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Applied Acoustics 69 (2008) 325–331

Comparisons between characteristic lengths and fibreequivalent diameters in glass fibre and melamine foam

materials of similar flow resistivity

Naoki Kino *, Takayasu Ueno

Shizuoka Industrial Research Institute of Shizuoka Prefecture, 2078 Makigaya, Aoi-ku, Shizuoka, Shizuoka 421-1298, Japan

Received 2 August 2006; received in revised form 20 November 2006; accepted 24 November 2006Available online 16 January 2007

Abstract

Flow resistivity, tortuosity, viscous characteristic length and thermal characteristic length of three melamine foam and two glass woolsamples have been measured. It has been found that a melamine foam sample with a bulk density 10.3 kg m�3 and a glass wool samplewith a bulk density 28.0 kg m�3 have almost the same flow resistivity, however, the bulk density is as much as three times different. Thecross-sectional pore shape factors, which are deduced with the non-acoustical parameters in the Johnson–Allard model, of the melaminefoam are smaller than those of the glass wool. This paper also discusses a new relationship between the flow resistivity, the fibre equiv-alent diameter and the bulk density in melamine foam.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Tortuosity; Viscous characteristic length; Thermal characteristic length; Melamine

1. Introduction

The fibrous structure of a glass wool is shown in Fig. 1aand the open cell structure of a melamine foam is shown inFig. 1b. The most obvious difference between melaminefoam and glass wool is that the former is an isotropic foamand the latter is an anisotropic layered fibrous material.Melamine foam has a cellular structure with open cells.Fig. 1b shows the absence of cell walls and short hexagonalcellular struts in the melamine foam. The short hexagonalstruts of the cells in the foam are regarded as ‘fibre equiv-alents’ for the purposes of this paper. The diameter ofglass fibre is almost equal to 7 lm. The melamine fibreequivalent diameter is slightly smaller than the diameterof the glass fibre. The cell size of melamine foam is about100–150lm (diameter).

Melamine foam is a light sound-absorbing material. Theflow resistivity for the melamine foam with a bulk density

0003-682X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apacoust.2006.11.008

* Corresponding author. Tel.: +81 54 278 3027; fax: +81 54 278 3066.E-mail address: nkino@s-iri.pref.shizuoka.jp (N. Kino).

10.3 kg m�3 is practically 13,100 Pa s m�2. According tothe Bies model (Eq. (1) [1]) the flow resistivity of glass fibrewith the same density and with mean fibre diameter 7 lm is2300 Pa s m�2.

rBiesd2Biesq

�1:53l ¼ 3:18� 10�9; ð1Þ

where rBies is the flow resistivity and dBies is the diameter ofa circular cross-sectional glass fibre.

To demonstrate the acoustical effects of the differentinternal structure of the melamine foam, the two character-istic lengths of the two materials (glass wool and melaminefoam) with the same flow resistivity are investigated. Addi-tionally, the bulk density and the flow resistivity of the twomaterials with a same viscous characteristic length are alsoinvestigated. The non-acoustical parameters of the mela-mine foam are found to be different from usual rigid-framed fibrous material. In Section 2, the non-acousticalparameters in the Johnson–Allard model are described.Subsequently, in Section 3, careful measurements of thenon-acoustical parameters of glass wool and melaminefoam materials are described. For the melamine foam the

Table 1Measured, estimated and predicted parameters of two glass wool samples

Sample number 1 2

Measurements

q1 (kg m�3) 28.0 31.8Thickness (mm) 25.0 25.0r (Pa s m�2) 11,900 16,800a1 1.0108 1.0093� (lm) 143 132� 0 (lm) 302 237

Estimations

/ 0.989 0.987

Predictions (Johnson–Allard model)

c 0.78 0.71c0 0.37 0.40

Predictions (Bies and Allard models)

dBies (lm) 6.6 6.1L · 108(m�2) 3.273 4.301�A (lm) 147 121^0A (lm) 294 242

Fig. 1. Digital microscope photographs: (a) glass fibre and (b) open cellstructure of melamine.

