7- Fundamentals of Turbine Design

5/22/2018 7- Fundamentals of Turbine Design

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7

Fundamentals 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 flow of

fluid into rotational mechanical energy of a shaft. Turbines are used in a

wide range of applications, in a wide variety of sizes. Large single-stage

turbines are used for power generation in hydroelectric dams, while large

multistage turbines are used in steam power plants. Aircraft propulsion

engines (turbofans, turbojets, and turboprops) use multistage turbines intheir power and gas generator sections. Other, less obvious uses of turbines

for aircraft are in auxiliary power units, ground power units, starters for

main engines, turboexpanders in environmental controls, and constant-

speed drives for electrical and hydraulic power generation. Rocket engines

use turbines to power pumps to pressurize the propellants before they reach

the combustion chamber. Two familiar consumer applications of turbines

are turbochargers for passenger vehicles and wind turbines (windmills).

Many excellent texts have been written regarding the design and

analyses 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.


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power plant and aircraft propulsion engine turbines. Here we make no

attempt to cover these areas. The purpose of this text is to familiarize a

mechanical or aerospace engineer who does not specialize in turbines with

basic turbine design and analysis. The emphasis will be on smaller turbines

for applications other than propulsion or electrical power generation.Further restricting our emphasis, detailed design activities such as geometric

specification of blades, vanes, etc. will not be covered. Our intent is to give

the system engineer the necessary information to choose the correct type of

turbine, estimate its performance, and determine its overall geometry

(diameter, blade height, and chord).

This chapter will first cover those equations and concepts that apply to

all types of turbines. Subsequently, the two main turbine types will be

discussed, specifically, axial-flow turbines and radial-inflow turbines.

BASIC TURBINE CONCEPTS

Flow Through a Turbine

Figure 1shows cross sections of generic single-stage axial-flow and radial-

inflow turbines. The figure shows the station notation used for subsequent

analyses. The high-pressure flow enters the turbine at station in, passes

through the inlet, and is guided to the stator inlet (station 0), where vanes

turn the flow in the tangential direction. The flow leaves the stator vanes andenters the rotor blades (at station 1), which turn the flow back in the

opposite direction, extracting energy from the flow. The flow leaving the

rotor blades (station 2), now at a pressure lower than inlet, passes through a

diffuser where a controlled increase in flow area converts dynamic head to

static pressure. Following the diffuser, the flow exits to the discharge

conditions (station dis).

The purpose of the inlet is to guide the flow from the supply source to

the stator vanes with a minimum loss in total pressure. Several types of inlets

are shown in Fig. 2. Most auxiliary types of turbines such as starters anddrive units are supplied from ducts and typically have axial inlets such as

that shown inFig. 2(a)or a tangential entry like that ofFig. 2(b). The axial

inlet acts as a transition between the small diameter of the supply duct and

the larger diameter of the turbine. No flow turning or significant

acceleration is done in this type of inlet. In the tangential-entry scroll of

Fig. 2(b), the flow is accelerated and turned tangentially before entering the

stator, reducing the flow turning done by the stator. Another type of inlet

for an auxiliary turbine is the plenum shown inFig. 2(c). For turbines that

are 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)].



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Figure 1 Cross sections of generic single-stage turbines: (a) axial-flow turbine, (b)

radial-inflow turbine.



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The sole purpose of the stator is to induce a swirl component to the

flow so that a torque can be imparted to the rotor blades. Stators are

typically equipped with numerous curved airfoils called vanes that turn the

flow in the tangential direction. Cross sections of an axial-flow turbine statorand a radial-inflow 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-entry

scroll inlet for axial-flow turbine, (c) plenum inlet with radial or axial entry, (d)

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



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respectively. Radial-inflow turbines supplied from a scroll, such as

turbocharger turbines, often have no vanes in the stator. For turbines

that must operate efficiently over a wide range of inlet flow conditions,

variable-geometry stators are used, typically with pivoting stator vanes. For

high-temperature applications, the stator vanes are cooled using lower-temperature fluid, usually compressor bleed air.

The purpose of the rotor is to extract energy from the flow, converting

it to shaft power. The rotor blades are attached to a rotating disk that

transfers the torque of the rotor blades to the turbine output shaft. Like the

stator, the rotor has a number of individual curved airfoils called rotor or

turbine blades. The blades are angled to accept the flow from the stator with

minimum disturbance when the turbine is operating at design conditions.

The flow from the stator is then turned back in the opposite direction in a

controlled manner, causing a change in tangential momentum and a force tobe exerted on the blades.Figure 4shows cross sections of generic axial-flow

and radial-inflow turbine blades. Axial-flow rotors have been constructed

with blades integral with the disk and with blades individually inserted into

the disk using a dovetail arrangement. Cooling is often used for rotors in

high-temperature applications. Exotic materials are sometimes used for both

rotors and stators to withstand the high temperatures encountered in high-

performance applications.

The flow leaving the turbine rotor can have a significant amount of

kinetic energy. If this kinetic energy is converted to static pressure in anefficient manner, the turbine can be operated with a rotor discharge static

pressure lower than the static pressure at diffuser discharge. This increases

the turbine power output for given inlet and discharge conditions. Diffusers

used with turbines are generally of the form shown in Fig. 5(a)and5(b)and

increase the flow area gradually by changes in passage height, mean radius

of the passage, or a combination of the two. Diffusers with a change in

radius have the advantage of diffusing the swirl component of the rotor

discharge velocity as well as the throughflow component.

Turbine Energy Transfer

The combined parts of the turbine allow energy to be extracted from the

flow and converted to useful mechanical energy at the shaft. The amount of

energy extraction is some fraction of the energy available to the turbine. The

following describes the calculation of the available energy for a turbine and

assumes familiarity with thermodynamics and compressible flow.

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 insignificant compared to the energy transferred as work.



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In the ideal case, the expansion is isentropic, as shown in Fig. 6(a) in an

enthalpyentropy (hs) diagram. The inlet to the turbine is at pressure p0inand the exit is at p0dis. The isentropic enthalpy change is the most specific

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



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energy that can be extracted from the fluid. Thus, if the inlet pressure and

temperature and the exit pressure from a turbine are known, the maximum

specific energy extraction can be easily determined from a state diagram forthe turbine working fluid. For an ideal gas with constant specific heats, the

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



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isentropic enthalpy drop is calculated from

DhisentropiccpT0in 1 pdisp0in g1=g" # 1

Figure 5 Turbine diffuser configurations: (a) constant mean-diameter diffuser, (b)

increasing mean-diameter curved diffuser.



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Where

T0ininlet absolute total temperature:

Opspecific heat at constant pressure:gratio of specific heats:

Figure 6 The expansion process across a turbine: (a) idealized isentropic

expansion, (b) actual expansion.



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The approximation of Eq. (1) is adequate for turbines operating on air and

other common gases at moderate pressures and temperatures. Total

conditions are normally used for both temperature and pressure at the

inlet 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 smaller

than the isentropic value due to irreversibilities in the flow. The actual

process is marked by an increase in entropy and is represented in the hs

diagram of Fig. 6(b) by a dotted line. The actual path is uncertain, as the

details of the entropy changes within the turbine are usually not known. Due

to the curvature of the isobars, the enthalpy change associated with an

entropy increase is less than that for an isentropic process. The degree of

entropy rise is usually described indirectly by the ratio of the actual enthalpy

drop to the isentropic enthalpy drop. This quantity is referred to as theisentropic (sometimes adiabatic) efficiency, Z, and is calculated from

ZOA hinhdisDhisentropic

2

The subscript OA indicates the overall efficiency, since the enthalpy drop is

taken across the entire turbine. The efficiency is one of the critical

parameters that describe turbine performance.

