ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Transient Heat Flow in Organic Materials Exposed to High Intensity Thermal Radiation HOYT C. HOTTEL AND CURTIS C. WILLIAMS Ill1 Massachusetts Institute o f Technology, C a m b r i d g e , Mass,
T
HE highly incendiary character of nuclear detonations has focused attention on the behavior of various materials exposed to short duration pulses of high intensity thermal radiation. Of particular interest are the combinations of irradiation intensity and exposure time needed to produce thermal damage in the target specimens. Previous quantitative studies on this general subject ( 1 , 2 , 4 , 1 0 , 1 4 , 1 6 )have been carried out a t heating rates substantially lower than most of those used in the present program, the current intensities ranging up to 6 cal./sq. em.-see. An exposure of only a few seconds to such a high radiaiit flux will produce severe burns on skin tissue, ignite a dry wood specimen, and severely damage many other organic materials. The initiation of such destruction, while an exceedingly complex phenomenon, is unquestionably related to the temperature behavior in the irradiated samples, and it would thus be desirable to have a quantitative means of predicting the temperature-space-time relationships during the first few seconds of the irradiation period. This report deals with the analytical picture developed for this purpose and its relation to experimental fact.
WHEN A NUCLEAR EXPLOSION OCCURS W h a t happens to exposed organic materials? This i s a study of the prediction of space-time-temperature relationships during the first seconds of irradiation
Analytical studies define factors affecting internal temperature patterns
A mathematical analysis has been made of the transient heat fdow in samples irradiated under these conditions and account taken of the following factors: 1. Slab thickness 2. Sample diathermancy 3. Convective and reradiative heat losses from the irradiated surface 4. Chemical damage prior to ignition In developing a model for the physical situation, it was necessary to make the following simplifying assumptions: 1. The specimens are one-dimensional-Le., the minimum dimension of the uniformly irradiated area is large relative to the depth below the irradiated surface under consideration. 2. The physical properties and thickness of the samples remain constant throughout the exposure. 1
3. The irradiation intensity is uniform over the target area and constant throughout the exposure. 4. The materials under irradiation are homogeneous and isotropic. 5 . The penetration of monochromatic radiant energy into the sample may be characterized by a Lambert’s law decay expression of the form
I,
=
Iae-
YZ
where ITz represents the net inward flux a t depth x, I o the intensity of unreflected radiation a t the surface, and y the absorption coefficient. The validity of these assumptions is discussed elsewhere ( 17’). I n view of the high intensities and short exposures employed and the low thermal conductivities of most organic materials, it was not surprising to find that, in the thickness range of primary interest, the specimens behaved as semi-infinite solids. Thus there would be no measurable temperature rise 5 mm. below the irradiated surface of an oven-dried Western pine sample exposed for 5 seconds to a 5 cal./sq. em.-sec. flux. If, however, one is alp0 interested in the behavior of thin materials such as “forest fuels”-i.e., leaves and blades of grass-he must extend this analysis by including an allowance for heat losses from the unirradiated surface. Bamford, Crank, and Malan ( 1 ) were able to correlate midplane temperature histories in wood samples exposed to irradiation intensities of less than 1 cal./sq. em.-sec. by postulatingafirstorder exothermic decomposition reaction. Lawrenre ( 9 ) , using essentially the same heat of reaction, activation energy, and specific reaction rate, extended their numerical analysis to the intensity range of present, interest and found chemical reaction effects to be negligible a t temperatures below 500” C. unless the activation energy was appreciably lower than that previously reported. Thus, the authors felt justified in neglecting factor 4 in the present analysis. Sample diathermancy (or transparency to heat radiation) was found to be a more significant effect, as it would tend to cause a redistribution of the energy throughout the solid resulting in lower temperatures near the surface and higher ones in the interior than would occur if the material were opaque. If one assumes, for the moment, that the penetration of radiant energy can be characterized by a single absorption coefficient, y (as would be the case with monochromatic radiation), then the conventional first law balance on an element of thickness, dz, takes the form (see Nomenclature section)
In the absence of heat losses from the irradiated surface, the boundary conditions are 1. At t 2.
