1. The Ultimate Importance of Invariant Property : Rothalpy

2. Another Crucial Zone of Concern

3. Flow in Stator-Rotor Inter stage Gaps

4. 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.

5. 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.

6. Radial Equilibrium Theory of Turbo-machines
P M V Subbarao
Professor
Mechanical Engineering Department
A Model for Stable Operation of A Machine
A guiding equation for distribution of load along blade length ….

7. Radial Variation of Blade Geometry

8. 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.

9. Radial Equilibrium Analysis
The centrifugal force = (rrdrdq)w2r
Vq = rw
The centrifugal force is
The pressure force on the element

10. If the two forces are the only ones acting (viscous and other effects neglected), the particle will move at constant radius if:

11. Equilibrium Condition for A Rotating Fluid
An equivalent equation for compressible flow can be developed by using the following thermodynamic relation:
The radial variation of whirl velocity should be according to above equation.
How to implement on a machine?

12. Total Energy Equation for A Rotating Fluid
Stagnation enthalpy should conserve, as there are not interactions with rotor at inlet or exit.

14. Radial component of velocity should be constant (zero) along radial direction for radial equilibrium of flow.

15. Constant in a turbo-machine along meridonial Plane
Stagnation enthalpy is Constant in a turbo-machine along radial direction at intake and discharge.

16. Twisted Blades for Large Turbines

17. Lessons from Nature THE VORTEX
In the case of a vortex, the flow field is purely tangential.
The complex potential function:

18. General Rules for Selection of Whirl Component
Free Vortex Whirl:
Forced Vortex Whirl :

19. More complex Models Weighted mean of free and forced vortices Inlet
Exit
General Whirl Distribution

20. Radial Variation of Flow Velocity in Real Machine
Discharge
Intake

21. Radial Variation of Whirl Velocity
Intake
Discharge

22. Radial Variation of Mass flow rate
Intake
Discharge