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Experimental mathematical model as a generalization of sensitivity analysis of high temperature spectral emissivity measurement method
Measurement 90 (2016) 475–482
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Experimental mathematical model as a generalization of sensitivity analysis of high temperature spectral emissivity measurement method Petra Honnerová ⇑, Zdeneˇk Vesely´ ⇑, Milan Honner New Technologies Research Centre (NTC), University of West Bohemia, Univerzitní 8, 306 14 Pilsen, Czech Republic
a r t i c l e
i n f o
Article history: Received 29 May 2015 Received in revised form 27 April 2016 Accepted 29 April 2016 Available online 30 April 2016 Keywords: Experimental Mathematical model Dimensionless analysis Emissivity Coatings High temperature
a b s t r a c t The development of an experimental mathematical model describing temperature state of the sample during high temperature spectral emissivity measurement is introduced. Dimensional analysis of the measurement process gives the physical dimensionless quantities and sensitivity analysis of the measurement process provides the large set of performed model experiments. Evaluated experimental mathematical models are presented including their accordance with model experiments. Established equations are generalization of sensitivity analysis of high temperature spectral emissivity measurement method and can be used for computation of spectral emissivity total uncertainty. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Emissivity is a physical quantity that defines how much infrared radiation is emitted by a real surface in comparison to an ideal blackbody. Direct [1–4] or indirect [1,5–7] methods are utilized for the emissivity measurement. A temperature dependence of spectral emissivity is usually measured by the direct radiometric methods [8–11]. The direct ones are based on the comparison of sample radiation to a reference body radiation at the same temperature, under similar geometrical and spectral conditions [12]. The emissivity measurements accuracy by direct radiometric methods is mainly affected by the uncertainty on the measurement of the real sample surface temperature. For example, the emissivity uncertainty of approximately 1% is the results of an uncertainty in the sample surface temperature measurement of 0.5 °C at the temperature 100 °C [13]. High temperature spectral emissivity measurement method [14] is primarily applied for the normal spectral emissivity measurement of high-temperature coating with different thickness up to temperature 1000 °C. The application of contact methods based on the surface temperature determination from temperatures measured inside the sample is limited. Measurement of surface temperature by contact method provided by thermocouple ⇑ Corresponding authors. E-mail addresses: [email protected] (P. Honnerová), [email protected] (Z. Vesely´), [email protected] (M. Honner). http://dx.doi.org/10.1016/j.measurement.2016.04.070 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
measurement in certain depth from surface does not include temperature drop across the thickness of measured coating, thus a high emissivity uncertainty is obtained. The difficulty in surface temperature measurement of coatings is solved with the help of an infrared camera system and a reference coating deposited on a part of the measured sample surface. On the other side, the reference coating influences the sample radiation behavior and sample surface temperature distribution [15,16]. The computer model giving the temperature distribution of the sample in a steady state is employed in the sensitivity analysis of high temperature spectral emissivity measurement method. Temperature distribution in the sample during emissivity measurement is evaluated for a considerable number of prescribed sample temperatures, emissivities of measured coating and thicknesses of reference and measured coatings [17]. These sample temperature distributions are outputs of the sensitivity analysis. Experimental mathematical model is used to summarize and generalize the results from sensitivity analysis. Experimental mathematical models (experimental models) are created using the mathematical apparatus called dimensional analysis and the number of experiments performed. Experimental models are used to describe physical processes in the cases where utilization of asymptotic mathematical models (equations of mathematical physics) are not possible because they are not known or it is too complicated to use them. Complicated mathematical models need a lot more time to be processed than simple equations called
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Nomenclature a, b, c, d, e unknown parameters of experimental mathematical model A dimensionless heat transfer coefficient Bi Biot number d thickness D dimensionless thickness Ki Kirpitchev number L reference length n number of computer model experiments q heat flux Q dimensionless heat flux r sample diameter R dimensionless sample diameter R2 determination index Sf Stefan number T temperature y quantity from computer model ^ y quantity from the experimental mathematical model y average value of the quantity
e h k K
r0
emissivity dimensionless temperature thermal conductivity dimensionless thermal conductivity Stefan–Boltzmann constant
Subscripts ave average A spot A B spot B C spot C e external i summing index mc measured coating rc reference coating rc/mc interface of reference and measured coating ref reference s substrate S surface 1, . . ., 7 resolving index
Greek symbols a heat transfer coefficient D difference
experimental models. Experimental models are only approximate expression of precise results, but in many cases it is sufficient. Using dimensional analysis [18–25] of high temperature spectral emissivity measurement method, dimensionless similarity quantities for measurement method description are obtained. These dimensionless quantities and a large number of computer model results for optional parameters of the sample and measurement method are inputs to the procedure for experimental mathematical model evaluation. As a result, the experimental mathematical model describing dependence of temperature difference of average temperature on the surface of measured coating and average temperature at the interface of reference and measured coatings is realized. The temperature difference from the experimental mathematical model can be used for more precise determination of measured coating surface temperature and for uncertainty evaluation of high temperature spectral emissivity measurement method.
The applied noncontact surface temperature measurement requires the knowledge of effective emissivity of the measured surface for the infrared camera and viewing angle. The problem is solved by the deposition of the reference coating (ZYP Coating Cr2O3) with known effective emissivity on the half of the measured
measuredcoating reference coating substrate
infrared camera
spectrometer
spot A 2. Emissivity measurement method
Trc,A
spot B =
Tmc,B
2.1. High temperature emissivity measurement method High temperature emissivity measurement method is briefly introduced in the section. A more detailed description of the method can be found in [14,26]. The method is based on the comparison of radiation of a blackbody and of the sample surface that are positioned opposite to each other. The radiation is collected by a rotary parabolic mirror situated halfway between the radiation sources. The spectral detection of radiation is performed by a FTIR spectrometer (Nicolet 6700). The optical path of radiation is covered by an optical box. The sample is clamped to a ceramic insulation and is heated to the required temperature by a fiber laser with a scanning head. The sample surface temperature is measured noncontactly by an infrared camera (FLIR A320) with wavelength range from 7.5 lm to 13 lm. The temperature dependence of normal spectral emissivity is evaluated in the wavelength range from 1.38 lm to 26 lm. The samples can be heated up to 1000 °C.
laser beam Fig. 1. Temperature and emission measurement positions in the emissivity measurement method (spot A is on reference coating, spot B is on measured coating) and temperature evaluation area in the model of the heat transfer in the sample.
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sample surface (Fig. 1). The sample’s front surface area is divided into two symmetrical parts. The sample surface temperature Trc,A (average temperature in spot A) is measured on the part of the sample with the reference coating. The radiation of the second part without the reference coating is detected by the spectrometer for the emissivity analyses (spot B). Assumption of the identical sample temperature on the part with the reference coating Trc,A and measured coating Tmc,B is considered.
Fig. 1). From these average temperatures, the temperature difference DTmc-rc/mc,ave is evaluated from the equation
2.2. Model of the heat transfer in the sample
The temperature difference DTmc-rc/mc,ave from the Eq. (1) is important value not only for uncertainty evaluation of high temperature emissivity measurement method (high temperature normal spectral emissivity measurement method) but it is also used for improving the preciseness of the measurement method. This temperature difference can be used for more precise calculation of emissivities. Utilization of computer model for full heat transfer analysis in the sample is not so comfortable because the analysis take a certain time to be performed. Simple experimental mathematical model for description of temperature difference DTmc-rc/is more convenient way. The number of various mc,ave experiments performed (that are needed for experimental model creation) are number of various variants of heat transfer model that have been computed. Experimental model creates generalization of the computer model results and give the values of temperature difference also for other variants than have not been computed. Temperature difference DTmc-rc/mc,ave is suggested due to requirement of spectral emissivity measurement method. In the method, temperature Trc,ave,A is obtained by infrared camera. The reference coating effective emissivity is evaluated for temperature at reference coating/substrate interface in the case of reference sample [14]. This effective emissivity of reference coating is applied for temperature evaluation also in the case of measured sample. It is supposed, that this temperature is identical with the temperature Trc/mc,ave,A of measured sample and this is identical with the temperature Tmc,ave,B. If the temperatures Trc/mc,ave,A and Tmc,ave,B are different, the emissivity uncertainty is increasing. The knowledge of the temperature difference can be included to the emissivity evaluation and thus emissivity uncertainty can be decreased. Therefore, the mathematical model operates with temperature difference DTmc-rc/mc,ave.
