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Rationally Speaking is a blog maintained by Prof. Massimo Pigliucci, a philosopher at the City University of New York. The blog reflects the Enlightenment figure Marquis de Condorcet's idea of what a public intellectual (yes, we know, that's such a bad word) ought to be: someone who devotes himself to "the tracking down of prejudices in the hiding places where priests, the schools, the government, and all long-established institutions had gathered and protected them." You're welcome. Please notice that the contents of this blog can be reprinted under the standard Creative Commons license.

Monday, June 11, 2012

Understanding Nuclear Power, Part 3: Fission reactors

by Ian Pollock

[Part 1 of this series, Whirlwind Nuclear Physics, appeared here; Part 2, Radiation and Health, appeared here.]

Today we discuss the general features of fission reactions and reactors. This is a pretty value-neutral post.

Thermal power stations in general

A standard nuclear reactor is basically a big tea kettle. It has this in common with most other thermal generation methods — coal, natural gas, geothermal, petroleum, nuclear fusion, and even the novel concentrated solar power (CSP). A source of heat (chemical reactions in the case of fossil fuels; nuclear reactions in the case of nuclear, geothermal and solar) is used to make steam, which turns a turbine (turbines look like this), which turns the shaft of a generator, which generates electrical current. Having passed through the turbine, the steam is condensed back into water in the condenser and then recycled (i.e., sent back to be converted to steam again by the source of heat). This complete cycle of conversion of heat into usable power is referred to as the Rankine cycle, and it lies behind essentially all thermal plants.

Just to drive home the fundamental similarity between all thermal power stations, see this basic diagram of a coal plant and this one of a nuclear plant.

A famous and highly relevant result from thermodynamics is that the efficiency of a heat engine (ranging between 0 and 1) is fundamentally limited by two temperatures: the temperature at which heat enters the engine (TH) and the colder temperature at which the heat engine rejects waste heat to the environment (TC). This relation (with temperatures in Kelvins, or some other absolute temperature scale) is given by:

efficiency ≤ 1 - TC/TH

Therefore, if you want to maximize efficiency, you can do so by both decreasing the temperature at which waste heat is rejected to the environment (e.g., build your plant in the arctic and use the ocean as a heat sink), and by running your engine at higher temperatures (e.g., superheat the steam before it goes through your turbines). Since it’s hard to vary TC given an already-chosen geographic location, increasing TH is a really important factor in increasing plant efficiency (for all thermal power stations).

The most conspicuous and iconic parts of a standard nuclear plant are the large hyperboloid cooling towers (think of the intro to The Simpsons), and these are necessary in order to reduce TC as much as practical. Their sole purpose is to allow the water (once steam) that has already gone through the turbine to cool back down to the ambient temperature of the environment. Despite their portentous pop-culture symbolism, they are themselves by far the most innocuous part of a plant.

Alert readers may have noticed the wastefulness inherent in rejecting heat to the environment at a power station, in order to generate electricity, in order (often) to create heat again for the end user, whether residential or industrial. It is in fact possible to direct this ‘waste’ heat straight to people who need it, in a process termed cogeneration. This can be done with any thermal power station, though there are technical challenges.

Given the ubiquity of steam turbines in power generation, it shouldn’t be surprising that engineers have worked out all sorts of tricks for optimizing the performance of steam turbines. All these tricks are available to all thermal generation methods, so that the relevant differences between nuclear generation and (say) gas generation are not to be found here so much as farther upstream, in the reactor, fuel cycle, and waste products.

Fission reaction

The heart of a nuclear plant is the controlled fission reaction.

Fission refers to the explosive breakup of a large nucleus, with the simultaneous release of a great deal of energy. It would help to quantify ‘a great deal.’

A typical exothermic chemical reaction, such as the burning of gasoline (octane), releases an energy of about 50 electron-volts (eV) per molecule of octane. The electron-volt is a tiny unit, useful for the amount of energy released in a single reaction (we don’t want to always have to write out a tiny fraction of a joule). Per kilogram of octane, this reaction generates about 3⋅10^26 eV, or 45 megajoules of energy. That’s quite a bit.

