Engineering & Scientific
Isentropic Choked Nozzle Critical Pressure Ratio & Sonic Mass Flow Rate
Determine whether gas discharge through a relief valve or converging nozzle is subsonic or choked at Mach 1 using critical ratio (2/(gamma+1))^(gamma/(gamma-1)) and unified Saint-Venant mass flow.
Why lowering downstream pressure below 0.5283 * P_0 cannot increase air mass flow
Applying incompressible orifice formulas to high-pressure compressed air, nitrogen, or steam relief valves dramatically overpredicts discharge! As back-pressure P_b drops relative to upstream stagnation pressure P_0, throat velocity accelerates only until reaching local speed of sound (Mach = 1) at critical pressure ratio r_crit = (2 / (gamma + 1))^(gamma / (gamma - 1))—equal to 0.5283 for diatomic gases (gamma = 1.4).
Once P_b / P_0 <= r_crit, acoustic pressure waves can no longer propagate upstream past the sonic throat: the flow is CHOKED, and mass flow locks onto its maximum ceiling regardless of whether downstream pressure is 1.0 bar or 0.5 bar! Binding effective ratio p_eff = MAX(P_b / P_0, r_crit) inside Saint-Venant’s compressible efflux equation seamlessly evaluates both choked sonic flow (P_b / P_0 <= r_crit) and subsonic flow (r_crit < P_b / P_0 < 1).
| Back-Pressure Regime (P0=600 kPa, T0=300 K, At=1 cm2) | Ratio P_b / P_0 vs 0.5283 | Mass Flow m_dot (kg/s) |
|---|---|---|
| Atmospheric Vent (P_b = 100 kPa) | 0.1667 <= 0.5283 (CHOKED_SONIC_M1) | 0.1400 kg/s |
| Deep Partial Vacuum (P_b = 50 kPa) | 0.0833 <= 0.5283 (CHOKED_SONIC_M1) | 0.1400 kg/s (Unchanged!) |
| High Back-Pressure (P_b = 480 kPa) | 0.8000 > 0.5283 (SUBSONIC_FLOW) | 0.1146 kg/s |
Isentropic Critical Pressure Ratio r_crit Formula
Evaluate (2 / (gamma + 1))^(gamma / (gamma - 1)) from specific heat ratio B5 (0.5283 for gamma = 1.4).
=ROUND((2 / (B5 + 1)) ^ (B5 / (B5 - 1)), 4)Unified Choked + Subsonic Compressible Gas Mass Flow Formula (kg/s)
Clamp p_eff = MAX(P_b / P_0, r_crit) when P_b < P_0 (returning 0 when P_b >= P_0) and evaluate Saint-Venant compressible mass flow across throat area B7.
=IF(B3 >= B2, 0, LET(gam, B5, r_cr, (2 / (gam + 1)) ^ (gam / (gam - 1)), p_eff, MAX(B3 / B2, r_cr), term_sv, ((2 * gam) / (B6 * B4 * (gam - 1))) * ((p_eff ^ (2 / gam)) - (p_eff ^ ((gam + 1) / gam))), ROUND(B7 * B2 * SQRT(MAX(0, term_sv)), 4)))Try the example
Use the original values or change them to test your understanding.