Short Answer:
Velocity triangles in a Francis turbine are graphical representations that show the relationship between different velocity components of water flowing through the turbine blades. These triangles are drawn at both the inlet and outlet of the runner blades to illustrate how water velocity changes as it passes through.
In simple words, velocity triangles help to understand the direction and magnitude of absolute velocity, relative velocity, and tangential velocity of water in the turbine. They are important for calculating the work done, efficiency, and performance of the Francis turbine.
Detailed Explanation :
Velocity triangles in Francis turbine
Velocity triangles are very important in understanding the flow of water through the runner blades of a Francis turbine. They represent the vector relationship between different velocity components, such as absolute velocity (V), relative velocity (Vr), and tangential velocity (u) of water. These triangles are drawn at the inlet and outlet of the runner blades to analyze how the water enters and leaves the runner. The study of velocity triangles helps engineers to design the turbine blades properly and calculate the hydraulic efficiency accurately.
The Francis turbine is a reaction turbine, meaning that both kinetic and pressure energies of water are converted into mechanical energy. The flow of water through the runner blades is not purely tangential or radial but a combination of both. Hence, the velocity of water has several components that must be understood clearly. The velocity triangles provide a graphical way to study these components and the direction of water flow inside the turbine.
- Components of velocity in Francis turbine
When water flows through the turbine, its velocity can be broken into several components:
- Absolute Velocity (V):
It is the velocity of water measured with respect to a fixed point (like the casing or ground). The absolute velocity makes an angle with the tangent to the runner, known as the angle of whirl (α). - Relative Velocity (Vr):
It is the velocity of water measured relative to the moving runner blade. It makes an angle with the tangent known as the blade angle (β). - Tangential Velocity (u):
It is the velocity of the runner at the point of contact with water. It acts tangentially to the circle traced by the runner blades. - Velocity of Flow (Vf):
It is the component of absolute velocity in the direction of the flow through the turbine, usually along the axis of the turbine. - Velocity of Whirl (Vw):
It is the component of absolute velocity in the tangential direction. This component is directly responsible for producing torque and hence power in the turbine.
The relationships among these velocities can be represented geometrically using velocity triangles.
- Inlet and outlet velocity triangles
There are two velocity triangles in a Francis turbine — one at the inlet (before water strikes the runner) and one at the outlet (after water leaves the runner).
- Inlet Velocity Triangle:
- At the inlet, water enters the runner after passing through the guide vanes.
- The absolute velocity of water (V₁) makes an angle α₁ with the tangent.
- The runner is rotating with tangential velocity u₁.
- The relative velocity (Vr₁) is the velocity of water relative to the blade and makes an angle β₁ with the tangent.
- The flow velocity (Vf₁) is perpendicular to the tangent and represents the axial or radial component of flow.
- The triangle is formed by the relationship:
V₁ = u₁ + Vr₁ (vector addition).
- Outlet Velocity Triangle:
- At the outlet, water leaves the runner blades with velocity V₂ at an angle α₂.
- The runner has tangential velocity u₂.
- The relative velocity Vr₂ makes an angle β₂ with the tangent.
- The flow velocity (Vf₂) again represents the axial or radial component of the flow.
- The triangle at the outlet shows how the water leaves the runner and how much energy has been transferred.
The difference in whirl velocity components at the inlet and outlet determines the energy transferred to the turbine.
- Importance of velocity triangles
The velocity triangles are used to determine the work done and efficiency of the turbine.
- The work done per second per unit weight of water is given by:
Work done = (Vw₁ ± Vw₂) × u / g
Here,- Vw₁ = velocity of whirl at inlet
- Vw₂ = velocity of whirl at outlet
- u = tangential velocity of the runner
- g = acceleration due to gravity
The sign of Vw₂ depends on the direction of outlet whirl. If water leaves the runner in the opposite direction to rotation, Vw₂ is taken as negative.
The hydraulic efficiency (ηh) of the turbine can be expressed as:
ηh = (Vw₁ ± Vw₂) × u / (g × H)
where H is the head of water on the turbine.
The accuracy of these calculations depends on the correct drawing and interpretation of velocity triangles.
- Design considerations
The design of runner blades in the Francis turbine depends directly on the shape of the velocity triangles. The blade angles β₁ and β₂ are determined from these triangles to ensure that water enters and leaves the runner without shock. If the velocity triangles are not designed properly, it may result in turbulence, energy loss, and cavitation inside the turbine.
By using velocity triangles, engineers can determine the most efficient angle of entry and exit, calculate the power output, and ensure smooth flow of water through the turbine.
In practical operation, the velocity triangles also help in adjusting the guide vane and runner blade angles for variable load conditions.
Conclusion:
Velocity triangles in a Francis turbine represent the velocity relationships between water and the runner blades at inlet and outlet. They are essential for calculating work done, hydraulic efficiency, and designing runner blades for smooth flow. Properly constructed velocity triangles ensure that water enters and leaves the runner without shock, leading to maximum efficiency and stable turbine operation.