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A look at the potential equations behind nuclear reactors
By Stillog
Long story short one day I got bored and, inspired by this game, wanted to learn more about nuclear reactor theory. I never intended to make a guide on this, but considering how awful and near non-existent the sources are on the topic, I figured I might just as well make a guide about it. I can say with reasonable confidence that most of the equations are similar if not exact to the ones the game uses.
If you are interested in the topic, and you have any amount of self respect, please do not waste your time on the six-factor formula. It has mostly been irrelevant since the 1980s. Check out the section called "OpenMC", if you are interested in anything actually serious related to the topic.
Steam also has a bug with disappearing images in guides. The guide should currently have 21 images, and I would appreciate it if anyone would notify me if this number differs.
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Created by
Stillog
Offline
Category: Gameplay Basics
Languages: English
Posted
Updated
Jul 4, 2022 @ 9:34am
Jun 8, 2023 @ 3:13pm
403 Unique Visitors
26 Current Favorites
Guide Index
Overview
Some basics 
Six-Factor Formula 
Thermal Fission Factor (aka Reproduction Factor), η 
Thermal Utilization Factor, f 
Resonance Escape Probability, p 
Fast Fission Factor, ε 
Thermal Non-Leakage Probability, P_{TNL} 
Fast Non-Leakage Probability, P_{FNL} 
The limitations of the model. 
Sources 
Data 
OpenMC 
Comments
Some basics
Important nomenclature and ideas before going any further:
-Fast neutrons are neutrons which have just been created, most commonly by fission.
-Thermal neutrons are neutrons which have been slowed down significantly via collisions with other nuclei.
-The probability of a nuclear reaction of any kind to happen is described as a specific microscopic cross section for each different element and their isotopes in existence. These microscopic cross sections are denoted as surface areas in units of barns.
-Thermal neutrons, due to quantum mechanical effects, are far more likely to cause different types of nuclear reactions in general. This is why most nuclear reactors in the world are thermal nuclear reactors and this is also why moderators exist in nuclear reactors.

https://www.nuclear-power.com/nuclear-power/reactor-physics/nuclear-engineering-fundamentals/neutron-nuclear-reactions/atomic-number-density/
https://www.nuclear-power.com/nuclear-power/reactor-physics/nuclear-engineering-fundamentals/neutron-nuclear-reactions/microscopic-cross-section/
https://www.nuclear-power.com/nuclear-power/reactor-physics/nuclear-engineering-fundamentals/neutron-nuclear-reactions/macroscopic-cross-section/

Cross-sections can be taken from the in-game data charts or from the JANIS database online.
Six-Factor Formula
The game models the multiplication factor for nuclear reactors using the 6 factor formula. The reactor uses a homogeneous version of the six factor formula, i.e the fuel, moderator, coolant... etc, are all mixed together. The 6 factor formula is composed of, as the name would suggest, 6 different factors which all have an influence on the multiplication factor.



Below is a convenient picture of the six factor formula and the starting/ending points of the neutrons in a reactor.


And below is a convenient educational video about the six factor formula.
Thermal Fission Factor (aka Reproduction Factor), η

Thermal Utilization Factor, f
Resonance Escape Probability, p


Moderating power and ratio:

https://www.nuclear-power.com/glossary/moderating-ratio-mr/
https://www.nuclear-power.com/glossary/macroscopic-slowing-down-power-msdp/

A video explaining why the resonance escape probability decreases strongly if the temperature of the thermal reactor increases:
Fast Fission Factor, ε
Thermal Non-Leakage Probability, P_{TNL}
Fast Non-Leakage Probability, P_{FNL}


Non-leakage constants for common moderators:
The limitations of the model.
First of all it is very easy to see where the module editor in Children of a Dead Earth runs directly into a wall. By taking a small amount, say 100 grams, of weapons grade uranium and having no reasonable neutron moderation, it is easy to get k values above 1 which doesn't make any sense considering that the lowest amount of uranium required for criticality is in the tens of kilograms.

Other things that aren't accounted for in-game and in these equations:

-These models do not account take any changes in the composition of the reactor into account, such as nuclear transmutations or chemical reactions. The stability of the reactor isn't taken into account, meaning that while the control "rods" may be able to regulate the reactor above and below a factor of k=1, the speed at which these rods would have to be inserted and extracted could be physically impossible, ex: faster than light. Delayed neutrons contribute to the stability of real nuclear reactors greatly.


-The build up of things such as neutron poisons isn't taken into account, which would mean that one may not be able to turn off the nuclear reactors in-game for a long period, because they might not be able to turn it back on until all of the neutron poison generated would have decayed away.


-The fact that the reactor is modeled with a homogeneous equation means that it would only be the most accurate when designing molten salt type nuclear reactors or aqueous homogeneous reactors. Heterogeneous pressurized water reactor values for the four-factor formula:


-The game also doesn't take changes in temperature into account. Thus things like thermal expansion aren't taken into account on the material level. Changes in temperature or pressure would yield changes in the densities of the materials in a reactor core.

-Talking about temperature, changes in temperature affect many things in nuclear reactors. Such as the average number of neutrons released in a fission reaction.


A convenient picture showing whether an increase in temperature changes the values of all the six factors, (and some leakage factors), to the positive or the negative direction.


-Nuclear reactions with smaller contributions are often ignored in these old models, such as neutrons generated by gamma rays, alpha particles, or neutrons naturally being unstable and decaying away.
Data
Different thermal cross-sections for natural elements, unless isotope is specified: ( ANL-5800 )





OpenMC
What is OpenMC and what can you do with it?

"OpenMC is a community-developed Monte Carlo neutron and photon transport code. It is capable of performing fixed source, k-eigenvalue, and subcritical multiplication calculations on models built using either a constructive solid geometry or CAD representation. A flexible and efficient tally system enables a wide variety of physical quantities to be tallied and analyzed."
( https://openmc.org/ )

Example: Molten salt nuclear reactor simulation video:
https://www.youtube.com/watch?v=pArhYbZQP0Q
How can I install and learn to use OpenMC?

Installation video:
https://www.youtube.com/watch?v=SFIUHU3J1eU
Installation guide:
https://docs.openmc.org/en/stable/quickinstall.html#installing-on-linux-mac-windows-with-docker
There are a set of learning tasks within that should teach the user the fundamentals of the program.
3 Comments
< >
Stillog  [author] Jul 14, 2022 @ 12:28am 
Thanks for the positive feedback. I might add more stuff to this guide as time goes by if I find or remember more things worth adding.
[EE] Kennith Jul 13, 2022 @ 7:49am 
Summary of CoaDE: One day I got bored and, inspired by this game, got a mechanical engineering degree
The Sewer Pig Jul 13, 2022 @ 4:37am 
Extremely good read, thanks for compiling all this data. At this rate it's cool even just on an abstract level, not merely as it relates to epic dead spess game.