diff --git a/doc/handbook/5_models.tex b/doc/handbook/5_models.tex index 8f4809ce6af70d40500a04b943120091018fa977..6a0422c519cf828eaac2e6b86062983746a841ba 100644 --- a/doc/handbook/5_models.tex +++ b/doc/handbook/5_models.tex @@ -1,6 +1,8 @@ \section{Models} Here the basic definitions, the general models concept, and a list of -models available in \Dumux are given. +models available in \Dumux are given. The actual differential equations +can be found in the localresiduals (see doxygen documentation of the +model's \texttt{LocalResidual} class). \subsection{Basic Definitions and Assumptions} The basic definitions and assumptions are made, using the example @@ -134,7 +136,7 @@ Dalton's law assumes that the gases in the mixture are non-interacting (with eac 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}). +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 @@ -154,7 +156,7 @@ or for an arbitrary number of gases: \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: +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. @@ -171,7 +173,7 @@ As an example, if two volumes of gas A and B at equal pressure are mixed, the pr % 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}}} = +\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} % @@ -182,7 +184,7 @@ or for an arbitrary number of gases: \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. +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} @@ -193,7 +195,7 @@ Using the assumption of ideal gases and either Dalton's law or Amagat's law lead % \begin{equation} \varrho = \frac{p}{RT} \sum_{i}^{}M_i x_i ; \quad \varrho_m = \frac{p}{RT}. -\end{equation} +\end{equation} % \subsection{Available Models} A list of all available models can be found @@ -203,7 +205,7 @@ The documentation includes a detailed description for every model. \subsubsection{Time discretization} -Our systems of partial differential equations are discretized in space and in time. +Our systems of partial differential equations are discretized in space and in time. Let us consider the general case of a balance equation of the following form \begin{equation}\label{eq:generalbalance}