# Computational Investigations on Soundproof Applications of Foam-Formed Cellulose Materials

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Background Approaches

#### 2.2. Materials

^{3}density. These FCM were obtained from bleached hardwood cellulose (BHCF), with 100–200 µm average fiber length, and 15–35 µm diameters from recycled fibers from recovered papers (RCF), dry grinded to obtain a fluffy material with high content of fines.

_{12}H

_{25}SO

_{4}Na are commonly used in cosmetics products and was added as anionic surfactant into the liquid before mixing.

#### 2.3. Methods

#### 2.3.1. Foam Forming Methods

_{foam}) to the mass of the fiber/water suspension. M

_{pulp}was required to obtain the same volume (1 − (m

_{foam}/m

_{pulp})) [56]. The foam and fibers suspension were filtered and dewatered using a Buchner funnel (15 cm diameter) with a filter paper at the bottom. The filtering was developed at a low level and vacuumed for approximately 20 min. Dewatered FCM with wet structure is presented in Figure 1 [36].

#### 2.3.2. Microscopic Studies and Structural Analysis

#### 2.3.3. Acoustics Evaluation

_{s}(ω) given by:

_{c}(ω) is [43]:

_{c}(ω) is [43]:

_{p}the porosity, ω the angular frequency, Z

_{0}= (ρ

_{0}c

_{0}) the impedance of the air saturating the pores, (ρ

_{0}) the density of air, c

_{0}the speed of sound in air, and d the sample thickness.

^{2}= −1, and σ denotes air flow resistivity. The eight constants (C

_{1..8}) acquire various values according to different materials. For different types of materials, the values presented in Table 2 can provide accurate predictions. Note that the first-row values correspond to the basic D-B model.

_{∞}is the tortuosity factor, ρ

_{f}is the fluid density, ε

_{p}is the porosity, σ is the airflow resistivity, μ is the dynamic viscosity, p

_{A}is the quiescent pressure, γ is the ratio of specific heats, Λ is the viscous characteristic length, Λ’ is the thermal characteristic length, and P

_{r}denotes the Prandtl number.

_{0}, and static thermal tortuosity τ’

_{0}(both dimensionless), respectively. Static viscous permeability is defined as k

_{0}= μ/σ and static thermal permeability, k’

_{0}(both have SI unit m

^{2}). Generally, τ

_{0}≥ τ

_{∞}; for porous materials it can be supposed that τ

_{0}= 3/4 τ

_{∞}and τ’

_{0}≈ τ

_{0}/τ

_{∞}. Regarding static thermal permeability, k’

_{0}≥ k

_{0}; it was assumed that for the cylindrical pores, k’

_{0}= k

_{0}, in the same time that for slits k’

_{0}= τ

_{∞}k

_{0}. It has to be mentioned that the JCA model is recovered by setting M’ = P = P’ = 1.

#### 2.3.4. Geometrical Properties

_{p}) is one of the most important parameters, along with material density and speed of sound, in tailoring the performance of an acoustic absorber [23].

_{pore}and V

_{total}denotes the pore and total volumes respectively (m

^{3}), ρ

_{a}is the density of porous material (kg/m

^{3}), and ρ

_{m}is the solid part density (skeletal density of the solid matrix) of the material (kg/m

^{3}).

_{pore}) and the surface area (S

_{pore}) of the pores

_{f}and open porosity, respectively [16]:

#### 2.3.5. Transport Properties

_{e}) traveled by a high frequency sound through the porous sample to the sample thickness d:

_{0}, the microphones diameter d

_{m}, the sample thickness d, and evaluated the time delay Δt, the tortuosity results by the expression:

_{m}= 6.35 × 10

^{−3}m, an exciting signal with 35 kHz frequency. The average values for samples tortuosity are shown in Table 3 for P1, P3S, and P4 materials. Considering the literature provided, values of 95% and 96% accuracy for the experimental method enable a good correlation to the values gained by the empirical expression (see Equation (23)).

