Difference between revisions of "Kharon theory manual"

Revision as of 14:52, 19 April 2018

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

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

(1)

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

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

(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.

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

(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

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

(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

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

(5)

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

- \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C \left| u_q \right| u_q - k\left(u_q - u_p\right) + 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.

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

(6)

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

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

(7)

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

Developing the left-hand side terms first

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

(8)

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

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

Adopting upwind discretization for the momentum fluxes on the cell faces

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

(9)

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

(10)

and rearranging terms

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

(11)

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

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

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

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

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

(12)

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

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

Identifying the coefficients of neighboring velocities

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

(13)

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

(14)

leads to a general form

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

(15)

= \left[a_{\text{e},q} + a_{\text{w},q} + \left(F_{\text{e},q} - F_{\text{w},q}\right)\right]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

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

(16)

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

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

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

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

(17)

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

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

Splitting the mass source term

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

(18)

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

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

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

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

(19)

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

-\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + 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, $\text{max}\left(\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right)$ , resulting from the subtraction of the mass equation times cell velocity, while the counterparts, $\left[\text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(-S_{\text{mass},q},0\right)\right]u_q$ , 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 $a_P = a_{\text{e},q} + a_{\text{w},q} + \text{max}\left(\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right) + \text{max}\left(-\Gamma_{pq},0\right)$ 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).

$\begin{bmatrix} a_{\text{P,c1},1} + K_{\text{c1}} & -K_{\text{c1}} & -a_{\text{e,c1},1} & 0 & \cdots & 0 \\ -K_{\text{c1}} & a_{\text{P,c1},2} + K_{\text{c1}} & 0 & -a_{\text{e,c1},2} & \ddots & \vdots \\ -a_{\text{w,c2},1} & 0 & a_{\text{P,c2},1} + K_{\text{c2}} & -K_{\text{c2}} & -a_{\text{e,c2},1} & 0 \\ 0 & -a_{\text{w,c2},2} & -K_{\text{c2}} & a_{\text{P,c2},2} + K_{\text{c2}} & 0 & -a_{\text{e,c2},2} \\ \vdots & \ddots & -a_{\text{w,c3},1} & 0 & a_{\text{P,c3},1} + K_{\text{c3}} & -K_{\text{c3}} \\ 0 & \cdots & 0 & -a_{\text{w,c3},2} & -K_{\text{c3}} & a_{\text{P,c3},2} + K_{\text{c3}} \end{bmatrix} \begin{pmatrix} u_{\text{c1,1}}\\ u_{\text{c1,2}}\\ u_{\text{c2,1}}\\ u_{\text{c2,2}}\\ u_{\text{c3,1}}\\ u_{\text{c3,2}} \end{pmatrix} = \begin{pmatrix} S_{\text{mom,c1,1}}\\ S_{\text{mom,c1,2}}\\ S_{\text{mom,c2,1}}\\ S_{\text{mom,c2,2}}\\ S_{\text{mom,c3,1}}\\ S_{\text{mom,c3,2}} \end{pmatrix}$

(20)

where:

S_{\text{mom},\text{c1},q} = \Big\{\big[\text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(-S_{\text{mass},q},0\right)\big]u_q + \text{max}\left(\Gamma_{pq},0\right)u_p

- \varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}\Big\}_{\text{c1}}

and

$\alpha_q$		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],
$\delta x$		distance between two adjacent cell centers [m],
$\varepsilon$		porosity, fluid fraction of cell volume [-],
$\mu_{\text{eff},q}$		effective viscosity of phase q [Ns/m²],
$\rho_q$		density of phase q [kg/m³],
$A$		face area [m²],
$C$		inertial loss coefficient [1/m],
$F$		face mass flow rate [kg/s],
$g$		acceleration of gravity [m/s²],
$k$		volumetric interphase drag coefficient [kg/m³s],
$K$		interphase drag coefficient [kg/s],
$n$		face normal pointing out of the cell (1 for east face, -1 for west face),
$p$		pressure [Pa],
$s_{\text{mass},q}$		volumetric mass source to phase q [kg/m³s],
$S_{\text{mass},q}$		mass source to phase q [kg/s],
$s_{\text{mom},q}$		volumetric momentum source to phase q [N/m³],
$S_{\text{mom},q}$		momentum source to phase q [N],
$u_q$		velocity of phase q [m/s],
$V$		cell volume [m³],
subscript c1		refers to first cell,
subscript c2		refers to second cell,
subscript c3		refers to third cell,
subscript e		east face (the face in the negative x-direction),
subscript E		east neighbor (the cell in the negative x-direction),
subscript w		west face (the face in the positive x-direction),
subscript W		west neighbor (the cell in the positive x-direction),
subscript P		central cell (the cell for which the equation is written),
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 Total Enthalpy

