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Fundamental similarity considerations

Fundamental similarity considerations. Similarity Considerations Reduced parameters Dimensionless terms Classification of turbines Performance characteristics. Similarity Considerations.

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Fundamental similarity considerations

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  1. Fundamental similarity considerations • Similarity Considerations • Reduced parameters • Dimensionless terms • Classification of turbines • Performance characteristics

  2. Similarity Considerations Similarity considerations on hydrodynamic machines are an attempt to describe the performance of a given machine by comparison with the experimentally known performance of another machine under modified operating conditions, such as a change of speed.

  3. Similarity Considerations • Valid when: • Geometric similarity • All velocity components are equally scaled • Same velocity directions • Velocity triangles are kept the same • Similar force distributions • Incompressible flow

  4. These three dynamic relations together are the basis of all fundamental similarity relations for the flow in turbo machinery. c = Const . 1 u F = = r × × 2 p c Const . 2 A 2 g H × × × 2 g H = Const . = Const . 3 2 u 2 c

  5. Velocity triangles c w 1

  6. Under the assumption that the only forces acting on the fluid are the inertia forces, it is possible to establish a definite relation between the forces and the velocity under similar flow conditions In connection with turbo machinery, Newton’s 2. law is used in the form of the impulse or momentum law:

  7. For similar flow conditions the velocity change Dc is proportional to the velocity c of the flow through a cross section A. It follows that all mass or inertia forces in a fluid are proportional to the square of the fluid velocities. 2

  8. By applying the total head H under which the machine is operating, it is possible to obtain the following relations between the head and either a characteristic fluid velocity c in the machine, or the peripheral velocity of the runner. (Because of the kinematic relation in equation 1) 3

  9. For pumps and turbines, the capacity Q is a significant operating characteristic.  c is proportional to Q/D2 and u is proportional to n·D.

  10. Affinity Laws This relation assumes that there are no change of the diameter D.

  11. Affinity Laws This relation assumes that there are no change of the diameter D.

  12. Affinity Laws This relation assumes that there are no change of the diameter D.

  13. Affinity Laws This relations assumes that there are no change of the diameter D.

  14. Affinity LawsExample Change of speed n1 = 600 rpm Q1 = 1,0 m3/s n2 = 650 rpm Q2 = ?

  15. Reduced parameters used for turbines The reduced parameters are values relative to the highest velocity that can be obtained if all energy is converted to kinetic energy

  16. Reference line Bernoulli from 1 to 2 without friction gives:

  17. Reduced values used for turbines

  18. Dimensionless terms • Speed • Speed number W • Specific speed NQE • Speed factor nED, n11 • Specific speed nq, ns • Flow • Flow factor QED, Q11 • Torque • Torque factor TED, T11 • Power • Power factor PED, P11

  19. ~ ~ Fluid machinery that is geometric similar to each other, will at same relative flow rate have the same velocity triangle. For the reduced peripheral velocity: For the reduced absolute meridonial velocity: We multiply these expressions with each other:

  20. Speed number D Geometric similar, but different sized turbines have the same speed number

  21. Speed number 1 2 From equation 1: Inserted in equation 2: D cm cm

  22. Speed Factorunit speed, n11 • If we have a turbine with the following characteristics: • Head H = 1 m • Diameter D = 1 m • we have what we call a unit turbine.

  23. Speed FactornED • If we have a turbine with the following characteristics: • Energy E = 1 J/kg • Diameter D = 1 m

  24. Energy h1 abs c1 ztw z1 Reference line

  25. Specific speed that is used to classify turbines

  26. Specific speed that is used to classify pumps nqis the specific speed for a unit machine that is geometric similar to a machine with the head Hq = 1 m and flow rate Q = 1 m3/s nsis the specific speed for a unit machine that is geometric similar to a machine with the head Hq = 1 m and uses the power P = 1 hp

  27. Flow Factorunit flow, Q11 • If we have a turbine with the following characteristics: • Head H = 1 m • Diameter D = 1 m • we have what we call a unit turbine.

  28. Flow FactorQED • If we have a turbine with the following characteristics: • Energy E = 1 J/kg • Diameter D = 1 m

  29. Exercise • Find the speed number and specific speed for the Francis turbine at Svartisen Powerplant • Given data: P = 350 MW H = 543 m Q* = 71,5 m3/s D0 = 4,86 m D1 = 4,31m D2 = 2,35 m B0 = 0,28 m n = 333 rpm

  30. Speed number:

  31. Specific speed:

  32. Performance characteristics NB: H=constant Efficiency [-] Speed [rpm]

  33. Kaplan

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