\chapter{Experimental (if relevant)}
\textit{Computer tools and simulations}

This section might be omitted, but it could also be used to describe how the raw data was treated, and how I arrived at the data that are represented in the results-section. \\

\section{Amount of leakage calculation}

The fraction of \chemfig{O_2} in the flue gas is measured both before the ECO/LUFO and after the flue gas fan (Figure xx). With this information it is possible to estimate the amount of air leaking into the flue gas between these two measurements as a fraction of the original amount of flue gas. This is done by creating both a total mass balance (Equation \ref{eq:massBalance}) and a mass balance of the amount of oxygen (Equation \ref{eq:O2-balance}): 

\begin{equation}
    \dot{n}_{Fan} = \dot{n}_{Air} + \dot{n}_{Flue}
    \label{eq:massBalance}
\end{equation}

\begin{equation}
    \dot{n}_{Fan}*x_{O_2, Fan} = \dot{n}_{Air}*x_{O_2, Air} + \dot{n}_{Flue}*x_{O_2, Flue}
    \label{eq:O2-balance}
\end{equation} 

$\dot{n}_{O_2,Air}$ is the molar flow of the air leaking into the flue gas, $\dot{n}_{O_2,Flue}$ is the molar flow of the flue gas before the ECO/LUFO, and $\dot{n}_{O_2,Fan}$ is the molar flow of the gas directly after the flue gas fan. $x_{O_2}$ represent the corresponding molar fractions of oxygen.

Combining Equation \ref{eq:massBalance} and Equation \ref{eq:O2-balance} and rearranging gives Equation \ref{eq:leakFraction}.

\begin{equation}
    \frac{\dot{n}_{Air}}{\dot{n}_{Flue}} = \frac{x_{O_2,Fan}-x_{O_2,Flue}}{x_{O_2,Air}-x_{O_2,Fan}}
    \label{eq:leakFraction}
\end{equation}

In this calculation it is assumed that there are no reactions (i.e no combustion) taking place between the two measurements. The sensors measure the the volume fraction of \chemfig{O_2} in dry air. The water content of the gases is therefore not taken into account, and it is assumed that the volume fractions that the sensors measure are equal to the molar fractions. \\

Equation \ref{eq:leakFraction} was applied to a data series that stretched several weeks with data points for each minute. This resulted in 21598 values for $\frac{\dot{n}_{Air}}{\dot{n}_{Flue}}$. These values where then averaged using the Matlab-function \textit{trimmean} that excludes a certain percentage of the outliers. \cite{trimmean} \\

Because there are no flow measurements of the flue gas, there is no direct way of knowing the absolute amount of leakage. It is also not possible to estimate it from a total mass balance due to uncertainties in the fuel composition (ref??), leaks in other parts of the process (ref??), and poor flow measurements of the air inlets (ref??). \\

The flue gas flow is therefore estimated based on the flue gas fan. There are however too little information to use compressor curves to estimate the flow at each instant. It is therefore assumed that fan operates at maximum capacity. This might not be as harsh of an assumption as it seems, because it is assumed that the flue gas fan is the main reason that boiler is constrained below rated capacity, and hence the fan should operate close to its maximum capacity (see appendix with compressor data?? or refer to topic somewhere else in report??)

The maximum capacity of the flue gas fan is given by the operating manual.

\section{Energy calculations}



\section{Notes}
What kind of calculations have I done:

\begin{itemize}
    \item Calculated amount of leakage between the two O2-measurements
    \item Calculated energy loss stemming from the amount of heat needed to heat the extra air
    \item Did a test of the Shunt pump. Show that nothing happened. 
    \item Have done some calculations to try to estimate economic loss
\end{itemize}

What kind of calculations should/could I do:

\begin{itemize}
    \item Calculate theoretical change in temperature because of the shunt pump.
    \item Use a better method for averaging, like Matlab instead of Excel?
    \item Graphs should be drawn in matlab
\end{itemize}

If accumulator is incuded, I think it would be good to show how the "friblåsning" is estimated in this section