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11. Turbomachine

Author: Jiří Škorpík, skorpik@fme.vutbr.cz : last updated 2014-02

Nomenclature and principle of operation; General classification and application of turbomachines; Difference between piston engine and turbomachine; Classification of turbomachinery according to stream direction; Construction features of turbomachines; Blade, blade passage and blade cascade; Power output/input of turbomachine; Turbomachine stage; Velocity triangle; Turbomachine losses

Turbomachines are wide group of machines (for example steam turbines, gas turbines, rotodynamic compressors, centrifugal pumps/rotodynamic pumps, water turbines and etc.). Their characteristic feature is a rotor and blades on its circumference. These blades forms channels, through these channels flows a working fluid. Energy is transformed through a force between the working fluid and the blades.

Symbols of used quantities

Used quantities are listed under equations and figures or in the text in this text format:

Quantity symbol  [Unit symbol]  Quantity name

The list of used quantities and abbreviations is shown at the first occurrence only.

Nomenclature and principle of operation

The rotation of the rotor is caused by the force on blades. If the working fluid transmits a energy on the rotor, then this machine is called a turbine (an action force from flow of the working fluid and a reaction force from the blades). Rotodynamic pumps, rotodynamic compressors, fans work opposite, the working fluid consumes the energy from the rotor (the action force from blades and the reaction force from flow of the working fluid).

For turbomachines is typical some pressure difference between the input and the output (pressure gradient) or some velocity difference or their combination. Next pictures show some general types of the turbomachines with descriptions their function:

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A Kaplan turbine as an example conversion of potential energy of water to work (hydraulic fluid machine).
Figure 1. A Kaplan turbine as an example conversion of potential energy of water to work (hydraulic fluid machine).
a the level in the top tank; b the level in low tank; c the guide vanes (the stator); d the reinforcement of the spiral casing.
H [m] the height difference between the top and the low level; ØD [m] the runner diameter (the external diameter of the rotor); ω [rad·s-1] the angular velocity. Some part of potential energy of water is converted to kinetic energy inside the stator, then water is flowed through the blade-to-blade passage of the runner and is making up the force on the blades.

- zl. 271 -
The cross-sectional view of a turbocharger as an example a heat turbomachine.
Figure 2. The cross-sectional view of a turbocharger as an example a heat turbomachine.
a the turbine rotor (the impeller); b the compressor rotor; c the double spiral casing (the volute); d the exhaust; e the input to the compressor; f the vaneless diffuser.
The waste gases with high temperature and high pressure flows to the turbine (the rotating channels), where expands. Work of the turbine feeds the turbocharger (compressor) which is located on the same shaft.

- zl. 193 -
An axial wind turbine.
Figure 3. An axial wind turbine.
c [m·s-1] the wind velocity far upstream of the turbine.
The wind velocity is very small, therefore wind turbines can be considered hydraulic machines. The wind turbine transforms of kinetic energy of wind on a work. The wind turbine is not inside any casing, therefore wind flow behind the turbine is affected by the parallel flow with higher kinetic energy.

The turbomachines can be classified by type of conversion energy on hydraulic and heat machines. Inside hydraulic turbomachines is not any change density of the working fluid during energy transformation (ρ≈const.). Inside heat turbomachines is changed density of the working fluid during energy transformation .

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General classification and application of turbomachines

There are many types and ways use of turbomachines, therefore this subchapter shows only their base applications.

Rotodynamic pumps are machines for transmission and increase pressure of a working liquids. Rotodynamic pumps can be subdivided to classes by work conditions on circulation pumps, pumps for pumping of a condensate and feed pumps. Circulating pumps are used for circulating of the working liquid in loops, they compensate a pressure drop inside a pipe. An increase of the working liquid energy inside the circulation pump is probably 100 J·kg-1. A power of circulation pumps can be up a few MW (main circulating pump of a nuclear power plant):

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The circulation pump and an example its application.
Figure 4. The circulation pump and an example its application.
a the heat exchanger; b the consumer of heat; c the circulation pump.
This figure shows small centrifugal pump with integrated motor. The liquid flows through the impeller from the center to its perimeter under the centrifugal forces. The working liquid exits from the impeller to the spiral casing and its to the exit.

