Rocket Propulsion by Ioncube khanz

2 Nozzle & Combustion Chamber Previous chapter deals with the basics of rocket flight & has defined some key points for analyzing the performance of a given rocket unit such as specific impulse & mass ratio. This chapter deals with inner rocket propulsion components in detail to further elaborate performance parameters while utilizing the principles exhibited in previous chapter. Thermodynamic relations of the processes inside a rocket nozzle and chamber furnish the mathematical tools needed to calculate the performance and determine several of the key design parameters of rocket propulsion systems. They are useful as a means of evaluating and comparing the performance of various rocket systems; they permit the prediction of the operating performance of any rocket unit that uses the thermodynamic expansion of a gas, and the determination of several necessary design parameters, such as nozzle size and generic shape, for any given performance requirement. This theory applies to chemical rocket propulsion systems (both liquid and solid propellant types), nuclear rockets, solar heated and resistance or arc heated electrical rocket systems, and to any propulsion system that uses the expansion of a gas as the propulsive mechanism for ejecting matter at high velocity.

2.1 Ideal Rocket An ideal rocket is one in which the following assumptions are valid: 1. 2.

The working substance (propellant chemical reaction products) is homogeneous. All the species of the working fluid are gaseous. Any condensed phases (solid or liquid) have negligible mass. 3. The working substance obeys the perfect gas laws. 4. There is no heat transfer across rocket walls; therefore, the flow is adiabatic. 5. The propellant flow is steady and constant. The expansion of the working fluid takes place in a uniform and steady manner without vibration. 6. Transient effects (start, stop) are of very short duration and can be neglected. 7. All the exhaust gases leaving the rocket nozzle have an axially directed velocity. 8. The gas velocity, pressure, temperature, or densities are uniform across any section normal to the nozzle axis. 9. Chemical equilibrium is established within the rocket chamber and the composition does not change in the nozzle. 10. There is no friction and boundary layer effects are neglected. 11. There are no shock waves or discontinuities in the nozzle.

2.2 Establishing Thermodynamic Relations The following thermodynamic relations, which are fundamental and important in analysis and design of rocket units, are introduced and explained in this chapter. The utilization of these equations should give the reader a basic understanding of the thermodynamic processes involved in rocket gas behavior and expansion. A knowledge of elementary thermodynamics and fluid mechanics on the part of the reader is assumed.

2.2.1 Derivation for exhaust velocity In the rocket engine, let vc be the velocity of gases inside combustion chamber and v be the velocity in any section of the nozzle such that vc «v. By the law of conservation of energy

eq. 2.1

When pressure at throat becomes zero velocity becomes maximum and is given as

eq. 2.2 Square root of RT is in it for velocity and hence given derivation is dimensionally correct. Thus if we can reduce pressure at output of nozzle to zero, we can get maximum velocity or maximum kinetic energy.

2.2.2 Derivation for Mach Number The velocity of sound or the acoustic velocity in ideal gases is independent of pressure. It is defined as 1

The Mach number is a dimensionless flow parameter and is defined as the ratio of the flow velocity to the local acoustic velocity. For an isentropic nozzle expansion process the pressure drops, the absolute temperature drops by a substantially smaller factor and the specific volume increases. When the flow of a compressible fluid is stopped or stagnated isentropically, the prevailing conditions are known as the stagnation conditions denoted by subscript o. The stagnation temperature or total temperature To is defined from the energy equation as

See reference 1

Substituting for Cp

from

where To is the total temperature and T is the static temperature of a stream flowing at Mach number M.

2.2.3 Nozzle Configuration The function of a nozzle is to reduce pressure; at the expense of pressure, velocity is increased. The required nozzle area decreases to a minimum and then increases again, Nozzles of this type consist of a convergent and a divergent section. •

The principle of conservation of matter in a steady flow process is expressed by equating the mass flow m at any section x to the flow at section y, and is known in mathematical form as the continuity equation. The first-order differential equation is expressed in terms of the specific volume V which is the reciprocal of the gas density ρ, the local gas velocity v, and the nozzle cross-section area A. •

When integrated, this gives a constant mass flow m at any cross-section x or y. For an isentropic flow process, the following relations hold between any points x and y:

Hence if ρ is the density at any point in the nozzle and ρc is the density at the throat of the nozzle,

Substituting for v from eq. 1.1 and ρ in the equation

eq. 2.3

Thus from the continuity equation, we have derived the relationship that the cross-sectional area A is inversely •

•

proportional to the ratio ρv or p/pc. From the equation for m / A , we can infer that the quantity is zero when m = 0 or A → ∞ • Also A = ∞ when p is zero or p= pc i.e., there is no change in pressure. The minimum nozzle area is called the throat area. The ratio of the nozzle exit area A2 to the throat area At is called the nozzle area expansion ratio denoted by ε: and is given by

The maximum gas flow per unit area occurs at the throat, and a unique gas pressure corresponding to this maximum flow density exists. This throat pressure Pt for a maximum flow in an isentropic expansion nozzle can be found by differentiating eqn 2.3 and setting the derivative equal to zero.

The first expression on the RHS indicates no flow condition when the equation is equated to zero. The second expression when equated to zero gives

pt ⎛ 2 ⎞ =⎜ ⎟ pc ⎝ γ + 1 ⎠

γ γ −1

eq. 1.4

The value of γ ranges from 1.2 to 1.4. When γ=1.24,

= 0.5568

0.5

Hence if pc = 100 then pt = 50 i.e., pt = pc/2

The throat pressure pt for which the isentropic weight flow is a maximum is called the critical pressure. Typical values are between 0.57 and 0.53. At the point of critical pressure, the values of specific volume and the temperature can be obtained from the isentropic relations

eq. 2.5

eq. 2.6 The nozzle inlet temperature Tt is usually also the combustion temperature and the nozzle flow stagnation temperature To. From equations 2.1 2.4 and 2.6, the critical velocity or throat velocity vt is obtained as

eq. 2.7 The first version permits the throat velocity to be calculated directly from the nozzle inlet conditions without any of the throat conditions being known. At the nozzle throat, the temperature is Tt and this is found to be identical to the equation for acoustic velocity. Therefore the critical throat velocity vt is always equal to the local acoustic velocity a for ideal nozzles in which critical conditions prevail; the Mach number at the throat of an ideal rocket is unity. The divergent portion of the nozzle permits a further decrease in pressure and an increase in velocity above the velocity of sound (supersonic). If the nozzle is cut-off at the throat section, the exit gas velocity is sonic or subsonic. There are therefore, essentially three types of nozzles: subsonic, sonic and supersonic. The supersonic nozzle is used in rockets. The ratio between the inlet and exit pressures in all rockets is sufficiently large to induce supersonic flow.

Only if the chamber pressure drops below approx. 2.17 atm will there be subsonic flow in the divergent portion of the nozzle when operating at sea level. The velocity of sound is equal to the velocity of propagation of a pressure wave within the medium, sound being a pressure wave. If, therefore, sonic velocity is reached at anyone point within a steady flow system, it is impossible for a pressure disturbance to travel upstream past the location of sonic or supersonic velocity. Therefore, any partial obstruction or disturbance of the flow downstream of the nozzle throat section has no influence on the flow at the throat section or upstream of the throat section provided that this disturbance does not raise the downstream pressure above its critical value. It is not possible to increase the throat velocity or the flow rate in the nozzle by lowering the exit pressure or evacuating the exhaust section. This condition is often described as choking of the flow. (Choking can also be stated as the condition when max mass flux is passing through the throat).

2.2.4 Aerodynamic Point of View •

Differentiating this equation and taking m constant

Euler's equation is given by 2

Substituting in previous eqn

It is found that when M=0.33, dp=l%. Hence compressibility effects can be ignored till M=0.3 which is the range for low subsonic flow. Substituting this value in main eqn

See reference 1

Thus min value for cross sectional area is when M=1. If v is to be positive, when M <1 i.e., we require negative dA which implies a reduction in nozzle throat area (convergent cross section). Whereas, if v is to be positive when M>1, we require positive dA i.e., increase in nozzle throat area (divergent cross section). A will reach its minimum when M= 1 and further reduction to A will not have any effect beyond this value. In short

Figure 2.1 Nozzle configurations 3

See reference 2

2.2.5 Mass Flow Rate at Throat

Thus mass flow rate is proportional to pc and At depends on burning of propellant i.e., mass of propellant supplied. In steady state condition, whatever mass of propellant is pumped in propulsion chamber is exited out as mass of combustion gases.

