Clase 4 turbomaquinas epn 2016-a

8/15/2019 Clase 4 turbomaquinas epn 2016-a

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Escuela Politécnica NacionalFacultad de Ingenieŕıa Mecánica

Turbomachinery Slides

Dr. Esteban Valencia, PhD, MSc, Eng

Semester 2016-A

(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN

17 de mayo de 2016 1 / 29

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Outline of the lecture

Velocity diagrams, design parameters and application of thermodynamic laws.

Losses, efficiencies and preliminary axial turbine design.

Effect of reaction on efficiency and correlation of Smith

Efficient design points turbines

Stresses in the rotor, cooling of the vanes and turbine flowcharacteristic

Homework(DIFFUSION WITHIN BLADE ROWS,TURBINE BLADECOOLING)

(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN

17 de mayo de 2016 2 / 29

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INTRODUCTION

Figure 1. Gas Turbin

Gas Turbin (https://www.youtube.com/watch?v=dc00xYsXgTQ)(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN

17 de mayo de 2016 3 / 29

https://www.youtube.com/watch?v=dc00xYsXgTQhttps://www.youtube.com/watch?v=dc00xYsXgTQhttp://find/


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The modern axial-flow turbine developed from a long line of inventions stretching back in time

In 1891 developed a multi-stage (15 stages) axial-flow steamturbine, which had a power output of 100 kW at 4800 rpm.

By 1920 General Electric was supplying turbines rated at 40 MW

for generating electricity. Now achieved 1000 MW

Thesimplest approach to their analysis is to assume that the flowconditions at a mean radius, called the pitchline

When ratio is large, as in the final stages of an aircraft or a steamturbine, a more elaborate three-dimensional analysis is necessary

Combustor can be at temperatures of around 16000C or more whilstthe material used to make turbine blades melt at about 12500C

(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 4 / 29

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VELOCITY DIAGRAMS

The axial turbine stage comprises a row of fixed guide vanes ornozzles (often called a stator row) and a row of moving blades orbuckets (a rotor row)

The sign convention is such that angles and velocities as drawn in

next Figure will be taken as positive throughout this chapter.When drawing the velocity triangles it is always worth sketching thenozzle and rotor rows beside them

within an axial turbine, the levels of turning are very high

flow is turned through the axial direction in both the rotors andnozzles

ρ1Ax 1c x 1 = ρ2Ax 2c x 2 = ρ3Ax 3c x 3 continuity uniform equation


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VELOCITY DIAGRAMS

Figure 1. Turbine Stage Velocity Diagrams


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TURBINE STAGE DESIGN PARAMETERS

STAGE LOADING COEFFICIENT

ψ = ∆h0

U 2 ; ∆h0 Stagnattion Enthalpy

In adiabatic turbine ∆h0 = U ∆c θ ⇒ ψ = ∆c θ

U

-High stage loading implies large flow turning and leads to highly“skewed” velocity triangles

-A high stage loading is desirable because it means fewer stages areneeded to produce a required work output


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TURBINE STAGE DESIGN PARAMETERS

STAGE REACTION

R = (h2 − h3) / (h1 − h3) ≈ (p 2 − p 3)/(p 1 − p 3)

R ≈ (p 2 − p 3)/(p 1 − p 3)

-The reaction is a statement of the blade geometries

-The reaction is more significant since it describes the asymmetry of the velocity triangles

-50 % turbine implies velocity triangles that are symmetrical andzero reaction turbine stage implies little pressure change through

the rotor(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 9 / 29

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THERMODYNAMICS OF THE AXIAL-TURBINE STAGE

Figure 1. Mollier Diagram for a Turbine Stage


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THERMODYNAMICS OF THE AXIAL-TURBINE STAGE

∆W =Ẇ

ṁ = h01 − h03 = U (c θ2 + c θ3) Rotor Work

In Nozzle no work is done ; h01 = h02

In axial turbin radial velocity is neligible;

h02 − h03 = (h2 − h3) + 1

2

c 2θ2 − c

2θ3

+

1

2

c 2x 2 − c

2x 3

= U (c θ2 + c θ3)

h2 + 12

w 22 = h3 + 12

w 23 or h02,real = h03,real

h2 + 1

2w 22 −

1

2U 22 = h3 +

1

2w 23 −

1

2U 23 or I 2 = I 3; radial considered


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REPEATING STAGE TURBINES

-Applications require turbines with high power output and highefficiency

-To allow for the reduction in fluid density that arises as the flowexpands through the turbine

-The blade height must be continuously increasing between blade

rows(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 12 / 29

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REPEATING STAGE TURBINES

Requirements for a repeating stage

c x = Constant , r = constant , α1 = α1

Starting with the definition of reaction

R = (h2 − h3) / (h1 − h3) = 1 − (h1 − h2) / (h01 − h03)

Development:

R = 1 − φ2

2ψ

tan2 α2 − tan

2 α1

If α1 = α3 or If α1 = α3

R = φ

2 (tan β3 − tan β2) If α1 = α3

-The choice of (φ, ψ, and R) are largely determined by best practice and previous experience


