Internal Combustion Engine & Turbomachinery-Wang

Internal Combustion Engine and TurbomachineryMCHE 562

Dr. Gongtao Wang

Policy and Outline Class policy

Mandatory attendance unless specially approved No late homework No makeup test/exams

Test schedule Floating within 2 weeks

Lecture Outline1. Introduction to Internal Combustion Engine2. Introduction to Gas Turbine Engine

• Definition and Applications• Thermal Cycles• Applications• Illustrations

3. Introduction to Turbomachinery Terms• Definition and classifications• Coordination systems and velocity diagrams• Variables and geometry

Lecture Outline4. Review of Aerodynamics and Fluidics

• Conservation: Mass, energy and Momentum• Gas Dynamics: Compressible flow

5. Dimensionless Analysis• Off Design Performance and specific speed• Buckingham -Theorem• Application in Turbomachinery

Lecture Outline6. Energy transfer between fluid and a rotor

• Euler’s Equation• Energy Transfer and velocity diagram• Reaction – Definition • Definition of total relative properties

7. Radial Equilibrium Theory• Derivation of Radial Equilibrium Equation• Free vertex• Problem

Lecture Outline8. Axial flow turbine

• Preliminary design of axial flow turbines• Detailed design• Final project

9. Axial flow compressor

10. Polytropic (small stage) efficiency

Introduction to Internal Combustion Engine Classification

Otto Cycle – Four stroke Clark Cycle – Two Stroke Diesel Cycle – Compression Ignition Wankel cycle – Rotary Engine

Latest 2-Stroke Engine

Wankel Engine

Clerk/Otto/Diesel Cycle Mechanism Thermal Cycle Design Issues

Reciprocating Mechanism

Piston Dynamics Exact piston acceleration

Piston Dynamics Approximate piston acceleration

Gas Force and Torque Gas force

Gas torque

Inertia and Shaking force Shaking = - inertia forces

Inertia and Shaking

Otto Cylce

Otto Cycle P-V & T-s Diagrams

Otto Cycle Derivation

Thermal Efficiency:

Air standard assumption (constant v + q)

Cold-air standard assumption (constant c)

Q

Q - 1 =

Q

Q - Q =

H

L

H

LHth

T C m = Q vin

1-TT

T

1 - TT

T-1 =

)T - T( C m

)T - T( C m - 1 =

2

32

1

41

23v

14vth

T C m = Q v Rej

For an isentropic compression (and expansion) process:

where: γ = Cp/Cv

Then, by transposing,

T

T = V

V = V

V = T

T

4

3

3

4

1-

2

1

1-

1

2

T

T = T

T

1

4

2

3


T

T-1 = 2

1thLeading to

The compression ratio (rv) is a volume ratio and

is equal to the expansion ratio in an otto cycle engine.

Compression Ratio

V

V = V

V = r3

4

2

1v

1 + v

v = rv

v + v = volume Clearance

volume Total = r

cc

sv

cc

ccsv

where Compression ratio is defined as


Then by substitution,

)r(

1 - 1 = )r( - 1 = 1-

v

-1vth

)r( = V

V = T

Tv

1

2

2

1 1

1

The air standard thermal efficiency of the Otto cycle then becomes:


Summarizing

Q

Q - 1 =

Q

Q - Q =

H

L

H

LHth T C m = Q v

1-TT

T

1 - TT

T-1 =

2

32

1

41

th

)r( = V

V = T

T -1v

1

2

-1

2

1

)r(

1 - 1 = )r( - 1 = 1-

v

-1vth

T

T = T

T

1

4

2

3

2

11T

T th

where

and then

Isentropic behavior


Determine the temperatures and pressures at each point in the Otto cycle. k=1.4

Compression ratio = 9:1

T1 temperature = 25oc = 298ok

Qin heat add in = 850 kj/kg

P1 pressure = 101 kPa

T2 = 717 p2 = 2189kpa

T3 = 1690k p3 = 5160kpa cv=1.205

T4 = 701k p4 =238kpa

Otto Cycle P & T Prediction

Diesel Cycle P-V & T-s Diagrams

Diesel Cycle Derivation

Thermal Efficiency (Diesel):

Q

Q - 1 =

Q

Q - Q =

H

L

H

LHth

T C m = Q p

For a constant pressure heat addition process;

For a constant volume heat rejection process;

T C m = Q v

Assuming constant specific heat:

1-TT

T

1 - TT

T - 1 =

)T - T( C m

)T - T( C m - 1 =

2

32

1

41

23p

14vth

where: γ = Cp/Cv

For an isentropic compression (and expansion) process:

However, in a Diesel

The compression ratio (rv) is a volume ratio and, in a diesel,

is equal to the product of the constant pressure expansion and the expansion from cut-off.

