The combustion of solid monopropellants and composite propellants

THE COMBUSTION OF SOLID MONOPROPELLANTS

AND COMPOSITE PROPELLANTS

M. W. BECKSTEAD, R. L. DERR, AND C. F. PRICE

Lockheed Propulsion Company, Redlands, California

A model describing the combustion of composite propellants has been applied to three different types of composite propellants, and to two oxidizers which burn as monopropellants. The oxidizers studied are AP, HMX, and KP (ammonium perchlorate, cyclotetramethylenetetranitramine, and potassium perchlorate), each of which has unique combustion characteristics. The monopropellant combustion of AP and HMX has been considered in detail, par- ticularly that of AP. Calculations have been compared with experimental data for the burning rate, the sensitivity of the burning rate to both pressure and initial temperature, and the surface temperature. The agreement between the cMculations and experimental data is very adequate. The results indicate that AP and HMX apparently burn with a considerable exothermic reaction at the burning surface as well as in the gas phase above the surface. Both reactions must be accounted for to give consistent results. The combustion of the three oxidizers burning with a fuel in composite propellants has also been simulated. Experimental data are presented for propellants in which the oxidizer has been systematically varied. A direct comparison has been made between the data and the calculations of the model. The calculations agree quite well with the combustion characteristics of the three different types of composite propellants.

Introduction

Several models have been proposed in the past attempting to describe the combustion of composite Propellants (see for example Refs. 1-6). For the most part, these models are qualitative in their description, and examine only a limiting condition or a single "controlling mechanism." The work by Hermance has considered the combustion problem on a sufficiently comprehensive scale as to be able to predict burning-rate be- havior a priori, while other models have only been somewhat successful in correlating experimental data after the fact. Recently, the present authors have developed a model 7 with the intent of arriv- ing at a greater understanding of the controlling factors of the combustion process of composite propellants.

This paper discusses the application of this combustion model to the combustion of solid composite propellants containing different oxidizers. The model is based on the assumption that the gas-phase heat release during combustion can be represented by multiple flamelets sur- rounding individual oxidizer crystals, the relationship between oxidizer crystals and the binder matrix being evaluated statistically. Three separate flames arc considered: 1) a flame (hereafter referred to as the "primary" flame) between the

decomposition products of the binder and the oxidizer; 2) a premixed oxidizer monopropellant flame; and 3) a final diffusion flame between the products of the monopropellant flame and the binder-decomposition products.

The results presented here are concerned with applying the model to mathematical limiting conditions which correspond to the combustion of three different oxidizers in various environ- ments. KP, HMX, and AP (potassium perchlorate, cyclotetramethylenetetranitramine, and ammonium perchlorate) have been used as the oxidizers. All three have unique combustion characteristics which can aid in the evaluation of a model. Specifically, AP and HMX burn as monopropellants, while KP does not. However, H MX is stoichiometrically balanced, while AP is oxidizer rich. Thus, the products of AP burning as a monopropellant can be expected to afterburn when fuel is present, while HMX would not be expected to do so.

For the purpose of comparison, three limiting conditions of the model have been studied. The first and most simple is the case of a solid monopropellant burning with a single premixed flame (corresponding to AP and HMX). The second case is a composite propellant burning with a primary flame; KP-composite propellants fit this category. The third case considered is a corn-

1047

1048 SOLID PROPELLANT COMBUSTION

posite propellant with the primary flame and a monopropellant flame, but no afterburning; t tMX-composi te propellants would seem to fit this category. The lat ter two cases have also been compared with the calculated results for an AP-composite propellant.

In comparing the combustion model with experimental observations, several separate cri- teria have been considered. The results have not only been compared with burning-rate data, but also with the particle-size dependence and concentration dependence of burning rates, with temperature sensitivity data, and with the variation of surface temperature and pressure exponent with pressure. This has permitted the evaluation of the results in terms of a very broad scope of combustion, rather than simply considering the observed burning rate as a function of pressure. Applying the model to describe the combustion of the different oxidizers, both as monopropellants and with fuel present, has also aided in evaluating the validity of the model on a broad basis.

Solid MonopropeUants

In reducing the model to the first limiting case, that of a monopropellant, the approach has been to apply the same form of equations to the monopropellant as used to describe the more-complex composite propellants. This differs from the more- common approach of developing a single complex monopropellant model, such as the Johnson- Nachbar model, s For the monopropeilant case, the model reduces to essentially three equations: one describing the surface kinetics; one resulting from an energy balance at the burning surface; and one related to the gas-phase kinetics.

