\chapter{Energy and economic calculations}
\label{sec:energyEconomicCalc}

This section presents the results of the most important energy and economic calculations. Section \ref{sec:amountLeak}, \ref{sec:heatingExtraAir} and \ref{sec:shuntPumpResults} are based on the mass and energy balances derived in Section \ref{sec:massEnergyBal}.

Almost all calculations are done using a Matlab script which or various reasons unfortunately are not included in this thesis.

The process parameters that are used in the calculations in Section \ref{sec:amountLeak}, \ref{sec:heatingExtraAir} and \ref{sec:shuntPumpResults} are based on a data series covering a period of little over two weeks. The data series contains data for every minute throughout the time period. When the process parameters are used in the calculation, they are averaged using the Matlab function \textit{trimmean} so that 10\% of the outliers are excluded.\cite{trimmean} The steam production of Boiler6 during these two weeks was usually between 10 and 14 kg/s as shown by Figure \ref{fig:steamProd2weeks}.

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{Figures/steamProd2weeks.png}}
\caption{Steam production by Boiler6 during a period of roughly two weeks.}
\label{fig:steamProd2weeks}
\end{figure}

There is also calculated some economic losses associated with the heat losses calculated in Section \ref{sec:heatingExtraAir} and \ref{sec:shuntPumpResults}. These calculations are based on the average electricity price that Norske Skog had to pay during the two week data period. During this period, the Electric boiler was used all the time, meaning that any efficiency increase in Boiler6 would lead to a corresponding reduction in steam produced (and hence electricity used) by the Electric boiler. Also, by assuming that the electricity tax (EL-avgift) and grid cost (nettleie) can be neglected (the cost of these are very low compared to the electricity price, and taking into account the already harsh assumption behind the energy loss calculations (see Section \ref{sec:massEnergyBal}), this should not have a huge impact on the results), the economic loss can simply be calculated by multiplying the heat loss with the energy loss.

\section{Amount of leakage}
\label{sec:amountLeak}

The mole fraction of air leaking into the flue gas was calculated using Equation \ref{eq:leakFractionRes} derived in Section \ref{sec:massBalance}:

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

$x_{O_2,Flue}$ and $x_{O_2,Mix}$ were estimated based on the data series described at the beginning of Section \ref{sec:energyEconomicCalc}. The averaged values for $x_{O_2,Flue}$ and $x_{O_2,Mix}$, together with the assumed value for $x_{O_2,Air}$ and calculated value for $x_{air}$ are presented i Table \ref{tab:massBalance}:

\begin{table}[H]
\caption{Mole fraction of air leaking into the flue gas}
\label{tab:massBalance}
\centering
\begin{tabular}{c l} 
 \hline \\ [-1.5ex]
 $x_{O_2,Flue}$ & 0.037 \\
 $x_{O_2,Mix}$ & 0.093 \\
 $x_{O_2,Air}$ & 0.21 \\ 
 $x_{Air}$ & 0.32 \\ [0.5ex] 
 \hline
\end{tabular}
\end{table}

Because there are no flow measurements of the flue gas, there is no direct way of knowing the absolute amount of leakage. It is neither possible to estimate it from a total mass balance due to uncertainties in the fuel composition (see Section \ref{sec:fuel}), leaks in other parts of the process (see Section \ref{sec:leaks}), and poor flow measurements of the air inlets (see Section \ref{sec:airExplained}). \\

The flue gas flow is therefore estimated based on the flue gas fan. There is however too few measurements regarding the flue gas fan to use compressor curves to estimate the flow at each instant. It is therefore assumed that the fan operates at its 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.

The maximum capacity of the flue gas fan is given by the operating manual to be 29.3 \si{Nm^3}. Assuming a density of 1.2922 kg/Nm\textsuperscript{3} and a Molar mass of 28.3 kg/kmol, Equation \ref{eq:molarFlowAirRes} (derived in Section \ref{sec:massBalance}) can be used to calculate a molar flow of the air leaking into the flue gas.

\begin{equation}
\label{eq:molarFlowAirRes}
    \dot{n}_{Air} = x_{Air}*\frac{q_{Mix}*\rho_{Mix}}{M_{Mix}}
    %n_air = x_air*m_flue/M_flue; %[mol/s]
\end{equation}

The molar flow of air leaking into the flue gas between the two points of \ce{O_2}-measurements (Figure \ref{fig:boiler6Schematic}) is calculated to be:

\begin{equation}
\label{eq:molarAirResult}
    \dot{n}_{Air} = 432 \text{ mol/s}
\end{equation}


\section{Heat loss because of air leaking into the flue gas}
\label{sec:heatingExtraAir}

