Problem Statement
The internal and external diameters of the impeller of a centrifugal pump are 300 mm and 600 mm respectively. The pump is running at 1000 r.p.m. The vane angles at inlet and outlet are 20° and 30° respectively. The water enters the impeller radially and velocity of flow is constant. Determine the work done by the impeller per unit weight of water.
Given Data & Constants
- Internal diameter, \(D_1 = 300 \, \text{mm} = 0.3 \, \text{m}\)
- External diameter, \(D_2 = 600 \, \text{mm} = 0.6 \, \text{m}\)
- Speed, \(N = 1000 \, \text{r.p.m.}\)
- Inlet vane angle, \(\theta = 20^\circ\)
- Outlet vane angle, \(\phi = 30^\circ\)
- Radial inlet: Whirl velocity at inlet, \(V_{w1} = 0\)
- Constant flow velocity: \(V_{f1} = V_{f2}\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Calculate Tangential Velocities (\(u_1\) and \(u_2\))
The tangential velocity of the impeller at the inlet and outlet is calculated from the rotational speed and diameter.
2. Determine Velocity of Flow (\(V_{f1}\)) from Inlet Triangle
Since the water enters radially, the inlet velocity triangle is a right-angled triangle.
3. Determine Whirl Velocity at Outlet (\(V_{w2}\))
Since the velocity of flow is constant, \(V_{f2} = V_{f1} = 5.717 \, \text{m/s}\). We use the outlet velocity triangle to find \(V_{w2}\).
4. Calculate Work Done by the Impeller
The work done by the impeller per unit weight of water is given by the Euler turbomachine equation.
The work done by the impeller per unit weight of water is approximately \(68.91\) meters (or N-m/N).
Explanation of Velocity Triangles & Euler's Equation
Velocity Triangles: In a centrifugal pump, the fluid has both a radial/flow velocity (\(V_f\)) and a tangential/whirl velocity (\(V_w\)). The impeller itself has a tangential velocity (\(u\)). A velocity triangle is a vector diagram that relates these components to the absolute velocity (\(V\)) of the fluid and its velocity relative to the moving vane (\(V_r\)). By analyzing the geometry of these triangles at the inlet and outlet, we can determine key velocity components.
Euler's Turbomachine Equation: The work done by an impeller on a fluid is fundamentally related to the change in the angular momentum of the fluid as it passes through. The equation \( \frac{1}{g} (V_{w2} u_2 - V_{w1} u_1) \) precisely quantifies this change in terms of energy per unit weight, also known as the "head" added by the pump.
Physical Meaning: Pump Head
The result of 68.91 m is the theoretical "head" added to the water by the pump impeller. This means that, under ideal conditions (no friction or other losses), the pump imparts enough energy to each newton of water to lift it vertically by 68.91 meters.
This value represents the ideal energy transfer from the motor, through the impeller, and into the fluid. In a real-world scenario, the actual head delivered by the pump would be lower due to hydraulic losses (friction, turbulence) within the pump casing. This theoretical head is a critical parameter for pump design and selection, as it defines the maximum energy-imparting capability of the impeller's geometry and speed.
