Fixed Bed Adsorption of Water Contaminants: A Cautionary Guide to Simple Analytical Models and Modeling Misconceptions

ABSTRACT This contribution is intended as a specific review of the modeling approaches used to describe breakthrough curves of water contaminants and as a cautionary guide to some common and persistent modeling misconceptions. Its intended audience is early career researchers working in the area of adsorptive water remediation. First, typical data- and mechanism-driven models are discussed. A brief description of numerical fixed bed models with multiple curve broadening mechanisms (axial dispersion; intraparticle diffusion; film mass transfer; reaction kinetics) is presented. Particular attention is paid to analytical fixed bed models constructed for linear and nonlinear trace systems. Because these models assume that a single curve broadening mechanism is operative, their functional forms are relatively simple. Examples of spreadsheet calculations with these models are provided. A summary describing several egregious misconceptions associated with the use of two phenomenological models based on reaction kinetics (Bohart–Adams and Thomas) and one empirical model (Yoon–Nelson) to fit breakthrough curves is presented. In many previous studies, the three models have been used blindly, resulting in meaningless findings. The author hopes that this article will help the early career researcher avoid repeating the same mistakes.


INTRODUCTION
The presence of trace amounts of contaminants in water and wastewater requires the use of highly efficient treatment technologies. Adsorption is often the best choice for water purification or decontamination owing to its unique ability to treat dilute solutions and to remove a diverse array of organic and inorganic contaminants. It is normally carried out in a column packed with porous solid particles because a fixed bed gives much better contaminant removal efficiency than a batch stirred tank. Because it is rarely possible to predict the performance of a given contaminant-adsorbent system from first principles, the analysis of fixed bed adsorption relies on information extracted from experimental studies. Initial testing is usually performed on small-scale laboratory columns to generate breakthrough data. A breakthrough curve is a plot of effluent concentration as a function of time.
The information contained within an experimental breakthrough curve is most conveniently captured using mathematical models. Although a variety of fixed bed models with varying degrees of complexity have been used to process breakthrough data taken at small scales, the most commonly used ones are those proposed by Bohart and Adams, Thomas, and Yoon and Nelson. The popularity of the three models owes much to the fact that their model equations can be linearized to allow parameter estimation by linear regression. Because the three models are highly simplified, they cannot be used to probe the mechanistic nature of fixed bed adsorption dynamics. Instead, they are used as empirical tools to correlate breakthrough profiles. Empirical relationships between their fitting parameters and operational conditions (e.g., flow rate, feed concentration) must be established if they are to be used for process design or optimization. A prominent example for such a modeling approach is the bed depth-service time (BDST) equation. The BDST equation uses breakthrough data acquired from multiple runs of fixed bed adsorption experiments to establish the relationship between bed depth and the time taken for a specified breakthrough concentration to occur, that is, the service or breakthrough time. According to the BDST equation, this relationship should be linear. The calibrated BDST equation can then be used to size full-scale adsorption columns. It is well known that the BDST equation is based on the Bohart-Adams model. It should be noted that the data analysis step in the BDST design methodology involves experimental data taken from the initial stage of a breakthrough curve where the service time resides. In contrast, in academic research, a model is often fitted to the entire profile of a breakthrough curve. Therefore, the modeling issues raised in this work have no direct bearing on the BDST design methodology.
Every year, scholarly journals churn out hundreds of research articles on the use of the Bohart-Adams, Thomas, and Yoon-Nelson models to fit fixed bed breakthrough data. Unfortunately, this substantial body of literature is marred by various modeling misconceptions and misapplications of the three seemingly simple models. This sad state of affairs reflects the collective failure of journal editors and manuscript reviewers to weed out low-quality modeling work. Despite repeated concerns raised by a handful of researchers, [1][2][3][4] the publication of such meaningless modeling results continues to grow unabated. This ever-growing body of fallacious literature is liable to mislead and confuse the interested researcher. Nonetheless, the previous efforts to sound the alarm bell had not been all in vain; the misuse of the three models is now slowly being recognized.
This article is intended to keep the faulty modeling practices in the spotlight and to help the early career researcher become a more competent user of simple fixed bed models. To this end, we firstly provide a specific review of the types of fixed bed models used in the field of adsorptive water decontamination. This introductory note puts the Bohart-Adams, Thomas, and Yoon-Nelson models in context and gives readers an appreciation for the diversity of fixed bed models so that they can better understand the modeling options available to them. Then, a comprehensive dissection of several persistent misconceptions associated with the use of the three models is presented. As background information on the Bohart-Adams, Thomas, and Yoon-Nelson models, readers are referred to the books by Cooney [5] and Tien. [6] The books by Ruthven, [7] Suzuki, [8] Tien, [9] and Worch [10] provide comprehensive information on more advanced models of fixed bed adsorption, some of which are briefly elucidated here.

FIXED BED MODELS
While it is not the purpose of this article to have an extensive discussion on fixed bed modeling, it is, nevertheless, useful to outline a brief survey of some recent developments that could serve as good starting points to help the novice modeler orient in the field and for interested readers to explore further. Generally speaking, one may broadly classify fixed bed models into (i) data-driven (or black-box) models, (ii) mechanismdriven (or white-box) models, and (iii) hybrid (or gray-box) models which are combinations of the previous two classes of models. We consider in turn the three classes of models and cite some references that are relevant to adsorptive water decontamination.
Data-driven models do not require a detailed understanding of the fixed bed adsorption process but can still capture its breakthrough characteristics within the design space of the model. A well-known example is the design of experiments (DoE) or response surface methodology (RSM) approach, which uses statistical methods to construct empirical relationships between breakthrough data and fixed bed variables such as feed concentration, bed length, flow rate, etc. This approach works quite efficiently if the number of variables is small and if the user has prior knowledge of which variables are significant. The use of RSM to model and optimize dye removal by various treatment methods, including batch adsorption, has recently been reviewed by Karimifard and Moghaddam. [11] Witek-Krowiak et al. [12] provide a more specific review focusing on the RSM approach to optimizing biosorption in batch adsorbers. As recent examples of the application of RSM to fixed bed adsorption one may cite the works of da Rosa Schio et al., [13] Mora et al., [14] Agani et al., [15] Schio et al., [16] and El Mouhri et al. [17] Another type of data-driven models is based on machine learning methodologies such as artificial neural networks, which are well-suited for a wide range of high-dimensional, nonlinear problems. In the era of big data, [18] it is difficult, if not impossible, to avoid reading about machine learning in scientific articles. A neural network consists of an input layer, one or more hidden layers, and an output layer of nodes that function to correlate the input and output data. After undergoing a learning process in which the neural network recognizes patterns and correlations in the input and output data, it may be used to predict system behavior. Therefore, the essential requirement for neural network modeling is sufficient data to support the learning process. In the context of fixed bed adsorption, breakthrough data can be generated from laboratory-scale columns by varying process and system variables such as feed concentration, flow rate, bed length, etc. Alternatively, synthetic breakthrough data may be generated by using a complex mechanistic model, which is often computationally expensive. A neural network trained on synthetic data is known as a surrogate model, which may be used for simulation or optimization studies because of its low computational cost. A review by Witek-Krowiak et al. [12] gives an overview of the modeling of batch and fixed bed biosorption processes by the neural network approach. Another review on the use of neural networks to model the batch and fixed bed adsorption of dyes has been given by Ghaedi and Vafaei. [19] These reviews, together with a rapidly growing literature, [16,[20][21][22][23][24][25] suggest that there is considerable excitement about the role of neural networks in adsorption modeling. However, the use of neural networks in this area is not new. As far back as the mid-1990s, neural networks were used by Basheer and Najjar [26,27] to model fixed bed adsorption processes.
Data-driven models do not require expert knowledge to be developed. However, because they merely seek to summarize or describe the data trends, they are almost always empirical and act as a black box with no theoretical representation of a fundamental mechanism. As a result, a data-driven model cannot be used to make predictions outside the experimental conditions that were used to develop the model, that is, it lacks extrapolation capabilities. Mechanism-driven or whitebox models, as the opposite of data-driven models, are based on general and accepted physical principles, such as conservation of mass, energy, and momentum. Because fixed bed adsorption is an important separation process in chemical processing, mechanistic models have long been used by chemical engineers to seek a detailed fundamental understanding of the various adsorption phenomena taking place in fixed bed adsorbers packed with porous, spherical particles. Most of the models have been developed for gas adsorption, but they are applicable to liquid phase adsorption with minor modifications. The phenomenological fixed bed models commonly used in adsorptive water decontamination research can be traced back to the chemical engineering literature. [7][8][9] A variety of such models with varying degrees of complexity can be constructed by considering mass and energy balances, thermodynamics, mass transport phenomena, and adsorption kinetics expressed as a system of differential and algebraic equations. A drawback of these models is that their solutions require the use of numerical methods and programming code. We will revisit phenomenological modeling in the next section.
Since the white-box and black-box modeling approaches have their merits as well as shortcomings, a logical next step is to bring the strengths of these two approaches together in one model. This modeling approach is called hybrid or gray-box modeling. There are many ways to combine data-driven and mechanismdriven models. For example, the use of simulated data generated by a complex phenomenological model to train a data-driven model may be considered as a type of graybox modeling. Some researchers attempted to improve the data fitting ability of the Thomas model by combining it with neural networks [28,29] or with an adaptive neuralbased fuzzy inference system. [30] Compared to the whitebox and black-box modeling approaches, gray-box modeling is not as frequent. For a more detailed discussion on gray-box modeling, readers are referred to a recent review, [31] which describes the use of hybrid models in chemical engineering.
The selection of the type of model depends on the goal of the modeling activity. To gain a deeper understanding of the dynamics of a fixed bed adsorber, one should use a phenomenological model. If one wishes to correlate or summarize experimental observations, a data-driven model is a good choice. Choosing the appropriate model may also depend on other factors such as the expertise of the modeler and the availability of software resources. The British statistician George Box once said that "all models are wrong, some are useful." So, in the end, what counts is to obtain a model that is easy to use and robust, and has sufficient accuracy for the intended application. Two of the three models considered in this work, Bohart-Adams and Thomas, belong to the class of phenomenological models. Before discussing these two models in detail, we give a broad overview of phenomenological fixed bed modeling to introduce some terminologies.

