Sustainable energy and the Kardashev scale

The y-axis is power; the x-axis is time. For a finite energy resource, the product [energy] is a constant, which is a diagonal line on this (log-log) plot. This represents, for a given civilization power consumption (y axis), the time scale over which it is maintainable (x axis). For renewable sources, the limiting factor is power - a flat horizontal line.

Energy potential by source

The units are in watts of thermal power equivalent. For some sources (e.g. wind), this does not make sense. So I've multiplied them by a factor of three. This is roughly an equal footing: for instance, 1/3rd is a typical thermal power plant efficiency. But vehicle engines are less efficient. And natural gas heat is more efficient (100%, actually). So the exact factor is debatable. Luckily this plot is logarithmic.

For the nuclear reactions, I use the following energy densities. For D+T fusion, 17.6 MeV over 4 amu = 424 TJ/kg. (Why 4 amu not 5? Because this is really D+D fusion, breeding tritium from its own neutron flux.) For proton-proton fusion, 26.73 MeV over 2 amu = 645 TJ/kg. For the fission breeding cycles, 190 MeV over 238 amu = 77 TJ/kg. For the once-through fuel cycle (the current method), we're limited to the fissile isotopes (U-235), and even then the limiting factor is not energy potential but the mechanical lifetime of fuel rods. Assuming enrichment from 0.7% to 3% U-235, and a burnup of 60 gigawatt days per metric ton U, this is 1.2 TJ/kg of the original, natural uranium.

For the fusion reactions, I use the mass of the earth's hydrosphere, 1.4 10^21 kg, as the supply of hydrogen fuel. 1/9th of the mass is hydrogen, and of that, 0.03% is deuterium. This excepts the "Jupiter" scenario, which is total fusion of the entire 1.9*10^27 kg mass of Jupiter (mostly pure H2).

There are three solar scenarios. One is the energy potential of the entire surface of the earth. Another is a Dyson ring, 10 kilometers in width at a solar radius of 10 million km. The last is a complete Dyson sphere, which is level II on the Kardashev scale for extraterrestial civilizations.

For the fission fuel supply, there are six lines, properly representing the wide range of figures represented in media sources, including the alarmist 'peak uranium' stories which use the lowest figure (overlaps natural gas). They are the permutations of three levels of ore - high-level conventional, phosphate rocks, and seawater - and closed vs. once-through fuel cycles (as above). From the IAEA red book, the first two numbers are estimated 4.7 million and 35 million metric tons uranium metal, respectively. The seawater figure is 4.5 billion tons, from the concentration figure of 3.2 ppb and the mass of the earths' oceans (given earlier). Note the seawater figure flattens horizontally at a certain level: this represents the rate of erosion of uranium from rivers, as referenced by Bernard Cohen in this Am. J. Phys. article. (32,000 tons/year). In this sense, fission is indeed a fully renewable resource, although at a far lower potential than wind or solar.

For wind, wave, geothermal, I simply stole figures from wikipedia [1] [2] [3]. The wind figure potential is slightly pragmatic, in that it ignores wind over open ocean (unlike the solar power figure). Same with the wave figure - apparently it only includes near-shore waves (but this makes small difference, since the "build-up" length for ocean waves is thousands of miles anyway: a mid-ocean wave farm would leave a wake thousands of miles long). The geothermal figure is the geological rate of heat dissipation out of the earth's crust, which is probably a huge overestimate. (But I'm ignoring near-surface pockets of heat, which can be exploited faster than natural conduction rates, but non-renewably so.)

I coudn't find a hydroelectric potential figure, so I made a quick estimate. From the histogram, I eyeballed an estimate of 500m for the average elevation of earth terrain (above sea level). Likewise, I eyeballed an average rainfall of 100 cm/year. Assuming they are uncorrelated, this represents a hydroelectric potential of 0.15 W/m^2, or an upper limit of 20 TW for the entire earth. This is the blue line between geothermal and ocean uranium breeders (they are all squished together).

Don't forget, I multiplied hydro, wind, and wave potentials by 3 to convert to thermal power equivalents. So the 20 TW of hydropower electricity are slightly greater than the 40 GW of geothermal heat.

The fossil fuel figures (coal/oil/gas) are from BP's statistical review. They are proven reserves, so they are underestimates.

Finally, the biomass figure is my own estimate, based on the IPCC figure for the carbon throughput of photosynthesis on earth (120 billion tons/year). This is the entire biosphere, not just anthropogenic farming.

The demand figures are (i) current world (thermal energy) consumption, and (ii) my guess at a near-future stablization point, which is a population of 10 billion with the same per-capita energy consumption as present-day USA.

Suggest improvements in the comments, and I may include them.