P - Technische Universiteit Eindhoven

6. Crystallization of sodium sulfate in porous media by drying 63
6.2 Drying experiments
During drying liquid moisture is transported to the surface where it can leave the material
by evaporation. The ions in the solution will be transported by two processes, i.e.,
advection with the moisture and diffusion within the moisture. Hence there will be a
competition between advection, which transports ions towards the surface, and diffusion,
which will try to level off the gradients in the ion concentration. For ion transport the
molar flux J (mol m −2 s −1 ) is given by [59, 78–80]:
(
J = 10 3 θ cU − D ∂c )
, (6.1)
∂x
Here, c is the concentration in mol/l instead of mol/m 3 (accounted for by the factor 10 3 ),
θ [m 3 m −3 ] is the moisture content, D [m 2 /s] the diffusion coefficient of the salt ions in a
porous material, which is given by D = D bulk T ∗ , where T ∗ is tortuosity and D bulk is the
bulk diffusion coefficient of the salt ions, t [s] is the time, x [m] the position and U [m/s]
the fluid velocity. The change in salt content is given by law of mass conservation as:
10 3 ∂cθ
∂t = −∂J ∂x . (6.2)
By combining equations (6.1) and (6.2) the ion transport can be described by the following
advection-diffusion equation [78]:
∂cθ
∂t = ∂ [ (
θ D ∂c )]
∂x ∂x − cU . (6.3)
This equation shows that there is a competition between advection and diffusion. The
first term on the right hand side represents the diffusion process and the second term
represents the advection process. The competition between the two processes can be
characterized by a Peclet number (P e). On a macroscopic level, for instance in a porous
material, P e is given by [81]:
P e ≡ |U|L
D , (6.4)
where L [m] is the length of the sample. Hence, for P e < 1 diffusion dominates and the
ion concentration profiles will be uniform, whereas for P e > 1 advection dominates and
ions will accumulate at the drying surface.
As soon as the concentration in the porous material reaches the maximum solubility
saturation of a certain crystal phase, crystallization will start. By measuring the concentration
in a porous material one can identify which crystalline phase of sodium sulfate is
being formed.
6.3 Materials and Methods
Nuclear Magnetic Resonance (NMR) has been used to measure the spatial distribution of
the salt and moisture concentrations inside a porous material. With NMR it is possible to