Problem Statement
An inward flow reaction turbine has external and internal diameters as 1.2 m and 0.6 m respectively. The velocity of flow through the runner is constant and is equal to 1.8 m/s. Determine : (i) Discharge through the runner, and (ii) Width at outlet if the width at inlet= 200 mm.
Given Data & Constants
- External (inlet) diameter, \(D_1 = 1.2 \, \text{m}\)
- Internal (outlet) diameter, \(D_2 = 0.6 \, \text{m}\)
- Velocity of flow, \(V_{f1} = V_{f2} = 1.8 \, \text{m/s}\)
- Width at inlet, \(b_1 = 200 \, \text{mm} = 0.2 \, \text{m}\)
Solution
(i) Discharge Through the Runner (Q)
The discharge is the product of the flow area at the inlet and the velocity of flow at the inlet.
(ii) Width at Outlet (\(b_2\))
Since the velocity of flow is constant, the discharge at the outlet is the same as at the inlet. We can use this to find the required width at the outlet.
(i) Discharge through the runner: \( \approx 1.357 \, \text{m}^3/\text{s} \)
(ii) Width at outlet: \( \approx 0.4 \, \text{m} \) or \(400 \, \text{mm}\)
Explanation of Inward Flow Reaction Turbine
An inward flow reaction turbine (like a Francis turbine) works differently from a Pelton wheel. Water enters the runner at the outer circumference, flows radially inwards through the vanes, and exits near the center. It extracts energy from both the pressure and the kinetic energy of the water.
- Discharge: The volume of water flowing through the turbine per second is determined by the circumferential area at the inlet (\(\pi D_1 b_1\)) and the component of velocity perpendicular to that area (the velocity of flow, \(V_{f1}\)).
- Constant Flow Velocity: The condition that the velocity of flow (\(V_f\)) is constant is a common design simplification. To maintain this constant radial velocity as the water moves from a large diameter to a smaller one, the flow area must be adjusted. Since the outlet diameter (\(D_2\)) is smaller than the inlet diameter (\(D_1\)), the width of the runner at the outlet (\(b_2\)) must be proportionally larger than the inlet width (\(b_1\)) to keep the flow area constant, which is why the outlet width is 400 mm compared to the inlet's 200 mm.
