A modified homogeneous surface diffusion model for the fixed-bed adsorption of 4,6-DMDBT on Ag–CeO_x/TiO₂–SiO₂

Abstract

The adsorption of 4,6-dimethyldibenzothiophene (4,6-DMDBT) from diesel fuel using a fixed bed column was investigated. A modified homogeneous surface diffusion model (HSDM) incorporating a new correlation to describe the “practical” equilibrium adsorption between the moving phase and solid phase was proposed to describe the adsorption kinetics of 4,6-DMDBT adsorption at different operation conditions of flow rate, bed length, adsorbent particle size and influent concentration. A sensitivity analysis indicates that the breakthrough curve is obviously sensitive to the change of the Freundlich constant K values, while film transfer coefficient k_f and surface diffusion coefficient D_s have little effect on the breakthrough curve. The value of the surface diffusion coefficient is 1.76 × 10⁻¹² m² s⁻¹. The axial dispersion coefficient and external mass transfer coefficient are evaluated using empirical correlation. The model equations are solved by the method of lines (MOL). The results prove that the modified HSDM model can predict the fixed bed breakthrough curves well.

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1. Introduction

Sulfur compounds in diesel fuel are considered as priority pollutants because they will be transformed to sulfur oxides (SO_x) and then do harm to environment and human health. As a result, appropriate desulfurization technologies need to be applied for the effective control of sulfur compounds in diesel fuel. There are many different methods such as hydrodesulfurization (HDS),^1,2 extraction,^3,4 oxidation,^5,6 and adsorption. Among them, adsorption desulfurization is one of the most common methods for the desulfurization.^7–12 Most of the recent works in adsorptive desulfurization studies were focused on the preparation of adsorbents as well as their modifications.^13–15 However, very few attempts have been made to predict the breakthrough curve using mathematical model, namely the homogeneous surface diffusion model (HSDM), taking into account film diffusion and surface diffusion.

The HSDM model has been popularly used to describe the breakthrough behaviour at different systems.^16–20 There are some model parameters, such as axial dispersion coefficient D_L, surface diffusion coefficient D_s, the external mass transfer coefficient k_f and equilibrium isotherm constant, which play decisive roles in predicting breakthrough curve. Most literature focused on the estimation of these model parameters, especially surface diffusion coefficient and the external mass transfer coefficient.^19,21–26 The axial dispersion coefficient is always obtained by empirical correlation.^27,28

However, few studies have paid attention to the prediction of the equilibrium correlation that connects the moving phase and solid phase, which is throughout the film diffusion from the moving phase to the stationary phase. It is considered that the liquid concentration at surface is in equilibrium with the concentration at solid phase. In the HSDM model, it is well known that the correlation to connect the moving phase and solid phase is one of the key factors that affect the prediction of the column's behaviour. In several fixed-bed mathematical models, the isotherm obtained by batch equilibrium experiment can not describe correctly the equilibrium situation of fixed-bed column. Brauch and Schlünder²⁹ found that the difference between theoretical and experimental data at high concentration can be due to the unfulfilled irreversible equilibrium assumption. Danny et al.^30,31 found that the “best-fit” equilibrium isotherm gained by batch equilibrium experiment was unsuitable for predicting the breakthrough curve using diffusional mass transport model and developed a new correlation to predict the breakthrough curve. Since few literatures have reported about the study of the equilibrium isotherm correlation on adsorption desulfurization, more detailed studies are demanded in this field.

The purpose of this paper is to investigate the breakthrough curve using a modified homogeneous surface diffusion model (HSDM) incorporating a new correlation to describe the “practical” equilibrium adsorption between the moving phase and solid phase for 4,6-DMDBT adsorption and tested on the single component adsorption of 4,6-DMDBT on Ag–CeO_x/TiO₂–SiO₂ adsorbents in fixed bed system. The axial dispersion coefficient and the external mass transfer coefficient are evaluated using empirical correlation. The surface diffusion coefficient and optimal Freundlich constant are evaluated by minimizing the sum of square of errors (SSEs) between the modeling output data and experimental points. A sensitivity analysis is used to discuss the effect of four mass-transfer parameters of model – the surface diffusion coefficient D_s, the external mass transfer coefficient k_f, the axial dispersion coefficient D_L and the Freundlich constant K-on the process adsorption behaviour. Modifications to the model have been made to describe the breakthrough curve under different operation conditions, namely, influent concentration, flowrate, adsorbent particle size and bed depth.

