\chapter{Process description}

This section gives a description of both the CFB technology in general, and the spesific CFB boiler at Norske Skog Skogn.





\section{Steam production at Norske Skog}
\label{sec:steamProdTotal}
The steam network at Norske Skog supplies steam to the the paper machines, as well as a range of other applications such as heating of water, buildings and other machines. Apart from heat recovery in the refining process, the paper mill has three different steam suppliers: One Electric boiler (Boiler4, 43 MW), one bark-fired boiler (Boiler5, 35 MW), and one biomass fired CFB boiler (Boiler6, 50 MW). 

Because of the high cost of using electricity to produces thermal energy, the Electric boiler is used as little as possible.

The bark-fired boiler burns bark that originates from debarking the wood logs prior to pulp production. When the factory uses a lot of virgin fiber (wood logs) in the paper production, bark is in plentiful supply, and it is relatively cheap to use the bark-boiler. However, when the paper mill uses a lot of recycled paper, the amount of bark produced at the factory is low, and the company is forced to buy bark from other suppliers. This is more expensive, and the cost of using the bark boiler increases accordingly. 

The biomass-fired CFB (Boiler6), which is the main focus of this thesis, burns a range of different fuels, including bio sludge, bark, demolition wood, oil (mainly for start-up) and reject from the deinked pulp plant mainly consisting of various plastics and foreign materials. The sludge burned in the CFB-boiler are either reject from the deinked pulp plant, or sludge from the waste water treatment plant at the site. Demolition wood is supplied from external sources and has a very low cost compared to electricity and, if the plant uses a lot of recycled paper, bark. If the plant use a lot of virgin fiber (wood logs), the cost of bark can almost be regarded as negative because the company has to pay to get rid of it if they do not burn it themselves. Anyways, the CFB boiler can, as mentioned, burn a range of different fuels, including bark. It is therefore usually desirable to use the CFB as much as possible at the expense of the other two boilers.




\section{The CFB technology}
%(Start with the applications of cfb? Like that it's used for coal fired power plants. That it has an advantage over other technologies. If it is an expanding or contracting technology (in the marketplace). This should perhaps come after the explanation of the technology instead?)

%Have comparisons with the boiler at Norske Skog during the description of the general technology? Yes. In describing the general technology and the different components, I should include the specific setup at Norske Skog at perhaps weigh in on the different advantages and disadvantages of this setup.

%Also, how general should the description of the cfb technology be? It is perhaps not that interesting to have a wide discussion about all the different types of cfb? No, not a wide discussion, but should include the most common versions of the cfb boiler setup and how it compares to the one at Norske Skog.

%The setup I'm thinking around this chapter:
%-Headline
%-Write some facts about the applications of the cfb-boiler. How it compares to alternative technologies in the marketplace. How the trendline is regarding its use (like future and recent past). This section should perhaps be partly in the introduction, and partly here.
%-Write about the general technology of fluidized bed. (Include pictures to explain). What implications does this have for its applications. Advantages/disadvantages.
%-Write about the specific components. This section should include comparisons to the cfb at Norske Skog.
%-End? Or should I include more?

%Explain cfb technology. Include pictures that help explain it. 1. Explain what fluidized bed means, then go on to explain circulating fluidized bed. \\ \\


%Although the CFB technology was invented around World War 2 for the catalytic cracking processes used in the petrochemical industry, it was not until 1982 that the first CFB boiler was built for the exclusive purpose of burning fuel to generate steam and heat. This boiler burned low grade coal, and coal is still the fuel that most CFB boilers use (Figure \ref{fig:coal_share}).



The \textit{criculating fluidized bed}-technology (or CFB) can be regarded as a special case of the general \textit{fluidized bed}-technology (or FB). FB is a phenomena that occurs when a fluid is passed through a bed of solid particles and the drag from the fluid offsets the gravitational pull on the particles. This makes the particles "float", and the solid/fluid-mixture is said to be a FB (Figure \ref{fig:FB}). \\

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=0.7\textwidth]{Figures/FB.png}}
\caption{Fluidized bed.}
\label{fig:FB}
\end{figure}

The FB behaves in much the same ways as a liquid. For example: \cite{BasuPrabir2015CFBB}

\begin{itemize}
\item The static pressure at any point is roughly equal to the weight of the particles per cross section above that point.
\item Objects will fall or rise in the FB depending on their relative density to the FB around them.
\item The particles are well mixed and the FB maintains a nearly uniform temperature when it is heated from any side.
\end{itemize}

The fluidized bed phenomena allows for a high contact surface between the fluid and the particles, and thus a higher reaction rate. \\

If the fluid flow through the particles is increased above the terminal velocity of the particles and the particles are carried away with the fluid, it is called a \textit{fast fluidization}. If this fluidized flow is recycled back to the bottom of the combustion chamber, the setup is called a \textit{circulating fluidized bed} (CFB). 

