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:

DEFINITION:

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.

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

NOTE:

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!

DEFINITION:

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.

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

NOTE:

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).

OUTCOME:

Calculate friction factors from correlations and read friction factors off charts

TEST YOURSELF

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.