Hydraulic turbines convert the energy of flowing or pressurized water into mechanical shaft power. A generator coupled to the shaft then converts this mechanical power into electrical energy. Hydropower remains important because it can provide high conversion efficiency, rapid load response, grid stability, and, in many installations, energy storage through reservoirs or pumped-storage systems.

Turbine selection is governed mainly by available head, discharge, desired rotational speed, cavitation conditions, sediment content, and variation in load. The principal machines considered in this chapter are the Pelton impulse turbine and the Francis and Kaplan reaction turbines.

Learning Objectives

1. Hydraulic Energy and Turbine Classification

where P_h is hydraulic power in watts, ρ is water density in kg/m³, g is gravitational acceleration in m/s², Q is discharge in m³/s, and H is the effective hydraulic head in m. For water near ordinary temperatures, ρ ≈ 1000 kg/m³ and g ≈ 9.81 m/s². The effective head is the energy head actually available across the turbine, not necessarily the gross difference in reservoir water levels.

where η_o is the overall efficiency, including hydraulic, volumetric, mechanical, and generator-independent turbine losses. Turbine efficiency alone is often defined as shaft power divided by hydraulic power entering the turbine.

Impulse turbines

In an impulse turbine, the entire available pressure head is converted approximately into kinetic energy in one or more nozzles before water reaches the runner. The runner operates near atmospheric pressure, and pressure change through the buckets is small. The Pelton turbine is the standard example. Impulse turbines are suited to high head and comparatively low discharge.

Reaction turbines

In a reaction turbine, water enters the runner with both pressure and velocity energy. Pressure changes as water passes through the runner, which must therefore be completely filled with water and enclosed in a casing. A draft tube is normally required at the outlet. Francis turbines are generally used for medium head, while Kaplan turbines are used for low head and high discharge.

Feature	Pelton	Francis	Kaplan
Principle	Impulse	Reaction	Reaction
Typical head range	High	Medium to high	Low to medium
Flow through runner	Tangential jet	Radial-inward to axial	Mainly axial
Runner pressure	Approximately atmospheric	Changes through runner	Changes through runner
Draft tube	Not required	Required	Required
Flow regulation	Nozzle needle and deflector	Guide vanes	Guide vanes and runner blades
Main advantage	High-head efficiency	Broad general application	High efficiency under varying discharge

The ranges overlap and depend on manufacturer design; they are guides rather than strict limits.

2. Pelton Turbine

A Pelton turbine uses one or more high-speed jets directed tangentially onto double-cup buckets. A splitter divides the jet so that water flows smoothly into two passages and leaves with substantially reduced whirl velocity. The bucket outlet edges are cut back to prevent interference with the following bucket.

where V₁ is jet velocity, C_v is the nozzle velocity coefficient, and H_n is the net head converted in the nozzle. The formula assumes negligible upstream velocity and represents an ideal energy conversion modified by nozzle losses.

For a frictionless bucket, the force in the direction of wheel motion is related to the change in whirl momentum. A useful ideal relation for a bucket moving at speed u is

when outlet whirl velocity is taken as opposite in sign to inlet whirl velocity and bucket speed effects are included in the relative-flow analysis. In general, force must be obtained from

with a consistent sign convention, where V_w1 and V_w2 are absolute whirl components at inlet and outlet. Runner power is P = F_t u.

For an ideal Pelton bucket with negligible friction and a jet entering tangentially, maximum power occurs approximately when

Real machines use a slightly different optimum because of bucket friction, deflection angle, nozzle losses, and mechanical losses. Pelton turbines may use multiple jets when the discharge is large. A spear or needle regulates jet area, while a deflector rapidly diverts the jet during load rejection to prevent excessive speed rise; the needle then closes more slowly to limit water hammer.

Pelton design considerations

Important practical factors include jet-to-runner diameter ratio, number of buckets, bucket spacing, nozzle alignment, abrasive sediment, and accessibility for inspection. Sediment erosion is particularly serious at high jet velocities. The runner and buckets are commonly manufactured from high-strength stainless alloys or other erosion-resistant materials.

