive method comes from the assumption that the initial axial strain is zero. Even though the assumption is compensated by utilizing Eq. (24), there is still an ineliminable error between the calculated and the actual results. According to the uniaxial behavior described by Eq. (24), the effect of axial stress on radial and hoop strains is neglected in the calculation. However, according to the stress-strain relationship of Eq. (25), radial and hoop strains vary during uniaxial tension to maintain constant radial and hoop stresses. In this condition, the maximum relative error of radial displacement is less than 0.55%, and the maximum relative error of hoop strain is less than 0.6%.
Fig. 5. Displacement and strains of the 6-layer model. (a) Radial displacement. (b) Radial strain. (c) Hoop strain. |
According to the above analysis, the calculation procedure of the recursive method has a minor effect on strains. Therefore, the results of the radial, hoop, and axial stresses are consistent for the three analytical methods and the numerical method. Compared to the published solution in [37], all results have good consistency, as shown in Fig. 6.
Fig. 6. Stresses of the 6-layer model. (a) Radial stress. (b) Hoop stress. (c) Axial stress. |
4.2.2. Case-2: Orthotropic materials
For CORC cables, superconducting tapes are spirally wound and composed of multiple constituent layers [20]. As shown in Fig. 7, the conductor layer of CORC cable can be regarded as an orthotropic material under the cylindrical coordinate system. Combined with the stiffness transformation matrix in Appendix A, the matrix equation can calculate the strains and stresses of the composite structure with orthotropic materials.
Fig. 7. Schematic diagram of superconducting layer and superconducting tape. (Other functional layers of CORC cable are not presented.). |
To verify the analytical model of matrix equation for estimating the mechanical behavior of orthotropic materials, a 4-layer composite model is built based on [38], in which the stacking sequence and ply angles are + 55°/-55°/+55°/-55°. The inner radius at 50 mm is subjected to internal pressure with 10 MPa. Each ply has a thickness of 0.5 mm and the same material. Unidirectional material properties are E1 = 141.6 GPa, E2 = 10.7 GPa, G12 = 3.88 GPa, v12 = 0.268, and v23 = 0.495.
A 3D numerical model is built based on COMSOL Multiphysics for comparison. Similarly, the axial height of the model is relatively small to improve the computational efficiency. Moreover, the model is treated as a half model to ensure computational convergence. At this time, the internal symmetrical cross-section is subjected to a roller boundary condition, which is more efficient than the symmetric boundary condition, and the external free cross-section is subjected to a normal-free symmetric boundary condition. According to the above-mentioned thermo-mechanical loads, temperature and boundary loads based on the surface are applied to the inner surfaces, respectively. Furthermore, the thermal and mechanical behaviors are fully coupled by thermal expansion. Most importantly, since the default global coordinate system for 3D models in COMSOL Multiphysics is the Cartesian coordinate system, a combined coordinate system is required to assign the orientation of orthotropic materials. Specifically, based on the cylindrical coordinate system, the material coordinate system along the 3D helix direction can be defined using the rotation of Euler angles.
According to the matrix method and 3D numerical model, the stress contour and calculation results of the orthotropic model are shown in Fig. 8. As shown in the stress contour in Fig. 8 (a), the red arrows indicate the 1-axis direction of the material. Similarly, the analytical and numerical methods are in good agreement, in which the stiffness transformation during post-processing of the 3D model is shown in Appendix B. Compared to published results in [38], [39], the relative error lies in the data gathering accuracy.
Fig. 8. Results of the orthotropic model. (a) Stress contour. (a) Displacement curves. (b) Strain curves. (c) Stress curves. |
5. Stresses and strains of CORC cable
According to Section 4, all analytical methods provide an exact solution under thermo-mechanical loads and a basis for analyzing spirally wound CORC cables. Therefore, this section investigates stresses and strains under thermo-mechanical loads for a hundred-meter scale CORC cable to evaluate its axial shrinkage at low temperatures.