So far we have not specified 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 ideal

enthalpy drop.) Usage depends on application. For applications where the

kinetic energy leaving the turbine rotor is useful, total pressure is used. Such

cases include all but the last stage in a multistage turbine (the kinetic energy

of the exhaust can be converted into useful work by the following stage) and

cases where the turbine exhaust is used to generate thrust, such as in a

turbojet. For most power-generating applications, the turbine is rated using

static exit pressure, since the exit kinetic energy is usually dissipated in theatmosphere. Note that the total-to-static efficiency will be lower than the

total-to-total efficiency since the static exit pressure is lower than the total.

With the energy available to the turbine established by the inlet and

exit conditions, lets take a closer look at the actual mechanism of energy

transfer within the turbine.Figure 7shows a generalized turbine rotor. Flow

enters 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 about

its centerline coincident with the x-axis with rotational velocity o. The

location 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



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the average for the gas flowing through the turbine. The net torque Gacting

on the rotor can be represented as the difference of two torques on either

side of the rotor:

Gr1Fy1r2Fy2 3

where Fy is the force in the tangential direction and r is the radius to the

point. 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.



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change of angular momentum:

FydmVydt

4

Performing the derivative, assuming constant Vy and mass flow rate _mm,

results in

G _mmVy1r1Vy2r2 5The energy transfer per time (power) is obtained by multiplying both sides

of Eq. (5) by the rotational velocity o:

PGoo _mmVy1r1Vy2r2 6

The power P can also be calculated from the hs diagram for the actualprocess as

P _mmhinhdis _mmDhactual 7Combining Eqs. (6) and (7) and defining the wheel speed Uas

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

DhactualU1Vy1U2Vy2 9

The Euler equation, as derived here, assumes adiabatic flow through the

turbine, since enthalpy change is allowed only across the rotor. The Euler

equation relates the thermodynamic energy transfer to the change in

velocities at the inlet and exit of the rotor. This leads us to examine these

velocities more closely, since they determine the work extracted from the

turbine.

Velocity DiagramsEulers equation shows that energy transfer in a turbine is directly related to

the velocities in the turbine. It is convenient to graphically display these

velocities at the rotor inlet and exit in diagrams called velocity or vector

diagrams. These diagrams are drawn in a single plane. For an axial-flow

turbine, they are drawn in the xy plane at a specific value ofr. At the inlet

of a radial-inflow turbine, where the flow is generally in the ry plane, the

diagram is drawn in that plane at a specific value ofx. The exit diagram for

a radial-inflow turbine is drawn in the xy plane at a specific value ofr.

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



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between the velocity diagram and the physical geometry of the turbine. The

flow leaves the stator at an angle ofa1 from the axial direction. The velocity

vector V!

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

andVy1 in the tangential direction. Note that the turbine work is controlled

by the tangential component, while the turbine flow rate is controlled by theaxial component (for an axial-flow turbine). The vector V

!1 is measured in

an absolute, nonrotating reference frame and is referred to as the absolute

rotor inlet velocity. Likewise, the anglea1is called the absolute flow angle at

rotor inlet. A rotating reference frame can also be fixed to the rotor.

Velocities in this reference are determined by subtracting the rotor velocity

from the absolute velocity. Defining the relative velocity vector at the inlet

to be W

!

1, we can write

W!1 V!1U1 10The vector notation is not used for the rotor velocity U1as it is always in the

tangential direction. The relative velocity vector is also shown in Fig. 8(a).

The relative flow angleb1is defined as the angle between the relative velocity

vector and the axial direction. Inspection of the diagram of Fig. 8(a) reveals

Figure 8 Velocity diagrams for an axial-flow turbine: (a) rotor inlet velocity

diagram, (b) rotor exit velocity diagram.



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several relationships between the relative velocity and absolute velocity and

their components:

V21

V2x1

V2y1

11

W21 W2x1W2y1 12Wy1Vy1U1 13Wx1Vx1 14

The sign convention used here is that tangential components in the direction

of the wheel speed are positive. This implies that botha1 andb1 are positive

angles.Figure 8(b)shows the vector diagram at the outlet of the rotor. Note

that in this diagram, both the absolute and relative tangential components

are 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 flow angles, we can also define

other relative quantities such as relative total temperature and relative total

pressure. In the absolute frame of reference, the total temperature is defined

as

T0TV2

2cp15

In the relative frame of reference, the relative total temperatureT00is definedas

T00TW2

2cp16

The static temperature is invariant with regard to reference frame.

Combining Eqs. (15) and (16), we have

T00T0W2 V22cp

17

The relative total temperature is the stagnation temperature in the rotating

reference; hence it is the temperature that the rotor material is subjected to.

Equation (17) shows that if the relative velocity is lower than the absolute

velocity, the relative total temperature will be lower than the absolute. This

is 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



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gas dynamics relation

p00

p T

00

T g=g1

18

Types of Velocity Diagrams

There are an infinite number of variations of the velocity diagrams shown in

Fig. 8. To help distinguish and classify them, the vector diagrams are

identified according to reaction, exit swirl, stage loading, and flow

coefficient. The reaction is the ratio of the change in static enthalpy across

the 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 in

static enthalpy (denoted as hs) can be found from

hs1hs2 h1V21

2

h2V

22

2

U1Vy1U2Vy212V21V22 19

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

UVy12V2 U2 W2 20

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

Rstg U21U22 W21W22

V21V22 U21U22 W21W22 21

Stage reaction is normally held to values greater than or equal to 0. For an

axial-flow turbine with no change in mean radius between rotor inlet androtor outlet,U1U2 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 flow separation associated with diffusion, negative

reaction is generally avoided. Diagrams with zero reaction (no change in

magnitude of relative velocity across the rotor) are referred to as impulse

diagrams and are used in turbines with large work extraction. Diagrams

with reactions greater than 0 are referred to as reaction diagrams. Stage

reaction is usually limited to about 0.5 due to exit kinetic energyconsiderations.



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Exit swirl refers to the value of Vy2. For turbines discharging to

ambient, the most efficient diagram has zero exit swirl. While a negative

value of exit swirl increases the work extraction, the magnitude of the

turbine discharge velocity increases, leading to a larger difference between

the exit static and total pressures. For turbines rated on exit static pressure,the tradeoff between increased work and lower exit static pressure results in

lower efficiency levels. Most turbines operating in air with pressure ratios of

3:1 or less use zero exit swirl vector diagrams.

The stage loading is measured by the loading coefficientl. The loading

coefficient is defined 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 loading

coefficient is usually calculated for an axial-flow turbine stage at either the

hub or mean radius. For a radial-inflow turbine, the rotor tip speed is used

in Eq. (23).

The stage flow is controlled by the flow coefficient, defined as

fVxU

24

These four parameters are related to each other through the vector diagram.

Specification of three of them completely defines the vector diagram.

Figure 9 presents examples of a variety of vector diagrams, with exit

swirl, reaction, and loading coefficient tabulated. Figure 9(a) shows a vector

diagram appropriate for an auxiliary turbine application, with relatively

high loading (near impulse) and zero exit swirl. A diagram more typical of a

stage in a multistage turbine is shown in Fig. 9(b), since the exit kinetic

energy can be utilized in the following stage, the diagram does show

significant exit swirl. Both Fig. 9(a) and 9(b) are for axial turbines; 9(c) is the

vector diagram for a radial-inflow turbine. The major difference is the

change in Ubetween 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



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Figure 9 Variations in turbine velocity diagrams: (a) axial-flow diagram for single-

stage auxiliary turbine, (b) axial-flow diagram for one stage in multistage turbine, (c)

radial-inflow turbine velocity diagram.



