41 Rocket Engines

Introduction[1]

Rocket engines are used to launch payloads, such as satellites and space probes, into Earth orbit. They are also used to propel spacecraft that are already in space. By imparting a time rate of change of momentum to the gas flow produced by some form of propellant, a force is applied from the rocket engine(s) to the vehicle. Before being expelled from the engine(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. Rocket engines are non-airbreathing, so they must be provided with a fuel and an oxidizer known as a propellant.

Four types of rocket engines can be used for different and relevant applications:

  1. Launch Vehicles: Rocket engines used in launch vehicles are designed to provide high thrust and momentum to lift the vehicle off the ground and into space. They must be powerful enough to overcome Earth’s gravity and give the required kinetic and potential energy for the spacecraft and its payload to reach orbit.
  2. Spacecraft: Rocket engines are designed to propel and maneuver vehicles already in space. They are typically smaller and less powerful than the engines used for launch vehicles. Still, they must be highly reliable, efficient, and versatile to enable the spacecraft to travel long distances and perform complex missions.
  3. Missiles: Rocket engines used in missiles usually operate on solid fuel and are designed to provide the speed and maneuverability to carry warheads or other payloads to their targets. Missiles must be highly precise and reliable and may need to operate in various flight conditions and environments.
  4. Miscellaneous: There are other applications for rockets, including suborbital flights, scientific experiments, and even recreational activities like model rocketry. These types of engines may have specialized requirements depending on their specific use.

Learning Objectives

  • Understand the basic principles of rocket propulsion systems.
  • Distinguish the different types of rocket engines and their purpose.
  • Appreciate the concept of specific impulse as a measure of rocket efficiency.
  • Know how to calculate the thrust of a rocket engine.

Rocket Propulsion Fundamentals

In a rocket engine, the propellants, i.e., a fuel and an oxidizer, undergo combustion at high pressures and temperatures to produce the needed thrust. Notice that, unlike an air-breathing engine, the oxidizer must be carried along with the fuel for a rocket engine. The fastest way to liberate energy is through combustion, which creates a high-speed gas flow that can be expanded through a nozzle at high velocities and Mach numbers to produce tremendous thrust values. An example of a rocket engine operating at full thrust is shown in the photograph below, which, in this case, is a solid propellant booster. Other types of rocket motors can use liquid propellants (fuel and an oxidizer) or some form of hybrid liquid/solid propellant.

In this case, the ferocity of a rocket engine at full thrust is that of a solid propellant booster, augmenting the launch of the ULA Vulcan Centaur rocket.  Photo courtesy of Henry Luniewski.

A rocket engine must be designed to withstand the pressures of combustion and the associated temperatures, plus some margins for safety.  The shape and length of the combustion chamber and exit nozzle are essential design parameters for a rocket engine. The combustion chamber must be long enough for complete propellant combustion before the hot gases enter the nozzle, ensuring efficient combustion and maximizing thrust production. The length of the combustion chamber is typically determined based on numerous factors, such as the type of propellant used, combustion efficiency, and the desired overall performance characteristics of the rocket engine relative to the mission profile.

Rocket engines operate at extremely high temperatures, usually between 2,500^{\circ}K to 3,000^{\circ}K. Therefore, to prevent overheating and structural failure, the combustion chamber and nozzle must be actively cooled to dissipate the heat generated during combustion. With liquid-fueled rocket engines, this process is accomplished by circulating cold fuel around channels or jackets within the walls of the combustion chamber and nozzle. The fuel absorbs the heat by conduction and convects it away, allowing the rocket to operate safely for extended times. A byproduct is that more efficient combustion is obtained by pre-heating the fuel before it is combined with the oxidizer. On solid-fuel rocket engines, which usually operate for shorter times, the nozzle is made of high-temperature materials that can be ablatively cooled, i.e., the material sheds its surface in layers along with the absorbed heat energy.

