A computational model for the sound absorption coefficients of multi-layer non-wovens

Abstract

Using the theory of Zwikker and Kosten for sound propagation through a porous flexible media and the sound propagation boundary conditions at adjacent interfaces, a general computational model for the sound absorption coefficients of multi-layer non-wovens, α_n, is presented in this paper. The iterative algorithm for calculating α_n is given using the back-stepping method. In order to yield the highest sound absorption coefficients in the audible frequency range the model has been used for numerical calculation of the intrinsic characteristics of non-woven materials. The simulation demonstrates that the multi-layer structure of non-wovens can have good sound absorption over a prescribed frequency range by choosing the appropriate parameters for each layer. This model can also be used to calculate the absorption coefficients of a multi-layer non-woven comprising different non-woven components, and this can then provide a theoretical support for product design.

Keywords

multi-layer non-wovens computational model absorption coefficient

Non-wovens are one of the most effective sound absorption materials and can be used for controlling noise in a wide range of applications, including wall claddings, acoustic ceilings and barriers, and carpets.^1–7 Experimental studies indicate that the sound absorption capacity of non-woven materials is comparable to that of traditional sound-proofing materials.^3–8 A computational model is required for the optimal design of acoustic assemblies made from non-woven materials, enabling the prediction of the sound absorption coefficients of the non-woven medium as a function of its thickness, porosity, fiber type, and other intrinsic characteristics. The sound absorption mechanism of non-wovens has attracted increasing attention and has achieved fruitful results.^1,9,10 The theoretical investigation of sound propagation through flexible porous media is of great importance for evaluating the sound absorption capacity of non-woven fiber webs.^1,2

The first monumental work on this subject was presented by Zwikker and Kosten,⁹ in which a porous medium was regarded as a mixture of two phases, air and solid fiber, in which sound waves behave differently. A mathematical analysis of the basic equations derived from this theory predicts two forward waves and two backward waves travelling through the medium. Based on the Zwikker and Kosten theory a more appropriate model was suggested by Dent, who examined the random mixture of fibers and air pores in a non-woven fiber web.¹¹ In addition, some investigations into the sound absorption of specific non-woven materials were carried out.^12,13 For example, a theoretical model was derived for calculating the sound absorption coefficient of weft knitted fabrics with complex structures, combining theoretical models for non-woven and plain knitted fabrics.¹² The microstructure of the nonwoven was modeled using a macroscopically homogeneous random system of straight cylinders in order to optimize the acoustic properties of a stacked fiber non-woven.¹³ In particular, the noise absorption coefficient calculated for some non-wovens, based on the work of Dent and Shosany, showed similar analytical results, demonstrating that the theory of Zwikker and Kosten for sound transfer through porous media was in agreement with experimental measurements on non-wovens.¹ The model was then used for the numerical calculation of the intrinsic characteristics of non-woven fiber webs, allowing the highest sound absorption coefficients over the audible frequency range to be achieved.² Furthermore, in order to calculate the sound absorption coefficient of non-woven materials containing a number of layers with different properties, a theoretical generalization of the Zwikker and Kosten model was developed,¹⁰ and the effect of variations in porosity on the sound absorption coefficient of three webs made from cotton, acrylic and polyester, respectively, were examined using this model.¹⁴

A layered absorption structure can obviously increase acoustic attenuation if it is constructed from different sound absorbing materials.¹⁵ The results show that a layered absorption structure can achieve satisfactory absorption over a target range of frequencies.^16,17 For example, by appropriate choice of porosity the sound absorption coefficients of three webs made from cotton, acrylic and polyester, respectively, can reach 0.99.¹⁴ A model for calculating the absorption coefficient of multi-layer porous structures was given by using the acoustic propagation equation of the layered media, and optimization of the parameters was conducted.¹⁶

Encouraged by this work, the present paper investigates more deeply the sound absorption of multi-layer structures of different non-woven materials. A more general model for calculating the absorption coefficient of such structures is presented, based on the theory of Zwikker and Kosten on sound propagation through the porous flexible media, and the boundary conditions at interfaces between adjacent layers. Compared with Shoshani’s theoretical generalization of the Zwikker and Kosten model,¹⁰ our approach is more general and can be used for calculating the absorption coefficients of multi-layer sound absorption structures containing different non-woven components.

Absorption mechanism for sound propagation through non-woven materials

In this section, we review the sound absorption process through the porous flexible media. When sound waves are propagated through non-woven materials the incident sound pressure may be denoted as $p + (x, t)$ . Some parts of the sound waves are reflected, and the reflected sound pressure is denoted as $p - (x, t)$ ; other regions of the sound waves are absorbed, and the remaining sound waves are transmitted; the transmitted sound pressure is denoted as $p_{t} (x, t)$ . This process is illustrated in Figure 1.

Figure 1.

Sound propagation model in non-woven materials.

For convenience, let us denote

p + (x, t) = P_{ai} e j (ω t - kx), p - (x, t) = P_{ar} e j (ω t + kx)

Then, $| r_{p} | = | \frac{P_{ar}}{P_{ai}} |$ is the reflection coefficient of the sound pressure and $r_{I} = | r_{p} | 2$ is the reflection coefficient of the sound intensity. Based on the law of conservation of energy, the absorption coefficient can be written as

α = 1 - r_{I} = 1 - | r_{p} | 2

(1)

If we denote the sound impedance at the front face (x = 0) as

z = \frac{p}{u}

we then have

α = \frac{4 {AW}_{0}}{(A + W_{0}) 2 + B 2},

(2)

where

p = p_{i} (0, t) + p_{r} (0, t)

is the total sound pressure, u is the total velocity at the front face (x = 0),

A = z (R)

and

B = z (I)

are the real and imaginary parts of z, respectively, and

W_{0} = ρ_{0} c_{0}

is the characteristic impedance of free air, and

ρ_{0}

and c₀ denote the density of free air and the speed of the sound in free air, respectively.

