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23.8.2 Mass and Momentum Transfer with Multiphase Species Transport

The FLUENT multiphase mass transfer model accommodates mass transfer between species belonging to different phases. Instead of a matrix-type input, multiple mass transfer mechanisms need to be input. Each mass transfer mechanism defines the mass transfer phenomenon from one entity to another entity. An entity is either a particular species in a phase, or the bulk phase itself if the phase does not have a mixture material associated with it. The mass transfer phenomenon could be specified either through the inbuilt unidirectional "constant-rate" mass transfer (Section  23.7.2) or through user-defined functions.

FLUENT loops through all the mass transfer mechanisms to compute the net mass source/sink of each species in each phase. The net mass source/sink of a species is used to compute species and mass source terms. FLUENT will also automatically add the source contribution to all relevant momentum and energy equations based on that assumption that the momentum and energy carried along with the transferred mass. For other equations, the transport due to mass transfer needs to be explicitly modeled by the user.

Source Terms due to Heterogeneous Reactions

Consider the following reaction:

 aA + bB \rightarrow cC + dD (23.8-2)

Let as assume that $A$ and $C$ belong to phase $1$ and $B$ and $D$ to phase $2$.

Mass Transfer

Mass source for the phases are given by:

$\displaystyle S_{1}$ $\textstyle =$ $\displaystyle {\cal R} (c M_c - a M_a)$ (23.8-3)
$\displaystyle S_{2}$ $\textstyle =$ $\displaystyle {\cal R} (d M_d - b M_b)$ (23.8-4)

where $S$ is the mass source, $M$ is the molecular weight, and ${\cal R}$ is the reaction rate.

The general expression for the mass source for the $i^{\rm th}$ phase is

$\displaystyle S_{r_{i}}$ $\textstyle =$ $\displaystyle -{\cal R}\sum_{r_{i}}\gamma^r_jM^r_j$ (23.8-5)
$\displaystyle S_{p_{i}}$ $\textstyle =$ $\displaystyle {\cal R}\sum_{p_{i}}\gamma^p_jM^p_j$ (23.8-6)
$\displaystyle S_{i}$ $\textstyle =$ $\displaystyle S_{p_{i}} + S_{r_{i}}$ (23.8-7)

where $\gamma$ is the stoichiometric coefficient, $p$ represents the product, and $r$ represents the reactant.

Momentum Transfer

Momentum transfer is more complicated, but we can assume that the reactants mix (conserving momentum) and the products take momentum in the ratio of the rate of their formation.

The net velocity, ${\vec u}_{net}$, of the reactants is given by:

 {\vec u}_{net} = \frac{a M_a {\vec u}_{1} + b M_b {\vec u}_{2}}{a M_a + b M_b} (23.8-8)

The general expression for the net velocity of the reactants is given by:

 {\vec u}_{net} = \frac{\sum_r\gamma_j^r M^r_j{\vec u_{r_j}}}{\sum_r\gamma_j^r M^r_j} (23.8-9)

where $j$ represents the $j^{\rm th}$ item (either a reactant or a product).

Momentum transfer for the phases is then given by:

$\displaystyle S^{\vec u}_{1}$ $\textstyle =$ $\displaystyle {\cal R}(c M_c {\vec u}_{net} - a M_a {\vec u}_{1})$ (23.8-10)
$\displaystyle S^{\vec u}_{2}$ $\textstyle =$ $\displaystyle {\cal R}(d M_d {\vec u}_{net}- b M_b {\vec u}_{2})$ (23.8-11)

The general expression is

 S^{\vec u}_{i} = S_{p_{i}}{\vec u_{net}} - {\cal R}\sum_{r_{i}}{ \gamma^r_jM^r_j{\vec u_{i}}} (23.8-12)

If we assume that there is no momentum transfer, then the above term will be zero.

Species Transfer

The general expression for source for $k^{\rm th}$ species in the $j^{\rm th}$ phase is

$\displaystyle S_{r^k_{i}}$ $\textstyle =$ $\displaystyle -{\cal R}\sum_{r^k_{i}}\gamma^{r^k}_jM^{r^k}_j$ (23.8-13)
$\displaystyle S_{p^k_{i}}$ $\textstyle =$ $\displaystyle {\cal R}\sum_{p^k_{i}}\gamma^{p^k}_jM^{p^k}_j$ (23.8-14)
$\displaystyle S^k_{i}$ $\textstyle =$ $\displaystyle S_{p^k_{i}} + S_{r^k_{i}}$ (23.8-15)

Heat Transfer

For heat transfer, we need to consider the formation enthalpies of the reactants and products as well:

The net enthalpy of the reactants is given by:

 H_{net} = \frac{a M_a (H_a + h_a^{f}) + b M_b (H_b + h_b^{f})}{a M_a + b M_b} (23.8-16)

where $h^f$ represents the formation enthalpy, and $H$ represents the enthalpy.

The general expression for $H_{net}$ is:

 H_{net} = \frac{\sum_r\gamma^r_j M^r_j (H_j^r + {h^{f}}_j^r)}{\sum_r\gamma^r_j M^r_j} (23.8-17)

If we assume that this enthalpy gets distributed to the products in the ratio of their mass production rates, heat transfer for the phases are given by:

$\displaystyle S^H_{1}$ $\textstyle =$ $\displaystyle {\cal R}(c M_c H_{net} - a M_a H^a - c M_c h^c_{f})$ (23.8-18)
$\displaystyle S^H_{2}$ $\textstyle =$ $\displaystyle {\cal R}(d M_d H_{net} - b M_b H^b - d M_d h^d_{f})$ (23.8-19)

The last term in the above equations appears because our enthalpy is with reference to the formation enthalpy.

The general expression for the heat source is:

 S^H_{i} = S_{p_{i}}H_{net} - {\cal R}\left(\sum_{r_{i}}\gamma^r_j M^r_j H_j^r +\sum_{p_{i}}\gamma^p_j M^p_j {h^f}^p_j\right) (23.8-20)

If we assume that there is no heat transfer, we can assume that the different species only carry their formation enthalpies with them. Thus the expression for $H_{net}$ will be:

 H_{net} = \frac{\sum_r\gamma^r_j M^r_j {h^{f}}_j^r}{\sum_r\gamma^r_j M^r_j} (23.8-21)

The expression $S^H_{i}$ will be

 S^H_{i} = S_{p_{i}}H_{net} - {\cal R}\sum_{p_{i}}\gamma^p_j M^p_j {h^f}^p_j (23.8-22)

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