Language:

Main >> Applications >> Sample problems >> Ficks laws of diffusion
An example is devoted to modeling the diffusion of water through a concrete wall. The diffusion equation (Fick's law) and the heat transfer equation have similarity. To simulate the diffusion process the heat transfer problem is calculated.
Problem Type
Planeparallel problem of transient heat transfer.
Geometry
Given
Initial concrete internal humidity: C0 = 10%;
Diffusion coefficient: D = 0.0001 cm^{2}/s;
Task
Estimate the water penetration into the concrete wall after 5 days exposure.
Solution
The diffusion equation* (Fick's law) for the onedimensional case is:
dC/dt = D * d^{2}C / dx^{2},
where C(x, t) is the water concentration, D is the diffusion coefficient.
Heat equation** for the onedimensional case is:
dT/dt = a * d^{2}T / dx^{2},
where T(x, t) is the temperature, a is the thermal diffusivity.
The diffusion equation (Fick's law) and the heat transfer equation have similarity. The role of concentration in the diffusion equation is the same as a role of the temperature in the heat transfer equation. And the role of the diffusion coefficient D is performed by the thermal diffusivity, which is calculated as a = λ /Cρ,
where λ is thermal conductivity, C is specific heat capacity and ρ is density.
We can choose any values of λ, C and ρ to achieve the desired thermal diffusivity value. In the thermal problem, the following values were taken to specify the thermal diffusivity 0.0001 cm^{2}/s: λ = 0.028 W/Km, C = 1273 J/kgK, ρ = 2200 kg/m^{3}.
a = 0.028 W/Km / (1273 J/kgK * 2200 kg/m3) = 1e8 m^{2}/s = 1e4 cm^{2}/s
Result
After 5 days, the humidity of concrete on the airside of the wall will be 56%, and the average volume moisture content will be 72%.
The distribution of humidity (temperature) inside the wall after 5 days exposure is shown in the figure.
Reference
*Wikipedia: Diffusion
**Wikipedia: Heat equation
Download simulation files (files may be viewed using any QuickField Edition).