A computer model of the sample steady state is utilized to describe the temperature distribution in the sample during the emissivity measurement at high temperature [17]. Usually the thicknesses of reference and measured coatings are very small. Because of possible difficulties with very small dimensions in some computer models, it is better to transform the real sample to the model sample with enlarged thicknesses of coatings. Along with the sample transformation, it is modified not only geometry, but also material properties and boundary conditions. Dimensionless analysis is performed to obtain the scale factors for all quantities. Using the quantities scale factors, real sample is transformed to the model sample. The temperature distribution is solved for model sample. The results are then transformed back to real sample. All the sample parameters introduced and all the temperature distribution results are concerning the real sample. Sample parameters and measurement method parameters are divided into two groups – prescribed and optional parameters. Prescribed parameters are constant parameters that are fixed for all variants solved. Constant parameters are sample radius 12.5 mm, substrate thickness 5.0 mm and substrate thermal conductivity 25 W m1 K1. Thermal conductivity of reference and measured coatings are also constant and equal to 2 W m1 K1. The side walls of the substrate, measured and reference coatings, are thermally isolated to have zero lossy heat flux. The front part of the sample has heat transfer coefficient equal to 10 W m2 K1 with external temperature 100 °C that includes only convection heat transfer. Reference coating emissivity is temperature dependent from 0.702 to 0.655. The optional parameters vary among the variants that are solved using computer modelling. Optional parameters are thickness of reference coating drc (30, 60, 90 lm), thickness of measured coating dmc (50, 100, 150, 300 lm), emissivity of measured coating emc (0.1, 0.5, 0.9) and temperature at heated sample surface Ts (400–1000 °C with the step 50 °C). Modelling results are average temperatures Trc,ave,A on the surface of reference coating (appropriate temperature of spot A, Fig. 1), average temperatures Trc/mc,ave,A at the interface of reference and measured coatings and average temperatures Tmc,ave,B on the surface of measured coating (appropriate temperature of spot B,
DT mc-rc=mc;av e ¼ T mc;av e;B T rc=mc;av e;A :
ð1Þ
3. Experimental mathematical model evaluation procedure 3.1. Experimental mathematical model idea
3.2. Dimensionless analysis Dimensional analysis employs simple mathematical operations with physical quantities units. The principle is the knowledge of physical process. Dimensional analysis works with units of physical quantities that occurred in the process. Dimensional analysis enables among others determination of number and form of dimensionless quantities in the process.
Table 1 Physical quantities describing the process. Quantity
Units
Description
ks, kmc, krc, kref ds, dmc, drc, r, L T, Ts, Tmc,ave, Trc,ave, Trc/mc,ave, DTmc-rc/mc,ave, Te, Tref qs, qmc, qrc, qref amc, arc, aref
W m1 K1 m K
Thermal conductivities of substrate, measured and reference coating, reference thermal conductivity Thicknesses of substrate, measured and reference coating, sample diameter, reference length Temperature, surface temperature of substrate, surface temperature of measured coating in the spot B, surface temperature of reference coating in the spot A, temperature at the interface of reference and measured coating under spot A, temperature difference, external temperature and reference temperature
W m2 W m2 K1 W m2 K4 –
Surface heat fluxes of substrate, measured and reference coating, reference heat flux Coefficients of heat transfer of measured and reference coating, reference heat transfer coefficient Stefan–Boltzmann constant Emissivities of measured and reference coating
r0
emc, erc
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Because the number of dimensionless quantities describing process is always less than number of dimensional quantities in the process, the transformation from dimensional to dimensionless quantities performs compression of results concerning the physical process analyzed. Generalization of results is possible because experimental mathematical models enable description of other processes that are physically similar. The process of steady state temperature distribution in the sample during measurement procedure is described by steady state heat transfer equation with boundary conditions of 1st (prescribed surface temperature), 2nd (prescribed surface heat flux) and 3rd (prescribed heat transfer coefficient, emissivity and external temperature) types. It gives totally 27 physical quantities, see Table 1. Dimensional analysis gives us 24 dimensionless quantities in Table 2 those can be used in experimental equation concerning this process. There are so called reference quantities among the physical quantities in Table 1. These reference quantities are constants with characteristic values. For example, initial sample temperature can be used as the value for reference temperature Tref. Reference length L means characteristic size in the model, in this case it is sample diameter. Reference thermal conductivity kref has been chosen as a value of the substrate material. Reference heat flux qref and reference heat transfer coefficient aref are set to constant values used in simulation model. 3.3. Experimental mathematical model development The aim is to evaluate experimental mathematical model that describes dimensionless temperature difference DHmc-rc/mc,ave dependence on dimensionless process parameters. There is an effort to establish only one equation of experimental mathematical model that is valid for all process parameters. Since the optional parameters are thicknesses of reference and measured coating, emissivity of measured coating and temperature at heated sample surface, the experimental model should include dimensionless quantities that characterize these parameters. Experimental model thus can contain only simple dimensionless quantities that correspond with variable parameters in computer models
DHmc-rc=mc;av e ¼ f ðemc ; Hs ; Dmc ; Drc Þ:
ð2Þ
Experimental mathematical model is evaluated to achieve very good agreement with the computer model. Evaluation is provided using determination index R2, that is defined in regression analysis by the equation
n X ^ i Þ2 R ¼1 ðyi y
,
2
ı¼1
! n X 2 Þ : ðyi y
ð3Þ
i¼1
It is evident that the better experimental model equation is, the ^ı with better agreement of quantity from experimental model y quantity from computer model yi is obtained. Quantity n denotes is the average of number of computer model experiments and y values yi from computer model. The greater value of determination index R2, the better agreement of experimental mathematical model with computer model. At the beginning of the experimental mathematical model development procedure, the form of experimental mathematical model equation is proposed. Then, initial values of equation parameters are suggested. These parameters values are refined by iteration method to find the maximum of determination index. Evaluation of parameters value is faster and easier when the experimental mathematical model is in the classic product form. Therefore, the first proposal of experimental model equation is in the product form and only when this form is not suitable, other forms are tested. Experimental mathematical model describing dimensionless temperature difference DHmc-rc/mc,ave was initially suggested in the product forms
DHmc-rc=mc;av e ¼ a ebmc Hcs Ddmc Derc :
ð4Þ
DHmc-rc=mc;av e ¼ a ebmc Hcs :
ð5Þ
