Fundamentals of Turbine Design

7Fundamentals of Turbine Design

David M. Mathis

Honeywell Aerospace, Tempe, Arizona, U.S.A.

INTRODUCTION

Turbines are used to convert the energy contained in a continuous ow ofuid into rotational mechanical energy of a shaft. Turbines are used in awide range of applications, in a wide variety of sizes. Large single-stageturbines are used for power generation in hydroelectric dams, while largemultistage turbines are used in steam power plants. Aircraft propulsionengines (turbofans, turbojets, and turboprops) use multistage turbines intheir power and gas generator sections. Other, less obvious uses of turbinesfor aircraft are in auxiliary power units, ground power units, starters formain engines, turboexpanders in environmental controls, and constant-speed drives for electrical and hydraulic power generation. Rocket enginesuse turbines to power pumps to pressurize the propellants before they reachthe combustion chamber. Two familiar consumer applications of turbinesare turbochargers for passenger vehicles and wind turbines (windmills).

Many excellent texts have been written regarding the design andanalyses of turbines [13]. There is also a large institutional body ofknowledge and practices for the design and performance prediction of

Copyright 2003 Marcel Dekker, Inc.

power plant and aircraft propulsion engine turbines. Here we make noattempt to cover these areas. The purpose of this text is to familiarize amechanical or aerospace engineer who does not specialize in turbines withbasic turbine design and analysis. The emphasis will be on smaller turbinesfor applications other than propulsion or electrical power generation.Further restricting our emphasis, detailed design activities such as geometricspecication of blades, vanes, etc. will not be covered. Our intent is to givethe system engineer the necessary information to choose the correct type ofturbine, estimate its performance, and determine its overall geometry(diameter, blade height, and chord).

This chapter will rst cover those equations and concepts that apply toall types of turbines. Subsequently, the two main turbine types will bediscussed, specically, axial-ow turbines and radial-inow turbines.

BASIC TURBINE CONCEPTS

Flow Through a Turbine

Figure 1 shows cross sections of generic single-stage axial-ow and radial-inow turbines. The gure shows the station notation used for subsequentanalyses. The high-pressure ow enters the turbine at station in, passesthrough the inlet, and is guided to the stator inlet (station 0), where vanesturn the ow in the tangential direction. The ow leaves the stator vanes andenters the rotor blades (at station 1), which turn the ow back in theopposite direction, extracting energy from the ow. The ow leaving therotor blades (station 2), now at a pressure lower than inlet, passes through adiffuser where a controlled increase in ow area converts dynamic head tostatic pressure. Following the diffuser, the ow exits to the dischargeconditions (station dis).

The purpose of the inlet is to guide the ow from the supply source tothe stator vanes with a minimum loss in total pressure. Several types of inletsare shown in Fig. 2. Most auxiliary types of turbines such as starters anddrive units are supplied from ducts and typically have axial inlets such asthat shown in Fig. 2(a) or a tangential entry like that of Fig. 2(b). The axialinlet acts as a transition between the small diameter of the supply duct andthe larger diameter of the turbine. No ow turning or signicantacceleration is done in this type of inlet. In the tangential-entry scroll ofFig. 2(b), the ow is accelerated and turned tangentially before entering thestator, reducing the ow turning done by the stator. Another type of inletfor an auxiliary turbine is the plenum shown in Fig. 2(c). For turbines thatare part of an engine, the inlet is typically integrated with the combustor[Fig. 2(d)] or the discharge of a previous turbine stage [Fig. 2(e)].


Figure 1 Cross sections of generic single-stage turbines: (a) axial-ow turbine, (b)radial-inow turbine.


The sole purpose of the stator is to induce a swirl component to theow so that a torque can be imparted to the rotor blades. Stators aretypically equipped with numerous curved airfoils called vanes that turn theow in the tangential direction. Cross sections of an axial-ow turbine statorand a radial-inow turbine stator are shown in Fig. 3(a) and 3(b),

Figure 2 Common types of turbine inlets: (a) in-line axial inlet, (b) tangential-entryscroll inlet for axial-ow turbine, (c) plenum inlet with radial or axial entry, (d)

turbine stage downstream of combustor, (e) turbine stage in multistage turbine.


respectively. Radial-inow turbines supplied from a scroll, such asturbocharger turbines, often have no vanes in the stator. For turbinesthat must operate efciently over a wide range of inlet ow conditions,variable-geometry stators are used, typically with pivoting stator vanes. Forhigh-temperature applications, the stator vanes are cooled using lower-temperature uid, usually compressor bleed air.

The purpose of the rotor is to extract energy from the ow, convertingit to shaft power. The rotor blades are attached to a rotating disk thattransfers the torque of the rotor blades to the turbine output shaft. Like thestator, the rotor has a number of individual curved airfoils called rotor orturbine blades. The blades are angled to accept the ow from the stator withminimum disturbance when the turbine is operating at design conditions.The ow from the stator is then turned back in the opposite direction in acontrolled manner, causing a change in tangential momentum and a force tobe exerted on the blades. Figure 4 shows cross sections of generic axial-owand radial-inow turbine blades. Axial-ow rotors have been constructedwith blades integral with the disk and with blades individually inserted intothe disk using a dovetail arrangement. Cooling is often used for rotors inhigh-temperature applications. Exotic materials are sometimes used for bothrotors and stators to withstand the high temperatures encountered in high-performance applications.

The ow leaving the turbine rotor can have a signicant amount ofkinetic energy. If this kinetic energy is converted to static pressure in anefcient manner, the turbine can be operated with a rotor discharge staticpressure lower than the static pressure at diffuser discharge. This increasesthe turbine power output for given inlet and discharge conditions. Diffusersused with turbines are generally of the form shown in Fig. 5(a) and 5(b) andincrease the ow area gradually by changes in passage height, mean radiusof the passage, or a combination of the two. Diffusers with a change inradius have the advantage of diffusing the swirl component of the rotordischarge velocity as well as the throughow component.

Turbine Energy Transfer

The combined parts of the turbine allow energy to be extracted from theow and converted to useful mechanical energy at the shaft. The amount ofenergy extraction is some fraction of the energy available to the turbine. Thefollowing describes the calculation of the available energy for a turbine andassumes familiarity with thermodynamics and compressible ow.

Flow through a turbine is usually modeled as an adiabatic expansion.The process is considered adiabatic since the amount of energy transferredas heat is generally insignicant compared to the energy transferred as work.


In the ideal case, the expansion is isentropic, as shown in Fig. 6(a) in anenthalpyentropy (hs) diagram. The inlet to the turbine is at pressure p0inand the exit is at p0dis. The isentropic enthalpy change is the most specic

Figure 3 Typical stator vane shapes: (a) axial-ow stator, (b) radial-inow stator.


energy that can be extracted from the uid. Thus, if the inlet pressure andtemperature and the exit pressure from a turbine are known, the maximumspecic energy extraction can be easily determined from a state diagram forthe turbine working uid. For an ideal gas with constant specic heats, the

Figure 4 Typical rotor blade shapes: (a) axial-ow rotor, (b) radial-inow rotor.


isentropic enthalpy drop is calculated from

Dhisentropic cpT 0in 1pdis

p0in

g1=g" #1

Figure 5 Turbine diffuser congurations: (a) constant mean-diameter diffuser, (b)increasing mean-diameter curved diffuser.


Where

T 0in inlet absolute total temperature:Op specific heat at constant pressure:g ratio of specific heats:

Figure 6 The expansion process across a turbine: (a) idealized isentropicexpansion, (b) actual expansion.


The approximation of Eq. (1) is adequate for turbines operating on air andother common gases at moderate pressures and temperatures. Totalconditions are normally used for both temperature and pressure at theinlet to the turbine, so that the inlet pressure is correctly referred to as p0in inEq. (1).

As discussed earlier, the actual energy transfer in a turbine is smallerthan the isentropic value due to irreversibilities in the ow. The actualprocess is marked by an increase in entropy and is represented in the hsdiagram of Fig. 6(b) by a dotted line. The actual path is uncertain, as thedetails of the entropy changes within the turbine are usually not known. Dueto the curvature of the isobars, the enthalpy change associated with anentropy increase is less than that for an isentropic process. The degree ofentropy rise is usually described indirectly by the ratio of the actual enthalpydrop to the isentropic enthalpy drop. This quantity is referred to as theisentropic (sometimes adiabatic) efciency, Z, and is calculated from

ZOA hin hdisDhisentropic

2

The subscript OA indicates the overall efciency, since the enthalpy drop istaken across the entire turbine. The efciency is one of the criticalparameters that describe turbine performance.

So far we have not specied whether the total or static pressure shouldbe used at the turbine exit for calculating the isentropic enthalpy drop.(Note that this does not affect the actual enthalpy drop, just the idealenthalpy drop.) Usage depends on application. For applications where thekinetic energy leaving the turbine rotor is useful, total pressure is used. Suchcases include all but the last stage in a multistage turbine (the kinetic energyof the exhaust can be converted into useful work by the following stage) andcases where the turbine exhaust is used to generate thrust, such as in aturbojet. For most power-generating applications, the turbine is rated usingstatic exit pressure, since the exit kinetic energy is usually dissipated in theatmosphere. Note that the total-to-static efciency will be lower than thetotal-to-total efciency since the static exit pressure is lower than the total.

With the energy available to the turbine established by the inlet andexit conditions, lets take a closer look at the actual mechanism of energytransfer within the turbine. Figure 7 shows a generalized turbine rotor. Flowenters the upstream side of the rotor at point 1 with velocity V

!1 and exits

from the downstream side at point 2 with velocity V!