Application of Conservation Principles

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 gases at high velocity out of an expansion nozzle, the gases being a byproduct of combustion of the fuel and the oxidizer, and so increasing the kinetic energy and momentum of the gases. Consequently, the gases and products of combustion are accelerated in one direction, and the resultant force on the vehicle is directed in the opposite direction, per Newton’s third law.

 

Control volume for the analysis of the thrust produced by a rocket engine.

If the flow is assumed to be steady, compressible, and inviscid, then the general form of the momentum equation is

(1)   \begin{equation*} - \oiint_S p \, d\vec{S} = \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V} \end{equation*}

The pressure integral on the left-hand side can be written as

(2)   \begin{equation*} \oiint_S p \, d\vec{S} = \iint_{A_e} p \, d\vec{S} + \iint_{\rm Nozzle} p \, d\vec{S} \end{equation*}

the throat area is much smaller than the exit area, so the pressure integral here can be assumed to be zero.

The force on the fluid to change its momentum, which by Newton’s 3rd law, is in the opposite direction to the thrust, T. Therefore, the momentum equation becomes

(3)   \begin{equation*} T = \oiint_S (\varrho \, \vec{V} \bigcdot d\vec{S}) \vec{V} + \iint_{A_e} p \, d\vec{S} \end{equation*}

If the flow can be assumed to be one-dimensional, which is a reasonable assumption for now, the thrust T produced by the rocket engine will be

(4)   \begin{equation*} T = \overbigdot{m} V_j + \left( p_e - p_a \right) A_e \end{equation*}

where p_e is the exit pressure at the nozzle, p_a is the ambient pressure, A_e is the exit area, V_j is the exit or jet velocity, and \overbigdot{m} is the propellant mass flow rate. Remember that there is no external mass flow into a rocket engine, i.e., no value of V_{\infty}, such as with an air-breathing engine.

The first term on the right-hand side of Eq. 4, i.e., the \overbigdot{m} V_j term, is a momentum flow rate or time rate of change of momentum, and the second term is the net force resulting from a pressure difference between the exit gases and the ambient pressure. For a rocket, the time rate of change of momentum is much larger than any pressure force, i.e., \overbigdot{m} V_j \gg (p_e - p_a) A_e. If p_e = p_a, the pressure force term is zero. In this case, the thrust produced is a maximum, and the nozzle is said to be ideally or optimally expanded; this condition is a design goal for a rocket engine.

For a rocket engine, especially when their performance characteristics are being compared, it is often useful to write for the net thrust that

(5)   \begin{equation*} T = \overbigdot{m} V_{\rm eq} \end{equation*}

where V_{\rm eq} is called an equivalent exhaust velocity that includes the pressure term, i.e.,

(6)   \begin{equation*} V_{\rm eq} = V_{j} + \left( p_e - p_a \right) \frac{A_e}{\overbigdot{m}} \end{equation*}

In practice, however, the pressure term is relatively small because the nozzle is designed for (or close to) ideal expansion where p_e = p_a, so the value of V_{\rm eq} is very close to V_j.

Check Your Understanding #1 – Calculation of rocket thrust

Compare the thrust and efficiency of a rocket engine versus a turbojet engine at a flight speed of 200 m/s. Both engines exhaust the flow with a mass flow of 40 kg/s. The rocket exhausts at a velocity of 3,000 m/s, and the turbojet at 800 m/s. Assume that the flow in the exhaust is ideally expanded in both cases.

Show solution/hide solution.

The thrust T produced by the rocket engine with ideal expansion will be

    \[ T = \overbigdot{m} \, V_j \]

where V_j is the jet or exit velocity and \overbigdot{m} is the propellant mass flow rate. Inserting the numerical values  gives

    \[ T = \overbigdot{m} \, V_j = 40.0 \times 3,000 = 120.0~\mbox{kN} \]

Remember that a rocket engine is not air-breathing, so there is no inlet mass flow rate.

