Chapter 3: Methods and Methodology
3.6 Evaluating the quality of the research process
3.6.1 Credibility
Some three term parametric isotherm models were equally investigated. They include the Redlich-Peterson isotherm, the Radke-Prausniiz Isotherm, the Sips Isotherm, the Toth isotherm, the Khan isotherm and the Koble-Carrigan Isotherm. Most of these isotherm models were explained in chapter two.
4.5.2.1 The Redlich-Peterson isotherm
The nonlinear analysis of the Redlich-Peterson isotherm was carried out using the Solver add-in function of the Microsoft excel in an iteration method. Redlich–Peterson isotherm is a hybrid isotherm featuring both Langmuir and Freundlich isotherms, which incorporate three parameters into an empirical equation (Prasad and Srivastava, 2009). A minimization procedure was adopted in solving the equation by maximizing the correlation coefficient between the experimental data points and theoretical model prediction.
The results are given in Tables 4.27 to 4.28. It was observed that the Redlich-Peterson isotherm constant A, decreased as the temperature increased for the adsorption of phenol using both CCAC and RHAC. The Redlich-Peterson isotherm exponent 𝛽 was gradually decreasing to zero as the temperature increased indicating that at very high temperature the Redlich-Peterson isotherm will approach Henry‘s isotherm (Nimibofa et al, 2017).
This was observed when using RHAC. The correlation coefficients were all high (>0.99) using both CCAC and RHAC adsorbents.
When 𝛽 = 1, The Redlich-Peterson isotherm reduces to Langmuir equation with 𝑏 = 𝐵 that is, Langmuir adsorption constant (Lmg-1) which is related to the energy of adsorption while 𝐴 = 𝑏qml where 𝑞𝑚𝑙 is Langmuir maximum adsorption capacity of the adsorbent (mg g−1). When 𝛽 = 0, the Redlich-Peterson isotherm reduces to Henry‘s equation with 1/(1 + 𝑏) representing Henry‘s constant (Nimibofa et al, 2017).
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Gimbert et al, (2008) reported that The Redlich-Peterson isotherm model has a linear dependence on concentration in the numerator and an exponential function in the denomination representing adsorption equilibrium over a wide range of concentration of adsorbate which is applicable in either homogenous or heterogeneous systems due to its versatility.
The Redlich-Peterson isotherm is a mix of the Langmuir and Freundlich isotherms. The numerator is from the Langmuir isotherm and has the benefit of approaching the Henry region at infinite dilution (Davoundinejad and Gharbanian, 2013). According to Soheila and Hasan (2017), the Redlich-Peterson isotherm approaches Freundlich isotherm model at high concentration and is in accordance with the low concentration limit of the ideal Langmuir condition.
Table 4.27: Nonlinear Redlich Peterson isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
A 0.03 0.03 0.025 0.024 0.021
B 0.23 0.25 0.26 0.26 0.3
y 0.26 0.25 0.25 0.25 0.27
R2 0.980347 0.998922 0.968333 0.996073 0.975062
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.078877 0.005877 0.039133 0.014052 0.036327 MPSD 0.360784 0.355871 0.370638 0.311817 0.218065 RMSE 0.048496 0.013104 0.0311 0.018268 0.025935
ARE 0.325293 0.089701 0.251654 0.153837 0.280131
SSE 0.009407 0.000687 0.003869 0.001335 0.002691
X2 1.687982 1.609393 1.555606 0.943907 0.310613
EABS 3.308304 3.197825 2.967801 2.140497 1.007205
Sre 1.544593 1.498193 1.384903 1.264202 0.990176
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Table 4.28: Nonlinear Redlich Peterson isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
A 0.1 0.078 0.07 0.056 0.055
B 0.14 0.18 0.2 0.245 0.247
y 0.3 0.29 0.288 0.26 0.26
R2 0.99956 0.998624 0.999687 0.971978 0.985504
Mean 9.516 8.516067 7.770733 6.677933 6.640733 HYBRID 0.048104 0.090699 0.017089 0.91619 0.634915 MPSD 0.063488 0.150934 0.128232 0.040666 0.127381 RMSE 0.060663 0.075561 0.03105 0.204397 0.173936
ARE 1.795717 0.240068 0.110074 -0.89648 0.730056
SSE 0.01472 0.022838 0.003856 0.167113 0.121015
X2 0.081099 0.413615 0.265856 0.020533 0.218854
EABS 0.903245 1.93773 1.466009 0.357027 1.214887
Sre 2.534089 2.499542 2.080089 2.153629 1.786047
4.5.2.2. Radke-Prausniiz Isotherm model
The Radke-Prausnitz isotherm parameters were obtained by nonlinear statistical fit of experimental data. The regression coefficient (R2) obtained at different temperatures were all greater than 0.99 suggesting good fit.
