This chapter introduces the general analytic theory of one-dimensional sound propagation in ducts and presents acoustic models for uniform, non-uniform and inhomogeneous ducts with hard or finite impedance walls and parallel sheared mean flow.

Chapter 11 describes calculation of the sound pressure level at a point in the acoustic field of an in-duct source. Insertion loss of a silencer is shown to be represented approximately by source-independent parameters under certain conditions. The discussion encompasses multi-modal sound propagation and radiation and the ASHRAE method of silencer sizing in ventilation and air distribution systems.

This chapter describes the basic analytic concepts and operations which are invoked throughout the book. Mathematical models of sound wave motion in ducts come from the solutions of the linearized forms of the basic fluid dynamic equations of unsteady fluid flow in frequency and wavenumber domains. The process of linearization is discussed in depth and the frequency and wavenumber transformations are defined rigorously. A quantity that is often of interest in duct acoustics is the acoustic power transmitted in a duct. Calculation of time-averaged acoustic power transmitted in ducts is described a unified manner. Finally, we describe the mathematical link with the analyses presented in the book and linear system dynamics. These topics are collected in this preliminary chapter as primer and also to avoid interruption of the continuity of discussions on the principal subjects.

In most duct systems, propagation of duct-borne sound terminates with a duct which opens to an exterior environment. Chapter 9 describes modeling of open ends of ducts and the acoustic field radiated from an open end. This enables acoustic model of a duct system to be extended from the source to the receiver.

In Chapters 3 to 7, the fluid is assumed to be inviscid. Effects of the viscosity and thermal conductivity of the fluid are considered in this chapter. The analysis is based on the low-reduced frequency theory and includes applications to catalytic converter and particulate filters.

Chapter 7 describes modal acoustic models of several coupled duct configurations. The acoustic models described in this chapter extend the one-dimensional area change, junction and perforate elements described in Chapters 3 to three dimensions.

Some coupled-duct configurations occur recurrently in practical systems. In view of the conditions of continuity and compatibility of the acoustic fields, ducts may or may not be connected as individual units. In this chapter we present one-dimensional intermediate acoustic models for basic coupling configurations such as area changes and junctions, and continuous and discrete acoustic models for packs of coupled perforated ducts. These are multi-port elements and they can be connected at their ports to the one-dimensional duct elements given in Chapter 3.

The design of the diffuser system immediately downstream of the impeller is considered. The diffuser transforms the kinetic energy at its inlet into a rise in the static pressure. Centrifugal compressors are usually fitted with either a vaned or a vaneless diffuser leading to a collector. The diffuser meridional channel comprises an annular channel extending radially outwards from the impeller outlet, usually of the same width as the impeller. The simplest diffuser system is a radial vaneless annular channel where the radial velocity component is reduced by the increase in the area of the channel with radius (conservation of mass) and the circumferential velocity component is reduced by the increase in radius in the diffuser (conservation of angular momentum). In a vaned diffuser, of which several types are considered, there is a small vaneless region upstream of the diffuser vanes. The vanes themselves form flow channels designed to decelerate the flow more than is possible in a vaneless diffuser by turning the flow in a more radial direction. The different zones of pressure recovery in vaned diffusers are examined and compared with the equivalent planar diffuser.

The laws of gas dynamics, that is, the fluid dynamics of compressible flows, that are relevant to understand compressible flow in channels of variable area and in turbocompressor blade rows are introduced. The theory of one-dimensional compressible flow in variable area ducts is developed. The mass-flow function or corrected flow per unit area is introduced. The variation of the pressure in a nozzle at different back pressures is described. The one-dimensional approach is used to describe the nature of choking, expansion waves and shock waves. Special emphasis is given on the nature of the transonic flow and shock structure at inlet to a radial compressor inducer and how this is affected by the blade shape and the operating conditions. The gas dynamics of flows of real gases are considered.

Fluid dynamic principles that are fundamental to understanding the motion of fluids in radial compressors are highlighted. These include the continuity and the momentum equations in various forms. These equations are then used to delineate the effect of the fluid motion on pressure gradients on the flow. The simple radial equilibrium equation for a circumferentially averaged flow is introduced. Special features of the flow in radial compressors due to the radial motion are considered, such as the effects of the Coriolis and centrifugal forces. The relative eddy, which gives rise to the slip factor of a radial impeller, is explained. A short overview of boundary layer flows of relevance to radial compressors is provided. The flow in radial compressor impellers is strongly affected by secondary flows and tip clearance flows, and an outline is provided of the current understanding of the physics related to these. The phenomenon of jet-wake flow in compressors is described.

A study of the Euler equation on the basis of one-dimensional velocity triangles provides insights into energy transfer in compressors, emphasising the importance of the centrifugal effect in the impeller, the diffusion of the flow and the degree of reaction. An introduction to thermodynamics is given leading to the steady flow energy equation (SFEE), which is the first law of thermodynamics applied to a fixed region with steady flow passing through it. The SFEE is used to account for the changes in fluid properties along the flow path and shows that the bookkeeping of the energy transfer needs to be carried out using the total enthalpy or the rothalpy. The study of compressors needs to consider the efficiency of processes concerned. The Gibbs equation, a form of the second law of thermodynamics, provides a rigorous way to do this through the thermodynamic state variable known as entropy. In the context of energy transfer, the entropy production characterises the lost work in the machine due to dissipation losses. Isentropic and polytropic compression processes are explained. The important concept of the aerodynamic work and the value of a polytropic analysis are considered.

The key aspects of the physics of unstable flows in compressors are described. Operating at part-load can cause serious instabilities in the compressor flow, even leading to damage to the compressor. Different types of unsteady flow can be categorised as surge, rotating stall and hysteresis, and these depend on both the compressor and the process to which it delivers the flow. The key parameter in the system dynamics that is used to measure the likelihood of rotating stall or surge is a stability parameter known as the Greitzer B parameter. The onset of instability can happen in two different ways, known as modes and spikes. The consequence of instability on the operating range is described, and field experience shows that the operating range reduces with higher tip-speed Mach numbers and larger work coefficients. The system requirements can be categorised in terms of the pressure versus volume characteristics of the process. Methods to extend the stable operating range of compressors by control with variable speed, variable geometry, passive recirculation systems and other regulation devices are described.