Principle of turbomachinery

Internal Combustion
Engine and
Turbomachinery
MCHE 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 Outline
1. Introduction to Internal Combustion Engine
2. 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

Lecture Outline
4. Review of Aerodynamics and Fluidics
• Conservation: Mass, energy and Momentum
• Gas Dynamics: Compressible flow
4. Dimensionless Analysis
• Off Design Performance and specific speed
• Buckingham P-Theorem
• Application in Turbomachinery

Lecture Outline
6. 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

Lecture Outline
8. Axial flow turbine
• Preliminary design of axial flow turbines
• Detailed design
• Final project
8. Axial flow compressor
9. 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

Clerk/Otto/Diesel Cycle
 Mechanism
 Thermal Cycle
 Design Issues

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

Otto Cycle Derivation
 Thermal Efficiency:
= 1 - Q
Q
= Q - Q
H L
Q
L
H
H
th h
 Air standard assumption (constant v + q)
Qin = m Cv T D
 Cold-air standard assumption (constant c)
ö çè
T T
4
- 1
T
ö çè
÷ø
æ
÷ø
æ
T T
3
-1
T
= 1-
= 1 - m C v (T 4 - T 1
)
m C (T - T )
2
2
1
1
v 3 2
th h
Q = m Cv DT Rej

 For an isentropic compression (and expansion)
process:
æ g æ
g
= V
= V
T
ö
2 ÷ ÷ø
1
 where: γ = Cp/Cv
 Then, by transposing,
= T
T
V
V
T
3
4
4
3
-1
2
-1
1
ö
ç çè
÷÷ø
ççè
= T
T
T
T
4
1
3
2
= 1- T
T
1
2
th Leading to h

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
r = V
4
3
1
2
v
r = v
s
+ 1
v
= v + v
s cc
v
= Total volume r
Clearance volume
cc
v
cc
v
where Compression ratio is defined as

 Then by substitution,
g
-
1
æ g
-
÷÷øö
1 1
= ( r )
= V
T
V
T
v
2
1
2
ççè
The air standard thermal efficiency of the Otto cycle
then becomes:
) = 1 - 1 = 1 - ( r -1
( r v
)
1-
h g
th v g

 Summarizing
= 1 - Q
th h Q = m Cv DT
Q
= Q - Q
H L
Q
L
H
H
ö çè
T T
4
- 1
T
ö çè
÷ø
æ
÷ø
æ
T T
3
-1
T
= 1-
2
2
1
1
th h
ö
æ
T
3
= T
T
2
T 1-
1 g
= ( r )
= V
T
V
v
2
1
1-
2
g
÷÷ø
ççè
T
4
1
T th h = -
1 1
T
) = 1 - 1 = 1 - ( r -1
( r v
)
1-
h g
th v g
2
where
and then
Isentropic
behavior

Otto Cycle P & T Prediction
 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

Diesel Cycle P-V & T-s Diagrams

Diesel Cycle Derivation
 Thermal Efficiency (Diesel):
= 1 - Q
Q
= Q - Q
H L
Q
L
H
H
th h
For a constant pressure heat
addition process;
Q = m Cp DT
For a constant volume heat
rejection process;
Q = m Cv DT
Assuming constant specific heat:
ö çè
T T
4
- 1
T
h where: γ = Cp/Cv
ö çè
÷ø
æ
÷ø
æ
T T
3
-1
T
= 1 -
= 1 - m C v (T 4 - T 1
)
m C (T - T )
2
2
1
1
p 3 2
th
g

 For an isentropic compression (and expansion) process:
æ g g
= V
T
ö
æ
ö
2 ÷ ÷ø
V
T
1
ççè
 However, in a Diesel
= T
T
V
V
3
4
4
3
-1
1
2
-1
ç çè
÷÷ø
V
V
V = V V
1 4 ¹
V
4
3
1
2
 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.

