Temperature sensitivity of propellant burning rates

COMBUSTION AND FLAME 30,267-276 (1977)

267

Temperature Sensitivity of Propellant Burning Rates J. A. CONDON, J. P. RENIE and J. R. OSBORN School of Mechanical Engineering, Purdue University, West Lafayette, Indiana 4 790 7

The effect of the solid propellant's initial temperature on its burning rate has long been recognized. This effect causes variations in the pressure, thrust, and burning time of solid propellant propulsion systems. An understanding of the effect of initial propellant temperature upon burning rate is essential for improving the performance of solid propellant propulsion systems. This paper presents the results of an experimental and theoretical investigation of the temperature sensitivity of the JANNAF standard composite propellant. Four combustion models were selected and calculations made using them for comparison to the experimental results. The models are: (1) the granular diffusion flame model based on the uniformly distributed heat release (KTSS) model, (2) the Beckstead, Derr, and Price (BDP) multiple flame model, (3) a modified BDP model, and (4) the petite ensemble model. The petite ensemble model is quite unusual since it describes the sizes of the propellant's oxidizer particles as a polydispersion of particle sizes. Experimentally, strands of the JANNAF standard composite propellant were burned in a modified Crawford type chimney burner with the propellant burning rate being measured as a function of pressure and initial temperature. The numerical values for the temperature sensitivity determined from the experimental results are compared to the results predicted by the theoretical models. It is concluded that the petite ensemble model provides better agreement with the experimental data because it models the size and size distribution of the oxidizer particles more realistically.

INTRODUCTION Knowledge of composite propellant combustion phenomena is an important aspect of solid propellant rocketry because it affects both propellant formulation and internal ballistics. It is wellknown that to date, the general combustion problem has not been solved theoretically. Therefore, the successes achieved in the past rest in large measure on empiricism, but as low exhaust signature constraints become more rigid, the ability of past empiricisms (based largely on metallized propellants with an AP oxidizer) to solve current and future propellant formulation and internal ballistic problems wanes. Thus, the need for a comprehensive combustion model for composite propellant combustion has never been more important. This paper is concerned with the description and evaluation of three of the more recent corn-

bustion models. In addition, a new combustion model is also described and evaluated. Each model is evaluated by a comparison to experimental combustion data. The first model described represents an early historical model, the granular diffusion flame model. The next two models are representative of the earlier attempts to describe the surface of the burning solid in a rudimentary statistical fashion, while the last model is unique in that it is the first model that attempts to characterize the size and size distribution of the oxidizer particles. While all models may achieve some modicum of success in either predicting the combustion trends for or the burning rate of a family of composite propellants, the true test of the ability of the model to represent the combustion processes lies in its ability to predict such phenomena as the temperature sensitivity of the burning rate of the Copyright © 1977 by The Combustion Institute Published by Elsevier North-Holland, Inc.

268

J.A. CONDON, J. P. RENIE and J. R. OSBORN

propellant. In this paper the four different models are evaluated on the basis of their ability to correctly predict the experimentally determined values of the temperature sensitivity of a composite propellant.

the granular diffusion flame models. The model was developed for predicting the dynamic burning rate, however, an expression for the burning rate, r, for the quasi-steady case can be derived. Thus,

ropG(T, - To) = ~b(p)/r + rppQs. TEMPERATURE SENSITIVITY A number of related parameters may be defined for describing the temperature sensitivity characteristics of a solid propellant. Each is useful for describing the effect of the initial propellant temperature on one aspect of the performance of a solid propellant propulsion system. Since all combustion models are concerned with the theoretical determination of the burning rate, the effect of the initial propellant temperature on the burning rate will be the only such parameter described herein. Thus, the temperature sensitivity of the propellant's burning rate at a particular value of pressure is given by the following relation: Op = 2-Z-_

dlo31nrp=constant

,

(1)

where r is the burning rate of the propellant whose initial temperature is To. A relationship for the above parameter can be derived in closed form for only one of the four models described herein. As a result, the method for determining the burning rate temperature sensitivity will be indicated for each of the other three models.

(2)

In the above equation, qS(p) is the heat feedback parameter defined in [8] andpp is thepropellant density. It is assumed that ~(p) is independent of the initial propellant temperature, To, and that the net surface heat release, Qs, and the propellant specific heat, Cs, are invariant with initial propellant temperature, To. It is assumed that the gas phase and solid phase specific heats are equal. It is also assumed that the propellant surface temperature, Ts, is related to the propellant burning rate through an Arrhenius expression for the surface reaction. That is,

I", = -Es/Rln(r/As).

