Hydrometallurgy 105 (2011) 314–320

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Hydrometallurgy

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Common data analysis errors in batch adsorption studies

Mohammad I. El-Khaiary ⁎, Gihan F. MalashChemical Engineering Department, Faculty of Engineering, Alexandria University, El-Hadara, Alexandria 21544, Egypt

⁎ Corresponding author. Fax: +20 3 5921853.E-mail address: elkhaiary@gmail.com (M.I. El-Khaiar

0304-386X/$ – see front matter © 2010 Elsevier B.V. Adoi:10.1016/j.hydromet.2010.11.005

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 March 2010Received in revised form 29 October 2010Accepted 2 November 2010Available online 21 November 2010

Keywords:AdsorptionRegressionIsothermLinearizationKineticsMechanism

Many models exist for describing the experimental results of batch adsorption which are used in research tostudy equilibrium, kinetics, and mechanisms of adsorption. In the process of statistically analyzing theexperimental data, the adsorption literature contains errors that render the results unreliable. These errorsinclude incorrect application of theoretical models and also incorrect application of statistical analysis. Someerrors are so abundant in the adsorption literature that they have actually gained credibility and mistakenlytaken for granted that these are sound scientific practices. This article highlights some common errors inadsorption data analysis that are frequently found in the literature and provides suggestions for more soundpractices.

y).

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© 2010 Elsevier B.V. All rights reserved.

1. Introduction

In the course of data analysis, statistical methods are applied to fitexperimental data to models and to assess statistical inferences. Thereis an abundance of software for statistical calculations which come asstand-alone commercial programs, as part of commercial spread-sheets or as shareware and freeware. However, the availability ofgood software does not ensure a correct statistical analysis. Pre-requisites for the correct use of software are a familiarity with thebasics of statistics; a knowledge of the assumptions and limitations ofeach statistical technique; and the ability to choose the appropriatemethod of analysis. Uncritical use of statistical software may lead toapplying the techniques incorrectly, or even using statistical methodsin cases where they are not appropriate.

In the field of batch adsorption research, certain errors in dataanalysis keep showing up over and over again. They show up in topranking journals and are so abundant in the adsorption literature thatthey have actually gained credibility. New comers to adsorptionresearch may take it for granted that these are sound scientificpractices and spread the errors evenmore. The objective of this articleis to highlight some of these errors and to provide suggestions formore sound practices. For clarity, a few specific examples are taken asprototypes from the literature. In the choice of these examples,articles were chosen to represent several journals and authors fromdifferent countries.

2. Linearization

The misuse of linearization is probably the most common error indata analysis. It originatedmanydecades agowhen computerswere notyet available and it has not lost its popularity today. There are problemsassociated with trying to linearize an inherently nonlinear equation byuse of various transformations. The main issue when transforming datato achieve a linear equation is knowledge of the error-structure of thedata and how this structure is affected by transformation. When theerrors are additive on the dependent variable (Y) and satisfy the usualassumptions of normality and homo-skedasticity (equal variance)throughout the range of the data, then transforming the dependentvariable with a nonlinear function can destroy the assumed distribu-tional properties. However, when the original error-structure does notsatisfy these assumptions, by judicious choice of a transformation, themodel can in some cases be transformed to satisfy these assumptions.For example, if the variance of errors increaseswith increasing values ofY, a square root or log transformation will often help, but if the varianceof the errors decreaseswith increasing Y, then the square or exponentialtransformation will often be appropriate. In the absence of data aboutthe error-structure,which is the case inmost adsorption studies, there isno point in applying linearization.

Typical examples are adsorption isotherms and adsorption kineticmodels. These equations are nonlinear, i.e. the observed response (de-pendent variable) does not depend linearly on the independent variable.The transformed (linearized) response function is used for the quantita-tive evaluation of the parameters by linear regression. For example, thenonlinear form of the well-known Langmuir (1916) isotherm is:

qe = qmKaCeð Þ= 1 + KaCeð Þ ð1Þ

Table 2Linearized forms of the pseudo-second-order kinetic model.

