4.2. Reactor Power

At any time $t$, the local power production at position $\overrightarrow{r}$ is assumed to be given by the space-time separated function:

(4.2-1) $$Q\left( \overrightarrow{r},t \right) = \psi_{\text{t}}\left( t \right)S\left( \overrightarrow{r} \right)$$

where $\psi_{\text{t}}\left( t \right)$ is the dimensionless, normalized power amplitude and $S\left( \overrightarrow{r} \right)$ is the steady-state reactor power in watts being produced in an axial node at location $\overrightarrow{r}$. In terms of input quantities, $S\left( \overrightarrow{r} \right)$ is given by the product of POW and PSHAPE. Initially, the power amplitude has a value of unity and $S\left( \overrightarrow{r} \right)$ is normalized to the total steady-state reactor power. Appendix 4.1 contains a description of the internal normalization of PSHAPE performed by SAS4A/SASSYS‑1. The time-dependent power amplitude is assumed to be made up of the sum of two components:

(4.2-2) $$\psi_{\text{t}}\left( t \right) = \psi_{\text{f}}\left( t \right) + \psi_{\text{h}}\left( t \right)$$

where $\psi_{\text{h}}\left( t \right)$ comes from the decay of fission and capture products. These two components have been separated to allow the simulation of both short- and long-term transients.

The direct fission component of the power amplitude is given by

(4.2-3) $$\psi_{\text{f}}\left( t \right) = \psi_{\text{f}}\left( 0 \right)\phi\left( t \right)$$

where $\phi \left( t \right)$ is the dimensionless, normalized fission power amplitude given by the point reactor kinetics model:

(4.2-4) $$\dot{\phi}\left( t \right) = \phi\left( t \right)\frac{\delta k\left( t \right) - \beta}{\Lambda} + \sum_{\text{i}}{\lambda_{\text{i}}C_{\text{i}}\left( t \right)}$$

with the initial condition $\phi\left( 0 \right) = 1$.

In Eq. (4.2-4), $\delta k \left( t \right)$ is the net reactivity, $\beta$ is the total effective delayed-neutron fraction, $\Lambda$ is the effective prompt neutron generation time, and $\lambda_{\text{i}}$ is the decay constant for the delayed-neutron precursor isotope whose normalized population is $C_{\text{i}}\left( t \right)$. The physical interpretation of the terms in the point reactor kinetics equation is made by Henry [4-2] and also by Bell and Glasstone [4-3].