MECHANISM-DRIVEN MODELS
Two common process configurations may be used to effect water decontamination by adsorption: stirred tank and packed bed. If the adsorbent is of a suitable size, one should opt for the fixed bed adsorber, which is normally much more efficient than the batch adsorber, resulting in a significant decrease in the amount of adsorbent needed for a given decontamination duty. However, this conventional wisdom has been challenged by Dichiara et al., [32] who demonstrated that under certain conditions adsorption in batch systems can be more efficient than adsorption in fixed bed columns.
We start our analysis of fixed bed adsorption by examining a plot of contaminant concentration in the effluent as a function of time (the so-called breakthrough curve). This curve is sometimes plotted as effluent concentration versus the number of bed volumes of the liquid phase passing through the bed. This quantity is given by the volumetric flow rate times the time divided by the bed volume (Qt/V b or εvt/L). The breakthrough behavior of a fixed bed adsorber defines its performance. An obvious point to address is what does a breakthrough curve look like if the performance of an adsorption system is at its maximum? The answer to this question is given in Figure  1, which shows the response of an initially contaminantfree column to a step-change in contaminant concentration at the column inlet. Figure 1 shows that the completely efficient breakthrough curve is a step function: at some breakthrough time, the effluent concentration suddenly jumps from zero to the value in the feed. A fixed bed adsorber with negligible axial dispersion and adsorption rate limitations will produce such an ideal breakthrough curve. We next turn to constructing a phenomenological model for such a fixed bed and use its solution to generate a breakthrough curve that looks like the one shown in Figure 1.

Equilibrium Nondispersive Model
Developing a phenomenological model of fixed bed adsorption involves setting up the differential mass balance equations for an element of the fixed bed column and for an adsorbent particle within such an element. To keep the exposition as simple as possible, we consider here that only one adsorbable contaminant is present in the feed. Other assumptions include: (i) the adsorption system is isothermal, (ii) the particles are spherical with uniform properties, (iii) plug flow (zero dispersion), (iv) the liquid velocity is constant, and (v) there is no variation of contaminant concentration in the radial direction.
The differential liquid phase mass balance equation is given by Eq. (1).
The first term on the left side of Eq. (1) represents contaminant convection in minus out. The second term on this side denotes the accumulation of the contaminant in the liquid phase, whereas the third term describes the accumulation of the contaminant in the solid phase. The mass balance for an adsorbent particle produces a rate expression (normally one or more diffusion equations), the general form of which may be written as Eq. (2).
The preceding equation is commonly based on a set of equations describing transport and kinetic processes within and outside the spherical particles. The rate expression incorporates the equilibrium isotherm relating the adsorbed and liquid phase concentrations to which Eq. (2) must reduce at long times when equilibrium is established. If all rate factors are sufficiently fast, a local interphase equilibrium exists at all points in the fixed bed column. Then, Eq. (2) can be written as Eq. (3).
The term dq*/dc in the last result is the derivative of the equilibrium isotherm. Substitution of Eq. (3) in Eq. (1) yields Eq. (4).
v @c @z þ @c @t As shown by Ruthven [7] and Cooney, [5] further mathematical manipulation followed by integration of the resulting expression leads to Eq. (5).
Since we are interested in the breakthrough curve, by setting z = L in Eq. (5) we obtain Eq. (6).
If the equilibrium isotherm is known, the preceding expression allows us to calculate the time at which a breakthrough curve exits from the column. Eq. (6) is called the equilibrium nondispersive model, [5] which is the simplest analysis of the dynamics of fixed bed columns. It predicts a perfectly sharp step function that is determined solely by the interplay of flow and thermodynamics. The equilibrium isotherm can be classified as linear, unfavorable, favorable, or irreversible, and the meaning of these terms is apparent in Figure 2. Experimental equilibrium data exhibiting the linear shape can be fitted to the one-parameter Henry law given by Eq. (7).
The nonlinear isotherm with a downward curvature (convex upward) is termed favorable, and the other nonlinear isotherm with an upward curvature (concave upward) is termed unfavorable. The unfavorable isotherm will not be discussed here because it is ineffective for adsorption in dilute solutions. The favorable isotherm is highly effective for adsorption in dilute systems, and experimental equilibrium data displaying this shape are usually fitted to the two-parameter Langmuir isotherm defined by Eq. (8).
In the limit of sufficiently small c e values, the Langmuir isotherm equation reduces to the linear Henry equation with q m K L = K. A limiting form of the favorable isotherm, called an irreversible isotherm, is also shown in Figure 2. This irreversible or rectangular isotherm, which does not exist in the real world, is quite useful because it allows nonlinear fixed bed models to be solved analytically. For the Henry and Langmuir equations, the equilibrium nondispersive model given by Eq. (6) becomes Eqs. (9) and (10).
Note that when the equilibrium isotherm is of favorable form Eq. (6) is no longer applicable and the term dq*/dc is replaced by q 0 /c 0 to account for the formation of shock fronts. [5] We illustrate the use of the equilibrium nondispersive model to fit a breakthrough curve in Example 1.