Condensate pump are used for pumping of the working liquid near its saturating (for an example a condensate). The transmitted energy to the working liquid inside these pumps is higher than in case of circulation pumps, because the condensate usually is pumped to higher pressure (500 J·kg-1 for the case of water).

For feed pumps is typical the pumping of the working liquid to high pressures. The transmitted energy to the working liquid inside feed pumps is approximately several tens of kJ·kg-1:

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A simplified section view through a multi-stage feedwater centrifugal pump.
Figure 5. A simplified section view through a multi-stage feedwater centrifugal pump.
In order to transfer to the working liquid so much energy is needed several impellers behind (multi-stage turbomachine).
Photo from [2].

Water wheels are machines able to convert potential and kinetic energy of water:

- zl. 294 -
An overshot, a breastshot and an undershot water wheel.
Figure 6. An overshot, a breastshot and an undershot water wheel.
c1 [m·s-1] the velocity of water stream in front of the wheel; c2 [m·s-1] the velocity of water stream behind the wheel.
Water wheels transform kinetic or potential energy of water stream to work.
An overshot water wheel cannot be considered purely the turbomachine.
More information in [3, 4].

Overshot water wheels exploits water potential gradient (this water potential gradient is a function of wheel diameter) and kinetic energy of water stream in the head race. Undershot water wheel exploits of kinetic energy of water stream only. This kinetic energy is very low (cca 3 to 5 J·kg-1), therefore for higher power is necessary bigger mass flow rate of water.

At the moment, for a processing of water potential gradient are used water turbines, which can have much more power output than the water wheels. The most used are three types of water turbines: the Pelton turbine, the Francis turbine and the Kaplan turbine. The water turbine need at least small water potential gradient for its function, if one is not turbine of a tidal power plant.

In a case of the Pelton turbine is transformed first potential energy of water on kinetic energy inside a nozzle (high velocity of water stream in output from nozzle). The water stream drives the runner of the Pelton turbine during touch with its blades, where is transformed the kinetic energy of the water stream to the work (principle of action and reaction).

The Francis and the Kaplan turbine are similar between themselves. In front of the quide vanes is pressure of water function of water potential gradient. Inside the quide vanes is increase of water velocity (it is through decrease of its flow area, which is formed blade-to-blade passages) and decrease of pressure of water. The water stream is entering to the blade-to-blade passages of the rotating runner. The blades inside quide vanes are able move round, this function enables a regulating of power output. The rotor blades of the Kaplan turbines are rotatingable also (unlike as Francis turbines). The water turbines are almost the most power output turbomachines with power output to 1 000 MW.

- zl. 295 -
A rotor of the Kaplan turbine (the runner).
Figure 7. A rotor of the Kaplan turbine (the runner).
The blade-to-blade passages are very well visible.
The rotor of the Kaplan turbine from the Orlík Dam (Czech republic), made in ČKD Blansko.

A typical feature of heat turbines is expansion of the working gas and a decrease its temperature. The most used types of the heat turbines are steam and gas turbines. On next figure is a section view through one-stage steam turbine (the Laval turbine) for purpose of the description of the heat turbine function. The steam expands from the state 0 to the state 1 through the Laval nozzle (stator). In this Laval nozzle is enthalpy transformed to kinetic energy (the velocity of steam is c1). This steam stream enters to the blade-to-blade passages of the rotor, where kinetic energy of steam is transformed to the work. Kinetic energy of the steam is less behind the rotor than in front of the rotor (the velocity of steam is c2), the difference of kinetic energy is the work shaft:

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A simplified section view through the Laval turbine (the steam turbine).
Figure 8. A simplified section view through the Laval turbine (the steam turbine).
a the nozzle (the guide vanes disk usually has several nozzles for higher mass flow rate and power); b the rotor; c the exit flange; d the gearbox; e the generator; f the direction of rotating.
0 the input of the steam; 1 the gap between the quide vanes and the rotor; 2 the exit of steam from the rotor; 3 the steam exit.
p [Pa] the pressure.
The Laval turbine is high-speed turbine with the convergent-divergent nozzles, which allows the supersonic velocity of the steam.