Figure \ref{fig:CFBsimple} gives a drawing of a simple CFB boiler to show the main concept of the technology. The primary air enters through the bottom of the fire chamber, while the secondary air enters at several places further up in the combustion chamber. This is done to make sure that there is a mostly uniform distribution of oxygen throughout the bed. The fuel is fed into the boiler near the bottom of the bed. The gas and solid particles are separated in the cyclone so that the solid particles are returned to the combustion chamber. \\

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=0.7\textwidth]{Figures/CFB.png}}
\caption{Simple example of a CFB boiler where solid fuel is burned with air. The orange arrows show what the main movements through the boiler are.}
\label{fig:CFBsimple}
\end{figure}

In addition to the solid fuel, there are also solid bed materials circulating in the CFB. This might be limestone if the fuel has a high sulfur content that needs to be removed, or it might be an inert material such as sand if the the fuel has a low sulfur and ash content (The boiler investigated in this thesis uses sand). The reason for having inert bed materials in the CFB is that they increase the thermal inertia of the CFB and contributes to the mixing of fuel and air. The CFB has a very good heat distribution, and the temperature is nearly constant (bounded between 800-900\si{\celsius}) throughout the combustion chamber. \cite{BasuPrabir2015CFBB}

The relatively low temperature (800-900\si{\celsius}), large combustion zone and high bed velocity gives the CFB technology a range of advantages over competing technologies. The major ones are:

\begin{itemize}
    \item The good mixing achieved by having a fast fluidization along with the thermal inertia provided by the bed material makes it possible to burn a wide range of fuels. 
    \item The large zone of combustion makes it easier to make more compact boiler designs, because the combustion practically takes place in the entire combustion chamber, recycle and cyclone. 
    \item The low temperature and somewhat uniform distribution of oxygen (because of the secondary air) gives the \ce{N_2} in the air relatively little chance to react with the \ce{O_2}, something that results in very low \ce{NO_x}-emissions compared to other technologies that operate with higher peak temperatures.
    \item If the fuel has a high sulfur content, the CFB-technology provides a cheap method for sulfur removal by simply letting limestone circulate with the fuel.
\end{itemize}

%The advantages of the CFB boiler compared to other boilers is that it 


%Begin with general introduction to cfb boilers and the advantages of it. Why should one use the cfb technology? What are the advantages compared to similar technologies (most important is bfb. Comparison to Pulverized  coal should perhaps be in the introduction) What are current applications of it? What is the current state of the art design of cfb's? Say something about emissions:(Different types, causes, remedies,). This subsection should also include an explanation of pressure loss over the bed and how the bed behaves during different air speeds (is it also possible to say something about how this affects the operation (like emissions, efficiency, and so on) \\

%Subsections: What is cfb?, comparison to similar technologies, state of the art, emissions:(CO, NOx, other emissions) \\

%Start with some facts (how many, importance and so on, 1-2 pages, including photo).(This should perhaps be in the introduction) \\

%\subsection{Emissions}
%\textit{Should I even write anything about this considering that I have not done any changes to it.}


%Might include something about how emissions could be reduced, but it should be clearly stated that it is out of scope for this work to do something about it. If included, it should also be a part of "Further work". \\

\section{The CFB boiler at Norske Skog}

The CFB-boiler at Norske Skog Skogn is a 50 MW Compact CFB. The Compact name stems from the design of the cyclones and integration of applicable surrounding equipment within the boiler construction. This essentially takes up less space compared to a traditional CFB boiler construction.  It was  delivered in 1997 by Foster Wheeler, one of the leading suppliers of CFB-boilers worldwide at the time. Foster Wheeler's CFB business is now a part of Sumitomo SHI FW. Figure \ref{fig:Kjel6_FW} gives an idea of the physical size and proportions of the boiler. It shows the cfb boiler along with the economizer and air preheater (described in Section \ref{sec:components})

\begin{figure}[h]
    \centering
    \makebox[\textwidth][c]{\includegraphics[width=0.5\textwidth]{Figures/Kjel6_FW.PNG}}
    \caption{Drawing of the CFB-boiler including economizer and air preheater at Norske Skog Skogn. \cite{CFB_Panna}}
    \label{fig:Kjel6_FW}
\end{figure}


A more schematic representation of the boiler with names of the most important components, as well as the flows of steam, water, air and flue gas, is given in Figure \ref{fig:boiler6Schematic}. \\

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=1.2\textwidth]{Figures/boiler6.png}}
\caption{Simplified schematic representation of Boiler6.}
\label{fig:boiler6Schematic}
\end{figure}

\clearpage

\subsection{Fuel}
\label{sec:fuel}
The fuel can either be various solid fuels or it can be fuel oil.

The fuel oil is mainly used for start ups, because the boiler needs some preheating before it can burn solid fuel. This is because the ignition of the solid fuel in a CFB happens by simply letting it come in contact with the already hot fluidized bed particles. The fuel oil enters the combustion chamber through oil lances. It has a high sulfur content compared to the solid fuels used. This means that there is an increased risk of sulfurous acid (\ce{H_2SO_3}) condensing on the walls in the flue gas system. This can have some consequences for how the shunt pump should be operated (see Section \ref{sec:shuntPump}).

During normal operation, the CFB usually burns a range of different solid fuels. This can be bark, demolition wood, biosludge from the waste water treatment, or other types of sludge derived from waste from the paper production. The wide range of different fuels, uneven mixing of the fuel before it enters the combustion chamber, and no good way of estimating the composition of the fuel, makes it challenging to operate the boiler so that the emissions are kept below their limits. 

