Kharon theory manual

Conservation of Mass

Figure 1: Conservation of mass for an interior cell

Steady one-dimensional conservation of mass (of phase q) is given through the following relation

$\nabla\cdot(\varepsilon \alpha_q \rho_q u_q) = \gamma_{pq} + s_{\text{mass},q}$

The above equation is discretized for an interior cell, shown in Figure 1, by volume integration over cell P.

$\Leftrightarrow \iiint\limits_V [\nabla\cdot(\varepsilon \alpha_q \rho_q u_q)]\, dV = \iiint\limits_V (\gamma_{pq} + s_{\text{mass},q})\, dV$

The divergence term on the left-hand side of the equation is transformed into a surface integral over the cell faces using the divergence theorem.

$\Leftrightarrow \oint\limits_A [(\varepsilon \alpha_q \rho_q u_q)\cdot n]\, dA = \iiint\limits_V (\gamma_{pq} + s_{\text{mass},q})\, dV$

Note that the first integral defines a closed surface integral over the cell faces, instead of what the notation normally stands for (which is a closed path integral). The current implementation of "Template:TeX" doesn't support the actual notation needed.

Throughout this document upwind discretization is used to evaluate the face mass flow rates, even though any discretization scheme could be chosen. The final discretized form reduces to

$\Leftrightarrow (\varepsilon \alpha_q \rho_q u_q A)_\text{e} - (\varepsilon \alpha_q \rho_q u_q A)_\text{w} = F_{\text{e},q} - F_{\text{w},q} = \Gamma_{pq} + S_{\text{mass},q}$

where:

$\alpha_q$	volume fraction of phase q [-],
$\gamma_{pq}$	volumetric mass transfer rate from phase p to q [kg/m3s],
$\Gamma_{pq}$	mass transfer rate from phase p to q [kg/s],
$\varepsilon$	porosity, fluid fraction of cell volume [-],
$\rho_q$	density of phase q [kg/m3],
$A$	face area [m2],
$F$	face mass flow rate [kg/s],
$n$	face normal pointing out of the cell (1 for east face, -1 for west face),
$s_{\text{mass},q}$	volumetric mass source to phase q [kg/m3s],
$S_{\text{mass},q}$	mass source to phase q [kg/s],
$u_q$	velocity of phase q [m/s],
subscript e	east face (the face in the negative x-direction),
subscript w	west face (the face in the positive x-direction),
subscript q	current phase for which the equation is written (1 = primary, 2 = secondary),
subscript p	the other phase (p = 2 for q = 1 and p = 1 for q = 2),
subscript pq	indicates exchange between phases (e.g. $\Gamma_{pq}$ mass transfer from phase p to q).

Conservation of Momentum

Figure 2: Conservation of momentum for an interior cell

Steady one-dimensional conservation of momentum (of phase q) is solved from the following relation

$\nabla\cdot(\varepsilon \alpha_q \rho_q u_q u_q) = \nabla\cdot(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q) + [\text{max}(\gamma_{pq},0)u_p - \text{max}(-\gamma_{pq},0)u_q]$

$- \varepsilon \alpha_q \nabla p + \varepsilon \alpha_q \rho_q g - \frac{1}{2} \varepsilon^3 \alpha_q \rho_q C \left| u_q \right| u_q - k(u_q - u_p) + s_{\text{mom},q}$

The above equation is discretized for an interior cell, shown in Figure 2, by volume integration over cell P. The diffusion term has been moved to the left-hand side, since it will be developed together with the divergence of momentum.

$\Leftrightarrow \iiint\limits_V [\nabla\cdot(\varepsilon \alpha_q \rho_q u_q u_q) - \nabla\cdot(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q)]\, dV = \iiint\limits_V [\text{max}(\gamma_{pq},0)u_p - \text{max}(-\gamma_{pq},0)u_q]\, dV$

$+ \iiint\limits_V (-\varepsilon \alpha_q \nabla p + \varepsilon \alpha_q \rho_q g - \frac{1}{2} \varepsilon^3 \alpha_q \rho_q C \left| u_q \right| u_q - k(u_q - u_p) + s_{\text{mom},q})\, dV$

As with the conservation of mass equation above, the divergence terms on the left-hand side of the equation are transformed into surface integrals over the cell faces using the divergence theorem.

$\Leftrightarrow \oint\limits_A [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA = [\text{max}(\Gamma_{pq},0)u_p - \text{max}(-\Gamma_{pq},0)u_q]$

$-\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \frac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K(u_q - u_p) + S_{\text{mom},q}$

Developing the left-hand side terms first

$\oint\limits_A [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

$= (\varepsilon \alpha_q \rho_q u_q u_q A)_{\text{e}} - (\varepsilon \alpha_q \rho_q u_q u_q A)_{\text{w}} - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x} A)_{\text{e}} + (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x} A)_{\text{w}}$

$= F_{\text{e},q} u_{\text{e},q} + F_{\text{w},q} u_{\text{w},q} - (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}} (u_{\text{e},q} - u_{\text{P},q}) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}} (u_{\text{P},q} - u_{\text{w},q})$

Adopting upwind discretization for the momentum fluxes on the cell faces

$F_{\text{e},q} u_{\text{e},q} = \text{max}(F_{\text{e},q},0)u_{\text{P},q} - \text{max}(-F_{\text{e},q},0)u_{\text{E},q}$

$F_{\text{w},q} u_{\text{w},q} = \text{max}(F_{\text{w},q},0)u_{\text{W},q} - \text{max}(-F_{\text{w},q},0)u_{\text{P},q}$

and rearranging terms

$\oint\limits_A [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

Conservation of Total Enthalpy

Figure 3: Conservation of total enthalpy for an interior cell