The gasdynamic calcualation of the axial turbine stage
The general law of circulation change across blade height. Determination of the axial turbine stages geometrical dimensions. Turbine stage calculation on the middle radius. Cinematic parameters determination on different turbine stage radiuses.
Рубрика  Производство и технологии 
Вид  методичка 
Язык  английский 
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MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE
National Aviation University
The gasdynamic calculation of THE axial turbine stage
Methodical guide for performing the course paper for students specialty 8.100106 “ Manufacturing, maintenance and repair of aircraft and engines”
Compiled by I.I. Gvozdetsky , V.V. Kharyton, S.I.Tkachenko
KYIV 2007
Contents
Introduction
The general law of circulation change across blade height
Determination of the turbine stage geometrical dimensions
Turbine stage calculation on the middle radius
Cinematic parameters determination on different turbine stage radiuses
Appendix 1 The example of gasdynamic calculation of the axial turbine stage
Introduction
The turbine serves to provide the power to drive the compressor and accessories. in a case of turboprop or turboshaft engine the turbine, in addition, provides the power to rotate propeller or rotor. It does this by extracting energy from the hot gases released from the combustion system and expanding them to a lower pressure and temperature. These processes take place when hot gases flow along specially shaped passages created by two rows of airfoils: stator vanes and rotor blades. These two rows of airfoils form a turbine stage. To produce the driving torque , the turbine unit may consist of one or several stages. The useful torque, created by turbine is transmitted to compressor by turbine shaft. Three stage turbine unite assembly is shown in fig.1.
Structurally this turbine unit can be divided into two main parts (fig.2): all rotating components (three bladed disks joined with shaft) are named turbine rotor, and all unmovable components (three turbine nozzle diaphragms and turbine casing) create the turbine stator.
The main objectives of turbine stage gasdynamic calculation are determination of stage geometrical dimensions, gas cinematic parameters and speed plans construction. In course paper cinematic parameters are determined in three sections: sleeve, middle and peripheral.
Stage scheme, sections designation and diametrical dimensions are shown in figure 3.
Fig. 3. Main geometrical dimensions of the turbine stage
The initial data for turbine stage calculation are taken from gasdynamic calculation of the designed engine. They are:
· full gas pressure and stagnated gas temperature at the entry to the turbine stage;
· mass gas flow rate ;
· turbine stage work ;
· circumferential velocity on the middle radius of the working wheel ;
· jet velocity of gas at the exit from the nozzle diaphragm ;
· reduced velocity at the nozzle diaphragm exit ;
· angle of the stream output from the nozzle diaphragm ;
· pressure recovery coefficient in the nozzle diaphragm ;
· external, middle and sleeve diameters at the entry to the working wheel ; ; .
All of these parameters are chosen for the first turbine stage of the designed engine.
T
The general law of circulation change across blade height
The gas work, the reactivity rate, the gas velocity, Mach numbers, efficiency, blade incidence angles and other parameters depend on law of circulation change across stage working wheel radius. Different laws of circulation change across radius are expressed by general equation
, (1)
where ; m  index rate.
If m=1 law of circulation constancy is implemented. This law of profiling is used for comparatively short blades (), because in this case reactivity rate across blade height is changed very essentially. And using long blades the reactivity rate can be negative near sleeve.
For longer blades profiling with index rate m<1 is applied. Particularly, for law of profiling with constant angle of the stream output from nozzle diaphragm is realized.
To obtain small m angle is increased. It causes increase of the axial gas velocity, which can reach local sonic speed at exit from working wheel. It will mean “choking” of the turbine stage. As a result, it is no point in increasing of angle more then on 2025 at first stages. At these values negative reactivity rate can occur near blade root, especially at high values of loading coefficient.
As a result of this, profiling on the base of equation (1) is common, because it gives possibility to avoid negative values of the reactivity rate near the blade root by matching of rate index m at the all values.
Determination of the turbine stage geometrical dimensions
Geometrical dimensions at the entry to the working wheel are determined in the gasdynamic calculation of the designed engine. At first area at the exit from the nozzle diaphragm is calculated
,
where and are stagnated temperature and full pressure of the flow at the exit from the nozzle diaphragm; ; ; mg  constant magnitude, which can be computed by the formula
for kg=1,33 and Rg=288 J/(kgK) we will have mg=0,0396 (kgK)/J.
Relative density can be determined from tables of gasdynamic functions using value of the reduced velocity or by the formula
.
At he given working wheel middle diameter other geometrical dimensions are computed by the following formulas:
; ; .
At he given relative sleeve diameter geometrical dimensions in the considered section are computed by the formulas:
Relative sleeve diameter for first stages is within the limits of , and for last stages .
