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Optimum Isotherms of Dyes Sorption by Activated Carbon: Fractional Theo-retical Capacity & Error Analysis
Gordon McKay, Alireza Mesdaghinia, Simin Nasseri, Mahdi Hadi, MehriSolaimany Aminabad
PII: S1385-8947(14)00486-0DOI: http://dx.doi.org/10.1016/j.cej.2014.04.054Reference: CEJ 12033
To appear in: Chemical Engineering Journal
Received Date: 4 March 2014Revised Date: 10 April 2014Accepted Date: 12 April 2014
Please cite this article as: G. McKay, A. Mesdaghinia, S. Nasseri, M. Hadi, M. Solaimany Aminabad, OptimumIsotherms of Dyes Sorption by Activated Carbon: Fractional Theoretical Capacity & Error Analysis, ChemicalEngineering Journal (2014), doi: http://dx.doi.org/10.1016/j.cej.2014.04.054
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1
Optimum Isotherms of Dyes Sorption by Activated Carbon: 1
Fractional Theoretical Capacity & Error Analysis 2
Gordon McKay1, Alireza Mesdaghinia2, Simin Nasseri2, Mahdi Hadi2*, 3
Mehri Solaimany Aminabad3 4
5
1. Department of Chemical and Biomolecular Engineering, Hong Kong University of 6
Science and Technology, Clearwater Bay, New Territory, HONG KONG SAR. 7
2. Center for Water Quality Research (CWQR), Institute for Environmental Research 8
(IER), Tehran University of Medical Sciences, Tehran, Iran. 9
3. Kurdistan Environmental Health Research Center, School of Health, Kurdistan 10
University of Medical Sciences, Sanandaj, Iran 11
12
Correspondence to: 13
Mahdi Hadi, Center for Water Quality Research (CWQR), Institute for Environmental 14
Research (IER), Tehran University of Medical Sciences, Tehran, Iran. E-mail: m- 15
[email protected], [email protected], Tel: +989189061738. 16
17
18
19
20
21
2
Abstract 22
The applicability of statistical Goodness-of-Fit Measures (GoFMs) and a new 23
measure, Fractional Theoretical Capacity (FTC), to finding the best fitting isotherm 24
model(s) in the adsorption of dyes have been assessed. The experimental data of 25
adsorption of three acid dyes; Acid Blue 80 (AB80), Acid Red 114 (AR114) and Acid 26
Yellow 117 (AY117) onto Granular Activated Carbon (GAC) type F400 were used in 27
the model selection. Three two-parameter, nine three and one four-parameter isotherm 28
models were used to fit the experimental data. In order to determine the best-fit 29
isotherm for each dye/sorbent system, the geometrical structure of the dyes was 30
optimized with a semi-empirical PM3 method. Thus the approximate maximum cross- 31
sectional area of dye molecules and then the minimum FTC were determined by 32
molecular calculations. The model with the highest FTC regarding to each dye was 33
chosen as the best descriptive model. Statistically eleven GoFMs were also applied to 34
evaluate and rank the feasibility of isotherm models. The model with the lowest 35
GoFM was chosen again as the best. The results showed that using GoFMs alone may 36
leads to wrong model selection but FTC can be a better measure for best descriptive 37
model selection. Based on the FTC measure the adsorption isotherm models fitted the 38
experimental data in the orders: Toth > Langmuir–Freundlich > Sips > UniLan, Toth 39
> UniLan > Langmuir–Freundlich > Sips and Toth > UniLan > Sips > Langmuir– 40
Freundlich for the dyes AY117, AR114 and AB80, respectively. The FTC measure 41
application is recommended in the isotherm model selection process. 42
Keywords: Adsorption, Error analysis, Fractional Theoretical Capacity, Isotherms, 43
Model selection 44
45
3
Nomenclature: 46
47
Ce equilibrium concentration of dye in solution (mg L-1) 48
C0 initial dye concentration (mg L-1) 49
m Sorbent mass, g 50
qe amount of dye adsorbed at equilibrium time(mg g-1) 51
R the universal gas constant (J mol-1 K-1) 52
T Temperature (K) 53
KT the Toth equilibrium isotherms constant (L g-1) 54
qmT maximum adsorption capacity in Toth model(mg g-1) 55
mT the Toth model exponent 56
mRPI the Radke---Prausnitz-I model exponent 57
mRPII the Radke---Prausnitz-II model exponent 58
mRPIII the Radke---Prausnitz-III model exponent 59
KRPI the Radke---Prausnitz-I equilibrium constant 60
KRPII the Radke---Prausnitz-II equilibrium constant 61
KRPIII the Radke---Prausnitz-III equilibrium constant 62
.
. RPI the Radke---Prausnitz-I maximum adsorption capacity (mg g−1) 63
.
. RPII the Radke---Prausnitz-II maximum adsorption capacity (mg g−1) 64
.
. RPIII the Radke---Prausnitz-III maximum adsorption capacity (mg g−1) 65
A the Fritz---Schlunder model parameter (L g−1) 66
B the Fritz---Schlunder model parameter (L mg−1).. 67
KLF the equilibrium constant for a heterogeneous solid 68
mLF the heterogeneity parameter, lies between 0 and 1 69
.
. LF the Langmuir---Freundlich maximum adsorption capacity (mg g−1) 70
.
. FS the Fritz---Schlunder maximum adsorption capacity (mg g−1) 71
KFS the Fritz---Schlunder equilibrium constant (L mg−1) 72
mFS the Fritz---Schlunder model exponent 73
b0 the Baudu isotherm equilibrium constant 74
qm0 the maximum adsorption capacity in the Baudu isotherm equation(mg g−1) 75
x the Baudu isotherm parameter 76
y the Baudu isotherm parameter 77
N the number of experimental points 78
GAC granular activated carbon 79
S the empirical parameter of UniLan model 80
qmu the maximum adsorption capacity in the UniLan isotherm equation(mg g−1) 81
Ku the empirical parameter of UniLan model 82
qmk the maximum adsorption capacity in the Khan isotherm equation(mg g−1) 83
bk the Khan isotherm equilibrium constant 84
qm Maximum adsorption capacity in Langmuir model (mg g-1) 85
b Langmuir constant related to the energy of adsorption (L mg-1) 86
Kj Jovanovic isotherm constant (L g-1) 87
qmj Maximum adsorption capacity in Jovanovic model (mg g-1) 88
Qs Theoretical monolayer saturation capacity in Dubinin-Radushkevich model (mg g-1) 89
BD Dubinin-Radushkevich model constant (mol2 kJ-2) 90
91
Greek letters 92
ε Polanyi potential 93
.
.
