# 35 Rocket Propulsion

# Introduction^{[1]}

Rockets launch payloads such as satellites and space probes into Earth orbit. By imparting a time rate of change of momentum to a gas flow produced by a propellant, a force is applied from the rocket motor(s) to the vehicle. Before being expelled from the motor(s), the propellant can be stored onboard as a compressed gas, in separate fuel and oxidizer tanks, or as a solid mixture of fuel and oxidizer.

A representative launch profile of a rocket is shown in the figure below. At the moment of initial launch, the thrust produced by the rocket motors will be greater than the vehicle’s weight, so the rocket accelerates away quickly from the pad. The rocket’s weight also rapidly decreases because of the high fuel consumption, so the rocket continues to accelerate as it gains altitude. As the rocket begins to exit the atmosphere, which is about 60,000 ft (about 20,000 m), it will be flying at supersonic speeds. It also begins to pitch over to a more horizontal flight path, so the rocket gains translational velocity for the payload to reach an equilibrium orbital altitude.

Several minutes into the ascent, staging will occur where the first stage is jettisoned and the rocket motor for the second stage is ignited. The first stage then falls back to the surface and either burns up in the atmosphere (depending on the staging altitude) or crashes into the ocean. In exceptional cases, the first stage may be recovered; solid rocket boosters are usually recovered by parachute and may be reused. The upper stage continues to accelerate into space, and at a predetermined altitude and speed, the rocket motors cut off as the payload reaches the necessary orbital velocity and altitude.

Objectives of this Lesson

- Understand the basic principles of rocket propulsion systems.
- Distinguish the different types of rocket motors and their purpose.
- Appreciate the concept of specific impulse as a measure of rocket efficiency.
- Know how to derive the rocket equation and use it to solve some simple rocket problems.
- Understand the process of staging and why it is used for lunch vehicles.

# Rocket Propulsion Fundamentals

Newton’s second and third laws are the basic physical principles that apply to rocket propulsion systems. Consider a typical rocket engine, as shown in the figure below. The principle of thrust generation for a rocket engine is from the reaction force associated with accelerating a mass of gas at high velocity out of an expansion nozzle, the gas being a byproduct of combustion of the fuel and the oxidizer, and so increasing the momentum of the gas. Notice that the oxidizer must be carried along with the fuel for a rocket motor. The gases and products of combustion are accelerated in one direction, and the resultant force on the vehicle is directed in the opposite direction in accordance with Newton’s third law.

Using the principles of conservation of momentum, then the thrust produced by the rocket engine will be

(1)

where is the exit pressure at the nozzle, is the ambient pressure, is the exit area, is the exit velocity, and is the propellant mass flow rate.

The first term on the right-hand side of Eq. 1, i.e., the term, is a momentum flow rate, and the second term is the net force resulting from a pressure difference. There is no external mass flow into the engine, such as with an air-breathing engine. It will be apparent that for a rocket, then . If and so the pressure force term goes to zero. In this case, the nozzle is said to be *ideally expanded*.

For a rocket engine it is useful to write for the thrust that

(2)

where is called an *equivalent exhaust velocity* that includes the pressure term, i.e.,

(3)

In practice, the pressure term is relatively small so the value of is very close to .

# Steering a Rocket

A rocket must be steered along a prescribed flight path so the payload can reach the necessary altitude. A modern rocket (launch vehicle) is usually steered along its flight path by gimbaling (rotating) the engine(s) to change the direction of the thrust vector. However, as summarized in the figure below, other means may be used.

Early rockets used movable aerodynamic surfaces or fins at the rear of the rocket, and this technique is also used on most air-to-air missiles. These surfaces create varying aerodynamic forces on the rocket, which can be used to control its trajectory. Some rockets have used additional vernier rocket engines to give control. However, because of this system’s additional weight and the different fuel needed, vernier rockets are not used as much. On some early rockets and ballistic missiles, small thrust vanes were placed directly in the exhaust stream of the rocket exhaust to produce forces that could be used for steering.

