NWL Report No. 1919

A HEAT TRANSFER MODEL STUDY

OF THE. HOT WIRE INITIATOR

byJ.M.b 14sey, Jr.

Warhead itnd Terminal Ballistics Laboratory -

DDC

JUL 2 4 1964

DDC-IRA B

U. S. NAVAL WEAPONS LABORATORY

DAHLGREN, VIRGINIA

U. S. Naval Weapons LaboratoryDahlgren, Virginia

A Heat Transfer Model Studyof the Hot Wire Initiator

by

J. M. Massey, Jr.Warhead and Terminal Ballistics Laboratory

NWL Report No. 1919

WEPTASK WOS. BMW)-33-020/210-1/FO08--01and R*o-33-o2/21o-1/FoO8-17-O1

6 July 1964

Qualified requesters may obtain copies of this report direct fron DDC.

NWL REPORT NO. 1919

CONTENTS

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiForeword ............ ........................... ivIntroduction ............ ........................ 1Mathematical Model . .......... ..................... 1

I. Eqvation o.& Conduction of Heat ............ 1II. Simplifyin1g A.sumptions ....... ............... 2

III. Final Equations ......... ................... 3Results ............. .......................... 9Concluding Remarks ....... ...................... .... 12References ........ .......................... . 13Appendices:

A. A;,-,.1oximation of First and Second PartialDerivatives and the Laplacian by FinieDifferences - Unequal Segments and CylindricalCoordinates

B. Distribu'ion

Figures:1. Cross Section of a Hot Wir-e &*.tiator (MARK 45 MOD 0

Cartridge)2. Detail of MARK 45 MOD 0 Bridgewire and Bead with

Coordinate System Superimposed3. Spatial Distribution of Temperature Stations in

Bridgewir t and Bead4. Intt.rxolaion Construction for ens5. Sen&-.tlvity of MARK 45 MOD Impulse Ca-tridge to

Constant DC Current6. Dete,-Lination of Heat Trapn.. Xr Film Coefficient and

Thermal Conductivity7. Bead Temperature (o .) vs Time8. Steady State Radial Temperature Distributio.: - 1=3 &zIps9. Steady State Axial Temperature Distribution - i-3 amps

10. Effect on Sensitivity of a 20% Increase in Ta11. Effect on Sensitivity of a 29% Increase in h

NWL REPORT IO. 1919

CC NsTS (Continued)

Figures (Continued):12. Effect on Sensitivity of a 20% increase in k13. Effect on Sensitivity of a 20% Decrease in pc14. Effect on Sensitivity of a 50% Increase in rbead15. Effect on Sensitivity of a 50% Increase in L

16. Finite Difference Construction for e e ' - e and V6r2 ar x2

Table:1. Input Parameters and Material Composition

INWL REPORT NO. 1919

ABSTRACT

A mathematical heat transfer model, applicable to determiningthe sensitivity to input current of a hot wire initiator, isdeveloped. Simplifying assumptions are introduced which make thern:del amenable to solution using a medium. size analog computer.The model is applied to the MARK 45 MOD 0 impulse cartridge. St!u-tions are compared to emr.r .al seisitivity data for this cartridgeconditioned at -80 0 F. 68 F and 225°F.

iii

NWL REPORT NO. 1919

FOREWORD

The development of the mathematical model, computations, andexperimental work suimmarized in this report was carried out underProblem Assignment 1 of WEPTASKS NC3S. I,4)-33-O20/21 )-I/F008-l1-O1and IBMQ-33-02/2l-l/fUO8-17-0l.

The author is indebted to Mr. George L. Poudrier for hisextensive suggestions and comments which proved beneficial todeveloping the mathematical model prezented here and for hisassistance in judiciously choosing the simplifying assumptions.Additionally, Mr. Poudrier assisted materially in the operation ofthe analog computer. The author's approc'tation is extended toMr. R. L. Montgomery for his time spent both in checking the analogcomputer program and in assisting in the operation of the computer.Finally, the author is indebted to Mr. S. E. Hedden, who suggestedthe problem and the approach to its solution.

