Civil Engineers with Experience Using the Colebrook Equation?
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02-08-2021, 05:56 PM
Post: #16
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RE: Civil Engineers with Experience Using the Colebrook Equation?
Here is another way, by shifting iterations to W(exp(x)).
Argument inside W might be huge, we combine W(exp(x)) ≡ Wexp(x) w*e^w = e^x → w + ln(w) = x If x → +∞, (x-ln(x)) + ln(x-ln(x)) ≈ x-ln(x)+ln(x) = x If x → −∞, (e^x) + ln(e^x) ≈ 0 + x = x We use these 2 asymptotic solutions for guess of W(e^z). Code: function Wexp(x) Colebrook Equation: x = log10(b-c*x) = ln(b-c*x)/ln(10), where b = eps/(3.7*D), c = 5.02/Re, x = -0.5/√f Let X = x*ln(10), C = c/ln(10), B = b/C, we have: X = ln(C*(B-X)) B-X = B - (ln(C) + ln(B-X)) = W(e^(B-ln(C))) Redo previous example: lua> eps, D, Re = 0.2, 105, 178500 lua> C = 5.02/(log(10)*Re) --> C = 1.2213771984057835e-005 lua> B = eps/(3.7*D*C) --> B = 42.14918335404198 lua> X = B - Wexp(B-log(C)) --> X = -7.4097831878678875 lua> f = (0.5*log(10)/X)^2 --> f = 0.024141285096278223 |
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