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 either doing 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. In the earlier editions of this book, open turbomachines, such as wind turbines, pro-pellers and unshrouded fans were deliberately excluded, primarily because of the conceptual difficulty of properly defining the mass flow that passes through the blades.
However, despite this apparent problem, the study of wind turbines has become an attractive and even an urgent task, not least because of the almost astonishing increase in their number. Wind turbines are becoming increasingly significant providers of electrical power and targets have even been set in some countries for at least 10% of power generation to be effected by this means by 2010. It is a matter of expediency to now include the aerodynamic theory of wind turbines in this book and so a new chapter has been added on the topic. It will be observed that the problem of dealing with the inde-terminate mass flow has been more or less resolved.Two main categories of turbomachine are identified: firstly, those that absorb powerto increase the fluid pressure or head (ducted fans, compressors and pumps); secondly,those that produce power by expanding fluid to a lower pressure or head (hydraulic,steam and gas turbines). Figure 1.1 shows, in a simple diagrammatic form, a selectionof the many different varieties of turbomachine encountered in practice. The reasonthat so many different types of either pump (compressor) or turbine are in use is becauseof the almost infinite range of service requirements. Generally speaking, for a given setof operating requirements one type of pump or turbine is best suited to provide optimum conditions of operation. This point is discussed more fully in the section of this chapter concerned with specific speed.
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. Figure 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 7.1,7.2, 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 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 aseither 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 turbomachine, together with the differing behaviour of individual types in oper-ation. 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.
Units and dimensionsThe International System of Units, SI (le Système International d’Unités) is a unifiedself-consistent system of measurement units based on the MKS (metre–kilogram–second) system. It is a simple, logical system based upon decimal relationships between units making it easy to use. The most recent detailed description of SI has been published in 1986 by HMSO. For an explanation of the relationship between, and use of, physical quantities, units and numerical values see Quantities, Units and Symbols (1975), published by The Royal Society or refer to ISO 31/0-1981.Great Britain was the first of the English-speaking countries to begin, in the 1960s,the long process of abandoning the old Imperial System of Units in favour of theInternational System of Units, and was soon followed by Canada, Australia, NewZealand and South Africa. In the USA a ten year voluntary plan of conversion to SIunits was commenced in 1971. In 1975 US President Ford signed the Metric ConversionAct which coordinated the metrication of units, but did so without specifying a schedule of conversion. Industries heavily involved in international trade (cars, aircraft, food and drink) have, however, been quick to change to SI for obvious economic reasons, but others have been reluctant to change.SI has now become established as the only system of units used for teachingengineering in colleges, schools and universities in most industrialised countries throughout the world. The Imperial System was derived arbitrarily and has no consistent numerical base, making it confusing and difficult to learn. In this book all numerical problems involving units are performed in metric units as this is more convenientthan attempting to use a mixture of the two systems. However, it is recognised that some problems exist as a result of the conversion to SI units. One of these is that many valuable papers and texts written prior to 1969 contain data in the old system of units and would need converting to SI units.Some SI unitsThe SI basic units used in fluid mechanics and thermodynamics are the metre (m),kilogram (kg), second (s) and thermodynamic temperature (K). All the other units used in this book are derived from these basic units. The unit of force is the newton (N), defined as that force which, when applied to a mass of 1 kilogram, gives an acceleration to the mass of 1 m/s2. The recommended unit of pressure is the pascal (Pa) which is the pressure produced by a force of 1 newton uniformly distributed over an area of 1 square metre. Several other units of pressure are in widespread use, however, foremost of these being the bar. Much basic data concerning properties of substances (steam and gas tables, charts, etc.) have been prepared in SI units with pres-sure given in bars and it is acknowledged that this alternative unit of pressure will continue to be used for some time as a matter of expediency. It is noted that 1 bar equals105 Pa (i.e. 105 N/m2), roughly the pressure of the atmosphere at sea level, and is perhaps an inconveniently large unit for pressure in the field of turbomachineryanyway! In this book the convenient size of the kilopascal (kPa) is found to be the most useful multiple of the recommended unit and is extensively used in most calculations and examples.In SI the units of all forms of energy are the same as for work. The unit of energyis the joule (J) which is the work done when a force of 1 newton is displaced through a distance of 1 metre in the direction of the force, e.g. kinetic energy (1/2 mc2) has the dimensions kg ¥ m2/s2; however, 1 kg = 1 Ns2/m from the definition of the newton given above. Hence, the units of kinetic energy must be Nm = J upon substituting dimensions.The watt (W) is the unit of power; when 1 watt is applied for 1 second to a systemthe input of energy to that system is 1 joule (i.e. 1 J).The hertz (Hz) is the number of repetitions of a regular occurrence in 1 second. Instead of writing c/s for cycles/sec, Hz is used. The unit of thermodynamic temperature is the kelvin (K), written without the ° sign, and is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. The degree celsius (°C) is equal to the unit kelvin. Zero on the celsius scale isthe temperature of the ice point (273.15 K). Specific heat capacity, or simply specific heat, is expressed as J/kg K or as J/kg°C. Dynamic viscosity, dimensions ML-1 T-1 , has the SI units of pascal seconds, i.e. Hydraulic engineers find it convenient to express pressure in terms of head of a liquid. The static pressure at any point in a liquid at rest is, relative to the pressure acting on the free surface, proportional to the vertical distance of the free surface above that point. The head H is simply the height of a column of the liquid which can be supported by this pressure. If r is the mass density (kg/m3) and g the local gravitational acceleration (m/s2), then the static pressure p (relative to atmospheric pressure) is p = rgH, where H is in metres and p is in pascals (or N/m2
). This is left for the student to verify as a simple exercise.