(3)

Differentiating Eq. (2) with respect to initial temperature at constant pressure yields

% l&

+2(r~-To-G/G

(4)

The above expression represents an improvement upon previously derived [3, 4] temperature sensitivity parameters which were based upon the granular diffusion flame models. In those models [3, 4], a constant surface temperature was assumed.

TIlE COMBUSTION MODELS

BDP Model

The combustion models were selected on the basis of their mathematical modeling of the physical and chemical processes for describing the steady state combustion of composite solid propellants. Each model is more or less representative of a class of models and is significant for its original treatment of at least one aspect of composite propellant combustion. The models are presented below in the chronological order in which they were developed.

The second model used herein for predicting the temperature sensitivity is the "Multiple Flame" (BDP) model of Beckstead, Derr, and Price [1]. The BDP model is based on the assumption that the gas phase conbustion process can be represented by multiple flames surrounding the individual oxidizer particles. Figure 1 illustrates the three separate flames considered. They are (1) the primary flame between the decomposition products of the binder and the oxidizer, (2) a premixed oxidizer monopropeUant flame and (3) a final diffusion flame between the products of the monopropeUant flame and the binder decomposition products.

KTSS Model

The first model selected is the KTSS model of Summerfield et al. [8, 9]. It is representative of

TEMPERATURE SENSITIVITY

269 FINAL DIFFUSION

Fig. 1. Multiple flame structure.

The BDP model assumes that both the oxidizer and fuel surface pyrolysis can be described by an Arrhenius rate expression with the oxidizer regression rate being the controlling step in the overall surface pyrolysis. Also, the oxidizer particles are assumed to be spherical and monodisperse. The geometrical relationship between the size of the oxidizer particles and the fuel binder matrix at the burning surface is evaluated statistically with the diameter of the statistically averaged oxidizer particle at the burning surface being D' = [2/3] 1/2D o

surface, Arrhenius kinetics and the flame standoff distances. The mass balance equation involves a description of the propellant surface subdividing it into two areas one consisting solely of oxidizer and the other solely of fuel. The mass flux from each area is then added to form the total mass flux from the propellant surface. An energy balance at the propellant surface yields the following expression for the average surface temperature:

(5)

QL

Ts = To - - a - - - - ( 1

where D o is some mean initial diameter characteristic of the size of all of the oxidizer particles. This statistical approach was first suggested by Hermance [7]. In this model, radiation heat transfer is neglected and the specific heat, Cp, is averaged over the solid and gas phases. Similarly, the mass diffusion coefficient and thermal conductivity are averaged over the gas phase reaction zone. Another simplification is the assumption that the fuel and oxidizer surface both have an overall average surface temperature, Ts. Unlike the KTSS model, the relationships for the burning rate for the BDP model cannot be reduced to a single equation. The determination of the burning rate involves an iteration of the basic equations establishing a surface temperature that satisfies all of them. The basic equations that comprise the model are the equations representing the mass and energy balance at the propellant

I-/Q.,,.\

×

--a)

Qr

+a(1 --/3F)

.

L ~-~-P /) exp (-~AP)

+(~'S) exp(--'~FF*)] (6) The determination of the surface temperature permits the computation of the surface regression rate since an Arrhenius expression for that rate is assumed in the development of the model. In the above equation the gas phase heat release associated with each flame, QAp, AP decomposition flame, QPF, primary flame, and QFF, final flame, is determined from an overall energy balance assuming equilibrium in each flame. The pa-

270

J.A. CONDON, J. P. RENIE and J. R. OSBORN

rameter To represents the initial propellant temperature, QL and Qt are the latent heat of the oxidizer and the heat of pyrolysis of the fuel binder, respectively, and a is the oxidizer weight fraction. The parameter/~F can be determined by assuming that the diffusion flame is parabolic in shape. From geometrical considerations, then, it can be shown that SAp

/3F -

$ -- XpF *

(7)

XpD*

The kinetic flame standoff distances for the oxidizer and primary flame, XAe* and XpF*, are determined by taking the product of the kinetic reaction time and the linear gas velocity of the reactants assuming the flow to be perpendicular to the propellant surface. In determining the kinetic reaction time, the original version of this model assumed that the reaction rate constants for the two flames were independent of the flame temperature, while the current version of the model accounts for variations in the rate constants with flame temperature. The diffusional mixing flame standoff distances for the primary and final flames, XpD* and XD*, are calculated by means of a truncated Burke-Schumann, two-dimensional diffusion flame analysis. The mathematical relationship for the BurkeSchumann diffusion flame [10] is given by the following series: v - c2(v + 1) 2(v+1)c

