study of the interaction mechanism between the metal and oxygen at the temperatures in question. According to the observations in the present experiments, in the case of titanium, this mechanism is in keeping with interaction with a crazed film on the metal with a considerable share of vapor-phase oxidation. The existence of a vapor-phase reaction was established from the observations in tests on the removal of condensed reaction products in a gas stream.

This fact, that ignition is due to reaction with a crazed oxide film on the surface of specimens, also is in accord with the results in [i], just like the existence of two characteristic temperatures (tcr and td) during ignition.

As a result of the research conducted the temperature at which titanium ignites in oxygen at 1 atm has been determined (1,605~ It has been found that the ignition of titanium in oxygen features the breakdown temperature of the thermal equilibrium and the deflagration temperature which differs by about 350 ~ . The critical conditions for the ignition of titanium in oxygen and, apparently, aluminum in oxygen during heating up are appropriate for a case of forced ignition. The conditions for spontaneous ignition in this case are not realized. Titanium ignites in oxygen in an area where the oxide film has crazed, in keeping with the conclusions in [i]. Such a situation with metals may be considered quite common.

LITERATURE CITED

i. N. M. Laurendeau and J. Glassman, Combust. Sci. Technol., 3, 77 (1971). 2. F. E. Littman, F. M. Church, and E. M. Kinderman, J. Less Common Metals, 3, 367 (1961). 3. A. G. Merzhanov, Yu. A. Gal'chenko, et al., in: Combustion and Explosion [in Russian], Izd.

Nauka, Moscow (1972), p. 245.

EXTINCTION OF A SOLID PROPELLANT ACCOMPANYING A FALL IN

PRESSURE AS A LOSS OF COMBUSTION STABILITY

V. A. Frost and V. L. Yumashev

i. STATEMENT .OF PROBLEM

It is shown in [i] that the extinction of a solid propellant accompanying a drop in pressure is described by the phenomenological theory of nonstationary combustion of a solid propellant in the general case of a variable surface temperature [2] without bringing in any additional hypotheses. It should be recalled that the theory of [2] is based on an as- sumption about the noninertia of the zone of dissociation of the k-phase and the gaseous flame. The nonstationary combustion is described by the following system of equations (in the usual dimensionless variables referring to the values of the parameters in the original

stationary state):

OT/OT--__O~T/OxZ--u.OT]Ox, --oo< x~O; OT/Oxl==o---~(p, u); T]~=o=T~(p, u)] (I.i)

T] , = o = e ~.

Here x and t are the dimensionless coordinate and time; T(x, t) is the distribution of temperature in a heated layer of the k-phase; u(t) is the rate of combustion; p(t) is the predetermined pressure; ~(p, u) and Ts(p, u) are functions characterizing a specific solid propellant which are considered to be known from the stationary experiment [2].

Translated from Fizika Goreniya i Vzryva, Vol. 12, No. 4, pp. 548-555, July-August,

1976. Original article submitted July ii, 1975.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 17th Street, New York, IV. Y. 10011. No part o f this publication may be reproduced, s tored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available from the publisher for $Z50.

496

~, ' ~ , ~ / 1,~ ~ .~ -C ' I - 'Y -~F \ . . t ' - .

I ~"\r w-"':':"-. >>r,014 ! ~--~-----d

0 2 5

Fig. I. Extinction and transition conditions

with converging pressure drops; PK = 0.6, v = 0.69, k = 1.4, and m = I0. i) u; 2) (~2T/

~X2)x = o.

In [i] the ~ and T s functions are given by the relations

U = e m ( T s - 1 ) , u = p V e ~ ( T s - - ~ l u ) , (1.2)

where v, k, and m are constant parameters; exponential drops in pressure are examined:

p (0 = p . + (1--p.)e-',"q (1.3)

Here PK is the dimensionless final pressure level, 0~PK~l , and At is the dimension- less characteristic drop time.

In [i] the extinction is revealed as a result of the numerical solution of the system of equations (1.1)-(1.3). It is found that given fairly rapid and profound drops in pressure, the usual transition to a new stationary state corresponding to p = PK is absent, the rate of combustion dropping smoothly instead, asymptotically approaching zero. It is characteris-

tic that at a certain moment during extinction the second temperature derivative in terms of the coordinate on the surface (32T/3x2) x = o becomes negative and a point of inflection ap- pears on the profile of the temperature in the k-phase denoting, according to [I], an in- tensive cooling of the k-phase surface layer.

