*2.1. Modelling Equations*

The transport of water and salts through a membrane are typically described according to the solution–diffusion model which can be used to calculate the flux of water (*J*W) and salt (*J*S):

$$J\_W = A\_W(\Delta P - \Delta \pi) \tag{1}$$

$$J\_{\rm S} = B\_{\rm S} \left( \mathcal{C}\_{\rm b} - \mathcal{C}\_{\rm P} \right) \tag{2}$$

where ∆*P* and ∆*π* are the transmembrane pressure and osmotic pressure, and *C*<sup>b</sup> and *C*<sup>p</sup> are the brine-side and permeate-side concentrations of salt. *A*<sup>w</sup> and *B*<sup>S</sup> are the water and salt transport coefficients.

Accounting for the effect of concentration polarization which causes the concentration of salt to increase at the membrane surface, these equations should be modified to use the concentration of salt at the membrane wall (*C*w). The concentration at the membrane wall can be calculated based on the following relation [30]:

$$\frac{\mathbf{C\_w} - \mathbf{C\_p}}{\mathbf{C\_b} - \mathbf{C\_p}} = \exp\left(\frac{J\mathbf{w}}{k}\right) \tag{3}$$

where *k* is the mass transfer coefficient and so that Equation (2) is modified:

$$J\_{\rm S} = B\_{\rm S} (\mathcal{C}\_{\rm W} - \mathcal{C}\_{\rm P}) \tag{4}$$

The pressure drop can be estimated based on Darcy's law which might be written as [30]:

$$\frac{dP}{d\mathfrak{x}} = bF(\mathfrak{x})\tag{5}$$

which gives the pressure drop as a function of volume flow rate multiplied by a fixed parameter *b*. Alternatively, if knowledge about feed spacer geometry is available, pressure drop can also be estimated using more complex equations suggested by Koutsou et al. [36].

The osmotic pressure is a function of salt concentration and temperature. For low concentrations, such as those used in seawater, it may be approximated by the van't Hoff relation in Equation (6). Thus the transmembrane osmotic pressure can be calculated using Equation (7):

$$
\boldsymbol{\pi} = \dot{\boldsymbol{\gamma}} \boldsymbol{\gamma} \mathbf{T} \mathbf{C} \tag{6}
$$

$$
\Delta \pi = i \gamma T \left( \mathbb{C}\_{\mathbb{W}} - \mathbb{C}\_{\mathbb{P}} \right) \tag{7}
$$

where *γ* is the gas law constant, *T* is temperature, and *i* is the number of ionic species formed. For the organic solutes considered by Sundaramoorthy, *i* is equal to 1 [31,32] but for NaCl the value of *i* is 2.

Combining and rearranging the above equations, the flux of water and the permeate salt concentration can be calculated as follows:

$$J\_W = \frac{A\_\text{W} \Delta P}{1 + \left(\frac{A\_\text{W} i \gamma}{B\_\text{S}}\right) T \mathbf{C}\_\text{P}} \tag{8}$$

$$\mathbf{C\_P} = \frac{\mathbf{C\_b}}{\left[1 + \frac{\left(^{\text{fw}/R\_s}\right)}{\exp\left(\text{bw}/\text{k}\right)}\right]} \tag{9}$$

The above equations can also be modified using a reflection coefficient, but for simplicity this will be assumed to be equal to 1.

To estimate the transport flux of boron an expression similar to Equation (4) can be used:

$$J\_{\rm B} = B\_{\rm B} \left( \mathbb{C}\_{\rm BW} - \mathbb{C}\_{\rm Bp} \right) \tag{10}$$

where *B*<sup>B</sup> is the boron transport coefficient and *C*Bw − *C*Bp is the difference in boron concentrations. The wall concentration of boron can also be estimated with an equation similar to Equation (3) which also requires a mass transfer coefficient *k*B:

$$\frac{\mathbf{C\_{Bw}} - \mathbf{C\_{Bp}}}{\mathbf{C\_{Bb}} - \mathbf{C\_{Bp}}} = \exp\left(\frac{I\_W}{k\_B}\right) \tag{11}$$

