# Absorption in a pseudo-first-order regime with two irreversible reactions: Influence of the second reaction on the transfer rate

Arnaud Delanney, Alain Ledoux, Lionel Estel

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Arnaud Delanney, Alain Ledoux, Lionel Estel. Absorption in a pseudo-first-order regime with two irreversible reactions: Influence of the second reaction on the transfer rate. Chemical Engineering Journal, 2024, pp.153724. 10.1016/j.cej.2024.153724. hal-04638127

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Submitted on 8 Nov 2024

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# Absorption in a pseudo-first-order regime with two irreversible reactions: Influence of the second reaction on the transfer rate

Arnaud Delanney, Alain Ledoux*, Lionel Estel

INSA Rouen Normandie, Univ Rouen Normandie, Normandie Univ, LSPC UR 4704, F-76000 Rouen, France

## ARTICLE INFO

### Keywords:

Gas-liquid mass transfer

Pseudo-first-order

Concentration profiles

Generalized equations

## ABSTRACT

Gas-absorbing liquids such as CO₂ often undergo two reactions: a main reaction and a secondary reaction. The solute flux transferred is a result of both reactions. Kinetics is one of the most important properties in the design of an absorber. To access the rate constant of the main reaction, the assumption of a pseudo-first-order (PFO) regime is often made, but the criteria for the system to be in a pseudo-first-order regime change between researchers. Furthermore, the impact of the secondary reaction is either neglected or overestimated. Recently, some researchers carried out simulations of multiple parallel gas-liquid reactions in the instantaneous regime and developed generalized criteria for this regime. In this study, to estimate the impact of the secondary reaction in a PFO regime, hundreds of simulations were carried out with two irreversible reactions within the framework of the film theory. This work proves that for $Ha_1 > 10$, $Ha_2 > 10$, $Ha_1/E_{\infty,1} < 0.5$, $Ha_2/E_{\infty,2} < 0.5$, normalized concentration profiles depend only on three factors: $Ha_1/E_{\infty,1}$, $Ha_2/E_{\infty,2}$, and $Ha_1/Ha_2$. Curves and generalized equations are provided enabling the estimation of the concentrations of reactants at the interface, the enhancement factor, and the influence of each reaction on the gas transfer flux (average absolute relative deviation ≤ 0.50 %). New criteria for determining whether the system can be considered in a pseudo-first-order regime are defined, giving accurate predictions with a success rate of 96.2 %.

## 1. Introduction

Mitigating climate change is one of the greatest challenges facing humankind today, with the principal cause being the greenhouse effect caused by human gas emissions, particularly CO₂. According to the Intergovernmental Panel on Climate Change [1], CO₂ capture is part of the solution, with chemical absorption being the most advanced process. Research into new absorption systems such as amino acid salts, mixtures of 2-amino-2-methyl-1-propanol and piperazine, or ionic liquids is very active, and precise determination of properties is therefore crucial. Indeed, inaccurate models or correlations can lead to underestimation or overestimation of absorber contactor size [2]. Among these properties, the rate constants are particularly important [3,4]. If the kinetics constant is high, the transfer of CO₂ is accelerated. To define the influence of the reactions on the transfer, the flux of species A transferred is expressed according to Eq. (1). In this expression, the enhancement factor can depend on the hydrodynamics conditions, the kinetics of the reaction, and the diffusivities.

$$ \phi = k_{liq} \cdot E_A \cdot C_{A,int} \quad (1) $$

Estimation of the rate constant is based on a system of equations and reaction mechanisms, and a transfer model. In the case of CO₂ absorption, the reaction system often consists of two parallel second-order reactions. According to the Zwitterion mechanism, the main reaction for alkanolamines [5], primary amines including MEA [6], and certain amino acid salts [5-7] is the reaction between CO₂ and the basic species introduced in large quantities. The frequent second reaction is a second-order reaction between CO₂ and OH⁻. The contribution of the second reaction to the transfer of CO₂ from the gas to the liquid is either neglected [6-8] or considered [5,9]. For instance, Thee et al. [5] state that for a pH above 10, the secondary reaction must be considered in the transfer flux. This is particularly true for strong bases. For example, the pH increases with the pKa of amino acid salts [5]. In this case, if the pH is high, the flux of A because of this reaction can be non-negligible. Not taking it into account may lead to an overestimation of the kinetics of the first reaction and its actual impact on solute transfer. In contrast, Portugal et al. [6] neglect the secondary reaction without calculating its effect but by indicating that the glycinate ion is a weak base.

Kinetics is a key when choosing a solvent, and this decision is based on the rate constant of the main reaction. This is one of the reasons why knowledge of the proportion of solute flux transferred by each reaction

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: alain.ledoux@insa-rouen.fr (A. Ledoux).</footnote>
<table><thead><tr><th colspan="2">Nomenclature</th></tr></thead><tbody><tr><td colspan="2"><strong>Symbol or abbreviation</strong></td></tr><tr><td>A</td><td>Normalized concentration of A (–)</td></tr><tr><td>A<sub>int</sub></td><td>Normalized concentration of A at the interface gas/liquid (–)</td></tr><tr><td>A<sub>m</sub></td><td>Average normalized concentration between x = 0 and δ<sub>r</sub> (–)</td></tr><tr><td>B</td><td>Normalized concentration of B (–)</td></tr><tr><td>B<sub>bulk</sub></td><td>Normalized concentration of B in the bulk of the liquid (–)</td></tr><tr><td>B<sub>int</sub></td><td>Normalized concentration of B at the interface gas/liquid (–)</td></tr><tr><td>B<sub>int,limit</sub></td><td>Lower limit of the normalized concentration of B at the interface gas/liquid (–)</td></tr><tr><td>C<sub>A</sub></td><td>Concentration of A (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>A,int</sub></td><td>Concentration of A at the interface gas/liquid (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>A,L</sub></td><td>Concentration of A in the bulk of the liquid (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>B</sub></td><td>Concentration of B (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>B,int</sub></td><td>Concentration of B at the interface gas/liquid (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>B,L</sub></td><td>Concentration of B in the bulk of the liquid (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>P</sub></td><td>Concentration of P (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>P,int</sub></td><td>Concentration of P at the interface gas/liquid (mol/m<sup>3</sup>)</td></tr><tr><td>C<sub>P,L</sub></td><td>Concentration of B in the bulk of the liquid (mol/m<sup>3</sup>)</td></tr><tr><td>D<sub>A,liq</sub></td><td>Diffusivity of A in the liquid (m<sup>2</sup>/s)</td></tr><tr><td>D<sub>B,liq</sub></td><td>Diffusivity of B in the liquid (m<sup>2</sup>/s)</td></tr><tr><td>D<sub>P,liq</sub></td><td>Diffusivity of P in the liquid (m<sup>2</sup>/s)</td></tr><tr><td>E<sub>A</sub></td><td>Enhancement factor (–)</td></tr><tr><td>E<sub>∞,1</sub></td><td>Enhancement factor limit for Reaction 1 (–)</td></tr><tr><td>E<sub>∞,2</sub></td><td>Enhancement factor limit for Reaction 2 (–)</td></tr><tr><td>Ha</td><td>Hatta number (–)</td></tr><tr><td>Ha<sub>1</sub></td><td>Hatta number associated with Reaction 1 (–)</td></tr><tr><td>Ha<sub>2</sub></td><td>Hatta number associated with Reaction 2 (–)</td></tr><tr><td>HL</td><td>Ratio between the lower limit of the normalized concentration at the interface not to be underpassed and the normalized concentration in the bulk (–)</td></tr><tr><td>HL<sub>B</sub></td><td>Ratio between the lower limit of the normalized concentration of B at the interface not to be underpassed and the concentration in the bulk (–)</td></tr><tr><td>HL<sub>P</sub></td><td>Ratio between the lower limit of the normalized concentration of P at the interface not to be underpassed and the concentration in the bulk (–)</td></tr><tr><td>k<sub>1</sub></td><td>Rate constant of the second-order Reaction 1 (mol/m<sup>3</sup>·s)</td></tr><tr><td>k<sub>2</sub></td><td>Rate constant of the second-order Reaction 2 (mol/m<sup>3</sup>·s)</td></tr><tr><td>P</td><td>Normalized concentration of P (–)</td></tr><tr><td>P<sub>bulk</sub></td><td>Normalized concentration of P in the bulk of the liquid (–)</td></tr><tr><td>P<sub>int,limit</sub></td><td>Lower limit of the normalized concentration of P at the interface gas/liquid (–)</td></tr><tr><td>PFO</td><td>Pseudo-first-order</td></tr><tr><td>P<sub>int</sub></td><td>Normalized concentration of P at the interface gas/liquid (–)</td></tr><tr><td>R<sub>A,1</sub></td><td>Proportion of A transferred reacted in Reaction 1 (%)</td></tr><tr><td>R<sub>A,2</sub></td><td>Proportion of A transferred reacted in Reaction 2 (%)</td></tr><tr><td>x</td><td>Distance from the interface (m)</td></tr><tr><td>X</td><td>Normalized distance from the interface (–)</td></tr><tr><td>δ</td><td>Liquid film thickness (m)</td></tr><tr><td>δ<sub>r</sub></td><td>Distance from which A = 0 (m)</td></tr><tr><td>φ</td><td>Molar flux of A transferring from the gas to the liquid (mol/m<sup>2</sup>·s)</td></tr></tbody></table>

is important.