 Compression Ratio
V
V
r = V
vc ¹
V
4
3
1
2
 Then by substitution,
V
v
r = r r = V
vc cp e · ·
V
3
4
2
3
ö
æ
T 1-
1 g
= ( r )
( )
ù
ú úû
é
( r -1)
ê êë
r - 1
= V
T
2
= 1 - 1
th g
( r )
cp
cp
-1
v
h
g
g
V
v
1
1-
2
g
÷÷ø
ççè

Diesel Cycle P & T Prediction
 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

Dual Cycle Efficiency
 Dual Cycle Thermal Efficiency
Qin = m Cv (T 2.5 - T 2 ) + m Cp (T3 - T 2.5 )
g g
h a b
V
3
V
2.5
= 1 - 1
ö çè
= P
é
a 3 b =
P
2
ù
úû
êë
÷ø
æ
- 1
( -1) + ( -1)
CR
( -1)
a ga b
where: γ = Cp/Cv
( ) Rej 4 1 Q = m Cv T -T

 Critical Relationships in the process include
ö
æ
T 1-
1 g
= ( r )
= V
T
V
v
2
1
1-
2
g
÷÷ø
ççè
F = m
Q
A
Q
cycle
æ
= V
P
ö
2 g
a fuel
= (r )
V
P
v
1
2
1
g
÷÷ø
ççè
Q = m Cp DT Q = m Cv DT
( )
ù
ú úû
é
( r -1)
ê êë
r - 1
= 1 - 1
th g
( r )
cp
cp
-1
v
h
g
g

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
m = ×V A = ×V A = const 1 1 1 2 2 2  r r
 Linear momentum equation
 Energy equation
q w h h V V g Z Z
+ = - + - + -
  
( ) 1
( ) ( )
shaft
+ = - + - + -
[( ) ( ) ( )]
2 1
2
1
2
1
2 2
2 1
2 1
2
1
2
2 2
2 1
Q W m h h V V g Z Z
Shaft
( ) ( ) x 2x 1x y 2 y 1y F = m × V - V F = m × V - V

Total Properties
 Isentropically convert all energy into enthalpy
 Total/Stagnational, local/static
= + 2 +
h ( h ) 1
V gZ
t
2
= =
h c T h c T
t p t p
P P
r rt
t

Ideal gas isentropic relations
 State
equation and
Constants
 Entropy
change of a
process
 Isentropic
process
p = RT R =
J
kg K
for compressor
for turbine
1.4
1.33
287
g
=
g
=
r 
g
c = × R c = × R
g - g -
D = × -
P
P v
s c T
R
ln( 2
) ln( 2
) 1
1
1
1
1
P
T
P
g g
1
T
2
1
r
2
1
P
2
1
-
ö
÷ ÷ø
æ
= ÷ ÷ø
ç çè
ö
æ
=
ç çè
g
r
T
P

Ideal Gas Adiabatic Relations
 Adiabatic means Tt = const.
s R P
T
g
/ ln
ö
æ
ö
2
æ
ö
æ
ö
æ
P
2
æ D -
T
2
P
s
t
q e P
2
2
2
t
g
 Adiabatic process is a better assumption for all
stationary turbo components
÷ ÷ø
ç çè
÷ ÷ø
ç çè
= =
÷ ÷ø
ç çè
- = D ÷ ÷ø
ç çè
=
-
÷ ÷ø ö
ç çè
-
1
1
1
1
1
1
1
1
/
T
P
P
T
P
P
cP
t
t
g
g

Sonic Speed and Mach Number
 Sonic speed
a dp g
 Mach Number
RT
= =
d
r
M = V
a

Isentropic Relations in Term of Mach
 Total to local
g
çè
2
ö g
g
-
ö çè
2 1
1
1
M
çè
2
ö 1 1
2
1 1
2
1 1
2
-
÷ø
T
P
t
r
=æ + -
÷ø
=æ + -
÷ø
=æ + -
g
g
r
g
M
P
M
T
t
t

Critical Property
 The local condition at
unity mach
ö
æ
+
cr t cr cr t T T V a R ×T
 Critical mach
g
+
= = × ÷ ÷ø
ç çè
=
1
2
1
2
g
g
)
M V
cr g
× + -
2 M 2
(1 1
1
2
2
1
M
R T
t
+
=
×
+
=
g
g g

Isentropic Flow in Critical Mach
2
g
g
-
ö
ö
2 1
1
1
2
æ
= - g
-
1 1
T T M
t cr
1
æ
g
= - g
-
1 1
P P M
t cr
1
æ
g
r r g
= - -
1 1
t cr
1
-
ö
÷ ÷ø
ç çè
+
÷ ÷ø
ç çè
+
÷ ÷ø
ç çè
+
g
g
M

