Darcy friction factor from Reynolds number and roughness.
1The Moody diagram
Is a graph in non-dimensional form that relates the Darcy-Weisbach friction factor f, Reynolds number Re, and surface roughness for fully developed flow in a circular pipe.
2Description
Moody's team used the available data (including that of Nikuradse) to show that fluid flow in rough pipes could be described by four dimensionless quantities (Reynolds number, pressure loss coefficient, diameter ratio of the pipe and the relative roughness of the pipe).
Colebrook’s Equation
Colebrook's equation is widely used to determine the friction factor for turbulent flows in pipes. Clearly this equation is implicit for the determination of the friction factor and the numeral solution can be obtained in different ways. \[{1\over\sqrt{f}} = {-2Log}{\left({{\varepsilon\over D}\over 3.7 }+{2.51\over Re\sqrt{f}}\right)}\]
In this work, it is used a numerical solution based on Newton’s method for the implicit equation for f. The numerical solution of Newton’s method is based on the following algorithm \[x_{i+1} = {x_i}-{f(x_i)\over df(x_i)}\]
In Newton’s method, the equation is used to calculate the friction factor from a known inicial value and it allows to estimate the following interaction value to find the solution for the friction factor’s equation. Colebrook's equation can be reorganized by replacing the value of F=1/√f and be expressed through a new relation g ( F ) as shown in the equation. \[g(f) = {1\over\sqrt{f}}+{2Log}{\left({{\varepsilon\over D}\over 3.7 }+{2.51\over Re\sqrt{f}}\right)}\]
If the equation is assumed as a function that is always continuous and differentiable, the
derivative of the function can be obtained in respect to f. The equation has the capability for rapid convergence, specially if there is an adequate estimation of the inicial value of the friction factor.The calculation is made for an error of 1E-10.
Solution of the implicit Colebrook equation for flow friction using JavaScript
The following session solves the Colebrook’s Equation using the Newton method explained above, to obtain the friction factor you must enter Reynolds number and Relative roughness
Below are explicit correlations that allow you to calculate the value of the friction factor and its error with respect to the value of Colebrook’s Equation.
Filonenko’s correlation
\[f = {\left({1.82Log\left(Re\right) -1.64}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Altshul I’s correlation
\[f = 0.11{\left({\varepsilon\over D}+{68\over Re}\right)}^{0.25}\]
| Friction factor | Relative error (%) |
|---|---|
Altshul II’s correlation
\[f = {\left({1.8Log\left(Re\over 0.135Re\left(\varepsilon\over D\right) +6.5\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Konakov’s correlation
\[f = {\left({1.8Log\left(Re\right) -1.5}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Shacham I’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7}-{5.02\over Re}Log\left({\left(\varepsilon\over D\right)\over 3.7}+{14.5\over Re}\right)\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Shacham II’s correlation
\[f = {{\left({{X\left(1-LnX\right)}-{\left(\varepsilon\over D\right)\over 3.7}\over {1.15129X}+{2.51\over Re}}\right)}}^{-2}\]
where \[X = {{\left(\varepsilon\over D\right)\over 3.7}-{5.02\over Re}Log\left({\left(\varepsilon\over D\right)\over 3.7}+{14.5\over Re}\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Chen’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7065}-{Y}\right)}\right)}^{-2}\]
where \[Y = {{5.0452\over Re}Log\left({{\left(\varepsilon\over D\right)}^{1.1098}\over 2.8257}+{Z}\right)}\]
and \[Z = {5.8506\left(Re\right)}^{-0.8981}\]
| Friction factor | Relative error (%) |
|---|---|
Churchill’s correlation
\[f = 8{\left({{\left(8\over Re\right)}^{12}+\left(A+B\right)}^{-\left({3}\over{2}\right)}\right)}^{{1}\over{12}}\]
where \[A = {\left({2.457}Ln\left({{1}\over {\left(7\over Re\right)}^{0.9}+{0.27\left(\varepsilon\over D\right)}}\right)\right)}^{16}\]
and \[B = {\left(37530\over Re\right)}^{16}\]
| Friction factor | Relative error (%) |
|---|---|
Swamee and Jain’s correlation
