Axial Turbine Design Steps

GAS TURBINE ENGINES - GENERAL CHAPTER

Contents: INTRODUCTION

PART 1. AERODESIGN OF THE MAINSTREAM FLOW PATH AND PRELIMINARY DESIGN PRACTICE (brief review, reference materials) PART 2. AEROPROPULSION REQUIREMENTS AND JET ENGINE PART 3. TURBOFAN ENGINES PART 4. TURBOSHAFT, TURBOPROP AND PROPFAN ENGINES

INTRODUCTION: MAIN CLASSIFICATION OF GAS TURBINE ENGINES

The following four sections are compiled to brief the readers on various configurations of gas turbine engines reflecting their applications for aircraft propulsion and also for the industrial power generation and mechanical drives. Presented below Part 1 provides classification of gas turbine engines and intends to refresh some basics of engine mainstream aerodynamics and discusses some aspects of the engine development process. Although initial turbine concepts go back a few centuries ago with early reaction steam turbine that followed in 19th century with Rankine cycle steam power turbine, the main activity in this field has started after the invention of jet engines operating as Brayton (gas) cycle with particular interest in their application for aero-propulsion.

Historically the early gas turbine engines have followed the path of steam turbines and reciprocating (piston) internal combustion engines used as mechanical driving power. Similarly to these engines, the first generation of gas turbines has focused on full utilization of the available energy to drive mechanical or electric power generation load. All attempts were made to minimize the energy losses resulting from the exhaust velocity of the working fluid. Only in the third decade of the 20th century with invention of the jet engine, the main benefits of gas turbine namely light weight, fast start and high power density have been fully realized for aircraft propulsion using reaction force (thrust) of high velocity of the exhaust gases. This followed extensive development and application of these engines for both military and civil aviation, adding jet engines to the already traditional engine that drives an output shaft. Thus, in addition to relatively slow aircrafts using the turboshafts and turboprops, the aviation led development of the jet and turbofan engines for subsonic and then supersonic aircrafts. Advancements of the gas turbine engine design and materials required for development of the military engines have benefited also industrial engines, providing significant improvement of their efficiency and durability. Parts 2, 3 and 4 discuss correspondingly jet, turbofan and turboshaft/ turboprop engines. The last group also includes the industrial heavy-duty and aero-derivative engines.

Gas turbine engines are the power plants generating thrust for most of aero (air borne) engines and sources of turboshaft power for various military and industrial land/marine applications such as tanks, marine vessels, heavy trucks/trains, electric power generation, drivers for natural gas compressors and pumping systems. In some cases aero derivative engines can be utilized for industrial and marine

(I. Newton- XVII Century)

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applications. Examples below show (clockwise from the upper left corner) some of the more recent applications for gas turbine engines as prime movers for tanks, helicopters, commercial and private aviation, solar energy conversion, electric power generation and pipeline gas compression, liquid rocket propulsion, heavy ground transportation, naval vehicles, drones and military jets.

Gas Turbine Engine

Applications

Figure below presents classification of gas turbine engines for aircraft application As mentioned above, the industrial applications are based primarily on turboshaft engine configuration with dedicated power turbine or turbine driving both engine compressor and output load. Invention of jet engines in the 1930th and their fast development have opened tremendous opportunity for their aircraft application, first allowing a high subsonic speed and later reaching and then exceeding the speed of sound.

Turbojets are based on reaction force of combustion products discharged from a gas producer turbine (GPT) through the exhaust nozzle. Their design can be classified as: - single spool (shaft) with turbine driving all stages of compressor, and - double spool configuration consisting of low pressure compressor (LPC), typically driven by a low pressure turbine stages (LPT), and high pressure compressor (HPC) driven by a faster high pressure turbine stages (HPT). The remaining energy of combustion products provides the useful thrust of the aircraft. Design of compressors can be further classified as axial (usually a relatively low pressure ratio per stage multistage compressor) and centrifugal (significantly higher pressure ratio per stage). Design specifics of the centrifugal compressors usually limit their application to relatively small engine thrust/shaft power. Combination of axial LPC with centrifugal HPC can significantly expand their application for a medium thrust/power range. Turbojets can also be classified as nonafterburning and afterburning additional fuel for achieving an increased thrust primarily for fighter aircraft where high speed and fast takeoffs becomes more critical than cost of extra fuel.

Turboprop engine design typically requires a free power turbine (PT) shaft, in addition to GPT, to drive a large propeller, thus utilizing the most of energy of the exhaust gases discharged from GPT. The products discharged from PT can generate only a small amount of propulsion thrust with the most of thrust produced by a propeller. The most efficient application of turboprops is for a relatively low speed (350-450 mph) aircrafts.

Turboshaft engines are similar to turboprops except they are designed to fully utilize expansion of the combustion products within the turbine, thus practically eliminating residual thrust of discharged gases. Their application is limited to even slower speed aircrafts, helicopters, for example.