The surface regression is described as a one-step Arrhenius-type function

mox = Aoxexp ( - -Eox/RT,) . (1)

The surface temperature is assumed to be uniform over the burning surface, and is derived from an energy balance at the surface. I t can be expressed as

T 8 = T o - - (QL/cp) + (Qox f/cp) exp (-~*ox ~),

(2)

where To corresponds to the initial energy in the system, the term involving QL corresponds to the energy released at the burning surface, and the final term represents the energy transported from the flame back to the surface. The heat released

in the flame is defined as

Qox ~ = e~,(To:, -- To) - - QL, (3)

where To~ f is the adiabatic flame temperature. For a premixed laminar flame, the nondimen-

sional flame-standoff distance can be represented by

~*ox = (cp/h )mo~,X*o~ = (c,/)~ ) (m2ox/kP~),

(4)

where the gas-phase rate constant k is an Arrhen- ius function of the flame temperature. The above equations represent the model as applied to monopropellants.

Pressure Exponent and Deflagration Limit

The problem Johnson and Nachbar had in fitting their analysis to experimental da ta was in accounting for the AP pressure deflagration limit where the slope of the burning-rate curve becomes infinite and the rate zero. An effective way of examining this aspect of the problem is to differentiate Eqs. (1), (2), and (4), and solve for the pressure exponent n. The result, after simplification, is

n = (d In m ) / ( d In P )

= 2 --~ [-c, RT2/Eo~Qox f~* exp ( - - ( * ) ] " (5)

An order-of-magnitude evaluation of the second term in the denominator is informative. For typical parameter values, the term has a value of --~0.2, and the pressure exponent is seen to be a little less than half of the gas-phase reaction order, which is the normally expected value.

I t is informative to examine n in detail. From Eq. (5), it can be seen that n is dependent on T, and ~*, which are variables. The variation of n with T, is demonstrated in Fig. 1 for AP, where QL is the varying parameter, and Eq. (2) has been used to evaluate T8 for a given value of ~* (a similar curve can be plotted for H M X or any other monopropellant).

I t can be seen that the pressure exponent can increase or decrease with pressure, depending on the value of QL and on which part of the curve applies to actual combustion. The surface temperatures, measured by Powling, 9 and extrap- olated to cover pressures from 20 to 200 atm, are on the order of 600 ~ to 720~ increasing with increasing pressure. For this range of temperatures, and for a large exothermic surface reaction

SOLID MONOPROPELLANTS AND COMPOSITE PROPELLANTS t049

0.5

0.4

C

0.3

QL : 0 " ' - ~

/ q~-12o

I I I I I " I I

20 ATM

Q L : 4 80

I I I |

DATA OF I POWLING --I

I I

200 ATM

500 600 I00 800

T S (~

F1G. 1. Variation of pressure exponent with surface temperature for AP (Eoz = 22 keal/mole).

(i.e., QL ,~-, --180 cal/g), the pressure exponent will increase with pressure and the burning rate curve will be concave upward. For a less-exothermic reaction, e.g., QL ~'~ 0, the exponent will decrease with increasing pressure and the burning- rate curve will be convex. The former case appears to correspond to HMX burning as a monopropellant, while the latter is more descriptive of AP combustion.

Considering the burning rate and slope of pure AP, as determined by Hightower, I~ the exponent varies from a value of about 0.8 at low pressure to about 0.7 at higher pressure. The ratio of the exponent at high pressure to the value at low pressure is, accordingly, 0.875. From Eq. (6), this same ratio can be calculated, independent of the reaction order for various values of QL (e.g., Fig. 1). Between 600 ~ and 720~ 9 this ratio is

QL (cal/g) n(T~ = 600~ = 720 ~

480 0.98 120 0.95

0 0.95 --120 0.96 --150 1.01

Values of QL on the order of 120 to 0 are higher than the experimental results, but optimum in this respect.

Figure 1 can also be examined to evaluate a reaction order consistent with the data and with the proposed model. Taking the surface temperature and exponent to be 600~ and 0.8, respectively, at low pressure, and 720 ~ and 0.7 at high

pressure, permits a reaction-order value to be calculated for various values of QL. In this manner, a reaction order of 1.7 to 1.8 has been determined. If a reaction order of 2 were assumed, an exponent of 0.9 is calculated, which is obviously too high.

