Sled Velocity Gradient
As a vehicle sled accelerates, the optimum ratio of sled magnet pitch to rotor pitch changes with velocity. When the vehicle is moving slowly, the ratio changes significantly over a sled length. How important is this?
The launch loop will be optimized for the most important task, extracting energy from the rotor for vehicle acceleration near the exit end of the acceleration track. If the vehicle is moving eastward at 10 km/s, and the sled is 50 meters long (WAG), it moves 200 sled lengths in one second. Accelerating at 30 m/s², it accelerates 0.15 m/s per sled length travelled, 15 ppm of the vehicle velocity. The optimum pitch of the sled magnets in the center Halbach array is very close to the same in front and back.
Note: the magnet sled rear end exciter and front-end de-exciter will have different pitches, designed to induce and suppress rotor circumferential fields entering and leaving the center zone, exact design TBD.
If the power needed to provide 200 kN of acceleration force ( 30 m/s², 5 tonne vehicle, 1.67 tonne sled ) at 10 km/s is 2 gigawatts. If slip is 50 m/s / 12.5 km/s or 0.4% (WAG), the rotor heating will be 8 MW, added to a rotor passing through at a relative speed of 2500 m/s; 3200 J/m . If the rotor is 0.64 kg/m of aluminum winding, that is 5000 J/kg instantaneous heating of the rotor. The specific heat of aluminum is
If the power needed to provide 200 kN of acceleration force ( 30 m/s², 5 tonne vehicle, 1.67 tonne sled ) at 10 km/s is 2 gigawatts. If slip is 50 m/s / 12.5 km/s or 0.4% (WAG), the rotor heating will be 8 MW, added to a rotor passing through at a relative speed of 2500 m/s; 3200 J/m . If the rotor is 0.64 kg/m of aluminum winding, that is 5000 J/kg instantaneous heating of the aluminum windings of the rotor. The specific heat of aluminum is 910 J/Kg-K, so the windings will flash-heat by 5.5°C, share their heat with the iron rotor over tens of seconds, then radiate that heat to the enclosing track over minutes. Most of the track is at 80 km altitude, where the ambient atmospheric temperature is 200K; with suitable sun-shielding the noontime temperature of the track can approach 200K or -70°C. Even with eight vehicles spaced along the track, the rotor may remain below 0°C.
The situation is different near the beginning of the acceleration run, and some compromise will be required. At 1 km per second, the vehicle is moving only 20 sled lengths per second, and accelerates 1.5 m/s per sled length travelled, 1500 ppm of the vehicle velocity. The optimum pitch changes significantly over one sled length. This may reduce rotor-to-vehicle power transfer efficiency, TBD. At 1 km/s, the acceleration power is 200 MW. If the slip is 0.4%, the rotor heating is 800 kW. The rotor passes the vehicle at 11.5 km/s, so the heating is 70 J/m, 109 J/kg aluminum, and 0.12°C flash winding heating. Working the other way, if we can tolerate flash heating of 5.5°C near the beginning, we can tolerate 45 times as much heating, 18% slip, or 2300 m/s(!) slip velocity in the rotor, absurdly high. The 1.5 m/s acceleration is a small fraction of that.
So - for now, the unanswered question is how we can exploit this "inefficiency tolerance" at the beginning of the acceleration run to simplify and bullet-proof the first 100 kilometers of the acceleration track. Another degree of freedom is that the initial acceleration can be low - if the first hundred kilometers accelerate at only 7.5 m/s² (50 kN acceleration force), resulting in a transition speed of 1225 m/s (instead of 2450 m/s for 30 m/s²), the length of the launch loop will increase by 75 km, but the design task will be much simpler. This first 100 km will be on the 480 kilometer arc from 30 km west station to 80 km full altitude; we will slow acceleration on this stretch anyway, to reduce aerodynamic heating in the stratosphere.
The very first portions of the track, between rest and a few hundred meters per second, may require a different approach than a velocity transformer; perhaps "100% slip" drag on the rotor, aided by power conversion through an electronically switched mass driver. At 7.5 m/s² and a transition velocity of 300 m/s, the first 40 seconds of acceleration may traverse 6 kilometers, with transition power levels of 15 megawatts, or 300 kilowatts per meter of sled (this ignores a few percent extra power for a 12° uphill climb from a 30 km high west station on a 2350 km radius curved incline rising from a 15 degree surface ramp).
If acceleration for this first section is completely supplied by a mass driver, the total instantaneous switched power would be 900 MW ... with a very low duty cycle, but instantaneous shock heating and voltage stress in the power switching semiconductor devices will be the design driver, not average system power. Scaling from this pessimistic analysis, the cost of the switching devices may be around $200 million, very high, but trivial compared to the absurd 50 trillion dollar cost of an end-to-end mass driver launching 5 tonne payloads through a magic self-levitating vacuum tunnel through the atmosphere.
That said, I would rather spend $200 million on schools and libraries, so if we can save capital cost at the expense of efficiency, perhaps we can downscale or eliminate the mass driver.
One of the first applications of a launch loop will be launching space power satellites, and most of those will power mid-ocean applications like hydrogen fuel synthesis, aluminum refining, and more launch loops. Overly optimistic SBSP satellite designs assume perhaps 8 kg/kW. If we pessimistically assume 40 kg/kW, a launch loop can launch its own power supply in less than a month. Front end efficiency may reduce system construction cost, but will have no significant effect on launch energy cost.
Many opportunities for optimization, by engineers more clever than I am.