Figure 3: Conservation of total enthalpy for an interior cell

Conservation of total enthalpy (of phase q) can be written as

$\frac{\partial \left(\varepsilon \alpha_q \rho_q h_{0,q}\right)}{\partial t} + \nabla\cdot\left(\varepsilon \alpha_q \rho_q h_{0,q} u_q\right)$

(21)

= \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right) + \left[\text{max}\left(\gamma_{pq},0\right) h_{0,\text{cre}} - \text{max}\left(-\gamma_{pq},0\right) h_{0,\text{des}}\right]

- \nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \varepsilon \alpha_q \frac{\partial p}{\partial t} + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q \right) + s_{h,q}

Disregarding the transient terms and moving the diffusion term to the left-hand side leads to

$\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_{0,q} u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right)$

(22)

= \left[\text{max}\left(\gamma_{pq},0\right) h_{0,\text{cre}} - \text{max}\left(-\gamma_{pq},0\right) h_{0,\text{des}}\right] - \nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q \right) + s_{h,q}

Total enthalpy is defined as the sum of specific enthalpy, kinetic energy and potential energy

$h_0 = h + \tfrac{1}{2} \left| u_q \right|^2 - \mathbf{g} \cdot \mathbf{r}$

(23)

where $\mathbf{g} \cdot \mathbf{r}$ is the dot product of gravity vector, $\mathbf{g}$ , and location vector, $\mathbf{r}$ , a vector pointing from the reference level to the evaluated location. The total enthalpies related to phase change can be defined as

$\begin{align} & h_{0,\text{cre}} = h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\\ & h_{0,\text{des}} = h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}} \end{align}$

(24)

In other words, when mass is transferred to phase q (“created”), it brings with it the saturation enthalpy of phase q, $h_{\text{P,sat,}q}$ , and the kinetic energy of phase p, $\tfrac{1}{2} \left| u_p \right|^2_{\text{P}}$ . When mass is transferred from phase q (“destroyed”), it leaves with the enthalpy and kinetic energy of phase q, $h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}}$ . It is chosen that the potential energy associated with phase change is evaluated at the center of the cell, $\left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}$ .

Substituting Equation 23 and Equation 24 to Equation 22 and rearranging terms

$\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right)$

(25)

= \left[\text{max}\left(\gamma_{pq},0\right) \left(h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\right) - \text{max}\left(-\gamma_{pq},0\right) \left(h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}} \right)\right]

-\nabla\cdot\left(\tfrac{1}{2}\varepsilon \alpha_q \rho_q \left| u_q \right|^2 u_q\right) +\nabla\cdot\left(\varepsilon \alpha_q \rho_q \left( \mathbf{g} \cdot \mathbf{r} \right)u_q\right) -\nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q \right) + s_{h,q}

At this point, an approximation is introduced: the diffusion term is expressed relative to the gradient of enthalpy, instead of temperature, as follows. This approximation is considered to be sufficiently accurate for small enthalpy differences.

$\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \nabla h_q \right)$

(26)

= \left[\text{max}\left(\gamma_{pq},0\right) \left(h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\right) - \text{max}\left(-\gamma_{pq},0\right) \left(h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}} \right)\right]

-\nabla\cdot\left(\tfrac{1}{2}\varepsilon \alpha_q \rho_q \left| u_q \right|^2 u_q\right) +\nabla\cdot\left(\varepsilon \alpha_q \rho_q \left( \mathbf{g} \cdot \mathbf{r} \right)u_q\right) -\nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q \right) + s_{h,q}

Equation 26 is discretized for an interior cell, shown in Figure 3, by volume integration over cell P.

Difference between revisions of "Kharon theory manual"

