**4.1 Metallic foil electrically exploding driving highvelocity flyers**

The code used to simulate the electrical explosion of metallic foil is improved based our SSS code[32], which is one dimensional hydrodynamic difference code based on Lagrange orthogonal coordinate. For the case of electrical explosion of metallic foil, the power of Joule

Magnetohydrodynamics of Metallic Foil Electrical

foil at different phase states. For solid state, there is

materials ,F()=2-1. For liquid state, there is

Explosion and Magnetically Driven Quasi-Isentropic Compression 361

forward by T.J. Burgess[33]. The Burgess's model can describe the electrical resistivity of the

3

*<sup>V</sup> C CT*

In equation (11), C1, C2 and C3 are fitting constants, is Gruneisen coefficient, for many

*L s <sup>T</sup> <sup>m</sup>*

For gas state, the electrical resistivity is related with both the impact between electrons and

9

0 11

(1 ) /( / )

*C V*

*X X*

Table 4 gives the parameters values of Burgess's model for Aluminum, which is used in our

*c V*

 

*C X m VV*

( )

*<sup>C</sup> m VV e*

1

*X X*

*mixed*

 

*c l*

*V C*

(1 )

In equation (13), i is the ionization fraction, C5, C6 , C7, C8 and C9 are fitting constants. In fact, there is mixed phase zone between liquid and gas states, a mass fraction m is defined. When m=0, all mass is condensed, and m=1, all mass is gas, and 0<m<1, the mass is

*C e VT*

/ 8 1/2 3/2

[1 ]

 

*<sup>C</sup> C VT*

*i C T*

1/2 1

7

*T C T*

 

*v ei en*

5 3/2 6

[1 ln(1 )

10 <sup>12</sup> /

0

1

*C T*

 

1 2

*<sup>C</sup> <sup>s</sup>*

latent heat, *T*m is melting point temperature and C4 is fitting constant.

*ei*

*i*

mixture states. Two mixture variants are also defined besides mass fraction.

*c*

 

The electrical resistivity of mixed phase zone can be expressed

In equation (12), for many materials, 0.069 / *L T F m*

ions and between electrons and neutrons. so,

Where C10, C11 and C12 are fitting constants.

experiments.

( )

( )

(11)

(12)

(13)

(15)

(16)

*ke* , k is a constant, *L*F is the melting

*F*

*C*

0

*V*

 <sup>4</sup> *m*

*T T*

heating is increase into the energy equation, and the magnetic pressure part is considered. In order to calculate the power of Joule heating and magnetic pressure, the discharging current history is needed which is detemined by the electric circuit equation (2) and equation (3). The resistance of foil varies from different phase states during dicharging process, so a precisionly electrical resistivity model is needed to decribe this change. The physical model is seen in Figure 1, and the Lagrange hydrodynamic equations are:

$$\begin{cases} V = \frac{\partial X}{\partial M} \\ \frac{\partial \mathcal{U}}{\partial t} = -\frac{\partial \sigma}{\partial M} + f\_{EM} \\ \frac{\partial E}{\partial t} = -\frac{\partial (\sigma \mathcal{U})}{\partial M} + \frac{\partial (\Delta P)}{\partial M} + \lambda \frac{\partial}{\partial M} \left(\frac{\partial T}{\partial X}\right) \\ \Delta P = I^2 R\_{foli} \\ f\_{EM} = \overline{f}(X) \times \overline{B}(X) / M \end{cases} \tag{9}$$

Where, *V* is specific volume, *M* is mass, *X* is Lagrange coordinate, *U* is velocity, *T* is temperature, is thermal conductivity, is the total pressure and *=p+q*, *p* is heating pressure, *q* is artifical viscosity pressure, *f*EM is magnetic pressure per mass, *E* is total specific energy and *E*=*e*+0.5*U*2, *e* is specific internal energy, *P* is power of Joule heating, *B* is magnetic flux density, is vacuum permeability, *k* is shape factor and *k*=0.65, *R*foil is resistance of metallic foil and I is the current flowing through metallic foil in the circuit, which can be expressed with equation (10).