# Types of Rocket Motors

Like all propulsion systems, rocket motors are energy conversion devices. The kinetic energy of the expelled propellant (hence the eventual gain in kinetic energy of the vehicle) comes from:

- The work of compressing the propellant into its tank.
- Liberating the chemical potential energy of a fuel and an oxidizer.
- An electrical or thermal power supply.
- Some combination of these latter methods.

Rocket motors can be broadly categorized according to their thrust and thrust-producing efficiency. Rocket propulsion systems are selected according to mission objectives. Generally, there is no “one-size-fits-all” solution, and several rocket propulsion systems could be used for a given space mission. There are two primary types, namely a liquid propellant rocket and a solid propellant rocket, as shown in the figure below, the latter type often being used as a secondary booster.

# High Thrust Rocket Propulsion Systems

High thrust systems are used to overcome gravity, as in a planetary launch vehicle, or to quickly accelerate a vehicle already in space, i.e., for an orbital ejection maneuver. These systems store energy in the propellant so that energy can be converted at a high rate, roughly proportional to the propellant flow rate.

## Bipropellant Systems

Bipropellant systems typically come to mind at the mention of rocket propulsion, i.e., one imagines flames and clouds of smoke, such as during a NASA Space Shuttle or SpaceX Falcon 9 launch. The propellant is the combustion product of a fuel and an oxidizer. Combustion is generally the fastest way to convert propellant energy. Bipropellant systems are further categorized as gas/liquid propellant systems, solid propellant systems, or hybrid systems.

## Gas/Liquid

Examples of this first type include the space shuttle main engine (SSME), which burned liquid hydrogen (LH) and liquid oxygen (LOX), and the Raptor engine of the SpaceX Falcon 9, which burns Rocket Propellant-One or RP-1 (a densified kerosene) and liquid oxygen or LOX. The process of mixing the fuel and oxidizer in the motor is shown in the figure below. The large volume flow rates require turbopumps, which are driven by burning a quantity of fuel and oxidizer tapped off from a bypass circuit.

Liquid methane (CH) is gaining in popularity as a fuel for commercial spacecraft uses because of its availability and performance. It is also far better for the environment than RP-1, which produces a lot of toxic byproducts. One advantage of a gas/liquid system is that the engine can be throttled by regulating the fuel flow. However, this capability comes at a price, including the complexity and associated weight of pumps, valves, pipes, and cryogenic fuel tanks.

Hypergolic propellants are those that combust spontaneously upon contact with one another. These are used for in-space applications rather than launch vehicles, partly because their performance is lower than a hydrocarbon and LOX system. Also, on the one hand, these chemicals tend to be highly toxic. However, on the other hand, their advantages are reliability, simplicity (no ignition system required), and ignition speed. For example, the Apollo lunar lander used Hydrazine (NH) and nitrogen tetroxide (NO), as did the Space Shuttle reaction control system (RCS).

## Solid

The Space Shuttle solid rocket booster (SRB) is an excellent example of a solid propellant system. In addition, solid propellants are favored for military applications, e.g., air-to-air missiles and intercontinental ballistic missiles (ICBMs), because they require little pre-launch processing.

Solid propellant fuel is usually a powdered metal, most commonly aluminum (Al) and sometimes magnesium (Mg). Typical oxidizers are ammonium perchlorate (AP) and ammonium nitrate (AN). The fuel and oxidizer are mixed with a binder, usually plastic or synthetic rubber, such as hydroxyl-terminated polybutadiene (HTPB) or polybutadiene acrylonitrile (PBAN). The Space Shuttle SRBs used Al, AN, and PBAN. Most commercially available motors for amateur use have Al, AP, and HTPB, because this combination is more efficiently and safely processed. Unlike liquid-fueled rocket motors, solid rockets cannot be throttled, and once ignited, they must burn until the fuel is exhausted.

