1. 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

2. Application of  theorem to hydraulic turbomachines (constant )
As shown in previous class, for geometrically similar machines :
Reynolds number
Torque coefficient
flow coefficient
Neglecting Re effects (fully turbulent flow):

3. Application of  theorem to hydraulic turbomachines (constant )
Torque L was arbitrarily chosen
For any other independent variable (P, H, ,…):
There is only one idependent non-dimensional coefficient for fully rough flows (large Re)
ect.

4. 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 cm
1200 rpm, D=20 cm
1350 rpm, D=15 cm
1500 rpm, D=15 cm H Q

5. Dynamical similaritudeIf
1000 rpm
1200 rpm
1350 rpm
1500 rpm
and
1 = 2
12
Points 1 and 2 are said to be dynamically similar points (same adimensional groups, same proportions of dynamic and kinematic quantities)

6. Dynamically similar points for the same machine at different rotations N
1000 rpm
1200 rpm
1350 rpm
1500 rpm
Same machine: D1=D2
12

7. Dynamically similar points for the same machine at different rotations N - D1=D2
Parables H=kQ2 H
Same machine P2 H2 P1 H1
N2 = 1200 rpm k
N1 = 1000 rpm Q1 Q2 Q
Points on the same parable in the H,Q diagram are dynamically similar points obtained at different rotations for the same machine.

8. 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.

9. System curve
Applying Bernoulli equation between the 2 free surfaces of the system in the figure:
Mechanical energy dissipated in the system
Mechanical energy accumulated in the form of pressure and potential energy
zB-zA Q pA pB
Required net mechanical energy supplied to the fluid in the pump
H=F(Q) is the system curve

10. 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. H
System curve k
If the flow is fully turbulent in the pipe f  f(Re)
H=F(Q)
Energy dissipated in the system
Mechanical energy accumulated by the fluid. Q

11. System curves Ventilation systemsQ H
H=F(Q)
System curve
Energy dissipation in the system
Ventilation systems
Closed circuit piping systems have similar curves (no energy storage). =0

12. System curves Hydroelectric plants H Q Energy dissipation System curve
H=F(Q)
System curve
Energy dissipation

13. Operation point
Flow and head to which the provided energy by the pump balances the system energy requirements:
System curve H 1 H1
Pump performance curve at rotation N Q1 Q

14. Maximum efficienty conditions
For which rotation is maximum efficiency achieved?
Point 2: Maximum efficiency point at original rotation
Maximum efficiency points for different N and same pump: H
System curve H2 2 1 H1
Pump curve at rotation N 3
Point 3: maximum efficiency point (at a different rotation speed) and also a point in the system curve Q2 Q1 Q3 Q

15. Series association of machines
What is the flow provided by the two pumps in series? Same flow, added heads BA BB Q H
Resultant curve of the series association
H=HA+HB
System curve
A+B A B
Pump A curve at rotation NA
Pump B curve at rotation NB Q

16. Parallel association of machines
What is the flow provided by the two pumps in parallel?
Same head, added flows H BA BB Q
System curve
A+B
Resultant curve of the parallel association A B
H=HA=HB
Pump A curve at rotation NA Q
Pump B curve at rotation NB
Q=QA+QB

17. Series and parallel association in hydraullic powered machines
Series association:
Parallel association:

18. Exercise: 1st Test
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. es
10,5 m
a) Compute the flow rate:
25 l/s l/s l/s l/s l/s l/s
b) And the dissipated energy in the pipe?
1,3 kW ,1 kW 4,5 kW
6,1 kW ,0 kW 8,5 kW

19. Exercise: 1st Test
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. es
10,5 m
c) Compute the rotational speed for pump maximum efficiecy?
1525 rpm rpm rpm
1936 rpm rpm rpm
c) Q RMX 0,03 m3/s H RMX 34,2 m K RMAX m/(m3/s)^2 Q'RMX 0, m3/s N' 1783,765 rpm

20. Bibliography Chapters 2 and 3
Turbomáquinas, A. F. O. Falcão, Folhas AEIST, 2004.