THERMODYNAMICS OF HEAT TURBINES

– page 13.3 –

● Problem 546: Example of calculation of basic energy balance of heat turbine, see art. 1 ● Problem 871: Calculation of steam expansion in steam turbine ● Problem 1000: Calculation of mass flow change in turbine when inlet pressure changes, see art. 15.

– page 13.4 –

Polytropic expansion

– page 13.5 –

Thermodynamic calculation of heat turbine stage

● Problem 188: Calculation of reaction stage ● Problem 1036: Calculation of internal losses of stage using empirical formulas, see art. 5. ● Problem 1035: Calculation of axial stage of heat turbine with twisted blades and consideration of losses, see art. 5.

– page 13.12 –

Blade cooling performance and methods to increase blade temperature stability

● Problem 1144: Calculation of thermal expansion of shafts with uneven temperature loads, see [Škorpík, 2025]

– page 13.16 –

References

– page 13.16-27 –

Appendices

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THERMODYNAMICS OF HEAT TURBINES

page 13.3

Expansion in heat turbine

Gas expansion in turbines smoothly transforms internal thermal, pressure, kinetic, and potential energy into work. Density and temperature change, necessitating knowledge of h-s and T-s diagram design for energy balance calculation. Usually the expansion is associated with high temperature, at least at the inlet, this puts additional special requirements, this time on the blade materials and their cooling.

Adiabatic expansion

The adiabatic expansion computational model is used in cases where the turbine is not expected to have a significant heat transfer to the surrounding, even if the temperature of the expanding gas is higher than the surrounding temperature, but they are also well thermally insulated and the expansion is too fast to have a significant heat transfer effect on the expansion.

1/2 Na obrázku je malá expanzní turbína pro klimatizace kabin dopravních letounů od společnosti Aviatron. Uvnitř expanduje vzduch odebraný z kompresorů hlavních motorů, expanzí se ochladí. pic.twitter.com/OPJ4waXBTg
— Jiří Škorpík (@jiri_skorpik) February 27, 2026

Definition of ideal adiabatic expansion: In adiabatic expansion, isentropic expansion serves as the ideal comparative change, identifying losses. In ideal expansion, the same exit velocity is usually expected as in actual expansion. This means that the real machine must have larger flow areas than the ideal machine because the loss heat increases the specific volume of the working gas.

Actual adiabatic expansion: A characteristic feature of expansion in a heat turbine is also the so-called re-used heat Δ. This is a part of the loss heat ([Škorpík, 2024]) generated by dissipation of energy L_q, which was transformed into internal work in another part of the turbine. Figure 307 shows an example of expansion in a turbine or in a single stage in the h-s and T-s charts. While in the h-s diagram only the internal losses as a whole can be identified, in the T-s diagram the individual types of losses can be identified.

– 307: –

Adiabatic expansion in h-s and T-s charts

THERMODYNAMICS OF HEAT TURBINES

page 13.4

h [J·kg^-1] enthalpy; L_q [J·kg^-1] loss heat, or sum of different types of energy transformed into internal energy of gas at expansion process; L_w [J·kg^-1] internal losses; s [J·kg^-1·K^-1] entropy; T [K] absolute temperature; w_is [J·kg^-1] internal work at isentropic expansion (adiabatic expansion with no losses); Δ [J·kg^-1] Re-used heat (part of L_q that has been transformed into work in another part of turbine); Δe_K [J·kg^-1] difference in kinetic energy between inlet and outlet (usually insignificantly large difference). The index _is denotes the isentropic compression states, the index _s the stagnation state. The T-s diagram is constructed when the kinetic energy difference is insignificant. The equations are derived in Appendix 307.

Analysis of multi-stage adiabatic expansion in turbine: The Re-used 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 116.

– 116: –

Multi-stage adiabatic expansion in the turbine

Z [-] number of stages; 1+f [1] reheat coefficient (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 116.

– Problem 871: –

Calculate the internal power of Steam turbines and the dryness of steam at the end of expansion. The steam flow through the turbine is 33 t·h^-1, the internal efficiency of the turbine is 75%, the specific isentropic work of the turbine is 1259.59kJ·kg^-1, the pressure at the turbine outlet is 3 kPa, at the turbine inlet are 3.5 MPa at temperature 450 °C. The solution of this problem is shown in Appendix 871.

Polytropic expansion

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.

THERMODYNAMICS OF HEAT TURBINES

page 13.5

Ideal vs. actual polytropic expansion: In polytropic expansion, the comparative process is usually reversible polytropic expansion. Polytropic expansion can be described by Equations 685. These equations can be derived from the general equation of the first law of thermodynamics.