## Hybrid

Hybrid systems have a solid fuel and a gaseous or liquid oxidizer, or rarely, the reverse. Experimentalists favor hybrids because small rocket systems can be relatively simple to construct. It is often argued that hybrid systems are safer than solid or liquid systems, although this is not true. Hybrid fuels are typically the same materials used as binders in solid propellants. Typical oxidizers are oxygen, nitrous oxide (NO), and hydrogen peroxide HO.

Hybrids have not found much favor in commercial applications because they have no performance advantage, and designing for optimal performance is primarily a cut-and-try process. A notable exception is Spaceship One, which uses HTPB and NO. Experimentalists often use polyvinyl chloride (PVC) or acrylonitrile butadiene styrene (ABS) as fuel because these are readily available and have low costs. NO is readily available at automotive stores for use in high-performance engines.

## Monopropellant Systems

Monopropellants do not burn but decompose exothermally in contact with a catalyst. Monopropellant systems are favored for small vehicles and reaction control systems because of their relative simplicity over bipropellants, although at the expense of reduced performance. Hydrogen peroxide (HO) decomposes into water and diatomic oxygen in contact with many metal oxides, especially silver oxide. A HO system propelled the Apollo lunar lander trainer. Hydrazine has been used more extensively because of its higher performance and ease of reaction initiation. The New Horizons spacecraft engine is an NH monopropellant system.

## Cold Gas Thrusters

Cold gas thrusters are the simplest by far, consisting of only a compressed gas tank and nozzle and proper piping and valves. Their disadvantage, of course, is relatively low performance. Therefore, they are appropriate for cubeSats, nanosats, and attitude control of small spacecraft. Any gas can be used as a propellant, but those with lower molecular weight, such as hydrogen (H) and helium (He), give better performance.

## High-Efficiency Rocket Propulsion Systems

In high-efficiency systems, the energy is not stored in the propellant but is generated by an onboard system. Therefore, the energy conversion rate is not proportional to the propellant flow rate but is limited by the power supply system’s capability. For example, solar panels or a nuclear source can generate either electrical or thermal power. Considering the energy supply rate (power) to the propellant to be fixed, a lower propellant flow rate will give higher efficiency but at a lower thrust, which is insufficient for use as a launch vehicle. High-efficiency systems are then used for long-duration missions to deep space or to raise orbits of Earth satellites.

In each case, this propulsion system does not have to oppose gravity directly (that is, to “lift” the vehicle). However, it increases its velocity gradually once the vehicle is already in space. The most straightforward system heats the propellant gas, which expands to high speed through a nozzle. A solar thermal system collects and focuses the sun’s rays onto the propellant flow path. A thermal electric system heats the gas with a resistive element or an electric arc. In other systems, electrical power is used to ionize the propellant gas and produce an electric and/or magnetic field, after which the charged particles are accelerated. Several configurations exist for such systems, including ion thrusters, Hall effect thrusters, and magnetoplasmadynamic thrusters.

# Total Impulse & Specific Impulse

Establishing the characteristics of a rocket motor requires some quantitative measures of performance. The *total impulse* is defined as the integral of the thrust over the “burnout time,” i.e.,

(4)

If and are constant, as is often a good approximation, then

(5)

where is the mass of propellant burned. The total impulse, therefore, is the net momentum imparted to the rocket during the burn.

The measure of efficiency used in most rocket performance calculations is the *specific impulse*, which is the thrust divided by the propellant flow rate. In general, one wishes to carry as little propellant as possible. The specific impulse, , is

(6)

where is acceleration under gravity at sea level. It is further apparent using Eq. 2 that

(7)

where is measured in units of time (seconds). Therefore, by definition, the specific impulse is the total impulse (or change in momentum delivered) per unit weight of propellant consumed, Notice then that its value is dimensionally equivalent to the generated thrust divided by the propellant flow rate in terms of weight of fuel per unit time, and so in some ways is equivalent to the inverse of the thrust specific fuel consumption used by a jet engine.

**Thrust or Specific Impulse?**

Remember that thrust is a

*force*supplied by the rocket engine and its value will depend on the amount of propellant flowing through the engine.

