# Investigation of the effect of dead volume on the shape of breakthrough curve – New method designed to correct this effect

Ahmed Grim, Laurent Perrier, Pierre Mocho

► To cite this version:

Ahmed Grim, Laurent Perrier, Pierre Mocho. Investigation of the effect of dead volume on the shape of breakthrough curve – New method designed to correct this effect. Journal of Environmental Chemical Engineering, 2024, 12 (6), pp.114929. 10.1016/j.jece.2024.114929. hal-05241292

HAL Id: hal-05241292

https://hal.science/hal-05241292v1

Submitted on 5 Sep 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_2.png" /></figure>

# Investigation of the effect of dead volume on the shape of breakthrough curve – New method designed to correct this effect

Ahmed Grim, Laurent Perrier, Pierre Mocho*

Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LFCR, 1 Allée du Parc Montaury, Anglet 64600, France

## ARTICLE INFO

### Keywords:

- Dead volume

- Breakthrough curve

- Mass transfer modeling

- Axial dispersion

## ABSTRACT

At laboratory scale, the reactor volume may be the same order of magnitude as the dead volume of the experimental set-up. The purpose of this study is to provide a new simple method to obtain a breakthrough curve on an inert porous media (without adsorption), corrected from the influence of the dead volume, to study only the effect of the axial dispersion. Indeed, our study parameters (pressure, low flow rate, fixed bed column filled with an inert porous media) did not allow us to use the classical and simple method, known as Point by Point (PBP). This new method (Elementary Surface Area ESA) is based on the subtraction of elementary surfaces of dead volume and raw breakthrough curves, whereas the PBP method is built on a point by point subtraction. Therefore, each elementary surface, defined by two points of the breakthrough curve, induces a smoothing of the fluctuations observed by the PBP method. So the ESA method could be considered as an improvement of PBP one. Analysis of breakthrough curve requires calibration of the measurement devices for both flow rate and gas stream composition. Some calibration results are also presented in this paper.

## 1. Introduction

Separation process, which transforms mixed substances into different products of various compositions, is a key issue in wide industrial fields [1–4]. In recent decades, a multitude of separation techniques have been developed and continually improved, given that the separation process accounts for 40–70 %, of the overall capital and

operating costs [4]. Gas separation techniques based on adsorption have been extensively employed in laboratory and industrial scale due to the variety of applications where it can be involved including carbon capture, hydrogen purification, biogas upgrading, air separation [5–7].

A deep understanding of adsorption mechanism can be effectively determined through various experimental methods. Equilibrium and kinetic data can be accurately estimated using static experiments, such

**Abbreviations:** C, gas-phase concentration of the solute methane in nitrogen mol.m⁻³; c, reference to corrected raw breakthrough curve; Cᵢ, gas-phase concentration increment i mol.m⁻³; Csat, inlet gas-phase concentration of methane in nitrogen mol.m⁻³; D, column diameter m; d, reference to dead volume curve; dp, particle diameter µm; H, column height m; P, column pressure bar; Pe, Peclet number =D.Uᵢ/D₂; Re, Reynolds number =dₚ.Uᵢρ/µ; Rᵢ⁽⁺/⁻⁾, elementary ratio of elementary surface area; (Sᵢ⁽⁺/⁻⁾ᵢ, to total surface area; (Sᵢ⁽⁺/⁻⁾ᵢ), for a given concentration increment i; Sᵢ⁽⁺/⁻⁾ᵢ, corrected surface areas mol.m-3.s-1; Sᵢ⁽⁺/⁻⁾ᵢ, elementary surface areas defined by a fraction of a concentration increment i to the total surface; (Sᵢ⁽⁺/⁻⁾ᵢ), positive or negative for dead volume d and raw r curves mol.m-3.s-1; Sᵢ⁽⁺/⁻⁾ᵢ, positive surface area located to the left of the plug flow time tp d/r, between the breakthrough curve and the line with equation C= 0 for dead volume d and raw r curves mol.m-3.s-1; Sᵢ⁽⁺/⁻⁾ᵢ, negative surface area located to the right of the plug flow time tp d/r, between the curve and the line with equation C= Csat for dead volume d and raw r curves mol.m-3.s-1; Sᵢ⁽⁺/⁻⁾ᵢ, corrected positive surface area (=; Sᵢ⁽⁺/⁻⁾ᵢ - Sᵢ⁽⁺/⁻⁾ᵢ), mol.m-3.s-1; Sᵢ⁽⁻/⁺⁾ᵢ, corrected negative surface area (=; Sᵢ⁽⁻/⁺⁾ᵢ - Sᵢ⁽⁺/⁻⁾ᵢ), mol.m-3.s-1; r, reference to raw breakthrough curve; t, time s; tᵢ d/r, time corresponding to the concentration increment Cᵢ in the breakthrough curve s; tᵢ cr, time corresponding to the concentration increment Cᵢ in the breakthrough curve of the fixed-bed column s; tᵢ d/r, breakthrough time equivalent to a plug flow s; t sat d/r, time corresponding to a concentration equal to the inlet concentration Csat in the breakthrough curve s; t₀ d/r, breakthrough time with C = 0 s; T, temperature °C; Uᵢ, interstitial velocity of gas stream m.s⁻¹; Vᵢ, column internal volume m³; Vd, dosing cell volume m³; Vdₜ, dead volume m³; Vf, porous media free volume m³; Vs, entire system volume dead volume+cell volume m³; y, volumetric fraction of methane in nitrogen; ε, porous media porosity; µ, dynamic viscosity of the fluid kg.m⁻¹.s⁻¹; ρ, density of the fluid kg.m⁻³; ρd²N₂, gas-phase nitrogen density within the dosing cell volume; ρgN₂, gas-phase nitrogen density within the entire system volume.

