Modi ﬁ cation of the Thomas model for predicting unsymmetrical breakthrough curves using an adaptive neural-based fuzzy inference system

The Thomas equation is a popular model that has been widely used to predict breakthrough curves (BTCs) when describing the dynamic adsorption of different pollutants in a ﬁ xed-bed column system. However, BTCs commonly exhibit unsymmetrical patterns that cannot be predicted using empirical equations such as the Thomas model. Fortunately, adaptive neural-based fuzzy inference systems (ANFISs) can be used to model complex patterns found in adsorption processes in a ﬁ xed-bed column system. Consequently, a new hybrid model merging Thomas and an ANFIS was introduced to estimate the performance of BTCs, which were obtained for Cd(II) ion adsorption on ostrich bone ash-supported nanoscale zero-valent iron (nZVI). The results obtained showed that the fair performance of the Thomas model ( NRMSE ¼ 27.6% and E f ¼ 64.6%) improved to excellent ( NRMSE ¼ 3.8% and E f ¼ 93.8%) due to the unique strength of ANFISs in nonlinear modeling. The sensitivity analysis indicated that the initial solution pH was a more signi ﬁ cant input variable in ﬂ uencing the hybrid model than the other operational factors. This approach proves the potential of this hybrid method to predict BTCs for the dynamic adsorption of Cd(II) ions by ostrich bone ash-supported nZVI particles.


INTRODUCTION
Cadmium compounds are extensively applied in many industries such as metal plating facilities, paint pigments, mining operations, stabilizers and silver-cadmium batteries (Boparai et al. ).The release of these compounds into the environment can cause adverse health effects for humans such as kidneys disease, high blood pressure, lung insufficiency and bone defects (Boparai et al. ).As a result, cadmium is considered as priority pollutant by the Agency for Toxic Substances and Disease Registry of the USA (www.atsdr.cdc.gov).The maximum concentration level of cadmium in drinking water is 0.005 mg L -1 as set by the Council of the European Communities in .
Adsorption has been considered to be an alternative method to conventional and modern wastewater treatments for the removal of Cd(II) from wastewater due to its high efficiency, simplicity, low-cost and adaptability (Boparai et al. ; Boparai et al. ).A great deal of attention has recently focused on the application of nanoscale zerovalent iron (nZVI) for the elimination of heavy metals (Zhang et al. ; Soleymanzadeh et al. ; Gil et al. ).However, nZVI has limited applications in wastewater treatment due to its tendency to aggregate and oxidize (Zhang et al. ; Gil et al. ).To resolve this problem, porous materials such as ostrich bone ash has been employed to support nZVI (Arshadi et al. ; Gil et al. ), which can subsequently be separated using an external magnetic field without the need for filtration and centrifugation.Although several researchers (Boparai et  In the present work, a new hybrid approach that combines both the Thomas and ANFIS models to estimate the performance of BTCs for the adsorption of Cd(II) ions by ostrich bone ash-supported nZVI is presented.The normalized root mean square error (NRMSE), efficiency (E f ) and linear regression were calculated for both observed and estimated data in order to evaluate the performance of this hybrid approach.

Preparation of the adsorbent
The ostrich bone waste used in these experiments was obtained from a local butcher's store.
Total amount of metal ions sent through the column Total quantity of metal ions adsorbed in the column where C o , C e and C ad are inlet, outlet and adsorbed metal ion concentrations, respectively, Q is the flow rate (mL min -1 ), t e is the bed exhaustion time (min), and M is the mass of the adsorbent in the column.

Modeling approach
The ANFIS model used was a five-layer ANN-FIS system.

Evaluation criteria
The performance of the hybrid model for predicting BTC was evaluated using different statistical criteria including the determination coefficient (R 2 ), efficiency (E f ) and where m and n are the regression coefficients.

Thomas model
The Thomas equation in linearized form is as follows (Thomas ): where C o (mg L -1 ) and C t (mg L -1 ) are the influent Cd(II) concentration and effluent Cd(II) concentration at time t, respectively, m is the mass of the adsorbent (g), Q is the volumetric flow rate (mL min -1 ), k Th (mL min -1 mg -1 ) is the Thomas rate constant and q ads (mg g -1 ) is the equilibrium Cd(II) uptake per g of the stabilized nZVI.

