Harnessing the stored energy of heavy elements
This post will explain how we can access the stored nuclear energy in heavy elements using nuclear reactors. This is fission.
Nuclear fuel is very energy dense, by volume or by mass, compared to anything else we've used in the past. The energy density of a nuclear fuel for fission is about one million times greater than fossil fuels using today's fission technologies and perhaps 10x further using next generation techniques. Fusion fuels are about 4-10x higher in energy density than fission fuels. But this is in terms of energy yield per mass of fuel. The energy delivered to the end-use per mass of fuel will likely be almost equivalent once a fusion reactor goes online. Fundamentally, nuclear fuel's higher energy density should make it much easier to build machines to harvest and use it.
What is energy by the way? In a fundamental way, it's the difference in the conditions of nearby systems. The universe appears to want to flatten things out or homogenize these differences, which creates flows from high energy states to low energy states. This can be water flowing down a gravitational potential. It is photons moving from hot to cold objects. It is the wind blowing from high pressure volumes to low pressure. We can use these flows, or equilibrations, to do things. What do we do? We humans usually like to create order from disorder, fighting the battle against chaos, building persistent pockets of beauty and structure.
Of all the known elements, iron is the most stable element around. Splitting isotopes heavier than iron or fusing lighter ones yields isotopes with lower binding energy per nucleon. Energy is conserved, and therefore some energy must be released. If all you have is iron, you’re out of luck. There’s no excess binding energy available.
Below, we see the binding energy per nucleon (protons and neutrons) for each element. Binding energy is the energy required to take apart the atom into its constituent particles.
When we look at an atom, its mass is less than the sum of the masses of its constituent particles. That missing mass is the binding energy keeping the particles together, as related by E=mc2, where E is the nuclear binding energy, m is the difference in mass, and c is the speed of light. Elements like Iron appear to be low energy or low mass relative to heavy elements like Uranium.
A higher binding energy per nucleon means the atom is more stable and efficient at keeping itself together. If we go from lower to higher binding energy, more energy is available for something else. Equivalently, if we go from high to low mass per nucleon, we are releasing stored energy.
First, we need some magic rocks – nuclear fuel. We have a lot of elements and isotopes to pick from, but on Earth today, the most readily available nuclear fuel is uranium, specifically Uranium-235, or Uranium with 92 protons and 143 neutrons. These are good fuels compared to other heavy elements because they are relatively easy to break apart, and they also release plenty of neutrons we can then use to break up other atoms. More about that later.
It’s just a rock, usually in the form of an oxide, Uranium dioxide. Yellowcake is just a convenient form of various uranium oxides. You can buy some here. It’s also slightly radioactive. Some of the Uranium isotopes are decaying (or transforming) into other isotopes, releasing some radiation (or energy) in the process. Some of its atoms are also spontaneously breaking apart into smaller atoms and, in the process, releasing neutrons and other particles, also known as radiation. There are many ways for an isotope to decay such as emitting an electron (beta), alpha particles, neutrons, or gammas.
For a given sample, we can measure the emission rate and energy of these particles. The particle emission rate is known as the activity, measured in Bequerels (Bq = particles/s) or Curies (3.7 x 1010 particles/s). Dividing by the sample mass and correcting for the detector's sampling area, we can estimate the specific activity of the sample.
By itself, the activity is not very meaningful. For practical considerations of radiation damage and biological harm, we need to know what kind of particles are being emitted and with what energy. Together with some assumptions about how a target is exposed in terms of its distance, effective thickness, and time, we can then calculate the absorbed dose (energy/mass). We can then apply various biological weighting factors depending on the organs exposed and radiation sources to find the equivalent or effective dose, which is measured by Sieverts. Various complexities arise from how different elements and isotopes interact with biological and natural processes. For example, some isotopes can be ingested and passed through urine relatively quickly, posing little danger even if they have a high activity. Other isotopes might lodge themselves inside an organ for very long periods of time.
If you look more closely, you’ll notice that not all the Uranium atoms are the same, and they behave differently. These are the different isotopes of Uranium, having the same number of protons but a different number of neutrons. Natural uranium ore is made of 99.3% U-238 and 0.7% U-235 and a tiny bit of U-234. Due to differences in isotope decay half-lives and mechanisms, radioactivity in Uranium ore is 2.2% from U-235, 48.6% from uranium-238, and 49.2% from the trace amounts of uranium-234.
