Modeling and experimental study on the solubility and mass transfer of CO₂ into aqueous DEA solution using a stirrer bubble column

Abstract

In this research, the chemical absorption rate and solubility of carbon dioxide into DEA aqueous solutions were investigated in a stirrer bubble column. Experiments envelopment the molarity of DEA, CO₂ partial pressures and stirrer speed are 0.2–2.0 M, 300–450 kPa and 0–600 rpm, respectively. The DEA and CO₂ concentration and gas–liquid flow rate effects on the mass transfer performance were measured in terms of CO₂ capture efficiency, loading, mass transfer flux and mass transfer coefficient. A thermodynamic model, according to the Pitzer's G^E model is provided for the equilibrium solubility of CO₂ in DEA aqueous solutions. The results have shown that the CO₂ loading in range of 0.16–0.40 and mass transfer rate in range of 0.36 × 10⁻⁵ to 0.97 × 10⁻⁵ kmol m⁻² s⁻¹ increases with DEA concentration and CO₂ partial pressure. Also the results indicate that the quantity of k_l increases with addition of DEA concentration, liquid and gas flow rate. Film parameter increases with increasing of DEA concentration and decreases with increasing of CO₂ partial pressure. Absorption percent of CO₂ was varied in range of 30–57%. The maximum absorption rate of CO₂ at partial pressure of 45 kPa has been obtaining 1.8 g min⁻¹ when the DEA concentration is 2.0 M.

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1. Introduction

Bubble column reactors are used in a wide range of industrial processes because of their simple design, very complex hydrodynamic behavior, selectivity for the desired product¹ and excellent heat and mass transfer properties, especially in the control of mass transfer processes in the liquid phase.² Stirrer bubble column reactors are used in a diversity of chemical processes, such as Fischer–Tropsch synthesis, oxidation, fermentation and alkylation reactions, effluent treatment and coal liquefaction.³ In the bubble columns, the gas phase exists as a dispersed bubble phase in a liquid phase; thus, the mixing behavior of the liquid phase is mainly affected by the agitating action of rising bubbles.⁴ Dispersed gas in bubble columns is known to be inherently unsteady and the gas phase moves in one of the homogeneous or heterogeneous regimes.⁵ Dispersion of gas bubbles in liquids is the fundament of many mass transfer operations in chemical engineering practice.⁶ The proficiency of stirrer bubble column reactors appertain on the physical bubble characteristic such as gas holdup, bubble size, bubble rise velocity, bubble–bubble interactions and mixing rate. Also the design parameters and the economy of bubble columns are gas–liquid specific interfacial zone, mass transfer coefficient, bubble size distribution, flow regime and coalescence of bubbles.⁷

Increasing greenhouse gas emissions, especially carbon dioxide, have become a challenging subject in recent decades. CO₂ basically to be available in the flue gas of the fossil-fuel fired powerhouse and sour natural gas flows.⁸ Various technologies including absorption, adsorption, distillation, membranes, separation and biological are available for removing acid gases.^9–11 One of the common technologies for CO₂ capture is the chemical absorptions or chemical solvents methods.¹² High efficiency, low cost, and mature technology are advantages of chemical absorption method as compared to the other methods.¹³ The most popular chemical solvents for capture of acid gaseous are aqueous alkanolamines, such as diethanolamine (DEA), monoethanolamine (MEA), methyl diethanolamine (MDEA), diisopropanolamine (DIPA), diglycolamine (DGA).^14–16 However, in contrast to their wide usage, these solvents have various disadvantages such as oxidative degradation, high volatility and regeneration energy of the primary and secondary amines, and low reaction rate of tertiary amines. Therefore, applying reactive solvents with great absorption capacity and low regeneration energy is very significant for CO₂ capture.^17,18

Mass transfer flux is one of the most important subjects in the reactive absorption processes. The absorption of CO₂ in aqueous solution couples physical and chemical absorption where both kinetics and thermodynamic equilibrium can play considerable impress in illustrating the gas loading.¹⁹ However the chemical absorption has a high CO₂ recovery capacity and it is suitable for atmospheric pressure. But some problems such as degradation, high energy regeneration, corrosion problem and side are there.²⁰ Several models are available for evaluate the CO₂ solubility in alkanolamines aqueous solutions.²¹ A Pitzer's G^E model was used to estimate the species concentrations. According to this model an activity coefficient is based on Debye–Hückel theory²² where the activity coefficients are calculated by the equation of Guggenheim, Kent and Eisenberg²³ and Chen and Evans model.²⁴

Reactive absorption of CO₂ in a thin layer can be explained as combination of chemical reaction and diffusion. In the film model, both gas and liquid phases can be districted into two different stagnant bulk and film regions. Fig. 1 shows the film resistance of CO₂ mass transfer from the gas phase into the liquid phase. CO₂ molecules will be partially transferred from the vapor to the liquid phase, wherever it is changed to various ionic and molecular species. The mass transfer is confined via resistances in the gas and liquid films. The resistances in both phases are specified by the diffusion coefficients and the film thickness. It is thought which at the interface equilibrium available between gas phase and liquid phase. A detailed description of the common dimensionless numbers in mass transfer of reactive absorption is given in Table 1.

Fig. 1 CO₂ mass transfer into the liquid phase with chemical reaction based on the film model.

Table 1 The common dimensional numbers in CO₂ chemical absorption

Dimensionless parameter	Parameter name	Definition
	CO2 loading	The ratio of total CO2 moles absorbed to total amine moles
	Film parameter	Maximum possible conversion in the film per maximum diffusional transport through the film
	Enhancement factor	The ratio of the mass flux with chemical reaction to the mass flux without chemical reaction
	CO2 mole fraction	CO2 partial pressure to total pressure ratio

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This work attempts to improve the study of CO₂ capture using DEA solution in a stirrer bubble reactor continuously due to the influence of the chemical structure of the liquid solution upon reaction mechanism, physical properties and mass transfer and hydrodynamic of the stirrer bubble columns. Also mass transfer flux and kinetics of CO₂ absorption into DEA solution will be discussed. The mass transfer performance was obtained by doing experiments and modeling in an atmospheric pressure and constant level via process parameters including DEA concentration, liquid and gas flow rate, stirrer speed and partial pressure of CO₂.

