# Simulation of Water Vapor Sorption Profiles on Activated Carbons in the Context of the Nuclear Industry

Felipe Carbral Borges Martins, Mouheb Chebbi, Celine Monsanglant Louvet, Bénoit Marcillaud, Audrey Roynette

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Felipe Carbral Borges Martins, Mouheb Chebbi, Celine Monsanglant Louvet, Bénoit Marcillaud, Audrey Roynette. Simulation of Water Vapor Sorption Profiles on Activated Carbons in the Context of the Nuclear Industry. Separations, 2025, 12 (5), pp.126. 10.3390/separations12050126. hal-05067961

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Article

# Simulation of Water Vapor Sorption Profiles on Activated Carbons in the Context of the Nuclear Industry

F. C. B. Martins, M. Chebbi *D, C. Monsanglant-Louvet, B. Marcillaud and A. Roynette

Autorité de Sûreté Nucléaire et de Radioprotection (ASNR), PSN-RES/SCA/LECEV, F-91400 Saclay, France; felipe.cabralborgesmartins@asnr.fr (F.C.B.M.); celine.monsanglant@asnr.fr (C.M.-L.); benoit.marcillaud@asnr.fr (B.M.); audrey.roynette@asnr.fr (A.R.)

* Correspondence: mouheb.chebbi@asnr.fr

**Abstract:** Activated carbons (ACs) are employed in the nuclear industry to mitigate the emission of potential radioactive iodine species. Their retention performances towards iodine are mainly dependent on the relative humidity due to the competitive effect induced by adsorbed water molecules. Thus, this work will focus on the prediction of AC behavior toward the capture of water vapor to better assess the poisoning effect on radiotoxic iodine removal. For the first time, H₂O breakthrough curves (BTCs) on nuclear grade ACs are predicted through a specific methodology based on the combination of transport phenomena with adsorption kinetics and equilibrium. Three ACs, similar to those deployed in the nuclear context, are considered within the present study. Our model is based on the Linear Driving Force Model (LDF), governed by an intraparticle diffusion mechanism, notably surface and Knudsen diffusions. In addition, the type V isotherms obtained for H₂O and the investigated carbon supports were described through the Klotz equation, taking into account the formation and progressive growth of H₂O clusters within the internal porosity. This methodology allowed us to successfully simulate the H₂O adsorption by a non-impregnated AC, where only physisorption phenomena are involved. In addition, promising results were highlighted when extrapolating to the two other impregnated ACs (AC 5KI and AC Nuclear).

**Keywords:** adsorption; breakthrough curves; equilibrium; Klotz equation; kinetics; LDF model; surface diffusion mechanism; nanopore diffusion

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check for
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Academic Editors: Guojie Zhang and
Mingheng Li

Received: 18 March 2025

Revised: 24 April 2025

Accepted: 12 May 2025

Published: 14 May 2025

**Citation:** Martins, F.C.B.; Chebbi, M.; Monsanglant-Louvet, C.; Marcillaud, B.; Roynette, A. Simulation of Water Vapor Sorption Profiles on Activated Carbons in the Context of the Nuclear Industry. *Separations* **2025**, *12*, 126. https://doi.org/10.3390/separations12050126

**Copyright:** © 2025 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

Some accidents, such as Chernobyl (1986) and Fukushima (2011), have induced disastrous consequences in terms of radioactive releases into the environment. In that respect, the limitation of radioactive substances release in all scenarios is a major issue for nuclear safety and radioprotection. Among the susceptible fission products to be released, the attention should be devoted to radioactive iodine isotopes [1]. The major issue lies both in the ability of iodine to be present under highly volatile species (namely molecular iodine I₂ and methyl iodide CH₃I) [2,3], as well as in high affinity with the human thyroid [4]. Among the radioactive isotopes of iodine, iodine 131 (half-life is 8.02 days [3]) is identified as the principal source of exposure during the first stages of a major nuclear accident.

Iodine traps (ITs) are widely employed within the ventilation networks of nuclear facilities to ensure the containment of these hazardous species. The majority of traps are composed of activated carbon (AC) [5] derived from a coconut shell. These adsorbents display a well-developed microporous structure ($d_{pore} < 2$ nm), enabling them to adsorb CH₃I molecules (kinetic diameter: 0.5–0.6 nm [6]) through physisorption phenomena. The
current nuclear grade AC is commonly co-impregnated with triethylenediamine (TEDA, ~5 wt.%) and potassium iodide (KI, ~1 wt.%) [5]. The impregnation purpose is to ensure a specific and strong trapping of iodine molecules that are present in traces. Therefore, two retention mechanisms exist in addition to the physical adsorption. On the one hand, the TEDA molecules interact with CH3I through a chemisorption mechanism [5]. On the other hand, an isotopic exchange reaction involves the KI reactivity toward radioactive iodine isotopes [7].

Several factors are reported to influence the iodine trapping efficiency. Among them, humidity has the greatest impact, notably for a relative humidity (RH) greater than 40% [5]. Indeed, the adsorbed water molecules show a competitive adsorption with the iodine species by preventing them from arriving to the active sites [8,9]. Thus, to better understand and predict this inhibiting effect, the attention should be given to the interaction between water vapor and nuclear grade AC. By now, most academic research studies have focused on H2O adsorption isotherms. Owing to their hydrophobic character, a type V isotherm is reported for the employed AC in the nuclear industry. A maximal water uptake of (30–40 mg/g) can be highlighted for these adsorbents at a RH of 40% (T = 30 °C) [10,11]. In comparison, more hydrophilic adsorbents such as silica gel composite [12], 13X zeolite [13], and MIL-160 MOF [14] display much higher water vapor adsorption capacities with values of about 200–300 mg/g in the same range of RH. At a higher water content (RH of about 95%), the water uptake is known to be dependent on the total accessible pore volume. In that respect, nuclear ACs with high specific surface areas (~1000 m²/g) exhibit an adsorption capacity of about 200–300 mg/g at a RH of 95%. This trend explains the drastic deterioration of retention efficiency toward iodine trapping in humid conditions [15]. Despite the availability of numerous data on H2O adsorption isotherms for nuclear AC, no experimental or theoretical works have been devoted to breakthrough curves for these adsorbents, to the best of our knowledge. Thus, the extrapolation of their behavior to the industrial scale is delicate. Furthermore, although several studies [16–19] have been performed to simulate the dynamic sorption profiles of pollutants (H2S, COV, etc.) on a wide range of adsorbents, no studies dedicated to activated carbons and H2O have been realized.