For higher power output are made multi-stage steam turbines. One stage contents: one row of stator blades, which forms nozzle row (the nozzle need not be only one, but the blades of stator may forms a few nozzles sorted on periphery of the rotor) and one row of rotor blades:

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A simplified longitudinal section view through the multi-stage steam turbine.
Figure 9. A simplified longitudinal section view through the multi-stage steam turbine.
S the stator blade row; R the rotor blade row.
The stator blades are fastened to the turbine casing, the rotor blades are fastened to the rotor. One row of the stator blades together with one row of the rotor blades is called the turbomachine stage.
The steam turbine 6 MW, 9 980 min-1, the admission steam 36,6 bar, 437°C, the exhaust steam 6,2 bar (for next used). Made in Alstom (factory PBS–CZ).
Information source: [7].

The working fluid in steam turbines is steam (water steam most often). The steam turbines have a very wide use: the steam power plants (in coal or nuclear power plants), industry etc.

The steam turbines with high power output are composed of several smaller turbines, which are arranged on shared shaft (together shaft may not for all cases) connected by couplings. These turbines are called multi–casing turbine:

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The multi-casing steam turbine (Temelin nuclear power plant cz).
Figure 10. The multi-casing steam turbine (Temelin nuclear power plant cz).
4 casing (1x high-pressure casing, 3x low-pressure casing). The last casing of turbine is closed. The length of turboset is 63 m it means the leght including generator, the length of the turbine rotor is 59,035 m and weighs 326,4 t. Made in Škoda (cz).
Photo from [8].

Other information about steam turbines are in the article Heat turbine and rotodynamic compressor.

The working fluid of Gas turbines is a gas or a combustion products. The gas turbines are most often used with the combustion chambers (therefore are called also combustion turbines). The combustion turbines contents a turbine section, a rotodynamic compressor section and a combustion chamber. Inside of the rotodynamic compressor is compressed the intake air. Inside the combustion chamber is run burning of the fuel and the air. During combustion are being created hot combustion products, which are feeded of the turbine section. The power output of the turbine section is consumed by the rotodynamic compressor section (bigger portion of the power output of the turbine section) and by the electric generator or other device. Power outputs of combustion turbines are from 30 kW (mikroturbines) to 500 MW. For combustion turbines is typical simplicity, because as a fuel is used natural gas or other fossil fuels, which are combusted inside the combustion chamber:

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The combustion turbines for industrial use.
Figure 11. The combustion turbines for industrial use.
a the air input; b the compressor; c the combustion chambers; d the turbine; e the exhaust.
Made in GE; 9F series; power output 300 MW.
Figure from [9], edited by author.

Combustion turbines are used for drive of jet engines. In this case is the power output of the turbine section equal to the power input of the rotodynamic compressor and surplus of enthalpy gradient inside the combustion products is used for expansion in a nozzle and it does a thrust of the jet engine. Gas turbines are used for drive a blower of internal combustion engines (the set of the gas turbine-blower is called the turbocharger Figure 2.). In this case the hot combustion products from exhaust of the internal combustion engine feeds the turbocharger, which compresses the air for the internal combustion engine see simple scheme connection of a turbocharger with an internal combustion engine in article Heat turbine and rotodynamic compressor.

Other information about gas turbines are in the article Heat turbine and rotodynamic compressor.

Rotodynamic compressors are turbomachines for compress of gases and steams. The Blade-to-blade passages inside the rotodynamic compressor forms diffusers, in which is transformed kinetic energy to enthalpy (an increasing of pressure and temperature). For higher compress are used multi-stage rotodynamic compressors:

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A simplified longitudinal section view through the multi-stage axial compressor.
Figure 12. A simplified longitudinal section view through an multi-stage axial compressor.
Photo: GE multi-stage axial compressor [10].

