The title is the question. Cheers.
There are denser fuels - the metal form U (dens = 19.05 g/cm3), UN, UC, and U3Si. They are not necessarily less expensive to produce, and generally a more expensive, considering that they are usually converted from UF6, which is commonly used in the enrichment process.
See - http://www.rertr.anl.gov/ADVFUELS/GHHD.html
Furthermore, one has to consider the properties such as thermal conductivity (particularly as a function of exposure (burnup)), melting point, thermal expansion coefficient (and anisotropy), chemical compatibility with cladding and coolant, fission product retention (particularly with respect to Xe, Kr), swelling as a function of exposure, and how these affect the dimensional and physical stability of the fuel system.
I have a little experience with different forms of Uranium fuel. I did a semester of undergraduate research comparing thermal conductivity between the ones Astro mentioned above (except for U3Si). Each has their own quirks (swelling, Hydrogen pickup, dislocations that can arise if you don't form the fuel properly). I didn't look into cost but that's why academia is so great.
I should mention that UN and UC have been proposed for fast reactor fuel, espeically high temperature fuel. They have higher thermal conductivity than UO2. In fast reactor designs, voids in the center of the fuel have been allowed, but in commercial LWR fuel, centerline melt (or void) is a no-no.
It appears that most applications of U3Si are as dispersed fuel, where the fuel portion is dispersed in a nonfuel metal matrix, e.g, Al.
Cermet fuels are another possibility.
It's a matter of finding the right balance of enriched U or Pu in the fuel and matrix.
Swelling and fission product migration are the significant challenges, especially at high exposure. Hydrogen pickup (or redistribution) is an issue in aqueous systems, or where the fuel is a metal hydride, e.g., U-Zr-H.
fast reactor? Is there a free reactor?
If there are warm or mild molten/liquid uranium alloys (with alkali metals or indium maybe) then maybe there wouldn't be a problem of fissile buildup.
Fissile/fissionable nuclides are necessary for 'fission' reactors. There are liquid fueled systems, but one still has to address fission product accumulation - it is inherent/inevitable in the process. The presence of alkali metals or indium does not change that.
But would they be lodged in the fuels?
Not if the uranium fuel is liquid. Fission products are gaseous (Xe, Kr), volatile (Cs, I, Br), or otherwise metal with various melting points.
The point of solid fuel is to retain the fission products, which accumulate with time/exposure.
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