The friction factor is a function of $f_d = fnc(Re_D, \frac{\epsilon}{D}, duct\ shape)$ For everything other than laminar flow, emperical correlations need to be used to obtain friction factor. ## 1. Laminar Flow For laminar flow, the friction factor can be calculated by a simple formula shown below. Note that friction factor is only a function of Reynolds number $f_D=\frac{64}{Re}$ ## 2. Turbulent Flow ### 2.1. Colebrook Equation It is also called Colebrook-White equation. It is the classic formulation for Darcy friction factor. However it is implicit ($f_D$ term is in both left and right sides of the equation) and needs iteration $\frac{1}{\sqrt{f_D}}=-2log_{10}\left[\left(\frac{\varepsilon /D}{3.7}\right) + \frac{2.51}{Re\sqrt{f_D}}\right]$ It was developed by Colebrook in 1939 extending work of Nikuradse who was a student of Prandtl. It was been plotted by Moody in 1944 into what is known as [[Moody chart]] ### 2.2. Haaland Equation Haaland equation provides the Darcy friction factor similar to Colebrook equation. This was derived for a full flowing circular pipe for both laminar and turbulent flows. It is easier to use than Colebrook equation since it can be solved explicitly $\frac{1}{\sqrt{f_D}}=-\frac{1.8}{n}log\left[\left(\frac{\varepsilon /D}{3.7}\right)^{1.11n} + \left(\frac{6.9}{Re}\right)^{n}\right]$ where, $\varepsilon /D$ is the pipes relative roughness $Re$ is the [[Reynolds number]] n=1 for liquids and n=3 for gases Reference: [Review of explicit approximations to the Colebrook relation for flow friction (hal.science)](https://hal.science/hal-01586547/document) ### 2.3. Petukhov equation Darcy Friction Factor for a *smooth* tube can be given by $f_D = (0.790 ln(Re) - 1.64)^{-2}$ for $10^4<Re<10^6$ ## 3. References - [wikipedia]([Darcy friction factor formulae - Wikipedia](https://en.wikipedia.org/wiki/Darcy_friction_factor_formulae))