1/sqrt(f)=-2*Log(Rr/3.7+2.51/Re*1/sqrt(f)) |
Since the 1930's this equation has been said to be unsolvable. Well, I have developed the most accurate solution. Excel will do a maximum of 15 decimals, and those will be easy!
The equation is to find the value of f, when you know Rr and Re. They are the Relative Roughness and Reynolds Number.
First let's "simplify" the equation. Let A = 2.51 / Reynolds Number and B = Relative Roughness / 3.7 . So A = 2.51/Re and B = Rr/3.7 Also let 1/(sqrt(f) be X.
Also f must be positive so lets write...X = Abs(2 * Log(B + A * X)).
Now lets say we computed A and B from Rr and Re and B= Rr/3.7 = 0.0022 and 2.51/Re = 0.00001, so now…X =ABS(2*LOG( 0.0022 + 0.00001 * X ))
To find the value of X, you can guess just about any number for the X and solve the right side of the equation. Let's say you used 1 for X, it would solve for 5.311215, If you said X was 100, it would solve to 4.9897, this means that X will be close to 5. But the importance is that each step will get much, much closer. I usually use 3 for a first estimate, but it make almost no difference. What ever number you get for a solution, use it again until you get as many decimal places as you need.
When I use X=3, I get 5.30339. Then I use that for X and get 5.294465. Then I use that for X and get 5.294499 and next I get the same result. That means I have quickly got 6 decimals of accuracy.
Now to find f, it is 1 / (5.294499^2) = 0.035674
If you go seven loops to find X, you will be averaging 15 decimal places of accuracy. If you go twenty steps in Excel you will have at least 15 decimal places for the f value. That is much more accurate that any of the Colebrook-White approximations.
To test you, let me give this. If Re=10,000 and Rr = 0.01 what would be f?
If you get X= 4.8153456499125 then get f= 0.0431265847068117, then you have solved for 15 decimals places, but usually about 6 decimal places is good enough depending on what you are designing.
Email me at Ragknot@gmail.com for several different solutions of the Colebrook-White equations.
