**4. Model Validation**

To show the prediction accuracy of the analytical model proposed here the model was tested both with the training set data and also data and conditions other than the training data.

For the FilmTec module the data from Table 2 is used in the model to predict the performance for the 32 data points used for training. The model is shown to predict reasonably well the permeate flow rate (Figure 6) and permeate salt concentration (Figure 7).

**Figure 6.** Permeate flow rate of FilmTec module comparison of experimental values [29] against model predictions for the 32 data points used to train the model.

Furthermore this model is then validated against two sets of experimental data and associated models from the literature (11 data points and analytical model results from Table 2 in the study of Avlonitis et al. [29] and 13 data points and numerical model results from Table 9 in the study of Senthilmurugan et al. [21]) which are labelled as run numbers 1–11 and run numbers 12–24 in Figures 8 and 9. This validation shows that the proposed model is reasonably accurate for all of the data points. The only exception is the permeate concentrations for run numbers 1–11 which are overpredicted by the model. The model of Avlonitis et al. is shown to predict these values slightly more accurately [29]. This is presumably because the model of Avlonitis et al. has been trained/fitted using a wider range of data which are not covered inside the training set of 32 data points [29]. The proposed model is shown to give similar accuracy compared to the numerical model of

Senthilmurugan et al. [21] in most of the run numbers 12–24. The overall accuracy of the proposed model is 6.3% for water permeation and 24.7% for permeate concentration. However, the accuracy of the model for runs 12–24 is 7.2% for water permeation and 8.9% for permeate concentration. If the model was retrained using data from run numbers 1–11 these errors in the permeate concentration could potentially be reduced. The more complex numerical model of Senthilmurugan et al. gives an accuracy for runs 12–24 of 8.8% for water permeation and 4.5% for permeate concentration.

Run numbers 1–11 show the experimental and model results from Avlonitis et al. [29] and run numbers 12–24 show the experimental and model results from Senthilmurugan et al. [21].

**Figure 7.** Permeate salt concentration of FilmTec module comparison of experimental values [29] against model predictions for the 32 data points used to train the model.

For the Saehan module the data from Table 2 is used in the model to predict the performance for the 5 data points used for testing/validation. The model is shown to predict reasonably well the permeate flow rate (Figure 10).

Furthermore, the boron rejection predicted by this model is compared against the experimental and model predictions of Mane et al. [23], as shown in Figure 11. In this Figure, the 10 data points denoted by empty circles/rings are those which were used for training and the squares are the data points which were used for testing. In this Figure, the literature model results are those given in the study of Mane et al. which were generated using a complex finite elements numerical model [23]. That model uses the boric acid (H3BO3) and borate ion (H2BO<sup>3</sup> −) transport coefficients given by Hyung and Kim [24], while the proposed model uses coefficients which are fitted to the experimental data of Mane et al. [23]. It can be seen that the model of Mane et al. fits well to the values at lower rejection data points (these values are for pH 7.5) but underpredicts the values at higher rejection (with pH at 8.5 and 9.5). Meanwhile, the proposed model gives a reasonably accurate prediction for all data points, except for some slight over- and underprediction at the lower pH values. For the testing data in Figure 11, the proposed model gives an absolute average error of 0.82% while the model of Mane et al. gives an absolute average error of 1.44%.

**Figure 8.** Permeate flow rate of FilmTec module comparison of experimental values [4,11] against model predictions for 24 different data points for testing/validation.

**Figure 9.** Permeate salt concentration of FilmTec module comparison of experimental values [4,11] against model predictions for 24 different data points for testing/validation. Run numbers 1–11 show the experimental and model results from Avlonitis et al. [29] and run numbers 12–24 show the experimental and model results from Senthilmurugan et al. [21].

**Figure 10.** Permeate volume flow rate. A comparison of experimental data [23] and model predictions.

**Figure 11.** Boron rejection for the Saehan module. A comparison of experimental data [23] and model predictions.