Other information about the rotodynamic compressors are in the article Heat turbine and rotodynamic compressor.

Fans are used for the transport of the gases (most often air) and smaller increasing of pressure (change of density is negligible). Increasing the pressure inside the fans is from 0 to 1 kPa (low pressure), to 3 kPa (middle pressure), to 6 kPa and higher (high pressure). The fans have wide use in industry and in households.

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A simplified section views through the radial–flow fan with forward curved vanes.
Figure 13. A simplified section views through the radial–flow fan with forward curved vanes.
b [m] width of impeller; h [m] width of spiral casing.
In this case the fan contents the impeller and the spiral casing. Inside impeller is increasing the velocity of the working gas.
Photo: Ebmpapst radial–flow fan [11].

Wind turbines are turbomachines without housing as airplane propellers or marine screw propeller. The change of specific energy of wind inside the wind turbine is about 100 J·kg-1.

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A rotor of the wind power plant.
Figure 14. A rotor of the wind power plant.
R [m] the rotor radius.
Vestas wind turbine V90, hub height 105 m; rotor diameter 90 m, nominal output 2 MW, locality Drahany (cz).

Other information about the wind turbines are in the article Use of wind energy.

Difference between piston engine and turbomachine

The working fluid flows through the turbomachine continuously, in the case a volume (reciprocating, piston) machine the working fluid is closed inside volume of the machine (the working volume). The working volume is formed by walls of machine part (piston, cylinder, head..) where at least one wall is moveable (the piston). In the case the working fluid does work, then the working volume of the volume machine is increasing. In the case the working fluid consumes work, then the working volume of the volume machine is decreasing. The work of the volume machine is transported through a movement of the piston (a piston engine; a piston compressor; a piston pump, a Wankel engine, a gear pump...):

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A piston engine versus a turbomachine.
Figure 15. A piston engine versus a turbomachine.
The figure shows a difference between principle of the steam piston engine (see article Steam piston engine) and the Laval turbine.
a the working fluid (steam).
c [m·s-1] the velocity of steam; F [N] force.
Subscript ok denotes surroundings.

There are a wide number of criteria for select between the turbomachine and the volume machine. The most significant can be power, weight, consumption (efficiency), reliability, frequency of maintenance, vibration, emission..., there are others criteria out technical criteria such as machine availability on the market, price and rate of return on investment atc. The efficiency of machine can be considered significant technical criterion. For the volume machines is characteristic higher efficiency at small power in several tens and hundreds kilowatts, opposite the turbomachines have higher efficiency than the volume machines at higher power:

- zl. 928 -
The comparison of the efficiencies of the volume machines and the turbomachines.
Figure 16. The comparison of the efficiencies of the volume machines and the turbomachines.
P [W] the power output of a machine; Q [W] a power input at the fuels; η [-] the efficiency of a machine; X [W] the point of the start higher efficiency of the turbomachine than the efficiency of the volume machine.
The index O denotes the volume machine, the index L denotes the turbomachine.
The point X is located 100..500 kW of the power output for case of the steam piston engine and steam turbine at this power output interval can be the efficiency of the steam turbine higher than the efficiency of steam piston engine. The reciprocating combustion engine and the combustion turbine is located the point X approximately the same as for the steam piston engine and the steam turbine.

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Classification of turbomachinery according to stream direction

The classification of turbomachines by a stream direction in relation with the axis of a shaft (meridional direction) informs about design of the machine:

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The classification of turbomachines by the stream direction in relation with the axis of a shaft.
Figure 17. The classification of turbomachines by the stream direction in relation with the axis of a shaft.
a to d are pumps, compressors or fans; e to j are turbines.
a axial–flow; b radial–with axial inlet; c mixed–flow; d radial–flow (centrifugal); e axial–flow; f radial–with axial outlet; g mixed–flow; h radial–flow (in case with alternate rotors rotating opposite); i radial–flow (centripetal); j tangential– (Pelton turbine).

