FEM-simulation of superconducting linear acceleration system for pellet injection

Abstract

In order to simulate the high-temperature superconducting (HTS) linear acceleration (SLA) system for the pellet injection, the integration method of the applied magnetic field generated from the acceleration coil has been proposed. To this end, the regularization technique is used in the evaluation of the improper integrals, and simultaneously, a FEM code is developed for analyzing the shielding current density in an HTS film. In addition, the SLA system has been simulated using the code. The results of the computations show that the accuracy of the applied magnetic field is considerably improved. In this sense, the regularization technique is a useful tool. Also by locating the outer coil, the acceleration time during which the pellet speed reaches 5 km/s is about 3.5 times shorter than that of the only use of the inner coil. These results mean that the outer coil is effective in the improvement of the acceleration performance for the SLA system.

Keywords

Accelerator magnets finite element method improper integral superconducting linear accelerators

1. Introduction

A pellet injection system using two types of hydrogen/deuterium pellets has been developed to supply fuel to the fusion reactor [1,2]. Pellets such as air guns using helium gas are accelerated and injected into high-temperature plasma. The pellet velocity by the system is about 1.2 km/s [2]. However, this system reduces the fuel efficiency of the helical fusion reactor because the pellets melt around the plasma before reaching the plasma core. In order to resolve this problem, a faster pellet injection system is desired.

Recently, a high-temperature superconducting (HTS) linear acceleration system for the pellet injection has been proposed to inject pellets into the plasma core [3]. Hereafter, this system is called the SLA system. This system can accelerate a pellet container made by a pellet and an HTS film electromagnetically (see Fig. 1(a)). Furthermore, two types of film are used for acceleration and levitation. According to their estimation, the speed of the SLA system is more than 5 km/s. However, since it is unclear how much pellet speed can be obtained, the SLA system has not yet been applied to pellet injection.

In our previous study [3], we analyze the time-dependent of a shielding current density in an HTS film by using the finite element method (FEM). Although we simulate the SLA system by modifying the code, it is recently found that the accuracy of the numerical integration for an applied magnetic flux density is remarkably degraded.

The purpose of the present study is to develop a FEM code for analyzing a shielding current density in an HTS film and a dynamic motion of the film. To this end, we implement the numerical integration method for accurately calculating the applied magnetic field. Moreover, we describe the enhanced performance of the SLA system.

Fig. 1.

A schematic view of a Superconducting Linear Acceleration (SLA) system for the axisymmetric model. Note that the outer coil is used in Section 4.

2. Governing equations and equation of motion

In an SLA system for a pellet injection, an applied magnetic field B ∕μ₀ generated from an acceleration coil and it is applied to an acceleration film. Simultaneously, a shielding current density j is induced in the film. By determining the characteristic of the superconducting properties, we use the J-E constitutive equation: E = E_C[| j |∕j_C]^N [ j ∕| j |] which is given by the relation between the shielding current density j and an electric field E . Here, E_C, j_C, and N are the critical electric field, the critical current density, and the non-negative integer-value.

A schematic view of the SLA system is shown in in Fig. 1(b). We consider a disk-shaped HTS film of radius R_H and thickness b, and we adopt an acceleration coil of radius R and length L. Note that the outer coil is also used in Section 4. Also, we simulate the SLA system in the cylindrical coordinate system. The symmetry axis and the origin O are the z-axis and the centroid of the coils, respectively.

As usual, we assume the thin-layer approximation [4]. This assumption is satisfied as follows: ∂ j ∕∂z = 0 and j ⋅ e _z = 0, where e _z denotes a unit vector in the z-direction. Under the above assumption, the shielding current density j can be expressed as j = (2∕b)(𝛻S × e _z) by using a scalar function S (r, t). The scalar function S is governed by the integro-differencial equations: $\begin{eqnarray}\displaystyle {\mu}_{0}\partial _{t}\int _{0}^{R_{H}}Q(r,r^{\prime })S(r^{\prime },t)r^{\prime }dr^{\prime }+{\displaystyle \frac{2}{b}}S+\partial _{t}\langle B_{z}\rangle +{\displaystyle \frac{1}{r}}\partial _{t}(rE_{{\theta}})=0. & & \displaystyle\end{eqnarray}$ (1) Here, B_z is the z-component of an applied magnetic flux density B , and 〈〉 is an average operator over the film thickness. Also, E_θ denotes the θ-component of an electric field E . We assume that the initial and boundary conditions are given by S (r,0) = 0 and S (R_H, t) = 0, respectively.