Revision as of 14:52, 19 April 2018

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@@ Line 9: / Line 9: @@
 <equation id="eqn:MassConservation">
-{{NumBlk|:|<math>\nabla\cdot(\varepsilon \alpha_q \rho_q u_q) = \gamma_{pq} + s_{\text{mass},q}</math>|<xr id="eqn:MassConservation" nolink />}}
+{{NumBlk|:|<math>\nabla\cdot\left(\varepsilon \alpha_q \rho_q u_q\right) = \gamma_{pq} + s_{\text{mass},q}</math>|<xr id="eqn:MassConservation" nolink />}}
 </equation>
@@ Line 15: / Line 15: @@
 <equation id="eqn:MassConservation2">
-{{NumBlk|:|<math>\Leftrightarrow \iiint\limits_V [\nabla\cdot(\varepsilon \alpha_q \rho_q u_q)]\, dV = \iiint\limits_V (\gamma_{pq} + s_{\text{mass},q})\, dV</math>|<xr id="eqn:MassConservation2" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \iiint\limits_V \left[\nabla\cdot\left(\varepsilon \alpha_q \rho_q u_q\right)\right]\, dV = \iiint\limits_V \left(\gamma_{pq} + s_{\text{mass},q}\right)\, dV</math>|<xr id="eqn:MassConservation2" nolink />}}
 </equation>
@@ Line 21: / Line 21: @@
 <equation id="eqn:MassConservation3">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MassConservation3" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q\right)\cdot n\right]\, dA = \iiint\limits_V \left(\gamma_{pq} + s_{\text{mass},q}\right)\, dV</math>|<xr id="eqn:MassConservation3" nolink />}}
 </equation>
@@ Line 27: / Line 27: @@
 <equation id="eqn:DiscretizedMassConservation">
-{{NumBlk|:|<math>\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}</math>|<xr id="eqn:DiscretizedMassConservation" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \left(\varepsilon \alpha_q \rho_q u_q A\right)_\text{e} - \left(\varepsilon \alpha_q \rho_q u_q A\right)_\text{w} = F_{\text{e},q} - F_{\text{w},q} = \Gamma_{pq} + S_{\text{mass},q}</math>|<xr id="eqn:DiscretizedMassConservation" nolink />}}
 </equation>
@@ Line 90: / Line 90: @@
 <equation id="eqn:MomentumConservation">
-{{NumBlk|:|<math>\nabla\cdot(\varepsilon \alpha_q \rho_q u_q u_q) = \nabla\cdot(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q)</math>|<xr id="eqn:MomentumConservation" nolink />}}
+{{NumBlk|:|<math>\nabla\cdot\left(\varepsilon \alpha_q \rho_q u_q u_q\right) = \nabla\cdot\left(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q\right)</math>|<xr id="eqn:MomentumConservation" nolink />}}
-::::<math>+ [\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</math>
+::::<math>+ \left[\text{max}\left(\gamma_{pq},0\right)u_p - \text{max}\left(-\gamma_{pq},0\right)u_q\right] - \varepsilon \alpha_q \nabla p + \varepsilon \alpha_q \rho_q g</math>
-::::<math>- \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}</math>
+::::<math>- \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C \left| u_q \right| u_q - k\left(u_q - u_p\right) + s_{\text{mom},q}</math>
 </equation>
@@ Line 99: / Line 99: @@
 <equation id="eqn:MomentumConservation2">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MomentumConservation2" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \iiint\limits_V \left[\nabla\cdot\left(\varepsilon \alpha_q \rho_q u_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \mu_{\text{eff},q} \nabla u_q\right)\right]\, dV = \iiint\limits_V \left[\text{max}\left(\gamma_{pq},0\right)u_p - \text{max}\left(-\gamma_{pq},0\right)u_q\right]\, dV</math>|<xr id="eqn:MomentumConservation2" nolink />}}
-::::<math> + \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 </math>
+::::<math> + \iiint\limits_V \left(-\varepsilon \alpha_q \nabla p + \varepsilon \alpha_q \rho_q g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C \left| u_q \right| u_q - k\left(u_q - u_p\right) + s_{\text{mom},q}\right)\, dV </math>
 </equation>
@@ Line 106: / Line 106: @@
 <equation id="eqn:MomentumConservation3">
-{{NumBlk|:|<math>\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]</math>|<xr id="eqn:MomentumConservation3" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x}\right)\cdot n\right]\, dA = \left[\text{max}\left(\Gamma_{pq},0\right)u_p - \text{max}\left(-\Gamma_{pq},0\right)u_q\right]</math>|<xr id="eqn:MomentumConservation3" nolink />}}
-::::<math> -\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} </math>
+::::<math> -\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q} </math>
 </equation>
@@ Line 113: / Line 113: @@
