The model contains two parameters: the maximum adsorption capacity (Qm), representing the upper limit of adsorbate loading in a monolayer, and the Langmuir constant (KL), the ratio of adsorption to desorption rate constants. These parameters are obtained through regression fitting of experimental data and provides critical insights into adsorbate-adsorbent interactions. Langmuir’s equation has extensive practical applications in adsorption column design (Murphy et al., 2023), environmental management decisions regarding fate of contaminants (Bolster and Hornberger, 2007), and catalysis and chemical kinetics (Králik, 2014; Swenson and Stadie, 2019).
Nonlinear regression methods estimate Langmuir parameters through iterative minimization procedures. Due to the complexity and potential computational challenges associated with nonlinear methods, linearized forms of the Langmuir equation have mostly been employed. However, significant limitations accompany linearization techniques, including discrepancies induced by data transformation, altered error structures in the regression models, and errors introduced into both the independent and dependent variables, thoroughly documented by Bolster and Hornberger (2007). Implications include discrepancies between fitted parameter values and erroneous conclusions. In fact, most errors and inconsistencies reported from adsorption studies were associated with approaches deployed to linearize the models for adsorption (Anako Opotu et al., 2022). Additionally, most adsorption studies also fail to rigorously validate the underlying statistical assumptions for regression. The validity of these regression assumptions has a critical influence on the accuracy and reliability of model parameters, forecasting, and scientific interpretations of the model (Flatt and Jacobs, 2019).
The lack of regression assumption validation reported in the literature undermines the robustness of results, especially when applying the Langmuir equation to emerging adsorbents. When traditional assumptions are violated, symbolic regression presents a robust alternative, enhancing the accuracy and predictive capability of adsorption models without strict adherence to assumptions. Symbolic regression, a machine learning-based method that can generate analytical equations without considering constraints and provides a physics-inspired overlook (Angelis et al., 2023). It does not require any theoretical knowledge of the system since it is ultimately data-driven, potentially offering a deeper understanding of physical laws or phenomena. In contrast, these expressions may provide a model that fits well, but the parameters may lack relevance to the subject matter.
The impact of regression assumptions on the inference of Langmuir parameters and their applicability to emerging biomass adsorbents has not been comprehensively explored. This study aims to address this gap by highlighting the limitations of the linearized Langmuir equations, demonstrating rigorous regression validation methods, and presenting symbolic regression as a novel corrective approach when traditional assumptions fail by evaluating the underlying assumptions of linear regression using Langmuir linearization methods (Hanes-Woolf, Lineweaver-Burk, Eadie-Hofstee, and Scatchard) against nonlinear regression for fitting adsorption isotherms of 2,4-dichlorophenol (DCP) on activated carbon and biomass-based adsorbents. Biomass-based adsorbents evaluated include cypress sawdust, peanut hulls, sugarcane bagasse, and rice hulls. Residual diagnostics included visual assessment and statistical tests for independence (Durbin-Watson test), normality (Shapiro-Wilk test), and homoscedasticity (White test). Additionally, symbolic regression was performed to further investigate the validity of the Langmuir equation.
Key findings indicate that the linearization methods gave vastly different Langmuir parameters for the adsorbents tested. For the activated carbon, the Qmax ranged from 102 to 239 mg/g, but also produced a Qm of -5539 mg/g in one case, which is invalid as a Langmuir parameter. The range of differences varied for the biomass adsorbents. For example, cypress sawdust had a Qmax range of 11.5 to 16.2 mg/g, but peanut hulls ranged from 9.4 to 39.1 mg/g. Hanes-Woolf (Type 1) and Lineweaver-Burk (Type 2) linearizations were the most suitable for most biomass adsorbents studied, while Eadie-Hofstee (Type 3) and Scatchard (Type 4) linearizations were generally invalid due to negative parameters or assumption violations. Residual testing revealed consistent violations of independence and normality assumptions across linear models for activated carbon, while biomass adsorption results varied individually. Notably, normality assumptions consistently failed in Hanes-Woolf linearizations for peanut hulls, sugarcane bagasse, and rice hull. Nonlinear regression demonstrated greater robustness with minimal assumption violations.
Symbolic regression identified Langmuir as the best-fitting equation from over one million tested expressions. Although the complete Langmuir equation was not directly generated for biomass adsorbents, linear (Henry’s law) expressions were commonly identified, reflecting Langmuir behavior at low adsorbate concentrations. Additionally, symbolic regression revealed a prevalent square root function across several biomass adsorbents, suggesting a significant mathematical relationship that warrants further exploration. The emergence of Freundlich isotherm expressions was also noted, particularly for sugarcane bagasse.
This study revealed shortcomings in relying solely on linearized Langmuir models. A proposed workflow recommends using nonlinear or weighted nonlinear regression, starting with Hanes-Woolf or Lineweaver-Burk linearization as initial values for parameter estimation. If assumptions remain violated with nonlinear techniques, novel methods like symbolic regression should be employed. This advanced regression technique can enhance the accuracy and predictive behavior of adsorption models without the stringent need for assumption checking. Symbolic regression can also aid in understanding mechanisms in novel adsorbents.
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Murphy, O.P., Vashishtha, M., Palanisamy, P., Kumar, K.V., 2023. A Review on the Adsorption Isotherms and Design Calculations for the Optimization of Adsorbent Mass and Contact Time. ACS Omega 8, 17407–17430. https://doi.org/10.1021/acsomega.2c08155
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