Model to Schedule A Flow Path through A Turbo-machine P M V Subbarao Professor Mechanical...

Model to Schedule A Flow Path through A Turbo-machine

P M V SubbaraoProfessor

Mechanical Engineering Department

Description of the Journey is More Important than the Outcome ….

Complete Fluid Dynamics of Turbo-machinery

1

1

1

1

T

V

p

2

2

2

2

T

V

p

IntakeDelivery

Conservation Laws for a Fluid

..

pVV

t

V

0.

Vt

wqVet

e

.

Conservation of Mass : A Law for Sizing

0.

Vt

Conservation of Mass:

Conservation of Mass for SSSF:

0. V

Integrate from intake to delivery over entire volume :

onstant. CVdVV

Gauss Divergence Theorem

Constantˆ.. AV

dAnVVdV

The entire flow is only due to normal component of the velocity to the flow area :

ConstantflowA

f dAV

Specific Speed of the machine dictates the variation of Flow area

along the fluid path!

Flow Area through A Turbo Machine

Along the flow path, define area averaged mean velocity as:

flow

flow

A

A

f

fmdA

dAV

V

Define area averaged mean density as:

flow

flow

A

A

mdA

dA

path flow alongstation any at

Constant flowfmm AV

222111 AVAV ff

2,2,2,1,1,1, flowfmmflowfmm AVAV

Order of Velocity Vectors

• The real flow through any turbomachine is three dimensional.

• Axial, Radial and Circumferential.

• Axi-symmetry: Inter blade row space.

• Axi-symmetry assumes an average value to represent the state of working fluid in the blade-to-blade plane.

Meridional Plane

The momentum balance is considered in a plane constructed through axis of rotation and radial axis.

This Plane is called Meridional Plane.

1122 rVrVm

Euler Turbo-machinery Equation.

Axial Momentum Equations

Axial Momentum Equation:

A turbomachine cannot tolerate this force doing any work.

The axial force should be completely absorbed by a thrust bearing.

Any remaining traces of this force can cause mechanical or aerodynamic damages.

inaxialoutaxialx VVmF ,,

Concurrent Designs for Minimum Axial Thrust

1122 rVrVmP

Euler Turbo-machinery Power Equation.

WgzV

hgzV

hm

1

2

2

2

22

Conservation of Energy for Turbo-machines

1

2

2

2

1122 22gz

Vhgz

VhrVrV

1111

21

12222

22

2 22gzrV

VhgzrV

Vh

For a Turbomachine:

constatnt2

2

gzrVV

h

gzrVV

hIRothalpy 2:

2

Combination of Euler and SFEE

Over an ideal turbomachine blade along flow directionrothalpy remains constant

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: