Introduction

Basic Principles

Publisher Summary

This chapter introduces the book on fluid mechanics and the thermodynamics of turbomachines. The book examines, through the laws of fluid mechanics and thermodynamics, the means by which the energy transfer is achieved in the chief types of turbomachines, together with the differing behavior of individual types in their operations. The turbomachines are devices in which energy is transferred continuously flowing fluid by the dynamic action of one or more moving blade rows. Two main categories of turbomachines are identified—those that absorb power to increase the fluid pressure or head (ducted and unducted fans, compressors, and pumps) and those that produce power by expanding fluid to a lower pressure or head (wind, hydraulic, steam, and gas turbines). Methods of analyzing the flow processes differ depending upon the geometrical configuration of the machine. The most fundamental and valuable principles in fluid mechanics is Newton’s second law of motion. The momentum equation relates the sum of the external forces acting on a fluid element to its acceleration, or to the rate of change of momentum in the direction of the resultant external force. The chapter presents the basic physical laws of fluid mechanics and thermodynamics, developing them into a form suitable for the study of turbomachines such as the continuity of flow equation, the first law of thermodynamics and the steady flow energy equation, the momentum equation, and the second law of thermodynamics. The energy and entropy equations are introduced in the chapter, along with all-important Euler work equation, which applies to all turbomachines. It also presents important efficiencies such as efficiency of turbines, steam and gas turbines, hydraulic turbines, and compressors and pumps.

1.1 Definition of a Turbomachine

We classify as turbomachines all those devices in which energy is transferred either to, or from, a continuously flowing fluid by the dynamic action of one or more moving blade rows. The word turbo or turbinis is of Latin origin and implies that which spins or whirls around. Essentially, a rotating blade row, a rotor or an impeller changes the stagnation enthalpy of the fluid moving through it by doing either positive or negative work, depending upon the effect required of the machine. These enthalpy changes are intimately linked with the pressure changes occurring simultaneously in the fluid.

Two main categories of turbomachine are identified: firstly, those that absorb power to increase the fluid pressure or head (ducted and unducted fans, compressors, and pumps); secondly, those that produce power by expanding fluid to a lower pressure or head (wind, hydraulic, steam, and gas turbines). Figure 1.1 shows, in a simple diagrammatic form, a selection of the many varieties of turbomachines encountered in practice. The reason that so many different types of either pump (compressor) or turbine are in use is because of the almost infinite range of service requirements. Generally speaking, for a given set of operating requirements one type of pump or turbine is best suited to provide optimum conditions of operation.

Turbomachines are further categorised according to the nature of the flow path through the passages of the rotor. When the path of the through-flow is wholly or mainly parallel to the axis of rotation, the device is termed an axial flow turbomachine [e.g., Figures 1.1(a) and (e)]. When the path of the through-flow is wholly or mainly in a plane perpendicular to the rotation axis, the device is termed a radial flow turbomachine [e.g., Figure 1.1(c)]. More detailed sketches of radial flow machines are given in Figures 7.3, 7.4, 8.2, and 8.3. Mixed flow turbomachines are widely used. The term mixed flow in this context refers to the direction of the through-flow at the rotor outlet when both radial and axial velocity components are present in significant amounts. Figure 1.1(b) shows a mixed flow pump and Figure 1.1(d) a mixed flow hydraulic turbine.

One further category should be mentioned. All turbomachines can be classified as either impulse or reaction machines according to whether pressure changes are absent or present, respectively, in the flow through the rotor. In an impulse machine all the pressure change takes place in one or more nozzles, the fluid being directed onto the rotor. The Pelton wheel, Figure 1.1(f), is an example of an impulse turbine.

The main purpose of this book is to examine, through the laws of fluid mechanics and thermodynamics, the means by which the energy transfer is achieved in the chief types of turbomachines, together with the differing behaviour of individual types in operation. Methods of analysing the flow processes differ depending upon the geometrical configuration of the machine, whether the fluid can be regarded as incompressible or not, and whether the machine absorbs or produces work. As far as possible, a unified treatment is adopted so that machines having similar configurations and function are considered together.

1.2 Coordinate System

Turbomachines consist of rotating and stationary blades arranged around a common axis, which means that they tend to have some form of cylindrical shape. It is therefore natural to use a cylindrical polar coordinate system aligned with the axis of rotation for their description and analysis. This coordinate system is pictured in Figure 1.2. The three axes are referred to as axial x, radial r, and tangential (or circumferential) r θ.

In general, the flow in a turbomachine has components of velocity along all three axes, which vary in all directions. However, to simplify the analysis it is usually assumed that the flow does not vary in the tangential direction. In this case, the flow moves through the machine on axi symmetric stream surfaces, as drawn on Figure 1.2(a). The component of velocity along an axi-symmetric stream surface is called the meridional velocity,

m=cx2+cr2. (1.1)

In purely axial-flow machines the radius of the flow path is constant and therefore, referring to Figure 1.2(c) the radial flow velocity will be zero and cm = cx. Similarly, in purely radial flow machines the axial flow velocity will be zero and cm = cr. Examples of both of these types of machines can be found in Figure 1.1.

The total flow velocity is made up of the meridional and tangential components and can be written

=cx2+cr2+cθ2=cm2+cθ2. (1.2)

The swirl, or tangential, angle is the angle between the flow direction and the meridional direction:

=tan−1(cθ/cm). (1.3)

Relative Velocities

The analysis of the flow-field within the rotating blades of a turbomachine is performed in a frame of reference that is stationary relative to the blades. In this frame of reference the flow appears as steady, whereas in the absolute frame of reference it would be unsteady. This makes any calculations significantly more straightforward, and therefore the use of relative velocities and relative flow quantities is fundamental to the study of turbomachinery.

The relative velocity is simply the absolute velocity minus the local velocity of the blade. The blade has velocity only in the tangential direction, and therefore the relative components of velocity can be written as

θ=cθ−U,wx=cx,wr=cr. (1.4)

The relative flow angle is the angle between the relative flow direction and the meridional direction:

=tan−1(wθ/cm). (1.5)

By combining eqns. (1.3), (1.4), and (1.5) a relationship between the relative and absolute flow angles can be found:

β=tanα−U/cm. (1.6)

1.3 The Fundamental Laws

The remainder of this chapter summarises the basic physical laws of fluid mechanics and thermodynamics, developing them into a form suitable for the study of turbomachines. Following this, some of the more important and commonly used expressions for the efficiency of compression and expansion flow processes are given.

The laws discussed are

All of these laws are usually covered in first-year university engineering and technology courses, so only the briefest discussion and analysis is given here. Some textbooks dealing comprehensively with these laws are those written by Çengel and Boles (1994); Douglas, Gasiorek, and Swaffield (1995); Rogers and Mayhew (1992); and Reynolds and Perkins (1977). It is...