$$\begin{cases} \frac{1}{\mathbf{C}\_{0}} \int\_{0}^{l} I(t)dt + RI + L\frac{dI}{dt} = Vol\_{0} \\ L = L\_{s} + L\_{d} \\ R = R\_{s} + R\_{foli} \\ L\_{d} = L(h + \Delta h) - L(h) = \frac{\mu l}{2\pi} \ln \frac{h + 1.23b}{h + \Delta h + 1.23b} \\ R\_{foli}(t) = \frac{l}{b} \cdot \frac{1}{\int\_{0}^{l} \int\_{0}^{l} \rho(X, t)dX} \end{cases} \tag{10}$$

In the equation (10), *C*0 is the capacitance of the experimental device, *L* is the total inductance of the circuit, *L*s is the fixed inductance of the circuit, *L*d is the variable inductance of the expansion of metallic foil caused by electrical explosion, *R* is the total resistance of the circuit, and *R*s is the fixed resistance and *R*foil is the dynamic resistance of the foil caused by electrical explosion, *b,h* and l is the width, thickness and length of the foil, is the electrical resistivity, which is variable and can be expressed by the model put forward by T.J. Burgess[33]. The Burgess's model can describe the electrical resistivity of the foil at different phase states.

For solid state, there is

360 Hydrodynamics – Advanced Topics

heating is increase into the energy equation, and the magnetic pressure part is considered. In order to calculate the power of Joule heating and magnetic pressure, the discharging current history is needed which is detemined by the electric circuit equation (2) and equation (3). The resistance of foil varies from different phase states during dicharging process, so a precisionly electrical resistivity model is needed to decribe this change. The

physical model is seen in Figure 1, and the Lagrange hydrodynamic equations are:

( ) ()

*EM*

*E UP T t M M MX*

Where, *V* is specific volume, *M* is mass, *X* is Lagrange coordinate, *U* is velocity, *T* is

pressure, *q* is artifical viscosity pressure, *f*EM is magnetic pressure per mass, *E* is total specific energy and *E*=*e*+0.5*U*2, *e* is specific internal energy, *P* is power of Joule heating, *B* is

resistance of metallic foil and I is the current flowing through metallic foil in the circuit,

0

1.23 ( ) ( ) ln <sup>2</sup> 1.23

is the total pressure and

is vacuum permeability, *k* is shape factor and *k*=0.65, *R*foil is

*hh b*

(9)

*=p+q*, *p* is heating

(10)

( ) ( )/

2

*P IR*

*EM*

 

0 0

 

*d*

  <sup>1</sup> ( )

*LL L RR R*

*s d s foil*

*C dt*

*foil h*

*<sup>l</sup> R t*

*t*

0

1

*dI I t dt RI L Vol*

( ,)

*dX X t*

In the equation (10), *C*0 is the capacitance of the experimental device, *L* is the total inductance of the circuit, *L*s is the fixed inductance of the circuit, *L*d is the variable inductance of the expansion of metallic foil caused by electrical explosion, *R* is the total resistance of the circuit, and *R*s is the fixed resistance and *R*foil is the dynamic resistance of the foil caused by electrical explosion, *b,h* and l is the width, thickness and length of the foil,

is the electrical resistivity, which is variable and can be expressed by the model put

*lh b L Lh h Lh*

<sup>1</sup> ( )

*b*

temperature, is thermal conductivity,

which can be expressed with equation (10).

 

magnetic flux density,

 

*<sup>X</sup> <sup>V</sup> M <sup>U</sup> <sup>f</sup> t M*

 

*foil*

*f jX BX M*

$$\eta\_s = \left(\mathbf{C}\_1 + \mathbf{C}\_2 T^{\mathbf{C}\_3}\right) \cdot \left(\frac{V}{V\_0}\right)^{F(\gamma)}\tag{11}$$

In equation (11), C1, C2 and C3 are fitting constants, is Gruneisen coefficient, for many materials ,F()=2-1.

