The effects of acceleration upon the linearburning rate of solid-rocket propellants

Item Type text; Thesis-Reproduction (electronic)

Authors Towson, Earl Raymond, 1938-

Publisher The University of Arizona.

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THE EFFECTS OF ACCELERATION UPON THE LINEAR BURNING RATE OF SOLID-ROCKET PROPELLANTS

byEarl Raymond Towson

A Thesis Submitted to the Faculty of theDEPARTMENT OF AEROSPACE ENGINEERING

In Partial Fulfillment of the Requirements For the Degree ofMASTER OF SCIENCE

In the Graduate CollegeTHE UNIVERSITY OF ARIZONA

1 9 6 5

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfill­ment of requirements for an advanced degree at The Univer­sity of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

without special permission, provided that accurate acknowledgment of source is made. Requests for permis­sion for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate Col­lege when in his judgment the proposed use of the material is in the interests of scholarship. In all other instan­ces, however, permission must be obtained from the author.

This thesis has been approved on the date shown below:

Brief quotations from, this thesis are allowable

SIGNED:

APPROVAL BY THESIS DIRECTOR

RUSSELLM * * . /oj cres.

DateAssociate Professor of Aerospace Engineering

ACKNOWLEDGMENTS

I wish to extend my appreciation to my advisor. Dr. Russell E. Petersen, Associate Professor of Aerospace Engineering, for his encouragement and review; to the Bureau of Naval Weapons; and, specifically, to the U. S„ Naval Ordnance Test Station, China Lake, California, and its Test Department for the opportunity to engage in this investigation.

iii

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS LIST OF TABLES . . . ABSTRACT . . . . . .CHAPTER I INTRODUCTION

vz « . viii

CHAPTER 2 CHAPTER 3

CHAPTER 4

THE MECHANICAL ACCELERATION INTERACTIONTHE CHAMBER PRESSURE ACCELERATION INTER­ACTION .A. Isentropic Flow with AccelerationB „ Flow Through the Plenum Chamber . ,C. Flow Through the NozzleD. The Shift of Equilibrium ConditionsE. Supporting Experimental Work . . . ,THE COMBUSTION ACCELERATION INTERACTIONA,

B<

C.

D<

Physical Evidence for the Combustion Model for Double-Base PropellantsChemical Evidence for the Combustion Model for Double-Base PropellantsThe Model of Burning for Double-Base Propellants . . O O & 0 0 0 O 0

The Rice and Ginell Combustion Theory for Double-Base Propellants . . . .

I X

15

88

1318223337

40

52

57

60

xv

CHAPTER 5 REFERENCES

vPage

E. Effect of Diffusion on the Combustion Model for Double-Base Propellants .- 68

F. The Effect of the Flame Upon the Combustion Model for Double-Base Propellants 75

G. Experimental Verification of theRice and'Ginell Theory 81

Hp Summary of the Rice and Ginell Com­bustion Model for Double-Base Propellants . . . . ............ 81

I. Acceleration Effects on Double-BaseP r O p e l l a n t S o o p o o o e o e e o p 85

Jp The Rice-Summerfield CombustionModel for Composite Propellants . , 97

K. Acceleration Effects Upon the Burn­ing- Rate of Composite Propellants . 108

Le The Effect of Acceleration UponMetal Additives.......... 113

CONCLUSIONS » „ „ . . . 0 » „ . , „ , » 121o e c p o o e o o e o e e o o e e e e o o 12A

LIST OF ILLUSTRATIONS

Figure Page1. Propellant Deformation as the Combined Result

of Acceleration and Pressure Drop . . . . . . . 62. The Location of the Sonic Plane as a Function

of Vehicle Acceleration in Schematic Form . . . 143. Notation and Parts of a Rocket Motor such as

the One Analyzed by the T e x t ............ 144» Nondimensional Nozzle Notation . . . . . . . . 2?5. Schematic Representation of Flow Through an a Accelerating Nozzle . . . . . . . . . . . . . . 305. Schematic Representation of Flow Through ab Decelerating Nozzle . . . . . . . . . . . . . . 306. Experimental Pressure versus Time Profiles . . 347. Burning-rate Deviation from that Obtained

Statically versus Acceleration . . . . . . . . 348. Sandia Tests, 10 KS 2$00 Test Motors under

Acceleration . . . . . . . . . . . 359° Burning-Rate versus Chamber Pressure, ReportedTest Data . . . . . . . . . . . . . . . . . . . 3510. Burning-Rate Data for Powder Composition 14400 4211. Change of Composition of Gas with Pressure . . 4312. Change in the Products of Combustion with Change

in the Initial Pressure of Inert Gas, Propellant HES 4016 . . . . . . . . . . . . . . . . . . . 43

13= Change in the Heat of Explosion with Change inthe Initial Pressure of Inert Gas, Propellant HES 4016 . . . . . . . . . . . . . . . . . . . 44

14. Variation of Dark Zone Length with Pressure . . 44vi

viiFigure Page15» Variation of Flame Zone Thickness with Pressure 4416. Schematic of Temperature Profile for Double-

Base Propellants. » „ 481?» Temperature Distribution in the Combustion Wave 4918. Temperature Distribution in the Combustion Wave 4919. Variation of Temperature Beneath the Surface and

through the Adjacent Surface Gas Region for Various Propellants . . . . . . . . . . . . . . 49

20. Schematic Representation of Combustion Zone . . 5921. Illustration of Reaction Rate Sensitivity to

Temperature for High and Low Activation Energies 5922. Schematic Diagram to Illustrate Construction of

the Fizz-Burning Curve . . . . . . . . . . . . 6923. Heat Conduction into a Moving Semi-Infinite

Slab with a Fixed Surface Temperature atVarious Velocities . . . » .............. 69

24. Comparison of the Rice and Ginell Burning-Rate Theory with Experimental Data for the Double-Base Propellant HES 4016- . . . . . . . . . . . 82

25. Heat Flow Notation into a Stationary Elementin a Moving Gas Stream . . . . . . . . . . . . 91

26. The Acceleration of an Element of Gas Flowingthrough a Temperature Profile . . . . . . . . . 93

27. Schematic Representation Illustrating Fitting aTemperature Profile of a Second Degree Polyno­mial . . . . . . . . . . . . . . . . . . . . . 93

28. Examples of Linear Pyrolysis Rates for SeveralOxidizers and Binders . . . . . . . . . . . . . 100

29. Schematic Representation of the Burning Surface 10130. Schematic Diagram Suggesting the Essential

Physical Assumptions. . . . . . . . . . . . . . 10331. Rice-Summerfield Burning-Rate Theory Adjusted

to Fit Several Composite Propellants . . . . . 103

LIST OF TABLES

Table Page1. Powder Composition 14400 . . . . . . . . . . . 422. Burning Rate Data for 14400 . . . . . . . . . . 423. Parameters for HES 4016 . . . . . . . . . . . . 80

viii

abstract

The performance of solid propellant rockets under acceleration may deviate from that obtained from static firings. This deviation may arise from three basically separate interactions i mechanical deformation, chamber pressure redistribution, and burning rate variation.

Mechanical deformation may be eliminated as a source of interaction by maintaining low propellant stress levels through proper mechanical support. Chamber pres­sure redistribution can be ignored as a source of perfor­mance deviation if chamber lengths are prevented from being abnormally long.

The burning rate may vary if a component of the acceleration is oriented orthogonal to any of the burning propellant surfaces. This occurs with both double-base and composite propellants and is the result of changing the heat flux returning to the solid portion of the pro­pellant from the gas-phase reactions. The exact manner by which the burning rate will vary under acceleration must await experimental testing, but an approximate technique for estimating the variation in buming-rate for compos­ite propellants is derived.

XFinally, it is shown that accelerations of engin­

eering interest (less than lOOOg) should not produce combustion inefficiency of propellants with powdered metal additives.

Through proper design, the deviations produced by acceleration can be eliminated and thus not be a "tech­nological bottleneck" which will limit the design of advanced chemical solid propulsion systems.

CHAPTER 1

INTRODUCTION

Recently, military and aerospace programs have ' developed requirements for high acceleration solid pro­pellant propulsion systems (1)*. The unusual behavior of certain prototype systems during their ballistic flights has caused concern over the possibility of an interaction between acceleration and the internal bal­listics and burn rates of the motors utilized.

Experimental investigations of the problem of acceleration interaction have produced conflicting results. When certain standard propulsion units were fired while attached to a centrifuge by the Alpha Division of Thiokol Corp. (2), wide variations in performance were obtained with the results statistically non-reproducible. Similarly, spin stabilized systems fired at the U. S.Naval Ordnance Test Station produced telemetry records unlike those obtained upon static firings. Other research groups at Douglas (3) and at United Technology Corp. (4)$ while working with smaller research motors designed

* Numbers in parentheses refer to REFERENCES.

1

2specifically to investigate the problem, did not produce any measurable variation in performance. Other tests care­fully conducted by Redel Corporation (5) did show perfor­mance changes. These contradictory results, while perplex­ing at first, do show a certain consistency when scruti­nized against the correct criteria.

First, it may have been that when the motors were of the internal burning grain type and had sufficient mechanical support to prevent grain slumpage no measur­able variations were recorded. Secondly, when the motors had poor mechanical support or the yield stress of the propellant was exceeded, erratic performance was obtained. Third, when the acceleration was oriented parallel to the burning surfaces no deviation was recorded if the yield stress was not exceeded and the propellant was adequately supported. Finally, when the acceleration was oriented with a component at right angles to the burning surfaces some deviation was recorded as noted in the spin stabi­lized internal burning rocket motors.

Thus there can be hypothesized three modes by which acceleration could interact to produce performance deviation:

1. Mechanical Mode: Acceleration can causepropellant deformation which in turn can pro­duce problems of internal choking, pressure

redistribution, and accentuated erosion«There may also exist a dependency of b u m rate upon propellant stress level. Also under this category may be included mechanical failure of the propellant with the production of grain cracking, inhibitor separation, or failure of the case bonding with the production in all cases of unintended burning surfaces.Pressure Model Acceleration can produce var­iable pressure gradients within the chamber such that the internal flow pattern differs from that obtained statically. This.change can produce variable localized changes in burn rate along the grain and affect the mass efflux by the nozzle such that the chamber - must seek a new equilibrium pressure.Combustion Mode: Acceleration can result incomplex changes in the heat transfer of the gas phase reactions back to the solid unburned propellant. Combustion inefficiency could be produced by carrying unburned particles such as metallic additives beyond the point of influ­encing heat feed-back to the propellant, or in the case of additives which require long stay times, acceleration could carry these particles outside the chamber before complete combustion.

4The purpose of this thesis is to examine the above

interaction modes in the light of present combustion and internal flow theories in an attempt to establish theore­tical support for them, and to attempt to predict to what extent performance would deviate from static firing records at a specified acceleration level.

CHAPTER 2

THE MECHANICAL ACCELERATION INTERACTION

Solid propellants are in general visco-elastic materials with mechanical properties not unlike that of rubber or common plastics. They, therefore, do not have a definite yield point and are time dependent in their abil­ity to yield under stress; also, they creep under load and may deform, i.e., slump, during storage by their own weight. Most propellants have a Poisson ratio of approx­imately 1/2; thus there is very little volumetric change with deformation under load. Thus when a tubular grain is subjected to compressive axial acceleration it shortens and deforms such that the interior port area decreases (6). This is illustrated schematically in Figure 1.

The flow through the deformed port produces a vari­able pressure distribution above that which normally occurs such that the chamber head end pressure is greater than that in the plenum section prior to the nozzle. This pres­sure differential further aids in deforming the propellant which, of course, further increases the pressure differen­tial. This problem was analyzed by Bartley and Mills (1956) (6).

6

P K O f - E l.i.ANXT "LEFOkhAG UNDER. Ac_CE_EkA. fi THIS PoSkTtOhX

P R O P E L L L A H rG»R AlNSUPPORTSG A S F L O W

<:-------------- V e h i c l e A c c e l e r a .'t i o h

Figure 1. Propellant Deformation as the Combined Result of Acceleration and Pressure Drop (after Bartley and Mills).

For the purposes of this thesis, it is sufficient to note that this is primarily a problem in stress analy­sis and that all production motors have designated accel­eration limits which should not be exceeded. In turn the design engineer must establish these limitations and, when needed, provide adequate bond strengths and mechani­cal supports, and, finally, must maintain stress levels well within the physical specifications of the propellants utilized.

Thus the problem of mechanical interaction with acceleration can be eliminated by adequate design and is not a phenomenon over which the designer has no control.

CHAPTER 3

THE CHAMBER PRESSURE ACCELERATION INTERACTION

A. Isentropic Flow with AccelerationThe combustion process of solid propellants is con­

fined to a zone immediately next to, and just within the surface of the propellant (7)» Considering an idealized motor where the acceleration is not allowed to influence the combustion process directly, or a real motor where no component of the acceleration is allowed to be orthogonal to any of the burning surfaces -- and thus has no effect upon the burning process (as will be demonstrated in a later section of this work) — the effect of the accelera­tion is to only accelerate the products of combustion down the chamber and to alter the flow process through the nozzle.

If certain assumptions are made, the problem may be analyzed by modifying the energy and momentum equations to introduce the effects of acceleration in conjunction with the isentropic flow equations. Thus it will be assumed that the products of propellant combustion are inviscid, behave adiabatically, and obey the perfect gas law.

8

Since the burning-rate of a oractical solid pro­pellant is on the order of 0.5 in. /sec. (0), the solid-gas interface does not change its oosition relative to the vehicle very rapidly, and thus may be considered stationary in a quasi-steady state manner.

If it is further stipulated that the changes in cross-sectional area of the combustion chamber and nozzle are gradual, i.e., (-r — « 1), then the flow may be treated as one-dimensional.

Under these limitations the flow of the combustion products is described by the work of Pottsepp and Wu (196/*) (51). The following equations describing the flow may be written:

Continuity; p U A = constant (1)

Momentum: pU-g; - p a + ^ ~ = 0 (2):j2Energy: Cp T + ^ - ax = constant (3)

State: P = p R T (4)

With the following notational definitions:p = density of the gaseous combustion productsU = velocity of products relative to chamberA = area of local cross-sectiona = vehicle acceleration positive when opposite

to flow

10P = pressure of gaseous productsT = temperature of gaseous products

Cp = specific heat at constant pressurex = distance along direction of flow relative to

chamberR = combustion oroducts gas constant.

