11.1. Definitions

The temperature coefficient of a reactor may be defined as the rate of change of reactivity per unit temperature change, and is in general a function of temperature.

A negative temperature coefficient leads to a self-stabilizing reactor. On the contrary, if the temperature coefficient is positive, the reactor is unstable and any flux distur­bance tends to amplify itself. In this case the reactor can only be operated if the control system counteractions stabilize it. A positive temperature coefficient imposes much more stringent requirements on the control system. On the other hand, a negative coefficient of very high absolute value, while improving the reactor safety, requires a large number of control rods to shut down the reactor in cold conditions. The temperature coefficient is the result of various effects and can be then subdivided in different ways.

It is customary to separate the fuel from the moderator contribution,

(11.1)

defining in this way a fuel and a moderator temperature coefficient.

Since following a transient the various contributions to the temperature coefficient may respond with different delays, it is also customary to speak of a prompt and a delayed temperature coefficient.

The two definitions are not necessarily the same, especially in high-temperature reactors where part of the moderator is intimately mixed with the fuel. The distinction between prompt and delayed coefficient depends on the fuel and moderator geometry and may be rather arbitrary. The distinction between fuel and moderator temperature coefficient is based on better-defined physical reasons.

The fuel temperature is determining the Doppler-broadening of resonances, while the moderator temperature defines the thermal spectrum. An interaction of these two effects can be given by low-lying resonances which can interact with the thermal spectrum (e. g. 240Pu). The effect of the thermal expansion and density change in the reactor is important in some reactor types, but negligible in high temperature graphite moderated reactors. In terms of the “four-factor formula” the temperature coefficient can be separated in the following way"’

where Р/ and P, h are respectively the fast and thermal non-leakage probability. This

expression is not used for calculating the temperature coefficient, for which multi-group computer codes are used, but this splitting provides a better physical insight.

Formula (11.2) can also be written

1 dke« 1 dkx 1 6Pnl,..

ійіг "Laf u ’

PNL being the total (fast and thermal) non-leakage probability. In this way it is possible to see the effect of temperature on к^ and on the leakage. The total leakage contribution is usually negative. The thermal leakage provides always a negative contribution because the thermal diffusion coefficient increases with temperature. The fast leakage has usually a small positive contribution because Doppler broadening may decrease the fast diffusion coefficient with temperature. The result is usually negative, so that a high leakage has a stabilizing effect on the reactor.

Sometimes a “power coefficient” is defined. This coefficient has only a meaning if the temperatures are in the equilibrium condition corresponding to a given power and if each power level corresponds to a given coolant mass flow. In this case, knowing the temperature coefficient, it is possible to calculate the rate of change in reactivity per unit power change.

During the reactor operation temperatures are space-dependent. Suitable average temperatures have to be defined in order to be able to define a temperature coefficient. Usually arithmetic averages are sufficient. Average fuel and moderator temperatures have to be defined either for the whole reactor (for zero-dimensional dynamics calculations) or for various reactor regions (for space-dependent dynamics calcula­tions). HTR temperature coefficients are of the magnitude KT5/°C, so that they can be calculated with sufficient accuracy as a difference of two multi-group static calcula­tions, performed at temperatures differing a few hundred degrees.