Lecture slide-General Analysis of Turbo Machines-edit.pdf

General Analysis
of
Turbo Machines

Learning objectives
• Explain the effects of machine parameters on the
performance parameters
• Explain the overall performance of machines as a
relationship between the specific work or head and the
capacity flow rate of the fluid
• Discuss the effects of variation of exit blade angle on
the energy transfer and degree of reaction

• The variations of the machine parameters e.g.
shape of the blade, angle of blades etc. affect the
general performance of turbo machines
• The performance parameters includes specific
work, the efficiencies, the reaction or the degree of
reaction, the utilization factor, etc.
• Analysis of the effects of the machine parameters
on the performance parameters is necessary in
order to enhance performance of the machines
3. General Analysis of Turbomachines
3.1 Introduction

3.2 Analysis of Radial Flow Machines
3. Cont…
• Turbo machines are classified as axial-flow,
mixed-flow, or radial-flow machines depending on
the pre-dominant direction of the fluid motion
relative to the rotor’s axis as the fluid passes the
blades

3. Cont…
• In a radial flow machine, the two ends of the
rotor blade have different linear velocities
2
2
60

D N
U

1
1
60

D N
U


3. Cont…
• Analysis on the performance parameters of radial flow
machines can be determined from the velocity triangles
• A velocity triangle is a representation of three
important velocities as the sides of a triangle
3.2.1. Velocity Triangles for Radial Flow Machines
1) the velocity of the rotor blade, U.
2) relative velocity of the fluid with respect to the rotor
blade Vr
3) absolute velocity of the fluid V

3. Cont…
• The three velocities form a triangle only when they
satisfy the condition V = U+Vr
• For the above condition to be satisfied, the relative
velocity must be tangential to the blade profile
regardless of the position chosen

3. Cont…
• The following points serve as guidelines in understanding
the velocity triangles
1) All velocities are considered as
vector quantities
2) The blade velocity U is invariably
tangential to the circular path of
the blade, with positive direction
in the direction of rotation

3. Cont…
3) relative velocity of the fluid (Vr)
with respect to the blades is
always tangential to blade profile
4) absolute velocity V of the fluid is
the vector sum of U and Vr Thus,
V = U+Vr
5) In radial flow, the plane of
velocity triangle is perpendicular
to the axis of rotation of the rotor

3. Cont…
6) In axial flow, the velocity
triangles are drawn on a plane
that is tangential to the rotor
7) The fluid angles (inlet a1, outlet
a2) and blade angles (inlet b1
and outlet b2) are specified with
respect to the blade velocity
vector U.

3. Cont…
• Fluid flow in a radial machine may be outward
or inward depending of the type of turbo
machine
• In a radially outward flow, the velocity triangle
is such that the smaller radius is made up of
lower velocities whereas the bigger radius if of
larger velocities
3.2.2 Radially Outward Flow Machines

3. Cont…
Solution
U1
V1
Vr1
a1
b1
U2
V2
Vr2
a2
b2
• Consider the blade profile of a radially outward flow
machine shown;

3. Cont…
• The absolute velocity V1 at the inlet is at 900 to the blade
velocity U1
• Tangential component of absolute velocity at inlet Vw1 = 0

3. Cont…
• Applying Euler turbine equation for power absorbing
and taking Vw1 = 0, the specific work gives;
2 2 1 1
 
E w w
W U V V U
• In terms of the head developed by the pump, this
equation can be written as;
2 2
 w
E
U V
H
g
Where Vw2 = tangential component of
absolute velocity at exit
2 2

E w
W U V

3. Cont…
• Vw2 can be obtained from the velocity triangle;
2 2 2 2
cot
  
w f
V U V b
2
2
1
tan
 
f
x V
b
Specific work is thus;
 
2 2 2 2
cot
 
E f
W U U V b
Head developed;
 
2
2
2
2
b
Cot
V
u
g
u
H f


In which
2 2
 
w
V U x

3. Cont…
• Since a pump, blower, or compressor are usually run
by a motor of constant speed N
• Therefore, the tangential speed of the rotor U is
constant 2
2
60

