Rocket Propulsion

 

Hardware

 

            A simplistic view of the rocket, without consideration of all the internal machinery that feeds the fuel to the combustion chamber is the best way to introduce the rocket engine. As seen in Figure 1, a rocket engine consists of a combustion chamber and nozzle. This differs from most other engines in that it is not air breathing. That is, it uses its own oxidizer to create combustion rather than atmospheric air. Also a rocket engine is a reaction engine as opposed to rotational engines. This means that a rocket uses the momentum of the exhaust gases as a means of propelling itself rather than some kind of rotational device such as a flywheel or crankshaft to produce motion.

 

The Combustion Chamber

 

   Figure 1                                       The combustion chamber is the place where the fuel and oxidizer meet to create a controlled explosion, which cause the gases to leave at a very high temperature and speed. The higher the flame temperature of the gases when they combust the more efficient the fuel  performance, but the lower molecular weight of the fuel and oxidizer the better. Therefore the trade off must be made between high temperature combustion and weight considerations. The best fuel for rockets is liquid oxygen and hydrogen, they provide a lower adiabatic flame temperature than liquid oxygen and kerosene but they but they have a much lower molecular weight. Because the fuel is fed at enormous pressures, the chamber must be built to withstand these pressures, and also to contain the heat produced from the combustion of the fuel. The chamber is also designed optimally for the complete burning of the fuel. An interesting note is that in the space shuttle main engine, over 3 million pounds of fuel are used in a couple of minutes.

The Nozzle

 

The function of the nozzle is to convert the chemical-thermal energy generated in the combustion chamber into kinetic energy. The nozzle converts the slow moving, high pressure, high temperature gas in the combustion chamber into high velocity gas of lower pressure and Figure 2                                                temperature. Since thrust is the product of mass and velocity, a very high gas velocity is desirable. Nozzles are usually converging-diverging nozzles. The reason for this is that a converging nozzle would only allow for the expansion of the gas to the speed of sound, whereas the converging-diverging nozzle allows for speed up to several times that amount. The minimum flow area between the convergent and divergent section is called the nozzle throat, and is the point at which the speed of the flow reaches the speed of sound (Mach 1). The flow area at the end of the divergent section is called the nozzle exit area. The nozzle is usually made long enough (or the exit area is great enough) such that the pressure in the combustion chamber is reduced at the nozzle exit to the pressure existing outside the nozzle. That is, (Figure 2-1) Pe=Pa where Pe is the pressure at the nozzle exit and Pa is the outside ambient pressure, and is the point that thrust is maximum and the nozzle is said to be at optimum or correct expansion. When Pe is greater than Pa, the nozzle is under-extended (Figure 2-3). When the opposite is true, it is over-extended (Figure 2-2).

Therefore, the nozzle is designed for the altitude at which it has to operate. At the Earth's surface, at the atmospheric pressure of sea level the discharge of the exhaust gases is limited by the separation of the jet from the nozzle wall. In a vacuum, this limitation does not exist. Therefore, there have to be two different types of engines and nozzles, those that propel the first stage of the launch vehicle through the atmosphere, and those that propel subsequent stages or control the orientation of the spacecraft in the vacuum of space. Nozzles in the atmosphere are much narrower than those in space.

The nozzle throat area, At, can be found if the total propellant flow rate is known and the propellants and operating conditions have been selected. Assuming perfect gas law theory, we have

(1)   At = (q / Pt) x SQRT[ (R' x Tt) / (M x k) ]

where q is the propellant mass flow rate, Pt is the gas pressure at the nozzle throat, Tt is the gas temperature at the nozzle throat, R' is the universal gas constant, M is the molecular weight and k is the specific heat ratio. Pt and Tt are given by

(2)   Pt = Pc x [1 + (k - 1) / 2] -k/(k-1)


 


(3)   Tt = Tc x [1 / (1 + (k - 1) / 2)]

where Pc is the combustion chamber pressure and Tc is the combustion chamber flame temperature. These equations also make up the isentropic flow equations found for jet engines. The combustion chamber conditions are typically known since the chamber is specifically designed for certain conditions.

The hot gases must be expanded in the diverging section of the nozzle to obtain maximum thrust. The pressure of these gases will decrease as energy is used to accelerate the gas. We must find that area of the nozzle where the gas pressure is equal to the outside atmospheric pressure. This area will then be the nozzle exit area.

Mach number Nm is the ratio of the gas velocity to the local speed of sound. The Mach number at the nozzle exit is given by the perfect gas expansion expression

(4)   Nm2 = (2 / (k - 1)) x [(Pc / Pa) (k-1)/k - 1]

where Pa is the pressure of the ambient atmosphere.

The nozzle exit area, Ae, corresponding to the exit Mach number is given by

(5)   Ae = (At / Nm) x [(1 + (k - 1) / 2 x Nm2)/((k + 1) / 2)] (k+1)/(2(k-1))

The section ratio, or expansion ratio, is defined as the area of the exit Ae divided by the area of the throat.

