Specific Static rotor work ( P P )

Transcription

1 The specific Static Rotor ork p 1 ρ Specific Static rotor work ( P P ) here P, P static pressures at points, (P P ) static pressure difference of the rotor ρ density, in case of a compressible medium average of ρ and ρ. p can be calculated from the energy difference of the flow medium between o and P blade m Z u P P ρ here blade 1 cosα 1 cos α 1

2 Applying the cos theorem of a triangle Applying the cos theorem of a triangle b a α α α cos a c b bc α cos α ( ) 1 blade It follows ( ) 1 It follows ( ) u u blade P Z Z m m 1 ( ) u u blade P

3 Bernoulli Equation of the Relative Flow Bernoulli Equation of the Relative Flow Bernoulli Equation of the Relative Flow Bernoulli Equation of the Relative Flow Neglecting the hydraulic loss, i.e. Z u, 1 P P P ρ It follows 1 P P The above formula applies to any points along the flow line ρ ρ passing the vane channel P Bernoulli Equation of the const P ρ Bernoulli Equation of the Relative Flow

4 Impulse and Reaction Type of Turbomachines onsidering P, the turbomachine can be grouped into: A. Turbomachines without pressure difference in front of and beyond the rotor, i.e (P P ) or p Impulse type of Turbomachines B. Turbomachines with pressure difference in front and beyond the rotor, i.e. (P PP ) p > Reaction type of Turbomachines 4

5 For impulse turbines The total head of the incoming fluid is converted into a large velocity head at the exit of the supply nozzle. Both ththe pressure drop across the bucket t(blade) and the change in relative speed of the fluid across the bucket are negligible. The space surrounding the rotor is not completely filled with fluid. The individual jets of fluid striking the buckets that generates the torque. 5

6 For reaction turbines There is both a pressure drop and a fluid relative speed change across the rotor. Guide vanes act as nozzle to accelerate the flow and turn it in the appropriate direction as the fluid enters the rotor. Part of the pressure drop occurs across the guide vanes and part occurs across the rotor, 6

7 Summary Impulse turbines: High head, head, low flowrate devices. Reaction turbines: Low head, high flowrate h devices. 7

8 Equal Pressure or Impulse Type of Turbomachines P P and P Example a. Single Stage Steam Turbine 8

9 The entirely available pressure difference (P P ) is converted into velocity while the flow passes through h the stationary tti guide vanes The velocity existing in the clearance between the stationary guide vanes and the rotor blades is the highest one which can be obtained from the available pressure difference, i.e. ie max attainable The kinetic energy of the flow entering the rotor is reduced d while the flow passes the blade channels, the absolute velocity is reduced from to. The specific static rotor work p is (for axial flow 1 ) ( ) Zu 1 P 9

10 Neglecting the hydraulic lose Z u of the rotor, it follows because p. onsidering i the loss: ϕ here the velocity coefficient φ takes in to account the drop in kinetic energy due to Z u ; Ф<1. The condition o demands rotor blades of the hookform type, i.e. β >9. Blades of a constant-pressure steam or Gas turbine. a is the channel width at all points approximately equal 1

11 Ifbladehasuniform thickness, the flow whilepassing the channel is first decelerated then accelerated. Such change in the flow velocity is undesirable as it leads to unnecessary losses. In order to obtain const. along the vane channel the blade be designed withstrong profiling; however, such blades are costly 11

12 The specific work blade of an impulse steam turbine stage as for a given velocity proportional to the velocity given velocity proportional to the velocity blade α For α 9 cos maxatt. Steam turbines are designed with approximately the same angle α 15 to degrees. As of impulse team turbines has highest possible value max att. The spec. work blade of these turbines has highest value blade impulse t. blademax. att. for a given The peripheral velocity will be lowest for a given blade if the turbine is designed as impulse turbine Impulse turbines are slow running turbines 1

13 Over-Pressure or Reaction Type of Turbomachine Example: Single Stage g Reaction Steam Turbine 1

14 hile the flow passes through the channels of stationary vanes (also called Guide Blades), only a portion of the available pressure difference (P D P S )is converted into velocity Thus < max attainable atta ab and, hence, the spec. work blade bade of the reaction turbine is smaller that that of the impulse turbine if the same velocity is assumed The velocity of reaction turbines has to be higher than that of impulse turbines if the same blade is to be obtained. Reaction turbines turbomachines. may be classified as fast running 14

