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Politecnico di Milano

SPACE PROPULSION NOTES

Master in Space Engineering

Year 2023-

Contents

• 1 INTRODUCTION
• 2 FLUIDODYNAMICS
  • 2.1 NON REACTING FLOWS
  • 2.2 REACTING FLOWS
    • 2.2.1 FLAMES
• 3 SOLID ROCKET MOTORS
• 4 LIQUID ROCKET MOTORS
• 5 HYBRID ROCKET ENGINES
• 6 MONO-PROPELLANTS
• 7 NUCLEAR PROPULSION
• 8 ELECTRIC PROPULSION

We have 3 main energy sources :

1. Chemical; 2. Nuclear; 3. Radiant; These are the only sources that can be used in a propulsion system. Chemical means combustion. Nuclear as nuclear reactor. Radiant as laser propulsion, solar panels. The primary energies sources are characterized by different energy-densities. It cannot be used directly but it requires a transformation. Once transformed, we have only 2 types of converted energy source that are thermal and electrical energies; in fact there are only 2 types of propulsion unit (thermal and electrical). Those sources are useful to generate the propulsion effect (kinetic energy) through the acceleration device. Every step has losses inside. In the propellant we have the storage and the feed system that make the propellant to flow into the acceleration device. There are 2 main families of propulsion, as illustrated in the figure:

Figure 2: Main families of propulsion.

They depend on the transformed energy source.

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Figure 3: Example of a rocket motor.

There is a heat generation into the combustion chamber and we obtain a thermal energy source.

Figure 4: Example of a rocket motor 2.

SPACE PROPULSION NOTES 5

MAIN PERFORMANCE PARAMETERS

Characterisation of an engine. We need to develop a methodology to make comparison of things that have extensive and intensive properties. Extensive properties:

1. Thrust; 2. Mass flow rate; 3. Total impulse

Important note: while designing an engine, when the thrust is fixed the size of the system is fixed as well; the thrust imposes, for example, the diameter of the nozzle and also the combustion chamber can be also decided on that basis. Intensive properties:

1. Gravimetric specific impulse; 2. Volumetric specific impulse; 3. Density; 4. Characteristics velocity; 5. Discharge coefficient; 6. Thrust coefficient;

The intensive properties do not depend on the size of the system.

Figure 7: Thrust.

SPACE PROPULSION NOTES 7

The thrust has a static and a dynamic part. The dynamic part is the one generated by the nozzle acceleration while the static part is relevant to the exit value of the pressure. A nozzle is called optimal if the exit pressure is equal to the environmental pressure. How to estimate the average thrust? We integrate in time the thrust and divide it by delta t. One important parameter is the value of c that is the effective exhausted velocity. This is not a real velocity but it has the dimension of a velocity. It comprises the effect of the dynamic part of the thrust and the static part. It is coincident with the exit velocity only for optimal nozzle.

Figure 8: Gravimetric specific impulse.

Is is needed to understand how much the rocket is consuming. The higher is the specific impulse the lower is the consumption of the rocket for the same thrust value. So the Is is the opposite of the fuel consumption. The higher is Is, the lower is the propellant needed. Chemical propulsion unit typically don’t exceed 500 seconds of Is because this value is for the cryogenic propulsion unit that is the best to have. In the case of 400 seconds, these are liquid propellant that can be partially storable (propellant that can exists at ambient temperature). A cryogenic system doesn’t exist at ambient temperature. There is a line placed at 25 °C; if a system exists at that temperature without boiling, then it is considered storable. Otherwise, if it boils below 25 °C, then it’s cryogenic.

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energy density storage. The gravimetric is related with the mass of the launcher (the higher is Is, the lighter is the launcher or propulsion unit). The higher is Iv, the more compact is the launcher; it refers to volume. So it has to be maximise for compact applications. Typically we maximise the gravimetric for LOx/LH 2 and we maximise the volumetric for solid propellant. The density of LOx is 1100 kg/m^3 while the density of LH 2 is 70-80 kg/m^3. The liquid hydrogen problem is the density (occupies too much space). The liquid methane allows a more compact design and has a similar boiling point with respect to liquid hydrogen and the thermal behaviour of the rocket is much better.

Figure 10: Total impulse.

The total impulse is the integral of the thrust in the time, in particular from the time of ignition to the time of burnout. This is an indication of the energy that can be provided from the propulsion unit.

When we talk about launcher or satellite the interest is in the Δ𝑣 because we want to

accelerate something or change an orbit, so the Δ𝑣 budget is a key point.

Δ𝑣 = 𝐼𝑠 · 𝑔 0 · ln

Where Mf is the final mass and is the inner mass of the satellite/launcher.

