Calibration and Uncertainty Analysis of Freundlich and Langmuir Isotherms Using the Markov Chain Monte Carlo (MCMC) Approach

Organic pollutants, such as dyes, widely used in textile, dyeing, and chemical industries, pose significant risks to human health and the environment if introduced into water resources. In modeling the transport of dissolved pollutants, three processes are commonly considered: advection, dispersion, and chemical mechanisms. Studies have shown that the third mechanism is often overlooked in practical modeling. Additionally, qualitative models are calibrated based on limited data, making their parameters subject to uncertainty. In this study, the Freundlich and Langmuir isotherms were analyzed. First, their parameters were estimated and calibrated using a simple optimization model. To analyze parameter uncertainty, a Bayesian approach employing the Markov Chain Monte Carlo method was adopted, utilizing the Metropolis-Hastings and Gibbs algorithms, and the results were compared. The study was conducted based on experimental data for the adsorption of Malachite Green onto activated carbon. While laboratory data allow for precise control of conditions, they may not fully represent field data due to scale and environmental constraints. Therefore, extending this research to field data could enhance the generalizability of the results and prove valuable in high-accuracy engineering designs and environmental management. The results indicated that the Langmuir isotherm performed better than the Freundlich isotherm at all temperatures, providing lower RMSE values. For instance, at 323 K, the Langmuir model demonstrated 13.55 units more accuracy than the Freundlich model. Confidence interval analysis revealed that the Metropolis-Hastings algorithm generally produced narrower and more symmetrical intervals, yielding more precise estimates. For example, for the KL parameter of the Langmuir model at 323 K, the 95% confidence interval obtained using the Metropolis-Hastings algorithm was [0.03,0.04], compared to [0.007,0.05] with the Gibbs algorithm. However, the Gibbs algorithm exhibited a higher convergence speed, making it suitable for scenarios where computational efficiency is a priority. The Metropolis algorithm’s runtime was 5.8 times that of the Gibbs algorithm. This study highlights the superiority of the Langmuir model and the Metropolis-Hastings algorithm in adsorption data analysis and uncertainty evaluation.