Based on the theory of Zwikker and Kosten for sound propagation through porous flexible media, a porous medium may be regarded as a mixture of two phases, air and solid material, in which the sound waves react differently in the two phases. The mathematical analysis of the basic equations derived from this theory predict that two forward and two backward waves are travelling through the medium. The sound pressure $p_{a} (x, t), p_{f} (x, t)$ and article velocity $u_{a} (x, t), u_{f} (x, t)$ in the air and fiber can, respectively, be solved and the sound impedance at the front face (x = 0) is given by

z = \frac{p}{u} = \frac{p_{a} (0, t) + p_{f} (0, t)}{v_{a} u_{a} (0, t) + v_{f} u_{f} (0, t)},

(3)

where the subscripts a and f refer to the air and fiber phases, respectively, and v_a and

v_{f} = 1 - v_{a}

are the volume fractions of air and fiber.

Sound absorption model of multi-layer non-wovens

In this section, according to the theory of Zwikker and Kosten on sound propagation through porous flexible media, and knowing the sound propagation boundary conditions at the interface between adjacent layers, a model for calculating the absorption coefficient of multi-layer non-woven can be derived.

The sound propagation process in a multi-layer sound absorption structure composed of non-woven materials is shown in Figure 2, in which n is the total number of layers, and $p_{s}^{-} (x, t)$ and $p_{s}^{+} (x, t)$ are the incident and reflected sound pressure in each layer, respectively. The subscript s = 0, 1, 2, … , n refers to the ordinal number of the layer, and $p_{t} (x, t)$ is the transmitted sound pressure. It is presumed that the multi-layer absorption structure ends in a rigid wall at x = l₁ + l₂ + ··· + l_n. For convenience, we will use the Shoshani notation.¹

Figure 2.

Sound propagation model in multi-layer absorbing structure.

Let $ρ_{a}$ and $ρ_{f}$ be the density of air and fiber per unit volume of the non-woven material, and $ρ_{p}$ be the density of polymeric material. Then, $ρ_{as} = v_{as} ρ_{0}$ and $ρ_{fs} = v_{fs} ρ_{ps}$ for s = 1, 2, … , n. Based on the analysis,¹ the total pressure $p_{as} (x, t) = p_{as}^{-} (x, t) + p_{as}^{+} (x, t)$ , $p_{fs} (x, t) = p_{fs}^{-} (x, t) + p_{fs}^{+} (x, t)$ , and the total particle velocity $u_{as} (x, t), u_{fs} (x, t)$ of air and fiber in each layer can be denoted as

{p_{as} (x, t) = \sum_{j = 1}^{4} C_{js} P_{as}^{k_{js}} e i (ω t - k_{js} x) p_{fs} (x, t) = \sum_{j = 1}^{4} C_{js} P_{fs}^{k_{js}} e i (ω t - k_{js} x) u_{as} (x, t) = \sum_{j = 1}^{4} C_{js} U_{as}^{k_{js}} e i (ω t - k_{js} x) u_{fs} (x, t) = \sum_{j = 1}^{4} C_{js} U_{fs}^{k_{js}} e i (ω t - k_{js} x)

(4)

where,

p_{as}^{-} (x, t)

and

p_{fs}^{-} (x, t)

are the incident sound pressure, and

p_{as}^{+} (x, t)

and

p_{fs}^{+} (x, t)

are the reflected sound pressure, in air and fiber, respectively, and satisfy the relationship

p_{s}^{-} (x, t) = p_{as}^{-} (x, t) + p_{fs}^{-} (x, t)

p_{s}^{+} (x, t) = p_{as}^{+} (x, t) + p_{fs}^{+} (x, t)

C_{js}

is the undetermined coefficient and

ω = 2

is the angular frequency and

k_{1 s} = - k_{3 s} = ω \sqrt{(\frac{b_{1 s}}{2} + \sqrt{\frac{b_{1 s}^{2}}{4} - c_{1 s}}) \frac{ρ_{as}}{v_{as} K_{as}}} k_{2 s} = - k_{4 s} = ω \sqrt{(\frac{b_{1 s}}{2} - \sqrt{\frac{b_{1 s}^{2}}{4} - c_{1 s}}) \frac{ρ_{as}}{v_{as} K_{as}}} b_{1 s} = (1 + v_{as} \frac{E_{s}}{R_{s}} + v_{fs} X_{s}) - i θ_{s} (1 + E_{s} + X_{s}) c_{1 s} = v_{as} \frac{E_{s}}{R_{s}} (1 - i θ_{s} (1 + R_{s})) R_{s} = \frac{ρ_{as}}{ρ_{fs}}, E_{s} = \frac{K_{a}}{K_{fs}}, X_{s} = \frac{v_{fs} (K_{as} - P_{0})}{v_{as} K_{fs}},

where

θ_{s} = \frac{φ_{s}}{ρ_{as} ω}

is the dimensionless coupling coefficient,

ϕ_{s} = i ω ρ_{as} (m_{s} - 1) + v_{as}^{2} δ

is the coupling parameter representing the drag between fiber and air phases, and m_s is the structure constant, δ is the resistance constant of air,

K_{fs}

is the fiber bulk modulus, K_a is the air bulk modulus, and

P_{0} = 101325 Pa

is the air pressure.

In the following, we consider the boundary conditions at the interface between any two adjacent layers.

Boundary condition $0$ : At the front face (x = 0), since the front is ‘open’ the pressure p is constant, and we then have

v_{f 1} p_{a 1} (0, t) = v_{a 1} p_{f 1} (0, t)

Boundary condition $s$ : At the adjacent interfaces between the s-th layer and the s+1-th layer (x = l₁ + l₂ + ··· + l_s) for s = 1, 2, … , n–1, the following boundary conditions are satisfied

v_{a (s + 1)} p_{as} (l_{1} + \dots + l_{s}, t) = v_{as} p_{a (s + 1)} (l_{1} + \dots + l_{s}, t) p_{as} (l_{1} + \dots + l_{s}, t) + p_{fs} (l_{1} + \dots + l_{s}, t) = p_{a (s + 1)} (l_{1} + \dots + l_{s}, t) + p_{f (s + 1)} (l_{1} + \dots + l_{s}, t) u_{as} (l_{1} + \dots + l_{s}, t) = u_{f (s + 1)} (l_{1} + \dots + l_{s}, t) v_{as} (u_{as} (l_{1} + \dots + l_{s}, t) - u_{fs} (l_{1} + \dots + l_{s}, t)) = v_{a (s + 1)} (u_{a (s + 1)} (l_{1} + \dots + l_{s}, t) - u_{f (s + 1)} (l_{1} + \dots + l_{s}, t))

