Definition. A rocket is defined as an ”engine or motor that develops thrust by ejecting a stream of matter rearward, or the missile or vehicle powered by such an engine”. Since the reaction principle involved assumes a self-contained source of energy, a rocket can operate in any medium including space outside the earth’s atmosphere, where there is no oxygen to support combustion (1). History. The Chinese are generally given credit for inventing the rocket because they appear to be the first to have employed black powder or solid rockets as weapons of war, somewhere between 1150 and 1350 a.d. (2). The Chinese attached a small rocket to the shaft of an arrow to extend its range (Fig. 1). The early history of rocket development is linked to their use as weapons; references to rockets as weapons appear from the fourteenth to eighteenth centuries (3). The technology spread to Europe and came to the attention of William Congreve of the Royal Laboratory at Woolrich, England. His stick-stabilized rocket designs were adopted by the British Navy’s arsenal and were employed in the attack on Fort McHenry in Baltimore, Maryland, that gave rise to the ”rocket’s red glare” phrase in the ”Star Spangled Banner.”
Modern treatments of rockets and spaceflight focus on the contributions of four men, Konstantin Eduardovich Tsiolkovsky (1857-1935), Dr. Robert Goddard (1882-1945), Dr. Hermann Oberth (1894-1989) and Dr. Wernher Von Braun (1921-1977).
The application of rocket technology to concepts of spaceflight originated with Konstantin Eduardovich Tsiolkovsky. It first appeared in 1903 in his treatise The Investigation of Outer Space by Means of Reaction Apparatus.A theoretician, his proficiency in mathematics and science enabled him to foresee and address such issues as escape velocities from the earth’s gravitational field,gyroscopic stabilization, and the so-called ”rocket equation” that establishes the relationship of the velocity increment added to a vehicle in terms of the exhaust velocity of the rocket device and the initial and final masses of the vehicle (4).
Chinese fire arrow
Figure 1. Chinese fire arrow
Dr. Robert Goddard was an American experimentalist who began his experiments in rocketry as a doctorate student at Clark University in Worcester, Massachusetts. His work was not widely accepted during his time. His report A Method of Reaching Extreme Altitudes, which offered scientifically sound concepts such as travel to the Moon, was criticized by the general public and in the press. Dr. Goddard built and successfully launched the first liquid propellant rocket on 16 March 1926. (Fig. 2). He went on to conduct additional experiments in Roswell, New Mexico, under the sponsorship of Daniel Guggenheim. He introduced the practical application of such concepts as gyroscopic stabilization of the rocket vehicle and movable deflector vanes in the rocket exhaust for directional control. He held 214 patents in rocketry (5).
Dr. Oberth was a physicist who wrote The Rocket into Interplanetary Space in 1923 to espouse his theories of space travel. Among his theories was the concept of staging to achieve higher velocities (6). His writings inspired Wernher Von Braun who later assisted Oberth in liquid rocket experiments. Von Braun eventually applied his knowledge to constructing liquid-fueled rocket-powered weapons during World War II. The launch of an A-4 rocket to an altitude of 50 miles (the altitude at which space is considered to begin) on 3 October 1942 might well be considered the beginning of the space age. Following the war, Von Braun and his Peenemunde team was reconstituted in the United States under the U.S. Army missile program, where its focus was again liquid-fueled rocketry. As Dr. Von Braun was developing the next generation ballistic missile, the Redstone, the United States was investing in the development of the Vanguard space launcher as a civilian launch vehicle. It was intended to inaugurate the U.S. exploration of space (7). The government intentionally avoided the use of Von Braun’s missiles as launch vehicles to emphasize the peaceful intent of space exploration. The Vanguard experienced a spectacular launch pad failure in December 1957 that forced the government to turn to Dr. Von Braun in the aftermath of the successful U.S.S.R. “Sputnik” launch. His successful January 1958 launch of the Explorer 1 atop a Jupiter C launch vehicle, based on missile technology, brought Dr. Von Braun into the nation’s spotlight and resulted in his becoming the leading U.S. figure in space launch technology and space exploration. The missile technology base, thus, became the foundation for the development of space launch vehicles. As the missions became more ambitious, the need emerged for higher energy reactants than the liquid oxygen/RP-1 (kerosene) typically used, and the cryogenic system of liquid oxygen and liquid hydrogen became the standard for the civilian space launch capability. The Saturn V was the first launch vehicle developed that was not based on a vehicle or rocket engine developed for weaponry. It used liquid oxygen and liquid hydrogen in the second and third stages. Presently, the Space Shuttle, the French Arianne V, and the Japanese H-II vehicles all use hydrogen and oxygen for space launch applications.
