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].