Determination of best-fit isotherm model for the sorption of Lead (II) and Manganese (II) ions onto acid-activated shale using selected non-linear error functions

The focus of this research is to apply the selected error function equation to establish the equilibrium isotherm model that best describes the adsorption of Pb2+ and Mn2+ onto acid-activated shale. Data collected from the batch experiment were analyzed using selected isotherm models (Langmuir, Freundlich, Temkin, Dubinin-Radushkevich, Sips and Redlich-Peterson). To compute the isotherm parameters used in choosing the best-fit isotherm model, selected non-linear error functions, namely, error sum of the square, normalized standard deviation, hybrid error function, root mean square error and Marquardt’s percent standard deviation were employed. From the scanning electron microscope results, it was observed that the surface characteristics of the shale change considerably with calcination and acid treatment but the acid-treated shale shows better uneven porous surface characteristics. Error function computation shows that the Dubinin-Radushkevich isotherm model had the least sum of normalized error of 0.3623 for Pb2+ adsorption and 0.5465 for Mn2+ adsorption; hence, it was selected as the best isotherm model for explaining the sorption of Pb(II) and Mn(II) ions unto acid-activated shale.

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Searching for optimum adsorption curve for metal sorption on soils: comparison of various isotherm models fitted by different error functions

SN Applied Sciences ◽

10.1007/s42452-021-04383-0 ◽

2021 ◽

Vol 3 (3) ◽

Author(s):

Péter Sipos

Keyword(s):

Experimental Data ◽

Error Function ◽

Isotherm Models ◽

Metal Sorption ◽

Error Functions ◽

Sorption System ◽

Geochemical Properties ◽

Function Combination ◽

Soil Metal ◽

Sorption Curve

AbstractStudies comparing numerous sorption curve models and different error functions are lacking completely for soil-metal adsorption systems. We aimed to fill this gap by studying several isotherm models and error functions on soil-metal systems with different sorption curve types. The combination of fifteen sorption curve models and seven error functions were studied for Cd, Cu, Pb, and Zn in competitive systems in four soils with different geochemical properties. Statistical calculations were carried out to compare the results of the minimizing procedures and the fit of the sorption curve models. Although different sorption models and error functions may provide some variation in fitting the models to the experimental data, these differences are mostly not significant statistically. Several sorption models showed very good performances (Brouers-Sotolongo, Sips, Hill, Langmuir-Freundlich) for varying sorption curve types in the studied soil-metal systems, and further models can be suggested for certain sorption curve types. The ERRSQ error function exhibited the lowest error distribution between the experimental data and predicted sorption curves for almost each studied cases. Consequently, their combined use could be suggested for the study of metal sorption in the studied soils. Besides testing more than one sorption isotherm model and error function combination, evaluating the shape of the sorption curve and excluding non-adsorption processes could be advised for reliable data evaluation in soil-metal sorption system.

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Adsorption of UV254 in Kerian River water onto ZeliacTM: Analysis using linear and non-linear forms of isotherm models

Global NEST Journal ◽

10.30955/gnj.001968 ◽

2017 ◽

Vol 19 (1) ◽

pp. 74-81 ◽

Cited By ~ 1

Keyword(s):

River Water ◽

Function Analysis ◽

Low Cost ◽

Error Function ◽

Linear Regression Analysis ◽

Freundlich Isotherm ◽

Linear Analysis ◽

Isotherm Models ◽

Freundlich Isotherms ◽

Non Linear

The composite media, ZeliacTM was developed with the initial aim to provide low cost adsorbent with promising adsorption capacity. This study was conducted to investigate the removal of UV absorbance at 254 nm (UV254) in Kerian river water using ZeliacTM as the media. Batch experiments study was carried out to determine the optimum removal of UV254 by ZeliacTM. The experimental data were fitted to Langmuir and Freundlich isotherms to investigate the adsorption mechanism. The results from batch study exhibit that ZeliacTM is capable to remove 74.4% UV254 at the dosage of 7g/100 ml. Linear isotherm analysis suggests that the best fitting linear line is Freundlich isotherm with R2 values of 0.9294 indicating multilayer adsorption. Similarly, non-linear regression analysis reveals that the adsorption of UV254 by ZeliacTM is attributed by physisorption. The non-linear Freundlich isotherm gives a better fit to the adsorption of UV254 than Langmuir isotherm with R2 values of 0.9488. The results are supported with low values of X2, ARE, HYBRID and MPSED from the error function analysis.  Additionally, it is noted that the linear analysis overestimates the constant parameters’ values for Freundlich isotherm, which cause larger errors as estimated by the error function analysis. Hence, non-linear analysis is more appropriate in explaining the batch experiment data.

