UDK 669.1:621.4:620.179
Original scientific article/lzvirni znanstveni članek
ISSN 1580-2949 MTAEC9, 45(1)21(2011)
A NEW METHOD FOR DETERMINING THE REMAINING LIFETIME OF COATED GAS-TURBlNE
BLADES
NOVA METODA ZA IZRAČUN PREOSTALE TRAJNOSTNE DOBE LOPATIC PLlNSKlH TURBIN
Leonid Borisovich Getsov1, P. G. Krukovski2, N .V. Mozaiskaja1, Aleksander I. Rybnikov1, K. A. Tadlja2
1NPO ZKTl Russian, 2lTTPh NAN Ukraina guetsov@yahoo.com
Prejem rokopisa - received: 2010-02-01; sprejem za objavo - accepted for publication: 2011-01-03
In the course of the exploitation of gas-turbine units (GTE) their turbine blades accumulate defects. These defects result from both the static stress caused by the centrifugal forces and by the thermo-cyclic stress. Since gas turbines usually operate under variable conditions, the time of exploitation cannot be taken as a reliable parameter for a determination of the remaining lifetime. Unfortunately, in many cases, at the moment when a GTE is stopped for maintenance, the thermal conditions of the previous exploitation of the blades are unknown.
ln this presentation a new method of determining the remaining lifetime for coated blades is considered. A long-term strength-decay evaluation is carried out on the basis of the equivalent exploitation temperature Teq, the known exploitation time r and the static stresses a.
Key words: Larson-Miller dependence, turbine blade, remaining lifetime, coating, diffusion, equivalent temperature
Med uporabo plinskih turbin (GTE) se v lopaticah kopičijo napake. Te so posledica statičnih napetosti zaradi centrifugalnih in termo-cikličnih napetosti. Ker plinske turbine obratujejo v spremenljivih razmerah, časa eksploatacije ni mogoče upoštevati kot zanesljivega parametra za določitev preostale trajnostne dobe. Na nesrečo v mnogih primerih, ko je GTE ustavljena zaradi vzdrževanja, niso poznane termične razmere prejšnje uporabe lopatice.
V tem sestavku predstavljamo novo metodo za določanje preostale trajnostne dobe za lopatice s prekritjem. Ocena dolgotrajnega zmanjšanja trdnosti je izvršena na podlagi ekvivalentne temperature uporabe Teq, znanega časa uporabe r in statičnih napetosti a.
Ključne besede: Larson-Miller odvisnost, turbinska lopatica, preostala trajnostna doba, pokritje, difuzija, ekvivalentna temperatura
1 MAIN THESES
Four different cases can be considered:
1.	Uncooled blades, working in constant conditions (at a constant temperature). Blades of this type (type 1) in a stationary mode of operation have a constant temperature and constant stress in each cross section;
2.	Cooled blades in a stationary mode of operation (at a constant gas temperature). ln this case blades have a specific distribution of stresses and temperatures in each cross-section. For the calculation of the remaining lifetime of these blades (type 2) the thermostatic stress is usually omitted and only the static centrifugal stress is used;
3.	Uncooled blades in a variable mode of operation (type 3) - for each mode of operation in each cross-section they have a constant temperature and stress;
4.	Cooled blades working under variable conditions (type 4).
In recent works 1-6 the surface-layer processes in the blades were investigated as a function of the long exposures to high temperature. By means of solving the direct and inverse diffusion problem, the authors have developed some models and methods of prediction for these processes. On the basis of these results the temperature distribution in the surface layer for the weak cross-sec-
tion can be found using an X-ray analysis and a quantitative metallographic analysis of the blade, taken out from the engine. These calculations are based on the temperature dependences of the coating elements' diffusion characteristics. ln this way, for example, an evaluation of the average operating temperatures of the coating, type NiCoCrAlY with the initial concentration of the c(Al) = 8.4 %, was performed6 - Figure 1 (cooled blades, 1N738 alloy; worked for 26 400 h in the turbine).