2. The rotor spins aboutits centerline coincident with the x-axis with rotational velocity o. Thelocation of points 1 and 2 is arbitrary (as long as they are on the rotor), asare the two velocity vectors. The velocity vectors are assumed to represent


the average for the gas owing through the turbine. The net torque G actingon the rotor can be represented as the difference of two torques on eitherside of the rotor:

G r1Fy1 r2Fy2 3

where Fy is the force in the tangential direction and r is the radius to thepoint. From Newtons second law, the tangential force is equal to the rate of

Figure 7 Velocities at the inlet and exit of a turbine rotor.


change of angular momentum:

Fy dmVydt

4

Performing the derivative, assuming constant Vy and mass ow rate _mm,results in

G _mmVy1r1 Vy2r2 5The energy transfer per time (power) is obtained by multiplying both sidesof Eq. (5) by the rotational velocity o:

P Go o _mmVy1r1 Vy2r2 6The power P can also be calculated from the hs diagram for the actualprocess as

P _mmhin hdis _mmDhactual 7Combining Eqs. (6) and (7) and dening the wheel speed U as

U or 8results in the Euler equation for energy transfer in a turbomachine:

Dhactual U1Vy1 U2Vy2 9The Euler equation, as derived here, assumes adiabatic ow through theturbine, since enthalpy change is allowed only across the rotor. The Eulerequation relates the thermodynamic energy transfer to the change invelocities at the inlet and exit of the rotor. This leads us to examine thesevelocities more closely, since they determine the work extracted from theturbine.

Velocity Diagrams

Eulers equation shows that energy transfer in a turbine is directly related tothe velocities in the turbine. It is convenient to graphically display thesevelocities at the rotor inlet and exit in diagrams called velocity or vectordiagrams. These diagrams are drawn in a single plane. For an axial-owturbine, they are drawn in the xy plane at a specic value of r. At the inletof a radial-inow turbine, where the ow is generally in the ry plane, thediagram is drawn in that plane at a specic value of x. The exit diagram fora radial-inow turbine is drawn in the xy plane at a specic value of r.

Figure 8(a) shows the velocity diagram at the inlet to an axial-owrotor. The stator and rotor blade shapes are included to show the relation


between the velocity diagram and the physical geometry of the turbine. Theow leaves the stator at an angle of a1 from the axial direction. The velocityvector V

!1 can be broken into two components, Vx1 in the axial direction

and Vy1 in the tangential direction. Note that the turbine work is controlledby the tangential component, while the turbine ow rate is controlled by theaxial component (for an axial-ow turbine). The vector V

!1 is measured in

an absolute, nonrotating reference frame and is referred to as the absoluterotor inlet velocity. Likewise, the angle a1 is called the absolute ow angle atrotor inlet. A rotating reference frame can also be xed to the rotor.Velocities in this reference are determined by subtracting the rotor velocityfrom the absolute velocity. Dening the relative velocity vector at the inletto be W

!1, we can write

W

!

1 V!1 U1 10

The vector notation is not used for the rotor velocity U1 as it is always in thetangential direction. The relative velocity vector is also shown in Fig. 8(a).The relative ow angle b1 is dened as the angle between the relative velocityvector and the axial direction. Inspection of the diagram of Fig. 8(a) reveals

Figure 8 Velocity diagrams for an axial-ow turbine: (a) rotor inlet velocitydiagram, (b) rotor exit velocity diagram.


several relationships between the relative velocity and absolute velocity andtheir components:

V21 V2x1 V2y1 11W21 W2x1 W2y1 12Wy1 Vy1 U1 13Wx1 Vx1 14

The sign convention used here is that tangential components in the directionof the wheel speed are positive. This implies that both a1 and b1 are positiveangles. Figure 8(b) shows the vector diagram at the outlet of the rotor. Notethat in this diagram, both the absolute and relative tangential componentsare opposite the direction of the blade speed and are referred to as negativevalues. The two angles are also negative.

In addition to the relative velocities and ow angles, we can also deneother relative quantities such as relative total temperature and relative totalpressure. In the absolute frame of reference, the total temperature is denedas

T 0 T V2

2cp15

In the relative frame of reference, the relative total temperature T 00 is denedas

T 00 T W2

2cp16

The static temperature is invariant with regard to reference frame.Combining Eqs. (15) and (16), we have

T 00 T 0 W2 V22cp

17

The relative total temperature is the stagnation temperature in the rotatingreference; hence it is the temperature that the rotor material is subjected to.Equation (17) shows that if the relative velocity is lower than the absolutevelocity, the relative total temperature will be lower than the absolute. Thisis an important consideration to the mechanical integrity of the turbine.

As with the static temperature, the static pressure is also invariant withreference frame. The relative total pressure can then be calculated from the


gas dynamics relation

p00

p T

00

T

g=g118

Types of Velocity Diagrams

There are an innite number of variations of the velocity diagrams shown inFig. 8. To help distinguish and classify them, the vector diagrams areidentied according to reaction, exit swirl, stage loading, and owcoefcient. The reaction is the ratio of the change in static enthalpy acrossthe rotor to the change in total enthalpy across the stage. In terms ofvelocities, the change in total enthalpy is given by Eq. (9). The change instatic enthalpy (denoted as hs) can be found from

hs1 hs2 h1 V21

2

h2 V

22

2

U1Vy1 U2Vy2 1

2V21 V22 19

Geometric manipulation of the vector diagram of Fig. 8 results in

UVy 12V2 U2 W2 20

Applying to Eqs. (9) and (19), the stage reaction can be expressed as

Rstg U21 U22 W21 W22

V21 V22 U21 U22 W21 W22 21

Stage reaction is normally held to values greater than or equal to 0. For anaxial-ow turbine with no change in mean radius between rotor inlet androtor outlet, U1 U2 and the reaction is controlled by the change in relativevelocity across the rotor. Negative reaction implies that W1 > W2,indicating diffusion occurs in the rotor. Due to the increased boundary-layer losses and possible ow separation associated with diffusion, negativereaction is generally avoided. Diagrams with zero reaction (no change inmagnitude of relative velocity across the rotor) are referred to as impulsediagrams and are used in turbines with large work extraction. Diagramswith reactions greater than 0 are referred to as reaction diagrams. Stagereaction is usually limited to about 0.5 due to exit kinetic energyconsiderations.


Exit swirl refers to the value of Vy2. For turbines discharging toambient, the most efcient diagram has zero exit swirl. While a negativevalue of exit swirl increases the work extraction, the magnitude of theturbine discharge velocity increases, leading to a larger difference betweenthe exit static and total pressures. For turbines rated on exit static pressure,the tradeoff between increased work and lower exit static pressure results inlower efciency levels. Most turbines operating in air with pressure ratios of3:1 or less use zero exit swirl vector diagrams.

The stage loading is measured by the loading coefcient l. The loadingcoefcient is dened here as

l DhactualU2

22

which can also be written as

l DVyU

23

for turbines with no change in U between inlet and outlet. The loadingcoefcient is usually calculated for an axial-ow turbine stage at either thehub or mean radius. For a radial-inow turbine, the rotor tip speed is usedin Eq. (23).

The stage ow is controlled by the ow coefcient, dened as

f VxU

24

These four parameters are related to each other through the vector diagram.Specication of three of them completely denes the vector diagram.

Figure 9 presents examples of a variety of vector diagrams, with exitswirl, reaction, and loading coefcient tabulated. Figure 9(a) shows a vectordiagram appropriate for an auxiliary turbine application, with relativelyhigh loading (near impulse) and zero exit swirl. A diagram more typical of astage in a multistage turbine is shown in Fig. 9(b), since the exit kineticenergy can be utilized in the following stage, the diagram does showsignicant exit swirl. Both Fig. 9(a) and 9(b) are for axial turbines; 9(c) is thevector diagram for a radial-inow turbine. The major difference is thechange in U between the inlet and exit of the turbine.

Turbine Losses

The difference between the ideal turbine work and the actual turbine work ismade up of the losses in the turbine. The losses can be apportioned to each


Figure 9 Variations in turbine velocity diagrams: (a) axial-ow diagram for single-stage auxiliary turbine, (b) axial-ow diagram for one stage in multistage turbine, (c)

radial-inow turbine velocity diagram.


component so that we may write

Dhideal Dhactual Linlet Lstator Lrotor Ldiffuser Lexit 25where the L terms represent losses in enthalpy in each component. Lossescan also be looked at from a pressure viewpoint. An ideal exit pressure canbe determined from

Dhactual cpT 0in 1pdisideal

p0in

g1=g" #26

The component losses are then represented as losses in total pressure, thesum of which is equal to the difference between the actual and ideal exitpressure:

Pdis pdisideal Dp0inlet Dp0stator Dp00rotor Dp0diffuser Dp0exit 27Most loss models incorporate the pressure loss concept.

