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Sensitivity to experimental errors in evaluating the thermal expansion coefficient of a thermal barrier coating by using coating system specimens
Vacuum 88 (2013) 93e97
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Vacuum journal homepage: www.elsevier.com/locate/vacuum
Sensitivity to experimental errors in evaluating the thermal expansion coefficient of a thermal barrier coating by using coating system specimens Hiroyuki Waki a, *, Izuru Nishikawa b, Akira Kobayashi c, Noriaki Ishii d a
Department of Mechanical Engineering, Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka, Iwate 020-8551, Japan Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1 Ohmiya, Asahi-ku, Osaka 535-8585, Japan c Joining and Welding Research Institute, Osaka University, 11-1 Mihogaoka, Ibraki, Osaka 567-0047, Japan d Department of Mechanical Engineering, Osaka Electro-Communication University, 18-8 Hatsu-cho, Neyagawa, Osaka 572-8530, Japan b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 November 2011 Received in revised form 24 February 2012 Accepted 6 March 2012
The coefficients of thermal expansion (CTEs) of coatings were necessary to evaluate the thermal stresses of the coated component. In general, the CTEs were measured from the free-standing coatings. It is useful if the CTEs of the coatings are able to be measured from the coating system specimens. The resulting CTE of a coating by the coating system is sensitive to experimental errors when compared with that for the free-standing coating. In this study, the sensitivity to experimental errors in evaluating the CTEs from the coating system specimens was investigated. The experimental study was also carried out on six kinds of specimens with different sensitivities. The surface strain during thermal cycling from 850 C to 300 C measured by a laser speckle method was used to evaluate the CTE. The CTEs of Yttria-stabilized-zirconia thermal barrier coating and CoNiCrAlY bond coatings were measured by a composite cylinder model. It was found from the sensitivity calculation that the errors of the CTE of the substrate and the measuring slope between strain and temperature caused a large error in the resulting CTE of the coating. The sensitivity calculation also showed that the increase of the thickness of the coating was effective in decreasing the sensitivities. A reasonable estimation was experimentally obtained if the specimens had low sensitivities. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Thermal barrier coating Coefficient of thermal expansion CoNiCrAlY Plasma spray Laser speckle
1. Introduction Thermal barrier coatings (TBCs) with low thermal conductivities are employed in gas turbine and aircraft engines in order to decrease the temperatures of super-alloy substrates [1]. The TBC system consists of a TBC, a bond-coat and a substrate. The bondcoat relaxes the thermal stresses due to the mismatch of the coefficient of thermal expansion (CTE) between the TBC and the substrate. The bond-coat has been also employed for the protective coating from oxidation and corrosion of the substrate. The CTEs of the coatings were necessary to evaluate the thermal stresses of the TBC system. In general, the CTEs of TBCs have been measured from the free-standing coatings and almost all the data reported is from the free-standing coatings [2e6]. Beghini et al. [7] measured the CTEs from the elongation of the coating system specimen with
* Corresponding author. Fax: þ81 19 621 6408. E-mail address: [email protected] (H. Waki). 0042-207X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2012.03.018
a length of 80 mm. The method is useful because the manufacturing process of the free-standing coating can be omitted. Strain measurement instead of elongation measurement could allow the specimen length to be short. In this study, the strain method of a composite cylinder model for the CTE of a coating from the small piece of the coating system specimen was investigated. There are two problems in the method: (1) The resulting CTE of a coating by the coating system is sensitive to the experimental errors as compared with that for the free-standing coating. (2) The composite cylinder model is satisfied over the area apart from the ends of the specimen because of the edge effect. First, the sensitivities to the errors on measuring the strainetemperature relation and the other necessary input parameters were calculated. The purpose of the calculation is to reveal what input parameter is the most important and how to decrease the sensitivity. Next the necessary minimum length of the specimen was calculated by FEM analysis. Finally, an experimental study was also carried out on six kinds of specimens with different sensitivities in order to investigate the accuracy of the method.
