Propellants and Explosives 6, 37-41 (1981) 37

Burning Rate Evaluation of Composite Solid Propellants. A Simplified Approach

V. Swaminathan and M. Soosaimarian

Propellant Engineering Division, Vikram Sarabhai Space Centre, Trivandrum - 695 022 (India)

Berechnung der Abbrandgeschwindigkeit von Composit-Festtreib- stoffen. Eine vereinfachte Naherung

Einer der Hauptparameter eines Festtreibstoffes ist seine lineare Abbrandgeschwindigkeit. Es sind schon viele Versuche unternommen worden die Abbrandgeschwindigkeit von Festtreibstoffen theoretisch zu berechnen unter Verwendung eines geeigneten Abbrandmodels. Ziel dieser Arbeit ist es nun, eine vereinfachte Theorie des Abbrandes fur Composit-Festtreibstoffe aufzustellen. Obwohl diese Arbeit sich im Grunde auf das Modell von Beckstead, Derr und Price stutzt, wurden gewisse vereinfachende Annahmen gemacht, um mit dem Modell leichter arbeiten zu konnen. Auch wurde der Versuch unter- nommen, das Modell auf aluminiumhaltige Festtreibstoffe auszudeh- nen unter Aufstellung einer speziellen Hypothese, die der Beeinflus- sung durch das Aluminium Rechnung tragt. Die relevanten transzen- denten Abbrandgleichungen wurden auf einem Computer berechnet . Die Abbrandgeschwindigkeiten und andere Charakteristika wurden nach diesem Verfahren berechnet fur zwei spezifische ammonium- perchlorathaltige Festtreibstoffe, einmal mit und einmal ohne Alumi- nium-Zusatze. Die so erhaltenen Ergebnisse stimmen annahernd mit den aufgefiihrten experimentellen Werten uberein.

Calcul de la vitesse de combustion des propergols composites solides; une methode d’approximation simplifiee

La vitesse de combustion lintaire constitue une des caracttristiques principales d’un propergol solide. De nombreuses tentatives ont Ctt faites dans le passe pour calculer thtoriquement la vitesse de combus- tion des propergols solides A partir d’un modde approprit dtcrivant le mtcanisme de la combustion. Le but de la prCsente etude est de pro- poser une thkorie simplifiCe de la combustion des propergols composi- tes solides. Tout en Ctant en principe fondCe sur le modkle de Beck- stead, Derr et Price, cette Ctude introduit certaines hypoth5ses simpli- ficatrices qui devraient faciliter I’application du modble. On a Cgale- ment cherchC a Ctendre le mod&le aux propergols solides contenant de l’aluminium en y introduisant une hypothbse sptciale qui tient compte de I’influence rCsultant de la prCsence de I’aluminium. Les Cquations transcendantes correspondantes ont CtC calculCes sur ordinateur. Cette m6thode a CtC appliqute pour calculer la vitesse de combustion et d’autres caracttristiques de deux propergols spCcifiques a base de perchlorate d’ammonium avec et sans aluminium. Les rtsultats obte- nus concerdent assez bien avec les valeurs dCterminCes expCrimentale- ment.

Summary

One of the principal parameters associated with a solid pro- pellant is its linear burning rate. Many attempts have been made in the past to determine theoretically the burning rates of solid propellants by the use of appropriate combustion mod- els. The object of the present paper is to propose a simplified theory of burning rate suitable for composite solid propellants. While the paper follows basically the scheme suggested for this purpose by Beckstead, Derr and Price using multiple flamelets, certain simplifying assumptions have been intro- duced with a view to make the model easier to operate. An attempt is also made in the paper to extend it to the case of aluminized solid propellants as well on the basis of a specific hypothesis regarding the role of aluminium. The relevant transcendental equations of combustion were solved on a digi- tal computer. The burning rates and related characteristics were evaluated by this technique for two specific ammonium perchlorate-based solid propellants, one aluminized and the other non-aluminized, and the results obtained agree reason- ably with the reported experimental trends.

