
Rocket Propulsion

The Rocket Motor
Pump, Combustion Chamber & Nozzle

Rocket Propulsion Principles

The Propellant Pump(s)
An essential component of liquid fuelled rocket engines is the means of delivering the propellants (the fuel and the oxidiser) to the combustion chamber. The simplest method used in low thrust rockets is by pressurising the fuel and oxidiser tanks with compressed air or a gas such as nitrogen, but for most liquid fuelled rockets, the high propellant flow rates required are provided by onboard turbopumps.
The Injector Plate
The injector plate is a passive device which has three purposes. It breaks up the liquid propellants into tiny droplets to aid and speed up combustion, it enables homogeneous mixing of the fuel with the oxidiser and it ensures stable, controlled burning of the fuel, preventing the explosive combustion of the propellants.
The Nozzle
The purpose of the nozzle is to promote the isentropic (constant entropy) expansion of the exhaust gas. As the gas expands, its pressure drops, but since there is no change in total energy, its velocity (kinetic energy) increases to compensate for the reduction in pressure energy.
There are thus two factors contributing to the engine thrust, namely, the kinetic energy of the gas particles ejected with high velocity from the exhaust and the pressure difference between the exhaust gas pressure and the ambient pressure of the atmosphere acting across the area of the nozzle exit. The relationship is shown in the following equation.
Engine Thrust F = dm/dt. V_{e} + A_{e}(P_{e}  P_{a})
Where
dm/dt = Propellant Mass Flow Rate per Second
V_{e} = Gas (Exhaust) Velocity at Nozzle Exit
A_{e} = Area of Nozzle Exit
P_{e} = Gas Pressure at Nozzle Exit
P_{a} = Ambient Pressure of the Atmosphere
The first term is known as the momentum thrust and the second term the pressure thrust.
Considering the pressure thrust alone, since the ambient pressure decreases with altitude, in the vacuum of free space where the pressure is zero, the rocket thrust will increase to a maximum of 15% to 20% more than the thrust at sea level.
(By contrast, the thrust of a jet engine decreases with altitude to zero in free space since it depends for its thrust on air as the oxidiser for the fuel. The rocket on the other hand carries its oxidiser with it.)
The momentum of the exhaust gas is however much more effective in creating thrust than the pressure difference at the exhaust exit, so that the more the pressure energy is converted into kinetic energy in the nozzle, the more efficient the nozzle will be. So paradoxically the maximum thrust occurs when the exhaust pressure is equal to the ambient pressure.
The effective exhaust velocity V_{e} is a function of the nozzle geometry such as the nozzle expansion ratio A_{e}/A_{t}
Where A_{t} = Area of Nozzle Throat
See also Steam Turbine Nozzles 
Rockets depend for their action on Newton's Third Law of Motion that: "For every action there is an equal and opposite reaction."
In a rocket motor, fuel and oxidiser, collectively called the propellants, are combined in a combustion chamber where they chemically react to form hot gases which are then accelerated and ejected at high velocity through a nozzle, thereby imparting momentum to the motor in the opposite direction.
A rocket can be considered as a large body carrying small units of propellant travelling with a velocity V.
The reaction due to expelling the propellant from the rocket exhaust causes the velocity of the rocket to increase.
Assuming no change in ambient pressure, the Conservation of Momentum for the rocket and the expelled propellant gives:
(M+dm)V = M(V+dv) + dm(V  V_{e})
Where
M = The total remaining mass of the rocket and its fuel
dm = The mass ejected rearwards through the exhaust nozzle or the change in mass during a given period.
V = The initial absolute forward velocity of the rocket just before the ejection of the propellant
dv = The increase in forward velocity of the rocket due to the ejection of the exhaust gases
V_{e} = The exhaust velocity, relative to the rocket, of the propellant leaving the rocket motor.
Simplifying we can derive the following:
dm.V_{e} = M.dv
or
dm/dt. V_{e} = M.dv/dt = M.a = F = The Force or Thrust acting on the rocket
Where
dm/dt = The mass flow rate
dM = The change (reduction) in the mass of the rocket due to the consumption of the fuel
a = The acceleration of the rocket
F = The instantaneous force or thrust acting on the rocket.
For the change in velocity over a longer period we must take into account the reduction in the mass of the rocket as its fuel is consumed and integrate the velocity over time for the duration of the period. Thus, from the above:
The mass expelled = The reduction in mass of the rocket and its propellant load
or
dm =  dM
and
∫dv = V_{e} ∫dm / M
so that
∫dv =  V_{e} ∫dM / M
Thus
V_{f}  V_{i} =  V_{e}(ln M)_{i}^{f}
=  V_{e} (lnM_{f}  lnM_{i})
= V _{e} ln (M_{i} / M_{f})
Where
V_{i} = The initial velocity of the rocket
V_{f} = The final velocity of the rocket
ln = The natural logarithmic function
M_{i} = The initial mass of the rocket including its payload all its propellant
M_{f} = The final mass of the rocket and its payload including its remaining propellant
M_{i} / M_{f} is known as the rocket's Mass Ratio
This is known as Tsiolkovsky's Equation
Note that although a greater initial mass (of propellant) which increases the Mass Ratio, will create a greater increase in velocity, the relationship is not linear and the increase in velocity due to the increased available fuel becomes proportionally less as the initial mass M_{i} increases. This is because some of the extra propellant must be used to accelerate the mass of the extra fuel itself.
Multistage Rockets, another of Tsiolkovsy's ideas, separate the propulsion into more than one stage, each stage with its independent rocket motor, propellant tanks and pumps or pressurisation systems. The stages may be "stacked" as in the Apollo space vehicle which took the astronauts to the moon, or "piggy backed" as in the Space shuttle. As the propellant in the first stage is used up, the stage is jettisoned and the propulsion taken over by the subsequent stage so that the later stages do not have to waste energy accelerating the useless mass of the jettisoned stages.
In this way higher velocity and range can be achieved with the same initial vehicle weight, payload weight and propellant capacity or alternatively a greater payload can be carried with a smaller initial weight.

