Numerical modeling-of-gas-turbine-engines

World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:11, 2008

Numerical Modeling of Gas Turbine Engines
А. Pashayev, D. Askerov, C.Ardil, and R. Sadiqov

International Science Index 23, 2008 waset.org/publications/2112

Abstract—In contrast to existing methods which do not take
into account multiconnectivity in a broad sense of this term, we
develop mathematical models and highly effective combination
(BIEM and FDM) numerical methods of calculation of stationary
and quasi-stationary temperature field of a profile part of a blade
with convective cooling (from the point of view of realization on
PC). The theoretical substantiation of these methods is proved by
appropriate theorems. For it, converging quadrature processes have
been developed and the estimations of errors in the terms of
A.Ziqmound continuity modules have been received.
For visualization of profiles are used: the method of the least
squares with automatic conjecture, device spline, smooth
replenishment and neural nets. Boundary conditions of heat
exchange are determined from the solution of the corresponding
integral equations and empirical relationships. The reliability of
designed methods is proved by calculation and experimental
investigations heat and hydraulic characteristics of the gas turbine
first stage nozzle blade.

Research has established that the temperature condition of
the blade profile part with radial cooling channels can be
determined as two-dimensional [2]. Besides, if to suppose
constancy of physical properties and absence of internal
sources (drains) of heat, then the temperature field under
fixed conditions will depend only on the skew shape and on
the temperature distribution on the skew boundaries. In this
case, equation (1) will look like:
ΔT =

I. INTRODUCTION

T

HE development of aviation gas turbine engines
(AGTE) at the present stage is mainly reached by
assimilation of high values of gas temperature in front of the
turbine ( T Г ). The activities on gas temperature increase are
conducted in several directions. Assimilation of high ( T Г ) in
AGTE is however reached by refinement of cooling systems
of turbine blades. It is especially necessary to note, that with
T Г increase the requirement to accuracy of results will
increase. In other words, at allowed values of AGTE metal
temperature Tlim = (1100...1300K ) , the absolute error of
temperature calculation should be in limits ( 20 − 30 K ), that
is no more than 2-3%.
This is difficult to achieve (multiconnected fields with
various cooling channels, variables in time and coordinates
boundary conditions). Such problem solving requires
application of modern and perfect mathematical device.
II. PROBLEM FORMULATION
In classical statement a heat conduction differential
equation in common case for non-stationary process with
distribution of heat in multi–dimensional area (FourierKirchhoff equation) has a kind [1]:
∂( ρCvT )
(1)
= div(λ grad T) + qv ,

∂x

2

+

∂ 2T
∂y 2

=0

(2)

When determining particular temperature fields in gas
turbine elements are used boundary conditions of the third
kind, describing heat exchange between the skew field and
the environment (on the basis of a hypothesis of a NewtonRiemann). In that case, these boundary conditions will be
recorded as follows:
α 0 (T0 − Tγ 0 ) = λ

Keywords—Multiconnected systems, method of the boundary
integrated equations, splines, neural networks.

∂ 2T

∂Tγ 0
∂n

(3)

This following equation characterizes the quantity of heat
transmitted by convection from gas to unit of a surface of a
blade and assigned by heat conduction in a skew field of a
blade.
−λ

∂Tγ i
∂n

= α i (Tγ i − Ti )

(4)

Equation (4) characterizes the heat quantity assigned by
convection of the cooler, which is transmitted by heat
conduction of the blade material to the surface of cooling
channels: where T0 - temperature of environment at i = 0 ;
Ti - temperature of the environment at i = 1, M
(temperature of the cooler), where M - quantity of outlines;
Tγ0 - temperature on an outline γi at i = 0 (outside outline

of blade); Tγi - temperature on an γi at i = 1, M (outline of
cooling channels); α 0 - heat transfer factor from gas to a
surface of a blade (at i = 0 ); α i - heat transfer factor from a
blade to the cooling air at i = 1, M ; λ - thermal
conductivity of the material of a blade; n - external normal
on an outline of researched area.
III. PROBLEM SOLUTION

where ρ , cv and λ - accordingly material density, thermal
capacity, and heat conduction; qv - internal source or drain
of heat, and T - is required temperature.

