Avoid use of \texttt in stream.tex

To get consistent formatting of subscripted variables inside Modelica source code, this involves changes in many places besides those where there used to be a \texttt

chapters/stream.tex

@@ -1,9 +1,5 @@
 \chapter{Stream Connectors}\doublelabel{stream-connectors}
 
-%TODO-FORMAT This chapter uses _ in code blocks and inline code.
-%This is no real Modelica code, but indicates a mathematical index.
-%We currently use \texttt{m\textsubscript j} instead of \lstinline!m_j!
-
 The two basic variable types in a connector -- \emph{potential} (or \emph{across})
 variable and \emph{flow} (or \emph{through}) variable -- are not sufficient to
 describe in a numerically sound way the bi-directional flow of matter
@@ -99,9 +95,9 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 from the stream connection equations of the flow variables and of the
 stream variables.
 
-For the following definition it is assumed that \lstinline!N! inside connectors
-\texttt{m\textsubscript{j}.c} (j=1,2,...,N) and \lstinline!M! outside connectors
-\texttt{c\textsubscript{k}} (k=1,2,...,M) belonging to the same connection set
+For the following definition it is assumed that $N$ inside connectors
+\lstinline[mathescape=true]!$m_{j}$.c! ($j = 1, 2, \ldots, N$) and $M$ outside connectors
+\lstinline[mathescape=true]!$c{k}$! ($k = 1, 2, \ldots, M$) belonging to the same connection set
 (see definition in \autoref{inside-and-outside-connectors}) are connected
 together and a stream variable \lstinline!h_outflow! is associated with a flow
 variable \lstinline!m_flow! in connector \lstinline!c!.
@@ -130,7 +126,7 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 end FluidSystem;
 \end{lstlisting}
 \begin{figure}[H]
-\caption{Examplary FluidSystem with N=3 and M=2}
+\caption{Examplary FluidSystem with $N$=3 and $M$=2}
 \begin{center}
 \includegraphics[width=2.3in,height=2.08125in]{fluidsystem}
 \end{center}
@@ -143,8 +139,8 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 \end{nonnormative}
 
 With these prerequisites, the semantics of the expression
-\texttt{inStream(m\textsubscript i.c.h\_outflow)} is given implicitly by
-defining an additional variable \texttt{h\_mix\_in\textsubscript{i}}, and by
+\lstinline[mathescape=true]!inStream($m_{i}$.c.h_outflow)! is given implicitly by
+defining an additional variable $\mathit{h\_mix\_in}_{i}$, and by
 adding to the model the conservation equations for mass and energy
 corresponding to the infinitesimally small volume spanning the
 connection set. The connection equation for the flow variables has
@@ -153,26 +149,29 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 
 \begin{lstlisting}[language=modelica,mathescape=true]
 // Standard connection equation for flow variables
-0 = sum(m_j.c.m_flow for j in 1:N) + sum(-ck.m_flow for k in 1:M);
+0 = sum($m_{j}$.c.m_flow for j in 1:N) + sum(-$c_{k}$.m_flow for k in 1:M);
 \end{lstlisting}
 
 Whenever the \lstinline!inStream()! operator is applied to a stream
 variable of an inside connector, the balance equation of the transported
 property must be added under the assumption of flow going into the
 connector
 
-\begin{lstlisting}[language=modelica]
-// Implicit definition of the inStream() operator applied to inside connector i
-0 = sum(mj.c.m_flow*(if mj.c.m_flow > 0 or j==i then h_mix_ini else mj.c.h_outflow)
-        for j in 1:N) +
-    sum(-ck.m_flow* (if -ck.m_flow > 0 then h_mix_ini else inStream(ck.h_outflow)
-         for k in 1:M);
-inStream(mi.c.h_outflow) = h_mix_ini;
+\begin{lstlisting}[language=modelica,mathescape=true]
+// Implicit definition of the inStream() operator applied to inside connector $i$
+0 =
+  sum($m_{j}$.c.m_flow *
+      (if $m_{j}$.c.m_flow > 0 or $j$==$i$ then $\mathit{h\_mix\_in}_{i}$ else $m_{j}$.c.h_outflow)
+    for $j$ in 1:N) +
+  sum(-$c_{k}$.m_flow *
+      (if -$c_{k}$.m_flow > 0 then $\mathit{h\_mix\_in}_{i}$ else inStream($c_{k}$.h_outflow)
+    for $k$ in 1:M);
+inStream($m_{i}$.c.h_outflow) = $\mathit{h\_mix\_in}_{i}$;
 \end{lstlisting}
 
 Note that the result of the
-\texttt{inStream(m\textsubscript{i}.c.h\_outflow)} operator is different
-for each port \lstinline!i!, because the assumption of flow entering the port is
+\lstinline[mathescape=true]!inStream($m_{i}$.c.h_outflow)! operator is different
+for each port $i$, because the assumption of flow entering the port is
 different for each of them.
 