Impulse, Thrust and Fuel Performance

Rocket Power and Dynamic Conversion Efficiency

Impulse
For a constant Thrust F, the Impulse I provided by a motor or a propellant over a specific Period t is defined as;
I = F.t
The Specific Impulse I_{s} is the ratio of the of thrust produced to the weight flow of the propellants (fuel plus oxidiser).
It is a measure of the potential effectiveness of a particular fuel and oxidiser combination in converting its chemical energy into useful work and is thus a convenient way of comparing fuel efficiencies.
It is defined (in Imperial units) as:
Thrust (lbs)
Propellant Consumption (lbs/s)

Specific Impulse =
Or
I_{s} = F / (dw/dt)
Where
I_{s} = The specific impulse is expressed in units of time (seconds)
F = The thrust
w = The combined weight of the fuel and oxidiser
dw/dt = The propellant consumption per second
In international SI or MKS units this relationship becomes:
I_{s} = F / (dm/dt).g_{0}
Rearranging, this becomes:
F = I_{s}(dm/dt).g_{0}
Where
F = The thrust in Newtons
m = The mass of propellant in Kg
g_{0} = The standard acceleration due to gravity at sea level (32.2 ft/s/s)
Thus increased thrust can be achieved by using propellants with a higher specific impulse and also by increasing the fuel burn rate.
From the equations of motion opposite, the exhaust velocity V_{e} is given by
V_{e} = F / dm/dt
Thus
V_{e} = I_{s}.g_{0}
The exhaust velocity, relative to the motor, is therefore directly proportional to the specific impulse. This is a simple way of determining the exhaust velocity from the specific impulse of the fuel / oxidiser combination.
Note: Due to the affect of the ambient air pressure, the specific impulse may be 15% to 20% lower at sea level than in the vacuum of space. (See the thrust equation in the diagram above)
Propellant Density
Fuel effectiveness also depends on its density as well as the density of its associated oxidiser. High density propellants, can be accommodated in smaller tanks and they can use smaller pumps for feeding the propellants to engine. This allows smaller lighter vehicle structures with less aerodynamic drag.
Taking density into account the effective specific impulse is given by:
I_{d} = ρ_{av} I_{s}
Where:
I_{d} = The Density Specific Impulse (Kg.secs/litre)
ρ_{av} = The average density of the fuel and the propellant mixture (kg/litre)