2. Theoretical background

2.1. Mathematical model

The HSDM model has been successfully applied to describe the breakthrough curve of different systems. Shao et al.¹⁶ studied the adsorption kinetics of p-nitrophenol in the structured column using HSDM model and confirmed that HSDM model can predict the breakthrough curves well at various conditions. Richard et al.¹⁹ found the adsorption breakthrough curves of complex phenolic compounds on active charcoal were well predicted with HSDM model. The HSDM model assumes that the adsorbate species diffuses into a homogeneous adsorbent sphere with constant surface diffusion coefficient D_s according to Fick's diffusion law with local adsorption equilibrium within the adsorbent particle and the main rate-limiting step is controlled by the surface diffusion plus the external film mass transfer.

The adsorption of 4,6-DMDBT molecules from bulk solution to particles can be considered as an irreversible process. The adsorption is a complicated process including following steps:^21,32–35 film diffusion from liquid phase to solid phase surface, intraparticle diffusion (pore diffusion or surface diffusion) into adsorbent and instantaneous adhesion on an internal site. Simultaneously, dispersion and advection play an important role in fixed bed process of 4,6-DMDBT adsorption. The following assumptions are made: (1) the instantaneous adsorption occurs on active site; (2) intraparticle diffusion is dominated by surface diffusion; (3) the adsorbent particles are spherical; (4) surface diffusion coefficient is constant.

The equations of HSDM model and initial and boundary conditions are summarized as follows:

The moving phase mass balance in the fixed bed is:

The particle mass balance can be represented as:

The liquid phase and solid phase concentration is correlated with Freundlich equation:

q(r = R, z, t) = Kc_s(z, t)^1/n (3)

The initial and boundary conditions are:

c(z > 0, t = 0) = 0 (4)

q(r, z, t = 0) = 0 (5)

c(z = 0, t > 0) = c₀ (6)

The external mass transfer coefficient represents the adsorbate mass transfer rate from the fluid phase to the solid phase surface, which is essential in the analysis of the measured values. The k_f is usually evaluated by empirical correlation, which is related to the Reynolds number, Re, the Sherwood, Sh, and the Schmidt number, Sc:³⁶

Sh = 5.4Re^1/3Sc^0.6, (0.04 < Re < 30) (11)

The axial dispersion coefficient^37,38 is obtained from empirical correlation:

2.2. Numerical method

The above equations are solved by method of lines (MOL), which discretizes the partial differential equations (PDEs) to ordinary differential equations (ODEs) by finite differences. The corresponding first-derivative radial group in eqn (2) is resolved by L'Hospital rule to eliminate the singularity. The radial is divided into nr increments, Δr = R/nr, and the jth radial distance inside the pellet is defined as, r_j = (j − 1)Δr. Similarly, we discretize the axial coordinate for bed, Δz = L/nz, z_i = (i − 1)Δz.

The first group on the right hand side of the eqn (2) is converted as follows:

At center of adsorbent:

At interior grid points j = 2, ^…, nr − 1:

At surface of adsorbent:

The second group on the right side of the eqn (2) is converted as follows:

At center of adsorbent:

At surface of adsorbent:

At interior grid points j = 2, ⋯, nr − 1:

Similarly, the resulting system for the axial fixed-bed is:

For the first grid line of fixed-bed:

For the last grid line of fixed-bed:

For the remaining grid lines of fixed-bed:

The above ODEs are solved by Matlab integrator ode15s.

The sum of square of error (SSE) between the modeling output data and experimental points can be represented as:

3. Experiment and materials

3.1. Materials

The model diesel was prepared by dissolving 4,6-dimethyldibenzothiophene (4,6-DMDBT) (98% purity, Sinopharm Chemical Reagent Co.,Ltd, AR) into n-dodecane (98% purity, Tianjin weiyi chemical technology Co., Ltd, AR) solvent. The Ag–CeO_x/TiO₂–SiO₂ was used as the adsorbent for removing the 4,6-DMDBT. This adsorbent was prepared according to our previous report.³⁹

3.2. Experiments

Column adsorption experiments were carried out in a quartz tube column with a height of 45.0 mm and an inner diameter of 5.0 mm. The required amounts of adsorbents were packed into column, and quartz wool was placed at both ends of adsorbents bed to cover both ends. The temperature of the column was maintained at 60 °C controlled by vertical thermostatic oven. A solution of the model diesel was then pumped through the fixed bed by a plunger pump to control the flowrate. The fixed-bed effluent was periodically sampled and analyzed by TS 3000 total sulfur analyzer (Jiangsu Jiangfen Electroanalysis Instrument Limited).

