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In the far future the posthuman successors of mankind have disassembled the stars and live on ultra slow and ultra efficient solar system sized computers, Matrioshka Brains. The last enemy of intelligent life is entropy, the harbinger of the universes inevitable heat death. Once there is no energy gradiant left to exploit entropys final order will purge the universe of all life.

For obvious reasons this fate should be delayed as long as possible, so the most efficient use of resources usable for energy generation is imperative. Leaving fission and the harvesting of the hawking radiation and rotational energy of black holes of the table; I'm interested in how much energy can be generated out of one kilogram of hydrogen using the following process.

The hydrogen is fused into helium using the NCO-cycle as efficiently as plausible followed by helium, carbon, neon, oxygen and silicon fusion. The iron produced by this process can't produce netto energy in a fusion process anymore, so it is fed to a black hole to harvest the frictional energy it will produce in the accretion disk. As I understand it this should be the most efficient way to generate energy from matter if I didn't miss anything.

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    $\begingroup$ Once there is no energy gradiant left to exploit entropys final order will purge the universe of all life. Only stored data will remain in skeletally stripped down storage facilities. Not really, maximum entropy means there will be no persistent gradient anywhere, so your data will be gone with the rest $\endgroup$ – nzaman May 28 '19 at 14:02
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    $\begingroup$ Most efficient energy production I know of is matter-antimatter annihilation. But that, of course, does not answer your question. $\endgroup$ – Mołot May 28 '19 at 14:11
  • $\begingroup$ @nzaman Good point, though there will be a delay between the end of all functional life and the final order because the last beeings cannot turn themselves into energy without seizing to function before. I removed the sentence in order to avoid confusion. Thank you for your feedback. $\endgroup$ – TheDyingOfLight May 28 '19 at 14:15
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    $\begingroup$ @TheDyingOfLight "Thus it must be produced in a very inefficient process" - today. Remember that the top of The Washington Monument was made out of aluminum because aluminum production was so inefficient that it was worth more than gold here. Now it is cheap enough to make cans out of it. You are talking about heat death of the universe time frame - today's efficiencies have nothing to do with the time you are writing about. $\endgroup$ – Mołot May 28 '19 at 14:25
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    $\begingroup$ Far from the 'Hard-Science' tag, please see Dr. Isaac Asimov's' excellent short story titled "The Last Question", describing an ultra-far future where Entropy management is everything. The spirit of your question is in his words. $\endgroup$ – Joe May 28 '19 at 14:45
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CNO Cycle: 4H $\rightarrow$ He

4 protons (i.e. Hydrogen nuclei) combine to form a Helium molecule while releasing 26.7 MeV energy. This is the net result of any of the various fusion pathways for hydrogen to helium.

Triple-$\alpha$ process: 3He $\rightarrow$ C

Three Helium molecules combine to form a Carbon molecule, while releasing 7.4 MeV of energy. Since it takes three CNO reactors (and 12 protons) to make a Carbon molecule, we now have a net gain of 87.5 MeV.

$\alpha$ process: C + 11He $\rightarrow$ Fe

See the link, this is a long chain of reactions, each successively adding a Helium to get to a bigger element, until we get to Fe-52. The total energy released by this chain is 80.6 MeV. Taking into account the 87.5 MeV to form the initial carbon and 11 $\times$ 26.7 MeV to form all of the Helium, the net energy gain from fusion is 462 MeV, divided by 52 initial protons.

Now, Fe-52 is not the most stable element, and you could theoretically get more energy by reacting up to Fe-56 or Ni-62 or something. But, I wasn't able to find a clear path for fusion up to that point. In the real world, creation of these elements is a result of an equilibrium between various fusion reactions and photodisintegration and such. I think this energy estimate is the best for your purposes.

Energy released by accretion

This is much more difficult to estimate, because there are a lot of factors here, and it depends strongly on the size of your black hole and shape of the accretion disk. However, reworking an estimate of luminosity based on mass transfer rate into a black hole gives: $$E = \frac{\mu m}{R},$$ where $\mu$ is the standard gravitational parameter for the black hole, $m$ is the mass of the object falling into it, and $R$ is the radius of the accretion disk.

Lets take the black hole at the center of our galaxy as an example. I calculate $\mu$ to be about $5.7\times10^{26}$ and $r$ about $7.5\times10^{12}$ meters (~ 50 AU). Therefore, each AMU generates about 0.8 MeV as it falls into the accretion disk. Consider this a pretty rough estimate. The problem here is that much of this kinetic energy is either a. carried into the event horizon by the falling particle or b. radiated into the event horizon by the accretion disk. Either way, much of the released energy is unusable.

Conclusion

You get about 9 MeV from fusion per AMU of protons that you throw into this process, and less than 0.8 MeV from accretion per AMU of protons. Converting to J and kg, we get 870 TJ per kg from fusion, and less than 77 TJ per kg from accretion. So, you are looking at something in the range of 900 TJ per kg of hydrogen.

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