This report has been reviewed by the following personnel of theWarhead and Terminal Ballistics Laboratory:

. E. HEDDEN, Head, Research Branch, CartridgeActuated Deices Division

J. J. GIANCY, Head, CartrIdge ActuatedDevices Division

W. E. McKENZIE, Assistant Director for TechnicalApplications, Warhead and TerminalBallistics Laboratory

R. I. ROSSBACHER, Director, Warhead and TerminalBallistics Laboratory

APPROVED FOR RELEASE:

ls/ RALPH A. IMANNActingTechnical Director

iv

NWL REPORT NO. 1919

INTRODUCTION

A hot wire initiator (see Figure 1) is a single discrete unitcontaining an explosive, propellant, or pyrotechnic materialcomonly referred to as the primary mix or bead material; hereafter*-eferred to as the bead. The bead usually surrounds a fine wireXnown as the bridgewire which spans or bridges two pins as shown inF.Sgure 1. The bead is ignited by ohmic heating of the bridgewJxewhen sufficient electric potential is applied across the pins. Theignited bead in turn ignites a secondary charge known as the boosteror main chat ge, as the case may be.

Of interest to the designer of a hot wire initiator is thecontrol of its sensitivity. Here sensitivity refers to the relationbetween the firing time of an initiator and the input firing stimulussuch as current or potential. The problem of determining the sensi-tivity of a newly designed hot wire initiator was treated, in thestudy reported here, as a heat transfer phenomenon. A mathematiualmodel was dt.veloped whose solutions yield the sensitivities of hotwire initiators. The model was then applied to the MP.RK 45 MOD 0Impulse Cartridge. In o-der to obtain values for two inaccuratelyknown parameters (the ther-mal conductivity of the bead and the heattransfer film coefficient of the bridgewii-e-bead interface), theywere systematically varied until, throuigh trial and error, the calcu-lated sensitivity matched the measursd sensitivity at two values ofinput current. la this way the model was calibrated against empiricalfiring time versus firing current data from cartridges conditioned atroom temperaturie;. The degree of fis between the calculated andeu.pirical data for other values of input curren. and at different con-ditioning temperatures was used as a measure of the adequacy of themodel.

Finally, certain of the inaccurately known parameters werearbitrarily changed, one at a time, to determine the sensitivity ofthe solutions to possible error in the parameter changed.

All solutions were obtained on an Electronic Associates analogccmpu':r consisting of an eighty amplifier M-I0 four cabinet complex.

MATHERNTICAL MODEL

1. Equation of Conduction of Heat -

Reference (a), page 10, gives the heat conduction equation foran isotropic, homogeneous solid with internal heat generation as

_ 2C. + A(Ax,y~zt)

1

NWL REPORT NO. 1919

where:

a is the thermal diffusivity,PC

A ±s the rate of heat produced per unit time per unit,volume at the point (x,y,z),

c is the specific heat,

k is the thermal conductivity,

p is the density,

e is the instantaneous temperature at the point x,y,z,

V is the Iaplacian operator.

II. SimlifYing Assumtions -

In order to obtain a mathematical model within reach of the capa-bilities of the computing equipment used, the following simplifyingassumptions were made.

A. Radial temperature distributions in the bridgewire wereneglected. Frcn equation (1), page 204 of reference (a), it wasconcluded that the differenre between the outer surface temperatureand the center temperature of the bridgewire of the MARK 45 MD 0 c.?r-tridge (the hot wire initiator used in this analysis) would neverexceed 1.340C. This difference was considered negligible since it isonly 0 .5% of the assumed autoignition temperature of the bead.

B. Although the MARK 45 MOD 0 cartridge is a four pin doublebridge hot wire initiator, only one of the bridge circuits was con-sidered in the study. Figure 2 is a sketch, with coordinate systemsuperimposed, of the bridgewire and bead of the MARK 45 MOD 0 car-tridge. Only one of the bridgewires is shown. The other bridgewireis parallel to the one shown and is electrically insulated from it.