At present for the solution of this boundary problem (2)(4) four numerical methods are used: Methods of Finite
Differences (MFD), Finite Element Method (FEM),
probabilistic method (Monte-Carlo method), and Boundary
Integral Equations Method (BIEM) (or its discrete analog ─
Boundary Element Method (BEM)).
Let us consider BIEM application for the solution of
problem (2)-(4).

Authors are with the National Academy of Aviation, AZ-1045, Baku,
Azerbaijan (phone: +99450 385 35 11; fax: +99412 497 28 29; e-mail:
sadixov@mail.ru).

3.1. The function T = T (x, y ) , continuous with the
derivatives up to the second order, satisfying the Laplace
equation in considered area, including and its

∂t

25


M

outline Г = U γi , is harmonic. Consequence of the Grin
i =0

integral formula for the researched harmonic function
T = T (x, y ) is the ratio:
1
∂( lnR)
∂Т
Т(x, y) =
− lnR Г ]ds
∫ [Т Г
2π Г
∂n
∂n

∂( lnRk )
∂Т
⎤
1 ⎡
ds − ∫ Г lnRk ds ⎥
∫ТГ
2π ⎢ Г
∂n
Г ∂n
⎣
⎦

(6)

With allowance of the boundary conditions (2)-(3), after
collecting terms of terms and input of new factors, the ratio
(6) can be presented as a linear algebraic equation,
computed for the point R :
ϕ k 1Tγ01 + ϕ k 2 Tγ02 + ... + ϕ kn Tγ0 m −
(7)
− ϕ kγ0 T0 − ϕ kγi Ti − 2πTk = 0


where n is the quantity of sites of a partition of an outside
outline of a blade l γ 0 (l γ i on i = 0 ) on small sections
ΔS 0 ( ΔS i at i = 0 ) , m is the quantity of sites of a partition of

outside outlines of all cooling channels l γ i (i = 1, M ) on
small sections ΔS i .
Let us note, that unknowns in the equation (7) except the
unknown of true value Tk in the k point are also mean on
sections of the outlines partition ΔS0 and ΔS i temperatures
Tγ 01 , Tγ 02 ,..., Tγ 0 m and Tγ i 1 , Tγ i 2 ,..., Tγ im
(total number
n + m ).
From a ratio (7), we shall receive the required temperature
for any point, using the formula (5):
1
T(x, y) = [ϕk1Tγ01 +ϕk2Tγ02 + ...+ϕknTγ0n +
2π
(8)
+ ...+ϕkmTγim −ϕkγ0Tcp0 −ϕkγiTcpi ]

where
ϕ k1 = ∫

ΔS01

.

α
∂( lnRk )
ds − 01 ∫ lnRk ds
λ1 ΔS01
∂n
.

.

.

T(x, y) = ∫ ρ lnR −1 ds ,

(5)

where R - variable at an integration of the distance between
point K (x , y ) and “running” on the outline k - point; T Г temperature on the outline Г . The temperature value in
some point k lying on the boundary is determined (as
limiting at approach of point K (x, y ) to the boundary)
Тk =

3.2. In contrast to [4], we offer to decide the given boundary
value problem (2)-(4) as follows. We locate the distribution
of temperature T = T (x, y ) as follows:

M

where Г = U γi -smooth closed Jordan curve; M -quantity of
i =0

cooled channels; ρ = M ρ i - density of a logarithmic potential
U
i =0

uniformly distributed on γi
i =0

ρ (s) −

R(s,ξ ) = ((x(s) − x(ξ ))2 + ( y(s) − y(ξ ))2 )1/ 2 .
For the singular integral operators evaluation, which are
included in (12) the discrete operators of the logarithmic
potential with simple and double layer are investigated.
Their connection and the evaluations in modules term of the
continuity (evaluation such as assessments by A. Zigmound
are obtained) is shown.
Theorem (main)
Let
ωξ ( x)
∫

