In the lattice burnup calculation, a combination of parameters such as moderator density and temperature, and fuel temperature is made at representative values (p0, Tm0, f) expected at normal operation of the reactor. In the core calculation, however, a different set (p, Tm, Tf) from the representative set is taken depending on position and time. Hence, a subsequent calculation called the branch-off calculation is performed after the lattice burnup calculation if necessary.

Figure 2.15 depicts an example branch-off calculation of moderator density. The calculation proceeds in the following order:

(i) Perform the lattice burnup calculation at a reference condition (p0, Tm0, Tf0). Designate the few-group homogenized cross section prepared from the lattice burnup calculation as £ (p0, Tm0, Tf0).

(ii) Perform the lattice calculation at each burnup step on the condition that only the moderator density is changed from p0 to pa, by using the same fuel composition at each burnup step. Designate the few-group homoge­nized cross section prepared from the lattice calculation (or the branch-off calculation) as E(p0 ! Pa, Tm0, f).

(iii) Carry out the branch-off calculation similar to (ii) at another moderator density of pb and give E(p 0 ! pb, Tm0, T0).

(iv) If the moderator density is instantaneously changed from p0 to an arbitrary p, calculate the corresponding cross section by the following approxima­tion (quadratic fitting).

S(p0 -> p, Tm0, Tf0) * 2(p0, Tm0, Tf0) + a(p — p0) + b(p — p0)2 (2-34)

(v) Determine the fitting coefficients a and b from the approximation at p = pa and p = рь.

Hence, the cross section at an arbitrary moderator density (an instantaneous moderator density) p away from the historical moderator density p0 can be expressed by the method mentioned above. If three moderator densities (p 1, p2, p3) are employed as p0 as shown in Fig. 2.14, their branch-off calculations can give X (p 1 ! p, Tm0, Tf0), X (p2 ! p, Tm0, Tf0), and X (p 3 ! p, Tm0, Tf0). By interpolation of the three points, the cross section at p resulting from an instantaneous change from a historical moderator density p can be obtained as X (p! p, Tm0, T0).

The cross section at an instantaneous change in fuel or moderator tempera­ture can also be described by the same function fitting as above. Since its change in cross section is not as large as a void fraction change (0-70 %), the following linear fittings are often used.

Z(Po, Tm0 Tm0> Tfo) = ^(р(П Tm0> Tfo) + c(Tm ~ Tmo) (2-35)

2(po, Tmo> Tfo Tf) = 2(po, Tmot TfQ) + ^(д/Т/ — V^/o) (2-36)

Equation (2.36) has square roots of fuel temperature in order to express the cross section change due to the Doppler effect by a lower-order fitting equation. An employment of a higher-order fitting equation can lead to a higher expres­sion capability and accuracy, but it gives rise to a substantial increase in the number of branch-off calculations and fitting coefficients and therefore results in an inefficient calculation. Thus, the reference cross sections and their fitting coefficients are stored into the few-group reactor constant library and used in the nuclear and thermal-hydraulic coupled core calculation.