Equations (1) through (4) may be all put into dif­ferential form. Thus the continuity equation becomes:

i e + * I + ii = 0 (5)

The momentum equation was already in differential form but may be modified to a more useful form by applying the defi­nitions of Mach number and sonic velocity, i.e.,

Mach Number: M = -1 (6)c

andSonic Velocity: c = ^ kRT (7)

where k is the ratio of specific heats. Thus applying equations (6) and (7) to equation (2) the momentum equa­tion becomes:

kM2 - ka ^ + IE = 0 (8)u c rThe energy equation (3) may be put into differential form while applying the relationships that

11

C = kRk-1to obtain:

+ (k-l) M2 ^ - (k-1) adx = oc~

The gas law in differential form is:

(9)

(10)

Finally, equations (6) and (?) become respectively:

dM _ dU + dc = 0M U c (ID

and dc 0 (12)

These equations contain eight differential variables; six of these may be solved in terms of the remaining two (— ■ and ^r^)* The following six equations are thusobtained:

dA . adx ^ c2 (13)

(M -1)M2 dA + adx

A (14)

and finally

dcc

12

M2 dA + adxr ^ (18)

Several interesting conclusions may be drawn fromexamination of these differential equations. First, when the acceleration is zero, the flow properties become only a func­tion of the area change in a manner normally derived in textson gasdynamics (9). Thus in each case, the (-~ -) term acts

cas a perturbation term of the area change. Secondly, when(— •) and (- ~~) have the same sign they aid each other in

A c

causing the flow to converge or divergence, and when they are of opposite sign they delay the process. Third, since the area is a function of the distance along the chamber, the relationship could be integrated if given a specified geo­metry and vehicle acceleration. The last point worth mention­ing here concerns the behavior of the location of the sonic

13pointo Since physical considerations require the flow to be continuous in its transition from subsonic to super­sonic through the throat, it may be noted that either equation (13) or (18) could be evaluated at the sonic point to yield:

- i = - 7 (19)

This leads to some interesting further conclusions. Ifthe acceleration is zero, the sonic point exists at (~)dx= 0 which is the point of minimum area, and which by defi­nition is the geometric throat; but if the acceleration is not zero, then the sonic point does not exist at the pointof minimum area. If a> 0, then (A&) < 0, and the sonicdxpoint exists upstream of the geometric throat, and if a < 0, then it must exist downstream of the throat. This is illustrated in Figure 2.

B, Flow Through the Plenum ChamberThe flow from the propellant surface to the exit

plane of the nozzle may usually be divided into two regions as illustrated in Figure 3, That portion between the burn­ing surface and the nozzle entrance is generally termed the plenum and has a constant cross-sectional area. The remaining section is the nozzle.

14 Posi + i ve V^nicie ucc.I crdu Li<sn.

Loi-&tian at soni plane tor & <iecelerat.'nj y e h r d e . a.<o, <M >o

iPvenLim Ch amoerKottVe Entrance

ELy.lt

— 4l

Figure 2. The Location of the Sonic Plane as a Function of Vehicle Acceleration in Schematic Form.

Mosaic Exit! U- Sonic PlaneMo t a l e ELntrance

^ Geo/netr.c n"r\raa+F i. a m e F r o n t

P r o p e l , l a m t

P o s i t i v e V e h i c l e / X c c e l e f d.ti o n

Figure 3. Notation and Parts of a Rocket Motor such as the One Analyzed by the Text.

15In a non-accelerating rocket, the products of com­

bustion move through the plenum chamber with low velocity and little measurable change in flow properties. Under acceleration, this is nearly so within certain limitations as will now be demonstrated.

With constant cross-sectional area, equations (13) through (18) reduce to the following:

fr = - 7 — U (M2-l)adx c:

(20)

fk-1

M2-l L cadx2 (21)

dMF"

k+12(M -1)

adx_2 (22)

adx2 (23)

and

adx_2

dc „ -(k-1) r adx2(M2 " 1 "2

(24)

(25)

Solving equation (22) for ( p^), it becomes:c

adx _ 2(M -1) dM7 1^1 r *

16which may now be substituted into the remaining expressions in order that they may be integrated to give the following between their limits of flow conditions at and X2:

2u, m 2 Tk+TT— = (— ) (26)ui Mi

-2(k-l)(HT)

and

■ Y 1271

- 2kp m ? Tk+TT— - (— ) (28)P1 M1

- 2Pp m Tk+TT

c M { M l— - (-2 ) (30)C1 M1

If a coordinate system is established such that x^ is that station where the sonic condition exists and is setto be zero by definition, then x0 = x is the distancewhere conditions denoted by subscript two exist with respect to the sonic point. Using this coordinate system and the above equations, the energy equation may be written in the form:

where the asterisk superscript denotes flow properties atthe sonic point.

(30) may be applied to the plenum chamber to describe the flow properties as they change under acceleration in their course from the propellant surface to the beginning of nozzle convergence.

be used to solve for the distance to the reference sonic point. This is similar to finding lengths in a channel with heat addition, where they are referred to the station where choking occurs - even though this reference point may be outside the physical length of the channel. After doing this, conditions could be solved for at the nozzle entrance, as its distance from the reference point would be known from the geometry of the motor (i.e., from the chamber length, L).

Thus the distance from the burning surface to the sonic point is:

^surface to “ sonic point

This last equation along with equations (26) through

Equation (31) in conjunction with equation (30) may

[<Msurf(32)

where the term in the last set of brackets is*2

l r18

Thus with Xgurf and the conditions at the sonic point (known from equation (31)), equation (32) can be used to give noz­zle entrance conditions by stating - where subscript N.E. stands for nozzle entrance:

"N.E. %surf to ~ L sonic

+ (Mn .e /Q___2a*2

(33)

This equation due to its form can not be solved directly for the Mach number at the nozzle entrance but would require iteration; however, certain arguments can be made for the purpose of this work without performing this step. With current propellant combustion gas properties (33), the distance from the surface to the sonic reference point for accelerations on the order of (10) g is on the order of (10)^ feet; thus, the change in station from Xgurf to Xjy E * which is on the order of (10)^ ’, does notresult in a measurable shift of flow properties for motors of current engineering interest.

C. Flow Through the NozzleThe flow through the nozzle can not be described by

equations (26) through (31) as there is a changing area

and the sonic condition at the throat shifts as was men­tioned previously.

This may be substituted into equations (13) through (18). These may now be integrated between the limits of local conditions to those at the sonic point - which will be des ignated by an asterisk superscript - to obtain the follow­ing:

In a similar manner to that used in the discussion of the plenum chamber, the energy equation can be modified with the previous set of equations to the following form:

Equation (16) may be put into the following form:

(34)

T (35)

and

(36)

(37)

(38)

where the sonic point is designated as x = 0.Again in a manner similar to that used in the plenum

chamber, equation (39) can be solved for the distance between local conditions and the sonic point to give:

2c (-Am ) A*

2a

2(kzl) i (- -2(kzl)k+l ' k+T

Mk+l k+l

k-1(40)

Since both x and A* are unknown, this equation can not be used in the above form to give us the complete pic­ture of how the motor behaves under acceleration.* How­ever, if the nozzle geometry is specified or already known, then there exists a second functional relationship between x and A*:

A* = f(x ) (41)

Thus since the flow properties are known at the nozzle entrance, a trial and error procedure between equa­tions (40) and (41) can be used to determine the correct value of A*. With the value of A* known, the discharge

* Under the coordinate system adapted here, with "a" being positive when opposite to the direction of flow, x will be negative as it will be upstream of the sonic plane.

characteristics of the nozzle under acceleration may be determined, and thus the chamber pressure shift and change in burning-rate may ultimately be determined as will now be demonstrated.

The energy equation may be modified again to give the sonic plane temperature in relation to the stagnation temperature. Since the kinetic energy of the flow leaving the plenum chamber is very small, due to its low velocity, in comparison to the flow heat energy, one can make the assumption that the distance "x" found in equation (40) gives the distance through which the flow "fell" from the stagnation point while converging to the sonic plane. Using this assumption and the convention that x = 0 at the sonic plane, the energy equation may be written as:

c < k+l' + FFI (42)v O

where the product of "a" and x is a positive term since x is upstream of the throat.

Using the isentropic relationships, the ratio of sonic pressure, density, and the sonic velocity may also be obtained:

D. The Shift of Equilibrium ConditionsThus now knowing the conditions at the sonic plane

and its area, the discharge rate may be found from thecontinuity equation. If it is assumed that the increase in free volume within the chamber is negligible, (the volume increase between the solid propellant and its gaseous com­bustion products is greater than 100:1), then the rate of gas generation by the burning propellant must equal that leaving the chamber through the sonic plane. Thus

M = Mgenerated exiting chamber (46)The burning rate can generally be described by a function such as

r - CP” (47)

where C and n are determined experimentally. Thus equation(45) can be put into the form:

Ab Pprop [CPo ] = A* U* (48)

where Ab is the area of the burning propellant, Ppr0p isnthe density of the solid propellant, CPQ is the burning

rate from equation (46), and where the asterisk super­script again denotes conditions at the sonic plane. The

23equilibrium chamber pressure may be obtained from (48) to be:

^equilibrium c ^b ^prop

1n (49)

This may now be written as a function of acceleration by applying equations (43) and (44) along with equation (4) to give:

A b p 'fRToK ‘ Z-2ay.\ f k -\ \ 2.

2(K-0j n-i

(50)

which would have a corresponding burning-rate of:

= c A ^ VRTo K K -rl

nn - i

(51)

The effect of acceleration upon equilibrium pressure and buming-rate is by no means clear when recalling that A* itself is also a function of acceleration and must be solved for from equations (40) and (41). Thus the exact solution is obtainable, but it gives little indication as to what order of magnitude changes may be expected as a result of the pressure interaction with acceleration, except for specific cases.

24To overcome this obscureness, certain further

simplifications of the above equations can be made if they are transformed by certain nondimensional parameters. Non- dimensionalizing the nozzle station and the acceleration can be obtained by the following parameters (52):

4 - I 1521

andM “ (53)

V owhere JL is the distance from where X = 0* (which under this new system of coordinates means the location where stag­nation conditions exist, and under the assumption of low plenum chamber velocity can be assigned to the nozzle entrance) to the exit plane of the nozzle.

With these parameters, equations (42) through (45) may be written as:

ip = (1+vu^ *) (54)

r 1 m k§r (1+'u$ * ) (55)

* Note that with X = 0 defined as the location of the stagnation condition, and thus the nozzle entrance under the assumptions made, then all X are positive. The acceleration of the vehicle is still positive when oppos­ite to the flow direction.

25

e (56)

and

(57)X*where £ * ■ i.e., the nondimensional distance from the

nozzle entrance to the sonic plane.Solving equation (39) for (-A) and using the non-A*

dimensional parameters just established, and the new def­inition for X, the energy equation will now acquire the form:

k+1A

I*1M k f r (1 + ¥ M >

i -v ju r *

2(k-l)(58)

The parameter JU. is usually very small for accelera­tions of engineering interest. This may be demonstrated by rearranging slightly:

J _ _ a I(k-1) = ai(k-l).CpTo KR T. =o2

and noting that for most solid propellant motors of c0 » ai(k-l). Thus some useful approximations may be obtained

26by expanding the flow variables in a power series in jjl as higher ordered terms in jjl quickly become insignificant.

To locate the sonic point first let the geometry of the nozzle be described by a third new parameter:

T =■ - j - = (59)

where r is the nozzle radius. Now equation (19) may be written as:

dA« - - £dx = ________ (60)

°* 2 ( M )(1+< * >

This may be solved by a series approximation after expand­ing A, , and in powers of .

First, may be expanded in a Taylor series about 4 = to obtain:

7 = q* + + -• (6D

where f( ) is restricted to be a function which has con­tinuous derivatives. Subscript t denotes geometric throatvalues, and f•( ) is zero as this is the point of minimumarea by definition as may be seen in Figure (4)•

27

- V E - H x CL e . A C L E . u B K / X T l O r 4

Figure 4 Nondimensional Nozzle Notation

The derivative of A with respect to £ may be put in terms of to get:

^ ^ t l 6. = ( f i r - " ■ ,6!’

Now if ^ * is expanded in powers of jll , it would become:

£ * bs ^ 1 *** ^ julz> jo.3 -*• — ^ (63 )

where ^ \ ^ \ etc. are unknown coefficients yet to be determined. Now if the right-hand sides of equations (60)

28and (62) are set equal to each other and the substitution for £ * made from equation (63 ), the coefficients of the series for ^ * may be determined by collecting coefficients of line powers of ^ . Equation (63) now becomes:

5 * = 1 1 ~ (64)

Equation (61), when evaluated at the sonic plane, now becomes:

(65)

and from the definition of as expressed in equation (59), the sonic area is:

A = A * [ 1 + (66)

As was previously discussed, to a first order approximation, higher ordered terras of yt may be neglected; thus equations (64), (65), and (66) reduce to:

V = -it >

29

and A* = A ., respectively.The interpretation of this phenomenon is rather

interesting; in effect, what happens is that the sonic plane moves upstream in a rocket with positive accelera­tion without any apparent increase in cross-sectional area. This is illustrated schematically in Figure (5).

Interpretation of the above results and equations (13) through (19) leads to the conclusion that the flow reacts as if the area had been perturbed by the accelera­tion. Physically, what happens is that during the finite time required for the flow to move towards the throat, the throat has advanced upstream relative to the initial starting position. Similarly, the reverse is true on the diverging portion of the nozzle in that the wall is pulled out from under the flow giving it more room in which to expand. For levels of acceleration of interest (i.e., less than lOOOg), the sonic throat moves upstream and the flow acts as if a tube moves through it. The net result is that the cross-sectional area of the throat "stream- tuben remains relatively constant. However, the fact that the nozzle does advance upstream causes a change in the relative momentum and energy of the flow and thus does

30

— cL

Son.'c t? t anc

— CSeomctri'c fmroatPhysical.N<a 4. *Le , Co n toar

P e r ti>rbe<i MoiaueCon'to'-tt-

Figure 5a Schematic Representation of Flow Through an Accelerating Nozzle.

a_F L O W

— S o m e P L an e

<£>comeVr,‘c Thr<aat"Per turbed No a 4Le C-<a r\-t-o <~^ • Physical

N o t t L eCor\-V«o uir

Figure 5b Schematic Representation of Flow Through a Decelerating Nozzle.

31effect the equilibrium pressure which ultimately changes the buming-rate. Recalling equation (45) and performing the necessary substitutions with equations (56) and (57), the mass flow rate through the sonic plane is:

= e * A*u * = P-V#r. (^ r ) r ' (6?)

This may be written in powers of A and compared with that obtained from static firings to give:

pl* = ( 1 + £ ^ 3 ^ 4 * + -) (68)0 St&tiC

where P0 static and M*static are the stagnation pressure and mass flow rates obtained statically.

Using equation (47), a second expression relating mass flow rates and stagnation pressures may be obtained, i.e.,

”* ■ "••-.ti- ,691

where r\ is the pressure dependency exponent of the burning- rate. Solving equations (68) and (69) simultaneously, the equilibrium pressure shift may be expressed in a power series in as:

32

L w w . - ' - z J n K h - - <70’

Finally, using equation (47), the shift in burning-rate may be written as:

r static = 1 - •••• (71)

Examination of equations (70) and (71) indicates that the nozzle discharge rate was assisted by the inter­action of the acceleration such that the equilibrium pres­sure shifted to a lower value and thus lowered the burning- rate.

For typical values of JL , k, and n *, the order of magnitude change in burning-rate and chamber pressure is

-4 -3(10) in and (10) psi for accelerations on the order of sec2(10) g; thus changes in ballistic performance, directly

attributed to the pressure interaction with acceleration, are far below present experimental accuracies. These changes would be masked out by the statistical variations found between motors and propellant batch mixes.

* For high acceleration rocket motors, values of JL are a function of optimizing case geometry to maximize propellant mass fraction and are on the order of (10)1 feet or less. Values of k run from 1.1 to 1.3 and values of n run from 0.6 to 0.85.

