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2. INTRODUCTION TO SPACE PROPULSION2. INTRODUCTION TO SPACE PROPULSION

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2.1 CLASSICAL MECHANICS2.1 CLASSICAL MECHANICS

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PROPULSIONPROPULSION

PROPULSION Changing the Motion of a Body

JET PROPULSION Reaction Force due to momentum of ejected matter

BODY

�

mB

�

dm

!v

ejected mass

!F

!v

Body

!!v

Body

!D

!F

Body

!v

Body

BODY

�

mB

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NEWTON’S LAWS OF MOTIONNEWTON’S LAWS OF MOTION

Newton’s Laws of Motion describe the relationships between forces acting ona body and the motion of the body.

Newton’s First Law: Law of Inertia Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi

quatenus a viribus impressis cogitur statum illum mutare.

Every body perseveres in its state of being at rest or of moving uniformly straightforward, except it is compelled to change its state by force impressed.

Newton’s Second Law: Law of Resultant Force Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua

vis illa imprimitur.

The rate of change of momentum of a body is proportional to the resultant force actingon the body and is in the same direction.

Newton’s Third Law: Law of Reciprocal Actions Actioni contrariam semper et aequalem esse reactionem: sive corporum duorum actiones in se

mutuo semper esse aequales et in partes contrarias dirigi.

All forces occur in pairs, and these two forces are equal in magnitude and opposite indirection.

Source: Wikipedia

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BODY IN MOTIONBODY IN MOTION

The motion and changing the motion of a body can be described with:

BODY

PROPULSION Changing the Motion of a Body

!I = !

!p =!p

f"!p

i=

!Fdt

ti

tf

# Impulse

Linear Momentum !p = m

!v

Kinetic Energy

E =1

2m v

2

!v

Body

Newton’s Laws

!F = m

!a

!F

!a

Body

!F

environment

!!v

Body

�

dm

!v

ejected mass

JET

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EQUATIONS OF MOTIONEQUATIONS OF MOTION

Equations of Motion describe the motion of a particle or body, onto which isexerted a force, as a function of time.

Newton’s Second Law:

Acceleration:

Velocity:

Position:

F! = m "a

a =F

ma =

dv

dt

v(t) = adt! = at +C v =

dx

dt

x(t) = v dt! = v

0t +

1

2at

2+ "C

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2.2 BASIC DEFINITIONS2.2 BASIC DEFINITIONS

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Jet Propulsion relies on expulsion of mass to generate thrust. The source forthis thrust is momentum exchange.

Conservation of System Momentum

Change in Linear Momentum:

Change in Momentum over time:

Newton’s Second Law: Momentum Thrust

THRUST EQUATIONTHRUST EQUATION

�

dm

Expelled PropellantResulting Force

BodyM

!v

Body

!F

Body

!v

ejected mass

dp = dm v

em

dp

dt=

dm

dtv

em

F

moment=

dp

dt=

dm

dtv

em= !m v

em

p

system= const.

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THRUST EQUATIONTHRUST EQUATION

As we will see in Chapter 4, an additional force term results from the sum ofpressure forces acting on the inside and outside over the entire affectedsurface area.

Sum of Forces:

Thrust Equation:

0 te

pape

Fpressure

Surface

! = pexit" p

amb( )Aexit

F! = Fmoment

+ Fpressure

T = !m ve

+ pexit" p

amb( )Aexit

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TOTAL IMPULSETOTAL IMPULSE

Impulse expresses the change of momentum of a system.

Definition:Total Impulse is defined as thrustintegrated over the burning time tb..

For negligible transients and F=const., equation reduces to:

Impulse is a requirement imposed on the propulsion system.Example: Minimum Impulse Bit (MIB) requirement for control system (CS)

Thrust can vary with time due to:Transients such as startup and shutdown,

Changing power level,

Changing throat area.

I

t= F dt

0

tb!

MIB = F t ! 1N "0.050 s = 50 mN # s

I

t= F t

b

Thrust Profile

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SPECIFIC IMPULSESPECIFIC IMPULSE

Definition:

Specific Impulse is defined as the total impulse per unit weight of propellant.

This parameter compares the thrust produced by a system with the propellantmass flow rate from which the thrust arises.

For negligible transients, F=const. and a constant mass flow rate, equationreduces to:

Isp=

F dt0

tb

!g

0!m dt

0

tb

!