_{f}of a material sample [23]:

_{v}(m

^{3}/s) denotes the volumetric flow rate through material sample. Taking into account the specific flow resistance, R

_{s}is as follows:

^{2}) denotes the cross-sectional area of the sample (normal to the flow direction) and results in air flow resistivity:

_{m}and the mean fiber diameter d

_{f}, could be estimated by the following empirical expression [34]:

_{1}and K

_{2}depends on material type. For K

_{1}, the literature proposes 1.404 (related to the Garai and Pompoli model, polyester fibers) and 1.530 (related to the Bies and Hansen model), respectivley. For glass-wool-like composites, the literature recommended that K

_{2}be valued at 3.18 × 10

^{−9}.

^{−5}(Pa s).

^{−6}m and the constant K

_{1}tuned to 1.59, the airflow resistivity for each FCM sample is presented in Table 3. In addition, the relative errors of the experiment and of the average value were evaluated. In order to perform the computational analysis, as well as taking into account the comparative values of relative errors in Table 4, the authors adopted the value of airflow resistivity for each sample as a rounded mean between the experimental and, respectively, the theoretical estimation according to Equations (29) and (30).

## 3. Results

#### 3.1. Comparison Between Experiments and Computational Approaches

#### 3.2. Extended Computational Investigations

_{0}were discretely adopted within the range of 0–60 degrees. For the sake of comparison, the graphs within Figure 8 were plotted with continuous line for normal incidence and with graphical markers for the other value of θ

_{0}(30°, 45°, 60°). The meanings of each diagram and symbol were mentioned on title and legend, respectively.

## 4. Discussion

_{1..9}coefficients according to this type of materials.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Snapshots of wet structures for the foam-formed cellulose materials within this study: (

**a**) sample P1; (

**b**) sample P3S; and (

**c**) sample P4. [36].

**Figure 2.**Optical micrographs of foam/water formed cellulose materials: sample P1 with magnification 350x (

**a**) and 80x (

**b**); sample P3S with magnification 350x (

**c**) and 80x (

**d**); sample P4 with magnification 350x (

**e**) and 80x (

**f**).

**Figure 3.**Schematization used for physical description of viscous and thermal characteristic lengths.

**Figure 4.**Experimental laboratory setup for tortuosity evaluation: (

**a**) Schematic diagram of the experimental setup; (

**b**) general view of the laboratory setup with ultrasonic generator and sample holder. Component parts are marked on pictures, where DAQ denotes the Data Acquisition system (National Instruments, Austin, Texas, USA).

**Figure 5.**Experimental laboratory setup for airflow resistivity evaluation: (

**a**) Basic principle; (

**b**) schematic diagram of the experimental setup underlining the main parts; and (

**c**) general view of laboratory setup with component parts marked on picture.

**Figure 6.**Comparative results between experimental evaluated and, respectively, computational estimated normal incidence absorption coefficients: (

**a**) Absorption for sample P1; (

**b**) Relative prediction error for sample P1; (

**c**) Absorption for sample P3S; (

**d**) Relative prediction error for sample P3S; (

**e**) Absorption for sample P4; (

**f**) Relative prediction error for sample P4. The significations of the symbols within graphs were mentioned on each diagram legend.

**Figure 7.**Comparative results between evaluated and computational estimated normal incidence reflection coefficients. (

**a**) Reflection for sample P1; (

**b**) relative prediction error for sample P1; (

**c**) reflection for sample P3S; (

**d**) relative prediction error for sample P3S; (

**e**) reflection for sample P4; and (

**f**) relative prediction error for sample P4. The significations of the symbols within graphs were mentioned on each diagram legend.

**Figure 8.**Parametrical evolutions of absorption α and reflection R coefficients respectively, with respect to the sound incidence angle θ

_{0}. (

**a**) α for sample P1; (

**b**) R for sample P1; (

**c**) α for sample P3S; (

**d**) R for sample P3S; (

**e**) α for sample P4; and (

**f**) R for sample P4. The significations of the symbols within graphs were mentioned on each diagram legend.

**Table 1.**Fibrous components of foam formed cellulose materials. FCM: foam-formed cellulose materials; BHCF: bleached hardwood cellulose; RCF: recycled fibers from recovered papers.

