Heat and Cubes: The secret math of SMRs

We’re going to do things a little differently this time. Normally I talk about a complex topic by drilling down to the fundamentals and then building back up to make the point about the original topic.

This time however we are going to start at the top and explore that layer before removing it and peering at what it is built on. Let’s see if this works as well, and hopefully saves me some time on getting articles out on the increasingly complex topics I keep getting interested in!

So today we are going to talk a bit about SMRs. Specifically the common claims about how they require fewer safety regulations such as smaller Emergency Planning Zones to operate since they cannot melt down.

Now we’ve talked about meltdowns before, about what they are not and to refresh everyone; Meltdowns are not explosions. They are a lack of ability to remove heat from a core faster than it is being produced in the core. Now all reactors have heat transfer systems to prevent this, lots of pumps and water, or molten salt/metal/helium in 4th gen designs. But these are all internal to the reactor core and require power to keep running.

But since meltdowns don’t tend to happen when everything is working fine, it’s more important to figure out what happens when there isn’t any power, and that’s where SMRs break out the secret weapon. Math.

So let’s imagine we have a perfectly Cubic (hold your offended gasps physicists, there is a reason we are not using a spherical cow in a vacuum) reactor core with no active cooling. Basically just a big lump of metal that is producing heat. It doesn’t matter how big the core is so let’s just say that it has dimensions of 1x1x1 meters/yards/parsecs/bloits/smoots (yes those are real). You can substitute in your favourite unit as you like.

So since it’s a 1x1x1 cube we can easily show that it has a volume of 1^3 and a surface area of 1×1+1×1+1×1+1×1+1×1+1×1… or 6x1x1 more simply. Or just 6 if you aren’t pedantic. Now this is the important part: the ratio of surface area to volume (lets call it the SAVR). In reactors and other similar systems, the amount of heat generated is related to the volume, while the rate at which heat leaves is related to the surface area. With our current cube we have a SAVR of 6, which seems trivially obvious, it’s a cube, it has 6 faces duh. Well here’s where things start to get weird… and useful.

Let’s cut the cube halfway along each of its faces so that now we have 8 identical reactor cubes of 0.5×0.5×0.5. We haven’t changed the total volume of the system, but we have increased the surface area, doubled it to be precise. So that means that the SAVR of all our 8 smaller reactors together is now 12. Here is the interesting (and slightly unexpected) part; each of the cubes Also now has a SAVR of 12, despite them still being perfect cubes just like the larger initial one. So how does something that has the exact same aspect ratio of another identical shape have a different ratio? Well this is where the SMRs secret math comes into play. We can see pretty easily that since we carved the first cube into 8 smaller cubes, each cube must have a volume of 1/8 of the original (0.5×0.5×0.5 = 0.125 if you prefer numerical examples) but the surface area is 6×0.5×0.5 = 1.5.

So we’ve actually reduced the amount of surface area per cube, but it has only been reduced by 4x compared to the volume reducing by 8x. Thus we’ve actually Doubled the SAVR. And this trend continues if we continue cutting all the cubes into smaller and smaller cubes. The SAVR goes from the initial value of 6 to 12 to 24 to 48 and so on with each consecutive halving of the dimensions of all the cubes. This is called the Square-Cube Law and it is what describes the math that allows SMRs to claim that they cannot melt down. Because they are Small. Remember that the amount of heat that is generated in a reactor is dependent on the volume while the rate of heat transfer out of the reactor is dependent on the surface area. So the higher we can drive our SAVR, the easier it is to get that heat out of the system faster than it can accumulate and potentially cause a meltdown.

The best part is that this property is agnostic to what form of cooling you want to use for this situation. Whether you are using air or water or something more exotic only changes how small you need to make the reactor to ensure adequate cooling in worst case scenarios, not whether cooling is possible or not. And in the next article we will dive a little more deeply into how the requirements for cooling inform the designs and mechanisms used.