The Reynolds number, \(Re\), is used to determine the flow regime.

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

where \(U\) = freestream velocity
\(\rho\) = fluid density
\(\mu\) = fluid dynamic viscosity
\(\nu\) = fluid kinematic viscosity
\(D\) = pipe diameter

The governing equation for pipe flow:

\[\frac{P_1}{\gamma} + \alpha _1 \frac{V^2 _1}{2g} + z_1 + h_P = \frac{P_2}{\gamma} + \alpha _2 \frac{V^2 _2}{2g} +z_2 +h_T + h_L\]

\[h_L = f\frac{L}{D} \frac{V^2}{2g}\]

where \(\gamma\) = specific weight of the fluid
\(P\) = pressure at each point in the pipe
\(V\) = velocity at each point in the pipe
\(g\) = gravitational constant
\(z\) = height at each point in the pipe, referenced to the same point at \(z = 0\)
\(\alpha\) = kinematic energy parameter
\(h_L\) = head loss
\(f\) = frictional factor, which can be determined using the Moody chart
\(L\) = length of the pipe

Although \(h_P\) is the energy of the pump, and \(h_T\) is the energy of the turbine, they are usually not included in the basic energy equation, but they must be used when pumps and/or turbines are part of the system.