Isentropic Flow in Varying Nozzle
 To increase the speed of fluid
 Converging the subsonic flow
 Diverging the supersonic flow
g
+
1
2( 1)
g M
1`
-
2
2
2
1
A
*
ö
æ + =
1 1 -
+
÷ ÷ø
ç çè
g
g
A M

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 Flow
with Shocks
 Flow in a Converging-Diverging Nozzle

Isentropic Flow of an Ideal Gas
– Area Variation
 Isentropic flow in a
converging-diverging nozzle

Definition of Turbomachinery Efficiency
 Total-to-total
efficiency
 Compressor
 Turbine
1
1
= D
h
( )
t ideal
( )
1
2
1
1
2
g
- ÷ø ö
çè æ
- ÷ø ö
çè æ
=
D
-
-
t
t
t
t
t actual
t t
T
T
P
P
h
g
h
1
1
h
= ( D )
- -
h
t actual
g ( )
1
1
2
1
2
- ÷ø ö
çè æ
- ÷ø ö
çè æ
=
D
g
t
t
t
t
t ideal
t t
P
P
T
T
h

Turbine Efficiency
 Total-to-static
Efficiency –
use in
applications
where exhaust
is counted as
waste, such as
power plant
h
= D
- -
ù
g
( g
-
1
2 ) 1
1 2
( )
é
2 2
P P
1
2
2 2
2
1
1
2
1
,
1
(1 )
1
1
g
-
+
-
= -
+
=
ú ú
û
ê ê
ë
÷ø ö
çè æ
-
-
g
g
g
g
g
g
h
t cr
t
t
P t
t actual
t s turbine
P M
M
P
c T P

Compressibility and Bernoulli
Equation
 Error of Bernoulli when used in compressible flow
é
g
- = æ + - -
1 1 1
ö çè
 M<= 0.3 incompressible
...
2 1
M
4 40 1600
1
ù
1
2
2 4 6
2
2
2
= + + + +
ú ú
û
ê ê
ë
- ÷ø
M M M
M
P P
t
V
g
r
g
g

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
f Q
gH
y r
,
f Q
gH
h r
 Power coefficient
ö
ö
÷ ÷ø
æ
æ
,
f Q
act
P P
h r
ç çè
æ
= = =
÷ ÷ø
ç çè
= =
ö
÷ ÷ø
ç çè
= =
m
m
m
2
3
2
3
2
2 3
ˆ ,
ND
ND
o
P
ND
ND
gH
ND
ND
U
i
P
ideal
H

Compressible-flow Turbomachine
æ
m RT
f
t in
ND
Pr, , t
, ,Re,
Pr : Total - to -Total Pressure ratio
: Total - to total efficiency
ö
: Total temperature change vs. inlet total temperature
h
t t
t
R : Gas constant
: ratio of specific heat of the gas mixture
g
Compressor 1.4
Turbine 1.33
,
, ,
2
,
,
=
=
D
÷ ÷
ø
ç ç
è
D =
-
-
g
g
h g
t in
t in t in
t in
t t
T
T
RT
D P
T
T 

Another Function and More Terms
æ
m T
f
t ,
in
h
Pr, , ,
N
, ,
,
P
t
T
t
t
ö
: Standard atmosphere temperature, i.e. 298K
: Standard atmosphere pressure, 101 kPa
T
STP
STP
STP
STP
t in t in
t in
t t
P
P
T
T
P
T
T
= =
÷ ÷
ø
ç ç
è
D =
-
q d


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
N Q
4
(gH)3
Ns =
 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
N Q
ex
 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
4
( h
)3 t ,ideal
Ns
D
=
t ideal h , D

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
m r V rV
= -
( )
q q
2 2 1 1

P = = m U V -
U V
( )
q q
2 2 1 1

q q
tw
2 2 1 1
t
p = U V -
U V

Component of Energy Transfer
 Typical velocity
diagram
 Vz1 = Vz2 = const
V Vz z
W - ( U - V )
= V -
V
q q
W - ( V - U )
= V -
V
q q
W - ( U + V - 2 U V )
= V -
V
q q q
U V = V + U -
W
( ) ( ) ( )
2
2
2
1
2
2
2
2
2
1
2
2
2
1
q
1 1 2 2
2
2
2
2
2
2
2 2
2
2
2
2 2 2
2
2
2
2
2
2
2
1
2
1
2
1 1
2
1
2
2
2
2
2
2 2
2
2
2
2
2
1
U V - U V = V - V + U - U + W -
W
=
q q

