A Modified Belter Model for Correlating Asymmetric Breakthrough Curves of Water Pollutants

Several theoretical and empirical models are available to correlate experimental breakthrough curves of water pollutants, one of which is the two-parameter Belter model. Although not as well known as the century-old Bohart–Adams model, the Belter model is being used with sufficient frequency to merit wider awareness of its strengths and weaknesses. Through a systematic analysis, it is shown that the two adjustable parameters of the Belter model are analogous to the equilibrium capacity and rate parameters of the Bohart–Adams model. Breakthrough curves predicted by the Belter model are perfectly symmetric because their inflection points are invariant and always correspond to the midpoint of the curves. As a consequence, the Belter model provides poor fits to asymmetric breakthrough curves. In this work, an improved version of the Belter model is introduced. The new model with a floating inflection point manifests excellent conformity with mildly and, more importantly, severely asymmetric breakthrough curves.

description of fixed bed dynamics, they are computationally complex. A simpler, commonly used fixed bed model employs a linear driving force (LDF) approximation, which lumps all dispersive effects into a single rate coefficient. This simplified modeling approach predicts a sigmoid breakthrough curve using the LDF rate coefficient as the sole curve-broadening factor and the equilibrium isotherm to shift the position of the curve on the time coordinate. A model with the correct functional form, together with an equilibrium parameter and a rate parameter, provides all that is needed to predict a sigmoid breakthrough curve. Indeed, several twoparameter models used in adsorptive water remediation research are of this type. Notable examples include the models of Bohart and Adams (1920) and Thomas (1944), both of which predict sigmoid curves with the help of an equilibrium capacity parameter and a rate parameter based on reaction kinetics. Empirical models proposed by Yoon and Nelson (1984) and Belter et al. (1988) have a scale parameter, which acts like an equilibrium parameter and a shape parameter, which behaves like a rate parameter. A major drawback of models of this type is that empirical relationships between model parameters and operational conditions (e.g., feed concentration, flow rate) must be established if they are to be used for process design.
The Bohart-Adams, Thomas, and Yoon-Nelson models, which are mathematically analogous (Chatterjee & Schiewer, 2011;Chu, 2020;Lee et al., 2015), have been widely used to correlate breakthrough curves of water contaminants. Their popularity owes much to the fact that the model equations can be easily linearized to allow parameter estimation by linear regression. However, the data fitting ability of the three models is rather limited. Restricted by their functional form, the three models are confined to describing symmetric breakthrough curves. Some attempts have been made to improve their ability to represent asymmetric breakthrough curves, as discussed by Apiratikul and Chu (2021). The model of Belter et al., by comparison, has attracted much less attention. Part of the reason for its obscurity lies in the mistaken belief that the model equation containing the error function cannot be linearized. (A linear version is given and tested in the present study.) The two-parameter model of Belter et al. was first used by Brady et al. (1999) to describe fixed bed adsorption of copper ions. Since then, it has been used to fit breakthrough data of several classes of water contaminants including metal ions Ghasemi et al., 2011;Lodeiro et al., 2006;Naji et al., 2020;Ramirez et al. 2007;Riazi et al., 2016;Saldaña-Robles et al., 2018;Stanley & Ogden, 2003;Sulaymon et al., 2010Sulaymon et al., , 2015Wong et al., 2003), ammonia Naji et al., 2020), organic compounds (Faisal et al., , 2021Knapik et al., 2020;Naji et al., 2020), dyes (Fernandez et al., 2014;Khoo et al., 2012;Lee et al., 2008;Teng & Lin, 2006), and oils (Srinivasan & Viraraghavan, 2014

The Belter Model
Equation 1 is the empirical model of fixed bed adsorption presented by Belter et al. (1988). In this equation, c is the effluent concentration at the bed outlet, c 0 is the feed concentration at the bed inlet, erf denotes the error function, t is time, and τ and σ are two adjustable parameters.
Equation 1 is known as the Belter model although that seems slightly unfair to Cussler and Hu. The Belter model uses two fitting parameters to predict a sigmoid curve with τ acting as a scale parameter and στ performing as a shape parameter. It should be noted that the τ term in the product στ is superfluous and can be removed.
The Belter model and its modified version were fit to published breakthrough data using a nonlinear (1) least-squares regression procedure. Model performance was evaluated by two statistical metrics: the coefficient of determination (R 2 ) and the residual root mean square error (RRMSE) given by Eqs. 2 and 3, respectively. In these two equations, n is the number of data points, M j is the j-th model value of c/c 0 , E j is the j-th experimental value of c/c 0 , E m is the mean of experimental values, and p is the number of adjustable parameters.
The Belter model can be linearized to facilitate parameter estimation by linear regression. Equation 4 is a linear form of the Belter model, where erf -1 stands for the inverse error function. Excel provides the error function but not the inverse error function. The latter can be approximated using the inverse normal cumulative distribution [erf -1 (x) = NORM.S.INV((x + 1)/2)/SQRT(2)].