The Goddard rocket
Figure 2. The Goddard rocket
Rocket-powered manned flight began in the 1930s in Germany. The Heinkel 176 was the first aircraft solely powered by a rocket engine. It used a 1320 lbf (297 N) engine that ran on decomposed hydrogen peroxide. In 1943, the Messerschmitt 163B rocket fighter became the first ”operational” rocket-powered fighter aircraft. It had dual combusters that gave a total thrust level of 4400 lbf (989 N) and operated on hydrogen peroxide oxidizer and a fuel mixture of  hydrazine hydrate, methyl alcohol, and water. One chamber could be shut down to achieve throttling down to 660 lbf (148 N). The U.S. developments in rocket-powered manned flight began with a rocket-powered flying wing, the Northrop MX 334 (8) and eventually led to the Bell X-1 in which Captain Charles Yeager broke the sound barrier in October 1947. Ever improved experimental aircraft were flown in attempts to increase the flight speed and altitude achievable. Mach 2 was exceeded in November 1953 in a Navy Douglas D-558-II and the record was increased to 2.5 only 22 days later by Captain Yeager in the Bell X-1A. (6) The North American Aviation X-15 dominated high-speed and high-altitude research in the late 1950s and 1960s. Its 60,000-lbf (13,489 N) thrust XLR-99 engine took the X-15 to new international records for speed (Mach 6.7) and altitude (354,200 feet). Many of the lessons learned from these experimental manned aircraft were subsequently applied in designing, constructing and operating the Space Shuttle orbiter.
Solid propellant rocket motors evolved in a role as strap-on boosters to liquid stages for many space launch applications. Solid propellant rocket history was closely linked to weaponry for its development. The first large high-performance motor design was for the Polaris submarine-launched ballistic missiles where the requirements for storability and logistics of shipboard operations made the solid propellant rocket very attractive. These same requirements led to its use in the Minuteman ballistic missile. These missile developments provided the base upon which the technology for very large solid propellant rocket boosters, suitable for space launch vehicles, was built (9). The industry explored yet even larger solid propellant rocket configurations and found that their efforts were constrained by the size of the mixing facilities and the ease of transport of the rockets. These problems were overcome by using a segmented construction method in which the solid propellant rocket was cast in sections or segments, cured, then assembled into the flight vehicle at the launch site. A segmented, 156-inch-diameter solid propellant motor demonstrated the feasibility and practicality of segmented, solid propellant rocket motors, a concept employed as a strap-on booster for the Titan III launch system. This led the way to using large strap-on solid propellant rockets for the recoverable solid propellant rocket motor boosters for the Space Shuttle. A parallel development of great significance was the movable nozzle that facilitated thrust vector control for solid propellant rockets. Previously, the thrust vector for large solid propellant motors was controlled by injecting liquids into the nozzle sidewall, generating side forces by the shock caused by the interaction of the liquid jet with the supersonic nozzle flow.

Governing Laws

The operation of rocket engines and motors and the vehicles that they propel are primarily governed by Newton’s laws of motion.
Newton’s first law, often called the law of inertia, states that there is no change in the motion of a body unless a resultant force acts on it. A number of forces act on a launch vehicle throughout its flight. The gravitational force (weight of the vehicle), lift, drag, and the thrust of the rocket engine all act on the vehicle to cause the resultant motion. The net amount of the resultant force and its direction determine the acceleration on the vehicle and the path of the flight trajectory, in accordance with Newton’s second law.
Newton’s second law of motion states that whenever a net (unbalanced) force acts on a body, it produces an acceleration in the direction of the force; the acceleration is directly proportional to the force and inversely proportional to the mass of the body:
This relationship is more typically seen in the form,
As stated in the definition, a rocket develops its thrust by expelling a mass rearward. Examining this in the context of this equation, a mass is accelerated rearward by some means that accelerates its velocity from near zero to thousands of meters per second. The force for this acceleration is proportional to the mass of the exhaust gases and the acceleration per Newton’s second law. The force acting on the exhaust gases is in the direction of the accelerating mass, but produces a thrust in accordance with Newton’s third law, which states that for every acting force, there is a reacting force that is equal in magnitude but opposite in direction. Therefore, the force of accelerating the fluid internal to the rocket has an equal but opposite external force which is the thrust produced by the rocket.