5.1. Parameters and models
Made of corrugated pipe, the inner and outer radii of the cable skeleton are about 20 mm and 24 mm, respectively. A semi-conducting layer is wounded outside the skeleton to distribute the electric field. Then, an insulating layer is wounded to prevent breakdown. Next, phases A, B, and C of the tri-axial cable are wounded outside the insulating layer in turn. Considering the current-carrying capacity of superconducting tapes, each cable phase includes two superconducting layers. Hence, two semi-conducting layers and an interlayer insulating layer are required. To prevent interphase breakdown, an insulating layer of 2.5 mm thick shall be wounded between phases. Moreover, even though the tri-axial CORC cable has environmentally friendly electromagnetic characteristics, a normal-conducting shielding layer is still needed. Made of copper tapes, the shielding layer is wrapped by two semi-conducting layers and an insulating layer. Finally, the cable core is tightly covered by a protective layer. Material parameters of each layer are listed in Table 2, in which the material parameters of superconducting tapes are calculated by the volume-weighted-average method [40].
Table 2. Material parameters of the tri-axial CORC cable. |
| Structure | Young’s modulus E/GPa | Poisson’s ratio v/1 | Density ρ/kg·m−3 | Thermal expansion α/K−1 | Thermal conductivity k/W·m−1·K−1 |
|---|---|---|---|---|---|
| Skeleton | 193 | 0.3 | 7850 | 1.23 × 10-5 | 44.5 |
| Semi-conducting layer | 100 | 0.3 | 420 | −0.8 × 10-6 | 1.7 |
| Insulating layer | 10 | 0.3 | 1100 | 3.2 × 10-5 | 0.5 |
| Interlayer insulating layer | 3.5 | 0.34 | 1420 | 2.5 × 10-5 | 0.125 |
| Superconducting layer | 86.8 | 0.335 | 8930 | 1.71 × 10-5 | 443.5 |
| Copper layer | 110 | 0.35 | 8960 | 1.7 × 10-5 | 400 |
| Protective layer | 34 | 0.3 | 3000 | 4.8 × 10-6 | 0.25 |
Even though the 2D and 3D numerical models in Section 4 have validated the analytical models, they both neglected the air gap between superconducting tapes. Therefore, a more detailed 3D numerical model is built in ABAQUS for a more precise comparison. Similar to the 2D and 3D numerical models, a short half model is used to improve efficiency and ensure convergence. Assembled as shown in Fig. 9, the tri-axial CORC cable has a total of 26 functional layers. Different from the models in COMSOL, 25 pairs of tie constraints without friction are required to create based on the perfectly bound assumption. The internal symmetrical cross-section is subjected to a symmetric boundary condition, while the external free cross-section is subjected to an equation constraint, ensuring that the axial displacement of each functional layer is consistent. As mechanical boundary loads are applied to the inner and outer surfaces of the cable core, thermal loads are imposed through pre-defined temperature fields to build the thermo-mechanical coupling relations.
Fig. 9. 3D numerical model of 26-layer tri-axial CORC cable in ABAQUS. |
5.2. Analytical results under thermo-mechanical loads
Generally, the rated operating temperature range of HTS power cables is between 70 and 76 K. Since the operating temperature is much lower than the initial room temperature, the axial temperature gradient can be ignored. Therefore, the tri-axial CORC cable can be regarded as a 2D axisymmetric structure, and the thermo-mechanical behavior at low temperatures can be analyzed by analytical methods. According to measurement results, the inner surface and outer surface of the cable core are about 70 K and 76 K, respectively. Thus, the radial temperature distribution under this condition is shown in Fig. 10. As the thermal conductivity of the insulating layer is small, the radial temperature gradient of the cable core is mainly concentrated in the main insulating layer and interphase insulating layers.
Fig. 10. Radial temperature distribution of the CORC cable. |
As the boiling point of liquid nitrogen at standard atmospheric pressure is about 77.3 K, the inflowing liquid nitrogen is pressurized to 6 atm to maintain the subcooled state. Considering the pressure drop along the cable, the pressure of outflowing liquid nitrogen is about 2 atm. For the cable core, the radial pressure on the inner surface is 0.6 MPa, and this value on the outer surface is 0.2 MPa. Fig. 11 presents the displacement, strains, and stresses of the CORC cable.
