# Water Turbines

**Contents**

- Water Turbines
- Stationary Curved Vanes
- A Series Of Moving Curved Vanes.
- Turbine With Curved Vanes And An Inward Radial Flow (francis Or Gerard Turbine)
- Efficiency Of Turbines
- Impulse Turbine Allowing For Friction
- Reaction Turbines
- Characteristic Curves And Iso-efficiency Curves For A Turbine Under All Operating Conditions
- Page Comments

## Water Turbines

Hydro electricity is a reliable form of renewable energy. Water turbines are highly efficient and easily controlled to provide power as and when it is needed. In addition, the only system currently available to store large quantities of electrical power is pumped storage.##### General Approach To Find Force, Work Done, Efficiency, Etc.

- Find the mass of liquid/sec. striking the vanes ( slugs/sec or kilograms/sec).
- Find the change in absolute (or relative) velocity of the liquid in the required direction. (ft/sec. or meters/sec.)
- Force on the vane in this direction (
**Newton'****s second Law**) equals mass/sec change of velocity. ( slugs/sec ft/sec = lb. or kg/sec mtr./sec = Newtons) - The vane exerts an equal and opposite reaction on the liquid and the supply nozzle. (Jet reaction)
- Work done / sec. by the liquid on the vane is the vane velocity force on plate in the direction of the velocity. (ft.lb./sec. or Watts)
- Horse Power output = Work done per sec.
- Efficiency, = Output Power/ Input power from jet

### Series Of Moving Plates

**turbine**is a rotary engine that extracts energy from a fluid flow and converts it into useful work. A

**water turbine**is a rotary engine that takes energy from moving water.

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- The weight of water hitting the plates per second . N.B. For a single plate the weight of water per second striking the plate is
- The change of velocity =
- Force on plate = mass/sec. Change of velocity
- Work done/second = Force Velocity
- The input = the Kinetic energy of the jet
- The efficiency of the system,
- For a given jet velocity,
- For maximum , and the value of the maximum efficiency .

### A Single Plate Inclined To The Jet.

**normal force**is the component, perpendicular to the surface of contact, of the contact force exerted on an object by, for example, the surface of a floor or wall, preventing the object from penetrating the surface.

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- The weight of water striking the plate per second will be, as before, given by:
- Change of velocity normal to the plate =
- Normal force on plate
- Component of in the direction of
- Work done on plate

## Stationary Curved Vanes

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## A Series Of Moving Curved Vanes.

**Relative velocity**is the vector difference between the velocities of two objects, as evaluated in terms of a single coordinate system.

**Absolute velocity**is the vector sum of the velocity of a fluid parcel relative to the earth and the velocity of the parcel due to the earth's rotation

- Relative velocity and is tangential to the blades.
- Blade velocity (Add to )
- Absolute velocity and the vector sum of and .

- The velocity of whirl (Component of in the direction of )
- The velocity of flow (Component of normal to direction of )

The

**suffix 1**refers to the

**outlet**triangle. are the inlet and outlet angles of absolute velocity (i.e.Jet 0) and are the inlet and outlet angles relative to the blade velocity.

##### Low Speed

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##### High Speed

- The outlet triangle remains the same but the inlet triangle is now:
Note: and if there is no friction
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### Pelton Wheel (circumferential)

- The two velocity triangles are for low and high flow (the inlet triangle is a straight line):
For both types of flow above:
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- Weight of water per second
- The blade speed,
- Force on the vanes = mass of water/second,
- Change in velocity
- Work done on the vanes = Force Velocity
- The Kinetic energy supplied
- The efficiency, = Work done / K.E. supplied

**For a Pelton Wheel Only**(if friction is ignored) But For the maximum at a given head and blade angle: Which occurs when i.e. The bucket speed is half the jet speed. This is a theoretical figure and in practice, due to frictional losses, the maximum efficiency is when

##### Example - Example 1

And From the outlet triangle: And

Also In the direction the Force on the vanes = mass of water change of velocity Force on vanes in the direction = Therefore, the resultant force The resultant is at i.e. at to the direction of motion.