4. Results and discussion

4.1. Model parameters

The evaluated model parameters under different conditions are summarized in Table 1. The axial dispersion coefficient²⁷ and external mass transfer coefficient³⁶ are evaluated using empirical correlation. The Freundlich constant, n, is based on our previous batch equilibrium experiments (in press).⁴⁰ The surface diffusion coefficient D_s and optimal Freundlich constant K are evaluated through a const optimization procedure until the sum of square of error is minimized.

Table 1 The estimated parameters at different operating conditions

c0 (mg L^−1)	F (mL min^−1)	L (m)	dp (cm)	K [(mg g^−1)/(L mg^−1)^1/n]	1/n	kf × 10^6 (m s^−1)	Ds × 10^12 (m^2 s^−1)	Bi	SSE
32	0.01	0.1	0.348	0.13	0.5115	2.937	1.76	7.71	0.0917
32	0.02	0.1	0.348	0.11	0.5115	3.700	1.76	11.13	0.4090
32	0.03	0.1	0.348	0.104	0.5115	4.236	1.76	13.30	0.1576
32	0.04	0.1	0.348	0.097	0.5115	4.662	1.76	15.93	0.4958
32	0.01	0.05	0.348	0.158	0.5115	2.937	1.76	6.19	0.1439
32	0.01	0.15	0.348	0.107	0.5115	2.937	1.76	8.94	0.2447
32	0.01	0.2	0.348	0.1	0.5115	2.937	1.76	9.47	3.3592
15.86	0.01	0.1	0.348	0.17	0.5115	2.937	1.76	4.01	0.2332
27.86	0.01	0.1	0.348	0.137	0.5115	2.937	1.76	6.60	0.1307
48.25	0.01	0.1	0.348	0.12	0.5115	2.937	1.76	9.78	0.0535
32	0.01	0.1	0.550	0.145	0.5115	2.937	1.76	10.97	0.1557
32	0.01	0.1	0.250	0.12	0.5115	2.937	1.76	6.14	0.4163
32	0.01	0.1	0.190	0.115	0.5115	2.937	1.76	4.80	0.3357

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As we can see from Table 1, the Freundlich isotherm constants K are a function of operation conditions, such as influent concentration, flowrate, bed length and particle diameter. The value of n is 1.955, which lies within 1–10, indicating a greater affinity for 4,6-DMDBT by this adsorbent.

Table 1 presents that D_s is not affected by changes in flowrate, influent concentration, bed depth and particle size. Therefore, D_s can be considered as independent of operation conditions, so a unique D_s value of 1.76 × 10⁻¹² m² s⁻¹ can be used for all cases. This conclusion can be supported by the sensitivity analysis results, as subsequently discussed.

We can see from Table 1 that the SSEs are between 0.03 and 4, demonstrating that the HSDM model is fairly consistent with the experimental data. The SSE reflects the error between the experimental and the simulated datas. The optimum values for the parameters of Freundlich constant K and surface diffusion coefficient D_s are obtained by minimizing the SSEs within the range of operating conditions studied.

4.2. Sensitivity analysis

The performance of model parameters-surface diffusion coefficient D_s, external mass transfer coefficient k_f, axial dispersion coefficient D_L and Freundlich constant K – on the column breakthrough curves can be employed by sensitivity analysis to determine the most significant model parameter.

