SPECIFICATIONS FOR OBTAINING MINIMUM DIMENSIONS OF THE PELTON WATER WHEEL, WHEN THE DEBIT AND FALL OF WATER HEAD ARE GIVEN

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1 Scientific Bulletin of the Politehnica University of Tiisoara Transactions on echanics Special issue The 6 th International Conference on Hydraulic achinery and Hydrodynaics Tiisoara, Roania, October -, 004 SPECIFICTIONS FOR OBTINING INIU IENSIONS OF THE PELTON WTER WHEEL, WHEN THE EBIT N FLL OF WTER HE RE GIVEN Ion GR, ssoc. Prof. uitru RSENIE, Prof. epartent of echanics epartent of Construction Ovidius University of Constanţa Ovidius University of Constanţa Georgeta NRESCU, ssoc. Prof. epartent of Physics, Cheestry and Oil Engineering Ovidius University of Constanţa *Corresponding author: Bv aaia4, Constanta, Roania Eail: g0@hotail.co BSTRCT In this paper the diensioning is accoplished for turbine with an even nuber of injectors diaetrical opposed so that the turbine axel will no longer be subitted to bending by the hydrodynaic forces. Those last ones are reduced in the axle tree to the wrench fored only by the torsion oent. The reduced journal diaeters leads to the increasing echanic efficiency of the Pelton turbine. KEYWORS Pelton turbine, echanic efficiency, journal diaeter diensioning NOENCLTURE [] arature diaeter of the Pelton turbine d [] journal diaeter F [N] force H [] the gross turbine fall g [/s ] gravity G [N] arature weight K v [-] velocity coefficient l [] length [kg] ass [N] oent N [N] reaction in bearings r [] position vector P [W] power R [N] resultant force Q [ /s]debit W [ ] resistent axial odul u [/s] cup velocity v [/s]] flush velocity [-] nuber of injectors α [rad] current return angle ω [s - ] angular velocity of the arature µ [-] friction coefficient η [%] echanic efficiency ρ [kg/ ] water density Subscripts and Superscripts c centrifuge f friction h hydrodynaic otor n noinal r resistant t total v velocity. INTROUCTION The injector debit is [,]: Qt Q () The relationship () is valid when it is used identical injectors which peranently assure sae by-pass sections, obtaining in this way, identical flushes. The hydrodynaic force F h developed by one flush turbine cup is [,, ]: 97

2 ( u)( ) F h ρq v cosα () The water velocity at the nole going out is [,,]: v K gh () v The cup transport velocity is: u ω (4) Figure. Pelton turbine ipeller - disc; - cup; - bolt; 4-axle For a axi efficiency, the flush return angle and the cup velocity [,, ] are: α π (5) v u (6) Fro () relationship and taking account of (5) and (6) relationships we obtain the hydrodynaic force for noinal regie of the turbine: F hn ρqv (7) The wrench of forces F h in 0 point (as shown in figure ) is: r r R Fhi i (8) τ 0 r r r i xfhi i The consued power at the Pelton arbor turbine is given by the relation [,, ]: P c ω (9) The position vectors respect the condition: r r i F hi (0) which is actual achieved.. VNCE FOR IPROVEENT OF THE ECHNIC EFFICIENCY The resultant R fro the relation (8) is taken on by the radial journals. Fro the arature equilibriu condition, F r 0 () results r r v R + G + N 0 () In each radial journal the reaction is: N N 0 () For the turbine axle with vertical axis, the reaction in each radial journal is: R N 0 (4) The reaction in the axial bearing is: N a G (5) It is valid the relationship [7]: d f d f f + f µ N0 + (6) The echanic efficiency is [,7]: P P P u c p p η (7) The lost power in journals P p is: P ω (8) p Fro (7) taking account of (7) and (8) obtain: η (9) Fro (9) results that we can increase the echanical efficiency by reducing the otor oent. Fro () and () results that for reducing the reaction N 0 it is adequate to reduce the coposite force and arature weight. We choose an even nuber of injectors. The (8) relation becoes [8]: r R 0 τ 0 r r (0) Fhi i Fro the otion equation for echaniss [, 7]: dω J0 r () dt Result that the otor oent ust be equal in odule with the resistant oent. P 98

3 () t r The resistant oent is: r r r r + + () r g fg The total inertial oent of the turbine aratures in regard to the rotation axis, fro () has the constant value [7, 8]. The charging schee of the turbine arature, for the case respecting (0) relationship, is given in figure. the journal in B bearing is torsion. accordingly we calculate the d f journal diaeter with the relation: t t τ af Wp f (8) 6. VNCE FOR OBTINING INIL TUINE BOUNRY LINE The turbine axle tree is deterined with the relations [5, 6]: σ a W e e i ax + ( α ) t (9) (0) The bending oent and the resistant axial odule are: i ax W G l () () Figure The charging schee of the Pelton turbine arature Fro figure results that in the journal the axle journal is subit on bending and shear. We select the length of the journal equal with its diaeter [4] and diension the journal with the relation [5, 6]: ai f σ e σ + 4τ σ ai (4) σ ai G d f (5) 4 f G τ f (6) f Fro (4) taking into account of (5) and (6) we obtained the end journal diaeter d f. The journal it is verified with the relation [4]: p ef G d pa (7) f In (7) the adissible pressure it is function of the axle box aterial [4]. The principal application of It is obvious that the distance between the turbine journals ust be the inial one. The torsion noinal is equal with the otor one tn n (figure ). (F h ) ax (F hax ) tn n f 0 n n,9n a n a n n n (ω) Figure The variation of the otor oent function of noinal speed Fro (9) results that we can reduce the torsion oent by increasing the angular velocity. The following relations can be written [,, ]: v v ρ Qt ρ Q () 99

