2.2.1. Step 1. Pressure Drop Parameter Estimation

As mentioned above the pressure drop on the feed/brine side can be estimated through Darcy's law as given by Equation (5) which is written in terms of the volume flow. However, this can also be written as [21]:

$$\frac{dP}{d\mathbf{x}} = k\_{\text{fb}} \,\mu\,\,\text{v}\_{\text{f}}^{n\_{\text{f}}} \tag{20}$$

where *v*<sup>f</sup> is the feed-side fluid velocity, *µ* is the fluid viscosity, *k*fb is a friction parameter, and *n*<sup>f</sup> is a constant which is commonly assumed to be 1, although some studies have considered other values. For example, Sentilmurugan at al. also considered *n*<sup>f</sup> = 1.5 and found that changing this value only had a small effect on results [21]. In this study it is assumed *n*<sup>f</sup> = 1 and hence Equation (20) is equivalent to Equation (5).

The estimation of *<sup>b</sup>* is possible through plotting *<sup>P</sup>*<sup>o</sup> <sup>−</sup> *<sup>P</sup>*<sup>i</sup> against *<sup>L</sup> φ*sinh*φ* [(*F*<sup>i</sup> + *F*o)(cosh(*φ*) − 1)] (from Equation (16)) and fitting a linear expression should give *b* as the gradient, as suggested by Sundaramoorthy et al. [30,31].

However, this requires knowledge of the feed/brine side outlet pressure which may not be provided or possibly not measured as part of experimental studies looking at reverse osmosis desalination. In these cases the pressure drop might be estimated based on the maximum pressure drop specified by the manufacturer. For example, we might estimate that the highest flow rate tested experimentally gives a pressure drop which is 100% of the maximum:

$$b = \frac{P\_{\text{dropmax}}}{L \, F\_{\text{i,max}}} \tag{21}$$

Alternatively, if a friction factor is available the value of *b* can be readily found:

$$b = \frac{k\_{\text{fb}}\,\mu}{A\_{\text{f}}} \tag{22}$$

where *A*<sup>f</sup> is the cross sectional area of the feed channel and the viscosity is calculated for a single typical experimental inlet value.
