Research Article
Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction
Table 3
Newton–Raphson procedure. Option 3: fixed initial starting point
x0 = 6.445695939
→
λ0 = 0.024069128765101 from Section
2.2.1, indirect calculation of
λ through the transmission factor
x: Equation (
10), and analytical derivative
f′(
x): Equation (
9).
| | Re = 5·106, ε/D = 2.5·10−5 | f(x), Equation (8) | f′(x), Equation (9) | x0 = 6.445695939 | λ0 = 0.024069128768719 |
| | Iteration 1 | −3.554956085 | 1.043635910 | 9.852014225862620 | 0.010302673560706 | | Iteration 2 | −0.011430857 | 1.037259804 | 9.863034470914730 | 0.010279663490514 | | Iteration 3 | −0.000000097 | 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 (9) | x0 = 6.445695939 | λ0 = 0.024069128768719 |
| | Iteration 1 | 1.391712393 | 1.024454486 | 5.087204750239650 | 0.038640395682209 | | Iteration 2 | −0.000651990 | 1.025427001 | 5.087840572945700 | 0.038630738577020 | | Iteration 3 | 0.000000000 | 1.025426528 | x = 5.087840573092420 | λ = 0.038630738574792 | | Control step | 0.000000000 | 1.046830475 | 5.087840573092420 | 0.038630738574792 |
|
|
