How the Orifice Flow Equation Works
An orifice is a defined opening — a drilled hole in a plate, a nozzle, or a valve port — that restricts flow and creates a pressure drop as fluid passes through it. The orifice flow equation, Q = Cd·A·√(2ΔP/ρ), predicts the volumetric flow rate Q through that opening from its cross-sectional area A, the pressure differential ΔP across it, the fluid density ρ, and an empirical discharge coefficient Cd that accounts for real-world losses. It is one of the most widely used relationships in fluid mechanics, underlying orifice-plate flow meters, hydraulic valve sizing, spray-nozzle design, and tank-drainage calculations.
Deriving Q = Cd·A·√(2ΔP/ρ)
Start from Bernoulli's equation between a point well upstream of the orifice (pressure P1, velocity ≈ 0 in a large reservoir or pipe) and the jet just past the opening (pressure P2, velocity v): P1 + ½ρv1² = P2 + ½ρv2². Neglecting the small upstream velocity and any elevation change gives an ideal, frictionless jet velocity of v_ideal = √(2ΔP/ρ), where ΔP = P1 − P2 — the same relationship behind Torricelli's law for a draining tank. Real flow always falls short of that ideal because of viscous friction and the vena contracta, the point just downstream of a sharp-edged orifice where the jet narrows to less than the hole's geometric area. The discharge coefficient Cd (typically 0.6–0.98) is the ratio of actual to ideal flow, so the actual flow rate becomes Q = Cd·A·√(2ΔP/ρ), with A = π/4 × d² for a circular orifice of diameter d.
Choosing Cd and where the equation breaks down
Cd depends on the orifice's edge geometry, its thickness-to-diameter ratio, and the Reynolds number of the flow: a thin sharp edge gives Cd ≈ 0.60–0.61, a short cylindrical tube (length about equal to its diameter) gives Cd ≈ 0.80, and a smooth, rounded bell-mouth entrance can reach Cd ≈ 0.95–0.98. This simple form also assumes the orifice sits in a large reservoir or a pipe whose diameter is much larger than the orifice, so upstream velocity is negligible; for an orifice plate installed in a pipe of comparable diameter, engineers apply an additional beta-ratio correction (per ISO 5167 / ASME MFC-3M). The formula further assumes steady, single-phase, incompressible flow — accurate for liquids and for gases only when ΔP stays small relative to the absolute upstream pressure (roughly ΔP/P1 below about 10%); beyond that, use compressible-flow or choked-flow equations. For liquids, also confirm the downstream pressure stays above the fluid's vapor pressure, or cavitation will reduce the real flow below the calculated value.