Fig. 11. Results of the CORC cable. (a) Radial displacement and radial stress. (b) Radial strain and hoop strain. (c) Axial stress and hoop stress. |
As shown in Fig. 11 (a), the radial displacement of the cable core is continuous, the inner surface shrinks 0.0459 mm inwards, and the outer surface shrinks 0.158 mm inwards. Satisfying continuity conditions described in Eqs. (15), (29), and (37), the radial stress of the cable core is also shown in Fig. 11 (a). Radial stress on the inner and the outer surfaces are −0.6 MPa and −0.2 MPa, respectively. In addition, the maximum radial stress is located at the outer surface of the skeleton.
Similar to radial temperature distribution, the radial displacement gradient is also concentrated in insulating layers. According to Eq. (11), the radial displacement gradient is radial strain, as shown in Fig. 11 (b). The radial strain of insulating layers is up to −0.939 ∼ -1.059%, the radial strain of superconducting layers is between −0.469 and −0.409%, and the radial strain of semi-conducting layers with a negative coefficient of thermal expansion is in the range of 0.282 to 0.323%. Equal to the ratio of radial displacement to the radius, it can be seen from the right axis of Fig. 11 (b) that hoop strain on the inner surface is −0.233%, and that on the outer surface is −0.411%. In addition, according to the calculated radial stress and radial strain of superconducting layers, superconducting tapes will not delaminate under the above-mentioned thermo-mechanical loads [41].
Fig. 11 (c) illustrates the axial stress of the cable core under axial shrinkage of −0.278%. The superconducting layer has a significant axial tensile stress of 130.8 MPa, within the tensile tolerance of superconducting tapes [42]. The semi-conducting layer with a negative coefficient of thermal expansion has a considerable axial compressive stress of up to −465.86 MPa. Finally, the hoop stress of the cable core is shown in the right axis of Fig. 11 (c). Like axial stress distribution, the maximum hoop tensile stress is about 124.5 MPa in the superconducting layer of phase A, and the maximum hoop compressive stress is about −569.4 MPa in the semi-conducting layer of the shielding layer. In addition, compared with the tangential friction stress in tension or bending [15], [16], [27], the thermal stress dominates the axial shrinkage of the cable at low temperatures. Therefore, it is reasonable to simplify the model considering each cable layer as perfectly bound, ignoring the contact behavior of layers.
The 3D numerical model is used in ABAQUS to verify calculation results. Due to air gaps between superconducting tapes, stress distribution in the circumferential direction is not uniform. Taking the average stress of each layer for comparison, calculation results of the 3D model considering air gaps and winding directions verify analytical methods, and relative error almost only comes from the elastic-plastic constitutive model used in the 3D model.
However, the above analysis only considers stresses and strains of the cable core in liquid nitrogen. To further evaluate the axial shrinkage of the cable at low temperatures, the cryostat shown in Fig. 1 also must be taken into account. As a vacuum insulation device, the inner pipe is in contact with outflow liquid nitrogen, while the outer pipe is always at room temperature. The cryostat is connected to both cable terminals and has the same axial strain as the cable core. Since there is no force between the cryostat and cable core in the cross-sectional direction, the action of the cryostat on the cable core can be considered as an axial external load Fz. According to the stress-strain relationship, the axial external force is:
$F_{z}=-E \varepsilon_{z z} S_{\mathrm{o}_{\mathrm{p}} \mathrm{ipe}}+E\left(\alpha \Delta T-\varepsilon_{z z}\right) S_{\mathrm{i}_{\mathrm{p}} \mathrm{ipe}}$
where So_pipe and Si_pipe denote cross-sectional areas of the outer and inner pipes of the cryostat, respectively.