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H. Waki et al. / Vacuum 88 (2013) 93e97
2. Sensitivity study of the composite cylinder model
CTCslope ¼
Es As as þ Eb Ab ab þ Ec Ac ac Ec Ac ac
(4)
CBCslope ¼
Es As as þ Eb Ab ab Eb Ab ab
(5)
2.1. Composite cylinder model The composite cylinder models shown in Fig. 1 are employed for the evaluation of the CTEs of the coatings. The ends of the cylinders are constrained by rigid bodies, and the interfaces between cylinders are not connected. Eq. (1) is obtained from the equilibrium of the axial normal forces and the equilibrium of the axial normal strains of the cylinders. The CTE of the TBC top-coat ac is evaluated by the three cylinders model with the top-coat, a bond-coat and a substrate as Eq. (1).
ac ¼
Es As þ Eb Ab þ Ec Ac D3 c Es As E A a b ba $ DT Ec Ac s Ec Ac b Ec Ac
(1)
Here E is Young’s modulus, A is area of cross-section, a is CTE, D3 /DT is the slope of axial normal strain range to the temperature range. The subscripts s, b and c denote the substrate, the bond-coat and the ceramic top-coat, respectively. In the calculation, the CTEs of the sub-surface cylinders, the Young’s modulus of every cylinder and the dimension of every cylinder with the measuring slope D3 c/DT are necessary. These input errors affect the resulting ac. The CTE of the bond-coat can also be evaluated from the two cylinders model with the bond-coat and a substrate as Eq. (2).
ab ¼
Es As þ Eb Ab D3 b Es As a $ DT Eb Ab s Eb Ab
(2) 2.3. Effect of input errors on the resulting CTE
2.2. Sensitivity study An error propagation theory [7,8] was employed in the sensitivity study. We consider the objective CTE as a function 4 of the independent variables xj. The Cxj shown in Eq. (3) indicates how strongly a relative error in xj affects the relative error in 4(x1, x2,.,xn).
Cxj ¼
The sensitivity coefficients were calculated for several dimensions shown in Table 1. The cylinders are circular tubes and h represents the thickness of each cylinder. The outer radius of the substrate was fixed as 5 mm. Only the thickness of the substrate was variable. The mechanical properties used are also shown in Table 1. The relationship between the sensitivity coefficients calculated and the area ratio Ac/(Ab þ As) for the TC model and Ab/As for the BC model is shown in Fig. 2. The sensitivity coefficients increase with a decrease of the area ratio. It is found from Fig. 2 that CTC-slope is much higher than CBC-slope. This means the top-coat is sensitive to the D3 c/DT error and it is mainly because the Young’s modulus of the top-coat is significantly lower than that of the substrate. The CTEs with a D3 /DT error were calculated from Eqs. (1) and (2) under the conditions shown in Table 1. The vertical axis of Fig. 3 shows the resulting a with a 5% D3 /DT error normalized by the true a without an error. It is understood from Fig. 3 that only the 5% slope error leads the critical error in the resulting a if the sensitivity coefficient is high.
xj v4ðx1 ; x2 ; .; xn Þ $ vxj 4ðx1 ; x2 ; .; xn Þ
(3)
The sensitivity coefficients Cxj are also called condition numbers. The sensitivity coefficients for the slope D3 /DT are derived as Eqs. (4) and (5) for the ceramic top-coat and the bond-coat, respectively.
Rigid body
As understood from Eqs. (1) and (2), the CTEs of sub-surface cylinders, the Young’s modus of every cylinder and the dimension of every cylinder as well as the measuring slope D3 /DT are necessary for the calculation. The CTEs are affected by these input errors. The CTEs with a certain error were calculated from Eqs. (1) and (2) under the conditions shown in Table 1. Here, a certain error is added on only a watching parameter under the condition that the other parameters are without errors. Figs. 4 and 5 show the evaluated a with a certain error normalized by the true a without an error. The errors are 5% on the slope D3 /DT, the CTEs of sub-surface cylinders and the Young’s modulus of every coating, respectively, and þ30 mm on the dimension of every coating. The horizontal axis of Figs. 4 and 5 shows the ratio of area Ac/(Ab þ As) for the TC model and the ratio Ab/As for the BC model. It is found from Figs. 4 and 5
Axial symmetry
Rigid body
Axial symmetry Bond coat
Bond coat
Top coat
Substrate
Substrate
a TC model
b BC model Fig. 1. Composite cylinder models.
H. Waki et al. / Vacuum 88 (2013) 93e97
95
Table 1 Input values used in the calculation.
BC model
Sub BC Sub BC TC
TC model
Thickness h [mm]
E [GPa]
a [106/K]
4.85, 1.1, 0.75, 0.4, 0.3 0.15 4.85, 2, 1.35, 0.7, 0.45 0.15 0.3
160 140 160 140 35
20 16 20 16 11
that the errors of as and D3 /DT affect data critically if the ratio of area is low. On the contrary, the other input values are insignificant to the resulting a. It is found that as and D3 /DT are very important input factors for accurate evaluation. The sensitivity coefficients for the as are derived as Eqs. (6) and (7) for the ceramic top-coat and the bond-coat, respectively.
