By Tom Murphy
05 January, 2012
Do The Math
A recent thrust on Do the Math has been to sort our renewable energy options into “abundant,” “potent,” and “niche” boxes. This is a reflection of my own mathy introduction to the energy scene, the result of which convinced me that we face giant—and ultimately insurmountable—hurdles in our quest to continue a growth trajectory. It is not obvious that we will even manage to maintain today’s energy standards. We have many more sources/topics to cover before moving on to the “now what” phase of Do the Math. Meanwhile, requests for me to address the nuclear story are mounting. So before readers become mutinous, I should interrupt the renewable thread to present my nuclear reaction. It’s a rich topic, and in this post I will only give a tutorial introduction and my big-picture take. A single post can’t possibly address all the nuances, so my main goal here is to demystify what nuclear is all about, build a vocabulary, and set a foundation for further discussion in later posts.
First of all, it is important to understand that a nuclear power plant operates in almost entirely the same way as a coal-fired plant. The chief difference is in the source of heat. For the coal plant, pulverized coal is combusted in a giant furnace to deliver heat to water, thereby generating steam that drives the turbines. Nuclear plants use energetic fission events to create heat, generating steam that drives the turbines. Did you pick up on the similarity?
Thus nuclear plants are merely heat engines that extract useful work from some fraction of the thermal energy flowing from a hot source (flame or fission) to a “cold” source (condenser). The hourglass-shaped cooling towers that came to symbolize nuclear power may just as often be found in modern coal-fired plants away from large bodies of water. Conversely, nuclear plants near large bodies of water (like San Onofre in SoCal) lack cooling towers and are rather compact installations.
Twin 1.1 GW reactors at San Onofre Nuclear Plant. No cooling towers necessary. From Southern California Edison.
Before we go on, let’s make sure we’re all on the same phonetic page. A branch of Fizzics deals with with the nukulous of atums, and this is where nukular energy comes from. I think there may be some operating plants in Texas. In this post, we’ll deal with the physics of nuclear energy arising from the nucleus of atoms. When in doubt, think “Let’s be clear—we’re dealing with nu-clear.”
Okay, I just had to get that off my chest—partly to confront my Southern heritage. The main point so far is that nuclear plants are only complicated in how they produce their heat. After that, it’s standard fare. And due to their similar design, both achieve something like 30–40% efficiency in converting thermal energy into useful electricity.
Heavy nuclei can be split apart by blasting them with high-energy particles and having extremely good aim. But this is no fun, since the requisite energetic particles can be hard to produce. Nature provides a subtle trick, however, in that three semi-stable nuclei are capable of splitting apart by the mere introduction of an extra (slow) neutron. These are 233U, 235U, and 239Pu. That’s uranium and plutonium to most of us. The preceding superscript denotes the total number of nucleons—another name for protons and neutrons—contained in the nucleus. Uranium always has 92 protons, while plutonium always has 94 protons (it’s what makes them what they are: chemically, it’s all about how the neutral atoms behave, containing 92 and 94 electrons in differing configurations, respectively). This leaves 141, 143, and 145 neutrons in 233U, 235U, and 239Pu, respectively.
As an aside, what pattern do you notice in the names uranium and plutonium? Hint: there is one element in between. Further hint: its name is neptunium. Considering that Pluto has been voted out of the planet club, perhaps it’s overdue that we vote plutonium off the periodic table! Hmmm. Astronomy textbooks are changing, but I suspect that Pluto’s planetary status will be immortalized in chemistry textbooks.
Left alone, 233U, 235U, and 239Pu will spontaneously decay (not by fission, but by emission of an alpha particle: two protons and two neutrons in a tidy chunk) on a timescale, or half-life, of 159,000, 704 million, and 24,000 years, respectively. But if we add one more neutron to any of these, they stand a significant chance of splitting into two large chunks and a few spare neutrons. The aggregate particle count does not change in the process. The extra neutrons are left to wander into other fissile nuclei and keep the process going—called a chain reaction. In order for this to work, one must control the likelihood that neutrons get absorbed by non-fissile material (including control rods), and have enough fissile nuclei about to keep the process humming (critical mass).
For instance, if we add a neutron to 235U, it momentarily becomes 236U before splitting into, for example, 97Rb and 137Cs plus two spare neutrons. Because heavier nuclei are increasingly neutron-rich (protons repel each other, so the fewer the merrier in a large nucleus), the “daughter” nuclei are exceedingly neutron rich for their tax bracket, and start converting neutrons to protons via “beta” decay, ejecting high-energy electrons in the process. Thus we say that the daughter nuclei are radioactive and emit damaging (ionizing) radiation in the form of fast electrons that can rip right through your DNA. In the above example, the daughter nuclei are short-lived: within a few seconds, the rubidium beta-decays several times through strontium and yttrium, pausing at zirconium for about a day, then spending a minute as niobium, finally retiring as a stable isotope of molybdenum (97Mo). The cesium lasts for about 30 years (half life) and beta-decays to a stable barium isotope (137Ba).
