# KLa Determination Using the Effectiveness-NTU Method: Application to Countercurrent Absorbers in Operation Using Viscous Solvents for VOCs Mass Transfer

Eric Dumont

► To cite this version:

Eric Dumont. KLa Determination Using the Effectiveness-NTU Method: Application to Countercurrent Absorbers in Operation Using Viscous Solvents for VOCs Mass Transfer. ChemEngineering, 2019, 3 (2), pp.57. 10.3390/chemengineering3020057. [https://doi.org/10.3390/chemengineering3020057](https://doi.org/10.3390/chemengineering3020057)

HAL Id: hal-02149326

https://hal.science/hal-02149326

Submitted on 13 Oct 2023

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>

Article

# KLa Determination Using the Effectiveness-NTU Method: Application to Countercurrent Absorbers in Operation Using Viscous Solvents for VOCs Mass Transfer

Éric Dumont

IMT Atlantique, GEPEA, UMR CNRS 6144, F-44307 Nantes CEDEX, France; eric.dumont@imt-atlantique.fr;
Tel.: +33-025-185-8266

Received: 4 February 2019; Accepted: 24 May 2019; Published: 1 June 2019

<figure><img src="image_3.png" /></figure>

check for
updates

**Abstract:** In this study, the Effectiveness-NTU method, which is usually applied to heat exchanger design, was adapted to gas-liquid countercurrent absorbers to determine the overall mass transfer coefficient, $K_{La}$, of the apparatus in operation. It was demonstrated that the $ε$-NTU method could be used to determine the $K_{La}$ using the Henry coefficient of the solute to be transferred ($H_{VOC}$), the gas flow-rate ($Q_G$), the liquid flow-rate ($Q_L$), the scrubber volume ($V$), and the effectiveness of the absorber ($ε$). These measures are calculated from the gaseous concentrations of the solute measured at the absorber inlet ($C_{Gin}$) and outlet ($C_{Gout}$), respectively. The $ε$-NTU method was validated from literature dedicated to the absorption of volatile organic compounds (VOCs) by heavy solvents. Therefore, this method could be a simple, robust, and reliable tool for the $K_{La}$ determination of gas-liquid contactors in operation, despite the type of liquid used, i.e., water or viscous solvents.

**Keywords:** absorption; mass transfer; heat exchanger; scrubber; effectiveness-NTU method

## 1. Introduction

The removal of volatile organic compounds (VOCs) from the air can be achieved using gas-liquid reactors. However, some VOCs are poorly soluble in water leading to mass transfer limitations. In response to the low solubility of hydrophobic pollutants, water can be replaced by heavy viscous organic solvents (silicone oils, alkanes, phthalates, [1]) to improve the mass transfer [2]. Basically, the absorption step is crucial, since VOCs have to be solubilized for further treatments. Usually, the mass transfer between the gas phase and the liquid phase (water) is performed in countercurrent gas-liquid packed absorbers, where the design procedure is clearly described in the extant literature [3–5]. Basically, the design of absorbers implies the use of hydrodynamic studies to determine the gas-liquid two-phase flows in the column and mass transfer studies, so as to calculate the absorption efficiency according to the height of the column. The overall volumetric mass transfer coefficient $K_{La}$, which depends on the local volumetric mass transfer coefficients ($k_{La}$, $k_{Ga}$) and on the Henry coefficient, is a key parameter in the calculation of the absorption efficiency, and subsequently in the reactor design. However, for viscous solvents, the physico-chemical properties of the couple COV/solvent (mainly the Henry coefficient and the liquid diffusivity) are not sufficiently known to enable an accurate $K_{La}$ prediction from empirical correlations. Additionally, in the case of VOCs absorption by mixtures of two immiscible liquids (i.e., multiphasic gas-liquid-liquid systems), the $K_{La}$ cannot be predicted. Several techniques for experimental $K_{La}$ determination exist; however, considerable problems concerning the accuracy of the measurements in relation to the high viscosity of the solvents have been encountered [6,7]. Moreover, these techniques cannot be applied to multiphasic gas-liquid-liquid systems. Therefore, the aim of this study was to develop a generally applicable method for experimentally determining
the overall volumetric mass transfer coefficient in the liquid phase, $K_{La}$, regardless of the solute to be transferred, the solvent and the packing material used, the operating conditions applied, and the hydrodynamic conditions encountered in the gas-liquid absorber. This method, based on the “effectiveness-NTU” method used for the design of heat exchangers [8], could then be considered as a complementary tool for $K_{La}$ determination to the empirical correlations usually applied.

## 2. Gas-Liquid Absorber Analysis

Although the countercurrent flow in a gas-liquid absorber occurs in vertical devices due to the significant density difference between gas and liquid, the mass balance between the phases was presented horizontally in this study. This was done for convenience, and to facilitate an easier analogy with the heat exchanger (Figures 1 and 2). For dilute concentrations, the gas side mass balance can be written as:

$$ \frac{Q_G}{V} C_G = d\phi + \frac{Q_G}{V} (C_G + dC_G) \qquad (1) $$

Then, the amount of the solute transferred from the gas phase is:

$$ d\phi = - \frac{Q_G}{V} dC_G \qquad (2) $$

Similarly for the liquid side:

$$ \frac{Q_L}{V} (C_L + dC_L) + d\phi = \frac{Q_L}{V} C_L \qquad (3) $$

$$ d\phi = - \frac{Q_L}{V} dC_L \qquad (4) $$

Consequently:

$$ dC_L = \frac{Q_G}{Q_L} dC_G \qquad (5) $$

This equation can be rearranged as:

$$ H_{VOC} dC_L = \frac{H_{VOC} Q_G}{Q_L} dC_G = \frac{1}{A} dC_G \qquad (6) $$

The flow-rate ratio ($H_{VOC}$ $Q_G/Q_L$) corresponds to the reciprocal of the absorption factor that is usually applied to the design of gas-liquid absorbers. Since the Henry coefficient can be expressed using different definitions, it is necessary to apply the appropriate units. For the purposes of this work, the parameter $H_{VOC}$ was defined as the dimensionless air-to-water concentration ratio ($H_{VOC} = C_G/C_L$), which is the most convenient measure for mass distribution calculations. The alternative forms of the Henry coefficient and corresponding conversion factors are given in Reference [9].