 Additional equations need to be generated for the stream variables of
@@ -181,38 +180,37 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 \begin{lstlisting}[language=modelica,mathescape=true]
 // Additional connection equations for outside connectors
 for q in 1:M loop
-  0 = sum($m_j$.c.m_flow*(if
-  $m_j$.c.m_flow > 0 then
-  h_mix_out_q
-  else $m_j$.c.h_outflow) for j
-  in 1:N) +
-  sum(-$c_k$.m_flow* (if
-  -$c_k$.m_flow > 0 or k==q
-  then h_mix_out$_q$
-  else inStream($c_k$.h_outflow)
-  for k in 1:M);
-    $c_q$.h_outflow = h_mix_out$_q$;
-  end for;
+  0 =
+    sum($m_j$.c.m_flow *
+        (if $m_j$.c.m_flow > 0 then $\mathit{h\_mix\_out}_{q}$ else $m_j$.c.h_outflow)
+      for j in 1:N) +
+    sum(-$c_k$.m_flow *
+        (if -$c_k$.m_flow > 0 or k==q then $\mathit{h\_mix\_out}_{q}$ else inStream($c_k$.h_outflow))
+      for k in 1:M);
+  $c_q$.h_outflow = $\mathit{h\_mix\_out}_{q}$;
+end for;
 \end{lstlisting}
 
 Neglecting zero flow conditions, the solution of the above-defined
-stream connection equations for inStream values of inside connectors and
+stream connection equations for \lstinline!inStream! values of inside connectors and
 outflow stream variables of outside connectors is (for a derivation, see
 \autoref{derivation-of-stream-equations}):
 \begin{lstlisting}[language=modelica,mathescape=true]
 inStream($m_i$.c.h_outflow) :=
-  (sum(max(-$m_j$.c.m_flow,0)*$m_j$.c.h_outflow for j in cat(1,1:i-1, i+1:N) +
-   sum(max( $c_k$.m_flow,0)*inStream($c_k$.h_outflow) for k in 1:M))/
-  (sum(max(-$m_j$.c.m_flow,0) for j in  cat(1,1:i-1, i+1:N) +
+  (sum(max(-$m_j$.c.m_flow,0)*$m_j$.c.h_outflow for j in cat(1, 1:i-1, i+1:N) +
+   sum(max( $c_k$.m_flow,0)*inStream($c_k$.h_outflow) for k in 1:M))
+  /
+  (sum(max(-$m_j$.c.m_flow,0) for j in cat(1, 1:i-1, i+1:N) +
    sum(max( $c_k$.m_flow ,0) for k in 1:M));
 
 // Additional equations to be generated for outside connectors q
 for q in 1:M loop
   $c_q$.h_outflow :=
     (sum(max(-$m_j$.c.m_flow,0)*$m_j$.c.h_outflow for j in 1:N) +
-     sum(max( $c_k$.m_flow,0)*inStream($c_k$.h_outflow) for k in cat(1,1:q-1, q+1:M))/
-    (sum(max(-$m_j$.c.m_flow,0) for j in  1:N) +
-     sum(max( $c_k$.m_flow ,0) for k in cat(1,1:q-1, q+1:M)));
+     sum(max( $c_k$.m_flow,0)*inStream($c_k$.h_outflow) for k in cat(1, 1:q-1, q+1:M))
+    /
+    (sum(max(-$m_j$.c.m_flow,0) for j in 1:N) +
+     sum(max( $c_k$.m_flow ,0) for k in cat(1, 1:q-1, q+1:M)));
 end for;
 \end{lstlisting}
 