Power
Rocket Engine power P = The maximum available kinetic energy delivered to the exhaust gas stream per second.
P = 1/2 dm/dt V_{e}^{2}
Vehicle Motive Power P_{m} = The power transmitted to the vehicle to drive it forwards
P_{m}= FV
This implies that the rocket power at any instant is dependent on its velocity and is zero when the forward velocity is zero as it would be at liftoff.
Once the rocket starts moving, the available kinetic energy and power are split between the exhaust stream and the rocket vehicle. Thus
P = P_{m} + P_{e}
So that
P_{e} = 1/2 dm/dt (V_{e}  V)^{2}
Where P_{e} is the remaining power in the exhaust stream
Efficiency
Ignoring parasitic efficiency losses such as propellant pumping power, frictional losses and nozzle design efficiency, the conversion efficiency of translating the energy in the exhaust gas flow into forward motion of the rocket is given by,
η = P_{m}/ P
Where η = The conversion efficiency
Thus
η = FV / ( FV + dm/dt (VV_{e})^{2}/2)
Note that the efficiency is dependent on the rocket's velocity and is maximum when V = V_{e}, that is when the forward velocity of the rocket is equal to the rocket's exhaust velocity.
Substituting F / V_{e} for dm/dt the above equation simplifies to:
η = 2 (V/ V_{e}) / (1+(V / V_{e})^{2})
This provides a measure of the rocket's efficiency in terms of velocity alone.
Ullage motors are rocket motors used to provide artificial gravity in multistage liquid fuelled rockets by momentarily accelerating the second stage forwards after the first stage burnout. This moment of forward thrust is required in the weightless environment of outer space to make certain that the second stage liquid propellant is in the proper position to be drawn into the pumps and that the gaseous zone above the liquid in the tank is not next to the pump input prior to starting the second stage engines.
The extra thrust also helps to make a clean separation between stages.
"Ullage" is an old brewers term meaning the air space above beer in a vat.
Rocket Fuels and Oxidisers

Liquid Fuels and Oxidisers
Liquid propellants pioneered in 1926 by Robert Goddard are relatively safe and easy to control and easy to start and stop. However they need a complex pumping system, pressure controls, valves and a feed system to deliver the propellants to the combustion chamber all of which reduce the mass ratio and hence the efficiency of the system.
Cryogenic Fuels and Oxidisers
Some of the highest energy liquid propellants have very low boiling points. Liquid Hydrogen (LH_{2}) fuel for example has a boiling point of 252.9°C and an oxidiser such as Liquid Oxygen (LOX) boils at 183°C. Using these high energy density propellants in gaseous form is impractical since the enormous onboard storage tanks and pumping systems they would require would be too big and heavy. Even in liquid form there are difficulties in using these propellants since the storage tanks may need to be insulated and the pumps must work at very low temperatures with a very high temperature gradient across the body of the pump. Safety, handling and storage are also issues of concern. Nevertheless, cryogenic propellants are used when controllable, maximum thrust is a priority.
Solid Fuels and Oxidisers
Solid propellant motors contain both the fuel and the oxidiser in a charge called the grain which is stored within the combustion chamber. Invented by the Chinese in 1150, the motors are compact and light weight and do not need pumps, valves or feed systems so they have a very high mass ratio and thrust per unit volume, but for the same reason they are difficult to control. Once the burn starts, it is difficult, if not impossible, to stop until all the fuel is consumed.
Hypergolic Propellants
Hypergolic propellants are fuel and oxidiser combinations, liquid at room temperature, which ignite spontaneously on contact with eachother. They are easy to control, start, stop and restart. Some combinations are extremely toxic and corrosive. Suitable for engines which must be ignited in space or reoperated numerous times. Elimination of the igniter removes a significant source of unreliability. 
Example  Saturn V S1C Engine Performance