Experimental mathematical model in the form (4) gives results that differ from computer model results. Even the form (5), that seems to be better for fitting of coefficients a, b, c individually for each combination of thicknesses of measured and reference coating, does not tend to better results. Therefore other forms of experimental model equations were prepared. Promising results were obtained using polynomial form that has been tested for emissivity of measured coating (6) and extended for variable dimensionless temperature Hs (7)
DHmc-rc=mc;av e ¼ ða1 emc þ a2 Þ; ð6Þ DHmc-rc=mc;av e ¼ a1 H2s þ a2 Hs þ a3 emc þ b1 H2s þ b2 Hs þ b3 : ð7Þ Experimental mathematical model in the form of Eq. (7) fits computer model correctly for low measured coating emissivities emc = 0.1 and 0.5. For high value emc = 0.9, it is necessary to design another equation. Polynomial form of sixth order is in good agreement with computer model
DHmc-rc=mc;av e ¼ a1 H6s þ a2 H5s þ a3 H4s þ a4 H3s þ a5 H2s þ a6 Hs þ a7 :
ð8Þ
Table 2 Dimensionless quantities describing the process. Quantity
Form
Ks, Kmc, Krc
ks kref
Ds, Dmc, Drc, R
ds L
H, Hs, Hmc,ave, Hrc,ave, Hrc/mc,ave, DHmc-rc/mc, ave,
He Qs, Qmc, Qrc
Description
;
kmc ks
;
dmc ; ds Ts T ref
;
T T ref
T rc;av e T ref
;
; ;
krc ks drc ; Lr ds T mc;av e T ref ,
T rc=mc;av e T ref ,
Dimensionless thermal conductivities of substrate, measured and reference coating Dimensionless thicknesses of substrate, measured and reference coating, dimensionless sample diameter Dimensionless temperatures local and substrate surface, dimensionless surface temperature of measured coating in the spot B, dimensionless surface temperature of reference coating in the spot A, dimensionless temperature at the interface of reference and measured coating under spot A, dimensionless temperatures difference and external temperature
DT mc-rc=mc;av e , T ref Te T ref qs qref
;
Amc, Arc
amc ; aref
emc, erc Ki
–
Bi
aL
Sf
er 0 T 3 L
qmc qref
arc aref
;
qrc qref
Dimensionless heat fluxes at the surface of substrate, measured and reference coating Dimensionless heat transfer coefficients of measured and reference coating Emissivities of measured and reference coating Kirpitchev number
qL kT
Biot number
k
k
Stefan number
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Good experimental models describing dependence of dimensionless temperature difference DHmc-rc/mc,ave on emissivity of measured coating and dimensionless sample temperature at heated side were evaluated. Eqs. (7) and (9) include the effect of emissivity of measured coating and dimensionless sample temperature. The influence of thicknesses of measured and reference coating Dmc and Drc is involved in subsequent equations. (A) Low value range of measured coating emissivity from emc = 0.1 to 0.5 One equation enables to include all other parameters in the form
DHmc-rc=mc;av e ¼ða1 emc þ a2 Þðb1 H2s þ b2 Hs þ b3 Þ ðc1 D2mc þ c2 Dmc þ c3 Þðd1 D2rc þ d2 Drc þ d3 Þ:
ð9Þ
Linear dependence for emissivity of measured coating and polynomial dependencies of second order for dimensionless sample temperature at heated side and dimensionless thicknesses of measured and reference coating are sufficient. There is a very good agreement of this experimental mathematical model Eq. (9) with corresponding results of computer model. The Eq. (9) is valid for low measured coating emissivities in the range from 0.1 to 0.5. It has been detected, that for higher measured coating emissivities close to 0.7, this equation loses its validity and another form of the experimental model equation is necessary. The equation (9) is applicable for all thicknesses of reference coating in the observed range from 30 to 90 lm and for all thicknesses of measured coating in the observed range from 50 to 300 lm.
Table 3 Parameters values of experimental mathematical model equations. Equation
(9)
(10)
(10)
(10)
(10)
emc (–) dmc (lm) drc (lm)
0.1–0.5 50–300 30–90
0.9 50 30–90
0.9 100 30–90
0.9 150 30–90
0.9 300 30–90
a1 a2 a3 a4 a5 a6 a7 b1 b2 b3 c1 c2 c3 d1 d2 d3
1.12778 0.90159 – – – – – 0.74134 0.42725 0.08331 5.44951 3.59402 0.24288 1.81206 0.42572 0.28809
1.53239 5.82965 8.88086 6.90821 2.89646 0.62247 0.05355 81.06340 14.80550 0.83600 – – – – – –
1.74359 6.64254 10.13569 7.89942 3.31906 0.71491 0.06165 53.95040 13.04250 0.85260 – – – – – –
1.93937 7.40131 11.31489 8.83697 3.72133 0.80345 0.06945 34.54080 10.61150 0.86690 – – – – – –
2.49797 9.56972 14.69048 11.52562 4.87717 1.05831 0.09194 17.15870 10.78750 0.86770 – – – – – –
Table 4 Determination index showing agreement between the results of experimental mathematical model and computer model. Equation
(9)
(10)
(10)
(10)
(10)
emc (–) dmc (lm) drc (lm)
0.1–0.5 50–300 30–90 0.9983
0.9 50 30–90 0.9986
0.9 100 30–90 0.9986
0.9 150 30–90 0.9986
0.9 300 30–90 0.9985
R2 (–)
(B) High value of measured coating emissivity emc = 0.9 One equation enabling to include all other parameters has not been found. It is necessary to formulate individual equations for each constant dimensionless thickness of measured coating. The form of these equations is the same
Polynomial dependency of sixth order for dimensionless sample temperature at heated side and polynomial dependency of second order for dimensionless thicknesses of reference coating are sufficient. Equation parameters are evaluated separately for each value of dimensionless thickness of measured coating. There is very good agreement of this experimental mathematical model Eq. (10) with corresponding results of computer model. The Eq. (10) is valid for high value of measured coating emissivity equal to 0.9 and for all thicknesses of reference coating in the observed range from 30 to 90 lm. Values of equation parameters have to be found for each individual observed value of measured coating thickness. 4. Results of the analysis 4.1. Experimental mathematical model equations Two forms of the experimental mathematical model equations are found to be in very good agreement with the computer models. The Eq. (9) is determined for the low value range of measured coating emissivity from 0.1 up to 0.5, the equation involves all
0.025 0.003
Δθ mc-rc/mc, ave (-)
ð10Þ
computer model
0.005
0.020 0.000 0.000
0.003
0.005
0.015 0.010 0.005 0.000 0.000
(a) 0.005
0.010
0.015
0.020
0.025
0.030
computer model Δθ mc-rc/mc, ave (-) experimental mathematical model
computer model
0.001
0.000
Δ θ mc-rc/mc, ave (-)
DHmc-rc=mc;av e ¼ a1 H6s þ a2 H5s þ a3 H4s þ a4 H3s þ a5 H2s þ a6 Hs þ a7 b1 D2rc þ b2 Drc þ b3 :
experimental mathematical model
0.030
-0.001 0.0005
-0.002 0.0003
-0.003 0.0000 0.0000
-0.004 -0.004
-0.003
-0.002
-0.001
0.0003
0.000
0.0005
(b)
0.001
computer model Δθ mc-rc/mc, ave (-) Fig. 2. The dimensionless temperature difference evaluated from the experimental mathematical model and computer model for the low emissivities of measured coating 0.1–0.5 (a), and high emissivity of measured coating 0.9 (b).
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ΔT ≤-5
0
5
10
15
≥17
ε = 0.1
ε = 0.5
ε = 0.9
100 80 drc (μm)
Ts = 400°C
2.75
0.40
0.8
2.50
0.35
0.9
60
1.0
2.25 2.00
40
0.30
1.1 0.25
1.2
1.75
20 100
drc (μm)
Ts = 600°C
5.5
80
2.25
5.0 0.5
2.00
60
4.5 0.4
1.75
40
4.0 0.3
1.50
3.5
20 100 0.6
3.5
80 drc (μm)
Ts = 800°C
12.5
0.5
4.0
11.0
4.5
60
0.4
9.5 5.0
40 8.0
5.5
0.3
20 100
drc (μm)
Ts = 1000°C
23.0
80 60
-1.7 8.0
19.0
-1.5
17.0
40 20
6.5
21.0
9.5 -1.3
15.5
50 100 150 200
250 300
50 100 150 200
dmc (μm)
dmc (μm)
250 300 50 100 150 200 250 300 dmc (μm)
Fig. 3. Temperature dependence of the temperature difference as a function of the reference and measured coatings thickness for three different values of measured coating total emissivity (0.1, 0.5 and 0.9).
dimensionless quantities that correspond with optional parameters in the computer models. The form of the Eq. (10) is evaluated for the high value of measured coating emissivity 0.9, whereas it is necessary to formulate the equation parameters values for each constant dimensionless thickness of measured coating. In total, five experimental mathematical model equations are established. The parameters values of equations are shown in Table 3. The values are related to the dimensional thickness of measured coating dmc for better clarity of results.
