The Bernoulli equation for a real liquid
For two sections of a flow of a liquid with density ρ the energy balance per unit weight is z₁ + p₁/(ρg) + α₁v₁²/(2g) = z₂ + p₂/(ρg) + α₂v₂²/(2g) + h_w. Each term is a head in metres of the liquid column: the elevation z, the pressure head p/(ρg) and the velocity head αv²/(2g). Their sum H is the total head and h_w is the energy loss between the sections (friction and local).
Hydraulic and energy grade lines
The energy grade line joins the points of the total head H and always falls along the flow by the amount of the losses. The hydraulic grade line lies below it by the velocity head and shows how high the liquid would rise in a piezometer. In a contraction the velocity grows, the velocity head increases and the pressure falls, so the hydraulic grade line goes down towards the pipe axis. If it goes below the axis there is a vacuum in the pipe.
Venturi meter and Pitot tube
In a Venturi meter the difference of piezometric heads between the wide and the narrow sections gives the flow: Q = Cd·A₂·√(2gΔh / (1 − (d₂/d₁)⁴)). For a differential manometer with a liquid of density ρ_m, Δh = (ρ_m/ρ − 1)·h_m. The coefficient Cd of real meters is about 0.95–0.99 and is taken from the data sheet or standards. A Pitot tube measures the difference between the total and static heads: v = Cp·√(2gΔh).
Worth remembering
- The pressure can be gauge or absolute, but it must be the same in both sections.
- The Coriolis coefficient α is 1 for developed turbulent flow and 2 for laminar flow in a round pipe.
- Choose sections where the flow changes smoothly, not in the eddies behind a local resistance.
Sources: Chugaev R. R. Hydraulics; Shterenlikht D. V. Hydraulics; Idelchik I. E. Handbook of Hydraulic Resistance; White F. M. Fluid Mechanics.