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Ultrasonic measurements of the two characteristic lengths in fibrous materials
Applied Acoustics 68 (2007) 1427–1438 www.elsevier.com/locate/apacoust
Ultrasonic measurements of the two characteristic lengths in fibrous materials Naoki Kino
*
Shizuoka Industrial Research Institute of Shizuoka Prefecture, 2078 Makigaya, Aoi-ku, Shizuoka 421-1298, Japan Received 29 September 2005; received in revised form 5 July 2006; accepted 7 July 2006 Available online 22 September 2006
Abstract In a method for measuring the two characteristic lengths in rigid frame porous media proposed by Leclaire et al. [Ph. Leclaire, L. Keiders, W. Lauriks. 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] ultrasonic transmission measurements are made through a sample of material when it is saturated with two different gases. It is shown that the accuracy of this method can be limited when the porous medium in saturated with helium as proposed originally. A considerable improvement in the signal-to-noise ratio can be achieved if argon is used instead of helium because the characteristic impedance is higher and the viscous skin depth in argon is 3 times less than that in helium. This paper also discusses the measurement of the tortuosity and the sound velocity in glass wool. 2006 Elsevier Ltd. All rights reserved. Keywords: Tortuosity; Viscous characteristic length; Thermal characteristic length
1. Introduction Leclaire et al. [1] have proposed the Sl method to determine the tortuosity and two characteristic lengths at the same time. In accordance with this approach a measurement system has been developed to determine the speed of ultrasonic signal propagation in rigid-framed fibrous materials saturated by different gases. While the measurements in *
Tel: +81 54 278 3027; fax: +81 54 278 3066. E-mail address: [email protected].
0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.07.007
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N. Kino / Applied Acoustics 68 (2007) 1427–1438
air were problem-free, problems were found with the measurements in helium. The acoustic attenuation increased considerably in helium even when there was no material between the broadband air-coupled transducers. When the material was placed between the transducers, the measurement of the transmitted ultrasonic signal through high flow resistance materials or thick material samples was difficult. It was difficult in the helium environment to obtain the necessary signal-to-noise ratio to ensure an accurate measurement. The problems have been traced to the acoustic coupling between the broadband air-coupled transducer and the gas and to the relatively high kinematic viscosity of helium. 2. Theoretical considerations According to the method proposed by Leclaire et al. [1] the tortuosity and characteristic lengths of porous medium can be resolved from the high-frequency behaviour of the refraction index. The wave number at high frequencies in rigid-porous materials is given by x pffiffiffiffiffiffi d 1 c1 kffi þ pffiffiffiffiffi 0 ; a1 1 þ ð1 iÞ ð1Þ c0 2 ^ Pr^ where sffiffiffiffiffiffiffiffi rffiffiffiffiffi 2g 2m ð2Þ ¼ d¼ xq0 x and k is the wave number, x is the angular frequency, c0 is the sound velocity in a gas, is 0 the viscous characteristic length, pffiffiffiffiffiffiffi is the thermal characteristic length, g is the viscosity of gas, a1 is the tortuosity, i ¼ 1, d is the viscous skin depth, Pr is the Prandtl number of gas, q0 is the density of a gas, c is the specific heat ratio of a gas, m is the kinematic viscosity of a gas, c is the sound velocity in the porous medium, f is the frequency of sound, and (c0/ c) is the refraction index. pffiffiffiffiffi 2 Using the approximation that d2 ð1= ^ þðc 1Þ=ð Pr^0 ÞÞ ffi 0 r ffiffiffi c 2 m 1 c1 1 0 þ pffiffiffiffiffi 0 pffiffiffi þ a1 : a1 ð3Þ p ^ c f Pr^ The right hand side of Eq. (3) is a linear function of f1/2. Initially the ultrasonic signal transmitted between the transducers was measured without the porous material present. This was used as the ‘‘reference signal.’’ Next, the porous material was inserted between the transducers and the transmitted ultrasonic signal was measured. This served as the ‘‘transmitted signal.’’ By analyzing the cross-spectrum of the ‘‘reference signal’’ and the ‘‘transmitted signal,’’ the delay time was obtained. The left hand side of Eq. (3) was calculated by substituting the measured delay time T in the following equation [2,3]: c0 ¼ 1 þ Tc0 =d; ð4Þ c where T is the delay time caused by inserting the porous material between the broadband air-coupled transducers and d the thickness of the porous material. Two simultaneous equations apply when analyzing the cross-spectrum for the material saturated by two different gases, e.g. air and argon:
N. Kino / Applied Acoustics 68 (2007) 1427–1438
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rffiffiffiffiffiffi mair 1 cair 1 þ pffiffiffiffiffiffiffiffiffi 0 ; Slopeair ¼ a1 p ^ Prair ^ ! rffiffiffiffiffiffiffiffiffiffiffi cargon 1 margon 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 : Slopeargon ¼ a1 ^ p Prargon ^
ð5Þ ð6Þ
