# Modelling of water vapor adsorption/desorption on hopcalite in an industrial medical air dry

C. Cardenas, D. Farrusseng, C. Daniel, C. Collomb

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C. Cardenas, D. Farrusseng, C. Daniel, C. Collomb. Modelling of water vapor adsorption/desorption on hopcalite in an industrial medical air dry. 33rd European Symposium on Computer-Aided Process Engineering, Jun 2023, Athens, Greece. [hal-04133203](https://hal.science/hal-04133203)

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Submitted on 30 Jun 2023

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# Modelling of water vapor adsorption/desorption on hopcalite in an industrial medical air dry

Cristian Cardenas, David Farrusseng, Cécile Daniel, Corentin Collomb

Institut de Recherches sur la Catalyse et l'Environnement de Lyon, UMR 5256 - CNRS / Université Lyon 1 - Villeurbanne Cedex, France / cristian.cardenas@ircelyon.univ-lyon1.fr

## Introduction

* The air treatment system is an essential part of medical air installations: it ensures the drying and purification of the compressed air produced, to provide healthcare facilities.

* Medical air must meet the following pharmacopoeia requirements: H₂O: 67 ppm, CO: 5 ppm, CO₂: 500 ppm, SO₂: 1 ppm, NO-NO₂: 2 ppm, Oil: 0.1 mg/m³, Oxygen: 20.4% ≤ O₂ ≤ 21.4%

* The production of dry medical air is ensured by air compressors and a column operating in PSA mode with a counter-current regeneration mode

<figure><img src="image_1.png" /></figure>

* The column is a packed bed with various moisture adsorbents such as alumina, hopcalite and activated carbon in series in the form of granules.

<figure><img src="image_2.png" /></figure>

* In the literature some models to describe the dynamic adsorption of water vapor on alumina are found, contrary to hopcalite where there is limited.

The objective is to analyze and model the water vapor adsorption and desorption breakthrough fronts on hopcalite as a function of gas flow rate, concentration and temperature in order to optimize the design of industrial medical air dry.

## Adsorption isotherms

Experimental adsorption isotherms of water vapor on hopcalite in a temperature range of 288-313 K and pressures up to the saturated concentration were measured using a manometric BelsorpMax 2 instrument.

### Toth-Aranovich-Donohue (Toth-AD) model

The temperature-dependent Toth-AD equation was applied to approximate the equilibrium relationship of water vapor on hopcalite. The model equation is as follows(1):

$$q_e = \frac{q_m \cdot b \cdot c}{\left[1 + (b \cdot c)^{t_o}\right]^{\frac{1}{t_o}}} \left[ \frac{1}{(1 - b_2 \cdot c)^{n_2}} \right]$$

where $q_e$ (mol.kg⁻¹) is the equilibrium amount adsorbed, c (mol.m⁻³) is the water vapor concentration, $q_m$ (mol.kg⁻¹) is the maximum amount adsorbed, b and $b_2$ (m³.mol⁻¹) are the equilibrium constants at low and high concentrations, respectively, $t_o$ is the Toth's heterogeneity factor and $n_2$ is an adjustable dimensionless factor. The temperature-dependent parameters $q_m$, b, $b_0$ and $t_o$ are detailed in the study by Cardenas et al.(1)

## Water sorption process modeling

The model developed consists of momentum and mass balance equations, along with relations for the prediction of **adsorption isotherms** and **kinetics**

### Fickian multiscale model

The multiscale model considers the bed scale and the particle scale. The particle is divided into two phases: a gas and a solid phase.

(i) In the bed column, the equation of the water vapor mass balance in the gas phase is expressed as follows:

$$\frac{\partial c}{\partial t} - D_{ax} \frac{\partial^2 c}{\partial z^2} + \frac{\partial(vc)}{\partial z} = - \frac{1 - \varepsilon_b}{\varepsilon_b} \frac{3}{r_{pe}} k_f \left( c - c_{pe} \right)_{r_p = r_{pe}}$$

(ii) In the particle, assuming the absence of lateral and azimuthal variations of the concentration, the mass balance in the radial direction can be written as:

$$\varepsilon_{pe} \frac{\partial c_{pe}}{\partial t} + (1 - \varepsilon_{pe}) \frac{\partial \bar{q}}{\partial t} = \frac{D_e}{r_{pe}^2} \frac{\partial}{\partial r_p} \left( r_p^2 \frac{\partial c_{pe}}{\partial r_p} \right)$$

(iii) In the particle, the water vapor adsorbed on the solid surface is approximated by the linear driving force LDF model and is written as:

$$\frac{\partial \bar{q}}{\partial t} = k_1 (q_e^* - \bar{q})$$

Because of this reversibility, we can propose to treat the desorption breakthrough curve as if it were an adsorption breakthrough curve. Only the initial conditions and the boundary conditions are different. More details see Cardenas et al.(2)

## Parameters identification

The experimental data used consist of breakthrough curve measurements carried out at gas flow rates of 90, 120, and 150 mL.min⁻¹ with a water vapor concentration of 0.67 mol.m⁻³, and at a gas flow rate of 135 mL.min⁻¹ with four water vapor concentrations of 0.19, 0.38, 0.48, 0.72, and 0.77 mol.m⁻³ i.e., eighth adsorption and desorption experimental measurements to identified parameters, using Toth-AD isotherm model.