T

T = V

V V

V = T

T

4

3

3

4

1-

2

1

1-

1

2

V

V V

V V = V3

4

2

141


Compression Ratio

Then by substitution, V

V V

V = r3

4

2

1vc

v

V V

V = r r = r4

3

3

2ecpvc

1)-r(

1 - r )r(

1 - 1 =

cp

cp

1-v

th

)r( = V

V = T

T -1v

1

2

-1

2

1


Determine the temperatures and pressures at each point in the Diesel Cycle

Compression Ratio = 20:1

Cut off ratio = 2:1

T1 temperature = 25oC = 298oK

Qin Heat added = 1300 kJ/kg

P1 pressure = 100 kPa

Diesel Cycle P & T Prediction

Otto-Diesel Cycle Comparison

Dual Cycle P-V Diagrams:

Dual Cycle Thermal Efficiency

5.2

3

V

V

P

P = 2

3

)T - T( C m + )T - T( C m = Q 2.53p22.5vin

1)-( + 1)-(

1 -

CR

1 - 1 =

1)-(

Dual Cycle Efficiency

where: γ = Cp/Cv

14Rej TT C m = Q v

Critical Relationships in the process include

)r( = V

V = T

T -1v

1

2

-1

2

1

Q A

F m =

cycle

Qfuela

r = V

V = P

Pv

2

1

1

2


T C m = Q p T C m = Q v

1)-r(

1 - r )r(

1 - 1 =

cp

cp

1-v

th

Design Issue Improve efficiency

Higher compression ratio Combustion control Ignition timing Exhaust recuperate

Minimize shaking force/torque Lubrication Pollution control Cost deduction – short stroke engine

MCHE 569 Project 1 Given a single cylinder internal combustion engine, r=2.6”, l=10.4”, m2=0.060 blob, rG2=0.4r, m3=0.12, rG3=0.36l, m4=0.16blob. Piston dia. is 5.18”. The crank rotates at 1850 rpm. Compression ratio is 8:1. Thermal condition: T1 = 20 deg. C, P1 = 101kpa, Qin = 810 kJ/kg

Calculate in Excel: Thermal condition of all 4 stroke Thermal efficiency Gas force Gas torque When theta = 0, 90, 180, 270, …720 calculate shaking force and torque Gas-fuel mixture mass flow rate If mass ratio of the mixture is 4 part air vs. 1 part fuel, calculate fuel consumption rate, and volumetric air

flow rate.

Gas Turbines - Definition

Definitions Thermal energy conversion device Fuel -> mechanical/electrical power Fuel -> Propulsion

Difference from ICE Absence of Reciprocating and Rubbing

Members Power/Weight ration

Gas Turbine – Components

Frame Casing Front / main

Gas generator Compressor – rotor/stator Combustor

Power conversion Turbine – rotor /stator/ exhaust

Gas Turbine / ICE Higher Efficiency, High power/weight Robust Combustion/Insensitive to fuel

condition Minimum Power output Complexity/Maintenance Higher Cost

Application of Turbine Power Generation

Lycoming TF-35 Garrett’s GTCP660 Auxiliary Power Unit

Propulsion Turbojet: GE J85-21 (F-5E/F) ; CJ610 Turbofan: Garatte F-109 (T-46 Twin-Shaft) Turboprop Garret’s TPE331-14

Turbine Configuration Shaft arrangement

Single: Fix speed and load Twin/Triple shafting

HPT drives compressor and LPT not need for gear reducer

High efficiency at variable speed High reliability at variable power

Multiple coaxial shaftes Complex control, high efficiency with more flexibility

Ch 2. Terminology of Turbomachinery Critical, challenging and special design

problem for turbomachinery is with blades. Definition of turbomachines

Energy conversion device Continues flow Dynamics acting Rotating blade rows

Classification of Turbomachine By function

Work absorber - Compressors, fans and pumps Worker - Turbines

By fluid Compressible Incompressible

By meridional flow path Axial Radial

Stage Definition -- Stator and rotor pair Stator

Convert fluid thermal to fluid kinetic energy No energy transfer to or from blade