Turbofan engines are designed as a compromise between the turboprop and turbojet engines by generating two streams of air: thrust generating ducted bypass air from the fan blades (internal propeller) and the GPT mainstream contributing to the engine thrust similarly to turbojet. Optimal bypass ratio (BR) depends on turbofan engine application: high BR for subsonic mainly civil aircrafts (such engines as GE90, CF6 family, RB211, PW 4000) and low BR for supersonic vehicles. The last group can be nonafterburning and afterburning configurations (GE F100, PW F119 and PW-RR JSF119-611, GE F404 family). Propfan engines combine the advantages of the turboprop propeller and the unducted fan of the turbofan with usually two counterrotating forward (D-27, Russia) or aft propellers (RR RB3011). Propfans are also known as ultra-high bypass engines. The design is intended to offer the speed and performance of a turbofan, with the fuel economy of a turboprop.

a) Afterburning turbojet d) Turbojet with dual entry centrifugal compressors b) Ramjet e) Propfan

Comparison between Axial Flow Compressor and Axial Flow Turbine Parameter Compressor Turbine Flow Meridianal passage Blade-to-blade passage Maximum blade thickness Direction of rotation

Number of stages Hub-to-tip ratio (first stage) Blade height Annulus area in a multistage Number of blades Temperature trend in the flow

PR IR and TR Max operating flow temperature Specific work Stage arrangement Flow deflection deg Maximum stage efficiency

Decelerating Divergent Vanes/Diffuser Small From suction to pressure surfaces 3-20 0.4-0.6 Reducing downstream

Decreasing Smaller than in turbines

Rises, small per stage Increase 350 600 C

U (Cu2 Cu1 ) Rotor--stator 20-35 0.88

Accelerating Convergent Nozzle Large From pressure to suction surfaces 1-4 > 0.7 Increasing downstream Increasing Larger than in compressor

Drops, large per stage Decrease 850-1700C U (Cu2 + Cu3) Stator-rotor 50-180 0.94

PART 1. AERODESIGN OF MAINSTREAM FLOW PATH AND PRELIMINARY DESIGN PRACTICE

The turbine blades operate in a much more hostile environment than compressor blades. Positioned just downstream of the combustor and nozzle vanes, the blades are exposed to an entry temperature of 950-1700C and very high stresses. Therefore, turbine blades have to use advanced materials and manufacturing methods. Most of the hottest first stages of turbine blades are hollow and require cooling by the air bled off the compressor.

COMPARISON OF AXIAL FLOW COMPRESSORS AND TURBINES

Through the turbine the flow is accelerated while it is decelerated for the compressor requiring the blade-to-blade passage to be convergent for turbine and divergent for compressor. However, due to changing flow pressure and correspondingly gas stream volume the meridional plane (axial-radial directions) is convergent for a compressor and divergent for a turbine. Other comparisons are listed below:

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Aerodynamics And Thermodynamics For Two-Dimensional Flow

= Ua2/Ca2=tan 2 - tan 2 and Ua3/C3= tan 3 - tan 3

If the axial velocity is constant at inlet and outlet of the stage, then Ca2 = Ca3 = Ca and

U2 = U3 = U.

The kinematical relations will be

U/Ca=- tan 2 - tan 2= tan 3 - tan 3

Thus, the following relation is reached:

tan 2 + tan 3 = tan 2 + tan 3

Assuming a constant rotational speed (U2 = U3 = U), the following kinematical relation between the absolute and relative velocities at the rotor inlet and outlet is valid:

Cu2 + Cu3 = Wu2 + Wu3

Ua2/Ca2=tan 2 - tan 2 and Ua3/C3= tan 3 - tan 3

Single stage of axial turbine is shown in the following page. The gases leaving the combustion chamber enter the stator nozzle with an absolute velocity (C1), typically in an axial direction thus, the absolute angle to the axial direction (l = 0). Since the passage forming a nozzle is either a convergent or convergent-divergent shape, the flow is accelerated resulting in an increased absolute velocity, so the flow leaves the stator at a speed (C2), where (C2 > C1) . The static pressure decreases while the gas flows through the nozzle. Moreover, the total pressure decreases due to skin friction and other sources of losses. Thus, P1 > P2, Po1>Po2. Correspondingly, the static enthalpy and total enthalpy drop toward the exit of the nozzle. The flow leaves the stator with an absolute speed of (C2) and an angle (2) when combined with the rotational speed (U); the relative velocity will be (W2), inclined at an angle (2) to the axial direction. The velocity triangles at the exit of the nozzle and exit of the rotor are shown in the following figure. The rotor blades extract work from the gases. As the rotor blade-to-blade passages are also converging, the relative velocity (W3) of the gas will increase (W3 > W2) while its absolute velocity C3 decreases. (C3 < C2) and the flow exit angle (3) will be greater than the flow inlet angle ( 3 > 2). The static and total pressures also drop in the rotor passage (P3 < P2, Po2 < Po3) with corresponding drop in the static enthalpy and total enthalpy toward the exit of the rotor. As the static pressure and temperature decrease in the turbine, the density decreases. The annulus area is increased to avoid excessive high velocities.

