Minimum propulsion for soft moon landing of instruments

CoA Note No

THE COLLEGE OF AERONAUTICS

CRANFIELD

MINIMUM PROPULSION FOR SOFT MOON

LANDING OF INSTRUMENTS

by

• to i »

T H E

'

NOTE NO, 94

J u l y . 1959

C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

Minimum Propulsion

for

Soft Moon Landing of

Instruments

- by «

D. S. Carton, A.F.R.Ae,S.

F . B . I . S . , M,MS.

COTITFIJITS P a g e L i s t of Symbols 1 , I n t r o d u c t i o n . 1 2, T r a j e c t c r i e s 1 2,-l, C o l l i s i o n Course 2 2 . 2 . H y p o r b o l i o - e l l i p t i c c o u r s e 2 2 . 3 . Tra.jeGtory a n a l y s i s 2 3 , S p e c i f i c a t i o n and a n a l y s i s 3 3 . 1 . S p e c i f i c a t i o n 3 3 . 2 . S p e c i f i c a t i o n a n a l y s i s 3 3 . 3 . P r o p u l s i o n a s p e c t s 3 4 , E x t e r n a l B a l l i s t i c s 3 5 , I d e a l Rocket Engine Performance 6

6 , P r o p e l l a n t S e l e c t i o n 7 6 . 1 , L i q u i d p r o p e l L a n t s 7 6 . 2 . S o l i d p r c p e l l a n t s 7 7, L i q u i d P r o p e l l a n t System Masses a n d S c a l i n g Laws 7

7 . 1 . Combustion cteuriber and n o z a l e 8

7 . 2 . Nozzle 8 7 . 3 . Valves and P i p e l i n e s 8

7 . 4 . T\arbo pump and g a s g e n e r a t o r 8

7 . 5 . Tanlcs 9

8, Solid Propellant Engine Masses and Scaling Lav/s 9

8.1. Engine case 9

8.2. Nozzle 10

9, Analysis - Scaling Constants and Complete Vehicle 10

1 0 , Disciossion 10 1 0 . 1 , S c a l i n g c o n s t a n t s 1 0

1 0 . 2 , R e s u l t s o b t a i n e d - V e h i c l e mass and performance 11

1 0 . 3 , D e f i c i e n c i e s i n t h e p r e s e n t r e s u l t s 12 1 1 , Ooncliosions 1 4 1 2 , Ackno?i/-ledgementa 1 4 Appendix 1 . 15 Appendix 2, 17 Appendix 3 . 19 Continued,

Contents (Cont Appendix 4. Appendix 5. Appendix é, References Table 1 , Table 2. Table 3. Table 4. Table 5. Table 6. Table 7. Figures. inued) • Page, 22 24 26 27 28 29 30 31 32 33 34

UST OP SUvKOIS

A a r e a f t ( i n )

a acceleration ft/sec P

c effective exhaust v e l o c i t y - — ft/sec m

P V

nozzle thrust coefficient e

°F

_X o X ^ c ^^t o p r o p e l l a n t c h a i ' a c t e r i s t i c v e l o c i t y f t / s e c &. P d d i a m e t e r f t ( I n ) P t h r u s t p d l (Lb) 2 g e a r t h s t a n d a r d g r a v i t y 3 2 , 1 7 4 f t / s e o 2

g_ local gravity ft/sec K scaling constants (Defined in Appendices 1 and 6)

L combustion chamber characteristic length ft (in) L length ft (in)

M mass lb Hi time rate of change of mass lb/sec

2 2 P pressure pdl/ft (Lb/in ) r prqoellant oxidant/fuel mixture ratio

T, n . J T T a . j . - T - j . - i- mass of proocllant burnt R solid propellant utilisation ratio -rr-'^—'—T^—r r ,

p J- X- niass of propellant carried 2 2

S material stress pdl/ft (Lb/in ) t thickness of material ft (in)

t- engine burning or firing time sec, V volume f t ^ ( i n ^ )

V v e l o c i t y f t / s e o

P density Ib/ft^ (ih/in^)

7? efficiency factor

L i s t of Symbols (Continued)

e area r a t i o s/A,

e solid propellant loading fraction - — E

p i:- i- o case volume a nozzle divergence half angle

Suffixes

0 refers t o state before f i r i n g 1 refers t o state after firing

c comb\istion chamber, t - throat, e - exit. p propellant ox - oxidant m - m a t e r i a l

T h e equations farmed in the analysis, and t h e values of t h e scaling

constants a r e a l l b a s e d o n t h e p o u n d a l , p o u n d m a s s , second unitary system. I n the text a n engineering system of units is u s e d based on the standard p o u n d force (Lb) a n d p o u n d m a s s (lb)

1, Introdviotion

This paper examines some of the problems that are posed in landing an instrumented package on the moon. The basic requirement is to put down this payload, in one piece, using a miiaimum of initial vehicle mass in the process. Very approximately, a thousand pounds of earth launching vehicle mass are meeded for each pound required by ths moon landing vehicle.

Because of its importance in this application a considei-able paxt of the paper is devoted to outlining an optimisation procedure that will ensure that thrust levels, and other propulsion parameters do in fact result in a minimum vehicle.

As there are no other papers specifically on the problem cf soft moon landings, a certain amount of apace has been devoted to eartiymoon

trajectories, and also various approach and landing techniques. Nevertheless, the main intention has been to look at this problem fran the propulsion

aspect. To this end much has had to be omitted, and many problems simplified or ignored. In particular no examination has been made of either the earth launching vehicle, or the earth-moon oarrier. Very little thought has been given to the problem of guidance and control. The author has not philosophised upon the possible contents of the instrumented package.

The optimisation procedure presented has been formed to permit the assessment of both solid and liquid propellant units in order to determine their efficiency in meeting the specification. Six liquid propellant combinations have been considered. Scaling rules have been derived in order to permit assessment of the relative merits of pressurisation and

turbo pumps, the selection of optiratim tank, and combuatian chamber pressures, expansion ratios, etc. In some instances engineering detail has been

considered to clarify particular points, but t M s paper is not intended as a design study. At a number of points, idiere insufficient data has been found, or i*ere a problem has been intransigent, "bulldozer" methods have been applied,

2 . T r a j e c t o r i e s

The minimum energy manoeuvre converting a circular orbit at 500 miles above the earth into an ellipse with an apogee just at moon radius is extremely susceptible to errors of velocity and direction. Poilowing the work carried out at Douglas (Ref, 1) a trajectory ha.s been selected Tising about A% excess energy which is much less touchy on errors. Figure 1 gives the basic details of such a trajectory.

Such a trajectory and velocity will result in the landing vehicle possessing an energ7 height very close to infinity with respect to the

moon. It is novv proposed to examine some possible moon approach and landing techniques. Broadly speaking, the moon may be approached in two manners -hit or miss,

H Energy height is the sum of potential and kinetic energies expressed in terms of height linits. Energy height equals real height at any instant v.'hen a body is at rest with respect to the gravitational centre,

- 2

2 . 1 , C o l l i s i o n Course

With such a course, without propulsion, the landing vehicle \70uld

impact with veiy nearly escape velocity. Propulsion is therefore required » in a vertical descent bringing the velocity to zero at the instant of

contact,

2.2, Hyperbolic - elliptic course (Pig, Z)

The landing vehicle approaches the moon on a predetermined hyperbola, At closest approach a thrust impulse parallel to the moon's surface can

convert the hyperbola into a grazing ellipse. Through a nijmber of circuits fine thrust control could bring the grazing approach down to a mile or even less. Final landing could then be achieved with reverse thrust reducing the elliptic velocity down to cir jular and on dovm to ZOIHD (Fig. 3 ) . A second rocket directed downv/ards talces most of the v/eight once the radial

velocity has been redxiced below circular,

2 . 3 , Tra,iectory a n a l y s i s

From the p u r e l y propulsion p o i n t of viev7, t h e h y p e r b o l i c - e l l i p t i c

course demands a somew^hat smaller t o t a l v e l o c i t y increment. At t h e same

time i t requires a number of periods of t h r u s t . E i t h e r t h r u s t l e v e l or

hurning time would have t o be f l e x i b l e t o allow for inaccuracies during

manoeuvres. This approach vrauld r e q u i r e accurate a t t i t u d e c a i t r o l T^rith

r e s p e c t t o spherical coordinates based on t h e centre cf the moon (becaiAse

of p o s s i b l e t h r e e dimensional e r r o r s ) . Up u n t i l the moment when t h r u s t i s

applied t h e v e h i c l e ' s energy height i s constant. I t s value can be p r e

-determined, and w i l l be t r u e r e ^ r d l e s s of a l l t r a j e c t o r y e r r o r s . Therefore,

information on t h e height a t c l o s e s t approach defines the i^-perbola and the

negative v e l o c i t y increment required t o convert hyperbola t o surface

grazing e l l i p s e . F u r t h e r height d a t a vrould be adequate for c o r r e c t i o n s and

I t n d i n g .

