Measurement of Energy Flux Density Distribution

in the Focus of an Arc Image Furnace

Rodney B. Beyer, Leonard McCulley, and Marjorie W. Evans

A calorimeter for mapping flux density in a focal volume is described. The design is based on an earlierdesign by Broido and Willoughby. It differs chiefly in two ways. First it is much smaller in order topermit its placement inside a test specimen chamber. Second the design provides for an aperture-containing shield which is placed in front of the receiving surface calorimeter in order to limit the area ofbeam which is intercepted. The decay constant is 2 X 10-3 sec'. Flux densities ranging from 5 to 120cal/cm2-sec can be conveniently measured over intercepted beam areas of from 0.03 to 0.2 cm2. Withthis calorimeter the flux-density distribution in the focal volume of a double ellipsoidal arc image furnacesystem was determined.

The arc image furnace is a useful device for supplyingenergy at controllable flux-density levels and for con-trollable times."2 For example, in work soon to bepublished, it has been used in studying the initiation ofdeflagration waves at solid surfaces. The radiationbeam from the double-ellipsoidal carbon arc imagefurnace used in this work, shown schematically in Fig.1, converges toward a 1-cm diam focus in a cone withvertex angle of 1200. Adjustable shutters (not shownin Fig. 1) placed near the sample reflector allow themagnitude of flux density at the focus to be adjusted.The focus is an ellipsoidal volume in which the flux-density varies. For the furnace to be a useful instru-ment in a quantitative study, one must know the flux-density distribution throughout this focal volume andespecially the maximum flux density. The purpose ofthis work was to design a suitable calibrating calo-rimeter, and with it to map the flux-density distribution.

1. Design

The calorimeter design was an adaptation of one byBroido and Willoughby. The receiver was constructedfrom a copper disk, 0.10 cm thick and approximately0.4 cm in radius. This was pressed into a sphericalsegment with an inner radius of 0.55 cm and an innerheight of 0.15 cm. The inner surface was blackenedwith electro-deposited platinum black. A plane silver

This work was carried out at Stanford Research Institute,Menlo Park, California. Rodney Beyer is now at UnitedTechnology Center, Sunnyvale, California.

Received 13 February 1963.This research was supported by Stanford Research Institute

and by the Office of Naval Research.

foil shield, 0.0127 cm thick, with an aperture which wasvaried between 0.025 and 0.508-cm diam, was placed infront of the concave surface to limit the area of the beamintercepted by the receiver. The shield was spaced0.025 cm from the rim of the receiver, with the lineconnecting the center of the aperture and the center ofthe receiver parallel to the optical axis of the system. A0.1 mm diam copper-constantan thermocouple wassilver-soldered to the back side of the receiver near oneedge to measure the temperature rise. The receiverwas mounted on bakelite posts for thermal insulationfrom the holder and the plate containing the aperture.A sketch of the receiver and the shield is shown in Fig.2. The physical constants of the receiver were

P =C =

X =K =

a =m =

density = 8.92 g/cm',specific heat = 0.093 cal/0 C g,thermal conductivity = 0.91 cal/cm C sec,X/pc = thermal diffusivity = 1.095 cm2/sec,thickness = 0.10 cm,mass = 0.406 g.

2. Theory

If heat losses can be neglected, if the gradient acrossthe thickness is small, and if the difference betweenthe temperature measured at the edge and the averagetemperature is small, the average flux density a at theaperture over an aperture area Aa is given over anytime interval At by

m C (Oav')IaA, At/ (1)

where Ma, is the increase of average temperature of thereceiver, and the temperature may be measured any-

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 131

CARBON ARC COAXIAL SLOTTED DISKS

(ARC REFLECTOR IMAGE OF ARC a SAMPLEPOSITION OF CALORIMETER REFLECTOR

Fig. 1. Sketch of double ellipsoidal arc image furnace system.

thermocouple at the edge of the calorimeter, 0E was

to good approximation the average temperature 06av.

This can be tested by considering the problem of a diskof diameter R1 and thickness 6, heated over a circulararea of radius R2 < R1 .

