Applied Acoustics 70 (2009) 595–604
Contents lists available at ScienceDirect
Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust
Investigation of non-acoustical parameters of compressed melamine foam materials Naoki Kino a, , Takayasu Ueno a, Yasuhiro Suzuki b, Hiroshi Makino b *
a b
Industrial Research Institute of Shizuoka Prefecture, 2078 Makigaya, Aoi-ku, Shizuoka 421-1298, Japan Inoac Corporation Automotive Technology Company, 3-1-36 Imaike-cho, Anjo 446-8504, Japan
a r t i c l e
i n f o
Article history: Received 13 December 2007 Received in revised form 10 March 2008 Accepted 1 July 2008 Available online 20 August 2008 Keywords: Tortuosity Viscous characteristic length Thermal characteristic length Melamine
a b s t r a c t
A series of careful careful non-acoustical non-acoustical parame parameters ters measureme measurements nts using 5 ‘Illtec’ melamine melamine foam foam and 10 ‘Basotect TG’ melami melamine ne foamsamples foamsamples have have been made. made. Flowresistivity, Flowresistivity, tortuosity tortuosity,, porosity, porosity, viscous viscous characte characterristic length and thermal characteristic length of two types of compressed melamine melamine foam materials materials with different foam magnifications have been investigated. It has been found that a relationship between the flow resistivity, fibre equivalent diameter and bulk density exists for each of the compressed melamine foam materials. materials. This paper also discusses relationships between the non-acoustical parameters and compression rates in the compressed melamine foam media. 2008 Elsevier Ltd. All rights reserved.
1. Introduction In this paper some investigation results are shown for the two types of compressible melamine foam materials. Melamine foam has a cellular structure with open cells. Fig. 1a 1a and b shows the absence of cell walls and short hexagonal cellular struts in the ‘Illtec’ melamine melamine foam. Those Those in the ‘Basotect ‘Basotect TG’ melamine melamine foam are shown in Fig. in Fig. 1c 1c and d. As in the previous study [1] [1],, the short hexago hexagonal nal struts of the cells in the foam are regard regarded ed as ‘fibre ‘fibre equivalents’ for the purposes of this paper. The diameter of glass fibre fibre is almost almost equal equal to 7 lm. The melam melamine ine fibre fibre equiva equivalent lent diameter is slightly smaller than the diameter of the glass fibre. The The cell cell size size of ‘Illte ‘Illtec’ c’ melam melamin ine e foam foam is abou aboutt 100– 100–20 200 0 lm (diame (diameter) ter).. That That of the ‘Basotect ‘Basotect TG’ melamine melamine foam is about about 150–300 lm (diameter). The following relationship (Eq. (1) (1))) for melamine fibre equivalents has been found in the previous study [1] 2 11 :5 109 ; rKino dKino q1 1:53 ¼ 11:
ð1Þ
where r Kino is the flow resistivity, d Kino is the diameter of a hexagonal onal cellul cellular ar strut strut in the melam melamine ine foam, foam, and and q1 is the the dens densit ity y of the the porous medium. medium. Castag Castagnèd nède e et al. [2] al. [2] studied studied the relation relationship ship betwee between n the physphysical parameters and the compression rates using the polyester fibres manufactured manufactured by the company company 3 M. Compressed Compressed melamine foam foam materi materials als are often often used used for contro controllin lling g autom automotiv otive e acousti acoustics. cs.
*
Corresponding author. Tel.: +81 54 278 3027; fax: +81 54 278 3066.
[email protected] ref.shizuoka.jp (N. (N. Kino). E-mail address:
[email protected]
0003-682X/$ - see front matter 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2008.07.002
The objective objective of this this work work is an investi investigat gation ion of non-aco non-acoust ustical ical parame parameter terss for the predict prediction ion of absorpti absorption on coefficie coefficients nts of such materials. Flow resistivity, tortuosity tortuosity [3] [3] and two characteristic characteristic lengths [3,4] of [3,4] of two types of melami melamine ne foam material materialss are investi investigat gated ed to demons demonstrat trate e the effects effects of variou variouss foam magnifica magnification tionss and compre compressio ssion n rates. rates. In Section Section 2, the relation relationship ship between between the acoustical acoustical properties properties and the non-acoustical non-acoustical parameters parameters are described by using the Johnson–Allard model. Subsequently, in Section 3 tion 3,, careful measurements of the non-acoustical parameters of melamine foam materials are described. Using the Eq. (1) (1) and and the ultrasonic ultrasonic measurements, measurements, it is shown the measurements measurements of the flow resistivity, the tortuosity and the two characteristic lengths for the ‘Illtec’ melamine foam are accurate. For the ‘Basotect TG’ melamine foam the accuracy of a new relation between the flow resistivity and bulk density is also shown. In Section 4 4,, it is shown that there is a relationship between between the non-acoustical non-acoustical parameters parameters and the compression rates. Additionally, the predicted absorption coefficient for the melamine foam is shown. Finally, Section 5 5 pre presents concluding concluding remarks. remarks.