Dimensional analysis and performance laws
The widest comprehension of the general behaviour of all turbomachines is, without doubt, obtained from dimensional analysis. This is the formal procedure whereby the group of variables representing some physical situation is reduced into a smaller number of dimensionless groups. When the number of independent variables is not too great, dimensional analysis enables experimental relations between variables to be found with the greatest economy of effort. Dimensional analysis applied to turbomachines has two further important uses: (a) prediction of a prototype’s performance from tests conducted on a scale model (similitude); (b) determination of the most suitable type of machine, on the basis of maximum efficiency, for a specified range of head, speed and flow rate. Several methods of constructing non-dimensional groups have been described by Douglas et al. (1995) and by Shames (1992) among other authors.
The subject of dimensional analysis was made simple and much more interesting by Edward Taylor (1974) in his comprehensive account of the subject. It is assumed here that the basic techniques of forming non-dimensional groups have already been acquired by the student.
Adopting the simple approach of elementary thermodynamics, an imaginary envelope (called a control surface) of fixed shape, position and orientation is drawn around the turbomachine (Figure 1.2). Across this boundary, fluid flows steadily, entering at station 1 and leaving at station 2. As well as the flow of fluid there is a flow of work across the control surface, transmitted by the shaft either to, or from, the machine. For the present all details of the flow within the machine can be ignored and only externally observed features such as shaft speed, flow rate, torque and change in fluid properties across the machine need be considered. To be specific, let the turbomachine be a pump (although the analysis could apply to other classes of turbomachine) driven by an electric motor. The speed of rotation N, can be adjusted by altering the current to the motor; the volume flow rate Q, can be independently adjusted by means of a throttle valve. For fixed values of the set Q and N, all other variables such as torque t, head H, are thereby established. The choice of Q and N as control variables is clearly arbitrary and any other pair of independent variables such as t and H could equally well
have been chosen. The important point to recognise is that there are for this pump, two control variables. If the fluid flowing is changed for another of different density r, and viscosity m, the performance of the machine will be affected. Note, also, that for a turbomachine handling compressible fluids, other fluid properties are important and are discussed later.
So far we have considered only one particular turbomachine, namely a pump of a given size. To extend the range of this discussion, the effect of the geometric variables on the performance must now be included. The size of machine is characterised by the impeller diameter D, and the shape can be expressed by a number of length ratios, l1/D, l2/D, etc.
Incompressible fluid analysis
The performance of a turbomachine can now be expressed in terms of the control variables, geometric variables and fluid properties. For the hydraulic pump it is convenient to regard the net energy transfer gH, the efficiency h, and power supplied P, as dependent variables and to write the three functional relationships as
By the procedure of dimensional analysis using the three primary dimensions, mass, length and time, or alternatively, using three of the independent variables we can form the dimensionless groups. The latter, more direct procedure requires that the variables selected, r, N, D, do not of themselves form a dimensionless group. The selection of r, N, D as common factors avoids the appearance of special fluid terms (e.g. mℋℋℋℳℳ, Q) in more than one group and allows gH, h and P to be made explicit. Hence the three relationships reduce to the following easily verified forms.
Energy transfer coefficient, sometimes called head coefficient
which is frequently used is the velocity (or flow) coefficient f = cx /U where U is blade tip speed and cx the average axial velocity. Since
Because of the large number of independent groups of variables on the right-hand side of eqns. (1.2), those relationships are virtually worthless unless certain terms can be discarded. In a family of geometrically similar machines l1/D, l2/D are constant and may be eliminated forthwith. The kinematic viscosity, - ℌ = m/r is very small in turbomachines handling water and, although speed, expressed by ND, is low the Reynolds number is correspondingly high. Experiments confirm that effects of Reynolds number on the performance are small and may be ignored in a first approximation. The functional relationships for geometrically similar hydraulic turbomachines are then,
This is as far as the reasoning of dimensional analysis alone can be taken; the actual form of the functions f4, f5 and f6 must be ascertained by experiment.
One relation between y, f, h and Pˆ may be immediately stated. For a pump the net hydraulic power, PN equals rQgH which is the minimum shaft power required in the absence of all losses. No real process of power conversion is free of losses and the actual shaft power P must be larger than PN. We define pump efficiency (more precise definitions of efficiency are stated in Chapter 2) h = PN/P = rQgH/P. Therefore
Thus f6 may be derived from f4 and f5 since Pˆ = fy/h. For a turbine the net hydraulic power PN supplied is greater than the actual shaft power delivered by the machine andthe efficiency h = P/PN. This can be rewritten as Pˆ = hfy by reasoning similar to theabove considerations





