###### Abstract  
A simple predictive analytical model for induction times in reverse osmosis (RO), both with and without an antiscalant (AS), has been developed based on the fundamental principles of mass and momentum balance. The simplicity of the model arises from the very low Reynolds number in the vicinity of the cluster surface, enabling the use and derivation of exact equations. The main assumption of the induction time without AS, t\({}_{\rm OA}\), is that the net growth of the cluster size results from the difference between adhesion and shear forces. With AS, the induction time, t\({}_{\rm A}\), is extended due to the competition between the AS and the scaling molecules on the cluster's surface ligands. The model was validated by fitting it to six independent datasets from experiments conducted with spiral-wound and tubular RO membranes under various operational conditions, resulting in an average difference of 8.0% (t\({}_{\rm OA}\)) and 8.7% (t\({}_{\rm A}\)) between predicted and experimental induction times. It was found that t\({}_{\rm OA}\) is governed by three dimensionless parameters: supersaturation ratio (S\({}_{\rm a}\)), shear (K\({}_{\rm u}\)), and scalar saturation (\(\kappa\)). t\({}_{\rm A}\) increases with t\({}_{\rm OA}\) and the AS concentration. reverse osmosis; mineral scaling; scale inhibition; induction time; crystal growth 2024 14 4700 10.3390/00000-0000-0000-0000-0000-0000-0000-0000-0000  
to simulate field conditions, such as reaction kinetics, wall shear stress, fluid velocity, water chemistry, temperature, and pressures. These methods include dynamic tube-blocking tests, rotating cylinder systems, scale coupons, and membrane tests [5]. In the latter, scaling is detected based on a rapid drop in the membrane permeability (flat sheet, tubular, or spiral-wound), a significant increase in feed water turbidity due to bulk precipitation of the scaling species [6; 7; 8; 9], and by real-time imaging of the membrane surface [10; 11; 12]. A comprehensive review of the characterization of incipient scaling on RO membranes can be found in [13; 14; 15; 16]. It can be concluded that the precision and reliability of the induction time determination depend on the type and sensitivity of the measurements. A theoretical model for predicting induction was based on classical nucleation theory developed by [17]. In this model, the \(\mathrm{t_{ind}}\) is regarded as the sum of three distinct crystallization times: the relaxation time (\(\mathrm{t_{r}}\)) required to achieve quasi-steady-state distribution of clusters, nucleation time (\(\mathrm{t_{n}}\)) responsible for critical size formation, and the time to crystal growth (\(\mathrm{t_{g}}\)) from the critical size to a detectable size. In solutions with low viscosity, it is assumed that \(\mathrm{t_{r}}<<\mathrm{t_{n}}\) leads to \(\mathrm{t_{ind}}\sim\mathrm{t_{n}}+\mathrm{t_{g}}\). The model takes into consideration three scenarios, \(\mathrm{t_{n}}>>\mathrm{t_{g}}\), \(\mathrm{t_{n}}\sim\mathrm{t_{g}}\), and \(\mathrm{t_{n}}<<\mathrm{t_{g}}\), each involving different possible nucleation and growth mechanisms. Another model, based on classical nucleation theory, which considers two distinct phases of crystallization, namely nucleation and crystal growth, was employed to calculate the induction time without the dosage of AS [18]. It was postulated that the AS influences both the molar volume and Gibbs free energy associated with nucleus formation. These effects were quantified to be proportional to the AS concentration. A different approach for predicting the induction time is based on [PERSON]'s coagulation theory, which describes the temporal evolution of the cluster size distribution [3]. Their model relies on the assumption that the collision of two clusters results in a larger cluster containing the combined entities of each, and the breakup of a large cluster leads to the formation of two smaller ones.