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Appendices
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.. _section-A4.1:

Appendix 4.1: Steady-State Power Normalization in |SAS|
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The power in axial fuel pin segment :math:`I` of channel :math:`J` is
given by the expression

.. math::

	{\text{PSHAPE}}\left( I,J \right)\  \times \ {\text{QMULT}} \times \ {\text{POW}}

Here POW is the maximum power of any axial fuel pin segment. QMULT is a
multiplier that is equal to one in steady state.

:math:`{\text{PSHAPE}}\left( I,J \right)` is the ratio of the
power of segment :math:`\left( I,J \right)` to the maximum power of any
segment. It is obvious that PSHAPE takes values between zero and one.

|SAS| will renormalize the input values of PSHAPE and PRSHAP.
This is done is subroutine PNORM. The user needs only supply
un-normalized data for these arrays.

PNORM will also compute POW from POWTOT or POWTOT from POW according to
the value of IPOWOP.

Input:

.. list-table::
    :header-rows: 1
    :align: center
    :widths: 1,4

    * - IPOWOP = 0
      - Calculate steady-state power in peak axial segment from total reactor power.
    * - IPOWOP = 1
      - Calculate steady-state total reactor power from the peak axial fuel pin segment.
    * - POWTOT
      - Total reactor power in steady state.
    * - POW
      - Steady-state power in the peak axial pin segment.
    * - FRPR
      - Fraction of total reactor power represented by sum of all |SAS| channels.
    * - NPIN(ICH)
      - Number of fuel pins in a subassembly of channel ICH.
    * - NSUBAS(ICH)
      - Number of subassemblies in channel ICH.
    * - MZ(ICH)
      - Number of axial nodes in channel ICH.
    * - PRSHAP(ICH)
      - The relative power per *subassembly* in channel ICH.

        PRSHAP will be normalized by PNORM.
    * - PSHAPE(I,ICH)
      - The relative power per axial *pin* segment of axial pin segment I and channel ICH. I = 1, … MZ(ICH).

        PSHAPE will be normalized by the PNORM routine.

Method:

The values of PRSHAP get renormalized as RELCHA for all channels:

.. math::

	{\text{RELCHA}}\left( \text{ICH} \right) = {\text{PRSHAP}}\left( \text{ICH} \right)\  \times \frac{\sum_{\text{I}}{{\text{NSUBAS}}\left( I \right)}}{\sum_{\text{I}}{{\text{PRSHAP}}\left( I \right) \times {\text{NSUBAS}}\left( I \right)}}

If we multiply the above equation by
:math:`{\text{NSUBAS}}\left( \text{ICH} \right)` and then sum
over all channels :math:`\text{ICH}`, we can show that

.. math::

	\sum_{\text{ICH}}{{\text{RELCHA}}\left( \text{ICH} \right)\  \times \ {\text{NSUBAS}}\left( \text{ICH} \right)} = \sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}

Thus, RELCHA is properly normalized.

The values of PSHAPE get normalized as RELSHP for each channel:

.. math::

	{\text{RELSHP}}\left( IZ,ICH \right) = \frac{{\text{PSHAPE}}\left( IZ,ICH \right)}{\sum_{\text{I}}{{\text{PSHAPE}}\left( I,ICH \right)}}

If we sum over all axial nodes :math:`\text{IZ}`, we can show that

.. math::

	\sum_{\text{IZ}}{{\text{RELSHP}}\left( IZ,ICH \right) = 1}

Thus, RELSHP is properly normalized.

The power of axial pin segment :math:`\left( I,J \right)` is

.. math::

	{\text{RELSHP}}\left( I,J \right)\  \times \frac{{\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)}\  \times \ \frac{{\text{FRPR}}\  \times \ {\text{POWTOT}}}{\sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}}

From the definition of POW we get the equation

.. math::

	{\text{POW}} = {\left\lbrack {\text{RELSHP}}\left( I,J \right)\  \times \frac{{\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)}\  \right\rbrack \times \ \frac{{\text{FRPR}}\  \times \ {\text{POWTOT}}}{\sum_{\text{ICH}}{{\text{NSUBAS}}\left( \text{ICH} \right)}}}

and the unknown POWTOT or POW can be found. Given
:math:`I_{\mathrm{\max}}` and :math:`J_{\mathrm{\max}}` to be the
indices of the peak power pin segment, we then redefine
:math:`{\text{PSHAPE}}\left( I,J \right)` as

.. math::

	{\text{PSHAPE}}\left( I,J \right) = \frac{{\text{RELSHP}}\left( I,J \right)\  \times \ {\text{RELCHA}}\left( J \right)}{{\text{NPIN}}\left( J \right)} \times \frac{{\text{NPIN}}\left( J_{\mathrm{\max}} \right)}{{\text{RELSHP}}\left( I_{\mathrm{\max}},J_{\mathrm{\max}} \right)\  \times \ {\text{RELCHA}}\left( J_{\mathrm{\max}} \right)}

This is the desired form of PSHAPE that is used by the subroutine SHAPE.