E s As a s CTCas ¼ Ec Ac ac
(6)
E s As a s CBCas ¼ Eb Ab ab
(7)
These coefficients are negative, which means a positive as error gives negative estimation in the CTE. The sensitivity coefficients for the slope D3 /DT are a positive effect as shown in Eqs. (4) and (5). The sensitivity coefficients for the other input values are significantly low as compared with Eqs. (4)e(7).
Δ ⁄Δ
Fig. 3. The influence of the measuring slope D3 /DT error on the accuracy of the resulting CTE plotted against the sensitivity coefficients of the D3 /DT.
were adopted for the BC model and the TC model, respectively. The outer radii of the substrates were fixed as 5 mm. 3.2. Necessary minimum length of the specimen The composite cylinder model is actually satisfied on the area apart from the ends of the specimen because of the edge effect. An FEM strain calculation was conducted by a commercial code,
3. Experimental study 3.1. Specimen
Δ
C BC−slope , C TC−slope
Examples of the composite cylinder specimens used in this study are shown in Fig. 6. The bond-coats were CoNiCrAlYs by atmospheric plasma spraying (APS) and high velocity oxy-fuel (HVOF) spraying. The TBC top-coat was 8 wt%-yttria-stabilizedzirconia (YSZ) by APS. Spraying conditions are shown in Table 2. Table 3 shows the mechanical properties of the specimen. The Young’s moduli and the Poisson ratios are cited values and the CTEs were measured from the free-standing coatings by the same laser speckle method described in Section 3.3. The free-standing coatings were manufactured from other specimens by dissolving out the mild steel substrate by nitric acid. Table 4 shows the dimensions of the specimens. Three thicknesses of the substrate, which means three types of sensitivities,
60
: BC model : TC model
Δ
⁄Δ
⁄Δ
Fig. 4. The influences of the input errors on the accuracy of the resulting ac plotted against the ratio of area Ac/(Ab þ As) on the TC model.
Δ
⁄Δ
40
20 Δ 0 0
0.2 0.4 A b/A s , A c/(A b+A s)
⁄Δ
0.6
Fig. 2. The relationship between the sensitivity coefficients calculated and the area ratios Ac/(Ab þ As) for the TC model and Ab/As for the BC model.
Fig. 5. The influences of the inputs error on the accuracy of the resulting ab plotted against the ratio of area Ab/As on the BC model.
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H. Waki et al. / Vacuum 88 (2013) 93e97 Table 3 Mechanical properties of the coatings.
E [GPa] (850e300 C)
n a [106/ C] (850e300 C)
Fig. 6. Examples of the composite cylinder specimens.
ANSYS. 2-dimensional 8-node element and linear isotropic elasticity were employed. The dimensions and mechanical properties are shown in Tables 3 and 4. Fig. 7 shows the relationship between the normalized FEM strain at the center surface and the normalized specimen length. The vertical axis of Fig. 7 is normalized by the axial strain of the composite cylinder model. The horizontal axis is normalized by the outer diameter of the substrate. If the value of vertical axis is near to 1.0, it means that the composite cylinder model is a reasonable model. Fig. 7 shows the center surface strain approximately satisfy the composite cylinder model if the ratio of the length of the specimen to the diameter of the substrate is l/d 1.5 for the BC model, or, l/d 2.0 for the TC model. Therefore, the specimen length was adopted as 15 mm for the BC model and 20 mm for the TC model when the diameter of substrate was 10 mm.
