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AcknowledgementsThe author is indebted to a number of people and manufacturing organisations fortheir help and support; in particular the following are thanked:Professor W. A. Woods, formerly of Queen Mary College, University of Londonand a former colleague at the University of Liverpool for his encouragement of theidea of a fourth edition of this book as well as providing papers and suggestions forsome new items to be included. Professor F. A. Lyman of Syracuse University, NewYork and Professor J. Moore of Virginia Polytechnic Institute and State University,Virginia, for their helpful correspondence and ideas concerning the vexed questionof the conservation of rothalpy in turbomachines. Dr Y. R. Mayhew is thanked forsupplying me with generous amounts of material on units and dimensions and thelatest state of play on SI Units.Thanks are also given to the following organisations for providing me with illustrative material for use in the book, product information and, in one case, usefulbackground historical information:Sulzer Hydro of Zurich, Switzerland; Rolls-Royce of Derby, England; VoithHydro Inc., Pennsylvania; and Kvaerner Energy, Norway.Last, but by no means least, to my wife Rose, whose quiet patience and supportenabled this new edition to be prepared.

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List of SymbolsAabCfCL , CDCpCpiCvCX , CYccoDDeqDhE, eFcfgHHEHfHGHShIiK, kKNLlMmNNSNSPNSSnpareasonic velocity, position of maximum camberpassage width, maximum cambertangential force coefficientlift and drag coefficientsspecific heat at constant pressure, pressure coefficient, pressure risecoefficientideal pressure rise coefficientspecific heat at constant volumeaxial and tangential force coefficientsabsolute velocityspouting velocitydrag force, diameterequivalent diffusion ratiohydraulic mean diameterenergy, specific energycentrifugal force in bladeacceleration, friction factorgravitational accelerationhead, blade heighteffective headhead loss fue to frictiongross headnet positive suction head (NPSH)specific enthalpyrothalpyincidence angleconstantsnozzle velocity coefficientlift force, length of diffuser wallblade chord length, pipe lengthMach numbermass, molecular ‘weight’rotational speed, axial length of diffuserspecific speed (rev)power specific speed (rev)suction specific speed (rev)number of stages, polytropic indexpressure

xvi Fluid Mechanics, Thermodynamics of TurbomachinerypapvQqRReRHRorSsTtUuV, vW WwXx, y, zYYidYkYpYSZatmospheric pressurevapour pressureheat transfer, volume flow ratedryness fractionreaction, specific gas constantReynolds numberreheat factoruniversal gas constantradiusentropy, power ratioblade pitch, specific entropytemperaturetime, thicknessblade speed, internal energyspecific internal energyvolume, specific volumework transferspecific work transferrelative velocityaxial forceCartesian coordinate directionstangential force, actual tangential blade load per unit spanideal tangential blade load per unit spantip clearance loss coefficientprofile loss coefficientnet secondary loss coefficientnumber of blades, Ainley blade loading parameter ˇ absolute flow anglerelative flow anglecirculationratio of specific heatsdeviation anglefluid deflection angle, cooling effectivenessenthalpy loss coefficient, total pressure loss coefficientefficiencyminimum opening at cascade exitblade camber angle, wake momentum thicknessprofile loss coefficientdynamic viscositykinematic viscosity, blade stagger angle, velocity ratiodensityslip factor, solidityblade cavitation coefficientThoma’s coefficient, centrifugal stresstorqueυε b c

List of Symbols xvii S SP SSωωflow coefficient, velocity ratiostage loading factorspeed of rotation (rad/s)specific speed (rad)power specific speed (rad)suction specific speed (rad)vorticitystagnation pressure loss coefficientSubscriptsavcDehiidismNnopRrrelssstvx, y, z averagecompressor, criticaldiffuserexithydraulic, hubinlet, impelleridealisentropicmean, meridional, mechanical, materialnozzlenormal componentstagnation property, overallpolytropic, constant pressurereversible process, rotorradialrelativeisentropic, stall conditionstage isentropicturbine, tip, transversevelocitycartesian coordinate componentstangentialSuperscriptÐ0*time rate of changeaverageblade angle (as distinct from flow angle)nominal condition

