Environmental Engineering and Management Journal [619151]

Environmental Engineering and Management Journal
July 2014, Vol.13, No. 7, 1743-1749
http://omicron.ch.tuiasi.ro/EEMJ/

“Gheorghe Asachi” Technical University of Iasi, Romania

MATHEMATICAL MODELLING OF SOUND PRESSURE LEVEL
ATTENUATION TRANSMI TTED BY AN ACOUSTIC SCREEN IN
INDUSTRIAL ENVIRONMENT

Claudia Tomozei1, Florin Nedeff1, Gigel Paraschiv2, Oana Irimia1,
Greta Ardeleanu1, Alina Con țu Petrovici1

“Vasile Alecsandri" University of Bac au, Faculty of Engineering, Calea M ărășești, 157 Bac ău, 600115, România
2UPB – Faculty of Biotechnical Systems, Splaiul Independen ței no. 313, S6, 060042 Bucharest, Romania

Abstract
This paper presents a three-dimensiona l mathematical model which characterizes the variation of sound pressure level
propagation and the variation of attenuation of the sound pressure level propagated in an enclosed space. The variable taken in to
consideration both in the experiments and in the mathematical modeling were: the positions of the acoustic screen to noise
source; the height of the microphone to record the sound pressure level for a variable number of walls. The math ematical model
is based on the experimental data obtained in laboratory, using an experimental setup, which comprise s a variable number of
walls as acoustic screen.

Key words: industrial noise, mathematical model, sound pressure level

Received: March, 2014; R evised final: July, 2014; Accepted: July, 2014

 Author to whom all correspondence should be addressed: e-mail: [anonimizat] 1. Introduction

Industrial environment is characterized by a
multitude of noise sources. The noise level generated
can affect the workers human health and can generate
impacts on their productivity. The noise level generated by industrial equipment and installations is
influenced by many factors related both to the nature
of the equipment and the environment in which it work. Such factors influencing the generated noise
are specific to the noise source, the propagation
medium and the ways of propagation (Chatillon,
2007; Dupont and Annick, 2009; Pinte et al., 2009;
Tomozei et al., 2011a; Tomozei, 2011; Tomozei et al., 2012). In order to reduce the noise level a series
of methods are applied directed towards the three-
main factors associated to noise, respectively source,
propagation paths and receiver (Chatillon, 2007;
Dupont and Annick, 2009; Pinte et al., 2009; Tomozei et al., 2011b). E xperimental determination
of the sound pressure level was based on the position
of the acoustic screen place ment to the noise source
and the recording positions of the microphone, but
also on the number of walls the acoustic screen was
made (Basaran and Lin, 2008; Law et al., 2011).
Measurement of sound pressure level is based
on the procedure which provides information on how
to use the equipment and placing the microphones, the time necessary for one measurement, the
determining the variation of the noise level
propagated inside the enclosure (Beranek and
Mellow, 2012; Bies and Hansen, 2009).
Mathematical modeling is a method to check
the viability of research method (Ani et al., 2012;
Bratu et al., 2011; Hatzigeorg ioub, 2008; Heinz et al.,
2003; Ioan and Ursu, 2010; Mitran et at., 2012; Petrescu et al., 2011; St erpu et al., 2010; Talamon
and Csoknyai, 2011).

Tomozei et al./Environmental E ngineering and Management Journal 13 (2014), 7, 1743-1749