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: pierre.mocho@univ-pau.fr (P. Mocho).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Sketch of the experimental setup.</figcaption></figure>

as volumetric and gravimetric techniques [8–10]. Nevertheless, it is
crucial to measure these parameters under the process conditions. Dy-
namic experiments provide the opportunity to investigate both equi-
librium and kinetic aspects as well as the study of the overall
performance of the adsorption column and validation of models. In this
way, dynamic experiments present a key step through the building of a
separation process based on adsorption.

Breakthrough curves represent the primary outcome of a dynamic
adsorption experiment [11,12]. These curves provide information about
the adsorbed quantities, adsorption equilibrium, and kinetics [13,14].
However, adsorption is not the only factor that affects the shape of the
breakthrough curves; diffusion and convection must also be taken into
account [15]. It should also be noted that these effects (diffusion and
convection) do not occur solely within the column but also within the
extra volume of the experimental setup, which includes tubes, valves,
flow controllers, pressure gauges... This extra volume is commonly
referred to as dead volume, its contribution in the residence time and the
broadening of breakthrough curves should be taken into account. This
influence becomes important in two scenarios. Firstly, in laboratory
scale, when the column volume is smaller than the dead volume, and
secondly, when the residence time within the dead volume cannot be
considered negligible [16]. Therefore, it is imperative to take into
consideration the influence of the dead volume on the shape of the
breakthrough curve and make the necessary corrections before using it
to determine various parameters related to the gas separation process.
Faroog et al. reported that dead volume contribution significantly im-
pacts the corrected breakthrough curve for gases with low adsorption
capacity [17]. On the other hand, for gases with high adsorption ca-
pacity, the dead volume correction results in only a minor difference
between the corrected curve and the raw breakthrough response [17].
Many studies have focused on the effect of dead volume in liquid
chromatography by pulse-response techniques [18–22]. However, few
studies are dedicated to breakthrough curve applications. The most
popular method to correct the effect of dead volume on breakthrough
curve is the Point by Point (PBP) method. In this method, the broadening
and the retention time in the dead volume and column responses are
considered as linearly additive [23,24]. This assumption of additivity of
dead volume and column effects is mainly relevant for inert porous
media considered non-adsorbent. In the case of adsorption experiments
performed to estimate mass transfer kinetics, this method is not appli-
cable because of the disturbance of solute flow in the mass transfer zone
due to adsorption as well as a potential thermal effect due to the
exothermic reaction. The models better suited to describe the effect of

dead volume on adsorption process are the tank in series (TIS) model
and dispersed plug flow model. In the TIS model, the dead volume is
described by ideal mixed reactors in series [23]. The dispersed plug flow
model is based on the convection diffusion model to represent the dead
volume with the introduction of an axial dispersion coefficient. To
improve their applicability in the field of non-ideal flow (fluid chan-
neling, stagnant regions), combination of dispersed flow tanks in series
and tank, to represent stagnant region, are implemented and known as
compartment models [25,26]. However, in these more complex models,
it should be noted that the dead volume is located downstream of the
adsorption column. So, the dead volume upstream of the column is
assumed in plug flow.

The development of a new experimental device to measure the
breakthrough curve under pressure led us to quantify the dead volume of
the set-up. This volume is of the same order of magnitude as the column
volume. It is therefore essential to take into account the effect of dead
volume on the shape of breakthrough curves. Initially, the aim is to
study the effect of fluid flow on the shape of the breakthrough curve by
filling the column with an inert porous material in order to evaluate the
axial dispersion. To achieve this, the main purpose of this paper is to
correct the breakthrough curve for the effect of dead volume. To do this,
the first step involves measuring the dead volume curve, by replacing
the column by a tubing of negligible volume, under the same experi-
mental conditions. The second step then consists in measuring the raw
breakthrough curve in the presence of the column. Among the dead
volume correction methods described above, the simplest to implement
and well suited to our experimental conditions (no adsorption and dead
volume evenly distributed upstream and downstream of the column) is
the PBP method. However, the application of this model to experimental
breakthrough curves has proved unsuitable, due to oscillations in the
corrected breakthrough curves. This result led us to propose a new dead
volume correction method, called the Elementary Surface Area (ESA)
method, which can be considered as an improvement on the PBP one.

Furthermore, this new experimental set-up also requires a wide
range of validation steps, measurement of the volumetric flow rate of
various gases, calibration of the gas analyzer, measurement of the dead
volume.