The magnitude of this force (thrust) can be determined by examining the change of momentum in the device and the sum of the forces that act on a closed duct or control volume, as shown in Fig. 3. The flow internal to the rocket experiences a change of momentum that is equal to the mass flow rate times the change in velocity of the gases:
Assuming that the inlet velocity into the device is low, the momentum at the inlet can be considered negligible. Thus, the momentum change is
tmp2A6_thumbPressure balance on the rocket chamber and nozzle wall
Figure 3. Pressure balance on the rocket chamber and nozzle wall
The sum of all pressures on the surfaces perpendicular to the flow axis of the device reduces to a resultant force due to the pressure differential between the pressure at the nozzle exit plane and the ambient pressure that acts on the exit area of the nozzle:
The sum of the forces that act on a rocket is equal to the change of momentum in accordance with Newton’s second law. By combining Equations 4 and 5 and rearranging the terms, we develop the following expression for the thrust of a rocket:
When pexit equals pamb, expansion is optimum and performance best. When the nozzle exit pressure is less than ambient, the nozzle is said to be over-expanded. If exit pressure is greater than ambient, the nozzle is said to be under-expanded. Because the rocket, generally, flies through the atmosphere, it experiences variations in the ambient pressure, so it operates at optimum expansion at only one altitude. The choice of the rocket exit area ratio then becomes the result of trading off a number of design and flight considerations.
Newton’s laws are applied in analyzing the acceleration of a vehicle propelled by a rocket as well. Examining vehicle flight in a vacuum free of gravitational forces, the rocket produces an unbalanced force and a resultant acceleration in accordance with Newton’s second law. Here, it is written in a way slightly different from that in Equation 2. The thrust is the net accelerating force that is equal to the instantaneous mass of the vehicle, and its instantaneous acceleration is written as the time rate of change of the velocity (10):
The thrust of the rocket F can also be expressed in terms of the mass flow rate m from the rocket and its effective exhaust velocity Ve assuming that nozzle exit pressure equals the ambient from Equation 6:
The negative sign indicates that the mass of the vehicle is decreasing as the propellant exits the engine.
Substituting for thrust F from Equation 8 and mass flow rate m from Equation 9 and rearranging,
Integrating, we obtain the result that is commonly called the ”rocket equation” that is used to calculate the velocity increment added to a stage in terms of its effective exhaust velocity and its initial and final masses:
This solution assumed flight in a vacuum free of any gravitational field; thus, the value calculated is an ideal velocity increment. Introducing the effects of atmospheric drag and gravity result in reducing the velocity increment achieved. The gravitational field has a component of force acting along the flight path of the vehicle (g cosine 8). The net loss due to the gravity field is computed by integrating this component along the path during the flight in the form of the equation,
The drag component is introduced as a drag coefficient Cd that is applied to the incompressible flow dynamic pressure 1/2PV2 and the cross-sectional area of the vehicle and evaluated along the flight path:
The drag coefficient is a function of the vehicle flight Mach number and its angle of attack (Fig. 4) (11).
Finally, the velocity increment added to a single stage during flight can be determined from the equation,
tmp2A17_thumbDrag coefficient vs. Mach number as a function of angle of attack
Figure 4. Drag coefficient vs. Mach number as a function of angle of attack
The initial mass of a stage includes the inert mass of the vehicle (structural dry mass plus engines), the mass of propellants and pressurant gases, and the pay-load mass:
The final mass of the stage includes the dry weight of the vehicle, any residual (unburned) propellant and pressurants, and the payload:
The rocket equation can be applied to multiple-stage vehicles by solving the equation for each stage and adding the velocity increments for all stages as follows:
Care must be exercised in considering the masses of the individual stages. The payload for the first stage of a multiple-stage vehicle is the mass of the stages above it. Only in the final stage is the payload the mass that is placed in orbit.

