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[handbook] add subsection about Dalton's and Amagat's law in chapter 5

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......@@ -122,6 +122,79 @@ $\boldsymbol{v}_\alpha$ & velocity (Darcy or free flow)& & \\
\label{fig:phaseMassEnergyTransfer}
\end{figure}
\subsection{Gas mixing laws}
Prediction of the $p-\varrho-T$ behavior of gas mixtures is typically based on the following two (contradicting) concepts: Dalton's law or Amagat's law.
In the following the two concepts will be explained in more detail.
%
\subsubsection{Dalton's law}
Dalton's law assumes that the gases in the mixture are non-interacting (with each other) and each gas independently applies its own pressure (partial pressure), the sum of which is the total pressure:
%
\begin{equation}
p = \sum_{i}^{}p_i.
\end{equation}
Here $p_i$ refers to the partial pressure of component i.
As an example, if two equal volumes of gas A and gas B are mixed, the volume of the mixture stays the same but the pressures add up (see Figure \ref{fig:dalton1}).
%
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{PNG/dalton1.png}
\caption{Dalton's law visualized}
\label{fig:dalton1}
\end{figure}
%
The density of the mixture, $\varrho$, can be calculated as follows:
\begin{equation}
\varrho = \frac{m}{V} = \frac{m_\mathrm{A} + m_\mathrm{B}}{V} = \frac{\varrho_\mathrm{A} V + \varrho_\mathrm{B} V}{V} = \varrho_\mathrm{A} + \varrho_\mathrm{B},
\end{equation}
%
or for an arbitrary number of gases:
\begin{equation}
\varrho = \sum_{i}^{} \varrho_i ; \quad \varrho_m = \sum_{i}^{} \varrho_{m,i}.
\end{equation}
%
\subsubsection{Amagat's law}
Amagat's law assumes that the volumes of the component gases are additive; the interactions of the different gases are the same as the average interactions of the components. This is known as Amagat's law:
%
\begin{equation}
V = \sum_{i}^{}V_i.
\end{equation}
%
As an example, if two volumes of gas A and B at equal pressure are mixed, the pressure of the mixture stays the same, but the volumes add up (see Figure \ref{fig:dalton2}).
%
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\textwidth]{PNG/dalton2.png}
\caption{Amagat's law visualized}
\label{fig:dalton2}
\end{figure}
%
The density of the mixture, $\varrho$, can be calculated as follows:
\begin{equation}
\varrho = \frac{m}{V} = \frac{m}{V_\mathrm{A} + V_\mathrm{B}} = \frac{m}{\frac{m_\mathrm{A}}{\varrho_\mathrm{A}} \frac{m_\mathrm{B}}{\varrho_\mathrm{B}}} =
\frac{m}{\frac{X_\mathrm{A} m}{\varrho_\mathrm{A}} \frac{X_\mathrm{B} m}{\varrho_\mathrm{B}}} = \frac{1}{\frac{X_\mathrm{A}}{\varrho_\mathrm{A}} \frac{X_\mathrm{B}}{\varrho_\mathrm{B}}},
\end{equation}
%
or for an arbitrary number of gases:
%
\begin{equation}
\varrho = \frac{1}{\sum_{i}^{}\frac{X_i}{\varrho_i}} ; \quad \varrho_m = \frac{1}{\sum_{i}^{}\frac{x_i}{\varrho_{m,i}}}.
\end{equation}
%
\subsubsection{Ideal gases}
An ideal gas is defined as a gas whose molecules are spaced so far apart that the behavior of a molecule is not influenced by the presence of other molecules.
This assumption is usually valid at low pressures and high temperatures. The ideal gas law states that, for one gas:
%
\begin{equation}
p = \varrho \frac{RT}{M} ; \quad p= \varrho_m RT.
\end{equation}
%
Using the assumption of ideal gases and either Dalton's law or Amagat's law lead to the density of the mixture, $\varrho$, as:
%
\begin{equation}
\varrho = \frac{p}{RT} \sum_{i}^{}M_i x_i ; \quad \varrho_m = \frac{p}{RT}.
\end{equation}
%
\subsection{Available Models}
A list of all available models can be found
in the Doxygen documentation at
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doc/handbook/SVG/dalton2.svg 0 → 100644
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