# A Turbulent Mass Diffusivity Model for Predicting Species Concentration Distribution in the Biodegradation of Phenol Wastewater in an Airlift Reactor

^{*}

## Abstract

**:**

_{c}. A good agreement is found between simulated and experimental results in the literature. It is not reasonable to assume a constant turbulent Schmidt number because the calculated distribution of turbulent mass diffusivity is not identical to that of turbulent viscosity. Finally, the hydrodynamic characteristics and biodegradation performance of the proposed model in a novel ALR are compared with that in the original ALR.

## 1. Introduction

_{2}on oxidation [22]. Mavaddat et al. investigated the hydrodynamics and production process of polyhydroxybutyrate in an ALR by using the CFD method, and in order to account for the effect of bubble coalescence and breakup phenomenons on the interfacial mass transfer process, a population balance model was adopted [23].

_{t}should be known to predict the mass transfer process in ALR. In the studies mentioned above, D

_{t}is obtained by experiments or assuming that the turbulent Schmidt number (Sc

_{t}) is fixed. In the biodegradation of wastewater in an ALR, the prediction of the distribution of species concentration is often simplified and case-specific. Sun et al. proposed a theoretical model to calculate D

_{t}in a distillation column by solving the auxiliary concentration variance and its dissipation rate equations [24], and it has been successfully used for simulating the turbulent mass transfer process in adsorption columns [25], circulating fluidized beds [26], and bubble columns [27].

_{c}model is adopted to enclose the turbulent mass transfer equations so that the distribution of species concentration in the ALR can be obtained by theoretical methods. Finally, the hydrodynamic characteristics and biodegradation performance of the proposed model in a novel ALR are compared with that in the original ALR.

## 2. Simulated Case

_{2}, H

_{2}O, and microbial cells. The batch biodegradation experiments are carried out in the ALR, the initial concentrations of phenol are 800 to 1600 mg/L, and the initial concentrations of cells are 34 to 102 mg/L. The temperature is maintained at 30 °C, and the initial pH of the liquid phase is 6.0. The liquid phase is sampled periodically to measure the concentrations of phenol and cells. The concentration of phenol is determined by high-performance liquid chromatography. The proposed model is verified by experimental results.

## 3. Model Equations

#### 3.1. Continuity Equations

**u**represents the velocity vector.

#### 3.2. Momentum Conservation Equations

_{k,eff}is the effective viscosity consisting of the turbulence-induced viscosity μ

_{t}and the molecular viscosity μ

_{lam}, and p is the pressure. μ

_{t}can be calculated by:

_{k}in Equation (2) accounts for the liquid–gas interfacial forces, including the lift force, drag force, and turbulent dispersion force. The drag force is caused by viscous stress and pressure distribution around the moving bubble in the liquid phase, which plays a major role in the interfacial forces [28]. The lift force arises from the rotation of the bubble in the liquid phase, which is perpendicular to the relative velocity. This study considers only drag and lift forces:

_{32}is the Sauter bubble diameter, C

_{L}is the lift coefficient (C

_{L}= 0.5 in this study) [3], and C

_{D}is the drag coefficient, which is related to the flow structure and liquid property and is calculated as follows [29]:

#### 3.3. Bubble Size Distribution (PBM)

_{j}is the source term related to bubble breakup and coalescence, and f

_{j}is the volume fraction of bubble class j:

_{m},v

_{n}) and b(v

_{j}) are the frequencies of bubble coalescence and breakup. Several mechanisms have been proposed to describe the bubble coalescence process, such as turbulent fluctuation, wake effect, eddy capture, and so on. In this work, only the turbulent fluctuation is considered and c(v

_{m},v

_{n}) is determined by the model of Luo et al. [32]. The turbulent collision between bubbles and turbulent eddies is considered the major contributor to bubble breakup, and the b(v

_{j}) is calculated by the model of Luo et al. [33]. Then, the equation for the Sauer bubble diameter d

_{32}is:

_{j}represents the diameter of each class.

#### 3.4. Species Conservation Equations

^{i}and ${D}_{t}^{i}$ represent the molecular and turbulent mass diffusivity, respectively:

_{ci}are calculated as follows [34,35]:

_{c}= 1.0, σ

_{εc}= 1.0, C

_{c}

_{0}= 0.11, C

_{c}

_{1}= 1.80, C

_{c}

_{2}= 2.20, and C

_{c}

_{3}= 0.80 [27].

#### 3.5. Microbial Kinetics

_{L}r). The bioreaction rates of phenol r

^{i}and biomass r

^{x}are calculated by:

_{max}= 0.48 h

^{−1}, K

_{I}= 207.9 g/m

^{3}, and K

_{s}= 11.7 g/m

^{3}) are determined by the batch experiments of Feng et al. [12].