C. Cylindrical symmetry was assumed.

D. As depicted by the broken lines in Figure 2, an arbitrarycyclindrical boundry was introduced. In the mathematical model thiswas assumed to be the outer boundary of the bead and was assumed toremain at ambient temperature. This assumption was based on theconcept that the temperature of the relatively large masses of the-pins and the ceramic plug probably does not differ appreciably fromambient.

2

SECONDARY CLOSURE

SECONDARY CHARroE

PRIMARY MIX BD

OR BEAD

9*s'* ~* ~*PRIMARY CLOSURE

o*.,s * ~BRI DGEWP

-SPACER

S C ALE. 4/1

FIG= 1

CAS SCYOK OF A Ur WII.n llrIA!OR

(K 4~5 MX) 0 C~mioZ)

CC

C,* I

00

_NWL REPORT NO. 1919

E. The thermal propeL'tics (e.g., thc na2 conductivity,specific heat, and diffusivity) of the initiator materials wereassumed to be independent of temperature. Approximately medianvalues for these properties were chosen consistent with the tempera-ture range encountered.

S F. The exothermic chemical self hieating of the bead wasassumed negligible until the bead reached a critical temperature T a.

Ignition was assumed to occur at this time and all sub,-quent calcula-tions were disregarded. The value chosen for Ta was based on experi-mental evidence of an exothermic reaction in the bead when theinitiator was subjected to various oven temperatures. in accord withthe experimental technique for determining the value of Tw it is

referred to as the autoignition temperature.

III. Final Equations -

Let primed symbols refer to the bridgewire and let unprimedsymbols refer to the bead. Also let a (6') now refer to the tempera-ture difference between the bead (bridgewire) and ambient. Asdeveloped in sections 4.2 and 4.10 of reference (a), equation (1)applied to the bridgewire becomes

3 = J(l e - P'h (e'-e5) (2)

where:

A' is the cross sectional area of the bridgewire,

-1 a' is Qhe diffusivity of the bridgewire,

h is the heaL transfer film coefficient of the bridgewire-bead interface,

I is the instantaneous current in the wire,

J is the mechanical equivalent of heat,

t is the time,

a is the temperature coefficient of resistivity,

&S is the surface temperature of the bead at the bridgewire-

bead interface.

i3

WL REPORT MD. 1919

Equation (1) applied to the bead simplifies to

V26. (3)

where a is the dJffusivity of the bead.

The initial, boundary, and tenrrinal conditions are

9=e'=0 " t t=Q, (4)

e --9 =0 at x =- r-rb.2 --ad-I

e (-x) ='(x) (6)

e(-x) = G(x), (7)

-k _ h(e'-O,) at r=rs, (8)

-<T~, (9)e < Ta -T,

where,

L is the length of the bridgewire and the bead,

rbead is the outer radius of the bead,

r. is the inrer radius of the bead (equal to the bridge-wire radius),

TC is the ambient temperature.

Equations (2) and (3) can be approximated by a set of ordinarydifferential equations with time as the independent variable. Thisis done by considering e' at discrete values of x and by consideringe at discrete values of x and r. The partial derivatives of 9' withrespect to x and of 0 with respect to x and r are then represented byappro-priate difference expressions (see Appendix A).

Two independent investigations were made to determine the distri-bution and the minimum number of discrete values of x and r at wh-chthe temperatures must be computed to obtain reasonable accuracy.

