**5. Conclusions**

An analytical model is proposed in this study for simultaneously predicting the removal of both salt and boron from seawater through reverse osmosis using spiral wound desalination membrane modules. This model and the fitting procedure is an extension of the methods proposed by Sundaramoorthy et al. for the removal of organic solutes [30,31] which is modified and extended to predict the removal of both salt and boron from seawater.

The fitting procedure proposed here is sequential, starting with the prediction of a pressure drop coefficient, followed by the fitting of water and salt transport coefficients. Subsequently the boron transport coefficients and the mass transfer correlation coefficients

can be fitted independently of each other. In all of these steps it is shown that the parameters can be obtained through linear fitting using experimental values and calculated parameters from previous steps. Hence, this approach offers a very simple method for obtaining all the parameters needed to build the predictive model.

The analytical model equations can be solved by following the steps given in Section 2.3 which offers a much simpler method for simulation of separation performance compared with the more complex numerical finite-difference type models which require solving much larger numbers of equations and more computational effort. A basic comparison of the CPU time required to simulate three modules 1000 times (optimization will typically require simulation of configurations at least 1000 times, in some cases much more) shows that the proposed analytical model required 7.3 s while a numerical model with 100 discrete points required 4 min 21.2 s using an i7 3.30 GHz intel computer.

Although the proposed model is simpler than numerical finite-difference type models it is shown to give similar accuracy when comparing the predicted outlet permeate flow rate, salt, and boron rejection. This type of model should be appropriate for the design and optimization of multistage desalination systems due to its simplicity and low computational requirements.

**Supplementary Materials:** The following are available online at https://www.mdpi.com/article/10 .3390/su13168999/s1.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article, in a supplementary file and in references. Experimental data used for fitting parameters and validation can be found in references [21,23,29]. Parameters fitted in this study can be found in Table 2 in this article. Model predictions from Figures 6–11 using the developed model can be found in the supplementary spreadsheet file.

**Conflicts of Interest:** The authors declare no conflict of interest.