Work done per second on the vanes = The force in the direction times blade velocity The Kinetic energy supplied by the jet per second Thus the efficiency of the turbine

- The
**efficiency**of the turbine is

## Turbine With Curved Vanes And An Inward Radial Flow (francis Or Gerard Turbine)

The following diagram shows the velocity triangles for both low and high speed.##### MISSING IMAGE!

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- Let the weight of water/second striking the vanes be lb/sec.
- Tangential momentum/second at entry
- Moment of momentum at entry
- Moment of momentum at outlet
- Torque on the vanes equals the change of moment of momentum per second

**Euler Equation**which can be applied to any type of turbine or centrifugal pumps.

## Efficiency Of Turbines

**Impulse Turbines**(No allowance for frictional losses)

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Jet Velocity = the exhaust velocity of water leaving the vanes. Work done per second on the vanes (per lb of water per second) The theoretical hydraulic efficiency of the Turbine is equal to:

## Impulse Turbine Allowing For Friction

**Friction**is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other.

**Actual**(

**overall efficiency**) is based on the useful work out divided by the water power input. It therefore makes allowance for the frictional losses.

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##### Example - Example 1

( one Horse Power (HP) is 550 ft.lb/sec. One cubic ft. of water weighs 62.4 lb., 32.2 ft/second squared)

Neglecting windage losses, find:

- a) The
**efficiency**of the runner. - b) The
**diameter**of each jet.

- a) The
**efficiency**of the runner is - b) The
**diameter**of each jet is or approx. inches.

## Reaction Turbines

Inward radial flow, mixed flow, or axial with a propeller shaft. They may be sited below the tail race or above it with a draft tube.##### MISSING IMAGE!

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- The absolute velocity at the entry of the runner
- The absolute velocity at the exit of the runner
- the head.

**Low Supply Head**Usually an open flume supply with a short Pinstock. e.g. Propeller Turbine. is the vertical height from the water level in the forebay to the level in the Tail Race.

**Large Head**Long Pinstock and smaller water through put e.g. Radial flow Francis Turbine. is now the total head (i.e. Pressure, Velocity and Datum) in the supply pipe just before entering the Turbine casing and is relative to the tail stock Datum. The Gross Head in this case is measured from the supply reservoir and includes Pinstock losses.

**Draft Tube**This is designed to convert Kinetic Energy of discharge into Pressure. This gives extra suction through the Turbine. i.e. It enables the height ( see diag) to be included into the Supply head .

**Bernoulli**at and (see diag) (neglecting friction) In practice the height is limited to avoid cavitation at . (If there are no losses in the runner, the supply system or draft tube.) Then the work done by the water on the runner (W.D.) (if there is no Draft tube or if it is parallel) The Theoretical Hydraulic Efficiency

**Notes:**

- For a Reaction Turbine
- For Axial Flow . For radial flow
- Velocity of Flow

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- Let be the number of blades, the blade thickness and the width of the

### Variations Of Pressure Head Across The Turbine Passage. Assuming No Losses

- Applying
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##### Example - Example 1

The runner is 24 inches outside diameter and 16 inches inside diameter and the widths at the entrance and exit are 2 and 3 inches respectively.

The pressure at entry to the guides is + 87 ft.head and the kinetic energy there can be neglected. The pressure at discharge is - 6 ft.head.

If the losses in the guides and moving vanes are taken as where is the radial component of flow calculate:

- a)The
**speed**of the runner in r.p.m. for tangential flow on to the running vanes - b) The
**horse-****power**given to the runner by the water.