4

NWL REORT NO. 1919

Computational experiments involving up to 6 axial temperature sta-tions in half the bridgewire length (equivalent to 12 axial sta-tiorR) showed that 3 equally spaced axial stations in half thebridgewire and bead length was sufficient. The error in computedtemperatures introduced by using only 3 axial stations was about3%. Similarly, computational experiments Involving up to 6 radialtemperature stations showed that 4 unequally spaced radial stationsin the bead were sufficient. The error in computed temperaturesintroduced by using only 4 radial stations was less than 10%. Thiswas provided the radial temperature station nearest the bridgewire-bead interface was chosen sufficiently close to the interface. Itwas determined that "sufficiently close" meant that the distancefrom the bridgewire-bead interface to the first radial temperaturestation should be less than 1% of the distance from the bridgewire-bead interface to the bead's outer radius.

In accordance with the conclusions reached, temperature sta-tions were selected within the bridgewire cid bead of the MARK 45MO)D 0 cartridge as indicated in Figure 3. Ea-h station isassigned the generic indices n,m where n is thL axial positionindex and m is the radial position index. n =o refers to any pointin a plane perpendicular to the bridgewire and passing through itscenter. m=o refers to any point on the surface of a circularcylinder whose axis coincides with the axis of the bricgewire andwhose radius is equal to the radial distance from the axis of thebridgewire to the row of stations within the bead nearest Zhebridgewire.

For the bridgewire

_ * (lC)

From equation (A 13) in Appendix A, equation (10) can be approxi-mated (with P=1) by

Va' 1 - 20 + 0'_. (11)X=X n q

5

NWL REPORT MO. 1919

Substituting from equation (11) into equation (2), combining terms,and imposing the initial, boundary, and symmetry conditio.is ofequations (4), (5), (6) and (7), the final equations for the bridge-wire are

d9~ 12 j(li1 e) a' 9 (2'aPh e

dt = p'(A') c a + + - z-P-'A C')

+- n_ + .~.P /henI p 'A'c 1en s

n 0 ,1,2 (12)

=0

e n = n,s -0 at t=O.

In order to calculate 9 n, s, a parabola lying in the r-e planeis fitted through the three points (-Poen_1), (O,On,o), and

(7pO, en,1 ) as shown in Figure 4. The equation of this parabola

with origin of the coordinate system arbitraaily chosen a- radialstation r= er is

, A(r-ro) 2 + B(r-ro ) + C. (13)

or

on,s Q 2 +B ( PO + C. (14)

The coefficients A, B, and C are derived in Appendix A. Substitutingthe values for A, B, and C from equations (A), (A9), and (A0) into

equation (14) results in the final expressions for the bead innersurface temperatures.

1 9 + 2y +1 + 2-+1en, s = " 4y(7+1) ' n, 1 4y n,o 4(y7) en,-I

(ULn = 0,1,2.

6

CIO~PS

ILo

0 onon to-

1~' F)

CY, C4 C/

/ /

;7717i m 0::z::)a W4/6C

o A

-'U, 60 UM N N N

o -O-

On,

2 0

I I I4

flamLmTom cowzz-r~mcm FoR ,

NWL REPORT NO). 1919

The final form of the boundary condition, equation (8), cannow be given. Substituting into equation (8) the finite difference

approximation for 6e derived in Appendix A (see equation (All)),"rra.d the expressions for the bead surface temperature (equation (15)),and combining terms results in

h _ _ _ NylK 4 17 - 4v k

o ,_- n, ,0 (+ n,1 + n, 0,

1 _ + h 2 +1 h +1 hep (~)+k+ 1 (16)

n = 0,1,2.

For the bead

2Le +.I g + (17)6r2 Cr C K2

From equation (A4) in Appendix A, equation (17) can be approximated(with P=i ) by

ve~ = (2rm+Pm r m i

- 2r +p)~

2 n, m+1

"\ rY+M r M)

(18)

+ (rm( Y) e~,l.. +Q ) en~(l+ ) @n) m-1 en+l., m

+ (eL)

7

NWL REPORT NO. 1919

Substituting from equation (18) into equation (3) and imposing theinitial, boundary, and symmetry conditions of equations (4), (5),(6), and (7); the final equations for the bead are

dn, ( (2r ) m n1

dt y l~y m r

(a(2rm+(-Y) P) 2a

2-1 '1 9n ml

eVmm rm

+ 1+yp r / nm- (19)

+ e1n+i, m +(~ en-1)m

= .4 1= m

e 3,M 0

enm =0 at t=O

n = 0,1,2.

m = 0,1,2,3.

