In the present studies we performed the analytical calculation of the average Dancoff factor for prismatic high-temperature reactors; in this type of core, the fuel elements consist of small fuel grains (TRISO particles) randomly dispersed in a moderator (graphite) matrix and confined to a cylindrical volume (fuel pin). By definition, the Dancoff factor is the probability that a neutron leaving a fuel kernel hits uncollided another fuel kernel in the same fuel pin, which represents the intrapin contribution, or in another pin, which represents the interpin contribution. Similar studies have already been performed for pebble bed high-temperature reactors, where spheres (fuel pebbles) play the role of the cylinders; consequently, we retained the physical model describing an infinite lattice of unit cells, each containing a pair of concentric spheres, where the inner sphere is filled with a mixture of fuel grains and moderator and the outer one is filled with pure moderator, and we derived the mathematical model for the case of concentric cylinders. The physical model is grounded on the chord theory and the concept of a pseudo cross section; the latter takes into account, when the medium consists of moderator and small fuel grains, the probability, per unit path length, that a neutron either collides with a moderator nucleus or hits a fuel surface. The above method possesses a general validity, and it is suitable for the treatment of spheres (fuel pebbles), cylinders (fuel pins), or cuboids (fuel prisms) filled by moderator and small fuel grains.
The predictions of the analytical method well match the results of the MCNP code; nevertheless, since in the case of prismatic cores the mathematical model involves the calculation of complicated double integrals, the CPU time required by the two different methods becomes comparable.