However, if the secondary reaction is considered, its contribution in practice is always overestimated, as the concentration of the species reacting with A at the interface is assumed to be equal to that of the liquid bulk due to the assumption of the pseudo-first-order (PFO) regime. The concentration of this species at the interface is lower than the concentration in the bulk of the liquid. Knowing the concentration at the interface makes it possible to calculate the rate constants of the main reaction more accurately. In addition, some researchers demonstrated that the absorption of CO₂ can be effective with a mixture of two active species. This mixture is made on purpose and the species can be introduced in almost identical proportions. In this case, the transfer caused by the second reaction cannot be neglected and it is crucial to know the impact of each reaction to determine the rate constants.

Indeed, to estimate rate constants, regardless of the technique, stirred reactor, or wetted-wall column, most researchers [5,7–13] impose conditions for the system to be in a PFO regime. In fact, in this regime, the CO₂ transfer rate does not depend on the partial transfer constant (k<sub>liq</sub>). Indeed, the relative uncertainty of this constant is far from negligible, and most researchers wish to avoid this uncertainty when assessing rate constants.

Westerterp et al. [14] studied PFO conditions in a system with two parallel irreversible reactions, i.e. a system similar to that of CO₂ absorption, with species A passing from gas to liquid to react with B or P (see Reactions 1 and 2). In the case of CO₂ capture, A would correspond to CO₂, while B and P are the base species.

$$A + \nu_B B \rightarrow \text{products}_1 \quad (\text{Reaction1}) \\ \text{with } r_1 = k_1 \cdot C_A \cdot C_B$$

$$A + \nu_P P \rightarrow \text{products}_2 \quad (\text{Reaction2}) \\ \text{with } r_2 = k_2 \cdot C_A \cdot C_P$$

They establish the molar balances of species A, B, and P and determine three asymptotic regimes, including the PFO regime. The PFO is a regime corresponding to quasi-uniform concentrations of reactants reacting with CO₂ in the liquid film. Concentration profiles and transfer rates depend on the physical movements (diffusion) of the different species and chemical processes (reactions). To take these phenomena into account, Westerterp et al. define a Hatta number which is the square of ‘the ratio of the maximum conversion of A in the film per unit area interface with reaction and the maximum diffusional transport through the film in the absence of a reaction’. In the case of two parallel reactions, the Hatta number is defined by Eq. (2). Moreover, they define two limits for the enhancement factors relating to reactions 1 and 2 according to the film theory (see Eqs. (3)–(4)). Westerterp et al. consider that, if Eqs. (5) and (6) are respected, the system is in the PFO regime. The condition of Ha > 3 corresponds to fast reactions. If one of the reactions is fast, A reacts in the film. However, if one of the reactions is too fast, B or P may not have a flat profile because their transfer is limited by their diffusivity. This is the meaning of the two upper inequalities, i.e. Eqs. (5) and (6). However, these conditions are sufficient but not necessary. Some systems may not fulfil one of the higher conditions but can be considered in a PFO regime, i.e., the concentrations of B and P can be considered quasi-uniform in the liquid film. Indeed, the profile of B is only impacted by the flux of A reacting with B (reaction 1). It does not depend on the flux of A reacting with P. In the same way, the profile of P is only directly impacted by the flux of A reacting with P, not B. The two upper conditions of Eqs. (5) and (6) do not distinguish between these two influences. To accurately estimate whether the transport of B or P is limited by diffusion, it is important to introduce specific Hatta numbers relating to the reactions (Eqs. (7–8)) and not exclusively to the rate of A. Ha₁ is the ratio of the maximum transferable flux of A due to the first reaction with C<sub>B</sub> = C<sub>B,L</sub> in the whole liquid film, to the maximum transferable flux of A if there were no reaction in the film. E<sub>∞,1</sub> is the maximum enhancement factor due to Reaction 1 (instantaneous regime). Similarly, Ha₂ is the ratio of the maximum transferable flux of A
due to the second reaction with $C_p = C_{p,L}$ in the whole liquid film, to the maximum transferable flux of A if no reaction occurs in the film. $E_{\infty,2}$ is the maximum increase factor due to Reaction 2 (instantaneous regime).

$$Ha = \frac{\sqrt{k_1 \cdot D_{A,liq} \cdot C_{B,L} + k_2 \cdot D_{A,liq} \cdot C_{P,L}}}{k_{liq}} \quad (2)$$

$$E_{\infty,1} = 1 + \frac{D_{B,liq} \cdot C_{B,L}}{\nu_B \cdot D_{A,liq} \cdot C_{A,int}} \quad (3)$$

$$E_{\infty,2} = 1 + \frac{D_{P,liq} \cdot C_{P,L}}{\nu_P \cdot D_{A,liq} \cdot C_{A,int}} \quad (4)$$

$$3 < Ha \ll E_{\infty,1} \quad (5)$$

$$3 < Ha \ll E_{\infty,2} \quad (6)$$

$$Ha_1 = \frac{\sqrt{k_1 \cdot D_{A,liq} \cdot C_{B,L}}}{k_{liq}} \quad (7)$$

$$Ha_2 = \frac{\sqrt{k_2 \cdot D_{A,liq} \cdot C_{P,L}}}{k_{liq}} \quad (8)$$

$$3 < Ha_1 \ll E_{\infty,1} \quad (9)$$

In practice, the criteria for determining whether a system is in a PFO regime vary among researchers and differ from those of Westerterp et al. [14]. Some consider that the system is in a PFO regime if Inequality (5) is satisfied, while others refer to Inequality (9), depending on whether the influence of the secondary reaction is neglected. In all cases, no comparison between $Ha_2$ or $Ha$ is made with $E_{\infty,2}$, unlike Westerterp et al. [14], and only one Hatta number is considered.

In addition, the upper limit of the condition varies from one researcher to another. Some consider that if $Ha < 0.2 E_{\infty}$ [7], the system is in a PFO regime, while others consider that $Ha < 0.5 E_{\infty}$ is a sufficient criterion [15]. Most of them do not indicate an upper limit but provide an experimental ratio of $Ha/E_{\infty,1}$. The meaning of this ratio and its origin are rarely remembered. This ratio comes from the study of a single second-order reaction ($A + \nu_B B \rightarrow \text{products with } r = k_1 \cdot C_A \cdot C_B$). Westerterp et al. [14] remind us that to consider a reaction as in a PFO regime, $C_B$ must be considered to be uniform over the entire thickness of the liquid film and equal to its value in the mass of the liquid $C_{B,L}$. The condition to be met between the concentration at the interface for B and the concentration in the liquid bulk is set up by considering the profiles of concentrations of the asymptotic case of one instantaneous reaction.

This is defined as follows: $\frac{C_{B,L}-C_{B,int}}{C_{B,L}} \ll 1$, which is equivalent to $\frac{\nu_B \cdot D_{A,liq} \cdot C_{A,int}}{D_{B,liq} \cdot C_{B,L}} \cdot Ha \ll 1$. The condition becomes: $Ha \ll \frac{D_{B,liq} \cdot C_{B,L}}{\nu_B \cdot D_{A,liq} \cdot C_{A,int}} = E_{\infty,1} - 1 \approx E_{\infty,1}$. Therefore, when one considers that if $Ha < 0.2 E_{\infty}$, the system is in the PFO regime, it means that $\frac{C_{B,L}-C_{B,int}}{C_{B,L}} < 0.2$. By extension of the single irreversible reaction, this criterion is used in practice for a system with two reactions. Fig. 1 shows an example of profiles of normalized concentrations.

On the one hand, the conditions of Westerterp et al. are sufficient, but very strict and difficult to meet. On the other hand, not taking the condition of upper Inequality (6) into account can lead to an error in the rate constant.

In addition, beyond the case of the parametric determination of rate constants, during the operation of an absorption column, it is essential to know or predict the proportion of A reacting in either of the two reactions, to estimate or predict the consumption of the various reagents.

There are many objectives of this study. Firstly, different concentration profiles and transfer rates for three case studies are provided, enabling the estimation of the error in the rate and the profile achieved with the PFO hypothesis. Secondly, one objective is to define universal factors leading to the concentration profiles and transfer rates. The final

<figure><img src="image_4.png" /><figcaption>Fig. 1. Normalized concentrations as a function of normalized distance X.</figcaption></figure>

objective is to establish easy-to-use generalized equations providing the concentration profiles, enhancement factor, and transfer rates without solving the partial differential equations of transfer both for systems in the PFO regime and for those which are close to the PFO regime. Some researchers have found analytical solutions to these equations. Van Swaaj and Versteeg [16] wrote an extensive overview of absorption with complex chemical reactions. In this overview, a trial-and-error method on several equations edited by Jhaveri [17] is mentioned, however, this method has been questioned by Alper [18]. Onda et al. [19] also developed a trial-and-error method on a complex system of equations, comparing their findings with the results of simulations. Their results aligned with simulation results with a maximum deviation of 6% for the enhancement factor. However, these equations do not use dimensionless terms such as $Ha$, $Ha_1$, and $Ha_2$. In addition, the system is quite complex and does not directly provide the proportion of the transfer rate due to each reaction. To the best of our knowledge, no easy-to-use correlation has been established since this overview.

The three objectives should enable researchers to precisely determine the proportion of the transfer rate due to each reaction. Moreover, this study helps researchers validate the hypotheses concerning the negligible or non-negligible nature of the secondary reaction, and allows for more precise estimations of the rate constant.