@@ -251,10 +249,9 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
   equation system identically (upto the usual numerical
   accuracy), provided the absolute value of every flow variable in
   the connection set is greater than a small value
-  (\texttt{\textbar{}m\textsubscript{1}.c.m\_flow\textbar{} \textgreater{} eps
-  \textbf{and} \textbar{}m\textsubscript{2}.c.m\_flow\textbar{}
-  \textgreater{} eps \textbf{and} ... \textbf{and}
-  \textbar{}c\textsubscript{M}.m\_flow\textbar{} \textgreater{} eps}).
+  ($\abs{m_{1}\text{\lstinline!.c.m_flow!}} > \text{\lstinline!eps!}$
+  and $\abs{m_{2}\text{\lstinline!.c.m_flow!}} > \text{\lstinline!eps!}$ and \ldots and
+  $\abs{m_{M}\text{\lstinline!.c.m_flow!}} > \text{\lstinline!eps!}$).
 \end{enumerate}
 
 \begin{nonnormative}
@@ -265,25 +262,25 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 \raggedright
 \item \textbf{N = 1, M = 0:}\newline
 \begin{lstlisting}[language=modelica,mathescape=true]
-inStream($m_1$.c.h_outflow) =$m_1$.c.h_outflow;
+inStream($m_1$.c.h_outflow) = $m_1$.c.h_outflow;
 \end{lstlisting}
 \item
 \textbf{N = 2, M = 0:}\newline
 \begin{lstlisting}[language=modelica,mathescape=true]
-inStream($m_1$.c.h_outflow) =$m_2$.c.h_outflow;
-inStream($m_2$.c.h_outflow) =$m_1$.c.h_outflow;
+inStream($m_1$.c.h_outflow) = $m_2$.c.h_outflow;
+inStream($m_2$.c.h_outflow) = $m_1$.c.h_outflow;
 \end{lstlisting}
 \item \textbf{N = 1, M = 1:}\newline
 \begin{lstlisting}[language=modelica,mathescape=true]
-inStream($m_1$.c.h_outflow) =inStream($c_1$.h_outflow);
+inStream($m_1$.c.h_outflow) = inStream($c_1$.h_outflow);
 // Additional equation to be generated
 $c_1$.h_outflow = $m_1$.c.h_outflow;
 \end{lstlisting}
 \item \textbf{N = 0, M = 2:}\newline
 \begin{lstlisting}[language=modelica,mathescape=true]
 // Additional equation to be generated
-c1.h_outflow = inStream(c2.h_outflow);
-c2.h_outflow = inStream(c1.h_outflow);
+$c_{1}$.h_outflow = inStream($c_{2}$.h_outflow);
+$c_{2}$.h_outflow = inStream($c_{1}$.h_outflow);
 \end{lstlisting}
 
 \item\textbf{All other cases:}\newline
@@ -322,37 +319,38 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 \end{itemize}
 
 The operator
-\texttt{positiveMax(-m\textsubscript{j}.c.m\_flow,s\textsubscript{i})}
+\lstinline[mathescape=true]!positiveMax(-$m_{j}$.c.m_flow, $s_{i}$)!
 should be such that:
 \begin{itemize}
 \item
-  positiveMax(-m\textsubscript{j}.c.m\_flow,s\textsubscript{i}) =
-  -m\textsubscript{j}.c\_m\_flow if
-  -m\textsubscript{j}.c.m\_flow\textgreater{}eps1\textsubscript{j}\textgreater{}=0,
-  where eps1\textsubscript{j} are small flows, compared to typical
-  problem-specific value,
+  \lstinline[mathescape=true]!positiveMax(-$m_{j}$.c.m_flow, $s_{i}$)! =
+  \lstinline[mathescape=true]!-$m_{j}$.c.m_flow! if
+  $-m_{j}\text{\lstinline!.c.m_flow!} > \mathit{eps1}_{j} \geq 0$,
+  where $\mathit{eps1}_{j}$ are small flows, compared to typical
+  problem-specific values,
 \item
-  all denominators should be \textgreater{} eps2 \textgreater{} 0,
-  where eps2 is also a small flow, compared to typical problem-specific
+  all denominators should be $> \mathit{eps2} > 0$,
+  where $\mathit{eps2}$ is also a small flow, compared to typical problem-specific
   values.
 \end{itemize}
 
