Introduction to fluid machines

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. 1000 rpm, D=25 cm H 1200 rpm, D=20 cm 1350 rpm, D=15 cm 1500 rpm, D=15 cm Q Application of  theorem to hydraulic turbomachines (constant ) • For the same family of gemetrically similar machines, the non-dimensional performance curves overlap:

5. 1000 rpm 1200 rpm 1350 rpm 1500 rpm Dynamical similaritude • If 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. 1000 rpm 1200 rpm 1350 rpm 1500 rpm Dynamically similar points for the same machine at different rotations N • Same machine: D1=D2 12

7. k 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 N1 = 1000 rpm Q2 Q1 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. pB zB-zA Q pA 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 Required net mechanical energy supplied to the fluid in the pump H=F(Q) is the system curve

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

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

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

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

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

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

18. 10,5 m es Exercise: 1st Test 2010-11 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. 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

19. 10,5 m es Exercise: 1st Test 2010-11 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. c) Compute the rotational speed for pump maximum efficiecy? 1525 rpm 1685 rpm 1784 rpm 1936 rpm 2352 rpm 2842 rpm

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