## 2. Material and methods

2.1. *Experimental setup*

A home-built experimental setup is designed to conduct dynamic
adsorption experiments under pressure for various gas mixtures primarily containing CO₂, CH₄, H₂ and N₂. Schematic of the experimental setup is shown in Fig. 1. A series of five mass flow controllers (BRONK-HORST Model F-201CV-050-RAD-11-V) is employed to monitor flow rate of each component and to adjust the gas mixture. The mass flow controllers provide a flow rate range of 0–30 mL.min⁻¹ for CO₂, CH₄, and H₂, while for N₂, the range extends to 0–90 mL.min⁻¹. To maintain the minimum required flow rate through the analyzer in case of adsorption of all gas mixture components, a fifth mass flow controller is situated at the column's exit. Its flow rate range is configured as 0–60 mL.min⁻¹ of N₂. A back pressure regulator (SWAGELOK Model KPB1L0A412P20000) with an inlet pressure range from 0 to 68 bar is installed downstream the column to regulate the pressure inside the column. The gas mixture exiting the column is analyzed using an infrared analyzer (AP2E Model PROCEAS) based on a direct intensity measurement scanning spectroscopy technique It is dedicated to multi-component (CH₄, CO₂, H₂) gas analysis. The analyzer's measurement frequency is one measure per second; that is why the plot of the experimental points appears as a continuous line. An electronic flow meter (AGILENT Flowmeter Mode ADM1000), installed downstream the back pressure regulator, allows to measure the flow rate of the gas mixture at the exit of the column. The mass flow controllers and the infrared analyzer are calibrated by comparing their measurements with those obtained from the electronic flow meter. Prior to this, the electronic flow meter is calibrated using a soap film flow meter (Hewlett Packard 0101–0113 Soap Film Flow Meter) to ensure its accuracy.

## 2.2. Fixed-bed column and porous media

The dynamic test are carried out in a stainless steel column with an internal diameter of 1.0 cm and height of 26.0 cm (internal volume V_c equal to 20.4 cm³). To set up an inert porous media, the column is filled with glass beads (SiLibeads Glass beads type S 5218–7) with a size range of 400–600 µm. The free volume of the porous media (V_p) is measured by a manometric method developed in 2.4 section (equal to 9.6 ± 0.2 mL) [27]. Then the bed's effective porosity is deduced, equal to 0.47 ± 0.01. The volume control of the column can be modeled by a dispersed plug flow, which takes into account convection and axial dispersion of the gas stream in the porous media.

## 2.3. Gas stream

The gas mixture used to investigate the contribution of dead volume in experimental breakthrough curves is composed of two components: a carrier gas, nitrogen (N₂), and a gas tracer, methane (CH₄). In order to explore a wide range of experimental conditions, this gas mixture will be generated under different pressure conditions (10, 20, 30 and 40 bar) and inlet volumetric fraction levels (0.10, 0.20 and 0.30 of CH₄ in N₂).

## 2.4. Dead volume measurement method

The measurement is carried out using a manometric technique previously developed in the laboratory [27,28]. The method's principle is based on a mass balance between a dosing cell and a measuring cell. An inert gas (nitrogen) contained in the dosing cell of known volume is expanded into the system (dead volume) for which the volume is to be measured.

The dead volume is quantified by using the following equations.

$$V_s = \frac{\rho_{N2}^d}{\rho_{N2}^s} V_d \qquad (1)$$

$$V_{dv} = V_s - V_d \qquad (2)$$

where V_s represents the total system volume (comprising the dead volume and dosing cell), V_d the dosing cell volume and V dv the dead

<figure><img src="image_4.png" /><figcaption>Fig. 2. Correction of dead volume effect on raw breakthrough curve by the point-by-point method.</figcaption></figure>

volume. Furthermore, ρ_s²N₂ is gas density within the dosing cell, while ρ_s²N₂ corresponds to the density of the gas released within the total system. Gas density is determined from the AGA8–92DC equation of state [29]. To validate repeatability tests, the dead volume measurement is repeated several times at different dosing pressure. The standard deviation of the different measurements is considered as the dead volume uncertainty.

## 2.5. Flow of a gas mixture across an inert porous media: determination of the dead volume curve and the raw breakthrough curve

Two experiments are carried out in parallel under identical experimental conditions on the experimental set-up illustrated in Fig. 1. In one experiment, the gas flow passes through a small-volume tubing, enabling the experimental dead volume curve to be plotted. In the other experiment, the tubing is replaced by the column containing the inert porous medium, which allows the raw breakthrough curve to be plotted. In this study, the gas mixture is composed of methane (solute) in an inert gas (nitrogen). First, the device is pressurized by introducing a nitrogen stream. Next, the methane stream is added to the nitrogen one to generate the gas mixture. Time zero of the experiment is defined by the start of methane injection into the gas stream.

# 3. Theory and calculation

Correction of the effect of dead volume on the raw breakthrough curve is usually achieved using the point-by-point method. This method is briefly described in the following section.

## 3.1. Point par point (PBP) method

The PBP method aims to correct the effect of the dead volume on experimental breakthrough curves. In summary, correction of the breakthrough curve shape is achieved by subtracting the time of dead volume curve (t_d) from the time of the raw breakthrough curve (t_r), as mentioned in Eq.3. This method is built upon the assumption that the contributions of the breakthrough time and broadening of both the column and the dead volume responses are linearly additive [23,24]. This assumption remains valid as long as Eq.4 is satisfied for all the experimental data points.

$$t_c = t_r - t_d \qquad (3)$$
<figure><img src="image_5.png" /><figcaption>Fig. 4. Calibration curves of mass flow meters for CO₂, CH₄, H₂ and N₂.</figcaption></figure>

the raw breakthrough curve ($t_{pr}$) as the reference time. The calculation is divided into two parts. In the direction of decreasing time, the first part starts from ($t_{pr}$) and ends with the breakthrough time of the raw breakthrough curve ($t_{or}$). In the direction of increasing time, the second part starts from ($t_{pr}$) to the saturation time ($t_{sat r}$). The calculation principle of shape correction of raw breakthrough curve is summarized in Table 1.