According to the specifications of the cryostat, the inner and outer diameters of the inner pipe are 100 mm and 115 mm, respectively. Moreover, the mentioned values for the outer pipe are 150 mm and 170 mm, respectively. As the cryostat is made of corrugated pipe, the wall thicknesses of the inner and outer pipes are only 1.5 mm. Thus, the relationship between axial strain and the equivalent inner diameter of two pipes is shown in Fig. 12. Separated by a vacuum layer, the inner pipe of the cryostat shrinks at a low temperature, while the outer pipe of the cryostat resists deformation at room temperature. Therefore, as seen in Fig. 12, when the wall thickness is constant, the smaller the equivalent diameter of the inner pipe, and the larger the equivalent diameter of the outer pipe the better the cryogenic axial shrinkage resistance of the cable. Considering the influence of cryostat, axial strain is reduced from the above-calculated −0.278% to −0.1568 ∼ -0.145%.
Fig. 12. Axial strain of the 360 m CORC cable. |
Based on the design parameters of the tri-axial CORC cable, the manufactured HTS power cable is about 360 m. Cryogenic trials were conducted in March 2022, and the test results showed an average axial displacement of 26.6 cm for one terminal after two cool-down and warm-up cycles. On the other hand, the axial shrinkage of the cable is predicted to be 52.2 ∼ 56.45 cm according to Fig. 12. Therefore, the calculation and relative errors are within 1.63 cm and 6.1%, respectively.
5.3. Efficiency comparison of three analytical approaches
Eventually, the computational efficiencies of the three analytical models are compared in Fig. 13. With increasing cable layers, the calculation time of analytical methods rises gradually. As shown in Fig. 13 (a), the fewer the variables are, the faster the axial strain is calculated. When the relative error of axial strain is less than 0.235%, the calculation time of the algebraic method is about 1.091 times greater than the recursive method, and the calculation time of the matrix method is about 1.214 times longer than the recursive method.
Fig. 13. Computation time of analytical methods. (a) Axial strain. (b) Strains and stresses. |
However, when values of each layer and each physical quantity are obtained from axial strain, the repeated call of symbolic functions will significantly increase calculation time. The calculation times for three analytical methods are shown in Fig. 13 (b) when seven functional values, including temperature, radial displacement, radial strain, hoop strain, radial stress, hoop stress, and axial stress, are calculated, and each layer is sampled at 0.05 mm radius intervals. Without repeated calls of symbolic functions, the calculation time of the matrix method is increased by 2.8 s less than axial strain. However, the calculation time of the algebraic method is 5.9 ∼ 8.6 times greater than the matrix method. For the recursive method, which is the fastest for axial strain calculation, calculation time can be up to 25.7 times longer than the matrix method. Unlike the numerical method requiring initialization, the analytical methods are independent of the initial value. In addition, further calculations show that the temperature difference related to the initial temperature has negligible impact on the computational efficiency.
In conclusion, if only the axial shrinkage of the cable must be evaluated, the recursive method without variables is the most efficient one. If the displacement, strains, and stresses of the cable core need to be solved, the matrix method with multi-variables has a significant efficiency advantage. Moreover, the algebraic method with a variable has a more balanced computational performance.
6. Conclusion
This paper developed analytical methods for stress and strain analysis of CORC cables under thermo-mechanical loads. The steady-state radial temperature distribution model applied to Dirichlet boundary and mixed boundary conditions, contributing to the thermal part of thermo-mechanical loads. Furthermore, three analytical solutions were provided for the mechanical properties of hollow CORC cables. Assuming the initial axial strain was zero, the recursive method without variables was the fastest to calculate axial strain. With axial strain as a variable, the algebraic method was employed to eliminate the error of the recursive method. Including 2 N + 1 functions for an N-layer CORC cable, the matrix method with multi-variables applied to isotropic materials and orthotropic materials. Based on the design parameters of the tri-axial CORC cable, the analytical analysis indicated that superconducting tapes could withstand stress caused by low temperatures. Compared to the test results of a 360 m cable, the calculation error of axial shrinkage was within 1.63 cm, and the relative error was within 6.1%. In this case, the calculation time of axial strain was about 12 s, and the matrix method was 5.9 ∼ 25.7 times faster than the other two methods in calculating stresses and strains.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work was supported by the National Key R&D Program of China under Grant 2018YFA0704300.
