Commit a566097d authored by Philipp Nuske's avatar Philipp Nuske
Browse files

- administered some formula cosmetics to the models section of the

  handbook


git-svn-id: svn://svn.iws.uni-stuttgart.de/DUMUX/dumux/trunk@7880 2fb0f335-1f38-0410-981e-8018bf24f1b0
parent 1cfe3114
......@@ -17,7 +17,7 @@ The advantage of the FE method is that unstructured grids can be used, while the
f(\tilde u(x^k_{ij})) \cdot \mathbf n^k_{ij} \: |e^k_{ij}| \qquad \textrm{with} \qquad \tilde u(x^k_{ij}) = \sum_i N_i(x^k_{ij}) \cdot \hat u_i .
\end{equation}
In the following, the discretization of the balance equation is going to be derived. From the \textsc{reynolds}s transport theorem follows the general balance equation:
In the following, the discretization of the balance equation is going to be derived. From the \textsc{Reynolds} transport theorem follows the general balance equation:
\begin{equation}
\underbrace{\int_G \frac{\partial}{\partial t} \: u \: dG}_{1} + \underbrace{\int_{\partial G} (\mathbf{v} u + \mathbf w) \cdot \textbf n \: d\varGamma}_{2} = \underbrace{\int_G q \: dG}_{3}
......@@ -26,8 +26,7 @@ In the following, the discretization of the balance equation is going to be deri
\begin{equation}
f(u) = \int_G \frac{\partial u}{\partial t} \: dG + \int_{G} \nabla \cdot \underbrace{\left[ \mathbf{v} u + \mathbf w(u)\right] }_{F(u)} \: dG - \int_G q \: dG = 0
\end{equation}
where term 1 describes the changes of entity $u$ within a control volume over time, term 2 the advective, diffusive and dispersive fluxes over the interfaces of the control volume and term 3 is the source and sink term. $G$ denotes the model domain and $F(u) = F(\mathbf v, p) = F(\mathbf v(x,t), p(x,t))$.\\
where term 1 describes the changes of entity $u$ within a control volume over time, term 2 the advective, diffusive and dispersive fluxes over the interfaces of the control volume and term 3 is the source and sink term. $G$ denotes the model domain and $F(u) = F(\mathbf v, p) = F(\mathbf v(x,t), p(x,t))$.
Like the FE method, the BOX-method follows the principle of weighted residuals. In the function $f(u)$ the unknown $u$ is approximated by discrete values at the nodes of the FE mesh $\hat u_i$ and linear basis functions $N_i$ yielding an approximate function $f(\tilde u)$. For $u\in \lbrace \mathbf v, p, x^\kappa \rbrace$ this means
......@@ -72,14 +71,13 @@ Application of the principle of weighted residuals, meaning the multiplication o
\begin{equation}
\int_G W_j \cdot \varepsilon \: \overset {!}{=} \: 0 \qquad \textrm{with} \qquad \sum_j W_j =1
\end{equation}
yields the following equation:
\begin{equation}
\int_G W_j \frac{\partial \tilde u}{\partial t} \: dG + \int_G W_j \cdot \left[ \nabla \cdot F(\tilde u) \right] \: dG - \int_G W_j \cdot q \: dG = \int_G W_j \cdot \varepsilon \: dG \: \overset {!}{=} \: 0 .
\end{equation}
Then, the chain rule and the \textsc{green-gaussian} integral theorem are applied.
Then, the chain rule and the \textsc{Green-Gaussian} integral theorem are applied.
\begin{equation}
\int_G W_j \frac{\partial \sum_i N_i \hat u_i}{\partial t} \: dG + \int_{\partial G} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_G + \int_G \nabla W_j \cdot F(\tilde u) \: dG - \int_G W_j \cdot q \: dG = 0
......@@ -92,11 +90,10 @@ A mass lumping technique is applied by assuming that the storage capacity is red
0 &i \neq j\\
\end{cases}
\end{equation}
where $V_i$ is the volume of the FV box $B_i$ associated with node i. The application of this assumption in combination with $\int_G W_j \:q \: dG = V_i \: q$ yields
\begin{equation}
V_i \frac{\partial \hat u_i}{\partial t} + \int_{\partial G} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_G + \int_G \nabla W_j \cdot F(\tilde u) \: dG- V_i \cdot q = 0
V_i \frac{\partial \hat u_i}{\partial t} + \int_{\partial G} \left[ W_j \cdot F(\tilde u)\right] \cdot \mathbf n \: d\varGamma_G + \int_G \nabla W_j \cdot F(\tilde u) \: dG- V_i \cdot q = 0 \, .
\end{equation}