-Trivial implementation of positiveMax guarantees continuity of \lstinline!inStream()!:
+Trivial implementation of \lstinline!positiveMax! guarantees continuity of \lstinline!inStream()!:
 \begin{lstlisting}[language=modelica,mathescape=true]
-postiveMax(-$m_j$.c.m_flow, $s_i$)= max(-m_j.c.m_flow, eps1); // so $s_i$ is not needed
+postiveMax(-$m_j$.c.m_flow, $s_i$) = max(-$m_j$.c.m_flow, $\mathit{eps1}$); // so $s_i$ is not needed
 \end{lstlisting}
 More sophisticated implementation, with smooth approximation, applied only when {all} flows are small:
 \begin{lstlisting}[language=modelica,escapechar=!,mathescape=true]
 // Define a "small number" eps (!\textbf{nominal}!(v) is the nominal value of v !--! see !\autoref{attributes-start-fixed-nominal-and-unbounded}!)
-  eps := relativeTolerance*min(nominal($m_j$.c.m_flow));
+eps := relativeTolerance*min(nominal($m_j$.c.m_flow));
 
 // Define a smooth curve, such that  alpha($s_i>=eps$)=1 and alpha($s_i<0$)=0
-  alpha := smooth(1, if $s_i$ > eps then 1 else
-                     if $s_i$ > 0 then  ( $s_i$/eps)^2*(3-2* $s_i$/eps)) else 0);
+alpha := smooth(1, if $s_i$ > eps then 1
+                   else if $s_i$ > 0 then ($s_i$/eps)^2*(3-2* $s_i$/eps)
+                   else 0);
 
-  // Define function positiveMax(v,s_i) as a linear combination of max (v,0)
-  // and of eps along alpha
-  positiveMax((-$m_j$.c.m_flow,s_i)  := alpha*max(-$m_j$.c.m_flow,0) +  (1-alpha)*eps;
+// Define function positiveMax(v,s_i) as a linear combination of max (v,0)
+// and of eps along alpha
+positiveMax((-$m_j$.c.m_flow,s_i) := alpha*max(-$m_j$.c.m_flow,0) + (1-alpha)*eps;
 \end{lstlisting}
 
 The derivation of this implementation is discussed in
@@ -366,44 +364,38 @@ \section{Stream Operator inStream and Connection Equations}\doublelabel{stream-o
 \begin{itemize}
 \item
   \lstinline!inStream(..)! is continuous (and differentiable),
-  provided that \texttt{m\textsubscript{j}.c.h\_outflow,
-  m\textsubscript{j}.c.m\_flow,
-  c\textsubscript{k}.h\_outflow}, and
-  \texttt{c\textsubscript{k}.m\_flow} are continuous and differentiable.
+  provided that \lstinline[mathescape=true]!$m_{j}$.c.h_outflow!,
+  \lstinline[mathescape=true]!$m_{j}$.c.m_flow!, \lstinline[mathescape=true]!$c_{k}$.h_outflow!, and
+  \lstinline[mathescape=true]!$c_{k}$.m_flow! are continuous and differentiable.
 \end{itemize}
 
 \begin{itemize}
 \item
-  A division by zero can no longer occur (since \texttt{sum(positiveMax(-m\textsubscript{j}.c.m\_flow,s\textsubscript{i}))\textgreater{}=eps2}
-  \textgreater{} 0), so the result is always well-defined.
+  A division by zero can no longer occur (since \lstinline[mathescape=true]!sum(positiveMax(-$m_{j}$.c.m_flow, $s_{i}$))! $\geq \mathit{eps2} > 0$),
+  so the result is always well-defined.
 \item
-  The balance equations are exactly fulfilled if the denominator
-  is not close to zero\\
-  (since the exact formula is used, if
-  \texttt{sum(positiveMax(-m\textsubscript{j}.c.m\_flow,s\textsubscript{i})) \textgreater{} eps}).
+  The balance equations are exactly fulfilled if the denominator is not close to zero (since the exact formula is used, if
+  \lstinline[mathescape=true]!sum(positiveMax(-$m_{j}$.c.m_flow, $s_{i}$))! $> \mathit{eps}$).
 \item
   If all flows are zero,
-  \texttt{inStream(m\textsubscript{i}.c.h\_outflow) =
-  sum(m\textsubscript{j}.c.h\_outflow for
-  j\textless{}\textgreater{}i and m\textsubscript{j}.c.m\_flow.min \textless{}
-  0)/Np}, i.e., it is the mean value of all the \lstinline!Np! variables
-  \texttt{m\textsubscript{j}.c.h\_outflow}, such that \texttt{
-  j\textless{}\textgreater{}i} and
-  \texttt{m\textsubscript{j}.c.m\_flow.min \textless{} 0}. This is a
+  \lstinline[mathescape=true]!inStream($m_{i}$.c.h_outflow)! =
+  \lstinline[mathescape=true]!sum($m_{j}$.c.h_outflow for $j \neq i$ and $m_{j}$.c.m_flow.min < 0) / $N_{\mathrm{p}}$!,
+  i.e., it is the mean value of all the $N_{\mathrm{p}}$ variables
+  \lstinline[mathescape=true]!$m_{j}$.c.h_outflow!, such that $j \neq i$ and
+  $m_{j}\text{\lstinline!.c.m_flow.min!} < 0$. This is a
   meaningful approximation, considering the physical diffusion effects
   that are relevant at small flow rates in a small connection volume
   (thermal conduction for enthalpy, mass diffusion for mass fractions).
 \end{itemize}
 