33E. Supporting Experimental Work

Careful experimental work by Landau and Cegielski (3), Iwaniciow, Lawrence, and Mertens (4), and by Horton (10), have demonstrated with both experimental motors and operational rounds that this effect was not measurable within the limits of their experimental measurements. In all instances, they insured sufficiently low propellant stress levels by using a high ratio of propellant bond area to propellant mass, or by staying below specified acceleration limits. In all cases, internal burning grains were utilized.

Figure (6) shows a typical group of chamber pres­sure versus time traces; these traces along with Figure (7), the corresponding variation in burning-rate with acceleration, were obtained by Iwanciow, et al.

Figure (8) shows the performance of the standard round 10 KS 2500. (This designation reads as follows; 10 seconds of b u m time, potassium perchlorate composite solid propellant, and 2500 lbs. of thrust). The deviation curves of + 3c and -3 6" are those experienced in statis­tical firing of static rounds; the acceleration fired rounds are well within this quantity.

Finally, Figure (9) is representative of the work done by Horton on the investigation of acceleration upon b u m rate of the Thiokol motor TE-243-1* It may be noted

34

Figure 6.

rTs+*V.c

Figure 7.

10 00

j)0-100,

u)Of3vllV)ulOfUt

300

tooo500

1000

500

<Iu 1050500O 0.10 0.2,0

T I M E , sec.

SOOg

O.iO

Experimental Pressure versus Time Profiles Taken from Identical Motors (UTX 1724 A with 0% Aluminum Additive). (After Iwanciow, et al.

.............. !

1---------- A <0----

-----------%

o 200 4.00 (300Acceleration Level, g

800

Buming-rate Deviation from that Obtained Statically versus Acceleration. (Based upon the Pressure Time Traces of Figure (6) and others Obtained by Iwanciow, et al.)

Figure 8

<4(tioOO

1 rrii e: , sec.

Sandia Tests, 10 KS 2500 Test Motors under Acceleration. (After Landau and Cegielski)

T f s T M O T O R.3*

e « r c M - T O . 6 A T L H

0V<091{UJ

S -30iI "om

8ftTCH-To-e«rcHTcst MoronL I M I T S

^ LEtiEklD: LlRCucRR C4K«iM q

S O

700

c h a m b e r p r e s s u r e :, psia.

Figure 9 Burning-Rate versus Chamber Pressure, Reported Test Data. (After Horton).

that the data scatter under acceleration is less than the statistical scatter of static firings as indicated by the deviation lines for firings made from the same propellant batch and by the batch-to-batch variation. Hence, it may be concluded that while the effects of the mechanical interaction mode could be circumvented by proper design, the effects of the pressure interaction mode may be neglected when considering motors with normal lengths at acceleration levels of practical value.

CHAPTER 4

THE COMBUSTION ACCELERATION INTERACTION

In order to understand how acceleration can affect the combustion processp the combustion process itself must be first understood. It is doubtful if the process will ever be completely understood due to the complexities of the chemical processes occurring. The truth of this may be appreciated if one considers the complexity of a sim­ple combustion process such as the burning of hydrogen in oxygen. The work of Lewis and von Elbe (1942) (11) is classical on this matter, but even their work, while a monumental milestone, fails to explain the complete pro­cess. This is primarily due to the lack of information on the reaction kinetics of the myriad number of simultan­eous interdependent reactions. Thus when something with the molecular complexity of a propellant is analyzed, there is scant available knowledge upon which to base the nature of the chemical kinetics.

Although the complete problem is insurmountably complex, physical chemists have succeeded in making strides towards analyzing the process by using certain artifices such as "over-all reaction rates" and average

37

38frequency coefficients and activation - energies. These, along with the concepts of activation centers, activated complexes, and the concepts of diffusion, conduction, and temperature from the kinetic theory of gases have allowed the formulation of combustion mechanism models (12).

Present solid propellants may be divided into two general categories,, double-base propellants and composite propellants. In general, for a propellant to be consid­ered acceptable, the burning process must occur in parallel layers and the burning surface must recede perpendicular to itself. Propellants may either detonate or deflagrate. From the standpoint of a propulsion engineer only the latter is of interest and specifically that phenomena which occurs between 10 and 200 atmospheres as dictated by propulsion system efficiency and structural limitations, A propellant suitable for a solid propulsion system must burn reliably and reproduce its characteristics, given any specified initial temperature and chamber pressure.The linear rate at which burning progresses through such a propellant can generally be made to satisfy equation (13) or by piecing together several such relations. In addi­tion to the strong dependency of bum-rate upon chamber pressure and initial temperature, a propellant may be influ­enced by changing its composition, or, for composites, by changing the oxidizer particle size; or it may be changed

by the addition of ballistic modifiers» the direction of extrusion, or even by the flow velocity (13) along the surface. To make the problem more tractable, these minor dependencies will be held invariant in the consideration of the combustion model and how it is affected by acceler­ation. In order to idealize the combustion model as much as possible, all nonsteady phenomena are discarded; the pressure distribution in the chamber is assumed constant; the process is assumed to be one-dimensional; the system is considered to be adiabatic and inviseid; and the burh- rate is considered to be of a modest level - most bura- rates are less than 1 in/sec. - such that the regression of the surface is slow enough in comparison to the expand­ing gaseous products of combustion that the burning sur­face may be considered as stationary in a quasi-steady- state manner. Thus, ideally, the combustion process can be treated as a one-dimensional steady-state process which consumes an advancing propellant surface at such a rate as to cause the solid-gas interface to remain stationary with respect to the nozzle throat (14)» (This idealiza­tion is approached by large end-burning motors).

The Crawford Strand Bomb (21), the standardized research tool in solid propellant combustion work, does not fit the idealized model above but is thought to be close enough to give data which can be\corrected to such

40an environment. It generally gives data which fall about 2 per cent below that obtained in complete system firings (21). Burning rate tests are conducted in the bomb at the desired chamber pressures with a nitrogen atmosphere. Fuze wires are imbedded in the propellant such that as the flame passes, these melt and stop electrical timers which enable the experimenter to determine the linear burning rate.It is mainly through the use of this piece of equipment that the experimental evidence was obtained upon which the combustion models have been formulated (15).

Historically, double-base propellants gained prom­inence first and the work which developed its combustion model laid the foundation for that on composites. Hence, following the sequence of development, the discussion of composites will be delayed to a later section.

A. Physical Evidence for the Combustion Model at Double- Base Propellants

Double-base propellants are composed of a mixture of fuel and oxidizer intimately mixed and bonded onto the same molecular structures which remain In thermodynamic metastability until ignition. They generally have as their principal constituents polycrystalline nitrocellulose, approximately 50 per cent, plasticized with nitroglycerin, from 30 to 40 per cent. To this is added other materials

in smaller proportions to serve as stabilizers, non­explosive plasticizers, coolants, extrusion lubricants, opacifiers, and ballistic modifiers (15). These additives are considered not to alter the fundamental steps of the combustion model, but only the rates at which they occur<, Many are neutral in their effects (16).

Using the Crawford Bomb technique for a typical double-base propellant, bum-rate versus pressure curves may be obtained as those shown in Figure (10) where the dependency on initial temperature may be seen as well as the logarithmic dependency on pressure (17). Analysis of the products of combustion remaining in the bomb are presented in Figure (11)(18) and Figure(12)(19). Note the transition which occurs indicating a change in com­bustion mechanism. Similarly, if the bomb had been placed inside a calorimeter and the heat of explosion measured, an increase in the heat would be observed par­alleling the change in chemical constituents as may be seen in Figure (13)(18).

Photographic records of the combustion indicate a definite visual change in the reaction accompanying the increase in heat of explosion (18)(20). The initial process which occurs at the lower pressures is a non- luminous reaction; as the chamber pressure is raised, a luminous flame appears in the gas stream at a considerable

42

Figure 10.

IT 3 .0

a i*>°c —

.O J.O 5.00.5 1.0

AVERAGE. PRESSURE ( |0 0 0 _ B ./ iN 2)

Burning-rate Data for Powder Composition 14400 (After Gibson).

TABLE 1 POWDER COMPOSITION 14400

Nitrocelluloseincluding per cent nitrogen

Nitroglycerin Ethyl centralite Total volatiles

59.^' 13.22%38.96%0.94%1.18%

TABLE BURNING-HATE

2DATA

Burning; Rate at Pressures1 ofTemperature n

H O O 1 lb/in2

2000 lb/in2

3000 _ lb/in2

4000 „ lb/in2

*C. in/sec in/aec inTsec in/sec50 0.76 3-70 0.71 1.20 1.64 2.0425 0.76 3.31 0.62 1.04 1.42 1.76

. 725...,. 0.78 2.24 0.49 0.84 ,1.1? 1.44n ........ 0.77ave.o ' ......... 0.595Tj . . . . 226

43° PREPRfcRTiOhJ £ONfc x. flame iout

co

10

oo zoo >0o 400 soo toOo tod o j zoo 30O ^oc 900 *00/700

Ci-i *-1-*

C H .

.zri- . l_ st?OJJ0 40I ic

lolo

a ‘OO 200 300 f‘X> 5cc faOr 7CO 0 100 100 JOO 4oo 500/ TOO

NO N.

3Do 100 >.00 joe <tx> r,oo bOo loo O loo 2Do JOO 44C S oo/700

lo - <7—rr'a-^ -

-1-- 1--_L_ -I K, ol 1 L. jf,O 100 160 100 4<30 fOO 6 0 0 700 O lOo 2.00 Soo H oc S o o / 7oo

PRE-SSulfte , pSi g.Fig. 11. Change of Composition of Gas with Pressure

(after Heller and Gordon)

co12to

No0 8

CO4-

z.

O0 000Pressure, lioZin.2-

Fig. 12. Change in the Products of Combustion with Change in the Initial Pressure of Inert Gas, Propellant HES 4016 (after Huggett)

LENG

TH

OF

D*\RK

iOblE

44

400 -

800AGO (cOOPRESSURE , psLa_

Fig. 13. Change in the Heat of Explosion withChange in the Initial Pressure of Inert Gas, Propellant HES 4016 (after Huggett)

to

EE

— o•2

zoo 5 0 0 tOOO 2000 PRESSURE., ps'.a

Fig. 14. Variation of Dark Zone Length with Pressure (after Heath and Hirst)

toEUJZ0HiIIII_1

O -

u,u.02zVuil-

1_

500 iooo 2600 P R E S S U R E , psia.

zoo

Fig. 15. Variation of Flame Zone Thickness with Pressure (after Heath and Hirst)

distance from the solid-gas interface. When the chamber pressure is further increased, the luminous flame rapidly approaches the surface and becomes thinner and more intense as can be seen in Figures (14) and (15) . At highpressures, the "dark zone" between the solid and the base of the flame becomes difficult to discern. Of major impor­tance to the combustion model is that while the combustion process changes from a non-luminous process to one which has a luminous flame and twice the heat of explosion, there is no deviation in the burning-rate versus pressure rela­tionship. Thus it may be deduced that very little addi­tional energy from the more energetic luminous reaction feeds back to.the burning surface. It may be further deduced that the non-luminous reaction which exists with­out the flame and precedes the flame during its presence is of primary importance in the rate-determining process except possibly at very high pressures where the flame approaches close enough to the surface to influence its rate of decomposition.

Further photographic work by Heath and Hirst (1962) (20) has resulted in obtaining surface phenomena at high magnification. It was observed that double-base propellants have a solid-gas interface which is non-luminous except for a number of scattered bright globules along with numerous

46small black spheres. The number of these bright “hot spots'8 was observed to increase with pressure, and thus with burning-rate, up to approximately 800 psia where the luminous flame obscured their observance. The size of these "hot spots" measured from 0.01 to 0.03 mm and was independent of their age or of chamber pressure.

These "hot spots" are interpreted to be gas bubbles in the surface foam which have ignited iptemally and are undergoing a reaction process similar to that occurring in the luminous reaction zone (20). Also photographically recorded are vertical luminous columns which appear to be the contents of globules after they have burst streaming upwards to the flame zone.

Photographic definition was insufficient to allow positive identification of the black spheres, but it is believed that these are not bubbles but molten spheroids of the inorganic constituents of the particular double- base propellant being investigated (20)*.

Of importance to the construction of the combus­tion model is that the majority of the surface is free of "hot spots" and that the surface appears to foam. This foaming is further substantiated by earlier work by Parr

* SU/K Cordite which contains 12.2 per cent nitra­ted nitrocellulose, 49 parts by weight; nitroglycerine,41 parts by weight; carbamite, 9 parts by weight; and potassium .cryolite, 2 parts by weight.

and Crawford (21) who found minute gas bubbles in the pro­pellant surface when examined microscopically after extinguishment by a stream of water. Also of importance to the combustion model is that there is a quiet prepara­tion zone separating the flame from the surface reactions; that the base of the flame is of fairly uniform height above the surface; and, finally, that the flame is nearly laminar.

The final physical evidence for formulating a com­bustion model is the determination of the temperature profile through the reaction zones by implanting extremely fine thermocouples. The reaction zones pass over the thermocouples up to the point where they melt - in the luminous flame (19) (13) (22).

The temperature profile for a double-base propel­lant is shown in Figure (16) in schematic form. The por­tion between T0 and T^ is shown in detail for nitrocel­lulose for various chamber pressures in Figure (14), and for various initial temperatures in Figure (15). Note that the surface temperature lies at about 250®C and that the temperature ceases to rise much above 1400°C. It may also be noted that the temperature gradient is steeper as the pressure increases and the reaction zones compress, thus more heat feeds back to the propellant and the rate

48

Hohl- LJM'MOVS — — L JMiNojS-

Ti

Figure 16. Schematic of Temperature Profile for Double-Base Propellants; Note Char­acteristic Steps

of decomposition is accelerated. Raising the initial tem­perature has the same effect (23).

Sabadell, Wenograd, and Summerfield (1964) (22) have found that additional information may be inferred from thermocouple work. In general, the one-dimensional flow of heat in a medium with heat liberation by reaction is given by:

5 7 f * § 7 + c p ( T - -ra)) = ^ (72)

49

'<600

„ IA0O 1 n111 100O

D l S T A H C E C E N T I H E r E k f S X l O 1- D I S T A N C E <_ENT l H E T c E ' ^ A l O A

Fig. 17 Fig. 18Fig. 17. Temperature distribution in the combustion

wave of 13.15 per cent nitrogen nitrocellulose plus 1 per cent ethyl centralite strands at 25°C. in the pressure region 350-700 p.s.i.g. (after Klein, et al.).

Fig. 18. Temperature distribution in the combustion wave of 13.15 per cent nitrogen nitrocellulose plus 1 per cent ethyl centralite at 400 p.s.i.g. in the temperature region 25-60cC. (after Klein, et al.).

0) loco

100

SOi ( b )

'00 ISOSO

A R B ' T K t A R M " D P B T A N C E * m i c r o n s .

Fig. 19Fig. 19. Variation of temperature beneath the surface and * through the adjacent surface gas region for various propel­lants. Position of "surfaces" and respective temperatures (Ts) are indicated. (after Sabadell, Wenograd, andSummerfield).(a) Double-base propellant, 150 psig chamber pressure.(b) Composite propellant PBAA at 70°F., 30 psig.(c) Composite propellant PBAA at 70*F., 100 psig.(d) Composite propellant PBAA at 70°F., 150 psig.

50where -m is the mass flux and c| is the heat liberated throughchemical reaction per unit volume as a function of distance from a fixed reference and T0 is the initial propellant temperature.

assumed constant, then equation (19) may be integrated to give:

Now, as chemical reaction rate studies indicate (24), the process of chemical reaction within the surface does not occur until just at the surface. Thus evaluating equation (20) for the nonreactive region of the solid:

Consequently, if the log of (T-Tq ) is plotted against x, a straight line of slope in Cp/K results. Deviations from linearity may be interpreted as chemical reactions as may be seen by equation (20), or as an abrupt solid-gas inter­face. If the reaction is exothermic, then the integral of equation (73) will cause the slope to decrease; if endo- thermic, the slope will increase.

Also, at the surface between the solid and gas phases, the interface condition is:

If the specific heat (0^) and conductivity (K) are

( T - T o) K(73)

(74)

51JK %as ~Cfr^out," (75)

where Q may be considered as the heat of sublimation.Since Ksolid is greater than K^as by approximately 4:1, asharp discontinuity in the propellant temperature profile would be expected unless the heat liberated by the process of gasification were:

'm Q = (Ksolid " Kgas) •IftnQ were small, a sharp upward break in the temperature profile would be expected (22).

Examination of Figure (19) now reveals some impor­tant insights into the combustion mechanism of double-base propellants and composites.

Note the smooth continuous trace for double-base propellants reflecting its homogeneity versus the erratic records of the various composites which reflect their heter­ogeneous structure. Delaying further discussion of com­posites until later, note that there was never a sharp upward break in the temperature profile of double-base propellants as expected between zones of high and low ther­mal conductivity. The smooth transition of double-base propellants at the non-discrete surface, see Figure 16a), (unlike that of the composites, Figure (19) b, c, and d) infers that the foam must gradually decrease in density while releasing its chemical energy. The foam zone is very narrow - less than 10^(14) - and the upper surface of the

52foam is approximately 300*0. The decreasing slope and lack of discontinuity imply a gradual exothermic reaction by equation (73). Once the volatile products leave the foam they are continuously heated to approximately 1250*0 where, as will be shown later, the fizz reaction begins.

The complete temperature profile. Figure (16), along with the deductions drawn from Figure (19) indicate three distinct reactions; foam, fizz, and flame. Also, there may be noted the presence of a quiet preparation zone separating the fizz reaction from the flame. In an attempt to study the details of these reactions and obtain reaction rates, orders, activation energies, etc., physi­cal chemists have worked with the isolated components of the propellant.

B. Chemical Evidence for the Combustion Model of Double- Base Propellants

The decomposition of nitrocellulose has been stud­ied (24)(13) up to and through the temperature at which it ignites and deflagrates rapidly. It has been observed that the reaction rate is very sensitive to impurities, and it has been observed that some of the products of decomposition are catalytic. If nitrocellulose is progressively purified, it will approach a minimum reaction rate, which increases with temperature and the degree of nitration. In tests where the products of decomposition were removed by a

53flow of inert gas over the surface, it was found that N02 constituted 50 percent of the gaseous products. If these products were allowed to remain in contact with the surface, the rate of reaction was found to obey a simple first- order reaction such that:

-E /RT"ff = A G e ‘ (90ec< T< 175°c) (76)

where c is the concentration of nitrocellulose, and where the activation energy, Fa, was determined to be approxi­mately 50 kcal/mole and the frequency factor. A, was 10^^/sec. Because of the low thermal conductivity of nitrocellulose and the exothermic nature of this decompo­sition, detonation may occur due to the continuous buildup of heat (13).

The nature of the decomposition reaction may be inferred from examination of the residue that can be obtained (the residue is a complex mixture of liquid prod­ucts known as the "red substance" and is often just refer­red to as the "RS"). The residue is rich in glyoxal, formic acid, formaldehyde, and water; traces may also be found of glyceraldyde, mesoxaldehyde, and other more com­plex substances. The reaction proposed which leads to the formation of the R3 is:

54

■CHIHCONO'o2nochHCO:!COHgCONOg

CHHC

HCOHCOx

H2C = 0

+ 3 N02

Cellulose Nitrate, (Press. = 100 mm Hg)

The original step is the breaking of the RO-NO^ bonds fol­lowed by the rearrangement of the resulting free radical with scission (splitting) of the 2-3 and 5-6 carbon-carbon bonds.

If the pressure is increased, all the aldehydic products will disappear. Fenimore (25) formulates that the formic acid, glyoxal, and some nitric oxide react to form molecular nitrogen, carbon monoxide, carbon dioxide, and water.

The decomposition of nitroglycerin £ (ONO^)^ ^begins with discoloring at 135°C with the color deepening as the temperature is raised until the decomposition reac­tion produces an effervescence at 145°C (13). This ebul­lition is due to the generation of volatile products. Continued heating causes the temperature to rise to 218°C where explosion occurs.

55Studies similar to those performed with nitrocel­

lulose show that again NC>2 is the primary end product. A first-order 155°G and 190°C are due presumably to the decreased solubility of the autocatalytic products. The activation energy was found to be 47.8 kcal/mole. Other studies which covered the temperature region below 155°C where autocatalism was important show an activation energy of 26 kcal/mole. The explosive decomposition, as with nitrocellulose, is therefore dependent upon thermal and catalytic action.

The high molecular weights of nitrocellulose and nitroglycerin make it difficult to conceive the combus­tion reaction in its entirety; therefore, investigators have studied other nitrate esters of simpler composition in order to obtain a clearer picture of the reaction kinetics (26).

Methyl nitrate was found to follow a first-order reaction between 210 and 240*0 (2?):

ch2ono2— ^ c h2o . + n o2

2CH^0* — OH + CH20

At higher temperatures, explosion would soon follow. Addi­tion of argon or nitrogen will facilitate the explosion of methyl nitrate vapors indicating a branching mechanism.

Similar studies with ethyl nitrate behave similarly and suggest an alternate scheme to stabilize the alkoxyl radical:

CHjCHgO + C2H50N02— *-CH3CH2OH + C^H^ONOg.

c2h4ono2. -^ch3cho + no2

This two-step mechanism is more probable than the simple disproportionation of alkoxyl radicals.

Huggett (13) notes that recent spectrophotometer studies of the vapor-phase decomposition of ethyl nitrate by Levy (1954)(28) indicate the following series of reac­tions :

c2h 5o n o 2 = ^ c2h 5o . + n o2

C2H5° ^ CH3e + CH2°

2CH20 + 3N0?--*-3NO + 2H20 + CO + C02

2CH3• + 7N0?— *-7N0 + 3H20 + 200^

CgH^O. + NO — ^C2H50N0

The trends which come out of these studies indi­cate that A and Ea increase with temperature and with increasing structural molecular complexity.

Explosives have, in general, large Afs and Ea 1s.The large activation energy being required to confer

51

adequate stability at low temperatures with rapid reaction rates at elevated "temperatures» The activation energy for the decomposition of nitrate esters has been found to com­pare reasonably well with the strength of the RO-NOg bond. The value of A is more difficult to interpret as it is larger than expected and since it indicates long stable reaction chains of lengths greater than experimentally found. One explanation advanced to explain this phenome­non is that complex molecules, such as RO-NOg, may receive energy from other degrees of freedom which weaken the NOg bond.

C. The Model of Burning for Double-Base Propellants/

The information obtained from the studies conducted with nitrocellulose and nitroglycerin normally came from reactions involving a single phase. The burning process of double-base propellants must bring into account the chemical and thermodynamic interaction of successive reac­tion steps. Since the burning has been found to proceed perpendicular to itself in parallel layers, one may describe the process as one-dimensional. The propellant, as men­tioned previously, has poor thermal conductivity and, there­fore, the combustion process affects only the surface of the propellant to a relatively shallow depth. As the burn­ing surface approaches a reference point within the

58propellant, the point becomes heated, first, by conduction, and then by local exothermic decomposition. As the temper­ature rises, volatile products are produced along with liquefaction such that a foam of decreasing density is formed. This foam finally decomposes completely into volatile fragments, which, in turn, react with one another until a critical concentration of propagation centers is formed. Finally, these propagation centers then become consumed by the flame. Schematically, the reactions are repeated as shown in Figure (20).

All three stages, excluding the preparation zone, are exothermic. The third stage, i.e., flame zone, does not appear until the pressure exceeds 20 atm., and apparently does not affect the rate of reaction at pres­sures less than 100 atm. The physical appearance of the combustion process (20) suggests that the second gas-phase reaction (i.e., the flame zone) resembles a branching- chain explosion.

At low pressures ( < 3.0 atm.), when only the sur­face reaction occurs, the surface temperature is T*. As the pressure is increased, more heat is generated in later reactions and some of this heat conducts back to raise the surface temperature to Ts. Below 100 atm. this latter heat conducts back from only the fizz zone, but above 100 atm. it comes from both the fizz and flame zones. The temper­ature at the end of the fizz zone, with no flame zone, is

59F e arn i o o e

' <=a> Phase KcActions o f +he F f i i i o n c

Fi * a z£ o neSo lid (■ LJnreac+e ^

" S u r f a c e '

FlameH.oncL u m . o o o s ziooe"

— Preparation Zone"Dark. Ho he —

[r 'c h o ] Tm o + c o [rONOiJ *|_ MOa j"*- [^HCHO+ HiO„ »o ... O ... .. Xt... DtS TAr-lC ES C

O .............. t , .Ts....r,.X V T / .Ws...W,..<K..... 1 • •

Vf o ••• Hy H i •

rNi+cO+ COij ■ |_ t H i + hlzo J

- To— T

... X 2. ...... X3 " * *T im G S , Ct. ) ..................... - t i t 3T e m p e r a t u r e s c t ) ......... T * . ............. T sTHEORETICAL HlHlMUM T...-Ti T J D l o u c C U L A R W g»T S . ( w ) . . .

r l T H t R M R L D l F F u S t V l T l E X ' j K j . . .

Elr4 TH ALPiES ( H )... H3

-V £50-

Figure 20. Schematic Representation of Combustion Zone Showing Temperature Profile with Chemical Reactions and Notation of the Rice and Ginell Theory.

L Case CO*H.oh A<-+W*t.'on ELnerj jifej5#x

111h6

§U1a

— Case C LOuj A c t l v a f . o o Eoc^ Reaction

AT

T E M P E R A T U k i E . ^ T

Figure 21. Illustration of Reaction Rate Sensitivity toTemperature for High and Low Activation Energies. (Based upon Conversations with C. A. Heller and R. H. Knipe of the Research Dept., U.S. N.O.T.S., China Lake, Cal.)

60T^8. At pressures above 100 atm., the temperature rises to above T^8 by conduction. The temperature at which the flame starts is Tg. (Tg > as it is closer to the flame; the flame zone does not start immediately after the fizz zone stops). The temperature leaving the flame is desig­nated as . This notation is illustrated in Figure (20).

D. The Rice and Ginell Combustion Theory for Double-Base Solid Propellants

There are two possibilities for the gas-phase reactions: (1) If the energy of activation is small, theaverage distance from the surface where the reaction occurs and the time which elapses until it occurs will be only a weak function of temperature. (2) If the energy of acti­vation is large, then the distance and time required will be strongly dependent upon temperature. Thus the reaction

. must await being carried into a region where the tempera­ture is high enough to accelerate the reaction. This may be more readily visualized by examination of Figure (21) where reaction rates of case (1) and case (2) are shown as a function of temperature^ Note that the reaction rate levels off for case (1) in the temperature region of inter­est and thus has only a weak dependency upon temperature and is controlled strongly by the rate coefficient. The reverse is true for case (2); since it has a high activa­tion energy, not much occurs at low temperatures, but once

61the temperature reaches a sufficient energy level to ini­tiate the reaction, it progresses at a very rapid rate and is strongly dependent upon temperature and only weakly upon the rate coefficient. (Note, of course, that this has meaning with respect to only one temperature region as the reverse was true at T*.)

Case (1) was considered by Rice and Ginell (14), and case (2) can be found in Boys and Corner1s work (23). Case (2) would have the gas-phase reactions be the rate- determining reaction in the burning-rate mechanism. This model would, in effect, have the overall reaction con­trolled by the flame speed of the gas-phase reactions in the fizz and/or flame zones. That is, if this were the rate controlling mechanism, then the gas-phase reactions should advance or recede from the surface until the heat balance was such that the rate at which the surface decom­posed and fed reactants to the gas-phase reactions matched the gas-phase reaction flame speed.

The luminous flame can be ruled out as the rate controlling reaction when it is recalled that very little or no heat is conducted back to the surface below 100 atm..; consequently, the rate of production of reactants would , either be too fast or too slow through much of the region of interest. If the rate of production were too fast, the

62flame would recede to infinity, and if too slow, the flamewould approach the surface until it dominated the rate ofheat transfer back to the surface.

If the gas-phase reactions in the fizz zone werethe rate controlling reaction of the burning mechanism,then as the pressure was lowered a point would be reachedwhere the surface temperature would try to drop below T*,This would result in a sudden shift in the burning-ratelaw as the temperature of the surface can not drop belowT® as this is determined from reactions at, and below, the ssurface. Since no such change is observed to occur, it is concluded that the gas-phase reactions of the fizz zone do not control the rate of the overall reaction mechanism.

Since neither of the two above possibilities occur, it may be concluded that the position of the flame is con­trolled by an initial rate controlling mechanism at the surface, and occurs in space at a distance and at a time which is a function of the overall reaction rate of the gas-phase reactions, and is only weakly dependent upon temperature as envisioned by case (1),

The gas-phase reactions may be analyzed based upon case (1) and their influence upon the surface rate con­trolling reaction determined, if certain simplifying assumptions are made. It is assumed that all of the reac­tions composing those of the gas-phase reactions can be

treated as a one-dimensional steady-state phenomenon. It is also assumed that the reactions of the fizz and flame zones may be confined to two planes at and x^, respec­tively. (This is a simplification of the studies made by Emmons (30) on gas-phase reactions in flame. It was demon­strated that most of the reactions occur in a narrow zone near the end of the flame). Also, if at first the analysis is limited to pressures below 100 atm. the effect of the flame may be neglected.

Taking the heat balance of an element at some x< x^, which is fixed in space above the surface, it is at equilibrium when the amount of heat being conducted upstream from the source at x^ is balanced by the heat being carried downstream by mass transport. Thus when equilibrium is achieved throughout the region between the surface and x^, the temperature, with respect to x, will be a constant. When there is no gas-phase reaction in thefizz zone, recall that the surface temperature is T9> Like-swise, when the flame may be ignored, the fizz reaction at x^ is at T^8. The enthalpy of an element at some point less than x^ is:

H =* Hg? + Gp (T - Ts8), (77)

where TL is the mean value of the specific heat up to thatP

point, H is the enthalpy, and the subscript s denotes

64surface conditions. The heat balance at this point may be written as:

since the reaction has not commenced.Neglecting diffusional effects until later, a small

element of gas rises with a velocity U, where

The rising element of gas experiences a change of tempera ture with time; this may be expressed as:

The density may be obtained from the perfect gas law:

where W is the mean molecular weight and R is the gas con­stant. It may be argued that the variations in X with tem­perature will offset those of p in equation (81) by their compensating behavior (see the Landolt-Bomstein tables*).

