loss between cross section 1, before the contraction, and cross section 3, the vena contracta at which the main flow is contracted most. But there are substantial losses in the flow expansion from cross section 3 to 2. These head losses can be expressed by using the Borda–Carnot equation, through the use of the coefficient of contraction μ: with A3 the cross-sectional area at the location of strongest main flow contraction 3, and A2 the cross-sectional area of the narrower part of the pipe. Since A3 ≤ A2, the coefficient of contraction is less than one: μ ≤ 1. Again there is conservation of mass, so the volume fluxes in the three cross sections are a constant (for constant fluid density ρ): with v1, v2 and v3 the mean flow velocity in the associated cross sections. Then, according to the Borda–Carnot equation (with loss coefficient ξ=1), the energy loss ΔE per unit of fluid volume and due to the pipe contraction is: The corresponding loss of total head ΔH can be computed as ΔH = ΔE/(ρg). According to measurements by Weisbach, the contraction coefficient for a sharp-edged contraction is approximately: Derivation from the momentum balance for a sudden expansion 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 volume's gain of momentum S1 at the inflow and loss S2 at the outflow. Besides, there is also the contribution of the force F by the pressure on the fluid exerted by the expansion's wall (perpendicular to the pipe axis): where it has been assumed that the pressure is equal to the close-by upstream pressure p1. 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