 <equation id="eqn:MomentumConservation4">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MomentumConservation4" nolink />}}
+{{NumBlk|:|<math>\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:MomentumConservation4" nolink />}}
-::::<math> = (\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}} </math>
+::::<math> = \left(\varepsilon \alpha_q \rho_q u_q u_q A\right)_{\text{e}} - \left(\varepsilon \alpha_q \rho_q u_q u_q A\right)_{\text{w}} - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x} A\right)_{\text{e}} + \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x} A\right)_{\text{w}} </math>
-::::<math> = 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}) </math>
+::::<math> = F_{\text{e},q} u_{\text{e},q} + F_{\text{w},q} u_{\text{w},q} - \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{e}} \left(u_{\text{e},q} - u_{\text{P},q}\right) + \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{w}} \left(u_{\text{P},q} - u_{\text{w},q}\right) </math>
 </equation>
@@ Line 122: / Line 122: @@
 <equation id="eqn:UpwindMomentumFlux1">
-{{NumBlk|:|<math>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}</math>|<xr id="eqn:UpwindMomentumFlux1" nolink />}}
+{{NumBlk|:|<math>F_{\text{e},q} u_{\text{e},q} = \text{max}\left(F_{\text{e},q},0\right)u_{\text{P},q} - \text{max}\left(-F_{\text{e},q},0\right)u_{\text{E},q}</math>|<xr id="eqn:UpwindMomentumFlux1" nolink />}}
 </equation>
 <equation id="eqn:UpwindMomentumFlux2">
-{{NumBlk|:|<math>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}</math>|<xr id="eqn:UpwindMomentumFlux2" nolink />}}
+{{NumBlk|:|<math>F_{\text{w},q} u_{\text{w},q} = \text{max}\left(F_{\text{w},q},0\right)u_{\text{W},q} - \text{max}\left(-F_{\text{w},q},0\right)u_{\text{P},q}</math>|<xr id="eqn:UpwindMomentumFlux2" nolink />}}
 </equation>
@@ Line 131: / Line 131: @@
 <equation id="eqn:MomentumConservation5">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MomentumConservation5" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:MomentumConservation5" nolink />}}
-::::<math> =[\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} </math>
+::::<math> =\left[\text{max}\left(F_{\text{e},q},0\right) + \text{max}\left(-F_{\text{w},q},0\right) + \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{e}} + \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{w}}\right]u_{\text{P},q} </math>
-::::::::<math> -[\text{max}(-F_{\text{e},q},0) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}}]u_{\text{E},q} </math>
+::::::::<math> -\left[\text{max}\left(-F_{\text{e},q},0\right) + \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{e}}\right]u_{\text{E},q} </math>
-::::::::<math> -[\text{max}(F_{\text{w},q},0) + (\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}}]u_{\text{W},q} </math>
+::::::::<math> -\left[\text{max}\left(F_{\text{w},q},0\right) + \left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{w}}\right]u_{\text{W},q} </math>
 </equation>
-Denoting <math>(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{e}} = D_{\text{e},q}</math> and <math>(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x})_{\text{w}} = D_{\text{w},q}</math> and modifying the central coefficient slightly, since <math>\text{max}(F_{\text{e},q},0) = F_{\text{e},q} + \text{max}(-F_{\text{e},q},0)</math> and <math>\text{max}(-F_{\text{w},q},0) = -F_{\text{w},q} + \text{max}(F_{\text{w},q},0)</math>, <xr id="eqn:MomentumConservation5"/> reduces to
+Denoting <math>\left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{e}} = D_{\text{e},q}</math> and <math>\left(\varepsilon \alpha_q A \frac{\mu_{\text{eff},q}}{\delta x}\right)_{\text{w}} = D_{\text{w},q}</math> and modifying the central coefficient slightly, since <math>\text{max}\left(F_{\text{e},q},0\right) = F_{\text{e},q} + \text{max}\left(-F_{\text{e},q},0\right)</math> and <math>\text{max}\left(-F_{\text{w},q},0\right) = -F_{\text{w},q} + \text{max}\left(F_{\text{w},q},0\right)</math>, <xr id="eqn:MomentumConservation5"/> reduces to
 <equation id="eqn:MomentumConservation6">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MomentumConservation6" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:MomentumConservation6" nolink />}}
-::::<math> = [\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}</math>