. . exponent in the Fritz---Schlunder model that lies between 0 and 1 94
4
β exponent in the Fritz-Schlunder and Khan models 95
96
1. Introduction 97
Among the physico-chemical treatment processes, adsorption technology is 98
considered to be one of the most effective and proven technologies having potential 99
application in both water and wastewater treatment [1]. Adsorption equilibria data is 100
the most important piece of information in understanding an adsorption process. The 101
adsorption equilibria of pure components are the essential ingredient for 102
understanding the amount of those components which can be accommodated by a 103
solid adsorbent [2]. Modeling of adsorption isotherm data is important for predicting 104
and comparing adsorption system performance. Isotherms data can be used to obtain a 105
rough estimate of the Carbon Usage Rate (CUR) and adsorbent bed life, which can be 106
useful in determining the applicability of adsorbent. Isotherm parameters can also be 107
used as input parameters for mathematical models to predict performance of an 108
adsorption process [3]. Using the data obtained from the batch isotherm studies, 109
prediction of the theoretical breakthrough curve in adsorption columns will be 110
possible. In another word the data of Continuously Mixed Batch Reactor (CMBR) are 111
prerequisite information for the detailed design of Fixed Bed Reactor (FBR) [3]. So 112
far, several isotherm models with different assumptions have been developed to 113
examine the adsorption mechanism. However, many models can not describe well the 114
experimental data. To finding the best isotherm model, statistical Goodness-of-Fit 115
Measures (GoFMs) were suggested and applied in literatures [4-6].The determination 116
of the best isotherm models for the sorption of reactive dyes from aqueous solutions 117
by furnace slag [7] and compost [8], Acid dyes by pine-cone derived carbon [5], 118
leather dye by tannery solid waste [9] and food dyes by chitosan films [10] were 119
5
assessed by statistical error analysis methods. Kumar et al.[11] imply that the size of 120
the GoFMs alone is not a deciding factor to choose the optimum isotherm, thus in 121
addition to the size of them, the theory behind the predicted isotherm should also be 122
verified with the help of experimental data while selecting the optimum isotherm. 123
The applicability of several GoFMs, for non-linear analysis, in determining the best 124
fitting isotherm model(s) and moreover considering the theoretical capacity of the 125
adsorbent (activated carbon) in the adsorption of adsorbate (dyes) molecules, will be 126
assessed in this study. The research is aimed to test several isotherm models to 127
describe the sorption data generated from acid dyes sorption by activated carbon 128
(commercial activated carbon F400). The experiments for the acidic dyes sorption, 129
namely Acid Blue 80 (AB80), Acid Red 114 (AR114) and Acid Yellow 117 (AY117) 130
were conducted by Choy et al. [12]. So far several papers regarding to these 131
experimental data have been published and the readers are referred to dyes sorption in 132
multi-component conditions [12], film-pore-surface diffusion model [13], Intraparticle 133
diffusion model [14] and equilibrium two-parameter isotherms models[5]. The 134
aforementioned experimental sorption data were provided by Prof. Gordon McKay 135
and used in this study too. Three two-parameter models - Langmuir, Jovanovic and 136
Dubinin-Radushkevich- nine three-parameter models - Langmuir-Freundlich, Sips, 137
Toth, Radke-Prausnitz (I, II and III), Fritz-Schlunder, Khan and UniLan - and one 138
four-parameter model - Baudu - were used to fit the experimental data. This study 139
aims to compare two approaches to determine which of the several isotherm models 140
fits best. The maximum cross-sectional surface area for each dye after optimizing the 141
dyes molecular geometry and then the Minimum Theoretical Adsorption Capacity 142
(MTAC) for the dyes were determined. The Fractional Theoretical Capacity (FTC) 143
measure of each isotherm model for the sorption of each dye was determined by 144
6
dividing the maximum adsorption capacity parameter of the model to the MTAC 145
derived for the dye. The model with highest FTC regarding to the theoretical capacity 146
for each dye was chosen as the best descriptive model. In another approach eleven 147
GoFMs used as a measure of the differences between values predicted by a model and 148
the values actually observed from the experimental data. The GoFMs were G2, Chi- 149
square (X2), Residual Root Mean Square Error (RMSE), Sum of the Squares of the 150
Errors (ERRSQ), Sum of the Absolute Errors (EABS), Composite Functional Error 151
(HYBRD), derivative of Marquardt’s Percent Standard Deviation (MPSD), Average 152
Relative Error (ARE), Average Percentage Error (APE), Akaike Information Criterion 153
(AICC) and Mallows CP statistic (CP). Application of all GoFMs by using them 154
together was assessed to finding the best isotherm model. These two approaches 155
(GoFMs and FTC) may be used to reduce errors when looking for the best isotherm 156
model to elucidating the adsorption mechanism but based on the results of this study it 157
is recommended that the best isotherm model be selected according to the FTC 158
measure. 159
160
161
162
163
164
165
166
2. Experimental section 167
2.1. Activated Carbon F400 168
7
The adsorbent used was a Granular Activated Carbon (GAC) type F400 and was 169
supplied by Chemviron Carbon Ltd.. Table 1 shows the physical properties of 170
Activated Carbon F400. The carbon particles were asssumed to be spheres having a 171
diameter given by the arithmetic mean value between respective mesh sizes (average 172
particle diameter was 605µm). For more details about the preparation of Activated 173
Carbon we refer the reader to [15]. 174
Table 1 Physical properties of Activated Carbon F400 175
Propertis Value
Total surface (N2 BET method) (m2 g
-1) 1150
Particle density (g cm-3
) 1.30
Particle voidage fraction 0.38
176
2.2. Dyes and dyes structure optimization and theoretical capacity 177
Three azo dyes, namely Acid Blue 80 (AB80), Acid Red 114 (AR114) and Acid 178
Yellow 117 (AY117), were used in this study. The dyestuffs were used as the 179
commercial salts. AB80 and AY117 were supplied by Ciba Specialty Chemicals and 180
AR114 was supplied by Sigma-Aldrich Chemical Company [15]. Some information 181
regarding the acid dyes, which were used to measure and prepare standard 182
concentration dye solutions, is listed in Table 2. The data include color index number, 183
molecular mass, the wavelengths at which maximum absorption of light occurs, λmax, 184
and PM3 optimization results. PM3, or Parameterized Model number 3, is a semi- 185
empirical method for the quantum calculation of molecular electronic structure in 186
computational chemistry. It is based on the Neglect of Differential Diatomic Overlap 187
integral approximation. The method was developed by J. J. P. Stewart [16]. In our 188
study PM3 optimization method is implemented in the Gaussian software. PM3- 189
8
optimized structures of dyes are shown in Fig. 1. In this study we assumed that each 190
molecule is placed on the sorbent surface from the largest cross-sectional area. Thus 191
after optimization of the molecular structure of dyes and determination of their 192
molecular lengths, widths and depths, the maximum cross-sectional surface area for 193
each dye was calculated by multiplying its largest dimensions together. Therefore the 194
minimum theoretical adsorption capacity for the dyes was determined by equation (1): 195
3
CA
20
th 10MWAN10BET
q ×××
×= (1) 196
where BET is the Brunauer, Emmett and Teller specific surface area(mg g-1), AC is 197
approximate maximum cross-sectional area (Å2) of dye, NA is the Avogadro's number 198
(6.022×1023) , MW is molecular weight of the dye (moles g-1) and qth is the minimum 199
theoretical adsorption capacity (mg g-1). 200
201
202
203
204
205
206
207
9
208 (a) 209
210 (b) 211
212 (c) 213
214
Fig. 1 PM3-Optimized dye structures of AB80 (a), AY117 (b), AR114 (c) 215
216
10
217
Table 2 Information regarding the optimization acid dyes structures 218
Characteristics Symbol Unit AB80 AR114 AY117 Molecular weight MW g mol-1 678.69 830.82 848.82 Color index number CI - 61585 23635 24820 Maximum wavelength λλλλmax nm 626 522 438 Optimization method - - PM3 PM3 PM3 Length L Å 16.93 30.92 33.16 Width W Å 16.28 13.6 17.17 Depth D Å 12.38 8.87 11.96 Maximum cross-sectional area Ac Å2 275.62 420.51 569.35 Total energy ET kcal mol-1 -307.85 -309.11 -96.88 Theoretical sorption capacity qth mg g-1 470 .39 377.35 284.79
219
2.3. Dye concentration measurement and equilibrium experiment 220
2.3.1. Preparation of acid dye solutions 221
Stock solutions of the three acid dyes were prepared in 2 liter volumetric flasks using 222
DI water. For each acid dye, two concentrations of solutions were used and the 223
concentrations were 100 mg L-1 and 250 mg L-1. The same procedure was used to 224
prepare solutions of all three acid dyes. For more details about the Preparation of 225
standard solutions we refer the reader to [15]. 226
2.3.2. Isotherm Adsorption Study 227
The 250 mg L-1 concentration dye solutions for AB80, AR114 and AY117 dyes were 228
used to determine the equilibrium contact time. For each acid dye system, eight jars of 229
fixed volume (0.05 L) of dye solutions were prepared and contacted with 0.05g 230
activated carbon F400. Then, the jars were put into the shaking bath with the same 231
conditions of the isotherm adsorption experiment (constant temperature 20◦C and 200 232
rev/min shaking rate). At three day intervals, one of the jars was taken from the shaker 233
and the dye concentration was measured. The equilibrium contact time for the 234
sorption equilibrium studies all three acid dyes has been shown to be 21 days 235
minimum [15]. The amount of dye adsorbed onto the sorbent, was calculated as 236
follows: 237
11
( )m
VCCq e0
e
−= (2) 238
For more details about the isotherm adsorption study we refer the reader to [15]. 239
240
2.4. DeterminingDeterminingDeterminingDetermining iiiisotherm sotherm sotherm sotherm pppparameters by arameters by arameters by arameters by nnnnonononon----llllinear inear inear inear rrrregressionegressionegressionegression 241
Due to the inherent bias resulting from linearization, the isotherm parameter sets were 242
determined by non-linear regression. This provides a mathematically rigorous method 243
for determining isotherm parameters using the original form of the isotherm equation 244
[6, 17]. Most commonly, algorithms based on the Levenberg-Marquardt or Gauss- 245
Newton methods [18, 19] are used. The adsorption equilibrium data for dyes sorption 246
onto GAC were analyzed by non-linear curve fitting analysis, using R software, to fit 247
the two, three and four-parameter isotherm models. 248
In order to evaluate the fit of the isotherm to the experimental data, the optimization 249
procedure requires the selection of a goodness-of-fit measure (GoFM). In this study, 250
eleven GoFMs were employed as Table 3 and minimized for the estimation of 251
isotherm parameters using the R programming language [20]. 252
253
254
255
256
257
258
259
260
261
Table Table Table Table 3333 Statistical Goodness-of-fit measures (GoFMs) 262
12
GoFMGoFMGoFMGoFM FormulaFormulaFormulaFormula Equation N.Equation N.Equation N.Equation N. Ref.Ref.Ref.Ref.