Substrate SUS304
Bond-coat CoNiCrAlY HVOF
APS
159 0.30 21.27
148 [9] 0.32 16.30
112 0.29 14.74
Top-coat APS YSZ 34.4 [9] 0.17 7.59
models should be located between those of the substrate and the free-standing coating. The strain of the BC model is located between those of the substrate and the coating. However, the strain of the TC model is unfortunately slightly higher than that of the substrate because of the strain measurement error at high temperature. The error will cause a critical error in ac. Fig. 9 shows the experimentally estimated CTEs plotted against the sensitivity coefficient CBC-slope, CTC-slope. The resulting a was normalized by the free-standing CTE afree-standing shown in Table 3. If the ratio is near to 1.0 it means that a reasonable estimation is obtained. The ratio a/afree-standing approaches to 1.0 when the
Table 4 The dimensions of the composite cylinder specimens. h [mm] BC model
Substrate Bond-coat (APS) Substrate Bond-coat (HVOF) Top-coat
TC model
5, 0.72, 0.29 0.145 5, 1.36, 0.41 0.2 0.31
a
3.3. Experimental procedure Thermal cycling was applied to the specimen by induction heating and natural cooling. The surface temperature at the center surface was measured by an emission thermo spot sensor. It was confirmed that the temperature of the substrate and the coatings was the same from 850 C to 300 C. The axial strain at the center surface during cooling process from 850 C to 300 C was measured by a laser speckle method [10,11]. The gauge area was inner circle of approximately 1.2 mm diameter. The measured strain was the average of the area. 3.4. Experimental result Examples of temperatureestrain curves during thermal cycling are shown in Fig. 8. The mean slope D3 /DT from 850 C to 300 C was evaluated by the least square method. The curves of composite
b
Table 2 Spraying conditions. Bond-coat CoNiCrAlY
Spraying gun Current [A] Voltage [V] O2 [MPa] Propylene [MPa] Air [MPa] Ar [MPa] H2 [MPa] He [MPa] Distance [mm]
HVOF
APS
Metco DJ e e 0.30 0.27 0.34 e e e 150
Metco 9MB 500 60 e e e 0.69 0.48 e 150
Top-coat APS YSZ
Plasma Dyne SG-100 850 35 e e e 0.41 e 0.62 100
Fig. 7. The relationship between the normalized FEM strain at the center surface and the normalized specimen length, on (a) the TC model, and (b) the BC model.
H. Waki et al. / Vacuum 88 (2013) 93e97
a
97
4. Conclusion In order to obtain the CTE of a thermal barrier coating from the coating system specimen, the strain method of a composite cylinder model was investigated. The resulting CTE of the coating by the coating system was sensitive to the experimental errors as compared with that of the free-standing coating. The sensitivities to experimental errors on the measuring strainetemperature relation and the other necessary input parameters were calculated. Experimental verification was also carried out on the six kinds of specimens with different sensitivities in order confirm the accuracy of the method.
b
(1) It was found from the sensitivity study that slight errors in as and D3 /DT caused a critical error in the resulting a when the sensitivity coefficient was high. The as and D3 /DT were very important input values for accurate evaluation. On the contrary, the other input values were insignificant to the resulting a. (2) Sensitivity study showed that low sensitivity coefficient was necessary to obtain the reasonable CTE. The increase of thickness of the coating as compared with the substrate was effective to decrease the sensitivity coefficient. (3) Reasonable estimation was experimentally obtained if the specimen had the low sensitivity to experimental errors. Acknowledgments
Fig. 8. Examples of temperatureestrain curves during thermal cycling, on (a) the TC model, and (b) the BC model.
This work was supported by KAKENHI (22760088). Authors also thank Messrs. T. Maekawa and Y. Ikebe (Graduate students of Osaka Electro-Communication University) for experimental works. References
Fig. 9. The accuracy of the experimentally estimated CTEs plotted against the sensitivity coefficient CBC-slope, CTC-slope.
sensitivity coefficient is low in both the TC and BC models. It is found that the BC model gives reasonable ab if the sensitivity coefficient is low, which means the area ratio Ab/As is high. On the contrary, a much lower sensitivity coefficient is necessary in the TC model. Approximately only 8% error in D3 /DT caused such a very high error of ac in the case of the TC model in Fig. 9.
[1] Feuerstein A, Knapp J, Taylor T, Ashary A, Bolcabage A, Hitchman N. Technical and economical aspects of current thermal barrier coating systems for gas turbine engines by thermal spray and EBPVD. J Therm Spray Technol 2008;17: 199e213. [2] Brindley WJ. Properties of plasma-sprayed bond coats. J Therm Spray Technol 1997;6:85e90. [3] Liang JJ, Wei H, Zhu YL, Sun XF, Hu ZQ, Dargusch MS, et al. Phase constituents and thermal expansion behavior of a NiCrAlYRe coating alloy. J Mater Sci 2011;46:500e8. [4] Chen H, Liu Y, Gao Y, Tao S, Luoz H. Design, preparation, and characterization of graded YSZ/La2Zr2O7 thermal barrier coatings. J Am Ceram Soc 2010;93: 1732e40. [5] Girolamo GD, Blasi C, Pagnotta L, Schioppa M. Phase evolution and thermophysical properties of plasma sprayed thick zirconia coatings after annealing. Ceramics Int 2010;36:2273e80. [6] Xie X, Guoa H, Gonga S, Xua H. Lanthanumetitaniumealuminum oxide: a novel thermal barrier coating material for applications at 1300 C. J Eur Ceram Soc 2011;31:1677e83. [7] Beghini M, Bertini L, Frendo F. Thermal expansion coefficient of a NiCoCrAlYeZrO2(Y2O3) coating by a digital image processing based dilatometer. Int J Mater Prod Technol 2000;15:78e90. [8] Stoer J, Bulirsch R. Introduction to numerical analysis. 3rd ed. New York: Springer-Verlag; 2002. p. 13. [9] Beghini M, Benamati G, Bertini L, Frendo F. Measurement of coatings’ elastic properties by mechanical methods: part 2. Application to thermal barrier coatings. Exp Mech 2001;41:305e11. [10] Nishikawa I, Ogura K, Yamagami M, Kuwayama K. J Soc Mater Science Jpn 1994;43:1290e6. [11] Waki H, Ogura K, Nishikawa I. Trans Jpn Soc Mech Engineers Ser A 2000;66: 923e30.
