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Energy and Rotalpy
E1  I1   E2  I2   g  H  
Where:
E1 =
E2 =
I1 =
I2 =
g =
H =
 =
Energy at the inlet of the turbine
Energy at the inlet of the turbine
Rotalpy at the inlet of the turbine
Rotalpy at the inlet of the turbine
Gravity constant
Head
Efficiency
[J/kg]
[J/kg]
[J/kg]
[J/kg]
[m/s2]
[m]
[-]
Energy and Rotalpy
E1  I1   E 2  I 2   g  H  

E1  I1   E 2  I 2   u1  cu1  u 2  cu 2
cu
cm
c
u
w
Rotalpy along the
streamline
I  E  u  cu  const.
Absolute and relative
acceleration
a abs
c
 c  c

t
c
0
t
We assume stationary flow
and introduce relative and peripheral velocity:
c  w  u  w   r
Where:
w
u

r
=
=
=
=
relative velocity
peripheral velocity
angular velocity
radius

[m/s]
[m/s]
[rad/s]
[m]
 
 
a abs    R  2  w  w   w
Centripetal
acceleration
Coriolis
Relative
acceleration acceleration
Acceleration along a streamline
w
2
as  w 
   R  cos 
s
Relative
acceleration
Centripetal
acceleration
Forces acting in a
rotating channel along
a streamline
Fs  m  a s
p
  ds  dn  b    dn  ds  b  a s
s
p
 w

2

  w 
   R  cos 
s
s



p
 w

2
 s   w 
   R  cos 

s



p
  w  w  2  R  s  cos

p
  w  w  2  R  s  cos

s  cos    R
By inserting the equation
And rearranging we obtain the following equation:
dp
 w  dw  2  R  dR  0

Rotalpy
dp
 w  dw  2  R  dR  0

If we integrate the equation above we get
the equation for rotalpy:
 dp

2
s    w  dw    R  dR ds   0

p w 2 2  R 2


 I  Const.
 2
2
Acceleration normal to a streamline
2
w
2
an  
 2    w    R  sin 
r
Coriolis
Centripetal
acceleration acceleration
Centripetal
acceleration
Forces acting in a
rotating channel
normal to a streamline
Fs  m  a n
p
  dn  ds  b    dn  ds  b  a n
n
p
  ds  dn  b    dn  ds  b  a n
n
We insert the equation for the normal acceleration
in to the equation above. We obtain the following
equation:
 w2

p
2

    
 2    w    R  sin  
n
 r

p w 2 2  R 2


I
 2
2
We derive the rotalpy equation above with respect
to the normal direction. We obtain the following
equation:
1 p
w
u

w
u
0
 n
n
n

p
w 
 u
 u   w 

n
n 
 n
We insert the equation the
equations to the right in to the
equation above. We obtain the
following equation:
n 
R
sin 
u   R
p
w 
 2
      R sin   w 

n
n 

We have the equation below from the
derivation from the Rotalpy equation
p
w 
 2
      R sin   w 

n
n 

We have the equation below from the
derivation from Newton’s second law
 w2

p
2

    
 2    w    R  sin  
n
 r

If we rearrange the equations above we
obtain the following equation:
w
w
 2
n
r
Pump
Pump-turbine
Francis turbine
in turbine mode
n
w
r
n

r
n
r
R

R

w
w
 2 
n
r

w
w
 2 
n
r
w

R
w

w
w
  2 
n
r
Pump

w
w
 2 
n
r
Pump-turbine

w
w
 2 
n
r
Francis turbine

w
w
  2 
n
r