The solid fuel enters the combustion chamber near the base through two separate screws that are situated on each side of the combustion chamber. As the fuel sometimes is unevenly mixed, the relative speed of the screws must sometimes be adjusted so that fuel and oxygen are evenly mixed in the bed (see Section \ref{sec:emissionControl}).

\subsection{Water/steam cycle}
The feed water comes in at around 60bar and 112\si{\celsius}. It is preheated to around 264\si{\celsius} by exchanging heat with the flue gas in the economizer. The econonmizer is split into two sections, ECO1 and ECO2. ECO1 is the first section the feed water enters. Between the two economizer sections, the water goes through the Dolezal heat exchanger. This is further explained in Section \ref{sec:dolezal}. There is also a shunt pump that drives a recycle loop around ECO1. This is further discussed in Section \ref{sec:shuntPump}.

After the economizer, the water enters the steam drum. From the steam drum, water falls from the bottom of the drum down through evaporator tubes that goes along the combustion chamber. Here, the water evaporates and becomes steam. The steam is returned to the steam drum (see Figure \ref{fig:boiler6Schematic}). From the top of the steam drum, saturated steam exits and flows into the superheaters (SH1 and SH2). Here, the steam is heated above its dew point. There is some water that enters the superheated steam through an attemperator between SH1 and SH2. This is saturated steam from the steam drum that has be condensed in the Dolezal (see Section \ref{sec:dolezal}). This water makes sure the steam enters the steam network with the desired temperature of 450\si{\celsius} (see Section \ref{sec:waterControl}). 

\subsection{Flue gas}
The flue gas forms in the combustion chamber, comes out at the top of the cyclones and enters the flue gas channel. There it exchanges heat with the superheaters (SH1 and SH2), before it enters the ECO/AP where it exchanges heat with the the feed water and primary and secondary air. After all this it exits the ECO/AP with an average temperature of around 158\si{\celsius}. It is desirable to keep the temperature of the flue gas above 130\si{\celsius} to reduce the risk of sulfourus acid (\ce{H_2SO_3}) condensing on the walls of the flue gas channel. According to the design, this is controlled by the steam air preheaters (see Section \ref{sec:SAP}), although they are not in use because of leaks (see Section \ref{sec:leaks}). After the ECO/AP, the flue gas goes through the electric filter where particles are filtered out(see Section \ref{sec:EL-filter}). After that it goes through the flue gas fan, which is the fan that propels the flue gas through the system.

After the flue gas fan, the stream splits into an exit stream and a flue gas recycle stream. The flue gas recycle stream recycles some of the flue gas back to the combustion chamber. It is propelled by the flue gas recycle fan. The reason for having this recycle, is because this relatively cool flue gas (around 150\si{\celsius}) can be used to control the bed temperature (around 860\si{\celsius}) and make sure that it does not become too high. 

The flue gas that does not enter the recycle passes an analyzer where environmental pollutants (such as CO and \ce{NO_x}) are measured. It finally exits through the chimney.

\subsection{Air}
\label{sec:airExplained}
Air is fed into the combustion chamber as either primary air, secondary air or tertiary air (high pressure air).

The primary air is supplied from the boiler room by the primary air fan. It is preheated to around 346\si{\celsius} by exchanging heat with the flue gas. This is done in the air preheater as shown in Figure \ref{fig:boiler6Schematic}. It finally enters the combustion chamber at its base where it is responsible for fluidizing the solid fuel.

The secondary air is supplied from the boiler room by the secondary air fan. It is also preheated in the air preheater by exchanging heat with the flue gas. After the air preheater it is split up into four different streams, each with it's own control valve (see Figure \ref{fig:boiler6RegAir} in Section \ref{sec:combustionControl}). The four streams represent an upper and a lower level with a left and a right hand side for both levels. The four streams are then split up further, so the secondary air finally enters the combustion chamber through a total of 22 separate openings. The number of openings used to be 32, but ten openings have been permanently shut down due to uneven distribution of oxygen. The reason for having the secondary air is to get a more even distribution of oxygen in the combustion chamber. 

The flow instruments used to measure the primary air and the different inlets of the secondary air are not dimensioned for this kind of use. The reason for this is probably some kind of miss that happened when the boiler was constructed. This means that there might be some unknown uncertainties in the air flow measurements.  

The third way air is supplied to the boiler is through the two high pressure fans. They are running at a constant speed, and the air is used for the "recycle knee", where the recycled material enters the combustion chamber again. The "recycle knee" is a tight opening, so there is a risk of having the recycled material blocking this opening. The high pressure air is used to blow away sand so that the opening stays open. The high pressure air that is not used enters the primary air through a valve (see Figure \ref{fig:boiler6Schematic}).

Apart from these three intended ways of adding air to the combustion chamber, there are also some unintended leaks that causes air to leak into the system at various places. This is further explained in Section \ref{sec:leaks}.

\subsection{Components}
\label{sec:components}

This subsection contains a list of the most significant components of the CFB. All process variables listed are averages over a period of two weeks(see Section \ref{sec:energyEconomicCalc}) and are included to give an idea of typical operating conditions. Some of the components are subjected to leaks, something that is further described in Section \ref{sec:leaks}.

