Problem Statement
A three-stage centrifugal pump has impeller 40 cm in diameter and 2.5 cm wide at outlet. The vanes are curved back at the outlet at 30° and reduce the circumferential area by 15%. The manometric efficiency is 85% and overall efficiency is 75%. Determine the head generated by the pump when running at 1200 r.p.m. and discharge is 0.06 m³/s. Find the shaft power also.
Given Data & Constants
- Number of stages = 3
- Outlet diameter, \(D_2 = 40 \, \text{cm} = 0.4 \, \text{m}\)
- Outlet width, \(b_2 = 2.5 \, \text{cm} = 0.025 \, \text{m}\)
- Outlet vane angle, \(\phi = 30^\circ\)
- Area reduction by vanes = 15% (Effective Area = 85%)
- Manometric efficiency, \(\eta_{\text{mano}} = 85\% = 0.85\)
- Overall efficiency, \(\eta_o = 75\% = 0.75\)
- Speed, \(N = 1200 \, \text{r.p.m.}\) (Assuming 1200, as 12000 is likely a typo for this pump size)
- Discharge, \(Q = 0.06 \, \text{m}^3/\text{s}\)
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)
Solution
1. Calculate Tangential Velocity at Outlet (\(u_2\))
2. Calculate Velocity of Flow at Outlet (\(V_{f2}\))
The flow velocity is based on the effective area at the outlet, which is reduced by the thickness of the vanes.
3. Determine Whirl Velocity at Outlet (\(V_{w2}\))
Using the outlet velocity triangle:
4. Calculate Head Generated per Stage (\(H_{\text{stage}}\))
The head per stage is the Euler head multiplied by the manometric efficiency. We assume radial inlet (\(V_{w1}=0\)).
5. Calculate Total Head Generated by the Pump (\(H_{\text{total}}\))
The total head is the head per stage multiplied by the number of stages.
6. Calculate the Shaft Power Required (\(P_s\))
The shaft power is the power delivered to the water divided by the overall efficiency.
Total Head Generated: \( H_{\text{total}} \approx 138.7 \, \text{m} \)
Shaft Power Required: \( P_s \approx 108.9 \, \text{kW} \)
Explanation of Key Concepts
Multi-stage Pump: A multi-stage pump consists of several impellers in series on a single shaft. The discharge from the first impeller is directed to the inlet of the second, and so on. This arrangement doesn't increase the flow rate, but it adds the head generated by each stage. It's a common method for achieving very high pressures and heads.
Vane Area Reduction: The theoretical outlet area of an impeller is a cylinder (\(\pi D_2 b_2\)). However, the physical thickness of the vanes themselves blocks a portion of this area. The problem states this blockage is 15%, so we must use the remaining 85% of the area to accurately calculate the velocity of flow (\(V_{f2}\)).
Overall Efficiency (\(\eta_o\)): This is the "wire-to-water" efficiency. It accounts for all energy losses, including hydraulic losses in the pump (covered by \(\eta_{\text{mano}}\)), mechanical losses in the bearings and seals, and electrical losses in the motor. It tells us how much of the electrical power drawn from the grid is converted into useful fluid power.
