Least-squares regression of adsorption equilibrium data: Comparing the options

https://doi.org/10.1016/j.jhazmat.2008.01.052Get rights and content

Abstract

Experimental and simulated adsorption equilibrium data were analyzed by different methods of least-squares regression. The methods used were linear regression, nonlinear regression, and orthogonal distance regression. The results of the regression analysis of the experimental data showed that the different regression methods produced different estimates of the adsorption isotherm parameters, and consequently, different conclusions about the surface properties of the adsorbent and the mechanism of adsorption. A Langmuir-type simulated data set was calculated and several levels of random error were added to the data set. The results of regression analysis of the simulated data set showed that orthogonal distance regression gives the most accurate and efficient estimates of the isotherm parameters. Nonlinear regression and one form of the linearized Langmuir isotherm also gave accurate estimates, but only at low levels of random error.

Introduction

Adsorption is the most commonly used technique for the treatment of industrial wastewaters. Activated carbon has been used widely for the removal of many pollutants; however, activated carbon is expensive and not easily regenerated [1]. Therefore, low-cost adsorbents that are able to bind pollutants have been extensively tested [1], [2], [3]. Equilibrium relationships between adsorbent and adsorbate are described by adsorption isotherms, usually the relationship between the quantity adsorbed and that remaining in solution at a fixed temperature. These equilibrium adsorption isotherms are important for the design of adsorption systems, and the constants of the isotherms express the surface properties and the capacities of the adsorbents. Although there are many adsorption isotherms in the literature, the most widely used by researchers are two of the oldest isotherms, namely Freundlich [4] and Langmuir [5] isotherms.

The Freundlich isotherm can be used for non-ideal adsorption on heterogeneous surfaces. It is expressed by the following empirical equation:qe=KFCe1/nwhere KF is the Freundlich adsorption constant ((mg/g)(L/g)n) and 1/n is a measure of the adsorption intensity.

The development of the Langmuir isotherm assumes monolayer adsorption on a homogenous surface. It is expressed asqe=(qmKaCe)/(1+KaCe)where Ce is the equilibrium concentration (mg/L), qe the amount adsorbed (mg/g), qm is qe for complete monolayer adsorption capacity (mg/g), and Ka is the equilibrium adsorption constant (L/mg).

As an alternative to the BET method, the adsorption of dyes from aqueous solution has been used to determine the specific surface area (SSA) of many substances [6], [7]. Assuming that the surface is homogenous and completely covered by dye molecules, the SSA (m2/g) can then be related to the first layer adsorption density (Γm) as described in Eq. (3):SSA=ΓmNAwhere N is the Avogadro's number (6.023 × 1023 molecules/mol) and A is the apparent surface area occupied by one dye molecule. The total multilayer adsorption capacity (Γ, mg/g) can be expressed by the following equation [8]:Γ=(ΓmK1Ce)/((1K2Ce)[1+(K1K2)Ce])where Γm is the monolayer adsorption capacity (mg/g), Ce the equilibrium MB concentration (mg/L), and K1 and K2 are the equilibrium adsorption constants of the first and second layers (L/mg). It is noted that in case of monolayer adsorption K2 will have a value of zero, and Eq. (4) is reduced to the monolayer Langmuir isotherm of Eq. (2).

There is no linear transformation for the multilayer adsorption (MLA) isotherm, but Freundlich isotherm can be linearized by taking the logarithm of both sides of Eq. (1), and also Langmuir isotherm can be linearized to at least three different linear forms as shown in Table 1 [9]. Linear regression has been the most commonly used technique to determine the adsorption isotherm parameters for Freundlich and Langmuir isotherms for many years. Linear regression was the easy and practical way when it was first suggested several decades ago, but it has become a custom principle nowadays [10] and is still widely used in spite of the availability of micro-computers and advanced statistical software. The mathematical linearization of nonlinear isotherm models leads to biased estimates of the isotherm parameters [11], [12], [13], [14], [15], [16], and therefore, some researchers apply iterative nonlinear regression to determine the best fitting isotherm model and to evaluate its parameters [17], [18], [19]. However, nonlinear regression is also not statistically correct because there are experimental errors in both the dependent and the independent variables in the isotherm equations.

