Engineering & Science
Rayleigh Flow Thermal Choking: P/P*, T/T* & Stagnation Temperature Ratio (T0/T0*)
Calculate static pressure ratio P/P*, static temperature ratio T/T*, and stagnation temperature ratio T0/T0* for compressible frictionless duct flow with heat addition (jet afterburners and ramjet combustors).
Model thermal choking in jet combustors, afterburners, and laser-heated ducts!
Adding heat q = cp * (T02 - T01) to a constant-area, frictionless compressible gas duct (Rayleigh flow) drives BOTH subsonic flows (M < 1) AND supersonic flows (M > 1) toward sonic Mach M = 1—where stagnation temperature reaches its maximum choked limit T0*! Attempting to add heat beyond T0* thermally chokes the combustor, spilling a shock upstream or reducing mass flow!
For an ideal gas with specific heat ratio gamma (1.4 for air) relative to sonic reference state (*) at M = 1: static pressure ratio is P / P* = (1 + gamma) / (1 + gamma * M^2), static temperature ratio is T / T* = (((1 + gamma) * M) / (1 + gamma * M^2))^2, and stagnation temperature ratio is T0 / T0* = ((1 + gamma) * M^2 * (2 + (gamma - 1) * M^2)) / (1 + gamma * M^2)^2! Notice a remarkable Rayleigh-line anomaly between M = 1/SQRT(gamma) (= 0.845) and M = 1: adding heat actually COOLS static temperature T even while raising stagnation temperature T0!
| Combustor Regime (gamma = 1.4) | Mach Number M | Pressure Ratio P / P* | Stagnation T0 / T0* & Static T / T* |
|---|---|---|---|
| Subsonic Ramjet Combustor | M = 0.50 | P / P* = 1.7778 | T0 / T0* = 0.6914 | T / T* = 0.7901 |
| Thermally Choked Throat | M = 1.00 | P / P* = 1.0000 | T0 / T0* = 1.0000 (Max Heat Limit!) |
| Supersonic Scramjet Burner | M = 2.00 | P / P* = 0.3636 | T0 / T0* = 0.7934 | T / T* = 0.5289 |
Rayleigh Stagnation Temperature Ratio T0 / T0* (Thermal Choking Fraction)
Computes T0 / T0* showing what fraction of the sonic choking stagnation temperature the flow currently carries.
=IF(OR(B2 <= 0, B3 <= 1), "Invalid Rayleigh Inputs", ROUND(((1 + B3) * (B2 ^ 2) * (2 + ((B3 - 1) * (B2 ^ 2)))) / ((1 + (B3 * (B2 ^ 2))) ^ 2), 4))Rayleigh Static Pressure Ratio P / P* & Static Temperature Ratio T / T*
Evaluates static pressure ratio (1 + gamma) / (1 + gamma * M^2) relative to sonic state P*.
=IF(OR(B2 < 0, B3 <= 1), "Invalid Rayleigh Inputs", ROUND((1 + B3) / (1 + (B3 * (B2 ^ 2))), 4))Try the example
Use the original values or change them to test your understanding.