An outward flow reaction turbine has internal and external diameters of the runner as 0.5 m and 1.0 m respectively. The guide blade angle is 15° and velocity of flow through the runner is constant and equal to 3 m/s. If the speed of the turbine is 250 r.p.m., head on turbine is 10 m and discharge at outlet is radial, determine : (i) The runner vane angles at inlet and outlet, (ii) Work done by the water on the runner per second per unit weight of water striking per second and (iii) Hydraulic efficiency.

Outward Flow Reaction Turbine Analysis

Problem Statement

Given Data & Constants

Internal (inlet) diameter, $D_1 = 0.5 \, \text{m}$
External (outlet) diameter, $D_2 = 1.0 \, \text{m}$
Guide blade angle, $\alpha = 15^\circ$
Velocity of flow, $V_{f1} = V_{f2} = 3 \, \text{m/s}$
Speed, $N = 250 \, \text{r.p.m.}$
Head, $H = 10 \, \text{m}$
Radial discharge at outlet: $V_{w2} = 0$
Acceleration due to gravity, $g = 9.81 \, \text{m/s}^2$

Solution

1. Calculate Tangential Velocities ($u_1, u_2$)

The tangential velocity of the runner at the inlet and outlet is calculated from the rotational speed and diameters.

$$ u_1 = \frac{\pi D_1 N}{60} = \frac{\pi \times 0.5 \times 250}{60} \approx 6.545 \, \text{m/s} $$ $$ u_2 = \frac{\pi D_2 N}{60} = \frac{\pi \times 1.0 \times 250}{60} \approx 13.09 \, \text{m/s} $$

2. Analyze Inlet Velocity Triangle

First, find the whirl velocity at the inlet ($V_{w1}$) using the guide blade angle.

$$ \tan(\alpha) = \frac{V_{f1}}{V_{w1}} \implies V_{w1} = \frac{V_{f1}}{\tan(\alpha)} $$ $$ V_{w1} = \frac{3}{\tan(15^\circ)} = \frac{3}{0.2679} \approx 11.196 \, \text{m/s} $$

(i) Runner Vane Angles at Inlet ($\theta$) and Outlet ($\phi$)

Inlet Vane Angle ($\theta$):

$$ \tan(\theta) = \frac{V_{f1}}{V_{w1} - u_1} $$ $$ \tan(\theta) = \frac{3}{11.196 - 6.545} = \frac{3}{4.651} \approx 0.645 $$ $$ \theta = \arctan(0.645) \approx 32.83^\circ $$

Outlet Vane Angle ($\phi$): Since the discharge is radial, $V_{w2}=0$.

$$ \tan(\phi) = \frac{V_{f2}}{u_2} $$ $$ \tan(\phi) = \frac{3}{13.09} \approx 0.229 $$ $$ \phi = \arctan(0.229) \approx 12.9^\circ $$

(ii) Work Done per Unit Weight of Water

This is calculated using the Euler turbomachine equation.

$$ \text{Work Done per Unit Weight} = \frac{1}{g} (V_{w1} u_1 - V_{w2} u_2) $$ $$ \text{Since } V_{w2} = 0, \text{ the equation simplifies to:} $$ $$ \text{Work Done} = \frac{V_{w1} u_1}{g} $$ $$ \text{Work Done} = \frac{11.196 \, \text{m/s} \times 6.545 \, \text{m/s}}{9.81 \, \text{m/s}^2} \approx 7.47 \, \frac{\text{N-m}}{\text{N}} $$

(iii) Hydraulic Efficiency ($\eta_h$)

The hydraulic efficiency is the ratio of the work done on the runner to the net head available on the turbine.

$$ \eta_h = \frac{\text{Work Done per Unit Weight}}{\text{Head on Turbine}} = \frac{(V_{w1} u_1)/g}{H} $$ $$ \eta_h = \frac{7.47 \, \text{m}}{10 \, \text{m}} = 0.747 $$

Final Results:

(i) Inlet Vane Angle: $ \theta \approx 32.8^\circ $, Outlet Vane Angle: $ \phi \approx 12.9^\circ $

(ii) Work done per unit weight of water: $ \approx 7.47 \, \text{m} $ (or N-m/N)

(iii) Hydraulic efficiency: $ \approx 74.7\% $

Explanation of Outward Flow Turbine

An outward flow reaction turbine operates by having water enter at the inner diameter and flow radially outwards towards the larger, external diameter. This is the reverse of the more common Francis (inward flow) turbine.

Velocity Triangles: The principles of constructing inlet and outlet velocity triangles are the same as for an inward flow turbine, but the roles of inlet (subscript 1) and outlet (subscript 2) are swapped with respect to the diameters. The smaller diameter ($D_1$) is now the inlet, and the larger diameter ($D_2$) is the outlet.
Radial Discharge: The condition "discharge at outlet is radial" is a key design feature for maximizing efficiency. It means the absolute velocity of the water leaving the runner has no tangential (whirl) component ($V_{w2} = 0$). This ensures that the maximum possible kinetic energy has been transferred from the water to the runner, as no energy is "wasted" in swirling the outlet flow.
Work Done: The work is calculated from the change in the moment of momentum of the water between the inlet and outlet, as defined by the Euler turbine equation.

Problem Statement

Given Data & Constants

Solution

1. Calculate Tangential Velocities (\(u_1, u_2\))

2. Analyze Inlet Velocity Triangle

(i) Runner Vane Angles at Inlet (\(\theta\)) and Outlet (\(\phi\))

(ii) Work Done per Unit Weight of Water

(iii) Hydraulic Efficiency (\(\eta_h\))

Explanation of Outward Flow Turbine

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Problem Statement

Given Data & Constants

Solution

1. Calculate Tangential Velocities (\(u_1, u_2\))

2. Analyze Inlet Velocity Triangle

(i) Runner Vane Angles at Inlet (\(\theta\)) and Outlet (\(\phi\))

(ii) Work Done per Unit Weight of Water

(iii) Hydraulic Efficiency (\(\eta_h\))

Explanation of Outward Flow Turbine

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