Specific Impulse

The usual measure of performance of a rocket engine is its specific impulse. This is a measure of how much thrust (lbf or Newtons) is generated by the engine when the flow rate is 1 unit (lbm/sec or kg/s):
The units of specific impulse are seconds in SI but lbf/lbm/sec in the U.S. Customary units (USCS). “Seconds” is still used as the terminology for specific impulse in the USCS system but is not correct technically and must have the correct units for use in solving any equations. Specific impulse is related to the exit velocity of the nozzle Ve through the relationship,
Specific impulse can also be determined as a function of the properties of the working fluid and the operating conditions of the rocket nozzle. In simplified terms, we see that the specific impulse is proportional to the square root of the temperature of the working fluid divided by its molecular weight:
The constant in this equation is a function of the thermodynamic properties of the combustion gases and the ratio of nozzle exit pressure to combustion chamber pressure. Therefore, the objective in any rocket engine is to achieve the highest possible temperature of the working fluid and the lowest possible molecular weight. In chemical rockets, this is typically achieved by using reactants that produce large quantities of hydrogen or steam at high temperature as their products. Beamed energy and nuclear rockets choose hydrogen for the working fluid because it is a good coolant and has the desired low molecular weight. The choice of a working fluid for electric rockets depends, in some cases, on other properties of the fluid such as ionization potential. Hydrogen is still of interest for an arcjet.

Vehicle Staging

The ideal velocity increment equation can be manipulated into the following form to examine the sensitivity of the stage performance to various aspects of rocket engine performance and stage design:
From this equation, we can see that the higher the engine specific impulse, the more closely the final mass approaches the initial mass of the stage (less pro-pellant is consumed).
A measure of stage design efficiency is the propellant mass fraction l defined by the equation,
The higher the value of l, the greater the structural efficiency of the stage. Now, we can manipulate Equations 18 and 19 into a form that presents the ratio of the initial mass of the vehicle to the payload delivered in terms of the propellant fraction, specific impulse, and ideal velocity increment:
We can illustrate the sensitivity of stage and vehicle performance to engine specific impulse and stage structural efficiency by conducting a parametric study of two vehicles, single-stage-to-orbit (SSTO) and two-stage-to-orbit (TSTO). First, one must estimate the ideal velocity increment for the mission. A low Earth orbit typically requires about 30,000 ft/sec (9144 m/s). In the SSTO case, three specific impulses were assumed, one representative of liquid hydrogen/liquid oxygen rocket performance (460 s), a second chosen as a conceivable increase in chemical rocket performance (500 s) and the third representative of predicted nuclear rocket performance (850 s). By assuming different values for the propellant fraction l, one can then solve Equation 23 and arrive at the family of curves presented in Fig. 5. The lower the value of the lift-off mass to payload ratio, the better the vehicle performance. The chemical rocket options approach a limit at a value in the vicinity of 10 (10% of the vehicle liftoff mass is effective payload delivered to orbit), whereas the nuclear option approaches a limit in the vicinity of about 3 (33% of liftoff mass). The lower specific impulse vehicle, is seen as quite sensitive to the propellant mass fraction. For example, at a propellant fraction of 0.89, the 40-second difference in specific impulse between 460 s and 500 s results in a twofold difference in the initial weight to payload ratio. So, if a vehicle design were to experience an increase in weight from its initial design to completion of fabrication (and they most always do), the payload capacity of the vehicle will be severely reduced and, conceivably, could become inconsequential, depending upon the magnitude of the growth in weight. For the higher specific impulse cases, the SSTO vehicle is seen as less sensitive to any decrease in propellant mass fraction, and the nuclear rocket is virtually insensitive to propellant mass fraction across the range studied. This illustrates that the higher the delivered specific impulse, the less sensitive the stage design is to its structural efficiency.
Another way in which the initial mass/payload ratio can be made less sensitive to the propellant mass fraction is through the concept of staging in which two (or more) rocket stages can be coupled, either in parallel or in tandem. As one stage is jettisoned after exhausting its propellant load, the subsequent stage ignites and continues on to complete the mission. Figure 6 presents the results of a parametric study to illustrate the effects of specific impulse and propellant fraction upon the lift-off mass to payload ratio for a tandem two-stage vehicle that has an ideal velocity increment of 30,000 ft/s (9144 m/s). In this case, it was assumed that the propellant mass fraction was the same for each stage, and the total propellant load was distributed, 80% in the first stage and 20% in the second stage. As in the SSTO case, there is a limit of initial to payload mass ratio of about 10. The maximum ratios calculated are 16 for a specific impulse of 460 s and 12 for a specific impulse of 500 s within the range of propellant mass fraction studied, as contrasted with values of 74 and 25, respectively, for the SSTO. It is easily seen that multiple stages can offer payload gains. The driver for minimizing the number of stages is cost considerations rather than performance.
Payload/initial mass vs. propellant mass fraction for SSTO.
Figure 5. Payload/initial mass vs. propellant mass fraction for SSTO.