Based on the assumption of dilute systems, i.e., the gas flow-rate and liquid flow-rate are constant along the apparatus (Figure 2), where the molar flux $\phi$ of the transferred solute is:

$$ \phi = \frac{Q_L}{V} (C_{Lout} - C_{Lin}) = \frac{Q_G}{V} (C_{Gin} - C_{Gout}) \qquad (7) $$

Consequently:

$$ C_{Lout} = C_{Lin} + \frac{Q_G}{Q_L} (C_{Gin} - C_{Gout}) \qquad (8) $$
<figure><img src="image_4.png" /><figcaption>Figure 1. Nomenclature for the material balances in a countercurrent gas-liquid absorber.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2. Concentration profile throughout the length of the gas-liquid countercurrent absorber.</figcaption></figure>

Change in the solute concentration throughout the length of the absorber can be expressed as:

$$Q_G \frac{dC_G}{dx} = - \Omega K_L a \left( \frac{C_G}{H_{VOC}} - C_L \right) \qquad (9)$$

To adequately integrate this equation, it is useful to make a change in the variables. Thus, using Equation (6):

$$d(C_G - H_{VOC} C_L) = dC_G - H_{VOC} dC_L = dC_G - \frac{1}{A} dC_G = \left(1 - \frac{1}{A}\right) dC_G \qquad (10)$$

Replacing Equation (10) in Equation (9):

$$\frac{Q_G}{\left(1 - \frac{1}{A}\right)} d(C_G - H_{VOC} C_L) = \frac{- \Omega K_L a}{H_{VOC}} (C_G - H_{VOC} C_L) dx \qquad (11)$$
Considering that concentrations of the values C<sub>G</sub> and C<sub>L</sub> are given at positions x = 0 and x = Z in Figure 2, the solution to this equation, integrated between x = 0 and x = Z, is:

$$ \ln \left[ \frac{C_{\text{Gout}} - H_{\text{VOC}} C_{\text{Lin}}}{C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lout}}} \right] = \frac{\Omega Z K_{\text{La}}}{H_{\text{VOC}} Q_G} \left(1 - \frac{1}{A}\right) \quad (12) $$

As (ΩZ) is the volume of the apparatus, Equation (12) can be rearranged under the following form:

$$ \frac{C_{\text{Gout}} - H_{\text{VOC}} C_{\text{Lin}}}{C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lout}}} = \exp\left(-NTU\left(1 - \frac{1}{A}\right)\right) \quad (13) $$

With:

$$ NTU = \frac{K_{\text{La}}}{H_{\text{VOC}}} \frac{V}{Q_G} \quad (14) $$

In Equation (14), the dimensionless number of transfer units (NTU) corresponds to the ratio of the gas residence time in the device (V/Q<sub>G</sub>) to the time needed for the mass transfer of the solute from the gas phase to the liquid phase (H<sub>VOC</sub>/K<sub>La</sub>). This NTU parameter can be directly compared to the NTU parameter used in the heat exchanger design (Table 1). Moreover, the reciprocal of the absorption factor (H<sub>VOC</sub> Q<sub>G</sub>/Q<sub>L</sub>) is also directly comparable to the relative magnitudes of the hot and cold fluid heat capacity rates that are used in the heat exchanger design. Consequently, the analogy with the heat exchanger design implies the definition of the maximum possible mass transfer rate φ<sub>max</sub> between the gas phase and the liquid phase. Assuming a countercurrent absorber of infinite length, the solute concentrations in the gas phase and in the liquid phase are at an equilibrium. Therefore, C<sub>Gout</sub> tends to (H<sub>VOC</sub> C<sub>Lin</sub>) at x→∞ and:

$$ \phi_{\text{max}} = \frac{Q_G}{V} (C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lin}}) \quad (15) $$

The effectiveness of the absorber, ε, can be defined as the ratio of the actual mass transfer rate φ (Equation (7)) to the maximum possible mass transfer rate φ<sub>max</sub>:

$$ \varepsilon = \frac{\phi}{\phi_{\max}} = \frac{(C_{\text{Gin}} - C_{\text{Gout}})}{(C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lin}})} \quad (16) $$

According to Equation (16), the solute gas concentration at the outlet of the absorber C<sub>Gout</sub> would tend toward zero only for a countercurrent absorber of infinite length using a solute-free liquid, i.e., C<sub>Lin</sub> = 0. It should be noted that the inlet liquid concentration is usually equal to zero, and consequently ε = (C<sub>Gin</sub> - C<sub>Gout</sub>)/C<sub>Gin</sub>, which corresponds to the equation that is generally applied to determine the absorber efficiency. Combining Equations (8), (13), and (16), the effectiveness of the absorber can be expressed using the following form (calculation details are given in Appendix A):

$$ \varepsilon = \frac{1 - \exp(-NTU \left(\frac{A-1}{A}\right))}{1 - \frac{1}{A} \exp(-NTU \left(\frac{A-1}{A}\right))} \quad (17) $$

Equation (17) becomes identical to the expression used for the countercurrent heat exchanger. The effectiveness versus the NTU is plotted in Figure 3, for various A values. This analogy between countercurrent devices, absorbers, and heat exchangers, can be extended to a gas-liquid stirred tank reactor (STR), as well as to a single stream heat exchanger (typically, a boiler or a condenser). In a gas-liquid STR, the gas flow-rate is dispersed through a known volume of a well-mixed liquid phase, and consequently Q<sub>L</sub> = 0. In this case, the effectiveness of the absorber is presented as (calculation details are given in Appendix B):

$$ \varepsilon = 1 - \exp(-NTU) \quad (18) $$

In the case of a single stream heat exchanger, A tends to ∞, and consequently, the heat exchanger behavior of the apparatus becomes independent of the flow arrangement. As a result, when A→∞,
Equation (17) is reduced to Equation (18). The analogies between the heat and mass transfer given in the current paper are summarized in Table 1.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Effectiveness (ε) of a countercurrent gas-liquid absorber versus the number transfer units (NTU).</figcaption></figure>

