Low-Pressure Tube Pod Lab

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Low-Pressure Tube Pod Lab

A pod in a partial-vacuum tube still pushes air through a narrow annular gap. Raise the speed or the pod's blockage and that gap chokes at the Kantrowitz limit; the cutaway shows the density field in the gap and the air column that builds up ahead of the pod. Inspired by the publicly published 2013 concept of tube-borne pods (the 'Hyperloop alpha' white paper).

Presets

Pod and tube

Tube pressure

Blockage ratio

Pod speed

Onboard compressor ingests the excess air

Compressor ratio

Curve radius

Bank angle

Line length

Power budget

Live readouts

Curve comfort

Graphs

Experiments

1: M_K for 14 blockage ratios from three routes: the closed-form A/A* relation, a numerical mass-flux maximisation, and a bisection of the simulation's own choke test. 2: drag power at the current speed over four decades of tube pressure with a log-log fit.

Run an experiment.

The science

Drag in thin air. At low Mach number the pod's aerodynamic drag is F = 1/2 rho C_d A v² with density from the ideal gas law rho = p / (R T). In a tube at 100 Pa (about 0.1 % of sea-level pressure) rho is about 1/1000 of normal, so for the same speed the force is about 1000 times smaller. The model uses C_d = 0.25 for the slender pod, amplified by the blockage factor 1 + phi/(1-phi)^2, and a compressibility term through the stagnation pressure: F = C_d,eff A_pod (p₀ - p).

Blockage and the Kantrowitz limit. Seen from the pod, tube air flows toward it at speed v and must squeeze through the annular gap of area A_gap = A_tube(1 - phi). With no shock and no losses (isentropic flow, same stagnation conditions) the gap flow reaches Mach 1 when

A_tube / A_gap = 1/(1-phi) = (1/M) [ (2/(gamma+1)) (1 + (gamma-1)/2 M^2) ]^((gamma+1)/(2(gamma-1)))

The Mach number M that solves this is the Kantrowitz limit M_K (gamma = 1.4). Example: phi = 0.36 gives M_K = 0.409, about 505 km/h at 20 C. Above it the gap cannot carry the whole mass flow rho v A_tube; the maximum it can pass is the choked flow m* = A_gap p₀ sqrt(gamma/R) (2/(gamma+1))^((gamma+1)/(2(gamma-1))) / sqrt(T₀).

Shock swallowing. For a supersonic approach a normal shock forms and the flow behind it needs the gap to pass the same mass flow at a lower stagnation pressure: (1 - phi) >= 1 / ((A/A*)(M) sigma(M)), sigma being the stagnation-pressure ratio across the shock. For realistic blockage this needs M of several; the sliders stop at M = 1.14, so a pod above the limit here is always choked.

Above the limit. The excess air cannot get by, so the pod behaves like a leaky piston: a column of compressed air builds up ahead. In the steady pod-frame picture the nose stagnation pressure must rise until the choked gap passes everything, p_0f = m_in sqrt(T₀) / (A_gap Phi), and the pod pays the extra (p_0f - p₀) A_pod. The column front is a shock moving at c = a sqrt(1 + (gamma+1)/(2 gamma) (p_r - 1)) in the tube frame; the pressure gradient quoted is (p_0f - p) / L_col with L_col growing as the front pulls away from the pod. Real columns are unsteady and 3D, so this is a lower bound on the drag penalty.

Compressor option. A fan at the nose ingests the excess mass flow m_c = m_in - m*. Ideal compressor power is P_c = m_c c_p T₀ (pi_c^((gamma-1)/gamma) - 1) / eta, with the default pressure ratio 1.5 and eta = 0.8 as illustrative values. The exhaust is not credited as thrust.

Power and energy. Cruise power is P = F v + P_c. Tube heating: every metre of tube receives the drag work F joules per pass; the 'steel' figure divides it over a 20 mm steel wall (about 1970 kg/m), the 'air only' figure over the tube air (a worst case, since thin air conducts heat poorly). Stagnation temperature rise at the nose is v^2 / (2 c_p).

Curves. Centripetal acceleration a = v^2 / R. On a bank of angle theta the passengers feel a cos(theta) - g sin(theta) sideways and g cos(theta) + a sin(theta) vertically. A perfectly banked curve has tan(theta) = a/g and no lateral force, but then the felt vertical load is sqrt(1 + (a/g)^2). Comfort bands used here: lateral felt 0.10 g good, 0.15 g tolerable.

Vacuum. The reversible isothermal work to pump a volume V from atmospheric pressure p_a to p is W = V [p_a - p (1 + ln(p_a/p))], almost independent of the final p because most of the air leaves in the first stage; real pumps cost several times more (pump efficiency 35 % assumed) and leaks need continuous pumping.

Sources and simplifications. Kantrowitz (1945) for the choking and shock-swallowing criterion; the 2013 'Hyperloop alpha' white paper for the compressor concept and tube-pod geometry; Braun et al. (2017) for the pod-in-tube flow regimes. Quasi-1D flow, constant temperature 293 K, no boundary layers, no pod motion dynamics, no Mach-wave reflections off the tube, no real-gas effects at very low pressure (the continuum assumption holds down to roughly 1 Pa for a 4 m tube). Drag constants and compressor parameters are illustrative; this is a teaching model, not a design tool.

Independent educational model, not affiliated with or endorsed by any company named here; trademarks belong to their owners. Numbers are textbook-level/publicly reported approximations.