VVI Solid Rocket Motor Design and Testing

8/6/2019 VVI Solid Rocket Motor Design and Testing

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SOLID PROPELLANT ROCKET MOTOR

DESIGN AND TESTING

Richard A. Nakka


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SOLID PROPELLANT ROCKET MOTOR

DESIGN AND TESTING

A Thesis

Presented to

The Department of Mechanical Ensineerins

The Faculty of Ensineerins

The Univer5it~ of Manitoba

In Partial Fulfillment

of the re~uirements for the deSree

Bachelor of Science in Mechanical Ensineerins

b~

Richard Allan Nakka

April 1984


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ABSTRACT

This thesis examines the theoretical performance of a

solid propellant rocket motor which was developed for launchinS

small amateur research rockets. The theoretical results are pre-

sented irithe form of two limiting cases concerning the behaviour

of the two-phase exhaust flow. Actual testing of the motor is

performed utilizinS a speciall~ desiSned test rig in order to

compare the results. As well, optimization of the motor's per-

formance is investigated.

The theoretical performance is found to be in sood asree-

ment with the test results, providinS a basis for future design

of larger engines.

No significant improvement in overall motor performance

as a result of modifications is foreseen.

ii


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ACKNOWLEDGEMENT

I would like to ex~ress sratitude to m~ thesis advisor

Dr.B.R. Sat~a~rakash for the time and patience he has devoted to

me in preparinS for this thesis. I would also like to thank Dr.

s . Balakrishnan for aauaintins me with proSramming the computer

N-C lathe. and Dr.D. M. McKinnon for his devotion to the

problems of propellant combustion. As well, special thanks to

Blair W. Nakka for supplying the microcomputer facilities and for

providing assistance with the softWare and circuit development.


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CHAPTER

ABSTRACT

TABLE OF CONTENTS

........ I I o f .. o f " ..

ACKNOWLEDGEMENTS

INTRODUCTION

1.1

1.2

1.3

2: THEORY

3 :

Propellant

Rocket Motor

11 .

Experimentation

................. 1

....................................................................Solid Propellant Rocket

Propellant

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I ..

Composition

Combustion

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Propellant Grain

NOZZLE THEORY t t of .

3.1

3.2

Nozzle Flow ................ t " .

3.2.1

Rocket Performance Parameters

Thrust ........ + , ..

.... t .

Thrust Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Characteristic Exhaust Velocit~ t + ..

Impulse

Chamber Pressure t .

Corrections for Two-Phase Flow

Corrections for Real Nozzles

. . . . . . . . . . . .. . . . . . . . . . . . . .

PAGE

ii

iii

1

1

1

2

4

4

5

5

6

12

16

16

22

23

24

25

26

28

30

34


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4 : EXPERIMENTAL TECHNIQUE

Motor .................................. I ..1

4.2

4.3

4.4

4.5

4.6

5 :

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cubic Nozzle Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Propellant Preparation ................... t to to

Var~in~ OIF Ratio ...... l' I .

Burnrate Testin~ ................. f to to .

Motor Static Testin~ ................ t ..

Test Ri~ Construction . .. . .. . .. .. . o fof t .... t ....

Calibration ...............................

RESULTS (Theoretical and Actual)

Nozzle Testin~ to f o f ..

........ ttt.tf

5.5.1

5.1.2

5.1.3

5.1.4

Motor Testins of Var~inS OIF Ratio Grains

Burnrate Testins ... .. ... ... ... to of o f ...

Performance Parameters

6.1

DISCUSSION OF RESULTS

Nozzle Testina .................. 1 ....... 1 .... 1

6.3

6.4

CONCLUSIONS

RECOMMENDATIONS

Motor Testins of Var~ing OIF Ratio Grains t ........

37

37

38

40

42

42

43

44

49

51

51

54

57

60

61

61

62

64

65

68

69

NOMENCLATURE 70

73EFERENCES

Burnrate Testing .11 .

Performance Paramete~s I to o f o f ..

..................................... 11 .

..... ...... ...... ...... ...... ..... ...... ...... ...... ...... .... +

........................................................... t .. t ......


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APPENDIX At Calculation of AFT for the 65/35 OIF ratio

APPENDIX B: Calculation of k for flow throush the nozzle

APPENDIX C: Derivation of eouation for exhaust velocit~, modifiedfor two-phase flow.

APPENDIX D: Derivation of eQuation for thrust, modified for two-phase flow.

APPENDIX E: Rocket motor design and specifications.

APPENDIX F: Strain gauge bridse-ampifier circuit diasram.


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essionall~ designed rockets. Therefo~e, there was no known data

available on the performance characteristics of this propellant.

This necessitated a theoretical anal~sis coupled with experiment-

ation on combustion product anal~sis, combustion temperature

measurements, burn rate ~easurement and the effects of varYing

the oxidizer-fuel (O /F ) ratio.

1.3 EXPERIMENTATION

Testin~ of the actual motor was conducted on a specially

desi~ned static testing stand. The static testin~ stand ~er-

mitted the motor thrust to be recorded continuously throughout

the firing, accomplished by converting the thrust to an electric

analo~ue siSnal which was amplified, di~itized by an AID converter,

then stored in a computer for processinS.

Optimization of the propellant involved testin~ the

effects upon its performance by var~inS O I F ratios. The motor

optimization consisted of modification to the nozzle desisn, inorder to attempt to increase the nozzle efficiency.


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FiSure 1.1. Rocket launch eauipped with small solid motor.


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CHAPTER 2: THEORY

2.1 Solid Propellant Rocket

A twpical solid propellant motor (fiSure 2.1) is of simple

construction, contain ins no movable parts. The maJor components

-SHEAR PIN-CASING

HEAD -NOZZLE

. SECTION A-A.

GASKETS PROPELLANT GRAIN

FiSure 2.1. A twpical solid propellant motor

include the combustion chamber, which contains the propellant

Srain of particular Seometr~ which in turn determines the surface

burninS area. Combustion of the Srain Senerates hiSh temperature

Sases, which are ejected throush an exhaust nozzle. designed to

accelerate the Sases to as hish a velocit as possible. The

throat is the point in the nozzle of least diameter, and the head

being the front of the motor. Combustion is initiated by passing

an electrical current through an isniter which contains a charge

,of black powder.

The theoretical anal~sis of the rocket motor necessitates

certain simplifications. that is. the assumptio~ of an ideal


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rocket. The usefulness of this concept is indicated b~ the ~act

that measured performance can be expected to be within 1 to 10

per cent of calculated ideal values [2].

An ideal rocket assumes the followinS:

(1) The propellant combustion is complete and nonvariant to that

assumed b~ the combustion eauation.

(2) The work inS substance obeys the perfect Sas law.

(3) There is no friction.

(4) The combustion and flow in the motor and nozzle is adiabatic.

(5) Stead~-state conditions (unless stated otherwise).

(6) Expansion of the workins fluid occurs in a uniform manner

without shock or discontinuities.

(7) Nozzle flow is one dimensional.

(8) The sas velocitw. pressure, and densitw is uniform across

anw cross section normal to the nozzle axis.

(9) Chemical eauilibrium is established in the combustion chamber

and does not shift in the nozzle.

An~ additional assumptions will be stated as necessarw.

2.2 Propellant

2.2.1 Composition

The important considerations for amateur rocket propellants

are the availabilitv of the constituentsr costr safet~ of hand-

lins csstabilitYr consistenc~ of performance, and adeQuate perfor-

mance. The propellant that is investisated here easily meets or


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exceeds these reQuirements.

The propellant consists of sucrose (suSar) fuel with potas-

sium nitrate as the oxidizer. The O/F ratio is chosen on the basis

of which ratio Sives the sreatest overall perfrmance for a siven

propellant Srain desiSn. For most propellants includins this one,

it is on the fuel rich side of stoichiometric. More accuratel~,

var~ins the O/F ratio affects both the characteristic exhaust

velocit~, CV, (essentiall~ relates to the ener9~ available), the

burnrate, r, and the Quantit~ of non-Saseous exhaust products.

These aspects are trealed in closer detail elsewhere.

Combustion

The assumed combustion eauation is based on the reaction of

potassium nitrate (KN0 3) with ox~~en (02) [3J, and upon the decom-

osition of sucrose (C12H22011) upon heatins [4J. The assumed

combUstion eauation is;

a C12

H22

011 + b KN03--> c CO + d CO

2+ e H

20 + f K

2C0

3+ 5 1 N2 +

h C + i 02

where K 2C03 represents potassium carbonate, where the coefficients

a throush i are dependant upon the OIF ratio with c,h or i often

beins zero.

This eQuation assumes that secondar~ exhaust products (such

as NO,K 20, etc.) are formed in negligible Quantities. As well, the

effects of dissociation are assumed neSlisible. This second assum-ption is ser~erall~ valid for combustion temperatures below 1700 K.

particularly at hi~her pressures [5J. It should be noted that


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there are two non-SaseOU5 products formed, K2 C0 3 and carbon, from

which no expansion work can be derived.

Knowled~e of the combustion eauation allows the calculation

of the adiabatic flame temperature (AFT), the maximum possible

balance:

combustion temperature. The AFT is determined from an enthalpv

+ .6 h ] = L: n [hOp e f

+i

where Rand P refer to the reactants and products, respectivel~,

nj and ne are the individual reactant and product molal numbers-0 -

respectivel~'Ahf ' the enthalp~ of formation per mole'Ah , the

standard enthalp~ at the specified temperature, per mole, and

where:

.6.h =

and .6.h =tr

T =1

T =K

C =p

T K

f .e dTT1 penthalpv of transition/mole

+ .6.htr

reference temperature, t~picall~ 300 K

AFT

specific heat at constant pressure/mole

O/F := 65/35.

