Proceedings of the Sixteenth Symposium on Explosives and Pyrotechnics, Essington, PA, April 1997.
Barry T. Neyer
EG&G Optoelectronics/Star City
Miamisburg, OH 45343-0529
Barry T. Neyer
1100 Vanguard Blvd
Miamisburg, OH 45342
(937) 865-5170 (Fax)
Electric currents are applied to an electro-explosive device for several reasons. One is to function the device. Another reason would be to check the electrical resistance of the circuit to ensure that the device will work as expected. When checking the resistance, it is important to ensure that the device does not function, and that it does not degrade. Thus, it is desirable to know the current-induced temperature rise in the bridgewire.
This report will describe calculations to compute the temperature rise. It uses a simplified physical model that nevertheless gives a good match to the physics of the device. Using this model, it is possible to construct an exact solution to the physical problem. This report will give the derivation, and show the temperature rise as a function of position along the bridgewire and the time. The temperature information will allow the design engineers to determine the allowable current to use when performing continuity tests, as well as to estimate the effects of various RF induced currents on the temperature of the device.
Technical Papers of Dr. Barry T. Neyer
Figure 1: Schematic Diagram of Electro-Explosive Device
Electro-explosive components are widely used to convert electrical energy into an explosive output. A schematic diagram of an electro-explosive device is shown in Figure 1. Current from an external source is applied via metallic pins to a resistive bridgewire. The current heats the bridgewire, which in turn heats the explosive, causing it to ignite.
A similar type of device is an exploding bridgewire device (EBW). A rapidly rising current from a capacitor discharge unit causes the wire to rapidly heat and burst. The bursting wire causes the explosive to either deflagrate or detonate.
The proper functioning of such a device requires that the bridgewire remains intact and not be degraded before the unit is used. Because the bridgewire can be broken during the explosive loading process, or can degrade due to incompatibility with the explosive mixture, it is often desirable to ensure that the bridgewire has not been damaged. The most commonly used method of determining that the bridgewire has not been degraded is to measure the resistance of the device before and after loading the explosive. These numbers should be approximately the same. The resistance can also be measured as the device ages to ensure that no degradation occurs.
Implicit in this measurement technique is the assumption that the measurement does not in any way damage the bridgewire - explosive interface. The requirement of no damage requires that the current used to perform the resistance measurement is low enough that it raises the temperature only minimally. Thus, current limiting resistors must be used.
This paper calculates the temperature rise in an electro-explosive device. The calculation is exact in the case of a electro-explosive device of a certain simple geometry. While detonators are not usually made in this simple geometric pattern, the calculation still provides a realistic temperature profile for most electro-explosive devices.
Figure 2: Current Flow
Consider a bridgewire of cross section area a with a current flow I, as shown in Figure 2. The change in energy of the small volume shown in Figure 2 generated by the current flow is given by the equation
where s is the bulk resistivity of the wire. Heat flows in the wire if there is a thermal gradient. The net heat flow into the small volume of wire is given by the equation:
where K is the heat diffusion constant. The temperature change in a wire is governed by the equation:
where r is the bulk density of the wire material. Combining the heat and temperature equations yields:
Taking the limit as Dt ® 0 yields the equation
Equation (5) is the well known one dimensional heat diffusion equation with a current source term. The general solution to the linear partial differential equation is found by finding a particular solution, and adding this to the general solution of the homogeneous equation. The general solution can be written as the sum of terms of the form:
The particular solution depends on the boundary conditions. Bridgewires used in explosive devices are usually thin resistive elements bonded to thicker low resistance pins. The pins are usually a factor of ten or more times the diameter of the bridgewire, and are often embedded in a composition that can dissipate heat. Thus, a good approximation is to assume that the pins remain at ambient temperature. (Calculations that follow will show that the assumption of ambient temperature of the pins is very realistic is most cases.) If the wire is of length l, then a particular solution to Equation (5) with the appropriate boundary conditions is
where z measures the distance from the mid point of the wire. That Equation (8) satisfies Equation (5) can be readily verified by substituting it into the differential equation.
The set of solutions to the homogeneous equation must allow for any initial condition. Assume that at time t = 0 the wire is at uniform temperature T0, when a current I "turns on." Because the endpoints are assumed to remain at the temperature T0 throughout, a convenient set of homogeneous solutions is to chose b n l/2=np/2. Because of the symmetry of the problem, the Bn terms in Equation (6) are identically zero. The boundary conditions further require the even An terms to also be zero. The Fourier coefficients are thus:
Because the temperature is required to be T0 at time t = 0, the Fourier coefficients are chosen to exactly cancel the parabolic term in brackets in Equation (8). Substituting the bracketed term for f(z) yields the general solution:
Inspection of Equation (10) shows that the solutions to the homogeneous equation all fall off exponentially. Thus, after a short time, the temperature profile becomes parabolic, determined by the first two terms in the bracket in Equation (10). The time constants for exponential decay are determined from the length and the diffusivity constant of the bridgewire. The thermal diffusivity constant, D, for nickel wire used in many EEDs is 23.5 mm2/s. A one millimeter long bridgewire has a thermal decay constant of 232 inverse seconds. Thus, all transient effects are gone in several tens of milliseconds.