**Bernoulli**at the inlet to the guide vanes and the outlet of the runner. From the

**velocity triangles**( above): Also . (Since ) Hence: But: Weight of water per second Work Done per Lb. of water is given by: W.D. Horse-power

- a)The
**speed**of the runner is - b) The
**horse-****power**given to the runner by the water is Kilo watts

### Specific Speed Of A Turbine

**Specific Speed**This is the speed (r.p.m.) at which a similar model of the Turbine would run under a head of 1ft. when of such a size as to develop 1 Horse Power. The suffix "s" is used to denote the values associated with the Specific Turbine. Each type of Turbine ( Pelton Wheel, Francis etc.) has it's own characteristic limits of . A similar model means :

- Geometrically similar - made from the same drawings but to a different scale.
- Dynamically similar - Operating conditions and equal efficiencies.

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##### Notes On Specific Speed

- is based on the values of , , and used at the design point. i.e. At maximum efficiency.
- is NOT dimensionless and there are different values in each of the measurement systems.

- Unless otherwise stated, is in r.p.m.
- is in Brake Horse Power(b.h.p.) ()
- The unit of are
- can be made dimensionless and still be a constant by dividing by and this is called the
**Speed Number**

- For a particular type of Turbine is constant.

- for different types of Turbine and a comparison of heads for a particular power and speed.

The head requirements for a turbine to develop 100 b.h.p. at 1000r.p.m.- For a Pelton Wheel Head required 520 ft.
- For a Turgot Turbine Head required 335 to 180ft
- For a Francis Turbine Head required 180 to 40ft
- For a Propeller Turbine Head required 40 to 23ft

- An example of the use of Specific Speed

What turbine would be used if there was a supply of 10 cu.ft/sec under a head of 225 ft. ? Assume an efficiency of 80%. Power Output = Water h.p.input Efficiency It would therefore be necessary to use a Turgot Turbine. However it might be possible to use a Pelton Wheel with two jets. Power per jet h.p. Therefore per Jet

Try a Pelton Wheel with four Jets: per Jet This would be a practical proposition but would result in some loss of efficiency due to interference between the jets. A better alternative would be to have two wheels on the same shaft with two jets per wheel.

### Unit Conditions

**Unit operating conditions**for a turbine are those under which that particular turbine would run when working under a head of one ft. ( or unit head in any other system) assuming there to be change in efficiency.

- Unit Speed

- Unit quantity of a Turbine is the flow through the turbine when operating under a head of one ft. assuming similar conditions.

- Unit Power of a given turbine is the power output of the turbine when operating under a head of one ft. assuming no change in efficiency

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**Note:**

##### Performance Curves Of A Turbine.

- These are plotted for a constant head and a constant Gate opening ( Or needle valve setting) and are on the basis of speed in r.p.m.
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## Characteristic Curves And Iso-efficiency Curves For A Turbine Under All Operating Conditions

The Turbine is tested under a constant head for each of several gate openings and the values of Power output and speed are reduced to unit conditions ( Equations (125) (132)). Suitable values for efficiency are the marked on the curve for the differing Gate openings and lines of isoefficiency are drawn.##### MISSING IMAGE!

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. Therefore, . Therefore,

But since , Unit Power,

And , . Therefore . Or .

Substitute for in the above equation: . Or

Or substitute for : Or

Power, if is unchanged .

From Equation (146), . Or

From Equation (148), . Or

i.e. (where is a Constant)

From equations (144) and (150) . Or

These seven expressions allow the performance of the prototype turbine to be estimated from tests on the model. Note that there are in fact only three independent equations.

The efficiency predicted for a large Turbine from test carried out on a model is usual lower than that obtained from the actual prototype. This is because of the relatively greater frictional losses in the smaller passages of the model.

Strictly speaking the surface finish of the model should be geometrically similar to that of the prototype. The reduction in efficiency is said to be due to scale effects and is corrected for in practice by the use of empirical equations such as:

**The Moody equation**

##### Example - Example 1

At what

**speed**must the model be run and if it develops 135 h.p. and uses 38cu.ft.of water per second at this speed, what

**power**will be obtained from the full scale Turbine, assuming that it's efficiency is 3% better than that of the model?

With a value for of 138 the Turbine must be a Propeller Turbine.

- The
**specific speed**is - The
**power**is