Rotor Energy transfer from or to the fluid -- fluid total

energy change

Coordinate System and Velocity Diagram

Coordination system Polar cylindrical system Radial – r, tangential θ, axial – z

Velocity diagram Total (absolute) velocity -- V Relative (fluid flow vs. blade) -- W Blade velocity due to rotation – U 1 – inlet, 2 -- exit V=W+U

Blade VD Stator

U = 0 V = W

Rotor V=W+U Impeller Compressor and turbine VD are reversed

Subscription convention Vr1 , …

Axial Flow Turbine Sign convention

Positive if along the rotation How to determine fluid acting surface

Turbine – Fluid acting on the convex side of blade airfoil

Compressor – Concave side

Comparison Between Axial and Radial Flow Turbine Signal stage efficiency

Radial is higher Loss between stages

Radial is higher Way to improve efficiency

Radial – make the diameter of the rotor larger Axial – add stages

Compressor Stall, Surge Stall

In axial compressors, gas density/pressure, sometime even temperature, may change sharply in certain stage

Low-speed, low-flow, high stagger, stall is imperceptible, and recoverable

Surge Domino stalls occur from last stage in high speed

compressor Non-recoverable, cause temperature rise, significantly

reduce the performance of the compressor, and often end up with blade damage

Turbine Choke / Blade Cooling Choke / shock

Relative velocity become supersonic Blade

High temperature alloy Intensive cooling Current technology – turbine temperature can be

25% high than the melting point of the blade

Variable Geometry in Compressor and Turbine Power = pressure * volume flow rate Recover from surge in compressor

Startup – ignition – surge Squeeze stall out

Different turbine work at different design point Keep pressure the same, reduce flow channel cross-

section area reduces volume flow rate reduce power and mass flow rate to maintain the pressure and less mass flow burn less fuel

Ch3. Aerodynamics of Flow Processes General flow governing equation Total properties Ideal gas isentropic properties Sonic speed and mach numbers Mach number expressed relations

Isentropic relation in term of local mach Critical velocity and critical properties Isentropic relation in term of critical mach

Continue Compressible flow in isentropic nozzle

Varying-area equation DeLaval nozzle - CD nozzle Unfavorable back pressure gradient

Other important relations for nozzle Choking flow

Shock equations

Continue Outline Definition of turbomachinery isentropic

efficiency Total-total efficiency

Compressor Turbine

Total-static efficiency Total condition of an incompressible flow Limitation of Bernoulli's equation

General Flow Governing Equation Continuity equation

Linear momentum equation

Energy equation

)]()()[(

)()()(

122

12

221

12

122

12

221

12

ZZgVVhhmWQ

ZZgVVhhwq

Shaft

shaft

)()( 1212 yyyxxx VVmFVVmF

constAVAVm 222111

Total Properties Isentropically convert all energy into enthalpy

Total/Stagnational, local/static

tt

ptpt

t

PP

TchTch

gZVhh

221)(

Ideal gas isentropic relations State

equation and Constants

Entropy change of a process

Isentropic process

turbinefor

compressorfor

RRTpKkg

J

33.1

4.1

287

)ln()ln(1

2

1

2

11

1

PP

TT

P

vP

Rcs

RcRc

1

1

2

1

2

1

2

T

T

P

P

Ideal Gas Adiabatic Relations Adiabatic means Tt = const.