Appendix A contains a sample calculation for the case of

flame temperatures [7].

Fi~ure (2.2) shows a comparison of theoretical to actual


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4000

3000

8

Fi~ure 2.2. The relationship between calculated and actualtemperature.

The AFT varies with the O/F ratio. particularl~ stronSl~

in the ranse close to stoichiometric. where the AFT is maximum.

as shown in fiSure (2.3).

The two points of sharp slope chBnse are at stoichiometric

tion eauation is

mixture (peak) where the O/F ratio is 73.3/26.7 and the combus-

C'2 H 22 0 11 + 9.6 KN0 3 --> 7.2 CO 2 + 4.8 N2 + 4.8 K 2 C0 3 + 11 H 2 0

(2.3)

and the point where partial oxidation of C to CO is completed,

with an O/F ratio of 63.9/36.1 , with the combustion eauation:


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2600

,---_200::.:::oI-< 1 :

514000..~wI-

1000

9

600

200~--~40~---5~O~--~6~O----~--~~--~9~O~~PERCENT

Figure 2.3. Adiabatic Flame Temperature as a function of OIF ratio.

The OIF ratio that is normall~ emplo~ed for this motor is

65/35. This gives the following combustion sauationl

C12 H22 011 + 6.28 KN03--> 8.29 CO + 0.56 CO 2 + 11 H 20 +

3.14 K 2C03 + 3.14 N2 (2.5)

The performance anal~sis of the rocket motor is based upon

this combustion ratio, unless stated otherwise.

Another parameter that is used extensivelw in ideal perfor-

manes analwsis is the average molecular weight of the gaseous


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exhaust products. 1'1'. This is calculated from the combustion

eauation and from the individual molecular weishts of the

products:

1'1' := M'i

t 1 ' 1 'k

t

where i, j and k are the individual constituents, n is the mole

number, and where t re~ers to the total number of moles.

The ratio of specific heats. k , is another important

~uantit~ in the anal~sis of compressible fluid flow, which is the

t~pe of flow encountered in rocket nozzles, where k is defined b~

k = (2.7)

where , for an ideal ~as, k is a function of temperature onl~.

The value of k can be determined from knowledse of the

specific heats' C p , of the individual exhaust sases, where

k :=C - R'

where R' is the universal ~as constant, and

n .I

(2.9)

However, complications arise in the flow throuSh a supersonic

nozzle where the temperature of the ~ases drop appreciabl~ (as

will be shown later), since C p can be a strons function of temp-

erature. It is possible to anal~se such a flow utilizin~ a vari-

able isentropic component e8J, however, a sufficientl~ accurate


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result can be obtained b~ calculatins an ave rase value of k for

flow throush the nozzle (see appendix B). Testins has found that

for a 150 conical nozzle with a small area ratio (6 to 8), the

variation in k does not have a pronounced effect [93. This is

simi liar to the nozzle t~pe used for the motor under consideration.

The remaininS combustion parameter to be determined is the

burnrate, r (also called the surface regression rate). This is

normall~ determined empirically, and is a function of the pro~el-

ant composition and certain conditions within the combustion

chamber. These conditions include propellant initial temperature,

chamber pressure, and the velocity of the gaseous combustion

products over the surface of the solid (erosive burninS). It is

necessary to combine a theoretical model together with empirical

data in order to particulariZe the burnrate for a siven propellant.

The usual model is to approximate the burnrate as a function of

pressure;

nr ::::a Po (2.10)

where a and n are empiricallY determined constants, and ~ is the

combustion chamber pressure. The pressure exponent n, associated, . '

with the slope of the pressure-burnratecurve is almost independant

of the propellant temperature. The coefficient a is a function of

the initial propellant temperature, but not of pressure. From

eQuation (2.10) it can be seen that the burnrate is very sen-

sitive to the exponent n. Hish values of n Sive a rapid chanse of

burnrate with pressure. This implies that even a small chanse in

chamber pre$su~e produces substantial chanSes in the auantit~ of


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hot Sases produced. As n approaches unit~, burnrate and chamber

pressure become ver~ sensitive to one another and disastrous rises

in chamber pressure can occur in a ~ew milliseconds. When the value

of n is low and becomes closer to zero, burnins can become unstable

and ma~ even extinsuish itself. Most producton propellants have a

pressure exponent ran~inS from 0.3 to 0.6 (10J.

A correction is senerall~ made for erosive burning where

the increased burnins can be accounted for b~ an empirical

correction of the form:

r = ro ( 1 + ku ) (2.11)where k is an empirical constant and u is the ~as velocit~.

However, for this report this form of correction will not be

applied. Instead, the burnrate is calculated with the erosive

term taken directl~ into account.

Propellant Grain

The srain is prepared b~ casting the molten propellant into

the desired shape. Several common confi~urations are shown in

figure (2.4). The main criterion for choosing the geometr~ is to

achieve the desired thrust- time characteristics. Thrust is a

function of the instantaneous bUrning area which itself is de-

pendant upon the initial Srain confisuration. The most common

desisns are those which achieve proSressive, regressive, or

neutral thrust- time curves, as shown in fisure (2.5).


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Bonded

S egmen te d r ub e ( ca rt ri dg e lo ad e d)

=.I~IEI~ e ~~t~( __ ~t nt em lll- ex temat b ur nin g r ub e(inhibited a t e nd s)

.I~~.I~~ntemlll burning tube ( case b onded and end restr_!ed) ~ Star (easebonded)

------

Figure 2.4. Several common srain confisurations

Tub.:iaT

.1Neutral

~UTimetar

FiSure 2.5.

~nime

DU31 thr us t

Multi-fin

~e

~rsDouble anchor TimeTwo-stepthrust

Dual c ompos iti on

Internal burning grain designs with their thrust-timecharacteristics.


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The srain confiSuraton of the motor beinS considered is an

internal- external burninS tube~ unrestricted at the ends. The

theoretical thrust- time characteristic is slishtl~ to moderatel~

resressive, depend inS upon the lensth/diameter ratio.

The initial burnins area, Ab is Siven bw:

(2.12)

where L is the srain lensth, Do and D j are the srain outside and

inside diameters, respectivel~.

A slishtl~ modified version of this Srain is also used in

the motor (fiSure 2.6), where Ab is siven b~:

rr { La -~~[31

(DI + Do) ]= (Do t DI ) + D -0b 2

1 1 2 2 _ 2D I2 ]![ - - - (D + D I ) + Do (2.13)4 4 0

These two expressions can be modified bw utili2inS eQuation

(2.11) to obtain the instantaneous burnins area, A ",() :b

rr [(L-n )2 ]A bet) = a t Po ) ( Do + DI > + 0.5 ( Do - D (2.14)1

and

T { (L an Lc

[ 3D o-1

+ Do> ](t):;;: - atP a ) (Do + D ) + - - - (D

b i 4 2 i

1

[1 2 2 2

] }- - - (D t [I ) + D 2D (2.15)4 4 0 i 0 i


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where t represents the time from initial burnin~. It is important

to recosnize that these expressions represent ideal burninS

which assumes that simultaneous ignition of all surfaces occurs

at the beSinning of the burn. As well the effects of increased

burnins surfaces due to flaws in the srain such as bubble holes

and other flaws, as well as the effects of erosive burnins are

nesUected.

These expressions are useful in calculatinS the chamber

pressure and therefore thrust, as will be shown subseauentl~.

Propellant density, Pp and srain density, PI are two

additional properties that will prove useful. Ideall~ these two

are identical but as a result of voids in the srain its density

is somewhat lower.

The ideal propellant density is a function of the O/F

ratio, since the density of the two constituents are different

(P KNO = 2.11 S/cm. tllH PSUCR05E= 1.58 S/cm. (12] ). The ideal3

propellant density can be expressed as:1 fo f f 3

-=-+-= 1.888 g/cm (2.16)p p p

where f 0 and f ~ r-efer 1:.0- tne --massfraction of the oxidizer and

fuel, respectively. For the 65/35 O/F ratio the actual propellant

density has been found to be about 5 percent lower than ideal.

--- -- ----------_----;..

- - - ~ -= ;- - - - - - - r -1 .43 3.36

--~_J __Figure 2.6. Modified Grain for the motor.


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CHAPTER 3: NOZZLE THEORY

The anal~sis of a rocket nozzle flow involves the stud~ of

stead~, one-dimensional compressible flow of an ideal ~a5. The

actual flow differs somewhat from this simplified model particu-

larl~ in regard to the presence of solid or liauid particles in

the flow stream. The necessar~ modifications will be dealt with

in the next chapter.

The analwsis of compressible flow involves four eouations of

particular interest: continuitw; momentum; eners~. and the

eauation of state. These eouations are applied to desisn a nozzlewith the objective of acceleratins the combustion Sases (and

particles) to as hish an exit velocitw as possible This is

achieved b~ designing the necessarw nozzle profile with the

condition that isentropic flow is to be aimed for This

necessitates minimizins frictional effects. flow disturbances.

and conditions which can lead to shock losses. As well, heat

losses would have to be minimized.

3.1 Nozzle Flow

In describing the state of s fluid at snw point in a flow

field it is convenient to emplow the stasnation state as a

reference state (the state characterized bw a condition of zero

velocitw ). Local isentropic stasnation properties are those

properties that would be attained at anw point in a flow field

if the fluid at that point were decelerated from local conditions

to zero velocitw followins a frictonless adiabatic. that is.


