Magnetoplasmadynamic acceleration of solid body in a railgun

Abstract

A physical model and a mathematical approach were developed to describe ponderomotive acceleration a body pushed by plasma column of high-current arc in a railgun. In contrast to the previously published studies, the arc plasma generation was considered by taking into account kinetics of heavy particle fluxes in a non-equilibrium layer near the vaporizating (ablating) rails and the phenomena of heat and mass transfer and electric sheath at rail surfaces. The self-consistent numerical analysis has been conducted for a wide range of arc currents (up to 2MA). It has been shown that the electron temperature was determined mainly by the Joule energy dissipation supporting high degree of ionization. It has been determined that the arc voltage is about 10² V. The rails temperatures are relatively low (~ 3000 K) and therefore, the electron flux emitted from the cathode manly due to thermionic emission enhanced by Schottky effect is lower than the flux of ions from the plasma, which supported the current continuity at the cathode region. The calculated velocity v_s of the plasma-solid body increased with the arc current (I) showing tendency to saturation. In range of I = 0.5–2 MA the v_s slightly increased from 5 × 10⁵ to 10 × 10⁵ cm/s when the rail length increased from 1 to 3 m respectively. The dependence of the velocity on the arc current and the calculated plasma parameters are in range of those values observed experimentally. The nature of the velocity with arc current saturation is explained by presence of the dynamic pressure resulting in predicted velocity dependence on the arc current as 4-th root.

Introduction

A railgun is a device designed in beginning of twenty century as a weapon to utilize electromagnetic force to launch a projectile to a high velocity. The projectile normally does not contain explosives. The railgun uses a pair of parallel conductive rails. While explosive-powered military guns cannot achieve an initial velocity of more than ≈2 km/s, a similar projectile in railguns can readily reach significantly larger velocity. Its kinetic energy determines the destructive force of a projectile, which may be much greater than the conventionally launched projectiles. High kinetic energy, absence of explosive materials and the low cost of projectiles compared to conventional weapon, determine the railgun advantages.

French inventor André Louis Octave Fauchon-Villeplée [1, 2] for the first time introduced a simple railgun device in 1917. Korol’kov [2, 3] in 1923 concluded that while the construction of a long-range electric gun was, in general, within the realm of possibility, the practical application of Fauchon-Villeplee’s railgun was hindered by its enormous electric energy consumption and need for a special electric generator. McNab [2] reported further development of the railguns design. Despite aforementioned difficulties at the early stages, the railgun construction and its technical characteristics continued to be improved up to present time. Some improved designs were described [4,5,6,7] summarizing many of the technical advances leading to the practical implementations.

Artsimovich et al. [8] developed a first method (named as electromagnetic) of acceleration of a plasma produced in an arc between parallel electrodes. The mechanism is based on a force due to interaction of the self-magnetic field with the arc current. Brast and Sawle [9] considered such mechanism of the plasma acceleration in a railgun accelerating a solid-state shells using the arc plasma as a pushing gas. Boxman [10] studied experimentally the motion of the arc column sustained between Cu electrodes in a vacuum when driven with currents of 15–35 kA and he observed the column velocities of about (0.1–3.5) × 10³ m/s. The effect of electrodynamic acceleration of the arc plasma was also used in a system of railgun configuration to high-speed acceleration of dielectric bodies. Such body was placed between the rails in a channel of a magnetoplasma accelerator. The acceleration of a dielectric body was carried out when the arc plasma is exposed to the dielectric projectile in a vacuum.

An analysis of published works for high current arc moving between elongated planar parallel electrodes at velocities of 1 km/s and higher was presented in a number of theoretical works [11]. It was noted that the efficiency of acceleration is limited not only by the strength of the railgun and mass of the working body, but also depends on the arc plasma properties. In some previous works, the research has been conducted in conditions of channels with length of one or few meter, height of 1 cm and width of about 1 cm [11]. Some plasma parameters have been determined from measured data of resistivity using Ohm’s law. In this estimate, the potential difference across the core of the plasma was taken as a certain fraction of the potential difference between the electrodes. The results showed that at current of about 10⁵-10⁶A, the electrical field in plasma column weakly dependent on the current and was in the order of 10² V/cm. Plasma propagated along the channel direction and it longitudinal size was about 10 cm with its dimension between the rails of about 1 cm. The plasma temperature was determined to be in range of (3–7) × 10⁴ K.

An important parameter of acceleration is the velocity of the solid-body (or projectile). In railgun, the plasma accelerates the attached solid-body and the degree of acceleration is determined by mechanism of electrical energy dissipation and its distribution in the arc plasma. The experimental value of the velocity was reported as (5–7) × 10⁵ cm/s and in a special cases as (8–11) × 10⁵ cm/s for body of 1–2 g and in a wide range of arc currents up to 10⁶ A [12,13,14].