* Landolt-Bornstein, Zahlenwerte Und Funktionen aus Physik, Chemie. Astronomie, Geophysik, und Technik.Springer-Verlag","Berlin , 1 $64. K

M Cp (T - Ts') - K dT/dx = 0 (78)

(80)

(81)

where the thermal diffusivity X is:

3c - K/cp e .

p = pw/rt (82)

65Thus holding p and X at their surface values of and X^, equation (81) may be integrated between the limits of t = 0 and T = Ts at x = 0 to t = t1 and T = at x = to give:

Tl-V Ts*

(83)

Substituting in the dummy variable where

ry 2 LnTi- Ts»T - T *s s (84)

into equation (83), and using the perfect gas law, it becomes:

P = (MRTg/WgZl) (ti/jCg)i (85a)

or in logarithmic form:

Ln(p) - Ln(M)- Ln(Zi) + Ln [ (R/Wg)(Ts2t1/Kg)5 j (85b)

The mass flux, M, which is controlled by the subsurface reactions of the foam zone, and which according to the experimental work of Daniels and others may be described by an Arrhenius type equation, can be represented as:

(-ES/RTg)M = Aexp (86)

where A is a "frequency factor" and Eg is the energy of activation.

66Using case (1) as was previously argued, and if the

reaction of the fragments leaving the surface is of order , then the time necessary for the reaction to occur is:

‘ i ■ i ^ r - 1871

where is a reaction rate coefficient which is nearly independent of temperature, and -n is the order of the reac­tion. From the previously mentioned studies of the chem­istry of the combustion process, it may be assumed that the fizz reaction is bimolecular, i.e., <1 = 2J and since ks is inversely proportional to the pressure and at least roughly proportional to Tg , it may be assumed that Ts t /)Cg can be considered a constant parameter.

Creating one more parameter:

(St - (Tg- Tg«)/(T-- Tg«) (88)

or using equation (84):

(31 = exp. (-Z^2) (89)

then the solution is reduced to solving four simultaneous equations, equations (85a), (88), (89), and (86), with five unknowns: Ts, P, M, Z^, and p * and several parameters: ws, (Tg t^/Kg), A, and Eg.

Since the effects of the flame are being ignored,T1 ” t2,» as previously stated. In addition, it may be

67stated that T^* and T^* are functions of TQ (this can be inferred from Figure (18)). From experimental work, the parameters can be evaluated and TQ can be specified, thus M can be found as a function of P. (Note that M, Ts, Z^, and are functions of both T0 and P, thus T0 must be specified and held fixed. For future use, also note that (3 1 is a function of both TQ and Ts.) Proceeding now to obtain M as a function of pressure, one must first evalu­ate the parameters from experimental work. Ts* and T^* may also be obtained experimentally at very low pressures where Tg is free of the influence of T^, and where is free of any influence of the flame. Using these as surface tem­peratures, corresponding values of M may be obtained from equation (86); these are designated as Mg 1 and M . Now on a log-log scale, a preliminary plot of M versus P is made by arbitrarily locating an interconnecting line AB (see Figure (22)) between the two levels of mass flow rates. (Assuming a straight line at this point is the same as assuming that Z^ has a constant value.) If a value of (3^ is now chosen, a corresponding value of Tg can be found from equation (88); this can now be used to find a corresponding value of M from equation (86) and located on the arbitrary line at point C. Using this same chosen value of |3j, , Z^ can be determined from equation (89). Observing the nature of equation (85b) it may be noted

that the right-hand side is composed of a sum of logarith­mic terms. The last term is a constant and served only to locate the curve’s position, i.e., it does not control the shape of the curve. The log (Z^) term serves as a correction factor in changing the shape of the curve from the original arbitrary straight line. Thus knowing log (Z^) for its corresponding chosen value of f3^t the cor­rection length CD is known. Finally, if some experimental point is known, then the curve can be shifted horizontally to pass through that datum point. This step evaluates the constant logarithmic term and eliminates the necessity ofhaving to evaluate k , W , and t^.s s 1E. Effect of Diffusion on the Combustion Model for Double-

Base PropellantsThe effect of diffusion is to provide a concentra­

tion potential which assists the mass transport flux. Diffusion of the active particles away from the place where they are formed tends to reduce their concentration, and thus to increase the time necessary for a build up of critical concentration; this consequently moves the reac­tion planes further away from the surface.

Recall that t^ was defined as the time to reach x^; with diffusion it would be:

69

M *i''ii-ii" Th®ore+i'c»t Misi V Flux ULl ithout- Flarrie E-tfeif'S Q

Curve Sh.ftc<i tb Kn*uun "Data Point

'— Sh'tt VouireA Evaluated C o n s ta n t Leg T e r m - ±

— --/ n [u /vAljVT t./Xs iS- / Lo j A p e r o x i m a t i' o A U s e d C o w / n o n l

N o t m e l R a n g e o< ^Preplsron InterCit in Linear...Mini mum Tneofetiotl Ma»s Flu*

Figure 22. Schematic Diagram to Illustrate Construction of the Fizz-Burning Curve. (After Rice and Ginell). The Zone of Normal Pressures used for Propulsion Systems and the Buming-Rate Law Are All Shown for Comparison.

= T;

ui riere x <-C - ( T j - 0

, efo ^ ° r-

—

$rp«

^ Tweoreti.calcT-O m in in rx u r n J

RCittiOftPlane

Figure 23. Heat Conduction into a Moving Semi-Infinite Slab with a Fixed Surface Temperature at Various Velocities.

where U is the average velocity of the flow. The actual time of the reaction increases to 8 > t^. During the time T ^ 1 the particles will be displaced by diffusion, an amount on the average of m T D"T * , where D5 is the root of the average value of the binary diffusion coefficient between x = 0 and x = x^, and >t, is a constant. (This is based upon an analogy to a planar source of heat into a semi-infinite slab, where the displacement distance is analogous to the centroidal location of the heat wave at time t**.) Thus the particles will be displaced by the streaming of an amount of U where Uj = U, where Uwas the average velocity between x = 0 and x = x^ without diffusion.

Now the distance to the gas-phase reaction plane of the fizz zone will be:

x = Ut1 = ^ JA Dt '' + UT]/ (91a)

This may be rewritten as:

Xx ” V (1 +<=«:) = 1 1 (9 1 b)oC

u ITr «iwhere ex = klV Z F

* See page 109 of L. R. Ingersoll, 0. J. Zobel, and A. C. Ingersoll, Heat Conduction. Univ. of Wisconsin Press,1954, for the analogous heat transfer problem.

pWith constant pressure, D varies as T ; thus sinceiU is proportional to T, the dependence of D'J and U on T is

_ isimilar, and thus it may be assumed that U and D in equa­tion (91b) may be replaced by their values at the surface.Therefore, equation (91b) becomes:

V i = V M DST1 • ' + (1 + « ) (91c)or

— ( 1 + )" s h = USTl’ (91d)

c< 1where T-, 1 = -------- (92)1 (1 + o< )

Another relationship between and t^ may be obtained from a census of active particles. The density of active particles between x = 0 and x = equals the number leaving a unit of area at the surface during time

divided by x^. The number of particles leaving is C0 Ug, where C0 is the constant of proportionality. Hence, the density is CQ U T^*/x^. Without diffusion this den­sity would be C0 Ust^/x^. Let (unprimed) be the aver­age life without diffusion; this lifetime is inversely proportional to the pressure for the assumed order of reaction of 2. That is, from equation (87) with n = 2:

72Since the pressure is directly proportional to the density, the time of reaction is, therefore, inversely related to the density of reactive particles. Since diffusion dilutes this density, then:

li = xiti* co us h

xiand, therefore,

^ 12 - (94)1

which is the second relationship between e and t^ that is needed for the solution.

This last relationship may now be substituted into the last grouping of equation (91c). Using equations (80) and (82) for Us, both sides of (91c) may be squared to obtain:

, i i f MRT 1 2 .> ( 1 + V i h ■ i p w ^ i b (95a)s

which upon rearranging gives:

,2. \ m \ l 2_ V /Zws JU(1 + o< )2 D T is 1

Now if equation (30) is cubed and then divided by (95b), the following expression is obtained:

Note now the similarity between the use of equation (84) and (85a) with the use of equation (97) and (96a) above. The system of equations is nearly identical, but now role of is replaced by that of Z]. Thus the prob­lem is reduced again to finding a certain parameter, this time as a function of (3-; but since is known as a function of (3- it may be used to find Z^ as a function of

provided cx can be determined.The dummy parameter can be evaluated by comparing

the right-hand side of (96a) with that of (85a), and by substituting for t^ from equations (92) and (94). Perform­ing these operations gives:

74therefore.

(98)

which is the expression desired if , and juu are known.Note that Dg and K s both vary with temperature and pressure in a similar manner and thus the right-hand side of equa­tion (98) may be assumed an invariant. Thus equation (98) can be evaluated for ex. and this used to find as a func­tion of from equation (97).

Rewriting equation (96b) in logarithmic form, itbecomes:

such as the one constructed in Figure (22) may be made using now equation (99) rather than (85b). (Rice and Ginell make a further refinement by finding the slight change in [S due to the spreading out of the reaction plane by dif­fusion, but for the most part the correction is minor throughout values of practical interest, and the error is minor compared with some of the assumptions made in hold­ing parameters invariant, etc.)

(99)

Now knowing as a function of Z^, a fizz burning curve

F, The Effect of the Flame upon the Combustion Model for Double-Base Propellants

At higher pressures, the flame physically approaches the gas-phase reaction plane (xj_). This causes an increase in temperature a t ' w h i c h , in turn, causes some increase of the surface temperature, which, in turn, accelerates the burning-rate by a small quantity over what it would be without the flame.

The preparation zone between and xg evolves no heat of its own; it appears to be a period where there is a build-up of active species to a critical concentration. These particles initiate the further flame reaction, which starts at xg and is completed at x^. The temperature at

proximity to the flame.The build-up of active particles in the preparation

zone appears to have an effective order of 3°5; the flame reaction following it behaves as a chain branched explo­sion and appears to have an effective order of 2. (This may be expected as it is a general rule of physical chem­istry that the apparent order of reaction in a chain branched explosion is nearly one-half of the order of the reactions which proceed it. Thus from equation (87) it may be inferred correctly that the flame reaction is not

76nearly as pressure sensitive as that in the preparation zone*.) Using this argument, it is felt that the length of the preparation zone eventually becomes narrower than that of the flame; this is felt necessary in order to explain why the rise in burning-rate is not sharper at higher pressures when the preparation zone becomes very narrow.

Provisionally ignoring the effects of diffusion in order to gain an overall insight into the problem without all the burden of details, and stipulating that all of the heat released by the flame occurs at ; then the tempera­ture at the surface may be expressed in terms of f3 heat feed-back ratios as was done before. Doing this, the surface temperature is:

Ts = v + ( Y - Ts*)p1 + (T3 -

(100)Since Ts1, T^*, and are experimentally obtainable, the problem of determining the surface temperature and thus burning-rate is reduced to one of determining the p heat feed-back ratios, which are by definition:

* Based upon conversations with R. A. Heller and R. H. Knipe of the Research Department, U. S. N.O.T.S., China Lake, Cal.

77

P 1 - (?s - Ts')/(Tl - Ts')

(3 2 = (?1 - Tl'V(T2 - T1' ) (101a,b,c)

^ 3 = (T2 - Ti')/(T3 - Tl,)

where the subscripts still obey the notation outlined by Figure (20). The conduction of heat back through each plane is essentially the same. It is desired to know the temperature at a fixed plane inside a moving medium when that plane is a given distance from another plane which is maintained at an elevated temperature. This is illustrated schematically in Figure (23).

Each |8j may be written in the form:

|3j = exp r U xj ) (102)

where U is the flow velocity, as before; A x j is the length of the jth reaction zone, and Xj is the corresponding thermal diffusivity. The ration (U/Xj) = (M Cpj^K^) is assumed constant within each zone, as was previously argued for the fizz zone.

The time required to traverse any particular zone may be used to express the length of the zone, i.e.,:

• . rtAXj —AtjU ™ Atj(M)(p- ) (103)

78The time A t j is the time necessary for the reactions required in that zone to go to completion, and is a func­tion of pressure, see equation (87). For the second order reaction of the fizz and flame zones, the reaction rate will be proportional to (l/P) and that of the preparation zone will be proportional to (l/p^*^) (31)**. The length of the reaction zone then, in general, will be:

a x , = MRTj (104)

where n is the order of the jth reaction, and kj is the average reaction rate constant. Substitution of equation (104) into (102) gives:

/3 j = expl W 3

(105)

ThejSj's, it may be noted, are functions of the pressure and certain rate constants; the surface tempera­ture is a function of the and finally the buming-rate is a function of surface temperature and is given by equation (86), which is repeated here for convenience:

M = A ex£"Es/RTs^ (86)

** There appears to be some difference of opinion on the exact value of (n-1); it seems to vary with the size of test strand used.

79Rice and Ginell generate a set of dummy variable

parameters in order to facilitate the solution. These can be represented in general form as:

z 2 U A X j _ M % j R T j {106)Kj KjEjWjP"

Now the pressure may be solved for in terms of this new parameter to give:

. 2 1 p = ( — ) j t~ -RT /fcr v D 1 J-) (CpjRTj/kjYj) J (1°7)

The pressure could be evaluated in terms of and this could be used to evaluate and z^. The values of the various |3j•s thus could be obtained in terms of z^ as by definition:

. = exp (-z 2 ) (108)J J

The surface temperature and, hence, the burning-rate can now be found by assigning values to z^ and iterating between (107) and (86).

(Rice and Ginell prefer to write equation (107) in logarithms; thus it would take the form:

( % ) Ln(P) = LnCb^-Ln ( + . Lrt [(#/

80which when evaluated for the fizz zone (j = 1) acquires the form of equation (85b), etc. The versatility of the loga­rithmic form in constructing Figure (22) was previously discussed.)

The solution to the problem of burning, when the influence of the flame can not be ignored, is thus an exten­sion of that used to explain fizz-burning. The work required to obtain a mass flux rate versus pressure curve is, however, compounded several fold. The solution is obtainable, though, if the necessary parameters can be determined experimentally. As a case for example, the pro­pellant HES 4016** was analyzed by Rice and Ginell in light of their theory. Some of the evaluated parameters for this propellant are listed in Table (3)•

TABLE 3CHEMICAL-PHYSICAL PARAMETERS FOR HES 4016 AT T0 = 300°kParameter Units Value

V °k 700V °k 1400t3 °k 3370(Ref .31,;"2 • • • 3 .5(Ref .31)A -2 -1 grm. cm *sec 1.4(10)4E /R degrees 8000s

** This propellant is composed of 54% nitrocellulose (with a 13.25% nitration level), 43% nitroglycerin, and 3% centralite (a wax added to improve workability).

81

Go Experimental Verification of the Rice and Ginell Theory Figure (24) shows the final comparison of Rice and

Ginell9s.theory for HES 4016 with experimental data. Note the excellent agreement throughout the range of 1 atm. to 200 atm. which encompasses the region of propulsion inter­est (which is usually limited to less than 200 atm. by structural limitations). Also note that if the effect of the flame had been ignored, no noticeable deviation from theory would be noted under 100 atm. Thus, for most pur­poses the Rice and Ginell fizz-burning model would describe the process with sufficient accuracy.