The thrust T produced by the turbojet engine with ideal expansion will be

    \[ T = \overbigdot{m} \, \left( V_j - V_{\infty} \right) \]

where V_{\infty} is the airspeed and inlet velocity. Inserting the numerical values  gives

    \[ T = \overbigdot{m} \, \left( V_j - V_{\infty} \right) = 40.0 \times \left( 800.0 - 200.0 \right) = 24.0~\mbox{kN} \]

Nozzle Shape

The nozzle is designed to accelerate the high-pressure, high-temperature gases generated in the combustion chamber to a very high supersonic exit or jet velocity. The nozzle typically has two sections, the convergent and divergent sections, as shown in the figure below. The resulting shape is often referred to as a nozzle “bell.” The convergent section narrows down the flow area, increasing the velocity of the gases as they pass through it and accelerating them to transonic and then supersonic conditions.

A rocket engine has a convergent/divergent nozzle designed to expand the exhaust gases to supersonic conditions.

The convergent section then leads to the throat, the narrowest part of the nozzle. Beyond the throat, the divergent area widens, allowing the supersonic gases to expand and accelerate to their final velocity, V_j. Ideally, this expansion continues smoothly and progressively until the hot gases reach the exit area of the nozzle.

Therefore, the design of the nozzle, including the length and shape of the divergent section, is critical in achieving optimal thrust performance from the rocket engine. Generally, the nozzle must be long enough, and the exit area must be large enough to ensure that the pressure at the exit is close to the ambient pressure outside the nozzle. This feature is essential for maximizing the propulsion system’s efficiency and achieving maximum thrust.

Nozzle Efficiency

Designing rocket engines for launch vehicles that can operate efficiently across a wide range of in-atmosphere altitudes is a significant engineering challenge that requires careful consideration of many factors. These factors include the shape and size of the nozzle, the materials used for the nozzle, the propellant flow rate, the combustion characteristics of the fuel and oxidizer, and the cooling system(s). Designers aim to achieve the best possible performance of the rocket engine across the entire altitude range of a rocket’s atmospheric flight trajectory to maximize its thrust and efficiency. Rocket engines must also be optimized for efficiency when they reach the vacuum of space.

When the exhaust pressure at the exit of the nozzle matches the ambient pressure of the surrounding environment, it is known as ideal or optimum expansion, as shown in the figure below. In this ideal state, there is a zero pressure gradient, and all the exhaust gases are directed away from the engine, resulting in maximum thrust generation because nearly all of the momentum of the exhaust gas is converted into thrust. This operating condition gives rocket engines their highest possible performance in terms of thrust and efficiency. However, achieving optimal expansion requires careful design and optimization of the nozzle’s shape, i.e., the shape of the bell.

 

Rocket engines’ nozzles must be designed to balance thrust and efficiency over various ambient pressures. Under- or over-expansion of the gas flow will result in a loss of thrust and efficiency.

Over-expansion means that the external (atmospheric) pressure, p_a, is higher than the exit pressure, p_e. When an over-expanded flow passes through the nozzle, the higher external pressure at the exit produces a positive (or adverse) pressure gradient that slows the jet flow, and the jet flux subsequently converges as it exits the nozzle. The pressure difference may be high enough even to cause the flow to separate from the nozzle’s walls. Over-expansion of the gas flow reduces the thrust and efficiency of the engine. The solution, in this case, is to use a shorter bell.

The opposite situation, where the atmospheric pressure is lower than the exit pressure and gives an adverse pressure gradient, is called an under-expanded flow. In this case, the flow continues to develop and expand outward after it exits the nozzle, and so this process also does not contribute to thrust production. The solution, in this case for thrust recovery, is a larger and longer bell. When designing rocket engines for launch vehicles that must fly in the atmosphere, the nozzle may be designed for a slight over-expansion at sea level, i.e., recognizing that the exhaust pressure at the exit of the nozzle will likely be lower than the ambient pressure of the surrounding air. This design approach can better optimize the rocket engine’s performance during more of the launch profile, allowing the engine to balance, on average, its overall thrust and efficiency throughout the atmosphere.