Here Slopeair and Slopeargon are the slopes of the line of (c0/c)2 as a function of f1/2, a1 is the intersection of the line of (c0/c)2 as a function of f1/2 in the ordinate axis, mair is the kinematic viscosity of air, Prair is the Prandtl number of air, margon is the kinematic viscosity of argon and Prargon is the Prandtl number of argon. The tortuosity and two characteristic lengths were deduced from data at the same time by solving the above simultaneous equations. High accuracy of measurement of the sound velocity in the material is important for the successful application of Eq. (3). There are gases with smaller or larger acoustic impedance than air. The value of the impedance determines the degree of acoustic coupling between the broadband transducer and the gas. It is necessary to select a gas with large characteristic acoustic impedance z0 ¼ q0 c0
ð7Þ
to ensure a good signal-to-noise ratio and perform an accurate measurement. Table 1 lists the parameters of the gases [4,5] used in this work. For helium, the acoustic impedance is 167 kg m2 s1. For air, the acoustic impedance is 412 kg m2 s1. For argon, the acoustic impedance is 530 kg m2 s1. The acoustic impedance of helium is much smaller than that of air and argon. Thus, if argon is used, the coupling between the ultrasonic transducers and the saturating gas is better than that observed in air. If helium is used, the coupling is poor. The kinematic viscosities of argon and helium (see Table 1) are different. The kinematic viscosity of helium is 7.8-fold greater than that of air. Eq. (2) suggests that a gas with a large kinematic viscosity will increase the amount of attenuation in the material, particularly in the low frequency range. The acoustic attenuation in porous medium depends upon the ratio between the pore size and the viscous skin depth. If the viscous skin depth is much less than the pore size, the attenuation is likely to be low. The fibrous structure of glass wool has been observed using a VHX-200 digital microscope of the Keyence Corporation (see Fig. 1). The diameter of the glass fibres is close to 7 lm. Table 2 lists the viscous skin depths in air, helium and argon at a temperature of 20 C. At 50 kHz for instance, the viscous skin depth in helium is 27.4 lm whereas that in argon is 9.2 lm. The viscous skin Table 1 Physical properties of gases at a temperature of 20 C q0 (kg m3) g (kg m1 s1) m (m2 s1) c Pr pffiffiffiffiffi ðc 1Þ= Pr z0 (kg m2 s1) c0 (m s1)
Air
Helium
Argon
1.20 1.82 · 105 1.51 · 105 1.4 0.71 0.475 412 343
0.166 1.96 · 105 11.78 · 105 1.67 0.67 0.819 167 1007
1.66 2.23 · 105 1.34 · 105 1.67 0.67 0.819 530 319
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Fig. 1. Digital microscope photograph of the fibrous structure of glass wool.
Table 2 Viscous skin depth in three different gases at a temperature of 20 C
df = 50 kHz (lm) df = 100 kHz (lm) df = 200 kHz (lm) df = 250 kHz (lm) df = 300 kHz (lm) df = 400 kHz (lm) df = 500 kHz (lm)
Air
Helium
Argon
9.8 6.9 4.9 4.4 4.0 3.5 3.1
27.4 19.4 13.7 12.3 11.2 9.7 8.7
9.2 6.5 4.6 4.1 3.8 3.3 2.9
depth in helium is about 3.9 times larger than the 7 lm of the diameter of the glass fibre. Similarly, the viscous skin depth at 100 kHz in helium is 19.4 lm whereas that in argon is 6.5 lm. The viscous skin depth at 100 kHz in helium is about 2.8 times larger than the fibre diameter. The viscous skin depth at 200 kHz in helium is 13.7 lm whereas that in argon is 4.6 lm. The viscous skin depth at 200 kHz in helium is about 2.0 times larger than the pore size. When the material is placed in helium, the size of the pores is less than the viscous skin depth in the considered frequency range. If a saturating a gas with smaller kinematic viscosity is used it is possible to lower the attenuation of the transmitted ultrasonic signal. Given that Argon is has a larger acoustic impedance and smaller kinematic viscosity than helium, it can be concluded that it was preferable to use argon. 3. Results and discussion 3.1. Material An experiment was conducted using glass wool samples (see Table 3) with different densities (41.7 kg m3 and 81.2 kg m3) and thickness (24.5 mm and 10.0 mm). The flow resis-
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Table 3 Characteristics of glass wool samples Sample
Sample 1
Sample 2
q1 (kg m3) d (mm) r (Pa s m2) /
41.7 24.5 21,500 0.983
81.2 10.0 49,000 0.968
tivity r was measured by a device developed in accordance with the ISO standard [6]. The porosity was calculated from the density of glass (2500 kg m3) using ð8Þ
/ ¼ 1 q1 =2500: where / is the porosity and q1 is the bulk density of glass wool. 3.2. Measurement system
Fig. 2 presents a sketch of the system used to measure the sound velocity of the ultrasonic signal propagated through the porous sample. The container in Fig. 2 has the gas exchange function, and can be filled with another type of gas besides air. Air, helium and argon were used in these experiments. Two air-coupled transducers made by Microacoustic Instruments Inc were used. The ultrasonic signals were recorded using a digital oscilloscope. The samples were inserted between the two transducers using the rotatingstick mounting mechanism shown in Fig. 2. The distance between the transducers was 57 mm. The sample was supported by the ring shown in Fig. 2. The personal computer and the digital oscilloscope were connected using a communication link. The cross-spectrum was calculated by the personal computer. The time window was rectangular. The analytical frequency range was set to between 100 kHz and 800 kHz because of a good signal-to-noise ratio.
