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Nuclear vs Chemical
Directly comparing the performance of a nuclear engine and a chemical one is not easy; the design of any rocket is a study in compromises and different ideas of what constitutes "better". In the outline below we will consider the NERVA-derived engine that was considered by NASA in the 1960s, comparing it with the S-IVB stage from the Saturn it was intended to replace.

For any given thrust, the amount of power that needs to be generated is defined by P = T * Ve / 2, where T is the thrust, and Ve is the exhaust velocity. Ve can be calculated from the specific impulse, Isp, where Ve = Isp * g, Using the J-2 on the S-IVB as a baseline design, we have P = 414 s * (1014 kN * 9.81) / 2 = 2,060 MW. This is about the amount of power generated in a large nuclear reactor.

However, as outlined above, even the simple solid-core design provided a large increase in Isp to about 850 seconds. Using the formula above, we can calculate the amount of power that needs to be generated, at least given extremely efficient heat transfer: P = 1014kN * (850 * 9.81) / 2 = 4,227 MW. Note that it is the Isp improvement that demands the higher energy. Given inefficiencies in the heat transfer, the actual NERVA designs were planned to produce about 5 GW, which would make them the largest nuclear reactors in the world.

The fuel flow for any given thrust level can be found from m = T / Ve. For the J-2, this is m = 1014 kN / (414 * 9.81), or about 250 kg/s. For the NERVA replacement considered above, this would be 121 kg/s. Remember that the mass of hydrogen is much lower than the hydrogen/oxygen mix in the J-2, where only about 1/6th of the mass is hydrogen. Since liquid hydrogen has a density of about 70 kg/m³, this represents a flow of about 1,725 litres per second, about three times that of the J-2. This requires additional plumbing but is by no means a serious problem; the famed F-1 had flow rates on the order of 25,000 l/s.

Finally, one must consider the design of the stage as a whole. The S-IVB carried just over 300,000 litres of fuel, 229,000 litres of liquid hydrogen (17300 kg), and 72,700 litres of liquid oxygen (86 600 kg). The S-IVB uses a common bulkhead between the tanks, so removing it to produce a single larger tank would increase the total load only slightly, for argument's sake, perhaps 2,000 litres. Assuming this for the moment, this means the new hydrogen-only nuclear stage would carry about 231,000 litres in total (231 m³), or about 16,500 kg (36,350 lb). At 1,725 litres per second, this is a burn time of only 135 seconds, compared to about 500 in the original S-IBV (although some of this is at a lower power setting).

The total change in velocity, the so-called delta-V, can be found from the rocket equation, which is based on the starting and ending masses of the stage:


Where m0 is the initial mass with fuel, m1 the final mass without it, and Ve is as above. The total empty mass of the J-2 powered S-IVB was 13,311 kg, of which about 1,600 kg was the J-2 engine. Removing the inter-tank bulkhead to improve hydrogen storage would likely lighten this somewhat, perhaps to 10,500 kg for the tankage alone. The baseline NERVA designs were about 15,000 lb, or 6,803 kg, making the total unfueled mass (m1) of a "drop-in" S-IVB replacement around 17,300 kg. The lighter weight of the fuel more than makes up for the increase in engine weight; whereas the fueled mass (m0) of the original S-IVB was 119,900 kg, for the nuclear-powered version this drops to only 33,800 kg.

Following the formula above, this means the J-2 powered version generates a of (414sec * 9.81) ln (119,900 / 13,311), or 8,925 m/s. The nuclear-powered version assumed above would be (850 * 9.81) ln (33,800 / 17,300), or 5,585 m/s. This drop in overall performance is due largely to the much higher "burnout" weight of the engine, and to smaller burn time due to the less-dense fuel. As a drop-in replacement, then, the nuclear engine does not seem to offer any advantages.

However, this simple examination ignores several important issues. For one, the new stage weighs considerably less than the older one, which means that the lower stages below it will leave the new upper stage at a higher velocity. This alone will make up for much of the difference in performance. More importantly, the comparison assumes that the stage would otherwise remain the same design overall. This is a bad assumption; one generally makes the upper stages as large as they can be given the throw-weight of the stages below them. In this case one would not make a drop-in version of the S-IVB, but a larger stage who's overall weight was the same as the S-IVB.

Following that line of reasoning, we can envision a replacement S-IVB stage that weighs 119,900 kg fully fueled, which would require much larger tanks. Assuming that the tankage mass triples, we have a m1 of 31,500 + 6,800 = 38,300 kg, and since we have fixed m0 at 119,900 kg, we get = (850 s * 9.81) ln (119,900 / 38,300), or 9,500 m/s. Thus, given the same mass as the original S-IVB, one can expect a moderate increase in overall performance using a nuclear engine. This stage would be about the same size as the S-II stage used on the Saturn.

Of course this increase in tankage might not be easy to arrange. NASA actually considered a new S-IVB replacement, the S-N, built to be as physically large as possible while still being able to be built in the VAB. It weighed only 10,429 kg empty and 53,694 kg fueled (suggesting that structural loading is the dominant factor in stage mass, not the tankage). The combination of lower weight and higher performance improved the payload of the Saturn V as a whole from 127,000 kg delivered to low earth orbit (LEO) to 155,000 kg.

It is also worth considering the improvement in stage performance using the more advanced engine from the STNP program. Using the same S-IVB baseline, which does make sense in this case due to the lower thrust, we have an unfueled weight of (m1) of 10,500 + 1,650 = 12,150 kg, and a fueled mass (m0) of 22,750 + 12,150 = 34,900 kg. Putting these numbers into the same formula we get a of just over 10,000 m/s – remember, this is from the smaller S-IV-sized stage. Even with the lower thrust, the stage also has a power-to-weight ratio similar to the original S-IVB, 34,900 kg being pushed by ~350 kN, as opposed to 253,000 lb pushed by ~250,000 lbf thrust. The STNP-based S-IVB would indeed be a "drop-n replacement" for the original S-IVB, offering higher performance from much lower weight.


[edit] Risks
There is an inherent possibility of atmospheric or orbital rocket failure which could result in a dispersal of radioactive material, and resulting fallout. Catastrophic failure, meaning the release of radioactive material into the environment, would be the result of a containment breach. A containment breach could be the result of an impact with orbital debris, material failure due to uncontrolled fission, material imperfections or fatigue and human design flaws. A release of radioactive material while in flight could disperse radioactive debris over the Earth in a wide and unpredictable area. The zone of contamination and its concentration would be dependent on prevailing weather conditions and orbital parameters at the time of re-entry. However given that oxide reactor elements are designed to withstand high temperatures (up to 3500 K) and high pressures (up to 200 atm normal operating pressures) it's highly unlikely a reactor's fuel elements would be reduced to powder and spread over a wide-area. More likely highly radioactive fuel elements would be dispersed intact over a much smaller area, and although individually quite lethal up-close, the overall hazard from the elements would be confined to near the launch site and would be much lower than the many open-air nuclear weapons tests of the 1950s.

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Originally posted by citizen:
The Reactor is going to be largely unshielded (with just a 'shadow shield' to protect the crew).

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Originally posted by Cybersnark:
"They're burning without core containment. Well, that's. . . kwong-juh duh. That's suicide."

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