RADIATION EFFECTS IN PROPELLAKT BURKIKG
917
tion, and equation 71 will still give an overestimate for (a In X j a In P ) ~ An ~ . approximate method of handling the case of moderately large values of El has recently been formulated by R. G. Parr. We believe, however, that the above calculation indicates that our neglect of E, is not likely to result in too great an error. It might result in E , and some of our temperature parameters being assigned incorrect values, in spite of the fact that the latter, in particular, appear so reasonable. However, we do not believe the effect will be as great as that caused by neglecting diffusion entirely. If diffusion is not considered we get (19) 2': = 770, T : = 1100, and E , = 17,000, but with these parameters the curves fit the data about as well as is indicated in figure 4. The values of T i and T i used here seem much more probable.
RADIATIOS EFFECTS I S PROPELLA4NTBURSING' W. H . AVERY T h e Applied Physics Laboratory, T h e J o h n s H o p k i n s Gnioersity, Silver S p r i n g , Maryland Received J a n u a r y 9 , 1950 INTRODCCTION
When the development of military rockets was begun in the United States in 1941, ballistic considerations required the use of powder grains2 having dimensions greatly exceeding those common for use in guns. Grains with web thicknesses3 of the order of half an inch and burning times of 1 sec. or more became necessary. Although English experience with Cordite SC, which is a transparent propellant, showed that good performance could be obtained with that powder in rockets, American experience with hotter (Le., higher flame temperature) compositions showed that transparent grains of these powders on burning developed a pitted, termitic looking surface, leading to a greatly increased gas production and excessive rocket pressures. The effect was particularly pronounced in propellants containing inorganic salts. After the usual mystical explanations of the phenomenon had been rejected, it was proved conclusively that the effect was caused by subsurface heating and ignition of powder in contact Ivith radiationabsorbing specks of foreign material unavoidably present in the propellant. Presented a t the Symposium on Kinetics of Propellants which was held under the auspices of the Division of Physical and Inorganic Chemistry a t the 112th Meeting of the American Chemical Society, S e w York City, September 15, 1947. * A powder grain is defined t o be one piece of propellant. A propellant grain for use in rockets may weigh upwards of 100 Ib. The web thickness is defined as the distance through which burning takes place in the consumption of a propellant grain.
918
W. H. AVERY
The obvious (and successful) solution t o the difficulty was to include in the propellant composition sufficient darkening material, e.g., carbon black, to prevent significant penetration of radiation beyond the burning surface. When this was done, it was found that powders containing only small amounts of darkening material displayed pressure-time curves with a pronounced tendency to curve up toward the end of burning, in some cases even rising to a high peak. Qualitatively it appeared evident that the phenomenon could be explained by internal heating of the powder by the hot propellant gases. I t was desirable, however, to investigate the nature of the effect quantitatively in order to make use of it in the design of neutral-burning charges and in the estimation of the effect of variations in salt content or amount of darkening agent in a service propellant. EXPERIMENTAL
In the charges employed in experimental rockets, the surface decreased nearly uniformly during burning, so that the pressure-time curve of a powder grain
FIG. 1
FIG. 2
FIG.1. Pressure-time plot of unsalted propellant FIG.2. Pressure-time plot of salted propellant
which did not show the radiation effect would be of the form shown in figure 1, while that typical of hot, salted powders with a small amount of darkening agent would be of the form shown in figure 2. For service rockets it was desired to have the more nearly rectangular pressure-time curve exhibited by the salted powders since, with this type of curve, a desired value of the average thrust could be obtained with a minimum strength of the rocket chamber. The propellant grains used in the studies described in this report were tubular, the dimensions being approximately those shown in figure 3. For the grain shown, the initial surface area is 22.3 in?, and the area at the end of burning is 20.1 ins2 The grains were burned in rocket motors, adapted for static experiments and equipped with instruments which recorded the pressure in the rocket chamber as a function of burning time (figures 1 and 2). From the pressure-time curve, the burning rate at any instant was obtained as follows: At any instant, the mass rate of consumption of propellant is ?iLs =
in which
= S = p = T
rsp
burning rate normal to burning surface, area of burning surface, and propellant density.
(1)
RADIATION EFFECTS I N PROPELLANT BURNIX0
919
The burning propellant generates gas at a pressure, P, part of which may remain within the rocket chamber, the remainder being discharged through the nozzle. That is, mb
= m,
+ md
The rate of change of mass within the rocket chamber is:
= porS
P + aVl ddt
FIG.3
in which
CY
= constant factor for converting density to pressure for the par-
ticular propellant composition and gas temperature, ps
= gas density, and
V t = V , - V,,
=
volume of rocket chamber minus propellant volume at time t.
The mass rate of discharge of gas from the rocket chamber is =
CDPA~
(3)
in which C D = discharge coefficient4 and A t = area of rocket nozzle throat. Equating equation 1 to equations 2 and 3 and solving for r :
r=
1
(4)
4 The discharge coefficient varies slightly with the pressure. For the pressure used in these studies the variation was small enough to be neglected.
920
W. H. AVERY
Equation 4 may be used to obtain the value of the burning rate a t any instant during burning if the surface area and instantaneous volume may be expressed as a function of the burning time. This may be done as follows: At the time, f b , at which the propellant is entirely consumed
in which Ti, = volume of the empty rocket chamber, P, = pressure at the end of burning, and M = initial mass of propellant. At any time, t, during burning the mass of propellant that has been burned will be :
Thus the fraction, f u l l of the mass burned at time t will
l l P dt
ma .fd
=
g
=
il*
P dt
+ aPVt
-
+ aP,B,
(7)
For a grain of the type shown in figure 3 it may easily be shown that
in which f w t is given implicity by the relation
in which fwt = fraction of the web burned a t time t, W = web thickness, 1 = length of the propellant grain, and
Equations 4 through 9 permit the burning rate at successive times during burning of a given round to be expressed as a function of the instantaneous pressures. If several rounds are fired at various pressures, it is possible also to obtain a plot by the usual means of average burning rate us. average pressure. If the instantaneous rates obtained for the given round are then compared with the aoerage rate for the same pressures, it is found that salted powders show a progressive increase in instantaneous rate relative to average rate, as the po\Tder burns. Typical curves obtained in this manner for two rounds of a standard salted powder are shown in figure 4.
92 1
RADIATION EFFECTS IN PROPELLANT BURNIXG
I t is apparent that there is an increase in rate (referred to constant pressure) of about 5 per cent. Owing to the sensitivity of the rocket pressure to changes in propellant burning rate, this 5 per cent rate increase produces an increase in pressure of about 20 per cent. THEORETICAL
To explain a change in the burning rate with burning time, it is necessary to find a factor, on which the rate depends, that does not reach an equilibrium value at the same time as the pressure does. Apart from the powder composi-
laso 1.025
1.wo .9?S
.950 0
.I
.2
w d
F R ~ S C T I O N . ~ ~BFU R ~ E D
.I
.e
.9
1.0
FIQ.4rt. Observed curve, round 1 FIQ.4b. Observed curve, round 2
tion, the variables which conceivably can affect the burning rate at constant pressures are: ( I ) the gas temperature, ( 2 ) the gas velocity parallel to the powder surface, (3) the powder temperature, (4)the intensity of radiation impinging on the poivder surface, and ( 5 ) the gas composition. The gas temperature and composition are constant at constant pressure. Since the port area increases, the gas velocity decreases as burning progresses and this should reduce the rate. However, the temperature of the powder can be expected to increase as burning progresses, because heat is conducted into the interior from the hot surface and because at least some of the radiation emitted by the powder gases penetrates the surface layer and is absorbed in the inside of the web; finally, the radiation intensity will, in general, increase because the thickness of the radiating layer increases. Of the possibilities enumerated, it,
922
W. € AVERY I.
appears that only the last two provide possible explanations of the progressive nature of the curves. WARMING O F POWDER BY HEAT CONDUCTION
I t can be shown that the temperature increment produced by conducted heat at a point inside a grain burning normal to the surface, a t a rate, T , is given by
T
- T ,e - b C / l l ) r r
(10)
provided a steady state has been reached. Here
T. = temperature at a depth of z centimeters below the burning surface, T , = surface temperature,
K
= thermal conductivity of the powder, C = heat capacity of the powder, p = density, and z = the distance of a given point in the powder from the surface.
TABLE 1 TempeTaluTe increment pToduced by heat conduction at various depths below the burning surface z DIFTK ~
cm.
1 0.1 0.01
0.002
'C.
10-10' 10-102
104.'
m.
0.001 0.0005
o.Ooo1
'C.
60 236 760
4
In table 1 are listed values for the temperature rise at various distances below the burning surface. The surface temperature is assumed to be 1000'K., the thermal conductivity to be 5 X lo-' cal./cm.a sec. deg./cm., the density to be 1.6 g./cm.8, and the burning rate to be 2.5 cm./sec. This table shows that the temperature increment is insignificant at depths greater than about 10 microns. Therefore, temperature equilibrium with respect to conducted heat is attained almost immediately after ignition, and heat conduction into the web does not provide an explanation of progressive burning. WARMING O F POWDER BY RADIANT ENERGY
Although insufficient data are available to permit an exact evaluation of the warming of the powder by radiation, it is possible to calculate rather simply the approximate change in burning rate to be expected. Consider a layer of powder parallel to the burning surface, of unit cross section and thickness dz, and a t a depth of b cm. (figure 5). Assume also that radiation of intensity 10 ergs/(cm? sec.) is impinging normally on the surface of the powder. We wish to determine the warming of the layer at an original depth b by radiation passing
RADIATION EFFECTS I N PROPELLANT BURNING
923
through the surface, during the time that the burning surface progresses from its position at zero time until the surface reaches the thin layer under consideration. If we let X represent the position of the surface at any time with respect to the position of the layer dz, then X=b-rt (11) where r is the burning rate of the powder and t is the time. Provided the radiation intensity and burning rate are constant, it is easily shown that the energy absorption per unit volume during the time the burning progresses to the depth b is
El = IO - (1 - e-'*)
-we b -
dx
FIG.5 The rise in temperature of the layer during this time is$
in which C = heat capacity and p = density. In a burning tubular grain of propellant, radiation penetrates the propellant from the perforation as well as from the outside surface. If the web thickness is W , then a surface initially at a depth b from one surface, is at a depth W - b from the other surface. The total energy that is absorbed during the time burning progresses to a depth b (where b is less than W / 2 ) is then
E
=
El
+ E2 = Io (e -
kb-l)[e--kb
+ e--k(W--b)
1
and the temperature rise is
The fraction of the web burned when the burning has proceeded to the depth b is f = 2b/W (12) Heat conduction is neglected.
924
w. n.
AVERS
Values of the quantity A Tr C p / I 0 are plotted in figure 6 as a function of j for values of IC ranging from 1 to 100. The web was taken to be 0.3 in. The figure shows that if I O and T remain constant during burning, the temperature just
FIG.6. ATrCpIIO us. fraction of web burned for various absorpt on coefficients
below the burning surface increases nearly linearly with the fraction of the web burned for low values of 12. But if k is large, the temperature rises rapidly to a value which may be much above the initial value, remains at this value until the r e b is nearly burned through, and then again rises steeply to a value twice the initial increase. In figure 7 the quantity ATrCp/Io is plotted as a function of k for values off varying in intervals of 0.1 from 0 to 1.0. I t is interesting to note that the temperature increase for the last sections of the web burned has a maximum for values of k in the neighborhood of 8. I n the above treatment of the temperature increment a t constant burning rate, two important factors have been neglected: (1) the dependence of k and 10 on the wave length of the radiation, and ( 2 ) the variation of Io with the pressure and depth of the emitting layer.