The c o l l i s i o n course approach r e q u i r e s acoiorate delivery by t h e '

c a r r i e r v e h i c l e . The landing vehicle r e q u i r e s accurate location of two

dimensions normal t o a moon radius and a coarse c o n t r o l on the other axis

t o prevent spinning. (A very criode s e l f - e n e r g i s i n g sim seeker would be

s u i t a b l e ) . I f the t h r u s t l e v e l and burning time a r e both prefixed,

extremely accurate determination of f i r i n g height i s needed. Decreased

donand on accurate i g n i t i o n height can be achieved by use of a very small

supplemental rocket f i r e d low down when height estimation e r r o r s are of «

small r e a l magnitude,

Difference i n height a t two p r e s e t i n s t a n t s towards the end of

f i r i n g , would define t r a j e c t o r y and t h e f i r i n g heiglit for t h e supplemental

r o c k e t , (See P i g , 4 ) .

- 3

-3. Stpecif ication and analysis

The collision course approach is selected. This is because it

requires less rigorous attitude control and is less demanding on propulsion, In fact it requires less operating parts.

Table 1 lists the pertinent moon data required from this work, 3.1. Specification

For a payload of 100 lb, it is required to design a landing vehicle

of minimum mass. No "in flight" acceleration limit is imposed on the „ payload. The landing vehicle, guidance has an acceleration limit of 400 ft,/sec ,

A single stage main propiolsion system is required. No thrust control of magnitude or direction is inquired from the main propulsion system. No arrangements are required for thrust cut-off, other than the normal end of burning,

3 . 2 . S p e c i f i c a t i o n Analysis

With the intention of ensuring that the vehicle T/ill, if anything, have something in hand, final performance estimation is based on meeting the full moon escape velocity of 7693 ft,/sec, assuming that the siirfaoe gravitational acceleration of 5,91 ft,/seo^ is constant at all heights, 3.3. Propulsion Aspects

I t i s r e q u i r e d t o examine i n d e t a i l t h e main propulsion system components, masses and performance, i n order t o determine t h e method which can meet t h e s p e c i f i c a t i o n i n terms of v e l o c i t y increment and minimum v e h i c l e mass. The study i s t o include t h e use of both p r e s e n t l y a v a i l a b l e , and higher performance p r c p e l l a n t s ,

4» E x t e r n a l B a l l i s t i c s

I n order t o a s c e r t a i n t h e approximate mass of t h e v e h i c l e , and i t s s e n s i t i v i t y both t o exhaust v e l o c i t y and coniponent masses, t h e concept of an i d e a l v e h i c l e i s introduced, (See Appendix l ) .

This v e h i c l e has a constant e j e c t i o n r a t e of p r o p e l l a n t i n terms of mass flow r a t e and exhatist v e l o c i t y r e s i i l t i n g i n a constant t h r u s t th-^oughout the f i r i n g p e r i o d . For such a v e h i c l e i n v e r t i c a l descent (v/ithout atmosphere) t h e v e l o c i t y r e l a t i o n s h i p •

-^o

Av = o logg J T - ©L *b f * - / s e c , 1.5 1

can be used. In order to relate the burning time (t, ) with the initial M

4

-effeots, the ideal vehicle definition is extended. The all burnt mass of the vehicle ( M ) is taken as the sum of the payload, a mass scaling

c

in direct proportion to propellant mass flow rate, and a mass scaling in direct proportion to the total mass of propellant carried,

\ = ^1 +*p*b ''•2

The Kp value will relate to tank mass in the liquid propellant engine or to the engine case n>ass in a solid, K. relates to pipelines, valves, and combustion chamber for liquids,

K

In terms of the payload to initiaL mass ratio rr- these equations

may "be brought together ; - o

i : = ^ - (1 -

= 1 - 0 -

ïrr-)

Trr-) (-r-

(T-

+ K , ^ . I ) 1.6

+ -2

o o/

where __o and t , a r e r e l a t e d as in. equation 1,5.

I t i s nov7 p o s s i b l e t o determine the payload t o i n i t i a l mass r a t i o

for any v a l u e s of the e f f e c t i v e exhaijat v e l o c i t y - ^ c ) , and t h e scaling

constants K. and K„, A nunjber of these ' i d e a l v e h i c l e ' curve» are

presented. In Figure 5 __o has been p l o t t e d against f i n a l v e h i c l e

a c c e l e r a t i o n for t h e following values •

-o = 10,000 arid 8,000 f t , / s e c ,

K, = 0

Kg = 0,08 and 0,16

I t i s assumed t h a t t h i s i s r e p r e s e n t a t i v e of a s o l i d p r o p e l l a n t landing

v e h i c l e both with reöpeot t o exhaust v e l o c i t y and K v a l u e s . The curves

a l l show t h e seme t r e n d - t h a t minimum i n i t i a l mass i s a t t a i n e d with

very high f i n a l -ÉLCOderation conditions obtained with conrparatively high

t h r u s t and short burning time. This i s due t o K. = 0, t h e r e being no

mass on t h e vehicle proportional t o flow r a t e , hence the v e l o c i t y l o s s

due t o g r a v i t y ("ET t, ) can be minimised. The other point t o be noted i s

t h e s t i s c e p t i b i l i t y of the MV^. r a t i o t o change of eodiaust v e l o c i t y and

i t s comparative i n s e n s i b i l i t y t o change of K. :

5

-M A

1 % change i n c 1.25^2 change i n ' l i

M A .

^% change i n K. 0,3% cliange i n ' TJ

Figure 6 r e p r e s e n t s t h e equivalent l i q u i d propellant curves for

t h e follov/ing :

-o = 10,000 and 8,000 f t , / s e -o ,

K. = 5 and 10

Kg = 0,02 and 0,04

ML.. A .

An immediate difference is apparent, ^ o is no longer minimum at vory high thrust level and final e.cceleration. This is due to K , there

being a mass on the vehicle proportional to propellant mass flov7 rate and thrust. Hence a minimum value of M j ^ is achieved at a point intermediate to a low thrust, high g loss and a high thrust, and a large mass proportional to fla; rate. It is v/orthy of note that at the higher values of K. on

optimisation procedure is most important, a minimum o/lvL is most sensitive to vehicle final acceleration and thrust level. Once again the large

sensitivity to e;-±iaust velocity shox-O-d be observed plus the comparative insensitivity to changes in either K. or Kp

:-M /. ^% change in c 1,25^ change in ^ X ^% change in K 0,105^ change in o/i»L. ^fo change in Kp 0,06^ change in ^o/K

Using these graphs permits the approximate thrust sizes at v/hich scaling rules and real constants can be determined. Values selected

are:-Solid P = 1i+29 lb. = 46,000 pdl, Liquid F = 267 lb.