A second-order gas-phase reaction does not give consistent results, while the calculations based on a reaction order of 1.8 do seem to give more consistent results. Therefore, a reaction order of 1.8 has been assumed for the AP-flame reaction. This is somewhat of an arbitrary assumption but, based on the discussion of the above paragraphs, it appears justified in that it does give consistent and reasonable results.

For a reaction order of 1.8, the value of the pressure exponent can be determined for the various values of the surface heat release

QL (cal/g) n(T~ = 600 ~ n(T , = 720 ~

120 0.83 0.80 0 0.82 0.78

-- 120 0.79 0.76 -- 150 0.75 0.76

The value of QL = -- 120 cal/g gives results most consistent with experimental data, and was therefore used in the following calculations.

If the solid were undergoing dissociative sub- limation, QL would be 480 cal/g. The fact that a value on the order of -- 100 cal/g is required for the surface heat release indicates that a very exothermic reaction is occurring at the surface of the AP, accounting for approximately 75% of


0.8

0.7

0.6

~0 .5

0.4

0..3

C,.2

/ / / , .

"~//X/////////A

_ . _ )A OE

20 40 60 PRESSURE (atm)

' FRIEDMAN

i '

QL "-150

"'~iQL=-~20

% 80 i00

FIG. 2. Comparison of calculated and experimental temperature sensitivity of AP.

the available energy. I t appears reasonable to assume that this energy release is occurring in the thin molten layer that covers the surface of the deflagrating AP} ~ These results are consistent with previous, somewhat heuristic arguments, indicating tha t the deflagration of AP is charac- terized by both a condensed-phase and a gas- phase energy release, u

I t is also informative to examine the case of n--~ ~ , which corresponas to the pressure deflagration limit (PDL). If a heat-loss term is introduced into Eq. (2), it enters into the exponent calculation very simply as

n/8 = 12 H- [RT~/EoxQox~* exp ( - -~*)]

X FcpT.- (qL/mo~)]} -~, (6)

seems to be that, as the surface temperature drops with decreasing pressure, it reaches a value below the melting point of AP. m Below this temperature, the thin liquid layer observed on AP crystals m is no longer present as a site for exothermic surface reactions. Without these reactions, deflagration ceases and the AP only de- composes by the so-called high-temperature sublimative mechanism (see, for example, the review by Hall and Pearsonla).

Temperature Sensitivity

The temperature sensitivity of the model can also be readily evaluated. Differentiating Eqs. (1)- (4) with respect to To yields (recalling that k is a function of To~ f, which in turn is dependent o n To)

a~, -- a in mox/a To 1 + (Eo,:JR To,:) (Qox ~/c~,)~* exp ( - ~*) (RT~2/Eox) + (2Qox/cp)~* exp ( - ~ * ) "

(7)

I t is apparent that ~, is pressure dependent also. Utilizing Eqs. (1), (2), (4), and (7), zp can be evaluated as a function of pressure with QL and Eox as parameters. In Fig. 2 there is a comparison of reported literature values 12,14,15 and calculated results. The calculated results are in excellent agreement with the experimental data, considering the wide variance of the experimental results. The calculations indicate that the more exothermic the surface reaction, the greater the temperature sensitivity. For QL values of 0 and -[-480, essentially constant ~p values of 0.47% and 0 .44%/~ were calculated.

4.0

2.0 where qL is the unidentified energy loss discussed by Johnson and Nachbar s (actually a flux). The pressure deflagration limit, considered by John- ~ 1.0 son and Nachbar, corresponds to an infinite 2 0.8 pressure exponent and, according to Eq. (6), can only result from a term such as the qL that has <~ 0. 5

2~ been included. For the denominator to go to zero ~ 0. 4 (and cause n to approach infinity), the surface temperature must decrease, the burning rate must decrease, and/or qL must increase. All of 0. 2 these trends are consistent with what actually occurs at a deflagration limit.

However, for n to approach infinity, qL must 0.1 be on the order of 200 cal/cm2-sec or greater (for 10 typical values). This is very unlikely. Thus, it appears tha t the PDL is not caused by an energy- loss mechanism. A more-consistent argument

y/

Io

01

o HMX OAP

20 40 60 80100 200 PRESSURE (aim)

FIG. 3. Calculated and experimental burning rates for AP and HMX monopropellants.