Δ𝑣 is also an energetic budget and is linked to a system. The total impulse is also an

energetic budget but is related to the propulsion system only. This is the difference between them.

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Figure 11: Specific impulse for non-constant thrust profile.

Let’s consider a throttleable propulsion unit, typical for liquid propulsion system. Liquid rockets can be conceived to change thrust or the specific impulse during the operation.

𝐼 s =

At the denominator we have the integral of the mass flow rate over the total time from which we obtain the total propellant mass Mp. So from the previous equation we get:

𝐼 s =

𝐼 tot

That is the average specific impulse. This formula is valid for constant and throttleable

operations, impulsive operations and so on. It can be inserted into the Δ𝑣 budget and get

provisions in terms of Δ𝑣 balance and mass estimation.

Two comments:

1. typically in a variable propulsion unit we change the mass flow rate but we can make in the principle the assumption that the specific impulse doesn’t change much if we keep the same oxidizer fuel ratio. If we introduce losses, we have a decrement of the specific impulse when we move outside the nominal behaviour of the engine and so it slightly changes. But in nominal case, we can assume that the throttleability of the engine can change only the thrust if we keep the same oxidizer fuel ratio.

SPACE PROPULSION NOTES 11

The characteristic velocity is a parameter of merit of the combustion chamber only. It is defined as:

𝑐∗^ =

Pc : pressure of the combustion chamber, Ac : throat area , 𝑚¤𝑝 : propellant mass flow rate.

The characteristic velocity is a matter of thermodynamics. There is another definition of this parameter that is also called discharge coefficient CD.

This is the opposite of 𝐶∗, so :

𝐶D =

The parameter of merit of the nozzle is called thrust coefficient that gives information on how the nozzle is performing. It is indicated with CT. The thrust coefficient is defined as:

𝐶D = 𝑃𝑇

The characteristic velocity is so called because it has the dimensions of a velocity but it is not, it hasn’t any physical meaning. The thrust coefficient is a non dimensional parameter given by the ratio of 2 forces.

Figure 13: Thrust coefficient.

SPACE PROPULSION NOTES 13

The characteristic velocity is a pure thermodynamic property that is function of the molar mass of the system Mmol, the gas inside, temperature of the gas Tgas and the specific heat

ratio of the gas 𝛾 =

, where 𝑐p is the specific heat at constant pressure and 𝑐v is the

specific heat at constant volume. 𝑇 gas is the flame temperature inside the combustion

chamber. On the other side, the thrust coefficient is function of the molar mass of the system Mmol,

the specific heat ratio of the gas 𝛾 and of the nozzle property 𝜀 = 𝐴 𝐴exit

𝑡

that is the ratio

between the exit area and the throat area of the nozzle. It is also called Nozzle Area Ratio as known as the Expansion Ratio.

Once we have these information, the ideal 𝑐∗^ is known. So these 2 parameters can be

derived ideally either thermodinamically that from gasdynamic. But rocket system is not ideal, so the propulsion unit won’t be efficient as expected. Thrust will be lower than expected and the first clue that something is wrong comes from the measurement of the specific impulse.

How can we obtain the specific impulse, 𝑐∗^ and CT from a real engine?

Consider this ideal experiment. We have a liquid propulsion unit that can measure instantaneously the mass flow rate of the propellant. Start from looking at the definition

of 𝑐∗. We measure 𝑐∗^ with the pressure Pc. We have information on At, even though there

might be erosion of the throat and this must be monitored by measuring the throat before and after the firing and then make an average of the 2 measurements and follow the evolution. Then the propellant mass flow rate is measured, obtained by a mass flow meter. Later it must be decided how to implement a pressure measurement with a transducer. Pc is a total pressure. The total pressure is :

where P is the static pressure. The pressure transducer is measuring a static property. Is the static pressure that generates the force on a wall. So we need a point where the static pressure sensed by the transducer and the total pressure, the one we want, coincide. That point exists when the Mach number is equal to zero, so in still condition. We place the transducer on that point that is at the head of the engine, where the speed is negligible (almost zero). Once in the combustion chamber, the flame heats up the gas that accelerate towards the nozzle. Now we can measure the real and effective characteristic velocity.

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MISSION AND PROPULSION TYPES

The specific impulse equation for an optimal nozzle :

𝐼 s = 𝑚 ¤𝑇𝑝 𝑔 0 = 𝑔𝑐 0 = 𝑣^ 𝑔exit 0

The effective exhausted velocity for an optimal nozzle is the exit velocity. The jet power :

𝑃 jet =

𝑚¤𝑝𝑣^2 exit =

𝑚¤𝑝 𝐼 s^2 𝑔 02 =

𝑇 𝐼 s 𝑔 0

The jet power is provided by an energy source. If we want high Is and high thrust, we need high jet power so we need high power availability on board. In order to have a long space exploration in reasonable time, it is needed a propulsion unit capable of high thrust because it’s necessary a high acceleration in short time, but also is needed high specific

impulse in order to have high Δ𝑣. If we have a limited budget on board (in terms of

power) as for S/C, we can’t indefinitely grow Is and thrust together. The solution is an intermediate point in which we balance both according to the specific need. So:

• High thrust to reduce the propulsion action and reach max velocity sooner
• High specific impulse to save propellant and allow higher peak velocity for the same propellant mass ratio.

