Borda–Carnot equation

This is in contrast with Bernoulli's principle for dissipationless flow (without irreversible losses), where the total head is a constant along a streamline.

[1] In other instances, the loss coefficient has to be determined by other means, most often from empirical formulae (based on data obtained by experiments).

For example, in case of a pipe expansion, the use of a gradual expanding diffuser can reduce the mechanical energy losses.

Due to mass conservation, assuming a constant fluid density ρ, the volumetric flow rate through both cross sections 1 and 2 has to be equal: Consequently – according to the Borda–Carnot equation – the mechanical energy loss in this sudden expansion is: The corresponding loss of total head ΔH is: For this case with ξ = 1, the total change in kinetic energy between the two cross sections is dissipated.

As a result, the pressure change between both cross sections is (for this horizontal pipe without gravity effects): and the change in hydraulic head h = z + p/(ρg): The minus signs, in front of the right-hand sides, mean that the pressure (and hydraulic head) are larger after the pipe expansion.

That this change in the pressures (and hydraulic heads), just before and after the pipe expansion, corresponds with an energy loss becomes clear when comparing with the results of Bernoulli's principle.

According to this dissipationless principle, a reduction in flow speed is associated with a much larger increase in pressure than found in the present case with mechanical energy losses.