# Numerical Simulation for the Sound Absorption Properties of Ceramic Resonators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Specimen Configuration

_{2}O

_{3}∙ bSiO

_{2}∙ cH

_{2}O. In addition to hydrated silicates, the clays also contain hydrated oxides, such as hydrated silica (SiO

_{2}∙ nH

_{2}O), hydrated alumina (Al

_{2}O

_{3}∙ nH

_{2}O), and hydrated ferric oxide (Fe

_{2}O

_{3}∙ nH

_{2}O). The hydrated aluminum silicates and, in part, the hydrated oxides are responsible for the plasticity of the clayey mixtures, that is, their ability to be modeled [21].

- Leaning action by the siliceous sands which, having particles considerably larger than hydrated aluminum silicates, allow the regulation of the shrinkage during drying and cooking, also favoring the stability of the raw material after forming
- Melting action of calcium carbonate. During the firing of ceramic materials, a liquid phase must be formed; this, upon cooling, solidifies, forming a compact glass that binds the grains of the material and fills the pores together. Calcium carbonate leads to the formation of a liquid phase at 950–1000 °C and is used as a flux for porous products obtained at low temperatures.

- Preparation of the dough. The raw materials are ground and mixed, and the resulting mixture is homogenized. The ground products are first sieved and then dry-mixed according to suitable weight or volume ratios. The dry process is applied to products obtained from a single raw material.
- Forming. By forming, the dough is given the shape of the desired product with the application of sufficient pressure to deform it plastically and stably. After forming, the blank must be strong enough to withstand its own weight and stresses during handling.
- Drying. The water contained in the raw must be removed before cooking by means of a delicate drying process, with which the water that forms a veil around the clay particles is removed, giving plasticity to the material. During this process, there is a contraction in volume due to a rapprochement of the particles with an increase in the mutual forces of attraction and, therefore, in the mechanical strength of the product. Artificial drying begins in a very humid environment (relative humidity = 70%, and temperature = 50 °C), where the product is heated without loss of water, favoring the water flow from the inside to the surface. The humidity is then gradually decreased, and the temperature increased, without exceeding 120 °C. Subsequently, the water contained in the internal pores of the clay particles is also removed, but this process is not associated with any contraction of the dough. The drying continues until the moisture content of the product is not less than 1%.
- Cooking. Through cooking, the dried doughs acquire their final physical-mechanical properties, dimensions, and appearance. The firing process with variable temperature between 900 °C and 1000 °C must be homogeneous to create a material with adequate porosity. During cooking, the clay disintegrates, releasing its oxides, namely silica (SiO
_{2}) and alumina (Al_{2}O_{3}). Above 900 °C, the limestone decarbonates and the formation of a compound between alumina and silica called “mullite” (3Al_{2}O_{3}∙ 2SiO_{2}) begin; in this phase, there is also the gradual formation of a liquid phase that fills the voids between the particles [24].

#### 2.2. Helmholtz Resonator-Based Setup

#### 2.3. Impedence Tube Measurements

#### 2.4. Artificial Neural Network-Based Model

_{1}, …, x

_{n}) as a set of independent variables, defined as network inputs, and y = (y

_{1}, …, y

_{k}) as a set of dependent variables, which represents the network output. Then, consider w = (w

_{1}, …, w

_{n}) as a set of weights. The association between output and input of the network can be exemplified through the following equation:

- y is the output
- f is the activation function
- w
_{j}is the jth weight - x
_{j}is the jth input - b is the bias

## 3. Results

- Frequency
- Specimen thickness
- Hole diameter
- Sound absorption coefficient

## 4. Discussion

^{−3}, we can say that the model is able to predict the values of the sound absorption coefficient for this configuration of the material with excellent performance.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Scheme of the principle of operation of a simple Helmholtz resonator: (

**a**) Description of Helmholtz resonator functioning and (

**b**) the equivalent mass-spring model.

**Figure 4.**Artificial neural network architecture with tree layers: Input layer, hidden layer, and output layer.

**Figure 6.**Architecture of the algorithm based on artificial neural networks used to simulate the acoustic behavior of a ceramic resonator. In the input layer, there are three inputs: Frequency, specimen thickness, and hole diameter. In the hidden layer, there are ten nodes. Finally, in the output layer, there is a single output, that is, the sound adsorption coefficient.

**Figure 7.**Comparison between the trend of the sound absorption coefficient as a function of frequency between the measurements carried out with the impedance tube and the simulations carried out with the model based on artificial neural networks: (

**a**) Ceramic resonator with a thickness of 0.6 cm and hole diameter of 0.2 cm; (

**b**) ceramic resonator with a thickness of 0.6 cm and hole diameter of 0.3 cm; (

**c**) ceramic resonator with a thickness of 0.6 cm and hole diameter of 0.6 cm; (

**d**) ceramic resonator with a thickness of 2.5 cm and hole diameter of 0.3 cm.

Specimen ID | Specimen Thickness, cm | Hole Diameter, cm | Percentage of Perforation, % |
---|---|---|---|

N1 | 0.6 | 0.2 | 3.5 |

N2 | 0.6 | 0.3 | 8 |

N3 | 0.6 | 0.6 | 32 |

N4 | 2.0 | 0.2 | 4 |

N5 | 2.5 | 0.6 | 50 |

N6 | 2.5 | 0.3 | 11 |

N7 | 3.0 | 0.5 | 25 |

Dataset | Regression R Value | Mean Squared Error |
---|---|---|

Training set | 0.991 | 1.01 × 10^{−3} |

Test set | 0.986 | 1.46 × 10^{−3} |

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Ciaburro, G.; Iannace, G.
Numerical Simulation for the Sound Absorption Properties of Ceramic Resonators. *Fibers* **2020**, *8*, 77.
https://doi.org/10.3390/fib8120077

**AMA Style**

Ciaburro G, Iannace G.
Numerical Simulation for the Sound Absorption Properties of Ceramic Resonators. *Fibers*. 2020; 8(12):77.
https://doi.org/10.3390/fib8120077

**Chicago/Turabian Style**

Ciaburro, Giuseppe, and Gino Iannace.
2020. "Numerical Simulation for the Sound Absorption Properties of Ceramic Resonators" *Fibers* 8, no. 12: 77.
https://doi.org/10.3390/fib8120077