trabalho adsorção

1. Hydrometallurgy 105 (2011) 314–320
Contents lists available at ScienceDirect
Hydrometallurgy
journal homepage: www.elsevi e r.com/locate/hydromet
Common data analysis errors in batch adsorption studies
Mohammad I. El-Khaiary ⁎, Gihan F. Malash
Chemical Engineering Department, Faculty of Engineering, Alexandria University, El-Hadara, Alexandria 21544, Egypt
a r t i c l e i n f o a b s t r a c t
Article history:
Received 16 March 2010
Received in revised form 29 October 2010
Accepted 2 November 2010
Available online 21 November 2010
Keywords:
Adsorption
Regression
Isotherm
Linearization
Kinetics
Mechanism
Many models exist for describing the experimental results of batch adsorption which are used in research to
study equilibrium, kinetics, and mechanisms of adsorption. In the process of statistically analyzing the
experimental data, the adsorption literature contains errors that render the results unreliable. These errors
include incorrect application of theoretical models and also incorrect application of statistical analysis. Some
errors are so abundant in the adsorption literature that they have actually gained credibility and mistakenly
taken for granted that these are sound scientific practices. This article highlights some common errors in
adsorption data analysis that are frequently found in the literature and provides suggestions for more sound
practices.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
In the course of data analysis, statistical methods are applied to fit
experimental data to models and to assess statistical inferences. There
is an abundance of software for statistical calculations which come as
stand-alone commercial programs, as part of commercial spread-sheets
or as shareware and freeware. However, the availability of
good software does not ensure a correct statistical analysis. Pre-requisites
for the correct use of software are a familiarity with the
basics of statistics; a knowledge of the assumptions and limitations of
each statistical technique; and the ability to choose the appropriate
method of analysis. Uncritical use of statistical software may lead to
applying the techniques incorrectly, or even using statistical methods
in cases where they are not appropriate.
In the field of batch adsorption research, certain errors in data
analysis keep showing up over and over again. They show up in top
ranking journals and are so abundant in the adsorption literature that
they have actually gained credibility. New comers to adsorption
research may take it for granted that these are sound scientific
practices and spread the errors even more. The objective of this article
is to highlight some of these errors and to provide suggestions for
more sound practices. For clarity, a few specific examples are taken as
prototypes from the literature. In the choice of these examples,
articles were chosen to represent several journals and authors from
different countries.
2. Linearization
The misuse of linearization is probably the most common error in
data analysis. It originatedmany decades agowhen computerswere not
yet available and it has not lost its popularity today. There are problems
associated with trying to linearize an inherently nonlinear equation by
use of various transformations. Themain issue when transforming data
to achieve a linear equation is knowledge of the error-structure of the
data and how this structure is affected by transformation. When the
errors are additive on the dependent variable (Y) and satisfy the usual
assumptions of normality and homo-skedasticity (equal variance)
throughout the range of the data, then transforming the dependent
variable with a nonlinear function can destroy the assumed distribu-tional
properties. However, when the original error-structure does not
satisfy these assumptions, by judicious choice of a transformation, the
model can in some cases be transformed to satisfy these assumptions.
For example, if the variance of errors increaseswith increasing values of
Y, a square root or log transformation will often help, but if the variance
of the errors decreaseswith increasing Y, then the square or exponential
transformation will often be appropriate. In the absence of data about
the error-structure,which is the case inmost adsorption studies, there is
no point in applying linearization.
Typical examples are adsorption isotherms and adsorption kinetic
models. 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, the
nonlinear form of the well-known Langmuir (1916) isotherm is:
qe = qmKaCe ð Þ= 1 + KaCe ð Þ ð1Þ
⁎ Corresponding author. Fax: +20 3 5921853.
E-mail address: elkhaiary@gmail.com (M.I. El-Khaiary).
0304-386X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.hydromet.2010.11.005

2. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 315
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 is
the equilibrium adsorption constant (L/mg). This nonlinear form
can be mathematically manipulated and linearized to at least three
linear forms as shown in Table 1 (El-Khaiary, 2008). Likewise, Table 2
(El-Khaiary et al., 2010) shows four linearized forms of the widely
used pseudo-second-order kinetic model of Ho (2004) which has the
following nonlinear form
q =
q2e
kt
1 + qekt
ð2Þ
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 time
t (mg/g), the independent variable.