Most of theoretical models involve the assumption that the moved plasma mass does not vary, and some theoretical studies show that surrounding gas and electrode ablation mass can influence the acceleration of the plasma in a railgun [15]. Moreover, measurements [16] show that a satisfactory description can be obtained utilizing the assumption that the mass increases in course of acceleration. Sometimes the rate of electrode ablation was used as an arbitrary parameter calculated from the armature pressure [17]. Another works determined the ablation rate of rail surface by assuming that either all energy flux [13], or only the excess over that which the rail material can conduct away was equal to the energy expended for ablation. In this case, the black-body radiation flux due to Stefan-Boltzmann law was arbitrary assumed as all absorbed energy flux by the wall material [18, 19]. Davidson [20] used two similar strong assumptions that difficult to justify: 1) the rate of ablation G per unit area was proportional to Joule energy dissipation in the plasma through some not justified constant α_g. 2) The velocity v_g of ablating mass flux was determined from relation G = ρ_gv_g, where ρ_g is the density, which determined from a continuity equation with a particle source assumed also proportional to Joule energy dissipation through α_g.

In general, there is a misunderstanding and even some confusion using the above-mentioned assumptions to determine the ablation rate and its mass velocity. First, the energy dissipated in the plasma volume is an energy source, should be considered to analyze the plasma energy balance in order to calculate the plasma electron temperature taking in account also the energy losses. The energy spend by the rail ablation should be calculated considering an electrode energy balance and cannot be related directly to the plasma Joule energy. Secondary, the rail evaporated (ablation) rate, the vapor mass velocity and density should be determined considering a non-equilibrium phenomena in a layer of few mean free path of evaporated particles. In additional, the previously published models not considered presence a dynamic (kinetic) pressure appeared by meeting the solid body with the plasma moving with high velocity along the rails.

It should be pointed out, that prior published studies were not contain any arc model, which take in account self-consistently processes by governing the current continuity at the plasma-electrode boundary and mutual phenomena related the arc motion during it rapid displacement with the mass and heat transfer near the walls. Also, still absent any approach to describe kinetics of the vaporization (ablation), which occurred during formation the arc plasma from material of the eroded rails. Modelling of the arc phenomena appeared in specific conditions of a railgun accelerator is the subject of this work. The main goal of the study was to devise a physical model for a high-current discharge migrating between long plane parallel electrodes, which allow understanding the coupling processes between the electrodes and in the plasma with plasma mass change, as well as phenomena in the layers near the electrodes. In additional, the subject of the present work was to develop a mathematical approach for numerical analysis of the self-consistent phenomena in the railgun, calculation parameters occurred in the moving column plasma, parameters of the heat- and mass change at the rail surfaces and determine of their coupling with the mechanism of plasma acceleration.

Physical model

Previously [21] the arc plasma column was described considering the processes in the electrode rails, the electrical layers near them, and plasma formation by ionization of the electrode vapor assuming presence of a single charged ions. The model employed was limited by fixing the cathode potential drop u_c, and by the heavy particle density derived from a simple particle balance in near wall region. The body acceleration and its velocity was calculated [11] using plasma parameters obtained from the first approximation model [21].

The physical model developed below takes in account the non-equilibrium heavy particle layer, in which a returned flux of the evaporated atoms is formed and consequently the resulting plasma mass is determined. The approach considers the kinetics of the cathode and anode evaporation, particle jump at the surface, presence of multi charged ions and acceleration of the solid body. The electrode-plasma transition is considered similar to such transition appeared in the cathode and anode spots of conventional vacuum arc with cylindrical electrodes. The specifics is in the geometry, i.e. in size difference of the classical minute spots and relatively large areas of the arc root contacted with the rails (Fig. 1). Thus, the model allows study the influence of the plasma mass change as dependence on arc current due to ablation of the rails.

Fig. 1

Schematic configuration of arc plasma-body assembly in a railgun accelerator with arc column length l_ar, width b and height h

The kinetic approach to study the plasma generation and plasma flow during the cathode evaporation into dense plasma of vacuum arc first was developed in 1982 [22] followed by mathematical description that has been formulated for phenomena in the cathode spots [23]. The main advance of this approach allows directly obtain the cathode potential drop u_c and the cathode erosion rate G. The details of the kinetic models for cathode and anode vaporization in the respectively spots were recently summarized [11, 23]. According to the model, the ionized vapor structure near the cathode surface consists of several partially overlapping physical regions separated by characteristic boundaries with corresponding gasdynamic parameters, temperature T, density, n and velocity v (Fig. 2).

Fig. 2

Schematic presentation of the plasma kinetic model for cathode rail. The parameters indexed by “0” indicated saturated values

As such the cathode region include a ballistic region comprising a space charge sheath, located between the cathode the surface (boundary 1) and it external boundary (2) of the sheath, a non-equilibrium Knudsen plasma layer located between boundaries 1 and external boundary of the Knudsen layer 3. A strong electric field is present within the ballistic zone. The cathode electron emission produces an electron beam, which is accelerated in the ballistic zone and the energy relaxation of the beam, occurs in a region with a boundary 4, where plasma begins to expand into the expansion region that continues until the anode. The equilibrium heavy particles n₀ and electron n_e0 densities were determined by the cathode spot temperature T_c. The plasma density decreases and the plasma velocity increases with a distance away from the electrode surface [11]. Two heavy particle fluxes (evaporated and returned) are formed in the Knudsen region. The difference between these fluxes determines the plasma velocity v₃ at the external boundary of the Knudsen layer (boundary 3) and the net rate of the mass evaporation.