2.2.6 Characteristic Velocity(c*) The characteristic exhaust velocity denoted by c* is defined as

eq. 2.8 The experimental or practical value of c* can be found as follows:

The combustion efficiency can be expressed in terms of c*(experimental) and c*(theoretical) as follows:

2.3 Thrust And Thrust Coefficient In the previous chapter we have defined thrust or reaction force on the rocket unit's structure is caused by the action of the pressure of the combustion gases against the rocket chamber, injector, and nozzle surfaces. Also from the basic thrust equation thrust acting on the vehicle is composed of two terms: the momentum thrust and the pressure thrust. eq. 2.9 where subscript 2 refers conditions at the cross-sectional area of the nozzle exit & p3 is the ambient pressure. The effective exhaust velocity c is the average equivalent velocity at which propellant is ejected from the vehicle is defined as eq. 2.10 The characteristic velocity c* is defined as •

c* = pc At / m

eq. 2.11

This c* is used in comparing the relative performance of different chemical rocket propulsion system designs and propellants and can be easily determined with above equation.

2.3.1 Ideal Thrust Equation Considering v2 as the exhaust velocity (vt), pressure in combustion chamber (p1) as pc, pressure at throat of nozzle (p2) as pe and pressure at exhaust (p3) which is normally equal to ambient or atmospheric pressure as pa, and •

substituting for m and v from eqn 2.1,2.5 and 2.7 in equation 2.9, we get\

eq. 2.12 This equation shows that the thrust is proportional to the throat area At and the nozzle inlet pressure pc and is a function of the pressure ratio across the nozzle pc/pe, the specific heat ratio γ and the pressure thrust. It is called the ideal thrust equation.

2.3.2 Thrust Coefficient CF The thrust coefficient CF is defined as the thrust divided by the chamber pressure pc and the throat area At. Thus CF is given by eq. 2.13 (From eq. 2.12)

eq.2.14

eqn above consists of two parts; when we talk about ideal nozzle whole of the pressure energy (created in combustion chamber) is converted to kinetic energy in other words pe=pa thus for an ideal CF denoted by CF* eqn 2.14 reduces to

= CF* The efficiency in terms of the thrust coefficient is given as

Isp can be written in terms of thrust coefficient as

2.3.3 Designing a Nozzle During flight-testing of an engine, thrust F can be measured by measuring pressure pc (using suitable instruments for measuring pressure). These practical values can then be compared with ideal theoretical values (denoted with asterisk superscript)

Assuming ideal conditions pe=pa

2.3.4 Types Of Nozzles •

Adapted nozzle (pe=pa). This is no flow or optimum condition.

•

Under expanding Nozzle(pe>pa) .An under expanding nozzle discharges the fluid at a pressure greater than the external pressure because the exit area is too small. The expansion of the fluid is therefore incomplete within the nozzle and continues outside. The nozzle exit pressure is higher than the local atmospheric pressure. The further expansion outside the nozzle causes the nozzle exhaust plume to expand.

•

Over expanding Nozzle (pe<pa). In an over expanding nozzle the fluid is expanded to a lower pressure than the external pressure; it has an exit area that is too large. The different possible flow conditions in an over expanding nozzle are as follows: (a) When the external pressure p3 is below the nozzle exit pressure p2, the nozzle will flow full but will have expansion waves at its exit. (b) For external pressures p3 slightly higher than the nozzle exit pressure p2, the nozzle will continue to flow full (p2>=0.4p3). Oblique shock waves will exist outside of the exit section. (c) For higher external pressures, a separation of the jet will take place in the divergent section of the nozzle. The flow separation is axially symmetrical and is accompanied by oblique shock waves. The point of separation travels upstream with increasing external pressure. The flow at the nozzle exit is supersonic. The nozzle portion of supersonic flow between the throat and the location of the separation diminishes in length with increasing back pressure until the shock wave and the point of separation reach the throat. (d) For nozzles in which the exit pressure is very close in value to the inlet pressure, subsonic flow prevails throughout the nozzle.

A rough criterion for jet separation is for p2<=0.4p3. The jet separation will depend not only on this relation between p2 and p3, but also on the pressure gradient, nozzle contour, boundary layer, and flow stability.

2.4 Real Nozzles Compared to an ideal nozzle, the real nozzle has energy losses and energy that is unavailable for conversion into kinetic energy of the exhaust gas. The principal losses are: (a) The divergence of the flow in the nozzle exit sections causes a loss, which varies as a function of the cosine of the divergence angle for conical nozzles. The losses can be reduced for bell-shaped nozzle contours. (b) Excessively low chamber or port area cross sections cause pressure losses in the chamber and reduce the exhaust velocity. (c) Reduced flow velocity in the boundary layer can reduce the effective exhaust velocity by 0.5 to 1.5%. (d) Solid particles or liquid droplets in the gas can cause losses up to 5%. (e) Unsteady combustion can account for a small loss. (f) Chemical reactions in nozzle flow change gas properties and gas temperatures, giving typically a 0.5% loss. (g) There is lower performance during transient pressure operation, for example during start, stop, or pulsing.

2.4.1 Conical nozzle In early rocket engine applications, the conical nozzle, which proved satisfactory in most respects, was used almost exclusively. A conical nozzle allows ease of manufacture and flexibility in converting an existing design to higher or lower expansion ratio without major redesign. The configuration of a typical conical nozzle is shown in Figure 2.2. The nozzle throat section has the contour of a circular arc with radius R, ranging from 0.25 to 0.75 times the throat diameter, Dt . The half-angle of the nozzle convergent cone section, θ, can range from 20 to 45 degrees. The divergent cone half-angle, α, varies from approximately 12 to 18 degrees. The conical nozzle with a 15-degree divergent half-angle has become almost a standard because it is a good compromise on the basis of weight, length, and performance. Since certain performance losses occur in a conical nozzle as a result of the nonaxial component of the exhaust gas velocity, a correction factor, γ, is applied in the calculation of the exit-gas momentum. This factor (thrust efficiency) is the ratio between the exit-gas momentum of the conical nozzle and that of an ideal nozzle with uniform, parallel, axial gasflow. The value of γ can be expressed by the following equation:

Figure 2.2 Geometry of Conical Nozzle

2.4.2 Bell nozzle To gain higher performance and shorter length, engineers developed the bell-shaped nozzle. It employs a fastexpansion (radial-flow) section in the initial divergent region, which leads to a uniform, axially directed flow at the nozzle exit. The wall contour is changed gradually enough to prevent oblique shocks. An equivalent 15-degree halfangle conical nozzle is commonly used as a standard to specify bell nozzles. For instance, the length of an 80% bell nozzle (distance between throat and exit plane) is 80% of that of a 15-degree half-angle conical nozzle having the same throat area, radius below the throat, and area expansion ratio. Bell nozzle lengths beyond approximately 80% do not significantly contribute to performance, especially when weight penalties are considered. However, bell nozzle lengths up to 100% can be optimum for applications stressing very high performance. One convenient way of designing a near optimum thrust bell nozzle contour uses the parabolic approximation procedures suggested by G.V.R. Rao 4 . The design configuration of a parabolic approximation bell nozzle is shown in Figure 2.3. The nozzle contour immediately upstream of the throat T is a circular arc with a radius of 1.5 Rt. The

G.V.R. Rao, "Exhaust Nozzle Contour for Optimum Thrust," Jet Propulsion 28, 377 (1958).

divergent section nozzle contour is made up of a circular entrance section with a radius of 0.382 Rt from the throat T to the point N and parabola from there to the exit E. Design of a specific nozzle requires the following data: throat diameter Dt, axial length of the nozzle from throat to exit plane Ln (or the desired fractional length, Lf, based on a 15-degree conical nozzle), expansion ratio ε , initial wall angle of the parabola θn, and nozzle exit wall angle θe. The wall angles θn and θe are shown in Figure 2.4 as a function of the expansion ratio. Optimum nozzle contours can be approximated very accurately by selecting the proper inputs. Although no allowance is made for different propellant combinations, experience has shown only small effect of the specific heat ratio upon the contour.

