I did some research a while back and went quite in depth.
My conclusion was that there is no analytical solution for maximising the vacuum produced by a venturi. By analytical, I mean working it out on a bit of paper using algebra only. There are quite simply too many variables.
Fluid dynamics is a horrifically complex subject with what I'd describe as 'improper' maths. As an engineer I like pure maths - situations you can analyse and describe in precise mathematical terms. Fluid dynamics has very little of this - it's all manipulation of experimental data.
So, by doing multiple experiments it might be possible to deduce a relationship between throat diameter and flow rate - that appears to be what Jim has done, although when I asked him for his data on which he based his formula I got a very short response.
You could potentially have a numerical solution - which rely on number crunching (i.e. computers) to produce an answer. This might produce a solution to a particular problem (combination of pipework, pump curve, venturi dimensions, discharge pipework, fluids used etc etc) but not a general one.
Anyhow, however Jim has ended up with the 'formula' it has been proved time and time again to produce venturi dimensions that work. How optimal they are is difficult to show without making several and testing them. I'd suggest that they are certainly in the right ballpark.
In any case, due to the approximate nature of the whole thing, I'd feel entirely justified in reducing the whole thing to sqrt(flow rate). The differences in dimensions will have negligible effect.
