Help, feeling sinusoidal

It`s enough to make one feel “Sinusoidal”

Could we be doing it all wrong?

The object always is to try and understand the fundamentals, the how and why some things work, why they work for you and others don`t, and then to work within the limitations imposed and of what can`t be changed. Above all is to accept that some of which one always knew as correct may not be so after all, and that there is still always a lot more to be learned. Undoubtedly the raison d`etra is knowledge, and knowledge is power!

Amongst any gathering of Bantam racing enthusiast’s huge amounts of interest and debate, even controversy is generated whenever race exhaust systems become the topic for discussion, and it nearly always does. This is particularly true concerning the various dimensions and their respective functions. Common to all of the popular published and software options is an empirical formula for calculating the pipe`s tuned length, some also offer scope for various diameters to be determined. However, not all of the authors of these formulas can even agree as to where the mean reflection point in the rear cone should really be. Actually there won`t even be a precise point but a zone or band, local differences in temperature, pressure and rpm will smear things around too much. More than one popular publication even offers a “one size fits all” for wave speed.
Lengths are a relatively straight forward calculation but suitable diameters are far more problematic, demanding more data input than is generally available. Implicit to all within the respective formulas however is an input for the “speed of sound”, (S of S) thereby suggesting that exhaust pipes are indeed acoustic devices, the only qualification here is that they do make a lot of noise! Further, the speed of sound as we know is strongly influenced by temperature and pressure. Neither of these are consistent in the atmosphere, the engine`s inlet tract and crankcase, rather variable in the transfer ducts and extremely variable, even chaotic, in the cylinder and exhaust pipe. So at best any calculation based on a temperature number can only ever be an approximation?

Acoustics is a sub-branch of gas dynamics that has been simplified by a whole set of assumptions which can be accepted if the sound pressure does not exceed certain limits. The wave pressures in your Bantam race engine will exceed those limits a thousand fold. Sound waves can therefore be characterized by their small amplitudes.

The universal formula for arriving at the required speed of sound in meters per second is as follows: square root (SxRxT): where S is the ratio of specific heats, R is the gas constant for exhaust gas, and T is the temperature. There are certain qualifications for both the first two, S has a nominal value of 1.4 for air at ambient temperature of 25*c so a compensating factor is assumed for hot exhaust gas at 500*K of 1.35. Similarly, the gas constant for air is approximately 278J/Kg-*K for ambient conditions, but 291J/Kg-*K for hot exhaust gas, T is quoted in Kelvin so is degrees C+273. Popping all of that into the speed of sound equation the only variable then becomes temperature T, but the speed of sound varies with changes in temperature. Chucking a further spanner into the works, the exhaust system can and does vary quite considerably in temperature along its length with the coolest region being at its largest expansion diameter. Strangely in our equation there is no provision for a pressure value particularly when one considers that the relatively static gas particles can`t move from the cylinder faster than the pressure differential with the header pipe giving signal for them to actually exit the cylinder?
After arriving at a speed of sound value this can then be factored into the formula for calculating the tuned length in mm units, which is: speed of sound x constant 83.3 x exhaust port duration/engine rpm at max power. This is the revised formula devised by Prof Gordon Blair, however, there are other such offerings by equally distinguished and respected practitioners which suggest a different constant value of; 88? Perhaps if undecided it might be a convenient to compromise with the average of the two competing numbers of 85.65, I doubt the difference could even be noticed.

The speed of sound under ambient conditions is around 320 mtrs/sec, (even that is dubious) strong pressure waves in hot exhaust gas can be almost double that figure. But, suction waves, even if they have a significant negative amplitude, can be as slow as 200 mtrs/sec. A strong wave can therefore overtake a weaker one. If that weak wave is a suction wave from the diffuser, which is moving slowly compared to a pressure wave from the reflector cone the ensuing collision shock wave is mutually destructive and the result is a temperature energy spike and short shock waves are no good at moving a mass of mixture so there is a negative benefit to the engine. All is not lost however, for the parallel belly section of suitable length in a pipe can be effective in separating the warring factions.
For the sake of the calculations within the formula it has always been assumed that the initial pulse from the cylinder at exhaust port opening and the residual pulse from the rear cone all move with the same speed of sound. This, as has been shown, is not the case, pressure pulses in exhaust waves move so quickly that acoustics no longer apply. We are dealing here with gas dynamics so the stronger the pulse the faster it moves. These pressure pulses are known as “finite amplitude waves” where compression waves squeeze the gas molecules together so increase density pressure within the wave, an expansion wave causes the molecules to move further apart and so decrease pressure in the wave, this then is the suction wave.

