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

Table 1

Newton–Raphson procedure. Option 1: starting point λ₀ depends on input parameters: Equation (2), calculation of λ: Equation (7), and analytical derivative f′(λ): Equation (6).

Re = 5·10^6, ε/D = 2.5·10^−5	f(λ), Equation (5)	f′(λ), Equation (6)	λ0 = 0.009352225155363
Iteration 1	0.495092014	−573.0134258	0.010216239839661
Iteration 2	0.031705666	−502.2190127	0.010279370993451
Iteration 3	0.000145453	−497.622807	0.010279663289327
Iteration 4	0.000000003	−497.6016902	λ = 0.010279663295529
Control step	0.000000000	−497.6016898	0.010279663295529
Re = 3·10^4, ε/D = 9·10^−3	f(λ), Equation (5)	f′(λ), Equation (6)	λ0 = 0.036588313752304
Iteration 1	0.143632267	−73.25157738	0.038549121591193
Iteration 2	0.005520057	−67.74092562	0.038630609361351
Iteration 3	0.000008725	−67.52696208	0.038630738574469
Iteration 4	0.000000000	−67.5266237	λ = 0.038630738574792
Control step	0.000000000	−67.5266237	0.038630738574792