# A PBM-Based Procedure for the CFD Simulation of Gas–Liquid Mixing with Compact Inline Static Mixers in Pipelines

^{*}

## Abstract

**:**

_{L}a, developed for partially segregated systems is applied finding k

_{L}a values comparable to those typically obtained with other static mixers. The proposed computational model allows us to locally analyze the oxygen transfer rate by observing the limitations due to gas accumulation behind the body of the static mixer, which leads to the local depletion of the driving force. Geometrical optimization of the static element is proposed, based on the analysis of gas–liquid fluid dynamics and of the interphase mass transfer phenomena.

## 1. Introduction

_{2}and/or O

_{3}are absorbed in the liquid phase for aerobic-activated sludge processes and advanced oxidation processes [3], respectively, to a number of chemical production processes, involving gas–liquid or gas–liquid–liquid chemical reactions [4].

_{L}a, and mass transfer rate. The local analysis of the interphase mass transfer phenomena is performed, highlighting the potential of the proposed CFD approach in troubleshooting and designing static mixers. Finally, conclusions are drawn.

## 2. Investigated System and Operative Conditions

_{L}= 998 kg/m

^{3}and viscosity μ

_{L}= 0.001 Pa∙s) enters the system, together with a gas phase consisting of air, assumed as a mixture of oxygen (mole fraction x

_{O2}= 0.21) and nitrogen (mole fraction x

_{N2}= 0.79), with a density ρ

_{G}= 1.225 kg/m

^{3}and a viscosity μ

_{G}= 1.7 × 10

^{−5}Pa∙s. Two operative conditions are considered in this study. In the first operative condition, the water volumetric flow rate is equal to 25 m

^{3}/h (6.94 × 10

^{−3}m

^{3}/s) and the air volume flow rate is equal to 3.75 L/min (6.25 × 10

^{−5}m

^{3}/s). The second operative condition consists of a water volumetric flow rate equal to 30 m

^{3}/h (8.33 × 10

^{−3}m

^{3}/s) and an air volume flow rate equal to 7.50 L/min (1.25 × 10

^{−4}m

^{3}/s). The gas volume fraction, α

_{G}, in the operative condition with lower fluids flow rates is equal to 0.0089, while in the operative condition with higher fluids flow rates, it is equal to 0.0148.

## 3. Computational Model

#### 3.1. RANS-TFM Equations

_{32}is the Sauter mean diameter of the bubble population, and C

_{D}is the drag coefficient, obtained from the correlation of Schiller and Naumann [27], as:

_{B}, is defined as:

#### 3.2. Interphase Mass Transfer Model

^{9}Pa, divided by the operative pressure equal to 1 atm, and $M{W}_{{O}_{2}}$ and $M{W}_{L}$ are the molecular weights of oxygen and of the liquid mixture, respectively. The oxygen transfer rate, OTR, is derived from the interphase mass flux of oxygen as:

_{L}a. The liquid side mass transfer coefficient is modelled according to the eddy cell model of Lamont and Scott [32], and it reads:

_{L}, equal to 2 × 10

^{−9}m

^{2}/s.

_{cell}being the volume of the computational grid cell. In a Eulerian–Eulerian description of the flow, below a local volume fraction of 0.3, the gas–liquid flow can be modelled as a mixture of bubbles in the continuous liquid medium, therefore the classical expression for the interfacial area can be adopted. For gas volume fractions higher than 0.3, coalescence increases [34] and therefore the average bubble diameter increases, while the gas phase is still in the dispersed phase. For gas volume fractions above 0.5, it was assumed that the liquid becomes the dispersed phase and the gas becomes the continuous phase. It is important to bear in mind that these considerations are limited to the Eulerian–Eulerian description of the flow with a constant bubble size and only for the calculation of the specific interfacial area. Further information on the interphase mass transfer model adopted in this investigation can be found in Maluta et al. [33].