2. The Johnson–Allard model The Johnson–Allard model, which involves flow resistivity, tortuosity tuosity,, porosit porosity y and two characte characterist ristic ic lengths, lengths, is describ described ed by Eqs. (2)–(5) Eqs. (2)–(5).. Eq. (2) Eq. (2) is is the equation for the effective density of rigid-framed materials as proposed by Johnson et al. [3] [3].. Eq. (5) (5) is is the equation of the bulk modulus of a rigid-frame porous material as proposed by Champoux and Allard [4,5] Allard [4,5].. Specifically, this model is described by the following set of equations:
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
596
Fig. 1. Digital microscope photographs of the open cell structure of melamine foams. (a) ‘Illtec’ of sample 51, (b) ‘Illtec’ of sample 54, (c) ‘Basotect TG’ of sample 61, (d) ‘Basotect TG’ of sample 69.
þ
qðxÞ ¼ q0 a1 1
r/ G J ðxÞ ; ia1 q0 x
ð2Þ
with
4ia21 gq0 x
ð Þ¼ þ ^¼ ," G J x
1
1 8a1 g c r/
K x
ð Þ ¼ cP
0
r2 K2 /2
1=2
1=2
ð3Þ
;
ð4Þ
;
c
ð Þ þ c
1 1
Þ #
8g G0 Pr x 0 2 i q0 Pr x J
^
ð
1
;
ð5Þ
Z 0 cosh Cb Z c sinh Cb Z Z c 0 ; Z sinh Cb Z c cosh Cb Z 0 iZ 0 cot k0 L0 ; Z = Z 0 1 r = 1 r ;
ð6Þ
where b is the material thickness, Z 0(q0c 0) is the characteristic impedance of air, c 0 is the sound speed in the air, k 0(2p f /c 0) is the wave number of air, and L 0 is the thickness of rear air gap. Finally, the absorption coefficient is
with
Þ¼ þ
G J0 Pr x
1
^0 ¼ 1c 0
8a1 g
ð
r/
iq0 02 Pr x 16g
^
1=2
1=2
;
ð7Þ
;
where q0 is the density of the air, a 1 is the tortuosity, r is the flow resistivity, / is the x (2p f ) is the angular frequency, f is the ffiffiffiffiffiffiporosity, ffi frequency, i 1, is the viscous characteristic length, 0 is the thermal characteristic length, g is the viscosity of the air, Pr is the Prandtl number of the air, c is the specific heat ratio of the air, P 0 is the atmospheric pressure, c and c 0 are the cross-sectional shape factors of the pore. Lafarge et al. [6] have extended the Johnson–Allard model by adding thermal permeability. Fellah et al. [7] have researched the inverse problem of porous material in the time domain using the Johnson–Allard and the Pride–Lafarge models. The characteristic impedance ( Z c) can be derived from the effective density (q(x)) and bulk modulus (K (x)), and the propagation constant (C) can be derived from the effective velocity ( c (x)). Eq. (8) shows the relationship between Z c, q (x) and K (x) and Eq. (9) shows the relation between c (x) and C
p ¼ ^
¼ qðxÞc ðxÞ ¼ ½K ðxÞqðxÞ C ¼ i½x=c ðxÞ:
Z c
^
1=2
;
The normal incidence absorption coefficient can be produced from the knowledge of Z c and C. The surface impedance ( Z ) can be calculated from Eq. (10). The impedance of the air gap behind the material layer ( Z 0 ) can be calculated from Eq. (11). The normal incidence reflection factor (r ) can be calculated from Eq. (12), and the normal incidence absorption coefficient (a) can be calculated from Eq. (13). Thus, the changes in the non-acoustical parameters appear as the changes in the normal incidence absorption coefficient
ð8Þ ð9Þ
ð Þþ ¼ ð Þþ ¼ ð Þ ¼ ð þ Þ ð Þ
ð Þ ð Þ
a ¼ 1 jr j2 :
ð10Þ ð11Þ ð12Þ
ð13Þ
3. Experiment method and results 3.1. Careful measurements of non-acoustical parameters The important non-acoustical parameters of 15 melamine foam samples (i.e. 5 ‘Illtec’ melamine foam and 10 ‘Basotect TG’ melamine foam samples) used in the experiment are listed in Tables 1–3, respectively. The melamine foam samples have been compressed by a heat press processing. Samples 51 and 61 are the reference materials, respectively. Measurements of the flow resistivity were made with a device in accordance with the ISO standard 9053 [8]. Measurements of the tortuosity and the two characteristic lengths were made by a method similar to that proposed by Leclaire et al. [9] involving saturation by two different gases, in this case air and argon [10] were used to improve the signal-to-noise ratio. Moussatov et al. [11,12] have researched the pressure variation technique to achieve the improved signal-tonoise ratio in the ultrasonic measurements.
