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 : Reynolds number Torque coefficient flow coefficient Neglecting Re effects (fully turbulent flow):

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.

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

Dynamical similaritude If 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)

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  

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.

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 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

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

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

System curves Hydroelectric plants H Q Energy dissipation System curve 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: System curve H 1 H1 Pump performance curve at rotation N Q1 Q

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

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

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

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

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. es 10,5 m 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 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 1685 rpm 1784 rpm 1936 rpm 2352 rpm 2842 rpm c) Q RMX 0,03 m3/s H RMX 34,2 m K RMAX 38000 m/(m3/s)^2 Q'RMX 0,017838 m3/s N' 1783,765 rpm

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