The isotopic abundance in natural uranium is a direct consequence of the history of the solar system and the universe. The uranium on Earth originates from super novae where energetically unfavorable r-process nuclear reactions turned iron into heavier elements including uranium and its parent isotopes, Pu-242 and Cm-247. It’s interesting to think that nuclear fission energy is a form of stored fusion energy that discharges on billion-year time scales. We can accelerate the energy release from the stardust by assembling a nuclear reactor. It’s pretty convenient for us humans given how difficult fusion energy may be to implement.
Both uranium isotopes decay into other elements relatively quickly compared to everyday materials. Uranium-235 has a decay half-life of 0.704 billion years, a bit shorter than U-238’s 4.468 billion years. As such, total Uranium and the U-235 fraction declines over time and there is less available today than in the past. This is unfortunate because while all the heavy atoms can be fissioned, it is much easier and practical to fission U-235 in a self-sustaining way, compared to the more stable U-238.
These decays produce about 40% of the Earth's total geothermal energy production of 44 TW. It is interesting to note that geothermal energy is just a form of nuclear decay energy. It drives plate tectonics and volcanic activity. Rerouting those energy flows could have interesting consequences for the Earth's geology, climate, and habitability. Today's human civilization consumes 18 TW of primary power and given the geographically limited distribution of geothermal resources, it is unlikely that geothermal energy would be the major contributor to human energy needs.
U-235 is special because it really likes to break apart and release nuclear energy when it gets hit by slow neutrons. U-235 is the nuclear fuel (fissile). In contrast, U-238 is more stable, and usually absorbs neutrons without releasing energy right away. When it does fission with fast neutrons, the neutrons collide inelastically with U-238, slowing to velocities that cannot cause further fissions in U-238. U-238 is not without its virtues, as it can capture neutrons and transform into Pu-239 which is more easily fissioned, similar to U-235. It is an effective neutron moderator or reflector by virtue of high inelastic reaction rates. It also has some beneficial neutron capture properties that improve the negative thermal feedback of a reactor, which we’ll get to later.
A Uranium fission in a nuclear reactor releases energy both instantaneously and after some time. Instantaneous energy release takes the form of kinetic energy carried away by the fission fragments and neutrons. That kinetic energy turns into thermal energy when the fragments interact with surrounding atoms. A small portion of the instantaneous energy release is contained in gamma rays that escape the core altogether.
The fission fragments are usually very unstable, having too few or too many neutrons, and will decay to more stable isotopes by various radioactive decay steps over the course of seconds, weeks, and years. The total energy that ends up as heat is roughly 202 MeV. Roughly 8 MeV escape the reactor and Earth as anti-neutrinos.
Increasing the percentage of U-235 in the material enhances the fuel’s ability to sustain fission reactions which means the reactor can be smaller, use less moderator, and less total Uranium. That said, reactors can be made with today’s natural uranium ore, but they are often quite large with lots of moderator material and fuel as in the heavy water CANDUs.
Isotopic enrichment methods take advantage of the slightly different properties of the two isotopes, such as slight mass differences, slightly different light absorption, or different reaction rates. Isotopic separation is also used for many other materials in the reactor where only certain isotopes are desired, for example in boron control rods which use the neutron absorbing B-10 instead of the less absorbing B-11.
The energy required to enrich the uranium is significant, though process and method improvements have steadily decreased the energy and infrastructure investment. Early methods for the Y-12 Manhattan project used a devices called calutrons to ionize uranium isotopes and separate them based on their mass difference – essentially a gigantic mass spectrometer. Curiously, they borrowed 13,300 metric tons of silver ($10.3B in 2024) from the US Treasury for their magnets. I wonder how effective calutrons would be with modern super conductors. Post-war enrichment at K-12, also at Oak Ridge, used gaseous diffusion taking advantage of the slightly different diffusion rates of the two isotopes in highly corrosive Uranium Hexafluoride gas. These were mile long, power hungry facilities. Current methods also take advantage of mass differences but use gas centrifuges spinning at around 50,000 rpm. These are more compact and energy efficient than previous methods. Other methods in development use lasers with a tight bandwidth to take advantage of slight differences in the energy absorption of different isotopes at particular wavelengths of light. This method can allow a laser to deposit the isotope of choice from a gas in a kind of chemical vapor deposition process.