Inlet Losses

Losses in inlets are usually modeled with a total pressure loss coefcient Ktdened as

Dp0inlet Ktinlet1

2rV2inlet

28

Where

Vinlet velocity at the upstream end of the inlet:r density of the working fluid:

The losses in an inlet primarily arise from frictional and turning effects.Within packaging constraints, the inlet should be made as large as possibleto reduce velocities and minimize losses. Axial inlets such as that of Fig. 2(a)have low frictional losses (due to their short length and relatively lowvelocities), but often suffer from turning losses due to ow separation alongtheir outer diameter. Longer axial inlets with more gradual changes in outerdiameter tend to reduce the turning losses and prevent separation, butadversely impact turbine envelope. Tangential entry inlets tend to havehigher losses due to the tangential turning and acceleration of the ow. Thespiral ow path also tends to be longer, increasing frictional losses.Typically, loss coefcients for practical axial inlets are in the range of 0.5 to2.0, while tangential inlets are in the range of 1.0 to 3.0. In terms of inlet


pressure, inlet losses are usually on the order of 13% of the inlet totalpressure. For turbines in engines, there is usually no real inlet, as they areclosely coupled to the combustor or the preceding turbine stage. In this case,the duct losses are usually assessed to the upstream component.

Stator Losses

The stator losses arise primarily from friction within the vane row, thesecondary ows caused by the ow turning, and exit losses due to blockageat the vane row trailing edge. The stator loss coefcient can be dened inseveral ways. Two popular denitions are

Dp0stator Ystator1

2rV21

29

or

Dp0stator Ystator1

2r

V20 V212

30

In either case, the loss coefcient is made up of the sum of coefcients foreach loss contributor:

Ystator Yprofile Ysecondary Ytrailing edge 31Prole refers to frictional losses. There can be additional loss contributionsdue to incidence (the ow coming into the stator is not aligned with theleading edge), shock losses (when the stator exit velocity is supersonic), andothers. Much work has been dedicated to determining the proper values forthe coefcients, and several very satisfactory loss model systems have beendeveloped. As loss models differ for axial-ow and radial-inow turbines,these models will be discussed in the individual sections that follow.

Rotor Losses

Rotor losses are modeled in a manner similar to that for stators. However,the pressure loss is measured as a difference in relative total pressures andthe kinetic energy is based on relative velocities. As with stators, the rotorloss is based on either the exit relative kinetic energy or the average of theinlet and exit relative kinetic energies:

Dp00rotor Xrotor1

2rW22

32


or

Dp00rotor Xrotor1

2r

W21 W222

33

With rotors, incidence loss can be signicant, so we include that contributorin the expression for the rotor loss coefcient:

Xrotor Xprofile Xsecondary Xtrailing edge Xincidence 34Other losses associated with the rotor are tip clearance and windage losses.Turbine rotors operate with a small clearance between the tips of the bladesand the turbine housing. Flow leaks across this gap from the high-pressureside of the blade to the low-pressure side, causing a reduction in the pressuredifference at the tip of the blade. This reduces the tangential force on theblade, decreasing the torque delivered to the shaft. Tip clearance effects canbe reduced by shrouding the turbine blades with a ring, but thisintroduces manufacturing and mechanical integrity challenges. The lossassociated with tip clearance can be modeled either using a pressure losscoefcient or directly as a reduction in the turbine efciency. The specicmodels differ with turbine type and will be discussed in following sections.

Windage losses arise from the drag of the turbine disk. As the diskspins in the housing, the no-slip condition on the rotating surface inducesrotation of the neighboring uid, establishing a radial pressure gradient inthe cavity. This is commonly referred to as disk pumping. For low-headturbines operating in dense uids, the windage losses can be considerable.Windage effects are handled by calculating the windage torque from a diskmoment coefcient dened as

Gwindage 2Cm12ro2r5disk

35

The output torque of the turbine is reduced by the windage torque. Valuesof the moment coefcient Cm depend on the geometry of the disk cavity andthe speed of the disk. Nece and Daily [46] are reliable sources of momentcoefcient data.

Diffuser Losses

Losses in the diffuser arise from sources similar to those in other owpassages, namely, friction and ow turning. The diffuser loss can beexpressed in terms of a loss coefcient for accounting in turbineperformance, but diffuser performance is usually expressed in terms of


diffuser recovery, dened as

Rp pdis p2p02 p2

36

The diffuser recovery measures how much of the kinetic energy at diffuserinlet is converted to a rise in static pressure. Recovery is a function of arearatio Adis=A2, length, and curvature. For an ideal diffuser of innite arearatio, the recovery is 1.0. Peak recovery of a real diffuser of given lengthtakes place when the area ratio is set large enough so that the ow is on theverge of separating from the walls of the diffuser. When the ow separateswithin the diffuser, the diffuser is said to be stalled. Once stalled, diffuserrecovery drops dramatically. Curvature of the mean radius of the diffusertends to decrease the attainable recovery, since the boundary layer on one ofthe diffuser walls is subjected to a curvature-induced adverse pressuregradient in addition to the adverse pressure gradient caused by the increasein ow area.

Even with the recent advances in general-use computational uiddynamics (CFD) tools, analytical prediction of diffuser recovery is notnormally performed as part of the preliminary turbine design. Diffuserperformance is normally obtained from empirically derived plots such asthat shown in Fig. 10. Diffuser recovery is plotted as a function of area ratioand diffuser length. The curvature of the contours of recovery shows thelarge fall-off in diffuser recovery after the diffuser stall. The locus ofmaximum recovery is referred to as the line of impending stall. Diffusersshould not be designed to operate above this line. Runstadler et al. [7, 8] andSovran and Klomp [9] present charts of diffuser recovery as a function ofinlet Mach number and blockage, as well as the three geometric factorsnoted earlier.

The total pressure loss across a diffuser operating in incompressibleow can be calculated using continuity and the denition of diffuserrecovery. The recovery for an ideal diffuser (no total pressure loss) is givenby

Rpideal 1A2

Adis

237

The total pressure loss for a nonideal diffuser in incompressible ow is givenby

p02 p0dis12rV22

Ktdiff Rpideal Rp 38


This can be used to calculate the diffuser loss when compressibility is notimportant. If the Mach number at the inlet to the diffuser is above 0.20.3,this can be used as a starting guess, and the actual value can be determinedby iteration. The diffuser recovery is a function of the inlet Mach number,blockage, and geometry (straight, curved, conical, or annular); it is criticalto use the correct diffuser performance chart when estimating diffuserrecovery.

Exit Losses

Exit losses are quite simple. If the kinetic energy of the ow exiting thediffuser is used in following stages, or contributes to thrust, the exit lossesare zero. If, however, the diffuser discharge energy is not utilized, the exitloss is the exit kinetic energy of the ow. For this case,

Dp0exit p0dis pdis 39

Nondimensional Parameters

Turbine performance is dependent on rotational speed, size, working uid,enthalpy drop or head, and ow rate. To make comparisons between

Figure 10 Conical diffuser performance chart. (Replotted from Ref. 8).


different turbines easier, dimensional analysis leads to the formation ofseveral dimensionless parameters that can be used to describe turbines.

Specic Speed and Specic Diameter

The specic speed of a turbine is dened as

Ns oQ2

p

Dhideal3=440

where Q2 is the volumetric ow rate through the turbine at rotor exit. Thespecic speed is used to relate the performance of geometrically similarturbines of different size. In general, turbine efciency for two turbines ofthe same specic speed will be the same, except for differences in tipclearance and Reynolds number. Maintaining specic speed of a turbine is acommon approach to scaling of a turbine to different ow rates.

The specic diameter is dened as

Ds dtipDhideal1=4

Q2p 41

where dtip is the tip diameter of the turbine rotor, either radial in-ow oraxial ow. Specic diameter and specic speed are used to correlate turbineperformance. Balje [3] presents extensive analytical studies that result inmaps of peak turbine efciency versus specic speed and diameter forvarious types of turbines. These charts can be quite valuable during initialturbine sizing and performance estimation.

Blade-Jet Speed Ratio

Turbine performance can also be correlated against the blade-jet speedratio, which is a measure of the blade speed relative to the ideal stator exitvelocity. Primarily used in impulse turbines, where the entire static enthalpydrop is taken across the stator, the ideal stator exit velocity, C0, is calculatedassuming the entire ideal enthalpy drop is converted into kinetic energy:

C0 2Dhideal

p42

The blade-jet speed ratio is then calculated from

U

C0 U

2Dhidealp 43

The value of blade speed at the mean turbine blade radius is typically used in


Eq. (43) for axial turbines; for radial-inow turbines, the rotor tip speed isused.

Reynolds Number

The Reynolds number for a turbine is usually dened as

Re rUtipdtipm

44

where m is the viscosity of the working uid. Sometimes odtip is substitutedfor Utip, resulting in a value twice that of Eq. (44). The Reynolds numberrelates the viscous and inertial effects in the uid ow. For mostturbomachinery operating on air, the Reynolds number is of secondaryimportance. However, when turbomachinery is scaled (either larger orsmaller), the Reynolds number changes, resulting in a change in turbineefciency. Glassman [1] suggests the following for adjusting turbine losses toaccount for Reynolds number changes:

1 Z0a1 Z0b

A B RebRea

0:245

where Z0 indicates total-to-total efciency and A and B sum to 1.0. That allthe loss is not scaled by the Reynolds number ratio reects that not all lossesare viscous in origin. Also, total-to-total efciency is used since the kineticenergy of the exit loss is not affected by Reynolds number. Glassman [1]suggests values of 0.30.4 for A (the nonviscous loss) and from 0.7 to 0.6 forB (the viscous loss).

