Modelling and Simulation of Burst Phenomenon in Electrically Exploded Foils

Terminal Ballistics Research Laboratory Electro Explosive Devices (EED) Division Project Report Modelling and Simulation of Burst Phenomenon in Electrically Exploded Foils

Abhishek Ghosh P2009ME1074 School of Mechanical, Material and Energy Engineering Indian Institute of Technology Ropar

Abstract Exploding Foil Initiators are used as detonation devices and employ the foil burst phenomenon in their working. The successful detonation of these devices depends on the foil and circuit parameters. A model has been constructed to describe the exploding foil process and has been utilized to write MATLAB code for simulating the process. This code has been verified by comparing it with experimental data points. Using this source code, a graphical interface has been developed for the end user. This GUI computes foil burst parameters based on certain input values and thus helps in determination of Exploding Foil Behaviour.

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Table of Contents 1. Introduction .............................................................................................................................................. 4 1.1 Exploding Foil Initiator ........................................................................................................................ 4 1.2 Burst Phenomenon ............................................................................................................................. 5 1.3 Objectives............................................................................................................................................ 6 2. Model for Exploding Foil ........................................................................................................................... 6 2.1 Assumptions ........................................................................................................................................ 6 2.2 Equations of model ............................................................................................................................. 7 3. Calculation of flyer velocity ....................................................................................................................... 9 4. MATLAB code .......................................................................................................................................... 10 4.1 Initialization....................................................................................................................................... 10 4.2 Function Files .................................................................................................................................... 11 4.2.1 Calculation of Flyer Velocity (flyerv) .......................................................................................... 11 4.2.2 Calculation of dynamic resistivity (res) ...................................................................................... 11 4.2.3 Calculation of specific heat (specificheat) ................................................................................. 11 4.2.4 Calculation of specific heat differential (diffheat) ..................................................................... 13 4.2.5 Functional form of equations of state (burst) ........................................................................... 14 4.2.6 Determination of the burst time (bursttime) ............................................................................ 15 4.2.8 Calculation of action integral (localaction) ................................................................................ 16 4.3 GUI Development.............................................................................................................................. 17 4.4 Script Files ......................................................................................................................................... 19 4.4.1 Solver script................................................................................................................................ 19 4.4.2 Plotting Graphs .......................................................................................................................... 20 5. Comparison with experimental data ...................................................................................................... 21 5.1 Copper Foil ........................................................................................................................................ 21 5.2 Gold Foil ............................................................................................................................................ 24 5.3 Aluminium Foil .................................................................................................................................. 25 6. Results and Discussion ............................................................................................................................ 28 7. Acknowledgements................................................................................................................................. 28 8. References .............................................................................................................................................. 29 9. Appendices .............................................................................................................................................. 30 2

9.1 List of Physical Constants used ......................................................................................................... 30 9.2 Table of Figures ................................................................................................................................. 31 9.3 List of Tables ..................................................................................................................................... 31

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1. Introduction 1.1 Exploding Foil Initiator An exploding foil initiator (EFI) also called slapper detonator, is a relatively recent kind of a detonator developed in Lawrence Livermore National Laboratory (1). Apart from its usage as a detonation device, this setup is also used for ultrahigh pressure research. It is an improvement of the earlier explodingbridge wire detonator; instead of directly coupling the shock wave from the exploding wire, the expanding plasma from an explosion of a metal foil drives another thin plastic or metal foil called a "flyer" or a "slapper" across a gap, and its high-velocity impact on the explosive (for example, PETN or hexanitrostilbene) then delivers the energy and shock needed to initiate a detonation. Normally all the slapper's kinetic energy is supplied only by the heating (and hence expansion) of the plasma (the former foil) by the current passing through it, though constructions with a "back strap" to further drive the plasma forward by magnetic field exist too. This assembly is quite efficient; up to 30% of the electrical energy can be converted to the slapper's kinetic energy.

Figure 1: Diagram of Slapper Detonator

The initial explosion is usually caused by explosive vaporization of a thin metal wire or strip, by driving several thousand amperes of electric current through it, usually from a capacitor charged to several thousand volts. The switching is performed by a spark gap. Usually the construction consists of an explosive booster pellet, against which a disk with a hole in the center is set. Over the other side of the disk, there is a layer of an insulating film, for example, Kapton or Mylar film, with a thin strip of metal (typically aluminum or gold) foil deposited on its outer side. A narrowed section of the metal explosively vaporizes when a current pulse passes through it, which shears the Mylar foil and the plasma ball pushes it through the hole, accelerating it to very high speeds (2-4 km/s). The impact then detonates the explosive pellet.