– 685: –

Polytropic expansion for case q>0

e_pol state of gas at the machine outlet during reversible polytropic expansion. w_pol [J·kg^-1] internal work during reversible polytropic expansion (expansion without losses) at same heat transfer with surrounding q - heat q must have the same impact on entropy and temperature as in the actual process. The index _pol denotes reversible polytropic expansion. The T-s diagram is constructed when the difference of kinetic energies is insignificant. The procedure for constructing the T-s diagram is described in Appendix 685.

– 686: –

Polytropic expansion for case q<0 (Cooled expansion)

(a) case when t_e,is>t_e; (b) case when t_e,is=t_e (apparently isentropic expansion). The T-s diagram is constructed when the effect of the kinetic energy difference is insignificant.

Thermodynamic calculation of heat turbine stage

For thermodynamic calculations of the heat turbine stage, findings from previous articles in these proceedings can be utilized (turbomachinery.education). Here, only special knowledge on heat turbine stage thermodynamics is summarized and supplemented: h-s stage charts; usual values of similarity coefficients; heat turbines stages with straight blades; heat turbines stages with twisted blades; conical stages and stages of radial turbines.

THERMODYNAMICS OF HEAT TURBINES

page 13.6

~ h-s charts of heat turbine stages: Figure 908 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 charts because the enthalpies are affected by the heat q. The energy balance of the whole stage is shown in blue.

– 908: –

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.

~ Usual values of similarity coefficients: The maximum Euler work efficiency is reached by the axial stages of heat turbines with a flow coefficient of around 0,65 and a head coefficient of 2,5-2,6 [Smith, 1965], [Dixon and Hall, 2010, p. 119].

~ General characteristics of straight blade turbine stages: 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, 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.

Typical reactions of straight blade turbine stages: Straight blade stages are designed with the reaction close to 0 (impulse stages) or with the reaction 0.5 (reaction stages). From the point of view of thermodynamic properties, the differences between these two types of stages can be seen from their dimensionless characteristics ψ-ϕ. Respectively, the impulse stage can be expected to do twice as much internal work as the reaction stage for the same blade length and mean radius and the same rotational speed. On the other hand, the dimensionless characteristic of the impulse stage will be more abrupt than that of the reaction stage, etc.

THERMODYNAMICS OF HEAT TURBINES

page 13.7

Impulse stage design with straight blades: Figure 350 shows a typical impulse stage design with reaction of approximately 0,03 to 0,05 and its velocity triangle according [Kadrnožka, 2004, p. 91]. The reaction should be such that it yields a reduction in profile losses, which are a function of velocity, while retaining the benefits of the impulse stage design. Impulse stages can also be implemented with twisted blades (reduction of loss due to incorrect angle of attack along the length of the blades), but in that case the blades are twisted significantly more than with variable reaction along the length of the blades.

– 350: –

Sh-shroud; LS-labyrinth seal. A [m²] flow area of blade passage; b [m] width of blade row; l [m] length; L_D [kg·s^-1] discharge of working fluid from gap between discs (it is loss); r [m] radii of blades (index _t denotes tip of blades, index _h denotes root of blades); U [m·s^-1] blade speed; W [m·s^-1] relative velocity; α [°] angle of absolute velocity; β [°] angle of relative velocity; γ [°] stagger angle; δ [m] sizes of axial gaps.

THERMODYNAMICS OF HEAT TURBINES

page 13.8

Reaction stage design with straight blades: Figure 353 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 (Problem 188, p. 13.9) and hence the same blade shape can be used for both stator and rotor rows, which is advantageous for manufacturing. The increase in blade length at the outlet is due to the requirement to maintain the meridional velocity as the density decreases during expansion.

– 353: –

Cylindrical cut through axial reaction stage and its velocity triangle

The blade roots are not drawn in the pictures.

Simplified calculation of profile losses of straight blades: Also typical of straight blades steam turbines is the simplified prediction of the magnitude of the profile losses using nozzle theory ([Škorpík, 2023]) 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 as shown in Figure 178.

– 178: –

Values of velocity coefficients of steam turbine blade cascades

a-pressure row (reaction 0,5); b-equal pressure row. Δβ [°] angle of camber of flow; φ [1] velocity coefficient in stator passage; ψ [1] velocity coefficient in rotor passage. Index ₁ indicates parameters upstream of the rotor blade row, index ₂ indicates parameters downstream of the rotor blade row, index _S indicates the stator blade row, index _R indicates the rotor blade row. Data source [Krbek, 1990, p. 82].

THERMODYNAMICS OF HEAT TURBINES

page 13.9

– Problem 188: –

Design the straight blade lengths, velocity triangles and calculate the axial force acting on the rotor of the reaction stage of steam turbines. The steam flow rate through the stage is 12 kg·s^-1, rotational speed is 50 s^-1, steam pressure at the stator inlet is 1,25 MPa at temperature 320 °C, mean diameter of the blade length is 650 mm, reaction is 0,5, the absolute velocity angle at the stator outlet is 20°, the stator and rotor velocity coefficient is the same 0,93. The isoentropic drop of the stage shall be 21,3 kJ·kg^-1. Compare the pressure coefficient with the pressure coefficient for an ideal reaction stage. The problem is taken from [Krbek, 1990 p. 110]. The solution of this problem is shown in Appendix 188.

Determination of density of profile cascade with straight blades: The camber line of profile of heat turbines with straight blades tends to be very camber. The optimal density of profile cascade with such blades can be determined using the Zweifel coefficient.

Mass flow through a blade passage with straight blades: The nozzle theory can also be used in the determination of the flow coefficient μ through the blade passage. For example, in [Kadrnožka, 2004, p. 110] the values of the flow coefficient for different cases of flow in blade rows are given.

Curtis stage width straight blades

THERMODYNAMICS OF HEAT TURBINES

page 13.10

– 913: –

The cylindrical section of the Curtis stage and its velocity triangle

Velocity triangle of Curtis stage in this figure is for the case of ideal flow without profile losses. w_C [kJ·kg^-1] Euler work of ideal Curtis stage; w_R=0 [kJ·kg^-1] Euler work of ideal impulse stage. The derivation of the ideal Curtis stage Euler work equation is shown in Appendix 913.

– 914: –

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].

EN: Pictured is the Curtis stage for the turbo-pump drive of the German V-2 (A-4) missile. Power output around 50 kW at 3800 to 4000 rpm. A mixture of water vapor and other gases was produced by reacting H2O2 and NaMnO4-now used on Soyuz, whose engines are improved V-2. pic.twitter.com/8L4n2ARLFi
— Jiří Škorpík (@jiri_skorpik) May 3, 2024

Comparison of individual types of stages with straight blades: 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.

Use of individual types of stages with straight blades: 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.

THERMODYNAMICS OF HEAT TURBINES

page 13.11

~ Twisted blades: The thermodynamic calculation of the axial stage of heat turbine with a twisted blades 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.

~ General reasons for building Conical stages: In conical stages, the flow area changes by varying the height of the blade edges so that the meridional velocity is maintained as the gas expands. Cone stages come in both twisted blade and straight blade designs. h-s diagram of the cone stage at the radius under investigation is identical to that of the radial stage.

Conical stage with straight blades: The conical stages with straight blades, or beveled blades in Figure 351, are typical of small industrial back pressure steam turbines. In the case of the stages in Figure 351(a), the advantage is that the l·r^-1 ratio decreases and the blades are the same.

– 351: –

ε [°] blade cutting at tip and root of blades.

Velocity triangle and h-s diagram of conical reaction stage with straight blades: Figure 547 shows the effect of increasing the output radius of the stage in the case of the straight bladed reaction stage of Figure 351. If the blade profile is maintained, the blade speed U changes and with it the other velocities, but the angles remain the same. As the blade speeds change, the energy distribution in the stage also changes according to the attached h-s diagram, which shows that increasing the blade speed leads to a decrease in the processed enthalpy drop.

– 547: –

Velocity triangle and h-s diagram of conical reaction stage with straight blades

THERMODYNAMICS OF HEAT TURBINES

page 13.12

~ General reasons for building Radial stages: 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 393. These stages are not suitable for water vapour with any moisture content, as the water droplets flow centrifugally due to centrifugal forces.

– 393: –

Radial single stage steam turbine

Reaction of 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.

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 blade cooling.

THERMODYNAMICS OF HEAT TURBINES

page 13.13

~ Reaction of Material of blade to temperature increse: At high temperatures, steel strength drop and susceptibility to corrosion rises - the effect of temperature on these material parameters is referred to as temperature stability. Ensuring steel strength at high temperatures involves various blade material ingredients, as depicted in Figure 1001 for desired operating temperatures. The alloy compositions of steels for rotors and blades of high temperature heat turbines are given in [Beneš et al., 1974, p. 194], [Koutský, 2005, p. 61], [Škopek, 2007, Appendix 20].

– 1001: –

Necessary ingredients of steel according operating temperature

Necessary ingredients of blades materials according operating temperature

t [°C] operating temperature of blades. According to data from [Anon., 2014].

Negative effects of Ingredients in blade materials: The use of admixtures is primarily intended to improve temperature resistance, but will usually degrade other properties of the steel, such as abrasion resistance – so a protective coating of harder metal is welded onto the blades using a laser [Míšek, 2014]. For example, Figure 782 shows a steam turbine blade made of a steel-titanium alloy, which reduces the density of the blade and hence the stress from centrifugal force (in the picture the blade is still without the protective coating), but the reduction of the surface abrasion resistance.