2 DETERMINATION OF T,
eq
Due to the differences in the temperature dependence for the diffusion coefficients and long-term strength, the temperature TeqD (obtained from the surface-layer elements redistribution) can be used for the remaining lifetime determination only in the case of Type-1 blades. In all the other cases a comparison of Teq" and TeqD is required (see Figure 2).
The method, described below, is based on the following assumptions:
1.	The remaining lifetime of the blade is determined by the remaining lifetime of its bulk metal;
2.	The temperature of the blade cross-section is supposed to be constant and it corresponds to the maximum
Figure 1: Working temperature (°C) distribution around the cross-section contour after exploitation for 26 400 h. The numbered points on the plot correspond to the numbered positions on the blade. Curve 1 - calculated; curves 2 - error range of measurements. Slika 1: Porazdelitev delovne temperature (°C) vzdolž oboda preseka po 26 400 h eksploatacije. Oštevilčene točke označujejo oštevilčene točke na lopatici. Krivulja 1 - izračunano, 2 - območje merilne napake.
temperature value in the maximally loaded cross-section;
3. The damages for different temperatures and different stresses are summed according to a linear law and at the moment of destruction the following relationship is true:
/r,) = 1	(1)
where Ti is the exploitation time at the temperature Ti and under the stress Oi; ti is the time of destruction under the same conditions. It is necessary to mention that for a traditional estimation, it is more appropriate to use "0.87" instead of "1" in the right-hand part of Eq.1.
		
	T2	
		
T,		
Duration at difBerent regimes. Total duiaticii of blade work
ig(i/n
Figure 2: The diagram of the change of temperature during operation (a) and the temperature dependences of long durability and the factors of diffusion (b)
Slika 2: Diagram spremembe temperature med operacijo (a) in temperaturna odvisnost dolge trajnosti in difuzijski faktorji (b)
Let us suppose that we have the following distribution of work-times for a gas-turbine unit: t1 : T2: t3 : T4 : T5 etc. under different conditions with the corresponding power N1, N2, N3, N4, N5 etc. and according to the blades temperatures T1, T2, T3, T4, T5 etc. In this case the time of exploitation can be found as t = aSTi.
Let us use the Larson-Miller dependence:
P(o) = T(C + lg t)
(2)
where t is the time of destruction at the temperature T and stress o. The value of the parameter C (in the first approximation C = 20) can be specified for the involved temperature interval from the experimental long-term strength of the blade material.
Using (2):
t1 = 10 [P(o1)/T1] - C ; t2 = 10 [P(o2)/T2] - C ; t3 = 10 [P(O3)/T3] - C ; t4 = 10 [P(O4)/T4] - C ; t5 = 10 [P(O5)/T5] - C ...
(3)
where Oi is determined by the rotation frequency in the regime Ni, P(Oi) and C can be obtained from the reference data for the corresponding material.
Taking into account that the damage summation is linear (1) and the relationship (3), we can determine the value a. It corresponds to the given time distribution for the different operation regimes T1 : T2: T3 : T4 : T5 etc.
The value of the equivalent temperature Teq can be obtained from the dependences:
T-T O _
T eq "
2,3P( o eq)
ln(Et, /Et, exp(-2,3P(oeq)/ T^)))
P(Oeq) = TeqO (C + lg Tp eq) t T.
E
P( o i) - CT^
= 1
(4)
which parameters are values of the operating time T1 : T2 : T3 : T4 : T5 etc. by the various power settings characterized by the values Ti and Oi. Here, P(oeq) - parameter of dependence of Larson-Miller. Thus, the remaining lifetime of the blades 2-4 after the turbine operation during the time t can be determined for the stress Oeq multiplied by the long-term strength coefficient (obtained from the initial turbine project or taken from the strength standards) as the destruction time at the temperature Teq0 minus the turbine operating time.