Many rocket engines, including the Merlin used in the Falcon 9, have a nozzle designed to work efficiently across a wide range of altitudes, from sea level to the stratosphere, where the pressure is very low. The RS-25 engines, first used for the Space Shuttle program, were optimized for sea-level operation during the initial phase of the launch but also when it transitioned to vacuum-optimized operation. To this end, the RS-25 engine has a movable nozzle extension to optimize its performance. When rocket engines operate at sea level, the nozzles are usually designed to give a slightly over-expanded condition, so they become more ideally expanded with increasing altitude in the atmosphere.

 

Rocket engine nozzles are designed so the pressure of the hot gases exiting the nozzle matches the external pressure at any given altitude or in the vacuum of space, providing maximum thrust and efficiency.

It will be noticed that second or upper-stage “vacuum-optimized” rocket engines have much larger nozzles than those used on sea-level (or atmospheric) optimized engines, as shown in the figure below. The Merlin second-stage engine is a good example. The second stage, “vacuum-optimized” Merlin, uses the biggest nozzle as practically possible to get an ideal expansion of the exhaust gases. The vacuum-optimized Merlin engine has a bigger exhaust section and a larger expansion nozzle ratio of 165:1, compared to the sea-level optimized version, which has a smaller 16:1 expansion ratio nozzle. The larger nozzle allows a more ideal and efficient expansion of the exhaust gases in the space vacuum, maximizing the propulsive thrust and efficiency.

“Over-Expanded” Versus “Under-Expanded”?

Remember that the nozzle is designed to accelerate the exhaust gases and convert the thermal energy of combustion into kinetic energy. At lower altitudes, the external atmospheric pressure is higher than at higher altitudes. This means that the exhaust gases from the rocket engine encounter a higher external pressure as they expand supersonically to reach the diameter of the nozzle exit. This higher external pressure resists the expansion of the flow, and so the flow “over-expands” itself within the limits of the nozzle. The external ambient pressure is lower at higher altitudes, allowing the flow to expand more quickly. However, as the flow reaches the nozzle exit, it may not have expanded enough, resulting in an “under-expanded” gas flow.

Types of Rocket Engines

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

  1. Compressing the propellant into its tank.
  2. Liberating the chemical potential energy of a fuel and an oxidizer.
  3. An electrical or thermal power supply.
  4. Some combination of these latter methods.

Rocket engines 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. As shown in the figure below, there are two primary types: a liquid propellant rocket and a solid propellant rocket. The latter type is often used as a secondary booster. Another type, the hybrid rocket engine, is considered later.

 

The two primary types of rockets are liquid-propellant rockets and solid-propellant rockets. Each has relative advantages.

High-Thrust 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 propulsion systems typically come to mind regarding 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 solid fuel/oxidizer systems.

Gas/Liquid Systems

Examples of this type of propulsion system include the Space Shuttle main engine (SSME), which burned liquid hydrogen (LH2) and liquid oxygen (LOX), and the Merlin engine used on the SpaceX Falcon 9, which burns Rocket Propellant-One or RP-1 (a densified kerosene) and LOX, the combination often being called Kerolox. The process of mixing the fuel and oxidizer in the engine is shown in the schematic below. The enormous volume flow rates require turbo-pumps, which are driven by burning a small quantity of fuel and oxidizer tapped off from a bypass circuit.

The internal flow paths of fuel and oxidizer inside a rocket motor have to flow through pipes at high volumetric rates created by turbo-pumps.

As shown in the photograph below, a Saturn V rocket launched Apollo 11, the first mission to land humans on the moon, lifted off from Kennedy Space Center in July 1969. The first stage of the Saturn V rocket used five F-1 engines. These engines burned RP-1 fuel with LOX as the oxidizer. The F-1 engine was the most powerful single-nozzle liquid-fueled rocket engine ever flown. The second stage used five J-2 engines; the propellant for these engines were LH_2 and LOX. The J-2 engine was designed to provide efficient propulsion in the higher levels of the atmosphere and into space. The third stage of the Saturn V rocket used one J-2 engine, which was also used for the trans-lunar injection maneuver, which set the spacecraft in the direction of the Moon.

The Saturn V three-stage rocket. The first stage rotor motors burned RP-1 and LOX.