Residual Root Mean Square Error (RMSE) ( )
2n
1icalc,eexp,e qq
2n
1∑
=
−−
(3) [21]
The chi-square statisticstatisticstatisticstatistic (X2) ( ) 2
n
1i calc,e
calc,eexp,e
q
qq∑
=
− (4) [22]
GGGG----square statistic square statistic square statistic square statistic (G(G(G(G2222)))) ∑=
×
n
1icalc,e
exp,e
exp,e q
qlnq2 (5) [23]
Sum Sum Sum Sum of the Sqof the Sqof the Sqof the Squares of the uares of the uares of the uares of the Errors (Errors (Errors (Errors (ERRSQERRSQERRSQERRSQ)))) ( )
2
i
n
1icalc,e exp,e q q∑
=
− (6) [24]
Hybrid Hybrid Hybrid Hybrid Fractional Error Fractional Error Fractional Error Fractional Error Function (HYBRFunction (HYBRFunction (HYBRFunction (HYBRIIIID)D)D)D)
( )
i
n
1i exp,e
2
calc,e exp,e
q
q q∑
=
− (7) [25]
Derivative of Marquardt’s Derivative of Marquardt’s Derivative of Marquardt’s Derivative of Marquardt’s Percent Standard Deviation Percent Standard Deviation Percent Standard Deviation Percent Standard Deviation (MPSD)(MPSD)(MPSD)(MPSD)
( )2
i
n
1i exp,e
calc,e exp,e
q
q q∑
=
− (8) [26]
The Average Relative Error The Average Relative Error The Average Relative Error The Average Relative Error (ARE)(ARE)(ARE)(ARE)
i
n
1iexp,e
calc,eexp,e
q
q q∑
=
−
(9) [27]
Sum of the Absolute Errors Sum of the Absolute Errors Sum of the Absolute Errors Sum of the Absolute Errors (EABS)(EABS)(EABS)(EABS)
i
n
1icalc,e exp,e q q∑
=
− (10) [25]
Average Percentage Errors (APE) 100
p
q
q qn
1ii exp,e
calc,e exp,e
×
−∑
=
(11) [28]
Corrected Akaike information criterion (AICC) 1pn
)1p(p2AICAIC c
−−
++= (12) [29]
Mallows CP statistic (CP) )p2(nRMSE
SSCP res ×+−= (13) [30]
“exp” and “calc” show the experimental and calculated values. 263 n is the number of observations in the experimental data. 264 p is the number of parameters in the model (plus one for CP). 265 SSres is the residual sum of squares for the model with p-1 variables. 266
267
3. Results and discussion 268
3.1. Dyes molecular characteristics 269
The information regarding the dyes molecules and their structures optimization by 270
PM3 method is summarized in Table 2. As can be seen there is a negative correlation 271
between the dyes molecular weight and the corresponding theoretical sorption 272
capacity in which by increasing the former the later be decreased considerably. Thus 273
13
theoretically as an initial perception we expected the molecules with higher molecular 274
weight show less adsorption capacity. In the following sections adsorption isotherms 275
of dyes were addressed by focusing on the comparison of adsorption capacity 276
estimated by models and theoretical values. Statistical error analysis was also used to 277
find the best descriptive model. 278
279
3.2. Model selection: Error analysis and Fractional Theoretical Capacity 280
The equilibrium isotherms describe how the adsorbent interacts with the adsorbate. 281
The correlation of experimental results to an adsorption model can help to understand 282
the mechanisms of adsorption and the heterogeneity of the adsorbent surface, and it is 283
also of importance in the practical design and operation of adsorption systems [31]. 284
Three two-parameter, eleven three-parameter and one four-parameter isotherm models 285
were employed to describe the adsorption of acid dyes. Eleven minimizing procedures 286
were also adopted to solve the isotherm equations by minimizing the errors between 287
the theoretical data for qe calculated from the equations and the experimental data. 288
Table 4 summarizes the corresponding two, three and four-parameter models used in 289
this study. The values of isotherm models parameters are summarized in Tables 5, 6 290
and 7. The results which are expressed as plots of solid-phase dye concentration 291
against liquid-phase dye concentration are shown in Figs. 2 to 4. As previously 292
mentioned, the aim of this study was to select the best descriptive isotherm model. 293
294
295
296
297
298
299
Table 4 Isotherm models equations 300
Parameter N. Isotherm model Formula Equation N. Ref.
14
Two
Langmuir
e
eme bC1
bCqq
+= (14)
[32]
Dubinin-Radushkevich
)Bexp(Qq 2DSe ε−=
)C
11ln(RT
e
+=ε
(15)
(16)
[33]
Jovanovic )CKexp(1(qq ejmaxe −= (17) [34]
Three
Sips mSeS
mSeSmS
e CK1
CKqq
+= (18) [35]
Langmuir-Freundlich
( )( )mLF
eLF
mLF
eLFmLFe
CK1
CKqq
+= (19) [35]
Fritz–Schlunder (F-S III) mFS
eFS
eFSmFSe CK1
CKqq
+= (20) [36]
Radke–Prausnitz (R-Pz) ( )mRPzI
eRPzI
eRPzImRPzIe
CK1
CKqq
+=
mRPzIIeRPzII
eRPzIImRPzIIe CK1
CKqq
+=
1mRPzIIIeRPzIII
mRPzIIIeRPzIIImRPzIII
e CK1
CKqq
−+=
(21)
(22)
(23)
[37]
Toth mT/1mT
eT
emTe
CK1
Cqq
+
= (24) [38]
Khan isotherm ( )β
+=
ek
ekmke
Cb1
Cbqq (25) [17]
UniLan
+
+=
−seu
seumu
e eCK1
eCK1ln
s2
(26) [39, 40]
Four Baudu ( )
( )x1e0
yx1e00m
e Cb1
Cbqq
+
++
+= (27) [41]
301
302 Table 5 Isotherm parameters of Two-parameter models 303
Model Parameter AB80 AR114 AY117
Langmuir qm 171.452 103.732 185.825 b 0.150 0.144 0.216
(qm/ qth)×100 36.45 27.48 65.27 Dubinin-Radushkevich B 0.006 0.007 0.003
Qs 145.300 88.511 158.337 (Qs/ qth)×100 30.89 23.45 55.61
Jovanovic kj 0.120 0.114 0.166 qmax 152.450 92.204 167.108
(qmax/ qth)×100 32.41 24.43 58.69
304
305
Table 6 Isotherm parameters of Three-parameter models 306 Model Parameter AB80 AR114 AY117
Toth mT 0.887 0.668 0.718
15
qmT 177.272 119.363 205.538 KT 0.203 0.359 0.426
(qmT/ qth)×100 37.69 31.63 72.19 Langmuir-Freundlich qmLF 174.902 113.863 198.966
KLF 0.142 0.110 0.178 mLF 0.944 0.797 0.828
(qmLF / qth)×100 37.18 30.17 69.88 Khan qmk 149.308 68.607 135.404
bk 0.186 0.280 0.362 β 0.953 0.875 0.904
(qmk / qth)×100 31.74 18.17 47.56 UniLaN s 0.967 1.989 1.709
qmu 174.963 114.207 197.377 Ku 0.142 0.109 0.182
(qmu / qth)×100 37.20 30.26 69.32 Fritz-Schlunder qmFS 146.812 66.283 134.175
KFS 0.193 0.312 0.388 mFS 0.965 0.899 0.921
(qmFS / qth)×100 31.21 17.56 47.12 Radke I qmRPI 149.309 68.603 135.397
KRPI 0.186 0.280 0.362 mRPI 0.953 0.875 0.904
(qmRPI / qth)×100 31.74 18.17 47.55 RadkeII qmRPII 146.814 66.289 134.175
KRPII 0.193 0.312 0.388 mRPII 0.965 0.899 0.921
(qmRPII / qth)×100 31.21 17.56 47.12 RadkeIII qmRPIII 28.316 20.713 52.006
KRPIII 5.182 3.197 2.579 mRPIII 0.035 0.102 0.079
qmRPIII / qth)×100 6.020 5.489 18.266 Sips qms 174.873 113.810 198.944
Ks 0.159 0.173 0.240 ms 0.944 0.798 0.828
(qms / qth)×100 37.18 30.16 69.87
307
308
Table 7 Isotherm parameters of Four-parameter model 309
Model Parameter AB80 AR114 AY117
Baudu
qm0 106 42.3 122 b0 0.241 0.335 0.429 x 0.155 0.594 -0.026 y 0.099 0.192 0.097
(qm0/ qth)×100 22.53 11.20 42.85
310
The descriptive models from the best to worst for each dye were sorted according to 311
GoFM values and shown in Tables 8, 9 and 10. The most visited models in each rows 312
of Tables 8, 9 and 10 were again sorted in overall terms in the last column of the 313
tables where reveals that the adsorption isotherm models fitted the experimental data 314
in the below orders: 315
AY117: 316
16
R-PzI>Khan>F-S III> R-PzII>R-PzIII>Toth&Baudu>UniLan>L-F>Sips>L>J>DR 317
AR114: 318
Baudu>Khan>R-PzI>R-PzIII>R-PzII>F-S III> Toth>UniLan>L-F>Sips>L>J>D-R 319
AB80: 320
Baudu>Khan>R-PzI>R-PzIII>R-PzII>F-S III> Toth & L>UniLan>L-F>Sips>J>DR 321
As shown above, R-PzI (3P), Baudu (4P) and Baudu(4P) models were estimated the 322
experimental values with lowest error for AY117, AR114 and AB80 dye-sorbent 323
systems, respectively. Although the best models have now been selected based only 324
