## 4 Results  
### Experimental Validation of the Induction Time Prediction  
Analytical models are often considered validated even when supported by limited experimental datasets [39; 40]. The presented analytical model was validated using datasets from two independent studies [24; 25], each employing different membranes, operating conditions, and types of antiscalants, as described in detail in the Experimental Section. It should be mentioned that in both studies, the RO systems were operated at a full recycle mode, where both permeate and concentrate are recycled to the feed vessel. In these tests, the time required for the onset of scaling was measured allowing for the evaluation of the lower scaling threshold limit. This limit is defined as the point at which precipitation is prevented or at least delayed for a sufficiently long period. The results presented by [PERSON] et al. [24] showed a correlation between the induction time data and the supersaturation level derived from nucleation theory. Due to the prevention of scaling onset in commercial RO systems, induction time data for such systems were not found in the literature. While the prediction of t\({}_{0A}\) is a standalone model, the t\({}_{\rm A}\) prediction depends on t\({}_{0A}\) as described in Equation (19). In other words, the prediction of t\({}_{\rm A}\) needs experiments without AS to determine k\({}_{\rm s}\) in Equation (15) for a specific RO system. Subsequently, further experiments employing AS based on the same RO system become imperative to determine the values of k\({}_{\rm a}\) and k\({}_{\rm d}\), as defined in Equation (19). It is worth recalling that k\({}_{\rm s}\) depends on the local shear stress exerted on each cluster. Cluster size decreases with k\({}_{\rm s}\)u\({}_{\rm d}\) (Equations (4) and (5)). The shear rate constant k\({}_{\rm s}\) is determined by fitting the model to the data, and u\({}_{\rm d}\) is determined from the balance equations (Equations (4)-(6)). Therefore, k\({}_{\rm s}\)u\({}_{\rm d}\) necessarily includes the resultant u\({}_{\rm d}\) of all shear velocity differences at various points in the channels and channel types, such as narrow, spacer-filled, or tubular ones. Different RO systems exhibit different local velocities, resulting in various shear stresses. Consequently, k\({}_{\rm s}\) inherently exhibits system-dependent characteristics. However, k\({}_{\rm a}\) and k\({}_{\rm d}\) are reaction rate constants that depend on each specific pair of scalant and AS molecules, independent of any particular desalination system. Consequently, k\({}_{\rm a}\) and k\({}_{\rm d}\) may be determined in advance for each pair of scalant and AS, applicable across all desalination systems. This leaves k\({}_{\rm s}\) as the sole fitting parameter for predicting t\({}_{0A}\) and t\({}_{\rm A}\) using Equations (15) and (19), respectively, for a given RO system. The experimental data used to validate the model were for calcium sulfate scaling, hence the exponent z in Equation (4) is 2 [41], indicating that the calcium sulfate cluster growth is surface reaction controlled. A cluster growth coefficient of k\({}_{\rm c}\) = 5.31 \(\times\) 10\({}^{-8}\) kg/ (s m\({}^{2}\) mM\({}^{2}\)) was calculated from Table 4 of [42]. Fit results of the model in Equations (15) and (19) to the experimental data of [24] are illustrated in Figure 1. Each of the experimental/prediction pair symbols is related to an individual experiment. The model's adjustable parameters used were k\({}_{\rm s}\) = 1.73 \(\times\) 10\({}^{4}\) s/m for t\({}_{0A}\) and k\({}_{\rm a}\) = 3.17 \(\times\) 10\({}^{-4}\) L/mg s and k\({}_{\rm d}\) = 4.00 \(\times\) 10\({}^{-4}\) 1/s for t\({}_{\rm A}\). As seen in Table 1 and Figure 1, a good fit has been obtained between the predicted and experimental induction times, with an average prediction error of 10.8% for both datasets, both with and without AS. Individual errors for both datasets range between 3.5% and 18.9%.