The analysis of a turbine stage is similar to that of a compressor stage. The same kinematical relation described in centrifugal and axial compressors is valid:

In a manner similar to the axial compressor, the following kinematical relations are

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(a) Common base velocity triangles. (b) Common apex velocity triangles. (c) Parallel rotational speed velocity triangles.

Detailed Velocity Diagrams

(c)

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Eulers equation for the specific work :

Ws =( U x C u ) 3 - ( U x C u ) 2

Layout of an axial turbine having variable mean radius.

In some cases, the turbine has a continuously varying mean radius as shown in figure below. The corresponding velocity triangles shown in Figure represent the most general case where neither the rotational speed nor the axial speed is equal at the inlet and outlet of the rotor. The specific work will be expressed in the general form

Ws= U2 Ca2 tan 2 + U3 Ca3 tan 3

If the axial velocities are unequal, but the mean radius and consequently the rotational speed are equal (U2 = U3 = U), then:

Ws = U (Ca2 tan 2 + Ca3 tan 3 ) - U (Ca2 tan 2 + Ca3 tan 3)

Since the swirl velocities C1 and C2 are in opposite directions,

Ws= (UCu ) 3 + (UCu ) 2

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the

below

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The velocity loss in the rotor is

it is also the ratio between the actual and ideal exit velocities for the rotor. The pressure loss may be measured in a cascade tunnel, while the enthalpy loss can be used in the blade row design. Both are nearly equal, or

The total-to-total isentropic efficiency can be rewritten as

1/s = 1+ (Ca/2U)[(R sec23+ (T3/T2) (N --sec2 2)/ (tan3+tan2 - (U/Ca))]

NONDIMENSIONAL QUANTITIES

I. Blade-loading or temperature drop coefficient 2. Flow coefficient 3. Degree of reaction

The temperature drop or blade loading is expressed as

.

the United Kingdom, defines this parameter as

The flow coefficient has the same definition as in the axial compressor, namely,

=Ca2/U

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739

13

Temperature loading versus flow coefficient for different degree of reactions.

Temperature loading versus flow coefficient for = 0.5.

(14.39) (14.40)

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Example 1. The object is a three-stage axial flow turbine where the total inlet temperature is 1100 K, the flow coefficient is 0.8, and the mean rotational speed is constant for all stages and equal to 340 m/s. Additional data presented in the following table:

Degree of Efficiency Inlet Temperature Stage Angles Reaction (s) Temperature (K) Drop [To K] 1 l=0 0.5 0.938 1100 145 2 I=2 0.5 0.938 ? 145 3 3 = 0 ? 0.94 ? 120

It is required to:

I. Sketch the turbine layout. 2. Draw the velocity triangles at the mean sections for each stage (assume constant

mean radius for all stages). 3.Calculate the degree of reaction at the mean section for the third stage. 4. Calculate the pressure ratio and temperatures of each stage. 5. Prove that for equal pressure ratios of the three stages, the efficiencies must have

the following relations:

s2 To1 xTo2 sl Tol x (Tol T0I)

s To1xTo3 _ sl To1 x (To1 To1 T02)

Solution 1. The sketch below illustrates the three-stage axial turbine having a constant mean radius.

2. To draw the velocity triangles. the angles (2, 3, 2, 3 ) must be calculated first. The appropriate governing equations are the temperature drop and degree of reaction.

Layout of the three-stage axial turbine.

----- =

----- =

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As described earlier: tan 1 + tan2 = CPTo/(UCa)

tan3 - tan2 = 2

Now solving these two equations simultaneously results in the angles 2 and 3 .

For Stage ( 1 ) Since T01 = 1100 K and Tos=145 K, To3 = 955 K.

Also, as m= 1/2 2 =3, 3 =2

tan 3 + tan 2= 1148 x 145/ (340x272 )= 1.8

tan b3 - tan b2 = 2x 0.5/ 0.8 =1.25

Then 2 =3 =15.376 and 3 =2 =56.746 deg.

Stage (2)

The same results can be obtained for stage (2) as both have the same temperature drop per stage, flow coefficient, and axial and rotational speeds.

2 = 3 = 15.376deg

3 - 2 = 56.746deg

Stage (3)

tan 3 + tan 2= Cp Tos/(U--Ca ) Since 3 =0 and Tos= 120K

tan 2 = Cp Tos/(U--Ca ) and 2 =56.125deg

To calculate the angles 2 and 3, we again use Equations 1 and 2

tan3 + tan 2 =1148 x 120/ ( 340x272) = 1.4896

Also tan 3 - tan 2 =2 x0.4042/0.8=1.0105

Solving both equations, we get 2 = 13.471 and 3 = 51.341deg.

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."-'

7

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Rotor Axial Thrust Force

above

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Dimensioning an axial turbine stage

reason is that the outlet speed from the nozzle C2 will be high and, consequently, the rotor inlet relative speed W2 will also be high enough to form shock waves within the rotor passage.

