Noise Calculation Method for Industrial Premises with ... - Science Direct

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ScienceDirect Procedia Engineering 176 (2017) 218 – 225

Dynamics and Vibroacoustics of Machines (DVM2016)

Noise calculation method for industrial premises with bulky equipment at mirror-diffuse sound reflection I. Tsukernikov1,3, A. Antonov2, V. Ledenev2, I. Shubin1, T. Nevenchannaya3* 1

Rersearch Institute of Building Physics, 12738, Moscow, Russian Federation 2 Tambov State Technical University, 393320, Tambov, Russian Federation Moscow State University of Printing Arts, 127550, Moscow. Russian Federation

3

Abstract The paper presents a method of sound energy characteristics calculation in premises with bulky equipment at mirror-diffuse character of sound reflection from enclosures. At such character of reflection a part of sound energy falling on an enclosure is reflected specularly, and the other part is dissipated under the Lamberts law. Sound pressure levels in reference points of a premise are determined by the sum of direct sound energy emitted by sound sources and energy of mirror and diffuse components of reflected sound field. The numerical method is proposed for calculation of sound energy density and the subsequent determination of sound pressure levels. The method is developed on the basis of the combined design model in which the mirror component of the reflected sound energy is determined by ray-tracing method and the diffuse energy component is calculated by numerical statistical energy method. The equations for determination of density of the direct, mirror-reflected and diffused sound energy are provided and technique of realization of the design model is given. The proposed method takes into account space-planning features of premises, presence of bulky equipment, sound absorption characteristics of enclosures and the nature of sound reflection from surfaces, and provide an opportunity to solve problems of estimating noise in industrial premises. The comparison of the calculated and experimental data obtained for rooms of different sizes and shapes in the absence and presence of bulky equipment confirmed the adequacy of the proposed calculation model for the description of noise field formation in such conditions. It is shown that divergences of calculation and experimental data do not exceed ±2÷3 dB in octave bands with central frequencies equal or more than 250 Hz. © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd. is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review Machines. under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines Keywords: noise, calculation method, sound pressure level, energy density, mirror-diffuse reflection

* Corresponding author. Tel.: +7-495-482-5093; fax: +7-495-482-4076 . E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the international conference on Dynamics and Vibroacoustics of Machines

doi:10.1016/j.proeng.2017.02.291

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1. Introduction It is known that the sound pressure level at any point in the room is determined by the components of direct and reflected sound energy propagating in a confined space. When calculating the direct sound energy, it is necessary to consider the shape and size of the sound source, and the distance from the calculated point. The reflected sound energy is influenced by space-planning parameters of room, sound absorption characteristics of walls and equipment surfaces, the nature of the sound reflected from the surface, location of noise sources, location and dimensions of the equipment in the premises. Typically, industrial facilities do not have a simple geometric shape and there is no sufficiently accurate information about the acoustic characteristics of the surfaces of walls. Besides, a number of other factors increase the uncertainty in the specification of initial and boundary conditions. In such circumstances, the most appropriate methods of solving the problem of sound energy distribution are methods based on statistical energy approach [1] and principles of computer simulation of trajectories and energy rays emitted by the sound source [2]. The choice of a particular method for each situation is largely determined by the nature of sound energy reflection. Sound reflection from the room walls and equipment occurs on complex dependencies, the description of which in general terms is not possible. Therefore, in the calculations we used idealized models with mirror or diffuse components of sound reflection. In developing more reliable calculation methods one can use closer to real, mirror-diffuse sound reflection, in which a part of the sound energy is reflected specularly, and the rest of the reflected energy is scattered diffusely. The calculation model is detailed below. 2. The combined calculation model of the sound field in industrial premises for the mirror-diffuse type of reflection We propose to use a combined calculation model, where energy distribution of direct sound and specularly reflected energy are determined by the method of ray tracing, and density distribution of diffusely reflected energy is measured by a numerical statistical energy method. The acoustic energy density at any i-th point of the room is determined by the sum of the energy densities of the direct sound and mirror and diffuse components of the reflected energy. Accordingly, the total sound pressure level Li, [dB], is calculated as

dif Li  10 lg[c( idir   imir   i ) / I0 ] ,

(1)

where с is the sound speed in air, [m/s];  i is the direct sound energy density, [J/m3];  imir and  idif are the energy densities of the mirror and diffuse components of the reflected energy, [J/m3]; I 0  10 12 [W/m 2 ] is the reference sound intensity. The essence of the proposed combined model is as follows. A sound source generates a number of sound rays in accordance with its free space pattern. Each of the rays carries a part of the source sound energy. The energy of all rays is equal to the total energy emitted by the source in a unit time, that is, its acoustic power. As the rays travel, they lose energy due to absorption in air and on surfaces of room boundaries or obstacles, which they are meeting. Each ray also loses energy, which is scattered at reflection, i.e. a part of reflected sound energy of the ray is transformed into diffuse component and a part of the energy is reflected specularly back to the room. Then the reflected ray energy is lost again when the ray hits a next obstacle. Each ray is traced until it loses its energy completely due to air and surface absorption and transformation of its mirror component into a diffuse one. Thus all rays emitted by source are traced and direct and mirror reflected sound energy of all rays passing through the reference point are summarized. Energy distribution of rays scattered diffusely is determined by the method based on the statistical energy approach [3] and used by us at noise calculations in industrial premises at diffuse reflection of sound from enclosures [7]. The basic construction principles of calculation methods included in a combined calculation model are considered below. When calculating the direct sound energy and mirror reflected energy we used an approach, in which the rays have infinitely small spatial angles of propagation. Sound power carried by each k-th ray after it has passed from the dir