F J1 (qSic)Jo(q~i~)q - ~ / L" ~

-

]

spectively. In the BDP model, only the first term of the series solution is considered. For the modified BDP model, the full series is used in the calculation of the parameter 77. The nondimensional flame standoff distances, ~AP*, ~PF*, and ~FF*, can be determined by taking into account both the kinetically limited flame standoff distances for the oxidizer and primary flames and the diffusional mixing flame standoff distances for the primary and final flames. The above equations are then solved numerically by iterating on the surface temperature for a given pressure. As a result of that calculation, the burning rate or surface regression rate can then be determined. The temperature sensitivity is determined by solving for the burning rate at a variety of initial temperatures and applying the definition of temperature sensitivity written above in Eq. (1). Modified BDP Model

The third model to be considered also involves the BDP model. It is modified only in respect to the method of calculation of the diffusion flame heights. In this case the full Burke-Schumann series is incorporated. At oxidizer loadings close to stoichiometric, the first term in the BurkeSchumann series actually does give a close approximation to the correct value for the flame standoff heights. However, as the oxidizer loading deviates from stoichiometric a great many more terms in the series must be considered to achieve acceptable accuracy in determining the diffusion flame heights. The modified BDP model is, therefore, a better model in that it correctly computes the flame heights for all stoichiometry. Petite Ensemble Model

X exp

2~ 2

,

(8) where ~ is a flame stoichiometry related coefficient, ~i is the ith zero of J1, a Bessel function of the first kind, c is the ratio of the oxidizer diameter at the surface to a diameter associated with the fuel binder, ~k is related to the gas diffusivity, gas velocity and surface geometry, and ~ and r/ are nondimensional radial and axial coordinates, re-

The final model [5] to be examined represents a completely new approach in that the size and size distribution of the oxidizer particles are treated properly. In the previous approaches, all of the oxidizer particles are assumed to be of one average diameter allegedly representative of the initial size of all of the particles. In reality, however, the oxidizer particles are a polydispersion of particles each particle having a different initial diameter. Therefore, a more realistic model of the combustion of a solid propellant should include an

TEMPERATURE SENSITIVITY

271 PLAN

k..,,..j//~

0 X I DI Z E R PARTICLE OF TYPE A

ELEVATION

NI N G

SURFACE

B/U/ R

FUEL

BINDER

Fig. 2. Polydisperse propellant burning surface.

accurate description of the particle size and size distribution. The petite ensemble model includes such a description. A fundamental concept of the petite ensemble model is that the actual burning surface of a polydisperse composite propellant can be mathematically rearranged according to the size and type of the oxidizer particle. This mathematical rearrangement results in a number of pseudo burning surfaces each with one oxidizer type and one particle size. These pseudo burning surfaces are called pseudo propellants in the petite ensemble model since their combustion characteristics may be different than the original, unrearranged propellant. For this model it is, therefore, important to examine a typical burning surface of a polydisperse composite propellant. Figure 2 depicts such a burning surface. It is comprised of individual oxidizer crystals protruding above or recessed below the planar burning surface depending upon the combustion pressure. A particle size distribution exists at the burning surface due to the random mixing of the various oxidizer particles during the propellant manufacturing process. At the burning surface each oxidizer particle

has associated with it some portion of the available fuel. As a result, each oxidizer/fuel surface pair will produce a unit flame. If it is assumed that all oxidizer/fuel surface pairs or unit flames burn independently of each other, the propellant surface can be rearranged into imaginary families of monodisperse propellants (pseudo propellants) containing one oxidizer type. This rearrangement is depicted in Fig. 3. The derivation of the petite ensemble model is initiated by applying the conservation of mass equation at the burning propellant surface. Thus, r~t, , = f

m ,, -d- S = r_p p ,

s.

(9)

where n~t" is the average mass flux per unit area from the burning surface, m" is the mass flux per unit surface area,S denotes surface area,iis the average burning rate, pp is the propellant density, and the subscripts b and p designate burning and planar areas respectively. By rearranging the burning surface area into Q monodisperse subareas (Fig. 3) with s oxidizer

272

J.A. CONDON, J. P. RENIE and J. R. OSBORN ('~

Oo0/ TYPE B, SIZE 3 ~ 0 ( " ~

00"dr"/

.,r _

TYPE A, Sl;ZE I

Qo /----x ok,_),...

