Principle of turbomachinery
-
Upload
walid-mohammed -
Category
Engineering
-
view
760 -
download
14
Transcript of Principle of turbomachinery
![Page 1: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/1.jpg)
Internal Combustion Engine and TurbomachineryMCHE 562
Dr. Gongtao Wang
![Page 2: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/2.jpg)
Policy and Outline Class policy
Mandatory attendance unless specially approved No late homework No makeup test/exams
Test schedule Floating within 2 weeks
![Page 3: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/3.jpg)
Lecture Outline1. Introduction to Internal Combustion Engine2. Introduction to Gas Turbine Engine
• Definition and Applications• Thermal Cycles• Applications• Illustrations
1. Introduction to Turbomachinery Terms• Definition and classifications• Coordination systems and velocity diagrams• Variables and geometry
![Page 4: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/4.jpg)
Lecture Outline4. Review of Aerodynamics and Fluidics
• Conservation: Mass, energy and Momentum• Gas Dynamics: Compressible flow
4. Dimensionless Analysis• Off Design Performance and specific speed• Buckingham Π-Theorem• Application in Turbomachinery
![Page 5: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/5.jpg)
Lecture Outline6. Energy transfer between fluid and a rotor
• Euler’s Equation• Energy Transfer and velocity diagram• Reaction – Definition • Definition of total relative properties
6. Radial Equilibrium Theory• Derivation of Radial Equilibrium Equation• Free vertex• Problem
![Page 6: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/6.jpg)
Lecture Outline8. Axial flow turbine
• Preliminary design of axial flow turbines• Detailed design• Final project
8. Axial flow compressor
9. Polytropic (small stage) efficiency
![Page 7: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/7.jpg)
Introduction to Internal Combustion Engine Classification
Otto Cycle – Four stroke Clark Cycle – Two Stroke Diesel Cycle – Compression Ignition Wankel cycle – Rotary Engine
![Page 8: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/8.jpg)
Latest 2-Stroke Engine
![Page 9: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/9.jpg)
Wankel Engine
![Page 10: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/10.jpg)
Clerk/Otto/Diesel Cycle Mechanism Thermal Cycle Design Issues
![Page 11: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/11.jpg)
Reciprocating Mechanism
![Page 12: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/12.jpg)
Piston Dynamics Exact piston acceleration
![Page 13: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/13.jpg)
Piston Dynamics Approximate piston acceleration
![Page 14: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/14.jpg)
Gas Force and Torque Gas force
Gas torque
![Page 15: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/15.jpg)
Inertia and Shaking force Shaking = - inertia forces
![Page 16: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/16.jpg)
Inertia and Shaking
![Page 17: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/17.jpg)
Inertia and Shaking
![Page 18: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/18.jpg)
Inertia and Shaking
![Page 19: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/19.jpg)
Inertia and Shaking
![Page 20: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/20.jpg)
Otto Cylce
![Page 21: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/21.jpg)
Otto Cycle P-V & T-s Diagrams
![Page 22: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/22.jpg)
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
LHthη
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
14vthη
T C m = Q v ∆Rej
![Page 23: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/23.jpg)
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
γγ
T
T = T
T
1
4
2
3
Otto Cycle Derivation
T
T-1 = 2
1thηLeading to
![Page 24: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/24.jpg)
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
![Page 25: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/25.jpg)
Then by substitution,
)r(
1 - 1 = )r( - 1 = 1-
v
-1vth γ
γη
)r( = V
V = T
Tv
1
2
2
1 1
1
−−
γ
γ
The air standard thermal efficiency of the Otto cycle then becomes:
Otto Cycle Derivation
![Page 26: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/26.jpg)
Summarizing
Q
Q - 1 =
Q
Q - Q =
H
L
H
LHthη T C m = Q v ∆
1-TT
T
1 - TT
T-1 =
2
32
1
41
thη
)r( = V
V = T
T -1v
1
2
-1
2
1 γγ
)r(
1 - 1 = )r( - 1 = 1-
v
-1vth γ
γη
T
T = T
T
1
4
2
3
2
11T
T th −=η
where
and then
Isentropic behavior