<table><caption>**Table 1.** The effectiveness-NTU method for the countercurrent devices. Analogy between the gas-liquid mass absorber and the heat exchanger (see nomenclature for details).</caption><thead><tr><th></th><th>Mass Absorber</th><th>Heat Exchanger</th></tr></thead><tbody><tr><td>Physical parameter</td><td>Concentration C (mol m<sup>-3</sup>)</td><td>Temperature T (K)</td></tr><tr><td>Transfer coefficient</td><td>K<sub>L</sub> (m s<sup>-1</sup>)</td><td>U (W m<sup>-2</sup> K<sup>-1</sup>)</td></tr><tr><td>Transfer surface area</td><td>a V (m<sup>2</sup>)</td><td>A (m<sup>2</sup>)</td></tr><tr><td>Transfer rate</td><td>φ = Q<sub>L</sub>/V (C<sub>Lout</sub> - C<sub>Lin</sub>)<br>φ = Q<sub>G</sub>/V (C<sub>Gin</sub> - C<sub>Gout</sub>)<br>φ = K<sub>L</sub>a ΔC<sub>ML</sub><br>(mol m<sup>-3</sup> s<sup>-1</sup>)</td><td>φ = M<sub>c</sub> C<sub>pc</sub> (T<sub>cout</sub> - T<sub>cin</sub>)<br>φ = M<sub>h</sub> C<sub>ph</sub> (T<sub>hin</sub> - T<sub>hout</sub>)<br>φ = U A ΔT<sub>ML</sub><br>(W)</td></tr><tr><td>Maximum possible transfer rate</td><td>φ<sub>max</sub> = Q<sub>G</sub>/V (C<sub>Gin</sub> - H<sub>VOC</sub>C<sub>Lin</sub>)<br>(mol m<sup>-3</sup> s<sup>-1</sup>)</td><td>φ<sub>max</sub> = C<sub>min</sub> (T<sub>hin</sub> - T<sub>cin</sub>)<br>(W)</td></tr><tr><td>Effectiveness<br>(dimensionless)<br>ε = φ/φ<sub>max</sub></td><td>ε = (C<sub>Gin</sub> - C<sub>Gout</sub>)/(C<sub>Gin</sub> - H<sub>VOC</sub>C<sub>Lin</sub>)<br>ε = (1 - exp(- NTU (1 - 1/A))) / (1 - 1/A) exp(- NTU (1 - 1/A))<br>1/A = H<sub>VOC</sub> Q<sub>G</sub>/Q<sub>L</sub></td><td>ε = max((T<sub>cout</sub> - T<sub>cin</sub>)/(T<sub>hin</sub> - T<sub>hout</sub>)) / (T<sub>hin</sub> - T<sub>cin</sub>)<br>ε = (1 - exp(- NTU (1 - R))) / (1 - R) exp(- NTU (1 - R))<br>R = C<sub>min</sub>/C<sub>max</sub></td></tr><tr><td>NTU (dimensionless)</td><td>K<sub>L</sub>a / H<sub>VOC</sub> Q<sub>G</sub></td><td>U A / C<sub>min</sub></td></tr></tbody></table>
### 3. Validation of the ε-NTU Method from the Data Reported in the Literature

The aim of this section is to demonstrate that the use of the ε-NTU method permits the calculation of the overall mass transfer coefficient, K<sub>L</sub>a, of an operating absorber when parameters Q<sub>L</sub>, Q<sub>G</sub>, V, H<sub>VOC</sub>, and ε are known. The validation is based on data from three studies [1,6,10], as detailed in Table 2. As observed, the experimental data are diversified in terms of solvent used, solutes to be transferred (i.e., various H<sub>VOC</sub> values), and operating conditions. From the absorption efficiency values reported in these studies, the NTU and K<sub>L</sub>a values were calculated using Equations (19) and (14), respectively.

$$NTU = \frac{A}{1-A} \ln\left(\frac{\varepsilon - 1}{\varepsilon/A - 1}\right) \qquad (19)$$

All experimental results are reported in Figure 3.