To remove the effect of mass accumulation in the dead volume of the experimental device, a shift of the corrected raw breakthrough curve on the x-axis (time) is carried out according to the following relationship (Eq. 9):

At each concentration $i$,

$$t_m = t_{icr} - t_p \quad (9)$$

where $t_p$ is the equivalent plug flow breakthrough time of the dead volume curve, $t_m$ the time of the breakthrough curve of solute (methane) in the inert gas (nitrogen) in the fixed-bed column, containing the porous media.

# 4. Results and discussion

Experimental study of breakthrough curves requires preliminary calibration of the various measuring devices (mass flows, composition of components in the gas stream) making up the experimental set-up.

## 4.1. Calibration of measurement apparatus

Mass balance equations require particularly an accurate quantification of flow rate for each component in the gas mixture. For this, the purpose of this section will be first the calibration of mass flow controllers, presented in Section 2.1.

The calibration is carried out with an electronic flow meter (AGILENT Flow Meter Model ADM1000), located in the column outlet, to measure the volumetric flow of each component under atmospheric pressure and at room temperature (1.013 bar and 22°C). These measurements, converted to normal conditions (0°C and 1.013 bar), are then compared to the displayed flow rates values set using the mass flow controllers. The absolute uncertainty of the electronic flow meter is 0.1 mL.min⁻¹. The results of calibration of CO₂, CH₄, H₂, N₂ are shown in Fig. 4. The dotted red line represents the displayed flow rate of the mass flow controllers, while the points correspond to the real flow rates measured by the electronic flow meter. The linearity of the measured points highlights the linearity of the calibration of mass flow controllers. However, there is a deviation between the measured and the set values. The relative deviation is equal to 34 % for CO₂, 11 % for CH₄, 5 % for H₂ and 2 % for N₂. To try to limit the deviation of CO₂, the pressure gauge upstream of flow controller is changed but without significant improvement. Therefore, for the breakthrough curve experiments, the flow rates of the studied gases will be measured with the electronic flow meter, located at the experimental device outlet.

Another important calibration procedure concerns the infrared gas analyzer (PROCEAS). To perform the calibration tests, an inert gas (N₂) is mixed with a gas tracer (CO₂, CH₄ and H₂) at different flow rates and compositions. Flow rates of each compound of the gas mixture are separately measured at the column's outlet with the electronic flow meter (AGILENT), considered as calibration reference, which leads then to the calculation of the composition of the gas stream (presently in volumetric fraction). Results of gas analyser calibration for CO₂, CH₄, H₂ are presented in Fig. 5.

The dotted red line represents the volumetric fraction of the studied gas in nitrogen, deduced from single gas flow measurements with the electronic flowmeter, considered as reference gas composition. Reference points are lined up on the first bisector of the various graphs. The
<figure><img src="image_6.png" /><figcaption>Fig. 5. Calibration curves of analyzer for CO₂ (a), CH₄ (b) and H₂ (c).</figcaption></figure>

<table><caption>**Table 2**
Main parameters of the experimental study.</caption><thead><tr><td>Porous media</td><td>Glass beads (SiLibeads)</td></tr></thead><tbody><tr><td>Particle diameter d<sub>p</sub> (μm)</td><td>300–600</td></tr><tr><td>Column height H (cm)</td><td>26</td></tr><tr><td>Column diameter D (cm)</td><td>1</td></tr><tr><td>Porous media porosity ε</td><td>0.47</td></tr><tr><td>Dead volume V<sub>dv</sub></td><td>30</td></tr><tr><td>Solute</td><td>Methane</td></tr><tr><td>Carrier gas</td><td>Nitrogen</td></tr></tbody></table>

analyzer response for the reference points is linear, confirming the linearity of mass flow controllers calibrations. These calibration tests show a quite good correlation between the analyzer and reference measurements. However, corrective relationships are calculated to adjust the compounds' volumetric fraction of the analyzer to those of the reference, as shown in Fig. 5.

## 4.2. Effect of the dead volume on the shape of the breakthrough curve

Based on the manometric method presented in Section 2.4, the dead volume ($V_{dv}$) is quantified by a value of 30 ± 1 mL. This value is of the same order of magnitude as the column volume, equal to 20.4 mL. For this experimental device, the dead volume may have a significant effect

<figure><img src="image_7.png" /><figcaption>Fig. 6. Correction of the dead volume effect on the raw breakthrough curve by PBP and ESA methods (case: 10 bar).</figcaption></figure>
<table><caption>**Table 3**
Experimental parameters to study the effect of pressure on breakthrough curve.</caption><thead><tr><th>Parameter</th><th>Case 1</th><th>Case 2</th><th>Case 3</th><th>Case 4</th></tr></thead><tbody><tr><td>Colum pressure <i>P</i> (bar)</td><td>10</td><td>20</td><td>30</td><td>40</td></tr><tr><td>Temperature <i>T</i> (°C)</td><td>22</td><td>22</td><td>22</td><td>22</td></tr><tr><td>Inlet volumetric fraction <i>y</i></td><td>0.15</td><td>0.15</td><td>0.15</td><td>0.15</td></tr><tr><td>Interstitial velocity <i>U</i><sub>i</sub> (×10<sup>3</sup> m/s)</td><td>3.39</td><td>1.59</td><td>1.10</td><td>0.74</td></tr><tr><td>Gas density <i>ρ</i> (kg.m<sup>-3</sup>)</td><td>10.72</td><td>21.51</td><td>32.36</td><td>43.24</td></tr><tr><td>Gas dynamic viscosity <i>μ</i> (×10<sup>5</sup> Pa.s)</td><td>1.77</td><td>1.79</td><td>1.81</td><td>1.83</td></tr><tr><td>Reynolds number <i>Re</i></td><td>1.03</td><td>0.96</td><td>0.98</td><td>0.87</td></tr></tbody></table>

on the breakthrough curve of the non-absorbed compounds. The purpose of this section is to investigate the influence of the dead volume on the breakthrough curves of compounds without adsorption. For this, the adsorbent is replaced by an inert granular material (glass beads).