15 β 1 should be small but not too small as leads to strong whirls in the discharge flow. The angle β of reaction turbines is 9 β and, thus, differs from that of impulse turbines. The blade does not have the hook form. As the relative velocity increases from to, the channel width decreases and no profile is necessary in order to obtain equal channel width. Reaction turbine has more stages because of the lower blade ofitssingle single stage. 15

16 Degree of Reaction In case of the reaction turbine, the driving force at the rotor is due to change of direction (impulse) and magnitude (reaction) of the relative velocity. A reaction effect, i.e. a change of the magnitude of the relative velocity in case of the axial flow machine, is only possible if a pressure difference exists between the entrance and discharge side of the rotor. Quota of reaction acting is found by comparing total energy, with pressure rotor energy ( p ). 16

17 Spec. Static rotor work Degree of reaction Spec. work betweeninlet and outlet( of the stage) P impulse machine : reaction machine : P P and R > and < R < 1 ( R 1in some special cases) The reaction effect exists also in case of radial or mixed flow rotors where 1 even for as shown by the equation P 1 ( ) m Z ( ) 1 u 1 1 m Z u 17

18 Blade Speed Ratio The blade speed ratio as defined below is widely used in the calculation l of turbines especially ill of steam turbines. Blade Speed Ratio is the velocity which could be obtained if the spec. work is converted without losses completely into velocity. ϕ 1 R here Ф is velocity coefficient of guide vanes (referring to velocity losses) After some derivation ηh 1 ϕ cosα 1 R 18

19 Assuming the following data: η h.85;φ.98; α. η h ϕ cosα 1 The blade speed ratio has the value R R for R for R.5 The following values of the blade speed ratio re obtained with actual machines: impuse steam turbines reaction steam turbines PeltonTurbines R R R>.44 to k' 1 R.5 { to.47 { heap Design. small power.47.5to 1 R.47 high quality design l arge power k' 19

20 (57) alculation of Mean Diameter and Peripheral Velocity using the Blade Speed Ratio

21 The Vane Angle β Three different axial flow vanes, namely form A, B, for which, m and β 1 are the same but the angle β differ 1

22 Β is chosen for the form A as β <9, for the form B as β 9 and for the form as β >9. A similar sketch for three different radial flow vanes with β <9 (form a), β 9 (form b) and β 9 (form c) is given below.

23 In case of the radial flow machine the vanes of the different forms are called Vane form a as backward curved vanes Vanes form b, c as forward curved vanes The following relation exists between β and ase:α 9 blade blade where, cot β, u blade and m ( cot β ) m u and it follows then m m blade tan β tan β ase:α 9 m m blade 1 tan tan β β O

24 The necessary peripheral p velocity for a given blade can be determined by these equation if the vane angle β is assumed. A large β decreases and the size of the rotor decreases, too, if the speed n is not altered: 4

25 Ranges of β with different types of turbomachines 5

26 Shape Number, Specific Speed The shape of the rotor is determined by the three related values n, and V as long as the vane angle β is unchanged. 1. Effect of Increase in speed n on the shape of the rotor (with unchanged dββ,v and ) The unchanged demands the same velocity triangle at. blade blade The unchanged velocity triangle can be obtained for increased speed n but same velocity as demanded by the unchanged velocity triangle only at a smaller outer diam. 6

27 7

28 . Effect of Increase in V on the shape of the slow running rotor (with unchanged β,n,d,and ) The larger volume V can be obtained only by increasing the channel width and the eye dia. Ds The meridian component of the velocity must remain unchanged because of the unchanged with same n and D Demanding and unchanged velocity triangle at. 8

29 The rotor shape is a function of n, V and. Shape number (N shape ) is a dimensionless number and is used to define the shape of the rotor by relating n, V and. It follows α β 1 m m N shape [ 1 ] n V, assume 1; s α s s 1 1 s 1 m s β m s γ m s γ 1 1/ 4 n V / m : β β γ Thus, N shape [ 1] n V 4 S: 1 βγ thus, 1 1 β or β γ or γ 4 nsh 1N shape 9

30 A relation which is based on the head H instead on the spec. work is called Specific Speed. n n q H here the values has a unit of n(rpm), (p V(m /s) /) and H(m). n q is not dimensionless for metric system n q has the following unit V 4 n q m s 4 6s 1min N shape N shape 4 m s.min For water turbines a specific speed derived from n, H and N is often used. n n s H N 5 4

31 Values of N shape,n q and n s : 1N shape n q n s Slow running rotor to 1 11 to 8 4 to 14 Medium running rotor 1 to 5 8 to 8 14 to Fast running rotor 5 to 5 8 to 164 to 6 axial flow rotor to to 5 4 to 18 1