turbomachinery theory

1. LECTURE II
BASIC GOVERNING EQUATIONS OF
TURBOMACHINES
HYDRAULIC MACHINERY

2. One Dimensional Theory of
Turbomachinery
• The real flow through an impeller is three dimensional
• That is to say the velocity of the fluid is a function of three
positional coordinates, say, in the cylindrical system, r, θ and
z, as shown in Fig. below.
• Thus, there is a variation of velocity not only along the radius
but also across the blade passage in any plane parallel to the
impeller rotation
• Also, there is a variation of velocity in the meridional plane,
i.e. along the axis of the impeller.
• The velocity distribution is therefore, very complex and
dependent upon the number of blades, their shapes and
thicknesses, as well as on the width of the impeller and its
variation with radius.
2

3. A Centrifugal Impeller in Relation to Cylindrical
Coordinates
3

4. Basic Assumptions
The one-dimensional theory simplifies the problem very considerably
by making the following assumptions:
 The blades are infinitely thin and the pressure difference across
them is replaced by imaginary body forces acting on the fluid and
producing torque.
 The number of blades is infinitely large, so that the variation of
velocity across blade passages is reduced and tends to zero.
Thus,
 There is no variation of velocity in the meridional plane, i.e. across
the width of the impeller. Thus,
 The flow through the impeller is frictionless
As a result of the above assumptions, the flow through, say, a
centrifugal impeller may be represented by a diagram shown in Fig.
below.
0



z
v
0




v
4

5. • The above assumptions enable us to limit our analysis to changes of
conditions which occur between impeller inlet and impeller outlet
without reference to the space in between, where the real transfer of
energy takes place.
• This space is treated as a ‘black box’ having an input in the form of an
inlet velocity triangle and an output in the form of an outlet velocity
triangle.
• Such velocity triangles for a centrifugal impeller rotating with a
constant angular velocity ω are shown in Fig. above.
Analysis
• At inlet, the fluid moving with an absolute velocity v1 enters the impeller
through a cylindrical surface of radius r1 and may make an angle α1 with
the tangent at that radius.
• At outlet, the fluid leaves the impeller through a cylindrical surface of
radius r2, absolute velocity v2 inclined to the tangent at outlet by the
angle α2.
Basic Assumptions Contd.
5

6. Analysis Contd.
The velocity triangles shown in Fig. above are obtained as follows.
 The inlet velocity triangle is constructed by first drawing the vector
representing the absolute velocity v1 at an angle α1.
 The tangential velocity of the impeller, u1 is then subtracted from it
vectorially in order to obtain vr1, the relative velocity of the fluid with
respect to the impeller blades at the radius r1.
 In this basic velocity triangle, the absolute velocity v1 is resolved into
two components:
 One in the radial direction, called velocity of flow vf1, and
 The perpendicular to it i.e in the tangential direction, vw1, sometimes called velocity
of whirl.
 These two components are useful in the analysis and, therefore, they
are always shown as part of the velocity triangles.
6

7. • From One-dimensional flow through a centrifugal impeller coupled
with Newton’s second law applied to angular motion,
• Torque = Rate of change of angular momentum
Now,
• Angular momentum = (Mass) x (Tangential velocity) x (Radius).
Analysis Contd.
7

8. ABSOLUTE AND RELATIVE VELOCITY RELATIONS
• In order to determine the force on moving blades and the energy
transfer between the blades and the fluid, the relative velocity
between the fluid and the blade becomes an important factor. The
blade may move in a direction at an angle to the velocity of the fluid.
• The relative velocity of a body is its velocity relative to a second
body which may in turn be in motion
relative to the earth.
• The absolute velocity V of the first body, is the vector sum of its
velocity relative to the second body v, and the absolute velocity of
the latter, u
Vectorially, V = u + v
This is easily determined by vector diagram called as velocity
triangle. Some possible diagrams are shown in the Figures below
8

9. 9
Sample Velocity diagrams
Some of the general relations are:
V sin α = v sin β
Vu = V cos α = u + v cos β
Vu is the component of the absolute velocity of the first body in
the direction of the velocity u of the second body.
ABSOLUTE AND RELATIVE VELOCITY RELATIONS

10. EULER TURBINE EQUATION
• The fluid velocity at the turbine entry and exit can have three
components in the tangential, axial and radial directions of the
rotor. This means that the fluid momentum can have three
components at the entry and exit.
• This also means that the force exerted on the runner can
have three components.
• Out of these, the tangential force only can cause the rotation
of the runner and produce work. The axial component
produces a thrust in the axial direction, which is taken by
suitable thrust bearings.
• The radial component produces a bending of the
shaft which is taken by the journal bearings.
10

11. • Thus it is necessary to consider the tangential component for
the determination of work done and power produced.
• The work done or power produced by the tangential force
equals the product of the mass flow, tangential force and the
tangential velocity.
• As the tangential velocity varies with the radius, the work
done also will be vary with the radius.
• It is not easy to sum up this work. The help of moment of
momentum theorem is used for this purpose.
• It states that the torque on the rotor equals the rate of
change of moment of momentum of the fluid as it passes
through the runner.
11
EULER TURBINE EQUATION