The sensitivity analysis is carried out by varying one parameter and maintaining other parameters constant. The D_L and the k_f are obtained from empirical correlation, and the Freundlich constant is based on batch equilibrium experiments. The parameters are listed in Table 1. As displayed in Fig. 1–4, the breakthrough curve is very sensitive to K and D_L, where K controls the breakthrough time of the column while D_L affects slightly the slope of the breakthrough curve. Conversely, the changes of the k_f and the D_s have not an apparent effect on the curve. The capacity of adsorbent increases with increasing K to adsorb more 4,6-DMDBT, which results in more breakthrough time to occur.^19,41 The sensitivity analysis for D_L in Fig. 4 shows that D_L will affect the shape of breakthrough curves, which increase with the increasing axial dispersion coefficient, and then keep constant. It implies that it is necessary to consider the transport of 4,6-DMDBT via axial dispersion around the column but not the convective transport.

Fig. 1 Sensitivity analysis-effect of Freundlich constant K – on the column breakthrough curves (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

Fig. 2 Sensitivity analysis-effect of surface diffusion coefficient D_s – on the column breakthrough curves (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

Fig. 3 Sensitivity analysis-effect of external mass transfer coefficient k_f – on the column breakthrough curves (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

Fig. 4 Sensitivity analysis-effect of axial dispersion coefficient D_L – on the column breakthrough curves (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

The reason for the insensitivity of the film transfer coefficient is explained by the dimensionless Biot number, which is defined as^42–44

The Biot number represents the contribution of intraparticle diffusion compared to film mass transfer. If Biot number is less than 1, film diffusion predominates the adsorption rate, while the major resistance is surface diffusion compared to external diffusion if the Biot number is large than 1. The Biot numbers in all experiments here are greater than 1, a value which suggests that the film mass transfer can be negligible compared to the surface diffusion. Therefore, it is reasonable to evaluate the film mass transfer coefficient by the empirical correlation.

The second parameter of the model is the surface diffusion coefficient, D_s. Once other parameters are determined, the D_s is calculated through the optimal procedure by minimizing the SSEs value between the experimental and model output data. The breakthrough curve is not affected by the changes in D_s within the range studied. According to the results in Table 1, we can know the estimated value of surface diffusion coefficient (D_s) is 1.76 × 10⁻¹² m² s⁻¹, which is unique and not a function of influent concentration, flowrate, particle size and bed depth. These results indicate that the surface diffusion is predominant rate controlling step.^41,45 Similar results were also obtained by other researchers.^41,46–50

The isotherm is one of the important parameters affecting the prediction of the breakthrough curve. The isotherm model used to describe the breakthrough curve is usually based on the batch experiment. Some investigators found that the isotherm had failed to predict correctly breakthrough curve and solid-phase loading.^{29–31,51,52} This reason may because it takes less time to get to equilibrium in the column experiments compared to that in the batch isotherm experiment. On the other hand, the reason for these dissimilarities may be attributed to the different approaches to get to equilibrium and the potentially irreversibility between batch and column experiment,^29,30 for example, the bulk liquid concentration around the solid phase is continuously increasing in the fixed-bed process, however, the bulk liquid concentration is continuously decreasing in the batch process. According to the results in Table 1, we can know the estimated values of Freundlich constant K are related to flowrate, influent concentration, particle size and bed depth.

Simultaneously, it is very important to know the relationship between parameters and sum of square of errors (SSEs). Fig. 5 presents the vary of the SSE with respect to different parameters values at 32 mg L⁻¹ influent 4,6-DMDBT concentration, 0.01 mL min⁻¹ flowrate, 0.1 m bed length, and mean particle diameter 0.348 mm. As can be seen from Fig. 5, small changes in K value have a more apparent effect compared to other parameters on sum of square of errors. That means the model is more sensitive to K. The surface diffusion coefficient D_s, the external mass transfer coefficient k_f, and axial dispersion coefficient D_L have little effect on the breakthrough curve.

Fig. 5 Sensitivity analysis-effect of k_f, D_s, D_L, K – on the sum of square of errors (SSEs) (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

So, we can conclude that Freundlich constant (K) has the greatest impact on the model predictions. The surface diffusion coefficient (D_s) is unique and not a function of flowrate, influent concentration, particle size, and bed depth. While the Freundlich constant (K) is related to flowrate, influent concentration, particle size, and bed depth. So we proposed a new Freundlich correlation to predict the breakthrough curve under different operation conditions.

