Slow reactivity insertion

As an example we consider the case of a PWR. The average normal temperature of the coolant is around 300 ${^{\circ}}C.$ For fresh fuel the reactivity change between zero and nominal power is close to -0.016[32]. 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 supposed to be 1 MWth^4.13. The initial value of the reactivity is taken to be $\rho$=0.016^4.14. The evolution of the power(see equation 3.65 is given by the set of equations:
$$\begin{align*}\frac{dW}{dt} & =\rho(T(W))W/\tau_{D}\\ T(W) & =T(W(0))+\left( T(... ...rho(T(W(0))\left( 1-\frac{T(W)-T(W(0))}{T(W_{nom} )-T(W(0))}\right) \end{align*}$$

where W(0) is the initial power, which we chose to be W(0)=1. W_nom is the nominal thermal power of the reactor, which we chose to be 3000 MW. The temperature at nominal power is T(W_nom) which we take to be $300{^{\circ }}C$ while the the initial temperature is $30{^{\circ}}C$. This set of equation reduces to

$$\frac{dW}{dt}=\frac{\rho(T(W(0))}{\tau_{D}}\left( W\left( 1+\... ...(0)}{(W_{nom}-W(0))}\right) -\frac{W^{2}}{W_{nom}-W(0)}\right)$$

The solution of this equation is found to be $W\left( t\right) =\frac {a}{b+e^{-at}\left( a-b\right) }$ with

$$a=\frac{\rho(T(W(0))}{\tau_{D}}\left( 1+\frac{W(0)}{(W_{nom}-W(0))}\right)$$ (4.63)

and

$$b=\frac{\rho(T(W(0))}{\tau_{D}\left( W_{nom}-W(0)\right) }$$ (4.64)

The solution of this schematic treatment for a typical PWR reactor is shown on figure 3.6

  

Figure 3.6: Evolution of the power of a reactor starting in a super-critical state at zero energy. The reactivity decreases due to the negative temperature coefficient. Criticality is reached at the nominal power

We see on the figure that power stabilization occurs within around 50 seconds.