Hydraulic Turbines

Introduction

A hydraulic turbine converts the energy of flowing or pressurized water into mechanical shaft power. The shaft commonly drives an electrical generator, although turbines may also power pumps, compressors, mills, and other machinery. Hydroelectric stations are valued because water is a renewable energy carrier, operating emissions are low, and modern units can respond rapidly to changes in electrical demand.

A turbine must be selected and designed for its available head, discharge, speed, efficiency, sediment content, operating range, and environmental conditions. The principal engineering distinction is between impulse and reaction turbines. Pelton turbines are the principal impulse machines, while Francis and Kaplan turbines are reaction machines.

Learning Objectives

After studying this chapter, the reader should be able to:

• classify hydraulic turbines according to energy conversion and flow direction;
• apply Euler’s turbomachinery equation and construct velocity-triangle relationships;
• explain the construction and operation of Pelton, Francis, and Kaplan turbines;
• evaluate draft-tube performance, cavitation risk, and turbine efficiencies;
• use unit quantities, specific speed, characteristic curves, and governing principles;
• select a suitable turbine for a proposed hydropower installation;
• describe standard testing procedures and practical hydropower applications.

1. Classification and Fundamental Terminology

A turbine may be classified in several ways.

In an impulse turbine, the entire available pressure head is converted into jet velocity before the water enters the runner. Pressure through the runner remains approximately atmospheric. In a reaction turbine, only part of the head is converted into velocity before the runner; the remaining pressure energy is converted within the runner. Consequently, a reaction turbine must be completely filled with water and normally requires a draft tube.

Important terms are:

• Gross head, H_g: elevation difference between upstream and downstream water surfaces.
• Net or effective head, H: head actually available at the turbine inlet after penstock and other hydraulic losses.
• Discharge, Q: volume flow rate, in m³/s.
• Runner: rotating element that receives energy from water.
• Nozzle: device that converts pressure head into a high-speed jet in an impulse turbine.
• Guide vanes or wicket gates: stationary passages that regulate discharge and set the inlet flow direction.
• Draft tube: expanding passage that conducts water from a reaction runner to the tailrace and recovers kinetic energy.
• Governing: automatic adjustment of flow so that turbine speed remains nearly constant as load changes.

The hydraulic power supplied to a turbine is

where ρ is water density in kg/m³, g is gravitational acceleration in m/s², Q is discharge in m³/s, and H is effective head in m. The result is in watts. This equation assumes steady flow and uses the net head across the turbine.

2. Euler’s Turbomachinery Equation

The fundamental energy-transfer equation for a turbomachine follows from angular momentum. Let V_u denote the whirl or tangential component of absolute velocity, u the blade speed, and subscripts 1 and 2 refer to runner inlet and outlet. The specific work transferred to the runner is

For a turbine, this expression is commonly written with a sign convention such that the positive value represents energy delivered by the fluid. Thus the power transferred to the runner is

The equation assumes one-dimensional mean flow at inlet and outlet, negligible shock and transient effects, and angular-momentum conservation. It is valid for impulse and reaction turbines.

Since blade speed is

where D is local runner diameter in m and N is rotational speed in revolutions per minute, the work depends strongly on runner speed and whirl velocity. Energy transfer is maximized when the inlet whirl component is large and the outlet whirl component is minimized, subject to acceptable blade angles and losses.

Velocity triangles

The absolute velocity V is the vector sum of blade velocity u and relative velocity V_r. The tangential component of absolute velocity is V_u, and the flow component is often denoted V_f or V_m. For a runner section,

In magnitude and component form, the velocity triangle gives

where α is the angle between absolute velocity and the direction of blade motion. Definitions of blade angles vary between textbooks; therefore, a consistent triangle convention must always be stated.

For an ideal turbine with no outlet whirl, V_u2=0, Euler’s equation becomes

This condition is called zero whirl at outlet and is a useful design target, although friction, finite blade number, and off-design operation prevent exact achievement.

3. Impulse Turbine: Pelton Wheel

Construction and operation

A Pelton turbine is used mainly for high heads and comparatively small discharges. Water passes through one or more nozzles and forms high-velocity jets. Each jet strikes a double-cup bucket mounted around the runner circumference. A splitter divides the jet, and the two streams are deflected through approximately 160°–170°, producing a large change in momentum.

