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 Const u Hg2. c Hg2. cp A F. u c      

5 Velocity triangles w c 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  c 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/D 2 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 Laws Example Change of speed n 1 = 600 rpmQ 1 = 1,0 m 3 /s n 2 = 650 rpmQ 2 = ?

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 Bernoulli from 1 to 2 without friction gives: Reference line

17 Reduced values used for turbines

18 Dimensionless terms Speed –Speed number  –Specific speedN QE –Speed factorn ED, n 11 –Specific speedn q, n s Flow –Flow factorQ ED, Q 11 Torque –Torque factorT ED, T 11 Power –Power factorP ED, P 11

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:

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

22 Speed number cmcm cmcm D 1 2 From equation 1: Inserted in equation 2:

23 Speed Factor unit speed, n 11 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.

24 Speed Factor n ED If we have a turbine with the following characteristics: Energy E = 1 J/kg Diameter D = 1 m

25 Energy Reference line z1z1 z tw h1h1 c1c1 abs

26 Specific speed that is used to classify turbines

27 Specific speed that is used to classify pumps n q is the specific speed for a unit machine that is geometric similar to a machine with the head H q = 1 m and flow rate Q = 1 m 3 /s n s is the specific speed for a unit machine that is geometric similar to a machine with the head H q = 1 m and uses the power P = 1 hp

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30 Flow Factor unit flow, Q 11 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.

31 Flow Factor Q ED If we have a turbine with the following characteristics: Energy E = 1 J/kg Diameter D = 1 m

32 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 m 3 /s D 0 = 4,86 m D 1 = 4,31m D 2 = 2,35 m B 0 = 0,28 m n = 333 rpm

33 Speed number:

34 Specific speed:

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

36 Kaplan

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