## 2. Methods

### 2.1. Molar balances

The reaction system is the same as Westerterp et al. [14] and a similar notation to that of Janssens et al. [20] is adopted.

$$A + \nu_B B \rightarrow \text{products}_1 \text{ with } r_1 = k_1 \cdot C_A \cdot C_B$$

$$A + \nu_P P \rightarrow \text{products}_2 \text{ with } r_2 = k_2 \cdot C_A \cdot C_P$$

The establishment of molar balances for components A, B and P over an elementary volume in the film liquid yields Eqs. (10)–(12) in an unsteady regime. The demonstration of Eqs. (10)–(12) is provided in Appendix A..
$$ \left(\frac{\partial^2 A}{\partial X^2}\right) - Ha_1^2 \cdot A \cdot B - Ha_2^2 \cdot A \cdot P = \frac{D_{A,liq}}{k_{liq}^2} \cdot \frac{\partial A}{\partial t} \quad (10) $$

$$ \left(\frac{\partial^2 B}{\partial X^2}\right) - \frac{Ha_1^2}{E_{\infty,1}-1} \cdot A \cdot B = \frac{D_{A,liq}^2}{D_{B,liq} \cdot k_{liq}^2} \cdot \frac{\partial B}{\partial t} \quad (11) $$

$$ \left(\frac{\partial^2 P}{\partial X^2}\right) - \frac{Ha_2^2}{E_{\infty,2}-1} \cdot A \cdot P = \frac{D_{A,liq}^2}{D_{P,liq} \cdot k_{liq}^2} \cdot \frac{\partial P}{\partial t} \quad (12) $$

A, B and P are the normalized concentrations at the instant t and at the normalized distance X from the gas-liquid interface, as defined according to Eqs. (13)-(16).

$$ X = \frac{\text{distance from the interface gas/liquid}}{\text{liquid film thickness}} = \frac{x}{\delta} \quad (13) $$

$$ A = \frac{C_A(x)}{C_A(x=0)} = \frac{C_A(x)}{C_{A,int}} \quad (14) $$

$$ B = \frac{C_B(x)}{C_B(x=\delta)} = \frac{C_B(x)}{C_{B,L}} \quad (15) $$

$$ P = \frac{C_P(x)}{C_P(x=\delta)} = \frac{C_P(x)}{C_{P,L}} \quad (16) $$

## 2.2. Numerical solutions

Eqs. (10)-(12) are parabolic partial differential equations and cannot be solved by analytical methods. To determine the concentration profiles and rate at the stationary regime, one technique consists of meshing space and discretizing time. Time is discretized into $\Delta t$ intervals and space into $\Delta x$ intervals. The initial and boundary conditions of the problem are given respectively by Eqs. (17)-(19) and (20)-(25).

Initial conditions, for $X \in [0; 1]$;

$$ A = 0 \quad (17) $$

$$ B = 1 \quad (18) $$

$$ P = 1 \quad (19) $$

Boundary conditions, for $t \ge 0$ and $X = 0$ (interface);

$$ A = 1 \quad (20) $$

$$ \frac{\partial B}{\partial X} = 0 \quad (21) $$

$$ \frac{\partial P}{\partial X} = 0 \quad (22) $$

Eqs. (21) and (22) are due to the non-transfer of B and P in the gas. Indeed, B and P are considered as non-volatile.

Boundary conditions, for $t \ge 0$ and $X = 1$;

$$ A = 0 \quad (23) $$

$$ B = 1 \quad (24) $$

$$ P = 1 \quad (25) $$

Time is integrated until the concentrations in the space do not evolve. This algorithm converges provided that the stability conditions are met (Eqs. (26)-(28)).

$$ \frac{k_{liq}^2 \cdot \Delta t}{D_{A,liq} \cdot \Delta x^2} < 0.5 \quad (26) $$

$$ \frac{D_{B,liq} \cdot k_{liq}^2 \cdot \Delta t}{D_{A,liq}^2 \cdot \Delta x^2} < 0.5 \quad (27) $$

$$ \frac{D_{P,liq} \cdot k_{liq}^2 \cdot \Delta t}{D_{A,liq}^2 \cdot \Delta x^2} < 0.5 \quad (28) $$

This first technique is accurate but has a considerable drawback: the integration time is too long to generate a large number of simulations. To reach the goals of the study, an approach is to run a large number of simulations covering a wide range of concentration conditions for three case studies, compiling results. The long time of simulations is a major barrier. One way of overcoming this drawback comes from Janssens et al. [20] who worked on the same system as this study but to define remarkably improved and generalized criteria for describing the instantaneous regime. To solve the partial differential equations, Janssens et al. [20] used a MATLAB's built-in pdepe function to find the stationary solution [21]. This function allows us to solve a system of parabolic equations by using ordinary space differential equations at different times. The originality of the approach of Janssens et al. [20] approach lies in the definition of a damping factor $\Delta$. For a given time, they increase this factor until the solution no longer changes: the stationary state is then achieved. The same method is used in this study with Eqs. (29)-(31), and the same initial and boundary conditions as those of the first method (Eqs. (17)-(25)).

$$ \left(\frac{\partial^2 A}{\partial X^2}\right) - Ha_1^2 \cdot A \cdot B - Ha_2^2 \cdot A \cdot P = \Delta \cdot \frac{\partial A}{\partial t} \quad (29) $$

$$ \left(\frac{\partial^2 B}{\partial X^2}\right) - \frac{Ha_1^2}{E_{\infty,1}-1} \cdot A \cdot B = \Delta \cdot \frac{\partial B}{\partial t} \quad (30) $$

$$ \left(\frac{\partial^2 P}{\partial X^2}\right) - \frac{Ha_2^2}{E_{\infty,2}-1} \cdot A \cdot P = \Delta \cdot \frac{\partial P}{\partial t} \quad (31) $$

Once the normalized concentrations as a function of X have been found for the steady state, the enhancement factor $E_A$ is determined in two ways (Eqs. (32), (33)), to ensure its value and to confirm that the molar balance is correct. Demonstrations of Eqs. (32) and (33) are provided in Appendix B.

$$ E_A = \left(\frac{\partial A}{\partial X}\right)_{X=0} \quad (32) $$

$$ E_A = (E_{\infty,1} - 1) \cdot \left(\frac{\partial B}{\partial X}\right)_{X=1} + (E_{\infty,2} - 1) \cdot \left(\frac{\partial P}{\partial X}\right)_{X=1} \quad (33) $$

The proportions of the mass transfer rate of A due to reaction with B or due to reaction with P are estimated by Eqs (34) and (35) respectively.

$$ R_{A1} = \frac{(E_{\infty,1} - 1) \cdot \left(\frac{\partial B}{\partial X}\right)_{X=1}}{(E_{\infty,1} - 1) \cdot \left(\frac{\partial B}{\partial X}\right)_{X=1} + (E_{\infty,2} - 1) \cdot \left(\frac{\partial P}{\partial X}\right)_{X=1}} \quad (34) $$

$$ R_{A2} = \frac{(E_{\infty,2} - 1) \cdot \left(\frac{\partial P}{\partial X}\right)_{X=1}}{(E_{\infty,1} - 1) \cdot \left(\frac{\partial B}{\partial X}\right)_{X=1} + (E_{\infty,2} - 1) \cdot \left(\frac{\partial P}{\partial X}\right)_{X=1}} \quad (35) $$

## 2.3. Simulation plan

First, to design the simulation plan, it is necessary to choose the relevant variables. As a reminder, $Ha_1$ is the ratio of the maximum transferable flux of A due to the first reaction with $C_B = C_{b,L}$ in the whole liquid film, to the maximum transferable flux of A if there were no reaction in the film. $E_{\infty,1}$, is the maximum enhancement factor due to Reaction 1. For an instantaneous reaction, $E_A = E_{\infty,1}$, i.e. Reaction 1 is so fast that the transfer is not driven by the reaction but by the diffusion of A and B. The concentration profiles are very different and are those of instantaneous reactions. The ratio $Ha_1/E_{\infty,1}$ is always the indicator used
for a single reaction system. This indicator seems relevant for a two-reaction system because its value can have an impact on the profiles of concentrations. In the same way, the ratio $Ha_2/E_{\infty,2}$ seems relevant for the profile of P. Finally, it is intuitive that the ratio between the maximum flux of transferable A due to reaction 1 and the maximum flux of transferable A due to Reaction 2 impacts the profiles. For instance, if $Ha_1 \gg Ha_2$, we can expect the P concentration profile to be close to that of the PFO regime profiles. Therefore, the $Ha_1/Ha_2$ ratio is introduced.

The simulation plan is designed to cover different concentration study cases. As this study focuses on the PFO regime, $Ha_1$ and $Ha_2$ are chosen greater than 3. The greater the number of Ha, the faster the reactions compared with the physical transfer.

The different conditions are given in Table 1. The plan is divided into three main conditions: Level 1 ($Ha_1 - 10$), Level 2 ($Ha_1 = 200$) and Level 3 ($Ha_1 = 500$). The number of simulations for levels 1, 2, and 3 are 72, 108, and 108, respectively.

In addition, different $Ha_1/Ha_2$ ratios are tested. $Ha_1/Ha_2 = 10$ corresponds to a strong influence of Reaction 1 over Reaction 2 on the mass transfer rate of A. Typically, this corresponds to the situation with a main reaction and a slower secondary reaction, or with low P concentrations.

