Tag Archives: nuclear

Fractal power laws and radioactive waste decay

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Here’s a fairly simple model for nuclear reactor decay heat versus time. It’s based on a fractal model I came up with for dealing with the statistics of crime, fires, etc. The start was to notice that radioactive waste is typically a mixture of isotopes with different decay times and different decay heats. I then came to suspect that there would be a general fractal relation, and that the fractal relation would hold through as the elements of the mixed waste decayed to more stable, less radioactive products. After looking a bit, if seems that the fractal time characteristic is time to the 1/4 power, that is

heat output = H° exp (-at1/4).

Here H° is the heat output rate at some time =0 and “a” is a characteristic of the waste. Different waste mixes will have different values of this decay characteristic.

If nuclear waste consisted of one isotope and one decay path, the number of atoms decaying per day would decrease exponentially with time to the power of 1. If there were only one daughter product produced, and it were non-radioactive, the heat output of a sample would also decay with time to the power of 1. Thus, Heat output would equal  H° exp (-at) and a plot of the log of the decay heat would be linear against linear time — you could plot it all conveniently on semi-log paper.

But nuclear waste generally consists of many radioactive components with different half lives, and these commpnents decay into other radioactive isotopes, all of whom have half-lives that vary by quite a lot. The result is that a semi-log plot is rarely helpful.  Some people therefore plot radioactivity on a log-log plot, typically including a curve for each major isotope and decay mode. I find these plots hardly useful. They are certainly impossible to extrapolate. What I’d like to propose instead is a fractal variation of the original semi-log plot: a  plot of the log of the heat rate against a fractal time. As shown below the use of time to the 1/4 power seems to be helpful. The plot is similar to a fractal decay model that I’d developed for crimes and fires a few weeks ago. 

Afterheat of fuel rods used to generate 20 kW/kg U; Top graph 35 MW-days/kg U; bottom graph 20 Mw-day /kg U. Data from US NRC Regulatory Guide 3.54 - Spent Fuel Heat Generation in an Independent Spent Fuel Storage Installation, rev 1, 1999. http://www.nrc.gov/reading-rm/doc-collections/reg-guides/fuels-materials/rg/03-054/

After-heat of nuclear fuel rods used at 20 kW/kg U; Top graph 35 MW-days/kg U; bottom graph 20 Mw-day /kg U. Data from US NRC Regulatory Guide 3.54. A typical reactor has 200,000 kg of uranium.

A plausible justification for this fractal semi-log plot is to observe that the half-life of daughter isotopes relates to the parent isotopes. Unless I find that someone else has come up with this sort of plot or analysis before, I’ll call it after myself: a Buxbaum Mandelbrot plot –Why not?

Nuclear power is attractive because it is a lot more energy dense than any normal fuel. Still the graph at right illustrates the problem of radioactive waste. With nuclear, you generate about 35 MW-days of power per kg of uranium. This is enough to power an average US home for 8 years, but it produces 1 kg of radioactive waste. Even after 81 years the waste is generating about 1/2 W of decay heat. It should be easier to handle and store the 1 kg of spent uranium than to deal with the many tons of coal-smoke produced when 35 MW-days of electricity is made from coal, still, there is reason to worry about the decay heat.

I’ve made a similar plot of decay heat of a fusion reactor, see below. Fusion looks better in this regard. A fission-based nuclear reactor to power 1/2 of Detroit, would hold some 200,000 kg of uranium that would be replaced every 5 years. Even 81 years after removal, the after-heat would be about 100 kW, and that’s a lot.

Afterheat of a 4000 MWth Fusion Reactor, from UMAC III Report. Nb-1%Zr is a fairly common high-temerature engineering material of construction.

After-heat of a 4000 MWth Fusion Reactor built from niobium-1%zirconium; from UWMAC III Report. The after heat is far less than with normal uranium fission.

The plot of the after-heat of a similar power fusion reactor (right) shows a far greater slope, but the same time to the1/4 power dependence. The heat output drops from 1 MW at 3 weeks to only 100 W after 1 year and far less than 1 W after 81 years. Nuclear fusion is still a few years off, but the plot at left shows the advantages fairly clearly, I. think.