| Parameter | Unit | Value |
| Enrichment | (% 235) | |
| Category | - | |
| SWU | SWU / kg | |
| Ore Cost | $/kg | |
| Conversion | $/kg | |
| Enrichment | $/kg | |
| Fabrication | $/kg | |
| Total Cost | $/kg | |
| Burn Up | MWd/kg | |
| Burn Up | % of fuel | |
| Fuel Cost | ¢/kWh | |
| Fuel Cost Diesel | ¢/kWh | [2, 25] |
We can look at how the fuel costs change with enrichment and burn-up for different reactor architectures. Typically, smaller reactors require higher enrichment and achieve lower burnups to the point that they are not economical.
We also incorporate some assumptions of the fuel fabrication costs.
Higher burnups have the added benefit of reducing the lifetime of the waste as well as the volume of waste per unit of energy produced. Of course, it requires fuel forms that can withstand greater irradiation damage and fission gas pressure, as well as reactor cores that can tolerate the extra fission product poisoning. In the short run, these fuels can be expected to cost more than traditional fuel forms, and currently only China and Russia are producing advanced fuels like TRISO fuel or fast reactor fuels for higher burn ups.
A nuclear reactor is just a system that can get heavy elements to break apart in an efficient manner. The reactor itself, the nuclear core, is not a complicated mechanical contraption. The core has no moving parts or complex mechanisms. It's just an arrangement of special materials, that permits nuclear reactions to occur. There are moving parts to transfer the heat and control the reactions, but it's basically a pile of bricks.
The goal is to get the fuel to keep splitting apart continuously and self sustainably in what is called a chain reaction. This generates heat that is carried away by a coolant, and which we can use for industrial processes like baking cookies or generating electrical power.
To split atoms and release nuclear binding energy, we need to launch neutrons at the fuel atoms. Where do we get these neutrons? We could use an accelerator, but that’s not very practical and poses a few safety issues. The spontaneous fission of U-235 or U-238 could be used. But these don’t happen very often. Indeed, 1 kg of natural Uranium generates just 10 such neutrons per second. Which is not very useful.
Fortunately, every time an isotope fissions, or breaks apart, it releases new neutrons that can cause more fissions. A U-235 fission consumes just 1 neutron while releasing an average of 2.43 fast-moving neutrons, with a slight dependence on the original neutron's energy and the target isotope. This gives us 2.43 neutrons to cause an extra fission reaction, which means we have a 1.43 neutron margin for inefficiencies in the system. We’re going to need that margin because we’re going to lose neutrons when they get absorbed by materials or escape from the core altogether.
To sum up, we will use whatever neutrons available to cause fission reactions in U-235 which release 2-3 neutrons each, which in turn cause fission reactions and produce more neutrons, and so on. We will try to get neutrons to multiply like bacteria, and carefully control the growth or decline in the population to sustain a desired number of fission reactions per unit time.
How much power is being produced? The thermal power produced is proportional to the number of fission reactions which is also proportional to the total number of neutrons at any given time.
What we’ve been doing here is trying to get a critical system. A critical system is just an arrangement of matter that can support a steady state neutron population. Neutron births are balanced by neutron deaths. Neutron births take place when a U-235 fissions or when some fission fragments decay. Neutrons “die” when they are absorbed, escape the reactor core, or are utilized to cause a fission reaction. A reactor core operating at constant power has a constant neutron population, and it is called “critical.”
At a constant 1 MWth, there are approximately 3.1x1016 fission reactions taking place per second (1 J/s divided by 200 MeV/ U fission). The neutrons causing these reactions move at roughly 2200 m/s and have lifetimes of about 20ms. This 1 MWth reactor would have a standing population of 1.5x1015 neutrons (fissions/sec / lifetime), but probably about 2.5x that to account for neutron absorptions.
We can also find the volumetric neutron density which is the neutrons per cm3 or neutrons per mL. For a low power density like 1 W/cm3, like the MMR, we would have 3x109 neutrons per cm3. For higher power density of a typical LWR, on the order 50 W/cm3, it would be 50x higher. This is not much compared to the nubmer of atoms in a cm3, roughly 1022. So neutrons don't collide with each other, but only with atoms.
