Difference between revisions of "Kharon theory manual"

Revision as of 16:14, 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.

$\Leftrightarrow \iiint\limits_V \left[\nabla\cdot\left(\varepsilon \alpha_q \rho_q h_q u_q\right) - \nabla\cdot\left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \nabla h_q \right)\right]\, dV$

(27)

= \iiint\limits_V \Big\{ \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} \Big\}\, dV

As with the conservation of mass and momentum 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.

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

(28)

= \iiint\limits_V \Big\{ \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} \Big\}\, dV

Developing the left-hand side first.

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

(29)

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

= F_{\text{e},q} h_{\text{e},q} + F_{\text{w},q} h_{\text{w},q} - \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} \left(h_{\text{E},q} - h_{\text{P},q}\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \left(h_{\text{P},q} - h_{\text{W},q}\right)

Adopting upwind discretization for the enthalpy fluxes on the cell faces

$\begin{align} & F_{\text{e},q} h_{\text{e},q} = \text{max}\left(F_{\text{e},q},0\right) h_{\text{P},q} - \text{max}\left(-F_{\text{e},q},0\right) h_{\text{E},q}\\ & F_{\text{w},q} h_{\text{w},q} = \text{max}\left(F_{\text{w},q},0\right) h_{\text{W},q} - \text{max}\left(-F_{\text{w},q},0\right) h_{\text{P},q} \end{align}$

(30)

and rearranging terms

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

(31)

= \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{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \right] h_{\text{P},q}

- \left[ \text{max}\left(-F_{\text{e},q},0\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} \right] h_{\text{E},q} - \left[ \text{max}\left(F_{\text{w},q},0\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \right] h_{\text{W},q}

Denoting $\left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} = D_{\text{e},q}$ and $\left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} = D_{\text{w},q}$ 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 31 reduces to

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

(32)

= \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] h_{\text{P},q}

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

Identifying the coefficients of neighboring velocities as

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

(33)

leads to a general form

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

(34)

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

Substituting the results of Equation 34 into Equation 28 and continuing with the development

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

(36)

= \iiint\limits_V \Big\{ \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} \Big\}\, dV

Integrating the right-hand side term by term. The phase change term reduces to

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

(36)

= \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]