\subsubsection{Dolezal}
\label{sec:dolezal}
The Dolezal is the name of the heat exchanger situated next to the economizer (Figure \ref{fig:boiler6Schematic}). The purpose of the exchanger is to condense saturated steam from the steam drum so that it can be used to control the temperature of the superheated steam. The feed water between Economizer 1 and 2 makes up the cold side of the exchanger. The hot side of the exchanger is made up of saturated steam that is condensed to water.
The two main challenges with this kind of setup for temperature control of the superheated steam temperature are:

\begin{itemize}
    \item Because there is no apparatus to drive the steam/water from the steam drum, through the Dolezal, and into the superheated steam after Superheater1, the only driving force is the the pressure difference between the steam drum and the steam after Superheater1. This $\Delta$P is around 1 bar. Because of the low $\Delta$P, it can be difficult to achieve turbulent flow through the attemperator, something that affects the distribution of the cooling water. Low turbulence lowers the atomization degree of steam which in turn leads to unequal attemperation and potentially phase separation (i.e., cooling water and steam co-existing over a distance before fully mixing). This increases wear on piping and could lead to corrosion problems.
    \item Because the cold side of the Dolezal is the feed water between Economizer 1 and 2, the temperature might be too high to achieve sufficient cooling. During normal operation, this cold side temperature is usually around 207 \si{\celsius} (This is a calculated value, see Section \ref{sec:shuntPumpResults})
\end{itemize}



\subsubsection{Economizer and Air Preheater}

The Economizer(ECO) and the Air preheater(AP) are used to recover the heat of the flue gas after it has passed the superheaters. The different parts of the ECO/AP are, sorted from high to low flue gas temperature, AP5, ECO2, ECO1, AP4, AP3, AP2 and AP1. Primary and Secondary air is preaheated in the AP from around 59 and 45\si{\celsius} to around 346 and 352\si{\celsius}. The feed water is preheated in the ECO from around 112\si{\celsius} to around 264\si{\celsius}. 

\subsubsection{Shunt pump}
\label{sec:shuntPump}
The shunt pump is used to pump the water recycle from after ECO1 to before ECO1 (see Figure \ref{fig:boiler6Schematic}). The water recycle has a higher temperature than the feed water, and thus by recycling the water, the water inlet temperature of ECO1 increases. 

When the inlet temperature of water increases, the temperature difference over the heat exchanger decreases, something that leads to less heat transferred from the flue gas to the water. This means that having the shunt pump turned on leads to less heat recovered from the flue gas, something that leads an energy loss if this heat is not recovered somewhere else.

There is also another reason for having the shunt pump running. Because the economizer is a heat exchanger with a liquid (water) on one side and a gas (flue gas) on the other side, the heat exchanger tubes will have a temperature much closer to that of the liquid than the temperature of the gas. \cite{2009Cg&s} This means that the tubes in the economizer will have a temperature much closer to the water, which without the shunt pump would have a temperature around 112\si{\celsius}. If the flue gas has a high \ce{SO_2} content, it means that there is an increased risk of sulfurous acid (\ce{H_2SO_3}) condensing and causing corrosion on the tubes. The instruction manual gives a temperature limit of 130\si{\celsius} which the flue gas should not fall below. Using the shunt pump at maximum capacity gives a water inlet of around 141\si{\celsius}.

The shunt pump is further investigated in Section \ref{sec:shuntPumpBal}. 

\subsubsection{Steam Air preheater}
\label{sec:SAP}

The steam air preheaters purpose is to make sure that the temperature of the flue gas does not drop below 130\si{\celsius}. By staying above this temperature the risk of corrosive acids (such as \ce{H_2SO_3}) condensating on the walls of the flue gas duct is minimized. The way they are supposed to function is that they heat the primary and secondary air before it enters the AP. This causes a smaller temperature difference across the AP, which results in less heat transferred from the flue gas and the temperature in it remains higher.
 
The SAP's are currently leaking, so they are not used during normal operation. They are usually not necessary either, because the flue gas temperature is usually above 130\si{\celsius}.

\subsubsection{Electrostatic precipitator (ESP)}
\label{sec:EL-filter}
The ESP is responsible for removing particulate matter from the flue gas. It functions by using high voltage to ionize the flue gas and its particulate matter. The ionized particles are directed towards charged ground plates, and thus are removed from the flue gas.


\subsubsection{Flue gas fan}
The flue gas fan is a speed controlled centrifugal fan that is situated after the ESP, but before the flue gas recycle split (see Figure \ref{fig:boiler6Schematic}). It has a power of 400 kW and a capacity of 29.3 Nm\textsuperscript{3}/s. The purpose of the flue gas fan is to make sure that the pressure inside the combustion is lower than outside, so that flue gas in the boiler room is avoided. 

The flue gas fan is an active constraint for the steam production. This is mainly due to leaks at various points throughout the flue gas system. This is further explained in Section \ref{sec:leaks}

\subsubsection{Primary air fan}
The primary air fan is a 400 kW centrifugal fan with a capacity of 9.6 Nm\textsuperscript{3}/s. It drives the primary air into the system. It is not speed regulated, and the primary air flow is instead controlled by a valve (see Figure \ref{fig:boiler6RegAir} in Section \ref{sec:combustionControl}).