Otto Cycle Derivation
![Page 27: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/27.jpg)
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
![Page 28: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/28.jpg)
Diesel Cycle P-V & T-s Diagrams
![Page 29: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/29.jpg)
Diesel Cycle Derivation
Thermal Efficiency (Diesel):
Q
Q - 1 =
Q
Q - Q =
H
L
H
LHthη
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-T
TT
1 - TT
T - 1 =
)T - T( C m
)T - T( C m - 1 =
2
32
1
41
23p
14vth
γη where: γ = Cp/Cv
![Page 30: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/30.jpg)
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
γγ
V
V V
V V = V3
4
2
141 ≠
Diesel Cycle Derivation
![Page 31: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/31.jpg)
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 γη
γ
γ
)r( = V
V = T
T -1v
1
2
-1
2
1 γγ
Diesel Cycle Derivation
![Page 32: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/32.jpg)
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
![Page 33: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/33.jpg)
Otto-Diesel Cycle Comparison
![Page 34: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/34.jpg)
Dual Cycle P-V Diagrams:
![Page 35: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/35.jpg)
Dual Cycle Thermal Efficiency
5.2
3
V
V
P
P = 2
3 =βα
)T - T( C m + )T - T( C m = Q 2.53p22.5vin
1)-( + 1)-(
1 -
CR
1 - 1 =
1)-(
βγααβαη
γγ
Dual Cycle Efficiency
where: γ = Cp/Cv
( )14Rej TT C m = Q v −
![Page 36: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/36.jpg)
Critical Relationships in the process include
)r( = V
V = T
T -1v
1
2
-1
2
1 γγ
Q A
F m =
cycle
Qfuela
( )r = V
V = P
Pv
2
1
1
2 γγ
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 γη
γ
γ
![Page 37: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/37.jpg)
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
![Page 38: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/38.jpg)
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.
![Page 39: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/39.jpg)
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
![Page 40: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/40.jpg)
Gas Turbine – Components
Frame Casing Front / main
Gas generator Compressor – rotor/stator Combustor
Power conversion Turbine – rotor /stator/ exhaust
![Page 41: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/41.jpg)
Gas Turbine / ICE Higher Efficiency, High power/weight Robust Combustion/Insensitive to fuel
condition Minimum Power output Complexity/Maintenance Higher Cost
![Page 42: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/42.jpg)
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
![Page 43: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/43.jpg)
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
![Page 44: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/44.jpg)
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
![Page 45: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/45.jpg)
Classification of Turbomachine By function
Work absorber - Compressors, fans and pumps Worker - Turbines
By fluid Compressible Incompressible
By meridional flow path Axial Radial
![Page 46: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/46.jpg)
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
![Page 47: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/47.jpg)
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
![Page 48: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/48.jpg)
Blade VD Stator
U = 0 V = W
Rotor V=W+U Impeller Compressor and turbine VD are reversed
Subscription convention Vr1 , …
![Page 49: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/49.jpg)
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
![Page 50: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/50.jpg)
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
![Page 51: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/51.jpg)
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
![Page 52: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/52.jpg)
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
![Page 53: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/53.jpg)
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
![Page 54: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/54.jpg)
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
![Page 55: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/55.jpg)
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