Results and Discussion
Three sets of previously published breakthrough curves with different curve characteristics are used to evaluate and compare the data fitting ability of the Belter model and its modified version.

Copper Breakthrough Data
Da Costa Rocha et al. (2020) have used two different adsorbents, geopolymer and zeolite, to remove copper from water. The adsorbent was loaded into a glass column with a diameter of 1.5 cm and a packed length of 15 cm. Figure 1a shows a set of (2) breakthrough data obtained with a fixed bed of geopolymer, which was used to treat a synthetic copper solution of 2 mmol L -1 . The flow rate used was 3 cm 3 min -1 , giving an empty bed contact time (EBCT) of 8.8 min.
As illustrated in Fig. 1a, the experimental breakthrough curve is fairly symmetric; the initial and saturation segments are of similar length. Figure 1a shows that the nonlinear fit of the Belter model is in satisfactory conformity with the experimental data, returning good fit statistics (R 2 > 0.99, Table 1). Plotting the copper data according to the linear form of the Belter model (Eq. 4) yields an apparent linear trend, as illustrated in Fig. 1b. The R 2 score for the linear regression analysis is 0.964. The Belter model curve computed using the parameter estimates of the linear fit (τ = 3403.8 min and

Reactive Red 141 Breakthrough Data
The uptake of the dye reactive red 141 by pyrrhotite ash in a fixed bed adsorber has been studied by Mouldar et al. (2020). In Fig. 2 a set of breakthrough data taken from the work of Mouldar et al. (2020) is shown. The breakthrough data were measured using a glass column with a diameter of 1.5 cm and a packed length of 4.7 cm. The feed concentration and flow rate used were 100 mg L -1 and 1.66 cm 3 min -1 , respectively. The EBCT calculated from the bed volume and flow rate was 5 min.
Because the initial segment (e.g., t = 300-400 min) is noticeably shorter than the saturation segment (e.g., t = 700-1100 min), the breakthrough curve depicted in Fig. 2 exhibits a moderate degree of asymmetry. This type of curve shape is known as tailing, which refers to a phenomenon in which the effluent exhibits a slow approach toward the influent concentration near column saturation. Various reasons have been put forward to explain the tailing phenomenon, including flow nonuniformity, nonspecific adsorption, heterogeneous particle size distribution, and gradual reduction in the intraparticle diffusion rate. The Belter model was fit to the data of Fig. 2. As can be seen, the predicted values of c/c 0 do not agree well with the experimental data. In particular, the initial and saturation stages of the observed breakthrough curve are poorly described by the Belter model. Table 1 shows that the R 2 and RRMSE scores for the dye fit are inferior to those for the copper fit, suggesting that the dye breakthrough curve is more asymmetric than the copper breakthrough curve.

Fluoride Breakthrough Data
Fixed bed experiments have been performed by Tovar-Gómez et al. (2013) to study the adsorption of fluoride on bone char. Figure 3 shows a data set taken from the work of Tovar- Gómez et al. (2013). A column with an internal diameter of 2.5 cm and a packed length of 7.5 cm was used. The feed concentration and flow rate used were 9 mg L -1 and 3.3 cm 3 min -1 , respectively. Based on the bed volume and flow rate, the EBCT was found to be approximately 11 min.  The fluoride breakthrough profile presented in Fig. 3 is very different from those of copper and reactive red 141 depicted in Figs. 1a and 2. The zero effluent concentration period was very brief; a sharp increase in the effluent concentration emerged quickly, reaching c/c 0 ≈ 60%. The subsequent tailing in the breakthrough data was very long. This severely asymmetric breakthrough curve resembles a hyperbolic profile rather than a sigmoid curve. From the results presented in Fig. 3, it is evident that the fit of the fluoride data by the Belter model is a failure. The level of agreement between the experimental and predicted values of c/c 0 is very low. In addition, the predicted curve intersected the vertical axis at t = 0, giving a conspicuous nonzero effluent concentration at time zero. Table 1 shows that the R 2 and RRMSE scores for this case are far inferior to those for the copper and reactive red 141 cases.