Appendix A.
For laminate layers, the stress-strain relationship of the ith layer with orthotropic material is [31]:
$\left[\begin{array}{c} \sigma_{z z}^{i} \\ \sigma_{\theta \theta}^{i} \\ \sigma_{r r}^{i} \\ \tau_{\theta r}^{i} \\ \tau_{z r}^{i} \\ \tau_{z \theta}^{i} \end{array}\right]=\left[\begin{array}{cccccc} \bar{C}_{11}^{i} & \bar{C}_{12}^{i} & \bar{C}_{13}^{i} & 0 & 0 & \bar{C}_{16}^{i} \\ \bar{C}_{12}^{i} & \bar{C}_{22}^{i} & \bar{C}_{23}^{i} & 0 & 0 & \bar{C}_{26}^{i} \\ \bar{C}_{13}^{i} & \bar{C}_{23}^{i} & \bar{C}_{33}^{i} & 0 & 0 & \bar{C}_{36}^{i} \\ 0 & 0 & 0 & \bar{C}_{44}^{i} & \bar{C}_{45}^{i} & 0 \\ 0 & 0 & 0 & \bar{C}_{45}^{i} & \bar{C}_{55}^{i} & 0 \\ \bar{C}_{16}^{i} & \bar{C}_{26}^{i} & \bar{C}_{36}^{i} & 0 & 0 & \bar{C}_{66}^{i} \end{array}\right]\left[\begin{array}{c} \varepsilon_{z z}^{i} \\ \varepsilon_{\theta \theta}^{i} \\ \varepsilon_{r r}^{i} \\ \gamma_{\theta r}^{i} \\ \gamma_{z r}^{i} \\ \gamma_{z \theta}^{i} \end{array}\right]-\left[\begin{array}{c} \xi_{z z} \\ \xi_{\theta \theta} \\ \xi_{r r} \\ 0 \\ 0 \\ \xi_{z \theta} \end{array}\right] \Delta T_{i}$
Where
$\left[\begin{array}{c} \xi_{z z} \\ \xi_{\theta \theta} \\ \xi_{r r} \\ \xi_{z \theta} \end{array}\right]=\left[\begin{array}{cccc} \bar{C}_{11}^{i} & \bar{C}_{12}^{i} & \bar{C}_{13}^{i} & \bar{C}_{16}^{i} \\ \bar{C}_{12}^{i} & \bar{C}_{22}^{i} & \bar{C}_{23}^{i} & \bar{C}_{26}^{i} \\ \bar{C}_{13}^{i} & \bar{C}_{23}^{i} & \bar{C}_{33}^{i} & \bar{C}_{36}^{i} \\ \bar{C}_{16}^{i} & \bar{C}_{26}^{i} & \bar{C}_{36}^{i} & \bar{C}_{66}^{i} \end{array}\right]\left[\begin{array}{c} \alpha_{z z} \\ \alpha_{\theta \theta} \\ \alpha_{r r} \\ \alpha_{z \theta} \end{array}\right]$
Considering material orientation, the transformed stiffness matrix from the cylindrical coordinate system is [39]:
$\left[\bar{C}^{i}\right]=\left[T_{\sigma}\right]^{-1}\left[C^{i}\right]\left[T_{\varepsilon}\right]$
Where
$\left[T_{\sigma}\right]=\left[\begin{array}{cccccc} m^{2} & n^{2} & 0 & 0 & 0 & 2 m n \\ n^{2} & m^{2} & 0 & 0 & 0 & -2 m n \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & m & -n & 0 \\ 0 & 0 & 0 & n & m & 0 \\ -m n & m n & 0 & 0 & 0 & m^{2}-n^{2} \end{array}\right]$
