Exhausting…Exhaust
Headers, extractors, bundle ‘o snakes! Despite headers being one of the most common modifications enthusiasts make to their vehicles, the science behind why they work is largely inaccessible. I’ll try to keep this simple, it’s very easy to make this an extremely complicated subject.
The first, and most obvious, question is: why does a long tube header make more power? If you step back and look at it from a pure flow standpoint, a long tube coming off of the exhaust port, particularly a tube with bends, will decrease flow. Is it easier to blow through a short straw or a really long straw? The answer is obvious without going into complicated fluid dynamics.
When the exhaust valve is open and the piston is pushing the exhaust gas out of the engine it is called the “blowdown” phase. The piston is moving fast, it actually pressurizes the exhaust gas much like an air compressor does. This process creates a high pressure “wave” that travels out of the exhaust system. When that wave reaches a large change in cross section area, for example a collector or open air, that wave is reflected back up the pipe. This process continues until the next blowdown phase, so you can have many reflections if the header primary pipe is short enough:
The “harmonics” shown here, 1st and 2nd, are just a fancy name acoustic engineers use for when the wave completes the journey. The pressure numbers on the y-axis are in bar, so 1.0 is the same pressure as open air. Notice that the blowdown peak surpasses 1.8 bar (26 psia), but there is a corresponding low pressure area that gets below 0.7 bar (10 psia). This low pressure point will actually suck the exhaust gas out of the cylinder, the exhaust valve is still open when that low pressure wave is in the exhaust port! Also notice that each harmonic gets smaller, the wave is losing energy as it makes the journey up and down the pipe. The pressure wave travels at the speed of sound, so you can see that as RPM rises it takes more crank angle degrees for the wave to arrive back at the valve.
First, you don’t want those high pressure waves traveling to another cylinder and causing poor exhaust flow. This is a problem with log manifolds or very short primary headers, the high pressure is going to affect adjacent cylinders that share a collector. A longer primary length will, at the very least, ensure that one of the later harmonics with less energy are affecting the adjacent cylinders..
What would be really great from a performance standpoint is to have one of those low pressure harmonics arrive at the exhaust valve when it is opening. This really helps the exhaust gas get going, it’s hard to get that exhaust mass moving initially. Due to the fact that the wave travels at the speed of sound we must choose what RPM we are going to design the primary pipe to be optimized for, as you can see in figure (a) the positions change with RPM. The math for the calculation is pretty basic:
Length = c/(2*h*f)
Where c = speed of sound, h = the harmonic, and f = frequency. The 2 is there because the wave has to travel down the pipe and then back up it, so we divide the length by 2. The frequency can be calculated from the RPM, it’s simply RPM/60 which converts revolutions per minute to revolutions per second. Thus the final equation looks like this:
Length = c/(2*h*(RPM/60))
If it’s so easy, why did I design a fancy calculator? The complication comes from the fact that c, the speed of sound, changes with temperature and pressure. The speed of sound at standard sea level conditions (20 C, barometric pressure) is 343 m/s or 1125 ft/s, let’s call that c1. If we’d like to know the the speed of sound at a different temperature and pressure we’d have to use the following equation:
c2 = c1 * (T2/T1)^0.5 * (P2/P1)^0.5 * (γ2/γ1)^0.5
The T’s are temperature, the P’s are pressure, and the y looking thing (greek letter gamma) is the ratio of specific heats. This equation looks complicated enough already, but you also need to convert your temperatures and pressures to absolute numbers, for example you’d need to convert your EGTs in Fahrenheit to Rankine. Basically, the speed of sound goes up as the temperature and pressure goes up, and I’m way too lazy to run that equation every time I want to play with header length!
Onto header primary diameter. This issue is much less clear-cut, there’s several considerations and schools of thought that conflict with each other. If the primary diameter is too large your wave reflection we previously calculated will be very weak. If it’s too small you’ll create too much exhaust backpressure. You also want to keep the diameter as small as possible so that your exhaust gas velocity produces a scavenging effect in adjacent cylinders. For a given exhaust gas volume/mass a larger primary diameter will slow down the flow velocity, and it also changes the pressure thanks to Bernoulli. This wouldn’t be so bad if we could treat the exhaust gas as an incompressible fluid, but the velocities in the exhaust combined with the pressure waves mean we can’t.
A very good starting point is to consider how much exhaust volume you need to get out of cylinder, and how much time you have to do it. To calculate time you need your exhaust lobe duration and RPM. To find the time it takes to complete one revolution of the engine:
t = 1/(RPM/60)
At 6000 RPM we’ve got 0.01s (10 ms) for one revolution. Let’s say our advertised exhaust duration is 250 degrees, now we’ve got less time because that’s not a complete revolution (360 degrees):
t = 1/(RPM/60) * (exdur/360)
We can simplify to:
t = exdur/(RPM*6)
This gives us how long the exhaust valve is open at a given RPM. The rest is simple, we calculate cross-section (A) from:
A = exVolume/(t*exVelocity)
Where “exVelocity” is the desired exhaust gas velocity and “exVolume” is the volume of the exhaust gas. The exhaust velocity number is very combination and exhaust design dependent, to finely tune this number you need to do testing. A good starting point is 300 ft/s (91.4 m/s) which gives a reasonable balance between backpressure and scavenging. Once we have the cross section we can easily calculate the diameter.
At the end of all this boring math, there’s some practical considerations. Having a 100” long primary tube is generally not practical in a vehicle, not to mention the fact that a really long primary tube will increase backpressure. You need a balance, generally the 3rd or 4th harmonic is used in a vehicle. With primary diameter, you need to consider the size of the exhaust port you’re using and commonly available tube sizes. Obviously if you’ve got a 2” diameter exhaust port exit (for whatever reason), you’re not going to throw a 1.5” primary diameter on it even if the math calls for it! Common sense has to prevail.
I’ve gotten more math heavy than I would have liked, but this subject is so complicated that entire books have been written on it. We didn’t touch on exhaust system length, collector length, tri-y headers, or venturi effect with convergent-divergent collectors. I’d encourage every enthusiast to check out “Scientific Design of Exhaust and Intake Systems” by Phillip H. Smith, it’s the most complete resource I’ve found on this subject.
I hope this wasn’t too….exhausting.