A sub-critical system has a declining neutron population. Deaths exceed births. A sub-critical core’s power is zero or declining. The core is fizzling out.
A super-critical system has a growing neutron population. Births exceed deaths. As in many biological growth models, the growth is exponential as each new neutron can give birth to generations of daughter neutrons. And just as in all growth models, the growth does not remain exponential indefinitely as negative feedback mechanisms curb the growth and turn exponential functions into logistic functions or even collapse functions.
For bacteria, the growth limiters are the walls of the petri dish and the food supply. For neutrons, growth is limited by many mechanisms, most importantly rising temperatures which either cause more neutrons to die or force the core to fall apart and become sub-critical - more on that later.
Nuclear reactors carefully take advantage of the limiting mechanisms to control the rate of energy release. On the other hand, the design of nuclear weapons aims to overcome the negative feedback mechanisms to achieve a rapid growth of the fissions and fully utilize the nuclear material in a very short period of time.
Let’s try to make a core. In each simulation, we’ll seed the sample with a starting population of neutrons and watch them propagate through the core, either achieving stability or going extinct. We are aiming to create a self sustaining neutron population in the core.
The first thing we’ll try to do is use the neutrons naturally emitted by the fuel itself. The fuel is just sitting there and spontaneously fissioning and we’re hoping these neutrons are enough to get a chain reaction going.
There’s a little problem. While there are some fissions, illustrated by the blue colored blips, most of the neutrons are escaping! And on top of that, the neutrons are not always doing what we want. We’d like all the neutrons to just cause fissions in the Uranium atoms. But nuclear reactions are probabilistic processes, and fissioning Uranium atoms is just one of the many possible things a neutron can do. The probability of those things occurring depends on the materials and the energy of the neutrons themselves.
The neutrons emitted from the U-235 spontaneous fissions are fast (high energy 2 MeV), moving at about 5% of the speed of light in random directions. At those speeds, the neutrons are not very likely to cause a fission of U-235 or U-238. Instead of causing a fission reaction, they can be scattered or absorbed. In a scattering reaction, the neutron interacts with an atom to slow down and change direction. In an absorption or capture, the target atom absorbs the neutron. So the neutrons travel around the core hitting atoms and slowing down until they are either absorbed, cause a fission, or escape from the core. Absorptions and escapes reduce the neutron population while fissions will grow the population.
There are lots of things to consider. The nuclear reactions are numerous and probabilistic. To control these reactions, we have to know their probabilities and reaction products as well as the energies and directions of the resulting neutrons. We have to know that U-238 and U-235 have very different reaction probabilities for different neutron energies. For example, U-238 is likely to be fissioned only by fast neutrons while U-235 is far more easily fissioned by slow neutrons. On the other hand, U-238 can absorb a neutron to eventually become Pu-239, which is fissile like U-235, easily fissioning with slow neutrons. In current reactors, the fissioning of U-238 contribute 5-10% of thermal power, and the fissioning of the Pu-239 derived from U-238 in the reactor contributes up to 50% of the total thermal power.
Measuring and predicting the probabilities of different reactions for different isotopes and neutron energies is the crowning achievement of nuclear physics. We use cross sections to describe the probabilities for each reaction, for each isotope or material, and across different neutron energies. We show only the fission, scattering, and absorption cross sections. For a given input neutron with a particular energy, we can read out the probability of a particular reaction occurring.
A cross section is basically the practical interaction area of the atoms when they are bombarded with neutrons. While an atom has a physical cross sectional area of a about 1 barn (10-24 cm2), the somewhat complex nuclear physics of nuclear forces and interactions leads to wide ranging cross sections for various neutron energies interacting with different isotopes.
The way to present cross sections for physical materials utilizes their density and is as a probability per unit length. This is called the macroscopic cross section. We can also consider the probability per unit time, which simply multiples the macroscopic cross section by the velocity, but this is less useful when implementing simulations. We can change the materials below.
Let’s use a neutron cannon to seed the core and observe what happens when we change neutron velocity (energy). We can change it from very slow neutrons with velocities of a few hundred m/s to fast neutrons at a few percent of the speed of light.
We can get a lot more fissions at lower neutron energies. But still, lots of the neutrons die by absorption or by escaping from the core. Also, we don't know how to make those slow neutrons yet.