on GoFM statistics, we can also find the best models in terms of the theoretical 325
adsorption capacity parameter. Thus the selection was also performed based on the 326
theoretical adsorption capacity provided by each model. We assumed that the best 327
model is a model that provides the maximum Fractional Theoretical Capacity (FTC) 328
regarding to each dye. Therefore, in this case, only the isotherm models those have 329
maximum adsorption capacity parameter in their formula were selected and assessed. 330
The FTC was obtained by dividing the maximum adsorption capacity parameter of 331
each model to the theoretical capacity value. In Table 11 the descriptive models based 332
on greatest FTC from the best to the worst were summarized. According to Table 11 333
the adsorption isotherm models fitted the experimental data in the below orders, 334
which are different from those obtained from the error analysis approach: 335
AY117: 336
Toth> L-F>Sips> UniLan 337
AR114: 338
Toth> UniLan> L-F>Sips 339
AB80: 340
Toth> UniLan> Sips>L-F 341
17
As can be seen above, unlike the results obtained from the error analysis approach, the 342
best model describing the adsorption of all three dyes is the Toth model. While 343
according to the error analysis approach, the Toth model was ranked 6 for AY117 and 344
7 for AR114 and AB80. The question is why these two approaches provide different 345
results in choosing the best isotherm model. Statistically a two-parameter model needs 346
less data points than those for three-parameter and the three-parameter model needs 347
less data points than those for the four-parameter model. Typically, if the number of 348
model parameters increases, the fitting error decreases because a better fit can be 349
obtained and at the same time, the generalization bias is expected to increase due to 350
the larger model variability [25]. But it should be noticed that smaller models may 351
tend to do better for the same data set. Thus, when models with different parameters 352
fit on the same set of data points, regardless of the models assumptions, the model that 353
has a sufficient number of data points in excess will provide the best conformity with 354
those data points. Based on the error analysis approach the Baudu model is in the 355
highest priority for AR114 and AB80 but not necessarily the best model for these 356
dyes. The estimated FTCs of the Baudu model for AR114 and AB80 dyes obtained 357
were 11.20% and 22.53%, while the estimated values by Toth were 31.63% and 358
37.69% respectively, which are closer to the experimental capacity. Based on the error 359
analysis approach Radke I was found as the best model to describe AY117 sorption 360
and the estimated FTC for Radke I was 47.55%; while based on the FTC ranking in 361
Table 11, the Toth is the best descriptive model for AY117 and its estimated FTC was 362
obtained equal to 72.19% which is significantly more than that for Radke I. Thus if 363
the isotherm models sort based on FTC model, the best will be the model which has 364
the highest theoretical capacity. Operationally the design of an adsorption process is 365
also based on the estimated capacity from the isotherm model [42, 43] thus if the 366
18
amount of this capacity is closer to the theoretical value, then the design status will be 367
closer to the optimum condition. Therefore, to receive an optimum design in an 368
adsorption process it is important that the optimum isotherm be selected accurately 369
and the FTC may be helpful measure in this case. 370
371
Fig. 2 Experimental and predicted two-parameter isotherms of dyes according to Langmuir 372
(a), Dubinin-Radushkevich (b), Jovanovic (c) models. 373
374
375
376
377
(b) (a)
19
(d) (c)
(g) (f)
(h)
Fig. 3 Experimental and predicted three-parameter isotherms of dyes according to L-F(a), 378
Sips(b), Radke(c), F-S(d), Toth(f), Khan(g), UniLan(h) models. 379
20
380
Fig. 4 Experimental and predicted four-parameter isotherm of dyes according to Baudu 381
model 382
383
Table 8 Models ranked from the best to worst based on GoFM values for the sorption of AY117 dye 384
G2 X2 RMSE HYBRD MPSD ARE APE(%) Mallows ERRSQ EABS AICc Most-visited
UniLan R-PzI R-PzI Khan F-S III R-PzIII R-PzIII R-PzI R-PzI R-PzIII R-PzI R-PzI
R-PzIII Khan Khan R-PzI R-PzII R-PzII R-PzII Khan Khan R-PzII Khan Khan
Toth F-S III Baudu F-S III R-PzIII F-S III F-S III R-PzIII Baudu F-S III R-PzIII F-S III
R-PzII R-PzII R-PzIII R-PzII R-PzI R-PzI R-PzI R-PzII R-PzIII Khan R-PzII R-PzII
F-S III R-PzIII R-PzII R-PzIII Khan Khan Khan F-S III R-PzII R-PzI F-S III R-PzIII
Sips Baudu F-S III Baudu Toth Toth Toth Baudu F-S III Baudu Toth Toth&Baudu
L-F Toth Toth Toth UniLan UniLan UniLan Toth Toth Toth UniLan UniLan
R-PzI UniLan UniLan UniLan Baudu Baudu Baudu UniLan UniLan UniLan L-F L-F
Khan Sips L-F Sips Sips Sips Sips L-F L-F Sips Sips Sips
Baudu L-F Sips L-F L-F L-F L-F Sips Sips L-F Baudu -
L L L L L L L L L L L L
J J J J J J J J J J J J
D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R DR
385
386
387
388
389
21
Table 9 Models ranked from the best to worst based on GoFM values for the sorption of AR114 dye 390
G2 X2 RMSE HYBRD MPSD ARE APE(%) Mallows ERRSQ EABS AICc Most-visited
Baudu Baudu Baudu Baudu Baudu Baudu Baudu Baudu Baudu Baudu Baudu Baudu
Sips R-PzI Khan R-PzI L L L Khan Khan Khan Khan Khan
L-F Khan R-PzI Khan R-PzI R-PzI R-PzI R-PzI R-PzI R-PzI R-PzI R-PzI
UniLan R-PzII R-PzIII R-PzII Khan Khan Khan R-PzIII R-PzIII R-PzIII R-PzIII R-PzIII
Toth F-S III R-PzII F-S III R-PzII R-PzII R-PzII R-PzII R-PzII R-PzII R-PzII R-PzII
R-PzII R-PzIII F-S III R-PzIII F-S III F-S III F-S III F-S III F-S III F-S III F-S III F-S III
F-S III Toth Toth Toth R-PzIII R-PzIII R-PzIII Toth Toth Toth Toth Toth
R-PzIII UniLan UniLan L Toth Toth Toth UniLan UniLan UniLan UniLan UniLan
Khan Sips L-F UniLan UniLan UniLan UniLan L-F L-F L-F L-F L-F
R-PzI L-F Sips Sips Sips Sips Sips Sips Sips Sips Sips Sips
L L L L-F L-F L-F L-F L L L L L
J J J J J J J J J J J J
D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R
391
Table 10 Models ranked from the best to worst based on GoFM values for the sorption of AB80 dye 392
G2 X2 RMSE HYBRD MPSD ARE APE(%) Mallows ERRSQ EABS AICc Most-visited
Baudu Baudu Baudu Baudu Baudu Baudu Baudu R-PzI Baudu Baudu R-PzI Baudu
L R-PzI Khan L L Khan Khan Baudu Khan Khan Khan Khan
Sips Khan R-PzI R-PzI R-PzI R-PzI R-PzI Khan R-PzI R-PzI L R-PzI
L-F L R-PzIII Khan Khan R-PzII R-PzII R-PzIII R-PzIII R-PzIII R-PzIII R-PzIII
UniLan R-PzII R-PzII R-PzII R-PzII F-S III F-S III R-PzII R-PzII R-PzII R-PzII R-PzII
Toth F-S III F-S III F-S III F-S III R-PzIII R-PzIII F-S III F-S III F-S III F-S III F-S III
R-PzIII R-PzIII Toth R-PzIII R-PzIII L L L Toth Toth Baudu Toth & L
R-PzI Toth UniLan Toth Sips Toth Toth Toth UniLan UniLan Toth UniLan
Khan UniLan L-F UniLan L-F UniLan UniLan UniLan L-F L-F UniLan L-F
R-PzII Sips Sips Sips Toth Sips Sips L-F Sips Sips L-F Sips
F-S III L-F L L-F UniLan L-F L-F Sips L L Sips -
J J J J J J J J J J J J
D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R D-R DR
393
Table 11 Ranking the best descriptive models based on FTC measure 394
dye Rank of models (1: best , 13:worst)
1 2 3 4 5 6 7 8 9 10 11 12 13
AB80 Toth UniLan Sips L-F L J R-PzI Khan R-PzII F-S III D-R Baudu R-PzIII
AR114 Toth UniLan L-F Sips L J D-R R-PzI Khan R-PzII F-S III Baudu R-PzIII