Example 1:
The fixed bed adsorption of a herbicide (glyphosate) by a modified activated carbon has been studied by Marin et al. [33] This study was chosen because the pertinent system and operational parameters are available in the article. First, we examine the equilibrium behavior of the adsorption system, which is shown in Figure 3. The four data points exhibiting a nonlinear adsorption trend were extracted from the breakthrough curve profiles obtained using different feed concentrations. Therefore, the liquid phase equilibrium concentrations correspond to the feed concentrations of the fixed bed experiments, i.e., c 0 = 5, 10, 15, and 20 mg/L. Marin et al. fitted the Langmuir isotherm, Eq. (8), to the four data points, obtaining q m = 6.78 mg/g and K L = 225 cm 3 /mg. Figure 3 shows that the Langmuir isotherm fit is excellent. However, it is not possible to determine the isotherm shape in the 0-5 mg/L concentration range due to a lack of data. If we assume linear adsorption behavior in the range of 0-5 mg/L, fitting Henry's law to the first data point at 5 mg/L yields K = 730 cm 3 /g, as shown in Figure 3. This linear adsorption assumption allows us to use the equilibrium nondispersive model given by Eq. (9) to describe the breakthrough curve obtained with c 0 = 5 mg/L. The conditions of this fixed bed experiment are listed in Table 1. With L, ε, and K known, the values of the remaining parameters in Eq. (9), v and ρ p , can be calculated. Using the data given in Table 1, the interstitial velocity v is estimated to be 26.12 cm/min. The adsorbent density ρ p is given by ρ b /(1ε). So, ρ p is 1.04 g/cm 3 . Plugging all these numbers into Eq. (9) yields t = 457 min. Figure 4 plots the perfectly sharp stepfunction profile predicted by Eq. (9). This step-function is of course unable to track the observed breakthrough trend which is S-shaped as a result of dispersion and mass transfer limitations. We can also use Eq. (10) to compute the time at which the step-function emerges. The quantity q 0 in Eq. (10), calculated from Eq. (8) with c e = c 0 and the q m and K L values given above, is 3.59 mg/g. With all the parameters known, Eq. (10) yields t = 449 min, which is similar to the prediction of Eq. (9). Example 1 vividly demonstrates that the equilibrium nondispersive model is practically useless in predicting breakthrough curves of real-life adsorption systems which tend to spread, resulting in sigmoid or S-shaped profiles. Any realistic dynamic model must allow for dispersive effects (axial dispersion and finite resistance to mass transfer), which lead to curve broadening, reducing the column's performance. We next turn to constructing such models.

Dispersion and Diffusion Model
To allow for the transport of the contaminant relative to the bulk liquid flow by axial dispersion arising from liquid mixing, the differential liquid phase mass balance equation given by Eq.
(1) is modified as follows: The preceding equation is subject to the following initial and boundary conditions: Eq. (11) is coupled with the mass balance equation for the solid phase given by Eq. (2), for which a variety of rate expressions may be formulated. Figure 5 illustrates schematically the locations of four commonly encountered mechanisms through which a contaminant is transported from the flowing liquid phase into the porous solid phase and is eventually adsorbed. These rate limitations include (i) external mass transfer in a stagnant liquid layer surrounding the porous particle; (ii) pore diffusion within the liquid phase contained inside the adsorbent pores; (iii) finite rates of adsorption and desorption at the liquid-adsorbent interface; and (iv) solid or surface diffusion along the surface of the solid phase. The emphasis here is on mass transfer processes, which, in most cases, control the overall rate of contaminant adsorption in fixed bed columns. Accordingly, we will not consider rate limitations caused by slow adsorption kinetics, which are rarely dominant in most practical cases. The mass balance expression accounting for the mass transfer effects is given by Eq. (16).
The preceding equation is subject to the following initial and boundary conditions:  [33] t � 0; r ¼ r p : In the above equations, the external mass transfer resistance is represented by the film mass transfer coefficient, k f . The pore diffusion mechanism is expressed in terms of the effective pore diffusion coefficient, D e , with the driving force for diffusion defined by the contaminant concentration gradient in the pore liquid (c p ). The solid diffusion process is defined by the adsorbed phase diffusion coefficient, D s , and the driving force for diffusion is expressed in terms of the adsorbed phase concentration, q. Eq. (16) states that the two intraparticle diffusion processes proceed in parallel. The local equilibrium assumption is used in Eq. (16) which contains the slope of the equilibrium isotherm, the general functional form of which is given by Eq. (20).
In this work, the set of coupled model equations, Eqs. (11) and (16), which account for axial dispersion, film mass transfer, and intraparticle pore and surface diffusion, is referred to as DFPSDM. The DFPSDM cannot be solved if the functional form of the equilibrium isotherm defined by Eq. (20) is unknown. If the equilibrium isotherm is nonlinear, the two partial differential equations are typically solved numerically. Standard numerical methods are implemented primarily through the method of lines, which relies on discretization techniques. [34] The commonly used discretization schemes are finite differences, finite elements, and finite volumes. In many cases, simplification is made to the combined diffusion equation [Eq. (16)] by assuming that either pore diffusion or solid diffusion predominates. The solid diffusion model, commonly known as the homogeneous surface diffusion model (HSDM), has been widely used to simulate organic contaminant adsorption in activated carbon columns. In this field of research, much of the pioneering work on the use of numerical diffusion models can be traced back to the research group of Walter J. Weber Jr. . [35,36] Some very recent examples of such numerical computations are listed in Table 2. As seen in this table, solving diffusion models with a nonlinear isotherm generally requires the use of specialized software and programming code. Interested readers with no access to commercial software will find the FAST software useful, which provides a numerical solution of the HSDM (film mass transfer plus surface diffusion). It is freely available for download from http://www.fast-software.de. In addition, computational fluid dynamics or CFD simulations are sometimes used to investigate the impact of hydrodynamic effects such as near-wall channeling on fixed bed adsorption. As examples, one may cite the works of da Rosa et al., [45] Esfandian et al., [46] and Vera et al. [47] In addition to the flow channeling phenomenon, other nonideal conditions such as enhanced adsorption under continuous flow and heterogeneous particle or pore size distribution can exert an impact on the performance of a fixed bed adsorber. When the equilibrium isotherm is linear, diffusion models can always be solved analytically. For example, Rasmuson and Neretnieks [48] have presented an exact solution of a model accounting for axial dispersion, film mass transfer, and surface diffusion. Their solution is given in terms of a semi-infinite integral involving the product of an exponentially decaying function and a periodic sine function, which converges slowly. The Rasmuson-Neretnieks solution is too complex to report here and is too cumbersome to be of much practical value. Fortunately, sigmoid or S-shaped breakthrough profiles can be generated by using models accounting for a single curve broadening effect. Analytical solutions of such models can be simplified to simple equations that are amenable to spreadsheet calculations. In the following, we discuss three such models.

