*Article* **Sound Absorbing and Insulating Low-Cost Panels from End-of-Life Household Materials for the Development of Vulnerable Contexts in Circular Economy Perspective**

**Manuela Neri 1,***∗***, Elisa Levi 1, Eva Cuerva 2, Francesc Pardo-Bosch 3, Alfredo Guardo Zabaleta <sup>4</sup> and Pablo Pujadas <sup>2</sup>**


**Abstract:** From a construction point of view, neighborhoods with residents living at or below the poverty threshold are characterized by low energy efficiency buildings, in which people live in acoustic discomfort with no viable options for home improvements, as they usually can not afford the materials and labor costs associated. An alternative to this is to use low-cost insulating elements made of non-conventional materials with acceptable acoustic properties. Given that household materials at their end-of-life (EoLHM) are free of costs and available also to the more disadvantaged population, they can be used to build acoustic panels for such contexts. This approach embraces several benefits since it reduces the amount of waste produced, the footprint deriving from the extraction of new raw materials and, by highlighting the potential of the EoLHM, discourages the abandonment of waste. In this paper, the acoustic properties of EoLHM, such as cardboard, egg-cartons, clothes, metal elements and combinations of them, are investigated by means of the impedance tube technique. The measured sound absorption coefficient and transmission loss have shown that EoLHM can be used for the realization of acoustic panels. However, since none of the analyzed materials shows absorbing and insulating properties at the same time, EoLHM must be wisely selected. This innovative approach supports the circular economy and the improvement for the living condition of low-income households.

**Keywords:** household end-of-life materials; building retrofitting; sound insulation; sound absorption; vulnerable houses; circular economy; egg-box; cardboard; textile waste; reuse

#### **1. Introduction**

The Sustainable Development Goals are a universal call embraced by all Member States of the United Nation in 2015 for eradicating poverty and protecting the environment. According to the call, for improving the living condition on a global scale, economy and social aspects must go hand-in-hand. For this reason, issues such as education, health, social protection, job opportunities, climate change and environmental protection must be taken into account through global, local and people actions [1]. Among the 17 Goals, number 11 deals with sustainable cities and communities, and it requires to ensure access for everyone to adequate, safe and affordable housing and basic services. Indeed, it is estimated that by 2030 the 60% of the world's population will live in cities that account for about the 70% of

**Citation:** Neri, M.; Levi, E.; Cuerva, E.; Pardo-Bosch, F.; Zabaleta, A.G.; Pujadas, P. Sound Absorbing and Insulating Low-Cost Panels from End-of-Life Household Materials for the Development of Vulnerable Contexts in Circular Economy Perspective. *Appl. Sci.* **2021**, *11*, 5372. https://doi.org/10.3390/ app11125372

Academic Editor: Luís Picado Santos

Received: 3 May 2021 Accepted: 2 June 2021 Published: 9 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

global carbon emission and 60% of resource use. The rapid urbanization that the world has been facing since 2007 is resulting in air pollution, unplanned urbanization, inadequate services and infrastructures. The growth and development of cities must be controlled, so to guarantee cities inclusive, safe, resilient and sustainable.

Since people with similar socioeconomic status tend to cluster in the same urban areas disadvantaged contexts can be easily identified in the urban fabric: in these places, the vulnerable population lives and it consists of refugees, migrants, elderly persons, people with disabilities and children [1]. In these specific contexts, people live below the threshold of poverty, and sometimes in conditions of great discomfort. This phenomenon is responsible of inequalities and it has been identified in several European cities, as in the case of Barcelona [2,3]. Between 2001 and 2011, in the city of Barcelona the migrant population increased from about 5% to 17%, and the new-low income immigrants reside mostly either in the historical center, usually on degraded 19th century buildings, or in peripheral districts characterized by poor quality houses built in the 1960s and in the 1970s [2]. In the case of the historical center, dwellings are in some cases small, overcrowded and lacking of openings with consequent insufficient natural light and poor air quality. When present, windows are crumbling and do not guarantee adequate acoustic insulation. Since many of these neighborhoods are also touristic destinations with noisy anthropic activities also during the night-time, the aspect related to the sound quality of dwellings should not be underestimated.

The urban environment is characterized by multiple simultaneous sounds due to transportation, industry and neighbors. When the sound pressure level exceeds a certain value it is perceived as noise and even if sound perception is subjective, noise control is very important. The exposure to excessive and prolonged levels of noise affects people's well-being, behavior, productivity, mental and physical health, with negative consequences such as sleep disturbances, stress, irritability and other health issues [2,4–6]. However, noise pollution and acoustic discomfort in buildings are not limited to disadvantaged neighborhoods. To depict the acoustic situation, the European Union, through the Environmental Noise directive [7], has requested to map the noise pollution sources and to define an action plan to reduce the effects on the population. It is estimated that 40% of the population does not experience acoustic well-being because of noise from neighbors and traffic [5], and the 65% of Europeans living in major urban areas are exposed to highnoise levels [8]. If in virtuous contexts the improvement of the indoor acoustic comfort is possible, in disadvantaged contexts this task is much more difficult to achieve. In fact, these interventions require specialized personnel and expensive insulating/absorbing acoustic solutions. Acoustic panels must be aesthetics, safe, acoustically efficient, easy to install and maintain, resistant to wear and environmental factors. Although inexpensive acoustic materials can be found on the market, in disadvantaged contexts they can not be easily purchased because people have little or no financial resources. An alternative can be the realization of no-cost panels featuring acoustic properties.

While on the one hand there is growing attention to the well-being of people, on the other it is necessary to define actions aimed at protecting the environment, optimizing the system that provides us with the raw materials necessary to make the products, and which houses the waste. To lighten the load on the environment, it is necessary to use raw materials in a conscious way, for example, by extending the life of the products as much as possible. Additionally, since many materials still possess exploitable properties when they are discarded, they can be reused for other purposes. This model is called Circular Economy (CE) which is in contrast to the linear economy model. In the linear economy model, the raw material is extracted, processed to make the product that, at the end of its life, is discarded. The circular economy model, on the other hand, is based on the 7R principle: reduce, reuse, recycle, repair, replace, recovery, remanufacture as shown in Figure 1.

**Figure 1.** Comparison between circular and linear economy models [9].

With a view to the circular economy and sustainable cities, household end-of-life materials (EoLHM), such as clothes or packaging, could be reused to realize acoustic panels. EoLHM can be defined as household waste materials which still possess exploitable properties, thus making them suitable for reuse. Many EoLHM still have properties when discharged and are largely available: for example, the estimated yearly global production is about 241 million tons for cardboard and paper packaging [10,11], 380,000 million tons for plastic packaging [12], and 92 million tons for textile waste [13] of which only the 12% is recycled [14]. The large availability of EoLHM and the problems related to recycling can support the approach of converting them into acoustic panels: as suggested in [15], this avoids the generation of waste, reduces the footprint due to raw materials extraction, and makes them accessible also to vulnerable population that can not afford commercial acoustic materials. Moreover, since EoLHM are largely available, this approach could incentive buildings renovation and facilitate the achievement of the energy and environmental international goals set by the European Parliament [16,17].

The study presented in this paper explores the possibility of converting EoLHM into panels featuring interesting acoustic properties, and it is focused mainly on those EoLHM that can be reused without any type of processing so that they are directly available to lowincome households. Indeed, any treatment would entail costs that would affect end-users and, consequently, the vulnerable population may not be able to afford them. Specifically, the aim of the experimental analysis presented in this paper is to understand which EoHLM can be used to make acoustic panels of limited thickness, and how these materials can be assembled to meet both sound insulation and sound absorption requirements. Since the panels are intended for the most disadvantaged population, they must be easy to be assembled and installed, so that these people, once trained, can collect the necessary EoLHM and assemble the panels independently. The first part of the study addresses the state of the art regarding the reuse of EoLHM to realize acoustic panels. There are several studies in the literature that address the recycling of these materials, but only a limited number analyzes their possible reuse. This highlights that the approach proposed in this paper is quite innovative. In the second part of the paper, EoHLM suitable for low-cost acoustic insulation panels for indoor comfort improvement are investigated by means of experimental tests performed with a 4-microphone impedance tube technique. Five sets of samples have been tested. In the first set of samples, the acoustic performance of egg-cartons has been evaluated. The second set of samples consists in egg-cartons coupled with fibrous materials and metal elements. In the third set of samples, cardboard has been

featured. Finally, in the fourth and fifth sets of samples, the acoustic properties of different fabrics coupled with egg-cartons and metal elements have been evaluated.

#### **2. Theoretical Background**

When a sound wave with a certain acoustic power *Wi* impinges a wall-partition, its energy is divided into three components. One portion of the power is reflected back (*Wr*), while another portion (*Wa*) is able to pass through the surface of the material. The energy that passes through the surface can be divided into two components *Wd* and *Wt*. The component *Wd* represents the part of the absorbed energy actually converted into heat due to the internal friction and viscoelastic effects. The component *Wt* represents the portion of the energy that passes through the partition and it is related to the power transmitted through the wall. The relation among incident, reflected and absorbed power is

$$\mathcal{W}\_i = \mathcal{W}\_r + \mathcal{W}\_d = \mathcal{W}\_r + \mathcal{W}\_d + \mathcal{W}\_t \tag{1}$$

as depicted in Figure 2. By dividing the single components for the incident power *Wi*, the sound reflection coefficient, the sound dissipation coefficient *δ*, and the sound transmission coefficient *τ* are defined:

$$r = \mathbb{N}\_r / \mathbb{N}\_i \tag{2}$$

$$
\delta = \mathcal{W}\_d / \mathcal{W}\_i \tag{3}
$$

$$
\pi = \mathbb{W}\_t / \mathbb{W}\_i \tag{4}
$$

**Figure 2.** Decomposition of a sound wave *Wi* impinging a wall into its reflected *Wr*, dissipated *Wd* and transmitted *Wt* components. The sum of the dissipated and transmitted components represents the absorbed *Wa* component.

Building acoustics usually investigates the frequency range going from 100 Hz to 3150 Hz [18]. The reason of the 100 Hz lower frequency limit is that, in general, the first speech tones range between 100 Hz and 125 Hz for men, and they are an octave higher for women. As concerns the emission due to traffic noise, the encompassed frequency range is 125–2500 Hz and depends on the vehicles' speed. The two main properties to be considered for indoor acoustic comfort are the apparent sound absorption coefficient *α* and the sound transmission loss *TL*. The apparent absorption coefficient is defined as:

$$\alpha = 1 - r \tag{5}$$

and it represents the portion of incident energy absorbed (or not reflected) by the partition. In practice, sound absorbing materials and structures reduce the possibility of multiple reflections and are able to 'clean' the indoor acoustic environment from the annoying effects of reverberation. Sound absorbing materials and structures can be classified as porous

materials, acoustic resonators (Helmholtz resonators that include perforated and microperforated panels respectively), vibrating panels and mixed systems (Figure 3).

**Figure 3.** Representation of (**a**) a double partition made of two rigid leafs with an internal porous layer, (**b**) Helmholtz resonator and (**c**) mixed system—multiple resonator.

#### *2.1. Sound Absorption*

The absorbing performance of a given material depends on the angle of incidence of the sound wave, on the frequency, on the material properties and thickness, and on the surface finishing. The absorption coefficient is usually measured in single reverberation rooms, that allows an evaluation of the absorption properties in diffuse field, or by two or four microphones impedance tubes that evaluate only the properties for a sound wave impinging normally on the sample surface. In spite of this, the impedance tube requires small samples and, for this reason, it is particularly suitable during the research and development phase. To easily compare the properties of different materials, the weighted Noise Reduction Coefficient (*NRC*) is one of the most used indicators [19].

$$NRC = \frac{a\_{125} + a\_{250} + a\_{500} + a\_{1000} + a\_{2000}}{5} \tag{6}$$

The *NRC* summarizes the absorption characteristic of a material through a single value ranging between 0 (perfectly reflective material) and 1 (perfectly absorbent material).

#### 2.1.1. Porous Materials

This kind of materials is characterized by high porosity, low density and, if possible, a high surface area. Porous materials include fibrous, cellular (foams) and granular materials. The absorption properties depend on a number of parameters including flow resistivity and tortuosity. The dissipation of sound energy is due to three phenomena that are the friction between air and material fibers, the compression and decompression of air, and viscous effects [20,21].

In the literature, several empirical and theoretical models have been proposed for the prediction of porous materials sound absorption. One of the first available models was proposed by Delany-Bazley [22] and requires only the flow resistivity *σ* as an input parameter, but since it neglects the thermal conductive effects, it is accurate in the 0.01 < (*ρ*<sup>0</sup> *f* /*σ*) < 1 range only, where *ρ*<sup>0</sup> is the air density and *f* is the sound frequency [23]. More accurate but, at the same time, more complex models were defined by several authors [24–27]. One of the most popular models was proposed by Johnson-Champoux-Allard (JCA) and takes into account the flow resistivity *σ*, open porosity *φ*, tortuosity *α*∞, the viscous characteristic length Λ and the thermal characteristic length Λ [20].

#### 2.1.2. Acoustic Resonators

A Helmholtz resonator consists of a cavity with one or more holes and necks, as represented in Figure 3b. The air inside the neck behaves like an oscillating piston (mass) while the air in the cavity behaves like an elastic element (spring). When the resonance frequency of the mass-spring system is equal to the frequency of the incident wave, the resonator express its maximum absorption. For these systems, the resonance frequency *f*<sup>0</sup> is defined as:

$$f\_0 = \frac{c\_0}{2\pi} \sqrt{\frac{r^2}{V(l + \frac{\pi}{2r})}}\tag{7}$$

where *c*<sup>0</sup> is the speed of sound in air, *r* is the radius of the hole, *V* is the volume of the cavity and *l* is the length of the neck. However, Helmholtz resonators do not express any sound absorption outside the resonance frequency region. The transmission loss *TL* and the absorption coefficient of a Helmholtz resonator are defined as:

$$TL = -10\log\left|\frac{p\_t}{p\_i}\right|^2 = 20\log\left|1 + \frac{1}{2}\frac{S\_0\rho c\_0}{SZ}\right|\tag{8}$$

$$\alpha = 1 - \left| \frac{p\_t}{p\_i} \right|^2 = \frac{4c\_0 \rho\_0 \frac{S}{S\_0} Z\_{Rc}}{(\frac{S}{S\_0} Z\_{Rc} + \rho\_0 c\_0)^2 + \frac{S^2}{S\_0^2} Z\_{Im}} \tag{9}$$

where *S* is the cross-sectional area of the neck, *S*<sup>0</sup> is the total areas of all necks, *Z* is the acoustic impedance of the resonator and represent the ratio between pressure amplitude and the particle velocity at the interface of the resonator [28].