4.2. Experimental mathematical model and computer model results comparison The parameters values of experimental mathematical model equations (Table 3) are verified by comparing the dimensionless temperature differences DHmc-rc/mc,ave that are evaluated from the experimental mathematical model and from the computer model. The verification is executed by calculating the determination index R2 defined by the Eq. (3). The determination index values for appropriate experimental mathematical models are
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shown in Table 4. Because the values are higher than 0.99 for all experimental mathematical models, very good agreement of these experimental mathematical models and computer model results is established. Comparison of dimensionless temperature difference DHmc-rc/mc,ave evaluated from the experimental mathematical model and computer model are graphically presented in Fig. 2. The dependence of dimensionless temperature differences for two models on the computer model temperature difference is presented. The results in Fig. 2a are related to the low emissivities of measured coating 0.1 to 0.5 including all thicknesses of measured coating and all thicknesses of reference layer and sample temperature level. In Fig. 2b, the comparison of dimensionless temperature differences for the experimental mathematical model and computer model are shown only for the high emissivity of measured coating 0.9, thickness of measured coating 150 lm and for all observed thicknesses of reference layer and sample temperature level. The similar results are obtained for other thicknesses of measured coating (50, 100 and 300 lm). Good agreement of the dimensionless temperature differences DHmc-rc/mc,ave evaluated from the experimental mathematical model and computer model are apparent for all process parameters. 4.3. Temperature differences The experimental mathematical model Eqs. (9) and (10) with the parameter values specified in Table 3 are used for the computation of temperature difference DTmc-rc/mc,ave between average surface temperature of measured coating and average temperature at the interface of reference and measured coatings. The results of the computer model for optional parameters are replaced by the results of experimental mathematical model and these results are extended by the temperature differences evaluated for any thickness of reference and measured coating, substrate surface temperature and total emissivity of measured coating. The results are applied for uncertainty evaluation of high temperature emissivity measurement method and improving the preciseness of the measurement method. The temperature dependence of temperature difference from the experimental mathematical model equations is shown in Fig. 3 for the selected values of substrate surface temperature in the temperature range from 400 °C to 1000 °C, three different values of total emissivity of measured coating 0.1, 0.5 and 0.9, thicknesses of reference coating in the range from 20 to 100 lm, and thicknesses of measured coating in the range from 25 to 300 lm. White color is displayed when the temperature difference is about 5 K. If the temperature difference diverges from 5 K, the colors are deeper. The difference over 17 K toward positive values and the difference 0 K are indicated by the deepest colors. Further increase in the temperature difference toward negative values is represented by the lighter shades. The first column of the matrix plot in Fig. 3 shows the temperature dependence of temperature difference for the emissivity of measured coating 0.1. It is clear that the temperature difference significantly depends on the sample temperature. With increasing temperature, temperature difference between average surface temperature of measured coatings and average temperature at the interface of reference and measured coating grows from 1.75 °C for sample temperature 400 °C to 25 °C for sample temperature 1000 °C. The temperature difference also increases with the sample temperature for the second (emc = 0.5) and the third (emc = 0.9) column of the matrix plot. However, with increasing emissivity, the differences in temperature decrease. For the emissivity of measured coating 0.5, the temperature difference about 1 °C is achieved for the temperature 400 °C, at the temperature 1000 °C it is up to 10 °C. The temperature dependence of
481
temperature difference is almost negligible for the emissivity of measured coating 0.9, it is about 1 °C. The dependence of temperature difference on the measured and reference coatings thicknesses is less significant that the dependence of temperature difference on substrate surface temperature. Generally, the higher temperature difference is reached for the greater thickness of the measured coating. The specific values are dependent on the emissivity of measured coating and sample temperature, these vary in the order of °C. The thickness of the reference coating affects the temperature difference just a few tenths of °C. The different trend of the dependence of temperature difference on the thickness of reference coating is proved for the emissivity up to 0.5 and emissivity 0.9. While for the emissivity up to 0.5 the temperature difference slightly decreases with increasing thickness of the reference coating, for the emissivity 0.9 it is opposite. 5. Conclusions Sample temperatures during high temperature spectral emissivity measurement are obtained from sensitivity analysis of the measurement method. Dimensionless physical quantities concerning the emissivity measurement process have been established utilizing dimensionless analysis technique. Experimental mathematical model have been compiled from these dimensionless quantities as a generalization of sensitivity analysis results. The equation describing the temperature difference DHmc-rc/mc, ave, between measured coating surface and interface of reference and measured coating, dependent on dimensionless process parameters has been evaluated. However, it is not possible to evaluate one equation comprising the effect of all observed parameters. Experimental mathematical model in the first form includes linear dependence to the emissivity of measured coating and polynomial dependencies of second order to the dimensionless sample temperature at heated side and dimensionless thicknesses of measured and reference coating. The model in this form is valid for low value range of measured coating emissivities emc = 0.1 up to 0.5 and for all thicknesses of measured and reference coatings including all measurement temperature levels. Experimental mathematical model in the second form comprises polynomial dependency of sixth order to the dimensionless sample temperature at heated side and polynomial dependency of second order to the dimensionless thicknesses of reference coating. The model in this form is used for high measured coating emissivity emc = 0.9. For each independent thickness of measured coating, the set of experimental model parameters has been derived for all thicknesses of reference coating including all measurement temperature levels. Absolute temperature differences DTmc-rc/mc,ave computed from the experimental mathematical models for observed thicknesses of measured and reference coatings, emissivity of measured coating and sample temperature level are in the range from negative values in the order of units up to positive values in the order of few tens. The emissivity of measured coating has the largest impact on the temperature difference, sample temperature level has lower effect and the thicknesses of the coatings have the minor influence. All evaluated experimental mathematical models have very good agreement with computer model results from sensitivity analysis of high temperature spectral emissivity measurement method. Experimental mathematical models are used for generalization of sensitivity analysis results and for computation of spectral emissivity total uncertainty that is subject of subsequent work. Acknowledgments The result was developed within the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, cofunded by the ERDF as part of the
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Ministry of Education, Youth and Sports OP RDI programme and, in the follow-up sustainability stage, supported through CENTEM PLUS (LO1402) by financial means from the Ministry of Education, Youth and Sports under the ”National Sustainability Programme I.”. References [1] H. Watanabe, J. Ishii, H. Wakabayashi, T. Kumano, L. Hanssen, Chapter 9 – spectral emissivity measurements, Exp. Methods Phys. Sci. 46 (2014) 333–366. [2] J. Manara, M. Arduini-Schuster, M. Keller, Infrared-optical characteristics of ceramics at elevated temperatures, Infrared Phys. Technol. 54 (2011) 395–402. [3] G. Neuer, G. Jaroma-Weiland, Spectral and total emissivity of hightemperature materials, Int. J. Thermophys. 19 (1998) 917–929. [4] R. Brandt, C. Bird, G. Neuer, Emissivity reference paints for high temperature applications, Measurement 41 (2008) 731–736. [5] J. Manara, M. Arduini-Schuster, L. Hanssen, Integrating sphere reflectance and transmittance intercomparison measurements for evaluating the accuracies of the achieved results, High Temp. – High Press 38 (2009) 259–276. [6] L.M. Hanssen, C.P. Cagran, A.V. Prokhorov, S.N. Mekhontsev, V.B. Khromchenko, Use of a high-temperature integrating sphere reflectometer for surfacetemperature measurements, Int. J. Thermophys. 28 (2007) 566–580. [7] J. Hameury, B. Hay, J. Filtz, Measurement of infrared spectral directional hemispherical reflectance and emissivity at BNM-LNE, Int. J. Thermophys. 26 (2005) 1973–1983. [8] L. del Campo, R.B. Pérez-Sáez, X. Esquisabel, I. Fernández, M.J. Tello, New experimental device for infrared spectral directional emissivity measurements in a controlled environment, Rev. Sci. Instrum. 77 (2006) 113111-1–113111-8. [9] C.H. Monte, J. Hollandt, The measurement of directional spectral emissivity in the temperature range from 80 °C to 500 °C at the Physikalisch-Technische Bundesanstalt, High Temp. – High Press 39 (2010) 151–164. [10] B. Hay, J. Hameury, N. Fleurence, P. Lacipiere, M. Grelard, V. Scoarnec, G. Davee, New facilities for the measurements of high temperature thermophysical properties at LNE, Int. J. Thermophys. 35 (2014) 1712–1724.