Linear Isotherm and One Curve Broadening Factor
The DFPSDM described above contains four curve broadening factors: axial dispersion, external film mass transfer, pore diffusion, and surface diffusion. When the equilibrium isotherm is linear, a model that accounts for any one of these four factors can be solved analytically. The asymptotic or approximate forms of these analytical solutions are fairly tractable, allowing fixed bed behavior to be described with a minimum of effort. However, the selected mechanism may not be the actual rate-limiting step. For example, one may use a model accounting for axial dispersion, which can usually be neglected for liquid phase systems, [49] to represent an adsorption system whose dynamic behavior is dominated by mass transfer resistance. In such a case the model lumps all effects into the axial dispersion coefficient to describe breakthrough curve broadening. A major drawback of the lumped or apparent dispersion coefficient is that it loses its original physical meaning and cannot be readily estimated from existing literature correlations. The assumption of a single dispersive mechanism greatly simplifies the problem of modeling fixed bed processes, but the results must be used carefully. It should be mentioned that the assumption of linear equilibrium behavior is frequently justified for adsorptive water decontamination. For instance, adsorption processes are often used to treat groundwater contaminated by arsenic. Such systems are generally operated at very low concentrations under linear conditions. Linear Isotherm and Axial Dispersion. This simplified model assumes that only axial dispersion is the operative curve broadening effect. The liquid phase mass balance equation, Eq. (11), is rewritten as Eq. (21).
As mentioned earlier, all curve broadening effects are lumped into the so-called apparent dispersion coefficient, D a,app . Under the assumption of linear adsorption equilibrium between the liquid and solid phase, we can rewrite Eq. (21) as Eq. (22).
By assuming that the column is infinitely long, the solution for the preceding equation, first derived by Lapidus and Amundson, [50] is given by Eq. (23). where Note that the Lapidus-Amundson solution is also known as the Ogata-Banks solution in some publications, although Ogata and Banks [51] derived the same solution in 1961, some nine years after Lapidus and Amundson in 1952. The Lapidus-Amundson solution is more commonly known as the equilibrium dispersive model in the adsorption literature. Under conditions where the Peclet number is not too large, the third term on the right side of Eq. (23) may be neglected. Then we get the following asymptotic solution: An example is given below showing how one calculates a breakthrough curve from the equilibrium dispersive model.  [37] Antibiotic; activated carbon 1, 2, 3, 4 Radke-Prausnitz Finite element Comsol Multiphysics [38] Mercury; activated carbon 1, 2, 3, 4 Langmuir Method of lines Matlab [39] Furan; resin 1, 2, 3 Langmuir Finite element and orthogonal collocation Matlab [40] Dye; malt bagasse 1, 2 Langmuir Finite difference Maple [41] Lithium; resin 2, 4 Langmuir Orthogonal collocation on finite element gPROMS [42] Vanadium; granular ferric hydroxide 2, 4 Freundlich Finite difference FAST [43] Uranium; granular ferric hydroxide 2, 4 Freundlich Finite difference FAST [44] Glimepiride; carbon nanotube 1 Freundlich Finite difference Mathematica Example 2: Using the data of Example 1, we fit the equilibrium dispersive model to the breakthrough data in Figure 4. The relevant variables are the following: L = 10 cm, ε = 0.39, v = 26.12 cm/min, ρ p = 1.04 g/cm 3 , and K = 730 cm 3 /g. The parameter to be fitted is D a,app . With the assumption of linear adsorption, the asymptotic form of the equilibrium dispersive model, Eq. (26), was fitted to the breakthrough data in Figure  4 by nonlinear regression. The best-fit value of D a,app is 15.94 cm 2 /min. Despite using only one curve broadening factor, the equilibrium dispersive model is quite effective in tracing the S-shaped profile of the breakthrough data, as shown in Figure 6a. Of course, the reasonably good fit does not imply that the axial dispersion mechanism is solely responsible for the curve broadening effect. Fitting the full solution given by Eq. (23) to the data produced a similar parameter estimate (D a,app = 13.07 cm 2 /min). There are however noticeable deviations between the model fit and experimental data in the saturation portion of the observed curve.
Most real-life breakthrough curves exhibit a phenomenon known as tailing, which describes a process in which the effluent exhibits a slow approach toward the influent concentration near column saturation. Many simple models including the equilibrium dispersive model are unable to track such asymmetric breakthrough profiles to a significant degree of precision.
Linear Isotherm and Solid Diffusion. For this model, the liquid phase mass balance equation is given by Eq. (1) Under the assumption of linear adsorption, Rosen has given an exact solution to the above model comprising Eqs. (1) and (27): [7,52] where However, Eq. (28) is in the form of a semi-infinite integral, which converges rather slowly and is not easy to use. In a subsequent paper, Rosen [53] gives an asymptotic form of Eq. (28): This approximate solution is much easier to handle than the exact solution and is valid for long columns, i.e., large L. An example of the use of Rosen's asymptotic solution to describe breakthrough data follows.
Example 3: Using the data of Example 1, we fit the asymptotic Rosen solution given by Eq. (33) to the breakthrough data in Figure 4 by nonlinear regression to estimate D s . The relevant variables are the following: L = 10 cm, r p = 0.007 cm, ε = 0.39, v = 26.12 cm/min, ρ p = 1.04 g/cm 3 , and K = 730 cm 3 /g. Figure 6b displays the fitted curve, calculated using the best-fit D s value of 1.76 × 10 -7 cm 2 /min. The quality of fit is similar to that of the equilibrium dispersive model shown in Figure 6a.
Linear Isotherm and Linear Driving Force Model. The linear driving force (LDF) model was originally proposed as a simple approximation for surface diffusion. [54] It is frequently used instead of the intraparticle diffusion model for practical analysis of adsorption column dynamics. [55,56] The appropriate equation is given by Eq. (34).
The LDF model is used as a proxy for the intraparticle diffusion model because it has similar accuracy and is computationally less complex. The LDF model, the liquid phase mass balance equation given by Eq. (1), and a linear isotherm together constitute a complete fixed bed model with the LDF rate coefficient k LDF acting as a single curve broadening factor. The solution to this fixed bed model, originally obtained by Anzelius [57] for an analogous heat transfer case, is given by Eq. (35).
The right side of Eq. (35) is known as the J-function, which is written as Eq. (36). where The J-function, which is computationally somewhat cumbersome, has been tabulated or graphically presented in several publications. For spreadsheet calculation, approximate representations of the J-function in the form of fairly simple equations are available. For example, an asymptotic expansion of the J-function, known as the Onsager approximation, [58,59] takes the following form: In most cases, the second term on the right side of Eq. (39) can be dropped, and the J-function is written as Eq. (40).
or c c 0 ¼ 1 2 1 þ erf ffi ffi ffi ffi ffi ffi ffiffi n l τ l p À ffi ffi ffi ffi n l p þ 1 8 ffi ffi ffi ffi ffi ffi ffiffi n l τ l p þ 1 8 ffi ffi ffi ffi n l p In Example 4, we redo Example 1 using the Onsager and Klinkenberg approximate expressions.
Example 4: First, we fit the Onsager approximation given by Eq. (42) to the breakthrough data in Figure 4 by treating k LDF as an adjustable parameter. From Example 1, the relevant variables are the following: L = 10 cm, ε = 0.39, v = 26.12 cm/ min, ρ p = 1.04 g/cm 3 , and K = 730 cm 3 /g. Figure 6c presents the resulting nonlinear fit, calculated with k LDF = 0.043 min -1 . Next, a nonlinear fit using the Klinkenberg approximation given by Eq. (43) produced k LDF = 0.048 min -1 , which agrees with the previous estimate obtained from the Onsager approximation. It is interesting to note that Marin et al. [33] have reported a similar k LDF value for the breakthrough data in Figure 4. These authors used a numerical model accounting for axial dispersion, intraparticle diffusion represented by the LDF model, and nonlinear equilibrium behavior described by the Langmuir equation to fit the data in Figure 4. The numerical solution was implemented using the Maple software. The resulting best-fit k LDF was 0.031 min -1 , which agrees with the estimates obtained from the Onsager and Klinkenberg approximations. When the feed concentration is small, the equilibrium behavior in the low concentration range can often be approximated by a linear isotherm. The assumption of linear adsorption allows one to use simple equations such as the Onsager expression to correlate breakthrough data. Hence, the practical value of fixed bed models with a linear isotherm is evident.
As noted earlier, the rate parameter of a model with a single curve broadening factor is a lumped parameter. In the context of the LDF model, the following relation lumps the four curve broadening factors of the DFPSDM into its rate coefficient: [7] 1 Eq. (44) suggests that the curve broadening effects of axial dispersion and the various mass transfer resistances are linearly additive. An important observation is that the LDF model with a single curve broadening factor and the DFPSDM with multiple curve broadening factors provide essentially the same breakthrough prediction. For linear adsorption systems, it is thus not necessary to use the mathematically complex DFPSDM to describe fixed bed adsorption. Eq. (44) also reveals the relations between the LDF model and the other two models with a single curve broadening factor analyzed above. In the case of the equilibrium dispersive model, Eq. (44) reduces to Eq. (45).
Eqs. (45) and (46) suggest that the three models with a single curve broadening factor are analogous to each other because their rate coefficients are related. In Example 4, the fitted k LDF value obtained from the Onsager approximation is 0.043 min -1 . According to Eq. (45), this k LDF value gives a D a value of 13.31 cm 2 /min, which agrees with the fitted D a value of 13.07 cm 2 /min obtained from the equilibrium dispersive model defined by Eq. (23), as shown in Example 2. Likewise, for the same k LDF value, Eq. (46) gives a D s value of 1.40 × 10 -7 cm 2 /min, which is in agreement with the best-fit D s value of 1.76 × 10 -7 cm 2 /min obtained from the asymptotic Rosen model given by Eq. (33), as shown in Example 3. It is evident that for linear adsorption systems the precise nature of the curve broadening factor is not important. Models based on vastly different transport mechanisms, such as axial dispersion or intraparticle solid diffusion, will typically provide nearly identical representations of fixed bed breakthrough data. [7]

