# Determination of a Bubble Drag Coefficient during the Formation of Single Gas Bubble in Upward Coflowing Liquid

^{*}

## Abstract

**:**

## 1. Introduction

- ${d}_{b}$—bubble diameter, m
- $g$—gravitational acceleration, m/s
^{2} - ${\rho}_{L}$—liquid density, kg/m
^{3} - ${\rho}_{G}$—gas density, kg/m
^{3} - ${d}_{i}$—inner nozzle diameter, m
- ${\sigma}_{L}$—liquid surface tension, N/m

- ${Q}_{VG}$—gas volumetric flow rate, m
^{3}/s - $f$—bubble detachment frequency, Hz

- ${c}_{D}$—drag coefficient, -
- ${A}_{D}$—area of influence of drag force, m
^{2} - ${u}_{L}$—liquid velocity, m/s
- $s$—displacement of the center of mass of the bubble, m
- $t$—time, s
- $M$—mass, kg

- ${Q}_{VG}$—gas volumetric flow rate, m
^{3}/s - $V$—volume, m
^{3}

^{−6}–5.11 $\times $ 10

^{−6}m

^{3}/s), and gas chamber volume (33.3 $\times $ 10

^{−6}–89.9 $\times $ 10

^{−6}m

^{3}). The nozzle outer diameter, ${d}_{o}$, was about 4 mm and analyzed bubble diameter, ${d}_{b}$, ranged between 3.5–12 mm.

- ${\mathrm{Fr}}_{mod}$—modified Froude number (Sada et al. [22]), -
- ${u}_{G}$—gas velocity, m/s
- ${u}_{0L}$—superficial liquid velocity, m/s

^{−6}–36.2 $\times $ 10

^{−6}m

^{3}/s) and superficial liquid flow velocity (0–1.549 m/s). The authors used two different gas orifices of 0.86/1.30 mm and 3.05/4.00 mm (${d}_{i}/{d}_{o}$). The generated bubble diameter, ${d}_{b}$, ranged between 3–12 mm.

- $\mathrm{Bo}$—Bond number,
- $\mathrm{Fr}$—Froude number,

^{−6}–4.5 $\times $ 10

^{−6}m

^{3}/s), and liquid velocity (0–0.4 m/s), on bubble formation rates (10–55 Hz) and bubble sizes (3.4–6.7 mm).

## 2. Materials and Methods

#### 2.1. Materials

#### 2.2. Experimental Setup

#### 2.3. Experimental Procedures

#### 2.4. Model Formulation

- ${u}_{L}\left(x,y\right)$—local liquid velocity at the point $\left(x,y\right)$, m/s
- ${u}_{L,mean}$—mean liquid velocity in the column, m/s
- $m$—coefficient depending on the velocity profile shape, -
- $l$—column width, m
- $x$, $y$—location coordinates in the column cross-section relative to the column axis, m

## 3. Results

#### 3.1. Velocity Profile

#### 3.2. Drag Coefficient

#### 3.3. Comparison with Drag Coefficient Calculated Using the Equation for Bubble Rising

#### 3.4. Validation of Experimental Drag Coefficient Correlation

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Symbols: | |

${A}_{D}$ | area of influence of drag force, m^{2} |

$a$, $b$ | coefficients, - |

$\mathrm{Bo}$ | Bond number, - |

${c}_{D}$ | drag coefficient, - |

${C}_{mp}$ | conversion factor for conversion diameter in pixels to meters, m/pixel |

${d}_{b}$ | bubble diameter, m |

${d}_{h}$ | horizontal bubble diameter, m |

${d}_{i}$ | inner nozzle diameter, m |

${d}_{o}$ | outer nozzle diameter, m |

$f$ | bubble detachment frequency, Hz |

$\mathrm{Fr}$ | Froude number, - |

${\mathrm{Fr}}_{mod}$ | modified Froude number (Sada et al. [22]), - |

$g$ | gravitational acceleration, m/s^{2} |

$l$ | column width, m |

$M$ | mass, kg |

$m$ | coefficient depending on the velocity profile shape, - |

${Q}_{VG}$ | gas volumetric flow rate, m^{3}/s |

$\mathrm{Re}$ | Reynolds number, - |

$S$ | bubble projection surface, pixels |

$s$ | displacement of the center of mass of the bubble, m |

$t$ | time, s |

${u}_{0L}$ | superficial liquid velocity, m/s |

${u}_{G}$ | gas velocity, m/s |

${u}_{L}$ | liquid velocity, m/s |

${u}_{L,mean}$ | mean liquid velocity in the column, m/s |

${u}_{t}$ | tracer settling velocity, m/s |

$V$ | volume, m^{3} |

$x$, $y$ | location coordinates in the column cross-section relative to the column axis, m |

