Applicability of the K F a v model in the prediction of fixed bed breakthrough curve

Abstract

Applicability of the K_Fa_v model in the prediction/simulation of the breakthrough curve of a fixed bed adsorber was studied through quantitative evaluation of the estimation error of the breakthrough curves calculated by the K_Fa_v model with those calculated by the phenomenological rigorous adsorption process model. The evaluation was carried out on the basis of dimensionless equations of each model as a function of the Biot number and the parameters of adsorption isotherms used in the phenomenological rigorous adsorption process models. Influence of the non-linearity of the adsorption isotherm and the Biot number to the estimation error of breakthrough curves obtained by the K_Fa_v model was clarified quantitatively. The application quality and/or degree of reliability of the K_Fa_v model were clarified systematically.

Introduction

To design a fixed bed adsorber, the transfer parameters of the target adsorbate are required, and considerable effort is undertaken to obtain them experimentally in advance. When the transfer parameters are obtained by the packed bed column method, numerical model calculation is also needed. In the case of the phenomenological rigorous model, the calculation itself is particularly time-consuming because of its mathematical complexity and the presence of many parameters of transport property describing the system. Typically, for practical industrial design of the adsorber, a simple and reliable parameter and a prediction method for the same are preferable (Fujiki et al., 2010; Furuya et al., 1996; Kawakita et al., 2013; Murota et al., 2013). Constant pattern concentration distribution and/or linear driving force (LDF) approximation for intraparticle transfer of the adsorbate have been frequently used to reduce the mathematical complexities and the amount of computation (Hashimoto and Miura 1976; Miura and Hashimoto 1977). Numerous studies have been concerned with the models for isothermal single component adsorption with film and/or intraparticle transfer that describes adsorption batch and column processes rigorously from a phenomenological perspective. The dynamic property of these models has been reviewed in the literature through various intercomparisons between them (Do, 1998; Froment and Bischoff, 1990; Guiochon et al., 1994; Hashimoto and Miura 1976; Jefferson, 1972; Miura and Hashimoto 1977; Ruthven, 1984; Suzuki, 1990; Xu et al., 2013). Ruthven (1984) performed the intercomparison of the BTCs between the LDF approximation model and the rigorous intraparticle transfer models as a function of nonlinearity of the Langmuir isotherm. Increase in nonlinearity of the isotherm caused increase in the matching gap between the BTCs of the models. Jefferson reported applicability of single parameter models in the prediction of BTCs in packed beds (Jefferson, 1972). Xu et al. (2013) reported the importance of simultaneous consideration of convenience and accuracy of the model because of the inhalant shortage included in the simplified model. In recent years, Shafeeyan et al. (2015a, 2015b) studied model applicability for BTCs in CO2 adsorption system based on average fitting error and nonlinear regression analysis using semi-empirical model to investigate reliability of kinetic model in the BTC estimation. Thus, quantitative evaluation of the application quality of a simplified model such as the LDF model should be performed in detail to sufficiently understand the applicability/reliability of the simplified adsorption model in order to optimally design the adsorber, that is, actually a voice from industrial sites. Actually, degree of application quality of the simplified model is hard to know in detail without its quantitative evaluation over a wide range of process conditions. While a systematic map of the precision in prediction of the simple model would be convenient and useful for engineering, few published articles provide them.

In the present study, quantitative applicability of the LDF model in the prediction of the BTC of the fixed bed adsorption process was examined through a comparison of the BTCs obtained by the phenomenological rigorous model describing the adsorption process with those obtained by the LDF model. The phenomenological rigorous model considers film and intraparticle transfer processes of the adsorbate. Surface or pore diffusion for the intraparticle transfer was considered in this study. The LDF model approximates the film and the intraparticle transfer processes in the rigorous models by using the LDF with overall mass transfer coefficient K_Fa_v (hereafter, K_Fa_v model). A dimensionless form of the fundamental equations describing the fixed bed adsorption process was used to minimize both the parameters used in the fundamental equations and a large amount of calculation (Fujiki et al., 2010; Furuya et al., 1996; Kawakita et al., 2013). However, this fails to maintain the equivalency of the dimensionless time and position variables among the models, causing difficulty in the comparison of BTCs on a common time axis. Thus, first, conversion equations that bridge the dimensionless position and time variables among the models were derived to enable the graphical comparison of the BTCs of each model on a common coordinate. The application quality of the K_Fa_v model in the BTC prediction was examined quantitatively and systematically on the basis of fitting (estimation) error analysis of the BTCs between the K_Fa_v model and the phenomenological rigorous process models as a function of Biot (Bi) number and adsorption parameters of each model.

