A turbine is a prime mover with rotary motion of the working unit, namely, the rotor, and with a continuous operating process of converting the potential energy of the working fluid supplied (steam, gas or their mixtures, or liquids such as water) into the mechanical work on the rotor shaft. Turbines are bladed machines, the energy conversion occurs in the blading consisting of guide (nozzle or strator) vanes mounted in the stationary casing, and moving blades fixed on the rotor and moving with the rotor. This combination of vanes and blades creates blade cascades — a system of channels where the operating process of the turbine takes place. Turbine blade design is one of the most important and complicated problems in the design process. The complexity is connected with the necessity of a complex solution of problems of gas dynamics, heat transfer, structural strength and the technology of manufacturing blades.
In the guide (nozzle) vane cascades, the flow of steam or gas is accelerated and spun. In the guide vane cascades (guiding apparatus) of a turbine with working fluid (for example, water), the required flow direction is provided and the fluid flow rate is controlled.
In the rotating cascade of moving blades stationed downstream of the guide (nozzle) vane cascade, the energy of the moving steam, gas or liquid is converted into mechanical work on the rotating rotor which in turn is used in overcoming the resistance forces of the driven machinery. This combination of rows of guide (nozzle) vanes and moving blades (arranged in series) is a turbine stage. The working fluid energy conversion may take place in one stage (in this case the turbine is referred to as a one-stage turbine) or in several successive stages (in this case the turbine is referred to as a multistage turbine).
Turbine blades and inlet and outlet units form the flow passage. Turbines are classified as axial and radial (radial-axial) depending on the direction of the flow relative to the rotor rotation axis (Figure 1).
In axial turbines (Figure 1a), the working fluid moves mainly along co-axial surfaces parallel to the turbine axis. The radial turbine differs from the axial one in that, in the nozzle unit and in the greater part of the rotor, the working fluid moves in the plane perpendicular to the axis of rotation. Depending on whether the flow is directed towards the circumference or to the axis of rotation, radial turbines are divided into centrifugal (Figure 1b) and centripetal (Figure 1c).
Turbine rotors are set into rotation by the change in the momentum of the working fluid as it flows through curved interblade channels which are formed by the moving blade surfaces. It is possible to choose the channels' cross sectional area in the rows of guide (nozzle) vanes and in the rotor, which affects the stage operating process. On the basis of the operating principles, there are impulse and reaction turbine stages (and turbines). In impulse stages (turbines), the potential energy of the working fluid is converted into kinetic energy only in stationary guide (nozzle) vanes. This kinetic energy is used to work the row of rotating blades. In reaction stages (turbines), a considerable part of the working fluid's potential energy is converted into mechanical work in the rotor wheel interblade channels. In steam and gas reaction turbines, the circumferential force applied to the rotor is created not only due to the change of the working fluid flow direction (as in an impulse turbine), but also because of the reaction force mixing when the working fluid in the rotor interblade channels expands.
In hydraulic reaction turbines, as the pressure of the fluid flowing along the gradually converging channels of the wheel decreases, its relative flow rate increases. The pressure p0 of the working fluid upstream of the nozzle vane cascade is higher than p1 both in impulse and reaction stages (Figure 2), therefore the flow in the nozzle accelerates: velocity c1 > c0. In an impulse stage, the static pressure p1 upstream of the rotor wheel is equal to p2 downstream of it (Figure 2a). In the reaction stage rotor wheel p2 < p1. Absolute velocity decreases in the rotor wheel of the impulse and reaction stages. Figure 2 shows the variation of relative velocity w and enthalpy i in turbine stages. The relationships between velocities and flow angles in turbine stages depend on the degree of reaction of the stage, which is the ratio of the rotating blade's theoretical temperature drop to the sum of the theoretical temperature drop of the nozzle vanes and rotating blade’s and is approximately equal to the temperature drop of the stage calculated using the stagnation parameters: ρ = .The temperature drop in this relation may be determined from "enthalpy-entropy" diagram as corresponding portions (Figure 3); stagnation parameters are marked with asterisks. The word "reaction" is used because at ρ > 0 the expansion of the working fluid occurs in the rotating blade row and an additional force of reaction arises which rotates the rotor wheel.