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sound source to the i-th reference point of the room is calculated: for direct sound in the points within the line-of-sight-propagation as

dir  W ki





W exp  m a R ki , N

(2)

for mirror reflected sound as

P D k kp , mir  W exp  m R Wki a ki  [(1   p ) p ] N p 1





(3)

where W is the sound power of the source, [W]; N is the number of rays emitted from the source, [-]; ma is the spatial coefficient of sound attenuation in the air, [m-1]; Rki is the distance traveled by the k-th ray from the source to the i-th elementary volume, [m]; р is the sound absorption coefficient of the p-th surface of the enclosure encountered by a traceable ray, [-]; Pk is the total number of reflection acts of the k-th ray from all reflective surfaces encountered on its way while travelling Rki distance to the i-th elementary volume, [-]; Dkp is the number of acts of the k-th ray falling on the p-th surface while travelling the distance Rki, [-]; kp is the share of energy directed by the k-th ray after its reflection from the p-th surface of the enclosure, [-]. The amount of mirror reflected energy in the elementary volume is equal to the sum of the energies of K rays passing through it, and its density is calculated as

K mir εimir   Wki cSred , k 1

(4)

where Sred is reduced cross-sectional area of the elementary volume, [m2]. In the combined model, in which the entire volume of the room is divided into elementary volumes in the form of cubes or parallelepipeds, we assume the square as cross-sectional area of the sphere equal to the volume of elementary cube or parallelepiped. A numerical statistical energy method used to determine diffusely reflected energy is based on the concept of quasi-diffuse reflection of the sound field. The reflected sound fields have a quasi-diffuse nature as they satisfy the isotropy condition of angular orientation of elementary flows, but they do not satisfy the uniformity condition [1]. For such fields, the isotropy condition enables to use the statistical theory of acoustics in the development of calculation model. At the same time, the lack of uniformity results in the need to incorporate in the model the resulting flow of reflected energy bound up with the potential magnitude of the reflected field, i.e. the sound energy density. Using general concepts of the phenomenological approach to the transfer phenomenon, the relationship between the density of flow calculated as follows [1]:

q    grad

dif ,

q , [W/m2], and the density of diffusely reflected sound energy  dif , [J/m3], is

(5)

where  is the ratio between the density of the resulting flow and density gradient in the energy of the reflected sound field of the room, [m2/s]. The value of the coupling coefficient is calculated using a statistical approach, in which the macroscopic properties of the reflected sound field are identified with some average statistical properties of field elements, the sound rays in this case [4]. It was found that in quasi-diffuse noise fields the coefficient can be determined as

  0,5cl ,

(6)

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and correspondingly,

q  0,5cl  grad

dif ,

(7)

where l is the mean free path of acoustic waves in the room, [m]. On the basis of the Equation (7) an equation of density distribution of diffusely reflected sound energy in quasidiffuse sound field was obtained in [4]. For the stationary quasi-diffuse sound field this equation has the form

 2 dif  cma dif  0

,

(8)

2

where  is the Laplace operator. The boundary conditions for the equation (8), according to [4] have the form dif  n

 S

s dif ,  (2   s )l S

(9)

where  s is a diffuse sound absorption coefficient in the point of the enclosure surface, [-]. Equation (8) with the boundary conditions (9) is a mathematical model of distribution of average density of frequency band of diffusely reflected sound energy in stationary excitation. The boundary conditions (9) make it possible to take into account the value of absorption of each section of volume surface, and hence its individual contribution to the formation of the reflected field. Therefore, the model can be useful for estimating the distribution of reflected energy in rooms with bulky equipment. For the first time the calculation model in a similar formulation was proposed in the 1970s and analog modeling method was developed for its implementation [5]. Later on in the 1980s, analytical [6] and numerical [3] methods for model implementation were proposed. It is necessary to notice that the similar approach to the estimation of diffusely reflected energy is used now also by other researchers at the solution of various acoustic problems [8, 9]. In the case of complex shapes of premises and in the presence of bulky equipment, the numerical method of energy balances is the most appropriate for the model implementation [1, 4]. A room is divided into elementary volumes (see Fig. 1), within which the variation of the reflected energy density can be adopted to be linear. The position and size of the equipment is also taken into account. For each elementary volume a balance equation of diffusely reflected sound energy per unit time is made. The overall distribution of the density of diffusely reflected energy is obtained by solving the system of algebraic equations. The balance of the reflected energy for each i, j, k-th elementary volume considering absorption of sound in air generally is written as