The convergent nozzle must be checked for choking. Thus if (P01/P2 < Po1 /Pc), then the nozzle is unchoked. Each ratio can be calculated as follows:

P01/P2 = (T01/T2)/(-1) where T2 =T2 -NC/2Cp and

For choked nozzle, the absolute velocity is equal to the sonic speed, as is illustrated later on. The annulus of turbine stages is generally flared (figure below). The annulus of turbine stages is generally calculated. The flare angle is calculated when the blade heights and widths are calculated first. The blade heights are calculated as follows. The axial velocities at stations (I), (2), and (3) are determined first. Next, the absolute velocities at the three states are determined. The static temperature and pressure are calculated as follows: T = To - (C2/2Cp) and P = Po(T /To)/(-1) ; thus, the density is = P/RT . The blade height is then obtained for a constant mean radius from the relation h = m/CaDm . The blade mean height is then calculated as follows: h nozzle = (hl + h2)/2 and h rotor = (h2 + h3)/2. The blade chord ranges from (h/4) to h (smaller for later stages). The spacing between the blade rows must be greater than 15% of the chord. The flare can then be calculated.

If the included angle of divergence of the walls is large, then a flow separation from the inner wall may be encountered. A safe value for the included angle is to be less than or equal to 20deg. Both the flare angles, height-to-width ratio of the stator/rotor blades, and the spacing between the stator and rotor blade rows are design parameters. They depend on aerodynamics and mechanical stresses of the stator and rotor blades. Rotor blades are subjected to vibration stresses as well as mechanical stresses. Vibration stresses arise as the rotor blades pass through the wake of the nozzle blades. Excessive vibration stresses arise when the space between the blade rows is less than 0.2 of the blade width. A value of nearly 0.5 is often used for downstream stages, which will reduce both the vibration stresses and the annulus flare.

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Example 2. A single-stage axial flow turbine has a mean radius of 30 cm and a blade height at free stator inlet of 6 cm. The hot gases enter the turbine stage at 1900 kPa and 1200 K and the absolute velocity leaving the stator (C2) is 600 m/s and inclined at an angle 65deg to the axial direction. The relative angles at the inlet and outlet of the rotor blade are 25 and 60deg, respectively. the stage efficiency is 0.88.

Calculate:

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5.

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Usually, engine design process is initiated by need recognition with the market environment defining specific customer requirements. The design itself begins with preliminary studies such as cycle selection and analysis, type of turbomachinery used (axial/mixed/radial), and conceptual layout. This preliminary procedure defines the inlet total conditions, expansion ratio, power required, mass flow rate and shaft speed.

If axial turbine is selected, then the design process starts with preliminary design that includes aerodynamic, mechanical, and thermal steps that are next followed by sophisticated analyses using computational fluid dynamics (CFD) for simulating the real flow to the best available accuracy. This last step is followed by numerous experimental testing for scaled and prototype models. Design procedure is known as an iterative procedure. Thus, several checks have to be carried out during the above steps and modifications are adopted when there is a necessity.

MAIN DESIGN STEPS

The following steps are used in preliminary analysis:

1. The number of stages (N) is first determined by assuming the temperature drop per stage. For aero engine it is between 120 and 250 K (larger usually for the first stage), thus:

Nst= (To)turbin/(To)stage 2. Aerodynamic design It may be subdivided into mean line design and variations along the blade span.

3. Blade profile selection The blade profile and the number of blades for both stator and rotor are determined.

4. Structural analysis It includes mechanical design for blades and discs, rotor dynamic analysis, and modal analysis. Normally, both aerodynamic and mechanical designs are closely connected and there is considerable iteration between them.

5. Cooling The different methods of cooling will be described in a separate chapter. In most cases, combinations of these methods are adopted to satisfy an adequate lifetime. Structural and cooling analyses determine the material of both stator and rotor blades.

6. Checking stage efficiency. 7. Evaluating off-design. 8. Rig testing

Aerodynamic Design It may be subdivided into mean blade design and variations along the blade span. (a) Mean line design analysis It starts by preliminary choices for the stage loading, flow coefficient, and degree of reaction aero engines, the following design parameters may be used as a guide: * Stage loading ranges from 1.5 to 2.5 (E3 up to 3.5) * Flow coefficient ranges from 0.8 to 1.0 * Stage efficiency ranges from 0.89 to 0.94

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0.7

0.6

0.5

40 50 60 70 Relative efflux angle (2 or 3)

Optimum pitch/chord ratio (solidity). (H.I.H. Saravanamuttoo, et al., Gas Turbine Theor).

