Finally, reactors are designed so that a power increase leads to a reactivity decrease. This has the interesting consequence that the reactor is auto­stable. Placing the control rods in a predetermined position ensures that the reactor power will converge to a given value.

Slow reactivity insertion

As an example we consider the case of a PWR. The average normal tempera­ture of the coolant is around 300 °C. For fresh fuel the reactivity change
between zero and nominal power is close to —0.016 [48]. The nominal power is taken to be 3 GWth. We make the simplifying assumptions that the tem­perature is proportional to the reactor’s power and that the power rise is slow enough for the equilibrium temperature to be reached at each power level. We neglect the contribution of radioactive processes to the power. The initial power is assumed to be 1 MWth.[20] The initial value of the reactivity is taken to be p = 0.016.^ The evolution of the power [see equation (3.101)] is given by the set of equations

— = p(T (W)) W/rD

W — W(0)

T(W) = T(W(0)) + (T(WnQm) T(W(0))) Wnom —

where W(0) is the initial power, which we chose to be W(0) = 1 MW. Wnom is the nominal thermal power of the reactor, which we chose to be 3000 MW. The temperature at nominal power is T(Wnom) which we take to be 300 °C while the initial temperature is 30 °C. This set of equations reduces to

dW = p(T(W(0))) ґ Ґ W(0) W2

dt td I I (WnQm — W (0))J WnQm — W (0)J

whose solution is found to be

a

b + e—at(a — b)

with

and

The result obtained with this approximate treatment for a typical PWR reactor is shown in figure 3.8. We see in the figure that power stabilization occurs within around 50 s.