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component so that we may write

Dhideal DhactualLinletLstatorLrotorLdiffuserLexit 25where the L terms represent losses in enthalpy in each component. Losses

can also be looked at from a pressure viewpoint. An ideal exit pressure can

be determined from

DhactualcpT0in 1 pdisideal

p0in

g1=g" # 26

The component losses are then represented as losses in total pressure, the

sum of which is equal to the difference between the actual and ideal exit

pressure:

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 coefficient Ktdefined as

Dp0inlet Ktinlet1

2rV2inlet

28

Where

Vinlet velocity at the upstream end of the inlet:rdensity 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 ofFig. 2(a)

have low frictional losses (due to their short length and relatively low

velocities), but often suffer from turning losses due to flow separation along

their outer diameter. Longer axial inlets with more gradual changes in outer

diameter tend to reduce the turning losses and prevent separation, but

adversely impact turbine envelope. Tangential entry inlets tend to have

higher losses due to the tangential turning and acceleration of the flow. The

spiral flow path also tends to be longer, increasing frictional losses.

Typically, loss coefficients 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



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pressure, inlet losses are usually on the order of 13% of the inlet total

pressure. For turbines in engines, there is usually no real inlet, as they are

closely 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, the

secondary flows caused by the flow turning, and exit losses due to blockage

at the vane row trailing edge. The stator loss coefficient can be defined in

several ways. Two popular definitions are

Dp0statorYstator1

2rV21 29

or

Dp0statorYstator1

2r

V20V212

30

In either case, the loss coefficient is made up of the sum of coefficients for

each loss contributor:

YstatorYprofileYsecondaryYtrailing edge 31Profile refers to frictional losses. There can be additional loss contributions

due to incidence (the flow coming into the stator is not aligned with the

leading edge), shock losses (when the stator exit velocity is supersonic), and

others. Much work has been dedicated to determining the proper values for

the coefficients, and several very satisfactory loss model systems have been

developed. As loss models differ for axial-flow and radial-inflow 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 and

the kinetic energy is based on relative velocities. As with stators, the rotor

loss is based on either the exit relative kinetic energy or the average of the

inlet and exit relative kinetic energies:

Dp00rotorXrotor 12 rW22

32



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or

Dp00rotorXrotor1

2r

W21W222

33

With rotors, incidence loss can be significant, so we include that contributor

in the expression for the rotor loss coefficient:

XrotorXprofileXsecondaryXtrailing edgeXincidence 34Other losses associated with the rotor are tip clearance and windage losses.

Turbine rotors operate with a small clearance between the tips of the blades

and the turbine housing. Flow leaks across this gap from the high-pressure

side of the blade to the low-pressure side, causing a reduction in the pressure

difference at the tip of the blade. This reduces the tangential force on theblade, decreasing the torque delivered to the shaft. Tip clearance effects can

be reduced by shrouding the turbine blades with a ring, but this

introduces manufacturing and mechanical integrity challenges. The loss

associated with tip clearance can be modeled either using a pressure loss

coefficient or directly as a reduction in the turbine efficiency. The specific

models differ with turbine type and will be discussed in following sections.

Windage losses arise from the drag of the turbine disk. As the disk

spins in the housing, the no-slip condition on the rotating surface induces

rotation of the neighboring fluid, establishing a radial pressure gradient inthe cavity. This is commonly referred to as disk pumping. For low-head

turbines operating in dense fluids, the windage losses can be considerable.

Windage effects are handled by calculating the windage torque from a disk

moment coefficient defined as

Gwindage 2Cm12ro2r5disk

35

The output torque of the turbine is reduced by the windage torque. Values

of the moment coefficientCmdepend on the geometry of the disk cavity andthe speed of the disk. Nece and Daily [46] are reliable sources of moment

coefficient data.

Diffuser Losses

Losses in the diffuser arise from sources similar to those in other flow

passages, namely, friction and flow turning. The diffuser loss can be

expressed in terms of a loss coefficient for accounting in turbineperformance, but diffuser performance is usually expressed in terms of



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diffuser recovery, defined as

Rppdisp2p02

p2

36

The diffuser recovery measures how much of the kinetic energy at diffuser

inlet is converted to a rise in static pressure. Recovery is a function of area

ratioAdis=A2, length, and curvature. For an ideal diffuser of infinite arearatio, the recovery is 1.0. Peak recovery of a real diffuser of given length

takes place when the area ratio is set large enough so that the flow is on the

verge of separating from the walls of the diffuser. When the flow separates

within the diffuser, the diffuser is said to be stalled. Once stalled, diffuser

recovery drops dramatically. Curvature of the mean radius of the diffuser

tends to decrease the attainable recovery, since the boundary layer on one ofthe diffuser walls is subjected to a curvature-induced adverse pressure

gradient in addition to the adverse pressure gradient caused by the increase

in flow area.

Even with the recent advances in general-use computational fluid

dynamics (CFD) tools, analytical prediction of diffuser recovery is not

normally performed as part of the preliminary turbine design. Diffuser

performance is normally obtained from empirically derived plots such as

that shown inFig. 10.Diffuser recovery is plotted as a function of area ratio

and diffuser length. The curvature of the contours of recovery shows thelarge fall-off in diffuser recovery after the diffuser stall. The locus of

maximum recovery is referred to as the line of impending stall. Diffusers

should not be designed to operate above this line. Runstadler et al. [7, 8] and

Sovran and Klomp [9] present charts of diffuser recovery as a function of

inlet Mach number and blockage, as well as the three geometric factors

noted earlier.

The total pressure loss across a diffuser operating in incompressible

flow can be calculated using continuity and the definition of diffuser

recovery. The recovery for an ideal diffuser (no total pressure loss) is givenby

Rpideal1 A2

Adis

237

The total pressure loss for a nonideal diffuser in incompressible flow is given

by

p02p0dis12rV22

Ktdiff RpidealRp 38



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This can be used to calculate the diffuser loss when compressibility is not

important. 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 critical

to use the correct diffuser performance chart when estimating diffuser

recovery.

Exit Losses

Exit losses are quite simple. If the kinetic energy of the flow exiting the

diffuser is used in following stages, or contributes to thrust, the exit losses

are zero. If, however, the diffuser discharge energy is not utilized, the exit

loss is the exit kinetic energy of the flow. For this case,

Dp0exitp0dispdis 39

Nondimensional Parameters

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

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



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different turbines easier, dimensional analysis leads to the formation of

several dimensionless parameters that can be used to describe turbines.

Specific Speed and Specific Diameter

The specific speed of a turbine is defined as

Ns offiffiffiffiffiffi

Q2p

Dhideal3=4 40

where Q2 is the volumetric flow rate through the turbine at rotor exit. The

specific speed is used to relate the performance of geometrically similar

turbines of different size. In general, turbine efficiency for two turbines of

the same specific speed will be the same, except for differences in tipclearance and Reynolds number. Maintaining specific speed of a turbine is a

common approach to scaling of a turbine to different flow rates.