Calculating first turbine stages the nozzle diaphragm is profiled to provide turbine blending with combustion chamber. In this case meridional profile of the nozzle diaphragm can be of arbitrary shape with the obligatory observance of sections areas.
To calculate section area at the exit from the stage (behind working wheel) it is necessary to compute gas parameters and behind calculated stage.
Stagnated gas temperature is determined from the energy equation:
Full gas pressure behind stage is calculated by the formula
where  stage efficiency.
Axial component of the jet velocity at the exit from the working wheel is assumed on 2080 m/s more then gas velocity at the entry to the working wheel, i.e.
; m/s,
where .
Section area at the exit from the working wheel is determined from following expression:
,
where is computed by the value of
.
Further for chosen profiling law determine main dimensions at the exit from turbine stage in a similar manner as have been done turbine stage entrance.
On the base of computed diameters values draw turbine stage scheme.
Turbine stage calculation on the middle radius
At given circumferential velocity value on the middle radius of the inlet edge calculate circumferential velocity behind working wheel from the relation
Performing approximate calculations it is possible to suppose.
Stage loading coefficient on the middle radius is determined by the formula
For the first turbine stage .
Gas jet velocity at the exit from the nozzle diaphragm is determined from the equation
and reduced velocity  by the formula
Amount of must not exceed 1,25. If it is possible to decrease it by increasing the circumferential velocity , decreasing of the angle of the stream output from nozzle diaphragm , increasing of the stage work, applying of airtwist at the exit from the working wheel in the opposite direction of rotation. If first three methods can not be used, can be obtained from the Euler's equation, have assigned:
,
where .
Value of must not exceed, otherwise it is necessary to decrease to meet this requirement.
As long as all of calculations are carried out for middle radius in what follows we will withdraw subscript “md”. Following formulas flow parameters calculation are legible for every blade section.
Circumferential components of the jet velocity at the entry to the working wheel and behind turbine stage, and parameters , and are calculated by the formulas:
; ; ;
.
Axial component of the jet velocity and parameter at the entry to the working wheel are calculated by the formulas:
; .
Axial component of the relative velocity and the relative velocity at the entry to the working wheel are calculated by the formulas:
; .
Angle of the stream inlet into the working wheel in the relative direction is computed by formula
.
Circumferential component of the relative velocity behind turbine stage is determined on the base of the following expression
axial turbine stage
Axial component of the jet velocity at the exit from the working wheel is assigned such that is on (2080) m/s more then.
Jet velocity and reduced velocity at the exit from the working wheel are calculated by the formulas:
; .
Stream output angle at the exit from the working wheel is calculated by the formula
.
Angle must be close to 90, but for first stages it can be n the range of (7585).
Stream output angle at the exit from the working wheel in the relative direction can be found by the formula
.
Relative velocity at the exit from the working wheel and its circumferential component are determined by the formulas:
; .
Stagnated temperature and critical gas velocity in the relative direction are computed from the equations:
Reduced velocities at the entry to the working wheel and behind turbine stage are calculated by the formulas:
; .
Cinematic reactivity rate is determined from the equation
,
where is airtwist in the working wheel.
Value of on the middle radius must be within limits of 0,20,35.
To verify calculations accuracy it is necessary to compute turbine stage work by the formula
.
If obtained value differs more than on 3% from accepted at the beginning it is necessary to look for mistake in previous calculations.
Cinematic parameters determination on different turbine stage radiuses
Speed plan calculation and construction (see pic. X.x) is done for chosen law of blade profiling across radius.
Pic. X.x
In approximate turbine stage calculations we can accept design sections diameters as average between diameters of inlet and output edge, i.e.
At flow parameters calculation first of all law of blade profiling across radius are chosen. For that reactivity rate are determined at rate index value m=1,0 () from the equation
.
If , the law of circulation constancy is accepted, which provide high stage efficiency. In this case angle of nozzle diaphragm blade incidence is changed across blade height.
If , blades can be profiled by the law of , which allows to obtain same profiles across the blade height and simplifies blade manufacturing technology. Disadvantage of these blades is nonconstant axial component of the jet velocity at the exit from the nozzle diaphragm. If value is obtained at index value m=1,0, then assigning , index value m can be found by the formulas:
, if ;
, if .
Velocities and on different radiuses in common case are calculated from the equations:
;
.
If the law of profiling is accepted, then:
; .
If the law of profiling is accepted, velocities and
are calculated from the equations (x.x) and (x.x) with index value .
Axial components of the jet velocity are determined by the formulas:
;
.
If the law of profiling is accepted, then:
;.
If the law of profiling is accepted, velocities and
are calculated from the equations (x.x) and (x.x) with index value .
Table 1
Stages cinematic parameters on different radiuses
Parameter and formula 
Section 
Note 