Liquid methane (CH_4) is gaining popularity for commercial rocket engines because of its availability, modest cost, and good performance. It is also far better for the environment than RP-1, which produces many 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 propellants are used for in-space applications rather than launch vehicles, partly because their performance is much lower than that of hydrocarbon and LOX or RP-1 and LOX systems. Also, on the one hand, these chemicals tend to be highly toxic. On the other hand, their advantages are reliability, simplicity (no ignition system required), and ignition speed. For example, the Apollo lunar lander used hydrazine (N_2H_4) and nitrogen tetroxide (N_2O_4), as did the Space Shuttle reaction control system (RCS).

Solid Fuel Systems

The Space Shuttle solid rocket booster (SRB) is one example of the use of solid propellant. Solid fuels are also favored for military applications, such as air-to-air missiles and intercontinental ballistic missiles (ICBMs) because they require little pre-launch processing. After ignition, the propellant burns until the target is reached or exhausted.

Solid fuel is usually powdered metal in a binder. Most commonly, aluminum (Al) and sometimes magnesium (Mg) are used. Typical oxidizers are ammonium perchlorate (AP) and ammonium nitrate (AN). The fuel and oxidizer are mixed with a binder, usually some plastic or (more commonly) synthetic rubber, such as hydroxyl-terminated polybutadiene (HTPB) or polybutadiene acrylonitrile (PBAN). The Space Shuttle SRBs used Al and AN, with PBAN as a binder.

Most commercially available fuels for amateur rocket builders are based on Al and AP with HTPB because this combination is more thermally efficient and safely stored and processed. Unlike liquid-fueled rocket engines, solid-fuel rockets cannot be throttled, and once ignited, like a firework, they must burn until the propellant is exhausted.

Hybrid Systems

Hybrid systems have a solid fuel and a gaseous or liquid oxidizer, or rarely, the reverse, as shown in the figure below. Experimentalists, amateur rocket builders, and small rocket companies favor hybrids because they can be relatively simple and inexpensive to construct. They can also be throttled and turned off and on again, unlike when a solid propellant is used.

Hybrid rocket systems use a solid fuel with a separate oxidizer.

It is often argued that hybrid systems are safer than solid propellants or liquid systems. However, this is only partially correct because the propellant used in the rocket still comprises the same fuel and oxidizer. Nevertheless, solid fuel by itself is more stable because it is not pre-mixed with an oxidizer, so it can be safely stored and has a longer shelf life. Typical oxidizers used in hybrid rockers are gaseous (compressed) oxygen (O_2), nitrous oxide (N_2O), and hydrogen peroxide H_2O_2.

Hybrid rocket motors, however, have not found many applications in commercial space applications because they have no performance advantage, i.e., they have relatively modest values in terms of thrust-producing efficiency, and designing for optimal performance is tentativa et error (i.e., trial and error). A notable exception is SpaceShipOne, which by design uses HTPB and N_2O.

Experimentalists and amateur rocket builders often use polyvinyl chloride (PVC) or acrylonitrile butadiene styrene (ABS) as a solid fuel because these are readily available at low cost. Nitrous oxide (N_2O) or “Nitrous”  can be used as an oxidizer with several different fuels, and it is popular for use in hybrid rockets. Nitrous oxide is readily available at modest cost at automotive stores for use in high-performance race car engines, which allows the engine to burn more fuel and create more power by providing more oxygen during combustion. N_2O is also used for medical purposes as a mild anesthetic known as “laughing gas.”

Monopropellant Systems

Monopropellants do not burn but decompose exothermally in contact with a catalyst. Monopropellant engines generate thrust from the propellant flowing through a valve into a catalytic decomposition chamber, where the propellant goes through a highly energetic decomposition process. The hot gases then accelerate through a nozzle, as shown in the figure below. These thrusters generally provide thrust levels up to about 3,000 N (674 lb).

