Considering fuel and moderator (coolant) for simplicity, the thermal utili­zation factor f is given by the next formula

(1.64)

where Xf and xjf are the macroscopic absorption cross sections for thermal neutrons of fuel and moderator, respectively, and VF and VM are their volume fractions. Z is called the thermal disadvantage factor [17] originat­ing from heterogeneity of the fuel lattice cell and it is defined as

Z — Фм/0F.

It is noted that the thermal disadvantage factor is also dependent on the fuel and moderator temperatures. Although structure materials were omitted in Eq. (1.64) for simplicity, the following discussions essentially do not change even if considering the structure.

In recalling the axial expansion of the fuel, it should be noted that, in the numerator and denominator of Eq. (1.64), there will be a variation in the atomic density in the macroscopic cross section of the fuel due to an increase in the fuel temperature. Considering the fuel temperature depen­dence of the thermal disadvantage factor additionally, the fuel temperature coefficient of the thermal utilization factor is given as

1 # _f1 n( 1 dN* 1

f dTF ‘ nf dTF z dTFJ

= (l-f’)(-eF-ai) .

The first term in the second parenthesis in the final expression is determined by the linear expansion coefficient and has a negative effect of about 10_5 Ak/k/K for a solid fuel. Physically, this shows the probability of the thermal neutron absorption in the fuel will decrease due to a decrease in the fuel density. The second term is discussed later with the thermal disadvantage factor term in the moderator temperature coefficient.

As mentioned above, the fuel temperature coefficient was discussed with the Doppler effect and the fuel expansion effects through p and f. A negative temperature coefficient by the Doppler effect is dominant among them. Furthermore, because the fuel temperature responds immedi­ately to changes in reactor power compared with the moderator tempera­ture, the fuel temperature coefficient is thus often described as the prompt temperature coefficient. In this connection, it is of greatest importance that the temperature coefficient becomes negative owing to the Doppler effect.