Francis Turbine MCQ Quiz - Objective Question with Answer for Francis Turbine - Download Free PDF

Concept:

The overall efficiency of a Francis turbine is defined as the ratio of useful power output at the runner shaft to the hydraulic power input from water striking the runner.

Formula:

Hydraulic Power Input = Weight of water per second × Net Head

\( \lambda_0 = \frac{\text{Power Output}}{\text{Hydraulic Power}} = \frac{P}{W \cdot H} \)

For Reaction turbine, the Degree of Reaction is given by:
\(R = 1 - \frac{\cot \alpha}{2(\cot \alpha - \cot \beta)}\)

Where, \(\alpha \rightarrow Guide~ blade~ angle \), \(\beta \rightarrow Vane~ angle~ at~ inlet \)

\(0 = 1 - \frac{\cot 12^\circ}{2(\cot 12^\circ - \cot \beta)}\)

\(⇒ \frac{1}{2} \cot 12^\circ = \cot 12^\circ - \cot \beta\)

\(⇒ \cot \beta = \frac{1}{2} \cot 12^\circ\)

⇒ \(\beta = 23.03^\circ\)

Concept:

Mixed Flow Turbine → Francis Turbine

• Flow Direction: In a mixed flow turbine, water enters radially (perpendicular to the shaft) and exits axially (along the direction of the shaft).
• Type: The Francis turbine is a classic example of a mixed flow reaction turbine.
• Working: It uses both pressure energy and kinetic energy of water, hence it's a reaction turbine.
• Application: Suitable for medium head and medium discharge.

Summary Table:

Explanation:

• High head turbine: In this type of turbines, the net head varies from 150 m to 2000 m or even more, and these turbines require a small quantity of water. Example: Pelton wheel turbine.
• Medium head turbine: The net head varies from 30 m to 150 m, and also these turbines require a moderate quantity of water. Example: Francis turbine.
• Low head turbine: The net head is less than 30 m and also these turbines require a large quantity of water. Example: Kaplan turbine.

Important Point:

• Pelton turbine – Low discharge and high head
• Francis turbine – Medium discharge and medium head
• Kaplan Turbine – High discharge and low head

Explanation:

According to the direction of flow through the runner, the turbine is classified as:

• Mixed flow turbine: The water enters the runner in the radial direction and leaves in an axial direction. Example: Modern Francis turbine
• Tangential flow turbines: In this type of turbine, the water strikes the runner in the direction of the tangent to the wheel. Example: Pelton wheel turbine
• Radial flow turbines: In this type of turbine, the water strikes in the radial direction. Accordingly, it is further classified as:
  • Inward flow turbine: The flow is inward from the periphery to the center (centripetal type). Example: old Francis turbine.
  • Outward flow turbine: The flow is outward from the center to the periphery (centrifugal type). Example: Fourneyron turbine.
• Axial flow turbine: The flow of water is in the direction parallel to the axis of the shaft. Example: Kaplan turbine and propeller turbine.

Concept:

The overall efficiency η_o of turbine = volumetric efficiency (η_v)× hydraulic efficiency (η_h)× mechanical efficiency (η_m)

\({\eta _o} = {\eta _v} \times {\eta _h} \times {\eta _m}\)

\({{\rm{\eta }}_{\rm{v}}} = \frac{{{\rm{volume\;of\;water\;actually\;striking\;the\;runner}}}}{{{\rm{volume\;of\;water\;actually\;supplied\;to\;the\;turbine}}}}\)

\({{\rm{\eta }}_{\rm{h}}} = \frac{{{\rm{Power\;deliverd\;to\;runner}}}}{{{\rm{Power\;supplied\;at\;inlet\;}}}} = \frac{{{\rm{R}}.{\rm{P}}}}{{{\rm{W}}.{\rm{P}}}}\)

\({{\rm{\eta }}_{\rm{m}}} = \frac{{{\rm{Power\;at\;the\;shaft\;of\;the\;turbine}}}}{{{\rm{Power\;delivered\;by\;water\;to\;the\;runner}}}} = \frac{{{\rm{S}}.{\rm{P}}}}{{{\rm{R}}.{\rm{P}}}}\)

Overall efficiency: \({\eta _o} = \frac{{S.P}}{{W.P}}\)

Water Power = ρ × Q × g × h 

Calculation:

Given:

η_o = 0.8, Head h = 10 m, and Q = 1 m³/s.