There is no controlling what the daughter products will be. In practice, they cover a distribution of middling-sized nuclei in two groups (one around 135 nucleons and the other around 95—sometimes called “fish and chips”—see figure below). As a rule, they will be neutron-rich and undergo beta decays on a whole range of timescales—some getting stuck for thousands of years. It is also possible that the absorbed neutron fails to spur fission, and the nucleus keeps absorbing neutrons to create “trans-uranic,” or actinide radioactive waste. The unstable daughter nuclei (fission products) and trans-uranic nuclei are what make spent nuclear fuel hazardous—often for very long times.
Fission yield probabilities for the three fissile nuclei, generally yielding a big nucleus and a small one. Yields for the three nuclei are more similar than different. The number of nucleons is plotted on the horizontal axis.
We’ve looked at the accounting of splitting atoms, but where does the heat come from? When these nuclei and neutrons erupt from the unstable nucleus, they fly apart, carrying kinetic energy. As they rudely bump into surrounding atoms, they deposit this energy as a cacophony of motion/vibration in the solid lattice. We call this heat. A typical fission event releases about 200 MeV of energy—the energetics of which we will revisit later.
Of the three slow-neutron fissile nuclei, only 235U is found naturally. The other two have substantially shorter half-lives, so the 233U and 239Pu provided in the initial supernova-generated stock of material for Earth has long since decayed away. Meanwhile, 235U has decayed much faster than its sister 238U, which has a half-life of 4.5 billion years (coincidentally about the age of Earth, so we have half of our original stock of 238U). Today, 99.3% of natural uranium is the impotent 238U variety, the remaining 0.7% being the fissile 235U. This is a very significant fact.
But there is a back door. Reactor fuel is typically enriched to be 3–5% 235U, leaving lots of 238U in the reactor. When 238U happens to gobble up one of the wandering neutrons, it becomes 239U, which is unstable and undergoes two beta decays in a matter of days to become—presto!—239Pu. Now we have two fissile contributors in the reactor, and in practice 239Pu contributes something like a third of the total energy in a typical reactor.
Some reactors are designed to be especially efficient at generating 239Pu, and these are called breeder reactors, built to make weapons material. On the energy front, breeders open up 140 times more uranium supply than is found naturally in 235U, by using the ubiquitous 238U nuclei. The principle problem with breeders is that plutonium is chemically distinct from uranium, making it very straightforward to isolate and make bombs. Conversely, the two isotopes of uranium are notoriously difficult to separate (enrich). Enrichment is a significant hurdle to those who strive to have nuclear weapons. Abundant plutonium would change the calculus considerably, tipping the scale toward weapons proliferation.
A similar back-door trick can be used to breed 232Th into fissile 233U (taking a month to work through the beta decays). Thorium is several times more abundant in Earth’s crust than uranium. It’s not fissile out of the ground like 235U is: some assembly required. In that sense, it is not terribly different from 238U. One of the main differences is that the bred 233U is often contaminated with 232U, which has a 69 year half-life and is a prodigious emitter of high-energy gamma radiation along its decay chain. After one year, the escalating radiation level from uranium that is 5 parts per million 232U is seven times higher than that of reactor-grade plutonium, and about 50 times worse than weapons-grade plutonium (reference). And because the gamma ray emission is higher-energy than the corresponding emission from plutonium, it is harder to shield.
The bottom line is that 232U contamination of thorium-produced 233U makes 233U very dangerous to handle, which is considered to deter proliferation (easy for terrorists to find uneducated martyrs, harder to find nuclear technicians willing to sacrifice their lives). Deliberate contamination with 232U would help curtail proliferation, and would make the substance easier to detect from its gamma emission.
I will no doubt get a lot of flak for not gushing over the advantages of thorium over conventional uranium reactors. It is not hard to find accounts extolling all the virtues—including chewing through our current stock of spent fuel (though no help on daughter products) and passive dumping in a loss-of-cooling emergency (if the salt plugs work properly). But it’s not all a bed of roses, and I sense that there’s a bit of “hopium” mixed in. I will need to spend time studying thorium more fully before I can say much more.
Because it’s fun to apply E = mc², I’ll use the example above of 235U plus a neutron producing 97Rb and 137Cs plus two neutrons. A neutron has a rest-mass of 1.0087 atomic mass units (a.m.u.: 1.6605×10−27 kg), and 235U has a rest-mass of 235.043 a.m.u. On the other side, 97Rb is 96.937 a.m.u. and 137Cs is 136.907 a.m.u. If we add up the initial mass, we get 236.05 a.m.u., and for the final mass we get 235.86 a.m.u. (235.83 after all beta decays are done).