Boundary condition $n$ : At the back reflecting face (x = l₁ + l₂ + ··· + l_n), the velocities of both phases vanish, so that

u_{an} (l_{1} + l_{2} + \dots + l_{n}, t) = 0 and u_{fn} (l_{1} + l_{2} + \dots + l_{n}, t) = 0

For solving $α_{n}$ based on the boundary conditions above

\frac{C_{11}}{C_{41}} U_{f 1}^{k_{11}} = A_{11}, \frac{C_{21}}{C_{41}} U_{f 1}^{k_{21}} = A_{21}, \frac{C_{31}}{C_{41}} U_{f 1}^{k_{31}} = A_{31} \frac{C_{1 s}}{C_{41}} U_{fs}^{k_{1 s}} = A_{1 s}, \frac{C_{2 s}}{C_{41}} U_{fs}^{k_{2 s}} = A_{2 s}, \frac{C_{3 s}}{C_{41}} U_{fs}^{k_{3 s}} = A_{3 s}, \frac{C_{4 s}}{C_{41}} U_{fs}^{k_{4 s}} = A_{4 s}

for s = 2, … , n, and

{α_{js} = \frac{U_{as}^{k_{js}}}{U_{fs}^{k_{js}}} asthevelocityratio β_{js} = \frac{P_{as}^{k_{js}}}{P_{fs}^{k_{js}}} asthepressureratio z_{ajs} = \frac{P_{as}^{k_{js}}}{U_{as}^{k_{js}}} astheairphaseimpedence z_{fjs} = \frac{P_{fs}^{k_{js}}}{U_{fs}^{k_{js}}} asthefiberphaseimpedence

Based on the results,¹ we have

{α_{js} = \frac{v_{as} (1 - c_{fs}^{2} (\frac{k_{js}}{ω}) 2) - {iR}_{s} θ_{s}}{(v_{fs} - i θ_{s}) R_{s}} β_{js} = v_{as} \frac{(1 - i θ_{s}) (1 - c_{fs}^{2} (\frac{k_{js}}{ω}) 2) - {iR}_{s} θ_{s}}{v_{fs} (1 - i θ_{s} (1 + R_{s})) - i θ_{s} v_{as} c_{fs}^{2} (\frac{k_{js}}{ω}) 2} z_{fjs} = ρ_{fs} \frac{ω}{k_{js}} \frac{v_{fs} (1 - i θ_{s} (1 + R_{s})) - i θ_{s} v_{as} c_{fs}^{2} (\frac{k_{js}}{ω}) 2}{v_{fs} - i θ_{s}} z_{ajs} = \frac{β_{js}}{α_{js}} z_{fjs} = ρ_{as} v_{as} \frac{ω}{k_{js}} \frac{(1 - i θ_{s}) (1 - c_{fs}^{2} (\frac{k_{js}}{ω}) 2) - {iR}_{s} θ_{s}}{v_{as} (1 - c_{fs}^{2} (\frac{k_{js}}{ω}) 2) - {iR}_{s} θ_{s}}

where,

c_{fs}^{2} = \frac{K_{fs}}{ρ_{fs}}

is the square of the fiber wave velocity.

Substituting the boundary conditions $0$ and n in Equation (4), we have

{\sum_{j = 1}^{4} α_{jn} A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n}) = 0 \sum_{j = 1}^{4} A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n}) = 0 \sum_{j = 1}^{3} (v_{a 1} - v_{f 1} β_{j 1}) z_{fj 1} A_{j 1} = - (v_{a 1} - v_{f 1} β_{41}) z_{f 41} U_{f 1}^{k_{41}}

(5)

Substituting the boundary condition s = 2, … , n–1 in Equation (4), we have

{v_{a 2} \sum_{j = 1}^{3} β_{j 1} z_{fj 1} A_{j 1} e - {ik}_{j 1} l_{1} - v_{a 1} \sum_{j = 1}^{4} β_{j 2} z_{fj 2} A_{j 2} e - {ik}_{j 2} l_{1} = - v_{a 2} β_{41} z_{f 41} U_{f 1}^{k_{41}} e - {ik}_{41} l_{1} \sum_{j = 1}^{3} (β_{j 1} + 1) z_{fj 1} A_{j 1} e - {ik}_{j 1} l_{1} - \sum_{j = 1}^{4} (β_{j 2} + 1) z_{fj 2} A_{j 2} e - {ik}_{j 2} l_{1} = - (β_{41} + 1) z_{f 41} U_{f 1}^{k_{41}} e - {ik}_{41} l_{1} \sum_{j = 1}^{3} A_{j 1} e - {ik}_{j 1} l_{1} - \sum_{j = 1}^{4} A_{j 2} e - {ik}_{j 2} l_{1} = - U_{f 1}^{k_{41}} e - {ik}_{41} l_{1} v_{a 1} \sum_{j = 1}^{3} (α_{j 1} - 1) A_{j 1} e - {ik}_{j 1} l_{1} - v_{a 2} \sum_{j = 1}^{4} (α_{j 2} - 1) A_{j 2} e - {ik}_{j 2} l_{1} = - v_{a 1} (α_{41} - 1) U_{f 1}^{k_{41}} e - {ik}_{41} l_{1}

(6)