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Figure 2: Flyer impact on primary explosive

Slapper detonator's flyer impacts an area of surface on the explosive output charge, and even though energy is lost to the sides of the area impacted, a cone of explosive is efficiently compressed. Advantages over explosive-bridge wire detonators include (2): 1. The foil does not come in contact with the explosive, which reduces the risk of corrosion of the foil or chemical reactions between the foil and explosive producing unstable compounds, and secondarily further reduces the risk of accidental electrical ignition of the explosive. 2. The energy to fire the detonator is quite low 3. The slapper pellet impacting an area of explosives rather than a single point as in an EBW is more reliable and efficient at initiating detonation. 4. The explosive can be pressed to higher density 5. Very insensitive explosives can be initiated directly. The slapper detonators are frequently used in modern weapon designs and aerospace technology.

1.2 Burst Phenomenon When a large amount of energy is deposited at very fast rate to fine bridge foil, the current heats the bridge through the melting, boiling and vaporisation phases up to the plasma state, giving off thermal energy and shock waves. This phenomenon is called the burst event (3). The resulting shock waves can be used for various applications such as Flyer plate acceleration, high explosive initiation or shock wave physics studies experimentation. In the present case, the burst event is aiding the detonation process in explosive foil initiators (EFI). These detonators are known to function correctly only under very special electrical conditions. The electrical pulse which bursts the bridge foil to accelerate the Flyer plate must have a very short rise time and be capable of delivering enough energy into the bridge to drive the Flyer plate to a velocity high enough to initiate the acceptor explosive on impact. This energy must be delivered in times comparable to the width of the voltage spike at burst, or it is wasted. These times, for small bridge slapper detonators, are typically a few tens of nanoseconds. Such short pulses can only be produced by an electrical circuit with low resistance, and extremely low inductance. Typical values for L, R and C are 300 5

nH, 150 mΩ and 3 µF. The burst time and current are a function of both the foil material properties and the circuit parameters. Determination of these quantities via simulation will aid in the testing of the detonators according to specific requirements.

1.3 Objectives A. To construct a model to describe the exploding foil process. B. Utilise the model to write Matlab code for simulating the process and verify it using experimental data. C. Construction of a graphical problem-solving interface employing the Matlab source code.

2. Model for Exploding Foil The following model was constructed based on the work of previous researchers (4). 2.1 Assumptions I. II. III. IV.

V.

VI.

The volume of the exploding foil is constant up to the burst time. The foil’s specific resistivity changes linearly with temperature up to the exploding temperature (Tb). The ratio of the burst resistance to the initial resistance (Rb/Ro) is characteristic constant for each foil material. The equation of state is based on a semi-empirical model for the internal energy ε = ε (T, V), where V is the specific volume and T is the temperature. The internal energy equation contains three components: the elastic energy, εc (low-temperature component), which was neglected in the numerical simulation, the thermal energy of atoms εt and the electronic contribution εe, especially important at high temperatures. The duration of the explosion process is much shorter than the time of energy dissipation in the foil. Heat conductance, radiation losses, and heat of phase transformation are all neglected. It is assumed that all the electrical energy delivered to the foil was converted to Joule heating in the foil. The metal foil is assumed to be at a uniform temperature at each point of time. The temperature gradients within the foil were neglected.

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2.2 Equations of model

Figure 3: The Equivalent Circuit of the EFI

The equations of the model are as follows:

(1)

( ) ( )

( )

( ( )

(

)

)

( )

( ) (∫ ( )

(

)

(2)

)

(3) (4)

- Heat capacity per unit mass of the metal foil -Current density through the foil’s cross-section -Electric conductivity of the foil material -Mast density of the foil material -The initial and dynamic resistances of the foil -Ambient and dynamic temperatures of the foil -Linear thermal coefficient of resistivity -Energy per unit mass (of the foil)

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Description of equations: (1) (2) (3) (4)

The rate of increase of energy is equal to ohmic heating occurring in the metal foil. The dynamic resistivity of the foil varies linearly with temperature. The current flow in the circuit is described by this equation. Equation of state of the foil material as a function of temperature and the specific volume V.