Fig. 3. Pseudo propellant subareas. types, Eq. (9) becomes

r~t"=l~_l(k=~lfASb,d,kmd,k"dS/Sp),

(10)

where ASh,d, k is the portion of S b occupied by particles of type k and size d, and md,~" is the mass flux from oxidizer particle/fuel surface pairs possessing oxidizer particles with diameters between D O and DO + dDo and oxidizer species k. By summing over all l as Q approaches infinity and by introducing a distribution function describing the oxidizer particle size distribution, the following equation results:

rnt" =N £ SD fflp,d,kSrZ~p,d,kFp,d,k dDo, k=l

0

(11) where N is the total number of particles on the burning surface, ;~p,d,k" is the mass flux based on the planar area, Z~p,a, k is the projection of the monodisperse pseudo propellant burning subarea Z~Sb,a, k on the propellant planar surface, and Fp,d,k is a particle size distribution function. Equation (11) is then modified relating the pseudo propellant parameters to propellant formulation variables. Thus, the following expression [5] for propellant burning rate can be derived:

i =

(mp,u,k -H/fa,k , )fk dDo,

PO,~ - 1 i le=l

0

(12)

where Po,k is the density of oxidizer species k, ~a,k* is the oxidizer volume fraction of a unit flame containing oxidizer particles of size between D O and Do + dDo and speciesk, r~p,a,k" is the mass flux per unit area from the planar surface, and F k is the oxidizer distribution function for species k. Equation (12) expresses the burning rate in terms of propellant formulation variables and the subarea mass fluxes. The mass flux of each individual pseudo propellant is determined through an appropriate unit flame combustion model. Any appropriate combustion model may be considered. For the discussion herein the combustion model is the modified BDP model, model (3). The integration of Eq. (12) over all particle sizes for a polydisperse propellant forms the petite ensemble model. The model described above permits consideration of a real propellant having an oxidizer particle size distribution. This model represents a significant improvement over the conventional models which require that a single particle size be selected to represent a polydispersion of oxidizer particle sizes. EXPERIMENTAL RESULTS The propellant selected for the experimental part of the program is the JANNAF standard composite propellant. The propellant formulation is provided in Table 1. Inhibited strands of the propellant measuring 6.3 mm by 6.3 mm by 152 mm long were burned in a modified Crawford chimney burner. The burning rate was measured as a function of the interior pressure and interior temperature of the burner. The burner was designed so that a nitrogen purge system supplied a small flow rate over the pressure range of from ambient pressure to 1.38 × 107 Pa. The interior pressure of the burner was controllable over that range. The interior temperature of the burner was also controllable from temperatures less than 273 K to temperatures greater than 327 K. As a result, the initial temperature of the strand was controllable over the same range of temperatures. With the temperature and pressure of the strand stabilized, the strand was ignited on its top surface

TEMPERATURE SENSITIVITY

273

I.O

0.8 o 0.6 \ :s t.) i 0.4 cc

(.9 z z

/ ~i /~

/k ~ ~

~°"

. ~ / / "

T =2q4 T~-294 K o = 273 K

0 ~-----T

0.2 I 03

I

I

I 0.6

I

I

I I 1.0

I

I

I

5.0

I

I 6.0

P R E S S U R E - MPA

Fig. 4. Experimental burning rates. using an electrically heated nichrome wire grid. Upon burning through the first timing wire, two Standard Electric Time Co. timers were started, one having an AC clutch; the other having a DC clutch, each having an indicated accuracy of 1/100 of a second. Upon burning through the second timing wire, the clocks were stopped. The timing wires were located 76 mm apart along the length of the propellant strand. The indicated times were checked against each other. The two times were averaged if the agreement was within 5/100 of a second. If they disagreed by more than 5/100 of a second, the test was rejected. A pressure-time trace was also recorded during the entire burning time. The pressure was measured with a Wianko pressure transducer, number P1671, with a range of 5.17 X 106 Pa. The output of the transducer was recorded on a Honeywell automatic balance potentiometer strip chart reTABLE 1

