The following data is given for a Francis turbine : Net head = 70 m, speed = 600 r.p.m., shaft power = 367.875 kW, overall efficiency = 85% , hydraulic efficiency = 95%, flow ratio = 0.25, breadth ratio = 0.1, outer diameter of the runner = 2 x inner diameter of runner. The thickness of vanes occupy 10% of the circumferential area of the runner. Velocity of flow is constant at inlet and outlet and discharge is radial at outlet. Determine : (i) Guide blade angle, (ii) Runner vane angles at inlet and outlet, (iii) Diameters of runner at inlet and outlet, and (iv) Width of wheel at inlet.

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Francis Turbine Design (Advanced)

Problem Statement

The following data is given for a Francis turbine : Net head = 70 m, speed = 600 r.p.m., shaft power = 367.875 kW, overall efficiency = 85% , hydraulic efficiency = 95%, flow ratio = 0.25, breadth ratio = 0.1, outer diameter of the runner = 2 x inner diameter of runner. The thickness of vanes occupy 10% of the circumferential area of the runner. Velocity of flow is constant at inlet and outlet and discharge is radial at outlet. Determine : (i) Guide blade angle, (ii) Runner vane angles at inlet and outlet, (iii) Diameters of runner at inlet and outlet, and (iv) Width of wheel at inlet.

Given Data & Constants

  • Net Head, \(H = 70 \, \text{m}\)
  • Speed, \(N = 600 \, \text{r.p.m.}\)
  • Shaft Power, \(P_s = 367.875 \, \text{kW} = 367875 \, \text{W}\)
  • Overall efficiency, \(\eta_o = 85\% = 0.85\)
  • Hydraulic efficiency, \(\eta_h = 95\% = 0.95\)
  • Flow ratio, \(K_f = 0.25\)
  • Breadth ratio, \(n = 0.1\)
  • Diameter ratio, \(D_1 = 2 D_2\)
  • Area blockage = 10% (Effective Area Factor = 0.90)
  • Constant flow velocity: \(V_{f1} = V_{f2}\)
  • Radial discharge: \(V_{w2} = 0\)
  • Density of water, \(\rho = 1000 \, \text{kg/m}^3\)
  • Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\)

Solution

1. Calculate Velocities and Discharge

First, we calculate the velocity of flow based on the head and flow ratio.

$$ V_{f1} = K_f \sqrt{2gH} = 0.25 \sqrt{2 \times 9.81 \times 70} \approx 9.26 \, \text{m/s} $$ $$ \text{Since flow velocity is constant, } V_{f2} = 9.26 \, \text{m/s} $$

Next, find the required discharge (Q) using the overall efficiency and shaft power.

$$ P_w = \frac{P_s}{\eta_o} = \frac{367875}{0.85} = 432794 \, \text{W} $$ $$ Q = \frac{P_w}{\rho g H} = \frac{432794}{1000 \times 9.81 \times 70} \approx 0.63 \, \text{m}^3/\text{s} $$

(iii) & (iv) Diameters and Width of Runner

The discharge is related to the effective flow area at the inlet.

$$ Q = (\text{Effective Area Factor}) \times (\pi D_1 b_1) \times V_{f1} $$ $$ \text{Given } b_1 = n D_1 = 0.1 D_1 $$ $$ 0.63 = 0.90 \times (\pi D_1 \times 0.1 D_1) \times 9.26 $$ $$ 0.63 = 0.90 \times \pi \times 0.1 \times 9.26 \times D_1^2 = 2.618 D_1^2 $$ $$ D_1^2 = \frac{0.63}{2.618} \approx 0.2406 \implies D_1 \approx 0.49 \, \text{m} $$ $$ \text{From the ratios, we find } b_1 \text{ and } D_2: $$ $$ b_1 = 0.1 \times D_1 = 0.1 \times 0.49 = 0.049 \, \text{m} $$ $$ D_2 = D_1 / 2 = 0.49 / 2 = 0.245 \, \text{m} $$

2. Calculate Peripheral Velocities and Whirl Velocity

$$ u_1 = \frac{\pi D_1 N}{60} = \frac{\pi \times 0.49 \times 600}{60} \approx 15.39 \, \text{m/s} $$ $$ u_2 = \frac{\pi D_2 N}{60} = \frac{\pi \times 0.245 \times 600}{60} \approx 7.70 \, \text{m/s} $$

Using the hydraulic efficiency and Euler's equation (with \(V_{w2}=0\)):

$$ \eta_h = \frac{V_{w1} u_1}{gH} \implies V_{w1} = \frac{\eta_h g H}{u_1} $$ $$ V_{w1} = \frac{0.95 \times 9.81 \times 70}{15.39} \approx 42.34 \, \text{m/s} $$

(i) Guide Blade Angle (\(\alpha\))

$$ \tan(\alpha) = \frac{V_{f1}}{V_{w1}} = \frac{9.26}{42.34} \approx 0.2187 $$ $$ \alpha = \arctan(0.2187) \approx 12.33^\circ $$

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

$$ \tan(\theta) = \frac{V_{f1}}{V_{w1} - u_1} = \frac{9.26}{42.34 - 15.39} = \frac{9.26}{26.95} \approx 0.3436 $$ $$ \theta = \arctan(0.3436) \approx 18.96^\circ $$ $$ \tan(\phi) = \frac{V_{f2}}{u_2} = \frac{9.26}{7.70} \approx 1.2026 $$ $$ \phi = \arctan(1.2026) \approx 50.25^\circ $$
Final Design Parameters:

(i) Guide blade angle: \( \alpha \approx 12.33^\circ \)

(ii) Runner vane angles: Inlet \( \theta \approx 18.96^\circ \), Outlet \( \phi \approx 50.25^\circ \)

(iii) Diameters of runner: Inlet \( D_1 \approx 0.49 \, \text{m} \), Outlet \( D_2 \approx 0.245 \, \text{m} \)

(iv) Width of wheel at inlet: \( b_1 \approx 0.049 \, \text{m} \) or \(49 \, \text{mm}\)

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