The energy equation (4) can be written as follows (5):

ε (V, T) = εc(V) + εt(V, T) + εe(V, T)

(5)

The calculation of the thermal energy of atoms is described by

εt = Cv (T).T

(6)

Where Cv (T) is the specific heat capacity and is only a function of T. At low temperatures, where we have six degrees of freedom

εt = 6 X 1/2 NKT = 3NKT

(7)

where K is the Boltzmann constant and N is the number of atoms per unit mass. At high temperatures we have three degrees of freedom and

εt = 3 X 1/2 NKT = 3/2 NKT

(8)

At the intermediate temperatures (semi-liquid phase), the following equation holds (6):

(

)

(

)

(

)

( )

where R is the gas constant, C is the speed of sound in the foil metal at ambient temperature, empirical parameter, is the internal energy at room temperature. The electronic contribution for T < 50,000 K is (5), (6) given by

εe where

( )

(10)

is 1.1 X 10-2 J/kg.K2 .

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is an

3. Calculation of flyer velocity The flyer velocity calculations are based on Gurney model and have been calibrated for the specific case of copper foil and Mylar flyer. The flyer velocity is given as: (

)(

)

(

)

- Flyer velocity, expressed in km/s - Burst current density, expressed in TA/m2 - Flyer mass per unit area - Copper foil mass per unit area Figure 4 plots these values for a particular specification.

2.6

Vf, Flyer Velocity [ km/s ]

2.4

2.2

2

1.8

1.6

1.4

0.5

0.6

0.7

0.8 0.9 1 J, current density [ TA/m2 ]

1.1

1.2

1.3

Figure 4: Flyer velocity vs. burst current density J; Copper foil thickness - 8 micrometer and Mylar flyer thickness - 76 micrometer

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4. MATLAB code Using the model described above, MATLAB code was written to solve the problem. The code consists of various functions which perform individual tasks. The gist of the code can be summarized by explaining the initialization, individual function files and the final script file which utilizes all these to reach the solution.

4.1 Initialization Various parameters were declared which were utilized throughout the program. These include various material properties and physical constants. The details are explained in the comments (statements starting with %). % declaring material properties % in order: current metal pointer, atomic mass, density, specific heat, % resistivity, thermal coeff. Of resistivity, speed of sound and burst % temperature global pointer atmass density spheat resistivity thres velocitys bursttemp; % default pointer : copper pointer=1; % in order of description: copper, gold, aluminium atmass=[63.546 196.96 26.98]; density=[8930 19300 2700]; spheat=[385 129 900]; resistivity =[1.72 2.214 2.82].*1e-8; % temperature averaged thermal coefficients of resistivity thres =[3 3.7 3].*1e-3; velocitys = [3921 2030]; bursttemp= [40000 25000 80000]; % computed by backtracking available data

The data presented here has been obtained using various sources including internet and text. A complete list of all the constants used is presented in the appendices. The burst temperature for different metals has been calculated by another computer code. It utilizes test data to arrive at the burst temperatures.

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4.2 Function Files 4.2.1 Calculation of Flyer Velocity (flyerv) Takes the burst current, foil thickness and flyer thickness as its input and provides the flyer velocity as its output. This calculation is according to Eq. (11) in section 3. function v = flyerv(J,thk) global flyden foithi density pointer; M=flyden*1e-6*thk; C=density(pointer)*foithi; v=((1.765*J)+.918)*(((M/C)+(1/3))^(-1/2)); end

v J thk flyden foithi

%flyer mass per unit volume %foil mass per unit volume

- Flyer velocity [km/s] - Burst current density [TA/m2] - Flyer thickness [µm] - Flyer material density [kg/m3] - Foil thickness [m]

4.2.2 Calculation of dynamic resistivity (res) Takes the temperature as its input and returns the resistivity of the specific metal at that temperature. This calculation is according to Eq. (2) in section 2.2. %resistivity of the foil material with respect to temperature function r = res(T) global pointer resistivity thres; i=resistivity(pointer); % initial resistivity at 20 degree celcius rate = thres(pointer); % thermal coefficient of resistivity r=i*(1 + (rate*(T-20))); end

r

-Resistivity of the foil material at the present temperature *Ωm+

4.2.3 Calculation of specific heat (specificheat) Takes the temperature, atomic mass, specific heat at room temperature and computes the specific heat for thermal energy using Eq. (9) in section 2.2. Using Equation (9), the specific heat at a given temperature T can be written as:

( (

) )

(

(

)

)

Where is constant for a particular material, R is the universal gas constant and M is the molar mass of the particular element. 11

Also, the specific heat at room temperature is a known quantity using which the factor calculated. ( ( where

) )

can be

(

)

(1
If the value of x exceeds these limits (not usual), then a default value can be assumed. This operation is performed by the following function with

~ factor (Eq. 13):

% finds the factor to be used in specificheat computation, in accordance % with the interpolation equation provided by altshuder global factor; R=8.314*1.5*1000; limit=(2*R)./atmass; for i=1:length(spheat) if spheat(i)>limit(i) spheat(i)=limit(i)-1; % in case of exceeding spheat, defaults end end f=atmass.*spheat; r=f./R; factor=((2-r)./(r-1))./300; %for obtaining semi liq phase specificheat

limit

- The boundary value of specific heat for the approximation to work

NOTE: This code finds the factor for all the element types and stores them in the corresponding array index.

Next, the specific heat is calculated using Eq. (12). % provides the thermal specific heat of foil material as a function of % temperature function Cv = specificheat(T) global pointer atmass factor; M=atmass(pointer); a=factor(pointer); t=T+273; %temp. in kelvins Cv=(2+(a*t))/(1+(a*t))*(1.5*8.314*1000/M); end

Cv

- Specific Heat (Thermal Energy)

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4.2.4 Calculation of specific heat differential (diffheat) In order to solve the differential equations in section 2.2, the time derivative of energy is required (Eq. 1) (

)

This can be written as (using Eq. 9 and 10):

( (( (

) )

(

)

)

(

(

)

)) (

=

Where,

)

R

During the burst event, the volume of the foil is assumed to be constant (assumption 1 of section 2.1). This implies V (t) = Vo. Hence the equation can be rewritten as:

((

(

)

(

) (

)

)

)

The first part of Eq. (15) corresponds to the derivative of the thermal energy. This part has been calculated using the below mentioned function. %provides the differential of the specific heat for use in burst function function Cd = diffheat(T) global pointer factor atmass; M=atmass(pointer); t=T+273; %temperature in kelvins R=8.31446; %gas constant a=factor(pointer); %parameter for specificheat calculation Ci=(1.5*R*1000/M); %limiting specific heat coeff=((a*t)+1)^2; Cd=Ci*((coeff+1)/coeff); end

Cd T R factor

- Time differential of the foil’s thermal energy - Temperature of the foil in °C - Universal Gas Constant - Computed earlier (section 4.2.3)

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(15)

4.2.5 Functional form of equations of state (burst) The equations of state (section 2.2) have been converted into a form which is required to solve the problem computationally. Eq. (1), (3) and (4) have been reformed as:

( ) ( ) ( ) ( ) ( )

(

(16) ( ))

(17)

( )

(18)

-Temperature of the metal foil -Current density through the foil’s cross-section -Differential of the Specific Heat of the material (Thermal + Electric Energy) [Eq. 15]

(

(

)

(

) (

)

)

-Electric conductivity of the foil material -Current through the foil -Charge in the capacitor -Circuit parameters -Resistance of the foil These three equations collectively form the set of simultaneous differential equations that need to be solved for reaching the solution. In the below written function, this set has been defined. NOTE: This function utilizes the earlier defined functions. % % % %

simulatneous governing equations dy(1)= Q < heat input per unit mass (Joules/kg)> dy(2)= I < current in the circuit(Amperes)> dy(3)= q < charge in the capacitor (coulumb) >

function dy = burst(t,y) global foiwid foithi foilen circap cirind cirres pointer density; dy= zeros(3,1); A= foiwid*foithi; % cross sectional area of foil l=foilen; % length of the foil den= density(pointer); % density of foil material C= circap; % cpacitor bank L=cirind; % inductance of the circuit R=cirres; % resistance of the circuit B=11e-3; %electronic factor r = res(y(1))*l/A; % dynamic resistance of foil