H. Summary of the Rice and Ginell Combustion Model for Double-Base Propellants

The Rice and Ginell combustion model applies to double-base propellants which are homogeneous mixtures of nitrocellulose and nitroglycerin. These components contain their fuel and oxidizers chemically bonded within the same molecules. Combustion progresses through a multi-step process which may be separated into three major zones, all of which evolve heat. The first step occurs just within and at the surface and produces unsaturated fragments which are ejected into a gas phase normal to the surface; this mechanism is a first order decomposition and accounts for the limiting minimum burn-rate at low pressures. In the

M AS

S, &

UR

NIN

3-R

at

e,

^nn

/cr

n^

-se

c.

82

2.0

S5T,'V/iL-Pfe aion— L _ p £ _ --------- o tP it i Cone

Oc. ne flame)

• o s

CH.

.05

5 10 to 50 too 160 500 1600

Pressure dtmospbco£?s

Figure 24. Comparison of the Rice and GinellBurning-Rate Theory with Experimen­tal Data for the Double-Base Propel­lant HES 4016. Numbers Near Curve Are the Values of |9 at the PointsIndicated. (After Rice and Ginell)

. 83second stage, the ejected fragments react in the gas phase and transmit heat back to the surface with increasing amounts as the pressure rises. The combination of the first two mechanisms is referred to as the foam and fizz zones and constitutes, in combination, what is called fizz burning. This is the rate-determining process up to 100 atm. The products which leave the fizz zone are still capable of further reaction, and if the pressures are high enough that after a quiet preparation period there is pro­duced a critical concentration of active particles, a third and final stage, termed the flame zone, is produced.As pressure increases, heat is conducted back through the preparation and fizz zones to the surface to further accel­erate the process.

This problem was analyzed by Rice and Ginell with the aid of experimental work of many others through the laws of chemical kinetics, heat transfer in a moving medium, and diffusion. The theory they developed does appear to be a near approximation of the phenomenon and does give good correlation with experiment. Their theory stands today as the single greatest contribution and insight into the combustion mechanism of double-base solid propellants. Some of the experimental evidence cited in the preceding sections was not available at the time it was formulated, but now serves to further substantiate the insight of its authors.

84Upon the basis of the Rice and Ginell theory, Parr

and Crawford (1950) (21) added the refinements of allowing some overlapping of reaction zones, a variable density foam, and distributed release of energy«, (Rice and Ginell also considered this latter point in their appendix but in a less accurate manner and did not use it in their basic treatment of the problem.) The greatest difference between the two efforts is that Parr and Crawford succeeded in integrating the deflagration equations of Boys and Corner (1949) (29). They did not consider the effects of diffu­sion nor the heat feed-back from the flame. Consequently, the results obtained follow experimental results more closely at very low pressures but break down at higher pressures. The results do not deviate from those of Rice and Ginell throughout the range of engineering value - 10 to 200 atm. The work of Parr and Crawford probably follows more closely the physical phenomenon which occurs. The difference from the Rice and Ginell theory lies not in a difference of opinion over the model of the combus­tion mechanism but in the details of its solution.

R. D. Geckler’s (33) review of the mechanism of the combustion of solid propellants contends that there is adequate information to judge the merits of the two theories. To paraphrase him: Rice and Ginell9s theoryrepresents a tour de force of heuristic argument and

85while it may not be adapted to furnish a basic fundamental understanding of the mechanism of burning, it can be con­sidered as an adequate basis for further modifications and extensions (33)»

I. Acceleration Effects on Double-Base PropellantsThat acceleration affects the burn-?rate of double­

base propellants follows as the logical conclusion to explain experimental firings of certain prototype hardware as discussed earlier. How the effect is produced is dif­ficult to conceive in its entirety, and first it is prob­ably easier to state what mechanisms of the combustion model would not be affected.

The application of an external acceleration may be considered as a body force only in its effect upon the gaseous flow rate of the products of combustion away from the propellant. The external acceleration can not be considered as a body force in regard to its effect upon diffusion coefficients, thermal conductivities, specific heats, activation energies, and reaction rates in that it affects all molecules equally through the pressure gradi­ents it creates and does not produce relative accelerations between molecules*. This may be seen by examining these

* The pressure gradient would produce weak changes in diffusion velocity but the magnitude of the gradient is so small that concentration and thermal gradients! remain the primary potentials.

86properties in the light of what is known from the kinetic theory of gases. The expressions derived for these proper­ties do contain provisions for body force effects (34) (35) only if there is a net difference in its effect upon the various species or ions which constitute the gas. Thus, since the acceleration can not produce a shift in the Maxwellian velocity distribution of any of the constitu­ents, and since the potential energies between atoms within the molecules of the constituents are unaltered, then the gas properties remain constant and the same energy as required statically is still needed to raise stable molecules to activated complexes. Therefore, the same reaction mechanisms which occurred statically will continue to occur under the influence of external acceler­ation.

The problem thus appears to reduce itself to one of determining how much the heat feed-back to the rate- governing foam reaction is altered, i.e., to one of deter­mining how the 9 s are altered. This is based on the assumption that since the foam reaction was by itself inde­pendent of pressure (14) it must possess sufficient rigidity to remain unaffected by acceleration, and can be influenced only by the amount of heat it receives above that which it generates. Provisionally ignoring the heat feed-back from the flame reaction - which has little influence below

87200 atm. - the problem reduces then to considering how the heat feed-back from the fizz reaction plane is altered.

Recall that there existed a minimum foam reaction rate at very low pressures where very little heat addi­tion was received from the fizz reaction; see Figure (20). Thus it may be deduced that it would be impossible to extinguish the propellant by application of an external positive acceleration, but that the bum-rate could reach a minimum value. There is no upper limiting value for the bum-rate indicated by Figure (20), but of course if all the heat generated by all three reactions were concen- -trated, the surface would reach an upper limit and thus a limit as to how fast it would decompose. Thus negative acceleration* should increase the rate of burning at all pressure levels.

The variable heat addition to the surface coming from the fizz reaction is transported primarily by conduction; the energy received by radiation - estimated at 10% by Huggett (13) - is not altered as may be noted from the equations of radiant heat transfer between two infinite parallel radiant surfaces (36)1

* Positive acceleration is defined here the same as it was when analyzing the chamber pressure interaction; i.e., the acceleration of the vehicle (or the unburned propellant) is positive when it is in the direction oppo­site to the flow.

88

e, e*. c (t*- t:*) (109)

where 02. and 0 2 are the emissivities of the two planes,<5* is the Stefan-Boltzman constant, and and Tg are the temperatures of the two planes. Equation (62) is independent of the distance between the two planes if the separating medium is non-absorbent. Considering the short distances involved between the reaction planes and the transparency noted in photographic work (20), the assump­tion of non-absorption should be valid.

within the gas stream between the solid-gas interface and the fizz reaction plane is dependent upon a balance between thermal conductivity and thermal transport (14). Acceleration, recall, can not alter the activation energy required by the fizz reaction. Thus the reaction may be still considered as limited to a plane, but now under the influence of acceleration the height at which this reac­tion plane occurs will be changed as the reactants will be carried through a different distance during the time required for the chemical kinetics to occur; this, in turn, will change the thermal gradient and thus the amount of heat feed-back.

Therefore, the thermal equilibrium of any point

89Consider the one-dimensional flow of a compres­

sible fluid from a cool zone, the surface, into one which is hotter, the fizz plane. Assuming the flow to be inviscid, steady, and confined to a constant area stream tube, the energy balance is as follows (48):

K / J V T \

the continuity equation:

3£_ > v • (p lT = o at v

These become:

(110a)

(111b)

and the momentum equation:

P = P - V P + —■ V (V • uJ) * > x C o (H2a)

e c^ ( u - E ) = ( v - E ) + ^ ( i n S ) (110b)

and

U = ■M.e

u 3u = F -dx ax

(lllb)

(112b)

Solving (112b) for substituting it into (110b) and then converting U to _ and simplifying, one may obtain:

(113)

90Setting the body force term F/p equal to "a", the external acceleration and regrouping (113) obtain (114):

M ( - a) = - Cp M (||) + K ( A ) (114)

This may be rewritten, when the external acceleration is zero, to give:

(^-f) - - c (^) + V ( A ) (115a)dtZ p dx M

Equation (67a) may be written as:2

M(^-4) = - ( - CD MT + K |1) (115b)dt^ 3x p 3x

where the forces on the element are equated and one canrecognize the terms on the right-hand side to be the change in heat transported into the element, and the change in heat conducted into the element.

At the risk of being somewhat repititious, equa­tion (115a) may be visualized more intuitively by examin­ing an element of fluid as shown in Figure (25), and then taking its heat balance.

First, considering the heat added to the element by mass transport:

C Heat added ? , C T ( U ^ ) C (U P ) (T + dx)I by transports t-I .i " .P 22 axin out

91

flow- ap = constant

Figure 25. Heat Flow Notation into a Stationary Element in a Moving Gas Stream

and, since = ^2^2 = ^

- - = > e ^ Hie.)

Now considering the heat added to the stationary element in the flowing fluid stream by conduction:

? Heat added by ? _ F y ST I conduction i ax in surface 1

L dx 3 d J out of surface 2therefore, the net heat added to the element by conduction is:

dx dx or dx (116b)dx ax

92Thus, assuming no heat loss, the net change in heat

content of the element, which, by its definition, has a fixed volume, is:

dQ = CvdT = (UdU) M

Combining equation (116a) through (116c):2, dx = (UdU) M

(116c)

(117a)

or.2^ )dt

- c (22) + % (“4)K /ATP dx M dx' (117b)

which is identical with equation (115a) previously derivedby manipulation of the equations of flow.

Several interesting interpretations may be obtainedfrom equation (117b). If (— ) was constant and positive,3 xthen the flow would continuously decelerate. Or, if the2region of flow had (2l_I) as a positive quantity as illu-

ax2 2strated in Figure (26a), and if 6(JLI) > CD ( — ), then theM dx2 3%

flow would continuously accelerate. The reverse is trueas shown in Figure (26b).

Therefore, if the first and second partials of tem­perature with respect to the distance from the surface are known, then the acceleration of an element of gas due to the pressure gradients acting upon it may be computed.

>o

Figure 26a. The acceleration of an element of gas flowing trough a temperature profile where (iHrh < (-#%), and thusfer> o , will be positive if *>1 I B x i / k D*. J

F l o w

Figure 26b. The acceleration of an element of gas flowing through a temperature profile where (ff )£ > (ir and thus( ^ ) < 0 , will be negative if k /vr\ . c /jgr\

>4 I V'P k d x )

Frcp&fd'tfonZoneU)(V

5mQuI01y tjs Efvd ot fizz aone’s purely

condvxt. ve r«<».*oo • ;e no cViemt'taii next released.

D i s t a n i c e f r o m S u r f a c e .

Figure 2?. Schematic representation illustrating fitting a temperature profile of a second degree polynomial to describe that obtained experimentally between x = 0 and x *= x^.

94These data were not found available directly for double­base propellants; however, it may be obtained for the pur­pose of argument here by fitting a polynomial curve to describe those shown in Figures (17) and (18). (This type of information was found for nitrocellulose (23), but it shows an inflection point not found in temperature profiles for double-base propellants (13) (22)). Assuming as simple a polynomial as possible for the purposes of argument, a second degree curve which fits the boundary conditions of T = Ts at x = 0 and T = T^ at x = x- is :

where the notation is illustrated in Figure (2?). The first and second derivatives with respect to x are, respec­tively :

T(x) = (- -i-) x2 + ( i) x + TT 2T

s (118)

(119)

and(120)

Rearranging equation (117b) to give the mass flux, and substituting in equations (119) and (120) gives;

Evaluation of equation (121a) at the surface gives:

Msurface (uM - a) + n (fli) 3x P

Since M „ is a small quantity, then surface '

(U~ - a) + C (fh) dx P x- > 2T-.- K (— - )

v- _ x-

1 J

(121b)

and since ^surjF*ace a positive real quantity, then

(ulE - a) 3x > 2T 2T,- K(_i) - c (_1)Xl2 P

Therefore, by examination of this last statement, it can be seen that statically (i.e., a = 0) U ~ , which is now

d2x 3xidentically __±, must be a large negative quantity. Thus

dt^returning to equation (121b), it may be inferred that as the natural "static" acceleration, which is a negative quantity, is changed by an increase in external positive acceleration - and thus the two add in the negative sense

* Values of C for double-base propellants run onthe order of 0.3 cal./gm.-°C., and values o f K run on the order of 0.0005 cal./cm.-°C. (33).

96the mass flux should decrease. The converse may also be argued; i.e., negative external acceleration changes the magnitude of the natural static acceleration at the surface in such a manner as to cause the mass flux to decrease.

initially in equation (112b). The superposition of an external acceleration upon the flow field causes a change in the pressure gradient through the field. This may assist or resist the pressure gradient produced by the temperature gradient. Exactly how this occurs depends upon the shape of the temperature profile and the sense of the accelera­tion; the summation of the two pressure gradients produces the change in mass flux through the field.

Using equations (121b) and (86), it may be shownthat:

from which it may be inferred that there must be an accom­panying change in surface temperature which varies with acceleration, and which, in turn, varies the mass flux being emitted by the surface.

Thus it appears that the rate at which the combus­tion process proceeds for double-base propellants is

What is happening here could have been observed

T = s (122)s

dependent upon the influence of external acceleration. Since the slopes of the various derivatives of temperature are different for each propellant, and are also a function of chamber pressure and initial grain temperature, it is difficult to develop an all-encompassing method of pre­dicting how much deviation in burning-rate may be expec­ted. (One must also bear in mind that the partials of temperature with respect to distance must also change with acceleration, and that this effect has been minimized in order to make the previous inferences as to burning behavior.) Thus, unfortunately, the prediction of how a propulsion system utilizing double-base propellants behaves under acceleration must await experimental data.

J. The Rice-’Summerfield Combustion Model for Composite Propellants

The historical foundation of the Rice and Ginell theory of combustion for double-base propellants has influenced efforts made on the construction of a combus­tion model for composites.

Composite propellants are made by suspending a finely divided solid oxidizing agent, potassium perchlor­ate or ammonium nitrate, in a matrix of a solid fuel binder such as a plastic, a resin, or an elastomer. Com­mon fuels are synthetic rubber, aldehyde-urea and phen­olic resins, and vinyl polymers. To this mixture is

98normally added powdered metals such as aluminum, magnesium, or zirconium. As with double-base propellants, several additives and modifiers may be present in trace amounts to impart improved or desirable properties, and as with double-base propellants these will be assumed to not change the basic steps of the combustion model (15).

Microscopic examination of a composite propellant would reveal a definite heterogeneous structure. This heterogeneity requires that the fuel and oxidizer vapors mix in the gaseous phase before combustion can occur; therefore, the burning-rate of composites depends strongly upon the diffusion rates of the volatile products produced by the gasification process. The pyrolysis (decomposition by heating) of most common fuels and oxidizers is an endo- thermic process; consequently, it is assumed that the rate of regression is completely governed by the heat exchange between the surface and the burning volatile products (37).