[11] M. Honner, P. Honnerová, Survey of emissivity measurement by radiometric methods, Appl. Opt. 54 (2015) 669–683. [12] R. Siegel, J. Howel, Thermal Radiation Heat Transfer, Taylor & Francis, Washington, 2001. [13] L.R. Koirala, FTIR-spectroscopic Measurement of Directional Spectral Emissivities of Microstructured Surfaces Ph.D. Thesis, University of the Federal Armed Forces, Hamburg, 2004. [14] P. Honnerová, J. Martan, M. Kucˇera, M. Honner, J. Hameury, New experimental device for high-temperature normal spectral emissivity measurements of coatings, Meas. Sci. Technol. 25 (2014) 095501. [15] J.F. Sacadura, Measurement techniques for thermal radiation properties, in: Proc. 9th International Heat Transfer Conference, 1990, pp. 207–222. [16] D.P. DeWitt, G.D. Nutter, Theory and Practice of Radiation Thermometry, John Wiley, New York, 1988. [17] Z. Vesely´, P. Honnerová, J. Martan, M. Honner, Sensitivity analysis of high temperature spectral emissivity measurement method, Infrared Phys. Technol. 71 (2015) 217–222. [18] T. Szirtes, Applied Dimensional Analysis and Modeling, McGraw-Hill, New York, 1998. [19] J.E. Baker, P.S. Westine, F.T. Dodge, Similarity Methods in Engineering Dynamics, Elsevier, Amsterdam, 1991. [20] G.I. Barenblatt, Dimensional Analysis, Gordon and Breach Science Publisher, New York, 1987. [21] P.W. Bridgman, Dimensional Analysis, AMS Press, New York, 1978. [22] H.E. Huntley, Dimensional Analysis, Dover Publications, New York, 1967. [23] R.C. Pankhurst, Dimensional Analysis and Scale Factors, Chapman and Hall, London, 1964. [24] E. Szücs, Similitude and Modelling, Elsevier, Amsterdam, 1980. [25] J. Zierep, Similarity Laws and Modeling, Marcel Dekker, New York, 1971. [26] M. Mauer, P. Kalenda, M. Honner, P. Vacíková, Composite fillers and their influence on emissivity, Phys. Procedia 44 (2013) 262–269.
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Measurement journal homepage: www.elsevier.com/locate/measurement
Experimental mathematical model as a generalization of sensitivity analysis of high temperature spectral emissivity measurement method Petra Honnerová ⇑, Zdeneˇk Vesely´ ⇑, Milan Honner New Technologies Research Centre (NTC), University of West Bohemia, Univerzitní 8, 306 14 Pilsen, Czech Republic
a r t i c l e
i n f o
Article history: Received 29 May 2015 Received in revised form 27 April 2016 Accepted 29 April 2016 Available online 30 April 2016 Keywords: Experimental Mathematical model Dimensionless analysis Emissivity Coatings High temperature
a b s t r a c t The development of an experimental mathematical model describing temperature state of the sample during high temperature spectral emissivity measurement is introduced. Dimensional analysis of the measurement process gives the physical dimensionless quantities and sensitivity analysis of the measurement process provides the large set of performed model experiments. Evaluated experimental mathematical models are presented including their accordance with model experiments. Established equations are generalization of sensitivity analysis of high temperature spectral emissivity measurement method and can be used for computation of spectral emissivity total uncertainty. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Emissivity is a physical quantity that defines how much infrared radiation is emitted by a real surface in comparison to an ideal blackbody. Direct [1–4] or indirect [1,5–7] methods are utilized for the emissivity measurement. A temperature dependence of spectral emissivity is usually measured by the direct radiometric methods [8–11]. The direct ones are based on the comparison of sample radiation to a reference body radiation at the same temperature, under similar geometrical and spectral conditions [12]. The emissivity measurements accuracy by direct radiometric methods is mainly affected by the uncertainty on the measurement of the real sample surface temperature. For example, the emissivity uncertainty of approximately 1% is the results of an uncertainty in the sample surface temperature measurement of 0.5 °C at the temperature 100 °C [13]. High temperature spectral emissivity measurement method [14] is primarily applied for the normal spectral emissivity measurement of high-temperature coating with different thickness up to temperature 1000 °C. The application of contact methods based on the surface temperature determination from temperatures measured inside the sample is limited. Measurement of surface temperature by contact method provided by thermocouple ⇑ Corresponding authors. E-mail addresses: [email protected] (P. Honnerová), [email protected] (Z. Vesely´), [email protected] (M. Honner). http://dx.doi.org/10.1016/j.measurement.2016.04.070 0263-2241/Ó 2016 Elsevier Ltd. All rights reserved.
measurement in certain depth from surface does not include temperature drop across the thickness of measured coating, thus a high emissivity uncertainty is obtained. The difficulty in surface temperature measurement of coatings is solved with the help of an infrared camera system and a reference coating deposited on a part of the measured sample surface. On the other side, the reference coating influences the sample radiation behavior and sample surface temperature distribution [15,16]. The computer model giving the temperature distribution of the sample in a steady state is employed in the sensitivity analysis of high temperature spectral emissivity measurement method. Temperature distribution in the sample during emissivity measurement is evaluated for a considerable number of prescribed sample temperatures, emissivities of measured coating and thicknesses of reference and measured coatings [17]. These sample temperature distributions are outputs of the sensitivity analysis. Experimental mathematical model is used to summarize and generalize the results from sensitivity analysis. Experimental mathematical models (experimental models) are created using the mathematical apparatus called dimensional analysis and the number of experiments performed. Experimental models are used to describe physical processes in the cases where utilization of asymptotic mathematical models (equations of mathematical physics) are not possible because they are not known or it is too complicated to use them. Complicated mathematical models need a lot more time to be processed than simple equations called
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Nomenclature a, b, c, d, e unknown parameters of experimental mathematical model A dimensionless heat transfer coefficient Bi Biot number d thickness D dimensionless thickness Ki Kirpitchev number L reference length n number of computer model experiments q heat flux Q dimensionless heat flux r sample diameter R dimensionless sample diameter R2 determination index Sf Stefan number T temperature y quantity from computer model ^ y quantity from the experimental mathematical model y average value of the quantity
e h k K
r0
emissivity dimensionless temperature thermal conductivity dimensionless thermal conductivity Stefan–Boltzmann constant
Subscripts ave average A spot A B spot B C spot C e external i summing index mc measured coating rc reference coating rc/mc interface of reference and measured coating ref reference s substrate S surface 1, . . ., 7 resolving index
Greek symbols a heat transfer coefficient D difference
experimental models. Experimental models are only approximate expression of precise results, but in many cases it is sufficient. Using dimensional analysis [18–25] of high temperature spectral emissivity measurement method, dimensionless similarity quantities for measurement method description are obtained. These dimensionless quantities and a large number of computer model results for optional parameters of the sample and measurement method are inputs to the procedure for experimental mathematical model evaluation. As a result, the experimental mathematical model describing dependence of temperature difference of average temperature on the surface of measured coating and average temperature at the interface of reference and measured coatings is realized. The temperature difference from the experimental mathematical model can be used for more precise determination of measured coating surface temperature and for uncertainty evaluation of high temperature spectral emissivity measurement method.