These linearized forms are used extensively in adsorption
literature. The structure of experimental error is transformed along
with the data (in some cases leading to loss of homo-skedasticity),
and a basic assumption of the least squares method (i.e. that the
independent variable has only a negligible error) is sometimes
violated. Moreover, the statistical tests used to check the goodness
of fit will often not detect that the parameters are biased.
Tables 1 and 2 present the effects of different linearizations to the
models of Langmuir and Ho. The effect of linearizing Ho's equation on
the accuracy of parameter estimates is graphically demonstrated in
Fig. 1. This figure was constructed by applying linear and nonlinear
least 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 the
untransformed data. It is clear that linear regression of the
three linearized forms of Ho's equation produced parameter
estimates (and consequently curve fittings) that vary wildly from
each other. It can be seen that nonlinear regression produced a better
fit that is closer to the data points. This should not be surprising as
there is a large body of literature that warns from the use of
linearization 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, modeling
using linearized equations have been published in literally thousands
of adsorption papers (Liu et al., 2010b; Yuan et al., 2010; Zhang et al.,
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 and
Sahu, 2006).
3. Abuse of R2 and model comparison
A common practice in research is to fit the experimental data
to several models, then perform some kind of test to compare and
decide which model fits the data better. Based on the choice of the
“best model”, conclusions about the intricate mechanism of the
system are often available. The most popular tool for model
comparison is the coefficient of determination, R2. It is the square of
Table 1
Linearized forms of Langmuir isotherm.
Type Linearized form Plot Effects of linearization
LR I Ce
qe
=
1
Kaqm
+
1
qm
Ce
Ce
qe
vs:Ce
Ce in both dependent and independent
variables, leading to spurious correlation
The error distribution of the dependent
variable, Ce/qe, is different from both the
error distributions of Ce and qe
Reversal of relative weights of data points
because of 1/qe in dependent variable
LR II 1
qe
=
1
qm
+
1
Kaqm
1
Ce
1
qe
vs:
1
Ce
Distortion of relative weights of data
points because of 1/qe and 1/Ce in
dependent and independent variables
Independent variable is 1/Ce, leading to
distortion of error distribution
LR III qe
Ce
= Kaqm−Kaqe
qe
Ce
vs:qe
qe in both dependent and independent
variables, leading to spurious correlation
The relative weights of data points are
distorted because the independent
variable is qe/Ce
The error distribution of the dependent
variable, qe/Ce, is different from both the
error distributions of Ce and qe
Table 2
Linearized forms of the pseudo-second-order kinetic model.
Type Linearized Form Plot Effects of linearization
Linear 1 t
q
=
1
kq2e
+
1
qe
t
t/q vs. t Reversal of relative weights of data
points because of 1/q in the
dependent variable
t in both dependent and
independent variables, leading to
spurious correlation
Linear 2 1
q
=
1
qe
+
1
kq2e
1
t
1/q vs. 1/t Reversal of relative weights of data
points because of 1/q in dependent
variable
Independent variable is 1/t,
leading to distortion of error
distribution
Linear 3
q = qe− 1
kqe
q
t
q vs. q/t qin both dependent and
independent variables, leading to
spurious correlation
The presence of q in the
independent variable (q/t)
introduces experimental error,
violating a basic assumption in the
method of least squares
1/t in independent variable,
leading to distortion of error
distribution
Linear 4 q
t
= kq2e
−kqeq
q/t vs.q qin both dependent and
independent variables, leading to
spurious correlation
The presence of q in the
independent variable introduces
experimental error, violating a
basic assumption in the method of
least squares
1 2 3 4 5 6
12
10
8
6
4
2
time, min
q, mg Ag/g
Experimental
— — N o n l in e ar
— — Lin 1
—. .— Lin 2
........ Lin 3
---- Lin 4
Fig. 1. Comparison of the pseudo-second-order parameters estimated by linear and
nonlinear regressions for the adsorption of silver thiocyanate complexes on anion
exchanger AV-17-10P.

3. 316 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320
the correlation coefficient, r, between the dependent variable and
the regression-predicted value of the dependent variable. In other
words, R2 is the fraction of the total variance of the dependent variable
that is explained by the model's equation. Mathematically, it is
defined as:
R2 = 1− SSE
SStot
ð3Þ
where SSE is the sum-of-squared deviations of the points from the
regression curve and SStot is the sum-of-squared deviations of the
points from a horizontal line where Y equals the mean of all the data
points.