The flow of heat in such a circular disk is representedby the following differential equation where it is as-sumed that the disk is sufficiently thin that only aradial flow exists:

where on the calorimeter. The absorptivity is denotedby a. The flux density versus time curve of a pulseof energy is obtained by differentiating the tempera-ture-time curve of the calorimeter.

Following Broido and Willoughby the characteristicsof the calorimeter are defined in terms of (a) the decayconstant A (sec'),

A = (/Apca), (2)

where h is the heat-loss coefficient of the receiver incal sec -C-1 and A, is the calorimeter surface area,and (b) a differentiated time constant T (sec),

T = (1.782/Kr2). (3)

They showed that, if At/2 < < 1, heat losses can beneglected. Again following Broido and Willoughbyh/A, was estimated by assuming conductive heat lossesto be negligible and convective heat losses to be equalto radiative losses, as calculated from the Stefan-Boltzmann law. In our measurements the calorimetertemperature rise was varied between 10 and 500C andwas normally 200C. For a rise of 200 from room tem-perature of 27CC, A = 2 X 10-3 sec1 . In agreementwith this estimate, the decay constant calculated from acooling curve slope was < 5 X 10-3 sec'. Thus forexposures to pulses even as long as our maximum of 10see the heat losses may be neglected. Broido andWilloughby further showed that, for T small with re-spect to the exposure time, the temperature gradientacross the thickness a is small, and the measure of therate of change of back surface temperature for a uni-formly exposed receiver is a valid measure of the rate ofchange of the average temperature. For this receiverT is 0.0016 sec, while flux measurements were made withexposure times from 0.030-10 sec. Thus AO/At can bemeasured at the back surface. AO/At was constantover the entire range of temperature rise and exposuretime for each of the apertures, showing that heat con-duction or convection from neighboring parts of the ap-paratus was negligible.

The received flux density in our case was not dis-tributed uniformly over the receiver surface. This wasbecause the flux density varied within the focal volumeand, more importantly, the area receiving the flux whichpassed through the aperture was in general smaller thanthe total surface area of the receiver. Thus it wasnecessary to show that the temperature measured by the

= aIR2 H(R2 - r)H(r - t) + K [rr + ±PCs rI (4)

