Talk:Darcy friction factor formulae

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[edit] A comment on notation

The Colebrook Equation and the Haaland's solution are written for 1/sqrt(f) rather than for 'f' and this is IMHO the right thing to do. However the Swamee-Jain and Serghide's solutions are written for 'f' instead of 1/sqrt(f) which makes them inconsistent with the first two equations and makes them more difficult to understand. BTW Swamee-Jain is very similar to Haaland, but that's hard to see it because of the way Swamee-Jain is written. Consider a change ?

I favor giving the most visibility to the versions introduced by the creators of the various formulae. Having shown the creator's version, then presentation of other, published forms seems useful to me. I suggest being bold if you have seen other forms and want them here.

If Swamee-Jain, etc., is usually written in terms of 'f' instead of '1/sqrt(f)' then it might cause confusion to _replace_ the usually-expected form with a revised form.

While the '1/sqrt(f)' form is traditional for the Colebrook equation, the other equations which calculate 'f' seem more user-friendly to me in that they don't require the user to unpack '1/sqrt(f)'.

About the similarity noted between Haaland and Swamee-Jain:

Interesting observation!

Playing fast and loose with the exponent, the constant in the Re term is 6.96 ( = 5.74^1.11 -- rounding 1/0.9 to 1.11) when Swamee-Jain is put into the Haaland form.

That is, the transformation from Swamee-Jain to almost-Haaland seems to involve this operation:

(a + b)^c = (a^c + b^c)

As one cannot simply move the exponent onto each term of a polynomial and have the result be _exactly_ the same, I would caution against showing the almost-Haaland form of Swamee-Jain in the article.

I realize that Colebrook may have done something comparable in deriving his formula, but his formula has historical acceptance. Putting the almost-Haaland form of Swamee-Jain in the article would show an approximation which is probably in the realm of original research.

--Ac44ck 00:28, 28 October 2007 (UTC)

[edit] Origin and approximations of Colebrook's equation

There is an intriguing statement in this document: http://www.psig.org/papers/2000/0112.pdf

"the Colebrook-White equation ... is nothing more than a combination of" the Nikuradse equation and the Karman-Prandtl equation.

If one pursues that notion, a more precise form of Colebrook's equation seems to be:

1/sqrt(f) = -2 * log [e/D / 3.71535... + 2.51188... / (Re * sqrt(f))]

Here are the details:

Nikuradse -- rough pipe law (infinite Reynolds number)

f approaches zero as e/D approaches zero.

1/sqrt(f) = 2*log(D/e) + 1.14

= 1.14 - 2*log(e/D)

= 2*log(3.71535...) - 2*log(e/D)

= -2*[log(e/D) - log(3.71535...)]

= -2*log(e/D / 3.71535...)

where:

3.71535... = 10^(1.14/2)

Karman and Prandtl -- smooth pipe law (e/D = 0)

f approaches zero as Re approaches infinity.

1/sqrt(f) = 2*log(Re * sqrt(f)) - 0.8

= 2*log(Re * sqrt(f)) - 2 * log(2.51188...)

= 2*log(Re * sqrt(f) / 2.51188...)

= -2*log[2.51188... / (Re * sqrt(f))]

where:

2.51188... = 10^(0.8/2)

Combining terms (which are mutually-exclusive at the limits) within the log by simple superposition:

1/sqrt(f) = -2 * log [e/D / 3.71535... + 2.51188... / (Re * sqrt(f))]

Which is the Colebrook equation -- with more "available" figures.

How "significant" those extra "available" figures are is a question which might be explored elsewhere.

Regardless of how significant those extra digits are in the end, it seems that attempts to create explicit formulae may not be aiming at quite the right target. They are trying to get as close as possible to the results of a formula which was derived from two others -- and _already_ contains rounding errors.

It seems that any curve-fitting to the Colebrook equation should be done with several more "available" figures in the Colebrook equation's constants.

The time to worry about significant digits should be _after_ the "best" constants have been found in the explicit formula. That is, after the results of the explicit formula provide the best fit to a precise form of Colebrook's equation.

Perhaps that has been happening, but 3.7 and 2.51 may have been treated as sacred in the development of most of the explicit formulae which are intended to approximate the Colebrook equation.

Exactly those constants certainly loom large in Serghide's solution. And exactly 3.7 appears in the other approximations listed in this article so far.

Perhaps this is the stuff of a Wikipedia article of its own. --Ac44ck 00:18, 21 October 2007 (UTC)