3. Francis Turbine

The Francis turbine is a mixed-flow reaction machine. Water enters a spiral casing, passes through stay vanes and adjustable guide vanes, and enters the runner approximately radially inward. It leaves in an axial or near-axial direction and then passes through a draft tube.

The guide mechanism controls discharge and imparts the appropriate whirl component. The runner extracts both pressure and velocity energy. The Euler turbine equation expresses the ideal specific work transfer:

where w is work per unit mass in J/kg, u₁,u₂ are blade speeds at runner inlet and outlet, and V_w1,V_w2 are absolute whirl components. The equation follows from angular momentum conservation and assumes steady flow with negligible changes in shaft elevation within the runner. Sign conventions must be maintained consistently.

For a well-designed Francis turbine, outlet whirl is often made small at the design condition, so the second term is near zero. At off-design conditions, residual outlet whirl can be substantial and represents lost recoverable energy or undesirable draft-tube swirl.

Francis runners contain fixed blades. Their performance depends on the relationship among guide-vane angle, runner speed, discharge, and head. Poor operation away from the best-efficiency point can cause pressure pulsations, vibration, and fatigue. Modern large Francis units may use adjustable guide vanes, optimized blade profiles, and computational fluid-dynamics analysis.

4. Kaplan and Propeller Turbines

Kaplan turbines are axial-flow reaction turbines with adjustable runner blades and adjustable guide vanes. They are especially effective at low heads and varying discharge. A fixed-blade axial turbine is called a propeller turbine; it is economical when head and discharge remain nearly constant.

The double-regulation feature of a Kaplan turbine allows the blade pitch to match the guide-vane setting. This maintains favorable incidence angles and high efficiency over a wide load range. In contrast, a fixed-blade propeller turbine may experience large efficiency losses when operating away from its design condition.

Kaplan turbines may be vertical, horizontal, or bulb-type. Bulb units place the generator in a streamlined housing within the water passage and are useful for low-head installations, although maintenance access can be more difficult.

5. Draft Tubes and Energy Recovery

A draft tube is a gradually enlarging passage connected to the outlet of a reaction turbine. It reduces exit velocity and converts part of the velocity head into pressure head, allowing the turbine to be installed above the tailwater level. A draft tube does not create energy; it recovers energy that would otherwise leave as kinetic energy.

η_d = (p₂/ρg + V₂²/2g) - (p₃/ρg + V₃²/2g)
-----------------------------------------
V₂²/2g

where sections 2 and 3 are the draft-tube inlet and outlet, respectively. A more complete expression includes elevation terms and uses total head; the simplified form assumes the elevation difference is treated separately and that inlet and outlet reference conditions are clearly defined.

The pressure at the runner outlet must not fall below the local vapor pressure. Excessive divergence, sharp bends, poor alignment, or strong swirl can cause separation, vibration, and low draft-tube efficiency. Common forms include straight conical, elbow, and Moody spreading draft tubes. The elbow type is compact but more difficult to design efficiently.

6. Cavitation

Cavitation occurs when local absolute pressure falls to or below the vapor pressure of water. Vapor bubbles form and subsequently collapse in regions of higher pressure. Collapse can produce noise, vibration, pitting, performance loss, and structural damage.

where H_a is atmospheric pressure head at the site, H_v is vapor-pressure head, H_s is turbine setting above tailwater (with the appropriate sign convention), and H is turbine head. The allowable value depends on turbine design and operating condition. Manufacturers normally provide a critical cavitation coefficient or recommended setting.

Cavitation control involves selecting an appropriate turbine elevation, maintaining proper draft-tube geometry, avoiding prolonged operation at unstable loads, using erosion-resistant materials, and monitoring vibration and acoustic emissions. Atmospheric pressure decreases with altitude, making cavitation more likely for the same turbine setting.

7. Governing and Speed Control

A turbine governor maintains speed and frequency as electrical load changes. If load decreases suddenly, the turbine accelerates unless water admission is reduced. If load increases, admission must increase to restore speed.