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%equations of state dy(1) = (y(2)^2)*(res(y(1)))/((A^2)*den*(diffheat(y(1))+(B*(y(1)+273)))); dy(2) = (y(3)/(C*L))-(((r+R)*y(2))/L); dy(3) = -y(2); end

dy

-Representing the differential elements

4.2.6 Determination of the burst time (bursttime) Once the simultaneous equations are solved, the output is in the form of arrays: three output arrays, one each for temperature (T), Current (I) and Capacitor charge (q); and one time array containing the corresponding time entries. These variables are synched in a manner such that the nth time array value corresponds to the nth output array value. Hence, if the 10th time element is 2 sec and the 10th T (temperature) element is 3000 °C, then this implies that the foil reaches 3000 °C at 2 sec interval. This is applicable to all the other output parameters as well. The following function takes as its input the output T (temperature) array and returns the index for the value closest to the burst temperature. % returns the index of the time flow matrix which is closer to the burst % conditions reaching in the circuits function t = bursttime(Tf) global pointer bursttemp; x=1; Tb=bursttemp(pointer); %burst temperature while Tf(x) abs(Tf(x-1)-Tb) t= x-1; else t=x; end end

Tf bursttemp t

- Solved Temperature Array (list of temperatures progressing with time) - Burst temperature of the respective element - Required array index

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4.2.7 Calculation of thermal and electrical energy (te & ee) Takes the temperature as its input and calculates the foils thermal and electric energy using Eq. (9) and (10). %thermal energy per unit mass as a function of temperature function Q=te(T) C=specificheat(T); Q=C*T; end

Q

-Thermal Energy of the foil [Joules]

%electric energy per unit mass as a function of temperature %volume of the foil considered constant throughout the burst event function Q = ee(T) B=11e-3; %electronic factor Q=B*((T+273)^2)/2; %electronic energy end

Q B

-Electric Energy of the foil [Joules] -Parameter (β from Eq. 10)

4.2.8 Calculation of action integral (localaction) The action integral is another parameter linked with the burst phenomenon. The current density J(t) and the burst time t satisfy the action integral (4): ∫

( )

(

This function takes the solved current array and time array as its input and performs the necessary integration to find out the action integral. %calculates the action integral at a specific time as a function of % time of inquiry, Time array and current array function g= localaction(T,If) global foiwid foithi Area= foiwid*foithi; %Cross-sectional area J2=(If./Area).^2; %integrand g= zeros(length(T)); for i=2:length(T) g(i)=trapz(T(1:i),J2(1:i)); %calculating local action array end g(1)=0; end

g T If trapz

- Action integral - Time output array - Current output array - Inbuilt trapezoidal integration function 16

)

4.3 GUI Development

A graphical user interface was developed to showcase the simulation outputs. This interface was prepared using Matlab guide. The developed interface is shown below:

Figure 5: Developed GUI [MATLAB guide]

The interface consists of three sections: Input section This section consists of text boxes and pop-up menus for data input by the user. It consists of (stating in the order of their appearance): a pop-up menu to select the foil material (copper, aluminium and gold); three text boxes to input the foil dimensions namely length, width and thickness; a pop-up menu to select the flyer material (Mylar or Kapton); text box for input of flyer thickness; four text boxes to input the circuit parameters namely capacitance, inductance, resistance and voltage. These variables are set to their defaults when the initial screen appears. Output section This section consists of text boxes representing the simulation outputs namely flyer velocity, burst current and burst time.

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Temporal Behaviour section This section consists of a plotting region which plots the temporal behaviour of output parameters. The various graphs that can be obtained include: current, voltage, foil resistance, energy, foil temperature and local action. In addition to these sections, the form consists of three push buttons Calculate: Displays the output after solving it. Reset: Resets the input parameters to their default values. About: Info about developer. The form also contains three tools for graphical interaction namely cursor data tool, zoom in and zoom out tools. The output form display for a random set of input parameters is shown below:

Figure 6: Interface output example

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4.4 Script Files 4.4.1 Solver script This piece of code executes when the user clicks the calculate button. It utilizes all the functions that have been described earlier (section 4.2). This code solves the simultaneous differential equations and presents the solution in the form of arrays containing the parameter values at each time step. Solving the burst function (section 4.2.5) provides temperature, current and the charge of the capacitor as a function of time. The other required parameters can be computed using these three. global foiwid foithi foilen circap cirvol; global pointer density; global T Tf If qf Vf Rf n mass; pointer=get(handles.popfoil,'val'); %this code solves the simultaneous differential eqauations to find the %current density as a function of time untill the burst time. Area= foithi*foiwid; %cross sectional area of foil len=foilen; %length of the foil den= density(pointer); %density of foil material C= circap; %cpacitor bank T0= 20; %ambient temperature I=0; %initial current in the circuit V= cirvol; %voltage of capacitor charge q =V*C; %initial charge in the capacitor mass = den*len*Area; %mass of the foil

%setting the maximum step size [ calibrated ] for the solver options = odeset('MaxStep',1e-8); %solving the system of equations [T,Y] = ode45(@burst,[0 1e-5],[T0 I q],options); %dynamic temperature, current and charge array Tf=Y(:,1); If=Y(:,2); qf=Y(:,3); n=bursttime(Tf); t=T(n);

%bursttime

Rf=arrayfun(@res,Tf)*(len/Area); J=If(n)/(Area*1e12);

%dynamic resistance of the foil %dynamic current Density in TA/m2

% flyervelocity as a function of current density and flyer thickness fv=flyerv(J,str2num(get(handles.editfly,'string'))); Vf=If.*Rf;

%dynamic foil voltage

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T Tf If qf Vf Rf n mass

- Time array - foil temperature array - foil current array - capacitor charge array - foil voltage array - foil resistance array - Index of the time step matching the burst time - mass of the foil

This part of the code handles the no burst event. Whenever the parameters are not sufficiently high to meet the burst criteria, an error message is displayed and reads “no burst event”. %Error handling during no burst event % error handling for burst times exceeding 10 micro seconds if n>=length(T) set(handles.textfly,'string','NO BURST EVENT'); set(handles.textbuc,'string','Insufficient power'); set(handles.textbut,'string','> 10'); else set(handles.textfly,'string',fv); set(handles.textbuc,'string',If(n)/1000); set(handles.textbut,'string',T(n)/1e-6); end

4.4.2 Plotting Graphs The following code plots data values based on the parameter selected for plotting. % --- Plots the choosen parameters temporal behaviour. %-------------------------------------------------------------------------global T Tf If Vf Rf n mass; switch get(handles.popgraph,'val') %fetches the selected parameter case 1 plot(T(1:n)./1e-6,If(1:n)./1e3); xlabel('Time [micro sec]'); ylabel('Current [kA]');title('I(t) vs t'); set(handles.textgraph,'string','Burst Current ='); set(handles.textgra,'string',If(n)/1000); set(handles.textgrau,'string','kilo amperes'); case 2 plot(T(1:n)./1e-6,Vf(1:n)./1e3);xlabel('Time [micro sec]'); ylabel('Voltage [kV]');title('V(t) vs t'); set(handles.textgraph,'string','Burst Voltage ='); set(handles.textgra,'string',Vf(n)./1e3); set(handles.textgrau,'string','kilo volts'); case 3 plot(T(1:n)./1e-6,Rf(1:n).*1e3);xlabel('Time [micro sec]'); ylabel('Foil Resistance [mohms]');title('Rf(t) vs t'); set(handles.textgraph,'string','Burst Resistance ='); set(handles.textgra,'string',Rf(n).*1e3);

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set(handles.textgrau,'string','milli ohms'); case 4 plot(T(1:n)./1e-6,(arrayfun(@te,Tf(1:n))+arrayfun(@ee,Tf(1:n)))... .*mass,'k');xlabel('Time [micro sec]'); ylabel('Total Energy [J]');title('E(t) vs t'); set(handles.textgraph,'string','Burst Energy ='); set(handles.textgra,'string',(te(Tf(n))+ee(Tf(n)))*mass); set(handles.textgrau,'string','joules'); case 5 plot(T(1:n)./1e-6,Tf(1:n)+273); xlabel('Time [micro sec]'); ylabel('Temperature [K]');title('T(t) vs t'); set(handles.textgraph,'string','Burst Temperature ='); set(handles.textgra,'string',Tf(n)+273); set(handles.textgrau,'string','kelvins'); case 6 g=localaction(T(1:n),If(1:n)); plot(T(1:n)./1e-6,g(1:n));xlabel('Time [micro sec]'); ylabel('Local Action [A2s/m2]');title('g(t) vs t'); set(handles.textgraph,'string','Action Integral ='); set(handles.textgra,'string',g(n)); set(handles.textgrau,'string','A2s/m2'); end

5. Comparison with experimental data The working code was tested against experimental results provided in various journals and papers. This comparison was made independently for each metal type.

