Estimate speed of wind through a hole in a spaceship hull

Question

Suppose that the pressure inside the ship is 1 atm and that the ship is so big relative to the hole size that escaping air doesn't change the pressure much over a short time. How do I estimate the speed of the wind through it?

Or, equivalently (I believe), how to compute the speed of air flow through an orifice between two volumes of constant but very different pressure?

My first thought is to use Bernoulli's equation, but as far as I can tell it is inapplicable because we cannot ignore compressibility of air here (since it disperses completely on the space side).

Answer 1

Your are right, as the air approaches the hole the speed is great enough to change the density, so it is compressible flow. If the hole was a nicely shaped nozzle you would calculate:

$${\dot {m}}=C_{d}P_{0}A{\sqrt {\frac{\gamma}{RT_0} \left({\frac {2}{\gamma +1}}\right)^{\frac {\gamma +1}{\gamma -1}}}}$$ Where: $\dot m$ is the mass flow rate, $C_d$ is the discharge coefficient, $P_0A$ is the product of pressure (away from the hole, called stagnation pressure where the speed is very low) and hole area, $\gamma $ is ratio of specific heats (use 1.4 if air at reasonable temperature), $R$ is gas constant for air ($286J/kg\cdot K$) and $T_0$ is temperature in Kelvin (Celsius plus 273.15).

The Wikipedia article (https://en.wikipedia.org/wiki/Choked_flow)brings up a fact I had not seen before: flow through a thin oriface does not choke, but I believe the above equation with $C_d$ of 1. would be the limiting amount of flow.