2.2.2. Step 2. Water and Salt Transport Parameter Estimation

The values of coefficients *A*<sup>w</sup> and *B*<sup>S</sup> can be estimated based on the equations given by Sundaramoorthy et al. which are as follows [30,31]:

$$\phi = \cosh^{-1}\left[\frac{(F\_{\text{i}} + F\_{\text{o}}) - \beta \, F\_{\text{o}}}{(F\_{\text{i}} + F\_{\text{o}}) - \beta F\_{\text{i}}}\right] \tag{23}$$

$$\beta = \frac{P\_{\text{i}} - P\_{\text{o}}}{P\_{\text{i}} - P\_{\text{p}}} \tag{24}$$

The value of *φ* can be calculated directly from inlet and outlet volumetric flow rates and pressures. Hence, a plot of 1/*φ* <sup>2</sup> against *TC*<sup>p</sup> should give a linear fitting which can be used to calculate the values of *A*<sup>w</sup> and *B*S:

$$\frac{1}{\phi^2} = \left(\frac{i\gamma}{L^2 \mathcal{W}bB\_\mathcal{S}}\right) T\mathcal{C}\_\mathcal{P} + \left(\frac{1}{L^2 \mathcal{W}bA\_\mathcal{W}}\right) \tag{25}$$

This is the same as the equation given by Sundaramoorthy et al. [30,31] except with the addition of *i* to account for the presence of NaCl.

If the outlet pressures *P*<sup>o</sup> are not measured then this can be estimated using the fitted *b* value and the following approximate expression:

$$P\_{\rm i} - P\_{\rm o} = L \text{ b } \frac{(F\_{\rm i} + F\_{\rm o})}{2} \tag{26}$$

It is also worth noting that the above fitting should utilize the inlet and outlet conditions for a single feed channel, accounting for the number of leaves and the number of feed channels per leaf. Additionally, while Sundaramoorthy et al. [30,31] assume the fitted constants are independent of temperature, this fitting can also be performed separately for each set of data at each temperature which can then be used to fit a temperature dependent term. For example, Arrhenius-type equations can be used [23,37]:

$$A\_{\rm W} = A\_{\rm W0} \exp\left[\frac{-E\_{\rm A}}{R} \left(\frac{1}{T} - \frac{1}{T\_0}\right)\right] \tag{27}$$

$$B\_{\rm S} = B\_{\rm S0} \exp\left[\frac{-E\_{\rm B}}{R} \left(\frac{1}{T} - \frac{1}{T\_0}\right)\right] \tag{28}$$

where *E*<sup>A</sup> and *E*<sup>B</sup> are apparent activation energies, *R* is the gas constant, and *A*w0 and *B*S0 are the values of water and salt transport coefficients at temperature *T*0. The above equations can be used to evaluate the values at other temperatures. Although Arrhenius equations are more commonly associated with chemical reactions, Mehdizadeh et al. have shown that this type of relation also works well for predicting fluxes through membranes at different temperatures as they argue it is a similar phenomenological process [37].
