Copyright©Jiří Škorpík, 2024
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Multi-stage expansion

Reheat factor

The Re-usable heat Δ directly increases the efficiency of multi-stage expansion compared to single-stage expansion, because part of the heat from the loss processes in the previous stage is used in the expansion of the next stage. This means that the internal efficiency of the stage part of multi-stage turbines η_i is greater than the mean internal efficiency of the individual stages η_j, see Figure 2.

2: Multi-stage adiabatic expansion in the turbine

Z [-] ] number of stages; 1+f [1] reheat factor (1.02 to 1.04 according to [Kadrnožka, 1991]); η_i [1] internal expansion efficiency between point 1-Z. The index _j denotes the j-th stage. The equations are derived for the assumption that all stages process the same enthalpy gradient and the expansion is adiabatic. For clarity, the absolute velocity kinetic energy is not plotted in the figure. The equations are derived in Appendix 4.

Polytropic expansion

Polytropic ideal expansion

Heat transfer

The computational model of polytropical expansion is used in cases where the expansion is affected by heat transfer with surrounding. This occurs, for example, in radial turbines with large disk area, in cooling of thermally exposed parts of the turbine, etc. In polytropic expansion, the comparative process is usually reversible polytropic expansion. Polytropic expansion can be described by Equations 3. These equations can be derived from the general equation of the first law of thermodynamics.

Thermodynamic calculation of heat turbine stage

For thermodynamic calculations of the heat turbine stage, findings from previous articles in these proceedings can be utilized. Here, only special knowledge on heat turbine stage thermodynamics is summarized and supplemented.

h-s diagrams

Euler work

Profile losses

Total losses

Figure 5 shows the h-s diagram of the heat turbine stage at the investigated radius for Euler work caculation. In the case of polytropic expansion, the individual kinetic energy cannot be plotted in the diagrams because the enthalpies are affected by the heat q. The energy balance of the whole stage is shown in blue.

5: h-s diagram of expansion in heat turbine stage

L_h [J·kg^-1] profile losses;ΣL [J·kg^-1] total losses of stage; V [m·s^-1] absolute velocity; q_E [J·kg^-1] heat transferred in the surroundings of the streamline under investigation.

Straight blades

Axial stage

1D calculation^1.

Axial stages with straight blades are used as a cheaper alternative to stages with twisted blades, especially in the case of very small ratios of blade length to mean blade diameter [Kadrnožka, 2004, p. 153], i.e. in cases where the spatial character of the flow is not so significant and the use of 1D blade claculation is adequate. Of course, lower efficiency of such stages compared to stages with twisted blades is to be expected. Straight blade stages are used in steam turbines which are manufactured in pieces.

Reaction stage

Symmetric blades

Meridian velocity^1.

Figure 7 shows a cylindrical section of the reaction stage and its velocity triangle for reaction R=0.5. In such the reaction, the profile losses are the lowest because the velocities V₁ and W₂ are the same or very similar. Hence the symmetrical shape of the velocity triangle for the stator and rotor blade rows and hence the same blade shape can be used for both stator and rotor rows, which is advantageous for manufacturing. When the symmetry of the velocity triangles is maintained, the condition V_2θ=0 cannot usually be met, see Problem 2. The increase in blade length at the outlet is due to the requirement to maintain the meridional velocity as the density decreases during expansion.

7: Cylindrical cut through axial reaction stage

The blade roots are not drawn in the pictures.

Profile losses

Outlet velocity

Velocity coefficient

Profile cascade

Also typical of straight blades steam turbines is the simplified prediction of the magnitude of the profile losses using nozzle theory instead of using the aerodynamic data of the profile cascades. This prediction consists of matching the blade passage to the nozzle, albeit curved, then the change in exit velocity from the passage and hence the profile loss can be predicted using the velocity coefficient (see the articel Flow of gases and steam through nozzles [Škorpík, 2023]) as shown in Figure 8.

8: Values of velocity coefficients of steam turbine blade rows

10: Single stage steam turbine for low flow and high enthalpy drop

The turbine is designed as the Curtis stage. The turbine contains only one Laval nozzle. Instead of a second stator row, there is a reversal passage that brings the steam back to the first rotor row. Figure from [Miller et al. 1972, p. 188].

