# Absorption and Chemisorption of Small Levitated Single Bubbles in Aqueous Solutions

^{*}

## Abstract

**:**

_{2}or CO

_{2}were investigated experimentally in aqueous and alkaline solutions. Different bubble sizes were studied ranging from 0.1 to 2.5 mm in alkaline concentrations of 0.1 mM to 1 M NaOH. The experiments were conducted in a device consisting of a converging microchannel with a down flowing liquid. Levitation positions of single bubbles were optically characterized. A correlation was developed for the drag force coefficient, C

_{D}, including wall effects based on the force equilibrium. A linear decrease of bubble diameters was identified with and without chemical reaction, which is referred to as a rigid bubble surface area. Measured Sherwood numbers agree well with the literature values for the investigated Reynolds number range.

## 1. Introduction

_{L}, represented in dimensionless form as the Sherwood number, Sh. Mass transfer depends on the surface of dissolving gas bubbles [13]. The surface attains different shapes according to the bubble’s size and rising velocity [14]. Additionally, the inner motion of the bubble improves the mass transfer [15]. This motion is influenced by the bubble size itself and its surface contamination. Air bubbles in water with diameters smaller than 0.2 mm act as solid spheres without any motion, due to dominant surface tension, generally [14,16,17]. Contamination is affected by the liquid purity in terms of salt and ion concentration. High electrical conductivity or low resistance favors contamination. In water purified by an ion exchanger bubbles with diameters up to 1.6 mm showed a rigid behavior [18]. The constant decay of the bubble radius due to absorption was observed for several gases in tap water with a low solubility in this range of bubble sizes. A lower liquid saturation resulted in a higher rate [19]. In tap water, too, comparable values of diameter decay were found for air bubbles with diameters of 8 mm. Two different sections of decreasing radii instead were identified using clean water for several gases with a low solubility. This rate change was related to impurities in the water acting as surfactants and creating a stagnant cap at the bubble’s back, which hinders inner motions, due to Marangoni convection [20]. The absorption of highly soluble gas bubbles, especially carbon dioxide, was observed in different liquids and concentrations. For water, an initially high mass transfer coefficient was observed followed by a constant period of a reduced value. The change occurred in a bubble diameter range of 1.0 to 1.5 mm. This was also related to surface contamination and transition to a rigid bubble behavior [21]. Only one regime could be observed for diameters from 0.2 to 1.0 mm in contaminated water [22]. Bubbles with radii from 0.1 to 0.5 mm decrease faster in sodium hydroxide solutions with concentrations of 0.01 to 1 M by chemical reaction. Sherwood numbers were calculated and show 10% deviations from a non-equilibrium reaction model [23]. The reduction of linear radius decay with time at a larger diameter was related to a shrinking surface, due to contamination [24]. The relation between motion and bubble size is also affected by the bubble age. Hence, the exact transition point from a rigid to a mobile surface is not identified clearly [20]. For low soluble gases, this point appeared after minutes, which exceeds the residence time of most classical devices, such as stripping columns. Two approaches are established to capture bubbles and to increase the observation time: a rotary chamber [18] or using a conical tube with down flowing liquid [20].

_{2}and CO

_{2}bubbles were analyzed in various liquids to investigate the mass transfer concerning the difference between a mobile and a rigid bubble surface.

## 2. Theory

#### 2.1. Bubble Rising Velocity and Drag Models

_{eq}, mainly dependent on size, which lets larger solid particles stay closer to the center line. This radius represents the distance between the solid particle center and the central axis [25].

_{D}[26]. Furthermore, buoyancy force gives a vertical force balance, written as:

_{1}and ρ

_{g}are liquid and gas densities, u the liquid velocity, A

_{b}the bubble surface normal to the flow, g the gravitational acceleration, d

_{b}the bubble diameter and Re the Reynolds number. Ellipsoids are described by their main diameters, a and b, due to the arrangement to the flow, also known as Feret diameters. The diameter of a volume-equivalent sphere, d

_{e}, is then calculated by Equation (3). For low aspect ratios, the deviation from an ideal sphere is negligible [20].

_{i}.

_{D}

_{,∞}is the drag force coefficient for an infinite flow field. Some empirical correlations modeling K

_{i}are listed in [26,27]. Most of these correlations have the same form as a division of two polynomials, where the denominator has a higher order. Each polynomial depends on the ratio of bubble diameter to channel width λ.

#### 2.2. Mass Transfer of Single Bubbles

_{L}the liquid side mass transfer coefficient, A the contact surface and c

_{0}and c

_{l}the mole concentrations at the gas-liquid interface and in the liquid bulk. If c

_{l}is negligibly small, Equation (5) can be transformed into the following using the ideal gas and Henry’s law.

_{b}and V

_{b}the pressure and volume of the gas bubble, R the universal gas constant and T the temperature in K. The Henry constant for electrolyte solutions is calculated with Equation (7) [28].