The secondary objective of this work is to provide additional equilibrium data for CO₂ absorption in DEA aqueous solution. In this regard, effort to inquire the profiles of some of the species in the system as a function of CO₂ loading was evaluated. A thermodynamic evaluate with the modified Pitzer's model has been carried out for systems combined of acidic gases and aqueous solution of DEA. The condition of vapor–liquid equilibrium is applied to calculate the total pressure and the composition of the gas phase by extended Raoult's and Henry's laws.

2. Theory

2.1. Reactive absorption mechanism

Absorption followed by reaction in the liquid phase often used to get more complete removal of a solute from a gas mixture. At first, component physically absorbed and then reaction with other components at bulk and film layer occurred. When the liquid film resistance is prevailing a quick chemical reaction in the liquid can allow to an increase in the mass transfer rate. The rapid reaction consumes much of the CO₂ very close to the gas–liquid interface, which causes the gradient for CO₂ steeper and enhances the mass transfer process in the liquid phase. The apparent value proportion of mass transfer coefficient to that for physical absorption defines an enhancement factor E, which ranges from 1.0 to 1000 or more. When absorption is accompanied by a very slow reaction, in this case reaction in the liquid phase in addition to being mass transfer has not increased, but also decreased the amount of mass transfer.²⁶

2.2. Two film theory

The easiest and oldest model which has been proposed for the statement of mass transport processes is the so-called film theory. To illustrate the process, a gas phase is brought in contact with a liquid phase.²⁷ According to this theory a stagnant film of thickness δ_L is supposed to exist at liquid and gas interface as sketched in Fig. 1. Mass transfer occurs by a steady molecular diffusion through the film. This theory predicts mass transfer rate on the basis of first order dependency. The absorption rate is determined by molecular diffusion in the surface layers. Although the diffusional and turbulence transport will apparently vary on continuous basis with depth below the surface. Therefore it is useful to take as a model a completely stagnant layer having an effective thickness δ_L, overlying liquid of uniform composition. In the film theory, mass transfer coefficient k_l and k_g are commensurate to the diffusion coefficient D_CO2–DEA and inversely proportional to the film thickness. k_l denotes the coefficient of mass transfer for liquid side and equals:

Similarly, the following equation can be obtained for the mass transfer coefficient in the gas phase:

The mass transfer coefficient depends on the diffusion coefficient computed by the film theory. However, a number of theoretical problems in the field of chemical absorption and desorption involve such mathematical problems as to allow their solution only for the simple film model.

2.3. Surface renewal theory

Danckwerts proposed the surface renewal theory which is an extension of the penetration theory. It is based on the assumption that the liquid elements do not stay on the same time at the phase interface surface.²⁸where N_CO2 is CO₂ absorption rate, k_l is the liquid-side mass transfer coefficient. The model has been applied in the present work because it is considered the more realistic theory²⁹ to analyze absorption processes since it includes the main part of the phenomena involved in these kinds of processes. The enhancement factor of chemical reaction to the surface renewal theory, for the absorption of CO₂ is obtained as follow:while:and:

Film parameter, M, were calculated as follow:

3. Experimental study

3.1. Materials

All aqueous DEA solutions used in this work were prepared with degassed distilled water and DEA. DEA was prepared from Sigma-Aldrich with a purity of ≥99%. The gas phase consists of CO₂ and air was used in the bubble column. CO₂ gas cylinder was of commercial grade with a minimum purity of 99.9% that was supplied by Hor Mehr Tab Gas. Air was supplied from air compressor with the specifications Mahak, AP-301, 300 L capacity. The gases were mixed and passed through pre-calibrated rotameters and a temperature controlled water bath equipped with an electric heater.

3.2. Experimental setup

The absorption experiments were carried out using the experimental setup as shown in Fig. 2. The stirrer column consists of 140 mm i.d in diameter and 0.5 m in height. In order to visualize the flow, the column is made up of transparent glass. The compressed air from the compressor and CO₂ gas mixed together in a mix tank and then spared in to the bubble column through a sparger. The sparger plate (75 mm) by six holes (1 mm diameter) is made up of stainless steel to make small bubbles at the bottom of the stirrer bubble column during the absorption process. Some physical properties and operation conditions of stirrer bubble column are given in Table 2. The absorbent temperature is kept constant at temperature 295.15 K. Gas sensor Testo 327-1 model was used to analyze the gas mixture. The physical properties and operation conditions are summarized in Table 3.

Fig. 2 Experimental setup for CO₂ absorption.

Table 2 Physical properties and operation conditions of CO₂ absorption in the stirrer bubble column

Index	Value	Index	Value
Diameter of test section (mm)	140	Partial pressure	0.3–0.45
Height of test section (mm)	500	R (m^3 Pa K^−1 mol^−1)	8.314
Shape	Cylindrical	DCO2–H2O (m^2 s^−1)	1.79 × 10^−9
Number of holes in sparger (number)	6	QCO2 (L min^−1)	1–2.2
Diameter of sparger (mm)	75	QAir (L min^−1)	2.8–5
Diameter of hole sparger (mm)	1	ρAir (kg m^−3)	1.22
Mixer type	Turbine	ρCO2 (kg m^−3)	1.98
Disc diameter (mm)	80	ρDEA (kg m^−3)	1970
DEA (M)	0.2–2.0	ρH2O (kg m^−3)	997.2
Volume of solution (L)	6	σ, surface tension (N m^−1)	0.0667
Rotational speed (rpm)	200, 400, 600	μH2O, viscosity of water (Pa s)	0. 85 × 10^−3
Pressure (kPa)	80	μAir, viscosity of air (Pa s)	1.983 × 10^−5

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Table 3 Operation conditions of the bubble column at d_B = 2.009 cm and 295.15 K

Con. (M)	μDEA × 10^−3 (Pa s)^34	DCO2–DEA (m^2 s^−1)	εG	S (m^2)	ΔV × 10^4 (m^3)	a (m^2 m^−3)	Stirrer speed (rpm)	yCO2-in	ΔG (g min^−1)
2.0	21.440	0.13 × 10^−9	0.061	0.078	2.60	19.50	—	0.30	1.102
							200	0.35	1.109
							400	0.40	1.112
1.4	10.340	0.24 × 10^−9	0.064	0.082	2.75	20.67	—	0.30	0.998
							200	0.35	1.013
							400	0.40	1.020
0.7	3.240	0.57 × 10^−9	0.067	0.085	2.86	21.45	—	0.30	0.822
							200	0.35	0.829
							400	0.40	0.837
0.2	1.255	1.31 × 10^−9	0.068	0.088	2.93	21.97	—	0.30	0.706
							200	0.35	0.715
							400	0.40	0.721