The specific diameter is defined as

DsdtipDhideal1=4ffiffiffiffiffiffi

Q2p 41

where dtip is the tip diameter of the turbine rotor, either radial in-flow or

axial flow. Specific diameter and specific speed are used to correlate turbine

performance. Balje [3] presents extensive analytical studies that result in

maps of peak turbine efficiency versus specific speed and diameter for

various types of turbines. These charts can be quite valuable during initial

turbine sizing and performance estimation.

Blade-Jet Speed Ratio

Turbine performance can also be correlated against the blade-jet speed

ratio, which is a measure of the blade speed relative to the ideal stator exitvelocity. Primarily used in impulse turbines, where the entire static enthalpy

drop is taken across the stator, the ideal stator exit velocity,C0, is calculated

assuming the entire ideal enthalpy drop is converted into kinetic energy:

C0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2Dhidealp

42The blade-jet speed ratio is then calculated from

U

C0

U

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Dhidealp 43The value of blade speed at the mean turbine blade radius is typically used in



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

used.

Reynolds Number

The Reynolds number for a turbine is usually defined as

RerUtipdtipm

44

wheremis the viscosity of the working fluid. Sometimes odtip is substituted

for Utip, resulting in a value twice that of Eq. (44). The Reynolds number

relates the viscous and inertial effects in the fluid flow. For most

turbomachinery operating on air, the Reynolds number is of secondaryimportance. However, when turbomachinery is scaled (either larger or

smaller), the Reynolds number changes, resulting in a change in turbine

efficiency. Glassman [1] suggests the following for adjusting turbine losses to

account for Reynolds number changes:

1Z0a1Z0b

AB RebRea

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whereZ0 indicates total-to-total efficiency and AandB sum to 1.0. That allthe loss is not scaled by the Reynolds number ratio reflects that not all losses

are viscous in origin. Also, total-to-total efficiency is used since the kinetic

energy of the exit loss is not affected by Reynolds number. Glassman [1]

suggests values of 0.30.4 forA (the nonviscous loss) and from 0.7 to 0.6 for

B (the viscous loss).

Equivalent or Corrected Quantities

In order to eliminate the dependence of turbine performance maps on the

values of inlet temperature and pressure, corrected quantities such as

corrected flow, corrected speed, corrected torque, and corrected power were

developed. Using corrected quantities, turbine performance can be

represented by just a few curves for a wide variety of operating conditions.

Corrected quantities are not nondimensional. Glassman [1] provides a

detailed derivation of the corrected quantities. The corrected flow is defined

as

wcorrwffiffiffiypd

46



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where

y T0in

TSTD47

and

d p0in

pSTD48

The standard conditions are usually taken to be 518.7 R and 14.7 psia.

Corrected speed is defined as

Ncorr Nffiffiffiyp 49Equation (5) shows torque to be the product of flow rate and the change in

tangential velocity across the rotor. Corrected flow is defined above;

corrected velocities appear with y1=2 in the denominator from the corrected

shaft speed. Therefore, corrected torque is defined as

GcorrGd

50

The form of the corrected power is determined from the product of

corrected torque and corrected speed:

Pcorr Pdffiffiffiy

p 51

These corrected quantities are used to reduce turbine performance data to

curves of constant-pressure ratio on two charts. Figure 11 presents typical

turbine performance maps using the corrected quantities. Figure 11(a)

presents corrected flow as a function of corrected speed and pressure ratio,

whileFig. 11(b)shows corrected torque versus corrected speed and pressure

ratio. Characteristics typical of both radial-inflow and axial-flow turbinesare presented in Fig. 11.

AXIAL-FLOW TURBINE SIZING

Axial-Flow Turbine Performance Prediction

Prediction methods for axial-flow turbine performance methods can be

roughly broken into two groups according to Sieverding [10]. The first

group bases turbine stage performance on overall parameters such as workcoefficient and flow coefficient. These are most often used in preliminary



26/62

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

corrected torque vs. pressure ratio and corrected speed.



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sizing exercises where the details of the turbine design are unknown. Smith

[11] and Soderberg [12] are both examples of this black box approach, as

are Baljes [3] maps of turbine efficiency as a function of specific speed and

specific diameter.

The second grouping is based on the approach outlined earlier whereturbine losses are broken down to a much finer level. In these methods, a

large number of individual losses are summed to arrive at the total loss.

Each of these loss components is dependent on geometric and aerodynamic

parameters. This requires more knowledge of the turbine configuration,

such as flow path and blading geometry, before a performance estimate can

be made. As such, these methods are better suited for more detailed turbine

design studies.

Among the loss component methods, Sieverding [10] gives an excellent

review of the more popular component loss models. The progenitor of afamily of loss models is that developed by Ainley and Mathieson [13]. It has

been modified and improved by Dunham and Came [14] and, more

recently, by Kacker and Okapuu [15]. A somewhat different approach is

taken by Craig and Cox [16]. All these methods are based on correlations of

experimental data.

An alternate approach is to analytically predict the major loss

components such as profile or friction losses and trailing-edge thickness

losses by computing the boundary layers along the blade surfaces. Profile

losses 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 technique

requires even more information on the turbine design; to calculate the

boundary layer it is necessary to know both the surface contour and the

velocities along the blade surface. Thus, this method cannot be used until

blade geometries have been completely specified and detailed flow channel

calculations have been made.

In addition to the published prediction methods just noted, each of the

major turbine design houses (such as AlliedSignal, Allison, General Electric,Lycoming, Pratt & Whitney, Sundstrand, and Williams) has its own

proprietary models based on a large turbine performance database. Of

course, it is not possible to report those here.

For our purposes (determining the size and approximate performance

of a turbine) we will concentrate on the overall performance prediction

methods, specifically Smiths chart and Soderbergs correlation. Figure 12

shows Smiths [11] chart, where contours of total-to-total efficiency are

plotted versus flow coefficient and work factor [see Eqs. (23) and (24)]. Both

the flow coefficient and stage work coefficient are defined using velocities atthe mean radius of the turbine. The efficiency contours are based on the



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measured efficiency for 70 turbines. All the turbines have a constant axial

velocity across the stage, zero incidence at design point, and reactions

ranging from 20% to 60%. Reynolds number for the turbines range from

100,000 to 300,000. Aspect ratio (blade height to axial chord) for the tested

turbines is in the range of 34. Smiths chart does not account for the effects

of blade aspect ratio, Mach number effects, or trailing-edge thickness

variations. The data have been corrected to reflect zero tip clearance, so theefficiencies must be adjusted for the tip clearance loss of the application.

Sieverding [10] considers Soderbergs correlation to be outdated but

still useful in preliminary design stages due to its simplicity. In Soderbergs

[12] correlation, blade-row kinetic energy losses are calculated from

Vo2idealV2oV2o

x

105Rth

1=4 1xref 0:9750:075 cxh 1 h in o 52

Figure 12 Smiths chart for stage zero-clearance total-to-total efficiency as

function of mean-radius flow and loading coefficient. (Replotted from Ref. 11

with permission of the Royal Aeronautical Society).



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where Rth is the Reynolds number based on the hydraulic diameter at the

blade passage minimum area (referred to as the throat) defined as

RthroVo

mo

2hs cos

ao

hs cosao 53

where h is the blade height and s is the spacing between the blades at the

mean radius. The blade axial chord is identified by cx. In both Eqs. (52) and

(53), the subscript o refers to blade-row outlet conditions, either stator or

rotor (for the rotor, the absolute velocity V is replaced by the relative

velocity W, standard practice for all blade-row relations). The reference

loss coefficient xrefis a function of blade turning and thickness and can be

found inFig. 13. Compared to Smiths chart, this correlation requires moreknowledge of the turbine geometry, but no more than would be required in

a conceptual turbine design. The losses predicted by this method are only

valid for the optimum blade chord-to-spacing ratio and for zero incidence.