Rotatable Stick Generator NF WF1943A Waveform :SinWaveBurst
Vacuum Pump
Power Amp Thamway T145-5714B
Gas Canister MicroAcoustic Air-coupledTransducer (Source) 40k Hz - 1.7M Hz
20kHz - 1MHz Porous Material (Ring support)
Digital Oscilloscope LeCroy 6051 SamplingRate:200MS/sec
MicroAcoustic Air-coupledTransducer (Receiver) 40k Hz - 2.25M Hz
Personal Computer Fig. 2. Measurement system.
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3.3. Measurement of the ultrasonic reception signal in the different gases Fig. 3 shows the measurement results for sample 2 when saturated with air (Figs. 3a and b), argon (Figs. 3c and d) and helium (Figs. 3e and f). The output level of the generator and the amplification level of the power amplifier were the same in all of the experiments. The time histories for the data shown in Fig. 3a–d were synchronized. The fact that the sound velocity in argon is slower than that in air is apparent. The sound velocity in helium 1.00
0.05
0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Amplitude [ V ]
Reference 0.50
0.01 -0.01 10
20
30
50
-0.05
(a) Air Sample 2
Time [ s ]
(b) Air Sample 2
Time [ s ]
0.05
1.00 0.50 0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Reference
Transmitted
0.03 0.01 -0.01 10
20
30
40
50
-0.03 -0.05
-1.00 (c) Ar Sample 2
(d) Ar Sample 2
Time [ s ]
1.00
Time [ s ]
0.05
0.50 0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Reference Amplitude [ V ]
40
-0.03
-1.00
Amplitude [ V ]
Transmitted
0.03
Transmitted
0.03 0.01 -0.01 10
20
30
40
50
-0.03
-1.00
-0.05 (e) He Sample 2
Time [ s ]
(f) He Sample 2
Time [ s ]
Fig. 3. Reference and transmitted signals when sample 2 is saturated by different gases: (a,b) air; (c,d) argon; (e,f) helium.
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Reception signal level [ dB ]
is about 2.9 times larger than that in air. The range of the time history was switched for the convenience of the synchronous addition processing of the digital oscilloscope, so the time histories presented in Figs. 3a and e were not synchronized. A clear decrease in the sound pressure is observed in helium [see Fig. 3e]. The same problem is apparent also in the transmitted signals when the material was inserted between the two broadband air-coupled transducers [see Fig. 3f]. The transmitted ultrasonic signal spectra and noise level spectra are shown in Fig. 4 for sample 1 (Fig. 4a) and sample 2 (Fig. 4b) saturated with argon, air and helium. The problem of acoustic coupling between the broadband air-coupled transducer and helium is demonstrated by the fact that the reference signal levels in helium are lower than that measured in air and argon by between 10 dB and 20 dB [see Figs. 4a and b]. The transmitted signal levels in air and argon in the frequency range from 50 kHz to 250 kHz [see Figs. 4a and b] are about 10 dB higher than that at 400 kHz. The transmitted signal levels in helium are low across the entire frequency range.
-30 Reference
-40 -50
Transmitted
-60 -70
Ar Air He Noise
-80 -90 -100 -110
Reception signal level [ dB ]
0 200 (a) Sample 1
400
600 800 Frequency [ kHz ]
-30 Reference
-40 -50
Transmitted
-60 -70
Ar Air He Noise
-80 -90 -100 -110 0
200 (b) Sample 2
400
600 800 Frequency [ kHz ]
Fig. 4. Reference and transmitted signal spectra obtained before and after glass wool samples are saturated by different gases. (a) sample 1; (b) sample 2.
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3.4. Analysis of the sound velocity in the different gases The temperature was 25 C and the atmospheric pressure was 1.013 · 105 Pa during the experiment. The sound velocity in the tested sample was determined using Eq. (4). Fig. 5 shows the sound velocity measured in the glass wool samples 1 and 2. The measurement results for air are shown in Fig. 5a, for argon in Fig. 5b and for helium in Fig. 5c. It can be clearly seen from Fig. 5c that the data for helium saturation are much less accurate than those obtained for saturation with air and argon. This result was due to the low signal-to-noise ratio mainly incurred by the problem of the acoustic coupling between the transducers and the helium. As suggested earlier, another factor which influenced the low signal-to-noise ratio in helium was related to the value of the kinematic viscosity of helium. Table 2 shows that the viscous skin depth of helium is much larger than that of air and argon. The measurement results for helium in the frequency range of 400 kHz or less [see Fig. 5c] are not especially accurate. 3.5. Determination of the two characteristic lengths, based on the measurements in air and argon 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 (c0/ c)2 (see Eq. (3)) are shown in Fig. 6 as a function of f1/2. The measurement results for sample 1 saturated by air and argon are shown in Fig. 6a and the corresponding measurement results for sample 2 are shown in Fig. 6b. An approximate linear fit was obtained for the data in the frequency range between 100 kHz and 800 kHz. The slopes of the linear fits shown in Fig. 6 were substituted in Eqs. (5) and (6) together with the tortuosity data. As a result the two characteristic lengths were obtained and their values are listed in Table 4. 3.6. Verification of the measurements of the tortuosity and two characteristic lengths By transposing Eq. (1), the following expression for the sound speed is obtained: c0 d 1 c1 ffiffiffiffi ffi p c ffi pffiffiffiffiffiffi 1 þ : ð9Þ 2 ^ a1 P r ^0 The values of the tortuosity and two characteristic lengths deduced from measurements and listed in Table 4 were substituted in Eq. (9), to enable calculation of values of sound velocity in the porous material samples. In Fig. 7, the predicted values of sound velocity in the materials are compared with the measured values for sample 1 (Fig. 7a) and sample 2 (Fig. 7b). The prediction error rate is defined by 100 jccal cmea j=cmea