Blade Geometry Definition

Originally, airfoil sections were laid out manually using a set of circular arcs and straight lines according to various empirical rules based on the experience of successful designs. Turbine airfoil sections are thicker than those of compressor to withstand very high temperatures and to allow for cooling passages. The airfoil is described by a set of 11 parameters including the airfoil radius, axial and tangential chords, inlet and exit blade angles, leading and trailing edge radii. together with a number of blades and throat. Modern airfoil section generator will typically define the blade pressure and suction surfaces

Usually, a minimum of three sections at hub, mean, and tip radii are used. This suffices only for the simplest blade geometries. For complex or three-dimensional shapes, 10 or more sections may be used. Once the blade sections have been generated, cascade testing or a simple computational two- dimensional blade-to-blade analysis may be used. Thus, the velocity or Mach number distribution over the suction and pressure surfaces is determined. The next step is to stack the airfoil section to create the three-dimensional blade shape. A stacking line is defined, which determines the relative position of each of the sections with one another. The stacking line need not be straight or radial, but it can be curved or leaned to produce complex blade shapes. Sections may be stacked on the center area of each section, or the leading, the trailing edge, or some other arbitrary line.

The blade profile and the number of blades for both stator and rotor are determined as follows. The mean heights for nozzle and rotor blade are determined as hN = (h1+h2)/2 and hR = (h2 + h3)/ 2 The blade aspect ratio (h/c) ranges from 0.7 to 6, lower value corresponds to first stage and higher to last stages. The optimum pitch/chord ratio (s/c) of the mean line flow is determined from known inlet and exit angles for stator (1 , 2) and rotor ( 2, 3). From industry experience: at the tip s/c 0.5 at the root. The corresponding chords for nozzle and rotor (CN, CR) are determined and thus the spacing for the nozzle and rotor blade rows (sN , sR ) are determined. Now, the number of blades is determined from the relation (see also Appendix for recommended formula for vane and blade count) nN= 2 r m/sN and nR = 2 rm/sR It is usual to avoid numbers with common multiples to reduce the possibilities of resonant frequencies.

It is a common practice to use an even number for the nozzle and a prime number for the rotor blades, particularly for stage 1.

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Mechanical And Structural Design Mechanical and structural designs have a direct impact on aerodynamic design. Both designs are interrelated to achieve high aerodynamic performance, and adequate life and structural integrity. Vibration analysis is very important for rotor blades to avoid resonance and the possibility of blade breakage. An assessment of stress level is considered here.

There are two types of stresses that must be considered, namely, steady and unsteady stresses. Steady-state stresses arise from the centrifugal, pressure loading, and thermal sources. Unsteady-

state stresses arise from the interaction of the rotating blade with stationary features in its vicinity (similar to any two successive blade rows), thermal cycling for turbine downstream combustion chambers as the products of combustion are not completely uniform in the tangential direction . There will also be some mechanical effects caused by residual imbalance and rub out of bearings. Failure in both cases has different features. Failure due to steady stresses causes the ductile blade material to creep or elongate and fail ultimately in plastic deformation. Failure due to unsteady stresses causes the blade to fail through fatigue due to crack initiation and growth.

Turbine blades in aero engines are designed to avoid creep. The allowable stress in turbines depends strongly on the operating temperature and specified temperature, because that stress will not exceed a creep extension of typically I % for 100,000 h at the specified temperature. Since 100,000 h) is a too long period for testing, it is common to compare materials for the stress causes rupture in accelerated 100 hour test. The rupture stress divided by blade material density is plotted versus the working temperature. An increase in the working temperature is associated with a decrease in the 100 h rupture stress/density (s/p) ratio. Ceramics have superior compressive strength-temperature characteristics. However, they are typically brittle in bending and tension and also unreliable in thermal cycling environment. Exception might be application of some ceramic composite materials. Adopting ceramic composites, for example, by including high-strength tin whiskers, may provide a solution to the problem where high strength and reliability could be achieved. For example, Si3N4 ceramic may have a 100 rupture stress/density ratio of 35 (kPa/kg/m3) at as high working temperature as 1200C. In general, there are three types of mechanical stresses for turbomachines. For turbines, there is an additional thermal stress due to high temperature gradient within the blades. The mechanical stresses are: 1.Centrifugal stresses 2.Gas bending stress 3.Centrifugal bending stress

Centrifugal stress (which is a tensile stress) is important for both blades and the discs that support them. It is the largest, but not necessarily the most important, since it is a steady stress. Bending stresses arise in the blades from the aerodynamic forces acting on them due to steady or unsteady conditions. Thermal stresses arise due to the temperature gradient within the blade, which may be caused by the surrounding high temperature gases and the cold air flowing into the cooling passages.

Centrifugal Stresses in Blades For a given hub/tip ratio, centrifugal stress at blade root depends on: 1. Blade density(b) 2.Hub-to-tip radius ratio ( ) 3.Blade tip speed (Ut) It increases with blade twist due to centrifugal loading. It can be reduced by tapering the blade cross section.