Theoretical considerations

Model equations

The adsorption process in a linear fixed bed adsorber was examined. Graphical and phenomenological representation of the adsorption process models and the k_Fa_v model under study are shown in Figure 1 (Crittenden, 1987). The model assumes a homogeneous packed bed column with porous spherical adsorbents. Only diffusion along flow direction of fluid is considered in the fluid phase. Surface diffusion with Freundlich-type adsorption isotherm and pore diffusion with Langmuir-type adsorption isotherm were employed in this study because these are the major combinations observed in various practical adsorbents. The surface diffusion model considers intraparticle transport of adsorbate via solid phase, and the driving force of the transfer is described by the adsorption gradient. The pore diffusion model considers the transport of the adsorbate via pore spaces in the adsorbent, whose driving force is described by the concentration gradient in the pores. The rigorous models also consider film transport process at the solid–fluid boundary. Film diffusion at solid/fluid interface and particle diffusion are considered in the phenomenological rigorous model. Constant mass transfer and particle diffusion coefficients are assumed. The K_Fa_v model expresses the adsorption process between fluid phase and adsorbent particle by LDF. The fundamental model equations are listed in Table 1. Mass balance in the adsorber is shown in equation (1), where q and c denote concentrations of adsorbate in the solid phase (adsorbent) and bulk fluid phase, respectively. ρ_B is the packed bed density defined by (1-ɛ_B)ρ_S, where ɛ_B is porosity of the packed bed and ρ_S is density of the solid phase (adsorbent), respectively. The film transport is shown in equation (2), where k_Fa_v and c_s denote the film mass transfer coefficient and the adsorbate concentration at the surface of adsorbent, respectively. The intraparticle mass balance of the surface diffusion and pore diffusion models are shown in equations (3b) and (3c), respectively. D_s and D_p, respectively represent the surface diffusion coefficient and the pore diffusion coefficient. Boundary conditions at the solid−fluid interface are shown in equations (4b) and (4c). Equations (5b) and (5c) show the adsorption isotherm, where q_∞ and c* denote the maximum adsorption capacity and the equilibrium concentration of the adsorbate, respectively. The LDF model used in this study is shown in equation (2a). The adsorption processes in both the film and the solid phase are simplified by the first order mass transfer rate equation in the LDF model by using the overall mass transfer coefficient (K_Fa _v ) with the concentration difference between fluid (bulk) phase c and c* as the driving force of transfer. c* is an equilibrium concentration with q (equation (2a)) as mentioned above. This model is referred to as the linear driving force model based on the concentration difference (LDFC) (Suzuki 1990). It is obvious that the LDFC model is suitable in the system in which fluid-to-solid mass transfer is the dominant resistance of the mass transfer. Dimensionless forms of the equations in Table 1 are listed in Table 2 along with the list of dimensionless parameters employed in the translation. As can be understood from the dimensionless equations of mass balance (5A) to (5C), comparison of the BTCs in the dimensionless form is not simple due to the difference of time (T) and position (Z) variables among the models in the dimensionless equations. The dimensionless adsorption isotherms are shown in equations (5B) and (5C), where n and γ denote the adsorption exponent and the separation factor, respectively. In addition, these parameters indicate the degree of non-linearity of the isotherms. C* denotes the concentration at the adsorption equilibrium. In the K_Fa_v model, equation (2a) is shown by equation (2A) in dimensionless form. Figure 2(a) and (b) show the dimensionless Freundlich and Langmuir isotherms for the series of n and γ, respectively. The profile of each isotherm changes from linear (n = 1, γ = 1) to rectangle nonlinear ones with increasing n and decreasing γ, respectively, as shown in Figure 2(a) and (b).

Figure 1.

Schematic diagram of the adsorption models.

Table 1.

Fundamental model equations.