Equations (12), (15), (16), and (.9) are the final equations forthe bridgewire, bead inner surface temperature, boundary condition,and bead, respectively.

8

NWL REPORT NO. 1919

RESULTS

Empi-ical data of firing time versus constant DC current,for the MARK 45 MCD 0 cartridge at three conditioning tempera-tures, is presented in Figure 5. The vertical bars indicate thespread in the data. A smooth curve (not shown) was visuallyfitted to the room temperature data, and from this curve, thesensitivity at 4 and 10 amps was determined. Several values forthe thermal conductivity k of the bead were chosen which wereexpected to bracket the true, but unknown, value. Using currentsof 4 and 10 amps in the computer program, selected values of heattransfer film coefficients h for the bridgewire-bead interfacewere determined by trial arid error. These values were selectedsuch that the computed sensitivity matched the empirical sensitiv-ity for each value of k chosen. By graphing the correspondingvalues of h vs k for 4 amps and for 10 amps respectively, it wasdetermined that a single value of h and of k would satisfy theempirical sensitivity requirement at both 4 and 10 amps. Thisresult is shown in Figure 6. This indirect method of measuring hand k is dependent on the adequacy of the mathematical model andthe accuracy of the other parameters. Due to the large number ofassumptions required in this study, no attempt was made to estimatethe accuracy of the values obtained for h and k.

Table 1 lists the input parameters used ii these and subsequentcalculations. It also lists the composition of the initiatormaterials of interest here and the sources for information given.

TABLE 1 IUJT PARAMETERS AND MATERIAL COMPOSITION

Parameter Symbol Value Source*

Bridgewire:

length L, 6q 0.2847 em 1

Radius rs 0.9D10 x c - cm 1

Cross sectional area A' 2.554 x l0 - 4 cm2 1

Perimeter P/ 5.661 x 10-2 cm 1

Electric conductivity a' 1.00 X 10 4 mhos/cm 2

Thermal conductivity k' 0.050 cal/sec-cm-0 C 3

Density P ' 8.50 gm/cm3 4

Thermal capacity c ' 0.126 cal/gm-0 C 4

Temperature coefficient a' 1.8 X 10 - 4 ( 0C)-1 2of resistivity

9

NWL REPORT NO. 1919

TABLE 1 (Continued)

Parameter ymbol Value Source*

Bead:

Length L, 6q 0.2847 cm 1

Inner radius r. 0.9010 x 10-2 cm 1

Outer radius rbead, r4 7.875 x 10 -2 cm 1

Thermal conductivity k 5.05 x I0 - 4 cal/sec-cm-0 C 5

Density p 2.52 gm/cm3 4

Thermal capacity c 0.2461 cal/gm-OC 4

Autoignition Ta 2700 C 6temperature

Bridgewire-bead interface:

Beat transfer h 0. 116 cal/sec-cm2 .- C 5film coefficient

Bridgevire composition:

Nickel Ni 80% 1

Chromium Cr 20% 1

Bead composition:

Zirconium Zr 7.00% 1

Ammonium dichromate (NH4) 2 Cr 2 O-, 14.08% 1

Ammonium perchlorate I04 CI0 4 44.92% 1

Barium chromate BaCrO4 34.00% 1

10

2000 -

1500.-.- .- a. --- _

2.000

800 .

MA8cP3 AM 0 Zwoso atAdw~st

600 . - _

.. .. .. .

100 2 10 -5----6--ox

C-a:rraa Aa~

:-k VI

60IIY0 AM.5*D0'NUE~CJR1G C

.4 .~4 ---.- ~ 4 .- .- ....7-Z :

00 . . .... ..IN t Tism sfer lila Opettleim tvs Tbezm1 Conktivity

CJ 1 10.0 so"

0.1

-~-1-449

I . -1-

Tt tI !

