# Identification of Material Parameters for the Simulation of Acoustic Absorption of Fouled Sintered Fiber Felts

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## Abstract

**:**

## 1. Introduction

## 2. Fouling Experiment

## 3. Material Characterization

#### 3.1. Structural Characterization

#### 3.1.1. Initial Material

#### 3.1.2. Fouled Material

#### 3.2. Acoustic Characterization

## 4. Numerical Investigation

#### 4.1. Modeling of Stapled Sintered Fiber Felts

**Viscous characteristic length Λ**is introduced by Johnson et al. as an extension of the hydraulic radius ${R}_{H}$ (equal with factor 2 for cylindrical pores) in 1986 [20]. The value is defined as two times the weighted pore’s volume integral divided by the weighted pore wall’s surface integral. The weighting is realised by the fluid’s velocity, respectively, within the volume and on the surface. For common porous materials, the following assumption is taken from literature [21]:$$\mathrm{\Lambda}=\frac{1}{c}{\left(\frac{8{\alpha}_{\infty}\eta}{{R}_{h}\mathrm{\Phi}}\right)}^{0.5},\phantom{\rule{30.0pt}{0ex}}0.3\le c\le 3.3$$**Thermal characteristic length**${\mathbf{\Lambda}}^{\prime}$ is defined by Champoux and Allard in 1991 [18] and is used to describe thermal dissipation effects at higher frequencies. ${\mathrm{\Lambda}}^{\prime}$ equals two times the pore’s volume divided by the pore wall’s surface and influences the bulk modulus of the material layer. For common porous materials, the following assumptions are taken from literature [21]:$$\begin{array}{ccc}\hfill {\mathrm{\Lambda}}^{\prime}& >& \mathrm{\Lambda}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathrm{\Lambda}}^{\prime}& =& \frac{1}{{c}^{\prime}}{\left(\frac{8{\alpha}_{\infty}\eta}{{R}_{h}\mathrm{\Phi}}\right)}^{0.5},\phantom{\rule{30.0pt}{0ex}}0.3\le {c}^{\prime}\le 3.3\hfill \end{array}$$

#### 4.2. Modeling of Fouling

#### 4.3. Inverse Parameter Identification

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CT | Computed Tomography |

SEM | Scanning Electron Microscope |

SFF | Sintered Fiber Felt |

TMM | Transfer Matrix Method |

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**Figure 1.**Light microscope images of Sintered Fiber Felts (SFF) before fouling. (

**a**) SFF 100; (

**b**) SFF 150.

**Figure 5.**Measured frequency-dependent absorption coefficients of clean (0 M) and fouled (12 M) SFF 100 for one layer and staple sizes of $0.01\phantom{\rule{0.166667em}{0ex}}$m and $0.02\phantom{\rule{0.166667em}{0ex}}$m.

**Figure 6.**Measured frequency-dependent absorption coefficients of clean (0 M) and fouled (12 M) SFF 150 for one layer and staple sizes of $0.01\phantom{\rule{0.166667em}{0ex}}$m and $0.02\phantom{\rule{0.166667em}{0ex}}$m.

**Figure 8.**Comparison of numerical and experimental data for a staple size of $0.02\phantom{\rule{0.166667em}{0ex}}$m. (

**a**) fitting of clean SFF 100; (

**b**) calculation (forward solution of the mechanical problem, without any model updating) of fouled SFF 100.

**Figure 9.**Comparison of clean and fouled SFF 100 for a staple size of $0.02\phantom{\rule{0.166667em}{0ex}}$m. (

**a**) experimental data; and (

**b**) numerical data.

**Table 1.**Structural parameters of sintered fiber felts in clean condition. Porosity ${\varphi}_{f}$ of the functional layer, thickness of the functional layers ${t}_{f}$, fiber diameter in the functional layer ${d}_{f}$, wire diameter of the support grid ${d}_{w}$, mesh size of the support grid w, porosity of the support grid ${\varphi}_{w}$, hydraulic radius of the support grid ${R}_{h}$.