Model	(a) KFav model		(b) Surface diffusion model  + Freundlich type isotherm	(c) Pore diffusion model  + Langmuir type isotherm
Mass balance in column			$$u(\frac{\partial c}{\partial z})+\varepsilon B(\frac{\partial c}{\partial t})+\rho B(\frac{\partial q}{\partial t})=0$$ (1)
Film transport	$$\rho B(\frac{\partial \overline{q}}{\partial t})=KFaV(c-c^{*})$$ (2a)		$$\rho B(\frac{\partial \overline{q}}{\partial t})=kFaV(c-cS)$$ (2)
Intraparticle transport			$$(\frac{\partial qm}{\partial t})=(\frac{DS}{r^2})\frac{\partial }{\partial r}(r^2\frac{\partial qm}{\partial r})$$ (3b)	$$\rho S(\frac{\partial qm}{\partial t})=(\frac{Dp}{r^2})\frac{\partial }{\partial r}(r^2\frac{\partial cm}{\partial r})$$ (3c)
Solid–fluid interface			$$(\frac{\partial qm}{\partial t})r=rp=-DSaP(\frac{\partial qm}{\partial r})r=rp$$ (4b)	$$\rho S(\frac{\partial qm}{\partial t})r=rP=-DPaP(\frac{\partial cm}{\partial r})r=rP$$ (4c)
Adsorption isotherm	$$q=k(c^{*}{)}^{1/n}$$ (5b)	$$q=\frac{q\infty {\mathrm{Kc}}^{*}}{1+{\mathrm{Kc}}^{*}}$$ (5c)	$$q=k(c^{*}{)}^{1/n}$$ (5b)	$$q=\frac{q\infty {\mathrm{Kc}}^{*}}{1+{\mathrm{Kc}}^{*}}$$ (5c)

Figure 2.

Dimensionless adsorption isotherms: (a) Freundlich type and (b) Langmuir type.

Table 2.

Dimensionless model equations.

Model	(A) KFav model (single parameter model)		(B) Surface diffusion model  + Freundlich type isotherm	(C) Pore diffusion model  + Langmuir type isotherm
Mass balance in column	$$(\frac{\partial C}{\partial Z1})+\frac{\varepsilon B}{\beta 0\rho B}(\frac{\partial C}{\partial T1})+(\frac{\partial Q}{\partial T1})=0$$ (1A)		$(\frac{\partial C}{\partial Z2})+\frac{\varepsilon B}{\beta 0\rho B}(\frac{\partial C}{\partial T2})+(\frac{\partial Q}{\partial T2})=0$ (1B)	$$(\frac{\partial C}{\partial Z3})+\frac{\varepsilon B}{\beta 0\rho B}(\frac{\partial C}{\partial T3})+(\frac{\partial Q}{\partial T3})=0$$ (1C)
Film transport	$(\frac{\partial \overline{Q}}{\partial T1})=(C-C^{*})$ (2A)		$$(\frac{\partial \overline{Q}}{\partial T2})=3\mathrm{Bi}B(C-CS)$$ (2B)	$(\frac{\partial \overline{Q}}{\partial T3})=3\mathrm{Bi}C(C-CS)$ (2C)
Intraparticle transport			$$(\frac{\partial Qm}{\partial T2})=(\frac{1}{R^2})\frac{\partial }{\partial R}(R^2\frac{\partial Qm}{\partial R})$$ (3B)	$$(\frac{\partial Qm}{\partial T3})=(\frac{1}{R^2})\frac{\partial }{\partial R}(R^2\frac{\partial Cm}{\partial R})$$ (3C)
Solid–fluid interface			$(\frac{\partial \overline{Q}}{\partial T2})=-3(\frac{\partial Qm}{\partial R})R=1$ (4B)	$$(\frac{\partial \overline{Q}}{\partial T3})=-3(\frac{\partial Cm}{\partial R})R=1$$ (4C)
Adsorption isotherm	$$Q=(C^{*}{)}^{1/n}$$ (5B)	$$Q=\frac{C^{*}}{\gamma +(1-\gamma )C^{*}}$$ (5C)	$$Q=(C^{*}{)}^{1/n}$$ (5B)	$$Q=\frac{C^{*}}{\gamma +(1-\gamma )C^{*}}$$ (5C)
Dimensionless parameters	$$T1=(\frac{KFaV}{\beta 0\rho B})t$$ $$Z1=(\frac{KFaV}{u})z$$		$$T2=(\frac{Ds}{r_P^2})t$$ $$Z2=(\frac{\beta 0\rho BDs}{{\mathrm{ur}}_P^2})z$$ $$\mathrm{Bi}B=\frac{kFrP}{DS\beta 0\rho S}$$	$$T3=(\frac{Dp}{\beta 0\rho S{r}_P^2})t$$ $$Z3=(\frac{1}{u})(\frac{DP}{r_P^2})(1-\varepsilon B)z$$ $$\mathrm{Bi}C=\frac{kFrP}{DP}$$

The set of dimensionless equations in Table 2 was solved numerically using the finite difference method. The calculation results were compared on a common time axis to evaluate the application quality of the K_Fa_v model after converting the different dimensionless variables of T and Z in the model according to the procedure shown in the following subsection.