Extinction conditions produced numerically for a close model of nonstationary combustion, under which the rate of combustion drops smoothly to zero, are also discussed in [3]. The authors of [3] note that the extinction is related to the attainment of the stability boundary combustion. At the same time examples are given in [i] of nonextinguishing con- ditions with a trajectory on the u vs ~ diagram which intersects the stability boundary and even the more distant boundary of free-oscillation existence. The latter result is natural if it is taken into account that these boundaries are valid only for stationary combustion and have no direct bearing on the nonstationary p~ocess under examination. The extinction can be explained by the attainment of the stability boundary only in the nase of a slow quasistationary transition.

The physical causes of extinction in the model under examination are explained in the present paper and the relationship between extinction and stability losses is established. The influence of the nonstationary nature of the process on stability criteria is also evaluated.

2. EXTINCTION AND STABILITY OF SOLID PROPELLANT C~{BUSTION

The extinction conditions produced in [i] recall the process of combustion stability loss (Fig. i). This will be demonstrated below.

There is known to be a boundary in terms of the profundity and rate of the drop in pressure which distinguishes the extinction conditions from the normal conditions of the transition to a new stationary state (in short, transition conditions). Near the boundary

497

U

S S 2

Fig. 2. Two patterns for the evolution of slight perturbations in the rate of combustion; S) stability boundary.

the slight change in the pressure drop gives rise to the replacement of the transition conditions by extinction. Consequently, slight changes in the input parameter (pressure) cause great changes in the output parameter (combustion rate). This behavior is character- istic of instability.

Figure 1 depicts extinction and transition conditions differing from each other in their pairs in terms of the characteristic pressure drop time by 0.i; 0.01; 0.001; and 0.0001 (in dimensionless variables). It is easy, to see that the smaller the difference in the pressure drop, the slower will be the growth in the discrepancy between the extinction and transition conditions and the greater will be the delay in the initiation of the extinction. This is also in agreement with theories concerning loss of stability, since it is known that the smaller the initial deviation from the equilibrium position of an unstable system, the more pYolonged will be the process of deviation growth.

Indirect signs also indicate a loss of stability. First, according to the results of [i], the closer the original stationary state to the boundary of stability, the easier it will be to achieve extinction (less rapid and profound drops in pressure are required). Secondly, during extinction the nature of the change in the rate of combustion in time is the same as in the case of unstable stationary conditions when p = const (modifications VII and VIII [i]). It is, however, difficult to establish the relationship between extinction and the attainment of a certain stability boundary. The study of stability is known to be reducible to the study of the behavior of slight perturbations in the fundamental process accompanying an unrestricted growth in time. Under stationary combustion conditions all the conditions are kept constant as long as necessary. In a certain vicinity of the stationary state, therefore, the slight perturbations either grow ever greater or drop ever lower. The behavior of the slight perturbations can be predicted give n infinitely long times from a knowledge of their local behavior, which is established from the solution of the linearized

equations of nonstationary combUstion [2].

Under nonstationary combustio~ conditions the behavior of fhe slight perturbations may differ at different moments. In a given case, therefore, the local behavior of the per- turbations does not govern the final result and cannot provide a conclusion about the

stability of the process as a whole.

In automatic control theory in the case of systems with variable coefficients the situation also arises when slight perturbations grow in some sections while being attenuated in others [4]. The concept of "stability at a given moment" and "stability for a given space of time" is introduced to denote the local behavior of the slight perturbations.

By analogy we shall refer to the local stability or instability of the combustion process if at a given moment the slight perturbations are subject to attenuation or growth, respectively. It is natural to call the geometrical location of the points on the trajec- tories of the process (the coordinates are immaterial), where the perturbation growth is replaced by attenuation (or vice versa), the local stability boundary.