When $Ha_1/Ha_2 = 1$, it is possible for us to assess the influence of the reaction at equivalent reaction kinetics. The case with $Ha_1/Ha_2 = 2$ is an intermediate case. These last two cases may correspond to a mixture of reagents introduced at near-equal concentrations and rate constants.

These three case studies cover a wide range of cases, going beyond the single case with a main reaction and a slower secondary reaction.

# 3. Results and discussion

## 3.1. Results of simulations: Normalized concentrations at the interface, rate of mass transfer, and enhancement factor

### 3.1.1. General analysis

All results are given in Appendix C.

One of the objectives of the study is to determine whether the system can be considered to be in a PFO regime. In this case, the concentrations of B and P are uniform over the thickness of the liquid film. For a strict PFO regime, this would imply that the normalized concentrations of B and P are equal to 1, including at the interface.

In addition, the enhancement factor $E_A$ would be equal to the Hatta number $Ha$. The means of comparison between the extreme case of the PFO regime and the simulated cases consists of evaluating the difference between $E_A$ calculated in the simulation and $Ha$, as well as plotting the normalized concentrations as a function of the factors.

The relative difference between $E_A$ and $Ha$ is calculated according to Eq. (36).

$$ \text{Relative difference} = \frac{E_A - Ha}{Ha} \qquad (36) $$

Finally, to estimate the influence of each reaction on the transfer, we estimate the flux of A transferred due to each reaction.

It is necessary to assess whether the Westerterp criteria can be used to determine whether the system can be considered in a PFO regime. If we set the maximum ratio $Ha/E_{\infty,1}$ and $Ha/E_{\infty,2}$ to 0.20 for the upper

<table><caption>**Table 1**

Parameters of the simulations.</caption><thead><tr><th>Ha<sub>1</sub></th><th>Ha<sub>2</sub></th><th>Ha<sub>1</sub>/Ha<sub>2</sub></th><th>Ha<sub>1</sub>/E<sub>∞,1</sub></th><th>Ha<sub>2</sub>/E<sub>∞,2</sub></th></tr></thead><tbody><tr><td>10</td><td>5; 10</td><td>2; 1</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td></tr><tr><td>200</td><td>20; 100;<br>200</td><td>10; 2; 1</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td></tr><tr><td>500</td><td>50; 250;<br>500</td><td>10; 2; 1</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td><td>0.05; 0.10; 0.20; 0.30;<br>0.40; 0.50</td></tr></tbody></table>

condition of Westerterp et al. (Eqs. (5) and (6)), it is expected that $\frac{C_{B,L}-C_{B,int}}{C_{B,L}} < 0.2$ and $\frac{C_{P,L}-C_{P,int}}{C_{P,L}} < 0.2$. In other words, $\frac{1-B_{int}}{1} < 0.2$ and $\frac{1-P_{int}}{1} < 0.2$ i.e $B_{int} > 0.80$ and $P_{int} > 0.80$. Therefore, in this study, a system is considered to be in the PFO regime if $B_{int} > 0.80$ and $P_{int} > 0.80$. If one of these conditions is not met, the system is not in the PFO regime. The 288 simulations are tested (Fig. 2).

Fig. 2 shows that if the criteria of Westerterp et al. are respected (red rectangle), the system is in a PFO regime. However, in many conditions, the system is in a PFO regime without fulfilling their criteria, confirming that the criteria of Westerterp et al. are too strict. Furthermore, according to the distribution of systems in a PFO regime (Fig. 2), it is difficult to find new relevant criteria that depend exclusively on $Ha/E_{\infty,1}$ and $Ha/E_{\infty,2}$.

The second conclusion is relatively surprising. The relative difference between $E_A$ and $Ha$, as well as the normalized concentration of the interface $B_{int}$ and $P_{int}$, depend solely on three parameters: the ratios $Ha_1/E_{\infty,1}$, $Ha_2/E_{\infty,2}$, and $Ha_1/Ha_2$ regardless of the values of $Ha_1$. In other words, the profiles of concentrations and the enhancement factor are similar for $Ha_1 = 10$, $Ha_1 = 200$, and $Ha_1 = 500$. Thus, in Subsections 3.1.2, 3.1.3, and 3.1.4, all comments are made for $Ha_1 = 200$ but are valid for $Ha_1 = 10$ and $Ha_1 = 500$. (see figures in Appendix C).

Moreover, it is confirmed that $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$ are more relevant than $Ha/E_{\infty,1}$ and $Ha/E_{\infty,2}$.

### 3.1.2. Results for $Ha_1 = Ha_2$

In all the following subsections, the results are analysed by discussing the concentration profiles and the relative difference between $E_A$ and $Ha$. The conditions of the PFO regime are established and the contributions of each reaction to the rate of transfer are discussed.

Considering Fig. 3, a higher $Ha_1/E_{\infty,1}$ value correlates to a reduced $B_{int}$ value. This trend is due to the difficulty for B to diffuse from the bulk to the interface, which corresponds to a higher value of $Ha_1/E_{\infty,1}$. In Fig. 4, a similar trend is observed for $P_{int}$ which decreases when $Ha_2/E_{\infty,2}$ increases. If $B_{int}$ and $P_{int}$ decrease, the concentration profiles deviate from the PFO profiles.

Fig. 5 shows that for $Ha_1$ and $Ha_2$, $E_A$ is lower than $Ha$, as expected. Indeed, $E_A$ would be equal to $Ha$ if $C_B = C_{B,L}$ and $C_P = C_{P,L}$ in the entire

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Test of the PFO regime for 288 simulations (○ PFO regime; ▲ not in PFO regime).</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 3. Normalized concentration of B at the liquid-gas interface as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 200$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ♦ 0.1; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Normalized concentration of P at the liquid-gas interface as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 200$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ♦ 0.1; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

liquid film, which is an ideal and unrealistic situation. Moreover, the absolute difference increases with $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$. This is explained by the influence of $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$ on $B_{int}$ and $P_{int}$. If one sets the maximum ratio $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$ to 0.20 for the upper condition of Westerterp et al. (Eqs. (5) and (6)), only four simulation conditions satisfy the two upper conditions. These conditions are illustrated in Fig. 5 in larger sizes.

Nevertheless, as mentioned previously, a system is considered to be in a PFO regime if $B_{int} > 0.80$ and $P_{int} > 0.80$. From Fig. 3, it can be seen

<figure><img src="image_8.png" /><figcaption>Fig. 5. Relative difference between $Ha$ and $E_A$ as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 200$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ♦ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

that the normalized concentration of B at the interface is greater than 0.8 when $Ha_1/E_{\infty,1} \le 0.30$ and it is quasi-independent of the ratio $Ha_2/E_{\infty,2}$. Concerning $P_{int}$, in Fig. 4, its value is greater than 0.80 for simulations with a $Ha_2/E_{\infty,2}$ value smaller than 0.30. Its value is quasi-independent of $Ha_1/E_{\infty,1}$. The observations for $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$ are analogous because $Ha_1 = Ha_2$.

Therefore, the conditions respecting the PFO regime are circled in red in Fig. 5 and correspond to $Ha_1/E_{\infty,1} \le 0.30$ and $Ha_2/E_{\infty,2} \le 0.30$. The criteria used by Westerterp et al. to ensure that the system is in the PFO regime are correct, but too strict. Indeed, some experiments are excluded by these criteria.

The proportion of transfer rate of A due to Reaction 1 (Fig. 6), is close to 0.5, which is logical given the equality between $Ha_1$ and $Ha_2$. This proportion increases as $Ha_2/E_{\infty,2}$ increases and decreases as $Ha_1/E_{\infty,1}$ increases. As $Ha_1/E_{\infty,1}$ increases, the diffusion of B to the interface for reaction increasingly limits the transfer rate.

### 3.1.3. Results for $Ha_1 = 2 \cdot Ha_2$

The observations in Figs. 7, 8, and 9 are similar to the findings presented in Section 3.1.2.

While with $Ha_1 = Ha_2$, the evolutions of B and P at the interface are symmetrical, the evolutions are a little different with $Ha_1 = 2 \cdot Ha_2$ (Figs. 7–8). Even with $Ha_2/E_{\infty,2} = 0.50$, the concentration of $P_{int}$ is close to 0.80. This can be explained by Reaction 1 being faster than Reaction 2 ($Ha_1 = 2 \cdot Ha_2$). The gas transferred reacts preferentially with B. Therefore, the concentration of $P_{int}$ is relatively high even with high $Ha_2/E_{\infty,2}$.

$E_A$ is lower than $Ha$, as expected (Fig. 9). Moreover, the difference increases with $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$. If one sets the maximum ratio of $Ha/E_{\infty,1}$ and $Ha/E_{\infty,2}$ to 0.20 for the upper condition of Westerterp et al. (Eqs. (5) and (6)), only two simulation conditions satisfy the upper conditions. These conditions are outlined in Fig. 9 in larger format. Fig. 7 illustrates that the normalized concentration of B at the interface is greater than 0.80 when $Ha_1/E_{\infty,1} \le 0.20$ and it is quasi-independent of the $Ha_2/E_{\infty,2}$ ratio. Concerning $P_{int}$, its value is greater than 0.80 for all the simulations except for $Ha_2/E_{\infty,2} = 0.50$. Its value is quasi-independent of $Ha_1/E_{\infty,1}$. Therefore, in this example, the conditions respecting the PFO regime are circled in red in Fig. 9 and correspond to $Ha_1/E_{\infty,1} \le 0.20$ and $Ha_2/E_{\infty,2} \le 0.40$.
<figure><img src="image_9.png" /><figcaption>**Fig. 6.** Relative proportion $R_{A1}$ of the flux of species A transferred due to Reaction 1 as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 200$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Normalized concentration of B at the liquid-gas interface as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 100$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

As shown in Fig. 10, the proportion of rate A transferred due to Reaction 1 decreases with $Ha_1/E_{\infty,1}$ and increases with $Ha_2/E_{\infty,2}$. The range of this proportion is [0.72–0.82] depending on $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$. This range of values is greater than 0.5 because $Ha_1 = 2 \cdot Ha_2$, although this was unpredictable before this study.