The present work aims to bridge this gap by reporting on the prediction of nuclear grade AC dynamic behavior toward water vapor. The employed methodology will be based mainly on H2O adsorption isotherms by gaining insights into both the adsorption equilibrium and kinetics. This model will be crucial to better quantify the poisoning effect induced by water toward the adsorption of radioactive iodine, which is of great interest to nuclear safety and radioprotection. In the following, the methodology for the model establishment will first be detailed. Then, the tested adsorbents and experimental methods will be described. Finally, the results will be presented by comparison between predicted and experimental sorption profiles.

## 2. Methodology for the Establishment of a Model for Breakthrough Curve Prediction

### 2.1. *Mass Balance Equation*

To model breakthrough curves (BTCs), phenomena involving the interactions between a pollutant and a given adsorption column [20] must first be modeled. Thus, the system displayed in Figure 1 was considered. A mass balance equation governing both gaseous and adsorbed species is required.
<figure><img src="image_4.png" /><figcaption>Figure 1. Schematic diagram of the modeled system.</figcaption></figure>

To decrease the complexity of this system, some approximations were undertaken. Firstly, a plug flow was considered. Therefore, there are no radial gradients of concentration, temperature, or velocity [19]. In addition, the pressure drop was disregarded due to the experiment's small scale. For the sake of simplification, the decrease in velocity due to the compound capture was also neglected [21]. Owing to the employed temperature, the sorption of O₂ and N₂ in the porous material can be ignored [22]. Lastly, the system was considered to be isothermal [19,21–23]. With these simplifications, the global transport equation can be written as the following:

$$ -D_{ax} \frac{\partial^2 C}{\partial Z^2} + u_i \frac{\partial C}{\partial Z} + \frac{\partial C}{\partial t} + \left( \frac{1 - \varepsilon_{bed}}{\varepsilon_{bed}} \right) \rho_{bed} \frac{\partial \bar{q}}{\partial t} = 0 \quad (1) $$

For the initial conditions, the porous material is in equilibrium with the residual humidity in the flowing air. In addition, the initial adsorption capacity q is estimated from the adsorption isotherm (see Section 3.2).

$$ t = 0 \rightarrow C = RH_{residual}, \bar{q} = \overline{q_{eq}}(RH_{residual}) \quad (2) $$

The water vapor concentration can be calculated through the ideal gas law:

$$ C = \frac{P_{vap}}{R \times T} \times M_{H_2O} \quad (3) $$

Using the definition of RH ($RH = P_{vap}/P_{sat} \times 100$), the H₂O concentration can be calculated through the following equation:

$$ C = \frac{P_{sat} \times RH}{R \times T \times 100} \times M_{H_2O} \quad (4) $$

Finally, the following boundary conditions were used in agreement with the literature [22–24]:

$$ Z = 0 \rightarrow \frac{\partial C}{\partial Z} = \frac{u_i}{D_{ax}} (C - C_{feed}) \quad (5) $$

$$ Z = L \rightarrow \frac{\partial C}{\partial Z} = 0 \quad (6) $$

Before finding a numerical solution for this model, certain parameters need to be established and estimated. Firstly, the axial dispersion ($D_{ax}$), which characterizes the axial spreading of the concentration profile observed in BTC. This parameter models the non-
ideal velocity profile of the flow [25,26] regarding its deviation from a plug flow. In this work, the correlation proposed by Rastegar et al. [25] was used:

$$D_{ax} = 0.7D_{m} + \frac{\epsilon_{bed} \times d_{p} \times u_{i}}{0.18+0.008 \times Re^{0.59}} \\ \text{for : } 1 < Re \times Sc < 200 \qquad (7)$$

The product of the Re and Sc numbers ranges between 2 and 14, which justifies the ad- equacy of the presented correlation with the employed conditions within the present study.

Then, specific models are needed to describe both the adsorption kinetics and equilib- rium. The focus will be devoted to the adsorption kinetics in the following part. The model retained for the H₂O adsorption isotherms will be presented in Section 4.1.

## 2.2. Adsorption Kinetics

The mass transfer global resistance can generally be supposed as the resultant from two main contributions: (i) external mass transfer and (ii) intraparticle mass transfer. It encompasses the path that the species needs to make across the outer particle layer, through the particle macroporosity, and finally inside its microporosity [27].

A global mass constant can be calculated by considering all the different mechanisms through the Linear Driving Force (LDF) model [28]. More specifically, the following equation adapted from the literature can be used to calculate the global mass transfer constant (K_LDF) [29]:

$$\frac{1}{K_{LDF}} = \frac{d_{p}^{2}}{60\epsilon_{bed}D_{p}}\left(\frac{q_{eq}}{C}\right) \qquad (8)$$

In the case of this study, the external mass transfer was neglected, since the predomi- nant resistances are due to the internal diffusion in microporous AC, in agreement with several literature studies [14,15]. More particularly, the pore diffusion resistance is known to be the governing mechanism for adsorption kinetics within ACs [30,31]. Thus, K_LDF was calculated from the pore diffusivity (D_p), as shown by Equation (8).

It is also notable that the adsorption equilibrium has an impact on the kinetics con- stant. The term q\_eq/C allows taking into account the non-linear character of the adsorption isotherm [32]. Moreover, the adsorbed quantity in equilibrium (q\_eq) is calculated using the adsorption isotherm model. This topic will be aborded in depth in Section 3.2.2.