The propulsion system is designed for a specific type of mission and typically can’t be used for another type of use.

MAIN PROPULSION TYPES

We can identify 2 main groups in propulsion:

• Primary propulsion , typically used for launchers, which is a high thrust propulsion. From a primary propulsion unit it is obtained an acceleration and a

substantial Δ𝑣 change in the mission.

• In-space propulsion , which is typically a subsystem with different roles and it is characterized by a not high thrust level and it doesn’t generate high acceleration

inside the spacecraft. The Δ𝑣 obtained from each fire of the propulsion system is a

minor Δ𝑣 contribution which is used typically for orbit trim, station keeping,

attitude control.

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When it come to the selection of propulsion , it can be considered the following picture:

Figure 14: Main propulsion types.

On the horizontal axis we have the acceleration obtained by the operation of the propulsion system or the thrust over weight ratio T/W if we consider a launcher. On the vertical axis there is the specific impulse. The oblique lines represent the power density (kW/Kg produced by the propulsion system). A launcher has to grant T/W larger than

unit. It can be built up only with systems in the area that goes from T/W = 1 to the right.

In the ovals there are different technologies. Chemical rockets is the typical system used for launchers. Fission stays for nuclear technologies for launcher; it has been proved in the ’60 that a Nuclear-Thermal Propulsion system NTP gives a T/W greater than one. We must consider the associate specific impulse that for chemical rockets and fission is low because thrust is high. In case of launchers we can expect that because it is a high thrust regime that is called primary propulsion regime. With electro propulsion we work with high specific impulse but low T/W, so this is the range of in-space propulsion.

Examples of primary propulsion: launch missions (as Space Shuttle with T/W = 3 ) and

planetary landing and return (as Apollo APS with T/W = 2 in lunar gravity field). N.B

primary propulsion unit may not be a primary propulsion system on the earth but they

SPACE PROPULSION NOTES 17

Assuming to be in vacuum 𝑃𝑎 = 0 (space engine or rocket case). So the thrust depends on

the mass flow rate: high 𝑚¤𝑝 means high thrust 𝑇.

Δ𝑣 budget is given by:

Δ𝑣 = 𝐼𝑠 𝑔 0 log

= 𝐼𝑠 𝑔 0 log

Where 𝑀𝑜 is the initial mass, 𝑀𝑓 is the final mass and 𝑀𝑝 is the propellant mass. So we

have to discharge 𝑀𝑝 in order to have Δ𝑣. This Δ𝑣 is obtained when all the 𝑀𝑝 is

discharged. If we take the assumption of constant propellant mass flow rate , the 𝑀𝑝

discharged is obtained as follow :

𝑀𝑝 = 𝑚¤p𝑡b

Where 𝑡𝑏 is the burning time. So:

𝑡b =

𝑚p

In case of electric propulsion, the propellant mass flow rate is low and so the action time^1

grows dramatically. The 𝑚¤𝑝 is low because the thrust is low and so the time to discharge

the amount of propellant 𝑀𝑝 is high and this means there is not an impulsive manoeuvre.

An impulsive manoeuvre is a manoeuvre that assumes that Δ𝑣 is obtained in an

instantaneous manoeuvre : the total propellant mass is discharged in a time which tends to zero and the propellant mass flow rates tends to infinity as the thrust tends to infinity; this goes beyond the electric propulsion capability. The impulsive manoeuvre works when the duration of the manoeuvre is small with respect to the orbit time. The kick motors are capable of doing the firing in a short leg of the trajectory. But this motors don’t fire along the trajectory like an electric propulsion does (they fire continuously). Let’s consider DEEP SPACE MISSIONS. These missions require high specific impulse. If we want to accelerate as much as possible and we have limited volume and mass available inside the S/C, so the amount of propellant on board must be used in the most efficient way that means high specific impulse. Also long term drag compensation and long term formation flights require high specific impulse due to the long term because we have to save and not waste propellant during the mission.

(^1) It is the corresponding burning time but it’s called action time because in the electric propulsion there is not a burn.

SPACE PROPULSION NOTES 19

Figure 16: Performance and technology 1.

Red : chemical, Yellow : electric, Pink : radiant, Green : nuclear.

Figure 17: Performance and technology 2.

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