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isentropic process.

The enersY eQuation for an adiabatic flow between points

x and y in which the decrease in enthalp~ is eQual to the increase

in kinetic enerS~ is siven as:

1 2 2h - h =x ~ 2

(v - V

'::I X=

where h. v. and T are the enthalpy, velocitv. and temperature.

respectivel!:l.

The stasnation temperature To is defined from the enersy

eQuation as:

2(3.2)

For an isentropic flow process, the followins relations for

stasnation conditions hold:

= (3.3)T

The local acoustic velocit!:l of an ideal sas is defined as:

a :::: ..JkRT

where R is the sas constant. The Mach number is defined as the

ratio of flow velocity to the local acoustic velocity:

v

H : : : :a


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F~om eauations (3.2), (3.4), and (3.5) the total te~pe~atu~e

Mach number relationship can be w~itten:

~2 =T

k-l 21 + M

2

It can be shown from the first and second laws of thermo-

d~namics [13] that for an ideal Sas undersoins an isentropic

process assuminS constant specific heats. that:

p

--k' = constantp

(3.7)

From this result and from the eauation of state, P = pRT,

the staSnation pressure~ densit~ - Mach number relationship

can be expressed as:

kp

[1 +k-l H2 ].~-!--~ =P 2

1

Po [1 +k-l H2] . - : - i=

p 2

(3.8)

(3.9)

The use of eauations (3.6), (3.8), and (3.9) allow each

propert~ (T,P, p) to be dete~mined in a flow field if the Hach

number and the stasnation properties are known. From the eners~

eauation for an adiabatic flow (3.1) the stasnation enthalp~ is

defined:2

h = h to (3.10)2


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PhYsically. the stagnation enthalpy is the enthalpy that

would be reached if the fluid were decelerated adiabatically to

zero velocity. We note that the stagnation enthalpy is constant

throughout an adiabatic flow field. Since the above stagnation

properties (Po' Po and To> are related to the stagnation enthalp!:I

by the specific heats and by the eQuation of state, it is

apparent that each of these stasnation properties is constant

throushout the adiabatic flowfield.

For stead~, one-dimensional flow, the continuitv eQuation

can be written

* * *A v = constant = P A v (3.11>

where A is the passage area, v is the velocity of the flow, and

a starred (*) variable indicates critical conditions, or where

M is unity.

From eQuations (3.4), (3.6), (3.9) and


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where M becomes unit~ is seen to exist at the throat (point of

minimum area) of the nozzle.

The variation of the properties durins flow throush the

nozzle is illustrated in fiSure (3.2). From eouation (3.1 ),

the nozzle exit yelocit~, va' can be found b~:

Ya= -/2(h - h ) + Y 2

" x e x

3,0

2.5

AA*

1.5

1.0

0.5

00 3.0

(3.13)

*iSyre 3.1. Variation of AlA with Mach number in isentropicflow for k = 1.4.

where h and v are the enthal~~ and velocitw at an~ pointx x

within the nozzle. This relation a~~lies to both ideal and

nonidaal rocket units. This eouation can be rewritten with the

aid of eQuations (2.8) and (3.3) ,(3.1)

vk-1

2k R'T o=e

(3.14)


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.. ..- .-- ... - .-_ . --. _ _ .. .-.. .---- . ... .. -- --.--.---~ x

l,Of-----.....---..

.9

.8

.7

.6

.5

.4

.3

.2

.1

O~--_L--~2~--~3~--~4--~~5~~~6~=-~-~-7~-=-~-=-~8~-=-~-~9

X (eM,)

L

\

\\

\\ J '/

IY p - ,/J;,'0 \

21

for flow between the chamber, where staSnaticn conditions are

Fis. 3.2 Variation of densit~ P, pressure p, andtempe~ature T, throush the rocket nozzle.

assumed to exist, and the nozzle exit.

From this e~uation it is seen that the maximum exit velocit~


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o ccu rs at an in fin i t e p res s u re ra t io Po /P g o r when exhau s t in S

in to a vacuum.

The r a t io between the th ro a t an d an ~ down s t ream area a t

which p res s u re P x p reva i l s can be ex p res s ed as a fu n c t io n o f the

p res s u re rat io an d k as fo l l ows , u s in S eQua t io n s ( 3 .3 ) ' (3.9>r(3~,},.(3A-)

an d ( 3 .14 ) ( n o t in S tha t a t the th ro at M is un i t~ ) :

* Px Vx= -~~r~*

x

1 1

( T '( - ;~ f~- ; -~ k + 1= ------ 1k - 1(3.15a)

(3.15b)

This eQu a t io n i s impo r tan t in tha t i t a l l ows the ex i t area,

Ae to be determin ed s u ch tha t the ex i t p res s u re i s eo ua l to the

ambien t p res s u re P a ( t ! : : l p i c a l l ! : : l a tm. ) . Th i s i s kn own as the

des is n co n d i t io n where i t wi l l l a ter be s hown tha t fo r s uchco n di t ion s max imum thru s t i s ach ieved . Fo r th i s des iSn , As / A*is

kn own as the op t imum exp an s ion ra t io .

3 .2 Rocket Per fo rman ce Parameter s

Thi s s ec t io n deal s wi th the var io u s p erfo rman ce p arameter s

tha t a re u s ed to determin e an d compare the p er fo rman ce o f s o l id

p ro pel l an t ro cket mo to r s . As wel l , modi fi ca t io n s to the s imp l i -

f ied model a re con s idered to co r rec t fo r rea l o r ac tu a l p s rfo r -

man ce. As wel l the effec ts o f two phas e f l ow are con s i d e r ed .


30/93

23

The thrust F, of s rocket motor can be shown to be ~iven

b~ [14]:

F ... f P dA ." " m v + ( p - P ) A (3.16)e e a e

where the first expression represents the inte~ral of the pressure

forces actin~ on the chamber and nozzle proJected on a plane normal

to the nozzle axis (fi~ure 3.3), m is the mass flowrate of theexhaust products and veis the exit velocit~. The second term of

the second expression is called the pressure thrust and is eaual

to zero for a nozzle with an optimum expansion ratio.

From continuit~, e~uation (3.16) can be rewritten:

* * *= p v A v +e

P - P )A (3.17)e a

and modified usins (3.14), (3.9), and


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24

INozzle Nozzlethroat exit

c b C D

Atmosphere C DP3

t't_ _ _ _ _ " I _ ~

III P, . Pl

Chamber

Pl

FiS. 3.3 Pressure balance on chamber and nozzle walls;internal Sas pressure is hishest inside chamberand decreases steadily in nozzle, while externalatmospheric pressure is uniform.

3.2.2 Thrust Coefficient

The thrust coefficient C f , is defined as the thrust divided

b~ the chamber pressure and throat area:

F

*o A

(3.19)

The thrust coefficient determines the amplification of the

thrust due to the Sas expansion in the nozzle as compared to that

would be exerted if the chamber pressure acted over the throat

area onl~. From eauation (3.18):

(P - p ) A+

e a e- - - - - - - - -

P o A*

(3.20)

For an~ fixed pressure ratio Pe/P o the maximum C f can be

found b~ takinS the derivitive = O. Therefore, the


32/93

25

maximym C f will occur when P = P , or when the nozzle is designede a

for optimum expansion (desi~n condition).

Eouation (3.19) is useful for comparins the measured Cf

to the theoretical value as determined from eauation (3.20).

3.2.3 Characteristic Exhaust Velocit~

The characteristic exhaust velocit~, c v , is defined as

CV = c I C f where c is the effective exhaust velocit~:

F

.III ... v eAe

+ ( p - P ) - -- (3.21)... .e a mThe CV can be expressed as a function of the ~as properties

in the combustion chamber using eauations (3.21), (3.19), (3.1 ) , (J .~

and (3.11):

CVR To

. . . . . . . . . . . . . ._ - - - - - _. . . .

(2 ) ! . . . !11-1

k+l

(3.22)

CV is usualls used as a fisure of merit of a propellant

combination an d combustion chamber design, and is essentiall~

independant of nozzle characteristics. This makes CV useful as

a comparison for different propellants.

It is interesting to note that cv -. rr' recallinS that-v -~ ~

R = R' I H' A hi~h value of CV is desirable~ however a hish value

of To might not be. High combustion temperatures ma~ cause ex-

cessive heating of the parts of the rocket motor that are exposed

directl~ to hot ~ases. Nozzle erosion can be severe with conven-


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26

tional materials particularl~ when burn durations exceed several

seconds. For this reason it ma~ be more desirable to increase

CV b~ reducin~ M I rather than increasin~ To This can often be

accomplished b~ usins a propellant that is on the fuel rich side

of stoichiometric.

The CV can also be considered an expression of the impetus

(n R T) of the propellant. This is useful for determinins a

propellant CV in the laborator~ b~ means of a closed bomb

(constant volume) measurement (15].

3.2.4 Impulse

The impulse (or total impulse),!, is the intesral of the

thrust over the operatins duration, t:

l ~ f to

or the area under the thrust- time curve (fi.ure 3.4).

I = (3.23)

Figure 3.4. Area under thrust-time curve represents thetotal impulse of the motor.