1744 2. Materials, methods and experimental results

The mathematical model presented in this
paper is based on the results of experiments addressing the determination of the sound pressure
levels emitted by a noise source in the laboratory.
During the propagation of acoustic waves it was interposed an acoustic scre en consistsing of a wall or
several walls.
The experimental setup was a cube with the
side of 1 m, with a metallic support, where walls
made of different materials: OSB (Oriented Strand Board), plasterboard, mineral wool, textile material,
polystyrene and corrugated cardboard are attached
(Fig. 1a). The number of walls can be varied so as to ensure different conditions for measuring sound
pressure level generated by the noise source through
a wall, two walls, three walls, three walls and lid, four walls and after cabin (five walls). The
experimental setup was located inside a laboratory,
whose interior was not treated with soundproofing materials, so that the energy of the acoustic waves
was not absorbed by the walls of room.
The noise source emitted a noise level of
about 90 dB; the noise generation mechanism was simple and consisted of a toothed wheel, which in
rotation touching an elastic lamella of metal (Fig.
1b). In Fig. 2 illustrates the points of the microphone
location (R – receiver) and positioning points of the
noise source (S). These points have been located on
the floor and at a distance of 0.5 m on vertical,
respectively in near the wall and at 0.5 m to the wall.
Measurements for determining sound pressure
level were based on working procedure that
determined the reference sound pressure level
generated by a noise source located inside an
enclosed space. These values of the reference sound
pressure level were compared with the sound
pressure level values measured during sound waves
propagation through the interposition of an acoustic
screen consists of a wall or more walls (Tomozei,
2011).
The values of reference sound pressure level
(SPL) were recorded for the four points of the noise
source position 0/0, 0.5/0, 0/0.5 and 0.5/0.5 (Fig. 2)
to the sixteen points of th e receiver location without
sound barrier. Reference sound pressure level values
are shown in Table 1, the va lue at which the reported
it was is the A-weighted sound pressure level, L
Aeq
(Tomozei, 2011).

(a) (b)

Fig. 1. The image of the screening cab of the noise source (a ) and the generating source of noise (b) (Tomozei, 2011)

Fig. 2. The graphical representation of the location of the measurem ent points, side view (Tomozei, 2011): S1, S2, S3, S4 – four
points for the location of the noise source; R – 16 points for the location of the receiver (microphone); E – Acoustic screen
S1, S2, S3, S4 – four points for the location of the noise sour ce; R – 16 points for the location of the receiver (microphone);
E – Acoustic screen

Mathematical modelling of sound pressure level attenuation tr ansmitted by an acoustic screen in industrial environment

1745
Reference values of sound pressure level
generated by the functioning of the noise source,
recorded the first point of placing the microphone (at
0.5 m away from the acoustic screen) were of 88.8
dB in point 0/0, of 86.6 dB in the point 0.5/0, of 87.7
dB in the point 0/0.5 and of 86.8 dB in the point
0.5/0.5 on positioning of the noise source (Tomozei,
2011). In each point of the noise source positioning it
was observed that the propagation of sound pressure
level showed a linear decrea se of noise at the four
heights of the microphone record (Tomozei, 2011).
Reference sound pressure level values were
useful in determining the sound pressure level,
respectively sound pressure attenuation level for the
measurements made with acoustic screen to obtain
calculated values of the sound pressure level
attenuation. The sound pressure level attenuation
values were calculated by reporting the reference
values of sound pressure level at the sound pressure
level values obtained in the first recording point by
interposing the acoustic screen at 0.5 m from screen
for all four heights registration (Tomozei, 2011).
The experiments for determination of sound
pressure level were carried out with the six types of
acoustic screen and the six materials mentioned above. These were carried out through the
interposition of acoustic screens after the noise
source at different distances (Fig. 2) and by varying
the recording position of the microphone (Fig. 2).
The values obtained as a result of sound
pressure level measurements propagated in enclosed
spaces were graphically represented, depending on
the position of the noise source and the number of
walls on the acoustical screen used for each set of
values recorded. One of the graphics can be seen in
Fig. 3. In the graphical representation, a linear
decrease of the sound pressure level was recorded,
both for reference values and for the values measured
after the acoustical screen.
The graphical representation of the attenuation
variation of sound pressure level was performed for
the six combinations of the wall, depending on the position of the noise source and of the recording
microphone position, through the acoustic screen
made of various materials (Fig. 4). The mathematical model was developed based on the sound pressure
level values and respectively the attenuation values
of the sound pressure level obtained experimentally and calculated.

