What is the relation between control rods surface exposed into a nuclear reactor and neutron energy? Is it linear? I mean, how do neutron absorbing rate change with the progressive immersion of control rods into the core?
[Physics] Nuclear reactor control rods
nuclear-engineeringnuclear-physics
Related Solutions
Per this article on the subject: http://theenergycollective.com/nathantemple/53384/how-shutdown-and-core-cooling-japanese-reactors-likely-functions
Even with rods inserted, the reactor continues to produce heat equivalent to about 3% of its full power level. This is not the same as taking a pot off the stove and letting it cool. There are still some atoms splitting and fission products decaying that produce heat. This drops off slowly and is why there needs to be layers of redundant cooling with backup power. During such an earthquake, power from outside the plant would not be expected to be available.
In case of overheating and melting of the core, the control rods may not be inserted any more leading to a disaster.
I have to interject that it's a little more complicated than that, although I like to discourage people from thinking about a reactor is a super complicated system. It's not super complicated, it's just a high-order system. The temperature of the fuel is proportional to the heat content of the fuel. As you put in more heat, the temperature goes up. But that makes power the derivative of temperature. If the thermal heat production rate equals the rate of removal, then temperature isn't changing either way. But if it's not balanced power is the derivative of temperature. Our goal is always to keep temperature low so that no part of it melts.
This is why reactivity trips people up. Reactivity is the derivative of power, via neutron flux levels. If reactivity is balanced, neutron flux stays constant, if it's not, it changes over time geometrically (by that I mean it changes by percent, like 10% per minute). This means that reactivity is the 2nd derivative of temperature.
As if that wasn't enough, decay heat leaves a constant heat source even after the chain reaction has been shut off. If you can safely shut things down, then you need a pathway for the heat to get out. If you're still worried about criticality, then you have two objectives, and sometimes they work against each other. This is generally the point that people start screaming "call in the nuclear engineers!"
A subcritical mass can be made critical by placing neutron reflective material all around it. So my question is why nuclear reactors are not operated with this kind of reflective shielding instead of control rods?
Yes, of course, like this:
You can see Reflector control labeled there. That is a piece of neutron reflector that can move up and down. The nuclear core is bunt from one end to the other.
Nuclear fuel burnup is another concept, but please don't let that scare you off either. Burnup only takes place over a long period of time because nuclear fuel holds a lot of energy in total. As we keep fissioning atoms, we will eventually run out of the atoms that can fission. In terms of our mental picture, this inserts negative reactivity. The reflector inserts positive reactivity. To start the reactor, you can move the reflector into place and it will start burning. But then over time that part of the reactor you moved it to will putter out. At that point, you move the reflect a little bit further along the core, and then new fuel comes to life.
This is a "candle" type, but that's not exclusive to reflector-controlled reactors. There are also control drums. Those are big cylinders that sit to the side of the reactor, which contain neutron reflectors on one side, and something on the other side that either doesn't reflect or eats neutrons on the other side. The only difference is that this moves the reflector to the side as opposed to up or down. Everything is pretty physically obvious. You could create a viable control system with any given way you could think of to move the reflector in place.
The shield could be designed in order to support a maximum temperature before crashing/burning/melting driving the reactor far from the criticality automatically in a totally passive way.
Obviously we want that sort of thing to happen, but it's a little vague as you've proposed it so far. In the image I posted above, they very likely planned for overheating to cause the moderator ring to drop.
In practice, they'll be more conservative than designing fail-safe when temperature goes out of control. When they can no longer input an active control signal to the reactor, it will drop the control elements to shut it off. This is actually done with control rods, but it would basically work the same with a moving reflector, except that control rods move in, reflector moves out to shut down the reactor.
In summary: "yeah, sure"
Best Answer
You mention control rod surface. Why? Do you understand how control rods work? They're inserted into the core and absorb neutrons. Now, why would the surface be a metric that matters? Perhaps this was just the most obvious thing to you at the time, so let me get into the complications here.