We suppose that an acceleration HTS film of the SLA system is moved to the direction of the positive z. For simulating the SLA system, it is necessary to determine the dynamic motion of the film. To this end, we adopt the Newton’s law of motion as follows: $\begin{eqnarray}\displaystyle {\displaystyle \frac{d^{2}z}{dt^{2}}}={\displaystyle \frac{4{\pi}}{m}}\int _{0}^{R_{H}}{\displaystyle \frac{\partial S}{\partial r}}\langle B_{r}\rangle rdr. & & \displaystyle\end{eqnarray}$ (2) Here, B_r and m denotes the r-component of B and the total mass of the pellet container and the film, respectively. Throughout the present study, the initial conditions of (2) are given by z = z₀ and v = 0 m/s, where z₀ and v is an initial position of the film and a pellet velocity, respectively.

Solving the initial-boundary-value problem of (1) and (2), we can obtain both the time evolution of the shielding current density and the dynamic motion of HTS film. In the present study, we discretize the problem with respect to space by using the FEM which in the interval from r = 0 to r = R_H is equally divided into the n finite elements. It is reduced to the simultaneous ordinary differential equations: $\begin{eqnarray}\displaystyle {\displaystyle \frac{d}{dt}}\left[\begin{array}{@{}l@{}}\boldsymbol{S}\\ v\\ z\end{array}\right]=\left[\begin{array}{@{}c@{}}-{\mu}_{0}W^{-1}U[\boldsymbol{e}(\boldsymbol{S})+v\boldsymbol{c}(z)+\boldsymbol{h}(z,t)]\\[3.0pt] -{\displaystyle \frac{4{\pi}}{m}}\boldsymbol{a}^{T}(z)\boldsymbol{S}\\ v\end{array}\right]. & & \displaystyle\end{eqnarray}$ (3) Here, the matrix W is defined by W ≡ UW^∗U + F, where W^∗ denotes the symmetric matrix obtained from the function Q (r, r′) and shape functions of FEM. In addition, the matrices, U and F, are determined from the boundary condition. On the other hand, S is the nodal vector corresponding to the scalar function S (r, t), and e ( S ) is the nodal vector corresponding the electric field E_θ. The nodal vector a (z) can be obtained from the r-component B_r of the applied magnetic field B , and we determine the vectors c (z) and h (z, t) by B_z. Note that B_z becomes the improper integral.

Throughout the present study, the geometrical and the physical parameters are fixed as follows: R_H = 4 cm, b = 1 mm, z₀ = 1 mm, m = 10 g, N = 20, j_C = 1 MA/cm², E_C = 1 mV/m, R = 5 cm, L = 10 cm, and n = 101.

3. Numerical method of applied magnetic field

In this section, we explain about the numerical method of B_z. The explicit form of B_z can be expressed as follows: $\begin{eqnarray}\displaystyle B_{z}(r,z)={\displaystyle \frac{{\mu}_{0}I}{4{\pi}Lr}}\left[\sqrt{{\displaystyle \frac{R}{r}}}\int _{-L/2}^{L/2}dz^{\prime }F(k)+2\sqrt{rR}\int _{-L/2}^{L/2}dz^{\prime }{\displaystyle \frac{F^{\prime }(k)}{2k}}{\displaystyle \frac{\partial k^{2}}{\partial r}}\right], & & \displaystyle\end{eqnarray}$ (4) where the integrand F (k) of the first term is given by F (k) = k[−2E (k)∕k² + (2∕k² −1)K (k)]. Here, K (x) and E (x) denote the complete elliptic integral of the first kind and that of the second kind, respectively. Its parameter k² is defined by k² ≡ 4rR∕[(r + R)² + (z − z^′)²]. On the other hand, F^′(k)∕(2k) and ∂k²∕∂r of the second integrand can be written by the following equations $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{F^{\prime }(k)}{2k}}=-{\displaystyle \frac{1}{k^{3}}}[K(k)-E(k)]+{\displaystyle \frac{1}{2k(1-k^{2})E(k)}},\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\displaystyle \frac{\partial k^{2}}{\partial r}}={\displaystyle \frac{4R}{(r+R)^{2}+(z-z^{\prime })^{2}}}-{\displaystyle \frac{8rR(r+R)}{[(r+R)^{2}+(z-z^{\prime })^{2}]^{2}}}.\end{eqnarray}$ (6) Note that these integrands in Eq. (4) are singular at r = R and z = z′. For this case, Eq. (4) becomes an improper integral.

