Choked Compressible Flow Rate Through an Orifice
Choked Compressible Flow Rate Through an Orifice
When gas discharges from a high-pressure vessel through an orifice or nozzle into a much lower-pressure region, the flow velocity at the throat cannot exceed the local speed of sound — once that limit is reached the flow is "choked," and further lowering the downstream pressure does not increase the mass flow rate any further, a critical fact for sizing relief valves, gas pipeline blowdowns, and control valve capacity. The choked mass flow rate depends only on the upstream stagnation conditions and the gas properties, not on the downstream pressure at all once choking occurs, which makes it one of the few compressible-flow results that can be computed directly without iterating on a pressure ratio.
The choked mass flow rate is mdot = Cd·A·P0·sqrt(k/(R·T0))·(2/(k+1))^((k+1)/(2(k-1))), a function of upstream stagnation conditions only. where Cd is the discharge coefficient, A is the orifice area, P0 and T0 are the upstream stagnation pressure and temperature, k is the gas's ratio of specific heats, and R is the specific gas constant.
Inputs
C_d := 0.95=0.95
A_o := 0.0005 m^2=5e-04 m^2
P_0 := 700000 Pa=700000 Pa
k_gam := 1.4=1.4
R_gas := 287 J/(kg·K)=287 m^2/K/s^2
T_0f := 350 K=350 K
This dimensionally homogeneous relation combines the orifice geometry, upstream stagnation state, and a dimensionless gas-property bracket to give the maximum possible mass flow rate the orifice can pass once flow chokes to sonic velocity at the throat.
mdot_choke := C_d·A_o·P_0·sqrt(k_gam/(R_gas·T_0f))·(2/(k_gam+1))^((k_gam+1)/(2·(k_gam-1)))=0.72 kg/s
Results
A choked mass flow rate on the order of half a kilogram per second through a 0.5 cm² orifice at 7 bar upstream pressure is a realistic value for a relief valve or blowdown line sizing calculation. Because this flow rate is independent of downstream pressure once choked (which occurs whenever the downstream-to-upstream pressure ratio drops below a critical value, about 0.528 for air), venting to a lower vacuum or a higher backpressure above that critical ratio makes no difference to this number — a counterintuitive result that trips up engineers new to compressible flow. If the flow is not actually choked (downstream pressure too close to upstream), this formula would overestimate the flow rate and a subsonic orifice equation should be used instead.
Assumptions & limitations
- Choked (sonic) flow at the throat assumed — downstream pressure ratio below the critical value.
- Ideal gas behavior with constant k and R.
- Isentropic flow from stagnation conditions to the throat, with losses captured only by Cd.
- Well-rounded or sharp-edged orifice with a known, constant discharge coefficient.