0

i =0

< +∞

x

And let the equation (12) have the solution f*∈CГ (the set of
continuous functions on Г). Then ∃Ν0∈Ν= {1, 2…} such
that the discrete system ∀N>N0, obtained from (12) by using
the discrete double layer potential operator (its properties
has
been
studied),
has
unique
solution
) (N)
{f j }, k = 1,m j ; j = 1,n;
ε N ω (x)ω (x)
) (N)
ξ
f*
*
| f jk − f jk |≤ С(Г)( ∫
dx +
x
0

+ε

L/2

∫

,

ωξ (x)ω * (x)
f

x

εN

+ τN

(9)

L/2

∫

εN

i =0

M

(12)

k

many discrete point and integrals that are included in the
equations as logarithmic potentials, was calculated
approximately with the following ratios:

(where ΔS γi ∈ L = U li ; li = ∫ ds )

,

where

In activities [2] the discretization of aniline Г = U γi by a

∫ lnRk ds ≈ lnRk ΔS γi

1
∂
∫ ( ρ ( s ) − ρ (ξ )) lnR( s, ξ )dξ =
2π Г
∂n

α
= i (T − ∫ ρ ( s )lnR −1 ds )
2πλ
Г

M

ΔS γi

s ∈ [0 , L ] ;

Г

α
α
= 01 ∫ lnRk ds + ... + im ∫ lnRk ds
λ1 ΔSi1
λm ΔSim

ΔS γi

y = y (s ) ;

Using BIEM and expression (11) we shall put problem (2)(4) to the following system of boundary integral equations:

.

∂( lnR k )
∂( lnR k )
ds ≈
ΔS γi
∂n
∂n

x = x(s ) ;

L = ∫ ds .

α01
α0n
∫ lnRk ds + ... +
∫ lnRk ds
λ1 ΔS01
λn ΔSn

∫

.

i =0

Г = U γi are positively oriented and are given

in a parametric kind:

α
∂( lnRk )
ds − 0m ∫ lnRk ds
λm ΔS0m
∂n
ΔS0m

ϕ kγii

M

S = U si

M

Thus curve

ϕ kn = ∫
ϕ kγ0 =

(11)

Г

L/2
0

f

ω * (x)
f

x

(10)

x

f

dx +

dx ),

where C ( Г ) is constant, depending only on

τN

∞
N =1

--the

sequence of partitions of Г ; {ε N }N =1 -- the sequence of
∞

positive numbers such that the pair (
− −1

,

ω * (x)

dx + ω * ( τ N ) ∫

τ N , , ε N ),

satisfies

the condition 2 ≤ ε τ ≤ p .
Let δ ∈ 0, d , where d is diameter Г, and the splitting τ is

(

2

)

that, which is satisfied the condition

p′ ≥ δ

γi

26

τ

≥2


ψ ∈ C Г ( C Г - space of all functions
continuous on Г ) and z ∈ Г , ( z = x + iy )

Then for all

(I τ δ f )( z ) − f ( z ) ≤ С ( Г )
,

⎛
⎜ f
⎝

C

δ ln

2d

δ

+ ω f ( τ ) + τ ln

2d

δ

+ f

C

⎞
⎠

ω Z ( τ )⎟;

Г
ω f ( x)ω l ( x)
⎛
~
f )( z ) − f ( z ) ≤ ⎜ С ( Г ) ∫
dx +
Ω, Г
⎜
x2
0
⎝
,
d
d
ω f ( x) ⎞
ω l ( x)
dx ⎟
+ ω f ( τ )∫
dx + τ ∫
⎟
x
x2
Δ
Δ
⎠