SOLID MONOPROPELLANTS AND COMPOSITE PROPELLANTS 1051

Burning Rate

Equations (1), (2), and (4) also can be solved for the burning rate, if the rate constant is known. A rate constant can be determined by using the actual burning-rate da ta for AP at a given pressure in the equations. This was done and a set of burning-rate curves were calculated varying QL. Figure 3 contains the computer results with experimental da ta points super- imposed, 1~ where the following parameters were used:

Aox = 0.3 X 106 g /cm 2 sec Eox -- 22 kcal/mole

% = 0.3 cal/g ~ QL = - - 120 cal/g

Eoxf = 30 kcal/mole h = 0.0003 cal/cm sec ~ ~t= 1.8

Tox = 1400~ k = 1.12 gm/cm 3 sec a tm ~.

The calculated surface temperatures ranged between about 570 ~ and 700~ giving values along the lower limit of Powling's da ta? Between 300 and 2500 psi, the standoff distance varied from 20 to 2.5 #, and from 10 to 1.3 # for QL= O. The activation energy of 22 kcal/mole was used used because this is the value tha t is most often reported 12 from low-temperature decomposition studies. Throughout the study, activation energies have been used for the oxidizers reported in the l i terature from low temperature studies.

The second monopropellant to be considered here is HMX; however it has not been studied as extensively as AP in the role of a monopropellant. In the case of AP, both the experimentally determined burning-rate data and surface temperature are very important in making a comparison with the calculations of the model. In the case of I ]MX, neither of these are known, nor is the temperature sensitivity. Some very limited data do exist for the burning rate of pressed strands of HMX. ~6 However, the strands were pressed to a density of only 1.66 g/cc while the density of the pure crystal is 1.91 g/cc. This fact tends to m'~ke the data somewhat suspect. However, due to a lack of any other da ta on the burning-rate characteristics of HMX, the available data have to be accepted as being representative. These da ta indicate that the pressure exponent increases with pressure, which is also the case observed with HMX-composite (not CMDB) propellants. ~ Actually, the exponent increases from a value of between 0.6 and 0.7 to a value greater than 0.9, approaching 1.0. Because of the high slope, it appears that the gas-phase reaction must be second order.

A plot of pressure exponent versus surface temperature that is virtually identical to Fig. 1

can also be constructed for HMX. The results of such a plot indicate that H M X also burns with an exothermic surface reaction. A value of QL between --200 and --250 was determined as necessary to provide results consistent with the known properties. The calculated results for H M X have been plotted in Fig. 3, along with the experimental data. The burning-rate characteristics are seen to be quite similar to those of AP. The parameter values used were:

Aox = 0.5 X 101~ g /cm 2 sec Eo~ = 50 kcal/mole

% = 0.3 cal/g OK QL = - - 225 cal /g

Tox--- 3275~ = 0.0003 cal/cm sec~ = 2.0

k = 0.246 g/cm 3 sec arm ~.

The adiabatic flame temperature was calculated for equilibrium, and the activation energy is a l i terature value. The calculated surface temperature varied from 820~ at 200 psi to 930~ at 2500 psi, and the standoff distance varied from 125 to 7/z over the same range. Again the rate constant k was evaluated from the experimental da ta at one pressure (70 a tm).

C o m p o s i t e P r o p e l l a n t s

KP-Composite Propellant

The second case to be considered is that of a composite propellant (such as KP oxidized propellants), whose oxidizer does not burn as a monopropellant. The only flame considered is that which results from the combustion of the oxidizer and binder-decomposition products. This flame is obviously ctependent on mixing, due to the physical separation of-the oxidizer and binder.

For conditions where mixing is rapid compared to reaction times, the flame will take on the characteristics of a premixed flame. A more- complete discussion of this primary flame, and the corresponding derivations associated with its relationship to composite propellants, is found in Ref. 7.

The limiting condition of the combustion model, applied to a primary flame-composite propellant, is summarized in the following: Equation (1) still applies, but the surface temperature now becomes an averaged temperature of the binder/oxidizer surface. Equation (2) becomes

T~ = To-- c~(QL/Cp) ;-- (1 - - ,u)[Q(fueI)/%]

Jr (QpF/Cp) exp (--~*PF), (8)


where the effect of the binder is introduced in the heat-of-pyrolysis term as well as in the flame- energy-release term. The latter becomes

Qpp -- c p ( T f - To) + aQL-}- (1 -- a)Q(fuel).