Let Us examine a certain nonstationary combustion process (Fig. 2, i). At a moment tl let the process fall into the region of local instability and at a moment t2 let it emerge from it. This behavior is characteristic of transition conditions not too far removed from the extinction boundary. Let slight perturbations be superimposed on the fundamental process (Fig. 2, 2 and 4). If the initial deviation from the fundamental process is fairly small, the disturbed process is negligibly removed from the fundamental process for as long as it remains in the region of instability and also emerges from the region of instability (Fig. 2,

498

2 and 3). From that moment the perturbations are attentuated and the perturbed process is reduced to the fundamental. Such behavior is characteristic of the transition conditions depicted in Fig. i. If the initial deviation is fairly great, the perturbed process manages to depart so far from the fundamental that it does not even emerge from the region of local

instability and the deviation grows further (Fig. 2, 4). This is the extinction process.

Thus, the end result of the evolution of the slight perturbations is dependent not only on their local behavior, but also on their amplitude and, in general, on the whole ~ourse of the perturbed process itself. This problem extends beyond the framework of the linearized theory of instability. It is for the same reason difficult to identify extinction criteria, i.e., the conditions whose achievement during the process of nonstationary combustion indi- cates that extinction will follow without fail. At the same time the presence of local instability is absolutely essential for the loss of stability of the process as a whole. Thus, the local instability of the combustion process is the cause and a necessary (but not sufficient) condition for extinction.

3. EXTINCTION MECHANISM

The pattern of extinction being examined is characteristic of the fact that the same laws governing nonstationary combustion act over a whole range of variations in parameters. Thus, the extinction can be explained without any additional assumption concerning limit points, i.e., parameter values such that the accepted model of nonstationary combustion [2, 5] ceases to be true for them. It may, on the other hand, turn out that the limit points do in fact exist and that the domain of the @(p, u) and Ts(P, u) functions is restricted. In this case the extinction looks different: the rate of combustion falls continuously to a certain limit value and thereafter falls in one step to zero.

The question arises of how the extinction proceeds in actual fact and t~is can be ex- plained experimentally.

The rate of nonstationary combustion of a solid propellant during extinction accompany- ing a fall in pressure is measured directly in [6, 7]. In [6] the rate of combustion is determined from the Doppler effect for microwave-band electromagnetic waves and in [7], from the variation in capacitance of a capacitor in which the dielectric is a fuel specimen. The qualitative results are obtained by other methods: the rate of combustion falls smoothly to zero during extinction.

Thus, there are no limit points in the ~(p, u) and Ts(P, u) relationships, at least in certain kinds of solid propellants, and the extinction mechanism examined in this paper occurs. At the same time in an experimental determination under stationary conditions the ~(p, u) and Ts(p, u) functions can be plotted only in the region of stability of stationary combustion, so they are known only in restricted sections of the parameter values [5]. In order to plot these functions over the whole range of parameter variations, experimental data on the nonstationary combustion and quenching of a solid propellant must be brought in.

4. LOCAL STABILITY ACCOMPANYING NONSTATIONARY COMBUSTION

The conditions for stability in stationary combustion and a restricted class of self- similar nonstationary combustion processes are currently under investigation [2]. The problem of the boundaries of local stability of a random nonstationary combustion process are of interest.

Let us examine the behavior of slight perturbations superimposed onto a random non- stationary combustion process. Let us designate the values of the physical magnitudes in the fundamental process by the index 0 and their perturbations, by the index I. The perturbed process is characterized by magnitudes

T ( x , O = T o ( x , t )q -T l (x , t). u ( t ) = u o ( t ) + u l ( t ) ,

etc. It is assumed that the pressure is not subject to perturbation. for the slight perturbations follow from (i.I):

OTl/Ot~O2T1/.4x2--uo . OT l /Ox - lq . oTo/o x, - - ~ ~ x ~ O :

OT1/Ox[~=o= t~ ( a ~ / O l ~ ) ~; TI I ~=o = t t! ( OTJO.) ~;

Tt] . . . . - -0 .

Linearized equations

(4.1)

499

The derivatives of the ~(p, u) and Ts(p, u) functions are taken to be for current values of the arguments in the fundamental procesS: u = uo(t) and p = po(t).