Figure 2.3 Geometry of Bell Nozzle

Figure 2.4 θn and θe as a function of ε

2.5 Combustion Chamber 5 The combustion chamber serves as an envelope to retain the propellants for a sufficient period to ensure complete mixing and combustion. The required stay time, or combustion residence time, is a function of many parameters. The theoretically required combustion chamber volume is a function of the mass flow rate of the propellants, the average density of the combustion products, and the stay time needed for efficient combustion. This relationship can be expressed by the following equation:

where Vc is the chamber volume, q is the propellant mass flow rate, V is the average specific volume, and ts is the propellant stay-time. A useful parameter relative to chamber volume and residence time is the characteristic length, L* (pronounced "L star"), the chamber volume divided by the nozzle sonic throat area:

eq. 2.15 The L* concept is much easier to visualize than the more elusive "combustion residence time", expressed in small fractions of a second. Since the value of At is in nearly direct proportion to the product of q and V, L* is essentially a function of ts. The customary method of establishing the L* of a new thrust chamber design largely relies on past experience with similar propellants and engine size. Under a given set of operating conditions, such as type of propellant, mixture ratio, chamber pressure, injector design, and chamber geometry, the value of the minimum required L* can only be evaluated by actual firings of experimental thrust chambers. Typical L* values for various propellants are shown in the table below. With throat area and minimum required L* established, the chamber volume can be calculated by eqn 2.15

Table 2.1: Chamber Characteristic Length, L*

All mathematical expressions in this section are obtained from reference 3

Three geometrical shapes have been used in combustion chamber design - spherical, near-spherical, and cylindrical with the cylindrical chamber being employed most frequently in the United States. Compared to a cylindrical chamber of the same volume, a spherical or near-spherical chamber offers the advantage of less cooling surface and weight; however, the spherical chamber is more difficult to manufacture and has provided poorer performance in other respects. The total combustion process, from injection of the reactants until completion of the chemical reactions and conversion of the products into hot gases, requires finite amounts of time and volume, as expressed by the characteristic length L*. The value of this factor is significantly greater than the linear length between injector face and throat plane. The contraction ratio is defined as the major cross-sectional area of the combuster divided by the throat area. Typically, large engines are constructed with a low contraction ratio and a comparatively long length; and smaller chambers employ a large contraction ratio with a shorter length, while still providing sufficient L* for adequate vaporization and combustion dwell-time. As a good place to start, the process of sizing a new combustion chamber examines the dimensions of previously successful designs in the same size class and plotting such data in a rational manner. The throat size of a new engine can be generated with a fair degree of confidence, so it makes sense to plot the data from historical sources in relation to throat diameter. Figure 2.5 plots chamber length as a function of throat diameter (with approximating equation). It is important that the output of any modeling program not be slavishly applied, but be considered a logical starting point for specific engine sizing.

Figure 2.5 Throat diameter versus chamber length

The basic elements of a cylindrical thrust-chamber are identified in Figure 2.2. In design practice, it has been arbitrarily defined that the combustion chamber volume includes the space between the injector face and the nozzle throat plane. The approximate volume of the combustion chamber can be expressed by the following equation:

Rearranging above equation we get the following, which can be solved for the chamber diameter via iteration:

References: 1) Cengel, Boles. “Engineering Thermodynamics” 5th ed. Chap#1 sub sect. 17.2 2) J M. Smith “Introduction To Chemical Engineering Thermodynamics” 6th ed. p 239-241 3) http://www.braeunig.us/space/propuls.htm#combustion

3 Elements Of Liquid Propulsion System The basic operation of both liquid- and solid-fuelled engines is the same, but behind the broad principles, technical issues have a significant impact on efficiency and performance. The design of two most vital propulsion components namely turbine & turbo-pump is beyond the scope of this text; thus only performance parameters are discussed here leaving the details to appropriate references. In this chapter we shall examine the technical issues pertaining to liquid-fuelled rocket engines, and see how they affect the performance.

3.1 The Basic Configuration Of The Liquid Propellant Engine A liquid propellant rocket engine system comprises the combustion chamber, nozzle, and propellant tanks, together with the means to deliver the propellants to the combustion chamber (Figure 3.1). In the simplest system, the propellant is fed to the combustion chamber by static pressure in the tanks. High-pressure gas is introduced to the tank, or is generated by evaporation of the propellant, and this forces the fuel and oxidizer into the combustion chamber. As we have seen in Chapter 2, the thrust of the engine depends on the combustion chamber pressure and, of course, on the mass flow rate. It is difficult to deliver a high flow rate at high pressure using static tank pressure alone, so this system is limited to low-thrust engines for vehicle upper stages. There is a further penalty, because the tanks need to have strong walls to resist the high static pressure, and this reduces the mass ratio. The majority of large liquid propellant engine systems use some kind of turbo-pump to deliver propellants to the combustion chamber. The most common makes use of hot gas, generated by burning some of the propellant, to drive the turbine.

Figure 3.1. Schematic of a liquid-propellant engine.

Since high combustion temperature is needed for high thrust, cooling is an important consideration in order to avoid thermal degradation of the combustion chamber and nozzle. The design of combustion chambers and nozzles has to take this into account. In addition, safe ignition and smooth burning of the propellants is vital to the correct performance of the rocket engine. Having discussed about the nozzle & combustion chamber we will discuss the following elements exclusive to liquid propulsion system •

Power cycles

•

Injector

•

Gas generator

•

Rocket Turbine

•

Turbo-pump

3.2 Power Cycles Liquid engines can be categorized according to their power cycles—that is, how power is derived to feed propellants to the main combustion chamber. Here are some of the more common types.

3.2.1 GasGenerator Cycle The gas-generator cycle taps off a small amount of fuel and oxidizer from the main flow (typically 3 to 7 percent) to feed a burner called a gas generator. The hot gas from this generator passes through a turbine to generate power for the pumps that send propellants to the combustion chamber. The hot gas is then either dumped overboard or sent into the main nozzle downstream. Increasing the flow of propellants into the gas generator increases the speed of the turbine, which increases the flow of propellants into the main combustion chamber (and hence, the amount of thrust produced). The gas generator must burn propellants at a less-than-optimal mixture ratio to keep the temperature low for the turbine blades. Thus, the cycle is appropriate for moderate power requirements but not high-power systems, which would have to divert a large portion of the main flow to the less efficient gas-generator flow.

3.2.2 Staged Combustion Cycle In a staged combustion cycle, the propellants are burned in stages. Like the gas-generator cycle, this cycle also has a burner, called a preburner, to generate gas for a turbine. The preburner taps off and burns a small amount of one propellant and a large amount of the other, producing an oxidizer-rich or fuel-rich hot gas mixture that is mostly unburned vaporized propellant. This hot gas is then passed through the turbine, injected into the main chamber, and burned again with the remaining propellants. The advantage over the gas-generator cycle is that all of the propellants are burned at the optimal mixture ratio in the main chamber and no flow is dumped overboard. The staged combustion cycle is often used for high-power applications. The higher the chamber pressure, the smaller and lighter the engine can be to produce the same thrust. Development cost for this cycle is higher because the high pressures complicate the development process.

3.2.3 Expander Cycle The expander cycle is similar to the staged combustion cycle but has no preburner. Heat in the cooling jacket of the main combustion chamber serves to vaporize the fuel. The fuel vapor is then passed through the turbine and injected into the main chamber to burn with the oxidizer. This cycle works with fuels such as hydrogen or methane, which have a low boiling point and can be vaporized easily. As with the staged combustion cycle, all of the propellants are burned at the optimal mixture ratio in the main chamber, and typically no flow is dumped overboard; however, the heat transfer to the fuel limits the power available to the turbine, making this cycle appropriate for small to midsize engines. A variation of the system is the open, or bleed, expander cycle, which uses only a portion of the fuel to drive the turbine. In this variation, the turbine exhaust is dumped overboard to ambient pressure to increase the turbine pressure ratio and power output. This can achieve higher chamber pressures than the closed expander cycle although at lower efficiency because of the overboard flow.

3.2.4 PressureFed Cycle The simplest system, the pressure-fed cycle, does not have pumps or turbines but instead relies on tank pressure to feed the propellants into the main chamber. In practice, the cycle is limited to relatively low chamber pressures because higher pressures make the vehicle tanks too heavy. The cycle can be reliable, given its reduced part count and complexity compared with other systems.