We’ve got to arrange things to make those fission reactions as likely as possible. So let’s keep adding Uranium by adding more blocks. This way, whatever direction the neutron scatters it will come across more Uranium atoms, and any neutrons that pass through one bit may cause a fission in another bit further along. And we can keep doing that. Pressing the reset button will unleash a burst of initial neutrons at the center of the core to test the conditions.
Enlarging the reactor reduces the leakage of neutrons. This has to do with the surface area to volume ratio. As the radius of the sphere increases, the volume increases faster than its surface area, and a smaller fraction of the core’s neutrons are able to escape. One reason reactors have been designed and built to be extremely large is to reduce this leakage of neutrons. Less leakage and the fuel can be used more efficiently, lowering the overall cost of energy.
Unfortunately, this won’t work even for an infinite amount of natural uranium. The U-238 fission cross section is much lower than the absorption cross section. Also the neutrons generated in the fission are too slow to appreciably tap into the fission cross sections on the far right of the spectrum for U-238. We’re losing too many neutrons from absorptions and the neutrons that aren’t absorbed aren't able to slow down to speeds that are likely to cause fission reactions in the tiny amount of U-235. But we have the tools to fix this and create self-sustaining chain reactions. These tools are enrichment, moderation, and size.
Using only the natural uranium available on Earth today, there’s just no way to get a stable, let alone growing, population of neutrons to sustain nuclear reactions. Natural reactors like those found in the Oklo region of Gabon were active roughly 1.8 billion years ago when the enrichment was close to 4% and could not work using today's natural enrichment level. The problem is the U-238 is absorbing a lot of neutrons and it doesn’t easily fission.
We can manually increase the U-235 content in the fuel by various enrichment processes. This is an extra expense but very doable. The U-235 displaces the less fissile U-238, and means fewer neutrons will be lost to the U-238 and more neutrons will be available to interact with the U-235 to cause fissions. U-235 is far easier to fission than U-238, and can utilize both fast and slow neutrons. Just look at the spectrum.
We can also slow the neutrons down, or moderate them, so they are more likely to cause fission reactions in the U-235. The U-235 fission releases neutrons moving at about 200 km/s (fast neutrons 2 MeV), and we need to slow them down to 2 km/s (thermal neutrons .1 eV) so they are more likely to cause fissions in U-235. There are several moderating materials that can slow down neutrons without absorbing them that are also resistant to radiation damage. We’ll just use graphite, the same material Fermi used in the Chicago pile and what high temperature gas-cooled reactors use nowadays.
All materials will slow down neutrons to a certain extent, but we want materials that are as efficient as possible per unit of volume. When neutrons hit heavy atoms like Uranium, it's like a ping pong ball hitting a bowling ball and the velocity change in the neutron is very small.Colliding with light atoms like hydrogen and carbon is like a ping pong ball hitting another ping pong ball, and is far more effective at slowing down neutrons. We also want the moderator to be atomically dense so that the chance of interaction is as high as possible. So hydrogen gas is not as effective as liquid water (H20).
All materials will slow down neutrons to a certain extent, but we want materials that are as efficient as possible per unit of volume. When neutrons hit heavy atoms like Uranium, it's like a ping pong ball hitting a bowling ball, and the velocity change in the neutron is very small. Colliding with light atoms like hydrogen and carbon is like a ping pong ball hitting another ping pong ball, and is far more effective at slowing down neutrons. We also want the moderator to be atomically dense so that the chance of interaction is as high as possible. So hydrogen gas is not as effective as liquid water (H20).
There are a few ways we might add graphite. We can add it to the fuel homogeneously so that the uranium and graphite are smeared together.
But mixing the fuel and moderator into a smeared material is not optimal as the U-238 present in the fuel tends to absorb the neutrons as they are slowing down through the resonance absorption region of the spectrum. It is more effective to keep the fuel and moderator separate, clumping the fuel in chunks. This clumping arrangement, also called heterogeneous, allows neutrons to slow down while traveling between fuel chunks and without getting absorbed by the U-238.
Below, you can see the neutron pass from one material to antoher. This is shown in both the 3d model and the 2 spectrum.
