# Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames

^{1}

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## Abstract

**:**

## 1. Introduction

^{−1}fuel injection rate. A DQMOM-based soot model targeting low-speed buoyancy-driven diffusion flame, which corresponds better to natural fire occurrences in open configuration, e.g., bushfire and compartment fire, has not been formulated and tested. Moreover, it has been reported that DQMOM algorithm can be numerically unstable and difficult to implement, i.e., generating unrealizable (negative) weights and nodes, if the system is not cautiously designed. Alternative approaches that were proposed by Mueller et al. and Chittipotula et al. [28,29] could potentially resolve the numerical issue. However, the alternative approaches often require solving additional sub-modules that are integrated to the framework or correction algorithms, which could significantly increase the computational cost or need the cumbersome coupling of the CFD code with an external mathematical package. This challenge could become more predominant when the investigated case has relatively small spatial-diffusion terms [29], like the flame of the interest in this work. Therefore, it is critical to systematically investigate the effect of all modelling aspects to ensure the numerical robustness and simulation accuracy.

- (i)
- comparison of three nucleation laws, including Moss, Leung, and Fairweather, and discuss their effectiveness in predicting the changes in particle size due to the particle generations;
- (ii)
- study of two different oxidation law including modified NSC and Said to investigate the appropriate soot particle reduction or disruption mechanisms for DQMOM;
- (iii)
- investigating the influences of fractal dimensions towards the aggregation and surface growth mechanisms, as well as the effect on particle size distribution;
- (iv)
- study the effect of diffusion coefficient for soot quantities and number density towards the dispersion of soot particles within the computational field; and,
- (v)
- provide a deeper understanding of the soot formation process, in particular, within the flaming/soot nucleating region.

## 2. Mathematical Model

#### 2.1. Governing Equations

#### 2.2. Turbulence and Combustion Modelling

_{2}H

_{4}) being selected as the parental fuel. Such an approach of resolving turbulence-chemistry interaction has been demonstrated in previous studies to provide a reasonable result with moderate computational burden [18,33].

^{−6}to 180. As can be seen, the key significant combustion products are oxygen, carbon monoxide, carbon dioxide, water vapour, hydrogen, and acetylene, while minor species include hydrocarbon compounds, oxygen, and hydrogen molecules.

#### 2.3. Direct Quadrature Method of Moments (DQMOM) Model

#### 2.4. Soot Formation Kinetics

#### 2.4.1. Nucleation

#### 2.4.2. Aggregation

_{c1}and R

_{c2}can be expressed with a single expression that interpolates the coagulation process between free molecular, continuum, as well as the transition regimes, as [41]:

#### 2.4.3. Molecular Growth

^{−1}). It should be noted that the term $\frac{6}{{D}_{f}{\rho}_{s}}{\left(\frac{{R}_{c}}{{R}_{c0}}\right)}^{\frac{3-{D}_{f}}{3}}$ is referred to as the soot surface area coefficient that is dependent on the size, as well as the fractal property of the soot aggregates [26,40].

#### 2.5. Numerical and Case Configurations

#### 2.6. Sensitivity Analysis

## 3. Results

#### 3.1. Flamelet and Combustion Process

_{2}H

_{4}) generally resides at low HAB i.e., regions close to burner front. It subsequently de-composed and reformed into other intermediate species via the pyrolysis process as it flows to the downstream. Acetylene (C

_{2}H

_{2}), the BSUs for soot particle nucleation and surface growth reaction, is capsulated within the flame sheet and its concentration increases with the heights at the low-intermediate flame region before approaching the high temperature oxidation zone. The concentration of oxidant (O

_{2}) peaks at regions that are away from the flame reaction zone and gradually decreases as the lateral position approaches the flame centreline. Unfortunately, due to the lack of available experimental resources, the validation of the concentration of reacting species is not carried out. However, the result presented above is in line with the general expectation of the spatial distribution of flammable gas mixture, as well as oxidant species of a co-flow diffusion flame configuration.

#### 3.2. Sensitivity Analysis of DQMOM Key Modelling Parameters

#### 3.2.1. Nucleation

#### 3.2.2. Oxidation

#### 3.2.3. Fractal Dimension

#### 3.2.4. Diffusivity

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Flamelet profiles applied in the combustion model for: (

**a**) Major chemical species; (

**b**) Minor and intermediate chemical species for scalar dissipation rates of 0.01 (near chemical equilibrium) and 180 (approaching flame extinction).

**Figure 2.**(

**a**) Temperature contour plot and (

**b**) comparisons of numerical and experimental temperature distributions at three vertical height levels above the burner.