If we suppose that the exploitation temperatures changed according to the requirements' specification (or according to the recorded exploitation parameters), then it is possible, as it is described above, to determine the remaining lifetime. Thus, the calculated Teq0 is based on the time ratio T1 : T2 : T3 : T4 : T5 etc., but in most cases this ratio is unknown. For a determination of the real (objective) ratio T1 : T2: T3: T4: T5 etc. the calculated TeqD can be used.
3	METHOD OF DETERMINING THE WORKING-TIME RATIO FOR DIFFERENT REGIMES
By solving the inverse diffusion problem for the experimental Al distribution in the surface layer, the value of the diffusion coefficient can be determined (for the real temperature-change law during the turbine exploitation). Then, if the diffusion temperature dependence is known for the selected element, the value of TeqD can be determined. This temperature (TeqD) will be equivalent to the average temperature determined using the Al diffusion parameters. It can be used as an equivalent temperature for the determination of the residual corrosion lifetime of the coating. But for the determination of the blade's remaining lifetime under applied stress (centrifugal bending stress, or especially thermo-cyclic stress), this temperature (TeqD) must be corrected according to the aforesaid statements.
Let us suppose that the element distribution in the diffusion layer does not depend on the consecution of the temperature-time conditions of the process. Then, by solving the direct diffusion problem, step by step (for different initial conditions), at different temperatures Ti and times of exploitation Xi, the distributions of the Al and oxide layer thickness can be obtained (corresponding to the exploitation times). By comparing this data with the calculated TeqD for the different points of the profile (Figure 1), the ratio of times x1 : x2 : x3 : x4 : x5 etc. can be obtained.
4	METHOD OF TeqD DETERMINATION
Let us consider the processes in the solid coating MeCrAlY (Figure 3).
The oxide Al2O3 is formed during the blade exploitation by a combination of Al and oxygen absorbed from the gas environment and diffused trough the oxide layer (X1 - X0) to the oxide-coating interface marked as x1. The diffusion of Al from the coating occurs in two directions:
• to the oxide-coating interface x1, where it reacts with
diffused oxygen;
Figure 3: Typical Al concentration distribution in the coating layer and the bulk alloy
Slika 3: Tipična porazdelitev koncentracije Al v plasti pokritja in v zlitini
• to the coating-bulk alloy interface x4, where it is accumulated in the inter-diffusion zone and then, depending on the Al concentration in the bulk metal, diffuses either into the bulk alloy orin to the coating. Due to the Al diffusion from the y + ß two-phase coating region, the depleted Al single-phase zones with a reduced concentration of Al (y-phase) are formed on both sides of the coating - on the side of the oxide and on the side of the bulk alloy (depleted zones I and II, Figure 2). All Al, diffused from the coating, is diffused from the y + ß two-phase coating region due to the disappearance (consumption) of the ß-phase. The Al concentration curve in the MCrAlY coating has the shape of a stepped line, and in the bulk alloy region - a curve with the maximum in the inter-diffusion zone.
The process of the Al mass-exchange within the calculation region xi < x < x» can be described with the following diffusion equation:
dC_ dx
d ''dx
D.
dC dx J

(5)
x>0 x, <x<x^ C = C(x,x)
Deff = Deff( x)
W = W(x, X)
x, = x1(x) x2 = x2(x) x3 = x3(x) x5 = x5(x) with the initial condition:
^c: 0<x<x4
C (x,0) = C (x) =
cO
x > x.