The applied noncontact surface temperature measurement requires the knowledge of effective emissivity of the measured surface for the infrared camera and viewing angle. The problem is solved by the deposition of the reference coating (ZYP Coating Cr2O3) with known effective emissivity on the half of the measured
measuredcoating reference coating substrate
infrared camera
spectrometer
spot A 2. Emissivity measurement method
Trc,A
spot B =
Tmc,B
2.1. High temperature emissivity measurement method High temperature emissivity measurement method is briefly introduced in the section. A more detailed description of the method can be found in [14,26]. The method is based on the comparison of radiation of a blackbody and of the sample surface that are positioned opposite to each other. The radiation is collected by a rotary parabolic mirror situated halfway between the radiation sources. The spectral detection of radiation is performed by a FTIR spectrometer (Nicolet 6700). The optical path of radiation is covered by an optical box. The sample is clamped to a ceramic insulation and is heated to the required temperature by a fiber laser with a scanning head. The sample surface temperature is measured noncontactly by an infrared camera (FLIR A320) with wavelength range from 7.5 lm to 13 lm. The temperature dependence of normal spectral emissivity is evaluated in the wavelength range from 1.38 lm to 26 lm. The samples can be heated up to 1000 °C.
laser beam Fig. 1. Temperature and emission measurement positions in the emissivity measurement method (spot A is on reference coating, spot B is on measured coating) and temperature evaluation area in the model of the heat transfer in the sample.
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sample surface (Fig. 1). The sample’s front surface area is divided into two symmetrical parts. The sample surface temperature Trc,A (average temperature in spot A) is measured on the part of the sample with the reference coating. The radiation of the second part without the reference coating is detected by the spectrometer for the emissivity analyses (spot B). Assumption of the identical sample temperature on the part with the reference coating Trc,A and measured coating Tmc,B is considered.
Fig. 1). From these average temperatures, the temperature difference DTmc-rc/mc,ave is evaluated from the equation
2.2. Model of the heat transfer in the sample
The temperature difference DTmc-rc/mc,ave from the Eq. (1) is important value not only for uncertainty evaluation of high temperature emissivity measurement method (high temperature normal spectral emissivity measurement method) but it is also used for improving the preciseness of the measurement method. This temperature difference can be used for more precise calculation of emissivities. Utilization of computer model for full heat transfer analysis in the sample is not so comfortable because the analysis take a certain time to be performed. Simple experimental mathematical model for description of temperature difference DTmc-rc/is more convenient way. The number of various mc,ave experiments performed (that are needed for experimental model creation) are number of various variants of heat transfer model that have been computed. Experimental model creates generalization of the computer model results and give the values of temperature difference also for other variants than have not been computed. Temperature difference DTmc-rc/mc,ave is suggested due to requirement of spectral emissivity measurement method. In the method, temperature Trc,ave,A is obtained by infrared camera. The reference coating effective emissivity is evaluated for temperature at reference coating/substrate interface in the case of reference sample [14]. This effective emissivity of reference coating is applied for temperature evaluation also in the case of measured sample. It is supposed, that this temperature is identical with the temperature Trc/mc,ave,A of measured sample and this is identical with the temperature Tmc,ave,B. If the temperatures Trc/mc,ave,A and Tmc,ave,B are different, the emissivity uncertainty is increasing. The knowledge of the temperature difference can be included to the emissivity evaluation and thus emissivity uncertainty can be decreased. Therefore, the mathematical model operates with temperature difference DTmc-rc/mc,ave.
A computer model of the sample steady state is utilized to describe the temperature distribution in the sample during the emissivity measurement at high temperature [17]. Usually the thicknesses of reference and measured coatings are very small. Because of possible difficulties with very small dimensions in some computer models, it is better to transform the real sample to the model sample with enlarged thicknesses of coatings. Along with the sample transformation, it is modified not only geometry, but also material properties and boundary conditions. Dimensionless analysis is performed to obtain the scale factors for all quantities. Using the quantities scale factors, real sample is transformed to the model sample. The temperature distribution is solved for model sample. The results are then transformed back to real sample. All the sample parameters introduced and all the temperature distribution results are concerning the real sample. Sample parameters and measurement method parameters are divided into two groups – prescribed and optional parameters. Prescribed parameters are constant parameters that are fixed for all variants solved. Constant parameters are sample radius 12.5 mm, substrate thickness 5.0 mm and substrate thermal conductivity 25 W m1 K1. Thermal conductivity of reference and measured coatings are also constant and equal to 2 W m1 K1. The side walls of the substrate, measured and reference coatings, are thermally isolated to have zero lossy heat flux. The front part of the sample has heat transfer coefficient equal to 10 W m2 K1 with external temperature 100 °C that includes only convection heat transfer. Reference coating emissivity is temperature dependent from 0.702 to 0.655. The optional parameters vary among the variants that are solved using computer modelling. Optional parameters are thickness of reference coating drc (30, 60, 90 lm), thickness of measured coating dmc (50, 100, 150, 300 lm), emissivity of measured coating emc (0.1, 0.5, 0.9) and temperature at heated sample surface Ts (400–1000 °C with the step 50 °C). Modelling results are average temperatures Trc,ave,A on the surface of reference coating (appropriate temperature of spot A, Fig. 1), average temperatures Trc/mc,ave,A at the interface of reference and measured coatings and average temperatures Tmc,ave,B on the surface of measured coating (appropriate temperature of spot B,
DT mc-rc=mc;av e ¼ T mc;av e;B T rc=mc;av e;A :
ð1Þ
3. Experimental mathematical model evaluation procedure 3.1. Experimental mathematical model idea
3.2. Dimensionless analysis Dimensional analysis employs simple mathematical operations with physical quantities units. The principle is the knowledge of physical process. Dimensional analysis works with units of physical quantities that occurred in the process. Dimensional analysis enables among others determination of number and form of dimensionless quantities in the process.