Tip clearance losses must also be added in the final determination of turbine

efficiency. Like Smiths chart, this correlation results in a total-to-total

efficiency for the turbine.

The optimum value of blade chord-to-spacing ratio can be found using

the definition of the Zweifel coefficient [17]:

z 2cx=s

cos ao

cos aisinaiao

54

where the subscript i refers to blade-row inlet. Zweifel [17] states that

optimum soliditycx=s occurs when z0:8.Tip clearance losses are caused by flow leakage through the gap

between the turbine blade and the stationary shroud. This flow does not get

turned by the turbine blade; so it does not result in work extraction. In

addition, the flow through the clearance region causes a reduction of thepressure loading across the blade tip, further reducing the turbine efficiency.

The leakage flow is primarily controlled by the radial clearance, but is also

affected by the geometry of the shroud and the blade reaction. Leakage

effects can be reduced by attaching a shroud to the turbine blade tips, which

eliminates the tip unloading phenomenon. For preliminary design purposes,

the tip clearance loss for unshrouded turbine wheels can be approximated by

Z

Zzc 1

Kc

rtip

rmean

cr

h 55



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Where

Zzc zero clearance efficiency:orradial tip clearance:

rtippassage tip radius:rmeanmean passage radius:

Kc empirically derived constant:

Based on measurements reported by Haas and Kofskey [18], the value ofKcis between 1.5 and 2.0, depending on geometric configuration. For

preliminary design purposes, the conservative value should be used. When

using Soderbergs correlation, the value of Kc should be taken as 1, since

Soderberg corrected his data using that value for Kc.

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

Figure 13 Soderbergs loss coefficient as function of deflection angle and blade

thickness. (Replotted from Ref. 12 with permission from Pergamon Press Ltd.)



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order to determine the overall turbine efficiency, it is necessary to include the

inlet, diffuser, and exit losses. These losses do not affect the turbine work

extraction, but result in the overall pressure ratio across the turbine being

larger than the stage pressure ratio. The overall efficiency can be calculated

from

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

56

The pressure losses in the inlet, diffuser, and exit are calculated from the

information presented earlier.

Mechanical, Geometric, and Manufacturing Constraints

Turbine design is as much or more affected by mechanical considerations as

it is by aerodynamic considerations. Aerodynamic performance is normally

constrained by the stress limitations of the turbine material. At this point in

the history of turbine design, turbine performance at elevated temperatures

is limited by materials, not aerodynamics. Material and manufacturing

limitations affect both the geometry of the turbine wheel and its operating

conditions.

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 the

disk). The allowable stress limit is affected by the turbine material, turbine

temperature, and turbine life requirements. Typical turbine materials for

aircraft auxiliary turbines are titanium in moderate-temperature applica-

tions (turbine relative temperatures below 1,000 8F) and superalloys for

higher temperatures.

Allowable blade-tip speed for axial-flow turbines is a complex function

of inlet temperature, availability of cooling air, thermal cycling (low cycle

fatigue damage), and desired operating life. In general, design point bladespeeds are held below 2,200 ft/sec, but higher blade speeds can be withstood

for shorter lifetimes, if temperatures permit. For auxiliary turbine

applications with inlet temperatures below 300 8F and pressure ratios of 3

or below, blade speed limits are generally not a design driver.

Both stress and manufacturing considerations limit the turbine blade

hub-to-tip radius ratio to values greater than about 0.6. If the hub diameter

is much smaller, it is difficult to physically accommodate the required

number 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



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performance reasons (secondary flow losses and tip clearance losses), it is

desirable to keep the hub-to-tip radius ratio below 0.8.

Manufacturing considerations limit blade angles on rotors to less than

608 and stator vane exit angles to less than 758. Casting capabilities limit

stator trailing-edge thicknesstteto no less than 0.015 in., restricting statorvane count. For performance reasons, the trailing-edge blockage should be

kept less than 10% at all radii. The trailing-edge blockage is defined here as

the ratio of the trailing-edge tangential thickness (b) to the blade or vane

spacing (s):

b

stte= cos ate

2pr=Z 57

whereZis the blade or vane count. Rotor blades are usually machined, butfor stress and tolerance reasons the trailing-edge thickness is normally no

less 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 to

close. Without any load, the turbine accelerates until the power output of

the turbine is matched by the geartrain and aerodynamic losses. Free-run

speed is roughly twice design-point speed for most aircraft auxiliary

turbines. This restricts the allowable design-point speeds and stress levelsfurther, since the disk and blade may be required to survive free-run

operation.

Hub-to-Tip Variation in Vector Diagram

Up to this point we have only considered the vector diagram at the mean

radius of the turbine. For turbines with high hub-to-tip radius ratios (above

0.85), the variation in vector diagram is not important. For a turbine with

relatively tall blades, however, the variation is significant.The change in vector diagram with radius is due to the change in blade

speed and the balance between pressure and body forces acting on the

working fluid as it goes through the turbine. Examples of body forces

include the centrifugal force acting on a fluid element that has a tangential

velocity (such as between the stator and rotor), and the accelerations caused

by a change in flow direction if the flow path is curved in the meridional

plane. The balance of these forces (body and pressure) is referred to as radial

equilibrium. Glassman [1] presents a detailed mathematical development of

the equations that govern radial equilibrium. For our purposes, we willconcentrate on the conditions that satisfy radial equilibrium.



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The classical approach to satisfying radial equilibrium is to use a free

vortex variation in the vector diagram from the hub to the tip of the rotor

blade. A free vortex variation is obtained by holding the product of the

radius 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 vortex

diagrams due to their simplicity. For preliminary design purposes, the free

vortex diagram is more than satisfactory.

Aside from its simplicity, the free vortex diagram has other

advantages. Holding rVy constant implies that the work extraction is

constant with radius. With Vx constant, the mass flow varies little with

radius. This implies that the mean section vector diagram is an excellent

representation of the entire turbine from both a work and mass flow

standpoint.When using a free vortex distribution, there are two key items to

examine in addition to the mean vector diagram. The hub diagram suffers

from low reaction due to the increase in Vy and should be checked to ensure

at least a zero value of reaction. From hub to tip, the reduction in Vy and

increase inUcause a large change in the rotor inlet relative flow angle, with

the rotor tip section tending to overhang the hub section. By choosing a

moderate hub-to-tip radius ratio (if possible), both low hub reaction and

excessive rotor blade twist can be avoided.

For a zero exit swirl vector diagram, some simple relations can bedeveloped for the allowable mean radius work coefficient and the hub-to-tip

twist of the rotor blade. For a zero exit swirl diagram, zero reaction occurs

for a work coefficient of 2.0. Using this as an upper limit at the hub, the

work coefficient at mean radius is found from

lm2 rhrm

258

For a turbine with a hub-to-tip radius ratio of 0.7, the maximum work

coefficient at mean radius for impulse conditions at the hub is 1.356. The

deviation in inlet flow angle to the rotor from hub to tip for a free vortex

distribution is given by

Db1b1hb1t

tan1 lmrm=rh2 1

fm

rm=rh

" #tan1 lmrm=rt

2 1fm

rm=rt

" # 59

For a vector diagram with lm1:356; rh=rt0:7, and



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fm0:6;Db156:1, 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 preceding

sections, an example is presented of the sizing of a typical auxiliary

turbine for use in an aircraft application. The turbine is to be sized to meet

the 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, flow rate,

and operating speed with a turbine design meeting the mechanical,

geometric, and manufacturing constraints outlined earlier. The following

procedure will be followed to perform this exercise:

1. Determine available energy (isentropic enthalpy drop).

2. Guesstimate overall efficiency to calculate flow rate.

3. Select the vector diagram parameters.

4. Calculate the vector diagram.

5. Determine the rotor overall geometry.6. Calculate the overall efficiency based on Smiths chart both with

and without a diffuser.