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
Porosity was estimated from Eq. (14)
Table 1
Measured, estimated and predicted parameters of the ‘Illtec’ samples Sample number
Measurements q1 (kg m3) Surface density (g m2) Thickness (mm) r ( Pa s m2)
a1
^ (lm) ^0 ( lm)
51
52
53
54
55
12.39 279 22.58 10,943 1.0090 230 460
14.88 241 16.22 17,314 1 .0101 169 373
20.99 241 11.48 28,582 1.0146 120 266
46.96 243 5.18 98,442 1.0267 48 97
82.64 253 3.06 232,798 1.0437 26 49
0.9921
Estimations 0.9905
0.9866
0.9701
0.9474
Predictions (Johnson–Allard model) c 0.51 c 0 0.25
0.55 0.25
0.60 0.27
0.83 0.41
0.99 0.53
Predictions (Kino and Allard models) 7.03 dKino ( lm) 2.033 L 108 (m2) 223 A ( lm) 0 ( lm) 445 A
6.43 2.919 170 339
6.51 4.015 122 243
6.50 9.026 54 108
6.51 1.582 31 62
/
^ ^
597
¼ 1 q =q ¼ 1 q =ðbq Þ;
/
1
m
s
ð14Þ
m
where q 1 and q m are the densities of the porous medium and the raw material, respectively, and qs is the surface density of the porous medium.The assumed densities of melamine were 1570 kg m3. The cross-sectional pore shape factors were predicted from the measurements of flow resistivity, tortuosity and two characteristic lengths according to the Johnson–Allard model. The measured data for a 1, r , and 0 were used in Eqs. (4) and (7) together with the estimated porosity value, / , so that the cross-sectional pore shape factors c and c 0 were predicted. The measurements of the flow resistivity, tortuosity and two characteristic lengths as a function of the bulk density are shown in Fig. 2. The predicted values of two cross-sectional pore shape factors as a function of the bulk density are shown in Fig. 3. The approximation curves, equations, and the R-Square were drawn on the graphs using a MATLAB curve fitting toolbox. They are also listed in Tables 4 and 5. By transforming Eq. (4), Eq. (15) was obtained. Similarly, by transforming Eq. (7), Eq. (16) was obtained. For the highly porous fibrous materials (a1 1 and / 1), the pore shape factors and the two characteristic lengths are important for the flow resistivity
^
^
ffi
Table 2
Measured, estimated and predicted parameters of the ‘Basotect TG’ samples Sample number
61
62
63
64
65
8.77 173 19.74 6,197 1.0101 271 572
9.29 168 18.07 7,314 1.0104 241 546
9.59 176 18.32 7,465 1.0110 250 512
10.86 167 15.36 9,261 1.0132 212 417
12.85 178 13.89 12,252 1.0151 181 350
0.9944
0 .9941
0.9939
0.9931
0.9918
Predictions (Johnson–Allard model) c 0.57 c 0 0.27
0.59 0.26
0.57 0.28
0.60 0.30
0.61 0.32
Predictions (Kino and Allard models) dKino(BTG) ( lm) 5.98 L 108(m2) 1.988 268 A ( lm) 0 ( lm) 535 A
5.75 2.276 243 486
5.84 2.283 239 478
5.76 2.652 208 417
5.70 3.210 174 348
Measurements q1 (kg m3) Surface density (g m2) Thickness (mm) r ( Pa s m2)
a1
^ (lm) ^0 ( lm) Estimations /
^ ^
r¼ r¼
8 a1 g ; / 2 c 2 8a1 g : / 02 c 02
Measured, estimated and predicted parameters of the ‘Basotect TG’ samples (Part 2) Sample number
Measurements q1 (kg m3) Surface density (g m2) Thickness (mm) r ( Pa s m2)
a1
^ (lm) ^0 ( lm)
66
67
68
69
70
15.72 176 11.23 17,395 1.0192 137 263
15.73 166 10.53 13,687 1.0199 163 309
27.90 160 5.75 36,937 1.0355 82 164
58.39 162 2.78 122,325 1.0733 37 69
71.57 161 2.25 160,828 1.0856 30 59
0.98999
0.98998
0.9822
0.9628
0.9544
0.64 0.34
0.79 0.39
0.99 0.53
1.08 0.55
Estimations /
Predictions (Johnson–Allard model) 0.68 c c 0 0.35 Predictions (Kino and Allard models) dKino(BTG) ( lm) 5.58 L 108(m2) 4.095 139 A ( lm) 0 ( lm) 279 A
^ ^
6.29 3.221 157 314
5.94 6.415 84 167
5.74 14.365 39 77
5.85 16.954 32 64
^
ð15Þ
^
ð16Þ
The two characteristic lengths of two types of melamine foams are close to each other in the density range over 50 kg m3 as shown in Fig. 2c and d, so in that density range it is found that the difference of the flow resistivities between the ‘Illtec’ and the ‘Basotect TG’ as shown in Fig. 2a is caused mainly from the difference of the two cross-sectional shape factors.