The Radke-Prausnitz maximum adsorption capacity qRP values were between 0.4 and 0.14mg/g and 0.071 and 0.05mg/g for the removal of phenol using RHAC and CCAC respectively as tabulated in Tables 4.29 and 4.30. These values were found to decrease as the temperature increased, becoming almost constant at temperatures above 50 oC.
The Radke-Prausnitz equilibrium constant KRPand Radke-Prausnitz model exponent m, equally decreased as the temperature decreased. The values of Radke-Prausnitz equilibrium constant KRPand Radke-Prausnitz model exponent m, obtained in this work
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were relatively close to that reported in the removal of picric acid by carboxylated multi-walled carbon nano tubes (Soheila and Hasan, 2017).
Nimibofa et al, (2017) reported that at low adsorbate concentration, the Radke-Prausnitz isotherm model reduces to a linear isotherm, while at high adsorbate concentration it becomes the Freundlich isotherm. When Radke-Prausnitz model exponent m = 0, it becomes the Langmuir isotherm. Important characteristic of this isotherm model includes the fact that it gives a good fit over a wide range of adsorbate concentration and it is used in the adsorption of organic solutes from dilute aqueous solutions (Soheila and Hasan, 2017; Vijayaraghavan et al, 2006). These important properties of Radke-Prausnitz isotherm model make it more preferred in most adsorption systems at low adsorbate concentration (Subramanyam and Ashutosh, 2012).
Table 4.29: Nonlinear Radke-Prausnizz isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
q 0.071 0.067 0.064 0.063 0.063
K 0.21 0.22 0.19 0.18 0.18
A 0.005 0.008 0.01 0.0092 0.0092
R2 0.998921 0.999439 0.998857 0.981598 0.981351
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.004329 0.00306 0.001412 0.065854 0.027166 MPSD 0.383615 0.002288 0.00169 0.011775 0.008565 RMSE 0.011361 0.009455 0.005908 0.039548 0.022428
ARE 0.076206 0.064725 0.047807 -0.33303 -0.24225
SSE 0.000516 0.000358 0.00014 0.006256 0.002012
X2 8.65778E-05 6.11996E-05 2.82453E-05 0.001317082 0.000543311 EABS 3.51766 3.427897 3.142612 0.079096 0.044856
Sre 0.049464 -10.7897 0.029812 0.206065 0.143551
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Table 4.30: Nonlinear Radke-Prausnizz isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
q 0.4 0.21 0.2 0.15 0.14
K 0.15 0.19 0.18 0.17 0.19
A 0.006 0.003 0.006 0.01 0.006
R2 0.992433 0.993567 0.990953 0.990825 0.991229
Mean 9.487333 8.516067 7.770733 6.677933 6.640733 HYBRID 0.843326 0.424002 0.494646 0.299988 0.38413 MPSD 0.023452 0.018351 0.020938 0.018137 0.020077 RMSE 0.254273 0.163374 0.167052 0.116959 0.135291
ARE 0.663322 -0.51906 0.592207 -0.51298 -0.56785
SSE 0.25862 0.106764 0.111625 0.054718 0.073215
X2 0.016866519 0.008480049 0.009892914 0.005999759 0.007682592 EABS -0.50855 0.326747 1.783144 0.233918 0.270583
Sre 0.585934 0.422903 0.463155 0.373449 0.419219
4.5.2.3 Sips Isotherm model
Tables 4.31 and 4.32 show the values of the different Sips isotherm parameters obtained at varying temperatures. The add-in solver in Excel was employed in analyzing the Sips isotherm. The Sips constant model (Ks ) values obtained using CCAC were lower (3.7 to 2.2 l/g) than that obtained using RHAC (7.5 to 5.6 l/g) though both of them decreased with increase in temperature. The Sips isotherm exponent 𝛽𝑠, values obtained were all less than unity suggesting that the experimental data fitted the Sips isotherm. In the removal of Cu (II), Kara et al (208) obtained significantly different values of the isotherm exponent. The correlation coefficients of the model were good (>0.9) at the experimental temperatures used.