(L

∂Tv ( x, y ) ∂Tv +1 ( x, y )
=
∂n
∂n

⎛ f ( z k ,e +1 ) + f ( z k ,e )
⎞
⎜
− f (z) ⎟ ⋅
⎜
⎟
2
z m , e∈τ ( z ) ⎝
⎠
− y k ,e )( x k ,e − x ) − ( x k ,e +1 − x k ,e )( y k ,e − y )

,

( y k ,e +1

z − z k ,e

2

3.4. For it is necessary at first to find solution of the heat
conduction direct problem with boundary condition of the
III kind from a gas leg and boundary conditions I kinds from
a cooling air leg

∂Tγ 0


f ( z k , j +1 ) + f ( z k , j )
2

z m , e∈τ ( z )

⋅ ln

1
zk, j − z

M

i =1 γ i

γ0

2

)

1

2

,

(i = 0 , M ) (18)

Using expression (17), (18) we shall put problem (2),
(15), (16), to the following system of boundary integral
equations Fredholm II

ρ 0 (s 0 ) −

,

z k ,e ∈ τ , z k ,e = xk ,e + iy k ,e

+

τ ( z ) = {z k ,e z k ,e − z > ε }
k

=

τ = max z k , j +1 − z k , j
j =1, mk

1
2π

1
2π

∂
∫ (ρ (s ) − ρ (ξ )) ∂n ln R (s
γ
0

0

0

0

, ξ 0 )dξ 0 +

0

∂ ∂

M

∑ ∫ μ (s ) ∂n ∂n ln R
i =1 γ i

−1
0

0

i

i

−1
i

ds i =

M
⎞
α0 ⎛
⎜ T − ρ (s ) ln R −1 ds − ∑ μ (S ) ∂ ln R −1 ds ⎟
0
∫ 0 0 0 0 i =1 γ∫ i i ∂n i i ⎟
2πλ ⎜
γ
⎝
⎠
0

i

(19)

Thus are developed effective from the point of view of
realization on computers the numerical methods basing on
constructed two-parametric quadratute processes for the
discrete operators logarithmic potential of the double and
simple layer. Their systematic errors are estimated, the
methods quadratures mathematically are proved for the
approximate solution Fredholm I and II boundary integral
equations using Tikhonov regularization and are proved
appropriate theorems [1].
3.3. The given calculating technique of the blade
temperature field can be applied also to blades with the
plug–in deflector. On consideration blades with deflectors in
addition to boundary condition of the III kind adjoin also
interfaces conditions between segments of the outline
partition as equalities of temperatures and heat flows
(13)
Tv ( x, y ) = Tv +1 ( x, y ) ,

∂
ln Ri−1 dsi
∂n

Ri = ( x − xi (si )) + ( y − y i (s i ))

on τ and δ parameters) for logarithmic potential simple
layer; &f&&(z ) -simple layer logarithmic potential operator;

τ k = {z k ,1 ,..., z k ,m }, z k ,1 ≤ z k , 2 ≤ ... ≤ z k ,m

(16)

2

⋅

(I τ ε f )( z ) - two-parameter quadrature formula (depending

)

, (17)

(

ω f ( x ) is

z k , j +1 − z k , j

k

(

(15)

Tγi -the unknown optimum temperature of a wall of a

a module of a continuity of functions f;
,

),

T ( x, y ) = ∫ ρ 0 (s 0 ) ln R0−1 ds 0 + ∑ ∫ μ i (si )

- two-parameter quadrature formula (depending
on τ and δ parameters) for logarithmic double layer
potential; ~ ( z ) - double layer logarithmic potential operator;
f
,

(I τ ε f )( z ) = ∑

α0
(T0 − Tγ
λ

blade from a leg of a cooling air.
We locate the distribution of temperature T ( x, y ) as
follows

(Lτ ε f )( z )

C ( Г ) – constant, dependent only from a curve Г ;

=

0
∂n
T ( x, y ) γ i = Tγi , i = 1, M

where
+ πf ( z )

