Hyperloop - what happens if the evacuated tube ruptures?

TL;DR Summary

Two questions - how fast would the atmosphere travel filling the tube in a rupture, and what happens to a pod that hits that wall of air?

I've been looking at the Hyperloop concept, where a low pressure sealed tube allows a pod to cruise intercity at around 700 mph, and Elon Musk's original whitepaper suggests excavating the track to about one sixth the pressure of the atmosphere on Mars.

Atmospheric pressure on the tubes has been described as 'the weight of a lorry' (I don't like that description, but do not know how to calculate the actual pressure on a segment), but my concern is:

1. If a tube ruptures at some point, how fast would the atmosphere fill up the entire length of the tube?
2. Would the atmosphere rushing back into the excavated tube travel at supersonic speed?
3. If a pod moving at 700 mph were to hit the air filling the tube, how catastrophic would the deceleration likely be?

I appreciate there will be vagaries around any answers, but hopefully an order of magnitude is possible.

 

[Baluncore]

If the rupture was complete and not blocked by wall fragments then the pressure in the tube would become the same as the atmosphere. A wave will travel along the tube at the speed of sound in air at that temperature.

When the wave reaches the capsule it will be reflected back with double the atmospheric pressure being applied to the front wall of the capsule. That will temporarily double the temperature against the capsule, maybe reaching 300°C.

The residual air in the evacuated tunnel will be at ground temperature. But it will be compressed by the introduced air by a greater ratio and so it may reach a much higher temperature.

The air will always travel at the speed of sound. If the air is very hot, a shock wave may form that travels at a high speed and temperature.

The deceleration of the capsule would be very rapid.

 

The deceleration of the capsule would be very rapid.

Thanks, @Baluncore, that's what I figured, likely in order of milliseconds, similar to a head on car crash. The heat on the capsule I had not considered, but the gee force from the deceleration would be extreme, and I'm assuming that seat belts and airbags would not provide sufficient protection for the passengers.

But at least the speed of sound is not so fast that capsules further down the line cannot safely come to a stop, it would only be the ones near to the rupture that would suffer catastrophic damage.

 

[TeethWhitener]

Thanks, @Baluncore, that's what I figured, likely in order of milliseconds, similar to a head on car crash.

I'm not an expert on shock waves, so I'll have to take @Baluncore 's word about the double atmospheric pressure, but the force is straightforward to calculate from that. Force is just pressure per unit area, and assuming double atmospheric pressure and that the front of the capsule has an area of ~10 m³, we get a force of 2 million Newtons on the capsule. According to this wikipedia page, that's roughly the force exerted by the main engine of the Space Shuttle at launch. According to the same page, it's about 20 times greater than the force applied to a passenger by a seatbelt/airbag combo in a 100 km/hr collision.

 

[boneh3ad]

A wave will travel along the tube at the speed of sound in air at that temperature.

Actually, given the pressure ratios in question, it would likely produce shock waves that travel through the tube faster than the speed of sound (relative to the still, rarefied air in the tube).

When the wave reaches the capsule it will be reflected back with double the atmospheric pressure being applied to the front wall of the capsule. That will temporarily double the temperature against the capsule, maybe reaching 300°C.

...

The air will always travel at the speed of sound. If the air is very hot, a shock wave may form that travels at a high speed and temperature.

The deceleration of the capsule would be very rapid.

None of this is true in general. This is, in essence, a shock tube problem. Making a few assumptions (100 Pa air in the tube at 300 K, 101325 Pa air outside at 300 K), then if there is a clean rupture of some kind, the initial shock will travel at roughly Mach 3.15 into the tube. The temperature in the air compressed by the shock will be about 860 K at a pressure of 6981 Pa and with an incoming (to the pod) flow velocity of 821 m/s (so quite hot and fast, but still quite low pressure).

The details of the shock reflection will depend on the movement speed of the hyperloop pod and its front design. I could assume it is a piston moving down the tube to get an easy answer, but that violates one of Musk's central ideas, which is that the front of the pod would ingest the incoming flow. In that case, there may or may not even be a shock reflection.

This is of course a 1D approximation and the actual situation would be a hemispherical shock at the point of rupture. That would propagate down the tube and normalize, then the two shocks would meet (or would meet a solid wall) reflect, and come back to the rupture, further compressing the gas. However, the general order of magnitude should be correct for the initial pod encounter.

