# Effects of Mesh Generation on Modelling Aluminium Anode Baking Furnaces

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{−2}. However, the nested scheme in model 8—it was initialised with the results of model 7—reached convergence at 10

^{−3}. Meshes were created with a smaller cell size only in critical areas, and a larger cell size in the rest of the geometry areas with the aim of reducing the computational load. Refining a mesh in the appropriate zones brings the Peclet number down. After observing the dissipation results, mesh 2—which is refined in the fuel jet stream zone—adequately models the split stream at the meeting of the fuel jet stream with the first tie-brick, in the incompressible isothermal turbulent flow. Regarding convergence and the turbulent viscosity ratio, mesh 3—which is refined in the whole combustion zone—adequately models the incompressible isothermal turbulent flow at the outlet of the fuel inlet pipes in the combustion zone.

## 2. Related Works

^{®}Multiphysics is based on the finite element method (FEM). Comparing the two approaches, Ansys Fluent—finite volume method—and COMSOL

^{®}Multiphysics—finite element method—may give an insight on the problem solution. In conclusion, vast modelling approaches are developed for anode baking furnace. However, the model for NOx reduction still requires significant attention.

## 3. Model Description

#### Standard κ–ε Turbulence Model

- ρ is the density of the fluid (SI units: kg/m
^{3}). - μ is the dynamic viscosity of the fluid (Pa·s or N·s/m
^{2}or kg/(m·s)). - ν is the kinematic viscosity of the fluid (m
^{2}/s). - ε is a small number added to avoid the division by zero.
- σk and σε are the turbulent Prandtl numbers for κ and ε.
- $\overline{p}=\overline{p}+\frac{2}{3}\rho k$ and the $\frac{1}{\rho}$ factor in front of the pressure term in the RANS equations are dropped. Then, if the true mean pressure field is sought, one has to take this into consideration.
- The default values of the model constants, ${\sigma}_{k}{,\text{}\sigma}_{\u03f5}{,\text{}C}_{\mu}{,\text{}C}_{1\u03f5}{,\text{}\mathrm{and}\text{}C}_{2\u03f5},$ have been determined from experiments with air and water for fundamental turbulent shear flows, including homogeneous shear flows and decaying isotropic grid turbulence. They have been found to work fairly well for a wide range of wall-bounded and free shear flows.
- Although the default values of the model constants are the standard ones, and the most widely accepted, one can change them (if needed).

## 4. Model Configurations

^{®}Multiphysics version 5.5 for solving the Navier–Stokes κ–ε turbulence model with the finite element method [9]. All solver parameters are set as default except for the linear solver. GMRES (as the Krylov subspace method), Algebraic Multigrid (as preconditioner) and Vanka (as pre- and post-smoother within Algebraic Multigrid) are selected for the linear solver.

#### 4.1. Geometry and Mesh

#### 4.2. Simulations

**u**,p) = 0.005. Finally, two incompressible isothermal turbulent flow simulations are conducted using mesh 3—models 7 and 8. Model 7 used artificial diffusion to achieve convergence, and the diffusion parameters were tuning at δ(

**u**,p) = 0.25 and δ(κ,ε) = 0.25.

**u**,p) = 0.25—was modelled as model 7, achieving convergence at 10

^{−3}. Then, the incompressible isothermal turbulent flow—using mesh 3 without artificial diffusion—was modelled as model 8, with initial values set from the results of model 7, achieving convergence at 10

^{−3}. On the other hand, the incompressible isothermal turbulent flow—using mesh 2 without artificial diffusion—were modelled as model 9, with initial values set from the results of model 8, achieving the lowest error at 10

^{−2}.

#### 4.3. Numerical Implementation

^{®}Xeon

^{®}CPU E5-2630 v3 @2.40GHz x32 cores, and 129GB RAM. Convergence was achieved when errors reached at least 10

^{−3}. Additionally, simulations’ accuracy were evaluated using the graphical results of the turbulent viscosity ratio plots, defined as μ

_{T}/μ

_{0}, at each element, as well as the physical interpretation by experts and the recent literature [4]. This paper is focused on the physical interpretation of the resulted velocity and the isothermal turbulent behaviour when using different types of mesh cells, refinement and parameters of artificial diffusion. In particular, this paper is focused on the behaviour of these phenomena in the combustion zones—where the air and fuel meet.

#### 4.3.1. Finite Element Method in CFD

- It is a very general method,
- There is more facility to increment the element order,
- Physical fields may be reproduced more accurately,
- Physics and mathematics often require different type of functions for a phenomenon. Different phenomena can be represented at the same time with FEM,
- To reach more accuracy, increase order of polynomials and refine the mesh.