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Blades

()rim

rAb+ d[(rAb)dr]/dr

Stresses in blades

Simplified layout of a rotor blade row is shown below. Consider an element on the blade at a radius (r) having a cross-sectional area, Ab , and thickness, dr. The element is subjected to a radial stress ( r ) and a centrifugal force due to its rotation with an angular speed .. Force balance of this element yields the following relation:

where o is the blade material density. If the cross-sectional area is assumed to be constant, then this relation is reduced to

Integrating from any radius to the tip radius and noting that the centrifugal stress is zero at blade tip, the stress at any radius will be

The maximum centrifugal stress occurs at the blade root radius (r = rh) or

where = rh/rt is the root-to-tip ratio ( industry recommends 0.5< < 0.85), N is the rotational speed and A is the annulus area. As a rough rule b=0.25x 501x(N/1000)2xA/(144x2240) tons/in2 , where A [sq.in], =501(density of steel). From industry experience AN2 should be < 4x1010 in2rpm2 for the unshrouded blades and < 3x1010 for the shrouded blades.

d

dr

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Blades

Outer rim (Cross-section area AD)

Inner rim/ hub (Cross-section area Ai)

Main disk stress related dimensions

The maximum blade stress factor, sb/pbUt2, has a value of 0.5 (at a fictitious hub -to-tip ratio of zero) and decreases rapidly as the hub-to-tip ratio increases.

Since turbine blades are tapered blades both in chord and thickness, the ratio between tip and root areas (at /ah) is between I /4 and 1/3; thus, it is accurate enough to assume that tapering of the blade reduces the stress to 2/3 of its value, or

For linear tapering, the exact expression

Centrifugal Stresses in Discs The centrifugal stress is checked in three locations, namely, the outer rim, inner rim, and in the disc material between both rims. The following assumptions are considered: 1. Centrifugal force applied by the blades to the rims can be considered equivalent to a

radially uniformly distributed load. 2. The disc is tapered such that its circumferential and radial stresses are constant everywhere and equal to (). 3. Rims are thin enough to have uniform tangential stress equal to ().

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The stress in the disc can also be rewritten as a stress factor in the following form:

s _ < 2 [1 _ ( ripb

2sAi - s (ridq) ti -

Force balance on outer rim

Stresses on outer rim and inner radius of disc.

Force balance on segment of a disk

1.Outer rim: Applying Newton's second law of motion over a small element on the outer rim the sum of force (on the left-hand side) will be equal to the mass multiplied by acceleration (right - hand side).

Dividing by A0 d bUt2 gives

s _ (Nb/2p)(Ab/Ao)(sb/bU12)+2_ pbU12 - 1+(rbtolAo)

2. Disc segment-. Also, apply Newton's second law to a segment on the disc. Refem.fig to Figure 14.31, a segment of the disc has a width of (dr), arc length of (rdO), and thickness (t). Force balance will give the relation

(2s tdr) - (strdO) dr - pb (trdqdr) W2r ` ' 2 dr ` ' '-` _

Since s is assumed to be constant, simplifying to get

d dt pbW2r2t t - (Ir) - -r dr - s--

dt pW2 ` - ~-' ~rdr t s

Inner rim/ hub (Cross-section area Ai)

Disc rim and hub stresses

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From the above equations

Gas Bending Forces

From industry experience the disc rim speed should be limited to 1300 ft/sec

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The corresponding gas bending stress is

-MxY MYX sgb - [xx--+ fYY--

which is tensile at the blade leading edge and compression at the blade trailing edge. Now to evaluate the order of magnitude of each term, the axial force is evaluated first. The axial force is developed from two sources:

* Momentum change which is the case when there is a change in axial velocity (Ca2 Ca3). The axial force per blade is then

lFal = m ICa2 - Ca3l n

where n is the number of blades.

* Pressure difference, thus the force per unit blade height is Fa = (P2- P3)(2r/n).

The tangential force is determined from the power, which is correlated by the relation to the specific work ws

P=mws = mU (Cu2m + Cu3m)

However, the power is also expressed as P = Ft x U. Then, the tangential force is Ft = m (Cum + Cu3m) The tangential force per blade is then

Mt= ( Cu2 + Cu3 ) F fl " -- ` um__ "I'

The tangential moment per blade is

Ml __ Vlt _h __ 'rn (Cu2m + Cu3m ) ` `.2 n 2

If the tangential momentum is far greater than the axial momentum and the angle f the gas bending moment is approximated as

where /xx/y = S = Zc3 is the smallest value of section modulus.

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Parameter Value

110 10.803 6.584 1600 1430.5 169.5 0.91

1

m (kg/s) Inlet pressure (bar) Outlet pressure (bar) Inlet temperature (K) Outlet temperature (K) Temperature drop (DTos) (K) Isentropic efficiency (ht) Suggested number of stages

CASE STUDY The following section represents a preliminary design of an axial turbine. The steps followed in the design are given below.

