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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

sonic 0.528 :

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 ≡ x downstream/uptream pressure ratio dimensionless ≡ 91 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 Ψmax sonic Ψmax + 40 black 1 5 mat 3 1 sys

ρv 6.153

Qmax Ao 2 Pv * ρv * sqrt * : 5.26217

x Q 0 x ≤ (x sonic ≤ (& (Qmax 1 Ao 2 x * ρv * x (1 k /^* sqrt * - (* sonic 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 :

sonic Q 4.728261

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

x Q 0.25 - Qmax sonic Qmax 2 2 mat sonic sonic Q 1 sonic Q 2 2 mat 0.528 sonic Q + 50 black 1 5 mat 0.125 6.25 SuperSonic 12 red 1 5 mat 0.75 4.75 SubSonic 12 green 1 5 mat 0.125 2 Air Discharge kg/sec wrt x 12 blue 1 5 mat 7 1 sys

Consider installed "C" discharge coeff's. For metering purpose, use specially designed devices from suppliers ... limit 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 [±].

MassFlow x 01 0.005 range : i 1 x rows range Q i el x i el Q eval : for x Q augment 31 line :

CWD C:\SmathFile CurrentDirectory :

MassFlow CWD MassFlowIsen wfile 1

Total CWD MassFlowIsen rfile :

data Σ Finite diff's Cumul Integration vx data 1 col : vy data 2 col : Y 1 el 0 : i 2 vx rows range Δx i el vx i el vx i 1 - el - : Y i el Y i 1 - el 12 / vy i 1 - el vy i el + (Δx i el * (* + : 21 line for vx Y augment 61 line :

Total Σ sonic 5 sonic 1 - 2 2 mat 1 4.32 . 14 black 1 5 mat 0.75 5.5 4.32 Σkg 12 black 1 5 mat 0.625 3 ~ 2.63 kg 12 black 1 5 mat sonic 2.63 + 40 black 1 5 mat 6 1 sys

Total Σ Total Σ rows row 1 4.32 12 mat