For intraparticle phenomena, three main intraparticle mechanisms can be distin- guished [18,19]: (i) molecular diffusion in macropores, (ii) Knudsen diffusion within mesopores and micropores, and finally (iii) the surface diffusion in micropores. The pore diffusivity was calculated with Equation (9) [29].

$$\frac{1}{D_p} = \tau \left( \frac{1}{D_m} + \frac{1}{D_K + \left( \frac{1-\epsilon_{bed}}{\epsilon_{bed}} \right) \frac{q_{eq}^0}{C_0} D_S} \right) \qquad (9)$$

The parameters presented in this equation were calculated as the following:

- The tortuosity (τ): given the lack of experimental measurements, a value of 3 was used, corresponding to a good order of magnitude in similar adsorption systems [33].

- The molecular diffusivity (D_m): a value of 2.6 × 10⁻⁵ m² s⁻¹ from a previous work [34] was used.

- The Knudsen diffusivity (D_K), was calculated using Equation (10)).

$$D_K = 48.5 \times d_{pore} \sqrt{T/M} \qquad (10)$$
- The surface diffusivity ($D_S$) was estimated using the Sladek et al. [35] correlation, seen in Equation (11).

$$D_S = 1.6 \times 10^{-6} \times \exp \left( -0.45 \frac{\Delta H_{ads}}{s \times R \times T} \right) \quad (11)$$

For the $D_S$ correlation, a typical value for the adsorption enthalpy of water in AC was chosen ($\Delta H_{ads} = 40$ kJ mol$^{-1}$ [36–38]). The parameter s of Equation (11) is based mainly on the interaction between the adsorbent and adsorbate. A value of s = 2 is recommended by Sladek et al. [35] when dealing with polar adsorbate/carbon interaction. The calculated diffusivities are presented in Section 4.2.

Incorporating the LDF model in Equation (1) results in Equation (12).

$$-D_{ax} \frac{\partial^2 C}{\partial Z^2} + u_i \frac{\partial C}{\partial Z} + \frac{\partial C}{\partial t} + \left( \frac{1 - \varepsilon_{bed}}{\varepsilon_{bed}} \right) \rho_{bed} K_{LDF} (q_{eq} - \bar{q}) = 0 \quad (12)$$

Since the instant concentration plays a role in the value of the kinetic constant, $K_{LDF}$ will be calculated during the model resolution. This parameter is evaluated for each step of the resolution based on the actual local concentration for a given time.

The numerical resolution process will be described in the following part.

## 2.3. Numerical Resolution

In this work, the method of lines [39] has been applied for the discretization and the integration of the transport Equation (1). More details about this method as well as the employed parameters are summarized in the Supplementary Materials (S1).

# 3. Experimental Methods

## 3.1. Tested Adsorbents and Presentation of Their Main Properties

Three commercial AC derived from coconut shells were studied in the presented work:

* a non-impregnated activated carbon denoted as AC nI;

* an AC impregnated by KI (5 wt.%) denoted as AC 5KI;

* a co-impregnated AC with a composition similar to the employed one for a nuclear grade AC: 1 wt.% of KI and 5 wt.% of TEDA. This adsorbent is designated in the following by AC Nuclear.

The textural properties of the tested materials as deduced from our previous investigation [40] are shown in Table 1.

<table><caption>**Table 1.** Characterization by N₂ porosimetry nitrogen at 77K of the investigated adsorbents. Presented uncertainties are related to the standard deviation of the mean expressed at coverage factor k = 2 [40].</caption><thead><tr><th></th><th>Impregnation (%wt)</th><th>S<sub>BET</sub> (m<sup>2</sup> g<sup>-1</sup>)</th><th>V<sub>pore</sub> (cm<sup>3</sup> g<sup>-1</sup>)</th><th>% Microporosity (%)</th></tr></thead><tbody><tr><td>AC nI</td><td>-</td><td>1142 ± 33</td><td>0.472 ± 0.016</td><td>96.0 ± 3.3</td></tr><tr><td>AC 5KI</td><td>5% KI</td><td>1132 ± 34</td><td>0.483 ± 0.017</td><td>94.4 ± 3.1</td></tr><tr><td>AC Nuclear</td><td>1% KI + 5% TEDA</td><td>976 ± 10</td><td>0.415 ± 0.005</td><td>93.5 ± 0.9</td></tr></tbody></table>

The investigated adsorbents are mainly microporous (fraction of micropores higher than 93%, Table 1), with large specific surface areas (>900 m²/g). These results are in line with type I of their N₂ adsorption/desorption isotherms (see Supplementary Materials (S2)). A very negligible contribution due to mesopores can also be noticed for the investigated materials, as shown from the very small hysteresis loop at high P/P₀ values (see Supplementary Materials (S2)). In addition, a certain contribution due to macropores is also presented, as observed by our previously published SEM data [5].
The presence of KI does not have a notable effect on the porosimetric properties of the materials (S<sub>BET</sub> and V<sub>micro</sub>), indicating therefore that KI entities are not located in the micropores. Indeed, our previous TEM and XRD characterizations have indicated the presence of well-dispersed KI nanoparticles on the external surface of AC 5KI [41]. Finally, previous XPS investigations [5] have shown the presence of two iodine forms: (i) a dominant part assigned to ionic I⁻ (such as in KI) and (ii) a less pronounced contribution due to covalent iodine I⁸⁻, resulting from a potential interaction with AC surface defects.

However, the presence of TEDA causes a decrease in the S<sub>BET</sub> and in the pore volume, due to a partial pore blockage by its molecules being present within the internal porosity [40], in agreement with the literature studies [1,42,43]. The nitrogen speciation from previous XPS results [5] has also shown two contributions: (i) a dominant one related to three-coordinated nitrogen N as in TEDA and (ii) a lower part presented by N-Csp2 bonds, as in quaternary ammonium, attributed probably to nitrogen interaction with the carbon support.

Finally, it is worth recalling the absence of any interaction between KI and TEDA within the AC Nuclear, as confirmed by similar iodine and nitrogen speciation between singly impregnated and co-impregnated ACs [5].

