A Spherical Cow

Moop is the mascot of my undergrad institution’s Society of Women in Physics. In my life before Nuclear engineering, I did my undergraduate degree in Physics. I like to say that I know way too much about waves for a fuel cyclist, but the background and training were invaluable in developing my general engineering toolkit. All that said, Moop has lived many lives, but she serves as a reminder that sometimes we can treat a cow as a perfect sphere to make our lives easier.

My training in physics probably mirrors that of many programs around the world where (allow me to wax philosophically for a second here), as you go through, you break down mundane processes around you, pedantisize them to bits until a circle is no longer

x^{2} + y^{2} = r^{2},

and is now "a straight line with an infinite radius of curvature." Moop represents a tool that a lot of science requires... approximation.

I’m going to quote one of my math professors here: the number of analytically solvable differential equations is countably small (she may have meant distinct, or maybe just in the time we had in class), and, when your degree is based around solving problems analytically, this is a huge bottleneck. Under the right circumstances, what really is the difference between a spherical cow and the normal one that our parents tried to tip back in the day? Not much!

Why is Moop an effective tool? Well, when assumptions are well-understood, we can pick apart the dominant physics of a problem. So when I see these approximations I ask myself three things:

What is the dominant behavior we are trying to examine?
Are the current approximations fair?
Is there a way to do the same with less or more approximation?

This sort of reflection is something my professors always encouraged, but I didn’t often follow through on at the time. I don’t think I’ve earned the right to be contemplative, after all it was only a couple years ago, but this is something I have picked up to make me more effective as a researcher. All of this to say, I hope the spherical cow is not a new concept, and I have a newfound appreciation for it.

Some examples

So, going back to the whole reason I wanted to write this, what is the nuclear spherical cow? I’ll break down some spherical cows I have interacted with.

Neutronics

In transport and diffusion I think there are two geometric examples that get used in a variety of contexts for teaching and exploring new methods. You can probably guess what they are if you’ve taking a class on either recently… a semi-infinite slab and a spherical generating point are (I’d argue) the canonical examples.

These two geometries allow you to visualize very rough cases in 1, 2, and 3 dimensions for either transport or diffusion problems. You can examine how neutrons will behave in either a roughly defined space by stringing together slabs or a bunch of points. More importantly, you can set up boundary conditions to look at things like shielding for just outside a reactor, or the activity of the reactor pressure vessel. The power of these two spherical cows is we can see how the neutronics will interact with materials, health statutes, and cooling requirements with something you can solve analytically (in many cases, maybe not all of them).

Fuel Cycles

Most of the fuel cycles work I have done is transaction-based, so my examples here involve keeping track of where fuel is traded to. To validate with back-of-the-envelope calculations, I will often make toy simulations that are A-B-C (source-facility/process-sink). This can be used at any level of fuel cycle analysis, inside a single facility or between facilities.

Another spherical cow in fuel cycles is when we make models that have one design and one fuel type. The reality of our fleet of reactors is that there are several designs, using fuel from a couple of companies. Depending on the metrics you are using to evaluate the fuel cycle, and the level of detail you're going into, these differences can fall within understandable uncertainty.

The final example we'll cover here is the abstraction of geographical informational data. For similar reasons as the reactor design, there are cases (like if you are tracking fuel composition leaving a reactor or separative work units) where the time it takes to transport fuel from fabrication to a reactor is a detail you can incorporate in successive rounds of inquiry.

Outreach

Outreach spherical cows are a little different than the ones we might expect in a class or model; in outreach we’re trying to give a non-expert a foothold for a broad subject, something they are already familiar with, to guide both their understanding and the conversation. One of the first abstractions of a nuclear power plant I learned for outreach in college was “the fancy tea pot.” Starting from this high level, slightly absurdist, picture of what’s going on gives people a foothold based in something they are intimately familiar with while talking about a nuclear power plant that they probably never have seen (and never will see) in their life. This spherical cow centers the conversation on the fuel and coolant, allowing you to expand to talk about the reactor pressure vessel or where the coolant goes in the system.

Another spherical cow that I have used in explaining my own work on the fuel cycle is that of a bakery (really it’s everything in the process of making a loaf of bread, but I always start with something to the effect of “think about a bread bakery”). Comparing the bakery to a reactor, you can draw connections to parts of the fuel cycle from growing the wheat, milling the grain, making the dough, firing the bread, and removing the bread from the oven. Especially after the pandemic when everyone was making sourdough, this has served me pretty well.

Making your own

At this point in the blog, I hope you’re starting to think to yourself “how can I make my own spherical cow?” because I have thoughts. First, we need to engage with how spherical cows are used. I think they have three strengths:

Visualize the rough case, quickly and cheaply.
Figure out which boundaries work, and what behavior looks like at the boundary.
See which terms/components dominate behavior.

The next thing I’d argue you need to do is to identify what part of the process you are trying to get insight into. Are you trying to understand the behavior of a system, or are you trying to understand the behavior of a component? This will help you decide what the dominant physics are, and what you can afford to ignore.

To complete your approximation, you just need to decide what scale you're looking at. This is how you will know when you have gone too far and are starting to lose important details. The final step is to put it in to action and refine as you go.