 Payload/initial mass vs. propellant mass fraction for TSTO.
Figure 6. Payload/initial mass vs. propellant mass fraction for TSTO.

Energy and Energy Conversion

Energy Conversion Mechanisms in a Rocket. The rocket engine is an energy conversion device that converts potential energy to the thermal energy of a high-temperature gas and then to the kinetic energy of a high-velocity exhaust gas. The energy sources for this conversion can be any of several types, chemical, electrical, beamed (solar or laser), or nuclear.
Chemical Rockets. In a chemical rocket, the potential energy is in the chemical bonds of the molecules that compose the fuel and the oxidizer. The class of reactants identified as “fuels” has large concentrations of hydrogen or carbon in the molecule and light metals such as aluminum, lithium, beryllium, magnesium, and boron. The class of reactants identified as oxidizers has large concentrations of oxygen, fluorine, or chlorine in the molecule (12). When the fuel reacts with an oxidizer in the combustion process, the resultant products are at a temperature significantly elevated over that at which they entered. This thermal energy is extracted for propulsion (thrust) by passing the gases through a converging-diverging (or DeLaval) nozzle that converts the thermal energy to kinetic energy. These chemical reactants are usually in either liquid or solid form.
A liquid-fueled rocket might be of either bipropellant or monopropellant type. In the former, fuel and oxidizer are introduced into the combustion device separately. They are atomized, vaporized, mixed, and combusted in the combustion chamber. The mixture ratio (proportions of oxidizer to fuel flow rate) at which the maximum combustion temperature occurs is called the stoichiometric ratio. This ratio does not yield the highest specific impulse (the ratio of thrust produced by the engine to the rate at which propellant is being consumed), however; this occurs when using fuel-rich mixtures. The excess fuel tends to reduce the combustion temperature somewhat, but more importantly, the molecular weight ofthe combustion products is reduced. The net effect is to increase the specific impulse, the measure of “goodness” of rocket performance. The most common oxidizers used to date are liquid oxygen, nitric acid (HNO3), and nitrogen tetroxide (N2O4). The most common fuels used to date have been liquid hydrogen, RP-1 (a kerosene-like hydrocarbon), and various amine-based fuels (hydrazine, unsymmetrical dimethyl hydrazine, monomethyl hydrazine, and mixtures of the foregoing). Liquid propellants can also be classified as cryogens or storables. Cryogens are liquefied gases, typically oxygen and hydrogen, and are more energetic reactants. Within the family of cryogens, there are some propellants referred to as ”space storable,” meaning that they are relatively mild cryogens (boiling point greater than — 238°F(— 150 K). These propellants can be stored in space for long periods of time and have acceptable levels of boil-off without engaging in major efforts to insulate the propellant containers (tanks) or refrigerate the propellants. Storable propellants are those that normally are liquids at standard temperature and pressure and are less energetic (13).
Monopropellants are generally introduced into a catalyst ”bed” or ”pack” where the propellant molecule is broken apart, liberates energy, and consequently increases in thermal energy. These reactions tend to be less energetic than bipropellant reactions and are frequently used in selected applications where specific impulse is not the most important characteristic. The most commonly used monopropellants are hydrazine (N2H4) and hydrogen peroxide (H2O2). The catalyst for hydrazine is iridium coated on an alumina substrate. The catalyst for hydrogen peroxide is silver, generally in the form of a screen.
The selection of a bipropellant combination depends on many factors, for example, the ignition characteristics of the propellant combination. Hypergolic combinations (nitrogen tetroxide and amine fuels, for example) provide a ready ignition source because the fuel and the oxidizer react upon contact. Such combinations are amenable to applications where engine restart may be required or for ignition at an altitude. Combinations such as hydrogen and oxygen require an external source of energy to provide the ignition energy for the main propellant flow. Multiple restarts are possible through appropriate design, as used on the RL-10 engine of the Centaur upper stage vehicle.
Another consideration for propellant selection is the choice of coolant for the rocket chamber. Specifically, the coolant should have a very high specific heat, low viscosity, and be thermally stable. Typically, fuels are used as engine coolants. Several designs have been tested in which oxidizers (liquid oxygen or nitrogen tetroxide) have been used as coolants (14).