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3.3. Experiments

In all experiments the volume of liquid has been set constant and equal to 6 liter, 39 cm height above the sparger. The gas flow of CO₂ and air, 5 L min⁻¹, were measured by two calibrated rotameters separately, and then entered the bottom of the bubble column continuously. A sparger produces bubbles that get in contact with the liquid phase and rise to the top. As the bubbles ascent, interface diffusion takes place. Eventually, equilibrium point is achieved for the specified gas–liquid flow rates. The absorbed gas flows out of the bubble column from the top. Once the gas flow rate becomes stable, the stirrer rotates at the assigned rotational speed. The change of speed is controlled accurately by a dimmer that reaches the desired speed in a sudden manner. The gas superficial velocity and dimensionless number of Reynolds is 14.15 m s⁻¹ and 3573 respectively. The inlet CO₂ to the bubble column was measured by a flow meter. The absorption rate was calculated as the difference between inflow and outflow of CO₂ concentration by CO₂ gas sensor. Due to the pressure and temperature is constant, can be acquired the adsorbed number of moles in the sample from the PQ_in/RT and also, all absorbed moles of carbon dioxide obtained from different inlet and outlet of CO₂ moles. Gas holdup and diameter of bubbles are obtained from the following semi-empirical equations:^30,31

d_B = 1.543 × u_G^−0.12 × D^−0.3, assumption d_s = d_B (9)

The gas–liquid interfacial area can be calculated using the Sauter mean diameter d_s and the gas holdup:³²

where d_B and d_s are means bubble size distribution and Sauter diameter of the bubble size distribution respectively. The diffusion coefficient of carbon dioxide in DEA solution obtained from the following equation:³³

In addition, the percentage of CO₂ removal efficiency was computed with the following equation:

Table 3 shows the bubble characteristics based on the above equations.

The gaseous CO₂ molecules diffuse from the gas film to the gas–liquid interface. The gaseous CO₂ in the gas–liquid interface dissolves agreeing the Henry's law. The dissolved CO₂ is significantly consumed near the interface according to reaction with the amine. The slope of the CO₂ concentration profile defines the mass transfer coefficients. To calculate the mass transfer flux in chemical absorption, the consequence of chemical reactions on the mass transfer is done using enhancement factor (E). The enhancement factor is given in eqn (14)³⁰ defined as the ratio of the mass flux CO₂ through the interface with chemical reaction and driving force to the mass flux through the interface without chemical reaction. However with the same driving force can be acquired from the calculated concentration profiles. Enhancement factor shows that it is function of components concentration and reaction type in process could be bigger or equal to one. Enhancement factor is usually acquired by laboratory result or theoretical evaluation via models that have shortening assumptions. This factor increases via increasing the reactions rate. For slow reactions its amount is one.³⁵

where N_CO2 is CO₂ mass transfer flux that can be achieved from division the number of moles adsorbed on the contact surface bubbles, k_l is mass transfer coefficient in liquid side and are total free CO₂ concentrations at the interface and liquid bulk, respectively.

4. VLE thermodynamic modeling

4.1. Chemical equilibrium in the liquid phase

When CO₂ is absorbed in DEA solutions, many chemical reactions happen in the liquid phase. The model used to predict the carbon dioxide solubility brings up the following equilibriums for the chemical species in the liquid phase.

Dissociation of the protonated DEA:

Formation of carbonate (primary and secondary amines):

Dissociation of CO₂:

Dissociation of bicarbonate ion:

Ionization of water:

The condition for chemical equilibrium for a chemical reaction x is:

where (i) is refers to the reaction number. Thus, the corresponding thermodynamic equilibrium constant expressions can be written.where γ_i is the activity coefficient of each species. The thermodynamic equilibrium constants, K_x(T) for all the chemical reactions are expressed in the form:

Constants for the calculation of the various K_x as a function of temperature as well as their sources for all of the reactions (21)–(25) are given in Table 4.

Table 4 Chemical equilibrium constant for the chemical reaction x, expressed on the molality scale

Parameter	C	B	A	Ref.
K1	6.7769	−3071.1	−48.759	36
K2	325.3375	105 892.2	−2221.140	25
K3	−36.7816	−12 092.1	235.482	37
K4	−35.4819	−12 431.7	220.067	37
K5	−22.4773	−13 445.9	140.932	37

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Moreover the above equilibrium equations, overall DEA and CO₂ concentrations (mol kg⁻¹) an also charge balance must be acquiescence. In the balance equations for DEA and CO₂ in the liquid phase, eqn (27) and (28), C⁰_DEA denotes the stoichiometric DEA concentration and α indicates the CO₂ loading, represented as total moles of CO₂ absorbed both physically and chemically per DEA moles.

Amine mass balance:

C⁰_DEA = (C_DEA + C_DEAH⁺ + C_DEACOO⁻) (27)

CO₂ mass balance:

α_CO2C⁰_DEA = (C_CO2 + C_HCO3⁻ + C_CO3²⁻ + C_DEACOO⁻) (28)

Electroneutrality:

C_H⁺ + C_DEAH2²⁺ = C_OH⁻ + C_HCO3⁻ + 2C_CO3²⁻ + C_DEACOO⁻ (29)

Solving this set of 8 independent equations (eqn (21)–(25) and (27)–(29)) for a given temperature, DEA overall concentration, and CO₂ loading consequence in the real liquid phase equilibrium composition, statement as the molality of each species (mol kg⁻¹), needed for solving the equations.

4.2. Vapor–liquid equilibrium

In the gas phase, the DEA vapor pressure is very little, so DEA in the gas phase could be neglected. The vapor–liquid equilibrium (VLE) condition is used to compute the total pressure and the gas phase composition. The extended Raoult's law is applied to express the VLE for water (eqn (30)), and the extended Henry's law is used to express the equilibrium for CO₂ (eqn (31)).

Assuming that the DEA in the gas phase is negligible, therefore the ratio of φ^sat_w/φ_w will be unit. The VLE calculation requires the knowledge of the Henry's constants for the CO₂ solubility in water. The vapor pressure p^sat_w and the molar volume V_w of pure water, pure component second virial coefficient according Table 7, the mixed second virial coefficients B_CO2 were calculated due to the correlations of Hayden and O'Connell.³⁸ The partial molar volumes of CO₂ dissolved at infinite dilution in water were calculated as recommended by eqn (33). The Henry's constant for CO₂ are expressed in the following form:³⁹

Values of the Henry's constant are taken from the literature as given in Table 5.