The process is iterative in that the efficiency determined in step 6 is then used

as the guess in step 2, with the process repeated until no change is found in

the predicted efficiency. We will also predict the turbine efficiency using

Soderbergs correlation.

The first step is to calculate the energy available to the turbine using

Eq. (1). For air, typical values for the specific heat and the ratio of specific

heats are 0:24 Btu=lbmR and 1.4, respectively. It is also necessary toconvert the inlet temperature to the absolute scale. We then have

Dhisentropic 0:24 BtulbmR

760R 1 1

3

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

lbm

Note that more digits are carried through the calculations than indicated, so

exact agreement may not occur in all instances. The vector diagram is

calculated using the work actually done by the blade row; therefore, we need

to start with a guess to the overall efficiency of the turbine. A good startingpoint is usually an overall efficiency of 0.8, including the effects of tip



35/62

clearance. Since tip clearance represents a loss at the tip of the blade, the rest

of the blade does more than the average work. Therefore, the vector

diagram is calculated using the zero-clearance efficiency. Since we do not

know 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-clearanceefficiency is 0.84. Note that the required flow rate is calculated using the

overall efficiency with clearance, since that represents the energy available at

the turbine shaft. Equation (2) is used to calculate the actual enthalpy drops:

DhOA 0:8 49:14Btulbm

39:31Btu

lbm

and

DhOA ZC 0:84 49:14Btulbm

41:28Btu

lbm

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

_mm PDhOA

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

1:798 lbm= sec

The mass flow rate is needed to calculate turbine flow area and is also a

system requirement.

We specify the vector diagram by selecting values of the turbine work

and flow coefficients. We also select a turbine hub-to-tip radius ratio of 0.7,

restricting the choice of mean work coefficient to values less than 1.356 in

order to avoid negative reaction at the hub. From Smiths chart(Fig. 12), we

initially choose a work coefficient of 1.3 and a flow coefficient of 0.6 to result

in a zero-clearance, stator inlet to rotor exit total-to-total efficiency of 0.94.

We apply these coefficients at the mean radius of the turbine. From Eq. (22)

we calculate the mean blade speed, Um:

UmffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiDhOA ZC

l

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi32:174 ftlbf=lbmsec2778:16ftlbf=Btu41:28 Btu=lbm

1:3

r891:6 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.



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

Vy1lUm 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 guidelines

presented earlier.

The rotor blade height and mean radius are determined by the

required rotor exit flow area and the hub-to-tip radius ratio. The rotor exit

flow area is determined from continuity:

A2r2Vx2_mm

The mass flow rate and axial velocity have previously been calculated;

the density is dependent on the rotor exit temperature and pressure. For a

turbine without a diffuser, the rotor exit static pressure is the same as the

discharge pressure, assuming the rotor exit annulus is not choked. For a

Figure 14 Mean-radius velocity diagrams for first iteration of axial-flow turbine

sizing example.



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turbine with an effective diffuser, the rotor exit static pressure will be less

than the discharge value. We will examine both cases.

Turbine Without DiffuserFirst we consider the turbine without a diffuser. Assuming perfect gas

behavior, the density is calculated from

r2 p2

RgasT2

where the temperature and pressure are static values and Rgas is the gas

constant. The rotor exit total temperature is determined from

T02T00 DhOA ZCCp 760 R 41:28 Btu=lbm

0:24 Btu=lbmR 588:0 R

The zero-clearance enthalpy drop is used because the local tempera-

ture over the majority of the blade will reflect the higher work (a higher

discharge temperature will be measured downstream of the turbine after

mixing of the tip clearance flow has occurred). Next we calculate the rotor

exit critical Mach number to determine the static temperature. The critical

sonic velocity is calculated from

acrffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2g

g1 gRgasT0

s

where g is a conversion factor. For air at low temperatures,

Rgas53:34 ft-lbf=lbmR, resulting in

acr2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

21:411:4 32:174

ftlbflbmsec2

53:34

ftlbflbmR

588R

s

1085 ft=secThe static temperature is found from

T2T02 1g1g1

V2

acr2

2" #

with zero exit swirl, V2Vx2 resulting in

T2 588R 11:4111:4 535 ft=sec1085 ft=sec

2" #564:2 R



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The density can now be determined:

r2

44:1 lbf=in2

3

144 in

2

ft2

53:34

ftlbf

lbmR 564:2 R 0:0703lbm

ft

3

and the required flow area:

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

in:2

ft2

6:882in:2

The rotor exit hub and tip radii cannot be uniquely determined until

either shaft speed, blade height, or hub-to-tip radius ratio is specified. Once

one parameter is specified, the others are determined. For this example, we

choose a hub-to-tip ratio of 0.7 as a compromise between performance and

manufacturability. If the turbine shaft speed were restricted to a certain

value or range of values, it would make more sense to specify the shaft

speed. The turbine tip radius is determined from

rt2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

A2

p1 rh=rt2

s

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6:882 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. and

a blade height of 0.622 in. The shaft speed is found from Eq. (8):

oUm=rm 891:6 ft=sec1:762in1 ft=12in6073 rad=sec

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

guidelines.

The next step is to calculate the overall efficiency. From Smiths chart,

a stator inlet to rotor exit total-to-total efficiency at zero clearance is

available. We must correct this for tip clearance effects, the inlet loss, andthe exit kinetic energy loss. At l1:3 and f 0:6, Smiths chart predicts

Z0020ZC0:94Assuming a tip clearance of 0.015 in., the total-to-total efficiency including

the tip clearance loss is calculated from Eq. (55) using a value of 2 for Kc:

Z0020 Z0020ZC 12rt

rm

d

h

0:94 12 2:073

1:762

0:015

0:622

0:8867

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



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ratiop0in=pdisis 3. The stator inlet to rotor exit total-to-total pressure ratiois calculated from

p02

p00 pdis

p0in p02

pdis p0in

p00 Based on earlier discussions, we assume an inlet total pressure loss ratio of

0.99. With no diffuser, the discharge and rotor exit stations are the same, so

the ratio of static to total pressure is found from the rotor exit Mach

number:

pdis

p02p2

p02 1g1

g1V2

acr2

2" # gg1 11

6

535

1085

2" #3:50:8652

We can now calculate the total-to-total pressure ratio from stator inlet

to rotor exit and the overall efficiency:

p02p00

13

1

0:8652

1

0:99

0:3891

and

ZOA

0:8867

1 0:38910:4=1:4

1 13 0:4=1:4 0:7779This completes the first iteration on the turbine size and performance for the

case without a diffuser. To improve the accuracy of the result, the preceding

calculations would be repeated using the new values of overall efficiency and

tip clearance loss.

Turbine with DiffuserFor an auxiliary type of turbine such as this, a diffuser recovery of 0.4 is

reasonable to expect with a well-designed diffuser. The rotor exit total

pressure is calculated from the definition of diffuser recovery given in Eq.