Sensitivity analysis for the hybrid model
Identification of the most important parameters for modeling BTCs is one of the key goals of this research.In this regard, the sensitivity analysis can provide important information regarding the modeling.As such, the influence index was calculated for each parameter as follows (Gontarski et al. ):

RESULTS AND DISCUSSION
A hybrid model (Thomas equation þ ANFIS model) The values of q ads and k Th in the Thomas equation can be determined using the linear regression approach, which may give unreliable results for unsymmetrical BTCs.As such, a new hybrid Thomas-ANFIS model has been developed for the dynamic adsorption of Cd(II) ions by ostrich bone ash-supported nZVI (see Figure 1).Consequently, the removal efficiency of Cd(II) by ostrich bone ash-supported nZVI increases upon increasing the pH from 2 to 9.This could be due to the point of zero charge (pH PZC ) and the degree of ionization of Cd(II) (Amiri et al. ).The pH PZC of the adsorbent was about 5.87 and the pH at the ostrich bone ash-supported nZVI surface is neutral.However, the surface charge of the adsorbent is positive at a pH below 5.87 and is negative at any pH above 5.87 (Gil et al. ).The lowest Cd(II) uptake was observed at pH 2 as a result of competition between hydrogen ions and Cd 2þ for binding sites on the adsorbent.
In addition, a higher electrostatic repulsion between the Cd(II) ions and sorbent surface is observed at pH < 5.87.
At pH > 5.87, increased deprotonation of the ostrich bone ash-supported nZVI surface causes a significant increase in the negatively charged sites, thus resulting in a higher electrostatic attraction between the adsorbent surface and

Comparison between the Thomas and hybrid models
The BTCs acquired upon dynamic adsorption of Cd(II) ions by ostrich bone ash-supported nZVI were predicted using the Thomas and hybrid models (see Figures 3(a), 3(b), 4(a) and 4(b)).To assess the goodness of fit, NRMSE, E f , linear regression and R 2 were calculated using measured and estimated data (see Table 2).As can be seen from Figures 3(a 2).Indeed, the fair performance of the Thomas model (NRMSE ¼ 27.6% and E f ¼ 64.6%) improved to excellent (NRMSE ¼ 3.8% and E f ¼ 93.8%) when combining this model with the ANFIS model to take advantage of the latter's unique advantages for nonlinear modeling.These findings are in a good agreement with those of Han et al. (), who reported that a nonlinear method is more effective than a linear method for predicting the parameters of the Thomas model.A comparison of the BTC estimated using the hybrid model and experimental data is depicted in Figure 5.The 95% prediction intervals, based on the distribution of points around the fitted line, exhibit an excellent reliability for the fitting and prediction of BTCs (see Figure 5).The error histogram of the BTC for the hybrid model is presented in Figure 6.As can be seen, the error

