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.
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·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.
|4.id719 i-s diagram of compressor 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.|
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 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.
Similar such as two different work of the stage are defined also two basic efficiencies of the stage:
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:
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] 1,6806Problem 1: results.
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:
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).
(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?
ŠKORPÍK, Jiří. Vztah mezi obvodovou a vnitřní prací stupně lopatkového stroje, Transformační technologie, 2009-10, [last updated 2016-02-07]. Brno: Jiří Škorpík, [on-line] pokračující zdroj, ISSN 1804-8293. Dostupné z http://www.transformacni-technologie.cz/vztah-mezi-obvodovou-a-vnitrni-praci-stupne-lopatkoveho-stroje.html. English version: Relation between specific shaft work and internal work of turbomachine stage. Web: http://www.transformacni-technologie.cz/en_vztah-mezi-obvodovou-a-vnitrni-praci-stupne-lopatkoveho-stroje.html.