34/93

27

The operating duration is approximatelw e~ual to the burn

time, t , for motors with lonser burn times (t b> several seconds)b

For short burntimes such as those associated with small motors

the duration of thrust would be considered the burntime plus the

time duration for the residual gases to exhaust from the combus-

ticn chamber after burnout. The total impulse would then be siven

b\:l[16]:

2C P A* Tf 0 (3.24)=

k+l

where T is a time constant siven bw

1 A*Po-=----l' P () V IJCV

where n is tFleOurnrate expone~aho---V:- ~-'S-,,-ne hamber volume.o

The specific impulse is one of the most important perfor-

manee parameters used in rocket research. It is defined as thethrust that can be obtained from an eQuivalent rocket which has

a propellant weisht flow of unit\:l.It is siven by:

F cI = = (3.25)sp

.w

With solid rocket motors it is difficult to measure the

weisht flowrate, sO the average specific impulse is usuallw

emplowed , where I = II t.SP

The ideal specific impulse can be calculated for a siven

propellant and motor combination from e~uations (3.25) aryd (3.14):


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28

I =sp

(3.26)

For comparison purposes Po is usuall~ taken as 1000 psia

and P e as 14.7 psia, b~ convention. A simplified semi-empirical

method for determining specific impulse was developed b~ Free

and Saw~er (17] with a claimed accuracw of 3 to 5 percent.

Recallins that CV is also a measure of the propellant

enerS~, the actual specific impulse can be determined b~ utili-

zins the ballistic bomb method as previousl~ mentioned, where

I is related to CV b~:SF'

r (3.27)-----

however, reQuirins the use of the ideal C f unless the actual value

is knowrl.

3.2.5 Chamber Pressure

The combustion chamber of a solid propellant rocket isessentiall~ a hish pressure tank containins the entire solid mass

of propellant. Combustion proceeds from the surface of the Srain

where the rate of Sas Seneration is eaual to the rate of consump-

tion of solid material, in the ideal case where onl~ gaseous pro-

ducts result {no solids or liQuids>!

(3.28)

.where m is the Sas Sene ration rate, Ab is the area of the burninS5 1

surface. and r is the surface reSression rate (burnrate). The rate

at which sas is stored within the combUstion chamber is given b~:


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29

dt

d= -- - - (Po Vo )

dt(3.29)

where pais the instantaneous gas densit~ and vois the instantaneous

chamber volume. The use of these two eauations as well as the

expression for the rate at which Sas flows throuSh the nozzle and

the expression for burnrate. leads to the followinS expression

for chamber pressure [19]:

= ~ - - - - - - - - - - - - - - -

For constant Po and therefore constant thrust it is clear

that the burning area must remain constant. such as with a neutral

burnins grain. The instantaneous chamber pressure can be siven b~

the same expression, in which Ab is the instantaneous burning

area. The variation of Ab with time depends uPon the burning rate

and the initial seometr~ or the propellanv srain.

For short burn time motors such as the one under consid-

eration (less than one second burntime) the duration under which

stead~-state conditions exist, where eauation (3.30) applies,

accounts for perhaps one- third to one- half of the total thrust

time. Prior to iSnition, the combustion chamber is filled with

cool air of ambient pressure. The start-up period where the

pressure builds UP to the stead~-state value is a complex

problem that has been studied bw Von Karman and Malina [19J.

The~ have developed an expression for the pressure as a function


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30

of time durins start-uP, siven by:

(3.31)

where P is the instantaneous pressure at time t and where T is

previously defined.

Immediately after burnout the combustion chamber remains

filled with hish pressure sas. The expression for chamber pressure

is siven as a function of time by [20]~

p =k - 1

(3.32)2

3.2.6 Corrections for Two Phase Flow

The ~revious analYsis of nozzle flow and performance psra-

meters considered the ideal case where the workins fluid is a pure

Sas. This analYsis is fine for the case where there is no solid

or linuid particles in the exhaust (such as for liauid propellants)but must be modified for the case where such particles are present.

The presence of particles is detrimental to the overall rocket

performance, with the extent dependant upon several factors.

One of these factors is motor size. It has been found by4

Gilbert, Allport, and Dunlop (21J that for larSe motors ( >10

pounds thrUst) the overall specific impulse losses are low, but2

for small motors (~10 pounds thrust) the losses can be sisni-

ficant. The motor under consideration is of this magnitude thrust

makinS corrections for two phase flow necessary.


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31

The lo s s es in p er fo rma~ce are du e to two facto rs :

i ) vel o c i t~ l as , where the p ar t ic l es a re n o t accel era ted

thro u~h the n o zzl e to the s ame vel oc i t~ as the Sas es ; a~d

i i ) thermal l aS , where the p ar t ic l es are n o t in thermal

eau i l ibr ium wi th the s u rro un d in s Sas es .

I t has been fo u n d tha t the l a t t er has a smal l effect , u s u a l l y

a l l owin s the as s ump t io n tha t the p ar t ic l es a re in thermal

eau i l ibr ium [22J .

The s ta r t in s p o in t in the con s idera t io n o f the effec t s o f

s o l id ( l i au id ) mat ter fo rmat io n i s wi th the eau at ion rel a t in s to

the Sas Sen era t io n ra te ( eQua t io n 3 .28) . Th i s eQua t io n becomes :

m

S

=(l-X)p A rp b ~

where i den o tes a p ar t icu l a r co n s t i tu en t . I f a l l the par t i c l es


39/93

32

are assumed to have the same velocity this can be expressed as!

1

I

=spTo accurately describe the erfects of particles on a

rocket's performance reauires knowledse of heat transfer proper-

ties and draS processes or the particles in order to express Vs

and Ts in terms of the overall flow properties. It is possible,

however, to derive the limitinS cases to determine the least and

most detrimental effects.

In the first model, the particles are assumed to be very

smallr where Ts ::::: gand Vs ~ Vg This wo;uld slive the followin!!i

expression for exhaust velocity (see appendix c for derivation>:

T [ c sx+(1 _ X ) ~ _~~ __ ] [1 _( _ ~ ~ _ )m ]o (k - l)M' P o

(3.37)

~ = [~~-~~;;~;~ + -;.-~ -~ ]- .

where Cs is the ave rase specific heat of the solid (liauid) matter.

where

Note that this eauation reduces to the familiar form (3.14) when

X = 0 (no particles).

The expression ror thrust would be modified to this form

(see appendix D for derivation):

F = * 1A Po-----1 - X (

~

~~II

.z.. [Xk + 1

c +s

fr . R' J [l -X) -- . : :----- 1(k-l) H'

+ ( P - P ) A e a e


40/93

33

Note asain that the eauation reduces to the familiar form (3.18)

when X = O.

The expression for the thrust coefficient is derived in the

same manner, ~ieldinS the expression:k+1

( ;_~- J -OI [~ : ;~ - :~+ ~:~ -= -~11:nc = -- - - -f 1 - X(3.39)

It is interestinS to note that the thrust coefficient in-

creases with the ~resence of particles compared to the Sas onl~

value. This eauation im~lies a 75 percent increase in the thrust

coefficient for a t~pical propellant where X = 0.45.The second limitins case models the particles Sych that the

particle velocit~ and temperature remain essentiall~ constant.

This implies a combination of relativel~ large ~h~sical size,

slow heat transfer, and low draS. With such a model. the ex~res-

sions for the performance parameters remain identical to theoriginal (gas onl~) expressions. i.e. the gas flow is not affected

b~ the presence of particles. However, the specific impYlse is

reduced. From eauation (3.36) where m v m v :SSg S

1I =Sf>

( 1 - X ) vst !Ii

(3.40)

Therefore, the specific impulse is reduced b~ a factor of i-X.

Research conducted into two phase flow also indicates that

the nozzle t~pe has a bearinS on the effect of particles in the

exhaust stream [23J. As well ,it has been found that particle


41/93

34

size (and dist~ibution) is independant of motor size, and is

therefore larsel~ a property of the particular propellant [241.

Another detrimental effect of liauid particles is the tendency

for the material to build UP on the nozzle surfaces, reducinS the

effective nozzle area. Additional losses are introduced bw the

resultinS rough surface which increases skin friction [251.

For the motor being considered, the losses due to particle

presence will be accounted for in the enerSY conversion factor,

discussed below.

3.2.7 Corrections for Real Nozzles

The precedins analwsis considers ideal rockets, which of

course do not exist. The ideal case represents the maximum per-

formance that can be attained, with the actual performance being

reduced by a number of factors. These factors are accounted for

in real n02zles bw using various correction factors.

Conical nozzles reQuire a factor to correct for the non-

axial component of the exit Sas velocit~ as a result of the

diver~ence ansle, defined as 2a (where a is the half anSle).

This factor ~ , is given bw:

1

2

As well. the flow in an actual nozzle differs from ideal

( 1 t cos a ) (3.41)

because of frictional effects, heat transfer. imperfect sases.

nonaxial flow, nonuniforrnity of the working fluid and flow

distribution. and the effects of particles. The degree of de-


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35

parture is indicated b~ the enerSw conversion efficienc~ of the

nozzle. This is defined as the ratio of the kinetic enersw per

unit of flow of the Jet leavinS the nozzle. to the kinetic enerSwper unit flow of a hwpothetical ideal Jet leavinS an ideal nozzle

that is supplied with the same workins fluid at the same initial

state and velocitw and expands to the same exit pressure as the

real nozzle. This is expressed as;

2v

e = --!!. (3.42)2

vaI

where the subscpipts i and a refer to the ideal and actual states

and e denotes the enerSw conversion efficiencw.

The velocitw correction factor,f v ' is defined as the sauare

root of e. For most production motors, the value lies between 0.85

and 0.98. This factor is also approximatel~ the ratio of the

actual to ideal specific impulse.

