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# 14. Relation between specific shaft work and internal work of turbomachine stage

Author: Jiří Škorpík, skorpik@fme.vutbr.cz : updated 2016-02-07

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 lu and a specific internal work of turbomachine stage ai.

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.id318 Difference between work shaft and internal work of stage.HST volume of stage; r [m] radius; lu [J·kg-1] specific work shaft of stage on radius r; c [m·s-1] absolute velocity; w [m·s-1] relative velocity; u [m·s-1] circumference velocity; ar [J·kg-1] rotor friction losses of stage*; lE [J·kg-1] specific work of working fluid during flow through rotor blade row without rotor friction losses; ai [J·kg-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.
*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:

2.id934 Subdividing of heat flow from rotor friction.
δ [-] coefficient subdividing of heat flow from rotor friction; δ·ar [J·kg-1] heat flow to part of machine (heat extraction to surrounding); (1-δ)ar [J·kg-1] heat flow to working fluid.

From equation of specific internal work of turbomachine is evident heat δ·ar increases reajected heat to surroundings and heat (1-δ)ar increases internal heat at the exit of stage ue respective enthalpy ie. This decreases the specific internal work and the specific shaft work same, therefore is true lu=ai.

## i-s diagram of stage with taking into account rotor friction losses

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

3.id936 i-s diagram of heat turbine stage.
The figure shows expansion inside a reaction axial stage on a tested radius r. (a) case for ar<<lE-negligible influence of rotor friction losses; (b) case for ar>0. i [J·kg-1] specific enthalpy; s [J·kg-1·K-1] specific entropy; p [Pa] pressure; zp [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.
*Remark
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 lE must be equal to the sum of the kinetic energy on the inlet of the stage and the difference of the kinetic energy of the relative velocities.

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.id719 i-s diagram of compressor stage.

## Other losses of stage

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* inside stages also e.g. a portion of the working fluid can flow outside the blade passages (leaks, construction gap) etc.:

 5.id1089 Example of flow through a leak of a turbine stage.
*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.id361 The specific internal work of the stage on a tested radius r.∑zost [J·kg-1] sum of specific other losses of stage.

## Total energy balance 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. It means the specific shaft work is not plotted, because it can be changed along the length of blade:

 7.id319 The specific internal work of a turbomachine stage.zst [J·kg-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.id947 Influence the other losses on the work shaft.zns [J·kg-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 and diagonals 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 zost.

## Efficiency of stage

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

9.id876 The specific shaft work efficiency and the thermodynamic efficiency of a turbine stage.
ηE [-] specific shaft work efficiency of turbine stage without other losses; e0 [J·kg-1] specific energy of working fluid at inlet of stage; (a) [J·kg-1] portion of specific kinetic energy of inlet velocity of working fluid, which is used inside stage*; (b) [J·kg-1] portion of specific kinetic energy of exit velocity of working fluid, which is used inside next stage**; ηst [-] internal (thermodynamic) efficiency of stage. Approximate is true c2,iz≐c2.
*Remark
The value of the coefficient κ0 is in interval from 0 to 1. Usually it is required κ0=1. The requirement κ0<1 there are if the losses between a measuring point of the velocity c0 (e.g. the exit previous stage) and a beginning of the blade row are not a portion of the losses of the stage. So, the value of the coefficient κ0 is connected with definition of the boundary stage. For example, the boundary of the wind turbine stage a designer can define tightly in front of the rotor then vortex losses, which are developed between the inlet to the stream-tube and the rotor are not included in energy balance of the rotor and the value of the coefficient is κ0<1.
**Remark
The value of the coefficient κ2 is in interval from 0 to 1. For cases multi-stage turbomachines is κ2=1 (in these cases the kinetic energy of the working fluid at the exit of the stage is not considered as a loss), only for case the last of the stages or one-stage machines is used κ2=0.

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:

10.id356 The effective efficiency of working machine stage and isentropic efficiency of a working machine stage.
ηef [-] effective efficiency of working machine stage (usually specific shaft work is without other losses in this case); ηiz [-] isentropic efficiency of stage; ηiz, c [-] isentropic efficiency of stage in relation to stagnation state on exit of stage (for usually case c2=c2, iz is true ηiz, ciz).
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.id923
```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                         ```
Problem 1: results.

## Power output/input of turboset

The power output/input a turboset derives from an efficiency of turboset η, which is calculated as a product of the internal efficiency of the machine, a mechanical efficiency of the machine, efficiency of gearbox (if it is inside the turboset) and an efficiency of generator or a drive:

11.id1027 The efficiency of the turboset and its power output/input.
(a) power output of turboset; (b) power input of turboset. 1 couplings of machine; 2 internal volumes of machine; 3 coupling; 4 převodovka; 4 gearbox; 5 generator/drive. P [W] power output/input of machine; Pi [W] internal power input/output of machine; Psp [W] power on coupling; Ppr [W] power on gearbox; Psv [W] power on contacts of generator/drive; mst [kg s-1] mass flow rate through stage; ηi [-] internal efficiency of machine; ηm [-] mechanical efficiency of the machine (e.g. losses inside bearings); ηpr [-] efficiency of gearbox; ηel [-] efficiency of electric generator/drive; η [-] efficiency of machine.

The parameters of turboset is given on a label of generator. On this label is shown nominal power Pn and optimal power Popt (power under maximal efficiency).

## Questions for study

```(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?               ```

## References

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