#### 4.3.2. Theoretical Definition of FEM

**h**is given Neumann boundaries.

**f**is external force.

**v**and q are test functions in the space V and Q, respectively. Additional details can be found at [34].

**u**and pressure p. It is known as mixed variational formulation. A solution may be addressed using Lagrange multipliers to determine the value of each variable. However, it is more efficient to use a penalisation model of p, simplifying the discrete problem into an equations system that only depends on

**u**. This system allows us to determine p once calculated

**u**.

**u**= (u, v, w) and p at each element will be given by:

**u**= (u, v, w), and ${\mathbf{q}}_{\mathit{p}}$ denote local values inside the pressure field. ${\mathbf{N}}_{\mathbf{u}}$, ${\mathbf{N}}_{\mathbf{v}}$,${\mathbf{N}}_{\mathbf{w}}$ and ${\mathbf{N}}_{\mathit{p}}$ are shape functions of the velocity and pressure and the unknown total vector of element e is given by ${\mathbf{q}}_{\mathbf{e}}^{\mathbf{T}}=\left[{\mathbf{q}}_{\mathbf{u}}^{\mathbf{T}},{\mathbf{q}}_{\mathbf{v}}^{\mathbf{T}},{\mathbf{q}}_{\mathbf{w}}^{\mathbf{T}},{\mathbf{q}}_{\mathit{p}}^{\mathbf{T}}\right]$.

**u**= (u, v, w) and p, in the problem, are no longer mathematical functions and become the values of these functions at the nodes. The complete problem solution follows the rules for discrete problems.

## 5. Results and Discussion

^{−3}. Table 5 shows the lowest error reached by the nine models. Models 4, 5 and 9 did not reach convergence according to the convergence criterium.

^{−3}, whilst model 4 has presented periodic oscillations around 10

^{−2}. There is an incorrect representation of the velocity field in both models. In fact, in Figure 7c, the fuel jet stream penetrates the furnace downwards, avoiding the first tie-brick obstacle. This implies that the flow will not be split and distributed along the furnace. Hence, the second subsection—when the flow goes up—will not have a uniform velocity with respect to the first subsection. This physical behaviour can be explained by the use of a higher viscosity in model 3. Thus, model 4 considers the lowest viscosity and the highest velocity required, as shown in Figure 7d. Nevertheless, the effect of having no refinement at the combustion zone is observed as the fuel jet stream goes to the right side in the leftmost fuel stream. This implies a remaining non-uniform velocity distribution in the subsequent section—where the flow goes up.

^{−3}, after 400 iterations. A lower viscosity and a higher velocity are required along with using a refined mesh that may increment the computer load. Additionally, the convergence plots showed repeated oscillations without signs of reaching a minimum error. This indicates that oscillations may occur [38] due to an incorrect meshing of the combustion zone. The lowest error reached was 10

^{−1}for the fluid flow variables and 10

^{−3}for the turbulence variables. Mesh 2 has to be refined in the whole combustion zone for modelling the isothermal turbulent flow.

^{−3}. Figure 9b shows the wall resolution in viscous units. Figure 9c shows the residual of velocity field calculation, and Figure 9d illustrates the turbulent viscosity ratio. Low residuals can be observed from the velocity field calculation. Nevertheless, the fuel inlet velocity is low, but not according to the incompressible isothermal turbulent flow. In particular, the turbulent viscosity ratio showed a dissipation in the right side of the combustion zone in Figure 9d. Therefore, the fuel jet stream does not penetrate downwards properly, as observed in Figure 9d. The fluid at the farthest down locations of the section of the ABF has not the desired velocities. Despite the above, the artificial diffusion scheme is an alternative to cases of high complexity and high computational load. Thus, the results of a simulation with artificial diffusion can be used as the initial value of a simulation without artificial diffusion [39]. This allows the solver to reach a solution starting from a value closer to the solution, as the Newton Raphson method assumed.

^{−2}for the velocity-pressure variables and 5 × 10

^{−3}for κ, ε. Figure 12 shows the velocity slide plot at the iteration with the lowest error value reached and the convergence solver plot. There is an increment of the velocity in all locations compared to model 8. Figure 12a shows the flow stream penetrating with high velocities in ever deeper locations. After 400 iterations, model 9 has not achieved convergence, with a minimum error of 10

^{−2}. Using an approximate solution as the initial value, without false linearity, is still useful for complex studies.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Test Using a Fourth Mesh

**Figure A1.**Mesh 4 generated by COMSOL: (

**a**) Mesh overview; (

**b**) Magnification at the leftmost fuel inlet pipe; (

**c**) Histogram of the skewness values.

**Figure A2.**Results of the model of Figure A1 using mesh 4 (Inlet velocity of fuel: 70 m/s. Viscosity 1.8 × 10

^{−5}Pa·s): (

**a**) Velocity results (m/s) (

**b**) turbulent viscosity ratio.

## Appendix B. Wall Resolution from Different Isotropic Diffusion δ Parameters

**Figure A3.**Comparison among wall resolution plots with different values of isotropic diffusion δ. Inlet velocity of fuel: 70 m/s. Viscosity 1.8 × 10