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According to Foo and Hamed (2010), at low adsorbate concentration, the Sips model reduces to the Freundlich model, but at high concentration of adsorbate, it predicts a monolayer adsorption capacity characteristics of the Langmuir model. The parameters of the Sips isotherm model are pH, temperature, and concentration-dependent and the isotherm constants differ by linearization and nonlinear regression (Elmorsi, 2011; Chen, 2012; Perez-Marin et al, 2007).
According to Travis and Etnier (1981), the Sips isotherm model is suitable for predicting adsorption on heterogeneous surfaces, thereby avoiding the limitation of increased adsorbate concentration normally associated with the Freundlich model.
Table 4.31: Nonlinear Sips isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
K 3.7 3.55 3.1 2.9 2.2
B 0.0098 0.0099 0.0094 0.0091 0.0092
a 0.32 0.34 0.32 0.34 0.35
R2 0.999008 0.995684 0.996982 0.996836 0.999304
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.003995 0.023326 0.003715 0.011247 0.001017 MPSD 0.002584 0.006346 0.002747 0.004883 0.001655 RMSE 0.010894 0.026223 0.009601 0.0164 0.004334
ARE -0.07307 0.179505 0.077692 0.138103 -0.04681
SSE 0.000475 0.002751 0.000369 0.001076 7.51E-05
X2 7.9609E-05 0.000470708 7.45955E-05 0.000226486 2.02902E-05 EABS 0.021788 -0.05245 -0.0192 -0.0328 0.008668
Sre 0.047432 0.115975 0.048447 0.085451 0.027741
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Table 4.32: Nonlinear Sips isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
K 7.5 6.9 6.5 5.8 5.6
B 0.023 0.02 0.018 0.015 0.012
a 0.38 0.35 0.32 0.29 0.34
R2 0.997211 0.998618 0.999304 0.995999 0.999432
Mean 9.487333 8.516067 7.770733 6.677933 6.640733 HYBRID 0.317222 0.090012 0.038393 0.128644 0.025067 MPSD 0.014238 0.008506 0.005809 0.011977 0.00511 RMSE 0.154372 0.075726 0.046349 0.077237 0.034436
ARE -0.40271 0.240591 -0.16431 0.338759 -0.14454
SSE 0.095323 0.022938 0.008593 0.023862 0.004743
X2 0.006216696 0.001821902 0.000761551 0.002616469 0.000497725 EABS 0.308744 -0.15145 0.092698 -0.15447 0.068872
Sre 0.355726 0.196022 0.128503 0.246616 0.106704
4.5.2.4 Toth isotherm model
The Toth isotherm was investigated using the Add-in solver of Excel spreadsheet and the parametric constants evaluated and summarized in Tables 4.33 and 4.34.
The Toth isotherm model showed a good fit to the adsorption data for the adsorption of phenol using both CCAC and RHAC adsorbents. The coefficients of correlation of the Toth isotherm model were good varying from 0.992 to 0.9999 for the adsorbents. The Toth‘s maximum adsorption capacity qm, were close to the experimental adsorption capacity.
Its correlation presupposes an asymmetrical quasi-Gaussian energy distribution, with most of its sites having adsorption energy lower than the peak (maximum) or mean value (Ho et al, 2002). The Toth exponent n, was almost constant (between 0.63 to 0.64mg/g)
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for adsorption using CCAC adsorbent but gradually decreased using RHAC adsorbent, with increase in the solution temperature.
This model is most useful in describing heterogeneous adsorption systems which satisfy both low and high end boundary of adsorbate concentration (Padder and Majunder, 2012). Jefari and Jefari, (2014) reported that the parameter 𝑛 characterizes the heterogeneity of the adsorption system; and that if 𝑛 = 1, this equation reduces to Langmuir isotherm equation but if it deviates further away from unity (1), then the system is said to be heterogeneous. Hence, this adsorption process cannot be said to be heterogeneous though it has been reported that the Toth isotherm model has been applied for the modeling of several multilayers and heterogeneous adsorption systems (Benzaoui et al, 2017). The Toth isotherm is another empirical modification of the Langmuir equation with the aim of reducing the error between experimental data and the predicted value of equilibrium data (Jefari and Jefari, 2014).