ð10Þ
where ccal is the frequency-dependent predicted sound velocity, and cmea is the frequencydependent measured sound velocity. For sample 1, the mean value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz was 0.15% in air and 0.19% in argon whereas the maximum value of the sound velocity prediction error rate in this frequency range
N. Kino / Applied Acoustics 68 (2007) 1427–1438
1435
350
Velocity [ m/s ]
330 310 290 270
Sample 1 Sample 2
250 230
0
200 (a) Air
400 600 800 Frequency [ kHz ]
350
Velocity [ m/s ]
330 310 290
Sample 1
270
Sample 2
250 230 0
200 (b) Ar
400 600 800 Frequency [ kHz ]
1000
Velocity [ m/s ]
980 960 940 920 900 880
Sample 1
860
Sample 2
840 0
200 (c) He
400 600 Frequency [ kHz ]
800
Fig. 5. Sound velocity as a function of frequency measured in glass wool samples saturated by different gases at a temperature of 25 C: (a) air; (b) argon; (c) helium.
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Squared refraction index
1.25
Air
x: 1000/ Frequency y: Squared refraction index
1.15 1.10 1.05
y = 0.0261x + 1.0124(Air)
1.00
0
1 (a) Sample 1
1.25 Squared refraction index
Ar
y = 0.0278x + 1.0115(Ar)
1.20
2
3
4
1000/
Frequency
y = 0.0523x + 1.0277(Ar)
1.20
x: 1000/ Frequency y: Squared refraction index
1.15 Ar
1.10
Air
1.05 1.00
y = 0.0492x + 1.0295(Air)
0
1 (b) Sample 2
2
3
1000/
Frequency
4
Fig. 6. Squared refraction index as a function of the square root of the inverse frequency measured in glass wool samples saturated by air and argon at a temperature of 25 C: (a) sample 1; (b) sample 2.
Table 4 Values of tortuosity and two characteristic lengths deduced from ultrasonic transmission measurements Sample
Sample 1
Sample 2
a1 (lm) 0 (lm)
1.0124 106 225
1.0295 57 123
is 0.47% in air and 0.56% in argon, corresponding to a maximum value of the sound velocity prediction difference of 1.59 m s1 in air and 1.72 m s1 in argon. For sample 2, the mean value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz is 0.30% in air and 0.44% in argon, whereas the maximum value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz was 0.93% in air and 1.06% in argon, corresponding to a maximum
N. Kino / Applied Acoustics 68 (2007) 1427–1438
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350 Air
Velocity [ m/s ]
340 330 320 310 Measured
300 Ar
290 280
0
200 (a) Sample 1
Predicted 400 600 Frequency [ kHz ]
800
350 Air
Velocity [ m/s ]
340 330 320
Ar
310 300
Measured
290
Predicted
280 0
200 (b) Sample 2
400
600
800
Frequency [ kHz ]
Fig. 7. Comparison between measured and predicted sound velocity as a function of frequency in glass wool samples saturated by air and argon at a temperature of 25 C: (a) sample 1; (b) sample 2.
value of the sound velocity prediction difference in this frequency range of 3.0 m s1 in air and 3.15 m s1 in argon. Since the difference between the measured values and the predicted values is small, it can be concluded that the measurement values of the tortuosity and two characteristic lengths in Table 4 are sufficiently accurate. 4. Concluding remarks A method for measuring the two characteristic lengths in a rigid-porous material, proposed by Leclaire et al. [1] and based on ultrasonic transmission measurements using different saturating gases has been studied. The influence of the gas properties on the signalto-noise ratio has been investigated. It has been found that the improved signal-to-noise ratios are obtained when using a gas with a reasonably high acoustic impedance and low kinematic viscosity, such as argon.
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When using helium in a high flow resistivity material, it has been found necessary to use thin samples. However, it is difficult to obtain uniform thin samples of a fibrous material. Also when the measurement is carried out on a thin layer the accuracy of the delay measurement is likely to be limited. There is a limit to the thickness of the material that can be measured even if argon is used. However, the range of applicability of the method is greater than when using helium. References [1] 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. [2] Nagy PB, Adler L, Bonner BP. Slow wave propagation in air-filled porous materials and natural rocks. Appl Phys Lett 1990;56:2504–6. [3] Nagy PB. Slow wave propagation in air-filled permeable solids. J Acoust Soc Am 1993;93:3224–34. [4] Gray DE (coordinating editor). American Institute of Physics Handbook. New York: McGraw-Hill; 1972. [5] Kaye GWC, Laby TH. Tables of physical and chemical constants and some mathematical functions. 14th ed. London: Longman Group Limited; 1973. [6] ISO 9053:1991. Acoustics – materials for acoustical applications – determination of airflow resistance.