{v_{a (s + 1)} \sum_{j = 1}^{4} β_{js} z_{fjs} A_{js} e - {ik}_{js} (l_{1} + \dots + l_{s}) - v_{as} \sum_{j = 1}^{4} β_{j (s + 1)} z_{fj (s + 1)} \times A_{j (s + 1)} e - {ik}_{j (s + 1)} (l_{1} + \dots + l_{s}) = 0 \sum_{j = 1}^{4} (β_{js} + 1) z_{fjs} A_{js} e - {ik}_{js} (l_{1} + \dots + l_{s}) - \sum_{j = 1}^{4} (β_{j (s + 1)} + 1) z_{fj (s + 1)} \times A_{j (s + 1)} e - {ik}_{j (s + 1)} (l_{1} + \dots + l_{s}) = 0 \sum_{j = 1}^{4} A_{js} e - {ik}_{js} (l_{1} + \dots + l_{s}) - \sum_{j = 1}^{4} A_{j (s + 1)} e - {ik}_{j (s + 1)} (l_{1} + \dots + l_{s}) = 0 v_{as} \sum_{j = 1}^{4} (α_{js} - 1) A_{js} e - {ik}_{js} (l_{1} + \dots + l_{s}) - v_{a (s + 1)} \sum_{j = 1}^{4} (α_{j (s + 1)} - 1) \times A_{j (s + 1)} e - {ik}_{j (s + 1)} (l_{1} + \dots + l_{s}) = 0

(7)

{v_{an} \sum_{j = 1}^{4} β_{j (n - 1)} z_{fj (n - 1)} A_{j (n - 1)} e - {ik}_{j (n - 1)} (l_{1} + \dots + l_{n - 1}) - v_{a (n - 1)} \times \sum_{j = 1}^{4} β_{jn} z_{fjn} A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n - 1}) = 0 \sum_{j = 1}^{4} (β_{j (n - 1)} + 1) z_{fj (n - 1)} A_{j (n - 1)} e - {ik}_{j (n - 1)} (l_{1} + \dots + l_{n - 1}) - \sum_{j = 1}^{4} (β_{jn} + 1) z_{fjn} A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n - 1}) = 0 \sum_{j = 1}^{4} A_{j (n - 1)} e - {ik}_{j (n - 1)} (l_{1} + \dots + l_{n - 1}) - \sum_{j = 1}^{4} A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n - 1}) = 0 v_{a (n - 1)} \sum_{j = 1}^{4} (α_{j (n - 1)} - 1) A_{j (n - 1)} e - {ik}_{j (n - 1)} (l_{1} + \dots + l_{n - 1}) - v_{an} \sum_{j = 1}^{4} (α_{jn} - 1) A_{jn} e - {ik}_{jn} (l_{1} + \dots + l_{n - 1}) = 0

(8)

Noting that $k_{3 s} = - k_{1 s}$ and $k_{4 s} = - k_{2 s}$ , we know that

α_{3 s} = α_{1 s}, β_{3 s} = β_{1 s}, z_{a 3 s} = - z_{a 1 s}, z_{f 3 s} = - z_{f 1 s} α_{4 s} = α_{2 s}, β_{4 s} = β_{2 s}, z_{a 4 s} = - z_{a 2 s}, z_{f 4 s} = - z_{f 2 s}

(9)

In the open face case, the sound impedance of the multi-layer non-woven sound absorption structure at the front face (x = 0), z_n, is given by

z_{n} = \frac{p_{n}}{u_{n}} = \frac{p_{a 1} (0, t) + p_{f 1} (0, t)}{v_{a 1} u_{a 1} (0, t) + v_{f 1} u_{f 1} (0, t)}

(10)

Combining Equation (4), Equation (10) can be further rewritten as

z_{n} = \frac{(β_{11} + 1) z_{f 11} (A_{11} - A_{31}) + (β_{21} + 1) z_{f 21} (A_{21} - U_{f 1}^{k_{41}})}{(v_{f 1} + α_{11} v_{a 1}) (A_{11} + A_{31}) + (v_{f 1} + α_{21} v_{a 1}) (A_{21} + U_{f 1}^{k_{41}})}

(11)

Therefore, according to Equations (2) and (11), we can see that the unknown quantities A₁₁, A₂₁, and A₃₁ should first be solved in order to calculate the absorption coefficients of the multi-layer non-woven α_n.

Based on the above analysis, we can calculate α_n by the following steps, using the back-stepping method:

Step 1

Based on the boundary $n - 1$ , i.e. Equation (8), we obtain

A_{1 n} = Φ_{1}^{n - 1} (n - 1, n) A_{1 (n - 1)} e {iK}_{1}^{n - 1} + Φ_{2}^{n - 1} (n - 1, n) \times A_{2 (n - 1)} e {iK}_{2}^{n - 1} + Φ_{3}^{n - 1} (n - 1, n) A_{3 (n - 1)} e {iK}_{3}^{n - 1} + Φ_{4}^{n - 1} (n - 1, n) A_{4 (n - 1)} e {iK}_{4}^{n - 1}

A_{3 n} = Φ_{5}^{n - 1} (n - 1, n) A_{1 (n - 1)} e {iK}_{5}^{n - 1} + Φ_{6}^{n - 1} (n - 1, n) \times A_{2 (n - 1)} e {iK}_{6}^{n - 1} + Φ_{7}^{n - 1} (n - 1, n) A_{3 (n - 1)} e {iK}_{7}^{n - 1} + Φ_{8}^{n - 1} (n - 1, n) A_{4 (n - 1)} e {iK}_{8}^{n - 1} A_{2 n} = Φ_{9}^{n - 1} (n - 1, n) A_{1 (n - 1)} e {iK}_{9}^{n - 1} + Φ_{10}^{n - 1} (n - 1, n) \times A_{2 (n - 1)} e {iK}_{10}^{n - 1} + Φ_{11}^{n - 1} (n - 1, n) A_{3 (n - 1)} e {iK}_{11}^{n - 1} + Φ_{12}^{n - 1} (n - 1, n) A_{4 (n - 1)} e {iK}_{12}^{n - 1} A_{4 n} = Φ_{13}^{n - 1} (n - 1, n) A_{1 (n - 1)} e {iK}_{13}^{n - 1} + Φ_{14}^{n - 1} (n - 1, n) \times A_{2 (n - 1)} e {iK}_{14}^{n - 1} + Φ_{15}^{n - 1} (n - 1, n) A_{3 (n - 1)} e {iK}_{15}^{n - 1} + Φ_{16}^{n - 1} (n - 1, n) A_{4 (n - 1)} e {iK}_{16}^{n - 1}

