## 2 Materials and Methods
### Exergy Analysis Methods  
The exergy rates of each stream throughout the MD systems and the exergy efficiencies of the MD modules are analyzed by taking the basic exergy definitions [1] and relationships for MD models as reported in the literature [5; 6; 9]. The intensive parameters governing the MD process are temperature and composition, and exergies related to kinetic and potential are not considered. Hence, the exergy rate is the sum of the physical and chemical exergies of streams as given by Equation (1). \[\text{Ex}=\text{Ex}_{\text{ph}}+\text{Ex}_{\text{ch}} \tag{1}\] \begin{table}
\begin{tabular}{l c c} \hline \hline \multirow{2}{*}{**Property**} & \multicolumn{2}{c}{**Value**} \\ \cline{2-3}  & **Xzero** & **Elixir500** \\ \hline Reference temperature, K & & 288 \\ Reference pressure, MPa & & 0.1 \\ Membrane area, m\({}^{2}\) & 4.6 & 0.19 \\ Feed rate, L/min & 20 & 3.8 \\ Coolant rate, L/min & 20 & 1.9 \\ Feed temperature, K & 353 & 338–353 \\ Coolant temperature, K & & 288–328 \\ TDS of feed and coolant, g/L & 0.25 & 0.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The experimental and reference state conditions for the two types of AGMD systems tested. Figure 4: Water and heat streams considered in the Elixir500 AGMD lab unit. Exph represents the thermomechanical exergy based on the temperature of the stream, and ExCh refers to the chemical exergy from the chemical potentials of the solute components in the stream. The physical exergy can be expressed in terms of heat enthalpies and entropies at the specified condition (h, s) and reference condition (h, s) by Equation (2). \[\text{Ex}_{\text{ph}}=(\text{h}-\text{h}_{\text{o}})-\text{T}_{\text{o}}\times (\text{s}-\text{s}_{\text{o}}) \tag{2}\]  
The individual exergies due to pressure and temperature are calculated independently and sum up to the physical exergy. Even though the working pressure in MD is generally very low, its effect could be detected in exergy destruction calculations due to pressure changes, especially in series configured modules. These pressure and temperature related exergies are given as follows, in Equations (3) and (4) [5,10]. \[\text{Ex}_{\text{p}}=\dot{\text{m}}\times(\text{p}-\text{p}_{\text{o}})/\varrho \tag{3}\]  
\[\text{Ex}_{\text{T}}=\dot{\text{m}}\times\text{c}_{\text{p}}\times[(\text{h}- \text{h}_{\text{o}})-\text{T}_{\text{o}}(\text{s}-\text{s}_{\text{o}})] \tag{4}\]  
Chemical exergy based on the chemical potentials or concentrations (C) of the components in the stream is calculated as in Equation (5). \[\text{Ex}_{\text{c}}=-\dot{\text{m}}\times\text{N}_{\text{s}}\times\text{R} \times\text{T}_{\text{o}}\times\text{ln}\text{X}_{\text{s}} \tag{5}\]  
\[\text{where}\text{N}_{\text{s}}=(1000-\sum\text{C}_{1}/\varrho)/\text{MW}_{ \text{s}} \tag{6}\]  
\[\text{and}\text{X}_{\text{s}}=\text{N}_{\text{s}}/[\text{N}_{\text{s}}+\sum( \beta_{\text{i}}\times\text{C}_{\text{i}}/\varrho\times\text{MW}_{\text{i}})] \tag{7}\]  
The exergy change across the MD module or any other component of the MD system is calculated from the differences in inlet and outlet exergies as  
\[\Delta\text{Ex}=\sum\text{Ex}_{\text{in}}-\sum\text{Ex}_{\text{out}} \tag{8}\]  
The second-law efficiency of the MD system is the ratio of the minimum exergy input required (which is equivalent to the minimum work of separation) to the total actual exergy input:  
\[\eta(\%)=(\text{W}_{\text{min}}/\text{E}_{\text{xin}})\times 100 \tag{9}\]  
The minimum work input required for the MD process is the difference between outgoing and incoming total exergies of the hot stream:  
Figure 4:0  
The exergy destruction due to irreversibility in the process is calculated by subtracting exergy determined from the second-law entropy generation [1] according to the Gouy-Stodola formula:  
Figure 4:1  
where To is the reference temperature, and Sgen is the entropy generated due to irreversibility of the process, given as  
Figure 4:2  
where the Qsource and Qsink represent the heat lost and gained from the heat source and sink, respectively.