I’m a math minor, and an equation like that is not a very appealing way to begin a blog post. But I got the unpleasant part out of the way early because the good news that this equation (the value equation) is totally central towards understanding how much separative work we need to separate isotopes from one another, and the even better news is that the value function makes a lot more intuitive sense than we might initially think.
Let me show you what the value function looks like when you plot it on a graph. The input value “x” can only range from zero to one. No negative numbers or values greater than one allowed here!
You can see that the value function looks like a deep valley with very steep sides. Reading the values of the graph, you can see that if the input “x” is one-half (0.5) then the value function is zero. And it’s hard to see, but you can imagine that if the value function is zero or one, then it has an infinite value. So what does all this actually mean?
To help answer this question recently I bought a bunch of red and white marbles online (www.megaglass.com if you’re interested). Red and white marbles are going to represent two different nuclear isotopes in my explanation of the value function. They’re pretty good at doing this because they’re the same size and only differ in color.
Let’s imagine that I only have red marbles in my “mixture” (x = 1.0):
But let’s say I have a half-and-half mixture of red and white marbles (x = 0.5):
Then my mixture doesn’t have any value at all?
Here was my first clue that there was something intuitive going on with the value function. Why does a half-and-half mixture have no value at all? Well, think about it. If you have a half-and-half mixture of marbles, you’re no closer to having a mostly white mixture than you are to having a mostly red mixture. It all depends on which kind of marble you start picking out.
But if you have a mixture that’s mostly white, or a mixture that’s mostly red, then those mixtures already have some “value”, and it’s easier to increase their “value” by removing the minority component. When it comes to enriching lithium, that’s exactly what we do. We remove more and more lithium-6 until the mixture is nearly all lithium-7.
But with uranium, surprisingly, we move the other direction on the value function. We “decrease” the value of the mixture as we “enrich” it! Hard to believe, huh?
Consider natural uranium. It’s only got 0.71% uranium-235. Let’s imagine that in white and red marbles:
Yup, because these equations don’t say anything about whether what you’re separating is good or bad or cheap or valuable. They just describe how “separated” it is. And natural uranium, with only 0.71% U-235, is more “separated” than enriched uranium at 3.5%.
Kinda wild, isn’t it?
What about the tails? What are they worth?
Now of course, we would not consider 50% enriched uranium to have a value of zero (in fact we would consider it QUITE valuable) but that’s how the equations come out. And that’s because 50% enriched uranium would be neither more U-235 or more U-238, but rather a mixture that could go either way.
Isn’t it interesting that when you plug all this into the SWU equation you get something that makes sense? It still amazes me, and I wrote the dang program that shows it.
Well, that’s probably enough fun with marbles and value functions for one night–I hope you liked it! Let me know if you did…