Advances in Civil Engineering

Research Article

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

Table 4

Newton–Raphson procedure. Option 4: starting point x₀ depends on input parameters: Equation (2), indirect calculation of λ through the transmission factor x: Equation (12), and the symbolic derivative f′(x): Equation (11).


Re = 5·10⁶, ε/D = 2.5·10⁻⁵	f(x), Equation (8)	f′(x), Equation (11)	x₀ = 10.34052343	λ₀ = 0.009352225155363

Iteration 1	0.495092014	1.036495031	9.862863625818000	0.010280019623455
Iteration 2	−0.000177305	1.037242471	9.863034564433310	0.010279663295576
Iteration 3	0.000000000	1.037242198	x = 9.863034564455800	λ = 0.010279663295529
Control step	0.000000000	1.037242198	9.863034564455800	0.010279663295529

Re = 3·10⁴, ε/D = 9·10⁻³	f(x), Equation (8)	f′(x), Equation (11)	x₀ = 5.227918429	λ₀ = 0.036588313752304

Iteration 1	0.143632267	1.025322691	5.087833489750430	0.038630846139210
Iteration 2	−0.000007263	1.025426533	5.087840573092400	0.038630738574793
Iteration 3	0.000000000	1.025426528	x = 5.087840573092420	λ = 0.038630738574792
Control step	0.000000000	1.025426528	5.087840573092420	0.038630738574792