Adiabatic process is a better assumption for all stationary turbo components

1

2

1

1

2

1

21

1

2

1

2

1

2

/

ln/

T

T

P

Peq

P

PRs

T

T

P

P

P

P

Pc

s

t

t

t

t

Sonic Speed and Mach Number Sonic speed

Mach Number

RTd

dpa

a

VM

Isentropic Relations in Term of Mach Total to local

1

1

2

12

2

2

11

2

11

2

11

M

MP

P

MT

T

t

t

t

Critical Property The local condition at

unity mach

Critical mach

tcrcrtcr TR

aVTT

1

2

1

2

)2

11(

12

12 2M

M

TR

VM

t

cr

Isentropic Flow in Critical Mach

1

1

2

12

2

1

11

1

11

1

11

crt

crt

crt

M

MPP

MTT

Isentropic Flow in Varying Nozzle To increase the speed of fluid

Converging the subsonic flow Diverging the supersonic flow

)1(2

1

2`1

22

1

*

11

M

MA

A

Nozzles in turbomachinery The most important feature Diffuser must be carefully designed so that

the flow remains attached to the wall Unfavorable pressure gradient makes the

design curve of diffuser

Other Important Features Choking flow

Normal Shocks-1 Control Volume

Normal Shocks-2 Basic Equations for a Normal Shock

Normal Shocks-3 Intersection of Fanno & Rayleigh Lines

Normal Shocks-4 Normal Shock Relations

Normal Shocks-5 Normal Shock Relations (Continued)

Supersonic Channel Flowwith Shocks

Flow in a Converging-Diverging Nozzle

Isentropic Flow of an Ideal Gas– Area Variation Isentropic flow in a

converging-diverging nozzle

Example 3-1

Example 3-2

Example 3-3

Definition of Turbomachinery Efficiency

Total-to-total efficiency Compressor

Turbine

1

1

)(

)(

1

2

1

1

2

t

t

t

t

actualt

idealttt

TT

PP

h

h

1

1

)(

)(1

1

2

1

2

t

t

t

t

idealt

actualttt

PP

TT

h

h

Turbine Efficiency Total-to-static

Efficiency – use in applications where exhaust is counted as waste, such as power plant

1221

122

221

22

1

1

21

,

1)1(

1

)(

1

crtt

ttP

actualtturbinest

MPM

PP

PPTc

h

Compressibility and Bernoulli Equation Error of Bernoulli when used in compressible flow

M<= 0.3 incompressible

...1600404

1

12

11

1

642

12

22

2

MMM

MM

PPV

t

Chapter 4 Dimensional analysis

Buckingham Π-Theorem Off-design performance of gas turbine

Dimensional analysis in turbomachinery Specific speed

Dimensional Analysis Buckingham π-theorem

Select all related as a set of n variables Determine k (either MLT 3, or MLTt 4) Select k most important variables as the central

group Multiply each of the rest n-k variables to solve

for n-k πs Set up the system of equation Arbitrarily set one variable’s exponential as unity Solve the rest exponentials

Application to Turbomachinery Geometric similarity

Dimensional proportional Dynamical similarity

Geometrical similar machines with each velocity vector parallel

Similarity principle Geometrically similar Non-dimensional term/number identical

Performance Characteristic Head coefficient

Head efficiency

Power coefficient

2

3

2

3

2

32

,ˆ

,

,

ND

ND

Qf

P

PP

ND

ND

Qf

gH

gH

ND

ND

Qf

U

gH

i

oP

ideal

actH

Compressible-flow Turbomachine

1.33 Turbine

1.4 Compressor

mixture gas theofheat specific of ratio :

constant Gas : R

re temperatulinlet tota vs.change re temperatuTotal :

efficiency totalto-Total :

ratio Pressure Total-to-Total :Pr

Re,,,,Pr,

,

,,2

,

,

int

t

tt

intint

int

int

ttt

T

T

RT

ND

PD

RTmf

T

T

Another Function and More Terms

kPa 101 pressure, atmosphere Standard :

298K i.e. re, temperatuatmosphere Standard :

,,Pr,,,

,

,

STP

STP

STP

t

STP

t

intint

int

int

ttt

P

T

P

P

T

T

T

N

P

Tmf

T

T

Map and Characteristics Turbine or compressor map – the plot Characteristic – the curves in the plot Design point of compressor is close to surge Design point for turbine is close to choke

Specific Speed – Incompressible

43

)(gH

QNNs

It was experimentally verified that certain type of turbomachinery (axial, radial, mixture) gives highest possible performance (efficiency) over certain range of specific speed value