4 v v ρ St ρs (4) π dot π d0 (5) 4 4 Fro (5) results: d0 t d0 (6) Function of turbine fall the relations are valid []: ( 8 )d 0 (7) 0t ( 8 ) d0t (8) Fro (6), (7), (8) result: 0 t (9) Fro (9) results that if we fit out the turbine with any injectors, we diinish the arature diaeter. Taking account of (), we can write the otor oent expression: Fh ρ Q( v u)( cosα ) (40) t the blank start of the turbine the resistant oent becoe equal with the shear oent (figure ) and in this case the expression of the angular velocity is [7]: ω ε t (4) The diensioning for the turbine arature is developed with the values: F hax ρ Qv (4) ax ρ Qv (4) Fro (7), (4) and (4) results: hax hn (44) F hax F hn (45) Fro () results: n f ε n J0 (46) The average acceleration in the regie phase is: ( ax + n ) f ε J0 (47) Ignoring the friction oent results: ε ε n (48) The start tie is: t n t (49) Fro the flush return angle α π, results: Q( t ) Q v v t ε n Finally we obtain the relation: S () t Q v ε n t Figure 4 The exit section of the flush fro the injector - needle nole (50) (5) d + δ S() t π B (5) B x(t) sinα (5) δ d-bcosα (54) Finally we obtain: Q v ε t n sin α π sinα d n x (55) Based on the relation (55) we deterine the low of the displaceent of the injector needle function of tie. In this way we calculate the hydrodynaic force in the noinal regie F hn, half of the axi hydrodynaic force F hax. In this way the turbine arature has ini diensions for a fall and debit given, and a nuber of injectors. In order to diension it is scheatic represented, the cup fro the figure, with a siple bea, according to figure 5a. We deterine the l and l [], []. Fro the cup equilibriu equation result the relations: 00

5 0 Fhn l l 0 Fhn l (56) F c F F x y 0 R 0 R x y F c F 0 hn + R B l (57) ωh (58) The reaction in the articulation is: x y R R + R (59) Figure 6 The cup attached on the turbine arature disc Figure 5a The charging schee of the Pelton turbine cup Figure 5b The section of the Pelton turbine ar The forces R and R B shear the bolts attachent of the cup on disc in two sections. The following relations are valid: R τ af (60) 4 τ af 4 B (6) Fro (58) and (59) deterine the bolts attachent of the cup on disc d in hinge and respectively d B in B bearing, knowing τ af (figure 6). The forces R and R B action on the arature disc which is crushed: R σ as (6) d bd Fro (6) results the disc thickness b d. The two ars of the cup are crushed in the bolts one and the relation are valid: R σ as (6) d b σ as (64) d BbB Result the b and b B ars thickness of the cup. The cup section in hinge is represented in figure 5b. It is valid the relation: i ax σ as W (65) where i ax F l (66) h hn b bd W (67) 6 6h Then we deterine the cup high h in the hinge one. In the bearing B (figure 5b and 6) the force R B apply shear stresses on ars and the relations are valid: τ as hb (68) 4bs h B (69) τ b as s 0

6 4. CONCLUSION It is possible to obtained a arature with ini diensions when the injectors are disposed in the arature plane so that the wrench of the hydrodynaic forces has the resultant null. The diensional calculus for the start fae of the arature it is based on the noinal hydrodynaic force which is half of the axi hydrodynaic force. So are obtained ini diaeters for the axle journals and in this way it is increased the turbine echanic efficiency. REFERENCES. nton I. (979) Turbine hidraulice, Editura Facla, Tiişoara. Constantinescu V.N. (980) Lagăre cu alunecare, Editura Tehnică, Bucureşti. anolescu N.I. (958), Teoria ecaniselor şi a aşinilor, Bucureşti 4. Florea J. (98), ecanica fluidelor şi aşini hidropneuatice, Bucureşti 5. Bărglăan. (95), aşini hidraulice, Institutul Politehnic Tiişoara 6. Budugan Gh. (974), Reistenţa aterialelor, Editura Tehnică, Bucureşti 7. Voinea R. (975) ecanica, Editura idactică şi Pedagogica, Bucureşti 8. răghici I. (980), Organe de aşini, Editura idactică şi Pedagogică, Bucureşti 0