Cleanup some formatting of math in derivationofstream.tex

chapters/derivationofstream.tex

@@ -127,15 +127,15 @@ \section{Rationale for the formulation of \lstinline!inStream!}\doublelabel{rati
 it is therefore the mixing enthalpy under the assumption of fluid
 flowing into said model.
 
-We establish this quantity using a dedicated operator $inStream(h_{outflow,i})=h_{mix}$ assuming that $\dot{m}_{i} >= 0$. This leads to
+We establish this quantity using a dedicated operator $\text{\lstinline!inStream!}(h_{outflow,i})=h_{mix}$ assuming that $\dot{m}_{i} >= 0$. This leads to
 three different incarnations of (n in the general case). This is
 illustrated in the figure below. For the present example of three
 components in a connection set, this means the following.
-\begin{eqnarray*}
-inStream(h_{outflow,1})&=&\frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_2,0)+\text{max}(-\dot{m}_3,0)}\\
-inStream(h_{outflow,2})&=&\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_3,0)}\\
-inStream(h_{outflow,3})&=&\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}
-\end{eqnarray*}
+\begin{align*}
+\text{\lstinline!inStream!}(h_{outflow,1}) &= \frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_2,0)+\text{max}(-\dot{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{outflow,2}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{outflow,3}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}
+\end{align*}
 \begin{figure}[H]
 \caption{Exemplary connection set with three connected components}
 \begin{center}
@@ -146,20 +146,20 @@ \section{Rationale for the formulation of \lstinline!inStream!}\doublelabel{rati
 In the general case of a connection set with \emph{n} components,
 similar considerations lead to the following.
 \begin{equation*}
-inStream(h_{outflow,i})=\frac{\sum_{j=1,...,n;j\neq i}\text{max}(-\dot{m}_j,0)h_{outflow,j}}{\sum_{j=1,...,n;j\neq i}\text{max}(-\dot{m}_j,0)}
+\text{\lstinline!inStream!}(h_{outflow,i})=\frac{\sum_{j=1,...,n;j\neq i}\text{max}(-\dot{m}_j,0)h_{outflow,j}}{\sum_{j=1,...,n;j\neq i}\text{max}(-\dot{m}_j,0)}
 \end{equation*}
 
 \section{Special cases covered by \lstinline!inStream! definition}\doublelabel{special-cases-covered-by-the-instream-operator-definition}
 \subsection{Stream connector is not connected ($N = 1$):}\doublelabel{stream-connector-is-not-connected-n-1}
 For this case, the return value of \lstinline!inStream! is arbitrary.
 Therefore, it is set to the outflow value.
 
-\subsection{Connection of 2 stream connectors, one to one connections (N=2):}\doublelabel{connection-of-2-stream-connectors-one-to-one-connections-n-2}
+\subsection{Connection of 2 stream connectors, one to one connections ($N = 2$):}\doublelabel{connection-of-2-stream-connectors-one-to-one-connections-n-2}
 
-\begin{eqnarray*}
-inStream(h_{outflow,1})&=&\frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_2,0)}=h_{outflow,2}\\
-inStream(h_{outflow,2})&=&\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}}{\text{max}(-\dot{m}_1,0)}=h_{outflow,1}
-\end{eqnarray*}
+\begin{align*}
+\text{\lstinline!inStream!}(h_{outflow,1}) &= \frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_2,0)}=h_{outflow,2}\\
+\text{\lstinline!inStream!}(h_{outflow,2}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}}{\text{max}(-\dot{m}_1,0)}=h_{outflow,1}
+\end{align*}
 
 In this case, \lstinline!inStream! is continuous (contrary to $h_{mix}$) and does not
 depend on flow rates. The latter result means that this transformation
@@ -177,15 +177,16 @@ \subsection{Connection of 3 stream connectors where one mass flow rate is identi
 The suggested implementation results in
 the following equations, and as indicated the last formula can be
 simplified further by using $\dot{m}_3=0$:
-\begin{eqnarray*}
-inStream(h_{outflow,1})&=&h_{outflow,2}\\
-inStream(h_{outflow,2})&=&h_{outflow,1}\\
-inStream(h_{outflow,3})&=&\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}\\
-&=&\begin{cases}
+\begin{align*}
+\text{\lstinline!inStream!}(h_{outflow,1}) &= h_{outflow,2}\\
+\text{\lstinline!inStream!}(h_{outflow,2}) &= h_{outflow,1}\\
+\text{\lstinline!inStream!}(h_{outflow,3}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}\\
+&=
+\begin{cases}
 h_{outflow,2}&\text{if $\dot{m}_1>=0$}\\
 h_{outflow,1}&\text{if $\dot{m}_1<0$ and $\dot{m}_3=0$}
 \end{cases}
-\end{eqnarray*}
+\end{align*}
 \begin{figure}[H]
 \caption{Example series connection of multiple models with stream connectors }
 \begin{center}
@@ -197,7 +198,7 @@ \subsection{Connection of 3 stream connectors where one mass flow rate is identi
 properties discussed for two connected components still hold. The
 connection set equations reflect that the sensor does not any influence
 by discarding the flow rate of the latter. In several cases a non-linear
-equation system is removed by this transformation. However, \lstinline!inStream(..)!
+equation system is removed by this transformation. However, \lstinline!inStream!
 results in a discontinuous equation for the sensor, which is consistent
 with modeling the convective phenomena only. The discontinuous equation
 is uncritical, if the sensor variable is not used in a feedback loop
@@ -228,20 +229,22 @@ \subsection{Connection of 3 stream connectors where two mass flow rates are posi
 \end{lstlisting}
 results in the following equation:
 \begin{equation*}
-inStream(h_{outflow,1})=\frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_2,0)+\text{max}(-\dot{m}_3,0)}=\frac{0}{0}
+\text{\lstinline!inStream!}(h_{outflow,1})=\frac{\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_2,0)+\text{max}(-\dot{m}_3,0)}=\frac{0}{0}
 \end{equation*}
 
 \lstinline!inStream! cannot be evaluated for a connector, on which
 the mass flow rate has to be negative by definition. The reason is that
 the value is arbitrary, which is why it is defined as follows.
 \begin{equation*}
-inStream(h_{outflow,1}):=h_{outflow,1}
+\text{\lstinline!inStream!}(h_{outflow,1}):=h_{outflow,1}
 \end{equation*}
 For the remaining connectors, \lstinline!inStream! reduces to a simple result.
-\begin{eqnarray*}
-inStream(h_{outflow,2})&=\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_3,0)}=h_{outflow,1}\\
-inStream(h_{outflow,3})&=\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}=h_{outflow,1}
-\end{eqnarray*}
+\begin{align*}
+\text{\lstinline!inStream!}(h_{outflow,2}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_3,0)}
+  = h_{outflow,1}\\
+\text{\lstinline!inStream!}(h_{outflow,3}) &= \frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}}{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)}
+  = h_{outflow,1}
+\end{align*}
 Again, the previous non-linear algebraic system of equations is removed.
 This means that utilizing the information about uni-directional flow is
 very important.