<table><caption>Table 2. Literature data used for the K<sub>L</sub>a determination using the ε-NTU method.</caption><thead><tr><th>Solvent</th><th>Solute</th><th>H<sub>VOC</sub></th><th>Operating Conditions</th><th>Ref</th></tr></thead><tbody><tr><td rowspan="4">Water 1 mPa s</td><td>Toluene</td><td>2.09 × 10<sup>-1</sup></td><td rowspan="4">V = 2.36 m<sup>3</sup><br/>Packed column (Pall rings)<br/>Q<sub>L</sub> = 3.06 × 10<sup>-3</sup> m<sup>3</sup>/s<br/>Q<sub>G</sub>: 1.20 m<sup>3</sup>/s<br/>(gas EBRT: 2 s)<br/>20 °C</td><td rowspan="4">Study n°1<br/>[1]</td></tr><tr><td>Dichloromethane</td><td>9.03 × 10<sup>-2</sup></td></tr><tr><td>Propanol</td><td>1.27 × 10<sup>-4</sup></td></tr><tr><td>Acetone</td><td>9.36 × 10<sup>-4</sup></td></tr><tr><td rowspan="4">DEHA<br/>(bis(di-2-ethylhexyl)adipate)<br/>12.5 mPa s</td><td>Toluene</td><td>3.12 × 10<sup>-4</sup></td><td rowspan="4"></td><td rowspan="4"></td></tr><tr><td>Dichloromethane</td><td>1.94 × 10<sup>-3</sup></td></tr><tr><td>Propanol</td><td>2.70 × 10<sup>-3</sup></td></tr><tr><td>Acetone</td><td>5.21 × 10<sup>-3</sup></td></tr><tr><td rowspan="4">Silicone oil 50 mPa s</td><td>Toluene</td><td>5.58 × 10<sup>-4</sup></td><td rowspan="4"></td><td rowspan="4"></td></tr><tr><td>Dichloromethane</td><td>9.04 × 10<sup>-3</sup></td></tr><tr><td>Propanol</td><td>8.79 × 10<sup>-3</sup></td></tr><tr><td>Acetone</td><td>2.17 × 10<sup>-2</sup></td></tr><tr><td rowspan="4">DEHP (di-2-ethylhexyl<br/>phthalate) 76 mPa s</td><td>Hexane</td><td>4.74 × 10<sup>-3</sup></td><td rowspan="4">V = 1.46 × 10<sup>-2</sup> m<sup>3</sup><br/>Cables-bundle contactor<br/>Q<sub>L</sub> = 1.00 × 10<sup>-5</sup>; 1.25 × 10<sup>-5</sup>;<br/>1.50 × 10<sup>-5</sup> m<sup>3</sup>/s<br/>Q<sub>G</sub>: 1.39 × 10<sup>-2</sup> m<sup>3</sup>/s<br/>(gas EBRT: 1 s)<br/>20 °C</td><td rowspan="4">Study n°2<br/>[6]</td></tr><tr><td>Toluene</td><td>3.19 × 10<sup>-4</sup></td></tr><tr><td>Octane</td><td>4.40 × 10<sup>-4</sup></td></tr><tr><td>Methylcyclohexane</td><td>9.79 × 10<sup>-4</sup></td></tr><tr><td rowspan="3">Silicone oil 50 mPa s</td><td rowspan="3">Toluene</td><td rowspan="3">1.17 × 10<sup>-3</sup></td><td rowspan="3">V = 1.13 × 10<sup>-2</sup> m<sup>3</sup><br/>Packed column (Raschig rings)<br/>Q<sub>L</sub> = 1.53 × 10<sup>-5</sup>–4.58 × 10<sup>-5</sup> m<sup>3</sup>/s<br/>Q<sub>G</sub>: 7.15 × 10<sup>-3</sup> m<sup>3</sup>/s<br/>(gas EBRT: 1.6 s)<br/>25 °C</td><td rowspan="3">Study n°3<br/>[10]</td></tr><tr><td>V = 1.13 × 10<sup>-2</sup> m<sup>3</sup><br/>Packed column (IMTP®)<br/>Q<sub>L</sub> = 6.62 × 10<sup>-6</sup>–1.16 × 10<sup>-4</sup> m<sup>3</sup>/s<br/>Q<sub>G</sub>: 1.10 × 10<sup>-2</sup> m<sup>3</sup>/s<br/>(gas EBRT: 1 s)<br/>25 °C</td></tr><tr><td>V = 1.13 × 10<sup>-2</sup> m<sup>3</sup><br/>Packed column (Flexipac®)<br/>Q<sub>L</sub> = 1.36 × 10<sup>-5</sup>–9.28 × 10<sup>-5</sup> m<sup>3</sup>/s<br/>Q<sub>G</sub>: 1.10 × 10<sup>-2</sup> m<sup>3</sup>/s<br/>(gas EBRT: 1 s)<br/>25 °C</td></tr></tbody></table>

#### 3.1. *Data from Study n°1*

Biard et al. [1] proposed a simulation of the absorption of hydrophilic and hydrophobic compounds (toluene, dichloromethane, propanol, and acetone) in pure water and pure organic solvents (silicone oil and DEHA, i.e., bis(di-2-ethylhexyl)adipate). A countercurrent column packed with metal Pall rings was used for this purpose. The authors calculated the K<sub>L</sub>a values according to the Billet–Schultes
correlation coupled with the Piché et al. correlation [1]. From the removal efficiencies reported in this study, the ε-NTU method was applied to the 12 cases considered. Results are detailed in Table 3. The KLa values calculated using the ε-NTU method for heavy solvents, i.e., silicone oil and DEHA, were compared to the values determined by the authors from the coupling of the Billet-Schultes and Piché theories (Figure 4). Logically, the results are in good agreement due to the similarity between the definitions used to express the absorber efficiency (i.e., Equation (2) given in Reference [1] and Equation (17) presented in the present paper). Consequently, it could be argued that the ε-NTU method could be a simple, robust, and reliable method to determine the KLa values of operating absorbers. Moreover, the calculation procedure highlights the fact that the KLa mainly depends on the HVOC (Figure 5). Such an expected result was consistent with the mathematical expression of the KLa given in Equation (14).

<figure><img src="image_7.png" /><figcaption>**Figure 4.** KLa parity plot obtained from the data analysis reported in Biard et al. [1] (circles: silicone oil; squares: DEHA; water data not provided by the authors).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 5.** KLa versus the Henry coefficient (HVOC) from the data analysis reported in Biard et al. [1]. For all the simulations: gas flow-rate = 1.20 m³/s (gas EBRT = 2 s); liquid flow-rate = 3.06 × 10⁻³ m³/s; and V = 2.36 m³ (T = 20 °C).</figcaption></figure>
<table><caption>**Table 3.** K<sub>L</sub>a determination using the data reported in Biard et al. [1]. For all the simulations: gas flow-rate = 1.20 m<sup>3</sup>/s (gas EBRT = 2 s); liquid flow-rate = 3.06 × 10<sup>-3</sup> m<sup>3</sup>/s; and V = 2.36 m<sup>3</sup> (T = 20 °C).</caption><thead><tr><th>Solute</th><th></th><th>H<sub>VOC</sub> (-)</th><th>A (-)</th><th>ε (-)</th><th>NTU (-)</th><th>K<sub>L</sub>a (s<sup>-1</sup>)</th></tr></thead><tbody><tr><td rowspan="3">Toluene</td><td>Water</td><td>0.20936</td><td>0.012</td><td>0.012</td><td>0.053</td><td>5.63 × 10<sup>-3</sup></td></tr><tr><td>DEHA</td><td>0.00031</td><td>8.162</td><td>0.949</td><td>3.251</td><td>5.17 × 10<sup>-4</sup></td></tr><tr><td>PDMS50</td><td>0.00056</td><td>4.561</td><td>0.857</td><td>2.225</td><td>6.33 × 10<sup>-4</sup></td></tr><tr><td rowspan="3">Dichloromethane</td><td>Water</td><td>0.09031</td><td>0.028</td><td>0.028</td><td>0.144</td><td>6.60 × 10<sup>-3</sup></td></tr><tr><td>DEHA</td><td>0.00194</td><td>1.311</td><td>0.676</td><td>1.695</td><td>1.68 × 10<sup>-3</sup></td></tr><tr><td>PDMS50</td><td>0.00904</td><td>0.282</td><td>0.181</td><td>0.325</td><td>1.50 × 10<sup>-3</sup></td></tr><tr><td rowspan="3">Propanol</td><td>Water</td><td>0.00013</td><td>20.009</td><td>0.994</td><td>5.331</td><td>3.46 × 10<sup>-4</sup></td></tr><tr><td>DEHA</td><td>0.00270</td><td>0.944</td><td>0.550</td><td>1.269</td><td>1.74 × 10<sup>-3</sup></td></tr><tr><td>PDMS50</td><td>0.00879</td><td>0.290</td><td>0.179</td><td>0.312</td><td>1.40 × 10<sup>-3</sup></td></tr><tr><td rowspan="3">Acetone</td><td>Water</td><td>0.00094</td><td>2.721</td><td>0.956</td><td>4.254</td><td>2.03 × 10<sup>-3</sup></td></tr><tr><td>DEHA</td><td>0.00521</td><td>0.488</td><td>0.343</td><td>0.756</td><td>2.01 × 10<sup>-3</sup></td></tr><tr><td>PDMS50</td><td>0.02174</td><td>0.117</td><td>0.077</td><td>0.132</td><td>1.46 × 10<sup>-3</sup></td></tr></tbody></table>