(14)

where ν - number of segments of the outline partition of
the blade cross-section; x, y- coordinates of segments. At
finding of cooler T best values, is necessary to solve the
inverse problem of heat conduction.

where

(Lτ ε f )( z ) = ∑

,

μ i (si ) −
+
+

1
2π

∂
∫ (μ (s ) − μ (ξ )) ∂n ln R (s , ξ )dξ
γ
i

i

i

1
2π

∂
∑ ∫ μ (s ) ∂n ln R

1
2π

∫ ρ (s )ln R
γ
0

i

i

+

i

i

M

i =1 γ 0
j ≠1

−1
i

i

j

j

0

−1
0

−1
j

ds j +

ds 0 = Tγi

(i = 1, M ),

0

(20)
where

(

Ri (s i , ξ i ) = ( x(si ) − xi (ξ i )) + ( y (si ) − y i (ξ i ))
and integral

27

2

(i = 0 , M )

2

)

1

2

,

(21)


∂ ∂

∫ μ (s ) ∂n ∂n ln R
γ
i

i

−1
i

dsi

(22)

2
3
⎡1 xe xe xe ⎤
⎢
⎥
2
A= ⎢0 1 2 xe 3 xe ⎥ ,
⎢
⎥
⎢0 0 2 6 xe ⎥
⎣
⎦

i

is not singular.
3.5. The multiples computing experiments with the using
BIEM for calculation the temperature fields of nozzle and
working blades with various amount and disposition of
cooling channels, having a complex configuration, is
showed, that for practical calculations in this approach,
offered by us, the discretization of the integrations areas can
be conducted with smaller quantity of discrete points. Thus
the reactivity of the algorithms developed and accuracy of
evaluations is increased. The accuracy of temperatures
calculation, required consumption of the cooling air, heat
flows, losses from cooling margins essentially depends on
reliability of boundary conditions, included in calculation of
heat exchange.


3.6. Piece-polynomial smoothing of cooled gas-turbine
blade structures with automatic conjecture is considered: the
method of the least squares, device spline, smooth
replenishment, and neural nets are used.
3.6.1. Let the equation of the cooled blade outline segments
is the third degree polynomial:
y (x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3
(23)
The equation of measurements of the output coordinate has a
kind:
Z y = a0 + a1 x + a 2 x 2 a 3 x 3 + δ y
(24)

where Zy=║z1y, z2y, …, zny║T - vector of measurements of
output coordinate, n-amount of the points in the
consideration interval. For coefficients of polynomial (23)
estimate the method of the least squares of the following
kind is used

)

θ

=(XTX)-1(XTZy),

(25)

Dθ) =(XTX)-1σ2 ,

(26)

where
1 x1 x12 x13

X=

2
3
1 x2 x2 x2
2
3
1 x3 x3 x3

- structural matrix;

............
2
3
1 xn xn xn

)

Dθ) - dispersion matrix of errors; θ =║a0, a1, a2, a3║T vector of estimated coefficients.
Estimations of coefficients for the first segment is
received with using formula (25). Beginning with second
segment, the θ vectors components is calculated on
experimental data from this segment, but with the account of
parameters found on the previous segments. Thus, each
subsequent segment of the blade cross-section outline we
shall choose with overlapping. Thus, it is expedient to use
the following linear connections between the estimated
parameters of the previous segment
for N-th segment:

AθN=V,

)

θN

1

and required

)

θN

(28)

)
)
)
)
2
3
⎡a0 N −1 + a1 N −1 xe + a 2 N −1 xe + a3 N −1 xe ⎤
⎢
⎥
)
)
)
2
⎥,
V= ⎢a1 N −1 + a 2 N −1 xe + 3a3 N −1 xe
(29)
⎢ )
⎥
)
⎢2a 2 N −1 + 6 a3 N −1 xe
⎥
⎣
⎦
e=(N-1)(n-L); L- number points of overlapping.
The expressions (27)-(29) describe communications,
which provide joining of segments of interpolation on
function with first and second degrees.
Taking into account the accuracy of measurements, the
problem of defining unknown coefficients of the model in
this case can be formulated as a problem conditional
extremum: minimization of the quadratic form (ZyXθ)Tσ2I(Zy-θ) under the limiting condition (27). Here I is a
individual matrix.
For the solution of such problems, usually are using the
method of Lagrange uncertain multipliers. In result, we shall
write down the following expressions for estimation vector
of coefficients at linear connections presence (27):