As described in Section \ref{sec:HeatFalseAirEqs}, there is a heat loss associated with air leaking into the flue gas. Using Equation \ref{eq:FalseAirResult} along with $Cp_{Air}$ (both derived in Section \ref{sec:HeatFalseAirEqs}), the values for $T_{Mix}$ and $T_{Room}$ from the data series described at the beginning of Section \ref{sec:energyEconomicCalc}, and the $\dot{n}_{Air}$ calculated in the previous section, $\dot{Q}_{Heat,Air}$ can be calculated.

\begin{equation}
    \label{eq:FalseAirResult}
    \dot{Q}_{Heat,Air} = \dot{n}_{Air}*Cp_{Air}*(T_{Mix}-T_{Room})
\end{equation}

\begin{table}[H]
\caption{Loss from heating of false air}
\label{tab:heat_loss}
\centering
\begin{tabular}{r l} 
 \hline \\ [-2ex]
 $T_{Mix}$ & 148 \si{\celsius} \\
 $T_{Room}$ & 36 \si{\celsius} \\ 
 $Cp_{Air}$ & 29.28 J/(K$\cdot$mol) \\
 $\dot{n}_{Air}$ & 432 mol/s \\ 
 $\dot{Q}_{Heat,Air}$ & 1.42 MW \\
 \textit{Economic Loss} & 496 000 NOK/month \\ [1ex] 
 \hline
\end{tabular}
\end{table}

Table \ref{tab:heat_loss} presents the value of the heat loss as well as the values of the parameters used to calculate it. 

There is also listed an \textit{Economic Loss} in Table \ref{tab:heat_loss}. This is calculated on the basis of the mean electricity price that Norske Skog had to pay during the same two weeks as the data series are from (see beginning of Section \ref{sec:energyEconomicCalc}). Provided that the assumptions behind the calculation of $\dot{Q}_{Heat,Air}$ holds (these are already some harsh assumptions, see Section \ref{sec:HeatFalseAirEqs}) the \textit{Economic Loss} listed in the table quite accurately represents the economic value of $\dot{Q}_{Heat,Air}$. The \textit{Economic Loss} is listed at a per monthly basis, although the underlying data only stretches a little over two weeks. This is because it is assumed that the two weeks that corresponds to the data points are quite typical for the winter season. Therefore, the \textit{Economic Loss} can be regarded as representative for the winter time.



\section{Energy/economic loss from shunt pump}
\label{sec:shuntPumpResults}

As explained in Section \ref{sec:shuntPump}, having the shunt pump running at its maximum capacity represents a heat loss resulting from reduced heat recovery in the economizer. Apart from fuel oil, the fuel burned in Boiler6 usually have a low sulfur content. This means that it is probably not necessary to have it running during normal operation.

By using the equations derived in Section \ref{sec:shuntPumpBal}, the temperatures of the water and flue gas streams in and out of ECO1 can be calculated. Table \ref{tab:shunt_pump_temp} shows the temperature results for these calculations.

\begin{table}[H]
\caption{Temperatures before and after the shutdown of the shunt pump. Red = Calculated values, Black = Measured values.}
\label{tab:shunt_pump_temp}
\centering
\begin{tabular}{l c c} 
 & Before shutdown & After shutdown \\ [0.5ex]
 \hline \\ [-1.5ex]
 $T_{f,out}$ & 250.1\si{\celsius} & \textcolor{red}{241.5\si{\celsius}}\\ 
 $T_{f,in}$ & \textcolor{red}{379.9\si{\celsius}} & \textcolor{red}{379.9\si{\celsius}} \\
 $T_{w,out}$ & \textcolor{red}{207.1\si{\celsius}} & \textcolor{red}{213.4\si{\celsius}}\\
 $T_{w,in}$ & 141.2\si{\celsius} & 111.5\si{\celsius} \\ [1ex]
 \hline
\end{tabular}
\end{table}

Based on these temperature values as well estimated mass flows and heat capacities (see Section \ref{sec:massEnergyBal}), an energy loss resulting from the lack of heat recovery can be calculated. Combining the heat loss and the cost of operating the pump with the electricity price (as described at the beginning of Section \ref{sec:energyEconomicCalc}) a corresponding economic loss is calculated. The results are listed in Table \ref{tab:shunt_pump_loss}.


\begin{table}[H]
\caption{Loss form heating of false air}
\label{tab:shunt_pump_loss}
\centering
\begin{tabular}{r l} 
 \hline \\ [-2ex]
 $\dot{Q}_{Heat,Shunt}$ & 0.371 MW \\
 $E_{Pump}$ & 0.003 MW \\
 \textit{Economic loss} & 131 000 NOK/month \\ [1ex] 
 \hline
\end{tabular}
\end{table}