6

-5, I d e a l Rocket Engine Performance

The d e f i n i t i o n of t h e i d e a l vehicle i s extended t o include the f a c t

t h a t i t i s po7rered by an i d e a l rocket engine. Such an engine i s defined

as follows : - i t operates v/ith perfect gases which a r e in equilibrium

i n the combustion chamber, and v/hose composition does not a l t e r through

t h e expansion. The cycle i s i s e n t r o p i c , and the gases leave the nozzle

Tirithovit divergence l o s s . There i s no e x t e r n a l pressure. (See Appendix 2),

The thmast of such a rocket operating i n space i s given by :

-^ = % -^e --^ -^ e -^ e -^-^'-^ 2.5

It is now required to relate equation 2,5 i.o the propellant performance

parameter - the characteristic ve^-ocity (c ) , the engine gecanetry parameters, throat area (A.) and exit/throat area ratio,

The relevant paranÊters based on the ideal rocket definition are

:--

P

A

.

c^ = - 2 — ^ ft./sec, 2.1

m

_p

°P = ^ 2.2

c

1 Jfti

/P ^

*• ••&)' 4 ( * r ••••••-'

These combine and simplify to give the effective exhaust velocity

:-o = ~ - = c''(Cj^, + ^ e ) ft,/sec 2.8

P °

from which may be derived directly the thriost and propellant mass flov7 rate relationships,

Also may bo derived the expression for throat area

:-Once a propellant has been selected these parameters may be xised to

7

-6. P r o p e l l a n t S e l e c t i o n 6 . 1 . Liquid p r o p e l l a n t s

It is required to examine the landing veliicle performance for a number of prcpellants including both presently available and possible future combinations. The fundamental need is that the combination shouJ.d be storable from the moment of loading on earth, through the launcliing pliase, and then from 2 to 5 days thereafter Nevertheless, it is felt that this should not override consideration of any attractive propellants as storage m.eans may be found. In consequence six propellant combinations have been considered

:-99.6^ hydrogen peroxide \rxth. kerosine and also hydrazine, Liquid oxygen \/ith kerosine, unsymmetrical dimethyl hydrazine and liquid hydrogen.

Liquid fluorine Vidth liquid hydrogen.

Hydrogen poroxide/hydrazine has been chosen as an excellent example of a normally storable propellant combination, 99.6?§ peroxide has a somevAiat better performance with hydrazine than the other storable oxidants - dinitrogen tetra oxide and nitric acid - and is thought to be available and in use, Peroxide/hyirazine are a self-igniting pair,

The other selections are a fairly obvious choice from the field of interest. Fluorine/hydrogen are the only other combination v/hich are self-ignit ing,

Table 2 lists the relevant data for these combinations, 6.2, Solid P r o p e l l a n t s

Insijfficicint u n c l a s s i f i e d d a t a a r e avai],able t o ex-bend t h i s study i n t o t h e r e a l f i e l d of s o l i d p r c p e l l a n t s . I n consequence i t has been found necessary t o define a n o n - e x i s t e n t p r o p e l l a n t liaving a performance thought t o be i.dthin reach of t h e b e s t p r c p e l l a n t s e x i s t i n g today. This i s t h e only s o l i d p r o p e l l a n t s t u d i e d i n t h i s paper. Table 3 l i s t s t h e r e l e v a n t d a t a ,

No suggestion i s made as t o i t s p o s s i b l e ccmpositian, 7. Liquid PropelD.ant S;ystem Masses and Scalin.fi; laws

Scalijig lav/s have been derived f o r the major components of tlie complete l i q u i d p r o p e l l a n t system. These a r e needed so t h a t a bett-i^r apxjreciation can be mad.e of a number of problems which a r i s e . The laws then permit a more a c c u r a t e determination of t h e v e h i c l e c o n s t a n t s ,

8

-7.1. Combustion chamber and nozzle (Sv.e Appendix 3)

Some consideration of a number of different combustion chamber shapes \ms made. Mainly, from the vier/point of simplifying the analysis a spherical shape v/as decided upon, A very interesting alternative, the diverging reactor (Ref, 2) was seriously considered (Fig. 7 ) .

Unfortunately the scaling rules for such a system did not appear reliable, In consequence it is not considered further,

A volume scaling lav; is required for the siiherical cliambers. The combustion chamber volume/throat area ratio L (the combustion chamber characteristic length) is used in this paper. This parameter has not been selected because of its reliability, but in the absence of any alternative. An L'^ of 5 ft. (60 inches) is used throughout this paper.

The basic design assumed is that the chamber is cooled by the usual double wall arrangement. It is assumed that the outer vrall carries the entire pressure load. The inner vjoll takes the biirsting load due to

excess coolant pressure. On the inside v/all of the chamber is an unstressed, flame sprayed heat barrier of zirconia. All chambers are made of stainless steel, and a safe working stress of 7^% (150,000 Lb/in^) of normal has been used because of possible outside wall heating. In addition, a safety

factor of 1,5 has been used. This uncertainty factor lias been used to make allovTanoe for the nozzle cut out.

7.2. Nozzle (See Appendix 3)

The ideal rocket definition includes the assimiption of no thrust loss due to divergence. It is therefore assumed that a "tulip" nozzle is used. As the profile of sioch a nozzle is complex, no attempt has been made to estimate its scaling laws. Instead a "similar nozzle" has been defined v/hose mass, throat, and exit areas are identical to the tulip t;?rpe. The "similar nozzle" is straight sided v/ith an expansion half angle of 15 , The nozzle is continuous with the combustion cl:iamber, has the same materiüls, total thickness and safety faj3tors,

7.3. Valves and Pipelines (See Appendix 4)

Precise scaling laws would require a number of separate system detail designs. In order to circumvent this it lias been vagiely assiAmed that all systems are similar and that the mass of valves and pipelines is proportional to the propellant volume flow rate,

7.4. Turbo pump and gas generator (See Appendix 4)

Again, it has been necessary to generalise. It is assumed that the masses of gas generator, turbine, case and sliaft are proportional to the total propellant pumping vrork rate. Pump masses are assumed proportional

s

-7 , 5 , Tarilrs (See Appendix 4)

A simple s c a l i n g law has been derived for tanks s t r e s s e d a s t h i n

spheres, ViTier^the same m a t e r i a l i s used for both t a n k s , and no problems

a r i s e from ' s t o r a b i l i t y ' a s p e c t s , t h i s i s d i r e c t l y a p p l i c a b l e ,

Consideration has been given t o the s p e c i a l problan of s t a r i n g t h e

l i q u i d s - oxygen, f l u o r i n e , and hydrogen. Attention has been given t o t h e

problem of tank volume requirements a t the end of t h e voyage when the

propellant teinperature w i l l be d i f f e r e n t from the loading temperature.

This problem i s extremely s e r i o u s v/ith t h e t h r e e p r o p e l l a n t s mentioned

due t o t h e i r loi/7 c r i t i c a l p r e s s u r e s and teirperatijres. Very l a r g e changes

of vapour pressure and d e n s i t y with temperature occur a t c e r t a i n p o i n t s ,

The main question t o be ansv/ered i s t o what extent heat b a r r i e r s shoxild be

used t o keep p r o p e l l a n t t e n p e r a t u r e down and hence keep down p r o p e l l a n t

d e n s i t y and r e q u i r e d tank volume. The importance of t h i s gan b e underlined

by mentioning t h a t f o r a temperature r i s e from 90TC t o 1 5 3 ^ t h e volume

requirement f o r l i q u i d oaygen doubles !

