Improve math formatting in derivationofstream.tex

chapters/derivationofstream.tex

@@ -18,19 +18,19 @@ \section{Reasons for avoiding the actual mixing enthalpy in connector definition
 \begin{equation*}
 \dot{H}_j=\dot{m}_j
 \begin{cases}
-h_{mix}&\text{if $\dot{m}_j>0$}\\
-h_{outflow,j}&\text{if $\dot{m}_j<=0$}
+h_{\mathrm{mix}}&\text{if $\dot{m}_j > 0$}\\
+h_{\mathrm{outflow},j}&\text{if $\dot{m}_j \leq 0$}
 \end{cases}
 \end{equation*}
 Herein, mass flow rates are positive when entering models (exiting the
 connection set). The specific enthalpy represents the specific enthalpy
 inside the component, close to the connector, for the case of outflow.
 Expressed with variables used in the balance equations we arrive at:
 \begin{equation*}
-h_{outflow,j}=
+h_{\mathrm{outflow},j}=
 \begin{cases}
 \frac{\dot{H}_j}{\dot{m}_j}&\text{if $\dot{m}_j<0$}\\
-\textrm{arbitrary}&\text{if $\dot{m}_j>=0$}
+\textrm{arbitrary}&\text{if $\dot{m}_j \geq 0$}
 \end{cases}
 \end{equation*}
 While these equations are suitable for device-oriented modeling, the
@@ -59,18 +59,18 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 \begin{split}
 0=&\dot{m}_1\cdot
 \begin{cases}
-h_{mix}&\text{if $\dot{m}_1>0$}\\
-h_{outflow,1}&\text{if $\dot{m}_1<=0$}
+h_{\mathrm{mix}}&\text{if $\dot{m}_1 > 0$}\\
+h_{\mathrm{outflow},1}&\text{if $\dot{m}_1 \leq 0$}
 \end{cases}\\
 +&\dot{m}_2\cdot
 \begin{cases}
-h_{mix}&\text{if $\dot{m}_2>0$}\\
-h_{outflow,2}&\text{if $\dot{m}_2<=0$}
+h_{\mathrm{mix}}&\text{if $\dot{m}_2 > 0$}\\
+h_{\mathrm{outflow},2}&\text{if $\dot{m}_2 \leq 0$}
 \end{cases}\\
 +&\dot{m}_3\cdot
 \begin{cases}
-h_{mix}&\text{if $\dot{m}_3>0$}\\
-h_{outflow,3}&\text{if $\dot{m}_3<=0$}
+h_{\mathrm{mix}}&\text{if $\dot{m}_3 > 0$}\\
+h_{\mathrm{outflow},3}&\text{if $\dot{m}_3 \leq 0$}
 \end{cases}
 \end{split}
 \label{eq:D1a}
@@ -82,38 +82,38 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 \label{eq:D1}
 \end{subequations}
 
-The balance equations are implemented using a max() operator in place of
+The balance equations are implemented using a $\operatorname{max}$ operator in place of
 the piecewise expressions, taking care of the different flow directions:
 \begin{subequations}
 \begin{equation}
 \begin{split}
-0=&\text{max}(\dot{m}_1,0)h_{mix}-\text{max}(-\dot{m}_1,0)h_{outflow,1}\\
-+&\text{max}(\dot{m}_2,0)h_{mix}-\text{max}(-\dot{m}_2,0)h_{outflow,2}\\
-+&\text{max}(\dot{m}_3,0)h_{mix}-\text{max}(-\dot{m}_3,0)h_{outflow,3}
+0=&\operatorname{max}(\dot{m}_1,0)h_{\mathrm{mix}}-\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}\\
++&\operatorname{max}(\dot{m}_2,0)h_{\mathrm{mix}}-\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}\\
++&\operatorname{max}(\dot{m}_3,0)h_{\mathrm{mix}}-\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}
 \end{split}
 \label{eq:D2a}
 \end{equation}
 
 \begin{equation}
 \begin{split}
-0=&\text{max}(\dot{m}_1,0)-\text{max}(-\dot{m}_1,0)\\
-+&\text{max}(\dot{m}_2,0)-\text{max}(-\dot{m}_2,0)\\
-+&\text{max}(\dot{m}_3,0)-\text{max}(-\dot{m}_3,0)
+0=&\operatorname{max}(\dot{m}_1,0)-\operatorname{max}(-\dot{m}_1,0)\\
++&\operatorname{max}(\dot{m}_2,0)-\operatorname{max}(-\dot{m}_2,0)\\
++&\operatorname{max}(\dot{m}_3,0)-\operatorname{max}(-\dot{m}_3,0)
 \end{split}
 \label{eq:D2b}
 \end{equation}
 \label{eq:D2}
 \end{subequations}
 