Example  Saturn V Fuel Choices 
Rocketdyne F1 Engine used in Saturn V S1C
Engine dimensions
Dry mass: 18,500 lbs
Length: 19 ft
Maximum diameter: 12ft 4in
Fuel: Kerosene (RP1), delivered at 1,754 lb/s (dm_{f}/dt)
Oxidizer: Liquid oxygen (LOX), delivered at 3,982 lb/s (dm_{o}/dt)
Total Propellant Flow (dm/dt): 5,736 lb/s
Mixture mass ratio (r): 2.27:1 oxidiser to fuel
The mixture mass ratio is the ratio of oxidiser to fuel for optimum combustion. This is not the same as the rocket's mass ratio which represents the efficiency of the mechanical construction of the rocket compared with its fuel load.
Turbopump: 5,550 rpm, 41,000 kW single turbine, powered by a gas generator requiring 1,694 lb/s propellants, driving fuel and oxidiser pumps on the same shaft with a total flow rate of 2,542 litres/sec (1,565 l/s of LOX and 976 l/s of RP1)
Thrust (F): 1,522,000 lbs at Sea Level
Specific Impulse (I_{s }): F /(dm/dt) = 265.3 secs at Sea Level, 305 secs in vacuum.
Exhaust Velocity (V_{e}): (I_{s}*g_{0}) = 8543 ft/s (5825 mph)
Expansion ratio: 16:1 with nozzle extension, 10:1 without
Combustion chamber pressure: 70 bars
Combustion chamber temperature: 3,300^{o}C
Burn time: rated at 165 seconds 
Hydrogen (LH_{2}) versus Kerosene (RP1)
The thrust provided by rocket fuel is proportional to the energy density of the fuel and its propellant and the rate at which the fuel is burned. While liquid hydrogen (LH_{2}) has the highest energy density (energy per unit mass) of all fuels, over 30% more than kerosene, it also has the lowest physical density (mass per unit volume), only one twelfth the density of RP1. Thus RP1 has a greater energy content per unit volume than LH_{2}, while LH_{2} has a greater energy per unit mass.
This means that to provide the same energy content as RP1, the fuel tanks, pipes and pumps and the structures needed to contain and transport the less physically dense LH_{2} will be disproportionately large compared with those needed for the kerosene fuel supply. This increases the final, (nonfuel) mass of the rocket, thus decreasing its mass ratio and hence its conversion efficiency.
Minimising this nonfuel mass at lift off is particularly important when maximum thrust is required which is why RP1 is considered as an alternative. For lower thrust levels however, the relatively high mass of the fuel supply system needed to supply the liquid hydrogen is less significant compared with the gains made by using the more energy dense hydrogen fuel and there is a crossover point which occurs as the required thrust decreases when the higher energy, though less physically dense, hydrogen becomes the more energy efficient option. This is because the volume of the fuel system needed to contain the less dense hydrogen increases as the cube of the linear dimensions, but the weight of its fuel containers and pipes, which depends roughly on their surface area, only increases as the square of the linear dimensions.
For very high, long duration thrusts such as those required from the S1C first stage of the Saturn V launch vehicle to get the heavy Apollo Space Vehicle off the ground, using the lighter hydrogen as the fuel would require an impractically large and heavy on board fuel supply system. For this reason kerosene with its lighter, more compact fuel supply system components was used to power the F1 rocket engines used in the S1C.
Once the heavy stage 1 has been jettisoned and the rocket is operating in much reduced gravity, the required thrust is reduced and hydrogen becomes the most efficient option for fuelling the J2 engines powering the lighter stage 2 (S11) and stage 3 (S1VB) of the Saturn V.

See also Missile Ballistics, Orbits and Aerodynamics
Some Liquid and Solid Fuel Characteristics
Fuel Type

Fuel

Fuel
Density
ρ_{f}
(g/cm^{3})

Fuel
Boiling
Point
(deg C)

Fuel
Specific
Impulse
(Secs)

Oxidiser

Oxidiser
Density
(g/cm^{3})

Oxidiser
Boiling
Point
(deg C)

Oxidiser
/ Fuel
Mix
Ratio (r) 
Density
Specific Impulse
of Mix
Kg.secs/L 
Density
of Mix
(g/cm^{3})
***

