Rate of air pressure drop in a reservoir with a hole to a vacuum

[superiority]

Homework Statement

A space capsule, which may be treated as a sphere of radius 10m, is hit by a micrometeorite which makes a hole of diameter 2mm in its skin. Estimate how long it will take for the air pressure to drop by 30%. Avogadro's number is about 6 \times 10^{23}

Homework Equations

PV^\gamma = k
J = -\D \frac{\partial \phi}{\partial x}
J = -\D\nabla\phi
\frac{\partial \phi}{\partial t} = D \frac{\partial^2 \phi}{\partial x^2}
Flux = -P \cdot A \cdot (c_1 - c_2)

The Attempt at a Solution

No gas would enter. Eventually, all the gas would leave. The conditions of the gas inside the capsule are unspecified; call pressure P, temperature T.

Volume of gas is \frac{4000\pi}{3} \approx 4190 \textrm{m}^3. Surface area of capsule is 400\pi \approx 1260 \textrm{m}^2. Area of hole is \approx \pi \times 10^{-6} \approx 3.14 \times 10^{-6} \textrm{m}^2. Area of hole is \approx 2.5 \times 10^{-7} \% of surface area of capsule.

Pressure outside the capsule is 0, volume of space may be treated as infinite (i.e. gas leaving the capsule will not be obstructed).

I'm pretty sure that the rate of transfer would be an exponential decay.

I'm not sure whether Fick's law for one dimension or many dimensions is appropriate; it's really only a movement in the direction perpendicular to the whole that will result in gas leaving the capsule. I think this is a diffusion problem, but I stuck adiabatic gas expansion law there just in case it's relevant.

I believe what I'm trying to find is \frac{\partial \phi}{\partial t}, the rate of change of concentration with respect to time. \frac{\partial \phi}{\partial t} = -\frac{\partial J}{\partial x} (I think). J is the amount of gas passing through the hole every second, which is 2.5×10^-7% of all gas hitting the sides of the capsule every second.

I'm afraid that from here, I have no clue where to go. I have a hunch that I would integrate something to find the total proportion of gas hitting the sides every second, work out the proportion that would leave through the hole based on that, assume that loss of gas to space results in instantaneous uniform pressure drop throughout the capsule, get some equation out of that and solve for 30% drop in pressure. But I'm stuck. Any pointers?

 

[Filip Larsen]

Perhaps you can determine the mass flow rate through the hole by considering with which very well-known characteristic speed (that is a function of pressure and density) the air would leave the hole with. Having the instantaneous mass rate should allow you to put up a differential relationship for pressure and (if you model the expansion as non-isothermal) temperature.

While I do agree with you that you will most likely end up with an exponential decay, I'm not sure you can get it from just looking at how likely air molecules are to hit the hole instead of the wall, since the mean free path of air at standard pressure is way lower than 10m, so air molecules are far more likely to hit each other than hitting the wall or the hole.

 

[superiority]

P = \frac{1}{3}\rho {v_{rms}}^2
\rho = \frac{m}{V}
P = \frac{m}{3V} {v_{rms}}^2
v_{rms} = \sqrt{\frac{3PV}{m}}
In some amount of time dt, if A_h is the area of the hole, the amount of mass dm that passes through the hole is
dm = A_h\cdot \sqrt{\frac{3PV}{m}}dt
Rearranging:
\sqrt{\frac{m}{P}}dm = A_h\cdot \sqrt{3V}dt
But I can't figure out how to write P in terms of m.

 

[superiority]

Wouldn't \frac{m}{P} actually be a constant? But that doesn't seem right at all.

Edit: Volume, not mass. Dammit.

 

[Filip Larsen]

I am by no means expert here, but assuming the volume of the air expands through the hole with the instantaneous speed of sound (like an expanding piston) and it does so adiabatically (pV^\gamma = p_0V_0^\gamma), I get a differential equation of the form

\dot{V} = A_h \sqrt{\frac{\gamma p_0 V_0^\gamma}{m}} V^{(1-\gamma)/2}

which seems solvable allowing the pressure to be calculated as a function of time. However, this also assumes \gamma is constant, which may be false when temperature varies.

Perhaps others here can provide some better advice on how to model and solve this problem?