AY117 Toth L-F Sips UniLan L J D-R Khan R-PzI R-PzII F-S III Baudu R-PzIII
395
22
Table 11 shows the Toth, UniLan, Sips and LF are the best models, based on FTC 396
measure, to describe the dyes sorption by activated carbon. Toth [38] has modified the 397
Langmuir equation to reduce the error between experimental data and predicted 398
values of equilibrium adsorption data [44]. It assumes an asymmetrical quasi-gaussian 399
energy distribution with a widened left-hand side, i.e. most sites have sorption energy 400
less than the mean value [45]. The Toth model in Tables 8, 9 and 10 has greater 401
precedence than UniLan, Sips and LF models. This indicates that first selected 402
isotherm model (Toth) by FTC measure has not only maximum fractional theoretical 403
capacity but also a minimum prediction error. The Sips equation is another three- 404
parameter model that is an extension of the Freundlich equation, modified such that 405
the amount adsorbed in the Sips equation has a finite limit at sufficiently high 406
concentration. The Sips model is sometimes called the Langmuir-Freundlich equation 407
in the literature because it has the combined form of Langmuir and Freundlich 408
equations [46]. The Freundlich equation is not valid at low and high ends of the 409
concentration range, and the Sips equation is not valid at the low end as both do not 410
possess the correct Henry law type behavior. The Sips equation, despite having the 411
correct finite capacity at sufficiently large concentration, has its applicability only in 412
the intermediate range of concentrations [47]. The Toth equation satisfies the two end 413
limits. The qmT is in Toth equation is the maximum adsorption capacity (mg g-1), KT is 414
the Toth equilibrium constant, and mT is the Toth model exponent. The mT is a 415
parameter which is usually less than unity. When mT =1, the Toth equation reduces to 416
the famous Langmuir equation; hence like the Sips equation the parameter mT is said 417
to characterize the system heterogeneity. The parameters mT in Toth, mS in Sips and 418
mLF in Langmuir-Freundlich models could be regarded as the parameters 419
characterizing the system heterogeneity, lies between 0 and 1. The parameter mT for 420
23
AB80, AR114 and AY117 dyes take values of 0.887, 0.668 and 0.718 respectively 421
(Table 6). As summarized in Table 12, the heterogeneity parameters for AR114 and 422
AY117 dyes are in a considerable distance from unity and this indicates the sorption 423
systems for these dyes may be heterogeneous. For AB80 sorption system, the 424
condition is more homogeneous than that for AR114 and AY117 systems. However, 425
this information does not point to what is the source of heterogeneity or homogeneity, 426
whether it can be the sorbent structural property, the sorbent energetic property or the 427
sorbate property. However, this may roughly point to the fact that the source of 428
heterogeneity or homogeneity may be the sorbate property because despite the same 429
adsorbent being used for all three dyes, there is yet a considerable difference between 430
the values of the heterogeneity parameters. As can be seen in the last column of Table 431
10, the Toth and Langmuir models have the same priority. It is consistent with the 432
Toth model formula which would expected to reduce to the Langmuir model for 433
mT=1≈0.887. If mT deviates further away from unity, the system is said to be more 434
heterogeneous. The Toth equation has correct limits when concentration approaches 435
either zero or infinity, because of its simplicity in form and its correct behavior at low 436
and high concentration. The prediction of adsorption isotherms of dyes onto GAC by 437
the Toth model is shown in Fig. 3(f). For favorable sorption, high qmT and a steep 438
initial isotherm slope (i.e. high KT) are desirable [48].The adsorption maximum 439
capacities (qmT) determined using the Toth model for the sorption of AB80, AR114 440
and AY117 were 177.2, 119.3 and 205.5 mg g-1 respectively (Table 6). These 441
capacities are higher than corresponding values that predicted by other models and 442
less than the theoretical values (470.39, 377.35 and 284.79 for AB80, AR114 and 443
AY117 respectively). This suggests that the dyes may be adsorbed flat on the carbon 444
surface. However in the case of AB80 this result has a greater certainty. 445
24
446
Table 12 Affinity constants, Heterogeneity constants and the sorption capacities of three- 447
parameter isotherm models 448
Isotherm model
Heterogeneity parameter Affinity parameter Sorption Capacity AB80 AR114 AY117 AB80 AR114 AY117 AB80 AR114 AY117
Toth* 0.887 0.668 0.718 0.203 0.359 0.426 177 119 205 L-F* 0.944 0.797 0.828 0.142 0.110 0.178 175 113 199 UniLan** 0.967 1.989 1.709 0.142 0.109 0.182 175 114 197 Sips* 0.944 0.798 0.828 0.159 0.173 0.240 174 113 199
*Heterogeneity parameter (lies between 0 and 1) 449 ** Heterogeneity parameter (≥ 0) 450
451
Unilan [39, 40] (the term UniLan comes from Uniform distribution and Langmuir 452
local isotherm) equation is also an empirical relation obtained by assuming a 453
patchwise topography on the surface and each patch is ideal such that the local 454
Langmuir isotherm is applicable on each patch. Based on the FTC measure, the 455
UniLan is the second priority model for the dyes AB80 and AR114. The UniLan 456
isotherm equation is shown in Table 4 where Ku is the Langmuir constant (L mg-1), 457
qmu, the amount of dye adsorbed (mg g-1) when the saturation is attained, s is a 458
constant and a parameter characterizes the heterogeneity of the system. The larger this 459
parameter is, the more heterogeneous is the system. If s=0, the UniLan equation 460
reduces to the classical Langmuir equation as in this limit the range of energy 461
distribution is zero. The parameter s for AB80, AR114 and AY117 dyes take values 462
of 0.967, 1.989 and 1.709 respectively (Table 6). The value of 0.967 for AB80 is 463
smaller than the same parameter for AR114 and AY117. On the other hand as can be 464
seen in the last column of Table 10 the UniLan model comes exactly after the Toth 465
and Langmuir models. This may be consistent with the UniLan model which would 466
be expected to be reduced to Langmuir model by closing the s parameter to zero. This 467
also shows that the sorption system of AB80 is more homogeneous than the systems 468
for AR114 and AY117 dyes. According to Table 2, the total energy of the molecule of 469
dyes AB80, AR114 and AY117 were obtained -307.85, -309.11 and -96.88 kcal mol-1 470
25
and the FTC according to the Toth model were found 37.69, 31.63 and 72.19 % 471
respectively. As can be seen in Fig.5, by increasing the amount of the total energy of 472
the molecule (internal energy), the FTC was increased such that among all models, 473
Toth and RPIII show the maximum and the minimum FTC, respectively. This is 474
consistent with the result in Table 11 and shows that the Toth model is the best model 475
for the description of the experimental data of all studied dye/carbon systems in this 476
study. 477
Fig. 5 Molecular total energy of dyes versus Fractional Theoretical Capacity obtained from 479
isotherm models 480
One important characteristic of the sorption isotherm curve is its initial slope. A curve 481