~T )T

θ

)

= θ +(VT- θ TAT)[A(XTX)-1AT]-1A(XTX)-1

(30)

T
-1 T
T
-1 T -1
T
-1 2
Dθ~ = Dθ) − (X X) A [A(X X) A ] A(X X) σ (31)
Substituting matrixes A and Х and vectors Zy and V in
expressions (25), (26), (30), and (31), we receive estimations
of the vector of coefficients for segment of the cooled blade
section with number N and also the dispersing matrix of
errors.
As a result of consecutive application of the described
procedure and with using of experimental data, we shall
receive peace-polynomial interpolation of the researched
segments with automatic conjecture.
Research showed that optimum overlapping in most cases
is the 50%-overlapping.

3.6.2. Besides peace-polynomial regression exist
interpolation splines which represent polynomial (low odd
degrees - third, fifth), subordinated to the condition of
function and derivatives (first and second in case of cubic
spline) continuity in common points of the next segments.
If the equation of the cooled gas-turbine blades profile is
described cubic spline submitted in obvious polynomial kind
(23), the coefficients а0, а1, а2, а3 determining j-th spline,
i.e. line connecting the points Zj=(xj, yj) and Zj+1=(xj+1, yj+1),
are calculating as follows:
a0 = z j ;

(32)

a1 = z 'j ;
a2 =

z 'j'

a3 =

z 'j''

/ 2 = 3( z j +1 − z
/ 6 = 2( z j −

−2
j )h j + 1

3
z j +1 )h −+1
j

where hj+1=|zj+1-zj|, j= 1, N

−

1
2 z 'j h −+1
j

+

2
z 'j h −+1
j

−

1
z 'j +1 h −+1 ;
j

2
+ z 'j +1 h −+1 ;
j

−1.

3.6.3. Let us consider other way smooth replenishment of
the cooled gas-turbine blade profile on the precisely
measured meaning of coordinates in final system of discrete

(27)

28


points, distinguishing from spline-function method and also
from the point of view of effective realization on computers.
Let equation cooled blades profile segments are described
by the multinomial of the third degree of the type (23). By
taking advantage the smooth replenishment method
(conditions of function smooth and first derivative are
carried out) we shall define its coefficients:
a0 = z j ;
1
a1 = ( z j +1 − z j )h −+1 ;
j

(33)

1
1
1
a 2 = −(( z j + 2 − z j +1 )h −+ 2 + ( z j +1 − z j )h −+1 )h −+1 ;
j
j
j
1
1
2
a 3 = (( z j + 2 − z j +1 )h −+ 2 − ( z j +1 − z j )h −+1 )h −+1 ;
j
j
j


j= 1, N −1 − S , S=1.
If it’s required carry out conditions of function smooth first
and second derivatives, i.e. corresponding to cubic splines
smooth, we shall deal with the multinomial of the fifth
degree (degree of the multinomial is equal 2S + 1, i.e. S =
2).
The advantage of such approach (smooth replenishment) is
that it’s not necessary to solve system of the linear algebraic
equations, as in case of the spline application, though the
degree of the multinomial is higher 2.
3.6.4. A new approach of mathematical models’ parameters
identification is considered. This approach is based on
Neural Networks (Soft Computing) [7-9].
Let us consider the regression equations:
n