## 3.2. Data from Study n°2

The main objective of this study was to investigate the absorption of four hydrophobic compounds in pure DEHP (Table 2) using a cables-bundle contactor. Experiments were performed at ambient temperature for three liquid flow-rates. The K<sub>L</sub>a results obtained using the ε-NTU method are summarized in Table 4. It was clear that regardless of the liquid flow-rate, the same order of magnitude of the K<sub>L</sub>a was observed for all the investigated compounds, even though the results were slightly higher for hexane (Figure 6). For this solute, the operating conditions led to very low removal efficiencies relative to the low absorption factors. For other compounds (toluene, octane, and methylcyclohexane), the absorption factors ranged from 0.7 to 3.4. Nonetheless, moderated absorption efficiencies were recorded (Figure 3), and these were potentially linked to the viscosity of the DEHP, which explained the low K<sub>L</sub>a values. According to Figure 6, a slight increase in the K<sub>L</sub>a relative to the liquid flow-rate was suggested by the authors. The weak influence of the liquid flow-rate in the case of viscous fluid was also reported in the literature [11]. Although the K<sub>L</sub>a values calculated using the ε-NTU method could not be directly validated in the present case, from the K<sub>L</sub>a values given by the authors, we observed that these results were still in agreement with the results of Biard et al. [1]. For instance, the influence of the Henry coefficient on the K<sub>L</sub>a value was confirmed (Figure 7). Moreover, from the solute diffusivities in DEHP given by the authors (toluene: 5.51 × 10<sup>-11</sup> m<sup>2</sup> s<sup>-1</sup>; hexane: 1.16 × 10<sup>-10</sup> m<sup>2</sup> s<sup>-1</sup>; octane: 6.23 × 10<sup>-11</sup> m<sup>2</sup> s<sup>-1</sup>; methylcyclohexane: 4.24 × 10<sup>-11</sup> m<sup>2</sup> s<sup>-1</sup>), and knowing that the specific interfacial area determined by the authors was 129 m<sup>-1</sup>, it was possible to calculate the thickness of the liquid film around the cables available for mass transfer (using the two-film theory) in each experiment. For the four solutes, all the calculated results were in good agreement. The thickness δ of the liquid film around the cables could be estimated from 30 to 56 µm (Table 4).