The main operating conditions are reported on Table 2.

The PBP and ESA methods, detailed in Sections 3.1 and 3.2, are applied on the raw breakthrough curves to remove the dead volume effect on these curves. Both methods are tested on methane breakthrough curves in nitrogen, as shown in Fig. 6. This figure shows fluctuations in the application of the PBP method to these experiments. This observation has led to the development of a new method to correct the effect of dead volume on the raw breakthrough curve.

The study's initial focus is on the influence of pressure on the shape of the breakthrough curve.

### 4.3. Influence of pressure on the shape of the breakthrough curve

This study is carried out in a pressure range from 10 to 40 ± 0.1 bar with an inlet volumetric (or molar) fraction of methane in nitrogen of 0.15±0.01 at a temperature of 22 ± 1°C. Four cases, summarized in Table 3, are investigated in this section.

The results of the application of the PBP and ESA methods are presented in Fig. 7. To highlight the spread of breakthrough curves as a function of pressure, all the curves in Fig. 7 are plotted on the same time scale. This figure shows clearly the influence of pressure on the slope of the breakthrough curve. The higher the pressure, the less steep the slope of the curve is. Similarly, the shorter the breakthrough time, the steeper the slope of the curve is [30]. In these conditions, Reynolds number (Re), the ratio of inertial forces to viscous forces, defined by $Re=d_p U_i \rho / \mu$, is around 1.0, less than 6, which corresponds to a laminar flow regime [31]. To calculate Re, the viscosity of the gas mixture is assumed to be equal to nitrogen viscosity because of gas composition (85 % N₂, 15 % CH₄). The viscosity calculations are based on the works of Lemmon et al. [32]. In this regime, viscous forces are predominant on inertial ones, which could explain the broadening of the breakthrough curve, due to shear forces between fluid layers, particularly at high pressure conditions. Morever, increasing pressure results in both an increase in gas density and a decrease in gas flow rate. A decrease of fluid velocity induces a decrease of Re (and an increase of contribution of viscous forces to inertia ones), which leads to a decrease of curve's slope. An increase of gas density has the same effect by an increase of the strengths of molecular interactions in the fluid and at the fluid/particle interface in the porous media, leading to a broadening of the breakthrough curve.

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** Influence of pressure on corrected breakthrough curves by ESA (red curves) and PBP (black curves) methods: 10 bar (Fig. 7a), 20 bar (Fig. 7b), 30 bar (Fig. 7c), 40 bar (Fig. 7d).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 8. Validity test of the assumption (Equ.4) of PBP method at different pressures: 10 bar (Fig. 8a), 20 bar (Fig. 8b), 30 bar (Fig. 8c), 40 bar (Fig. 8d).</figcaption></figure>

Morever, Edwards et al. have established an empirical correlation to estimate the axial dispersion coefficient [33]. Applied to methane in nitrogen, this value varies between $10^{-6}$ and $10^{-7}$ m$^2$.s$^{-1}$. So, the associated Peclet number (Pe), ratio of rate of convection to the rate of axial diffusion ($=D.U'_i/D_z$), is around 30, showing the major influence of convection on axial dispersion [34].

This same figure makes in evidence fluctuations in the correction of breakthrough curve by the PBP method. This method's applicability requires the slope of the dead volume curve to be greater than that of the raw breakthrough one for all concentration values, according to Eq.4. This assumption is not always valid in cases where the shapes of the two curves are almost identical, as in this study where the porous media is constituted of inert material. The study of the validity of this assumption is shown in Fig. 8. Negative values represent the concentration points where the assumption is not satisfied. Respectively, 32 % of concentration points in Case 1, 23 % for Case 2, 13 % for Case 3 and 7 % for Case 4 do not satisfy this assumption. It can be seen that an increase in pressure leads to a decrease in the percentage of unfavorable cases. This can be explained by the fact that an increase in pressure favors a broadening of the breakthrough curve due to stronger molecular interactions within the fluid and at the fluid/particle interface in the porous media. This broaden of breakthrough curve allows better differentiation of the slopes of the dead volume curve and the raw breakthrough one. To apply the PBP method, the variation of the subtraction point by point of the dead volume and raw curves must be at least constant or generally an increasing function of the concentration C (or $C/C_o$). When the subtraction point by point is constant, then the corrected breakthrough has a plug flow shape. A slight variation of subtractions point by point around this constant results in slight fluctuations in the corrected curve.