Yi = ∑aij x j ;i =1,m

(34)

r s
Yi = ∑arsx1 x2 ;r = 0,l;s = 0,l;r + s ≤ l

(35)

j=1

r ,s

where ars are the required parameters (regression
coefficients).
The problem is put definition of values aij and ars
parameters of equations (34) and (35) based on the statistical
experimental data, i.e. input x j and x1 , x2 , output
coordinates Y of the model.
Neural Network (NN) consists from connected between their
neurons sets. At using NN for the solving (34) and (35)
input signals of the network are accordingly values of
variables X = ( x1 , x2 ,..., xn ) , X = ( x1 , x2 ) and output Y .
As parameters of the network are aij and ars parameters’
values.
At the solving of the identification problem of parameters
aij and ars for the equations (34) and (35) with using NN,
the basic problem is training the last.
We allow, there are statistical data from experiments. On
the basis of these input and output data we making training
pairs ( X , T ) for network training. For construction of the
model process on input of NN input signals X move and
outputs are compared with reference output signals Т .
After comparison, the deviation value is calculating by
formula

E=

training (correction) parameters of a network comes to end
(Fig. 1). In opposite case it continues until value Е will not
reach minimum.
Correction of network parameters for left and right part is
carried out as follows:
н
c
ars = ars + γ

∂E
,
∂ars

c
н
where ars , ars are the old and new values of NN parameters
and γ is training speed.
The structure of NN for identifying the parameters of the
equation (34) is given on Fig. 2.

IV. RESULTS
The developed techniques of profiling, calculation of
temperature fields and parameters of the cooler in cooling
systems are approved at research of the gas turbine Ist stage
nozzle blades thermal condition. Thus the following
geometrical and regime parameters of the stage are used:
step of the cascade - t = 41.5 мм , inlet gas speed to cascade
- V1 = 156 м / s , outlet gas speed from cascade -

V2 = 512 м / s , inlet gas speed vector angle - α1 = 0.7 0 , gas
flow temperature and pressure: on the entrance to the stage Tг* = 1333 K ,

*
p г = 1.2095 ⋅ 10 6 Pа , on the exit from

p г 1 = 0.75 ⋅ 10 6 Pа ; relative gas
speed on the exit from the cascade - λ1аd = 0.891 .

stage - Tг1 = 1005 K ,

The geometrical model of the nozzle blades (Fig. 3),
diagrams of speed distributions V and convective heat
exchange local coefficients of gas α г along profile contour
(Fig. 4) are received.
The geometrical model (Fig. 5) and the cooling tract
equivalent hydraulic scheme (Fig. 6) are developed. Cooler
basics parameters in the cooling system and temperature
field of blade cross section (Fig. 7) are determined.
V. CONCLUSION
The reliability of the methods was proved by
experimental
investigations
heat
and
hydraulic
characteristics of blades in "Turbine Construction"
(Laboratory in St. Petersburg, Russia). Geometric model,
equivalent hydraulic schemes of cooling tracks have been
obtained, cooler parameters and temperature field of "Turbo
machinery Plant" enterprise (Yekaterinburg, Russia) gas
turbine nozzle blade of the 1st stage have been determined.
Methods have demonstrated high efficiency at repeated and
polivariant calculations, on the basis of which has been
offered the way of blade cooling system modernization.
The application of perfect methods of calculation of
temperature fields of elements of gas turbines is one of the
actual problems of gas turbine engines design. The
efficiency of these methods in the total influences is for
operational manufacturability, reliability of engine elements
design, and on acceleration characteristics of the engine.

1 k
∑ (Y j − T j ) 2
2 j =1

If for all training pairs, deviation value Е less given then

29

REFERENCES
[1]

[2]
[3]

[4]
[5]
[6]

[7]

[8]


[9]
[10]

[11]
[12]
[13]