Nonlinear Isotherm and One Curve Broadening Factor
Most adsorption systems exhibit nonlinear equilibrium over a large concentration range, usually of the convex upward or favorable isotherm shape (see Figure 2). This isotherm shape is typically represented by the Langmuir or Freundlich equation. When a fixed bed adsorber is subjected to a large feed concentration, its breakthrough behavior is likely to be governed by a nonlinear isotherm. Models constructed to describe such nonlinear adsorption are not amenable to analytical solutions; they are typically solved numerically (see Table 2). There is, however, one exception: a fixed bed model based on second-order reversible kinetics, which reduces to a Langmuir isotherm at equilibrium, has been solved analytically by Thomas. [61] When the equilibrium isotherm is highly favorable, it may be approximated by a rectangular or irreversible isotherm (see Figure 2). Models with the assumption of irreversible adsorption equilibrium can be solved analytically. A pertinent example is the fixed bed model based on second-order irreversible kinetics proposed by Bohart and Adams, [62] which reduces to an irreversible isotherm at equilibrium. Another useful approximation is the concept of constant pattern. Under the assumption of constant pattern, it is possible to derive analytical solutions to nonlinear fixed bed models. In the following, we will discuss the Thomas and Bohart-Adams models and several other models based on the constant pattern approximation. We will also take a brief look at the fixed bed model proposed by Yoon and Nelson, [63] which often appears together with the Thomas and Bohart-Adams models in this field of research.

Langmuir Isotherm and Reaction Kinetics (Thomas Model).
In the modeling of adsorption systems, a reaction kinetic mechanism is sometimes used to describe the resistance to binding at a fluid-adsorbent interface. For physical adsorption, this type of kinetic mechanism is generally much faster than other mass transfer effects. As such, it was not included as a curve broadening factor in the DFPSDM given by Eq. (16). Nonetheless, in the early days of fixed bed modeling, reaction kinetic mechanisms were commonly used as a single rate factor to account for all curve broadening effects under nonlinear equilibrium conditions. Notable examples include the fixed bed models developed by Bohart and Adams [62] and Thomas. [61] Although the assumption of reaction kinetics as the only rate-determining mechanism is unrealistic, the analytical solutions of such models are easy to use because they are often given in the form of simple equations or can be simplified.
The following reversible kinetic equation, which is second order in the forward direction and first order in the reverse direction, was first used by Thomas [61] to describe liquid phase ion exchange processes in column operation: Because the preceding equation corresponds to a Langmuir isotherm at equilibrium, it is known as the Langmuir kinetics model. The solution to Eqs. (1) and (47), derived by Thomas [61] and generalized by Hiester and Vermeulen, [59] is given by Eq. (48).
As mentioned before, the J-function in Eq. (49) may be estimated using the Onsager approximation given by Eq. (40) or the Klinkenberg approximation defined by Eq. (41). A value of R between 0 and 1 corresponds to a favorable isotherm. The value of R depends on the feed concentration, as may be seen in Eq. (50). The Thomas solution is of great significance as the first analytical solution for nonequilibrium conditions and a nonlinear isotherm. This is why it can still be found in the latest edition of Perry's Chemical Engineer's Handbook, [64] even though it has largely been superseded by more realistic mass transfer-based models. The derivation of other analytical solutions for nonlinear adsorption has to invoke various simplifying assumptions, such as irreversible adsorption equilibrium or constant pattern behavior. In Example 5, we use the Thomas model to fit an experimental breakthrough curve.

Example 5:
We have previously analyzed an experimental breakthrough curve taken from the work of Marin et al., [33] see Examples 1-4. That breakthrough curve was obtained with the lowest feed concentration of 5 mg/L. The modeling work has relied on the assumption of linear equilibrium behavior. In this example, we select another breakthrough curve from the same source, which has been obtained with the highest feed concentration of 20 mg/L. We see from Figure 3 that the adsorption equilibrium behavior in the 0-20 mg/L range is nonlinear. The parameters of the Langmuir isotherm fitted to the data in Figure 3 are q m = 6.78 mg/g and K L = 225 cm 3 /mg. Table 1 presents the conditions of the selected fixed bed experiment. Note that the flow rate for this example (4 cm 3 /min) is one-half of the flow rate for Example 1. We fit the Thomas model given by Eq. (48) to the selected breakthrough data, shown in Figure 6d, by nonlinear regression. The only unknown in Eq. (48) is the rate parameter k T ; all the other parameters in Eqs. (49)-(52) are known. The J-function in Eq. (49) was approximated by the Klinkenberg equation in the data fitting process. The bestfit value of k T is 0.483 cm 3 /mg⋅min. Figure 6d plots the fitted curve (solid line). There are noticeable deviations between the Thomas fit and experimental data. Another attempt was made to fit the breakthrough data by treating both k T and q m as adjustable parameters. The resulting parameter values are k T = 0.838 cm 3 /mg⋅min and q m = 5.43 mg/g. This two-parameter fit displays much better agreement with the experimental data, as indicated by the broken line. The smaller q m derived from the Thomas fit is believed to be the result of an inaccurate fit of the saturation region which displays a gradual and extended approach to saturation. Figure 2, is a limiting form of a favorable equilibrium isotherm.

Irreversible Isotherm and Reaction Kinetics (Bohart-Adams Model). An irreversible or rectangular isotherm, sketched in
The assumption of an irreversible isotherm means that it is generally possible to derive analytical expressions for the breakthrough curve. The available solutions are well covered in the monograph by Ruthven. [7] Solutions for two broadening mechanisms are also possible. A solution for the case of an external film rate expression plus a pore diffusion model has been presented by Weber and Chakravorti, [65] while a solution for the case of an external film rate law plus the LDF model for the solid phase has been given by Yoshida et al. [66] These two solutions are however somewhat complex.
In the context of a single curve broadening factor, a solution for the breakthrough curve given by Bohart and Adams [62] is much simpler. The Bohart-Adams modeling approach is based on the following second-order irreversible kinetic model: The preceding rate equation reduces to an irreversible isotherm at equilibrium, � q ¼ q s , where q s corresponds to the horizontal part of the isotherm. The solution of Eqs. (1) and (53), first derived by Bohart-Adams [62] and generalized by Amundson, [67] is given by Eq. (54). where Note that in this case, the q 0 -term in the expression for τ [Eq. (51)] should be replaced by q s . The century-old Bohart-Adams solution is unique in that it is the first solution for the breakthrough curve. It should be mentioned that an erroneous version of Eq. (54) has been given by Hu and Zhang, [68] as pointed out by Chu. [69] The following example illustrates the use of the Bohart-Adams solution to fit an experimental breakthrough curve.