Defining the weighting function $W_j$ to be piecewisely constant over a control volume box $B_i$
......
......@@ -55,7 +55,7 @@ $\varrho_{\alpha}$ & mass density of phase $\alpha$ & $h_\alpha$ & specific enth
$k_{\text{r}\alpha}$ & relative permeability & $c_\text{s}$ & specific heat enthalpy \\
$\mu_\alpha$ & phase viscosity & $\lambda_\text{pm}$ & heat conductivity \\
$D_\alpha^\kappa$ & diffusivity of component $\kappa$ in phase $\alpha$ & $q^h$ & heat source term \\
$v_\alpha$ & Darcy velocity & $v_{a,\alpha}$ & advective velocity
$\boldsymbol{v}_\alpha$ & Darcy velocity & $\boldsymbol{v}_{a,\alpha}$ & advective velocity
\end{tabular}
......@@ -67,21 +67,20 @@ us to drop source/sink terms for describing the mass transfer between
phases. Then, the
molar mass balance can be written as:
%
\begin{eqnarray}
\begin{multline}
\label{A3:eqmass1}
&& \phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha}
x_\alpha^\kappa S_\alpha )}{\partial t} \nonumber
\phi \frac{\partial (\sum_\alpha \varrho_{\text{mol}, \alpha}
x_\alpha^\kappa S_\alpha )}{\partial t}
- \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
\alpha}}{\mu_\alpha} \varrho_{\text{mol}, \alpha}
x_\alpha^\kappa K (\grad p_\alpha -
\varrho_{\alpha} \boldsymbol{g}) \right) \nonumber \\
\varrho_{\alpha} \boldsymbol{g}) \right) \\
%
%
\nonumber \\
%
&& - \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mol},
\alpha} \grad x_\alpha^\kappa \right) \nonumber
- \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mol},
\alpha} \grad x_\alpha^\kappa \right)
- q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}.
\end{eqnarray}
\end{multline}
The component mass balance can also be written in terms of mass fractions
by replacing molar densities by mass densities and mole by mass fractions.
......@@ -92,46 +91,42 @@ C^\kappa = \sum_\alpha \phi S_\alpha \varrho_{\text{mass},\alpha} X_\alpha^\kapp
\end{displaymath}
Using this definition, the component mass balance is written as:
\begin{eqnarray}
\begin{multline}
\label{A3:eqmass2}
&& \frac{\partial C^\kappa}{\partial t} =
\frac{\partial C^\kappa}{\partial t} =
\sum\limits_\alpha \Div \left( \frac{k_{\text{r}
\alpha}}{\mu_\alpha} \varrho_{\text{mass}, \alpha}
X_\alpha^\kappa K (\grad p_\alpha +
\varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right) \nonumber \\
\varrho_{\text{mass}, \alpha} \boldsymbol{g}) \right) \\
%
\nonumber \\
%
&& + \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mass},
\alpha} \grad X_\alpha^\kappa \right) \nonumber
+ \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mass},
\alpha} \grad X_\alpha^\kappa \right)
+ q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}.
\end{eqnarray}
\end{multline}
In the case of non-isothermal systems, we further have to balance the
thermal energy. We assume fully reversible processes, such that entropy
is not needed as a model parameter. Furthermore, we neglect
dissipative effects and the heat transport due to molecular
diffusion. The heat balance can then be
diffusion. The energy balance can then be
formulated as:
%
\begin{eqnarray}
\begin{multline}
\label{A3:eqenergmak1}
&& \phi \frac{\partial \left( \sum_\alpha \varrho_{\alpha}
\phi \frac{\partial \left( \sum_\alpha \varrho_{\alpha}
u_\alpha S_\alpha \right)}{\partial t} + \left( 1 -
\phi \right) \frac{\partial \varrho_{\text{s}} c_{\text{s}}
T}{\partial t} \nonumber
T}{\partial t}
- \Div \left( \lambda_{\text{pm}} \grad T \right)
\nonumber \\
%
\nonumber \\
%
&& - \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
\\
- \sum\limits_\alpha \Div \left( \frac{k_{\text{r}
\alpha}}{\mu_\alpha} \varrho_{\alpha} h_\alpha
K \left( \grad p_\alpha - \varrho_{\alpha}
\boldsymbol{g} \right) \right) \nonumber
\boldsymbol{g} \right) \right)
- q^h \; = \; 0.
\end{eqnarray}
\end{multline}
In order to close the system, supplementary constraints for capillary pressure, saturations and mole
fractions are needed, \cite{A3:helmig:1997}.
......
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