an exclusive conservation equation for ideal turbo-machines p m v subbarao professor mechanical...
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An Exclusive Conservation Equation for Ideal Turbo-machines
P M V SubbaraoProfessor
Mechanical Engineering Department
Invention of New Property for CVs with Work Transfer….
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Conservation of Rothalpy
• A cornerstone of the analysis of steady, relative flows in rotating systems has, for many years, been the immutable nature of the fluid mechanical property rothalpy.
• "In a moving passage the rothalpy is therefore constant provided:– the flow is steady in the rotating frame;– no friction from the casing;– there is no heat flow to or from the flow.
gzUVV
hIRothalpy blade 2:
2
or
gzUVhIRothalpy blade 0:
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gzUVVVV
hIRothalpy bladerx
2:
222
gzUVVVVhIRothalpy bladerx 22
1: 222
gzUUVVVhIRothalpy bladebladerx 2222
2
1:
Ideas for creation of a variety in turbo-machine.
Novel Idea for Creation of Variety
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Blade Velocity Vs Tangential Component of Fluid Velocity
Ub
Ub
Vwi
Vai
Vfi
gzUUVVVhI bladebladerx 2222
2
1
Vri
In maridional plane at mean radius of rotor
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Ub
Vwi
Vai
VfiVri
Ub
Vwi
Vai Vfi
Vri
VwiUb
Vai
Vfi
Vri
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gzUVVVhIRothalpy bladerelrx 22,
22
2
1:
gzUVhIRothalpy bladerel 22
2
1:
Relative Angular Velocity
gzUVhgzU
hIRothalpy bladeblade
rel 0
2
,0 2:
Constant in an ideal turbo-machine
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For stator Ublade =0
constant: 0,0 gzhgzhEnthalpyStagnation rel
For rotors :
constant2
: 0
2
,0 gzUVhgzU
hIRothalpy bladeblade
rel
For a true axial flow machines: Ublade constant
constant: ,0 gzhIRothalpy rel
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Complex Geometrical Features of A Turbo-Machinne
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A turbomachine working with incompressible fluid will be isothermal and hence U(T) is constant throughout the machine.
constant22
12
22 gzUVVp
gzUVp
bladebladerel
constant2
2
, gzUVpgzU
p bladestagnationblade
relstagnation
For an Ideal Hydro Power Plant :
constant2
2
, bladestagnationblade
relstagnation UVgzpU
gzp
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A Two-Way Welfare for the Globe
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Hydro Electric Plant with High Headspatm
H
gHpV
p atmpenstock
static
2
2
gHV jet
2
2
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Option for High Head Hydro Station
In an ideal Penstock constant2
2
gHpV
p atmpenstock
static
In an ideal Nozzle constant2
2
jetstatic
Vp
In an ideal turbo-machine constant2
2
relstatic
Vp
constant2
1 22 gzUVp
bladerel
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U
Vri
Vre
Vai
UVri
Vai
Inlet Velocity Triangle
U
VreVae
Exit Velocity Triangle
Vri
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More Ideas
For an Ideal Hydro Power Plant :
constant2
1 22 gzUVp
bladerel
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Turbo-machines working with Vapors/Gas
constant2
: 0
2
,0 bladeblade
rel UVhU
hIRothalpy
constant,2
,0,0
0
2
0
blade
T
pblade
T
p UVdTTpcU
dTTpcIrel
For an ideal gas:
constant2
0,0
0
2
0
blade
T
pblade
T
p UVdTTcU
dTTcIrel
constantgz
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For simple compressible fluid: Like Inert Gas
constant2
: 0
2
,0 bladepblade
relp UVTcU
TcIRothalpy
constant2
0,0
0
2
0
blade
T
pblade
T
p UVdTcU
dTcIrel
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The Fourth Generation Nuclear Power Plants
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An Advanced Nuclear Power Plant
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Geometrical Details along the Third Direction
• True flow through a turbo-machinery is three-dimensional.
• Flow and tangential flow velocities are very important for better operation of a turbo-machine.
• The third component, which is normal to flow and tangential direction is in general of no use.
• This direction can better represented as blade height direction.
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Third Direction of an Axial Flow Turbo-Machines
• The third direction in an axial flow machine is the radial direction.