– 782: –

Steam turbines blade made of steel and titanium alloy

Doosan Škoda Power 1375 mm steam turbines last stage blade.

Blade materials with Ceramic matrix: Ceramic matrix composite (CMC) materials are also used to increase the high temperature stability of the blades (reducing the effect of creep). This material also has a relatively low density, which reduces stresses in the blade from centrifugal forces [Hocko, 2012, p. 55].

Do you know what goes into the composition of CMCs (Ceramic Matrix Composites)? We’re breaking it down in 27 seconds or less! pic.twitter.com/dJbUWsNwV7
— GE Aerospace (@GE_Aerospace) June 13, 2024

Corrosion resistance of blade materials

THERMODYNAMICS OF HEAT TURBINES

page 13.14

~ Methods of Blade 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 682), 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].

Blade cooling through Air cooling: 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.

– 682: –

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

Reducing number of cooled blades by using rapid expansion in first stage: 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.

THERMODYNAMICS OF HEAT TURBINES

page 13.15

References

ŠKORPÍK, Jiří, 2023, Flow of gases and steam through nozzles, fluid-dynamics.education, Brno, [online], ISSN 1804-8293, https://fluid-dynamics.education/flow-of-gases-and-steam-through-nozzles.html.

ŠKORPÍK, Jiří, 2024, Technická termomechanika, engineering-sciences.education, Brno, ISSN 1804-8293, https://engineering-sciences.education/technicka-termomechanika.html.

ŠKORPÍK, Jiří, 2025, Meze použití materiálů, engineering-sciences.education, Brno, https://engineering-sciences.education/meze-pouziti-materialu.html.

ANON., 2011, MS 5002 Gas Turbine a Through D Evolution, ge.com.

ANON., 2014, Co nám může v budoucnosti nejvíc chybět? Na čem jsem závislí, Technický týdeník, 01/2014, Business Media CZ, Praha, ISSN 0040-1064.

BENEŠ, Antonín, DRASTÍK, František, HOSTINSKÝ, Zdeněk, KOUTSKÝ, Jaroslav, NĚMEC, Josef, 1974, Nauka o kovech, SNTL, Praha.

DIXON, S., HALL, C., 2010, Fluid Mechanics and Thermodynamics of Turbomachinery, Elsevier, Oxford, ISBN 978-1-85617-793-1.

DOKOUPIL, Eduard, 2015, Turbíny pro Luftwaffe: zrod a popis prvních německých proudových motorů, Dokoupil Eduard, Světlá, ISBN 978-80-260-8153-1.

HOCKO, Marián, 2012, Transformace leteckých lopatkových motorů na spalovací turbíny, Západočeská univerzita v Plzni, Plzeň, ISBN 978-80-261-0218-2.

KADRNOŽKA, Jaroslav, 1991, Teorie lopatkových strojů, Vysoké učení technické v Brně, Brno, ISBN 80-214-0275-X.

KADRNOŽKA, Jaroslav, 2004, Tepelné turbíny a turbokompresory, Akademické nakladatelství CERM, s.r.o., Brno, ISBN 80-7204-346-3.

KOUTSKÝ, Jaroslav, 2005, Development and application of Original Special Steels-Base of World Famous Level of Skoda-Works Steam turbines, Energetické stroje-termomechanika-mechanika tekutin , Fakulta strojní Západočeské university v Plzni, Plzeň.

KRBEK, Jaroslav, 1990, Tepelné turbíny a turbokompresory, Vysoké učení technické v Brně, Brno, ISBN 80-214-0236-9.

MILLER, Rudolf, HOCHRAINER, A., LÖHNER, K., PETERMANN, H., 1972, Energietechnik und Kraftmaschinen, Rowohlt taschenbuch verlag GmbH, Hamburg, ISBN 3-499-19042-7.

MÍŠEK, Tomáš, 2014, Vývoj ultra dlouhé lopatky Doosan Škoda Power pro kondenzační Steam turbines, Technický týdeník, 10/2014, Business Media CZ, Praha, ISSN 0040-1064.

POLSTER, Burkard, 2014, Q.E.D. Krása matematického důkazu, Dokořán s.r.o., Praha, ISBN 978-80-7363-532-9.

ŠKOPEK, Jan, 2007, Parní turbína-tepelný a pevnostní výpočet, Západočeská univerzita v Plzni, Plzeň, ISBN 978-80-7043-256-3.

SMITH, F., 1965, A simple correlation of turbine efficiency, Journal of the Royal Aeronautical Society, 69, 467–470.

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