# Determination of Pore and Surface Diffusivities from Single Decay Curve in CSBR Based on Parallel Diffusion Model

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## Abstract

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## 1. Introduction

_{f}) was determined from the initial slope of the decay curves in the CSTR, since the intraparticle diffusion influence is minimal at this stage. The pore diffusion coefficient (D

_{p}) was independently calculated from the molecular diffusion coefficient obtained from Wilke and Chang’s equation [21] and the tortuosity factor of the adsorbent. Subsequently, they determined the surface diffusion coefficient (D

_{s}) by curve fitting of the decay curve simulated using the determined k

_{f}and D

_{p}values.

_{f}using the Wilson–Geankoplis correlation equation. The D

_{s}and D

_{p}, were determined from batch adsorption experiments. The concentration decay curves obtained by the batch adsorption experiments were fitted and compared with the model predictions using the single pore diffusion, single surface diffusion, and parallel diffusion models. The influence of the diffusion coefficient towards the concentration decay curves in the models was observed. The parallel diffusion model simulated the column separation of levulinic acid efficiently using the kinetic parameters determined from batch adsorption experiments.

_{f}and D

_{s}without calculating the diffusion equations in a batch adsorption system [24]. The k

_{f}was determined by fitting the early-stage concentration decay curve obtained from a batch adsorption experiment with an early-stage kinetic equation. The D

_{s}was then determined by fitting the late-stage decay curve with an approximate equation for the analytical solution of the diffusion equation. The obtained kinetic parameters were applied to other adsorption systems to validate the accuracy of the method.

_{p}was first determined using frontal analysis. BTC prediction was performed using the parallel diffusion model with D

_{p}obtained from frontal analysis. The D

_{s}was calculated using the best fitting obtained from the model predictions to the experimental BTC data.

_{0.2}/T

_{0.8}= T (C/C

_{0}= 0.2)/T (C/C

_{0}= 0.8)) and the diffusivity ratio between pore and surface diffusion (R

_{D}) were successfully correlated based on the parallel diffusion model. The time ratio characterizes the decay curve as a single parameter. This correlation was used to determine the diffusivities from the single concentration decay curve. The effectiveness of the method was confirmed using the adsorption data of a p-nitrophenol/activated carbon system obtained by the CMBR method. The adsorbate/adsorbent system is known to be surface-diffusion-dominant, with large Freundlich n values [28,29].

## 2. Parallel Diffusion Model in Dimensionless Form

^{1/n}, with high k and n values. Assuming that the porosity and packed density of the adsorbents are ε and ρ

_{s}, respectively, the amount of adsorbate in the pores and the solid in a certain volume of the adsorbent V are denoted by εVc and Vρ

_{s}kc

^{1/n}, respectively. The ratio of the amounts of adsorbates in the solid to the pores is expressed by ρ

_{s}kc

^{1/n−1}/ε. Figure 1 shows the ratio as a function of n and c. The value of ρ

_{s}k/ε is not small but more than ten in general; thus, the ratio is very large in the adsorbents with n values greater than unity, indicating the small contribution of pore diffusion transfer.

_{m}and D

_{s}are the amount of adsorption at radius r in the adsorbent and the surface diffusion coefficient, respectively.

_{f}, a

_{p,}c

_{t}and c

_{s}are the film mass transfer coefficient, surface area, concentration in the vessel and fluid concentration at r = r

_{p}, respectively.

_{t}= 0 and c

_{t}= c

_{0.}

_{t}is the amount of adsorbate in the adsorbent.

_{m}= 0 at t = 0 and q

_{m}= f(c

_{t}) at r = r

_{p}.

_{m}= 0 at T = 0.

_{D}denotes the ratio of diffusion resistance owing to surface diffusion and pore diffusion, and V/mβ is the dimensionless fluid-to-solid ratio.

## 3. Results and Discussion

#### 3.1. R_{D} Dependence of Adsorption Profiles and Decay Curves

_{m}) and the solid (Q

_{m}) differ significantly depending on the R

_{D}value. As seen in these figures, the adsorption profiles changed significantly from a gentle to a sharp distribution with a steep adsorption front depending on the R

_{D}value. This is attributed to the difference in transfer resistance between the pores and the adsorbent surface. In the large R

_{D}system, adsorption profiles with a steep adsorption front appear because of the large surface diffusion resistance, as shown in Figure 2b,d. In contrast, the small R

_{D}system results in an adsorption profile penetrating deep inside the adsorbent because of the faster surface diffusion.

_{D}= 0.01 (n = 3, 5) and 10 (n = 3, 5). Faster transport through pore diffusion in the case of a large R

_{D}results in a rapid concentration decay. A smaller R

_{D}system results in a slow concentration decay with long tailing. The total adsorbate flux in the adsorbent decreases with an increase in the n value due to the increase in adsorption and slow surface diffusion, and thus the decay becomes slow with the increase in the n value.