However, both per single reaction, and per mass of fuel, nuclear reactions are much more energetic than chemical reactions. The fission of a single U-235 atom generates about 200 MeV (mega electron-volts) — that’s a factor of 4 million more energy than a single octane molecule combustion. Looked at on a per-weight basis, 1 kilogram of pure U-235 generates 80 terajoules — this time a factor of about 2 million more than 1 kg of gasoline.* One can see why, when physicists first saw these numbers, they used phrases like “too cheap to meter.” Although that proved to be an exaggeration, this tremendous difference in energy density is a big part of nuclear’s advantage over other generation methods.

When bombarded with a neutron, a U-235 nucleus will undergo fission, exploding into two daughter nuclei of varying types (e.g., Barium-141 and Krypton-92) and releasing the aforementioned 200 MeV of energy, as well as, on average, 2.4 neutrons per reaction. This is great, because it means that those neutrons can go on to bombard other U-235 nuclei in a self-sustaining chain reaction.

Now let’s put ourselves in the shoes of somebody who, knowing this, wants to generate power from nuclear fuel.

Obstacle number one is that the neutrons need to actually hit other U-235 nuclei in order to cause further fissions. Whether they do this or not depends on how closely spaced the U-235 nuclei are, and on the geometry of the piece of nuclear fuel. When the geometry and density of U-235 are such that one fission reaction generates, on average, one or more subsequent fission reactions, it is said to have reached “criticality.” In a nuclear weapons context, one speaks of “critical mass” — really a combination of mass and geometry in which one fission leads immediately to substantially more than one further fission in an exponential progression.

Obstacle number two is that U-238, which makes up the vast majority of the atoms in naturally occurring Uranium (99.3% by weight), is not fissile. In fact, U-238 tends to absorb neutrons, preventing them from impacting the U-235 nuclei and causing new fission events (however, neutron absorption by U-238 is not, on balance, a bad thing, as we shall see). Because of U-238’s neutron-absorbing effect, many reactors require enriched Uranium fuel (about 3% U-235). However, enrichment is difficult and expensive.

Obstacle number three is that not all neutrons cause U-235 to fission equally well. Slow, or “thermal” neutrons are most effective in doing this, and are also absorbed less by U-238. However, the neutrons ejected from a fission event are fast neutrons. Accordingly, they need to be slowed down before they are very effective in causing further fissions, and this requires the use of a moderator — in essence, a material that slows neutrons down. Commonly used moderators are ordinary or “light” water (e.g., United States), “heavy” (neutron-enriched) water (e.g., Canada), and graphite (e.g., former Soviet Union). Heavy water and graphite have an advantage over light water in that, because they are more effective moderators, unenriched Uranium can be used as a fuel. However, graphite presents a fire risk (Chernobyl), and heavy water is more expensive than light water. We’ll talk more about this later.

As a given piece of nuclear fuel spends more and more time in the reactor, its composition changes. More and more atoms of U-235 fission, creating the aforementioned fission fragments, and U-238 is transmuted into Pu-239, which also undergoes fission. Both the fission products and Pu-239 represent a mixed blessing.

On the one hand, the fission products decay radioactively, which releases heat that contributes to steam generation. Likewise, the generation and subsequent fission of Pu-239 releases large amounts of useful heat.

On the other hand, some fission products, for example Xenon-135, are “reactor poisons,” meaning they absorb neutrons very easily, slowing or stopping the reaction. Because Xe-135 builds up to especially high levels after a reactor is shut down, it precludes easy re-start of reactors, so while a natural gas plant can power up and down at the drop of a hat, typical nuclear plants are less time-flexible (you have to wait around 10 hours for the Xenon to decay away). Eventually, buildup of reactor poisons in the fuel requires a particular fuel assembly to be removed from the reactor as “spent fuel.”

Likewise, while from a pure engineering perspective, the generation of Pu-239 in a reactor is great (more fission, more power output per kg of fuel), it leads to nuclear weapon proliferation concerns (Pu-239 is relatively easy to chemically separate from the spent reactor fuel and purify into weapons-grade plutonium).

Sometimes, reactor poisons such as Boron are pumped into the reactor coolant intentionally by reactor operators to control reaction rates, or as emergency shutdown measures.

Common features of most modern reactors

Most operational modern reactors have a rather similar generic design at the level of the reactor core, which is worth understanding before we branch out into the specifics of particular reactor types.