(9)

Assuming that the characteristic diffusion and reaction distances are additive, the standoff distance becomes

~%F = (c,m,#x) (x% + x%). (10)

One additional equation is needed to complete the set. I t is 7

mr = (m,~/a) (So~/So) (11)

The term So,,/So represents the fraction of surface area occupied by oxidizer. Equations (1) and (8)-(11) must be solved iteratively for the burning rate.

KP-composite propellants are known to burn with a relatively high and uniform slope (~0.7 to 0.9) over a wide range of pressures. They usu- ally show little effect of particle size or oxidizer concentration. Results of a concurrent experimental program ls,19 have shown that the KP melts and agglomerates on the surface of actual propellants. This indicates that the surface temperature is approximately that of the melting point of the KP (~560~ Within the frame- work of the model, the oxidizer was assumed to melt, but not necessarily mix with the binder. No attempt was made to model the agglomeration, and the surface was assumed to be essentially planar. Thus, the surface area of the oxidizer, Sox, was taken as being a constant for a given propellant loading. I t can be shown 7 that, for a planar surface, the ratio of So=/aSo is unity.

In order to solve Eqs. (8)-(11) and Eq. (1), the diffusional distance X*D must be evaluated. In the original development of the model/ two different forms were considered for this quantity, both of which resulted from an analysis of diffusion-type flames. One corresponds to the laminar diffusion of typical tall flames 2~ and can be written

X*DZ ~ m (�89 (12)

where the diffusivity is

~D = ~Do ( T1.75 / P ) . (13)

Equation (12) is essentially the form used by Hermance 4 and others, 2~ and is dependent on burning rate only. I t can also be shown that, for

very short diffusion flames/ the characteristic dimension is simply

x*Dn o: �89 (14)

Calculations for AP-composite propellants have shown that Eq. (12) (i.e., tall diffusion flames) leads to inconsistent results, and that Eq. (14) (i.e., short flames) appears to give more reasonable results. Comparing Eqs. (12) and (14) with (10) and (4), it is apparent that there are regions where the premixed nature of the flame is controlling (X*R))X*D), and there are regions where diffusion is controlling (X'D))X'R) . The former will normally occur at low pressures, and the latter will eventually result at higher pressures. At high pressures, the burning rate will eventually become independent of pressure, which is the conclusion reached by Nachbar a in his attempt to model composite propellants utilizing an analysis based on a diffusion flame.

Because the slope of KP propellants is uniform over a very wide pressure range and does not approach zero, a third condition has been considered. The characteristic mixing distance has been taken as

X*DIII ~x m (~Do)2/P. (15)

This is equivalent to assuming a pressure-independent diffusion coefficient, or more likely that the diffusion is turbulent rather than laminar. Although the Reynolds number for the propellant surface is lower than that normally associated with turbulence, the presence of solid material, such as KC1, in the gas stream, could trigger turbulence or it could be generated at the surface by the gasifying ingredients.

6.0 J I

I ~1.0 \

0.6 I I I I I \ �9 ~. O. 4 ~ ;R.R' "~ LAMINAR FLAME ~

o.2 I i l l m ~( 78% 50.# KP

u~ 22~/o P.U

J l O. 08 0.06

O. 04 1 6 8 I0 20 40 60 80 I00 200 PRESSURE (atm)

FIG. 4. A comparison of calculated results and experimental data for a KP-composite propellant with three different assumed mixing mechanisms.


Utilizing the three different forms FEqs. (12), (14), and (15)] for the mixing length, the combustion of KP-composite propellants have been studied. Typical results are shown in Fig. 4 for a 78% KP/22% polyurethane propellant. The data that are shown were obtained in this study utilizing unimodal oxidizer of ~-~50/~ diam. Parameter values were taken from the literature where possible, calculated where possible (e.g., Tf, W, etc.), and estimated when not otherwise available. The values used are listed below:

Eox = 70 kcal/mole Aox = 1017 g/cm~-sec QL = 15 cal/g

Q (fuel) = 100 cal/g kpF = 30 g/cm 3 sec atm *

t3= 2.0 cv = 0.3 cal/g ~ h = 0.0003 cal/cm-sec ~ 20 = 0.16 cm2/sec (at room tempera-

ture and pressure) a = 0.78

Tf = 2667~ W = 36.2 g/mole Do = 50/.t po,~ = 2.52 g/cc pf = 1.0 g/cc

ERR = 60 kcal/mole.