Examination of t h i s problem ha.a not yet been t a k e n ftarther. I n

t h i s paper, s c a l i n g r u l e s have been developed for t h e t h r e e p r c p e l l a n t s ,

based on a crude assimption; for l i q u i d oxygen, and f l u o r i n e , t h e tank

mass i s double the mass r e q u i r e d f o r storage \7hen t h e vapour pressiare i s

equal t o one atmosphere. This i s p e r f e c t l y reasonable for t h e p r e s s u r i s e d

tank requirement since vapour p r e s s u r e can be used f o r motivation. For

hydrogen, because of i t s very low c r i t i c a l ternperature and p r e s s u r e , a

value of 4 tes been teJcen. I t i s roughly assumed t h a t t h i s value accounts

f o r both e x t r a tanlc volume r e q u i r e d due t o temperature r i s e of t h e l i q u i d ,

p l u s a tank i n s u l a t i o n . I n order t h a t comparison can be made, s c a l i n g

r u l e s have a l s o been derived for the t h r e e p r c p e l l a n t s , assuming storage

a t one atmosphere vapour pressure v/ithout excess volume or heat b a r r i e r ,

8, Solid Propellant Engine Masses and Scaling Laws (See Appendix 5)

Scaling 3^ws have been derived for both engine case and nozzle,

8 , 1 . Eng^ine case

» I l é l l M I I • W L W C ^ M l ^ M *

The engine case i s a c y l i n d e r , and i n t h e s c a l i n g , allovjance has been

made for both c i g a r e t t e burning and case bonded i n t e r n a l burning a r r a n g e

-ments, Both have been s t r e s s e d a s t h i n c y l i n d e r s ,

With the case bonded arranganent i t i s assumed t h a t very l i t t l e case

heating occurs, and a value of S = 75^ = 150,000 L b , / i n has been used,

The propellant u t i l i s a t i o n r a t i o (R) = 0 , 9 , t h e loading f r a c t i o n 0,8

and t h e s a f e t y f a c t o r a t 2,

I n comparison t h e w a l l heating e f f e c t s i n t h e c i g a r e t t e burning

10

-heat b a r r i e r , but i t should be appreciated t h a t the -heat e j e c t i o n r a t e of

t h e outer ca.se i s due e n t i r e l y t o r a d i a t i o n i n t h i s p a r t i c u l a r a p p l i c a t i o n ,

A value for S of 54^ = 108,000 Lb./in^ has been taken. Coupled v/ith 77 = 2,

t h i s r e s u l t s i n a case mass for the c i g a r e t t e burning a p p l i c a t i o n tvooe

t h a t of the case bonded one.

8.2. Nozzle

An alternative model has been used to derive the scaling relation ship, As the nozzle is tincooled, allov/anoe his been made for a zirconia heat

barrier. Material strength is assumed to be dovm to 54% of maximum working stress. This is the same figure that was used in a similar situation for the cigarette burning arrangement of case.

9, Analysis - Scaling Constants and Complete Vehicle

The analysis of the scaling rules of the various engine components has revealed tha^t in both liquid and solid propellant systems, nozzle mass is a function of the propellant mass flay rate to the 4 rd power. This means

3

that the simplified analysis of Appendix 1 has to be extended to include this factor. It is important to note that in space operation the nozzle mass is several orders of magnitude greater than the conbustion chamber alone, Considerable errors can, therefore, arise from the assumption that the complete thrust chamber scales in proportion to propellant mass flow rate,

The complete vehicle analysis is treated in Appendix 6. A simple

graphical method of determining vehicle characteristics for a given propellant combination is described.

10. Discussion

10,1. Scaling constants

The scaling laws discussed in sections 7 and 8 have been used to derive the scaling constants K,, K_, and K, for the solid propellants and all the liquid propellant combinations for ijoch pressurised and tiorbo/punrped systems. These are all listed in Table 7. These tr.bles ajre enlightening in that they give immediate information on component sizes. The K. values detail component mass per unit mass floxv rate, the Kp values the tank (or solid propellant case) mass per unit mass of propellant carried. The K, values give the nozzle mass for unit mass floi7 rate (but scales away from this proportional to mtp •^) ,

^J^ilst it is interesting to note these values, particularly the effects of using lew density propellants, they cannot be used for dij?ect comparison in tha.t they do not take into accoxint either propellant performance, or the optimisation procedure,

11

-10.2, Results obtained - Vehicle mass and performance

At the moment of cairpletion of this note, it has been found possible to complete the analysis on six sets of 'vehicle systan/propellant combination'

scaling constants. These are for the propellant combinations of hydrogen peroxide and hydrazine, and also liquid oxygen and kerosine, Each is

considered for both pressurised, and turbo pumped systems. In addition, with the liquid oxygen combination, both systems have been calculated for the tv/o

storage methods lased for cryogenic liquids in this paper One is based on storage at a maintained temperature of 90 K throughout the journey, v/ithout any mass allo-vvance for preventing heating, the second, and more realistic case, assumes that the oxygen teraperature rises through the journey to 153 K and has in consequence a density one half of that taken in the previous case.

The results are plotted on an initial to payload mass ratio — against vehicle final acceleration (a ) in Pig, 8. The most remarkable

• M ^

point about the results so far obtained is the narrow range of minimum rr-M

covered by all the groups, there being less than 0,1 — between them. The actual minimum values are as f ollov/s :

-Oxygen and kerosine. Pressurised (at 153 K ) rr- = 2,396

Oxygen and kerosine. Pressurised (at 90IC) = 2.316 Hydrogen peroxide and hydrazine. Pressurised = 2,320 Oxygen and kerosine. Pumped (at 153 K ) = 2,310 Oxygen and kerosine. Pumped (a^t 90 K ) = 2.305 Hydrogen peroxide and hydrazine. Pumped = 2.302

Consider first the pressurised systems, and accept the scaling procedure as correct. The storable propellant combination shows the expected advantage over the other v/hen allov/ance is made for extra oxygen tankage. I'Then no allo\7ance is made for the extra tank volume (again, is a most unlikely possibility) the oxygen combination aharrs an extremely small advantage.

With the turbo pumped systems, tank mass is a very small part of the complete system mass. Because of this the differences betv/een the three systems are ins ignif icant,

In conclusion it is felt that considerable caution should be exercised in using these results to make comparison betv/een pressurised and turbo pumped systems. The small apparent improvement obtained using tvirbo pumps

is entirely d.ependent on the precise val.ues given to the scaling constants, and oould be upset by very small changes in their magnitude,

12

-The solid propellant analysis has not been proceeded v-lth at the present moment, as a spot check revealed that it Vi'as considera.bly outside the range

of interest. The value derived is

:-M

Solid propellant (Case bonded) — = 2,70

Ti

10,3, Deficiencies in the present results

Each of the 22 sets of constants given in Ta.ble 7 represents a different 'vehicle systen/propellant combination' group. Each group may be analysed in order to determine the engine operating values of thrust and biiming time that together meet the specification, and result in a minimum vehicle mass for a given payload. S\jch an appr-ach v/ill only result in an absolute minimum mass vehicle for a given group v/hen all the possible changes in

operating parameters have been considered. The more important of these operating parameters are nov/ discussed.

10.3.1. Ccmbustion Pressure.

With the scaling rioles it is possible to derive further sets of constants for a number of combustion pressures. It is then possible to ascerta,in the optimum pressure for each 'vehicle systen/propellant ccmbina.tion' group.

Because the data relating to the combustion pressure, propellant chara.cteristic velocity available to the author at the present moment is ina.dequate it has not been possible to proceed further. IThilst inter-rela.ted data is required over a v/ide pressure ra-nge, partictilar interest centres on the possibility

of operation at very lew pressure values. In this paper liquid propellant p sjratons are given a value of 500 Lb/in^, and the solid propellants at 1000 Lb/in . There is no rea.son for assuming tha.t these represent optimum for the conditions

considered.

10.3.2. Expansion Ratio

Again the scaling rules permit the examination of the relative effects of changes in nozzle mass and effective exhaust velocity. Ignoring problems of nozzle rigidity there v/ill exist an optimum expansion ratio which v/ill be different for each set of scaling constants and propellant parameters,. Data is available to do this, but it has not been found possible to include the problem vri.thin the investigations so far undertaJ:en.

Throughout this paper an expansion ratio of 10,000 has been used. Tables 5 and 6 indicate chanp-es of nozzle parameters a.nd effective exhaust velocity for a constant thrust nozzle based on the propellants used in this paper, 10.3.3. Mixture ratio - liquid propellants

Examination in this paper lias been limited to a single mixtxrre ratio for ea.ch propellant combination. The ra.tio used is the one producing maximum

~ 13

-effective exhaust velocity in an eaxth surface environment T.dth expansion to one a.tmosphere. Higher effective exha.ust velocities may be attainable v;ith slightly different mixtures due to the larger expansion ratios being used. More important th.an this is the effect of mixttire ratio on component mass. For example, \7ith the use of liquid hydrogen it appears obvious that a mixt'jre ratio deficient in hydrogen v.'-ould result in an optimum combination of tank mass and effective exhaust velocity, To a lesser extent tliis will be truefor all propellant combinations.