Type Linearized Form Plot Effects of linearization

Linear 1 tq=

1kq2e

+1qe

tt/q vs. t Reversal of relative weights of data

points because of 1/q in thedependent variablet in both dependent andindependent variables, leading tospurious correlation

Linear 2 1q

=1qe

+1kq2e

� �1t

1/q vs. 1/t Reversal of relative weights of datapoints because of 1/q in dependentvariableIndependent variable is 1/t,leading to distortion of errordistribution

Linear 3q = qe−

1kqe

� �qt

q vs. q/t q in both dependent andindependent variables, leading tospurious correlationThe presence of q in theindependent variable (q/t)introduces experimental error,violating a basic assumption in themethod of least squares1/t in independent variable,leading to distortion of errordistribution

Linear 4 qt= kq2e−kqeq

q/t vs. q q in both dependent andindependent variables, leading tospurious correlationThe presence of q in theindependent variable introducesexperimental error, violating abasic assumption in the method ofleast squares

315M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320

where the independent variable Ce is the equilibrium concentration(mg/L), the response qe is the amount adsorbed at equilibrium (mg/g),qm is qe for complete monolayer adsorption capacity (mg/g), and Ka isthe equilibrium adsorption constant (L/mg). This nonlinear formcan be mathematically manipulated and linearized to at least threelinear forms as shown in Table 1 (El-Khaiary, 2008). Likewise, Table 2(El-Khaiary et al., 2010) shows four linearized forms of the widelyused pseudo-second-order kinetic model of Ho (2004) which has thefollowing nonlinear form

q =q2e kt

1 + qektð2Þ

where k is the rate constant of pseudo-second-order adsorption(g/mg min), qe is the amount of solute adsorbed at equilibrium (mg/g),and the dependent variable q is the amount of solute adsorbed at timet (mg/g), the independent variable.

These linearized forms are used extensively in adsorptionliterature. The structure of experimental error is transformed alongwith the data (in some cases leading to loss of homo-skedasticity),and a basic assumption of the least squares method (i.e. that theindependent variable has only a negligible error) is sometimesviolated. Moreover, the statistical tests used to check the goodnessof fit will often not detect that the parameters are biased.

Tables 1 and 2 present the effects of different linearizations to themodels of Langmuir and Ho. The effect of linearizing Ho's equation onthe accuracy of parameter estimates is graphically demonstrated inFig. 1. This figure was constructed by applying linear and nonlinearleast squares regressions to published experimental kinetic data(Kononova et al., 2007) to estimate the kinetic parameters, qe and k,then using the estimated parameters to plot Eq. (2) with theuntransformed data. It is clear that linear regression of thethree linearized forms of Ho's equation produced parameterestimates (and consequently curve fittings) that vary wildly fromeach other. It can be seen that nonlinear regression produced a betterfit that is closer to the data points. This should not be surprising asthere is a large body of literature that warns from the use oflinearization in adsorption studies (El-Khaiary, 2008; Ho et al., 2005;Kumar, 2006; Bolster and Hornberger, 2007; Kundu and Gupta, 2006;Hamdaoui, 2006; Badertscher and Pretsch, 2006; Crini et al., 2008;Tsai and Juang, 2000). In spite of warnings in the literature, modelingusing linearized equations have been published in literally thousandsof adsorption papers (Liu et al., 2010b; Yuan et al., 2010; Zhang et al.,

Table 1Linearized forms of Langmuir isotherm.

Type Linearized form Plot Effects of linearization

LR I Ce

qe=

1Kaqm

+1qm

CeCe

qevs:Ce

Ce in both dependent and independentvariables, leading to spurious correlationThe error distribution of the dependentvariable, Ce/qe, is different from both theerror distributions of Ce and qeReversal of relative weights of data pointsbecause of 1/qe in dependent variable

LR II 1qe

=1qm

+1

Kaqm

1Ce

1qe

vs:1Ce

Distortion of relative weights of datapoints because of 1/qe and 1/Ce independent and independent variablesIndependent variable is 1/Ce, leading todistortion of error distribution

LR III qeCe

= Kaqm−KaqeqeCe

vs:qe qe in both dependent and independentvariables, leading to spurious correlationThe relative weights of data points aredistorted because the independentvariable is qe/CeThe error distribution of the dependentvariable, qe/Ce, is different from both theerror distributions of Ce and qe

2009; Rengaraj et al., 2007; Wan Ngah and Fatinathan, 2010;Hubicki and Wołowicz, 2009; Unuabonah et al., 2008; Acharya et al.,2009; Nadeem et al., 2009; Kamal et al., 2010; Akperov et al., 2009;Abd El-Ghaffar et al., 2009; Cox et al., 2005; Atia, 2005; Agrawal andSahu, 2006).