with a steep initial slope indicates a sorbent which has a capacity for the sorbate in the 482
low residual concentration range. This means that in this case the sorbent has a high 483
affinity for the sorbed species. This affinity is indicated by the affinity parameter. The 484
higher the value of the affinity parameter the higher the sorption affinity. As shown in 485
Table 12, based on Toth model, the highest affinity constant (KT = 0.426) was 486
obtained for the sorption system of AY117 in comparison with other dyes. This may 487
26
be due to the AY117 molecular structure and the contribution of the hydroxyl groups, 488
which can lead to stronger connections between the dye molecule and the surface of 489
sorbent. The affinity parameter for AR114 (KT = 0.359) is more than that for AB80 490
(KT = 0.203) and lower than that for AY117, this shows that in spite of higher 491
sorption capacity which was obtained for AB80 in comparison with AR114, there are 492
stronger connections between AR114 dye molecules and the surface of the sorbent. 493
3.3. Fractal dimension analysis 494
The fractal dimension concepts developed by Mandelbrot [49] was applied to the 495
determination of surface ruggedness. Fractal analysis has become a powerful tool to 496
describe the surface heterogeneity, geometric and structural properties of fractal 497
surfaces and pore structures. The larger the value of the surface fractal dimension, the 498
more irregular and rougher the pore surface [50]. Different sized dye molecules will 499
have different access to surfaces that are rugged or indeterminate. The optimization of 500
the dyes molecular geometry and estimation of theoretical sorption capacity presented 501
in the current work can also be used to determine of the value of the fractal dimension 502
parameter developed by Farin and Avnir [51] : 503
)2/Df(m .kN −σ=
(28) 504
where Nm is the number of moles in the completed monolayer (mmol g-1), σ is the 505
cross sectional adsorptive molecular area (Å2) and Df is the fractal dimension of the 506
accessible surface [51]. The value of Df is expected to have a value between 2 and 3. 507
In principle, a lower limit of Df = 2 is obtained with a perfectly smooth surface on the 508
molecular scale. Fig. 6 showed the fractal plot for the adsorption of dyes on the 509
activated carbon. It is noteworthy that Df=2.0 was obtained for the Granular Activated 510
Carbon (GAC) type F400 used in this study. The fact that Df=2.0 appeared, confirms 511
that the activated carbon surface was smooth and regular. The k is a prefactor, which 512
27
contains the necessary dimensional conversions and is called the lacunarity [49]. This 513
term is the monolayer value for unit σ and carries information about the connectivity 514
and porosity of the surface (larger values of k correspond to a greater extent of 515
porosity) [52]. The value of 191.1 was obtained for the k parameter. 516
517
Fig. 6 Fractal plot for the adsorption of dyes on the activated carbon 518
3.4. Incorporation the temperature effect 519
In the present experimental and statistical work, we did not investigate the effect of 520
the temperature but it is an important parameter with particular reference to 521
elucidating mechanisms. Thermodynamic parameters such as change in Gibb’s free 522
energy, ∆Go, the enthalpy (∆Hads) and the entropy of adsorption (∆Sads) can be 523
assessed using the following equation: 524
eq
eq
Td C
qK = (29)
525
where KTd is the apparent equilibrium constant, qeq is the amount of dye adsorbed on 526
the unitary sorbent mass (mg g-1) at equilibrium and Ceq is the equilibrium 527
concentration of the dye in solution (mg L-1), when amount adsorbed is equals qeq. 528
28
The thermodynamic equilibrium constants (KTd) can be calculated by the method 529
suggested by [53] from the intercept of the plots of ln (qeq/Ceq) vs. qeq. 530
The temperature dependent rate constant, kTd, can be used in the Arrhenius equation 531
(Eq. 30), to evaluate the activation energy of adsorption, EA, and the Arrhenius pre- 532
exponential factor, A, as shown [54]: 533
RT
EAlnkln A
Td −= (30) 534
where R is the universal gas constant (8.314 J mol-1 K-1) and T is the solution 535
temperature in degrees K. Then, the Gibb’s free energy, ∆Go, enthalpy (∆Hads) and the 536
entropy of adsorption (∆Sads) can be assessed with the Van’t-Hoff equation [55]: 537
Tdoads KlnRTG =∆ (31) 538
The slope and intercept of the Van’t-Hoff plot (Eq. 32) of lnKTd vs. 1/T can be used to 539
determine the values of ∆Hads and ∆Sads , 540
RT
H
R
SKln AdsAds
Td
∆−
∆= (32) 541
Then, the enthalpy and the entropy influence on the system may be evaluated using 542
the equation: 543
AdsAds
0
ads STHG ∆−∆=∆ (33) 544
In most dye adsorption systems, ∆Go ads, is negative, indicating the spontaneous nature 545
of the adsorption process; and, ∆Hads can be negative indicating an exothermic 546
favorable reaction, but implying that the adsorption capacity will decrease with 547
increasing temperature, or, ∆Hads can be positive indicating an endothermic reaction, 548
in which case the adsorption capacity increases with increasing temperature. However 549
several papers have shown the opposite trend can occur with temperature and the dye 550
29
adsorption is classified as endothermic and the capacity increases with increasing 551
temperature [55]. 552
The thermodynamic parameters of the adsorption can also be calculated by using the 553
isotherm models constants such as Toth (KT), UniLan (Ku), Sips (Ks) or Langmuir- 554
Freundlich (KLF) for the equations (31-33) instead of KTd. The obtained data on 555
thermodynamic parameters then can be compared. The best constant or the best 556
isotherm model for the evaluation of the thermodynamic parameters may be examined 557
by comparing the obtained Gibb’s free energy, ∆Go, with the total energy (ET) of the 558
structure-optimized dye molecule as an only available energy-related variable in the 559
system. However further studies are needed to clarify the possibility of these 560
comparisons. 561
3.5. Literature review 562
The literature review revealed that several studies have been performed so far to 563
investigate some approaches for the selection of best isotherm model. Table 13 564
implies some of these studies and their conclusions. According to it, non-linear 565
method has been proposed for estimating the isotherm model parameters instead of 566
linear methods. One of these studies has also been implied an important point that the 567
size of error statistics are not a good decision measure in choosing the best model 568
[11]. With respect this point, in our study, not only non-linear parameter estimation 569
method but also a new proposed measure namely FTC were used for the best isotherm 570
model selection. The application of the FTC with further investigations is recommend 571
for the best isotherm model selection process in the future adsorption studies. 572
573
574
575
30
Table 13 Aim(s) and conclusion(s) of some studies on isotherm model selection 576
Aim(s) Sorbate / Sorbent Conclusion(s) Ref.
Comparison of linear regression and Chi-square analysis
Cadmium / Tree fern
Non-linear Chi-square analysis could be a better method in comparison with linear regression for the isotherm model selection.
[56]
Comparison of linearized and non-linearized isotherm models.