+::::<math> = \left[\text{max}\left(-F_{\text{e},q},0\right) + D_{\text{e},q} + \text{max}\left(F_{\text{w},q},0\right) + D_{\text{w},q} + \left(F_{\text{e},q} - F_{\text{w},q}\right)\right]u_{\text{P},q}</math>
-::::::::<math>-[\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}</math>
+::::::::<math>-\left[\text{max}\left(-F_{\text{e},q},0\right) + D_{\text{e},q}\right]u_{\text{E},q} - \left[\text{max}\left(F_{\text{w},q},0\right) + D_{\text{w},q}\right]u_{\text{W},q}</math>
 </equation>
@@ Line 150: / Line 150: @@
 Identifying the coefficients of neighboring velocities
 <equation id="eqn:MomentumNeighborCoefficient1">
-{{NumBlk|:|<math>a_{\text{e},q} = \text{max}(-F_{\text{e},q},0) + D_{\text{e},q}</math>|<xr id="eqn:MomentumNeighborCoefficient1" nolink />}}
+{{NumBlk|:|<math>a_{\text{e},q} = \text{max}\left(-F_{\text{e},q},0\right) + D_{\text{e},q}</math>|<xr id="eqn:MomentumNeighborCoefficient1" nolink />}}
 </equation>
 <equation id="eqn:MomentumNeighborCoefficient2">
-{{NumBlk|:|<math>a_{\text{w},q} = \text{max}(F_{\text{w},q},0) + D_{\text{w},q}</math>|<xr id="eqn:MomentumNeighborCoefficient2" nolink />}}
+{{NumBlk|:|<math>a_{\text{w},q} = \text{max}\left(F_{\text{w},q},0\right) + D_{\text{w},q}</math>|<xr id="eqn:MomentumNeighborCoefficient2" nolink />}}
 </equation>
@@ Line 159: / Line 159: @@
 <equation id="eqn:MomentumConservation7">
-{{NumBlk|:|<math>\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</math>|<xr id="eqn:MomentumConservation7" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow\iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q u_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \mu_{\text{eff},q} \frac{\Delta u_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:MomentumConservation7" nolink />}}
-::::<math> = [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}</math>
+::::<math> = \left[a_{\text{e},q} + a_{\text{w},q} + \left(F_{\text{e},q} - F_{\text{w},q}\right)\right]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}</math>
 </equation>
@@ Line 166: / Line 166: @@
 <equation id="eqn:MomentumConservation8">
-{{NumBlk|:|<math>\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}</math>|<xr id="eqn:MomentumConservation8" nolink />}}
+{{NumBlk|:|<math>\Rightarrow\left[a_{\text{e},q} + a_{\text{w},q} + \left(F_{\text{e},q} - F_{\text{w},q}\right)\right]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}</math>|<xr id="eqn:MomentumConservation8" nolink />}}
-::::<math> = [\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</math>
+::::<math> = \left[\text{max}\left(\Gamma_{pq},0\right)u_p - \text{max}\left(-\Gamma_{pq},0\right)u_q\right] - \varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g</math>
-::::::::<math>- \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}</math>
+::::::::<math>- \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}</math>
 </equation>
@@ Line 174: / Line 174: @@
 <equation id="eqn:MomentumConservation9">
-{{NumBlk|:|<math>\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}</math>|<xr id="eqn:MomentumConservation9" nolink />}}
+{{NumBlk|:|<math>\Rightarrow\left[a_{\text{e},q} + a_{\text{w},q} + \left(F_{\text{e},q} - F_{\text{w},q}\right) - \left(F_{\text{e},q} - F_{\text{w},q}\right)\right]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}</math>|<xr id="eqn:MomentumConservation9" nolink />}}
-::::<math> = [\text{max}(\Gamma_{pq},0)u_p - \text{max}(-\Gamma_{pq},0)u_q] - (\Gamma_{pq} + S_{\text{mass},q})u_{\text{P},q}</math>
+::::<math> = \left[\text{max}\left(\Gamma_{pq},0\right)u_p - \text{max}\left(-\Gamma_{pq},0\right)u_q\right] - \left(\Gamma_{pq} + S_{\text{mass},q}\right)u_{\text{P},q}</math>
-::::::::<math> -\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}</math>
+::::::::<math> -\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}</math>
 </equation>
@@ Line 182: / Line 182: @@
 <equation id="eqn:MomentumConservation10">
-{{NumBlk|:|<math>\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}</math>|<xr id="eqn:MomentumConservation10" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow\left[a_{\text{e},q} + a_{\text{w},q}\right]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}</math>|<xr id="eqn:MomentumConservation10" nolink />}}
