Advances in Civil Engineering

Research Article

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

Table 5

Halley procedure. Option 5: fixed initial starting point x₀ = 7.990256504 → λ₀ = 0.015663210285978339 from Section 2.2.2, indirect calculation of λ through the transmission factor x: Equation (13), and the symbolic derivatives f′(x) and f″(x): Equations (11) and (14).


Re = 5·10⁶, ε/D = 2.5·10⁻⁵	f(x), Equation (8)	f′(x), Equation (11)	f″(x), Equation (14)	x₀ = 7.990256504	λ₀ = 0.015663210285978

Iteration 1	−1.945484250	1.040493788	−0.001887828	9.863203600915390	0.010279310950983
Iteration 2	0.000175332	1.037241928	−0.001596798	x = 9.863034564455800	λ = 0.010279663295529
Control step	0.000000000	1.037242198	−0.001596821	9.863034564455800	0.010279663295529

Re = 3·10⁴, ε/D = 9·10⁻³	f(x), Equation (8)	f′(x), Equation (11)	f″(x), Equation (14)	x₀ = 7.990256504	λ₀ = 0.015663210285978

Iteration 1	2.973246188	1.023435376	−0.000632309	5.087698791122220	0.038632891696967
Iteration 2	−0.000145387	1.025426633	−0.000744326	x = 5.087840573092420	λ = 0.038630738574792
Control step	0.000000000	1.025426528	−0.000744320	5.087840573092420	0.038630738574792