Figure 2. Pressure, enthalpy and velocity changes in a turbine: (a) impulse stage, (b) reaction stage.
The degree of reaction of stages at the mean diameter is usually chosen depending on the relative blade length 1/D (D is the mean diameter of the flow passage) so that ρr ≥ 0.05 to 0.1 at the blade root. In multistage turbines, the degree of reaction at the mean diameter gradually increases from the first to the last stage.
The turbine is evaluated using two main parameters: specific work and efficiency. These values vary depending on which turbine losses are taken into account.
The working fluid specific work on the rotor wheel circumference may be determined from Euler equation: lu = c1uu1 + c2uu2, where u1, u2, c1u and c2u are velocities at the turbine flow passage mean diameter, which are usually determined from velocity triangles at this diameter at the rotor wheel inlet and outlet u1 is the circumferential velocity at the rotor inlet; u2 is the circumferential velocity at the rotor outlet; c1u is the circumferential projection of absolute velocity c1 of the working fluid at the rotor inlet; c2u is the circumferential projection of the absolute velocity c2 at the rotor outlet. The other parameters of the velocity triangles are axial projections c1a and c2a of absolute velocities c1 and c2, relative velocities of the working fluid w1 and w2 in the rotor wheel and their circumferential projections w1u and w2u.
where ξn = hn/H0, ξb = hb/H0, ξe = he/H0 are corresponding relative losses. Usually there are radial clearances δn and δb between tbe blades and the casings in the nozzle cascade and the wheel, and the working fluid leakage occurs through their annular areas and thus the work lu at the wheel circumference decreases. If total specific losses in the radial clearances are hc, the corresponding relative losses ξc = hc/H0. Taking into account losses in the radial clearances the turbine specific work at the wheel circumference luc = lu – hc, power efficiency and blade efficiency ηb = . If the kinetic energy of the flow issuing from the stage is used in the next stage, the losses may be estimated with the help of the stage efficiency from stagnation parameters so that
where is the total theoretical heat drop in accordance with stagnation parameters (see Figure 3).
Specific work 1u is less than the enthalpy drop H0 by the amount of energy losses in the flow passage (it is a sum of specific losses hn in the nozzle unit and hb in the moving blades) and the working fluid kinetic energy he at the stage outlet. These losses may be evaluated using the efficiency at the wheel circumference
To estimate a turbine work or shaft power losses besides the above losses, it is necessary to determine relative losses ξfv caused by the friction of the disk and the working fluid and by the ventillation of the gas in the rotor wheel interblade channels. Ventilation losses occur in partial admission turbines in which nozzle channels occupy only a part of the total circumference. The degree of partial admission ε = z1t1/(πd1), where z1 and t1 are the number and the pitch of the nozzle vanes, d1 is the mean diameter at the nozzle cascade outlet. The stage power efficiency taking into account friction and ventilation losses ηT = = ξfv; the efficiency in accordance with stagnation parameters, taking into account these losses = ηT( ) and the blade efficiency ηb = ηT + ξe. It is not possible to obtain a large heat drop in one stage, because there is no possibility to keep at the optimum level the velocity ratio in the flow passage which provides minimum possible losses and thus maximum possible turbine efficiency. Multistage turbines create conditions for operation of each stage at velocity ratios close to optimal, furthermore, the energy lost in the proceeding stage is used in the following stage.
Multistage turbines are built with velocity stages and pressure stages. In turbines with velocity stages almost all the total heat drop is used in the nozzle cascade. The kinetic energy acquired by the working fluid is then converted into work in two or three cascades of active rotating blades, with guide vane cascades being stationed between them. In turbines with pressure stages, the theoretical heat drop is divided between stages so that optimal velocity ratio is reached at each stage. Usually the heat drop is distributed proportionally to the square of the circumferential velocity at the mean diameter of the flow passage of each stage. When determining the heat drop in each stage, with the exception of the first, it is necessary to take into account the velocity of the working fluid flow at the inlet to the nozzle cascade of this stage.
