\section*{Problem 4}

\subsection*{Problem Statement}

A pressurized water reactor (with highly enriched fuel) is initially at a steady state of 25\% steam load. The operators are directed to raise power (picking up electrical load) by increasing steam flow to 50\% in a single motion.

\textbf{Initial conditions:}
\begin{itemize}
    \item $T_{ave} = 500$ \degree F
    \item $T_h = 510$ \degree F (hot leg)
    \item $T_c = 490$ \degree F (cold leg)
    \item Power = 25\%
    \item No automatic control forcing changes in average temperature
\end{itemize}

\subsection*{Part A}

\subsubsection*{Solution}

\textbf{Chronological list of physical impacts on the primary system and reactor:}

\textbf{Secondary Side (Given):}
\begin{enumerate}
    \item Turbine throttle opened (operators pick up electrical load)
    \item Steam flow increases from 25\% to 50\%
    \item Pressure in steam generator drops
    \item Aggressive boiling in steam generator begins
    \item SG water cools (maintains saturation conditions)
    \item Primary side cold leg ($T_c$) cooled by steam generator
\end{enumerate}

\textbf{Primary Side and Reactor:}
\begin{enumerate}
    \setcounter{enumi}{6}
    \item Cold leg temperature ($T_c$) decreases
    \item Average temperature ($T_{ave}$) decreases (no automatic control)
    \item \textbf{Moderator temperature reactivity feedback} ($\alpha_{mod} < 0$):
    \begin{itemize}
        \item Cooler moderator $\rightarrow$ increased water density
        \item Better neutron moderation and thermalization
        \item Positive reactivity insertion: $\rho = \alpha_{mod} \times \Delta T_{ave}$
        \item Since $\alpha_{mod} < 0$ and $\Delta T_{ave} < 0$, then $\rho > 0$
    \end{itemize}
    \item Reactor power begins to increase
    \item Fuel temperature increases $\rightarrow$ negative fuel feedback (Doppler effect)
    \item Hot leg temperature ($T_h$) increases
    \item Core $\Delta T$ increases: $\Delta T = \frac{\text{Power}}{\dot{m} \times c_p}$
    \item Average temperature begins to rise back toward initial value
    \item \textbf{Power levels off at 50\% when:}
    \begin{itemize}
        \item Heat removal rate matches heat generation rate
        \item $\Delta T$ establishes new equilibrium: $\Delta T_{new} = \Delta T_{old} \times \frac{P_{new}}{P_{old}}$
        \item $T_{ave}$ returns to approximately initial value
        \item Net reactivity returns to zero (reactor critical)
    \end{itemize}
\end{enumerate}

\textbf{Impacts on six factors in $k_{eff} = \varepsilon \times p \times f \times \eta \times P_{FNL} \times P_{TNL}$:}

\begin{enumerate}
    \item \textbf{$\varepsilon$ (fast fission factor):} Negligible change
    \begin{itemize}
        \item Minimal U-238 present for fast fission
        \item No significant change
    \end{itemize}

    \item \textbf{$p$ (resonance escape probability):} Slight increase
    \begin{itemize}
        \item Limited U-238 resonance absorption with high enrichment
        \item Cooler moderator $\rightarrow$ higher density $\rightarrow$ more scattering
        \item Faster neutron slowing down through resonance region
        \item Less time spent at resonance energies $\rightarrow$ less absorption
        \item $p$ increases slightly
    \end{itemize}

    \item \textbf{$f$ (thermal utilization factor):} Possible slight increase
    \begin{itemize}
        \item $f = \frac{\Sigma_{a,fuel}}{\Sigma_{a,fuel} + \Sigma_{a,mod}}$
        \item Cooler moderator $\rightarrow$ better thermalization efficiency
        \item More neutrons successfully reach thermal energies where fuel cross section dominates
        \item Competing effect: higher moderator density increases $\Sigma_{a,mod}$
        \item Net effect: likely small increase
    \end{itemize}

    \item \textbf{$\eta$ (reproduction factor):} Minimal change
    \begin{itemize}
        \item $\eta = \nu \frac{\sigma_f}{\sigma_a}$ for U-235
        \item Temperature effects on U-235 cross sections are small
        \item Essentially constant
    \end{itemize}

    \item \textbf{$P_{FNL}$ (fast non-leakage):} Minimal change
    \begin{itemize}
        \item Large reactor $\rightarrow$ already high fast non-leakage
        \item Small temperature changes don't significantly affect
    \end{itemize}