## 4. Simulation Setup

^{−4}. The velocity inlet condition is prescribed to the ALR gas distributor. The gas-phase velocity is specified based on experimental conditions. The degassing condition is prescribed to the outlet of the ALR [6,36]. A no-slip condition is ascribed to the walls for both gas and liquid phases. The initial values for $\overline{{c}^{2}}$ and ε

_{c}are described as follows [26]:

## 5. Results and Discussion

#### 5.1. The Hydrodynamic and Mass Transfer Performance of Rectangle ALR

_{t}of dissolved phenol and liquid phase turbulent viscosity, respectively. Similar distribution patterns are observed for D

_{t}and turbulent viscosity, and both D

_{t}and turbulent viscosity increase with increasing ALR height. At a fixed ALR height, D

_{t}decreases quickly in the near-wall region. This could be attributed to the constraint of turbulence in this region. In the previous works, D

_{t}is often calculated by using Sc

_{t}and assuming that D

_{t}is proportional to the turbulent viscosity. Huang et al. adopted Sc

_{t}= 0.75 to simulate the direct coal liquefaction process in an ALR [21], while Talvy et al. predicted the absorption process in an ALR by assuming Sc

_{t}= 1 [38]. Figure 10 displays the profiles of calculated Sc

_{t}at different heights, which shows that Sc

_{t}varies throughout the ALR. So the traditional Sc

_{t}method may not be feasible for the wastewater biodegradation process in the ALR. The $\overline{{c}^{2}}$-ε

_{c}model proposed in this work is more rigorous, as it takes into account the impact of concentration fluctuation on the turbulent mass transfer.

#### 5.2. Comparison between Different Structures of ALRs

## 6. Conclusions

_{t}is decided by the $\overline{{c}^{2}}$-ε

_{c}model, so the empirical methods for simulating the turbulent mass transfer process can be avoided. The simulation results agree well with the experimental data, indicating that the proposed model is able to characterize the biodegradation of phenol in the ALR. The predicted distributions of D

_{t}and turbulent viscosity are not identical, which indicates that it may not be reasonable to assume a constant Sc

_{t}in the ALR. Additionally, a novel structure of ALR is proposed and it has better hydraulic and biodegradation performance than the original one.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A schematic of the simulated ALRs: (

**a**) rectangle ALR; (

**b**) cylinder ALR with six cylindrical draft tubes.

**Figure 2.**Schematic view of the grids: (

**a**) rectangle ALR; (

**b**) cylinder ALR with six cylindrical draft tubes.

**Figure 3.**The profiles of (

**a**) gas holdup and (

**b**) liquid axial velocity in the XZ−plane at Y = 0.1 m in the ALR when the superficial gas velocity is 0.02 m/s.

**Figure 4.**The contours of gas holdup obtained at different superficial gas velocities in the XZ−plane at Y = 0.1 m in the ALR: (

**a**) ug = 0.01 m/s, (

**b**) ug = 0.02 m/s.

**Figure 5.**The contours of the Sauter bubble size distribution (m) in the XZ−plane at Y = 0.1 m in the ALR at u

_{g}= 0.02 m/s.

**Figure 6.**Distributions of shear stress (kg m

^{−1}s

^{−2}) in the XZ−plane at Y = 0.1 m in the ALR at (

**a**) u

_{g}= 0.01 m/s and (

**b**) u

_{g}= 0.02 m/s.

**Figure 7.**Comparison between simulated cell concentrations and experimental results: (

**a**) initial cell concentration: 34 mg/L; (

**b**) initial cell concentration: 102 mg/L.

**Figure 8.**Comparison between simulated dissolved phenol concentrations and experimental results: (

**a**) initial cell concentration: 34 mg/L; (

**b**) initial cell concentration: 102 mg/L.

**Figure 9.**The contour of (

**a**) turbulent mass diffusivity of dissolved phenol (m

^{2}/s) and (

**b**) turbulent viscosity (kg/(m·s)) in the XZ−plane at Y = 0.1 m in the ALR.

**Figure 11.**The contour of the (

**a**) gas holdup, (

**b**) Sauter bubble size distribution (m), and (

**c**) shear stress in the XZ−plane at Y = 0.12 m in the novel ALR at u

_{g}= 0.02 m/s.

**Figure 12.**The variation of dissolved phenol concentration at an initial cell concentration of 102 mg/L.

**Table 1.**Comparison of bubble size, volume-averaged gas holdup, and shear stress between original and novel ALR.

Type of ALR | Volume-Averaged Gas Holdup | Volume-Averaged Sauter Bubble Size (m) | Volume-Averaged Shear Stress (kg·m ^{−1}s^{−2}) |
---|---|---|---|

original | 0.053 | 0.0092 | 3.798 |

novel | 0.073 | 0.0079 | 1.370 |

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**MDPI and ACS Style**

Li, L.; Hao, R.; Jin, X.; Hao, Y.; Fu, C.; Zhang, C.; Gu, X.
A Turbulent Mass Diffusivity Model for Predicting Species Concentration Distribution in the Biodegradation of Phenol Wastewater in an Airlift Reactor. *Processes* **2023**, *11*, 484.
https://doi.org/10.3390/pr11020484

**AMA Style**

Li L, Hao R, Jin X, Hao Y, Fu C, Zhang C, Gu X.
A Turbulent Mass Diffusivity Model for Predicting Species Concentration Distribution in the Biodegradation of Phenol Wastewater in an Airlift Reactor. *Processes*. 2023; 11(2):484.
https://doi.org/10.3390/pr11020484

**Chicago/Turabian Style**

Li, Liang, Runqiu Hao, Xiaoxia Jin, Yachao Hao, Chunming Fu, Chengkai Zhang, and Xihui Gu.
2023. "A Turbulent Mass Diffusivity Model for Predicting Species Concentration Distribution in the Biodegradation of Phenol Wastewater in an Airlift Reactor" *Processes* 11, no. 2: 484.
https://doi.org/10.3390/pr11020484