Ultrasonic measurements of the two characteristic lengths in fibrous materials Naoki Kino
*
Shizuoka Industrial Research Institute of Shizuoka Prefecture, 2078 Makigaya, Aoi-ku, Shizuoka 421-1298, Japan Received 29 September 2005; received in revised form 5 July 2006; accepted 7 July 2006 Available online 22 September 2006
Abstract In a method for measuring the two characteristic lengths in rigid frame porous media proposed by Leclaire et al. [Ph. Leclaire, L. Keiders, W. Lauriks. 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] ultrasonic transmission measurements are made through a sample of material when it is saturated with two different gases. It is shown that the accuracy of this method can be limited when the porous medium in saturated with helium as proposed originally. A considerable improvement in the signal-to-noise ratio can be achieved if argon is used instead of helium because the characteristic impedance is higher and the viscous skin depth in argon is 3 times less than that in helium. This paper also discusses the measurement of the tortuosity and the sound velocity in glass wool. 2006 Elsevier Ltd. All rights reserved. Keywords: Tortuosity; Viscous characteristic length; Thermal characteristic length
1. Introduction Leclaire et al. [1] have proposed the Sl method to determine the tortuosity and two characteristic lengths at the same time. In accordance with this approach a measurement system has been developed to determine the speed of ultrasonic signal propagation in rigid-framed fibrous materials saturated by different gases. While the measurements in *
Tel: +81 54 278 3027; fax: +81 54 278 3066. E-mail address: [email protected].
0003-682X/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2006.07.007
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N. Kino / Applied Acoustics 68 (2007) 1427–1438
air were problem-free, problems were found with the measurements in helium. The acoustic attenuation increased considerably in helium even when there was no material between the broadband air-coupled transducers. When the material was placed between the transducers, the measurement of the transmitted ultrasonic signal through high flow resistance materials or thick material samples was difficult. It was difficult in the helium environment to obtain the necessary signal-to-noise ratio to ensure an accurate measurement. The problems have been traced to the acoustic coupling between the broadband air-coupled transducer and the gas and to the relatively high kinematic viscosity of helium. 2. Theoretical considerations According to the method proposed by Leclaire et al. [1] the tortuosity and characteristic lengths of porous medium can be resolved from the high-frequency behaviour of the refraction index. The wave number at high frequencies in rigid-porous materials is given by x pffiffiffiffiffiffi d 1 c1 kffi þ pffiffiffiffiffi 0 ; a1 1 þ ð1 iÞ ð1Þ c0 2 ^ Pr^ where sffiffiffiffiffiffiffiffi rffiffiffiffiffi 2g 2m ð2Þ ¼ d¼ xq0 x and k is the wave number, x is the angular frequency, c0 is the sound velocity in a gas, is 0 the viscous characteristic length, pffiffiffiffiffiffiffi is the thermal characteristic length, g is the viscosity of gas, a1 is the tortuosity, i ¼ 1, d is the viscous skin depth, Pr is the Prandtl number of gas, q0 is the density of a gas, c is the specific heat ratio of a gas, m is the kinematic viscosity of a gas, c is the sound velocity in the porous medium, f is the frequency of sound, and (c0/ c) is the refraction index. pffiffiffiffiffi 2 Using the approximation that d2 ð1= ^ þðc 1Þ=ð Pr^0 ÞÞ ffi 0 r ffiffiffi c 2 m 1 c1 1 0 þ pffiffiffiffiffi 0 pffiffiffi þ a1 : a1 ð3Þ p ^ c f Pr^ The right hand side of Eq. (3) is a linear function of f1/2. Initially the ultrasonic signal transmitted between the transducers was measured without the porous material present. This was used as the ‘‘reference signal.’’ Next, the porous material was inserted between the transducers and the transmitted ultrasonic signal was measured. This served as the ‘‘transmitted signal.’’ By analyzing the cross-spectrum of the ‘‘reference signal’’ and the ‘‘transmitted signal,’’ the delay time was obtained. The left hand side of Eq. (3) was calculated by substituting the measured delay time T in the following equation [2,3]: c0 ¼ 1 þ Tc0 =d; ð4Þ c where T is the delay time caused by inserting the porous material between the broadband air-coupled transducers and d the thickness of the porous material. Two simultaneous equations apply when analyzing the cross-spectrum for the material saturated by two different gases, e.g. air and argon:
N. Kino / Applied Acoustics 68 (2007) 1427–1438
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rffiffiffiffiffiffi mair 1 cair 1 þ pffiffiffiffiffiffiffiffiffi 0 ; Slopeair ¼ a1 p ^ Prair ^ ! rffiffiffiffiffiffiffiffiffiffiffi cargon 1 margon 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi 0 : Slopeargon ¼ a1 ^ p Prargon ^
ð5Þ ð6Þ