Properties of the Belter Model
The results presented above reveal that the Belter model could handle the copper data with a slight degree of asymmetry but was practically useless when challenged with the fluoride data with a pronounced degree of asymmetry. Before we discuss possible reasons for the poor performance, it is instructive to first explore how the two parameters τ and σ control the behavior of the Belter model.
The nonlinear fit of Fig. 1a is used as an example, which was calculated using τ = 3320 min and σ = 0.11. Figure 4a depicts three curves generated with different values of τ while holding σ constant at 0.11. Curve 2 corresponds to the original nonlinear fit of Fig. 1a. Decreasing τ (3320 min) by 50% yields curve 1; increasing τ by 50% produces curve 3. It is clear that the smaller τ value shifts curve 2 to the left while the bigger τ value moves it to the right. So, the parameter τ functions very much like the equilibrium capacity parameters of the Bohart-Adams and Thomas models, which control the position of a breakthrough curve on the time coordinate.
The effect of σ is illustrated in Fig. 4b. As in Fig. 4a, curve 2 in Fig. 4b corresponds to the original nonlinear fit of Fig. 1a. Curves 1 and 3 were calculated by holding τ constant at 3320 min and changing σ (0.11) by ± 50%. Curve 1, produced by the smaller σ value, is much sharper than curve 2. Curve 3, generated by the bigger σ value, is a more spread-out profile compared to curve 2. The parameter σ is therefore similar to the rate parameters of the Bohart-Adams and Thomas models, which control the spread of a breakthrough curve. However, in contrast to σ, the value of the Bohart-Adams/Thomas rate parameter needs to be increased in order to obtain a sharper curve.
The unsatisfactory fits of the reactive red 141 and fluoride breakthrough curves are due to an inflexible mathematical property of the Belter model. Every sigmoid curve predicted by the Belter model contains a single inflection point. A sigmoid curve is perfectly symmetric if its inflection point is located at the midpoint, and asymmetric if it does not. A three-step procedure may be used to find the location of the inflection point for a Belter sigmoid curve. First, the second derivative of the Belter model is derived, given here by Eq. 5. Next, the left-hand member of Eq. 5 is set equal to zero and the resulting equation is solved for t. The result is t = τ. Finally, substitution of this t value into the Belter model (Eq. 1) leads to Eq. 6. Equation 6 states that the location of the inflection point for a Belter breakthrough curve corresponds to the midpoint of the curve, that is, c/c 0 = 0.5. As demonstrated in Fig. 4B, all three simulated curves with different σ values pass through the midpoint at c/c 0 = 0.5 and t = τ = 3320 min, which is indicated by the intersection point of the two dashed lines. It is clear that the location of the inflection point for a Belter breakthrough curve is invariant and always matches the midpoint. This means that it is impossible to alter the inflection point location (c/c 0 ) by changing τ or σ. This is the reason why the Belter model was unable to track the strongly asymmetric fluoride data (Fig. 3) and the moderately asymmetric reactive red 141 data (Fig. 2). To fit such breakthrough curves, the Belter model must shift the location of the inflection point away from the midpoint. We describe in the next section a modification procedure that can convert the invariant inflection point of the Belter model to a floating one, allowing the Belter model so modified to track asymmetric breakthrough curves to a significant degree of precision.

Logarithmic Transformation
A logarithmic transformation method (Apiratikul & Chu, 2021) is used to modify the Belter model. The main idea is to convert the parameter t in the Belter model to ln(t). Since we can take the logarithm only of dimensionless numbers, the parameter t, which has the dimension of time (min), must be made dimensionless. To this end, we define a new entity, T, which is equal to 1 min. Next, we multiply each term having the dimension of time in the argument of the error function by (T/T), as shown in Eq. 7. We simplify Eq. 7 to Eq. 8.
Finally, taking the logarithm of the dimensionless term (t/T) in Eq. 8 yields Eq. 9. Since T is equal to 1 min, it may be omitted from Eq. 9. Equation 10 is the final form of the modified Belter model. It should be noted that t and τ in Eq. 10 are now dimensionless quantities. A linear form of the modified Belter model is given by Eq. 11.
As an aside, we note that the functional form of the modified Belter model defined by Eq. 10 is consistent with an analytical solution for a phenomenological model of fixed bed adsorption, given here by Eq. 12 (Sigrist et al., 2011;Xiu et al., 1997). The symbols in Eq. 12 are defined by Eqs. 13-21, where R p is the adsorbent radius, ρ p is the adsorbent density, L is the length of packed bed, ε is the bed voidage, v is the interstitial velocity, K is the Henry's law equilibrium constant, k f is the external film mass transfer coefficient, D s is the surface diffusion coefficient, and D a is the axial dispersion coefficient. The two empirical parameters of the modified Belter model may thus be interpreted in terms of these physical parameters. It is important to note that Eq. 12 is valid for linear adsorption. Fixed bed adsorbers designed for water remediation are generally operated under nonlinear conditions.