Example 6:
This example fits the Bohart-Adams solution to the breakthrough data in Figure 6d by nonlinear regression. The equilibrium behavior of the fixed bed experiment followed the Langmuir isotherm, as shown in Figure 3. The relevant variables are the following: c 0 = 20 mg/L, L = 10 cm, ε = 0.39, ρ p = 1.04 g/cm 3 , and v = 13.06 cm/min. We see from Figure 3 that the favorable isotherm is not sharp enough to justify the assumption of irreversible adsorption equilibrium, so the Bohart-Adams model is not strictly valid. Nevertheless, in practical applications the irreversible adsorption assumption is often relaxed, that is, q s is treated as a free parameter. In such a modeling approach, the two parameters, k BA and q s , are obtained by matching the data in Figure 6d to Eq. (54). The best-fit values are k BA = 0.877 cm 3 /mg⋅min and q s = 4.41 mg/g. Figure 6e shows that the Bohart-Adams model appears to provide a reasonably good representation of the observed breakthrough curve.
Aside: Yoon-Nelson Model. At this juncture, we digress to make a brief observation on the Yoon-Nelson model which frequently appears together with the Bohart-Adams and Thomas models in the literature of environmental adsorption. Unlike the Bohart-Adams and Thomas models, which are based on mass balance considerations, the Yoon-Nelson model is based on a probabilistic analysis of air contaminant breakthrough in respirator cartridges packed with activated charcoal. The solution to the model proposed by Yoon and Nelson [63] is given by Eq. (56).
Although the Yoon-Nelson model is often used to correlate breakthrough data, it is devoid of typical fixed bed variables such as flow rate and bed length. As such, it cannot be used to design fixed bed columns. According to Yoon and Nelson, [63] the rate coefficient k YN is given by Eq. (57).
The last result links k YN to several fixed bed variables, but it is rarely mentioned in studies that employ the Yoon-Nelson model. In addition, the validity of Eq. (57) cannot be verified because it was formulated based on experimental observations reported in a single study. [63] The Yoon-Nelson model is highly empirical in this sense and its use in the modeling of breakthrough curves should be discouraged.

Solutions for Constant Pattern Behavior.
When the isotherm is favorable, it is often observed in column operation that the shape of the concentration front becomes essentially column length-independent after a certain distance from the entrance. In other words, the concentration front in a sufficiently long column approaches a constant pattern, giving rise to an asymptotic breakthrough curve. This behavior was first identified by Bohart and Adams. [62] The shape of a constant pattern breakthrough curve is determined by both the equilibrium and the kinetics of adsorption. At constant pattern and under the condition that axial dispersion, as well as accumulation in the void volume, can be neglected, the continuity relation, Eq. (1), reduces to Eq. (58).
The preceding equation indicates that the liquid phase and average solid phase concentrations no longer have to be considered independently. Therefore, the assumption of constant pattern greatly simplifies the mathematical treatment of the adsorption process. Since the ratio c 0 /q 0 is known from the isotherm, Eq. (58) allows one to derive the constant pattern breakthrough curve by integrating the rate expression describing the kinetics of the adsorption process. Analytical constant pattern solutions for the Langmuir isotherm and a single curve broadening factor have been derived by various investigators. Michaels [70] has solved the relevant equation for the breakthrough curve for external film mass transfer. Glueckauf and Coates [54] have solved the problem for the LDF rate equation. When the Langmuir kinetics alone predominates, the solution given by Walter [71] can be used. These three constant pattern solutions are given by Eqs. External film : Langmuir kinetics : where The parameters R, τ, and n T in the above equations are defined by Eqs. (50)-(52), respectively. The condition of constant pattern is quickly established when R is very small, even in relatively short columns. Analytical constant pattern solutions for the Freundlich isotherm are also available in the literature. [72][73][74] In Example 7, we illustrate the use of the constant pattern solution of Walter in data fitting.

Example 7:
In this example, we fit the Walter equation given by Eq. (61) to the breakthrough data in Figure 6d. The constant pattern solution is appropriate because the fixed bed experiment was operated under the control of the Langmuir isotherm. First, The Walter equation can be rearranged to the following form: Note that Thomas also simplified his original solution given by Eq. (48) to the form of Eq. (64) in a separate publication. [58] Following Example 5, both k T and q m are treated as free parameters. The fit of Eq. (64) gives k T = 0.877 cm 3 /mg⋅min and q m = 5.39 mg/g, which are very similar to the values of k T and q m of the Thomas model determined in Example 5. The Walter equation fit shown in Figure 6f is practically identical to the Thomas model fit shown in Figure 6d (broken curve). This agreement confirms the validity of the constant pattern assumption and suggests that the Walter equation is a practical approximation of the Thomas model, which is somewhat tedious to evaluate owing to the presence of the J-function.

MISCONCEPTIONS
Here, we turn our focus to some persistent misconceptions about the Bohart-Adams, Thomas, and Yoon-Nelson models. From the previous section, we can see that the Bohart-Adams and Thomas models have phenomenological merit, while the Yoon-Nelson model is highly empirical without typical fixed bed variables. Although these three simple models are immensely popular, they suffer from several misconceptions that appear with troublingly high frequency throughout the literature of adsorptive water decontamination. The following is a list of what are, in our estimation, the most common conceptual misunderstandings in published research: (i) the Bohart-Adams, Thomas, and Yoon-Nelson models possess different data fitting abilities; (ii) it is OK to compare the "Adams-Bohart" model to the Thomas and Yoon-Nelson models; (iii) the bed depth-service time (BDST) equation is an independent model; and (iv) use of the Bohart-Adams and Thomas models requires breakthrough data. Despite concerns raised by several investigators over the past 10 years, [1][2][3][4] these misguided modeling practices continue to infiltrate how researchers model breakthrough curves of water contaminants. Such low-quality modeling results and mistaken conclusions can be found in numerous previous studies. This growing body of fallacious literature, if left unchecked, poses a significant risk to researchers who may be misled into repeating the same mistakes. The message needs to be repeated. The purpose of this section is to outline and dispel the misconceptions so deeply ingrained in our research literature. They represent a form of scholarly pollution that requires urgent decontamination.