\subsubsection{Secondary air fan}
The secondary air fan is is a 200 kW speed controlled centrifugal fan with a capacity of 10.5 Nm\textsuperscript{3}/s. In maintains the pressure of the secondary air. The flow of secondary air is instead controlled by the four different valves that distribute the secondary air into the different parts of the combustion chamber. The distribution of secondary air is further explained in Section \ref{sec:airExplained}.


\subsubsection{Flue gas recycle fan}
The flue gas recycle fan is a 288 kW speed controlled centrifugal fan with a capacity of 10.0 Nm\textsuperscript{3}/s. It drives the recycle gas stream and, because the the recycle gas stream has a much lower temperature than the flue gas bed, it regulates the temperature of the flue gas bed and makes sure that it does not become to high.

\subsubsection{High pressure fans}
The high pressure fans runs at constant speed and supplies tertiary air to the boiler. The tertiary air is used to keep the recycle knee open as explained in Section \ref{sec:airExplained}.


\subsection{Leaks}
\label{sec:leaks}

Figure \ref{fig:boiler6Leaks} gives a representation of where air leaks into the flue gas system. Notice that there are leaks from both the primary and secondary air into the flue gas channel.

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=1.1\textwidth]{Figures/boilerLeaks.png}}
\caption{Schematic representation of where air leaks into the boiler.}
\label{fig:boiler6Leaks}
\end{figure}

The leaks lead to range of problems:


\begin{itemize}
    \item Perhaps the most important effect of the air leaking into the flue gas system is the increased mass flow through the flue gas fan, something that forces the fan to operate at maximum capacity. This is considered the main reason why steam production is limited below rated capacity. The amount of air leaking into the flue gas between the first O2-measurment and the flue gas fan is estimated to constitute roughly 32\% of the resulting mix of air and flue gas (calculated in Section \ref{sec:amountLeak}). This means that the air leaking in represents an increase of almost 50\% of the total amount of gas that has to pass through the flue gas fan.
    \item Another effect is the fact that the air leaking in has a temperature of around 36\si{\celsius}, and thus ends up cooling down the flue gas. This lowers the temperature of the flue gas and reduces the amount of heat that can be recovered in the ECO/AP.
    \item Air leaking into the flue gas also causes the oxygen level in the recycle stream to be considerably higher than it should be. The level is >9.3vol\%, compared to 3.72vol\% (which is the \chemfig{O_2}-level before the ECO/AP) if there had been no leaks.  The recycle is used to cool down and control the temperature of the bed. Any oxygen in this stream is therefore undesired because extra oxygen leads to increased combustion and thus extra heat, rendering the recycle stream less effective as a cooling mechanism. To compensate for this, the recycle stream has to be larger, something that leads to an increased load on both the recycle gas fan and the flue gas fan. This adds to the problem of having the flue gas fan as an upper constraint for the steam production.
    \item There is also a leakage in the steam preheaters (not shown in Figure \ref{fig:boiler6Leaks}). This is not a contributor to air leaking into the flue gas, but rather a leakage of steam into the primary and secondary air streams and the boiler room. This leakage renders the steam preheaters useless, and they are therefore not used during normal operation. This means that the flue gas temperature after the economizer and air preheater is not regulated by the steam preheaters as intended in the design (Section \ref{sec:SAP})
    \item Corrosion in the AP have prompted the company to close down some of the tubes in the heat exchangers. This has reduced the heat exchanger surface and hence the heat recovery of the system. If the leaks were to be fixed, the heat recovery (and boiler efficiency) would increase. Another problem arising because some of the heat exchanger tubes are closed down, is that the overall resistance in the heat exchanger is increased. This resistance makes some of the secondary air valves go into maximum position, reducing the degrees of freedom available to control the distribution of secondary air. 
\end{itemize}




\chapter{Mass and Energy Balances}
\label{sec:massEnergyBal}
This section describes how the equations used in Section \ref{sec:energyEconomicCalc} are derived.
\section{Mass balance of \ce{O_2}}
\label{sec:massBalance}
%Heat loss calculations From Matlab
%Show where the measurements are in the control structure


The vol\% of \chemfig{O_2} in the flue gas is measured both before the ECO/AP and after the flue gas fan (Figure \ref{fig:boiler6Schematic}). 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 resulting mixture of air and flue gas. This can be done by creating both a steady-state mass balance of the total amount of gas (Equation \ref{eq:massBalance}) and a steady-state mass balance of the amount of oxygen (Equation \ref{eq:O2-balance}): 

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

\begin{equation}
    \dot{n}_{Mix}\cdot x_{O_2, Mix} = \dot{n}_{Air}\cdot x_{O_2, Air} + \dot{n}_{Flue}\cdot 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/AP, and $\dot{n}_{O_2,Mix}$ is the molar flow of the mixture of air and flue 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 a formula for the mole fraction of air in the mixture (Equation \ref{eq:leakFraction}):

\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:leakFraction}
\end{equation}

