Question 6 of 7: Stability, Air-Breathing Engines and Rocket Propulsion

Question 6: Stability, Air-Breathing Engines and Rocket Propulsion (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — stability, static and dynamic. An aircraft is stable if, after a disturbance from a trimmed equilibrium condition, it tends to return to that condition without action by the pilot. Static stability concerns only the initial tendency: the aircraft is statically stable in some degree of freedom if the disturbance immediately generates a force or moment that acts to reduce it. Pitching the nose up must generate a nose-down moment; if it generates a nose-up moment the aircraft is statically unstable, and if it generates no moment at all it is neutrally stable. Dynamic stability concerns the subsequent time history of the motion. A statically stable aircraft will develop a restoring moment, but it will overshoot the equilibrium and oscillate; whether that oscillation dies away, persists at constant amplitude or grows depends on the damping in the system, and only if it dies away is the aircraft dynamically stable. The relationship between them is one-way: static stability is necessary for dynamic stability but not sufficient. An aircraft can be statically stable yet dynamically unstable — a divergent phugoid or a divergent Dutch roll are the classic examples — whereas a statically unstable aircraft cannot be dynamically stable.

Part (b) — how inherent stability about all three axes is obtained. Longitudinal (pitch) stability is obtained by placing a horizontal tailplane on a long moment arm behind the centre of gravity and by keeping the centre of gravity ahead of the neutral point of the complete aircraft. If the nose rises, the tailplane meets an increased angle of attack, generates additional upload and produces a nose-down restoring moment; the distance between the centre of gravity and the neutral point, expressed as a fraction of the mean chord, is the static margin, and a positive static margin of roughly 5–15 per cent is conventional. The tailplane is usually set at a small negative incidence relative to the wing, the longitudinal dihedral, so that the aircraft can be trimmed at a positive lift coefficient.

Directional (yaw) stability comes from concentrating side area well behind the centre of gravity: the vertical fin, aided by a long rear fuselage and sometimes by a dorsal fin or ventral strakes. A sideslip presents the fin with an angle of attack, and the side force it develops acts behind the centre of gravity, yawing the nose back into the relative wind — the weathercock effect. Lateral (roll) stability is obtained mainly through wing dihedral: in a sideslip, the lower wing meets a larger effective angle of attack than the upper one and generates more lift, rolling the aircraft back towards wings-level. Wing sweep and a high wing position both contribute a similar dihedral effect, which is why high-wing and swept-wing aircraft often need little or even negative geometric dihedral. Because roll and yaw are strongly coupled, these three are not designed independently: too much dihedral effect relative to fin area produces a poorly damped Dutch roll, and too little produces spiral divergence, so the designer balances them and adds a yaw damper where the natural damping is inadequate.

Part (c) — why by-pass engines are used. The reason is propulsive efficiency. Thrust is the rate of change of momentum imparted to the air, $F = \dot m\,\Delta V$, whereas the kinetic energy wasted in the jet — energy that does no useful work on the aircraft — goes as $\tfrac{1}{2}\dot m\,(\Delta V)^2$. A given thrust can therefore be produced either by giving a small mass flow a large velocity increment or by giving a large mass flow a small one, and the second is far more efficient. The propulsive efficiency of an ideal jet is $\eta_p = 2/(1 + V_j/V_\infty)$, which approaches unity only as the jet velocity approaches the flight velocity. A by-pass or turbofan engine exploits this by using the core turbine to drive a large fan that accelerates a mass of air several times greater than the core flow, bypassing it around the combustor at a modest velocity increment.

The practical gains are large. Specific fuel consumption at subsonic cruise falls by roughly a third to a half relative to a pure turbojet, which is decisive for airline economics. Jet noise scales with a very high power of the exhaust velocity — the eighth power in Lighthill’s classical result — so halving the jet velocity reduces jet mixing noise dramatically, and the cool by-pass stream also shields the hot core jet. Static and low-speed thrust improve, which shortens take-off. The penalties are the frontal area and weight of the large fan and its nacelle, and the fact that the advantage disappears at high supersonic flight speeds, where the flight velocity approaches the jet velocity of a low by-pass engine anyway; this is why airliners use by-pass ratios of 5–12 and rising, while supersonic combat aircraft use ratios below about 1.

Part (d) — why afterburning is used. Only a fraction of the oxygen in the air passing through a gas-turbine engine is consumed in the main combustor, because the mixture must be kept lean enough to hold the turbine entry temperature within the limits of the turbine blade materials. Downstream of the turbine there is no such limit, so additional fuel can be injected and burned in the jet pipe, raising the gas temperature to $1700\text{--}2000\ \text{K}$ and hence the exhaust velocity. With a variable-area nozzle to accommodate the increased volume flow, this produces a thrust increase of the order of 50 per cent, and considerably more at high flight Mach number, from an engine of unchanged frontal area and only modestly increased weight. That is the whole justification: afterburning buys a large, instantly available thrust margin for take-off from short or hot runways, for transonic acceleration through the drag rise, and for combat manoeuvring. It is used only in short bursts because the thermal efficiency of burning fuel at low pressure is poor: the specific fuel consumption roughly doubles or triples, and the infrared signature increases sharply.