Misconception 1. Comparing the Bohart-Adams, Thomas, and Yoon-Nelson Models Is OK
A common feature of many papers reporting the analysis of water contaminant breakthrough data is the comparison of the Bohart-Adams, Thomas, and Yoon-Nelson models. Such comparisons are often made based on linearized versions of the three models, as described below.
Bohart À Adams : ln Thomas : ln Yoon À Nelson : ln It is straightforward to obtain the linear form of the Yoon-Nelson model given by Eq. (67) from the nonlinear form given by Eq. (56) by simple rearrangement. However, deriving the linear forms of the Bohart-Adams and Thomas models from the corresponding nonlinear forms requires some mathematical manipulation. First, we show the steps involved in deriving the linear version of the Bohart-Adams model given by Eq. (65) from the nonlinear form given by Eq. (54), which may be rewritten as Eq. (68).
Since the two exponential terms on the right side of the preceding equation are usually much greater than unity, the unity term on the same side is often neglected. Accordingly, Eq. (68) is rewritten as Eq. (69).
Dividing each term in the last result by exp(n BA τ) and taking the natural logarithm of each side, we obtain Eq. (70).
Note that in Eq. (51) q 0 is replaced by q s . It is straightforward to show that the entity ρ p q s (1 -ε) is equivalent to the mass of adsorbed contaminant divided by the bed volume, which matches the definition of N 0 . Furthermore, the product εv is equal to u. Accordingly, Eq. (71) is rewritten as Eq. (72).
Since t is much greater than L/v in typical column operation, the preceding equation simplifies to Eq. (73).
The last result is formally identical to the linear version of the Bohart-Adams model given by Eq. (65). Next, we show how one may obtain the linear form of the Thomas model given by Eq. (66) from the nonlinear version given by Eq. (48). In this case, it is more convenient to use the constant pattern solution of Walter given by Eq. (64) as our starting point, which may be rewritten as Eq. (74).
Given that the entity ρ p (1 -ε) represents the mass of adsorbent divided by the bed volume, that is, M/AL, Eq. (76) may be written as Eq. (77).
Since εvA is equal to the volumetric flow rate, Q, the last result is rewritten as Eq. (78).
Since t is much greater than L/v, the preceding equation reduces to Eq. (79).
Eq. (79) is the same as the linear form of the Thomas model given by Eq. (66). The linearized Bohart-Adams model, Eq. (65), was given prominence in several papers published in the chemistry literature in the 1940s. [75][76][77] Eq. (65) made its appearance in the environmental engineering literature in the 1960-70s when Eckenfelder [78,79] and Ramalho [80] included it in their books on water and wastewater treatment. It began to attract the attention of environmental researchers after it was mentioned in the 1973 paper of Hutchins, [81] in which he proposed the BDST method for sizing activated carbon columns. From the mid-1970s onward, the linearized Bohart-Adams model has been used by McKay and colleagues to describe breakthrough curves of metal ions and dyes. [82][83][84] The linearized Thomas model given by Eq. (66) was also introduced to the environmental engineering domain via the textbook route. For example, Reynolds included the linearized Thomas model in his 1982 textbook. [85] From the late 1980s onward, Eq. (66) began to appear in a series of papers published by Viraraghavan and co-workers who used it to correlate breakthrough curves of metal ions. [86][87][88][89][90] There was little interaction between the McKay and Viraraghavan groups, with the former favoring the Bohart-Adams model and the latter championing the Thomas model. From the mid-2000s onward, articles comparing the performance of the Bohart-Adams, Thomas, and Yoon-Nelson models began to appear in the literature of adsorptive water decontamination. The three models were listed together in a review article published in 2005, [91] which has since garnered more than 2000 citations. Several subsequent review articles also listed the three models separately, creating the impression that they are independent models. [92][93][94] As a result, the research literature is replete with articles comparing the performance of the three models in data fitting.
So why is it wrong to compare the three models? The answer to this question has been given in a handful of papers published over the last 10 years. [1][2][3][4] From the mathematical perspective, the three models are analogous to each other. This mathematical equivalence can be illustrated using the following generic equation: The preceding equation contains two general parameters, a and b. It is possible to derive the three models from Eq. (80) by assigning the following parameters and variables to a and b: One may estimate the values of a and b by fitting Eq. (80) to breakthrough data. With known values of a and b, the unknown parameters of the Bohart-Adams (N 0 and k BA ), Thomas (q 0 and k T ), and Yoon-Nelson (t 0.5 and k YN ) can be calculated from Eqs. (81)- (83). It is therefore a fallacy to estimate these unknown parameters by fitting the three models one by one to the same set of breakthrough data and compare their quality of fit on the basis of the R 2 metric. In some data fitting cases, one may obtain different R 2 scores for the three model fits. Such statistical results are nothing more than artifacts arising from the linear regression method employed. Note that the functional form of Eq. (80) is analogous to the logistic equation, a well-known symmetric function commonly used to model sigmoid growth curves. It is often presented in the following mathematical form: The next question to ask is this: How widespread is this misconception? The novice researcher may be surprised to know that the problem runs deep. A light search from Google Scholar will return hundreds of papers reporting such fallacious modeling results. Since it is impractical to cite all these papers here, Table 3 presents a selection of some recent ones published across a broad spectrum of journals.

Misconception 2. Fitting the Adams-Bohart Model to Breakthrough Data Is OK
A sizable body of research has been claiming that a fixed bed model, called "Adams-Bohart", is not as good as the Thomas and Yoon-Nelson models in correlating breakthrough data of water contaminants. Is the Adams-Bohart model the same as the Bohart-Adams model? It turns out that the former can be obtained from the latter by dropping the unity term on the left side of Eq. (65), as shown below.
Eliminating the unity term from Eq. (65) is equivalent to saying that the unity term can be neglected when c 0 /c is much greater than 1, which is true when the effluent concentration c is very small. The Adams-Bohart model as defined by Eq. (85) is therefore valid for small effluent concentrations; it breaks down when c is large. This characteristic can be seen by rewriting Eq. (85) as Eq. (86).
The preceding equation states that the dimensionless effluent concentration c/c 0 increases exponentially with t. In fact, c/c 0 easily exceeds unity with increasing t, a physically impossible situation in a single contaminant adsorption system. Eq. (85) first emerged in the 1940s in the chemistry literature. It was devised to describe the initial part of a breakthrough curve, that is, the region of low effluent concentrations. It was given the label "Adams-Bohart" when it was introduced to the environmental adsorption domain. Eq. (85) became well known to environmental researchers mainly through the highly cited review article discussed above. [91] It should be stressed that the restrictive condition (valid for small effluent concentrations) was stated in the review article. Unfortunately, this condition has been ignored by many researchers, leading to extensive misuse of Eq. (85). More often than not, investigators use Eq. (85) to fit the entire profile of a sigmoid or S-shaped breakthrough curve, without realizing that it is an exponentially growing function incapable of fitting such a curve shape. Thus, it is no surprise that the Thomas and Yoon-Nelson models easily outperform Eq. (85) in fitting typical breakthrough curves of water contaminants which are sigmoid in shape. Table 3 lists some studies reporting the relative performance of Eq. (85) and the models of Thomas and Yoon-Nelson. Additional studies of this type can be found in the environmental adsorption literature. [107][108][109][110][111][112] It seems that these researchers were oblivious to the existence of the more versatile Bohart-Adams model given by Eq. (65), which can describe sigmoid breakthrough curves. It is unfortunate that there are two versions of the Bohart-Adams model in circulation, causing needless and endless confusion.

Misconception 3. The BDST Equation Is an Independent Model
The BDST equation proposed by Hutchins uses breakthrough data obtained from laboratory or pilot tests to size full-scale activated carbon columns. [81] In his paper, Hutchins clearly stated that the BDST equation is based on the Bohart-Adams model defined by Eq. (65). If t b is the breakthrough time and c b is the breakthrough concentration, a simple rearrangement of Eq. (65) gives us the BDST equation: The preceding equation says that the breakthrough time t b is a linear function of the bed depth L. The BDST equation has worked well since its introduction in the 1970s because it only makes use of the initial part of a breakthrough curve where the breakthrough time resides. Astonishingly, some investigators assume that the BDST equation is an independent model and use it to fit the entire profile of a breakthrough curve. For example, several researchers put forward the following BDST equation: [95,[113][114][115][116] or its linear form: Since Eq. (88) is based on the Bohart-Adams model, the rate coefficient k BDST must equal the Bohart-Adams rate coefficient k BA . Functionally, fitting Eq. (88) to a breakthrough curve is equivalent to fitting the Bohart-Adams model to the same data.
Treating the BDST equation given by Eq. (88) and the equations of Thomas and Yoon and Nelson as independent models, as were done by several researchers, [95,[113][114][115][116] is therefore a fallacy because these equations are mathematically equivalent. The creation of the rate coefficient k BDST is unnecessary, which could lead the unsuspecting modeler astray.

