A survival analysis framework for predicting gypsum scaling risk in dynamically concentrating desalination systems

Gypsum scaling remains a persistent challenge in high-recovery desalination systems due to supersaturation and subsequent crystallization. Although gypsum crystallization kinetics are commonly characterized using static induction-time experiments, such measurements do not capture the evolving nucleation risk in dynamic systems where concentration changes continuously. In this study, we introduce a survival-analysis framework to quantify the time-dependent probability of gypsum nucleation in dynamically concentrating systems. The approach leverages static induction-time data to estimate the evolution of crystallization risk as concentration and time progress. We validated the approach using benchtop dynamic experiments in which gypsum nucleation was monitored by a UV-Visible spectrophotometer under controlled CaSO₄ concentration rates ranging from 0.11 to 4.67 mMmin^-1. Using experimentally determined nucleation kinetics, the framework predicted dynamic induction times with a mean error of 12.4%. When empirical models for nucleation rate and growth time were coupled with the survival-analysis framework, the model captured dynamic induction times across experiments conducted at 25 and 50 °C, yielding an R² value of 0.95 and a mean absolute error of 1.7 min. The framework was evaluated using batch reverse osmosis and membrane distillation case studies from the literature. It predicted a low nucleation probability of 18% in the reverse osmosis system, consistent with the absence of observed scaling, and captured the observed scaling onset in the membrane distillation system. This work provides a probabilistic framework for predicting scaling risk and supporting high-recovery operation in dynamic desalination systems while accounting for the full concentration history.