The characteristics of the fundamental process of nonstationary combustion, which is fully determined if the initial temperature distribution in the k-phase at a given moment and the law governing the change in pressure are present, appear as coefficients in the equation for slight perturbations (4.1), In an analysis of the local stability for a given subsequent change in pressure, therefore, the nature of the pertflrbations development is dependent only on the shape of the temperature profile in the k-phase.

Any pair of u and p values corresponds to a set of stationary combustion conditions for a given initial temperature with given values of ~ and T s. It is convenient to introduce local k' and r' parameters characterizing the stability of the stationary conditions in accordance with the Novozhilov criterion [2]. The dash is used to differentiate from the corre- sponding parameters in relation (1.2) characterizing the stability of the initial stationary state. The following relations are valid:

( k P + r ' - - l ) / h ' = ( 0 In ~/0 in u),; r'lk'=u/~. (.OTdO In u) p . ( 4 . 2 )

The complexity of the solution of the system of equations in (4.1) comprises the fact that all the coefficients in it are functions of time. The "coefficient freezing method" [4] is used in automatic control theory to study systems with variable coefficients. The method is not distinguished by particular strictness but gives correct results in a number ol cases. An analogous approach is employed here.

Let us examine the system in (4.1) in the vicinity of a certain moment to. Let us ex-

pand the coefficients of the system into a series:

OTo (x, t)/Ox-~- OTo (x, to) /Ox+ ( t - - to ) . 02To(x, to) ] O x O t + . . . ,

etc. Near to the first term in the expansion exceeds all the others in order of magnitude. By virtue of the continuous dependence of the solution of the system on the coefficients, the influence of the terms of the expansion on the solution must be proportional to their order of magnitude. It can be assumed, therefore, that the local stability is dependent, in the first place, on the current values of To(x) and the other coefficients and that their derivatives exert an influence of the next orde~ of triviality. Let us restrict ourselves to an investigation into the influence of the current values only and let us fix values for

the coefficients in the system of (4.1) when t = to:

OT1/Ot---O2Tl/Ox~--uo (to)" OT1/Ox--ut. 01"o (x, to)/Ox, - -oo < x<~ 0;

OT~/Oxlx=o= u~ ( ~ lOu) p I ,=,0, ( 4 . 3 ) TII~=_~---~ul (OT,]Ou)~[ ,=,o; Tll . . . . = 0 .

F o l l o w i n g on f r o m [2] l e t us s e e k a s o l u t i o n i n t h e f o r m

T l = O ( x ) e ~ u l = v e " . ( 4 . 4 )

By i n c o r p o r a t i n g ( 4 . 4 ) . i n t o ( 4 . 3 ) , we o b t a i n

d~-O/dx2--uo �9 dO/dx--oO ~ v . OTo/Ox, - -oo < x ~ f l ;

dO/clx l . = o = V ( O~l Ou ) p; (4.5) Olx=o=O,=v(OT, /Ou)p; O[ . . . . -~-0.

The identification of the moment to is omitted, since time is excluded from the examination.

The solution of the differential equation in the system of equations in (4.5) which satisfies the boundary condition in terms of temperature when x = 0 and x = -- ~ takes the

form

O(x) = e ',x O~ - - e(uo -2~,)~' Oro " ~--oo v - 0 7 e("--~'o)~"dx' dx' . ( 4 . 6 )

500

o

f

Fig. 3. Region of combus-

tion instability.

Here z~ is the wave number related to frequency by the characteristic equation

z2--uoz--o-----O. ( 4 . 7 )

It follows from the convergence condition of the improper integral in (4.6) for the most. common form of To(x) that Re(zl -- Uo) > 0. On the other hand, the roots of Eq. (4.7) are re- lated by the correlation z~ + z2 = uo; therefore, Rez= < 0. Thus, of the two roots of Eq.