Here Slopeair and Slopeargon are the slopes of the line of (c0/c)2 as a function of f1/2, a1 is the intersection of the line of (c0/c)2 as a function of f1/2 in the ordinate axis, mair is the kinematic viscosity of air, Prair is the Prandtl number of air, margon is the kinematic viscosity of argon and Prargon is the Prandtl number of argon. The tortuosity and two characteristic lengths were deduced from data at the same time by solving the above simultaneous equations. High accuracy of measurement of the sound velocity in the material is important for the successful application of Eq. (3). There are gases with smaller or larger acoustic impedance than air. The value of the impedance determines the degree of acoustic coupling between the broadband transducer and the gas. It is necessary to select a gas with large characteristic acoustic impedance z0 ¼ q0 c0
ð7Þ
to ensure a good signal-to-noise ratio and perform an accurate measurement. Table 1 lists the parameters of the gases [4,5] used in this work. For helium, the acoustic impedance is 167 kg m2 s1. For air, the acoustic impedance is 412 kg m2 s1. For argon, the acoustic impedance is 530 kg m2 s1. The acoustic impedance of helium is much smaller than that of air and argon. Thus, if argon is used, the coupling between the ultrasonic transducers and the saturating gas is better than that observed in air. If helium is used, the coupling is poor. The kinematic viscosities of argon and helium (see Table 1) are different. The kinematic viscosity of helium is 7.8-fold greater than that of air. Eq. (2) suggests that a gas with a large kinematic viscosity will increase the amount of attenuation in the material, particularly in the low frequency range. The acoustic attenuation in porous medium depends upon the ratio between the pore size and the viscous skin depth. If the viscous skin depth is much less than the pore size, the attenuation is likely to be low. The fibrous structure of glass wool has been observed using a VHX-200 digital microscope of the Keyence Corporation (see Fig. 1). The diameter of the glass fibres is close to 7 lm. Table 2 lists the viscous skin depths in air, helium and argon at a temperature of 20 C. At 50 kHz for instance, the viscous skin depth in helium is 27.4 lm whereas that in argon is 9.2 lm. The viscous skin Table 1 Physical properties of gases at a temperature of 20 C q0 (kg m3) g (kg m1 s1) m (m2 s1) c Pr pffiffiffiffiffi ðc 1Þ= Pr z0 (kg m2 s1) c0 (m s1)
Air
Helium
Argon
1.20 1.82 · 105 1.51 · 105 1.4 0.71 0.475 412 343
0.166 1.96 · 105 11.78 · 105 1.67 0.67 0.819 167 1007
1.66 2.23 · 105 1.34 · 105 1.67 0.67 0.819 530 319
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N. Kino / Applied Acoustics 68 (2007) 1427–1438
Fig. 1. Digital microscope photograph of the fibrous structure of glass wool.
Table 2 Viscous skin depth in three different gases at a temperature of 20 C
df = 50 kHz (lm) df = 100 kHz (lm) df = 200 kHz (lm) df = 250 kHz (lm) df = 300 kHz (lm) df = 400 kHz (lm) df = 500 kHz (lm)
Air
Helium
Argon
9.8 6.9 4.9 4.4 4.0 3.5 3.1
27.4 19.4 13.7 12.3 11.2 9.7 8.7
9.2 6.5 4.6 4.1 3.8 3.3 2.9
depth in helium is about 3.9 times larger than the 7 lm of the diameter of the glass fibre. Similarly, the viscous skin depth at 100 kHz in helium is 19.4 lm whereas that in argon is 6.5 lm. The viscous skin depth at 100 kHz in helium is about 2.8 times larger than the fibre diameter. The viscous skin depth at 200 kHz in helium is 13.7 lm whereas that in argon is 4.6 lm. The viscous skin depth at 200 kHz in helium is about 2.0 times larger than the pore size. When the material is placed in helium, the size of the pores is less than the viscous skin depth in the considered frequency range. If a saturating a gas with smaller kinematic viscosity is used it is possible to lower the attenuation of the transmitted ultrasonic signal. Given that Argon is has a larger acoustic impedance and smaller kinematic viscosity than helium, it can be concluded that it was preferable to use argon. 3. Results and discussion 3.1. Material An experiment was conducted using glass wool samples (see Table 3) with different densities (41.7 kg m3 and 81.2 kg m3) and thickness (24.5 mm and 10.0 mm). The flow resis-
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Table 3 Characteristics of glass wool samples Sample
Sample 1
Sample 2
q1 (kg m3) d (mm) r (Pa s m2) /
41.7 24.5 21,500 0.983
81.2 10.0 49,000 0.968
tivity r was measured by a device developed in accordance with the ISO standard [6]. The porosity was calculated from the density of glass (2500 kg m3) using ð8Þ
/ ¼ 1 q1 =2500: where / is the porosity and q1 is the bulk density of glass wool. 3.2. Measurement system
Fig. 2 presents a sketch of the system used to measure the sound velocity of the ultrasonic signal propagated through the porous sample. The container in Fig. 2 has the gas exchange function, and can be filled with another type of gas besides air. Air, helium and argon were used in these experiments. Two air-coupled transducers made by Microacoustic Instruments Inc were used. The ultrasonic signals were recorded using a digital oscilloscope. The samples were inserted between the two transducers using the rotatingstick mounting mechanism shown in Fig. 2. The distance between the transducers was 57 mm. The sample was supported by the ring shown in Fig. 2. The personal computer and the digital oscilloscope were connected using a communication link. The cross-spectrum was calculated by the personal computer. The time window was rectangular. The analytical frequency range was set to between 100 kHz and 800 kHz because of a good signal-to-noise ratio.