Table 1. Reference values of the sound pressure level (dB) measured in the four points of locatio n of the noise source without
acoustic screen (Tomozei, 2011)

The position of the nois e source in the point
0/0 0.5/0 0/0.5 0.5/0.5 Position of the
microphone from
the noise source 0.5
m 1 m 2 m 4 m 0.5
m 1 m 2 m 4 m 0.5
m 1 m 2 m 4 m 0.5
m 1 m 2 m 4 m
0 88.8 86.5 83.8 80.9 86.6 84. 4 82.3 80.2 88.7 86.6 86.1 84.6 86.8 86.1 85.5 84.7
0.6 88 86.4 83.5 80.5 86.2 83.8 82.5 80.3 87. 6 86.9 86.2 84.9 87.2 86.6 85.6 84.9
1.2 86.1 84.6 82.6 80.2 86.5 83. 1 81.8 81.1 88.3 86.6 86.1 84.2 87.1 85.9 85.1 84.4 Microphone
position on
the vertical
(m)
1.8 83.9 83.4 81.8 80.2 85.7 82. 7 80.7 79.9 87.7 86.5 85.9 83.7 86.9 85.4 84.5 83.8

75808590
Distance of the microphone from the noise source, D (m)The microphone
position at height:
reference values
0 m
0,6 m
1,2 m
1,8 m
the values det. after acoustical screen
0 m
0,6 m
1,2 m
1,8 mSound pressure level, LAeq (dB)
0 4 2 1 0.5Position of the noise source
0/0.5 0.5/0.5
0.5/0 0/0

Fig. 3. Graphical representation of the sound pr essure level variation depending on th e noise source position and recording
position of the microphone, through the acoustic screen of OSB, variant with one wall, the noise source position in the point
0.5/0.5 (Tomozei, 2011)

Tomozei et al./Environmental E ngineering and Management Journal 13 (2014), 7, 1743-1749

1746
1 w all 2 w alls 3 w alls 3_1 w alls 4 w alls 5 w alls024681012The acoustic attenuation (dB )
The position of source in the point 0.5/0.5 0 m
0.6 m
1.2 m
1.8 m
Fig. 4. Graphical representation of the attenuation variation
of sound pressure level for th e six combinations of walls,
depending on the noise source position (point 0.5/0.5) and
recording position of the microphone, with the acoustic
screen of OSB (Tomozei, 2011)
3. Mathematical model

3.1. Elaboration of the mathematical model

The mathematical modeling was performed
using with the 3D Table Curve program, which
generated the linear and nonlinear equations of the
propagation variation of the acoustic wave according to the location of the nois e source and the microphone
height positioned for the recording of the sound
pressure level. Thus, it was obtained a surface characterizing the propagation variation of the sound
pressure level depending on the positions of the noise
source and the height of th e recording microphone for
a variable number of walls.
The equation which characterizes the surface
obtained is in the form of Eq. (1) (Tomozei, 2011).

yjx ixy hy gx fxy ey dx cy bxaz2 2 3 3 2 2
(1)
where: x is the position of the noise source in the
points 0/0; 0/0.5; 0.5/0; 0.5/0.5; y is the microphone
height to recording (the r eceiver height): 0 m, 0.6 m,
1.2 m and 1.8 m;
The corresponding values of the coefficients
calculated by Eq. 1 are given in Tables 2 and 3. These
values describe the mathematical model corresponding to sound pressure level or attenuation
of sound pressure level for a variant working with
each of the six types of material. By means of the values of coefficients a, b, c, d, e, f, g, h, i and j in Eq.
(1) it is described the variation of sound pressure
level (Table 2) respectively the attenuation of the
sound pressure level (Table 3). These values are
obtained by measurements after the acoustic screen depending of noise source position and the height of
the microphone record the sound pressure level.
Then, it was calculated the correlation coefficient for the mathematical model, in which the values of sound
pressure level respectively the attenuation values of
the sound pressure level for each variant working with the six types of materials are used. The
correlation coefficients are listed in Tables 4 and 5,
and ranges between 0.69 – 0.98. They differ according to the recorded values of sound pressure level for
each type of wall, the material used and the walls
number of the acoustic screen.
Mathematical models were graphically
represented by surfaces characterizing the
propagation variation of the sound pressure level, depending on the positions of the noise source and the
height of the microphone record (receiver). Fig. 5
shows the surface of the sound pressure level
variation for the case of wall with mineral wool. The
second surface obtained shows the attenuation variation of the sound pressure level depending on the
positions of the noise source and height of the
microphone record for a variable number of walls. The response surface for this mathematical model in
the variant with two walls of OSB is shown in Fig. 6.

