Inverse physics-informed neural network for CaCO3 crystallization fouling: Integrating initial rate and asymptotic resistance across multi-source data

Abstract

Crystallization fouling of CaCO₃ in cooling water systems degrades heat transfer efficiency, yet the transient initial fouling rate and the asymptotic fouling resistance are traditionally evaluated using separate empirical correlations developed on distinct experimental rigs. Reconciling both descriptors within a single data-driven formulation is challenging because experimental datasets are scarce, heterogeneous across flow geometries, and sensitive to reporting conventions. This study proposes a compact Inverse Physics-Informed Neural Network (CaCO₃-PINN) with two outputs and 9703 trainable parameters, informed by dimensionless supersaturation, temperature, and wall shear stress. The network is trained on 25 experimental points compiled from three independent sources (Watkinson 1983; Choi, via Liu 1999; Arsenyeva 2013) with a physics-constrained loss coupling rate and resistance via the Arsenyeva relationship $R_f^{\ast }=b/{\tau }_w^{\alpha }$. Across five deterministic seeds, the model achieves $R^2=0.9959$–0.9985 for asymptotic resistance and $R^2=0.853$–0.910 for initial rate. The inversely recovered coefficient $b=1.30\times 10^{-4}$ K s/m matches the published plate-geometry value ($1.45\times 10^{-4}$ K s/m) within −10%, while the effective shear exponent ($\alpha \approx 1.43$) reflects the localized operating envelope rather than a universal physical revision. Leave-One-Out cross-source validation shows that the physics constraint enables partial structural transfer across disparate geometries ($R^2>0.78$ on retained sources), though predictive skill remains constrained by domain boundaries. Operational scenario evaluations demonstrate that the model functions reliably as an engineering screening tool for cooling-water heat exchangers, with demonstrated validity strictly confined to CaCO₃ precipitation under the calibrated operational envelope.