3.2. Verification of the measurements of two characteristic lengths using fibre equivalent diameter Allard et al. [13] showed that sound propagation in rigid-framed fibrous materials depends on the total length of fibres per unit volume of a material. Eqs. (17)–(19) show the relationship of the two characteristic lengths and the total length of fibres per unit volume of a material. They are applicable to a case where the velocity of the air is perpendicular to the direction of the fibres 2
¼ 4q =pd q ; ^ ¼ 1=2pRL; where R ¼ d=2; ^0 ¼ 2^ ; L
Table 3
ffi
1
ð17Þ ð18Þ ð19Þ
m
A
A
A
where d is the diameter of a fibre, L is the total length of fibres per unit volume of a material, A is the viscous characteristic length, and 0A is the thermal characteristic length. The flow resistivities r of the ‘Illtec’ samples in Table 1 were measured. By using Eq. (1) with rKino = r, the fibre equivalent diameter dKino in Table 1 was predicted. The melamine fibre equivalent diameter dKino predicted from Eq. (1), d = dKino was substituted for Eq. (17) so that the total length of fibre equivalents per unit volume L was obtained. The total length of fibre equivalents per unit volume was substituted in Eq. (18) so that the two characteristic lengths A and 0A were predicted. The measurements of and 0 shown in Table 1 are compared with the predictions of A and 0A shown in Table 1. The predictions are close to the measurements. The discrepancies between the measurements of and 0 and the predictions of A and 0A are examined in detail. For the viscous characteristic lengths of the 5 ‘Illtec’ samples shown in Table 1 the discrepancy is represented
^
^
^
^
^
^
^
^
^
^
^
^
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
598
a
b
c
d
Fig. 2. Approximations as a function of the bulk density obtained using the measurements of five ‘Illtec’ and ten ‘Basotect TG’ samples. (a) Flow resistivity; (b) tortuosity, (c) viscous characteristic length, (d) thermal characteristic length.
a
b
Fig. 3. Approximations as a function of the bulk density obtained using the predicted values of five ‘Illtec’ and ten ‘Basotect TG’ samples. (a) Cross-sectional pore shape factor c; (b) cross-sectional pore shape factor c 0 .
as 100 / . The mean value for the ‘Illtec’ data is 6.89%. For A the thermal characteristic lengths of the 5 ‘Illtec’ samples shown in 0 0 = 0 . The Table 1 the discrepancy is represented as 100 A mean value is 11.56%. The predictions are close to the measurements, so that the predictions of the two characteristic lengths are judged to be accurate. The following relationship (Eq. (20)) for ‘Basotect TG’ melamine fibre equivalents was obtained from measurements of flow resistivity, bulk density and two characteristic lengths
j^ ^j ^
j^ ^ j ^
rKinoðBTGÞ d2KinoðBTGÞ q1 1:53 ¼ 8 109 ;
ð20Þ
where rKino(BTG) is the flow resistivity, d Kino(BTG) is the diameter of a hexagonal cellular strut in the ‘Basotect TG’ melamine foam and the constant ‘‘8 109” was obtained from a manual fit as described in the following paragraph. The flow resistivities r of the ‘Basotect TG’ samples in Tables 2 and 3 were measured. The melamine fibre equivalent diameter dKino(BTG) predicted from Eq. (20), d = dKino(BTG) was substituted for
N. Kino et al./ Applied Acoustics 70 (2009) 595–604 Table 4
Approximations as a function of the bulk density obtained using the measured values of five ‘Illtec’ samples Non-acoustical parameter
Approximated equation
r ( Pa s m2) a1
268:6q1:533 1 0.0004911q1+1.003 1:19 4457q 1 7925q1 1:129 0:2081q0:3551 1 0:07823q0:4322 1
^ (lm) ^0 ( lm) c c 0
Table 5
Approximations as a function of the bulk density obtained using the measured values of ten ‘Basotect TG’ samples Non-acoustical parameter
Approximated equation
r ( Pa s m2)
238:3q1:528 1 0.001226q1+0.9998 1:016 2421q 1 1:153 6908q 1 0:2849q0:3092 1 0:1317q0:3381 1
a1
^ (lm) ^0 ( lm) c c 0
Eq. (17) so that the total length of fibre equivalents per unit volume L was obtained. The total length of fibre equivalents per unit volume was substituted in Eq. (18) so that the two characteristic lengths A and 0A were predicted. Then, the right side value of Eq. (20) was adjusted, so that the value of A and 0A were suitable
^
^
^
599
for the measured characteristic lengths of and 0 shown in Tables 2 and 3. For the viscous characteristic lengths of the 10 ‘Basotect TG’ samples shown in Tables 2 and 3 the discrepancy is represented as 100 / . The mean value for the ‘Basotect TG’ data is A 3.15%. For the thermal characteristic lengths of the 10 ‘Basotect TG’ samples shown in Tables 2 and 3 the discrepancy is repre0 0 = 0 . The mean value is 5.18%. For the sented as 100 A ‘Basotect TG’ samples, it is also found that the two characteristic lengths are derivable from the fibre equivalent diameter as shown in Eq. (20). The predicted two characteristic lengths are close to the measured ones, so that the measurements of the two characteristic lengths in Tables 1–3 are judged to be highly accurate.