326 N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331

accuracy of a new relation between the flow resistivity andbulk density is also shown. Using the Delany and Bazleymodel and the ultrasonic data, it is shown the measure-ments of the flow resistivity, the tortuosity and the twocharacteristic lengths are accurate. In Section 4, usingtwo glass wool examples predicted by the Bies and theAllard models, it is shown the measured non-acousticalparameters of the melamine foam are greatly different fromthe glass wool. Additionally, the accuracy of the predictedabsorption coefficient for the melamine foam is discussed.Finally, Section 5 presents concluding remarks.

2. The Johnson–Allard model

According to the Johnson–Allard model, the flow resis-tivity [2–6], porosity [7], tortuosity [8], viscous characteris-tic length [8] and thermal characteristic length [9] areindispensable non-acoustical parameters, used to deter-mine the effective density and the bulk modulus of rigid-framed fibrous materials. Eq. (2) is the equation of theeffective density of rigid-framed materials, at the audiblefrequency, as proposed by Johnson et al. [8]. Eq. (5) isthe equation of the bulk modulus of rigid-framed materi-als, at the audible frequency, as proposed by Allard et al.

[10]. The viscous characteristic length � depends only onthe geometry of the frame. Johnson et al. [8] pointed outEq. (4) as the relation between � and (8a1g/r/)1/2. Thethermal characteristic length � 0 is the surface to volumeratio of pores with no weighting. Champoux and Allard[9] proposed Eq. (7) as the relation between � 0 and(8a1g/r/)1/2.

qðxÞ ¼ q0a1 1þ r/ia1q0x

GJðxÞ� �

; ð2Þ

with

GJðxÞ ¼ 1þ 4ia21gq0x

r2K2/2

� �1=2

; ð3Þ

^ ¼ 1

c8a1gr/

� �1=2

; ð4Þ

KðxÞ ¼ cP 0 c� ðc� 1Þ 1þ 8gi^02q0Prx

G0JðPrxÞ� ��1

" #,;ð5Þ

with

G0JðPrxÞ ¼ 1þ iq0^02Prx16g

� �1=2

; ð6Þ

^0 ¼ 1

c08a1gr/

� �1=2

; ð7Þ

where q0 is the density of the air, a1 is the tortuosity, r isthe flow resistivity, / is the porosity, x is the angular fre-quency, i ¼

ffiffiffiffiffiffiffi�1p

, g is the viscosity of the gas, P0 is theatmospheric pressure, c is the specific heat ratio of thegas, Pr is the Prandtl number of the gas, and c and c 0 arethe cross-sectional shape factors of the pore.

Eqs. (4) and (7) in the Johnson–Allard model show thatthe two characteristic lengths are related to the pore shapefactors, tortuosity, porosity and flow resistivity. So materi-als with the same flow resistivity but different pore struc-tures should have different characteristic lengths.

Table 2Measured, estimated and predicted parameters of three melamine foamsamples

Sample number 31 32 33

Measurements

q1 (kg m�3) 8.6 10.3 13.27Thickness (mm) 32.0 25.5 10.5r (Pa s m�2) 10,500 13,100 17,500a1 1.0059 1.0053 1.0055� (lm) 240 199 161�0

(lm) 470 445 375

Estimations

/ 0.995 0.993 0.992

Predictions (Johnson–Allard model)

c 0.49 0.53 0.57c 0 0.25 0.24 0.25

Predictions (Kino and Allard models)

dKino (lm) 5.43 5.58 5.86L · 108(m�2) 2.367 2.684 3.135�A (lm) 248 213 173^0A (lm) 495 425 347

N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331 327

By transforming Eq. (4), Eq. (8) was obtained. Similarly,by transforming Eq. (7), Eq. (9) was obtained. For thehighly porous fibrous materials (a1 @ 1 and / @ 1), thepore shape factors and the two characteristic lengths areimportant for the flow resistivity.

r ¼ 8a1g/^2c2

; ð8Þ

r ¼ 8a1g/^02c02

: ð9Þ

3. Experiment method and results

3.1. Material

The important non-acoustical parameters of two glasswool and three melamine foam samples used in the exper-iment are listed in Tables 1 and 2, respectively. The bulkdensities of glass wool samples were 28 kg m�3 and31.8 kg m�3. The bulk densities of melamine foam sampleswere all about 10 kg m�3. Measurements of the flow resis-tivity were made with a device in accordance with the ISOstandard 9053 [11]. Measurements of the tortuosity and thetwo characteristic lengths were made by a method similarto that proposed by Leclaire et al. [12] involving saturationby two different gases, in this case air and argon [13].