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CHAPTER 1Introduction: DimensionalAnalysis: SimilitudeIf you have known one you have known all. (TERENCE, Phormio.)Definition of a turbomachineWe classify as turbomachines all those devices in which energy is transferredeither to, or from, a continuously flowing fluid by the dynamic action of one ormore moving blade rows. The word turbo or turbinis is of Latin origin and impliesthat which spins or whirls around. Essentially, a rotating blade row, a rotor or animpeller changes the stagnation enthalpy of the fluid moving through it by eitherdoing positive or negative work, depending upon the effect required of the machine.These enthalpy changes are intimately linked with the pressure changes occurringsimulataneously in the fluid.The definition of a turbomachine as stated above, is rather too general for thepurposes of this book as it embraces open turbomachines such as propellers, windturbines and unshrouded fans, all of which influence the state of a not readilyquantifiable flow of a fluid. The subject fluid mechanics, thermodynamics of turbomachinery, therefore, is limited to machines enclosed by a closely fitting casing orshroud through which a readily measurable quantity of fluid passes in unit time.The subject of open turbomachines is covered by the classic text of Glauert (1959)or by Duncan et al. (1970), the elementary treatment of propellers by general fluidmechanics textbooks such as Streeter and Wylie (1979) or Massey (1979), and theimportant, still developing subject of wind turbines, by Freris (1990).Two main categories of turbomachine are identified: firstly, those which absorbpower to 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 diagrammaticform, a selection of the many different varieties of turbomachine encountered inpractice. The reason that so many different types of either pump (compressor) orturbine are in use is because of the almost infinite range of service requirements.Generally speaking, for a given set of operating requirements there is one type ofpump or turbine best suited to provide optimum conditions of operation. This pointis discussed more fully in the section of this chapter concerned with specific speed.Turbomachines are further categorised according to the nature of the flow paththrough the passages of the rotor. When the path of the through-flow is wholly or mainlyparallel to the axis of rotation, the device is termed an axial flow turbomachine (e.g.1

2Fluid Mechanics, Thermodynamics of TurbomachineryFIG. 1.1. Diagrammatic form of various types of turbomachine.Figure 1.1(a) and (e)). When the path of the through-flow is wholly or mainly in a planeperpendicular 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 inthis context refers to the direction of the through-flow at rotor outlet when both radialand axial velocity components are present in significant amounts. Figure 1.1(b) showsa mixed flow pump and Figure 1.1(d) a mixed flow hydraulic turbine.One further category should be mentioned. All turbomachines can be classifiedas either impulse or reaction machines according to whether pressure changes are

Introduction: Dimensional Analysis: Similitude 3absent or present respectively in the flow through the rotor. In an impulse machineall the pressure change takes place in one or more nozzles, the fluid being directedonto 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 mechanicsand thermodynamics, the means by which the energy transfer is achieved in thechief types of turbomachine, together with the differing behaviour of individualtypes in operation. Methods of analysing the flow processes differ depending uponthe geometrical configuration of the machine, on whether the fluid can be regardedas incompressible or not, and whether the machine absorbs or produces work. Asfar as possible, a unified treatment is adopted so that machines having similarconfigurations and function are considered together.Units and dimensionsThe International System of Units, SI (le Système International d’Unités)is a unified self-consistent system of measurement units based on the MKS(metre kilogram second) system. It is a simple, logical system based upon decimalrelationships between units making it easy to use. The most recent detaileddescription of SI has been published in 1986 by HMSO. For an explanation ofthe relationship between, and use of, physical quantities, units and numerical valuessee Quantities, Units and Symbols, published by The Royal Society (1975) or referto ISO 31/0-1981.Great Britain was the first of the English-speaking countries to begin, in the1960s, the long process of abandoning the old Imperial System of Units in favourof the International System of Units, and was soon followed by Canada, Australia,New Zealand and South Africa. In the USA a ten year voluntary plan of conversionto SI units was commenced in 1971. In 1975 US President Ford signed the MetricConversion Act which coordinated the metrication of units, but did so withoutspecifying a schedule of conversion. Industries heavily involved in internationaltrade (cars, aircraft, food and drink) have, however, been quick to change to SI forobvious 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 countriesthroughout the world. The Imperial System was derived arbitrarily and has noconsistent numerical base, making it confusing and difficult to learn. In this bookall numerical problems involving units are performed in metric units as this is moreconvenient than attempting to use a mixture of the two systems. However, it isrecognised that some problems exist as a result of the conversion to SI units. Oneof these is that many valuable papers and texts written prior to 1969 contain datain the old system of units and would need converting to SI units. A brief summaryof the conversion factors between the more frequently used Imperial units and SIunits is given in Appendix 1 of this book.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 otherunits used in this book are derived from these basic units. The unit of force is the