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STAGE LOSSES AND EFFICIENCY

Loss coeficients of energy can be defined:

h2 − h2s = 12

c 22ζ N ; Nozzle Row

h3 − h3s = 1

2w 23 ζ R ; Rotor Row

Combining Equations

ηtt =

1 +

ζ R w 23 + ζ N c

22

T 3T 2

2 (h1 − h3)

−1

When the exit velocity is not recovered,totalto-static efficiency forthe stage is:

ηts = (h01 − h03) / (h01 − h3ss ) =

1 +

ζ R w 23 + ζ N c

22

T 3T 2

+ c 212 (h1 − h3)

−1


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Cosidering the static temperature drop through the rotor is notlarge, T 3 = T 2

ηtt =

1 + ζ R w

2

3 + ζ N c

2

22 (h1 − h3)

−1

ηts = 1 + ζ R w 23 + ζ N c

22 + c

21

2 (h1 − h3)−1


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Remembered that loss coefficients in cascade testing is on twodimensional, however three effects are significant when can contributemore than 50 % of total losses

So these estimates can be made of the efficiency of a proposed

turbine by Semi-empirical methods such us: Soderberg(1949),Horlock(1966) and Mathieson(1951)

Although CFD can often accurately predict trends in efficiency

CFD can be applied only once, detailed turbine rotor and stator

geometries have been createdFor a design use preliminary design methods before carrying out thefinal design refinements using computational fluid dynamics.


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PRELIMINARY AXIAL TURBINE DESIGN

Either fix the shapes of the velocity triangles or choosing values for

the three dimensionless design parameters, φ, ψ , and R

Number of Stages:

nsttage ≥Ẇ

˙mψU 2Blade Height and Mean Radius:Given that the axial velocity remains constant throughout each stage.

ρ1Ax 1 = ρ2Ax 2 = ρ3Ax 3 = constant

Ax = ṁ

ρφU ≈ 2π × r mH ; m = mean

Ax = π × r 2t 1− (r h/r t )2 ; h = hub and t = tip


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STYLES OF TURBINE

Zero Reaction Turbine

Figure 5. Velocity Diagram and Mollier Diagram for a Zero Reaction Turbine Stage

R = φ

2 (tan β3 − tan β2) =⇒ β2 = β3; If R = 0 and h2 = h3


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STYLES OF TURBINE

50 % Reaction Turbine

Figure 6. Velocity Diagram and Mollier Diagram for a 50 % Reaction Turbine Stage

R = 1 − φ

2 (tan α2 − tan α1) =⇒ 1 = φ

tan β2 +

1

φ − tan α1

⇒ β2 = α1 = α3

Check Example 4.1, Dixon; Six Edition


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EFFECT OF REACTION ON EFFICIENCY

To more than preliminary parameters is considered, Reaction is consideres like a preliminarydessign parameter:

Figure 7. Velocity Diagram and Mollier Diagram for a 50 % Reaction Turbine Stage


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THE EFFICIENCY CORRELATION OF SMITH (1965)

Figure 8. Smith Chart for Turbine Stage Efficiency.(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 23 / 29

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THE EFFICIENCY CORRELATION OF SMITH

Dimensionless Velocity Triangles for a 50 % Reaction Turbine Stage:

f s = ∆h0/ c 21 + c

22

= ∆h0/U 2

c 21 /U 2 + c 22 /U 2Solving Velocities Triangle:

f s = ψ

φ2 +ψ+1

2

2+ φ2 +

ψ−1

2

2 = 2Ψ4φ2 + ψ2 + 1


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Dimensionless Velocity Triangles for a 50 % Reaction Turbine Stage:

f s = ∆h0/

c 21 + c 22

=

∆h0/U 2

c 21 /U

2

+

c 22 /U 2

Solving Velocities Triangle:

f s = ψ

φ2 + ψ+12

2+ φ2 + ψ−1

2

2 =

2Ψ

4φ2 + ψ2 + 1


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THE EFFICIENCY CORRELATION OF SMITH

Figure 8. Smith’s Kinetic Energy Coefficient fs and the Optimum Stage Loading

For optimum stage:

∂ f s ∂ψ

= 2

4φ2 − ψ2 + 1

(4φ2 + ψ2 + 1)

ψopt =

4φ2 + 1

ψopt .exp = 0,65 4φ2 + 1(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 25 / 29

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STRESSES IN TURBINE ROTOR BLADES

Centrifugal Stresses

Figure 9. Centrifugal Forces Acting on Rotor Blade Element(ESCUELA POLITÉCNICA NACIONAL) AXIAL-FLOW TURBINES: MEAN-LINE ANALYSIS AND DESIGN17 de mayo de 2016 26 / 29

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dF c = −Ω2 rdm

d σc ρm

= dF c

ρmA = −Ω2 rdr

σc

ρm= Ω2

rt rh

rdr = U 2t

2

1−

r h

r t

2

K = stress at root of tapered blade

stress at root of untapered blade


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Figure 11. Effect of Tapering on Centrifugal Stress at Blade Root

T b = T 2 + 0,85 w 22/ (2 C p ) ; Blade Temperature Estimate

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Clase 4 turbomaquinas epn 2016-a