What's an orifice equation good for?

The orifice equation Q = C_d × A × √(2gh) predicts flow through a sharp-edged opening — used for tank drains, low-level outlets on retention ponds, flow-measurement devices, and culvert inlet control.

Why is C_d less than 1.0?

C_d combines two effects: vena contracta (jet contracts to ~62% of orifice area) and friction loss (~98% efficient). Net C_d for a sharp-edged round orifice ≈ 0.62. For rounded entrances, C_d approaches 1.0.

When does orifice flow apply vs weir flow?

Orifice flow when the opening is fully submerged on both sides. Weir flow when there's free surface above and atmospheric below. Transition occurs near opening; check both at design conditions to find which governs.

How is the head H measured?

For a free orifice (jet discharges to atmosphere), H is measured from the upstream water surface to the orifice centroid. For a submerged orifice, H is the difference in water-surface elevations on either side of the orifice. The same equation applies; only the head definition changes.

How long does it take to drain a tank through an orifice?

For a constant-area tank draining through a sharp orifice: t = 2·A_tank·(√H₁ − √H₂) / (C_d·A_orifice·√(2g)). For a 10-ft diameter tank with 4 ft of head draining through a 6-inch orifice: t ≈ 165 seconds (2.75 min).

What's the difference between vena contracta and discharge coefficient?

The contraction coefficient C_c is the ratio of the actual jet area to the orifice area (~0.62 for sharp-edged). The velocity coefficient C_v accounts for friction (~0.97). The discharge coefficient C_d = C_c · C_v ≈ 0.61 combines both.

Does orifice equation apply to gates and sluice gates?

Yes, with C_d typically 0.55–0.65 for vertical sluice gates. For radial gates, C_d ≈ 0.65–0.75 because the curved face approximates a rounded entrance. Always verify with manufacturer rating curves for design submittals.

Orifice Flow Calculator — Discharge Through Orifice

Discharge through an orifice from upstream head and orifice area. The classical Torricelli formula plus a discharge coefficient for the contracted jet.

Discharge coefficient table

The discharge coefficient C_d = C_c · C_v combines the contraction coefficient (how much the jet shrinks below the orifice area — the vena contracta) and the velocity coefficient (velocity reduction due to friction).

Discharge coefficients C_d for common orifice and short-tube configurations
Configuration	C_d	C_c	C_v
Sharp-edged orifice, thin plate	0.61	0.62	0.98
Slightly rounded entrance (r > 0.15 d)	0.95	0.97	0.98
Well-rounded entrance (bellmouth)	0.98	1.00	0.98
Borda mouthpiece (re-entrant tube)	0.51	0.52	0.98
Standard short tube (length 2–3 d)	0.82	1.00	0.82
Conical converging tube (13.5° total)	0.94	0.99	0.95
Square-edged hole in thick wall (l/d ≈ 1)	0.73	0.75	0.97
Vertical sluice gate (typical)	0.55–0.65	—	—
Radial (Tainter) gate	0.65–0.75	—	—
Drop inlet, square edge (FHWA HDS-5)	0.60	—	—

Source: Brater, King, Lindell & Wei (1996), Handbook of Hydraulics, 7th ed., Table 4-1; USBR (1997) Water Measurement Manual; FHWA HDS-5.

Worked examples

Example 1 — Detention basin low-flow outlet (sharp-edged orifice)

Given: Round 4-inch sharp-edged orifice (C_d = 0.61) on a stormwater detention basin riser, 5.0 ft of head from WSE to orifice centroid.

Find: Discharge Q.

A = π·(4/12)²/4 = π·(0.333)²/4 = 0.0873 ft²

Q = 0.61 · 0.0873 · √(2 · 32.2 · 5.0) = 0.0532 · √322

Q = 0.0532 · 17.94

Q = 0.955 cfs ≈ 1.0 cfs

Example 2 — Tank drainage time

Given: 10-ft diameter cylindrical tank, initial depth 4.0 ft above orifice, drains through a 6-inch sharp-edged orifice in the bottom (C_d = 0.61). Drain to empty.

Find: Drain time.

A_tank = π·(10)²/4 = 78.54 ft²; A_orifice = π·(0.5)²/4 = 0.1963 ft²

t = 2·A_tank·(√H₁ − √H₂) / (C_d·A_orifice·√(2g))

t = 2·78.54·(√4.0 − 0) / (0.61·0.1963·√64.4)

t = 314.2 / (0.61·0.1963·8.025) = 314.2 / 0.961

t = 327 seconds ≈ 5.5 minutes

Free vs. submerged orifice

For a free orifice (jet discharges into atmosphere), H is measured from the water surface upstream to the orifice centroid. For a submerged orifice (downstream water level above the orifice), H is the difference in water surface elevations on either side of the orifice. The discharge equation is the same; the head definition changes.

Where this calculator applies

This is the standard equation for: gate openings on small dams, drop inlets and sluice gates, orifice plate flow meters in pipes, low-flow outlet works for stormwater detention basins, valve-gallery openings, and tank drainage time problems. For very large gates where velocity head upstream is non-negligible (drawdown over 10% of total head), use the more general energy equation rather than this simplified form.

Tank drainage time

If you're computing the time to drain a tank through an orifice, you can integrate dV/dt = −Q over the head range. For a constant-cross-section tank: t = 2A_tank · (√H₁ − √H₂) / (C_d·A·√(2g)). That's a separate but related calculation.

Reference: Brater, E.F., King, H.W., Lindell, J.E., Wei, C.Y. (1996). Handbook of Hydraulics (7th ed.), Chapter 4. Original: Torricelli's law (1643).