3. Abuse of R2 and model comparison

A common practice in research is to fit the experimental datato several models, then perform some kind of test to compare anddecide which model fits the data better. Based on the choice of the“best model”, conclusions about the intricate mechanism of thesystem are often available. The most popular tool for modelcomparison is the coefficient of determination, R2. It is the square of

1 2 3 4 5 62

4

6

8

10

12

time, min

q, m

g A

g/g

Experimental

— — Nonlinear— — Lin 1—. .— Lin 2

........ Lin 3

---- Lin 4

Fig. 1. Comparison of the pseudo-second-order parameters estimated by linear andnonlinear regressions for the adsorption of silver thiocyanate complexes on anionexchanger AV-17-10P.

316 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320

the correlation coefficient, r, between the dependent variable andthe regression-predicted value of the dependent variable. In otherwords, R2 is the fraction of the total variance of the dependent variablethat is explained by the model's equation. Mathematically, it isdefined as:

R2 = 1− SSESStot

ð3Þ

where SSE is the sum-of-squared deviations of the points from theregression curve and SStot is the sum-of-squared deviations of thepoints from a horizontal line where Y equals the mean of all the datapoints.

3.1. Abuse of R2

In spite of the apparent simplicity of interpreting R2, it is notalways suitable for evaluating the goodness of fit and comparingmodels. The most obvious issues with R2 are:

• R2 is sensitive to extreme data points, resulting in misleadingindication of the quality of fit. This is further complicated when thedata points are subjected to linear transformations; existingextreme-points may disappear and new extreme-points may becreated.

• R2 is influenced by the range of the independent variable. R2

increases as the range of independent variable increases anddecreases as the range decreases.

• R2 can be made artificially large by adding more parameters to themodel. In other words, R2 value increases with the decrease indegrees of freedom for error.

The first two issues with R2 can be avoided by fitting the data to themodel without any transformations and by examination of extreme-points, while the third issue is more subtle and is discussed further inSections 3.1 and 3.2.

3.1.1. Spurious goodness of fit concluded from high R2 values when thefitting has a small degree of freedom

This is especially common in adsorption studies when it comes toestimating thermodynamic parameters and it is also found in othercases such as isotherm and diffusion calculations (Wan Ngah andHanafiah, 2008; Abd El-Ghaffar et al., 2009; Unuabonah et al., 2008;Agrawal and Sahu, 2006). To illustrate, Fig. 2 is reproduced from arecent article (Unuabonah et al., 2008) where three data pointswere used to plot the natural logarithm of b, the Langmuir constantrelated to energy, against the reciprocal absolute temperature, 1/T, to

0.0031 0.00315 0.0032 0.00325 0.0033 0.00335

-5.8

-5.7

-5.6

-5.5

-5.4

-5.3

1/T (K-1)

ln b

Fig. 2. Plot of ln b vs. 1/T for cadmium adsorption onto unmodified kaolinite clay.

calculate ΔH and ΔS from the slope and intercept of a straight lineaccording to the linearized equation:

ln b =ΔSRc

− ΔHRcT

ð4Þ

where Rc is the universal gas constant (8.314 J/K mol), ΔH is theenthalpy change (J/mol) and ΔS is the entropy change (J/K mol).Linear regression analysis was performed and some of the results areshown in Table 3. The linear plot in Fig. 2 seems good, with littledeviation of the points from the regression line, and also the value ofR2 (which represents the proportion of the variation in log b that canbe accounted for by variation in 1/T) is 0.9415, which is relatively highand may suggest an acceptable fit. However, the numbers in Table 3tell another story.