Sound waves are a special case of finite wave of very low amplitude. Typical pressure in a Bantam engine might just approach a couple of bar but a loud sound wave to the ear will have the amplitude of a tiny fraction of one bar.
All gas movements within an engine, as in transfer ports and exhaust ports, start their journey at zero velocity, accelerate up to its maximum velocity and depletes thereafter. Waves are similar, initial amplitude is modest becoming stronger as it develops towards its maximum amplitude, large amplitude waves overtake small amplitude ones of similar sign but of lower velocity and can join forces and move more rapidly, this is typical behaviour in a super position situation at the exhaust port.
Quite a lot of reference then about waves but nothing about the oscillating mass flow of the gas itself within the pipe…..the non-acoustic Helmholtz resonance. From what I have been able to glean from popular literature over the years this feature is almost totally neglected in the exhaust pipe context, but it is an important part of wave action. Both wave and Helmholtz theories are useful in trying to get a handle on what happens in an exhaust system. Lengths are by and large the domain of waves; Helmholtz theory helps in determining suitable diameters and volumes in addition to these lengths, with all of this functioning at a molecular level then the two systems become one gas dynamic event. Where did acoustics go?

A lot has mentioned about waves so what form do they assume within an engine exhaust context. They have more or less the profile of sine waves, those regular up and down traces showing movements of hills and valleys, the peaks are pressure pulses with the valleys being suction pulses, usually depicted either side along a horizontal median line. They can elongate and compress and become steep fronted before collapse but fundamentally remain sinusoidal.

Returning to our initial formula, there seems to be no provision for a specific input for the value of pressure. The mass of mixture movement from the cylinder into the exhaust duct and pipe is subject to the pressure release at initial exhaust port opening. For any gas flow from cylinder and pipe there must be a pressure difference ratio. At exhaust port opening the cylinder pressure may still be at 5 bar, the best your pipe can muster with a reflected pulse is, may be, 2 bar. Once the ratio reaches > 2 sonic flow velocity will occur, Mach1 the speed of sound. Attempting to raise the differential any higher will not result in higher flow velocity. Before this can happen the cylinder pressure begins to fall as piston motion moves toward transfer opening and gas flow velocity rapidly goes sub-sonic. Raising the cylinder (upstream) pressure will not increase velocity but will increase the density of the flowing gas giving rise to increase in mass flow. It is quite easy to visualize these events occurring in 15bar bmep GP engines like the Aprilia but what about 8, 9 or 10 bar engines such as our Bantams?
At “first light” exhaust port opening, sonic flow will happen but with the low BMEP involved it is comparatively short lived providing that is that the critical cylinder to exhaust pipe pressure ratio of 2 has been achieved. In the example of 5bar cylinder and 2bar exhaust that criteria has been achieved, but by a narrow margin only.

There has been a lot of detail research during the last few years, starting with the incredible Aprilia engine, then latterly and principally in the Karting world which still uses two-stroke engines. The goal has been to improve the efficiency of the first section of the total exhaust pipe system, namely the exhaust port interface at the bore and the exhaust duct to where it connects with the pipe header itself. From the instant the exhaust port opens there is a finite quantity of energy with which to work with, so what is lost here at the beginning can`t be made up for later. It must also be remembered the exhaust duct is the only port/duct combination which must be encouraged to efficiently flow in both directions with spent exhaust gas going out and over-scavenged fresh mixture hopefully stuffed back in. Things never work out quite as easily as that, at some point the old and new end up confronting each other. The boundary between these two gasses of different composition and/or temperature is known as a “contact discontinuity” and when a wave passes through it alters shape, as a consequence has an effect on how pipes work. Of course it is most tricky to factor this into our pipe formulas or even simulation software, but is none the less important and interesting to know and be aware of.

One potential ministerial candidate out campaigning got so fed up with telling journalists when repeatedly asked as to what was his most pressing policy announcement he reportedly shouted out in utter frustration, “it’s the economy stupid”! How prophetic that outburst has prover to be.
In similar vein we might today heavily emphasise the expression, “its blowdown stupid”!
All events in the exhaust and scavenge cycles of a race motor are today predicated by blowdown efficiency, with perhaps less than half the exhaust port height being effectively used. We might even go so far as to suggest the total exhaust port is unimportant but blowdown is crucially important.

Cheers, Trevor