#### 2.1.3. Vibrating Panels

Another type of sound absorbing mechanism is the one involving vibrating panels placed at a distance from a rigid wall. Vibrating panels are thin, rigid and flat leafs and the absorption mechanism is again of the mass-spring type. The resonance frequency, at which the maximum absorption occurs, is determined as:

$$f\_0 = 60 / \sqrt{\mu'' d} \tag{10}$$

where *μ* is the mass per unit area of the panel and *d* is the thickness of the panel as depicted in Figure 3c.

#### *2.2. Sound Transmission Loss*

The Transmission Loss represents the ability of a structure to block the sound propagation in neighboring ambient and is defined as

$$TL = 10 \cdot \log \frac{1}{\tau} \tag{11}$$

Materials characterized by a low transmission coefficient have a high *TL*. The sound insulation properties depend mainly on the mass per unit area of the structure, the angle of incidence and the frequency of the impinging wave. Other factors influencing the transmission of the sound are the nature of the partition (single, double), the internal losses and the boundary conditions [29]. The acoustic insulation performance of a homogeneous wall can be divided into four regions as shown in Figure 4.

**Figure 4.** Sound transmission loss of a single panel: (1) stiffness controlled region, (2) resonances region, (3) mass law region, (4) coincidence region.

In the low frequency region, the *TL* is governed by the material stiffness and it decreases 6 dB/oct. The behavior of the panel is then dominated by the modes of the specimen, which depend on the elastic and geometric properties of the wall. When the modal density is sufficiently high, the wall behaves according to the mass law [30]. In this region, the transmission loss has a linear trend and increases 6 dB/oct. The mass-law region is limited by the coincidence effect that occurs when the wavelength of the sound in the air is the same as the wavelength of the bending waves in the partition. In this region the partition does offer a weak opposition to sound propagation. The coincidence frequency *fc* of a homogeneous board is related to its size, thickness, Young's modulus, and surface density:

$$f\_c = \frac{c^2}{2\pi} \sqrt{\frac{\mu''}{D}} \tag{12}$$

where *D* is the bending stiffness calculated as

$$D = E \cdot I\_b \tag{13}$$

*E* is the Young modulus, and *Ib* is the moment of inertia. The bending stiffness can be computed also for complex structures, once the Young's modulus and the moment of inertia of the elements are known [31].

The sound insulation of a wall can be significantly increased if it is built as a multiple structure. One common way to increase the sound insulation of a partition, without increasing the mass per unit area, is to build it with two or more layers separated by an air gap, possibly filled by sound absorbing material. In this case the wall behaves like a multiple mass-spring-mass system. When an acoustic wave passes through such a construction, the total transmission factor *τtot* is:

$$
\pi\_{tot\_{n=1}}^{n=N} = \pi\_1 \cdot \pi\_2 \cdot \dots \cdot \pi\_n \tag{14}
$$

in which the assumption is that the *N* layers have a transmission factor *τn*. The equation holds at sufficiently high frequency (above the mass-spring-mass frequency *f*0), for large enough distances between the layers and when the damping of the gap, in the form of sound absorbing material, is sufficiently high. Below the mass-spring-mass resonance standing waves between the layers modify the transmission factor. The most common case is the one featuring a double wall. If the mass law holds, then the following equation can be applied:

$$R\_{\perp} = 20 \log(\mu'' \cdot f) - 42 \text{ for } f < f\_{\mathcal{E}} \tag{15}$$

In case of a finite double wall, the sound transmission loss can be computed as:

$$R\_{\perp}^{double} = 20 \log \left[ \frac{2 \cdot \pi \cdot f \cdot \mu\_1^{\prime\prime}}{(2\rho\_0 \cdot c\_0)} \right] + 20 \cdot \log \left[ \frac{(2 \cdot \pi \cdot f \cdot \mu\_2^{\prime\prime})}{(2\rho\_0 \cdot c\_0)} \right] \text{ for } f < f\_c \tag{16}$$

The mass law has a lower bound given by the lower mechanical resonance of the system. This resonance corresponds to the mass-spring-mass resonance of the wall, where the air enclosed in the gap acts as a spring, while the walls act like two masses. A two degrees of freedom system has a resonance frequency equal to:

$$f\_0 = \frac{\pi}{2} \left( \frac{\rho\_0 \cdot c^2 \cdot (\mu\_1^{\prime \prime} + \mu\_2^{\prime \prime})}{(\mu\_1^{\prime \prime} \cdot \mu\_2^{\prime \prime} \cdot h)} \right)^{1/2} \tag{17}$$

Below the mass-air-mass frequency the wall behaves like a single wall with a total mass per unit area equal to the sum of the mass per unit areas of the two walls composing the entire wall. As concerns the coincidence effects, the discussion made for single walls also applies to double walls. For double walls, the coincidence frequency is determined by the mass per unit area and thickness of each element, while the *TL* is higher than that predicted by the mass-law for a single panel of the same mass. As suggested in [32], it can be an advantage to realize the double panel with two panels having different thicknesses to avoid that the coincidence effect takes place at the same frequency.

#### *2.3. Acoustic Performance of EoLHM in the Literature*

In this section, the acoustic performances reported in the literature of some EoLHM are collected. It is worth noting that the performances of acoustic materials deriving from agriculture have not been analyzed, because they are not directly available to disadvantaged people. Neither organic waste has been analyzed, even if in the literature several studies, such the one presented in [33], can be found.

#### 2.3.1. Textile Waste

Textile waste includes clothes, carpets, tablecloths and pieces from the textile sector. In literature, a very recurring classification is between woven (WF) and non-woven (NWF) fabrics: WF are obtained by threading fibers together perpendicularly, whereas NWF are bounded together by using heat, chemical, or mechanical treatment. Textile waste have been widely investigated from the acoustic point of view because they are largely available and their porous structure makes them suitable for acoustic absorption. The sound absorption of NWF waste was investigated in relation to the fiber content and the fiber diameter [34], and NWF shows higher sound reduction than WF [35]. A panel made of waste wool and polyamide fibers was designed in [36], and it presented a sound absorption coefficient equal to 0.91 and *NRC* equal to 0.56. The study pointed out that the sound absorption coefficient in the low-frequency range is affected by the thickness, while the volume density affects the absorption properties in the middle-frequency range. Blankets for building roofing and internal walls insulation were realized with polyester fabrics of different sizes and they showed an *NRC* ranging between 0.54 and 0.74 [37]. The study in [38] investigated the correlation between the humidity content and the transmitted wave through cotton fabric: for moisture content between 0 and 100%, the transmitted wave ranges between 31% and 7%.

#### 2.3.2. Cardboard

According to [39], cardboard panels from the packaging industry present promising acoustic insulation performance but slightly lower than common insulation panels. Cardboard performance intended as the combination of acoustic properties, transportability, lightweight, cost and recyclability was evaluated for several cardboard design options in [40], and honeycomb panels filled with cellulose fiber presented the best performance. To evaluate the conservation status of the beer during transport by trucks, the acoustic

properties of beer packaging was investigated in [41]. The sound absorption of a cover made of porous sponge and cardboard was 0.58 [42]. In [43], sound absorber obtained by mixing recycled paper and a blowing agent showed a *NRC* of 0.75.

#### 2.3.3. Plastic Bottles and Metal Cans

Plastic bottles of different sizes (500 mL, 750 mL, 1 L, 1.25 L and 2 L) can be easily found on the market [18]. In the literature, bottles have been analyzed mainly from the structural point of view when incorporated in the construction of walls since, if compared to ceramic and concrete blocks, they are faster to build, require less water and cement and do not produce waste [44]. For these reasons, plastic bottles are continuously investigated. According to [45], polyethylene terephthalate-based material shows good sound-absorbing characteristics, especially at high frequencies. Panels made of recycled PET and sheep wool showed an absorption coefficient *α* higher than 0.7 in the range of 50–5700 Hz regardless of the humidity content [46]. The sound absorption of light-soft-plastic bottles with net capacity from 7 to 2000 mL is affected by the capacity in the range of 100 and 1000 Hz [47]. Plastic bottles are often used to hold materials that lack structural strength. For example, a slightly lower *TL* than those of traditional construction materials was measured for PET bottles filled with plastic bags [18]. End-of-life PET bottles were incorporated in a wall 12 cm thick and tests showed a reduction between 29.8 dB and 55.8 dB than the wall without bottles [48]. Additionally, the acoustic properties of aluminum cans were investigated [49]: a sandwich panel made of polystyrene, pressed aluminum cans, rockwool and corrugated cardboard showed a better acoustic performance than gypsum panels, but lower than panels made of rock-wool and egg-boxes.

#### 2.3.4. Egg-Boxes and Trays

Egg-boxes and trays can be made of different materials such as plastic, recycled paper, cardboard, but what distinguishes them is their shape. For a long time, egg-cartons have been considered good sound absorbing materials, and they have been widely used for this purpose since they are inexpensive, easy-to-install and easily available [50]. However, their acoustic performances have been recently questioned. It was pointed out that egg-boxes provide good sound absorption only at high frequencies, their *NRC* equal to 0.4 is too low for considering them sound absorbing elements, and the sound absorption coefficient profile is irregular [51]. The experimental tests presented in [52] showed that the sound absorption coefficient of egg-boxes and fruit trays is affected by the material, orientation of the boxes, and by if they are closed or open. Experimental tests showed that egg-cartons can reduce the reverberation time at mid-frequency [53]. In spite of this, researches have been looking for a way to improve their sound performance by coupling them with other materials. A sound absorbent made of egg-boxes pulp showed an optimized *NRC* equal to 0.5 [54]. A non-standardized test method showed that filling egg-boxes with mineral wool blocks a percentage of sound ranging between 14.42% and 17.71% depending on the frequency. Egg-boxes were filled with shredded rice straw paper and textile waste [50], and with polyurethane foam [52]. The panels proposed in [50] showed higher sound absorption coefficients than common egg-boxes cartons at all frequencies, and those presented in [52] featured a *NRC* equal to 0.87.

#### **3. Methods**

The review presented in the previous section shows that, even if a limited number of papers investigated the EoLHM acoustic performances, these materials have exploitable properties for the improvement of the indoor acoustic quality. An ideal panel suitable to be used as a façade element posses both good sound absorption and high transmission loss. As concerns the transmission loss, it must be remembered that the final acoustic performance will also depend on the basic wall on which they will be installed. In this study, the acoustic properties of different panels, realized by coupling different EoLHM, are experimentally investigated to understand whether further studies are required. Tests have

been performed on samples made of easily obtainable EoLHM such as cardboard, textile waste, egg-boxes, metallic elements and their combination. Since the acoustic conditions of the environment in which these panels will be installed are unknown, configurations with high sound absorption coefficient and good *TL* are considered interesting and worth to be further investigated. Since this analysis is exploratory, the experimental tests have been performed by means of the impedance tube method that requires small samples and gives reproducible results. However, this technique allows the determination of the properties for sound waves impinging normally on the sample surface.

The experimental investigation of the acoustic properties was performed following the standard procedure given by the ASTM E2611 [55] that required the use of a fourmicrophone impedance tube (Figures 5 and 6). This device consists of two tubes of equal internal cross section connected to a test sample holder. Four microphones were placed along the tube (two on either side of the specimen). A source emitting a pink noise was placed at one end of the tube. A multi-channel Fast Fourier Transform (FFT) analyzer acquired the signals captured by the microphones. The second endpoint of the tube could be equipped with an anechoic or a reflecting termination, allowing us to perform the tests with two different boundary conditions. The pressure and particle velocity of the traveling waves and of reflected waves could be determined by means of a MATLAB script implemented on the basis of the E2611 ASTM standard [55]. The frequency range investigated went from 100 Hz to 3150 Hz.

**Figure 5.** Schematic drawings of a four-microphones impedance tube. *A* represents the energy emitted by the loudspeaker, *C* is the component that crosses the sample, *D* is the component reflected by the termination, and *B* is the component reflected by the sample and/or that crosses the sample after being reflected by the termination.