4.3. Development of the correlation

Several investigators^31,52 have demonstrated the isotherm can not provide the correct adsorbent capacity and a “practical” adsorbent capacity value obtained by the equilibrium isotherm has been usually used.³¹ So a new Freundlich correlation to predict the breakthrough curve under different operation conditions is necessary. Devi et al.⁵³ reported that the different sizes of silica NPs had different isotherm behaviors. Ali and Mahmud⁵⁴ found that the Langmuir adsorption coefficient Y and K were as a function of salinity by a linear and a quadratic relationship, respectively. Mckay and Al-Duri⁵⁵ used a modified Freundlich isotherm to predict multicomponent dye isotherms, and found that the Freundlich constants varied with different concentration ranges. Similar results were also obtained by other researchers.^56–58

The constant K is a parameter which is related to operation conditions and eqn (29) can be used to evaluate the K value. The equilibrium isotherm correlation varies with different operation conditions, but each factor has its own degree of dependence. As the possible factors affecting the K value will be flowrate F, bed depth L, initial influent concentration c₀ and particle size d_p, the following empirical correlation is proposed to predict the K value.

K = θ × F^α × L^β × c₀^γ × d_p^δ (29)

The constants α, β, γ and δ can be determined by varying one of parameters and keeping the others constant. For example, the constant α can be decided by getting different K values at different flowrates and keeping L, c₀ and d_p fixed, eqn (29) simplifies to:

K = θ₁F^α (30)

where

θ₁ = θ × L^β × c₀^γ × d_p^δ (31)

Taking the logarithm of both sides, eqn (30) can be rewritten as follows:

ln K = α ln F + ln θ₁ (32)

The constant α can be obtained from the slope of a graph of ln K against ln F₁. The other constants β, γ and δ can be obtained by above similar solution. The constant term, θ, can be obtained by eqn (29) once α, β, γ and δ are known, which should be consistent, so a mean value of θ can be used for all cases. The constants α, β, γ and δ are dependent on operation system. Once they are known, the correlation eqn (29) can be applied to HSDM model and to predict the breakthrough curve.

Fig. 6 shows the relationship between the ln K and different variables. The constants α, β, γ and δ can be obtained from the slope of lines. Based on the method discussed above, the constants can be obtained and the empirical correlation is summarized as:

K = 0.087385 × F^−0.2068 × L^−0.3408 × c₀^−0.3153 × d_p^0.2221 (33)

Fig. 6 Relationship between ln K and different variables: (a) ln K versus ln F (L = 0.1 m, c₀ = 32 mg L⁻¹, mean d_p = 0.348 mm); (b) ln K versus ln L (F = 0.01 mL min⁻¹, c₀ = 32 mg L⁻¹, mean d_p = 0.348 mm); (c) ln K versus ln c₀ (F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm); (d) ln K versus ln d_p (F = 0.01 mL min⁻¹, L = 0.1 m, c₀ = 32 mg L⁻¹).

4.4. Effect of flow rate

Fig. 7 demonstrates the effect of flowrate on the breakthrough curve of 4,6-DMDBT adsorption. As we know, the changes in flow rate will have an apparent effect on the external mass transfer but not the intraparticle diffusion. Therefore, there is no change in D_s with the change of the flowrate as shown in Table 1. The k_f varies with the flowrate, which increases with increasing flowrate due to the decreasing film resistance to mass transfer. The higher flow rate the higher Biot number indicating that the adsorption rate limiting step is not external mass transfer compared to the surface diffusion. Biot number represents the contribution of intraparticle diffusion compared to film mass transfer. If the surface diffusion is rate controlling step, the residence time increases with the decreasing flowrate resulting in the higher bed capacity.

Fig. 7 Effect of flowrates on the breakthrough curves of 4,6-DMDBT adsorption (c₀ = 32 mg L⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

4.5. Effect of bed height

Fig. 8 shows the effect of bed height on the breakthrough curves of 4,6-DMDBT adsorption. As clearly shown from Fig. 8, the higher bed height the more breakthrough time compared with the smaller bed height. The higher bed height means more adsorbent usage than for the smaller one. The larger bed height resulting in larger adsorption capacity will improve the bed efficiency.⁵⁹ Simultaneously, the residence time increases with the increasing bed depth resulting in the adsorbate molecules diffusing deeper inside the adsorbent.

Fig. 8 Effect of bed height on the breakthrough curves of 4,6-DMDBT adsorption (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, mean d_p = 0.348 mm).