Comments about the Fuel

Liquid
Bipropellant
Petroleum 
Kerosene
Paraffin
(RP1) 
0.820 
216.3 
265
(Sea level)
305
(Vacuum) 
Liquid
Oxygen
(LOX) 
1.14 
183.0 
2.29 
264 
1.03 
Inexpensive, Practical.
As with most liquid fuels, relatively easy to control, start and stop.
Stable at room temperature.
Complex ignition process.
Low explosion hazard.
Less energy per unit mass than hydrogen.
More energy per unit volume than hydrogen
Lower specific impulse than cryogenic fuels, but more than hypergolic propellants.
Low temperature oxidiser needs insulation. 
Liquid
Bipropellant
Cryogenic 
Liquid
Hydrogen
(LH_{2})

0.071 
252.9 
425 (Vacuum) 
Liquid
Oxygen
(LOX)

1.14 
183.0 
5.0 
294 
0.29 
Very high specific impulse 30% to 40% higher than most other fuels
Low temperature means difficult to store and handle.
Needs insulated tanks.
Very low density fuel needs large storage tanks and pumps. 
Hypergolic 
Hydrazine 
1.004 
113.5 
286 
Nitrogen
tetroxide 
1.45 
21.15 
1.08 
342 

Fuels and oxidizers ignite spontaneously on contact with each other.
Easy to start, stop and restart
Highly toxic and must be handled with extreme care.
Remain liquid at room temperature.
Relatively easy to control. 
UDMH
Unsymmetrical
dimethyl
hydrazine 
0.791 
63.9 
277 
Nitrogen
tetroxide 
1.45 
21.15 
2.10 
316 

Aerozine 50
5050 Hydrazine
with UDMH 


280 
Nitrogen
tetroxide 
1.45 
21.15 
1.59 
326 

MMH
Monomethyl
hydrazine 
0.866 
87.5 
280 
Nitrogen
tetroxide 
1.45 
21.15 
1.73 
325 

Solid 
Aluminium with
HTPB
Hydroxy
terminated Polybutadiene



277 
Ammonium
perchlorate 


2..12 
474 

Fuel contained in the combustion chamber. No tanks or pumps required.
Compact, lightweight motor designs with a very high mass ratio.
Safe, Easy to store, Quick to start
Difficult to control.
Needs an ignition system.
Low specific impulse, but high thrust per unit volume. Allows lighter, simpler, and more reliable casing / combustion chamber designs. 
Aluminium with
PBAN
Polybutadiene Acrylonitrile 


277 
Ammonium
perchlorate 


2.33 
476 

*** Average Density ρ_{av} is given by:
ρ_{av} = ρ_{o}ρ_{fv}(1+r) / (ρ_{f}r+ρ_{o})
Where
ρ_{o} = The density of the oxidiser
ρ_{f} = The density of the fuel
r = The ratio of oxidiser mass to fuel mass

Apollo 11 Rocket Motors and their Propellants 
Application 
Motor 
Number
Used 
Fuel
Type 
Fuel 
Oxidiser 
Propellant
Feed 
Specific
Impulse
(Secs) 
Thrust
(lbs) 
Gross
Weight
(lbs) 
Propellant Weight
lbs / (%) 
Burn
Time
(Secs) 
Comments 
Saturn V
Launch Vehicle
Stage 1
(SIC)
(Uprated version) 
F1
Propulsion 
5 
Petroleum 
Kerosene
RP 1
(Paraffin) 
Liquid Oxygen LOX 
Turbopump 
289 
1,530,000 
4,792,000
(Stage1) 
4,492,000 (93.7%)
(Stage 1) 
150 
5 F1 engines giving the S1C a total thrust of 7,650,000 lbs 
Retrorockets 
8 
Solid 
Composite of polysulphides 
Ammonium
perchlorate 
NA 
277 
87,913 
504 
278 
0.633 
Stage 1 2 separation 
Saturn V
Launch Vehicle
Stage 2
(SII)

