Modern computer methods are integrated into a simulation scheme to predict the sound attenuation in complicated exhaust systems.
Numerical acoustic simulation is being used effectively in industries as diverse as HVAC, heavy equipment and automotive. Certainly, the most popular numerical acoustics method is the boundary element method(BEM). The BEM is a numerical approximation used to solve the acoustic wave equation. This application is well documented in the literature. The BEM is similar to the finite element method but with an important difference that makes it especially advantageous for NVH problems. The boundary surface is discretized or meshed instead of the acoustic domain. This saves considerable modelling effort when compared to domain discretization methods such as the finite element and finite difference methods.
However, it is infeasible to model complete systems using the BEM because of the CPU-intensive nature of the analyses. Nonetheless, individual components (i.e. expansion chambers, Helmholtz resonators, perforated elements etc.) can be modelled, and the BEM results for individual components can be integrated into a process to predict insertion loss for built-up exhaust systems. It should be noted that a similar process could be used for HVAC systems in automobiles as well. The technique described in this article is known as the transfer matrix approach. The chief assumption and limitation is that of plane acoustic waves at the inlet and termination of each component. The scheme for integrating BEM calculations into a procedure for predicting transmission or insertion loss of complicated exhaust systems is described here.
Acoustic waves propagate in ducts for a wide range of applications. For small mufflers and silencers, the duct cross-sectional dimensions are normally small compared to the acoustic wavelength simplifying the analysis. Plane wave models are appropriate up to some cut-off frequency. This frequency can be estimated easily for ducts and is equal to c/2d where c is the speed of sound (343m/s in air) and d is a characteristic dimension of the duct cross-section (i.e. diameter for a circular duct, or width or height of a square duct). Munjal's classic text summarises transfer matrix theory, and includes the transfer matrix for many common muffler and silencer components.
A transfer matrix is composed of four-pole parameters A, B, C and D. Figure 1 illustrates the fourpole parameters for an exhaust component. These four-pole parameters are defined according to the matrix equation where p1 and p2 are sound pressures an.d v1 and v2 are acoustic particle velocities as defined in Figure 1.
The four-pole parameters for certain components like rigid-walled straight pipes or ducts are well-known. However, numerical or experimental methods must be used to determine the fourpole parameters of more sophisticated components like expansion chambers and Helmholtz resonators at high frequencies. The four-pole parameters are easily determined using the BEM via two successive analyses.
One metric for measuring sound attenuation in mufflers is transmission loss. Transmission loss refers to the sound attenuation independent of the engine source and the exhaust pipe termination condition. Another metric, insertion loss, is used to account for the attenuation muffler in conjunction with the engine source and exhaust pipe termination. Only transmission loss will be considered in this article. However, the insertion loss may also be discerned if the source and termination conditions are known.
Transmission loss can be determined by multiplying transfer matrices together to find the total four-pole parameters for the built-up system. The total fourpole parameters (AT, BT, CT, and DT) for the system shown in Figure 2 is determined as
Once the overall four-pole parameters have been determined using the above equation, transmission loss can be determined using the equation
where Si and So are the inlet and outlet duct cross-sectional areas respectively. ρ is the fluid mass density (1.21kg/m3 for air) and c is the speed of sound (343m/s for air). If the ratio of the surface areas (Si/So) at the inlet to the outlet is unity, the second term in the above equation may be neglected.
The procedure is illustrated via an example. Figure 3 shows a schematic of the pipe system that was tested and simulated. It should be noted that the experiment though simple is representative of many muffler systems. BEM analyses were conducted to find the four-pole parameters for both expansion chambers (i.e. both drums). The BEM meshes are indicated in Figure 3. Note that the BEM approach can be selectively applied to larger muffler components since plane wave approximations will not be valid. However, the transfer matrix for the straight pipe connecting the two expansion chambers can be handled using equations like those in Munjal's text since the pipe diameter is small compared to an acoustic wavelength. Thus, it can be assumed that acoustic waves propagate along the pipe but not in a direction transverse to the pipe. The transmission loss for the system (Figure 3) was measured and compared to simulation (Figure 4). Notice the excellent agreement between the two.
In summary, numerical methods provide means by which the mechanisms controlling the acoustical characteristics of muffler components and systems can be understood and, ultimately, controlled in design. This article demonstrates that a practical level of accuracy can be readily obtained via numerical simulation in exhaust problems. An approach for using numerical techniques to determine transmission loss for multi-component exhaust systems has been summarised. It should be noted that state-of-the-art software like LMS Virtual Lab and other commercial software has the ability to model perforated elements and sound absorbing materials in complex arrangements. Certainly, the science of muffler design is sufficiently mature to minimise prototype development.