3.2. Verification of the mathematical model

After the mathematical modeling performed
using the software TableCurve 3D have resulting two
mathematical models. These mathematical models
were checked for both cases as follows:

Table 2. Values of the equation constants that describe the ma thematical model corresponding to sound pressure level for
experimental variants with five walls (Tomozei, 2011)

Band Equation’s
constants OSB Polystyrene Plaste rboard Mineral wool Textil
material Corrugated
cardboard
1. a 76.1779703 72.91576733 78.82101485 72.64616337 69.22217822 71.3735396
2. b 0.322536054 3.164660366 0.226055206 -7.66929678 -11.2925042 -2.84368612
3. c 0.477447745 -5.15572057 -1.04803355 -9.48367024 -3.57198157 -4.51718234
4. d -0.59066667 -1.302 -0.088 2.089666667 6.381 0.918333333
5. e 0.985973597 13.51141364 4.065250275 5.715587184 1.551670792 6.233670242
6. f 0.330726073 -1.07630363 0.683729373 2.03310231 0.470165017 1.444884488
7. g 0.105050505 0.140606061 0.004848485 -0.17838384 -0.81515152 -0.08707071
8. h -1.35030864 -5.63271605 -2.81635802 -1.63966049 -0.44367284 -2.83564815
9. i 0.247524752 0.149889989 -0.08938394 -0.48748625 -0.02131463 -0.25921342
10. j -0.14 0.18 -0.08 -0.10333333 -0.09 -0.15

Mathematical modelling of sound pressure level attenuation tr ansmitted by an acoustic screen in industrial environment

1747Table 3. Attenuation of the sound pressure level represented by th e values of equation constants described on the mathematical
model for experimental varian t with a wall (Tomozei, 2011)

Band Equation’s
constants OSB Polystyrene Plaste rboard Mineral wool Textil
material Corrugated
cardboard
1. a 7.051212871 8.00720297 6. 959455446 9.997549505 10.56420792 10.49648515
2. b 0.628069057 -1.92724047 3.476278828 -2.7814675 -3.24989379 -5.348535
3, c -0.96652228 -1.80114824 -1 .41988449 -2.91514714 -4.54686469 0.889872112
4. d -0.92166667 0.145 -2 .38933333 0.419666667 0.144 0.183666667
5. e -2.7927324 -0.93148377 -2 .30679318 0.348081683 0.617436744 -7.66656353
6. f 2.79009901 2.816831683 3. 322079208 3.558184818 4.386435644 6.17019802
7. g 0.138383838 0.017575758 0. 314343434 -0.00989899 0.054545455 0.113737374
8. h 1.099537037 0.366512346 1.041666667 -0.13503086 -0.00000001 2.488425926
9. i 0.043316832 -0.06531903 -0. 27227723 -0.22758526 -0.13613861 0.161578658
10. j -0.48333333 -0.45 -0.46 -0.53666667 -0.69333333 -1.06333333

Table 4. Correlation coefficient values of the equations that describe the mathematical model given by the recorded values of the
sound pressure leve l (Tomozei, 2011)