![Page 56: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/56.jpg)
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
![Page 57: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/57.jpg)
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 ρρ
![Page 58: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/58.jpg)
Total Properties Isentropically convert all energy into enthalpy
Total/Stagnational, local/static
ρρtt
ptpt
t
PP
TchTch
gZVhh
==
++= 221)(
![Page 59: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/59.jpg)
Ideal gas isentropic relations State
equation and Constants
Entropy change of a process
Isentropic process
turbinefor
compressorfor
RRTpKkg
J
33.1
4.1
287
==
==
γγ
ρ
)ln()ln(1
2
1
2
11
1
PP
TT
P
vP
Rcs
RcRc
−⋅=∆
⋅=⋅= −− γγγ
1
1
2
1
2
1
2−
=
=
γγγ
ρρ
T
T
P
P
![Page 60: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/60.jpg)
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
γγ
γγ
![Page 61: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/61.jpg)
Sonic Speed and Mach Number Sonic speed
Mach Number
RTd
dpa γ
ρ==
a
VM =
![Page 62: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/62.jpg)
Isentropic Relations in Term of Mach Total to local
1
1
2
12
2
2
11
2
11
2
11
−
−
−+=
−+=
−+=
γ
γγ
γρρ
γ
γ
M
MP
P
MT
T
t
t
t
![Page 63: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/63.jpg)
Critical Property The local condition at
unity mach
Critical mach
tcrcrtcr TR
aVTT ⋅+
==⋅
+
=1
2
1
2
γγ
γ
)2
11(
12
12 2M
M
TR
VM
t
cr −+⋅+
=⋅
+
=γ
γγγ
![Page 64: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/64.jpg)
Isentropic Flow in Critical Mach
1
1
2
12
2
1
11
1
11
1
11
−
−
+−−=
+−−=
+−−=
γ
γγ
γγρρ
γγ
γγ
crt
crt
crt
M
MPP
MTT
![Page 65: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/65.jpg)
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 −+
+
−
+=γ
γ
γ
γ M
MA
A
![Page 66: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/66.jpg)
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
![Page 67: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/67.jpg)
Other Important Features Choking flow
![Page 68: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/68.jpg)
Normal Shocks-1 Control Volume
![Page 69: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/69.jpg)
Normal Shocks-2 Basic Equations for a Normal Shock
![Page 70: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/70.jpg)
Normal Shocks-3 Intersection of Fanno & Rayleigh Lines
![Page 71: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/71.jpg)
Normal Shocks-4 Normal Shock Relations
![Page 72: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/72.jpg)
Normal Shocks-5 Normal Shock Relations (Continued)
![Page 73: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/73.jpg)
Supersonic Channel Flowwith Shocks
Flow in a Converging-Diverging Nozzle
![Page 74: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/74.jpg)
Isentropic Flow of an Ideal Gas– Area Variation Isentropic flow in a
converging-diverging nozzle
![Page 75: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/75.jpg)
Example 3-1
![Page 76: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/76.jpg)
Example 3-2
![Page 77: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/77.jpg)
Example 3-3
![Page 78: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/78.jpg)
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
hγ
γ
η
1
1
)(
)(1
1
2
1
2
−
−
=∆∆= −−
γγη
t
t
t
t
idealt
actualttt
PP
TT
h
h
![Page 79: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/79.jpg)
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
−+−
−
−−
−=+
=
−
∆=
−
γγ
γγ
γ
γγ
γγ
η
crtt
ttP
actualtturbinest
MPM
PP
PPTc
h
![Page 80: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/80.jpg)
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
tγγ
ρ
γγ
![Page 81: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/81.jpg)
Chapter 4 Dimensional analysis
Buckingham Π-Theorem
Off-design performance of gas turbine Dimensional analysis in turbomachinery
Specific speed
![Page 82: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/82.jpg)
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
![Page 83: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/83.jpg)
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
![Page 84: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/84.jpg)
Performance Characteristic Head coefficient
Head efficiency
Power coefficient
===
==
==
µρη
µρη
µρψ
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
![Page 85: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/85.jpg)
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