As shown in Fig. 7, by taking into account the effect of dead volume using the ESA method, we obtain a well-defined sigmoidal breakthrough curve, suitable for quantifying the axial dispersion coefficient of

methane in the fluid flowing through the porous media. This method is particularly based on the assumption that the ratio of the elementary surface to the total surface in the dead volume curve is equivalent to that observed in the raw breakthrough curve, according to Eq.8. The validity of this assumption is tested in Fig. 9 by comparing the ratios of the dead volume and raw breakthrough curves. The ratio values lie on the first bisector, showing the validity of this assumption. This similarity of ratio $(R_i^{+/-})$ of dead volume and raw curves supports the assumption of additivity of the dead volume and raw curves, like in the PBP method. So, in the ESA method, the variation of the subtraction of the elementary surfaces of the dead volume and raw curves is an increasing function of the concentration C (or $C/C_o$) in these experimental conditions. This is why the breakthrough curve corrected by the ESA method does not show fluctuations as in the PBP method. This could be explained by the laminar flow and high pressures of these experimental conditions, which are favorable to a spreading of the breakthrough curves, particularly for the raw curve by the addition of the free volume of the porous media.

### 4.4. Influence of gas composition on the shape of the breakthrough curve

This study is carried out in an inlet volumetric fraction of methane in nitrogen range from 0.1 to 0.3 ±0.01 with a constant column's pressure of 30 ±0.1 bar at a temperature of 22 ± 1°C.

Three cases, summarized in the Table 4, are investigated in this section.

The application of the PBP and ESA methods to the three cases is shown in Fig. 10. Observation of Fig. 10a shows an increase in the amplitude of the curves as composition increases, accompanied by an increase in the steepness of the slope of the curves. Like previously, fluctuations in correction of breakthrough curves are observed by the PBP method. So, in the same approach than Figs. 8, 8 % of concentration points in Case 5, 20 % for Case 6, 30 % for Case 7 do not satisfy the
<figure><img src="image_10.png" /><figcaption>Fig. 9. Validity test of the assumption (Equ.8) of ESA method at different pressures: 10 bar (Fig. 9a), 20 bar (Fig. 9b), 30 bar (Fig. 9c), 40 bar (Fig. 9d).</figcaption></figure>

<table><caption>**Table 4**

Experimental parameters to study the effect of gas composition on breakthrough curve.</caption><thead><tr><th>Parameter</th><th>Case 5</th><th>Case 6</th><th>Case 7</th></tr></thead><tbody><tr><td>Colum pressure <i>P</i> (bar)</td><td>30</td><td>30</td><td>30</td></tr><tr><td>Temperature <i>T</i> (°C)</td><td>22</td><td>22</td><td>22</td></tr><tr><td>Inlet volumetric fraction <i>y</i></td><td>0.1</td><td>0.2</td><td>0.3</td></tr><tr><td>Interstitial velocity <i>U</i><sub>i</sub> (×10<sup>3</sup> m/s)</td><td>1.03</td><td>1.15</td><td>1.32</td></tr><tr><td>Gas density <i>ρ</i> (kg.m<sup>-3</sup>)</td><td>33.04</td><td>31.68</td><td>30.32</td></tr><tr><td>Gas dynamic viscosity <i>μ</i> (×10<sup>5</sup> Pa.s)</td><td>1.81</td><td>1.81</td><td>1.81</td></tr><tr><td>Reynolds number <i>Re</i></td><td>0.94</td><td>1.01</td><td>1.11</td></tr></tbody></table>

assumption of Eq.4 respectively, which corresponds to an average of about 10% assumption invalidity by volume fraction range of 0.1. This may be explained by the steeper slope of the curves as composition increases, making it more difficult to differentiate the dead volume curve from the raw one.

Plotting y/y<sub>o</sub> versus time in Fig. 10b highlights quite similar curve shapes in all three cases, which seems to demonstrate the preponderant effect of pressure on the broadening of the breakthrough curve and not that of the relative composition.

Input composition affects breakthrough time, as shown in Fig. 10. Indeed, an increase in composition generates a decrease in breakthrough time at constant pressure, because of a higher methane flux, resulting in faster apparition of methane breakthrough curve. This phenomenon is also enabled by a slight increase in interstitial velocity of gas stream in three cases, experimentally required to set the inlet methane volumetric fraction. But, this change in interstitial velocity has a minor effect on reducing breakthrough time.

On the other hand, the corrected curves obtained using the ESA method exhibit a smoother profile in Fig. 10, like in Fig. 7, due to the validity of the assumption of Eq.8. This method is well adapted to the determination of axial dispersion coefficient in a porous media without adsorption.

### 4.5. Comparison of PBP and ESA methods

In the PBP method, the dead volume response is subtracted point by point from the raw breakthrough response, considering their linear additivity. In ESA method, the same assumption of linear additivity is required but not expressed on a single point but on an elementary surface. Each elementary surface is built with two points, which corresponds to a concentration increment instead of a single concentration (PBP method), thus smoothing out the differences observed with the PBP method. Therefore, the ESA method could be considered as an improvement of the PBP one.

## 5. Conclusion

In this new experimental device, the column volume (equal to 20.4 mL) is the same order of magnitude as the dead volume (30 mL). At laboratory scale, it is essential to correct the effect of dead volume on the raw breakthrough curve. The corrected curve will be then used to determine the axial dispersion in the fixed bed filled with an inert porous media.

