In fluid dynamics the Borda–Carnot equation is an empirical description of the mechanical energy losses of the fluid due to a (sudden) flow expansion. It describes how the total head reduces due to the losses. This is in contrast with Bernoulli's principle for dissipationless flow (without irreversible losses), where the total head is a constant along a streamline. The equation is named after Jean-Charles de Borda (1733–1799) and Lazare Carnot (1753–1823). This equation is used both for open channel flow as well as in pipe flows. In parts of the flow where the irreversible energy losses are negligible, Bernoulli's principle can be used. Formulation The Borda–Carnot equation is: where ΔE is the fluid's mechanical energy loss, ξ is an empirical loss coefficient, which is dimensionless and has a value between zero and one, 0 ≤ ξ ≤ 1, ρ is the fluid density, v1 and v2 are the mean flow velocities before and after the expansion. In case of an abrupt and wide expansion the loss coefficient is equal to one. In other instances, the loss coefficient has to be determined by other means, most often from empirical formulae (based on data obtained by experiments). The Borda–Carnot loss equation is only valid for decreasing velocity, v1 > v2, otherwise the loss ΔE is zero – without mechanical work by additional external forces there cannot be a gain in mechanical energy of the fluid. The loss coefficient ξ can be influenced by streamlining. For example, in case of a pipe expansion, the use of a gradual expanding diffuser can reduce the mechanical energy losses. Relation to the total head and Bernoulli's principle The Borda–Carnot equation gives the decrease in the constant of the Bernoulli equation. For an incompressible flow the result is – for two locations labelled 1 and 2, with location 2 downstream to 1 – along a streamline: with p1 and p2 the pressure at location 1 and 2, z1 and z2 the vertical elevation – above some reference level – of the fluid particle, and g the gravitational acceleration. The first three terms, on either side of the equal sign are respectively the pressure, the kinetic energy density of the fluid and the potential energy density due to gravity. As can be seen, pressure acts effectively as a form of potential energy. In case of high-pressure pipe flows, when gravitational effects can be neglected, ΔE is equal to