3 6 8 10

DWOKMau1uo Of mW TFAMwu nxm SwnMC AND =VL M =1"ff

NWL REPORT NO. 1919

TABLE 1 (Continued)

*Source key1. Either obtained directly or derived from information

given in reference (d).

2. Obtained directly from reference (e).

3. Derived from information given in refer]ee (f).

4. Derived from information given in reference (e). Thederivation was based on values given for each component of thecomposition if it was available. 11 it was not available, thevalues for chemically similar compounds were used.

5. Derived by graphically fitting computed results toempirical data at two input currents.

6. Based on experimental evidence of an exothermicreaction in the bead when the initiator was subjected to variousoven temperatures.

Using the values of k = 5.05 x 10 - 4 cal/cm-0 C and h = 0.116

cal/cm2 -0 C (the point of intrsection in Figure 6), the bead tempera-ture eo, s of the MARK 45 WD0 cartridge was calculated for values

of input current ranging from 3.0 to 20.0 amps. The o,s point wasselected because it, being in the center and at the bead's inner sur-face (nearest the bridgewire), is always the hottest point in thebead. The initiator sensitivity for a given current and condition-ing temperature, was then determined by observing the time reqairedfor -, , to reach a value equal to the difference between the assumedautoignition temperature of 2700C and the conditioning temperature.Conditioning temperatures of -80, 68, and 2250F were investigated.These correspond to -62.22, 20.00, and 107.20 C, respectively. Thecomputed sensitivities at the three conditioning temperatures arepresented by the three curves in Figure 5.

Figure 7 is a plot of eo, s versus time for various currents.The intersections of these curves with the horizontal lines givesthe times at which ignition is assumed to occur. The sensitivitycurves of Figure 5 are derived from this infozmation.

In Figure 7, the curve of e0 , s versus time, for 1=3 amps,0 indicates that a steady state temperature below the autolgnition

temperature would be reached at condit! &)ning temperatures below-10 C (14.0 F). Hence, as an example, steady state temiperatures

11

NWL REPORT NO. 1919

were computed for the bridgewire and bead for I=3 amps.. Figure 8is a plot of these steady state temperatures versus radius r.Figure 9 is a plot of the same steady state temperatures versusaxial position x.

'Table 1 shows that values for h, k, p, and c (not beingavailable in the literature) were estimated from related data. Thegeometry of the outer boundary was arbitrary and the autoignitiontemperature Ta was based on a gross experiment. To determine thesensitivity of the solutions .o each of these parameters, the sen-sitivity of the MARK 45 MOD 0 cartridges was recalculated with eachuf the above parameters changed one at a time. Since p and c appearonly as a product, only the product pc was changed. The inputparameters Ta, h, and k were each increased 20%. The product pc wasdecreased 20%. The bead radius and the bridgewire and bead lengthvere each increased 50%. Each of these changes is reflected in thesensitivity of the initiator. In Figures 10 through 15, the newsensitivity is compared to that originally calculated and shown inFigure 5. The comparisons are made for a conditioning temperatureof 680F.

CONCUJING REMARKS

A mathematical heat transfer model has been developed which isamenable to solution using a medium size analog computer. The modelis applicable to determining the sensitivity of a conventional hotwire initiator. While the model has been applied only to the MARK 45XDD 0 impulse cartridge, its ability to faithfully reproduce the roomtemperature sensitivity of this cartridge over a range of firingtimes from 15 to 1000 milliseconds demonstrates the adequacy of themodel to account for the important parameters adsociated'with a hotwire initiator of this basic design. Additionally, the model predictswith reasonable accuracy the sensitivity of the MARK 45 MOD 0 impulsecartridge conditioned at 225 F and at -80 0 F.

