For a sudden expansion in a pipe, see the figure above, the Borda–Carnot equation can be derived from mass- and momentum conservation of the flow. The momentum flux S (i.e. for the fluid momentum component parallel to the pipe axis) through a cross section of area A is – according to the Euler equations:
Consider the conservation of mass and momentum for a control volume bounded by cross section 1 just upstream of the expansion, cross section 2 downstream of where the flow re-attaches again to the pipe wall (after the flow separation at the expansion), and the pipe wall. There is the control volumes gain of momentum S at the inflow and loss S at the outflow. Besides, there is also the contribution of the force F by the pressure on the fluid exerted by the expansions wall (perpendicular to the pipe axis):
where it has been assumed that the pressure is equal to the close-by upstream pressure p.
Adding contributions, the momentum balance for the control volume between cross sections 1 and 2 gives:
Consequently, since by mass conservation :
in agreement with the pressure drop Δp in the example above.
The mechanical energy loss ΔE is:
which is the Borda–Carnot equation (with ξ = 1).