Two Fluid Design
Summary: This picture shows a two-fluid LFTR but the heat exchanger loop is not shown.
I want to start this discussion from the design proposed for the two fluid design involving a fissile sale and a fertile Th salt.

I'll echo as accurately I can the popular proposed design from the Google talks. I welcome any corrections to my understanding, this is all nothing more than my current mental impressions.
A separate Th blanket is necessary so that there is no need to separate the Th from the fission products, among other reasons. But for the design above, we have the added advantage that there is no particular need to remove the Pa continuously, as the large volume of the Th blanket will keep it at a low concentration so that neutrons won't often disturb it before it decays. Then the useful U-233 is continuously processed out.
In terms of desired attributes of a reactor, we want a low volume of fissile material. This is a cited reason for a thermal reactor, but a fast reactor could be much more sustainable. In terms of the core critical configuration, we have an engineering limitation on the volumetric heat rate; in particular the wall separating the reactor and blanket has a limitation of the flux passing through it. Since we are surrounding the core with the blanket, the only rational option is to increase the surface area of the reactor (I will offer another option later), thus a strong proposal is to elongate the critical configuration and employ basically a cylindrical design. Thus the core will have a set diameter but can be as long as the power output dictates.
The heat is produced in the fissile salt and both the fissile salt and the blanket salt are continuously processed. But in addition to the fissile salt being processed, it is pumped into the steam generator where it exchanges heat with the thermal cycle plant. Now here is where I feel like there should be room for improvement. The proposal is to have a fissile salt / coolant heat exchanger, but this requires a larger volume of the fissile salt.
"Cauldron" Design
Summary: Hyperion had an excellent design with a bubbling liquid fissile core that had tubes going through it to remove the heat
I've often pondered what is the best theoretical design for a nuclear reactor given different objectives. The original design of the Hyperion reactor provided some valuable reflection for me.
This is what I would dub a "cauldron" design. Why? Because it is as nearly as possible a big blob of fissile material - close to a sphere. The fuel is also a liquid, which is advantageous for many reasons everyone here understands. Liquids don't loose integrity by melting down, there are no structural concerns other than the container itself, the volume of fissile material is minimized, and convection allows for better heat distribution and flux distribution. But there's a problem - if you make a blob of nuclear fuel, how do you remove the heat?! Note closely the rods coming out of the fuel pool in the image above. Those are cooling channels. They run cooling channels through a critical mass, thus the fissile mass never needs to go anywhere and doesn't need any processing until the standard reload procedure.
The other advantage of the cauldron design is that gases can be easily boiled off or dissolved in. Hyperion understood this and called for Hydrogen gas (if my memory serves me correctly) to be mixed in with the fissile salt as well as another sink that allowed material transfer as the temperature changed, giving a negative feedback that was the mother of all negative feedbacks. This is a very good design (that Hyperion itself abandoned), but like any design it's not perfect.
Mainly, because the coolant material is very different from the core material, we should expect a major event if one of the pipes failed. The pipes contain water/steam and the core contains liquid fuel. There is a risk of the water/steam leaking out into the core giving greater moderation or the fuel leaking into the pipe, causing a more compact shape and thus adding reactivity. I'm not sure which one increases power actually but I see this as a problem with the design because it has the potential to constitute a major contingency.
Fissile Salt + Fertile Salt cooling tubes?
Summary: Use a cauldron of fissile liquid fuel but run tubes through it containing fertile salt, exchange heat using the fertile salt, profit
Now I'll pick and choose my favorites to make a proposal. I like the cauldron design and I like the two-fluid design. Starting from the two-fluid LFTR design, why not have pipes containing the fertile salt permeating the fissile salt, allowing the heat exchanger to use the fertile salt?
The volume of the fertile salt can be large, thus it presents no added proliferation risk by having a large heat exchanger. The core itself, however, is a heat exchanger and is a more challenging environment. The barrier between the fissile salt and the fertile salt must withstand large neutron fluxes, and in this design, a temperature gradient as well. We would also require a large surface area between the two fluids (for the heat transfer).
However, given this design the reactor shape would logically be roughly spherical, like a cauldron. The fertile salt acts as a poison so adjusting the total volume as well as the ratio of fertile to fissile fluids would allow it to match whatever power requirement was demanded. Consider an infinitely large reactor. There will be a set ratio of the fertile to fissile materials giving k = k_infinity = 1. This infinite reactor would have pipes with the fertile material in the tubes and the fissile material outside the tubes with the engineered limit of neutron flux and heat flux going over the pipes.
The pressure of the fertile salt would have to be kept greater than that of the fissile salt so in the event of failure reactivity would decrease - just like in the current two-fluid proposal. There would still be a need for a blanket in order to use the leaked neutrons, so in addition to the pipes the fertile salt would be flowing through a downcomer in parallel.
The fissile salt would still have the "Chemical Separator" loop as shown in the first picture, but that would be the only loop and my understanding is that this loop can have a very low volume. This is fantastic because there is scarcely any more fissile material more than what's needed to have the core critical. A severe reactivity event could still occur if the fissile salt for some reason leaked into the fertile salt tubes, giving a tighter configuration with less poisons. But it's the same thing that can be said for the two-fluid design as it stands, and there would be less fissile material present at the site due to the heat exchanger using the benign fertile salt.
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Well I hope that kicks things off well. I just always thought to myself but it makes so much sense to use the fertile salt in the heat exchanger! Maybe there's something I haven't considered which kills the idea. Who knows? I thought this forum look cool, like people were actually discussing what the best possible design for a nuclear reactor would be. I would like to hear how this possibility fits into the picture.