Contents lists available at SciVerse ScienceDirect
Vacuum journal homepage: www.elsevier.com/locate/vacuum
Sensitivity to experimental errors in evaluating the thermal expansion coefficient of a thermal barrier coating by using coating system specimens Hiroyuki Waki a, *, Izuru Nishikawa b, Akira Kobayashi c, Noriaki Ishii d a
Department of Mechanical Engineering, Faculty of Engineering, Iwate University, 4-3-5 Ueda, Morioka, Iwate 020-8551, Japan Department of Mechanical Engineering, Osaka Institute of Technology, 5-16-1 Ohmiya, Asahi-ku, Osaka 535-8585, Japan c Joining and Welding Research Institute, Osaka University, 11-1 Mihogaoka, Ibraki, Osaka 567-0047, Japan d Department of Mechanical Engineering, Osaka Electro-Communication University, 18-8 Hatsu-cho, Neyagawa, Osaka 572-8530, Japan b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 November 2011 Received in revised form 24 February 2012 Accepted 6 March 2012
The coefficients of thermal expansion (CTEs) of coatings were necessary to evaluate the thermal stresses of the coated component. In general, the CTEs were measured from the free-standing coatings. It is useful if the CTEs of the coatings are able to be measured from the coating system specimens. The resulting CTE of a coating by the coating system is sensitive to experimental errors when compared with that for the free-standing coating. In this study, the sensitivity to experimental errors in evaluating the CTEs from the coating system specimens was investigated. The experimental study was also carried out on six kinds of specimens with different sensitivities. The surface strain during thermal cycling from 850 C to 300 C measured by a laser speckle method was used to evaluate the CTE. The CTEs of Yttria-stabilized-zirconia thermal barrier coating and CoNiCrAlY bond coatings were measured by a composite cylinder model. It was found from the sensitivity calculation that the errors of the CTE of the substrate and the measuring slope between strain and temperature caused a large error in the resulting CTE of the coating. The sensitivity calculation also showed that the increase of the thickness of the coating was effective in decreasing the sensitivities. A reasonable estimation was experimentally obtained if the specimens had low sensitivities. Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Thermal barrier coating Coefficient of thermal expansion CoNiCrAlY Plasma spray Laser speckle
1. Introduction Thermal barrier coatings (TBCs) with low thermal conductivities are employed in gas turbine and aircraft engines in order to decrease the temperatures of super-alloy substrates [1]. The TBC system consists of a TBC, a bond-coat and a substrate. The bondcoat relaxes the thermal stresses due to the mismatch of the coefficient of thermal expansion (CTE) between the TBC and the substrate. The bond-coat has been also employed for the protective coating from oxidation and corrosion of the substrate. The CTEs of the coatings were necessary to evaluate the thermal stresses of the TBC system. In general, the CTEs of TBCs have been measured from the free-standing coatings and almost all the data reported is from the free-standing coatings [2e6]. Beghini et al. [7] measured the CTEs from the elongation of the coating system specimen with
* Corresponding author. Fax: þ81 19 621 6408. E-mail address: [email protected] (H. Waki). 0042-207X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2012.03.018
a length of 80 mm. The method is useful because the manufacturing process of the free-standing coating can be omitted. Strain measurement instead of elongation measurement could allow the specimen length to be short. In this study, the strain method of a composite cylinder model for the CTE of a coating from the small piece of the coating system specimen was investigated. There are two problems in the method: (1) The resulting CTE of a coating by the coating system is sensitive to the experimental errors as compared with that for the free-standing coating. (2) The composite cylinder model is satisfied over the area apart from the ends of the specimen because of the edge effect. First, the sensitivities to the errors on measuring the strainetemperature relation and the other necessary input parameters were calculated. The purpose of the calculation is to reveal what input parameter is the most important and how to decrease the sensitivity. Next the necessary minimum length of the specimen was calculated by FEM analysis. Finally, an experimental study was also carried out on six kinds of specimens with different sensitivities in order to investigate the accuracy of the method.