Each of the tools discussed can be tuned to achieve different sized cores that achieve criticality. The nuclear core designer will balance these design parameters to achieve a critical core with desired metrics of size, fuel loading, fuel utilization, and ultimately cost.
There are significant limitations on the materials and geometries available for nuclear reactor environments and there’s not much wiggle room on the materials we can use. Most materials are neutron killers and cannot tolerate any significant radiation damage, high temperatures, or corrosive environments. And you have to keep an eye on the materials activation - which refers to the habit of materials to become radioactive when subject to radiation either by absorbing radioactive substances or by reactions with the radiation.
The research and development required to understand materials for use in the complex nuclear environment is often a decades long endeavor with limited success. The knobs we can play with are the ratios and arrangements of materials, the coolant, the enrichment of the Uranium, and the size of the reactor. Later we will be able to choose the power level.
To make the reactions self-sustaining, we need to at least achieve a critical system. But in practice we want a system that can be super-critical by simply moving the control rods out of the reactor. This way, we can change the power level and keep burning uranium over time.
The latter point about burning uranium over time concerns the fact that as we burn the fuel, the core conditions are changing. First, the U-235 content is decreasing over time, reducing criticality. In addition, many of the fission products produced by the nuclear reactions will absorb neutrons, so that over time, the reactor by itself becomes less and less conducive to neutron multiplication. On the other hand, some of the fission or decay products are fissionable materials like U-233 which will help extend the life of the core. To account for all this, we have to make the starting configuration sufficiently super-critical that it can tolerate an increasing number of fission products that poison the neutron environment. How super-critical does it have to be? Enough that we can burn an economical fraction of the fuel without compromising safety.
The spontaneous fission of U-235 is not typically relied upon to start-up up a fresh reactor core. The spontaneous fission half-lives for U-235 and U-238 are 3.5*1017 and 8.4*1015 years and produces an average of 1.86 neutrons per fission. 1kg of natural Uranium can be expected to produce about 14 neutrons per second depending on the enrichment, which can be sufficient to start up a nuclear core. However, in practice the startup of a nuclear reactor is aided by a more emissive neutron starter like Cf-252 or Pu-238 and Be so that operators can be really sure there's enough neutrons to get things going.
To change the power level we need the ability to change the core conditions to our liking, making it critical, sub-critical, or super-critical; in other words, changing the environment so that the neutron populations are stable, declining, or growing. The control rods are neutron killers and moving them in and out of the core allows us to control the neutron population. For the MMR, temperature can also be used as the primary population control mechanism.
To increase the power, we want to increase the number of neutrons available for fissions. The core is made super-critical by withdrawing the control rods, which allows more neutrons to survive and propagate. The neutrons will continue to multiply, and their population exponentially increases unless we put the control rods partially back in and make the reactor critical again. Once critical, the reactor power level will remain constant. To shut it down, we must reinsert the control rods, making the reactor sub-critical and exterminating neutrons in the core.
We’ve mentioned that the super-critical configuration is an exponential growth environment. So you might ask, why doesn’t a reactor just produce exponentially more power and blow up or behave like a nuclear weapon?
And fast things are generally hard to control. The lifetime of a neutron from birth to fission can range from a few microseconds to milliseconds depending on the moderator. When a fission reaction occurs, most of the resulting neutrons are released very quickly, on the order of 10-14 seconds. These are the prompt neutrons. This short timescale implies that neutrons breed really fast. It's like a rabbit species that reproduces every few milliseconds. There’s little you can do to control it or apply active feedback mechanisms to slow things down. You certainly can’t move the control rods quickly enough to make a difference. The dynamics are too quick for active control.
Fortunately, a small fraction of the fission neutrons, typically around 0.1%, are released more than 0.1 seconds later. These neutrons breed slowly enough to be observed and can be controlled with mechanical actuators. So reactors try to operate in the regime where these delayed neutrons are driving the criticality. The fraction of neutons that are delayed is higher for thermal spectrum reactors as they have more U-238, which produces 4x more delayed neutrons per fission than U-235.
It should be noted, that some reactors like MMR don’t particularly need delayed neutrons because the control is accomplished primarily by temperature feedback mechanisms which happen very quickly.
