| The law of physics on which rocket propulsion is | | | | Although gravity has nothing whatever to do with |
| based is called the principle of momentum. According | | | | the rocket propulsion chemistry, it has entered into |
| to this principle, the time rate of change of the total | | | | the definition of specific impulse because in past |
| momentum of a system of particles is equal to the | | | | engineering practice mass was expressed in terms of |
| net external force. The momentum is defined as the | | | | the corresponding weight on the surface of the |
| product of mass and velocity. If the net external | | | | earth. By inspection of the equation, it can be seen |
| force is zero, then the principle of momentum | | | | that the specific impulse Isp is physically equivalent to |
| becomes the principle of conservation of momentum | | | | the effective exhaust velocity c, but is rescaled |
| and the total momentum of the system is constant. | | | | numerically and has a different unit because of |
| To balance the momentum conveyed by the | | | | division by g. Some manufacturers now express |
| exhaust, the rocket must generate a momentum of | | | | specific impulse in newton seconds per kilogram, |
| equal magnitude but in the opposite direction and | | | | which is the same as effective exhaust velocity in |
| thus it accelerates forward. | | | | meters per second. |
| The system of particles may be defined as the sum | | | | Two other important parameters are the thrust |
| of all the particles initially within the rocket at a | | | | coefficient CF and the characteristic exhaust velocity |
| particular instant. As propellant is consumed, the | | | | c*. The thrust coefficient is defined as |
| exhaust products are expelled at a high velocity. The | | | | CF = F / At pc = m(dot) c / At pcwhere F is the |
| center of mass of the total system, subsequently | | | | thrust, At is the throat area, and pc is the chamber |
| consisting of the particles remaining in the rocket and | | | | pressure. This parameter is the figure of merit of the |
| the particles in the exhaust, follows a trajectory | | | | nozzle design. The characteristic exhaust velocity is |
| determined by the external forces, such as gravity, | | | | defined asc* = At pc / m(dot) = c / CF |
| that is the same as if the original particles remained | | | | This parameter is the figure of merit of the |
| together as a single entity. In deep space, where | | | | propellant. Thus the specific impulse may be written |
| gravity may be neglected, the center of mass | | | | Isp = CF c* / gwhich shows that the specific impulse |
| remains at rest. | | | | is the figure of merit of the nozzle design and |
| ROCKET THRUST | | | | propellant as a whole, since it depends on both CF |
| The configuration of a chemical rocket engine | | | | and c*. However, in practice the specific impulse is |
| consists of the combustion chamber, where the | | | | usually regarded as a measure of the efficiency of |
| chemical reaction takes place, and the nozzle, where | | | | the propellant alone. |
| the gases expand to create the exhaust. An | | | | LAUNCH VEHICLE PROPULSION SYSTEMS |
| important characteristic of the rocket nozzle is the | | | | In the first stage of a launch vehicle, the exit |
| existence of a throat. The velocity of the gases at | | | | pressure of the exhaust is equal to the sea level |
| the throat is equal to the local velocity of sound and | | | | atmospheric pressure 101.325 kPa (14.7 psia) for |
| beyond the throat the gas velocity is supersonic. | | | | optimum expansion. As the altitude of the rocket |
| Thus the combustion of the gases within the rocket | | | | increases along its trajectory, the surrounding |
| is independent of the surrounding environment and a | | | | atmospheric pressure decreases and the thrust |
| change in external atmospheric pressure cannot | | | | increases because of the increase in pressure thrust. |
| propagate upstream. | | | | However, at the higher altitude the thrust is less than |
| The thrust of the rocket is given by the theoretical | | | | it would be for optimum expansion at that altitude. |
| equation : | | | | The exhaust pressure is then greater than the |
| F = lm(dot) ve + ( pe - pa ) Ae | | | | external pressure and the nozzle is said to be |
| This equation consists of two terms. The first term, | | | | underexpanded. The gas expansion continues |
| called the momentum thrust, is equal to the product | | | | downstream and manifests itself by creating |
| of the propellant mass flow rate m(dot)and the | | | | diamond-shaped shock waves that can often be |
| exhaust velocity ve with a correction factor l for | | | | observed in the exhaust plume. |