_{w}is the Henry constant for water calculated by the method given in [29], h the contribution referring to positive and negative ions and to the gas and z the valence electrons. Differentiation of Equation (6) gives (corresponding to [30]):

_{b}, of a sphere and the bubble surface, A, are inserted into Equation (8), it can be rewritten using the definition of Sh to:

_{b}= p

_{atm}+ ρ

_{l}gh + 4σ / d

_{b}

_{atm}is the atmospheric pressure, h the height of the water head and σ the surface tension. If bubbles are levitated at the same position and the fluid properties stay constant, pressure change in the bubble depends only on variation of d

_{b}. In the range of diameters from 2.0 to 0.05 mm, the surface tension pressure change can be calculated to differences of 58 mbar for a water/air-system and is therefore negligibly small. This result is valid for all aqueous solutions in this work. Hence, Equation (9) can be written as follows:

_{1}Re

^{1/2}Sc

^{1/2}

^{1/2}Sc

^{1/3}

## 3. Experimental Methods

#### 3.1. Apparatus and Procedure

**∙**cm), deionized water (0.34 MΩ

**∙**cm) and Merck Millipore (Merck KGaA, Germany) treated water (5.0 MΩ

**∙**cm) are used. Sodium hydroxide is purchased from Merck KGaA, Germany. Nitrogen (5.0, 99.999% pure) and carbon dioxide (technical grade, 99.5% pure; 4.5, 99.995% pure) from Messer Group GmbH, Germany, as well as pressurized air are investigated in the set-up. An Advance ICD (10×)–(160×) transmitted-light microscope from Bresser GmbH, Germany, is used for visualization with a length scale mounted next to the glass channel. A D7000 digital camera from Nikon GmbH, Germany, with an AF Nikkor 24–85 mm f/2.8-4D IF macro-objective and a MotionXtra NR4 Speed 2 high-speed camera from ImagingSolutions GmbH, Germany, are connected to the microscope for documentation.

**Figure 1.**Drawing of the experimental device with details of one polyacrylic cube, the microvalve and a sketch of the inside of the glass channel with all the forces acting on the bubble.

#### 3.2. Bubble Diameter Determination with ImageJ

## 4. Results and Discussion

#### 4.1. Levitation Position

_{eq}, of the equilibrium position increases, as seen in Figure 2b. Higher flow rates imply the same tendency regarding larger drag forces. Bubbles stay in the middle of the channel for λ > 0.5. Differences to this central position are identified for smaller ratios in an increasing manner; thus, each equilibrium radius is larger for smaller bubbles. This general trend corresponds with the solid particles studied in [33]; even equilibrium radii are notably smaller here.

**Figure 2.**(

**a**) The horizontal levitation position with the vertical levitation position of the bubbles; (

**b**) The non-dimensional equilibrium radius with the non-dimensional bubble diameter for the levitated bubbles.

#### 4.2. Modeling Drag Force Coefficient C_{D}

_{D∞}and offers the best results in the investigated bubble range. Constants K

_{I}and K

_{II}are calculated by the least-squares method. Figure 3a,b shows the determined values and their correlations. An exponential and a double-linear approach are used as the best-fitting curves for each constant. The intersection points of the straight lines are λ = 0.62 and λ = 0.648. The walls have to be considered for r

_{eq}< 2d

_{b}[35], which is equivalent to λ larger than 0.2. No elongation of the bubbles was observed for these λ values. Therefore, larger constants and, hence, drag force coefficients are shown for higher values of λ, but further investigations have to clarify this observation. Deviations are smaller than 0.5% for all four fits, so similar results are given using either the exponential or linear modeling. In Figure 3c, good conformity is shown for the experimental results and the two modeling approaches according to Equation 4 for three λ exemplarily. The second approach multiplying K

_{II}with C

_{D∞}approximates the calculated drag force coefficients even better. Thus, in Figure 3d, five modeled drag force coefficient curves with this approach are shown additionally to the experimental results mentioned before. The maximum deviations are lower than 8% for each λ in total, which involves an accurate correlation.

**Figure 3.**(

**a**,

**b**) Constants K

_{I}and K

_{II}with λ; (

**c**) Calculated and correlated drag coefficients C

_{D}with Re for three λ values; (

**d**) Calculated and correlated drag coefficients C

_{D}with Re for λ values from 0.1 to 0.9.