Porosity was estimated from Eq. (10).

/ ¼ 1� ql=qm; ð10Þwhere ql and qm are the densities of the porous medium andthe raw material, respectively.

The assumed densities of glass and melamine were2500 kg m�3 and 1570 kg m�3, respectively.

The cross-sectional shape factors were predicted fromthe measurements of flow resistivity, tortuosity and two

characteristic lengths according to the Johnson–Allardmodel. Measured and estimated values of a1, r, �, � 0and / were substituted in Eqs. (4) and (7) and the cross-sectional shape factors were predicted.

Allard and Champoux [14] showed that sound propaga-tion in rigid-framed fibrous materials depends on the totallength of fibres per unit volume of a material. Eqs. (11)–(13) show the relationship of the two characteristic lengthsand the total length of fibres per unit volume of a material.They are applicable to a case where the velocity of the air isperpendicular to the direction of the fibres.

L ¼ 4ql=pd2qm; ð11Þ^A ¼ 1=2prL; where r ¼ d=2; ð12Þ^0A ¼ 2^A; ð13Þ

where d is the diameter of a fibre, L is the total length offibres per unit volume of a material, �A is the viscous char-acteristic length, and ^0A is the thermal characteristiclength.

The flow resistivities r of the glass wool samples inTable 1 were measured. By using Eq. (1) with rBies = r,the glass fibre diameter dBies in Table 1 was predicted.The glass fibre equivalent diameter dBies predicted fromEq. (1), d = dBies was substituted for Eq. (11) so that thetotal length of fibre per unit volume L was obtained. Thetotal length of fibres per unit volume was substituted inEq. (12) so that the two characteristic lengths �A and ^0Awere predicted.

The following similar relationship (Eq. (14)) for mela-mine fibre equivalents was obtained from measurementsof flow resistivity, bulk density and two characteristiclengths.

rKinod2Kinoq

�1:53l ¼ 11:5� 10�9; ð14Þ

where rKino is the flow resistivity, dKino is the diameter ofa hexagonal cellular strut in the melamine foam and theconstant ‘‘11.5 · 10�9’’ was obtained from a manual fit asdescribed in the following paragraph.

The flow resistivities r of the melamine foam samples inTable 2 were measured. The melamine fibre equivalentdiameter dKino predicted from Eq. (14), d = dKino wassubstituted for Eq. (11) so that the total length of fibreequivalents per unit volume L was obtained. The totallength of fibre equivalents per unit volume was substitutedin Eq. (12) so that the two characteristic lengths �A and ^0Awere predicted. Then, the right side value of Eq. (14) wasadjusted, so that the value of �A and �A were suitablefor the measured characteristic lengths of � and � 0 shownin Table 2.

By using Eq. (14) with rKino = r, the fibre equivalentdiameter dKino in Table 2 was predicted. The predicteddiameter of melamine fibre equivalents in Table 2 is about5.5 lm. It is in good agreement with visual inspection [seeFig. 1b]. The measurements of � and � 0 shown in Table 2are compared with the predictions of �A and ^0A shown inTable 2. The predictions are close to the measurements.

Fig. 3. Squared refraction index as a function of the square root of theinverse frequency in melamine foam samples saturated by air and argon ata temperature of 22.5 �C.

328 N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331

The discrepancies between the measurements of � and� 0 and the predictions of �A and ^0A are examined indetail. For the viscous characteristic lengths of the threemelamine foam samples shown in Table 2 the discrepancyis represented as 100 · j � A � �j/ � . The mean valuefor the melamine foam data is 5.9%. For the thermalcharacteristic lengths of the three melamine foam samplesshown in Table 2 the discrepancy is represented as100�j ^0A � ^0 j =^0. The mean value for the melaminefoam data is also 5.9%. The predictions are very closeto the measurements, so that the predictions of the twocharacteristic lengths are judged to be accurate. For themelamine foam samples, it is found that the two charac-teristic lengths are derivable from the fibre equivalentdiameter as shown in Eq. (14).

