*3.4. Determination of Mass Transfer Correlation Coefficients*

The value of the Sherwood number can be calculated for each experimental point. If both brine inlet and outlet data are available then the mass transfer coefficient and Sherwood number can be calculated for both points. However, in this case since the outlet pressures are not given in the case study references, only the inlet conditions are used to estimate mass transfer correlations:

$$k\_{\rm i} = \frac{J\_{\rm Wi}}{\ln\left[ \left( \frac{I\_{\rm Wi}}{B\_{\rm S}} \right) \left( \frac{C\_{\rm P}}{C\_{\rm i} - C\_{\rm P}} \right) \right]} \tag{42}$$

$$sh\_{\rm i} = \frac{k\_{\rm i} D\_{\rm i}}{d\_{\rm e}} \tag{43}$$

**Spiral Wound Module FilmTec FT30 Saehan RE4040-SR** *b* atm m−<sup>4</sup> s −1 2.9760 <sup>×</sup> <sup>10</sup><sup>3</sup> 1.0126 <sup>×</sup> <sup>10</sup><sup>3</sup> *T*<sup>0</sup> (K) 293.15 298.15 *A*w0 m atm−<sup>1</sup> s −1 2.5258 <sup>×</sup> <sup>10</sup>−<sup>7</sup> 2.7550 <sup>×</sup> <sup>10</sup>−<sup>7</sup> *B*S0 m s−<sup>1</sup> 4.0699 <sup>×</sup> <sup>10</sup>−<sup>8</sup> 1.7062 <sup>×</sup> <sup>10</sup>−<sup>8</sup> *E*<sup>A</sup> J mol−<sup>1</sup> 1.4192 <sup>×</sup> <sup>10</sup><sup>4</sup> *EB* J mol−<sup>1</sup> 4.2116 <sup>×</sup> <sup>10</sup><sup>4</sup> *B*(H3BO3)<sup>0</sup> m s−<sup>1</sup> 5.4306 <sup>×</sup> <sup>10</sup>−<sup>7</sup> *B*(H2BO<sup>3</sup> −)0 m s−<sup>1</sup> 5.3760 <sup>×</sup> <sup>10</sup>−<sup>8</sup> *A* −1.2604 5.619 *B* 0.35923 *C* 0.65885 0.5641 *D* 0.86483

**Table 2.** Estimated and fitted parameters for case study of spiral wound membrane modules.

**Figure 5.** Plot of *B*<sup>B</sup> (calculated using experimental data [23] and fitted parameters *b*, *A*w, and *B*s) against *α*<sup>0</sup> together with linear fit.

These values can be calculated for the two case studies considered here which can be used to fit values of *A*, *B*, *C*, and *D* in Equation (34). In principle, all four of these parameters can be fitted simultaneously using linear fitting, but some of these may be

statistically insignificant, in particular for the Saehan membrane data which are based on 10 data points [23] all with very similar salt rejection and with the same value of *Sc* at the inlet. For this reason initially *A*, *B*, and *C* were fitted simultaneously. However, when used in the prediction code, this sometimes led to erroneous calculations of *C<sup>p</sup>* = *C<sup>i</sup>* ; perhaps because the correlation is overfitted to a narrow range of conditions. Hence, a simpler expression is tested using either *A* and *B* or *A* and *C*. In other words, trying to fit a linear relation between either ln(*sh*) and ln(*Re*f) or ln(*sh*) and ln *Re*p and in this case ln *Re*p was found to more statistically significant; the values of *A* and *C* are given in Table 2.

For the FilmTec membrane module a larger set of data with 32 data points [29] was used to fit all four parameters as given in Table 2.