Conversion equations for the dimensionless variables

The definition of dimensionless time T and position Z are different among the models (Table 2). Conversion equations that correlate the dimensionless variables were derived in this study to enable the comparison of BTCs on a common dimensionless coordinate. The relationship between Bi number and ζ was derived at first by using Glueckauf’s equations (6) and (7), which show the equivalence relationship between the intraparticle diffusion coefficient (D_s or D_p) and the intraparticle mass transfer coefficient k_s (Glueckauf 1955). a_v and r_p denote the surface area and the radius of adsorbent, respectively. β shows the slope of the adsorption isotherm. Here, the ζ is defined by the ratio of mass transfer coefficients in the film and the solid phase as shown in equation (8). By substituting k_sa_v of equation (6) or equation (7) into equation (8), ζ can be given as a function of Bi number (i.e., the ratio between the mass transfer coefficient of film k_F and the diffusion coefficient in particle D_s, D_p), as shown in equations (9) and (10), by using a_v = 3ρ_B/r_pρ_S. Bi_B and Bi_C are defined in Table 2. Equation (9) was derived under the assumption of β = β₀.

$$DS=(\frac{kSaV{r}_P^2}{15\rho B})$$

(6)

$$DP=(\frac{\beta kSaV{r}_P^2}{15(1-\varepsilon B)})$$

(7)

$$\zeta =(\frac{kFaV}{\beta kSaV})$$

(8)

$$\zeta =(\frac{kFrP}{5DS\rho S\beta 0})=(\frac{\mathrm{Bi}B}{5})$$

(9)

$$\zeta =(\frac{1}{5})(\frac{kFrP}{DP})=(\frac{\mathrm{Bi}C}{5})$$

(10)

Then, the dimensionless time T of each model was correlated with the other by using ζ and the relation in mass transfer resistances shown by equation (11) (Suzuki and Kawazoe 1975). Substituting equation (11) into equation (8), the relationship between K_Fa_v and ζ was derived under assumption of b = 1 as shown in equation (12). The conversion equation between T₁ and T₂ was derived using ζ as in equation (13). The conversion equation between T₁ and T₃ was also derived by the same manner as shown in equation (14). Using equation (13) and equation (14), T₂ and T₃ can be expressed by T₁ as a function of Bi number, which enables the comparison between the BTCs of the rigorous and the K_Fa_v models on a common time (T₁) axis. The conversion equations for Z were also obtained on the basis of manner shown in equations (15) and (16).

$$\frac{b}{KFaV}=\frac{1}{kFaV}+\frac{1}{\beta kSaV}$$

(11)

$$\frac{\beta kSaV}{KFaV}=\frac{1+\zeta }{\zeta }$$

(12)

$$T1=(\frac{15\zeta }{1+\zeta })T2$$

(13)

$$T1=(\frac{15\zeta }{1+\zeta })T3$$

(14)

$$Z1=(\frac{15\zeta }{1+\zeta })Z2$$

(15)

$$Z1=(\frac{15\zeta }{1+\zeta })Z3$$

(16)

Evaluation of applicability of the K_Fa_v model

The breakthrough behavior of each model was characterized by the dimensionless time ratio (T − T_0.5)_0.8/(T − T_0.5)_0.2. Here, T_0.5 denotes the dimensionless time at half breakthrough, and (T − T_0.5)_0.2 and (T − T_0.5)_0.8 denote the time at C = 0.2 (C_0.2) and C = 0.8 (C_0.8) in BTC, respectively, as shown in Figure 3(a). The dimensionless time ratio (T − T_0.5)_0.8/(T − T_0.5)_0.2 indicates the slope and sharpnessof BTC. The transfer property in the rigorous models can be characterized by the Bi number. Figure 3(b) indicates the relationship between the Bi number and the dimensionless time ratio (T − T_0.5)_0.8/(T − T_0.5)_0.2. Note that (T − T_0.5)_0.8/(T − T_0.5)_0.2 correlates with the Bi number as shown in Figure 3(b). The breakthrough curve tends to show tailing (small (T − T_0.5)_0.8/(T − T_0.5)_0.2 value) in case of small Bi number and leading (large (T − T_0.5)_0.8/(T − T_0.5)_0.2 value) in case of large Bi number, as can be understood from the trend in Figure 3(b). The slope is moderate in the range of Bi = 0.5 − 5 meaning that the BTC curve changes largely in that range. The Bi number examined in this study was selected on the basis of the trend in Figure 3(b).

