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Isentropic discharge [Air example]

k 1.4 : Air Isentropic coefficient T 293 K ≡

Ψmax k k 1 - / sqrt 2 k 1 + / ln k k 1 - / * exp (2 k /^2 k 1 + / ln k k 1 - / * exp (k 1 + k /^- sqrt * :

Ψmax 0.484

x Ψ 0 x ≤ (x 0.528 ≤ (& (Ψmax 0.528 x ≤ (x 1 ≤ (& k k 1 - / sqrt x 2 k /^x k 1 + k /^- sqrt * if if :

T 293 : Air temperature 293 K ≡ ρo 1.293 : Air kg/m^3 STP ≡ Ao 0.05 : Area orifice m^2 ≡ Pv 900 : Vessel Pressure kPa ≡ p2 101.325 : Outlet backup P kPa ≡ ρv 1.293 Pv 101.325 / * : Static Air ρv kg/m^3 ≡ ρv ρo Pv P2 / (1 k /^* : Dynamic kg/m^3 ≡ x Q Ao 2 Pv * ρv * sqrt * x Ψ * : MassFlow rate kg/sec ≡ 81 sys

Isentropic Discharge Coefficient wrt p1/po where p1 =inlet P , po = outlet P p1/po => choke flow pressure ratio @ sonic velocity for Air @ 293 K ... p1/po = 0.528

x Ψ 0.484 0.528 0.484 . 15 black 1 5 mat 3 1 sys

ρv 11.485

Qmax Ao 2 Pv * ρv * sqrt * :

Qmax 7.189

x Q 0 x ≤ (x 0.528 ≤ (& (Qmax 1 Ao 2 x * ρv * x (1 k /^* sqrt * - (* 0.528 x ≤ (x 1 ≤ (& Qmax 1 Ao 2 x * ρv * x (1 k /^* sqrt * - (* k k 1 - / sqrt x 2 k /^x k 1 + k /^- sqrt * Ψmax / (* fully discharged if if :

0.528 Q 6.192509

7.189 6.193 - 0.996

Q[kg/sec] MassFlow Isentropic Discharge wrt the measured pressure ratio P2 [backup outlet]/P1 [initial reservoir].

x Q 7.189 0.528 6.193 + 30 red 1 5 mat 0.5 8 Initial Qmax 15 green 1 5 mat 0.125 7 SuperSonic 12 black 1 5 mat 0.75 4 SubSonic 12 black 1 5 mat 0.125 5 Air Discharge kg/sec wrt x 12 blue 1 5 mat 7 1 sys

Supersonic discharge vs linear

x Q 7.189 1.887 x * - 0.528 6.193 + 30 black 1 5 mat 0.275 6.75 Choked flow 12 blue 1 5 mat 4 1 sys

Consider installed "C" discharge coeff's. For metering purpose, use specially designed devices from suppliers ... limt Qmax by Ao. It is easy to "meter for tonnage" by FD integration wrt "x" pressure ratio, close the valve on the actual measured P2/P1 => 'x' A metering "Flow Nozzle" C = 0.99 [±].

0 0.528 21 mat Qmax 0.528 Q 21 mat Slope 1.887 -

0 0.528 21 mat Qmax 0.528 Q 21 mat Intercept 7.189