Closed Conduit Hydraulics – Bernoulli Equation

The Bernoulli Equation is used to analyze flow in closed pipe systems and is one of the most used equations in hydraulics (that I can remember!).

The base form of the equation relates energy between two or more points in a system. I think it is easier to remember and use in terms of head loss(ft or m).

$$ h_f = \frac{V_1^2}{2g}+\frac{p_1}{\rho g} + z_1 = \frac{V_2^2}{2g}+\frac{p_2}{\rho g} + z_2 $$

\(p\) is Pressure \(lb/ft^2\)  or \(N/m^2\) (Pa)

\(\rho\) is Density \(lb / ft^3\) or \(kg/m^3\)

\(V\) is Velocity \(ft/s\) or \(m/s\)

\(z\) is the elevation in \(ft\) or \(m\)

The reason I like to use the head loss form is that it fits well with the Darcy-Weisbach, Hazen-Williams, and minor loss head loss equations.

You can find the initial head at some point A and set it equal to the head at B + or – other head changes (friction in pipe, bends, connections, pumps, generators).

$$A = B – \text{friction} – \text{connections} + \text{pump head} $$

It is also useful to set variables in the equation to zero depending on the problem. For example if one end of the system being analyzed is open to the atmosphere ( or both ends) then pressure on that side (or both!) \(p\) is zero.

The elevation on one side of the equation can be set to zero for most problems due to most problems covering a drop between two points, I like to show the elevation as positive energy on the higher side of the equation (makes the most sense). Both elevations can be zero if the pipe system is entirely flat.

The velocity is rarely zero on either side.