## Peter Fortescue 1975 Paper on Burners/Breeders

Yesterday I was going through some old papers and I came across one that I must have photocopied quite some time ago. It was called “Advanced HTGR Systems” and was written by Peter Fortescue of General Atomics in 1975, and published in the Annals of Nuclear Energy, Vol. 2, pages 787-799.

As I looked at it, I tried to ascertain what it was that had caught my attention years ago that would have caused me to photocopy it and preserve it. I’m not particularly interested in high-temperature, gas-cooled reactors (HTGRs) that General Atomics was pushing back in the 1970s. I dug through the paper and found some mention of closed-cycle gas turbines, which are interesting, and some discussion of high-temperature hydrogen production which is interesting–but then I found it–the reason why I must have been interested in this paper.

Fortescue did a very interesting analysis of the rate of growth of a nuclear-powered world consisting of both breeder reactors and burner reactors. By breeders, I mean reactors that can produce more fissile material from fertile material than they consume (conversion ratio > 1.0) and by burners I mean reactors that produce less fissile material than they consume (conversion ratio < 1.0). I’ve seen lots of analyses of the rates of growth possible in an all-breeder world, but this one was interesting because it mixed up the two types. The particular reactors Fortescue was interested in were HTGRs that ran on thorium and U-233 and had a high conversion ratio, roughly 0.85, and gas-cooled fast reactors (GCFRs) that could have a rather high conversion ratio, about 1.45. Fortescue wanted to know how many fast reactors he would need to produce the extra fissile needed to keep the thermal reactors going, because they weren’t producing enough fissile to meet their own needs. All of this analysis assumed that reprocessing gas-cooled reactor fuels was basically a given, and in reality it’s really hard, because gas-cooled reactor fuels tend to be SUPER chemically stable and super hard to break down for reprocessing, but for a moment we won’t worry about that. Fortescue found that the ratio of fast reactors to thermal reactors tended to be about 2-3 thermal reactors per fast reactor, if you just wanted to maintain steady state power production, or less if you wanted to “grow” your nuclear enterprise. Which kind of makes sense if you think about it–if you have one kind of reactor that doesn’t make enough fuel to keep itself going and another kind that makes more than it needs, then the one that makes more is going to be favored in a growth scenario. So I began wondering, as I must have wondered years ago, about what the scenarios would look like if we thought about the whole thing in terms of fluoride (LFTR) and chloride reactors instead of HTGRs and GCFRs. The main difference in the analysis would be that LFTR can be safely assumed to have a conversion ratio of one, so that once started, a LFTR wouldn’t need makeup fissile material to keep it going. That would mean that any excess fissile U233 produced by the chloride reactors could be applied to starting more and more LFTRs rather than keeping existing ones going. So I started to plunge into Fortescue’s equations and it wasn’t long before I ran into trouble. A bunch of his equations had a term like (1 – C) in the denominator, where C is the conversion ratio of the thermal reactor, in our case, LFTR. Well, you can see that if your conversion ratio is one, then (1 – 1) = 0, and zero in the denominator is a big no-no mathematically, causing the solution to be undefined. So to figure out the answer to my question using Fortescue’s equations probably means that I’m going to have to go back to the drawing board and try to re-derive the equations in a way where this numerical “explosion” doesn’t happen. But it’s exciting. I was so excited about it last night that I kept glancing at Fortescue’s equations while my wife and I were watching “LOST”, which caused her to emphatically remind me to PAY ATTENTION! (you can’t ignore “LOST” or you miss important things) I woke up in the middle of the night and thought about the equations while lying there in bed. It’s pretty hard to do any derivations while the lights are out and your eyes are closed. My mental algebra skills are only so good. Hopefully I’ll have more on this subject to report soon. UPDATE: Here’s a scan of Fortescue’s original derivation.

When you published this post, I remember responding that I found the problem with the equations: time is assumed Infinite so that the conversion ratio =1. In any finite time, conversion < 1. The point of the exercise was to find the steady-state solution. The fix is to develop differential equations and find their asymptotes. My offer to help work on this stands.