12. 12
One-dimensional flow through a centrifugal
impeller

13. • V1 = absolute fluid velocity at inlet
• V2 = absolute fluid velocity at outlet
• w1 = tangential velocity at inlet
• w2 = tangential velocity at outlet
• Vw1 = tangential component of the absolute velocity of the fluid at
inlet
• Vw2 = tangential component of the absolute velocity of the fluid at
outlet.
• Let r1 and r2 = the radii at inlet and outlet.
• vf1 = velocity of flow, component of absolute velocity in radial
direction for centrifugal turbomachine
• vw1 = perpendicular component of the absolute fluid velocity in the
tangential direction, sometimes called the velocity of whirl 13
BASIC ASSUMPTIONS

14. EULER TURBINE EQUATION
•
14

1


15. Analysis Contd.
15

16. Analysis Contd.
16

17. • Equation (1.4) is known as Euler’s equation.
• From its mode of derivation it is apparent that Euler’s equation
applies to a pump (as derived) and to a turbine.
• In the case of a turbine, however, since
• E would be negative, indicating the reversed direction of energy
transfer.
• It is, therefore, common for a turbine to use the reversed order
of terms in the brackets to yield positive E.
• Since the units of E reduce to metres of the fluid handled, it is
often referred to as Euler’s head.
• It is useful to express Euler’s head in terms of the absolute fluid
velocities rather than their components.
• From the velocity triangles of Fig. above,
Analysis Contd.
17

18. Analysis Contd.
18

19. In the above expression,
 The first term denotes the increase of the kinetic energy of the
fluid in the impeller.
 The second term represents the energy used in setting the fluid in
a circular motion about the impeller axis (forced vortex).
 The third term is the regain of static head due to a reduction of
relative velocity in the fluid passing through the impeller.
Analysis Contd.
19

20. Application of Euler’s Equation to Centrifugal and
Axial Flow Machines
Centrifugal Flow Machine
• For centrifugal flow machine, the velocity triangles are as shown
in Fig. above.
• In addition, the following relationships hold.
• In general, u = ωr, it follows that the tangential blade velocities
at inlet and outlet are given by
20

21. Application of Euler’s Equation to Centrifugal
Flow Machines Contd.
21

22. Application of Euler’s Equation to
Centrifugal Flow Machines Contd.
22

23. Components of Power Produced
• The power produced can be expressed as due to three effects.
These are the dynamic, centrifugal and acceleration effects.
Consider the general velocity triangles at inlet and exit of turbine
runner, shown in figure below
•
23

24. Components of Power Produced
24

25. Example
Determine the diameters and blade angles of a Francis turbine running at
500 rpm under a head of 120 m and delivering 3 MW. Assume flow ratio as
0.14 and D2 = 0.5D1 and b1 = 0.1 D1. The hydraulic efficiency is 90% and the
overall efficiency is 84%.
25
Solution

26. Solution Contd.
26

27. Application of Euler’s Equation to Centrifugal
Machines
27
(a) Radial inflow impeller (b) Velocity Triangle

28. Application of Euler’s Equation to Centrifugal
Machines
28
above
(3)
(1)
(2)

29. Application of Euler’s Equation to Centrifugal
Machines
29
(4)
(5)

30. Application of Euler’s Equation to Centrifugal
Machines
30
(6)
From triangle geometry of the above Figure
(7)

31. Application of Euler’s Equation to Centrifugal
Machines
31
(8)
8

32. Application of Euler’s Equation to Centrifugal and
Axial Flow Machines
32

33. Centrifugal Flow Machine
At inlet the usual assumptions are as follows:
• The absolute velocity is radial,
• v1 can be calculated (1.9) and
• This condition does not apply when there is a pre-whirl (vw1)
component present, perhaps due to inlet vanes or unfavourable
inlet conditions. In that case vf1 is calculated from equation (1.9)
and 1 can be determined only if v1 is known.
• To minimize entry losses, the blade angle at the inlet is made
equal to (the angle between the relative velocity and the
tangential direction).
33
0
and 1
1
1 
 w
f v
v
v

 90


34. Centrifugal Flow Machine
34
Outlet Diagram
• It is assumed that the fluid leaves the impeller with a relative velocity
tangential to the blade at the outlet, i.e.
• vf2 can be calculated from equation (1.9).
• Draw vf2 vector. Draw the direction vr2 starting from the same origin.
Now draw u2 starting from the intersection with the direction of vr2,
then
 Substituting these into Euler’s equation
• The total amount of energy transferred to the impeller is
2
2 
 

2
2 r
u 

  2
2
2
2
cot f
w v
v
u 


2
2
2
2 cot 
f
w v
u
v 

 
2
2
2
2
cot 
f
v
u
g
u
H 

 
2
2
2
2 cot 
f
t v
u
u
m
gH
m
WH
E 


 


35. Example 1
A centrifugal pump with a 700 mm diameter impeller runs at 1800
rev/min. The water enters without whirl, and 2 = 60o. The actual head
produced by the pump is 17 m. Calculate the theoretical head and the
hydraulic efficiency when V2 = 6 m/s.
Solution:
35
   s
m
97
.
65
60
1800
35
.
0
2
60
2
2 



rN
u
m
g
V
u
H
o
18
.
20
81
.
9
92
.
197
81
.
9
60
cos
6
97
.
65
cos 2
2
2







%
24
.
84
%
100
20.18
17



h

%
100


head
l
theoretica
head
actual
h


36. Example 2
36

37. Solution
37

38. 38

39. 39

40. Worked Example 3
• Consider the velocity diagram shown in Figure
below. The magnitude of the absolute velocity
is V1 = 240m/s, and the flow angle is α = -20°.
The blade speed is U = 300 m/s. Find the
magnitude of the relative velocity and its flow
angle.
40

41. 41

42. Solution
42