Table 1 Physical quantities describing the process. Quantity
Units
Description
ks, kmc, krc, kref ds, dmc, drc, r, L T, Ts, Tmc,ave, Trc,ave, Trc/mc,ave, DTmc-rc/mc,ave, Te, Tref qs, qmc, qrc, qref amc, arc, aref
W m1 K1 m K
Thermal conductivities of substrate, measured and reference coating, reference thermal conductivity Thicknesses of substrate, measured and reference coating, sample diameter, reference length Temperature, surface temperature of substrate, surface temperature of measured coating in the spot B, surface temperature of reference coating in the spot A, temperature at the interface of reference and measured coating under spot A, temperature difference, external temperature and reference temperature
W m2 W m2 K1 W m2 K4 –
Surface heat fluxes of substrate, measured and reference coating, reference heat flux Coefficients of heat transfer of measured and reference coating, reference heat transfer coefficient Stefan–Boltzmann constant Emissivities of measured and reference coating
r0
emc, erc
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Because the number of dimensionless quantities describing process is always less than number of dimensional quantities in the process, the transformation from dimensional to dimensionless quantities performs compression of results concerning the physical process analyzed. Generalization of results is possible because experimental mathematical models enable description of other processes that are physically similar. The process of steady state temperature distribution in the sample during measurement procedure is described by steady state heat transfer equation with boundary conditions of 1st (prescribed surface temperature), 2nd (prescribed surface heat flux) and 3rd (prescribed heat transfer coefficient, emissivity and external temperature) types. It gives totally 27 physical quantities, see Table 1. Dimensional analysis gives us 24 dimensionless quantities in Table 2 those can be used in experimental equation concerning this process. There are so called reference quantities among the physical quantities in Table 1. These reference quantities are constants with characteristic values. For example, initial sample temperature can be used as the value for reference temperature Tref. Reference length L means characteristic size in the model, in this case it is sample diameter. Reference thermal conductivity kref has been chosen as a value of the substrate material. Reference heat flux qref and reference heat transfer coefficient aref are set to constant values used in simulation model. 3.3. Experimental mathematical model development The aim is to evaluate experimental mathematical model that describes dimensionless temperature difference DHmc-rc/mc,ave dependence on dimensionless process parameters. There is an effort to establish only one equation of experimental mathematical model that is valid for all process parameters. Since the optional parameters are thicknesses of reference and measured coating, emissivity of measured coating and temperature at heated sample surface, the experimental model should include dimensionless quantities that characterize these parameters. Experimental model thus can contain only simple dimensionless quantities that correspond with variable parameters in computer models
DHmc-rc=mc;av e ¼ f ðemc ; Hs ; Dmc ; Drc Þ:
ð2Þ
Experimental mathematical model is evaluated to achieve very good agreement with the computer model. Evaluation is provided using determination index R2, that is defined in regression analysis by the equation
n X ^ i Þ2 R ¼1 ðyi y
,
2
ı¼1
! n X 2 Þ : ðyi y
ð3Þ
i¼1
It is evident that the better experimental model equation is, the ^ı with better agreement of quantity from experimental model y quantity from computer model yi is obtained. Quantity n denotes is the average of number of computer model experiments and y values yi from computer model. The greater value of determination index R2, the better agreement of experimental mathematical model with computer model. At the beginning of the experimental mathematical model development procedure, the form of experimental mathematical model equation is proposed. Then, initial values of equation parameters are suggested. These parameters values are refined by iteration method to find the maximum of determination index. Evaluation of parameters value is faster and easier when the experimental mathematical model is in the classic product form. Therefore, the first proposal of experimental model equation is in the product form and only when this form is not suitable, other forms are tested. Experimental mathematical model describing dimensionless temperature difference DHmc-rc/mc,ave was initially suggested in the product forms
DHmc-rc=mc;av e ¼ a ebmc Hcs Ddmc Derc :
ð4Þ
DHmc-rc=mc;av e ¼ a ebmc Hcs :
ð5Þ
Experimental mathematical model in the form (4) gives results that differ from computer model results. Even the form (5), that seems to be better for fitting of coefficients a, b, c individually for each combination of thicknesses of measured and reference coating, does not tend to better results. Therefore other forms of experimental model equations were prepared. Promising results were obtained using polynomial form that has been tested for emissivity of measured coating (6) and extended for variable dimensionless temperature Hs (7)
DHmc-rc=mc;av e ¼ ða1 emc þ a2 Þ; ð6Þ DHmc-rc=mc;av e ¼ a1 H2s þ a2 Hs þ a3 emc þ b1 H2s þ b2 Hs þ b3 : ð7Þ Experimental mathematical model in the form of Eq. (7) fits computer model correctly for low measured coating emissivities emc = 0.1 and 0.5. For high value emc = 0.9, it is necessary to design another equation. Polynomial form of sixth order is in good agreement with computer model
DHmc-rc=mc;av e ¼ a1 H6s þ a2 H5s þ a3 H4s þ a4 H3s þ a5 H2s þ a6 Hs þ a7 :
ð8Þ
Table 2 Dimensionless quantities describing the process. Quantity
Form
Ks, Kmc, Krc
ks kref
Ds, Dmc, Drc, R
ds L
H, Hs, Hmc,ave, Hrc,ave, Hrc/mc,ave, DHmc-rc/mc, ave,
He Qs, Qmc, Qrc
Description
;
kmc ks
;
dmc ; ds Ts T ref
;
T T ref
T rc;av e T ref
;
; ;
krc ks drc ; Lr ds T mc;av e T ref ,
T rc=mc;av e T ref ,
Dimensionless thermal conductivities of substrate, measured and reference coating Dimensionless thicknesses of substrate, measured and reference coating, dimensionless sample diameter Dimensionless temperatures local and substrate surface, dimensionless surface temperature of measured coating in the spot B, dimensionless surface temperature of reference coating in the spot A, dimensionless temperature at the interface of reference and measured coating under spot A, dimensionless temperatures difference and external temperature
DT mc-rc=mc;av e , T ref Te T ref qs qref
;
Amc, Arc
amc ; aref
emc, erc Ki
–
Bi
aL
Sf
er 0 T 3 L
qmc qref
arc aref
;
qrc qref
Dimensionless heat fluxes at the surface of substrate, measured and reference coating Dimensionless heat transfer coefficients of measured and reference coating Emissivities of measured and reference coating Kirpitchev number
qL kT
Biot number
k
k
Stefan number
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Good experimental models describing dependence of dimensionless temperature difference DHmc-rc/mc,ave on emissivity of measured coating and dimensionless sample temperature at heated side were evaluated. Eqs. (7) and (9) include the effect of emissivity of measured coating and dimensionless sample temperature. The influence of thicknesses of measured and reference coating Dmc and Drc is involved in subsequent equations. (A) Low value range of measured coating emissivity from emc = 0.1 to 0.5 One equation enables to include all other parameters in the form
DHmc-rc=mc;av e ¼ða1 emc þ a2 Þðb1 H2s þ b2 Hs þ b3 Þ ðc1 D2mc þ c2 Dmc þ c3 Þðd1 D2rc þ d2 Drc þ d3 Þ:
ð9Þ
Linear dependence for emissivity of measured coating and polynomial dependencies of second order for dimensionless sample temperature at heated side and dimensionless thicknesses of measured and reference coating are sufficient. There is a very good agreement of this experimental mathematical model Eq. (9) with corresponding results of computer model. The Eq. (9) is valid for low measured coating emissivities in the range from 0.1 to 0.5. It has been detected, that for higher measured coating emissivities close to 0.7, this equation loses its validity and another form of the experimental model equation is necessary. The equation (9) is applicable for all thicknesses of reference coating in the observed range from 30 to 90 lm and for all thicknesses of measured coating in the observed range from 50 to 300 lm.