**Figure 6.** Impedance tube used for the determination of the acoustic properties of EoLHM.

Defining the wave number in air, *k* = 2*π* · *f* /*c*0, the traveling and reflected components of the plane wave propagation in the tube (A, B, C and D) can be calculated using the following correlations, once the complex acoustic transfer functions *Hi*,*ref* between the *i*th microphone and the reference microphone are measured:

$$A = 0.5 \times j(H\_{1,ref} \text{e}^{-j \text{k} \text{L}\_1} - H\_{2,ref} \text{e}^{-j \text{k} (\text{L}\_1 + \text{s}\_1)}) / \sin(\text{ks}\_1) \tag{18}$$

$$B = 0.5 \times j(H\_{2,ref}e^{+j\mathbf{k}\left(L\_1 + s\_1\right)} - H\_{1,ref}e^{+j\mathbf{k}\left(L\_1\right)}) / \sin\left(\mathbf{ks\_1}\right) \tag{19}$$

$$\mathcal{C} = 0.5 \times j(H\_{3,ref}e^{+j\mathbf{k}(L\_2 + s\_2)} - H\_{4,ref}e^{-j\mathbf{k}(L\_2)}) / \sin(\mathbf{ks\_2}) \tag{20}$$

$$D = 0.5 \times j(H\_{4,ref}e^{-j k L\_2} - H\_{3,ref}e^{-j k (L\_2 + \varepsilon\_2)}) / \sin(k \varepsilon\_2) \tag{21}$$

where, in the case at hand, microphone 1 was selected as the reference microphone. For a given boundary condition, it is possible to determine the acoustic pressure *p* and the particle velocity *u* on each face of the specimen using the following equations:

$$p\_0 = A + B \qquad \qquad p\_d = \mathcal{C}e^{-jkd} + De^{+jkd} \tag{22}$$

$$
\mu\_0 = \frac{A - B}{\rho\_{0\text{f}\odot 0}} \qquad \mu\_d = (\text{C}e^{-j\text{kd}} - \text{D}e^{+j\text{kd}})/\rho\_0 c\_0 \tag{23}
$$

where *ρ*<sup>0</sup> is the density of air. In general, the elements of a transfer matrix *T*, putting into relation pressures and particle velocities at either side of the specimen under test, can be calculated from the acoustic pressures and particle velocities measured during two different experimental sessions performed using an anechoic (*a*) and a reflecting (*b*) termination:

$$\begin{aligned} \begin{bmatrix} T \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} = \begin{bmatrix} \frac{p\_{0a}u\_{db} - p\_{0b}u\_{da}}{p\_{da}u\_{db} - p\_{db}u\_{da}} & \frac{p\_{0b}p\_{da} - p\_{0a}p\_{db}}{p\_{da}u\_{db} - p\_{db}u\_{da}} \\\\ \frac{u\_{0a}u\_{db} - u\_{0b}u\_{da}}{p\_{da}u\_{db} - p\_{db}u\_{da}} & \frac{p\_{da}u\_{0b} - p\_{db}u\_{0a}}{p\_{da}u\_{db} - p\_{db}u\_{da}} \end{bmatrix} \end{aligned} \tag{24}$$

The absorption coefficient can be computed as:

$$\alpha = 1 - \left| \frac{T\_{11} - \rho c T\_{21}}{T\_{11} + \rho c T\_{21}} \right|^2 \tag{25}$$

The sound transmission loss *TL* is expressed as:

$$TL = 20 \times \log\_{10} \left| \frac{T\_{11} + (T\_{12}/\rho c) + T\_{21}\rho c + T\_{22}}{2e^{jkd}} \right| \tag{26}$$

#### *Samples*

The analyzed samples were made putting together different types of EoLHM and are shown in Figures 7 and 8, where the sequence described in the caption starts with the material nearest to the sound source. The layers of the samples were only placed close to and not connected to each other. The samples were 50 mm long, except for samples 13 and 14 that had a length of 24.2 mm, and sample 26 that was 100 mm long. The weight of the samples is reported in Table 1. For each sample, three repetitions were performed and the results of the experimental tests were averaged. This has made it possible to evaluate how manual skills influenced the panel acoustic performances.

**Table 1.** Weight in grams of the samples.


**Figure 7.** Configurations analyzed with the impedance tube: (**a**) samples made of eggboxes and polyester, and (**b**) samples made of egg-box, polyester and a metallic element. Samples 04 and 05 are made of the same elements but in sample 04 the polyester faces the loudspeaker. Sample 08 has an additional perforated cardboard layer.

**Figure 8.** Configurations analyzed with the impedance tube: (**a**) samples made of several layers of cardboard, (**b**) samples made of clothes (one fabric at a time), (**c**) samples made of egg-boxes and clothes.

Since the acoustic properties of egg-boxes have been questioned in a number of papers [51,52,56], to analyze this aspect, the first set of samples (03, 22) was made at least by one egg-carton 2.79 g in weight made of recycled paper with a density of 355 kg/m3. In sample 03, the egg-carton facing the sound source was coupled with loose polyester. The egg-carton was perforated and the holes were less than 1 mm in diameter. This element was used for realizing the other samples that included egg-boxes, which exception of sample 22 that was made of two not-perforated spaced egg-cartons whose cavity faced the sound source.

To improve the acoustic performance, in the following set of samples (04, 05, 07, 08) the perforated egg-carton was coupled to other EoLHM such as loose polyester, a metallic element and cardboard 197 kg/m3 in density. The cardboard was made of two external linear boards 0.11 mm thick and an internal board with 130 flutes/m. The metal element was a steel sphere with an external diameter of 13 mm, and a weight of 8.95 g.

In particular, samples 04 was made of a box-carton, polyester and a metallic sphere. Sample 05 was similar but had mounted reverse. Samples 07 and 08 were realized to exploit the double panel characteristics. Indeed, a plane wave impinging a double-panel system saw the impedance of the panel closest to the sound source, the impedance of the airspace, the impedance of the second panel, and finally the impedance of the air beyond. The cavity acted as a spring element reducing significantly the TL, especially at higher frequencies. In the cavity the absence of absorptive material contributed to the transmission of sound, while the addition of damping elements such as fibrous materials attenuated the modes of the cavity. For this reason, in samples 07 and 08 cardboard layers were added to create a sort of cavity. In sample 07, to reduce the permeability, only a cardboard was added on the back of the sample. In sample 08 a perforated layer was added in the front of the sample: the first panel being perforated allowed the passage of a certain quantity of sound and behaved like a Helmholtz resonator.

To characterize cardboard panels, a third set of samples (13, 14, 17) was prepared. In the last few years, cardboard is largely available at domestic level as a result of the ecommerce. For this reason, highlighting its acoustic properties would encourage its conversion into a building element. Since very sound reflective materials could have a negative impact on acoustic indoor comfort, especially in very crowded ambient such as homes in disadvantaged contexts, the first cardboard layer of the samples was perforated to increase the sound absorption capability. Sample 13 and 14 were made of nine cardboard elements: the first two were perforated, the internal five disks presented a central hole 17 mm in diameter, while the last two disks were not perforated. In sample 14 a metal sphere was housed in the central layers. The weight of the whole cardboard disk was 0.77 g. Sample 17 was similar to samples 08 except for the presence of the metal sphere and, consequently, the weight of the sample was (13.33 g for sample 08, and 4.36 g for sample 17).

In the fourth set of samples (26–32) different fabrics were tested. Samples 26 and 27 were made of cotton, while samples 28, 29 and 30 were respectively made of polyester, plush cotton, and viscose. Since the fabrics had no structural strength, it was necessary to fold them inside the impedance tube. This revealed that the installation of the fabrics was strongly influenced by the operator's skills.

In the fifth and last set of samples (32–36), textile waste was coupled with other EoLHMs to improve their insulation performance. To reach a certain degree of stiffness, these samples were realized with a perforated egg-carton facing the sound source. Sample 32 was made of cotton, while the others are made of viscose. By means of tests performed on samples 34–36, the influence of metallic elements was investigated. In particular, a metallic sphere was inserted in sample 34, while a metal cap 1.75 g in weight and 26 mm in diameter was included in samples 35 and 36 but only in sample 35 there was contact with the egg-carton.

#### **4. Results and Discussion**

The results of the tests described in the previous section are reported in Figures 9–13 showing the sound absorption coefficient *α* and the transmission loss *TL* obtained by means of the impedance tube measurements. For each sample, the NRC has been calculated and reported in Table 2.

**Table 2.** *NRC* of the samples tested in this paper.


Figure 9 shows that, for samples 03 and 22, the most interesting sound insulation performance was given by sample 22, with a *TL* following the mass law up to 800 Hz and then a behavior typical of double walls, with a coincidence frequency around 1800 Hz. Sample 03 was not able to reach the same performances. As regards the absorption coefficient, it was characterized by wide peaks at given frequencies due to holes in the eggcardboard behaving like Helmholtz resonators. In the case of sample 03, the peak around 700 Hz was very wide due to the presence of sound absorbing polyester fibers inside the main volume. This result suggests that to reach a good *TL* it was important to arrange the egg boxes upside down, but they still had to be coupled with other materials to improve their performances. The presence of the holes improved the absorption characteristics.

**Figure 9.** Transmission loss (**a**) and absorption coefficient (**b**) of samples made of at least on egg-box and polyester.

**Figure 10.** Transmission loss (**a**) and absorption coefficient (**b**) of samples made of egg-box, cardboard, polyester and a metallic element.

**Figure 11.** Transmission loss (**a**) and absorption coefficient (**b**) of samples made of several layers of cardboard.

**Figure 12.** Transmission loss (**a**) and absorption coefficient (**b**) of samples made of clothes (one fabric at a time).

**Figure 13.** Transmission loss (**a**) and absorption coefficient (**b**) of samples made of egg-boxes and clothes.

The *TL* curves just analyzed were very similar to the curves obtained for the second set of samples and reported in Figure 10. This behavior can be explained by the nature of the samples which were built with the same elements: the egg-box, the polyester foam and steel spheres. The only variables were the orientation of the samples and the presence of a cardboard disk. The TLs of samples 04 and 05 were characterized by the typical mass law behavior due to the single egg cardboard, except a dip around 1200 Hz. Samples 04 and 05 showed an interesting absorption coefficient, and sample 04 featured better performances since it did not present valleys at high frequency. However, since these samples were permeable to air, the *TL* was very weak. Reducing the permeability, in samples 07 and 08 the additional cardboard layer caused an increase of the curve slope due to the massspring-mass behavior of the layers. Sample 08 exhibited very interesting performances at low frequencies and the highest *TL* of the group also at high frequencies. In sample 08 the additional perforated cardboard layer facing the sound source captured sound energy at some specific frequency bands. Part of the energy that passed through the egg-box was absorbed by the fibrous material. Finally, the final last cardboard reduced the transmitted energy. For these samples, the absorption coefficient was characterized by multiple peaks typical of the resonators featured in the cardboard portions. The width of the peaks depended again on the presence of polyester fibers. The best performance belonged to sample 08, characterized by a rather good sound absorption coefficient also at high frequency, with a maximum of 0.95 at 2 kHz.

As regards the third set of samples, Figure 11 shows very similar *TL* curves for all the samples. This group of TL was the highest among the entire group of tested materials. Such behavior is due to the high density of the samples that made the samples similar to sandwich materials featuring non compressible cores. For this reason the trend was characterized by a mass-law behavior followed by the typical coincidence dip. For samples 13 and 14, the sound absorption coefficient was marked out by a maximum around 600 Hz due to the resonance of the Helmholtz resonators featured in the cardboard. For sample 17, the graph of the sound absorption was very similar to the one of sample 08 (having a very similar structure), with three peaks reaching a value of 0.8.

The results of the fourth set of test samples, which were made of fabrics, are shown in Figure 12. The absorption curves were similar for all the samples and typical of porous materials. The best performance was given by sample 30 having the highest density of the group. Additionally, sample 26 showed a good performance but since it was 100 mm thick it could not be directly compared with the other samples. However, this result shows that better performances could be achieved by increasing the thickness of the panels. Additionally, in this case, the sound absorption coefficient had a shape typical for porous materials, featuring an *S* shape, with low values at low frequencies and values approaching 1 at high frequencies. By comparing the *NRC* measured for samples 26 and 27 and reported in Table 2, it emerged that the operator skills affected the acoustic performances of the panels. Indeed, even if the samples were made of the same material and the same *NRC* should be obtained, a higher value of *NRC* was measured for the thinner sample (26) and this is probably due to the assembling mode.

The fifth and last set of samples was a combination of fabrics, egg-boxes and metal parts. As can be observed in Figure 13, all the *TL* curves had very similar trends. If compared to the *TL* in Figure 12, for samples 32–36, values were generally higher at low frequency due to the higher mass per unit area of the samples. Sample 36 had the best performances for this group and this is probably due to the fact that the metal element could vibrate because it was not in contact with the rigid egg-carton. As concerns the sound absorption coefficient, the behavior was dominated by the Helmholtz resonator featuring a peak around 400 Hz followed by an increase of the coefficient due to the presence of the tissues.

#### **5. Conclusions**

The study presented in this paper has shown that EoLHM, such as cardboard, eggboxes, clothes and metal elements, can be reused to realize low-cost acoustic panels for the improvement of the indoor comfort. Since these panels are easy realizable and cheap, they can be used in disadvantaged contexts where low-income people live and can not afford commercial acoustic panels. By wisely coupling EoLHM, good acoustic performances can be obtained for panels of limited thickness. Measurements performed with the impedance tube technique have shown that samples made of fabrics present a sound absorption coefficient greater than 0.8 in the range 300–3500 Hz. The higher insulation performance has been measured for samples made of perforated cardboard that present a *TL* of

25–30 dB in the range 100–300 Hz, and 30–40 dB in the range 300–2000 Hz. To reach interesting performances from both the insulation and the absorption point of view, it is necessary to couple egg-cartons, cardboard, polyester and metal elements. For this configuration, *NRC* is higher than 0.54 and the *TL* varies between 25 and 40 dB in the range 250–2000 Hz. Since in this study only the performance related to normal waves has been analyzed, in future investigation the most performing configurations will be tested in a reverberation room. Since the acoustic panels will be realized by not-skilled personnel and the manual skills affect the panels acoustic performance, it will be necessary to provide courses and guidelines for illustrating how the panels must be realized and installed.

**Author Contributions:** All authors conceived the presented idea. M.N. and E.L. designed the experimental campaign and wrote the paper with input from all authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like thank the Department of Mechanical and Industrial Engineering of University of Brescia for funding the research through the MetATer PRD project.

**Acknowledgments:** All the authors would like to thank the Applied Acoustics Laboratory of University of Brescia and Edoardo Alessio Piana for the opportunity to carry out the experimental tests and for the support in designing the experimental campaign.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Optimization of Shunted Loudspeaker for Sound Absorption by Fully Exhaustive and Backtracking Algorithm**

**Zihao Li, Xin Li and Bilong Liu \***

School of Mechanical & Automobile Engineering, Qingdao University of Technology, No. 777 Jialingjiang Road, Qingdao 266520, China; zihao\_li0401@163.com (Z.L.); jz03-4lx@163.com (X.L.) **\*** Correspondence: liubilong@qut.edu.cn

**Abstract:** The shunted loudspeaker with a negative impedance converter is a physical system with multiple influencing parameters. In this paper, a fully exhaustive backtracking algorithm was used to optimize these parameters, such as moving mass, total stiffness, damping, coil inductance, force factor, circuit resistance, inductance and capacitance, in order to obtain the best sound absorption in a specific frequency range. Taking the maximum average sound absorption coefficient in the range of 100–450 Hz as the objective function, the optimized parameters of the shunted loudspeaker were analyzed. Simulation results indicated that the force factor and moving mass can be sufficiently reduced in comparison with that of a typical four-inch loudspeaker available on the market. For a given loudspeaker from the market as an example, the four optimized parameters of the shunted loudspeaker were given, and the sound absorption coefficient was measured for verification. The measured results were in good agreement with the predicted results, demonstrating the applicability of the algorithm.