The primary-flame rate constant was arbitrarily varied to give consistent burning-rate magnitudes at 1 arm, and a second-order reaction was assumed.

The equations describing the propellant can be differentiated, as was done for the monopro-

pellant case. Four equations relating to pressure, temperature, concentration, and particle-size dependence, can be derived. Utilizing the parameter values given above, the equations can be simplified to give

n = 0 In m/O In p

-- (X*D/X*a) (0 In z*D/O In P ) 2 + (x*/z%) (16)

ap -- (0 In m/OTo)v

~-- E1 "4- cp (EpF/RTI)]/2%T: (17)

aD ~ (0 In m/O In Do) ~'~ -- (x'T) -1 (Ox*D/ODo)

(18)

a , = (0 In talon)

0.2 + (X*R/2X*T) (EpF/RT])

X (0 In T//Oa) - - (x*D/2x*z) (0 In V/Oa).

(19)

The final form of these equations depends on the form of the diffusional distance that is assumed. I t is informative to evaluate these equations numerically and compare the results with available literature data. Utilizing several computer calculations, Eqs. (16)-(19) have been numerically evaluated over a range of pressures. The results, summarized in Table I, indicate that Eq. (15), assuming a turbulent form of diffusion, leads to the most-consistent agreement of the model with experimental results.

TABLE I

KP-composite propellant dependencies

Partical-size Concentration Temperature dependence, dependence, dependence,

,~ ,~ ,~A%/~

10 68 170 10 68 170 10 68 170 10 68 170 Assumptions atm ar atm arm atm atm atm atm arm atm atm atm

I Tall flame, 0.9 0.1 0.03 0.95 0.2 0.06 18 0.2 --1 Eq. (12)

II Short flame, 0.95 0.33 0.2 0.5 0.3 0.2 18 9 3.5 Eq. (14)

III Turbulent 0.75 0.6 0.5 0.7 0.2 0.06 9 0.2 - 1 flame, Eq. (15)

Propellant data 0.9 to 0.7 0.621 ( 0.322 -* 7 ~4 3 i 0.1 ~ 0 ~3 32~ 0 + O--

0.21 0.16 0.14

0.21 0.19 0.20

0.19 0.16 0.14

0.16-0.125.u

Superscript numbers refer to references where the experimental data were found.


H M X-Composite Propellants

The third ease considered here corresponds to the combustion of HMX-composite propellants. The monopropellant characteristics of H M X have already been discussed above. I t has been proposed 18 that nitramine composite propellants burn with the binder acting only as a coolant, and not actually entering into the flame reactions. However, the adiabatic flame temperature of pure H M X is 3275~ while the flame tempera- Lure for a 78% H M X - P U propellant is 1669~ I t appears unlikely that the binder could cool the flame down to tha t extent, acting only as an inert coolant. I t would seem more likely that a primary type of flame is established between some of the H M X decomposition products and the binder products, reacting in a less-energetic manner than would the H M X products by themselves. In either case, the binder does act as a coolant, reducing the burning rate. In one case, the binder is inert, and in the other i t is reactive. Both of these cases have been considered as limiting conditions of the model.

From surface-structure studies, ~s i t appears as though the H M X and the binder melt and, to a degree, mix at the propellant surface. However, within the model as presently structured, this mixing would be very difficult to describe mathe- matically. Therefore, it was again assumed that the surface was planar (i.e., possibly molten), but that the ingredients were not mixed (i.e., c~Sox/So = 1). If the degree of mixing is small, this assumption is probably quite good. Con- densed-phase reactions are considered to occur at the surface and not in depth, the same as before.

Considering the binder as an inert coolant, the cooling effect of the binder enters in the term Q (fuel), the binder heat of pyrolysis. Utilizing input parameters corresponding to a 78% H M X - P U propellant, calculations were made on the burning rate. To fit the experimental data, a value of Q ( f u e l ) = 1800 cal/g was necessary. For smaller values, the calculated burn rates were too high. A value this large for the heat of pyrolysis is highly unlikely, and, therefore, the inert cooling of the binder seems to be inadequate to produce the observed results.