The estimated slope is highly uncertain because zero is within its95% confidence interval, and consequently, there is no evidence thatthe slope is different from zero. Therefore, the apparent variation of lnb with changes in 1/T may be due to random variation (noise). Bycalculating the 95% confidence interval of ΔH it turns out to be from−22.5 to +50.8 kJ/mol, a very wide range that renders the estimatedvalue of ΔH (+14.1 kJ/mol) virtually useless. The insignificance of theslope is further corroborated by the result of t-test, the significancelevel of this test is 0.1283 which means that there is a 12.83%probability that the apparent slope is caused by noise, and since0.1283 is much larger than the conventional 0.0500 cut-off value, thehypothesis that the slope is zero is not rejected. The same discussionapplies to the statistical insignificance of the intercept, and accord-ingly, any conclusions based on the values or signs of ΔH and ΔS areunsupported.

The previous analysis does not prove that ln b is independent from1/T, it only shows that fitting the straight line to three data points didnot give enough evidence – from a statistical point of view – tosupport a hypothesis that there is a linear correlation between the twovariables.

3.2. Comparing models that have different degrees of freedom

For a fixed sample size, increasing the number of regressionparameters leads to a decrease in the degrees of freedom, and almostuniversally decreases SSE. The value of R2, as calculated from Eq. (3),has no consideration for the degrees of freedom. Consequently,models with more regression parameters will tend to have higher R2

values. Therefore, the goodness of fit cannot be based solely on SSE(and R2) but must also include a penalty for the decrease in thedegrees of freedom.

It is customary in batch adsorption studies to fit the equilibriumuptake data to several isotherms, then to use R2 to compare thegoodness of fit and select the best isotherm model. With the bestisotherm supposedly identified, conclusions are usually presentedregarding the homogeneity of the adsorbent surface and themechanism of adsorption. However, a common pitfall is that somestudies use R2 to compare isotherms that have two, three, and fourparameters (Nadeem et al., 2009; Lu et al., 2009; Gunay et al., 2007;Chan et al., 2008; Wang et al., 2005; Debnath and Ghosh, 2008).

Akaike's Information Criterion (AIC) (Burnham and Anderson,2002) is a well established statistical method that can be used tocompare models. It is based on information theory and maximum

Table 3Linear regression results of the experimental data shown in Fig. 2.

Intercept 95% confidencelimits of intercept

Slope 95% confidencelimits of slope

Probabilityvalue for slope(t-test)

R2

−0.12 −14.30 to 14.07 −1698 −6106 to 2711 0.1283 0.9415

Table 4Estimated parameter-values for Langmuir, Freundlich, and Redlich–Peterson isotherms.

Langmuir isotherm Freundlich isotherm Redlich–Peterson isotherm

qm(mg/g)

Ka

(L/mg)R2 SSE Kf n R2 SSE KR

(L/g)aR(L/mg)

β R2 SSE

129.7 0.134 0.993 3.20 33.1 3.28 0.934 8.98 22.1 0.242 0.921 0.995 2.34

317M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320

likelihood theory, and as such, it determines which model is morelikely to be correct and quantifies how much more likely. For a smallsample size, AIC is calculated for each model from the equation:

AIC = N lnSSEN

� �+ 2Np +

2Np Np + 1� �

N−Np−1ð5Þ

where N is the number of data points, and Np is the number ofparameters in the model.

AIC values can be compared using the Evidence ratio which isdefined by:

Evidence ratio =1

e−0:5Δ ð6Þ

where Δ is the absolute value of the difference in AIC between the twomodels.

This comparison method is illustrated using isotherm data from arecent study (Gunay et al., 2007) where several two and three-parameter isotherms were fitted to the data. The two-parametermodels are Langmuir and Freundlich (1906) isotherms, while thethree-parameter model is Redlich and Peterson (1959) isotherm.

Freundlich isotherm : qe = Kf C1 = ne ð7Þ

Redlich–Peterson isotherm : qe =KRCe

1 + aRCβe

ð8Þ

The results of nonlinear regression, published in the study, arepresented in Table 4. The study concluded, on the basis of R2

comparison, that the three-parameter isotherm is a better fit.However, AIC would be a more sound method to compare thegoodness of fit to Langmuir and Redlich–Peterson isotherms.Accordingly, AIC values were calculated for Langmuir (1.521) andRedlich–Peterson (6.330) isotherms. Having a smaller AIC valuesuggests that Langmuir isotherm is more likely to be a better fit. TheEvidence ratio of 11.07 means that it is 11.07 times more likely to bethe correct model than the Redlich–Peterson isotherm.