**Figure 3.**Contour plot of the mole fraction of reacting species: (

**a**) ethylene, ${X}_{{C}_{2}{H}_{4}}$; (

**b**) acetylene, ${X}_{{C}_{2}{H}_{2}}$; and, (

**c**) oxygen, ${X}_{{O}_{2}}$.

**Figure 4.**Contour plot of nucleation rate, evaluated from: (

**a**) Moss law; (

**b**) Leung law; and, (

**c**) Fairweather law.

**Figure 5.**Contour plot of soot number density, evaluated from: (

**a**) Moss law; (

**b**) Leung law; and, (

**c**) Fairweather law.

**Figure 6.**X-Y plot of soot particle size at selected height above burners (HABs), evaluated from: (

**a**) Moss law; (

**b**) Leung law; and, (

**c**) Fairweather law as compared with experimental result.

**Figure 7.**X-Y plot of soot volume fraction at selected HABs, evaluated from: (

**a**) Moss law; (

**b**) Leung law; and, (

**c**) Fairweather law compared with experimental result.

**Figure 9.**X-Y plot of oxidation rate at selected HABs, as evaluated from modified NSC law and Said law.

**Figure 10.**Contour plot of soot particle size, evaluated from: (

**a**) modified NSC law and (

**b**) Said law.

**Figure 11.**Contour plot of soot volume fraction, evaluated from: (

**a**) modified NSC law and (

**b**) Said law.

**Figure 12.**Contour plot of oxidation rate, evaluated from: fractal dimension of (

**a**) 2.0, (

**b**) 1.8, and (

**c**) 2.2.

**Figure 13.**Contour plot of surface growth rate evaluated from: fractal dimension of (

**a**) 2.0, (

**b**) 1.8, and (

**c**) 2.2.

**Figure 14.**Contour plot of soot particle size, evaluated from: fractal dimension of (

**a**) 2.0, (

**b**) 1.8, and (

**c**) 2.2, and X-Y plot of soot particle size at selected HABs, evaluated from: fractal dimension of (

**d**) 2.0, (

**e**) 1.8, and (

**f**) 2.2 as compared with experimental result.

**Figure 15.**Contour plot of soot volume fraction, evaluated from: fractal dimension of (

**a**) 2.0, (

**b**) 1.8, and (

**c**) 2.2, and X-Y plot of soot volume fraction at selected HABs, evaluated from: fractal dimension of (

**d**) 2.0, (

**e**) 1.8, and (

**f**) 2.2 as compared with experimental result.

**Figure 16.**Contour plot of soot number density, evaluated from: Schmidt number of (

**a**) 0.7, (

**b**) 0.5, and (

**c**) 0.9.

**Figure 17.**X-Y plot of soot number density at selected HABs: (

**a**) 20mm HAB, (

**b**) 30mm HAB, and (

**c**) 40mm HAB.

**Figure 18.**Contour plot of soot particle size, evaluated from: Schmidt number of (

**a**) 0.7, (

**b**) 0.5, and (

**c**) 0.9, and X-Y plot of soot particle size at selected HABs, evaluated from: Schmidt number of (

**d**) 0.7, (

**e**) 0.5, and (

**f**) 0.9 as compared with experimental result.

**Figure 19.**Contour plot of soot volume fraction, evaluated from: Schmidt number of (

**a**) 0.7, (

**b**) 0.5, and (

**c**) 0.9, and X-Y plot of soot volume fraction at selected HABs, evaluated from: Schmidt number of (

**d**) 0.7, (

**e**) 0.5, and (

**f**) 0.9 as compared with experimental result.

Case | Nucleation | Oxidation | Fractal dimension | Schmidt Number |
---|---|---|---|---|

A | Moss | modified NSC | 2.0 | 0.7 |

B | Leung | modified NSC | 2.0 | 0.7 |

C | Fairweather | modified NSC | 2.0 | 0.7 |

D | Moss | Said | 2.0 | 0.7 |

E | Moss | modified NSC | 1.8 | 0.7 |

F | Moss | modified NSC | 2.2 | 0.7 |

G | Moss | modified NSC | 2.0 | 0.5 |

H | Moss | modified NSC | 2.0 | 0.9 |

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**MDPI and ACS Style**

Wang, C.; Yuen, A.C.Y.; Chan, Q.N.; Chen, T.B.Y.; Yang, W.; Cheung, S.C.-P.; Yeoh, G.H. Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames. *Energies* **2019**, *12*, 910.
https://doi.org/10.3390/en12050910

**AMA Style**

Wang C, Yuen ACY, Chan QN, Chen TBY, Yang W, Cheung SC-P, Yeoh GH. Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames. *Energies*. 2019; 12(5):910.
https://doi.org/10.3390/en12050910

**Chicago/Turabian Style**

Wang, Cheng, Anthony Chun Yin Yuen, Qing Nian Chan, Timothy Bo Yuan Chen, Wei Yang, Sherman Chi-Pok Cheung, and Guan Heng Yeoh. 2019. "Sensitivity Analysis of Key Parameters for Population Balance Based Soot Model for Low-Speed Diffusion Flames" *Energies* 12, no. 5: 910.
https://doi.org/10.3390/en12050910