            Since a rocket nozzle’s shape cannot be changed in mid-flight without the use of stages, an engineer will usually design the nozzle for a specific atmospheric pressure and accept the thrust loss as an acceptable performance loss.

 

Propellant

Propellant is the chemical mixture burned to produce thrust in rockets and consists of a fuel and an oxidizer. A fuel is a substance that burns when combined with oxygen producing gas for propulsion. An oxidizer is an agent that releases oxygen for combination with a fuel. Propellants are classified according to their state.

Two major types of propellant are available for rockets, solid fuel and liquid bipropellants. Bipropellant means that two fuels, usually a fuel and oxidizer, are used to create combustion. Solid rocket fuels are fuels that contain solid compounds and are used to produce specific kinds of thrust.

 

Solid Propellant

 

Solid propellant motors are the simplest of all rocket designs. They consist of a casing, usually steel, filled with a mixture of solid compounds (fuel and oxidizer), which burn at a rapid rate, expelling hot gases from a nozzle to produce thrust.

A solid fuel's geometry determines the area and contours of its exposed surfaces, and thus its burn pattern. There are two main types of solid fuel blocks as seen in Figure 3. These are cylindrical blocks, with combustion at a front, or surface, and cylindrical blocks with combustion within a channel. In the first case, the front of the flame travels from the nozzle end of the block towards the top of the casing. This produces constant thrust throughout the burn. In the second case the combustion surface develops along the length of a central channel. Sometimes the channel has a star shaped, or other, geometry to moderate the growth of this surface.

Figure 3.

The shape of the fuel block for a rocket is chosen for the particular type of mission it will perform. Since the combustion of the block progresses from its free surface, as this surface grows, geometrical considerations determine whether the thrust increases, decreases or stays constant.

Figure 4

Fuel blocks with a cylindrical channel (1) develop their thrust gradually. Those with a pipe shape rather than a channel (2) produce a relatively constant thrust, since the inner surface area decreases as the outer increases, which reduces to zero very quickly when the fuel is used up. The five-pointed star profile (3) develops a relatively constant thrust that decreases slowly to zero as the last of the fuel is consumed. The 'cruciform' profile (4) produces progressively less thrust due to constant reduction in fuel surface area. Fuel in a block with a 'double anchor' profile (5) produces a decreasing thrust, which drops off quickly near the end of the burn. The 'cog' profile (6) produces a strong initial thrust, followed by an almost constant lower thrust. This is produced because of the large amount of surface area that the cog initially contains, which burns up very quickly.

Unlike liquid bipropellant engines, solid propellant motors cannot be shut down. Once ignited, they will burn until all the propellant is exhausted.

There are two families of solids propellants: homogeneous and composite. Both types are dense, stable at ordinary temperatures, and easily storable.

Homogeneous propellants are either simple base or double base. A simple base propellant consists of a single compound, usually nitrocellulose, which has both an oxidation capacity and a reduction capacity. Double base propellants usually consist of nitrocellulose and nitroglycerine, to which a plasticiser is added to create a solid and increase the safety of the nitroglycerine, which can ignite with simple vibrations.

Homogeneous propellants do not usually have specific impulses greater than about 210 seconds (see Specific Impulse) under normal conditions. Their main asset is that they do not produce traceable fumes and are, therefore, commonly used in tactical weapons. The ability to not produce traceable fumes is substantial in that it is harder for early alert systems cannot pick them up and allow for preemptive strikes against enemies. They are also used to jettison spent parts or separating one stage from another. Modern composite propellants are heterogeneous powders, which use a crystallized or finely ground mineral salt as an oxidizer, often ammonium perchlorate. The fuel itself is aluminum. A polymeric binder holds the propellant together. Additional compounds are sometimes included, such as a catalyst, which speeds the speed of reaction.

Solid propellants are used to power the final stages of communications satellites and also to place satellites into geosynchronous orbit. According to NASA, the Space Shuttle uses the largest solid rocket motors ever built and flown. Each booster contains 1,100,000 pounds (499,000 kg) of propellant and can produce up to 3,300,000 pounds (14,680,000 Newtons) of thrust.

 

Liquid Bipropellant

 

In a liquid propellant rocket, the fuel and oxidizer are stored in separate tanks, and are fed through a system of pipes, valves, and turbo pumps to a combustion chamber where they are combined and burned to produce thrust. Liquid propellant engines are more complex then their solid propellant counterparts, however, they offer several advantages. By controlling the flow of propellant to the combustion chamber, the engine can be throttled, stopped, or restarted. This is great for safety considerations. If a leak or crack or other malfunction were to develop the engines could be stopped so that no explosions or other problems would occur, such as the tragedy of the Challenger.

A good liquid propellant is one with a high speed of exhaust gas ejection. This implies a high combustion temperature and exhaust gases with small molecular weights, as was discussed earlier. Liquid propellants used by NASA and in commercial launch vehicles can be classified into three types: petroleum, cryogenics, and hypergolics.