#### 3.2. Determination of the Time Ratio T_{0.2}/T_{0.8} from Concentration Decay Curves

_{t}vs. T graph was plotted, as shown in Figure 4a. Then, the vertical axis of the concentration decay curve was transformed into a new normalized variable, (C

_{t}– C

_{e})/(C

_{0}– C

_{e}). Irrespective of the experimental conditions, the values of the numerical concentration decay curve (NCDC) can be expressed between zero and unity by transforming the data (Figure 4b). The values of T

_{0.8}and T

_{0.2}are defined by reading the decay curve at (C

_{t}– C

_{e})/(C

_{0}– C

_{e}) = 0.8 and 0.2, respectively, as shown in Figure 4b.

#### 3.3. Relationship between T_{0.2}/T_{0.8} and R_{D}

_{0.8}and T

_{0.2}, which were used as an index of the concentration variation range of the NCDC, the T

_{0.2}/T

_{0.8}value was determined. Thus, concentration decay curves were characterized using the dimensionless time ratio T

_{0.2}/T

_{0.8}. The NCDCs under various R

_{D}values were calculated for the series of n and c

_{0}, and the relationship curve between T

_{0.2}/T

_{0.8}and R

_{D}values was obtained (Figure 5). The R

_{D}value was obtained using the relationship shown in Figure 5 by comparing the numerical calculation results with the T

_{0.2}/T

_{0.8}value determined experimentally by the CMBR method. This relationship also means that the value of T

_{0.2}/T

_{0.8}depends on the diffusion coefficient in the phase with a larger adsorbate distribution. It can be seen from Figure 5 that the smaller the value of C

_{e}, the more accurate the diffusion coefficient. For a more accurate determination of the pore and surface diffusivities, the R

_{D}= D

_{p}/D

_{s}βρ

_{s}values should be between 0.1 and 10. The procedure for using the experimental values is described as follows.

#### 3.4. Determination Procedure with Experimental Data

_{0.2}/T

_{0.8}

_{0}= 1447 mg/L, n = 6.2, V = 1 L, m = 1 g, mβ/V = 1.0) under negligible film mass transfer resistance conditions, which are reproducible and reliable from an experimental point of view, were used in this study. The experimentally obtained concentration decay curve is shown in Figure 6a, and the dimensional equilibrium concentration c

_{e}was determined from the decay curve. The dimensionless equilibrium concentration in the CMBR vessel was obtained as c

_{e}/c

_{0}= 0.188 for the system under study. Figure 6b shows a plot of (C

_{t}– C

_{e})/(C

_{0}– C

_{e}) as the vertical axis. The values of T

_{0.8}and T

_{0.2}are given in Figure 6b, and it was found to be 41.1.

_{0.2}/T

_{0.8}values based on the NCDCs. The rate-limiting process in the system under consideration can be calculated from the T

_{0.2}/T

_{0.8}value (=41.1) obtained from the experimental NCDC (dotted line in Figure 7). As seen in Figure 7, the mass transfer resistance in the fluid-to-solid film is negligible because of the very large Biot number condition of the adsorption experiment at T

_{0.2}/T

_{0.8}= 41.1. This step can be skipped when the film transfer is not obviously rate-limiting in the adsorption process. Nonetheless, this step is effective to ensure that the decay curve (adsorption data) is obtained under appropriate conditions for estimating the intraparticle diffusion coefficient, and the estimated diffusion coefficient is reliable.

_{D}

_{D}(=D

_{p}/D

_{s}βρ

_{s}) using the parallel diffusion model. The T

_{0.2}/T

_{0.8}value was determined for each NCDC to obtain the relationship between R

_{D}and T

_{0.2}/T

_{0.8}. The relationship between R

_{D}and T

_{0.2}/T

_{0.8}is shown in Figure 8, along with the T

_{0.2}/T

_{0.8}value determined from the experiment. The value of R

_{D}in the experimental system was found to be 1.75.

_{s}and D

_{p}

_{D}= D

_{p}/D

_{s}βρ

_{s}= 1.75. The obtained NCDC values were compared with the experimental data, as shown in Figure 9. Then, the time ratio (T/t) of 3.9 × 10

^{−4}min

^{−1}was obtained (Figure 9). The value of D

_{s}was estimated using Equation (14), based on the definition of the dimensionless variable T. Constant 60 is included in the equation for the conversion of time units from minutes to seconds.

_{p}can be obtained from the R

_{D}and D

_{s}values, as shown in Equation (15).

_{0}/c

_{0}= kc

_{0}

^{1/n}/c

_{0}[L/g], and ρ

_{s}[g/L] denotes the apparent solid density.

