Research Article
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 = 5·106, 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 = 3·104, 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 |
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