THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

(1 + m)A I 1=A 314.14 Mufflers and Silencers 401the inlet and outlet of the expansion chamber, resulting in the complex ratio(1 + 1 )( ) 1e ikC + (m − 1)mm − 1 e −ikC= cos(kC) + i (m + 1 )sin(kC) (14.60)2 mIn order to establish the transmission loss, the ratio of sound intensity at the expansionchamber inlet to transmitted sound intensity is needed:I I 1= p2 rms(I 1)I 3 prms(3)2 =A 2 I 1∣A 2 ∣ = cos2 (kC) + 1 (m + 1 ) 2sin 2 (kC)34 m= 1 + 1 (m − 1 ) 2sin 2 (kC) (14.61)4 mBecause transmission loss is a function of the ratio of power incident on the mufflerto the power transmitted, i.e.,TL = 10 log4( )WincidentW transmittedand if the inlet and outlet areas are equal, we obtain( ) ( ) [WI 1II 1TL = 10 log = 10 log = 10 log 1 + 1 W 3 I 3 4(m − 1 ) ]2sin 2 (kC)m(14.62)where k = 2π/λ is the wave number. Equation (14.62) becomes invalid if any ofthe lateral dimensions of the expansion chamber exceeds 0.8λ.Figure 14.20 displays a plot of the theoretical transmission loss versus C/λfor various area ratios. Equation (14.62) forecasts a transmission loss of zerowhen the argument of the sine function is 0, π, 2π, .2nπ, ... and a maximumtransmission loss when the argument is π/2, 3π/2, ... (2n + 1) π/2, ..., wheren = 0, 1, 2, 3, etc. It then follows that the expansion chamber works best whenlength C constitutes an odd number of quarter-wavelengths, i.e., C = λ(2n + 1)/4for maximal TL. But the expansion chamber becomes ineffective when the chamberlength C measures out to an integer number of half-wavelengths, i.e., C = nλ (i.e.,λ/2, λ, 3λ/2, etc.). For extremely low frequencies C/λ approaches zero, and thetransmission loss likewise approaches zero.Example Problem 11A reactive muffler is to be designed to produce a transmission loss of 20 dB at150 Hz. The inlet and outlet pipes are 60 mm in diameter and the average gas (air)temperature is 100 ◦ C. Determine the size of the expansion chamber.