An Exclusive Conservation Equation for Ideal Turbo-machines

P M V SubbaraoProfessor

Mechanical Engineering Department

Invention of New Property for CVs with Work Transfer….

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:

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

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

Ub

Vwi

Vai

VfiVri

Ub

Vwi

Vai Vfi

Vri

VwiUb

Vai

Vfi

Vri

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

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

Complex Geometrical Features of A Turbo-Machinne

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

A Two-Way Welfare for the Globe

Hydro Electric Plant with High Headspatm

H

gHpV

p atmpenstock

static

2

2

gHV jet

2

2

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

U

Vri

Vre

Vai

UVri

Vai

Inlet Velocity Triangle

U

VreVae

Exit Velocity Triangle

Vri

More Ideas

For an Ideal Hydro Power Plant :

constant2

1 22 gzUVp

bladerel

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

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

The Fourth Generation Nuclear Power Plants

An Advanced Nuclear Power Plant

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.

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.

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 ….

Radial Variation Blade Geometry

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.

Radial Equilibrium Analysis

The centrifugal force = (rdrd)2r V = r

The centrifugal force is

The pressure force on the element

drdVF lcentrifuga2

rdpdFpressure

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

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?

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

0222

2222

0

VVV

dr

drVdh rf

0222

22220

VVV

dr

d

r

V

dr

dh rf

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

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.

Twisted Blades for Large Turbines

Lessons from Nature

• In the case of a vortex, the flow field is purely tangential.

ziiW ln2

The complex potential function:

THE VORTEX

•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

More complex Models

• Weighted mean of free and forced vortices

• General Whirl Distribution

Inlet Exit

Radial Variation of Flow Velocity in Real Machine

Intake

Discharge

Radial Variation of Whirl Velocity

Intake

Discharge

Radial Variation of Mass flow rate

Intake

Discharge

Design of Compact Machine

Kaplan Turbine

DESIGN OF THE BLADE

Two different views of a blade

90% or better in efficiency

Basic Rules for Design of An Ideal Turbo-machine

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 ……

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.