Somebody stole 0.2 atomic mass units! Putting this into E = mc² (with mass in kg and c ≈ 3×108 m/s), we find the missing amount to be 3×10−11 Joules. It might not sound like much, but a gram of 235U contains 2.6×1021 atoms, for a net of 76 GJ of energy. This amounts to about 20 million kilocalories per gram—over a million times more potent than chemical energy. This is why nuclear is such hot stuff—literally.
To put a scale on things, the world’s energy appetite, at 13 TW, would require the expenditure of about 20 grams of 235U per second: a seemingly paltry amount. That works out to about 600 tons per year. In terms of total uranium (235U is only 0.7%), we would therefore need to dig out about 800,000 tons yearly. At present, the world uses about 60,000 tons of uranium per year (correcting for the fact that 85% of the yellowcake ore—U3O8—is actual uranium), even though only 0.75 TW (thermal) of power currently derives from nuclear reactors worldwide. Scaling this number to 13 TW implies a need for 1 million tons of uranium per year, in reasonable agreement with our first estimate.
So how much uranium resource exists in the world? The World Nuclear Association puts the number at about 5.4 million tons of ore (4.6 t U). At present rates of use and no more exploration, this suggests 80 years of resource. But nuclear is just 6% of global energy production. In the spirit of stacking each energy resource up against our total demand—as I have done for solar, wind, tidal, hydroelectric, etc.—the implication is that we would deplete our resource in a mere 6 years if we required conventional nuclear power to be our sole source of energy!
That’s the scary/extreme version of the story. We would no doubt identify additional resources (at higher extraction cost) if push came to shove. Presently, nuclear fuel is not the cost driver for generating nuclear electricity, so there is room for mining costs to increase before nuclear gets more expensive (already loses economic competitions, though). Still, doubling or tripling the resource does not break us out of the few-decade regime. Therefore conventional nuclear cannot provide a long-term replacement for fossil fuels—and it’s not nearly as convenient in any case.
If it became economical, we could harvest uranium from the ocean, where each cubic meter of seawater contains about 3 mg of uranium (3 parts per billion). At a volume of 1.3 billion cubic kilometers, this translates to 4 billion tons, extending the supply by a factor of nearly 1000.
We also must consider the breeder angle. Only 0.7% of natural uranium is fissile, so breeders can extend the scope by a factor of 140. Now our decade scale is a millennium scale. Likewise for thorium, which is a breeder program from the start, and arrives on the scene more abundant than uranium.
So Do We, or Don’t We?
You are indeed a patient lot, bearing with me through a lengthy pedagogical exposition when all along you just want to know whether nuclear will be a major component of our future and can scale sufficiently to put our minds at ease with respect to energy scarcity.
And my answer is: I could imagine it going either way. After all, I would likely not be motivated to “do the math” on our various energy options if I thought that nuclear would simply take care of business without worry.
Even though nuclear fission does not represent a renewable resource (at best we’re dealing with millennia, not eons), breeder programs can make it last long enough for me to throw it into the “abundant” box. Absent breeders, I must put conventional nuclear employing conventional mining into the “niche” box, since it can’t supply over a quarter of our demand for a meaningfully long period. Ocean harvesting allows conventional nuclear to jump straight over “potent” back to “abundant.” It’s worth emphasizing that we can’t simply ramp up the nuclear we know and—ahem—love, and expect to get much out of it.
So the prospects for nuclear are all over the map. As such, economic, sociological, and practical concerns—more so than physics/math—will determine how large a role nuclear will play.
Let’s start with the obvious. Nuclear represents yet another way of creating electricity. Between solar photovoltaics, solar thermal, wind, hydroelectric, geothermal, tidal, wave, ocean currents, etc.—granted, some of these are puny—we are not running short of ways to generate electricity. Nuclear has no special power over the others to alleviate the liquid fuels crunch we will soon experience. Like all the other options, it’s no panacea. It’s not the silver bullet we might hope to find.
Nuclear power does not tolerate rapid changes in power output. It can adjust over a matter of hours, not minutes or seconds. It is therefore not useful for balancing short-term intermittency arising demand fluctuations or from wind and solar, etc. On the flip side, it does not have problems with intermittency. It could therefore be a useful baseload source, which accounts for about a third of peak electricity demand, and about two-thirds of total electricity produced.
