MATTHEW MOXON explains the theory of hypersonic propulsion and its practical applications to aircraft.
This is a full article published in Aerospace International: May 2012
This article was originally intended to cover the subject of hypersonic propulsion. However, the main concepts involved are general, and therefore it seems sensible to provide a general treatment. A small amount of mathematics will be used to concisely illustrate the basic concepts involved. This is included because I consider it to be helpful rather than out of a sadistic desire to punish the less mathematically inclined, for I number myself amongst them. Inevitably, a certain amount of “hand waving” will be resorted to in order to compress a massive subject into a 2,500 word article; the interested reader is directed to the many excellent textbooks which cover the subject without this constraint.
All propulsion eventually reduces to Newton’s laws. From conservation of momentum, we may write:
If object 1 is the vehicle we seek to propel, imparting an opposite momentum change to some reaction mass will produce the desired effect. The kinetic energy of an object is:
The efficiency with which energy imparted to reaction mass is transferred to the vehicle being propelled is therefore a function of the velocities of vehicle and reaction mass.
This is quite reasonable if you think about it, because it implies that the propulsive efficiency is 100% when the reaction mass is brought to rest. Clearly, this is no help for accelerating a vehicle from rest.
This equation applies equally to land vehicles, but they have the advantage that their reaction mass is the earth itself.
Aeroplanes are in the less fortunate position that they must either carry their reaction mass with them, or else use the air around them.
Although the efficiency of a propulsive system increases with the reaction mass, so does its size and mass. This means that there is a finite optimum reaction mass flow rate, because there is a point beyond which the extra mass and drag of making the system larger outweighs the benefit of increased propulsive efficiency.
The propulsion system design having set the rate at which kinetic energy must be imparted to the reaction mass, the means of providing this power must now be considered.
With a few exceptions which fall beyond the scope of this article, this power is generally supplied by a heat engine.
The maximum possible efficiency of a heat engine in classical thermodynamics is that of the Carnot cycle(1):
The source of high temperature is the combustion of fuel; the sink of low temperature is the environment. A reasonable maximum ratio of hot to cold temperature is 10 to 1. This imposes a limiting value of thermal efficiency of about 90%(2) ; a figure which could not be exceeded, even by an ideal engine, without increasing the temperature ratio.
The temperature provided by combustion is set by chemistry; in most cases, however, the maximum usable temperature is actually set by the somewhat tedious requirement that the engine should not melt.
The inexorable demand to increase the power:mass ratio of engines has driven most designers of reasonably large engines towards steady-flow machines, as pipe-work of a given size and mass can obviously process more mass in a given time if it is in constant use.
The most common steady-flow thermodynamic cycle is the Brayton cycle(3). This consists of
a. Steady-flow compression
b. Steady-flow heat addition
c. Steady-flow expansion
d. Steady-flow heat rejection
(The last step is virtual in most practical applications, in that it takes place outside the engine as the exhaust cools to ambient conditions.)
A simple explanation of how this cycle works is that the compression and expansion processes through approximately the same Pressure Ratio (PR) produce approximately the same temperature ratio. Adding heat after the compression process means that the absolute temperature drop produced by the expansion process is larger than the absolute temperature rise produced by the compression process, allowing for the extraction of useful work. This is an extremely useful magic trick, which, in various forms, powers most of the modern world.
- The efficiency of an ideal Brayton cycle is that of an ideal Carnot cycle operating between the ambient temperature and its compressor delivery temperature.
- The useful work produced by an ideal Brayton cycle is the product of its efficiency and the heat input.
- The compressor temperature ratio which maximises the specific power of the cycle the square root of the ratio between the peak cycle temperature and the ambient temperature.
- Although the high enthalpy gas produced by the cycle may be used directly for propulsion, it will usually be too energetic for high propulsive efficiency to be obtained. It is therefore common to use a turbine and fan (termed a turbofan engine) or propeller (termed a turboprop engine) to improve the engine’s overall efficiency. The ratio of the mass flow of air sent through this fan or propeller to that used in the power-generating cycle is termed the By-Pass Ratio (BPR). An engine with a BPR of zero relies upon its exhaust jet for propulsion, and is termed a turbojet; such engines are generally obsolete for subsonic applications.
The effects of speed
Looking back at the kinetic energy equation, it may be seen that the energy of a ‘packet’ of air varies as the square of its velocity relative to our frame of reference.
Assuming a constant specific heat capacity, this means that the temperature of the air, were it to be brought to rest adiabatically(4), would also vary as the square of its velocity(5) .
This means that the optimum compressor temperature ratio decreases as the cruise Mach number increases, and that above a certain Mach number, no mechanical compression is required – indeed, at very high Mach numbers, it becomes extremely difficult to add heat to the working fluid at all. In addition, the intake momentum drag (that is, the momentum of the air captured by the propulsion system) increases linearly with velocity, causing the ratio of net thrust:gross thrust to fall.
Putting it all together
The impact of the various factors discussed is shown graphically in Figure 1 and Figure 2, which look rather more complicated than they really are.
- There is little intake ram effect(6). Mechanical compression is therefore required to achieve acceptable thermodynamic cycle efficiency, and so we have a gas turbine engine.
- The exhaust energy of this cycle is far too high for acceptable propulsive efficiency, and so some sort of fan or propeller is clearly required to achieve acceptable overall efficiency.
- A ducted fan engine might achieve a propulsive efficiency of about 70% at Mach 0.8ish with a BPR of about 10-15
- An unducted fan engine might achieve a propulsive efficiency of about 80% (roughly 14.3% better) with a BPR of about 35; a very powerful law of diminishing returns takes effect from here on – a BPR of about 80 would be required to achieve 90% propulsive efficiency, and this is only about 12.5% better!