Type of turbomachine according to stream direction is usually chosen through an assumption its specific speed and working conditions.

Construction features of turbomachines

Parts of turbomachines are different by type of the turbomachine. Nevertheless can be identified common construction features of the turbomachines:

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Main construction features of the turbomachines.
Figure 18. Main construction features of the turbomachines.
The most of the turbomachines contain an inlet section-inlet branch (the working fluid enters to the machine); an exit section–exit branche (the working fluid exits from the machine); blades/vanes (rotor blades, stator blades); the shaft; the turbomachine casing; shaft bearings. The turbomachines usually contains a regulate of quality and quantity of the working fluid; an oil system, etc.
Kaplan turbine: 1 inlet of water to the turbine through the spiral casing; 2 stator blades–are rotating for a regulation of mass flow rate; 3 rotor blades–are rotating–for a regulation of efficiency; 4 suction pipe–exit section; 5 radial bearing–absorbs of forces which are perpendicular to the axis of rotation; 6 axial bearing–absorbs of forces which are parallel to the axis of rotation.

Blade, blade passage and blade cascade

Blades are usually made separately. The blades are built into the stator and the rotor and they forms the row blade-to-blade passages (blade cascade) with a required size. Some turbomachines contain the move round blades (these blades enable a change the size of the flow area of the passages or close of the flow) for an example the Kaplan turbines.

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The blade and the blade cascade (the Laval turbine).
Figure 19. The blade and the blade cascade (the Laval turbine).
a blade; b the blade passages are formed by the blades; c turbine rotor with blades; d the cylindrical sectional view through the blade passages at the radius r (blade cascade); e shroud (may not always be).
u [m·s-1] circumference velocity at the radius r; s [m] pitch; ØD [m] mean diameter of the blade length.

The flow area of the blade-to-blade passage is function the radius of the cylindrical sectional view as it is shown on the Figure. In the case the blades are short in relation with the radius of the rotor then the variation size of the area flow is not significant, this type of blade is called straight blade. For higher efficiency are used so-called twisted blades (the variations size and the shape with radius, for an example Figure 7; Figure 12; Figure 14). Straight blades (same the shape with length of the blade) are usually used as stator blades for hydraulic machines (Figure 18) or for the cases heat turbomachines with short blades (Figure 19).

The blade-to-blade passage is bordered by the shroud on the end of blades (Figure 19), or the cylindrical surface of the casing and the rotor (Figure 12). The blade-to-blade passage for the case of radial machines is bordered by disk disk of the rotor or the stator (Figure 13).

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The basic terminology of the blade.
Figure 20. The basic terminology of the blade.
NH leading edge; OH trailing edge; SS suction surface; PS pressure surface (the term is derived from a variation of pressure on the blade profile).

Power output/input of turbomachine

The significant parameter of the turbomachine is its internal power output/input. The internal power output is a power of the working fluid which flow through the turbomachine:

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The power output/input of a turbomachine.
Equation 1. The power output/input of the turbomachine.
Pi [W] the power output/input is transfered power between the working fluid and the rotor inside turbomachine; ai [J·kg-1] the specific internal work of the turbomachine (transfered energy between the working fluid and the rotor); m [kg·s-1] the mass flow rate through the turbomachine.
The term "output" is used for cases turbines (the working fluid generates of work), the term "input" is used for pumps/compressors (the working fluid consumes of work). If the working fluid consumes work, then ai and Pi inside turbomachine be negative, but this sign "negative" is not usually used and is used term "input".

Enthalpy, kinetic energy and potential energy of the working fluid can change during flow through the turbomachine. The working fluid temperature inside of the turbomachine can change (the working fluid can be interaction of heat through a border of the control volume-walls of the turbomachine; or heat produced inside the control volume, for an example chemical reaction in the volume of the working fluid). For calculation of the specific internal work of the turbomachine is necessary thus used the equation of the First law of thermodynamics for open system. This equation take into account the all these forms of energy.