#### 3.3. PBM-Based Procedure for Bubble Size Determination

_{min}~(11.4 ÷ 31.4)μ

_{L}

^{0.75}ρ

_{L}

^{−0.75}ε

^{−0.25}[35]. Equation (15) describes the surface energy increase constraint, ${c}_{f}$, and in the expression for the breakage frequency function, Equation (16), the term multiplying the integral accounts for binary breakage.

^{I}, is employed in the PBM to obtain a first-guess bubble Sauter mean diameter, d

_{32}

^{I}. This bubble size is then adopted in the gas–liquid CFD simulations to obtain a first-guess gas–liquid flow field from which the updated value of the average turbulent dissipation rate in the static mixer is calculated, ε

^{II}. A new solution of the PBM is performed, and the resulting updated Sauter mean diameter, d

_{32}

^{II}, is compared with the first-guess bubble size, d

_{32}

^{I}. If relevant differences are observed between these two values, then a new gas–liquid simulation is performed with the updated bubble size, repeating the procedure until the first-guess and updated bubble sizes are sufficiently similar. At this point, the gas–liquid simulation with the converged Sauter mean diameter is run coupling the interphase mass transfer model.

## 4. Numerical Solution

^{−5}in each case. Moreover, gas and liquid velocity profiles and gas volume fraction were monitored in several points of the domain, and the simulations were stopped once those values reached a plateau. It was verified that the total mass flow entering the system equalled the mass at the outlet, to rule out mass imbalance.

^{−8}. Steady state was assumed when the moments evolution in time reached a plateau and the final resolution time underwent a sensitivity study to confirm that the solution did not depend on the integration time.

## 5. Results and Discussion

#### 5.1. Bubble Size and Gas Volume Fraction Distribution

^{3}/h and 30 m

^{3}/h, respectively, to obtain the mean turbulent dissipation rate in the static mixer zone, ε

_{SM}. The results of the procedure are shown in Table 1.

_{32}obtained from the ε

_{SM}calculated from single-phase simulations is equal to 0.74 mm and 0.52 mm for the case of a liquid flow rate equal to 25 m

^{3}/h and 30 m

^{3}/h, respectively, and the corresponding bubble terminal velocity, Ut, is equal to 8.4 cm/s and 5.8 cm/s. Two gas–liquid simulations were then run with these bubble sizes and with gas flow rates equal to 3.75 L/min and 7.5 L/min, respectively, as described in Section 2. The turbulent dissipation rate shown in Table 1 is obtained by averaging the dissipation of the fluids in the static mixer region. In this region, turbulent dissipation rates considerably higher than in the bulk of the system are found, as similarly observed in stirred tanks [24]. The dissipation rates obtained from gas–liquid simulations were found to be close to the values obtained in single-phase conditions, with deviations of a few % in both the investigated cases. Consequently, a further iteration of the procedure to determine the bubble size returned d

_{32}and terminal velocities that differed less than 5% from the results obtained in single-phase conditions. For these reasons, additional gas–liquid simulations with updated bubble sizes were not performed. The results of the gas–liquid simulations obtained with d

_{32}derived from single-phase simulations were then analyzed and are described in the following section.

_{i}on the i-th section is defined as:

#### 5.2. k_{L}a and Mass Transfer Rate

_{L}a, calculated with the modelling approach described in Section 3.2 is shown on radial cutaway surfaces in Figure 6, for the two operative conditions considered.