D N
U

• The flow velocity Vf2 can be determined by;
2
2

f
Q
V
A
where A2 is the exit area of the impeller
and Q is the volume flow rate

3. Cont…
• The equation for specific work can therefore be
expressed as;
1 2
 
E
W C C Q
where 2 2
2
1 2 2
2
cot
 
U
C U and C
A
b
• In terms of the head:
Q
gA
Cot
u
g
u
H 

















2
2
2
2
2 b  Q
C
C
H 2
1 


3. Cont…
• The blade outlet angle β2 in a radial flow machine
significantly affects the work done and the degree
of reaction
• Its effect can be studied by making the following
assumptions;
3.3 Effects of Blade Outlet Angle b2 on Energy Transfer
i. Centrifugal effect at outlet = 2 x centrifugal effect at inlet (u2 = 2 u1)
ii. Flow velocity is constant (Vf1 = Vf2 = Vf )
iii. No tangential component at inlet (Vw1= 0)
iv. Inlet blade angle is right angle (α1= 90˚)
v. Outlet blade angle β2 is variable

3. Cont…
• There are three possible orientation of the blade at the
outlet:
• Backward curved blades β2 < 90°
• Radial blades β2 = 90°
• Forward curved blades β2 > 90°

3. Cont…
• For backward curved vanes, β2 < 90° (and α1 = 90°,
Vw1 = 0, Vf1 = V1 as assumed)
1) Backward Curved Vanes
2
2 2 2
1 2
2
cot
 
U U
C and C
g gA
b
where
Q
C
C
H 2
1 

Backward
Flow, Q
Head,
H
• In a backward curved vanes,
 C2 is positive
 H-Q line has negative slope
 Outlet tip of the blade is in the
direction opposite to that of
rotation.
 Flow and wheel rotation are in
the opposite direction.
 As Q increases, specific work
and head develops reduces

3. Cont…
• For radial blades, β2 = 90° Vw2 = u2 (and α1 = 90°,
2) Radial blades
2
2 2 2
1 2
2
cot
 
U U
C and C
g gA
b
where
Q
C
C
H 2
1 

Flow, Q
Head,
H
• In a radial blade,
 Head and specific work remains
constant
 C2 = 0, H = C1 = =
Constant
 Outlet tip of the blade is in the radial
direction
 Flow and wheel rotation are in the
same direction.
g
u2
2
β2 = 90˚
Radial

3. Cont…
• For forward curved blades, β2 > 90° (and α1 = 90°,
3) Forward curved blades
2
2 2 2
1 2
2
cot
 
U U
C and C
g gA
b
where
Q
C
C
H 2
1 

Flow, Q
Head,
H
• In a forward curved blades,
 C2 is negative
 H-Q line has positive slope
 Flow and wheel rotation are in the
same direction
 Outlet tip of the blade is in the
direction of rotation
 As Q increases, specific work and
head developed also increases
Forward

3. Cont…
• The variation of the blade exit angle b2 affects the
degree of reaction
3.4 Effects of the Blade Exit Angle on Degree of Reaction
• Degree of reaction, is the ratio of the energy transfer
due to change in fluid pressure to the total energy
transfer
 
Total work - Kinetic energy component

R
Total energy transfer work
Dynamic change that occur due
to differences in V1 and v2
Energy transfer/Specific work

3. Cont…
2 2
2 1
2 2
2 2
2
 

 
 

w
w
V V
U V
R
U V
• The value of the degree of reaction depends on the
inlet blade angle b1 and the ratio of the diameters
D2/D1, along with the blade exit angle b2
• By definition, degree of reaction can be expressed as;
where v2
2= vw2
2 + vf2
2

3. Cont…
 
2 2 2
2 2 1
2 2
2 2
2
 
 
 

 
 

w f f
w
w
V V V
U V
R
U V
Replacing these in the degree of reaction equation;
Since the flow velocity is assumed to remain constant
2 2 2
2 2 1
 
w f
V V V
1 1 2
 
f f
V V V
We have Hence:
2 2 2
2 1 1
2 2
2 2
2
 
 
 