the boundary condition for infinity:
dC (x ^, X)
dx
=0
(6)
(7)
and the boundary condition on the moving interface x1, describing the diffusion flux (due to the concentration gradient) of the Al from the coating to the left (Figure 3), forming the oxide:
dCfx, ,x)
Jok(XI + , X)	^^^
The flux (8) conjointly with the flux C(x,, x)(dx, /dx) (formed by moving the interface x1 to the right) produce the total Al flux, which creates an oxide film with a thickness of Ax. The kinetics of the oxide film growth can be described with two parabolic equations:
Ax(x) = x, (x) - x o(x) =
k *
k *

0 < x<x* X* < x<^
(9,10)
where kox*, kox** - are the coefficients of the oxide-film-growth intensity (the so-called constants of oxidation), X* - boundary time point, before it the kinetics obey the oxidation law (9), after - the oxidation law (10). The curves (9) and (10) for intersect at the time point X* ; X0 - time point (negative value) of the intersection for the curve (10) and the time axis X. The
necessity of using two parabolic laws (9) and (10) for the description of the Al-oxide growth can be explained by the faster formation (growth) of the 0-oxide phase before the time point r*, and by the slower growth of the Al2O3 a-oxide phase for r > r*. The coefficients ^ok*, kOK**, r0 and the value r* in (9) and (10) are determined from the experimental data of the oxide growth as a function of time.
The following concentration conditions are taken for the moving boundaries X2 and Xs:
C (X 2 _, r) = C (X 3 +, r) = Cy C(X2+, r) = C(X,_, r) = Cy+^	(11)
Thus, by analogy with the boundary condition (8) for the fixed boundary X4, the diffusion flux of the Al from the coating to the left (which forms the diffusion zone Ay = X5 - x4) and the diffusion of Al to the bulk alloy can be written as:
J Jx X 4 -, r) =
dC(x 4 _ ,r)
dX
(12)
According to the physical model (Figure 3), the flux of the Al (12) from the coating induces a new phase formation in the diffusion zone having the thickness Ay = X5 - X4, i.e.:
Joc(X 4 -, r) = W(X, r) •Ay( r)	(13)
where W is the Al mass, which diffused from the coating to the diffusion zone X4 < X < X5 depending on Cy+ß - Cy and determined by the relationship
Ay(r) = X5(r)-X4 = kD ^r-rD	(14)
The coefficient kD and the value rD0 in (14) are determined from the experimental data of the diffusion-zone increase with time. Ay = X5 - X4 - the diffusion-zone width, which increases with time due to boundary X5 shift to the right according to the parabolic law:
W = W (X, r) =
k w • (C y+ß -C y )m 0
X, < X < X4, X > X5
(15)
where kw is the coefficient of the Al precipitation in the diffusion zone.
Since in the system the coating/oxide film/bulk alloy mass balance should be fulfilled, according to (8) and (12) the total flux of Al from the two-phase coating will take the form:
J2 (r) = JoK (X1 +, r)+Jo, (X4 -, r)
(16)
According to the physical model, all the Al outgoing from the coating, actually outgoes from the y + ß-two-phase coating zone by consuming the ß-phase. Then the movement of the boundaries X2 and X3, and the Al concentration decrease in the y + ß-two-phase zone can be described by the equation of the mass balance between the Al diffusion fluxes through these boundaries
and for the diffusion fluxes due to the Al concentration differences in the y + ß- and y-phases (AC = C y+ß - C y):
J2 = Jy (X 2 -, r)+Jy (X 3 +, r) =
= AC •
dX
dr
■+AC •
dX
where Jy (x 2 -, r) =-D
and Jy (x 3 +, r) =-De,
dr
dC(x 2 - ,r)
eff dX
dC(x 3 + ,r) dx
+ (X3 -X,)-
dC
y+ ß
dr
(17)
are the diffusion fluxes towards the oxide and bulk alloy due to the Al concentration gradients to the left and to the right of the boundaries X2 and X3, respectively.
The expression (17) after dividing by J2 takes the form:
dx,	dx 3
1 = AC ^/ J2 +AC ^ dr 2	dr
/ J 2 +
+(X3 -X2)-
dC
y+ß
dr
J2 = ^ 2 + ^ 3 + ^ 2
(18)
where g2 and g3 are the portions of the total Al mass, which left the coating due to the boundaries X2 and X3 shifting correspondingly; g2,3 = (1 - g2 - g3) - the portion of the total Al mass, which left the coating due to the decrease of the Al ß-phase concentration in the region X2 < x < X3. The values g2 and g3 directly affect the speed of the shifting of the borders X2 and X3 and are taken as functions of the concentration differences
AC = C y+ß - C y:
= k.