_{L}a is observed downstream of the static mixer. In particular, two distinct zones are discernible, the first being the zone between the static mixer and around two pipe diameters downstream of the device and the second being at higher axial coordinates. In the first zone, the gas is still largely segregated, and it has not undergone a substantial distribution. In this zone, the main contribution to the volumetric mass transfer coefficient is given by the central zone of the pipe and the gas accumulated in the top part of the tube that starts rotating due to the swirling motion of the static mixer. Local high k

_{L}a values are present alongside zones of nearly zero k

_{L}a values. After about two pipe diameters, the gas is more homogeneously distributed leading to a subsequent increase in the volumetric mass transfer coefficient in the whole pipe volume. At axial coordinates between 4–6 pipe diameters, almost all the pipe section encounters significant k

_{L}a values. In this latter section, inhomogeneities in the volumetric mass transfer coefficient are present and k

_{L}a maximum values are found in correspondence to the highest values of the gas volume fraction, as it can be observed when comparing the gas volume fraction distribution in Figure 3.

_{L}, and the specific interfacial area, a, for the two operative conditions considered are reported in Figure 7.

_{L}and a in the sections downstream of the static mixer. At lower axial coordinates, high liquid side mass transfer coefficients are found due to the high turbulence dissipation rates generated by the static mixer, and relatively low specific interfacial areas are present due to the low dispersion of the gas phase. Conversely, at higher axial coordinates the inverse is obtained, since the turbulent energy is mostly dissipated close to the mixer, but in turn, the gas phase is more homogeneously distributed downstream. These trends limit the volumetric mass transfer coefficient, which is alternatively controlled either by a or k

_{L}. Nonetheless, the k

_{L}a values found in the system are comparable to those obtained with other static mixer designs [40].

_{L}a. At axial coordinates around six pipe diameters downstream of the mixer, the concentration profiles reach a plateau due to the reduced mass transfer driving force. In fact, the small differences between the equilibrium and the liquid oxygen mole fraction observable at high axial coordinates, together with the significant values of k

_{L}a in the same region observed in Figure 6, suggest that the oxygen transfer rate at high axial coordinates is limited by the driving force, rather than the interphase mass transfer coefficient.

_{L}a, and driving force are shown for the two operative conditions considered in Figure 9.

#### 5.3. Local Analysis of the Interphase Mass Transfer Phenomena

_{L}a. The local k

_{L}and a distributions in the proximity of the static mixer are shown in Figure 10.

_{L}are found in the static mixer zone and downstream, due to high local turbulent dissipation rates. In the operative conditions with the lower fluids flow rates, as shown in Figure 10a, a larger zone with low k

_{L}values is found towards the pipe center, with respect to the other operative condition, Figure 10c. In fact, the higher liquid flow rate generates more turbulence, while interacting with the static mixer, which thanks to its dissipation is more evenly distributed in the section. The turbulent dissipation also causes an axial turbulence decrease, as observable in both conditions, which in turn generates lower k

_{L}as the fluids move towards higher axial coordinates. For this reason, the pipe section with the highest liquid side mass transfer coefficient is the one closer to the static mixer.

_{L}and a leads to high volumetric mass transfer coefficients in the close proximity of the mixer.

_{L}a shown in Figure 9, a corresponding sudden valley is found in the driving force profile. This reduction derives from the local drop in oxygen concentration in the gas phase coupled with the increase in the liquid phase, which drives the interphase mass transfer driving force to zero, as observable in Figure 11.

_{L}values are found, an improvement in the OTR can be simultaneously driven by the enhanced volumetric mass transfer coefficient and by a local high interphase mass transfer driving force.

## 6. Conclusions

- Upstream of the static mixer, the gas bubbles accumulate towards the top of the pipe, resulting in a high coefficient of variations of the gas hold-up and low volumetric mass transfer coefficients, thus determining negligible oxygen transfer rates despite a high interphase mass transfer driving force;
- Just downstream of the static mixer, the lighter phase starts to rotate following the liquid phase and gradually reduces the segregation, which lasts to axial coordinates up until 4–5 pipe diameters downstream of the static mixer. Despite the relatively high liquid side mass transfer coefficient, the oxygen transfer rate is limited by the available specific interfacial area;
- At higher axial coordinates, the reduced interphase mass transfer driving force limits the oxygen transfer rate, even though very low coefficients of variations of the gas hold-up are found, indicating sufficient mixedness of the gas–liquid dispersion, together with relatively high values of k
_{L}a; - Modification of the hub geometry is suggested in order to better exploit the low-pressure region downstream of the static mixer to improve the gas distribution in the zone where k
_{L}is higher.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) Cutaway drawing of the pipe with the compact static mixer in red. The gas–liquid mixture enters from the left pipe opening. (

**b**) Details of the mesh close to the static mixer.

**Figure 2.**Schematic diagram of the PBM-based iterative procedure for the bubble size determination and gas–liquid simulations.