4Fluid Mechanics, Thermodynamics of Turbomachinerynewton (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 isthe pascal (Pa) which is the pressure produced by a force of 1 newton uniformlydistributed 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 concerningproperties of substances (steam and gas tables, charts, etc.) have been prepared in SIunits with pressure given in bars and it is acknowledged that this alternative unit ofpressure will continue to be used for some time as a matter of expediency. It is notedthat 1 bar equals 105 Pa (i.e. 105 N/m2 ), roughly the pressure of the atmosphere atsea level, and is perhaps an inconveniently large unit for pressure in the field ofturbomachinery anyway! In this book the convenient size of the kilopascal (kPa) isfound to be the most useful multiple of the recommended unit and is extensivelyused 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 througha distance of 1 metre in the direction of the force, e.g. kinetic energy ( 21 mc2 ) has thedimensions kg ð m2 /s2 ; however, 1 kg D 1 N s2 /m from the definition of the newtongiven above. Hence, the units of kinetic energy must be Nm D J upon substitutingdimensions.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 instead.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 triplepoint of water. The degree celsius ( C) is equal to the unit kelvin. Zero on thecelsius scale is the 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.kgN.s2M D 2 D Pa s.LTm.sm. sHydraulic engineers find it convenient to express pressure in terms of head of aliquid. The static pressure at any point in a liquid at rest is, relative to the pressureacting on the free surface, proportional to the vertical distance of the free surfaceabove that point. The head H is simply the height of a column of the liquid whichcan be supported by this pressure. If is the mass density (kg/m3 ) and g the localgravitational acceleration (m/s2 ), then the static pressure p (relative to atmosphericpressure) is p D gH, where H is in metres and p is in pascals (or N/m2 ). This isleft for the student to verify as a simple exercise.Dimensional analysis and performance lawsThe widest comprehension of the general behaviour of all turbomachines is,without doubt, obtained from dimensional analysis. This is the formal procedurewhereby the group of variables representing some physical situation is reduced

Introduction: Dimensional Analysis: Similitude 5into a smaller number of dimensionless groups. When the number of independent variables is not too great, dimensional analysis enables experimental relationsbetween variables to be found with the greatest economy of effort. Dimensionalanalysis applied to turbomachines has two further important uses: (a) predictionof 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 maximumefficiency, for a specified range of head, speed and flow rate. Several methods ofconstructing non-dimensional groups have been described by Douglas et al. (1995)and by Shames (1992) among other authors. The subject of dimensional analysis wasmade 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 formingnon-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 drawnaround 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 thereis a flow of work across the control surface, transmitted by the shaft either to, orfrom, the machine. For the present all details of the flow within the machine canbe 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 bespecific, let the turbomachine be a pump (although the analysis could apply to otherclasses of turbomachine) driven by an electric motor. The speed of rotation N, canbe adjusted by altering the current to the motor; the volume flow rate Q, can beindependently adjusted by means of a throttle valve. For fixed values of the set Qand N, all other variables such as torque , head H, are thereby established. Thechoice of Q and N as control variables is clearly arbitrary and any other pair ofindependent variables such as and H could equally well have been chosen. Theimportant point to recognise is, that there are for this pump, two control variables.If the fluid flowing is changed for another of different density , and viscosity , the performance of the machine will be affected. Note, also, that for a turbomachine handling compressible fluids, other fluid properties are important and arediscussed later.So far we have considered only one particular turbomachine, namely a pump ofa given size. To extend the range of this discussion, the effect of the geometricFIG. 1.2. Turbomachine considered as a control volume.

6Fluid Mechanics, Thermodynamics of Turbomachineryvariables 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 numberof length ratios, l1 /D, l2 /D, etc.Incompressible fluid analysisThe performance of a turbomachine can now be expressed in terms of the controlvariables, geometric variables and fluid properties. For the hydraulic pump it isconvenient to regard the net energy transfer gH, the efficiency , and power suppliedP, as dependent variables and to write the three functional relationships as l1 l2gH D f1 Q, N, D, , , , , . . . ,.1.1a/D D l1 l2.1.1b/ D f2 Q, N, D, , , , , . . . ,D D l1 l2.1.1c/P D f3 Q, N, D, , , , , . . . ,D DBy the procedure of dimensional analysis using the three primary dimensions, mass,length and time, or alternatively, using three of the independent variables we canform the dimensionless groups. The latter, more direct procedure, requires that thevariables selected, , N, D, do not of themselves form a dimensionless group. Theselection of , N, D as common factors avoids the appearance

2. Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency 23 Introduction 23 The equation of continuity 23 The first law of thermodynamics internal energy 24 The momentum equation Newton’s second law of motion 25 The second law of thermodynamics entropy 29 Definitio