The solid-propellant rocket motor is a device in which the fuel and oxidizer are premixed to form a combustible mixture which, when ignited, burns until all of the propellant is consumed. It burns at a rate that depends on the combustion pressure p and is a function of the propellant type (burn rate exponent n) according to the following relationship:
The burning rate r generally ranges in value from 0.3-0.5 inches per second (0.762 to 1.27 cm/s), and the burning rate exponent n ranges from 0.2-0.5 for modern propellants (15). For a detailed description of solid-propellant rockets, see the article Solid Fuel Rockets by Donald Sauvigeau on page 531 of this topic.
The hybrid is another chemical rocket embodiment. It typically consists ofa solid fuel grain and a liquid (or gaseous) oxidizer. Its advertised advantages are that the fuel grain is less susceptible to safety and handling problems because the grain contains no oxidizer and, thus, cannot detonate or sustain combustion by itself. Additionally, using a liquid (or gaseous) oxidizer provides a very simple throttling scheme through oxidizer flow regulation and also provides for thrust termination by simply stopping the oxidizer flow. The fuel grain is typically designed to have several flow passages or ports through it to maximize the fuel surface exposed to the oxidizer, and, consequently, the burning surface and the attendant flow of combustion products. The hybrid has also achieved some attention because it permits eliminating the ammonium perchlorate oxidizer typical of most solid-fueled rockets and the attendant HCl in the exhaust products. The HCl presents a small but persistent environmental concern (16). The burn rate of a hybrid motor is similar in form to that of the solid-propellant rocket:
In this case, r is the burn rate, a is a constant that depends on the reactants, Go is the oxidizer mass flow rate per port, and n is the burning rate exponent whose value is from 0.5 to 0.7. Hybrid rockets have been successfully fired in motors of up to 250,000 lbf (1.11 x 106N) (17). The problems that exist are achieving high burning rates to minimize the size of the grain and complete consumption of the grain. Residual propellant can amount to 5 to 30% of the initial fuel load depending on the port configuration.
Nonchemical Rockets. In an electric rocket, the energy is collected from some power source (nuclear or solar) and converted to electrical energy. The energy conversion for thrusting purposes may occur by striking an arc between an anode and a cathode and passing a working fluid through the arc to heat it (arcjet). The hot gases are then passed through a nozzle to convert the thermal energy to kinetic energy. Another energy conversion technique might be to use the electrical energy to ionize an easily ionized gas (xenon, for example) and accelerate the resultant ionized particles across a potential (electrostatic thruster). Another approach may use the interaction of the current and its induced electromagnetic field that produces Lorentz forces to accelerate the gas (electromagnetic thruster) (18).
Beamed energy depends on capturing a high-energy beam from outside the vehicle and transferring its energy to a working fluid. Solar energy is one such beaming mechanism. Concentrating lenses (or mirrors) located on the vehicle capture the solar radiation and focus it on a blackbody absorber at the focal point. The absorbed energy is then used to heat a working fluid that is subsequently expelled through the nozzle at a high velocity. A laser beam might apply equally for this purpose. The working fluid also cools the absorber/thruster structure to maintain structural integrity at elevated temperatures.
Finally, a nuclear energy source may be used in place of any of the previously mentioned energy sources to heat the working fluid. Here also, the working fluid serves the dual purpose of cooling the structure and generating the thrust, as it is expelled through a nozzle.

Thermodynamics of Rockets

As previously discussed, rocket exhaust velocity is related to the specific impulse through the gravitational constant gc as follows when the nozzle exit pressure is equal to the ambient:
The exhaust velocity must be maximized to maximize the specific impulse. The first law of thermodynamics (conservation of energy) for bulk flow requires that the total energy entering the engine must equal the total energy leaving the engine. In equation form this is:
The average stagnation enthalpy per unit mass ho of all of the flows entering the engine equals the kinetic energy at the nozzle exit plus the static enthalpy h of the exhaust at the exit. The static enthalpy for a chemically reacting system is given in terms of the static temperature T, entropy s and Gibbs free energy f by the equation,
The Gibbs free energy is a summation of the chemical binding energy (19). Substituting and rearranging, the kinetic energy can be expressed as a function of the stagnation enthalpy Ts and the Gibbs free energy:
By differentiating the kinetic energy with respect to entropy s and Gibbs free energy f, we can find that the exhaust velocity for a given exit pressure increases as exit entropy decreases and Gibbs free energy decreases. From the second law of thermodynamics, the minimum exit entropy is equal to the inlet entropy for bulk flow. Therefore, the maximum exhaust velocity occurs for an exit entropy per unit mass that is equal to the inlet entropy per unit mass. Additionally, the minimum Gibbs free energy occurs under conditions of chemical equilibrium. Chemical equilibrium is achieved as the chemical constituents of the exhaust gases alter their relative proportions in response to the pressure and temperature changes that occur as the gases flow through the nozzle. Energy is liberated to the expansion process as a result. Therefore the maximum specific impulse is achieved for an ideal one-dimensional flow when the exhaust products are expanded to ambient pressure in chemical equilibrium and have total enthalpy and entropy equal to the inlet conditions (20).
