The gas-dynamic calcualation of the axial turbine stage
MINISTRY
OF EDUCATION AND SCIENCE OF UKRAINEAviation University
The
gas-dynamic 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”
by I.I. Gvozdetsky , V.V. Kharyton, S.I.Tkachenko
KYIV
2007
Contents
Introductiongeneral
law of circulation change across blade heightof the turbine stage geometrical
dimensionsstage calculation on the middle radiusparameters determination on
different turbine stage radiuses1 The example of gas-dynamic calculation of the axial turbine stage
Introduction
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.
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 gas-dynamic 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.
.
3. Main geometrical dimensions of the turbine stage
initial data for turbine stage calculation are taken
from gas-dynamic 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.
The general law of circulation change across blade height
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); 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.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.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 20-25° at first stages. At these values negative reactivity
rate can occur near blade root, especially at high values of loading
coefficient.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
dimensions at the entry to the working wheel are determined
in the gas-dynamic calculation of the designed engine. At first area at the
exit from the nozzle diaphragm is calculated
,
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
kg=1,33 and Rg=288 J/(kg×K) we will have mg=0,0396 (kg×K)/J.density can be determined from tables of
gas-dynamic functions using value of the reduced velocity or by the formula
.
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:
sleeve diameter for first stages is within the limits
of , and for last stages .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.calculate section area at the exit from the stage
(behind working wheel) it is necessary to compute gas parameters and behind
calculated stage.gas temperature is
determined from the energy equation:
gas pressure behind stage is calculated by the formula
-
stage efficiency.component of the jet velocity at the exit from the working
wheel is assumed on 20-80 m/s more then gas velocity at the entry to the
working wheel, i.e.
;
m/s,.
area at the exit from the working wheel is determined
from following expression:
,
is
computed by the value of
.
for chosen profiling law determine main dimensions at
the exit from turbine stage in a similar manner as have been done turbine stage
entrance.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.loading coefficient on the middle radius is
determined by the formula
the first turbine stage .jet velocity at the exit from the nozzle diaphragm is
determined from the equation
reduced velocity - by the formula
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 .
of must
not exceed, otherwise it is necessary to decrease to meet this requirement.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.components of the jet velocity at the entry to the working wheel
and behind turbine stage, and parameters ,
and are
calculated by the formulas:
;
; ;
.
component of the jet velocity and parameter at the entry to the working wheel are calculated by
the formulas:
;
.
;
.
of the stream inlet into the working wheel in the
relative direction is computed by formula
.
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 (20-80) m/s more then.velocity and reduced velocity at the exit from the
working wheel are calculated by the formulas:
;
.
output angle at the exit from the working wheel is
calculated by the formula
.
must
be close to 90°, but for first stages it can be n the
range of (75-85)°.output angle at the
exit from the working wheel in the relative direction can be found by the
formula
.velocity
at the exit from the working wheel and its circumferential component are
determined by the formulas:
;
.
temperature and critical gas velocity in the relative
direction are computed from the equations:
velocities at the entry to the working wheel and
behind turbine stage are calculated by the formulas:
;
.
reactivity rate is determined from the equation
,
is
airtwist in the working wheel.of on
the middle radius must be within limits of 0,2-0,35.verify calculations
accuracy it is necessary to compute turbine stage work by the formula
.
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
plan calculation and construction (see pic. X.x) is done for
chosen law of blade profiling across radius.
.
X.x
approximate turbine stage calculations we can accept
design sections diameters as average between diameters of inlet and output
edge, i.e.
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
.
,
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., 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 non-constant 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 .
and
on different radiuses in common case are calculated
from the equations:
;
.
the law of profiling is
accepted, then:
;
.
the law of profiling is
accepted, velocities and
are
calculated from the equations (x.x) and (x.x) with index value .components of the jet velocity are determined by the
formulas:
;
.
the law of profiling is
accepted, then:
;.
the law of profiling is
accepted, velocities and
are
calculated from the equations (x.x) and (x.x) with index value .