**Keywords:** shunted loudspeaker; optimal sound absorption; fully exhaustive method

#### **1. Introduction**

Low-frequency sound absorption within a limited space is always a challenge in noise control engineering. Traditional passive acoustic structures usually have the disadvantage of being large in size, but active noise control technology also has drawbacks, such as instability and high cost. In recent years, the semi-active structure of a shunted loudspeaker (SL) for sound absorption has attracted much attention. For an SL with a negative impedance converter (NIC), the circuit parameters, such as resistance, capacitance and inductance, are transformed due to the negative impedance converter. This can effectively adjust the acoustic impedance of the coupled system to match that of the air in a wide frequency range [1]. Initially, Forward [2] proposed a preliminary experiment on the feasibility of using shunted damping in optical systems. Lissek et al. [3–6] introduced shunt circuits to loudspeakers and used the SL to control the acoustic impedance of walls for indoor sound absorption. Good sound absorption for low frequencies can be achieved in a relatively narrow frequency band. In their later research, analogous analysis, experimental optimization of the SL and active control theory were also carried out. Due to the low-frequency sound absorption properties of the SL, many structures relevant to the SL that have better sound absorption performance have been reported [7–10].

Some references can be found for the optimal design of an SL. Lissek et al. [5] established a low-order polynomial function and the effect of four parameters on sound absorption was investigated by using the response surface method (RSM). These four parameters were the moving mass of the loudspeaker, the enclosure volume, the filling density of mineral fiber within the enclosure and the electrical load value to which the loudspeaker was connected. Rivet et al. [11] introduced the SL for interior damping optimization and they determined the interior eigenfrequency by using a finite element model established in COMSOL Multiphysics. They also calculated the optimal location and orientation of the

**Citation:** Li, Z.; Li, X.; Liu, B. Optimization of Shunted Loudspeaker for Sound Absorption by Fully Exhaustive and Backtracking Algorithm. *Appl. Sci.* **2021**, *11*, 5574. https://doi.org/10.3390/ app11125574

Academic Editor: Yoshinobu Kajikawa

Received: 9 May 2021 Accepted: 13 June 2021 Published: 16 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

loudspeaker by establishing the linear equations of the system. Liu et al. [12] applied the SL to the pipe by means of a polar configuration of the system's characteristic equations. The optimal resistance, inductance and location of the SL were derived. This method effectively improved the insertion loss of the pipe. Zhang et al. [13] analyzed the effect of the circuit resistance, inductance and capacitance (RLC) on the acoustic impedance and absorption coefficient of the SL in detail. They provided an experimental procedure for achieving effective broadband sound absorption from the low to the middle frequency range. An array of 64 SLs was experimentally investigated by Qiu et al. [14], and the optimal array alignment spacing, to control 100 Hz and 200 Hz tone noise, was also discussed.

The loudspeaker in a reported SL is oriented for sound generation. This means that this type of loudspeaker would not be suitable for optimal sound absorption due to its large force factor and moving mass. Designing a loudspeaker from the perspective of sound absorption has not been reported in the literature. To achieve this task, the loudspeaker and shunt circuit parameters must be taken into account. Since the SL is a coupled field consisting of electrical, mechanical and acoustic components, the system contains a large number of parameters and potential interactions among these parameters. The problem of multi-parameter optimization is rather complex.

The fully exhaustive backtracking algorithm (EBA) is a programming method frequently used in programming design. EBA is often applied to solve the problems that cannot be solved by conventional mathematical methods [15]. The fully exhaustive algorithm allows multivariate functions, with potential interactions, to be solved numerically according to a combined enumeration [16]. After the multi-dimensional database is created by the fully exhaustive method, the backtracking algorithm is then used to search for the target value by using loop traversal according to the optimal conditions [17]. Genetic algorithms (GA) and simulated annealing algorithms (SAA) are stochastic optimization algorithms that are based on probabilistic convergence [18]. In the optimization process of multi-peaked objective functions, GA and SAA may converge to a local optimal solution prematurely, and there is no effective quantitative analysis method for the convergence and reliability of the solution [19,20]. The global search feature of the EBA can effectively avoid these disadvantages. Although the EBA has the advantage of a simple computational process, it usually has the disadvantage of requiring a large amount of computing resources. In SL optimization problems, the amount of computation required is very limited; therefore, the EBA is well suited for SL multi-parameter optimization.

In this paper, six main parameters of the SL, namely moving mass Δ*Mm*, system stiffness Δ*K*m, force factor *Bl*, total resistance Δ*R*, total inductance Δ*L* and capacitance *C*e, are considered in an optimization algorithm. In the following section, the principle of the SL is introduced briefly; then, an optimized sound absorption algorithm based on a six-dimensional EBA is described, and the simulation and analysis of the loudspeaker parameters suitable for sound absorption are demonstrated. For a given loudspeaker from the market, the experimental method to determine the key parameters of the loudspeaker by an impedance tube is provided, and, finally, the optimization results of the four parameters are verified by an experiment.

#### **2. Theoretical Model of an SL**

The layout of a typical SL is shown in Figure 1 and the technical date of the loudspeaker used in the experiment are listed in Table 1. The SL with an NIC is assembled at the end of the impedance tube, and the effective absorption can be achieved after reasonable adjustment of the electrical parameters. From an energy perspective, it can be understood that the sound energy is dissipated in the form of mechanical and electrical energy, reducing the reflected sound energy and achieving the purpose of sound absorption.

**Figure 1.** Schematic of the shunted loudspeaker with an NIC.

**Table 1.** Technical data for the loudspeaker Hivi–M4N.


When the SL is in an open-circuit state, it can be considered a single-degree-of-freedom, second-order system, which consists of stiffness, mass and damping [21]. The mechanical impedance under the case of an air cavity of depth *D* can be expressed as:

$$Z\_{\rm m} = \delta\_{\rm m} + j\omega M\_{\rm m} + \frac{1}{j\omega K\_{\rm m}} + \frac{\rho\_0 c\_0 A}{i \tan(kD)} \tag{1}$$

where *ω* is the angular frequency, *c*<sup>0</sup> is the speed of sound in air, *Mm*, *Km*, *δm* are the moving mass, suspension diaphragm stiffness and damping of the moving-coil loudspeaker, respectively. *Zair* = *ρ*0*c*0*A* is the acoustic impedance of air, and A is the cross-section area of the impedance tube. *k* is the wavenumber and *D* is the cavity depth, where a more specific impedance expression can be obtained after making a second-order approximation to tan (*kD*) <sup>−</sup><sup>1</sup> [22]:

$$Z\_m(\omega) = \delta\_m + j\omega (M\_m + \frac{\rho\_0 A D}{3}) + (K\_m + \frac{\rho\_0 c\_0^2 A}{D}) / j\omega \tag{2}$$

From Equation (2), it can be seen in the acoustic model of the SL with an air cavity that the total stiffness is the sum of the suspension diaphragm stiffness and the air spring of the air cavity, where the second item dominates. For example, a cylindrical cavity with a depth of 10 cm and radius of 5 cm can produce a stiffness of 11 KN/m, while a four-inch loudspeaker's diaphragm stiffness is generally 1 KN/m. The total vibrating mass is the sum of the mechanical vibrating mass and one-third of the cavity air mass; the latter is usually negligible.

An NIC can generate the equivalent value of a negative electrical parameter between the in-phase input and ground [23]. It can flexibly adjust the impedance caused by a larger resistance and inductance of the loudspeaker itself, enabling impedance of the SL to match with air over a wider frequency band. When connecting the SL with an NIC, the impedance of the circuit is:

$$Z\_{\varepsilon}(\omega) = (R\_{\varepsilon} - R) + j\omega(L\_{\varepsilon} - L) + \frac{1}{j\omega \mathcal{C}} \tag{3}$$

The following is a derivation of the electrical force and impedance analogy. When the sound waves are transmitted to the loudspeaker's diaphragm, it will produce a vibration with speed of *v*. The loudspeaker's coil will cut the magnetic field of the permanent magnet, producing an induced electrical potential *Blv*. As the induced current is *Blv*/*Ze*, the electromagnetic force applied to the coil is *F*<sup>e</sup> = *B*2*l* <sup>2</sup>*v*/*Ze*. The equivalent mechanical impedance induced by the circuit can be obtained by *Z*Δ*m*(*ω*) = *F*e/*v* = (*Bl*) 2 /*Ze*(*ω*). Here, the total impedance of the SL can be expressed as:

$$Z\_{\rm sys}(\omega) = Z\_{\rm m}(\omega) + (Bl)^2 / Z\_{\rm t}(\omega) \tag{4}$$

The normal incident absorption coefficient of the SL is:

$$a(\omega) = 1 - |\frac{Z\_{\rm sys} - Z\_{\rm air}}{Z\_{\rm sys} + Z\_{\rm air}}|^2 = \frac{4Z\_{\rm air} \text{Re}(Z\_{\rm sys})}{\left[Z\_{\rm air} + \text{Re}(Z\_{\rm sys})\right]^2 + \text{Im}(Z\_{\rm sys})^2} \tag{5}$$

Equation (5) shows that the sound absorption of the SL depends on the acoustic impedance of the system. The impedance matching condition should be satisfied when the sound is completely absorbed:

$$\operatorname{Re}(Z\_{\text{sys}}) = \delta\_m + \frac{\left(Bl\right)^2 \Delta R}{\Delta R^2 + \left(\omega \Delta L - 1/\omega \text{C}\right)^2} = \rho\_0 c\_0 A \tag{6}$$

$$\operatorname{Im}(Z\_{\text{sys}}) = \omega M\_m - \frac{\Delta K\_{\text{ff}}}{\omega} - \frac{\left(Bl\right)^2 \left(\omega \Delta L - 1/\omega \text{C}\right)}{\Delta \text{R}^2 + \left(\omega \Delta L - 1/\omega \text{C}\right)^2} = 0\tag{7}$$

where <sup>Δ</sup>*<sup>R</sup>* = *Re* − *<sup>R</sup>*, <sup>Δ</sup>*<sup>L</sup>* = *Le* − *<sup>L</sup>* and <sup>Δ</sup>*Km* = *Km* + *<sup>ρ</sup>*0*c*<sup>2</sup> <sup>0</sup>*A*/*D*. The connection of the shunt circuit introduces new mechanical resistance and reactance. These parameters are mostly constant for the actual device, but the total impedance of the system changes with frequency. It is impractical to achieve an exact theoretical match, so a comprehensive optimization of sound absorption, based on experimental and theoretical calculation, is needed.

#### **3. Algorithm Model and Simulation of the EBA**

#### *3.1. Procedure of the Algorithm Model*

Loudspeakers used in the SL are for sound absorption, not sound generation. In contrast, loudspeakers available on the market are always used for sound generation. From the perspective of sound absorption, the loudspeaker in the SL must be redesigned. Through the simple analysis of the loudspeaker parameters, a certain trend of sound absorption can be obtained. However, the parameters influencing the sound absorption are coupled with each other and are difficult to analyze from a numerical point of view. The EBA is a method to obtain the ideal solution by calculating and analyzing all possible scenarios within the constraint. It can be expressed as enumerating all possible combinations of parameters within the boundaries, according to the step size of each variable, and then performing numerical analysis. Therefore, Matlab's powerful matrix solving capability can be used to perform the EBA. The optimal parameters for Δ*R*, Δ*L*, *C*e, Δ*K*m, *Mm*, *Bl* can be calculated by the EBA and then used as design values for the SL. The following describes the EBA optimization algorithm for six parameters.

#### 3.1.1. Parameter Boundary

Since the algorithm corresponds to the actual physical system, a realistic boundary condition should be set for Δ*R*, Δ*L*, *C*e, Δ*K*m, *Mm*, *Bl*. The characteristic equations of this system can be obtained by stability analysis. According to the Rouse criterion, Δ*R*, Δ*L* and *C*<sup>e</sup> must be positive, which determines the lower boundary of the Δ*R*, Δ*L* and *C*<sup>e</sup> [24]. Usually, the resistance of a typical four-inch loudspeaker does not exceed 25 Ω. Considering the actual component size, Δ*L*, *Ce* should be limited to the magnitude of *mF* and *mH*, respectively.

The upper boundary of the mechanical parameters can be set reasonably, according to the actual size of the speaker and the assembly model. Here, *M*<sup>m</sup> is limited to 10 g, Δ*K*<sup>m</sup> is limited to 1.95 × 104 N/m, and *Bl* is limited to 6 *Tm*. It is necessary to set a reasonable step size for these six parameters in this calculation. If the step size is small, it will lead to long computation time or even be impossible to compute. By using multiple iterations of the EBA to improve the computational efficiency of the program, sufficiently accurate solutions can be obtained in a relatively short time.

Under excitation of sound pressure, the SL generates an output voltage. The transfer function of the circuit section is shown in Equation (8). The maximum amplification can be obtained at the resonant frequency of the circuit. The actual output voltage at resonance can be calculated by multiplying the output signal obtained from the experimental test with Equation (9). When the actual transmission voltage can maximize the op-amp saturation value, the balancing resistance *Rb* can be determined [22].

$$G(S) = \frac{R\_b + R + sL}{R - R\_c + s(L - L\_c) - \frac{1}{sC\_c}}, \ \omega\_o = \frac{1}{\sqrt{(L\_c - L) \cdot \overline{C}\_c}} \tag{8}$$

$$|G(\omega\_0)| = \frac{\sqrt{\left(R\_b + R\right)^2 + \frac{L^2}{\left(L\_c - L\right) \cdot C}}}{R\_c - R} \tag{9}$$

#### 3.1.2. Database Creation

The nonlinearity of the damping *δ<sup>m</sup>* is usually difficult to predict accurately after assembly. To obtain an accurate theoretical calculation, the damping corresponding to the open circuit should be sampled for replacing the damping in Equation (2). The absorption coefficient at each frequency in any group within the boundary can be calculated in a nested cycle using Equations (2)–(5). Then, the absorption coefficients of each group are stored after taking the average values, and thus the database of the average absorption coefficients of the six-dimensional parameters of the SL is established.

