The law of physics on which rocket propulsion is based is called the principle of momentum. According to this principle, the time rate of change of the total momentum of a system of particles is equal to the net external force. The momentum is defined as the product of mass and velocity. If the net external force is zero, then the principle of momentum becomes the principle of conservation of momentum and the total momentum of the system is constant. To balance the momentum conveyed by the exhaust, the rocket must generate a momentum of equal magnitude but in the opposite direction and thus it accelerates forward.

The system of particles may be defined as the sum of all the particles initially within the rocket at a particular instant. As propellant is consumed, the exhaust products are expelled at a high velocity. The center of mass of the total system, subsequently consisting of the particles remaining in the rocket and the particles in the exhaust, follows a trajectory determined by the external forces, such as gravity, that is the same as if the original particles remained together as a single entity. In deep space, where gravity may be neglected, the center of mass remains at rest.

**ROCKET THRUST**

The configuration of a chemical rocket engine consists of the combustion chamber, where the chemical reaction takes place, and the nozzle, where the gases expand to create the exhaust. An important characteristic of the rocket nozzle is the existence of a throat. The velocity of the gases at the throat is equal to the local velocity of sound and beyond the throat the gas velocity is supersonic. Thus the combustion of the gases within the rocket is independent of the surrounding environment and a change in external atmospheric pressure cannot propagate upstream.

The thrust of the rocket is given by the theoretical equation :

*F = lm(dot) ve + ( pe – pa ) Ae*

This equation consists of two terms. The first term, called the momentum thrust, is equal to the product of the propellant mass flow rate m(dot)and the exhaust velocity *ve* with a correction factor l for nonaxial flow due to nozzle divergence angle. The second term is called the pressure thrust. It is equal to the difference in pressures *pe* and *pa* of the exhaust velocity and the ambient atmosphere, respectively, acting over the area *Ae* of the exit plane of the rocket nozzle. The combined effect of both terms is incorporated into the effective exhaust velocity *c*. Thus the thrust is also written

*F* = m(dot)* c*

where an average value of *c* is used, since it is not strictly constant.

The exhaust exit pressure is determined by the expansion ratio given by

*e= Ae / At*

which is the ratio of the area of the nozzle exit plane *Ae* and the area of the throat *At* . As the expansion ratio e increases, the exhaust exit pressure *pe* decreases.

The thrust is maximum when the exit pressure of the exhaust is equal to the ambient pressure of the surrounding environment, that is, when *pe* = *pa*. This condition is known as optimum expansion and is achieved by proper selection of the expansion ratio. Although optimum expansion makes the contribution of the pressure thrust zero, it results in a higher value of exhaust velocity *ve* such that the increase in momentum thrust exceeds the reduction in pressure thrust.

A conical nozzle is easy to manufacture and simple to analyze. If the apex angle is 2a , the correction factor for nonaxial flow is

- = ½ (1 + cos a )

The apex angle must be small to keep the loss within acceptable limits. A typical design would be a = 15° , for which l = 0.9830. This represents a loss of 1.7 percent. However, conical nozzles are excessively long for large expansion ratios and suffer additional losses caused by flow separation. A bell-shaped nozzle is therefore superior because it promotes expansion while reducing length.

**ROCKET PROPULSION PARAMETERS**

The specific impulse *Isp* of a rocket is the parameter that determines the overall effectiveness of the rocket nozzle and propellant. It is defined as the ratio of the thrust and the propellant weight flow rate, or

*Isp = F / m(dot) g = c / g*

where *g* is a conventional value for the acceleration of

gravity (9.80665 m/s2 exactly). Specific impulse is expressed in seconds.

Although gravity has nothing whatever to do with the rocket propulsion chemistry, it has entered into the definition of specific impulse because in past engineering practice mass was expressed in terms of the corresponding weight on the surface of the earth. By inspection of the equation, it can be seen that the specific impulse *Isp* is physically equivalent to the effective exhaust velocity *c*, but is rescaled numerically and has a different unit because of division by *g*. Some manufacturers now express specific impulse in newton seconds per kilogram, which is the same as effective exhaust velocity in meters per second.

Two other important parameters are the thrust coefficient *CF* and the characteristic exhaust velocity *c**. The thrust coefficient is defined as

*CF = F / At pc = m(dot) c / At pc*

where *F* is the thrust, *At* is the throat area, and *pc* is the chamber pressure. This parameter is the figure of merit of the nozzle design. The characteristic exhaust velocity is defined as

*c* = At pc / m(dot) = c / CF*

This parameter is the figure of merit of the propellant. Thus the specific impulse may be written

*Isp = CF c* / g*

which shows that the specific impulse is the figure of merit of the nozzle design and propellant as a whole, since it depends on both *CF* and *c**. However, in practice the specific impulse is usually regarded as a measure of the efficiency of the propellant alone.

**LAUNCH VEHICLE PROPULSION SYSTEMS**

In the first stage of a launch vehicle, the exit pressure of the exhaust is equal to the sea level atmospheric pressure 101.325 kPa (14.7 psia) for optimum expansion. As the altitude of the rocket increases along its trajectory, the surrounding atmospheric pressure decreases and the thrust increases because of the increase in pressure thrust. However, at the higher altitude the thrust is less than it would be for optimum expansion at that altitude. The exhaust pressure is then greater than the external pressure and the nozzle is said to be underexpanded. The gas expansion continues downstream and manifests itself by creating diamond-shaped shock waves that can often be observed in the exhaust plume.

The second stage of the launch vehicle is designed for optimum expansion at the altitude where it becomes operational. Because the atmospheric pressure is less than at sea level, the exit pressure of the exhaust must be less and thus the expansion ratio must be greater. Consequently, the second stage nozzle exit diameter is larger than the first stage nozzle exit diameter.

For example, the first stage of a Delta II 7925 launch vehicle has an expansion ratio of 12. The propellant is liquid oxygen and RP-1 (a kerosene-like hydrocarbon) in a mixture ratio (O/F) of 2.25 at a chamber pressure of 4800 kPa (700 psia) with a sea level specific impulse of 255 seconds. The second stage has a nozzle expansion ratio of 65 and burns nitrogen tetroxide and Aerozene 50 (a mixture of hydrazine and unsymmetrical dimethyl hydrazine) in a mixture ratio of 1.90 at a chamber pressure of

5700 kPa (830 psia), which yields a vacuum specific impulse of 320 seconds.

In space, the surrounding atmospheric pressure is zero. In principle, the expansion ratio would have to be infinite to reduce the exit pressure to zero. Thus optimum expansion is impossible, but it can be approximated by a very large nozzle diameter, such as can be seen on the main engines of the space shuttle with e = 77.5. There is ultimately a tradeoff between increasing the size of the nozzle exit for improved performance and reducing the mass of the rocket engine.

In a chemical rocket, the exhaust velocity, and hence the specific impulse, increases as the combustion temperature increases and the molar mass of the exhaust products decreases. Thus liquid oxygen and liquid hydrogen are nearly ideal chemical rocket propellants because they burn energetically at high temperature (about 3200 K) and produce nontoxic exhaust products consisting of gaseous hydrogen and water vapor with a small effective molar mass (about 11 kg/kmol). The vacuum specific impulse is about 450 seconds. These propellants are used on the space shuttle, the Atlas Centaur upper stage, the Ariane-4 third stage, the Ariane-5 core stage, the H-2 first and second stages, and the Long March CZ-3 third stage.

Continue Reading Here