the loss Δ(p+½ρv2): For open channel flows, ΔE is related to the total head loss ΔH as: with H the total head: where h is the hydraulic head – the free surface elevation above a reference datum: h = z + p/(ρg). Examples Sudden expansion of a pipe The Borda–Carnot equation is applied to the flow through a sudden expansion of a horizontal pipe. At cross section 1, the mean flow velocity is equal to v1, the pressure is p1 and the cross-sectional area is A1. The corresponding flow quantities at cross section 2 – well behind the expansion (and regions of separated flow) – are v2, p2 and A2, respectively. At the expansion, the flow separates and there are turbulent recirculating flow zones with mechanical energy losses. The loss coefficient ξ for this sudden expansion is approximately equal to one: ξ ≈ 1.0. 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: so 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. Sudden contraction of a pipe In case of a sudden reduction of pipe diameter, without streamlining, the flow is not able to follow the sharp bend into the narrower pipe. As a result, there is flow separation, creating recirculating separation zones at the entrance of the narrower pipe. The main flow is contracted between the separated flow areas, and later on expands again to cover the full pipe area. There is not much head