22-Mec-B7 Aero and Space Flight · December 2016
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Paper format. 07-Mec-B7 Aero and Space Flight, National Examinations, December 2016. Three hours, open book, any non-communicating calculator permitted. Seven questions of equal value (20 marks each); any six constitute a complete paper, and the grade is (mark obtained / 120) × 100. Some questions require an essay answer, where clarity and organisation are marked. All seven questions are solved below.
Reference texts. J. D. Anderson, Introduction to Flight, 9th ed. (standard atmosphere, altitude definitions, Pitot-static measurement, airplane performance, take-off and landing, atmospheric entry); J. D. Anderson, Fundamentals of Aerodynamics, 6th ed. (finite-wing theory, induced drag, critical Mach number and drag divergence, wave drag and area ruling); W. F. Phillips, Mechanics of Flight, 2nd ed. (parabolic drag polar, minimum-drag speed, maximum rate of climb, jet range and endurance); H. J. Allen and A. J. Eggers, A Study of the Motion and Aerodynamic Heating of Ballistic Missiles Entering the Earth's Atmosphere at High Supersonic Speeds, NACA Report 1381 (1958) (ballistic entry, maximum deceleration).
Check: standing assumptions. The paper's page-1 instruction is that "if doubt exists as to the interpretation of any question, the candidate is urged to submit… a clear statement of any assumptions made." Three assumptions are used throughout and are stated once here: (i) the International Standard Atmosphere with sea-level values $T_0 = 288.15\ \text{K}$, $p_0 = 101.325\ \text{kPa}$, $\rho_0 = 1.225\ \text{kg}/\text{m}^3$, tropospheric lapse rate $0.0065\ \text{K}/\text{m}$, $R = 287.05\ \text{J}/(\text{kg}\cdot\text{K})$ and $\gamma = 1.4$, giving the exponents $g/(LR) = 5.2559$ for pressure and $4.2559$ for density; (ii) where a question needs the variation of thrust with altitude but does not state it, the fixed-geometry jet assumption $T = T_0\,(\rho/\rho_0)$ is used; (iii) ground-run accelerations are evaluated once at $V/\sqrt{2}$, the speed at which $V^2$ takes its mean value, so that $s = V^2/(2a)$. Aircraft weights use $g = 9.81\ \text{m}/\text{s}^2$; the atmosphere model itself uses the defining value $9.80665\ \text{m}/\text{s}^2$.
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.
The by-pass ratio of a turbo-fan is the ratio of the mass flow of air that passes through the fan duct only, bypassing the core, to the mass flow that passes through the core (compressor, combustor and turbine):
$$BPR = \frac{\dot{m}_{by\text{-}pass}}{\dot{m}_{core}}.$$A by-pass ratio of 10:1, typical of a modern wide-body engine, means ten kilograms of air are accelerated modestly by the fan for every kilogram that is burnt with fuel in the core. Early civil turbo-fans of the 1960s had ratios near 1:1; military fighter engines, which must retain high specific thrust and accept afterburning, use ratios of 0.3–1.0; current geared and ultra-high-by-pass civil engines reach 12–16:1. The trend to higher by-pass ratio is essentially a trend towards better propulsive efficiency, which part (b) explains.
The governing physics is that thrust is the product of mass flow and velocity increment, $F = \dot{m}\,\Delta V$, whereas the wasted kinetic energy left in the jet goes as $\tfrac{1}{2}\dot{m}(\Delta V)^2$. The propulsive (Froude) efficiency of an ideal propulsor is
$$\eta_p = \frac{2}{1 + V_j/V_\infty},$$which approaches unity as the jet velocity $V_j$ approaches the flight speed. A given thrust can therefore be produced either by giving a small mass of air a large velocity increment (a turbo-jet) or a large mass of air a small increment (a high by-pass turbo-fan), and the second is far more efficient at subsonic flight speeds. The practical advantages that follow are:
(i) Much lower specific fuel consumption at subsonic cruise — typically 0.55–0.60 kg/(h·kg thrust) for a modern turbo-fan against 0.85–1.0 for a pure turbo-jet — which translates directly into range and operating cost. (ii) Greatly reduced noise: jet mixing noise scales roughly as the eighth power of jet velocity, so halving $V_j$ cuts jet noise by some 24 dB, and the by-pass stream also shields the core jet. This, more than fuel burn, made large turbo-fans mandatory for airline operation. (iii) Higher take-off thrust for a given core size, because the fan contributes most of the thrust and its performance falls off less rapidly with forward speed at low Mach numbers. (iv) Better thrust retention at low speed and a cooler, slower exhaust that is less erosive to runways and less hazardous to ground crew. The corresponding disadvantages — large frontal area and nacelle drag, weight, and a rapid loss of efficiency above about $M = 0.9$ — are why supersonic aircraft still use low by-pass ratios.
In a liquid-propellant rocket engine the fuel and the oxidiser are stored separately as liquids in tanks and are delivered to the combustion chamber by pumps or by tank pressurisation. Typical combinations are liquid oxygen with kerosene or with liquid hydrogen, or storable hypergolic pairs such as nitrogen tetroxide with monomethylhydrazine. Because the propellant flow is controlled by valves and pumps, the engine can be throttled, shut down and (with a suitable ignition system) restarted, and the mixture ratio can be trimmed in flight. Specific impulse is high — 350 s for LOX/kerosene at altitude, over 450 s for LOX/hydrogen. The penalties are complexity, cost, and the need for turbopumps, cryogenic handling and long launch preparation.
