# Estimation of the Acoustic Properties of the Random Packing Structures of Granular Materials: Estimation of the Sound Absorption Coefficient Based on Micro-CT Scan Data

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## Abstract

**:**

## 1. Introduction

## 2. Samples and Measuring Equipment Used for Measuring the Sound Absorption Coefficient

#### 2.1. Sample for Measuring the Sound Absorption Coefficient

#### 2.2. Equipment for Measuring the Sound Absorption Coefficient

## 3. Methods and Results of Measuring Tortuosity

#### 3.1. Tortuosity Measurement Sample

_{∞}can be expressed using the speed of sound c

_{0}in air and the apparent speed of sound c in the filling structure, as shown in Equation (1) [13]:

#### 3.2. Tortuosity Measurement

_{∞}of the packing structure.

_{∞}= 1.45.

## 4. Theoretical Analysis

#### 4.1. Analysis Unit and Element Division of the Packing Structure

_{n}of the grain and the volume V

_{n}of the clearance were equal. The number of images used for the theoretical analysis n was 2000, 1000, and 500 for the grain size d = 1 mm, d = 2 mm, and d = 4 mm, respectively, for 20 mm in the x-direction. Thus, the thickness l of the element in Figure 5c corresponded to the pitch in the x-direction of the image, which was 10 µm, 20 µm, and 40 µm for the grain sizes of d = 1 mm, 2 mm, and 4 mm, respectively. Thus, for each grain size d = 1 mm, 2 mm, and 4 mm, the number of analysis units was n = 100 in the packing structure partitioning method [7,8], as shown in Figure 5c, which was a value at which the theoretical values of the normal incident sound absorption coefficient sufficiently converged.

_{n}, as shown in Figure 5c. Similarly, multiplying the total circumference of the cross-section by l yielded S

_{n}, as shown in Figure 5c. Thus, as shown in Figure 5d, using Equation (2), the thickness of the clearance between the two planes, b

_{n}, could be obtained for a single image with a thickness l. Here, F

_{n}is the correction factor for obtaining the true surface area, explained in the next section.

#### 4.2. Numerical Analysis of the Tomograms in Random Packing

_{n}of the cylindrical element in the nth division is expressed as in Equation (3), where k is the number of divisions of the hemisphere in the x-direction.

_{n}of the cylindrical element could be expressed as in Equation (4):

_{n}of the divided cylindrical elements obtained from the CT images and the true surface area S

_{correct}(= 1/2·πd

^{2}) of the hemisphere is shown in the following equation, where F

_{1}is the correction factor.

_{1}is unity (i.e., they are equal). However, when the number of divisions k is infinite, the correction factor F

_{∞}asymptotically approaches ~1.273.

_{50}for k = 50 is ~1.259.

_{50}= 1.259 for k = 50 by S

_{n}in Equation (2).

#### 4.3. Propagation Constants and Characteristic Impedance Considering the Tortuosity

_{s}, and compressibility C

_{s}were derived, as shown in Equations (8) and (9), respectively [14]. For several atmospheric properties, including the density (ρ

_{0}) of air, λ

_{s}is the parameter of mediation, b

_{n}is the clearance thickness between the two planes, ω is the angular frequency, η is the viscosity of air, κ is the specific heat ratio of air, P

_{0}is the atmospheric pressure, and N

_{pr}is the Prandtl number.

_{s}multiplied by the tortuosity α

_{∞}, the propagation constant and characteristic impedance considering the tortuosity could be obtained [15]. Therefore, the propagation constant γ and characteristic impedance Z

_{c}when considering the tortuosity α

_{∞}could be expressed using the effective density ρ

_{s}and compression ratio C

_{s}, as follows [15]:

#### 4.4. Transfer Matrix

_{n}and four-terminal constants A~D of the acoustic tube element could be calculated, as shown in Equation (12):

_{1}and u

_{1}, respectively, and the sound pressure and particle velocity at Plane 2 are p

_{2}and u

_{2}, respectively, the transfer matrix can be expressed as in Equation (13):

_{unit}and T

_{top}for the analysis unit at the top end of the sample, respectively, by cascading the transfer matrices of each divided element.