To use this equation some assumptions have to be made:

\begin{itemize}
    \item It is assumed that the gas is perfectly mixed at both points of measurement.
    \item It is assumed that there are no reactions (i.e. no combustion) taking place between the two points of measurement. 
    \item The \chemfig{O_2}-sensors measure the \chemfig{O_2}-concentration as a volume fraction of \chemfig{O_2} in dry air. The water content of the gases is therefore not taken into account, and it is also assumed that the volume fractions that the sensors measure are equal to the molar fractions.
    \item The molar fraction of \ce{O_2} in air is assumed to be 0.21. 
\end{itemize}

If the volume flow, $q_{Mix}$, mass density, $\rho_{Mix}$, and molar mass, $M_{Mix}$, of the mixture of air and flue gas are known, the absolute molar flow of the air leaking in can be found using Equation \ref{eq:molarFlowAir}:

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

\section{Heat loss from cold air leaking in}
\label{sec:HeatFalseAirEqs}

As mentioned in Section \ref{sec:leaks}, there are numerous problems arising as a consequence of the fact that air is leaking into the flue gas. One of these is the fact that the air leaking in has a much lower temperature than the flue gas. This means that the air cools down the flue gas, and hence reduces the amount of heat that can be recovered in the ECO/AP. \\

It is difficult to estimate how much this cooling affects the amount of heat recovered in the ECO/AP. One approach to estimate this is to first calculate heat transfer coefficients (UA's) for the different parts of the ECO/AP using current temperature and flow data. If these heat transfer coefficients are assumed to be constant, it is then possible to calculate how changes in temperature and flow of the flue gas affects the heat transfer. This might seem like a straightforward approach, but the somewhat complicated setup of the ECO/AP, few temperature measurements, and possible leakage from the primary and secondary air into the flue gas,  makes it impossible to use this approach. \\

In order to still get an idea of the energy loss, a simpler approach can be chosen instead. It creates an upper estimate of the loss in heat transfer. The method is simply to calculate the energy needed to heat the air from the temperature of the boiler room (c. 36\si{\celsius}) to the temperature of the flue gas before the flue gas fan (c. 148\si{\celsius}). Estimating the energy loss this way assumes that all of the energy used to heat the air leaking in could be recovered in the ECO/AP if the leaks were to be fixed. This is probably an assumption that is too harsh, but the method still creates an upper bound for the energy loss, and thus gives an idea of what order of magnitude the loss is. \\

In order to calculate the energy needed to heat the air leaking in, an energy balance needs to be constructed. Equation \ref{eq:heatAir} shows a simplified version where the heat capacity is assumed to be constant:

\begin{equation}
    \label{eq:heatAir}
      \dot{Q}_{Heat,Air} = \dot{n}_{Air}*Cp_{Air}*(T_{Mix}-T_{Room})
    %E_heatLoss = n_air*Cp_air*(T_elfilt-T_kjelrom)/(10^6); %[MJ/s]=[MW]
\end{equation}

where $\dot{Q}_{Heat,Air}$ is the energy needed to heat the air, $\dot{n}_{Air}$ is the molar flow of air and $T_{Mix}$ and $T_{Room}$ is the temperatures right before the flue gas fan and in the boiler room respectively. $Cp_{Air}$ is the molar heat capacity of air and is assumed to be constant and equal to the average of the molar heat capacity of air at 36\si{\celsius} and 148\si{\celsius} (Equation \ref{eq:aveCp}):

\begin{equation}
    \label{eq:aveCp}
    Cp_{Air} = \frac{Cp_{Air, 36\si{\celsius}} + Cp_{Air, 148\si{\celsius}}}{2}
\end{equation}

In order to calculate the molar heat capacities of air at different temperatures, it is assumed that the molar heat capacity of air can be approximated as a linear combination of the molar heat capacities of the components of air (Equation \ref{eq:CpAir}): 

\begin{equation}
    \label{eq:CpAir}
    Cp_{Air,T} = 0.21*Cp_{O_2,T} + 0.78*Cp_{N_2,T} + 0.01*Cp_{Ar,T}
    %result = dot([0.21 0.78 0.01], [Cp_O2(T) Cp_N2(T) Cp_Ar()])
    %Assume Cp_air is constant and equal to the average of the Cp at 36 and 148 degC
\end{equation}

where $Cp_{X, T}$ is the molar heat capacity of component $X$ at temperature $T$. The molar heat capacities for each component were obtained using the general formula given in Equation \ref{eq:CpFormulaBook}. \cite{PolingBruceE2001Tpog}

\begin{equation}
\label{eq:CpFormulaBook}
    \frac{Cp}{R} = a + b \cdot 10^{-3} \cdot T + c \cdot 10^{-5} \cdot T^2 + d \cdot 10^{-8} \cdot T^3 + e \cdot 10^{-11} \cdot T^4
\end{equation}

where $Cp$ is the heat capacity, $R$ is the universal gas constant, $T$ is the temperature in kelvin, and $a$-$e$ are constants specific for each component (The values for these are given in Table \ref{tab:CpParameters:app} in Appendix \ref{app:heatCapacity}). The equation and constants are obtained from the literature. \cite{PolingBruceE2001Tpog}