RADIATIOX EFFECTS I N PROPELLANT BURNING
925
The first factor may be dealt with rather easily by dividing the radiation spectrum into segments narrow enough so that the radiant energy and absorption coefficients of the powder may be considered constant within a given segment. The temperature increment is computed from energy absorption in each
0
0
J
2
ABSORPTION COEFFICIENT K(CM7‘)
3 .4 .5 .6 .7 .8 FRPCTON OF wEa BURNED
.9
1.0
FIG.7 FIG.8 FIG.7 . A T T C ~ /uIs~. absorption coefficient for various fractions of the web burned. FIG.8. Curves of pressure u s . fraction of the web burned at various intensities of radiation.
segment and the temperatures so obtained are summed over all segments. When this is done equation 13 becomes: AT
1,Ij-q
1,[1 - e-‘”’*][1
-
I
(15)
For this calculation it is necessary to know the energy distribution and the absorption coefficient of the powder as functions of the wave length. If the powder emits as a “gray” or black body, the energy distribution is given by Planck’s function. Average values of the energy emitted by a black body at 3000°K. for wavelength intervals covering the spectrum are listed in table 2 along nith the absorption coefficients of a typical propellant for the same spectral regions. The data are taken from curves obtained by Dr. J. Beek (0.04-
w. n.
926
AVERY
1.0 p ) and Professor Farrington Daniels (1.0-5.0 in the region 5-10 p has been estimated.
p).
The absorption coefficient
PRESSURE FRACTION OF WEB CURVES AT CONSTANT RADIATION DENSITY
The data obtained above may be used to calculate the change in the shape of a pressure us. time curve (or more conveniently, pressure os. fraction-of-the-webburned curve) that would be associated with different intensities of radiation striking the powder surface. This is done as shown below. TABLE 2 Radiation emitted bv a black bodu at 3000'K. and absorption coefficients of JPT powder BLACX-BODY PADIATION AT
3wO'K. ABSOIPIION COEFFICIENT'
E II
0.3- 0 . 4 0.4- 0 . 5 0.5- 0 . 6 0.6- 0 . 7 0.7- 0 . 8 0.8- 1 . 0 1.0- 2 . 0 2.0- 3 . 0 3.0- 4 . 0 4.0- 5 . 0 5.0-10.0
srgr/cm.'
SCC.
1.27 x 101' 5.38 12.50 20.30 26.45 30.20 21.56 7.76 2.74 1.16 0.40
ergs
calories
cm.-1
1 . 3 x 107 5.4 12.5 20.3 26.5 60.4 215.6 77.6 27.4 11.6 20.0
0.30 1.29 2.99 4.85 6.32 14.43 51.51 18.54 6.55 2.77 4.78
56 34 23 17 12 13 42 64 31 40
* These values for k = d-1 log.L/I are approximate because corrections for light scat. tering, reflections, and refraction at the powder surface have been neglected. Dr. Beek has estimated that reflection would decrease the intensity of the radiation penetrating the surface by about 10 per cent and that the effect of refraction could be approximated by increasing the value of k by 13 per cent. The uncertainty in the absorption coefficient makes these refinements superfluous. Studies of a large number of propellants of the type under discussion show that the dependence of the average burning rate on the average ambient pressure and initial propellant temperature may be represented by the equation
in which C', It, and T1are constants characteristic of the propellant composition and T p is the initial propellant temperature. Since the effect of radiation is to raise the temperature of the burning layer of propellant, the instantaneous rate may be expressed by the equation
927
RADIATION EFFECTS I N PROPELLAKT BURNINQ
in which AT, is the increase in propellant temperature at the burning layer and C”, n’, and T:will have slightly different values from those found by use of the average dependence of rate on average pressure and initid temperature.‘ Assuming that the rate is given by equation 17, and equating this to equation 4. we have:
In this equation C’, n, Ti,p , a , and Co are dependent only on the propellant composition and, for the propellant considered, had the values:
C’ = 2.085 (cm./sec.)(lb. in?)”,
n = 0.72, p =
1.6 g./cm.3, and
a = i.25 X 10-eP (g./cm?).