Pashaev A.M., Sadykhov R.A., Iskenderov M.G., Samedov A.S.,
Sadykhov A.H. The Efficiency of Potential Theory Method for Solving
of the Tasks of Aircraft And Rocket Design. 10th National Mechanic
Conference. Istanbul Technical University, Aeronautics and
astronautics faculty, Istanbul, Turkey, July, 1997, p. 61-62.
R.A. Sadigov, A. S. Samedov, Modelling of temperature fields of gas
turbines elements. Scientists of a slip АzTU, Baku, 1998, Vol.VI, №
5, p.234-239.
Pashaev A. M., Sadykhov R. A., Samedov A.S. Highly Effective
Methods of Calculation Temperature Fields of Blades of Gas
Turbines. V International Symposium an Aeronautical Sciences “New
Aviation Technologies of the XXI Century”, A collection of technical
papers, section N3, Zhukovsky, Russia, August, 1999.
L.N. Zisina-Molojen and etc., Heat Exchange in Turbo Machines. M.,
1974
G. Galicheiskiy, A Thermal Guard of Blades, М., МАI, 1996.
A.M.Pashaev, R.A.Sadiqov, C.M.Hajiev, The BEM Application in
development of Effective Cooling Schemes of Gas Turbine Blades. 6th
Bienial Conference on Engineering Systems Design and Analysis,
Istanbul, Turkey, July, 8-11, 2002.
Abasov M.T., Sadiqov A.H., Aliyarov R.Y. Fuzzy Neural Networks in
the System of Oil and Gas Geology and Geophysics // Third
International Conference on Application of Fuzzy Systems and Soft
computing/ Wiesbaden, Germany, 1998,- p.108-117.
Yager R.R., Zadeh L.A. (Eds). Fuzzy Sets, Neural Networks and Soft
Computing. VAN Nostrand Reinhold. N.-Y. - № 4, 1994.
Mohamad H. Hassoun. Fundamentals of Artificial Neutral Networks /
A Bradford Book. The MIT press Cambridge, Massachusetts, London,
England, 1995.
Pashaev A. M., Sadykhov R. A., Samedov A.S., Mamedov R.N. The
Solution of Fluid Dynamics Direct Problem of Turbo Machines
Cascades with Integral Equations Method. Proceed. NAA. Vol. 3,
Baku, 2003.
V.S. Beknev, V.M. Epifanov, A.I. Leontyev, M.I. Osipov and ets.
Fluid Dynamics. A Mechanics of a Fluid and Gas. Moscow, MGTU
nam. N.E. Bauman, 1997, 671 p.
S.Z. Kopelev, A.F. Slitenko Construction and calculation of GTE
cooling systems. Kharkov, “Osnova”, 1994, 240 p.
L.V. Arseniev, I. B. Mitryayev, N. P. Sokolov The Flat Channels
Hydraulic Resistances with a System of Jets in a Main Stream.
“Energetika”, 1985, № 5, p.85-89.

30


APPENDIX

Correction algorithm

X

Input
signals

NN

Parameters

Deviations

Random-number

Target
signals

Y

Training
quality

generator

Fig. 1 System for network-parameter (weights, threshold) training (with feedback)

ai


x1

1

x2

ai

xj

aij

Yi

2

Fig. 2 Neural network structure for multiple linear regression equation

у,
mm

αg,
Vt

100

2

m K

95

1

λ
2

90
85
80
75
70
65
60
55
50
45
40
35
30

Fig. 4 Distribution of the relative speeds λ (1)
and of gas convective heat exchange coefficients
α Г (2) along the periphery of the profile contour

25
20
15
10
5
0
0

5 10 15 20 25 30 35 40 45

x, mm

Fig. 3 The cascade of profiles of the
nozzle cooled blade

31


у, mm


x, mm
Fig. 5 Geometrical model with foliation of
design points of contour (1-78) and
equivalent hydraulic schemes reference
sections (1-50) of the experimental nozzle
blade

Fig. 6 The equivalent hydraulic scheme of experimental nozzle blade cooling system
ТB , К
1180
1160
1140
1120
1100
1080
1060
1040
1020
1000

0

10

20

30

40

Fig. 7 Distribution of temperature along outside ( ) and internal ( ) contours of
the cooled nozzle blade

32

Numerical modeling-of-gas-turbine-engines

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