## 3.2. Determination of H₂O Adsorption Isotherms

### 3.2.1. Experimental Part

The H₂O adsorption was investigated with a sorption microbalance (DVS Vacuum, photo in S3), using a sample mass of 50 mg with a grain size between 1.0 mm and 1.4 mm. Firstly, an in situ pretreatment is performed to limit the presence of residual humidity within the tested adsorbent. Mild conditions were chosen for the degassing process (60 °C, 15 h, vacuum of 10⁻⁵ torr) to avoid significant modification to the adsorbent surface chemistry, in agreement with recent works of L.F. Velasco et al. [44]. Then, the adsorption phase was conducted at 30 °C using gravimetric measurements from the RH of 0% up to 95%, with increments of 5%. The thermodynamic equilibration for each tested RH was assumed for a maximal duration of 400 min per step or when the following criterion was satisfied:

$$ \frac{\Delta m}{m \times \Delta t} \le 0.0004\% \min^{-1} \qquad (13) $$

### 3.2.2. Data Processing

An appropriate isotherm model needs to be used to describe the interactions between AC and water. In this work, two models were explored:

* Henry model: applied only for a RH < 40%;

$$ q_{eq} = K_H \cdot x \qquad (14) $$

* Klotz equation (Equation (15)): developed in a previous work related to the interaction between proteins and particles [45]. This equation is applied to our study to mathematically consider the sigmoidal nature of the obtained isotherms. This equation was applied in the whole RH range (5–95%).

$$ q_{eq} = \frac{q_m \cdot C_K \cdot K \cdot x \cdot \left\{ 1 - (1 + m_K)(K \cdot x)^{m_K} + m_K(K \cdot x)^{m_K+1} \right\}}{(1 - K \cdot x) \left\{ 1 + (C_K - 1) \cdot K \cdot x - C_K(K \cdot x)^{m_K+1} \right\}} \qquad (15) $$

where

* x molar fraction of the species in the gas (-);

* q<sub>m</sub> H₂O adsorbed quantity in a monolayer (g g⁻¹);
### 3.3.2. Investigated Experimental Conditions

A summary of the tested experimental configurations (operating conditions and tested adsorbents) is reported in Table 2. All tests were performed with a temperature of 30 °C and a RH of 40%, in agreement with the expected conditions in the normal functioning of IT [47]. For the non-impregnated AC, the effect of linear velocity was studied. More particularly, three flow rates were studied: 1.0, 1.5, and 6.5 L min⁻¹ (NTP), resulting in linear velocities of about 4, 6, and 25 cm s⁻¹, respectively.

<table><caption>**Table 2.** Overview of tested operating conditions.</caption><thead><tr><th>Materials</th><th>Gas Flow Rate (L min<sup>-1</sup> NTP)</th><th>Temperature (°C)</th><th>RH (%)</th><th>[H<sub>2</sub>O] (g m<sup>-3</sup>)</th></tr></thead><tbody><tr><td rowspan="3">AC nI</td><td>1.0</td><td rowspan="3">30</td><td rowspan="3">40</td><td rowspan="3">12</td></tr><tr><td>1.5</td></tr><tr><td>6.5</td></tr><tr><td>AC 5KI</td><td>1.0</td><td></td><td></td><td></td></tr><tr><td>AC Nuclear</td><td>1.0</td><td></td><td></td><td></td></tr></tbody></table>

## 4. Results

### 4.1. H₂O Adsorption Isotherms

The adsorption isotherms obtained, presented in Figure 3 for the tested adsorbents, display a type V characteristic of hydrophobic materials. The first derivative of these experimental curves is shown in the Supplementary Materials (figure in Supplementary Materials (S4)) to better localize the inflection points indicating the beginning of capillary condensation phenomena [48,49].

<figure><img src="image_5.png" /><figcaption>**Figure 3.** H₂O adsorption isotherms at 30 °C for the three studied materials (1.0 mm < dₚ < 1.4 mm); dashed lines represent the fitting according to the Klotz equation.</figcaption></figure>

Let us begin by discussing the behavior of non-impregnated AC. Three main stages can be distinguished. In the first stage (RH < 40%), the material presents a small mass gain due to the hydrophobic surface (2.6% for the AC nI, Table 3, in line with former works on similar ACs [10]). Few water molecules adsorb only at the available heteroatoms at the surface [50]. These heteroatoms are mainly attributed to the residual oxygenated groups
after the activation treatment. Indeed, oxygenated species (carbonyls, quinones, alcohols, etc.) were identified on the tested adsorbents from our already published XPS data [5]. Then, a rapid gain of mass occurs for RHs ranging from 40% to 60%, where water clusters are formed [49]. They are assembled by the creation of hydrogen bonds between new water molecules and absorbed ones. The size of these clusters increases progressively before reaching a maximal value and entering the pores [49]. Then, the saturation of the pores takes place, and the rate of mass gain decreases until the material can no longer accommodate them.

<table><caption>**Table 3.** H₂O adsorption capacities at the equilibrium for the studied materials of different RHs.</caption><thead><tr><th>Adsorbent</th><th>q<sub>eq</sub> (g g<sup>-1</sup>)<br>[RH = 20%]</th><th>q<sub>eq</sub> (g g<sup>-1</sup>)<br>[RH = 40%]</th><th>q<sub>eq</sub> (g g<sup>-1</sup>)<br>[RH = 70%]</th><th>q<sub>sat</sub>(g g<sup>-1</sup>)<br>[RH =95%]</th></tr></thead><tbody><tr><td>AC nI</td><td>0.011</td><td>0.026</td><td>0.204</td><td>0.256</td></tr><tr><td>AC 5KI</td><td>0.029</td><td>0.076</td><td>0.204</td><td>0.236</td></tr><tr><td>AC Nuclear</td><td>0.017</td><td>0.045</td><td>0.191</td><td>0.223</td></tr></tbody></table>

Micropore saturation is known to occur at a RH of 70% [51]. However, a threshold of 60% was obtained in the present study, leading to a water uptake of about 20% for AC nI (Table 3). Then, the water adsorption is reported to occur according to capillary condensation phenomena within mesopores (RH between 60 and 95% for the present study) and macropores (RH > 95%) [51]. Owing to the negligible fraction of mesopores, as stated from the N₂ adsorption isotherms (Table 1 and figure in Supplementary Materials (S2)), a small increase in water uptake is observed in the [60%, 95%] range of RH. A final mass gain of about 26% was obtained for the AC nI (Table 3), in accordance with the literature [37,49].

