อัปเดต 2026-09-07

Compressible pipe flow

Crane TP-410 §1-9 Eq 1-7 isothermal

Sizes/rates compressible single-phase gas pipe flow (isothermal Crane method, plus Weymouth/Panhandle A/B pipeline correlations) with a Peng-Robinson real-gas Z-factor correction.

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Crane TP-410 Eq 1-7 isothermal-compressible-flow momentum balance, iterated 2–3× on Z at the mean pressure; choking checked against the standard critical-pressure-ratio criterion; Z from the Peng-Robinson (1976) cubic EOS solved in closed form via Cardano's method.

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Isothermal compressible flow (Crane Eq 1-7)

m ˙^{2} = \frac{( P _{1}^{2} - P _{2}^{2} ) A ^{2} M W}{Z _{a v g} R T [ f \frac{L}{D} + 2 ln \frac{P _{1}}{P _{2}} ]}

P1,P2 = absolute upstream/downstream pressure; A = flow area; MW = molecular weight; Z = mean compressibility; f·L/D = friction term; 2ln(P1/P2) = kinetic/acceleration term

Peng-Robinson EOS (compressibility Z)

Z^{3} - (1 - B) Z^{2} + (A - 3 B^{2} - 2 B) Z - (A B - B^{2} - B^{3}) = 0

A = aP/(RT)², B = bP/(RT); a,b from Tc, Pc, acentric factor ω; solved via Cardano's cubic formula for the vapour root

Choked-flow criterion

\frac{P _{2}}{P _{1}} < (\frac{2}{k + 1})^{\frac{k}{k - 1}}

k = adiabatic index (cp/cv); ≈0.528 for air (k=1.4), ≈0.546 for natural gas/steam (k=1.30)

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