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^{235}U) in the form of a ceramic, UN. By defining a fuel rod geometry (pellet diameter, cladding inner and outer diameter) and fuel rod pitch (typically P/D is ~1.25, but could be as much as 1.3), one will calculate the number of fuel rods in the system.

However, a core is composed of fuel assemblies, each containing a set of fuel rods.

Now, in a hexagonal array, one starts with a central location (row 1) of either an assembly or fuel rod. Immediately, or directly, surrounding the one central are six (6) in row 2, then around the six (6), is twelve (12), and so on. For each successive row, one adds 6 to the number in the previous row.

So, the counting goes as: 1, 6, 12, 18, 24, 30, 36, 42, 48, . . . .

and the cumulative or total number of elements in the array goes as: 1, 7, 19, 37, 61, 91, 127, 169, 217, . . .

See - https://en.wikipedia.org/wiki/Centered_hexagonal_number

The total number of fuel rods, NR, in an assembly, can be defined by 3(r

^{2}-r)+1, where r is the number of rows of fuel rods, and similarly, the number of assemblies, NA, in a core is given by 3(a

^{2}-a)+1, where a is the number of rows of assemblies.

Then one is faced with selecting the number of rods per assembly and the number of assemblies. In other words, the product of NR and NA has to equal the total number of fuel rods from the neutronics calculation. For example, one could use NR = 127, NA = 169, or NR = 169, NA = 169, since 127 x 169 = 169 x 127 = 21463.

One the other hand, one could decide NR = NA = 169, and 169 x 169 = 28561.

One also has to decide on the fuel pellet diameter (and density of the ceramic fuel, or metal fuel), the cladding inner and outer diameter, and the pitch (distance between the centers of adjacent fuel rods or assemblies). The fuel pellet diameter affects the criticality and power density, the inner and outer cladding diameters affect the stress in the cladding depending on the differential pressure across the cladding wall, the outer diameter affects the heat flux for a given power density (or linear heat rate, kW/m), and the cladding diameter and fuel rod pitch affects the hydraulic resistance and various thermal-hydraulic characteristics of the lattice.

It's an interesting design problem, but I thought the mathematics of 'centered hexagonal numbers' is interesting. Note that I didn't address control rods and their interaction with the fuel assembly.