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Euler equation for water turbine is derived on the basis of
- a)conservation of mass
- b)rate of change of linear momentum
- c)rate of change of angular momentum
- d)rate of change of velocity
Correct answer is option 'C'. Can you explain this answer?
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Euler equation for water turbine is derived on the basis ofa)conservat...
- The Euler turbine equation relates the power added to or removed from the flow, to characteristics of a rotating blade row. The equation is based on the concepts of conservation of angular momentum and conservation of energy.
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Euler equation for water turbine is derived on the basis ofa)conservat...
Derivation of Euler equation for water turbine
Euler equation is derived for water turbine on the basis of the principle of conservation of angular momentum. The derivation is as follows:
- Consider a water turbine with blades of length R and rotating at an angular velocity ω.
- Let the water enter the turbine at a distance r from the axis of rotation with a velocity Vr and leave the turbine at a distance R from the axis of rotation with a velocity Vθ.
- Let the mass flow rate of water be dm/dt.
- Applying the principle of conservation of angular momentum, we have:
Initial angular momentum = Final angular momentum
- The initial angular momentum of the water entering the turbine is given by:
L1 = r(Vr)dm/dt
- The final angular momentum of the water leaving the turbine is given by:
L2 = R(Vθ)dm/dt
- The change in angular momentum of the water is given by:
ΔL = L2 - L1 = R(Vθ)dm/dt - r(Vr)dm/dt
- The torque produced by the water turbine is equal to the rate of change of angular momentum of the water. Therefore, we have:
T = dΔL/dt = R(Vθ)(dm/dt)(dω/dt) - r(Vr)(dm/dt)(dω/dt)
- Simplifying the above equation, we get:
T = (Vθ)(dm/dt)(Rdω/dt - rdω/dt)
- Dividing both sides of the equation by the mass flow rate of water, we get:
T/dm = (Vθ)(Rdω/dt - rdω/dt)
- This equation is known as Euler equation for water turbine.
Conclusion
Thus, Euler equation for water turbine is derived on the basis of the principle of conservation of angular momentum. It relates the torque produced by the turbine to the angular velocity of the turbine and the velocity of the water entering and leaving the turbine.
Euler equation is derived for water turbine on the basis of the principle of conservation of angular momentum. The derivation is as follows:
- Consider a water turbine with blades of length R and rotating at an angular velocity ω.
- Let the water enter the turbine at a distance r from the axis of rotation with a velocity Vr and leave the turbine at a distance R from the axis of rotation with a velocity Vθ.
- Let the mass flow rate of water be dm/dt.
- Applying the principle of conservation of angular momentum, we have:
Initial angular momentum = Final angular momentum
- The initial angular momentum of the water entering the turbine is given by:
L1 = r(Vr)dm/dt
- The final angular momentum of the water leaving the turbine is given by:
L2 = R(Vθ)dm/dt
- The change in angular momentum of the water is given by:
ΔL = L2 - L1 = R(Vθ)dm/dt - r(Vr)dm/dt
- The torque produced by the water turbine is equal to the rate of change of angular momentum of the water. Therefore, we have:
T = dΔL/dt = R(Vθ)(dm/dt)(dω/dt) - r(Vr)(dm/dt)(dω/dt)
- Simplifying the above equation, we get:
T = (Vθ)(dm/dt)(Rdω/dt - rdω/dt)
- Dividing both sides of the equation by the mass flow rate of water, we get:
T/dm = (Vθ)(Rdω/dt - rdω/dt)
- This equation is known as Euler equation for water turbine.
Conclusion
Thus, Euler equation for water turbine is derived on the basis of the principle of conservation of angular momentum. It relates the torque produced by the turbine to the angular velocity of the turbine and the velocity of the water entering and leaving the turbine.
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Euler equation for water turbine is derived on the basis ofa)conservation of massb)rate of change of linear momentumc)rate of change of angular momentumd)rate of change of velocityCorrect answer is option 'C'. Can you explain this answer?
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