3.3 Injection The injector has to fulfill three functions: it should ensure that the fuel and oxidizer enter the chamber in a fine spray, so that evaporation is fast; it should enable rapid mixing of the fuel and oxidiser, in the liquid or gaseous phase; and it should deliver the propellants to the chamber at high pressure, with a high flow rate. Figure 3.2 shows schematically the vaporisation, mixing and combustion zones in the combustion chamber. The specific injector design has to take into account the nature of the propellants. For cryogenic propellants such as liquid oxygen and liquid hydrogen, evaporation into the gaseous phase is necessary before ignition and combustion. In this case a fine spray of each component is needed. The spray breaks up into small droplets which evaporate, and mixing then occurs between parallel streams of oxygen and hydrogen. For hypergolic or self-igniting propellants such as nitrogen tetroxide and Unsymmetrical dimethylhydrazine (UDMH), the two components, which react as liquids at room temperature, should come into contact early, and impinging sprays or jets of the two liquids are arranged. In some cases pre-mixing of the propellants in the liquid form is needed, and here the swirl injector is used, in which the propellants are introduced together into a mixing tube. They enter the chamber pre-mixed, and are exposed to the heat of combustion. In all cases, the heat of the gases undergoing combustion is used to evaporate the propellant droplets. The heat is transferred to the droplets by radiation, and conduction through the gas. The propellant passing through the combustion chamber has a low velocity, and does not speed up until it reaches the nozzle. The requirement for a fine spray, together with a high flow rate, is contradictory, and can be realised only by making up the injector of many hundreds of separate fine orifices. Good mixing requires that adjacent jets consist of fuel and oxidiser. Thus, the hundreds of orifices have to be fed by complex plumbing, with the piping for two components interwoven. The design of the injector is a major issue of combustion chamber design.

Figure 3.2. Injection and combustion.

3.3.1 Types of injector The simplest type of injector is rather like a shower head, except that adjacent holes inject fuel and oxidant so that the propellants can mix. Improved mixing can be achieved with the use of a coaxial injector. Here each orifice has the fuel injected through an annular aperture which surrounds the circular oxidant aperture, and this is repeated many times to cover the area of the injector. These types of injector are shown in Figure 3.3. The above injectors are used for propellants which react in the vapour phase. The fine sprays quickly form tiny droplets, which also evaporate quickly. The impinging jet injector is shown in two forms in Figure 3.4. The first is designed to make sure that propellants mix as early as possible, while still in the liquid phase, and is useful for hypergolic propellant combinations. In the second form, jets of the same propellant impinge on one another. This is useful where fine holes are not suitable. The cross section of the jets can be larger, while the impinging streams cause the jets to break up into droplets. The injector can be located across the back of the combustion chamber, as indicated in Figure 3.2, or it can be located around the cylindrical wall of the rear end of the combustion chamber. The choice depends on convenience of plumbing, and the location of the igniter, where used.

3.3.2 Ignition Secure and positive ignition of the engine is essential in respect of both safety and controllability. The majority of engines are used only once during a mission, but the ability to restart is vital to manned missions, and contributes greatly to the flexibility of modern launch vehicles. For single-use engines, including all solid propellant engines, starting is usually accomplished by means of a pyrotechnic device. The device is set off by means of an electric current, which heats a wire set in the pyrotechnic material. The material ignites, and a shower of sparks and hot gas from the chemical reaction ignites the gaseous or solid propellant mixture. Pyrotechnic igniters are safe and reliable. They have redundant electrical heaters and connections, and similar devices have a long history, as single-use actuators, for many applications in space. For this reason, they are often the preferred method of starting rockets. They are clearly one-shot devices, and cannot be used for restarting a rocket engine. An electrical spark igniter, analogous to a sparking plug is generally used to ignite LH2/L02 engines, which in principle provides the possibility of a restart. However, there is a difficulty in that the electric spark releases less energy than a pyrotechnic device, and there is also the possibility of fouling during the first period of operation of the engine, which may then put the restart at risk.

Figure 3.3. Types of injector.

Figure 3.4. The impinging jet injector.

3.3.3 Combustion instability While the key performance characteristics of a rocket engine depend on the propellant combination and the design of the nozzle, many of the practical problems with rocket engine development are associated with combustion and its instabilities. So far, we have assumed that the propellants are injected and vaporised, and that this is followed smoothly by combustion, where all the chemical energy is converted into heat. This heat is then converted by the nozzle into work, which produces the high-velocity exhaust stream. However, for all practical engines, this process is not smooth; and resonant fluctuations in chamber pressure can develop, which produce vibration, and in extreme cases, destruction of the engine. Even if damage does not ensue, the fluctuating pressure, and the associated fluctuation in combustion temperature, reduce the efficiency with which the propellant energy is converted into thrust. Thus, particular effort has to be made in the design to remove or limit the magnitude of these fluctuations in chamber pressure. These efforts are made more difficult because there is still uncertainty about the mechanisms that result in instability; however some basic rules of thumb have been established which allow successful engine development. Much trial and error is still required in particularly difficult cases. Any combustion process has smalI fluctuations, which lead to smalI, random pressure changes. These manifest themselves mainly in relatively low amplitude noise on the steady thrust of the engine. In general, if this is below about 3%, then the engine is regarded as acceptable. The true instability develops when these small random fluctuations are amplified, by some resonant feedback mechanism, to much higher amplitudes capable of causing damage in various ways. The danger of a particular instability also depends on its frequency: some frequencies are less damaging than others. For a resonant feedback situation to develop, there has to be a delay somewhere in the system, which can generate the resonance. There also has to be a process that feeds energy into the resonance; the only available energy source is combustion, and it is fluctuations in combustion, amplified by pressure changes, that feed the resonance. Thus, a smalI local pressure change causes a change in energy supply, which, with a delay, causes a change in pressure, and so on.

3.4 Gas Generator The gas generator which provides the high-pressure gas to drive the pumps is a miniature combustion chamber, burning part of the propellant supply. It needs a separate igniter, and it has to be supplied with propellant, usually by a branch line from the turbo-pump itself. The propellant burned in the gas generator represents loss of thrust, and is included in the mass ratio. The mass decreases as a result of the gas generator operation, but no thrust is produced. Some of this loss can be recovered by exhausting the gas, after it has driven the turbine, through a proper miniature nozzle in order to develop some additional thrust. The natural temperature of the burning propellant in the gas generator would be close to that in the main combustion chamber, rather too high for the turbine blades. Sometimes water is injected into the gas generator to reduce the temperature of the emerging gas, or a very fuel-rich mixture is used, which achieves the same result. A fuel-rich mixture is also less corrosive. Basically, the former measure requires a water tank on board, and the latter implies a waste of propellant; both reduce the efficiency of the rocket. Some rocket engines with turbo-pumps make use of propellant evaporated in the cooling of the combustion chamber to drive the pump. This saves the mass of the gas generator, but generally results in a lower inlet pressure, and is suitable for low thrust engines. For modern high-thrust engines, the inlet pressure needs to be of the order of 50 bar, and this requires a gas generator powered turbo-pump. Turbines are most efficient when the hot gas inlet and exhaust pressures are very similar. When used for electricity generation or on ships, for example, many stages are used with different sized turbines, each with a small pressure drop to make the most efficient use of the energy. If the turbine exhaust of a rocket turbine were to go directly to the ambient, then the pressure ratio would be too large, and the efficiency would be low. This can be overcome by utilising a multi-stage turbine, but the extra stages add weight. It is therefore necessary to reach a compromise. An important variant of the gas generator system is the staged combustion system. The exhaust from the turbine enters

the combustion chamber, instead of the ambient. This has two advantages: the pressure ratio for the turbine is more compatible with high efficiency, and the remaining energy in the turbine exhaust contributes to the main combustion chamber energy and ultimately to thrust.

3.5 Rocket Turbine Representative engine cycles and installations for which rocket turbines are developed appear in figure 3.5.

Figure 3.5. Typical rocket engine cycles and turbine installations.

To date, most rocket engines have used the gas-generator (GG) cycle (figs. 3.5 (a) and (b)), in which the turbine working fluid is derived by combustion of the main propellants in the GG at a temperature below the turbine temperature limits. If the turbine exhaust is afterburned by the introduction of additional oxidizer, higher performance can be obtained from the GG cycle. A new turbine design is evolved, as part of the overall turbopump design, from studies of candidate configurations and component schemes that best satisfy the requirements of the turbine gas path, the turbomachinery, and the rocket engine propellant-feed system. Development of a turbine design includes the following major phases: 1. 2. 3. 4.

Preliminary design analysis Aerothermodynamic design of the turbine gas path Design of nozzle, stator, and blade profiles Mechanical design of the turbine assembly and selection of materials.

Table 3.1 Advantages and Disadvantages of Major Turbine Drive Cycles

In the succeeding text only the preliminary design analysis procedure is dealt with. In-depth discussions of the design & geometry of turbines are presented in reference 1 & 2.