The physics of kinetic approach consists in consideration of the electron emission as an evaporation process together with the atom vaporization with following atom ionization and determination of the plasma velocity at the Knudsen layer (boundary 3) using the quasineutrality condition at this boundary. At the sheath boundary 2, the plasma electrons are returned to the cathode, while the ions and emitted electrons are accelerated with energy determined by the cathode potential drop eu_c. The returned flux of the charge particles depends on value of potential barrier u_c in the space charge sheath, which in turn produced by the charged particles generated due to high power dissipation in the current carrying cathode region. As the charge particle motion is coupled self-consistently with the potential barrier, the corresponding height of the barrier, i.e. u_c can be obtained studying the energy and momentum equations of charge particle fluxes in the Knudsen layer including the space charge layer. Similar kinetic model was described also for the anode vaporization [11]. In the anode region was take in account that the current mainly supported by the electron flux to the anode.

The arc plasma column produced between rails is bounded by the inertial boundary at the solid body on one side and the magnetic wall on the other behind the plasma. Thus, in the railgun device the rails evaporate essentially in the closed volume, which is bounded by the electrodes and the insulating walls (Fig. 1). During the rails erosion two fluxes are produced: 1) from the cathode G_c and 2) from the anode G_a, which are determined as mnvF, with atom mass m, heavy particle density n, and velocity v for the cathode and anode respectively. These two erosion fluxes produce common plasma in the mentioned closed volume with resulting heavy particle density n_h. The resulting erosion mass flux G (in grams per second) from both electrodes is given by G = G_c + G_a. The atoms are ionized by electrons emitted from the cathode and mainly by plasma electrons, which were heated by the electron beam and by Joule energy dissipation in the regions of plasma column. The presence of multiple ionizations of the evaporated atoms are determined by a level of the electron temperature. The system of Saha equations determine a state of the multi charged ions in the plasma.

System of equations

The system includes equations described the electrode and arc plasma parameters and equations to describe velocity of the arc plasma column and the attached insulator body moved along the rails as dependence on the arc current. As in the above physical model, the electrode-plasma transition is considered similar to such transition appeared in the cathode and anode spots of conventional vacuum arc with cylindrical electrodes.

Equations for the arc parameters

It is known, the arc motion is affected by the current. To simplify description of this influence and to analyze the arc parameters, we consider the current as set and constant. The task of mathematical approach is to determine the physical conditions characterized the discharge parameters such that the electrode heating provides for carrier charge reproduction in the amount necessary to maintain the set current at a high migration speed of the plasma column. This involves a mathematical description by taking into account the coupled phenomena determined the energy balance for the cathode and anode, their heat conduction, the plasma energy balance in the arc column, and the particle fluxes produced in the electrode layers. The previous estimations showed [11] that one could use the equations for the plasma column at conditions subject to the thermodynamic equilibrium and the kinetic relations to characterize the near electrode regions and the electrode erosion rate. Let us describe the equations for the electrodes and arc plasma.

A negative potential drop u_an at the anode surface is produced when the flux of thermal electrons supported the current to the anode exceeds the arc current I. This potential drop u_an is determined taking into account the equation conservation of the electron and ion fluxes to the anode, and areas F_c at the cathode and F_a at the anode contacted with the plasma of the arc root respectively by the following expression [11]:

$$u_{an} = T_{e} Ln\left[ {\frac{{j_{etha} F_{a} }}{{I + (j_{ia} + j_{ema} )F_{a} - j_{eb} K_{h} F_{c} }}} \right]$$

(1)

where T_e is the electron temperature in eV, j_etha is the thermal electron current density at the anode side, j_ia is the back ion current density to the anode, j_ema is the electron emission current density from the hot anode, j_eb is the cathode electron emission current density reached the anode surface. K_h is the coefficient of cathode electron beam absorption in the plasma column at the anode side, which determined by the cross sections of electron beam collisions with plasma atoms, ions, and electrons.

For experimental arc velocity of 1–10 km/s, the time of arc location within the characteristic length scale l ~ 10 cm is ~ 10^–4–10^–5 s. During that time, for Cu rails the heat wave from the plasma heat source expands in the electrode with a layer having thickness ≤ (at)^0.5 ~ 10^–2 cm (a is thermal diffusivity, which is ~ 1 cm²/s for Cu). This thickness is much less than the electrode thickness (~ 1 cm), so one can take the electrodes as infinitely extended in the thermal-wave direction in order to consider the heat transfer processes. The energy balances at the cathode and anode are represented by.