Fig. 3. Pore-diffusion plots for the removal of Pd(II) complexes from NaCl–HCl solutionscontaining 100 μg/cm3 Pd (II) determined for the weakly basic Amberlyst A 21.

4. Incorrect application of models

4.1. Incorrect application of Webber's pore-diffusion model

Webber's pore-diffusion model (Weber and Morris, 1963) iscommonly used in adsorption studies. It is defined by the equation:

q = kit0:5 + c ð9Þ

where ki (mg/g min0.5) is the pore-diffusion parameter, and c (mg/g)is an arbitrary constant.

It can be seen from Eq. (9) that if pore-diffusion is the rate limitingstep in the adsorption process, then a pore-diffusion plot (q vs t0.5) isexpected to be a straight line with a slope that equals ki. In practice,things are not that simple because pore-diffusion plots often showseveral linear segments. It has been proposed that these linear segmentsrepresent pore-diffusion in pores of progressively smaller sizes (Ho andMcKay, 1998; Allen et al., 1989, 2005; Koumanova et al., 2003; Cheunget al., 2007). Eventually, equilibrium is reached and q stops changingwith time; and a final horizontal line is established at qe.

It follows from the previous discussion that it would be a goodpractice to examine pore-diffusion plots and decide how many linearsegments exist. When a group of points are identified as belonging toa linear segment, linear regression can then be applied to these pointsand the corresponding ki is estimated. In some cases the linearsegments are strikingly obvious, but in others they are obscured and/or a group of points may form a curved segment. What a researchermay do when faced with uncertainty in identifying segments is amatter of judgment. The linear segments can be either chosenvisually, or determined numerically by piecewise linear regression(PLR) (Malash and El-Khaiary, 2010). Some common errors frequent-ly occur in the application of Webber's pore-diffusion model, theseerrors are discussed next.

4.1.1. Extending the linear regression of pore-diffusion plots to includepoints after equilibrium

After equilibrium is reached q remains constant and the datapoints represent a horizontal line. If the data points after equilibrium

2 3 4 5 6 7 8 9

3

4

5

6

7

t 0.5 (min0.5)

q (m

g/g)

······· Line in original publication—— Suggested linear segments

10 mg/L20 mg/L

Fig. 4. Pore-diffusion plots for the adsorption of chromium(VI) onto activated carbonfor different initial feed concentrations.

0 2 4 6 80

0.2

0.4

0.6

0.8

1

1.2

t 0.5 (min0.5)

q (m

g/g)

······· Line in original publication—— Suggested linear segments

60 mg/L20 mg/L

Fig. 5. Pore-diffusion plots for the adsorption of methylene blue onto fly-ash fordifferent initial feed concentrations.

318 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320

are lumped with pre-equilibrium data (Bhattacharyya and Gupta,2008; Hubicki andWołowicz, 2009) to make one regression line, thenthe quality of fit will seem poor.

For the purpose of this illustration, Fig. 3 is a partial reproductionof a published pore-diffusion plot (Hubicki and Wołowicz, 2009). Inthis plot the last four data points to the right are at (or near)equilibrium; and they obviously don't belong to the same straight linewith the rest of the points. By excluding the first point to the left andfitting the data by PLR, the linear segments plotted in solid lines wereobtained. The details of PLR analysis is presented elsewhere (Malashand El-Khaiary, 2010).

4.1.2. Ignoring the presence of linear segmentsFig. 4 shows a published pore-diffusion plot (Acharya et al., 2009),

where it is easy to visually separate the data points into segments.Clearly the estimated slopes, and consequently ki values, differ greatlywhen segmentation is applied. Many papers (Abd El-Ghaffar et al.,2009; Atia, 2005; Debnath and Ghosh, 2008) present unsegmentedpore-diffusion plots, the result is either a faulty estimate of ki or awrong conclusion that the pore-diffusion model does not apply to thesystem.

4.1.3. Segmenting the data and discarding the first linear segmentAnother common practice is to detect and acknowledge segments,

then automatically dismiss the first segment(s) as a period wherefilm-diffusion is controlling the rate of adsorption (Sarkar et al., 2003;Kumar et al., 2005; Liu et al., 2010a). In most cases this practice isassociated with a common misconception of Boyd's diffusion models,which will be discussed later in Section 4.2. A typical case is shown inFig. 5.

