[1] Diffusion equation

In core calculations such as the nuclear and thermal-hydraulic coupled core calculation, core burnup calculation, or space-dependent kinetics calculation, high-speed calculation is the biggest need. Especially since large commercial reactors have a considerably larger core size compared with critical assemblies or experimental reactors, the neutron transport equation (2.1) has to take many unknowns and hence need a high computing cost.

For most core calculations, the angular dependence (J) of neutron flux is disregarded and the neutron transport equation is approximately simplified as Eq. (2.37).

(2.37)

v dt

Its physical meaning is simple. As an example, a space of unit volume (1 cm3) like one die of a pair of dice is considered. The time variation in neutron population within the space [the LHS of Eq. (2.37)] is then given by the balanced relationship between neutron production rate (S), net neutron leakage rate through the surface (V-J), and neutron loss rate by absorption (£аф).

J is referred to as the net neutron current and expressed by the following physical relation known as Fick’s law

] = —DV<p (2.38)

where D is called the diffusion coefficient; it is tabulated together with few-group cross sections in the lattice calculation to be delivered to the core calculation. Inserting Eq. (2.38) into Eq. (2.37) gives the time-dependent diffusion equation.

(2.39)

The time-independent form of Eq. (2.39) is the steady-state diffusion equa­tion and all types of the core calculation to be mentioned thereafter are based on the equation

V-DV(/)-i:a(t)+S = 0 (2.40)

Thus, solving the diffusion equation in critical reactors is to regard the reactor core as an integration of subspaces like the dice cubes and then to find a neutron flux distribution to satisfy the neutron balance between production and loss in all the spaces (corresponding to mesh spaces to be described later).