Rotatable Stick Generator NF WF1943A Waveform :SinWaveBurst
Vacuum Pump
Power Amp Thamway T145-5714B
Gas Canister MicroAcoustic Air-coupledTransducer (Source) 40k Hz - 1.7M Hz
20kHz - 1MHz Porous Material (Ring support)
Digital Oscilloscope LeCroy 6051 SamplingRate:200MS/sec
MicroAcoustic Air-coupledTransducer (Receiver) 40k Hz - 2.25M Hz
Personal Computer Fig. 2. Measurement system.
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3.3. Measurement of the ultrasonic reception signal in the different gases Fig. 3 shows the measurement results for sample 2 when saturated with air (Figs. 3a and b), argon (Figs. 3c and d) and helium (Figs. 3e and f). The output level of the generator and the amplification level of the power amplifier were the same in all of the experiments. The time histories for the data shown in Fig. 3a–d were synchronized. The fact that the sound velocity in argon is slower than that in air is apparent. The sound velocity in helium 1.00
0.05
0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Amplitude [ V ]
Reference 0.50
0.01 -0.01 10
20
30
50
-0.05
(a) Air Sample 2
Time [ s ]
(b) Air Sample 2
Time [ s ]
0.05
1.00 0.50 0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Reference
Transmitted
0.03 0.01 -0.01 10
20
30
40
50
-0.03 -0.05
-1.00 (c) Ar Sample 2
(d) Ar Sample 2
Time [ s ]
1.00
Time [ s ]
0.05
0.50 0.00 10
20
30
40
50
-0.50
Amplitude [ V ]
Reference Amplitude [ V ]
40
-0.03
-1.00
Amplitude [ V ]
Transmitted
0.03
Transmitted
0.03 0.01 -0.01 10
20
30
40
50
-0.03
-1.00
-0.05 (e) He Sample 2
Time [ s ]
(f) He Sample 2
Time [ s ]
Fig. 3. Reference and transmitted signals when sample 2 is saturated by different gases: (a,b) air; (c,d) argon; (e,f) helium.
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Reception signal level [ dB ]
is about 2.9 times larger than that in air. The range of the time history was switched for the convenience of the synchronous addition processing of the digital oscilloscope, so the time histories presented in Figs. 3a and e were not synchronized. A clear decrease in the sound pressure is observed in helium [see Fig. 3e]. The same problem is apparent also in the transmitted signals when the material was inserted between the two broadband air-coupled transducers [see Fig. 3f]. The transmitted ultrasonic signal spectra and noise level spectra are shown in Fig. 4 for sample 1 (Fig. 4a) and sample 2 (Fig. 4b) saturated with argon, air and helium. The problem of acoustic coupling between the broadband air-coupled transducer and helium is demonstrated by the fact that the reference signal levels in helium are lower than that measured in air and argon by between 10 dB and 20 dB [see Figs. 4a and b]. The transmitted signal levels in air and argon in the frequency range from 50 kHz to 250 kHz [see Figs. 4a and b] are about 10 dB higher than that at 400 kHz. The transmitted signal levels in helium are low across the entire frequency range.
-30 Reference
-40 -50
Transmitted
-60 -70
Ar Air He Noise
-80 -90 -100 -110
Reception signal level [ dB ]
0 200 (a) Sample 1
400
600 800 Frequency [ kHz ]
-30 Reference
-40 -50
Transmitted
-60 -70
Ar Air He Noise
-80 -90 -100 -110 0
200 (b) Sample 2
400
600 800 Frequency [ kHz ]
Fig. 4. Reference and transmitted signal spectra obtained before and after glass wool samples are saturated by different gases. (a) sample 1; (b) sample 2.
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3.4. Analysis of the sound velocity in the different gases The temperature was 25 C and the atmospheric pressure was 1.013 · 105 Pa during the experiment. The sound velocity in the tested sample was determined using Eq. (4). Fig. 5 shows the sound velocity measured in the glass wool samples 1 and 2. The measurement results for air are shown in Fig. 5a, for argon in Fig. 5b and for helium in Fig. 5c. It can be clearly seen from Fig. 5c that the data for helium saturation are much less accurate than those obtained for saturation with air and argon. This result was due to the low signal-to-noise ratio mainly incurred by the problem of the acoustic coupling between the transducers and the helium. As suggested earlier, another factor which influenced the low signal-to-noise ratio in helium was related to the value of the kinematic viscosity of helium. Table 2 shows that the viscous skin depth of helium is much larger than that of air and argon. The measurement results for helium in the frequency range of 400 kHz or less [see Fig. 5c] are not especially accurate. 3.5. Determination of the two characteristic lengths, based on the measurements in air and argon 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 (c0/ c)2 (see Eq. (3)) are shown in Fig. 6 as a function of f1/2. The measurement results for sample 1 saturated by air and argon are shown in Fig. 6a and the corresponding measurement results for sample 2 are shown in Fig. 6b. An approximate linear fit was obtained for the data in the frequency range between 100 kHz and 800 kHz. The slopes of the linear fits shown in Fig. 6 were substituted in Eqs. (5) and (6) together with the tortuosity data. As a result the two characteristic lengths were obtained and their values are listed in Table 4. 3.6. Verification of the measurements of the tortuosity and two characteristic lengths By transposing Eq. (1), the following expression for the sound speed is obtained: c0 d 1 c1 ffiffiffiffi ffi p c ffi pffiffiffiffiffiffi 1 þ : ð9Þ 2 ^ a1 P r ^0 The values of the tortuosity and two characteristic lengths deduced from measurements and listed in Table 4 were substituted in Eq. (9), to enable calculation of values of sound velocity in the porous material samples. In Fig. 7, the predicted values of sound velocity in the materials are compared with the measured values for sample 1 (Fig. 7a) and sample 2 (Fig. 7b). The prediction error rate is defined by 100 jccal cmea j=cmea
ð10Þ
where ccal is the frequency-dependent predicted sound velocity, and cmea is the frequencydependent measured sound velocity. For sample 1, the mean value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz was 0.15% in air and 0.19% in argon whereas the maximum value of the sound velocity prediction error rate in this frequency range
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350
Velocity [ m/s ]
330 310 290 270
Sample 1 Sample 2
250 230
0
200 (a) Air
400 600 800 Frequency [ kHz ]
350
Velocity [ m/s ]
330 310 290
Sample 1
270
Sample 2
250 230 0
200 (b) Ar
400 600 800 Frequency [ kHz ]
1000
Velocity [ m/s ]
980 960 940 920 900 880
Sample 1
860
Sample 2
840 0
200 (c) He
400 600 Frequency [ kHz ]
800
Fig. 5. Sound velocity as a function of frequency measured in glass wool samples saturated by different gases at a temperature of 25 C: (a) air; (b) argon; (c) helium.
