A hydraulic turbine converts the potential energy of a flowing liquid to rotational energy for further use. In principle, there is no restriction on either the liquid or the use for the energy developed. However, in most cases, these are respectively water and electrical generation. Hence, hydraulic turbines have become synonymous with hydro electric power.

The rate of doing work (power) developed in a hydraulic turbine is:

where Δp is the drop in total pressure across the turbine, is the volumetric flow rate and η the efficiency of the turbine. It is common practice to quote Δp in terms of the difference between the upstream and downstream total heads, called the turbine head which equals Δp/gρ.

The basis of the design of the turbine hydraulic passages is the velocity diagrams at the entry and exit of the turbine rotating element (called the runner). These lead to the *Euler equation* for theoretical torque and to the theoretical *Euler efficiency* of the turbine (see Flow of Fluids). Although elemental velocity triangles are employed for preliminary design of the hydraulic passages, for large turbines, model testing is necessary for verification of performance. Because of the cost and time involved in developmental model testing, more recently, a computerized finite element solution of the inviscid flow equations in the hydraulic passages (see Computational Fluid Dynamics), cross correlated with general data from model test results, is employed for advanced design. In particular, the *efficiency of the hydraulic turbine* must be optimized and established for contractual purposes. The peak efficiency of a properly designed large hydraulic turbines can be as high as 95%, with typically every point of improved efficiency involving considerable monetary benefits in operation.

Testing cannot model the losses due to hydraulic friction. Hence model test efficiencies are converted to full scale values with established Reynolds Number based formulas. Confirmatory site efficiency testing of hydraulic turbines is possible using current meters, ultrasonic, salt velocity tracer, water column inertia (Gibson method) and thermodynamic methods to evaluate flow [IEC (1991)]. However, because of cost and the measuring inaccuracy inherent in site testing, there is a general trend to rely solely on model test results.

Depending on the use, the amount of water and level difference available, the power of hydraulic turbines can be a few kilowatts up to hundreds of Megawatts. However, regardless of size, their performance can be equated through similarity laws; hence the applicability of tests on models to predict the performance of large turbines. From nondimensional considerations the similarity laws are:

where is the power, ρ is the water density, u is the water velocity, and D is a characteristic diameter of the runner from which all other dimensions of the hydraulic passages follow.

Hydraulic turbines are classified according to specific speed. Specific speed is defined as the rotational speed (revolutions per minute) at which a hydraulic turbine would operate at best efficiency under unit head (one meter) and which is sized to produce unit power (one kilowatt). The equation for specific speed derived from non- dimensional considerations is therefore:

The historical development of hydraulic turbines has culminated in two distinct types namely *impulse* (or constant pressure) and *reaction*. Reaction turbines are further divided into radial and axial flow and variable and fixed runner blade. In the impulse turbine, flow is directed through a nozzle to impact on a series of buckets attached to the periphery of the runner. The total transfer of energy is from the change of momentum of the fluid jet; there is no change in hydrostatic pressure once the fluid exits the jet. Impulse turbines (known as *Pelton turbines* after their inventor) are typically used for heads above 100 m and reasonably low flows. They can have up to six jets to better utilize larger flows, as shown in Figure 1.

Reaction turbines for heads in the range 600 m to 30 m are known as* Francis turbines* (after their inventor). These are radial flow units in which the flow enters the runner radially and discharges axially, as illustrated in Figure 2.

The runner of a reaction turbine is equipped with blades which contain and direct the flow. Therefore, in addition to energy derived from the momentum changes of the fluid as it passes through the runner, it is also generated from the changes in hydrostatic pressure of the fluid within the runner passages. Below design heads of approximately 50 m axial flow turbines are used, known as *propeller units* because of their similarity to a ships propeller. A subsection of propeller units known as *bulb turbines* are used for heads below approximately 10 m. Small hydraulic turbines can be arranged with a horizontal shaft for ease of maintenance, but the larger units used for hydroelectric power installations are almost universally vertical. The exception is the bulb turbine, which is only arranged horizontally.

Utilization of the three basic turbine types are within the following ranges of specific speed (calculated from rpm, kW and m):

Single jet Pelton: 3 < Specific Speed < 36

Multiple jet Pelton: 36 < Specific Speed < 60

Francis: 60 < Specific Speed < 400

Propeller: 300 < Specific Speed < 1200

Reversible pump turbines are a special type of reaction turbine. These change direction of rotation to operate both as a pump and a turbine and are used for pumped storage applications. Hydraulically, a reversible pump turbine is designed as a pump with only minor modifications to accommodate its role as a turbine.

The stability of operation and the internal hydraulic forces (both static and dynamic), are directly dependent on the velocity of flow through the turbine. For a given design head and flow velocity there is the unique specific speed. This leads to the relationship:

where K is a constant.

There are strong commercial benefits in using as small a turbine (hence high flow velocities) as possible for any given application. However, this is restricted by the state of the art in respect of vibration and performance. In 1994, the generally accepted maximum value for K was about 2300.

In all three general groups of hydraulic turbine, flow is directed to the periphery of the runner of the turbine via a spiral casing and discharges from the runner through a draft tube. In reaction turbines, the rotation of the liquid commences as a free vortex in the spiral casing; it is directed through the fixed stay vanes and then through the adjustable turbine wicket gates, such that the angle of approach of the flow to the runner at the design conditions is precisely the runner blade angle (shockless entry). Flow through the reaction turbine to obtain the required power is regulated by the turbine wicket gates. Thus, shockless flow is only obtained at the output for best efficiency and at the design head. In a reaction turbine the draft tube is designed for maximum recovery of hydrostatic pressure. This is especially critical for low head turbines.