Isp=

It

g0m

p

=F

g0!m

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VELOCITIESVELOCITIES

Actual exhaust velocity is NOT uniform over the entire cross section of thenozzle.

We define Exit Velocity ue (or ve, v2)as the average actual velocity of the exhaust material at the exit plane of thenozzle.

The Effective (Equivalent) Exhaust Velocity c (or ueq)is the average equivalent velocity at which exhaust material is ejected from thevehicle:

c = I

spg

0=

F

!m= u

e+

(p2! p

3)A

2

!m

pe

ve

e

F

cpc (or p1)

apa (or p3)

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IDEAL ROCKET EQUATIONIDEAL ROCKET EQUATION

Assumptions: No external forces like gravity or drag.

Change of Momentum:

Integration:

Ideal Rocket Equation:

The Ideal Rocket Equation estimates the amount of propellant needed tochange a vehicle’s velocity (∆v).Further, it relates mission requirements to system requirements.

Rocket MvR vF

Propellant Massdm

(M ! dm) dv

R= !dm v

F

dvR= !v

F

dm

M

!v = dv

0

!v

" = #vF

dm

MMi

Mf"

!v = "vF

lnM

f

Mi

#

$%%

&

'((

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MASS NOMENCLATUREMASS NOMENCLATURE

Mass Ratio MR ( ) is a direct result of the Rocket Equation:

Initial Vehicle Mass m0 (mi):

Final (Burn–Out) Vehicle Mass mf:

The basic components of any vehicle type (launch vehicle, ballistic missile, etc.)are: Inert Components: non-propellant components of a vehicle, stage or propulsion

system (engine, tank(s), pumps, structure, insulation, etc.),

Propellant: fuel & oxidizer or working fluid,

Payload Components: represents the object or system which is accelerated.

The definitions of mass and mass ratios are applicable to either an entirevehicle, a stage or a propulsion system. It is very important to define thesystem and identify the masses with the proper notation/indices.

!

MR =m

f

m0

=m

f

mi

or ! =m

0

mf

m

0= m

P+m

f= m

P+m

PL+m

inert

m

f= m

PL+m

inert

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MASS RATIO NOMENCLATUREMASS RATIO NOMENCLATURE

Propellant Mass Fraction ζ is a measure of the structural efficiency of thesystem, or what proportion of the propulsion system mass goes towarddeveloping thrust and moving a payload.

High values of ζ are very desirable. ζ is generally limited by propellant type andmaterial issues.

Important Note:

! =m

P

mP+ m

inert

< 0.5

< 0.5

400 - 1500

1000 - 10,000

Electric Propulsion

Arcjet

Ion Thruster

700 - 1000Nuclear Thermal

0.85 - 0.92300 - 350Hybrid Rocket Engine (HRE)

< 0.5

0.7 - 0.9

150 - 250

300 - 460

Liquid Rocket Engine (LRE)

Monoprop

Biprop

0.8 - 0.95150 - 300Solid Rocket Motor (SRM)

ζIsp [s]System

MR and ζ can be applied to a vehicle, a stageof a vehicle or to a rocket propulsion system.Depending on the system of interest, MR andζ can differ substantially. It is important todefine the system!!

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MASS RATIO NOMENCLATUREMASS RATIO NOMENCLATURE

Inert Mass Fraction is another performance parameter and measures theefficiency of the structure. Small numbers are desirable indicating highperformance:

Payload Mass Fraction:

Payload Mass Ratio:

Structural Coefficient:

For a chemical rocket is a measure of the designer’s skill in designing a very lightpropulsion system, tank and support structure.

! =m

PL

mP+m

inert

=m

PL

m0"m

PL

! =m

inert

mP+m

inert

=m

inert

m0"m

PL

! =m

PL

m0

=m

PL

mP+m

PL+m

inert

=m

PL

mP+m

f

! =m

inert

mPL+m

inert

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Other Important DefinitionsOther Important Definitions

Impulse–to–Weight Ratio: A high value indicates an efficient design.

Thrust–to–Weight Ratio is a measure of the acceleration (in multiples of g0)that the engine is capable of providing to its own mass.

Launch Vehicle:

Orbit-Transfer vehicle:

Orbit-Maintenance Vehicle:

F

W=

F

mvehicle

g0

It

W=

It

mvehicle

g0

=I

t

mP+m

f( ) g0

=I

sp

mf

mP+1

F W

initial>1.0

F W > 0.2

F W > 0.1