.. _section-4.2:

Reactor Power
-------------

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

(4.2‑1)

.. _eq-4.2-1:

.. math::

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

where :math:`\psi_{\text{t}}\left( t \right)` is the dimensionless, normalized
power amplitude and :math:`S\left( \overrightarrow{r} \right)` is the
steady-state reactor power in watts being produced in an axial node at
location :math:`\overrightarrow{r}`. In terms of input quantities,
:math:`S\left( \overrightarrow{r} \right)` is given by the product of
:sasinp:`POW` and :sasinp:`PSHAPE`. Initially, the power amplitude has a value of unity and
:math:`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)

.. _eq-4.2-2:

.. math::

	\psi_{\text{t}}\left( t \right) = \psi_{\text{f}}\left( t \right) + \psi_{\text{h}}\left( t \right)

where :math:`\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)

.. _eq-4.2-3:

.. math::

	\psi_{\text{f}}\left( t \right) = \psi_{\text{f}}\left( 0 \right)\phi\left( t \right)

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

(4.2‑4)

.. _eq-4.2-4:

.. math::

	\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 :math:`\phi\left( 0 \right) = 1`.

In Eq. :ref:`4.2-4<eq-4.2-4>`, :math:`\delta k \left( t \right)` is the net reactivity,
:math:`\beta` is the total effective delayed-neutron fraction,
:math:`\Lambda` is the effective prompt neutron generation time, and
:math:`\lambda_{\text{i}}` is the decay constant for the delayed-neutron
precursor isotope whose normalized population is
:math:`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].