Design steps 1. Calculation of flow properties at mean sections: flow angles, absolute and relative velocities, as well as total and static pressures and temperatures 2. Calculation of annulus dimensions 3. Calculation of the values of flow properties at different sections along the blade height from hub to tip, including airflow angles, absolute and relative velocities, together with the static pressures and temperatures 3. Estimation of the number of blades from data at mean section for nozzles and rotors 4.Calculation of the chord length at any section along blade height for nozzles and rotors 5. Selection of blade material that withstand thermal and mechanical stresses 6.Calculation of stresses on rotor blades 7. Turbine cooling 8.Estimation of design point performance 9. Calculation of turbine efficiency

14Design Point The turbine to be designed is a part or a module of either an aero engine or an industrial gas turbine. The following data represent the values of the design variables determined from the performance analysis of the main engine sections. Example

Only one stage is needed for this temperature drop. The rotational speed of the turbine driving the compressor has to match the compressor speed of 150 rev/s. Corresponding mean radius is assumed to be 0.4 m. The following values of some parameters are suggested.

m m 1m (deg.) N 0.45 0.8 0 0.05

The specific heat ratio and specific heat are assumed to be = 1.333 and Cp = 1148 J/kg K. From the temperature drop Tos = 169.5 K, the stage-loading coefficient is m = 1.5717. The design assumes a constant mean radius along the nozzle (stator) and rotor blade rows.

.

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Station (1) Station (2) Station (3) m deg m deg Um (m/s) Cam (m/s) Cum Cm/s) Wm (m/S) TO (K) Tm (K) PO (bar) Pm (bar) m (kg/m3) A(m2) h{m) rt(m) rm (m) rh(m)

0

293.4656

1600 1562.5 10.803 9.8250 2.1910 0.1711 0.0681 0.4340 0.4 0.3660

59.0841 22.7740

351.8584 281.4867 470.0339 305.2871

1600 1469.3

10.612 7.5460 1.7895 0.2184 0.0869 0.4434 0.4 0.3566

16.4269 57.0841

351.8584 281.4867

82.9897 518,0035

1430.5 1393

6.5847 5.9208 1.4810 0.2639 0.1050 0,4525 0.4 0.3475

Wc s/c No. of airfoils

2 0.8490

78

2 0.8015

67

Meanline flow The governing equations for mean flow calculations are described previously. The results of the mean flow analysis are given in the following table for the three stations upstream of the stator and rotor and downstream of the rotor.

Summary of the Geometry, Aerodynamic, and Thermodynamic Properties of a Turbine Stage:

Number of airfoils for nozzle vanes and rotor blades: From calculations at mean section, the blade heights at stations I, 2, and 3 are determined. Next, assuming that the aspect ratio (h/c) is in the range (1- 4), the blade chord at mean section (c) is calculated.

From figure below we can get the optimum pitch/chord ratio (s/c) as a function of the flow angles at the mean section. Then the number of blades is estimated from the relation:

Number of blades = 2 rm/s

This results in calculations of the following number of the nozzle vanes and rotor blades after rounding: Nozzles Blades

Turbine Map

The overall turbine performance is defined the same way as for the compressor in terms of similar non-dimensional parameters, namely, mass flow parameter (mT0i /Poi), speed parameter (N / T0i), pressure ratio, and thermal efficiency. If the turbine is composed of more than one stage then the efficiency previously calculated is considered as a polytrophic efficiency and the turbine efficiency is calculated from the relation:

1 - (P0e / P0i) s( -1)/ t=

1 - (P0e / P0i) ( -1)/

The maximum mass flow rate defines the choking conditions, which is constant for a given geometry and certain range of turbine pressure ratios, as shown in fig. below. The efficiency is also plotted versus the turbine pressure ratio. It could be noticed that the variations in mass flow parameter and efficiency are les significant for the turbine speed compared to compressor.

Turbine characteristics: a) mass flow versus pressure ratio and b) efficiency versus pressure ratio

Nondimensional parameters for sizing compressor and turbine stages (simplified method)

Compressor or turbine stage performance is dependent on flow rate, enthalpy change (or equivalent pressure head, rotational speed, size and working fluid. Two nondimensional similarity parameters combining these factors can be quite valuable for preliminary sizing and performance estimation of geometrically similar stages: specific diameter Ds=DHad1/4/ Q1/2 and specific speed Ns= NQ1/2/Had3/4 . From the Bernoulli equation, the added in a compressor or reduced in a turbine pressure difference is often called the pressure head h, defined as a difference between total exit pressure and total inlet pressure minus total pressure losses experienced during the compression/expansion process. The pressure head in the British units can be expressed as a height of the column of the flowing media measured in ft, so h =(P2-P1) lbf/in2 144 in2/ft2 / ( lbm/ft3 g ft2/s), where is a density of the fluid. For a preliminary design of compressor and turbine stages the designers often use so called n-d performance diagrams (figures below). These diagrams assist in selecting a diameter and a rotational speed for required flows and pressure ratios at optimal efficiency The coordinates of the diagrams are the mentioned above nondimensional specific diameter Ds=DHad1/4/ Q1/2 and specific rotor speed Ns=NQ1/2/ Had3/4, where H= hg [ ft-lbf/lbm] - another measure of the pressure head often used in engineering, Q[ft3/sec] - flow rate, D [ft] - diameter, N [rpm]- rotor speed. For example, the pressure head of a stage compressing the ambient air adiabatically with a pressure ratio 3:1 is: Had= (P2-P1)/ = (44.1-14.7) lbf/in2 144in2/ft2 / 0.07 lbm/ft 3 = 60,472 ft-lbf/lbm . Testing this head for a centrifugal stage at a flow rate of 2lb/s and selecting a realistically high efficiency corresponds (figure below) to Ds~2.0. With Ds=2.0, Had=60,472 and Q=2lbm/s /0.07 lbm/ft3 = 28.6 ft3/s, calculation gives the impeller outer diameter of D=2.0 x28.61/2/60,4721/4 =0.68ft. Similarly, for the corresponding Ns= 80, we get N=80x60,4723/4/28.61/2 =59,592 rpm. GP turbine (see turbine map below): assuming a combustor P=1.5psi (3.5%), PR=2.2, turbine operating at TIT=2460R and the speed matching compressor, calculate: Had=(44.1-1.5-19.4) x 144/ 0.04 =83,520 ftlbf/lbm; Q=2/0.04 =50ft3/s; Ns=59,592x501/2 /83,5203/4=85 and D=1.3 (from the map) x50 1/2/ 83,5201/4= 0.54ft (mean blade diameter) that results in the blade height of 0.05 ft for the hot gas rate Q=50 ft3/s and assumed gas velocity of 600 ft/s.