By estimating $Cp_{Air, T}$ using Equation \ref{eq:CpAir}, it is assumed that the air can be regarded as an ideal gas. Ideal gas law is also assumed when using Equation \ref{eq:heatAir}, because the air is mixed with the flue gas at the higher temperature, so any mixing effects of the air and the flue gas has to be disregarded.


\begin{equation}
    %Monthly_Loss = E_heatLoss*24*31*El_price;
    %Yearly_Loss = E_heatLoss*24*365*El_price_year;
\end{equation}


\newpage
\section{Loss from having shunt pump running at full capacity}
\label{sec:shuntPumpBal}

As explained in Section \ref{sec:shuntPump}, the purpose of the the shunt pump is to reduce the amount of heat recovered in ECO1 (Figure \ref{fig:eco1ShuntPump}). \\

\begin{figure}[h!]
\centering
\makebox[\textwidth][c]{\includegraphics[width=1.0\textwidth]{Figures/shuntDiagram.png}}
\caption{Schematic representation of ECO1 and the shunt pump. Blue colored streams refer to water, black colored streams refer to flue gas.}
\label{fig:eco1ShuntPump}
\end{figure}

 When the pump is running at full capacity, the amount of heat recovered from the flue gas is minimized. This might be desired under certain circumstances, such as when the flue gas temperature falls under a certain temperature (130 \si{\celsius}) where the risk of corrosion increases, or the fuel has a high sulfur content (see Section \ref{sec:shuntPump}). Under normal operation however, the shunt pump should run at it's minimum capacity so that the amount of heat recovered is maximized. 
 
 Before the work with this thesis started, the pump was running at full capacity all the time. This is exactly the opposite of how it should have been operated if the goal is to recover as much heat as possible. The reduction in heat recovery constitutes an energy loss. \\

It would seem that the simplest way to estimate this heat loss is to just turn off the shunt pump and see how the outlet temperature of the flue gas is affected. From the temperature change and heat capacity of the flue gas, an energy loss could then be calculated. (The logic is that turning of the shunt pump results in a lower flue gas temperature. This means that more heat must have been transferred from the flue gas to the feed water. This added energy to the feed water ultimately results in more steam produced.) In reality, the temperature change of the flue gas is so small compared to the general fluctuations of the flue gas temperature that it is not possible to easily measure it. \\

Instead, another approach can be chosen to estimate the energy loss. By using a few known temperatures and heat capacities, it is possible to estimate a UA-value for ECO1. If this value is assumed to be constant during changes in temperature and water flow, it is possible to get an estimate of the energy loss. \\

First, in order to calculate the water temperature after it has passed ECO1 ($T_{w,out}$), a steady state energy balance for the mixing of water recycle and feed water can be constructed (Equations \ref{eq:waterMixBalance}).

\begin{subequations}
\label{eq:waterMixBalance}
\begin{equation}
    \Delta \dot{H}_{w,in} = \Delta \dot{H}_{w,feed} + \Delta \dot{H}_{w,recycle}
\end{equation}
\begin{equation}
\begin{aligned}
    & \left( \Delta h_{F,w}+\int_{T_0}^{T_{w,in}}Cp_{w}dT\right)\cdot (\dot{m}_{w,recycle}+\dot{m}_{w,feed}) = \\ & \left( \Delta h_{F,w}+\int_{T_0}^{T_{w,feed}}Cp_{w}dT\right)\cdot \dot{m}_{w,feed} + \\ & \left( \Delta h_{F,w}+\int_{T_0}^{T_{w,recycle}}Cp_{w}dT\right)\cdot \dot{m}_{w,recycle}    
\end{aligned}
\end{equation}
\end{subequations}

The subscripts $w,feed$, $w,recycle$ and $w,in$ refers to the feed water stream, the water recycle stream and the water stream entering the heat exchanger (ECO1) respectively. $\Delta H$ refers to the enthalpy of a stream, $\Delta h_{F,w}$ is the specific enthalpy of formation of water, $Cp_{w}$ is the specific heat capacity, $T_0$ is the standard temperature and $\dot{m}$ is the mass flow of a stream. \\
Since the only component in Equations \ref{eq:waterMixBalance} is water, the enthalpy of formation, $\Delta h_{F,w}$, cancels out. Also, because $\int_{T_0}^{T_{w,feed}}Cp_{w}dT$ cancels out, $T_0$ can be switched out with $T_{w,feed}$. If $Cp_{w}$ is assumed to be temperature independent, it also cancels out. This simplifies Equations \ref{eq:waterMixBalance} to Equation \ref{eq:waterMixSimple}:

\begin{equation}
\label{eq:waterMixSimple}
    (T_{w,in}-T_{w,feed})\cdot (\dot{m}_{w,recycle} + \dot{m}_{w,feed}) = (T_{w,recycle}-T_{w,feed})\cdot \dot{m}_{w,recycle}
\end{equation}