For a typical rocket the gas density is less than 0.5 per cent of the propellant density, and the rate of change of mass in the chamber is less than 0.5 per cent of the rate of discharge. Neglecting the small terms, the pressure may then be expressed as a function of the fraction of the web burned by the formula
in which So is the initial surface area. Curves for pressure verws fraction of web burned may then be constructed by selecting initial conditions to give a desired pressure, computing the appropriate rate of equation 17, determining the temperature rise when a small fraction f is consumed, calculating the pressure and rate corresponding to this temperature, and repeating at appropriate intervals until the powder is consumed. Figure 8 presents a series of pressure us. fraction-of-the-web-burned curves obtained by the above method. The different curves correspond to different values of the radiation intensity, Le., different fractions of the energy that would be produced by a black body at 3000°K. The curves bring out three important points: ( I ) radiation can have an appreciable effect on the pressure-time curve even when the emissivity of the powder gases is as low as 20 cal./(cm> sec.); ( I )all of the curves except the first, in which there is no radiation, exhibit the “end-peaks” characteristic of salted powders; (3) the assumption of constant emissivity is not a good approximation to the actual conditions, since the formula derived on this basis does not predict pressure-time curves that are markedly progressive over the entire course of the curve even for the case of maximum possible radiation.
+
6 It is evident t h a t the principal change will be in the value of T I ,since 7’: TI aT,. The values of n and C‘ change slightly since the burning time, and hence the average value of AT,, changes with pressure. To the approximation considered here i t has been assumed t h a t n‘ = I I , C“ = C’.
928
w. n.
AVERY
The treatment of the radiation effect may be further extended to take account of changes in radiation density due to increase in the depth of the emitting layer and to changes in pressure. The resulting formulas are complicated and of too specialized applicability to make it worthwhile to present them here. However, incorporation of these refinements leads to satisfactory qualitative agreement between the form of the observed dependence of rate on fraction burned and that calculated. I t appears that quantitative agreement would necessitate the use of somewhat lower values of the gas emissivity than those that were employed in the calculations. SUMMARY
In an effort to explain the action of added potassium salts upon the burning rates of rocket propellants, an examination is made of the factors which can affect the shape of the pressure-time curve of a rocket fired statically under standard conditions. Of the variables considered, it is most reasonable to assume that warming of the powder (1) through heat conduction or ( 2 ) through radiation causes the burning rate, and hence the pressure, to increase with time of burning. A roughly quantitative treatment is made which shows that the effect of heat conduction is negligible but that the warming of the powder by radiation may be appreciable. The radiation effect is studied in some detail and the changes in shape of the pressure-time curves investigated theoretically for two cases in which the postulated conditions are: (a) a constant radiation intensity and constant absorptivity of the powder; ( b ) a radiation intensity constant with time but distributed spectrally according to Planck's radiation law for a black body at 3000"K., the absorption coefficient of the powder being given as a function of the wave length from experimental data. Pressure-time curves calculated by the formulas developed are presented for radiation intensities corresponding to several different fractions of the radiation from a black body at 3000°K.
I wish to express my thanks to Professors Farrington Daniels and Bryce L. Crawford, Jr., for supplying data on the absorption coefficient of the propellant and the emissivity of the gas. I am also indebted to Drs. F. T. McClure and John Beek for helpful discussions. The work described was done at the Allegany Ballistics Laboratory operated by George Washington University under Contract OEMsr-273 with the Office of Scientific Research and Development.