Considering the binder as an active coolant, the equations become considerably more com- plicated. To calculate the surface temperature, written for H M X with no final diffusion flame (the H M X combustion products are slightly fuel rich and therefore do not react with the binder products), Eq. (8) becomes

T~ = T o - a(Qn/cp) -- (1 -- v~)[Q(fuel)/c~J

+ (1 - ~F)C~(Qo~/C~,) exp (--~*o~)J

'Jr flF(QpF/c~) exp (--~*PF), (20)

where f~p is the fraction of oxidizer reacting in the primary flame, t t is calculated as in Ref. 7.

The parameter values used for the HMX calculations were determined from the literature, where available. Some were inferred from the results of the monopropellant calculations, and some were calculated. The remainder were estimated. The burning rates calculated for these propellants are higher than the experimental data, and to reduce the calculated burning rates to compare favorably, a value of Q(fuel) on the order of 1000 eal /g would be necessary, which seems to be unreasonable. In spite of the lack of quantitative agreement, a burning rate considerably lower than that of the pure monopropellaut is calculated and the general character of the burning rate curve is correct.

A P-Composite Propellant

By way of summarizing the application of the model to several different types of propeltants~ calculations have been made for composite propellants containing AP, KP, and HMX, respectively, as the oxidizer. All of the parameters associated with the binder were held constant, and values of the oxidizer parameters were used consistent with the calculations for the various limiting cases. The calculations are compared to experimental data obtained for the corresponding propellants in Fig. 5. The propellants all had 78 % weight fraction oxidizer and 22% polyurethane binder. The nominal oxidizer size was 50 ~ in all three propellants. The comparison is very en- couraging. In each case, the model describes the general combustion characteristics of the three different classes of propellants.

5.0 4.O

2.0

-~ 1.0

~o.4

0.1

O. 06 O. 04

i

t

4 6 10 20 40 60 100 200 400 PRESSURE (atm)

FIG. 5. A comparison of calculated results and experimental data for three types of composite propellants containing 78% 50~, AP, KP, and HMX, respectively, in a polyurethane binder.


Conclusion

A simple model has been developed to describe the combustion of solid monopropellants and composite propellants. In comparing the known combustion characteristics of A P with the calculations of the model, the results indicate that there is a considerable energy release at the surfaee of the condensed phase as well as in the gaseous phase.

A detailed examination of the model, as applied to KP-composite propellants, indicates that the assumption of a simple, laminar mixing mechanism is not adequate to describe the combustion characteristics of this type of propellant. How- ever, the assumption of a turbulent-type of mixing mechanism gives good agreement between the model and the known characteristics of KP propellants.

Nomenclature

A prefactor c~ specific heat, averaged for solid and gas ~D diffusivity Do oxidizer diameter E activation energy k gas-phase rate constant,

A exp (--Eox f /RTf ) m mass burning rate n burning-rate pressure dependence,

d In m/d In P P pressure Q energy release QL, Q (fuel) oxidizer and fuel energy of decom-

position, respectively Qox i, QPF gas-phase energy release for oxidizer

and primary flame, respectively qL energy flux loss from surface R gas constant To initial temperature Ts average surface temperature Tox, Tf adiabatic flame temperature for mono-

propellant and propellant Sox, So surface area of oxidizer and total burning

surface W molecular weight a weight fraction oxidizer BF fraction of oxidizer reacting in primary

flame 6 reaction order

stoichiometric related coefficient (see Ref. 7)

X average gas-phase thermal conductivity Po~, Of density of oxidizer and fuel binder (*o~ f, ~*PF nondimensional flame-standoff dis-

tance of monopropellant and primary flame

ap temperature sensitivity of burning, O in m/O To

~ concentration dependence of burning, (0 In m/Oa)p

~D particle size dependence of burning, (0 in m/O In Do)7,

Acknowledgments

This work was sponsored in part by the Air Force Rocket Propulsion Laboratory under Contract AFO4611-67-C-0089, and in part by the Lockheed Propulsion Company.

REFERENCES

1. SUMMERFIELD, M., ET At.: ARS Progress in Astronautics and Rocketry, Vol. I: Solid Pro- pellant Rocket Research, p. 141, Academic Press, 1960.

2. CHAIKEN, R. F. AND AbID]ERSON, W. H.: ARS Progress in Astronautics and Rocketry, Vol. I: Solid Propellants Rocke t Research, p. 227, Academic Press, 1960.

