On the Anechoic Termination Assumption When Modeling Exit Pipes

The anechoic assumption has long been used for the modeling of long exit pipes or ducts of acoustic elements [1, 2, 3, 4, 5, 6, 7]. Especially in compressor discharge line modeling, the exhaust pipe from the compressor manifold leading all the way to the condenser is usually assumed to be anechoic since it is very long. Also, experimental efforts have been made by investigators using different acoustical testing rigs to simulate anechoic terminations. In this study, the effect of the discharge pipe length of a muffler model on the transfer function is discussed on a theoretical basis. The transfer functions of a muffler model with different discharge pipe lengths and dampings are simulated and compared. The results mimic actual data which would be measured experimentally if the pipe length is finite instead of infinitely long or if absorption material is used. Furthermore, a horn type exit is investigated. The permissibility of the anechoic termination assumption in realistic modeling is also discussed. BASIC MODELS In this study, a muffler model which appeared in the authors' companion paper [8] was used as an example. The simulations were performed for the muffler model, which was attached to a finite length exhaust pipe, a finite exit pipe containing absorption material and a horn shape exhaust pipe respectively, and compared to the ideal case of anechoic termination. The analysis utilized the concept of four pole parameters technique. For a review of the basic discussions of the concept and derivation of four poles of some acoustic elements, the readers are referred to references [9, 10, 11, 12]. Muffier with Anechoic Termination For the muffler as shown in Figure 1, the four pole parameters A, B, C, D of this muffler are known [8]. If an anechoic termination assumption is applied, the transfer function of this muffler can be expressed as 1 s , A-+B pc (1) where P2 is the acoustic pressure at muffler exit, Q1 is input volume velocity, S the cross sectional area of the exit pipe, p gas density and c sound speed. Muffier with Finite Length Exhaust Pipe In contrast, if the length of the exhaust pipe is finite, the four poles of this pipe can be modeled by a 1-D method [10]: Ap = cosh(JL), (2) Bp jwS . (3) = 2smh ('y L), pc 'Y Cp pC~/ . (4) --:--s smh (J L), JW Dp cosh(1L), (5) where 1 = e 2 -..; + j~, ~ is the damping ratio, d the diameter of the pipe, and L the length of this pipe.