Curtis stage

Impulse stage

Reaction stage

The Curtis stage under the same conditions can handle a larger enthalpy difference in the stage than the impulse or reaction stage (in the ratios of 8:2:1 as shown in the above paragraphs), but at a lower internal efficiency because the velocities, and hence the profile losses, are very high. To increase the internal efficiency at the mean radius of the Curtis stage, individual blade rows are constructed with a slight overpressure.

Impulse stage

Curtis stage

Reaction stage

Impulse stages are used in single-stage turbines called Laval turbines as well as in multi-stage steam and gas turbines. Curtis stages are used where the emphasis is on high power in a small volume. The Curtis stage was the driving turbine of the 50 kW turbopumps of the German V-2 rocket engine, as well as being found in the Russian RD 108 rocket engines for powering Soyuz etc. Reaction stages are more common in multistage steam turbines with an emphasis on simplicity and smaller purchase requirements.

Twisted blades

Twisted blades^5.

The thermodynamic calculation of the axial stage of a twisted-blade heat turbine is presented in one of the problem assignments in the article Internal losses of turbomachines and their influence on turbomachine calculation. The procedure for calculating the blade height reaction and the types of losses to be taken into account are also described in that article.

Radial turbines

Radial stages are required to have at least a minimum reaction and even equal pressure. For example, for turbines (centripetal stage), a zero reaction stage would result in a deceleration of the relative velocity at the outlet of the rotor row due to centrifugal forces, and therefore to reduce the Euler work. The minimum reaction of centripetal turbines can be determined by imposing the condition W₁=W₂.

Radial vs. Axial stages

Steam turbine

Radial stages are used as a more expensive but more efficient single-stage alternative to impulse and Curstis stages, because the Euler work of the radial stage is increased by the tangential velocity difference and so flow velocities can be low even with large enthalpy drop. They are used, for example, in turbochargers and turboexpanders. Radial straight bladed heat turbines are also marginally produced, see Figure 13. These stages are not suitable for water vapour with any moisture content, as the water droplets flow centrifugally due to centrifugal forces.

13: Radial single stage steam turbine

Blade cooling performance and methods to increase blade temperature stability

The high temperature of the working gas also allows for high thermal efficiency of the cycle in which the turbine operates. However, this places high demands on the blade material and surface treatment. Another possibility is to cool the blades.

Polishing

Surface treatments such as polishing can in turn increase corrosion resistance. Especially at high temperatures in the presence of oxygen, there is a risk of increased scale formation (formation of hard oxides - rust).

Blade cooling

Blade roots

Cooling passages

Boundary layer cooling

If high-quality materials aren't enough, especially for initial heat turbine stages, blades require active cooling. This can involve cooling blade roots, entire blades via passages (Figure 16), or using a film of cold gas blown into the boundary layer through small holes on the pressure or suction side. Air (for combustion turbines) or water (for steam turbines) serves as the cooling medium [Miller et al., 1972, p. 931]. Some early combustion turbine blade cooling concepts, though unsuccessful, are discussed in [Dokoupil, 2015, p. 221].

Air cooling

Anti-corrosion coating

Combustion turbines

Thermal conductivity

Ceramic coatings

For air cooling, cooling passages need anti-corrosion coating. In combustion turbines, cooling air is drawn from the compressor section at slightly higher pressure than around the blade to be cooled. It exits through holes in the blade's trailing edge into the exhaust stream. The cooling efficiency is increased by the outer surface layer of the blades with a low thermal conductivity value, such as ceramic coatings, etc.

16: GE MS5002 series combustion turbine blades with cooling passages [Anon., 2011]

Entalpic drop

First stages

Steam turbines

It is typical for the enthalpy drop distribution in a heat turbine that the first stages are designed to process a larger enthalpy drop than the next stages. While this may degrade the thermodynamic efficiency of the first stages, it will reduce the number of stages operating in the high temperature region and hence the cost of the turbine - such a distribution of enthalpy drops is common in steam turbines.