Table 5 Henry's constant for the DEA, CO₂ and H₂O system³⁷

A	B	C	D
94.4914	−6789.04	−0.010454	−11.4519

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The water vapor pressure is calculated from the following equation:

where T_c and p_c are 647.096 K and 22.064 MPa, respectively. The constants of eqn (34) are presented in Table 6.

Table 6 Constants of eqn (34) (ref. 40)

a1	a2	a3	a4	a5	a6
−7.8595	1.8441	−11.7866	22.6807	−15.9619	1.8012

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Fugacity coefficient of CO₂ in the gas phase can be calculated as the virial equation of state:³⁹

4.3. Pitzer's G^E model for activity coefficients

Several models are presented to characterize the VLE of CO₂ in aqueous amines solutions in the literature. Among them, Kent and Eisenberg²³ and the Deshmukh and Mather²¹ models are frequently used. However the former one does not take into account the activity coefficients in solution and the latter is limited to low concentration because the activity coefficients are calculated with Guggenheim's equation. Activity coefficients of both molecular and ionic species were calculated using a modified Pitzer model for the excess Gibbs energy of aqueous electrolyte solutions:⁴¹where f₁(I) is a modified Debye–Hückel term depending on ionic strength (I), temperature, and solvent properties, λ_ij(I) is the ionic strength dependent second virial coefficient:where x = 2(I)^1/2. To estimate the activity coefficients of the solute species follows from the appropriate derivative of G^E, the equation proposed by Guggenheim⁴² which is an extension of Debye–Hückel²² theory is used:

Here, the first term indicative the solvent electrostatic effects on the solute species at infinite dilution; the remaining term takes into account short-range van der Waals forces and β_ij is the interaction parameter for two solute species. The quantity A_φ is related to the dielectric constant of the solvent that a function of temperature as proposed by Lewis et al.⁴³ and B equals to 1.2 as suggested by Pitzer.⁴⁴ The ionic strength, I, of the solution was calculated using the following equation:

The quantity A_φ is the Debye–Hückel parameter for the osmotic coefficient. It is related to the dielectric constant of the solvent and the taken as a function of temperature:⁴³

A_φ = −1.306568 + 0.01328238 × T − (0.355080 × 10⁻⁴) × T² + 0.338197 × 10⁻⁷) × T³ (43)

The activity of water is determined from Gibbs–Duhem equation as follows:

where, M_w is 0.018 kg mol⁻¹. The Pitzer's model has been used to estimate the species concentrations in the liquid bulk and solubility data. The set of eqn (21)–(25) and (31) are solved together with the electric charge balance (29) and the mass balance for the amine (27) to determine the concentrations of all components, provided all the equilibrium constants K₁–K₅, the Henry's constant for CO₂, the activity coefficients of all the different species, the activity of water which can be calculated using Gibbs–Duhem equation, the total concentration of amine and the partial pressure of CO₂ are known. Once the concentrations of all species are calculated, the mass balance equation for CO₂ (28) permits the calculation of CO₂ loading.

Interaction parameters, β_ij, between the different ionic and molecular species in the system are represented by Pitzer and Mayorga⁴⁵ in the following form:

β_ij = β⁽⁰⁾_ij = a_ij + b_ij × T (K) and β⁽¹⁾_ij = 0.018 + 3.06 × β⁽⁰⁾_ij (45)

Parameters a_ij, b_ij are summarized in Table 7. In this system, eight species are available in the liquid phase: CO₂, CO₃²⁻, HCO₃⁻, DEACOO⁻, DEAH⁺, DEA, H⁺ and OH⁻. Due to the very low concentration of H⁺ and OH⁻ with respect to the other species, their interactions with all other species were ignored. Therefore the corresponding interaction parameters were set to zero. All ternary interaction parameters between species were also set to zero. To additionally reduce the number of parameters, all binary interaction parameters involving species with the same sign of charge were ignored. Only the parameters which were found to have a significant effect on the liquid phase distribution of the species were optimized based on the experimental data. The values of interaction parameters are given in Table 7. A sensitivity study showed that all other possible interaction parameters that appear in the expressions for the activity coefficients (eqn (41) and (45)) can be neglected without reducing the accuracy of the VLE representation of this system.

Table 7 Species interaction parameters in Pitzer's G^E equation for DEA–CO₂–H₂O system⁴⁶

Ions/molecules interactions (L mol^−1)	aij (L mol^−1)	bij (L K mol^−1)
DEAH^+–DEA	0.801 × 10^−3	−0.150 × 10^−3
DEAH^+–CO2	0.398	−0.199 × 10^−8
DEAH^+–DEACOO^−	4.700	−0.116 × 10^−1
DEAH^+–HCO3^−	0.377	−0.678 × 10^−6
DEA–DEA	0.703	−0.316 × 10^−7
DEA–CO2	0.805 × 10^−5	−0.130 × 10^−6
DEA–DEACOO^−	1.919	−0.491 × 10^−2
DEA–HCO3^−	4.521	−0.129 × 10^−1
CO2–DEACOO^−	0.184 × 10^−5	−0.651 × 10^−7
CO2–HCO3^−	0.661 × 10^−3	−0.679 × 10^−3

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The CO₂ loading (α) was computed using eqn (46). The quantity of absorbed CO₂ was specified using the inlet and outlet gas flow rate by integration.⁴⁷

4.4. Mass transfer regime in chemical absorption

The influence of a reaction on the absorption rate can be expressed by a dimensionless ratio Hatta number or film parameter and enhancement factor. Extent of reaction can also be identified by the relative rates of diffusion and reaction. M is film parameter which is used to see the effect of chemical reactions in mass transfer flux.⁴⁸ Film parameter is maximum possible conversion in the film per maximum diffusional transport through the film. In order to compute the film parameter, all reactions of CO₂ should be considered. Reactions of DEA with CO₂ contain hydrolysis of monocarbamate DEA, as a second order reaction rate mechanism.⁴¹where the total reaction rate k_app is supposed thoroughly being specified by the following equation.

