2. Theoretical analysis 2.1 Analytical model of clearance in a double cylinder

Transcription

1 Bulletin of the JSME Journal of Advanced Mechanical Design, Systems, and Manufacturing Vol.9, No.3, 5 Estimation and experiment for sound absorption coefficient of cross-sectional shape of clearance by concentric cylinder Shuichi SAKAMOTO*, Arata NAKANO**, Hirohiko TANIKAWA** and Yuki MARUYAMA** * Department of Mechanical and Production Engineering, Niigata University -85 Ikarashi, Nishi-ku, Niigata, 95-8, Japan sakamoto@eng.niigata-u.ac.jp ** Graduate School of Science and Technology, Niigata University -85 Ikarashi, Nishi-ku, Niigata, 95-8, Japan Received April 5 Abstract There are engineering benefits if we can predict the acoustic properties such as the sound absorption coefficient of a sound incident on the clearance of such an annular cross-section based on the geometric dimensions of the tubes and the physical properties of the gas. However, a successful theory for the sound absorption coefficient of narrow tubes of the abovementioned shapes has not yet been presented, and the comparison between experimental and theoretical values has not been made. Further, the propagation constant and characteristic impedance in the clearance in a double cylinder have not yet been derived in the cylindrical coordinate system or by other methods. Therefore, in this paper, the propagation constant and characteristic impedance were derived by approximating clearance of an annular cross-section with clearance between two parallel planes. And the sound absorption coefficient was calculated, then compared with experimental results. The theoretical values, according to this method, agreed well with the experimental values, and practically sufficient accuracy was obtained. It was experimentally confirmed that the eccentricity of a round bar in a circular hole does not appreciably influence the sound absorption coefficient. Even when this calculation method is applied to a circular hole, the results agree with the theoretical and experimental values for the circular hole. Therefore, this method is effective even if the diameter of the internal cylinder is considerably smaller than that of the external cylinder. Key words : Sound and acoustics, Noise control, Sound absorption coefficient, Modified cross-section tube, Silencer designs. Introduction Double cylinder structures (Yamamoto, et al., 4), (Tamura, et al., 999) are frequently used as machine elements and tube elements of equipments such as reactors (Fujiwara, et al., ), catalysts, heat exchangers (Alkan and Al-Nimr, 999), and thermal insulators. A clearance exists between double cylinders of annular cross-section. This has engineering benefits if we can predict the acoustic properties such as the sound absorption coefficient of a sound incident on the clearance of such an annular cross-section based on the geometric dimensions of the tubes and the physical properties of the gas. With respect to the propagation of a sound wave in a narrow tubule structures, several studies have been performed on the propagation constant and particle velocity distribution (Tijdeman, 975), complex density and complex compressibility (Stinson, 99), complex density and complex bulk modulus (Allard and Atalla, 9) in a circular hole, in an equilateral-triangular tube, or in between two planes (Stinson and Champou, 99). Further, analysis using the finite element method of the vibration of the air in a rectangular tube (Beltman, et al., 998), and on the propagation of a sound wave in the clearance created by the close packing of cylinders has also been conducted (Matsuzawa, 963). With respect to naturally derived narrow tubes, studies have been performed on the acoustic absorption properties of straw (McGinnes, et al., 5) and rice straw (Sakamoto, et al., ). The authors experimentally clarified that acoustic absorption effect in the clearance of an annular cross-section of a rice straw was useful (Sakamoto, et al., Paper No.5-6 [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers

2 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) ). However, a successful theory for the sound absorption coefficient of narrow tubes of the abovementioned shapes has not yet been presented, and the comparison between experimental and theoretical values has not been made. Therefore, in a previous study, the authors conducted theoretical analysis on structures in which circular holes are bundled, and a clearance created by closely packed cylindrical tubes by considering the attenuation in the tube and then compared the obtained sound absorption coefficient and transmission loss with experimental results (Sakamoto, et al., Under review), (Sakamoto, et al., 3). Further, the propagation constant and characteristic impedance in the clearance in a double cylinder have not yet been derived in the cylindrical coordinate system or by other methods. Therefore, in this paper, the propagation constant and characteristic impedance were derived by approximating clearance of an annular cross-section with clearance between two parallel planes. And the sound absorption coefficient was calculated, then compared with experimental results.. Theoretical analysis. Analytical model of clearance in a double cylinder Clearance in a double cylinder (name as clearance) is analyzed using the transfer matrix method (Sasao, 7), (Suyama and Hirata, 979) for sound pressure and volume velocity based on the one-dimensional wave equation. The blue portion of Fig. shows the cross section of the clearance. Figure shows the analysis unit for the clearance. Because it was difficult to use the cylindrical coordinate system to analyze the shape as shown in Fig., by approximating the shape to that shown in Fig. 3 (a) (Allard and Atalla, 9), the propagation constant and characteristic impedance were derived and applied to the transfer matrix (Suyama and Hirata, 979), given in Eq. (). In addition, to show the usefulness of the approximation in Fig. 3 (a), the calculation results using a simple equivalent circular approximation (Sakamoto, et al., 3) shown in Fig. 3 (b) are discussed in Chapter 3. Here, the area of clearance S in Fig. and Figs. 3 (a) and (b) are equal to each other. In addition, clearance has the same size b in Figs. and 3 (a). Furthermore, the sum of the circumferences of the circular hole and the round bar in Fig. and the sum of the two lengths of the opposed wall surfaces in Fig. 3 (a) are equal to each other. These facts suggest that the approximation in Fig. 3 (a) is good. Conversely, the size of clearance b and the wall surface length are not considered in the equivalent circular approximation of Fig. 3 (b). Assuming cross section S, sample length l, characteristic impedance Z c, and propagation constant γ for each element, the transfer matrix T of an analysis unit is given by Eq. (). A, B, C, and D in this equation are four-terminal constants. The theoretical value of the sound absorption coefficient is obtained by substituting the matrix into Eq. (9). Because it was necessary to make an open-end correction for the inlet end (Bolt, et al., 949), (Benade, 967), the sample length l was corrected by adding.4 times the value of b. A T C cosh( l) B D S sinh( l) Zc Zc sinh( l) S cosh( l) (). Propagation constant and characteristic impedance considering attenuation As for the attenuation of a sound wave in the tube, the attenuation constant and phase constant are experimentally derived for the tube with an inside diameter of mm or more (Suyama and Hirata, 979). For the attenuation constant and phase constant considering the viscosity of the air in the tube, studies were performed on a circular hole (Tijdeman, 975), (Stinson, 99) and an equilateral-triangular tube (Stinson and Champou, 99). Studies were also performed on the clearance between parallel planes (Stinson and Champou, 99), (Beltman, et al., 998) to derive the propagation constant. However, it was difficult to derive the propagation constant and characteristic impedance of a double cylinder in the cylindrical coordinate system. Therefore, in this paper, a double cylinder was applied to the rectangular coordinate system for the analysis. [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers

3 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) D d b S D Fig. Cross section and analysis unit Fig. Analysis unit of the test sample (N=3) b S D S D (a) Clearance between two planes (b) Equivalent circle Fig. 3 Two approximation methods for the clearance in Fig. In this section, we applied the method of expressing the element of clearance between two planes to analyze the sound wave in the clearance between two planes considering the sound wave attenuation (Beltman, et al., 998). In the rectangular coordinate system shown in Fig. 4, the three-dimensional analysis are performed by using Navier Stokes equations, a gas state equation, a continuity equation, an energy equation, and a dissipation function, representing the heat transfer due to internal frictions (Beltman, et al., 998). We assumed the particle velocity in the x-, y-, and z-directions, acoustic sound pressure, and the density and temperature of air. x is the position in the long direction of the rectangular coordinate system. (y, z) are the coordinates in the cross section of the clearance. j is an imaginary unit. ω is the angular frequency. P s is the atmospheric pressure. c is the atmospheric speed of sound. is the density of air. T s is the air temperature. u, v, and w are minute amounts of dimensionless amplitude of the velocities in the x-, y-, and z-direction. p,, and T are minute amounts of the dimensionless amplitude of pressure, density, and temperature perturbations. P P p x,y,z e s u c u x,y,z e jt jt,, v c v x,y,z e jt jt ρ ρ x,y,z e, T T T x,y,z e jt, w c w x,y,z e s jt () [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 3