Fig. 2.

Spatial distributions of B_z(r, z) for (a) L = 10 cm and (b) L = 30 cm. Here, N_G = 16.

Let us investigate the influence of the singularity on the spatial distribution of B_z. In order to evaluate Eq. (4), we adopt the Gauss–Legendre quadrature. Here, the number of the integration points for this quadrature is denoted by N_G. At two observation points, r = 0 and r = 4R∕5, the spatial distributions of B_z are calculated for the lengths L = 10 cm and L = 30 cm of the coil and are depicted in Fig. 2. We see from Fig. 2(a) that, for L = 10 cm, the value of B_z monotonously decreases with increasing z for two cases. As is shown in Fig. 2(b), B_z becomes a shape like a wave for 0 ≤ z ≲ 13 cm. This is mainly because the function is strongly singular when the observation point is located near the coil radius R.

In order to evaluate the two integrals accurately, we use the regularization technique. Hereafter, we define $\begin{eqnarray}I_{1}\equiv \int _{-L/2}^{L/2}dz^{\prime }F(k)\quad \text{and}\quad I_{2}\equiv \int _{-L/2}^{L/2}dz^{\prime }{\displaystyle \frac{F^{\prime }(k)}{2k}}{\displaystyle \frac{\partial k^{2}}{\partial r}},\end{eqnarray}$ as the parts of Eq. (4). As a result, I₁ is factored into the sum of two parts as follows: $\begin{eqnarray}\displaystyle I_{1}=\int _{-L/2}^{L/2}dz^{\prime }[F(k)-P_{1}(r,z,z^{\prime })]+\int _{-L/2}^{L/2}dz^{\prime }P_{1}(r,z,z^{\prime }). & & \displaystyle\end{eqnarray}$ (7) Here, P₁ is the asymptotic form for F (k) and its explicit form is given by $\begin{eqnarray}\displaystyle P_{1}(r,z,z^{\prime })\equiv -{\displaystyle \frac{1}{2}}\log [(r-R)^{2}+(z-z^{\prime })^{2}]\sim F(k). & & \displaystyle\end{eqnarray}$ (8) The first term of Eq. (7) indicates nonsingular integrals which can be accurately evaluated by using the Gauss–Legendre quadrature, whereas we can obtain the second term of Eq. (7) is an analytical evaluation. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle I_{2}=\int _{-L/2}^{L/2}dz^{\prime }\left[{\displaystyle \frac{F^{\prime }(k)}{2k}}{\displaystyle \frac{\partial k^{2}}{\partial r}}-P_{2}(r,z,z^{\prime })\right]+\int _{-L/2}^{L/2}dz^{\prime }P_{2}(r,z,z^{\prime }),\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{2}(r,z,z^{\prime })\equiv -{\displaystyle \frac{r-R}{(r-R)^{2}+(z-z^{\prime })^{2}}}-{\displaystyle \frac{3}{16R^{2}}}(r-R)\log [(r-R)^{2}+(z-z^{\prime })^{2}].\end{eqnarray}$ (10)

Let us investigate the accuracy of the integration for B_z. As a measure of the accuracy, we define a relative error: ϵ ≡|I_N − I_A|∕I_A. Here, I_A is an approximate value of B_z with 8-digit precision, whereas I_N is an approximate value of B_z obtained by the regularization technique.

Fig. 3.

Dependence of the relative error ϵ on the number N_G of the integration points. Here, L = 30 cm, r = 4R∕5, and z = 0.