Here the published study (Kumar et al., 2005) passed a pore-diffusion line through two points only just prior to equilibrium,arguing that the data that precede these two points are in a film-diffusion controlled period. This argument was based on theobservation that the first linear segment does not have a zero in-tercept. However, the first (and sometimes the only pre-equilibrium)segment of a pore-diffusion plot does not necessarily need to have a

Table 5The results of piecewise linear regression analysis of the kinetic data shown in Figs. 5 and

Slope of first segment Intercept of first segment

Pore-diffusion plot (Fig. 5) 0.164 (0.144–0.183) 0.002 (−0.069–0.070)Boyd plot (Fig. 6) 0.044 (0.035–0.054) −0.096 (−0.022–0.032)

zero intercept. A zero intercept of the first linear segment that startsfrom t=0 would imply that pore-diffusion is rate controllingthroughout the entire adsorption period. That would be a specialcase, possibly when the system is very vigorously agitated so that theresistance in the boundary layer is negligible at all times. Moreover,the first data point in this study was taken after 5 min; and duringthese 5 min 18% and 35% of qe were adsorbed for the initialconcentrations of 20 and 60 mg/L, respectively. During this 5 minperiod anything could have happened, maybe there are more linear orcurved segments, it is simply unknown because there is no data. It isnot correct to extrapolate a pore-diffusion line and base conclusionson the extrapolation. In addition, even if strong evidence existsagainst the pore-diffusion hypothesis, one cannot automaticallyconclude that film-diffusion is in control, other mechanisms may bein control, such as the rate of chemical reaction.

By analyzing the data in Fig. 5 by piecewise linear regression, thelinear segments plotted in solid lines were obtained. The numericalvalues of regression parameters (in case of initial concentration20 mg/L) are listed in Table 5. It can be seen that the confidenceinterval of the intercept of the first segment embraces zero, thus theintercept is not significantly different from zero. The break point is thepoint where two linear segments meet. By defining break-time as thetime a break point occurs, it is noticed that the first linear segmentends at a break-time of 23.2 min. These results are very different fromthose presented in the original study, and are based on chemical andstatistical theories.

4.2. Incorrect application of Boyd's diffusion models

In 1947 Boyd et al. published their legacy series of papers, wherethey presented theoretical models for ion-exchange that simulateequilibrium (Boyd et al., 1947a), kinetics (Boyd et al., 1947b), and nonequilibrium conditions (Boyd et al., 1947c). The adsorption commu-nity found that these kinetic models also apply to adsorption systemsand Boyd's diffusion models have been applied in numerousadsorption studies. However, a distorted version of Boyd's pore-diffusion model is circulating the literature and was used in manyrecent research papers.

4.2.1. Boyd's diffusion modelsIf diffusion inside the pores is the rate limiting step, the following

equation was derived (Boyd et al., 1947b):

F = 1− 6= π2� �

∑∞

n=11 = n2

� �exp −n2Bt

� �ð10Þ

where F is the fractional attainment of equilibrium, at different times,t, and Bt is a function of F

F = qt = qe ð11Þ

where qt and qe are the dye uptakes (mg/g) at time t and atequilibrium, respectively, and B is defined as:

B = π2Di

� �= r2o : ð12Þ

From Eq. (10), it is not possible to estimate directly the values of Bfor each fraction adsorbed. Reichenberg (1953)managed to obtain the

6. The values in parentheses represent the 94% confidence interval.

Slope of second segment Intercept of second segment Break time(min)

0.119 (0.084–0.155) 0.215 (0.015–0.416) 23.20.097 (0.074–0.120) −0.135 (−2.11–0.596) 23.9

0 10 20 30 40

-0.5

0

0.5

1

1.5

2

2.5

3

Time (min)

Bt

······· Line in original publication—— Suggested linear segments- - - - Linear segments from distorted Boyd equation

Bt from correct Boyd equationBt from distorted Boyd equation

Fig. 6. Boyd plot for the adsorption of methylene blue onto fly-ash and initial feedconcentration 20 mg/L.