with

0= 0 at t = 0

0, = 0 at r = 0, R,

and

~~~~~~~~ 1 > H(x) =

0 X<0.

Here t and r are time and radial independent variablesand the subscripts t and r indicate partial derivatives.i- is exposure time and IR, is the flux density at thecalorimeter over the circle of radius R2. Equation(4) is solved by reducing it to two partial differentialequations:

Fig. 2. Calorimeter for calibrating radiant flux density.

132 APPLIED OPTICS / Vol. 3, No. 1 / January 1964

0 = K 0, + - r) for R2 < r < RI

with

0 = 0 at t =

0, = 0 at r = R

and

= - 2H(T - t) + K(rr + )

with

for 0S r<R 2

(5) = 2 + I(qr 1) [I(qR 2)Ki(qR) - I(qRI)K(qR 2 )I

0 < r< R2. (13)

Set r = R, in Eq. (12) and r = 0 in Eq. (13). Then

a R2 11 (qR 2)q2 R 11 (qR2 )

(6)

0 = 0 at t =

at the edge (12a)

and

0 = + -[ Ii(qqR2) KI((R - KI(qR2)] at the center.

(13a)

0, = 0 at r = 0.

On taking the Laplace transform

j(s,r) =fj1 e-`t0(t,r)dt

Laplace inversion5 of Eqs. (12a) and (13a) gives in-tegrals which may be evaluated using Cauchy's theoremof residues.6 LetR = R2/R1 . Define

(7)

of Eqs. (5) and (6) along with their boundary condi-tions, one obtains the following ordinary differentialequations in the variable r:

Si = K (°rr +-r) R2 < r < RI (8)

with

° = 0 at r = R,

and

s = - + K (rr +-0,6), 0 < r < R2PCs S r (9)

I) = aIR2 2(R 22 - R12) 2R El (14)

PesaL 8K ±Rt

and

aJR2 F 2 R2 (R2 2 - R 12)(t,0) [ I -ln R + R 2 + R2t -2R2c]

(15)

where

.e-a2' JI( CeiR2)2E(t) = e

i=l Kaj2 aiRJo(aRI)

and

MCM = 2 + e -Ka2t J((aiR2)

icl = Ka~2

Jo(aiRi)

with

r =0 at r = 0.

Define q2 = S/K and a = (aIR2 /Kpc6)(l - e-vr)/S.Then the solutions4 of Eq. (8) and (9) are

0 = CI [ IR(qr) ± IKi,,i) Ko(qr) ], R 2 < r < RI

= + C2Io(qr), 0 < r < R2,

X azRiZlE F ( (+a1)l) 27/r+ mr =OL r!(r + 1)' + - '=1

and Jo, J are the ordinary Bessel functions. The a1are the non-negative solutions of J(aRj) = 0. Then

(10)0(t,RI) = (t,R) - H(t - r)p(t - Rj)

0(t,O) = p(t,O) - H(t - r)(t - r,O).(11)

(16)

(17)

Assume t - < 0. Then Eqs. (16) and (17) becomewhere C1 and C2 are arbitrary constants and I,, and K.are the solutions of the modified Bessel equation. Theconstants, C and C2, are determined by applying thefurther conditions that , 0, in (10) and (11) must agreeidentically at r = R2. Equations (10) and (11) be-come

aft 2 I1(qR)q = I-(qR) [Io(qr)Kl(qR,) + I(qR,)Ko(qr)]

Re < r < R (12)

and

0(t,RI) = (t,R) = O at the edge

0(t,O) = sp(t,O) at the center.

(16a)

(1 7a)

At time t, the average temperature of the disk is givenby

oav(t) = pa1R2tPCs

(18)

The ratio of the temperature at the edge of the disk tothe average temperature at any time, t, is found by com-bining Eqs. (14), (16a), and (18) to give

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 133

1.0 I I

AXIAL 0TRAVERSE

0.90

0.80 APERTURE DIA. 0.127 cm

0.75 | 1 | I I I I

1.0 I

HORIZONTAL0 TRAVERSEQ90~~~~~~~~~

0 APERTURE DIA.= 0.127 cmG80 0 APERTURE DIA.- 0.064cm

I7 I l l l l l

1.0

0.90I ..

oasoI mow

0.80

/ APERTURE DIA. 0.127cm

070 l l l l I l I-0.25 -0.15 0.05 0 005 015 0.25

DISTANCE FROM FOCAL VOLUME CENTER-cmx

Fig. 3. Flux density divided by maximum flux density vsdistance from center of focal volume.

R = 1- - _R2 22E (19)

Note that lim Eg(t) = 0. It is therefore possible to

find a to such that VI VI',. as closely as desired for allt > to, where

RI 2 - R22 (20)8K

The arc length along a geodesic line across the calori-meter surface is 0.828 cm. For an aperture diameter of0.25 cm the arc length subtended by the beam is 0.604cm. In approximation take RI = 0.414 cm and R2 =

0.302 cm. For a typical case with this aperture Ia =20 cal/cm 2 -sec, and At = 1.17 sec for AO = 20CC.

From Eq. (20) ip. = 0.992 for K = 1.095 cm 2 /sec.

Considerations of the asymptotic forms for the Besselfunctions and inspection of the leading terms allow oneto bound the series 2E(t) so that

22-E(t) < e-'1' at t = 1.17 sec.t

Thus the error in approximating V, by q/. in this case isnegligible, and the measurement of temperature at theedge of the calorimeter gives a good approximation tothe average temperature.

In summary the analysis shows that the average fluxdensity at the plane of the aperture and over the aper-ture area may be calculated from Eq. (1), where AO,,/ At

is to good approximation the rate of temperature rise atthe thermocouple.

3. Flux Density Distribution in the Focal Volume

