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A Different View Of Harrell's Colebrook-White Solution
01 October 2011
For some reason, most people still do not understand how my Colebrook-White solution works.
It has been tested, and re-tested over and over, and always gets the right answers if you don't make
a weird error.
This may help those who are questioning "Why does it work?"
In the Wikipedia you can read about the Lambert W function. Here's a quote near the beginning...
"The Lambert W relation cannot be expressed in terms of elementary functions. It is useful in combinatorics, for instance in the enumeration of trees. It can be used to solve various equations involving exponentials and also occurs in the solution of delay differential equations, such as y'(t) = a y(t − 1)."
The "a" represents the numbers that don't change as we move toward the answer, like the numbers for Re, Rr, 2,51, 3.7. We are computing "1/sqrt(f)" which is the y in that quote. y(t-1) should by y to the sub (t-1), the y value of y just before the y'(t). To appear simpler to show in Excel I write is like this y2=a*y1. In other words we assume an initial value for y1 and multiply by a, the result is y2, Then do another step where we use the y2 result as y1 and get another y2. The "funnel" that directs the solution is the value of "a".
Colebook-White Equation written for Excel is 1/sqrt(f)=-2*Log(Rr/3.7+2.51/Re*1/sqrt(f)). Above I said about the Lambert W function y2=a*y1. y1 is our initial guess, "a" is the complex part "-2*Log(Rr/3.7+2.51/Re*". .... y1 is part of that result like Lambert showed as "a*y1".
Some argue that this is not a solution, just like pi() is not 3.14159. They said pi() has an infinity of digits following the "3." I would not disagree about the math, but for engineering designs a specific number of decimals places works for specific designs. In Excel we can easily get 15 decimal places which exceeds the needs of all the designs engineers normally do.
The attached XLS version of Excel shows how simple my method is.to find the Darcy Friction Factor.
Please comment and ask questions. Your questions will help me know how to explain it,
My Easy and True solution of Colebook-White is little this.
In Excel enter a guess number from 0 to 100. This will compute the 1/Sqrt(f) and I call it X. Below the Guess number inter this equation. =-2(Log(Rr/3.7+2.51/Re*X))
But enter the Rr and Re numbers, and for the X, just use the Excel to point the cell above where you guess and X. Then copy that equation and down to about 20 cells, and you will see the solution stops changing. Then below the bottom cell where the X stopped changing and enter this =1/X/X, but in Excel just point to the bottom X two times, and the solution will be the Easy and True solution to f.
Then it enter the two equations to see that is is right.
=1/Sqrt(f)
=-2*Log(Rr/3.7+2.51/(Re*sqrt(f))
Both will solve for the X you computed.
Example:
Rr= 0.02886 is B1 Re= 5,754,334 is B2 Guess X= 1000 Equations for B4 then copy it and the X (B3) will change. 1 4.1685469739455100 FALSE =-2*LOG($B$1/3.7+2.51/$B$2*B3) 2 4.2156083377835900 FALSE =-2*LOG($B$1/3.7+2.51/$B$2*B4) 3 4.2156060523896700 FALSE =-2*LOG($B$1/3.7+2.51/$B$2*B5) 4 4.2156060525006500 FALSE =-2*LOG($B$1/3.7+2.51/$B$2*B6) 5 4.2156060525006500 TRUE =-2*LOG($B$1/3.7+2.51/$B$2*B7) 6 4.2156060525006500 TRUE =-2*LOG($B$1/3.7+2.51/$B$2*B8) f= 0.0562703946738773 =1/B9/B9