(35):

p02 pdis

Rp1p2=p02 p2=p02 44:1 psia=3

0:410:8652 0:865215:99 psia

and the rotor exit static pressure is

p2 15:99 psia0:8652 13:84 psia



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This is a considerable reduction compared to the discharge pressure of

14.7 psia. From this point, the rotor exit geometry is calculated in the same

way as that presented for the case without the diffuser. The following results

are obtained:

r20:0662 lbm=ft3A27:311 in:2rt22:136 in:rh21:495 in:

rm21:816 in:h20:641 in:N56;270 rpm

The tip speed is the same as the turbine without the diffuser, since the mean

blade speed is unchanged, as is the hub-to-tip radius ratio of the rotor. The

efficiency calculations also proceed in the same manner as the earlier case

with the following results (using the same inlet pressure loss assumption):

Z0020 0:8882

p02p00

0:3663ZOA0:8224

Since this result differs from our original assumption for overall

efficiency, further iterations would be performed to obtain a more accurate

answer. Note the almost 6% increase in overall efficiency due to the

inclusion of a diffuser. This indicates a large amount of energy is contained

in the turbine exhaust. The efficiency gain associated with a diffuser is

dependent on diffuser recovery, rotor exit Mach number, and overallpressure ratio and is easily calculated. Figure 15 shows the efficiency

benefit associated with a diffuser for an overall turbine pressure ratio

(total-to-static) of 3. Efficiency gains are plotted as a function of rotor exit

critical Mach number and diffuser recovery. As rotor exit Mach number

increases, the advantages of including a diffuser become larger. This

tradeoff is important to consider when sizing the turbine. For a given flow

or power level, turbine rotor diameter can be reduced by accepting high

rotor exit velocities (high values of flow coefficient); however, turbine

efficiency will suffer unless a diffuser is included, adversely impacting theaxial envelope.



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Automation of Calculations and Trade Studies

The calculations outlined in this example can be easily automated in either a

computer program or a spreadsheet with iteration capability. An example of

the latter is presented inFig. 16,which contains the iterated final results forthe example turbine when equipped with a diffuser. The advantage of

automation is the capability to quickly perform trade studies to optimize the

turbine preliminary design. Prospective variables for study include work and

flow coefficients, 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 results

shown in the spreadsheet of Fig. 16. Soderbergs correlation [Eq. (52)]

requires the vane and blade chords in order to calculate the aspect ratio

cx=h. We first 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 efficiency at an overall pressure ratio of 3.0.



42/62

stator at 0.020 in. These values are selected based on the guidelines given

earlier in the chapter. Solving Eq. (57) for the blade number results in

Zb=s2prmtte= cosate

For the stator, the flow anglea1is used forate; for the rotor, the relative flow

angle b2 is substituted for ate. The blade angle is slightly different from the

flow angle due to blockage effects, but for preliminary sizing, the

approximation is acceptable. For the stator, we have

Zstator 0:12p1:773 in0:020 in= cos65:2223:35

Figure 16 Spreadsheet for preliminary axial-flow turbine sizing showing iterated

results for example turbine.



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and for the rotor

Zrotor 0:12p1:773 in

0:020 in

= cos

59:04

28:65

Of course, only integral number of blades are allowed, so we choose 23

vanes for the stator and 29 rotor blades, resulting in a blade spacing of

0.484 in. for the stator and 0.384 in. for the rotor. Normal practice is to

avoid even blade counts for both the rotor and stator to reduce rotor blade

vibration response. The blade chord is now calculated from Zweifels

relation given in Eq. (54) using the optimum value of 0.8 for the Zweifel

coefficient:

cx 2

z=s

cos

ao

cosai sinaiao For the stator,

cxstator 2

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

cos0 sin65:22

0:460 in:

and for the rotor,

c

xrotor 2

0:8=0:384 in:cos

59:04

cos26:57 sin

26

: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 are

required to calculate the density. The viscosity is calculated using the total

temperature to approximate the temperature in the boundary layers where

viscous effects dominate. For the stator, the exit total temperature is the

same as the inlet temperature. We assume a 1% total pressure loss across the

stator. Using the stator exit Mach number, the static pressure is calculated:

p144:1 psia0:990:99 1161:05172

3:5 21:18 psia

as is the static temperature:

T1 760R 1161:05172

619:9 R

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



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from that presented by ASHRAE [19]:

m ffiffiffiffiT

p

1:34103

306:288=T

13658:3=T2

1; 239; 069=T3

6106 lbm

ft-sec

The original expression was in SI units. For the stator, the viscosity

m11:59986105 lbm=ft-sec. The Reynolds number is then calculated usingEq. (53):

Rth

stator

0:09225 lbmft3

1297:4 ftsec

1:59986

105 lbm

ft-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 (station

2) are used. The viscosity is calculated using the relative total temperature

determined using Eq. (17). For the rotor, the Reynolds number is

1:23566105.

The reference value of the loss coefficient x is found fromFig. 13as afunction of the deflection across the blade row. The deflection is the

difference between the inlet and outlet flow angles. For the stator, the

deflection is 65.228, and for the rotor it is 85.618, resulting in xref s0:068andxref r0:083, assuming a blade thickness ratio of 0.2. The adjusted losscoefficients are calculated from Eq. (52):

xstator 105

1:91036105 1=4

10:068 0:9750:075 0:4600:626 1

0:0852

and for the rotor

xrotor 105

1:23566105

1=410:083 0:9750:075 0:551

0:626

1

0:1209

The stator inlet to rotor exit total-to-total efficiency is calculated from theratio of the energy extracted from the flowUDVy divided by the sum of



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the energy extracted and the rotor and stator losses:

Z0020ZC UDVy

UDVy

12

V22xstator

12

W22xrotor

Numerically, we have

Z0020ZC 1:3906:15 ft

sec2

1:3906:15 ftsec

2 0:08522 1297:39 ft

sec2 0:1209

2 1056:75 ft

sec2

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 1 1

0:85

0:015

0:626

0:8597

and correcting for overall pressure ratio using the total-to-total pressure

ratio fromFig. 16results in the overall efficiency:

ZOA0:85971 1=2:7201g1=g

1 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 developed

for blades with high aspect ratios (h=cxin the range of 34), which will resultin higher efficiency than lower aspect ratios, such as in this example. For

preliminary sizing purposes, the conservative result should be used.

Partial Admission Turbines

For applications where the shaft speed is restricted to low values or the

volumetric flow rate is very low, higher efficiency can sometimes be obtainedwith a turbine stator that only admits flow to the rotor over a portion of its

circumference. Such a turbine is called a partial-admission turbine. Partial-

admission turbines are indicated when the specific speed of the turbine is

low. Balje [3] indicates partial admission to be desirable for specific speeds

less than 0.1. Several conditions can contribute to low specific speed.

