Problem Statement
A centrifugal pump is to discharge 0.12 m³/s at a speed of 1400 r.p.m. against a head of 30 m. The diameter and width of the impeller at outlet are 25 cm and 5 cm respectively. If the manometric efficiency is 75%, determine the vane angle at outlet.
Given Data & Constants
- Discharge, \(Q = 0.12 \, \text{m}^3/\text{s}\)
- Speed, \(N = 1400 \, \text{r.p.m.}\)
- Manometric Head, \(H_m = 30 \, \text{m}\)
- Outlet diameter, \(D_2 = 25 \, \text{cm} = 0.25 \, \text{m}\)
- Outlet width, \(b_2 = 5 \, \text{cm} = 0.05 \, \text{m}\)
- Manometric efficiency, \(\eta_{\text{mano}} = 75\% = 0.75\)
- 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 (\(V_{f2}\))
3. Calculate the Theoretical (Euler) Head (\(H_e\))
4. Determine Whirl Velocity at Outlet (\(V_{w2}\))
Assuming radial entry (\(V_{w1}=0\)), the Euler head equation simplifies, allowing us to solve for \(V_{w2}\).
5. Determine the Vane Angle at Outlet (\(\phi\))
The outlet vane angle is found from the geometry of the outlet velocity triangle.
The vane angle at outlet is approximately \(44.7^\circ\).
(The negative sign indicates forward-curved vanes, where the angle relative to the tangent is obtuse, i.e., \(180^\circ - 44.7^\circ = 135.3^\circ\))
Explanation of the Result
The calculation for the outlet vane angle (\(\phi\)) yields a negative result. This is mathematically correct based on the given data but indicates an unusual pump design.
- Velocity Triangle: The formula \( \tan(\phi) = V_{f2} / (u_2 - V_{w2}) \) is derived from the outlet velocity triangle. In our case, the whirl velocity (\(V_{w2} \approx 21.4\) m/s) is greater than the impeller's tangential velocity (\(u_2 \approx 18.3\) m/s).
- Forward-Curved Vanes: This condition (\(V_{w2} > u_2\)) is characteristic of a pump with forward-curved vanes. Most centrifugal pumps use backward-curved vanes (\(\phi < 90^\circ\)) for more stable operation. Forward-curved vanes (\(\phi > 90^\circ\)) can produce a very high head but are often less stable and less efficient.
The calculated angle of -44.7° corresponds to an absolute angle of \(180^\circ - 44.7^\circ = 135.3^\circ\), which is consistent with a forward-curved design. While atypical, it is the correct answer derived from the problem's parameters.
