Advances in Acoustics and Vibration

Volume 2016, Article ID 6053704, 15 pages

http://dx.doi.org/10.1155/2016/6053704

## Sound Radiation Characteristics of a Rectangular Duct with Flexible Walls

Department of Mechanical & Aerospace Engineering, Indian Institute of Technology Hyderabad, Kandi, Telangana 502285, India

Received 30 July 2016; Revised 10 October 2016; Accepted 19 October 2016

Academic Editor: Marc Thomas

Copyright © 2016 Praveena Raviprolu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Acoustic breakout noise is predominant in flexible rectangular ducts. The study of the sound radiated from the thin flexible rectangular duct walls helps in understanding breakout noise. The current paper describes an analytical model, to predict the sound radiation characteristics like total radiated sound power level, modal radiation efficiency, and directivity of the radiated sound from the duct walls. The analytical model is developed based on an equivalent plate model of the rectangular duct. This model has considered the coupled and uncoupled behaviour of both acoustic and structural subsystems. The proposed analytical model results are validated using finite element method (FEM) and boundary element method (BEM). Duct acoustic and structural modes are analysed to understand the sound radiation behaviour of a duct and its equivalence with monopole and dipole sources. The most efficient radiating modes are identified by vibration displacement of the duct walls and for these the radiation efficiencies have been calculated. The calculated modal radiation efficiencies of a duct compared to a simple rectangular plate indicate similar radiation characteristics.

#### 1. Introduction

The most commonly used duct cross sections are rectangular, flat oval, and circular in heating, ventilation, and air-conditioning (HVAC) applications. Ducts transport the conditioned air from an air-handling unit (AHU) to an occupied space. Similarly, the sound produced from an AHU propagates to the receiver (a room or space) in longitudinal and transverse direction of the duct. The radiated noise from the duct walls traveling in the transverse direction is called “breakout noise.” Breakout noise is one of the common paths of sound transmission in HVAC thin wall ducts. Cummings [1] discussed the role of various duct cross section geometries on the breakout and break-in noise. Further, Cummings provided research review progress for the past two decades on “duct breakout noise.”

Of all the geometries, rectangular ducts had maximum breakout noise at lower frequencies, which means minimum wall transmission loss. Breakout noise causes the structural-acoustical coupling between the flexible duct wall structure and the acoustic domain inside the duct volume. Airborne and structure-borne sounds contribute to duct breakout noise, which is more dominant at low frequencies [1]. The hydrodynamic force and the acoustic pressure waves excite the duct wall and cause structure-born noise. These forces and pressure waves are associated with the flow and sound propagation through the duct, respectively. The excitation due to hydrodynamic force generates lower frequency vibration, which could be a focus for fatigue analysis.

An acoustic wave excites the duct walls strongly and induces vibrations. The produced vibrations of the structure will induce acoustic pressure inside the duct. This phenomenon continues under coupling and it is efficient at the strongly coupled modes. Coupling depends on the acoustic, structural natural frequency and spatial distribution of mode shapes. Transfer factor identifies the strongly coupled structural and acoustical modes [2]. Vibration displacement of the flexible structure due to acoustic excitation produces sound radiation outside the duct volume. The radiated sound power depends on vibration velocity, surface area, and radiation efficiency [3]. In the present paper, mean flow effects are not considered because of lower Mach numbers in the HVAC ducts.

Analytical methods are available in the literature for calculating the sound radiation from different ducts, such as a finite-length line source, equivalent finite-length cylindrical radiator, equivalent plate model, and finite and boundary element methods [1–5]. Cummings estimated the breakout noise from a rectangular duct using line source model and equivalent cylinder model. Astley and Cummings discussed a finite element scheme and an experimental setup to study the acoustic transmission through walls of a rectangular duct. Venkatesham et al. developed an analytical solution for prediction of breakout noise from the rectangular duct [4] and a plenum with four compliant walls [5] based on an equivalent plate model for sound radiation. In this model, the acoustic pressure and the vibration velocity vectors were expressed in terms of uncoupled acoustical and structural subsystems. The radiated sound power was expressed in terms of radiation impedance matrix and average velocity vector. Quadruple integrals are involved in the radiation impedance matrix. Venkatesham et al. [5] discussed a procedure for solving quadruple integrals based on the coordinate transformation. The same method is extended in the current manuscript to determine rectangular duct sound radiation characteristics.

Modal radiation efficiency calculation helps in finding efficient sound radiation modes in the total sound power radiated from a duct. Wallace [6] discussed an analytical formula for calculating radiation efficiencies at low frequencies from a baffled rectangular plate. Ran Lin and Pan [7] studied the vibration and sound radiation characteristics of box structures. It provides a basis for understanding the vibration energy flows between the panels of the box, grouping of various modes, radiation efficiencies, and the various kinds of sound sources.

In this paper, the uncoupled structural modes of the rectangular duct and acoustical cavity modes are calculated by the analytical method and validated with the numerical results. These rectangular duct modes are classified into four different groups and are similar to box structures as discussed in [7]. In these four groups, the most efficient radiating modes are estimated based on the symmetries between the panel pairs and the net volume displacements in a particular mode. The modal radiation efficiencies of different groups of a rectangular duct are estimated and compared analytically and numerically. These modal radiation efficiencies of the rectangular duct of the four groups are compared to that of simple rectangular plate. This comparison shows a similarity between duct sound radiation behaviours in terms of plate modes. As a part of the study, the total sound power radiated from the duct walls is estimated by using finite element method (FEM) and boundary element method (BEM). The sound radiation behaviour of a duct is also studied to understand its equivalence with monopole and dipole sources. The total radiation efficiencies of coupled acoustic and structural modes are calculated. These results are used for validation of the proposed analytical model.

#### 2. Theoretical Formulation

The outline of the formulation of an analytical model of sound radiation from the rectangular duct is shown in Figure 1. The objective here is to calculate the total sound power radiation and radiation efficiency from the flexible duct walls. Assumptions made in this model are (i) strong coupling amongst inside duct volume and flexible duct wall area; (ii) weak coupling between flexible wall area and outside environment; (iii) that coupled behaviour can be expressed in terms of a finite number of uncoupled acoustic and structural modes.