3.5.1 Preliminary Design Analysis Algorithm 1 • • • • • • • •

Final turbine design state conditions and geometry shall be based on tradeoff studies of design control parameters. Perform design optimization studies with the parameters that influence the selection of turbine type, arrangement, size, number of stages, and performance. Establish the effect of pressure ratio, inlet temperature, number of stages, pitchline velocity, and velocity ratio on turbine Performance. Determine how variations in mass flowrate, inlet temperature, pressure ratio, and speed influence developed turbine horsepower. Investigate blade height requirements for changes in mass flow, inlet pressure, and number of stages. Study the influence of pitchline velocity on pitch diameter, velocity ratio, and turbine efficiency. Establish preliminary blading stresses. If the primary concern is maximum performance, special care should be directed to the limiting parameters of staging, pitch diameter, speed, and blading stress. Establish parametric data plots.

3.5.2 Turbine Performance The power balance implies that the power of turbine PT equals the power consumed by pumps and auxiliaries. The power supplied by the turbine is given by

All mathematical expressions are adopted from reference 4

eq. 3.1

eq. 3.2 The power delivered by the turbine PT is proportional to the turbine efficiency t the flow through the turbine mT, and the available enthalpy drop per unit of flow Δh. The units in this equation have to be consistent (1 Btu = 778ftlbf- 1055J). This enthalpy is a function of the specific heat cp, the nozzle inlet temperature T1, the pressure ratio across the turbine, and the ratio of the specific heats k of the turbine gases. For gas generator cycles the pressure drop between the turbine inlet and outlet is relatively high, but the turbine flow is small (typically 2 to 5% of full propellant flow). For staged combustion cycles this pressure drop is very much lower, but the turbine flow is much larger. For very large liquid propellant engines with high chamber pressure the turbine power can reach over 250,000 hp, and for small engines this could be perhaps around 35 kW or 50 hp. For gas generator engine cycles, the rocket designer is interested in obtaining a high turbine efficiency and a high turbine inlet temperature T1 in order to reduce the flow of turbine working fluid, and for gas generator cycles also to raise the overall effective specific impulse, and, therefore, reduce the propellant mass required for driving the turbine.

3.6 Tubopump The assembly of a turbine with one or more pumps is called a turbopump. Its purpose is to raise the pressure of the flowing propellant. Its principal subsystems are a hot gas powered turbine and one or two propellant pumps. It is a high precision rotating machine, operating at high shaft speed with severe thermal gradients and large pressure changes, it usually is located next to a thrust chamber, which is a potent source of noise and vibration. Details on the design of the various components including inducers, gears, and bearings may be found in the reference 1 & 3.

Figure 3.6. Simplified diagrams of different design arrangements of turbopumps. F is fuel pump, O is oxidizer pump, T is turbine, G is hot gas, and GC is gear case.

The centrifugal pump is generally considered the most suitable for pumping propellant in large rocket units. For the large flows and high pressures involved, they are efficient as well as economical in terms of mass and space requirement.

3.6.1 Preliminary Design Analysis AlSgorithm 2 Final pump design state conditions and geometry shall be based on tradeoff studies of design control parameters •

The headrise and flowrate delivered by the pump shall be adequate for the engine to produce its design thrust. To determine the pump discharge-pressure requirement, add the engine-system pressure drops that occur downstream of the pump discharge. For gas-generator cycles, add to the chamber pressure the pressure drops due to line losses, valve losses (if any), the regenerative jacket (if applicable), and the injector. For staged-combustion cycles, include the pressure drops across the preburner injector and the turbine and the line losses between them. Estimate the pump headrise from the expression eq. 3.3

where (Po) 2 = pump discharge total pressure, psia (Po) l = pump inlet total pressure, psia ρ1 = pump inlet propellant density, lbm/ft 3 For a high-pressure hydrogen pump (above 2000 psi pressure rise), obtain the required isentropic enthalpy rise from the propellant properties and calculate the corresponding head rise from

eq. 3.4 where Hisen= headrise for an isentropic compression from (Po) l to (Po) 2, ft (hid) 2 = ideal specific enthalpy at (Po) 2, Btu/lbm h1 = inlet specific enthalpy, Btu/lbm J = 778 ft-lbf/Btu To determine the volume flowrate requirements, obtain the total weight flowrate requirement from

eq. 3.5 where wE = engine total weight flowrate, lbm/sec

All mathematical expressions are adopted from reference 5

F = engine thrust, lbf (IS) E = specific impulse, lbf-sec/lbm Then use the following equations to obtain the volume flowrates for the oxidizer and the fuel pumps: eq. 3.6

eq. 3.7

where (Qo) e = oxidizer pump volume flowrate, gpm (Qf) P = fuel pump volume flowrate, gpm ρo = oxidizer density, lbm/ft 3 ρe = fuel density, lbm/ft 3 MR = engine mixture ratio, ratio of oxidizer to fuel 448.8 = factor for converting ft 3/sec to gpm For preliminary design estimates, use the inlet density. For more detailed flow-passage sizing, use the average or the local density. •

The pump net positive suction head shall be suitable for the particular application, shall be adequate for stable and predictable pump performance, and shall minimize vehicle overall weight. By definition, net positive suction head NPSH is the difference, at the pump inlet, between the head due to total fluid pressure and the head due to propellant vapor pressure; it is expressed in feet of the propellant being pumped. In the preliminary design phase of the vehicle, the NPSH is determined from an optimization that considers the weights of the vehicle tank, the tank pressurization system, the pressurization gas, and the feed line and, in addition, the system cost, the pump efficiency, and the turbopump weight. The trade often is made without the last two items because, in most cases, vehicle considerations far outweigh engine considerations. If the NPSH is less than a certain critical value, cavitation will occur in the pump inlet, and the pump headrise will be less than the design value. This critical NPSH is usually the value where the headrise is 2 percent less than the noncavitating value. Three methods have been used to correct the problem of an NPSH insufficient for the pump to meet design requirements: (1) increasing the tank pressure, which increases the NPSH supplied but also increases the required tank wall thickness and the tank weight; (2) decreasing the pump design speed, which decreases the NPSH required but also decreases the pump efficiency and increases the turbopump weight; and (3) redesigning the pump inlet by increasing the diameter and lowering the flow coefficient, a step that decreases the NPSH required but can decrease the pump efficiency.

Figure 3.6. Effects of variations in pump NPSH on various design factors. •

The turbo-pump design shall reflect the impact of the properties of the individual propellants and of the propellant combination.

Because propellant properties have a major influence on all aspects of turbopump system design, care must be taken in acquiring adequate physical & chemical properties of respective species.

•

The turbo-pump shall be compatible with the turbine drive cycle.

For optimum performance the energy output from turbine must coincide with the energy input of turbo-pump plus the auxiliaries. The turbine power output can be calculated via eq. 3.2

•

The turbopump efficiency shall be adequate for the engine to meet its requirements.

For engine cycles in which the turbine is in parallel with the thrust chamber (gas generator and tapoff cycles), determine the maximum allowable turbine flowrate from the expression

eq. 3.8

where (Is) T/c = thrust chamber specific impulse, lbf-sec/lbm Then obtain the minimum allowable turbopump efficiencies from

eq. 3.9

where

• The weight and size of the turbopump system shall be minimal consistent with other requirements. Maximize the turbopump-system design rotational speeds within the limitations of life, reliability, NPSH, and performance.

3.7 Liquid Propellants The propellants, which are the working substance of rocket engines, constitute the fluid that undergoes chemical and thermodynamic changes. The term liquid propellant embraces all the various liquids used and may be one of the following: 1. 2. 3. 4.

Oxidizer (liquid oxygen, nitric acid, etc.) Fuel (gasoline, alcohol, liquid hydrogen, etc.). Chemical compound or mixture of oxidizer and fuel ingredients, capable of self-decomposition. Any of the above, but with a gelling agent.

3.7.1 Desirable properties of Liquid Propellants The liquid propellants (i.e. fuels and oxidizer) should exhibit:(a) Low freezing point (less than -400 deg Celsius ) (b) High Boiling Point/High decomposition temperature (c) High specific gravity (d) High specific heat and thermal conductivity (e) Low vapour pressure and low viscosity (f) Low temperature variation of viscosity and vapour pressure and low coefficient of thermal expansion (g) Good physical and chemical stability (h) Hypergolic combustion with ID less than 50 milliseconds (i) Smooth and stable combustion (j) No smoke at exhaust (k) Less toxicity and safety in handling (l) Easy availability (m) High performance

3.7.2 COMBUSTION AND THE CHOICE OF PROPELLANTS Having examined the practicalities of propellant distribution in the liquid-fuelled engine, we shall now discuss the different types of propellant and the combustion process. Referring for the moment to Chapter 2, we recall that the exhaust velocity and thrust are related to the two coefficients c*, the characteristic velocity, and C f , the thrust coefficient. The thrust coefficient is dependent on the properties of the nozzle, while the characteristic velocity depends on the properties of the propellant and the combustion. It is defined by

eq. 2.xx The exhaust velocity and thrust defined by

For a given rocket engine the performance depends on the value of c*, defined above in terms of the molecular weight, the combustion temperature, and the ratio of specific heats, all referring to the exhaust gas. Different propellant combinations will produce different combustion temperatures and molecular weights. The exhaust velocity will also depend on the nozzle and ambient properties, but the primary factor is the propellant combination.