_{p}) to molecular diffusivity (D

_{AB}) is independent of the adsorption system and depends on the adsorbate used [1]. Wilke and Chan’s equation [31] was used to estimate the molecular diffusivity. For p-nitrophenol at 293.2 K, the value was 6.85 × 10

^{−6}, and the value of D

_{p}/D

_{AB}was calculated to be 0.204. This value is similar to that of 0.22 reported by Furuya [8]. The efficiency of this method for determining the kinetic parameters in the parallel diffusion model was validated.

#### 3.5. Summary of the Determination Procedure

_{D}vs. T

_{0.2}/T

_{0.8}and Biot number vs. T

_{0.2}/T

_{0.8}, as shown in Figure 5 and Figure 7, respectively, can be prepared in advance for the conditions used in practice. The kinetic parameters examined in this study can be determined simply using graphs with a few batch adsorption experiments.

## 4. Conclusions

_{D}value is less than 0.1, and the pore diffusion controlling model should be applied when the R

_{D}value is greater than 10.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

a_{p}: | Surface area based on solid particle | [cm^{2}/cm^{3}] |

B_{i}: | Biot number = k_{f}r_{p}/(D_{s}βρ_{s}) | [-] |

c_{e}: | Equilibrium concentration at time = infinity | [mg/L] |

c_{s}: | Fluid concentration at r = r_{p} | [mg/L] |

c_{t}: | Concentration within the vessel at time t | [mg/L] |

c_{0}: | Concentration within the vessel at time = 0 | [mg/L] |

C_{m}: | Dimensionless concentration | [-] |

D_{p}: | Pore diffusivity | [cm^{2}/s] |

D_{s}: | Surface diffusivity | [cm^{2}/s] |

k_{f}: | Fluid film mass transfer coefficient | [cm/s] |

m: | Weight of adsorbent | [g] |

1/n: | Freundlich exponent | [-] |

q_{e}: | Amount adsorbed in equilibrium with c_{e} | [mg/g] |

q_{t}: | Average amount adsorbed within the adsorbent at time t | [mg/g] |

q_{0}: | Amount adsorbed at equilibrium with fluid concentration c_{0} | [mg/g] |

Q_{m}: | Dimensionless amount of adsorption | [-] |

r: | Internal radial length (length from the solid center) | [cm] |

r_{p}: | Particle radius | [cm] |

R_{D}: | Ratio of diffusion resistance = D_{p}/(D_{s}βρ_{s}) | [-] |

t: | Time | [min] |

V: | Volume of the vessel | [L] |

β: | q_{0}/c_{0} | [L/g] |

ε: | Porosity | [-] |

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**Figure 2.**Distribution of adsorbate in the adsorbent particles. (

**a**) C

_{m}, n = 3, (

**b**) Q

_{m}, n = 3, (

**c**) C

_{m}, n = 5, (

**d**) Q

_{m}, n = 5. The bold lines indicate the distribution in the case of R

_{D}= 0.01 at T = 4 × 10

^{−3}, 4 × 10

^{−2}and 1 × 10

^{−1}, respectively. The dotted lines indicate the distribution in the case of R

_{D}= 10 at T = 4 × 10

^{−4}, 4 × 10

^{−3}and 1 × 10

^{−2}.

**Figure 4.**Typical numerical concentration decay curve. (

**a**) Dimensionless concentration decay curve simulated based on parallel diffusion model, (

**b**) reduced dimensionless concentration decay curve.

**Figure 6.**Experimental concentration decay curve. (

**a**) Dimensional decay curve, (

**b**) dimensionless decay curve.

**Figure 9.**Comparison of the concentration decay curves obtained by the experiment and the numerical calculation of the theoretical model.

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**MDPI and ACS Style**

Seida, Y.; Sonetaka, N.; Noll, K.E.; Furuya, E. Determination of Pore and Surface Diffusivities from Single Decay Curve in CSBR Based on Parallel Diffusion Model. *Water* **2022**, *14*, 3629.
https://doi.org/10.3390/w14223629

**AMA Style**

Seida Y, Sonetaka N, Noll KE, Furuya E. Determination of Pore and Surface Diffusivities from Single Decay Curve in CSBR Based on Parallel Diffusion Model. *Water*. 2022; 14(22):3629.
https://doi.org/10.3390/w14223629

**Chicago/Turabian Style**

Seida, Yoshimi, Noriyoshi Sonetaka, Kenneth E. Noll, and Eiji Furuya. 2022. "Determination of Pore and Surface Diffusivities from Single Decay Curve in CSBR Based on Parallel Diffusion Model" *Water* 14, no. 22: 3629.
https://doi.org/10.3390/w14223629