Part (e) — the ram-jet engine. A ram-jet is the simplest air-breathing engine: a duct with no moving parts, consisting of an inlet diffuser, a combustion chamber with flame holders and fuel injectors, and a propelling nozzle. All of the compression is achieved by the ram effect — the incoming air is decelerated in the diffuser, and at supersonic flight speeds through a system of shock waves, so that its kinetic energy is converted into a rise in static pressure and temperature. Fuel is then burned at that elevated pressure and the products are expanded through the nozzle to produce thrust. Because there is no compressor, the pressure ratio depends entirely on flight Mach number: a ram-jet produces no static thrust at all and must be accelerated to high subsonic or supersonic speed by a booster or a carrier aircraft before it will run. It becomes efficient above about $M = 2$ and is generally superior to a turbojet from roughly $M = 3$ to $M = 5$, above which the temperature rise in the diffuser becomes so severe that combustion must be carried out in a supersonic stream instead — the supersonic-combustion ram-jet, or scramjet. Its simplicity, light weight and high speed capability make it the natural choice for missiles and high-speed research vehicles.

Part (f) — the liquid-propellant rocket engine. A liquid-propellant rocket engine is a rocket in which the fuel and the oxidiser are stored separately as liquids in tanks and are fed on demand into a combustion chamber, where they burn to produce high-pressure, high-temperature gas that is expanded through a converging–diverging nozzle to a high exhaust velocity. Because it carries its own oxidiser it is independent of the atmosphere and works in vacuum. The defining advantage over a solid motor is control: the engine can be throttled, shut down and in many designs restarted, and the propellants can be loaded shortly before use.

The main components are shown above. The propellant tanks hold fuel (kerosene RP-1, liquid hydrogen, hydrazine derivatives) and oxidiser (liquid oxygen, nitrogen tetroxide) separately. The feed system delivers them to the chamber at a pressure above the chamber pressure — either by pressurising the tanks with an inert gas such as helium, which is simple but demands heavy tanks, or, for any large engine, by turbopumps driven by a turbine. The turbine of a gas-generator cycle is supplied by a gas generator that burns a small fraction of the propellants very fuel-rich to keep the turbine inlet temperature down, and exhausts overboard; staged-combustion and expander cycles route that flow back into the main chamber instead. The injector forms the propellants into fine, well-mixed sprays with the correct mixture ratio and distribution, and is the component that most strongly governs combustion efficiency and combustion stability. The combustion chamber provides the volume and residence time for complete combustion at pressures typically from 2 to 25 MPa. The nozzle converges to a throat, where the flow chokes at $M = 1$ and which therefore sets the mass flow, and then diverges to accelerate the gas supersonically to the exhaust velocity; the area ratio is chosen for the ambient pressure at which the engine will spend most of its working life. Finally, a cooling jacket or set of cooling passages carries the fuel around the chamber and nozzle throat before injection — regenerative cooling, which protects the structure and recovers the absorbed heat into the propellant.

Part (g) — why shaped charges are used with solid-propellant rocket engines. In a solid-propellant motor the fuel and oxidiser are pre-mixed into a single rubbery grain cast directly into the motor case, and combustion proceeds inward from whatever surface is exposed. The mass flow, and hence the thrust, is the product of the burning surface area, the propellant density and the linear burning rate: $\dot m = \rho_p A_b r$, where $r \approx a p_c^{\,n}$. The burning rate is a property of the chemistry and can only be adjusted in manufacture; the one variable the designer controls freely is the burning surface area as a function of how far the flame front has travelled. Shaping the internal cavity of the grain — the “shaped charge” of the question — is therefore the only way to tailor the thrust–time curve of a motor that cannot be throttled.

The three canonical outcomes are progressive, regressive and neutral burning. A plain cylindrical internal perforation has a burning area that grows as the bore enlarges, so the thrust rises steadily — progressive burning. An end-burning or externally burning grain has a shrinking or constant area and gives regressive or nearly constant thrust but a low mass flow. A star-shaped or wagon-wheel perforation is the classic compromise: as the flame front advances, the star points burn away and shorten the perimeter while the valleys burn outward and lengthen it, and with the number and geometry of the points chosen correctly the two effects cancel, giving an almost constant burning area and therefore near-neutral thrust over most of the burn. Other geometries — slots, cones, dendrites, dual-thrust grains with a fast boost phase followed by a long sustain phase — are all variations on the same idea. A second and important benefit of an internally burning shaped grain is thermal: the unburnt propellant remains pressed against the motor case throughout most of the burn and insulates it from the combustion gases, so the case can be made of thin, light material rather than being designed to survive direct exposure to a $3000\ \text{K}$ flame.