\({\eta _o} = \frac{{S.P}}{{W.P}} = \frac{{S.P}}{{\rho \times Q \times g \times h}}\)

\(0.8 = \frac{{S.P}}{{1000 \times 1 \times 9.81 \times 10}} \Rightarrow S.P = 78480\;W \approx 78\;kW\)

Explanation:

Flow ratio

The flow ratio of Francis turbine is defined as the ratio of the velocity of flow at the inlet to the theoretical jet velocity.

\(Flow\;Ratio = \frac{{{V_{f1}}}}{{\sqrt {2gH} }}\)

In the case of Francis turbine,

Flow ratio varies from 0.15 to 0.3

Speed ratio varies from 0.6 to 0.9

Calculation:

\(Flow\;Ratio = \frac{{7}}{{\sqrt {2\times10\times62} }}=0.2\)

Concept:

In order to predict the behaviour of a turbine working under a varying condition of head, speed output, etc, the results are expressed in terms of quantities that may be obtained when the head on the turbine is reduced to unity.

The following are three important unit quantities.

Explanation:

Flow ratio

The flow ratio of Francis turbine is defined as the ratio of the velocity of flow at the inlet to the theoretical jet velocity.

\(Flow\;Ratio = \frac{{{V_{f1}}}}{{\sqrt {2gH} }}\)

In the case of Francis turbine,

Flow ratio varies from 0.15 to 0.3

Speed ratio varies from 0.6 to 0.9

Explanation:-

Fransis turbine is a radial flow inward reaction turbine.

Inlet triangle:

U₁ = velocity of blade at inlet, V₁  = absolute velocity of entering water, V_r1 = relative velocity of entering water, V_f1  = velocity of  flow at inlet

Outlet triangle:

U₂ = velocity of blade at outlet, V_r2 = relative velocity of leaving  water, V_f2  = V₂ = velocity of flow at outlet

We know that, 

Work Equivalent head

\(W = \frac{{V_1^2 - V_2^2}}{{2g}} + \frac{{V_{r2\;}^2 - \;V_{r1}^2}}{{2g}} + \frac{{U_1^2 - U_2^2}}{{2g}}\)

From inlet and outlet velocity triangles, 

\({V_1} = {U_1}\sec \alpha \)

\({V_2} = {V_{f2}}\; = \;{U_1}\tan \alpha = {U_2}\tan \beta \;\;\)

Using these relations into the expression of W, we have \(W = \frac{1}{{2g}}\;\left[ {U_1^2{{\sec }^2}\alpha \; - \;U_1^2{{\tan }^2}\alpha + \;U_2^2{{\sec }^2}\beta - \;U_1^2{{\tan }^2}\alpha + U_1^2\; - \;U_2^2} \right]\)

\(W = \frac{1}{{2g}}\;\left[ {U_1^2 + U_2^2 + U_2^2{{\tan }^2}\beta \; - U_1^2{{\tan }^2}\alpha + U_1^2\; - \;U_2^2} \right]\)

\(W = \frac{1}{{2g}} \times 2U_1^2 = \frac{{U_1^2}}{g}\)

Available head

H = work head + energy rejected from turbine 

\(H = \frac{{U_1^2}}{g} + \frac{{V_2^2}}{{2g\;}}\)

\(H = \;\frac{1}{{2g}}\left( {\;2U_1^2 + V_2^2\;} \right)\)

\(H = \;\frac{1}{{2g}}(\;2U_1^2 + U_1^2{\tan ^2}\alpha \;)\)