These different stages of water vapor adsorption are illustrated in Figure 4.

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Mechanisms of pore filling depending on the RH.</figcaption></figure>

The impregnation has an impact on the adsorption isotherms, notably by the presence of KI, which increases the hydrophilic nature of the AC [40]. This is observed at a low RH, where the adsorbed amount is greater than that in the non-impregnated material. Indeed, the inclusion of KI on the surface serves as nucleation site for the water molecules.

This phenomenon increased the adsorbed H₂O amount at a RH = 40% to 4.5% and 7.6% for AC Nuclear and AC 5KI, respectively (Table 3). In addition, a shift in the start of capillary condensation toward a lower RH was noted for the impregnated AC, in line with the hydrophilicity enhancement. However, the inclusion of these impregnations creates a partial blockage of the pores, decreasing their accessibility by adsorbate molecules. Therefore, a reduction in the maximum adsorbed amount can be highlighted after impregnation. More particularly, the maximal adsorption capacities of about 22–24% are obtained for impregnated AC (Table 3). This is due to the presence of KI nanometric particles (an
average size between 2 and 10 nm) within the mesopores, as proven in our recent study of TEM characterizations [41].

To sum up, it is worth mentioning that H₂O adsorption isotherms are highly dependent on the KI content in the present study. The TEDA molecules appear to have a less prominent effect compared with KI, possibly due to the pretreatment temperature and its higher volatility [41]. Indeed, previous investigations have shown a constant water uptake at a low RH (30–40%) for different TEDA impregnated ACs (content ranging from 3 to 10 wt%) [5].

Consequently, an intermediary behavior between AC nI and AC 5KI is obtained for the AC Nuclear, consistent with its lower KI concentration on the surface (%KI = 1%).

The Klotz equation was employed to mathematically describe the complex process involved in the H₂O adsorption by carbonaceous materials. The obtained parameters for each material at 30 °C are reported in Table 4.

<table><caption>**Table 4.** Computed parameters of the Klotz equation and Henry's model for the H₂O adsorption isotherms obtained on the three tested adsorbents (T = 30 °C, 1.0 mm < dₚ < 1.4 mm). The presented uncertainties are related to the residuals obtained between the experimental and fitted data.</caption><thead><tr><th></th><th>AC nI</th><th>AC 5KI</th><th>AC Nuclear</th></tr></thead><tbody><tr><td colspan="4">Klotz's equation parameters (Equation (15))</td></tr><tr><td>q<sub>m</sub> (g g<sup>-1</sup>)</td><td>0.0104 ± 0.0002</td><td>0.025 ± 0.002</td><td>0.0130 ± 0.0004</td></tr><tr><td>K</td><td>1.604 ± 0.003</td><td>1.75 ± 0.03</td><td>1.772 ± 0.007</td></tr><tr><td>m<sub>K</sub></td><td>26.3 ± 0.5</td><td>10.7 ± 0.6</td><td>18.3 ± 0.5</td></tr><tr><td>C<sub>K</sub></td><td>4.6 ± 0.9</td><td>8 ± 3</td><td>8 ± 2</td></tr><tr><td>R<sup>2</sup></td><td>0.99986</td><td>0.99774</td><td>0.99963</td></tr><tr><td colspan="4">Henry's model parameters (Supplementary Materials (S5))</td></tr><tr><td>K<sub>H</sub> (-)</td><td>0.059 ± 0.002</td><td>0.179 ± 0.006</td><td>0.097 ± 0.004</td></tr><tr><td>R<sup>2</sup></td><td>0.99371</td><td>0.97564</td><td>0.98582</td></tr></tbody></table>

The increase in the hydrophilicity of the surface has a notable effect on the parameters of the model (for example, on the parameter *m*<sub>K</sub>, which describes the maximum number of water molecules in a cluster). The presence of hydrophilic species on the surface facilitates the formation of these clusters, thus decreasing the number of molecules per nucleation site. Indeed, a reduction in the *m*<sub>K</sub> number was observed progressively from 26.3 to 10.7 for increasing KI content (Table 4). This happens since, on a hydrophobic surface, the water molecules tend to agglomerate together, rather than distribute between the hydrophilic sites. A significant increase in the monolayer capacity (*q*<sub>m</sub>) for the AC 5KI can also be depicted (Table 4). However, a constant evolution seems to be observed for *K*, which refers to the intermolecular interactions. Finally, *C*<sub>K</sub> expresses the affinity between the water molecules and the hydrophilic sites on the surface. A great increase is obtained after KI impregnation (1%). Then, a quasi-similar value is recorded for the AC 5KI.

## 4.2. Adsorption Kinetics (LDF Model)

Following the equations discussed in Section 2.2, the diffusivities were calculated and are presented in Table 5.

<table><caption>**Table 5.** Calculated diffusivities in the present work (T = 30 °C; d<sub>pore</sub> = 0.5 nm). The same D<sub>s</sub> value was employed for the three investigated adsorbents, assuming a similar H₂O adsorption enthalpy of 40 kJ mol<sup>-1</sup>.</caption><thead><tr><th>Adsorbent</th><th>D<sub>K</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>D<sub>m</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>D<sub>S</sub> (m<sup>2</sup> s<sup>-1</sup>)</th><th>D<sub>p</sub> (m<sup>2</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>AC nI</td><td></td><td></td><td></td><td>7.24 × 10<sup>-8</sup></td></tr><tr><td>AC 5KI</td><td>3.03 × 10<sup>-8</sup></td><td>2.63 × 10<sup>-5</sup></td><td>4.50 × 10<sup>-8</sup></td><td>2.02 × 10<sup>-7</sup></td></tr><tr><td>AC Nuclear</td><td></td><td></td><td></td><td>1.24 × 10<sup>-7</sup></td></tr></tbody></table>
In the literature, the overall order of magnitude when dealing with microporous activated carbons is 10⁻⁵~10⁻⁶ m² s⁻¹ for Dm [52], 10⁻⁷ m² s⁻¹ for DK [52], and 10⁻⁸ to 10⁻¹⁵ m² s⁻¹ for Ds [35]. The calculated diffusivities seem to be in the correct range, as a comparison with former research studies. It is interesting to note that the transport phenomenon is governed mainly by Knudsen and surface diffusion mechanisms. This trend is in line with the porous character of the tested AC, dominated mainly by the presence of micropores (Table 1).