<figure><img src="image_3.png" /><figcaption>Table 1. Optimized parameter values for water vapor adsorption on hopcalite</figcaption></figure>

<table><thead><tr><th>Parameters</th><th>c<sub>m2</sub></th><th>c<sub>m3</sub></th><th>c<sub>m4</sub>×10<sup>-5</sup></th><th>D<sub>M</sub>, (m<sup>2</sup>.s<sup>-1</sup>)</th><th>k<sub>1,ads</sub> (s<sup>-1</sup>)</th><th>k<sub>1,des</sub> (s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Value</td><td>5.93</td><td>0.005</td><td>1.01</td><td>4.3×10<sup>-5</sup></td><td>0.065</td><td>0.0018</td></tr><tr><td>CI (95%)</td><td>±1.26</td><td>±0.002</td><td>±0.09</td><td>±8.5×10<sup>-6</sup></td><td>±0.005</td><td>±0.0006</td></tr></tbody></table>

## Model validation

The identified model is then validated with additional measurements of water vapor adsorption and desorption breakthrough curves obtained on hopcalite at different temperatures, i.e., 15, 30, 40, and 50 °C, corresponding to the water vapor concentrations of 0.48, 0.47, 0.46, 0.45 mol.m⁻³ and a gas flow rate of 150 mL.min⁻¹

<figure><img src="image_4.png" /></figure>

The Kolmogorov-Smirnov statistical test is used and is based on the calculation of the divergence ($D_n$) between two distributions. For more details, see the study by Cardenas et al.(5)

## Implementation and simulation

The external mass transfer coefficient is estimated using the Wakao and Funazkri (1978) correlation(3):

$$k_f = \frac{D_m Sh}{2r_{pe}} \quad \text{with} \quad Sh = c_{m1} + c_{m2} Re_p^{0.6} Sc^{1/3} \quad \text{and} \quad c_{m1} = 2$$

The axial dispersion coefficient is calculated using Rastegar and Gu (2017) correlation(4):

$$D_{ax} = \frac{2r_{pe}v}{Pe'} \quad \text{with} \quad \frac{1}{Pe'} = \frac{0.7D_m}{2r_{pe}v} + \frac{\varepsilon_b}{c_{m3} + c_{m4} (Re_p)^{0.5}}$$

The effective diffusion coefficient is linked to the mixture diffusion:

$$D_e = \varepsilon_{pe}^{4/3} D_M$$

Unknown parameters are be estimated: $k_1$, $D_M$, $c_{m2}$, $c_{m3}$, $c_{m4}$

Comsol Multiphysics® is used to implement and solve the set of partial differential equations of the model.

## Conclusion

The accuracy of the one-dimensional dynamic adsorption and desorption model using the temperature-dependent Toth-Aranovich-Donohue equation developed, identified, and validated for the simulation of a packed bed column in which water vapor is physically adsorbed on hopcalite was demonstrated. The fickian multiscale model allowed the finding that the adsorption process is controlled by the diffusion of water vapor in the macropores by a value of the material Biot number higher than 3. The statistical validation and the performance of the model performance indices ensure the quality of the model predictions, which is not always the case in modeling studies. This model coupled with an energy balance to improve the design and optimization of commercial adsorption columns using multilayer beds.

<footnote>(1) C. Cardenas, D. Farrusseng, C. Daniel, R. Aubry, *Fluid Phase Equilib.* **2022**, 561, 1132520.</footnote>

<footnote>(2) C. Cardenas, D. Farrusseng, C. Daniel, *SSRN J.* **2022**, 4259051.</footnote>

<footnote>(3) N. Wakao, T. Funazkari, *Chem Eng Science.* **1978**, 33, 1375-1384.</footnote>

<footnote>(4) S. Rastegar, T. Gu, *Journal of Chromatography A.* **2017**, 1490, 133-137.</footnote>

<footnote>(5) C. Cardenas, A. M. Latifi, C. Vallières, S. Marsteau, *Comput. Aided Chem. Eng.* **2022**, 51, 457-462.</footnote>