3.7.3 Combustion temperature The exhaust velocity and thrust depend on the square root of the combustion temperature. The temperature itself varies a little depending on the expansion conditions, but the main dependence is on the chemical energy released by the reaction: the more energetic the reaction, the higher the temperature. Table 3.2 shows the combustion temperature under standard conditions for a number of propellant combinations. The data in Table 3.2, which are calculated for adiabatic conditions, provide an insight into the effects of chemical energy. The combustion temperatures directly reflect the chemical energy in the reaction. With oxygen as the oxidant, hydrogen produces a lower temperature than the hydrocarbon fuel RP1, the molecules of which contain more chemical energy. Fluorine and hydrogen produce a still higher temperature. This combination produces the highest temperature of any bi-propellant system. The corrosive nature of fluorine has prevented its use except with experimental rockets. If the temperature is calculated theoretically for the complete reaction for example, the combustion 2H2 + O2 →2H2O then a much higher value of about 5,000 K is predicted. In fact at this temperature, and for pressures prevalent in combustion chambers, much of the water fonned by the reaction dissociates and absorbs some energy, lowering the temperature to the values shown in Table 3.2. If we deliberately introduce additional fuel, which cannot be burned without additional oxygen, then these atoms have to be heated by the same amount of chemical energy, and the temperature will be lowered further.

Table 3.2 Combustion temperature and exhaust velocity for different propellants.

3.7.3 Molecular weight The expression for exhaust velocity also shows that De depends inversely on the square root of molecular weight: lower molecular weight produces a higher velocity. This is very obvious in Table 3.2, comparing hydrogen and the hydrocarbon RPI as fuels, with oxygen as oxidant. Although the RPI, with its greater chemical energy, produces a much higher combustion temperature, the carbon atoms produce heavy carbon dioxide molecules which raise the mean molecular weight of the exhaust gases. The net result is a significantly lower exhaust velocity for the RPI fuel. In fact, except for fluorine which has very high chemical energy the H2 : O2 combination

Figure 3.7 The variation of exhaust velocity, temperature and molecular weight for different propellant combinations. produces the highest exhaust velocity, largely due to the low molecular weight of theexhaust gases. Additional hydrogen can be added to the mixture, in which case the exhaust velocity is actually raised, although the mean chemical energy and therefore the combustion temperature is reduced by the addition. Figure 3.7 shows, for different propellant combinations, how the exhaust velocity (represented as specific impulse ve/g) varies, together with temperature, molecular weight, and γ, as the mixture ratio is changed. It is remarkable how the maximum exhaust velocity is shifted away from the stoichiometric value, in the direction of lower molecular weight for each mixture. This fact is made use of when choosing the mixture ratio for maximum exhaust velocity; the fuel rich mixture in the Space Shuttle main engines (SSME) contributes directly to the high exhaust velocity. It might be asked: Why use a propellant with a high molecular weight? If exhaust velocity were the only criterion, then this is a valid question. We should not forget, however, that thrust also depends on mass flow rate, and a heavy propellant may give a higher mass flow rate, for an engine of a given throat area, than a low-mass propellant. The ultimate velocity may be lower in this case, because the exhaust velocity is lower, but the overall thrust will be increased. This may be appropriate for the first stage of a rocket where the main objective is to raise it off the launch pad and gain some altitude; it is particularly applicable to strap-on boosters.

References: 1) GEORGE P. SUTTON “Rocket Propulsion Elements” 7th ed. Chap#4 JOHN WILEY & SONS, INC. 2) Jack D. Mattingly “Elements of Propulsion: Gas Turbines and Rockets” 2nd ed. Chap#9 sub sect. 9.4-5 AIAA EDUCATION SERIES 3) Ahmad Nourbakhsh · André B. Jaumotte “Turbopumps and Pumping Systems” c Heidelberg 2008

Springer-Verlag Berlin

4) NASA monograph NASA SP-8110 January 1974 5) NASA monograph NASA SP-8107 August 1974

4 Elements Of Solid Propulsion System Considering the complexities of the liquid propellant rocket engine, it does not seem remarkable that so much attention has been given to the design and development of the much simpler solid propellant motor. This chapter It discusses the burning rates, motor performance, grain configurations, and structural analysis. In solid propellant rocket motors; and the word "motor" is as common to solid rockets as the word "engine" is to liquid rockets; the propellant is contained and stored directly in the combustion chamber, sometimes hermetically sealed in the chamber for long-time storage (5 to 20 years). Motors come in many different types and sizes, varying in thrust from about 2 N to over 4 million N (0.4 to over 1 million lbf). Historically, solid propellant rocket motors have been credited with having no moving parts. This is still true of many, but some motor designs include movable nozzles and actuators for vectoring the line of thrust relative to the motor axis.

4.1 Basic Configuration Thermodynamically a solid-fuelled rocket motor is identical to a liquid-fuelled engine. The hot gas produced by combustion is converted to a high-speed exhaust stream in exactly the same way, and so the nozzle, the throat and the restriction in the combustion chamber leading to the throat are all identical in form and function.The thrust coefficient is calculated in the same way as for a liquid-fuelled engine, as is the characteristic velocity. The theoretical treatment in Chapter 2 serves for both. The hot gas is produced by combustion on the hollow surface of the solid fuel block, known as the charge, or grain. In most cases the grain is bonded to the wall of the combustion chamber to prevent access of the hot combustion gases to any surface of the grain not intended to burn, and to prevent heat damage to the combustion chamber walls. The grain contains both fuel and oxidant in a finely divided powder form, mixed together and held by a binder material. Figure 4.1 shows a typical solid-motor configuration. In comparison with the liquid rocket combustion chamber it is very simple. It consists of a casing for the propellant, which joins to a nozzle of identical geometry to that of a liquid-fuelled engine. Once the inner surface of the grain is ignited, the motor produces thrust continuously until the propellant is exhausted.

4.2 Ignition Solid motors are used for two applications, both of which require ignition which is as stable and reliable as for a liquid-fuelled engine. The main propulsion unit for a rocket stage or satellite orbital injection system requires timely ignition in order to achieve the eventual orbit. A booster forms one of a group of motors which must develop thrust together. In all cases a pyrotechnic igniter is used. Pyrotechnic devices have an extensive and reliable heritage for space use, in a variety of different applications. To ignite a solid motor, a significant charge of pyrotechnic material is needed to ensure that the entire inner surface of the grain is simultaneously brought to the ignition temperature: 25 kg of pyrotechnic is used in the Ariane 5 solid boosters. It is itself ignited by a redundant electrical system.

4.3 Thrust profile and grain shape 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 used in the space industry. These are cylindrical blocks, with combustion at a front, or surface, and cylindrical blocks with internal combustion. In the first case, the front of the flame travels in layers from the nozzle end of the block towards the top of the casing. This so-called end burner produces constant

thrust throughout the burn. In the second, more usual 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 4.1. Schematic of a solid-fuelled rocket motor.

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 4.2. Cross sections of grains.