If the heat added to the water recycle by the shunt pump and the heat loss from the water recycle to the surroundings is assumed to be negligible, it follows from Figure \ref{fig:eco1ShuntPump} that  the water temperature out of the heat exchanger, $T_{w,out}$, is equal to the temperature of the recycle, $T_{w,recycle}$. Putting this relation into Equation \ref{eq:waterMixSimple} and rearranging gives Equation \ref{eq:waterMixFinal}:

\begin{equation}
\label{eq:waterMixFinal}
    T_{w,out} = T_{w,feed} + (T_{w,in}-T_{w,feed})\cdot \frac{\dot{m}_{w,recycle} + \dot{m}_{w,feed}}{\dot{m}_{w,recycle}}
    %(Tw_out-Tw_feed)*Cp_w*m_recycle = (Tw_in-Tw_feed)*Cp_w*(m_recycle + m_feed)
    %Tw_out*m_recycle = Tw_in*m_recycle + (Tw_in-Tw_feed)*m_feed
    %Tw_out = Tw_in + (Tw_in-Tw_feed)*m_feed/m_recycle;
\end{equation}

Next, the inlet temperature of the flue gas ($T_{f,in}$) needs to be calculated. This can be achieved by creating an energy balance over ECO1 and assuming that the heat added to the water is equal to the heat lost from the flue gas. If the specific heat capacities are assumed to be constant over the temperature range, the energy balance simplifies to Equation \ref{eq:energyBalanceEco1}:

\begin{equation}
\label{eq:energyBalanceEco1}
  (T_{w,out}-T_{w,in})\cdot Cp_w \cdot (\dot{m}_{w,feed}+\dot{m}_{w,recycle})= (T_{f,in}-T_{f,out})\cdot Cp_f \cdot \dot{m}_{f}
\end{equation}

$Cp_f$ is the specific heat capacity of the flue gas, $\dot{m}_f$ is the mass flow of the flue gas and $T_{f,out}$ is the temperature of the flue gas after ECO1.

Rearranging Equation \ref{eq:energyBalanceEco1} gives an expression for $T_{f,in}$ (Equation \ref{eq:TfinExpression}):

\begin{equation}
\label{eq:TfinExpression}
    T_{f,in} = T_{f,out}+(T_{w,out}-T_{w,in})\cdot \frac{Cp_w\cdot (\dot{m}_{w,feed}+\dot{m}_{w,recycle})}{Cp_f\cdot \dot{m}_{f}}
    %(Tw_out-Tw_in)*Cp_w*(m_feed + m_recycle) = (Tf_in-Tf_out)*Cp_f*m_flue
    %Tf_in = Tf_out + (Tw_out-Tw_in)*(Cp_w/Cp_f)*(m_feed + m_recycle)/m_flue;
\end{equation}

Knowing the temperatures and flows of both water and flue gas on both sides of ECO1 makes it possible to calculate the heat transfer coefficient ($UA$). This is done by assuming that ECO1 can be regarded as an ideal counter current heat exchanger and using the equation for an ideal heat exchanger in combination with the enthalpy change of water to create the following energy balance (Equation \ref{eq:UAenergyBal}):

\begin{equation}
\label{eq:UAenergyBal}
    UA\cdot \Delta T_m = (T_{w,out}-T_{w,in})\cdot Cp_w\cdot (\dot{m}_{w,feed}+\dot{m}_{w,recycle})
    %(Tw_out-Tw_in)*Cp_w*(m_feed + m_recycle) =
    %UA*((Tf_in-Tw_out)-(Tf_out-Tw_in))/ln((Tf_in-Tw_out)/(Tf_out-Tw_in))
    %UA = (Tw_out-Tw_in)*Cp_w*(m_feed + m_recycle)/deltaTm(Tf_in, Tf_out, Tw_in, Tw_out);
\end{equation}

$\Delta T_m$ is the logarithmic mean temperature difference (Equation \ref{eq:LMTDeco1}):

\begin{equation}
\label{eq:LMTDeco1}
    \Delta T_m = \frac{(T_{f,in}-T_{w,out})-(T_{f,out}-T_{w,out})}{ln(T_{f,in}-T_{w,out})-ln(T_{f,out}-T_{w,out})}
\end{equation}

If $UA$ is assumed to be constant during flow and temperature changes, the effect of turning off the shunt pump (i.e. setting $\dot{m}_{w,recycle}=0$) can then be estimated numerically by using equations \ref{eq:energyBalanceEco1} and \ref{eq:UAenergyBal} and the fact that the only two unknowns during this change are $T_{f,out}$ and $T_{w,out}$ ($T_{w,in}$ is simply equal to $T_{w,feed}$ when the shunt pump is turned off).\\

The results of these calculations are presented in Section \ref{sec:shuntPumpResults}.
%\begin{equation}
    %syms Tw_out_new Tf_out_new

%eqns = [Cp_w*m_feed*(Tw_out_new-Tw_feed) == UA*deltaTm(Tf_in, Tf_out_new, Tw_feed, Tw_out_new), ...
    %Cp_w*m_feed*(Tw_out_new-Tw_feed) == Cp_f*m_flue*(Tf_in-Tf_out_new)];
    
    %vars = [Tw_out_new, Tf_out_new];
%init_guess = [Tw_out, Tf_out];

%[T_water, T_flue] = vpasolve(eqns, vars, init_guess)

%\end{equation}

