section_4.tex

\newpage 
\section{Numerical Results} \label{s4}
With the theoretical background described in the previous sections, this section demonstrates the capabilities of the proposed framework in addressing multi-physics problems using several test cases, ranging from two-dimensional pin-cell problems to whole-core problems. Most of the problems presented in this thesis use material compositions and geometry from the Virtual Environment for Reactor Applications (VERA) core physics benchmark \cite{godfrey}. Additionally, ENDF/B-VII.1 was used for the MCS cross section library.

This section is divided into three major subsections. Subsection \ref{sec40} briefly describes the VERA reactor benchmark used for the numerical test problems in this thesis. Subsection \ref{sec41} provides the solutions for MC coupled multi-physics with spatially continuous material properties. While the solutions for thermal expansion are presented in subsection \ref{sec42}. Lastly, subsection \ref{sec43} presents the solutions for assembly and whole-core problems using both spatially continuous material and thermal expansion methods.

\subsection{Benchmark Description} \label{sec40}

The reactor operational data for the VERA benchmark were taken from Watts Bar Generating Station Unit 1, operated by the Tennessee Valley Authority (TVA) since 1996. This reactor is a 3411 MWth Westinghouse pressurized water reactor (PWR) with 193 fuel assemblies and an active core height of 365.76 cm. Each fuel assembly contains a \(17 \times 17\) array of pins, consisting of 264 fuel pins, 24 guide tubes, and 1 instrumentation tube as shown in Figure \ref{f3}. Eight spacer grids are used for each assembly to maintain its structural integrity. Radially, the fuel-assembly pitch is 21.5 cm, with fuel pins within an assembly having pitch of 1.26 cm. In a \(17 \times 17\) array of pins, this results in a 0.04 cm inter-assembly gap. These inter-assembly gaps are essential to accommodate fuel assembly deformation due to thermal expansion. The fuel pin consists of a fuel pellet with a radius of 0.4096 cm, along with a gap and cladding, as illustrated in Figure \ref{f2}.

\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{figs/asm.png}
    \caption[Fuel assembly radial layout.]{Fuel assembly radial layout. Adapted from \cite{albagami}.}
    \label{f3}
\end{figure}

During the first cycle, the reactor employed three different fuel enrichments: 2.11\%, 2.62\%, and 3.10\% by weight of U-235, and Pyrex burnable absorbers were used. Reactor regulation is provided by 57 control rod assemblies, grouped into 8 banks, as illustrated in Figure \ref{f1}. Pyrex is a discrete burnable neutron absorber typically used in Westinghouse reactors, made from borosilicate glass (B$_2$O$_3$-SiO$_2$), and inserted into assembly guide tubes. These inserts can be placed in any assembly not located in a control rod position. The control rod is an axial stack of Silver-Indium-Cadmium (AIC) and boron carbide (B$_4$C). A set of 24 control rods are clustered into a reactor cluster rod assemblies (RCCAs) to control and ensure a safe core shut down. The tips of the control rods are made from AIC and the remaining portions up to the plenum are made from B$_4$C. 

\begin{figure}
    \centering
    \includegraphics[width=0.4\textwidth]{figs/pin.png}
    \caption[Fuel rod radial layout.]{Fuel rod radial layout. Adapted from \cite{albagami}.}
    \label{f2}
\end{figure}

\begin{figure}
    \centering
    \includegraphics[width=0.9\textwidth]{figs/core.png}
    \caption[VERA benchmark core configuration and control rods layout.]{VERA benchmark core configuration and control rods layout. Adapted from \cite{godfrey}.}
    \label{f1}
\end{figure}

\input{section_41}
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\input{section_42}
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\input{section_43}