where

Φ_{11}^{n - 1} (n - 1, n) = Φ_{1}^{n - 1} (n - 1, n) = W_{1} (n - 1, n) + W_{3} (n - 1, n) Φ_{12}^{n - 1} (n - 1, n) = Φ_{2}^{n - 1} (n - 1, n) = W_{2} (n - 1, n) + W_{4} (n - 1, n) Φ_{9}^{n - 1} (n - 1, n) = Φ_{3}^{n - 1} (n - 1, n) = W_{3} (n - 1, n) - W_{1} (n - 1, n) Φ_{10}^{n - 1} (n - 1, n) = Φ_{4}^{n - 1} (n - 1, n) = W_{4} (n - 1, n) - W_{2} (n - 1, n) Φ_{15}^{n - 1} (n - 1, n) = Φ_{5}^{n - 1} (n - 1, n) = W_{5} (n - 1, n) + W_{7} (n - 1, n) Φ_{16}^{n - 1} (n - 1, n) = Φ_{6}^{n - 1} (n - 1, n) = W_{6} (n - 1, n) + W_{8} (n - 1, n) Φ_{13}^{n - 1} (n - 1, n) = Φ_{7}^{n - 1} (n - 1, n) = W_{7} (n - 1, n) - W_{5} (n - 1, n) Φ_{14}^{n - 1} (n - 1, n) = Φ_{8}^{n - 1} (n - 1, n) = W_{8} (n - 1, n) - W_{6} (n - 1, n)

W_{1} (n - 1, n) = \frac{1}{2 (β_{1 n} - β_{2 n}) z_{f 1 n}} (\frac{v_{an}}{v_{a (n - 1)}} β_{1 (n - 1)} z_{f 1 (n - 1)} - ((1 - \frac{v_{an}}{v_{a (n - 1)}}) β_{1 (n - 1)} + 1) β_{2 n} z_{f 1 (n - 1)}) W_{2} (n - 1, n) = \frac{1}{2 (β_{1 n} - β_{2 n}) z_{f 1 n}} (\frac{v_{an}}{v_{a (n - 1)}} β_{2 (n - 1)} z_{f 2 (n - 1)} - ((1 - \frac{v_{an}}{v_{a (n - 1)}}) β_{2 (n - 1)} + 1) β_{2 n} z_{f 2 (n - 1)}) W_{3} (n - 1, n) = \frac{1}{2 (α_{1 n} - α_{2 n})} \times (1 - \frac{v_{a (n - 1)}}{v_{an}} + \frac{v_{a (n - 1)}}{v_{an}} α_{1 (n - 1)} - α_{2 n}) W_{4} (n - 1, n) = \frac{1}{2 (α_{1 n} - α_{2 n})} \times (1 - \frac{v_{a (n - 1)}}{v_{an}} + \frac{v_{a (n - 1)}}{v_{an}} α_{2 (n - 1)} - α_{2 n}) W_{5} (n - 1, n) = \frac{1}{2 (β_{2 n} - β_{1 n}) z_{f 2 n}} (\frac{v_{an}}{v_{a (n - 1)}} β_{1 (n - 1)} z_{f 1 (n - 1)} - ((1 - \frac{v_{an}}{v_{a (n - 1)}}) β_{1 (n - 1)} + 1) β_{1 n} z_{f 1 (n - 1)}) W_{6} (n - 1, n) = \frac{1}{2 (β_{2 n} - β_{1 n}) z_{f 2 n}} (\frac{v_{an}}{v_{a (n - 1)}} β_{2 (n - 1)} z_{f 2 (n - 1)} - ((1 - \frac{v_{an}}{v_{a (n - 1)}}) β_{2 (n - 1)} + 1) β_{1 n} z_{f 2 (n - 1)})

W_{7} (n - 1, n) = \frac{1}{2 (α_{2 n} - α_{1 n})} \times (1 - \frac{v_{a (n - 1)}}{v_{an}} + \frac{v_{a (n - 1)}}{v_{an}} α_{1 (n - 1)} - α_{1 n}) W_{8} (n - 1, n) = \frac{1}{2 (α_{2 n} - α_{1 n})} \times (1 - \frac{v_{a (n - 1)}}{v_{an}} + \frac{v_{a (n - 1)}}{v_{an}} α_{2 (n - 1)} - α_{1 n})

K_{1}^{n - 1} = (k_{1 n} - k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{2}^{n - 1} = (k_{1 n} - k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{3}^{n - 1} = (k_{1 n} + k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{4}^{n - 1} = (k_{1 n} + k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{5}^{n - 1} = - (k_{1 n} + k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{6}^{n - 1} = - (k_{1 n} + k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{7}^{n - 1} = (k_{1 (n - 1)} - k_{1 n}) (l_{1} + \dots + l_{n - 1}), K_{8}^{n - 1} = (k_{2 (n - 1)} - k_{1 n}) (l_{1} + \dots + l_{n - 1}) K_{9}^{n - 1} = (k_{2 n} - k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{10}^{n - 1} = (k_{2 n} - k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{11}^{n - 1} = (k_{2 n} + k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{12}^{n - 1} = (k_{2 n} + k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{13}^{n - 1} = - (k_{2 n} + k_{1 (n - 1)}) (l_{1} + \dots + l_{n - 1}), K_{14}^{n - 1} = - (k_{2 n} + k_{2 (n - 1)}) (l_{1} + \dots + l_{n - 1}) K_{15}^{n - 1} = (k_{1 (n - 1)} - k_{2 n}) (l_{1} + \dots + l_{n - 1}), K_{16}^{n - 1} = (k_{2 (n - 1)} - k_{2 n}) (l_{1} + \dots + l_{n - 1})

Step 2

Keeping the iterative calculation process based on the boundary s for s = n − 2, … , 1. In the general case, if we suppose