As stated before, the adsorption equilibrium has an impact on the kinetic constant. Using the Klotz equation, the K<sub>LDF</sub> constant was calculated for a RH ranging from 5% to 40%. The results are presented in Figure 5 for the three investigated adsorbents.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Evolution of calculated K<sub>LDF</sub> kinetic constant as function of RHs.</figcaption></figure>

A similar evolution of K<sub>LDF</sub> as a function of RH can be outlined for the tested adsorbents (Figure 5). More particularly, it is characterized by two opposite trends:

(i) An initial increase in adsorption kinetics occurs as a function of the water vapor concentration. The RH threshold for this augmentation is about 24% for AC NI and AC Nuclear. This value is shifted to a lower RH of about 18% for AC 5KI in agreement with its more pronounced hydrophilicity, as commented before.

(ii) A decrease in K<sub>LDF</sub> is, however, noticed for the further RH increase.

This antagonist effect played by RH on the H₂O adsorption kinetics is in accordance with previous investigations [53–55]. At a low RH, the adsorption is known to occur mainly through water vapor interaction with hydrophilic sites (functional AC groups, KI or TEDA impregnates) playing the role of nucleation centers [5,42]. Initially, the concentration of these hydrophilic sites is in excess at the expense of incoming H₂O molecules. Thus, the increase in concentration increases the concentration gradient and consequently the K<sub>LDF</sub> constant [56–58]. However, the RH increase reduces the availability of nucleation sites [59]. Hence, the adsorption will take place mainly between water molecules. Due to the lower energy of these interactions, a slower kinetics takes place [60,61].

In the following, the performance of the simulation model will be discussed.

### 4.3. Simulation of Breakthrough Curves

The inlet parameters for the prediction model may be classified into three categories:

* The ones linked to adsorbent column conditioning: bed density, bed porosity, and the initial RH, which determines the initial water quantity adsorbed on the AC;
* The second category related to the air flow parameters: water vapor concentration, face velocity, and axial dispersion;

* The last one, regarding the adsorption reaction: constants from kinetics and equilibrium models. The parameters used for the first two categories are shown in Table 6.

<table><caption>**Table 6.** Inlet parameters for the simulation models associated with the different investigated configurations.</caption><thead><tr><th></th><th></th><th>AC nI</th><th>AC 5KI</th><th>AC Nuclear</th></tr></thead><tbody><tr><td>T (°C)</td><td>30</td><td>30</td><td>30</td><td>30</td></tr><tr><td>ρ<sub>bed</sub> (kg m<sup>-3</sup>)</td><td>605</td><td>605</td><td>605</td><td>640</td></tr><tr><td>RH<sub>0</sub> (%)</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>RH<sub>feed air</sub> (%)</td><td>40</td><td>40</td><td>40</td><td>40</td></tr><tr><td>Porosity—ε (%)</td><td>35</td><td>35</td><td>35</td><td>35</td></tr><tr><td>Q (L min<sup>-1</sup>) NTP</td><td>1.0</td><td>1.5</td><td>6.5</td><td>1.0</td></tr><tr><td>D<sub>ax</sub> × 10<sup>4</sup> (m<sup>2</sup> s<sup>-1</sup>)</td><td>2.5</td><td>3.6</td><td>13.0</td><td>2.5</td></tr></tbody></table>

### 4.3.1. Model Established for Non-Impregnated AC

The model was first studied for AC nI, due to its simplicity in comparison with impregnated materials. Two configurations were investigated, depending on the employed adsorption isotherm model for equilibrium (see Table 7). The corresponding results (comparison between experimental and simulated BTC) are presented in Figure 6 and in the Supplementary Materials (S6).

<table><caption>**Table 7.** Tested model configurations for the AC nI.</caption><thead><tr><th>Version</th><th>Isotherm Equation</th></tr></thead><tbody><tr><td>1</td><td>Henry</td></tr><tr><td>2</td><td>Klotz</td></tr></tbody></table>

Attention was firstly devoted to the model performance at a gas flow rate of 1.0 L min⁻¹ (NTP). The associated results are presented in Table 8 for the different model versions.

<table><caption>**Table 8.** Correlation coefficient (R²), breakthrough time (t₅%), saturation time (t₉₅%), and prediction errors for gas flow equal to 1.0 L min⁻¹ (NTP) in both program versions.</caption><thead><tr><th rowspan="2">Version</th><th colspan="5">Q = 1.0 L min<sup>-1</sup> (NTP)</th></tr><tr><th>R<sup>2</sup></th><th>t<sub>5%</sub> model (min)</th><th>Error (%) *</th><th>t<sub>95%</sub> model (min)</th><th>Error (%) *</th></tr></thead><tbody><tr><td>1</td><td>0.8798</td><td>9.8</td><td>58</td><td>26.4</td><td>57</td></tr><tr><td>2</td><td>0.9909</td><td>8.5</td><td>35</td><td>58.8</td><td>4</td></tr></tbody></table>

Regarding version 1, the application of the Henry model ($K_H = 0.059 \pm 0.002$) led to poor prediction performances (R² of about 0.87, Table 8). In addition, huge prediction errors for breakthrough (t₅%) and saturation times (t₉₅%) can be noticed (around 60%, Table 8).

