1. Dr. Szlivka: Fluid Mechanics 7.1 ÓBUDA UNIVERSITY Dr. Ferenc Szlivka professor

2. Dr. Szlivka: Fluid Mechanics 7.2 Momentum equation and it’s applications Chapter 7.

3. Dr. Szlivka: Fluid Mechanics 7.3 Momentum equation Integral form With single (loads) forces

4. Dr. Szlivka: Fluid Mechanics 7.4 Force acting on a flat plate A horizontal water jet with "A" cross section area and "v" absolute velocity acting on a flat plate that is perpendicular to the jet. The flat plate moving with "u" horizontal velocity. How big force acting on the flat plate from the water jet ?

5. Dr. Szlivka: Fluid Mechanics 7.5 Force acting on a flat plate Solution: Investigate the case when the flat plate is moving with "u" velocity. The relative velocity of the yet to the flat plate is:

6. Dr. Szlivka: Fluid Mechanics 7.6 Solution: I. Step: Calculate the outgoing velocity magnitude with the help of Bernoulli ‘s equation in a relative coordinate system. Neglect the height difference among the points 1,2 and 3. 2 3 1 z2z2 z3z3 w2w2 w3w3 w1w1

7. Dr. Szlivka: Fluid Mechanics 7.7 II. Step draw a control surface -the flow should be steady (pl. the rigid body does not go out from the surface), -if we are searching for the force acting on the rigid body, the body should be inside the surface -where fluid is going in or coming out the surface should be perpendicular to the yet or parallel with it Control surface Moving together with the fat plate

8. Dr. Szlivka: Fluid Mechanics 7.8 III. Step Write the momentum equation The left integral argument is not equal to zero where the fluid is crossing the control surface. Denote these parts of integral with The first integral on the right hand side is the result of the pressure forces acting on the control surface. It can be calculated also a sum of parts integrals. Control surface Moving together with the fat plate

9. Dr. Szlivka: Fluid Mechanics 7.9 IV. Step The closed surface momentum integral can be calculated sum of part integrals. The only term is and w 1 is opposite than dA so the direction of momentum vector is directed out from the surface. In usual case the momentum vector is directed out from the surface. The result of the pressure forces is zero, because the pressure is everywhere constant, p 0. I1I1 dI 2 dI 3 Control surface Moving together with the fat plate

10. Dr. Szlivka: Fluid Mechanics 7.10 V. Step "x" direction: I1I1 dI 2 dI 3 Control surface Moving together with the fat plate

11. Dr. Szlivka: Fluid Mechanics 7.11 Pelton-turbine

12. Dr. Szlivka: Fluid Mechanics 7.12 Pelton-turbine

13. Dr. Szlivka: Fluid Mechanics 7.13 Pressure tube Járókerék Turbine casing Quick Jet regulator Valve Regulator Nose elv. Downstream data: Questions: a./ Calculate the force acting on one blade of the turbine! b./ Calculate the average force acting on the wheel! c./ Calculate the power of the turbine with the given data! d./ Calculate the function of power respect to „u” (circumferential speed) !

14. Dr. Szlivka: Fluid Mechanics 7.14 Solution: a./ Force acting on one blade w 2 ’’ w2’w2’ w1w1 Using the momentum equation Control surface I1’I1’ I 2 ’’ I2’I2’ X component of the momentum law

15. Dr. Szlivka: Fluid Mechanics 7.15 w 2 ’’ w2’w2’ w1w1 I1’I1’ I 2 ’’ I2’I2’ R 1x a./ Force acting on one blade

16. Dr. Szlivka: Fluid Mechanics 7.16 b./ Force acting on the weel Using the momentum equation Control surface X component of the momentum law

17. Dr. Szlivka: Fluid Mechanics 7.17 b./ Force acting on the weel Using the momentum equation Control surface X component of the momentum law v 2x

18. Dr. Szlivka: Fluid Mechanics 7.18 b./ Force acting on the weel The average circumferential force acting on the weel: Force acting on one blade : The average force is bigger than the one blade force. The jet can act not only one blade but two or more blades too.

19. Dr. Szlivka: Fluid Mechanics 7.19 b./ Force acting on the weel A kerületi sebesség:

20. Dr. Szlivka: Fluid Mechanics 7.20 c./ T he function of power respect to „u” (circumferential speed) 0 100 200 300 400 500 600 700 800 900 1000 020406080100120 u [m/s] P e [kW]

21. Dr. Szlivka: Fluid Mechanics 7.21 c./ The power with the given data (P e ) and the maximum power (P emax ) The maximum power Circumferential speed

22. Dr. Szlivka: Fluid Mechanics 7.22 d./ The circumferential force change In the same time more than one blade is working. Sometimes "1",and sometimes "2" are the acting force. The working time of two blades is denoted by „t 2 ",and the working time of one blade is „t 1 ".

23. Dr. Szlivka: Fluid Mechanics 7.23 d./ The circumferential force change

24. Dr. Szlivka: Fluid Mechanics 7.24 Airscrew theory The air screw make a pressure jump in the air. The incoming and outgoing flow is approximately ideal flow without losses.

25. Dr. Szlivka: Fluid Mechanics 7.25 Airscrew theory The pressure changing around the airscrew.

26. Dr. Szlivka: Fluid Mechanics 7.26 Airscrew theory (with the momentum equation)

27. Dr. Szlivka: Fluid Mechanics 7.27 Airscrew theory (with the momentum equation) Continuity equation

28. Dr. Szlivka: Fluid Mechanics 7.28 Airscrew theory (with the momentum equation)

29. Dr. Szlivka: Fluid Mechanics 7.29 Airscrew theory (with the momentum equation)

30. Dr. Szlivka: Fluid Mechanics 7.30 Propulsion efficiency Usefull power Total power

31. Dr. Szlivka: Fluid Mechanics 7.31 a./Calculate the force acting on an aircrew, with cross section area „A l „. The aircraft is moving with „ v 1 " velocity. The air velocity after the aircrew is „ v 2 " in the coordinate system fixed to the aircraft! b./Calculate the ideal propulsion efficiency of the screw! data:

32. Dr. Szlivka: Fluid Mechanics 7.32 Solution:

33. Dr. Szlivka: Fluid Mechanics 7.33 Windmill

34. Dr. Szlivka: Fluid Mechanics 7.34 Windmill Inverse airscrew

35. Dr. Szlivka: Fluid Mechanics 7.35 Maximum power of a windmill The question is the optimum velocity after the windmill „v 2 " if the „v 1 " is constant? How big is the maximum power of the windmill?

36. Dr. Szlivka: Fluid Mechanics 7.36 Maximum power of a windmill The expression shows that the maximum power of the windmill is only the 16/27 (59,26 %) part of the power crossing through the A sz area!

37. Dr. Szlivka: Fluid Mechanics 7.37 A Windmill on a farm has 2 m diameter. Calculate the maximum ideal power of it at wind velocity. The windmill at Kulcs in Hungary The height of the pile is 60 m, the diameter of the impeller is 50 m. The maximum power calculated with the upper formula is 689 kW. The effective power is approximately 65-90 %-a, so the power is 600 kW.

38. Dr. Szlivka: Fluid Mechanics 7.38 Windmill in Kulcs in Hungary http://www.winfo.hu