319M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320

following approximations by applying the Fourier transform and thenintegration:

for F values N 0:85 Bt = −0:4977− ln 1−Fð Þ ð13Þ

and for F values b 0:85 Bt =ffiffiffiπ

p−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiπ− π2F =3

� �q� �2: ð14Þ

In order to apply this model to experimental data, the right-handsides of Eqs. (13) and (14) are calculated from the available q vs. t dataand a knowledge of qe. The resulting Bt values are then plotted againstt (Boyd plot). If the plot is linear, the slope is equal to B and it can beconcluded that pore-diffusion is the rate controlling step. The effectivediffusion coefficient,Di, (cm2/s) can be calculated from Eq. (12). Linearsegments can also be encountered in Boyd plots and in such casesevery segment is analyzed separately to obtain the correspondingdiffusion coefficient.

4.2.2. Distorted Boyd's diffusion modelsThe following distorted equation of Boyd's pore-diffusion model is

found in many recent publications:

for all F values Bt = −0:4977− ln 1−Fð Þ: ð15Þ

Possibly because a 1947 publication is not available in many librariesand databases, many studies copied Eq. (15) from each other (Beheraet al., 2008; Acharya et al., 2009; Ofomaja, 2010; Kamal et al., 2010;Liu et al., 2009; Sarkar et al., 2003) , including studies by one ofthe present authors (El-Khaiary, 2007). The use of Eq. (15) for allvalues of F leads to erroneous values of Btwhen F is less than 0.85, themagnitude of error increases as F becomes smaller.

This is graphically illustrated in Fig. 6 where Boyd's pore-diffusionmodel is applied to the same kinetic data of Fig. 5, the numericalresults of piecewise linear regression are presented in Table 5.Although the points clearly show two linear segments, the originalstudy passed a single straight line through all the points, resulting in apoor fit and an intercept far from zero. Accordingly, the originalpublished study considered this as an affirmation of its previousconclusion obtained from Fig. 5, confirming that pore-diffusion doesnot control the rate of adsorption in the time period from 5 to 40 min.Conversely, the results obtained from the correct model of Boyd, byacknowledging the presence of segmentation, lead to the oppositeconclusion. Interestingly, the break point in Fig. 6 is 23.9 min, aremarkably close value to the 23.2 min obtained from Fig. 5.

5. Discussion and conclusions

Statistical analysis and hypothesis testing are universally acceptedas the basic fundamentals of experimental science, and accordingly,research papers are supposed to present conclusions that aresupported by sound statistical tests. Excellent textbooks are nowfreely available online (Motulsky and Christopoulos, 2004; NIST,2010).

The misuse of linearization (linear transformation) is probably themost common error in batch adsorption literature. In the past,researchers needed to transform their data into a form suitable forsimple linear regression, but in the present age computers andsoftware are easily available to do this instead. In order to justify atransformation two conditions should be met:

1. The error-structure of the experimental data is known to violatesome assumptions of the least squares method.

2. A specific transformation is expected to change the error-structureto better satisfy these assumptions.

If these conditions are not met, then there is no point in linearizingthe data.

R2 is generated in the regression output of virtually all spread-sheets and statistical software. An issue with R2 is that its value can beartificially large when a model has a small degrees of freedom forerror. Therefore, one should not rely solely on R2 in assessing thegoodness of fit. The significance of estimated regression parametersshould also be tested with conventional statistical tests. In addition, R2

is often incorrectly used for comparing models that have differentdegrees of freedom. Akaike's Information Criterion is very easy tocompute and provides a sound basis for comparing such models.

Webber's pore-diffusionmodel is often abused in batch adsorptionstudies. This is mainly manifested in the disregard of segments ormismanagement of segmented data. It is recommended that pore-diffusion plots are examined carefully and segments, if present,identified numerically by PLR. It would also be beneficial to have asmany kinetic data points as possible if Webber's model is a candidatefor data analysis. This would ensure having a reasonable number ofpoints in each segment and thus obtaining statistically significantestimates of the diffusion parameters.

A distorted version of Boyd's pore-diffusion model is widelyspread in the literature. Using this distorted model leads to wrongestimates of Bt from the beginning of adsorption up to 85% attainmentof equilibrium; consequently, it leads to wrong conclusions about therate limiting step.

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