\[f = 0.25{\left({Log\left({\left(\varepsilon\over D\right)\over 3.7}+{5.74\over Re}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Pavlov’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7}+{\left(6.81\over Re\right)}^{0.9}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Round’s correlation
\[f = {\left({-1.8Log\left({0.27\left(\varepsilon\over D\right)}+{6.5\over Re}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Barr’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7}+{4.518Log\left(Re\over 7\right)\over {Re\left({1}+{{Re}^{0.52}{\left(\varepsilon\over D\right)}^{0.9}\over 29}\right)}}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Zigrang-Sylvester’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7}-{5.02\over Re}Log\left({\left(\varepsilon\over D\right)\over 3.7}-{5.02\over Re}Log\left({\left(\varepsilon\over D\right)\over 3.7}+{13\over Re}\right)\right)\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Haaland’s correlation
\[f = {\left({-1.8Log\left(\left({\left(\varepsilon\over D\right)\over 3.7}\right)^{1.11}+{6.9\over Re}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Manadilli’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7}+{95\over {Re}^{0.983}}-{96.82\over Re}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Romeo et al’s correlation
\[f = {\left({-2Log\left({\left(\varepsilon\over D\right)\over 3.7065}-{5.0272\over Re}A\right)}\right)}^{-2}\]where \[A = {Log\left({\left(\varepsilon\over D\right)\over 3.827}-{4.567\over Re}Log\left({\left(\left(\varepsilon\over D\right)\over 7.7918\right)}^{0.9924}+{\left(5.3326\over {208.815+Re}\right)}^{0.9345}\right)\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Sonnad and Goudar’s correlation
\[f = {\left({0.8686Ln\left({0.4587Re\over {G}^{G\over {G+1}}}\right)}\right)}^{-2}\]where \[G = {{0.124Re\left(\varepsilon\over D\right)}+Ln\left({0.4587Re}\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Buzzelli’s correlation
\[f = {\left(B-\left({{{B+2Log\left(C\over Re\right)}\over {1+\left(2.18\over C\right)}}}\right)\right)}^{-2}\]where \[B = {{0.774Ln\left(Re\right)-1.41}\over \left({1+1.32\sqrt{\left(\varepsilon\over D\right)}}\right)}\]
and \[C = {{Re\left(\varepsilon\over D\right)\over 3.7}+{2.51B}}\]
| Friction factor | Relative error (%) |
|---|---|
Avci and Karagoz’s correlation
\[f = {{6.4}\over \left(Ln\left(Re\right)-Ln\left(1+0.01Re\left(\varepsilon\over D\right)\left({1+10\sqrt{\left(\varepsilon\over D\right)}}\right)\right)\right)^{2.4}}\]
| Friction factor | Relative error (%) |
|---|---|
Saeed Samadianfard’s correlation
\[f = {\left({{Re^{\left(\varepsilon\over D\right)}}-0.6315093}\over{{Re^{1\over 3}}+Re\left(\varepsilon\over D\right)}\right)}+{0.0275308\left({{6.929841\over Re} +\left(\varepsilon\over D\right)}\right)^{1\over 9}}+{{\left({10^{\left(\varepsilon\over D\right)}}\over{\left(\varepsilon\over D\right)+4.781616}\right)}\left({{\sqrt{\left(\varepsilon\over D\right)}+{9.99701\over Re}}}\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Papaevangelo et al’s correlation
\[f = {{0.2479-0.0000947{\left(7-LogRe\right)}^{4}}\over\left({Log\left({\left(\varepsilon\over D\right)\over 3.615}+{7.366\over {Re^{0.9142}}}\right)}\right)^{2}}\]
| Friction factor | Relative error (%) |
|---|---|
Brkic I’s correlation
\[f = {\left({-2Log\left({10^{\left(-0.4343\varLambda\right)}}+{\left(\varepsilon\over D\right)\over 3.71}\right)}\right)}^{-2}\]where \[\varLambda = {Ln\left({Re}\over {1.816Ln\left(1.1Re\over {Ln\left(1+1.1Re\right)}\right)}\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Brkic II’s correlation
\[f = {\left({-2Log\left({2.18\varLambda\over Re}+{\left(\varepsilon\over D\right)\over 3.71}\right)}\right)}^{-2}\]where \[\varLambda = {Ln\left({Re}\over {1.816Ln\left(1.1Re\over {Ln\left(1+1.1Re\right)}\right)}\right)}\]
| Friction factor | Relative error (%) |
|---|---|
Fang et al’s correlation
\[f = 1.613{\left({Ln\left({0.234\left(\varepsilon\over D\right)^{1.1007}}-{60.525\over {Re}^{1.1105}}+{56.291\over {Re}^{1.0712}}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|
Li et al’s correlation
\[f = {\left({-2Log\left({1.25603\over {Re\sqrt{{-0.0015702\over LnRe}+{0.3942031\over {\left(LnRe\right)}^{2}}+{2.5341533\over {\left(LnRe\right)}^{3}}}}}+{\left(\varepsilon\over D\right)\over 3.71}\right)}\right)}^{-2}\]
| Friction factor | Relative error (%) |
|---|---|