 

w f f
w
w
V V V
U V
R
U V

2 2 2
2 2 2
2
 
  
 
w w
w
U V V
R
U V U

3. Cont…
Solving for Vw2 from the outlet triangle

2 2 2
2 2 2
2
 
  
 
w w
w
U V V
R
U V U
2
2
1
1
2
 
   
 
w
V
R
U
2
2
tan 
f
V
x
b 
2
2 2
2
cot
tan
 
f
f
V
x V b
b
2 2
 
w
V x U
2 2
  
w
V U x  2 2 2 2
cot
 
w f
V U V b
Replacing Vw2 into
2
2
2
1
1 cot
2
 
 
 
 
f
V
R
U
b
2
2
1
1
2
  u
V
R
U
 Valid for b1 450, a1  900, V1 = Vf1 = Vf2 = U1; Vw1=0

3. Cont…
• The variation of degree of reaction with respect to
blade exit angle is illustrated below;
2 1 2
2.5 2.5
  f
U U V
2
2
1/ 2.5

f
V
U

3. Cont…
In a radially outward flow pump, the impeller has its
smaller diameter of 5 cm and bigger diameter of 12.5
cm. Its speed is 1500 rpm. The inlet blade angle is 500.
The fluid enters the impeller without any whirl velocity
component. The flow component remains constant.
a) Find the specific work and degree of reaction at a
blade outlet angle of 700.
b) Also find at what outlet angle the impeller becomes
a zero-work impeller.
Example 1

3. Cont…
Velocity triangles at inlet and outlet
Velocity triangle at inlet Velocity triangle at outlet
From the inlet velocity
triangle, we have
From outlet velocity
triangle, we have

3. Cont…
Also, the whirl component at outlet is given by
The specific work
For zero work, V2 is the same as V1 (magnitude and direction)
2
2 2 2 2
cot
 
E f
W U U V b
2
2 2 2 2
0 cot
  f
U U V b 2
2
2
tan 
f
V
U
b

2 2 9.82 1.7 8.12 /
    
w
V U x m s
79.74 /
  J kg

3. Cont…
In a radially inward flow turbine, the diameter of the
runner at the inlet is 50 cm and the diameter at
the outlet is 15 cm. The speed of the machine is 1500
rpm. The fluid at a velocity of 35 m/s enters the runner
at 200 to the tangent and leaves the runner without any
whirl component. The flow component of the fluid
velocity remains constant in the runner. Calculate:
a) the blade angles at the inlet and outlet,
b) the specific work, and the
c) degree of reaction.
Example 2

3. Cont…
The velocity triangles

3. Cont…
From inlet velocity triangle From the outlet velocity triangle

3. Cont…
The degree of reaction

3.3 Analysis of Axial-Flow Machines
3. Cont…
• Axial flow machines e.g. compressors and pumps have the fluid
flowing parallel to the axis of the rotor
Self Study

3. Cont…
• In axial flow machines, the blade velocities at the inlet and
outlet are equal, U1 = U2 = U.
• The two sets of the triangle are
identical and can be drawn on
the same base

3. Cont…
• Generally for a compressor the angles are defined with
respect to axial direction (known as air angles)
• It can be seen that the fluid turning angle is low in case
of compressors as compared to centrifugal pumps
Flow across a turbine blade
Inlet and exit velocity triangles for turbine

3. Cont…
• Velocity triangles are drawn for axial flow machines in
which U1 = U2 = U assuming 𝑉f1=𝑉f2
3.3.1 Velocity triangles for different values of degree of reaction
2 2 2
1 1 1
 
r w f
V V V
2 2 2
2 2 2
 
r w f
V V V
 
1 2
 
w w
W U V V
Where

3. Cont…
When 𝑅<0 (𝑅 𝑖𝑠 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒)
R becomes negative when 𝑉𝑟1>𝑉𝑟2
A negative reaction means that a high-velocity
fluid stream enters the flow passages between the
rotor blades, imparts some energy to the shaft, and
gets itself compressed to a higher pressure.

