Abstract
A novel stochastic model is proposed to characterize the adsorption kinetics of pollutants including dyes (direct red 80 and direct blue 1), fluoride ions, and cadmium ions removed by calcium pectinate (Pec-Ca), aluminum xanthanate (Xant-Al), and reed leaves, respectively. The model is based on a transformation over time following the Ornstein–Uhlenbeck stochastic process, which explicitly includes the uncertainty involved in the adsorption process. The model includes stochastic versions of the pseudo-first-order (PFO), pseudo-second-order (PSO), and pseudo-[Formula: see text]-order (PNO) models. It also allows the estimation of the adsorption parameters, including the maximum removal capacity ([Formula: see text]), the adsorption rate constant ([Formula: see text]), the reaction pseudoorder ([Formula: see text]), and the variability [Formula: see text]. The model fitted produced [Formula: see text] values similar to those of the nonstochastic versions of the PFO, PSO, and PNO models; however, the obtained values for each parameter indicate that the stochastic model better reproduces the experimental data. The [Formula: see text] values of the Pec-Ca-dye, Xant-Al-fluoride, and reed leaf-Cd+2 systems ranged from 2.0 to 9.7, 0.41 to 1.9, and 0.04 and 0.29 mg/g, respectively, whereas the values of [Formula: see text] ranged from 0.051 to 0.286, 0.743 to 75.73, and 0.756 to 8.861 (mg/g)1- n/min, respectively. These results suggest a variability in the parameters [Formula: see text] and [Formula: see text] inherent to the natures of the adsorbate and adsorbent. The obtained [Formula: see text] values ranged from 1.13 to 2.02 for the Pec-Ca-dye system, 1.0–3.5 for the Xant-Al-fluoride system, and 1.8–3.8 for the reed leaf-Cd+2 system. These ranges indicate the flexibility of the stochastic model to obtain fractional [Formula: see text] values, resulting in high [Formula: see text] values. The variability in each system was evaluated based on [Formula: see text]. The developed model is the first to describe pollutant removal kinetics based on a stochastic differential equation.
Highlights
Water pollution is currently a transcendental issue
The proposed model was applied in three adsorbent–adsorbate systems: direct red 80 (DR80) and direct blue 1 (DB1) dyes adsorbed by calcium pectinate (Pec-Ca); the removal of fluorides using aluminum xanthanate (Xant-Al); and cadmium removal by reed leaves
These systems are denominated as PecCa-DR80 and Pec-Ca-DB1, respectively
Summary
Biological, and physical methods have been used to remove different contaminants that affect aquatic ecosystems Among these methods, adsorption is relatively inexpensive and easy to scale up to industrial levels and shows great potential to remove specific pollutants from aqueous media [1,2,3,4]. The Ornstein– Uhlenbeck (OU) model has been applied in various areas such as finance, environmental modeling, and biological systems [15,16,17,18] This novel TOU adsorption model explicitly includes the uncertainty (randomness) that exists in the adsorbate removal data and provides additional information that can be used to reduce the cost of adsorption-based pollutant removal. The proposed model was applied in three adsorbent–adsorbate systems: direct red 80 (DR80) and direct blue 1 (DB1) dyes adsorbed by calcium pectinate (Pec-Ca); the removal of fluorides using aluminum xanthanate (Xant-Al); and cadmium removal by reed leaves
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