## 5 Conclusions  
When small clusters of sparingly soluble salt move at a constant velocity in the feed solution of RO systems, a small relative velocity, \(\mathrm{u_{d}}\), flowing over their surface is generated, resulting in very low Reynolds numbers (Re = \(\mathrm{u_{d}R/\
u<<1}\)) close to the cluster surface. This characteristic has been utilized to develop a simple analytical model for predicting the induction times in both AS-free and AS-present RO systems. The predictive model is based on principles of mass and momentum balance. The very low Reynolds number allows for the derivation of precise equations for drag and shear forces.  
The main assumption for predicting t0A is that the net growth of the cluster size arises from the interplay between adhesion and shear forces, yielding an exact equation for the induction time t0A. For tA, the primary assumption is the competition between AS  
Figure 4: Impact of \(\mathrm{S_{a}}\) and \(\mathrm{K_{u}}\) on the normalized induction time \(\tau_{0A}\). It should be noted that the impact of the cluster critical radius (which is proportional to \(\mathrm{n_{c}}^{1/3}\)) on \(\tau_{0A}\) is similar to that of \(\mathrm{K_{u}}\).  
and the scalar molecules on the surface ligands of clusters, which results in a prolonged induction time.  
The model was validated by fitting it to six independent datasets generated from experiments conducted using spiral-wound and tubular membranes under various operational conditions. Average differences of 2.4% to 10.8% were obtained for \(t_{0A}\), with 5.2% to 10% for \(t_{A}\), between the experimental and predicted induction times, with a range from 1.5% to 20%.  
The predicted induction time, \(t_{0A}\), was normalized (\(\tau_{0A}\)), yielding three independent and dimensionless parameters that exert opposite effects on \(\tau_{0A}\). The shear variable \(K_{u}\) extends \(\tau_{0A}\), while the supersaturation ratio \(S_{a}\) and the saturation parameter \(\kappa\) shorten it. Specifically, for \(S_{a}<1.6\), \(\tau_{0A}\) sharply decreases as \(S_{a}\) increases. At higher \(S_{a}\) values, \(\tau_{0A}\) asymptotically decreases with \(S_{a}\), tending towards zero, rendering the impact of all three parameters negligible. An analysis of the \(t_{A}\) behavior indicates a monotonic increase with \(t_{0A}\) and \(C_{a}\).  
The analytical model provides simple quantitative insight into the induction times of RO systems. To facilitate the model's prediction of the induction time for pairs of scalars and antiscalants, the parameters \(k_{a}\) and \(k_{d}\) need to be a priori determined experimentally. These parameters are common to all sizes of desalination systems.  
Conceptualization, A.S.; methodology, A.S. and H.S.; formal analysis, A.S., R.S. and H.S.; resources, R.S.; data curation, H.S.; writing--original draft preparation, A.S. and H.S.; writing--review and editing, R.S. and H.S.; project administration, H.S. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.