94
H. Waki et al. / Vacuum 88 (2013) 93e97
2. Sensitivity study of the composite cylinder model
CTCslope ¼
Es As as þ Eb Ab ab þ Ec Ac ac Ec Ac ac
(4)
CBCslope ¼
Es As as þ Eb Ab ab Eb Ab ab
(5)
2.1. Composite cylinder model The composite cylinder models shown in Fig. 1 are employed for the evaluation of the CTEs of the coatings. The ends of the cylinders are constrained by rigid bodies, and the interfaces between cylinders are not connected. Eq. (1) is obtained from the equilibrium of the axial normal forces and the equilibrium of the axial normal strains of the cylinders. The CTE of the TBC top-coat ac is evaluated by the three cylinders model with the top-coat, a bond-coat and a substrate as Eq. (1).
ac ¼
Es As þ Eb Ab þ Ec Ac D3 c Es As E A a b ba $ DT Ec Ac s Ec Ac b Ec Ac
(1)
Here E is Young’s modulus, A is area of cross-section, a is CTE, D3 /DT is the slope of axial normal strain range to the temperature range. The subscripts s, b and c denote the substrate, the bond-coat and the ceramic top-coat, respectively. In the calculation, the CTEs of the sub-surface cylinders, the Young’s modulus of every cylinder and the dimension of every cylinder with the measuring slope D3 c/DT are necessary. These input errors affect the resulting ac. The CTE of the bond-coat can also be evaluated from the two cylinders model with the bond-coat and a substrate as Eq. (2).
ab ¼
Es As þ Eb Ab D3 b Es As a $ DT Eb Ab s Eb Ab
(2) 2.3. Effect of input errors on the resulting CTE
2.2. Sensitivity study An error propagation theory [7,8] was employed in the sensitivity study. We consider the objective CTE as a function 4 of the independent variables xj. The Cxj shown in Eq. (3) indicates how strongly a relative error in xj affects the relative error in 4(x1, x2,.,xn).
Cxj ¼
The sensitivity coefficients were calculated for several dimensions shown in Table 1. The cylinders are circular tubes and h represents the thickness of each cylinder. The outer radius of the substrate was fixed as 5 mm. Only the thickness of the substrate was variable. The mechanical properties used are also shown in Table 1. The relationship between the sensitivity coefficients calculated and the area ratio Ac/(Ab þ As) for the TC model and Ab/As for the BC model is shown in Fig. 2. The sensitivity coefficients increase with a decrease of the area ratio. It is found from Fig. 2 that CTC-slope is much higher than CBC-slope. This means the top-coat is sensitive to the D3 c/DT error and it is mainly because the Young’s modulus of the top-coat is significantly lower than that of the substrate. The CTEs with a D3 /DT error were calculated from Eqs. (1) and (2) under the conditions shown in Table 1. The vertical axis of Fig. 3 shows the resulting a with a 5% D3 /DT error normalized by the true a without an error. It is understood from Fig. 3 that only the 5% slope error leads the critical error in the resulting a if the sensitivity coefficient is high.
xj v4ðx1 ; x2 ; .; xn Þ $ vxj 4ðx1 ; x2 ; .; xn Þ
(3)
The sensitivity coefficients Cxj are also called condition numbers. The sensitivity coefficients for the slope D3 /DT are derived as Eqs. (4) and (5) for the ceramic top-coat and the bond-coat, respectively.
Rigid body
As understood from Eqs. (1) and (2), the CTEs of sub-surface cylinders, the Young’s modus of every cylinder and the dimension of every cylinder as well as the measuring slope D3 /DT are necessary for the calculation. The CTEs are affected by these input errors. The CTEs with a certain error were calculated from Eqs. (1) and (2) under the conditions shown in Table 1. Here, a certain error is added on only a watching parameter under the condition that the other parameters are without errors. Figs. 4 and 5 show the evaluated a with a certain error normalized by the true a without an error. The errors are 5% on the slope D3 /DT, the CTEs of sub-surface cylinders and the Young’s modulus of every coating, respectively, and þ30 mm on the dimension of every coating. The horizontal axis of Figs. 4 and 5 shows the ratio of area Ac/(Ab þ As) for the TC model and the ratio Ab/As for the BC model. It is found from Figs. 4 and 5
Axial symmetry
Rigid body
Axial symmetry Bond coat
Bond coat
Top coat
Substrate
Substrate
a TC model
b BC model Fig. 1. Composite cylinder models.
