HAL Id: hal-01104123
https://hal-supelec.archives-ouvertes.fr/hal-01104123
Submitted on 16 Jan 2015
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Eddy-current NDE of combustion turbine blade
coatings. Determination of conductivity profiles in the
presence of a diffusion process
Frédéric Nougier, Marc Lambert, Riadh Zorgati
To cite this version:
Frédéric Nougier, Marc Lambert, Riadh Zorgati. Eddy-current NDE of combustion turbine blade
coatings. Determination of conductivity profiles in the presence of a diffusion process. ENDE’08, Jun
2008, Séoul, South Korea. �hal-01104123�
Context of the study Theoretical formulation Validation
Eddy-current NDE of combustion turbine blade
coatings. Determination of conductivity profiles in
the presence of a diffusion process
Frédéric Nougier Marc Lambert Riadh Zorgati
Département de Recherche en Électromagnétisme,
Laboratoire des Signaux et Systèmes (UMR8506 CNRS-SUPELEC-Univ. Paris 11), 3, rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France
ENDE 2008
Context of the study Theoretical formulation Validation
Context and configuration of the study
Eddy-Current measurements over combustion turbine blade coatings affected by depletion of aluminium;
Model taking inward and outward depletion of aluminum inside the coating into account;
Conductivity profile follows a two-hyperbolic-tangent law;
Analytical formulation of the variation of impedance obtained combining the approaches found in [1, 2, 3]
Air
Substrat Interdiffusion zone Coating (reservoir of aluminium)
Protective oxide zone z
σ(z)
r1 r2 h2
h1
−r Zone 1a
Zone 1b Zone 1c
Zone 2
Zone 3
0
Context of the study Theoretical formulation Validation
General formulation
Formulation for a two-tanh profile
General formulation
A(r) = A(r ,z) uθ+ Aec(r ,z) = R(r )W (z)– lead to
∂2
∂r2R(r ) +1 r
∂
∂rR(r ) + µ
a2− 1 r2
¶
R(r ) = 0 (1)
∂2
∂z2W(z) =h
a2+j ωµσ(z)i
W(z) (2)
Following [2] the general solution given by
W(z) = CF1(f (z)) + BF2(f (z)) (3) where F1and F2typical mathematical functions related to σ(z) Expression of
A1c(r ,z) = Z+∞
0
µNII(r1,r2)e−azJ1(ar )³
e−ah1−e−ah2´ 2a3(h2−h1)(r2−r1) da +
Z+∞
0 C1e−azJ1(ar )da, with I (r1,r2) = Zar2
ar1
xJ1(x)dx ∀z ∈[0;+∞[
A2(r ,z) = Z+∞
0 [C2F1(f (z)) + B2F2(f (z))] J1(ar )da ∀z ∈[−r ;0]
A3(r ,z) = Z+∞
0 B3F3(g (z))J1(ar )da ∀z ∈] − ∞;−r ]
Context of the study Theoretical formulation Validation
General formulation
Formulation for a two-tanh profile
A1c(r,z) known, then Z deduced as
Z=K
+∞
Z
0 I(r1,r2)2
a6
·
2³e−a(h2−h1)−1+a(h2−h1)´+³e−ah2−e−ah1´2φ(a)
¸ da
(4)
withφ(a)=C1 K;K =
µNII(r1,r2)³e−ah1−e−ah2´
2a3(h2−h1)(r2−r1) ;K= −jωπµN2 (h2−h1)2(r2−r1)2. Continuity conditions of the quantities and/or their derivatives with respect z and/or their cancellation at±∞expressφ(a)as
φ(a)=(aM−O)RT−(aL−N)ST+[a(LQ−MP)−NQ+OP] U (aM+O)RT−(aL+N)ST+[a(LQ−MP)+NQ−OP] U (5) where
L=F1(f (z=0)); M=F2(f (z=0)); N=F1′(f (z))¯¯z=0; O=F2′(f (z))¯¯z=0; P=F1(f (z= −r)); Q=F2(f (z= −r)); (6) R=F1′(f (z))¯¯z=−r; S=F2′(f (z))¯¯z=−r; T=F3(g (z= −r));
U=F3′(g (z))¯¯z=−r
′means derivative with respect to z
Context of the study Theoretical formulation Validation
General formulation
Formulation for a two-tanh profile
Formulation for a two-tanh profile
Conductivity profile given by
σ(z)=
σ12+σ1−σ12 2
· 1+tanh
µz+c1 2v1
¶¸
∀z∈[−r,0]
σ2+σ12−σ2 2
· 1+tanh
µz+c2
2v2
¶¸
∀z∈]− ∞, −r] (7)
Particular functions F1,F2and F3are
F1(y2(z))=y2µ(z) [1−y2(z)]νF(µ + ν, µ + ν +1,2µ +1;y2(z)) (8) F2(y2(z))=y2−µ(z)[1−y2(z)]νF(ν − µ +1,ν − µ, −2µ +1;y2(z)) (9) F3(y3(z))=y3λ(z) [1−y3(z)]τF(λ + τ, λ + τ +1,2λ +1;y3(z)) (10)
with y2(z)= µ
1+e−
z+c1 v1
¶−1
, y3(z)= µ
1+e−
z+c2 v2
¶−1
µ =v1 q
a2+jωµ0σ12, ν =v1 q
a2+jωµ0σ1 λ =v2
q
a2+jωµ0σ2, τ =v2 q
a2+jωµ0σ12
(11)
F(α, β, γ;x) is the hypergeometric function
Context of the study Theoretical formulation Validation
Single tanh-profile Two-tanh-profile
Description of the configuration
r
11.3 mm
r
23.3 mm
h
10.5 mm
h
27.8 mm
N
turn580
σ1
1.883 10
7S m−1 σ123.766 10
7S m−1c
10.3 mm
v
10.1857 mm
1 1.5 2 2.5 3 3.5 x 1074−15
−10
−5 0
x 10−4
Conductivité
Profondeur
Profil de conductivité
Comparison with the results given in [1]
Real part of Z , (b=NR2) Imaginary part of Z , (b=NR2) Frequency from [1] N=10 N=20 from [1] N=10 N=20 1kHz 0.00817 0.008169 0.008165 −0.00828 −0.008267 −0.00828 10kHz 0.02583 0.02585 2.5823 −0.22571 −0.22557 −0.22566 100kHz -0.68836 -0.68799 -0.6882 −1.49719 −1.49645 −1.496769
Context of the study Theoretical formulation Validation
Single tanh-profile Two-tanh-profile
Description of the configuration
r1=2.0mm σ1=6 105S m−1 r2=4.0mm σ12=9 105S m−1 h1=0.5mm σ2=8 105S m−1 h2=7.3mm Nturn=200 c1=0.2mm v1=0.03mm c2=0.8mm v2=0.1mm
4 5 6 7 8 9 10
x 105
−14
−12
−10
−8
−6
−4
−2 0
x 10−4
Conductivite
Profondeur
Comparison with the results obtained from a multi-layer model
0 2 4 6 8 10
x 106
−100
−90
−80
−70
−60
−50
−40
−30
−20
−10 0
Our model N = 1000 N = 10000 N = 100000 N = 1000000
Realpartof∆Z
Frequency (Hz) 0 2 4 6 8 x 10106
0 100 200 300 400 500 600 700 800 900 1000
Our model N = 1000 N = 10000 N = 100000 N = 1000000
Imaginarypartof∆Z
Frequency (Hz)
Context of the study Theoretical formulation Validation
Single tanh-profile Two-tanh-profile