AC
'c 0
= AC '3 = k 3 • c 0
(19)
From the equations (15) and (16) it is possible to obtain the laws of the borders X2 and X3 shifting and the law of the decrease of concentration of Al Cy+ß in the two-phase zone:
dx 2 k 2
IT = J 2 ^
dx 3 = J
dr J2 C!
dC
y+ ß
dr
= J2 (X 3 - X 2)(1-g2 - g3)
(20)
the constants ki and ki in (19) can be obtained from the equation (20) using the experimental data of the borders X2 and X3 shifting dynamics and of the C y+ ß plateau value in the region X2 < x < X3 for different times of the tests or exploitation.
The total concentration of Al C y+ ß depends on the
ß-phase concentration C ß (r) in the coating as:
C y+ß (r) = C ß (r) •C^' +[1-C ß (r)]Cy
(21)
where C - concentration of Al in the ß-phase.
The diffusion coefficient Deff, the coefficient kw and the power m in the mathematical model (5)-(21) can be obtained from the experimental data by means of the inverse-problem solution.
The diffusion coefficient Deff in (5) is correct for the whole solution region, excluding the sub-region X2 < x < x3, where it was taken as a greater value due to the absence of the Al-concentration space gradient.
Al accumulated in the diffusion zone partially diffuses back into the coating, due to the absence of the Al-concentration space gradient.
The Al accumulated in the diffusion zone partially diffuses back into the coating, due to the gradient in the concentration to the right of the border x4. The related diffusion flux:
Jmd(x4 +, r) =-De
dC(x 4 + ,T) dx
returns back into the intra-diffusion zone and is added to the main flux (13).
The program complex COLTAN created in the ITTPh NAN Ukraina allows us to calculate the Al concentration distribution in the coating and in the base blade material, TeqD and time ratios r1 : r2 : 73: r4 : r5 etc. in the cooled blades for different operation modes. In the case of the uncooled blades (type 3) in order to obtain the time ratios the inverse problem must be solved for the blades taken from the GTE for different times of operation r1 : r2 : r3 : r4 : r5 etc.
5 ALGORITHM FOR THE CALCULATION OF THE RESIDUAL LIFE TIME
On the basis of the aforesaid the following several steps are used for the determination of the residual lifetime of the coated blade removed from the turbine that has been running for the operation time r:
1)	determination of the parameters of diffusion for the blade coating by means of experiments and calculations,
2)	experimental determination of the distribution of aluminium in different points of the surface layer of the coated blade taken out from the turbine,
3)	calculation of the equivalent temperature TeqD according to a special method based on the solution of the reverse diffusion problem for experimental data on the distribution of aluminium (see step 2),
4)	calculation of the allocation of the time of GTE operation in different modes r1 : r2 : r3 : r4 : r5 etc. = r1 : r2 : r 3 : r 4 : r 5 etc. , corresponding to the power values N1, N2, N3, N4, N5 etc. (operation time r = a Sr,, a ti = art. according to the data obtained during the step 2,
5)	calculation of the equivalent temperature Teq" using equation (4)
6)	calculation of residual lifetime of the set of blades (after the operation during the time t).
The algorithm of the solution of the problem mentioned in step 4 consists of a number of successive calculations of the direct diffusion problem (with different initial conditions) for different temperatures Ti (within the real range for the examined GTE) and operation
times ri (in different modes) as a result of which we obtain the relevant distributions of aluminium and the width of the oxide layer, corresponding to the operation times. Having these set of points for different conditions we chose those in which:
•	the calculated Al distribution coincides with its
experimental distribution,
•	the equivalent temperature coincides with Teq®.