The three effects, of combustion pressure, expansion ratio and mixture ratio, require integrated consideration v/ith the system in order to determine their combined effects on vehicle mass,

10,3.^4. Solid propellant burning rate

The question of v/hether to use the cigarette burning a.rrangement, or the less ma.ssive case bonded internal bijrning conduit system cannot be

decided \;lthout a^dequate data on the surface bvuning rates of real propellajnts. As tliose ajre not available in the open literature (for the high performance propella.nts dema.nded by the moon landing application) the problem has ha^d to be left unsnalysed.

There is, hov/ever, one aspect v.'orthy of attention, a^nd that arises from the possible need for physical methods of controlling burning rate, The discussion here is based on the assumption that high performance

propellants tend to ha»ve very fast burning ra.tes, and the fact that in tliis application, v/ith burning times of 30 seconds or so, slov/ burning ra^tes are needed to utilise the case bonded layout. It is reqiaired to examine the situation in order to determine the possibility of decreasing burning rate by environment al c ont rol,

Two parameters are available v/hich appear v/orthy of consideration, Since bi.irning rate is a function of combiostion pressure, lev/ pressure opera.tion should help towards a solution. In itself the lov/ combustion pressure need not affect performance since a potentially infinite expansion ratio is a.vallable. It also a.ppears possible to achieve low burning rates by use of propellant grain at lov/ temperature at the mom.ent of firing, It might in consequence be v/orth v/hile storing the grain in a refrigerated state during the intervening journey,

14

-11, Conclusions

The main aim of this paper hr^s been to investigate the approximate size scale for a moon landing vehicle and its propulsion system. The v/ork so far canpleted, based on a 100 lb. pa,yload, indicates that some 230 - 240 lb, of landing vehicle are required. Such a project v/ould need an earth launching vehicle in the 250,000 - 350,000 lb. mr.ss range. Although these figures arc based on a 100 lb. payload the proportions are scalable over quite a v/ide range, and can in consequence be applied to other proposals made in the light

of more detailed study on a.ctual payloa^d requirements, for example Gatland's 'Migrant' study vehicle,

In order to meet the need^s of tliis paper it v/as found necessary to develop a vehicle and propulsion system scaling procedure, A great deal of work is still required to be carried out ir. order to fully utilise the method derived. At present, 22 sets of scaling consta.nts have been obtained for six liquids,

and one hypothetical solid propellant. The full procedure for deterniining the minimum vehicle mass has only been applied to six sets of these scaling constants. In addition to the completion of the study discussed in thia paper more v/ork is required on the conbined effects of combustion pressure, expansion ratio and mixture ratio. It is hoped that computer time may be made available in order to continue ajid i^dden the various aspects still requiring consideration,

The work so far completed shov.'s that there is a considerable mass

advantage obta.ined by using liquid propellants. In spite of this the possible reliability advantages of the solid unit cannot be ignored. From the liquid propellant aspect, therefore, studies are needed in the field of lütra simple

systems, probably based on single explosive valve operation.

Because of the inadequate data availa.ble on the subject of high performance solid propellants existing today, and their possible improvements in the future,

it has been found impossible to do justice to this side of the study. In consequence the inter-related problems met in trying to obtain the optimum balance between performance requirements, and component masses has had to be

ignored.

12, Acknowledgement s

The author v/ishes to express his tha.nlcs to Professor A. G, Smith for permission to initiate the moon landing studies, and to Mr. T. G, Andrev/s for his very considerable help in the calculations involved,

1 5

-/•PPENDIK 1,

SBglJFIED AMALrSIS. IDEAL VliHICLE, VERTICAL DESCENT

The ideal vehicle is defined as having constant mass flov/ rate of prooellcint, effective exhaust velocity and thrust. The all burnt mass of the vehicle is expressed as the sum of the payload, a mass scaling in direct proportion to propellant mass flow rate (e,g. analogous to a. liquid propellant thrust chamber, pipe lines and valves) and a mass scaling in direct proportion to the total mass of propellant carried (e. g, liquid propellant system tanks or a solid propellant combxistion chamber case),

Then

M^ = M^ + K^ ihp + K2 Ap t^ (lb.) 1,1

M^ = M^ + ftp t^ (lb.) 1.2

Since engine thrust F = ft c (pdl) the final or maximum acceleration of the vehicle is

s ' k = V- ^^•/'•^^^ '-^

1 1 ^ -E (ft./sec'') 1.4 M - A t, o P b

For a v e r t i c a l descent t h e v e h i c l e v e l o c i t y increment may be -v/ritten

\ —

Av = c log^ ~

-

W (f-fc./sec) 1.5

1

It is required to determine the variation of the intial/payload mass ratio for various values of the constants c, K. and K^. This

' ^ 2

may bo done a t a number of a v a l u e s , (Alternati.vely values of F , or ft coxild be s p e c i f i e d ) . The i n i t i a l / p a y load mass r a t i o may be expressed i n terms of t h e constants and a. using equations 1,2 and 1.4.

"t.

_{= 1 - - ^ \ t^}

[ K I * \ ( K 2 . I ) ]

... 1.S

M - c o

The initial/final mass ratio ma-y be treated in the same manner using equations 1.1, 1,3 arjd 1.6

M^ r a^ -t M

1 6

-Por each selected value of a the biorning time (t. ) value must satisfy equation 1,5 for the velocity increment (^) and mean local gravitational acceleration (gT)involved. This step may be carried out graphically or otherv/ise,

M

Plots cf ~ (and M for a payload of 100 lb,) vs, a for the following

\

° ^

values of the constants

Av = 7600 ft,/sec.

c = 8000 and 10,000 f t , / s e c .

K, = 0 , 5 and 10 sec

1 '

K2 = 0,02, 0,04, 0,08 and 0.16 are given in figures 6 and 7.

17

APPEl^OrX 2, IDK1L ROCEir SFGIKE

In this paper an ideal rocket engine is defined as operating v/ith a perfect gas. The gas is in equilibrium in the combustion chamber. Its composition does not alter througli the expansion process. The usual ideal cycle assuniptions of zero friction and heat loss are made. Also the gas l&aves the exit v/ithout divergence loss and enters an environment of zero pressure.

It is required to determine the mass flov/ rate of propellant, the effective exhaust velocity, throat area and exit/throat area ratio for engines of specified thrust, A number of operating conditions in terms of

p

combustion pressure (P ) and expansion pressure ratio o/P are to be ° X ^ investigated. The characteristic velocity o and y have been used as baèic parameters, as a number of different propellant combinations are to be considered.

The follov/ing equations are derived from nozzle flo\/ and the ideal engine def init ion

:-P A. , o t * A pdl. ft^ ft sec. lb. ft./sec. 2.1

P

_e X 2,2

which expands to

M. r

id;"-i

1

-{'r

a/ 2.3

Also

7 - = e

^^t

1 J ^

2.4

e/ S'

V^V

The t o t a l thrust of an ideal engine operating in space i s given by :

18

-Which, expanding and substituting from 2,1, 2,2, and 2,4 gives

:-P

P = A c"" (0^+ ~ e ) pdl, 2.6

^ c

/ . A =

^--

Ib./seo, 2.7

c

p

and o = ~ = c^ (C_ + ~ e) ft./sec. 2,8

%

Also from 2.1 and 2.6

19

-APPSIVDIX: 3.

MASS OF THRUST CHAtiBER - UQUID PROPELLANT ENGINES

The term thrust chamber is taken to include combustion chamber, nozzle and injector,

1 . Combvxstipn Chamber Shape

A l l chambers SJTO s p h e r i c a l v/ith smoothed e n t r y i n t o t h e n o z z l e .

2. Combustion CIvamber Volume

At the present moment adequate scaling methods are not available.