At present, fundamental data required as input to the model isonly partially available. However, the usefulnes. of the model isnot lost since, as was done in this study, the model can be calibratedto reproduce the available sensitivity data. The sensitivity for otherconditions can then be predicted. For example, the model is notrestricted to DC input currents. However, if the sensitivity toDC input currents is known then the model can be calibrated to repro-duce this data. Subsequently, the model could serve as a powerful t o1to predict the sensitivity to any time varying input current, e.g.sinusoidal. Or, it could be used to predict (or investigate) theeffects on sensitivity of changing one or more design parameters suchas the bridgewire material or bridgewire diameter.

12

$nut LA tu T Twill"I'Villu 14, 1 -; --41F-r- r U. M,

oil.IH 1H

F. W

40 77:77-HE-

oj4j, . : : : . mm:

0

3 laq 41.40 0

v4

TI40

W4ow

0ram

::z .. :l

40

71

z z

A N p

MN

2,

-++ + -:I cy

i's s, 1: 11:

0 0 -i 0 0g 8 co \0 cu

cu

(30) 240

ao

2. BrIdgewire *ad Dead Towakwo

-2. kidgovire sad Dead Tmp.t

&to X, - .4 X Ia-

.31.

4- - 4---±:

01135 67r (six lot)

Fla= 8

hYU.7 RLM UDIAL TIMM AMMONfIUM - I - 3 AM

170 -Z& -W

15 2. -- & isaertuz. at r.- 0.90O102 2 -~ 43. Dea Taesa at r. -. 95101 102

Dea ?wetiare at rl 1.20711. 1r2 a

5.0: ~ II BW' ?mperstzro at re - 1.89650 X 10-214 - 6 . B a Ia s u t r -5 2 X 1 -

130 2 . -v-:--

0

70--:

IIlk

30--

0 2 46 8 1 21

x (ca x 10?)

FIGURE 9

8TMDY STAT1E AXIAL TD4PWATURE DISTRIDUYCU 1-3 AMPs

im 1111RAIK'! :UL nrl" Tim To cavrmtWOO i wm 45 mm 0 impa" owtridip

27CPC1,H

14.. Rm T V40C (imroamd 20%)- 1-1 IL

T 411 64

if 444 i 11HH4300- H111

ismOR' "JI1-U

Z7'

T

"It T.

100 I f

80

6140iq "I HOW

-. L1 1,7;j if V -2i

A" :zti-'. U

30T2 !J,'

;M T, $W.

ti: ---t V 11 4

.f+tt0iff", fT4 + jt 4

4 kt + 44T ; 4

4 4 1, 1 .. ; I

+ 4H444 +

2 3 6 8 10 20 30 50 TO 100

am-rmt (emp)

no= 10ZMM OR SMIM.M. 07 A 20% MMMN 12 Ta

AM

1 0 0 0 i : 7 .. . . V/ - -v 2800 _____ 4 _

- - '-- . . ._ i :i I ! I I I

SFiring Time vs urrent 4SmRK 45 MO2DOImapul~se Cartridge

i- ,O'-'-t,: . .... 116 caJ- - sec----2- - F-F- t400 (increaed 20%)

. I , ' ' ' i T - i •-

TT

40.

200 __-41 --

10 ----. . , --- -

*- 1 21 3 4_ _ 6t 02 0 0 8 0

; ' uret (amps) "

60 i727' l l, __ --.

4o , ; k I i I 1 , I i

FIGURE 1.1

RYFWr ON SEI'fV17Y OF A 20% INCRESE IN h

00

600~m 4I 5 mm 0 IUZNL1SO catW14

600

k 1.00 __0_CkO

201

10 2 FI6802 3)0 6 oo.. . . . . . .a. .)

445M 2

60UTc BTIIYO 0%ICUII

TI1000ji~U 1~i

100

2V 3f. 6 802 3 0 81

Firing (mpg) (VXsa

3-UR 13H7U Jula SUTIIT MM A impls DDCRUSF

-rb.4mh18T11:11

T..

80

2t 3U~l mm 55 M7 10 30~d 10 6 8010

FI~X 10-

3 T~ S 8~IIY1Y!OF A 0% I.81,1IIN-2

300_

1000 tz4.