3.1. Abuse of R2
In spite of the apparent simplicity of interpreting R2, it is not
always suitable for evaluating the goodness of fit and comparing
models. The most obvious issues with R2 are:
• R2 is sensitive to extreme data points, resulting in misleading
indication of the quality of fit. This is further complicated when the
data points are subjected to linear transformations; existing
extreme-points may disappear and new extreme-points may be
created.
• R2 is influenced by the range of the independent variable. R2
increases as the range of independent variable increases and
decreases as the range decreases.
• R2 can be made artificially large by adding more parameters to the
model. In other words, R2 value increases with the decrease in
degrees of freedom for error.
The first two issues with R2 can be avoided by fitting the data to the
model without any transformations and by examination of extreme-points,
while the third issue is more subtle and is discussed further in
Sections 3.1 and 3.2.
3.1.1. Spurious goodness of fit concluded from high R2 values when the
fitting has a small degree of freedom
This is especially common in adsorption studies when it comes to
estimating thermodynamic parameters and it is also found in other
cases such as isotherm and diffusion calculations (Wan Ngah and
Hanafiah, 2008; Abd El-Ghaffar et al., 2009; Unuabonah et al., 2008;
Agrawal and Sahu, 2006). To illustrate, Fig. 2 is reproduced from a
recent article (Unuabonah et al., 2008) where three data points
were used to plot the natural logarithm of b, the Langmuir constant
related to energy, against the reciprocal absolute temperature, 1/T, to
calculate ΔH and ΔS from the slope and intercept of a straight line
according to the linearized equation:
ln b =
ΔS
Rc
−ΔH
RcT
ð4Þ
where Rc is the universal gas constant (8.314 J/K mol), ΔH is the
enthalpy change (J/mol) and ΔS is the entropy change (J/K mol).
Linear regression analysis was performed and some of the results are
shown in Table 3. The linear plot in Fig. 2 seems good, with little
deviation of the points from the regression line, and also the value of
R2 (which represents the proportion of the variation in log b that can
be accounted for by variation in 1/T) is 0.9415, which is relatively high
and may suggest an acceptable fit. However, the numbers in Table 3
tell another story.
The estimated slope is highly uncertain because zero is within its
95% confidence interval, and consequently, there is no evidence that
the slope is different from zero. Therefore, the apparent variation of ln
b with changes in 1/T may be due to random variation (noise). By
calculating 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 estimated
value of ΔH (+14.1 kJ/mol) virtually useless. The insignificance of the
slope is further corroborated by the result of t-test, the significance
level of this test is 0.1283 which means that there is a 12.83%
probability that the apparent slope is caused by noise, and since
0.1283 is much larger than the conventional 0.0500 cut-off value, the
hypothesis that the slope is zero is not rejected. The same discussion
applies to the statistical insignificance of the intercept, and accord-ingly,
any conclusions based on the values or signs of ΔH and ΔS are
unsupported.
The previous analysis does not prove that ln b is independent from
1/T, it only shows that fitting the straight line to three data points did
not give enough evidence – from a statistical point of view – to
support a hypothesis that there is a linear correlation between the two
variables.
3.2. Comparing models that have different degrees of freedom
For a fixed sample size, increasing the number of regression
parameters leads to a decrease in the degrees of freedom, and almost
universally 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 the
degrees of freedom.
It is customary in batch adsorption studies to fit the equilibrium
uptake data to several isotherms, then to use R2 to compare the
goodness of fit and select the best isotherm model. With the best
isotherm supposedly identified, conclusions are usually presented
regarding the homogeneity of the adsorbent surface and the
mechanism of adsorption. However, a common pitfall is that some
studies use R2 to compare isotherms that have two, three, and four
parameters (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 to
compare models. It is based on information theory and maximum
0.0031 0.00315 0.0032 0.00325 0.0033 0.00335
-5.3
-5.4
-5.5
-5.6
-5.7
-5.8
1/T (K-1)
ln b
Fig. 2. Plot of ln b vs. 1/T for cadmium adsorption onto unmodified kaolinite clay.
Table 3
Linear regression results of the experimental data shown in Fig. 2.
Intercept 95% confidence
limits of intercept
Slope 95% confidence
limits of slope
Probability
value for slope
(t-test)
R2
−0.12 −14.30 to 14.07 −1698 −6106 to 2711 0.1283 0.9415

4. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 317
Table 4
Estimated parameter-values for Langmuir, Freundlich, and Redlich–Peterson isotherms.