<table><caption>**Table 4.** K<sub>L</sub>a determination using the data reported in Bourgois et al. [6]. For all the experiments, solvent: DEHP (di-2-ethylhexyl phtalate); gas flow-rate = 1.39 × 10<sup>-2</sup> m<sup>3</sup>/s (gas EBRT = 1 s; V = 1.46 × 10<sup>-2</sup> m<sup>3</sup>).</caption><thead><tr><th>Solute</th><th>H<sub>VOC</sub> (-)</th><th>Q<sub>L</sub> (m<sup>3</sup>/s)</th><th>A (-)</th><th>ε (-)</th><th>NTU (-)</th><th>K<sub>L</sub>a (s<sup>-1</sup>)</th><th>δ (m)</th></tr></thead><tbody><tr><td rowspan="5">Hexane</td><td rowspan="5">4.7 × 10<sup>-3</sup></td><td>1.00 × 10<sup>-5</sup></td><td>0.152</td><td>0.059</td><td>0.077</td><td>3.48 × 10<sup>-4</sup></td><td>4.30 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>0.190</td><td>0.068</td><td>0.087</td><td>3.94 × 10<sup>-4</sup></td><td>3.80 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>0.190</td><td>0.067</td><td>0.086</td><td>3.87 × 10<sup>-4</sup></td><td>3.87 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>0.228</td><td>0.051</td><td>0.059</td><td>2.67 × 10<sup>-4</sup></td><td>5.59 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>0.228</td><td>0.083</td><td>0.108</td><td>4.87 × 10<sup>-4</sup></td><td>3.07 × 10<sup>-5</sup></td></tr></tbody></table>
<table><caption>Table 4. *Cont.*</caption><thead><tr><th>Solute</th><th>H<sub>VOC</sub> (-)</th><th>Q<sub>L</sub> (m<sup>3</sup>/s)</th><th>A (-)</th><th>ε (-)</th><th>NTU (-)</th><th>K<sub>L</sub>a (s<sup>-1</sup>)</th><th>δ (m)</th></tr></thead><tbody><tr><td rowspan="8">Toluene</td><td rowspan="8">3.2 × 10<sup>-4</sup></td><td>1.00 × 10<sup>-5</sup></td><td>2.257</td><td>0.345</td><td>0.462</td><td>1.40 × 10<sup>-4</sup></td><td>5.07 × 10<sup>-5</sup></td></tr><tr><td>1.00 × 10<sup>-5</sup></td><td>2.257</td><td>0.335</td><td>0.444</td><td>1.35 × 10<sup>-4</sup></td><td>5.27 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>2.821</td><td>0.346</td><td>0.455</td><td>1.38 × 10<sup>-4</sup></td><td>5.15 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>2.821</td><td>0.351</td><td>0.464</td><td>1.41 × 10<sup>-4</sup></td><td>5.05 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>2.821</td><td>0.35</td><td>0.462</td><td>1.40 × 10<sup>-4</sup></td><td>5.07 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>3.386</td><td>0.367</td><td>0.486</td><td>1.48 × 10<sup>-4</sup></td><td>4.82 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>3.386</td><td>0.386</td><td>0.520</td><td>1.58 × 10<sup>-4</sup></td><td>4.50 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>3.386</td><td>0.386</td><td>0.520</td><td>1.58 × 10<sup>-4</sup></td><td>4.50 × 10<sup>-5</sup></td></tr><tr><td rowspan="6">Octane</td><td rowspan="6">4.4 × 10<sup>-4</sup></td><td>1.00 × 10<sup>-5</sup></td><td>1.636</td><td>0.272</td><td>0.349</td><td>1.46 × 10<sup>-4</sup></td><td>5.50 × 10<sup>-5</sup></td></tr><tr><td>1.00 × 10<sup>-5</sup></td><td>1.636</td><td>0.284</td><td>0.369</td><td>1.54 × 10<sup>-4</sup></td><td>5.21 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>2.045</td><td>0.304</td><td>0.394</td><td>1.65 × 10<sup>-4</sup></td><td>4.87 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>2.045</td><td>0.286</td><td>0.364</td><td>1.53 × 10<sup>-4</sup></td><td>5.27 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>2.455</td><td>0.298</td><td>0.379</td><td>1.58 × 10<sup>-4</sup></td><td>5.07 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>2.455</td><td>0.298</td><td>0.379</td><td>1.58 × 10<sup>-4</sup></td><td>5.07 × 10<sup>-5</sup></td></tr><tr><td rowspan="6">Methylcyclohexane</td><td rowspan="6">9.8 × 10<sup>-4</sup></td><td>1.00 × 10<sup>-5</sup></td><td>0.735</td><td>0.144</td><td>0.174</td><td>1.62 × 10<sup>-4</sup></td><td>3.38 × 10<sup>-5</sup></td></tr><tr><td>1.00 × 10<sup>-5</sup></td><td>0.735</td><td>0.131</td><td>0.155</td><td>1.44 × 10<sup>-4</sup></td><td>3.79 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>0.919</td><td>0.158</td><td>0.189</td><td>1.76 × 10<sup>-4</sup></td><td>3.10 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>0.919</td><td>0.143</td><td>0.168</td><td>1.57 × 10<sup>-4</sup></td><td>3.49 × 10<sup>-5</sup></td></tr><tr><td>1.25 × 10<sup>-5</sup></td><td>0.919</td><td>0.166</td><td>0.201</td><td>1.87 × 10<sup>-4</sup></td><td>2.92 × 10<sup>-5</sup></td></tr><tr><td>1.50 × 10<sup>-5</sup></td><td>1.103</td><td>0.163</td><td>0.193</td><td>1.80 × 10<sup>-4</sup></td><td>3.04 × 10<sup>-5</sup></td></tr><tr><td></td><td></td><td>1.50 × 10<sup>-5</sup></td><td>1.103</td><td>0.164</td><td>0.194</td><td>1.81 × 10<sup>-4</sup></td><td>3.02 × 10<sup>-5</sup></td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Figure 6. K<sub>L</sub>a versus liquid flow-rate (from the data analysis reported in Reference [6]).</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7. KLa versus Henry coefficient (from the data analysis reported in Reference [6]). Diamond: gas flow-rate = 1.0 × 10⁻⁵ m³/s; circle: gas flow-rate = 1.25 × 10⁻⁵ m³/s; square: gas flow-rate = 1.50 × 10⁻⁵ m³/s.</figcaption></figure>

### 3.3. Data from Study n°3

The purpose of this study was to compare the toluene absorption using two silicone oils with different dynamic viscosities (5 mPa s and 50 mPa s, respectively) in a countercurrent gas-liquid absorber filled with three types of packing material (Raschig rings, IMTP®, and Flexipac®). The KLa values calculated using the ε-NTU method were compared to the values determined by the authors using the following equation reported in Heymes et al. [11].

$$ K_{La} = \frac{\frac{Q_G}{V} \frac{(C_{Gin} - C_{Gout})}{\frac{\ln\left[\frac{(C_{Gin}/H_{VOC}-C_{Lout})-(C_{Gout}/H_{VOC}-C_{Lin})}{(C_{Gout}/H_{VOC}-C_{Lin})}\right]} } (20) $$

From Figure 8, it appears that the KLa parity plot is validated for the solvent with low dynamic viscosity (5 mPa s). Conversely, for the solvent with the high dynamic viscosity (50 mPa s), the KLa determined using the ε-NTU method was usually 30% higher than the values calculated using Equation (20). According to this equation, the KLa determination required the measurement of solute concentrations in the gas phase and in the liquid phase. As the concentration measurement of a solute in a viscous fluid could be relatively inaccurate, the KLa determination using Equation (20) should be considered cautiously. Consequently, the KLa determination using the ε-NTU method should be preferred, since only the inlet and outlet gaseous concentrations are needed. Results from this study also confirmed that the viscosity of the liquid, as well as the type of packing material filling the column, has a considerable impact on the volumetric mass transfer coefficient of the operating absorber (Figure 9).
<figure><img src="image_11.png" /><figcaption>**Figure 8.** K<sub>L</sub>a parity plot obtained from the data analysis reported in Reference [10], (V = 1.13 × 10<sup>-2</sup> m<sup>3</sup>; T = 25 °C).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 9.** K<sub>L</sub>a versus the liquid flow-rate (calculated from Reference [10], (V = 1.13 × 10<sup>-2</sup> m<sup>3</sup>; T = 25 °C).</figcaption></figure>

**4. Conclusions**

We adapted the ε-NTU method, which is usually applied to heat exchanger design, to gas-liquid countercurrent absorbers. We demonstrated that this method could be successfully adapted to determine the overall mass transfer coefficient of the absorber in operation, the K<sub>L</sub>a, using advance knowledge of the physical parameters: H<sub>VOC</sub>, Q<sub>G</sub>, Q<sub>L</sub>, V, C<sub>Gin</sub>, and C<sub>Gout</sub>. Therefore, the ε-NTU method may be a robust and reliable tool for the K<sub>L</sub>a determination of gas-liquid contactors used in environmental applications for treating the removal of air pollutants. Consequently, new empirical correlations for the prediction of K<sub>L</sub>a for heavy viscous solvents may be developed. Moreover, since this method only requires the measurement of solute concentrations in the gas phase, it should be efficient in determining the K<sub>L</sub>a values of complex systems, such as Two-Phase Partitioning bioreactors (TPPB) used to remove hydrophobic pollutants using mixtures of immiscible liquids. In such a case,
the main difficulty lies in the knowledge of the H<sub>VOC</sub>. As a result, the ε-NTU method must be combined with the “equivalent absorption capacity” concept [12,13], thereby enabling calculation of the physical properties of the mixture [14].

