In Fig. 1 a schematic representation of a combined wicking and evaporation experiment is given. In case the sample stays saturated, as is the case in the one dimensional experiments performed in this study and there is no crystallization, the ion transport can be described by an advection–diffusion equation (see, e.g., [11]):
$$\frac{\partial c}{\partial t} = \frac{\partial }{\partial x}\left( {D^{*} \frac{\partial c}{\partial x} + uc} \right)$$
(1)
where c is the ion concentration, D* the macroscopic diffusivity of the ions within the porous material and u the macroscopic velocity of the liquid in the porous material, i.e., the Darcy speed.
The macroscopic diffusivity of the ions within a porous material is related to the microscopic diffusivity of the ions through the pores by the tortuosity T*, i.e.;
$$D^{*} = \frac{D}{{T^{*} }}$$
(2)
where T* > 1 is a correction factor for the increase in path length due to the tortuous nature of pores.
Therefore the first part of the right hand side of Eq. 1 is describing the ion flux due to diffusion, whereas the second part describes the advection of the ions along with the liquid flow. Here advection will give rise to accumulation, whereas diffusion will give rise to levelling off of the concentration. These two competing processes can be characterized with a dimensionless number, i.e., the Peclet number. In this case, based on Eq. 1 it can be defined as [12]:
$$P_{e} = \frac{uL}{{D^{*} }}$$
(3)
where L is a so-called characteristic length scale, which in this case was chosen as the length of the sample. In the case Pe > 1, advection will be dominant and there will be a concentration gradient and as a result the salts will accumulate near the surface. Whereas in the case of Pe < 1, diffusion is dominant and we expect a homogenous distribution of salt. This Pe-number was also found to be very useful for giving an indication of the effect of poulticing [13]. In this paper we only address the case of Pe > 1, where ions will be advected towards the surface and there will be a concentration build up near the interface giving rise to crystallization.
As the sample is completely saturated the liquid speed, u, will be constant throughout the sample. The liquid speed will be completely determined by the drying condition at the top, i.e., the liquid flux ql at the surface is given by:
$$q_{l} = \beta \left( {h_{\text{air}} - h_{\text{m}} \left( {c,\theta } \right)} \right)$$
(4)
where β is the mass transfer coefficient which is dependent on many parameters such as the air velocity, porosity and surface roughness, ha the relative humidity of the air and hm the relative humidity of the materials at the interface, which is a function of both the moisture content θ and the salt concentration. As the salt concentration is slowly changing near the drying surface the liquid speed will change slowly in time even if the drying airflow is kept constant. Therefore the salt concentration will have a large influence on the evaporation [14, 15].
As long as there is no crystallization at the drying surface the boundary condition for the ion transport is determined by the no flux condition, i.e.;
$$D^{*} \frac{\partial c}{\partial x} + uc = 0$$
(5)
At the wicking surface where the salt solution is absorbed the boundary condition is given by a constant flux, i.e.;
$$q_{\text{salt}} = uc_{o}$$
(6)
where co is the concentration of the liquid being absorbed at the wicking surface If we now assume that in the first-order approximation the liquid flow throughout the sample and the concentration at the inflow are constant, we can solve the partial differential Eq. 5, to as arrive at a first-order approximation of the ion concentration profiles in the experiment:
$$c\left( {x,t} \right) = \alpha \left( t \right)e^{{\frac{{ - D^{*} }}{u}x}} + c_{o}$$
(7)
where α(t) is a constant, which is a function of the time reflecting the increase of the concentration at the boundary. If we assume that the liquid velocity is constant, the increase of total ion content will be linear in time due to the regular advection of ions. Therefore by integrating Eq. 7 over the sample length l (here it is assumed that l > 4D*/u) we can get an expression for α(t), i.e.;
$$\alpha \left( t \right) = c_{o} \frac{{u^{2} }}{{D^{*} }}t + c_{o} \frac{u}{{D^{*} }}l$$
(8)
Hence it can be seen that as long as the liquid speed is constant, the concentration rise at the surface will increase linearly with time.
The solution of the concentration as function of time and space indicates that the ion concentration profiles can be described by a simple exponential decay which is dependent on the ratio between the diffusion and the liquid velocity. This dependence reflects the competition between advection and diffusion. One can define a peak width over which the concentration drops to about 2% of the original peak height α(t)-co as (see also Eq. 7):
$$\Delta = 4\frac{{D^{*} }}{u}$$
(9)
Hence as the ion diffusivity is constant, the peak width is fully determined by the liquid velocity and as the liquid velocity increases the peak width near the surface will decrease.
As soon as the ion concentration has reached the threshold level for crystallization, this mechanism will start and in this case the ion transport can be described by adding a sink term to Eq. 1 to account for crystallization, i.e.;
$$\frac{\partial c}{\partial t} = \frac{\partial }{\partial x}\left( {D^{*} \frac{\partial c}{\partial x} + uc} \right) + \gamma \left( {c - c^{*} } \right)$$
(10)
where c* is the threshold concentration at which the crystallization starts and γ is a crystallization rate. In this case the crystallization term does not only indicate whether the crystallization will take place either inside near or outside at the surface of the sample.