The equation for First law of thermodynamics for open system can be simplified with species of the working fluids and the type of the turbomachine. For an example: for a case ideal liquid (hydraulic machine) can be derived this equation:

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Specific work of the working fluid inside hydraulic turbomachine (ρ=constant; working fluid is liquid) and a change of specific total energy of the working fluid (without losses and with losses).
Equation 3. Specific work of the working fluid inside hydraulic turbomachine (ρ=constant; working fluid is liquid) and a change of specific total energy of the working fluid (without losses and with losses).
ρ [kg·m-3] density of the liquid; Yi, e [J·kg-1] specific total energy of the liquid at the inlet and at exit; ΔY [J·kg-1] change of specific total energy of the liquid between the inlet and the exit; ΔYz [J·kg-1] specific losses arise during flowing of the liquid between the inlet and the exit; q [J·kg-1] specific heat of the working fluid transfer with the surroundings; H [m] the level of the inlet / the exit flange.
The index i denotes the inlet, index e denotes the exit*.
This equation is called Bernoulli equation for incompressible flow.
The change of internal energy of the working liquid is considered to be a loss for hydraulic turbomachinery (it be reducing the work of the liquid). The change of internal energy of the working liquid arise during the flowing (the usable energy is transformed to the heat, which can not use in hydraulic turbomachine). External heat transfer inside hydraulic turbomachine only increases the internal energy of liquid and does not affected to the work machine.

Remark
For description energy equations of the hydraulic machines is usual marking of specific energy by letter Y.

In a case of heat machines can be simplify of the equation for First law of thermodynamics for open system to the form:

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Specific internal work of the turbomachine (a working fluid is the gas).
Equation 4. Specific internal work of the turbomachine (the working fluid is the gas).
Assumptions solution of this equation is: negligible influence of potential energy of the working gas.
More information about energy balance of of heat turbomachines are shown for cases adiabatic processes (q=0) in chapters Energy balance of heat turbine[13] and Energy balance of rotodynamic compressor[13] and for cases polytropic processes (q≠0) in chapters Expansion inside of turbomachine at heat transfer with surroundings[TT14] and Compression inside of turbomachine at heat transfer with surroundings[TT14].

The Equations 3. and the Equations 4. can be use for a simple calculation of basic parameters of a turbomachine:

- zl. 545 -
20 t·h-1 of water is pumped from a lower tank to a higher tank through rotodynamic pump. Pressure is 1 bar in the lower tank, the pressure is 40 bar in the higher tank. The height difference between levels in tanks is 7 m. What is approximate power input of the pump?
Problem 1.

ai  [J·kg-1] -4168,7
Pi  [W]       23159 
Results of Problem 1.

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Steam enters to a steam turbine at 36,6 bar and 437 °C. The exit pressure is 6,2 bar. Find the specific internal work of this steam turbine.
Problem 2.

ii  [kJ·kg-1·K]  3306,04
ie  [kJ·kg-1·K]  2845,51
ai  [kJ·kg-1]    460,53 
Results of Problem 2.

Turbomachine stage

The turbomachine stage contains the stator (stator blade row) and the rotor (rotor blade row):

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The definition of the turbomachine stage (for an example reversible Francis turbine).
Figure 21. The definition of the turbomachine stage (for an example reversible Francis turbine).
Total energy of the working fluid can be transformed in work only inside rotor, therefore for states of the working fluid are used subscript 1 in front of the rotor blade row and subscript 2 behind the rotor blade row. For a case turbines is used subscript 0 in front of the stator blade row. For a case pumps, compressors etc. is used subscript 3 behind the stator blade row (because the stator blade row is in front of the rotor blade row):

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The denote of the working fluid state for a case multi-stage turbomachine (longitudinal section view).
Figure 22. The denote of the working fluid state for the case multi-stage turbomachine (longitudinal section view).
a turbine stage; b compressor stage.