The correlation coefficient of the equation s generated by mathematical modeling
Band Number of
walls OSB Polystyrene Plaste rboard Mineral wool Textile
material Corrugated
cardboard
1. 1 wall 0.81 0.85 0.76 0.91 0.96 0.83
2. 2 walls 0.75 0. 87 0.79 0.84 0.95 0.95
3. 3 walls 0.85 0. 88 0.87 0.95 0.91 0.95
4. 3+1 walls 0.74 0. 83 0.78 0.98 0.95 0.95
5. 4 walls 0.96 0. 97 0.98 0.98 0.94 0.95
6. 5 walls 0.82 0. 95 0.98 0.95 0.93 0.87

Table 5. Attenuation values of the sound pressure level given by the correlation coefficien t of the equations that describe the
mathematical model ob tained (Tomozei, 2011)

The correlation coefficient of the equation s generated by mathematical modeling
Band Number of
walls OSB Polystyrene Plaste rboard Mineral wool Textile
material Corrugated
cardboard
1. 1 wall 0.90 0.95 0.94 0.98 0.96 0.83
2. 2 walls 0.90 0. 92 0.94 0.96 0.98 0.97
3. 3 walls 0.90 0. 98 0.95 0.97 0.97 0.94
4. 3+1 walls 0.85 0. 93 0.83 0.95 0.97 0.88
5. 4 walls 0.92 0. 96 0.95 0.95 0.92 0.91
6. 5 walls 0.69 0. 93 0.93 0.94 0.97 0.98

Fig. 5. Variation of sound pressure level depending to the noise
source positions and recordin g microphone height (receiver)
(Tomozei, 2011)
Fig. 6. Variation of attenuation of the sound pressure level
depending to the noise source positions and recording
microphone height (Tomozei, 2011)

The mathematical model describing the
variation of sound pressure level according to the
noise source positions and the height of the
microphone record for a variable number of walls. The mathematical model describing the variation of
attenuation of the sound pressure level depending to
the noise source positions and the height of the
microphone record for a variable number of walls.

Tomozei et al./Environmental E ngineering and Management Journal 13 (2014), 7, 1743-1749

1748 In this paper, we presented only some of the
experimental values together with those resulted from
models. It was found that the relative deviation
calculated is between -1.75 – 1.79. In Fig. 7 and 8 there are plotted the math ematical model of the
relative deviations calculated by reporting at the
number of experiments.
0 2 04 06 08 0 1 0 0-2.0-1.5-1.0-0.50.00.51.01.52.0 The error of the mathematical modelThe error of the mathematical model
Experiment number

Fig. 7. The graphical representation of the mathematical
model relative deviation de pending on the experiments
number of the sound pressure level
0 2 04 06 08 0 1 0 0-2.0-1.5-1.0-0.50.00.51.01.52.0 The error of the mathematical modelThe error of the mathematical model
Experiment number
Fig. 8. The graphical representa tion of the mathematical
model relative deviation de pending on the experiments
number regarding the attenuation of the sound pressure
level

4. Conclusions

Noise control strategies are quite different, as
the depollution techniques are, due to the type and
complexity of soundproofing materials and type noise generated. The reference values of the sound pressure
level obtained by measurements showed a decrease of
noise level at all four heights of the microphone recording.
The attenuation values of sound pressure level
calculated by comparing the reference values of the sound pressure level at the values of sound pressure
level obtained with acoustic screen have varied
depending on the type of material used and by the number of walls of the acoustical screen.
Mathematical modeling was based on the
recorded values of sound pressure level and calculated values of the sound pressure level
attenuation. In this work was we have verified the real
data through a mathematical model to present the real
model of the sound propagation. In the same time it was expecting that the differences between the two
models, real and modeled, to be as small.
The mathematical modeling performed has
revealed that the recorded values of sound pressure
level respectively the calculated attenuation values of
the sound pressure level fit very well. This research
demonstrates the viability of the model, demonstrated
by the fact that the calculation relative deviations are situated in a very restri cted range of variation.

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