• Cathode:

  $$Iu_{ef}^{c} + F_{c} \sigma_{SB} (T_{e}^{4} - T_{c}^{4} ) - G_{c} (\lambda_{s} + \frac{{2kT_{c} }}{m}) = Q_{Tc}$$

  (2)

• Anode:

  $$Iu_{ef}^{a} + F_{a} \sigma_{SB} (T_{e}^{4} - T_{a}^{4} ) - G_{a} (\lambda_{s} + \frac{{2kT_{a} }}{m}) = Q_{Ta}$$

  (3)

Here m is atomic mass, σ_sB is the Stefan’s constant, λ_s is the latent heat of evaporation, k is the Boltzmann’s constant, and F is the total area of the arc contact with the electrode. Index ‘c’ and ‘a’ is related to the cathode and anode respectively. The effective potentials $u_{ef}^{c}$ and $u_{ef}^{a}$ characterize energy fluxes to the cathode by ion flux bombardment and to the anode by electron flux bombardment respectively. The fluxes being dependent on the cathode potential drop u_c and the anode one u_a, as well as the fraction of the electron component in the current and effects associated with the returned ion and electron fluxes (toward the cathode) and ions (toward the anode) are formed in the ballistic and Knudsen layers [11]. The electrode rails are heated ohmically. The effective potentials $u_{ef}^{c}$ and $u_{ef}^{a}$ also included the effects of cathode cooling due to ion neutralizations at the surface and the heat of electron condensation at the anode. The specific heat of heavy particles ablation and condensation are described by the last terms at the cathode and at the anode as result of their erosion rate. All the mentioned effects heating and cooling are detailed previously considering the electrode balance [11, 23]. Q_T is the total heat flux due to electrode heat conduction. The electrode thermal regimes and their temperatures were determined from solution of two-dimensional nonstationary heat conduction equation for a heat source modeled with normally distributed heat flux at top rail surface within a characteristic length l_ar for moved source along this surface. The model was described early [24] and the solution to calculate temperatures of the cathode T_c and anode T_a surfaces as dependence on time t and source velocity v_s is presented in following form [11]:

$$T_{c,a} = \frac{{Q_{Tc,a} }}{\pi \lambda b}\int_{0}^{{\sqrt {\frac{t}{{t_{0} }}} }} {\frac{d\omega }{{\sqrt {\left( {1 + \omega^{2} } \right)} }}} \exp \left[ {\frac{{v_{s} t_{0} }}{2a}\left( {1 - \frac{1}{{2\left( {1 + \omega^{2} } \right)}} - \frac{{\left( {1 + \omega^{2} } \right)}}{2}} \right)} \right]\,\,\,\,\,\,\,\,t_{0} = \frac{{l_{ar}^{2} }}{4a}\,\,\,\,\,\,l_{ar} = \frac{I}{jb\sqrt \pi }$$

(4)

where j is the arc current density, λ is the heat conduction coefficient, t₀ is the parameter characterized concentration of the normally distributed heat source [24], which acts on the lateral side of the electrodes rails of width b, modeling the electrodes heating by the heat flux from arc column.

The important plasma parameter is the electron temperature T_e, which determined the degree of vapor ionization and the plasma characteristics. To calculate this parameter the plasma energy balance can be used in following form [11]:

$$sj\left( {u_{c} + \frac{{2kT_{c} }}{e}} \right) + ju_{pl} = \sum\limits_{i} {\Gamma_{i} } u_{i} + \xi j\frac{{kT_{e} }}{e} + \sigma_{SB} T_{e}^{4} + + q_{a} { + }q_{c} (T) + q_{bs} (T)$$

(5)

$$q_{a} = j_{etha} {\text{e}}^{{{( - }\frac{{{\text{e}}u_{an} }}{{T_{e} }}{)}}} (2T_{e} + u_{an} ),\,\,\,\,\,\,\,\,\,q_{bs} = j_{eth} {\text{e}}^{{{\text{( - e}}\frac{{{\text{e}}u_{c} }}{{T_{e} }}{)}}} (2T_{e} + u_{c} )$$

where s is the electron current fraction at the cathode, ξ is a coefficient that corrects for the electron energy transport [25], while q_c, q_a and q_bs represent transport of the convective energy flux by the heavy particles, by the electron flux to the anode surface and by the returned plasma electrons to the cathode surface respectively. The second term at left in Eq. (5) indicated the Joule energy dissipation in the plasma column, u_pl = jh/σ_pl is the voltage drop in the arc column of height h, and σ_pl is the Spitzer electrical conductivity of the highly ionized plasma. The first term on the right in Eq. (5) describes the total energy loss by ionization by the ions of charge i-th, while u_i are the ionization potentials for ions of charge i-th. The total current in accordance with the normally distributed heat source was expressed as [24]:

$$I = Fj = \sqrt \pi jbl_{ar}$$

(6)

Velocity of accelerating solid body pushed by the plasma motion of the arc in a railgun

Conversion of the electric energy into kinetic energy determine the mechanism of the arc plasma acceleration in railgun devices. The arc plasma column is accelerated by ponderomotive force and the column pushed the attached body (Fig. 1). Different reasons could limit the velocity of accelerating bodies in the accelerator. The breaking phenomena include the mechanical strength of the device, thermal loads inside it volume, instability of the discharge, the power system forming the current pulse [14]. In addition, an increase of arc plasma mass due to wall erosion can also limit the acceleration process. At the same time the mechanism supported the arc operation in vacuum determine the electrical energy dissipation between two rails and the acceleration parameters. When a certain density of electrode vapor is reached in order to support an increase of the arc current the vacuum arc could be ignited and developed. As result, self-consistent processes of the plasma-electrode transition, of the electrode erosion and plasma column heating determine the relation between arc plasma parameters and the plasma column acceleration.