^{−5}Pa·s. Mesh 2. (

**a**) Wall resolution. Artificial diffusion with δ = 0.5; (

**b**) Wall resolution. Artificial diffusion with δ = 0.05; (

**c**) Wall resolution. Artificial diffusion with δ = 0.05.

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**Figure 1.**Illustration of the two geometries used for representing the z-component of the single section of an anode baking furnace (ABF). (

**a**) Geometry 1: a full representation of the z-axis; (

**b**) Geometry 2: a half representation of the z-axis.

**Figure 2.**Magnification of the section at the left fuel inlet pipe zone. (

**a**) Geometry 1: fuel inlet pipes in the z-axis, and (

**b**) Geometry 2: fuel inlet pipes in the z-axis.

**Figure 3.**Illustration of the meshes created using the two geometries: (

**a**) Mesh 1 generated using cfMesh and geometry 1; (

**b**) Mesh 2 generated using cfMesh and geometry 2; (

**c**) Mesh 3 generated using COMSOL Multiphysics and geometry 2.

**Figure 4.**Magnification at one of the combustion zones (the leftmost top zone) of meshes: (

**a**) mesh 2; (

**b**) mesh 3.

**Figure 5.**Histograms of skewness quality measures: (

**a**) Mesh 1 generated from geometry 1; (

**b**) Mesh 2 generated from geometry 2; (

**c**) Mesh 3 generated from geometry 2.

**Figure 6.**Visual representation of the Finite Element Method (FEM) with a finite element triangular mesh.

**Figure 7.**Velocity field plot of models 1 to 4 (dimensions m/s): (

**a**) Model 1. Inlet velocity of fuel: 0. Viscosity 8.9 × 10

^{−4}Pa·s. (

**b**) Model 2. Inlet velocity of fuel: 0. Viscosity 1.8 × 10

^{−5}Pa·s. (

**c**) Model 3. Inlet velocity of fuel: 70 m/s. Viscosity 8.9 × 10

^{−4}Pa·s. (

**d**) Model 4. Inlet velocity of fuel: 70 m/s. Viscosity 1.8 × 10

^{−5}Pa·s.

**Figure 9.**Model 6: isotropic diffusion with δ = 0.005, using mesh 2. Inlet velocity of fuel: 70 m/s. Viscosity 1.8 × 10

^{−5}Pa·s. (

**a**) Velocity field (dimensions m/s); (

**b**) Wall resolution in viscous units; (

**c**) Residual plot; (

**d**) Turbulent viscosity ratio.

**Figure 10.**Model 7: artificial diffusion δ = 0.25

**u**,p using mesh 3. Initial value for the Newton Raphson method: zero. Dimensions m/s.

**Figure 11.**Model 8: Without artificial diffusion using mesh 3. Initial values for the Newton Raphson method are the results of model 7. (

**a**) Velocity field (dimensions m/s); (

**b**) Segregated solver plot.

**Figure 12.**Model 9: No artificial diffusion using mesh 2. Initial value for Newton Raphson method: results of model 8. (

**a**) Velocity field (dimensions m/s) at iteration 400; (

**b**) Segregated solver plot.

**Figure 13.**Comparison between velocity field results (m/s) at the combustion zone using (

**a**) mesh 2 used in model 9 and (

**b**) mesh 3 used in model 8.

**Figure 14.**Comparison between turbulent viscosity ratio plots at the outlet of the leftmost fuel inlet pipe zone using (

**a**) mesh 2 in model 9, and (

**b**) mesh 3 in study 8.

Authors | Year | Objectives | Combustion Model | Detailed Kinetics | Radiation Model |
---|---|---|---|---|---|

Ping. et al. | 2002 | Influence of the baffles on the flowing field | Non-reactive flow | Not included | Not included |

Severo et al. | 2005 | Developing a 3D CFD model for flue-wall design modification | EDM | Not included | P1 |

Ordronneau et al. | 2006 | Application of CFD simulation for crossover design off-gas cleaning system optimisation training purposes | Not specified | Not specified | Not specified |

Gregoire et al. | 2011 | Comparison of two modelling approaches to predict variability | Hot air jet approximation | Not specified | DO method |

Kocaefe et al. | 2013 | Different modelling approaches on anode baking furnace | Not mentioned | Not included | Not specified |

Baiteche et al. | 2015 | Effects of flue-wall deformation and employing different radiation models | Empirical kinetic expression | Not included | - P1 - Monte Carlo method |

Ghaui et al. | 2016 | Implementation of baffleless flue-wall technology | Not mentioned | Not included | Not specified |