Sensitivity analysis
The sensitivity of the hybrid model to input variables is presented in Table 3.The changes in influence index (%) upon elimination of each input parameter from the hybrid model are presented in Table 3.It is obvious that initial solution pH is the most sensitive parameter, followed by Q, H and C o .All four operating parameters in Table 3 are considered to be important for the hybrid model as the lowest value of the influence index was significant.Thus, Table 3 shows that an increase in pH, Q, H and C o is significant at 17.082, 13.26, 9.33 and 3.52, respectively.
Calculation of k Th and q ads using the Thomas and hybrid models The q ads values for Cd(II) dynamic adsorption using experimental data are compared with those calculated using the Thomas and hybrid models in Table 4.The q ads calculated using the linear regression of the Thomas model is acceptable at the greatest H and lowest Q and C o (see Table 4).
However, the linear regression of the Thomas model, which is the most widely used approach for modeling BTCs, provides a poor estimate of q ads at higher Q and C o (see Table 4).Thus, although the Thomas model is suitable for data modeling in the case of symmetrical BTCs, it may fail to correctly describe the performance of unsymmetrical BTCs.However, the hybrid model is able to predict both symmetrical and asymmetrical experimental BTCs.For instance, the q ads values obtained using experimental data, and the Thomas and hybrid models were found to be 5.24, 5.39 and 5.31 mg g -1 , respectively, at pH ¼ 7; Q ¼    experimental conditions (see Table 4).In addition, the relative differences between the calculated q ads of hybrid model 4.This hybrid model can be employed for data modeling of any unsymmetrical BTC or natural surface and groundwater, as it provides a powerful tool for accurate nonlinear modeling and is also capable of learning from the environment.
5. The suggested adsorbent is a versatile and eco-friendly material for attenuating cadmium ions in a fixed-bed column.
al. ; Boparai et al. ; Zhang et al. ; Soleymanzadeh et al. ) have shown the potential of stabilized nZVI as an adsorbent for Cd(II) removal under laboratory batch conditions, the performance of nZVI supported on porous materials for Cd(II) ion removal in a fixed-bed column has not been investigated.The performance of fixed-bed adsorption is assessed by plotting an effluent concentration-time profile or breakthrough curve (BTC).However, as fixed-bed adsorption experiments are costly, difficult and time-consuming, an ability to predict BTCs could be a good and speedy alternative to the measurable column experiments for describing the adsorption process in fixed-bed systems.Therefore, a variety of mathematical empirical equations such as Bohart-Adams (Bohart & Adams ), Thomas (Thomas ), bed-depth service time (BDST) (Hutchins ) and Yoon-Nelson (Yoon & Nelson ) have been widely used to describe BTCs.Of these, the Thomas model is the most widely used due to its simplicity and applicability for different pollutants.This model is appropriate to predict the adsorption process when internal and external diffusion resistances can be ignored.Furthermore, this model supposes that the adsorption process is described by pseudo-second-order reversible reaction kinetics and Langmuir isotherm at equilibrium (Han et al. ; Hasan et al. ; Chowdhury & Saha ).This model considered that the adsorption process is not limited by the chemical reaction but controlled by the mass transfer at the interface (Hasan et al. ; Chowdhury & Saha ).Although these traditional methods, particularly the Thomas model, are able to well predict symmetrical BTCs, they are not particularly suitable for data fitting with unsymmetrical BTCs (Tovar-Gomez et al. ).In recent years, artificial neural networks (ANNs) have been successfully used to estimate the performance of BTCs (Cavas et al. ; Chowdhury & Saha ; Tovar-Gomez et al. ; Oguz & Ersoy ; Masomi et al. ; Oguz ).To date, no studies have considered the adaptive neural-based fuzzy inference system (ANFIS) even though this model can be employed as an alternative when modeling complex input-output dependencies, like estimation of the elastic constant of rocks (Singh et al. ), evapotranspiration estimation (Kisi & Ozturk ), prediction of nitrate (Mousavi & Amiri ), predicting lead removal from aqueous media (Amiri et al. ), leaf area prediction (Amiri & Shabani ), etc.The ANFIS can model the complex patterns seen in adsorption processes in a fixed-bed column system.

Further
details of the five-layer ANFIS architecture are presented in the supplementary data.The ANFIS structure with two inputs, two rules based on Sugeno fuzzy rules and one output are presented in Figure S6 (available online) (Singh et al. ).Determination of the number and type of membership functions for input parameters, and definition of the type of membership functions for output data, are the most important steps for modeling by ANFIS.In this regard, the optimal ANFIS construction was selected using trial and error.Of the eight membership functions (i.e.trimf, dsigmf, psigmf, gbellmf, pimf, trapmf, gaussmf and gauss2mf) Gaussian has the best performance in terms of the mean square error (MSE) due to its smoothness, concise notation and non-zero at each point.Furthermore, a hybrid algorithm combining gradient descent and least-squares method was employed when learning the model.The complexity of the ANFIS model is determined using the number of membership functions for each input value.The results showed that the error was not significantly changed when the membership function was increased from three to four.Thus, three membership functions were selected for each input value for the rest of the modeling.Finally, the linear membership functions were chosen for the output parameters to increase the model accuracy.The performance of the ANFIS model should be controlled using an individual dataset as this model is based on trial and error.In this regard, the experimental data gathered from dynamic adsorption of Cd(II) in a packed bed column were divided into two groups: 200 of the available data points (70% of the observations) were randomly selected for training and the remaining 85 (30% of the observations) were used for testing.A hybrid model combining ANFIS and the Thomas equation was employed in MATLAB software Version 8.1.In this hybrid model, four operational parameters including inflow rate (Q), initial Cd(II) concentration (C o ), bed height (H ), and initial solution pH, were taken as input data for the ANFIS model and the q ads and k Th for the Thomas equation was taken as output (see Figure 1).Subsequently, q ads and k Th which were calculated initially, were used as input data the for Thomas equation and C t C o was calculated as the final output.The flowchart of computations in the ANFIS-Thomas model is presented in Figure 2.