Monopropellant rocket engines 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 (H2O2) is often used for monopropellant engines because it decomposes into water and diatomic oxygen in contact with many metal oxides, especially silver oxide. An H2O2 system propelled the Apollo lunar lander trainer. Hydrazine has been used more extensively because of its higher performance and ease of reaction initiation. The rocket engine on the New Horizons spacecraft is an N2H4 monopropellant system.

Cold Gas Thrusters

Cold gas thrusters are rocket engines that use compressed gas, typically nitrogen or helium, as a propellant. They release the pressurized gas through a nozzle to generate thrust, as shown in the figure below. Because they do not involve any combustion, cold gas thrusters have a relatively low specific impulse, which means they provide less thrust per unit of propellant than other rocket engines. Their simplicity makes them less efficient and powerful than engines that use monopropellants or bipropellants.

Cold gas thrusters have a straightforward design, consisting only of a fuel tank, a regulating valve, a propelling nozzle, and some plumbing.

Cold gas thrusters are commonly used for small spacecraft or subsystems requiring small, precise movements or adjustments. 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 (H2), nitrogen (N2), and helium (He), will give better performance.

High-Efficiency 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 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 the orbits of satellites.

In each case, this propulsion system does not have to oppose gravity directly (that is, to “lift” the spacecraft). However, it increases its velocity gradually once the spacecraft 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.

Electric rocket engines, such as ion thrusters or Hall effect thrusters, are becoming increasingly attractive for use on spacecraft. These engines cannot produce much thrust but are highly efficient and maintain thrust production for long periods, making them well-suited for deep space missions.

Total Impulse & Specific Impulse

Establishing a rocket engine’s thrust and efficiency characteristics requires some quantitative performance measures. The total impulse is defined as the integral of the thrust over its “burnout time,” i.e., the time the engine runs. 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 a property of the chemical composition of the propellant and can be evaluated using a thermodynamic analysis.

Impulse

The impulse is given by

(7)   \begin{equation*} I = \int_0^{t_b} T \, dt = \int_0^{t_b} \overbigdot{m} V_{\rm eq} \, dt \end{equation*}

If \overbigdot{m} and V_{\rm eq} are constant, as is often a good approximation, then

(8)   \begin{equation*} I = T \int_0^{t_b} dt = T \, t_b = (\overbigdot{m} \, t_b) V_{\rm eq} = M_P \, V_{\rm eq} \end{equation*}

where M_P is the mass of the propellant used. The total impulse, therefore, will be equivalent to the net momentum imparted to a rocket during the engine burn.

Specific Impulse

The specific impulse, I_{\rm sp}, is

(9)   \begin{equation*} I_{\rm sp} = \frac{\mbox{\small Total~impulse}}{\mbox{\small Weight~of~propellant~burned}} = \frac{I}{M_P \, g_0} \end{equation*}

where g_0 is acceleration under gravity at sea level on Earth, which is used as a reference. Therefore, the higher the value of I_{\rm sp}, the more efficient the rocket motor will produce thrust.

It is further apparent using Eq. 5 that

(10)   \begin{equation*} I_{\rm sp} = \frac{V_{\rm eq}}{g_0} = \frac{T}{\overbigdot{m} \, g_0} \end{equation*}

where it will be noticed that I_{\rm sp} is measured in units of time (seconds). Therefore, the specific impulse is the total impulse (or change in momentum delivered) per unit weight of the propellant consumed. Its value is dimensionally equivalent to the generated thrust divided by the propellant flow rate in terms of fuel weight per unit of time. So, in some ways, it is equivalent to the inverse of the thrust-specific fuel consumption used by a jet engine.

Standard Gravity

The standard acceleration under gravity or “standard gravity,” denoted by g_0, is the nominal gravitational acceleration of an object at the surface of the Earth. The value of g_0 is defined as 9.80665 m/s^2 or 32.17405 ft/s^2. The symbol “g” should not be confused with “G” for the universal gravitational constant or “g” used as the symbol for gram.