Fort
=
0, for all x, T = TO
>0
Present address, Shell Development Co., Emeryville, Calif.
1136
~
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 6
~
.
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
I ym-w
0. 8
6
4
2
0 0.4
0.8
x/2
X*
Figure 1 , Effect of diathermancy on dimensionless temperature histories in semi-infinite solids
As x approaches
1.6
1 2
at
2.0
2.4
\iot
Figure 2. Effect of diathermancy on dimensionless temperature profiles in semi-infinite solids
m
Both Reuter ( I @ , who studied the melting of snow, and Van der Held ( 1 6 ) , in his work on heat conduction in fibrillar materials, arrived a t Equation 1, albeit with somewhat different boundary conditions. The present writers, in attacking the problem with Laplace transform techniques ( l 7 ) , followed an approach which can more readily be extended to other cases of interest than either of the methods heretofore employed. The solution thus obtained for the present situation may be written in either of the dimensionless forms
or
Figure 3. Energy transmission vs. depth for sample diathermancy witb different values in two spectral regions e Z ( Y d Z t ) ( x / 2 d ; ) erfc
[
z/22/al
+
Equations 2A and 2B, shown graphically in Figures 1 and 2, describe the effect of diathermancy on the dimensionless temperature patterns in semi-infinite solids. The reader will recognize the abscissa groups on these two plots as forms of the dimensionless time-position combination characteristic of unsteadystate diffusional processes and will note in the ordinate groupings the linear relationship between A T and the intensity of unreflected radiation (which applies in the absence of fourth-power heat losses from the irradiated surface). It is also apparent that, as the parameter group (either yx or y 4 2 )increases, the temJune 1955
perature behavior approaches that of the opaque solid-Le., yz or y-\/\/olt equal to m . Hence it can be concluded that a t sufficiently large values of the time or depth, the effect of diathermancy becomes negligible. This analysis may readily be extended to cover the case where various portions of the incident flux are characterized by different y ' s (as occurs when the diathermancy of the sample material is spectrally variant). In such situations, one merely computea the temperature rise a t any depth and time produced by each fraction of the incident flux acting independently, and sums them arithmetically. Expressed mathematically in terms of the dimensionless groups used above, this result becomes
INDUSTRIAL AND ENGINEERING CHEMISTRY
1137
ENGINEERING. DESIGN. AND PROCESS DEVELOPMENT only the former necessitated the iterative solution of a Volterra integral equation, while finite difference techniques were required when both factors were considered. The latter computations (which assumed steady-state convection coefficients and a surface emissivity of 0.9) showed that the diminution in the internal AT’S due to surface losses would be less than 10% at temperature levels below those characteristic of damage initiation in most organic materials. Graphite resistance furnace and solar-focusing mirror provide irradiated samples
The experimental phase of this study included the irradiation, with two high intensity sources, of test samples in which fine-wire thermocouples were embedded, I n addition, the energy transmission of one of the test materials was measured as a function of sample thickness and source spect,ralquality. Radiant S o u r c e s . T w o r a d i a n t sources were used in this program in order to determine the effect of variations in the source spectral quality, a t constant intensity, on the behavior of the irradiated specimens. One of these units was a segmented-mirror solar-focusing device capable of concentrating solar radiation to an irradiation intensity of 5.7 cal./sq. cm.-sec. and of delivering this flux uniformly (within 10%) over a 2-inchFigure 4.
Vertical section of graphite resistance furnace 1.
Insulating spacer, alumina
C. Conducting spacer, carbon
(3) where Fi represents the fraction of the total incident flux characterized by the absorption coefficient, y i , and the functional notation, 6,refers to the solution for ( A T ) k presented as Equa-
lox
tion 2A. t Under these conditions, the net inward flux by radiation a t any depth, x, is given by the expression (4)
Figure 3 represents a plot of Equation 4 for the simplified case in which only two absorption coefficients, 7 1 and 7 2 , are required t o describe the internal penetration of the radiant flux. The graph shows that a spectral variation in sample diathermancy is equivalent to a spatial variation in the local absorption coefficient, given by the slope of the curve. Allowance has also been made in this treatment for convective and reradiative losses from the irradiated surface. Inclusion of
1138
0.I
I. 0
at X I
Figure
5. Temperatures in plastic samples exposed
to
solai radiation Fi = 0.85, yi = = 4 ern.-'
Curves based o n Equation 3 and energy partition:
16 an.-’;
F2
= 0.15;
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
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Vol. 47, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
- _-
pared in a two-stage casting operation, the end result of which was the location of the rare-metal thermocouples (lap welded from 0.001-inch diameter platinum, 10% rhodium and 50% gold, 40% palladium, 10% rhodium wires) a t several known depths
Figure 6.
