## Appendix A
### Short Review on Research Work with Solution-Diffusion-Based Models
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Objective** & **Osmotic** & **Type of** \\
**Process** & **Membrane** \\ \hline
[43] &
\begin{tabular}{c} **Presents equations for the volumetric flow considering the** \\ membrane orientation, DECP, and CECP and evaluates the \\ model against experimental data. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[44] &
\begin{tabular}{c} **Presents a mathematical model of a scale-up PRO process** \\ capable of considering the increase in process scale and the \\ detrimental effects, like CICP, DECP, and RSF. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[45] &
\begin{tabular}{c} **Presents an analog electric circuit model representing PRO.** \\ The model considers RSF, CICP, and salt storage capacity of water. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[46] & \
\begin{tabular}{c} **Presents equations for a simplified square channel counter** \\ flow configuration. Present an equation for the diffusional resistance with pure water as feed solution and \(\Delta P=0\). \\ \end{tabular} & PRO &
\begin{tabular}{c} **Tubular** \\ Membrane \\ \end{tabular} \\ \hline
[47] &
\begin{tabular}{c} **Presents a model for an integrated seawater reverse osmosis** \\ (SWRO) and pressure-retarded osmosis (PRO). It considers components such as the SWRO unit, energy recovery device (ERD), PRO unit, high-pressure and low-pressure pumps, turbine, and others. \\ \end{tabular} & SWRO- PRO & Flat Sheet \\ \hline
[48] &
\begin{tabular}{c} **Model for forward osmosis (FO) and pressure-retarded osmosis (PRO) based on irreversible thermodynamics (IT), specifically the Spiegler-Kedem (SK) model. This model includes three phenomenological parameters: the hydraulic permeability \(A\), the solute permeability \(B\), and a reflection coefficient \(\sigma\) that represents the convective effects or coupling effects in mass transport in the active layer of the membrane.** \\ \end{tabular} & FO, PRO & Flat Sheet \\ \hline
[49] &
\begin{tabular}{c} **Presents equations for flat sheet, hollow fiber, and** \\ double-skinned hollow fiber membranes, considering \\ membrane orientation, DECP, and CICP. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[50] &
\begin{tabular}{c} **Presents one-dimensional (1D) analytical equations for a** \\ parallel flow configuration in pressure-retarded osmosis \\ (PRO) exchangers by using actual brine and feed salinity values from the Kuwait desalination industry. Uses mass transfer units. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[51] &
\begin{tabular}{c} **P****Proposes a model to represent the difference between** \\ experimental and theoretical water flow and explain this \\ difference with cavitation. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[52] &
\begin{tabular}{c} **P****Proposes a model that considers cavitation and** \\ independence between the effects of feed and draw \\ hydraulic pressures. \\ \end{tabular} & PRO & Flat Sheet \\ \hline
[53] & \
\begin{tabular}{c} **Depicts the limitations of a stand-alone PRO process and the** \\ **effectiveness of a RO-PRO hybridized process.** \\ \end{tabular} & PRO- RO &
\begin{tabular}{c} **Flat Sheet,** \\ Hollow Fiber \\ \end{tabular} \\ \hline
[54] &
\begin{tabular}{c} **Presents a model that describes an osmotic power station. It** \\ considers the relevant water and salt flows in the system. It** \\ also accounts for irreversible losses in the flow across the \\ membrane as well as through the membrane unit, and in the surrounding pump–turbine system. \\ \end{tabular} & PRO & Flat Sheet \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cont. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Objective** & **Osmotic** & **Type of** \\
**Process** & **Membrane** \\ \hline
[55] & \ \begin{tabular}{c} **Presents models for scaled-up dual-stage PRO power plants** \\ compared to a single-stage PRO system with the same \\ membrane area. These models consider the pressure and \\ flow drop as well as the salinity change along the membrane. \\ \end{tabular}  & PRO & Flat Sheet \\  &