# Scaling nuclear reactors

Summary:
A project to apply a chemical engineering scaling approach to nuclear reactors
Hi all,

For my studies I chose a course on scaling up and down of industrial processes (mostly focussed on the chemical industry), but for our project we (a group of students who knew almost nothing about nuclear reactors) chose to look if the approach (dimensional analysis) can be applied to nuclear reactors as well. Through thorough reading we all have most of the basics down already we think.

I have a few questions right now (although if this proves usefull I might ask some more :) ). I am mostly looking for good sources to read since I'm having difficulty finding good sources for very specific questions, but direct answers are always welcome as well of course.

The questions:

- For the neutron mediation (so the slowing down of the neutrons to have them become thermal neutrons as far as I understand): is the volume of mediating fluid (like water) very important?

I've read that only about 16 collisions are needed to halve the energetic value of the neutron, so doing some calculations using the same formula's you would need about 420 collisions with ordinary water in order to reduce the energy from 2 MeV to 0.025 eV (the general energy of a thermal neutron). This amount seems quite low so I would guess the amount of fluid is not very important for this to happen.

- Is the heat generated in this process, slowing of neutrons, relevant compared to the heat produced in the rods?

With kind regards,
Jurgen

Summary:: A project to apply a chemical engineering scaling approach to nuclear reactors

I've read that only about 16 collisions are needed to halve the energetic value of the neutron, so doing some calculations using the same formula's you would need about 420 collisions with ordinary water in order to reduce the energy from 2 MeV to 0.025 eV (the general energy of a thermal neutron). This amount seems quite low so I would guess the amount of fluid is not very important for this to happen.
It's important to slow down the neutrons before the fast neutrons get captured by U238.
The moderator/fuel ration will be important.
- Is the heat generated in this process, slowing of neutrons, relevant compared to the heat produced in the rods?
Almost all of the energy of fission will go into the fission fragments. The energy of the neutrons will go mostly into the water, where you want it anyway.
The heat produced in the fuel rods is the most important, because the center of the rods can't be too hot, or the rods can get damaged. This is the reason the rods have to be rather thin.

One problem with big reactors is that they are hard to control. It will be possible for a part of the reactor to go critical, you'd need separate control rod systems for parts of the reactor.
Another thing is that you can't shut down a part of the reactor. You still need to do maintenance and refueling.
You might have to invest a lot in the power grid to be able to shut down your big reactor and get energy from somewhere else. If you had 2 reactors of half the size this would be much easier. Other kinds of reactors also do not get bigger than about 1Gb.

hutchphd
Astronuc
Staff Emeritus
One problem with big reactors is that they are hard to control.
Large reactors are relatively easy to control.
- For the neutron mediation (so the slowing down of the neutrons to have them become thermal neutrons as far as I understand): is the volume of mediating fluid (like water) very important?
If one wishes to design a 'thermal' reactor, in which the neutrons are slowed from fast (fission) energies to thermal equilibrium in the reactor, the fuel-to-moderator (fuel to hydrogen) ratio is important. One can design a compact fast reactor, but the enrichment must be increased, and one would normally use mixed oxide ((U,Pu)O2) fuel, or more likely, mixed nitride (U,Pu)N or even metal fuel (U,Pu)Zr or (U,Pu)Mo fuel.

Small reactors leak more neutrons than large reactors, so one might add a reflector around the core.

Reactors can be over or under moderated depending on the volume (mass) of the moderator to fuel. Typical large PWR (17x17 lattice) uses UO2 pellets of 8.19 mm diameter clad in Zr-alloy tubing of 9.5 mm OD and 0.57 mm wall thickness (under moderated), although some use cladding (tubing) with 9.14 mm OD and 0.57 mm wall thickness with pellets of 7.84 mm OD. The pitch of a square lattice is 12.6-12.7 mm. Some of the small modular PWRs have been designed around the 17x17 lattice fuel, but with shorter fuel columns (core height).

One could use a triangular or hexagonal lattice like that used in VVER or fast reactor fuel, which would be better for a small reactor.

Last edited:
berkeman

If one wishes to design a 'thermal' reactor, in which the neutrons are slowed from fast (fission) energies to thermal equilibrium in the reactor, the fuel-to-moderator (fuel to hydrogen) ratio is important. One can design a compact fast reactor, but the enrichment must be increased, and one would normally use mixed oxide ((U,Pu)O2) fuel, or more likely, mixed nitride (U,Pu)N or even metal fuel (U,Pu)Zr or (U,Pu)Mo fuel.

Just for my understanding: I thought the fast neutrons can induce fission in U-238 instead of only in U-235 and therefore this might be an advantage right? Even if you need to enhance the fuel in other ways?