Transmissivities of plastic samples exposed to solar radiation
Curve based on Equation 4 and y's predicted from temperature measurements
c
square area. The reader is referred to the report of Gardon (6)who designed and operated this mirror. The majority of the results presented here were obtained with EL 2000" K. graphite resistance furnace. A section view of this source, taken in a vertical plane parallel to the target surface, is shown in Figure 4. The resistor array is made up of 1 X 1/8 inch cross-section giaphite heaters stacked one on top of the other with alumina (insulating) arid carbon (conducting) spacers alternating a t opposite ends of each element, the total effect being a series flow of current through the radiating grid. By mounting t h e resistors in this way and carefully choosing the element spacing relative t o their width and thickness, it was hoped to achieve substantially black body radiation from the furnace port. The maximum attainable temperature level in this furnace (2000" to 2100" K.) was dictated by the breakdown characteristics 01 the alumina insulators. At these temperatures, it was of course necessary to bathe the grid in a protective nitrogen atmosphere. Even with this precaution, however, marked oxidation of the elements occurred, and the useful operating life of a single set was consequently limited to approximately 20 minutes at 2000" IC. With this furnace temperature, varying the locatiori of the samples relative to the heater array resulted in a n lrradistion intensity range for these experiments of 0.26 t o 3.5 Gal./ sq. cm.-sec., these fluxes being uniform (within 15%) over a 1lnch diameter target area. The irradiation intensities from the two sources were measured with a circular foil radiometer of the type described by Gardon
below the exposed surface. An attempt was made to vary plastic diathermancy by blending the liquid polyester with different amounts of tar. While the differences thus produced could be detected by transmissivity measurements (Figures 6 and 8) they were apparently masked by the scatter of the temperature data. Further details on the preparation of the plastic samples may be found in the S.M. thesis of Hopkins (8),done in association with the present study. The location of thermocouples in specimens of the two test woods presented somewhat more of a problem. The technique deemed the most satisfactory of the several tested in this program involved the insertion of No. 30 gage (0.010-inch diameter) butt welded iron-constantan thermocouples in holes drilled with a No. 80 (0.013-inch diameter) drill parallel t o the irradiated surface. The wood samples, containing the thermocouples, were oven-dried a t 105" C. for 24 hours and then kept in a desiccator over calcium chloride until they were to be irradiated. A detailed evaluation of this and other techniques for wood sample preparation is presented elsewhere (17'). The rapidly changing e.m.f. values from the embedded fine-wire thermocouples were recorded on a 12-channel Heiland AR-500 oscillograph. Transmissivity Measurements. The energy transmitted by several thicknesses of the polyester plastic was measured as a function of the source spectral quality. A simple radiometer with a wide field of view was used to determine the total energy transmission of samples exposed to solar radiation and that from a low temperature (1650" K.) carbon-tube resistance furnace. A Perkin-Elmer Model 12C infrared spectrometer furnished monochromatic transmissivities over a limited portion of the spectrum. Unfortunately, the lower limit of the spectral range
v)
iz0 2 2
(6).
