Dilute Solutions

Dilute electrolytic solutions consist of minor quantities of ionic species which exist in a non-ionized solvent, excessing the ions. Transport models for such solutions model the transport primarily independent of the interactions between different kinds of ions, that is, only dependent on the concentration and electric potential gradients of the electrochemical species under consideration (diffusion, migration).

Following the dilute assumption, electrochemical species mixtures which are considered by the electrochemical species and solid ion models in Simcenter STAR-CCM+ will not affect the density of a mixture.

Coupling Electrochemical Species and Electrodynamic Potential Models

The equations in this section are primarily solved when modeling applications which enforce electroneutrality within the solution, such as batteries, corrosion, etching, and electrochemical deposition. For this case, the electrodynamic potential model provides the electric potential in the transport equation for electrochemical species.

The transport equation for the molar concentrations $c_{i}$ of the electrochemical species $i$ can be written as:

Figure 1. EQUATION_DISPLAY

\frac{\partial}{\partial t} \int_{V} χ c_{i} d V + \oint_{A} c_{i} v d a = \oint_{A} D_{eff, i} χ^{β} \nabla c_{i} \cdot d a + \oint_{A} z_{i} F u_{i} c_{i} χ^{β} \nabla ϕ \cdot d a + \int_{V} S_{c_{i}} d V

(4072)

Continuous form:

Figure 2. EQUATION_DISPLAY

\frac{\partial}{\partial t} (c_{i} χ) = - \nabla \cdot N_{i}

(4073)

Figure 3. EQUATION_DISPLAY

N_{i} = {- D}_{eff, i} χ^{β} \nabla c_{i} - z_{i} F u_{i} c_{i} χ^{β} \nabla ϕ + c_{i} v

(4074)

where, for species $i$ , $c_{i}$ is concentration, $D_{eff}$ is the effective diffusivity, $z_{i}$ is the charge number, $u_{i}$ is the mobility, and $N_{i}$ represents the molar concentration flux. $F$ is Faraday's constant, $S_{c_{i}}$ is the electrochemical species source term, $t$ is time, $V$ is the cell volume, $d a$ is the surface vector, and $χ^{β}$ represents the porosity.

The effective diffusivity is given by:

D_{eff, i} = D_{i} + \frac{μ_{t, i}}{ρ_{i} {Sc}_{t, i}}

(4075)

where

D_{i}

is the diffusivity,

μ_{t, i}

is the turbulent viscosity, and

{Sc}_{t, i}

is the turbulent Schmidt number.

The convective term $c_{i} v$ uses the superficial velocity for porous regions, which is advection velocity that is multiplied by porosity, as given by Eqn. (1838).

The Electrodynamic Potential model solves a transport equation for the ionic space charge density $ρ_{i o n}$ :

Figure 4. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{i o n} χ) = - \nabla \cdot J = 0

(4076)

Figure 5. EQUATION_DISPLAY

J = - κ_{i} χ^{β} \nabla ϕ - F \sum_{i} z_{i} D_{eff, i} χ^{β} \nabla c_{i} + ρ_{i o n} v

(4077)

The first term on the right-hand side represents the electrical current due to migration, while the second term represents the diffusion current density. Assuming electroneutrality ( $ρ_{i o n} = 0$ ), the third term vanishes and the integral form reads:

Figure 6. EQUATION_DISPLAY

0 = \int_{A} κ_{i} χ^{β} \nabla ϕ \cdot d a + F \sum_{i} z_{i} \int_{A} D_{eff, i} χ^{β} \nabla c_{i} \cdot d a + \int_{V} S_{ρ_{i o n}} d V

(4078)

where $ϕ$ is the electric potential, and $S_{ρ_{i o n}}$ represents user-defined sources. Substituting expressions for $κ_{i}$ the ionic conductivity of species $i$ :

Figure 7. EQUATION_DISPLAY

κ_{i} = F^{2} \sum_{i} z_{i}^{2} u_{i} c_{i}

(4079)

the Nernst-Einstein relation:

Figure 8. EQUATION_DISPLAY

u_{i} = \frac{D_{i}}{R_{u} T}

(4080)

(where $R_{u}$ is the universal gas constant and $T$ is temperature).

and the relation between the ionic space charge density and molar concentrations:

Figure 9. EQUATION_DISPLAY

ρ_{i o n} = F \sum_{i} z_{i} c_{i}

(4081)

the following flux expressions are derived for the molar concentration flux $N_{i}$ of species component $i$ and the electrical current density $J$ :

Figure 10. EQUATION_DISPLAY

N_{i} = - D_{eff, i} χ^{β} \nabla c_{i} - z_{i} F \frac{D_{i}}{R_{u} T} χ^{β} c_{i} \nabla ϕ + c_{i} v

(4082)

Figure 11. EQUATION_DISPLAY

J = - F^{2} \sum_{i} z_{i}^{2} \frac{D_{i}}{R_{u} T} χ^{β} c_{i} \nabla ϕ - F \sum_{i} z_{i} D_{eff, i} χ^{β} \nabla c_{i} + F \sum_{i} z_{i} c_{i} v

(4083)

Comparison of Eqn. (4082) and Eqn. (4083) produces a relation between the molar species fluxes and the electric current densities:

Figure 12. EQUATION_DISPLAY

J = F \sum_{i} z_{i} N_{i}

(4084)

Due to the electroneutrality ( $ρ_{i o n} = 0$ ) that is generally assumed in liquid electrolytes, the convective term in Eqn. (4077) is zero.

Coupling Electrochemical Species and Electrostatic Potential Models

When electroneutrality cannot be assumed such as in ionic winds or in certain kinds of plasmas, the Electrochemical Species and the Electrostatic Potential models must be coupled. The electrostatics potential equation:

Figure 13. EQUATION_DISPLAY

\nabla \cdot E = ρ_{i o n} / ε_{0}

(4085)

replaces Eqn. (4076) to determine the electric potential. $E$ is the electric field, and $ε_{0}$ is the vacuum permittivity.

The integral form is written as:

Figure 14. EQUATION_DISPLAY

\oint_{A} ε_{0} \nabla ϕ \cdot d a = \int_{V} ρ_{i o n} d V + \int_{V} S_{E s t a t} d V

(4086)

Here, $ρ_{i o n}$ is non-zero and is computed from Eqn. (4081), and $S_{E s t a t}$ is the electrostatic potential source term.

The transport equations for the electrochemical species are often expressed in terms of a species number density

n_{p}

Figure 15. EQUATION_DISPLAY

\frac{\partial}{\partial t} (n_{P} χ) = - \nabla \cdot N_{P}

(4087)

where

N_{p}

is the number flux:

Figure 16. EQUATION_DISPLAY

N_{P} = - n_{P} K_{P} χ^{β} \nabla ϕ - D_{P} χ^{β} \nabla n_{P} + n_{P} v

(4088)

See Cagnoni et al. (2013) [824].

Where $K_{p}$ is ion mobility and $D_{p}$ is diffusivity.

The species number density $n_{P}$ is equal to the molar concentration of the ionic species $c$ multiplied by Avogadro’s number $N_{A}$ . Therefore, number density represents the number of ionic species particles per volume.

Often, only a single ionic species with a unity charge number ( $z = 1$ ) is considered.

For a single ionic species $i$ , the relation:

Figure 17. EQUATION_DISPLAY

q n_{P} = ρ_{i o n} = q N_{A} c = F c

(4089)

(by definition of Faraday’s constant), where $q$ is the elementary charge, is used to rewrite Eqn. (4087) and Eqn. (4088) as a transport equation for charge density:

Figure 18. EQUATION_DISPLAY

\frac{\partial}{\partial t} (ρ_{i o n} χ) = - \nabla \cdot J

(4090)

Figure 19. EQUATION_DISPLAY

J = - ρ_{i o n} K χ^{β} \nabla ϕ - D χ^{β} \nabla ρ_{i o n} + ρ_{i o n} v

(4091)

as in Neimarlija et al. (2009) [832].

You can also obtain Eqn. (4090) and Eqn. (4091) by multiplying Eqn. (4073) and Eqn. (4074) with Faraday’s constant

F

, and charge number

z_{i}

, applying Eqn. (4081), and substituting ionic mobility as:

Figure 20. EQUATION_DISPLAY

K_{i} = F {| z_{i} | u}_{i}

(4092)

Although $K_{i}$ and $u_{i}$ are commonly termed ion mobility, their dimension and units differ by the dimension of Faraday’s constant ( $[m^{2} / V s]$ vs $[m^{2} kmol / J s]$ ).