Let us consider the system of equations to study the acceleration of arc plasma column and the attached body as a function of arc current. The arc plasma is collision dominated and therefore its parameters can be described by magnetohydrodynamic equations. In general, the problem should be solved in 2D approximation considering the processes of plasma generation (Sec. 2), distribution of the plasma parameters and plasma pressure along the rails. However, the problem in such formulation is very complicated. Therefore, in order to develop main scaling characteristics and to understand the main phenomena influencing railgun characteristics we are presenting a reduced model. Such model can be used to understand the principal mechanism of the acceleration for different arc current and influence of the pressure (static and dynamic). The considered approach was simplified assuming a uniform distribution of current, plasma density, temperatures and others plasma parameters in the arc column. This approach takes in account separately the momentum equations for the plasma and the solid body, assuming that the arc velocity v_s is steady ($\frac{{\partial v_{s} }}{\partial t} = 0$) during the arc motion along the rails x [11, 20, 26]. Really, Baker [15] showed that v_s sharply increased at beginnig of the rails and then the v_s is saturated to a constant value during the arc motion.

Equations of momentum for plasma column:

$$\frac{{d(\rho v_{s} )}}{dt} = - \nabla p + jB\,\,\,\,\,\,\,\, \to \,\,\,\,v\,\frac{\partial (\rho )}{{\partial t}}\, + \rho v_{s} \frac{{\partial (v_{s} )}}{\partial x} = - \frac{\partial p}{{\partial x}} + jB$$

(7)

where ρ is the mass density, p is the pressure, B is the magnetic field. According to Boynton and Huerta [27], the boundary of the plasma, whose pressure p, acting over the rear surface of the insolated body of area S_b, provides the force F_ex = $pS_{b} \,$ that pushes the body forward. The dielectric body acceleration caused by the pressure formed in the plasma due to ponderomotive force. During the motion static and dynamic pressures occur. The dynamic pressure is the kinetic energy of the accelerated mass per unit volume and equal to the difference between braking pressure and static pressure [28]. The dynamic pressure can reach substantial value when the plasma acceleration occurred to relatively high velocity v_s that is typical for a railgun. Thus, equation of momentum for body of mass m_b take in account that the force of body acceleration is due to the force F_ex:

$${m}_{b}=\frac{d{v}_{s}}{dt}={F}_{ex}\;\text{or}\;{m}_{b}\left(\frac{\partial {v}_{s}}{\partial t}+{v}_{s}\frac{\partial {v}_{s}}{\partial x}\right)={F}_{ex}\mathrm{\,and\,then\,}{m}_{b}\frac{\partial \left({v}_{s}^{2}\right)}{2\partial x}=p{S}_{b},{ S}_{b}=bh$$

(8)

Let us multiply both sides of Eq. (7) by a volume hbl_ar (Fig. 1) 

$$\,\,\,\,\,\,v_{s} \,\,\frac{\partial \rho }{{\partial t}}l_{ar} bh + \rho l_{ar} bh\frac{{\partial (v_{s}^{2} )}}{2\partial x}\, = \,\, - \frac{\partial p}{{\partial x}}l_{ar} bh + jBl_{ar} bh\,\,\,$$

Taking into account that B is the self-magnetic field due to arc current I and assuming that l_ardp/dx = Δp≈p, (i.e. the pressure change occurred at the size l_ar), the equation described the mass m = ρl_arbh acceleration is

$$\,v_{s} \,\frac{\partial m}{{\partial t}}\, + m_{{}} \frac{{\partial (v_{s}^{2} )}}{2\partial x} = - pS_{b} + \frac{{\mu_{0} I^{2} }}{2\pi }$$

(9)

Considering for simplicity steady motion of the arc, the equation for acceleration of total mass can be obtained after sum of Eqs. (8) and (9) in form:

$$\,(m_{b} + m)\frac{{d(v_{s}^{2} )}}{2dx}\, = \,\frac{{\mu_{0} I^{2} }}{2\pi }\,\,\,$$

(10)

Furthermore, let us take in account that action of the dynamic pressure can be characterized as the rate of an increase in density of a flow due to its motion when it is exerted on a braking surface [29]. This density increase due to the dynamic pressure can be described by some effective density n_ef and, consequently, an effective mass m_ef (by analogy with plasma heavy particle density and corresponding plasma mass m_pl at static pressure). As result, in order to describe the body pushing it is obvious that the arc mass m in Eq. (10) can be characterized by some equivalent mass m_ec, which include both m_pl and m_ef. Let us determine the expression for mass m_ec via an equation of state for total plasma pressure as p = n_eck(T + αT_e) used similarly by Hueta & Bounton [26], where n_ec is an equivalent heavy particle density, T is the heavy particle temperature, T_e is the electron temperature in the plasma column, and α is ionization degree. With this expression for p, the modified Eq. (8) is obtained in form:

$$\,m_{b} \frac{{\partial (v_{s}^{2} )}}{2\partial x}\, = \,n_{ec} kT\,\left[ {1 + \alpha \theta_{e} } \right]hb,\,\,\,\,\,\,\,\,\theta = \frac{{T_{e} }}{T}$$