Unlike solar or wind, nuclear will not be distributed, but will be centralized into large, expensive, high-tech facilities. Successful operation of a large fleet of nuclear power plants requires a highly functional, stable society with an educated workforce—you know, like Homer Simpson and buddies Lenny and Carl (I was surprised to learn that Lenny “actually” has a master’s degree in nuclear physics). So nuclear looks fine if our transition to the future is smooth and orderly. If we get caught off guard by energy scarcity—possibly leading to hoarding, resource wars, severe economic crashes, and other surprises—then it may be very difficult to mount a strong nuclear response. In other words, nuclear is the most technologically challenging of our primary energy options, so that we risk not having it together enough to pursue the nuclear road when the need becomes apparent. Things would have to get pretty bad before this self-limiting effect kicks in, and I cannot easily assess how likely such a scenario might be. As such, this consideration is not the reason I’m not hyped about nuclear, but it is certainly a factor to bear in mind. Nuclear requires advanced planning, and is not going to be an effective pinch-hitter in a crunch.
Then of course we have the pesky issues of waste storage (unsolved worldwide after five decades), proliferation of weapons-grade material (relevant for breeders), and safety. True, radiation levels outside a coal-fired plant are higher than outside a nuclear plant—until they’re not. Three-mile Island, Chernobyl, and Fukushima certainly violated this “truism.” When nuclear plants have trouble, the trouble can be pretty unsettling.
This brings us to the societal angle. People fear nuclear power, almost as much as they fear nukular power. First, we form a mental connection with the most destructive weapons known to man. The connection becomes real when proliferation is considered. Second, when nuclear plants have trouble, it’s big news—leading to out-sized fear. More people die from bee stings than from shark attacks, yet our fear of the shark is vastly greater than our fear of bees. Crazier yet, we don’t run screaming from cars—as deadly as they are—and will actually wade into a parking lot packed with these killers. Third, we have not been able to implement a waste storage scheme yet, holding spent fuel indefinitely at nuclear plants in swimming pools that were meant to provide temporary storage only.
Spent fuel rods stored in a "temporary" pool at a nuclear plant.
Public acceptance of nuclear power plants tends to run counter to their acceptance of Congressional Representatives. Huh? We (in the U.S.) tend to despise Congress, giving approval ratings as low as 10% for the job they do. Meanwhile, our local Representative has a 90% chance of re-election because they’re perceived to be one of the good eggs. Conversely, we might overall begrudgingly accept nuclear power at a national level, but not in our district, please. We invoke the NIMBY (not in my back yard) response to almost any new proposed facility (wind, solar, coal-fired plant, etc.), but our aversion to nuclear is on a whole different level.
As a scientist who understands what nuclear energy is all about—that a power plant cannot detonate, that nominal radiation levels outside the plant are quite low, that natural radiation is far greater (especially during air travel), and that nuclear fuel is hyper-efficient compared to chemical sources—I don’t personally harbor an intrinsic knee-jerk fear of nuclear power. Nor do I discount the waste storage complications, the potential for proliferation, or the finite nature of the resource. These are legitimate societal concerns (see the Union of Concerned Scientists’ super-solid analysis on the pros and cons). I am also aware of the complexity inherent in building and maintaining a nuclear plant, and that the energy derived is never going to be “too cheap to meter,” as was once promised.
Sure, molten salt thorium reactors could ease some of these concerns (and bring about new ones). As I said, I need to look at this option in more depth. One thing I know already: nuclear is far more complicated than many of our other other options (e.g., solar, wind), relies on rare materials for fuel, and has serious societal barriers. That said, it has some very nice properties as well: stable power output, low carbon, potentially abundant fuel (if economies allow), and decades of experience (on certain reactor types).
In the end, I think nuclear is likely to play an increasing role in our energy story. Energy hardship will trump concerns over waste, proliferation, and safety. As long as such hardship does not bind us in an Energy Trap or plunge us into dysfunction, we will likely build more plants. But because nuclear does not smack the primary problem right on the kisser (fossil fuel substitute), I doubt it will be heralded as the answer to our prayers, and imagine that its role will be correspondingly modest.
Are you as disappointed in my non-committal answer as I am? Perhaps I could have saved us all a lot of time, and just said: “Meh.”
We’ll see fusion next week.
Tom Murphy is an associate professor of physics at the University of California, San Diego. An amateur astronomer in high school, physics major at Georgia Tech, and PhD student in physics at Caltech, Murphy has spent decades reveling in the study of astrophysics. He currently leads a project to test General Relativity by bouncing laser pulses off of the reflectors left on the Moon by the Apollo astronauts, achieving one-millimeter range precision. Murphy’s keen interest in energy topics began with his teaching a course on energy and the environment for non-science majors at UCSD. Motivated by the unprecedented challenges we face, he has applied his instrumentation skills to exploring alternative energy and associated measurement schemes. Following his natural instincts to educate, Murphy is eager to get people thinking about the quantitatively convincing case that our pursuit of an ever-bigger scale of life faces gigantic challenges and carries significant risks.
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