(4.7) the zl root must be selected with a large real part. It follows from (4.6) that

0

O0 @v Ox ~=0 =zl0~ ~ @ e(: l -~~ . . (4.8)

By i n c o r p o r a t i n g t h e b o u n d a r y c o n d i t i o n s ( 4 . 5 ) i n t o ( 4 . 8 ) and t a k i n g i n t o a c c o u n t ( 4 . 2 ) , we o b t a i n

0

Uo z-L r ' - - k' - - r ' -r' 1 @ k' ~u~ ~ ~aT" e(:~_~o)~dx = 0. ( 4 . 9 )

E q u a t i o n ( 4 . 9 ) i s an a l g e b r a i c e q u a t i o n w i t h r e s p e c t t o z~. A l l t h e r o o t s o f t h i s e q u a t i o n mus t be f o u n d i n o r d e r t o a n a l y z e t h e s t a b i l i t y , and t h e c o r r e s p o n d i n g v a l u e s o f w mus t be d e t e r m i n e d u s i n g ( 4 . 7 ) . I n s t a b i l i t y o c c u r s i f r o o t s a r e f o u n d s a t i s f y i n g t h e c o n d i t i o n

Re co>O, Re(zl--Uo) > 0 . (4.10)

The first condition denotes a growth in the amplitude of the slight perturbations with time, and the second condition ensures the convergence of the improper integral in (4.6) and the existence of a solution in the form of (4.4). It should be noted that in special cases with a specific temperature distribution the second condition can be made milder.

In view of the fact that the solution of Eq. (4.9) can be linked with considerable difficulties and that the roots of the equation cannot be expressed in a clear form, it is expedient to exclude ~ from the examination and to obtain the stability condition directly from the values of zl instead of (4.10). Representing zl in the form of Re zl +ilmz~, let us use (4.7) and reduce the condition (4.10) to the form of (Re2zl -- Im2zl)u~ -- Rezl/uo > 0:

Re z!lus> I. (4. Ii)

The range of zl values satisfying the conditions of (4.11) is depicted in Fig. 3 in the complex zl/uo plane (cross-hatched). If any root of Eq. (4.9) lies within this area, in- stability occurs. If the root lies on the abscissa, then the slight perturbations grow monotonically and if not, the growth in the slight perturbations is oscillatory.

5. CASE OF EXPONENTIAL TEMPERATURE DISTRIBUTION

The problem of investigating stability becomes more specific if the temperature

501

I

o , 3 ~ ~ ~. �9 I

~ > u o

\\i

'GZ c

.~. s _ _ _ o

0,5 I,G 1,5 k'

Fig. 4. Displacement of boundary of local combustion stability (S) and natural oscillation boundary (0) ac- companying deviations of the temperature distribution from the stationary; N is the new location of the boundary when i -- ~/Uo = • 0.3.

distribution in the k-phase is expressed in terms of elementary functions. In particular, the real temperature distribution can be approximated by elementary functions. Let us examine the case in which the temperature distribution in the k-phase takes the form

To(x ) - -~T,e =x. ( 5 . 1 )

It should be stressed that (5.1) is Dot a stationary Michelson distribution in as much as the general case ~ # uo.

With an exponential temperature distribution the convergence condition of the integral in (4.9) takes the form

Rezl--uo~-~>O. (5.2)

Thus, the second condition in (4.11) should be replaced by the less severe condition (5.2).

By incorporating (5.1) into (4.9) and taking into account the fact that @ = ~T s, we

arrive at a quadratic equation

z j u o " r' + 1 - - t e ' - - r ' - ] - u o k ' / ( a - 4 - z l - - U o ) = O, (5.3)

the roots of which are

(Z rr - k ' + r ' - - l + r " 1 - - a ( k ' - F r ' - - l ) 2 - - 4 k ' r ' - ~ - r "2 t - - r~-~-~2--2r' [---~-~- (k '+ 1, ( z , ) : ' u o ,

"~o !,2 2r' ~- 2 r ' (5.4)

From (4.11), (5.2), and (5.4) when k' > i follows the local stability condition

( k ' - - l ) 2 - - r t (k-F l ) - - r ,-2. a/go. (l--G./'do) "~0 (5.5)

and the condition for the oscillatory nature of the evolution of the slight perturbations

( (k' + r ' - - 1) 2 - - 4 U r ' -p r 's 1 - - 7 o - - 2r ' 1 - - ( k ' ~-' r ' - - I) r 0 when k ' < 1,

k ' q - r t - - l - - r r ( l - -cz/uo,) ~>0 when k" < 1.