Mach 1 – Mach 3ish
- Intake ram effect begins to rapidly reduce the optimum compressor pressure ratio.
- The exhaust energy becomes progressively better matched to the requirements of propulsion; this reduces the BPR required for any given level of propulsive efficiency.
- At cruise Mach numbers above about 2, the optimum BPR is sufficiently low that one might reasonably question whether the efficiency increase is worth the added cost, mass, drag, and complexity.
Mach 3ish – Mach 6ish
- Intake ram effect is now so powerful that the compressor does more harm than good to the specific power of the engine. Having discarded the compressor, we now have a ramjet engine.
- Cycle efficiency rises at the expense of specific power as the cruise Mach number increases.
- This is not, on balance, a happy state of affairs. The specific thrust of the engine is falling away extremely rapidly (it is inversely proportional to velocity, even at constant specific power), and so the excess of thrust over engine pod drag and the drag due to engine mass becomes more and more marginal.
At about Mach 6.6, the intake total temperature equals the peak cycle temperature, and it is no-longer possible to add heat to the cycle.
In order to avoid the ram temperature rise which kills the subsonic combustion Brayton cycle at high Mach numbers, we must add heat to the flow without bringing it to rest; yet to attain acceptable thermodynamic cycle efficiency, it is clearly necessary to attain some degree of ram compression.
The obvious thing to do is to design the intake system to produce the optimum ram temperature rise, allowing combustion to take place at whatever Mach number this implies (though for reasons beyond the scope of this article, it is inadvisable to attempt to add heat at Mach numbers close to unity). This performance of the resulting supersonic combustion ramjet is displayed in the Figures.
The main problem (there are many others) with this is that it is extremely difficult to sustain stable combustion in high Mach number flow.
If stable combustion is achieved, the next problem is that, because of our friend the v2 term, the heat energy we are able to add to the flow by combustion becomes progressively less important in relation to the kinetic energy of the intake flow.
This means that, at very high Mach numbers, quite small flow losses (due to internal or external drag, imperfect combustion and so on) can easily swamp the thrust produced by the engine; by Mach 15 or so, the gross thrust produced by the engine is only about 105% of the intake momentum drag, such that even a few percentage points of unanticipated losses might easily swamp the thrust produced.
As a result, the supersonic combustion ramjet tends to produce a ‘flying intake’, with somewhat marginal acceleration capability. Such a machine is perhaps acceptable for a cruise missile, but its utility for other applications is questionable.
Why not fit it to an airliner?
The obvious answer is that, like any ramjet, the supersonic combustion ramjet cannot start from rest. Indeed, it probably needs to be flying in excess of Mach 4 to get started.
Whilst it would be theoretically possible to construct a combined-cycle propulsion system to overcome this limitation, this would imply that considerable dead mass and volume would have to be carried along by the (already marginal) supersonic combustion ramjet in the cruise; the expendable booster rocket favoured by the cruise missile designer is not available to the airliner designer!
At low Mach numbers, the defining problem of aerospace propulsion is the need to achieve a low exhaust gas velocity in order to achieve high propulsive efficiency. This often drives the designer towards turboprop or turbofan concepts. At high Mach numbers, the problem is reversed, with naturally high propulsive efficiency from the pure jet coming at the price of an extremely low ratio of nett:gross thrust, which renders engine performance extremely vulnerable to small uncertainties in intake momentum drag. Simultaneously, the high ram temperature rise erodes the ability of combustion to add heat to the cycle.
This article has sought to provide a broad overview of what is an extremely large subject. Future articles will zoom in on smaller Mach number ranges to provide more detailed analysis of the technologies now in use, as well as those in various stages of development.
- Named after the French engineer, Nicolas Léonard Sadi Carnot (1796-1832), who first proposed it..
- The temperature of the stratosphere is 216.65K on a standard day; stoichiometric steady-flow combustion of hydrocarbon fuels produces a flame temperature of about 2,500K; a temperature ratio of 10 is therefore a reasonable approximation after the need for combustor cooling air is accounted for.
- Named after the American engineer, George Brayton (1830-1892), who built stationary piston engines using steady-flow combustion.
- i.e. without heat transfer.
- In reality, the specific heat capacity of air increases as a function of its temperature, and therefore the actual stagnation temperature of air increases more slowly than the kinetic energy equation would suggest.
- Pressure and temperature rise due to deceleration of the flow from the free-stream flow velocity to that required by the engine (it is conventional to analyse steady-state flight problems from the frame of reference of a stationary vehicle with air flowing around it). The model used to produce the figures assumed the flow to be completely stagnated for the power generation cycle, other than in the supersonic combustion ramjet case.
Aerospace International Contents - May 2012
News Roundup – p4
Libya air war: the verdict p 12
report on the RAeS/ISS Air Power Group seminar
Open for business p 16
Profile of the world’s first biz-jet shop
A bluffer’s guide to aerospace propulsion - p 18
Technical focus on hypersonic propulsion
Inside story- p 21
News from the 2012 Aircraft Interiors show
On the brink - p 22
What future for the European military aircraft industry?
Command lessons from QF32- p 26
Report on RAeS Aircraft Commander in the 21st Century conference
Space debris - p 30
The problems of in-orbit space debris
Letters – p 34
Composite materials – lessons from the de Havilland Comet
The last word – p 35
Keith Hayward on UK future carriers
This is a full article published in Aerospace International: May 2012. As a member, you recieve two new Royal Aeronautical Society publications each month – find out more about membership.