Langmuir isotherm Freundlich isotherm Redlich–Peterson isotherm
qm
(mg/g)
likelihood theory, and as such, it determines which model is more
likely to be correct and quantifies how much more likely. For a small
sample size, AIC is calculated for each model from the equation:
AIC = N ln
SSE
N
+ 2Np +
2Np Np + 1
N−Np−1
ð5Þ
where N is the number of data points, and Np is the number of
parameters in the model.
AIC values can be compared using the Evidence ratio which is
defined by:
Evidence ratio =
1
e−0:5Δ
ð6Þ
where Δ is the absolute value of the difference in AIC between the two
models.
This comparison method is illustrated using isotherm data from a
recent study (Gunay et al., 2007) where several two and three-parameter
isotherms were fitted to the data. The two-parameter
models are Langmuir and Freundlich (1906) isotherms, while the
three-parameter model is Redlich and Peterson (1959) isotherm.
Freundlich isotherm : qe = Kf C1 = n
e ð7Þ
Redlich–Peterson isotherm : qe =
KRCe
1 + aRCβ
e
ð8Þ
The results of nonlinear regression, published in the study, are
presented 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 the
goodness of fit to Langmuir and Redlich–Peterson isotherms.
Accordingly, AIC values were calculated for Langmuir (1.521) and
Redlich–Peterson (6.330) isotherms. Having a smaller AIC value
suggests that Langmuir isotherm is more likely to be a better fit. The
Evidence ratio of 11.07 means that it is 11.07 times more likely to be
the correct model than the Redlich–Peterson isotherm.
4. Incorrect application of models
4.1. Incorrect application of Webber's pore-diffusion model
Webber's pore-diffusion model (Weber and Morris, 1963) is
commonly 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 limiting
step in the adsorption process, then a pore-diffusion plot (q vs t0.5) is
expected to be a straight line with a slope that equals ki. In practice,
things are not that simple because pore-diffusion plots often show
several linear segments. It has been proposed that these linear segments
represent pore-diffusion in pores of progressively smaller sizes (Ho and
McKay, 1998; Allen et al., 1989, 2005; Koumanova et al., 2003; Cheung
et al., 2007). Eventually, equilibrium is reached and q stops changing
with time; and a final horizontal line is established at qe.
It follows from the previous discussion that it would be a good
practice to examine pore-diffusion plots and decide how many linear
segments exist. When a group of points are identified as belonging to
a linear segment, linear regression can then be applied to these points
and the corresponding ki is estimated. In some cases the linear
segments are strikingly obvious, but in others they are obscured and/
or a group of points may form a curved segment. What a researcher
may do when faced with uncertainty in identifying segments is a
matter of judgment. The linear segments can be either chosen
visually, 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, these
errors are discussed next.
4.1.1. Extending the linear regression of pore-diffusion plots to include
points after equilibrium
After equilibrium is reached q remains constant and the data
points represent a horizontal line. If the data points after equilibrium
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
Fig. 3. Pore-diffusion plots for the removal of Pd(II) complexes from NaCl–HCl solutions
containing 100 μg/cm3 Pd (II) determined for the weakly basic Amberlyst A 21.
10 mg/L
20 mg/L
2 3 4 5 6 7 8 9
7
6
5
4
3
t 0.5 (min0.5)
q (mg/g)
······· Line in original publication
—— Suggested linear segments
Fig. 4. Pore-diffusion plots for the adsorption of chromium(VI) onto activated carbon
for different initial feed concentrations.

5. 318 M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320
1.2
1
0.8
0.6
0.4
0.2
are lumped with pre-equilibrium data (Bhattacharyya and Gupta,
2008; Hubicki and Wołowicz, 2009) to make one regression line, then
the quality of fit will seem poor.
For the purpose of this illustration, Fig. 3 is a partial reproduction
of a published pore-diffusion plot (Hubicki and Wołowicz, 2009). In
this plot the last four data points to the right are at (or near)
equilibrium; and they obviously don't belong to the same straight line
with the rest of the points. By excluding the first point to the left and
fitting the data by PLR, the linear segments plotted in solid lines were
obtained. The details of PLR analysis is presented elsewhere (Malash
and El-Khaiary, 2010).
