Change notation for mass flow rate from \dot{m} to \tilde{m}.

chapters/derivationofstream.tex

@@ -5,21 +5,22 @@ \chapter{Derivation of Stream Equations}\label{derivation-of-stream-equations}
 
 \section{Reasons for avoiding the actual mixing enthalpy in connector definitions}\label{reasons-for-avoiding-the-actual-mixing-enthalpy-in-connector-definitions}
 
-Consider a connection set with \emph{n} connectors. The mixing enthalpy
-is defined by the mass balance
+Consider a connection set with $n$ connectors, and denote the mass flow rates \lstinline!m_flow! by $\tilde{m}$.
+The mixing enthalpy is defined by the mass balance (the general mass-balance for a component has
+$\dot{m}=\sum\tilde{m}$ which simplifies for the mixing enthalpy where $m=0$ and thus $\dot{m}=0$)
 \begin{equation*}
-0=\sum_{j=1}^n\dot{m}_j
+0=\sum_{j=1}^n\tilde{m}_j
 \end{equation*}
-and the energy balance
+and similarly the energy balance
 \begin{equation*}
-0=\sum_{j=1}^n\dot{H}_j
+0=\sum_{j=1}^n\tilde{H}_j
 \end{equation*}
 with
 \begin{equation*}
-\dot{H}_j=\dot{m}_j
+\tilde{H}_j=\tilde{m}_j
 \begin{cases}
-h_{\mathrm{mix}}&\text{if $\dot{m}_j > 0$}\\
-h_{\mathrm{outflow},j}&\text{if $\dot{m}_j \leq 0$}
+h_{\mathrm{mix}}&\text{if $\tilde{m}_j>0$}\\
+h_{\mathrm{outflow},j}&\text{if $\tilde{m}_j<=0$}
 \end{cases}
 \end{equation*}
 Herein, mass flow rates are positive when entering models (exiting the
@@ -29,8 +30,8 @@ \section{Reasons for avoiding the actual mixing enthalpy in connector definition
 \begin{equation*}
 h_{\mathrm{outflow},j}=
 \begin{cases}
-\frac{\dot{H}_j}{\dot{m}_j}&\text{if $\dot{m}_j<0$}\\
-\textrm{arbitrary}&\text{if $\dot{m}_j \geq 0$}
+\frac{\tilde{H}_j}{\tilde{m}_j}&\text{if $\tilde{m}_j<0$}\\
+\textrm{arbitrary}&\text{if $\tilde{m}_j \geq 0$}
 \end{cases}
 \end{equation*}
 While these equations are suitable for device-oriented modeling, the
@@ -57,26 +58,26 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 \begin{subequations}
 \begin{equation}
 \begin{split}
-0=&\dot{m}_1\cdot
+0=&\tilde{m}_1\cdot
 \begin{cases}
-h_{\mathrm{mix}}&\text{if $\dot{m}_1 > 0$}\\
-h_{\mathrm{outflow},1}&\text{if $\dot{m}_1 \leq 0$}
+h_{\mathrm{mix}}&\text{if $\tilde{m}_1>0$}\\
+h_{\mathrm{outflow},1}&\text{if $\tilde{m}_1 \leq 0$}
 \end{cases}\\
-+&\dot{m}_2\cdot
++&\tilde{m}_2\cdot
 \begin{cases}
-h_{\mathrm{mix}}&\text{if $\dot{m}_2 > 0$}\\
-h_{\mathrm{outflow},2}&\text{if $\dot{m}_2 \leq 0$}
+h_{\mathrm{mix}}&\text{if $\tilde{m}_2>0$}\\
+h_{\mathrm{outflow},2}&\text{if $\tilde{m}_2 \leq 0$}
 \end{cases}\\
-+&\dot{m}_3\cdot
++&\tilde{m}_3\cdot
 \begin{cases}
-h_{\mathrm{mix}}&\text{if $\dot{m}_3 > 0$}\\
-h_{\mathrm{outflow},3}&\text{if $\dot{m}_3 \leq 0$}
+h_{\mathrm{mix}}&\text{if $\tilde{m}_3>0$}\\
+h_{\mathrm{outflow},3}&\text{if $\tilde{m}_3 \leq $}
 \end{cases}
 \end{split}
 \label{eq:D1a}
 \end{equation}
 \begin{equation}
-0=\dot{m}_1+\dot{m}_2+\dot{m}_3
+0=\tilde{m}_1+\tilde{m}_2+\tilde{m}_3
 \label{eq:D1b}
 \end{equation}
 \label{eq:D1}
@@ -87,18 +88,18 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 \begin{subequations}
 \begin{equation}
 \begin{split}
-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}
+0=&\operatorname{max}(\tilde{m}_1,0)h_{\mathrm{mix}}-\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}\\
++&\operatorname{max}(\tilde{m}_2,0)h_{\mathrm{mix}}-\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}\\
++&\operatorname{max}(\tilde{m}_3,0)h_{\mathrm{mix}}-\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}
 \end{split}
 \label{eq:D2a}
 \end{equation}
 
 \begin{equation}
 \begin{split}
-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)
+0=&\operatorname{max}(\tilde{m}_1,0)-\operatorname{max}(-\tilde{m}_1,0)\\
++&\operatorname{max}(\tilde{m}_2,0)-\operatorname{max}(-\tilde{m}_2,0)\\
++&\operatorname{max}(\tilde{m}_3,0)-\operatorname{max}(-\tilde{m}_3,0)
 \end{split}
 \label{eq:D2b}
 \end{equation}
@@ -107,13 +108,13 @@ \section{Rationale for the formulation of inStream}\label{rationale-for-the-form
 