The principal components are the penstock, nozzle, spear or needle, runner, buckets, casing, brake nozzle, shaft, bearings, and governing system. The casing does not contain pressure; it prevents splashing and directs water to the tailrace.

For a nozzle supplied by head H, the ideal jet speed is

The actual jet speed is

where C_v is the nozzle velocity coefficient, normally less than unity. The formula neglects upstream velocity and assumes the head is converted principally into jet kinetic energy.

For a simple ideal bucket model, let V₁ be jet speed, u bucket speed, and k the ratio of relative outlet speed to relative inlet speed. If the outlet relative velocity is k(V₁−u), and the outlet is deflected through angle φ, the tangential force is approximately

for the stated angle convention. The ideal expression assumes a smooth bucket, steady jet, no bucket friction other than the velocity reduction represented by k, and negligible jet interference.

For a flat plate with complete reversal and no loss, maximum power occurs near u=V₁/2. Real Pelton buckets operate close to, but not exactly at, this ratio because of bucket friction, splitter losses, jet contraction, and leakage.

Pelton efficiencies

The hydraulic efficiency is

or, symbolically,

The overall efficiency is

Mechanical, volumetric, hydraulic, and generator efficiencies may be separated when a detailed loss assessment is required.

4. Reaction Turbines

In a reaction turbine, water enters the runner with both kinetic and pressure energy. Pressure changes across the runner, and the runner must be enclosed in a pressure casing. The draft tube permits the runner to be located above the tailwater while recovering part of the exit velocity head.

Francis turbine

The Francis turbine is a medium-head, mixed-flow reaction turbine. Water enters through a spiral casing, passes through stay vanes and adjustable guide vanes, and flows radially inward and axially outward through the runner. The runner discharges into a draft tube.

Its major parts are the spiral casing, stay ring, guide-vane mechanism, runner, shaft, draft tube, and governor. Guide vanes regulate flow and establish the inlet whirl component. The runner converts both pressure and velocity energy into shaft work.

Francis runners are designed for a particular head and speed. Excessively low discharge can cause unstable flow and pressure pulsations; excessive discharge can cause separation, vibration, and cavitation. Modern large Francis units often use vertical shafts and removable runner assemblies for maintenance.

Kaplan and propeller turbines

A Kaplan turbine is an axial-flow reaction turbine for low heads and large discharges. It has adjustable-pitch runner blades and adjustable wicket gates. The double-regulation system allows the blade angle to match the guide-vane setting over a broad range of load.

A fixed-blade axial turbine is called a propeller turbine. It is efficient near its design point but loses efficiency more rapidly than a Kaplan turbine when the operating condition changes. Kaplan units are commonly installed in river barrages, low-head dams, and tidal or estuarine schemes.

The draft tube is particularly important in axial turbines because the runner exit velocity may be substantial. Common forms include straight conical, elbow, Moody spreading, and bulb-turbine draft tubes.

5. Draft Tubes

A draft tube is an expanding conduit between the reaction runner outlet and tailrace. Its principal function is to reduce exit velocity and recover kinetic energy as pressure head. Applying the steady-flow energy equation between runner outlet and draft-tube outlet shows that recovery is limited by friction, separation, and nonuniform velocity profiles.

A draft-tube efficiency may be expressed as

A commonly used form is

when elevation terms are treated separately and sections 2 and 3 are selected consistently. Here, p is gauge pressure, V is mean velocity, and sections 2 and 3 denote draft-tube inlet and outlet. The exact form must follow the selected datum and energy equation.

A draft tube should expand gradually. A very large expansion angle causes boundary-layer separation and loss; a very small angle produces an unnecessarily long and expensive tube. Flow straighteners or carefully designed elbows may be used to control swirling and pressure pulsations.

6. Cavitation

Cavitation occurs when local absolute pressure falls to or below the vapour pressure of water. Vapour bubbles form in low-pressure regions and collapse when carried into higher-pressure zones. Collapse produces noise, vibration, surface pitting, efficiency loss, and potentially severe runner damage.

Risk increases with high temperature, high runner elevation above tailwater, excessive speed, poor draft-tube design, and operation away from the design point. Cavitation is assessed using the Thoma cavitation parameter:

where H_a is atmospheric pressure head at the site, H_v is vapour-pressure head, H_s is suction elevation of the runner reference point above tailwater, and H is turbine head. Depending on the chosen datum, velocity and local loss terms may also be included in a more detailed pressure calculation.