A_{1 n} = Φ_{1}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{1}^{s + 1} A_{1 (s + 1)} + Φ_{2}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{2}^{s + 1} A_{2 (s + 1)} + Φ_{3}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{3}^{s + 1} A_{3 (s + 1)} + Φ_{4}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{4}^{s + 1} A_{4 (s + 1)} A_{3 n} = Φ_{5}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{5}^{s + 1} A_{1 (s + 1)} + Φ_{6}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{6}^{s + 1} A_{2 (s + 1)} + Φ_{7}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{7}^{s + 1} A_{3 (s + 1)} + Φ_{8}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{8}^{s + 1} A_{4 (s + 1)} A_{2 n} = Φ_{9}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{9}^{s + 1} A_{1 (s + 1)} + Φ_{10}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{10}^{s + 1} A_{2 (s + 1)} + Φ_{11}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{11}^{s + 1} A_{3 (s + 1)} + Φ_{12}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{12}^{s + 1} A_{4 (s + 1)} A_{4 n} = Φ_{13}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{13}^{s + 1} A_{1 (s + 1)} + Φ_{14}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{14}^{s + 1} A_{2 (s + 1)} + Φ_{15}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{15}^{s + 1} A_{3 (s + 1)} + Φ_{16}^{s + 1} (s + 1, \dots, n - 1, n) e {iK}_{16}^{s + 1} A_{4 (s + 1)}

Then, based on the boundary s, i.e. Equation (7), we obtain

A_{1 n} = Φ_{1}^{s} (s, \dots, n - 1, n) e {iK}_{1}^{s} A_{1 s} + Φ_{2}^{s} (s, \dots, n - 1, n) e {iK}_{2}^{s} A_{2 s} + Φ_{3}^{s} (s, \dots, n - 1, n) e {iK}_{3}^{s} A_{3 s} + Φ_{4}^{s} (s, \dots, n - 1, n) e {iK}_{4}^{s} A_{4 s} A_{3 n} = Φ_{5}^{s} (s, \dots, n - 1, n) e {iK}_{5}^{s} A_{1 s} + Φ_{6}^{s} (s, \dots, n - 1, n) e {iK}_{6}^{s} A_{2 s} + Φ_{7}^{s} (s, \dots, n - 1, n) e {iK}_{7}^{s} A_{3 s} + Φ_{8}^{s} (s, \dots, n - 1, n) e {iK}_{8}^{s} A_{4 s} A_{2 n} = Φ_{9}^{s} (s, \dots, n - 1, n) e {iK}_{9}^{s} A_{1 s} + Φ_{10}^{s} (s, \dots, n - 1, n) e {iK}_{10}^{s} A_{2 s} + Φ_{11}^{s} (s, \dots, n - 1, n) e {iK}_{11}^{s} A_{3 s} + Φ_{12}^{s} (s, \dots, n - 1, n) e {iK}_{12}^{s} A_{4 s} A_{4 n} = Φ_{13}^{s} (s, \dots, n - 1, n) e {iK}_{13}^{s} A_{1 s} + Φ_{14}^{s} (s, \dots, n - 1, n) e {iK}_{14}^{s} A_{2 s} + Φ_{15}^{s} (s, \dots, n - 1, n) e {iK}_{15}^{s} A_{3 s} + Φ_{16}^{s} (s, \dots, n - 1, n) e {iK}_{16}^{s} A_{4 s}

where

Φ_{t}^{s} (s, \dots, n - 1, n) e {iK}_{t}^{s} = \sum_{j = 1}^{4} Φ_{tj}^{s} (s, \dots, n - 1, n) e {iK}_{tj}^{s} Φ_{t 1}^{s} (s, \dots, n - 1, n) = Φ_{1 + l}^{s + 1} (s + 1, \dots, n - 1, n) Φ_{t}^{n - 1} (n - 1, n) Φ_{t 2}^{s} (s, \dots, n - 1, n) = Φ_{2 + l}^{s + 1} (s + 1, \dots, n - 1, n) Γ_{t}^{n - 1} (n - 1, n) Φ_{t 3}^{s} (s, \dots, n - 1, n) = Φ_{3 + l}^{s + 1} (s + 1, \dots, n - 1, n) Δ_{t}^{n - 1} (n - 1, n) Φ_{t 4}^{s} (s, \dots, n - 1, n) = Φ_{4 + l}^{s + 1} (s + 1, \dots, n - 1, n) Ψ_{t}^{n - 1} (n - 1, n)

For t = 1, 2, … , 16, $l = 4 [\frac{t - 1}{4}]$ and

K_{(e + d) 1}^{s} = K_{1 + d}^{s + 1} + (k_{1 (s + 1)} - (- 1) [\frac{e}{3}] k_{ts}) \sum_{j = 1}^{s} l_{j} K_{(e + d) 2}^{s} = K_{2 + d}^{s + 1} + (k_{2 (s + 1)} - (- 1) [\frac{e}{3}] k_{ts}) \sum_{j = 1}^{s} l_{j} K_{(e + d) 3}^{s} = K_{3 + d}^{s + 1} + (- k_{1 (s + 1)} - (- 1) [\frac{e}{3}] k_{ts}) \sum_{j = 1}^{s} l_{j} K_{(e + d) 4}^{s} = K_{4 + d}^{s + 1} + (- k_{2 (s + 1)} - (- 1) [\frac{e}{3}] k_{ts}) \sum_{j = 1}^{s} l_{j}

for e = 1, 2, 3, 4, and d = 0, 4, 8, 12.

Here, $[x]$ is the integral function, defined as $[x] = m$ ; if $m \leq x < m + 1$ , m is an integer.

