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.

Aspects of impeller design are explained taking into account the constraints from mechanical and aerodynamic considerations. A one-dimensional steady flow analysis is used to obtain a general understanding of the effects of the impeller design parameters on the geometry. This analysis provides some clear design guidelines for values of specific nondimensional flow parameters for optimum performance. The effects of the impeller blade inlet design on the inlet relative Mach number are considered together with that of the throat on flow capacity. The effect of the outlet velocity triangle on the work input and degree of reaction is explored. The considerations that lead to the choice of backsweep at the impeller outlet are explained. The steps required to adapt an impeller designed for one task to fulfil other requirements by means of trimming or flow cuts are explained. Guidance on the selection of mixed flow impellers is given. Some important differences are explained between the velocity triangles in radial flow compressor impellers and those in the rotors of centrifugal pumps, axial compressors and radial turbines.

The systematic definition of efficiency introduces isentropic, polytropic and isothermal efficiencies. The isentropic efficiency compares the actual work transfer to that which would take place in an ideal isentropic adiabatic process (with no losses and no heat transfer). Unfortunately, this does not represent the real thermodynamic process of a compressor very well. For example, a two-stage turbocharger using two stages each with a pressure ratio of 2 and an isentropic efficiency of 80%, has a pressure ratio of 4, but an isentropic efficiency of 78.1%. The polytropic efficiency overcomes this issue, and the two-stage compressor has the same polytropic efficiency as its individual stages. The kinetic energy present at the inlet and outlet of a stage can be identified by the difference between total and static states. The value of the kinetic energy in these planes is taken into account by comparing the total-to-total or total-to-static efficiencies. Care is needed as a radial compressor impeller may have a total-to-total polytropic impeller efficiency of over 90% but a static-to-static isentropic efficiency of well below 60%.

The essential aspects of the modern design and development process for radial flow turbocompressors are described. The different phases of the design process are described, including the conceptual design of the compressor type, the preliminary design of the components, the geometry specification of the ducts and blade rows, the blade-to-blade design, the throughflow design, 3D CFD performance analysis and FEM mechanical analysis. The final decision about the quality of a design is made through compressor testing. The chapter concludes with a section on the testing of compressors; this includes a discussion of the different types of tests, testing methods, standards, guidelines and procedures. Information about suitable instrumentation is also provided. Finally, there is a short review of recent experimental studies from some of the most active research groups with experimental rigs.

The compressor performance map with variable speed is described. A relatively simple 1D method for map prediction for a single stage based on the compressor duty is described. This provides good estimates of the expected performance map of a well-designed stage with only limited information about the geometry and gives a useful guide as to the performance map that can typically be achieved. The extension of the typical compressor performance map, with positive flow, positive rotation and positive pressure rise, into four regions with negative flow, negative pressure rise and reverse rotation, is described. The prediction of the performance of multistage compressors is described based on the principle of stage-stacking calculations. Matching of an impeller with a vaned diffuser at different tip-speed Mach numbers is considered. The matching of a compressor with a turbine in a micro gas turbine is described. The matching of a centrifugal compressor with a radial inflow turbine in a turbocharger is examined and demonstrates the advantages of the use of a turbine bypass valve or a turbine with variable inlet guide vanes to improve the performance.

This chapter describes a streamline curvature throughflow method which has been developed specially for radial machines, such as centrifugal compressor stages, pumps and radial turbines. The method can also be used for axial machines, both compressors and turbines, and for axial compressors with a centrifugal rear stage. The terms in the governing equation are described in some detail to illustrate how the geometry controls the pressure and flow gradients both along and normal to the streamlines in the compressor. Such codes have been in use since the 1960s as they a offer a good overview of the ideal velocity and pressure distributions through a component. They also highlight blade loading issues and identify risks of choking. In the design process, the throughflow method weeds out the weakest design options very quickly and thus eliminates the need for more complex, time-consuming CFD simulations on poor designs. A comparison of a throughflow calculation with the test results on the Eckardt impeller A shows the limits of the method and the efficacy of this procedure for preliminary design.