radial equilribrium

7/31/2019 Radial Equilribrium

1/23

Radial Equilibrium Theory

P M V Subbarao

Professor

Mechanical Engineering Department

Long Blades are more complex in shape .


2/23

Geometrical Details along Radial Direction

True flow through a turbo-machinery is three-dimensional.

The effect of the strong centrifugal forces are exerted by/on

blades 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 low hub: tip ratio designs.

An approach known as the radial equilibrium method, widely

used for three-dimensional design calculations.


3/23

Radial Variation Blade Geometry


4/23

Radial Equilibrium Theory

Assumes that flow is in radial equilibrium before and aftera blade row.

Radial adjustment takes place through the row.

More important for Axial Flow Machines.


5/23

Radial Equilibrium Analysis

The centrifugal force = (rrdrdq)w2r

Vq= r

The centrifugal force is

The pressure force on the element

qr q drdVF lcentrifuga2

qrdpdFpressure


6/23

If the two forces are the only ones acting (viscous and

other effects neglected), the particle will move at

constant radius if:

lcentrifugapressure FF

rV

drdp

2

qr

rdrVdp 2q

r


7/23

Radial Equilibrium Analysis of Compressible

Machines

An equivalent equation for compressible flow can be

developed by using the following thermodynamic relation:

0r

dpdhvdpdhTds

r

dpdh


8/23

2222

2222

0

qVVVhVhh r

f

r

drV

dp 2q

r

0222

222

0

qVVVhddh rf

No Interactions: Conservation of Stagnation Enthalpy

r

dpdh


9/23

0222

2222

0

qqVVV

dr

drVdh r

f

0

222

2222

0

qqVVV

dr

d

r

V

dr

dh rf


10/23

02

0 dr

dVV

dr

dVV

dr

dVVr

V

dr

dh rr

f

fq

qq

Radial component of velocity should be constant (zero)

along radial direction for radial equilibrium of flow

02

0 dr

dVV

dr

dVVr

V

dr

dh ff

qq

q


11/23

021

2

00

drrVd

rV

drdV

drdh

drTcd fp qq

For inert Gas

02

12

0

dr

rVd

r

V

dr

dV

dr

dT

c

f

p

qq


12/23

0

0

0

0

1 T

dT

p

dp

For an isentropic process:

0

2

112

0

0

0

dr

rVd

r

V

dr

dV

dr

dp

p

T

c

x

p

qq


13/23

01 0 dr

rVd

r

V

dr

dVV

dr

dp xx

qq

r

Radial Equilibrium Equation for

Incompressible Fluid Machine


14/23

gzUVhU

hIRothalpy bladeblade

rel q0

2

,02

:

Constant in a turbo-machine along meridonial Plane

02

12

00

dr

rVd

r

V

dr

dV

dr

dh

dr

Tcd fp qq

Stagnation enthalpy is Constant in a turbo-machine

along radial direction at intake and discharge.


15/23

Lessons from Nature

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

ziiW ln2

The complex potential function:

THE VORTEX


16/23

Free Vortex Whirl:

Forced Vortex Whirl :

General Rules for Selection of Whirl Component

r

CV q

constantfV

rCV q

2

21C rCVf

0

q

q fV

r

rV


17/23

Twisted Blades for Large Turbines


18/23

Radial Variation of Flow Velocity

100

120

140

160

180

200

220

240

260

280

300

0.75 0.85 0.95 1.05 1.15 1.25

Flowvelocities(m

/s)

Radius ( in m)

Intake

Discharge


19/23

Radial Variation of Whirl Velocity

-100

0

100

200

300

400

500

0.75 0.85 0.95 1.05 1.15 1.25

whirlvelocities(m

/s)

Radius ( in m)

Intake

Discharge


20/23

Radial Variation of Mass flow rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.750 0.850 0.950 1.050 1.150 1.250

massfowrate

Radius ( in m)

Intake

Discharge


21/23

Kaplan Turbine


22/23

DESIGN OF THE BLADE

Two different views of a blade

90% or better inefficiency


23/23

radial equilribrium

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