Peak differential pressure Δpmax (A.30)
–
–
Venting time tc (A.43)––
Time ratio K = tc/tp (A.54)–
Sonic (choked) phase (A.61)––
Design load FS·Δpmax––
Ascent snapshot
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t = 0 s
Pressure history
Compartment vs fairingDifferential pressure Δp(t)
Positive = burstingGas density
Compartment ρ0 vs fairing ρeGas temperature
Implied by the polytropic law, Eq. (A.13)Sonic-phase check r = pe/p0
Choked when r ≤ rcrDesign chart δmax vs K
Gaussian profile, same n and γVent sizing
Set an allowable Δp in section 5Detailed results
Equation numbers refer to A. Pagani et al., Appunti di Strutture per Veicoli Spaziali, PoliTo, 2026| Quantity | Symbol | Value | Ref. |
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Model assumptions
- Zero-dimensional (lumped) compartment: uniform p, ρ, T; rigid walls (constant V); single equivalent vent to the fairing (Eqs. A.2–A.7).
- Polytropic closure p/ρn = const (Eq. A.9), in both models. n = γ (isentropic) and n = 1 (isothermal) bound the real behaviour. The isothermal case gives the higher peak Δp (up to about 40 % more in the UPM-Sat 1 cases; Δpmax ∝ 1/n when K ≪ 1), so check both, or use n = 1, for design.
- Fairing pressure is imposed: Gaussian law of Eq. (A.53), or the user table. Real profiles should come from the launcher user's manual.
- Pagani & Carrera, AASS, 2016 model, compressible orifice flow with choking: quasi-steady isentropic flow through the vent, choked (sonic) or subcritical (Eqs. A.22/A.24), with S/√ξ as effective area. Valid at any pressure ratio. Reverse flow (pe > p0) is handled with the fairing gas as upstream.
- Sanz-Andrés et al., JSR, 1997 model, incompressible orifice loss law (small Δp): quadratic loss Δp = ½ρ0Uh2ξ with constant density across the vent (Eqs. A.32–A.36). Only the vent flow law is incompressible; the gas in the compartment still expands polytropically. Valid for small pressure differences. It has no sonic limit and can predict vent velocities above the speed of sound, so once the vents choke it overestimates the outflow and underestimates Δp. For small Δp it coincides with the compressible orifice model.
- Temperatures are not physical predictions. T0 = Tinit·ρ̄n−1 is what the polytropic hypothesis implies, not the result of an energy balance. With n = γ it is a reversible adiabatic expansion, the coldest the gas could get. In reality heat from the walls and equipment (large thermal mass, venting over tens of seconds), humidity, condensation or freezing, viscous losses and incomplete mixing keep the gas warmer, between the isentropic and isothermal values. The fairing curves are only an assumption: the fairing gas is taken to follow the same polytropic law, while its real temperature depends on the launcher (including heating through the fairing wall). This law is also used for the inflowing gas in reverse flow.
- Not modelled: outgassing, heat exchange beyond the polytropic law, acoustic/wave effects (valid only if the acoustic time ≪ tc), multiple interconnected compartments (Eq. A.14), and wall flexibility.