,
,
==
∆
=∆
−
−
γγ
γ
η
γη
int
t
tt
intint
int
int
ttt
T
T
RT
ND
PD
RTmf
T
T
![Page 86: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/86.jpg)
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
==
=∆
−
δθ
η
![Page 87: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/87.jpg)
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
![Page 88: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/88.jpg)
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
![Page 89: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/89.jpg)
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 ,∆
![Page 90: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/90.jpg)
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
![Page 91: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/91.jpg)
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
![Page 92: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/92.jpg)
Euler’s Equation Torque Power Specific work
1122
1122
1122
)(
)(
θθ
θθ
θθ
τωτ
VUVUp
VUVUmP
VrVrm
−=−==
−=
![Page 93: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/93.jpg)
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
−+−+−=−
−+=
−=−+−
−=−−
−=−−
=
θθ
θ
θθθ
θθ
θθ
![Page 94: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/94.jpg)
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
![Page 95: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/95.jpg)
Enthalpy Across A Stage Absolute
Relative
Rothalpy
RothalpyUVhI
totalrelativeTch
totalabsoluteTch
etemperaturStaticLocalTs
MMTT
MMTT
t
rtprt
tpt
aW
rsrt
aV
st
r
−−=
−=
−=
=+=
=+=−
−
θ
γ
γ
,,
22
1,
22
1
)(:
)1(
)1(
![Page 96: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/96.jpg)
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
−+−+−−+−=
−+−+−−+−=
![Page 97: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/97.jpg)
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
![Page 98: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/98.jpg)
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
![Page 99: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/99.jpg)
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
![Page 100: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/100.jpg)
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
![Page 101: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/101.jpg)
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
−−
−−
+
+−
+−
+−
+−
+−
+−
−=−=
−=−=
−=−=
==
=+=
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
γγ
ρρρ MM
MPMPP
MTMTT
M
rotoracrossconstTc
WTT
ttr
ttr
ttr
RT
WWW
crr
rtp
str
crr
crr
crr
trcr
![Page 102: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/102.jpg)
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
−
=
=
γγ
t
t
t
t
TT
PP
tr
tr
trtr
P
P
TT
![Page 103: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/103.jpg)
Graphic Shown For Turbine
P2 < Pt2 <P1< Ptr2 <=Ptr1<Pt1<=Pt0
For Compressor Po<P1<Pto <= Pt1 < P2<Ptr1<=Ptr2<Pt2
![Page 104: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/104.jpg)
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
![Page 105: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/105.jpg)
Derivation Pressure force, and
mass of the differential control elements
[ ] θρπθππρ
θ
θθ
θ
rdrdd
rdrrm
ddprFFFF
rpF
prdF
ddrrdppF
sideundertopp
ddrdpside
under
top
=−+=
⋅⋅=++=++=
=
++−=
2)(
)sin())((2
))((
22
222
![Page 106: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/106.jpg)
Acceleration Centrifigal
Meridional curvature
Convective )sin(
)cos(2
2
mmconvective
mm
mlcentrifigameridional
lCentrifiga
Va
r
Va
r
Va
α
α
θ
=
=
=
−
![Page 107: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/107.jpg)
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
ααρ
ααρ
ααθρθ
θ
θ
θ
++=
−−=
−−=⋅⋅
++= −
![Page 108: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/108.jpg)
Simplified cases Vm = const
Vr=0 Invoke
total enthalpy
r
V
dr
dp 21 θ
ρ=
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
θθ
θ
γγ
θ
θ
γρ
ρργγ
θ
γρρρ
ρρ
ρρργ
γθ
ργγ
θθ
++=
−++=
=⇔=−⇔=
−++=
+
++=++=+=
−
−
−
−
![Page 109: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/109.jpg)
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
=⇔−=
++=
θ
θ
θθ
θθ2
00
![Page 110: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/110.jpg)
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
![Page 111: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/111.jpg)
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
![Page 112: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/112.jpg)
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.)