Enthalpy vs. entropy for flow in a rocket engine
Figure 7. Enthalpy vs. entropy for flow in a rocket engine
The reality of rocket operation is that none of the processes occurs ideally. The combustion process is irreversible and has entropy increases attendant to it. Viscous boundary layer effects and nonequilibrium expansion of the combustion products also introduce irreversibilities. Figure 7 presents the enthalpy versus entropy diagram for an oxygen-hydrogen reaction in chemical equilibrium at 3000 psia (20.7 MPa). Process 1 to 2 is the irreversible combustion of oxygen-hydrogen reactants at 3000 psia (20.7 MPa). The equilibrium products are then expanded in a constant entropy (isentropic) process to local ambient pressure (0.101 MPa) from point 2 to point 3.

Nozzle Theory

The nozzle is the mechanism for accelerating the hot gases in all of the devices described, except those that use electrostatic or electromagnetic forces. Applying the first law of thermodynamics across the nozzle, the change in enthalpy equals the kinetic energy of the exhaust gases at the exit:
Applying the idealization of isentropic, one-dimensional flow, the exhaust velocity of the nozzle is
The nozzle is typically identified by its geometric area ratio (exit area/throat area), but the pressure ratio across the nozzle is important in determining its performance in accelerating the gas flow, as seen by examining the equation for exit velocity. The term g is the ratio of specific heats of the hot gases, “R” is the universal gas constant, and M is the molecular weight of the gases that exit the nozzle. Optimum expansion of the gases occurs when the exit pressure of the nozzle equals the local ambient pressure.
The typical shape of a DeLaval nozzle is shown in Fig. 8. In the subsonic region (flow velocities less than Mach 1), the gases are accelerated by decreasing the area of the flow passage. Continuing the decrease of the flow area increases the gas velocity until a point is reached at which the maximum mass flow rate per unit area is achieved. At this condition, the flow is at the speed of sound or sonic (Mach number equal to 1). This location is called the throat of the nozzle, and the flow is referred to as “choked.” The ratio of chamber pressure to pressure at the throat (critical pressure ratio) is approximately 2:1 for this condition. From that point on, the flow passage must increase in area to permit continuing acceleration of the flow in the supersonic regime (Mach numbers greater than 1). Once the nozzle achieves the choked condition, the chamber pressure remains  constant regardless of the back-pressure from the flight altitude. If the exit pressure exceeds the local ambient, it is underexpanded; if it is less than the local ambient, it is overexpanded. Selection of the nozzle area/pressure ratio is a compromise to provide the best performance across the vehicle’s flight regime. One way to examine this design choice is through the nozzle thrust coefficient Cf. The thrust coefficient is a measure of nozzle performance and can be used to determine the thrust of a rocket engine as a function of the throat area and the chamber pressure from the equation,
tmp2A40_thumb The DeLaval Nozzle
Figure 8. The DeLaval Nozzle
For a given ratio of specific heats, the optimum geometric area ratio corresponds to a given pressure ratio across the nozzle. For example, at a nozzle area ratio of 10, the design pressure ratio for a ratio of specific heats of 1.3 is 100, and the corresponding thrust coefficient is 1.6. If a rocket has a nozzle of area ratio of 10, it is operating at its optimum condition only when the pressure ratio is 100. Because a rocket flies through the atmosphere and is subject to varying ambient pressure, the delivered performance will be less than optimum at all other points in the trajectory than when the pressure ratio is 100. Therefore, the choice of design area ratio (pressure ratio) is a compromise in which the designer knowingly accepts less than optimum performance at some portions of the trajectory.
The nozzle designer must make compromises in choosing the design pressure ratio and also in the shape of the nozzle. However, the most common nozzle design practice uses the Rao optimum contour nozzle also known as the ”bell” nozzle (21). This design yields a shorter design than a simple 15° half-angle conical shape and has lower divergence losses because the gases are exiting the nozzle at a divergence angle of less than 8°.
