A parsimonious Langmuir-Freundlich loading-response framework for sonochemical degradation of nonvolatile organic...

A physically motivated Langmuir-Freundlich loading-response framework was formulated to describe the sonochemical degradation of nonvolatile organic contaminants in water. Acoustic cavitation was represented as a heterogeneous oxidative microenvironment in which contaminant degradation depends on apparent occupation of the bubble-liquid interfacial region and the finite oxidative capacity generated under fixed operating conditions. The framework was evaluated using naphthol blue black (NBB) and furosemide as chemically distinct nonvolatile contaminants. For NBB, increasing the initial concentration from 3 to 120 mg/L increased the initial degradation rate from 0.22268 to 2.1502 mg/L·min, while the 30-min removal efficiency decreased from 100 % to 43.46 % and the net H₂O₂ formation rate decreased from 5.4794 to 3.9161 μmol/L·min. Initial-rate calibration gave r_max = 3.8324 mg/L·min, K_LF = 1.1298 × 10^-2 L/mg, and n = 0.8256. To reduce parameter redundancy, all concentration-time profiles for each compound were described using four global parameters, r_0,ref,T, β_T, K_LF,T, and n_T, without concentration-specific fitted coefficients. The calibration RMSE values were 0.0196 for NBB and 0.0399 for furosemide. Blocked leave-one-concentration-out validation gave pooled RMSE values of 0.0391 and 0.0566, respectively. For NBB, 30-min removal measurements at 7, 10, 40, and 120 mg/L, which were excluded from time-resolved parameter estimation, were calculated with an RMSE of 1.79 percentage points. Comparison with simpler formulations showed that the loading-dependent pseudo-first-order model gave the lowest cross-validated RMSE for NBB, whereas the classical Langmuir and Langmuir-Freundlich formulations gave comparable cross-validated performance for furosemide. Parameter confidence intervals and covariance-based correlation analysis identified strong compensation among the initial-rate parameters but lower correlations in the four-parameter time-resolved formulation. The fitted coefficients are therefore interpreted as condition-specific apparent descriptors rather than direct measurements of interfacial adsorption, radical flux, or molecular cooperativity. The framework provides a reduced-order tool for interpolation and treatment-time calculations within a calibrated operating domain, but application to other compounds, reactors, or solution matrices requires recalibration and independent validation.