Analysis of breakthrough curves is based on exact knowledge of the quantities of solute contained in the gas stream as a function of time, which requires calibration of the measurement devices for both flow rate and gas stream composition.
<figure><img src="image_11.png" /><figcaption>Fig. 10. Corrected breakthrough curves using the ESA method (dotted lines) and PBP method (solid lines) to study the influence of gas composition.</figcaption></figure>

Calibration tests of mass flow controllers reveal perfect linearity of flow rates in the calibration range, although there is a deviation between the set and the real flow rates. The relative deviation is equal to 34 % for CO₂, 11 % for CH₄, 5 % for H₂ and 2 % for N₂. So the real flow rate is therefore measured systematically for each experiment at the column outlet using an electronic flowmeter.

Concerning the analyzer, the calibration study shows a quite good correlation between the analyzer responses and reference values, deduced from each gas flow measurement with an electronic flow meter. This study confirms the linearity of flow rates monitored with the mass flow controllers.

The study of the pressure effect in the shape of breakthrough curves, in a range from 10–40 bar, highlight the spread of breakthrough curves as a function of pressure. The increase of pressure results in both an increase in gas density and a decrease in gas flow rate. An increase of gas density leads to an increase of the strengths of molecular interactions in the fluid and at the fluid/particle interface in the porous media, leading to a broadening of the breakthrough curve. Corrected breakthrough curves by the PBP method show some fluctuations, which can be explained by a shape of dead volume and raw curves almost identical.

When the subtraction point by point of the PBP method is constant, then the corrected breakthrough has a plug flow shape. A slight variation of subtractions point by point around this constant results in slight fluctuations in the corrected curve.

The study of the gas composition in the shape of breakthrough curves shows that the relative composition (C/C₀ or y/y₀) seems to have no effect on the broadening of these curves.

In this study, the ESA method proved efficient to correct breakthrough curves. Like in PBP method, this one has the same assumption of linear additivity of dead volume and raw curves but not expressed on a single point but on an elementary surface. Each elementary surface is built with two points, which corresponds to a concentration increment instead of a single concentration for the PBP method, thus smoothing out the differences observed with the PBP method. To conclude, the ESA method could be considered as an improvement of PBP one.

### CRediT authorship contribution statement

Pierre Mocho: Conceptualization, Laurent Perrier: Methodology, Ahmed Grim: Funding acquisition.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

No data was used for the research described in the article.

### References

[1] F. Rainone, O. D'Agostino, A. Erto, M. Balsamo, A. Lancia, Biogas upgrading by adsorption onto activated carbon and carbon molecular sieves: experimental and modelling study in binary CO₂/CH₄ mixture, J. Environ. Chem. 9 (5) (2021) 106256, https://doi.org/10.1016/j.jece.2021.106256.

[2] F.G. Kerry, Industrial gas handbook: gas separation and purification, CRC press, 2007, https://doi.org/10.1201/9781420008265.

[3] O.W. Awe, Y. Zhao, A. Nzihou, D.P. Minh, N. Lyczko, A review of biogas utilisation, purification and upgrading technologies of biogas utilisation, Waste Biomass--. Valoriz. 8 (2017) 267-283, https://doi.org/10.1007/s12649-016-9826-4.

[4] P. Pullumbi, F. Brandani, S. Brandani, Gas separation by adsorption: technological drivers and opportunities for improvement, Curr. Opin. Chem. Eng. 24 (2019) 131-142, https://doi.org/10.1016/j.coche.2019.04.008.

[5] D.M. Ruthven, S. Farooq, S. Brandani, Work of separation in CO₂ capture: applicability of the value function, Chem. Eng. Sci. 126 (2005) 604–607, https://doi.org/10.1016/j.ces.2015.01.001.

[6] R.M.A. Roque-Malherbe, Adsorption and diffusion in nanoporous materials, CRC press, 2007, https://doi.org/10.1201/9781420046762.

[7] V.G. Gomes, K.W.K. Yee, Pressure swing adsorption for carbon dioxide sequestration from exhaust gases, Sep. Purif. Technol. 28 (2002) 161–171, https://doi.org/10.1016/j.ces.2015.01.001.

[8] S. Sircar, Recent developments in macroscopic measurement of multicomponent gas adsorption equilibria, kinetics, and heats, Ind. Eng. Chem. Res. 46 (2007) 2917-2927, https://doi.org/10.1021/ie0601293.

[9] L. Gao, Z. Wang, M. Liang, Y. Yu, L. Zhou, Experimental study of the relationship between particle size and methane sorption capacity in shale, J. Vis. Exp. (138) (2018) e57705, https://doi.org/10.3791/57705.

[10] D. Broom, Characterizing adsorbents for gas separations, Chem. Eng. Prog. 114 (3) (2018) 30–37.

[11] M. Bastos-Neto, A. Möller, R. Staudt, J. Böhm, R. Gläser, Dynamic bed measurements of CO adsorption on microporous adsorbents at high pressures for hydrogen purification processes, Sep. Purif. Technol. 77 (2) (2011) 251–260, https://doi.org/10.1016/j.seppur.2010.12.015.

[12] L.F. Zafanelli, E. Aly, A. Henrique, A.E. Rodrigues, G. Mouchaham, J.A. Silva, Green hydrogen recovery from natural gas grids by vacuum pressure swing adsorption, Ind. Eng. Chem. Res. 63 (14) (2024) 6333–6346, https://doi.org/10.1021/acs.iecr.3c04532.

[13] N.S. Wilkins, A. Rajendran, S. Farooq, Dynamic column breakthrough experiments for measurement of adsorption equilibrium and kinetics, Adsorption 27 (2021) 397–422, https://doi.org/10.1007/s10450-020-00269-6.