From your answers can I conclude that cooling the fuel tubes is more important than the heat dissipating from the neutrons slowing down? I would also still like to know if there is some relation to determine the average length that neutrons travel.

Astronuc
Staff Emeritus
Just for my understanding: I thought the fast neutrons can induce fission in U-238 instead of only in U-235 and therefore this might be an advantage right? Even if you need to enhance the fuel in other ways?
The probability of fission of U-235 by fast neutrons is much lower than for thermal neutrons. Fast reactor fuel usually contains Pu-239 (and some Pu-240,-241) as fissile material.

From your answers can I conclude that cooling the fuel tubes is more important than the heat dissipating from the neutrons slowing down? I would also still like to know if there is some relation to determine the average length that neutrons travel.
Most of the thermal energy in a thermal spectrum reactor is deposited in the fuel, with a small fraction deposited directly into the coolant. In a thermal reactor, about 7% of fissions occur in U-238 by fast fission. U-238 capture resonance and fast neutrons and is transformed into Pu-239, -240, -241, and they contributed to both thermal and fast fissions.

The most efficient core geometry (from a nuclear/neutronic standpoint) is a sphere, which is rather impractical for a power reactor (heat transfer). The next most efficient geometry is a right circular cylinder, which is difficult to achieve with square arrays of assemblies, more the larger the array. A triangular/hexagonal lattice is a better geometry for core.

The other consideration is the enrichment and batch size. The batch is the set of assemblies manufactured and irradiated as a group. One may have subgroups of different enrichments in a so-called split batch.

For large power reactors, a fraction of the used fuel is normally discharged and 'fresh' fuel added. In the past, a fresh core would be loaded with three batches of fuel, with each batch approximately 1/3 of the core. One batch would be discharged annually, so the whole initial core would be replaced after 3 years. One could use 1/4 reloads, but that requires a greater enrichment. Over time, utilities decided that it was more economical to have longer cycles of 1.5 to 2 years (18 to 24 months), and batch sizes are more like 37-50% of the core depending on a number of factors.

In order to produce a given amount of energy in a reactor, one must have excess reactivity, which means enrichment above that which is needed to achieve and maintain criticality. Excess reactivity must be balanced by burnable absorbers which compete with the fuel in absorbing neutrons. Burnable/consumable poisons include boron, and gadolinium or erbium. Boron may be dissolved as boric acid (buffered by LiOH or KOH) in the water (in PWRs), or integrated into the fuel (coated on the surface of fuel pellets as ZrB2) or mixed into the UO2 as gadolinia or erbia). If the fuel is metal, then gadolinium or erbium could be alloyed with the U-alloy. In the case of boiling water reactors (BWRs), soluble boron is not used in the water, because of the boiling, so BWRs must use control rods (control blades) actively in the core; BWR control blades sit between assemblies. Scaling a PWR design would be easier than scaling a BWR design.

One could develop a compact through away core, i.e., but that requires greater enrichment and judicious use of burnable poisons, or use of active control rods to control power and reactivity distribution in the core.

Another challenge in fuel design is the accommodation of fission products in the fuel, which leads to fuel swelling and internal pressurization. Each fission produces two atoms, and a substantial fraction of fission products are isotopes of Xe and Kr, noble gases, or volatiles such as I, Br, which decay to Xe, Kr, respectively. Some of the Xe, Kr isotopes decay to Cs and Rb, respectively, and some of those to La and Y, respectively. It is the isotopes of Xe and Kr that contribute to pressurization of the fuel and enhanced swelling in the UO2 fuel matrix. Fission products, particularly the noble gases have poor thermal conductivity, so as fission products accumulate, they degrade the thermal conductivity of the fuel, and that causes fuel temperature to increase for a given heat load.

I think it might be easier if you describe exactly what equations you are trying to scale. Are you looking at the diffusion equations? 6-factor formulas? transport equations? something else?

You ask about the number of scattering collisions to slow down a neutron, but this is never used as an input to an equation (that I know of). It is more a value that is edited to help understand the moderating properties of a material. In reality, the number of scattering collisions is a distribution.

All of the physical quantities should scale very easily, with the exception of the neutron leakage. The leakage term is going to cause you the most problems because there is not a linear relationship between the leakage and the dimensions. In your research you might have come across terms like "buckling".

anorlunda
Staff Emeritus
@jugren, Isn't the real purpose of your course of study to learn what limits scaling? Scaling up or down. That's the question you should be asking. Others have already hinted that neutron leakage may be the key.

What have you read about the conditions necessary to maintain a nuclear chain reaction?