Vane Angle at Inlet from Degree of Reaction Solution

STEP 0: Pre-Calculation Summary
Formula Used
Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction))))
θi = acot(cot(αf)*(1-1/(2*(1-R))))
This formula uses 2 Functions, 3 Variables
Functions Used
cot - Cotangent is a trigonometric function that is defined as the ratio of the adjacent side to the opposite side in a right triangle., cot(Angle)
acot - The ACOT function calculates the arccotangent of a given number which is an angle given in radians from 0 (zero) to pi., acot(Number)
Variables Used
Vane Angle at Inlet - (Measured in Radian) - Vane Angle at Inlet is the angle made by the relative velocity of jet with the direction of motion at inlet.
Guide Blade Angle for Francis Trubine - (Measured in Radian) - Guide Blade Angle for Francis Trubine is defined as the angle between the direction of the jet and the direction of motion of the plate.
Degree of Reaction - Degree of Reaction is defined as the ratio of pressure energy change inside a runner to the total energy change inside the runner.
STEP 1: Convert Input(s) to Base Unit
Guide Blade Angle for Francis Trubine: 11.03 Degree --> 0.192509816494938 Radian (Check conversion here)
Degree of Reaction: 0.45 --> No Conversion Required
STEP 2: Evaluate Formula
Substituting Input Values in Formula
θi = acot(cot(αf)*(1-1/(2*(1-R)))) --> acot(cot(0.192509816494938)*(1-1/(2*(1-0.45))))
Evaluating ... ...
θi = 1.13440226038694
STEP 3: Convert Result to Output's Unit
1.13440226038694 Radian -->64.9964617902845 Degree (Check conversion here)
FINAL ANSWER
64.9964617902845 64.99646 Degree <-- Vane Angle at Inlet
(Calculation completed in 00.004 seconds)

Credits

National Institute of Technology Calicut (NIT Calicut), Calicut, Kerala
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18 Francis Turbine Calculators

Volume Flow Rate of Obtuse Angled Outlet Bladed Francis Turbine given Work Done per Second
Go Volume Flow Rate for Francis Turbine = Work Done per Second by Francis Turbine/(Density of Fluid in Francis Turbine*(Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine-Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine))
Volume Flow Rate of Acute Angled Francis Turbine given Work Done Per Second on Runner
Go Volume Flow Rate for Francis Turbine = Work Done per Second by Francis Turbine/(Density of Fluid in Francis Turbine*(Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine+Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine))
Work Done per Second on Runner by Water for Acute Angled Outlet Blade
Go Work Done per Second by Francis Turbine = Density of Fluid in Francis Turbine*Volume Flow Rate for Francis Turbine*(Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine+Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine)
Work Done per sec on Runner by Water for Obtuse Angled Outlet Blade
Go Work Done per Second by Francis Turbine = Density of Fluid in Francis Turbine*Volume Flow Rate for Francis Turbine*(Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine-Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine)
Hydraulic Efficiency of Francis Turbine with Obtuse Angled Outlet Blade
Go Hydraulic Efficiency of Francis Turbine = (Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine-Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine)/(Acceleration due to Gravity*Net Francis Turbine Head)
Hydraulic Efficiency of Francis Turbine with Acute Angled Outlet Blade
Go Hydraulic Efficiency of Francis Turbine = (Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine+Whirl Velocity at Outlet of Francis Turbine*Velocity of Vane at Outlet for Francis Turbine)/(Acceleration due to Gravity*Net Francis Turbine Head)
Volume Flow Rate of Right Angled Outlet Bladed Francis Turbine given Work Done per Second
Go Volume Flow Rate for Francis Turbine = Work Done per Second by Francis Turbine/(Density of Fluid in Francis Turbine*Velocity of Vane at Inlet for Francis Turbine*Whirl Velocity at Inlet of Francis Turbine)
Work Done per Second on Runner by Water for Right Angled Outlet Blade Angle
Go Work Done per Second by Francis Turbine = Density of Fluid in Francis Turbine*Volume Flow Rate for Francis Turbine*Velocity of Vane at Inlet for Francis Turbine*Whirl Velocity at Inlet of Francis Turbine
Degree of Reaction of Turbine with Right Angled Outlet Blade
Go Degree of Reaction = 1-cot(Guide Blade Angle for Francis Trubine)/(2*(cot(Guide Blade Angle for Francis Trubine)-cot(Vane Angle at Inlet)))
Hydraulic Efficiency of Francis Turbine with Right Angled Outlet Blade
Go Hydraulic Efficiency of Francis Turbine = (Whirl Velocity at Inlet of Francis Turbine*Velocity of Vane at Inlet for Francis Turbine)/(Acceleration due to Gravity*Net Francis Turbine Head)
Velocity of Vane at Inlet given Speed Ratio Francis Turbine
Go Velocity of Vane at Inlet for Francis Turbine = Speed Ratio of Francis Turbine*sqrt(2*Acceleration due to Gravity*Head at Inlet of Francis Turbine)
Francis Turbine Speed Ratio
Go Speed Ratio of Francis Turbine = Velocity of Vane at Inlet for Francis Turbine/(sqrt(2*Acceleration due to Gravity*Head at Inlet of Francis Turbine))
Francis Turbine Flow Ratio
Go Flow Ratio of Francis Turbine = Velocity of Flow at Inlet of Francis Turbine/(sqrt(2*Acceleration due to Gravity*Head at Inlet of Francis Turbine))
Velocity of Flow at Inlet given Flow Ratio in Francis Turbine
Go Velocity of Flow at Inlet of Francis Turbine = Flow Ratio of Francis Turbine*sqrt(2*Acceleration due to Gravity*Head at Inlet of Francis Turbine)
Pressure Head given Speed Ratio in Francis Turbine
Go Head at Inlet of Francis Turbine = ((Velocity of Vane at Inlet for Francis Turbine/Speed Ratio of Francis Turbine)^2)/(2*Acceleration due to Gravity)
Pressure Head given Flow Ratio in Francis Turbine
Go Head at Inlet of Francis Turbine = ((Velocity of Flow at Inlet of Francis Turbine/Flow Ratio of Francis Turbine)^2)/(2*Acceleration due to Gravity)
Guide Blade Angle given Degree of Reaction
Go Guide Blade Angle for Francis Trubine = acot(cot(Vane Angle at Inlet)/(1-1/(2*(1-Degree of Reaction))))
Vane Angle at Inlet from Degree of Reaction
Go Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction))))