Fig. 4.

Spatial distribution of B_z(r, z) for L = 30 cm, N_G = 16, r = 4R∕5.

Figure 3 shows the dependence of the relative error ϵ on the number N_G of the integration points. From this figure, the accuracy is considerably improved by using the regularization. In Fig. 4, we show the spatial distributions of B_z without and with the regularization. By using the regularization, the function of B_z becomes smooth for 0 cm ≤ z ≲ 13 cm. In this sense, the regularization technique is a useful tool for analyzing the shielding current density in an HTS film. Hereafter, we use N_G = 16.

4. Simulation of SLA system for pellet injection

On the basis of the above method, let us simulate the SLA system for the pellet injection. In this section, we investigate the acceleration time during which pellet speed reaches 5 km/s. Incidentally, this value is almost the same speed as advanced two-stage pneumatic acceleration [5]. We perform the simulation of the SLA system focusing on this value.

In the previous study, we found that the following is effective in the improvement of the speed by using a single acceleration coil.

–
A short in the distance between the initial position and the origin
–
An increase in the film radius and the coil current.

Because it was impossible to reach a speed of 5 km/s with the above items, we employed multiple coils. In the present study, we use not only the inner coil but also the outer coil of radius R_out and L_out to enhance the performance of the SLA system. Note that the centroid of the outer coil is also taken at the origin. We use a plurality of the inner/outer coils which are equally placed in the z-direction. A current I of the inner/outer coils is controlled as follows: $\begin{eqnarray}\displaystyle I(z_{\text{mod}},t)=\left\{\begin{array}{@{}ll@{}}{\alpha}(t-t_{\text{min}}) & (0\leq z_{\text{mod}}\leq z_{\text{limit}})\\ 0 & (\text{otherwise}).\end{array}\right. & & \displaystyle\end{eqnarray}$ (11) Here, 𝛼 denotes the increasing rate of the inner coil current, and z_limit is the limit of acceleration region. Also, t_min is the time at z_mod = 0 m. In addition, z_mod is defined by z_mod ≡ mod(z + z_p∕2, z_p) − z_p∕2, where z_p denotes a coil interval. The above parameters are fixed as follows: 𝛼 = 20 MA/ms, z_limit = 25 cm, L_out = 30 cm, and z_p = 50 cm. In addition, the current of the outer coil is given by I_out = I.

Fig. 5.
Time dependence of the pellet velocity 𝜈 for the case with R_out = 10 cm. Here, τ is the acceleration time during which the velocity reaches 5 km/s.

Figure 5 indicates the time dependence of the pellet velocity v. We can see from this figure that, by using the outer coil, the pellet velocity v drastically increases compared with the only inner coil. The acceleration time during which the velocity reaches 5 km/s is 2.6 s for this case. This value is about 3.5 times shorter than that of the only use of the inner coil.

We investigate the influence of the radius R_out of the outer coil on the pellet velocity v. In Fig. 6, we show the dependences of the time ratio τ∕τ₀ and the applied magnetic field B_r on the radius R_out, where τ₀ is the acceleration time for the only inner coil. From this figure, the time ratio τ∕τ₀ monotonously decreases with the length L_out. In addition, the acceleration time is reduced to about 1/3 or less for R_out = 10 cm. This is because the value of B_r increases with the radius R_out, and the performance of the SLA system is further accelerated.

Fig. 6.
Dependences of the acceleration ratio τ∕τ₀ and the dimensionless applied magnetic field B_r on the radius R_out out of the outer coil.
5. Conclusion

The accuracy of the applied magnetic field is considerably improved by using the regularization technique. As a result, the behavior of the magnetic field becomes smooth. In this sense, the regularization technique is a useful tool for analyzing the shielding current density in an HTS film. Also by locating the outer coil, the acceleration time during which the pellet speed reaches 5 km/s is about 3.5 times shorter than that of the only use of the inner coil. These results mean that the outer coil is effective in the improvement of the acceleration performance for the SLA system. However, the SLA system will be larger than a fusion reactor. In this sense, this result means that it is difficult to apply the SLA system to a fuel supply of the fusion reactor without solving the above problem.

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