Figure 3.

(a) Definition of (T − T_0.5)_0.2 and (T − T_0.5)_0.8 in the dimensionless breakthrough curve and (b) relationship between the (T − T_0.5)_0.2/(T − T_0.5)_0.8 and Bi number.

The application quality of the K_Fa_v model in the prediction of BTCs was evaluated as follows. The BTCs calculated by the phenomenological rigorous models were fitted by the K_Fa_v model as shown in Figure 4 for the series of adsorption parameters and Bi numbers. The application quality of the K_Fa_v model was determined from the sum of fitting error (hatched area shown in Figure 4 defined as the estimation error) by using equation (17), where C_rigo and C_KFav denote concentrations of effluent calculated by the rigorous model and the K_Fa_v model. The conditions that are examined in this study are listed in online Tables S1 and S2. To investigate the influence of nonlinearity of the series of adsorption isotherms to the application quality of the K_Fa_v model based on a common index, deviation D of each isotherm as defined in Figure 5 was introduced in this study as the common index of the isotherms showing nonlinearity. Its value increases with the increase in nonlinearity of adsorption isotherms. In the case of rectangle isotherm, the D value becomes large. The D value becomes small upon reaching the linear-type isotherm. Large n or small γ in each adsorption isotherm results in large D value. The estimation error of each fitting was reviewed as the function of the D value. The Bi number examined in this study was selected according to the relation in Figure 3(b).

$$\mathrm{Estimationerror}={\int }_0^{\infty }\lfloor C\mathrm{rigo}-CKFaV\rfloor \text{d}T$$

(17)

Figure 4.

Comparison of the calculated BTCs and the definition of the estimation error (hatched area).

Figure 5.

Definition of the deviation D in the adsorption isotherm.

Results and discussion

Figure 6(a) and (b) indicate the estimation error of the K_Fa_v model for the BTCs of the surface diffusion with Freundlich-type adsorption isotherm model (Figure 6(a)) and the pore diffusion with Langmuir-type isotherm model (Figure 6(b)), respectively. The application quality of the K_Fa_v model for the rigorous models being studied can be overviewed systematically as the function of the Bi number and the type of adsorption isotherm. In the case of a large Bi number, namely the intraparticle diffusion control condition, the estimation error of the K_Fa_v model becomes large for the n range examined in this study in the surface diffusion with Freundlich isotherm model as shown in Figure 6(a). The estimation error increases monotonically with the increase of n and Bi number when n ≥ 3. When n ≤ 3, minimum estimation errors appears in the vicinity of Bi = 10 condition. When Bi ≥ 25, minimum estimation error appears at around n = 1.5 − 2.0. It is not surprising that the estimation error is small in the case of the linear adsorption isotherm (n→1) with a small Bi number.

Figure 6.

Estimation error of (a) surface diffusion with Freundlich-type adsorption isotherm model and (b) pore diffusion with Langmuir-type adsorption isotherm model.

The estimation error increases monotonically with the increase of separation factor γ and Bi number in the case of pore diffusion with Langmuir isotherm model, as shown in Figure 6(b). The predictivity of the K_Fa_v model is effective to the system with a small Bi number. The reason is that such a system is under film transfer control. Thus, the mathematical description of the rate controlling process is close to the linear form, as is the case with the K_Fa_v model. The estimation error maps examined for a series of Z are shown in online Figures S1 and S2.