(11)

Taking into account that $m_{ec} = \,m_{a} n_{ec} h\frac{I}{j}$(m_a is the atom weight), it can be derived from Eq. (11):

$$\,\,m_{ec} = \frac{{\partial (v_{s}^{2} )}}{\partial x}\,\frac{{m_{b} m_{a} }}{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]b}}\,\frac{I}{j}$$

(12)

Substitute m_ec from Eq. (12) into Eq. (10), taking into account m = m_ec (according to above definition), after integrating that equation one can obtained:

$$\,m_{b} v_{s}^{2} + \frac{{m_{b} v_{s}^{2} m_{a} }}{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]bx}}\frac{I}{j}v_{s}^{2} \, = \,\frac{{\mu_{0} I^{2} }}{\pi }x\,\,\,$$

(13)

After some simple manipulation, Eq. (13) can be expressed in form:

$$\,v_{s}^{4} + \frac{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]jbx}}{{m_{a} I}}v_{s}^{2} = \frac{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]jbx^{2} }}{{m_{b} m_{a} }}\,\frac{{\mu_{0} I}}{\pi }\,\,\,$$

(14)

Let us represent Eq. (14) for plasma-body velocity in following form:

$$\Psi^{2} + B_{b} \Psi - C_{b} = 0,\,\,\,\,\,\,\,\,\,\,\,\Psi = v_{s}^{2} \,$$

(15)

where

$$B_{b} = \frac{{M_{b} }}{I};\;M_{b} = \frac{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]bxj}}{{m_{a} }}$$

(16)

$$C_{b} = N_{b} I;\;N_{b} = \frac{{2kT\,\left[ {1 + \alpha \theta_{e} } \right]bx^{2} j\mu_{0} }}{{m_{b} m_{a} \pi }}$$

(17)

Finally, the expression for the velocity is:

$$v_{s} = \sqrt { - \frac{{M_{b} }}{2I} + \sqrt {\frac{{M_{{_{b} }}^{2} }}{{4I^{2} }} + N_{b} I} }$$

(18)

The Eqs. (16)–(18) indicate that the body velocity depends on the body mass, rail width b, rail length x = L_R, arc current and plasma parameters.

Brief outline of calculation approach and unknown parameters

The above-described kinetic model for cathode and anode with corresponding equations summarized in Ref. [11] and the Eqs. (1)–(18) may be supplemented by equations for cathode electron emission, for electric field at the cathode surface E, Saha’s system of equations and equations for saturated pressure at the electrode temperatures in order to obtain a complete system of equation [23]. The unknown parameters are: cathode temperature T_c, anode temperature T_a, and plasma temperature T_e, heavy-particle density n₃₀ = n₃/n₀ degree of ionization α, current density j, fraction of electron current s, arc longitudinal dimension l_ar, electrode erosion rates G_c and G_a. in g/C, potential drop in regions at the cathode u_c, at the anode u_an and across plasma column u_pl. The mentioned parameters are calculated as dependence on arc current by assumption that these parameters not changed due to the relatively short time during the plasma column motion at rail length and thus can be used to calculate the arc column velocity v_s from equations of plasma column motion by mathematical model presented in Velocity of accelerating solid body pushed by the plasma motion of the arc in a railgun section.

Results of numerical calculations

The calculations were conducted for Cu rails of length x = L_R = 1,2 and 3 m and mainly for time close to time t of arc motion between the rails (about 100 µs). The attached body mass was used m_b = 1 g and two values of rail width b = 1.2 and 3 cm were used. The plasma parameters were calculated as dependences on arc current I for static (v = 0) in range of I = 0.07–2 MA and moved (v = v_s) arc in range of I = 0.2–2 MA. Table 1 illustrate the characteristic values for static arc and for moved arc, at different rail length. Let us consider the calculated data. The calculation shows that for L_R = 1 m the plasma parameters for static arc mostly do not depend on current, while for moved arc the plasma parameters are varied with arc current, but weakly. This variation is indicated for two limited values I = 0.2 MA and I = 2 MA of the considered range of I (Table 1). The influence of the arc current is enhanced for larger rail length. As example, the results are presented for I = 0.7, 0.8 and 1 MA at L_R = 2 m as well for I = 0.8 and 1 MA at L_R = 3 m.

Table 1 Cu arc plasma and electrode parameters at different arc currents in range of 0.07–2 MA for static arc and in range of 0.2–2 MA for moved arc for rail width b = 0.12 cm and rail length L_R = 1 m. Data for L_R = 2 and 3 m were indicated for 0.7–1 MA. Mass arc m_pl, m_ec are indicated for static and moved arc respectively

In general, the calculations indicate that the arc voltage consists of potential drops in cathode u_c and anode u_an regions as well in the interelectrode plasma u_pl and their sum is about 200 V. The values u_c, and u_an are larger for moved arc in comparison to that calculated for static arc and they decrease with arc current. The plasma potential drop u_pl decrease for moved arc and increase with arc current. These parameters are changed weakly with rail length increasing for moved arc. According to the data of Table 1, the cathode and anode temperatures have comparable values, which are not relatively large and therefore a low vapor density is produced. As result, plasma density n₀ is low in comparison to that in cathode spots arising at cylindrical cathodes. The mass of plasma column m_pl, calculated using relatively low plasma density in form m_pl = n₀n₃₀bl_arh is significantly lower than the body mass m_b. Contrary, the equivalent mass m_ec calculated using Eq. (12) exceeds by few order of magnitude the values m_pl and m_b. It should be noted that relation between m_ec and m_pl could be changed due to instability of the discharge in case when the current distribution is not uniform. However, this case requests additional experiments to study the areas with localize of the current, which is not subject of this work.