Nomenclature

A Arrhenius frequency factor BDP Beckstead, Derr, Price-model C, E Activation energy

0 Verlag Chemie GmbH, D-6940 Weinheim, 1981

Average mean heat capacity for the solid and gases

hf k m mT P Q Q€Ud

QL r R S SO T TO a B F

6 ?&

E* t

Heat of fusion of aluminium Rate constant, A exp(- EIRT) Mass flux associated with propellant components Total mass flux of propellant Pressure Heat release associated with combustion steps Heat of pyrolyiis of fuel binder Heat of gasification of oxidizer Linear burning rate Gas constant Surface area Total surface area Temperature Initial temperature of propellant Weight fraction of oxidizer Fraction of oxidizing reactants involved in primary diffusion flame Reaction order Thermal conductivity Density Non-dimensional flame stand-off distance Volume fraction of oxidizer in propellant

Subscripts

A1 Aluminium AP AP flame conditions f Fuel binder FF Final flame conditions OX Oxidizer PF Primary flame conditions s Propellant surface conditions

0340-746218110204-0037$02.5010

38 V. Swaminathan and M. Soosairnarian Propellants and Explosives 6, 3741 (1981)

1. Introduction

The a priori determination of the linear burning rate of a solid propellant is an aspect of considerable significance in propellant technology. The problem is quite complex and requires a knowledge of the mechanism of combustion of solid propellants which, again, involves an interplay of a variety of scientific disciplines. The burning rate of a solid propellant is primarily influenced by its formulation and it assumes impor- tance since it controls directly the internal ballistics of the propellant. In the past, several attempts have been made to develop models of combustion suitable for double-base and composite solid propellants, and to deduce therefrom the most important parameters of interest such as propellant burning rate and surface temperature. While some of the earlier mod- els were oriented towards the development of empirical equa- tions with adjustable parameters to fit the experimentally determined values of the burning rates, the recent ones, such as the Hermance and BDP models('4) have considered the more challenging task of evaluating a priori the burning rate of a composite propellant. The BDP model, in particular, ap- pears to be rather satisfying from many points of view and has also been used subsequently in some research papers.

There appear, however, certain complexities in the BDP model such as, for instance, the determination of the stand-off distances of the multiple flamelets, which sometimes limit somewhat its day-to-day utility. The difficulty here arises as much from the nature of the relevant mathematical expres- sions as from a lack of precise information regarding the values of the parameters involved. Under such circumstances, it seems worthwhile to make the approach a little more sim- plified, which will perhaps enable the model to liberate itself from its emphasis on the flame stand-off distances. The object of the present paper is to describe a modification of the BDP model suitable for such a simplified approach. No doubt in this process, the technique is liable to become less sophisticated, but it will be easier to operate and this is precisely what may be warranted when new propellant compositions are developed.

This paper also makes an effort to extend the above model to the case of aluminized propellants on the basis of certain simplifying assumptions. Generalized computer programs have been written on the IBM-360 computer for solving the relevant combustion equations and obtaining the values of the required parameters under a wide range of conditions. Burn- ing rate values, got by the above technique for certain typical propellant combinations, appear to be in reasonable accord with the reported experimental trends.

2. Basic Equations

Following Beckstead et al.('), the mass burning rate can be expressed as a one-step Arrhenius function:

wx = A,, exp(- Eox/RTS) , (1)

where T, is taken to be uniform on the propellant surface and is obtained from an energy balance at the surface. Thus,

SOX Sf SO So

mTCp(Ts - TO) = - mox - QL - mf - Qfuei +

Terms involving the exponentials on the right side of Eq. ( 2 ) describe the heat transferred from the various flame fronts (viz, the premixed AP monopropellant flame, the primary flame between the decomposition products of the oxidizer and the binder, and the final diffusion flame between the products of the monopropellant flame and the binder decomposition products) to the solid surface. The multiplying factor stands for the heat flux generated by the flame front, while the expo- nential function represents the fraction of generated heat con- ducted back to the propellant surface. To overcome the uncer- tainties in the determination of the various parameters occur- ring in the flame equations, it may be assumed, as suggested by Summerfield et al.(') that the overall heat feedback from the flamelets to the surface in the combustion of a non- aluminized propellant is approximately 1.25 times that from the AP monopropellant flame. Eq. ( 2 ) , under these condi- tions, takes the simpler form:

QL Qfuel +

CP CP T, = T o - a - - ( 1 - a ) -

(3 )

LP

where sip is given by the gas phase kinetic equation:

k being an Arrhenius function of the flame temperature. The values of PF for a typical polysulphide propellant, with cc = 0.7 and corresponding to different chamber pressure con- ditions, taken from Ref. 6 (p. 103), are shown in Fig. 1. The value of PF (say &) corresponding to any other a value (say a ' ) could be obtained from:

This equation, though purely empirical, throws sufficient light on the problem and may be expected to reflect ade- quately the overall pattern. Eqs. (l), (3) and (4) are to be solved simultaneously for the oxidizer mass flux, surface tem-

0 200 400 600 800 1(

PRESSURE [psi] - p q 0

Figure 1. Variation of PF with respect to pressure.