Least squares is arguably the most common method for fitting data to a model when there are errors in the observations [20]. For example, given the data pairs (xi, yi); i = 1,2, …, n, where xi is the independent variable and yi is the dependent variable, suppose that xi and yi are related by a smooth, possibly nonlinear function f, i.e.,yi=f(xi;β)where β is the regression parameters vector. If the function is linear, the relationship takes the formyi=β0+β1x

The above equations state that if y and x could be measured with no errors in either xi or yi, they would be exactly related. Typical examples where this might be thought to be the case occur in the physical sciences when the variables are related by fundamental physical laws [21]. In classical least squares, it is assumed that xi is known exactly and yi is observed with error. Although it is often the case that xi have errors, these errors can be safely ignored if they are much smaller than the corresponding errors in yi. Thus, taking the error in yi to be given by ɛi, we writeyi+εi=f(xi;β)

We now seek the values of the parameters β that minimize the sum of the squares of the residuals (S):S=i=1n(yif(xi;β))2This can be interpreted as minimizing the sum of the squares of the vertical distances from the data points to the fitted curve.

If f(x) is linear in parameters, the solution is simple and reduces to a system of linear equations. However, when f(x) is not linear in parameters, the solution is either performed as a general unconstrained optimization problem, or by an iterative algorithm that is developed especially to solve least squares problems [22], such as the Gauss–Newton algorithm or the Levenberg–Marquardt algorithm.

Based on the Gauss–Markov theorem, least-squares regression makes various assumptions about the errors in a regression model. The basic assumptions are [20], [23]:

  • (1)

    The error, ɛ, is uncorrelated with x, the independence assumption.

  • (2)

    The error has the same variance (S2) across the different levels of x, i.e. the variance of ɛ is homoskedastic and not heteroskedastic.

  • (3)

    The values of ɛ are independent of each other, i.e. not autocorrelated or serially correlated.

  • (4)

    The error is normally distributed.

  • (5)

    The independent variable x is fixed, i.e. there is no measurement error in x.

If these assumptions are met, then the estimates of the regression constant and the regression coefficients are unbiased and efficient. Violation of one or more of these assumptions may lead to biased and/or inefficient estimates.

Linearization is used extensively in least-squares regression and model testing of experimental data, possibly because regression analysis of linear models can be carried out graphically and also because linear regression software is readily available. Another reason that explains the affinity to linear models is the simplicity of statistical estimation and hypothesis testing. Algorithms for estimating parameters of linear models are straightforward, direct solutions are available, and iteration is not required.

Unfortunately, linearization may lead to false conclusions [9], [10], [11], [12], [13], [24], [25], and the statistical tests used to check the goodness of fit will often not detect that the parameters are incorrect. In other words, statistical tests performed to check the quality of the fit between the data and the calculated curve can be meaningless if they are performed using transformed data. Nonlinear transformation distorts the experimental error. Linear regression assumes that the vertical scatter of points around the line follows a normal distribution, and that the standard deviation is the same at every value of x. These assumptions are usually not true with the transformed data. A second problem is that some transformations alter the relationship between x and y. Since the assumptions of linear regression are violated, the results of linear regression are incorrect. The values derived from the slope and intercept of the regression line are not the most accurate determinations of the variables in the model [25].

It is not widely appreciated in the adsorption community that a special treatment of the least squares problem is required when there is more than one observation having error per equation of condition. This is commonly referred to as the measurement error model or the errors-in-variables problem. Failure to formulate the problem correctly may result in an asymptotically biased estimator, even when fitting a straight line.

Orthogonal distance regression provides one method for fitting these error-in-variables models. If the error in xi cannot be ignored and δi denotes the error in xi, then Eq. (9) becomesyi+εi=f(xi+δi;β)and it is reasonable to approximate the parameter β by minimizing the sum of the squares of the orthogonal distances from the data points to the curve yi = f(xi; β). As shown by Boggs et al. [26] this gives rise to the orthogonal distance regression (ODR) problem given byminβ,δ12i=1n[(f(xi+δi;β)yi)2+δi2]Note that ODR is easily seen to be equivalent tominβ,δ,ε12i=1nεi2+δi2subject toyi+εi=f(xi+δi;β),i=1,,nfrom which it is easy to see that ODR is, indeed, minimizing the sum of the squares of the orthogonal distances. In ordinary least squares, we try to minimize the sum of the vertical squared distances between the observed points and the fitted line. In ODR, we try to fit a line which minimizes the sum of the squared distances between the observed points and the fitted line, as measured perpendicular to that line.