This plot was really designed to look at the statistics of crime, fires, and the need for servers / checkout people.

Dr. R.E. Buxbaum, January 2, 2014, edited Aug 30, 2022. *A final, final thought about theory from Yogi Berra: “In theory, it matches reality.”

Hormesis, Sunshine and Radioactivity

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It is often the case that something is good for you in small amounts, but bad in large amounts. As expressed by Paracelsus, an early 16th century doctor, “There is no difference between a poison and a cure: everything depends on dose.”

Aereolis Bombastus von Hoenheim (Paracelcus)

Phillipus Aureolus Theophrastus Bombastus von Hoenheim (Dr. Paracelsus).

Some obvious examples involve foods: an apple a day may keep the doctor away. Fifteen will cause deep physical problems. Alcohol, something bad in high doses, and once banned in the US, tends to promote longevity and health when consumed in moderation, 1/2-2 glasses per day. This is called “hormesis”, where the dose vs benefit curve looks like an upside down U. While it may not apply to all foods, poisons, and insults, a view called “mitridatism,” it has been shown to apply to exercise, chocolate, coffee and (most recently) sunlight.

Up until recently, the advice was to avoid direct sun because of the risk of cancer. More recent studies show that the benefits of small amounts of sunlight outweigh the risks. Health is improved by lowering blood pressure and exciting the immune system, perhaps through release of nitric oxide. At low doses, these benefits far outweigh the small chance of skin cancer. Here’s a New York Times article reviewing the health benefits of 2-6 cups of coffee per day.

A hotly debated issue is whether radiation too has a hormetic dose range. In a previous post, I noted that thyroid cancer rates down-wind of the Chernobyl disaster are lower than in the US as a whole. I thought this was a curious statistical fluke, but apparently it is not. According to a review by The Harvard Medical School, apparent health improvements have been seen among the cleanup workers at Chernobyl, and among those exposed to low levels of radiation from the atomic bombs dropped on Hiroshima and Nagasaki. The health   improvements relative to the general population could be a fluke, but after a while several flukes become a pattern.

Among the comments on my post, came this link to this scholarly summary article of several studies showing that long-term exposure to nuclear radiation below 1 Sv appears to be beneficial. One study involved an incident where a highly radioactive, Co-60 source was accidentally melted into a batch of steel that was subsequently used in the construction of apartments in Taiwan. The mistake was not discovered for over a decade, and by then the tenants had received between 0.4 and 6 Sv (far more than US law would allow). On average, they were healthier than the norm and had significantly lower cancer death rates. Supporting this is the finding, in the US, that lung cancer death rates are 35% lower in the states with the highest average radon radiation levels (Colorado, North Dakota, and Iowa) than in those with the lowest levels (Delaware, Louisiana, and California). Note: SHORT-TERM exposure to 1 Sv is NOT good for you; it will give radiation sickness, and short-term exposure to 4.5 Sv is the 50% death level

Most people in the irradiated Taiwan apartments got .2 Sv/year or less, but the same health benefit has also been shown for people living on radioactive sites in China and India where the levels were as high as .6 Sv/year (normal US background radiation is .0024 Sv/year). Similarly, virtually all animal and plant studies show that radiation appears to improve life expectancy and fecundity (fruit production, number of offspring) at dose rates as high as 1 Sv/month.

I’m not recommending 1 Sv/month for healthy people, it’s a cancer treatment dose, and will make healthy people feel sick. A possible reason it works for plants and some animals is that the radiation may kill proto- cancer, harmful bacteria, and viruses — organisms that lack the repair mechanisms of larger, more sophisticated organisms. Alternately, it could kill non-productive, benign growths allowing the more-healthy growths to do their thing. This explanation is similar to that for the benefits farmers produce by pinching off unwanted leaves and pruning unwanted branches.