Propellants and Explosives 6,3741 (1981) Burning Rate Evaluation of Composite Solid Propellants 39

perature and flame stand-off distance. The mass flux mT, aver- aged over the entire propellant surface, is clearly

Table 1. Burning Rates and Surface Temperatures of 3 Non-Alumi- nized Propellants at Different Pressure Levels.

Pressure Burning Rate [ids] Surface Temperature [K] a = 0.70 a = 0.75 a = 0.80 a = 0.70 a = 0.75 a = 0.80 (5) [psi]

200 0.035 0.044 0.051 766 779 787 SOX 300 0.061 0.070 0.077 794 805 811 where - may, for simplicity, be taken to be 5, the volume 400 o.084 o.094 O.lol 817 823 828

500 0.105 0.115 0.124 830 836 840 600 0.124 0.136 0.145 841 846 851 fraction of the oxidizer in the propellant as elucidated in Ref. 1

(p. 2204), and the linear burning rate r of the propellant is 7oo o.143 o,155 o,166 850 855 860 800 0.161 0.175 0.187 857 863 868 given by

so

900 0.178 0.193 0.206 864 870 874 1OOO 0.194 0.211 0.225 870 876 880

(6)

An obvious limitation of this model is that, on account of the various simplification that have been effected, it cannot accommodate the finer variations in the burning rate and related parameters caused by changes in AP particle size. This limitation should not, however, be of any serious consequence if we address ourselves to the more limited objective of deter- mining theoretically the relative variations in the burning rate of the propellant rather than finding an a priori estimate of the burning rate. Burning Rate [ink] Surface Temperature [K]

Table 2. Burning Rates and Surface Temperatures of Aluminized Propellant at Different Pressure Levels

Pressure [Psi] = 0.75

0.068 798 2oo 300 0.113 829

Differentiating Eqs. (l), ( 2 ) and (4) and solving for the pressure exponent n, we obtain after some simplification

400 0.155 849 n = (dlnm)/(dInP)

,s

500 0.191 0.226 0.258

800 900

1000

v - -

(1.25)(1 - OF) a s i p exp(- sip) 0.290 0.319 0.347

863 874 884 892 899 905

while the temperature sensitivity can, in an analogous manner, be evaluated as

up = 3(lnm,,)/3T~ (8)

QAP E 1 + ( 1 . 2 5 ) ~ - pF) - . a . EX^ ~ x P ( - 5iP) -

- CP RTOX

Eox CP

- - RT: + (1.25)(1 - PF) - QAP . a . 2& exp(- s i p ) 0.3

a = 0.75 (ALUMINIZ

Eqs. (7) and (8) describe respectively, the dependence of the 10.2 burning rate of the propellant on the pressure and the initial temperature. - L.l . ._ -

w t-

2 0.1 C.7 2

3. Results of the Computations -

A computer program was set up on the IBM-360 computer for obtaining, by the above described method, the linear burn- ing rate and the surface temperature of solid propellants. The combustion equations were solved iteratively and no particular difficulty was encountered in the convergence of the solution. The calculations were actually carried out for a typical propel- lant combination (whose combustion parameters, taken from

$ 2

Ref. 1, are given in Appendix 1) for a = 0.7, 0.75 and 0.8 at different chamber pressure levels and the results exhibited in 200 400 800

0.03 I00

Table 1. Figs. 2 and 3 present a comparative study of the variations of the burning rates and surface temperatures of the 3 propellants with respect to pressure.

PRESSURE [psi] - Figure 2. Variation of burning rate with respect to pressure.

V. Swaminathan and M. Soosaimarian Propellants and Explosives 6, 37-41 (1981) 40

I c V v

; 700

d E 8 50c 2

3 I-

w 600 n

I-

a 3 v)

4M

u = 0.75 (ALUMINIZED) & u = 0 . 8

-u=0.75

200 400 800 1000 - PRESSURE [psi] - IPIE333 1

spelt out in Appendix 2, are shown in Table 2 and the relevant graph included in Figs. 2 and 3. The value of the thermal conductivity was taken from Ref. 7 (p. 730) for this purpose. Burning rate results, obtained as above, appear to be in rea- sonable agreement with the experimental trend available in the literature (Ref. 8, p. 430).

5. Conclusion

A simplified numerical procedure, based essentially on the BDP model, has been suggested in this paper for the evalua- tion of the linear burning rate and surface temperature of non- aluminized propellants. The technique may prove helpful in the a priori determination of approximate burning rates of newly developed propellant compositions. An indication is also given in the paper of a possible extension of the model to aluminized composites on the basis of certain simplifying as- sumptions.