Experimental work on the pyrolysis of fuel and oxidizer components, taken separately, has advanced the concept of the "two-temperature postulate." (38) (39)(42). This is based upon the evidence that the regression rate of the fuel and oxidizer for all purposes is the same, and since both can be made to fit simple reaction rate equations, it follows that:

Since Q%idizer ^fuel*99

(123)

then-Efuel

Afuel exp (R^ , Tfuel surf, of fuel —E

= A O X . eXP (n T ) (124)ox. 1surface of oxidizer

Now since the two activation energies and frequency coef­ficients are generally unequal, then the surface tempera­ture of the fuel and oxidizer can not be equal; i.e.,Tgf ^ Tgo. This may be visualized readily by examination of Figure (28) where, for a surface regression rate of .06 cm^sec, the binder polystyrene would have a surface temperature of 742°k, and, depending on whether ammonium nitrate or ammonium perchlorate was used, the oxidizer surface temperature would have been 470°k or 862°k, respec­tively (50).

This difference between oxidizer and fuel surface temperatures has suggested the surface combustion models illustrated in Figures (29a) and (29b) (50) (38).

Thermocouple data were obtained by Sabadell, Wenograd, and Summerfield (1962) (40) for polysulfide- ammonium perchlorate propellant with bimodal oxidizer distribution; (i.e., the oxidizer, which was by weight 80 per cent of the propellant, contained 70% coarse particles

100

A. Amrrtoniij'VN kiitrate3 VouS'bTy KEHEC . A rn nr»o n i'urv%Pcronlof *fc

l.L \.4 l<c t a Z.0 ^eeciprocil Sur4ace T e m p e f a t u f e

Figure 28. Examples of Linear Pyrolysis Rates for Several Oxidizers and Binders (after Chaiken and Andersen)

Vn't •; r5 6 0 °x - Am<txof\i't>f<\ 'I iN".t<&te^SuH»ce Temperature

> < 1 | ■' — 0>-.'di£er /(. I S t r e a m /f)

FueL

FyeL * streams

Am moOiumKJit-ta*eP a .r t* c le s

Figure 29a. Schematic Representation of the Burning Surface of an Ammonium Nitrate Propellant.

<3a* Ph&se Reaction* 04 O*; A iter £ fcirvier

Figure 29b. Schematic Representation of the Burning Surface of an Ammonium Perchlorate Propellant.

102with a mean size of 5 4 > and 30$ fine particles with a mean size of 9/t). The data show fluctuations as may be seen in Figure (19)= The temperature profile fluctua­tions below the surface are believed due to the hetero­geneous nature of the propellant* The sharp discontinuity at the surface and the increasing slope are the result of the endothermic nature of the surface reaction. The sur­face has a mean temperature of 600*0.

Photographic examination of the surface reveals a cratered effect where the smaller ammonium perchlorate particles have left their impression having volatilized ahead of this particular fuel* Some of the largest ammo­nium perchlorate particles, due to their size and heat sink capacity, were observed riding on the dry surface with a haze about them. Finally, it was noted that the reaction zone is very thin,, being on the order of 10" cm at 200 psi (40).

Based on the above experimental data and the work of Rice and Ginell, Summerfield has proposed a ,egranular diffusion flame model18 (37) based on the premise that the combustion speed is a function of the time required for diffusional transport mixing of fuel and oxidizer pockets and of chemical reaction.

Schematically, the model suggested may be seen in Figure (30) (21). Summerfield notes that for small

103

Combuitfoft Process

BinderFigure 30

Schematic diagram suggesting the essential physical assump­tion in Summerfield1s granular diffusion flame model for the burning of composite propellants. The illustration has been drawn as though fuel pockets of variable dimensions were burning in an atmosphere of oxidizing gas, reaction products, and some fuel vapor (after PennerJ.

\ 2S fo vcr^ f«/ie oxid.'derTS'Zo co»Ose exi'df*e<*

r = -Y ISOOPV1 +5 d o 4-mt eniAiter

coarse cx.j-icf

tocoPRESSURE 3 psia-Figure 31

Rice-Summerfield burning-rate theory adjusted to fit sev­eral composite propellants (after Penner).

104oxidizer particles the bum-rate process is time limited by the chemical reaction time, whereas, for large oxidizer particles, the bum-rate is time limited by the diffusional process. Summerfield also notes that the burning rates of pure ammonium perchlorate as a monopropellant are compar­able to those of the burning rates of ammonium perchlorate composite propellants and thus suggests that the gas phase reaction rates of the oxidizer determine the controlling reaction rate for the decomposition of the propellant.

The pressure dependency of the burning-rate based upon the Rice-Summerfield model, as with the Rice and Ginell theory, starts with a statement of the energy equa­tion ; thus the energy equation at the surface is (38):

(K|l) = m[ Ahg - G (Tf a - Ts )l (125)ax o+ L p 1 a Jwhere Ahg is the total energy released per unit mass of the propellant, Tf a is the adiabatic flame temperature,Ts is the surface temperature and the remaining terms have the same meaning as before.

For (K aT/ax)0> the following crude approximation is utilized:

(K5l)0 = K (TV Ts ) (126)d x v € h

where the flame zone has been replaced by a wall of thick­ness h with a linear temperature gradient. The factor €

is a correction factor of order unity for any non-lineari­ties in the temperature profile through the flame zone which may exist. The thickness of the reaction zone, h, is composed of two parts, that required for mixing by dif­fusion and that required for the chemical reactions to occur. Note that with composites the reaction is not lim­ited to a plane but occurs continuously throughout the whole reaction zone, thus the replacement by a homogene­ous wall with a temperature differential impressed upon it is not too extreme an approximation. Also note that the diffusion thickness and chemical thickness does not refer to two separate regions but to the fact that the flame zone is that distance traversed during the time it took for all the "pockets" to be consumed by mixing and burning.

Thus the thickness of the reaction zone is

h = Cje ^diffusion + C2V ^chemical (127)

where c^* and C29 are constants of the order of unity and^diffusion and hchemical are the flame thickness requiredfor the diffusional mixing and the gas-phase reactions.

From the conservation of mass, it follows that:

hchem. = (M/f) tchem- (128a)

and hdiff. = (%/(>) tdiff. (128b)

106Recall that if the reaction is assumed to be second order that the time required for the reaction kinetics is:

Similarly, if the solid particles embedded in the propel­lant gasify very rapidly after ejection from the surface or at the surface to form a granular pocket of fuel or oxidizer prior to combustion by a gaseous diffusion-flame mechanism, then the dimension of the pocket can be stated

where ju, denotes the mass contained by the pocket, which is assumed to be constant, and d is the diameter charac­terizing the gas pocket. Hence, the time required for diffusion to occur and consume the pocket could be related to pressure as follows:

t 1 (129)chem kpn-l

thus for n = 2: chem

as:

(130)

diff.'*1/3

(13D

where D is the coefficient of binary diffusion

Defining: T* = Ahg/Cp 107

equations (125, 126, 12?) may be solved in terms of the mass flow rate M by substitution of equations (128) and (132) to give:

M = 1 g e

e{T*/(Tf>a-Ts) - . U pJ . (°1 tdiff + c2* tchem")_

which may be regrouped to give:JKC,

(133a)

(M)2 =6 ^ / ( T f>a-Tg) -ig cl' tdiff + °2' tchem.

(133b)Multiplying equation (133b) by (25) gives:CD(M)2 -

fD€ \T*/(Tf>a - Ts)e 2p

Cl' tdiff. + c2' tchem,and recalling that:

then

(m )

eD' - e0Do (p/po)'

& = e02 d0(p/p0)

Substitution of (135) into equation (134) gives:

(135)

108

i . t v r r „ - T , v i l c:tjc;t . „ „ (136.)2Letting the group of terms in the brackets equal o< , a

parameter to be determined experimentally for the particu­lar propellant being investigated, equation (136a) becomes:

M cl' tdiff.+ c2* tchem. (136b)

2where it is to be noted that is essentially independent of pressure.

Now recalling equations (125) and (131), and solv­ing (131b) for the burning-rate instead of mass flow rate, it becomes:

^ ~ r T i r - ? * 3

where the arbitrary constants a1 and b1 are selected to adjust equation (137) to fit the experimental data obtained for each specific propellant as illustrated in Figure (31).

K . The Effect of Acceleration upon the Buming-Rate of Composites

The combustion model for composites has some impor­tant differences from that proposed for double-base propel-

109lants that would make the effect of acceleration somewhat different. The major differences are that the propellant decomposition is endothermic and, therefore, is always dependent on.the reception of energy from the gas-phase reactions; that the rate process is diffusion dependent and thus a function of particle size; and, finally, that the combustion process can no longer be confined to a plane but occurs continuously throughout the combustion zone on the boundaries of the oxidizer and fuel pockets by a diffusion-flame process. Thus, as an approximation, the combustion zone was analyzed above as a homogeneous wall with its thickness a function of the time lapses required for diffusional mixing and chemical kinetics to occur. The heat flux to the surface was then analyzed as if this wall had been subjected to.a temperature dif­ference such that the temperature gradient through the wall was nearly linear. Any errors produced by this approximation were absorbed by the selection of constants particular to each propellant as determined by experimen­tation.

Based upon the acceptance of this model, certain inferences as to the effect of acceleration may be.made. First, as with double-base propellants, external accelera­tion cannot change the molecular potentials between reactants and thus cannot directly change the rates at

110which chemical reactions occur. External acceleration also cannot change the gas properties such as thermal conduc­tivity or the diffusion rates of the various reactants. (This may be inferred by examination of the laws of the kinetic theory of gases; the expressions derived in works on this topic (41) do include the effects of body forces but not those of external accelerations, as they can only produce pressure gradients.)

Hence, it may be deduced that the effects of exter­nal acceleration upon the Rice-Summerfield model do not change the reactions which occur, nor their rate. Using the concept of the "thermal-wall", it may be seen that the same quantity of gas will liberate the same amount of heat as statically, but that the acceleration will cause a stretch-out or compression of the wall depending upon the sense of that acceleration. Using this concept, to a first approximation the thickness of the thermal wall will now be:

h = L ( -p ) + 1/"e U38)

Substitution of this into equation (126) would give an equation analogous to (133a); i.e.:

Ill

* L ' 6 i T 7 C T « , r T > i ^ ( c J (139>

^ /'H:i['t +(5 )1 1 7 ^

This can now be written as:

cX-'P« t t „ ♦ 1 1 4 1

Applying the continuity equation to the surface:

eProp r = egas U (U1)

where p prop and Pgas are the densities of the solid pro­pellant and surface gas, respectively, while r and Usurf are the burning-rate and surface gas flow velocity. Now also from the continuity equation,

I = (?) (U2)M

therefore:/ Ka--r = (?) (H3)xprop M

and, letting rl = fpropV* gas

(144)

112then, finally.

(145)

Substitution of equation (145) into (140) gives:

and if the same operations are employed that were used to obtain equation (137), equation (146) becomes:

which would reduce to equation (137) if the acceleration were zero. Finally, solving equation (147) for the burning-rate explicitly and realizing that only positive burning rates have any physical meaning:

(148)

the desired expression.

113Equation (148), with the constants a1 and b* deter­

mined experimentally by static firings, should allow an estimation of the order of magnitude of change that may be expected under acceleration. Trial calculations show that several propellants listed in Figure (31) may be quite sensitive to the effects of acceleration, but, of course, judgment must await experimental verification.

L. The Effect of Acceleration upon Metal Additives

often loaded with powdered metal additives such as alumi­num, zirconium, or even heavier metals such as tungsten.The purpose of this is usually three-fold. Aluminum, magnesium, etc. when added as part of the fuel result in a sharp reduction of high frequency combustion instabil­ity, raise the flame temperature, and when added in opti­mum amounts raise the specific impulse. The heavier metals not only produce the above effects but can be optimized to deliver the best "density-specific impulse." Heavy metals may decrease the specific impulse, Is, slightly by increasing the overall molecular weight, Mc, of the exhaust products faster than they raise the flame temperature, as may be appreciated by examination of equation (97) (43):

where subscripts c and e refer to inside and outside chamber.

Both double-base and composite propellants are

(149)

114but, they result in a greater energy storage on a volu­metric basis. The "density-specific impulse" of a propel­lant is of importance in small volume-limited tactical weapons (44). The importance of this factor may be seen in equation (98) for the burned-out velocity of a rocket in a vacuum:

Vge0e = c Ln (1 + Pp/e) (150)

where Vg q is the burned-out velocity, c is the effectiveexhaust velocity; i.e., c = gl^^, is the density of the propellant, and £ is the mass of the missile at burnout divided by the propellant volume.

Several investigators have studied the problem of metallic powder combustion in propellants. Markstein (1963 ) (45) discusses photographic observations of alumi­num-loaded composite propellants. The observations show that the point of ignition for a particle is dependentupon its size. 5 particles of aluminum were observed toignite at the surface while 40/4. particles ignited shortly after leaving the surface, and particles larger than 80jx. were observed to still be unignited at 1 cm above the surface. Propellants loaded with magnesium displayed a similar dependency upon particle size, but both ignition distance and burn times were shorter. It was also observed that both aluminum and magnesium particles burn with flashes and streamers indicative of vapor-phase

115burning, and that the particles were often observed to end their combustion with a burst of sparks like those obtained when grinding high carbon steel. The burning particles were observed to leave in their wakes oxide particles whose size was not a function of the chamber pressure (2—►3 a*' )• The burning particles appear to increase in size, and based upon the observations of Fassell, et al. (I960) (46), it has been proposed that what is occurring is that the molten metal droplet is expanding its oxide layer. The particle expands, since the boiling point of the metal is less than that of the oxide. While the burning particle increases in size, the droplet decreases in size as the oxidizer diffuses inward and consumes the boiled off vapors; finally, when the oxide bubble reaches a critical size it bursts.

Based on the photographic work discussed by Markstein (45), it appears that the propellant receives a certain amount of heat feed-back from the finer parti­cles. However, the larger particles, due to their igni­tion lag, probably return very little energy back to the surface. The energy addition resulting from the combus­tion of these larger particles is added to the flow as it passes down the chamber.

Thus the influence of acceleration upon the larger particles will be different than upon the smaller ones.

116The influence of acceleration upon the smaller particles will be analogous to the effects produced upon the rising fuel pockets coming from the binder; that is, the smaller particles will be consumed within the "thermal-wall." (Acceleration, recall, produces a change in the "thermal- wall" thickness, which influences the thermal gradient through the wall and thus causes a change in burning-rate). The larger particles are consumed beyond the "therma1-wal1" and thus do not influence the burning-rate. They do, how­ever, add energy to the system and thus raise the overall performance of the propellant. Consequently, if their combustion is incomplete, a certain amount of enthalpy will not be available to the flow when it reaches the nozzle and will thus result in a loss of performance by the system.

The larger particles, due to their low character­istic Reynold’s numbers, behave as dictated by Stoke*s laws. The fact that they are burning results in some error, but the investigations of Crowe, Nicholls, and Morrison (53) upon minute burning particles indicate that the difference may be ignored.

The work of Watermeir, Aungst, and Pfaff (34) indicates that the particles have velocities only slightly less than the flow velocity at the surface and that the overall fluctuations in diameter during combustion are

are small enough to allow the use of a ,,mean-diameter" without loss of meaning for the purposes of argument here.