Specific Speed - Compressible

Qex is the volumetric flow rate at stage exit, which is not the same as that at the inlet due compressible flow

is the idea specific work extracted from or to the turbomachine

43

)( ,idealt

ex

h

QNNs

idealth ,

Ch5. Euler’s Equation Energy transfer between fluid and rotors

Force/torque generated through momentum change

Energy transfer happens while these force/torque do works

Momentum Change at All Directions Axial velocity change

Axial load on to the shaft – no works Radial velocity change

Radial load bending moment vibration Destructive works

Both of above should be minimized Tangential direction – effective works

Euler’s Equation Torque Power Specific work

1122

1122

1122

)(

)(

VUVUp

VUVUmP

VrVrm

Component of Energy Transfer Typical velocity

diagram Vz1 = Vz2 = const

2

)()()(

2

)2(

)(

)(

21

22

22

21

22

21

2211

22

22

22

22

22

2222

22

22

22

21

21

211

21

22

22

222

22

22

21

WWUUVVVUVU

WUVVU

VVVUVUW

VVUVW

VVVUW

VV zz

Heads Dynamic Head (Absolute V)

Total kinetic energy lost/gain in fluid flow Effective shaft works

Convective Head (U) Annual expansion/shrinkage Small

Static Head (relative W) Action of fluid flow to stages

Enthalpy Across A Stage Absolute

Relative

Rothalpy

RothalpyUVhI

totalrelativeTch

totalabsoluteTch

etemperaturStaticLocalTs

MMTT

MMTT

t

rtprt

tpt

aW

rsrt

aV

st

r

,,

22

1,

22

1

)(:

)1(

)1(

Reaction Definition

)()()(

)()(

)()()(

)()(

22

21

21

22

21

22

22

21

21

22

21

22

22

21

22

21

21

22

22

21

WWUUVV

WWUUR

WWUUVV

WWUUR

Compressor

Turbine

Stage Blade Design vs. Reaction Inlet and exit angles for stator

α0, α1 Inlet and exit angles for rotor

β0, β1 Deviation angle

difference of flow and metal Swirl angle

local absolute angles

Axial Turbomachine Zero-reaction stage – Impulse stage

W1=W2, β1= -β2 50% reaction (symmetric) turbine stage

V1=W2, V2=W1 α1= -β2, α2 = - β1

50% reaction (symmetric) compressor stage V1=W2, V2=W1 α1= -β2, α2 = - β1

Incidence and Deviation Angles Incidence angle

Flow angle to leading edge metal angle Always exists like attacking angle Positive or negative

Deviation angle Insufficient flow momentum change A very important controlled feature in compressor A measure to adverse/unfavorable pressure gradient

Real-life Flow path in Axial Turbo Explain with isentropic and γ / (γ-1)>>1

Total pressure drop much faster than temperature Total density decrease across rotor If Mach change over rotor is neglected,

Static density decreases across the rotor

To keep Vz constant, the annular cross area Decreasing for compressor Increasing for turbine

Flow passage over stator, due to significant M increase Converging for compressor Diverging for Turbine

Definition of Total Relative Properties in the Rotor Sub-domain Relative properties can be modeled as flow through nozzle

at speed W across

11

11

,

11,

,

12

)1()1(

)1()1(

)1()1(

2

2112

11

2112

11

2112

11

,

,

2

MM

MPMPP

MTMTT

M

rotoracrossconstTc

WTT

ttr

ttr

ttr

RT

WWW

crr

rtp

str

crr

crr

crr

trcr

Continue General term

Isentropic – Total relative pressure is constant across rotor

Other process total relative pressure decrease

1

1

2

1

2

1

2

21

t

t

t

t

TT

PP

tr

tr

trtr

P

P

TT

Graphic Shown For Turbine

P2 < Pt2 <P1< Ptr2 <=Ptr1<Pt1<=Pt0

For Compressor Po<P1<Pto <= Pt1 < P2<Ptr1<=Ptr2<Pt2

Ch6 Radial Equilibrium Theory Background

Study for thermal properties as traverses a stage Pitch line analysis How properties (except U) vary at a given axial location