Figure 3 shows a typical temperature rise above ambient as a function of time, assuming the above diffusivity constant. The real temperature at any point is found by multiplying the curve by the coefficient in front of the bracket in Equation (10). A different diffusivity constant would produce similar curves, but at different times.
The above discussion assumed a constant current. However, if the current changes slowly compared to the thermal decay constant, the upper curve in Figure 3 will also represent the instantaneous temperature increase as a function of the current.
Figure 3: Temperature Rise in Constant Current Bridgewire
The curve and analysis show that the peak temperature is given by the equation:
where R = ls/a is the resistance, and m = rla is the mass of the bridgewire. Equation (11) represents the final temperature at the center of the wire. Inspection of Figure 3 shows that the temperature rise is a linear function of time for short times. Taking the limit t®0 in Equation (10) yields the short time behavior of the temperature at the center of the wire:
. (center of wire, short time) (12)
Equation (12) is also the form of the temperature that would be derived assuming there was no heat flow. (The limit t®0 is the same as D®0, because D and t only appear as a product.)
Figure 3 shows that the temperature is essentially at the maximum after 40 ms of current flow. If a constant current of 0.3 Amps flows for 40 milliseconds, in a 2 mil bridgewire, the above calculations yield a temperature rise of:
|Bridgewire length||l||1 mm|
|Bridgewire cross section||a||2.027 10-3 mm2 (2 mil diameter)|
|Bridgewire density||r||8.9 g/cm3|
|Bridgewire mass||m||1.804 10-5 g|
|Bridgewire specific heat||Cp||0.105 cal/g C = 0.4395 J/g C|
|Heat Diffusion Constant||D||23.5 mm2/s|
|Resistivity||s||6.32 cm = 38 / cmf|
|Temp Peak||1.9 C|
|Temp Peak (no heat flow, 40 ms)||14 C|
|Temp Peak (no heat flow, 1 s)||352 C|
The importance of heat flow in the calculation is clear if one considers what would happen for longer duration currents. Neglecting heat flow would lead to a predicted temperature rise of 350 C for a one second 0.3 Amp current, and a 1000 C rise for a three second pulse, all from a 3 milliwatt heat source! The calculation with heat flow would show the realistic 2 C rise. If the current was on for an extremely long time, then the temperature of the entire device would start to rise. However, since the mass of a typical device is 10 or more grams, or one million times the mass of the bridgewire, the temperature rise would be only a few thousands of a degree.
If the current turns off at a later time, the temperature will decay with the same speed as it rose to the steady state solution. Thus, approximately 8 ms after the current stops, the temperature will be reduced to 36% of the peak value, to 14% after 16 ms, etc. The temperature fall is shown in Figure 4.
Figure 4: Temperature Fall After Removing Current
Repeated pulsing of such a system will not result in any appreciable temperature rise in the bridgewire compared to the rest of the component. Thus, there is no need to wait between current pulses to ensure that there is no large temperature rise in the system.
The above calculations have assumed no heat transfer to the explosive powder surrounding the bridgewire. Because the thermal diffusion constant for explosive powder is usually many orders of magnitude lower than the constant for the bridgewire, the approximation is a good one. The three dimensional heat diffusion equation can be solved for the full problem. Because of the geometry, a full calculation is rather difficult. However, a reasonable approximation would be to calculate the heat flow radially at any point along the wire. Such a calculation is beyond the scope of this paper. A full calculation would reduce the temperature of the bridgewire, but by a negligible amount.
The above calculations also were performed under assumption that the pins remained at the ambient temperature. Calculation of the temperature rise in the pins is much more complicated, than the rise in the bridgewire, because the pins are in intimate contact with header material. The header material will generally conduct heat much more readily than the explosive powder. If heat conduction to the header material can be neglected, then the solution is given by Equation (10), with the 8 replaced by a 2 and where l represents the length of one pin.. The pins used in EEDs are usually at least a factor of 10 larger in diameter than the bridgewire, which results in a factor of 10-4 smaller temperature, while a factor of 10 or more times the length results in a factor of 100 larger temperature. Combined, the two factors result in a temperature rise of at most a few percent of the temperature rise in the bridgewire. Thus, for even this unrealistic case, the temperature rise in the pins can safely be ignored. For the more realistic case of the pins imbedded in a glass or ceramic header, the temperature rise of the pins would be drastically reduced by several orders of magnitude.
The temperature rise in electro explosive devices was calculated analytically. The formula allows EED designers to determine the temperature rise in such devices. Proper utilization of current limiting resistors allows experimenters to test the EED without any concern about generating too large of a temperature rise in the EED.
This work was initiated to determine the temperature rise in the High Voltage Detonator (HVD) during continuity tests. The HVD is manufactured for Lockheed Martin Aerospace Corporation. This study was partially funded by Lockheed Martin.