Pelton governing uses nozzle needles and jet deflectors. Deflectors act quickly, while gradual needle movement reduces pressure transients in long penstocks. Francis and Kaplan turbines use guide-vane servomotors; Kaplan units additionally adjust runner-blade pitch. The relationship between guide-vane opening and blade pitch is called the cam relationship.

A governor system must balance speed regulation, response time, stability, and hydraulic transients. Rapid closure may cause water hammer, while excessively slow response may permit unacceptable overspeed or frequency deviation. Modern governors use electrohydraulic control, feedback sensors, supervisory protection, and programmable operating limits.

8. Unit Quantities and Similarity

Unit quantities permit comparison of geometrically similar turbines operating under different heads. For a turbine tested at head H, the unit speed, unit discharge, and unit power are commonly defined as

where N is rotational speed, Q is discharge, and P is power. Units must be retained consistently; these quantities are not dimensionless.

where D is a characteristic runner diameter. These relationships assume similar flow coefficients, comparable efficiency, and dynamic similarity. They become less accurate when Reynolds number, roughness, cavitation, or relative clearances differ significantly.

Specific speed is a compact classification parameter. One commonly used metric form is

where N is in rpm, P is in kW, and H is in metres. Because definitions vary by unit system and whether power or discharge is used, a specific-speed value must never be compared without confirming its definition. Low specific speed generally indicates a high-head, low-flow design such as Pelton; high specific speed generally indicates a low-head, high-flow design such as Kaplan.

9. Performance Testing

A turbine performance test measures head, discharge, speed, torque, and power. The main efficiencies are:

In practice, torque may be measured with a dynamometer or torque meter, discharge by calibrated flow measurement, and head using pressure taps corrected for elevation and velocity. Electrical output is measured separately when generator performance is also assessed.

A complete test should specify the measurement locations, instrument calibration, uncertainty, operating head, speed, gate opening, and stabilization time. Performance curves commonly show efficiency, power, discharge, and cavitation limits against load or guide-vane opening. Acceptance testing should follow an agreed standard, such as an applicable International Electrotechnical Commission code, with traceable instruments and documented uncertainty.

Worked example: hydraulic power and output

A Francis turbine receives Q = 18 m³/s under an effective head of H = 42 m. If turbine overall efficiency is 91%, determine hydraulic power and shaft output.

Therefore, the available hydraulic power is approximately 7.42 MW and the shaft output is approximately 6.85 MW. This calculation assumes that the stated head is the effective turbine head and that the efficiency includes all losses between hydraulic input and shaft output.

10. Selection and Hydropower Applications

Selection begins with a reliable site assessment. The designer determines net head, design discharge, seasonal flow duration, sediment load, water temperature, grid requirements, environmental constraints, and expected operating range. A single design point is insufficient for rivers with strongly variable flow.

The best choice also depends on number of units. Several smaller units can improve part-load operation and maintenance flexibility, while one large unit may reduce civil construction and equipment cost. Life-cycle evaluation must include civil works, penstock, powerhouse, controls, sediment management, maintenance, outage consequences, and environmental mitigation.

Hydropower schemes may be storage, run-of-river, diversion, pumped-storage, or small hydro. Pumped-storage plants operate reversible pump-turbines: water is pumped uphill during low-demand periods and released through turbines during high-demand periods. Their round-trip efficiency is less than 100%, but they provide valuable peak power, frequency regulation, reserve, and renewable-energy balancing.

11. Illustrative Case Study: Low-Head River Plant

Background

This illustrative case concerns a proposed run-of-river plant with an effective head of 16 m, a design discharge of 75 m³/s, and large seasonal variation. The site has limited reservoir storage and requires frequent operation away from the nominal design flow.

Problem

A fixed-blade propeller turbine, a Francis turbine, and a double-regulated Kaplan turbine are considered. The selection must address energy yield, part-load efficiency, cavitation, fish passage, sediment, and maintenance.

Analysis and approach

The low head and high discharge indicate an axial-flow machine. Because seasonal discharge varies widely, adjustable runner blades are valuable. The estimated hydraulic power at design flow is

If the expected overall efficiency is 90%, the shaft output near the design condition is approximately 10.59 MW. A Kaplan turbine is preferred because coordinated guide-vane and blade-pitch control can preserve efficiency over a wider operating range than a fixed-blade propeller machine. A Francis alternative would require careful evaluation of flow passage size, speed, and off-design hydraulic stability.

