Here, an explanation is given as to the reason for placing a design point on the under-moderated region being discussed. The moderator tem­perature coefficient is mainly determined by the balance of the tem­perature coefficients of p in Eq. (1.66) and f regarding the first term in the second parenthesis of the right-hand side in Eq. (1.72). It can be then written as Eq. (1.76).

(1.76)

When p approaches 1, the following holds

and the moderator temperature coefficient can be approximated as

aTM^SOM(p—f) . (1.77)

If f exceeds p, it may provide a negative moderator temperature coefficient.

Figure 1.13 illustrates the infinite multiplication factor and the four factors as a function of the ratio (denoted by x) of the atomic number densities of fuel and moderator. A decrease in moderator density due to a rise in moderator temperature corresponds to the increase of x. As mentioned up to now, the figure indicates that the variation of the effective multiplication factor due to an increase in moderator temper­ature is determined by p andf (n in the four-factor formula is out of the range of this figure, but almost constant with x without the resonance effect of nuclides such as 239Pu).

To make the moderator temperature coefficient negative in Fig. 1.13, in other words, to decrease the infinite multiplication factor with an increase of x, x should be larger than xmax at the maximum of the infinite multiplication factor.

The region above xmax, in which the ratio of the moderator atomic density is smaller than that at xmax, is called the under-moderated region. As an optimal point, a reactor may be designed with an as-small-as-possible fuel concentration which gives the largest value of the infinite multiplication factor. However, it is usual to place the design point slightly toward the under-moderated region from the maximum in order to make the moderator temperature coefficient negative from the viewpoint of reactor safety. In the region below xmax (the over-moderated region), the infinite multiplication factor increases with x, that is, the moderator temperature coefficient is positive.

Further, it is easy to confirm that Fig. 1.13 includes the positive or negative relation of the moderator temperature coefficient in Eqs. (1.76) or (1.77).

(i) 239Pu buildup effect on n and f [21]

The multiplication of n andf in uranium fuel, including 239Pu produced during burnup, can be written as Eq. (1.78).

vzp35+vz;u239

^U235 _|_ ^U238 _|_ ^Pu239_|_ ^Others

The right-hand side were partially substituted by F and A, which represent the contribution rates of 239Pu to fission neutron production and absorption of thermal neutrons, respectively. Further, referring to Eq. (1.75) and applying the temperature coefficient of n to 239Pu gives Eq. (1.80).

a%=(F-А)аадтГ+Ааа^

A rough approximation is

1.:vPu239 v^Pu239

VLf_____ ____________

VZ^+vSf"239 ~ £U235_|_2Pu239

and then

Inserting Eq. (1.82) into Eq. (1.80) gives

• • аТм ^FaKl /fissile) (%Tfn /fissile ^гГ39] . (1:83)

Therefore, the moderator temperature coefficient of nf in 239Pu buildup can be expressed in an easy form by the thermal utilization factor of fissile nuclides ffisile through the approximation of Eq. (1.81).

Fig. 1.14 An example of the shift of moderator temperature coefficient to positive with fuel burnup in a LWR [22]

For example, consider

aff239=—lAX10~[1]Ak/k/K aft-239=-5 X ~A Ak/k /К.

According to both temperature coefficients, it is necessary that > 0.74 for af < 0 in Eq. (1.83). The thermal absorption rate of fissile nuclides should be large to make the reactivity effect negative against 239Pu build up. By contrast, a small thermal absorption rate of fissile nuclides for natural uranium or very low-enriched fuel may lead to emergence of factors to make the moderator temperature coefficient positive due to 239Pu buildup. Figure 1.14 shows an example that the moderator temperature coefficient in a LWR shifts to positive values with fuel burnup.

The coolant temperature coefficient is also used in fast reactors with no moderator. The mechanism of reactivity change is different from that in thermal reactors (see Sect. 1.4).