Table 4.33: Nonlinear Toth isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
qm 7.9 7.7 6.6 6.3 4.8
K 0.03 0.03 0.027 0.028 0.034
n 0.638 0.642 0.645 0.643 0.63
R2 0.996889 0.997629 0.998637 0.998886 0.997632
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.012484 0.012927 0.001684 0.003987 0.00345 MPSD 0.004575 0.004703 0.001846 0.002897 0.003052 RMSE 0.019293 0.019434 0.006452 0.009731 0.007993
ARE 0.129414 0.133033 0.052207 0.081946 0.086329
SSE 0.001489 0.001511 0.000167 0.000379 0.000256
X2 0.000249685 0.000258534 3.3684E-05 7.9743E-05 6.89988E-05 EABS 0.038587 0.038868 0.012904 0.686284 0.015985
Sre 0.045414 0.047083 0.019652 0.031242 0.035172
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Table 4.34: Nonlinear Toth isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
qm 17.9 16.3 15.6 14.3 14.5
K 0.04 0.028 0.025 0.018 0.019
n 0.82 0.75 0.69 0.61 0.638
R2 0.99964 0.996749 0.995513 0.999874 0.997592
Mean 9.487333 8.516067 7.770733 6.677933 6.640733 HYBRID 0.040148 0.21431 0.245329 0.004122 0.105468 MPSD 0.005117 0.013047 0.014745 0.002126 0.01052 RMSE 0.05548 0.11615 0.117646 0.013711 0.070891
ARE 0.14473 -0.36902 -4.82396 0.060135 -0.29755
SSE 0.012312 0.053963 0.055362 0.000752 0.020102
X2 0.000802961 0.004286208 0.004906572 8.24491E-05 0.002109353 EABS -3.82928 -0.2323 -0.23529 0.027421 -0.14178
Sre 0.016885 0.068362 0.090885 0.016357 0.077883
4.5.2.5 Khan isotherm model
The Khan isotherm model constants were evaluated by the nonlinear regression of the isotherm model and presented in Tables 4.35 and 4.36. The high values of the correlation coefficient (above 0.99) signify that the experimental data is in close agreement with the predicted data. The Khan isotherm maximum adsorption capacity qm obtained using CCAC was much closer to the experimental value than that obtained using RHAC.
Hence, the Khan isotherm described the adsorption of phenol using CCAC adsorbent
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better than using RHAC. Kara et al, (2018) reported that if the Kahn isotherm model exponent a is equal to unity, then the Khan isotherm reduces to Langmuir isotherm.
Equally, When b value is much bigger than unity, Khan isotherm turns into the Freundlich isotherm. Adsorption data using Khan isotherm has been reported for phenol and phenol derivatives in aqueous medium for single and bisolute systems (Khan et al, 1997; Kara et al, 2018).
The Kahn isotherm model is a generalized model suggested for adsorption from pure solutions (Amrhar and Nassali, 2015).