Figure 1 |
Figure 1 | ANFIS architecture applied for predicting the Ct Co using ostrich bone ash supported-nZVI composite.
Cd(II) ions(Soleymanzadeh et al. ; Amiri et al. ).Moreover, the higher removal efficiency of Cd(II) ions at pH 9 may be due to the precipitation of cadmium(II) as Cd(OH) 2 .Another key factor for dynamic Cd(II) adsorption in a fixed-bed column is related to the influent flow rate.As such, the BTC at six different Q (0.5, 1, 1.5, 10, 20 and 30 mL min À1 ) was examined, while maintaining the C o , H and pH.An increase in Q from 0.5 to 30 mL min À1 shifted the BT time and t e to lower values, thus resulting in a lower adsorption capacity (Chowdhury & Saha ; Oguz ).The removal efficiency of Cd(II) by ostrich bone ashsupported nZVI increased in the order Q 5 mL min À1 .The increase in Cd(II) removal as the Q decreased can be related to the longer mass transfer time for the sorption of Cd(II) by the binding sites (Cavas et al. ).Furthermore, a decrease in external film mass resistance on the ostrich bone ash-supported surface was observed at higher Q (Cavas et al. ).Thus, C t C o values of 0 and 1 were observed for Q of 0.5 and 30 mL min À1 , respectively, obtained in the interval of 4 h (see Figure 3(b)).
Figure 4(a); C o ¼ 50 mg L -1 in Figure 4(b)).However, the performance of this model worsens for unsymmetrical BTCs ( pH ¼ 2 in Figure 3(a); Q ¼ 30 mL min À1 in Figure 3(b); H ¼ 8 cm in Figure 4(a); C o ¼ 150 mg L -1 in Figure 4(b)) and the difference between the experimental data and data estimated using the model increases.As can be seen from Figures 3(a) to 4(b), the new hybrid model is able to predict both the symmetric and asymmetric experimental BTCs under various key operating conditions.Indeed, higher R 2 and E f values and smaller values of NRMSE are obtained when using the hybrid model compared to the Thomas model, thus indicating the higher accuracy of the hybrid model (see Table2).Indeed, the distribution is very close to zero and is almost symmetrically bell-shaped, thus indicating that hybrid model satisfies the suppositions of normality.As such, this hybrid model can be a fast and accurate alternative to the mathematical empirical equations currently available.

1
mL min -1 ; H ¼ 16 cm and C o ¼ 50 mg L -1 .Similarly, q ads values of 3.03, 4.02 and 2.98 mg g -1 were obtained using experimental data, and the Thomas and hybrid models, respectively, at pH ¼ 7; Q ¼ 10 mL min -1 ; H ¼ 8 cm and C o ¼ 150 mg L -1 .Similar results have been observed for k Th with values ranging from 0.24 to 1.56 mL min -1 mg -1 for the Thomas model and 0.21-1.42mL min -1 mg -1 for the hybrid model.Clearly, the obtained values of q ads and k Th by Thomas and hybrid models vary dramatically under

Figure 6 |
Figure 6 | Error histogram of the hybrid model.

Figure 5 |
Figure 5 | Comparison of the BTC predicted by hybrid model and experimental data.
were smaller than those in the Thomas model for each operating condition.As a result, the hybrid model provides the flexibility of reliably estimating the performance of fixed bed column at various operational conditions.Similar results were reported by Tovar-Gomez et al. ().Furthermore, the estimated k Th by Thomas and hybrid models increased with increasing the influent Cd(II) concentration and inlet flow rates demonstrating that the driving force of Cd(II) mass transfer is also raised.Similar trends were also observed by Cavas et al. () for adsorption of methylene blue by a beach waste dead leaves.CONCLUSIONS The present study reveals a novel hybrid model that combines both Thomas and ANFIS models, to predict BTCs for Cd(II) ions in a fixed-bed column.The dynamic adsorption of Cd(II) ions on ostrich bone ash-supported nZVI was studied as a function of initial solution pH, influent flow rate, bed height and initial Cd(II) concentration.The results indicate the following: 1.The fair performance of the Thomas model (NRMSE ¼ 27.6% and E f ¼ 64.6%) improves to excellent (NRMSE ¼ 3.8% and E f ¼ 93.8%) when combining this model with ANFIS.2. The hybrid model provides more reliable results as regards predicting the performance of BTC compared to the Thomas model.3. The sensitivity analysis indicates that initial solution pH is a more significant input variable for the hybrid model than the other operational factors.

Table 1 |
Statistical performance evaluation criteria for the final models

Table 2 |
The performance of Thomas and Thomas-ANFIS models to predict BTC

Table 3 |
The sensitivity of hybrid model to each input variable

Table 4 |
Calculated design parameters using Thomas and hybrid models for dynamic adsorption of Cd(II) ions by OBA/nZVI