^
^
j^ ^j ^
j^ ^ j ^
3.3. Verification of the measurements of micro-structural parameters using ultrasonic velocity The measurement results for the ultrasonic propagation in air and argon have been used to determine the two characteristic lengths and the tortuosity. The results for the squared refraction index as a function of the square root of the inverse frequency are shown in Fig. 4. The accuracy of the measurements of tortuosity and two characteristic lengths was tested by using the measured values to predict the sound velocity through Eq. (21). By transforming the wave number equation at high frequencies [14], Eq. (21) is obtained. Subsequently the predicted and measured sound velocities were compared as shown in Fig. 5. The temperature was 22.5–22.8 C during the experiment
^
a
b
c
d
Fig. 4. Squared refraction index as a function of the square root of the inverse frequency in ‘Illtec’ and ‘Basotect TG’ samples saturated by air and argon at a temperature of 22.5–22.8 C. (a) sample 51, (b) sample 54, (c) sample 61, (d) sample69.
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
600
a
b
c
d
Fig. 5. Comparison between measured and predicted sound velocities saturated by air and argon as a function of frequency at a temperature of 22.5–22.8 (b) sample 54, (c) sample 61, (d) sample 69.
" # ffiffi ffi ffi ^ þ p ffiffi ffi ^ ^ þ p ffiffi ffi ^ ¼ p s ffiffi ffi ffi ffi ffi
c high d
¼
c 0
a1
1
d
1
2
2g ; xq0
c
1
Pr 0
2
1
d
4
1
c
1
Pr 0
2
;
ð21Þ
The predicted sound velocities are very close to the measured one, so that the measurements of the tortuosity and the two characteristic lengths in Tables 1–3 are judged to be highly accurate.
ð22Þ
4. Discussion
where c high is the sound velocity in the materials at high frequencies, and d is the viscous skin depth. The discrepancy between measured and predicted sound velocities is calculated as follows:
100
c m
j
c high =c m ;
j
C. (a) sample 51,
23
ð Þ
where c m is the measured frequency-dependent sound velocity and c high is the predicted frequency-dependent sound velocity. For the ‘Illtec’ sample 51 in air, the mean value of the sound velocity discrepancy between measurement and prediction in the frequency range between 100 kHz and 800 kHz was 0.039%. For the sample 51 in air, the maximum value of the sound velocity prediction difference was 0.24 ms1. For the ‘Illtec’ sample 54 in air, the mean value of the sound velocity prediction discrepancy in the frequency range between 100 kHz and 800 kHz was 0.286%. For the sample 54 in air, the maximum value of the sound velocity prediction difference was 2.65 ms1. For the ‘Basotect TG’ sample 61 in air, the mean value of the sound velocity discrepancy between measurement and prediction in the frequency range between 100 kHz and 800 kHz was 0.016%. For the sample 61 in air, the maximum value of the sound velocity prediction difference was 0.18 ms1. For the ‘Basotect TG’ sample 69 in air, the mean value of the sound velocity prediction discrepancy in the frequency range between 100 kHz and 800 kHz was 0.425%. For the sample 69 in air, the maximum value of the sound velocity prediction difference was 3.75 ms1.
4.1. Relationships between non-acoustical parameters and compression rates The measurements of non-acoustical parameters were shown in the Section 3. The relationship between the non-acoustical parameters and the compressionrates are shown in Table 6 to investigate the compressed ‘Illtec’ samples in detail. Castagnède et al. [2] showed the relationship between the physical parameters and the compression rate ( n). For the two characteristic lengths caused by the compression ( (n) and 0ðnÞ , they showed Eqs. (24) and (25) as a case for an uncompressed fibre equivalents of radius R , surrounded by an ‘‘homogenization” volume in the form of a concentric cylinder, having the same length
^
^ Þ
Table 6
Predicted parameters using compression rate of 4 compressed ‘Illtec’ samples Sample number
52
53
54
55
n
1.3921 165 330 0.9890 1.0125 1.0109 1 5,233 21,207
1.9669 116 232 0.9845 1.0177 1.0139 21,524 42,335
4.3591 50 103 0.9656 1.0392 1.0269 47,701 207,933
7.3791 29 60 0.9418 1.0664 1.0437 80,749 595,856
^(n) (lm), using Eq. (24) ^0ðnÞ ( lm), using Eq. (25) /(n), using Eq. (26)
a1(n), using Eq. (27) a1(n), using Eqs. (28) and (29) r(n) ( Pa s m2), using Eq. (30) r(n) ( Pa s m2), using Eq. (31)
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
601
from those of the fibre in the Appendix [2]. The radius ( R) 3 lm of melamine fibre equivalents was used in this experiment. For the porosity caused by the compression ( /(n)), Eq. (26) was obtained on a condition that the values of surface density hardly changed as shown in Tables 1–3
100 /(n) / //. The mean value for the ‘Illtec’ data is 0.36%. Eqs. (24)–(26) are judged to be accurate for the compressed ‘Illtec’ samples. Castagnède et al. [2] also showed Eq. (27) as the tortuosity model caused by the compression