#### 3.1.3. Optimal Results of Δ*R*, Δ*L*, *C*e, Δ*Km*, *Mm*, *Bl*

Once the database is created, the optimal average absorption coefficient values, and structural parameters such as Δ*R*, Δ*L*, *C*e, Δ*Km*, *Mm*, *Bl*, can be searched in the database by the backtracking method. Thus, the circuit parameters *R*, *L*, *Ce*, and the equivalent depth *D* of the air cavity required for the design, can be calculated. Then, the sound absorption performance of the multi-parameter SL can be analyzed.

#### *3.2. Optimal Sound Absorption for a Six-Parameter SL*

In this simulation, the frequency range was 100–450 Hz and the inner diameter of the impedance tube was 10 cm. The mechanical damping as a constant was used in calculations and had a value of 1.74 sN/m. The optimized absorption coefficient and acoustic impedance of a six-parameter SL are shown in Figure 2. As observed in Figure 2a, the optimized six-parameter SL had an excellent sound absorption coefficient close to 1 in a wide frequency range of 100–450 Hz. By the EBA optimization search, a matched acoustic resistance close to 1 and a flat acoustic reactance trending toward 0 can be obtained in the specified frequency range, as shown in Figure 2b,c, respectively. In addition, it can be found that the first resonance occurs at 120 Hz, which is due to the resistance being close to 1 and the reactance being close to 0 at this frequency. Similarly, the second resonance is located at approximately 320 Hz.

**Figure 2.** Optimized sound absorption of the SL by a six–dimensional EBA, (**a**) the sound absorption coefficient, (**b**) the specific acoustic resistance, (**c**) the specific acoustic reactance.

The optimized mechanical and electrical parameters of the SL by the six-dimensional EBA are listed in Table 2. The optimized results indicated that the moving mass and force factor were smaller than that of a typical loudspeaker available on the market. The decrease in force factor can effectively reduce the cost and the weight of the loudspeaker magnet. This would be an obvious potential benefit for practical applications. The results also revealed that the total stiffness was smaller than that of a typical SL reported in the literature. Lower stiffness suggests that the backing air cavity needs to be larger; for example, when the loudspeaker suspension diaphragm stiffness is 900 N/m, a cubic cavity with a side length of 23.5 cm is needed to provide the remaining stiffness.


**Table 2.** Parameter upper bounds and step size settings.

#### *3.3. Optimal Sound Absorption for a Five-Parameter SL*

The optimization results of the six parameters showed that the loudspeaker suitable for optimal sound absorption has the advantage of smaller *Bl*. In practical applications, the thickness of the loudspeaker should be as small as possible, which means that the *Bl* should be as small as possible. As an example, for the value of *Bl* set to 0.5 Tm, is taken into account in the algorithm; thus, the optimization procedure becomes an EBA of the remaining five variables. The optimized mechanical and electrical parameters of the SL by

the five-dimensional EBA are listed in Table 3. The theoretical absorption coefficients under this condition are shown in Figure 3, and the average absorption coefficient is up to 0.96.


**Table 3.** Parameter upper bounds and step size settings.

**Figure 3.** Optimized sound absorption of the SL by a five–dimensional EBA, (**a**) the sound absorption coefficient, (**b**) the specific acoustic resistance, (**c**) the specific acoustic reactance.

Below 150 Hz, the system resistance is less than half of that of the air, and the absolute value of sound reactance deviates from the zero point, which together leads to a lower value of the sound absorption coefficient in this frequency band. The peak of sound absorption occurs around 300 Hz, where the system resistance is close to 1 and the system reactance trends towards 0, as shown in Figure 3b,c.

The optimized total stiffness shown in Table 3 is relatively small. In this case, when the loudspeaker's suspension diaphragm stiffness is 900 N/m, a cubic cavity with a side length of 31.8 cm is required to provide the remaining stiffness. If the volume of the air cavity needs to be sufficiently reduced, the loudspeaker's suspension diaphragm stiffness must be set relatively low. Compared with Table 2, when the *Bl* becomes smaller, the total resistance and inductance are relatively reduced, and the required capacitance is increased.

#### **4. Experiment of Optimal Sound Absorption for a Four-Parameter SL**

Since there were no loudspeakers available that were specifically suitable for optimal sound absorption, a commercial loudspeaker was used for the experimental verification.

Thus, only four parameters of the SL, namely Δ*R*, Δ*L*, *Ce*, Δ*K*m, needed to be optimized. The experimental setup is shown in Figure 4 and a photograph of the setup is shown in Figure 5. The inside diameter of the impedance tube (SW422 (BSWA, Beijing, China)) was 100mm, and the noise signal generated by the computer was amplified by a power amplifier (PA50 (BSWA, Beijing, China)). When the loudspeaker is excited to emit a sound source, the end of the impedance tube uses a dual microphone (BSW416 (BSWA, Beijing, China)) to pick up the sound signal. The four-channel digital collector (MC3242 (BSWA, Beijing, China)) samples the signal and sends it to the computer for data processing. The loudspeaker (M4N (HiVi, Zhuhai, China)) was fixed at the end of the pipe by an air cavity equipped with a piston.

**Figure 4.** Experimental setup of the SL.

**Figure 5.** Photograph of the experimental setup.

For the accurate establishment of an SL absorption model, the exact mechanical and electrical parameters of the loudspeaker need to be known. Generally, the factory-calibrated parameters of the loudspeaker are accurate, but the actual parameters will change after it is assembled due to the coupling influence in the impedance. After assembly, the damping of the loudspeaker is nonlinear and difficult to predict accurately [22]. This will lead to a mismatch between the results of the theoretical predictions and the actual experiments, so, in our experiment, the actual damping was used in the calculation. The following describes how to experimentally determine the values of Δ*δ*, Δ*K*m, *Mm*, *R*c, *Lc*, *BL*.

#### *4.1. The Experiment of Parameter Determination*

#### 4.1.1. Determination of Mechanical Parameters

The loudspeaker was assembled at the end of the standing wave tube, in accordance with the open-circuit state. The cylinder piston was placed on the leftmost side. The

resistance and reactance diagram is shown in Figure 6a,b. The theoretical equation of acoustic impedance is:

**Figure 6.** Experimental impedance of the open circuit: (**a**) the specific acoustic resistance; (**b**) the specific acoustic reactance.

As shown in Figure 6a, the actual resistance of the system was not a constant value, but a nonlinear function that is dependent on the frequency. There was a damping peak at 380 Hz, which was caused by the resonance of the mechanical system. According to Equation (10a), the reactance of the system was only related to the moving mass and stiffness. Therefore, the equivalent moving mass and stiffness can be calculated by fitting the measured acoustic reactance using the least squares method. The equivalent moving mass of the loudspeaker in this experiment was 7.5 g and the system stiffness was 14,724 N/m; therefore, the three mechanical parameters of the loudspeaker could be accurately determined.

#### 4.1.2. Determination of Electrical Parameters

The coil resistance *R*c can be measured directly using a Digital Multi-Meter. In contrast, the coil inductance *Lc* has a frequency-dependent nonlinearity, so its value as determined by multi-meter measurement would be inaccurate. *Lc* can be fitted under short-circuit states, and when the *R*c, *Lc*, *Bl* are introduced, the system acoustic impedance can be expressed as:

$$Z\_{\rm short} = j\omega \left[ M\_{\rm m} - \frac{(Bl)^2 L\_{\rm c}}{R\_{\rm c}^2 + \omega^2 L\_{\rm c}^2} \right] + \left[ \delta\_{\rm m} + \frac{(Bl)^2 R\_{\rm c}}{R\_{\rm c}^2 + \omega^2 L\_{\rm c}^2} \right] + \frac{\Delta K\_{\rm m}}{j\omega} \tag{11}$$

First, according to the frequency at the resonance peak and the absorption coefficient, the *Bl* and the coil *Lc* can be counted out at the resonance frequency *f*0, as shown in Equations (12) and (13).

$$f\_0 = \sqrt{\frac{K\_{\rm sys}}{M\_{\rm sys}}} / 2\pi = \sqrt{\frac{\Delta K\_{\rm w}}{M\_{\rm m} - \frac{(Rl)^2 \cdot L\_c}{R\_c \cdot ^2 + \omega\_0 \cdot ^2 L\_c \cdot ^2}}} / 2\pi$$

$$a\_0 = 1 - |\frac{\mathbf{Z}\_{\rm sys} - \mathbf{Z}\_0}{\mathbf{Z}\_{\rm sys} + \mathbf{Z}\_0}|^2 = 1 - |\frac{j\omega\_0(M\_{\rm int} - \frac{(Bl)^2 \cdot \mathbf{I}\_c}{\mathbf{R}\_c^2 + \omega\_0^2 \cdot \mathbf{I}\_c^2}) + (\delta\_{\rm int} + \frac{(Bl)^2 \cdot \mathbf{R}\_c}{\mathbf{R}\_c^2 + \omega\_0^2 \cdot \mathbf{I}\_c^2}) + \frac{\Lambda \mathbf{K}\_{\rm ext}}{j\omega\_0} - \rho\_0 c\_0 \mathbf{S}}|^2 \tag{13}$$

$$\omega\_0(M\_{\rm int} - \frac{(Bl)^2 \cdot \mathbf{I}\_c}{\mathbf{R}\_c^2 + \omega\_0^2 \cdot \mathbf{I}\_c^2}) + (\delta\_{\rm int} + \frac{(Bl)^2 \cdot \mathbf{R}\_c}{\mathbf{R}\_c^2 + \omega\_0^2 \cdot \mathbf{I}\_c^2}) + \frac{\Lambda \mathbf{K}\_{\rm ext}}{j\omega\_0} + \rho\_0 c\_0 \mathbf{S}}|^2 \tag{13}$$

The resistance measured from the three sets of experiments for the short circuit and the open circuit are shown in Figure 7. According to Equations (10b) and (11), the resistance difference between the short circuit and the open circuit is expressed as (*Bl*) 2 *Re*/ *Re* <sup>2</sup> + *ω*2*Le* 2 . For a small signal input, *Bl* can be regarded as a constant value. By using least squares method at each frequency, the theoretical value of *Lc* can be calculated. The calculated *Bl* was 4 Tm, and *Lc* was 0.68 *mH*. The actual parameters of the loudspeaker were all obtained. In Figure 8, the theoretical calculation results and the actual sound absorption results are shown to match better in the short-circuit state, which verifies the parameters obtained.

**Figure 7.** Short-circuit and open-circuit resistance comparison.

**Figure 8.** Short-circuit absorption coefficient comparison.

#### *4.2. Experimental Results*

The resistance obtained from the experiments in the open-circuit state was sampled and used in the optimization program as the actual damping. *Mm* and *Bl* obtained in the previous section were taken as fixed parameters. The optimization procedure thus becomes EBA about Δ*R*, Δ*L*, *C*e, Δ*Km*. The upper bound settings, step size settings and optimal parameters are shown in Table 4.


**Table 4.** Parameter upper bounds and step size settings.

The experimental balance resistance *Rb* is 1 Ω, and the selected operational amplifier is *OPA*552 − *PA* with a ±15 V power supply. The experimental and theoretical predictions were in good agreement in the overall frequency band. As shown in Figure 9a, in the target frequency band, the average absorption coefficient was 0.65, and the overall absorption coefficient was improved compared with the short-circuit condition. However, due to the larger *Mm* and *Bl* of the loudspeaker, the SL was less adjustable. Compared with the open-circuit case in Figure 6b, the total reactance of the SL shown in Figure 9c was significantly lower over a wide frequency band, especially in the 150–300 Hz band, where it trended toward zero, which allowed the reactance to better meet the matching conditions. As shown in Figure 9b, the resistance below 450 Hz was much larger than the air acoustic resistance, so the SL over-damping limited further improvement of the sound absorption level.

**Figure 9.** Comparison of the four–dimensional EBA simulation and measurement: (**a**) the sound absorption coefficient, (**b**) the specific acoustic resistance, (**c**) the specific acoustic reactance.

#### *4.3. Discussion*

The experimental results of the four–parameter optimization showed that loudspeakers on the market have large values of moving mass and force factor, which limit the sound absorption performance improvement of the SL. The EBA optimization simulation indicated that a loudspeaker suitable for sound absorption should be characterized by small moving mass and force parameters. This would facilitate the miniaturization and design of a lightweight SL, as well as allowing for innovation in the loudspeaker structure. As shown in Equations (6) and (7), the introduction of the shunt circuit inevitably increases the acoustic resistance, while reducing the system acoustic reactance. Since excessive damping is a disadvantage for sound absorption, designing a loudspeaker with small amounts of linear damping should be the focus of future research. In addition, designing a large-area SL for diffuse field sound absorption would also be worth studying.

#### **5. Conclusions**

In order to obtain excellent sound absorption in the frequency range of 100–450 Hz, a fully exhaustive backtracking algorithm was proposed for optimizing the loudspeaker and shunt circuit parameters. For a given loudspeaker, the experimental method to determine its parameters was provided, and the optimal sound absorption algorithm under four parameters was verified by measurement. Through multiple-parameter optimization, it was found that the force factor and moving mass can be sufficiently reduced in comparison with that of a typical four-inch loudspeaker available on the market. The results imply that if an air cavity is properly sized, the SL can be redesigned to achieve good sound absorption, while also significantly reducing the weight and volume of the loudspeaker.

**Author Contributions:** Conceptualization, Z.L. and B.L.; Methodology, Data curation and Investigation, Z.L.; Validation and Software, Z.L. and X.L.; Resources and Supervision, B.L.; Writing—Original Draft Preparation, Z.L.; Writing—Review and Editing, Z.L. and X.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** Financial support was given by NSFC through Grant No. 11874034, as well as the Taishan Scholar Program of Shandong (No.ts201712054). Both are highly appreciated.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **Acoustic Characterization of Some Steel Industry Waste Materials**

**Elisa Levi \*, Simona Sgarbi and Edoardo Alessio Piana**

Department of Industrial and Mechanical Engineering, University of Brescia, via Branze 38, 25123 Brescia, Italy; s.sgarbi001@studenti.unibs.it (S.S.); edoardo.piana@unibs.it (E.A.P.)