Depending on the value of M and the ratio between M and E, three absorption regimes can be distinguished. For M < 0.3 and E = 1, the reaction is too slow and the rate of reaction is faster than diffusional transfer of solute to liquid phase. This transfer of solute is rate controlling step. Mass transfer coefficient and absorption flux will depend on liquid flow rate.⁴⁹ If 3 < M ≪ E, the reaction will be fast, diffusion and chemical reactions are proceeding parallel. The liquid phase retention time is not of considerable importance because all reaction takes place in reaction film. When 3 < E_∞ ≪ M and E = E_∞ applies for instantaneous reaction regime. The reaction is too fast in case of irreversible reaction, the reactants cannot co-exist in film.⁵⁰

4.5. Chemical reactions

The CO₂ reaction with alkanolamines is of enormous significance. In aqueous solution of DEA, the absorption of CO₂ can be investigated as an incorporation of various reactions^32,51,52 inclusive CO₂–OH⁻, CO₂–DEA and CO₂–DEACOO⁻.

CO₂ + OH⁻ ↔ HCO₃⁻ (49)

where:

−r_CO2–OH⁻ = k_OH⁻C_OH⁻C_CO2 (50)

The bicarbonate formation reaction rate constant in eqn (49) is apparent by Pinsent et al.⁵³

Two mechanisms Zwitterion and single-step termolecular are presented in the literature to describe the chemical interactions available between primary or secondary amines and CO₂. Zwitterion mechanism was presented for the first time by Caplow⁵⁴ and reintroduced by Danckwerts.⁵⁵ The zwitterion mechanism inclusive of two steps: firstly, the reaction between carbon atom of the CO₂ molecule and the free electronic of amine proceeds through the produce of intermediate called zwitterions:

Second step, the deprotonated of the zwitterion complex by any base B present in the solution, such as hydroxyl ion, water or an amine-functionality, to generate a combine called carbamate.

In the zwitterion deprotonation, the portion of each basis depends on its concentration, basicity and steric hindrance. In aqueous solutions, the corresponding reactions are as follows:

The overall forward reaction rate can be derived using the quasi-steady-state condition to the intermediate zwitterion concentration:⁵⁶

Versteeg and van Swaaij⁵⁷ concluded that portion of hydroxyl ion in aqueous DEA solution to the remove protonation of the zwitterions can be ignored as far it's very low concentration. So that eqn (57) reduces to:

In a general form:

where the kinetic constant ∑k_bC_B demonstrator the contribution to the proton removal of the zwitterion by any base such as CO₃²⁻, HCO₃⁻, H₂O, OH⁻ or DEA, as also by compound of bases. For the absorption of CO₂ into aqueous DEA solution, the overall reaction rate can be expressed as follows:

−r_CO2 = r_CO2–DEA + r_CO2–OH⁻ = k_appC_CO2 (60)

Thus:

The contributions of the CO₂ reactions with hydroxyl ion to the remove proton in the overall reaction rate is supposed to be negligible.⁵⁸ The apparent kinetic rate constant (k_app) can be specified as:

k_app = k_ov + k_OH⁻C_OH⁻ (62)

where k_app is the apparent rate of the reaction calculated as follows:

The kinetic constants of CO₂ into DEA solutions are provided in Table 8.

Table 8 Literature data on the reaction between CO₂ and DEA

Constant	Unit	Value	Ref.
	L^2 mol^−2 s^−1	1101	59
	L^2 mol^−2 s^−1	1.71	57
k2	L mol^−1 s^−1	3977	59
kOH^−	L mol^−1 s^−1	6698	53

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In the solution of the equations, the Newton's iteration technique was employed for determination of the activity and concentration of species in the CO₂–DEA–H₂O systems. The method can be extended to complex functions and to systems of equations. Newton's method is an algorithm for finding the roots of differentiable functions that uses iterated local linearization of a function to approximate its roots.

5. Results and discussion

5.1. CO₂–DEA–H₂O solubility

The solubility of CO₂ in 0.2–2.0 M DEA aqueous solutions were obtained at fixed temperature of 295.15 K at partial pressures of CO₂ ranging from 1 to 10 kPa. In applying the Pitzer's model, the values of K₁–K₅ that are reported in the literature (Table 4) were used to analyze the data. As indicated earlier only ten interaction parameters in the CO₂–DEA–H₂O system have been found to be significant. These parameters are given in Table 7. The equilibrium constant K_x was computed at different temperatures and fitted according to eqn (26) to give the various constants as shown in Table 4. The interaction coefficients were obtained using reliable solubility data from literature for the model, able to cover large temperature, pressure, and amine composition ranges. The partial pressure of CO₂, in the vapor phase was as a result simply derived as the difference between the total measured pressure and the vapor pressure of the CO₂-free, aqueous DEA solution. A comparison of solubility of CO₂ in DEA solution is shows in Fig. 3. All data repeated three times and the average is reported. Fig. 3 shows that near 0. 3 loading the solution of 2 M have a higher CO₂ partial pressure than the 1.4 M DEA data, which is higher than the 0.7 M and 0.2 M DEA CO₂ partial pressure. In other hand, with increasing DEA concentration in the solution, CO₂ loading decreases at a given temperature. This means that DEA has a positive effect on CO₂ solubility.

Fig. 3 Effect of CO₂ partial pressure on the solubility of DEA solution at 295.15 K.

5.2. Species activity and concentration

The concentration profiles of the different species in the CO₂–DEA–H₂O system and their activity coefficient using the modified Pitzer model were shown respectively in Fig. 4 and 5. Since H⁺, OH⁻, and DEAH⁺ concentrations are very less than the concentrations of all other species, the respective curves were not demonstrated. The free DEA concentration decreases quickly by increasing loading, while this reduction is accompanied by a steady increase in the concentrations of protonated DEA as well as that of HCO₃⁻. It was found that at low partial pressure and loading, further of CO₂ absorbed into the solution is in the form of carbamate by a small value in the form of bicarbonate. When the loading increases, the trend is reversed where the attendance of bicarbonate is considerable. Open the whole range of CO₂ loading, free CO₂ is only present in trace amounts and it only appears in the solution at high loadings regularly above 0.7. The formation of carbamate from DEA is rapid and the equilibrium is arrived quickly. It shows that at low pressure, CO₂ is rather absorbed into DEA. When the concentration of CO₂ stoichiometric is less than that of DEA, CO₂ is actually entirely chemically dissolved and mainly converted preferentially into DEA carbamate, DEA dicarbamate, and protonated DEA species. In addition, the carbamate concentration is very low, which is compatible by the treatment of the sterically prevented amines.

Fig. 4 species concentrations on the liquid phase at 0.7 M DEA aqueous solution at y_CO2-in = 0.3, obtained using the modified Pitzer model.