Equivalent or Corrected Quantities

In order to eliminate the dependence of turbine performance maps on thevalues of inlet temperature and pressure, corrected quantities such ascorrected ow, corrected speed, corrected torque, and corrected power weredeveloped. Using corrected quantities, turbine performance can berepresented by just a few curves for a wide variety of operating conditions.Corrected quantities are not nondimensional. Glassman [1] provides adetailed derivation of the corrected quantities. The corrected ow is denedas

wcorr wy

p

d46


where

y T0in

TSTD47

and

d p0in

pSTD48

The standard conditions are usually taken to be 518.7R and 14.7 psia.Corrected speed is dened as

Ncorr Ny

p 49

Equation (5) shows torque to be the product of ow rate and the change intangential velocity across the rotor. Corrected ow is dened above;corrected velocities appear with y1=2 in the denominator from the correctedshaft speed. Therefore, corrected torque is dened as

Gcorr Gd 50

The form of the corrected power is determined from the product ofcorrected torque and corrected speed:

Pcorr Pdy

p 51

These corrected quantities are used to reduce turbine performance data tocurves of constant-pressure ratio on two charts. Figure 11 presents typicalturbine performance maps using the corrected quantities. Figure 11(a)presents corrected ow as a function of corrected speed and pressure ratio,while Fig. 11(b) shows corrected torque versus corrected speed and pressureratio. Characteristics typical of both radial-inow and axial-ow turbinesare presented in Fig. 11.

AXIAL-FLOW TURBINE SIZING

Axial-Flow Turbine Performance Prediction

Prediction methods for axial-ow turbine performance methods can beroughly broken into two groups according to Sieverding [10]. The rstgroup bases turbine stage performance on overall parameters such as workcoefcient and ow coefcient. These are most often used in preliminary


Figure 11 Typical performance maps using corrected quantities for axial-ow andradial-inow turbines: (a) corrected ow vs. pressure ratio and corrected speed; (b)

corrected torque vs. pressure ratio and corrected speed.


sizing exercises where the details of the turbine design are unknown. Smith[11] and Soderberg [12] are both examples of this black box approach, asare Baljes [3] maps of turbine efciency as a function of specic speed andspecic diameter.

The second grouping is based on the approach outlined earlier whereturbine losses are broken down to a much ner level. In these methods, alarge number of individual losses are summed to arrive at the total loss.Each of these loss components is dependent on geometric and aerodynamicparameters. This requires more knowledge of the turbine conguration,such as ow path and blading geometry, before a performance estimate canbe made. As such, these methods are better suited for more detailed turbinedesign studies.

Among the loss component methods, Sieverding [10] gives an excellentreview of the more popular component loss models. The progenitor of afamily of loss models is that developed by Ainley and Mathieson [13]. It hasbeen modied and improved by Dunham and Came [14] and, morerecently, by Kacker and Okapuu [15]. A somewhat different approach istaken by Craig and Cox [16]. All these methods are based on correlations ofexperimental data.

An alternate approach is to analytically predict the major losscomponents such as prole or friction losses and trailing-edge thicknesslosses by computing the boundary layers along the blade surfaces. Prolelosses are then computed from the momentum thickness of the boundarylayers on the pressure and suction surfaces of the blades or vanes. Glassman[1] gives a detailed explanation of this method. Note that this techniquerequires even more information on the turbine design; to calculate theboundary layer it is necessary to know both the surface contour and thevelocities along the blade surface. Thus, this method cannot be used untilblade geometries have been completely specied and detailed ow channelcalculations have been made.

In addition to the published prediction methods just noted, each of themajor turbine design houses (such as AlliedSignal, Allison, General Electric,Lycoming, Pratt & Whitney, Sundstrand, and Williams) has its ownproprietary models based on a large turbine performance database. Ofcourse, it is not possible to report those here.

For our purposes (determining the size and approximate performanceof a turbine) we will concentrate on the overall performance predictionmethods, specically Smiths chart and Soderbergs correlation. Figure 12shows Smiths [11] chart, where contours of total-to-total efciency areplotted versus ow coefcient and work factor [see Eqs. (23) and (24)]. Boththe ow coefcient and stage work coefcient are dened using velocities atthe mean radius of the turbine. The efciency contours are based on the


measured efciency for 70 turbines. All the turbines have a constant axialvelocity across the stage, zero incidence at design point, and reactionsranging from 20% to 60%. Reynolds number for the turbines range from100,000 to 300,000. Aspect ratio (blade height to axial chord) for the testedturbines is in the range of 34. Smiths chart does not account for the effectsof blade aspect ratio, Mach number effects, or trailing-edge thicknessvariations. The data have been corrected to reect zero tip clearance, so theefciencies must be adjusted for the tip clearance loss of the application.

Sieverding [10] considers Soderbergs correlation to be outdated butstill useful in preliminary design stages due to its simplicity. In Soderbergs[12] correlation, blade-row kinetic energy losses are calculated from

Vo2ideal V2oV2o

x

105

Rth

1=41 xref 0:975 0:075

cx

h 1

h in o52

Figure 12 Smiths chart for stage zero-clearance total-to-total efciency asfunction of mean-radius ow and loading coefcient. (Replotted from Ref. 11

with permission of the Royal Aeronautical Society).


where Rth is the Reynolds number based on the hydraulic diameter at theblade passage minimum area (referred to as the throat) dened as

Rth roVomo2hs cosaoh s cosao 53

where h is the blade height and s is the spacing between the blades at themean radius. The blade axial chord is identied by cx. In both Eqs. (52) and(53), the subscript o refers to blade-row outlet conditions, either stator orrotor (for the rotor, the absolute velocity V is replaced by the relativevelocity W, standard practice for all blade-row relations). The referenceloss coefcient xref is a function of blade turning and thickness and can befound in Fig. 13. Compared to Smiths chart, this correlation requires moreknowledge of the turbine geometry, but no more than would be required ina conceptual turbine design. The losses predicted by this method are onlyvalid for the optimum blade chord-to-spacing ratio and for zero incidence.Tip clearance losses must also be added in the nal determination of turbineefciency. Like Smiths chart, this correlation results in a total-to-totalefciency for the turbine.

The optimum value of blade chord-to-spacing ratio can be found usingthe denition of the Zweifel coefcient [17]:

z 2cx=s

cos aocos ai

sinai ao

54

where the subscript i refers to blade-row inlet. Zweifel [17] states thatoptimum solidity cx=s occurs when z 0:8.

Tip clearance losses are caused by ow leakage through the gapbetween the turbine blade and the stationary shroud. This ow does not getturned by the turbine blade; so it does not result in work extraction. Inaddition, the ow through the clearance region causes a reduction of thepressure loading across the blade tip, further reducing the turbine efciency.The leakage ow is primarily controlled by the radial clearance, but is alsoaffected by the geometry of the shroud and the blade reaction. Leakageeffects can be reduced by attaching a shroud to the turbine blade tips, whicheliminates the tip unloading phenomenon. For preliminary design purposes,the tip clearance loss for unshrouded turbine wheels can be approximated by

ZZzc

1 Kc rtiprmean

cr

h55


Where

Zzc zero clearance efficiency:or radial tip clearance:rtip passage tip radius:

rmean mean passage radius:Kc empirically derived constant:

Based on measurements reported by Haas and Kofskey [18], the value of Kcis between 1.5 and 2.0, depending on geometric conguration. Forpreliminary design purposes, the conservative value should be used. Whenusing Soderbergs correlation, the value of Kc should be taken as 1, sinceSoderberg corrected his data using that value for Kc.

With the information above, the turbine efciency (total-to-total) canbe determined from the stator inlet (station 0) to rotor exit (station 2). In

Figure 13 Soderbergs loss coefcient as function of deection angle and bladethickness. (Replotted from Ref. 12 with permission from Pergamon Press Ltd.)


order to determine the overall turbine efciency, it is necessary to include theinlet, diffuser, and exit losses. These losses do not affect the turbine workextraction, but result in the overall pressure ratio across the turbine beinglarger than the stage pressure ratio. The overall efciency can be calculatedfrom

ZOA Z0020 1 p02=p00g1=g1 pdis=p0ing1=g

56

The pressure losses in the inlet, diffuser, and exit are calculated from theinformation presented earlier.

Mechanical, Geometric, and Manufacturing Constraints

Turbine design is as much or more affected by mechanical considerations asit is by aerodynamic considerations. Aerodynamic performance is normallyconstrained by the stress limitations of the turbine material. At this point inthe history of turbine design, turbine performance at elevated temperaturesis limited by materials, not aerodynamics. Material and manufacturinglimitations affect both the geometry of the turbine wheel and its operatingconditions.

Turbine blade speed is limited by the centrifugal stresses in the diskand by the tensile stress at the blade root (where the blade attaches to thedisk). The allowable stress limit is affected by the turbine material, turbinetemperature, and turbine life requirements. Typical turbine materials foraircraft auxiliary turbines are titanium in moderate-temperature applica-tions (turbine relative temperatures below 1,000 8F) and superalloys forhigher temperatures.

Allowable blade-tip speed for axial-ow turbines is a complex functionof inlet temperature, availability of cooling air, thermal cycling (low cyclefatigue damage), and desired operating life. In general, design point bladespeeds are held below 2,200 ft/sec, but higher blade speeds can be withstoodfor shorter lifetimes, if temperatures permit. For auxiliary turbineapplications with inlet temperatures below 300 8F and pressure ratios of 3or below, blade speed limits are generally not a design driver.

Both stress and manufacturing considerations limit the turbine bladehub-to-tip radius ratio to values greater than about 0.6. If the hub diameteris much smaller, it is difcult to physically accommodate the requirednumber of blades on the hub. Also, the twist of the turbine blade increases,leading to sections at the tip not being directly supported by the hub section.This leads to high bending loads in the blade and higher stress levels. For


performance reasons (secondary ow losses and tip clearance losses), it isdesirable to keep the hub-to-tip radius ratio below 0.8.