**\*** Correspondence: elisa.levi@unibs.it; Tel.: +39-0303715571

**Abstract:** From a circular economy perspective, the acoustic characterization of steelwork by-products is a topic worth investigating, especially because little or no literature can be found on this subject. The possibility to reuse and add value to a large amount of this kind of waste material can lead to significant economic and environmental benefits. Once properly analyzed and optimized, these byproducts can become a valuable alternative to conventional materials for noise control applications. The main acoustic properties of these materials can be investigated by means of a four-microphone impedance tube. Through an inverse technique, it is then possible to derive some non-acoustic properties of interest, useful to physically characterize the structure of the materials. The inverse method adopted in this paper is founded on the Johnson–Champoux–Allard model and uses a standard minimization procedure based on the difference between the sound absorption coefficients obtained experimentally and predicted by the Johnson–Champoux–Allard model. The results obtained are consistent with other literature data for similar materials. The knowledge of the physical parameters retrieved applying this technique (porosity, airflow resistivity, tortuosity, viscous and thermal characteristic length) is fundamental for the acoustic optimization of the porous materials in the case of future applications.

**Citation:** Levi, E.; Sgarbi, S.; Piana, E.A. Acoustic Characterization of Some Steel Industry Waste Materials. *Appl. Sci.* **2021**, *11*, 5924. https://doi.org/10.3390/app11135924

Academic Editor: César M. A. Vasques

Received: 7 May 2021 Accepted: 23 June 2021 Published: 25 June 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Keywords:** steel industry by-products; circular economy; sound absorption; sound reduction index; granular materials; inverse method

#### **1. Introduction**

The steel industry is one of the main global economic sectors providing raw materials for a wide variety of manufacturing processes. During the various activities, a steel plant produces large amounts of waste under different forms. In recent decades, this type of industry is also trying to gradually leave the linear economy model and aim for the global and ambitious "zero waste" target [1], focusing its efforts on the development of innovative and sustainable production schemes. The emerging principle of the circular economy supports the reuse and recycling of industrial by-products, creating a symbiosis [2] which encourages collaboration and synergy with different sectors. The final goal is to develop new business opportunities through the conversion of waste into valuable raw materials or secondary materials exploitable in other sectors [3]. These activities allow the steel industry to reduce its environmental impacts: indeed, the reduction in waste materials can be ensured by providing an alternative solution to safe and environmentally friendly disposal of polluting industrial wastes and by avoiding the extraction of new natural resources. In this way, it is possible to achieve both environmental and economic benefits for all the industries involved in the symbiosis.

The main waste product of the steel industry is represented by slags. There are different types of slag, depending on the type of furnaces, raw materials and process adopted during production. The slag deriving from melting the scrap iron by an electric arc furnace (EAF—Figure 1a) is generically defined as "black slag". Such a type of waste material results from the oxidation of the scrap and includes impurities and compounds generated by the additives used to control the chemical processes. These elements form a layer that floats on top of the molten steel in the furnace, insulating the liquid part from the external environment and helping maintain the temperature inside the furnace at the right set-point. At the end of the process, the floating layer is collected and cooled down, resulting in "black slag". The melted steel is then processed and refined in a ladle furnace (LF—Figure 1b). The slag deriving from this process is defined as "white dross", or "white slag", and has completely different chemical and physical properties if compared to the black slag.

**Figure 1.** Schematic drawings of EAF and LF furnaces: (**a**) EAF furnace; and (**b**) ladle furnace.

The two slag types are kept separate as they have a different chemical composition and must be treated differently. Black slag can be assimilated to natural effusive rocks of volcanic origin, takes on granular characteristics and mainly consists of a ternary mixture of calcium oxide (CaO), silicon dioxide (SiO2) and iron oxides (FeO), to which heavy metals and other components, in percentages, are mixed. The white dross chemically differs from the black slag, particularly for the content in iron oxides and calcium; therefore, this dross, after cooling, undergoes a transformation of the crystalline lattice which leads to the formation of a fine and lightweight material.

Some studies are actually investigating the physical properties and the environmental compatibility of these materials, to establish the environmental impact of the slags and how to treat them. Depending on the application field, several studies have been carried out to characterize steel slags. One of the first practical applications of steel slags outside the steel production cycle is as sustainable (alternative) aggregates in pavement layers for road construction. They have been used not only for unbound layers, like road bases and sub-bases, but also for bituminous mixtures in surface layers [4]. In [5], the EAF steel slag was preliminarily investigated from chemical, leaching, physical and mechanical points of view. The bituminous conglomerates have also been characterized to verify their potential application in high performance asphalt concretes for road and airport pavements. The comparison with the corresponding traditional natural aggregates shows that using slags as coarse under-pavement material brings both technical and environmental advantages [5,6]: the mixtures with EAF slag improve the mechanical properties and prevent the depletion of raw materials.

Several studies also investigated the suitability of steel slags for civil engineering applications in cement-based materials. In particular, they were used for replacing natural sand [7] in the production of concrete [8–13], as armor stones for hydraulic engineering constructions [14,15] (during the restoration of marine environments and stabilization of shores), and finally, as an agricultural fertilizer. In [2], steel slags were investigated as to help in the removal of harmful elements and wastewater treatment. More recently, slags have been used as green resource in ceramic tile production and for biomedical applications.

The development of innovative sustainable solutions, by means of already existing or new technologies, is a goal that the steel industry is willing to pursue in order to further reduce its environmental impacts. Recently, some critical environmental aspects emerged regarding the use of steel slags [2,14]. The concerns are about volume instability and leaching behavior, the latter being a crucial aspect for environmental considerations, especially in terms of possible water and soil pollution caused by the release of heavy metals. Such aspects must be deeply investigated and solved, depending on the characteristics of the specific chemical composition of the recycled slag and the exposure to atmospheric elements.

The study reported in [16] aimed to find an inertization process for the recovery of steelwork slags and granite cutting waste as raw materials for the production of rockwool, which is a good thermal insulator and acoustic absorber for the construction and automotive sectors. It was found that the partial replacement of traditional raw materials does not influence the thermal insulation and fireproof properties of rockwool.

In [17,18], a recent improvement consisting of a new production method was introduced: high-pressure cold air is passed through the molten slag and the result is a material consisting of slag granules characterized by an almost spherical shape. The studies provide a comprehensive experimental characterization, in terms of fundamental and durability properties. The outcomes of both studies confirmed that a fine aggregate of spheric slags is a promising and advantageous alternative to natural sand in concrete pavement, also in terms of workability, water content and cement mechanical requirements.

Granular materials are emerging as an interesting alternative to the more popular and conventional sound absorbers. This trend is also encouraged by the large amount of industrial waste or by-products available in granular shape. If properly treated, these materials could become a valuable "second raw" resource, instead of using them as waste material for landfill, with all the related costs in terms of money and environmental impact. In this way, these materials can re-enter the production cycle and can be addressed in different application fields. Of course, this depends on how much their properties and potentialities are investigated and optimized. In this perspective, the steel industry is continuing its path towards the "zero waste" and circular economy goals by funding studies on the waste reuse and sustainable recycling and developing new technological solutions in an effort to find new fields of application. For instance, the traditional mineral wool production process can be applied to steelwork slag: by means of spin dryers and a high-speed air flow, the white dross molten slag forms long fibers and a sort of wool that could represent a partial or complete substitute to the traditional rockwool.

This paper aims to analyze the acoustic behavior of some steelwork waste materials. In particular, slags shaped as wool, granules and spheres. As previously mentioned, the literature describes many studies focused on the investigation of the chemical, mechanical and thermal properties of steelwork slags, focusing on their reuse for various outcomes, especially when combined with other materials and mixtures, such as cement, concrete and soils. The novelty of the present study lies in the fact that, to the knowledge of the authors, steelwork slags have never been acoustically characterized before, especially in the form of wool, granules or spheres. It can be highlighted that the process adopted to obtain the spheres is relatively recent. Once the acoustic properties of these waste slags are obtained, the aim is to evaluate and optimize them as a function of the specific noise control application at hand modifying their non-acoustical parameters. Their acoustic characterization will be performed by means of a four-microphone impedance tube. This technique allows one to obtain the complex acoustical properties of the tested samples. This study will mainly be focused on the sound absorption properties of the slags. However, the sound transmission loss (*TL*) will also be reported for the sake of completeness and because it is included in the acoustic properties retrieved from the four-microphone impedance tube method. Finally, the Johnson–Champoux–Allard (JCA) model will be employed to

better understand the relations between the acoustic behavior and the microstructure of the investigated materials. In particular, this theoretical model was based on the knowledge of five intrinsic properties of the material. Such properties are usually determined using specific laboratory equipment. In order to have a rough estimation of these parameters, a well-established inverse characterization method was applied to find the main non-acoustic characteristics of the materials.

The paper is organized as follows: Section 2 describes the samples analyzed and the experimental set up, including the laboratory equipment, providing the methodology for the experimental and analytical investigation; in Section 3, the experimentally obtained results are reported and discussed, including the comparison with the predicted results; finally, Section 4 draws the conclusions and highlights future research directions/perspectives.

#### **2. Materials and Methods**

#### *2.1. Porous Materials*

Porous materials are the most used sound-absorbing materials in many engineering and industrial applications. The Biot theory [19] describes how acoustic and elastic waves propagate and dissipate energy inside a porous medium characterized by air-saturated open-cell structures. When excited by a sound wave, the solid skeleton of the material can be considered as acoustically rigid (i.e., motionless) over a wide frequency range. Consequently, the compression and shear waves in the solid phase can be neglected. Thus, only a compression wave is able to propagate in the fluid phase and the porous material can be assumed to behave like an equivalent fluid. The absorption mechanism is possible thanks to the structure of the porous medium: it is made by a large number of small pores that are interconnected with each other and with the external air, thus allowing the sound wave to enter and propagate within the cavities. During the propagation process, the viscosity of air in the pores causes viscous losses. The conversion of sound energy in internal energy and the subsequent dissipation caused by the viscosity of air enables obtaining a certain sound absorption [20].

Recently, increasing interest has emerged in granular porous materials [20–24], considered to be a promising alternative to the more traditional fibrous or foam sound absorbers, thanks to their advantage of merging a good sound absorption with interesting mechanical properties and low production costs [25,26].

Granular materials are made of assemblies of particles that can have the same or different shape and diameter. The grains, that can be hollow, porous or solid, represent the rigid frame of the medium while the fluid (i.e., air), saturating the interconnected cavities, can be assumed as an equivalent homogeneous fluid, characterized by two effective (or equivalent) properties: the equivalent dynamic density *ρeq* and equivalent dynamic bulk modulus *Keq*. At the macroscopic level, the viscous and thermal losses that occur in porous media and are responsible of the energy sound dissipation, can be related to the so-called transport (or non-acoustic or macroscopic) parameters: depending on the model chosen to characterize the acoustic performance of the investigated materials, these parameters differ in number and type. The appropriate knowledge of the relationships relating the acoustic behavior to the microstructure is of importance to customize the material for specific target frequencies. As effectively summarized and described in [27,28], these models can be mainly sorted into empirical, phenomenological, and semi-phenomenological/microstructural models. The Delany-Bazley model [29], designed for fibrous and cellular materials and based on airflow resistivity as relevant parameter, and the Miki model [30], which improved the previous one with the inclusion of two additional non-acoustic parameters, porosity and tortuosity, belong to the first group. The Voronina–Horoshenkov model [31], suitable for loose granular materials, is of empirical type as well, and considers the characteristic particle dimension and specific density of the grain base in addition to porosity and tortuosity. In [32], the authors assumed that pore geometry and pore size distribution obey an approximately statistical distribution. The Hamet–Berengier [33] and Attenborough [34] models are located in the phenomenological group: the first results useful for porous pavements, the latter for fibrous and granular materials and is based on five parameters (airflow resistivity, porosity, tortuosity, steady flow shape factor and dynamic shape factor). The Johnson–Champoux–Allard [35,36] and the Champoux–Stinson [37] models fall into the semi-phenomenological/microstructural group and involve five non-acoustic parameters: porosity, airflow resistivity and tortuosity are common to both, whereas the JCA model uses thermal and viscous characteristic lengths, and the Champoux–Stinson model considers viscous and thermal shape factors.

Subsequent implementations of the JCA model, such as the six-parameter Johnson– Champoux–Allard–Lafarge (JCAL) model or the eight-parameter model of Johnson–Champoux –Allard–Pride–Lafarge (JCAPL), involve more parameter, such as viscous and thermal tortuosities and permeabilities. Compared to the JCA model, they provide more precision at low frequencies [38]. In general, the more sophisticated models require more parameters and have better performances. Nevertheless, as a counterpart, they are more complex and demanding. All the aforementioned models require physical techniques to measure the non-acoustic parameters. Some of them involve expensive set-ups and even complex or destructive tests. Recently, multiscale analyses have been developed to compute non-acoustic parameters by means of numerical simulations at the microstructural level [21,22]. The multiscale approach, which establishes micro–macro relationships, bypasses the difficulty of direct measurements by developing specific finite-element analyses.

Each model has its application field, related to the type of material its development is based on, and respective limitations and advantages. More details about these aspects and model comparisons can be found in [21,22,25,27,28,39–41].

In this work, the five-parameter JCA model, better described in Section 2.2, was selected to perform the investigation: it is one of the most known generalized models, suitable for the accurate description of the wide-band sound propagation in porous materials. It is a robust model as it is applicable to the random geometry of porous materials, it allows rapid calculation and the five parameters, having a physical meaning, and can be directly measured by experiments. The JCA model, coupled with the four-microphone impedance tube and inversion methods, results to be a well-established and fast technique to investigate the intrinsic properties of a material, thus being a valuable alternative whenever direct measurements are not available.

#### *2.2. JCA Model and Inverse Method*

The JCA model assumes that rigid-frame open-cell porous media can be seen as an equivalent fluid of effective, or equivalent, dynamic density *ρeq* and equivalent dynamic bulk modulus *Keq*. These equivalent properties depend on five transport (or macroscopic– non-acoustic) parameters: open porosity *Φ*; static airflow resistivity *σ*; tortuosity *α*∞; viscous characteristic length *Λ;* and thermal characteristic length *Λ'*. These parameters are referred to the geometry of the porous material and describe the complexity of the porous network.

