The Ultimate Importance of Invariant Property : Rothalpy

Another Crucial Zone of Concern

Flow in Stator-Rotor Inter stage Gaps

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 directions 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 of 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

drVdp 2

drdVrdpd 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