Manufacturing considerations limit blade angles on rotors to less than608 and stator vane exit angles to less than 758. Casting capabilities limitstator trailing-edge thickness tte to no less than 0.015 in., restricting statorvane count. For performance reasons, the trailing-edge blockage should bekept less than 10% at all radii. The trailing-edge blockage is dened here asthe ratio of the trailing-edge tangential thickness (b) to the blade or vanespacing (s):

b

s tte= cos ate

2pr=Z57

where Z is the blade or vane count. Rotor blades are usually machined, butfor stress and tolerance reasons the trailing-edge thickness is normally noless than 0.015 in. The 10% limitation on blockage is also valid for rotors.

Auxiliary turbines often are required to survive free-run conditions.Free run occurs when the turbine load is removed but the air supply is not.This can happen if an output shaft fails or if an inlet control valve fails toclose. Without any load, the turbine accelerates until the power output ofthe turbine is matched by the geartrain and aerodynamic losses. Free-runspeed is roughly twice design-point speed for most aircraft auxiliaryturbines. This restricts the allowable design-point speeds and stress levelsfurther, since the disk and blade may be required to survive free-runoperation.

Hub-to-Tip Variation in Vector Diagram

Up to this point we have only considered the vector diagram at the meanradius of the turbine. For turbines with high hub-to-tip radius ratios (above0.85), the variation in vector diagram is not important. For a turbine withrelatively tall blades, however, the variation is signicant.

The change in vector diagram with radius is due to the change in bladespeed and the balance between pressure and body forces acting on theworking uid as it goes through the turbine. Examples of body forcesinclude the centrifugal force acting on a uid element that has a tangentialvelocity (such as between the stator and rotor), and the accelerations causedby a change in ow direction if the ow path is curved in the meridionalplane. The balance of these forces (body and pressure) is referred to as radialequilibrium. Glassman [1] presents a detailed mathematical development ofthe equations that govern radial equilibrium. For our purposes, we willconcentrate on the conditions that satisfy radial equilibrium.


The classical approach to satisfying radial equilibrium is to use a freevortex variation in the vector diagram from the hub to the tip of the rotorblade. A free vortex variation is obtained by holding the product of theradius and tangential velocity constant rVy constant. When this is done,the axial velocity Vx is invariant with radius. Until the widespread use ofcomputers in turbine design, almost all turbines employed free vortexdiagrams due to their simplicity. For preliminary design purposes, the freevortex diagram is more than satisfactory.

Aside from its simplicity, the free vortex diagram has otheradvantages. Holding rVy constant implies that the work extraction isconstant with radius. With Vx constant, the mass ow varies little withradius. This implies that the mean section vector diagram is an excellentrepresentation of the entire turbine from both a work and mass owstandpoint.

When using a free vortex distribution, there are two key items toexamine in addition to the mean vector diagram. The hub diagram suffersfrom low reaction due to the increase in Vy and should be checked to ensureat least a zero value of reaction. From hub to tip, the reduction in Vy andincrease in U cause a large change in the rotor inlet relative ow angle, withthe rotor tip section tending to overhang the hub section. By choosing amoderate hub-to-tip radius ratio (if possible), both low hub reaction andexcessive rotor blade twist can be avoided.

For a zero exit swirl vector diagram, some simple relations can bedeveloped for the allowable mean radius work coefcient and the hub-to-tiptwist of the rotor blade. For a zero exit swirl diagram, zero reaction occursfor a work coefcient of 2.0. Using this as an upper limit at the hub, thework coefcient at mean radius is found from

lm 2 rhrm

258

For a turbine with a hub-to-tip radius ratio of 0.7, the maximum workcoefcient at mean radius for impulse conditions at the hub is 1.356. Thedeviation in inlet ow angle to the rotor from hub to tip for a free vortexdistribution is given by

Db1 b1h b1t

tan1 lmrm=rh2 1

fmrm=rh

" # tan1 lmrm=rt

2 1fmrm=rt

" #59

For a vector diagram with lm 1:356; rh=rt 0:7, and


fm 0:6;Db1 56:1, which is acceptable from a manufacturing viewpoint.Large negative inlet angles at the blade tip are to be avoided.

An Example of Turbine Sizing

In order to demonstrate the concepts described in this and precedingsections, an example is presented of the sizing of a typical auxiliaryturbine for use in an aircraft application. The turbine is to be sized to meetthe following requirements:

1. Generates 100 hp at design point.2. Operates at an overall pressure ratio of 3:1 in air.3. Inlet pressure is 44.1 psia, and inlet temperature is 300 8F.

The object of this exercise is to determine the turbine size, ow rate,and operating speed with a turbine design meeting the mechanical,geometric, and manufacturing constraints outlined earlier. The followingprocedure will be followed to perform this exercise:

1. Determine available energy (isentropic enthalpy drop).2. Guesstimate overall efciency to calculate ow rate.3. Select the vector diagram parameters.4. Calculate the vector diagram.5. Determine the rotor overall geometry.6. Calculate the overall efciency based on Smiths chart both with

and without a diffuser.

The process is iterative in that the efciency determined in step 6 is then usedas the guess in step 2, with the process repeated until no change is found inthe predicted efciency. We will also predict the turbine efciency usingSoderbergs correlation.

The rst step is to calculate the energy available to the turbine usingEq. (1). For air, typical values for the specic heat and the ratio of specicheats are 0:24Btu=lbm R and 1.4, respectively. It is also necessary toconvert the inlet temperature to the absolute scale. We then have

Dhisentropic 0:24 Btulbm R

760R 1 1

3

0:4=1:4" # 49:14Btu

lbm

Note that more digits are carried through the calculations than indicated, soexact agreement may not occur in all instances. The vector diagram iscalculated using the work actually done by the blade row; therefore, we needto start with a guess to the overall efciency of the turbine. A good startingpoint is usually an overall efciency of 0.8, including the effects of tip


clearance. Since tip clearance represents a loss at the tip of the blade, the restof the blade does more than the average work. Therefore, the vectordiagram is calculated using the zero-clearance efciency. Since we do notknow the turbine geometry at this point, we must make another assumption:we assume that the tip clearance loss is 5%, so that the overall zero-clearanceefciency is 0.84. Note that the required ow rate is calculated using theoverall efciency with clearance, since that represents the energy available atthe turbine shaft. Equation (2) is used to calculate the actual enthalpy drops:

DhOA 0:8 49:14 Btulbm

39:31 Btu

lbm

and

DhOA ZC 0:84 49:14 Btulbm

41:28 Btu

lbm

The required turbine ow is found using Eq. (7):

_mm PDhOA

100 hp:7069Btu=sec=hp39:31Btu=lbm

1:798 lbm= sec

The mass ow rate is needed to calculate turbine ow area and is also asystem requirement.

We specify the vector diagram by selecting values of the turbine workand ow coefcients. We also select a turbine hub-to-tip radius ratio of 0.7,restricting the choice of mean work coefcient to values less than 1.356 inorder to avoid negative reaction at the hub. From Smiths chart (Fig. 12), weinitially choose a work coefcient of 1.3 and a ow coefcient of 0.6 to resultin a zero-clearance, stator inlet to rotor exit total-to-total efciency of 0.94.We apply these coefcients at the mean radius of the turbine. From Eq. (22)we calculate the mean blade speed, Um:

Um DhOA ZC

l

r

32:174 ft lbf=lbm sec2778:16 ft lbf=Btu41:28Btu=lbm

1:3

r 891:6 ft= sec

The axial velocity is calculated from Eq. (51):

Vx2 0:6891:6 ft= sec 535:0 ft= sec

In order to construct the vector diagram, we make two more assumptions:(1) there is zero swirl leaving the turbine stage in order to minimize the exitkinetic energy loss, and (2) the axial velocity is constant through the stage.


By assuming that Vy2 is zero, Eq. (23) reduces to

Vy1 lUm 1:3891:6 ft= sec 1159:1 ft= sec

Using Eqs. (10) through (14) results in the vector diagram shown in Fig. 14.Note that the critical stator and rotor exit angles are within the guidelinespresented earlier.

The rotor blade height and mean radius are determined by therequired rotor exit ow area and the hub-to-tip radius ratio. The rotor exitow area is determined from continuity:

A2 r2Vx2_mm

The mass ow rate and axial velocity have previously been calculated;the density is dependent on the rotor exit temperature and pressure. For aturbine without a diffuser, the rotor exit static pressure is the same as thedischarge pressure, assuming the rotor exit annulus is not choked. For a

Figure 14 Mean-radius velocity diagrams for rst iteration of axial-ow turbinesizing example.


turbine with an effective diffuser, the rotor exit static pressure will be lessthan the discharge value. We will examine both cases.

