Rocket Propulsion Basics

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 on-board 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 react chemically to form hot gases which expand rapidly and 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 and 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

Dividing by dt to get the instantaneous rates and substituting Newton's Laws

a = dv/dt The acceleration of the rocket

F = M.a The force or thrust acting on the rocket

This gives

dm/dt. V_e = M.dv/dt = M.a = F = The Force or Thrust

Where

dm/dt = The mass flow rate of the burning fuel mass ejected.

Thus the instantaneous thrust on the rocket is directly proportional to the fuel mass flow or burn rate.

For the change in velocity over a longer period we must take into account the reduction dM in the total mass of the rocket as its fuel is consumed and integrate the velocity over time for the duration of the period.

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.

See also Missile Ballistics, Orbits and Aerodynamics

Multi-stage 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:

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).g₀

Rearranging, this becomes:

F = I_s(dm/dt).g₀

Where

F = The thrust in Newtons

m = The mass of propellant in Kg

g₀ = 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.g₀

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_e²

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 lift-off.

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 (V-V_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 multi-stage 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.

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 (LH₂) 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 on-board 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 re-start. Some combinations are extremely toxic and corrosive. Suitable for engines which must be ignited in space or re-operated numerous times. Elimination of the igniter removes a significant source of unreliability.

Example - Saturn V S-1C Engine Performance

Example - Saturn V Fuel Choices

Rocketdyne F-1 Engine used in Saturn V S-1C

Engine dimensions

Dry mass: 18,500 lbs

Length: 19 ft

Maximum diameter: 12ft 4in

 

Fuel: Kerosene (RP-1), 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 RP-1)

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*g₀) = 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^oC

Burn time: rated at 165 seconds

Fuel for Saturn V Main Engines - Hydrogen (LH₂) versus Kerosene (RP-1)

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 (LH₂) 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 RP-1. Thus RP-1 has a greater energy content per unit volume than LH₂, while LH₂ has a greater energy per unit mass.

This means that to provide the same energy content as RP-1, the fuel tanks, pipes and pumps and the structures needed to contain and transport the less physically dense LH₂ will be disproportionately large compared with those needed for the kerosene fuel supply. This increases the final, (non-fuel) mass of the rocket, thus decreasing its mass ratio and hence its conversion efficiency.

Minimising this non-fuel mass at lift off is particularly important when maximum thrust is required which is why RP-1 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 S-1C 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 F-1 rocket engines used in the S-1C.

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 J-2 engines powering the lighter stage 2 (S-11) and stage 3 (S-1VB) of the Saturn V.

Details of the specialised propellants chosen for the the various other thrusters used on Saturn V and the rocket motors used on the Apollo 11 spacecraft are given in the table about Apollo11 Rocket Motors below.

Some Liquid and Solid Fuel Characteristics

*** Average Density ρ_av is given by:

ρ_av = ρ_oρ_fv(1+r) / (ρ_fr+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