Table 4.35: Nonlinear Khan isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
qm 5.5 5.3 4.5 4.2 3.1
b 0.028 0.026 0.03 0.033 0.37
a 0.84 0.82 0.83 0.81 0.83
R2 0.9981 0.997485 0.998989 0.998932 0.999817
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.007627 0.013716 0.001249 0.003822 0.000266 MPSD 0.003576 0.004845 0.00159 0.002837 0.000848 RMSE 0.01508 0.020018 0.005557 0.009527 0.002221
ARE 0.101152 3.448376 -0.04497 1.656087 -0.02399
SSE 0.00091 0.001603 0.000124 0.000363 1.97E-05
X2 0.000152538 0.000274319 2.49885E-05 7.64405E-05 5.32649E-06 EABS 0.03016 0.040037 -0.01111 -0.01905 -0.00444
Sre 0.035496 0.048499 0.016926 0.030588 0.009772
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Table 4.36: Nonlinear Toth isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
qm 11.2 9.6 9.5 8.2 8.3
b 0.53 0.48 0.41 0.33 0.35
a 0.72 0.76 0.84 0.88 0.88
R2 0.999845 0.999715 0.999577 0.997946 0.995981
Mean 9.487333 8.516067 7.770733 6.677933 6.640733 HYBRID 0.017233 0.018762 0.023119 0.067142 0.176039 MPSD 0.003352 0.00386 0.004527 0.00858 0.013591 RMSE 0.036348 0.034367 0.036115 0.055333 0.091588
ARE -0.09482 0.109187 0.12803 -0.24269 0.384418
SSE 0.005285 0.004724 0.005217 0.012247 0.033553
X2 0.000344651 0.000375239 0.000462378 0.001342849 0.003520787 EABS -0.0727 0.068733 0.07223 -0.11067 0.183175
Sre 0.011062 0.020227 0.0279 0.066011 0.100621
4.5.2.5 Koble-Carrigan Isotherm model
The add-in solver function in excel was employed in determining theKoble-Carrigan Isotherm parameters which are tabulated in Tables 4.37 and 4.38.
At all temperatures, the correlation coefficients were high for both CCAC and RHAC adsorbents. The Koble-Carrigan‘s isotherm exponent, p, was greater than unity at lower temperatures with its value decreasing as the temperature increased even becoming less than unity above 50 oC.Nimibofa et al (2017) reported that when the Koble-Carrigan‘s isotherm exponent, p, is less than unity, it signifies that the model is incapable of defining
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the experimental data even if it has high correlation coefficient and low error value. It is only valid when it is greater than or equal to 1.Therefore, at temperatures above 50oC, Koble-Carrigan isotherm cannot describe the adsorption of phenol unto these adsorbents.
According to Kara et al, (2018), when the adsorption experiments are carried out at high adsorbate concentrations, the Koble-Carrigan isotherm model approaches the Freundlich isotherm. The isotherm has an exponential dependence on concentration in the numerator and denominator. It is usually used with heterogeneous adsorption surfaces (Hossein et al, 2013).
Table 4.37: Nonlinear Koble-Corrigan isotherm parameters using CCAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
A 6 5.9 5 4.8 3.7
B 0.6 0.6 0.8 2.1 2.3
p 1.2 1.1 0.8 0.8 0.8
R2 0.998199 0.997012 0.999491 0.99696 0.99776
Mean 5.271467 5.045067 4.5938 4.166933 3.374867 HYBRID 0.00723 0.016293 0.00063 0.01088 0.003263 MPSD 0.003482 0.00528 0.001129 0.004786 0.002968 RMSE 0.014682 0.021818 0.003945 0.016075 0.007773
ARE 0.098485 -0.14936 0.031919 0.135368 -0.08395
SSE 0.000862 0.001904 6.22E-05 0.001034 0.000242
X2 0.000144599 0.000325867 1.25913E-05 0.000217603 6.52524E-05 EABS -0.02936 -0.04364 -0.00789 -0.03215 0.015545
Sre 0.03456 0.052859 0.012015 0.051609 0.034203
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Table 4.38: Nonlinear Koble-Corrigan isotherm parameters using RHAC
Parameter TEMPERATURE (o C)
30 35 40 50 60
A 15.2 12.8 11.2 9.3 9.5
B 1.3 1.1 0.92 0.87 0.87
p 1.5 1.3 1.1 0.8 0.77
R2 0.999459 0.997509 0.999125 0.99878 0.99742
Mean 9.487333 8.516067 7.770733 6.677933 6.640733 HYBRID 0.060285 0.164175 0.047838 0.03988 0.112978 MPSD 0.00627 0.011419 0.006511 0.006613 0.010888 RMSE 0.067984 0.10166 0.05195 0.042644 0.073372
ARE -0.17735 -0.32299 -0.18417 0.187037 -0.30796
SSE 0.018487 0.041339 0.010795 0.007274 0.021534
X2 0.001205699 0.003283504 0.000956755 0.000797607 0.002259567 EABS 0.135968 -0.20332 0.103901 -0.08529 0.146744
Sre 0.020691 0.059833 0.040133 0.050874 0.080609
4.5.3 Four terms parametric isotherm