4 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Then, we derived the following equations by substituting the assumed variables in the Navier Stokes equations, a gas state equation, a continuity equation, and an energy equation. Where a is the width of the plane, l is the length of the test sample, k is the wave number, is the specific heat ratio, and µ is the viscosity. c ju l x p s u z c a y p v p, jv,, s z k z s b (3) b u b v w b j l x a y z c (4) p T (5) jt s T z j p (6) We calculated the sound pressure and particle velocity in the x-axis using the Navier Stokes equations for plane wave propagation under isothermal conditions. The propagation constant γ in the two planes, considering attenuation and using the as-derived variables, as shown below (Beltman, et al., 998). Here, σ is the square root of the Prandtl number. jk Bs Bs, B cosh jx x jxsinh (7) From the ratio of average particle velocity of the traveling wave and sound pressure, the characteristic impedance Z c in the clearance with attenuation is given by Z c j B s (8) Thus, the propagation constant γ and characteristic impedance Z c were obtained in the clearance, considering attenuation. The three-dimensional analysis results on the clearance between two planes can be introduced into the one-dimensional transmission matrix by substituting them into γ and Z c in Eq. (). These calculation results (Sakamoto, et al., 4) were consistent with the calculation using the Stinson s method (Stinson and Champou, 99). [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 4

5 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) x z b l a y Fig. 4 Cartesian coordinate system for parallel space surrounding by a pair of planes.3 Derivation of sound absorption coefficient Assuming the sound pressure p, p and particle velocity u, u at the incident face and the end of the sound wave of the analysis unit, the transfer matrix is expressed by Eq. (9), considering that the end of the clearance is rigid and that u = holds for the particle velocity. Further, S is the cross section of an impedance tube divided by the number of holes. p Su A C B p D Su Ap Cp (9) Figure 5 shows the thick portion of an analysis unit. Here, S/S is equal to the aperture ratio of the sample shown in Table. Assuming the sound pressure p and particle velocity u just outside plane in Fig. 5, the specific acoustic impedance Z of the inside, viewed from the incident face, is expressed using p = p, S u = Su, and by using Eq. (9) as follows: Z p p S S S u us us C p A () Using the specific acoustic impedance Z and the specific acoustic resistance of the air ρ c, the sound absorption coefficient α is given by Z Z c c () [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 5

6 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Rigid wall S S Plane Plane Fig. 5 Analysis unit S for each narrow clearance Table Typical specifications of test sample elements Length l [mm] Number of clearance N Inner diameter of hole D [mm] Outer diameter of rod d [mm] Correspon dence to Fig. 6 Area of each clearance S [mm ] Circumfer ence of each clearance D d [mm] Clearance b [mm] Aperture ratio S/S (a) (b) (c) (g) (4.).8.5 (d) (e) (h).6.6 (.).66.5 (f) (i) (.).6 [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 6

7 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) 3. Comparison between calculation and measurement 3. Dimensions for analysis and corresponding measured samples Table shows the dimensions for analysis, namely, the specifications of the measured samples. Figures 6(a) (f) shows the pictures of the measured samples. Clearance of an annular cross-section was created between a circular hole made in the brass sample and a stainless steel round bar. The cut surface of the round bar is polished flat and fixed to the bottom of the circular hole using modeling clay. Note that the eccentricity of the round bar in the sample will be discussed later. To evaluate the influence of clearance on the sound absorption coefficient, experiments were conducted by changing the size of the circular hole and the diameter of the round bar. For comparison, calculations and experiments were also conducted for the case without the round bars in the circular holes (Sakamoto, et al., 3) as shown in Figs. 6 (g)-(i). (a) D=8. mm, d=4. mm (b) D=8. mm, d=.5 mm (c) D=8. mm, d=.5 mm (d) D=4. mm, d=.5 mm (e) D=4. mm, d=.5 mm (f) D=. mm, d=.5 mm (g) D=8. mm, d= mm (h) D=4. mm, d= mm (i) D=. mm, d= mm Fig. 6 Test sample elements in measurement tube (outer diameter: approximately 9 mm) [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 7

