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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
Inertia and Shaking
Inertia and Shaking
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
Otto Cycle Derivation
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
Otto Cycle Derivation
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:
Otto Cycle Derivation
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
Otto Cycle Derivation
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
Diesel Cycle Derivation
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
Diesel Cycle Derivation
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
Diesel Cycle Derivation
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