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, usually expressed in seconds. Notice that these two definitions differ by a factor of g_0. 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 shown in the table below. While it can be seen that H2/O2 (LH2/LOX) has the highest specific impulse, liquid hydrogen needs a much larger (volumetric) fuel tank than RP-1, and it is also more expensive and challenging to transport and store. Liquified methane is fast becoming the rocket fuel of choice, but it must be stored cryogenically, so it is subject to the same issues as LH2.

Some propellants and their specific impulse values.
Propellant Molecular weights Specific Impulse (secs.)
H2/O2 (LH2/LOX) 2/32 445
RP-1/O2 (Kerosine/LOX) or Kerolox 170/32 295
H2O2 (Hydrogen Peroxide) 34 300
CH4/O2 (Methane/LOX) or Metholox 16/32 320

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. To this end, the impulse gives a measure of the integrated thrust applied over time. Specific impulse is not a force because its value measures the impulse or momentum produced per unit of propellant and will be proportional to the exhaust velocity. Both metrics are helpful in the performance characterization of rocket engines.

Thermodynamic Considerations

Rockets use combustion or a chemical reaction to generate hot gases. These gases are then expanded through a supersonic nozzle to create high flow speeds in the exhaust stream. Rocket engines work at temperatures between 2,500 K and 3,000 K, and then only because regenerative cooling protects the combustion chamber and nozzle walls from melting.

Thermodynamic principles can be used to derive an expression for the exhaust velocity. The approach uses the conservation energy with the assumptions of isentropic, compressible, two-phase flow. The exhaust velocity is given by

(11)   \begin{equation*} V_{\rm eq} = \sqrt{ 2 g_0 \, R \left( \dfrac{\gamma}{\gamma - 1} \right) \left( \dfrac{ {\cal{T}}}{M_r} \right) \bigg( 1 - \left( \dfrac{p_e}{p_c} \right)^{\left( \frac{\gamma -1}{\gamma} \right)} \bigg) } \end{equation*}

where p_e is the exit pressure, p_c is the chamber or combustion pressure, \gamma is the ratio of the specific heats of the propellant gas, R is the universal gas constant, and M_r is the relative molecular mass of the propellant. Because the combustion process happens so quickly, it can be considered isothermal, meaning that the combustion chamber and exhaust gas temperatures are equal, i.e., {\cal{T}} \approx constant.

For in-space operations, i.e., in a vacuum, Eq. 11 can be approximated by

(12)   \begin{equation*} V_{\rm eq} \approx k \sqrt{ \dfrac{{\cal{T}}}{M_r}} \end{equation*}

It has been found that k \approx 0.25 for most rocket engines, but its value does depend on the specific engine design. If the combustion temperature, {\cal{T}}, is expressed in units of Kelvin, the units of V_{\rm eq} in Eq. 12 will be in terms of kilometers per second (km/s).

The specific impulse can also be obtained from Eq. 11, i.e.,

(13)   \begin{equation*} I_{\rm sp} = \frac{V_{\rm eq}}{g_0} = \sqrt{ \left( \dfrac{ 2 \, R}{g_0} \right) \left( \dfrac{\gamma}{\gamma - 1} \right) \left( \dfrac{ {\cal{T}}}{M_r} \right) \bigg( 1 - \left( \dfrac{p_e}{p_c} \right)^{\left( \frac{\gamma -1}{\gamma} \right)} \bigg) } \end{equation*}

Equations 12 and 13 show that high temperatures and low molecular weight are essential to achieve the largest equivalent exhaust velocity and highest specific impulse. The effect of the specific heat ratio upon specific impulse is not as significant as the effect of the combustion temperature and the mean molecular weight of the propellant. To this end, the best-known propellants are liquid hydrogen (LH2) with M_r = 2 and liquid oxygen (LOX) with M_r = 32, which in gaseous form combust to produce superheated steam with M_r = 18. For cold gas thrusters, helium and nitrogen are good choices, in part because they are inert. Helium (He) has M_r = 4, and nitrogen (N2) has M_r = 28, helium being a preferred choice even though it is expensive.

Check Your Understanding #2 – Using specific impulse to calculate thrust

A rocket engine is to be tested on a test stand. The burning of the propellent occurs at a steady rate of 150.0 kg/s, and the specific impulse of the propulsion system is 240.0 seconds. What thrust does the rocket engine develop?