Turbine Without Diffuser

First we consider the turbine without a diffuser. Assuming perfect gasbehavior, the density is calculated from

r2 p2

RgasT2

where the temperature and pressure are static values and Rgas is the gasconstant. The rotor exit total temperature is determined from

T 02 T 00 DhOA ZC

Cp 760R 41:28Btu=lbm

0:24Btu=lbm R 588:0R

The zero-clearance enthalpy drop is used because the local tempera-ture over the majority of the blade will reect the higher work (a higherdischarge temperature will be measured downstream of the turbine aftermixing of the tip clearance ow has occurred). Next we calculate the rotorexit critical Mach number to determine the static temperature. The criticalsonic velocity is calculated from

acr 2g

g 1 gRgasT0

s

where g is a conversion factor. For air at low temperatures,Rgas 53:34 ft-lbf=lbm R, resulting in

acr2 21:41 1:4 32:174

ft lbflbm sec2

53:34

ft lbflbm R

588R

s 1085 ft=sec

The static temperature is found from

T2 T 02 1g 1g 1

V2

acr2

2" #

with zero exit swirl, V2 Vx2 resulting in

T2 588R 1 1:4 11 1:4

535 ft=sec

1085 ft=sec

2" # 564:2R


The density can now be determined:

r2 44:1 lbf=in

2

3

144 in

2

ft2

53:34 ftlbf

lbmR

564:2R 0:0703 lbm

ft3

and the required ow area:

A2 1:798 lbm=sec0:0703 lbm=ft3535:0 ft=sec144

in:2

ft2

6:882 in:2

The rotor exit hub and tip radii cannot be uniquely determined untileither shaft speed, blade height, or hub-to-tip radius ratio is specied. Onceone parameter is specied, the others are determined. For this example, wechoose a hub-to-tip ratio of 0.7 as a compromise between performance andmanufacturability. If the turbine shaft speed were restricted to a certainvalue or range of values, it would make more sense to specify the shaftspeed. The turbine tip radius is determined from

rt2

A2

p1 rh=rt2

s

6:882 in:2

p1 0:72

s 2:073 in:

This results in a hub radius of 1.451 in., a mean radius of 1.762 in. anda blade height of 0.622 in. The shaft speed is found from Eq. (8):

o Um=rm 891:6 ft=sec1:762 in1 ft=12 in 6073 rad=sec

or 57,600 rpm. The tip speed of the turbine is 1,049 ft/sec, well within ourguidelines.

The next step is to calculate the overall efciency. From Smiths chart,a stator inlet to rotor exit total-to-total efciency at zero clearance isavailable. We must correct this for tip clearance effects, the inlet loss, andthe exit kinetic energy loss. At l 1:3 and f 0:6, Smiths chart predicts

Z0020ZC 0:94Assuming a tip clearance of 0.015 in., the total-to-total efciency includingthe tip clearance loss is calculated from Eq. (55) using a value of 2 for Kc:

Z0020 Z0020ZC 1 2rt

rm

dh

0:94 1 2 2:073

1:762

0:015

0:622

0:8867

Equation (56) is used to determine the overall efciency including inlet andexit losses. From the problem statement, we know that the overall pressure


ratio p0in=pdis is 3. The stator inlet to rotor exit total-to-total pressure ratiois calculated from

p02p00

pdisp0in

p02pdis

p0inp00

Based on earlier discussions, we assume an inlet total pressure loss ratio of0.99. With no diffuser, the discharge and rotor exit stations are the same, sothe ratio of static to total pressure is found from the rotor exit Machnumber:

pdis

p02 p2p02

1 g 1g 1

V2

acr2

2" # gg1 1 1

6

535

1085

2" #3:5 0:8652

We can now calculate the total-to-total pressure ratio from stator inletto rotor exit and the overall efciency:

p02p00

13

1

0:8652

1

0:99

0:3891

and

ZOA 0:88671 0:38910:4=1:4

1 13 0:4=1:4 0:7779

This completes the rst iteration on the turbine size and performance for thecase without a diffuser. To improve the accuracy of the result, the precedingcalculations would be repeated using the new values of overall efciency andtip clearance loss.

Turbine with Diffuser

For an auxiliary type of turbine such as this, a diffuser recovery of 0.4 isreasonable to expect with a well-designed diffuser. The rotor exit totalpressure is calculated from the denition of diffuser recovery given in Eq.(35):

p02 pdis

Rp1 p2=p02 p2=p02 44:1 psia=30:41 0:8652 0:8652 15:99 psia

and the rotor exit static pressure is

p2 15:99 psia0:8652 13:84 psia


This is a considerable reduction compared to the discharge pressure of14.7 psia. From this point, the rotor exit geometry is calculated in the sameway as that presented for the case without the diffuser. The following resultsare obtained:

r2 0:0662 lbm=ft3A2 7:311 in:2rt2 2:136 in:rh2 1:495 in:rm2 1:816 in:h2 0:641 in:N 56;270 rpm

The tip speed is the same as the turbine without the diffuser, since the meanblade speed is unchanged, as is the hub-to-tip radius ratio of the rotor. Theefciency calculations also proceed in the same manner as the earlier casewith the following results (using the same inlet pressure loss assumption):

Z0020 0:8882p02p00

0:3663

ZOA 0:8224

Since this result differs from our original assumption for overallefciency, further iterations would be performed to obtain a more accurateanswer. Note the almost 6% increase in overall efciency due to theinclusion of a diffuser. This indicates a large amount of energy is containedin the turbine exhaust. The efciency gain associated with a diffuser isdependent on diffuser recovery, rotor exit Mach number, and overallpressure ratio and is easily calculated. Figure 15 shows the efciencybenet associated with a diffuser for an overall turbine pressure ratio(total-to-static) of 3. Efciency gains are plotted as a function of rotor exitcritical Mach number and diffuser recovery. As rotor exit Mach numberincreases, the advantages of including a diffuser become larger. Thistradeoff is important to consider when sizing the turbine. For a given owor power level, turbine rotor diameter can be reduced by accepting highrotor exit velocities (high values of ow coefcient); however, turbineefciency will suffer unless a diffuser is included, adversely impacting theaxial envelope.


Automation of Calculations and Trade Studies

The calculations outlined in this example can be easily automated in either acomputer program or a spreadsheet with iteration capability. An example ofthe latter is presented in Fig. 16, which contains the iterated nal results forthe example turbine when equipped with a diffuser. The advantage ofautomation is the capability to quickly perform trade studies to optimize theturbine preliminary design. Prospective variables for study include work andow coefcients, diffuser recovery, shaft speed or hub-to-tip radius ratio,inlet loss, tip clearance, exit swirl, and others.

Soderbergs Method

We conclude this example by calculating the turbine performance usingSoderbergs correlation. We will use the iterated turbine design resultsshown in the spreadsheet of Fig. 16. Soderbergs correlation [Eq. (52)]requires the vane and blade chords in order to calculate the aspect ratiocx=h. We rst determine the blade number by setting the blockage level atmean radius to 10% and the trailing-edge thickness for both the rotor and

Figure 15 Effect of diffuser on turbine efciency at an overall pressure ratio of 3.0.


stator at 0.020 in. These values are selected based on the guidelines givenearlier in the chapter. Solving Eq. (57) for the blade number results in

Z b=s2prmtte= cosate

For the stator, the ow angle a1 is used for ate; for the rotor, the relative owangle b2 is substituted for ate. The blade angle is slightly different from theow angle due to blockage effects, but for preliminary sizing, theapproximation is acceptable. For the stator, we have

Zstator 0:12p1:773 in0:020 in= cos65:22 23:35

Figure 16 Spreadsheet for preliminary axial-ow turbine sizing showing iteratedresults for example turbine.


and for the rotor

Zrotor 0:12p1:773 in0:020 in= cos59:04 28:65

Of course, only integral number of blades are allowed, so we choose 23vanes for the stator and 29 rotor blades, resulting in a blade spacing of0.484 in. for the stator and 0.384 in. for the rotor. Normal practice is toavoid even blade counts for both the rotor and stator to reduce rotor bladevibration response. The blade chord is now calculated from Zweifelsrelation given in Eq. (54) using the optimum value of 0.8 for the Zweifelcoefcient:

cx 2z=scosaocosai sinai ao

For the stator,

cxstator 2

0:8=0:484 in:cos65:22cos0 sin65:22

0:460 in:

and for the rotor,

cxrotor 2

0:8=0:384 in:cos59:04cos26:57 sin26:57

59:04

0:551 in:

The Reynolds number for each blade row is calculated from Eq. (53).At the exit of each blade row, the static temperature and pressure arerequired to calculate the density. The viscosity is calculated using the totaltemperature to approximate the temperature in the boundary layers whereviscous effects dominate. For the stator, the exit total temperature is thesame as the inlet temperature. We assume a 1% total pressure loss across thestator. Using the stator exit Mach number, the static pressure is calculated:

p1 44:1 psia0:990:99 1 161:05172

3:5 21:18 psia

as is the static temperature:

T1 760R 1 161:05172

619:9R

Using the perfect gas relation, the stator exit density r1 is calculated to be0:09225 lbm=ft

3. The viscosity is determined using an expression derived


from that presented by ASHRAE [19]:

m T

p

1:34103 306:288=T 13658:3=T2 1; 239; 069=T3

6106lbm

ft-sec

The original expression was in SI units. For the stator, the viscositym1 1:59986105 lbm=ft-sec. The Reynolds number is then calculated usingEq. (53):

Rthstator 0:09225 lbm

ft31297:4 ft

sec

1:59986105 lbmft-sec

20:626 in:0:484 in: cos65:2212 ft

in0:626 in: 0:484 in: cos65:22

resulting in a Reynolds number of 1:91036105. A similar procedure is usedfor the rotor, except the relative velocity and angle at the rotor exit (station2) are used. The viscosity is calculated using the relative total temperaturedetermined using Eq. (17). For the rotor, the Reynolds number is1:23566105.