Cavitation analysis must use the site atmospheric pressure, water temperature, vapor pressure, turbine setting, and manufacturer’s critical coefficient. The draft tube should be tested or validated for swirl, pressure recovery, and stability. A fish-management plan, trash rack, bypass or fish passage, and safe intake velocity are included in the civil and environmental design.

Outcome

The illustrative recommendation is a Kaplan unit, or multiple Kaplan units if staged operation materially improves seasonal energy capture. The final decision should be based on guaranteed performance curves, life-cycle cost, sediment resistance, environmental licensing, and maintainability rather than peak efficiency alone.

Learning points

Turbine selection is a systems decision. Head and discharge identify the likely turbine family, but variation in flow, cavitation margin, environmental conditions, transient behavior, and maintenance requirements determine the final design.

12. Limitations, Safety, and Contemporary Practice

Hydraulic machines operate under high pressure, high rotational speed, and large stored-water energy. Safe practice requires isolation and lockout procedures, controlled dewatering, confined-space precautions, overspeed protection, emergency shutoff, pressure-relief provisions, and inspection of lifting and rotating equipment.

Environmental governance includes minimum ecological flows, sediment continuity, fish protection, water-quality requirements, downstream ramping limits, and community consultation. Regulatory approval commonly requires dam safety review, emergency action planning, and monitoring of structural and hydraulic performance.

Digital monitoring increasingly combines pressure sensors, vibration analysis, oil-debris monitoring, thermography, and machine-learning-based anomaly detection. Artificial intelligence can assist predictive maintenance by identifying deviations from a learned normal pattern, but it does not replace engineering judgment, calibrated instruments, cybersecurity controls, or physical inspection. Training data may be incomplete during rare failure events, so alarm thresholds must be validated and fail-safe protections retained.

Best practice is to operate near the best-efficiency region when system demands permit, avoid prolonged unstable zones, monitor cavitation indicators, maintain clean intakes, control sediment, verify governor response, and use documented performance tests after overhaul.

Chapter Summary

Hydraulic turbines transform water-head energy into shaft power through impulse or reaction processes. Pelton turbines use high-velocity jets and are suited to high head. Francis turbines use mixed-flow reaction action and serve a broad medium-head range. Kaplan turbines use axial flow and adjustable blades, making them effective for low head, high discharge, and variable flow.

The Euler turbine equation relates angular momentum change to specific work. Draft tubes recover outlet kinetic energy in reaction turbines, while cavitation analysis protects against vapor formation and collapse. Governing systems regulate speed but must balance response with water-hammer control. Unit quantities, affinity laws, specific speed, characteristic curves, and standardized testing support comparison and selection. Successful hydropower design integrates hydraulic performance with civil works, environmental obligations, safety, maintainability, and life-cycle economics.

Key Terms

Impulse turbine: A turbine in which pressure head is converted mainly to jet velocity before the runner.

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

Pelton turbine: A high-head impulse turbine using tangential jets and double-cup buckets.

Francis turbine: A mixed-flow reaction turbine with radial-inlet and axial-outlet flow tendencies.

Cavitation: Vapor formation and collapse caused by local pressure falling to vapor pressure.

Thoma coefficient: A nondimensional indicator used to assess cavitation conditions.

Euler turbine equation: Angular-momentum relation for ideal specific work transfer.

Net head: Effective head available at the turbine after relevant hydraulic losses.

Pumped storage: A hydropower arrangement that pumps water uphill for later generation.

Review Questions

Short Answer Questions

Long Answer Questions

Analytical / Application Questions

Multiple Choice Questions with Answers

Question 1

Question 2

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

Question 3

Question 4

Which turbine is generally most suitable for low head, high discharge, and widely varying flow?

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

If a turbine has hydraulic input power of 5 MW and shaft output of 4.25 MW, its overall efficiency is:

Chapter 12 Hydraulic Turbines and Energy Conversion

Hydraulic Turbines and Energy Conversion

Introduction