3. NACHBAR, W. : ARS Progress in Astronautics and Rocketry, Vol. I : Solid Propellants Rocket Research, p. 207, Academic Press, 1960.

4. HERMANCE, C. E.: AIAA J. 4, 1629 (1966). 5. HERMANCE, C. E.: Proc ICRPG/AIAA 2nd

Solid Propulsion Conference, p. 89, Anaheim, Calif., 1967.

6. F~NN, J. B.: Combust. Flame 12, 201 (1968). 7. BECKSTEAD, M. W., DERR, R. L., AND PR1CE,

C. F.: AIAA J. 8, 220 (1970). 8. JOHNSON, W. E. AND NACHBAR, W.: Eighth

Symposium (Inlernational) on Combustion, p. 678, Williams and Wilkins, 1962.

9. POWLING, J.: Eleventh Symposium (Interna- tional) on Combustion, p. 447, The Combustion Institute, 1967.

10. HIGHTOWER, J. D. AND PRICE, E. W. : Eleventh Symposium (International) on Combustion, p. 1373, The Combustion Institute, 1967.

11. WENOGRAD, J. AND SHINNAR, R.: AIAA J. 6, 964 (1968).

12. BECKSTEAD, M. W. AND HIGHTO~.VER, J. D.: AIAA J. 5, 1785 (1967).

13. HALL, A. R. AND PEARSON, G. S.: "Ammonium Perchlorate: A Review of its Role in Composite Propellant Combustion," R.P.E. Tech Report 67/1, Wescott, 1967.

14. FRIEDMAN, R., ET AL.: Sixth Symposium (Inter- national) on Combustion, p 612, Reinhold, 1957.

15. SHANNON, L. J.: "Effects of Particle Size and Initial Temperature on the Deflagration of Ammonium Perchlorate," Ph.D. thesis, Uni- versity of California, Berkeley, 1963.

16. TAYLOR, J. W.: Combust. Flame 6, 103 (1962). 17. ZIMMER-GALLER, R.: AIAA J. 6, 2107 (1968).


18. DERR, R. L. AND BOGOS, T. L.: Combust. Sci. Technol. 1, 369 (1970).

19. BOGGS, T. L., DERa, R. L., AND BECKSTEAD, iV[. W. : AIAA J. 8, 370 (1970).

20. BUaKE, S. P. AND SCHUMANN, T. E. W.: Ind. Eng. Chem. 20, 998 (1928); see also First/ Second Symposium on Combustion, p. 2, The Combustion Institute, Reprinted 1965.

21. IRWIN, 0. R., SALZMAN, P. K., AND ANDERSON, W. H.: Ninth Symposium (International) on Combustion, p. 358, Academic Press, 1963.

22. BAKHMAN, N, N, AND BELYAEV, A. F.: Dokl. Acad. Nauk SSSR 133, 866 (1960).

23. Lockheed Propulsion Company, unpublished data, 1969.

24. FRIEDMAN, R., LEVY, I. R., AND RUMBEL, K. E.: Bulletin of the 15th JANNAF Solid Propellant Group Meeting, u IV, p. 99, 1959.

25. SUTTON, G. P.: Rocket Propulsion Elements, 2nd ed., p. 312, Wiley, 1956.

26. GECKLER, R. D. AND SPRENGLER, D. F.: Jet Propulsion 2~, 22 (1954).

COMMENTS

Y. Manheimer-Timnat, Technion--Israel In- stitute of Technology. Did you find any experimental evidence from high-speed photographic measurements of the turbulent nature of the KP propellant? Your model for KP-propellant combustion suggests the possibility of using a spherical geometry to describe heat transfer, rather than a one-dimensional one. This should also be amen- able to mathematical treatment.

Authors' Reply. The high-speed movies of the KP propellants did indicate a possible source of turbulence at low pressures only. This source of

turbulence appears in the form of molten agglomerates on the surface of the burning propellant. These agglomerates occur predominately at low pressures (i.e., ( 5 0 atm), and translate laterally across the surface in much the same way as do aluminum agglomerates. This could be a source of turbulence or premixing for the conditions described, but no direct attempt was made to look for turbulence by classical methods.

The possibility of employing spherical geometry to describe heat transfer from the flame appears to have merit, but was not considered in the original study.

Documents

The combustion of solid monopropellants and composite propellants