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

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

(1)

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

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

(2)

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.

$\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset [(\varepsilon \alpha_q \rho_q u_q)\cdot n]\, dA = \iiint\limits_V (\gamma_{pq} + s_{\text{mass},q})\, dV$

(3)

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

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

(4)

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

$\nabla\cdot(\varepsilon \alpha_q \rho_q u_q u_q) = \nabla\cdot(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q)$

(5)

$+ [\text{max}(\gamma_{pq},0)u_p - \text{max}(-\gamma_{pq},0)u_q] - \varepsilon \alpha_q \nabla p + \varepsilon \alpha_q \rho_q g$

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

Equation 5 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.

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

(6)

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

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.

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

(7)

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

Developing the left-hand side terms first

$\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

(8)

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

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

Adopting upwind discretization for the momentum fluxes on the cell faces

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

(9)

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

(10)

and rearranging terms

$\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

(11)

$=[\text{max}(F_{\text{e},q},0) + \text{max}(-F_{\text{w},q},0) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}} + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}}]u_{\text{P},q}$

$-[\text{max}(-F_{\text{e},q},0) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}}]u_{\text{E},q}$

$-[\text{max}(F_{\text{w},q},0) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}}]u_{\text{W},q}$

Denoting $(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}} = D_{\text{e},q}$ and $(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}} = D_{\text{w},q}$ and modifying the central coefficient slightly, since $\text{max}(F_{\text{e},q},0) = F_{\text{e},q} + \text{max}(-F_{\text{e},q},0)$ and $\text{max}(-F_{\text{w},q},0) = -F_{\text{w},q} + \text{max}(F_{\text{w},q},0)$, Equation 11 reduces to

$\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

(12)

$= [\text{max}(-F_{\text{e},q},0) + D_{\text{e},q} + \text{max}(F_{\text{w},q},0) + D_{\text{w},q} + (F_{\text{e},q} - F_{\text{w},q})]u_{\text{P},q}$

$-[\text{max}(-F_{\text{e},q},0) + D_{\text{e},q}]u_{\text{E},q} - [\text{max}(F_{\text{w},q},0) + D_{\text{w},q}]u_{\text{W},q}$

Identifying the coefficients of neighboring velocities

$a_{\text{e},q} = \text{max}(-F_{\text{e},q},0) + D_{\text{e},q}$

(13)

$a_{\text{w},q} = \text{max}(F_{\text{w},q},0) + D_{\text{w},q}$

(14)

leads to a general form

$\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset [(\varepsilon \alpha_q \rho_q u_q u_q)\cdot n - (\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x})\cdot n]\, dA$

(15)

$= [a_{\text{e},q} + a_{\text{w},q} + (F_{\text{e},q} - F_{\text{w},q})]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}$

Substituting the results of Equation 15 into Equation 7 and continuing with the development

$\Rightarrow[a_{\text{e},q} + a_{\text{w},q} + (F_{\text{e},q} - F_{\text{w},q})]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}$

(16)

$= [\text{max}(\Gamma_{pq},0)u_p - \text{max}(-\Gamma_{pq},0)u_q] - \varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g$

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

At this point, the conservation of mass equation times cell velocity, $u_{\text{P},q}$, is subtracted from Equation 16

$\Rightarrow[a_{\text{e},q} + a_{\text{w},q} + (F_{\text{e},q} - F_{\text{w},q}) - (F_{\text{e},q} - F_{\text{w},q})]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}$

(17)

$= [\text{max}(\Gamma_{pq},0)u_p - \text{max}(-\Gamma_{pq},0)u_q] - (\Gamma_{pq} + S_{\text{mass},q})u_{\text{P},q}$

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

Splitting the mass source term

$\Leftrightarrow[a_{\text{e},q} + a_{\text{w},q}]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}$

(19)

$= [\text{max}(\Gamma_{pq},0)u_{\text{P},p} - \text{max}(-\Gamma_{pq},0)u_{\text{P},q}]$

$-[\text{max}(\Gamma_{pq},0) - \text{max}(-\Gamma_{pq},0) + \text{max}(S_{\text{mass},q},0) - \text{max}(-S_{\text{mass},q},0)]u_{\text{P},q}$

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

$\Leftrightarrow[a_{\text{e},q} + a_{\text{w},q} + \text{max}(\Gamma_{pq},0) + \text{max}(S_{\text{mass},q},0) + \text{max}(-\Gamma_{pq},0)]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}$

(19)

$= [\text{max}(-\Gamma_{pq},0) + \text{max}(-S_{\text{mass},q},0)]u_q + \text{max}(\Gamma_{pq},0)u_p$

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

Since the right-hand side contains only references to the central cell, subscripts P have been dropped from right-hand side to simplify the notation. Note that the diagonal element of the coefficient matrix contains the split terms, $\text{max}(\Gamma_{pq},0) + \text{max}(S_{\text{mass},q},0)$, resulting from the subtraction of the mass equation times cell velocity, while the counterparts, $[\text{max}(-\Gamma_{pq},0) + \text{max}(-S_{\text{mass},q},0)]u_q$, remain on the right-hand side. These are not considered as momentum source terms, as such, since they are only artifacts of the derivation procedure.

Finally, denoting the central coefficient by $a_P = a_{\text{e},q} + a_{\text{w},q} + \text{max}(\Gamma_{pq},0) + \text{max}(S_{\text{mass},q},0) + \text{max}(-\Gamma_{pq},0)$ and casting Equation 19 into a vector form (over the phases) the final form, in which the equations are actually solved, is obtained. For simplicity’s sake, the equation is written only for the first 3 cells (c1, c2 and c3).

Conservation of Total Enthalpy

Figure 3: Conservation of total enthalpy for an interior cell