$\dot{Q}_{Heat}$ represents the energy loss caused by the lack of heat recovery, $E_{Pump}$ represents the energy needed to operate the pump. The \textit{Economic loss} calculated here indicates a substantial loss caused by having the pump running. It is however important to keep in mind the assumptions behind the calculation:

\begin{itemize}
    \item The heat transfer coefficient, $U$, is assumed to be constant, but in reality it decreases when the flow through the heat exchanger decreases.\cite{GeankoplisChristieJ2003Tpas} This is exactly what happens when the shunt pump is turned off, meaning that the possible heat recovery resulting from turning off the pump is less then the result in Table \ref{tab:shunt_pump_loss}.
    \item It is assumed that the lower flue gas temperature after the Eco does not affect the heat recovery in AP1-4. The flue gas exiting the Eco is the same that goes into to AP1-4 (See Figure \ref{fig:boiler6Schematic}), and a lower flue gas temperature here will in reality reduce the heat recovery in AP1-4 (because of a lower $\Delta T$ over the heat exchanger). This also reduces the possible savings listed in Table \ref{tab:shunt_pump_loss}. How large this effect is is hard to know, because there are no temperature measurements of the primary and secondary air after AP4, so the $\Delta T$ over the hot side of AP4 is unknown.  However, because there is not drastic shift in temperature of the flue gas, it indicates that this effect is small.
    \item The \textit{Economic Loss} corresponding to $E_{Heat,Shunt}$ is calculated using the average price of electric energy for the data period. This is considered an accurate choice for this period, because the electric heater is used every day during the period which means that increased energy efficiency in Boiler6 leads to less need for electricity. For the rest of the year however, this might not be the case. Taking into account that also the electricity price varies through the year it is clear that extrapolating this cost over the whole year should be done with caution. 
\end{itemize}

The uncertainties listed here indicates that the \textit{Economic Loss} is less than listed in Table \ref{tab:shunt_pump_loss}. The economic loss will however be substantially higher than zero, and simply turning off a pump is not considered an expensive thing to do. \\

Another thing worth mentioning is that the reason for having the shunt pump there is (as mentioned in Section \ref{sec:shuntPump}) to avoid that the flue gas temperature becomes too low when oil is used as a fuel. Oil has a higher sulfur content, so it becomes more important to keep a high enough flue gas temperature to avoid that \chemfig{SO_2} condenses in the system and cause corrosion. 


\section{Loss because Boiler6 is constrained}

As mentioned in Section \ref{sec:leaks}, the leaks throughout the flue gas system makes the flue gas fan become a bottleneck that limits the maximum power output from the boiler. This means that in addition to the direct cost of having to operate Boiler6 with leaks, there is also an alternative cost associated with the fact that Boiler5 and the electric heater need to be utilized to cover the "gap" left by the Boiler6 that is constrained below capacity. This cost stems from the fact that the cost of fuel for Boiler6 is considerably less than the the cost of electricity (for the Electric boiler). \\

Figure \ref{fig:boiler_use} shows the daily average power output from Boiler6 and the electric heater. The data series starts at the end of 2013 for Boiler6 and the middle of 2015 for the Electric heater. Both series end in June 2018. The data is modified such that for the days when Boiler6 has a power output lower than 20MW$_{\text{th}}$, both the data for Boiler6 and the electric heater are removed. This is done because it is assumed that for these days there are other reasons for the lower output of Boiler6 (e.g. maintenance, low steam demand, lack of fuel).

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=1.3\textwidth]{Figures/graphBoilerUse2.png}}
\caption{Daily average power output of Boiler6 and the Electric heater.}
\label{fig:boiler_use}
\end{figure}

As can be clearly seen from Figure \ref{fig:boiler_use}, the power output of Boiler6 has a higher output during the winter season compared to the summer season; at least for the years 2014-2016. This is expected since the steam demand of the factory is higher during the winter. For the years 2017 and 2018, the power output does not show the same peaks during the winter season, suggesting that the boiler is constrained below capacity. 
The data for the electric heater does not go as far back in time as the data for Boiler6. Nevertheless, it is clear that the utilization is higher during the winter of 2017 and 2018 than the winter of 2016. This suggests that the electric heater now acts as a "peaker" during periods of high steam demand because Boiler6 cannot perform at the same level as before. \\

As mentioned above, the cost of the fuel used in Boiler6 is considerably less than the cost of electricity. This suggest that using the electric heater as a "peaker" might constitute a significant economic loss. 

To estimate the economic loss associated with the use of the electric heater instead of Boiler6, the daily average power output of the two boilers (the same data that is shown in Figure \ref{fig:boiler_use}) together with the daily average electricity price for the corresponding days were used. 