The losses in a rocket nozzle consist of the divergence loss (nonaxial velocity vector for the exiting gases), finite-rate kinetic losses, and drag losses, in accordance with the equation,
Inserting typical values into this equation, we can find the overall efficiency of a modern rocket nozzle design (22):
There are nozzle designs that are intended to compensate for altitude as the rocket flies through the constantly varying pressure of the atmosphere. This includes mechanical means in which nozzle ”skirts” are moved into place at selected times in the flight profile; each increases the area/pressure ratio to approximate optimum expansion more closely. There are also aerodynamic designs such as the ”aerospike” shown in Fig. 9 that have a free jet boundary that adjusts to the local ambient pressure through a Prandtl-Meyer or corner expansion (23). Such an expansion process constrains the nozzle gases to expand to the local ambient at the outside lip of the nozzle. Consequently, the flow cannot overexpand, and the nozzle flow is always optimum until the nozzle design pressure ratio is exceeded. In practice, these nozzle designs do not operate with perfect altitude compensation. A procedure has been developed to determine the degree to which the aerospike type of nozzle approaches optimum expansion. This procedure is illustrated in Figs. 10 and 11. Figure 10 presents the thrust efficiency CT (the ratio of measured CF to ideal CF) of both an altitude compensating and bell nozzle versus pressure ratio. Both nozzles have the same area ratio (design pressure ratio), the same thermodynamic properties of the flowing gases, and the same maximum thrust efficiency. At a given pressure ratio, the percentage of maximum thrust efficiency CT achieved by the altitude compensating nozzle is compared to that of the fully flowing bell nozzle. The results of this comparison are presented in Fig. 11. The 45° line across the graph represents the performance of a bell nozzle in which the flowing gases do not separate. All points above that line represent some degree of altitude compensation. The point identified by CTADV/CTMAX = 1 and CTN-S/CTMAX = 1 represents the design point for both nozzles. The line where CTADV/CTMAX = 1 represents perfect altitude compensation. The crosshatched area is representative of the altitude compensation performance of the aerospike type of nozzle. In general, the data indicate a compensation capability in excess of 50%, as determined by the ratio of the distance from the aerospike family of curves to the line where CTADV/ CTMAX = 1 divided by the distance from the 45° line (nonseparating bell performance). The line identified as the ”80% bell” illustrates the fact that a bell nozzle can have some separation of the flow at the low end of its operating regime, and the result is that it has some altitude compensation. The breadth of the aerospike band results from the range of nozzle lengths from 16% of an isentropic spike at the upper end to 0% at the lower end.
Aerospike nozzle and its operating modes.This figure is available in full color at http://www.mrw.interscience.
Figure 9. Aerospike nozzle and its operating modes.
Nozzle altitude compensation determination.
Figure 10. Nozzle altitude compensation determination.
Nozzle altitude compensation comparison.
Figure 11. Nozzle altitude compensation comparison.
Additionally, the aerospike nozzle length is about 25% that of the Rao optimum nozzle length for the same thrust capability. This allows reducing in vehicle length or including more propellant in the vehicle.
The same Prandtl-Meyer expansion process occurs in the external flow aft of the flight vehicle that produces a pressure at the nozzle lip (the flow controlling pressure) that is lower than the ambient at the flight altitude. Thus, the nozzle behaves as though it were operating at an altitude higher than actual. This is most noticeable in the Mach 1 to Mach 3 range of flight operations. In that flight regime, the time spent in off-design conditions is considered negligible compared to bell nozzle performance capabilities. The added performance of optimized nozzle expansion across the full flight trajectory opens the possibility of single-stage-to orbit operations.
Rocket Engine Efficiencies. The overall efficiency of a rocket engine is generally presented as a percentage of the theoretical specific impulse. The calculated theoretical value is based on shifting equilibrium, a case in which the reaction products remain in equilibrium (their proportions change) as the pressure and temperature decrease during the expansion process in the nozzle. This efficiency is composed of the combustion and nozzle efficiencies, generally, it is greater than 90% and can be as high as 95-96%. The nozzle efficiency depends on design, but can be expected to be close to the 98% value previously shown. The combustion efficiency of the propellants is quite sensitive to the propellant combination used and plays the major role in overall efficiency. The liquid oxygen/ hydrocarbon combination of the early space launch engines typically had combustion efficiencies in the low 90s% range. The efficiencies of today’s hydrogen/ oxygen engines are much higher and approach 97%.

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