*Specific impulse*is not a force because it value is a measure of the impulse or momentum produced per unit of propellant and will be proportional to the exhaust velocity.

Notice that if mass (slugs or kg) is used as the unit of propellant, then the specific impulse has units of velocity. If the weight (lb or N) is used, which is much more common, then the specific impulse has units of time. Notice that these two definitions differ by a factor of . The higher the specific impulse, the less propellant is needed to produce a given thrust during a given time. Some propellants and their specific impulse values are given in the table below. While it can be seen that H2/O2 (LOX) has the highest specific impulse, liquid hydrogen needs a much larger (volumetric) fuel tank than RP-1, and it is also more challenging to transport and store.

Propellant | Specific Impulse (secs.) |

H_{2}/O_{2 }(LOX) |
445 |

RP-1/O_{2} (LOX) |
295 |

H_{2}O_{2} |
300 |

CH_{4}/O_{2(}LOX) |
320 |

# Rocket Equation

Consider an accelerating rocket vehicle where the engine’s thrust is used to propel a vehicle of mass . Therefore, it can be written that

(8)

where is the acceleration of the vehicle. Here, the forces from atmospheric drag have been neglected relative to the vehicle’s weight, so the equation is strictly valid for a vehicle in space. However, it is not an unreasonable approximation otherwise, including for flight in the atmosphere.

With substitutions and some algebra, then

(9)

and because the decrease in mass is then

(10)

which is an ordinary differential equation. After integration of the equation then the change in the velocity of the vehicle is

(11)

where is the initial mass of the vehicle and is the final or *burnout* mass. This latter equation is called the* rocket equation* and is very useful in mission performance analysis and vehicle sizing. In some ways, it is analogous to the Breguet equations used for aircraft performance.

If gravity is included (but no aerodynamic drag) in the case of a pure vertical launch, then the rocket equation is modified to

where is the burnout time, although the second (gravity) term is usually relatively smaller than the first term, which is usually referred to as the *gravity loss*. For a rocket going up vertically, then

(12)

where is the burnout time. If the rocket follows a curved trajectory and pitches over at a local trajectory angle (with respect to the horizon) as it increases altitude, then

(13)

where when the rocket flies vertically and when it flies horizontally. The proper evaluation of this latter term, therefore, requires specific information about the launch profile.

The initial mass of the rocket vehicle can be written as the sum

(14)

where is the mass of the propellant, is the structural mass, and is the mass of the payload. The burnout mass is reached when all of the propellant is exhausted, so is given by

(15)

The mass ratio is defined as

(16)

The payload ratio is defined by

(17)

Finally, the structural mass coefficient is defined by

(18)

In light of these foregoing definitions then it can be shown that the mass ratio is given by

(19)

so in terms of the mass and payload ratios and the structural coefficient then

(20)

Payload ratios can vary considerably from mission to mission. However, structural coefficients are found to be relatively constant based on historical data for various types of launch vehicles. Also, the value of can be fairly similar for different vehicles, which is convenient for preliminary design purposes when this value needs to be estimated.

Example #1

A rocket must provide a speed of 6,000 m/s to a payload of 12,000 kg. The structural coefficient of the vehicle is 0.06. The propellants used in the engines have a specific impulse of 325 secs. What must be the initial mass and propellant mass be to meet these requirements?

The equivalent exhaust velocity is

(21)

so the mass ratio is

(22)

and the payload ratio is

(23)

Therefore, the initial mass is

(24)

and the burnout mass is

(25)

Finally, the propellant mass is

(26)

Example #2

The rocket equation gives the change in the velocity of the vehicle , i.e.,

where is the initial mass of the vehicle and is the final or burnout mass. If gravity is included (but no aerodynamic drag), then

where is the burnout time, which is given as 60 seconds in this case.