### 3.1.4. Results for $Ha_1 = 10 \cdot Ha_2$

The observations of Figs. 11, 12, and 13 are similar to the findings presented in Section 3.1.2. The concentration of $P_{int}$ is greater than 0.90

<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Normalized concentration of P at the liquid-gas interface as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 100$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 9.** Relative difference between $Ha$ and $E_A$ as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 100$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

(Fig. 12). This can be explained by Reaction 1 being faster than Reaction 2 ($Ha_1 = 10 \cdot Ha_2$). The gas transferred reacts preferentially with B. Therefore, the concentration of P is relatively high even with a high $Ha_2/E_{\infty,2}$ value.

$E_A$ is lower than $Ha$, as expected (Fig. 13). Moreover, the absolute difference increases with $Ha_1/E_{\infty,1}$ but is almost independent with $Ha_2/E_{\infty,2}$. No condition achieves the criteria of Westerterp et al.[14]. However, in Figs. 11 and 12, the normalized concentration of B at the
<figure><img src="image_13.png" /><figcaption>Fig. 10. Relative proportion $R_{A1}$ of the flux of species A transferred due to Reaction 1 as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 100$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Normalized concentration of P at the interface liquid-gas in function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 20$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 11. Normalized concentration of B at the liquid-gas interface in function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 20$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

interface is greater than 0.8 when $Ha_1/E_{\infty,1} \le 0.20$ and is quasi-independent of the ratio $Ha_2/E_{\infty,2}$. Concerning $P_{int}$, its value is greater than 0.90 for all the simulations. Therefore, in this example, the conditions respecting the PFO regime are framed in a red rectangle in Fig. 13 and correspond to $Ha_1/E_{\infty,1} \le 0.20$.

Fig. 14 shows that the relative proportion $R_{A1}$ of the flux of species A

<figure><img src="image_16.png" /><figcaption>Fig. 13. Relative difference between $Ha$ and $E_A$ in function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 20$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ◆ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

transferred due to reaction 1 decreases with $Ha_1/E_{\infty,1}$ and increases with $Ha_2/E_{\infty,2}$. The range of this proportion is [0.98–0.99] depending on $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$. This range of values is close to 1 because $Ha_1 = 10 \cdot Ha_2$. This case is close to the case with a secondary relatively slow reaction and is common during the absorption of CO₂ by alkanolamines or amino acid salts. Calculating $Ha_1/E_{\infty,1}$ is sufficient to know if the system is in a PFO regime in this case study. These curves are useful for estimating the error if we use the PFO regime assumption and if the
<figure><img src="image_17.png" /><figcaption>Fig. 14. Relative proportion $R_{A1}$ of the flux of species A transferred due to Reaction 1 as a function of $Ha_2/E_{\infty,2}$ for $Ha_1 = 200$ and $Ha_2 = 20$ (values of $Ha_1/E_{\infty,1}$: ■ 0.05; ♦ 0.10; □ 0.20; ▲ 0.30; ● 0.40; — 0.50).</figcaption></figure>

secondary reaction is negligible or not. In this case, the main reaction is involved in 98%–99% of the A transferred.

## 3.2. Estimation of normalized concentrations at the interface and enhancement factor for the PFO regime

### 3.2.1. Establishment of the expressions of $B_{inb}$, $P_{inb}$, $E_A$, $R_{A1}$ and $R_{A2}$

In this section, generalized equations are established to directly estimate the concentrations of reactants at the interface, the enhancement factor, and the influence of each reaction on the gas transfer flux for a system in, or close to, the PFO regime. These equations are valid in the conditions of the simulations of this study and do not allow researchers to estimate the data for the instantaneous regime nor for moderately fast reactions for which part of A is consumed in the bulk of the liquid.

These variables are linked to the concentration profiles of each reagent. Like Westerterp et al. [14], the first step is to approximate the concentration profiles by a simplified profile. This method was introduced by Van Krevelen and Hoftijzer [22] who were the first to approximate concentration profiles. This technique was used again by Onda [19], who approximated the profiles using a quadratic function.

The enhancement factor can then be estimated based on these simplified profiles. Flow expressions are used to obtain expressions for B and P at the interface.

Normalized concentration profiles for all simulations are roughly the same (Fig. 15), with a sharp decrease in A near the interface. B and P diffuse without reaction until they reach A near the interface. Over a short distance $\delta_r$, B and P react with A and the normalized concentrations of B and P do not vary. This distance increases when the Hatta numbers decrease and vice versa. To estimate the flux, the following assumptions are made:

* B and P diffuse from the bulk ($X = 1$) to $X = \delta_r$ (in the example of Fig. 16, $\delta_r = 0.01$). Then between $X = 0$ and $X = \delta_r$, their concentrations are constant and equal to those at the interface, i.e. $B_{int} = B$ ($X = 0$) and $P_{int} = P(X = 0)$.

* A is approximated by $A = e^{-\delta_r/2}$. The flux of A transferring through the interface is calculated using the slope at the interface $\left(\frac{dA}{dX}\right)_{X=0} =$

<figure><img src="image_18.png" /><figcaption>Fig. 15. Normalized concentrations as a function of the normalized distance X ($Ha_1 = 200$; $Ha_2 = 20$; $Ha_1/E_{\infty,1} = 0.05$; $Ha_2/E_{\infty,2} = 0.30$).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Fig. 16. Normalized concentrations as a function of normalized distance X ($Ha_1 = 200$; $Ha_2 = 20$; $Ha_1/E_{\infty,1} = 0.05$; $Ha_2/E_{\infty,2} = 0.30$).</figcaption></figure>

$$ \frac{\partial^2}{\partial \delta_r^2} \bullet e^{-\frac{X}{\delta_r/2}} = \frac{\partial^2}{\partial \delta_r^2} \text{ The mean of normalized concentrations of A between } X = 0 \text{ and } X = \delta_r \text{ is evaluated as} $$

$$ A_m = \frac{1}{\delta_r} \cdot \int_0^{\delta_r} e^{-\frac{X}{\delta_r/2}} \cdot dX = -\frac{1}{2} \cdot \left[ e^{-\frac{\delta_r}{\delta_r/2}} - e^0 \right] = 0.43 \approx 0.5 \qquad 5 $$

By expressing the ratio between the rate of A transferred due to reaction 1 to the rate of A transferred due to Reaction 2, we obtain the expression of the normalized concentration of P at the interface as a
function of the normalized concentration of B at the interface (Eq. 37).

$$P_{\text{int}} = \frac{B_{\text{int}}}{F + (1 - F) \bullet B_{\text{int}}} \qquad (37)$$

With:

$$F = \frac{(E_{\infty,1} - 1) \bullet Ha_2^2}{(E_{\infty,2} - 1) \bullet Ha_1^2}$$

By setting a molar balance between X = 0 and δᵣ, we obtain a non- explicit equation of the normalized concentration of B at the interface (Eq. 38). The demonstrations of Eqs. (37) and (38) are given in Appendix D.

$$\frac{1 - B_{\text{int}}}{B_{\text{int}}} = \frac{Ha_1^2}{E_{\infty,1} - 1} \bullet \frac{1}{((E_{\infty,1} - 1) \bullet (1 - B_{\text{int}}) + (E_{\infty,2} - 1) \bullet (1 - P_{\text{int}}))} \qquad (38)$$

The definition of Eₐ (Equation 1) and the expressions of flux lead to the expression of Eₐ (Eq. 39).

$$\phi = k_{liq} \bullet E_{\text{A}} \bullet C_{A,\text{int}} \qquad (1)$$

$$E_{\text{A}} = (E_{\infty,1} - 1) \bullet (1 - B_{\text{int}}) + (E_{\infty,2} - 1) \bullet (1 - P_{\text{int}}) \qquad (39)$$

The proportion of A transferred reacting in Reaction 1 (Rₐ₁) and in Reaction 2 (Rₐ₂) are estimated with Eqs. (40) and (41).

$$R_{A1} = \frac{(E_{\infty,1} - 1) \bullet (1 - B_{\text{int}})}{(E_{\infty,1} - 1) \bullet (1 - B_{\text{int}}) + (E_{\infty,2} - 1) \bullet (1 - P_{\text{int}})} \qquad (40)$$

$$R_{A2} = \frac{(E_{\infty,2} - 1) \bullet (1 - P_{\text{int}})}{(E_{\infty,1} - 1) \bullet (1 - B_{\text{int}}) + (E_{\infty,2} - 1) \bullet (1 - P_{\text{int}})} \qquad (41)$$

### 3.2.2. Validation of the expressions of Bᵢₙₜ, Pᵢₙₜ, Eₐ, and Rₐ₁

To validate the expressions established in Section 3.2.1., it is necessary to compare the results obtained by Eqs (37)–(41) with the results of the simulations. The parity curves are plotted and the distribution of relative differences is set. The root mean square error (RMSE) and the average absolute relative deviation (AARD) are calculated for each expression (Eqs. (42) and (43)). The results are compiled for all 288 simulations in Table 2.

$$RMSE = \sqrt{\frac{\sum_{1}^{N} (y_{i,exp} - y_{i model})^2}{N}} \qquad (42)$$

<caption>**Table 2**</caption>

Comparison between the results of the simulations and the results predicted with Equations (37–41).