By definition, open porosity *Φ* is a measure of the volume fraction of air (*Vfluid*) in the total volume (*Vtot*) or the complement to unit of the ratio between the solid volume of the frame (*Vsolid*) on the total volume [42]:

$$\Phi = \frac{V\_{fluid}}{V\_{tot}} = 1 - \frac{V\_{solid}}{V\_{tot}} \tag{1}$$

Airflow resistivity expresses the resistance opposed to the airflow while passing through the material. It can be calculated as [42]

$$
\sigma = \frac{\Delta p}{v\_{airflow} d} \qquad \left[ \text{Ns/m}^4 \right] \tag{2}
$$

with Δ*p* as the pressure drop across the medium, *vairflow* the amount of airflow passing through the material and *d* its thickness.

Tortuosity *α*∞ is an intrinsic property of the porous frame, related to the microgeometry of the interlinked cavities. It is a dimensionless quantity that expresses the tortuous fluid paths through the porous material. It can be calculated as [42]

$$\mathfrak{a}\_{\infty} = \frac{1}{V} \int\_{V} v^2 \,dV / \left| \frac{1}{V} \int\_{V} v \,dV \right|^2 \tag{3}$$

where *v* is the microscopic velocity of an ideal inviscid fluid within the pores and *V* a homogenization volume that expresses the volume of free fluid contained in the cavities. Tortuosity cannot be lower than 1.

Viscous characteristic length *Λ* is used to describe the viscous forces generating within the cavities at high frequencies and is related to the characteristic dimension of the connection between pores—particularly to the mean diameter of the hole connecting two adjacent cells, expressed in micrometers. It is given by [42]

$$
\Lambda = 2 \int\_{V} |v|^2 dV / \int\_{S} |v|^2 dS \qquad [\mu \text{m}] \tag{4}
$$

where *S* is the specific surface that denotes the total contact surface between the frame and the pores.

Thermal characteristic length *Λ'* describes the thermal exchanges between the solid frame and its saturating fluid at high frequencies and it is related to the pores dimension, especially to the mean diameter of the cell in micrometers; it can be expressed as [42,43]:

$$
\Lambda' = 2 \int\_{V} dV \slash \ \int\_{S} dS = 2V / S \qquad \left[ \mu \text{m} \right] \tag{5}
$$

Alternatively, *Λ* and *Λ'* can be calculated in function of the above-described parameters, as follows [38]:

$$
\Lambda = \frac{1}{c\_1} \left[\frac{8\alpha\_{\infty}\eta}{\sigma\Phi}\right]^{1/2} \tag{6}
$$

$$
\Lambda' = \frac{1}{c\_2} \left[ \frac{8a\_\infty \eta}{\sigma \Phi} \right]^{1/2} \tag{7}
$$

where *η* is the viscosity of air, *c1* and *c2* are pore shape parameters, related, respectively, to the viscous and thermal dissipation, and they can assume values in the following ranges:

$$0.3 \le c\_1 \le 3.3\tag{8}$$

$$0.3 \le c\_2 \le c\_1 \tag{9}$$

In the case of the granular material shaped in spheres, the calculation of the macroscopical parameters can be simplified in function of the porosity *Φ* and particle radius *r* as follows [21,23]:

$$
\sigma = \frac{45(1-\Phi)(1-\theta)\eta}{2\Phi^2 r^2 \left(5-9\theta^{1/3} + 5\theta - \theta^2\right)}\tag{10}
$$

$$\kappa\_{\infty} = 1 + \frac{1 - \Phi}{2\Phi} \tag{11}$$

$$
\Lambda = \frac{4(1-\theta)\Phi\alpha\_{\infty}}{9(1-\Phi)}r\tag{12}
$$

$$
\Lambda' = \frac{d}{3} \left( \frac{\Phi}{1 - \Phi} \right) \tag{13}
$$

where *θ* is expressed as

$$\theta = \frac{3(1-\Phi)}{\pi 2^{1/2}}\tag{14}$$

The purpose of the model is to finally obtain the acoustic behavior of the analyzed material, so the procedure to compute the sound absorption coefficient is made with the following steps:


$$\rho\_{e\eta} = \frac{\alpha\_{\infty}\rho\_0}{\Phi} + \frac{\sigma}{i\omega} \left( 1 + \frac{4i\alpha\_{\infty}^2\eta\rho\_0\omega}{\sigma^2\Lambda^2\Phi^2} \right)^{1/2} \quad \left[ \text{kg/m}^3 \right] \tag{15}$$

$$K\_{\eta} = \frac{\text{x}P\_0/\Phi}{\text{x} - (\text{x} - 1)\left[1 + \frac{8\eta}{i\rho\_0\omega N\_p\Lambda^2} \left(1 + \frac{i\rho\_0\omega N\_p\Lambda^2}{16\eta}\right)^{1/2}\right]^{-1}} \quad \left[\text{kg/ms}^2\right] \tag{16}$$

where *ρ<sup>0</sup>* is the density of air, *ω* = 2*πf* is the angular frequency, *η* is the air viscosity, *κ* is the specific heat ratio and *Np* is Prandtl number of the saturating air.

Once the effective properties are obtained, it is possible to determine the complex acoustical parameters [44]; the characteristic impedance *Zc*:

$$Z\_{\mathbf{c}} = \left(\rho\_{\mathbf{c}\mathbf{q}} Z\_{\mathbf{c}\mathbf{q}}\right)^{1/2} \quad \left[\text{Ns/m}^3\right] \tag{17}$$

and the complex wave number *kc*:

$$k\_c = \omega \left(\rho\_{eq} / Z\_{eq}\right)^{1/2} \qquad \left[\text{m}^{-1}\right] \tag{18}$$

From these acoustic properties, the surface impedance *Zs* can be derived as follows:

$$Z\_s = Z\_\circ \cdot \cot(k\_\circ d) \left[ \text{m}^{-1} \right] \tag{19}$$

Finally, the normal incidence sound absorption coefficient *α* is calculated as

$$\alpha = \frac{4\text{Re}\{\mathbf{Z}\_{\text{s}}\}\rho\_{0}\mathbf{c}\_{0}}{\left|\mathbf{Z}\_{\text{s}}\right|^{2} + 2\rho\_{0}\mathbf{c}\_{0}\text{Re}\{\mathbf{Z}\_{\text{s}}\} + \left(\rho\_{0}\mathbf{c}\_{0}\right)^{2}} \quad \text{or} \quad \mathbf{a} = 1 - \left|\frac{\mathbf{Z}\_{\text{s}} - \rho\_{0}\mathbf{c}\_{0}}{\mathbf{Z}\_{\text{s}} + \rho\_{0}\mathbf{c}\_{0}}\right|^{2} \tag{20}$$

where *c0* is the speed of sound in air.

Classical methods to estimate the non-acoustic properties can be mainly classified in three groups [45]:


Indirect and inverse methods are based on impedance tube measurements or ultrasound measurements. The indirect method uses the equivalent properties, *ρeq* and *Keq*, obtained from measured *Zc* and *kc* values by using an impedance tube, and combine (17) with (18) as follows [45]:

$$
\rho\_{c\eta} = \frac{Z\_c k\_c}{\omega} \quad \text{and} \quad \quad K\_{c\eta} = \frac{Z\_c \omega}{k\_c} \tag{21}
$$

At this point, it is possible to extract non-acoustic parameters from the limit behavior of the effective properties [44]:

$$\Phi = \frac{\rho\_0 \mathfrak{a}\_{\infty}}{\left(\lim\_{\omega \to \infty} \mathcal{Re}\{\rho\_{c\eta}\}\right)}\tag{22}$$

$$\sigma = -\lim\_{\omega \to 0} \left[ \operatorname{Im} \{ \rho\_{\text{eq}} \} \omega \right] \tag{23}$$

$$\mathfrak{a}\_{\infty} = \left\{ \lim\_{\omega^{-1/2} \to 0} (\mathfrak{c}/\mathfrak{c}\_0) \right\}^{-2} \tag{24}$$

$$\Lambda = \lim\_{\omega \to \infty} \left( \alpha\_{\infty} \left( \frac{2 \rho\_0 \eta}{\omega \Phi \operatorname{Im} \{ \rho\_{c\eta} \} \left( \rho\_0 u\_{\omega} - \Phi \operatorname{Re} \{ \rho\_{c\eta} \} \right)} \right)^{1/2} \right) \tag{25}$$

$$\Lambda' = \left[ \frac{(N\_P)^{1/2}}{\kappa - 1} \left( \lim\_{\omega \to \infty} \left\{ \frac{\operatorname{Re}\{k\_c c\}}{\operatorname{Im}\{k\_c c\}} \left( \frac{2\eta}{\omega p\_0} \right)^{1/2} \right\} - \frac{1}{\Lambda} \right) \right]^{-1} \tag{26}$$

where *c* is the speed of sound within the material.

Alternatively, in [45], a straightforward procedure is proposed where, in addition to the effective properties *ρeq* and *Keq*, the direct measurement of the open porosity *Φ* is necessary. In this case, the analytical solutions suitable to obtain the macroscopic parameters starting from the effective properties are reported below [45,46]:

$$\sigma = -\frac{1}{\Phi\_{\omega \to 0}} \lim\_{\omega \to 0} \left[ \operatorname{Im} \{ \rho\_{c\eta} \omega \} \right] \tag{27}$$

$$a\_{\infty} = \frac{1}{\rho\_0} \left[ \text{Re}\{\rho\_{c\eta}\} - \left( \text{Im}\{\rho\_{c\eta}\}^2 - \left(\frac{\sigma \Phi}{\omega}\right)^2 \right)^{1/2} \right] \tag{28}$$

$$\Lambda = \mathfrak{a}\_{\infty} \left[ \frac{2 \rho\_0 \eta}{\omega \operatorname{Im} \left\{ \rho\_{\varepsilon \eta} \right\} \left( \rho\_0 \mathfrak{a}\_{\infty} - \operatorname{Re} \left\{ \rho\_{\varepsilon \eta} \right\} \right)} \right]^{1/2} \tag{29}$$

$$\Lambda' = \left(\frac{2\eta}{\rho\_0 \omega}\right)^{1/2} \left[ 2 \left( -\operatorname{Im} \left\{ \left( \frac{1 - \operatorname{K}\_{tq} / \operatorname{K}\_d}{1 - \kappa \operatorname{K}\_{tq} / \operatorname{K}\_d} \right)^2 \right\} \right)^{-1} \right]^{-1/2} \tag{30}$$

where *Ka = κP0/Φ*, with *P*<sup>0</sup> static pressure, is the equivalent adiabatic bulk modulus of the equivalent fluid.

Inverse methods generally need a surface acoustic property to start with, such as the sound absorption coefficient or surface impedance, both obtained from impedance tube measurements. The optimization process is based on the fact that the unknown parameters (in this paper, the five non-acoustic parameters) are adjusted so that the estimated surface acoustic property is as close as possible to the one experimentally obtained. The objective function is designed as a cost function where small values mean close agreement.

There are different optimizing methods: for instance, the group of global optimization techniques includes the simulated annealing [20], based on Monte Carlo iteration, and the class of evolutionary algorithms, such as genetic algorithms [42,44] and differential evolution algorithms [38]; moreover, there are standard minimization procedures, such as nonlinear best-fit [42,47], which is a direct search method that requires an initial trial guess of the parameters and operates within a research domain set on the lower and upper bound constraints for all the variables.

#### *2.3. Experimental Characterization—Four-Microphone Impedance Tube*

In this paragraph, the experimental set-up used for the characterization of the samples is described. The measurements of the acoustic properties have been performed by means of the four-microphone impedance tube method, following the process given by the ASTM E2611 standard [48]. On one end of the apparatus features, a loudspeaker generates a plane wave field inside the tube. The other end can be configured with two different types of termination (anechoic and/or reflecting), to perform the investigation with two different boundary conditions. Two microphones are mounted in front of the sample, at the "emitting side" of the tube, and the other two microphones are placed close to the sample at the "receiving side" of the tube.

A transfer matrix approach can be used, allowing to relate the particle velocities (*ui*) and the sound pressures (*pi*) at both surfaces of the tested sample. Denoting the front surface of the sample with the coordinate *x* = 0 and the back surface with *x* = *d*, the resulting transfer matrix can be written as

$$
\begin{bmatrix} p\_0 \\ u\_0 \end{bmatrix} = \begin{bmatrix} T\_{11} & T\_{12} \\ T\_{21} & T\_{22} \end{bmatrix} \begin{bmatrix} p\_d \\ u\_d \end{bmatrix} \tag{31}
$$

Thanks to the comparison between the signals measured by the four microphones, it is possible to apply the decomposition technique: referring to Figure 2, the upstream and downstream sound field can be distinguished in two forward travelling waves (A and C) and two backward travelling waves (B and D).

**Figure 2.** Schematic drawing of a four-microphone impedance tube.

The wave components A–D represent the complex amplitudes of the incident and reflected waves on both sides of the sample and can be derived from the complex transfer functions *Hi,ref* measured between the -*i*th microphone (*i =* 1, ... , 4) and the reference (*ref*) microphone. In this study, the first microphone was selected as the reference microphone, but generally any of the four microphones can be chosen for this role. At this point, an interchanging procedure must be applied between the transducers to correct the measured transfer functions for amplitude and phase mismatches. Once the corrected transfer functions are obtained by dividing the measured transfer functions by relative correction transfer functions, the four components A, B, C and D can be obtained. These coefficients are used for the derivation of the transfer matrix terms. Pressures and particle velocities at both sides of the sample can be determined in terms of incident and reflected plane wave components. In the case of geometrically symmetric specimens, since the physical properties are the same on either side, reciprocity and symmetry can be applied and a single set of measurements is sufficient to characterize the material. The acoustical properties of the sample can thus be calculated as a function of the transfer matrix elements, the acoustic impedance of air, the sample thickness and the wavenumber in air. In particular, the following properties are obtained:


The Applied Acoustic Laboratory impedance tube at the University of Brescia is composed of two 1200 mm long-segments, with an internal diameter of 46 mm, determining a cross-section that ensures that the plane-wave assumption is verified up to approximately 3700 Hz. The loudspeaker is installed in an isolated and sealed volume at the source endpoint of the tube. Through the connection to the generator of a multichannel analyzer, a wide-band white noise test signal (50 Hz–5 kHz) is created inside the tube. As the samples tested in this article are symmetric, it was not necessary to use a double boundary condition, and the second endpoint of the tube was equipped with an anechoic termination.