As a result of this calculation, it is possible to obtain the working fluid parameters and the geometry parameters of nozzle vanes and rotating blades at a certain (usually mean) radius of the flow passage.
When the blades are relatively short (1/D < 1/7 to 1/8), it is possible to assume that the gas pressure and the nozzle cascade exhaust velocity do not change along the radius at α1 = constant. The stage designed without taking into account these variations will have efficiency which differs little from the efficiency obtained when these factors are taken into consideration. To provide high efficiency when the blades are relatively long, the stage flow passage should be designed taking into account variations of the pressure upstream of the rotor blades with the radius. The, geometrical parameters of the blades vary along the radius. According to the design predictions under these conditions, the rotating blades — and sometimes the nozzle vanes — should be twisted which is just how they are made in metal. There are no definite recommendations concerning the values of 1/D at which the twist of a blade does not significantly increase the efficiency of the stage but blades are in practice usually short. It is only known that, for stages with high degree of reaction, the stages with twisted blades and with cylindrical blades may have high efficiency if 1/D < 1/4.5 to 1/ 6. According to some data, twisting of blades in stages with low degree of reaction (0.15 on the mean diameter) results in approximately 2 per cent increase of the efficiency at 1/D = 1/8 and 5 per cent at 1/D = 1/5.
To obtain maximum efficiency of the stage at the wheel circumference it is necessary to increase the rotor wheel heat drop, the degree of reaction of the stage, and pressure at the entrance to the wheel as the radius of the blade increases. This variation of the stage parameters is possible because special methods of designing long blades have been developed. One of the first methods was to design stages along the radius at constant circulation. In this case, the axial component of the absolute velocity of flow upstream of the rotor wheel is constant along the radius: c1a = const and the circumferential component of the absolute velocity of flow c1u upstream of the wheel varies relative to 2πrc1u = Г1u = const, where Г1u is the circulation velocity. This design method is sometimes termed designing on "the free vortex law" or at constant circulation. When this method is used, the absolute velocity c1 in the axial clearance between the nozzle cascade and the rotor wheel decreases with the increase of the radius. The pressure in this clearance increases considerably with radius. The working fluid density increases, and the mass flow rate through the unit section area increases at c1a constant. In accordance with the design law, the inlet and outlet angles of nozzle vanes and rotating blades also vary with the radius. The angle α1 decreases with the decrease of the radius. This variation may cause some difficulties when profiles at the root of long nozzle vanes are designed, which is connected with large relative thickness of the trailing edge at the root section. As a result, profile losses increase. Therefore, the general tendency is to have angle α1 ≥ 13 to 14°, and to avoid possible partial admission effect, α1 ≈ 11 to 12° is taken only for low capacity turbines or turbines with low volume flow rate (at high working fluid pressure). Angle β1 increases considerably along the radius and it may be more than 90° when the blades are long.
There are also some other methods besides designing at Г1u = const. One of the most widely used is the designing the nozzle cascade at α1 = const. This method has some advantages. When this design method is used, the velocities c1u and c1 increase less intensively with the decrease of the radius than when designing is done for Г1u. Pressure p1 and the degree of reaction at the root section of the stage is somewhat higher and therefore the profile losses may decrease. The moving blades twist where designing for α1 = const is weaker.