[14] E. Clatworthy, S. Ghojavand, R. Guillet-Nicolas, J.P. Gilson, P. Llewellyn, N. Nesterenko, S. Mintova, Dynamic adsorption of CO₂ by CHA zeolites – size matters, Chem. Eng. J. 471 (2023) 144557, https://doi.org/10.1016/j.cej.2023.144557.

[15] P.G. Aguilera, F.J. Gutiérrez Ortiz, Prediction of fixed-bed breakthrough curves for H2S adsorption from biogas: importance of axial dispersion for design, Chem. Eng. J. 289 (2016) 93–98, https://doi.org/10.1016/j.cej.2015.12.075.

[16] F. Gritti, A. Felinger, G. Guiochon, Influence of the errors made in the measurement of the extra-column volume on the accuracies of estimates of the column efficiency and the mass transfer kinetics parameters, J. Chromatogr. A 1136 (1) (2006) 57–72, https://doi.org/10.1016/j.chroma.2006.09.074.

[17] S. Farooq, M.J. Purdue, P. Goyal, Adsorption and diffusion of N₂ and CO₂ and their mixture on silica gel, Ind. Eng. Chem. Res. 58 (42) (2019) 19611–19622, https://doi.org/10.1021/acs.iecr.
[18] R. Delley, Modifying the gaussian peak shape with more than one time constant, Anal. Chem. 58 (11) (1986) 2344-2346, https://doi.org/10.1021/ac00124a051.

[19] K.J. Fountain, U.D. Neue, E.S. Grumbach, D.M. Diehl, Effects of extra-column band spreading, liquid chromatography system operating pressure, and column temperature on the performance of sub-2-µm porous particles, J. Chromatogr. A 1216 (32) (2009) 5979-5988, https://doi.org/10.1016/j.chroma.2009.06.044.

[20] S. Katsuo, C. Langel, P. Schanen, M. Mazzotti, Extra-column dead volume in simulated moving bed separations: theory and experiments, J. Chromatogr. A 1216 (7) (2009) 1084-1093, https://doi.org/10.1016/j.chroma.2008.12.031.

[21] M. Martin, G. Blu, C. Eon, G. Guiochon, Optimization of column design and operating parameters in high speed liquid chromatography, J. Chromatogr. Sci. 12 (8) (1974) 438-448, https://doi.org/10.1093/chromsci/12.8.438.

[22] T. Tsuda, M. Novotny, Band-broadening phenomena in microcapillary tubes under the conditions of liquid chromatography, Anal. Chem. 50 (4) (1978) 632-634, https://doi.org/10.1021/ac50026a023.

[23] A. Rajendran, V. Kariwala, S. Farooq, Correction procedures for extra-column effects in dynamic column breakthrough experiments, Chem. Eng. Sci. 63 (10) (2008) 2696-2706, https://doi.org/10.1016/j.ces.2008.02.023.

[24] Guntuka, S. Farooq, A. Rajendran, A- and B-site substituted lanthanum cobaltite perovskite as high temperature oxygen sorbent. 2. column dynamics study, Ind. Eng. Chem. Res. 47 (1) (2008) 163-170, https://doi.org/10.1021/ie070860p.

[25] O. Levenspiel. Chemical Reaction Engineering, 3rd Edition, Wiley, New York, 1999.

[26] L. Joss, M. Mazzotti, Modeling the extra-column volume in a small column setup for bulk gas adsorption, Adsorption 18 (5-6) (2012) 381-393, https://doi.org/10.1007/s10450-012-9417-z.

[27] L. Perrier, F. Plantier, D. Grégoire, A novel experimental setup for simultaneous adsorption and induced deformation measurements in microporous materials, Rev. Sci. Instrum. 88 (3) (2017) 035104, https://doi.org/10.1063/1.4977595.

[28] J. Rouquerol, F. Rouquerol, P. Llewellyn, G. Maurin, K.S.W. Sing, Adsorption by powders and porous solids: principles, methodology and applications, Academic press, 2013, https://doi.org/10.1016/C2010-0-66232-8.

[29] AGA8-DC92, Compressibility and super compressibility for natural gas and other hydrocarbon gases, Transm. Meas. Comm. Rep. 8 (1992).

[30] P. Serra-Crespo, R. Berger, W. Yang, J. Gascon, F. Kapteijn, Separation of CO2/CH4 mixtures over NH2-MIL-53 - an experimental and modelling study, Chem. Eng. Sci. 124 (2015) 96-108, https://doi.org/10.1016/j.ces.2014.10.028.

[31] R.E. Treybal, Mass-Transfer Operations, McGraw-Hill Book Company, 1980. ISBN: 0-07-065176-0.

[32] E.W. Lemmon, R.T. Jacobsen, Viscosity and thermal conductivity equations for nitrogen, oxygen, argon and air, Int. J. Thermophys. 25 (1) (2004) 21-69, https://doi.org/10.1023/B:IJOT.0000022327.04529.f3.

[33] M. Edwards, J. Richardson, Gas dispersion in packed beds, Chem. Eng. Sci. 23 (2) (1968) 109-123, https://doi.org/10.1016/0009-2509(68)87056-3.

[34] F.J. Gutiérrez Ortiz, M. Barragán Rodríguez, R.T. Yang, Modeling of fixed-bed columns for gas physical adsorption, Chem. Eng. J. 378 (2019) 121985, https://doi.org/10.1016/j.cej.2019.121985.