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Squared refraction index
1.25
Air
x: 1000/ Frequency y: Squared refraction index
1.15 1.10 1.05
y = 0.0261x + 1.0124(Air)
1.00
0
1 (a) Sample 1
1.25 Squared refraction index
Ar
y = 0.0278x + 1.0115(Ar)
1.20
2
3
4
1000/
Frequency
y = 0.0523x + 1.0277(Ar)
1.20
x: 1000/ Frequency y: Squared refraction index
1.15 Ar
1.10
Air
1.05 1.00
y = 0.0492x + 1.0295(Air)
0
1 (b) Sample 2
2
3
1000/
Frequency
4
Fig. 6. Squared refraction index as a function of the square root of the inverse frequency measured in glass wool samples saturated by air and argon at a temperature of 25 C: (a) sample 1; (b) sample 2.
Table 4 Values of tortuosity and two characteristic lengths deduced from ultrasonic transmission measurements Sample
Sample 1
Sample 2
a1 (lm) 0 (lm)
1.0124 106 225
1.0295 57 123
is 0.47% in air and 0.56% in argon, corresponding to a maximum value of the sound velocity prediction difference of 1.59 m s1 in air and 1.72 m s1 in argon. For sample 2, the mean value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz is 0.30% in air and 0.44% in argon, whereas the maximum value of the sound velocity prediction error rate in the frequency range from 100 kHz to 800 kHz was 0.93% in air and 1.06% in argon, corresponding to a maximum
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350 Air
Velocity [ m/s ]
340 330 320 310 Measured
300 Ar
290 280
0
200 (a) Sample 1
Predicted 400 600 Frequency [ kHz ]
800
350 Air
Velocity [ m/s ]
340 330 320
Ar
310 300
Measured
290
Predicted
280 0
200 (b) Sample 2
400
600
800
Frequency [ kHz ]
Fig. 7. Comparison between measured and predicted sound velocity as a function of frequency in glass wool samples saturated by air and argon at a temperature of 25 C: (a) sample 1; (b) sample 2.
value of the sound velocity prediction difference in this frequency range of 3.0 m s1 in air and 3.15 m s1 in argon. Since the difference between the measured values and the predicted values is small, it can be concluded that the measurement values of the tortuosity and two characteristic lengths in Table 4 are sufficiently accurate. 4. Concluding remarks A method for measuring the two characteristic lengths in a rigid-porous material, proposed by Leclaire et al. [1] and based on ultrasonic transmission measurements using different saturating gases has been studied. The influence of the gas properties on the signalto-noise ratio has been investigated. It has been found that the improved signal-to-noise ratios are obtained when using a gas with a reasonably high acoustic impedance and low kinematic viscosity, such as argon.
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When using helium in a high flow resistivity material, it has been found necessary to use thin samples. However, it is difficult to obtain uniform thin samples of a fibrous material. Also when the measurement is carried out on a thin layer the accuracy of the delay measurement is likely to be limited. There is a limit to the thickness of the material that can be measured even if argon is used. However, the range of applicability of the method is greater than when using helium. References [1] 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. [2] Nagy PB, Adler L, Bonner BP. Slow wave propagation in air-filled porous materials and natural rocks. Appl Phys Lett 1990;56:2504–6. [3] Nagy PB. Slow wave propagation in air-filled permeable solids. J Acoust Soc Am 1993;93:3224–34. [4] Gray DE (coordinating editor). American Institute of Physics Handbook. New York: McGraw-Hill; 1972. [5] Kaye GWC, Laby TH. Tables of physical and chemical constants and some mathematical functions. 14th ed. London: Longman Group Limited; 1973. [6] ISO 9053:1991. Acoustics – materials for acoustical applications – determination of airflow resistance.