Investigating Radiator-Assisted Pulsed Nuclear Propulsion
Already discussed in the narrative so far are:
1. Implementing crankshafts to control pusher-plate movement
2. Electromagnetic launchers to launch pulse units
3. Stronger, lighter and higher temperature supporting pusher-plate materials
These together make the propulsion system more reliable, lighter and more resilient than the initial 1950s designs.
An additional possibility is the replacement of the ablative with extremely high heat radiators. While not so much of a problem with small pusher-plates, as ship sizes get bigger the ablative layer becomes a problematic amount of mass. Could we swap the ablative for extremely hot radiators?
In principle, a pusher-plate system could operate with no ablative coating; Dyson considered such a design back in the 1960s as part of his evaluation of using Hydrogen fusion bombs to propel an interstellar Orion ship. The Physics Today article is available
here.
Dyson's proposal considers energy limited and momentum limited concepts. The first relies on allowing the pusher-plate to cool between pulses. This meant very high wait times between pulses; his copper concept relied on the dissipation of 4 PJ of thermal energy naturally via thermal radiation, and meant pulses, rather than on the order of 0.8 to 3 second waits between pulses described in real world Orion applications, would be on the order of once every few minutes.
While tolerable for interstellar applications, interplanetary use has some substantial problems with such slow accelerations, especially for military or fast transit applications.
We will now consider the requirements for such radiators.
The equation for an ideal blackbody radiator is P/A =σT^4, where σ is the Stephen-Boltzmann constant; 5.67×10^−8 W⋅m^−2⋅K^−4.
Power is the input energy from the pulse unit, with A being the area of radiators needed. Therefore, A = P/σT^4. Because we can safely assume double sided radiators, we can divide that by 2 again to actually have an amount of area of radiator to bolt on the side of our ship. We also need to allow for the fact that materials aren't always good blackbody radiators, which we can do by allowing larger radiators.
One highly advantageous part of Dyson's proposal was to use copper as the material for the pusher-plate; copper is limited to a melting temperature of 1358K. And he considered putting ~4PJ through it. Copper is not exactly a brilliant material, although a thermal conductivity of 369W/m K is very good, and part of why he suggested it. It's certainly not some sci-fi unobtainium or classified military material, and exactly as he was proposing, it's a nice easily worked metal we do a lot of things with. But it's by no means the best material available, even back in the '60s when he proposed it; his point was to prove feasibility after all, and if a low grade material could manage it, his point was achieved.
We are looking for vastly improved performance however. A safe usage temperature of 1290K is unacceptably low, it's density of 8960kg/m³ is rather high, it's specific heat is rather middling and it's strength is really rather low.
Now, we are going to push the temperature much higher. 3000K or better means we can cut the dimensions of our end radiator to a fifth of the size of the initial copper proposal.
A few candidates come to mind:
Tantalum Hafnium Carbide. Anything that has a safe usage temperature of 4050K immediately demands consideration as a radiator, although it is let down by it's massive density at 14,700kg/m³, and half the specific heat. Of note is that it has a fraction of the thermal conductivity. (in space you can't get rid of heat by conduction anyway, so, all this does is limit heat transfer from the pusher-plate into the rest of the ship) It's much stronger than copper, so in principle we can get away with a thinner pusher-plate. But there's no getting away from the mass penalty.
Hafnium Carbide. Not quite as high safe usage temperature, but lower density.
Amorphous Carbon. A natural choice of Children Of A Dead Earth players for radiators because it's light, cheap and has an extremely high safe usage temperature. It isn't as strong in terms of ultimate yield strengths as copper, although acceptably close otherwise. It has better than four times copper's specific heat and best of all, a density of 2100kg/m³ and a safe usage temperature of 3719K. Let down by the fact it'll vaporise at 3915K.
Glass-like Carbon. Not quite as good as Amorphous Carbon, but is only 1500kg/m³.
Tungsten. While only usable up to 3461K and one of the densest materials, it is already well invested in for mining and manufacturing by our protagonists and is extremely strong, and therefore very suitable for pusher-plates.
How can we choose between them?
The obvious answer is use whichever has the lowest radiator mass. To find this we bung the numbers into the equation. Assuming a single square radiator, side length is √((P/σT^4)/2)
Using 100 degrees below each materials safe usage temperature as factor of safety gives in order, 256m, 268m, 305m, 311m, and 354m. (
Copper, for context, gives 2812m)
Using a 1cm thickness gives...
Well, the answer is really bad. THC has 9.6 kilotons of radiator required to dissipate that much heat in a second. We can therefore immediately discount THC and Tungsten, because that is denser and lower temperature. But what about the least dense material, Glass-like Carbon? That works out at a much more feasible 1.56kt of radiator mass. Still far too heavy for military applications; ablative systems would operate for thousands of pulse units for that.
But equally, this assumes that the entire nuke is used as thermal energy to heat the pusher-plate, which it isn't; 80% goes to kinetic energy of the ship. If we allow a little margin of safety and drop the power dissipated to 33% of the nuke, we can use radiators 33% of the size. ~500 ton radiators are not ideal, but they aren't impossible to work with either.
In conclusion, it seems reasonable that a mixed approach of using smaller radiators and longer waits between pulses could lead to a feasible solution for freight operations that would eliminate one source of maintenance cost, although it isn't suitable for all pulsed nuclear propulsion applications.