Floating Inflection Point
The advantage of the modified Belter model over the original model is that the former has a floating or variable inflection point. The location of the inflection point can be derived using the three-step procedure discussed earlier. Briefly, the second derivative of the modified Belter model, given here by Eq. 22, is set equal to zero and solved for ln(t). The result is ln(t) = τ -(στ) 2 . Substitution of the last result into Eq. 10 leads to Eq. 23, which defines the location of the inflection point. According to Eq. 23, the modi-

Data Correlation Using the Modified Belter Model
The modified Belter model defined by Eq. 10 was fit to the copper data. Values of the parameters required to fit Eq. 10 to the experimental data are listed in Table 1, and a comparison of the fitted and experimental results is shown in Fig. 5a. As can be seen, the modified Belter model provides a quantitatively correct description of the experimental data. Table 1 shows that the fit statistics (R 2 and RRMSE) are slightly better than those for the original model fit. Because the copper breakthrough curve is only slightly asymmetric, the superiority of the modified model versus the original model is not obvious in this case. Figure 5b reveals that the copper data can be regressed using the linear form of the modified Belter model (Eq. 11). Figure 5a shows that the linear fit is marginally inferior to the nonlinear fit. The fits of the modified Belter model to the reactive red 141 and fluoride data are presented in Figs. 6 and 7, respectively. The agreement between the fitted and experimental results is excellent in these two cases, demonstrating that the modified Belter model has the ability to accurately track the two asymmetric breakthrough curves. The R 2 and RRMSE scores for these two fits are vastly superior to those for the original model fits, as can be seen in Table 1. Comparisons of Figs. 2 and 6 and Figs. 3 and 7 indicate that the modified Belter model is the best performing model.
As noted above, the modified Belter model has a floating inflection point, which allows it to track the shape of an asymmetric breakthrough curve. The inflection point locations for the three predicted breakthrough curves depicted in Figs. 5, 6 and 7 can be calculated from Eq. 23. In Fig. 8, the inflection point locations calculated from Eq. 23 using the relevant parameter estimates listed in Table 1 are shown. To fit the mildly asymmetric copper data, the modified model placed the inflection point slightly below the midpoint (c/c 0 = 0.46). To handle the moderately asymmetric reactive red 141 data, the modified model moved the inflection point further away from the midpoint (c/c 0 = 0.39). To track the strongly asymmetric fluoride data, the modified model placed the inflection point close to the (0,0) origin (c/c 0 = 0.17). The flexibility of inflection point placement allows the modified Belter model to fit a diverse array of asymmetric breakthrough curves. In contrast, the location of the inflection point for a breakthrough curve predicted by the original Belter model is always fixed at c/c 0 = 0.5. Consequently, the original Belter model is confined to fitting highly symmetric breakthrough curves, which are rarely observed in fixed bed adsorption experiments.

Conclusions
The results presented here have shown that the scale and shape parameters of the Belter model, τ and σ (or στ), are analogous to the equilibrium capacity and rate parameters of the widely used Bohart-Adams and Thomas models. The parameter τ controls the position of a Belter breakthrough curve on the time coordinate, whereas the parameter σ (or στ) dictates the spread of the curve. Through a detailed mathematical analysis, this work has shown that a Belter breakthrough curve is characterized by an invariant inflection point, which is always located at the midpoint of the curve (c/c 0 = 0.5). This property restricts the Belter model to correlating perfectly symmetric breakthrough curves. Consequently, the Belter model can only track highly symmetric breakthrough data, as demonstrated by the correlation of the copper breakthrough curve. Another goal of the present study has been to develop a new version of the Belter model capable of handling breakthrough curve asymmetry. The new Belter model with a floating inflection point has been shown to correlate the asymmetric breakthrough curves of reactive red 141 and fluoride to a significant degree of precision.
Funding The author received no specific funding for writing this article.