Fuel blocks with a cylindrical channel (1) develop their thrust progressively. Those with a channel and also a central cylinder of fuel (2) produce a relatively constant thrust, which reduces to zero very quickly when the fuel is used up. The five pointed star profile (3) develops a relatively constant thrust which decreases slowly to zero as the last of the fuel is consumed. The 'cruciform' profile (4) produces progressively less thrust. 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 inital thrust, followed by an almost constant lower thrust. The pressure in the chamber, and hence the thrust, depends on the rate at which the grain is consumed. The pressure depends on the recession rate and on the area of the burning surface, and the mass flow rate depends on the volume of propellant consumed per second. The shape of the charge can be used to preset the way the area of the burning surface evolves with time, and hence the temporal thrust profile of the motor. The pressure and the thrust are independent of the increase in chamber volume as the charge burns away, and depend only on the recession rate and the area of the burning surface. The simplest thrust profile comes from linear burning of a cylindrical grain (as with a cigarette): a constant burning area produces constant thrust. This shape, however, has disadvantages: the burning area is limited to the cylinder cross section, and the burning rim would be in contact with the wall of the motor. Active cooling of the wall is of course not possible with a solid motor, and this type of charge shape can be used only for low thrust and for a short duration because of thermal damage to the casing. The most popular configuration involves a charge in the form of a hollow cylinder, which burns on its inner surface. This has two practical advantages: the area of the burning surface can be much larger, producing higher thrust, and the unburned grain insulates the motor wall from the hot gases. In the case of a simple hollow cylinder, the area of the burning surface increases with time, as do the pressure and the thrust. If a constant thrust is desired, the inner cross section of the grain should be formed like a cog, the teeth of which penetrate part way towards the outer surface. The area of burning is thus initially higher, and the evolving surface profile corresponds roughly to constant area and hence constant thrust. Other shapes for the grain produce different thrust profiles, depending on the design. Figure 4.2 illustrates some examples.

4.4 Burn Rate The burning surface of a rocket propellant grain recedes in a direction perpendicular to this burning surface. The rate of regression, typically measured in millimeters per second (or inches per second), is termed burn rate. This rate can differ significantly for different propellants, or for one particular propellant, depending on various operating conditions as well as formulation. Knowing quantitatively the burning rate of a propellant, and how it changes under various conditions, is of fundamental importance in the successful design of a solid rocket motor. Propellant burning rate is influenced by certain factors, the most significant being: combustion chamber pressure, initial temperature of the propellant grain, velocity of the combustion gases flowing parallel to the burning surface, local static pressure, and motor acceleration and spin. These factors are discussed below. •

Burn rate is profoundly affected by chamber pressure. The usual representation of the pressure dependence on burn rate is the Saint-Robert's Law,

where r is the burn rate, a is the burn rate coefficient, n is the pressure exponent, and Pc is the combustion chamber pressure. The values of a and n are determined empirically for a particular propellant formulation

and cannot be theoretically predicted. It is important to realize that a single set of a, n values are typically valid over a distinct pressure range. More than one set may be necessary to accurately represent the full pressure regime of interest. Example a, n values are 5.6059* (pressure in MPa, burn rate in mm/s) and 0.35 respectively for the Space Shuttle SRBs, which gives a burn rate of 9.34 mm/s at the average chamber pressure of 4.3 MPa. * NASA publications gives a burn rate coefficient of 0.0386625 (pressure in PSI, burn rate in inch/s). •

Temperature affects the rate of chemical reactions and thus the initial temperature of the propellant grain influences burning rate. If a particular propellant shows significant sensitivity to initial grain temperature, operation at temperature extremes will affect the time thrust profile of the motor. This is a factor to consider for winter launches, for example, when the grain temperature may be lower than "normal" launch conditions.

•

For most propellants, certain levels of local combustion gas velocity (or mass flux) flowing parallel to the burning surface leads to an increased burning rate. This "augmentation" of burn rate is referred to as erosive burning, with the extent varying with propellant type and chamber pressure. For many propellants, a threshold flow velocity exists. Below this flow level, either no augmentation occurs, or a decrease in burn rate is experienced (negative erosive burning). The effects of erosive burning can be minimized by designing the motor with a sufficiently large port-tothroat area ratio (Aport/At). The port area is the cross-section area of the flow channel in a motor. For a hollow-cylindrical grain, this is the cross-section area of the core. As a rule of thumb, the ratio should be a minimum of 2 for a grain L/D ratio of 6. A greater Aport/At ratio should be used for grains with larger L/D ratios.

•

In an operating rocket motor, there is a pressure drop along the axis of the combustion chamber, a drop that is physically necessary to accelerate the increasing mass flow of combustion products toward the nozzle. The static pressure is greatest where gas flow is zero, that is, at the front of the motor. Since burn rate is dependant upon the local pressure, the rate should be greatest at this location. However, this effect is relatively minor and is usually offset by the counter-effect of erosive burning.

•

Burning rate is enhanced by acceleration of the motor. Whether the acceleration is a result of longitudinal force (e.g. thrust) or spin, burning surfaces that form an angle of about 60-90o with the acceleration vector are prone to increased burn rate.

It is sometimes desirable to modify the burning rate such that it is more suitable to a certain grain configuration. For example, if one wished to design an end burner grain, which has a relatively small burning area, it is necessary to have a fast burning propellant. In other circumstances, a reduced burning rate may be sought after. For example, a motor may have a large L/D ratio to generate sufficiently high thrust, or it may be necessary for a particular design to restrict the diameter of the motor. The web would be consequently thin, resulting in short burn duration. Reducing the burning rate would be beneficial. There are a number of ways of modifying the burning rate: decrease the oxidizer particle size, increase or reduce the percentage of oxidizer, adding a burn rate catalyst or suppressant, and operate the motor at a lower or higher chamber pressure. These factors are discussed below.

•

The effect of the oxidizer particle size on burn rate seems to be influenced by the type of oxidizer. Propellants that use ammonium perchlorate (AP) as the oxidizer have a burn rate that is significantly affected by AP particle size. This most likely results from the decomposition of AP being the ratedetermining step in the combustion process.

•

The burn rate of most propellants is strongly influenced by the oxidizer/fuel ratio. Unfortunately, modifying the burn rate by this means is quite restrictive, as the performance of the propellant, as well as mechanical properties, are also greatly affected by the O/F ratio.

•

Certainly the best and most effective means of increasing the burn rate is the addition of a catalyst to the propellant mixture. A catalyst is a chemical compound that is added in small quantities for the sole purpose of tailoring the burning rate. A burn rate suppressant is an additive that has the opposite effect to that of a catalyst – it is used to decrease the burn rate.

•

For a propellant that follows the Saint-Robert's burn rate law, designing a rocket motor to operate at a lower chamber pressure will provide for a lower burning rate. Due to the nonlinearity of the pressure-burn rate relationship, it may be necessary to significantly reduce the operating pressure to get the desired burning rate. The obvious drawback is reduced motor performance, as specific impulse similarly decays with reducing chamber pressure.

Figure 4.3. The Different Operation Regimes. The pressure curve of a rocket motor exhibits transient and steady state behavior. The transient phases are when the pressure varies substantially with time – during the ignition and start-up phase, and following complete (or nearly

complete) grain consumption when the pressure falls down to ambient level during the tail-off phase. The variation of chamber pressure during the steady state burning phase is due mainly to variation of grain geometry with associated burn rate variation. Other factors may play a role, however, such as nozzle throat erosion and erosive burn rate augmentation. A worked example of calculating the combustion temperature for KN/Sucrose, 65/35 O/F ratio, is given in Appendix A

4.4.1 Product Generation Rate The rate at which combustion products are generated is expressed in terms of the regression speed of the grain. The product generation rate integrated over the port surface area is

where q is the combustion product generation rate at the propellant surface, ρp is the solid propellant density, Ab is the area of the burning surface, and r is the propellant burn rate. It is important to note that the combustion products may consist of both gaseous and condensed-phase mass. The condensed-phase, which manifests itself as smoke, may be either solid or liquid particles. Only the gaseous products contribute to pressure development. The condensed-phase certainly does, however, contribute to the thrust of the rocket motor, due to its mass and velocity.

4.4.2 Rocket Motor Performance

Figure 4.4

Model rocket performance (how far, how high, how fast) depends a great deal on the rocket engine performance. But there are several different ways to characterize rocket engine performance. Model rocket engines come in a variety of sizes, a variety of weights, with different amounts of propellant, with different burn patterns which effects the thrust profile, and with different values of the delay charge which sets the amount of time for the coasting phase of the flight. On this page, we discuss all of the engine performance factors that affect the flight of a model rocket. At the top of the page we show typical performance curves for several different rocket engines. We are plotting the thrust of the engine versus the time following ignition for each engine. You will notice that when comparing engines, there is a great difference between the levels and shapes of the plots. And for any single engine, the thrust changes from time to time. At the bottom of the page, we show a typical engine schematic which we will use to explain why the thrust changes so much for a given engine. The thrust of any engine depends on how fast and how much hot gas exhaust passes through the nozzle. Solid rocket fuel only burns on the surface and the surface burns away as it turns into a gas. You can then imagine the flaming surface moving with time through the propellant. The flaming surface is called the flame front. At any time and at any location the amount of hot gas being produced depends on the area of the flame front. The greater the area, the greater the thrust. As the propellant burns away the shape and the area can change.

