The engineering to do a 5 mile section is pretty straightforward. It's only 5 miles long (25000 feet) and around 250 joints.
1. Giant foundations and just handle the thermal stress by not letting anything move
2. Figure out the slip joints really well to soak up the ~125 feet of travel and still hold a good vacuum
3. Figure them out OK and just install extra vacuum pumps since there are only ~250 joints
4. Try out some/all of these options on 500 feet of tube in parallel to see how it all performs and don't make a final decision on the whole 5 miles until you have real cost numbers
The other thing I'll mention is that you don't need the steel to be continuous in order to hold a vacuum. You need the inside face of the tube to be smooth in order to not jerk around the vehicles, but all the sealing could be done on the outside with clamp-on seals. If the average continuous tube piece is 100 feet long and the max thermal expansion is 0.5% then you only need a half-inch gap between the tube pieces.
If your air bearings are say 3 feet long each and divided into 10 sections internally which are fed through orifices so that no one section can rob all the pressurized air flow then you're never going to lose more than 10% of your bearing force and you should be able to glide right over these 1/2" gaps with no problems. And if there are some problems a few accumulators (plain air tanks or pressurized bladders) inline with the supply lines would probably increase the momentary recharge capabilities enough to negate the problem.
700mph is 1000 feet per second or 12 inches per millisecond. That means a 1/2" gap is crossed in just 40 microseconds or so.
1. Giant foundations and just handle the thermal stress by not letting anything move
2. Figure out the slip joints really well to soak up the ~125 feet of travel and still hold a good vacuum
3. Figure them out OK and just install extra vacuum pumps since there are only ~250 joints
4. Try out some/all of these options on 500 feet of tube in parallel to see how it all performs and don't make a final decision on the whole 5 miles until you have real cost numbers
The other thing I'll mention is that you don't need the steel to be continuous in order to hold a vacuum. You need the inside face of the tube to be smooth in order to not jerk around the vehicles, but all the sealing could be done on the outside with clamp-on seals. If the average continuous tube piece is 100 feet long and the max thermal expansion is 0.5% then you only need a half-inch gap between the tube pieces.
If your air bearings are say 3 feet long each and divided into 10 sections internally which are fed through orifices so that no one section can rob all the pressurized air flow then you're never going to lose more than 10% of your bearing force and you should be able to glide right over these 1/2" gaps with no problems. And if there are some problems a few accumulators (plain air tanks or pressurized bladders) inline with the supply lines would probably increase the momentary recharge capabilities enough to negate the problem.
700mph is 1000 feet per second or 12 inches per millisecond. That means a 1/2" gap is crossed in just 40 microseconds or so.