-The value of relativeTolerance should be larger than the relative
+The value of \lstinline!relativeTolerance! should be larger than the relative
 tolerance of the nonlinear solver used to solve the implicit algebraic
 equations.
 
 As a final remark, further symbolic simplifications could be
 carried out by taking into account equations that affect the flows in
-the connection set (i.e., equivalent to m\textsubscript{j}.c.m\_flow =
-0, which then implies m\textsubscript{j}.c.m\_flow.min \textgreater{}=
-0). This is interesting, e.g., in the case of a valve when the stem
+the connection set (i.e., equivalent to $m_{j}\text{\lstinline!.c.m_flow!} =
+0$, which then implies $m_{j}\text{\lstinline!.c.m_flow.min!} \geq 0$). This is interesting, e.g., in the case of a valve when the stem
 position is set identically to closed by its controller.
 \end{nonnormative}
 
@@ -425,27 +417,27 @@ \section{Stream Operator actualStream}\doublelabel{stream-operator-actualstream}
 The \textbf{actualStream}(v) operator is typically used in two
 contexts:
 \begin{lstlisting}[language=modelica]
-  der(U) = c.m_flow*actualStream(c.h_outflow);  // (1)energy balance equation
-  h_c = actualStream(c.h);                      // (2)monitoring the enthalpy at port c
+  der(U) = c.m_flow*actualStream(c.h_outflow);  // (1) energy balance equation
+  h_c = actualStream(c.h);                      // (2) monitoring the enthalpy at port c
 \end{lstlisting}
-In the case of equation (1), although the actualStream() operator
+In the case of equation (1), although the \lstinline!actualStream()! operator
 is discontinuous, the product with the flow variable is not, because
-actualStream() is discontinuous when the flow is zero by construction.
-Therefore, a tool might infer that the expression is smooth(0, ...)
+\lstinline!actualStream()! is discontinuous when the flow is zero by construction.
+Therefore, a tool might infer that the expression is \lstinline!smooth(0, ...)!
 automatically, and decide whether or not to generate an event. If a user
 wants to avoid events entirely, he/she may enclose the right-hand side
-of (1) with the noEvent() operator.
+of (1) with the \lstinline!noEvent()! operator.
 
 Equations like (2) might be used for monitoring purposes (e.g.
 plots), in order to inspect what the actual enthalpy of the fluid
 flowing through a port is. In this case, the user will probably want to
 see the change due to flow reversal at the exact instant, so an event
 should be generated. If the user doesn't bother, then he/she should
-enclose the right-hand side of (2) with noEvent(). Since the output of
-actualStream() will be discontinuous, it should not be used by itself to
+enclose the right-hand side of (2) with \lstinline!noEvent()!. Since the output of
+\lstinline!actualStream()! will be discontinuous, it should not be used by itself to
 model physical behaviour (e.g., to compute densities used in momentum
-balances) - inStream() should be used for this purpose. The operator
-actualStream() should be used to model physical behaviour only when
+balances) --- \lstinline!inStream()! should be used for this purpose. The operator
+\lstinline!actualStream()! should be used to model physical behaviour only when
 multiplied by the corresponding flow variable (like in the above energy
 balance equation), because this removes the discontinuity.
 \end{nonnormative}

preamble.tex

@@ -275,6 +275,7 @@
   \fi%
 }
 \makeatother
+\newcommand{\abs}[1]{\left\lvert #1{} \right\rvert}
 
 % Text mode additions
 \newcommand{\textgreatereq}{$\geq$}