Transport of Diluted Species (tds)

Diffusion, convection and first-order reactions of a dilute species. Elmer AdvectionDiffusion (streamline diffusion) or ModelPDE (Galerkin).

Category

Chemical Species Transport

Name (tag)

tds

Space dimensions

3D, 2D, 2D axisymmetric

Studies

Stationary, Time Dependent

Dependent variables

c Concentration (mol/m^3)

Default features

Transport Properties, Initial Values, No Flux

\(\frac{\partial c}{\partial t} + \nabla\cdot(-D\nabla c) + \mathbf{u}\cdot\nabla c = R\)

Interface settings

Setting Symbol Unit Default Choices / type

Discretization

Element order

Quadratic

Linear, Quadratic

Dependent Variables

Species (dependent variable name)
Each Transport of Diluted Species interface solves one species; a second interface needs its own name, e.g. c2, and streamline diffusion (AdvectionDiffusion keeps the species' keywords apart)

c

Text

Consistent Stabilization

Streamline diffusion
Consistent stabilization of convection-dominated transport (element Péclet number > 1), which otherwise oscillates from node to node: streamline-upwind Petrov-Galerkin (SUPG). Consistent: it vanishes for the exact solution. Elmer AdvectionDiffusion; without it, Galerkin ModelPDE. A first-order rate constant (Reactions) needs the Galerkin form.

On

Check box

Elmer Solver

Additional Elmer solver keywords
Lines appended verbatim to this interface’s Solver section in the .sif file

Text (multi-line)

Domain features

Transport Properties

Selection: Domains (all by default) · Physics ribbon › Domains, or right-click the interface in the Model Builder

\(\frac{\partial c}{\partial t} + \nabla\cdot\mathbf{J} + \mathbf{u}\cdot\nabla c = R,\quad \mathbf{J}=-D\nabla c - z\frac{D}{RT}F c\nabla\phi_l\)

Setting Symbol Unit Default Choices / type

Convection

Velocity field

User defined

User defined, Velocity field (spf)

u x

m/s

0

Expression

u y

m/s

0

Expression

u z

m/s

0

Expression

Diffusion

Diffusion coefficient

D_c

m^2/s

1e-9

Expression

Migration

Migration in electric field
Nernst-Planck transport in the electrolyte potential of the Secondary Current Distribution interface: flux -z (D/RT) F c grad(phil) (Nernst-Einstein mobility)

Off

Check box

Charge number

z

1

1

Expression

Temperature

T

K

298.15[K]

Expression

Reactions

Selection: Domains · Physics ribbon › Domains, or right-click the interface in the Model Builder

\(R = -k\,c + R_0\)

Setting Symbol Unit Default Choices / type

Reaction Rates

First-order rate constant

k

1/s

0

Expression

Reaction rate (source)

R₀

mol/(m^3*s)

0

Expression

Initial Values

Selection: Domains (all by default) · Physics ribbon › Domains, or right-click the interface in the Model Builder

Setting Symbol Unit Default Choices / type

Initial Values

Concentration

c

mol/m^3

0

Expression

Boundary features

Concentration

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(c = c_0\)

Setting Symbol Unit Default Choices / type

Concentration

Concentration

c₀

mol/m^3

1[mol/m^3]

Expression

Flux

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{J} = J_0 + k_c(c_b - c)\)

Setting Symbol Unit Default Choices / type

Inward Flux

Flux type

General inward flux

General inward flux, External convection (mass transfer)

Inward flux

J₀

mol/(m^2*s)

0

Expression

Mass transfer coefficient

k_c

m/s

1e-5

Expression

Bulk concentration

c_b

mol/m^3

1

Expression

Electrode Surface Coupling

Species flux from the electrode reaction of a Current Distribution interface: Faraday’s law.

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{J} = \frac{\nu\, i_{loc}}{n F}\)

Setting Symbol Unit Default Choices / type

Electrode Surface

Electrode surface (name of a Current Distribution feature)
e.g. es1; its kinetics give the local current density i_loc

es1

Text

Stoichiometry

Stoichiometric coefficient
Moles of this species released into the electrolyte per mole of reaction in the anodic direction (negative if consumed)

ν

1

1

Expression

Number of participating electrons

n

1

2

Expression

No Flux

Selection: Boundaries (all by default) · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{J} = 0\)

No settings.

Symmetry

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot\mathbf{J} = 0\)

No settings.

Outflow

Selection: Boundaries · Physics ribbon › Boundaries, or right-click the interface in the Model Builder

\(-\mathbf{n}\cdot D\nabla c = 0\)

No settings.

Result variables

Name Description Unit

c

Concentration

mol/m^3

tds.tflux

Diffusive flux

mol/(m^2*s)

tds.tfluxMag

Diffusive flux magnitude

mol/(m^2*s)

tds.ntflux

Normal total flux (outward)

mol/(m^2*s)