<table><thead><tr><th></th><th>RMSE</th><th>AARD</th><th>Proportion of values with a relative deviation of less than 1 %</th><th>Absolute maximum relative deviation</th></tr></thead><tbody><tr><td>Normalized concentration of B at the interface B<sub>int</sub></td><td>0.0078</td><td>0.50 %</td><td>83.7 %</td><td>4.4 %</td></tr><tr><td>Normalized concentration of P at the interface P<sub>int</sub></td><td>0.0062</td><td>0.36 %</td><td>85.1 %</td><td>3.0 %</td></tr><tr><td>Enhancement factor E<sub>A</sub></td><td>0.481</td><td>0.31 %</td><td>89.6 %</td><td>2.5 %</td></tr><tr><td>Proportion of A reacting in Reaction 1 (R<sub>A1</sub>)</td><td>0.0023</td><td>0.16 %</td><td>95.1 %</td><td>1.8 %</td></tr></tbody></table>

$$AARD(\%) = \frac{100}{N} \times \sum_{1}^{N} \left| \frac{y_{i,exp} - y_{i,mode}}{y_{i,exp}} \right| \qquad (43)$$

All estimates of Bᵢₙₜ, Pᵢₙₜ, Eₐ, and Rₐ₁ are within 5 %, with most of estimates within 1 % (Table 2). The estimations with Eqs. (37)–(41) are accurate with an AARD ≤ 0.50 %. These equations can predict the values of Bᵢₙₜ, Pᵢₙₜ, Eₐ, and Rₐ₁ for the simulations in the PFO regimes and also in the other situations simulated. As shown in Fig. 17, for high Ha₁ values, the Eqs. (37)–(41) very accurately predict the results from the simulations. For Ha₁ = 10, a minor deviation is attributed to the concentration profiles. The concentration of A has the same shape for all the conditions but δᵣ increases when Ha₁ decreases, with the approximate curve deviating from the actual. However, the relative difference is still within 5 %.

To test the robustness of Eqs. (37)–(41), we extend the range of Ha₁ and simulate different conditions of Ha₂/Ha₁, Ha₁/E₄,₁, and Ha₂/E₄,₂ (Table 3). These 54 new simulations confirm the capacities of these equations to predict Bᵢₙₜ, Pᵢₙₜ, Eₐ, and Rₐ₁. Indeed, as shown in Table 4 and Fig. 18 the difference between the results from the simulations and the results of these equations is relatively low.

All 342 data points are then classified between systems in the PFO regime and systems in the non-PFO regime. Tables 5 and 6 show that the generalized equations perform better for systems in the PFO regime and less well for those in the non-PFO regime. This can be explained by the assumption of A concentration profiles being valid mainly for a PFO regime. Nevertheless, these generalized equations correctly predict all the conditions of all 342 simulations.

### 3.3. New criteria for a pseudo-first-order regime with two irreversible reactions

To consider a system in the PFO regime, the concentrations of reactants B and P are relatively constant in the liquid film. We saw above that the limits depend on the researchers. We can define limits (HL) for the concentration. These HL are defined as the ratio between the lower limit of the normalized concentration at the interface not to be under-passed and the concentration in the bulk. (See Eqs. (44)–(45)). If HL<sub>B</sub> =

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Parity plot for the values of the normalized concentration of B at the interface (▲ Ha₁ = 10; ● Ha₁ = 200; □ Ha₁ = 500).</figcaption></figure>
<table><caption>**Table 3**
Parameters of the simulations for validation (54 simulations).</caption><thead><tr><th>Ha<sub>1</sub></th><th>Ha<sub>2</sub></th><th>Ha<sub>1</sub>/E<sub>∞,1</sub></th><th>Ha<sub>2</sub>/E<sub>∞,2</sub></th></tr></thead><tbody><tr><td>1000</td><td>50; 250; 500</td><td>0.20; 0.50</td><td>0.20; 0.50</td></tr><tr><td>750</td><td>37.5; 187.5; 375</td><td>0.20; 0.50</td><td>0.20; 0.50</td></tr><tr><td>100</td><td>5; 25; 50</td><td>0.20; 0.50</td><td>0.20; 0.50</td></tr><tr><td>50</td><td>2.5; 12.5; 25</td><td>0.20; 0.50</td><td>0.20; 0.50</td></tr><tr><td>10</td><td>1</td><td>0.05</td><td>0.05; 0.10; 0.20; 0.30; 0.40; 0.50</td></tr></tbody></table>

<table><caption>**Table 4**
Comparison between the results of the simulations and the results predicted with Equations (37–41) for validation data (54 simulations).</caption><thead><tr><th></th><th>RMSE</th><th>AARD</th><th>Proportion of values with a relative deviation of less than 1 %</th><th>Absolute maximum relative deviation</th></tr></thead><tbody><tr><td>Normalized concentration of B at the interface B<sub>int</sub></td><td>0.0033</td><td>0.31 %</td><td>92.6 %</td><td>1.2 %</td></tr><tr><td>Normalized concentration of P at the interface P<sub>int</sub></td><td>0.0021</td><td>0.13 %</td><td>100 %</td><td>0.9 %</td></tr><tr><td>Enhancement factor E<sub>A</sub></td><td>0.332</td><td>0.18 %</td><td>100 %</td><td>0.7 %</td></tr><tr><td>Proportion of A reacting in Reaction 1 (R<sub>A1</sub>)</td><td>0.0004</td><td>0.03 %</td><td>100 %</td><td>0.3 %</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Parity plot for the values of the enhancement factor, E<sub>A</sub>, as calculated with Equation 41 (■ First 288 simulations; ● 54 validation data set).</figcaption></figure>

0.80, this means that the concentration of B at the interface is equal to 80 % of the concentration of B in the mass of the liquid.

$$HL_B = \frac{B_{\text{int.limit}}}{B_{\text{bulk}}} = \frac{B_{\text{int.limit}}}{1} = B_{\text{int.limit}} \qquad (44)$$

$$HL_P = \frac{P_{\text{int.limit}}}{P_{\text{bulk}}} = \frac{P_{\text{int.limit}}}{1} = B_{\text{int.limit}} \qquad (45)$$

It seems logical to take the same limit for B and P. Thus, in the

<table><caption>**Table 5**
Comparison between the results of simulation and the results predicted with Equations (37–41) for simulations in a PFO regime.</caption><thead><tr><th></th><th>RMSE</th><th>AARD</th><th>Proportion of values with a relative deviation of less than 1 %</th><th>Absolute maximum relative deviation</th></tr></thead><tbody><tr><td>Normalized concentration of B at the interface B<sub>int</sub></td><td>0.0043</td><td>0.25 %</td><td>92.3 %</td><td>1.8 %</td></tr><tr><td>Normalized concentration of P at the interface P<sub>int</sub></td><td>0.0044</td><td>0.24 %</td><td>91.1 %</td><td>2.2 %</td></tr><tr><td>Enhancement factor E<sub>A</sub></td><td>0.600</td><td>0.21 %</td><td>98.8 %</td><td>1.0 %</td></tr><tr><td>Proportion of A reacting in Reaction 1 (R<sub>A1</sub>)</td><td>0.0010</td><td>0.07 %</td><td>100 %</td><td>1.0 %</td></tr></tbody></table>

<table><caption>**Table 6**
Comparison between the results of simulation and the results predicted with equations (37–41) for simulations in a non-PFO regime.</caption><thead><tr><th></th><th>RMSE</th><th>AARD</th><th>Proportion of values with a relative deviation of less than 1 %</th><th>Absolute maximum relative deviation</th></tr></thead><tbody><tr><td>Normalized concentration of B at the interface B<sub>int</sub></td><td>0.0093</td><td>0.68 %</td><td>78.2 %</td><td>4.4 %</td></tr><tr><td>Normalized concentration of P at the interface P<sub>int</sub></td><td>0.0068</td><td>0.40 %</td><td>83.9 %</td><td>3.0 %</td></tr><tr><td>Enhancement factor E<sub>A</sub></td><td>0.8579</td><td>0.38 %</td><td>83.9 %</td><td>2.5 %</td></tr><tr><td>Proportion of A reacting in Reaction 1 (R<sub>A1</sub>)</td><td>0.0028</td><td>0.20 %</td><td>92.0 %</td><td>1.8 %</td></tr></tbody></table>

following, we consider HL<sub>B</sub> = HL<sub>P</sub> = HL. The analytical solution leading to the value of B<sub>int</sub> (Eq. (38)) can be arranged in Eq. (46).

$$\frac{B_{\text{int}}}{1 - B_{\text{int}}} = \frac{E_{\infty,1} - 1}{Ha_1} \cdot \left( \left( \frac{E_{\infty,1} - 1}{Ha_1} \right) \cdot (1 - B_{\text{int}}) + \left( \frac{E_{\infty,2} - 1}{Ha_2} \right) \cdot \frac{Ha_2}{Ha_1} \cdot \left( 1 - \frac{B_{\text{int}}}{F + (1 - F) \cdot B_{\text{int}}} \right) \right) \qquad (46)$$

This way, it can be seen that B<sub>int</sub> and P<sub>int</sub> depend on three terms: the ratios Ha<sub>1</sub>/(E<sub>∞,1</sub>−1), Ha<sub>2</sub>/(E<sub>∞,2</sub>−1), and Ha<sub>1</sub>/Ha<sub>2</sub>. These ratios are close to the parameters found in the first conclusion following the first series of simulations. Indeed, for the frequent case where E<sub>∞,1</sub> and E<sub>∞,2</sub> ≫ 1, Ha<sub>1</sub>/(E<sub>∞,1</sub>−1) ≈ Ha<sub>1</sub>/E<sub>∞,1</sub>, Ha<sub>2</sub>/(E<sub>∞,2</sub>−1) ≈ Ha<sub>2</sub>/E<sub>∞,2</sub>.

