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This article follows the article 12. Essential equations of turbomachines and 13. Energy balances of turbomachines. In these articles are defined quantities a specific shaft work **l _{u}** and a specific internal work of turbomachine stage

The specific internal work of stage is corresponded work of the working fluid inside stage and it is calculated from difference of stagnation states between the inlet and the exit of stage. The specific work shaft is corresponded work of the working fluid transformed to the torque of the shaft and it is calculated from a velocity triangels in front of and behind the rotor and from a friction between the rotor and the working fluid so called __rotor friction losses__:

The specific shaft work of real machines is influenced by friction between the rotor and the working fluid so called __rotor friction losses__:

1.318 Difference between work shaft and internal work of stage.HST volume of stage; r [m] radius; l [J·kg_{u}^{-1}] specific work shaft of stage on radius r^{1}; c [m·s^{-1}] absolute velocity; w [m·s^{-1}] relative velocity; u [m·s^{-1}] circumference velocity; a [J·kg_{r}^{-1}] rotor friction losses of stage^{2}; l [J·kg_{E}^{-1}] specific work of working fluid during flow through rotor blade row without rotor friction losses; a [J·kg_{i}^{-1}] specific internal work of stage; A, B areas of development friction losses under friction between rotor and working fuid. S stator blade row; R rotor blade row. |

^{1}Remark- The specific shaft work can be changed along the lenght of blade. Also the pressure is changed along the lenght of blade under pressure gradient.

^{2}Rotor friction losses- The friction is function of a construction of the stage, it is the biggest in stages with a disc rotor (e.g. the one-stage Laval turbine or radial stages), this friction is usually negligible for stages with
__drum rotors__.

The rotor friction losses of the stage is a consumed work, which is transformed on heat. This friction heat heatings the working fluid in surrounding:

From equation of specific internal work of turbomachine is evident heat *δ·a _{r}* increases reajected heat to surroundings and heat

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For turbine stages be can constructed i-s diagram according the chapters 13. Adiabatic expansion inside heat turbine, 13. Polytropic expansion inside heat turbine:

The figure shows expansion inside a reaction axial stage on a tested radius *r*. **(a)** case for *a*_{r}<<l_{E}-negligible influence of rotor friction losses^{3}; **(b)** case for *a*_{r}>0. **i** [J·kg^{-1}] specific __enthalpy__; **s** [J·kg^{-1}·K^{-1}] specific __entropy__; **p** [Pa] pressure; **z**_{p} [J·kg^{-1}] specific blade __profile loss__ (friction between working fluid and surface of blades). Subscript **iz** denotes the state of the working fluid at exit of the blade rows for case __isentropic process__, subscript denotes **c** stagnation state.

For the indication of the kinetic energy of the relative velocities in the i-s diagram can to help equation for the specific shaft work. From this equation is evident, the sum of the specific shaft work

For stages of working machines be can constructed i-s diagram according the chapters 13. Adiabatic compression inside compressor, 13. Polytropic compression inside compressor:

4.719 i-s diagram of compressor stage on a tested radius r. |

Here descripted energy balances of the stages assumed the working fluid flow only through blade passage at development only the profile losses and rotor friction losses without the other losses of the stage. But the turbomachine stage is a classically engineering product, which usually is not perfectly, therefore there are other losses^{4} inside stages also e.g. a portion of the working fluid can flow outside the blade passages (leaks, construction gap) etc.:

5.1089 Example of flow through a leak of a turbine stage. |

^{4}Other losses of the stage- These losses are depend by type of a construction of the stage and its quality (inside one stage can be a few types of the other losses). More information is shown in the article
__17. Losses in turbomachines__.

The internal work is possible calculated by a simple equation if the rotor friction losses is negligible in relation to the other losses:

6.361 The specific internal work of the stage on a tested radius r.∑z [J·kg_{ost}^{-1}] sum of specific other losses of stage. |

Total energy balance of the stage in i-s diagram shows all losses, enthalpy and the work of the working fluid at the exit for case ideal mixing:

7.319 The specific internal work of a turbomachine stage on a tested radius r.z [J·kg_{st}^{-1}] total losses of stage. The other losses of stage increases enthalpy from state 2 on the state 2' at the exit. |

If the other losses influences state of the working fluid inside core flow at exit first blade row then it can be influenced proces inside second blade row:

8.947 Influence the other losses on the work shaft.z [J·kg_{ns}^{-1}] specific loss through leaks of stator. This is case from Figure 5. The flow inside core of stage is influenced by leaks on the stator blade row in this case. The working fluid from seal of the stator increases enthalpy at the inlet rotor blade row. |

The shape of i-s diagram of flow through blade row of the stage corresponds to construction of stage. The i-s diagrams axial and diagonal stages are shown in article __19. Design of axials turbomachine stages__, and in article __20. Design of radials turbomachine stages__ are shown i-s diagrams of radial stages.

