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36 lines
2.8 KiB
TeX
36 lines
2.8 KiB
TeX
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% This file has been autogenerated from the LaTeX part of the %
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% doxygen documentation; DO NOT EDIT IT! Change the model's .hh %
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% file instead!! %
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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A generic compositional multi-\/phase model using primary-\/variable switching.
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This model assumes a flow of $M \geq 1$ fluid phases $\alpha$, each of which is assumed to be a mixture $N \geq M$ chemical species (denoted by the upper index $\kappa$).
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By default, the standard multi-\/phase Darcy approach is used to determine the velocity, i.\-e. \[ \mathbf{v}_\alpha = - \frac{k_{r\alpha}}{\mu_\alpha} \mathbf{K} \left(\text{grad}\, p_\alpha - \varrho_{\alpha} \mathbf{g} \right) \;, \] although the actual approach which is used can be specified via the {\ttfamily Velocity\-Module} property. For example, the velocity model can by changed to the Forchheimer approach by
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\begin{lstlisting}[style=eWomsCode]
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SET_TYPE_PROP(MyProblemTypeTag, VelocityModule,
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Dumux::BoxForchheimerVelocityModule<TypeTag>);
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\end{lstlisting}
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The core of the model is the conservation mass of each component by means of the equation \[ \sum_\alpha \frac{\partial\;\phi c_\alpha^\kappa S_\alpha }{\partial t} - \sum_\alpha \text{div} \left\{ c_\alpha^\kappa \mathbf{v}_\alpha \right\} - q^\kappa = 0 \;. \]
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To determine the quanties that occur in the equations above, this model uses {\itshape flash calculations}. A flash solver starts with the total mass or molar mass per volume for each component and, calculates the compositions, saturations and pressures of all phases at a given temperature. For this the flash solver has to use some model assumptions internally. (Often these are the same primary variable switching or N\-C\-P assumptions as used by the other fully implicit compositional multi-\/phase models provided by e\-Woms.)
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Using flash calculations for the flow model has some disadvantages\-:
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\begin{itemize}
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\item The accuracy of the flash solver needs to be sufficient to calculate the parital derivatives using numerical differentiation which are required for the Newton scheme.
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\item Flash calculations tend to be quite computationally expensive and are often numerically unstable.
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\end{itemize}
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It is thus adviced to increase the target tolerance of the Newton scheme or a to use type for scalar values which exhibits higher precision than the standard {\ttfamily double} (e.\-g. {\ttfamily quad}) if this model ought to be used.
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The model uses the following primary variables\-:
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\begin{itemize}
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\item The total molar concentration of each component\-: $c^\kappa = \sum_\alpha S_\alpha x_\alpha^\kappa \rho_{mol, \alpha}$
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\item The absolute temperature \$\-T\$ in Kelvins if the energy equation enabled.
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\end{itemize}
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