The data was modified such that the days where Boiler6 had a power output below 20MW$_{\text{th}}$ were excluded from the calculation (same as in Figure \ref{fig:boiler_use}). For the days where the sum of the power output from Boiler6 and the electric heater exceeded 50MW$_{\text{th}}$ (corresponding to the maximum capacity of Boiler6), the overshooting power output from the electric heater was not included (an interesting point is that there is only two days over the three year period where this happens). Both modifications were performed such that only the power output of the electric heater that could have been replaced by a well functioning Boiler6 was included in the cost calculation. \\

The cost of the fuel used in Boiler6 is very low compared to the electricity price, and it is assumed to be in the same price range as the sum of the electricity tax (EL-avgift) and grid cost (nettleie). By letting these two costs, associated with operating Boiler6 and the electric heater respectively, cancel out, the calculation can be done using only the electricity price. This simplification is done such that the cost of the fuel for Boiler6 remains secret to some degree. \\

Table \ref{tab:loss_constrained} lists the results over a few chosen time periods. The period from 5.jun 2015 - 2.jun 2018 represents the whole data period. The next three time periods represent periods of one year, beginning and ending in June. 

\begin{table}[H]
\caption{Economic loss over different time periods resulting from using the electric heater instead of utilizing the full capacity of Boiler6}
\label{tab:loss_constrained}
\centering
\begin{tabular}{c  r} 
 Time period & Economic loss [MNOK] \\
 \hline \\ [-2ex]
 5.jun 2015 - 2.jun 2018 & 12.96 \\
 5.jun 2015 - 4.jun 2016 & 1.50 \\
 3.jun 2016 - 2.jun 2017 & 2.89 \\
 3.jun 2017 - 2.jun 2018 & 8.57 \\ 
 21.feb 2018 - 8.mar 2018 & 1.49 \\ [1ex]
 \hline
\end{tabular}
\end{table}

As one would expect from Figure \ref{fig:boiler_use}, the economic loss shows a rising trend. Especially comparing the years 2015-2016 to 2017-2018 shows an increase of over 7 MNOK. In fact, only 16 days during the winter of 2018 (corresponding to the data period used in the energy calculations in Section \ref{sec:heatingExtraAir}) and \ref{sec:shuntPumpResults} shows almost the same loss as the whole 2015-2016 year. \\

A few things to keep in mind when evaluating these data:

\begin{itemize}
    \item The calculations are based on daily averages of the power output. This means that for days where it seems that a well functioning Boiler6 could have rendered the Electric heater redundant, it might have been necessary to use the electric heater anyway to cover peaks during the day. The effect of this is that the Economic loss is somewhat overestimated.
    \item As the previous item, the electricity price is also based on daily averages. This means that a varying power output from the electric heater coupled with a varying electricity price could mean that the calculated economic loss deviates from the actual economic loss. The effect of this might lead to both an overestimation and an underestimation of the economic loss.
    \item It assumed that a well functioning Boiler6 can have a power output of 50MW$_{\text{th}}$ regardless of the fuel. This might be a bit optimistic considering the large variation in fuel quality (see Section \ref{sec:fuel}). However, it can be seen from Figure \ref{fig:boiler_use} that the boiler regularly had daily average power outputs in the 40-45 MW$_{\text{th}}$ range in the past, indicating that it can perform not too far from the rated capacity even when running on fuel with lower quality. 
    \item The effect of a possible fix of the steam turbine is not taken into account in these calculations. If the steam turbine where to be fixed, it would be possible to generate electricity by producing extra steam for the turbine  it will add one more component to the economic loss: The lost income resulting from not being able to produce extra steam for electricity generation in the steam turbine.
    \item The effect on the use of Boiler5 (the bark boiler) is not considered. The cost of bark might be higher than the average cost of fuel used in Boiler6, depending on the overall bark situation at the paper mill (see Section \ref{sec:steamProdTotal}). This means that if Boiler6 was able to perform at its rated capacity, it might to some degree reduce the need for Boiler5.
\end{itemize}

Although this list of items includes assumptions that suggest that the economic loss is overestimated, it is still clear from Table \ref{tab:loss_constrained} that the economic loss increases roughly 7 MNOK over the course of two years. \\

It is not clear that it is the leaks that are the main reason for the change in the output of Boiler6 that, from Figure \ref{fig:boiler_use}, seems to happen sometime during 2016, although it is a plausible explanation. There was also performed adjustments on Boiler6 by Foster Wheeler during the summer of 2016, and the results of these changes might very well be the cause of the abrupt change in maximum capacity that happens during 2016. A further investigation into this matter is out of scope for this thesis, but should be seriously considered for future work regarding the performance of the boiler.