## 3 Dynamics Model of Unsteady Mass Transfer  
It is assumed that there is sufficient contact between the salt and water at the interface. When the solid phase is a mixed salt, it is assumed that the contact area between each salt component and water at the salt-water interface is proportional to the dry-based mole fraction of the different salt components. Assuming that the solution behaves ideally, solute transfer is only due to diffusion and electron migration under a concentration gradient. Because diffusion is very slow, the interfacial mass transfer resistance between salt and water is neglected, and only the mass transfer process in the bulk solution is considered [22], as shown in Figure 2.  
At this point, solute transfer is an unsteady diffusion mass transfer process, and the bulk water solution is not flowing. According to [PERSON]'s second law [23], the mass transfer can be calculated as follows:  
\[\frac{\partial c_{A}}{\partial\theta}=D_{AB}\bigg{(}\frac{\partial^{2}c_{A}}{ \partial x^{2}}+\frac{\partial^{2}c_{A}}{\partial y^{2}}+\frac{\partial^{2}c_ {A}}{\partial z^{2}}\bigg{)} \tag{1}\]  
where \(\theta\) is the diffusion time; \(x\), \(y\), and \(z\) are the diffusion directions; \(c_{A}\) is the salt concentration in the liquid phase; and \(D_{AB}\) is the diffusion coefficient of the salt component in water. Assuming that the diffusion direction is one-dimensional (i.e., diffusion only occurs in the vertical \(y\) direction), Equation (1) can be transformed into the following form:  
\[\frac{\partial c_{A}}{\partial\theta}=D_{AB}\frac{\partial^{2}c_{A}}{\partial y ^{2}} \tag{2}\]  
Figure 2: Schematic diagram of the diffusion model.  
The initial conditions and boundary conditions of Equation (2) are as follows:  
\[\theta=0,c_{A}=c_{A0}\ (\text{for any }y) \tag{3}\]  
\[y=0,c_{A}=c_{AS}\ (\text{when }\theta>0) \tag{4}\]  
\[y\rightarrow\infty,c_{A}=c_{A0}\ (\text{when }\theta>0) \tag{5}\]  
where \(c_{A0}\) is the initial concentration in the bulk liquid phase, \(c_{AS}\) is the salt concentration at the salt-water interface, \(c_{A0}\) has a value of zero, and \(c_{AS}\) is assumed to be equal to its saturation concentration.  
Equation (2) can be solved using the synthetic variable method to substitute the partial derivatives of synthetic variables with respect to \(y\) and \(\theta\) into Equation (2). Furthermore, the partial differential equation of Equation (2) can be transformed into an ordinary differential equation:  
\[\frac{d^{2}c_{A}}{d\left(\frac{y}{\sqrt{4D_{AB}\theta}}\right)^{2}}+2\eta\frac {dc_{A}}{d\left(\frac{y}{\sqrt{4D_{AB}\theta}}\right)}=0 \tag{6}\]  
The boundary conditions can thus be transformed as follows:  
\[\left(\frac{y}{\sqrt{4D_{AB}\theta}}\right) = \infty,\ c_{A}=\ c_{A0} \tag{7}\]  
\[\left(\frac{y}{\sqrt{4D_{AB}\theta}}\right) = 0,\ c_{A}=\ c_{AS} \tag{8}\]  
Equation (6) is integrated twice, and the integral constants that appear in the integration process are determined according to Equations (3) and (4). Finally, the error function can be used to obtain Equation (9):  
\[\frac{c_{A}-c_{AS}}{c_{A0}-c_{AS}}=erf\left(\frac{y}{\sqrt{4D_{AB}\theta}}\right) \tag{9}\]  
Equation (9) shows the relationship between diffusion time, diffusion location, and local salt concentration.  
The diffusion coefficient (\(D_{AB}\)) can be roughly derived using the Stokes equation [9]. In this calculation, some of the basic physicochemical constants are used, including [PERSON]'s constant (\(k=1.380649\times 10^{-23}\) J/K), thermodynamic temperature (\(T\)), the Stokes-Einstein radius (\(r_{A}\)) of solute \(A\), and the dynamic viscosity of the solvent (\(\eta_{0}\)).