Figure 3. Variation of surface temperature with respect to pressure. Appendix 1. Standard Parameter Values‘’) for Non-Aluminized Poly- sulphide Propellant

4. Proposed Modification for Aluminized Composite Propellants

The problems that one is faced with in developing a model of combustion appropriate for metallized propellants are many, and Ref. 5 (pp. 69-71) brings into bold relief the major hurdles in this direction. The effect of aluminium addition on the burning rate of a propellant is difficult to predict on account of the several competing factors that are called into play and any combustion modelling effort in respect of an aluminized propellant has, of necessity, to be on the basis of a specific hypothesis regarding the role of aluminium. In the present paper, we proceed on the assumption [cf. Ref. 5 , (p. 70)] that during the combustion of aluminized propellants, aluminium particles melt, get heated up to the propellant sur- face temperature and the molten metal is present in the gas- eous zone in an agglomerated state, its combustion being rather slow, taking place probably at some distance down- stream of the gaseous reaction zone of the propellant. Under such conditions, the addition of aluminium in any propellant is likely to result in a higher heat flux from the gaseous zone to the solid surface. The increased heat flux arises owing to the steeper temperature gradient, increased thermal conductivity of the gases and the lowered average heat capacity of the solid. When this effect is incorporated into Eq. (3), it gets modified into:

Qfuel + (1 - a ) 7 %

(9) QAP

c* = (1.25) (1 - PF) u - exp(- E i p )

where u represents the ratio of the oxidizer to oxidizer-fuel combination. Eqs. ( l) , (4) and (9) can be solved by iterative techniques for the mass flux and surface temperature of the propellant.

A computer program was again set up for the evaluation of the burning rates of aluminized propellants by using the above equations and the results obtained at different pressure levels for a typical metallized propellant, whose characteristics are

Parameter Magnitude Dimensions

1.12 75 50

405 1.8 0.3

3 x 10-4

3 x 105 22

300 0.70, 0.75, 0.80 1.56, 1.60, 1.64

g/(cm3 . s . a d ) callg calig cal/g

cal/(g . K) cal/(cm . s . K) kcal/mol g/(cm2 . s) K

g/cm3

. . .

. . .

Appendix 2. Standard Parameter Values for Aluminized Polysulphide Propellant*

Parameter Magnitude Dimension

Percentage of A1 in the 15 % propellant

0.285 cal/(g . K)

0.75 . . . 10.02 x cal/(cm . s . K) ?

a e 1.72 g/cm’ C*I 0.215 cal/(g . K) hf 93 cal/g

* The values of the remaining parameters were taken to be the same as in Appendix 1 .

6. References

(1) M. W. Beckstead, R. L. Derr, and C. F. Price, “A Model of Composite Solid Propellant Combustion Based on Multiple Flames,” AZAA J. 8, 22W2207 (1970).

(2) M. W. Beckstead, R. L. Derr, and C. F. Price, “The Combustion of Solid Monopropellants and Composite Propellants”, Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, Pa., 1971, Proceedings pp. 1047-1056.

(3) C. E. Hermance, “A Model of Composite Propellant Combustion Including Surface Heterogeneity and Heat Generation”, AIAA J. 9, 1625c1637 (1966).

Propellants and Explosives 6, 37-41 (1981) Burning Rate Evaluation of Composite Solid Propellants 41

(4) J. A. Condon, J. P. Renie, and J. R. Osborn, “Temperature Sensitivity of Propellant Burning Rates”, Combust. Flame 30,

(5) J. A. Steinz, P. L. Stang, and M. Summerfield, “The Burning Mechanism of Ammonium Perchlorate-Based Composite Solid Propellants”, Aerospace and Mechanical Sciences Report No. 830, Princeton University, 1969.

(6) D. Baker, “Investigation of Composite Solid Propellant Burning Rate Temperature Sensitivity”, Ph. D. Thesis, Purdue University, 1973.

(7) D. Gross and A. B. Amster, “Thermal Explosions: Adiabatic Self-Heating of Explosives and Propellants”, Eighth Symposium (Internationat) on Combustion, The Williams and Wilkins Com- pany, Baltimore, Md., 1962, Proceedings pp. 728-734.

267-276 (1977).

(8) F. A. Williams, M. Barrtre, and N. C. Huang, “Fundamental Aspects of Solid Propellant Rockets”, AGARDograph 116, Tech- nivision Services, Slough, England, 1969.

AcknowZedgement

The authors are grateful to Dr. Vasant R. Gowariker, Director, Chemicals and Materials Group, Vikram Sarabhai Space Centre, Tn- vandrum for his kind encouragement of this work.

(Received October 25, 1978, revised January 22, 1980)