First of all, we have fuel pellets, which are basically Uranium dioxide compacted and sintered into a ceramic pellet that looks like this. These can contain Uranium at various enrichment levels, but a typical level would be 3% U-235. The fuel pellets are stacked inside a long cylindrical shell of (usually) Zirconium alloy cladding, which does not absorb many neutrons and is highly corrosion-resistant. We now have a so-called fuel rod. These fuel rods, about 0.5-3m long, are then assembled into a fuel bundle, a grouping of up to a few hundred rods that looks like this.

A single reactor might take hundreds to thousands of fuel bundles and assemble them in a reactor core, which schematically looks something like this.

Obviously, the nuclear chain reaction is something that needs to be carefully controlled if it is to be used safely in a power reactor. Depending on the reactor, various safety features to prevent a criticality accident may be in place, but direct control of the rate of the chain reaction is accomplished using the control rods, which look like this. Control rods are simply neutron absorbers (typically containing some combination of Silver, Indium, Cadmium and Boron) that can be partially or wholly inserted into or removed from the reactor core. Since they prevent neutrons from going on to produce further fission events, the farther they are inserted into the core, the less critical the reaction gets. In this way, the reactor is controlled by negative feedback from the operators.

All of this takes place inside a reactor pressure vessel designed to handle the moderate to high pressures involved in a nuclear plant, typically 7 to 15 MPa, which looks something like this. It includes lots of sensors that are required by power station operators to monitor the reaction and the core conditions.

Coolant water driven by a pump circulates through the reactor core, transporting heat away from it and towards a secondary water loop or directly to a turbine. A coolant loss will have the immediate effect of stopping the chain reaction (because water is typically the moderator), but it will also mean that the core is no longer cooled, and the fission products caused by the reaction will generate a lot of heat for a long time after the reaction has stopped. For this reason, it is vital for the core to be continuously cooled. Accordingly, backup generators and backup pumps are typical.

In almost all plants, all critical components of a reactor that are radioactive are contained inside a large concrete structure called a containment building, which looks like this. The idea behind the containment building is to prevent the release of radioactive material (chiefly contaminated steam) to the environment in the event that a nuclear accident is not stopped by other safety features; the containment building is thus a last line of defense. We will discuss this more in an upcoming post on nuclear accidents.

Here is a video showing the whole process in a way that will probably help you to picture it; in this case, the reactor is a specific type called a Pressurized Water Reactor, which we will discuss in the next post, on reactor types and fuel cycles.


Useful sources:

David Bodansky, “Nuclear Energy: Principles, Practices and Prospects.” 2004, Springer.

John R. Lamarsh and Anthony J. Baratta, “Introduction to Nuclear Engineering.” 2001, Prentice Hall.

Anthony V. Nero, Jr, “A Guidebook to Nuclear Reactors.” 1979, University of California Press.

* It’s a little bit tricky to compare things here, because gasoline’s burning also requires oxygen, and because the number for U-235 only counts the U-235 fission events, not the decay heat released by fission products, or the fact that U-235 fissions also help to transmute fertile U-238 into fissile Pu-239, which itself fissions in the reactor. Moreover, nuclear fuel is mainly U-238 by weight, not U-235 — a further complication. However, no matter how you slice it, the energy density for nuclear is a lot higher than fossil fuels.


  1. Quote: "However, both per single reaction, and per mass of fuel, nuclear reactions are much more energetic than chemical reactions. The fission of a single U-235 atom generates about 200 MeV (mega electron-volts) — that’s a factor of 4 million more energy than a single octane molecule combustion. Looked at on a per-weight basis, 1 kilogram of pure U-235 generates 80 terajoules — this time a factor of about 2 million more than 1 kg of gasoline.* One can see why, when physicists first saw these numbers, they used phrases like “too cheap to meter.” Although that proved to be an exaggeration [in other words, a big fat lie], this tremendous difference in energy density is a big part of nuclear’s advantage over other generation methods."

    Not counted here is the energy required to mine the uranium and process it so that the energy can be extracted. Also not counted is the energy required to construct the plant. Decommissioning costs, transportation costs are also not included here.

    As Caldicott has written,

    "... the actual costs of nuclear energy are consistently misstated and incomplete. Nuclear power is(also)heavily subsidized by taxpayers (through programs that benefit the industry, but are excluded from their cost estimates). Developed countries ostensibly wedded to the principles of economic rationalism and the 'free market,' are inexplicably enthusiastic about nuclear power, which cannot be sustained without huge government subsidies and handouts from its very inception. This socialization of electricity within a capitalist society has never been called into question, nor has it been critically scrutinized by the general public and their elected representatives."