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
g
-
T T M M
= + =
(1 )
g
-
T T M M
= + =
(1 )
Ts Local Static temperature
h = c T -
absolute total
t p t
h = c T -
relative total
t r p t r
I h UV Rothalpy
t
a W
t r s r
V
a
t s
r
= - -
q
, ,
2
2
1
,
2
2
1
: ( )

Reaction
 Definition
2
1
2
2
2
2
2
1
R = U - U + W -
W
( ) ( )
2
1
2
2
2
2
2
1
2
2
2
1
V - V + U - U + W -
W
( ) ( ) ( )
2
2
2
1
2
1
2
2
Turbine
R = U - U + W -
W
( ) ( )
2
2
2
1
2
1
2
2
2
1
2
2
V V U U W W
( ) ( ) ( )
Compressor
- + - + -

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
T const across rotor
1
T T W
M
c
W
RT
W
W
cr tr
T T M T M
g
-
g
-
= + =
= =
g
g
= - = -
tr t
g
g
r cr
P = P - M = P -
M
g
g
- -
1
2
2 1
-
1
(1 ) (1 )
2 1
-
1
(1 ) (1 )
g
-
tr r cr
t
2 1
1
1
,
g
1 1
,
,
1
2
2
g
-
1
(1 ) (1 )
1
1
2
1
1
2
1
1
,
,
2
- -
+
+
+
+
+
+
+
= - = -
g g
g
g
g
g
g
g
g
r r M r M
tr t
r cr
t r
p
tr s
r cr

Continue
 General term
 Isentropic – Total
relative pressure is
constant across rotor
 Other process total
relative pressure
decrease
2
T =
T
tr tr
1 2
P
t
( 2
) 1
1
1
2
1
g
-
=
g
T
t
t
t
T
P
P
tr
P
tr

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
F p dp r dr d
top
= - + +
( )( )
F prd
dp dr d
under
=
F p r
side
= + +
2( )( )sin( )
2 2 2
F = F + F + F = r × dp ×
d
p top under side
[ ] r q
r p p q
p
q
q
q
q
m = r + dr - r d =
rdrd
2
( )
2 2

Acceleration
 Centrifigal
a V
Centrifigal
 Meridional curvature
a V
m
 Convective sin( )
cos( )
2
2
convective m m
m
m
meridional centrifigal
a V
r
r
a
a
q
= 
=
=
-

Radial Equilibrium Theory
 F=ma
a a a
= + + -
Centrifigal meridional centrifigal convective
× × = - -
cos( ) sin( )
2 2
V
2 2
V
q
V
F
dm
r dp d
dp
V
m
1 = q
- m
cos( ) -
sin( ) ( )
2 2
V
V
dp

1 m
cos( ) V sin( ) ( Converging
)
r
r
dr
V diverging
r
r
dr
V
r
r
rdrd
m m m
m
m m m
m
m m m
m
a a
r
a a
r
a a
r q
q
q


= + +

Simplified cases
 Vm = const
Vr=0
 Invoke
total
enthalpy
1 dp V
2 q
r
r
dr
=
2 2 1
h h c T V V V V
= + = + + = + +
( ) ( )
1
+
meridional centrifigal convective
dV
dV
V V
= + + -
t z
( )
q
g
-
p dp
= Û - = Û =
g g
g
= + + -
V
r
dV
q
dV
dr
const
dV
V V
t z
dV
dr
dh
dh
dh
dr z
p d
r r r
dp
p
dr
2
0
p
dp
d
dp
dr
dr
dr
dr z
dp
dr
d
dr p
dr
dr
dr
dr
dr
dr
dr z
p
p z z
V
t
V V
a
t z
2
2
2
r
( 1
)
1
1
1
1
g
1
2 2
2
2
2
q q
q
g
g r r
q
g
r r
r
g r r
q
g r
q q
= + +
-
-
-

Continue Simplification
 dVz / dr = 0 dht / dr = 0 
 Free Vertex
0 = 0
+ +
2 V
dV
dV
 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
V
rV const
V
r
dr
r
dr
= - Û =
q
q
q q
q q

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
c T
= = D = - = - = -
W W
V V
U V V
( ) 1 2 1 2 )
W
y s p t
q 1 q 2
q q q q
2 2
2
y = V
b -
b
(tan tan ) 1 2
U
 Define Flow coefficient
U
U
U
U
U
z
Vz
f
= U
= -
(tan tan ) 1 2 y f b b

