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4.2. 1 : A basic approach to modelling the problem -a first order model 4.2: Simple analytical models

4.2. 1 : A basic approach to modelling the problem - a first order model

A simple analytical model is required for the preliminary analysis of the plasma accelerator. Because the voltage trace from the experimental results in sections 3.3.4 to 3.3.6 of chapter 3 shows the electrical circuit as being underdamped, the derived analytical model is only required to reflect this behaviour.

A full nonsteady state solution to Maxwell ' s equations (discussed in section 4.3 .2) cannot be made, but a simple "lump sum" analytical model based on a few assumptions can be derived. Most of the analytical models that have been derived in the literature were created during the 1 960 to 1 970 period. Even though both the plasma block [9] and the snowplough [ 1 0] models have been considered, the majority of the models that have been derived are mainly concerned with the properties of the circuit rather than the motion of the plasma and are often l imited to specific ranges; however in ( 1 1 ] it was the motion of the plasma that was of prime interest.

In [ 1 1 ], which is a one dimensional model, a filament current was assumed to be accelerated by the action of the magnetic pressure upon it, accreting mass as it was accelerated down the length of the plasma accelerator in the snowplough mode of operation (as discussed in section 1 .2 of chapter I ). In this model the equations of motion were derived by solving the circuit equation for charge q , substituting this result into the equation of force and then integrating the resulting acceleration equation with respect to both volume and time. Using this model as a basis, a one dimensional plasma block model for an underdamped system can be derived. The first step involves solving the circuit equation for charge:

.. R ·

₁

q+_o q+--q =O _{Lo LoC}

(4. 1 )

In equation (4. 1 ) R

o

is the resistance,

Lo

is the inductance and

C

is the capacitance of the circuit. The underdamped solution to equation (4. 1 ) is given as:

q =

Q) -:�:)cos(ru)+ Q(�)) -:�:) Sin(nt)

2

nLo

(4.2)

In equation (4.2)

Q

is the initial charge on the capacitor and Q is defined by equation (4.3).

4.2. 1 : A basic approach to modelling the problem -a first order model

The current i is defined as:

.

aq

.

l = --= - q

at

Substituting equation (4.2) into equation (4.3) results in :

(4.3)

(4.4)

(4.5)

The force F on the plasma sheath in [ 1 1 ] was derived from the magnetic pressure on a filament current (refer to section l . l ), as shown in equation (4.6) below (where ri and ro are the inner

and outer radii of the coaxial electrodes).

(4.6)

Because the plasma block mode of operation is being considered the mass of the plasma m can be assumed to be a constant property, it is at this point that the derivation (of the simple first order plasma block model) departs from that described in [ 1 1 ]; the equation of motion in the z

direction (the positive direction of motion along the electrodes) for the plasma block mode of operation is simply derived from Newton's law as:

F = m z

The acceleration z is then derived by combining equations (4.6) and (4.7) to obtain :

.. f '

2

z = - q m

(4.7)

4.2. 1 : A basic approach to modelling the problem -a fIrst order model

Integrating this equation with respect to both volume and time, in which

v

0 is defIned as the

initial velocity and the terms

d

and D are constants given by equations (4.9) and (4. 1 0), results in the velocity (of the plasma mass)

v(t)

being defIned by equation (4. 1 1 ).

Q2 2 Q 2

R

4 D = Q2 Q 2 + Ro + 0 2Lo 2 1 6Q 2 Lo 4 �

v(t) =

Vo

+ �

-

e-dl +

20Sin2nt) -

402

(4.9) (4. 1 0) (4. 1 1 )

integrating this velocity equation with respect to time produces equation (4. 1 2), which is the equation for the distance travelled by the plasma block

z(t)

as a function of time.

[(

1

d

J

1 I

de-dl

I

- -

₂

t + -2 e-dl - -2 +

(

(2QSin20t -dCos20t)

( )

jD

2d 2 d + 40

2d

2d

2 d2 + 402

z t = v t + -

o m

d2

Oe-dl

₂₀₂

+ (

) + (

) (dSin20t + 20Cos20t) - (

)

2 d2 + 402

d2 + 402

d2 + 402

(4. 1 2)

The derived plasma block model has a terminal velocity, which is realised when the value for time

t

approaches infinity. The terminal velocity and distance for the plasma block model are then derived as equations (4. 1 3 ) and (4. 1 4).

(4. 1 3 )

(4. 1 4)

Equation (4. 1 3 ) gives an upper limit for the velocity, which would result if the acceleration distance was infInite. The derived plasma block model has inherent limitations due to the simplifIcations made in its derivation. It does not have the correct current and magnetic fIeld distribution across the plasma, magnetohydrodynamic effects resulting from the compression of

4.2. 1 : A basic approach to modelling the problem - a fIrst order model

the plasma have been neglected and the effects of ablation have not been considered. These simplifIcations mean that the velocity predicted by this model will probably be higher than the experimentally obtained values.

