I've been sitting here looking at all of the performance related books that I have for exhaust primary length and diameter formulas.
Performance Automotive Engine Math by John Baechtel and Performance Exhaust Systems by Mike Mavrigian both give the same formula for primary pipe cross section area. Both sound like they're based on formulas by Jim McFarland.
Ac/s = (cylinder volume in cubic inches x RPM) ÷ 88,200
Then to get the pipe size form the cross-section area, they give two versions of the same formula. The only difference is that Mike Mavrigian saves you one piece of math.
Pipe Diameter = sqrt(A x 1.273)
where A = cross section area
A. Graham Bell offers a slightly different formula for pipe size in Four-Stroke Performance Tuning.
Pipe ID = sqrt[cylinder volume cc ÷ [(P + 3) x 25]] x 2.1
where P = primary length inches (so you must know length before calculating diameter in this case)
Bell also mentions this one :
primary pipe Ac/s in mm2 = (peak torque RPM x cylinder volume cc) ÷ 2,240
It seems like there should be only 1 formula for primary length, but there are 2. This is where I am a little lost.
So, in Bell's book he gives this formula :
primary length = [(850 x ED) ÷ RPM] - 3
where ED = exhaust valve opening BBDC + 180
Then Baechtel and Mavrigian give what sounds like it's supposed to be the same as Bell's, because Bell is cited in an earlier version of the book that I have. It doesn't match though, and it gives different results.
Here is their formula directly over Bell's formula, just with ED in Bell's changed to what it means.
primary length inches = [850 x (360 - EVO)] ÷ RPM] - 3 (Baechtel & Mavrigian)
primary length inches = [850 x (180 + EVO)] ÷ RPM] - 3 (Bell)
where EVO = exhaust valve opening
They give a variation of Bell's diameter formula as well, but it's just done to use cubic inches vs cc. These formulas are not noted to be modified, but both are. As said, one is obvious. I don't know why they chose to change how EVO is dealt with vs Bell though. There is no explanation and it is stated as if it came directly from Performance Tuning In Theory and Practice, so I looked up that old version. It has the same formula that my newer version has.
I did the calculations for all of this stuff, with two different RPM. I used 8,500RPM and 7,500RPM. 8,500RPM is where I initially planned to tune for, based on the idea of trying to extend power either above or below the curve a little. 8,500RPM tuning may be able to help on the underside of the HP peak if I'm correct. 7,500RPM is roughly the peak torque of the engine. This stuff is all based on the dyno from a YouTube video that I found and posted a few pages ago. Some seem to say that the exhaust should be tuned for torque peak, so that's why I did those numbers as well.
I'll start with primary tube inside diameter. I got 1.0316", 1.032", 1.056", 1.0981", 1.0986", and 1.1242". So the whole range is roughly from 1" to 1 1/8" pipe ID or 26.2 to 28.55mm. The pipe that I initially chose and now have is 1 1/8" ID. Larger IDs may sway RPM tuning higher or favor larger engines. That's fine with me. I'm after a close sort of ballpark, and that should make whatever I end up with fine if I did go with a BBK or maybe it would just tune it a little nearer HP peak that initially intended. It sounds like 28mm is very common for aftermarket stuff. Plus, mine will taper, so it will start nearer 7/8" (0.875") to more closely match the port if I can get it right.
Then I started looking at length. There's much more of a range there. Using Bell's 180 + EVO version of the formula, I get 19.5 to 22.5" for primary length. The longer numbers being lower RPM tuning. Using 360 - EVO, I get 28.5 to 32.7". So now I've got a range from 19.5" to 32.7", or even using the same 8500RPM figure only, the range is 19.5" to 28.5". Hmmm.
So then I figure, why not check the specs based on info that John linked me to from KiwiBiker about the tapered header and see if they are obviously using the same formulas. Info there said 450mm length (17.7165") for roughly 12,000RPM tuning on a FXR150. I looked up FXR150 cam info and found someone that measured it at 24 BBDC EVO.
I came up with 11.45" and 20.8" for primary length, depending which of the formulas I used... so neither matches the 17.7" that they mentioned using. I'm not sure if there are changes based on the taper or what's up there.
I don't think it matters that much to be perfect for this, but it does bother me that I can't get a closer approximation of length. At this point, I'd probably just follow Bell's guidelines since he's the one credited for the length formulas anyway. I don't know if I'm done there or not yet.
Competition Engine Building by and the previously mentioned Performance Automotive Engine Math books both recommend PipeMax software as an affordable and trusted solution for exhaust design. Affordable is relative. It's $150 for the latest version here :
maxracesoftware.com/PipeMaxPro400.htm It looks like older versions may be found for less, like $75-100, but I'm not sure how much you miss out on. I don't really want to spend much ATM and I'm not sure how much of a point there is to it so I doubt that I'll get it. Just makes me curious because multiple sources speak highly of it for exhaust design.