^ð Þ ¼ ^nð Þ þ R 1n 1 0 ^ 0 ^ð Þ ¼ nð Þ þ R 1n 1
a1ðnÞ ¼ 1 nð1 a1ð1Þ Þ:
1
n
1
n
qsð1Þ
/ðnÞ
b1 n
ðÞ
¼ 1 q ¼ 1 m
;
ð24Þ
;
ð25Þ
nqsð1Þ bqm
¼ 1
nq1ð1Þ
qm
¼ 1 nð1 /ð ÞÞ; 1
ð26Þ
where n = b(1)/b(n), b(1) is the thickness of the reference material (samples 51 and 61), b(n) is the thickness of a compressed material, of the references , 0ð1Þ is 0 of the references , and /(1) is / of (1) is the references. For the viscous characteristic lengths of the 4 compressed ‘Illtec’ samples 52–55 shown in Table 6 the discrepancy is represented as 100 / . The mean value for the ‘Illtec’ data is 4.95%. For (n) the thermal characteristic lengths of the ‘Illtec’ samples shown in 0 0 = 0 . The Table 6 the discrepancy is represented as 100 ðnÞ mean value is 12.91%. The predictions are close to the measurements as shown in Fig. 6. For the porosity of the ‘Illtec’ samples 52–55 shown in Table 6 the discrepancy is represented as
^
^
^
^
j^ ^j ^
j^ ^ j ^
a
j
j
ð27Þ
For the tortuosity of the 4 compressed ‘Illtec’ samples 52–55 shown in Table 6 the discrepancy is represented as 100 (a1(n) 1) (a1 1) /(a1 1). The mean value for the ‘Illtec’ data is 36.05%. The predictions are far from the measurements as shown in Fig. 7a, so that the prediction by Eq. (27) is judged to be poor for the compressed ‘Illtec’ samples. The structure of a pore in the porous materials is very complex as shown in Fig. 1, so for another model of tortuosity caused by the compression, Eqs. (28) and (29) are considered on a condition that the sinuosity propagation route is shortened by the compression rate (N 3)
a1ðnÞ
j
n N 3
2
¼
N 3ðILLTECÞ
j
a1ð1Þ ; 2
¼ 0:002674n þ 1:003n 0:0003189:
ð28Þ ð29Þ
The measurements of tortuosity data a1, the compression rates n of 5 ‘Illtec’ samples 51–55 in Table 1 were used. The values of
b
Fig. 6. Comparison between measured and predicted non-acoustical parameters as a function of the squared compression rate of five ‘Illtec’ samples. (a) Viscous characteristic length as a function of the squared compression rate, with predictions by Eq. (24), (b) thermal characteristic length, with predictions by Eq. (25).
a
b
Fig. 7. Comparison between measured and predicted tortuosity of five ‘Illtec’ samples. (a) predictions were deduced using Eq. (27), (b) predictions were deduced using Eqs. (28) and (29).
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
602
a1(1) = a1(sample 51), a1(n) = a1(sample) and n were substituted for Eq.
(28) so that N 3(ILLTEC) was obtained. The approximated equations (Eqs. (28) and (29)) were obtained using a MATLAB curve fitting toolbox. For the tortuosity of the 4 compressed ‘Illtec’ samples 51–55 shown in Table 6 the discrepancy is represented as 100 (a1(n) 1) (a1 1) /(a1 1). The mean value for the ‘Illtec’ data is 1.49%. The predictions are close to the measurements as shown in Fig. 7b, so that the prediction by Eqs. (28) and (29) is judged to be highly accurate. It is found that the compression rate of sinuosity propagation route ( N 3) is a little smaller than that of material thickness (n). Castagnède et al. [2] showed Eq. (30) as the flow resistivity model caused by the compression ( r(n))
j
j
rðnÞ ¼ nrð1Þ ;
ð30Þ
where r (1) is r of the references (samples 51 and 61). For the flow resistivity of the 4 compressed ‘Illtec’ samples 52– 55 shown in Table 6 the discrepancy is represented as 100 r(n) r /r. The mean value for the ‘Illtec’ data is 38.39%. Castagnède et al. [2] also showed Eq. (31) as the flow resistivity model caused by the compression ( r(n))
j
j
rðnÞ ¼ n2 rð1Þ :
ð31Þ
For the flow resistivity of the 4 compressed ‘Illtec’ samples 52–55 shown in Table 6 the discrepancy is represented as 100 r(n) r /r. The mean value for the ‘Illtec’ data is 84.45%. The predictions are far from the measurements as shown in Fig. 8a and b, so that the prediction by Eqs. (30) and (31) is judged to be poor for the compressed ‘Illtec’ samples. The measurements of the flow resistivity were approximated using a MATLAB curve fitting toolbox, as shown in Fig. 9. For the 5 ‘Illtec’ samples 51–55 Eq. (32) is obtained
j
j
r ¼ 2903n2 þ 10520n 2856:
ð32Þ
Accuracy shortage of Eqs. (30) and (31) is clearly understood from Eq. (32). Next, for another melamine medium (i.e. the compressed ‘Basotect TG’ samples), the relationship between the non-acoustical parameters and the compression rate are investigated in detail as well as the compressed ‘Illtec’ samples. For the two characteristic lengths of the 9 compressed ‘Basotect TG’ samples 62–70, the predictions are deduced using Eqs. (24) and (25). The discrepancy is represented as 100 / . The (n)
j^ ^j ^
a
Fig. 9. Approximations as a function of the compression rate obtained using the measured flow resistivity of five ‘Illtec’ and ten ‘Basotect TG’ samples.