For cases turbomachines without casing the flow mass of the working fluid beside the stream-tube of rotor are not included to the other losses *z _{ost}*.

Similar such as two different work of the stage are defined also two basic efficiencies of the stage:

The value of the coefficient

The value of the coefficient

For cases stages of working machines is usually defined the effective efficiency respectively isentropic efficiency of the stage. These efficiency are connected with static state of the working fliud at isentropic processes:

The difference of the enthalpy of steam inside one stage of a steam turbine for isentropic process is *21,3 kJ·kg*^{-1}, the velocity of steam at the inlet to the stage is *147,47 m·s*^{-1}, the velocity of steam at the exit of the stage is the same. The calculated blade profile losses of the stator blade row are *1,6985 kJ·kg*^{-1}, the calculated blade profile losses of the rotor blade row are *1,6985 kJ·kg*^{-1} (the rotor blade row is geometrically same as the stator blade row). The calculated other losses of the stage are *1,6806 kJ·kg*^{-1}. Calculate work shaft efficiency and the internal efficiency of this stage. This stage is the first stage of a multi-stage steam turbine.

**Problem 1.**923

lE [kJ·kg-1] 17,9030 e0 [kJ·kg-1] 21,3000 ai [kJ·kg-1] 16,2224 κ0 [-] 1 ηE [-] 0,8405 ηi [-] 0,7616 κ2 [-] 1 Σzost [kJ·kg-1] 1,6806

The sum of the energy balance all stages and the energy balance of other section of the turbine must be equal the energy balance of the stage section of the turbine as one entire:

11.116 A multistage adiabatic expansion inside turbine.η [-] mean internal efficiency of individual stages; _{j}1+f [-] reheat factor (coefficient of the re-usable heat, 1,02 to 1,04 according [3]); Δ [J·kg^{-1}] re-usable heat of turbine; 1+f [-] reheat factor for case heat turbine with infinite number of stages; _{∞}z [-] numbers of stages. The efficiency of stage section of multi-stage turbines η e.g. multi-stage steam turbines is higher than _{i}η of individual stages. These equation are derived for the same the enthalpy difference for all stages and for adiabatic expansion. For better illustration is not drawn the absolute velocity _{j}c. The derivation of this equation is shown in the Appendix 116. |

It is evident, a portion of the heat from the loss processes inside previous stage is used inside next stage during expansion. Only the heat from the loss processes inside last stage of the heat turbine are not used.

In case multi-stage turbocompressors, the resulting compression is composed of several sub-compression, their number is equal number of the stages inside turbocompressor:

12.121 An multistage compression inside turbocompressor.1+f [-] preheat factor (coefficient of additional losses); Δ [-] additional losses of one stage; _{j}1+f [-] preheat factor in theoretical case turbocompressor with infinite number of stages. The efficiency _{∞}η of multi-stage turbocompressor is less than _{i}η. These equation are derived for the same the enthalpy difference for all stages and for adibatic compresion. For better illustration is not drawn the absolute velocity _{j}c. The derivation of these equations is shown in the Appendix 121. |

It is evident, the internal losses of one turbocompressor stage reduces the internal efficiency of next stage.

The air with temperature *15 °C* and pressure *0,1013 MPa* enters to a turbocompressor, on exit of the turbocompressor the air has temperature *293 °C* and pressure *0,802 MPa*. Calculate work *a*_{i}, *a*_{iz}, internal efficiency *η*_{i} and factor *1+f*. Number of stages of the turbocompressor is *12*. Use simplification *c*_{p}=const..

**Problem 2.**122

This problem is published in [4].

This problem is published in [4].

ai [kJ·kg-1] -279,251 aiz [kJ·kg-1] -233,2122 ηi [-] 0,8355 1+f [-] 1,0437 |
Problem 2: results. |

(1) Define the velocity coefficient of the turbomachine stage. (2) What is there difference between the specific shaft work and the specific internal work of the turbomachine stage?

- KADRNOŽKA, Jaroslav.
*Tepelné turbíny a turbokompresory I*, 2004. 1. vydání. Brno: Akademické nakladatelství CERM, s.r.o., ISBN 80-7204-346-3.

ŠKORPÍK, Jiří. Vztah mezi obvodovou a vnitřní prací stupně lopatkového stroje, *Transformační technologie*, 2009-10, [last updated 2017-01-30]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/14.html. English version: Relation between specific shaft work and internal work of turbomachine stage. Web: http://www.transformacni-technologie.cz/en_14.html.

©Jiří Škorpík, LICENCE

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