 Equation \eqref{eq:D2a} is solved for $h_{\mathrm{mix}}$
 \begin{equation*}
-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)}
+h_{\mathrm{mix}}=\frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}
+{\operatorname{max}(\tilde{m}_1,0)+\operatorname{max}(\tilde{m}_2,0)+\operatorname{max}(\tilde{m}_3,0)}
 \end{equation*}
 Using \eqref{eq:D2b}, the denominator can be changed to:
 \begin{equation*}
-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)}
+h_{\mathrm{mix}}=\frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}
+{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{m}_2,0)+\operatorname{max}(-\tilde{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 +128,14 @@ \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 $\text{\lstinline!inStream!}(h_{\mathrm{outflow},i})=h_{\mathrm{mix}}$ assuming that $\dot{m}_{i} \geq 0$. This leads to
+We establish this quantity using a dedicated operator $\text{\lstinline!inStream!}(h_{\mathrm{outflow},i})=h_{\mathrm{mix}}$ assuming that $\tilde{m}_{i} \geq 0$. 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_{\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)}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}) &= \frac{\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\tilde{m}_2,0)+\operatorname{max}(-\tilde{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{m}_3,0)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{m}_2,0)}
 \end{align*}
 \begin{figure}[H]
   \begin{center}
@@ -146,7 +147,7 @@ \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_{\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)}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},i})=\frac{\sum_{j=1,...,n;j\neq i}\operatorname{max}(-\tilde{m}_j,0)h_{\mathrm{outflow},j}}{\sum_{j=1,...,n;j\neq i}\operatorname{max}(-\tilde{m}_j,0)}
 \end{equation*}
 
 \section{Special cases covered by inStream definition}\label{special-cases-covered-by-the-instream-operator-definition}
@@ -157,8 +158,8 @@ \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_{\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}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1}) &= \frac{\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\tilde{m}_2,0)}=h_{\mathrm{outflow},2}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},2}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}}{\operatorname{max}(-\tilde{m}_1,0)}=h_{\mathrm{outflow},1}
 \end{align*}
 
 In this case, \lstinline!inStream! is continuous (contrary to $h_{\mathrm{mix}}$) and does not
@@ -168,23 +169,23 @@ \subsection{Connection of 2 stream connectors, one to one connections (N = 2)}\l
 case is treated directly.
 
 \subsection{Connection of 3 stream connectors where one mass flow rate is identical to zero}\label{connection-of-3-stream-connectors-where-one-mass-flow-rate-is-identical-to-zero-n-3-and}
-The case where $N=3$ and $\dot{m}_3=0$ occurs when a one-port sensor (like a temperature sensor) is
+The case where $N=3$ and $\tilde{m}_3=0$ occurs when a one-port sensor (like a temperature sensor) is
 connected to two connected components. For the sensor, the \lstinline!min! attribute
 of the mass flow rate should be set to zero (no fluid exiting the
 component via this connector).
 This simplification (and similar ones) can also be used if a tool determines that a mass flow rate is zero or non-negative.
 It is also possible to generalize this to the case where more than one sensor is connected.
 The suggested implementation results in
 the following equations, and as indicated the last formula can be
-simplified further by using $\dot{m}_3=0$:
+simplified further by using $\tilde{m}_3=0$:
 \begin{align*}
 \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)}\\
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{m}_2,0)}\\
 &=
 \begin{cases}
-h_{\mathrm{outflow},2}&\text{if $\dot{m}_1 \geq 0$}\\
-h_{\mathrm{outflow},1}&\text{if $\dot{m}_1 < 0$ and $\dot{m}_3 = 0$}
+h_{\mathrm{outflow},2}&\text{if $\tilde{m}_1 \geq 0$}\\
+h_{\mathrm{outflow},1}&\text{if $\tilde{m}_1 < 0$ and $\tilde{m}_3 = 0$}
 \end{cases}
 \end{align*}
 \begin{figure}[H]
@@ -228,7 +229,7 @@ \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_{\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}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},1})=\frac{\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\tilde{m}_2,0)+\operatorname{max}(-\tilde{m}_3,0)}=\frac{0}{0}
 \end{equation*}
 
 \lstinline!inStream! cannot be evaluated for a connector, on which
@@ -239,9 +240,9 @@ \subsection{Connection of 3 stream connectors where two mass flow rates are posi
 \end{equation*}
 For the remaining connectors, \lstinline!inStream! reduces to a simple result.
 \begin{align*}
-\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},2}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_3,0)h_{\mathrm{outflow},3}}{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{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)}
+\text{\lstinline!inStream!}(h_{\mathrm{outflow},3}) &= \frac{\operatorname{max}(-\tilde{m}_1,0)h_{\mathrm{outflow},1}+\operatorname{max}(-\tilde{m}_2,0)h_{\mathrm{outflow},2}}{\operatorname{max}(-\tilde{m}_1,0)+\operatorname{max}(-\tilde{m}_2,0)}
   = h_{\mathrm{outflow},1}
 \end{align*}
 Again, the previous non-linear algebraic system of equations is removed.