It is possible to plot contours of nozzle vanes and rotating blades for known working fluid parameters at a definite radius of the flow passage. Designing turbine blades may be considered as plotting a number of sections which determine the blade shape. The shape of nozzle vanes and rotating blades is determined by the function of the turbine stage, that is, to turn the flow with the least losses and to extract the required work at the prescribed circumferential velocity. For plotting the profile contours of nozzle vanes and rotating blades, the blade length is divided into several sections stationed at different radii. To select the optimum profile contour at each radius, one can select blades with different profiles and study them experimentally. This method is rather complicated. In a situation where there is a profile which has been well investigated, satisfactory results may be obtained by redesigning this profile. The necessary condition in this case is an insignificant difference in the structural inlet and outlet angles of the reference blade and the blade being designed. In the absence of such reference blades (with parameters close to those of the blade being designed), computer and graphic methods of blade design are used, also. The application of these methods allows plotting blade profile shapes with a smoothly changing curvature, for example, along the square parabola, arc of circles, hyperbolic spirals, Bernoulli's lemniscates. The optimum cascade pitch is determined from the chord length value b. There are analytical and graphic relationships which allow the determination of the optimum relative cascade pitch topn = t/b = 0.75 to 0.85. The larger values are for reaction cascades of blades (and nozzle vanes).
Blade profiles obtained at several sections along the blade length are arranged in the blade drawing in such a way as to provide the prescribed strength of the blade and also high technological standard of its manufacture.
At identical working fluid parameters upstream and downstream of the stage and at one and the same degree of reaction, velocity w2 of a centripetal turbine is much lower than the velocity of an axial or centrifugal turbine. This is explained by the movement of the working fluid from a large radius to a smaller radius. In a radial turbine, the working fluid parameters in the stage flow passage are determined in the same way as in an axial turbine, namely at u2 ≠ u1. For the same working fluid flow rate, the losses in the rotor wheel hb of a centripetal turbine may be lower compared with the losses in an axial turbine rotor wheel.
Turbines which are components of engines and plants operating in a wide range of load variations operate at varying working fluid parameters and varying speeds of rotation. As a result, at operation modes differing from the design mode parameters and velocity triangles change in each stage. The flow conditions in interblade channels also change. Consequently, the specific work and the efficiency of stages and the whole turbine change. These deviations can be evaluated using the turbine characteristics — that is, a combination of analytical and graphical dependences of its general parameters on the operation modes. In other words, the dependence of external parameters determining the conditions of the turbine operation (pressure at the inlet of the turbine and at the outlet pT or , temperature at the inlet of the turbine , speed of rotation n) on its operating parameters (flow rate G, efficiency ηT or , power N) or on the values which allow to determine the operating parameters.
Relative or complex parameters including the above external and operating parameters are used to plot characteristics. When a turbine is operating in the range of parameters that differ slightly from the design parameters (that is at low losses) the turbine characteristics may be integrated by one curve. For example, Figure 4 gives typical characteristics of one-, two-, and three-stage turbines (curves 1, 2 and 3) as dependency of ratios of total pressures on the flow rate parameter . This flow rate parameter or the discharge capacity decrease at a prescribed value of the pressure ratio and when additional stages are added. One of the first attempts to evaluate variation of flow parameters at off-design flow conditions in multistage turbines was made be A. Stodola who formulated the law of ellipse used in turbomachinery. Curve 4 in Figure 4 for multistage turbines satisfies the elliptical law:
where k is a constant. In multistage turbines at subcritical flow conditions in the stages, the heat drop in the last stages increases more rapidly than in the other stages with the increase of pressure ratio . After the critical value is reached the flow of gas or steam in the preceeding stages stabilizes and does not vary with .
Turbines are classified into steam, gas (and also steam-gas operating on a mixture of steam and the fuel combustion products) and hydraulic turbines. Within the class of gas turbines there is a special group of wind turbines using the energy of wind. Steam turbines belong to steam power units, gas turbines are components of gas turbine units and engines. Steam-gas turbines are component parts of steam-gas units.
The major advantages of turbines are high economy, compactness, reliability and the possibility to use them as component parts of power units with unit capacity from hundreds of watt to thousands of megawatt. (See also Gas Turbines, Steam Turbines)
Dixort, S. L. (1975) Fluid Mechanics, Thermodynamics of Turbomachinery, Pergamon Press, Oxford.
Stodola, A. (1945) Steam and Gas Turbines, 6th edn., Peter Smith, New York.
Traupel, Walter (1966) Termische turbomaschinen, bd. 1–2, Berlin, Springer, 1966–1968.