Propellant Ingredients Ingredient

% Weight

PBAN Copper chromite Carbon black Ammonium perchlorate Epoxy curative

19.1

1.0 0.5 75.0 4.4

corder. If the measured pressure varied by more than -+1.72 X 104 Pa from the selected test pressure, the test was rejected. A detailed description of the experimental procedures can be found in Condon [2]. All tests were conducted in the pressure range from 0.24 X 106 to 4 X 106 Pa and at initial temperatures of 273 K, 294 K, and 327 K. Figure 4 presents a plot of the burning rate versus pressure for those initial temperatures. The curves were obtained by a third order polynomial least squares fit of the data. These third order polynomial fits to the experimental data had standard deviations of 0.00897, 0.00947 and 0.01188 cm/s for the three initial temperatures of 273 K, 294 K and 327 K, respectively. All of the experimental data within the range of pressures Of 0.3 and 2.5 MPa were within two standard deviations of the polynomial curves with the "majority of the data within one standard deviation. The experimental value for the temperature sensitivity for the JANNAF propellant was determined from the above curves representing the experimental results. THEORETICAL RESULTS

Temperature sensitivity calculations are presented below for the four theoretical combustion models: (1) the KTSS model, (2) the BDP model, (3) a

J. A. CONDON, J. P. RENIE and J. R. OSBORN

274

TABLE 2 Model Reference Parameters Symbol

Description

Value

M DO Qf pf Af Ef MpF

Molecular weight of final combustion products Initial diameter of oxidizer crystals Heat of pyrolysis of the fuel binder Density of fuel binder Arrhenius frequency factor of fuel binder Activation energy of the fuel binder Molecular weight of primary flame combustion products Latent heat of vaporization of the oxidizer Density of the oxidizer Arrhenius frequency factor of the oxidizer Activation energy of the oxidizer Average heat capacity for solids and gases Thermal conductivity of the combustion gases

20.78 80.0 microns 50.0 cal/g .975 g/cm a 2.7 × 10 a g/cm2-s 1.5 X 104 cal/mol 28.00 -150.0 cal/g 1.95 g/cm 3 1.0 × 106 g/cm2-s 2.6 x 104 cal/mol .3 cal/g-K .0003 cal/cm-s K

QL Pox Aox Eox Cp h

modified BDP model, and (4) the petite ensemble model. The numerical values of the combustion parameters selected for input to the computer were typical for the propellant burned in the experimental program, the JANNAF standard composite propellant. The values were appropriately adjusted within the limits commonly accepted in the literature. They were selected so that the burning rates predicted by the combustion model would match those obtained experimentally at an initial propellant temperature of 294 K. A discussion of this type of parameter adjustment is given by Condon [2]. A listing of some of the input reference parameters used in the BDP model calculations is given in Table 2 above. The adiabatic flame temperatures were calculated using the NASA Equilibrium Thermochemistry Computer Program [6]. The numerical values of the combustion parameters for the KTSS model are listed with the discussion of that model. Comparisons of the theoretical burning rate data with the experimental burning rate data at 294 K are given in Fig. 5. Both BDP models described herein are represented by the one curve.

Following the theoretical determination of the burning rate versus pressure curve at 294 K, each model with its appropriate numerical values was used for predicting the burning rate versus pressure

curve for two different initial propellant temperatures, 273 K and 327 K. From these calculations the temperature sensitivity was determined. For the KTSS model the temperature sensitivity was calculated directly using Eq. (4). KTSS Model For the KTSS model and the JANNAF propellant, values for the surface activation energy of 16,Q00 cal/g and a pre-exponential frequency factor'\of 4600 cm/s were assumed along with average values of surface heat release and specific heat of 140 cal/g and .3 cal/g-K, respectively. These necessary numerical values were established by fitting the model to the experimental burning rate data obtained at 294 K. The model was unsuccessful in predicting the experimental temperature sensitivity data especially at the low pressures. BDP Model and Modified BDP Model Figure 5 presents the results of the parameter adjustment described above; that is, a comparison of the experimental and the calculated burning rate of the composite propellant selected for the experimental part of the program. At low pressures, the value for the burning rate predicted by the BDP model is controlled almost entirely by the primary flame. As the pressure is increased to about 1.0 X 106 Pa, the burning rate curve tends to flatten, indicating the dominance of the diffu-

TEMPERATURE SENSITIVITY

275

1.0 /"2

0.8

To = 294 K

/ , . / . ~

0

0.6 :5 L) I

uJ 0.4 ~ " / ~ /

w

,A

EXPERIMENT

BoP-MoNoo,s s

(.9 Z Z n.m

0.2

o.3

0.6

,.o

3.0

PRESSURE

6.0 -

MPA

Fig. 5. Comparison o f burning rates.

sional aspects of the primary flame. Further increases in pressure cause the slope of the curve to increase, indicating a greater influence of the AP flame on the burning rate. Figure 6 presents three theoretical curves for three different BDP models. The lowest curve, that marked BDP (original), represents the results for the BDP model for which the reaction rate constants were independent of the flame temperature and the truncated Burke-Schumann series was