The discharse correction factor ~ , is defined as the ratio

of the mass flow rate in a real nozzle to that of an ideal ~zzle

that expands an identical work inS fluid from the same initial

conditions to the same exit pressure:

r ::::cI

ma

.m ca

(3.43). .ffi.

IF

The value of the discharSe factor is often larser than

unitw. because the actual flow can be larser than theoretical

for the followins reasons:

i) the molecular weiSht of the Sases increases slishtlw


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36

when ~lowin~ through a nozzle, thereb~ chanSinS their densit~.

ii) some heat is transferred to the nozzle walls. lowerins

the Sas temperature, increasins its den$it~.

iii) the specific heat and other Sas properties chanse in

an actual nozzle in such a wa~ as to slishtl~ increase the value

of the discharSe coefficient.

tv) incom~lete combustion increases the densit~ of the

exhaust Sases.

These correction factors result in a thrust lower than for

the ideal casel

F := ! !d X Fa v i

*:= r rx c p Av d f 0a aI

(3.44)

(3.45)


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3 7

CHAPTER 4 : EXPERIMENTAL TECHNIQUE

4.1 Motor

The motor under consideration is illustrated in the fiSure

below (fi~ure 4.1), with a complete description Siven in appendix

( E ).

- 2 .5 em .

-" ' -

3.8 c m .

I

/nozzle grain

//

/ head

head gas kets (5) s afety bol t

FiSure 4.1. Actual motor used in testins.

The standard n0221e has a conical profile with a 12 degree

divergence half angle. It was fabricated from mild steel bar stock

turned and bored to the desired dimensions on a standard lathe.

The nozzle contour is rounded at the throat to avoid sharp dis-

continuities which could lead to shock losses. As well, the entire

n0221e inside surface is polished to reduce friction losses.

The nozzle is retained to the combustion chamber bw siw high

strength allen screws.


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38

The head of the motor is readil~ removable to allow for

eas~ loadins of the Srain. Constructed as well from mild steel,

the head is retained b~ a sinsle safet~ bolt(3/16

X 1112

in.;Srade 8) desisned to shear if the chamber pressure should attain

an unsafe level (17 MPa.). The combustion Sases are sealed from

escapins around the head b~ five asbestoes-fibre composite Sas-

kets. The nozzle is sealed aSainst the combustion chamber b~

utilizins a tool which rolls a circular die under hish pressure

around the perimeter of the chamber outside wall esainst the

nozzle section forward of the retention srcove. This effectivel~

seals the chamber wall asainst the nozzle to reduce sas leakase.

4.2 Cubic Nozzle Profile

Althoush the exact nozzle profile is not critical for

sood performance as the flow occurs in a resion of favourable

pressure Sradient, optimum performance is achieved bw desisnins

the nozzle b~ the method of characteristics. A thoroush dis-

cussion of this techni~ue is presented in references [263 and

[27]. This method is lensthw and complex and will not be consid-

ered here. However, an approximation to this desisn is that of a

cubic profile where the nozzle contour follows the curve of a

cubic eGuation [28].

A nozzle was constructed with the aid of a computer numer-

icall~ controlled lathe. The criterion for desisn was based on the

same throat, entrance and exit areas as the conical nozzle. The

conver~ent section was based on a maximum half an~le of 30 deSrees+


46/93

and a maximum diver~ent angle of 20 degrees. F i au re (4.2 )

39

shows the dimensions of the nozzle where the eouations of the

contours are:2 3

R = 1.748 - 0.762 x + 0.226 x em. (Region I)

2 30.037 x em. (Resion II)= 0.461 t 0.201 x

where R is the inside radius.

. . . ! rTC~~~: ' - - '!; , . . . . . . ' - - ,+-- ' -c -+c- i" - ' -++_ . : .c . - ; Id- ,LU- j+ + _ ...: 'j T ~

-~ - i~~-=f i ji I-' --h~~:_ ;,~ .. '- , .".j.1 "- '- . ~ - , , .. . . . - . - :I

._J_ - - ~ r ' "T I!, . . , '~; , ; 1 1 '- , '.l- ~'-I--I'I .

Fi~ure 4.2. Cubic nozzle profile.

( 4.1>


47/93

4 0

This nozzle was constructed of cold rolled mild steel

bar stock, and was hi~hl~ polished on the flow surfaces. The

diver~ence exit an~le was machined to zero deSrees, to eliminate

radial flow losses (non-truncated).

It should be noted that while the critical dimensions

of the cubic nozzle are coincident with the conical nozzle, the

overall lensth is about 17 percent shorter.

4.3 Propellant Preparation

As stated previousl~, the standard propellant com-

bination is 65? oxidizer (KN0 3 ) and 35~ fuel (sucrose). The

propellant


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41

The commerciall~ obtained potassium nitrate oxidizer has

an average particle size of 250 ~. The oxidizer is subseouentl~

around down to an averaSe particle size of 100~. The sucrose is

obtained in powder form with t~pical particle size of approxi-

matel~ 10 ~

It has been found that the particle size of the oxidizer

has an important influence on both the burnins rate and more

importantl~ on the impulse delivered. Therefore, careful attempt

was made in the experimentation to obtain consistenc~ in oxidizer

particle size.After accuratel~ weiShinS the desired Quantities of both

constituents, the~ are placed in an electric rotating drum mixer

where mixinS occurs for twentw hours.

The Srain cast ins procedure involves heatins the mixture

in a container maintained at a temperature of 190 to 200 desrees

celcius bw immersion in an electricallw heated oil bath. The

sucrose, (mt? 182 desrees C.) combines with the non-meltinS KN0 3

(mtp. 441 C.), to form a slurr~. Decomposition of the sucrose

initiates as well at this temperature being both a function of

of temperature and time. Therefore, the temperature has to be

maintained within this ranSe, and for the minimum time duration

necessarw for the entire mixture to melt.

This decomposition (caramelization), if kept in check, has been

found to have no detrimental effect on the propellant performance.

Investigation of the effects of low desree caramelization of

sucrose indicates that there is no sisnificant mass chanse. which


49/93

42

is i~portant since the occurance of such would alter the effective

OfF ratio. As well, it has been found that less than 1 percent

moisture is absorbed b~ the finel~ divided suSar, which is again

an important consideration for an~ moisture initiall~ present

would be Siven UP upon heating.

Once the slurrY has become sufficientlw fluid, it is

cast into the desired shape. This shape (hollow cylindrical)

is achieved bw pouring the slurry into a lubricated cYlindrical

mould, of slightl~ smaller diameter than the combustion chamber.

A lubricated bore rod is inserted down the central axis. After

allowins to cool and harden for approximatelw forty-five minutes,

the grain is removed, trimmed to siZe, weiShed and measured. The

grain is then stored in a sealed container. This is to prevent ex-

posure to the open air for the propellant in its final form is

verY hysroscopic.

4.4 Varying OfF Ratio

A number of Srains were prepared to determine the effect

of varyins OfF ratios upon the motors impulse. As well, several

small cylinders (1.5 x 10 cm) of varying OfF ratios were prepared

for burn rate testinS.

4.5 Burnrate testing

In order to auantify burnrates under conditions of one

atmosphere and at room temperature, several samples of different

OfF ratios were prepared. The samples were cast into cylinders


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43

as indicated earlier. These c~linders were mounted u~ri~ht b~

adhering one end to a base. A specified length was marked off.

tspicall~ 7 cm. The top end was then ignited b~ a flat surfaced

tool, heated prior in a flame. As the sample burned down, the

elapsed time was measured between marks. This t~pe of burnrate

test ins sives results for conditions of constant temperature

propellant and non-erosive burning.

More complete burn rate testing would involve determining

the dependance of burnrate on pressure. In order to conduct such

testins , it would be necessar~ to construct a suitable vesselthat would allow burning to occur at constant pressure. Such an

investigation has not ~et been condUcted, however. Research of

this tspe has indicated that the burnrate measured b~ this tech-

ni~ue is not in complete aSreement with the actual burnrate

that occurs in the actual motor, being about 7 percent hisher

(29J. No explanation could be found for this discrepanc~.

4.6 Motor Static Testins

Static testing of the motor (fisure 4.3) to obtain the

thrust-time characterics is conducted on a specialls built test

ris (figure 4.4). A remotels located Heath H-B microcomputer is

used for data acauisition. with the measured signal sent to it

via a 50 metre shielded cable. The ac~uisition swstem is potent-

ialls capable of handling UP to four channels of data input (e.g.

thrust, chamber pressure, temperature. and wall 5tress). For this

series of tests. onls thrust measurements were attempted, however.


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4 4

FiSure 4.3. Static ~irinS of the rocket motor mountedin test rig.

4.6.1 Test Ris Construction

The frame of the rig is constructed of heavw steel I-beam

to eliminate undesired deflections under hish thrust. Althoush

the maximum thrusts endured bw the motor under consideration

were 1.1 kN (250 lbs.), the riS can handle much larser values of

thrust.

The motor is mounted verticallw. nozzle upward, in a

holder desisned to allow for the small vertical motion encountered

b~ the motor durins firins.


52/93

1. Deflection bar 7. Electrical islniter2. Lower stoP 8. Deflection bar tensionel'3. Upper stop 9. Frame4. Strain transducer 10. Hwd~aulic damper5. Motor mount 11. Damper adJust valve6. Rocket motor

Fi~ure 4.4. Rocket motor static testins riS.