Literature comparison
The expanding of the nonlinear isotherms represents a viable and powerful tool and leading to the superior improvement in the area of adsorption science.
[57]
Comparison of linear and non-linear regression methods
Basic red 9 / Activated carbon
Non-linear regression was found to be a better way to obtain the parameters and the size of the error function alone is not a deciding factor to choose the optimum isotherm.
[11]
Optimum sorption isotherm selection by linear and non-linear methods
Malachite green / Lemon peel
Non-linear method is a better way to obtain the isotherm parameters.
[58]
Comparison of Chi-square (X 2) and Log-likelihood (G 2) analysis
Dyes / Activated carbons
The G2 could be better than X2 statistic when the number of model parameters is three.
[6]
Comparison of statistical Goodness-of-Fit Measures (GoFMs) and the Fractional Theoretical Capacity (FTC) measure to finding the best fitting isotherm model(s)
Dyes / activated carbon
Using GoFMs alone may lead to wrong model selection. FTC measure can be a better measure for best descriptive model selection.
This study
577
4. Conclusions 578
Adsorption isotherms of three acid dyes on a commercial activated carbon were 579
studied using nine three-parameter, three two-parameter and one four-parameter 580
isotherm models. A best-fit isotherm for each dye-sorbent system was assessed by 581
geometrical molecular structure optimization of dyes and estimation of minimum 582
Fractional Theoretical Capacity (FTC) by each isotherm model. The model with 583
highest FTC regarding each dye was chosen as the best descriptive model. 584
Statistically eleven Goodness-of-fit Measures (GoFMs) were also applied to evaluate 585
and rank the feasibility of isotherm models. The model with the lowest GoFM was 586
chosen again as the best. The classification of the isotherm models according to the 587
FTC measure for the sorption of AB80 was: Toth> UniLan> Sips>L-F. This 588
31
classification for the sorption of AY114 was: Toth> UniLan> L-F>Sips and for 589
AY117 was: Toth> L-F>Sips> UniLan. The successful application of the Toth model 590
to the present data supports the fact that the sorption systems for all studied dyes were 591
heterogeneous. However in the terms of heterogeneity of the systems, the order of 592
dyes was: AR114 >AY117 >AB0. This means that the adsorption system for AR114 593
is more heterogeneous than that for other two dyes. The Toth model is the best model 594
to describe the adsorption mechanism of dyes AB80, AR114 and AY117. From the 595
FTC measure, the Toth model estimates the maximum sorption capacity and 596
maximum affinity parameter for all carbon/dye systems. Activated carbon adsorption 597
affinity for the dyes was determined as follows: AY117 >AB80 >AR114. This means 598
that there may be some stronger connections between the AY117 dye molecules and 599
the surface of carbon and desorption probability of this dye is lower than that for the 600
other dyes. The optimization of the dyes molecular geometry then the estimation of 601
theoretical sorption capacity presented in the current work can also used to determine 602
the value of the fractal dimension parameter as a measure of surface heterogeneity of 603
the sorbent. The value of 2.0 for the fractal dimension obtained in this study, 604
confirmed that the Granular Activated Carbon surface was smooth and regular. 605
The results obtained using the four-parameter isotherm equations do not provide 606
suitable correlations for the description of experimental data. Using GoFMs alone 607
may lead to wrong model selection and FTC measure can be a better measure for best 608
descriptive model selection. As a recommendation, the FTC measure was suggested 609
as a more reliable measure instead of statistical error measures in isotherm model 610
selection process. 611
612
5. Acknowledgments 613
32
This work was supported by the Center for Water Quality Research (CWQR) of 614
Tehran University of Medical Sciences. This support is gratefully acknowledged. The 615
authors would like also to extend thank to the Department of Chemical and 616
Biomolecular Engineering of Hong Kong University of Science and Technology, for 617
the provision of some experimental data. 618
619
6. References 620
[1] S. Venkat Mohan , S. Krishna Mohan, J. Karthikeyan, Adsorption mechanism of 621
acid-azo dye from aqueous solution on to coal/coal based sorbents and activated 622
carbon: A mechanist study, In: S. Jayarama Reddy (Eds.), Analytical Techniques 623
in Monitoring the Environment, Student offset printers, Tirupathi, India, 2000, 624
pp. 97-103. 625
[2] D. Do Duong, Adsorption analysis: equilibria and kinetics, Imperial College 626
Press, 1998. 627
[3] M. Hadi, M.R. Samarghandi, G. McKay, Simplified fixed bed design models for 628
the adsorption of acid dyes on novel pine cone derived activated carbon, Water 629
Air Soil Poll. 218 (2011) 197-212. 630
[4] G. McKay, M. Hadi, M.T. Samadi, A.R. Rahmani, M. Solaimany Aminabad, F. 631
Nazemi, Adsorption of reactive dye from aqueous solutions by compost, 632
Desalination and Water Treatment 28 (2011) 1-10. 633
[5] M. Hadi, M.R. Samarghandi, G. McKay, Equilibrium two-parameter isotherms of 634
acid dyes sorption by activated carbons: Study of residual errors, Chem. Eng. J. 635
160 (2010) 408-416. 636
33
[6] M. Hadi, G. McKay, M.R. Samarghandi, A. Maleki, M. Solaimany Aminabad, 637
Prediction of optimum adsorption isotherm: comparison of chi-square and Log- 638
likelihood statistics, Desalination and Water Treatment 49 (2012) 81-94. 639
[7] Y. Xue, H. Hou, S. Zhu, Adsorption removal of reactive dyes from aqueous 640
solution by modified basic oxygen furnace slag: Isotherm and kinetic study, 641
Chem. Eng. J. 147 (2009) 272-279. 642
[8] G. McKay, M. Hadi, M.T. Samadi, A.R. Rahmani, M.S. Aminabad, F. Nazemi, 643
Adsorption of reactive dye from aqueous solutions by compost, Desalination and 644
Water Treatment 28 (2011) 164-173. 645
[9] J.S. Piccin, C.S. Gomes, L.A. Feris, M. Gutterres, Kinetics and isotherms of 646
leather dye adsorption by tannery solid waste, Chem. Eng. J. 183 (2012) 30-38. 647
[10] G.L. Dotto, J.M. Moura, T.R.S. Cadaval, L.A.A. Pinto, Application of chitosan 648
films for the removal of food dyes from aqueous solutions by adsorption, Chem. 649
Eng. J. 214 (2013) 8-16. 650
[11] K.V. Kumar, K. Porkodi, F. Rocha, Comparison of various error functions in 651
predicting the optimum isotherm by linear and non-linear regression analysis for 652
the sorption of basic red 9 by activated carbon, J. Hazard. Mater. 150 (2008) 158- 653
165. 654
[12] K.K. Choy, J.F. Porter, G. McKay, Langmuir isotherm models applied to the 655
multicomponent sorption of acid dyes from effluent onto activated carbon, J. 656
Chem. Eng. Data 45 (2000) 575-584. 657
[13] K.K. Choy, J.F. Porter, G. Mckay, A film-pore-surface diffusion model for the 658
adsorption of acid dyes on activated carbon, Adsorption 7 (2001) 231-247. 659
34
[14] K.K. Choy, J.F. Porter, G. Mckay, Intraparticle diffusion in single and 660
multicomponent acid dye adsorption from wastewater onto carbon, Chem. Eng. J. 661
103 (2004) 133-145. 662
[15] K.K.H. Choy, J.P. Porter, G. McKay, Single and multicomponent equilibrium 663
studies for the adsorption of acidic dyes on carbon from effluents, Langmuir 20 664
(2004) 9646-9656. 665
[16] J.J. Stewart, Optimization of parameters for semiempirical methods I. Method, J. 666