The Henry linear isotherm model was chosen for its simple mathematical form. However, as seen in Figure 3, the H₂O adsorption isotherm on AC is not linear. Thus, using an appropriate adsorption model (Klotz's equation presented in Equation (15)) allowed better accordance with experiments (R² = 0.99). This improvement can be seen in the t₉₅% and t₅% errors (Table 8).

<footnote>* the prediction error is calculated from the relative deviation as a comparison with the experimental breakthrough time.</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Comparison of experimental and simulated BTCs by different model versions of H₂O BTC for AC nI applied to the following: 1.0 L min⁻¹ TPN ((a) version 1, (b) version 2); 1.5 L min⁻¹ TPN ((c) version 1, (d) version 2), and 6.5 L min⁻¹ TPN((e) version 1 and (f) version 2).</figcaption></figure>

The model was then applied to two other flow rates. It presented good results ($R^2$ = 0.9746 and 0.9866 at 1.5 and 6.5 L min⁻¹ NTP, respectively). In addition, prediction errors (<30%) were obtained for t₅% and t₉₅% errors (table in Supplementary Materials (S6)). In Figure 6, both simulation versions are presented for the three gas flow rates.

Even though these diffusivities were calculated using general values from the literature (notably Dₛ), the resulting values led to relatively good accordance with experimental sorption profiles. Thus, specific data for the mass transport with the employed AC would be able to lower the prediction errors even more.

Hence, the established model well describes the involved phenomena (mainly physisorption) for the interaction between water vapor and non-impregnated AC. Having validated the model for the AC nI for different gas flows, it was then extrapolated to the other investigated adsorbents through the same methodology.

### 4.3.2. Extrapolation of AC Used in Nuclear Context

The two materials (AC 5KI and AC Nuclear) were studied according to the same approach as AC nI. A flow rate of 1.0 L min⁻¹ (NTP) was chosen to increase the experimental
BTC retention phase. Otherwise, the breakthrough would be immediate, and a precise experimental BTC would not be possible. In Figure 7, the simulated and experimental curves are compared.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Simulated and experimental BTCs for the AC used in a nuclear context.</figcaption></figure>

For both materials, there is an overall agreement with the experiments ($R^2 > 0.87$). More particularly, acceptable prediction errors for the breakthrough time ($t_{5\%}$) can be highlighted. However, higher deviations can be noticed for the prediction of the time needed to AC saturation ($t_{95\%}$), as displayed in Table 9.

<table><caption>**Table 9.** $R^2$, breakthrough time ($t_{5\%}$), saturation time ($t_{95\%}$), and prediction errors for AC 5KI and AC Nuclear (flow rate of 1.0 L min$^{-1}$ NTP).</caption><thead><tr><th>Material</th><th>R<sup>2</sup></th><th>t<sub>5% model</sub> (min)</th><th>Error (%) *</th><th>t<sub>95% model</sub> (min)</th><th>Error (%) *</th></tr></thead><tbody><tr><td>AC 5KI</td><td>0.8736</td><td>19.2</td><td>15</td><td>100.6</td><td>51</td></tr><tr><td>AC Nuclear</td><td>0.9156</td><td>11.9</td><td>23</td><td>160.4</td><td>44</td></tr></tbody></table>

The observed deviation at high breakthrough levels may be caused by the KI reactivity with water. The kinetics of this interaction are not accounted for in the present model. A second possible origin would be the non-isothermal character of adsorption, neglected in the present work. Indeed, it is known that energy balance will have an impact on the local adsorption equilibrium, especially close to the adsorbent saturation [22,62].

Nevertheless, it is worth noting that the developed model was mainly dedicated to describing H₂O physisorption phenomena on the non-impregnated AC. The extrapolation of this model to KI/TEDA impregnated adsorbents gives promising results, especially at the breakthrough level.

## 5. Conclusions

The objective of this study was the simulation of the BTCs from the data at equilibrium. Three ACs were studied, akin to the materials used in the nuclear industry. The focus of the simulation was the capture of water vapor, since it is the principal poison in the capture of iodine in the nuclear industry.

For the implemented model, several parameters have been determined. One of them was adsorption kinetics, where a global approach called LDF was picked. The kinetic constant was estimated by accounting for all the intraparticle mass transfer mechanisms.

Another important parameter is the adsorption isotherm. All considered materials had an isotherm type V, characteristic of the low affinity between the water molecules and the AC surface. However, the presence of KI in the material had a considerable effect on the hydrophilicity of the material. This increased the mass gain in low RH but decreased the pore accessibility, decreasing the maximum adsorbed amount. Afterward, the adsorption

<footnote>* the prediction error is calculated from the relative deviation as a comparison with the experimental breakthrough time.</footnote>
isotherms were modeled using the Klotz equation, which allowed the consideration of the cluster's progressive formation. This equation was also applied to determine the surface diffusivity.

For the validation of the simulation, experimental curves were necessary. They were measured in conditions equivalent to the representative operation of IT. The BTCs were simulated with success. There was a remarkable agreement between the simulated and experimental curves for the non-impregnated material. This demonstrates that the model has a good description of the physisorption phenomenon toward H₂O. This model also gives promising results for the prediction of impregnated AC behavior toward H₂O at the breakthrough level (5%). However, deviations are noticed mainly at AC saturation, owing to the interaction between KI and water vapor molecules and the non-isothermal character of the adsorption process.

This established model on physisorption phenomena from data at equilibrium will have great potential to be applied in the simulation of the BTCs of other molecules, such as iodine species and volatile organic compounds (VOCs). It would allow for the prediction of the IT behavior toward iodine species as well as its poisoning by VOCs. Moreover, it is important to highlight the modular capacity of the developed model as a comparison with commercial codes. Complementary efforts will be performed in order to improve the presented methodology to take into account particular reactions involved in trapping, in addition to physisorption. Specifically, attention will be paid to the prediction of IT interactions toward volatile iodine species by modeling the chemisorption (TEDA) and isotopic exchange (KI) reactions.