3.2. Verification of the measurements of non-acoustical

parameters

In this section, the measurements of the flow resistivitydata, the tortuosity data and the two characteristic lengthsdata in Tables 1 and 2 are verified. The measured flowresistivities were used to evaluate the characteristic imped-ance (Zc) and propagation constant (C) of the porous med-ium using the Delany and Bazley expressions [5].

ZcðxÞ ¼ q0c0½1þ 0:0571ðq0f =rÞ�0:754 � i0:087ðq0f =rÞ�0:732�;ð15Þ

CðxÞ ¼ xc0

½0:189ðq0f =rÞ�0:595 þ if1þ 0:0978ðq0f =rÞ�0:700g�;

ð16Þ

where c0 is the sound velocity in the air, and f is thefrequency.

The normal incidence plane wave absorption coefficient(a) for a hard-backed porous layer was then calculatedfrom

a ¼ 1� jrj2; ð17Þ

Fig. 2. Comparison between measured and predicted normal incidence absoBazley model as a function of frequency: (a) glass wool sample 2 and (b) mel

with

Zc cothðCbÞ=q0c0 ¼ ð1þ rÞ=ð1� rÞ; ð18Þwhere b is the thickness of the sample, and r is the pressurereflection factor.

The absorption coefficient was also measured using animpedance tube in accordance with the ISO standard trans-fer-function method [15]. The predicted normal incidenceabsorption coefficients are compared with the measureddata in Fig. 2. The predictions and measurements are close.Thus the flow resistivity data in Tables 1 and 2 were judgedto be sufficiently accurate.

The measurement results for the ultrasonic propagationin air and argon have been used to determine the two char-acteristic lengths and the tortuosity. The results for thesquared refraction index as a function of the square rootof the inverse frequency are shown in Fig. 3. The accuracyof the measurements of tortuosity and two characteristiclengths was tested by using the measured values to predictthe sound velocity through Eq. (19). By transforming thewave number equation at high frequencies [16], Eq. (19)is obtained. Subsequently the predicted and measuredsound velocities were compared as shown in Fig. 4. Thetemperature was 22 �C during the experiment.

rption coefficients for hard-backed material samples, by the Delany andamine foam samples 32 and 33.

Fig. 4. Comparison between measured and predicted sound velocities saturated by air as a function of frequency: (a) glass wool sample 2 and (b)melamine foam samples 32 and 33.

N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331 329

chigh ¼c0ffiffiffiffiffiffia1p 1� d

2

1

^ þc� 1ffiffiffiffiffi

Prp^0

� �� �1� d2

4

1

^ þc� 1ffiffiffiffiffi

Prp^0

� �2" #,

;

ð19Þ

d ¼ffiffiffiffiffiffiffiffi2gxq0

s; ð20Þ

where chigh is the sound velocity in the materials at high fre-quencies, and d is the viscous skin depth.

The discrepancy between measured and predicted soundvelocities is calculated as follows:

100 � jcm � chighj=cm; ð21Þwhere cm is the measured frequency-dependent soundvelocity and chigh is the predicted frequency-dependentsound velocity.

For the glass wool sample 2, the mean value of thesound velocity discrepancy between measurement and pre-diction in the frequency range between 100 kHz and800 kHz was 0.06%. For the sample 2, the maximum valueof the sound velocity prediction difference was 0.88 m s�1.For the melamine foam sample 32, the mean value of thesound velocity prediction discrepancy in the frequencyrange between 100 kHz and 800 kHz was 0.06%. For thesample 32, the maximum value of the sound velocity pre-diction difference was 0.54 m s�1. The predicted soundvelocities are very close to the measured ones, so that the

Fig. 5. Comparison between measured and predicted normal incidence absorpmodel as a function of frequency: (a) glass wool sample 2 and (b) melamine f

measurements of the tortuosity and the two characteristiclengths in Tables 1 and 2 are judged to be highly accurate.