Fig. 5 Species activity coefficients in the H₂O–CO₂–DEA system at 0.7 M DEA solution and y_CO2-in = 0.3, using the modified Pitzer model.

Also, Fig. 6 illustrate the influence of free DEA in various concentrations of amine on CO₂ mass transfer flux. It is obvious that mass transfer flux extremely belongs on the free DEA existing in the solution. Also increase in the amine concentration conduce enhance in free DEA and this afford mass transfer flux to increase.

Fig. 6 The effect of free DEA concentration (calculated from the presented model) on CO₂ mass transfer flux (experimental data) at CO₂ volume fraction 0.3–0.45 (% v/v).

5.3. Absorption rate

Fig. 7 shows the effect of different partial pressure on CO₂ removal efficiency at various DEA concentrations. It can be seen that the performance of 2.0 M of DEA solution was better than another one. With the increase in the DEA concentration, from 0.2 to 2.0 M, the capture of CO₂ enhanced from 30 to 56%. This is because at higher DEA concentration, higher value of free amine molecules was disposal to react with CO₂ in the liquid phase. As well as reduction of CO₂ loading capacity in the liquid phase which reduces the resistance in the liquid film. Where the CO₂ capture in DEA solution is controlled by the resistance in the liquid phase, the mass transfer efficiency was increased in the process. Hence, the increase in liquid concentration has a considerable influence on the mass transfer performance.

Fig. 7 Effect of different DEA concentrations on CO₂ removal efficiency.

To survey the absorption system in the stirrer bubble column, the evaluation in the absorption rate of DEA solution were conducted out in two cases; with and without stirrer working for the test section. Fig. 8 shows CO₂ absorption rate into DEA solution versus absorption time at 295.15 K and atmospheric pressure. It shows the absorption rates at different DEA aqueous solution from 0.2 to 2.0 M. It can be seen that in the beginning, there is a high rate of absorption which is dependent of DEA concentration. This is due to the consumption of the convenience existing dissolved alkalinity. After that, the number of active sites available in solution for CO₂ perch there was less and causes the decrease of driving force. By decreasing the driving force the absorption rate reduce until the reaction has arrived a respectively steady level and also absorption rates for all solution are lower than 2.0 kmol m⁻² s⁻¹. It is clearly implied that the speed of stirrer enhance the absorption performance in all dissolved. It is also found that the absorption performance at high concentration of DEA with stirrer is better than that without stirrer.

Fig. 8 Effect of stirrer speed on absorption rate at different DEA solution: (A) 0.2 M, (B) 0.7 M, (C) 1.4 M and (D) 2.0 M.

Fig. 9 indicates the evolution of pH profile at the continuous mode examine. It is revealed that all the solutions have similar initial pH values (∼10.6). In the beginning, all the dissolved carbon dioxide gas instantly reacts with hydroxide ions (OH⁻) and is conversion to carbonate (CO₃²⁻) and pH drops very fast. Therefore, only small traces of dissolved carbon dioxide gas placed near the bubbles are apperceived while due to the utilization of initial total DEA the bicarbonate (HCO₃⁻) concentration is little. With increasing of solution conversion, the concentrations of DEAH⁺ and HCO₃⁻ rise, this can be described via a base catalyst mechanism. At the lower conversion, the carbamate is formed rapidly and then decreases slowly at the higher conversion. When the pH value becomes low sufficient, the DEA conversion to HCO₃⁻ becomes higher and postpone the formation of DEACOO⁻. Therefore due to conversion of DEA to DEA carbamate and protonated DEA, the concentration of free DEA becomes low but the value of DEA is still important for absorption of CO₂.

Fig. 9 pH profile of 4 different concentrations of DEA solutions at gas and liquid flow rate 5.0 L min⁻¹ and 5.0 L h⁻¹ respectively and y_CO2-in = 0.3.

Fig. 10 shows the effect of CO₂ loading on mass transfer flux at 295.15 K. As it can be seen by increasing the CO₂ partial pressure, mass transfer flux of CO₂ increases and this is obvious because the increase in the partial pressure of CO₂ in gas phase leads to increase in the driving force of mass transfer. In a constant pressure ratio, increase in loading leads to decrease in mass transfer flux which is because of decrease in driving force of mass transfer. Also Fig. 10 shows that when the concentration is increased at constant temperature and interfacial CO₂ partial pressure, the flux has been augmented, as well. This enhancement is commonly ascribed to the presence of more free and active DEA in the solution to react by CO₂.

Fig. 10 Mass transfer fluxes of CO₂ at different partial pressures with gas flow rate of 5.0 L min⁻¹ and liquid flow rate of 5.0 L h⁻¹.

In order to study the performance of difference DEA solutions in parameters of loading capacity, the CO₂ solubility, the constants mass flow and the relative between model predictions and experimental data was evaluated at ambient temperature and atmospheric pressure as listed in Tables 9–11. It is observed that at the same CO₂ partial pressure, the CO₂ loading increased with increasing DEA concentration in the solution. Also in Table 9, highest CO₂ mass transfer rates were achieved in 2.0 M DEA solution over the entire range of CO₂ loadings.

Table 9 Experimental data and calculation results of CO₂ absorption into aqueous DEA solutions, at 80 kPa, 295.15 K and H_CO2,H2O = 131.136 × 10² (kPa kg kmol⁻¹)

Con. (M)	yCO2,in	COH^− × 10^4 (M)	NCO2 × 10^5 (kmol m^−2 s)	CDEA (M)	CDEACOO (M)	pCO2,b (kPa)	pCO2,i (kPa)	(kPa)	(M)	(M)
2.0	0.30	45.7	5.89	1.141	0.123	0.151	0.114	0.065	0.444	0.513
2.0	0.35	46.0	7.12	1.139	0.123	0.139	0.136	0.081	0.482	0.513
2.0	0.40	46.7	8.28	1.13	0.124	0.164	0.160	0.092	0.504	0.522
2.0	0.45	48.1	9.69	1.159	0.121	0.190	0.185	0.095	0.511	0.518
1.4	0.30	18.9	5.18	0.868	0.077	1.96	1.90	0.852	0.294	0.353
1.4	0.35	19.4	6.35	0.867	0.077	2.34	2.27	0.996	0.319	0.355
1.4	0.40	19.5	7.42	0.866	0.077	2.73	2.65	1.089	0.340	0.356
1.4	0.45	20.3	8.69	0.867	0.077	3.12	3.03	1.155	0.339	0.357
0.7	0.30	0.4	4.27	0.470	0.019	2.18	2.12	0.431	0.050	0.353
0.7	0.35	0.4	5.05	0.467	0.020	2.57	2.49	0.503	0.053	0.355
0.7	0.40	0.4	5.99	0.468	0.020	2.97	2.88	0.533	0.054	0.356
0.7	0.45	0.4	6.90	0.461	0.020	3.36	3.26	0.563	0.055	0.357
0.2	0.30	0.1	3.63	0.142	0.001	2.33	2.26	0.144	0.054	0.092
0.2	0.35	0.2	4.38	0.147	0.001	2.74	2.65	0.168	0.057	0.097
0.2	0.40	0.2	5.12	0.144	0.001	3.13	3.03	0.171	0.058	0.099
0.2	0.45	0.3	6.62	0.142	0.001	3.53	3.42	0.174	0.058	0.094

