The moderator temperature coefficient of the thermal utilization factor can be approached in the same way. A variation in the moderator atomic density due to an increase in the moderator temperature (the moderator volume fraction is constant) can be described with the linear expansion coefficient of the moderator and also by considering the moderator temperature dependence of the thermal disadvantage factor. In addition, the thermal absorption cross sections of both the fuel and moderator change in order to shift the thermal neutron spectrum due to an increase in the moderator temperature. The moderator temperature coefficient can be therefore obtained by Eq. (1.68).

The temperature dependence of the thermal cross sections is given by Eq. (1.69) [18].

o — (Tn) = f<? (TO (2f )1/2 a (To) (1 -69)

This equation can be obtained by integrating a 1/u cross section for the thermal neutron spectrum in a Maxwellian distribution characterized by effective neutron temperature Tn of the system. g(Tn), called the non-1/u factor (Westcott factor), represents the extent of the difference from the 1/u behavior of resonance cross sections in the range of thermal neutron energies. The non-1/u factor depends on the neutron temperature because the thermal neutron spectrum and resonance cross section vary with tem­perature. a(T0) is the cross section at the neutron speed of и = 2200 m/s and it is employed as a reference mark. The corresponding neutron energy and temperature (T0) are 0.0253 eV and 293.61 K, respectively.

The neutron temperature Tn may be considered to be the same as the moderator temperature TM to determine the thermal neutron spectrum. However, since the thermal neutron spectrum is practically hardened by thermal neutron absorption (absorption hardening), the neutron tempera­ture is somewhat different from the moderator temperature. The neutron temperature can be regarded as approximately proportional to the moder­ator temperature by using a proportionality constant a, given by

Tn = aTM (1.7°)

where a is about 1.2-1.3 for a light water-moderated reactor [14, 19].

Thus the moderator temperature coefficient of the thermal cross section can be written as

JL = JL дд дТп _ 1 дТп = 9 _ 1 d 71)

* ^

Applying Eq. (1.71) to Eq. (1.68) gives the moderator temperature coeffi­cient concerning the thermal utilization factor

a£v=(l— f)(36M—а^+ааф (172)

where the temperature coefficient of the non-1/u factor was removed for a 1/u absorber such as the light water moderator. The first term in the second parenthesis on the right-hand side is positive. Physically, this results from the effect to raise the absorption probability of thermal neutrons in the fuel due to a decrease in moderator density caused by a moderator temperature increase. Its magnitude is large, about 10~4Ak/k/K for liquid moderators. For the second term in the second parenthesis on the right-hand side, let us consider a special case in which the fuel and moderator temperatures uniformly change; then the moderator temperature coefficient of the dis­advantage factor in Eq. (1.72) can be summed with the fuel temperature coefficient of the disadvantage factor in Eq. (1.65), that is,

CCt=CCtf~^~(Xtm, (1:73)

It turns out that aT is always negative [14]. This is due to the fact that the thermal diffusion length increases with temperature. As the diffusion length increases, the neutron flux in the lattice fuel cell tends to flatten, that is, the depression of the flux across the cell becomes less pronounced, and this leads to a smaller value of the disadvantage factor. As a result, there is a positive reactivity effect on the temperature coefficient of the thermal utilization factor. Further details are not discussed here.

The third term in the second parenthesis on the right-hand side of Eq. (1.72) is the effect of the non-1/u absorption cross section of fuel.

Table 1.3 n values of major fuel nuclides [18]

The temperature coefficient is generally quite small, on the order of 10_6AklklK for 233U and 235U. However, 239Pu has a high dependence of its non-1/u factors on the neutron temperature, as shown in Table 1.2, and the moderator temperature coefficient of f has a large positive reactivity effect (order of 10_4AklklK). Hence, this effect should not be neglected as 239Pu builds up with fuel burnup.