• The direction of Centrifugal forces!
• Strong centrifugal forces are exerted on blades & fluid in radial direction.
• The centrifugal field distorts the flow velocity profiles considerably.
• Fluid particles tend to move outwards rather than passing along cylindrical stream surfaces as classically assumed.
• Particularly in tall blade (low hub: tip) ratio designs.
• An approach known as the radial equilibrium method, widely used for three-dimensional design calculations in a an axial flow machine.
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Radial Equilibrium Theory of Turbo-machines
P M V SubbaraoProfessor
Mechanical Engineering Department
A Model for Stable Operation of A Machine
A guiding equation for distribution of load along blade length ….
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Radial Variation Blade Geometry
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Radial Equilibrium Theory
• Assumes that flow is in radial equilibrium before and after a blade row.
• Radial adjustment takes place through the row.
• More important for Axial Flow Machines.
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Radial Equilibrium Analysis
The centrifugal force = (rdrd)2r V = r
The centrifugal force is
The pressure force on the element
drdVF lcentrifuga2
rdpdFpressure
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If the two forces are the only ones acting (viscous and other effects neglected), the particle will move at constant radius if:
lcentrifugapressure FF
r
V
dr
dp 2
r
drV
dp 2
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Equilibrium Condition for A Rotating Fluid
An equivalent equation for compressible flow can be developed by using the following thermodynamic relation:
0dp
dhvdpdhTds
dp
dh r
drVdh 2
The radial variation of whirl velocity should be according to above equation.
How to implement on a machine?
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2222
2222
0VVV
hV
hh rf
0222
222
0
VVV
hddh rf
Total Energy Equation for A Rotating Fluid
Stagnation enthalpy should conserve, as there are not interactions with rotor at inlet or exit.
r
drVdh 2
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0222
2222
0
VVV
dr
drVdh rf
0222
22220
VVV
dr
d
r
V
dr
dh rf
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02
0 dr
dVV
dr
dVV
dr
dVV
r
V
dr
dh rr
ff
Radial component of velocity should be constant (zero) along radial direction for radial equilibrium of flow.
02
0 dr
dVV
dr
dVV
r
V
dr
dh ff
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gzUVhU
hIRothalpy bladeblade
rel 0
2
,0 2:
Constant in a turbo-machine along meridonial Plane
0
2
12
dr
rVd
r
V
dr
dV f
Stagnation enthalpy is Constant in a turbo-machine along radial direction at intake and discharge.
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Twisted Blades for Large Turbines
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Lessons from Nature
• In the case of a vortex, the flow field is purely tangential.
ziiW ln2
The complex potential function:
THE VORTEX
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•Free Vortex Whirl:
•Forced Vortex Whirl :
General Rules for Selection of Whirl Component
r
CV
constantfV
rCV
221C rCV f
0
2
12
dr
rVd
r
V
dr
dV f
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More complex Models
• Weighted mean of free and forced vortices
• General Whirl Distribution
Inlet Exit
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Radial Variation of Flow Velocity in Real Machine
Intake
Discharge
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Radial Variation of Whirl Velocity
Intake
Discharge
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Radial Variation of Mass flow rate
Intake
Discharge
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Design of Compact Machine
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Kaplan Turbine
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DESIGN OF THE BLADE
Two different views of a blade
90% or better in efficiency
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Basic Rules for Design of An Ideal Turbo-machine
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Basic Rules for Design of An Ideal Turbo-machine
• Enumerate the details of source or demand.
• Calculate Specific speed and identify the fundamental concept of operation.
• X1 (Impulse)+X2(Reaction)+(1-X1-X2)(centrifugal)
• Y1 (Radial)+(1-Y1 )(Axial)
• Design of Flow Path using Conservation of rothalpy.
• Design blade cascade using conservation of mass and momentum.
• Design of Radial Geometry using Radial Equilibrium Theory.
• A design of an Ideal Machine…..
• Real Performance will be lower ……
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Basic Rules for Design of A Real Turbo-machine
• More customized rules along with the general rules.
• Customized rules are specific to application:• Power consumption Vs Power Generation.• Radial Vs Axial.• Incompressible flow Vs Compressible.• In Reality:• Design analysis of A Real Machine is an
Exclusive Scientific Art.