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Non-linear Regression Analysis for the Adsorption Kinetics and Equilibrium Isotherm of Phenacetin onto Activated Carbons

Current Journal of Applied Science and Technology ◽

10.9734/cjast/2019/v36i430246 ◽

2019 ◽

pp. 1-18 ◽

Cited By ~ 1

Author(s):

Christian Sadeu Ngakou ◽

Gabche Solomon Anagho ◽

Horace Manga Ngomo

Keyword(s):

Experimental Data ◽

Activated Carbon ◽

Activated Carbons ◽

Linear Regression Analysis ◽

Xrd Analysis ◽

Strong Interactions ◽

Isotherm Models ◽

Linear Isotherm ◽

Non Linear ◽

Best Fit

Activated carbon obtained from ayous sawdust, Cucurbitaceae (egussi) peelings and the mixture of the two were studied for the adsorption of phenacetin. Characterisation of activated carbon by SEM and XRD analysis shows that the mixture of precursors combine the properties of activated carbon obtained separately. The well-knownbatch sorption models– Langmuir (one and two sites), Freudlich, Tempkin, Elovich, Langmuir-Freudlich, Redlich Peterson, Radke-Prausnitz, Fritz Shlunder)—were tested with experimental data for the adsorption of phenacetin to estimate adsorption equilibrium parameters—rate constantsand adsorption capacities. The model with the best fit was identified from extensive statistical analysis of the results of nonlinear regression of the experimental data. Comparison of the statistical errors in parameter estimation between linear and non-linear isotherm models shows that transformation of non-linear isotherm equations to linear forms implicitly alter their error structure. The much smaller size of the various error indicators —Determination Coefficient, R2; Sum of Square Errors, SSE; Chi Test, χ2; Average Relative Errors, ARE—, calculated for the case of non linearization when compared to linearization, indicate the greater accuracy in the application of non linearization. The Langmuir model (one site) gave the best fit and thus the values of adsorption capacity for each activated carbon were calculated from it. Kinetic models show that weak and strong interactions are involved in the adsorption process and that the controlling mechanism may not be limited to intra particle diffusion. The lower value of the boundary layer thickness in the case of activated carbon obtained from the mixture, justified the higher adsorbed quantity of this activated carbon compared to those of activated carbon from each precursor.

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Removal of basic dyes from aqueous solutions by activated carbon derived from Eichornia crassipes: equilibrium and kinetic studies

Water Quality Research Journal ◽

10.2166/wqrjc.2014.050 ◽

2014 ◽

Vol 49 (2) ◽

pp. 163-178 ◽

Cited By ~ 1

Author(s):

Syed Usman Nasrin Banu ◽

G. Maheswaran

Keyword(s):

Activated Carbon ◽

Linear Regression ◽

Second Order ◽

Isotherm Models ◽

Batch Adsorption ◽

Non Linear ◽

Equilibrium Data ◽

Pseudo Second Order ◽

Eichornia Crassipes ◽

Best Fit

The feasibility of preparing activated carbon from Eichornia crassipes by chemical activation was investigated. Batch experiments were carried out for the sorption of Methylene Blue (MB) and Rhodamine B (RB) onto the prepared activated carbon. The variables studied were initial dye concentration, pH, adsorbent dose, and contact time. Equilibrium data for the adsorption of the dyes onto activated carbon were obtained from batch adsorption experiments. Two-parameter isotherm models including Langmuir, Freundlich, Temkin, and Dubinin–Radushkevich were employed for fitting equilibrium data. Three-parameter isotherm models including Redlich–Peterson, Toth, and Koble–Corrigan models were also employed for fitting the equilibrium data. Linear and non-linear regression methods were used to determine the best fit model to the equilibrium data. It was found that non-linear regression is a better method for determining isotherm parameters. The data were fitted to pseudo-first-order, pseudo-second-order, intraparticle diffusion model, and Elovich equation. The pseudo-second-order model gave the best fit to the equilibrium data as seen from correlation coefficient values. Fourier transform infrared spectroscopy and scanning electron microscopic investigations were carried out to confirm the morphological characteristics of the adsorbent. The prepared activated carbon had greater affinity for adsorbing MB when compared to RB.