For safe operation,

where σ_c is the critical value determined from model tests or manufacturer data, with an engineering safety margin. Cavitation control includes lowering the runner, improving hydraulic passages, limiting speed, maintaining adequate tailwater, and avoiding prolonged operation in unstable zones.

7. Efficiencies and Performance

The main efficiencies are:

• Hydraulic efficiency: runner power divided by water power supplied to the runner.
• Mechanical efficiency: shaft power divided by runner power.
• Volumetric efficiency: useful flow through the runner divided by supplied flow, accounting for leakage.
• Generator efficiency: electrical output divided by mechanical shaft input.
• Overall efficiency: electrical or shaft output divided by available hydraulic power.

For a generating unit,

where P_e is generator electrical output. If generator efficiency is excluded, P_shaft should replace P_e. Efficiencies should never be multiplied unless their definitions refer to compatible input and output stages.

A turbine characteristic curve commonly plots efficiency, power, discharge, or torque against speed, head, or gate opening. The best-efficiency point is important, but annual energy depends on the complete operating pattern, not only peak efficiency.

8. Unit Quantities and Specific Speed

Unit quantities compare geometrically similar turbines operating under unit head. They are useful for model testing and preliminary design.

Unit speed is

unit discharge is

and unit power is

These relationships assume geometric similarity and approximately similar flow conditions. N is in rpm, Q in m³/s, P in kW or W consistently, and H in m.

Specific speed is commonly defined as

where P is turbine power, usually in kW in conventional turbine practice. Because unit conventions differ, a stated definition and unit system are essential. Specific speed is an index rather than a universal physical constant; it helps classify machines and select preliminary runner geometry.

Low-specific-speed turbines are generally associated with high head and low discharge, such as Pelton turbines. Medium values commonly indicate Francis turbines, while high values indicate Kaplan or propeller turbines. Boundaries overlap and depend on the chosen convention.

9. Governing

Electrical generators require nearly constant rotational speed to maintain frequency. Governing changes turbine discharge as load changes. In a Pelton turbine, a spear or needle changes nozzle area, while a deflector may temporarily divert the jet during rapid load rejection to prevent excessive pressure rise in the penstock. In Francis and Kaplan turbines, wicket gates regulate discharge; Kaplan machines also adjust runner blade pitch.

A governor measures speed or frequency and commands an actuator. The control system must balance response speed, stability, water-hammer limitation, and mechanical constraints. Rapid closure of gates can cause dangerous pressure transients, whereas excessively slow control produces poor frequency regulation.

Modern governors use digital controllers, position feedback, hydraulic power units, and protective interlocks. Control laws must be commissioned carefully, and failure modes should move the system toward a safe condition.

10. Selection of a Turbine

Selection begins with effective head, design discharge, seasonal variation, speed requirement, sediment load, environmental constraints, and grid operating requirements.

Preliminary selection must be followed by manufacturer-specific hydraulic design, model testing, structural analysis, transient analysis, sediment assessment, and life-cycle costing. A unit with a slightly lower peak efficiency may yield more annual energy if it operates efficiently over the actual flow-duration curve.

11. Testing of Hydraulic Turbines

Acceptance testing determines whether the installed unit meets guaranteed performance. Measurements commonly include head, discharge, rotational speed, torque, shaft power, electrical output, pressure, vibration, and noise.

Discharge may be measured using current meters, acoustic methods, pressure-time methods, tracer dilution, or calibrated hydraulic structures. Head measurements require carefully located pressure taps and elevation corrections. Torque and speed determine shaft power:

where N is rpm and T is torque in N·m, giving power in watts.

Tests should define steady operating points, instrument accuracy, calibration procedures, uncertainty, water temperature, and correction methods. Model tests are used to estimate prototype efficiency, cavitation behavior, pressure pulsations, and runaway speed. Similarity is imperfect because Reynolds number, surface roughness, and scale effects do not remain identical between model and prototype.

12. Applied Case: Illustrative Selection for a Small Hydropower Site

Background

Consider an illustrative run-of-river site with an effective head of 42 m and a design discharge of 18 m³/s. The plant is connected to a grid requiring stable frequency and must operate over seasonal flows.

Problem

Estimate hydraulic power, compare likely turbine types, and identify an appropriate preliminary choice. Assume water density ρ=1000 kg/m³, g=9.81 m/s², and an expected turbine-generator overall efficiency of 0.88.

