Interphase Reactions

Interphase Reactions for Multiphase Models

For Eulerian Multiphase (EMP), Mixture Multiphase (MMP), Volume of Fluid Method (VOF) and wall film reacting flows, the user-defined reaction rate is calculated in the form:

Figure 1. EQUATION_DISPLAY

ω_{i} = k_{i} Π {[X_{j}]}^{r_{j}} a_{c d}

(3587)

based on the following user inputs:

$k_{i}$ , the user reaction coefficient for reaction $i$ .
The units of $k_{i}$ . The unit of $k_{i} Π {[X_{j}]}^{r_{j}}$ must take the form [Quantity/Length²/s] as an option in the GUI.
$r_{j}$ , the rate exponent, which is specified for each reactant and product.

$X_{j}$ is the species concentration and $a_{c d}$ is the interaction area density, see Interaction Area Density. The units for reaction rate should be per unit surface area [kmol/m²/s]. If the units of the user reaction rate are per unit volume, this expression should be divided by $InteractionAreaDensityPhaseInteraction1.

Diffusion Limited Reaction Rate

In general, intephase reactions are limited by the rate of diffusion of species from either side of the interface between the different phases.

Consider an interphase reaction between gas (g) and liquid (l) phases:

Figure 2. EQUATION_DISPLAY

α^{(g)} + β^{(l)} \to ω^{(g)} + γ^{(l)}

(3588)

where $α^{(g)}$ and $β^{(l)}$ are reactant species in the gas and liquid phases, respectively and $ω^{(g)}$ and $γ^{(l)}$ are product species in the gas and liquid phases, respectively.

The reaction rate $R$ of equation Eqn. (3588) is computed from chemical kinetics.

Reaction-Diffusion Balance

Following is a time-scale based formulation to limit the reaction rate based on the rate of diffusion of individual species to the interface.

The maximum rate of diffusion of gas reactant

α^{(g)}

to the interface is given as:

Figure 3. EQUATION_DISPLAY

Γ (α^{(g)}) {= ρ}^{(g)} K_{D}^{α^{(g)}} Y^{α^{(g)}} A

(3589)

where $ρ^{(g)}$ is the density in the gas-phase, $K_{D}^{α^{(g)}}$ is the mass transfer coefficient of species $α^{(g)}$ , $Y^{α^{(g)}}$ is the mass fraction of species $α^{(g)}$ and $A$ is the interfacial area density.

The maximum rate of diffusion of liquid reactant

β^{(l)}

to the interface is given as:

Figure 4. EQUATION_DISPLAY

Γ (β^{(l)}) = ρ^{(l)} K_{D}^{β^{(l)}} Y^{β^{(l)}} A

(3590)

The mass transfer coefficient

K_{D}

is estimated as:

Figure 5. EQUATION_DISPLAY

K_{D} = S h (\frac{D}{L})

(3591)

where

S h

is the Sherwood number,

D

is molecular diffusivity and

L

is the Interaction Length Scale. See Interaction Length Scale and Interaction Area Density.

Net Rate of Reaction

The net rate of the reaction is the minimum of the reaction rate

R

, calculated from the chemical kinetics, and the diffusion rates of the reactants. In addition, the user-specified diffusion multiplier

F

, scales diffusion rates as needed. Finally, the reaction rate is given by:

Figure 6. EQUATION_DISPLAY

ω = \min (R, F ρ^{(g)} K_{D}^{α^{(g)}} Y^{α^{(g)}} A, F ρ^{(l)} K_{D}^{β^{(l)}} Y^{β^{(l)}} A)

(3592)