Cross sections are higher at thermal versus fast energies. The fast and thermal neutron flux both matter a great deal for typical light water reactors. The main difference is that the average path length of a neutron (before interacting) is much longer for fast neutrons than thermal neutrons. Now, since we've made this distinction, we can ask if the control rods absorb a significant fraction of the neutrons incident on its surface. The answer is mostly "yes" for thermal energies and "no" for fast energies. See, because the thermal flux has a much smaller mean free path, it is much bumpier because of the presence of different materials, including fuel, moderator, and absorbers. It would be mostly correct to suppose that some significant fraction of a thermal neutron beam is absorbed after entering the control rod surface in terms percentage. As an order of magnitude guidance, it would be more than 1%. The same is probably not true for the fast flux. Furthermore, according to transport/diffusion physics, if you were to reduce the fast flux by a significant percentage, you would necessarily suppress the fast flux in the entire region of the core.
Anyway, let me address your question by saying that under the unphysical assumption that a control rod absorbed all neutrons incident on its surface, its reactivity contribution would be proportional to its surface area, relying on a few other assumptions as well. Notably, if you have too many control rods in the same small area they will depress the flux around each other so its not 100% valid. As I've already pointed out, this doesn't fit current reactors.
Let's look at the other extreme. Say, for instance, that the control rods absorbed a negligible fraction of the flux relative to the total flux. This is much closer to normal reactor conditions. In this case, the reactivity contribution would be proportional to the volume of the control rods times the local flux value, and this is shown mathematically in several nuclear engineering text books using perturbation theory.
Again, use perturbation theory and it's linear for small movements. But what does this physically correlate to? My answer is that it is with respect to the total amount of absorbing material times the local flux where it exists. This is because the control rods don't significantly affect the shape of the flux. In the thermal energies, however, this is a mediocre assumption, which is why people use actual computer codes to design reactors.
(math warning) variables:
Let's speak of an isotropic neutron flux of a single energy. Then the flux will be of a certain physical shape $\phi(\vec{r})$ and the entire point of using the words "perturbation theory" is to refer to the assumption that $\phi(\vec{r})$ doesn't change with the insertion of the control rod. If the end of a control rod is at some given $\vec{r}$ in the core, then the change in reactivity due to movement of the rod will be $\phi(\vec{r}) N \sigma A \Delta l$ which will come out to units of $1/s$, representing the number of neutrons absorbed per second from the volume of the absorber element introduced. This affects the power of the core through a concept called the multiplication factor, which is how much the neutron chain reaction grows or declines for each neutron generation.
$$k = \frac{ \text{Fission neutrons created} }{\text{Neutrons born} }$$
The absorber removes neutrons from the population that could cause a fission and thus create move neutrons, so it can be said to subtract from the numerator of that equation. The reactor changes flux (and thus power) over time as:
$$\phi(t) \propto e^{t \frac{k-1}{\Lambda} }$$
You need not concern yourself with the details other than the fact that inserting absorbers causes the flux to decrease over time.
(end math)
This is the most objective part of the question.
According to my prior arguments, the differential reactivity contribution depends on the flux value at the end of the rod being inserted.
So let me go back to the linearity question. The reactivity contribution from control rods is highly nonlinear with respect to control rod position, because the flux in the core changes dramatically with vertical position. AdamRedwine points out, correctly, that the flux is well-smoothed by core design, but that statement is specific to the xy-plane. This is not true for the z-direction, where it is nearly a cosine shape for a PWR, something else for the complicated thermal-hydraulic and neutronic feedback of the BWR.
Here is an example of a differential control rod worth curve.
It is greater than zero at the top and bottom of the core because some neutrons do leave the core, bounce of Hydrogen, and then enter the core again to cause fission.
For any small control rod movement it's linear. Sure. That follows from the fact that the above graph is continuous. The fact also implies that the integral control rod worth curve is smooth.