Temperature Measurement and Sample Preparation. A polyester plastic, KO. 28C of the Marco Chemical Co., Linden, N. J., tw** woods^ a (ponderosa) pine and a sapwood birch, were used as sample materials. The plastic specimens were pre-
June 1955
Figure
7. Spectral qualities of radiation in Polyester plastics
(Upper) monochromatic transmissivity measurementsi (lower) radiation from carbon-tube and graphite-resistance furnacer
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1139
'ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table 1. Material Polyester plastic
Oven-dried western (Ponderosa) pine Oven-dried sapwood Birch
'
Maple (8.3% Moisture)b
Physical Properties of Test Materials Property and Symbol
Valuea
Thermal conductivity, k Density, p Specific heat, Cp Reflectance, T
0.00064 1.36 0.28 0.95
Experiment (8)and (17) Experiment, this work
Thermal conductivity Density Specific heat Reflec tance
0,00025 0.38 0.327 0.45
MacLean ( 1 2 ) Experiment ( 1 7 ) Dunlap ( 5 ) Experiment, this work
Thermal conductivity Density Specific heat Reflectance
0.00035 0.59 0,327 0.66
MacLean ( 1 1 ) Experiment ( 1 7 ) Dunlap (5) Experiment, this work
Thermal conductivity Density Specific heat Reflectance
0.00045
}
0.63 0.4
0.5
Source
MacLean ( 1 1 ) Dunlap
(i)' ... '
a CaLgram-sec. system of units used throughout. b Naval Material Laboratory experiments; moisture taken into account in calculating k and C p from literature correlations; no allowance made for moisture migration effects.
1.0
I
'SURFACE I
I
I
I
I
I
I
I
I 0.6
AL?SOfff T/VlTV
figure has been redrawn on Figure 5 as a dotted curve. The experimental results cut across the dotted line in a manner shown by Figure 1 to be characteristic of a diathermanous medium. While an approximate correlation of the data can be achieved by postulating a single absorption coefficient of 16 cm.-' and using Equation 2A, a somewhat better fit results if one assigns a lower y to a small portion (about 1501,) of the incident flux. The correlating curves developed from this assumption and Equation 3, and shown on Figure 5, lie reasonably close to their corresponding data points. It is plain, however, that each depth rises in temperature, in the latter stages of irradiation, less rapidly than the rise that corresponds t o the mathematical relation used. The explanation is unknowii; a possibility is some endothermic polymerization reaction occurring in the material at the higher temperatures. Given values of y for different portions of the incident flux, one can, from Equation 4, express IJIO or, with Table I, I J H as a function of depth. The I J H relationship predicted from the temperature data appears as a curve on Figure 6, for com-
0.1
0.03
I
0
I
0.2
I
0.4
0.6
x.
10
1.0
CM.
Figure 8. Transmissivities of plastic samples irradiated in carbon-tube resistance furnace at 1620-1720' K. Curve bared on Equation 4 and energy partition: Fl = 0.77, 71 = m ; FZ = 0.23; y2 = 2 cm.-'
covered by the calibration of the rock salt prism in this instrument was 2.5 microns, and wave lengths reported below this value are somewhat suspect.
1.0
a 4x 0 1
Heat flow analyses give fair prediction of temperature behavior of plastic and woods
Polyester Plastic. Data points taken from the temperaturetime traces from samples exposed to the solar furnace are shown on Figure 5 in dimensionless form. The physical properties assigned to this plastic and their sources are given in Table I. The format of this plot is particularly useful when data are available a t a few fairly well defined depths (as in this case). I n instances where the depth measurements are somewhat less precise (as in the studies on oven-dried woods), a clearer picture of the interior temperature patterns results from their presentation as dimensionless temperature gradients (see Figure 2). For comparison with Figure 1,the opaque-solid curve from that
1140
0.01 0.1
1.0
cct -
10
It
X2
Figure
9.
Temperatures in plastic samples exposed to 2000" K. radiation
Curves based on Equation 3 and energy partition: FI = 0 . 6 5 , ~ =~m; F2 = 0.35, 7 2 = 2 a n.-'
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 6
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT $
eo
expressed on a per unit irradiation intensity basis, were found to be substantially independent of the incident flux over the range, 0.26 to 3.5 cal./sq. cm.-sec., or in mathematical terms
8 g $ z
z Fb g
B s \
l S
(g)
x, t
I O
f(H)
3.5
> H > 0.26 (5)
0 5
Q.
ilZ
0 0,
7
0 4
:
;
05
~
06
+
0 1
w
Od
'
s
i
a
t
I
0
09
i
r
~
/ J
-
N
!,
IS
c
*
,6
a
,7
i d
is
so
i i