4 5


53/93

46

The frame sUPPorts a double cantilever deflection bar

assembl~. It is asainst this bar that the motor acts during firing

resulting in the bar deflecting. The thrust is therefore trans-

duced into a lineal deflection. In order to obtain high resolution

of the thrust transduction, it is desirable to have a relatively

large defection of the bar. This maximum deflection is determined

bw the transducer unit, havins been determined to be 0.635 cm.

(1/4 in.). Bw knowing the maximum expected thrust of motor it is

possible to choose the bar-cantilever assembl~ to achieve this

(for an~ motor).

The deflection w for a double cantilever assembl~ is given

bw [301:3

F L(4.3)

192 E I

where F is the force (thrust), E is the YounS's modulus of the

bar material, I is the moment of inertia of the bar, and L is the

bar lensth. For a rectanSular bar of thickness d and width b, this

can be rewritten as:3

F L

316 E b d

The deflection is therefore proportional to LID cubed

and inverselw proportional to the width. Therefore, the choice

of the bar gives great flexibilit~ in achievins the maximum

deflection, ~ max

A lower stop is placed directlw below the centre of the


54/93

4 7

bar to prevent damage to the transducer unit in case of over-

thrusting, or a blowout of the motor head (see figure 4.5).

The thrust induced displacement of the transducer bar

results in an eaual displacement of the end of the transducer

unit, located directly under the motor mount. This transducer

unit uses a strain SauSe bridSe fastened to its fixed end. This

strain sause bridse forms the fundamental component of the data

aauisition s~stem, illustrated in fiSure (4.6). The amplifier

BA RTEN510NER

\

/i

L : JWERSTOP

Ef jO SUPPORT (2:( AO-USTABL )8EA~

Figure 4.5. Deflection bar assembly of static test ris.

COMPUTER

HEATH He

PRINTERj----.-.-~DECWRI TER

LA-36

AID C'I---~8-81 T

STRAIN

GAUGEBRIDGE ZERO

ADJU 5 T

TERMINAL

HEATH H19

Fisure 4.6. Data acauisition sYstem used to measuremotor thrUst-time characterics.


55/93

4 8

circuit is reproduced in Appendix ( F). The A/D converter is an

8-bit, four channel unit capable of samplins 581 points per second

(samples ever~ 1.72 milliseconds), These points are stored in the

computer memor~ for post-proces5in~.

Due to the sprin~-mass nature of the deflection bar assembl~,

it was found necessarw to provide adeauate dampins to reduce

oscillations of the deflection bar. A variable h~draulic damper

was desisned and built to allow a larse degree of flexibilit~ of

dampins. The damper is detailed in tisure (4.7). The operation of

CONNECT.ROD /'0' RING (2)

FiSure 4.7. Hydraulic damper used with test ria.

HEAD

AaH- -- '0' RINGVP--.L_,L..J......(_~ I

JET

df'-:~L1\"'-LJ.:. . .LLL.I:!II----"_'_-"_"'_-FL U lD

CHANNELS

the device is straightforward. A connecting rod with a piston at

~MOUNT.HOLE (4)

BLEEDSCREW

\\

NEEDLEVALVE

one end is attached to the deflection bar. The reversed vertical


56/93

4 9

motion of the rod alternatel~ forces fluid from and back into the

fluid chamber throush a small Jet. The effective area of the Jet

is controlled b~ an adjustable needle valve, the adjustment of

which Soverns the desree of dampins. This allows for a wide ranSe

of damping from essentiall~ nil


57/93

50

The force applied at the deflection bar F, is Siven bw:

F =Wa + W

2

where W is the total weisht huns at the bar end, Wa is the arm

weisht. Since Wa ' L" L2 are constant for a siven arm, this

can be written as:

F = C,W + C2

where C, = L2/ L, and C 2 = C,Wa /2 Calibration is carried out b~ hansins a series of weights and

recordins the output voltase. The calibration data is presented

in the form of voltase V, as a function of force, havins fitted

the points throuSh a second order curve (through the oriSin)

to yield a calibration curve of the following form:

2F = C3V + C4 V

where C 3 and C 4 are constants.


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51

CHAPTER 5: RESULTS (Theoretical and Actual)

5.1.1 Nozzle Testins

The thrust-time curves for the conical and cubic nozzle

are presented in fiSure (5.1) and (5.2), respectivel~. The time

period between successive points is 8'0 milliseconds (i.e. ever~

5th point is printed). For comparison purposes, both results are

presented on a sinsle curve illustrated in fi~ure (5.3) (note:

different time base).

A summar~ of the performance for both of these nozzles

is presented in Table 5.1.

Conical Cubic

Total impl .. Jlse (N-s) 2se 281Specific impulse ( s ) 130.5 127.3Max. thrust ( N ) 1155 1075Thrust duration ( s ) 0.38 0.40Propellant mass 10)

makinS the motors indefinitel~ reusable.


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52

0-J: III ...=:I'"E(Ij

....7..._-..... - ...- - - ....- .... - .... + :; - .......- - - ...- - .... - - ..... - .......-- - - - - ..- - -I

)(In[In I ++ I I I:1:.... I + ! 1 ! +++ I. . . I +1 ! I J I J J I

I I I I I I I I I ++++ II I I I I I I I I I I I +I + I I I I I ! I I I I I 1+ II I I I I I I I I I I I 1 1++ I I I I I I I , I I I I I I I II I I I 1 I I I I I I I I I I I +I I I I I I I , , I I I I I I I I ++ Ir I I , I I I 1 I I I I I I I I I I I

+ I I I I I I I I I I I I I I I I I +L I j 1 I I I I I I I I I I 1 I I I II I I I I I I I I I I I I I I I I I I +I I I I I I I I I I I I , I I I I I I I II I I I I I I I I I I 1 L I I I 1 I I + L I I I 1 I I I I I I I I I I I 1 I I I \I I I I I I I I I I I J I 1 I I I I I II I I I I I I I I I I I I I I I I I I I

,,I + I I t I I I I I I I I I I I I I I I I + ,

1 . 11 I t I I , I I I I I I I I I I I I 1 I 1 I I II I I I I 1 I I I I I ! I I I I 1 I I I I I ,

E I I I J I I I I I I I I I I I I I I I I I + ,I I , I I I I I I I I , I I I I I I I I I I- I I I t I I I I I I I I I I I 1 I I I I I I ,I I I I I I I I I I I I I I I I I I I I I I I

. . . . . I I I I I I , I , I I I I I I I I I I I ,I I I I I I I I I I I I I I I I I I I I I I I I I , I I I I I I I I t I I I I ! I I I I + tI I I I , I I I I I I I I I I 1 I I I I I 1 I I IUl , I I I , I I I I I I 1 I I I I I I I I I 1 1 I I I ,I + I I I I I I I I I I 1 I I J I I I I I I I I I I I

I I I I I I I I I I I I I I J I I J I I I I I I I 1I , I I I I I I I I I J I I I I I I I I I I I I I I II I I I , , I I I I I I I I I I I I I I I I I I I I tI I I I I I I I I I I I I I I I I I I I I I I I ,. . . . I 1 I I 1 I I I , I I I I I I t I I I I I I I I + ,I I I 1 I I I I I I I I I I 1 I I ! I I I I I I I 1on I I 1 I I I I I I I I I I I I I I I 1 I I 1 I , I II I I I I I I I I I I I 1 I I I I I I I I ! I I I ,

;::. I I I I I I I I I I I I ! I I I I I I I 1 I I , I II I I I I I I I I I I I I I I I , I I I I I I I I + I

i:l! I I I ! I I ! ! , I I I t I I ! I I I I I I I I I II I I I 1 I I I , , I ! I I I I I I I I I I I I I:I: I I I 1 I I ! I I I I I I I I I I , I I I I I I I II ! ! I I ! I I I I I I I I I I I I I I I I I I I ,. . . I 1 1 L I I I ! 1 1 I I I I I ! I I I I I I 1 I II + 1! I I I I I I I I I I ( , I I I 1 I I I I I II , I I I I I I I I I I I I I I I I I I ! I I I II ! ) I I I I I I 1 I I I 1 I I I I I 1 I I ! I I.. . I I I I ! I I I I I I I ! I I I I I I I I I I I I! , I I I I I I I I I I ! I I I I I I I I I j I I

w I 1 I I 1 ! ! , I I ! I I I I I I I I I I I I I II I I I I I I 1 I I I I I I I I I I I I 1 I I I +:: . : I I I I I I I I I I I I I I I I 1 I I I I I I I I I II I I I I I I 1 I I 1 I I I I I 1 I I 1 I ( I ! I I I , I


60/93

0-E........: '0-:Cl?

- .-----------------+-+---------------------------

)(In + +1


61/93

C ONIC AL --CUBIC

54

Figure 5.3. Comparison of the conical and cubic ~ozzlemotor thrust-time curves.

!-(f)

:Ja:If---

JIII

,-,\ ,

Ii TIME

5.1.2 Motor Testing of Var~inS OIF Ratio Grains.

The results of this series of tests are prese~ted in

Table 5.2. Onl~ the values for the averaSe specific impulse are

presented because it is felt that this would provide the most

useful base for comparison. It should be noted that a non-

standard conical nozzle was used for this series of tests. result-

ins in lower overall values of specific impulse as compared to

the standard conical nozzle as described earlier. Therefore, the

results should be used for comparison purposes onl~.

The impulse for the 75/25 OIF ratio was not measurable due


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55

to poor burnins resultins from the hiSh oxidizer percentame. A

larme Guantit~ of liQuid matter (potassium carbonate> was ejected

from the nozzle, and burninS continued for about half a minute.