Step 3

Based on the result of Step 2 and the boundary conditions $0$ and n, i.e. Equation (6), we obtain the following equations

A_{11} = \frac{det (D_{1})}{det (D)} U_{f 1}^{k_{41}}, A_{21} = \frac{det (D_{2})}{det (D)} U_{f 1}^{k_{41}}, A_{31} = \frac{det (D_{3})}{det (D)} U_{f 1}^{k_{41}}

(12)

where

D = [a_{11} a_{12} a_{13} a_{21} a_{22} a_{23} a_{31} a_{32} a_{33}], D_{1} = [b_{1} a_{12} a_{13} b_{2} a_{22} a_{23} b_{3} a_{32} a_{33}] D_{2} = [a_{11} b_{1} a_{13} a_{21} b_{2} a_{23} a_{31} b_{3} a_{33}], D_{3} = [a_{11} a_{12} b_{1} a_{21} a_{22} b_{2} a_{31} a_{32} b_{3}]

a_{12} = (α_{2 n} - α_{1 n}) (Φ_{10}^{1} (1, \dots, n - 1, n) e i (K_{10}^{1} - k_{2 n} \sum_{j = 1}^{n} l_{j}) + Φ_{14}^{1} (1, \dots, n - 1, n) e i (K_{14}^{1} + k_{2 n} \sum_{j = 1}^{n} l_{j})) a_{13} = (α_{2 n} - α_{1 n}) (Φ_{14}^{1} (1, \dots, n - 1, n) e i (K_{11}^{1} - k_{2 n} \sum_{j = 1}^{n} l_{j}) + Φ_{15}^{1} (1, \dots, n - 1, n) e i (K_{15}^{1} + k_{2 n} \sum_{j = 1}^{n} l_{j})) b_{1} = (α_{1 n} - α_{2 n}) (Φ_{12}^{1} (1, \dots, n - 1, n) e i (K_{12}^{1} - k_{2 n} \sum_{j = 1}^{n} l_{j}) + Φ_{16}^{1} (1, \dots, n - 1, n) e i (K_{16}^{1} + k_{2 n} \sum_{j = 1}^{n} l_{j})) a_{21} = (α_{2 n} - α_{1 n}) (Φ_{1}^{1} (1, \dots, n - 1, n) e i (K_{1}^{1} - k_{1 n} \sum_{j = 1}^{n} l_{j}) + Φ_{5}^{1} (1, \dots, n - 1, n) e i (K_{5}^{1} + k_{1 n} \sum_{j = 1}^{n} l_{j})) a_{22} = (α_{2 n} - α_{1 n}) (Φ_{2}^{1} (1, \dots, n - 1, n) e i (K_{2}^{1} - k_{1 n} \sum_{j = 1}^{n} l_{j}) + Φ_{6}^{1} (1, \dots, n - 1, n) e i (K_{6}^{1} + k_{1 n} \sum_{j = 1}^{n} l_{j})) a_{23} = (α_{2 n} - α_{1 n}) (Φ_{3}^{1} (1, \dots, n - 1, n) e i (K_{3}^{1} - k_{1 n} \sum_{j = 1}^{n} l_{j}) + Φ_{7}^{1} (1, \dots, n - 1, n) e i (K_{7}^{1} + k_{1 n} \sum_{j = 1}^{n} l_{j})) b_{2} = (α_{1 n} - α_{2 n}) (Φ_{4}^{1} (1, \dots, n - 1, n) e i (K_{4}^{1} - k_{1 n} \sum_{j = 1}^{n} l_{j}) + Φ_{8}^{1} (1, \dots, n - 1, n) e i (K_{8}^{1} + k_{1 n} \sum_{j = 1}^{n} l_{j})) a_{31} = (v_{a 1} - v_{f 1} β_{11}) z_{f 11}, a_{32} = (v_{a 1} - v_{f 1} β_{21}) z_{f 21} a_{33} = - (v_{a 1} - v_{f 1} β_{11}) z_{f 11}, b_{3} = (v_{a 1} - v_{f 1} β_{21}) z_{f 21}

Step 4

Substituting Equation (11) in Equation (12), we obtain

z_{n} = \frac{{(β_{11} + 1) z_{f 11} (det (D_{1}) - det (D_{3})) + (β_{21} + 1) z_{f 21} (det (D_{2}) - det (D))}}{{(ν_{f 1} + α_{11} ν_{α 1}) (det (D_{1}) - det (D_{3})) + (ν_{f 1} + α_{21} ν_{α 1}) (det (D_{2}) - det (D))}}

(13)

Then, the absorption coefficient of multi-layer non-woven $α_{n}$ is given by

α_{n} = \frac{4 z_{n} (R) W_{0}}{(z_{n} (R) + W_{0}) 2 + (z_{n} (I)) 2}

(14)

Numerical simulation

In this section, various properties of the absorption coefficient α_n will be examined and plotted using MATLAB software. For double-layered non-wovens with polyester in the outer layer and nylon in the inner layer, the model obtained above may be used for numerical evaluation of the intrinsic characteristics of non-wovens to yield the highest absorption coefficient. The programs which calculate the sound absorption coefficient α₂ from Equation (14) are written as a function of l₁, l₂, $v_{a 1}$ and $v_{a 2}$ .

The constants involved in the calculation are as follows: $ρ_{0} = 1.293 kg / m 3$ , $c_{0} = 340.29 m / s$ , $P_{0} = 101325 Pa$ , $K_{a} = 1.01 \times 105$ , $K_{f 1} = 1.1 \times 1010$ , $K_{f 2} = 4.7167 \times 1010$ , $ρ_{p 1} = 1380 kg / m 3$ , $ρ_{p 2} = 1140 kg / m 3$ , $δ = 4168.6$ , $m_{1} = m_{2} = 1$ .

Taking $l_{1} = 4.5 cm$ and $l_{2} = 2.5 cm$ , the corresponding calculated sound absorption coefficients $α_{2}$ are shown in Figures 3 –7 as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at $f = 250 Hz$ , $f = 500 Hz$ , $f = 1000 Hz$ , $f = 2000 Hz$ , and $f = 4000 Hz$ , respectively. Similar graphs can be obtained for other values of thickness, l₁ and l₂. Each figure has a maximal point of $α_{2}$ for well defined values of $v_{a 1}$ and $v_{a 2}$ , and the corresponding values of $v_{a 1}$ and $v_{a 2}$ yielding the maximal value of $α_{2}$ are given in Tables 1 and 2, which can serve as guidelines for optimal design.

Figure 3.

Calculated sound absorption coefficient $α_{2}$ at l₁ = 4.5 cm and l₂ = 2.5 cm as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at f = 250 Hz.

