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Addressing optimal underwater electrical explosion of a wire

A. Virozub, V. Tz. Gurovich, D. Yanuka, O. Antonov, and Ya. E. KrasikPhysics Department, Technion, Haifa 32000, Israel

(Received 24 May 2016; accepted 6 September 2016; published online 22 September 2016)

The underwater electrical explosion of a wire in the timescale 107106 s is characterized by

different phase transitions at extreme values of deposited energy density, allowing one to obtain

warm dense matter using rather moderate pulse power generators. In order to achieve maximal

energy density deposition, the parameters of the wire and the pulse generator should be optimized

to realize an overdamped explosion where most of the initially stored energy is delivered to the

exploding wire during a time comparable with the quarter of the discharge period. In this paper, the

results of 1D magneto-hydrodynamic modeling, coupled with the copper and water equations of

state, of the underwater electrical explosion of Cu wires having an identical length and average

current density but different discharge current rise time are analyzed and compared with those of a

simplified model of a conductivity wave, the propagation velocity of which determines the mode of

the wires explosion. In addition, it is shown that in the case of extreme high current densities, a

scaling from a single wire to a wire array having the same total cross-sectional area of wires cannot

be used to obtain an optimal wire explosion. Published by AIP Publishing.[]


The underwater electrical explosion of a wire allows

one to achieve an extremely high energy density deposition

(>200 eV/atom) during 106 s.1 Such high energy densitydeposition becomes available because of the low compress-

ibility of water, resulting in a rather slow wire radial expan-

sion velocity 105 cm/s, and because of the high value of thebreakdown electric field in water (300 kV/cm), preventingthe formation of a shunting plasma channel along the wires

surface, typical for wire explosions in vacuum or gas.

In order to achieve high energy density deposition, a wire

explosion characterized by an overdamped discharge, i.e.,

when the damping parameter 1 0:5RffiffiffiffiffiffiffiffiffiC=L

p 1, should be

realized. Here, R(t) is the time-dependent resistance of theexploding wire and C and L(t) are the capacitance and induc-tance of the discharge circuit, respectively. In this case, a sig-

nificant (>60%) part of the energy primarily stored in thecapacitor bank is transferred to the exploding wire during a

time shorter than the discharges half of a period. To obtain an

overdamped discharge, one can apply a simplified model of

the wires explosion and use similarity parameters2 to define

the parameters of the electrical circuit, namely, capacitance,

inductance, and charging voltage, as well as the parameters of

the wire, i.e., radius, length, and material. This model considers

uniform wire evaporation and conductivity of the wire propor-

tional to the deposited energy density. However, wire electrical

explosion is a rather complicated phenomenon, characterized

by phase transitions (solid stateliquidvaporplasma) and

radial expansion of the wire due to pressure gradients. These

processes lead to fast changes in the wires electrical and ther-

mal conductivities, density, pressure, and temperature, which

cannot be considered to be independent of each other. In recent

years, a self-consistent magneto-hydrodynamic (MHD) model-

ing, coupled with Ohms law, electrical circuit equations,

equations of state (EOS), and electrical conductivity models of

an underwater wire explosion was conducted in a broad range

of wire and current pulse parameters.3 This modeling allows

one to obtain the resistive voltage and discharge current wave-

forms in the RLC circuit (here R is the total resistance of thedischarge circuit) and, consequently, the energy density deposi-

tion into the wire for a known charging voltage of the capacitor

bank. In experiments with Cu wire underwater electrical explo-

sions, the wires resistance was changed from its resistance of

102 X to several Ohms during 107106 s, depending onthe pulse generators parameters.

In the case of an optimal wire explosion characterized by

an overdamped mode of the electrical discharge, the discharge

current reaches its maximal amplitude of 0.8Isc, where Isc isthe amplitude of the current in the case of a pure inductive