5.1 Copper Foil A. Reference (4): Pages 145 – 147 Table 1: Input parameters for Cu Exp. A

Foil thickness (µm) Foil length (µm) Foil Width (µm) Capacitance (µF) Inductance (nH) Charging Voltage (kV)

8 1000 1000 3 200 4

21

Thermal coefficient of Resistivity = 0.003 (K-1) Circuit resistance has been approximated to 100 mΩ.

Figure 7: Solution screen for Cu Exp. A data

The percentage error in the simulation results are presented below: Table 2: Data comparison - Cu Exp. A

Burst Parameters Time (µs) Current (kA) Voltage (kV) Foil Resistance (mΩ) Foil Energy (J)

Experimental Results 0.6 8.4 2.6 259 1.245

Simulation Results 0.6024 8.6532 2.226 257.25 1.267

22

Percentage Error 0.4 3.014286 14.38462 0.675676 1.767068

B. Reference (7): Page 625 [Table 2] Table 3: Input parameters - Cu Exp. B

Foil thickness (µm) Foil length (µm) Foil Width (µm) Capacitance (µF) Inductance (nH) Charging Voltage (kV) Circuit Resistance (mΩ)

51 25400 25400 56 40 40 6

Figure 8: Solution screen for Cu Exp. B data

Table 4: Data comparison - Cu Exp. B

Burst Parameters Time (µs) Current (kA) Action (A2s/m2)

Experimental Results 1.25 932.69 3.12 X 1017

Simulation Results 1.3172 902.63 3.141 X 1017

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Percentage Error 5.376 3.222936 0.673077

5.2 Gold Foil Reference (7): Page 625 [Table 2] Table 5: Input parameters for Au Exp.


25 25400 25400 56 40 40 6

Figure 9: Solution screen for Au Exp. data

Table 6: Data comparison - Au Exp.



Simulation Results 0.6423 502.85 1.789 X 1017 24


5.3 Aluminium Foil Reference (7): Page 625 [Table 2] A. Table 7: Input parameters for Al Exp. A


25 25400 25400 56 40 40 6

Figure 10: Solution screen for Al Exp. A data Table 8: Data comparison - Al Exp. A



Simulation Results 0.6023 365.67 1.37 X 1017 25

Percentage Error 7.338462 17.73453 0

B. Table 9: Input parameters for Al Exp. B


51 25400 25400 56 40 40 6

Figure 11: Solution screen for Al Exp. B data

Table 10: Data comparison - Al Exp. B




26


C. Table 11: Input parameters for Al Exp. C


51 35600 35600 56 40 40 6

Figure 12: Solution screen for Cu Exp. C data

Table 12: Data comparison - Al Exp. C




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6. Results and Discussion The algorithm is able to solve for the burst parameters and provides realistic results. Further, the graphical interface allows for visualization of patterns in the behaviour of parameters involved in burst phenomenon. Comparison with experimental values shows that the code works best for copper, then for gold and least accurate for aluminium foils. This is due to the fact that the available property data was specific to copper, while the same parameters had to be reverse engineered from the experimental data in the case of gold and aluminium. Various simulation parameters can be modified to improve the accuracy of the simulator centred on a particular metal type. By testing with more data points it is possible to obtain more accurate figures for burst temperature, which is the sole determiner of the burst event. The linear variation of resistance with temperature is an over simplified assumption and requires a temperature averaged thermal coefficient of resistivity. This is different from the available data (valid only at ambient temperature) and calls for a non – linear resistivity model. The model used is a simple one dimensional model. The assumptions are valid for low burst times but at higher burst times the heat dissipation factors alter the results. Hence, this model is best suited for Copper foils with burst times below 1 µs.

7. Acknowledgements The help of Dr. Dinesh Pal throughout the code development and testing phase is gratefully acknowledged. I also thank Dr. T. K. Ray Chaudhuri for his support and motivation. The work was performed under the auspices of the Electro Explosive Devices (EED) division of Terminal Ballistics Research Laboratory.