The sample holder is a detachable unit, made of separate segments of tube of appropriate length, which can be usually chosen to be 50, 100 or 200 mm long. Once carefully filled with the material to be tested, the holder is placed in the central section of the tube, between two microphone pairs, and it is additionally sealed to the main parts of the tube by means of O-rings and petroleum jelly for assuring air tightness. Four PCB microphones Type 130F22 are inserted in openings sealed with O-rings and flush mounted with the inner surface of the tube. The microphone pairs are spaced 500 mm for low-frequency measurements and 45 mm for high-frequency measurements. It is worth noting that this study focuses on high-frequency characterization (200–3150 Hz). This choice was made because the JCA model used for the inverse characterization is less accurate in the low frequency range, as discussed in [38]. The transducers are connected to an OROS OR 36 multichannel analyzer which measures the complex transfer functions between the microphones. All the microphones were calibrated before the test by using a Bruel and Kjaer pistonphone Type 4228. In Figure 3, some details of the four-microphone impedance tube used for the experiments are shown.

**Figure 3.** Details of the four-microphone impedance tube: (**a**) the sound source; and (**b**) central part with the inserted sample holder and the two microphone pairs.

To determine the transfer matrix elements, it is necessary to measure the complex sound pressure, including amplitude and phase, at four positions. Once microphone 1 is chosen as a reference, the standard procedure requires a first measurement with all the microphones placed in the port corresponding to their respective number, and then three other measurements are made by physically switching the location of each microphone with the reference microphone 1. This enables obtaining the correction of the transfer functions for phase and amplitude mismatches. In this way, for each tested sample, four measurements have to be executed. A self-built MATLAB® code allows one to post-process the measured transfer functions and to describe the acoustic behavior of the tested material, giving as an output the normal incidence sound absorption coefficient *α*, the normal

incidence sound transmission loss *TL*, the characteristic acoustic impedance *Zc*, the speed of sound *c* and the propagation wavenumber *kc* in the tested material. To correct the speed of sound in the air and the air density values, the temperature and atmospheric pressure were measured before each test and then considered during the post-processing phase.

#### **3. Tested Samples**

Among the different types of waste resulting from the steel production, slags probably represent the main (90% by mass) and most hazardous one, due to the possible content of heavy metals such as chromium, manganese and iron. In order to make slag suitable for recycling and reuse, a deep knowledge of its composition and physical properties is needed, to apply appropriate stabilization and inertization methods that allow environmentally sustainable applications of slags.

In this work, three types of steelwork waste materials were analyzed: wool derived from white dross, spheres derived from black slag, and spheres encapsulated in an inert material. The first material is a white wool, made of long fibers, similar to mineral wool or glasswool. This material is derived from a centrifuge process of the white dross and it features inclusions of transparent spheres and thin dark flakes, as shown in Figure 4.

**Figure 4.** Sample of wool derived from white dross.

The second material is a conglomerate of spheres derived from black slag, as shown in Figure 5. Three diameters (∅) ranges were obtained by using progressive sieves on a sample of unselected byproduct. The samples are categorized as "BIG" (<sup>∅</sup> <sup>∈</sup> [1.4; 2.0) mm), "MEDIUM" (<sup>∅</sup> <sup>∈</sup> [0.71; 1.4) mm) and "SMALL" (<sup>∅</sup> <sup>∈</sup> (0; 0.71) mm) depending on the dimension of the spheres. The composition of the sample is approximately: 15% BIG, 35% MEDIUM and the remaining 50% SMALL spheres.

The third material is made of spherical black slag embedded in inert material, resulting in an irregular granular assembly, as shown in Figure 6. After the spherification process, a fluid cement consisting of mixtures of hydraulic binders (lime, silica and alumina) is mixed with the slag spheres. This mixture completely covers the granules and makes them inert. Table 1 reports the samples thickness, the net weight and the density of the different materials considered in this work, together with the diameter ranges of slag spheres specimens.

**Figure 5.** Conglomerate of spheres derived from black slag: (**a**) sample "BIG", with <sup>∅</sup> <sup>∈</sup> [1.4; 2.0) mm; (**b**) sample "MEDIUM" with <sup>∅</sup> <sup>∈</sup> [0.71; 1.4) mm; and (**c**) sample "SMALL" with <sup>∅</sup> <sup>∈</sup> (0; 0.71) mm.

**Figure 6.** Sample of slag spheres encapsulated in inert material.



The second and third material specimens were prepared in the following way:

1. The front surface of the sample holder cylinder was terminated with a protective layer, sealed by glue along the perimeter to contain loose granules and guarantee flat surface;


Previous separate measurements had confirmed that the protective layer, Figure 7, has no influence on the acoustic properties of the tested samples.

#### **4. Results and Discussion**

In Figure 8, the measured sound absorption coefficient and transmission loss of the white dross wool are presented. This material features an "S-shaped" absorption curve with the characteristic behavior typical of porous–fibrous materials, that is low values at low frequencies and values approaching a unit value at high frequencies: in particular, α starts with a value of 0.13 at 200 Hz, it linearly increases and around 1700 Hz, it reaches the unit value. The *TL* curve is also typical for fibrous materials, it does not reach high values but at about 1250 Hz, it shows a change in slope with an increasing trend.

In Figure 9, the acoustic performances of all the samples made of slag spheres of the three diameter ranges are shown in the same graph. The absorption coefficient curves of BIG and MEDIUM samples can be referred to the typical quarter wavelength resonance behavior of granular materials: the oscillations of the sound absorption for granular materials are caused by the air gap around the granules. If the gap between granules is too small or too large, not enough friction and subsequent heat transfer can develop between the air and the solid skeleton of the pore wall during the propagation of sound waves [20]. The first peaks are, respectively, at about 580 Hz for sample BIG of 100 mm length, 1330 Hz for sample BIG with 50 mm of thickness and about 1190 Hz for sample MEDIUM, which is 50 mm long. As the dimension of the sample increases, a more complex gap distribution occurs together with longer channels. While the acoustic wave propagates, the air particle collisions and the flow volume raise within the pores, resulting in a higher dissipation of energy. When the natural frequency of the spheres mix decreases, the sound absorption peak shifts to a lower frequency.

**Figure 8.** Acoustic properties of white dross wool sample: (**a**) sound absorption coefficient; and (**b**) transmission loss.

**Figure 9.** Acoustic properties of slag spheres samples BIG, MEDIUM and SMALL: (**a**) sound absorption coefficient; and (**b**) transmission loss.

It can be observed that for the SMALL sample, a smooth absorptive behavior is present where the resonance peaks and throughs are suppressed. As stated in [31], for large grain mixes, the absorption coefficient spectrum shows an oscillating trend, corresponding to resonance maxima and minima. On the contrary, small grain mixes lose the resonant behavior, featuring a less pronounced trend. For this reason, the transmission loss of SMALL sample is higher than the one measured for the BIG and MEDIUM samples throughout the whole frequency range of interest. This may be caused by the nature of the SMALL sample, featuring spheres with <sup>∅</sup> <sup>∈</sup> (0; 0.71) mm. Such structure can be considered since compact sand and its higher density results in a very high airflow resistance. This gives rise to a reflective behavior. In the frequency range between 200 and 300 Hz, a drop in both α and *TL* values of the SMALL sample were observed, probably because of the rigid frame resonance of the system.

Figure 10 shows the absorption curve of the slag spheres sample embedded inside an inert material. Additionally, in this case, the graph shows the typical quarter wavelength resonance behavior of granular materials. The first peak almost reaches a unit value around 1390 Hz. The transmission loss remains quite low throughout the whole frequency range of interest.

**Figure 10.** Acoustic properties of slag spheres encapsulated in inert material: (**a**) sound absorption coefficient; and (**b**) transmission loss.

In order to easily compare the performances of different materials, the weighted noise reduction coefficient (*NRC*) and the sound absorption average (*SAA*) [49] are used to summarize the absorption characteristics of the tested samples by single rating numbers: they range between 0 and 1, in the case of perfectly reflective or perfectly absorptive materials, respectively. As stated by the standard, *NRC* is rounded off to the nearest multiple of 0.05, while *SAA* is rounded off to the nearest multiple of 0.01. Table 2 reports the sound absorption coefficients for the twelve one-third octave bands from 200 to 2500 Hz of the investigated waste materials and their respective *NRC* and *SAA* values.

**Table 2.** Sound absorption coefficients for the twelve one-third octave bands from 200 to 2500 Hz of the tested materials and the respective noise reduction coefficient (*NRC*) and sound absorption average (*SAA*).


In this paragraph, the inverse method based on the standard minimization approach is applied in order to derive the main non-acoustic parameters of the different materials considered in this article. For this analysis, the selected optimization objective function is the difference between the sound absorption coefficient measured by means of the fourmicrophone impedance tube and the absorption coefficient predicted by using the JCA model. Thus, the investigated cost function is defined as

$$\mathbb{C}F\{|a|\} = \sum |a\_{measured} - a\_{fCA\_{model}}|\tag{32}$$

The purpose is to determine the best solution of the unknown parameters to minimize the cost function. According to the literature [20,26,38], the intervals of the five non-acoustic parameters are set as

⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ *Φ* ∈ [0.1 ; 0.9] *σ* ∈ [1000 ; 150000] *α*<sup>∞</sup> ∈ [1; 4] *c*<sup>1</sup> ∈ [0.3 ; 3.3] *c*<sup>2</sup> ∈ [0.3 ; *c*1] (33)

To better understand the degree of agreement between measurements and predictions, using the method described in [44], the relative error *E%* was estimated for all the predictions as

$$E\% = \left| \frac{Measured - Predicted}{Measured} \right| \tag{34}$$

As shown in Table 3, the error was evaluated for each computed third octave band and then the average value of the relative error for the single material is given in the last column.

**Table 3.** Relative errors for third octave bands and average values of the relative errors.


Figure 11 refers to the wool sample and shows the comparison between the absorption experimentally obtained coefficient and the one estimated by using the JCA model achieved by means of the iterative minimization method. It can be noted that there is a good agreement between the two curves with respect to the frequency range considered, except for a slight overestimation upstream of 630 Hz in the predicted curve, and a little drop downstream of 1600 Hz which is not present in the measured curve.

**Figure 11.** Comparison between experimental and predicted sound absorption coefficient for the wool sample derived from white dross.

In Figure 12, the experimental and predicted curves are depicted for the three slag spheres samples: BIG, MEDIUM and SMALL. The predictions referring to BIG and MEDIUM samples are in good agreement with the experimental curves, showing the classic resonant behavior. In particular, sample BIG—50 features a high degree of agreement between the two curves above 500 Hz. In the high frequency range, a slight discrepancy can be observed for sample BIG—100. For the sample SMALL, the estimated curve fairly approximates the measured curve, but it does not follow the trend in an optimal way along the entire frequency range. This is probably caused by the nature of the SMALL sample, which is neither an absorbing nor an insulating material.

**Figure 12.** Comparison between experimental and predicted sound absorption coefficients for slag spheres samples BIG, MEDIUM and SMALL. Blue lines are referred to the respective JCA fittings.

Figure 13 shows, overall, a good correspondence between the estimated and the measured absorption curves relative to the sample made of encapsulated slag spheres.

**Figure 13.** Comparison between experimental and predicted sound absorption coefficient for the sample made of encapsulated spheres.

Table 4 summarizes the values of the five non-acoustic parameters obtained by applying the optimization procedure. The achieved parameters seem to be consistent with what can be found in the literature for similar materials. Nevertheless, in [21], the authors stated that the porosity of the random close packing of spherical beads should remain constant to a value of approximately 0.36, when the ratio between the sample holder diameter and the tested spheres diameter exceeds the value of 10. This corresponds to the characteristics of the case at hand, since the internal diameter of the sample holder is 46 mm and the diameter of the largest spheres is 2 mm. However, the discrepancies may be due to the fact that, in this study, the spherical particles have not a single diameter value, but they are indeed assembled in diameter ranges, thus, the internal arrangement may be different from the one described in [21]. In order to completely validate the optimal parameters identified by the inverse technique, the next step of the research will be the direct experimental measurement of the five non-acoustic parameters.

**Table 4.** Inversely determined non-acoustic parameters.


#### **5. Conclusions**

In this paper, the acoustic characterization of some steel industry waste materials derived from black and white slags is provided. The measurements performed by using a four-microphone impedance tube allowed us to obtain the acoustic properties of the tested samples. As a result, the analyzed materials can be mainly considered as porous media featuring interesting sound absorption and insulation characteristics. The wool derived from white dross exhibits a trend of the sound absorption which is typical of fibrous material, while slag spheres and encapsulated spheres behave as granular materials, with an oscillating tendency whose peaks are due to the resonance of the particle frame at a frequency corresponding to the one of a quarter wavelength resonator having the same thickness. Only the SMALL sample showed a more insulating than absorptive behavior, due to its higher density and airflow resistance values. It can be said that the SMALL sample acoustically behaves like compact sand. In order to determine the non-acoustic parameters of the samples, without the possibility of performing direct measurements, an inverse characterization technique was applied. Based on the JCA model, the inverse technique used relies on a standard iterative optimization procedure: the minimization is performed between the sound absorption coefficient measured in a four-microphone impedance tube and the one estimated by optimizing the inversion values into the JCA model. The optimization intervals were set according to the literature data. The five resulting non-acoustic parameters are compatible and comparable with the ones which can be found in other studies dealing with porous and granular media. The fact that some samples are made by spheres assembled by ranges of diameters and not by single diameter values explains possible discrepancies with literature data. The next step of the research will be focused on the validation of the inversion procedure and of the optimized non-acoustic parameters, by means of specific experimental measurements. Further investigations on the microstructure and the particle arrangements will allow the optimization and customization of the material for specific noise control applications.

**Author Contributions:** Investigation, data curation, writing—original draft preparation, visualization, E.L. Methodology, resources, supervision, E.A.P. Software, S.S. Conceptualization, writing review and editing, E.A.P. and E.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors would like to thank ORI Martin (Brescia), particularly Maurizio Zanforlin, for providing the material and technical insight on the manufacturing and inertization processes.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