Analysis

The hydraulic power is

More accurately, 9.81×18=176.58 and 176.58×42=7,416.36, so

The estimated electrical output is

The head is intermediate and the discharge is substantial. A Pelton turbine would be technically possible only under a less favorable arrangement because its normal application is high head and relatively low discharge. A Kaplan turbine is more suitable for low heads, while a Francis turbine is a strong preliminary choice for this medium-head site.

Approach and solution

A vertical Francis unit, or several smaller Francis units for improved seasonal flexibility, would be investigated. The final design should examine sediment abrasion, cavitation parameter, pressure transients, grid requirements, fish passage, minimum ecological flow, and access for maintenance. If seasonal discharge varies greatly, multiple units may produce more annual energy than one large unit.

Outcome and learning points

The preliminary electrical capacity is approximately 6.5 MW, but this is not a final plant rating. The example demonstrates that head and discharge determine hydraulic power, while turbine selection also depends on operating range, environmental obligations, civil works, and life-cycle cost.

13. Applications, Limitations, and Contemporary Practice

Hydraulic turbines are used in large dams, run-of-river plants, pumped-storage stations, irrigation canals, water-supply pipelines, and small community systems. Pumped-storage plants use reversible pump-turbines to store electricity as elevated water and recover it during peak demand.

Important limitations include drought, sediment erosion, debris blockage, fish mortality, reservoir impacts, resettlement, downstream flow alteration, and greenhouse-gas emissions associated with some reservoirs. Hydropower development therefore requires environmental impact assessment, water rights, dam-safety regulation, occupational safety systems, and consultation with affected communities.

Digital monitoring now combines vibration sensors, pressure transducers, fiber-optic measurements, and machine-learning tools for condition assessment. Artificial intelligence can support anomaly detection and predictive maintenance, but it does not replace hydraulic analysis, calibrated instruments, engineering judgment, or safety governance. Training data must be representative, false alarms must be controlled, and operators must retain authority over protective decisions.

Best practice includes operating near efficient regions when possible, maintaining intake screens and gates, inspecting runner surfaces, monitoring cavitation indicators, calibrating instruments, reviewing transient events, and documenting changes in control settings. Sustainable operation integrates energy production with ecological flow, sediment management, dam safety, and community responsibilities.

Chapter Summary

Hydraulic turbines convert water energy into mechanical power through momentum and pressure changes. Impulse turbines, especially Pelton wheels, receive a high-speed jet at approximately atmospheric runner pressure. Reaction turbines, including Francis and Kaplan machines, undergo pressure change through a fully flooded runner and normally require a draft tube.

Euler’s turbomachinery equation, P=ρQ(u₁V_u1−u₂V_u2), connects angular momentum change with runner power. Velocity triangles establish the relationship between absolute velocity, relative velocity, blade speed, whirl, and flow components. Pelton turbines suit high heads, Francis turbines generally suit medium heads, and Kaplan turbines suit low heads with large discharges.

Draft tubes recover exit kinetic energy, while cavitation analysis ensures that local pressure remains above vapour pressure. Efficiency, characteristic curves, unit quantities, and specific speed support performance evaluation and preliminary selection. Governing systems regulate discharge and speed, and acceptance tests verify head, discharge, power, efficiency, vibration, and cavitation performance. Final turbine selection must include technical, economic, environmental, safety, and life-cycle considerations.

Key Terms

Hydraulic turbine: Machine that converts hydraulic energy into mechanical shaft energy.

Impulse turbine: Turbine in which available pressure head is converted into jet velocity before the runner.

Reaction turbine: Turbine in which pressure and velocity energy change through the runner.

Pelton turbine: High-head impulse turbine using one or more jets and buckets.

Francis turbine: Medium-head mixed-flow reaction turbine.

Kaplan turbine: Low-head axial-flow reaction turbine with adjustable blade pitch.

Runner: Rotating element that receives energy from water.

Whirl velocity: Tangential component of absolute fluid velocity.

Velocity triangle: Vector representation of absolute, relative, and blade velocities.

Draft tube: Expanding passage that recovers kinetic energy downstream of a reaction runner.

Cavitation: Formation and collapse of vapour cavities caused by low absolute pressure.

Governing: Regulation of turbine flow to maintain approximately constant speed.

Hydraulic efficiency: Ratio of runner power to hydraulic power supplied to the runner.

