Hyperloop heat balance

Key points:

• The Hyperloop proposal implies a steam tank almost as large as the entire vehicle, and at least 3200 kg, which is not accounted for in the Hyperloop mass budget.  This is a design breaker.
• The capsule as configured is using most of it's power to drive a refrigerator which cools the air in the tube.  This refrigerator is not mentioned as such.
• The refrigerator, as configured, will not cool the tube air back to it's ambient temperature.  But refrigeration is unnecessary.

I understand that Elon commutes from Los Angeles to San Francisco every week, and that the high speed rail project will not help him.  I imagine that his frustration with this commute is exactly what has kept bringing his attention back to this Hyperloop thing for so long.  I'm sure the boards at both SpaceX and Tesla have threatened to fire him if he starts another high-risk startup.  The guy has a tough life.  :)

The Hyperloop proposal has one big idea that I like: the air bearings, rather than the usual magnetic levitation.  Air bearings are a well developed technology, and require much less capital in the track.  I had not realized that air bearings were a feasible idea for high speed transport, as this is the first I've read about their air requirements at high speed.  (I called three air bearing companies to verify Elon's numbers, and none of them could offer any guidance on air consumption or even stable operation at near-sonic velocity.)  Elon is implying a 2000:1 lift/drag ratio for the passenger version, and a 2500:1 lift/drag ratio for the vehicle version.  That's incredibly good.  Maglev lift/drag ratios are generally rise to 200:1, which is considered really good.  The Livermore maglev requires substantial forward speed before it gets that efficient.  If Elon's numbers are real then air bearings are awesome... but I'd like to see some evidence.

I don't quite understand Elon's bearing drag power numbers.  Drag power is usually just vehicle speed times the drag force.  He's got an extra factor in there which adds about 10%, and I'm not sure where that comes from.  It's not compressor power, because that is much greater.  If we add the compressor power to the drag power, we get L/D ratios of 426:1 and 634:1 for the two configurations.  Those are still game changing numbers.  These numbers matter because if the air demand of the bearings is higher, the compressor power will increase.  If the compressor power increases too much, the battery will get too big and the Hyperloop idea will not work.

The Hyperloop proposal has another big idea that I don't understand: the Kantrowitz limit.  It's clearly a big problem, because the drag coefficient implied by Elon's numbers here is 3.95.  (That number is suspiciously close to an integer, and suggests that the drag numbers are back-of-envelope and not the result of simulation.)  Compare that to a Tesla S (0.24), or a bullet (0.29).  The only saving grace is that his predicted aerodynamic drag (910 N for the vehicle-carrying capsule) isn't too much more than the bearing drag (187 N), even if you include the prorated compressor power in the latter number (740 N).

At one point, Elon insists that propulsive power should be delivered through the track, rather than from the vehicle.  But his propulsive power (628 kW) is less than the compressor power (868 kW) that the vehicle is already signed up to provide.  It might be simpler to just let the vehicle do the whole job and let the track be passive.  He's already suggested using huge battery packs to deliver the propulsion power, so the only change is to put those packs on the vehicle.  This is a nit, let me get on to my main point.

The proposed bypass refrigerator scheme is fascinating but flawed.  Here's the flow diagram for the vehicle-carrying Hyperloop.  There is a typo: the air coming out of the second compressor should be at 594 K, not 59 K, assuming that second compressor is 72.4% efficient like the first one is.

There are much bigger problems than typos here.

• The air in the tube is not going to be 19 C.  Those Hyperloop capsules dissipate 868 kW of compressor power and 285 kW of propulsive power.  As drawn above, that propulsive power is going to be primarily dissipated into the air in the tube, heating it tremendously.  There is also 59 kW of bearing drag that will be dissipated into the tube wall and somewhat into the air.
• As drawn above, the intercooler is actually acting as a refrigerator.  The air expanded out the back would leave at around 126 Kelvin... 147 degrees C below freezing!  This will suck 209 kW of heat out of the tube air (49 kW for the passenger-only version).
• The intercooler refrigerator does not balance the propulsive drag heating.  Combining the two, I find that the vehicle will still dump 76 kW into the air, heating the air in the tube by 24 C (35 C for the passenger-only version).  By increasing the intermediate pressure and intercooler heat load a bit, this problem can be fixed.
• There is no way the output temp of a lightweight intercooler will be within 7 C of the incoming water temperature.
• The water is absorbing 852 kW, or 2185 J/g.  The water flow rate should be 327 g/sec, not 390 g/sec, to fully boil the water at normal atmospheric pressure.  But that's wrong too...
• To make the steam tank reasonably small, the steam will have to be held at high pressure.  Boiling 20 C input water at 450 psi would absorb 2800 J/g and produce steam at 507 K.  The water rate would then be 304 g/s, the tank would be 638 kg and 42.5 m³.  That's clearly far too large (it's as big as the whole vehicle), but that's the largest pressure my steam table goes to.  Even if the steam is heated to 857 K (supercritical), I don't think the steam tank will be small enough to work.
• The steam tank will be heavy.  Assuming carbon composite overwrap construction and 100 MPa working strain, the steam tank will weigh 3200 kg.  Changing the steam temperature and pressure conditions will vary this a little, but not much.

As a first step, I would take the bypass air directly from the axial compressor, which will remove about 80% of the heat load on the intercooler.  A 5x reduction in the size of the steam tank will get the mass budget back under control, but the steam tank will still be awkwardly large.  As a bonus this should deliver 70% more thrust, but that's still too small to matter much.

It should be possible to dump a fair bit of compressor heat from a radiator on the surface of the capsule.  I think this can reduce the heat load on the intercooler and steam tank by another factor of two, which should bring the steam tank volume into a more reasonable range.  It will still be a major component in the vehicle design.

As a next step, the vehicle can be slowed, perhaps by 10%, which allows some of the air in front of the vehicle to actually accelerate up to the alpha proposal mach .91 number as it goes around.  The compressor inlet and mass flow can then be downsized since it need only handle a portion of the airflow.  I think it should be possible to cut the compressor power numbers in half this way.

In my next post I'll work the numbers on these suggestions.