@@ Line 136: / Line 136: @@
 ::::::::<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>\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">
@@ Line 146: / Line 144: @@
 ::::::::<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>
 Identifying the coefficients of neighboring velocities
@@ Line 424: / Line 421: @@
 <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}}
+::::<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>
-\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)
@@ Line 434: / Line 430: @@
 <equation id="eqn:TotalEnthalpyConservation4">
-{{NumBlk|:|<math>\Leftrightarrow\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:TotalEnthalpyConservation4" nolink />}}
+{{NumBlk|:|<math>\Leftrightarrow \iiint\limits_V \left[\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)\right]\, dV </math>|<xr id="eqn:TotalEnthalpyConservation4" 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}}
+::::<math>= \iiint\limits_V \Big\{  \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>
-\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} \Big\}\, dV </math>
+</equation>
+As with the conservation of mass and momentum 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.
+<equation id="eqn:TotalEnthalpyConservation5">
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q h_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:TotalEnthalpyConservation5" nolink />}}
+::::<math>= \iiint\limits_V \Big\{  \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} \Big\}\, dV </math>
+</equation>
+Developing the left-hand side first.
+<equation id="eqn:TotalEnthalpyConservation6">
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q h_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:TotalEnthalpyConservation6" nolink />}}
+::::<math> = \left(\varepsilon \alpha_q \rho_q h_q u_q A\right)_{\text{e}} - \left(\varepsilon \alpha_q \rho_q h_q u_q A\right)_{\text{w}} - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x} A\right)_{\text{e}} + \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x} A\right)_{\text{w}} </math>
+::::<math> = F_{\text{e},q} h_{\text{e},q} + F_{\text{w},q} h_{\text{w},q} - \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} \left(h_{\text{E},q} - h_{\text{P},q}\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \left(h_{\text{P},q} - h_{\text{W},q}\right) </math>
+</equation>
+Adopting upwind discretization for the enthalpy fluxes on the cell faces
+<equation id="eqn:UpwindEnthalpyFluxes">
+{{NumBlk|:|<math>
+\begin{align}
+& F_{\text{e},q} h_{\text{e},q} = \text{max}\left(F_{\text{e},q},0\right) h_{\text{P},q} - \text{max}\left(-F_{\text{e},q},0\right) h_{\text{E},q}\\
+& F_{\text{w},q} h_{\text{w},q} = \text{max}\left(F_{\text{w},q},0\right) h_{\text{W},q} - \text{max}\left(-F_{\text{w},q},0\right) h_{\text{P},q}
+\end{align}
+</math>|<xr id="eqn:UpwindEnthalpyFluxes" nolink />}}
+</equation>
+and rearranging terms
+<equation id="eqn:TotalEnthalpyConservation7">
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q h_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:TotalEnthalpyConservation7" nolink />}}
+::::<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{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \right] h_{\text{P},q}</math>
+::::<math> - \left[ \text{max}\left(-F_{\text{e},q},0\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} \right] h_{\text{E},q} - \left[ \text{max}\left(F_{\text{w},q},0\right) + \left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{w}} \right] h_{\text{W},q}</math>
+</equation>
+Denoting <math>\left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,q}\delta x}\right)_{\text{e}} = D_{\text{e},q}</math> and <math>\left(\varepsilon \alpha_q A \frac{\lambda_{\text{eff},q}}{c_{p,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:TotalEnthalpyConservation7"/> reduces to
+<equation id="eqn:TotalEnthalpyConservation8">
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q h_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:TotalEnthalpyConservation8" nolink />}}
+::::<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] h_{\text{P},q} </math>
+::::<math> - \left[ \text{max}\left(-F_{\text{e},q},0\right) + D_{\text{e},q} \right] h_{\text{E},q}
+- \left[ \text{max}\left(F_{\text{w},q},0\right) + D_{\text{w},q} \right] h_{\text{W},q} </math>
+</equation>
+Identifying the coefficients of neighboring velocities as
+<equation id="eqn:EnthalpyNeighborCoefficients">
+{{NumBlk|:|<math>
+\begin{align}
+& a_{\text{e},q} = \text{max}\left(-F_{\text{e},q},0\right) + D_{\text{e},q}\\
+& a_{\text{w},q} = \text{max}\left(F_{\text{w},q},0\right) + D_{\text{w},q}
+\end{align}
+</math>|<xr id="eqn:EnthalpyNeighborCoefficients" nolink />}}
+</equation>
+leads to a general form
+<equation id="eqn:TotalEnthalpyConservation9">
+{{NumBlk|:|<math>\Leftrightarrow \iint\limits_{A}\!\!\!\!\!\!\!\!\!\!\!\subset\!\supset \left[\left(\varepsilon \alpha_q \rho_q h_q u_q\right)\cdot n - \left(\varepsilon \alpha_q \frac{\lambda_{\text{eff},q}}{c_{p,q}} \frac{\Delta h_q}{\delta x}\right)\cdot n\right]\, dA</math>|<xr id="eqn:TotalEnthalpyConservation9" nolink />}}
+::::<math> = \left[ a_{\text{e},q} + a_{\text{w},q} + \left( F_{\text{e},q} - F_{\text{w},q} \right) \right] h_{\text{P},q} - a_{\text{e},q} h_{\text{E},q} - a_{\text{w},q} h_{\text{W},q}</math>
+</equation>
+Substituting the results of <xr id="eqn:TotalEnthalpyConservation9"/> into <xr id="eqn:TotalEnthalpyConservation5"/> and continuing with the development
+<equation id="eqn:TotalEnthalpyConservation10">
+{{NumBlk|:|<math>\Rightarrow \left[ a_{\text{e},q} + a_{\text{w},q} + \left( F_{\text{e},q} - F_{\text{w},q} \right) \right] h_{\text{P},q} - a_{\text{e},q} h_{\text{E},q} - a_{\text{w},q} h_{\text{W},q}\, dA</math>|<xr id="eqn:TotalEnthalpyConservation10" nolink />}}
+::::<math>= \iiint\limits_V \Big\{  \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>
+-\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} \Big\}\, dV </math>
+</equation>
+Integrating the right-hand side term by term. The phase change term reduces to
+<equation id="eqn:TotalEnthalpyConservation10">
+{{NumBlk|:|<math>\iiint\limits_V \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] dV</math>|<xr id="eqn:TotalEnthalpyConservation10" 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>
 </equation>

Difference between revisions of "Kharon theory manual"

Revision as of 16:14, 19 April 2018

Conservation of Mass

Conservation of Momentum

Conservation of Total Enthalpy

Navigation menu

Personal tools

Namespaces

Variants

Views

More

Search

Navigation

Tools