When a = Uo (stationary temperature distribution) conditions (5.5) and (5.6) make the transition to the well-known Novozhilov conditions for stationary combustion [2]. When a < uo the boundaries of stability and natural oscillation existence in the (k', r') coordinates are displaced to the right and the region of stability is extended (Fig. 4). When e > Uo the boundaries are displaced to the left and the region of stability is constricted. The new locations of these boundaries are shown in Fig. 4 displaced relative to the original

position when a = Uo.

The results obtained have a simple physical significance; the narrower the heated layer in the k-phase, the less stable will be the combustion.

502

LITERATURE CITED

i. V. A. Frost and V. L. Yumashev, Zh. Prikl. Mekh. Tekh. Fiz., No. 3 (1973).

2. B. V. Novozhilov, Nonstationary Combustion of Solid Rocket Fuels [in Russian], Nauka,

Moscow (1973). 3. M. Summerfield et al., J. Spacecr. Rockets, No. 3 (1971). 4. N. T. Kuzovkov, Automatic Control Theory Based on Frequency Methods [in Russian],

Oborongiz, Moscow (1960). 5. S. S. Novikov and Yu. S. Ryazantsev, Zh. Prikl. Mekh. Tekh. Fiz., No. 2, (1969). 6. L. D. Strand, A. L. Schultz, and G. K. Reedy, J. Spacecr. Rockets, No. 2 (1974). 7. C. F. Yin and C. E. Hermance, AIAA Paper No. 71-173.

EXPERIMENTAL INVESTIGATION INTO THE INFLUENCE OF OSCILLATIONS ON THE

RATE OF COMBUSTION OF CONDENSED SUBSTANCES

B. N. Fedorov

The rate of combustion of condensed substances in an acoustic field with a given field strength is known to differ substantially from the rate of combustion without an acoustic field. In a standing acoustic wave the rate of combustion at the antinode differs from the rate of combustion at the acoustic velocity node [i, 3]. The aim Of this paper is an investigation into the influence of oscillations on the rate of combustion of condensed substances of various different compositions given different mean pressures and specimen locations in the acoustic wave.

Special sets of apparatus are designed (Fig. i). The operating principle of the ap- paratus is based on the artificial generation of pressure oscillations using a reciprocal action siren (i) in a chamber (2) containing the specimen under investigation of a con- densed substance (3), plated along the side surface and burning from the end. The siren, activated by an electric motor (4) via a reducing gear (5), covers the "nozzle-eye"* at regular intervals introducing oscillations into the chamber. The frequency of the oscil- lations introduced can be varied from 50 to 5000 Hz. The pressure oscillations are measured using a piezoelectric transducer (6).

On apparatus I the specimen of condensed substance is fixed into a bracket which can be moved by means of a screw along the length of the chamber, thus altering the acoustic properties of the chamber. The mean pressure level is determined by the effective cross section of the "nozzle-eye." The rate of combustion of the condensed substance is measured with the aid of fuse wires (7).

The parameters are measured with the following errors (maximum): mean pressure 2%, oscillation frequency 1%, condensed substance combustion rate 2%, and pressure oscillation amplitude 12%. The specimens of ballistite composition without any catalyzing or metallic additives (composition A) which are being investigated measure 120 mm in diameter and 80 ram in length and burn from one end.

On apparatus II the mean pressure level is established by the nitrogen and maintained by the reducing gear. The rate of combustion is measured using cine photography through transparent windows (8) in the chamber walls. The condensed substance specimen, plated along the side surface and measuring 20 mm in diameter and 50 mm in length, burns from the end in the nitrogen flow. The thickness of the transparent plating is selected on the basis

*The term "nozzle-eye" means a nozzle without an expanding funnel-shaped outlet.

Translated from Fizika Goreniya i Vzryva, Vol. 12, No. 4, pp. 555-559, July-August, 1976. Original article submitted April 21, 1975.

This material is protected by copyright r ~ i s ~ r e d in m e name o f Plenum Publishing Co~orat ion, 227 West ! 7th S~eeg N e w York, ~ 10011. No part of thispubticat ion may be reproduced, stored in a retrieval system, or transmitted, in any fo rm or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available from the publisher for $7.50.

503