Misconception 4. Use of the Bohart-Adams and Thomas Models Requires Breakthrough Data
As discussed earlier, the Bohart-Adams and Thomas models, originally developed in the chemistry research literature, were introduced to the environmental adsorption research literature through textbooks written in the 1960s-80s for environmental engineering students. The two models were presented in these textbooks in the form of simplified, linearized equations for the design of full-scale adsorption columns. This design approach requires the estimation of model parameters from breakthrough data obtained from laboratory or pilot tests. This correlative modeling approach has been followed by researchers as well. Virtually all research articles report parameters estimated by the direct fitting of the models to breakthrough data. A disadvantage of the correlative modeling approach is the general lack of opportunities for extrapolation outside the experimental area of parameter estimation.
There is an alternative approach, known as predictive modeling, that has attracted little attention in this field of research. In this approach, the model parameters are determined from independent sources, that is, no breakthrough data are used to estimate their values. For example, adsorption isotherm parameters, which appear in the Bohart-Adams and Thomas models, are estimated from batch equilibrium data. A fixed bed model with independently estimated parameters can be used to provide a priori predictions of breakthrough behavior over a wide range of operating conditions.
Here, we illustrate how the Thomas model may be used for predictive modeling. Referring to the full Thomas model given by Eq. (48) or the constant pattern solution given by Eq. (64), we see that the unknown parameters are the following: (i) the Langmuir isotherm parameter q m , which appears in q 0 and n T , (ii) the Langmuir isotherm parameter K L , which appears in q 0 and R, and (iii) the rate coefficient k T , which appears in n T . To determine q m and K L , it is straightforward to fit the Langmuir isotherm to equilibrium data obtained from a batch adsorber. However, determining the rate coefficient k T is somewhat tricky because we know that the reaction kinetic mechanism assumed by Thomas is not realistic for most adsorption systems of interest; their rate of adsorption is controlled by mass transport. Fortunately, k T can be viewed as an overall rate coefficient that lumps several mass transport processes. This treatment of the reaction kinetic mechanism of Thomas is similar to the treatment of the LDF model, where several dispersive effects are lumped into its rate coefficient k LDF [see Eq. (44)].
As an example, we assume that the rate of adsorption is limited by the intraparticle pore diffusion resistance and the mass transfer resistance in the laminar fluid boundary layer surrounding an individual adsorbent particle. An approximate relation has been found that lumps the film mass transfer coefficient k f and the pore diffusion coefficient D e into k T . [117] This expression is given by Eq. (90).
Therefore, it is possible to calculate a value for k T if the two mass transport parameters k f and D e are known.
To estimate k f one may use a suitable engineering correlation for the fixed bed configuration, many of which are available in the literature. [6,10,64] To estimate D e one may use Eq. (91).
Although several correlations are available to calculate an approximate value for the tortuosity factor, τ p , [64] they are highly empirical and prone to error. Values of τ p between 2 and 6 are typical for granular activated carbons, [10] but a value of ~10 has been reported for liquid phase adsorption on a commercial carbon. [6] A more reliable way to estimate D e is to extract it from batch kinetic data. However, under nonlinear adsorption conditions, the film-pore diffusion model for a batch adsorber requires a numerical solution. Approximate methods are available that circumvent the use of numerical methods. Here, we illustrate one such method, which is based on the reaction kinetic equation of Thomas. We assume that the rate of adsorption for the system of interest in a batch adsorber is controlled by external film mass transfer and intraparticle pore diffusion. The steps involved are as follows: [118] • Estimate D e by plugging the values of k T and k f into Eq. (90). Note that the k f value cannot be used to model a fixed bed column. It is used in Eq. (90) to eliminate its effect on the calculation of D e , which is independent of the mode of operation and can therefore be used to model a fixed bed adsorber.
With k f determined from a suitable engineering correlation and D e estimated from the batch kinetic data, it is simple to calculate a value for k T from Eq. (90). With known values of k T , q m , and K L , we can use the Thomas model given by Eq. (48) or the constant pattern solution given by Eq. (64) to predict breakthrough curves. No breakthrough data are needed for parameter estimation in this predictive modeling approach.
The predictive modeling approach described above is directly applicable to the Bohart-Adams model. The rate coefficient k BA is related to the mass transport parameters by the following approximate expression: As before, k f is obtained from a suitable engineering correlation, while D e is extracted from batch kinetic data. An analytical solution for a batch adsorber that follows the reaction kinetic mechanism of Bohart and Adams is given by Eq. (97). [119] c With k f and D e estimated from independent sources, k BA for fixed bed operation is readily obtained from Eq. (96). The q sterm in Eq. (96) can be obtained from the horizontal part of the equilibrium isotherm. One can then simulate breakthrough curves using Eq. (54). However, it should be noted that the equilibrium isotherm must be sharp enough for this modeling method to work owing to the assumption of irreversible adsorption. The predictive modeling approach is not applicable to the Yoon-Nelson model due to its highly empirical nature. There are no links between its rate parameter, k YN , and the mass transport coefficients.

LIMITATIONS
Having commented on the misconceptions surrounding the Bohart-Adams, Thomas, and Yoon-Nelson models, it is perhaps worthwhile closing with a summary of their data fitting ability. Because the three models are by design symmetric, their data fitting ability is somewhat limited. The symmetry of a sigmoid breakthrough curve is determined by the location of its inflection point, at which the curvature of the sigmoid curve changes from concave to convex. A sigmoid breakthrough curve is symmetric if its inflection point corresponds to the midpoint, that is, c/c 0 = 0.5, and asymmetric if it does not. The inflection point of the sigmoid curves predicted by the three models is always located at c/c 0 = 0.5. In consequence, they are confined to predicting perfectly symmetric breakthrough curves.
This limitation is a serious drawback because most breakthrough curves of water contaminants exhibit varying degrees of curve asymmetry, which is largely a result of fronting or tailing. Fronting refers to the early leakage of contaminant in the effluent while tailing describes a phenomenon in which the effluent exhibits a slow approach toward the influent concentration near column saturation. The three models are therefore unable to fit the entire profile of an asymmetric breakthrough curve. A strongly asymmetric breakthrough curve tends to make the three models fit poorly to both its initial and saturation parts, resulting in grossly inaccurate estimates of the breakthrough and exhaustion times.
The popularity of the three models has little to do with performance, but very much with practical considerations such as easy data fitting using linear regression and minimal requirement of system information. For instance, fitting the models to breakthrough data does not require knowledge of bed void fraction or adsorbent density. The three models have recently been modified in an attempt to improve their ability to track asymmetric breakthrough curves. [120] Another limitation concerns the behavior of the three models at time zero. At t = 0, the generic model given by Eq. (84) reduces to Eq. (98).
The three models have the undesirable property of not being constrained to go through the (0,0) origin, predicting a nonzero effluent concentration at time zero. Depending on the value of a, the nonzero effluent concentration at time zero can be quite conspicuous. This nonzero effluent concentration problem usually shows up when the three models are used to fit asymmetric breakthrough data. If the nonzero effluent concentration at time zero is greater than the desired breakthrough concentration, it becomes impossible to estimate the corresponding breakthrough time. Nevertheless, this limitation does not affect the column design methodology based on the BDST equation. This is because the BDST equation is not used to fit the entire profile of a breakthrough curve. The effluent concentration at time zero predicted by the BDST equation is always close to zero because the equation fit is not distorted by breakthrough curve asymmetry. Therefore, the problem of nonzero effluent concentration at time zero has no bearing on the application of the BDST equation in process design.

CONCLUDING REMARKS
The emphasis of the present review has been on the use of simple phenomenological models to correlate water contaminant breakthrough data. Over the past 40 years, the modeling domain has been dominated by the Bohart-Adams and Thomas models, which are based on the unrealistic concept of reaction kinetics. Numerous studies have compared the relative performance of these two models (plus the highly empirical Yoon-Nelson model) in data fitting. Despite their prevalence in the literature of adsorptive water decontamination, a large percentage of the published research on the three models is marred by misconceptions and meaningless findings. The predominance of the three models has also led to the situation today where very few researchers are aware of other equally simple models, such as the Lapidus-Amundson, Rosen, Anzelius, and constant pattern models. The latter models are less well investigated and exploited in this field of research. In addition to the modeling misconceptions discussed here, the environmental adsorption literature is replete with many more questionable data analysis methods. [121][122][123][124][125][126][127][128][129][130][131][132][133][134][135][136] A large part of the modeling research in this field has been mostly misleading. Poor quality research continues to be published in massive amounts. The peer review system is weak in a publish-or-perish environment. Our field deserves better than this. The early career researcher is advised to approach the substantial literature in this area with caution. Many pitfalls await the unsuspecting modeler.