Define velocity triangle at impeller.

Short Answer:

The velocity triangle at the impeller is a graphical representation used to show the relationship between different velocities of the fluid at the inlet and outlet of an impeller in a centrifugal pump or turbine. It helps in understanding how the velocity of the fluid changes as it passes through the impeller blades.

The velocity triangle includes three main velocity components — the absolute velocity of the fluid, the velocity of the impeller (blade velocity), and the relative velocity of the fluid with respect to the impeller blade. It is used to calculate energy transfer, head developed, and efficiency of pumps and turbines.

Detailed Explanation :

Velocity Triangle at Impeller

The velocity triangle at the impeller is an essential concept in turbomachinery such as centrifugal pumps, compressors, and turbines. It helps to understand the motion of fluid relative to the moving blades of the impeller. Since the impeller rotates at a certain speed, the fluid entering and leaving the impeller experiences different velocity directions and magnitudes. The velocity triangle provides a simple and effective graphical way to represent these velocities and their relationships.

The velocity triangle is called a “triangle” because when the velocity vectors — absolute velocity, relative velocity, and blade velocity — are drawn head to tail, they form a closed triangle. This triangle can be drawn for both the inlet (entry) and outlet (exit) of the impeller. It helps in analyzing the energy transfer between the impeller and the fluid.

Components of Velocity Triangle

The velocity triangle at the impeller has three main velocity components:

  1. Absolute Velocity (V):
    This is the actual velocity of the fluid measured with respect to a stationary observer. It represents the direction and speed with which the fluid approaches or leaves the impeller. The absolute velocity has two components:

    • Vw: The whirl or tangential component (responsible for energy transfer).
    • Vf: The flow or radial component (responsible for flow rate).
  2. Blade Velocity (u):
    It is the linear velocity of the impeller blade at the point where the fluid enters or leaves. It depends on the rotational speed of the impeller and the radius at that point.

where
= diameter of impeller (m),
= speed of impeller (rpm).

  1. Relative Velocity (Vr):
    This is the velocity of the fluid relative to the impeller blade. It is the vector difference between the absolute velocity and the blade velocity.

It shows how the fluid moves over the rotating blade surface.

Inlet and Outlet Velocity Triangles

There are two velocity triangles — one at the inlet and another at the outlet of the impeller.

  1. Inlet Velocity Triangle:
    At the inlet, the fluid enters the impeller eye with an absolute velocity , which makes an angle  with the tangential direction. The blade velocity is , and the relative velocity  makes an angle  with the tangential direction of the impeller blade.
    The inlet triangle helps to determine how the fluid enters the impeller and whether it matches the blade angle properly for smooth flow.
  2. Outlet Velocity Triangle:
    At the outlet, the fluid leaves the impeller with absolute velocity  making an angle  with the tangent. The blade velocity at the outlet is , and the relative velocity  makes an angle  with the tangent to the impeller.
    The outlet triangle determines the energy transfer from the impeller to the fluid and helps calculate the head developed.

Mathematical Relations

From the velocity triangles, the energy or head developed by the impeller can be found using Euler’s Equation for turbomachines:

Where:

  •  = Head developed (m)
  •  = Tangential velocity component at outlet (m/s)
  •  = Tangential velocity component at inlet (m/s)
  •  = Blade velocities at outlet and inlet respectively (m/s)
  •  = Acceleration due to gravity (9.81 m/s²)

This equation shows that the energy transferred to or from the fluid depends mainly on the whirl velocity components at the inlet and outlet.

Importance of Velocity Triangle

  1. Helps in Design:
    It is used to design the shape and angle of impeller blades for efficient energy transfer.
  2. Energy Calculation:
    Engineers can calculate the head developed or absorbed by the machine using the velocity triangle.
  3. Determines Efficiency:
    The triangle helps analyze hydraulic and mechanical efficiency by showing velocity losses.
  4. Flow Analysis:
    It explains how fluid moves through the impeller, which is useful in performance testing and troubleshooting.
  5. Blade Angle Setting:
    It helps in setting the correct inlet and outlet blade angles to ensure smooth flow and avoid shock losses.

Example

Suppose the impeller of a centrifugal pump rotates at 1000 rpm with an outlet diameter of 0.3 m.
Blade velocity at outlet,

If the whirl component at outlet  and at inlet , then

Thus, the head developed by the impeller is 16 meters.

Conclusion

The velocity triangle at the impeller shows the relationship between the fluid velocity, blade velocity, and relative velocity at the inlet and outlet of the impeller. It is very important in understanding fluid motion and calculating the head and efficiency of pumps and turbines. The proper design and analysis of velocity triangles ensure smooth fluid flow, higher efficiency, and effective energy transfer in turbomachinery.