![Page 113: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/113.jpg)
Preliminary Design -- Find Meridional flow path Flow condition along pitch line Hub and tip velocity diagram (assuming free-
vortex stages)
![Page 114: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/114.jpg)
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
![Page 115: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/115.jpg)
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
ββψ
ψ θθθθθθ
−=
===== −−−∆
UV
UWW
UVV
U
VVU
U
Tc
U
W
z
tps
)tan(tan 21 ββφψφ
−== U
Vz
![Page 116: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/116.jpg)
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
ββφββ
ββθθ
θθ
θθ
θθ
θθ
θθθθ
θθθθ
+=+=
===
+=−
−=−
−=∴
−=−−+=−
=−=−
U
VR
VWWW
U
WW
WWU
WW
VVU
WWR
WWWWWWWW
UWWVV
z
zz
ZZ
![Page 117: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/117.jpg)
Coefficient Design-2
φβα
φβα
ψφ
β
ψφ
β
ββφψ
ββφ
1tantan
1tantan
)2(2
1tan
)2(2
1tan
)tan(tan
)tan(tan2
11
22
2
1
21
21
+=
+=
+−=
−=⇔
−=
+=
R
RR
![Page 118: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/118.jpg)
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
γ
α
![Page 119: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/119.jpg)
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
![Page 120: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/120.jpg)
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
![Page 121: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/121.jpg)
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
ψγγ
η
ψ
γγ
![Page 122: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/122.jpg)
Φ Use Φ and α2 to set R close to 0.5
Try α2 = 0 R=? and α2 =-15 R=?
+=
+−=
+=
−=
φβα
ψφ
β
φβα
ψφ
β
1tantan
)2(2
1tan
1tantan
)2(2
1tan
22
2
11
1 R
or
R
![Page 123: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/123.jpg)
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
![Page 124: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/124.jpg)
Blade Design at 0,1,2 Density From mass flow rate
mmhub
mmtip
hubtipm
crt
rV
mrr
rV
mrr
rrrV
mA
V
VVAm
××−=
××+=
−××==⇒
+−−=⇒=
−
πρπρ
πρ
γγρρρ
γ
2222
)(2
1
11
1
12
![Page 125: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/125.jpg)
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
![Page 126: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/126.jpg)
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
![Page 127: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/127.jpg)
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
![Page 128: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/128.jpg)
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
![Page 129: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/129.jpg)
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
![Page 130: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/130.jpg)
Typical Blade Load
0102030405060708090100
0 1 2 3 4 5
![Page 131: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/131.jpg)
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
=−=
Γ=−=
−=Γ=
)1(
)(
)(_.
212
ρ
πθθ
![Page 132: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/132.jpg)
Real Fluid Effects Pitch/axial chord ratio s/c Aspect ratio h/c Incidence Tip clearance Viscosity and friction
![Page 133: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/133.jpg)
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
![Page 134: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/134.jpg)
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
![Page 135: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/135.jpg)
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
![Page 136: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/136.jpg)
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
![Page 137: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/137.jpg)
Reynolds Number - Viscosity Similar to a plate Re > 10^5 Ypconstant Re > 10^5 Yp change rapidly
![Page 138: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/138.jpg)
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
![Page 139: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/139.jpg)
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
λ
![Page 140: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/140.jpg)
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
![Page 141: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/141.jpg)
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 …
![Page 142: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/142.jpg)
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
![Page 143: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/143.jpg)
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
![Page 144: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/144.jpg)
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 +
−= − βββ
![Page 145: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/145.jpg)
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
βββσ
σ
βββψ
−=
=
−
=
s
cRatioSolidity
c
s
z
zT
![Page 146: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/146.jpg)
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
![Page 147: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/147.jpg)
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
![Page 148: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/148.jpg)
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
![Page 149: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/149.jpg)
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
![Page 150: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/150.jpg)
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%
![Page 151: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/151.jpg)
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
![Page 152: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/152.jpg)
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
![Page 153: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/153.jpg)
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.
![Page 154: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/154.jpg)
Effect of Mach
![Page 155: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/155.jpg)
![Page 156: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/156.jpg)
![Page 157: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/157.jpg)
![Page 158: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/158.jpg)
![Page 159: Principle of turbomachinery](https://reader031.fdocuments.us/reader031/viewer/2022012306/55a3a9921a28ab62658b45ba/html5/thumbnails/159.jpg)