-OIF Ratio Isp ( s >

50/50 49.955/45 93.660/40 10065/35 10670/30 10675/25 ---

Table 5.2 Co~parison of var~inS OIF ratio motor tests.

The 50/50 OIF ratio Srain also resulted in poor firins.

Larse volumes of carbon were ejected with much remainins in the

combustion chamber after firinS in the form of a sinsle porous

mass.

Each of these test firinms were recorded on 8mm colour

movie film for post tirins examination of the test. The results

showed that the exhaust flame varied in size and colour with

differinS OIF ratios. For the more fuel rich ratios, the flame

was smaller and more oranse in colour. For the more oxidizer

rich ratios, the flame increased in size with a more purple

coloured hue, a result of the potassium compound in the exhaust.

As well, it was found that the nozzle throat became hotter with

increasins oxidizer percentaSe, with the exception of the 75/25

O/F ratio. The throat colour ranSed from a dull red (50/50 OIF


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56

ratio) to a bri~ht ~ellow (70/30 O/F ratio). This res~lt is

consistent with the theoretical prediction for combustion temp-

erature as a function of O/F ratio as disc~s$ed earlier.

A comparison of the propellant properties To ' H' , CV

and X for var~inS O/F ratios as predicted b~ theor~ is shown in

f i S~ re (5. 4) t

f.,cC)

2000

1800

1600T o

M CV(m,s) 1400

J O

1200

25

900

20 ao o CV

700 X

600 0,6

500 X

o.~

0,2

440 48 642 686 720 76 80PERCENT OXID IZER

Fi~ure 5.4. Comparison of theoretical properties:(1) Combustion temperature, T o(2) Avera~e molecular weisht of exhaust

gases, H'(3) Characteristic velocit~, CV(4) Particle mass fraction, X.


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57

5.1.3 Burnrate Testing

The results of burnrate testins for ~arious O IF ratios

are presented in figure (5.5). The conditions of testing were

atmospheric pressure and room temperature. The burnrate results

encountered.

for the 75/25 O I F ratio are not shown due to the erratic burning

As mentioned earlier, burnrate testing has not ~et been

conducted for conditions of elevated pressures to determine the

burnrate-pressure relationship. However, it is possible to est-

05

~02. .a

zB:;

ii l0.1

55 60 65 7Q

)!; CXIQIZER

Figure 5.5. Comparison of atmospheric burnrate testinSfor var~ing O IF ratios.

imate the pressure dependence in order to determine an appro~imate

value for both the coefficients a and n (from eauation 2.10),

usinS the results of the motor testing. This method involves


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58

estimatins the time reauired for the Srain web (wall thickness)

to burn through under ~otor operating conditions. The validit~

of this method lies in the assumption that essentiall~ all the

surface regression occurs at high pressure. This pressure, taken

as Po' occurs during the stead~-state operation of the motor,

allowing an estimation of this time period from the thrust-time

curve (see figure 5.6>

. . . .lI)

~0 : : -I. . . .

TIME

Figure 5.6. Estimation of web burnthroush timefrom actual thrust-time curve.

Since burning progresses from both the inside and out-

side surfaces of the grain, the web expension rate is effectivelw

ast

doubled. The burnrate at a pressure of Pocan then be expressed

rR - R-~---!..2 tw

=::


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59

where Ro and Ri are the srain-outside and inside radii.

Combinins this e~uation with the results of the atmos-

pheric burnrate testins:

r l = aP=P atm

and the stead~-state pressure e~uation


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60

5.1.4 Performance Parameters

The theoretical performance parameters of the motor based

upon the two particle flow models, as well as the actual results

are presented in Table 5.2. These results are based upon the

motor eouipped with the standard conical nozzle and the 65 /35 O I F

ratio propellant.

THEORE TICAL THEORE TICAL ACTUALPARAMETER

PAR TIC LE MODEL PARTlCLE MODEL

Ts"..T~ VS"'Vg Ts""const. VsVg VALUE

THRUST (MAX), N, 1691 1153 1155

THRUST TIME, SEC, - -- 0.318V " m/s. 1490 1821 --T o ," c 1406 1406 1347Po , MPa 10.55 10.55 --Ct 2.36 :2:.00 --1, N-s. 408 226 288I sp , sec. 152 103 131"x 0.989 0.989 ----..td 0.998 ~.998 --tv -- -- 0.833e -- -- 0.401

"average

Table 5.2. Comparison of theoretical performanceparameters to actual performance parameters.

The ener~~ conversion efficienc~ is based upon the actual

performance of the motor as compared to the maximum performance

possible, which would be attained for the first particle flow

model.


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61

CHAPTER 6 : DISCUSSION OF RESULTS

6.1 Nozzle Testing

Contrarw to expectations, the performance of the cubicnozzle was somewhat lesser to that of the conical nozzle, deliv-

ering a specific impulse 2.5 percent lower. This result however,

is tentative for onl~ a sins Ie firinS of the cubic nozzle was

performed due to time limitations. Clear'lw, however, no signifi-

cant increase in nozzle efficiency is implied. This suggests

that the conical nozzle is certainly a satisfactor~ desiSn. Con-

siderins the far greater simplicity of fabrication, the conical

desisn would have to be considered the superior choice for the

amateur rocket enthusiast.

If further testing reinforces the present findings, the

lesser performance of the cubic nozzle could be a result of its

greater maxi~um divergence angle (20 degrees v.s. 12) and/or its

shorter length (17X shorter). The latter of these would be con-

sistent with the particle flow theory where a decrease in perfor-

mance would be a direct result of particle las. In a shorter

nozzle the particles would have less time to accelerate to the

gaS yelocit~ before exiting the nozzle. This would suggest that

for a small motor with a high mass fraction of particles, the

nozzle should have a small divergence angle in order to increase

the length. The e>:istingdivergence angle should possibl~ be re-

duced to 10 or 8 degrees. As well, the effects of reducinS the

conversence ansle in order to increase the overall lensth of the

nozzle should be investiSated. Possibl~ the best approach to the


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62

design of a nozzle for such conditions would be to sha~e the con-

tour in a manner such that the maximum flow acceleration is re-

duced, to minimize ~article lag.

6.2 Motor Testin~ of Var~inS O/F Ratio Grains

Maximum performance for the hollow c~lindrical ~rain is

achieved with a moderatel~ fuel rich O/F ratio. Testing indicates

that the specific impulse is highest in the ranse of 65 to 70

percent oxidizer, taperinS off with a lower oxidizer percentage,

and being drasticallw lowered at high oxidizer ratios. The sen-

sitivit~ at high oxidizer ratios is shown b~ the results that a

maximum impuse is achieved at 70 percent oxidizer. but b~ incr-

easins the ratio b~ 5 percent results in an immeasurable impulse.

The hishl~ fuel rich ratios 70/30). the reduction in

performance could as well be explained bs this particle flow model.

Examination of the exhaust blast area revealed that the potassium

carbonate particles ejected from the nozzle were actuall~ in the


70/93

63

form of sizeable droplets. This would create the conditions of

v ~ constant, where the relativel~ larse particle size would nots

allow acceleration of the particles to a hish velocit~. The for-

mation of the droplets could be explained thuslw: durins combus-

tion a voluminous auantitw of potassium carbonate is formed, beinS

in the liauid state. As flow occurs throuSh the nozzle multiple

collisions occur between the liauid particles. resultins in co-

hesion and effective srowins in particle size until either the

maximum stable size is achieved, or exit occurs from the nozzle.

The practical ranse, therefore, for O/F ratio selectionappears to be an oxidizer percentage of between 55 and 70 percent.

Coupling this result with the realization that actual combustion

temperatures appear to parallel theoretical. would indicate that

since lower combustiOn temperatures might be more desirable, the

lower OIF ratios misht be more suitable. If however, the combus-

tion temperatures encountered at the upper end of the O/F ratio

pose no practical problem, the ranse for best performance could

be reduced to the 65 to 70 percent oxidizer renSe. Since no

erosion or other problems have been encountered due to these tem-

peraturesr this would appear to be the optimal ranse. One final

factor to be considered is the castabilitw of the srain. Since

a more sucrose rich propellant is easier to cast, the best over-

all ratio would then appear to be the existin~ 65/35 OIF ratio.

It should be noted that since testinS occurred in $te~s of 5

percentase points, that the optimum ratio is therefore within

a certain plus-minus ranee of this value.


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64

6.3 Burnrate Testins

The results of testins indicate that burnrate is Guite

sensitive to the O/F ratio. with the maximum occurrins at the

65/35 O/F ratio. For small rocket motors. a Sreater rate of bur-

nins is desirable to develop the high chamber pressures that such

motors can readil~ operate at. With lower burnrates. it would be

necessar~ to increase the srain surface (burnins) area to compen-

sate for the reduced rate of burnins in order to maintain the Sas

Seneration rate. A decrease in throat area would as well be a

solution to developing the desired pressure, however. a corres-

pondins thrust loss would result.

Increased srain surface area can be achieved b~ modif~ins

the seometr~ of the central bore, b~ moulding a star or other

shape as discussed in section 2.2.3. This result can also be

achieved b~ increasinS the lenSth/diameter ratio of the srain.

However, both methods eomplicate the grain castins procedure and

increase the likelihood of srain casting flaws. Therefore, practi-

cal considerations make the hisher burnrate that can be achieved

with the 65/35 O/F ratio a desirable trait.