mean value for the ‘Basotect TG’ data is 5.33%. The discrepancy is 0 0 = 0 . The mean value is 8.23%. represented as 100 ðnÞ For the porosity of the 9 compressed ‘Basotect TG’ samples 62– 70, the predictions are deduced using Eq. (26). The discrepancy is represented as 100 /(n) / //. The mean value is 0.10%. The predictions are close to the measurements, so that the predictions by Eqs. (24)–(26) are also judged to be accurate for the compressed ‘Basotect TG’ samples. For the tortuosity of the 9 compressed ‘Basotect TG’ samples 62–70, the predictions are deduced using Eq. (27). The discrepancy is represented as 100 (a1(n) 1) (a1 1) /(a1 1). The mean value is 3.79%. The predictions are close to the measurements, so that the prediction by Eq. (27) is judged to be accurate for the compressed ‘Basotect TG’ samples. The measurements of tortuosity data a1, the compression rates n of 10 ‘Basotect TG’ samples 61–70 in Tables 2 and 3 were used. The values of a1(1) = a1(sample61), a1(n) = a1(sample) and n were substituted for Eq. (28) so that N 3(BASOTECT TG) was obtained.
j^ ^ j ^ j
j
j
N 3ðBASOTECTTGÞ
j
2
¼ 0:004324n þ 1:002n þ 0:00287:
ð33Þ
For the tortuosity of the 9 compressed ‘Basotect TG’ samples 62–70, the predictions are also deduced using Eqs. (28) and (33). The discrepancy is represented as 100 (a1(n) 1) (a1 1) / (a1 1). The mean value is 4.31%. The predictions are close to
j
j
b
Fig. 8. Comparison between measured and predicted flow resistivity of five ‘Illtec’ samples. (a) predictions were deduced using Eq. (30), (b) predictions were deduced using by Eq. (31).
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
603
Table 7
Predicted non-acoustical parameters for the compressed ‘Illtec’ sample 53
q1 (kg m3)
b (mm)
r ( Pa s m2)
a1
/
d ( lm)
^ (lm)
^0 ( lm)
20.99
11.48
33,657
1.0139
0.9845
6
112
224
Table 8
Predicted non-acoustical parameters for the compressed ‘Basotect TG’ sample 66
q1 (kg m3)
b (mm)
r ( Pa s m2)
a1
/
d ( lm)
^ (lm)
^0 ( lm)
15.72
11.23
15,046
1.0182
0.9902
6
150
300
a
b
c
d
e
f
Fig. 10. Comparison between measured and predicted normal incidence acoustical properties. Z 0 = q0c 0. (a) Absorption coefficient for ‘Illtec’ sample 53 in Table 7, (b) absorption coefficient for’ Basotect TG’ sample 66 in Table 8, (c) attenuation constant for sample 53 in Table 7, (d) imaginary part of Z c/ Z 0 for sample 53 in Table 7, (e) attenuation constant for sample 66 in Table 8, (f) imaginary part of Z c/ Z 0 for sample 66 in Table 8.
N. Kino et al./ Applied Acoustics 70 (2009) 595–604
604
the measurements, so that the prediction by Eqs. (28) and (33) is judged to be highly accurate. For the flow resistivity of the 9 compressed ‘Basotect TG’ samples 62–70, the predictions are deduced using Eq. (30). The discrepancy is represented as 100 r(n) r /r. The mean is also 31.69%. The predictions are also deduced using Eq. (31). The mean value of the discrepancies is also 59.59%. The predictions are far from the measurements, so that the prediction by Eqs. (30) and (31) is also judged to be poor for the compressed ‘Basotect TG’ samples. For the 10 ‘Basotect TG’ samples 61–70 Eq. (34) is also obtained as shown in Fig. 9
j
j
r ¼ 1149n2 þ 9007n 4144:
ð34Þ
Accuracy shortage of Eqs. (30) and (31) is clearly understood from Eq. (34).