In a solid-propellant rocket motor the fuel and oxidiser are pre-mixed into a rubbery grain that is cast directly into the motor case, most commonly ammonium perchlorate oxidiser with aluminium fuel in an HTPB binder. Combustion proceeds on the exposed surface of the grain, so the thrust–time history is fixed at manufacture by the geometry of the internal port — a star-shaped port gives near-constant thrust, a cylindrical port a progressive one. The motor is simple, has almost no moving parts, can be stored ready to fire for years, and gives very high thrust density; but it cannot in general be throttled or restarted, is difficult to stop once ignited (only by blowing out the forward closure), and delivers a lower specific impulse, typically 240–270 s. The two are therefore used for different jobs: solids for boosters, military missiles and any application requiring instant readiness; liquids for upper stages, orbital manoeuvring and any mission needing controlled or repeated burns. Hybrid motors, with a solid fuel grain and a liquid or gaseous oxidiser, are an attempt to combine throttleability with storage simplicity.
Afterburning (reheat) is the injection and combustion of additional fuel in the jet pipe, downstream of the turbine and upstream of the propelling nozzle. It is possible because a gas turbine runs with very large excess air — the overall air/fuel ratio is of the order of 60:1 against a stoichiometric 15:1 — so the turbine exhaust still contains ample oxygen. Burning that fuel raises the jet-pipe total temperature from perhaps 900 K to 2000 K, which raises the exhaust velocity approximately as $\sqrt{T}$ and hence the thrust. The nozzle must be variable in area, opening as reheat is selected so that the turbine and compressor working line is unchanged.
The gain is large and immediate: thrust augmentation of 50 % at static conditions and, because ram effects assist at speed, up to 100 % at supersonic Mach numbers, all from an engine of unchanged frontal area and only modest added weight. The cost is equally large: specific fuel consumption roughly triples, so reheat can only be used in short bursts. Afterburning engines are therefore fitted where a brief, very high thrust matters more than efficiency — military combat aircraft needing rapid acceleration, high rates of climb, and the ability to pass through the transonic drag rise of Figure 2.1, plus catapult launch and combat manoeuvre; and historically on the supersonic transport Concorde, which used reheat for take-off and for transonic acceleration only, cruising dry at $M = 2$.
A propeller blade is a rotating wing, and the angle of attack of each blade element is the difference between the blade's geometric pitch angle and the helix angle of the resultant flow, $\phi = \arctan(V/\omega r)$. With a fixed-pitch blade this helix angle changes with every change of forward speed and rotational speed, so a fixed-pitch propeller can be efficient at exactly one combination of airspeed and rpm. At low speed (take-off) the blade angle of attack is too high and the blade approaches stall; at high speed it is too low and the propeller produces little thrust while the engine over-speeds.
A variable-pitch propeller rotates the blades about their own axes so that a favourable angle of attack, and hence a good efficiency, is maintained across the whole speed range; a constant-speed unit does so automatically through a governor that varies pitch to hold a selected engine rpm as flight condition and power change. Two further blade positions are equally important operationally. Feathering turns the blades edge-on to the flow after an engine failure, which reduces the drag of the dead engine's propeller by an order of magnitude and prevents the engine being windmilled; on a twin, this is often the difference between a controllable aircraft and an uncontrollable one. Reverse pitch takes the blades past fine to a negative angle, producing reverse thrust for landing and ground manoeuvring. Together these are the reason nearly every propeller aircraft above light-trainer class has a variable-pitch unit.
A ram-jet is a duct engine with no rotating machinery. Air enters an intake designed to decelerate it and thereby compress it — at supersonic flight speeds through a system of shock waves, giving compression ratios of 10:1 or more at $M = 3$ purely from the ram effect. Fuel is injected into the resulting high-pressure, subsonic stream and burnt in a flame-holder region, and the hot gas is expanded through a convergent–divergent nozzle to a velocity greater than the flight speed. The thrust is the resulting momentum increment. Since compression is done entirely by the intake, no compressor is needed, and therefore no turbine is needed to drive one.
Its advantages follow from that simplicity: no moving parts, hence low weight, low cost and high reliability; no turbine, hence no turbine-entry temperature limit, so the combustion temperature can be much higher than a gas turbine allows; and excellent performance at high supersonic Mach numbers, roughly $M = 2$ to $M = 5$, where a turbojet's compressor becomes a liability and its intake temperature becomes prohibitive. Its disadvantages are equally fundamental: it produces no static thrust at all, since with no forward speed there is no ram compression, so it must be accelerated to roughly $M = 0.5$–1.0 by another means (a rocket booster, or a carrier aircraft); it is inefficient at low supersonic speeds compared with a turbojet; and above about $M = 5$ the deceleration of the flow to subsonic speed produces dissociation and stagnation temperatures the structure cannot tolerate, which is where the supersonic-combustion variant, the scram-jet, must take over. Ram-jets are consequently used for missiles and boosted high-speed vehicles rather than for aircraft that must take off under their own power.
| Term | Essence of the answer |
|---|---|
| By-pass ratio | $\dot{m}_{by\text{-}pass}/\dot{m}_{core}$; 10:1 typical of a modern civil turbo-fan |
| Turbo-fan advantages | Higher propulsive efficiency $\eta_p = 2/(1+V_j/V_\infty)$: lower SFC, far less noise, better low-speed thrust |
| Liquid vs solid rocket | Throttleable, restartable, $I_{sp}$ 350–450 s vs simple, storable, fixed thrust history, $I_{sp}$ 240–270 s |
| Afterburning | Fuel burnt in the jet pipe; +50 % thrust static, up to +100 % supersonic; SFC roughly triples |
| Variable pitch | Holds blade angle of attack across the speed range; adds feathering and reverse pitch |
| Ram-jet | Compression by intake shocks alone; efficient at $M = 2$–5; no static thrust, needs boosting |