Coefficient Design-1
V - V = W - W =
U
q q q q
W - W = W + W - W - W = W -
W
Z Z
q q q q
2
1
1 2 1 2
2
1
W W
= +
q q
W W
= -
q q
-
R W W
-
= -
2 ( ) 2 ( ) 2
q q
(tan tan )
2
q q
W q = W q
= W =
V
b b
tan tan
2
2
(tan tan )
R V
2
1 2 1 2
1
1
1 2
1 2
2
2
1 2
2
2
2
1
2
2
2 2
1
2 2
2
2
1
2
2
b b f b b
= + = +
U
U
U W W
U V V
z
z z


Coefficient Design-2
f b b
= +
(tan tan )
2
1 2
tan = tan +
1
f
2 2
a b
f
a b
tan 1
= -
R
( 2 )
2
tan 1
y
f
b
y
f
b
y f b b
tan tan 1
( 2 )
2
(tan tan )
1 1
2
1
1 2
= +
ì
ï ïî
ï ïí
= - +
Û
ïî
ïí ì
= -
R
R

Example 7-1
Flow inlet angle 0
Mass flow rate m 20kg/s
³
Stage efficiency 90%
Inlet total temperature 1100 K
Inlet total pressure 4 bars
Total Pressure ration 1.873
Rotational speed 15000 rpm
£
=
=
Mean blade speed 340 /
Stage work coefficient ( h
t
= = -
Assume 1.333, R 287 /
Find one stage turbine
m s
kJ kg K
-
D £
:
) 1.5
U
Given :
gas
2
0
g
a



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
Cp T
=D = ´D
t t
2 2
D = - = - = -
P
T
U
h
U
t
t
ö
æ
g
ö
æ
,2
,2
U use the given m s
T
t
,2
One stage may be fine
¬
g
Cp R
P
T
T
T T T T T T
t
tt
t
t
t t t t t t
-
=
1
= ----
÷ ÷ ÷
ø
ç ç ç
è
÷ ÷ø
ç çè
- = -
-
1.427
340 /
1 1
(1 )
1
1 ,
1 ,
1 ,
,0 ,2 ,1 ,2 ,1
y
g
h
y
g

Φ
 Use Φ and α2 to set R close to 0.5
ì
tan 1
= -
R
1 y
f
b
( 2 )
2
 Try α2 = 0 R=?
 and α2 =-15 R=?
ì
ï ïî
ï ïí
tan 1
= - +
y
f
( 2 )
2
= +
ï ïî
ï ïí
= +
f
b
2
a b
f
a b
tan tan 1
tan tan 1
2 2
1 1
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
m
r r m
 
hub m
æ
æ
+
m r VA r r 1 g
1
V
1
r r r

Þ = = ´ ´ -
2 ( )
m
A m
p
r r m
tip m
cr
m tip hub
t
V r
V r
V
V
´ ´
= -
´ ´
= +
ö
÷ ÷
ø
ç ç
è
ö
÷ ÷ø
ç çè
= Þ = - -
1
-
1
2
r p r p
r
g
g
2 2 2 2


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
100
90
80
70
60
50
40
30
20
10
0
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
S V V
S 2
r
= G = -
F P P S F V
= - = G
( )
x in exit y z
F Rp P S Rp P
exit
in
x in
P
no blades
= - =
(1 )
( )
._
2 1
r
p
q q

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 Ypconstant
 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
0.25
ö
P
P
t exit t V
1
max ( )
ö
P
t V
max ( )
£
ö
÷ ÷ø
-
÷ ÷ø
- ÷ ÷ø
=
Vcr
Vcr
P
P
P
l

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
é
ö çèb = - æ b - b
0.95 tan 1 tan 1 tan 1 + úû
 Note:
 Stator design use α instead of β
 One of the two angle is negative
5
2
ù
êë
÷ø

Optimum Spacing and Chord Ratio
 Definition of Zweifel’s loading coefficient
 Zweifel’s law
 Optimum Zweifel’s coefficient is 0.8
ö
- ÷ ÷ø
æ
=
ç çè
s
y b b b
2 cos (tan tan )
c
z
Solidity Ratio :
s
=
c
z
s
0.8 2cos (tan tan )
2 1 2
2
2 1 2
2
s = b b -
b
T

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; 3vane 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.

Principle of turbomachinery

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