Assumption – axi-symmetric flow Note – Wake at gap is negligible The Problem

Find the relationship among fluid properties, annual geometry, and velocity

Derivation Pressure force, and

mass of the differential control elements

rdrdd

rdrrm

ddprFFFF

rpF

prdF

ddrrdppF

sideundertopp

ddrdpside

under

top

2)(

)sin())((2

))((

22

222

Acceleration Centrifigal

Meridional curvature

Convective )sin(

)cos(2

2

mmconvective

mm

mlcentrifigameridional

lCentrifiga

Va

r

Va

r

Va

Radial Equilibrium Theory F=ma

)()sin()cos(1

)()sin()cos(1

)sin()cos(

22

22

22

ConvergingVr

V

r

V

dr

dp

divergingVr

V

r

V

dr

dp

Vr

V

r

V

rdrd

ddpr

aaadm

F

mmmm

m

mmmm

m

mmmm

m

convectivelcentrifigameridionallCentrifiga

Simplified cases Vm = const

Vr=0 Invoke

total enthalpy

r

V

dr

dp 21

rV

dr

dV

drdV

zdrdh

drdp

pp

drdp

dr

dV

drdV

zdrdh

drdp

pdrd

drd

drdpp

drdp

drdp

dr

dV

drdV

zdrdh

convectivelcentrifigameridional

pzzp

Vt

VV

VV

const

VV

a

VVVVTchh

zt

zt

zt

2

2

2

2

)(

0

)(

)()(

11

1

11

122

2122

21

2

Continue Simplification dVz / dr = 0 dht / dr = 0 Free Vertex

Nature fluid flow Flow vorticity – flow particles spinning around

its own axis Least vorticity in free vortex flow Free vortex blade design is most desired in

aerodynamics, but unrealistic Disadvantage in structural design and

manufacturing Boundary layer and tip leakage cancel the idea

effect of free-vortex

constrV

V

r

V

dr

dV

rV

dr

dV

2

00

Chapter 7 Axial Flow Turbine Steam Turbine

Superheated Region Wet Mixture Region

Gas Turbine Similar to superheat steam turbine High temperature alloy

Basic gas turbine design process

Stage Definition Stator followed by rotor

Stator airfoil cascades – vanes Rotor airfoil cascades – blades

Design process Preliminary phases

Compressor/combustor exit, inlet path/nozzle, Stage 1,2,3,4, Casing, pitch line, interstage axial gap

Detailed phases Blade geometry design Real flow effects

Empirical equation Stacking vanes and blade sections CAD Approach to axial turbine

Preliminary Design of Axial-Flow Turbines Given conditions

Turbine inlet conditions (p, t,α,β) Rotary speed min. tip clearance, max tip Mach Envelope radial constrains (casing), max axial

length, max diverging angle Interstage Tt, max exit flow rate (A*N^2), Mach Other, (such as overall efficiency, etc.)

Preliminary Design -- Find Meridional flow path Flow condition along pitch line Hub and tip velocity diagram (assuming free-

vortex stages)

Design Processes Step 1 -- Justify axial turbine type

Ns = N*Q^0.5/(Δht)^0.75 > = 0.775 Δht is enthalpy change over a single stage, you change the number of stages

to make the Ns to be optimum (usually “1”) Step 2 –Split work across turbine individual stages (Δht1, Δht2…),

according to experience Efficiency Off-design, and operation conditions usually 60:40, 55:45,50:50

Step 3 According to the experienced work split, and efficiency, determine interstage total condition Too small axial gap triggers strong and dangerous flow interaction Too large axial gap increases end-wall friction loss Stator/rotor gap is more critical that interstage because large swirl velocity

Formulating an Simplified Approach Calculate specific speed

Find optimum number of stage Estimate turbine efficiency

Define a stage work coefficient

Define Flow coefficient

)tan(tan 21

))( 21212

2122

UV

UWW

UVV

U

VVU

U

Tc

U

W

z

tps

)tan(tan 21

UVz

Coefficient Design-1

)tan(tan2

)tan(tan2

tantan

2)(2)(2

2121

2

2

1

1

21

21

21

22

21

21

22

21

22

221

222

21

22

2121

U

VR

VWWW

U

WW

WWU

WW

VVU

WWR

WWWWWWWW

UWWVV

z

zz

ZZ

Coefficient Design-2

1tantan

1tantan

)2(2

1tan

)2(2

1tan

)tan(tan

)tan(tan2

11

22

2

1

21

21

R

RR

Example 7-1

turbinestageoneFind

KkgkJ

sm

:

/287R 1.333, Assume

5.1)U

h(t coefficien work Stage

/340speed bladeMean

rpm 15000speed Rotational

1.873ration Pressure Total

bars 4pressure lInlet tota

K 1100re temperatulInlet tota

90%efficiency Stage

20kg/sm rate flow Mass

0 angleinlet Flow

:Given

gas

2t

0

Solution Calculate specific speed

As a rule of thumb, you may assume the density of the fluid is 1kg/m^3

It may invoke too much error if calculate isentropic process, why? -- rotor

This is just an initial calculation, so it is not wise to spend too much time and effort to make your result very accurate

Step 1. From density; mass flow rate volumetric flow rate From inlet total temperature; inlet/exit total pressure

ratio outlet temperature assuming isentropic process Inlet/exit temperature and Cp total enthalpy change

over the turbine stages Calculate Ns using N*Qex^1/2 / (Δht)^0.75 Increase number of stages to make Ns per each stage to

be > 0.775

Design the stages ψ

finebemaystageOne

RCp

smgiventheuseU

P

P

T

T

T

TTTTTTT

U

TCp

U

h

t

ttt

t

t

t

ttttttt

tt

427.1

1

/340

11

)1(

1

1,

2,

1,

2,

1,

2,1,2,1,2,0,

22

Φ Use Φ and α2 to set R close to 0.5

Try α2 = 0 R=? and α2 =-15 R=?

1tantan

)2(2

1tan

1tantan

)2(2

1tan

22

2

11

1 R

or

R

Other parameters U=340 m/s and N – 1500rpm

rm = 0.216m α1= atan (tanβ1+1/Φ)=?

Sketch the velocity diagram Calculate V1, W1, V2, W2 Check Mcr

None of the Mach can be greater than 1

Blade Design at 0,1,2 Density From mass flow rate

mmhub

mmtip

hubtipm

crt

rV

mrr

rV

mrr

rrrV

mA

V

VVAm

2222

)(2

1

11

1

12

Stage Configuration Symmetric design (Config 1.)

Simplest for design calculation Rotor rubbing

Descendent (Configuration 2) No rotor and simple enough Hub weakening

Optimized (Config. 3) Theoretically optimum

Design for blade shape Aspect ratio

Chord (the axial projective length of blade) Cz_vane, Cz_blade

Gap between rotor and stator Gap = 0.25*(Cz_vane+Cz_blade)/2 1/8 of the stage solidity length

Detail turbine airfoil cascades Select an airfoil Camber the center line to achieve the inlet

and exit flow Consider other factors that affects the

efficiency of the flow The detailed design procedure

Detail Design Procedure With the velocity diagram Design for the efficiency of flow deflection

Blade geometry parameters Iterative process

Given inlet/exit condition Find the most efficient shape of blade

Real flow considerations Some CAD packages

Blade Geometry Geometry to be determined -- page 120 Suction side (SS) and pressure side (PS) Design Principle

Higher loaded – larger P/V difference between SS and PS

Real fluid consideration

Typical Blade Load

0102030405060708090

100

0 1 2 3 4 5

Force Applied To The Blade Cascade x-y coordination r- θ - z

X Z (axial direction) Y θ direction

S - pitch of blades Circulation around each blade

in

exitinx

zyexitinx

P

PRpSPRpF

VFSPPF

VVSbladesno

rS

)1(

)(

)(_.

212

Real Fluid Effects Pitch/axial chord ratio s/c Aspect ratio h/c Incidence Tip clearance Viscosity and friction

Pitch/axial chord ratio s/c Definition of s and c

s: circular pitch of at given radius, usually the meridional

c: tip to trail linear distance, not counting the curvature of the blade

Figure 7.14 on Page 124 Conclusion: larger deflection smaller s/c

Aspect Ratio h/c Definition

h: tip-hub distance (delta-R) c: tip to hub distance of blade

Design perference - smaller the better <<1.0 boundary layer affects performance >6.0 vibration and bending stress Old optimum value is 3.0 ~~ 4.0 Modern design is around 1.0

Incidence Gas (attacking) angle and metal angle Profile (pressure) loss coefficient Yp

Yp = ( Total pressure loss ) (exit total to local pressure Difference) Reaction blade (momentum absorber – both

velocity magnitude and direction change counts) has lower Yp than Impulse blade (direction only)