#### 4.3. Absorption of N_{2} and CO_{2}

**Figure 4.**(

**a**) The nitrogen bubble diameter with time in water of different purities; (

**b**) The carbon dioxide bubble diameter with time. Technical (techn.) grade and quality 4.5.

#### 4.4. Chemisorption of CO_{2}

**Figure 5.**(

**a**) The carbon dioxide bubble diameter with time in sodium hydroxide solutions of different concentrations; (

**b**) The mass transfer coefficient depending on the flow rate and the bubble diameter, represented by Sh and Re.

_{1}are presented in the figure, too. For comparison, Equation (14) and Equation (15) of Higbie and Froessling are also given. Sh values for N

_{2}in water align to the Froessling curves with only a low deviation. Sh values of CO

_{2}in water are located between Higbie and Froessling with a minor tendency to Froessling’s equation. This confirms the observation of a rigid surface area. Regarding Equation (11), the Sherwood numbers are calculated with several parameters dependent on the concentration. The diffusion coefficient, density and Henry constant vary only slightly up to a concentration of 1 M; hence, the Sherwood numbers and k

_{L}increase, due to the steeper bubble decrease. The determined values are in the same order as the values from Takemura and Matsumoto [23], even for a smaller range of Re. The differences in the Reynolds numbers are refer to the device used in their work, since the rising velocities of free bubbles and, hence, the Reynolds numbers are larger due to a smaller drag force coefficient compared to this work. Madhavi et al. [24] determined E as a function of time in the range from 13 to 19 for 1 M NaOH. E can be calculated with k

_{L}assuming a negligible concentration in the bulk and an identical surface and concentration in the bubble. The ratio of the mass transfer coefficient with chemisorption to the one without depends on the bubble diameter decrease. For a value of −1.0 mm/s, E is determined to be 13, and for a value of −1.4 mm/s, E is 18, which corresponds to the data of Madhavi et al. very well.

## 5. Conclusions

_{2}/CO

_{2}in aqueous and alkaline solutions. Different bubble sizes ranging from 0.1 to 2.5 mm and concentrations from 0.1 mM to 1 M NaOH were studied. The experiments were conducted in a novel device consisting of a converging microchannel according to the design of Vasconcelos et al. [20] to increase the observation time of single bubbles. The device is more than one order of magnitude smaller compared to existing devices. Therefore, the inner volume of the complete set-up only has a few milliliters, which make this set-up favorable for very costly substances. A special feature is an inserted needle into the smallest cross-section of the channel to generate single, very small bubbles.

_{2}showed a linear decreasing bubble diameter with time until complete extinction of the bubbles. The range of an immobilized bubble surface was enlarged in terms of water purity and bubble diameters. Absorption and chemisorption of CO

_{2}bubbles showed two linear regimes of diameter decreases independent of the gas composition. The first regime was related to pure CO

_{2}and the second regime to impurities in the initial gas, mainly N

_{2}. Larger concentrations of NaOH result in faster bubble decreases. Mass transfer coefficients represented by the Sherwood numbers in correlation with Reynolds numbers and the enhancement factor were calculated showing comparable values from the literature. Comparison of Sh with the Froessling equation indicates a rigid bubble surface area for N

_{2}and CO

_{2}.

## Nomenclature

a | maximum Feret diameter, mm |

A | surface area, m ^{2} |

Ar | Archimedes number, − |

b | minimum Feret diameter, mm |

c | concentration, mol∙m ^{−3} |

C_{1} | constant, − |

C_{D} | drag force coefficient, − |

d | diameter, mm |

D | diffusion coefficient, m ^{2}∙s^{−1} |

E | enhancement factor, − |

h | Henry constant model parameter, − |

h | height, mm |

H | Henry constant, bar∙m ^{3}∙mol^{−1} |

k | mass transfer coefficient, m∙s ^{−1} |

K | drag force model constant, − |

n | amount of substance, mol |

p | pressure, barg (gauge) |

r | radius, mm |

R | universal gas constant, J∙mol ^{−1}∙K^{−1} |

Re | Reynolds number, − |

Sc | Schmidt number, − |

Sh | Sherwood number, − |

t | time, s |

T | temperature, K |

u | channel flow velocity, m∙s ^{−1} |

V | volume, m³ |

z | valence electrons, − |

## Greek symbols

λ | ratio of bubble diameter to channel width, − |

ρ | density, kg∙m ^{−3} |

σ | surface tension, N∙m ^{−1} |

ν | kinematic viscosity, m ^{2}∙s^{−1} |

## Variable index nomenclature

atm | atmosphere |

b | bubble |

D | drag |

e | volume equivalent |

eq | equilibrium |

g | gas |

l | liquid |

L | liquid |

w | water |

0 | gas-liquid interface |

∞ | infinite flow field |

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Tollkötter, A.; Kockmann, N.
Absorption and Chemisorption of Small Levitated Single Bubbles in Aqueous Solutions. *Processes* **2014**, *2*, 200-215.
https://doi.org/10.3390/pr2010200

**AMA Style**

Tollkötter A, Kockmann N.
Absorption and Chemisorption of Small Levitated Single Bubbles in Aqueous Solutions. *Processes*. 2014; 2(1):200-215.
https://doi.org/10.3390/pr2010200

**Chicago/Turabian Style**

Tollkötter, Alexander, and Norbert Kockmann.
2014. "Absorption and Chemisorption of Small Levitated Single Bubbles in Aqueous Solutions" *Processes* 2, no. 1: 200-215.
https://doi.org/10.3390/pr2010200