\

0.4

i >I--

>_- 0.5 I.MOO, ,EO

Z

ENSEMOLE

used. The next curve, that marked BDP, represents the results for a similar BDP model except that the reaction rate constants are now functions of the flame temperature. The last curve, that marked modified BDP, represents the results for a BDP model using the full Burke-Schumann series. In each of the above, the oxidizer particle size distribution is assumed to be monodisperse. It should be pointed out that for all of the monodisperse BDP models discontinuities exist in the temperature sensitivity predictions at pressures of about 1.0 X 106 Pa. The discontinuity occurs at the pressure at which the burning rate versus pressure curve flattens due to the dominance of the diffusional aspects of the primary flame. The effect is strictly an artificial one arising from the effect of a monodisperse oxidizer particle size distribution on the calculation of the parameter fie, Eq. (7).

o~ W orE) I< nr"

Petite Ensemble Model

0.2

..___.__~S

uJ 0.1

8 DP (ORIGINAL)

(3_ W F

0.0

0

I

i

i

}

2

5 PRESSURE- MPA

Fig. 6. Temperature sensitivity.

Figure 5 indicates that the burning rate calculated with the petite ensemble model agrees with the experimental data. The reason that the dominance of the primary flame at low pressure and oxidizer and final flames at higher pressures is not observed is due to the method of the petite ensemble model in describing the polydisperse nature of the size of the oxidizer particles. When the propellant is viewed as polydisperse, then at all pressures a

276 broad range of sizes of oxidizer particles exist each having different flame characteristics. Therefore, since all flame characteristics exist, no one flame characteristic can dominate the burning process. As a result, the pressure versus burning rate curve does not exhibit plateau characteristics for the pressures considered. Figure 6 presents the theoretical temperature sensitivity predictions for the petite ensemble model. The results indicate a significant improvement over the other models in predicting the experimental values of the temperature sensitivity. This improved correlation results from a more realistic mathematical description of the oxidizer particle size and size distribution.

J.A. CONDON, J. P. RENIE and J. R. OSBORN particle size and size distribution are treated much more realistically. Thus, when the monodisperse character of the propellant burning surface assumed in the BDP models was replaced with the polydisperse description of the oxidizer particles of the petite ensemble model, the prediction of the temperature sensitivity was improved. The authors wish to express their sincere gratitude to Dr. R. L. Glick o f the Thiokol Chemical Corporation for his many helpful comments and discussions during the preparation o f that part o f the paper concerned with the petite ensemble model. The work reported herein was supported by a grant from the Thiokol Chemical Corporation.

CONCLUSIONS The modified BDP model and the petite ensemble model are more effective in predicting the temperature sensitivity than any of the other models. The differences in the calculated values of the temperature sensitivity for the various models can be attributed to two separate effects. On the one hand, differences can be attributed to differences in the assumptions and simplifications used in computing flame standoff distances. For example, when the full Burke-Schumann series was employed in the modified BDP model instead of the truncated version, an improved temperature sensitivity correlation resulted. On the other hand, the differences in the calculated values of the temperature sensitivity can be attributed to differences in the techniques used for modeling the physical characteristics of the burning surface. The petite ensemble model represents a departure from all the others in that the oxidizer

REFERENCES 1. Beckstead, M. W., Derr, R. L. and Price, C. F., AIAA Journal 8, 2200-2207 (1970).

2. Condon, J. A., Interim technical report for Thiokol Chemical Corporation, Pennsylvania, 1975. 3. Ewing, D. L. and Osborn, J. R., J. Spacecraft 8,

290-292 (1971). 4. Glick, R. L.,AIAA Journal 5,586-587 (1967). 5. Glick, R. L. and Condon, J. A., 13th JANNAF Combustion Meeting, 1976. 6. Gordon, S. and McBride, B. J., NASA SP-273, 1971. 7. Hermance, C. E., AIAA Journal 4, 1629-1637 (1966). 8. Krier, H., T'ien, J. S., Sirignano, W. A. and Summerfield, M.,AIAA Journal 6, 278-285 (1968). 9. Summerfield, M., Caveny, L. H., Battista, R.A., Kubota, N., Gostintsev, Yu. A. and lsoda, H., J. Spacecraft 8, 251-258 (1971). 10. Williams, F. A., Combustion Theory, Addison

Wesley, Massachusetts, 1966. Received 16 December 1976; revised 26 May 1977