From Eqs. (37) and (46), new criteria of the PFO regime are determined for the upper limit. For B ≥ HL and P ≥ HL, the criteria are given by Inequalities (47) and (48):

$$\frac{Ha_1}{E_{\infty,1} - 1} \le \frac{1 - HL}{HL^{1/2}} \cdot \left( 1 + \left( \frac{Ha_2}{Ha_1} \right)^2 \right)^{\frac{1}{2}} \qquad (47)$$

$$\frac{Ha_2}{E_{\infty,2} - 1} \le \frac{Ha_1}{Ha_2} \cdot \frac{1 - HL}{HL^{1/2}} \cdot \left( 1 + \left( \frac{Ha_2}{Ha_1} \right)^2 \right)^{\frac{1}{2}} \qquad (48)$$

For instance, with HL = 0.80, the criteria to respect are:
$$ \frac{Ha_1}{E_{\infty,1}-1} \le 0.2236 \cdot \left( 1 + \left( \frac{Ha_2}{Ha_1} \right)^2 \right)^{\frac{1}{2}} (47\text{bis}). $$

$$ \frac{Ha_2}{E_{\infty,2}-1} \le \frac{Ha_1}{Ha_2} \cdot 0.2236 \cdot \left( 1 + \left( \frac{Ha_2}{Ha_1} \right)^2 \right)^{\frac{1}{2}} (48\text{bis}). $$

The results of simulations are tested according to these criteria to predict if the regime is in the PFO regime or not. These new criteria predict 96.2 % of the 342 simulations. The simulations that do not match the test (3.8 %) correspond to a limiting concentration of P very close to 0.8.

### 3.4. Utilization of generalized equations to estimate the maximal value of $k_{ov}$

To illustrate the usefulness of these generalized equations, an analysis of the data from Portugal et al. [6] was carried out. One aspect of the work by Portugal et al. [6] consists of measuring fluxes at different partial pressures of CO₂ for different concentrations of aqueous solutions of potassium glycinate in a reactor operated in semi-batch mode.

The main reaction for this is the reaction between carbon dioxide and the glycinate ion (Gly⁻). According to Portugal et al. [6], and, Guo et al. [9], the mechanism for this reaction can be explained by the zwitterion mechanism given by Caplow. The two steps of the mechanism are the formation of the zwitterion followed by the deprotonation of the zwitterion.

Zwitterion mechanism (reaction between carbon dioxide and glycinate ion):

**Step 1: formation of the zwitterion**

$$ \frac{k_1}{\mathrm{NH}_2\mathrm{CH}_2\mathrm{COO}^- + \mathrm{CO}_2} \rightleftharpoons \frac{-\mathrm{OOC}\mathrm{NH}_2^+ - \mathrm{CH}_2\mathrm{COO}^-}{k_{-1}} $$

**Step 2: deprotonation of the zwitterion and formation of the carbamate.**

$$ \frac{-\mathrm{OOC}\mathrm{NH}_2^+ - \mathrm{CH}_2\mathrm{COO}^- + \mathrm{B}}{\rightarrow} \frac{-\mathrm{OOC}\mathrm{NH}_2\mathrm{CH}_2\mathrm{COO}^- + \mathrm{BH}^+}{\rightarrow} $$

As mentioned by Portugal et al. [6], the deprotonation of primary amines is relatively fast in comparison with the reversed reaction of the formation of the zwitterion. Thus, the direct formation reaction of the zwitterion is the limiting step and the kinetic of the reaction is given by Eq. (49).

$$ r_1 = k_1 \cdot C_{gly}^{-} \cdot C_{CO_2} \qquad (49) $$

In solutions with low CO₂ loading, this reaction can be considered as an irreversible second-order reaction. The second reaction taken into consideration is the reaction between the carbon dioxide and the hydroxide ions. This reaction is:

$$ \frac{k_2}{\mathrm{CO}_2 + \mathrm{OH}^-} \rightleftharpoons \frac{\mathrm{HCO}_3^-}{k_{-2}} $$

This reaction is almost irreversible and according to Vaidya et al. [7], its rate is given by Eq. (50):

$$ r_2 = k_2 \cdot C_{OH}^{-} \cdot C_{CO_2} \qquad (50) $$

The system can be considered as a system with two irreversible reactions and can be treated with the generalized equations of this study. Thus, in comparison with the previous equations, B corresponds to Gly⁻ (glycinate ion) and P corresponds to OH⁻ (hydroxide ion).

Portugal et al. [6]. introduced $k_{ov}$ defined according to Eq. (51).

$$ k_{ov} = \frac{-r_{CO_2}}{C_{CO_2}} \qquad (51) $$

When $k_{ov}$ is used to express an overall number Hatta number (Eq. (52)), it is often considered as the ratio of the rate of consumption of CO₂, to the concentration of CO₂ under the conditions where $C_B = C_{B,L}$ and $C_P = C_{P,L}$ throughout the entire film.

$$ Ha = \frac{\sqrt{k_{ov} \cdot D_{CO_2}}}{k_l} \qquad (52) $$

Portugal et al. [6]. treat their results in two ways. The first is to consider that $E_A = Ha$ within the strict framework of the PFO, i.e. the glycinate ion concentration is the same throughout the liquid film ($C_{gly-int} = C_{gly-L}$). By measuring the slope of the CO₂ transfer flux curve as a function of the CO₂ partial pressure for low values of CO₂ partial pressure, $k_{ov}$ is determined through Eq. (53).

$$ N_{CO_2} = \frac{R \cdot T}{V_g} \cdot \frac{\sqrt{k_{ov} \cdot D_{CO_2}}}{He} \cdot P_{CO_2,b} \qquad (53) $$

This equation is often used in the determination of $k_{ov}$. This equation is only valid when the partial pressure of CO₂ tends to 0. However, in practice, the slope is difficult to estimate when the partial pressure of CO₂ is equal to 0. This is because the higher the partial pressure of CO₂, the higher $E_\infty$ is and the system deviates from the PFO regime. In this way, $E_A$ increasingly differs from $Ha$. By measuring the slope for $Ha/E_{\infty,1} > 0$, this technique underestimates $k_{ov}$.

In this study, the data from Portugal et al. [6] are processed, using generalized equations to frame the values of $k_{ov}$. Unlike Portugal et al. [6], the secondary reaction with hydroxide ions is not neglected. The maximum value of $Ha/E_\infty$ indicated by Portugal et al. [6] is considered to estimate the slope. The normalized concentrations at the interface are determined with these assumptions. At the maximum value of $Ha/E_\infty$, the system deviates most significantly from the PFO regime. By applying the generalized equation from this study under these conditions, we obtain the value of the normalized concentrations at the interface and deduce the maximum value of $k_{ov}$. The influence of Reaction 2 on CO₂ transfer is then estimated. See Appendix F for the details of the calculation. For this calculation, references [23–26] are used.

In their study, Portugal et al. [6] neglect the participation of the second reaction in CO₂ transfer because glycinate is a weak base. However, Thee et al. [5] state that for a pH greater than 10, the secondary reaction must be taken into account in the transfer flux. With the application of the generalized equations of this study, it is shown that the proportion of CO₂ reacting with OH⁻ is comprised between 0.1 % and 2.1 % (Tables 7, 8, and 9). Some may consider this value to be negligible, while others may not. Nonetheless, it must be calculated on a case-by-case basis because the pKa value is not necessarily sufficient to determine whether the secondary reaction with OH⁻ can be considered to be negligible. Indeed, if the pKa value is low and $k_1$ is not very, the contribution of the second reaction is not negligible.

When $Ha/E_\infty$ is at its maximum, the normalized concentration of glycinate ion is between 0.551 and 0.813, and the normalized concentration of hydroxide ions is close to 1. The carbon dioxide reacts more with the glycinate ions than with the hydroxide ions, which explains why the concentration of hydroxide ions at the interface is close to that in the bulk of the liquid. The generalized equations are applied and the conditions $Ha_1/E_{\infty,1} \le 0.5$ and $Ha_2/E_{\infty,2} \le 0.5$ are met, except the experiment for which $B_{int} = 0.550$ ($Ha_1/E_{\infty,1} = 0.604$). It should be noted that the values of $B_{int}$ are relatively low and that the system deviates from the PFO regime. Portugal et al [6] study this system according to the penetration theory. In this study, we use the film theory. The expression of $Ha$ does not change from one theory to the other, but the expression of $E_\infty$ does. $E_{\infty,1}$ is smaller in the film theory for this system than in the penetration theory. Thus, with the film theory, $Ha_1/E_{\infty,1}$ and $Ha_2/E_{\infty,2}$ become larger and the system deviates from the PFO regime. Given the hydrodynamic behaviour of the cell used by Portugal et al, governed by Equation (54), the system behaves between the penetration model and the film model ($0 < C_4 = 0.4076 < 0.5$). Under the penetration theory, the system is closer to the PFO regime's condition.

$$ Sh = 0.1018 \cdot Re^{0.7279} \cdot Sc^{0.4076} \qquad (54) $$
<caption>**Table 7**</caption>

Comparison between the results of Portugal et al. [6] who used a strict PFO interpretation, and the results obtained using the generalized equations of the present study (T = 293 K).