FCM Sample Codes | Composition |
---|---|

P1 | 100% (BHCF) |

P3S | 100% (RCF) |

P4 | 50% (BHCF) + 50% (RCF) |

**Table 2.**Empirical constants for Delany–Bazley (D-B) model [30].

Material Type | C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 |
---|---|---|---|---|---|---|---|---|

material 1 | 0.0571 | 0.745 | 0.087 | 0.732 | 0.0978 | 0.700 | 0.189 | 0.595 |

material 2 | 0.078 | 0.623 | 0.074 | 0.660 | 0.159 | 0.571 | 0.121 | 0.530 |

material 3 | 0.114 | 0.369 | 0.0985 | 0.758 | 0.168 | 0.715 | 0.136 | 0.491 |

material 4 | 0.212 | 0.455 | 0.105 | 0.607 | 0.163 | 0.592 | 0.188 | 0.544 |

Tortuosity | Sample P1 | Sample P3S | Sample P4 |
---|---|---|---|

theoretical | 1.008097 | 1.007585 | 1.00861 |

experimental (averaged values) | 1.04721428 | 1.036407 | 1.049156122 |

Airflow Resistivity [Pa s m^{−2}] | Sample P1 | Sample P3S | Sample P4 | |
---|---|---|---|---|

Theoretical estimation—Equation (29) | 5936.869 | 5685.805 | 6161.118 | |

Theoretical estimation—Equation (30) | 5179.872 | 4538.770 | 5865.454 | |

Experimental investigation (averaged values) | 6104.445 | 5849.473 | 6332.070 | |

Average value per sample | 5740.395 | 5358.016 | 6119.547 | |

Relative error [%] of experimental value related to | Equation (29) | 2.75 | 2.80 | 2.70 |

Equation (30) | 15.15 | 22.41 | 7.37 | |

Relative error [%] of average value related to | Experiment | 5.96 | 8.40 | 3.36 |

Equation (29) | 3.31 | 5.77 | 0.67 | |

Equation (30) | 10.82 | 18.05 | 4.33 |

Parameter | Sample P1 | Sample P3S | Sample P4 |
---|---|---|---|

Bulk density [kg m^{−3}] | 37.3 | 36.3 | 38.18 |

Foam porosity [-] | 0.984 | 0.985 | 0.983 |

Flow resistivity [Pa s m^{−2}] | 6000 | 5770 | 6200 |

Tortuosity [-] | 1.0081 | 1.0076 | 1.00861 |

Static viscous tortuosity [-] | 1.3441 | 1.3435 | 1.3448 |

Static thermal tortuosity [-] | 1.3333 | 1.3333 | 1.3333 |

Viscous characteristic length [m] | 157.24 × 10^{−6} | 160.22 × 10^{−6} | 154.80 × 10^{−6} |

Thermal characteristic length [m] | 314.48 × 10^{−6} | 320.44 × 10^{−6} | 309.60 × 10^{−6} |

Static thermal permeability [m^{2}] | 3.02 × 10^{−9} | 3.14 × 10^{−9} | 2.92 × 10^{−9} |

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**MDPI and ACS Style**

Debeleac, C.; Nechita, P.; Nastac, S.
Computational Investigations on Soundproof Applications of Foam-Formed Cellulose Materials. *Polymers* **2019**, *11*, 1223.
https://doi.org/10.3390/polym11071223

**AMA Style**

Debeleac C, Nechita P, Nastac S.
Computational Investigations on Soundproof Applications of Foam-Formed Cellulose Materials. *Polymers*. 2019; 11(7):1223.
https://doi.org/10.3390/polym11071223

**Chicago/Turabian Style**

Debeleac, Carmen, Petronela Nechita, and Silviu Nastac.
2019. "Computational Investigations on Soundproof Applications of Foam-Formed Cellulose Materials" *Polymers* 11, no. 7: 1223.
https://doi.org/10.3390/polym11071223