---

Table 10 Experimental data and calculation results for DEA solution at 295.15 K

Con. (M)		kg × 10^5	δl × 10^5	δg × 10^5	kapp	KG × 10^7	kl × 10^5	k′l × 10^9	−rCO2
2.0	0.30	196.33	1.10	0.0070	1168	1.13	1.19	1.20	4.06
2.0	0.35	237.33	0.91	0.0057	1165	1.23	1.43	1.29	4.59
2.0	0.40	207.00	0.76	0.0067	1153	1.15	1.71	1.22	5.22
2.0	0.45	193.80	0.67	0.0070	1201	1.02	1.93	1.08	6.05
1.4	0.30	0.49	2.22	4.855	727	4.68	1.08	4.90	3.36
1.4	0.35	0.50	1.65	4.815	727	4.72	1.46	4.98	3.78
1.4	0.40	0.47	1.35	5.049	725	4.52	1.78	4.75	4.27
1.4	0.45	0.46	1.15	5.178	727	4.42	2.08	4.63	4.86
0.7	0.30	7.12	8.05	0.800	248	7.12	0.71	2.53	1.49
0.7	0.35	6.31	6.79	0.902	245	6.32	0.84	2.54	1.73
0.7	0.40	6.65	5.73	0.867	245	6.65	0.99	2.55	1.73
0.7	0.45	6.90	4.97	0.825	240	6.90	1.15	2.56	2.16
0.2	0.30	5.18	1.39	2.527	34	5.18	9.40	1.72	0.23
0.2	0.35	4.86	1.20	2.694	34	4.86	10.91	1.76	0.27
0.2	0.40	5.117	1.06	2.560	33	5.12	12.36	1.79	0.31
0.2	0.45	6.021	0.71	2.175	33	6.02	18.55	2.04	0.34

---

Table 11 Dimensionless numbers results of CO₂ reactive absorption in DEA solution

Con. (M)			α	δg/δl	M	EFM	ESR
2.0	0.30	0.11	0.23	0.006	33	133.0	5.8
2.0	0.35	0.12	0.28	0.006	27	118.4	5.3
2.0	0.40	0.14	0.32	0.008	23	93.1	4.9
2.0	0.45	0.15	0.38	0.01	21	73.3	4.6
1.4	0.30	0.13	0.21	2.209	39	6.0	6.3
1.4	0.35	0.14	0.26	3.006	29	4.5	5.4
1.4	0.40	0.16	0.30	3.923	23	3.5	4.9
1.4	0.45	0.18	0.36	4.333	20	2.9	4.6
0.7	0.30	0.16	0.18	0.099	53	4.7	7.4
0.7	0.35	0.19	0.21	0.132	45	4.0	6.7
0.7	0.40	0.21	0.25	0.151	38	3.4	6.2
0.7	0.45	0.24	0.29	0.164	32	2.9	5.8
0.2	0.30	0.18	0.16	1.698	2	0.2	1.8
0.2	0.35	0.21	0.19	2.242	2	0.2	1.7
0.2	0.40	0.24	0.26	2.415	2	0.2	1.6
0.2	0.45	0.27	0.29	3.100	2	0.1	1.5

---

Fig. 11 shows the variation in enhancement factor with film parameter in various DEA concentrations. From this figure can be consummate that, by increasing the concentration of DEA, the slope of enhancement factor alterations increases and this means that the absorption process occur faster in higher concentrations. Moreover, in lower concentrations, enhancement factor decreases and as this factor is commensurate to mass transfer flux. Also by increasing the concentration, mass transfer flux increases; this is obvious because absorption driving force is increase.

Fig. 11 The effect of film parameter on enhancement factor at various DEA concentrations at CO₂ volume fraction 0.3–0.45 (% v/v) and gas and liquid flow rate 5.0 L min⁻¹ and 5.0 L h⁻¹ respectively.

Diversity of the film parameter with CO₂ loading is described at three different concentrations in Fig. 12. The consequence represent that in constant of DEA concentration, solution with less loading has great film parameter. Also the result shows that the film parameter was decreased by increasing of DEA concentration in the constant loading.

Fig. 12 Variation of film parameter with loading at different DEA concentrations, CO₂ volume fraction 0.3–0.45 (% v/v), gas flow rate 5.0 L min⁻¹, liquid flow rate 5.0 L h⁻¹.

The effect of CO₂ partial pressure on the reaction rate was considered in Fig. 13. It can be derived that, by increasing the concentration of DEA, the numerous of activate site increases and it because of the reaction rate increase. Also, in DEA constant concentration, solution with less CO₂ partial pressure has low activate molecules, it because of reaction rate to decrease.

Fig. 13 Effect of the presence of CO₂ on the overall reaction rate, gas flow rate 5.0 L min⁻¹ and liquid flow rate 5.0 L h⁻¹.

6. Conclusion

The mass transfer performances, solubility and absorption rate of CO₂ into aqueous DEA solution have been investigated experimentally and theoretically using a stirrer bubble column. In the experiments, the increase in stirring speed consequence in increasing the mass transfer rate, indicative the diffusive mass transfer regime. The reaction between CO₂ and free DEA in aqueous solutions was described with zwitterion mechanism. A mathematical model is provided based on the Pitzer's G^E model has been studied to evaluate the solubility of CO₂ and carbamate concentration for the absorption of CO₂ in DEA aqueous solution. The activity coefficients were obtained using modified Pitzer's thermodynamic model and experimental data for CO₂–DEA–H₂O system. The experimental analysis of the film parameters are near to the quantity regarding to the fast reaction and the rate of reaction is faster than diffusional transfer of solute. The absorption rates of CO₂ into DEA aqueous solution increases by increasing the stirrer speed. It was also observed that the effect of stirrer in high concentration of DEA is more than dilute solution.