4.6. Effect of influent concentration

As shown in Table 1, the surface diffusion coefficient and the external mass transfer coefficient are not related to the influent concentration.

Typical breakthrough curves at three different influent concentrations for the adsorption of 4,6-DMDBT on the packed bed are illustrated in Fig. 9. The breakthrough time decreases with the increasing influent concentration, which may because the mass transfer zone moved slower than that at higher concentration.⁶⁰

Fig. 9 Effect of influent concentration on the breakthrough curves of 4,6-DMDBT adsorption (F = 0.01 mL min⁻¹, L = 0.1 m, mean d_p = 0.348 mm).

4.7. Effect of particle diameter

The effect of adsorbent particle size on the breakthrough curve of 4,6-DMDBT adsorption is shown in Fig. 10. The breakthrough curve is insensitive to the change of adsorbent particle diameter. The reason for that may because the change of outer surface area which varies with adsorbent particle size is not apparent compared to the total internal pore surface area.

Fig. 10 Effect of particle diameter on the breakthrough curves of 4,6-DMDBT adsorption (c₀ = 32 mg L⁻¹, F = 0.01 mL min⁻¹, L = 0.1 m).

Above results suggest the simulated breakthrough curves are in agreement with the experimental datas, indicating that the modified HSDM model is able to correctly predict the adsorption kinetics of 4,6-DMDBT in column.

5. Conclusions

The adsorption of 4,6-DMDBT from diesel fuel using fixed bed column at different operation conditions was investigated. A sensitivity analysis indicates that K affects the breakthrough time and the D_L controls slightly the slope of the breakthrough curve. Namely, the breakthrough curve is very sensitive to the changes in the K values, while the k_f and the D_s have little effect on breakthrough curve. A new isotherm correlation to predict the breakthrough curve at different operation conditions is proposed and the predictive correlation is summarized as:

K = 0.087385 × F^−0.2068 × L^−0.3408 × c₀^−0.3153 × d_p^0.2221

The modified HSDM model can predict the breakthrough curve quite well under different operation conditions such as influent concentration, flowrate, adsorbent particle size and bed depth. The breakthrough time increases with the decrease in influent concentration, flow rate, and with the increase in bed height, while adsorbent particle size just has little effect on the breakthrough curve. The value of optimized surface diffusion coefficient D_s is 1.76 × 10⁻¹² m² s⁻¹, which is constant and not as a function of operation conditions.

Nomenclature

Bi	Biot number (dimensionless)
c(z, t)	Liquid phase concentration at axial position z and time t (mg L^−1)
c0	Influent liquid phase concentration (mg L^−1)
cs(z, t)	Liquid phase concentration at adsorbent surface (mg L^−1)
DL	Axial dispersion coefficient (m^2 s^−1)
dp	Adsorbent diameter (mm)
Ds	Surface diffusion coefficient (m^2 s^−1)
F	Flowrate (mL min^−1)
K	Freundlich constant ((mg g^−1) (L mg^−1)^1/n)
kf	External mass transfer coefficient (m s^−1)
L	Bed length (m)
nr	The radial increments
nz	The axial increments
1/n	Heterogeneity factor (dimensionless)
q0	Adsorbed amount at equipment with influent liquid phase concentration (mg g^−1)
q(r, z, t)	Solid phase concentration at radial position r, axial position z and time t (mg g^−1)
r	Radial coordinate (m)
R	Adsorbent radius (m)
Re	Reynolds number (dimensionless)
Sc	Schmidt number (dimensionless)
Sh	Sherwood number (dimensionless)
t	Time (s)
u	Superficial velocity (m s^−1)
v	Interstitial velocity (m s^−1)
z	Axial position z (m)

Greek letters

α	Flowrate index (dimensionless)
β	Bed length index (dimensionless)
γ	Influent concentration index (dimensionless)
δ	Particle diameter index (dimensionless)
θ	Constant (dimensionless)
ε	Porosity of adsorbent (dimensionless)
ρp	Density of adsorbent particle (g mL^−1)

Subscripts

cal	Calculated value
exp	Experimental value
i	The ith axial distance
j	The jth radial distance

Acknowledgements

M. Z. sincerely acknowledges the support of the National Natural Science Foundation of China (No. 21376055).

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