**Figure 3.**Gas volume fraction distribution on radial pipe cutaway surfaces for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The static mixer is placed at −0.14 < z/D < 0 and the fluids are moving from left to right. The initial and final part of the pipe are omitted to improve readability.

**Figure 4.**Iso-surfaces of a gas volume fraction downstream of the static mixer obtained at a value of α

_{G}= 0.025 for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The iso-surfaces are colored with the ratio between the centripetal acceleration and the gravitational acceleration.

**Figure 5.**Axial profiles of gas volume fraction CoV for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min.

**Figure 6.**Volumetric mass transfer coefficient on radial pipe cutaway surfaces for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The static mixer is placed at −0.14 < z/D < 0 and fluids move from left to right. The initial and final part of the pipe are omitted to improve readability.

**Figure 7.**Axial profiles of k

_{L}and a, for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The initial and final part of the pipe are omitted to improve readability.

**Figure 8.**Axial profiles of oxygen mole fraction in water and equilibrium oxygen mole fraction for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The initial and final part of the pipe are omitted to improve readability.

**Figure 9.**Axial profiles of OTR, k

_{L}a, and interphase mass transfer driving force, y

_{O}

_{2}/m-x

_{O}

_{2}, for the operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min; (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min. The initial and final part of the pipe are omitted to improve readability.

**Figure 10.**k

_{L}and a distributions in the proximity of the static mixer for the two operative conditions considered: (

**a**,

**b**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min and (

**c**,

**d**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min.

**Figure 11.**Interphase mass transfer driving force in the proximity of the static mixer for the two operative conditions considered: (

**a**) Q

_{L}= 25 m

^{3}/h and Q

_{G}= 3.75 L/min and (

**b**) Q

_{L}= 30 m

^{3}/h and Q

_{G}= 7.5 L/min.

Q_{L} (m^{3}/h) | Q_{G} (L/min) | ε_{SM} (m^{2}/s^{3}) | d_{32} (m) | Ut (m/s) |
---|---|---|---|---|

25 | 0 | 3.78 | 7.4 × 10^{−4} | 0.084 |

25 | 3.75 | 3.56 | 7.7 × 10^{−4} | 0.087 |

30 | 0 | 10.07 | 5.2 × 10^{−4} | 0.058 |

30 | 7.5 | 9.81 | 5.4 × 10^{−4} | 0.060 |

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**MDPI and ACS Style**

Maluta, F.; Paglianti, A.; Montante, G.
A PBM-Based Procedure for the CFD Simulation of Gas–Liquid Mixing with Compact Inline Static Mixers in Pipelines. *Processes* **2023**, *11*, 198.
https://doi.org/10.3390/pr11010198

**AMA Style**

Maluta F, Paglianti A, Montante G.
A PBM-Based Procedure for the CFD Simulation of Gas–Liquid Mixing with Compact Inline Static Mixers in Pipelines. *Processes*. 2023; 11(1):198.
https://doi.org/10.3390/pr11010198

**Chicago/Turabian Style**

Maluta, Francesco, Alessandro Paglianti, and Giuseppina Montante.
2023. "A PBM-Based Procedure for the CFD Simulation of Gas–Liquid Mixing with Compact Inline Static Mixers in Pipelines" *Processes* 11, no. 1: 198.
https://doi.org/10.3390/pr11010198