| nonaxial flow due to nozzle divergence angle. The | | | | The second stage of the launch vehicle is designed |
| second term is called the pressure thrust. It is equal | | | | for optimum expansion at the altitude where it |
| to the difference in pressures pe and pa of the | | | | becomes operational. Because the atmospheric |
| exhaust velocity and the ambient atmosphere, | | | | pressure is less than at sea level, the exit pressure |
| respectively, acting over the area Ae of the exit | | | | of the exhaust must be less and thus the expansion |
| plane of the rocket nozzle. The combined effect of | | | | ratio must be greater. Consequently, the second |
| both terms is incorporated into the effective exhaust | | | | stage nozzle exit diameter is larger than the first |
| velocity c. Thus the thrust is also written | | | | stage nozzle exit diameter. |
| F = m(dot) cwhere an average value of c is used, | | | | For example, the first stage of a Delta II 7925 launch |
| since it is not strictly constant. | | | | vehicle has an expansion ratio of 12. The propellant is |
| The exhaust exit pressure is determined by the | | | | liquid oxygen and RP-1 (a kerosene-like hydrocarbon) |
| expansion ratio given bye= Ae / Atwhich is the ratio | | | | in a mixture ratio (O/F) of 2.25 at a chamber |
| of the area of the nozzle exit plane Ae and the area | | | | pressure of 4800 kPa (700 psia) with a sea level |
| of the throat At . As the expansion ratio e increases, | | | | specific impulse of 255 seconds. The second stage |
| the exhaust exit pressure pe decreases. | | | | has a nozzle expansion ratio of 65 and burns nitrogen |
| The thrust is maximum when the exit pressure of | | | | tetroxide and Aerozene 50 (a mixture of hydrazine |
| the exhaust is equal to the ambient pressure of the | | | | and unsymmetrical dimethyl hydrazine) in a mixture |
| surrounding environment, that is, when pe = pa. This | | | | ratio of 1.90 at a chamber pressure of |
| condition is known as optimum expansion and is | | | | 5700 kPa (830 psia), which yields a vacuum specific |
| achieved by proper selection of the expansion ratio. | | | | impulse of 320 seconds. |
| Although optimum expansion makes the contribution | | | | In space, the surrounding atmospheric pressure is |
| of the pressure thrust zero, it results in a higher | | | | zero. In principle, the expansion ratio would have to |
| value of exhaust velocity ve such that the increase in | | | | be infinite to reduce the exit pressure to zero. Thus |
| momentum thrust exceeds the reduction in pressure | | | | optimum expansion is impossible, but it can be |
| thrust. | | | | approximated by a very large nozzle diameter, such |
| A conical nozzle is easy to manufacture and simple to | | | | as can be seen on the main engines of the space |
| analyze. If the apex angle is 2a , the correction | | | | shuttle with e = 77.5. There is ultimately a tradeoff |
| factor for nonaxial flow is | | | | between increasing the size of the nozzle exit for |
| - = ½ (1 + cos a ) | | | | improved performance and reducing the mass of the |
| The apex angle must be small to keep the loss within | | | | rocket engine. |
| acceptable limits. A typical design would be a = 15° , | | | | In a chemical rocket, the exhaust velocity, and hence |
| for which l = 0.9830. This represents a loss of 1.7 | | | | the specific impulse, increases as the combustion |
| percent. However, conical nozzles are excessively | | | | temperature increases and the molar mass of the |
| long for large expansion ratios and suffer additional | | | | exhaust products decreases. Thus liquid oxygen and |
| losses caused by flow separation. A bell-shaped | | | | liquid hydrogen are nearly ideal chemical rocket |
| nozzle is therefore superior because it promotes | | | | propellants because they burn energetically at high |
| expansion while reducing length. | | | | temperature (about 3200 K) and produce nontoxic |
| ROCKET PROPULSION PARAMETERS | | | | exhaust products consisting of gaseous hydrogen |
| The specific impulse Isp of a rocket is the parameter | | | | and water vapor with a small effective molar mass |
| that determines the overall effectiveness of the | | | | (about 11 kg/kmol). The vacuum specific impulse is |
| rocket nozzle and propellant. It is defined as the ratio | | | | about 450 seconds. These propellants are used on |
| of the thrust and the propellant weight flow rate, or | | | | the space shuttle, the Atlas Centaur upper stage, the |
| Isp = F / m(dot) g = c / gwhere g is a conventional | | | | Ariane-4 third stage, the Ariane-5 core stage, the H-2 |
| value for the acceleration ofgravity (9.80665 m/s2 | | | | first and second stages, and the Long March CZ-3 |
| exactly). Specific impulse is expressed in seconds. | | | | third stage. |