So the pod impacts a shock wave and some portion is ingested and some portion reflect. For now I will just assume perfect ingestion (not likely a great assumption but it simplifies things greatly and we have no other data to go on). The pod travels at 313 m/s (700 mph) meets a flow coming at it at 821 m/s while it travels at 313 m/s for a total relative velocity of 1134 m/s. At the compressed temperature, that equates to Mach 1.93. The resulting dynamic pressure is 18.18 kPa. The dynamic pressure before it encountered the shock was about 0.35 kPa.

Basically, it's a 50-fold increase in dynamic pressure and therefore a roughly 50-fold increase in drag. That said, it's still not a huge force. I'd imagine it would be felt as a jolt but nothing catastrophic. Far worse would be encountering whatever debris was leftover for the rupture.

 

[russ_watters]

Thanks, @Baluncore, that's what I figured, likely in order of milliseconds, similar to a head on car crash.

No, if @Baluncore 's calculation is right, that's more like a couple of g's for a 100,000 kg train or a fraction of a g for a heavier one. It's hard braking, not a crash.

And yeah, if there's a sudden hole capable of causing that in rush, there's debris with it.

 

[Baluncore]

None of this is true in general. This is, in essence, a shock tube problem.

I believe the wave that travels ahead through the rarefied air in the tube would be a shock wave, but it will be of a similar magnitude to that encountered by the capsule during normal operation of the tube.
For that reason I was considering only the transmission line effect of the atmospheric bolt of gas that enters the evacuated tube following the rupture.

 

[boneh3ad]

it will be of a similar magnitude to that encountered by the capsule during normal operation of the tube.

What does that mean? The capsule, which is traveling subsonic (albeit high subsonic) will not encounter shock waves during normal operation.

For that reason I was considering only the transmission line effect of the atmospheric bolt of gas that enters the evacuated tube following the rupture.

The atmospheric gas that enters the tube after rupture cannot be decoupled from the effects of the shock. It's the same process. The burst occurs and a shock accelerates and compresses the gas in the tube up to the same velocity and pressure as the air rushing into the tube.

 

[Baluncore]

It comes down to details of the pressure and temperature profile at the front of the atmospheric bolt.
Given the flexible nature of the English language, I am inclined to agree with you.

What does that mean? The capsule, which is traveling subsonic (albeit high subsonic) will not encounter shock waves during normal operation.

I assumed there would be air pinched between the tube and capsule that would involve compression and heating, to the same extent that the rarefied air at the front of the traveling atmospheric bolt would have a leading layer of compressed and heated tube air. That leading layer would radiate, and gradually mix into the following atmospheric wall, which would regulate the temperature profile by speed of sound.

The burst occurs and a shock accelerates and compresses the gas in the tube up to the same velocity and pressure as the air rushing into the tube.

While at the same time … The burst occurs and the leading wall of the atmospheric bolt expands into the tube, cooling and slowing slightly.
The rarefied still air in the tube is compressed and heated by the advancing wall, so part of it can keep ahead of the atmospheric bolt that follows immediately.

 

[mfb]

I'm not an expert on shock waves, so I'll have to take @Baluncore 's word about the double atmospheric pressure, but the force is straightforward to calculate from that. Force is just pressure per unit area, and assuming double atmospheric pressure and that the front of the capsule has an area of ~10 m³, we get a force of 2 million Newtons on the capsule. According to this wikipedia page, that's roughly the force exerted by the main engine of the Space Shuttle at launch. According to the same page, it's about 20 times greater than the force applied to a passenger by a seatbelt/airbag combo in a 100 km/hr collision.

a) that is higher than the actual pressure, as discussed later
b) that is the force on the overall capsule. If the capsule has a mass of 50 tonnes then it leads to a deceleration of 40 m/s², or 4 g. A 100 kg person would experience a force of 4 kN. Downwards this is typical in roller coasters, as "eyeballs out" force it is unpleasant but not immediately dangerous. A three-point seat belt is advisable, alternatively people can sit in reverse. If the capsule is lighter then the acceleration will be larger, but it will stay better than the car crash - even with the overestimated force.

 

[TeethWhitener]

a) that is higher than the actual pressure, as discussed later
b) that is the force on the overall capsule. If the capsule has a mass of 50 tonnes then it leads to a deceleration of 40 m/s², or 4 g. A 100 kg person would experience a force of 4 kN. Downwards this is typical in roller coasters, as "eyeballs out" force it is unpleasant but not immediately dangerous. A three-point seat belt is advisable, alternatively people can sit in reverse. If the capsule is lighter then the acceleration will be larger, but it will stay better than the car crash - even with the overestimated force.