-::::<math> = [\text{max}(\Gamma_{pq},0)u_{\text{P},p} - \text{max}(-\Gamma_{pq},0)u_{\text{P},q}]</math>
+::::<math> = \left[\text{max}\left(\Gamma_{pq},0\right)u_{\text{P},p} - \text{max}\left(-\Gamma_{pq},0\right)u_{\text{P},q}\right]</math>
-::::::::<math>-[\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}</math>
+::::::::<math>-\left[\text{max}\left(\Gamma_{pq},0\right) - \text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right) - \text{max}\left(-S_{\text{mass},q},0\right)\right]u_{\text{P},q}</math>
-::::::::<math> -\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}</math>
+::::::::<math> -\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}</math>
 </equation>
-<equation id="eqn:MomentumConservation10">
+<equation id="eqn:MomentumConservation11">
-{{NumBlk|:|<math>\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}</math>|<xr id="eqn:MomentumConservation10" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow\left[a_{\text{e},q} + a_{\text{w},q} + \text{max}\left(\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right) + \text{max}\left(-\Gamma_{pq},0\right)\right]u_{\text{P},q} - a_{\text{e},q}u_{\text{E},q} - a_{\text{w},q}u_{\text{W},q}</math>|<xr id="eqn:MomentumConservation11" nolink />}}
-::::<math> = [\text{max}(-\Gamma_{pq},0) + \text{max}(-S_{\text{mass},q},0)]u_q + \text{max}(\Gamma_{pq},0)u_p</math>
+::::<math> = \left[\text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(-S_{\text{mass},q},0\right)\right]u_q + \text{max}\left(\Gamma_{pq},0\right)u_p</math>
-::::::::<math> -\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}</math>
+::::::::<math> -\varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}</math>
 </equation>
-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, <math>\text{max}(\Gamma_{pq},0) + \text{max}(S_{\text{mass},q},0)</math>, resulting from the subtraction of the mass equation times cell velocity, while the counterparts, <math>[\text{max}(-\Gamma_{pq},0) + \text{max}(-S_{\text{mass},q},0)]u_q</math>, 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.
+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, <math>\text{max}\left(\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right)</math>, resulting from the subtraction of the mass equation times cell velocity, while the counterparts, <math>\left[\text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(-S_{\text{mass},q},0\right)\right]u_q</math>, 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 <math> a_P = a_{\text{e},q} + a_{\text{w},q} + \text{max}\left(\Gamma_{pq},0\right) + \text{max}\left(S_{\text{mass},q},0\right) + \text{max}\left(-\Gamma_{pq},0\right)</math> and casting <xr id="eqn:MomentumConservation11"/> 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).
-Finally, denoting the central coefficient by <math> 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)</math> and casting <xr id="eqn:MomentumConservation10"/> 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).
+<equation id="eqn:DiscretizedMomentumConservation">
+{{NumBlk|:|
+<math>
+\begin{bmatrix}
+a_{\text{P,c1},1} + K_{\text{c1}} &            -K_{\text{c1}}         &          -a_{\text{e,c1},1}       &                   0               &                \cdots             &                   0               \\
+          -K_{\text{c1}}          & a_{\text{P,c1},2} + K_{\text{c1}} &                   0               &          -a_{\text{e,c1},2}       &                \ddots             &                \vdots             \\
+         -a_{\text{w,c2},1}       &                   0               & a_{\text{P,c2},1} + K_{\text{c2}} &            -K_{\text{c2}}         &          -a_{\text{e,c2},1}       &                   0               \\
+              &            -a_{\text{w,c2},2}     &            -K_{\text{c2}}         & a_{\text{P,c2},2} + K_{\text{c2}} &                   0               &          -a_{\text{e,c2},2}       \\
+               \vdots             &                \ddots             &          -a_{\text{w,c3},1}       &                   0               & a_{\text{P,c3},1} + K_{\text{c3}} &            -K_{\text{c3}}         \\
+               &                \cdots             &                   0               &          -a_{\text{w,c3},2}       &             -K_{\text{c3}}        & a_{\text{P,c3},2} + K_{\text{c3}}