J2
Propulsion 
5 
Cryogenic 
Liquid Hydrogen
LH_{2} 
Liquid
Oxygen
LOX 
Turbopump 
381 
225,000 
1,037,000 
942,000
(90.8%)
(Stage 2) 
359 
5 nonrestartable J2 engines giving the S11 a total thrust of 1,125,000 lbs 
Retrorockets 
4 
Solid 
Composite of polysulfides 
Ammonium
perchlorate 
NA 
277 
34,810
each 
377.5 each 
268.2 
1.52 
Stage 23 separation 
Ullage rockets 
4 
Solid 
Flexadyne
Polybutadiene (CTPB) 
Ammonium
perchlorate 
NA 
277 
22,700
each 
504 each 
336 
3.7 
Stage 2 ullage 
Saturn V
Launch Vehicle
Stage 3
(SIVB) 
J2
Propulsion 
1 
Cryogenic 
Liquid Hydrogen
LH_{2} 
Liquid
Oxygen
LOX 
Turbopump 
381 
225,000 
262,000 
228,000
(87%)
(excluding reserves) 
480 
1 restartable J2 engine
2 burns 
Ullage
rockets 
2 
Solid 
Composite of polysulfides 
Ammonium
perchlorate 
NA 
277 
3390 

58.8 
3.8 
Main third stage ullage 
Saturn V
Launch Vehicle
Stage 3
(SIVB)
Auxiliary Propulsion System
(APS) 
Ullage 
2 
Hypergolic 
MMH 
Nitrogen Tetroxide 
Helium pressurised tanks 
280 
70 

303 
50 
2 APS in stage 3
1 Ullage motor in each APS
Used during third stage restart 
Attitude
Control 
6 
Hypergolic 
MMH 
Nitrogen Tetroxide 
Helium pressurised tanks 
280 
150 

303 
0.07 
2 APS in stage 3
3 Attitude control thrusters in each APS 
Apollo
Command Module 
Reaction Control System (RCS) 
12 
Hypergolic 
UDMH 
Nitrogen Tetroxide 
Helium pressurised tanks 
280 
92 

270 
Variable 

Apollo
Service Module 
Service Module
Propulsion 
1 
Hypergolic 
Aerozine 50
(See note ** below) 
Nitrogen Tetroxide 
Helium pressurised tanks 
311 
20,500 
55,000 
40,974

Variable 
Nonthrottleable
But can be switched on and off 
Reaction Control System (RCS) 
16 
Hypergolic 
MMH 
Nitrogen Tetroxide 
Helium pressurised tanks 
280 
100 

1,362 
Variable 
16 used in groups of 4 
Apollo
LM Descent 
LM Descent
Motor
Propulsion 
1 
Hypergolic 
Aerozine 50 
Nitrogen Tetroxide 
Helium pressurised tanks 
311 
10,125max
Variable
1,020 to
6,800 
25,600* 
19,500 

Throttleable thrust
*Gross weight without crew 
Apollo
LM Ascent 
LM Ascent
Motor
Propulsion 
1 
Hypergolic 
Aerozine 50 
Nitrogen Tetroxide 
Helium pressurised tanks 
311 
3,500 
9,900* 
5,200 

*Gross weight without crew 
Reaction Control System (RCS) 
16 
Hypergolic 
MMH 
Nitrogen Tetroxide 
Helium pressurised tanks 
290 
100 

605 
Variable 
16 used in groups of 4 
Apollo
Launch Escape System 
Escape
Motor 
1 
Solid 
Composite of polysulfides 
Ammonium
perchlorate 
NA 
277 
147,000 
Total
8,910


4 
Ejects Command Module from a dangerous launch 
Tower Jettison 
1 
Solid 
Composite of polysulfides 
Ammonium
perchlorate 
NA 
277 
31,500 


Jettisons tower after safe launch or when it is no longer required 
Launch
Vehicle
Pitch
control 
1 
Solid 
Composite of polysulfides 
Ammonium
perchlorate 
NA 
277 
2,400 


Provides an initial pitch manoeuvre away from the launch pad toward the Atlantic Ocean in case of an abort 
Total Apollo Rockets 
87 

Note ** Aerozine 50 is a 50/50 mix by weight of hydrazine and unsymmetrical dimethylhydrazine (UDMH) with nitrogen tetroxide oxidiser

V2 Missile

V2
Rocket 
Motor 
1 
BioEthanol
or
Petroleum 
Ethyl
Alcohol
(Ethanol) 
Liquid Oxygen LOX 
Turbopump 
269 
56,000 
27,500
Including
payload 
19,301
(70.0%) 
65 
Reference
(German V2 Missile) 

See also:
Ballistics and Aerodynamics
Aerodynamics and the Theory of Flight
Gyroscopes and Navigation
Satellites and Orbits
