*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
*

*Contact Address
Barry T. Neyer
PerkinElmer Optoelectronics
1100 Vanguard Blvd
Miamisburg, OH 45342
(937) 865-5586
(937) 865-5170 (Fax)
Barry.Neyer@PerkinElmer.com
*

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

, (1)

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:

, (2)

where K is the heat diffusion constant. The temperature change in a wire is governed by the equation:

, (3)

where r is the bulk density of the wire material. Combining the heat and temperature equations yields:

. (4)

Taking the limit as Dt ® 0 yields the equation

, (5)

where

.

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:

, (6)

where

. (7)

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

. (8)

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 T_{0}, when a current I
"turns on." Because the endpoints are assumed to remain
at the temperature T_{0} throughout, a convenient set of
homogeneous solutions is to chose b _{n}*
l/*2*=n*p*/*2. Because of
the symmetry of the problem, the *B*_{n}
terms in Equation (6) are identically zero.
The boundary conditions further require the even *A*_{n}
terms to also be zero. The Fourier coefficients are thus:

. (9)

Because the temperature is required to be T_{0} 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:

. (10)

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 mm^{2}/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:

, (11)

where *R = l*s*/a* is the
resistance, and *m = *r*la*
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:

Parameter | Symbol | Value |

Bridgewire length | l |
1 mm |

Bridgewire cross section | a |
2.027 10^{-3} mm^{2} (2
mil diameter) |

Bridgewire density | r | 8.9 g/cm^{3} |

Bridgewire mass | m |
1.804 10^{-5} g |

Bridgewire specific heat | C_{p} |
0.105 cal/g C = 0.4395 J/g C |

Heat Diffusion Constant | D |
23.5 mm^{2}/s |

Resistivity | s | 6.32 cm = 38 / cmf |

Resistance | ls/a |
31 m |

Current | I |
0.3 Amp |

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.