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8. References 1. Weingart, R.C., et al. The Electric Gun: A new tool for ultrahigh-pressure research. 1979. 2. Slapper Detonator. Wikipedia. [Online] [Cited: 20 July 2012.] http://en.wikipedia.org/wiki/Slapper_detonator. 3. Pal, Dinesh Kumar. Theoratical Calculation for Energy required to explode gold bridge foil . 4. Measurement of Shock Initiation Threshold of HNAB by Flyer Plate Impact. Hasman, E., M.Gvishi and Y.Carmel. 1989, Propellents, Explosives, Pyrotechnics 11, pp. 144-149. 5. Physics of Shock Waves and High Temperature Hydrodynamic Phenomena. Zeldovich, Ya B. and Raizer, Ya P. New York : Academia Press Inc., 1966. 6. Altshuder, L. V. Use of Shock Waves in High Pressure Physics. Sov Phys. Uspekhi (Engl. Transl.). 1965, Vol. 8, 52. 7. Calculation of heating and burst phenomena in electrically exploded foils. Logan, J.D., et al. 1977, J. Appl. Phys. 48, pp. 621-628.

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9. Appendices

9.1 List of Physical Constants used

Table 13: Physical Properties of metals

Metal Molar mass [gram/mol] Density [kg/m3] Specific Heat (20 °C)[ J/kg.K] Resistivity (20 °C)[Ω.m] Thermal Coefficient of Resistivity [K-1] Burst Temperature [K]

Copper 63.546

Gold 196.96

Aluminium 26.98

8930

19300

2700

385

129

900

1.72 X 10-8

2.214 X 10-8

2.82 X 10-8

3 X 10-3

3.7 X 10-3

3 X 10-3

40000

25000

80000

Physical Constants: Boltzmann Constant = 1.3806503 × 10-23 m2kg s-2K-1 Avogadro number = 6.02214X×1023 mol-1 Gas Constant = 8.3144621 J.mol-1K-1 (Eq. 10) = 1.1 X 10-2 J/kg.K2

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9.2 Table of Figures Figure 1: Diagram of Slapper Detonator ....................................................................................................... 4 Figure 2: Flyer impact on primary explosive ................................................................................................. 5 Figure 3: The Equivalent Circuit of the EFI .................................................................................................... 7 Figure 4: Flyer velocity vs. burst current density J; Copper foil thickness - 8 micrometer and Mylar flyer thickness - 76 micrometer ............................................................................................................................ 9 Figure 5: Developed GUI [MATLAB guide] .................................................................................................. 17 Figure 6: Interface output example ............................................................................................................ 18 Figure 7: Solution screen for Cu Exp. A data ............................................................................................... 22 Figure 8: Solution screen for Cu Exp. B data ............................................................................................... 23 Figure 9: Solution screen for Au Exp. data .................................................................................................. 24 Figure 10: Solution screen for Al Exp. A data .............................................................................................. 25 Figure 11: Solution screen for Al Exp. B data .............................................................................................. 26 Figure 12: Solution screen for Cu Exp. C data ............................................................................................. 27

9.3 List of Tables Table 1: Input parameters for Cu Exp. A ..................................................................................................... 21 Table 2: Data comparison - Cu Exp. A ......................................................................................................... 22 Table 3: Input parameters - Cu Exp. B ........................................................................................................ 23 Table 4: Data comparison - Cu Exp. B ......................................................................................................... 23 Table 5: Input parameters for Au Exp. ........................................................................................................ 24 Table 6: Data comparison - Au Exp. ............................................................................................................ 24 Table 7: Input parameters for Al Exp. A ...................................................................................................... 25 Table 8: Data comparison - Al Exp. A .......................................................................................................... 25 Table 9: Input parameters for Al Exp. B ...................................................................................................... 26 Table 10: Data comparison - Al Exp. B ........................................................................................................ 26 Table 11: Input parameters for Al Exp. C .................................................................................................... 27 Table 12: Data comparison - Al Exp. C ........................................................................................................ 27 Table 13: Physical Properties of metals ...................................................................................................... 30

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Modelling and Simulation of Burst Phenomenon in Electrically Exploded Foils

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