    1. "Not counted here is the energy required to mine the uranium and process it so that the energy can be extracted. Also not counted is the energy required to construct the plant. Decommissioning costs, transportation costs are also not included here."

      True, but I also didn't include drilling, transport, cleanup etc. energy for gasoline, so the comparison is still pretty much apples-to-apples.

    2. Nonsense! How can you say that using sloppy procedure in one case will necessarily balance out with sloppy procedure used in another case?

      The not calculated costs for nuclear may be much higher than the not calculated costs for gasoline, particularly since the cost for nuclear *should* (but often doesn't) include costs for disposal and seclusion of waste that can remain dangerous for thousands of years -- a problem that remains, as yet, unsolved. You can't even put a price tag on it because no one knows how to solve the problem at any price.

  2. Replies
    1. @rdbrown: Thanks for pointing out the "Nuclear Options" article. The author's opinion on nuclear is mainly "meh", but even this is being overly optimistic!

      The article states:

      "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."

      Um, that's a pretty big difference! Like saying that the only difference between us is that you get your source of money from being a pharmacist, and I get my source of money from dealing black tar heroin.

      Also, the article never discusses the percentage of uranium in the ore that is needed to be economical. It even discusses extracting uranium from seawater! (While, at least, acknowledging that it is *not* economical!)

      Many decades ago, German scientists had a plan to pay off Germany's war debt: they would extract minute amounts of gold from seawater and use the gold to pay off the debt. You may notice that, decades later,
      there are no gold extraction facilities located on the seashore. Of course, the cost to extract the gold from seawater far far far exceeds the value of gold so obtained.

      Likewise, it is plain silly to discuss the energy in a kilogram of U235 without discussing the energy required to *obtain* the kilogram and the energy required for all the proper measures that should be taken to safely release that energy.

      Discussions of obtaining uranium from seawater is just a pie-in-the-sky "too cheap to meter" fairy tale. I wouldn't hold my breath waiting for it to happen.

      Thanks for the article. I may have some more comments on it when I digest the article further.

    2. Speaking of "doing the math", I had some problems with the math in the article.

      Tmurphy states that .7% of natural Uranium is U235. He then states that to get 600 tons of U235 we need to dig 800,000 tons. But when I divide 600 by .007, I get 85,714.

      "That works out to about 600 tons [of U235] per year. In terms of total uranium (235U is only 0.7%), we would therefore need to dig out about 800,000 tons yearly."

      Now perhaps tmurphy is talking about the total amout of *ore* you'd have to dig, not the amount of uranium oxide / yellowcake. In the U.S. the percentage of uranium oxide in the ore is about .05% to .3%. You'd have to dig about 333 to 2,000 tons of ore for each ton of uranium oxide (and get only .007 ton of U235 out of that). To come up with 800,000 you'd have to have an ore with about 10% Uranium Oxide in it (an un-heard of amount). It is hard for me to see where tmurphy gets his figure of 800,000.

      Tmurphy also states that 20 grams per second works out to 600 tons per year. I get 695 tons (20*60*60*24*365/907185) -- closer to 700 tons. Unless tmurphy is referring to metric tons, which would be about 630 tons per year.

      In my math, 20 grams of U235 per second requires 630 metric tons of U235 per year, which requires 90,000 metric tons of natural uranium per year [a Scientific American article puts the figure at 70,000 metric tons per year, using different initial asumptions] which requires that you dig and process anywhere from 30,000,000 to 180,000,000 tons of ore per year.

    3. RE: "Nuclear Options" / "Do the Math":

      The "Do the Math" slogan goes up my nose a bit: it implies that people who disagree with the conclusions have *not* done the math.

      In truth, math works on a "garbage in, garbage out" principle. Math can only calculate the factors that are input. It is no more infallible than the assumptions you make.

      Sec. of State McNamara committed this error in the conduct of the Vietnam War. He stubbornly pointed to statistics "proving" that he was winning the war
      without considering that the statistics he relied upon were garbage based upon cherry-picked facts and distorted information. This was satirized in the movie
      "Full Metal Jacket", when Private Joker asks if the blood trails found after a fire-fight should be reported as their platoon "killing a NVA General".

