Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F....
-
Upload
ethan-fitzgerald -
Category
Documents
-
view
215 -
download
1
Transcript of Cascade theory The theory in this lecture comes from: Fluid Mechanics of Turbomachinery by George F....
Cascade theory
The theory in this lecture comes from:Fluid Mechanics of Turbomachinery
by George F. WislicenusDover Publications, INC. 1965
.konst2
cpp
2
0
c = c∞+c
0dt
dc
c∞
FY
FX
ds
X
Y
Contour
The contour is large compared to the dimensions of the vane
c is the change of velocity due to the vane
Decompose the velocity in the normal and the tangential direction
of the contour
2sn
22
2s
2nsn
2222
2s
2n
2
csinccoscc2cc
ccsinccoscv2sincoscc
csincccoscc
Bernoulli’s equation
2sn2
0
2
0
csinccoscc2c2
pp
2
cpp
Forces in the x-direction
The forces in the x-direction acting on the element ds can be calculated as a force coming from pressure and impulse.
sincsincdsccosc
cosccoscdsccosc
cosdspdF
sn
nn
x
Flow Rate, Q Velocity in x-direction, cx
Forces in the x-direction
2sn2
0 csinccoscc2c2
pp
We insert the equation for the pressure, p from Bernoulli’s equation.
sincsincdsccosc
cosccoscdsccosc
cosdspdF
sn
nn
x
Forces in the x-direction
2sn2
0 csinccoscc2c2
pp
We insert the equation for the pressure, p from Bernoulli’s equation.
sincsincdsccosc
cosccoscdsccosc
cosdscsinccoscc2c2
cosdspdF
sn
nn
2sn
2
0x
Forces in the x-direction
2nssn
22
2n
2n
32
2
s2
n
2
0x
sinccsincosccsinccsincoscds
coscc2cosccoscds
cos2
csincoscccoscccos
2
cds
cosdspdF
Forces in the x-direction
2nssn
22
2n
2n
32
2
s2
n
2
0x
sinccsincosccsinccsincoscds
coscc2cosccoscds
cos2
csincoscccoscccos
2
cds
cosdspdF
The change of velocity, c is very small because the large distance from the airfoil to the contour. We neglect the terms that has the second order of c.
Forces in the x-direction
dsccdscos2
ccosdspdF
sincosccdssincos2
1coscdscosdspdF
n
2
0x
22n
2220x
This is the force acting in the x-direction on a small element, ds of the contour.
Forces in the x-direction
dsccdscos2
ccosdspdF n
2
0x
By integrating around the contour, we will find the total force acting in the x-direction.
dsccF
dsccdscos2
cdscospF
nx
n
2
0x
=0 =0
d’Alembert paradox
The term cn·ds is the flow rate through the contour. If the flow is incompressible, the integral of the term cn·ds around the contour will be zero.
A body in a two-dimensional and non-viscous flow with constant energy will not exert a force in the direction parallel undisturbed flow, c∞
0dsccF nx
Forces in the y-direction
The forces in the y-direction acting on the element ds can be calculated as a force coming from pressure and impulse.
cosccoscdsccosc
sinccoscdsccosc
sindspdF
sn
nn
y
Forces in the y-direction
dsccdssin2
csindspdF s
2
0y
This is the force acting in the y-direction on a small element, ds of the contour.
Forces in the y-direction
By integrating around the contour, we will find the total force acting in the y-direction.
dsccF
dsccdssin2
cdssinpF
sy
s
2
0y
=0 =0
dsccdssin2
csindspdF s
2
0y
Lift
dsccF sy
Circulation
dscs
Lift
cFy
The law of the circulatory flow about a deflecting body
In the absence of any deflecting body inside the hatched area of the contour the force in y-direction must necessarily be zero. This leads to the theorem that:
For a flow of constant energy, the circulation around any closed contour not enclosing any force-transmitting body must be zero.
The law of the circulatory flow about a deflecting body
dscs1
Let the circulation around the outer contour in the figure be:
cs Let the circulation around the inner contour in the figure be:
dscs2
The law of the circulatory flow about a deflecting body
Let the circulation around the inner and outer contour be connected along the line A-B.
The circulation around the hatched area can now be written as:
D
C
s2
B
A
s121 dscdsc
cs
The law of the circulatory flow about a deflecting body
2121
From the figure we can see that:
The circulation around the hatched area can now be written as:
D
C
s
B
A
s dscdsc
cs
The law of the circulatory flow about a deflecting body
02121
Since we do not have any body inside the hatched area:
Which gives:
21 cs
The law of the circulatory flow about a deflecting body
21
cs
This leads to the theorem:
For a given flow condition (with constant energy), the circulation around the deflecting body is independent of the size and shape of the contour along which the circulation is measured.
The law of the circulatory flow about a deflecting body
dsccs ssm
cs
The mean velocity for the circulation around a contour having the length s is:
For a constant value of the circulation, the mean velocity, csm has to decrease if the length s increases.
The circulation is in inverse ratio to the distance of the contour
Circulation about several deflecting bodies
We have 3 wing profiles in a two-dimensional cascade and makes a contour around the whole cascade. This contour is marked ABGDEF.
A
E
s
E
A
s
AEF
s1 dscdscdsc
A
E
s
E
D
s
D
B
s
B
A
s
ABDE
s2 dscdscdscdscdsc
B
D
s
D
B
s
BGD
s3 dscdscdsc
A
E
s
E
A
s dscdsc
B
D
s
D
B
s dscdsc
Circulation about several deflecting bodies
From the figure we can see that:
0321
Circulation about several deflecting bodies
E
D
s
D
B
s
B
A
s
A
E
s321 dscdscdscdsc
Circulation around 3 wing profiles in a cascade becomes:
Cascade in an axial flow turbine
Let us look at the cylindrical section AB through the axial flow turbine.
Cascade in an axial flow turbine
By unfolding the cylindrical section AB from the last slide, we can look at the blades in a cascade
Cascade in an axial flow turbineCirculation around the blades is: (where Z is the number of blades)
b
a
s
a
a
s
a
b
s
b
b
si dscdscdscdscZ
Cascade in an axial flow turbineFrom the figure we can see that:
1u
a
a
s
2u
b
b
s
cr2dsc
cr2dsc
Cascade in an axial flow turbine
b
a
s1u
a
b
s2ui dsccr2dsccr2Z
Cascade in an axial flow turbine
From the figure we can see that:
b
a
s1u
a
b
s2ui dsccr2dsccr2Z
a
b
s
b
a
s dscdsc
Cascade in an axial flow turbine
1u2u cr2cr2
The circulation becomes:
Cascade in an axial flow turbine
1u2u crcr2
Z
The change of angular momentum is related to the vane circulation by the equation:
Cascade in an axial flow turbine
1122
12 2
uu
uu
cucuE
ZcrcrE
By multiplying the change of angular momentum from the upstream to the downstream side of a turbine runner is the torque acting on the turbine shaft with the angular velocity of the runner we will recognize Euler’s turbine equation.