In a case of adiabatic process, a change of sum of energy of the working fluid is being done in blade passage of the rotor only (input/output work). The sum of energy of the working fluid is constant in blade passage of the stator.

In a case of hydraulic machines, the energy equilibrium can be derived from the Equation 3. for stator (individual types of energy can be transformed between themselves, but their sum is constant and reduced by losses):

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The energy balance equation of the stator blade row of the hydraulic turbomachine stator (derived from the equation 3, zl. 543 for a=0).
Equation 5. The energy balance equation of stator blade row of the hydraulic turbomachine stator (derived from the Equation 3. for a=0).
ΔYz,stator [J·kg-1] specific losses arise during flowing of liquid through blade passages of the stator; 0 state of the working liquid in front of the stator row; 1 state of the working liquid behind the stator row.

In a case of heat machines, the equilibrium of stagnation enthalpy can be derived from the Equation 4.:

- zl. 547 -
The energy balance equation of the stator blade row of the heat turbomachine.
Equation 6. The energy balance equation of the stator blade row of the heat turbomachine.
q1-2 [J·kg-1] heat transffered to working gas in stator.
Derived from Equation 4.

The sum of energy of the working fluid is changed inside rotor blade row. In a case of turbines, energy is extracted from the working fluid (sum of energy at he exit is lesser than at the inlet of the rotor blade row). In a case, pump/compressor, energy is consumed by the working fluid (sum of energy at the inlet is lesser than at the exit of the rotor blade row). Inside rotor blade row is true a≠0 from the equation of First law thermodynamics for open system.

Velocity triangle

The rotor of turbomachine is a rotating mechanism. The blade passages of the rotor rotates around the axis of the rotor. The working fluid flows with the velocity c1 to these passages and flows with the velocity c2 from these passages.

The velocity of the working fluid c is called absolute and it has three spatial components. The component of the absolute velocity in direction of the axis is called an axial component and it is denoted by index a. The component of the absolute velocity in direction of the rotating is called a circumference/tangential component and it is denoted by index u. The component of the absolute velocity in direction of a perpendicular on the axis is called radial component and it is denoted by index r:

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Absolute velocity and its components.
Figure 23. Absolute velocity and its components.
c absolute velocity of flow; a axial direction; u tangential direction; r radial direction.

The absolute velocity of the working fluid c is a vector summation of a relative velocity w and a circumference velocity u (the blade velocity). The relative velocity w is velocity of flow, which measured with respect to the rotating system (the move of the observer is with the rotating system). The relative velocity has three spatial components as absolute velocity:

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The explanation of the relative velocity.
Figure 24. The explanation of the relative velocity.
A the cyclist; B the stationary observer.
c [m·s-1] the absolute velocity of wind; v the velocity of the cyclist; w [m·s-1] the velocity of wind respect to the cyclist, this velocity is called relative velocity of wind.

The circumference velocity is function of the rotating radius r and the angular velocity ω. It has not any components in axial and radial direction as the absolute velocity. The circumference velocity lies in the plane which is perpendicular on the axial and the radial direction:

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The circumference velocity of rotor.
Equation 7. The circumference velocity of the rotor.

A scheme, which shows of the absolute, relative and circumference velocity of the working fluid is called the velocity triangle:

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The velocity triangle of the Laval turbine.
Figure 25. The velocity triangle of the Laval turbine.
The working gas (steam) flowing with the velocity c1to the rotor blade passages and flowing from rotor blade passages with the velocity c2.

The velocity triangle is being usually portrayed separately from the picture of blade row (for better a overview and need of the calculations):

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The dimensioning of the angles of the velocity triangle (for a case the axial stage).
Figure 26. The dimensioning of the angles of the velocity triangle (for a case the axial stage).
α [deg] the angle of the absolute velocity; β [deg] the angle of the relative velocity .
The inlet and the exit velocity triangle is portrayed in the meridian plane. The positive direction any components of the velocities is at the direction of the circumference velocity. The angle between the absolute velocity and the circumference velocity is denoted by α. The angle between the relative velocity and the circumference velocity is denoted by β. The angles are dimensioned counterclockwise (so there is no need to examine to the positive direction of the velocity), there are others ways of the angle dimensioning for an example [1, p. 26 (CZ)].

