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"What are the maths of dilution of radioactive material collecting above ground", is the title of Yahoo energy resources post. They're somewhat difficult, and I am not entirely on top of them yet, but I found this pdf document at www.rertr.anl.gov enlightening.

Its Figure 2, shows how the radioactivity, measured as a proportion of a fission reactor's heat production, varies with time after the reactor is shut down. At the left side the graph cuts off at 200 days post-shutdown, i.e., doesn't show the heat production fraction for earlier times.

If the missing trace from 200 days to zero days were there, it would require a screen or a piece of paper 140 times taller, because at the instant of shutdown the fraction, I happen to know, is 0.07, not 0.0005. But in those early days it's also dropping very fast, so to plot them, we would make the divisions hours or days rather than the hundreds of days in the chart we have, and we would make the vertical units larger. What this, I think, means is that the chart would look the same.

Three equations are plotted; they all give close-enough results. The one that I find useful, although I recognize that it doesn't look very nice, is the one labelled U. & W., Untermyer and Weills:

Delayed power/in-service power =
   0.1*{ (t+10)^(-0.2) - (t + T_0 + 10)^(-0.2) -0.87*[(t + 20000000)^(-0.2) - (t + 20000000 + T_0)^(-0.2)] }

... where 'T_0' is how long the reactor was on and 't' is how long ago it shut down, both in seconds. This can be simplified if we approximate the time the reactor is on. 'T_0', as infinity; this causes the "^(-0.2)" terms that include it to become zero, so we can cross them out:

Delayed power/in-service power =
   0.1*{ (t+10)^(-0.2) - xxxxxxxxxxxxxxxxxxxx -0.87*[(t + 20000000)^(-0.2) - xxxxxxxxxxxxxxxxxxxxxxxxxxx] }

... and get this:

Delayed power/in-service power =
   0.1*{ (t+10)^(-0.2) -0.87*(t + 20000000)^(-0.2) }

What good is that?

Well, we can interpret the post-shutdown time in seconds 't' as the time a man-made radioactive nucleus takes to escape and become diluted. Suppose it takes ten years, i.e. 316 million seconds. Then the delayed power in the escaped, diluted man-made radioactive material is this fraction of the power of the reactor that we suppose to have been running, and leaking, forever:

0.1*{ (316,000,000+10)^(-0.2) -0.87*(316,000,000 + 20,000,000)^(-0.2) }

... and that's:

0.1 * {0.0199555 0.87*0.0197121}

... and that's:

0.0002806.

If you are able to put the whole equation, with a finite value for 'T_0', into a spreadsheet you will find that reactors that have run less than forever will have built up slightly less ten-years-delayed leakage. After infinite time, the radioactive nuclei that have escaped to the wild are decaying, and emitting their radiation, exactly as fast as new leakage replaces them; after long but finite times, they are decaying almost as fast, and therefore their activity is building up slowly.

So, for a collection of reactors that over many decades averages 1 trillion watts of total heat production, and have spent fuel pools that are only big enough for ten years' accumulation, and spent fuel older than that is all totally leaked and dispersed, we know the environmental radioactivity due to them will asymptotically approach 0.00028 trillion watts, 280 megawatts. It will forever get closer but never quite there.

To decide whether that can be "put safely into the world system on a dilute basis", you could compare it to the megawattage of radioactivity naturally dilute in, um, nature. That's actually the easy part because the dilute radioactivity in nature doesn't noticeably vary in a human lifetime. I'll return to this if there is some interest.

3.4 Enrichment At its natural concentration of 0.7%, 235U92 can be used as a reactor fuel only in particular reactor types (heavy-water reactors and high-temperature reactors). In order to be able to maintain a nuclear chain reaction in typical light water reactors, the concentration of 235U92 in the uranium isotope mix has to be increased to about 3%. At present there exist a range of enrichment methods using UF6 as feed. Since uranium isotopes do not differ in their chemical behaviour, enrichment techniques exploit their mass difference as a means for separating them [25]. These methods are: • Gaseous diffusion: The heavier 238U92 isotope diffuses more slowly than the lighter 235U92 : Enrichment from 0.7% to 3% 235U92 requires in the order of 1,000 consecutive separation cascades. In 2002, 40% of all enrichment plant used gaseous diffusion (mostly France and USA). This percentage is decreasing in favour of the centrifuge method. • Gas centrifuge: The partial pressure of two gases (contained as a gas mixture in a rotating cylinder) depends on their masses. Centrifugal forces cause a radial concentration gradient, with the heavier isotope concentrated outside, and the lighter isotope concentrated inside. Enrichment from 0.7% to 3% 235U92 requires in the order of 10 consecutive separation cascades. In 2002, 60% of all enrichment plants used the centrifuge method (mostly Russia, Germany, UK, Netherlands, China, and Japan). • Electromagnetic Isotope Separation (EMIS): Uses the magnetic separation principle of a mass spectrometer, albeit at a larger scale. Used for building the Hiroshima bomb, and in Iraq’s nuclear program, but now outdated. • Aerodynamic (jet nozzle) method: Exploits the same physical principle as the gas centrifuge, but creates a rotating gas mixture by injection into a circular jet. Demonstration plants built in Brazil and South Africa. • Laser: The energy spectra, and therefore the ionisation energies of different isotopes depend on their masses. Using mono-energetic laser beams, one isotope can be preferentially ionised, and filtered out using an electrostatic field. At the end of this stage, the enriched UF6 is converted into uranium oxide (UO2). The energy needed for enrichment is partly dependent on the incremental enrichment factor for one cascade, which in turn determines the number of cascades necessary to achieve enrichment to around 3%. Gaseous diffusion needs more cascades than the gas centrifuges, and additionally requires the energy-intensive compression of UF6 at the entry point of each cascade (Table 3.4). Gas centrifuges only require electrical energy for the rotation of the cylinders, and some heat in order to maintain an axial convection of the UF6. Atomic laser techniques require the normally metallic uranium to be evaporated (using considerable heat energy), and then transferred into a vacuum, so that ions can be electrostatically filtered [25]. The Australian laser technique is based on molecular rather than atomic laser separation. Instead of having to maintain uranium atoms in a hot gas, the technique uses the already gaseous UF6, and preferentially excites UF6 molecules.6 [ ... ] The two tables above require an explanation of the unit SWU. Amounts of enriched uranium are usually expressed as Separative Work Units (for example tonne SWU).8 There is a trade-off between the amount of natural uranium feed and the number of SWUs needed to produce enriched uranium. For example: in order to produce 10kg of uranium at 4.5% 235U92 concentration while allowing a tails assay of 0.3% requires 100 kg of natural uranium and 62 SWU. Asking for the tails to have only 0.2% assay limits the amount of natural uranium needed to 83 kg, but it also increases the separative work to 76 SWU. Hence, the optimal (tails assay) compromise between uranium feed and separative work depends on the price of natural uranium versus the cost of enrichment operating inputs. During times of cheap uranium, an enrichment plant operator will probably choose to allow a higher 235U92 tails assay, and vice versa. In terms of the energy balance of the nuclear fuel cycle this means that lower tails assays mean that less energy is spent on mining, milling and conversion, and more on enrichment, and vice versa ([17] pp. 26-36 & 43). Storm van Leeuwen and Smith [18] summarise studies undertaken between 1974 and 2003, averaging 2,600 kWh/SWU for gas diffusion, and 290 kWh/SWU for gas centrifuges.9 These values agree well with most of the additional references (Table 3.4).