Overall efficiency: Ratio of useful output power to hydraulic power based on effective head.

Unit speed: Speed corresponding to unit head for geometrically similar operation.

Unit discharge: Discharge corresponding to unit head.

Specific speed: Performance index used to compare and preliminarily classify turbines.

Runaway speed: Maximum speed reached when load is suddenly removed while flow remains available.

Water hammer: Pressure transient caused by rapid change in water velocity.

Effective head: Head available across the turbine after relevant upstream losses.

Review Questions

Short Answer Questions

1. Distinguish between an impulse turbine and a reaction turbine.
2. What is the physical meaning of whirl velocity in Euler’s equation?
3. State two principal functions of a draft tube.
4. Why are Kaplan turbines fitted with adjustable runner blades?
5. Define cavitation and identify one method of controlling it.

Long Answer Questions

1. Derive the Euler turbine equation from angular-momentum principles and explain the associated velocity triangles.
2. Describe the construction, operation, and applications of a Pelton turbine.
3. Explain the construction and working of a Francis turbine, including the functions of its guide vanes and draft tube.
4. Compare Pelton, Francis, Kaplan, and propeller turbines with respect to head, discharge, flow direction, and regulation.
5. Explain turbine governing, characteristic curves, unit quantities, and specific speed.

Analytical / Application Questions

1. A site has an effective head of 60 m and discharge of 12 m³/s. Estimate hydraulic power using ρ=1000 kg/m³ and g=9.81 m/s². If overall efficiency is 0.86, estimate electrical output.
2. A proposed plant has low head, large discharge, and substantial seasonal variation. Select a suitable turbine and justify whether single or double regulation is preferable.
3. A reaction turbine develops vibration and pitting near full load. Analyze the possible roles of cavitation, draft-tube separation, sediment, and off-design operation, and propose a diagnostic plan.

Multiple Choice Questions with Answers

Question 1

Which turbine is primarily classified as an impulse turbine?

A. Pelton turbine

B. Francis turbine

C. Kaplan turbine

D. Propeller turbine

Correct Answer: A. Pelton turbine

Question 2

In Euler’s turbine equation, which velocity component directly represents angular-momentum transfer?

A. Radial velocity

B. Whirl velocity

C. Vertical velocity

D. Leakage velocity

Correct Answer: B. Whirl velocity

Question 3

A Kaplan turbine is generally most suitable for which condition?

A. High head and very small discharge

B. Medium head and moderate discharge

C. Low head and large discharge

D. Zero head and closed conduit flow

Correct Answer: C. Low head and large discharge

Question 4

What is the principal hydraulic function of a draft tube?

A. Convert shaft power into pressure power

B. Recover part of the runner-exit velocity head

C. Increase the runner inlet velocity by contraction

D. Separate sediment from the water

Correct Answer: B. Recover part of the runner-exit velocity head

Question 5

Cavitation begins when local absolute pressure approaches or falls below the:

A. Atmospheric pressure only

B. Critical pressure of steel

C. Vapour pressure of water

D. Penstock stagnation pressure

Correct Answer: C. Vapour pressure of water

Question 6

The hydraulic power available from water of density ρ, discharge Q, and effective head H is:

A. ρQH/g

B. ρgQH

C. ρgH/Q

D. QH/(ρg)

Correct Answer: B. ρgQH

Question 7

A turbine receives Q=10 m³/s under H=20 m head. Using ρ=1000 kg/m³ and g=9.81 m/s², its hydraulic power is approximately:

A. 196.2 kW

B. 1.962 MW

C. 19.62 MW

D. 0.01962 MW

Correct Answer: B. 1.962 MW

Question 8

Which device commonly regulates the jet discharge of a Pelton turbine?

A. Spear or needle

B. Draft-tube cone

C. Runner-pitch hub only

D. Tailrace weir only

Correct Answer: A. Spear or needle

Question 9

The expression N_u=N/√H represents:

A. Unit speed

B. Specific speed

C. Unit power

D. Hydraulic efficiency

Correct Answer: A. Unit speed

Question 10

Why may several smaller turbine units be preferred at a site with strongly varying seasonal discharge?

A. They eliminate all hydraulic losses

B. They can improve operation over a wider range of available flows

C. They make cavitation impossible

D. They require no governing system

Correct Answer: B. They can improve operation over a wider range of available flows