load.1 During this rise time of the current, the wire experien-

ces fast Joule heating accompanied by solid state-liquid and

partial liquid-vapor phase transitions characterized by a signif-

icant increase in the wires total resistance. Typically, in the

case of an optimal wire explosion, 30% of the initiallystored energy W0 in the capacitor banks is deposited into thewire during this discharge phase. Later in the discharge, a

much faster increase in the wires resistance is realized, lead-

ing to a fast decrease in the discharge current and generation

of voltage along the exploding wire e ItRwt LwtdIt=dt ItdLwt=dt, where Lw(t) is the time-dependentinductance of the wire due to its radial expansion and I(t) isthe discharge current. During this phase of the discharge, a

low-ionized, non-ideal plasma is formed as a result of partial

ionization of the wire vapors. The wires maximal resistance

is realized close to the peak of the generated voltage, and later

in the discharge one obtains a decrease in the wires resis-

tance. However, this decrease in the resistance should not be

sufficient to transfer the discharge into an under-damped


1070-664X/2016/23(9)/092708/7/$30.00 Published by AIP Publishing.23, 092708-1

PHYSICS OF PLASMAS 23, 092708 (2016)

The wires cross section S, necessary to obtain an elec-trical explosion at 0.8Isc, can be estimated using the actionintegral g as S2 g1


I2tdt having tabulated data fordifferent metals.4 Considering this value of S and 0.3W0energy deposited into the wire to achieve its evaporation

within the time interval of texp, one can estimate the wireslength. These calculations do not account for a fast change in

the wires resistance during the heating and evaporation of

the wire and can be considered only as rough estimates of

the wire and electrical circuit parameters necessary to obtain

an optimal wire explosion.


MHD modeling shows that, for the same average current

density, electrical explosions of a single wire with a cross-

sectional area S0 and an array of wires having the samelength and total cross section as the single wire result in dif-

ferent time-dependent resistances and, consequently, energy

density depositions. Let us consider the underwater electrical

explosions of a 10 mm in diameter array of 40 Cu wires each

of which is 40 mm in length and 100 lm in diameter with areturn current path having a diameter of 80 mm, and of a sin-

gle Cu wire having the same length but a 632 lm diameter,so that it has the same cross-sectional area as the array wires.

The pulse generator has a capacitor C 10 lF charged up to28 kV, total inductance of 75 nH, and load inductance of39 nH. Here, for simplicity the inductances of the singlewire and the array of wires are considered to be equal. The

1D MHD modeling was the same as that explained in detail

in Ref. 5; the SESAME EOS6 was used for copper and water

and the BKL7,8 model was applied for electrical conductivity

calculations. The waveforms of the discharge current and

resistive voltage and temporal evolution of the deposited

power and resistance (the latter is calculated based on spatial

integration of the data of the conductivitys radial distribu-

tion and assuming axial and azimuthal uniformities) for the

single wire and the wire array are shown in Figs. 1(a) and

1(b). One can see that, despite the fact that the maximal

amplitude of the discharge current in both cases is almost the

same, the rate of the current decrease, dI/dt, and, conse-quently, the amplitude of the resistive component of the

voltage (48 kV), maximal deposited power (10 GW), andmaximal resistance (0.25 X) are significantly larger for thewire array than for the single 632 lm diameter wire. Here,let us note that the resistance of a single 100 lm diameterwire in the array reaches 10 X.

To understand the differences in the deposited power

during these explosions, let us analyze the parameters of

the wires that are realized at different discharge times. In

Table I, the average radial values of expansion velocity, den-

sity, and radii of the thin (i.e., a single wire of the wire array)

and thick (632 lm in diameter) wires at the maximal ampli-tude of the discharge current, resistive voltage, and resis-

tance are presented. Here, n0 is the density of Cu at normalconditions. One can see that the average expansion velocity

of the thin wire is smaller, but the decrease in density is

larger than that obtained for the thick wire.

The radial distributions of temperature and conductivity

that are realized at t


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