Petroleum fuels are those refined from crude oil and are a mixture of complex organic compounds, are those that are made of hydrogen and carbon. The petroleum used as rocket fuel is kerosene, or a type of highly refined kerosene called RP-1 (refined petroleum). This fuel was used on the Jupiter S-3D and H-1 engines. It is used in combination with liquid oxygen as the oxidizer. The disadvantage is that it has a high molecular weight and therefore cannot produce thrust for very long because it is used up quickly.

Cryogenic propellants are liquefied gases stored at very low temperatures, namely liquid hydrogen (LH2) as the fuel and liquid oxygen (LO2) as the oxidizer. Because of the low temperatures of cryogenic propellants, they are difficult to store over long periods of time, without costly storing tanks and facilities. For this reason, they are less desirable for use in military rockets, which must be kept launch ready for months at a time. Despite this drawback, the high efficiency of liquid hydrogen/liquid oxygen makes these fuels the best choice for most rocket engines. Liquid hydrogen delivers a thrust to fuel burning rate about 40% higher than other rocket fuels.

Hypergolic propellants are fuels and oxidizers that ignite spontaneously on contact with each other and require no ignition source. The easy start and restart capability of hypergolics make them ideal for spacecraft maneuvering systems. Once the fuel is shut off, combustion stops and vice versa. Also, hypergolics remain liquid at normal temperatures; they do not pose the storage problems of cryogenic propellants.

 

Specific Impulse

 

            Specific impulse is one method by which the efficiency of a rocket can be measured. Given in seconds, specific impulse is how long one kilogram of fuel can produce on Newton of thrust. The higher the specific impulse the longer fuel can be used to produce a constant thrust. Propellants are typically rated by their specific impulse, though the exact value can vary slightly due to rocket design or operating conditions. Chemical rockets usually have specific impulses between 200 and 450 seconds. Liquid Oxygen/Hydrogen have the highest specific impulse at 450 seconds. More advanced vehicles such as ion propulsion and nuclear thermal rockets can have specific impulses in the thousands. The exact equation for specific impulse is given in Eq. 6.

 

(6)        Isp = T/(g*dm/dt) = ((dm/dt)*Ve)/((dm/dt)*g) = Ve/g

 

T is thrust, g is the acceleration due to gravity, dm/dt is the rate that the fuel burns, and Ve is the exhaust velocity.

 

The Rocket Equation

 

            The velocity of a rocket is given by the solution of the momentum equation describing the rocket. The rocket is expelling mass, in the form of fuel, so it is losing mass, therefore the momentum equation is given by equation 7 and solved in equation 8.

 

L1 + FextDt= L2 

(7)        MrVr + Fext*Dt = (Mr – DM)(Vr +DV) +DM(Vr – Vex)

 

            The subscript r represents rocket, ext is external, ex is exhaust, the letter V represents velocity, M is mass, F is force and t is time. D represents a small finite change, or the differential.

 

            MrVr + Mr*-g*Dt = MrVr –DM*Vr +Mr*DV + DM*Vr – DM*Vex

 

            MrVr + Mr*-g*Dt = MrVr+MrDV-DM*Vex

 

            -g*Dt = DV – (dM/Mr)*Vex

 

(8)        Vf – Vi = -g*t + ln(Mi/Mf)*Vex

 

            This gives us the fact that the final velocity is directly related to the ratio of final mass to the initial mass. This means that the more fuel you have the higher end velocity a rocket can achieve. On the other hand though the amount of propellant you need goes up exponentially as the amount of change in velocity increases. Which means that the amount of mass you need goes to infinity quickly, which is impossible to build. So a higher exhaust velocity and lower DV are sought after by engineers. Since it is very important that the exhaust velocity is very high the equation for the velocity should be given. This equation is obtained by evaluating the energy equation from the combustion chamber to the nozzle exit.

            The energy equation is given by equation 9 and solved in equation 10.

 

(9)        Hc = V2/2 + He

            cpTc – cpTe = V2/2

 

            cp = gR/(g-1)

 

            Te/Tc = (Pe/Pc)(g-1)/g

 

            cp*Tc(1-Te/Tc) = V2/2

 

            ((2*g*R*Tc)/(g-1))*(1- (Pe/Pc)(g-1)/g) = V2

 

(10)      Vex = (((2*g*R*Tc)/(g-1))*(1- (Pe/Pc)(g-1)/g))0.5

 

            T is temperature, P is pressure, R is the gas constant, g is specific heat ratio, V is velocity. The exhaust velocity is therefore proportional to the combustion chamber parameters and the exit pressure. The exhaust velocity levels out when the exit pressure is equal to the atmospheric pressure, and stays relatively constant as the exit pressure decreases relative to the atmospheric pressure.

 

Other Related Sites:

How Stuff Works: Rocket Engines

Principles of Rockets