Typically, drive turbines operate most efficiently at shaft speeds higher than

the loads they are coupled to, such as generators, hydraulic pumps, and, in

the case of an air turbine starter, the main engine of an aircraft. For low-cost

applications, it may be desirable to eliminate the speed-reducing gearboxand couple the load directly to the turbine shaft. In order to attain adequate



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blade speed at the reduced shaft speed, it is necessary to increase the turbine

diameter, which causes the blade height to decrease. The short blades cause

an increase in secondary flow losses reducing turbine efficiency. With partial

admission, the blade height can be increased, reducing secondary flow

losses. In a low-flow-rate situation, maintaining a given hub-to-tip radiusratio results in an increase in the design shaft speed and a decrease in the

overall size of the turbine. However, manufacturing limits restrict the radial

tip clearance and blade thickness. With a small blade height, tip clearance

losses are increased. With a limitation on how thin blades can be made, it is

necessary to reduce blade count in order to keep trailing-edge blockage to a

reasonable level. Fewer blades result in longer blade chord and reduced

aspect ratio, leading to higher secondary flow losses. The taller blades

associated with partial admission can increase turbine performance. For

high-head applications a high blade speed is necessary for peak efficiency.With shaft speed restricted by bearing and manufacturing limitations, an

increase in turbine diameter is required, resulting in a situation similar to the

no-gearbox case discussed earlier. Here, too, partial admission can result in

improved turbine efficiency.

The penalty for partial admission is two additional losses not found in

full-admission turbines. These are the pumping loss and sector loss. The

pumping loss accounts for the drag of the rotor blades as they pass through

the inactive arc, the portion of the circumference not supplied with flow

from the stator. The sector loss arises from the decrease in momentumcaused by the mixing of the stator exit flow with the relatively stagnant fluid

occupying the blade passage just as it enters the active arc. Instead of being

converted into useful shaft work, the stator exit flow is used to accelerate

this stagnant fluid up to the rotor exit velocity. An additional loss occurs at

the 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 flow into the rotor blade passage is reduced. This reduced flow has the

entire blade passage to expand into. The sudden expansion causes a loss in

momentum resulting in decreased power output from the turbine. Lossmodels for partial-admission effects are not as well developed as those for

conventional, full-admission turbines. As a historical basis, Glassman [1]

presents Stodolas [20] pumping loss model and Stennings [21] sector loss

model in an understandable form and discusses their use. More recently,

Macchi and Lozza [22] have compiled a number of more modern loss

models and exercised them during the design of partial-admission turbines.

The reader is referred to those sources for detailed information regarding

the estimation of partial-admission losses.



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RADIAL-INFLOW TURBINE SIZING

Differences Between Radial-Inflow and Axial-Flow Turbines

Radial-inflow turbines enjoy widespread use in automotive turbochargers

and in small gas turbine engines (auxiliary power units, turboprops, andexpendable turbine engines). One advantage is their low cost relative to

machined axial turbines, as most of these applications use integrally bladed

cast radial-inflow turbine wheels.

The obvious difference between radial-inflow and axial-flow turbines is

easily seen inFig. 1; a radial-inflow turbine has a significant change in the

mean radius between rotor inlet and rotor outlet, whereas an axial-flow

turbine has only a minimal change in mean radius, if any. Because of this

geometric difference, there are considerable differences in the performance

characteristics of these two types of turbines. Referring to the typicalradial-inflow vector diagram of Fig. 9(c), the radius change causes a

considerable decrease in wheel speed Ubetween rotor inlet and outlet. For

zero exit swirl, this results in a reduced relative exit velocity compared to an

axial turbine with the same inlet vector diagram (since U2&U1 for an axial

rotor). Since frictional losses are proportional to the square of velocity, this

results in higher rotor efficiency for the radial-inflow turbine. However, the

effect of reduced velocity level is somewhat offset by the long, low-aspect-

ratio blade passages of a radial-inflow rotor.

Compared to the axial-flow diagram of Fig. 9(a), there is a much largerdifference between the rotor inlet relative and absolute velocities for the

radial-inflow diagram. Referring to Eq. (17), this results in a lower relative

inlet total temperature at design point for the radial-inflow turbine. In

addition, due to the decrease in rotor speed with radius, the relative total

temperature decreases toward the root of radial-inflow turbine blades (see

Mathis [23]). This is a major advantage for high inlet temperature

applications, since material properties are strongly temperature-dependent.

The combination of radial blades at rotor inlet (eliminating bending stresses

due to wheel rotation) and the decreased temperature in the high-stressblade root areas allows the radial-inflow turbine to operate at significantly

higher wheel speeds than an axial-flow turbine, providing an appreciable

increase in turbine efficiency 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 not

constrained by stress considerations and the radial-inflow turbine is at a size

disadvantage. 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



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radial-inflow turbine will be higher than that for an axial-flow turbine,

which can have Vy1=U1 > 1 with only a small impact on efficiency. Thisassumes zero exit swirl. For a fixed shaft speed, this means that the radial-

inflow turbine will be larger (and heavier) than an axial-flow turbine. Stage

work can be increased by adding exit swirl; however, the radial-inflowturbine is again at a disadvantage. The lower wheel speed at exit for the

radial-inflow turbine means that more Vy2 must be added for the same

amount of work increase, resulting in higher exit absolute velocities

compared to an axial-flow turbine. In addition, high values of exit swirl

negatively impact obtainable diffuser recoveries.

Packaging considerations may lead to the selection of a radial-inflow

turbine. The outside diameter of a radial-inflow turbine is considerably

larger than the rotor tip diameter, due to the stator and inlet scroll or torus.

Compared to an axial-flow turbine, the radial-inflow package diameter maybe twice as large or more. However, the axial length of the package is

typically considerably less than for an axial turbine when the inlet and

diffuser are included. Thus, if the envelope is axially limited but large in

diameter, a radial-inflow 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-inflow turbines have the advantage of lower free-run speed than an

axial turbine of comparable design-point performance. Figure 11shows the

off-design performance characteristics of both radial-inflow and axial-flowturbines. At higher shaft speeds, the reduction in mass flow for the radial-

inflow turbine leads to lower torque output and a lower free-run speed.

Because of the change in radius in the rotor, the flow through the rotor must

overcome a centrifugal pressure gradient caused by wheel rotation. As shaft

speed increases, this pressure gradient becomes stronger. For a given overall

pressure ratio, this increases the pressure ratio across the rotor and

decreases the pressure ratio across the stator, leading to a reduced mass flow

rate. A complete description of this phenomenon and its effect on relative

temperature at free-run conditions is presented by Mathis [23]. However, therotor disk weight savings from the lower free-run speed of a radial-inflow

turbine is offset by the heavier containment armor required due to the

increased length of a radial-inflow turbine rotor compared to an axial

turbine.

Radial-Inflow Turbine Performance

The literature on performance prediction and loss modeling for radial-

inflow turbines is substantially less than that for axial-flow turbines. Wilson[2] states that most radial-inflow turbine designs are small extrapolations or



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interpolations from existing designs and that new designs are executed using

a cut-and-try approach. Rodgers [24] says that minimal applicable

cascade test information exists (such as that used to develop many of the

axial-flow turbine loss models) and that exact analytical treatment of the

flow within the rotor is difficult due to the strong three-dimensionalcharacter of the flow. Glassman [1] presents a description of radial-inflow

turbine performance trends based on both analytical modeling and

experimental results and also describes design methods for the rotor and

stator blades. More recently, Rodgers [24] has published an empirically

derived performance prediction method based on meanline quantities for

radial-inflow turbines used in small gas turbines. Balje [3] presents analytical

performance predictions in the form of efficiency versus specific speed and

specific diameter maps.

For our purposes, we will use the results of Kofskey and Nusbaum[25], who performed a systematic experimental study investigating the effect

of specific speed on radial-inflow turbine performance. Kofskey and

Nusbaum used five different stators of varying flow area to cover a wide

range of specific speeds (0.2 to 0.8). Three rotors were used in conjunction

with these stators in an attempt to attain optimum performance at both

extremes of the specific speed range. Results of their testin

7- Fundamentals of Turbine Design

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