The obtained relatively large electron temperature provided significant electron flux returned to the cathode from the adjacent plasma. As result, the returned electron current can exceed the electron emission current from the cathode (possible a negative value of s, not shown in the table). The plasma is highly ionized (degree of atom ionization approached unity) and the most contribution is by presence of one-, two-charge ions for static arc and in addition of three-charge ions for moved arc. The erosion rate of the rails for static arc is both by the cathode mass loss and by the anode mass loss, which is lower by coefficient of twice in comparison to the cathode loss. However, for moved arc the anode mass loss is significantly decrease and can be negligible in comparison with cathode mass loss. In this case, the anode is a collector. While the normalized plasma velocity b₃ at the external boundary of Knudsen layer at the cathode is subsonic, nevertheless it is enough large for static and moved arcs. When the anode is not a collector this parameter is about b_3a = 0.3 at the anode side.

An essential calculated parameter is the longitudinal arc length l_ar. The dependences of l_ar on I for static (rail width b = 1.2 and 3 cm) and moved (b = 1.2 cm) arcs are presented in Fig. 3. It can be seen that l_ar increase with I, and this length is lower than 10 cm for I < 3 × 10⁵A, but l_ar increase very strong for I > 5 × 10⁵A. The length l_ar decreases with b for static arc and for moved arc, but in last case the decrease is stronger in comparison with that for static state. The velocity of the plasma and attached solid body v_s is increased with arc current for the considered here rail length is demonstrated for static and moved arcs at b = 1.2 cm (Fig. 4) and for static arc at b = 3 cm (Fig. 5). These dependences indicate in both figures that the plasma and body velocity increase linearly with I for I < 2 × 10⁵A, but this dependence slightly deviate from linear indicating lower values of the velocity for larger arc currents. At current larger than 5 × 10⁵A this deviation is enhanced and the velocity slightly changed with current, which indicate tendency to a velocity dependence similar to saturation value. The calculated dependences for moved arc are very close to that data obtained for static arc.

Fig. 3

Dependence of longitudinal arc length l_ar on arc current I at static and moved arc

Fig. 4

Velocity of the arc column with attached body (mass of 1 g) as dependence on arc current for different rail length at b = 1.2 cm at static and moved arc

Fig. 5

Velocity of the arc column with attached body (mass of 1 g) as dependence on arc current for different rail length at b = 3 cm at static arc

Discussion

The above-considered physical model and the numerical analysis, conducted according to the developed mathematical approach, show that the phenomena in the plasma column and at the electrode rails of the arc in railgun are coupled and influence on acceleration character of the plasma and the attached dielectric body during their moving along the rails. The important parameter is the current density that determines the intensity of the energy dissipation in the arc sufficient for the intense electrode evaporation, vapor ionization and therefore for supporting the arc operation. The presented self-consistent calculations show relatively low current density of j ~ 10⁴A/cm² weakly changed in wide range of arc currents (Table 1) and relatively high longitudinal size (l_ar, Fig. 3), at which the current distributed along the rails length. So, in difference to the low current vacuum arc with cylindrical cathode, the mentioned low value of j is obtained due to a relatively large electrode area S occupied by the discharge in the railgun. This area S increases due to increase of the arc column size in longitude direction of the rails (l_ar grow) with arc current in a dependence, which supported weakly variation of j.

The obtained low current density produce relatively low heat flux to the electrodes and as result, low electrode temperature (~ 3000 K, Table 1). This explains relatively small vapor density well agreed with Baker's also low values measurements [15] and the low electron current density emitted from the cathode manly due to thermionic emission enhanced by Schottky effect in comparison to the ion current. Therefore, the main contribution in the cathode total current is mainly due to ion current. Note that the mentioned parameters are small is intended in comparison to that obtained for spots appeared in vacuum arc with cylindrical electrodes. Nevertheless, the ability of self-consistent processes generation of the plasma near electrode in railgun possible to support existing of the arc and the current continuity between the rails because the vapor density is highly ionized. This highly ionization state is supported by relatively high electron temperature T_e. Although that the cathode electron emission is relatively small, Joule energy dissipation due to high sum voltage at the electrodes and in the plasma gap (Table 1) provide the high value of T_e. This mechanism of significant plasma electron heating in railgun is also different from that in cathode spot at cylindrical cathode, in which the energetic beam of electron emission from the cathode significantly heats the plasma electrons [11, 23].