Appendix A

The adsorbed number of moles in the column can be acquired from the following equation:

The absorption rate is calculated by the following equation:

G_CO2,in = y_CO2,in × G_in and G_Air,in = G_in − G_CO2,in and G_Air,in = G_Air,out (A.2)

where G_in, G_CO2-in and G_CO2-out represent absorption rate, initial and final mass of the solution, respectively. Also:

y_Air,out = 1 − y_CO2,out (A.3)

All models of absorbed CO₂ are calculated as follow:

ΔG = G_CO2,in − G_CO2,out (A.5)

Appendix B

Two film theory

The mass balance in the film area was obtained as follow:

N_AzS|_z − N_AzS|_z+Δz − r_ASΔz = 0 (B.1)

where r_A is the rate of CO₂ reaction in the film area.

From combining the eqn (B.2) and (B.3) for physical adsorption:

While the boundary limits are:

z = 0, C_CO2 = CO_2,i (B.5)

z = δ_L, C_CO2 = CO_2,b (B.6)

Integrating and applying boundary conditions yields.

The mass transfer flux will then be:

k_l denotes the coefficient of mass transfer for liquid side and equals:

Similarly, the following equation can be obtained for the mass transfer coefficient in the gas phase:

Eqn (B.11) describes the mass transfer flux equations which can be written using the variety mass transfer coefficients.

where, p_CO2,b is the operational partial pressure of CO₂ in the bubble column, which was calculated by the logarithmic average:

p_CO2,i is equilibrium CO₂ partial pressure that is measured by classification both absorption and desorption data near equilibrium. The CO₂ partial pressure at the gas–liquid interface is calculated by:

p_CO2,i = (p − p^sat_(w+DEA))y_BM (B.13)

where:

The flux in eqn (B.11) is constant and combining these equations results a series resistance relationship among the mass transfer coefficients. In the mass transfer process between the gas and the liquid phase, the total resistance to mass transfer was modeled as the sum of gas film and liquid film resistance:

k′_l can also be expressed as:

Appendix C

Surface renewal theory

The following equations between the various parameters are established:

From combining the eqn (C.2) and (C.3) for physical adsorption:

Boundary conditions include:

z = 0, C_CO2 = CO_2,i (C.5)

z = ∞, C_CO2 = 0 (C.6)

The initial condition is defined as follows:

t = 0, C_CO2 = 0 (C.7)

Take Laplace from eqn (C.1):

By combining the above equations:

General solution of the equation will be as follows:

Take Laplace the boundary conditions:

By substituting eqn (C.11) in (C.9):

From the initial condition as will be following:

According to the Danckwerts assumptions:

By substituting the eqn (C.15) in to the eqn (C.17):

By substituting film parameter in above equation will be:

By writing Fick's law at the interface:

The enhancement factor can be calculated from the surface renewal theory:

While

And:

Nomenclature

a	Specific gas–liquid area, m^2 m^−3
aw	Activity of water
BCO2	Constant of virial eqn, cm^3 mol^−1
CCO2	Total CO2 concentration, M
CDEA	Total DEA concentration, M
CCO2,b	Free CO2 concentrations at the liquid bulk, M
	Total CO2 concentrations at the bulk, M
CCO2,i	Free CO2 concentrations at the interface, M
	Total CO2 concentrations at the interface, M
dB	Diameter of bubbles, cm
d0	Diameter of sparger holes, mm
ds	Sauter diameter of the bubble, cm
DEA	Diethanolamine
DCO2–DEA	CO2 diffusion coefficient in DEA, m^2 s^−1
DCO2–H2O	CO2 diffusion coefficient in water, m^2 s^−1
E	Enhancement factor
EFM	Two film theory enhancement factor
ESR	Surface renewal enhancement factor
Gin	Gas molar flow rate, mol s^−1
HCO2,H2O	Henry's constant, atm mol^−1 kg^−1
kapp	Apparent rate constant, s^−1
kg	Gas film physical mass transfer coefficient, m s^−1
KG	Total gas phase mass transfer coefficient, kmol m^−2 s^−1 Pa^−1
k′l	Liquid film mass transfer coefficient, kmol m^−2 s^−1 Pa^−1
kl	Liquid side mass transfer coefficient, m s^−1
k^0l	Liquid film physical mass transfer coefficient, m s^−1
kobs	Observed kinetic rate constant
Kx(T)	Equilibrium constant, m s^−1
mi	Molality of species, mol kg^−1
M	Molarity, mol L^−1
Mi	Molarity of species, mol L^−1
Mw	Molecular weight of water, kg mol^−1
NCO2	Mass transfer flux, kmol m^−2 s^−1
P	Total system pressure, kPa
pCO2,out	Outlet CO2 partial pressure, kPa
pCO2	Partial pressure of CO2, kPa
pCO2,b	Gas bulk partial pressure of CO2, kPa
pc	Critical pressure, MPa
	Equilibrium CO2 partial pressure of the bulk solution, kPa
pCO2,i	Interface partial pressure of CO2, kPa
p^satw	Saturated vapor pressure of water, kPa
Qin	Gas flow rate, L s^−1
rCO2	Rate of reaction, mol L^−1 s^−1
Re0	Reynolds number
S	Contact surface, m^2
t	Time, s
T	Temperature, K
Tc	Critical temperature, K
uG	superficial velocity, m s^−1
yCO2-in	Mole fraction of inlet CO2
zi	Charge of ion i

Greek letters

αCO2	CO2 loading, mole CO2 per mole amine
γi	Activity coefficient of component i
ρL	Density of liquid, kg m^−3
ρg	Density of gas, kg m^−3
σ	Surface tension of liquid, N m^−1
μDEA	DEA solution viscosity, Pa s
β^0,1i,j	Binary interaction parameters between species i and j in the Pitzer's equation, L mol^−1
εG	Gas holdup
γi	Activity coefficient of species
λij(I)	Second virial coefficient in Pitzer's equation
Tijk	Ternary interaction parameter in Pitzer's equation
φ^satw	Saturated fugacity coefficient of water
φw	Fugacity coefficient of water
ΔG	Absorption rate, g min^−1
ΔV	Differences in fluid volume, m^3

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