From ec1df1978d82390c6cc3a7588105cbeaebeb67aa Mon Sep 17 00:00:00 2001 From: Jochen Fritz Date: Thu, 4 Jun 2009 13:01:21 +0000 Subject: [PATCH] --- doc/handbook/models.tex | 25 +++++++++++++++++++++++++ 1 file changed, 25 insertions(+) diff --git a/doc/handbook/models.tex b/doc/handbook/models.tex index 4dd639fea..aa4c29442 100644 --- a/doc/handbook/models.tex +++ b/doc/handbook/models.tex @@ -82,6 +82,31 @@ molar mass balance can be written as: - q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}. \end{eqnarray} +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. +To obtain a single conserved quantity in the temporal derivative, the total +concentration, representing the mass of one component per unit volume, is defined as +\begin{displaymath} +C_\alpha^\kappa = \phi S_\alpha \varrho_{\text{mass},\alpha} X_\alpha^\kappa \; . +\end{displaymath} +Using this definition, the component mass balance is written as: + +\begin{eqnarray} + \label{A3:eqmass2} + && \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 \\ + % + \nonumber \\ + % + && + \sum\limits_\alpha \Div \left( \tau \phi S_\alpha D_\alpha^\kappa \varrho_{\text{mass}, + \alpha} \grad X_\alpha^\kappa \right) \nonumber + + q^\kappa = 0, \qquad \kappa \in \{\text{w,a,c}\}. +\end{eqnarray} + + 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