4.1.2. Ignoring the presence of linear segments
Fig. 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 greatly
when segmentation is applied. Many papers (Abd El-Ghaffar et al.,
2009; Atia, 2005; Debnath and Ghosh, 2008) present unsegmented
pore-diffusion plots, the result is either a faulty estimate of ki or a
wrong conclusion that the pore-diffusion model does not apply to the
system.
4.1.3. Segmenting the data and discarding the first linear segment
Another common practice is to detect and acknowledge segments,
then automatically dismiss the first segment(s) as a period where
film-diffusion is controlling the rate of adsorption (Sarkar et al., 2003;
Kumar et al., 2005; Liu et al., 2010a). In most cases this practice is
associated with a common misconception of Boyd's diffusion models,
which will be discussed later in Section 4.2. A typical case is shown in
Fig. 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 the
observation 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
zero intercept. A zero intercept of the first linear segment that starts
from t=0 would imply that pore-diffusion is rate controlling
throughout the entire adsorption period. That would be a special
case, possibly when the system is very vigorously agitated so that the
resistance in the boundary layer is negligible at all times. Moreover,
the first data point in this study was taken after 5 min; and during
these 5 min 18% and 35% of qe were adsorbed for the initial
concentrations of 20 and 60 mg/L, respectively. During this 5 min
period anything could have happened, maybe there are more linear or
curved segments, it is simply unknown because there is no data. It is
not correct to extrapolate a pore-diffusion line and base conclusions
on the extrapolation. In addition, even if strong evidence exists
against the pore-diffusion hypothesis, one cannot automatically
conclude that film-diffusion is in control, other mechanisms may be
in control, such as the rate of chemical reaction.
By analyzing the data in Fig. 5 by piecewise linear regression, the
linear segments plotted in solid lines were obtained. The numerical
values of regression parameters (in case of initial concentration
20 mg/L) are listed in Table 5. It can be seen that the confidence
interval of the intercept of the first segment embraces zero, thus the
intercept is not significantly different from zero. The break point is the
point where two linear segments meet. By defining break-time as the
time a break point occurs, it is noticed that the first linear segment
ends at a break-time of 23.2 min. These results are very different from
those presented in the original study, and are based on chemical and
statistical theories.
4.2. Incorrect application of Boyd's diffusion models
In 1947 Boyd et al. published their legacy series of papers, where
they presented theoretical models for ion-exchange that simulate
equilibrium (Boyd et al., 1947a), kinetics (Boyd et al., 1947b), and non
equilibrium conditions (Boyd et al., 1947c). The adsorption commu-nity
found that these kinetic models also apply to adsorption systems
and Boyd's diffusion models have been applied in numerous
adsorption studies. However, a distorted version of Boyd's pore-diffusion
model is circulating the literature and was used in many
recent research papers.
4.2.1. Boyd's diffusion models
If diffusion inside the pores is the rate limiting step, the following
equation was derived (Boyd et al., 1947b):
F = 1− 6 = π2
∞
Σ
n=1
1 = 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 at
equilibrium, respectively, and B is defined as:
B = π2Di
= r2
o : ð12Þ
From Eq. (10), it is not possible to estimate directly the values of B
for each fraction adsorbed. Reichenberg (1953) managed to obtain the
0 2 4 6 8
0
t 0.5 (min0.5)
q (mg/g)
······· Line in original publication
—— Suggested linear segments
60 mg/L
20 mg/L
Fig. 5. Pore-diffusion plots for the adsorption of methylene blue onto fly-ash for
different initial feed concentrations.
Table 5
The results of piecewise linear regression analysis of the kinetic data shown in Figs. 5 and 6. The values in parentheses represent the 94% confidence interval.
Slope of first segment Intercept of first segment Slope of second segment Intercept of second segment Break time
(min)
Pore-diffusion plot (Fig. 5) 0.164 (0.144–0.183) 0.002 (−0.069–0.070) 0.119 (0.084–0.155) 0.215 (0.015–0.416) 23.2
Boyd plot (Fig. 6) 0.044 (0.035–0.054) −0.096 (−0.022–0.032) 0.097 (0.074–0.120) −0.135 (−2.11–0.596) 23.9

6. M.I. El-Khaiary, G.F. Malash / Hydrometallurgy 105 (2011) 314–320 319
following approximations by applying the Fourier transform and then
integration:
for F values N 0:85 Bt = −0:4977− lnð1−FÞ ð13Þ
and for F values b 0:85 Bt =
q
ffiffiffi
π
p
−
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π− π2F =3
2
: ð14Þ
In order to apply this model to experimental data, the right-hand
sides of Eqs. (13) and (14) are calculated from the available q vs. t data
and a knowledge of qe. The resulting Bt values are then plotted against
t (Boyd plot). If the plot is linear, the slope is equal to B and it can be
concluded that pore-diffusion is the rate controlling step. The effective
diffusion coefficient, Di, (cm2/s) can be calculated from Eq. (12). Linear
segments can also be encountered in Boyd plots and in such cases
every segment is analyzed separately to obtain the corresponding
diffusion coefficient.
