## 4 Results
### Impact of the Model's Parameters on \(t_{0A}\) and \(t_{A}\)  
#### 4.2.1 Induction Time without AS (t\({}_{0A}\))  
Validating the model enables an analysis of the impact of its components on its magnitude. Such an analysis has the potential to optimize the efficiency of RO systems with a crossflow velocity significantly greater than the permeate velocity. For this purpose, a feed velocity of V\({}_{\text{f}}\) = 0.189 m/s was selected. Any other velocity within the performance envelope of RO membranes, such as the 8-inch spiral-wound brackish water membrane [44], is also valid for analysis. The shear rate coefficient k\({}_{\text{s}}\) = 2.43 \(\times\) 10\({}^{5}\) s/m (Table 1) and standard surface energy of 9.25 mJ/m\({}^{2}\)[24] were used in the analysis.  
Since dimensionless numbers reduce the number of variables and increase the physical interpretability [45], the induction time in Equation (15) was normalized by the following:  
\[\tau_{0A}=t_{0A}/t_{\text{h}} \tag{20}\]  
where th is any time magnitude at the same unit as t\({}_{0A}\) in Equation (15). Since t\({}_{0A}\) is calculated or measured in seconds, it is convenient to normalize it using th as a unit of either 60 s or 3600 s, which provides dimensionless values \(\tau_{0A}\) numerically identical to minutes or hours, respectively.  
Next, a dimensionless shear rate K\({}_{\text{u}}\) is set as follows: K\({}_{\text{u}}\) = k\({}_{\text{s}}\)u\({}_{\text{f}}\). The dimensionless parameter \(\kappa\) (Equation (21)) representing the denominator in Equation (15), is introduced as the target dimensionless concentration at saturation.  
\[\kappa=(t_{\text{h}}/M_{\text{w}})k_{\text{c}}N_{\text{A}}A(1)c_{\text{e}} \text{{}^{2}} \tag{21}\]  
where A(1) is defined in Equation (13) for n = 1. For calcium sulfate, the exponent in Equation (21) z = 2 [41]. Finally, the model can be expressed in a dimensionless form:  
\[\tau_{0A}=\frac{3(4+\pi K_{\text{u}})n_{\text{c}}^{1/3}}{4\kappa(S_{\text{a}} -1)^{2}} \tag{22}\]  
The normalized induction time (\(\tau_{0A}\)) in Equation (22) depends on three dimensionless parameters S\({}_{\text{a}}\), K\({}_{\text{u}}\), and \(\kappa\) specific to the scalar in question. Notice that, according to Equation (1), the critical cluster size (n\({}_{\text{c}}\)) depends on S\({}_{\text{a}}\). The analyzed experimental data used in the present study show that K\({}_{\text{u}}\)\(>>\) 4/\(\pi\). Therefore, practically, \(\tau_{0A}\) is directly proportional to K\({}_{\text{u}}\). Since S\({}_{\text{a}}\) appears in both Equations (1) and (22), its impact on \(\tau_{0A}\) is illustrated graphically in Figure 4. As seen in the Figure, the most influential parameter affecting \(\tau_{0A}\) is S\({}_{\text{a}}\). \(\tau_{0A}\) exhibits a monotonic decrease with S\({}_{\text{a}}\). For S\({}_{\text{a}}\) < 1.6, \(\tau_{0A}\) decreases sharply as S\({}_{\text{a}}\) increases, while for S\({}_{\text{a}}\) > 1.6, \(\tau_{0A}\) decreases asymptotically with S\({}_{\text{a}}\). At a low S\({}_{\text{a}}\) (<1.6), the impact of K\({}_{\text{u}}\) and \(\kappa\) on \(\tau_{0A}\) is considerable. Its impact decreases asymptotically for a high S\({}_{\text{a}}\) (>1.6) until it becomes negligible for S\({}_{\text{a}}\) > 2. This dimensionless analysis simplifies the understanding of the impact of all variables and constants on \(\tau_{0A}\) (Equations (1)-(15)) by reducing it to only three variables: S\({}_{\text{a}}\), K\({}_{\text{u}}\), and \(\kappa\). This dimensionless analysis provides the following quantitative conclusion: longer induction times are achieved at a high feed velocity, high local velocity in the vicinity of the cluster surface (resulting from spacers, for example), and low scalar species concentrations, and vice versa.