Zaidani et al. | 2017 | Effects of flue-wall deformation | Non-reactive flow | Not included | Not specified |

Chaodong et al. | 2018 | Optimisation and development of the furnace structures, process parameters and firing control system | Not specified | Not specified | Not specified |

Nakate et al. | 2018 | Develop a mathematical 2D model to reduce NOx emissions considering turbulent flow, combustion model and radiation | EDM | - κ–ε - Spalart Allmaras | - P1 - DO |

Talice | 2018 | Develop a 2D model to analyse flow behaviour | Not used | Spalart Allmaras | Not used |

Nakate et al. | 2019 | Develop a 3D model to analyse flow behaviour | Not used | κ–ε | Not used |

Talice | 2019 | Develop a 3D model to analyse flow behaviour | Not used | κ–ε | Not used |

Nakate et al. | 2021 | Establish an analysis in 3D flow with a high rate of fuel injection | Energy equation | - Standard κ–ε - Realizable κ–ε | Not used |

Parameter | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 | Model 8 | Model 9 |

Fluid properties | |||||||||

Density (k/m^{3}) | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 | 1.2 |

Dynamic viscosity (Pa·s) | 8.9 × 10^{−4} | 1.8 × 10^{−5} | 8.9 × 10^{−4} | 1.8 × 10^{−5} | 1.8 × 10^{−5} | 1.8 × 10^{−5} | 1.8 × 10^{−5} | 1.8 × 10^{−5} | 1.8 × 10^{−5} |

Initial values for Newton’s iteration | |||||||||

U_{x} (m/s) | 70 | 0 | 70 | 0 | 0 | 0 | 0 | 0 | 0 |

U_{y} (m/s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

U_{z} (m/s) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Pressure (Pa) | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Boundary conditions | |||||||||

Wall | No slip | ||||||||

Inlet | Fully developed flow | ||||||||

Outlet | Pressure | ||||||||

Geometry and mesh | |||||||||

Geometry | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |

Mesh | 1 | 1 | 1 | 1 | 2 | 2 | 3 | 3 | 2 |

Mesh generator tool | cfM | cfM | cfM | cfM | cfM | cfM | COMS | COMS | cfM |

Artificial diffusion scheme | |||||||||

δ(u,p) | Off | Off | Off | Off | Off | 0.005 | 0.25 | Off | Off |

δ(κ,ε) | Off | Off | Off | Off | Off | Off | 0.25 | Off | Off |

Results used as initial value for the Newton Raphson method | |||||||||

Initial value | 0 | 0 | 0 | 0 | 0 | δ(u,p) = 0.01 | 0 | Model 7 | Model 8 |

Mesh | Mesh 1 | Mesh 2 | Mesh 3 |
---|---|---|---|

Generator | cfMesh | cfMesh | COMSOL |

Symmetry | No | Yes | Yes |

Length x-axis (m) | 5.5 | 5.5 | 5.5 |

Length y-axis (m) | 5.0 | 5.0 | 5.0 |

Length z-axis (m) | 0.54 | 0.27 | 0.27 |

Location of fuel inlet pipes (z-axis) (m) | 0.27 | 0.27 | 0.27 |

Cell shape | Cartesian | Cartesian | Tetrahedral |

Mesh | Number of Cells | Minimum Skewness | Average Skewness |
---|---|---|---|

Mesh 1 | 2,424,973 | 0.00 | 0.79 |

Mesh 2 | 545,694 | 0.23 | 0.86 |

Mesh 3 | 4,924,080 | 0.08 | 0.66 |

Model | Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | Model 6 | Model 7 | Model 8 | Model 9 |
---|---|---|---|---|---|---|---|---|---|

Lowest error reached (u,p) | 10^{−3} | 10^{−3} | 10^{−3} | 10^{−2} | 10^{−1} | 10^{−3} | 10^{−3} | 10^{−3} | 10^{−2} |

Lowest error reached (κ,ε) | 10^{−3} | 10^{−}^{3} | 10^{−3} | 10^{−2} | 10^{−3} | 10^{−3} | 10^{−3} | 10^{−3} | 10^{−2} |

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**MDPI and ACS Style**

Libreros, J.; Trujillo, M.
Effects of Mesh Generation on Modelling Aluminium Anode Baking Furnaces. *Fluids* **2021**, *6*, 140.
https://doi.org/10.3390/fluids6040140

**AMA Style**

Libreros J, Trujillo M.
Effects of Mesh Generation on Modelling Aluminium Anode Baking Furnaces. *Fluids*. 2021; 6(4):140.
https://doi.org/10.3390/fluids6040140

**Chicago/Turabian Style**

Libreros, Jose, and Maria Trujillo.
2021. "Effects of Mesh Generation on Modelling Aluminium Anode Baking Furnaces" *Fluids* 6, no. 4: 140.
https://doi.org/10.3390/fluids6040140