In 1996, Schulthess and Dey published an article that describes a nonlinear least-squares regression analysis of the Langmuir equation that is based on minimizing the sum of the normal distance of the data to the isotherm [27]. The authors have noted that this regression method yields different Langmuir constants when compared with linear and nonlinear regression methods. However, they just pointed out that their regression method should be less biased than linear and nonlinear regression methods and they did not give recommendation for using one method of regression, quoting “None of the regressions are endorsed per se since they should all agree if the isotherm is Langmuirian”. This subject has not been investigated again, and the work of Schulthess and Dey is rarely referenced in adsorption literature.

The objectives of this paper are (1) to demonstrate the differences in estimated isotherm parameters arising from the application of different regression methods to the adsorption equilibrium data, (2) to discuss the causes of these differences, and (3) to systematically assess the accuracy of predictions from different regression methods. For demonstration, the adsorption of Methylene Blue (MB) onto Water Hyacinth (WH) is considered. The methods applied are linear regression, nonlinear regression, and orthogonal distance regression. A comparison of the best fitting model and the predicted parameter values obtained from each method is presented. In order to assess the accuracy of each regression method in the presence of measurement errors, simulated Langmuir-type data were simulated then random errors were added to the data set, the simulated data were subsequently analyzed by different least-squares regression methods, and the accuracy of predict isotherm parameters were compared.

Section snippets

Adsorbent preparation

Live WH was collected from El-Mahmoudeya Canal, Alexandria, Egypt. Live WH consists of 94–95% water and barely contains 50–60 g total solid/kg [28]. The plants were thoroughly washed with water, the roots were cut out and disposed, and then the leaves and stems were left to dry in the sun for 14 days. In a recent publication, the sun-dried WH of El-Mahmoudeya Canal near Alexandria was analyzed [29]. It was found to contain 19% crude fiber, 18.2% ash, 21.1% crude protein, 1.0% crude lipids, and

Nitric acid treated water hyacinth (NWH)

The experimental equilibrium data for the adsorption of MB at 40 °C were fitted to Freundlich isotherm, Langmuir isotherm, and the multilayer adsorption Langmuir isotherm. The fitting of experimental data to the isotherms was performed by the methods of linear regression (LR), nonlinear regression (NLR), and ODR. The different regression methods resulted in different estimates of the parameters of adsorption isotherms as shown in Table 2 and Fig. 1, Fig. 2.

The values of the Kf parameter of

Conclusions

Based on the results of this study, it is not recommended to use Langmuir II and III methods in the estimation of Langmuir isotherm parameters. Nonlinear regression gives more accurate estimates than LR by Langmuir I method, especially when the experimental error is large. The results of this study also show that orthogonal distance regression gives the most accurate estimates of the isotherm parameters among the different methods compared. A further point to consider is that the challenge for

Acknowledgments

The author thanks Prof. Dr. Mohammad H. Abdel-Magid and Prof. Dr. Roshdy R. Zahran for providing the laboratory facilities. The author is also grateful to the Egyptian Academy for Scientific Research and Technology for financial support.

References (39)

  • P.S. Ganesh et al.

    Extraction of volatile fatty acids (VFAs) from water hyacinth using inexpensive contraptions, and the use of the vfas as feed supplement in conventional biogas digesters with concomitant final disposal of water hyacinth as vermicompost

    Biochem. Eng. J.

    (2005)
  • A.M. El-Sayed

    Effects of fermentation methods on the nutritive value of water hyacinth for nile tilapia, Oreochromis niloticus (L.) Fingerlings

    Aquaculture

    (2003)
  • S.J. Allen et al.

    Comparison of optimized isotherm models for basic dye adsorption by Kudzu

    Bioresour. Technol.

    (2003)
  • G. McKay et al.

    The removal of dye colours from aqueous solutions by adsorption on low-cost materials

    Water Air Soil Pollut.

    (1999)
  • H.M.F. Freundlich

    Uber Die Adsorption in Losungen

    Z. Phys. Chem.

    (1906)
  • I. Langmuir

    Constitution and fundamental properties of solids and liquids. I. Solids

    J. Am. Chem. Soc.

    (1916)
  • F. Guzel et al.

    Determination of the micropore structures of activated carbons by adsorption of various dyestuffs from aqueous solution

    Turk. J. Chem.

    (2002)
  • C. Kaewprasit et al.

    Application of methylene blue adsorption to cotton fiber specific surface area measurement. Part I. Methodology

    J. Cotton Sci.

    (1998)
  • P. Persoff et al.

    Estimating Michaelis–Menten or Langmuir isotherm constants by weighted nonlinear least-squares

    Soil Sci. Soc. Am. J.

    (1988)
  • Cited by (148)

    View all citing articles on Scopus
    View full text