A nuclear reactor can operate at whatever power level desired as long as criticality can be maintained. But there are many mechanisms that reduce criticality as power increases. These are based on the materials in the core (control rods), the temperatures, and the geometry of the core. Power can be increased as long as heat can be removed from the reactor core and temperatures remain safe and stable. If the power generation exceeds the limitations of the fuel and reactor, various physical mechanisms will cause the reactions to stop. It’s just a matter of how calmly or violently the negative feedback takes place. The fundamental idea is that super-critical conditions are not sustainable for long because they will lead to higher temperatures, the destruction of the reactor, and will ultimately achieve equilibrium with the environment.
For reactors with high power density, conditions can arise by which the only way the reactor can reach equilibrium is by melting or exploding. For low power density reactors, reaching equilibrium and subcritical conditions is a subdued affair.
While the control rods appear to be the main mechanism for controlling the reactor there can be stronger mechanisms that overpower the control rods.
Control rods are usually neutron killers. So putting them into the core will kill neutrons, while taking them out of the core will allow more neutrons to live. For this reason, control rods are usually designed to fail by falling into the core with the help of gravity, thereby shutting down the nuclear reactions.
The main exceptions are boiling water reactors like BWRX-300 and ABR, in which control rods are placed below the reactor core because they wouldn't survive the hot steam conditions on top of the reactor – not ideal. The other exceptions are control drum reactors which can be configured for a gravity driven automatic shutdown.
A core design has to consider the effects of accidental control rod ejection as well as the replacement of that empty space with water which will provide extra moderation and criticality. Failure to account for these effects is responsible for the power spikes and explosion at Cernobyl.
Most nuclear reactor designs only consider accidents with a handful of control rod ejections and would probably not be able to cope with the ejection of a majority or all the control rods. Ideally, reactors would be able to tolerate ejection of all the control rods with a simultaneous loss of coolant.
In the fuel, raising the temperature causes doppler broadening which allows more neutrons to be absorbed by U-238, thereby reducing the reactivity. This effect is dominant and instantaneous as fuel temperatures rise. Higher temperatures indicate faster vibrating atoms and a higher velocity relative to incoming neutrons, which means a given reaction is accessible at more neutron energies. Visually, the high probability peaks are smeared.
Notice the peaks in the U-238 absorption probabilities. If a neutron has one of those energies, there is a very large chance of absorption. At lower temperatures, the U-238 resonance absorption peaks (the spikininess) are tight, and there is small chance of neutrons having those exact energies, allowing many neutrons to pass through unscathed. At higher temperatures, the peaks broaden and the probability of neutrons having energies susceptible to absorption increases, effectively culling the neutron population.
For this mechanism to work, the fuel cross sections must have significant spikiness or resonances. Comparing U-235 and U-238 probabilities reveals the importance of U-238. It contributes significant resonance absorption probabilities that ultimately give rise to negative temperature feedback. Keeping a low enrichment (high U-238 fraction) is needed to achieve a strong negative temperature feedback.
Another feature of the U-235 resonances is that they occur at lower neutron energies. So to access the negative feedback of these resonances, the core should significantly rely on slow neutron fissions (aka thermalized or moderated). This means that thermal reactors that use quite a bit of moderator have superior negative temperature feedback in the fuel compared to fast fission reactors.
Moderator feedback mechanisms tend to be delayed compared to fuel feedback mechanisms because the moderators take some time to heat up from the fuel's power changes. In general, moderators will expand with rising temperature, thereby reducing moderation and the number of slow neutrons available for fissions.
If things get very bad, the reactor will achieve temperatures and conditions that lead to geometric changes. The reactor will eventually reach an equilibrium with its environment, and it's just a matter of how gently or violently it gets there. For example, there could be a sufficient build up of pressure and hydrogen (in water-cooled reactors) that leads to an explosive release of energy. This could spread out the core from a tidy, critical core into a spread out jumble that is sub-critical. Similarly, the core can melt itself, morphing from a compact cylinder into a flat puddle.
Needless to say, this is not a great feedback mechanism to achieve a sub-critical system because the fuel has basically failed, releasing many fission products outside of the pressure boundary. At a minimum, the powerplant will be trashed with 10s of $B in losses and cleanup efforts. Worst case, the containment fails or leaks and there is a release of fission products into the atmosphere or ground.