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Comparative Analysis of the Adsorption of Methylene Blue Using Magnetised and Non-Magnetised Coconut Shell Biochar

UNIOSUN Journal of Engineering and Environmental Sciences ◽

10.36108/ujees/9102.10.0250 ◽

2019 ◽

Vol 1 (2) ◽

Author(s):

A Musa

Keyword(s):

Methylene Blue ◽

Nonlinear Regression ◽

Error Function ◽

Coconut Shell ◽

Isotherm Models ◽

P Value ◽

Regression Methods ◽

Best Fit ◽

Coconut Shell Biochar ◽

Mb Adsorption

It is envisaged that magnetisation might alter the sorption behaviours of magnetised biochars due to some variation in the physicochemical properties from their precursor. This study evaluated the adsorption behaviours of a coconut shell biochar produced at 600 °C, CSB600, and its magnetised pair, MCSB600, in the adsorption of methylene blue (MB) from aqueous solutions. Langmuir, Freundlich and Redlich-Peterson isotherm models were used to describe the experimental isotherm using linear and nonlinear regression methods to determine the best fit for MB adsorption from the batch experiments conducted. The Langmuir model proved to be the best fit to explain the experimental data as it had the highest R2 (0.9684 and 0.9855) from linear regression and the lowest hybrid fractional error function, HYBRID (4.58, 1.145) and marquardt’s percent standard deviation, MPSD (10.61, 5.04) error function values from the nonlinear regression methods with maximum monolayer adsorption capacities of 5.590 and 5.229 mg/g for CSB600 and MCSB600 respectively. The magnetised biochar exhibited similar adsorption characteristics to what was observed for the non-magnetised biochar and only about 6.46% lower MB adsorption capacity was recorded. A p-value of 0.088 obtained suggested the isotherms were similar and therefore, magnetisation did not affect the adsorption of MB.

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Adsorption of ferrocyanide onto raw and acid-activated clinoptilolite and sepiolite: equilibrium modelling by error minimization

Clay Minerals ◽

10.1180/claymin.2013.048.4.15 ◽

2013 ◽

Vol 48 (4) ◽

pp. 613-626 ◽

Cited By ~ 1

Author(s):

V. Önen ◽

E. Yel

Keyword(s):

Linear Models ◽

Error Function ◽

Linear Optimization ◽

Error Minimization ◽

Non Linear ◽

Initial Adsorption ◽

Adsorption Rates ◽

Function Models ◽

Best Fit ◽

Modelling Approach

AbstractThe experimental data on adsorption of Fe and CN of a ferrocyanide complex onto raw and acid-activated clinoptilolite/sepiolite on the basis of detention time and particle size was modelled by a linear and a non-linear approach. The linearized best-fit isotherm selection method and non-linear error minimization was applied through Freundlich, Langmuir and Temkin isotherms. ERRSQ, MPSD, HYBRID and ARE error functions were minimized by a developed MATLAB script to determine the isotherm parameters in non-linear optimization. The complex was not adsorbed as whole anions but the Fe and CN were adsorbed separately. 0.65 mg Fe/L. min and 4.84 mg CN/L. min initial adsorption rates were achieved with acid activated clinoptilolite. The Fe adsorption was not as successful as CN. The adsorption of Fe and CN was described by Freundlich and Langmuir isotherms respectively. The differences between the predicted isotherm parameter sets of linear models and minimized error function models indicated that both the best-fit isotherm selection and the isotherm constant determinations can be performed properly by error minimization as well as by conventional linear best fit modelling approach.

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Competitive adsorption of organic solvents using modified and unmodified calcium bentonite clay mineral

10.31221/osf.io/w9dj6 ◽

2019 ◽

Author(s):

Chem Int

Keyword(s):

Adsorption Isotherm ◽

Adsorption Capacity ◽

Organic Solvents ◽

Competitive Adsorption ◽

Correlation Factor ◽

Isotherm Model ◽

Isotherm Models ◽

High Concentration ◽

Dual Purpose ◽

Calcium Bentonite

Dodecyltrimethylammonium bromide (DTAB)–modified and unmodified calcium bentonite were both used for the competitive adsorption of aromatics (xylene, ethylbenzene and toluene) and petroleum products (gasoline, dual purpose kerosene and diesel) from their aqueous solution. Infrared spectroscopy (IR) and expansion tests (adsorption capacity and Foster swelling) measurement were performed in order to evaluate the performance of the adsorbents. The Foster swelling index and adsorption capacity of the DTAB modified calcium bentonite in the organic solvents follow the trend: xylene > ethylbenzene > toluene > gasoline > dual purpose kerosene (DPK) > diesel > water. However, the adsorption capacity of the adsorbent in diesel outweighed the adsorption capacity in DPK at high concentration of DTAB indicating that diesel has higher affinity for high DTAB concentration than DPK. The percentage removal of the solvent is directly proportional to the concentration of DTAB used in modifying the bentonite as well as the contact time between the adsorbent and the solvent, hence modified calcium bentonite adsorbed a higher percentage of organic solvents than the unmodified calcium bentonite. The adsorption characteristics of both adsorbents improved remarkably after proper agitation of the organic solvents, the unmodified calcium bentonite however adsorbed more water than the modified bentonite. Data obtained from adsorption isotherm models confirms that Freundlich adsorption isotherm model was favored more than Langmuir adsorption isotherm model with the correlation factor (R2) of the former tending more towards unity. The adsorption of ethylbenzene using DTAB modified and unmodified calcium bentonites follow a pseudo second order kinetics mechanism, suggesting that the rate determining step of adsorption involves both the adsorbent and the organic solvent.