Vane Angle at Inlet from Degree of Reaction Formula

Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction))))
θi = acot(cot(αf)*(1-1/(2*(1-R))))

What is the significance of degree of reaction?

Degree of reaction is an important factor in designing the blades of a turbine, compressors, pumps and other turbo-machinery. It also tells about the efficiency of machine and is used for proper selection of a machine and it others for a required purpose. The degree of reaction contributes to the stage efficiency and thus used as a design parameter. Stages having 50% degree of reaction are used where the pressure drop is equally shared by the stator and the rotor for a turbine. This reduces the tendency of boundary layer separation from the blade surface avoiding large stagnation pressure losses. Stage having reaction less than half suggest that pressure drop or enthalpy drop in the rotor is less than the pressure drop in the stator for the turbine. Stage having reaction more than half suggest that pressure drop or enthalpy drop in the rotor is more than the pressure drop in the stator for the turbine.

What if degree of reaction is zero?

This is special case used for impulse turbine which suggest that entire pressure drop in the turbine is obtained in the stator. The stator performs a nozzle action converting pressure head to velocity head. It is difficult to achieve adiabatic expansion in the impulse stage, i.e. expansion only in the nozzle, due to irreversibility involved, in actual practice.

How to Calculate Vane Angle at Inlet from Degree of Reaction?

Vane Angle at Inlet from Degree of Reaction calculator uses Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction)))) to calculate the Vane Angle at Inlet, The Vane angle at inlet from degree of reaction formula is used to find the vane angle at inlet of a francis turbine with right angled outlet blade angle. Vane Angle at Inlet is denoted by θi symbol.

How to calculate Vane Angle at Inlet from Degree of Reaction using this online calculator? To use this online calculator for Vane Angle at Inlet from Degree of Reaction, enter Guide Blade Angle for Francis Trubine f) & Degree of Reaction (R) and hit the calculate button. Here is how the Vane Angle at Inlet from Degree of Reaction calculation can be explained with given input values -> 3724.023 = acot(cot(0.192509816494938)*(1-1/(2*(1-0.45)))).

FAQ

What is Vane Angle at Inlet from Degree of Reaction?
The Vane angle at inlet from degree of reaction formula is used to find the vane angle at inlet of a francis turbine with right angled outlet blade angle and is represented as θi = acot(cot(αf)*(1-1/(2*(1-R)))) or Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction)))). Guide Blade Angle for Francis Trubine is defined as the angle between the direction of the jet and the direction of motion of the plate & Degree of Reaction is defined as the ratio of pressure energy change inside a runner to the total energy change inside the runner.
How to calculate Vane Angle at Inlet from Degree of Reaction?
The Vane angle at inlet from degree of reaction formula is used to find the vane angle at inlet of a francis turbine with right angled outlet blade angle is calculated using Vane Angle at Inlet = acot(cot(Guide Blade Angle for Francis Trubine)*(1-1/(2*(1-Degree of Reaction)))). To calculate Vane Angle at Inlet from Degree of Reaction, you need Guide Blade Angle for Francis Trubine f) & Degree of Reaction (R). With our tool, you need to enter the respective value for Guide Blade Angle for Francis Trubine & Degree of Reaction and hit the calculate button. You can also select the units (if any) for Input(s) and the Output as well.
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