Comput. Chem. 10 (1989) 209-220. 667
[17] A.R. Khan, I.R. Al-Waheab, A. Al-Haddad, A generalized equation for 668
adsorption isotherms for multi-component organic pollutants in dilute aqueous 669
solution, Environ. Technol. 17 (1996) 13-23. 670
[18] T.F. Edgar, D.M. Himmelblau, Optimization of Chemical Processes, McGraw- 671
Hill, New York, 1989. 672
[19] O.T. Hanna, O.C. Sandall, Computational Methods in Chemical Engineering, 673
Prentice-Hall International, New Jersey, 1995. 674
[20] R Core Team, R: A language and environment for statistical computing R 675
Foundation for Statistical Computing, Vienna, Austria, 2013, http://www.R- 676
project.org/. 677
[21] J.S. Armstrong, F. Collopy, Error measures for generalizing about forecasting 678
methods: Empirical comparisons, International Journal of Forecasting 8 (1992) 679
69-80. 680
[22] Y.S. HO, Selection of optimum sorption isotherm, Carbon 42 (2004) 2113-2130. 681
[23] R.R. Sokal, F.J. Rohlf, V. Cairns, U. Keil, A. Doering, W. Koenig, J. Stieber, 682
Biometry: the principles and practice of statistics in biological research 683
International Journal of Epidemiology 14 (1985) 389-395. 684
35
[24] K.H. Choy. Equilibrium sorption studies of acid dyes on activated carbon [Thesis 685
(M.Phil.)]: Hong Kong University of Science and Technology, 1999. 686
[25] J. Porter, G. McKay, K. Choy, The prediction of sorption from a binary mixture 687
of acidic dyes using single-and mixed-isotherm variants of the ideal adsorbed 688
solute theory, Chem. Eng. Sci. 54 (1999) 5863-5885. 689
[26] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear 690
parameters, J. Soc. (Ind.) Appl. Math. 11 (1963) 431-441. 691
[27] A. Kapoor, R. Yang, Correlation of equilibrium adsorption data of condensible 692
vapours on porous adsorbents, Gas Sep. Purif. 3 (1989) 187-192. 693
[28] J.M. Zachara, C.C. Ainsworth, C.E. Cowan, B.L. Thomas, Sorption of binary 694
mixtures of aromatic nitrogen heterocyclic compounds on subsurface materials, 695
Environ. Sci. Technol. 21 (1987) 397-402. 696
[29] H. Akaike, A new look at the statistical model identification, IEEE Transaction 697
on Automatic Control 19 (1974) 716-723. 698
[30] C.L. Mallows, Some comments on Cp, Technometrics 42 (2000) 87-94. 699
[31] A. Ip, J. Barford, G. McKay, Reactive Black dye adsorption/desorption onto 700
different adsorbents: effect of salt, surface chemistry, pore size and surface area, 701
J. Colloid Interf. Sci. 337 (2009) 32-38. 702
[32] I. Langmuir, The adsorption of gases on plane surfaces of glass, mica, and 703
platinum, J. Am. Chem. Soc. 40 (1918) 1361-1403. 704
[33] M.M. Dubinin, L.V. Radushkevich, The equation of the characteristic curve of 705
activated charcoal, Dokl. Akad. Nauk SSSR 55 (1947) 327-329. 706
[34] D.S. Jovanovic, Physical adsorption of gases I: isotherms for monolayer and 707
multilayer adsorption, Colloid Polym. Sci. 235 (1969) 1203-1214. 708
[35] R. Sips, On the structure of a catalyst surface, J. Chem. Phys. 16 (1948) 490-495. 709
36
[36] W. Fritz, E.U. Schlunder, Simultaneous adsorption equilibria of organic solutes in 710
dilute aqueous solution on activated carbon, Chem. Eng. Sci. 29 (1974) 1279- 711
1282. 712
[37] C.J. Radke, J.M. Prausnitz, Adsorption of organic solutions from dilute aqueous 713
solution on activated carbon, Ind. Eng. Chem. Fund. 11 (1972) 445-451. 714
[38] J. Tóth, Calculation of the BET Compatible Surface Area from Any Type I. 715
Isotherms Measured below the Critical Temperature, J. Colloid Interface Sci. 225 716
(2000) 378-383. 717
[39] J.M. Chern, C.Y. Wu, Desorption of dye from activated carbon beds: effects of 718
temperature, pH, and alcohol, Water Res. 35 (2001) 4159-4165. 719
[40] I. Quinones, G. Guiochon, Extension of Jovanovic-Freundlich isotherm model to 720
multicomponent adsorption on heterogeneous surfaces, J. Chromatogr. A 796 721
(1998) 15-40. 722
[41] M. Baudu, Etude des interactions solute-fibres de charbon actif. Application et 723
regeneration [Ph.D. Thesis]: Universit´e de Rennes I, 1990. 724
[42] Z. Aksu, D. Akpınar, Modelling of simultaneous biosorption of phenol and nickel 725
(II) onto dried aerobic activated sludge, Sep. Purif. Technol. 21 (2000) 87-99. 726
[43] D.O. Cooney, Adsorption design for wastewater treatment, CRC Press, 1999. 727
[44] A.R. Khan, R. Ataullah, A. Al-Haddad, Equilibrium adsorption studies of some 728
aromatic pollutants from dilute aqueous solutions on activated carbon at different 729
temperatures, J. Colloid Interface Sci. 194 (1997) 154-165. 730
[45] Y.S. Ho, J.F. Porter, G. McKay, Equilibrium isotherm studies for the sorption of 731
divalent metal ions onto peat: Copper, nickel and lead single component systems, 732
Water Air Soil Poll. 141 (2002) 1-33. 733
37
[46] P. Húmpola, H. Odetti, A. Fertitta, J. Vicente, Thermodynamic analysis of 734
adsorption models of phenol in liquid phase on different activated carbons, 735
Journal of the Chilean Chemical Society 58 (2013) 1541-1544. 736
[47] O. Moradi, Thermodynamics of Interfaces, In: J. Carlos Moreno-Pirajan (Eds.), 737
Thermodynamics - Interaction Studies - Solids, Liquids and Gases, InTech, 2011. 738
[48] K. Vijayaraghavan, J. Mao, Y.-S. Yun, Biosorption of methylene blue from 739
aqueous solution using free and polysulfone-immobilized Corynebacterium 740
glutamicum : Batch and column studies, Bioresource Technol. 99 (2008) 2864- 741
2871. 742
[49] B.B. Mandelbrot, The fractal geometry of nature, Macmillan, 1983. 743
[50] S.-I. Pyun, C.-K. Rhee, An investigation of fractal characteristics of mesoporous 744
carbon electrodes with various pore structures, Electrochimica acta 49 (2004) 745
4171-4180. 746
[51] D. Farin, D. Avnir, The Fractal Approach to Heterogeneous Chemistry, Wiley, 747
New York, 1989. 748
[52] J. Rouquerol, F. Rouquerol, P. Llewellyn, G. Maurin, K.S. Sing, Adsorption by 749
powders and porous solids: principles, methodology and applications, Academic 750
press, 2013. 751
[53] A.A. Khan, R. Singh, Adsorption thermodynamics of carbofuran on Sn (IV) 752
arsenosilicate in H+, Na+ and Ca2+ forms, Colloids and Surfaces 24 (1987) 33-42. 753
[54] D.K. Mahmoud, M.A.M. Salleh, W.A.W.A. Karim, A. Idris, Z.Z. Abidin, Batch 754
adsorption of basic dye using acid treated kenaf fibre char: equilibrium, kinetic 755
and thermodynamic studies, Chemical Engineering Journal 181 (2012) 449-457. 756
38
[55] Y.S. Al-Degs, M.I. El-Barghouthi, A.H. El-Sheikh, G.M. Walker, Effect of 757
solution pH, ionic strength, and temperature on adsorption behavior of reactive 758
dyes on activated carbon, Dyes and Pigments 77 (2008) 16-23. 759
[56] Y.-S. Ho, Selection of optimum sorption isotherm, Carbon 42 (2004) 2115-2116. 760
[57] K. Foo, B. Hameed, Insights into the modeling of adsorption isotherm systems, 761
Chem. Eng. J. 156 (2010) 2-10. 762
[58] K.V. Kumar, Optimum sorption isotherm by linear and non-linear methods for 763
malachite green onto lemon peel, Dyes and Pigments 74 (2007) 595-597. 764
765
766
767
39
768
Highlights:
• Fractional Theoretical Capacity measure was proposed for isotherm model 771
selection. 772
• Using error analysis statistics alone may lead to wrong isotherm model 773
selection. 774
• The sorption system for all dyes was heterogeneous based on the Toth 775
model. 776
• Fractal dimension can be determined based on the theoretical sorption 777
capacity. 778
• The sorbent (GAC) surface was smooth according to the fractal dimension 779
parameter. 780
782