The method of estimatins the burnrate-pressure relation-

ship appears to be satisfactor~ for a first approximation. However,

it is difficult to Sause the accurac~ at this point. Knowledse of

the actual chamber pressure would certainl~ increase the precision

of this techniQue. Due to the nonlinearit~ of the eauations, the

value for the chamber pressure, however, could be subject to app-

reciable error.


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6 5

6.4 Pe~formance Parameters

Fronltable 5.2 it can be seen that there is an ap?reci-

able difference in ideal thrust for the two particle flow models.

In the second case where the particles are assumed to have no app-

reciable velocit~, the thrust is 32 percent lower than the first

case where all the particles are assumed to have a velocit~ e~ual

to that of the Sas flow. As a result of this rather sisnificant

variance, the choice of model will importantl~ affect the expected

perfo~mance of a siven motor desisn. Neither model would be ex-

pected to accuratel~ represent the actual flow conditions since

these are the two limitinS cases. The actual particle flow cond-

itions would reauire a consolidaten of the two models, where the

expected performance would lie somewhere between these two bounds.

As well. the actual flow would be difficult to model since the

real particle size would certainl~ be distributed, where the smal-

ler particles misht follow closel~ the Sas flow velocit~ whereas

the larger pa~ticles misht have a sisnificant velocit~ las.

The actual thrust produced b~ the metor would therefore

be expected to lie somewhere between these two ideal values. which

it in fact does. The actual maximum thrust value (1155 N.) almost

coincides with the ideal value for the second particle flow model

(1153 N.), sussestins that the actual flow would be accuratel~

described b~ this model. However, such a conclusion would be hi~hl~

tentative, is not incorrect, since the calculated thrust values

for these two models are almost directl~ proportional to chamber

pressure. The chamber pressure used for these two calculations


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66

(10.55 MPa.) is based on the value arrived at throuSh the pressure-

burnrate calculations described in the previous section which, as

mentioned, could be subject to appreciable error. Ans such error

would be almost directl~ reflected in the ideal thrust compu-

tations. This factor renders the thrust parameter a poor basis of

comparison of the actual to theoretical flow conditions. A far

better comparison would be made uti IizinS the specific im.pulse,

which is insensitive to chamber pressure variations.

The value of the actual avera_e specific impulse (131 5.)

is in fact close to the midpoint of the values predicted b~ the

two particle flow models (152: first model9 103: second model).

This implies that a consolidation of the two models would closel~

predict the behaviour of the actual flow.

The ene rS'ldeff icienc'lde, and the ve Ioc it'ldcoeff i cient r,v

are emplo~ed to relate the actual thrust to the ideal thrust that

would be achieved, as explained in section 3.2.7. The values ob-

tained appear to be on the low side, asain possibl~ as a result

of error in the chamber pressure assumed.

The actual value for the thrust coefficient C , wouldf

also likel'ldlie between the two values as shown. This result

demonstrates that the thrust can be more than doubled with a

properl~ desisned nozzle, with the importance of the diverSent

section apparent.

The nozzle exit velocit'ld v ,is shown to be siSnificantl~e

hiSher for the second particle flow model, where this represents

the Sas only value. The exit velocity for the fi~st model where


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6 7

the particles are assumed to have the same velocit~ as the ~as

flow is reduced as a result of acceleratin~ the particles.

The actual combustion temperature (1347 C.) was found

to be in close a~reement with the theoretical value (1406 C.) beinS

4 percent lower. The sli~htl~ lower combustion temperature would

be a result of incomplete combustion and nonadiabatic conditions.

The final parameters to be considered are the dischar~e

coefficient td and the diver~ence correction A. It can be seen

that onl~ minor thrust losses are a result of these two factors.


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68

CHAPTER 7: CONCLUSIONS

Theoretical anal~si$ of a small solid propellant rocket

motor was conducted, with the performance results presented in

the form of two limiting models. The actual findinss were ex-

pected to lie within this range, and experimental measurements

found this to be the case. This result demonstrated the impor-

tance of considering the effects of two-phase flow for small

solid propellant rocket motors.

Theoretical anal~sis suggested that the propellant per-

formance should remain essential1~ constant over a fairl~ wide

range of oxidizer/fuel ratios, however, other factors such as burn-

rate and castabilit~ reduced the acceptable ranSe. The best over-

all ratio was found to be the presentl~ emplo~ed ratio of 65/35.

Significant improvement of the motor1s performance does

not appear likelw to be achieved bw variation of the nozzle de-

sign. The conical n022le emersed as a hiShl~ satisfactor~ designwhen the simplicit~ of fabrication is considered. Marsinal in-

crease in nozzle efficien~ might be obtained b~ reducing the con-

vergence and/or diveraence anales.

A thrust-time curve of the rocket motor's performance was

obtained, providing a basis upon which to determine the expected

acceleration, velocity, and altitude of the roc~et in actual flight.


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6 9

RECOMMENDATIONS

The followins recommendations are suggested to provide a

more complete understanding of the rocket motor's and propellant's

performance, and to provide a sound basis for the desi~n of

larger rocket motors:

1. Conduct testing in order to determine the actual cham-

ber pressure during firing, possibl~ utilizins a pressure trans-

ducer s~stern to obtain a pressure-time curve. As well' a method

of determinins burnrate as a function of chamber pressure should

be investigated utilizing this data.

2. Other methods of determining the pressure-burnrate rel-

ationship should be considered, such as construction of a constant

pressure vessel to studw the rate of burning at elevated pressures.

3. Investigation into modif~ins the conical nozzle design

in order to reduce particle acceleration misht be attempted.

4. Further stud~inS of two-phase flow in an attempt to con-

solidate the particle flow models would certainl~ prove useful.

As well, it is suggested that a computer prOSram be de-

veloped to solve the eQuations of motion involved with a rocket

vehicle in flisht, utilizins the obtained thrust-time data in con-

Junction with vehicle draS data (either experimentall~ obtained or

theoreticall~ modeled>.


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70

NOMENCLATURE

a burnrate coefficient

a acoustic velocit'ol

A area

A burninsi areab

A/D analosiue I disital

AFT adiabatic flame temperature

* critical throat areac effective exhaust velocit~

c. Celsius

C Coefficient of thrustf

C Specific heat at constant pressurep

C Specific heat of solid (limuid>s

C Specific heat at constant volumev

CV characteristic velocit'ol

D srain inside (bore) diameteri

D Srain outside diametero

e enerSy conversion efficienc~

f mass fraction

F thrust


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71

S acceleration of gravit~

h enthalp~

hO enthalp~ of formationf

I total impulse

I specific impulsesp

k ratio of specific heats

K Kelvin

L grain lensth

m mass flow rate.m Sas mas s flowrate

5 1

.m particle (solid) mass flowrates

m.p. melting point

M Mach number

M' molecular weisht

n burn rate exponent

N Newton

OfF oxidizer/fuel

p pressure

P ambient pressurea

P nozzle exit pressureE ll

P stasnation pressureo

r burn rate

R molar gas constant


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72

R' IJ nivers al !las cons t an t

t t ime

t blJ rn timeb

T temperature

T s t a sna t i on temperature0

v veloci t l : l

v n ozzle ex it ve loc i t l : le

v S ia s v el oc it l: l

s

v par t ic le (so l id) ve loc i t l : l5

V chamber volumec

X part icle (s o l id) mas s fraction

a. divers en ce an s l e

dis chars e coefficien t

v el o ci tl : l c o ef fi ci en t

divers en ce correc tion fac tor

p dens i t l : l

gas den sit l : l

ppo

propel lan t den sit l : l

po

s tas n at ion den s i t l : l


80/93

--- --_ - -- - - _.

7 3

REFERENCES

1. Lancaster, O.E., edt Jet Propulsion EnSines. Princeton, N.J:Princeton Universit~ Press, 1959. Section H:Solid Propellant Rockets, bw Bartlew, C.E. andMills, H.M., P. 613

2. Sutton, G.P. and Ross, D.M., Rocket Propulsion Elements, 4th.ed., New York: John Wile~ and Sons, 1976. P. 47

3. Johnson, C.A. ed Encsclopedia of Chemical Reactions, 2nd.ed New York: Reinhold Publishing Corp., 1953.

4 . McKinnon, D.M., Private Communication, Orsanic Chemistr~Department, Universit~ of Manitoba. 1984.

5. Smith J. and Stinson, Fuels and Combustion, Maple Press, p.100

6. Van W~len, G.J. and Sonntas, R.E., Fundamentals of ClassicalThermod~namics, 2nd. ed., New York: John Wilesand Sons, 1978, P. 502.

7. Free, B.A. and Sarner, S.F., A Rapid Method For Estimationof Specific Impulse, ARS Journal, Januarw, 1959.

8. Gventert, E.C. and Neumann, H.E., Design ofAxis~mmetric Ex-haust Nozzles b~ Method of Characteristics In-corporating a Variable Isentropic Component, NASATR R-33. 1959.

9. Oliver, R.C. and Stephanov, S.E., 'Y Larse or Sn,all". ARSJournal, Oct. 1961.

10. Sutton, G.P Rocket Propulsion Elements, P. 396.

11. The Merck Index of Chemicals and DruSs, 6th ed 1952

12. Ibid

13. Fox, R.W. and McDonald, A.T Introduction to Fluid Mechanics,2nd. ed New York: John Wilew and Sons. 1978. p490

14. Sutton,G.P Rocket Propulsion Elements, p.36

15. Griffin. D.N Turner, C.F., and Angeloff, G.T., A ballisticBomb Method For Determinins the Experimental Per-formance of Rocket Propellants, ARS Journal,Jan. 1959

Documents

VVI Solid Rocket Motor Design and Testing