OK, speaking only of lower altitude, low-to-moderate speed (non QED-ARC) operation, let's assume that we have a heat exchanger that hermetically separates a REB, in vacuum, from the flowing air (+ any propellant), but still transfers enough thermal energy into the flow to provide sufficient thrust for lift, hover and forward acceleration until QED-ARC territory is reached. Also assume that the REB is magnetically or electrostatically diffused or defocused and that the intercepted power per unit area is low enough to avoid disintegrating the heat exchanger.
No good. Relativistic electrons are going to spall atoms off the surface no matter how diffuse they are. You have to thermalize the energy somehow, for instance by firing the beam at a fluid rather than at a solid.
Also, the heat exchanger is going to consist of a lot of little tubes and/or fin-type structures, in order to maximize surface area. Directly REB-heating the inside of the metal structure evenly is probably close to impossible.
While I've not done any calculations (it's been many, many years since thermodynamics class, and I've never used it since), intuitively it seems like such a heat exchanger would have to be, at a minimum, white hot, more likely blue (UV?) hot, to get sufficient energy into the flow to lift/propel a massive vehicle.
If you want huge power and thrust output, you have to make the engine bigger. The temperature won't be that high - remember that the turbine blades have to be able to take it.
The catch is that while the turbine blades can be cooled, it's silly to try to cool a heat exchanger. On the other hand, the turbine is under more stress, so creep is a bigger problem...
So I'm wondering, what kind of fluid are you considering for the closed loop, a liquid metal such as sodium, or something else? Maybe my intuition is failing to give me a clear picture of just how much energy needs to be transferred to the flow to be practical for flight.
I don't know, actually. Maybe helium would do it? It has almost six times the thermal conductivity of air, so the heat exchanger shouldn't get that much bigger relative to a liquid metal one... REB-heating liquid metal could be problematic. On the other hand, you might be able to use resistance heating in a ceramic-coated element rather than a REB... heck, you could forget the exchanger loop and just run a resistance heater through the engine. But that wipes out the regen cooling advantage...
Megavolt-range resistance heating? Why doesn't that strike me as a bad idea? Maybe it's WELL PAST TIME I WENT TO BED...
500,000 lb? A fully shielded 6 GW BFR weighs twice that on its own.
pound shielded BFR? (I think less than 6 GW might do, but let's go with it.) Is this a generally accepted value? That's about the max takeoff weight of a 747-8. Polywells are basically just big spherical vacuum tubes. I know, it's mostly shielding and cooling system weight. MSimon and other nukes -- is this a reasonable weight number for full shielding?
I calculated it upthread. >350 tonnes for just shielding - 5 metre vessel radius, 6" water followed by a thin layer of boron-10 followed by 3½" of lead.
Now that I've been reminded about the high levels of gamma radiation (600 kW at up to 16 MeV), the lead doesn't really seem thick enough, so that number might go up...
Actually it might go up a lot... this might be less feasible than I thought. Drop the gamma intensity by 7 orders of magnitude, and the dose from sitting next to it for a year is about 30 rem. According to the data on Wikipedia's "Gamma_ray" page, that takes about ten inches of lead
(By the way, this will completely obliterate the 300 MW of bremsstrahlung coming off the core; X-rays are easy to stop... and you probably won't need to bother with the water and boron-10 either, since 10" of lead will handle the neutrons nicely too...)
So a fully-shielded 6 GW BFR weighs maybe twice what I said it did. Two million pounds.
Please remember that the power-to-weight ratio goes down VERY FAST if you go for a less powerful reactor, because aside from the obvious fact that the reactor itself isn't that much smaller, you still need almost the same shield thickness. So the huge reactor is about as good as it gets, unless you want to go to 10 GW (which people are apparently dubious about)...
Maybe the shielding thickness can be varied a bit depending on how likely it is that someone's going to be standing next to the vehicle in that particular direction...
Note that even with the new shielding calculations, using the thrust-to-power ratio of a high-bypass turbofan, the generated power is still sufficient to lift the reactor straight up off the ground, along with more than a million pounds of additional structure and payload...