The BTCs calculated by the rigorous model and the K_Fa_v model are shown in Figure 7(a) and (b) for some comparative common Bi number and the adsorption parameters. The BTCs of surface diffusion with Freundlich isotherm model show convex shape in the middle region of BTC uptake, as can be seen in Figure 7(a). Leading of BTC appears with smaller Bi numbers and tailing of BTC with larger Bi numbers in the rigorous model (Hashimoto and Miura 1976; Miura and Hashimoto 1977). Further, the BTC sharpens with larger n and smaller γ. The BTCs obtained by the surface diffusion with Freundlich isotherm model show steeper initial uptake and smoother saturation pattern of BTCs than the BTCs simulated by the K_Fa_v model do. The K_Fa_v model hardly simulates the steep initial uptake with gentle tailing and convex pattern of BTC due to the use of average intraparticle concentration and adsorption in its model, as can be seen the large estimation error in the initial and middle regions of BTC uptake at around T₁ = 0.75, 1.0, and 1.3 in Figure 7(a), which is the reasons for the increase in the estimation error of the K_Fa_v model in the prediction of the BTC of surface diffusion with Freundlich isotherm system. The estimation error increases over the whole range of BTC uptake with the increase of n when n ≥ 3, especially at the middle region of the BTC. The estimation error at the initial and middle stage of BTC uptake increases when n is small below n = 3, resulting in an inflection point of the estimation error at around n = 2.

Figure 7.

Estimation error of (a) surface diffusion with Freundlich isotherm system and (b) pore diffusion with Langmuir isotherm system.

In Figure 7(b), the BTCs obtained from pore diffusion with Langmuir type isotherm model are compared with the BTCs obtained from the K_Fa_v model. The derivative pattern of the BTCs simulated by the K_Fa_v model is similar to the BTC of the pore diffusion with Langmuir isotherm model. This is why the estimation error is smaller in the pore diffusion with Langmuir isotherm model than in the surface diffusion with Freundlich isotherm model over a wide range of adsorption parameter conditions. The BTCs calculated by the pore diffusion with Langmuir isotherm model become sharper in case of small γ and large Bi number as can be seen in Figure 7(b). The estimation error increases in the case of a sharp BTC because of the decreased predictivity of the K_Fa_v model for the steep initial and saturating stages of BTC in the rigorous model. The K_Fa_v model predicts a similar pattern of the BTCs, but the BTCs obtained by the pore diffusion with Langmuir isotherm model show more gentle BTCs rather than the K_Fa_v model does. Suzuki (1990) reported that the spreading tendency of the adsorption profile in adsorber column due to the nonequilibrium nature (mass transfer resistance) of adsorption and/or dispersion is balanced with the sharpening tendency due to the nonlinearity (larger n or smaller γ) of the isotherm when a constant pattern profile of adsorption zone is established in the column. The influence of the Bi number and the parameter of adsorption isotherm on the estimation quality of the K_Fa_v model is simply identified by the procedure performed in this study in the dimensionless form.

Figure 8 indicates the estimation errors as a function of the deviation D of each isotherm. The estimation error increases with the increase of deviation D in the surface diffusion with Freundlich isotherm model but the opposite trend is true in the case of pore diffusion with Langmuir-type adsorption isotherm model. In the latter model, the estimation error is small for the γ and Bi number ranges examined in this study. The opposite trend between the two rigorous models is due to the reason stated above. The BTCs simulated by the K_Fa_v model reveal simple sigmoidal curve with a gentle rising edge of breakthrough followed by short tailing, which hardly simulates BTCs with steep rising edge and gentle tailing and thus causes deviation in the fitting of BTCs simulated by the surface diffusion with Freundlich-type adsorption isotherm model at large Bi number condition.

Figure 8.

The estimation error as a function of deviation, D, of each isotherm.

In conclusion, the K_Fa_v model is sufficiently reliable in the system with an adsorption isotherm that is closer to linear isotherm and with the film transfer control system (small Bi number condition). The application quality and the mechanism of the K_Fa_v model was clearly understood in the systematic comparison of the estimation error patterns carried out in this study. The application quality of the K_Fa_v model should be examined extensively for other combinations of diffusion mechanism and adsorption isotherm and for the case of multi-mode diffusion mechanism.

Conclusion

In the present study, the applicability of the K_Fa_v model in the prediction of BTCs of linear column fixed bed adsorber was examined on the basis of a dimensionless model. The conversion equations for the dimensionless time and position variables between the K_Fa_v and the phenomenological rigorous adsorption process models were derived to enable simple and easy evaluation of applicability of the K_Fa_v model on the common dimensionless coordinate. The estimation error of the K_Fa_v model was evaluated as a function of the Bi number and the adsorption parameters in the rigorous models. Influence of type and/or nonlinearity of the adsorption isotherms and the Bi number in the prediction of BTC were qualitatively clarified in detail.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Supplemental Material

The online tables and figures are available at .

Appendix