It should be noted that the arc parameters, j, T_c, T_e, v_s, l_ar, u_pl calculated as dependences on arc current are obtained in range, which agrees with measured and calculated data reported previously. For comparison, some published data were discussed (in particular, limitation of the measured velocity and others) in Introduction. In addition, it can be pointed out that previous works [17, 30,31,32,33,34,35,36,37] studied at current 10⁵-10⁶A, reported that the voltage between the electrodes is constant in the dependence on arc current, being 150–300 V, plasma size in the propagation direction ~ 10 cm and arc transverse dimension about 1 cm, equal to rail width. It has been found that the plasma temperature was in range of present calculations and that the current is observed to distribute itself from the front towards the end face of the plasma armature. It can be seen that the mentioned data are similar to that calculated in present work (Table 1).

An essential result of the present calculation is the velocity v_s dependence of the arc plasma with attached solid body on arc current showing saturation. Such dependence is obtained for considered here range of rail length as well for static and moved arcs and, in general, it is agreed with a number of experiments. Namely, the measurements show that although the discharge current significant change in wide region up to ~ 1MA, nevertheless the observed plasma and attached body velocity remain limited and mainly was in range of (5–10) × 10⁵ cm/s (see Introduction, [12]). Let us consider a possible reason for such behavior (Figs. 4 and 5).

The numerical analysis show that limitation of the velocity cannot be caused by the electrode ablation (indicated as possible mechanism by Parker et al. [16]) because the column mass of the discharge is significantly lower than the body mass, even at significant grow of the arc current. This result was obtained by calculation of the electrode erosion rate from mathematical model self-consistently described the plasma-electrode phenomena in static and moving arc. On the other hand, the static pressure, which is due to the potential energy of a pressurized flow, is relative small to sufficiently influence on deceleration of the body motion. However, the dynamic pressure (head pressure), which is due to the kinetic energy of the moving flow, can be considered as mechanism, which could significantly influence on the plasma-body velocity limitation when this velocity is sufficiently high. Let us consider how can be understood the influence of the dynamic pressure.

When the arc plasma column pushed, the dielectric body a drag force appears that is directed against velocity of the movement and a value of this force is proportional to the body area S, to the plasma density ρ and to the square of the speed, i.e. proportional to the dynamic pressure. Physics of this effect is expressed by the calculated relative large mass m_eq, introduced as equivalent characteristic of influence of the dynamic pressure using the equation of state of the pressure. As the mass m_eq is larger than m_b by a few order of magnitude (Table 1), the dynamic pressure action occurred as reaction to the ponderomotive force, could be considered as a possible mechanism limited the velocity of plasma-with attached body, when the arc current increases, i.e. significantly rise of the input energy in the discharge. This result allows understanding the measured values of the limiting velocity by the weak dependence of the calculated velocity on arc current. Namely, according to the theory (Eq. 18) the velocity is proportional to 4-th root of arc current (i.e. weakly) after some certain it value of I, while the calculated arc voltage change slightly.

Finally, it can be emphasized that recent publications [38,39,40,41,42], indicate the advantage of the speed and overall cost of the electromagnetic railguns. Nevertheless, the lifetime of the rails has become a bottleneck restricting the development of electromagnetic railgun technology. It is noted that the rails should withstand huge thermal and arc action during the operation to meet the increasing requirement of highly technologies and facilities for the development of electromagnetic railgun.

Conclusions

Let us summarize the main results obtained according to the physical model and a mathematical approach developed here to describe the self-consistently phenomena of a mega-ampere arc, and acceleration of the arc plasma pushing dielectric body in a railgun. The main feature of the calculating model is ability to determine the arc column parameters by considering the rail vaporization (ablation), vapor ionization and the character of the plasma generation. The numerical analysis show:

• 1. The temperature of the electrode rails determined mainly due to ion energy flux to the cathode is T ~ 3000 K and by electron energy flux to the anode is T = 2600–3000 K. The electron temperature occurs in range of T_e = 2.5–3.5 eV and determined mainly by the Joule energy dissipation in the plasma column. The plasma voltage is in range 120–150 V, while the potential drop in the space charge layer is 20–30 V at the cathode and 15–20 V at the anode region.

• 2. The velocity of the plasma and attached solid body v_s is increased with arc current I for static and moved arcs with linear dependence for I < 2 × 10⁵A, and for larger I the velocity slightly deviate from linear dependence. At current larger than 5 × 10⁵A this deviation is enhanced, so that the velocity weakly increased with I (up to 2 × 10⁶A) to values from 5 × 10⁵ to10 × 10⁵ cm/s when size L_R increase from 1 to 3 m respectively showing to be similar to limited velocity observed experimentally.

• 3. While the arc plasma is accelerated in the railgun due to ponderomotive force, the solid body motion is limited by the presence dynamic pressure of high speed plasma column, which can be considered as a possible mechanism explaining the measured in railguns saturating dependence of the velocity on I in range of significant arc currents.

• 4. Despite simplifications used in this model, a good agreement was obtained between calculated parameters (T_e, T, v_s, arc voltage) and published data measured for similar rail geometry and range of arc currents. The measured limiting velocity good agree with the calculated plasma-solid body velocity, which is explained by the theoretical dependence on arc current, namely, the velocity is proportional to 4-th root of arc current, i.e. weakly dependence at large currents.