4.2. Prediction of absorption coefficient Here, the absorption coefficients of the melamine foam deduced from the predicted non-acoustical parameters are demonstrated. The ‘Illtec’ sample 53 in Table 7 and the ‘Basotect TG’ sample 66 in Table 8 are prepared to investigate the absorption coefficients. Using 6.0 lm as the melamine fibre equivalent diameter, the flow resistivity is predicted by the Kino model [Eqs. (1) and (20)]. Two characteristic lengths are predicted by using the Allard model [Eqs. (17)–(19)]. Porosity is predicted by using the Eq. (26). Tortuosity is predicted by using the Eqs. (28),(29) and (33). The predicted absorption coefficients for hard-backed material samples by the 3 models (the Delany and Bazley, the Johnson–Allard and the New models) and the measured one are shown in Fig. 10a and b. The absorption coefficient was measured using an impedance tube in accordance with the ISO standard transfer-function method [15]. The Delany and Bazley model [16] is shown in Eqs. (35) and (36)
h
0:754
0 0
0
h
0
0
0:595
0
i oi
0:732
¼ q c 1 þ 0:0571ðq f =rÞ i0:087ðq f =rÞ x 0:189ðq f =rÞ þ i 1 þ 0:0978ðq f =rÞ C ¼ c
Z c
n
0:700
0
; :
ð35Þ ð36Þ
The new model [17] is shown in Eqs. (37)–(40). The two correction factors for the ‘Illtec’ samples 53 are 8.5(N 1) and 250(N 2). These values are the same as the one in the previous study [1]. The two correction factors for the ‘Basotect TG’ samples 66 are 10(N 1) and 800(N 2)
r/ qðxÞ ¼ q0 a1 1 þ GN ðxÞ ; ia1 q0 x
ð37Þ
with
ffi ffi ffi ffi p p ffi ffi ffi ffi ffi ffi ffi ffi ffi ð þ Þ ð Þ¼ ^ ," ð Þ¼ ð Þ þ a1 1 i 2gq0 x r/
GN x K x
cP 0
c
c
1 1
N 1
1=2
ð38Þ
;
Þ #
1 8g 0 Pr x G ; i 02 q0 Pr x N
ð
^
ð39Þ
with
G0N ðPr xÞ
ffi ffi ffi ffi p p ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ^ ð þ Þ ¼ 2gq0 Pr x
01
i
8g
N 2
1=2
;
ð40Þ
where N 1 andN 2 are the correction factors. Fig. 10a and b shows that the absorption coefficients predicted by both the Delany and Bazley and the New models are close to the measured ones. However, Fig. 10c–f shows that the attenuation
constant and the imaginary part of characteristic impedance predicted by the New model are closest to the measured values. Consequently it is found that the New model is able to predict the acoustical properties of the compressed melamine foam media rather well.
5. Concluding remarks It has been shown that the flow resistivity and the two characteristic lengths are derivable from the fibre equivalent diameter and the bulk density in the compressed melamine foam materials. It has been found that the two characteristic lengths of the compressed melamine foams are predictable by using the characteristic lengths of reference foam and the compression rates shown by Castagnède et al. [2]. However the tortuosity and the flow resistivity of the compressed melamine foams were not predictable by using the parameters of reference foam and the compression rates shown by Castagnède et al. [2]. We showed the possibility that the tortuosity is predictable using the tortuosity of reference foam, the compression rates and the new correction factors. For the flow resistivity two empirical equations as a function of the compression rate were also shown. The flow resistivity and the two characteristic lengths of the compressed melamine foams are predictable from the fibre equivalent diameter and the compression rates by using a relationship between the compression rate and the bulk density. The resulting New model has been shown to predict the acoustical properties of compressed melamine foam materials well.
References [1] Kino N, Ueno T. Comparisons between characteristic lengths and fibre equivalent diameters in glass fibre and melamine foam materials of similar flow resistivity. Appl Acoust 2008;69:325–31. [2] Castagnède B, Aknine A, Brouard B, Tarnow V. Effects of compression on the sound absorption of fibrous materials. Appl Acoust 2000;61:173–82. [3] Johnson DL, Koplik J, Dashen R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. J Fluid Mech 1987;176:379–402. [4] Champoux Y, Allard JF. Dynamic tortuosity and bulk modulus in air-saturated porous media. J Appl Phys 1991;70:1975–9. [5] Allard JF. Propagation of sound in porous media – modelling sound absorbing materials. London: Chapman and Hall; 1993. [6] Lafarge D, Lemarinier P, Allard JF, Tarnow V. Dynamic compressibility of air in porous structures at audible frequencies. J Acoust Soc Am 1997;102: 1995–2006. [7] Fellah ZEA, Depollier C, Fellah M, Laurinks W, Chapelon JY. Influence of dynamic tortuosity and compressibility on the propagation of transient waves in porous media. Wave Motion 2005;41:145–61. [8] ISO 9053:1991. Acoustics – materials for acoustical applications – determination of airflow resistance –. [9] Leclaire Ph, Keiders L, Lauriks W. Determination of the viscous and thermal characteristic lengths of plastic foams by ultrasonic measurements in helium and air. J Appl Phys 1996;80:2009–12. [10] Kino N. Ultrasonic measurements of the two characteristic lengths in fibrous materials. Appl Acoust 2007;68:1427–38. [11] Ayrault C, Moussatov A, Castagnède B, Lafarge D. Ultrasonic characterization of plastic foams via measurements withstatic pressure variations. Appl Phys Lett 1999;74:3224–6. [12] Moussatov A, Ayrault C, Castagnède B. Porous material characterization – ultrasonic method for estimation of tortuosity and characteristic length using a barometric chamber. Ultrasonics 2001;39:195–202. [13] Allard JF,Champoux Y. Newempirical equations for sound propagationin rigid frame fibrous materials. J Acoust Soc Am 1992;91:3346–53. [14] Allard JF, Castagnède B, Henry M, Laurinks W. Evaluation of tortuosity in acoustic porous materials saturated by air. Rev Sci Instrum 1994;65: 754–5. [15] ISO 10534-2:1998. Acoustics – determination of sound absorption coefficient and impedance in impedance tubes – Part 2: transfer-function method. [16] Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Appl Acoust 1970;3:105–16. [17] Kino N, Ueno T. Improvements to the Johnson–Allard model for rigid-framed fibrous materials. Appl Acoust 2007;68:1468–84.