Greek Symbols: | |

${\mu}_{L}$ | liquid dynamic viscosity coefficient, Pa·s |

${\rho}_{G}$ | gas density, kg/m^{3} |

${\rho}_{L}$ | liquid density, kg/m^{3} |

${\sigma}_{L}$ | liquid surface tension, N/m |

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**Figure 1.**Schematic drawing of the experimental setup. 1—bubbling zone, 2—liquid valve, 3—rotameter, 4—water inlet, 5—Raschig rings, 6—flow straighteners, 7—air compressor, 8—air valve, 9—orifice, 10—water and air outlet, 11—high fps camera connected to PC, and 12—LED backlight.

**Figure 2.**Liquid velocity profiles for various mean liquid flow velocity; the points represent experimental data and the solid lines represent the calculated velocity profiles.

**Figure 5.**Drag coefficient, ${c}_{D}$, (28) versus Reynolds number (32) for two nozzles with identical inner diameter (${d}_{i}=$ 1.6 mm) and two different values of outer diameter (${d}_{o}=$ 1.8 mm and ${d}_{o}=$ 2.5 mm).

**Figure 6.**Comparison of drag coefficient, ${c}_{D}$, calculated using Equation (28) versus drag coefficient calculated using Equation (39).

Parameter | Symbol | Value |
---|---|---|

Gas density | ${\rho}_{G}$ | 1.23 kg/m^{3} |

Liquid density | ${\rho}_{L}$ | 1000 kg/m^{3} |

Liquid viscosity | ${\mu}_{L}$ | 1.31 $\times $ 10^{−3} Pa·s |

Liquid surface tension | ${\sigma}_{L}$ | 7.42 $\times $ 10^{−2} N/m |

Nozzle Number | Symbol | ${\mathit{d}}_{\mathit{i}}\left[\mathbf{mm}\right]$ | ${\mathit{d}}_{\mathit{o}}\left[\mathbf{mm}\right]$ |
---|---|---|---|

I | 0.8/1.0 | 0.80 ± 0.01 | 1.00 ± 0.01 |

II | 1.0/1.2 | 1.00 ± 0.01 | 1.20 ± 0.01 |

III | 1.2/1.4 | 1.20 ± 0.01 | 1.40 ± 0.01 |

IV | 1.4/1.6 | 1.40 ± 0.01 | 1.60 ± 0.01 |

V | 1.6/1.8 | 1.60 ± 0.01 | 1.80 ± 0.01 |

VI | 1.8/2.0 | 1.80 ± 0.01 | 2.00 ± 0.01 |

VII | 1.1/2.0 | 1.10 ± 0.01 | 2.00 ± 0.02 |

VIII | 1.6/2.5 | 1.60 ± 0.01 | 2.50 ± 0.02 |

IX | 2.1/3.0 | 2.10 ± 0.01 | 3.00 ± 0.02 |

**Table 3.**Coefficient of determination, ${r}^{2}$, for different combinations of drag coefficient, ${c}_{D}$, and Reynolds number, $\mathrm{Re}$, definitions.

${\mathit{r}}^{2}$ | ${\mathit{c}}_{\mathit{D}}$ | |||
---|---|---|---|---|

(28) | (29) | (30) | ||

$Re$ | (31) | 0.916 | 0.861 | 0.709 |

(32) | 0.994 | 0.987 | 0.903 | |

(33) | 0.727 | 0.772 | 0.936 |

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**MDPI and ACS Style**

Luty, P.; Prończuk, M.
Determination of a Bubble Drag Coefficient during the Formation of Single Gas Bubble in Upward Coflowing Liquid. *Processes* **2020**, *8*, 999.
https://doi.org/10.3390/pr8080999

**AMA Style**

Luty P, Prończuk M.
Determination of a Bubble Drag Coefficient during the Formation of Single Gas Bubble in Upward Coflowing Liquid. *Processes*. 2020; 8(8):999.
https://doi.org/10.3390/pr8080999

**Chicago/Turabian Style**

Luty, Przemysław, and Mateusz Prończuk.
2020. "Determination of a Bubble Drag Coefficient during the Formation of Single Gas Bubble in Upward Coflowing Liquid" *Processes* 8, no. 8: 999.
https://doi.org/10.3390/pr8080999