Figure 4.

Calculated sound absorption coefficient $α_{2}$ at l₁ = 4.5 cm and l₂ = 2.5 cm, as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at f = 500 Hz.

Figure 5.

Calculated sound absorption coefficient $α_{2}$ at l₁ = 4.5 cm and l₂ = 2.5 cm as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at f = 1000 Hz.

Figure 6.

Calculated sound absorption coefficient $α_{2}$ at l₁ = 4.5 cm and l₂ = 2.5 cm as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at f = 2000 Hz.

Figure 7.

Calculated sound absorption coefficient $α_{2}$ at l₁ = 4.5 cm and l₂ = 2.5 cm as a function of $v_{a 1}$ and $v_{a 2}$ with the frequency set at f = 4000 Hz.

Table 1.

Optimal porosity v_a yielding the highest sound absorption coefficient when $l_{1} = 4.5 cm$

f (Hz)
l₂ (cm)	250	500	1000	2000	4000
0.5	$α_{max} = 0.0226$	$α_{max} = 0.174$	$α_{max} = 0.6935$	$α_{max} = 0.9837$	$α_{max} = 0.9832$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.13$	$v_{a 1} = 0.36$
	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
1	$α_{max} = 0.0547$	$α_{max} = 0.265$	$α_{max} = 0.7615$	$α_{max} = 0.9952$	$α_{max} = 0.9906$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.14$	$v_{a 1} = 0.37$
	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
1.5	$α_{max} = 0.0752$	$α_{max} = 0.2939$	$α_{max} = 0.7757$	$α_{max} = 0.9977$	$α_{max} = 0.9927$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.14$	$v_{a 1} = 0.37$
	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
2	$α_{max} = 0.0869$	$α_{max} = 0.3057$	$α_{max} = 0.7808$	$α_{max} = 0.9986$	$α_{max} = 0.9937$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.14$	$v_{a 1} = 0.37$
	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
2.5	$α_{max} = 0.0939$	$α_{max} = 0.3117$	$α_{max} = 0.7833$	$α_{max} = 0.999$	$α_{max} = 0.9942$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.14$	$v_{a 1} = 0.37$
	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$

Table 2.

Optimal porosity v_a yielding the highest sound absorption coefficient when $l_{2} = 4.5 cm$

f (Hz)
l₁ (cm)	250	500	1000	2000	4000
0.5	$α_{max} = 0.0233$	$α_{max} = 0.058$	$α_{max} = 0.1657$	$α_{max} = 0.4842$	$α_{max} = 0.9548$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$
	$v_{a 2} = 0.97$	$v_{a 2} = 0.95$	$v_{a 2} = 0.95$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
1	$α_{max} = 0.034$	$α_{max} = 0.0942$	$α_{max} = 0.2875$	$α_{max} = 0.7419$	$α_{max} = 0.9995$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.13$
	$v_{a 2} = 0.97$	$v_{a 2} = 0.97$	$v_{a 2} = 0.97$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
1.5	$α_{max} = 0.0447$	$α_{max} = 0.1307$	$α_{max} = 0.3946$	$α_{max} = 0.8815$	$α_{max} = 0.9995$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.18$
	$v_{a 2} = 0.98$	$v_{a 2} = 0.97$	$v_{a 2} = 0.98$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
2	$α_{max} = 0.056$	$α_{max} = 0.166$	$α_{max} = 0.487$	$α_{max} = 0.9546$	$α_{max} = 0.9994$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.22$
	$v_{a 2} = 0.98$	$v_{a 2} = 0.97$	$v_{a 2} = 0.98$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$
S	$α_{max} = 0.0672$	$α_{max} = 0.1999$	$α_{max} = 0.5663$	$α_{max} = 0.9888$	$α_{max} = 0.9989$
	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.1$	$v_{a 1} = 0.26$
	$v_{a 2} = 0.98$	$v_{a 2} = 0.97$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$	$v_{a 2} = 0.99$

From the simulation results, we can see that the effects of volume fraction of outer layer $v_{a 1}$ and inner layer $v_{a 2}$ on the acoustic properties of double-layered non-wovens are different. Specifically, with a small value of ν_a₁ (<0.15 for $f = 250 Hz$ and $f = 500 Hz$ , <0.2 for $f = 1000 Hz$ , <0.3 for $f = 2000 Hz$ and <0.4 for $f = 4000 Hz$ ); the absorption coefficient increases sharply with increase in ν_a₂, whereas for large values of ν_a₁ (>0.8), the absorption coefficient changes little with different values of ν_a₂.

It is seen that a multi-layer sound absorption structure made from non-woven materials has good sound absorption and can give a satisfactory absorption level over a given frequency range by choosing the appropriate parameters for each layer. The model developed in this paper for calculating the absorption coefficients of multi-layer non-woven components can provide a theoretical basis for high performance non-woven absorbent design and manufacture.

Conclusions

A general computational model for the absorption coefficient of multi-layer non-wovens, α_n, and the sound propagation boundary conditions at the adjacent interfaces has been presented based on the theory of Zwikker and Kosten for sound propagation through porous flexible media. An iterative algorithm for calculating the absorption coefficient α_n has been developed using the back-stepping method. Compared with the theoretical generalization of the Zwikker and Kosten model presented by Shoshani, the present model is more general and can be used to calculate the sound absorption coefficients of multi-layer absorption structures containing a variety of non-woven materials, and provides a theoretical support for product design. The numerical simulations show that by selecting the appropriate parameters for each layer, a multi-layer structure based on suitable non-woven materials can give satisfactory sound absorption over the required frequency range.

Footnotes

Funding

This work was supported by the National Natural Science Foundation of P. R. China (grant number 11102072), the Fundamental Research Funds for the Central Universities (grant number JUSRP21104), the Natural Science Foundation of Jiangsu Province (grant number BK2012254), Prospective industry–university research project of Jiangsu Province (grant numbers BY2011117 and BY2012065), the China Postdoctoral Science Foundation (grant number 20110490098), the Special Financial Grant from the China Postdoctoral Science Foundation (grant number 2012T50754) and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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