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

## Nomenclature

### absorber

a specific interfacial area (m⁻¹)

A absorption factor

C concentration (mol m⁻³)

EBRT Empty Bed Residence Time (s)

H Henry coefficient

k<sub>G</sub>, k<sub>L</sub> local mass transfer coefficients (m s⁻¹)

K<sub>Ga</sub>, K<sub>La</sub> overall mass transfer coefficients (s⁻¹)

NTU Number of Transfer Units

Q flow-rate (m³ s⁻¹)

V packing volume (m³)

Z packing height (m)

### Greek letters

δ film thickness (m)

ε effectiveness

Ω cross sectional area (m²)

φ molar flux (mol m⁻³ s⁻¹)

### Subscripts

G gas

in inlet

L liquid

Max maximum

out outlet

VOC Volatile Organic Compound

### heat exchanger, Table 1

A heat-transfer area (m²)

C heat capacity rate of the fluid = M C<sub>p</sub> (W K⁻¹)

C<sub>p</sub> specific heat capacity (J kg⁻¹ K⁻¹)

M flow rate (kg s⁻¹)

NTU Number of Transfer Units

T temperature (K)

U heat transfer coefficient (W m⁻² K⁻¹)

### Greek letters

ΔT<sub>ML</sub> log mean temperature difference (K)

ε effectiveness

φ power, heat transferred per unit time (W)

### Subscripts

c cold

h hot

in inlet

Max maximum

Min minimum

out outlet
## Appendix A $\varepsilon$-NTU Relationship for a Gas-Liquid Absorber

Equations (8), (13), and (16) are rewritten:

$$C_{\text{Lout}} = C_{\text{Lin}} + \frac{Q_G}{Q_L}(C_{\text{Gin}} - C_{\text{Gout}}) \quad (\text{A1})$$

$$\frac{(C_{\text{Gout}} - H_{\text{VOC}} C_{\text{Lin}})}{(C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lout}})} = \exp\left(-\text{NTU}\left(1 - \frac{1}{A}\right)\right) = \beta \quad (\text{A2})$$

$$\varepsilon = \frac{\Phi}{\Phi_{\max}} = \frac{(C_{\text{Gin}} - C_{\text{Gout}})}{(C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lin}})} \quad (\text{A3})$$

Multiplying Equation (A1) by $(-H_{\text{VOC}})$ gives:

$$-H_{\text{VOC}} C_{\text{Lout}} = -H_{\text{VOC}} C_{\text{Lin}} - \frac{H_{\text{VOC}} Q_G}{Q_L} (C_{\text{Gin}} - C_{\text{Gout}}) \quad (\text{A4})$$

$$-H_{\text{VOC}} C_{\text{Lout}} = -H_{\text{VOC}} C_{\text{Lin}} - \frac{1}{A} (C_{\text{Gin}} - C_{\text{Gout}}) \quad (\text{A5})$$

Introducing Equation (A5) in Equation (A2):

$$\frac{(C_{\text{Gout}} - H_{\text{VOC}} C_{\text{Lin}})}{(C_{\text{Gin}} - H_{\text{VOC}} C_{\text{Lin}} - \frac{1}{A} (C_{\text{Gin}} - C_{\text{Gout}}))} = \beta \quad (\text{A6})$$

Rearranging Equation (A6):

$$H_{\text{VOC}} C_{\text{Lin}} = C_{\text{Gout}} \left( \frac{\beta/A - 1}{\beta - 1} \right) + C_{\text{Gin}} \beta \left( \frac{1 - 1/A}{\beta - 1} \right) \quad (\text{A7})$$

Introducing Equation (A7) in Equation (A3):

$$\varepsilon = \frac{(C_{\text{Gin}} - C_{\text{Gout}})(\beta - 1)}{(\beta - 1)C_{\text{Gin}} - [C_{\text{Gout}}(\beta/A - 1) + C_{\text{Gin}}\beta(1 - 1/A)]} \quad (\text{A8})$$

Rearranging Equation (A8):

$$\varepsilon = \frac{(1 - \beta)(C_{\text{Gout}} - C_{\text{Gin}})}{(1 - \beta/A)(C_{\text{Gout}} - C_{\text{Gin}})} \quad (\text{A9})$$

As $\beta = \exp(-NTU(1-1/A))$, Equation (A9) can be rewritten under the final form:

$$\varepsilon = \frac{1 - \exp(-NTU \left(\frac{A-1}{A}\right))}{1 - \frac{1}{A} \exp(-NTU \left(\frac{A-1}{A}\right))} \quad (\text{A10})$$

## Appendix B Gas-Liquid Stirred Tank Reactor (STR)

In a gas-liquid stirred tank reactor (STR), a gas flow-rate, $Q_L$, continuously bubbles through a known volume, V, of the well-mixed liquid phase. The solute mass transfer from the gas to the liquid phase is expressed as:

$$\frac{dC_G}{dt} = - K_L a \left( \frac{C_G}{H_{\text{VOC}}} - C_L \right) \quad (\text{A11})$$