Table 1cinematic 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
|
|
|
|
|
, K
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
,J/kg
|
|
|
|
|
, %
|
|
|
|
|
Appendix 1
THE EXAMPLE OF gas-dynamic CALCULATION OF THE axial turbine stage
The initial data for the axial turbine stage gas-dynamic
calculation are gas parameters and geometrical dimensions at the entry to the turbine obtained
during gas-dynamic calculation of the designed engine.main goals of the turbine
stage gas-dynamic calculation are geometrical dimensions determination of the turbine
stage, speed
plans construction in three sections across the blade height.the engine gas-dynamic
calculation following parameters are known:
· gas pressure at the entry to the high-pressure turbine
;
· gas temperature at the entry to the high-pressure
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 gas-dynamic calculation:
.
relative
density and then;
.dimensions
before the working wheel will be found on the basis of the next formulas:
· blade height
;
· external diameter
;
· sleeve diameter
;
· relative sleeve diameter
;
determine geometrical dimensions at the exit from the
working wheel, first of all, it is necessary to compute flow parameters in this
section.temperature at the exit from the turbine stage is calculated from the
equation
;
.
pressure at the exit from the turbine stage is
calculated by the formula
;
,
.component
of the jet velocity is assigned at the exit from the working wheel:
;
;;.
velocity is calculated by the formula
;
,.
area at the exit from the working wheel is calculated
from the equation
;
.
,
we assume.dimensions at the working wheel exit:
;
;
;
stage flow duct are drown in scale x:x (see pic x.x)
.
X.x Turbine stage flow duct
is necessary to determine stage loading coefficient
for turbine stage calculation on the middle radius:
;
velocity and reduced velocity at the exit from the
nozzle diaphragm is computed supposing axial flow output, i.e. :
;
;
;
.
component of the jet velocity and parameter at the entry to the working wheel are calculated by
the formulas:
;
;
;
.
components of the jet velocity at the entry to the
working wheel and behind turbine stage, and parameters , and
are calculated by the formulas:
;
;
;
;
;
;
;
;
;
.
gas velocity and its circumferential component can be
found from the expressions:
;;
;
.
of the stream inlet into the working wheel in the
relative direction is computed by formula
;
.
velocity and reduced velocity at the exit from the
working wheel are calculated by the formulas:
;
.
output angle at the exit from the working wheel in the
relative direction can be found by the formula
;
.
velocity at the exit from the working wheel and its
circumferential component are determined by the formulas:
;
;
;
.
temperature and critical gas velocity in the relative
direction are computed from the equations:
;
;
;
.
is calculated by the formula
;
.
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
;
.
velocities at the entry to the working wheel and
behind it are accepted the same ().
Turbine stage work is verified by the equation
;
;
;
.
,
all calculations are correct.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:
;
.
,we
choose law of circulation constancy across the radius .parameters calculations on different radiuses are
summarized in the table x.x.
Table X.x
Parameter and
formula
|
Section
|
|
sleeve
|
middle
|
peripheral
|
, m0,6860,7180,75
|
|
|
|
, m/s382400417,8
|
|
|
|
1,681,531,4
|
|
|
|
, m/s222,9222,9222,9
|
|
|
|
,m/s641,1612,5586,4
|
|
|
|
, m/s678,7651,8627,3
|
|
|
|
,966
,927
,
deg
.3
.3
22.3
|
|
|
|
, m/s259,1212,5168,6
|
|
|
|
, m/s341,8308279,5
|
|
|
|
,
deg
,7
,4
52,9
|
|
|
|
, m/s263,6263,6263,6
|
|
|
|
, m/s000
|
|
|
|
, m/s263,6263,6263,6
|
|
|
|
,
deg
,4
,4
,
deg
,6
,4
,
m/s
,2
494,7
|
|
|
|
, m/s382,1400418,6
|
|
|
|
,
K
1364
,51
,46
,696
,716
,16
,234
,J/kg
,
%
,04
,013
the basis of foregoing calculation we plot speed plans for
three sections on blade height (see pic. X).
)
)
).
X.x. Axial turbine stage speed plans:- sleeve section; b - middle section; c-
peripheral section.
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