38

fig. below.

in next figure.

40

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41

REFERENCE MATERIAL: A. Terminology

46

Ref. C

48

APPENDIX The following is an example of a practical conceptual and preliminary design task. (Discussed in details during lectures through IV)

HP Turbine: Turbine inlet flow = [9.2 (9.2x.15)]1.02=7.98lb/s at PIT=125.4/1.04=120.58psia, where 1.02 indicates addition of fuel flow and 1.04 - the combustor pressure drop; then st.1 turbine nozzle inlet volumetric flow is 7.98/.114=69.88ft3/s. Assuming that 65% of total available energy goes to HP turbine, the total HPT delta T=(2000-900)x.65= 715 F, where 2000F is TRIT for assumed temperature drop of 150Ffrom st.1 nozzle cooling) . Stage 1 temperature drop should exceed 450F to avoid cooling of the st.2 blade, thus the temperature drop is 450F and 265F correspondingly for st.1 and 2. This corresponds to pressure ratios of 2.25 and 1.72 for st.1 and st.2 turbines. Then Pst1=120.58-66.99=53.59 psi (flow rate 125.78ft3/s at nozzle exit) and P st2=66.99-38.95= 28.04psi. Now using the above flow of 69.88 ft3/s, determined earlier compressor speed of 35,245 RPM and NS-Ds turbine map (fig) one can calculate mean diameters of these stages: Htst1= 53.59 x 144/ .114 =67,693 ftlbf/lbm; Q= 69.88 ft3/s after selecting for calculated Ns=70.2 and corresponding Ds=1.3 find Dst1=.674 ft (8.08). For the calculated flow rate of 125.78ft3/s, the st1. nozzle total throat area for Ma=.9 is 125.78:1800 ft/s = .0699 ft2.This corresponds to the blade height of .066 ft for calculated mean diameter .674ft and plane projection of the throat at 2=30deg. Correcting for 10% trailing edge blockage results in the nozzle exit height of .0726ft (.87). Assuming the aspect ratio h/c=1.0, find c=.87. From the following figure assuming te=.015 and 1=60 deg, find number of vanes Nn=Dm /(c tan 1- te/cos2 1)= 8.08 /(.87x 1.73/1.0 -.015/.25)=17.7 round to a higher even number of 18 nozzle vanes. Similarly for st.1 blades choosing aspect ratio 1.0, te=.015 and 2 =50deg, obtain Nb=Dm /(c tan 2 - te/cos2 2 ) = 8.08 /(.87x 1.19/1.0 -.015/.413)= 25.4 to be rounded to the nearest preferably prime number of 29 blades.

LP Turbine: From defined temperature/pressure ratios through the turbine stages the remaining stages now can be dimensioned as well. The number of vanes and blades will increase downstream with expansion and larger gas path heights (larger aspect ratios). A lower speed of LPT helps to reduce centrifugal stresses in significantly longer blade with the last stage setting the tip speed-stress limit. For the considered case with three stage LP turbine driving output shaft assume equal enthalpy split between stages and exhaust pressure 5% above ambient =15.5 psi. The last stage nozzle exit area required for 9.2x1.02 =9.384 lb/s at Ma~.8 or1350ft/s, pressure 15.5 psi and temperature ~1500R is equal to 9.384/(.0255x1350) = 0. 273 ft2. Applying AN2< 2x1010 in2rpm2 , obtain Nmax=22,500 rpm with 25% margin N=18,000 rpm.

Hst5= (20.46 -15.5)x144/.0255=28,009, Q= 368ft3. Then Ns=18,000x3681/2/28,009 =159 and Ds=1.2, so D5=1.2x3681/2/28009 =1.33ft at blade height 2.24. As before, use formula to get number of ~60 blades.

Mainstream layout has to be followed with a preliminary design of rotor and rotor support static structures with thermo-mechanical analyses of major components (see pages 30-33 above with more details presented in the lectures).

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Axial Turbine Design Steps