During the mean residence time of the gas in the liquid phase, $t = V/Q_G$, the solute concentration in the rising gas bubbles changes from $C_{\text{Gin}}$ to $C_{\text{Gout}}$, and consequently:

$$H_{\text{VOC}} \int_{C_{\text{Gin}}}^{C_{\text{Gout}}} \frac{dC_G}{(C_G - H_{\text{VOC}} C_L)} = - K_L a \int_{t=0}^{t=V/Q_G} dt \quad (\text{A12})$$

$$\ln\left[\frac{(C_{\text{Gout}} - H_{\text{VOC}} C_{\text{L}})}{(C_{\text{Gin}} - H_{\text{VOC}} C_{\text{L}})}\right] = - \frac{K_L a}{H_{\text{VOC}}} \frac{V}{Q_G} \quad (\text{A13})$$
Defining NTU = (K_L a V)/(H_{VOC} Q_G), Equation (A13) becomes:

$$C_{\text{Gout}} = H_{\text{VOC}} C_{\text{L}} (1 - \exp(-\text{NTU})) + C_{\text{Gin}} \exp(-\text{NTU}) \quad (\text{A14})$$

Moreover, the molar flux $\phi$ of the transferred solute is:

$$\phi = \frac{Q_G}{V} (C_{\text{Gin}} - C_{\text{Gout}}) \quad (\text{A15})$$

Introducing the expression of $C_{\text{Gout}}$ (Equation (A14)) in Equation (A15) gives:

$$\phi = \frac{Q_G}{V} (1 - \exp(-\text{NTU})) (C_{\text{Gin}} - H_{\text{VOC}} C_{\text{L}}) \quad (\text{A16})$$

where the term $(1-\exp(-\text{NTU}))$ is the solute fraction that can be transferred from the gas phase to the liquid phase. Basically, for industrial applications, the initial concentration of the liquid phase is $C_L = 0$. Consequently:

$$\phi = \frac{Q_G}{V} (1 - \exp(-\text{NTU})) C_{\text{Gin}} \quad (\text{A17})$$

According to the NTU expression, the maximum solute fraction that can be transferred is obtained for a long residence time of the gas phase in the liquid phase (i.e., using a large liquid volume and a small gas flow-rate). In that case, the NTU is significantly higher than 1 and $\exp(-\text{NTU})$ tends to 0. As a result:

$$\phi_{\text{max}} = \frac{Q_G}{V} C_{\text{Gin}} \quad (\text{A18})$$

The effectiveness of the STR is then:

$$\varepsilon = \frac{\phi}{\phi_{\text{max}}} = 1 - \exp(-\text{NTU}) \quad (\text{A19})$$

## References

1. Biard, P.-F.; Couvert, A.; Giraudet, S. Volatile organic compounds absorption in packed column: Theoretical assessment of water, DEHA and PDMS 50 as absorbents. *J. Ind. Eng. Chem.* **2018**, *59*, 70–78. [CrossRef]

2. Dumont, E.; Delmas, H. Mass transfer enhancement of gas absorption in oil-in-water systems: A review. *Chem. Eng. Process. Process Intensif.* **2003**, *42*, 419–438. [CrossRef]

3. Mackowiak, J. *Fluid Dynamics of Packed Columns: Principles of the Fluid Dynamic Design of Columns for Gas/Liquid and Liquid/Liquid Systems*; Chemische Technik/Verfahrenstechnik; Springer: Berlin/Heidelberg, Germany, 2010; ISBN 978-3-540-88780-5.

4. Fair, J.; Steinmeyer, D.; Penney, W.; Corcker, B. Gas absorption and gas-liquid system design. In *Perry's Chemical Engineers' Handbook*, 8th ed.; Perry, R.H., Green, D.W., Eds.; McGraw-Hill: New York, NY, USA, 2008; pp. 14-1-14-98.

5. Sinnott, R. *Coulson and Richardson's Chemical Engineering Volume 6 (Design)*; Pergamon Press: Oxford, UK, 1994; Volume 6, ISBN 0-08-041865-1.

6. Bourgois, D.; Vanderschuren, J.; Thomas, D. Study of mass transfer of VOCs into viscous solvents in a pilot-scale cables-bundle scrubber. *Chem. Eng. J.* **2009**, *145*, 446–452. [CrossRef]

7. Garcia-Ochoa, F.; Gomez, E. Bioreactor scale-up and oxygen transfer rate in microbial processes: An overview. *Biotechnol. Adv.* **2009**, *27*, 153–176. [CrossRef] [PubMed]

8. Russell, T.W.F.; Robinson, A.S.; Wagner, N.J. *Mass and Heat Transfer: Analysis of Mass Contactors and Heat Exchangers*; Cambridge Series in Chemical Engineering; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2008; ISBN 978-0-521-88670-3.

9. Staudinger, J.; Roberts, P.V. A critical compilation of Henry's law constant temperature dependence relations for organic compounds in dilute aqueous solutions. *Chemosphere* **2001**, *44*, 561–576. [CrossRef]

10. Guillerm, M.; Couvert, A.; Amrane, A.; Norrant, E.; Lesage, N.; Dumont, É. Absorption of toluene in silicone oil: Effect of the solvent viscosity on hydrodynamics and mass transfer. *Chem. Eng. Res. Des.* **2016**, *109*, 32–40. [CrossRef]
11. Heymes, F.; Manno Demoustier, P.; Charbit, F.; Louis Fanlo, J.; Moulin, P. Hydrodynamics and mass transfer in a packed column: Case of toluene absorption with a viscous absorbent. *Chem. Eng. Sci.* **2006**, *61*, 5094–5106. [CrossRef]

12. Dumont, E.; Darracq, G.; Couvert, A.; Couriol, C.; Amrane, A.; Thomas, D.; Andrès, Y.; Le Cloirec, P. Determination of partition coefficients of three volatile organic compounds (dimethylsulphide, dimethyldisulphide and toluene) in water/silicone oil mixtures. *Chem. Eng. J.* **2010**, *162*, 927–934. [CrossRef]

13. Dumont, E.; Darracq, G.; Couvert, A.; Couriol, C.; Amrane, A.; Thomas, D.; Andrès, Y.; Le Cloirec, P. VOC absorption in a countercurrent packed-bed column using water/silicone oil mixtures: Influence of silicone oil volume fraction. *Chem. Eng. J.* **2011**, *168*, 241–248. [CrossRef]

14. Dumont, É. Mass Transfer in Multiphasic Gas/Liquid/Liquid Systems. KLa Determination Using the Effectiveness-Number of Transfer Unit Method. *Processes* **2018**, *6*, 156. [CrossRef]

<figure><img src="image_13.png" /></figure>

© 2019 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).