3.2: The Darcy-Weisbach Friction Factor

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The value of the wall friction factor \(\ \mathrm{\lambda_{l}}\) depends on the Reynolds number:

\[\ \mathrm{R} \mathrm{e}=\frac{\mathrm{v}_{\mathrm{l s}} \cdot \mathrm{D}_{\mathrm{p}}}{v_{\mathrm{l}}}=\frac{\rho_{\mathrm{l}} \cdot \mathrm{v}_{\mathrm{l} \mathrm{s}} \cdot \mathrm{D}_{\mathrm{p}}}{\mu_{\mathrm{l}}}\]

For laminar flow (\(Re<2320\)) the value of \(\ \lambda_{l}\) can be determined according to Poiseuille:

\[\ \lambda_{1}=\frac{64}{\mathrm{Re}}\]

For turbulent flow (\(Re>2320\)) the value of \(\ \mathrm{\lambda_{l}}\) depends not only on the Reynolds number but also on the relative roughness of the pipe \(\ \mathrm{\varepsilon}\)/D_p, which is the absolute roughness \(\ \mathrm{\varepsilon}\) divided by the pipe diameter D_p. A general implicit equation for \(\ \mathrm{\lambda_{l}}\) is the Colebrook-White (1937) equation:

\[\ \lambda_{1}=\frac{1}{\left(2 \cdot \log _{10}\left(\frac{2.51}{\operatorname{Re} \cdot \sqrt{\lambda_{1}}}+\frac{0.27 \cdot \varepsilon}{D_{p}}\right)\right)^{2}}\]

For very smooth pipes the value of the relative roughness \(\ \mathrm{\varepsilon}\)/D_p is almost zero, resulting in the Prandl & von Karman equation:

\[\ \lambda_{1}=\frac{1}{\left(2 \cdot \log _{10}\left(\frac{2.51}{\operatorname{Re} \cdot \sqrt{\lambda_{1}}}\right)\right)^{2}}\]

At very high Reynolds numbers the value of \(\ 2.51 /(\mathrm{Re} \cdot \sqrt{\lambda_{1}})\) is almost zero, resulting in the Nikuradse (1933) equation:

\[\ \lambda_{1}=\frac{1}{\left(2 \cdot \log _{10}\left(\frac{0.27 \cdot \varepsilon}{D_{p}}\right)\right)^{2}}=\frac{5.3}{\left(2 \cdot \ln \left(\frac{0.27 \cdot \varepsilon}{D_{p}}\right)\right)^{2}}\]

Because equations (3.2-3) and (3.2-4) are implicit, for smooth pipes approximation equations can be used. For a Reynolds number between 2320 and 10⁵ the Blasius equation gives a good approximation:

\[\ \lambda_{1}=0.3164 \cdot\left(\frac{1}{R e}\right)^{0.25}\]

For a Reynolds number in the range of 105 to 108 the Nikuradse (1933) equation gives a good approximation:

\[\ \lambda_{\mathrm{l}}=\mathrm{0 . 0 0 3 2}+\frac{\mathrm{0 . 2 2 1}}{\mathrm{R e}^{\mathrm{0 . 2 3 7}}}\]

Figure 3.2-1 gives the so called Moody (1944) diagram, in this case based on the Swamee Jain (1976) equation.

Screen Shot 2020-07-07 at 12.47.13 PM.png

Over the whole range of Reynolds numbers above 2320 the Swamee Jain (1976) equation gives a good approximation:

\[\lambda_{1}=\dfrac{1.325}{\left(\ln \left(\dfrac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\dfrac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}}=\frac{0.25}{\left(\log _{10}\left(\dfrac{\varepsilon}{3.7 \cdot \mathrm{D}_{\mathrm{p}}}+\dfrac{5.75}{\mathrm{Re}^{0.9}}\right)\right)^{2}} \]