H. Waki et al. / Vacuum 88 (2013) 93e97
95
Table 1 Input values used in the calculation.
BC model
Sub BC Sub BC TC
TC model
Thickness h [mm]
E [GPa]
a [106/K]
4.85, 1.1, 0.75, 0.4, 0.3 0.15 4.85, 2, 1.35, 0.7, 0.45 0.15 0.3
160 140 160 140 35
20 16 20 16 11
that the errors of as and D3 /DT affect data critically if the ratio of area is low. On the contrary, the other input values are insignificant to the resulting a. It is found that as and D3 /DT are very important input factors for accurate evaluation. The sensitivity coefficients for the as are derived as Eqs. (6) and (7) for the ceramic top-coat and the bond-coat, respectively.
E s As a s CTCas ¼ Ec Ac ac
(6)
E s As a s CBCas ¼ Eb Ab ab
(7)
These coefficients are negative, which means a positive as error gives negative estimation in the CTE. The sensitivity coefficients for the slope D3 /DT are a positive effect as shown in Eqs. (4) and (5). The sensitivity coefficients for the other input values are significantly low as compared with Eqs. (4)e(7).
Δ ⁄Δ
Fig. 3. The influence of the measuring slope D3 /DT error on the accuracy of the resulting CTE plotted against the sensitivity coefficients of the D3 /DT.
were adopted for the BC model and the TC model, respectively. The outer radii of the substrates were fixed as 5 mm. 3.2. Necessary minimum length of the specimen The composite cylinder model is actually satisfied on the area apart from the ends of the specimen because of the edge effect. An FEM strain calculation was conducted by a commercial code,
3. Experimental study 3.1. Specimen
Δ
C BC−slope , C TC−slope
Examples of the composite cylinder specimens used in this study are shown in Fig. 6. The bond-coats were CoNiCrAlYs by atmospheric plasma spraying (APS) and high velocity oxy-fuel (HVOF) spraying. The TBC top-coat was 8 wt%-yttria-stabilizedzirconia (YSZ) by APS. Spraying conditions are shown in Table 2. Table 3 shows the mechanical properties of the specimen. The Young’s moduli and the Poisson ratios are cited values and the CTEs were measured from the free-standing coatings by the same laser speckle method described in Section 3.3. The free-standing coatings were manufactured from other specimens by dissolving out the mild steel substrate by nitric acid. Table 4 shows the dimensions of the specimens. Three thicknesses of the substrate, which means three types of sensitivities,
60
: BC model : TC model
Δ
⁄Δ
⁄Δ
Fig. 4. The influences of the input errors on the accuracy of the resulting ac plotted against the ratio of area Ac/(Ab þ As) on the TC model.
Δ
⁄Δ
40
20 Δ 0 0
0.2 0.4 A b/A s , A c/(A b+A s)
⁄Δ
0.6
Fig. 2. The relationship between the sensitivity coefficients calculated and the area ratios Ac/(Ab þ As) for the TC model and Ab/As for the BC model.
Fig. 5. The influences of the inputs error on the accuracy of the resulting ab plotted against the ratio of area Ab/As on the BC model.
96
H. Waki et al. / Vacuum 88 (2013) 93e97 Table 3 Mechanical properties of the coatings.
E [GPa] (850e300 C)
n a [106/ C] (850e300 C)
Fig. 6. Examples of the composite cylinder specimens.
ANSYS. 2-dimensional 8-node element and linear isotropic elasticity were employed. The dimensions and mechanical properties are shown in Tables 3 and 4. Fig. 7 shows the relationship between the normalized FEM strain at the center surface and the normalized specimen length. The vertical axis of Fig. 7 is normalized by the axial strain of the composite cylinder model. The horizontal axis is normalized by the outer diameter of the substrate. If the value of vertical axis is near to 1.0, it means that the composite cylinder model is a reasonable model. Fig. 7 shows the center surface strain approximately satisfy the composite cylinder model if the ratio of the length of the specimen to the diameter of the substrate is l/d 1.5 for the BC model, or, l/d 2.0 for the TC model. Therefore, the specimen length was adopted as 15 mm for the BC model and 20 mm for the TC model when the diameter of substrate was 10 mm.