The reference value of the loss coefcient x is found from Fig. 13 as afunction of the deection across the blade row. The deection is thedifference between the inlet and outlet ow angles. For the stator, thedeection is 65.228, and for the rotor it is 85.618, resulting in xref s 0:068and xref r 0:083, assuming a blade thickness ratio of 0.2. The adjusted losscoefcients are calculated from Eq. (52):

xstator 105

1:91036105

1=41 0:068 0:975 0:075 0:460

0:626

1

0:0852

and for the rotor

xrotor 105

1:23566105

1=41 0:083 0:975 0:075 0:551

0:626

1

0:1209

The stator inlet to rotor exit total-to-total efciency is calculated from theratio of the energy extracted from the ow UDVy divided by the sum of


the energy extracted and the rotor and stator losses:

Z0020ZC UDVy

UDVy 12V22xstator 12W22xrotorNumerically, we have

Z0020ZC 1:3906:15 ft

sec2

1:3906:15 ftsec2 0:08522 1297:39 ftsec2 0:12092 1056:75 ftsec2

0:8846which is considerably lower than the 0.94 value from Smiths chart.Correcting for tip clearance using a value of 1 for Kc in Eq. (55) yields

Z0020 0:8846 11

0:85

0:015

0:626

0:8597

and correcting for overall pressure ratio using the total-to-total pressureratio from Fig. 16 results in the overall efciency:

ZOA 0:85971 1=2:7201g1=g1 1=3:0g1=g

0:7935

This value is 0.025 lower than the value of 0.8187 from Fig. 16 predictedusing Smiths chart. Sieverding [10] notes that Smiths chart was developedfor blades with high aspect ratios (h=cx in the range of 34), which will resultin higher efciency than lower aspect ratios, such as in this example. Forpreliminary sizing purposes, the conservative result should be used.

Partial Admission Turbines

For applications where the shaft speed is restricted to low values or thevolumetric ow rate is very low, higher efciency can sometimes be obtainedwith a turbine stator that only admits ow to the rotor over a portion of itscircumference. Such a turbine is called a partial-admission turbine. Partial-admission turbines are indicated when the specic speed of the turbine islow. Balje [3] indicates partial admission to be desirable for specic speedsless than 0.1. Several conditions can contribute to low specic speed.Typically, drive turbines operate most efciently at shaft speeds higher thanthe loads they are coupled to, such as generators, hydraulic pumps, and, inthe case of an air turbine starter, the main engine of an aircraft. For low-costapplications, it may be desirable to eliminate the speed-reducing gearboxand couple the load directly to the turbine shaft. In order to attain adequate


blade speed at the reduced shaft speed, it is necessary to increase the turbinediameter, which causes the blade height to decrease. The short blades causean increase in secondary ow losses reducing turbine efciency. With partialadmission, the blade height can be increased, reducing secondary owlosses. In a low-ow-rate situation, maintaining a given hub-to-tip radiusratio results in an increase in the design shaft speed and a decrease in theoverall size of the turbine. However, manufacturing limits restrict the radialtip clearance and blade thickness. With a small blade height, tip clearancelosses are increased. With a limitation on how thin blades can be made, it isnecessary to reduce blade count in order to keep trailing-edge blockage to areasonable level. Fewer blades result in longer blade chord and reducedaspect ratio, leading to higher secondary ow losses. The taller bladesassociated with partial admission can increase turbine performance. Forhigh-head applications a high blade speed is necessary for peak efciency.With shaft speed restricted by bearing and manufacturing limitations, anincrease in turbine diameter is required, resulting in a situation similar to theno-gearbox case discussed earlier. Here, too, partial admission can result inimproved turbine efciency.

The penalty for partial admission is two additional losses not found infull-admission turbines. These are the pumping loss and sector loss. Thepumping loss accounts for the drag of the rotor blades as they pass throughthe inactive arc, the portion of the circumference not supplied with owfrom the stator. The sector loss arises from the decrease in momentumcaused by the mixing of the stator exit ow with the relatively stagnant uidoccupying the blade passage just as it enters the active arc. Instead of beingconverted into useful shaft work, the stator exit ow is used to acceleratethis stagnant uid up to the rotor exit velocity. An additional loss occurs atthe other end of the active arc as the blade passages leave the active zone.Just as a blade passage is at the edge of the last active stator vane passage,the ow into the rotor blade passage is reduced. This reduced ow has theentire blade passage to expand into. The sudden expansion causes a loss inmomentum resulting in decreased power output from the turbine. Lossmodels for partial-admission effects are not as well developed as those forconventional, full-admission turbines. As a historical basis, Glassman [1]presents Stodolas [20] pumping loss model and Stennings [21] sector lossmodel in an understandable form and discusses their use. More recently,Macchi and Lozza [22] have compiled a number of more modern lossmodels and exercised them during the design of partial-admission turbines.The reader is referred to those sources for detailed information regardingthe estimation of partial-admission losses.


RADIAL-INFLOW TURBINE SIZING

Differences Between Radial-Inow and Axial-Flow Turbines

Radial-inow turbines enjoy widespread use in automotive turbochargersand in small gas turbine engines (auxiliary power units, turboprops, andexpendable turbine engines). One advantage is their low cost relative tomachined axial turbines, as most of these applications use integrally bladedcast radial-inow turbine wheels.

The obvious difference between radial-inow and axial-ow turbines iseasily seen in Fig. 1; a radial-inow turbine has a signicant change in themean radius between rotor inlet and rotor outlet, whereas an axial-owturbine has only a minimal change in mean radius, if any. Because of thisgeometric difference, there are considerable differences in the performancecharacteristics of these two types of turbines. Referring to the typicalradial-inow vector diagram of Fig. 9(c), the radius change causes aconsiderable decrease in wheel speed U between rotor inlet and outlet. Forzero exit swirl, this results in a reduced relative exit velocity compared to anaxial turbine with the same inlet vector diagram (since U2&U1 for an axialrotor). Since frictional losses are proportional to the square of velocity, thisresults in higher rotor efciency for the radial-inow turbine. However, theeffect of reduced velocity level is somewhat offset by the long, low-aspect-ratio blade passages of a radial-inow rotor.

Compared to the axial-ow diagram of Fig. 9(a), there is a much largerdifference between the rotor inlet relative and absolute velocities for theradial-inow diagram. Referring to Eq. (17), this results in a lower relativeinlet total temperature at design point for the radial-inow turbine. Inaddition, due to the decrease in rotor speed with radius, the relative totaltemperature decreases toward the root of radial-inow turbine blades (seeMathis [23]). This is a major advantage for high inlet temperatureapplications, since material properties are strongly temperature-dependent.The combination of radial blades at rotor inlet (eliminating bending stressesdue to wheel rotation) and the decreased temperature in the high-stressblade root areas allows the radial-inow turbine to operate at signicantlyhigher wheel speeds than an axial-ow turbine, providing an appreciableincrease in turbine efciency for high-pressure-ratio, high-work applica-tions.

For applications with moderate inlet temperatures (less than 500 8F)and pressure ratios (less than 4:1), the blade speed of an axial wheel is notconstrained by stress considerations and the radial-inow turbine is at a sizedisadvantage. Due to bending stress considerations in the rotor blades,radial blades are used at the inlet to eliminate bending loads. This limits theVy1=U1 ratio to 1 or less, meaning that the tip speed for an equal work


radial-inow turbine will be higher than that for an axial-ow turbine,which can have Vy1=U1 > 1 with only a small impact on efciency. Thisassumes zero exit swirl. For a xed shaft speed, this means that the radial-inow turbine will be larger (and heavier) than an axial-ow turbine. Stagework can be increased by adding exit swirl; however, the radial-inowturbine is again at a disadvantage. The lower wheel speed at exit for theradial-inow turbine means that more Vy2 must be added for the sameamount of work increase, resulting in higher exit absolute velocitiescompared to an axial-ow turbine. In addition, high values of exit swirlnegatively impact obtainable diffuser recoveries.

Packaging considerations may lead to the selection of a radial-inowturbine. The outside diameter of a radial-inow turbine is considerablylarger than the rotor tip diameter, due to the stator and inlet scroll or torus.Compared to an axial-ow turbine, the radial-inow package diameter maybe twice as large or more. However, the axial length of the package istypically considerably less than for an axial turbine when the inlet anddiffuser are included. Thus, if the envelope is axially limited but large indiameter, a radial-inow turbine may be best suited for the application,considering performance requirements can be met.

For auxiliary turbine applications where free run may be encountered,radial-inow turbines have the advantage of lower free-run speed than anaxial turbine of comparable design-point performance. Figure 11 shows theoff-design performance characteristics of both radial-inow and axial-owturbines. At higher shaft speeds, the reduction in mass ow for the radial-inow turbine leads to lower torque output and a lower free-run speed.Because of the change in radius in the rotor, the ow through the rotor mustovercome a centrifugal pressure gradient caused by wheel rotation. As shaftspeed increases, this pressure gradient becomes stronger. For a given overallpressure ratio, this increases the pressure ratio across the rotor anddecreases the pressure ratio across the stator, leading to a reduced mass owrate. A complete description of this phenomenon and its effect on relativetemperature at free-run conditions is presented by Mathis [23]. However, therotor disk weight savings from the lower free-run speed of a radial-inowturbine is offset by the heavier containment armor required due to theincreased length of a radial-inow turbine rotor compared to an axialturbine.