Utilizing available data for air, since none was found available for the combustion products of propel­lants, it can be shown that the drag coefficient for a particle obeying Stoke’s law is (48):

where Re is the Reynold’s number, and Cq is the drag coefficient. Expanding equation (151) to a more elemen­tary form, it becomes:

CD = ^52

where p is the density of the immersion fluid, U is the particle velocity relative to the fluid, >i is the abso­lute viscosity, and d is the particle diameter. The drag of the particle in the fluid is thus:

(151)

D = CD3(i e u 2)

(153)

where S is the particle’s reference area, and r is its

118radius. Thus the drag is directly proportional to the relative velocity of the particle to the flow.

Recalling that the external acceleration is not a body force, such as gravity, then the only force acting upon a particle is its drag. Thus a summation of forceson the particle yields:

m$ + KUrel. = 0 (154)

where M is the mass of the particle, and Ure is the rel­ative velocity. Solving this between the limits of t^ = 0 to t2 = t, and Ui = (Uflow - Uparticle)surface to U2 =

^relative* equation (154) becomes:-ht

"'relative *" Uflow ' ^particle) 6 m (15 5)surface

where the quantity in brackets is the relative velocity atthe surface. Evaluating the mass, m, in terms of volumeand density and solving equation (155) for t, it becomes:

t - | (-^p) Ln(Uflow " Uparticle>gurfac

^relative(156)

The time required for the particle to attain flow velocity is, of course, infinite. The time required for the com­bustion to occur is, however, finite; thus, for a properly designed motor, the time required for the flow to pass

119from the propellant surface to the nozzle should be greater than the combustion time. This required time for complete combustion is referred to as the minimum chamber residence time and is used to stipulate the minimum length of the plenum chamber.

Therefore, equation (156) may be set equal to the residence time as follows:

(uflow - uparticle)surfaee ‘relative at nozzle entrance

(157)

Recalling those arguments used to analyze the influence of acceleration upon the flow through the plenum chamber from Chapter 3> it was demonstrated that there would be no measurable change in flow properties between the surface and the nozzle entrance for acceleration levels of inter­est. Consequently, since the flow velocity did not change measurably, then the change in relative velocity between the metal additive particles and the flow would also be too small to measure. Therefore, the residence time would remain essentially constant and no loss in performance due to incomplete combustion of the larger powdered additives should be expected. (Most useful metal additives are typi­fied by high combustion rates and are seldom larger than

^residence 9

120150 microns, thus their combustion is usually complete within a few centimeters of the surface (49).)

Iwanciow, Lawrence, and Mertens (4) examined the exhaust products of aluminized propellants fired both statically and in centrifuges and found no change in com­position as suggested by the above reasoning.

Thus it may be concluded that combustion ineffi­ciency of metallic additives due to acceleration interac­tion may be discarded as a cause for performance loss if the chamber was designed for adequate stay-time (resi­dence time) when statically fired. This does not, how­ever, preclude the existence of such problems as centrif­ugal depositing of the oxides by acceleration in high spin-rate motors; this will result in a reduction of mass flow rate leaving the chamber and, therefore, a loss of performance.

CHAPTER 5

CONCLUSIONS

The concern of advanced propulsion systems groups over the possible performance deviation of high accelera­tion systems from that obtained statically appears, on the basis of theoretical investigation and review of past experimental work, not unjustified; but, the problem may be circumvented.

Operation at high acceleration levels can produce problems arising from three modes of interaction: mechan­ical deformation, pressure redistribution, and combustion rate deviation. The problem of mechanical deformation can be prevented by proper design by maintaining modest propellant stress levels through sufficient grain support. The problem area of chamber pressure redistribution, upon close examination, reveals that it is not an interaction of concern unless the motors to be utilized have abnor­mally long combustion chambers,

The interaction of acceleration upon the combus­tion mechanism can produce significant deviations in pro-ipellant buming-rate if the acceleration is oriented such that it has a component orthogonal to the burning

121

122propellant surface. The exact way by which this occurs depends on the type of propellant utilized.

Externally applied acceleration does not directly affect the combustion mechanisms which occur, but only the rate at which they occur, by influencing the heat feed­back from the gas-phase reactions to the surface. This was argued for both double-base and composite propellants, and in the case of composites, an approximate solution was derived which should allow a prediction as to the magni­tude of this effect.

Concern over the combustion inefficiency of metal­lic powdered additives due to the effects of acceleration does not appear warranted. Current additives, in the particle sizes utilized, have sufficiently short required "stay-times" to not be affected.

The precise information as to how a propellant will perform under acceleration must, unfortunately, be obtained from experimental work. It is the intent of the author to complete such an investigation, which was being undertaken at the U. 3. Naval Ordnance Test Station, to obtain data for an aluminized polyurethane composite, a fluorocarbon, and a double-base propellant.

In conclusion, it may be stated that, by the use of proper design, where burning surfaces are maintained parallel to the action of the acceleration, where

123sufficient mechanical support is provided, and where the chamber length is of normal proportions, the problems which might arise from acceleration can be circumvented and need not inhibit the operation of solid propellant rocket motors at acceleration levels of engineering interest.

REFERENCES

1, Industrial Observer. Aviation Week and Space Tech­nology, Volume 80, No. 6, February 10, 1964, p. 19.

2. Overall, R. E., and Rogstron, E. D., Analysis of Exper­imental Results Obtained from Solid Propellant MotorsTested Under High Acceleration Conditions (U). Preliminary Report,. Thiokol fihemical Corp., Alpha Division, Feb. 1963, (Confidential).

3« Landau, H., and Cegielski, J. M., The BallisticBehavior of Solid Propellant Rocket Grains in a High Acceleration Environment.XTaA Paper No . 64-228. presented to the 1st AIAa Annual Meeting in Washington, D. C., June 29-July 2, 1964«

4. Iwanciow, B. L., Lawrence, W. J., and Mertens, J., The Effect of Acceleration on Solid Composite Propellant Combustion. AIAA Paper No. 64-227, presented to the 1st AIAA Annual Meeting in Washington, D. C., June 29- July 2, 1964.

5. An Investigation of the Effects of Acceleration on the Burning Characteristics of Solid Propellants fu).Report No. KPR-5, Final Report, Redel, Inc., Feb. 1961, (Confidential).

6. Bartley, C. E., and Mills, M. M., Solid Propellant Rockets. Jet Propulsion Engines, Vol. XII of the Princeton University Series on High Speed Aerodynamics and Jet Propulsion, Princeton University Press, Princeton, N. J., 1959.

7. Zucrow, M. J., Aircraft and Missile Propulsion. Vol. II, Wiley, N. Y., 1958.

8. Sutton, G. P., Rocket Propulsion Elements, Sec. Ed..Wiley, N. Y., 19551

9. Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow. Vol. I, Ronald, N. Y., 1953.

124

12510. Horton, J. G., Jr., Experimental Evaluation of Solid

Propellant Rocket Motors Under Acceleration Loads,AIAA Paper No. 64-133» presented to the Solid Propel­lant Rocket Conference in Palo Alto, Calif., Jan. 29- 3.1, 1964.

11. Lewis, B., and vonElbe, G., Combustion. Flames. and Explosions of Gases. Academic Press, N. I., 1951.

12. Smith, L. S., and Stinson, K. W., Fuels and Combus­tion. McGraw-Hill, N. Y., 1952. _13. Huggett, C., Combustion of Solid Propellants. Com­

bustion Processes, Vol. II of the Princeton Univer­sity Series on High Speed Aerodynamics and Jet Pro­pulsion, Princeton University Press, Princeton, N. J.,1956.

14» Rice, 0. K., and Ginell, R., The Theory of Burning of Double-Base Rocket Powders, Journal, of Physical and Colloidal Chemistry, Vol. 54, pp. 885-917, 1950.

15. Warren, F. A., Rocket Propellants. Reinhold. N. Y., 1958.

16. Wimpress, R. N., Internal Ballistics of Solid-Fuel Rockets. McGraw-Hill, N. Y., 1950.

17. Gibson, R. E., Introductory Remarks to the Symposium on Kinetics of Propellants. Jour, of Phys. and Colloidal Chem., Vol. 54, pp. 847-853» 1950.

18. Crawford, B. L., Jr., Huggett, C., and MeBrady, J. J., The Mechanism of Burning of Double-Base Propellants. ibid., pp. 854-862.

19. Heller, C. A., and Gordon, A. S., Structure of the Gas Phase Combustion Region of a Solid Double-Base Propellant, Jour, of Phys. and Colloidal Chem.,Vol. 59, pp. 773-777, 1955.

20. Heath, G. A., and Hirst, R., Some Characteristics of the High Pressure Combustion of Double-Base^'Propel­lant * Eighth Symposium (International) on Combustion, Williams and Wilkins Co., Baltimore, Md., 1962,pp. 711-720.

21. Parr, R. G., and Crawford, B. L., Jr., A Physical Theory of Burning of Double-Base Rocket Propellants, Jour. oT~Phys. and Colloidal Chem., Vol. 54, pp. 929- 954, 1950.

12622. Sabadell, A. J., Wenograd, J., and Summerfield, M.,

The Measurement of Temperature Profiles Through Solid Propellant Flames Using Fine ThermocouplesT^IAA Paper No. 64-1067 presented to the Solid Propellant Rocket Conference in Palo Alto, Calif., Jan. 29-31, 1964.

23• Klein, R., Menster, M., vonEble, G., and Lewis, B., Determination of the Thermal Structure of a Combus­tion Wave by Fine Thermocouples. Jour, of Phys. and Colloidal Chem., Vol. 54, pp.- 877-884, 1950.

24. Wilfong, R. E., Penner, S. S., and Daniels, F., An Hypothesis for Propellant Burning. ibid., pp. 863-872.25. Fenimore, C., The Final Reactions of the Burning of

Nitrocellulose. Ballistic Research Lab. Report No.464, 1944. ~*

26. Adams, 0. K., and Wiseman, L. A., The Combustion of Double-Base Propellants9 AGARD Selected Combustion Problems, pp. 277“5857™Butterworths Scientific Pub­lications, Ondon, 1954.

27. Appin, A ., Todes, 0., and Khariton, Y., Journal ofPhys. Chem. U.S.S.R. 8, 1^36.

28. Levy, J. B., Journal of the Am. Chem. Soc., Vol. 76, pp. 3254s 3790; 1954.

29. Boys, S, F,, and Corner, J., The Structure of the Reaction Zone in a Flame, Proc. Roy. Soc. London,Il9f, pp.~W:l05, 1949; Corner, J., The Effect of Diffusion of the Main Reactants on Flame Speeds in Gases, Proc. Roy. Soc. London, A198, pp. 388-405, B 4 9 7

30. Petersen, R. E., Lecture Notes, (Unpublished), Advanced Propulsion, AME 352, The University of Arizona, 1963.

31. Crawford, B. L., Jr., Huggett, C., and Parr, R. G., University of Minnesota Report, 1944.

32. Huggett, C., University of Minnesota Report, 1945.33. Geckler, R. D...The Mechanism of Combustion of Solid

Propellants, AGARD Selected Combustion Problems, pp. 283-339# Butterworths Scientific Publications,London, 1954.

12?34* Penner, S. S., Introduction to the Study of Chemical

Reactions in Flow Systems, Rev. Ed., Macmillan, N. Y.,1957. ~~™~

35. Hirschfelder, J. 0., Curtiss, C. F., and Bird, R. B., Molecular Theory of Gases and Liquids. Wiley, N. Y., 1954.

36. Condon, E., and Odishaw, H., Handbook of Physics. McGraw-Hill, N. Y., 1958.

37. Summerfield, M., Sutherland, G. S., Webb, M* J.,Taback, H. J., and Hall, K. P., Burning Mechanism of Ammonium Perchlorate Propellants. Progress in Astro- nautics and Rocketry: Solid Propellant Rocket Research, pp. 227-259, Academic Press, N. Y., i960.

38. Penner, S. S., Chemical Rocket Propulsion and Combus­tion Research. Gordon and Breach, N. Y., 19&2.

39. Barrere, M., Jaumotte, A., De Veubeke, B. F., Vandenkerckhove, J., Rocket Propulsion. Elsevir Pub. Co., Amsterdam, I960.

40. Cole, R. C., Wenograd, J., and Summerfield, M.,Solid Propellant Combustion Mechanism Studies. Fourteenth Progress Report, Aero. Engr. Report No. 466n, Princeton Univ., 1962.

41. Jeans, Sir James, The Dynamical Theory of Gases,4th ed., Dover, N. Y., 1954»'”"~~

42. Schultz, R., Green, L., Jr., and Penner, S. S.,Studies of the Decomposition Mechanism, Erosive Burn­ing. Sonance and Resonance for Solid ComposTte~Pro- pellants. Combustion andPropulsion, Third AGARD Colloquium, pp. 367-420, Pergamon Press, Limited, ■ London, 1958.

43. Wilkins, R. W., Theoretical Evaluation of Chemical Propellants. Prentice-Hall, N. J., 1963.

44« Kaufman, M. H., Unanimity in P-Isp, Please. Depart­mental Memo., Propulsion Department, U. S. Naval Ordnance Test Station, China Lake, Gal., I960.

45* Markstein, G. H., Combustion of Metals. ProjectSQUID, Tech. Report Cal-86-lP, University of Virginia, 1962.

12846. Fassell, W. M., Papp, C. A., Hildenbrand, D. L., and

Sernka, R, P ., The Experimental Nature of the Combus­tion of Metallic Powders, Progress in Aeronautics andRocketry: Solid Propellant Rocket Research, pp. 259-269, Academic Press, N. Y., I960.

47. Lamb, Sir Horace, Hydrodynamics. 6th ed., Dover, N. Y., 1945.

48. Schlichting, H., Boundary Layer Theory. 4th ed., McGraw-Hill, N. Y ., i960.49. Wood, W. A., Metal Combustion in Deflagrating Propel­

lant . pp. 287-291 of RefT 4&.50. Chaiken, R. P., and Andersen, W. H., The Role of

Binder in Composite Propellant Combustion,,, pp. 227- 249 of Ref. 46.

51. Wu, J. C., and Pottsepp, L., Flow of a Compressible Fluid in an Accelerating Nozzle. Douglas ReportSM-38824, Missile and Space Systems Division, Douglas Aircraft Co., Inc., April 6, 1962 (Unclassified).

52. Wu, J. C., and Pottsepp, L., Quasi-Steady Flow and - Thrust in Accelerating Nozzles, and Channels, presented to the A.S.M.E. Aviation and Space Division Meetingin Los Angeles, March 16-18, 1965.

53• Crowe, C. T., Nicholls, J. A., and Morrison, R. B.,Drag Coefficients of Inert and Burning Particles Accelerating in Gas Streams. Ninth Symposium (InternationaTT on Combustion, pp. 395-406, Academic Press, N. Y., 1963.

54. Watermeir, L. A., Aungst, W. P., and Pfaff, S. P.,An Experimental Study of the Aluminum Additive Role in Unstable Combustion of Solid Rocket Propellants. Ninth Symposium (Internetionai} onCombustion, pp. 316-327, Academic Press, N. Y., 1963«