#### 4.5. Vertical Incident Sound Absorption Coefficient

_{2}= 0, and Equation (13) could be transformed into Equation (14):

_{0}and u

_{0}, respectively, the specific acoustic impedance Z

_{0}looking inward from the plane of incidence of the sample can be expressed as follows:

_{0}= p

_{1}and Equation (15), the specific acoustic impedance Z

_{0}of the sample can be expressed as follows:

_{0}and reflectance R can be expressed as follows:

## 5. Comparison of the Measured and Theoretical Values

_{n}between the two planes in Equation (2), resulting in a larger proportion of the boundary layer in the voids and a larger calculated attenuation of the sound waves by viscosity. However, for the grain size d = 4 mm, as shown in Figure 10c, the peak frequency of the theoretical value was even lower than the measured value when considering the tortuosity and surface area corrections, and the reasons for this result are discussed below.

## 6. Conclusions

- 1.
- It is difficult to construct a mathematical model for random packing, as it has no structure periodicity. Therefore, the sound absorption coefficient was estimated using a theoretical analysis based on cross-sectional CT scan images.
- 2.
- For the theoretical values considering the tortuosity, the peak sound absorption values were higher, and the peak frequency moved to lower frequencies compared with the case without considering the tortuosity. As a result, in all cases, the theoretical values were closer to the measured values. Therefore, the measured tortuosity values are reasonable.
- 3.
- Regarding the theoretical values, when both the surface area and tortuosity were considered, the peak sound absorption frequency moved to a lower frequency compared with the theoretical value when only considering tortuosity, and was in general agreement with the measured values for the particle diameters of d = 1 mm and d = 2 mm. Therefore, the method of estimating the vertical incident sound absorption coefficient using computed tomographic images is useful. Moreover, this model can be applied even if the material changes, provided the granular material can be considered rigid. Additionally, the model can be applied without problems for general granular densities and grain sizes in the order of mm.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Conflicts of Interest

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**Figure 2.**Scheme of two microphone impedance tubes for the sound absorption coefficient measurement.

**Figure 5.**Divided element approximated to the clearance between the two planes: (

**a**) binarization of the cross-sectional image; (

**b**) calculation of the circumference of the sphere and the cross-sectional area of the clearance; (

**c**) divided element; (

**d**) approximated clearance between the two planes.

**Figure 6.**Divided element for the half sphere: (

**a**) cylindrically approximated surface area; (

**b**) true surface area.

**Figure 8.**Cross-sectional shape at any position of the analysis unit (sound incident area and aperture area).

**Figure 9.**Equivalent circuit in the analysis (cascade connection of the transfer matrix of each element).

Frequency [kHz] | Inverse of the Square Root of Frequency $[1/\sqrt{\mathit{H}\mathit{z}}]$ | Distance between Sensors [mm] | Transmission Time [ms] | Tortuosity |
---|---|---|---|---|

32.7 | 0.00559 | 395 | 1.229 | 2.09 |

58 | 0.004152 | 395 | 1.199 | 1.53 |

110 | 0.003015 | 345 | 1.044 | 2.13 |

150 | 0.002582 | 204 | 0.626 | 2.00 |

200 | 0.002236 | 204 | 0.618 | 1.67 |

300 | 0.001826 | 204 | 0.615 | 1.33 |

∞ | 0 | - | - | 1.45 |

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**MDPI and ACS Style**

Sakamoto, S.; Suzuki, K.; Toda, K.; Seino, S.
Estimation of the Acoustic Properties of the Random Packing Structures of Granular Materials: Estimation of the Sound Absorption Coefficient Based on Micro-CT Scan Data. *Materials* **2023**, *16*, 337.
https://doi.org/10.3390/ma16010337

**AMA Style**

Sakamoto S, Suzuki K, Toda K, Seino S.
Estimation of the Acoustic Properties of the Random Packing Structures of Granular Materials: Estimation of the Sound Absorption Coefficient Based on Micro-CT Scan Data. *Materials*. 2023; 16(1):337.
https://doi.org/10.3390/ma16010337

**Chicago/Turabian Style**

Sakamoto, Shuichi, Kyosuke Suzuki, Kentaro Toda, and Shotaro Seino.
2023. "Estimation of the Acoustic Properties of the Random Packing Structures of Granular Materials: Estimation of the Sound Absorption Coefficient Based on Micro-CT Scan Data" *Materials* 16, no. 1: 337.
https://doi.org/10.3390/ma16010337