Flux density in the focal volume was mapped bytraversing along axial, horizontal, and vertical axeswith constant apertures in the range 0.025-0.508 cmdiam. In further experiments the calorimeter was cen-tered at a selected point, and the flux density was meas-ured for different apertures over the same range.Figure 3 shows flux density normalized by maximumflux density, as a function of vertical, horizontal, andaxial distance from the point of maximum flux. Theaperture was 0.127 cm. For the horizontal traverse,points for an aperture of 0.064 cm are also shown; theshape of the surfaces of constant flux density was thesame for each aperture. The dispersion is believed duechiefly to wandering of the arc and hence of the imageover the aperture.

A typical curve of measured flux density versus aper-ture diameter, with the calorimeter in this case centeredat the point of maximum flux density, is given in Fig. 4.An absorptivity a of 0.97 is assumed. For aperturediameters smaller than 0.250 cm the apparent fluxdensity rises sharply. For any given aperture, overthe entire range of apertures used, the flux densitydrops off as one moves away from the focal volumecenter. Therefore the increase in apparent flux densitywith decrease in aperture at given calorimeter positionmust be related to aperture thickness; it cannot be dueto the existence of local regions of extremely high flux.At an aperture of 0.250 cm the aperture thickness beginsto be a significant fraction of the radius, the ratio being0.1. A thickness-related increase in apparent fluxdensity can arise from reflections from the walls of an

300

280

260

240E

I. 220

1 200

>. 180en

w 1600

D 1401jIL

120

100

0) 0.10 0.20 0.30

APERTURE DIAMETER-cm0.40 0.50

Fig. 4. Apparent flux density at center of focal volume vsaperture diameter.

134 APPLIED OPTICS / Vol. 3, No. 1 / January 1964

aperture placed in a focal volume. From geometricoptics it is clear that proper placing of a thick aperturewithin the focal volume can substantially increase theflux passing through the aperture above the amountwhich would pass if the shield were infinitely thin.Auxiliary experiments confirmed that apparent fluxdensity was dependent on foil thickness at small aper-tures.

The data, which include Figs. 3 and 4 as well astraverses with larger apertures, are consistent under theassumption that effects due to reflection from the wall ofthe aperture are negligible at apertures of 0.25 cmand greater. They depict an image with roughlyellipsoidal surface of constant flux, the maximum fluxdensity being at the center. The flux density falls offto 97-99 % of maximum at 0.05 cm from the center, andto 88-95 % of maximum at 0.125 cm from center. Themaximum value is 5% higher than that measured bythe 0.250 cm aperture centered at the focal center.

I. A. Illing performed the experimental measure-ments. Thanks are due to numerous others for sug-gestions and cooperation. Among these are C. P.Butler of the U. S. Naval Radiological Defense Labora-tory in San Francisco; John H. Hett of Hett As-sociates, Inc., Cresskill, N. J.; and C. M. Ablow,Norman Fishman, and Henry Wise of Stanford Re-search Institute.

References1. M. R. Null and W. W. Lozier, Rev. Sci. Instr. 29, 163 (1958).2. N. K. Hiester and R. E. De La Rue, Am. Rocket Soc. J. 30,

928 (1960).3. A. Broido and A. B. Willoughby, J. Opt. Soc. Am. 48, 344

(1958).4. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids

(Oxford University Press, New York, 1959), 2nd ed., Ap-pendix III.

5. Ref. 4, Appendix I.6. See for example E. T. Copson, An Introduction to the Theory

of Functions of a Complex Variable (Clarendon Press, Oxford,1935), 1st ed., p. 117.

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January 1965

Books continued from page 14

optical transfer function, corrections of aberrations, and testing ofoptical components and instruments. Another completely newchapter is called "The Limitations of Optical Instruments",and its content ranges over a variety of topics, including "ac-cidental and fundamental limitations", information theory,noise, and idealized and practical receptors. The major short-coming of the book remains the same; reading it, one often getsthe feeling that there is nothing more to be said on the subjectunder consideration, while, in fact, the treatment is usually super-ficial (as it must of necessity be, in such a one-volume under-taking as this). The diligent reader, however, can make use ofthe adequate and timely references.

The book will continue to be an up-to-date encyclopedia,indispensable to anyone with an interest in optics.

SUMNER P. DAVIS

January 1964 / Vol. 3, No. 1 / APPLIED OPTICS 135