text: 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 v, the pressure is p and the cross-sectional area is A. The corresponding flow quantities at cross section 2 – well behind the expansion (and regions of separated flow) – are v, p and A, 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.