|Fig. 1: Famous 1803 caricature by James Gillray showing King George III holding a tiny Napoleon in the palm of his hand. (Source: Wikimedia Commons)|
When a danger gets small enough is it still a danger at all? If we take one of the most potentially good and simultaneously dangerous devices ever made - the nuclear power plant - and we shrink it down small enough, is there a size below which it can actually be inherently safe? The satirical political cartoon shown in Fig. 1 depicts a tiny Napoleon in the hands of King George III. If we just make him small enough, could he possibly pose any threat at all?
The idea of small distributed nuclear reactors (usually about 50 MW) is not a new one by any means. The possibility has even been discussed recently in the media when the Department of Energy was investing in some of these technologies.  Some of the advantages often cited are benefits of assembly line production and ship-ability for dramatic cost reduction as well as standardization (compared to building everything on-site and having a multitude of site-specific designs that could have design flaws). This model of distributed power could be a big benefit for areas without extensive power distribution lines already in place. It also of course comes with the cost of putting nuclear material in more places that could be a problem closer to population centers, and the proliferation risk of having nuclear material in more sites.
As discussed in a previous report, since there is a natural "background radiation," there is also therefore a natural threshold for the amount of radiation that really matters.  The question then comes up: how big could a reactor be for it not to appreciably change the amount of radiation naturally present in the event of an accident.
We will choose the background radiation of interest to be that found in Colorado, or about 9 mSv/year because it is slightly elevated and yet does not appear to have severely detrimental effects on the population.  Incidentally, this is the sort of data that is has led some to believe that there might be a nonlinear relationship between amount of radiation and cancer causation with lower levels of radiation being perhaps not as bad as was once assumed by linear interpolation of extreme data. 
We will use data from Chernobyl, and make some admittedly poor assumptions for the sake of discussion along the way. The first of which, is that we really care about the radiation after a little bit of time - say about 10 years. The magnitudes of the radiation right after any event will undoubtedly be very high, but we are trying to find a reactor that won't make a permeant black hole on maps for the rest of human history, not prevent even any short term problems.
We will further assume that chernobyl was the worst sort of event that can be expected - this is probably not a terrible assumption given how bad it was. We will assume further that the energy of the event will be about constant, independent from reactor power (as it is easy to have a mistake that would blow apart the reactor, but much harder to make an efficient bomb), but that the actual material spread in the event will be proportional to the power capacity of the reactor (ie nuclear material mass proportional to power).
From the previous report we estimate a region of about 50 × 50 miles as the primary area of radiation fallout around the reactor.  There is another 50 mile by 100 mile zone of sporadic hot spots as well, so for the sake of discussion, we are looking at 7500 square miles. There isn't data on exactly how bad these areas are, but the lower threshold limit of the worst spots is 40 Ci/ km2 with most of that being from Cs-137. We will assume the worst of the radiation to be twice that lower bound.
It was found in the previous report that the radiation of these hotter spots resulted in a 186 mSv/year radiation dose, or 186mSv/year / 9mSv/year = 20 times the yearly dose of Colorado. With the assumptions that the area of the speed is independent of power plant size, and that the amount of material spread is proportional to the power rating of the reactor, we simply find that the max power of the reactor that would double the Colorodo background radiation is as the 1gigawatt rating of the reactor in Chernobyl divided by the ratio of the resulting radiation to the level we are calling background:
(186 mSv/year / 9 mSv/year)
This is surprisingly close to the value that these mini reactors are pushing for - about 50 Megawatts.
To take this logic one step further, this is the reactor size that we hypothesize would about double the background radiation of colorado if the worst case should happen, but that background radiation covered 7500 square miles. So the be very conservative, we should not have more than one of these 50 Megawatt reactors every 7500 square miles. The earth has a total land area of 57 million square miles, allowing for 57 million square miles / 7500 square miles = 7600 isolated zones for our "safe" reactors, assuming there are not any concentration mechanisms in place (which is not true).
With 40 megawatts per zone we get 0.3 TerraWatts of nuclear power operating as isolated, "safe" units. This is less than the total human use for 2008 of 15 TerraWatts, but is at least approaching the same order of magnitude.  Obviously there would be insane power excesses in uninhabited areas, and shortages in New York that would make this idea impractical but interesting to consider. Of course harkening back to the idea expressed in Fig. 1, Napoleon in real life was already quite small, and yet still cost France alone 916,000 men. 
© David Christensen. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 B. Bradford, "Are Mini-Reactors the Future of Nuclear Power?" National Public Radio, 4 Feb 13.
 D. Christensen, How Much Worse Is Chernobyl Than Background Radiation," Physics 241, Stanford University, Winter 2014.
 B. L. Cohen, "Cancer Risk From Low-Level Radiation," Am. J. Roentgenol. 179, 1137 (2002).
 "BP Statistical Review of World Energy," British Petroleum, June 2009, p. 13.
 D. Gates, The Napoleonic Wars 1803-1815 (Hodder Education Publishers, 1997), p272.