4.4.3 Rocket Motor Burn Sequence

In this sequence, we show the shape and location of the flame front for a C6-4 engine. The schematic is two dimensional while the real engine is three dimensional. So a three dimensional cone surface will appear as a two dimensional angle on the schematic. The flame front is shown as a red line and it moves through the propellant as the engine burns. The hot exhaust is shown in yellow. The time is noted on the plot by a moving red line.

On a typical model rocket engine, a small cone is formed in the propellant on the nozzle end of the engine. As the fuel burns, the size of the cone increases until it hits the engine casing (about time = .2 on this engine).

Between time = .2 and .5, the shape of the cone flattens out and the area and thrust decreases because the burn rate also depends on the curvature of the surface. By time = .5, the cone has become a flat flame front which proceeds on down the engine until the propellant is used up at time = 2. Between .5 and 2, the thrust is constant.

At time = 2, the thrust goes to zero and the delay charge begins to burn. Even though the delay charge is shorter (smaller) than the propellant, it burns longer because it is made of a different material. For this engine we have a 4 second delay (the "4" of C6-4 denotes the delay time)

At time = 6, the ejection charge is reached and ignited and blows out the front of the engine.

Considering the various engine plots, we see a burn pattern similar to the previously discussed C6-4, but with some variations in the amount of thrust. We have seen that the shape of the thrust curve is affected by the shape of the flame front. Designers of solid rockets can produce the given thrust curves by changing the total amount of propellant placed in the engine, by varying the the angle of the cone in the propellant, and by varying the diameter of the propellant (and casing). Considering a single engine plot, the thrust varies greatly with time. We can specify a time-averaged thrust of the engine by adding up the product of the thrust over some small time increment times the amount of the time increment and then dividing by the total time. The number designation of an engine indicates the average thrust in Newtons. A C6-4 has an average thrust of 6 Newtons. The average thrust times the length of the engine burn in time is called the total impulse of the engine. The letter designation of an engine tells the maximum total impulse of that class of engine. An "A" engine has a maximum impulse of 2.5 Newton-seconds, a "1/2A" has 1.25 N-sec, a "B" has 5.0 N-sec, a "C" has 10.0 N-sec, and a "D" has 20.0 N-sec. If we compare the curves for B6 and the C6, we find that both engines have the same average thrust (6 Newtons), but the "C" engine burns almost twice as long for double the total impulse. As mentioned above, the engine designer can affect the thrust and the total impulse of an engine by changing the diameter of the propellant (and casing). Typical "1/2A" engines are 13 mm in diameter, typical "A", "B" and "C" engines are 18 mm in diameter, and typical "D" engines are 24 mm in diameter. This is important to remember because a model rocket designed for a "B" engine will not accept a "1/2A" or a "D". The engines will not fit into the fixed engine mount of the rocket.

4.5 Preliminary Design Analysis Algorithm Design begins with the total impulse required, this determines the fuel/oxidizer mass. Grain geometry and chemistry are then chosen to satisfy the required motor characteristics. The following are chosen or solved simultaneously. The results are exact dimensions for grain, nozzle and case geometries: • • • •

The grain burns at a predictable rate, given its surface area and chamber pressure. The chamber pressure is determined by the nozzle orifice diameter and grain burn rate. Allowable chamber pressure is a function of casing design. The length of burn time is determined by the grain 'web thickness'.

One basic performance relation is derived from the principle of conservation of matter. The propellant mass burned per unit time has to equal the sum of the change in gas mass per unit time in the combustion chamber grain cavity and the mass flowing out through the exhaust nozzle per unit time.

The term on the left side of the equation gives the mass rate of gas generation . The first term on the right gives the change in propellant mass in the gas volume of the combustion chamber, and the last term gives the nozzle flow according to section 2.2.5. The burning rate of the propellant is r; Ab is the propellant burning area; ρb is the solid propellant density; P1 is the chamber gas density; V1 is the chamber gas cavity volume, which becomes larger as the propellant is expended; At is the throat area; Pl is the chamber pressure; T1 is the absolute chamber temperature, which is usually assumed to be constant; and k is the specific heat ratio of the combustion gases. For preliminary performance calculations the throat area At is usually assumed to be constant for the total burning duration. For exact performance predictions, it is necessary also to include the erosion of the nozzle material, which causes a small increase in nozzle throat area as the propellant is burned; this causes a slight decrease in chamber pressure, burning rate, and thrust. The gas volume V1 will increase greatly with burn time. If the gas mass in the motor cavity is small, and thus if the rate of change in this gas mass is small relative to the mass flow through the nozzle, the term d(ρ1V1)/dt can be neglected. Then a relation for steady burning conditions can be obtained

Substituting Saint-Robert's Law gives

The ratio of the burning area to the nozzle throat area is an important quantity in solid propellant engineering and is given the separate symbol K. Above equation show the relation between burning area, chamber pressure, throat area, and propellant properties. For example, this relation permits an evaluation of the variation necessary in the throat area if the chamber pressure (and therefore also the thrust) is to be changed. For a propellant with n = 0.8, it can be seen that the chamber pressure would vary as the fifth power of the area ratio K. Thus, small variations in burning surface can have large effects on the internal chamber pressure and therefore also on the burning rate. The formation of surface cracks in the grain (due to excessive stress) can cause an unknown increase in Ab. A very low value of n is therefore desirable to minimize the effects of small variations in the propellant characteristics or the grain geometry.

4.6 Desirable Properties of a Solid Propellant (a) High heating value. (b) Low molecular weight combustion products. (c) Combustion products in gaseous state. (d) Chemical and physical stability. (e) Compatibility with construction materials. (f) Non-toxicity. (g) Storage and Handling ease. (h) Availability of raw materials. (i) Processing techniques should be simple and safe.

4.6.1 Classification

Two main propellants will be listed below

4.6.1.1 DBRPs • • •

•

They do not contain metallic parts So the exhaust products are 100 % gaseous. Main disadvantage is specific Impulse is limited to a maximum of 220 seconds. The basic ingredients of a DBRP are: (a) Fuel + Oxidiser (80-90 %) (b) Additives (15-10 %) There are three types of additives that can be used in DBRP.

Essential Additives (a) Energetic material pyrotechnic compositions, especially solid rocket propellants and smokeless powders for guns, often employ plasticizers to improve physical properties of the propellant binder or of the overall propellant, to provide a secondary fuel, and ideally, to improve specific energy yield (e.g. specific impulse, energy yield per gram of propellant, or similar indices) of the propellant. Other non-explosive (non-energetic) plasticizers are added to permit adjustment of the fuel-oxidizer ratio. (b) Stabilizer stabilizes propellant from undergoing chemical decomposition or deterioration or degradation. This is because if degraded propellant is used the burning will not be adequate resulting in insufficient thrust build-up.

4.6.1.2 Composite Propellants •

These are also called heterogeneous propellants since they are a mixture of organic, inorganic and metallic compounds. Here the oxidizer and fuel are present in separate phases. They decompose to give alternately oxygen rich and fuel rich streams. Burning is believed to occur with the oxidizer particles decomposing in the midst of the decomposing fuel.

•

Besides the energetics, which is related to high combustion energy and low average molecular weight of the products, other factors to be considered in choosing a suitable binder are: (a) It should provide sufficient strength to the grain so that it does not suffer mechanical failure during combustion or storage. (b) It should have high density so that it can occupy small chamber volume. (c) In case of case bonded propellants, the coefficient of thermal expansion o the binder and that of the chamber material should be more or less same, so that thermal stress during combustion are minimized. (d) It should have suitable end groups, which could be cross-linked easily. (e) The binder (case bonded propellant) should possess adhesive capacity for metallic case/heat resistant liners.

(f) The temperature sensitivity coefficient and temperature coefficient of the burning rate should be low. (g) The binder should be stable for long periods and should not deteriorate chemically or physically during storage. (h) It should have negligible void volume in the cast, the polymer should adhere strongly to the crystalline oxidizer and the other solid fillers. (i) The viscosity of the pre-polymer should be low for castability but without sedimentation of solids. (j) The processing should give no split out products. (k) It should have ample pot life. (1) The curing process should have low exothermic heat. (m) It should have sufficient reactivity for complete low temperature cure. (n) It should exhibit low shrinkage for case bonding.

Figure. 4.5 Combustion in heterogeneous propellants

Table 4.1. Characteristics of some operational solid propellants 1 1

GEORGE P. SUTTON â&#x20AC;&#x153;Rocket Propulsion Elementsâ&#x20AC;? 7th ed. p.479 JOHN WILEY & SONS, INC.