## Appendix A  
### Analytical Method for Obtaining the Solution-Diffusion Model  
In [21], the authors use an energy balance method and Fick's law to obtain the solution-diffusion model. The authors consider that the chemical potential of the \(i\)-th element in a solution \(\mu_{i}\) mainly comprises a concentration component and a convective component. \[\mu_{i}=\mu_{i}^{0}+R\ T\ln\left(\frac{\gamma_{i}\ C_{i}}{\rho_{i}\ m_{i}} \right)+\hat{\
u}\ (P_{i}-P_{sat,i}) \tag{1}\]  
where \(R\) is the gas constant, \(T\) is the temperature, \(\gamma\) is the activity coefficient, \(C\) is the concentration, \(M\) is the molecular weight, \(\rho\) is the molecular density, \(\hat{\
u}\) is the molar volume, and \(P\) is pressure. The superscript \(0\) denotes that the quantity corresponds to a pure substance, the subscript \(i\) denotes the i-th element in a solution, and the subscript \(sat\) refers to the saturation vapor pressure. Figure 11 shows a membrane in an osmotic process. It shows the parameters necessary to establish a difference between the chemical potential of the bulk of the feed and draw solutions (\(\mu_{Db}\) and \(\mu_{Fb}\)) and the chemical potential on the area near the membrane (\(\mu_{Dm}\) and \(\mu_{Fm}\)). Now, for the draw solution side, regarding the velocities, concentrations, and pressures between the bulk of the feed and draw solutions and these parameters in the area near the membrane, considering a closed system that has no losses, the chemical potential should remain balanced. We have, then, the following equation: \(\mu_{Db}=\mu_{Dm}\). \[\mu_{D}^{0}+R\ T\ln\left(\frac{\gamma_{Db}C_{Db}}{\rho_{Db}m_{Db}}\right)+\hat{ \
u}_{Db}(P_{Db}-P_{sat,Db})=\mu_{D}^{0}+R\ T\ln\left(\frac{\gamma_{Dm}C_{Dm}}{ \rho_{Dm}m_{Dm}}\right)+\hat{\
u}_{Dm}(P_{Dm}-P_{sat,Dm}) \tag{2}\]  
where subscripts \(D\) and \(F\) correspond to quantities in the draw and feed solution, and the subscripts \(b\) and \(m\) correspond to the zone near the membrane and the zone. From Equation (A2), when considering that the water is an incompressible fluid and that the membrane is a solid dense element (\(P_{Db}=P_{Dm}\), \(\vartheta_{Db}=\vartheta_{Dm}\)), we arrive at the following expression:  
\[C_{Dm}=C_{Db}\ G_{D}\] (A3)  
where the auxiliary variable \(G_{D}\) is defined as follows:  
\[G_{D}=\frac{\gamma_{Db}\ C_{Fb}}{\gamma_{Dm}\ C_{Db}}\]  
Similarly, when we perform an energy balance at the feed solution, we have the following: \(\mu_{Fm}=\mu Fb\). \[\mu_{\rm F}^{0}+R\ T\ln\biggl{(}\frac{\gamma_{\rm FB}C_{\rm FB}}{\rho_{\rm FB} m_{\rm FB}}\biggr{)}+\vartheta_{Fb}(P_{\rm Pb}-P_{sat,Fb})=\mu_{\rm F}^{0}+R\ T\ln\biggl{(}\frac{\gamma_{\rm Fm }C_{\rm Fm}}{\rho_{\rm Fm}m_{Fm}}\biggr{)}+\vartheta_{Fm}(P_{\rm Fm}-P_{sat, Fm})\] (A4)  
In this case, in addition to the considerations related to volumes and liquid incompressibility, we consider that \(P_{Fm}=P_{Db}\), and after rearranging expression (A4), we obtain the following equation:  
\[C_{\rm Fm}=C_{\rm Fb}\ G_{\rm F}\ e^{-\frac{\zeta_{F}}{RT}(P_{Db}-P_{Fb})}\] (A5)  
where the auxiliary variable \(G_{F}\) is defined as follows:  
\[G_{D}=\frac{\gamma_{Db}\ C_{Fb}}{\gamma_{Dm}\ C_{Db}}\]  
We have, then, a semipermeable membrane like the one presented in Figure A1. If it is considered that the solvent flow on its majority is due to the diffusive effect, we make use of [PERSON]'s law and arrive at the following expression:  
\[J_{w}=\frac{D_{w}}{l}(C_{Dm}-C_{Fm})\] (A6)  
where \(l\) is the length of the cross-section of the membrane, and \(D_{w}\) is the diffusivity of the water. When we replace Equations (A3) and (A5) in Equation (A6), we obtain the following:  
\[J_{w}=\frac{D_{w}}{l}\biggl{(}C_{Db}\ G_{D}-C_{Fb}\ G_{F}\ e^{\frac{\zeta_{F} }{RT}(P_{Db}-P_{Fb})}\biggr{)}\] (A7)  
Up to this point, we have considered a generic osmotic process.