\(H = \;\frac{{U_1^2}}{{2g}}(\;2 + {\tan ^2}\alpha \;)\)

∴ Hydraulic Efficiency 

\({\eta _h} = \frac{{work\;equivalent\;head}}{{available\;head}}\)

\({\eta _h} = \frac{{\frac{{U_1^2}}{g}}}{{\frac{{U_1^2\;\left( {2 + {{\tan }^2}\alpha \;} \right)}}{{2g}}}} = \frac{2}{{2 + {{\tan }^2}\alpha }}\)

Explanation:

Characteristic curves in turbine:

Characteristic curves of a hydraulic turbine are the curves, with the help of which the exact behaviour and performance of the turbine under different working conditions, can be known. These curves are plotted from the results of the tests performed on the turbine under different working conditions.

• Main Characteristic Curves or Constant Head Curve.
• Operating Characteristic Curves or Constant Speed Curve.
• Muschel Curves or Constant Efficiency Curve.

Main Characteristic Curves or Constant Head Curve:

• Main characteristic curves are obtained by maintaining a constant head and a constant gate opening on the turbine. The speed of the turbine is varied by changing the load on the turbine.
• For each value of the speed, the corresponding values of Power (P) and discharge (Q) are obtained. Then the overall efficiency (η_o) for each value of the speed is calculated.
• From these readings the values of the unit speed (N_u) unit power (P_u) and unit discharge (Q_u) are determined.

Taking N_u as abscissa, the values are Q_u are plotted for the Kaplan and Francis turbine in the following figures.

Explanation:

Flow ratio

The flow ratio of the Francis turbine is defined as the ratio of the velocity of flow at the inlet to the theoretical jet velocity.

\(Flow\;Ratio = \frac{{{V_{f1}}}}{{\sqrt {2gH} }}\)

In the case of the Francis turbine,

The flow ratio varies from 0.15 to 0.3

The speed ratio varies from 0.6 to 0.9

Concept:

In order to predict the behaviour of a turbine working under a varying condition of head, speed output, etc, the results are expressed in terms of quantities that may be obtained when the head on the turbine is reduced to unity.

The following are three important unit quantities.

Explanation:

According to the direction of flow through the runner, the turbine is classified as:

• Mixed flow turbine: The water enters the runner in the radial direction and leaves in an axial direction. Example: Modern Francis turbine
• Tangential flow turbines: In this type of turbine, the water strikes the runner in the direction of the tangent to the wheel. Example: Pelton wheel turbine
• Radial flow turbines: In this type of turbine, the water strikes in the radial direction. Accordingly, it is further classified as:
  • Inward flow turbine: The flow is inward from the periphery to the center (centripetal type). Example: old Francis turbine.
  • Outward flow turbine: The flow is outward from the center to the periphery (centrifugal type). Example: Fourneyron turbine.
• Axial flow turbine: The flow of water is in the direction parallel to the axis of the shaft. Example: Kaplan turbine and propeller turbine.

Explanation:

Characteristic curves in turbine:

Characteristic curves of a hydraulic turbine are the curves, with the help of which the exact behaviour and performance of the turbine under different working conditions, can be known. These curves are plotted from the results of the tests performed on the turbine under different working conditions.

• Main Characteristic Curves or Constant Head Curve.
• Operating Characteristic Curves or Constant Speed Curve.
• Muschel Curves or Constant Efficiency Curve.

Main Characteristic Curves or Constant Head Curve:

• Main characteristic curves are obtained by maintaining a constant head and a constant gate opening on the turbine. The speed of the turbine is varied by changing the load on the turbine.
• For each value of the speed, the corresponding values of Power (P) and discharge (Q) are obtained. Then the overall efficiency (ηo) for each value of the speed is calculated.
• From these readings the values of the unit speed (Nu) unit power (Pu) and unit discharge (Qu) are determined.

Taking Nu as abscissa, the values are Qu are plotted for the Kaplan and Francis turbine in the following figures.