The value of the Reynold’s number, RE, determines the flow regime in a pipe.

\[Re = \frac{\rho VD}{\mu} = \frac{VD}{\nu}\]

where \(\rho\) = density of the fluid, \(V\) = average velocity, \(D\) = diameter of the pipe, \(\mu\) = dynamic viscosity, and \(\nu\) = kinematic viscosity.

The energy equation is the governing equation for viscous pipe flow.

\[\frac{p_1}{\gamma} + \frac{V^2_1}{2g} + z_1 = \frac{p_2}{\gamma} + \frac{V^2_2}{2g} + z_2 + h_L\]

where \(p\) = pressure, \(V\) = velocity, \(g\) = gravity, \(z\) = height measured from some origin, and \(h_L\) = head loss, which consists of:

\[h_{L,major} = f \frac{L}{D} \frac{V^2}{2g}\]

\[h_{L,minor} = \frac{\Sigma K_L V^2}{2g}\]

where \(f\) = friction factor and \(K_L\) = loss coefficient.

The Colebrook equation is a method to solve for the friction factor.

\[\frac{1}{\sqrt{f}} = -2.0 log \left( \frac{\frac{\epsilon}{D}}{3.7} + \frac{2.51}{Re \sqrt{f}} \right)\]

where \(\epsilon\) = surface roughness