![Francis Turbine – Velocity Triangle
Force exerted by Jet on the Bucket (β<90O )
Now, Angular Momentum = Momentum x Radius
i. momentum /sec at the inlet = (mass/sec) x (velocity
component in tangential dirn)
= (ρ a V1) x [Vw1]
ii. momentum /sec at the outlet= (mass/sec) x (velocity
component in tangential dirn)
= (ρ a V1) x [-Vw2]
iii. Angular momentum at inlet = (ρ a V1) x [Vw1] x R1
iv. Angular momentum at outlet = (ρ a V1) x [-Vw2] x R2
Prepared By Prof. S. G. Taji](https://image.slidesharecdn.com/5-200618172719/85/Francis-Turbine-8-320.jpg)
![Francis Turbine – Velocity Triangle
Force exerted by Jet on the Bucket (β<90O )
∴ Fx= Mass of water striking the vane per second ×
(Initial angular momentum – final angular momentum)
Fx = (ρ a V1) x [Vw1 R1] - (ρ a V1) x [-Vw2] x R2
= (ρ a V1) [Vw1 R1 + Vw2 R2]
When, β > 90o Fx (ρ a V1) [Vw1 R1 - Vw2 R2]
In general,
Fx = = (ρ a V1) [Vw1 R1 ± Vw2 R2]
When, β = 90o (Radial discharge at outlet )
Fx = = (ρ a V1) [Vw1 R1]
Prepared By Prof. S. G. Taji](https://image.slidesharecdn.com/5-200618172719/85/Francis-Turbine-9-320.jpg)
![Francis Turbine – Velocity Triangle
Work Done by Jet on the runner / sec
Work Done /sec = Fx x Angular Velocity (ω)
= (ρ a V1) x [Vw1 R1 ± Vw2R2] x ω
= (ρ a V1) x [Vw1 ω R1 ± Vw2 ω R2]
but, u1 = ωR1 & u2 = ωR2
Work Done /sec = (ρ a V1) x [Vw1 u1 ± Vw2 u2] N-m/s
Power = (ρ a V1) x [Vw1 u1 ± Vw2 u2] N-m/s or watts
When, β = 90o (Radial discharge at outlet )
Work Done /sec = (ρ a V1) x [Vw1 u1]
Prepared By Prof. S. G. Taji](https://image.slidesharecdn.com/5-200618172719/85/Francis-Turbine-10-320.jpg)
Francis Turbine
- 1. Francis Turbine Main Components Velocity Triangle Efficiencies Design Aspects Satish G. Taji Assistant Professor Civil Engineering Department SRES’s Sanjivani College of Engineering, Kopargaon1 Hydraulic Turbines 1
- 2. Francis Turbine – Main Components Prepared By Prof. S. G. Taji Image Courtesy: Wikipedia (Google); gmdu.net (Google); pinterest (Google), learnengineering.org
- 3. Francis Turbine – Velocity Triangle R1, R2= Radius of wheel at the inlet and outlet of vanes V1, V2 = Absolute velocities of the jet at the inlet and outlet respectively, u1, u2 = Peripheral velocities of the vane at the inlet and outlet respectively, Vr1, Vr2 = Relative velocities at the inlet and outlet respectively, Vf1, Vf2 = Velocities of the flow at the inlet and outlet respectively, Vw1, Vw2 = Velocities of the whirl at inlet and outlet θ, φ = Tip angles at the inlet and outlet respectively, α, β = Angles made by absolute velocities at the inlet and outlet Prepared By Prof. S. G. Taji
- 4. Francis Turbine – Velocity Triangle β < 90o Prepared By Prof. S. G. Taji Wheel Tangent at outlet Tangent at inlet 1 2 2 1 1 2
- 5. Francis Turbine – Velocity Triangle β =90o Prepared By Prof. S. G. Taji Wheel Tangent at outlet Tangent at inlet
- 6. Francis Turbine – Velocity Triangle β > 90o Prepared By Prof. S. G. Taji Wheel Tangent at outlet Tangent at inlet
- 7. Francis Turbine – Velocity Triangle Uniform velocity at inlet and outlet Tip (vane) u1 = π D1 N / 60 u2 = π D2 N / 60 where, D1 =Dia. Of Wheel at the inlet D2 =Dia. Of Wheel at the outlet; N= Speed in rpm = 60f/p Force exerted by Jet on the Bucket (β<90O ) Force exerted in the direction of motion Fx = Rate of Change of Momentum in the same dirn But, in this case, momentum is angular momentum ∴ Fx= Mass of water striking the vane per second × (Initial angular momentum – final angular momentum) Prepared By Prof. S. G. Taji m R VM = m x v Angular momentum = (m x v) x R
- 8. Francis Turbine – Velocity Triangle Force exerted by Jet on the Bucket (β<90O ) Now, Angular Momentum = Momentum x Radius i. momentum /sec at the inlet = (mass/sec) x (velocity component in tangential dirn) = (ρ a V1) x [Vw1] ii. momentum /sec at the outlet= (mass/sec) x (velocity component in tangential dirn) = (ρ a V1) x [-Vw2] iii. Angular momentum at inlet = (ρ a V1) x [Vw1] x R1 iv. Angular momentum at outlet = (ρ a V1) x [-Vw2] x R2 Prepared By Prof. S. G. Taji
- 9. Francis Turbine – Velocity Triangle Force exerted by Jet on the Bucket (β<90O ) ∴ Fx= Mass of water striking the vane per second × (Initial angular momentum – final angular momentum) Fx = (ρ a V1) x [Vw1 R1] - (ρ a V1) x [-Vw2] x R2 = (ρ a V1) [Vw1 R1 + Vw2 R2] When, β > 90o Fx (ρ a V1) [Vw1 R1 - Vw2 R2] In general, Fx = = (ρ a V1) [Vw1 R1 ± Vw2 R2] When, β = 90o (Radial discharge at outlet ) Fx = = (ρ a V1) [Vw1 R1] Prepared By Prof. S. G. Taji
- 10. Francis Turbine – Velocity Triangle Work Done by Jet on the runner / sec Work Done /sec = Fx x Angular Velocity (ω) = (ρ a V1) x [Vw1 R1 ± Vw2R2] x ω = (ρ a V1) x [Vw1 ω R1 ± Vw2 ω R2] but, u1 = ωR1 & u2 = ωR2 Work Done /sec = (ρ a V1) x [Vw1 u1 ± Vw2 u2] N-m/s Power = (ρ a V1) x [Vw1 u1 ± Vw2 u2] N-m/s or watts When, β = 90o (Radial discharge at outlet ) Work Done /sec = (ρ a V1) x [Vw1 u1] Prepared By Prof. S. G. Taji
- 11. Francis Turbine – Efficiencies Hydraulic Efficiency (ήh) ήh = (WD/sec) / (KE/sec) OR RP / WP Prepared By Prof. S. G. Taji
- 12. Francis Turbine – Efficiencies Mechanical Efficiency (ήm) ήm = (Shaft Power) / (WD/sec) OR SP / RP Overall Efficiency (ή0) Prepared By Prof. S. G. Taji
- 13. Francis Turbine – Design Aspects 1. Discharge through reaction Turbine Prepared By Prof. S. G. Taji
- 14. Francis Turbine – Design Aspects 2. Speed Ratio (Ku): 3. Flow Ratio (Kf): 4. Ratio of Width to Diameter (n) Prepared By Prof. S. G. Taji