The actual nuclear fuel is not expensive at all. The online calculator [1] shows that to enrich uranium to 20% (HALEU grade, which is what DARPA is looking into), you pay about $6500/kg. At this concentration, Uranium has an energy density of about 800 GJ/kg [2]. The NERVA engine had a power of about 1.1 GW and provided about 246 kN of thrust, which is about 3.4 times the thrust envisioned for the Starship upper stage [3], so assuming you burn only 50% of the uranium at only 50% efficiency (that's very conservative), you can run the engine for 3 minutes. If you want to run this engine for 5 hours (how many trips to Mars and back would that be?), the fuel cost would be $650k. Most likely this would be well below the fuel cost for the Starship upper stage. The cost of the cryogenic hydrogen is a different story, but even that one should not be a dealbreaker.
> 246 kN of thrust, which is about 3.4 times the thrust envisioned for the Starship upper stage
AFAICT Raptor is planned to have ~2,000 kN thrust [0], and Starship has more than one.
And your metrics for fuel use are a bit off to me - better to assume that all of the fuel in the reactor is used with no possibility to recycle, because no one's going to be happy with you landing with a used engine. (And the value of a used engine is pretty significantly negative anyways - just look at the decommissioned naval reactors at Hanford, or the used fuel rods sitting in pools across the country) Similarly, a 50% burn rate is incredibly optimistic - commercial reactors normally manage around 6.5%, and while I believe naval reactors are higher, it's by a factor closer to two than ten. For NERVA I'd expect something more like 0.1%, because it'd be operating for such a short time. (minutes or hours instead of years) Combined, that puts the fuel cost in the $10,000,000-$100,000,000 range to operate for 5 hours, instead of $100,000-$1,000,000.
Or, coming from a completely different direction, from reading this [1], specifically page 99, NRX/EST had 176kg of highly enriched uranium for a 1,055MW reactor. At ~$30k/kg for 85% enrichment from your calculator, that means $5,280,000. Yes, modern NTR designs don't use HEU, but that means much lower power densities and thus more uranium.
You are right, I made quite a number of mistakes in my calculations. In particular, the Raptor engine indeed provides 2000 kN of thrust, and the upper stage has 6 of those, for a total of 12000 kN. The Nerva engine is only about 250 kN, so you'd need 48 of them to match the 6 Raptors.
Your find [1] is fantastic. If it's $5.3 MM per engine, that would be about $250 MM for the 48 engines.
However, that could be used several times, at least 4.5 by my calculation: The NRX A6 was able to run for more than 1 hour [2]. The fuel flow was 32 kg/s, so that's about 1.5 tons/s for 48 engines, or about 800s to burn 1200 tons (fuel mass of Starship upper stage). That makes the uranium fuel cost per trip about $55MM.
Now if instead of 85% one uses 20% enriched uranium, the price per kg decreases by a factor of 4.6 (from 30k to 6.5k) but the burnup decreases too, but probably by less than that. Still, a ballpark figure of about $50MM per trip, only for the uranium fuel.
Some updates after some further googling:
- the NRX A6 engine was tested for 1 hour, but based on the observed corrosion, NASA extrapolated that the lifetime of the engine could be 2-3 hours ([1] page 10). That would reduce the fuel cost that I estimated above from $50MM per trip to Mars to $20MM per trip.
- nuclear thermal rockets can use different propellants. In [2] Nasa looked at some other propellants that could be available on Mars, such as CO2, H2O, CH4, CO, N2, Argon, NH3. They conclude that CO2 would be very convenient, H2O even more so (but were not sure at the time how easy it would be to source water on Mars). CH4 would be very appealing, but one would need to overcome some coking concerns. The other fuels were all inferior (except for H2, obviously).
- nuclear thermal rockets are not intended for takeoff from the Earth [3]. The intention is to be carried to space with chemical rockets, and use them for trips starting in LEO.
The actual nuclear fuel is not expensive at all. The online calculator [1] shows that to enrich uranium to 20% (HALEU grade, which is what DARPA is looking into), you pay about $6500/kg. At this concentration, Uranium has an energy density of about 800 GJ/kg [2]. The NERVA engine had a power of about 1.1 GW and provided about 246 kN of thrust, which is about 3.4 times the thrust envisioned for the Starship upper stage [3], so assuming you burn only 50% of the uranium at only 50% efficiency (that's very conservative), you can run the engine for 3 minutes. If you want to run this engine for 5 hours (how many trips to Mars and back would that be?), the fuel cost would be $650k. Most likely this would be well below the fuel cost for the Starship upper stage. The cost of the cryogenic hydrogen is a different story, but even that one should not be a dealbreaker.
[1] https://www.uxc.com/p/tools/FuelCalculator.aspx
[2] https://energyeducation.ca/encyclopedia/Energy_density
[3] https://en.wikipedia.org/wiki/NERVA