    \item \textbf{$P_{TNL}$ (thermal non-leakage):} Slight increase
    \begin{itemize}
        \item Cooler moderator $\rightarrow$ higher density $\rightarrow$ shorter diffusion length
        \item Reduced thermal neutron leakage
        \item However, large reactor already has high $P_{TNL}$ (low leakage baseline)
        \item Effect is present but modest in absolute terms
    \end{itemize}
\end{enumerate}

\textbf{Overall mechanism:} Net positive reactivity from moderator cooling
results from the combined contributions of increased $p$ (faster slowing through
resonances), increased $f$ (better thermalization), and increased $P_{TNL}$
(reduced leakage). For a large reactor with high enrichment, multiple effects
contribute to the reactivity rather than a single dominant mechanism. The
cooler, denser moderator improves neutron economy across several factors,
causing power to increase until new equilibrium at 50\%.

\subsection*{Part B}

\subsubsection*{Python Code}
\begin{lstlisting}[language=Python,
    basicstyle=\ttfamily\small,
    keywordstyle=\color{blue},
    commentstyle=\color{gray},
    stringstyle=\color{red},
    showstringspaces=false,
    numbers=left,
    numberstyle=\tiny,
    frame=single,
    breaklines=true]
# Initial conditions
T_ave_initial = 500  # F
T_h_initial = 510    # F
T_c_initial = 490    # F
P_initial = 25       # %
P_final = 50         # %

# Calculate DeltaT
DeltaT_initial = T_h_initial - T_c_initial  # 20 F
DeltaT_final = DeltaT_initial * (P_final / P_initial)  # 40 F

# Estimate final temperatures (assume T_ave approximately constant)
T_ave_final = T_ave_initial
T_h_final = T_ave_final + DeltaT_final / 2
T_c_final = T_ave_final - DeltaT_final / 2
\end{lstlisting}

\subsubsection*{Solution}

At steady state, reactor power is proportional to the temperature rise across the core:
\[P \propto \dot{m} \times c_p \times \Delta T\]

For constant flow rate (no pump speed change):
\[\frac{P_{new}}{P_{old}} = \frac{\Delta T_{new}}{\Delta T_{old}}\]

\textbf{Initial state (25\% power):}
\begin{itemize}
    \item $\Delta T_{initial} = T_h - T_c = 510 - 490 = 20$ \degree F
\end{itemize}

\textbf{Final state (50\% power):}
\[\Delta T_{final} = \Delta T_{initial} \times \frac{P_{final}}{P_{initial}} = 20 \times \frac{50}{25} = \boxed{40 \text{ \degree F}}\]

Assuming strong moderator feedback returns $T_{ave}$ to approximately its initial value:
\[T_{ave,final} \approx 500 \text{ \degree F}\]

\[T_{h,final} = T_{ave} + \frac{\Delta T}{2} = 500 + 20 = \boxed{520 \text{ \degree F}}\]

\[T_{c,final} = T_{ave} - \frac{\Delta T}{2} = 500 - 20 = \boxed{480 \text{ \degree F}}\]

\textbf{Comparison table:}

\begin{center}
\begin{tabular}{lccc}
\hline
\textbf{Parameter} & \textbf{Initial (25\%)} & \textbf{Final (50\%)} & \textbf{Change} \\
\hline
Power & 25\% & 50\% & $+25\%$ \\
$T_h$ (\degree F) & 510 & 520 & $+10$ \degree F \\
$T_c$ (\degree F) & 490 & 480 & $-10$ \degree F \\
$T_{ave}$ (\degree F) & 500 & $\sim$500 & $\sim$0 \degree F \\
$\Delta T$ (\degree F) & 20 & 40 & $+20$ \degree F \\
\hline
\end{tabular}
\end{center}

\subsection*{Part C}

\subsubsection*{Solution}

For a reactor with low enrichment fuel (3-5\% U-235), the final condition will
be significantly different due to the large quantity of U-238 present. Low
enrichment fuel contains approximately 95\% U-238, compared to only about 7\% in the highly enriched case.

As power increases and fuel temperature rises, Doppler broadening of U-238 resonances creates a strong negative fuel temperature coefficient. This negative fuel feedback counteracts the positive moderator temperature feedback from the cooler water. The net reactivity insertion becomes much smaller than in the high enrichment case, and the power increase may be insufficient to reach 50\%.

To restore proper operation, operators must withdraw control rods to add positive reactivity and overcome the strong negative Doppler feedback. This allows the reactor to reach the desired 50\% power level matching the steam demand.