Since all chambers are the same shape L is assumed a valid scaling concept. In the absence of adequate data a value

X "^o

L = T— = 60 in. 3,1 ^t

is used throughout the calculations,

3 . Chamber Construction

A l l chambers are cooled. Construction i s assijmed t o be of

conventional double wall v/ith coolant enclosed. Both walls a r e taken

as of equal tliickness. Because walls are very t h i n , nucleate b o i l i n g

might occur. A l l chambers a r e t h e r e f o r e assumed t o have an inner heat

b a r r i e r coating of z i r c o n i a having a mass per u n i t area equal t o t h e

s i n g l e Trail onto v/hich i t i s a t t a c h e d (by flame spraying),

4 . S t r e s s i n g

It is assumed that the chamber outer wall carries the entire pressure load and is everywhere below 550 JK, The inside \7all takes the inward bursting load due to the excess pressure of the coolant. The heat barrier

is unstressed. 5. Injector

No specific allowance has been made for the injector. Rather hopefully it has been assumed that the absent mass at the throat could be acoovmted to this. This is not serious since both injector and combustion chamber scale approximately on propellant mass flow rate. It is assumed that adequate allovranoe has been made,

6. Ccmbustion Chamber Mass

I t i s r e q u i r e d t o r e l a t e combustion chamber mass with t h e performance

parameters, t h r r i s t , nozzle t h r u s t c o e f f i c i e n t , combustion p r e s s u r e ,

area and p r e s s u r e expansion r a t i o s and the p r o p e l l a n t c h a r a c t e r i s t i c

v e l o c i t y . From geometry of a sphere :

20 -1 . 3

^C = V""^/ ^'^

/6 V^ \ ^

, * . surface = v \-—^ J 3,3

/ 6 V \ ^

, • , mass = '"'V'Tr') P^^„. 3 . 4

\ ir / m m

f o r a single t h i c k n e s s ,

S t r e s s i n g as tliin sphere

P T? d «5 o c , c

m

S u b s t i t u t i n g 3 , 1 , 3.2 and J>,^ i n 3 , 4

D

mass = 1.5A. L P r ; - ^ 3.6

"6 C o

for a single thickness.

Since in the combustion chamber there will be tv/o sheets of metal and one of zirconia, each of equal mass per unit area

M^ = 4.5 A ^ L ^ P^ r, 'f 3.7

•which, using equation 2,7 t o eliminate A. gives

V 4 . 5 L^ 77 F p^

c

I t should be noted tlïxt t h e value of S should be an a v a i l a b l e maximum

v/orking s t r e s s for a teraperature of 550 K, The value of n v/as chosen

a f t e r some thought as 2 , 5 , I t was intended with t b i s r a t h e r high value

t o make allov/ance for the f a c t t h a t t h e e f f e c t s of s t r e s s i n g t h e i n j e c t o r

and nozzle had not been taken i n t o account,

7. Nozzle Mass

In order to eliminate thrust losses due to divergence a "tulip" nozzle profile has been assumed in the performance calculations. At a given combustion pressure the profile v/ill change in a complex manner v/ith different gas temperatures and conqposition. It is not possible

21

-therefore to malce a reasonable assessment of nozzle mass for the actual profile. It has therefore been a.ssumed that the real nozzle mass is equal to a "similar nozzle" having the sam.e throat ajid exit areas, but vdth a complete divergence tuigle of 30 ( 2 a ) .

Fi-om geometry it can be shewn that the surface area of such a shape is

A.

The mass of the nozzle has been based on t h e assumption tha.t t h e

equivalent tliickness i s the same as the combustion chamber, i . e . t h r e e

times t h e c a l c u l a t e d "cold s t r e s s " t h i c k n e s s . No allowance has been

made for the p o s s i b i l i t y cf decrea,sing thicloiess tov/ards the e x i t . Nor

has any allovvrance been made for the fact t h a t cooling may not be required

neaj? the e x i t . Equation 3.9 niay be rev/ritten using equations 2 , 7 , 3.2

£^d 3,5 :

-M

n

0.75^ ( e - 1 ) Pm

_{s i n a S y-rr T ) '" ^'^^}

( L É S

3 10

8, Thrust Chamber Mass

The mass of t h e complete t h r u s t chamber i s the sum of equations 3.8

and 3.1 O.Throu^iout t h i s paper s t a i n l e s s s t e e l , vidth a v/orking s t r e s s of

150,000 Ik>,/in.'^ a t 550 K, has been lised. Combustion chambers f o r p r e s s u r i s e d

and turbo-pumped systems have been taJcen as i d e n t i c a l ,

A l l chambers have been based on a dimensional design t h r u s t of 267 l b ,

for P^ = 500 L b , / i n 2 ,

22

-APPSNpn: 4.

COMPONEMT mSSES - LIQUID PRQPEI.LANP SÏSTE^.^S

It is required to calculate the masses of the various components which may occur in a liquid propellant system, A method is required which will permit an assessment of the performance of vehicles operating with varioiAS propellants at selected combustion pressures, using either pressurised tamks or turbo pumps for propellant motivation,

1. Tank Mass

Tanks a r e s p h e r i c a l . Using an i d e n t i c a l method t o t h a t o u t l i n e d

i n Appendix 3 , equations 3,2 t o 3 . 5 .

1.5 Pn, rj p

^ = %*b — ^ -I ^'

Equation 4.1 can be used d i r e c t l y f a r the mass of both tanks i f t h ^

are of t h e same m a t e r i a l ,

Equation 4.1 i s used as i t stands for p r o p e l l a n t s which present

no thermal insula.tion problems, and use t h e same material for each

p r o p e l l a n t . ( e . g . Hydrogen peroxide and hydrazine both use s t a i n l e s s

s t e e l ) ,

li/here d i f f e r e n t materials are required the eqiHtion i s modified t o

^ = ''^W^T "

r-l \-p Sj^^* r-1 VP sy^^^j

r I ^ ^m\ 1 / 1 ^m^

4.2

Throughout t h i s paper tank pressvu^es ha.ve been taken a s

P j = (200 + P^) Lb,/in^

= 700 Lb./in^

2

for p r e s s u r i s e d systems and 50 L b , / i n for pumped systems. V has been

taken as 1,5 i n a l l c a s e s .

Equation 4.2 may a l s o be ijsed i n the c a l c u l a t i o n of tank mass for t h e

p r o p e l l a n t s , oxygen, f l u o r i n e , and hydrogen, where e i t h e r considerable

d e n s i t y cliange or heat b a r r i e r i s recxuired. In t h i s paper i t i s assumed

t h a t for oxygon and f l u o r i n e , the d e n s i t y at the end of the voyage v/ill be

h a l f t h e loading d e n s i t y , v/ithout any heat b a r r i e r . For hydrogen, a f i g u r e

of l / 4 loading d e n s i t y has been used, of v/hich h a l f the e x t r a tank mass

i s considered due t o r e a l d e n s i t y change, the remainder due t o some unspecified

heat b a r r i e r .

23

-2, Valve and Pipeline Mass

It is assumed that this is directly proportional to propellant

volume flov/, then

:-A

IL. = 90 -rE lb. ... 4.3

P

The value of t h i s constant and that i n 4 . 4 have been based on f i g u r e s

given by Baxter (Ref, 3 ) .

3 . T\jrbo/pump and Gas Generator Mass

I t i s assumed t h a t t h e gas generator, t u r b i n e , shaft and casing have

a mass dir'ectly proportional t o the r a t e a t v/Hoh pumping v/ork i s being

done on the p r o p e l l a n t s

M = 1,83 A A P X I O ' l b . 4 . 4

g P

The pumps and t h e i r cases axe assirned t o have a mass proportional t o

propellant volume f Icn// ra.te

A

M = 10 -E lb, 4.5

PU P . -r.^

P

2

The value of AP in equation 4.4 has been taken as (P + 200 - P_)Lb/in

C JL 2

24

APPETOIX 5 .