N 6 N dif 6 N dif  q n S n   Wm   q( )m S m  cma ε i, j ,k Vi, j,k  0 , n1 m1 m1

(10)

where q n is the density of diffuse energy flows, [W/m2], among i, j, k-th volume and adjacent volumes dif contacting through the joining surfaces Sn, [m2 ]; Wm is the diffuse component of sound power, [W], coming into i, j, k-th volume after rays reflection from the m-th surface of the volume with the area S m , which is the surface of the enclosure or equipment, [m2]; q( ) m is the density of diffuse energy flow, [W/m2], absorbed on the m-th surface of i, j, k-th volume, which is the surface of enclosure or equipment with the area S m , [m2]; N is the number of elementary volumes contacting with i, j, k-th volume, [-]; 6-N is number of faces of i, j, k-th volume, which are dif surfaces of enclosure or equipment, [-]; Vi,j,k is the parallelepiped elementary volume, [m3];  i , j , k is the density of 3 diffusely reflected energy in i, j, k-th volume, [J/m ].

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Fig. 1. Schemes of dividing space into elementary volumes.

For internal volumes, which are not in contact with enclosures and equipment, the reflected energy balance is written as

6 dif  qn Sn  cma εi, j,kVi, j ,k  0 . n1

(11)

The last terms in equations (10) and (11) show the energy loss in the i, j, k-th volume caused by air absorption. Densities of energy flows qn, [W/m2], are determined as

q n  η(ε

dif dif ε )/hn , i, j , k n

(12)

where index n  i  1, j, k ; i  1, j, k ; i, j  1, k ; i, j  1, k ; i, j, k  1; i, j , k  1, [-]; hn is the distance between centres of i, j, k-th volume and contacting volumes in the direction n, [m]. The value of flow density q(α)m, [W/m2], is calculated by the formula

dif  m  cεi, j, k

, q( )m  2( 2  αm )

(13)

where αm is the sound absorption coefficient of m-th surface of the i, j, k-th volume, [-]. dif

The value Wm , [W], is the sum of the energy of rays converted into diffuse component when rays are reflected from the m-th surface of the i, j, k-th volume, which is part of the surface of enclosures or equipment, calculated as K dif Wm  (1   m )(1   m )  W mir , k 1 ki

(14)

where K is the number of rays incident on the m-th surface of the i, j, k-th volume, [-]; m is the share of mirror energy directed along the ray after it is reflected from the m-th surface of the enclosures or equipment, [-].

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To implement the calculation model a computer program was developed to perform calculations of sound pressure levels in the industrial premises of any shape and containing production equipment. The program is written in language Visual Basic 6, constructed by a modular-block principle, contains the block of input of the initial data, calculation modules, the block of the analysis of results and report formation [10]. The algorithm of calculation of a sound field mirror component is based on results of works [10-16]. The diffusely reflected energy is defined as a result of the decision of system of the algebraic linear equations of the statistical power approach by a method of simple iterations with use of Zejdel’s principle [17]. Now the calculation model and the computer program are used as well for premises without the equipment, for example, for noise calculations in air channels [18]. 3. Experimental verification of the calculation model To assess the reliability of the combined calculation model we carried out special experiments on real objects of complex shape with bulky equipment, structures and other objects scattering the sound energy. Fig. 2 shows the scheme of ventilation shaft inside the library building and input data for the calculation of the system. The ceiling and floor in the shaft were made of concrete; the walls were made of concrete and brick. Diffusers were large volumes of building structures made of concrete and brick.

Fig 2. Scheme of the ventilation shaft of the library building

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Fig. 3 shows the experimental and calculated sound pressure levels in the shaft. Calculations were made using the combined method of mirror-diffuse reflection model in which the quantity of specularly reflected energy was 85% (=0.85) of the total reflected energy. There was good agreement between the calculated and experimental data. The divergence between calculation and experiment in the octave band with central frequency 1000 Hz does not exceed 2.0 dB. In the lower frequency bands the divergences make the big sizes. For example, in the octave band with central frequency 250 Hz they reach 3 dB. Divergences increase as sound diffraction on scatterers is not considered in calculation model. Diffraction influence increases at low frequencies with increase of wave length. Similar results have been received and at other positions of the sound source in the given ventilating shaft. As a whole, the series of experimental researches executed by us and the comparative analysis for calculation and experimental data of premises with the large-sized equipment have shown that the method has sufficient accuracy in octave bands with central frequencies equal or more than 250 Hz. In these cases discrepancy between the calculated and experimental data did not exceed ± 2 ÷ 3 dB.

Fig 3. The experimental and calculated sound pressure levels in the shaft.

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4. Conclusions 1. The proposed combined calculation model and its program implementation provide an opportunity to solve problems of estimating noise in industrial premises. It takes into account space-planning features of premises, presence of bulky equipment, sound absorption characteristics of enclosures and the nature of sound reflection from surfaces. 2. The comparison of the calculated and experimental data obtained for rooms of different sizes and shapes in the absence and presence of bulky equipment confirmed the adequacy of the proposed calculation model for the description of noise field formation in such conditions. Divergences of calculation and experimental data do not exceed ±2÷3 dB in octave bands with central frequencies equal or more than 250 Hz. References [1] Ledenev, V.I. 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