$\left[C^{i}\right]=\left[\begin{array}{cccccc} 1 / E_{x x}^{i} & -v_{x y}^{i} / E_{x}^{i} & -v_{x y}^{i} / E_{x}^{i} & 0 & 0 & 0 \\ -v_{x y}^{i} / E_{x}^{i} & 1 / E_{y}^{i} & -v_{y z}^{i} / E_{y}^{i} & 0 & 0 & 0 \\ -v_{x y}^{i} / E_{x}^{i} & -v_{y z}^{i} / E_{y}^{i} & 1 / E_{y}^{i} & 0 & 0 & 0 \\ 0 & 0 & 0 & 2\left(1+v_{y z}^{i}\right) / E_{y}^{i} & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 / G_{x y}^{i} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 / G_{x y}^{i} \end{array}\right]^{-1}$
$\left[T_{\varepsilon}\right]=\left[\begin{array}{cccccc} m^{2} & n^{2} & 0 & 0 & 0 & m n \\ n^{2} & m^{2} & 0 & 0 & 0 & -m n \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & m & -n & 0 \\ 0 & 0 & 0 & n & m & 0 \\ -2 m n & 2 m n & 0 & 0 & 0 & m^{2}-n^{2} \end{array}\right]$
where, m = cos(φ) and n = sin(φ). φ is the angle between the 1-axial direction of the material and the z-axial direction of the cylindrical coordinate system.
Appendix B.
As the default global coordinate system of the 3D model in COMSOL Multiphysics is the Cartesian coordinate system, the stress transformation relationship between the cylindrical coordinate system and the Cartesian coordinate system is:
$\left[\begin{array}{c} \sigma_{x x} \\ \sigma_{x y} \\ \sigma_{x z} \\ \sigma_{y y} \\ \sigma_{y z} \\ \sigma_{x z} \end{array}\right]=\left[\begin{array}{cccccc} \frac{x^{2}}{x^{2}+y^{2}} & -\frac{2 x y}{x^{2}+y^{2}} & 0 & \frac{y^{2}}{x^{2}+y^{2}} & 0 & 0 \\ \frac{x y}{x^{2}+y^{2}} & \frac{x^{2}-y^{2}}{x^{2}+y^{2}} & 0 & -\frac{x y}{x^{2}+y^{2}} & 0 & 0 \\ 0 & 0 & \frac{x}{\sqrt{x^{2}+y^{2}}} & 0 & -\frac{y}{\sqrt{x^{2}+y^{2}}} & 0 \\ \frac{y^{2}}{x^{2}+y^{2}} & \frac{2 x y}{x^{2}+y^{2}} & 0 & \frac{x^{2}}{x^{2}+y^{2}} & 0 & 0 \\ 0 & 0 & \frac{y}{\sqrt{x^{2}+y^{2}}} & 0 & \frac{x}{\sqrt{x^{2}+y^{2}}} & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{c} \sigma_{r r} \\ \sigma_{r \varphi} \\ \sigma_{r z} \\ \sigma_{\varphi \varphi} \\ \sigma_{\varphi z} \\ \sigma_{z z} \end{array}\right]$
Further, the stress transformation relationship between the material coordinate system and the cylindrical coordinate system with additional rotation is:
$\left[\begin{array}{c} \sigma_{z z} \\ \sigma_{\varphi \varphi} \\ \sigma_{r r} \\ \sigma_{\varphi z} \end{array}\right]=\left[\begin{array}{cccc} \cos ^{2} \theta & \sin ^{2} \theta & 0 & 2 \sin \theta \cos \theta \\ \sin ^{2} \theta & \cos ^{2} \theta & 0 & -2 \sin \theta \cos \theta \\ 0 & 0 & 1 & 0 \\ \sin \theta \cos \theta & -2 \sin \theta \cos \theta & 0 & \sin ^{2} \theta-\cos ^{2} \theta \end{array}\right]\left[\begin{array}{l} \sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{12} \end{array}\right]$