8 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) 3. Apparatus used for measurement of the sound-absorption coefficient The sound absorption coefficient was measured using a Brüel & Kjær type 46 -microphone impedance measurement tube. The experimental setup is shown in Fig. 7. The test sample was placed inside the impedance tube. The sinusoidal signals from an internal signal generator were analyzed with a fast Fourier transform (FFT) analyzer, and the transfer functions between the sound pressure signals of the two microphones attached to the impedance tube were measured. Using the measured transfer functions, we determined the normal incident sound absorption coefficient according to the ISO 534- standard test method. The threshold wavelength of the plane waves is a function of the internal diameter of the acoustic tube. The test sample did not have high sound absorption coefficient in a low-frequency range; hence, we used a small tube with an internal diameter of 9 mm. The measurement range was 5 64 Hz. Loudspeaker Dummy mic. Amplifier Signal out FFT Analyzer A-ch B-ch Test sample Microphone amplifier A-ch B-ch Impedance measurement tube Microphones Fig. 7 Two microphone impedance tube for absorption coefficient measurement 3.3 Sound absorption coefficient Figure 8 shows the measured results of the sound absorption coefficient changing circular hole diameter D and the diameter of round bar d in the sample of length l = 5 mm. In addition, Figs. 9-7 compare the calculated and measured values of the sound absorption coefficient. Figures 9- correspond to the diameter of round bar: d = 4. mm,.5 mm,.5 mm, and mm, respectively, where D = 8. mm. Similarly, Figs. 3-5 and Figs. 6-7 correspond to D = 4. mm and D =. mm, respectively. The calculated results are shown for two kinds of analysis: () with clearance assumed between two planes as shown in Fig. 3 (a); () with clearance assumed as an equivalent circle as shown in Fig. 3 (b). Then the sound-absorption coefficient was calculated using 5 Hz steps. In the experimental values in Fig. 8, the sound absorption coefficient increases as the value of b decreases. This will be discussed in the below. The thickness of the laminar boundary layer produced by particle velocity of the sound wave is said to be depended on the frequency and be about several tens of µm at khz (Wesley, 958). The samples used in this paper have a clearance sufficiently larger than the thickness of this boundary layer. Therefore, the influence of the viscosity of the boundary layer formed near the tube wall-and so the sound absorption coefficient increases as the value of b decreases (Sakamoto, et al., 3). Generally, an acoustic absorption peak appears at the frequency in which the length of the sample l agrees with the quarter wavelength of the sound wave. Therefore, the peak frequency in each condition in Fig. 8 almost agrees with each other. However, a decrease in the peak frequency is seen only in the case of sample (f), namely, b =.5 mm. It is reported that is because a decrease in the speed of sound (Koshiroi and Tateishi, ) occurs due to the increase of the ratio of the boundary layer in the clearance. [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 8

9 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Then, the results in Figs. 9-7 are discussed. The difference between the experimental values and the values calculated by approximating clearance as the clearance between two planes, slightly increases as the value of d decreases in case of Figs. 9,,, 3, 4, and 6. This is because of the assumption in the analysis used in this method; the assumption was that the size of the clearance between the two planes is very small in comparison with the circumference of the tube wall. (Beltman, et al., 998). Further, even in the case of d = mm, the calculated and experimental values almost agree with each other. However, the differences between the calculated and measured values in Figs., 5, and 7 are attributed to a lack of the round bar inside the circular tubes; the round bar was approximated as a concentric cylinder (f), b=.5 mm (d), b=.75 mm (i), b=. mm (e), b=.5 mm (g), b=. mm (h), b=. mm (b), b=.75 mm (c), b=3.5 mm (g), b=4. mm Theoretical value (with attenuation of section.) Fig. 8 Comparison between experiments Fig. 9 (D=8. mm, d=4. mm) Theoretical value (with attenuation of section.) Theoretical value (with attenuation of section.) Fig. (D=8. mm, d=.5 mm) Fig. (D=8. mm, d=.5 mm) [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 9