For design of the turbomachine stage is first, the velocity triangle is computed, which is the base for the design of the blades. There are three possible procedures for the calculation of the turbomachine stage:

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(1) 1D calculation of main streamline only (the mean diameter of the stage).
(2) 2D calculation of several streamlines (several diameters along          
    length of the blade).                                                   
(3) 3D calculation whole volume of stage (Finite element methods; CFD).     
List 1. Basic calculation methods of the turbomachine stage.
1D calculation of the turbomachine stage
In this case is used a many simplifications to simplify of the calculation. This method of calculation is used for the turbomachine stages with negligible influence of spatial character of the stream (the change of the velocity triangle along the length of the blade is negligible) or for an approximate calculation. The mean diameter or a quadratic diameter (flow area of the stage is the same over the diameter and under the diameter) is used as the reference diameter. The Figure 17. shown these reference streamlines. This type of calculation is described in the article Design of turbomachine stage with negligible influence of spatial character of stream.
2D calculation of the turbomachine stage
This method is the same as previous method with the difference, that the calculation of the velocity triangle is being performed on several diameters along the length of the blade. This method is used for calculations of turbomachine stages with a big influence on spatial character of flow inside stage (twist blades). This calculation is described in the article Design of turbomachine stage with taking into account spatial character of flow.
3D calculation of the turbomachine stage
The completely numerically calculation of the turbomachine stage using advanced CFD software. This method usually taking into account changes of the velocity triangle near blade profiles (influences of boundary layer). Before use the 3D calculation is known approximate geometry of the turbomachine stage from the 1D or the 2D calculation.

Turbomachine Losses

Inside of the turbomachine occurs losses, which influence its power output. There is an friction inside the stream of the working fluid and on the surface of the parts machine. The working fluid can be flows from the working volumes through seals and others gaps and etc. Others losses are in mechanical parts of the turbomachine (mechanical losses). The Losses are usually increased during no-normal state of the turbomachine*. The turbomachine losses is possible subdivided to the classes, which are influenced between them:

- zl. 550 -
(1) Mechanical losses (friction between mechanical parts of machine).  
(2) Aerodynamic losses (change of forces on blade profiles during flowing    
    fluid).                                                            
(3) Energy losses (decreasing of enthalpy /pressure/ potential gradient
    and kinetic energy of the working fluid).                          
(4) Losses is caused by change properties of working fluid (for example
    condensation during steam expansion).                              
(5) Losses is caused by leakages (there is internal /leakage between   
    stages/ and external /escape working fluid outside machine/)       
List 2. AThe classification of the turbomachine losses.

*Remark
The normal state is a state of the working fluid (pressure, gradient, temperature, density...), for which the turbomachine designed to operate as efficiently as possible.

At the calculation start of the turbomachine or its parts the losses usually need estimate (because the geometry of the machine is not know). At the calculation end, these estimates are checked by control calculations. If the results of the control calculations are not same as the estimates (they are outside required interval), then new calculation an estimate of losses is necessary.

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Questions for study

(1)  Describe methods of calculation of the turbomachine stages         
     (3 methods).                                                       

(2)  Base classification of the turbomachine losses.                    

References

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Citation this article

This document is English version of original in Czech language: ŠKORPÍK, Jiří. Lopatkový stroj, Transformační technologie, 2009-08, [last updated 2014-02]. Brno: Jiří Škorpík, [online] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/lopatkovy-stroj.html. English version: Turbomachine. Web: http://www.transformacni-technologie.cz/en_lopatkovy-stroj.html.

©Jiří Škorpík, [LICENCE]