-Equation \eqref{eq:D2a} is solved for $h_{mix}$
+Equation \eqref{eq:D2a} is solved for $h_{\mathrm{mix}}$
 \begin{equation*}
-h_{mix}=\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}
-{\text{max}(\dot{m}_1,0)+\text{max}(\dot{m}_2,0)+\text{max}(\dot{m}_3,0)}
+h_{\mathrm{mix}}=\frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}
+{\operatorname{max}(\dot{m}_1,0)+\operatorname{max}(\dot{m}_2,0)+\operatorname{max}(\dot{m}_3,0)}
 \end{equation*}
 Using \eqref{eq:D2b}, the denominator can be changed to:
 \begin{equation*}
-h_{mix}=\frac{\text{max}(-\dot{m}_1,0)h_{outflow,1}+\text{max}(-\dot{m}_2,0)h_{outflow,2}+\text{max}(-\dot{m}_3,0)h_{outflow,3}}
-{\text{max}(-\dot{m}_1,0)+\text{max}(-\dot{m}_2,0)+\text{max}(-\dot{m}_3,0)}
+h_{\mathrm{mix}}=\frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}
+{\operatorname{max}(-\dot{m}_1,0)+\operatorname{max}(-\dot{m}_2,0)+\operatorname{max}(-\dot{m}_3,0)}
 \end{equation*}
 Above it was shown that an equation of this type does not yield properly
 formulated model equations. In the streams concept we therefore decide
@@ -127,14 +127,14 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 it is therefore the mixing enthalpy under the assumption of fluid
 flowing into said model.
 
-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
+We establish this quantity using a dedicated operator $\text{\lstinline!inStream!}(h_{\mathrm{outflow},i})=h_{\mathrm{mix}}$ assuming that $\dot{m}_{i} \geq 0$. 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{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)}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}) &= \frac{\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\dot{m}_2,0)+\operatorname{max}(-\dot{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\dot{m}_1,0)+\operatorname{max}(-\dot{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\dot{m}_1,0)+\operatorname{max}(-\dot{m}_2,0)}
 \end{align*}
 \begin{figure}[H]
   \begin{center}
@@ -146,7 +146,7 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 In the general case of a connection set with \emph{n} components,
 similar considerations lead to the following.
 \begin{equation*}
-\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)}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},i})=\frac{\sum_{j=1,...,n;j\neq i}\operatorname{max}(-\dot{m}_j,0)h_{\mathrm{outflow},j}}{\sum_{j=1,...,n;j\neq i}\operatorname{max}(-\dot{m}_j,0)}
 \end{equation*}
 
 \section{Special cases covered by inStream definition}\label{special-cases-covered-by-the-instream-operator-definition}
@@ -157,11 +157,11 @@ \subsection{Stream connector is not connected (N = 1)}\label{stream-connector-is
 \subsection{Connection of 2 stream connectors, one to one connections (N = 2)}\label{connection-of-2-stream-connectors-one-to-one-connections-n-2}
 
 \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}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}) &= \frac{\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\dot{m}_2,0)}=h_{\mathrm{outflow},2}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}}{\operatorname{max}(-\dot{m}_1,0)}=h_{\mathrm{outflow},1}
 \end{align*}
 
-In this case, \lstinline!inStream! is continuous (contrary to $h_{mix}$) and does not
+In this case, \lstinline!inStream! is continuous (contrary to $h_{\mathrm{mix}}$) and does not
 depend on flow rates. The latter result means that this transformation
 may remove nonlinear systems of equations, which requires that either
 simplifications of the form $a * b / a = b$ must be provided, or that this
@@ -178,13 +178,13 @@ \subsection{Connection of 3 stream connectors where one mass flow rate is identi
 the following equations, and as indicated the last formula can be
 simplified further by using $\dot{m}_3=0$:
 \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)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}) &= h_{\mathrm{outflow},2}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= h_{\mathrm{outflow},1}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\dot{m}_1,0)+\operatorname{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 $\dot{m}_3=0$}
+h_{\mathrm{outflow},2}&\text{if $\dot{m}_1 \geq 0$}\\
+h_{\mathrm{outflow},1}&\text{if $\dot{m}_1 < 0$ and $\dot{m}_3 = 0$}
 \end{cases}
 \end{align*}
 \begin{figure}[H]
@@ -228,21 +228,21 @@ \subsection{Connection of 3 stream connectors where two mass flow rates are posi
 \end{lstlisting}
 results in the following equation:
 \begin{equation*}
-\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}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1})=\frac{\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\dot{m}_2,0)+\operatorname{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*}
-\text{\lstinline!inStream!}(h_{outflow,1}):=h_{outflow,1}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}):=h_{\mathrm{outflow},1}
 \end{equation*}
 For the remaining connectors, \lstinline!inStream! reduces to a simple result.
 \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}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\dot{m}_1,0)+\operatorname{max}(-\dot{m}_3,0)}
+  = h_{\mathrm{outflow},1}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\dot{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\dot{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\dot{m}_1,0)+\operatorname{max}(-\dot{m}_2,0)}
+  = h_{\mathrm{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