The equivalent exhaust velocity is given in terms of the specific impulse, i.e.,

and the burnout mass is given by

So now we have the burnout velocity, which is

so

if we do not include the gravity term. With the gravity term included, then

so at the burnout is reduced to

Assuming the rate of fuel consumption is constant, then the mass of the rocket varies over time as

The velocity then is

The height achieved at the burnout time is then

which after some rearrangement gives

Inserting the values gives

so

The final additional coasting height of the rocket can then be determined by equating the rocket’s kinetic energy at its burnout time with its change in potential energy between that point and the maximum obtained height, which is left as an exercise.

Staged Rocket Vehicles

Staging a launch vehicle aims to maximize the payload ratio that can be launched into space. The goal is to launch the largest payload to the required burnout velocity using the least amount of non-payload mass (defined as the structural weight of the rocket plus the fuel). As shown in the figure below, there are two types of staging:

- Serial staging, where the stages are ignited, used, and jettisoned in serial sequence.
- Parallel staging, where all stages are ignited and used, but the stages are jettisoned as they burn out, e.g., solid rocket boosters.

To reach an optimal staging of the launch vehicle, there are four basic considerations:

- The initial stages should have lowest values of , and later stages should have highest .
- The stages with the lower should contribute more to .
- Each successive stage should be smaller than the previous stage.
- Similar stages should provide similar increments to .

For a staged rocket vehicle, then

(27)

where the index refers to the stage number. Also

(28)

For a staged vehicle overall, then the values are added for each stage, i.e.,

(29)

For a staged vehicle then the structural mass coefficients can be similar to those of a single stage. However, payload ratios are generally higher for a staged vehicle. In some cases, it may be desirable to find the maximum allowable structural mass to meet a certain set of payload requirements, which sets some goals for the purposes of design.

# Summary & Closure

Gas/liquid fuel-based rotor motors have been employed for many spaceflight applications, including most launch vehicles. Such systems have good efficiency and have the advantage that the engine can be throttled by regulating the fuel flow, e.g., to limit dynamic pressure loads on the vehicle during its launch. However, this capability comes at a price, including mechanical complexity and associated weight. For some launch vehicles, the thrust from the liquid fuel engines is augmented with solid rocket boosters, e.g., the Space Shuttle concept. Solid motors are also used on missiles and for other in-space applications. The lower performance and non-throttling characteristics of solid propellant motors are acceptable because of their operational simplicity.

Since the dawn of human space flight in the 1960s, advancements have continued to be made by using improved propellants and rocket motor designs. Today, most space missions use a combination of different engines and fuels, which are selected to optimize the thrust capability needed at each mission stage, including the launch and when in space. Rocket motors produce extremely high thrusts and operate near their safe limits, albeit relatively for relatively short times. However, the reliability of rocket motors is still a concern, especially when they are recovered and reused to save on launch costs. In addition, the environmental compatibility of rocket fuels has become a more significant concern in recent years, and the move toward considering alternative propellants such as propane and methane will continue.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

- Discuss and identify safety hazards associated with liquid, solid, and hybrid rocket propulsion systems.
- What type of fuel is used for the SpaceX Raptor engine? What are the advantages of this type of fuel?
- Research high specific impulse propulsion systems. What values for specific impulse are attainable?
- Consider a staged vehicle for which all of the values are the same. How can be maximized?
- What sort of mass is an astronaut?
- What types of thrusters are used for cubesats, nanosats, and small spacecraft attitude control?

Other Useful Online Resources

To learn more about rocket propulsion, check out these helpful online resources:

- A series of videos on how rocket motors work: The playlist is here.
- A good video on how rockets work.
- An article on the history of rockets by NASA.
- A simple guide on how rockets work by NASA.
- Learn more about solid fuel rockets from Northrop-Grumman.
- Some good resources on solid fuel rockets by AeroJet-RocketDyne.
- The Smithsonian Air and Space Museum entry on the X-15 a rocket-powered aircraft.
- An article from AIAA: X-15 Propulsion System
- Elon Musk explains how the Raptor 2 rocket engine works.

**Initial drafts of this lesson were written by Prof. Eric "Rick" Perrell at ERAU.**↵