Material | ${\mathit{\varphi}}_{\mathit{f}}$ | ${\mathit{d}}_{\mathit{f}}$ $(\mathsf{\mu}$m) | ${\mathit{t}}_{\mathit{f}}$ $(\mathsf{\mu}$m) | ${\mathit{d}}_{\mathit{w}}$ $(\mathsf{\mu}$m) | w $(\mathsf{\mu}$m) | ${\mathit{\varphi}}_{\mathit{w}}$ | ${\mathit{R}}_{\mathit{h}}$ $(\mathsf{\mu}$m) |
---|---|---|---|---|---|---|---|

SFF 100 | 0.92 | 430 | 26 | 254 | 432 | 0.69 | 142 |

SFF 150 | 0.94 | 270 | 24 | 252 | 424 | 0.69 | 140 |

**Table 2.**Specific flow resistance, thickness and flow resistivity of clean and fouled SFF 100 and SFF 150.

Parameter | SFF 100 | SFF 150 | ||
---|---|---|---|---|

Clean | Fouled | Clean | Fouled | |

${R}_{S}$ (Ns/m${}^{3}$) | $51.26$ | $55.56$ | $15.13$ | $15.48$ |

${h}_{f}$ (m) | $0.00043$ | $0.00043$ | $0.00027$ | $0.00027$ |

r (Ns/m${}^{4}$) | 119,208 | 129,209 | 56,036 | 57,333 |

Material | Staple Size ${\mathit{h}}_{\mathit{s}}$ (m) | Number of Specimens ${\mathit{N}}_{\mathit{s}}$ (-) |
---|---|---|

SFF 100 | $0.0009\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.01\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.02$ | $1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}11\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}21$ |

SFF 150 | $0.0007\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.01\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}0.02$ | $1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}12\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}25$ |

Parameter | Symbol | Unit | Functional Layer | Support Grid |
---|---|---|---|---|

layer thickness | ${h}_{S}$ | (m) | $0.00043$ | $0.00047$ |

Young’s modulus | E | (N/m^{2}) | $2\xb7{10}^{11}$ | - |

Poisson ratio | ν | (-) | $0.3$ | - |

structural density | ${\rho}_{S}$ | (kg/m^{3}) | 8000 | - |

loss factor | ${\eta}_{l}$ | (-) | $0.01$ | - |

fluid density | ${\rho}_{F}$ | (kg/m^{3}) | $1.21$ | $1.21$ |

speed of sound | ${c}_{F}$ | (m/s) | 343 | 343 |

porosity | Φ | (-) | $0.915$ | $0.691$ |

flow resistivity | r | (Ns/m^{4}) | 119,208 | 2610 |

tortuosity | ${\alpha}_{\infty}$ | (-) | 1–1.6 | 1 |

viscous length | Λ | (µm) | $11\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}123$ | 409 |

thermal length | ${\mathrm{\Lambda}}^{\prime}$ | (µm) | $11\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{3.33333pt}{0ex}}123$ | 204 |

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**MDPI and ACS Style**

Lippitz, N.; Blech, C.; Langer, S.; Rösler, J.
Identification of Material Parameters for the Simulation of Acoustic Absorption of Fouled Sintered Fiber Felts. *Materials* **2016**, *9*, 709.
https://doi.org/10.3390/ma9080709

**AMA Style**

Lippitz N, Blech C, Langer S, Rösler J.
Identification of Material Parameters for the Simulation of Acoustic Absorption of Fouled Sintered Fiber Felts. *Materials*. 2016; 9(8):709.
https://doi.org/10.3390/ma9080709

**Chicago/Turabian Style**

Lippitz, Nicolas, Christopher Blech, Sabine Langer, and Joachim Rösler.
2016. "Identification of Material Parameters for the Simulation of Acoustic Absorption of Fouled Sintered Fiber Felts" *Materials* 9, no. 8: 709.
https://doi.org/10.3390/ma9080709