The efficiency and operational stability of turbines with low design heads is a strong function of the inlet approach angle, efficiency dropping rapidly with decrease in output and, to a lesser extent, with change in head. To maintain efficiency over the operating range, low head units often have adjustable runner blades, the runner inlet angle changing with wicket gate position and, if required, with operating head. These units are said to be double regulating and include the semi radial flow *Deriaz turbines*, axial flow *Kaplan turbines* (both named after their inventors) and bulb turbines.

Reaction turbine runners can suffer Cavitation at the blade inlet (due to off design flow conditions), in the runner hydraulic channels at part load operation and at the runner exit on the suction side of the runner blades. The latter is the most critical and is a function of the back pressure on the runner. Suction side cavitation in a hydraulic turbine is accordingly related to downstream (tailwater) level through the *Thoma Coefficient* defined as:

where z is the height of the runner exit plane above tailwater level, p_{a} is atmospheric pressure and p_{vap} vapor pressure of the fluid. Height z is commonly called the setting of the turbine.

Typically, it is too expensive to set a large hydraulic turbine deep enough below tailwater level to completely eliminate cavitation and, in any particular application, an economic balance between cavitation repair and cost of excavation has to be established. For preliminary design, operating experience is used to establish an acceptable σ_{Th} . One such criterion is:

For major installations, the cavitation performance of the hydraulic turbine should be established with model testing. Cavitation model testing of medium to tow head hydraulic turbines is often conducted with Froude Number similarity to the prototype.

The wetted surfaces of hydraulic turbines are also prone to damage due to Erosion by transported silt and sand and corrosion from aggressive fluids. Damage is particularly problematical when silt erosion, corrosion and cavitation act in conjunction (synergistic effects). In applications where the silt content is extreme the hydraulic design may sacrifice efficiency for partial immunity from silt erosion (contouring of surfaces and thickening of runner blades, for example).

Stainless steel, which has a far better resistance to cavitation and erosion than carbon steel, is extensively employed in susceptible areas.

The economic pressures to increase specific speed and hence flow velocities, for a particular head, have led to operational problems with flow induced vibrations from runner blades and stay vanes. These are particularly problematical when their forcing frequency coincides with the natural frequency of any other part of the mechanical, hydraulic or electrical system thus leading to resonance. Also, at part load operation reaction turbines suffer from draft tube pressure oscillations, which can result in unacceptable power swings. Air admitted naturally to areas of low pressure or force fed from compressors can be effective in curing oscillations due to part load operation.

The speed of hydraulic turbines is regulated by a governor. The governor senses the speed of the turbine and adjusts the wicket gate opening to maintain speed within close limits. Speed sensing and the associated feed back control systems are typically digital. On all other than very small hydraulic turbines, the amplification from the digital governor to the wicket gates (and runner blades if required) is through a high pressure oil servomotor system. If the hydraulic turbine is operating on a large integrated network then its speed is controlled by the network and the governor is used to change output via its permanent speed droop. The governor feed back and gain have to accommodate the water column and generator inertias. The turbine and all equipment connected to it, both mechanically and electrically, are designed for runaway speed of the turbine (speed at zero torque) resulting from governor failure. To aid regulation, high to medium head turbines are often equipped with pressure relief valves. For the same reason, the jets of impulse turbines are often equipped with flow diverters. For security, isolation valves or hydraulic gates are commonly installed at spiral casing inlets.

#### REFERENCES

Brekke, H. (1994) State of the Art in Pelton Turbine Design, *The International Journal on Hydropower and Dams*, Aqua Media International, 1-2.

Gummer, J. H. and Hensman, P. C. (1992) A Review of Stayvane Cracking in Hydraulic Turbines, *International Water Power and Dam Construction*, Reed Business Publishing, 44-8.

International Electrotechnical Commission (IEC) 41 (1991) Field acceptance tests to determine the hydraulic performance of hydraulic turbines, storage pumps and pump turbines, *Bureau Central de la Commission Electrotechnique Internationale*, Geneva, Switzerland.

Raabe, J. (1985) *Hydro Power, The Design, Use and Function of Hydromechanical Hydraulic and Electrical Equipment*, VDI, Verlag, Dusseldorf, Germany. ISBN 3 18 400616 6.

Vivier, L. (1966) *Turbines Hydrauliques et Leur Regulation*, Editions Albin Michel, Paris.

#### References

- Brekke, H. (1994) State of the Art in Pelton Turbine Design,
*The International Journal on Hydropower and Dams*, Aqua Media International, 1-2. - Gummer, J. H. and Hensman, P. C. (1992) A Review of Stayvane Cracking in Hydraulic Turbines,
*International Water Power and Dam Construction*, Reed Business Publishing, 44-8. -
International Electrotechnical Commission (IEC) 41 (1991) Field acceptance tests to determine the hydraulic performance of hydraulic turbines, storage pumps and pump turbines,

*Bureau Central de la Commission Electrotechnique Internationale*, Geneva, Switzerland. - Raabe, J. (1985)
*Hydro Power, The Design, Use and Function of Hydromechanical Hydraulic and Electrical Equipment*, VDI, Verlag, Dusseldorf, Germany. ISBN 3 18 400616 6. - Vivier, L. (1966)
*Turbines Hydrauliques et Leur Regulation*, Editions Albin Michel, Paris.