Yes, I didn't include the acceleration calculation because I was too lazy to go looking for the estimated mass of the hyperloop car. I had initially estimated it at no more than 10 tons (typical for a freight train car), which would make the acceleration 20g. Bad, but not necessarily deadly.

 

Thanks for all the thoughts and calculations on this, I had (naively) thought a breach would be a generally catastrophic event for hyperloop, but it seems that unless you're unlucky enough to be right on top of the incident, the system should have time to slow the majority of pods down in time. Whether one ever gets built is another question, but on the basis of safety, I'd be happy to give one a try.

 

[boneh3ad]

I assumed there would be air pinched between the tube and capsule that would involve compression and heating, to the same extent that the rarefied air at the front of the traveling atmospheric bolt would have a leading layer of compressed and heated tube air.

Certainly there would be some of that, but exactly how much depends on the details of the capsule. Musk's original design pitched the idea that the front of the capsule would essentially be a turbine inlet that ingests a lot of the incoming air and uses it to provide some kind of power to the capsule. It was not a well-fleshed-out part of the idea so the details were a bit sketchy. However, if a decent portion of that air is ingested, there would be very little actual compression in front of the vehicle or around the edges.

Air moving around the edges generally would speed up, not slow down, which means it would not be compressing but actually becoming more rarefied. Now, if that acceleration was enough to produce locally sonic (or supersonic flow, if the capsule shape supported it), then you could end up with shocks that locally compress the flow, but the overall effect would be a rarefaction.

That leading layer would radiate, and gradually mix into the following atmospheric wall, which would regulate the temperature profile by speed of sound.

It is not clear to me what you mean by this. Yes, "information" about the moving vehicle travels upstream and downstream of it at the speed of sound, but for a constant-speed vehicle, this is a steady, continuous process. You would have a more or less steady and smoothly-varying compressed region in the front of the vehicle. The unsteady portion of the problem is when it would encounter the shock generated by the rupture, but given the ingestion of the air by the vehicle, the effect of this leading compressed region on my previous rough calculations is likely to be fairly minor.

While at the same time … The burst occurs and the leading wall of the atmospheric bolt expands into the tube, cooling and slowing slightly.
The rarefied still air in the tube is compressed and heated by the advancing wall, so part of it can keep ahead of the atmospheric bolt that follows immediately.

Mass can only pass through a given sized hole with a given pressure ratio and temperature so fast. That "atmospheric bolt" is not going to be at atmospheric pressure or temperature because it had to accelerate to that given speed, which necessitates a temperature and pressure drop. The atmosphere is effectively infinite volume compared the the volume of the tube, so the slowing effect will be completely negligible and what is produced is a quasi-steady expansion into the tube once the initial wave system clears the hole and before any of those waves return.

As I stated before, the shock generated by the burst will accelerate the air in the tube to exactly the same velocity as the bolt of air coming into the tube from the atmosphere. There will still be a so-called contact surface between those two air masses across which velocity and pressure are constant but temperature is not.

Below is some figures illustrating the conditions in the simple shock tube approximation I used before (description below figure):

So basically, to describe that process, the two gases are initially at 300 K and separated by a diaphragm (the tube wall in the case of hyperloop) as shown in the top figure. The diaphragm bursts causing a shock (red) to propagate into the tube and an expansion wave (blue) to set up in order to accelerate air into the tube (second figure). A contact surface forms (gray) and travels downstream slower than the shock. Across the contact surface, pressure and velocity (figures 3 and 5) are constant but temperature (figure 4) varies due to the different reservoir condition and acceleration method from which each region was derived. That temperature interface will blur over time as diffusion (i.e. conduction in this case) occurs, but we are ignoring that effect here.

 

[Vigardo]

While at the same time … The burst occurs and the leading wall of the atmospheric bolt expands into the tube, cooling and slowing slightly.

I just wanted to add here that the expansion of the atmospheric air entering into the tube would be considered at some point as an adiabatic expansion, and so, it would cool enough to condense water if the air is sufficiently wet. This mist may contribute to further reduce visibility in case of such a catastrophic event and it should be accounted for by emergency services. (Watch out Elon ;-)