4. Discussion

The predicted absorption coefficient for hard-backedmaterial samples by the Johnson–Allard model and themeasured one are shown in Fig. 5. The predictions are closeto measurements. The different pore structures of the twomaterials (glass fibre and melamine foam) are discussedusing the Johnson–Allard model. For the two glass woolsamples in Table 1, the values of c are predicted to be0.71 and 0.78. For the melamine foam samples in Table 2the values of c are predicted to be between 0.49 and 0.57.According to the Johnson–Allard model, it is found thatthe melamine foam is an efficient material that achieveslarge flow resistivity by lowering the cross-sectional shapefactors, though the two characteristic lengths are much lar-ger than those for glass wool.

Next, two glass wool examples in Table 3 are prepared toinvestigate the melamine foam sample 32 more in detail.Using 7.0 lm as the diameter of a glass fibre, the flow resis-tivity of the glass wool examples is predicted by the Biesmodel [Eq. (1)]. The two characteristic lengths are predictedby using the Allard model [Eqs. (11)–(13)]. Using 1.0 as thetortuosity, the cross-sectional shape factors are predictedby using the Johnson–Allard model [Eqs. (4) and (7)].

tion coefficients for hard-backed material samples, by the Johnson–Allardoam sample 32.

330 N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331

By comparing melamine foam sample 32 and glass woolexample 1 with a same flow resistivity 13,100 Pa s m�2, it isfound that the bulk density for melamine foam is 0.32 timessmaller than that for glass wool. It is also found that theviscous characteristic length for melamine foam is 1.46times larger than that for glass wool and the thermal char-acteristic length for melamine foam is 1.63 times largerthan that for glass wool. Additionally, it is found thatthe cross-sectional shape factor c for melamine foam is0.68 times smaller than that for glass wool and c 0 for mel-amine foam is 0.62 times smaller than that for glass wool.

By comparing melamine foam sample 32 and glass woolexample 2 with a same viscous characteristic length199 lm, it is found that the bulk density of melamine foamis 0.47 times smaller than that for glass wool. It has beenalso found that the flow resistivity for melamine foam is1.79 times larger than that for glass wool. Additionally, itis found that the cross-sectional shape factor c for mela-mine foam is 0.75 times smaller than that for glass wooland c 0 for melamine foam is 0.69 times smaller than thatfor glass wool. As a result, it is found that the structureof melamine foam is clearly different from that of glasswool.

The predicted absorption coefficient for hard-backedmaterial samples by the Johnson–Allard model for themelamine foam sample 32 shown in Fig. 5b is less accurate

Table 3Predicted parameters of two glass wool examples

Example number 1 2

Predictions (Bies and Allard models)

q1 (kg m�3) 32.104 21.95rBies (Pa s m�2) 13,100 7322a1 1.0 1.0/ 0.987 0.991dBies (lm) 7.0 7.0L · 108 (m�2) 3.34 2.28�A (lm) 136 199^0A (lm) 273 399

Predictions (Johnson–Allard model)

c 0.78 0.71c 0 0.39 0.35

Fig. 6. Comparison between measured and predicted normal incidence absofunction of frequency. Correction factors were 8.5 (N1) and 250 (N2). (a) har20 mm.

than the glass wool sample 2 shown in Fig. 5a. The predic-tions by our new model [17] as shown in Eqs. (22)–(25)have been executed and are shown in Fig. 6.

qðxÞ ¼ q0a1 1þ r/ia1q0x

GNðxÞ� �

; ð22Þ

with

GN ðxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi2gq0x

p a1ð1þ iÞffiffiffiffiffiffiN 1

p

r/^

� �1=2

; ð23Þ

KðxÞ ¼ cP 0 c� ðc� 1Þ 1þ 8gi^02q0Prx

G0N ðPrxÞ� ��1

" #,;

ð24Þ

with

G0N ðPrxÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gq0Prx

p ^0ð1þ iÞffiffiffiffiffiffiN 2

p

8g

� �1=2

; ð25Þ

where N1 and N2 are the correction factors.The predicted and measured absorption coefficients for

hard-backed material are shown in Fig. 6a. Those with rearair layer of 20 mm are shown in Fig. 6b. The two correctionfactors were 8.5(N1) and 250(N2). The discrepancy betweenmeasurement and prediction in Fig. 6 was calculated by

100� jam � apj=am; ð26Þwhere am is the measured frequency-dependent narrow bandabsorption coefficient and ap is the predicted frequency-dependent narrow band absorption coefficient.