M ^ S OF Tlg^UST CauoBER - SOLID PROPEUANT UNIT 1 , Case Mass

N e i t h e r s h a p e , nor len^\\/d2BiD.eter r a . t i o a.re d e f i n e d . I t i s t h e r e f o r e assumed t h a t t h e r e a l c a s e mass, e x c l u d i n g t h e n o z z l e d i v e r g e n t s e c t i o n , i s e q u a l t o t h e mass of a c y l i n d e r c o n t a i n i n g t h e same volume, s t r e s s e d t o t h e same p r e s s u r e , of t h e same m a t e r i a l and temperatur"e a s t h e r e a l c a s e ,

Then c a s e volume may be e x p r e s s e d a s : -A t ^ ^P ^ ' ' p where e i s t h e p r o p e l l a n t l o a d i n g f r a c t i o n and R t h e p r o p e l l a n t u t i l i s a t i o n r a t i o and b o t h e q u a l one i n t h e c i g a j r e t t e b u r n i n g a r r a n g e m a i t , From geometry of c y l i n d e r : -ir o c ^ c V„ = 7j D : L 5 . 2 Then b a s e d on s u r f a c e Mass = •?rD„ L t ^ P„ 5 . 3 c m m S u b s t i t u t i n g 5.1 and 5 . 2 i n t o 5 . 3 :

-^-^ =

P ^ ' R

È- *m''m 5.4

P P p c S t r e s s i n g a s t h i n c y l i n d e r P ?] D

^ = i - t r 5.5

m o P 13 p e R " s 5 . 6 P P P

Equation 5.6 is directly valid for a case bonded internal burning grain. A value ^ = 2 has been used. Stainless steel case v/ith S = 75^ of normal maximum v/orking stress,

25

Por a cigarette burning charge v/ith an internally sprayed zirconia heat barrier of mass per unit area eqvrl to case material

:-^c ''m

^c = % * b ^ "^ — ~§ 5.7

P

Again r? = 2. A value of S = 5^ci has been used in t h i s a p p l i c a t i o n .

2, Nozzle Mass

Nozzle i n uncooled andhas a heat b a r r i e r of z i r c o n i a . As i n Appendix 3 ,

the nozzle i s t u l i p shaped t o eliminate divergence l o s s e s . The estimation

of mass i s based on t h e same " s i m i l a r nozzle" v»lth an expansion half angle

of 15°.

From 3.9 :

-K

Mass = ~ i - — ( e - i ) p t ^ . . . 5.8

s i n a * m m

In t h i s a p p l i c a t i o n nozzle thickness i s based on a t h i n cylinder

P 7? D

t - - 2 S 5 9

m - 2S ^ ' ^

where D^ = \ ^—^J 5,10

Introducing a factor of 2 to take into account the zirconia

M - ( c ^ m ) 3 (^ - ^ ) ^

(-^^^ ^

5 11

n -

^°

V P sin« \ 2 / S

^'^^

v/hich may be w r i t t e n i n terms of thn.ist ajid the nozzle c o e f f i c i e n t s

4 ± ^

M -/ £ V i l r J l l L (±±SY ^ 5 12

^ " L !e J Pe^^" V 2 ; S 5.12

^p + P c

This method of estimation of nozzle mass gives values within a fev/ per cent of the alternative method, the resiolt of which is derived in

26

-AFEEIOIX 6.

COLOPIETE IDEAL VEEilCIE ANALI5IS

The propiilsion system scaling rules indicate that tlie full definition of the ideal vehicle must include a mass scaling in proportion to

V3

ft , From this the all burnt mass of the vehicle is expressed as

^^ =

l^^S % * ^2%%^h

K = Mr + K, ft 4. Ko ft_ t^ + K, ft/^ 6,1

V e l o c i t y increment and burning time a r e given by M

^ = °=^^ge ÏT - &L % ^.5

1 M„ - M. + o 1 g 2 ^ A P M

Rewriting 6,1 and 6,2 i n terms of •?.—, and e l i m i n a t i n g M and M.

1

% M ' ^2 = - : i + K + K, ftj , , , 6.3 m T -J P P

M ^ IVL A graphical method p e r m i t t i n g the determinaticn of r j - , t , , rr-, etc :

1 ' c for a given •propellant canbinatior/vehicle system' groijp (for known values of

as follows:

values of Av, c, K., K„ and K,) a t various ft (or t h r u s t ) l e v e l s i s

M

Graph 1, P l o t equation 1,5 as ïrr- v s . t, . 1

Graph 2. P l o t l e f t hand s i d e of equation 6,3 v s , t , (using t h e r e l a t i o n s h i p betv/een t, and _ o given i n equation 1,5, and p l o t t e d i n

IS

graph l). Values of the right hand side of equation 6,3 are then

determined for various A values. Then from graph 2 t, can be obtained. P M - -1^ b

Graph 2 can then be used to give o , H. and M are then determined ° rj- 1 o

REFERENCES 2 7 -1, Hvmter, M,ïï,, Klemperer, Vf.B,, and Gunkel, R.J, 2, Gill, G,, Eckell, E,F,,

Williams, F.A,, and Penner, S.S,

3, Baxter, A,D.

4.

Impiolsive Midcourse Correction of a

Lunar Shot.

9th International Astronautical Congress 1958. Determination of Rocket Motor Parameters

by Means of a Diverging Reactor, 7th International Symposium on Combustion 1958.

Some Propulsion Problems of High Altitude Rockets,

High Altitude and Satellite Rockets. (The proceedings of a Symposium held at Cranfield, England, July 1957).

Published jointly by the Royal Aeronautical Society and the British Interplanetary Society.

Theoretical Performance of Several Rocket Propellant Combinations, Chart prepared by Rocketdyne, a

Division of North American Aviation Inc. 1958 5. Pocket Data for Rocket Designers. Published by Rockets Division,

28

-RELEVANT E/iRTH AMD MOON lulTA

Moon E a r t h

2159

5.19

7693

7926

32,17

36,677

Diameter (miles)

2

Surface g r a v i t y ( f t / s e c )

Escape v e l o c i t y ( f t / s e c )

Mirdmum v e l o c i t y ( a t e a r t h surface) t o reach moon 34,800 ( f t / s e c )

C i r c u l a r v e l o c i t y a t 500 miles above eajrth stirface 24,800 ( f t / s e c )

Minimum v e l o c i t y increment (500 c i r c u l a r t o moon) 10,000 ( f t / s e c )

29

-T/^LB 2

UQUID PROR'H-.LANT DATA

C

99.6% Hj'drogen Perozide and Kerosine ( =r = 6)

99.6fo Hydrogen Peroxide and 100% Hydrazine

n

Liquid Oxygen and Kerosine ( — = 6)

Liquid Oxygen and Unsymmetrical Dimethyl Hydrazine Liquid Oxygen and Liquid Hydrogen

Liquid Fluorine and Liquid Hydrogen

Self-igniting combination

Performance data based on P = 500 Lt/in'

P r o p e l l a n t Ccmbination X 0 y T c m ^Puel'' %x ' ' P u e l p , P r o p - 1 5320 1.2 2939 22 6 . 5 8 7 . 0 4 9 . 8 7 9 . 9 2 5660 1.22 2861 19 1.7 8 7 . 0 6 2 , 4 7 7 . 4 3 5740 1.24 3461 22 2 . 3 7 1 . 3 4 9 . 8 6 1 . 2 4 5975 1.24 3394 20 1 . 4 7 1 . 3 4 8 , 9 5 9 . 9 5 7950 1,26 2790 9 3 . 5 7 1 . 3 4,41 1 6 , 2 3 6 8275 1.33 3033 8,9 4 . 5 9 4 . 3 4.41 1 9 . 9 8 f t / s e o \ l b l b , mol l b f t ' l b f t ' l b f t ' References 4 axid 5 Combination 1

2^

3

4

5

6^

30

-T/^IE 3.

SOLID PROPELLANT DATA

Values are assumed typical of best available at present time. They

are not representative of any specific type of propellant,

2

Performance data based on P = 1 0 0 0 Lb/in

C^ ss 4920 ft/sec

y = 1.26

T^ a 2750 °K

P

s 100 Ib/ft^

31

-TABLE 2)..