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An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials

Mathematica Slovaca ◽

10.1515/ms-2017-0374 ◽

2020 ◽

Vol 70 (3) ◽

pp. 599-604

Author(s):

Şahsene Altinkaya

Keyword(s):

Error Function ◽

Trigonometric Polynomials ◽

Error Functions ◽

The Family ◽

Coefficient Inequalities

AbstractIn this present investigation, we will concern with the family of normalized analytic error function which is defined by$$\begin{array}{} \displaystyle E_{r}f(z)=\frac{\sqrt{\pi z}}{2}\text{er} f(\sqrt{z})=z+\overset{\infty }{\underset {n=2}{\sum }}\frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}. \end{array}$$By making the use of the trigonometric polynomials Un(p, q, eiθ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.

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Association between serum uric acid and triglycerides in Chinese children and adolescents with short stature

Lipids in Health and Disease ◽

10.1186/s12944-020-01429-x ◽

2021 ◽

Vol 20 (1) ◽

Author(s):

Yuntian Chu ◽

Qianqian Zhao ◽

Mei Zhang ◽

Bo Ban ◽

Hongbing Tao

Keyword(s):

Uric Acid ◽

Standard Deviation ◽

Children And Adolescents ◽

Linear Relationship ◽

Short Stature ◽

Serum Uric Acid ◽

Standard Deviation Score ◽

Short Children ◽

Non Linear ◽

Males And Females

Abstract Background Elevated triglyceride (TG) levels are a biomarker for cardiovascular disease (CVD) risk. The correlation between serum uric acid (SUA) and TG concentrations in adults or obese children is well established. However, studies on SUA and TG in children with short stature are limited. Aim To determine the relationship between SUA and TG levels in short children and adolescents. Method This was a cross-sectional evaluation of a cohort of 1095 patients with short stature (720 males and 375 females). The related clinical characteristics, including anthropometric and biochemical parameters, were determined. Results Smooth curve fitting, adjusted for potential confounders was performed, which indicated the existence of a non-linear relationship between these measures. Piecewise multivariate linear analysis revealed a significant positive relationship between SUA and TG at SUA concentrations over 7 mg/dL (β = 0.13, 95% CI: 0.05–0.22, P = 0.002) but no significant correlation at lower SUA levels (β = 0.01, 95% CI: 0.01–0.04, P = 0.799). Furthermore, a stratified analysis was performed to appraise changes in this relationship for different sexes and standard deviation levels of body mass index (BMI). The non-linear relationship remained consistent in males and females with BMI standard deviation scores (BMI SDS) ≥ 0, with inflection points of 6.71 mg/dL and 3.93 mg/dL, respectively. Within these two groups, SUA and TG levels showed a positive association when SUA levels were higher than the inflection point (β = 0.21, 95% CI: 0.11–0.31, P < 0.001 for males and β = 0.1, 95% CI: 0.03–0.17, P = 0.005 for females). However, a specific relationship was not observed at lower SUA levels. No significant relationships were found between SUA and TG levels in males and females with BMI SDS < 0. Conclusion The present study identified the non-linear association of SUA and TG levels with short children and adolescents. This relationship was based on BMI status. This finding suggests that health status should be considered for short stature children with high SUA levels, especially in children with a high BMI standard deviation score.

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Design of a New Synthetic Acceptance Sampling Plan

Symmetry ◽

10.3390/sym10110653 ◽

2018 ◽

Vol 10 (11) ◽

pp. 653 ◽

Cited By ~ 2

Author(s):

Saeed Dobbah ◽

Muhammad Aslam ◽

Khushnoor Khan

Keyword(s):

Standard Deviation ◽

Normal Distribution ◽

Operating Characteristic ◽

Linear Optimization ◽

Sampling Plan ◽

Quality Characteristic ◽

Acceptance Sampling ◽

Acceptance Sampling Plan ◽

Non Linear

In this paper, we propose a new synthetic sampling plan assuming that the quality characteristic follows the normal distribution with known and unknown standard deviation. The proposed plan is given and the operating characteristic (OC) function is derived to measure the performance of the proposed sampling plan for some fixed parameters. The parameters of the proposed sampling plan are determined using non-linear optimization solution. A real example is added to explain the use of the proposed plan by industry.

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