10 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Theoretical value (with attenuation of section.) Theoretical value (with attenuation of section.) Theoretical value (with attenuation of circle) Fig. (D=8. mm, d= mm) Fig. 3 (D=4. mm, d=.5 mm) Theoretical value (with attenuation of section.) Theoretical value (with attenuation of section.) 5 5 Fig. 4 (D=4. mm, d=.5 mm) Fig. 5 (D=4. mm, d= mm) Theoretical value (with attenuation of section.) Theoretical value (with attenuation of circle) Theoretical value (with attenuation of section.) 5 5 Fig. 6 (D=. mm, d=.5 mm) Fig. 7 (D=. mm, d= mm) 64 [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers

11 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Then, in Figs. 9-7, the calculated values derived by approximating clearance using an equivalent circle are compared with the experimental values. It is shown that the difference between the calculated and experimental values increases as the value of d approaches D, especially in the cases of Figs. 9 and 3. In Fig. 6, the derived acoustic absorption coefficient at peak frequency is close to the measured value. However, the peak frequency shifted to the high-frequency side and the difference between the measured and calculated values in the wide-frequency range are significant. The b of the test sample in Fig. 6 is smaller than that in the other test samples because of the increase in the ratio of the boundary layer in the clearance. This shows that the method of approximating clearance by an equivalent circle has poor accuracy when b is small. In addition, in Figs., 5, and 7 the two kinds of calculated values are almost identical when assuming d = (i.e., a circular hole). Thus, this method is applicable for predicting the sound absorption coefficient of a circular hole by assuming d =. Therefore, this method of approximating clearance using the clearance between two planes is effective for predicting the sound absorption coefficient in a wide range of d < D. 3.4 Experimental study on the eccentricity of round bars It is difficult to precisely place a round bar in the center of a circular hole by hand. Therefore, the influence of eccentricity of a round bar on the sound absorption coefficient was tested using the samples shown in Fig. 6 (a) and Table (a). Figure 8 shows the samples used in the measurement. Figure 8 (a) to (e) show the samples with the eccentricity. Figure 9- show the measured results. There was hardly any difference between the experimental values with and without eccentricity. This shows that the sound absorption coefficient can be predicted by using the clearance b averaged over the entire circumference. (a)eccentric: mm (Fig. 6(a), Table (a)) (b)eccentric: mm (Fig. 6(a), Table (a)) (c)eccentric: mm (Fig. 6(b), Table (b)) (d)eccentric: mm (Fig. 6(b), Table (b)) (e)eccentric:.75 mm (Fig. 6(d), Table (d)) Fig. 8 Test sample of eccentric [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers

12 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Measured Value Centered Eccentric; mm from center Eccentric; mm from center Measured Value Centered Eccentric; mm from center Eccentric; mm from center Fig. 9 Comparison between experiments of eccentricity (Fig. 6(a), Table (a)) Fig. Comparison between experiments of eccentricity (Fig. 6(b), Table (b)) Measured Value Centered Eccentric;.75 mm from center Fig. Comparison between experiments of eccentricity (Fig. 6(d), Table (d)) 4. Conclusion Clearance of a double cylinder was approximated by the clearance between two planes. Using three-dimensional analysis, the propagation constant and characteristic impedance were derived, and the sound absorption coefficient was calculated. As a result, the following became clear. The theoretical values, according to this method, agreed well with the experimental values, and practically sufficient accuracy was obtained. It was experimentally confirmed that the eccentricity of a round bar in a circular hole does not appreciably influence the sound absorption coefficient. Even when this calculation method is applied to a circular hole, the results agree with the theoretical and experimental values for the circular hole. Therefore, this method is effective even if the diameter of the internal cylinder is considerably smaller than that of the external cylinder. [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers

13 Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.3 (5) Acknowledgment This work was supported by JSPS KAKENHI Grant Number References Alkam, M. K., Al-Nimr, M. A., Improving the performance of double-pipe heat exchangers by using porous substrates, International Journal of Heat and Mass Transfer, Vol. 4, (999), pp Allard, J. F., Atalla, N., Sound propagation in porous media: Modeling sound absorbing materials, Second edition, pp , Wiley, 9. Beltman, W. M., Van der Hoogt, P. J. M., Spiering, R. M. E. J., Tijdeman, H., Implementation and experimental validation of new viscothermal acoustic finite element for acousto-elastic problems, Journal of Sound and Vibration, Vol. 6, No. (998), pp Benade,A. H., Measured end corrections for woodwind toneholes, Journal of Acoustical Society of America, Vol. 4, No. 6 (967), p. 69. Bolt, H. R., Labate, S. and Ingård, U., The acoustic reactance of small circular orifices, Journal of the Acoustical Society of America, Vol., No. (949), pp Fujiwara, T., Somiya, I., Tsuno, H., Okuno, Y., Application study of a draft-tube-type reactor to nitrification-denitrification process, Doboku Gakkai Ronbunshu, Vol. No. 657 (), pp (in Japanese). Koshiroi, T., Tateishi S., Features of sound absorption data in porous material caused by elastic frame structure, Journal of Acoustical Society of Japan, Vol. 68, No. 9 (), pp (in Japanese). Matsuzawa, K., Sound propagation in a tube of arbitrary cross-sectional shape, Journal of the Acoustical Society of Japan, Vol. 9, No. (963), pp. -8 (in Japanese). McGinnes, C., Kleiner, M., Xiang, N., An environmental and economical solution to sound absorption using straw, Journal of the Acoustical Society of America, Vol. 8, No. 3 (5), pp Sakamoto, S., Hoshino, A., Sutou, K., Sato, T., Estimating sound-absorption coefficient and transmission loss by the dimentions of bundle of narrow holes (Comparison between theoretical analysis and experiments), Transactions of the Japan society of Mechanical Engineers, Series C, Vol. 79, No. 87 (3), pp (in Japanese). Sakamoto, S., Sugahara, R., Nagumo, T., Nakano, A., Kawase, H., Theretical analysis and measurement of sound transmission loss in louver elements with a sound attenuating function using a Helmholtz resonator array, Journal of Advanced Mechanical Dsign Systems and Manufacturing, Vol. 8, No. 3 (4), Paper No. 4-6, 6 pages. Sakamoto, S., Sutou, K., Nakano, A., Tanikawa, H., Azami, T., Experiment and estimation of sound-absorption coefficient and transmission loss for longitudinal clearances between close-packed cylinders, Journal of Advanced Mechanical Design Systems and Manufacturing, Paper No.4-5, Under review. Sakamoto, S., Takauchi, Y., Yanagimoto, K., Watanabe, S., Study for sound absorbing materials of biomass tubule etc. (Measured result for rice straw, rice husks, and buckwheat husks), Journal of Environment and Engineering, Vol. 6, No., (), pp Sasao, H., A guide to acoustic analysis by Excel - Analysis of an acoustic structural characteristic - (4) Analysis of the duct system silencer by Excel, Journal of the Society of Heating, Air-Conditioning and Sanitary Engineers of Japan, Vol. 8, No. (7), pp (in Japanese). Stinson, M. R., The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape, Journal of the Acoustical Society of America. Vol. 89, No. (99), pp Stinson, M. R. and Champou, Y., Propagation of sound and the assignment of shape factors in model porous materials having simple pore geometries, Journal of the Acoustical Society of America, Vol. 9, No. (99), pp Suyama, E., Hirata, M., Attenuation constant of plane wave in a tube : Acoustic characteristic analysis of silencing systems based on assuming of plane wave propagation with frictional dissipation part, The Journal of the Acoustical Society of Japan, Vol. 35, No. 4 (979), pp (in Japanese). Tamura, H., Fujii, S., Yuasa, K., Tanaka, T., Evaluation method of VOC emissions from cleanroom materials using the double-cylinder chamber, Journal of Architecture, Planning and Environmental Engineering, No. 5 (999), pp (in Japanese). Tijdeman, H., On the propagation of sound waves in cylindrical tubes, Journal of Sound and Vibration, Vol. 39, No. (975), pp Wesley, L. N., Acoustic streaming near a boundary, Journal of the Acoustical Society of America, Vol. 3, No. 4 (958), pp Yamamoto, K., Kato, S., Hayashi, N., Yamashita, H., Flow field and interaction of lifted flames in a triple port burner, Transactions of the JSME, Vol. 8, No. 8 (4), Paper No.4-35, 3 pages (in Japanese). [DOI:.99/jamdsm.5jamdsm4] 5 The Japan Society of Mechanical Engineers 3