<table><caption>**Table 8**</caption><thead><tr><th rowspan="2">Cs (M)</th><th colspan="2">Portugal et al. [6]</th><th rowspan="2">B<sub>int</sub> (normalized concentration of glycinate ion at the interface)</th><th colspan="3">Generalized equations (this study)</th></tr><tr><th>Minimal value of k<sub>ov</sub> (s<sup>-1</sup>)<br>(E<sub>A</sub> = Ha and B<sub>int</sub> = 1, P<sub>int</sub> = 1)</th><th>Maximal value of k<sub>ov</sub> (s<sup>-1</sup>)</th><th>P<sub>int</sub> (normalized concentration of hydroxide ion at the interface)</th><th>Proportion of CO<sub>2</sub> reacting with glycinate ion</th><th>Proportion of CO<sub>2</sub> reacting with hydroxide ion</th></tr></thead><tbody><tr><td>0.299</td><td>2340</td><td>4259</td><td>0.551</td><td>0.975</td><td>0.991</td><td>0.009</td></tr><tr><td>0.587</td><td>2640</td><td>4030</td><td>0.655</td><td>0.966</td><td>0.979</td><td>0.021</td></tr><tr><td>0.999</td><td>9390</td><td>15,580</td><td>0.606</td><td>0.981</td><td>0.996</td><td>0.004</td></tr><tr><td>1.984</td><td>22,800</td><td>30,573</td><td>0.748</td><td>0.990</td><td>0.997</td><td>0.003</td></tr></tbody></table>

Comparison between the results of Portugal et al. [6], who used a strict PFO interpretation, and the results obtained using the generalized equations of the present study (T = 298 K).

<table><caption>**Table 9**</caption><thead><tr><th rowspan="2">Cs (M)</th><th colspan="2">Portugal et al. [6]</th><th rowspan="2">B<sub>int</sub> (normalized concentration of glycinate ion at the interface)</th><th colspan="3">Generalized equations (this study)</th></tr><tr><th>Minimal value of k<sub>ov</sub> (s<sup>-1</sup>)<br>(E<sub>A</sub> = Ha and B<sub>int</sub> = 1)</th><th>Maximal value of k<sub>ov</sub> (s<sup>-1</sup>)</th><th>P<sub>int</sub> (normalized concentration of hydroxide ion at the interface)</th><th>Proportion of CO<sub>2</sub> reacting with glycinate ion</th><th>Proportion of CO<sub>2</sub> reacting with hydroxide ion</th></tr></thead><tbody><tr><td>0.0994</td><td>732</td><td>1189</td><td>0.613</td><td>0.955</td><td>0.984</td><td>0.016</td></tr><tr><td>0.299</td><td>2540</td><td>3522</td><td>0.721</td><td>0.971</td><td>0.992</td><td>0.008</td></tr><tr><td>0.587</td><td>5590</td><td>9241</td><td>0.603</td><td>0.964</td><td>0.990</td><td>0.010</td></tr><tr><td>0.999</td><td>16,200</td><td>25,823</td><td>0.627</td><td>0.980</td><td>0.998</td><td>0.002</td></tr><tr><td>1.984</td><td>36,100</td><td>46,796</td><td>0.773</td><td>0.989</td><td>0.998</td><td>0.002</td></tr><tr><td>3.005</td><td>86,300</td><td>108,634</td><td>0.795</td><td>0.994</td><td>0.999</td><td>0.001</td></tr></tbody></table>

Comparison between the results of Portugal et al [6], who used a strict PFO interpretation and the results obtained using the generalized equations of the present study (T = 303 K).

<table><thead><tr><th>Cs (M)</th><th>Minimal value of k<sub>ov</sub> (s<sup>-1</sup>)<br>(E<sub>A</sub> = Ha and B<sub>int</sub> = 1)</th><th>Maximal value of k<sub>ov</sub> (s<sup>-1</sup>)</th><th>B<sub>int</sub> (normalized concentration of glycinate ion at the interface)</th><th>P<sub>int</sub> (normalized concentration of hydroxide ion at the interface)</th><th>Proportion of CO<sub>2</sub> reacting with glycinate ion</th><th>Proportion of CO<sub>2</sub> reacting with hydroxide ion</th></tr></thead><tbody><tr><td>0.299</td><td>3930</td><td>4988</td><td>0.788</td><td>0.974</td><td>0.995</td><td>0.005</td></tr><tr><td>0.587</td><td>9490</td><td>13,724</td><td>0.691</td><td>0.969</td><td>0.997</td><td>0.003</td></tr><tr><td>0.999</td><td>20,000</td><td>26,482</td><td>0.756</td><td>0.980</td><td>0.998</td><td>0.002</td></tr><tr><td>1.984</td><td>68,000</td><td>83,810</td><td>0.813</td><td>0.991</td><td>0.999</td><td>0.001</td></tr></tbody></table>

Aware of the limitations of the first method, which underestimates k<sub>ov</sub>, Portugal et al. [6]. employed a second method. In this second method, they used a De Coursey equation in the intermediate regime, which lies between the PFO regime and the instantaneous regime. The value of k<sub>ov</sub> with this second method lies between the minimum value (k<sub>ov</sub> determined by the first method) and the maximum value calculated previously with the generalized equations of this study (Table 10). In this example, the generalized equations of the study correctly predict the upper limit of k<sub>ov</sub>.

In summary, this approach helps researchers find the maximum

value of k<sub>ov</sub> or k<sub>1</sub> and, determine the normalized concentrations at the interface. From this, it is possible to evaluate if the secondary reaction can be neglected. Furthermore, it also provides a framework for the value of k<sub>ov</sub>.

## 4. Conclusions

In this study, more than 300 simulations representing an extensive number of cases of absorption with two irreversible reactions were performed. The results of these normalized cases are provided and can

<caption>**Table 10**</caption>

Comparison between the results of Portugal et al [6], who used a strict PFO interpretation, the De Coursey equation, and the results obtained using the generalized equations of the present study.

<table><thead><tr><th rowspan="2">Cs (M)</th><th colspan="2">293 K</th><th colspan="2">298 K</th><th colspan="2">303 K</th></tr><tr><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(strict PFO method)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(DeCoursey method)</th><th>This study<br>k<sub>ov</sub> (s<sup>-1</sup>)<br>(generalized equations)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(strict PFO method)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(DeCoursey method)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(generalized equations)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(strict PFO method)</th><th>k<sub>ov</sub> (s<sup>-1</sup>)<br>(DeCoursey method)</th><th>This study<br>k<sub>ov</sub> (s<sup>-1</sup>)<br>(generalized equations)</th></tr></thead><tbody><tr><td>0.0994</td><td></td><td></td><td>732</td><td>881</td><td>1189</td><td></td><td></td><td></td><td></td></tr><tr><td>0.299</td><td>2340</td><td>2860</td><td>4259</td><td>2540</td><td>2710</td><td>3522</td><td>3930</td><td>4360</td><td>4988</td></tr><tr><td>0.587</td><td>2640</td><td>3130</td><td>4030</td><td>5590</td><td>6420</td><td>9241</td><td>9490</td><td>11,000</td><td>13,724</td></tr><tr><td>0.999</td><td>9390</td><td>11,500</td><td>15,580</td><td>16,200</td><td>19,600</td><td>25,823</td><td>20,000</td><td>22,000</td><td>26,482</td></tr><tr><td>1.984</td><td>22,800</td><td>24,200</td><td>30,573</td><td>36,100</td><td>38,600</td><td>46,796</td><td>68,000</td><td>69,800</td><td>83,810</td></tr><tr><td>3.005</td><td></td><td></td><td></td><td>86,300</td><td>93,900</td><td>108,634</td><td></td><td></td><td></td></tr></tbody></table>
be reused to estimate the concentrations of each component at the interface, the enhancement factor, and the participation of each reaction in the transfer flow of A. These data are useful to precisely determine the rate constants of the main reaction.

Secondly, the analysis of the results demonstrates that the relevant factors are the ratios $Ha_1/E_{\infty,1}$, $Ha_2/E_{\infty,2}$, and $Ha_1/Ha_2$. Indeed, the relative difference between $E_A$ and $Ha$, and the normalized concentrations of the interface $B_{int}$ and $P_{int}$, depend on these three parameters. Current criteria for determining if the system is in the PFO regime do not include these ratios and are too strict.

Next, equations were set to estimate the values of $B_{int}$, $P_{int}$, $E_A$, and $R_{A1}$ for both the PFO regimes and other situations close to the pseudo-first-order regime. Researchers can use directly these equations without performing additional calculations. These equations give accurate predictions with an AARD less than 0.50 % for $B_{int}$, $P_{int}$, $E_A$, and $R_{A1}$. Finally, new criteria can accurately predict if a system is in a PFO regime using an accurate test.

## 5. Formatting of funding sources

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

### 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

Data will be made available on request.

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.cej.2024.153724.

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