MATERIALS DATA USED IN Fi^PER

1 , S t a i n l e s s s t e e l (Armco PH 15-7 M )

Maximum v/orking s t r e s s a t 20 C

Maximum v/orldng s t r e s s a t 300 C

Density

Strength/Mass a t 20°C

2 , Aluminium Alloy

Maximum v/orking s t r e s s a t 20 C

Density

Strength/ÏVlass a t 20°C

3 , T i t a n i m Alloy (6A1 - 4V)

Max:imum working s t r e s s a t 20 C

Density

Strengtly^Iass a t 20°C

200,000 Lb/in^

160,000 Lb/in^

0.277

7.25 X

70,000

0,10

7 X 1o5

I b / i j i ^

105 L b , i n , / l b

Lb/in^

I b / i n ^

L b , i n / l b

160,000 Lb/in^

0,16

10^

I b / i n ^

Lb,in/l'b.

Reference 5 and Manufacturers data.

- 32

TA3LE 5 .

NOZZLE EKIT PRESSURE ~ SIZE RELATIONSHIP FOR CONSTANT THRUST AND CCl/IBUSTION PRESSURE

2

Data based on P = 1000 Lb/in

c

P = 1429 Lb

C^ = 4920 f t / s e c

y = 1.26

Pg Lb/in^

1 ^e

; ^

V f t / s e c

e '

1

""^

\ - '

^ e ^ ^

^ t ^

\ ^

2

Surface area i n

A l b / s e c

c f t / s e c

14.7

68

1.575

7750

7 . 8

0.845

6,59

1.036

2.90

22.2

5.53

8300

10

100

1.61

7930

10,5

0.833

8.75

1.029

3.28

30.5

5.44

8450

2

500

1.75

8620

36

0.784

28.2

0.998

6.00

105.7

5.11

9000

1

1000

1.79

8820

60

0.772

46.4

0.991

7.68

176

5.05

9120

0.1

10000

1.9

9350 1

200

0,745 1

149.0 1

0.973

13.76

571

4,86 1

9460 1

1

33

-TABLE 6.

PERFaRItAICE Ijm NOZZLE PAR/a,IETSRS FOR YI^RIOJS LIQUID PROPELIANTS FOR COl^lSTAJMT THSUST AND COMBUSTION HffiSSURE

Data based on 500 Lb/in^

0.05 Lb/in^ 267 Lb

Propellant combination••

°F

v^ f t / s e c

A

e = _ ^

• 2

^ t ^

. 2

A^ on

_e

D^ i n

D i n

e

Surface area

in2

ft l l / s e c

P ^

c f t / s e o

•

1

1.98

10530

459

0.263

120,8

0.578

12.4

466

0.797

10780

2

1.95

11040

416

0,268

111.6

0,584

11,9

430

0.76

11,300

2^

1.84

10420

67.1

0.28

18.8

0.597

4.89

71.5

0.795

10,800

3

1.92

11010

383

0,273

105

0,589

11.53

403

0.765

11,220

4

1.92

11480

383

0.273

105

0.595

11.53

403

0,734

11,710

5

1.885

14990

344

0,278

103

0.595

11.44

369

0,565

15,260

6

1,8

149C0

251

1

0.293 1

1

73.5

0,61

9.67 j

'1

283

0,569 i

15,100.

3 4

-TABIB 7 ,

SCALEMG CONSTANESFOR LIQUID PROPELLAMT S I S T H S

Propellant Ccmbination—->

S

1

S

'

^1

'S 1

K, ( K, ( K j (

K j l

K ^ l

K j '

•S'

; chamber)

[valves

[Pressurised system)

[gas generator)

[pumps)

[T\irbo/pianped system)

[at 700 Lb/in^)''

[at 700 Lb/in^)^

[at 50 Lb/in^)""

[a± 50 Lb/in^)^

P„

[ ^ = 10,000)

e

1

0.206

1.126

1.332

5.525

0.125

6,982

«w

0,0456

-0,00326

3.425

2

0.215

1,162

1.381

5.525

0.129

7.035

-0.0496

-0.00354

3.350

- •

3

0,222

1.471

1.693

5.525

0.163

7.381

0.0773

0,0478

0.00553

0.00342

3.15

4

0,231

1.502

1.733

5.525

0,167

7.425

0.0914

0,0618

0.00654

0,00/|42

3.32

5

0,308

5.560

5.868

5.525

0,616

12,009

0.674

0,185

0,0481

0.0132

4.23

6

0.320

4.510

4,830

5.525

0.501

10.856

0,563

0,157

0,0483

0,0112

3.36

Allov/ance made f a r t e m p e r a t u r e r i s e , a n d / o r h e a t b a r r i e r " ^ a s e d on d e n s i t y a t 273°K o r b o i l i n g p o i n t a t 1 atmosphere (lovrest v a l u e ) . No allov/ance made f o r h e a t b a r r i e r maps,

SCALING CONSTANTS - THE SOLID PROHILLAW SYSTMi

K^ = 0 Kg = 0.36 \ 0 . 1 8 ^ K^ = 0 . 4 8

C i g a r e t t e b u r n i n g arrangement

TOTAL VELOCITY INCREMENT EARTH SURFACE TO MOON 35,2SO ft./s«c. VELOCITY INCREMENT SOO MILE CIRCULAR ORBIT TO MOON lO.SOO ft./sec.

1-7 DAY

O DAY

FIG. I. TYPICAL EARTH-MOON TRAJECTORY BASED ON REEL

l' CONVERT HYPERBOLA TO i\ ELLIPSE. / 2'^ CONVERT , ELLIPSE TO I 6RAZMG ELLIPSE

FIRING OF •VERTICAL 'CONTROL UNIT VELOCITY = CIRCULAR / VELOCITY FIRING STARTS VELOCITY 7690 ft/s«c.

FIG. 3. GRAZING ELLIPSE LANDING TECHNIQUE.

2 DIMENSIONAL GYRO PLATFORM ENERGISED SIGNAL HEIGHT. FOR IGNITION. ZERO VELOCITY WORST CASE' ZERO VELOCITY BEST CASE LANDING APPROACH VELOCITY sa 7690 ft/see. ^ S U N SEEKER IN OPERATION.

Lin

- IGNITION.

IGNITION OF LANDING ENGINE.

MOON S SURFACE

4 0 BO I20 I60 2 0 0 240 280 320 360 4 0 0 INITIAL MASS

FOR lOOIb PAYLOAD.

ft VEHICLE MAXIMUM ACCELERATION at j^2

FIG. 5. IDEAL VEHICLE! MASS/ACCELERATION K," O

O 4 0 8 0 I20 I60 2 0 0 2 4 0 2BO 320 360 4 0 0 VEHICLE MAXIMUM ACCELERATION a, ^ INITIAL MASS

FOR lOOIb PAYLOAD

DATA QUOTED FOR A 99 6 % H.TP / HYDRAZINE UNIT THRUST 267 Lb. MASS FLOW RATE 0 76 lb/sec. COMBUSTION PRESSURE 5 0 0 Lb/in.» EXIT PRESSURE OOS Lb/ln.2

L ^ - Ó O l n . ^ - 4 1 6 - 5 o C - l s " Dt 0-584 in

D( 11-92 in.

At 0-268 in.2 A. Ill-6in.2

a. STANDARD TYPE COMBUSTION CHAMBER.

EFFECTIVE THROAT

INJECTOR.

NOT TO SCALE.

b. A DIVERGING REACTOR TYPE COMBUSTION CHAMBER.

FIG. 7 LIQUID PROPELLANT COMBUSTION CHAMBERS.

Ol -i2S5 2I2 O 2 5 " 2 4 5 in ^ 2 4 g 2 3 5 - I i 2-3 5 2-25 Z 2 2 2SS 2 SO 24S 2 4 0 235 230 225 I. LOX./KERO. PRESSURISED 2. H.TP/HYD. PRESSURISED.

3. BOTH SYSTEMS USING TURBO/PUMPS.

--LOX/KERO. (No Allowance for Temperature Rise.)

^ \ \ \ \ \ \

I \ \ I

f SO 6 0 70 80 9 0 100 no 120 I30 I40 ISO INITIAL MASS

FOR POIb. FWLOAD VEHICLE MAXIMUM ACCELERATION a, ^ 2