$$\mathbf{C\_{Bp}} = \frac{\mathbf{C\_{Bb}}}{\left[1 + \frac{\left(\frac{\left(\mathbf{b} \mathbf{v}/\mathbf{a}\right)}{\exp\left(\mathbf{b} \mathbf{v}/\mathbf{a}\right)}\right)}\right]} \tag{12}$$

$$B\_{\rm B} = \frac{\{H^{+}\}}{\{H^{+}\} + K\_{\rm a1}} B\_{\rm (H\_{3}\rm BO\_{3})0} e^{(0.067(T - T\_{0}))} + \frac{K\_{\rm d1}}{\{H^{+}\} + K\_{\rm a1}} B\_{\rm (H\_{2}\rm BO\_{3}^{-})0} e^{(0.049(T - T\_{0}))} \tag{13}$$

To estimate the boron transport coefficients Hyung and Kim [24] proposed Equation (13) where they found the temperature dependence follows the same trend for all the membranes they tested. The effect of pH is included through the calculation of the fraction of boric acid (H3BO3) and borate ion (H2BO<sup>3</sup> <sup>−</sup>) which have different transport coefficients: *B*(H3BO3)<sup>0</sup> and *B*(H2BO<sup>3</sup> −)0 (the values at *T* = *T*0).

Various correlations have been proposed in the literature for estimating the mass transfer coefficient, although in most cases these correlations predict the Sherwood number (*sh*) as a function of the feed-side Reynolds number (*Re*f) and Schmidt number (*Sc*) and also sometimes consider the permeate-side Reynolds number *Re*p [31]. In this study the following general expression is considered:

$$\text{sh} = e^A \left( \text{Re}\_{\not\!\!\!}^{B} \right) \left( \text{Re}\_{\not\!\!\!}^{C} \right) \left( \text{Sc}^{D} \right) \tag{14}$$

To model the performance of a spiral wound reverse osmosis membrane for purification of seawater and boron, we made the following assumptions:


Based on these assumptions, Sundarmoorthy et al. showed that analytical solutions can be obtained for the pressure *P*, volume flow rate *F*, and water flux *J*<sup>W</sup> [30,31]. The permeate-side fluid velocities are much lower than those on the retentate side and thus the permeate-side pressure drop should be significantly lower, which is why it is often neglected, allowing for the development of Equations (15)–(19) [30]. Additionally, it has been shown by Taniguchi et al. that the mass transfer coefficient for salt is very close to that of boron and so for simplicity they are considered equal in this study [27]. The equations for pressure and volume flow can be used to calculate the outlet pressure and outlet volume flow rate as given below:

$$F\_0 = F\_{\text{i}} \cosh \phi - \frac{\phi \sinh \phi}{bL} \Delta P\_{\text{i}} \tag{15}$$

$$P\_{\rm o} = P\_{\rm i} - \frac{bL}{\phi \sinh \phi} [(F\_{\rm i} + F\_{\rm o})(\cosh(\phi) - 1)] \tag{16}$$

$$\mathcal{C}\_{\rm o} = \mathcal{C}\_{\rm p} + \frac{F\_{\rm i}(\mathcal{C}\_{\rm i} - \mathcal{C}\_{\rm p})}{F\_{\rm o}} \cosh \phi - \frac{\phi \sinh \phi}{bL} \Delta P\_{\rm i} \tag{17}$$

where ∆*P*<sup>i</sup> = *P*<sup>i</sup> − *P*<sup>p</sup> is the transmembrane pressure at the inlet and the *φ* is given by the following equation:

$$\phi = L \sqrt{\frac{WbA\_{\rm W}}{\left(1 + A\_{\rm W} \left(\frac{i\gamma}{B\_{\rm S}}\right) T \mathbf{C\_{p}}\right)}}\tag{18}$$

This parameter *φ* is a dimensionless number which is defined by Sundaramoorthy et al. in the following equation relating the second order derivative of the feed channel volume flow rate with respect to distance along the module [30]:

$$\frac{d^2F(\mathbf{x})}{d\mathbf{x}} = \frac{\phi^2}{L^2}F(\mathbf{x})\tag{19}$$

**Figure 1.** Spiral wound membrane geometry (unwound diagram).
