1

Principles of turbomachinery

2

Flow Mechanism in the impeller of a turbomachine

Notations:

U: peripheral velocity.

C: absolute velocity

W: relative velocity

ALL VELOCITIES ARE IN m/s

3

t=t0

t=t1

t=t2

Relative velocity and the path of a particle

Index:Velocity relative to an observer sitting on the green carriage

Velocity of the green carriage itself

Absolute velocity (velocity relative to an observer on the ground)

4

Relative velocity and the path of a particle inside the blade passage

1 21

2

1

2Absolute Path

5

Schematic of vane congruent flow

6

Velocity triangle(in case of vane congruent flows)

7

Velocity triangle(in case of vane congruent flows)

8

Velocity triangle

C: absolute velocity

W: relative velocity

U: peripheral velocity of the blade

Cu: peripheral (whirl) component of the absolute velocity

Cm: meridional component of the absolute velocity

1) Angle (β) is measured between positive direction of W and negative direction of U.

C W

UCu

Cm

α β

2) Angle (α) is measured between positive direction of C and positive direction of U

NOTE:

INDEX:

9

Velocity trianglePump/Compressor

U1

W1C1=Cm1

α1 β1

U2

C2 W2

Cm

2

Cu2

β2α2

Velocity triangle at 1(Suction Side)

Velocity triangle at 2(Pressure Side)

Usually for pump/compressor, Cu1 is zero. Cu1 in this case is the pre-whirl. So C1=Cm1. The velocity triangle thus looks like

InletOutlet

10

Velocity triangleTurbine

U1

W1C1=Cm1

α1 β1U2

C2 W2

Cm

2

Cu2

β2α2

Velocity triangle at 1(Suction Side)

Velocity triangle at 2(Pressure Side)

Ideally Cu1 is zero for improved efficiency. Cu1 in this case is the exit whirl. So C1=Cm1. The velocity triangle thus looks like

Inlet Outlet

11

Energy Transfer Equation/Euler’s Energy Equation

Conservation of angular momentum: rate of change of angular momentum is equal to the applied torque.

1. Control volume considered includes all the blade passages responsible for the energy transfer.

m Qρ=2. The mass flow rate entering and leaving the control volume are equal

and is given by

3. It is assumed that the velocity C is uniform from blade to blade, i.e. circumferential direction and also from shroud to shroud.

12

W2

C2

U2

W1C1

U1

Control Volume

r1L1

r2

L2

2

1

13

Derivation of Euler’s Energy Equation

Torque (T): 2 2 1 1( )T m C L C L= −

cos , 1,2i i iL r iα= =&

( )2 2 2 1 1 1

2 2 1 1

( cos cos )

u u

T m C r C rm r C rC

α α= −

= −

L2L1

14

Euler’s Energy Equation

2 2 1 1

2 2 1 1

( )( )

bl

u u

u u

P Tm r C rCm U C U C

ωω

∞ =

= −

= −

Neglecting friction

Specific Work:

2 2 1 1( )blbl u u

PW U C U Cm∞

∞ = = −

Euler’s energy equation (also called Euler’s turbine equation)

Actual flow patternFactors causing deviation of actual flow from vane congruent flow are:

Factors affecting specific work Factors affecting flow angle but not specific work

Effect of vane number:

a) Non-viscous effect

b) Viscous effect

i) Pressure difference effect

ii) Relative circulation effect

Effect of vane thickness

Actual flow patternPressure difference effect

s

1

2

+-

++

++

-

--

-

- ++

++

-

--

-

-

2

1

Actual flow patternRelative circulation effect(in radial impellers only)

Vane Congruent Flow Circulatory Flow

Resultant actual flow in a pump/compressor

Relative circulation effect

d

Estimation of slip

bl

bl

blblbl

WWp

pWWW

∞

∞

=+

=−

1

Pfleiderer’s method: an empirical formulation based on experiments.

U2

C2 W2

∆Cu2

Ideal

∆Cu2: Slip deviation of the actual flow (with finite number of vanes) from the ideal flow (with infinite number of vanes)

p: Slip power factor where,1

22'

Zsrp Ψ= ∫=

2

1

11

r

rrdss

Ψ′: slip power coefficient

Estimation of slip

Special cases:

a) Radial flow impeller: ds1=dr

b) Axial flow impeller: r is constant

2

2

11

1'2

−

Ψ=

rrZ

p

Zerp 'Ψ=

Estimation of slip

Ψ′ is a function of β2b, the impeller type (radial/axial) and the system at the exit of the impeller.

bbk 2

2 ,60

1' ββ

+=Ψ is in degrees.

For radial and mixed flow impellers:

with guide vanes after impeller, k=0.6

with spiral casing after impeller, k=0.65 to 0.85

with vaneless diffuser after impeller, k=0.85 to 1.0,

For axial flow impellers, k=1.0 to 1.2

Stodola’s Method of determining Slip

2reldC ω ′

=

2

2

2UD

ω =

2

2rel

U dCD

′=

where,

so,

From ∆ABC,

22

sin bd d DZβπ′ = =

22 2

22

2

sinsin

b

bU D

ZCrel UD Z

βπ π β= = SLIP

22

DSZ

π=

2uC2uC ∞

Linear relative velocity of this eddy is:

Source: Shepherd

SLIP

U2

C2 W2

∆Cu2

Ideal

Stodola’s slip factor (s)'2

2

u

u

CsC

=

22 2

2

sin bu

u

C UZ

C

π β∞

∞

−=

2 2

2

sin1 b

u

UC Z

π β

∞

= −

2 2

2 2 2

sin1cot

b

m b

UsU C Z

π ββ

= − −

Thus 2 conclusions can be drawn: as Z →∞, s → 1

as Q increases, Cm2 increases and s reduces

Can you find expressions for Wbl∞ and Wbl?

2

2

u

u

CC ∞

=

Actual flow patternViscous effect

Actual flow patternVane thickness effect

β1b

tu1

bu

ttβsin

=

Applying continuity across suction edge (1)

&ZDS π

=No. of blades

S1 tu1

tu2

β1b

β2b

1

II

I

2

S2

( )1 1 1 1.1. .1.mI u mV S C S t C= = −

1m mIC C>

11

1 1m mI

u

SC CS t

= −

Actual flow patternVane thickness effect

Along pressure edge (2)

22

2 2m mII

u

SC CS t

= −

2m mIIC C>

What is the effect of changing Cm on flow directions?

S1 tu1

tu2

β1b

β2b

1

II

I

2

S2

Actual flow patternVane thickness effect

CmI

Cm1 W1

WI

β1 βI

β1 > βI

Suction edge

U1

Cm

II

Cm

2U2

W2

WIIC3

C2

βIIβ2

β2 > βII

Pressure edge

(Z)

Actual flow pattern

Combined effect of all the factors mentioned:

For pump/compressor

C2 W2

U2

β2

−

+++

+++

∆

∆+∆

∆

βββ

β

β2b

Ideally flow should leave (considering direction of W2) at an angle β2b but actual flow leaves at an angle β2. The different causes for changes in angle are as follows:

+++∆β : due to viscous effect+++ ∆+∆ ββ : due to non-viscous effects

−∆β : due to vane thickness