It is not conclusive radiation improved human health in any of these studies. It is possible that exposed people happened to choose healthier life-styles than non-exposed people, choosing to smoke less, do more exercise, or eat fewer cheeseburgers (that, more-or-less, was my original explanation). Or it may be purely psychological: people who think they have only a few years to live, live healthier. Then again, it’s possible that radiation is healthy in small doses and maybe cheeseburgers and cigarettes are too?! Here’s a scene from “Sleeper” a 1973, science fiction, comedy movie where Woody Allan, asleep for 200 years, finds that deep fat, chocolate, and cigarettes are the best things for your health. You may not want a cigarette or a radium necklace quite yet, but based on these studies, I’m inclined to reconsider the risk/ benefit balance in favor of nuclear power.

Note: my company, REB Research makes (among other things), hydrogen getters (used to reduce the risks of radioactive waste transportation) and hydrogen separation filters (useful for cleanup of tritium from radioactive water, for fusion reactors, and to reduce the likelihood of explosions in nuclear facilities.

by Dr. Robert E. Buxbaum June 9, 2013

Chernobyl radiation appears to cure cancer

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In a recent post about nuclear power, I mentioned that the health risks of nuclear power are low compared to the main alternatives: coal and natural gas. Even with scrubbing, the fumes from coal burning power plants are deadly once the cumulative effect on health over 1000 square miles is considered. And natural gas plants and pipes have fairly common explosions.

With this post I’d like to discuss a statistical fluke (or observation), that even with the worst type of nuclear accident, the broad area increased cancer incidence is generally too small to measure. The worst nuclear disaster we are ever likely to encounter was the explosion at Chernobyl. It occurred 27 years ago during a test of the safety shutdown system and sent a massive plume of radioactive core into the atmosphere. If any accident should increase the cancer rate of those around it, this should. Still, by fluke or not, the rate of thyroid cancer is higher in the US than in Belarus, close to the Chernobyl plant in the prime path of the wind. Thyroid cancer is likely the most excited cancer, enhanced by radio-iodine, and Chernobyl had the largest radio-iodine release to date. Thus, it’s easy to wonder why the rates of Thyroid cancer seem to suggest that the radiation cures cancer rather than causes it.

Thyroid Cancer Rates for Belarus and US; the effect of Chernobyl is less-than clear.

Thyroid Cancer Rates for Belarus and US; the effect of Chernobyl is less-than clear.

The chart above raises more questions than it answers. Note that the rate of thyroid cancer has doubled over the past few years, both in the US and in Belarus. Also note that the rate of cancer is 2 1/2 times as high in Pennsylvania as in Arkansas. One thought is test bias: perhaps we are  better at spotting cancer in the US than in Belarus, and perhaps better at spotting it in Pennsylvania than elsewhere. Perhaps. Another thought is coal. Areas that use a lot of coal tend to become sicker; Europe keeps getting sicker from its non-nuclear energy sources, Perhaps Pennsylvania (a coal state) uses more coal that Belarus (maybe).

Fukushima was a much less damaging accident, and much more recent. So far there has been no observed difference in cancer rate. As the reference below says: “there is no statistical evidence of a difference in thyroid cancer caused by the disaster.” This is not to say that explosions are OK. My company, REB Research, makes are high pressure, low temperature hydrogen-extracting membranes used to reduce the likelihood of hydrogen explosions in nuclear reactors; so far all the explosions have been hydrogen explosions.

Sources: for Belarus: Cancer consequences of the Chernobyl accident: 20 years on. For the US: GEOGRAPHIC VARIATION IN U.S. THYROID CANCER INCIDENCE, AND A CLUSTER NEAR NUCLEAR REACTORS IN NEW JERSEY, NEW YORK, AND PENNSYLVANIA.

R. E. Buxbaum, April 19, 2013; Here are some further, updated thoughts: radiation hormesis (and other hormesis)

Nuclear Power: the elephant of clean energy

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As someone who heads a hydrogen energy company, REB Research, I regularly have to tip toe about nuclear power, a rather large elephant among the clean energy options. While hydrogen energy looks better than battery energy in terms of cost and energy density, neither are really energy sources; they are ways to transport energy or store it. Among non-fossil sources (sources where you don’t pollute the air massively) there is solar and wind: basically non-reliable, low density, high cost and quite polluting when you include the damage done making the devices.

Compared to these, I’m happy to report that the methanol used to make hydrogen in our membrane reactors can come from trees (anti-polluting), even tree farming isn’t all that energy dense. And then there’s uranium: plentiful, cheap and incredibly energy dense. I try to ignore how energy dense uranium is, but the cartoon below shows how hard that is to do sometimes. Nuclear power is reliable too, and energy dense; a small plant will produce between 500 and 1000 MW of power; your home uses perhaps 2 kW. You need logarithmic graph paper just to compare nuclear power to most anything else (including hydrogen):

log_scale

A tiny amount of uranium-oxide, the size of a pencil will provide as much power as hundreds of train cars full of coal. After transportation, the coal sells for about $80/ton; the sells for about $25/lb: far cheaper than the train loads of coal (there are 100-110 tons of coal to a train-car load). What’s more, while essentially all of the coal in a train car ends up in the air after it’s burnt, the waste uranium generally does not go into the air we breathe. The coal fumes are toxic, containing carcinogens, carbon monoxide, mercury, vanadium and arsenic; they are often radioactive too. All this is avoided with nuclear power unless there is a bad accident, and bad accidents are far rarer with nuclear power than, for example, with natural gas. Since Germany started shutting nuclear plants and replacing them with coal, it appears they are making all of Europe sicker).

It is true that the cost to build a nuclear plant is higher than to build a coal or gas plant, but it does not have to be: it wasn’t that way in the early days of nuclear power, nor is this true of military reactors that power our (USA) submarines and major warships. Commercial nuclear reactors cost a lot largely because of the time-cost for neighborhood approval (and they don’t always get approval). Batteries used for battery power get no safety review generally though there were two battery explosions on the Dreamliner alone, and natural gas has been known to level towns. Nuclear reactors can blow up too, as Chernobyl showed (and to a lesser extent Fukushima), but almost any design is better than Chernobyl.

The biggest worry people have with nuclear, and the biggest objection it seems to me, is escaped radiation. In a future post, I plan to go into the reality of the risk in more detail, but the worry is far worse than the reality, or far worse than the reality of other dangers (we all die of something eventually). The predicted death rate from the three-mile island accident is basically nil; Fukushima has provided little health damage (not that it’s a big comfort). Further, bizarre as this seems the thyroid cancer rate in Belarus in the wind-path of the Chernobyl plant is actually slightly lower than in the US (7 per 100,000 in Belarus compared to over 9 per 100,000 in the USA). This is clearly a statistical fluke; it’s caused, I believe, by the tendency for Russians to die of other things before they can get thyroid cancer, but it suggests that the health risks of even the worst nuclear accidents are not as bad as you might think. (BTW, Our company makes hydrogen extractors that make accidents less likely)

The biggest real radiation worry (in my opinion) is where to put the waste. Ever since President Carter closed off the option of reprocessing used fuel for re-use there has been no way to permanently get rid of waste. Further, ever since President Obama closed the Yucca Mountain burial repository there have been no satisfactory place to put the radioactive waste. Having waste sitting around above ground all over the US is a really bad option because the stuff is quite toxic. Just as the energy content of nuclear fuel is higher than most fuels, the energy content of the waste is higher. Burying it deep below a mountain in an area were no-one is likely to live seems like a good solution: sort of like putting the uranium back where it came from. And reprocessing for re-use seems like an even better solution since this gets rid of the waste permanently.

I should mention that nuclear power-derived electricity is a wonderful way to generate electricity or hydrogen for clean transportation. Further, the heat of hot springs comes from nuclear power. The healing waters that people flock to for their health is laced with isotopes (and it’s still healthy). For now, though I’ll stay in the hydrogen generator business and will ignore the clean elephant in the room. Fortunately there’s hardly any elephant poop, only lots and lots of coal and solar poop.