LTR: Friction Factors and Drag Coefficients

In performing drag experiments, as with any transport experiments, it is useful to calculate/measure dimensionless quantities (for reasons of generality, scale invariance, etc.).

Taking the frictional force per unit surface area to represent the shear stress, we can make a dimensionless group using the fluid density and velocity to get:


The coefficient of skin friction or the Fanning friction factor is the ratio of the total normalized (i.e., dimensionless) shear stress acting on the surface of a solid.

$\displaystyle{f_f = C_f = \frac{F_s/A_S}{\frac12 \rho v_\infty^2}}$


The Fanning friction factor is the one most often used by chemical engineers

If instead we base our dimensionless group more on the "head losses" (a pressure-related drag to be discussed next) we get the Darcy friction factor, $f_D$. While the Darcy factor is more common in general engineering, it can simply be shown to be 4 times as big as the Fanning friction factor, so be careful which is which!


The drag coefficient is a more general dimensionless group that is the ratio of the total normalized drag (i.e., not simply skin drag) acting on the surface of a solid. Here, we use the total projected area rather then the surface area.

$\displaystyle{C_D = \frac{F_d/A_p}{\frac12 \rho v_\infty^2}}$


Typically in internal flows (pipe flows and friends) we are interested primarily in the skin drag, and use friction factors. In external flows, form drag plays a larger role and we tend to use the (total) drag coefficient.

Correlations and Charts

For laminar flow in a pipe, we can analytically show that the (Fanning) friction factor is given by:

$\displaystyle{f_f = \frac{16}{Re}}$

In turbulent flows, however, things are more difficult, as the friction factor depends strongly on the pipe roughness:

For a very smooth pipes a correlation is given as:

$\displaystyle{\frac{1}{\sqrt{f_f}} = 4 \log_{10} \left [Re\sqrt{f_f} \right ] - 0.40}$

For turbulent flow in a rough pipe:

$\displaystyle{\frac{1}{\sqrt{f_f}} = 4 \log_{10} \frac{D}{e} +2.28}$

An expression for transitional flow is given by:

$\displaystyle{\frac{1}{\sqrt{f_f}} = 4 \log_{10} \frac{D}{e} +2.28 - 4\log_{10}\left ( 4.67\frac{D/e}{Re\sqrt{f_f}} + 1\right)}$

Perhaps the easiest way to obtain friction factors is to use the popular Moody charts (on page 173 of your text).


Calculate friction factors from correlations and read friction factors off charts


Calculate the friction factor for a water flow though a smooth 1in ID pipe at (an average of) 10 m/s.

First we calculate the Reynolds number:

$\displaystyle{Re = \frac{V_{avg}D}{\nu} = \frac{(10m/s)(1in)(2.54x10^{-2}m/in}{1x10^{-6}m^2/s} = 2.54x10^5}$

For flow through a circular pipe, this would be considered a turbulent flow. Using the Moody charts, we get $f = 0.0036$.

Alternatively, we can use the correlation for a turbulent flow in a very smooth pipe:

$\displaystyle{\frac{1}{\sqrt{f_f}} = 4 \log_{10} \left [Re\sqrt{f_f} \right ] - 0.40}$

Using "solver" (or similar) and/or trial-and-error, we get $f \approx 0.0038$ using this method.