The ability to rapidly change the power level of a reactor seems scary, and one would expect that criticality accidents or rapid energy releases would be the main cause of nuclear accidents. But history has shown the main cause of nuclear accidents is the inability to remove heat from the fuel after the nuclear chain reactions have been stopped. Heat piles up and temperatures rise until the fuel fails, releasing fission products and radiation. The major accidents have to do with what happens after the reactor has shut down.
As the fissions take place, uranium atoms are splitting and making periodic-table soup. The uranium isotopes usually split into fragments, and the distribution of new isotopes resembles a boob curve . Many of these isotopes are not stable and will eventually decay to more stable isotopes through various decay processes that also release energy. This is the decay heat, and it is an appreciable amount of energy that can do a lot of damage if not carefully dealt with. The problem is that it cannot be stopped even after the nuclear chain reaction has been shutdown. In most cases, the decay heat is about 6-10% of the reactor power after shutdown, and will slowly fall away over the next few hours and weeks according to a power law.
10% of operating power right after shutdown doesn't seem like a big deal. But 10% of a 3 GWth reactor is 300 MWth - or 6 Boeing 777s (12 GE-90-115B engines) worth of power, pumped into the core. After 80 hours, even 0.2% decay heat presents a sizeable challenge because that heat needs to be removed or temperatures will rise.
Almost all nuclear reactor designs in use today use active systems to remove the decay heat of a reactor after it has shut down. This usually involves keeping the coolant flowing with electrically powered pumps. Those pumps can be powered by local electical grid or from emergency diesel genstes. Unscheduled interuptions in that emergency power supply are typically national news.
The problem is that maintaining that cooling capability is a finicky thing with many ways to fail. Possible failure modes include coolant leakage, partial or total; clogging; pump failure; power interruptions if the grid connection goes down or emergency back-up generators stop working for whatever reason. Any of these things can happen by accident such as during external events like weather, flooding, or earthquakes. Operators could accidentally trigger some events. They can also be intentionally triggered through malicious actions.
In the below accident simulation, we assume the reactor manages to shut down but loses all coolant and forced cooling capability. It has to deal with all the decay heat using only conductive and radiative heat transfer to the surrounding earth or building structures. This is also known as Class A passive cooling as opposed to Class B passive cooling which relies on natural convection of a fluid or moving parts.
The advantages of high surface area to power ratio become clear – you can keep the core at safe temperatures with simple passive cooling! Not obvious from the visual above, is that if we constrain the reactor to a power level that can be safely cooled during accidents, smaller cores can have higher power densities. This suggests that smaller cores can have cost advantages.
It also becomes clear that some reactor types are much better suited to handling decay heat than others. In particular, the below simulations show how water cooled reactors experience rapid temperature increases in loss of coolant accidents.
Now that we have a collection of materials capable of sustaining nuclear reactions, we have to package the core into a reactor and extract its heat. The basic function of the reactor is to change the power level and extract heat from the core for use elsewhere, either in a turbine or for process heat applications.
We place the nuclear core into a pressure vessel that holds pressurized helium or other coolants. The helium flows to the core though a double walled pipe (aka concentric tube). The cold inlet flows on the outside to insulate the vessel walls. It then flows up the sides of the reactor and then down through the warm core, picking up heat. The hot outlet fluid then flows through the inside of the pipe to the end use.
For this type of high temperature gas-cooled reactor, we basically have just two controls on the reactor. The coolant flow rate and the position of the control rods. Together, these determine the criticality, the reactor power, and core temperatures. Control rods are moved in and out of the core to change the criticality which then allows the neutron population to grow or shrink. The neutron population is directly proportional to the power. The criticality is also affected by the core temperatures.
The temperature of the core remains constant as long as we remove heat at the same rate as it is being generated by the nuclear reactions. Temperatures rise if more heat is produced than removed. As temperatures rise, heat transfer becomes more efficient and more power is removed until the reactor reaches a new equilibrium. Also, as temperatures rise, the reactor produces less power due to the negative feedback mechanisms described above.
I have given a simplified view of how nuclear reactors work. But there are many subtleties with significant effects, and very many different approaches to designing nuclear reactors. I hope readers will appreciate some of the extraordinary, nearly magical, phenomenon associated with nuclear energy. Our civilization somehow discovered these obscure mechanisms, learning to control and exploit the nuclear binding energy, but we have yet to fully unleash its Promethian potential.