      I think the most interesting (but perhaps unwitting) point in the article is this:

      "Unlike solar or wind, nuclear will not be distributed, but will be centralized into large, expensive, high-tech facilities."

      The implications for democracy are not good.** When you have a large concentration of wealth and power the tendency is for the interests of the wealthy investors to run roughshod over the interests of the public. If you have billions invested in a power plant, you want that plant to operate and make a return on investment. You can't have some hippy environmentalist standing in the way and threatening to shut the plant down. The fact that the hippy environmentalist may be in the *right* is irrelevant. The money and power behind a nuclear project will take on a life of its own and, like a run-away freight train, it will crush anyone who dares stand in the way. What inevitably happens is that facts are re-interpreted, ignored, or kept secret if those facts interfere with profits. It is the avowed principle of the Nuclear Regulatory Commission that the Nuclear Industry be mainly "self policing". The NRC's approach to questions of safety is to evaluate them on a "cost/benefit" basis (their own words).

      The "cost/benefit" approach puts the cart before the horse: it assumes, a priori, that nuclear energy is cost effective, and then uses that assumption to evaluate the worth of safety regulations. If a regulation stands in the way of making a profit, then negative profits is taken as evidence that the proposed regulation must be unreasonable and too burdensome.

      It should be the other way around. Responsible people decide on regulations that *must* be met -- whether it is profitable to the investors or not. But it's not what happens in real life.

      ** "In the mid-1980s the citizens of the three counties surrounding Three Mile Island voted by a margin of 3:1 to permanently retired TMI Unit One, which had been shut when Unit Two melted. The Reagan Administration trashed the vote and re-opened the reactor..." (Harvey Wasserman)

  3. Since there seems to be a paucity of comments here, allow me to buttress my previous comments with two recent newspaper articles.

    (1) "Energy Department Steps In to Help Uranium Enrichment Company" [New York Times, 06/14/2012]

    Via the U.S. Department of Energy, the taxpayers are taking over tons of left-over uranium from a company called USEC. This uranium is considered to be WASTE and is worthless on the open market. Indeed, USEC lists the left-over uranium as a LIABILITY, and rightly so, since it would cost money to dispose of it. Taxpayers will pay USEC $88 million dollars for this trash, and perhaps an additional $190 million in aid to the same company. The government handouts going to the company are more than the entire company is worth!

    This is the manner in which nuclear fuel is "cheap" -- enormous government subsidies.

    (2) "San Onofre's Cloudy Future" [Los Angeles Times editorial, 06/24/2012]

    Several points are made in this editorial. San Onofre is my local nuclear power plant. There is a "flaw" in the generators that cost the ratepayers $671 million to build. The tubes vibrate too much, which causes excessive wear, and it is a very expensive replacement. Who will pay for that? Not Mitsubishi Industries (maker of the generators). Mitshubishi EXPLICITLY EXCLUDES themselves from liability in their contract. Strange, that people who build generators and claim that they are safe and cheap will not accept the liability. The costs are always fobbed off on the ratepayers, or taxpayers in general, while the industry takes the profits (see the "Price Anderson Act").

    The tube problem occurred because the new generators were represented as being nearly identical to the old ones -- allowing them to be "fast tracked". The NCR was only too happy to agree, without bothering to note that the tube design was, in fact, very different -- hence the vibration problem.

  4. Some additional points made in the "San Onofre" editorial:

    * more sophisticated techniques have discovered that there are greater seismic threats to San Onofre than previously believed (at the time of licensing).

    * use of seawater by the plant is damaging marine life. This is really a story in itself (not in the editorial). Environmentalists warned of this result. San Onfre got licensed over the objections of environmentalists on the promise that it would create a marine life refuge to "offset" the sea life that they would be killing. After the license was granted, they subsequently wormed their way out of the expense of the "offset". In a closed door meeting, the compliant people at the NRC were only too happy to agree to let them off the hook: Let the PUBLIC bear the cost of lower quality environment, while the industry takes the profits and crows about how "cheap" nuclear power is.

    * the editorial noted "there's still no solution to the problem of how to store nuclear waste.

  5. I like Helen Caldicott and her data may be reliable, but it's nuke or the vege patch (sustainability), and it will take a while to wind down. 6 billion, half in deprivation rising every day, nuke may need to be a buffer, but its too balanced to be anything more than an attitude on my part.


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