**Supplementary Materials:** The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/separations12050126/s1, S1: Detailed description of the employed method for numerical resolution: "the method of lines", S2 N2 adsorption/desorption isotherms at 77 K of the tested activated carbons: (A) AC nI and (B) AC 5KI, (C) AC Nuclear., S3: Photos of the experimental apparatus, S4: First derivative of the adsorption isotherms, S5: Adsorption isotherms modeled with henry's model, S6: correlation coefficient (R²), breakthrough time (t5%), saturation time (t95%) and prediction errors for the gas flow of 1.5 and 6.5 L.min⁻¹ (NTP) and program versions.

**Author Contributions:** F.C.B.M.: methodology, investigation, software, validation, visualization, data curation, writing—original draft preparation. M.C.: supervision, conceptualization, methodology, validation, data curation, writing—review and editing. C.M.-L.: project administration, funding acquisition, writing—review and editing. B.M. and A.R.: investigation, experimentation. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research leading to these results is funded by the French Authority for Nuclear Safety and Radiation protection (ASNR). This work was performed within the "Iodine" research program of the ASNR.

**Data Availability Statement:** Data will be made available on request.

**Acknowledgments:** The research leading to these results is funded by the French Authority for Nuclear Safety and Radiation protection (ASNR).

**Conflicts of Interest:** The authors declare no conflicts of interest.

## Nomenclature

The meanings of the following parameters are presented below:

C water concentration in the gas phase (kg m⁻³)

Z coordinate in the axis following the column length (m)

D internal diameter of the adsorption column (m)

Dax axial dispersion (m² s⁻¹)
<table>
  <tr>
    <td>u<sub>0</sub></td>
    <td>bulk flow speed (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>ε<sub>bed</sub></td>
    <td>bed porosity (-)</td>
  </tr>
  <tr>
    <td>u<sub>i</sub></td>
    <td>interstitial velocity, u<sub>i</sub> = u<sub>0</sub>/ε<sub>b</sub> (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>t</td>
    <td>time (s)</td>
  </tr>
  <tr>
    <td>ρ<sub>bed</sub></td>
    <td>fixed bed density (kg m<sup>-3</sup>)</td>
  </tr>
  <tr>
    <td>q̄</td>
    <td>average adsorbed quantity (g of adsorbed H<sub>2</sub>O per g of adsorbent)</td>
  </tr>
  <tr>
    <td>D<sub>m</sub></td>
    <td>molecular diffusion of water vapor in air at 30 °C and 1 atm, 2.630 × 10<sup>-5</sup> m<sup>2</sup> s<sup>-1</sup> [34]</td>
  </tr>
  <tr>
    <td>d<sub>p</sub></td>
    <td>particle diameter (m)</td>
  </tr>
  <tr>
    <td>ν</td>
    <td>dynamic viscosity of air at 30 °C and 1 atm, 1.5921 × 10<sup>-5</sup> m<sup>2</sup> s<sup>-1</sup> [63]</td>
  </tr>
  <tr>
    <td>Re</td>
    <td>Reynolds number, Re = d<sub>p</sub>·u<sub>0</sub>/ν</td>
  </tr>
  <tr>
    <td>Sc</td>
    <td>Schmidt number, Sc = ν/D<sub>m</sub></td>
  </tr>
  <tr>
    <td>m</td>
    <td>sample mass used in the sorption microbalance (mg)</td>
  </tr>
  <tr>
    <td>RH</td>
    <td>relative humidity (%)</td>
  </tr>
  <tr>
    <td>q<sub>eq</sub></td>
    <td>adsorbed quantity at equilibrium for a given RH (g g<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>K<sub>LDF</sub></td>
    <td>constant associated with the LDF model (s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>D<sub>p</sub></td>
    <td>pore diffusivity (m<sup>2</sup> s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>D<sub>K</sub></td>
    <td>Knudsen diffusivity (m<sup>2</sup> s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>D<sub>S</sub></td>
    <td>surface diffusivity (m<sup>2</sup> s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>τ</td>
    <td>tortuosity (-)</td>
  </tr>
  <tr>
    <td>C<sub>0</sub></td>
    <td>feed concentration (kg m<sup>-3</sup>)</td>
  </tr>
  <tr>
    <td>q<sup>0</sup><sub>eq</sub></td>
    <td>adsorbed quantity in equilibrium with feed concentration (g g<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>d<sub>pore</sub></td>
    <td>pore diameter (m)</td>
  </tr>
  <tr>
    <td>ΔH</td>
    <td>adsorption enthalpy (J mol<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>s</td>
    <td>model parameter based on the interaction (s = 2) [35]</td>
  </tr>
  <tr>
    <td>P<sup>sat</sup></td>
    <td>water vapor saturation pressure for a given temperature (bar)</td>
  </tr>
  <tr>
    <td>R</td>
    <td>ideal gas constant (8.314 J mol<sup>-1</sup> K<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>T</td>
    <td>gas temperature (K)</td>
  </tr>
  <tr>
    <td>M<sub>H<sub>2</sub>O</sub></td>
    <td>molecular mass of water (18 g mol<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>x</td>
    <td>molar fraction of the species in the gas (-)</td>
  </tr>
  <tr>
    <td>q<sub>m</sub></td>
    <td>H<sub>2</sub>O adsorbed quantity in a monolayer (g g<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>K<sub>H</sub></td>
    <td>Henry constant (-)</td>
  </tr>
  <tr>
    <td>K</td>
    <td>constant of the adsorption equilibrium in the Klotz equation (-)</td>
  </tr>
  <tr>
    <td>C<sub>k</sub></td>
    <td>constant linked to the first association between the surface and the adsorbent in the Klotz equation (-)</td>
  </tr>
  <tr>
    <td>m<sub>K</sub></td>
    <td>maximum molecules association number in the clusters in the Klotz equation (-)</td>
  </tr>
  <tr>
    <td>t<sub>5%</sub></td>
    <td>time for C/C<sub>0</sub> = 5%</td>
  </tr>
  <tr>
    <td>t<sub>95%</sub></td>
    <td>time for C/C<sub>0</sub> = 95%</td>
  </tr>
</table>

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