Substrate SUS304
Bond-coat CoNiCrAlY HVOF
APS
159 0.30 21.27
148 [9] 0.32 16.30
112 0.29 14.74
Top-coat APS YSZ 34.4 [9] 0.17 7.59
models should be located between those of the substrate and the free-standing coating. The strain of the BC model is located between those of the substrate and the coating. However, the strain of the TC model is unfortunately slightly higher than that of the substrate because of the strain measurement error at high temperature. The error will cause a critical error in ac. Fig. 9 shows the experimentally estimated CTEs plotted against the sensitivity coefficient CBC-slope, CTC-slope. The resulting a was normalized by the free-standing CTE afree-standing shown in Table 3. If the ratio is near to 1.0 it means that a reasonable estimation is obtained. The ratio a/afree-standing approaches to 1.0 when the
Table 4 The dimensions of the composite cylinder specimens. h [mm] BC model
Substrate Bond-coat (APS) Substrate Bond-coat (HVOF) Top-coat
TC model
5, 0.72, 0.29 0.145 5, 1.36, 0.41 0.2 0.31
a
3.3. Experimental procedure Thermal cycling was applied to the specimen by induction heating and natural cooling. The surface temperature at the center surface was measured by an emission thermo spot sensor. It was confirmed that the temperature of the substrate and the coatings was the same from 850 C to 300 C. The axial strain at the center surface during cooling process from 850 C to 300 C was measured by a laser speckle method [10,11]. The gauge area was inner circle of approximately 1.2 mm diameter. The measured strain was the average of the area. 3.4. Experimental result Examples of temperatureestrain curves during thermal cycling are shown in Fig. 8. The mean slope D3 /DT from 850 C to 300 C was evaluated by the least square method. The curves of composite
b
Table 2 Spraying conditions. Bond-coat CoNiCrAlY
Spraying gun Current [A] Voltage [V] O2 [MPa] Propylene [MPa] Air [MPa] Ar [MPa] H2 [MPa] He [MPa] Distance [mm]
HVOF
APS
Metco DJ e e 0.30 0.27 0.34 e e e 150
Metco 9MB 500 60 e e e 0.69 0.48 e 150
Top-coat APS YSZ
Plasma Dyne SG-100 850 35 e e e 0.41 e 0.62 100
Fig. 7. The relationship between the normalized FEM strain at the center surface and the normalized specimen length, on (a) the TC model, and (b) the BC model.
H. Waki et al. / Vacuum 88 (2013) 93e97
a
97
4. Conclusion In order to obtain the CTE of a thermal barrier coating from the coating system specimen, the strain method of a composite cylinder model was investigated. The resulting CTE of the coating by the coating system was sensitive to the experimental errors as compared with that of the free-standing coating. The sensitivities to experimental errors on the measuring strainetemperature relation and the other necessary input parameters were calculated. Experimental verification was also carried out on the six kinds of specimens with different sensitivities in order confirm the accuracy of the method.
b
(1) It was found from the sensitivity study that slight errors in as and D3 /DT caused a critical error in the resulting a when the sensitivity coefficient was high. The as and D3 /DT were very important input values for accurate evaluation. On the contrary, the other input values were insignificant to the resulting a. (2) Sensitivity study showed that low sensitivity coefficient was necessary to obtain the reasonable CTE. The increase of thickness of the coating as compared with the substrate was effective to decrease the sensitivity coefficient. (3) Reasonable estimation was experimentally obtained if the specimen had the low sensitivity to experimental errors. Acknowledgments
Fig. 8. Examples of temperatureestrain curves during thermal cycling, on (a) the TC model, and (b) the BC model.
This work was supported by KAKENHI (22760088). Authors also thank Messrs. T. Maekawa and Y. Ikebe (Graduate students of Osaka Electro-Communication University) for experimental works. References
Fig. 9. The accuracy of the experimentally estimated CTEs plotted against the sensitivity coefficient CBC-slope, CTC-slope.
sensitivity coefficient is low in both the TC and BC models. It is found that the BC model gives reasonable ab if the sensitivity coefficient is low, which means the area ratio Ab/As is high. On the contrary, a much lower sensitivity coefficient is necessary in the TC model. Approximately only 8% error in D3 /DT caused such a very high error of ac in the case of the TC model in Fig. 9.
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