The discrepancy using our new model is 5% or less in thefrequency range between 400 and 5 kHz, so it has beenfound that our prediction model with the two correctionfactors is effective for the melamine foam.

5. Concluding remarks

According to the Johnson–Allard model, it has beenfound that the melamine foam achieves large flow resistiv-ity by lowering the cross-sectional pore shape factors,despite the fact that the two characteristic lengths are muchlarger than those of the glass wool with a diameter of a

rption coefficients for melamine foam sample 32 by our new model as ad-backed material sample and (b) material sample with rear air layer of

N. Kino, T. Ueno / Applied Acoustics 69 (2008) 325–331 331

glass fibre of 7 lm. We also showed the possibility that thetwo characteristic lengths for the melamine media arederivable using the measured flow resistivity data and bulkdensity data.

Acknowledgement

The authors thank Mr. Yasuhiro Suzuki.

References

[1] Bies DA, Hansen CH. Flow resistance information for acousticaldesign. Appl Acoust 1980;13:357–91.

[2] Zwikker C, Kosten CW. Sound absorbing materials. NewYork: Elsevier; 1949.

[3] Biot MA. Theory of propagation of elastic waves in a fluid-saturatedporous solid. I. Low-frequency range. J Acoust Soc Am1956;28:168–78.

[4] Biot MA. Theory of propagation of elastic waves in a fluid-saturatedporous solid. II. Higher frequency range. J Acoust Soc Am1956;28:179–91.

[5] Delany ME, Bazley EN. Acoustical properties of fibrous absorbentmaterials. Appl Acoust 1970;3:105–16.

[6] Miki Y. Acoustical properties of porous materials – generalizations ofempirical models. J Acoust Soc Jpn 1990;11(E):25–8.

[7] Champoux Y, Stinson MR, Daigle GA. Air-based system for themeasurement of porosity. J Acoust Soc Am 1991;89:910–6.

[8] Johnson DL, Koplik J, Dashen R. Theory of dynamic permeabilityand tortuosity in fluid-saturated porous media. J Fluid Mech1987;176:379–402.

[9] Champoux Y, Allard JF. Dynamic tortuosity and bulk modulus inair-saturated porous media. J Appl Phys 1991;70:1975–9.

[10] Henry M, Lemarinier Pavel, Allard JF, Bonardet JL, Gedeon A.Evaluation of the characteristic dimensions of porous sound-absorb-ing materials. J Appl Phys 1995;77:17–20.

[11] ISO 9053, Acoustics – materials for acoustical applications –determination of airflow resistance; 1991.

[12] Leclaire Ph, Keiders L, Lauriks W. Determination of theviscous and thermal characteristic lengths of plastic foams byultrasonic measurements in helium and air. J Appl Phys1996;80:2009–12.

[13] Kino N. Ultrasonic measurements of the two characteristic lengths infibrous materials. Appl Acoust 2006, doi:10.1016/j.apacoust.2006.07.007.

[14] Allard JF, Champoux Y. New empirical equations for soundpropagation in rigid frame fibrous materials. J Acoust Soc Am1992;91:3346–53.

[15] ISO 10534-2, Acoustics – determination of sound absorption coeffi-cient and impedance in impedance tubes – part 2: transfer-functionmethod; 1998.

[16] Allard JF, Castagnede B, Henry M, Laurinks W. Evaluation oftortuosity in acoustic porous materials saturated by air. Rev SciInstrum 1994;65:754–5.

[17] Kino N, Ueno T. Improvements to the Johnson–Allard model forrigid-framed fibrous materials. Appl Acoust 2006, doi:10.1016/j.apacoust.2006.07.005.

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