4.2.2. Distorted Boyd's diffusion models
The following distorted equation of Boyd's pore-diffusion model is
found in many recent publications:
for all F values Bt = −0:4977− lnð1−FÞ: ð15Þ
Possibly because a 1947 publication is not available in many libraries
and databases, many studies copied Eq. (15) from each other (Behera
et al., 2008; Acharya et al., 2009; Ofomaja, 2010; Kamal et al., 2010;
Liu et al., 2009; Sarkar et al., 2003) , including studies by one of
the present authors (El-Khaiary, 2007). The use of Eq. (15) for all
values of F leads to erroneous values of Bt when F is less than 0.85, the
magnitude of error increases as F becomes smaller.
This is graphically illustrated in Fig. 6 where Boyd's pore-diffusion
model is applied to the same kinetic data of Fig. 5, the numerical
results of piecewise linear regression are presented in Table 5.
Although the points clearly show two linear segments, the original
study passed a single straight line through all the points, resulting in a
poor fit and an intercept far from zero. Accordingly, the original
published study considered this as an affirmation of its previous
conclusion obtained from Fig. 5, confirming that pore-diffusion does
not control the rate of adsorption in the time period from 5 to 40 min.
Conversely, the results obtained from the correct model of Boyd, by
acknowledging the presence of segmentation, lead to the opposite
conclusion. Interestingly, the break point in Fig. 6 is 23.9 min, a
remarkably close value to the 23.2 min obtained from Fig. 5.
5. Discussion and conclusions
Statistical analysis and hypothesis testing are universally accepted
as the basic fundamentals of experimental science, and accordingly,
research papers are supposed to present conclusions that are
supported by sound statistical tests. Excellent textbooks are now
freely available online (Motulsky and Christopoulos, 2004; NIST,
2010).
The misuse of linearization (linear transformation) is probably the
most common error in batch adsorption literature. In the past,
researchers needed to transform their data into a form suitable for
simple linear regression, but in the present age computers and
software are easily available to do this instead. In order to justify a
transformation two conditions should be met:
1. The error-structure of the experimental data is known to violate
some assumptions of the least squares method.
2. A specific transformation is expected to change the error-structure
to better satisfy these assumptions.
If these conditions are not met, then there is no point in linearizing
the data.
3
2.5
2
1.5
1
0.5
0
-0.5
······· Line in original publication
—— Suggested linear segments
- - - - Linear segments from distorted Boyd equation
Bt from correct Boyd equation
Bt from distorted Boyd equation
0 10 20 30 40
Time (min)
Bt
Fig. 6. Boyd plot for the adsorption of methylene blue onto fly-ash and initial feed
concentration 20 mg/L.
R2 is generated in the regression output of virtually all spread-sheets
and statistical software. An issue with R2 is that its value can be
artificially large when a model has a small degrees of freedom for
error. Therefore, one should not rely solely on R2 in assessing the
goodness of fit. The significance of estimated regression parameters
should also be tested with conventional statistical tests. In addition, R2
is often incorrectly used for comparing models that have different
degrees of freedom. Akaike's Information Criterion is very easy to
compute and provides a sound basis for comparing such models.
Webber's pore-diffusion model is often abused in batch adsorption
studies. This is mainly manifested in the disregard of segments or
mismanagement 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 as
many kinetic data points as possible if Webber's model is a candidate
for data analysis. This would ensure having a reasonable number of
points in each segment and thus obtaining statistically significant
estimates of the diffusion parameters.
A distorted version of Boyd's pore-diffusion model is widely
spread in the literature. Using this distorted model leads to wrong
estimates of Bt from the beginning of adsorption up to 85% attainment
of equilibrium; consequently, it leads to wrong conclusions about the
rate limiting step.
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