Dissolution

The Dissolution Mass Transfer Rate model calculates the mass of a component that is transferred from one multi-component phase to another.

The rate of this mass transfer is assumed to be slow and independent of the rate of transfers between any other species. Diffusion driven by the concentration gradient is assumed to be the mechanism that moves species between the phase bulk region and the phase boundary. This movement can occur in either direction.

Gas-Liquid Interactions

For gas-liquid interactions, the equilibrium between species at the interface is defined as:

Figure 1. EQUATION_DISPLAY

{X^{*}}_{g a s, i} = K_{i}^{0} {X^{*}}_{l i q, i}

(2159)

where ${X^{*}}_{g a s, i}$ and ${X^{*}}_{l i q, i}$ are the equilibrium mole fractions of component $i$ at the gas side and liquid side of the interface respectively, and $K_{i}^{0}$ is the equilibrium constant. With this equilibrium definition, the value of $K_{i}^{0}$ is independent of the choice of the continuous and dispersed phase.

The equilibrium constant can be specified using the following techniques:

Raoult’s Law

Raoult’s law states that the equilibrium partial pressure of a component above the surface of a liquid is equal to the vapor pressure of a pure sample of the component multiplied by the mole fraction of the component in the liquid. Hence Raoult’s law gives an equilibrium constant:

Figure 2. EQUATION_DISPLAY

K_{i}^{0} = \frac{p_{v a p}}{p_{a b s}}

(2160)

Raoult’s Law is appropriate to volatile liquid species evaporating into or out of a gas.

Henry’s Law

For cases where only small amounts of a species are able to dissolve into a liquid, for instance when a gas such as oxygen dissolves in trace amounts into water, Raoult’s law no longer holds. In this case Henry’s law can be used. Henry’s law states that the equilibrium amount of a gas species that dissolves in a liquid is directly proportional to the partial pressure of that species in the gas:

Figure 3. EQUATION_DISPLAY

K_{i}^{0} = \frac{H_{i}^{0}}{p_{a b s}}

(2161)

where $H_{i}^{0}$ is Henry’s coefficient and is unique to any gas and liquid solvent pair. The default value of 4.2E9 Pa is appropriate for oxygen dissolving in water at room temperature.

The rate of transfer of the species from one phase to the other is governed by the rate at which the species can diffuse between the phase boundary and the interior of the phase.

For gas-liquid interactions, the species mass balance at the interface is:

Figure 4. EQUATION_DISPLAY

{\dot{m}}_{l, i} + {\dot{m}}_{g, i} = 0

(2162)

Figure 5. EQUATION_DISPLAY

{\dot{m}}_{l, i} = k_{l} a_{l g} ρ_{l} ({Y^{*}}_{l, i} - Y_{l, i})

(2163)

Figure 6. EQUATION_DISPLAY

{\dot{m}}_{g, i} = k_{g} {a_{l g} ρ}_{g} ({Y^{*}}_{g, i} - Y_{g, i})

(2164)

where:

The subscript $l$ refers to the liquid phase and $g$ to the gas
$k$ is a mass transfer coefficient for either phase
$a_{l g}$ is the interfacial area density
$ρ$ is the phase density
$Y_{l, i}$ and $Y_{g, i}$ are the mass fractions of the two phases
$Y^{*}$ refers to the equilibrium mass fraction at the interface of the two phases.

Liquid-Liquid Interactions

For liquid-liquid interactions, the equilibrium between species at the interface is defined as:

Figure 7. EQUATION_DISPLAY

{X^{*}}_{c o n t, i} = K_{i}^{0} {X^{*}}_{d i s p, i}

(2165)

Where ${X^{*}}_{c o n t, i}$ and ${X^{*}}_{d i s p, i}$ are the continuous phase and dispersed phase equilibrium mole fractions at the interface respectively. In this case, the value of the equilibrium constant $K_{i}^{0}$ depends on the choice of the continuous and dispersed phase. In particular, if the continuous and disperse phases are swapped, the value of $K_{i}^{0}$ (either constant or field function) becomes $1 / K_{i}^{0}$ .

The equilibrium constant can be specified using the following techniques

Constant value
Field function

For liquid-liquid interactions, the species mass balance at the interface is:

Figure 8. EQUATION_DISPLAY

{\dot{m}}_{c, i} + {\dot{m}}_{d, i} = 0

(2166)

Figure 9. EQUATION_DISPLAY

{\dot{m}}_{c, i} = k_{c} a_{c d} ρ_{c} ({Y^{*}}_{c, i} - Y_{c, i})

(2167)

Figure 10. EQUATION_DISPLAY

{\dot{m}}_{d, i} = k_{d} {a_{c d} ρ}_{d} ({Y^{*}}_{d, i} - Y_{d, i})

(2168)

where the subscript $c$ refers to the continuous phase and $d$ to the dispersed phase. The meaning of the symbols is the same as in the liquid-gas interaction case above.