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Introduction to fluid machines. Contents: Dynamical similaritude Same machine at different rotations System curves Operation point Operation optimization of turbomachine and system Maximum efficiency conditions Exercise. Application of theorem to hydraulic turbomachines (constant ). - PowerPoint PPT Presentation
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Introduction to fluid machines Contents:
Dynamical similaritude Same machine at different rotations
System curves Operation point Operation optimization of turbomachine and
system Maximum efficiency conditions
Exercise
Application of theorem to hydraulic turbomachines (constant ) As shown in previous class, for geometrically
similar machines :
2
352,
ND
ND
QF
DN
L
Torque coefficientflow coefficient
Reynolds number
352 ND
QF
DN
L
Neglecting Re effects (fully turbulent flow):
For any other independent variable (P, H, ,…):
322 ND
QF
DN
gHH
353 ND
QF
DN
PP
3ND
QF
Torque L was arbitrarily chosen
ect.
XfY There is only one idependent non-dimensional coefficient for fully rough flows (large Re)
Application of theorem to hydraulic turbomachines (constant )
For the same family of gemetrically similar machines, the non-dimensional performance curves overlap:
1000 rpm, D=25 cm1200 rpm, D=20 cm1350 rpm, D=15 cm1500 rpm, D=15 cm
22DN
gH
3ND
Q
H
Q
Application of theorem to hydraulic turbomachines (constant )
Dynamical similaritude
If 1000 rpm1200 rpm1350 rpm1500 rpm
22DN
gH
3ND
Q
23
13
ND
Q
ND
Q
222
122
DN
gH
DN
gHand 1 = 2
Points 1 and 2 are said to be dynamically similar points (same adimensional groups, same proportions of dynamic and kinematic quantities)
12
Dynamically similar points for the same machine at different rotations N Same machine: D1=D2
1000 rpm1200 rpm1350 rpm1500 rpm
22DN
gH
3ND
Q
32
23
1
1
DN
Q
DN
Q
222
222
1
1
DN
gH
DN
gH
122
1
21
2
1
2
1
N
N
H
H
Q
Q
Dynamically similar points for the same machine at different rotations N - D1=D2 Same machine
Points on the same parable in the H,Q diagram are dynamically similar points obtained at different rotations for the same machine.
2
1
21
2
1
2
1
N
N
H
H
Q
Q
H
Q
N1 = 1000 rpm
N2 = 1200 rpm
Parables H=kQ2
P1
P2
Q1
H1
Q2
H2
221
1 QQ
HH
k
Exercise 1
Take the Francis turbines of Cabora Bassa plant (Mozambique): H=113,5m; N=107,1rpm, P=415MW, D=6,56m.
It is inteded to test a 1/20 scale model in the laboratory with a head of 22m.
What is the rotational speed, shaft power and volume flow rate in the model for nominal conditions? Neglect the influence of Re; assume an efficiency of 95%.
Answer: N’ = 943 rpm, P’ = 88 kW, Q’ = 0,43 m3/s.
System curve
zB-zA Q
pA
pB
Applying Bernoulli equation between the 2 free surfaces of the system in the figure:
222
1Q
gAd
lfzz
g
ppH
eqAB
AB
Mechanical energy accumulated in the form of pressure and potential energy
Required net mechanical energy supplied to the fluid in the pump
Mechanical energy dissipated in the system
H=F(Q) is the system curve
System curve Curve that gives the mechanical energy per unit weight
H requied to be provided to the fluid if a given flow rate Q is expected to flow in the system.
Q
HH=F(Q)
System curve
Energy dissipated in the system
Mechanical energy accumulated by the fluid.
222
1Q
gAd
lfzz
g
ppH
eqAB
AB
k
If the flow is fully turbulent in the pipe f f(Re)
System curves
Ventilation systems
Closed circuit piping systems have similar curves (no energy storage).
222
1Q
gAd
lfzz
g
ppH
eqAB
AB
=0Q
H
H=F(Q)
System curve
Energy dissipation in the system
System curves
Hydroelectric plants
222
1Q
gAd
lfzzH
eqBA
Q
H
H=F(Q)
System curve
Energy dissipation
Operation point
Flow and head to which the provided energy by the pump balances the system energy requirements:
Q
HSystem curve
Pump performance curve at rotation N
Q1
H11
Maximum efficienty conditions For which rotation is maximum efficiency achieved?
222
2 QQ
HH
Q
HSystem curve
Pump curve at rotation N
Q1
H11
Q2
H2
Maximum efficiency points for different N and same pump:
Q3
2
3
2
3´Q
QNN
Point 2: Maximum efficiency point at original rotation
Point 3: maximum efficiency point (at a different rotation speed) and also a point in the system curve
Series association of machines What is the flow provided by the two pumps in series?
Same flow, added heads
Q
HResultant curve of the series association
Pump B curve at rotation NB
H=HA+HB
Pump A curve at rotation NA
System curve
BA
A+B
BA
BB
Q
Parallel association of machines What is the flow provided by the two pumps in parallel?
Same head, added flows
Q
H
Resultant curve of the parallel association
Pump B curve at rotation NB
H=HA=HB
Pump A curve at rotation NA
System curve
Q=QA+QB
BA BB
Q
B
AA+B
Series and parallel association in hydraullic powered machines Series association:
Parallel association:B
B
A
A
BA
B
B
A
A
BA
r
s
HHHH
HHgQ
HHgQ
P
P
)(
)(
B
B
A
A
BA
B
B
A
A
BA
r
s
QQQQ
QQgH
QQgH
P
P
)(
)(
Exercise: 1st Test 2010-11
es
10,5 m
A radial hydraullic pump, pumps water ( = 1000 kg/m3; = 10-6 m2/s) from a river to a reservoir at atmospheric pressure, as shown in the figure. The equations of the the pump curves at 3000 rpm are :
and
with H in meters and Q in m3/s. The flow in the pipes can be taken as fully turbulent, with a overal friction coefficient (suction and discharge pipes) of 5000 m/(m3/s)2.
21200045 QH 2780004670 QQ
a) Compute the flow rate:
25 l/s 31 l/s 40 l/s 45 l/s 52 l/s 60 l/s
b) And the dissipated energy in the pipe?
1,3 kW 2,1 kW 4,5 kW
6,1 kW 7,0 kW 8,5 kW
Exercise: 1st Test 2010-11
es
10,5 m
c) Compute the rotational speed for pump maximum efficiecy?
1525 rpm 1685 rpm 1784 rpm
1936 rpm 2352 rpm 2842 rpm
A radial hydraullic pump, pumps water ( = 1000 kg/m3; = 10-6 m2/s) from a river to a reservoir at atmospheric pressure, as shown in the figure. The equations of the the pump curves at 3000 rpm are :
and
with H in meters and Q in m3/s. The flow in the pipes can be taken as fully turbulent, with a overal friction coefficient (suction and discharge pipes) of 5000 m/(m3/s)2.
21200045 QH 2780004670 QQ
Bibliography
Chapters 2 and 3
Turbomáquinas, A. F. O. Falcão, Folhas AEIST, 2004.