3. Cont…
When R=0; 𝑉𝑟1=𝑉𝑟2; it is an impulse
turbine; there is no static pressure
change across rotor
When R=0.5; 𝑉1=𝑉𝑟2 𝑎𝑛𝑑 𝑉2=𝑉𝑟1;
Impulse turbine; 50% by impulse and
50% by reaction, Symmetrical velocity
triangle; 𝛼1=𝛽2;𝛼2=𝛽1
When 𝑅>1.0;𝑉2>𝑉1;energy transfer can be
negative or positive
R > 1 means that the fluid, at a higher pressure at
the inlet to the rotor expands in the blade passage
and gets accelerated when the shaft is driven by
external energy and contributes to accelerate the
fluid flow

3. Cont…
When R=1; 𝑉1=𝑉2; purely reaction turbine;
energy transfer occurs purely due to
change in relative K.E of fluid
When 𝑅>1.0;𝑉2>𝑉1;energy
transformation can be negative or
positive

3. Cont…
• In an axial flow turbine, the exit fluid velocity V2 must be
minimum in order to obtain the maximum utilization factor.
• V2 must be axial, that is, V2 must be at 900 to the blade
velocity U
3.3.2 Effect of Blade Angles on the Specific Work and Degree of Reaction on Turbines
2
1
2 tan  
  
 
f
V
U
b
• Exit blade angle b2 is
calculated by;

3. Cont…
• At b1 = b2, R = 0, 2
 
u
V U we have Wmax = U(2U) = 2U2
• At b1 = 900, R = 0.5,  
u
V U we have W = U2
• As b1 tends to reach 180 - b2 R tends to be 1

3. Cont…
• In an axial flow compressor, the inlet fluid angle a1 is taken
as 900 for axial entry
• Inlet blade angle b1 has to be constant at b1 = tan-1 (Vf1/U)
3.3.3 Effect of Blade Angles on the Specific Work and Degree of Reaction: Compressors
• b2 can vary from a minimum
value equal to b1 up to (180 - b1,
for the values of degree of
reaction R to be between 0 and 1
• R and W are set at Rmin = 0.5
and Wmax = U2
(a) R = 0 (not desirable)
R > 0.5 (generally acceptable)

3. Cont…
In an axial flow turbine, the blade velocity is 60 m/s. The
fluid enters at 300 to the plane of the wheel at a velocity of
80 m/s. Calculate the blade inlet angle. If the blades are to
be designed for
a) R=0.25 and
b) R = 0.5,
calculate the blade outlet angle and specific work in each
case.
Example 4

3. Cont…
Data
Solution
The inlet velocity triangle is shown

3. Cont…
Blade inlet angle is the angle b1 of relative velocityVr1 (=BC)
with blade velocity U (=AB),
a) For the required value of R = 0.25

3. Cont…
The outlet blade angle
Specific work, W,
where

3. Cont…
b) For the required value of R = 0.5, the triangles are required to
be symmetrical
Therefore specific work

3. Cont…
The impeller diameter of an axial flow pump is 50 cm. The
impeller speed is 750 rpm. The fluid enters the impeller at a
velocity of 15 m/s, without any whirl component.
a) Determine the blade inlet angle.
b) Calculate the specific work and reaction when the outlet
blade angle is 650
Assume that the flow component remains constant.
Example 5

3. Cont…
Solution Cont.…
The velocity triangles are as shown;
In the outlet velocity triangles, the projection of
Vr2 on U is designated as x

3. Cont…
Solution Cont…
b) When b2 = 650

3. Cont…
Solution Cont…
degree of reaction R:

ru mean
V
R
U
Where Vru mean
hence

1. An Introduction to energy conversion, Volume III –
Turbo machinery, V. Kadambi and Manohar Prasad,
New Age International Publishers (P) Ltd.
Acknowledgements
2. Turbines, Compressors & Fans, S. M. Yahya, Tata
McGraw HillCo. Ltd., 2nd edition, 2002
3. Turbomachine, B.K.Venkanna PHI, New Delhi 2009.

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