Show solution/hide solution.

The equivalent exhaust velocity V_{\rm eq} is given in terms of the specific impulse, i.e.,

    \[ V_{\rm eq} = I_{\rm sp} \, g = 240.0 \times 9.81 = 2,354.4~\mbox{m/s} \]

Therefore, the thrust produced T will be

    \[ T = \overbigdot{m}_P \, V_{\rm eq} = 150.0 \times 2,354.4 = 353.16~\mbox{kN} \]

Check Your Understanding #3 – Nitrogen or helium?

You go for an interview with SpaceX. They ask you during the interview whether nitrogen or helium is better for a cold gas thruster to control the new versions of their StarLink satellites, and why. What is your response?

Show solution/hide solution.

You first explain that for cold gas thrusters, helium and nitrogen offer distinct advantages and trade-offs. You write down and explain that the thrust produced by the thruster will be

    \[ T = \overbigdot{m}_p \, V_e \]

where \overbigdot{m}_p is the mass flow rate, and V_e is the exhaust (exit) velocity. Therefore, for a given mass flow rate of propellant the thrust produced will related to the achievable exhaust velocities, but every AE graduate knows that.

However, they begin to get impressed when you go on to say that the best propellants are those with low molecular mass, which give higher values of specific impulse. To compare the potential exhaust velocities of two gases, the exhaust rate of a gas at a constant temperature will be inversely proportional to the square root of its relative molecular mass, i.e., in this case

    \[ \frac{\text{\small Flow rate of He}}{\text{\small Flow rate of N}_2} = \sqrt{ \frac{M_{r_{\text{N}_2}}}{M_{r_{\text{He}}}} } \]

where M_r is the relative molecular mass of the gas. Fortunately, you remember from your chemistry classes that helium (\text{He}) has a relative molecular mass of about 4, and nitrogen (\text{N}_2) has a relative molecular mass of about 28. Therefore, you write down that

    \[ \frac{\text{\small Flow rate of He}}{\text{\small Flow rate of N}_2} = \sqrt{\frac{28}{4}} = \sqrt{7} \approx 2.65 \]

You then explain that this result indicates that helium will exhaust at a speed approximately 2.65 times faster than nitrogen because of its lower relative molecular mass. Therefore, the use of helium will produce 2.65 times as much thrust for a given mass flow rate.

But you are not done! You go on to explain that on the one hand, for the same total mass of propellant, M_P, helium will require seven times the volume of nitrogen at the same pressure and temperature. On the other hand, seven times less mass of helium needs to be carried to obtain the same total impulse from the thruster, i.e., to obtain the same values of I, where

    \[ I = M_P \, V_{\rm eq} \]

To clinch the job, you finally explain that helium is also more expensive and requires careful thermal management. In contrast, nitrogen is much cheaper, more readily available, and easier to handle. Therefore, nitrogen as a gas is often chosen for thrusters where performance demands are moderate. In the end, you say that the choice between the two gases will be mission-specific, balancing performance and cost.

Summary & Closure

Gas/liquid fuel-based rocket engines 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, which can produce more than half of the initial thrust on leaving the launch pad, e.g., the Space Shuttle concept. Solid fuel engines are also used on missiles and for other in-space applications. The lower performance and non-throttling characteristics of solid propellant engines are acceptable because of their operational simplicity, although solid rocket boosters are by no means simple propulsion systems.

Since the dawn of human space flight in the 1960s, advancements have continued to be made by using improved propellants and rocket engine 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 engines produce extremely high thrusts and operate near their safe limits, albeit relatively for relatively short times. However, the reliability of rocket engines 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 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?
  • 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 engines 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.
  • A fantastic explanation of the Saturn 5 rocket.
  • 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.
  • A video on how to start up a rocket engine!
  1. Prof. Eric "Rick" Perrell wrote the initial draft of this chapter. His expertise, knowledge, and contributions to this ebook are greatly appreciated. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2024 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.