+\end{bmatrix}
+\begin{pmatrix}
+u_{\text{c1,1}}\\
+u_{\text{c1,2}}\\
+u_{\text{c2,1}}\\
+u_{\text{c2,2}}\\
+u_{\text{c3,1}}\\
+u_{\text{c3,2}}
+\end{pmatrix}
+=
+\begin{pmatrix}
+S_{\text{mom,c1,1}}\\
+S_{\text{mom,c1,2}}\\
+S_{\text{mom,c2,1}}\\
+S_{\text{mom,c2,2}}\\
+S_{\text{mom,c3,1}}\\
+S_{\text{mom,c3,2}}
+\end{pmatrix}
+</math>
+|<xr id="eqn:DiscretizedMomentumConservation" nolink />}}
+</equation>
-[[File:MomentumConservationEquation.png|thumb|800px| ]]
+where:
+:<math>S_{\text{mom},\text{c1},q} = \Big\{\big[\text{max}\left(-\Gamma_{pq},0\right) + \text{max}\left(-S_{\text{mass},q},0\right)\big]u_q + \text{max}\left(\Gamma_{pq},0\right)u_p </math>
+::::::<math>- \varepsilon \alpha_q V \nabla p + \varepsilon \alpha_q \rho_q V g - \tfrac{1}{2} \varepsilon^3 \alpha_q \rho_q C V \left| u_q \right| u_q - K\left(u_q - u_p\right) + S_{\text{mom},q}\Big\}_{\text{c1}}</math>
+and
+:{|
+|<math> \alpha_q </math>
+|
+|volume fraction of phase ''q'' [-],
+|-
+|<math> \gamma_{pq} </math>
+|
+|volumetric mass transfer rate from phase ''p'' to ''q'' [kg/m<sup>3</sup>s],
+|-
+|<math> \Gamma_{pq} </math>
+|
+|mass transfer rate from phase ''p'' to ''q'' [kg/s],
+|-
+|<math> \delta x </math>
+|
+|distance between two adjacent cell centers [m],
+|-
+|<math> \varepsilon </math>
+|
+|porosity, fluid fraction of cell volume [-],
+|-
+|<math> \mu_{\text{eff},q} </math>
+|
+|effective viscosity of phase ''q'' [Ns/m<sup>2</sup>],
+|-
+|<math> \rho_q </math>
+|
+|density of phase ''q'' [kg/m<sup>3</sup>],
+|-
+|<math> A </math>
+|
+|face area [m<sup>2</sup>],
+|-
+|<math> C </math>
+|
+|inertial loss coefficient [1/m],
+|-
+|<math> F </math>
+|
+|face mass flow rate [kg/s],
+|-
+|<math> g </math>
+|
+|acceleration of gravity [m/s<sup>2</sup>],
+|-
+|<math> k </math>
+|
+|volumetric interphase drag coefficient [kg/m<sup>3</sup>s],
+|-
+|<math> K </math>
+|
+|interphase drag coefficient [kg/s],
+|-
+|<math> n </math>
+|
+|face normal pointing out of the cell (1 for east face, -1 for west face),
+|-
+|<math> p </math>
+|
+|pressure [Pa],
+|-
+|<math> s_{\text{mass},q} </math>
+|
+|volumetric mass source to phase ''q'' [kg/m<sup>3</sup>s],
+|-
+|<math> S_{\text{mass},q} </math>
+|
+|mass source to phase ''q'' [kg/s],
+|-
+|<math> s_{\text{mom},q} </math>
+|
+|volumetric momentum source to phase ''q'' [N/m<sup>3</sup>],
+|-
+|<math> S_{\text{mom},q} </math>
+|
+|momentum source to phase ''q'' [N],
+|-
+|<math> u_q </math>
+|
+|velocity of phase ''q'' [m/s],
+|-
+|<math> V </math>
+|
+|cell volume [m<sup>3</sup>],
+|-
+|subscript c1
+|
+|refers to first cell,
+|-
+|subscript c2
+|
+|refers to second cell,
+|-
+|subscript c3
+|
+|refers to third cell,
+|-
+|subscript e
+|
+|east face (the face in the negative x-direction),
+|-
+|subscript E
+|
+|east neighbor (the cell in the negative x-direction),
+|-
+|subscript w
+|
+|west face (the face in the positive x-direction),
+|-
+|subscript W
+|
+|west neighbor (the cell in the positive x-direction),
+|-
+|subscript P
+|
+|central cell (the cell for which the equation is written),
+|-
+|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. <math> \Gamma_{pq} </math> mass transfer from phase ''p'' to ''q'').
+|}
 == Conservation of Total Enthalpy ==
@@ Line 209: / Line 374: @@
 [[File:EnthalpyConservation.png|thumb|400px| <xr id="fig:EnthalpyConservation"/>: Conservation of total enthalpy for an interior cell]]
 </figure>
+Conservation of total enthalpy (of phase ''q'') can be written as
+<equation id="eqn:TransientTotalEnthalpyConservation">
+{{NumBlk|:|<math>\frac{\partial \left(\varepsilon \alpha_q \rho_q h_{0,q}\right)}{\partial t} + \nabla\cdot\left(\varepsilon \alpha_q \rho_q h_{0,q} u_q\right)</math>|<xr id="eqn:TransientTotalEnthalpyConservation" nolink />}}
+::::<math>= \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right) + \left[\text{max}\left(\gamma_{pq},0\right) h_{0,\text{cre}} - \text{max}\left(-\gamma_{pq},0\right) h_{0,\text{des}}\right]</math>
+::::::::<math>- \nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \varepsilon \alpha_q \frac{\partial p}{\partial t} + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q  \right) + s_{h,q}</math>
+</equation>
+Disregarding the transient terms and moving the diffusion term to the left-hand side leads to