Lead edge thickness reduces sensitivity of incidence effect on Yp

Tip Clearance Tip leakage

Direct leakage axial leakage Indirect leakage tangential from pressure side

to suction side Leakage prevention

Direct leakage prevention slot in casing Indirect leakage prevention Full or partial

shroud

Reynolds Number - Viscosity Similar to a plate Re > 10^5 Ypconstant Re > 10^5 Yp change rapidly

Guideline For Blade Design Criterion for Acceptable Diffusion Downstream turning angle of cambered airfoil Location of front stagnation point Trailing edge thickness Effect of Endwall contouring

Criterion for Acceptable Diffusion Diffusion – expansion or de-compression Velocity decline Diffusion aversive pressure

(with large deflection) boundary layer separation large loss

Diffusion factor

25.0

1)(max

)(max

Vcr

Vt

Vcr

Vtexitt

PP

PP

PP

Downstream Turning Angle Definition:

A build-in camber angle of airfoil centerline – design for camber curve of airfoil

Reasoning: straight portion of latter half camber line in airfoil

The purpose is to control diffusion With the angle δ build into blades squeeze the

subsonic flow path increase flow momentum decrease diffusion

However, if too much Mach ~~ 1.0 supersonic pocket shock abrupt total pressure drop

With M~~0.8, δ = [8.0, 12] deg

Location of Front Stagnation Point Front Stagnation Point the point where

flow hit metal surface at 90deg Actual stagnation point s can be far from the

theoretically point a With high flow velocity separation

Correction Negative incidence angle leading edge radius, arc length …

Trailing Edge Thickness Trailing edge of airfoil Flow from different blades mixed after

trailing edge sudden expansion duct flow Thinner the better, but

Strength consideration Coolant pass

Endwall Contouring Contour of surface of either casing or hub Purpose of the contouring -- to improve blade

aerodynamic loading Form a nozzle to change the flow property

Accelerate the flow at rear portion of suction side Force the boundary layer thinner

Gather/collect the scatter fluid

Useful Equations Choice of stagger angle

Stagger angle between the connecting line airfoil front tip to trailing edge and the axial direction

Note: Stator design use α instead of β One of the two angle is negative

52

tantantan95.0 111

Optimum Spacing and Chord Ratio Definition of Zweifel’s loading coefficient Zweifel’s law

Optimum Zweifel’s coefficient is 0.8

)tan(tancos28.0

:

)tan(tancos2

2122

2122

s

cRatioSolidity

c

s

z

zT

Staking of 2D Sections Blade design is first done by design sections at each

radius Staking these 2d Sections to form a 3D blade Experiment and and reworking

Problems: secondary flow – flow crossed original design path into other plane

Method of staking Fix a staking axis Rotate each design 2d airfoil to optimize

Chapter 8 Axial Flow Compressors Introduction

Centrifugal compressor is first used Axial flow compressor is much more efficnet Axial turbine can be used as a compressor if

reversed, at price of significant efficiency loss

Axial compressor vs turbine Turbine

Fluid flow from high pressure to low pressure naturally

Accelerating though passage Compressor

Fluid flow from low pressure to high pressure Convert kinetic energy to pressure potential Compression must be a slow decelerating flow

Multi-stage Compressors and Stage Definition Multi-staging is necessary

Pressure ratio vs performance Compressor stages

Inlet Guide vane – nozzle axial flow to tangential flow

Rotor-stator for each stage Subscription 1 rotor inlet; 2 rotor

outlet/stator inlet; 3vane outlet V3=V1; α3=α1

Compressor Blade Simpler than turbine blade Selected from standard

British C4 – design from pressure distribution but no definite form Base profile and camber line Standard parameter – t/c 10% above appr. 40%

US NACA Series Classified according to CL

The amount of cambers 4, 5, 6, 7 series Most commonly used is 65xxx Deflection angle ε Solidity c/s

Real Flow Effect Incident and deviation

Total pressure loss coefficient (PLC) ΔPt/(ρV^2/2)

Deflection angle Stalling

PLC is twice as minimum Nominal e* is 0.8 of stalling es

Positive incident angle cause high loss

Reynolds Number Lower than 2x10^5 leads to high profile loss Higher than 3x10^5 does not change much Critical Re is 3x10^5 This effect is partially affected by the

turbulence.

Effect of Mach

Internal Combustion Engine & Turbomachinery-Wang

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