600~JU 45 MW 0 INVA150 *Stri4W .. .

Pi L 3 .987 ~(xrs~~%

. . .. ... ----... .... L~ 1.4 ca -.-.. 5.... .......

100"1

300 V m gx..1T

IF

100 2 8 1 0 0 h 6 00

43r1t aua

GoUR 1

60~ V. vIITT OFAt~ica l

NWL REPORT ND. 1919

The usefulness of the model as a starting point for thedesign of new hot wire initiators is yet to be demonstrated. How-ever, the work reported here indicates the model is adequate topredict representative firing times versus input stimuli of conven-tional hot wire initiators.

REFERENCES

(a) Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids,Oxford University Press, Amen House, London E.C.4, 1959

(b) Cook, M. A., The Science of High Explosives, Reinhold PublishingCorporation, New York, New York, 1958

(c) Zinn, John and Mader, Charles L., Jcurnal of Applied Physics 31,323 (1960)

(d) BUWEPS LD 537926 end all drawings listed thereon - "Cartridge,Impulbe, MARK 45 MDD 0"

(e) Handbook of Chemistry and Physics, 37th edition, 1955(f) Fine, Morris E., "Correlation Between Electrical and Thermal

Conductivity in Nickel and Nickel Alloys", JourrRL of Metals 2,951 (1950)

13

APPENDIX A

NWL. REPORT ND. 1919

APPROXIMATION OF FIRST AND SECOND PARTIAL DERIVATIVES AND THE IAPIACIANBY FINITE DIFFERENCES - UNEQUAL SEMENIS AND CYLINDRICAL SMETRY

The Iaplacian in cylindrical coordinates for a system havingcylindrical symmetry is

S219 = ± __e + 1: Le + e. (Al)3r2 r 3r )X2

To evaluate - and Ler a parabola lying in the r-e plane with x=x n is

fitted through the three points (-pmOn m-1)) (Oen0m), and (yPm)0nm+1)

as shown in Figure 16. The equation of this parabola with origin ofthe coordinate system arbitraaily chosen at radial station r=rm is

el = A(r-rm) 2 + B(r-rm) + C, (A2)

=~eB, (A3)

r:rm

62e 2A. (A4)

r X=n

Also,

en,m_ = A(-p n B(-p ) + C, (.A5)

m M = A(0) 2 + B(O) + C, (A6)

and

e A( 7p ) 2 + B(ypm) + C. (A7)

n 1

-- ....VL REFORT NO. 1919

Simultaneous solution of equations (AS), (A), and (A7) for A, B, and Cyields

2 On., m+1 (Y+1) On) m Yn (A)Pm 7(7+I ) [en7n, -i

B+) en,m+l " (1_Y2) enm 2 2 (A91

C = en,- (A10)

Combining equations (AB) and (A) with equations (A3) and (A4)results in

=r ..... [ - (.nr 2 ) e - 7e (All)

X n

r~r

r [ - (i-(7) + + M+1 . (AJ 2X=:Xn -

r=r m

To evaluate e a parabola lying in the x, 0 plane with r=rm is fittedthrough the three points (-qn, en ) (oe and ( neenlm asshown in Figure 16. Analogous to the approximation obtained forFe. a similar approximation for e isC3r 2 X2

( -2 on+lm - (-V) On,,m + P - (A13)

r=; m

2

6 n, rn-i Or P

f~m +

(9 n+IM I

IL

Qr= rm')Ficanm 16

MIhTE Dn7FMCX ODNTJTON FOR r- ) 2

14WI REPORT NO. 1919

Combining equations (All), (A12), and (A13) with equation (Al) results

in

x~x ( 2rnPr"O = On, m+1'X=Vn 7(Y-. ) Pm rm

- (rm 1-y) m + 2 )n, m, (A'

+ ( 2 e

+ 2 OnnlI m'

3

APPENDIX B