Radial-Inow Turbine Performance

The literature on performance prediction and loss modeling for radial-inow turbines is substantially less than that for axial-ow turbines. Wilson[2] states that most radial-inow turbine designs are small extrapolations or


interpolations from existing designs and that new designs are executed usinga cut-and-try approach. Rodgers [24] says that minimal applicablecascade test information exists (such as that used to develop many of theaxial-ow turbine loss models) and that exact analytical treatment of theow within the rotor is difcult due to the strong three-dimensionalcharacter of the ow. Glassman [1] presents a description of radial-inowturbine performance trends based on both analytical modeling andexperimental results and also describes design methods for the rotor andstator blades. More recently, Rodgers [24] has published an empiricallyderived performance prediction method based on meanline quantities forradial-inow turbines used in small gas turbines. Balje [3] presents analyticalperformance predictions in the form of efciency versus specic speed andspecic diameter maps.

For our purposes, we will use the results of Kofskey and Nusbaum[25], who performed a systematic experimental study investigating the effectof specic speed on radial-inow turbine performance. Kofskey andNusbaum used ve different stators of varying ow area to cover a widerange of specic speeds (0.2 to 0.8). Three rotors were used in conjunctionwith these stators in an attempt to attain optimum performance at bothextremes of the specic speed range. Results of their testing are presented inFig. 17, which shows the maximum efciency envelopes for both total-to-total and total-to-static efciencies. These efciencies were measured fromscroll inlet ange to rotor exit and include the effects of tip clearance. Axialtip clearance was approximately 2.2% of the inlet blade height, while theradial tip clearance was about 1.2% of the exit blade height. Efcienciesabove 0.90 were measured for both total-to-total and total-to-staticefciencies. The turbine tested was designed for maximum efciency andlikely represents a maximum attainable performance level. For predictingthe performance of new turbine designs, the efciency obtained from thisdata should likely be derated to account for nonoptimum factors in the newdesign such as constraints on scroll size, different blade counts, etc.

Tip clearance losses in a radial-inow turbine arise from two sources:axial clearance at the rotor blade inlet, and radial clearance at the rotorblade exit. Of the two, the radial clearance is by far the more important.Futral and Holeski [26] found that for axial clearances in the range of 17%of inlet blade height, an increase in clearance of 1% (say from 2% to 3% ofinlet blade height) caused a decrease in total-to-total efciency of only0.15%. For radial clearances in the range of 13% of exit blade height,Futral and Holeski measured a 1.6% decrease in total-to-total efciency fora 1% increase in radial clearance, roughly 10 times greater than the changefor axial clearance. In a radial-inow turbine, the majority of ow turning inthe rotor is done in the exit portion of the blading, called the exducer.


Because of radial clearance in the exducer, some fraction of the ow isunderturned and does less work (similar to the situation at the tip of anaxial-ow turbine blade). Since little ow turning is done in the inlet portionof the blade, the axial clearance has a smaller effect.

As with axial-ow turbines, peak total-to-static efciency in radialturbines usually occurs when there is no exit swirl Vy2 0. Rodgers [27]reports that the exit vector diagram is optimized for maximum total-to-static efciency when the exit ow coefcient f2, dened as

f2 Vx2

U160

has a value between 0.2 and 0.3. Rodgers [27] also reports that the geometryof the exit is optimized when the ratio of the rotor inlet radius to the rotorexit root mean squared radius is 1.8. Regarding the rotor inlet vectordiagram, maximum efciency occurs when the mean rotor inlet ow entersthe normally radially bladed rotor at some incidence angle. According to

Figure 17 Effect of specic speed on radial-inow turbine efciency. (Replottedfrom Ref. 25.)


Glassman [1], the optimum ratio of Vy1 to U1 is given by

Vy1

U1 1 2

Zr61

where Zr is the rotor blade count at the inlet (includes both full and partialblades). The optimum blade speed occurs for U=C0 0:7 [see Eq. (43)]according to empirical data from Rodgers [27] and analytical results fromRohlik [28]. Specication of the optimum rotor inlet vector diagram iscompleted by choosing a stator exit angle of approximately 758 (measuredfrom radial) based on data from Rohlik [28].

Due to the change in radius through the rotor, local blade solidity (theratio of blade spacing to chord) changes appreciably. At the rotor inlet,more blades are needed than at the rotor exit if uniform blade loading is tobe maintained. This situation can be treated by adding partial blades at therotor inlet. These partial blades, called splitters, end before the exducer. Theintent of adding the splitter blades is to reduce the blade loading in the inletportion of the rotor and so reduce the boundary-layer losses. However, thesplitters increase the rotor surface area, counteracting some of the benet ofreduced loading. Futral and Wasserbauer [29] tested a radial-inow turbineboth with and without splitters (the splitters were machined off for thesecond test) and found only slight differences in turbine performance. In thisparticular case, the benets of reduced blade loading were almost completelyoffset by the increased surface area frictional losses. It is not clear that thisresult can be universally extended, but it does indicate that splitters shouldnot always be included in a radial-inow turbine design.

For low-cost turbines such as those in automotive turbochargers, nonozzle vanes are used, with all ow turning being done in the scroll. Thisincreases the scroll frictional losses due to the increased velocity and alsodecreases the obtainable rotor inlet absolute ow angle. Balje [3] hascalculated the efciency ratio for radial-inow turbines with and withoutnozzles and found it to be approximately 0.92, regardless of specic speed.

Adjustments for the effects of diffusers and Reynolds number changesare similar to those previously presented for axial turbines.

Mechanical, Geometric, and Manufacturing Constraints

Radial-inow turbine design is as much affected by mechanical considera-tions as axial-inow turbines. As with axial-ow turbines, turbine efciencyfor high-temperature applications is limited by materials, not aerodynamics.Material and manufacturing limitations affect both the geometry of theturbine wheel and its operating conditions.


When Rohlik performed his analytical study in 1968, he limited therotor exit hub-to-tip radius ratio to values greater than 0.4. The turbineinvestigated by Kofskey and Nusbaum [25] had a hub-to-tip radius ratio atthe exit of 0.53. However, with the desire for smaller and less expensiveturbine wheels, hub-to-tip radius ratios now are seen as low as 0.25 and less.Along with inertia and stress considerations, this limits rotor blade countfrom 10 to 14 (Rodgers [27]).

Typical materials for radial-inow turbine wheels are cast superalloysfor high-temperature applications and cast or forged steel for lowertemperatures. Ceramics have been used in production turbochargers andare in a research stage for small gas turbines. Radial-inow turbine wheelshave three critical stress locations: inlet blade root, exducer blade root, andhub centerline. Rodgers [27] notes that the tip speed of current superalloyradial-inow turbine wheels is limited to approximately 2,200 ft/sec. Theexact value is dependent on both operating temperature and desired life. Formoderate inlet temperatures and pressure ratios T 0in < 500 F andp0in=pdis < 4, stress considerations, while they must be addressed in themechanical design, usually do not constrain the aerodynamic design of theturbine. This includes free-run operation.

As previously mentioned, radial-inow turbine blades are usuallyradial at the inlet to eliminate bending loads. At the exit, the rotor bladeangle is limited to about 608 from axial for manufacturing reasons. Withcasting being the preferred method of construction, rotor trailing-edgethickness should be greater than 0.020 in. Limitations on the radial-inowstator are similar to those for an axial-ow stator: exit blade angle should beless than 758 (for a radial-inow stator, this is measured from the radialdirection) and trailing-edge thickness should be 0.015 in. or greater.Signicantly thicker trailing edges are needed if the stator vanes are cooled.Trailing-edge blockage for both stators and rotors should be kept below10% for best performance. With low hub-to-tip radius ratios at rotor exit,this guideline is frequently violated at the hub, where the blade spacing issmallest and the trailing-edge thickness is large for mechanical reasons.

Overall package diameter is determined by rotor tip diameter, radiusratio across the stator, and the size of the scroll. In addition, there isnormally a vaneless space between the stator and rotor, similar to the axialgap between the stator and rotor in an axial-ow turbine. The vaneless spaceradius ratio is usually held to 1.05 or less. Stator vane radius ratio iscontrolled by stator vane count and stator turning. In most radial-inowturbines, a scroll provides a signicant amount of tangential component atstator inlet, resulting in relatively low amounts of ow deection in thestator vane row. This results in reduced solidity requirements, so that fewerand shorter stator vanes can be used. Rodgers [24] states that a common


design fault in the radial-inow turbine stator is too high a value of solidity,resulting in excessive frictional losses. Based on turbine designs presented byRodgers [27] and the turbine used by Kofskey and Nusbaum [25], statorvane radius ratios range from 1.2 to 1.3. For preliminary sizing exercises, avalue of about 1.25 may be taken as typical. The radius to the centerline ofthe scroll inlet of the turbine from Kofskey and Nusbaum [25] is twice theradius at stator inlet. Cross-section radius at scroll inlet is approximatelytwo thirds the stator inlet radius, so the maximum package radius is roughly2.67 times the stator inlet radius. This represents a fairly large scroll,commensurate with the high efciency levels obtained during testing. For areduction in efciency, the scroll size can be reduced.

An Example of Radial-Inow Turbine Sizing

To demonstrate the concepts and guidelines described in this and precedingsections, we will size a radial-inow turbine for the same application as theaxial-ow turbine example presented earlier. The design requirements forthat turbine were:

1. Generates 100 hp at design point.2. Operates at an overall pressure ratio of 3:1 in air.3. Inlet pressure is 44.1 psia, and inlet temperature is 300 8F.

A procedure similar to that used in the axial-ow turbine sizing example willbe used here with a few modications:

1. Determine available energy (ise

Fundamentals of Turbine Design

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