+<equation id="eqn:TotalEnthalpyConservation">
+{{NumBlk|:|<math>\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_{0,q} u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right)</math>|<xr id="eqn:TotalEnthalpyConservation" nolink />}}
+::::<math>= \left[\text{max}\left(\gamma_{pq},0\right) h_{0,\text{cre}} - \text{max}\left(-\gamma_{pq},0\right) h_{0,\text{des}}\right] - \nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q  \right) + s_{h,q}</math>
+</equation>
+Total enthalpy is defined as the sum of specific enthalpy, kinetic energy and potential energy
+<equation id="eqn:TotalEnthalpyDefinition">
+{{NumBlk|:|<math>h_0 = h + \tfrac{1}{2} \left| u_q \right|^2 - \mathbf{g} \cdot \mathbf{r}</math>|<xr id="eqn:TotalEnthalpyDefinition" nolink />}}
+</equation>
+where <math>\mathbf{g} \cdot \mathbf{r}</math> is the dot product of gravity vector, <math>\mathbf{g}</math>, and location vector, <math>\mathbf{r}</math>, a vector pointing from the reference level to the evaluated location. The total enthalpies related to phase change can be defined as
+<equation id="eqn:EnthalpyPhaseChangeTerms">
+{{NumBlk|:|<math>
+\begin{align}
+& h_{0,\text{cre}} = h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\\
+& h_{0,\text{des}} = h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}
+\end{align}
+</math>|<xr id="eqn:EnthalpyPhaseChangeTerms" nolink />}}
+</equation>
+In other words, when mass is transferred to phase ''q'' (“created”), it brings with it the saturation enthalpy of phase ''q'', <math>h_{\text{P,sat,}q}</math>, and the kinetic energy of phase ''p'', <math>\tfrac{1}{2} \left| u_p \right|^2_{\text{P}}</math>.  When mass is transferred from phase ''q'' (“destroyed”), it leaves with the enthalpy and kinetic energy of phase ''q'', <math>h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}}</math>. It is chosen that the potential energy associated with phase change is evaluated at the center of the cell, <math>\left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}</math>.
+Substituting <xr id="eqn:TotalEnthalpyDefinition"/> and <xr id="eqn:EnthalpyPhaseChangeTerms"/> to <xr id="eqn:TotalEnthalpyConservation"/> and rearranging terms
+<equation id="eqn:TotalEnthalpyConservation2">
+{{NumBlk|:|<math>\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \lambda_{\text{eff},q} \nabla T_q \right)</math>|<xr id="eqn:TotalEnthalpyConservation2" nolink />}}
+::::<math>= \left[\text{max}\left(\gamma_{pq},0\right) \left(h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\right) - \text{max}\left(-\gamma_{pq},0\right) \left(h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}
+\right)\right]</math>
+::::<math> -\nabla\cdot\left(\tfrac{1}{2}\varepsilon \alpha_q \rho_q \left| u_q \right|^2 u_q\right)
++\nabla\cdot\left(\varepsilon \alpha_q \rho_q \left( \mathbf{g} \cdot \mathbf{r} \right)u_q\right)
+-\nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q  \right) + s_{h,q}</math>
+</equation>
+At this point, an approximation is introduced: the diffusion term is expressed relative to the gradient of enthalpy, instead of temperature, as follows. This approximation is considered to be sufficiently accurate for small enthalpy differences.
+<equation id="eqn:TotalEnthalpyConservation3">
+{{NumBlk|:|<math>\Rightarrow\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \nabla h_q \right)</math>|<xr id="eqn:TotalEnthalpyConservation3" nolink />}}
+::::<math>= \left[\text{max}\left(\gamma_{pq},0\right) \left(h_{\text{P,sat,}q} + \tfrac{1}{2} \left| u_p \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}\right) - \text{max}\left(-\gamma_{pq},0\right) \left(h_{\text{P,}q} + \tfrac{1}{2} \left| u_q \right|^2_{\text{P}} - \left(\mathbf{g} \cdot \mathbf{r}\right)_{\text{P}}
+\right)\right]</math>
+::::<math> -\nabla\cdot\left(\tfrac{1}{2}\varepsilon \alpha_q \rho_q \left| u_q \right|^2 u_q\right)
++\nabla\cdot\left(\varepsilon \alpha_q \rho_q \left( \mathbf{g} \cdot \mathbf{r} \right)u_q\right)
+-\nabla\cdot\left(\varepsilon \alpha_q \mathbf{q} \right) + \nabla\cdot\left(\varepsilon \alpha_q \boldsymbol{\tau} \cdot \mathbf{u}_q  \right) + s_{h,q}</math>
+</equation>
+<xr id="eqn:TotalEnthalpyConservation3"/> is discretized for an interior cell, shown in <xr id="fig:EnthalpyConservation"/>, by volume integration over cell P.