The analytical plasma block model is further limited by the assumption that the circuit has constant resistance and inductance. These values are time dependent quantities, which are dependent upon the position of the plasma block within the plasma accelerator, especially the resistance of the plasma as it exits the plasma accelerator. The resistance of the plasma will increase as the plasma exits the accelerator, because there will be less surface area (and conducting mass) to conduct across.

Because of the above limitations, and the perceived difficulty of analytically deriving a model that takes into account these limitations, a numerical approach is required. A Runge-Kutta solver is typically used to solve the simultaneous equations of (4. 1 ), (4.4), (4.8) and the velocity equation (4. 1 5) [4] to obtain a simple fIrst order one dimensional numerical model.

Oz

V =-

at

(4. 1 5)

Using a fIfth order Runge-Kutta scheme [78], the numerical procedure begins with specifying the initial conditions for charge, current, velocity and distance travelled by the plasma block. Then the system of equations is solved for a given time step, the results are extracted and the initial conditions are reset using the newly calculated values. The process is repeated until the plasma has left the plasma accelerator. Most mathematical package programs such as MATLAB contain a variety of Runge-Kutta solvers. Using the above procedure with a program written in MATLAB equations (4. 1 1 ) and (4. 1 2) were confIrmed as being the correct analytical answers. Ablation is likely to be the biggest factor affecting the performance of the plasma accelerator, rather than the time dependent circuit parameters of resistance and inductance. A simple ablation model was proposed in [40] and [ 1 04], where the ablated mass from the electrodes is assumed to be equal to the energy flowing in these conductors multiplied by an ablation constant a . The ablation constant is a material dependent property and is equal to the reciprocal of the energy required to ionize one gram of material. S ince the dominant factor in determining the ablation constant will be proportional to the average atomic weight of the material being ablated, the ablation constant in [40] and [1 04] was approximated by the following equation:

3Mr

a = --

4.2. 1 : A basic approach to modelling the problem -a first order model

In equation (4. 1 6) Mr is the average atomic weight and

a

has the units of g/MJ. The ablated

mass ma removed from the electrodes as a function of time (assuming this proposed

mechanism) given in equation (4. 1 7) was derived by considering the electrical power dissipated in the electrodes at any given instant.

ama

1 2 R

1 2 R

-- = _at

a!

electrode! + a2

electrode2

( 4. 1 7)

In equation (4. 1 7) subscripts 1 and 2 relate to the positive and negative electrodes of the

accelerator and Relectrode is the resistance of an electrode. This resistance will be taken as a fixed

value (for preliminary purposes) calculated over the entire length of the plasma accelerator. Rewriting equation (4. 1 7) in terms of charge, as equation (4. 1 8), allows it to be directly used in the one dimensional numerical model.

oma

· 2 R

· 2 R

8t

=

a ! q electrode! + a2q electrode 2

(4. 1 8)

The effect of this ablated mass will be considered in terms of the worst case scenario, in which

the ablated mass is directly incorporated into the plasma mass mp [40]; hence equation (4.8) is modified to

(4. 1 9)

The effects of arcing/plasma restrike [4 1 ] , [45] are beyond the scope of the proposed model, because a detailed ionlbreakdown model would be required to model such effects.

4.2.2: Results of the first order model

The Lica was modelled using both the analytical plasma block and the one dimensional ablation plasma block model, in an attempt to explain the experimental results. In both models the resistance and inductance were calculated using the known dimensions of the plasma accelerator and its associated components (mentioned in section 3.3. 1 ), including the material properties (at room temperature) in which the skin effect was taken into account [96]. The

4.2.2: Results of the first order model

resistance

Ra

and inductance

La

of the plasma accelerator were calculated as constant values

(using the equations for a coaxial cable given in [82], [95]), taken over the entire plasma acceleration distance. The resistance of the plasma was calculated in accordance to the method described in section 3.2.2, in which the conductivity profile calculated in section 3.3.8 at 5000K (graph 3. 1 8) was used in the calculation. The acceleration distance mentioned in tables 4. 1 and 4.2 relates to the distance over which the plasma block was accelerated over. The calculations in both models were conducted until the rear of the plasma block had reached this distance, effectively leaving the accelerator.

The results from the analytical plasma block model (equations (4. 1 1 ) and (4. 1 2)) are presented in table 4. 1 , together with the parameters used to generate them. The calculated velocity and voltage trace for the acceleration distance of O.05m are shown in graphs 4. 1 and 4.2 respectively.

Acceleration Mass: mg Final Ro: n Lo: nH C: uF Charging Time period

distance: m velocity: voltage: V of the

km/s acceleration:

us

0.05 0.0001 4 456 0.0086 64.6 1 7.5 1 0000 0.42

0. 1 0.0001 4 705 0.0087 76.0 1 7.5 1 0000 0.55

0.2 0.0001 4 1 027 0.0089 98.8 1 7.5 1 0000 0.75

Table

4. 1 :

Results from the analytical plasma block model, obtained using the stated

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