sleeve 
middle 
peripheral 

, m 

, m/s 

, m/s 

,m/s 

, m/s 

, deg 

, m/s 

, m/s 

, deg 

, m/s 

, m/s 

, m/s 

, deg 

, deg 

, m/s 

, m/s 

, K 

,J/kg 

, % 

Appendix 1
THE EXAMPLE OF gasdynamic CALCULATION OF THE axial turbine stage
The initial data for the axial turbine stage gasdynamic calculation are gas parameters and geometrical dimensions at the entry to the turbine obtained during gasdynamic calculation of the designed engine.
The main goals of the turbine stage gasdynamic calculation are geometrical dimensions determination of the turbine stage, speed plans construction in three sections across the blade height.
From the engine gasdynamic calculation following parameters are known:
· gas pressure at the entry to the highpressure turbine ;
· gas temperature at the entry to the highpressure turbine ;
· mass gas flow rate ;
· HPT stage work ;
· circumferential velocity on the middle radius ;
· angle of the stream output from nozzle diaphragm ;
· external, middle and sleeve diameters at the entry to the working wheel , , ;
· jet velocity of gas at the exit from the nozzle diaphragm ;
· reduced velocity at the nozzle diaphragm exit ;
· pressure recovery coefficient in the nozzle diaphragm .
Section area at the entry to the HPT first stage working wheel was determined in the engine gasdynamic calculation:
.
For relative density and then;
.
Geometrical dimensions before the working wheel will be found on the basis of the next formulas:
· blade height
;
· external diameter
;
· sleeve diameter
;
· relative sleeve diameter
;
To determine geometrical dimensions at the exit from the working wheel, first of all, it is necessary to compute flow parameters in this section.
Gas temperature at the exit from the turbine stage is calculated from the equation
; .
Gas pressure at the exit from the turbine stage is calculated by the formula
; ,
where .
Axial component of the jet velocity is assigned at the exit from the working wheel:
;
;;.
Reduced velocity is calculated by the formula
; ,
and .
Section area at the exit from the working wheel is calculated from the equation
;
.
Supposing, we assume.
Diametric dimensions at the working wheel exit:
;
;
;
Turbine stage flow duct are drown in scale x:x (see pic x.x)
Pic. X.x Turbine stage flow duct
It is necessary to determine stage loading coefficient for turbine stage calculation on the middle radius:
;
Absolute velocity and reduced velocity at the exit from the nozzle diaphragm is computed supposing axial flow output, i.e. :
; ;
; .
Axial component of the jet velocity and parameter at the entry to the working wheel are calculated by the formulas:
; ;
; .
Circumferential components of the jet velocity at the entry to the working wheel and behind turbine stage, and parameters , and are calculated by the formulas:
; ;
; ;
; ;
; ;
; .
Relative gas velocity and its circumferential component can be found from the expressions:
;;
; .
Angle of the stream inlet into the working wheel in the relative direction is computed by formula
; .
Jet velocity and reduced velocity at the exit from the working wheel are calculated by the formulas:
; ;
; .
Stream output angle at the exit from the working wheel in the relative direction can be found by the formula
; .
Relative velocity at the exit from the working wheel and its circumferential component are determined by the formulas:
; ;
; .
Stagnated temperature and critical gas velocity in the relative direction are computed from the equations:
;
;
; .
Airtwist is calculated by the formula
; .
Reduced velocities at the entry to the working wheel and behind turbine stage are calculated by the formulas:
;
;
Cinematic reactivity rate is determined from the equation
; .
Circumferential velocities at the entry to the working wheel and behind it are accepted the same (). Turbine stage work is verified by the equation
;
;
; .
As , all calculations are correct.
To calculate flow parameters on different radiuses, first of all, it is necessary to choose law of profiling across blade height. To do it, calculate reactivity rate an rate index value m=1:
;
.
As ,we choose law of circulation constancy across the radius .
Flow parameters calculations on different radiuses are summarized in the table x.x.
Table X.x
Parameter and formula 
Section 

sleeve 
middle 
peripheral 

, m 
0,686 
0,718 
0,75 

, m/s 
382 
400 
417,8 

1,68 
1,53 
1,4 

, m/s 
222,9 
222,9 
222,9 

,m/s 
641,1 
612,5 
586,4 

, m/s 
678,7 
651,8 
627,3 

0,966 
0,927 
0,892 

, deg 
20.3 
21.3 
22.3 

, m/s 
259,1 
212,5 
168,6 

, m/s 
341,8 
308 
279,5 

, deg 
40,7 
46,4 
52,9 

, m/s 
263,6 
263,6 
263,6 

, m/s 
0 
0 
0 

, m/s 
263,6 
263,6 
263,6 

, deg 
90 
90 
90 

0,4 
0,4 
0,4 

, deg 
34,6 
33,4 
32,2 

, m/s 
464,2 
479 
494,7 

, m/s 
382,1 
400 
418,6 

, K 
1352 
1358 
1364 

0,51 
0,46 
0,42 

0,696 
0,716 
0,738 

0,16 
0,234 
0,3 

,J/kg 
244092 
244968 
245314 

, % 
0,04 
0,013 
0,13 

On the basis of foregoing calculation we plot speed plans for three sections on blade height (see pic. X).
a)
b)
c)
Pic. X.x. Axial turbine stage speed plans:
a  sleeve section; b  middle section; c peripheral section.
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