2.2.3. Step 3. Boron Transport Parameter Estimation

The transport coefficients of boron can be estimated based on the following equation suggested by Hyung and Kim which accounts for the transport of boric acid (H3BO3) and borate ion (H2BO<sup>3</sup> −) [24]:

$$B\_{\rm B} = \alpha\_0 B\_{\rm (H\_3BO\_3)} + \alpha\_1 B\_{\rm (H\_2BO\_3{-})} \tag{29}$$

In this equation, *α*<sup>0</sup> and *α*<sup>1</sup> represent the fractions of boric acid and borate ion which can be estimated using the apparent dissociation constant *K*a1 and the *H*<sup>+</sup> ion concentration [24]:

$$\mathfrak{a}\_{\mathbf{0}} = \frac{\{H^{+}\}}{\{H^{+}\} + \mathrm{K}\_{\mathbf{a}1}} \tag{30}$$

$$\alpha\_1 = \frac{K\_{\rm a1}}{\{H^+\} + K\_{\rm a1}} \tag{31}$$

The value of *K*a1 can be determined by a correlation in terms of salt concentration and temperature as given by the correlation of Edmond and Gieskes as presented by Nir and Lahav [38]:

$$
\log\_{10}{K\_{\rm a1}} = \frac{2291.90}{T} + 0.01756 \, T - 3.3850 - 0.32051 \left( \frac{\text{S}}{1.80655} \right)^{1/3} \tag{32}
$$

where *T* is the temperature in kelvin, and *S* is the concentration of salt in g/L. It was also noted by Nir and Lahav [38] that a number of authors have missed the temperature dependence from the 0.01756 *T* term when writing this correlation.

Although in principle the temperature dependent factors can also be estimated here through fitting expressions similar to Equation (27), the temperature-dependent expressions determined by Hyung and Kim can also be used since they show that their fitted parameters fit well for a number of different membranes tested [24]. The work of Hyung and Kim also gives values for the boric acid and borate ion transport coefficients for those types of membranes, and these values are used by Mane et al. as part of their numerical simulation model [23]. However, the simulation results of Mane et al. underpredict the rejection of boron for higher pH values (8.5 and 9.5) compared with their experimental results [23]. This difference could be due to the fact that the transport coefficient values determined by Hyung and Kim were based on a flat sheet membrane [24] while Mane et al. utilized a spiral wound module with the same material [23]. To account for this, it is suggested here that the values of *B*(H3BO3)<sup>0</sup> and *B*(H2BO<sup>3</sup> −)0 should be fitted for each membrane material and for each design of membrane module.

The values of *B*<sup>B</sup> can be determined from experimental measurements and calculation with Equation (12) rearranged as (assuming that *k*<sup>B</sup> = *k*):

$$B\_{\rm B} = \frac{\mathsf{C\_{Bp}} \, f\_{\rm W}}{\left(\mathsf{C\_{Bb}} - \mathsf{C\_{Bp}}\right) \, e^{\left(f\_{\rm W}/k\right)}}\tag{33}$$

Since *α*<sup>0</sup> + *α*<sup>1</sup> = 1 then for each temperature measured the values of *B*<sup>B</sup> can be plotted against *α*<sup>0</sup> which should give a linear fit with intercept *B*(H2BO<sup>3</sup> −) and gradient equal to *B*(H3BO3) − *B*(H2BO<sup>3</sup> −) so that the transport coefficients of boric acid and borate ion can be determined. If values are fitted at each temperature the temperature dependence can also be included.