The accord between these data and those obtained with the help of the calculations for the definition TeqD in different points of the blade profile allows us to calculate the ratio of times r1: r2 : r3 : 74 : t5 etc. (see Figure 1, included as an example of the solution of the problem of finding TeqD distributions) allows us to calculate the ratio of times r1 : r2 : r3 : r4 : r5 etc.
ln other words:
1.	Knowing TeqD and the total operation time of the examined blade, we are able to write the following equation representing the sum of solutions of reverse problems for different operating modes contributing to the resulting distribution curve
F1(r, TeqD) =2F1(ri, TD)	(22)
2.	Knowing TeqD, we obtain a number of possible values ti - TiD from the equation (22).
3.	Knowing different values TeqjD for different blade points we find the true values ri - TiD from the equation (22).
4.	Having found the true values ti - TiD (step 3), we determine the value Teq" using the formula (4).
The residual lifetime of the set of blades (after the operating time r) is calculated as the time value before fracture at the temperature Teq" under the stress "eq, multiplied by the strength safety factor (obtained in the course of the strength calculations for blades at the design stage or by the safety factor chosen from the strength standards) minus the elapsed operating time r.
100 150 200 250
Distance, d/|j
300 350 400
Figure 4: Al concentration distribution across the coating layer [NiCoCrAlYRe (Sicoat 2464) coating after exposition at 950 °C] Slika 4: Porazdelitev koncentracije Al na prerezu plasti pokritja [NiCoCrAlYRe (Sicoat 2464) pokritje po žarjenju pri 950 °C]
Figure 5: /S-phase volume fraction distribution across the coating layer (different duration of exposure at 950 °C)
Slika 5: Porazdelitev volumskega deležaS-faze (različno trajanje žar-jenja pri 950 °C)
This experimental data should be obtained for at least three different temperatures in the range of the real gas-turbine operating conditions with several time exposures that should bring meaningful results.
For the coatings of a different type, an element sensible to the working-temperature history should be taken as a diffusing element.
Some illustrative examples of the NiCoCrAlReY coating study on the alloy Rene80 can be seen in Figure 4-5.
The samples were isothermally exposed for times up to 20 000 h at the temperatures 900-980 °C. From Figure 4 it can be seen that on the surface (up to 40 pm depth) and at the interface coating-alloy (at 180-220 pm) the formation of a de-alloyed layer is observed.
7 CONCLUSION
As a basis for the above-mentioned calculations for MeCrAlY coatings the experimental data obtained in the present work can be used, namely:
•	determination of the Al concentration distribution on the basis of the coating-layer thickness using the X-ray microanalysis method;
•	determination of the volumetric share of the S-phase on the basis of the coating-layer thickness using digital optical metallography (for example, with the help of software made by "IstaVideoTest");
•	measurement of the oxide film thickness on the coating surface and the thicknesses of inner and outer de-alloyed layers.
These experimental data must be obtained for not less than three temperatures from the real GTE temperature range and for several exposures that must produce meaningful results for the sought parameters.
6 X-RAY AND METALLOGRAPHIC ANALYSIS METHOD
The experimental data required for the above calculations for the MeCoCrAlY coatings include:
•	a determination of the Al concentration distribution across the coating layer using the X-ray microana-lysis method;
•	a determination of the S-phase volume-fraction change across the coating layer using Digital Optical Metallography (for example with the software made by "IstaVideoTest");
•	measurements of the oxide film thickness on the surface of the coating, and the inner and outer de-alloyed layers thicknesses (Figure 3).
A new method of determining the remaining lifetime of coated blades was developed for stationary conditions. This method requires one blade's removal from the turbine after a long operating time, metallographic analysis and calculations using the data obtained from the laboratory experiments performed in advance.
The method is based on the Larson-Miller dependence, the law of linear summation of damages, and the assumption that the elements-distribution in the diffusion layer of the coating does not depend on the sequence of the temperature-time conditions.
8 REFERENCES
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