For liquid state, there is

$$
\eta\_L = \Delta \eta \cdot \left(\eta\_s\right)\_{T\_m} \cdot \left(\frac{T}{T\_m}\right)^{C\_4} \tag{12}
$$

In equation (12), for many materials, 0.069 / *L T F m ke* , k is a constant, *L*F is the melting latent heat, *T*m is melting point temperature and C4 is fitting constant.

For gas state, the electrical resistivity is related with both the impact between electrons and ions and between electrons and neutrons. so,

$$\begin{cases} \boldsymbol{\eta}\_{\boldsymbol{v}} = \boldsymbol{\eta}\_{\boldsymbol{e}i} + \boldsymbol{\eta}\_{\boldsymbol{en}} \\ \boldsymbol{\eta}\_{\boldsymbol{e}i} = \frac{\mathbf{C}\_{5}}{T} [1 + \ln(1 + \mathbf{C}\_{6} VT^{3/2}) \\ \boldsymbol{\eta} = \mathbf{C}\_{7} T^{1/2} [1 + \boldsymbol{\alpha}\_{i}^{-1}] \\ \boldsymbol{\mathcal{O}}\_{i} = (1 + \frac{\mathbf{C}\_{8} e^{\mathbf{C}\_{6}/T}}{VT^{3/2}})^{-1/2} \end{cases} \tag{13}$$

In equation (13), i is the ionization fraction, C5, C6 , C7, C8 and C9 are fitting constants. In fact, there is mixed phase zone between liquid and gas states, a mass fraction m is defined. When m=0, all mass is condensed, and m=1, all mass is gas, and 0<m<1, the mass is mixture states. Two mixture variants are also defined besides mass fraction.

$$\begin{cases} m = (V - V\_0) \frac{\mathcal{C}\_{10}}{\mathcal{C}\_{11}} e^{-\mathcal{C}\_{12}/T} \\ X\_c = (1 - m) / (V / V\_0) \\ X\_V = 1 - X\_{\mathcal{C}} \end{cases} \tag{15}$$

Where C10, C11 and C12 are fitting constants.

The electrical resistivity of mixed phase zone can be expressed

$$\begin{cases} \eta\_{\text{mixed}} = \left( \frac{X\_{\text{C}}}{\eta\_{c}} + \frac{X\_{V}}{\eta\_{V}} \right)^{-1} \\ \eta\_{c} = \eta\_{l} \end{cases} \tag{16}$$

Table 4 gives the parameters values of Burgess's model for Aluminum, which is used in our experiments.

Magnetohydrodynamics of Metallic Foil Electrical

Explosion and Magnetically Driven Quasi-Isentropic Compression 363

Fig. 14. The calculated and experimental results of flyer velocities for different flyer sizes.

Fig. 15. The experimental and calculated results of discharging current.

appropriate to the electrical explosion of metallic foils.

**4.2 Magnetically driven quasi-isentropic compression** 

The results presented in Fig.12 through Fig.15 show that the physical model here is

In order to simplify the problem, the one dimensional model of magnetically driven quasiisentropic compression can be described by the model shown in Fig.16. The changes of


Table 4. The parameters values of Burgess's model for Aluminum

The calculated results are presented in from Fig.12 through Fig.15. In Fig.14 and Fig.15, the experimental and calculated results are compared.

Fig. 12. The calculated pressure and flyer velocity history results of electrical explosion of Aluminum and Copper foils.

Fig. 13. The calculated results of pressure and specific volume of aluminum foil when exploding.


The calculated results are presented in from Fig.12 through Fig.15. In Fig.14 and Fig.15, the

Fig. 12. The calculated pressure and flyer velocity history results of electrical explosion of

Fig. 13. The calculated results of pressure and specific volume of aluminum foil when

LF(Mbarcm3/mole

C1(m-cm) C2 C3 C4 C5 C6 <sup>0</sup>

Table 4. The parameters values of Burgess's model for Aluminum

experimental and calculated results are compared.

Aluminum and Copper foils.

exploding.

Fig. 14. The calculated and experimental results of flyer velocities for different flyer sizes.

Fig. 15. The experimental and calculated results of discharging current.

The results presented in Fig.12 through Fig.15 show that the physical model here is appropriate to the electrical explosion of metallic foils.