Table 3 Parameters values of experimental mathematical model equations. Equation
(9)
(10)
(10)
(10)
(10)
emc (–) dmc (lm) drc (lm)
0.1–0.5 50–300 30–90
0.9 50 30–90
0.9 100 30–90
0.9 150 30–90
0.9 300 30–90
a1 a2 a3 a4 a5 a6 a7 b1 b2 b3 c1 c2 c3 d1 d2 d3
1.12778 0.90159 – – – – – 0.74134 0.42725 0.08331 5.44951 3.59402 0.24288 1.81206 0.42572 0.28809
1.53239 5.82965 8.88086 6.90821 2.89646 0.62247 0.05355 81.06340 14.80550 0.83600 – – – – – –
1.74359 6.64254 10.13569 7.89942 3.31906 0.71491 0.06165 53.95040 13.04250 0.85260 – – – – – –
1.93937 7.40131 11.31489 8.83697 3.72133 0.80345 0.06945 34.54080 10.61150 0.86690 – – – – – –
2.49797 9.56972 14.69048 11.52562 4.87717 1.05831 0.09194 17.15870 10.78750 0.86770 – – – – – –
Table 4 Determination index showing agreement between the results of experimental mathematical model and computer model. Equation
(9)
(10)
(10)
(10)
(10)
emc (–) dmc (lm) drc (lm)
0.1–0.5 50–300 30–90 0.9983
0.9 50 30–90 0.9986
0.9 100 30–90 0.9986
0.9 150 30–90 0.9986
0.9 300 30–90 0.9985
R2 (–)
(B) High value of measured coating emissivity emc = 0.9 One equation enabling to include all other parameters has not been found. It is necessary to formulate individual equations for each constant dimensionless thickness of measured coating. The form of these equations is the same
Polynomial dependency of sixth order for dimensionless sample temperature at heated side and polynomial dependency of second order for dimensionless thicknesses of reference coating are sufficient. Equation parameters are evaluated separately for each value of dimensionless thickness of measured coating. There is very good agreement of this experimental mathematical model Eq. (10) with corresponding results of computer model. The Eq. (10) is valid for high value of measured coating emissivity equal to 0.9 and for all thicknesses of reference coating in the observed range from 30 to 90 lm. Values of equation parameters have to be found for each individual observed value of measured coating thickness. 4. Results of the analysis 4.1. Experimental mathematical model equations Two forms of the experimental mathematical model equations are found to be in very good agreement with the computer models. The Eq. (9) is determined for the low value range of measured coating emissivity from 0.1 up to 0.5, the equation involves all
0.025 0.003
Δθ mc-rc/mc, ave (-)
ð10Þ
computer model
0.005
0.020 0.000 0.000
0.003
0.005
0.015 0.010 0.005 0.000 0.000
(a) 0.005
0.010
0.015
0.020
0.025
0.030
computer model Δθ mc-rc/mc, ave (-) experimental mathematical model
computer model
0.001
0.000
Δ θ mc-rc/mc, ave (-)
DHmc-rc=mc;av e ¼ a1 H6s þ a2 H5s þ a3 H4s þ a4 H3s þ a5 H2s þ a6 Hs þ a7 b1 D2rc þ b2 Drc þ b3 :
experimental mathematical model
0.030
-0.001 0.0005
-0.002 0.0003
-0.003 0.0000 0.0000
-0.004 -0.004
-0.003
-0.002
-0.001
0.0003
0.000
0.0005
(b)
0.001
computer model Δθ mc-rc/mc, ave (-) Fig. 2. The dimensionless temperature difference evaluated from the experimental mathematical model and computer model for the low emissivities of measured coating 0.1–0.5 (a), and high emissivity of measured coating 0.9 (b).
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ΔT ≤-5
0
5
10
15
≥17
ε = 0.1
ε = 0.5
ε = 0.9
100 80 drc (μm)
Ts = 400°C
2.75
0.40
0.8
2.50
0.35
0.9
60
1.0
2.25 2.00
40
0.30
1.1 0.25
1.2
1.75
20 100
drc (μm)
Ts = 600°C
5.5
80
2.25
5.0 0.5
2.00
60
4.5 0.4
1.75
40
4.0 0.3
1.50
3.5
20 100 0.6
3.5
80 drc (μm)
Ts = 800°C
12.5
0.5
4.0
11.0
4.5
60
0.4
9.5 5.0
40 8.0
5.5
0.3
20 100
drc (μm)
Ts = 1000°C
23.0
80 60
-1.7 8.0
19.0
-1.5
17.0
40 20
6.5
21.0
9.5 -1.3
15.5
50 100 150 200
250 300
50 100 150 200
dmc (μm)
dmc (μm)
250 300 50 100 150 200 250 300 dmc (μm)
Fig. 3. Temperature dependence of the temperature difference as a function of the reference and measured coatings thickness for three different values of measured coating total emissivity (0.1, 0.5 and 0.9).
dimensionless quantities that correspond with optional parameters in the computer models. The form of the Eq. (10) is evaluated for the high value of measured coating emissivity 0.9, whereas it is necessary to formulate the equation parameters values for each constant dimensionless thickness of measured coating. In total, five experimental mathematical model equations are established. The parameters values of equations are shown in Table 3. The values are related to the dimensional thickness of measured coating dmc for better clarity of results.
4.2. Experimental mathematical model and computer model results comparison The parameters values of experimental mathematical model equations (Table 3) are verified by comparing the dimensionless temperature differences DHmc-rc/mc,ave that are evaluated from the experimental mathematical model and from the computer model. The verification is executed by calculating the determination index R2 defined by the Eq. (3). The determination index values for appropriate experimental mathematical models are
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shown in Table 4. Because the values are higher than 0.99 for all experimental mathematical models, very good agreement of these experimental mathematical models and computer model results is established. Comparison of dimensionless temperature difference DHmc-rc/mc,ave evaluated from the experimental mathematical model and computer model are graphically presented in Fig. 2. The dependence of dimensionless temperature differences for two models on the computer model temperature difference is presented. The results in Fig. 2a are related to the low emissivities of measured coating 0.1 to 0.5 including all thicknesses of measured coating and all thicknesses of reference layer and sample temperature level. In Fig. 2b, the comparison of dimensionless temperature differences for the experimental mathematical model and computer model are shown only for the high emissivity of measured coating 0.9, thickness of measured coating 150 lm and for all observed thicknesses of reference layer and sample temperature level. The similar results are obtained for other thicknesses of measured coating (50, 100 and 300 lm). Good agreement of the dimensionless temperature differences DHmc-rc/mc,ave evaluated from the experimental mathematical model and computer model are apparent for all process parameters. 4.3. Temperature differences The experimental mathematical model Eqs. (9) and (10) with the parameter values specified in Table 3 are used for the computation of temperature difference DTmc-rc/mc,ave between average surface temperature of measured coating and average temperature at the interface of reference and measured coatings. The results of the computer model for optional parameters are replaced by the results of experimental mathematical model and these results are extended by the temperature differences evaluated for any thickness of reference and measured coating, substrate surface temperature and total emissivity of measured coating. The results are applied for uncertainty evaluation of high temperature emissivity measurement method and improving the preciseness of the measurement method. The temperature dependence of temperature difference from the experimental mathematical model equations is shown in Fig. 3 for the selected values of substrate surface temperature in the temperature range from 400 °C to 1000 °C, three different values of total emissivity of measured coating 0.1, 0.5 and 0.9, thicknesses of reference coating in the range from 20 to 100 lm, and thicknesses of measured coating in the range from 25 to 300 lm. White color is displayed when the temperature difference is about 5 K. If the temperature difference diverges from 5 K, the colors are deeper. The difference over 17 K toward positive values and the difference 0 K are indicated by the deepest colors. Further increase in the temperature difference toward negative values is represented by the lighter shades. The first column of the matrix plot in Fig. 3 shows the temperature dependence of temperature difference for the emissivity of measured coating 0.1. It is clear that the temperature difference significantly depends on the sample temperature. With increasing temperature, temperature difference between average surface temperature of measured coatings and average temperature at the interface of reference and measured coating grows from 1.75 °C for sample temperature 400 °C to 25 °C for sample temperature 1000 °C. The temperature difference also increases with the sample temperature for the second (emc = 0.5) and the third (emc = 0.9) column of the matrix plot. However, with increasing emissivity, the differences in temperature decrease. For the emissivity of measured coating 0.5, the temperature difference about 1 °C is achieved for the temperature 400 °C, at the temperature 1000 °C it is up to 10 °C. The temperature dependence of
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temperature difference is almost negligible for the emissivity of measured coating 0.9, it is about 1 °C. The dependence of temperature difference on the measured and reference coatings thicknesses is less significant that the dependence of temperature difference on substrate surface temperature. Generally, the higher temperature difference is reached for the greater thickness of the measured coating. The specific values are dependent on the emissivity of measured coating and sample temperature, these vary in the order of °C. The thickness of the reference coating affects the temperature difference just a few tenths of °C. The different trend of the dependence of temperature difference on the thickness of reference coating is proved for the emissivity up to 0.5 and emissivity 0.9. While for the emissivity up to 0.5 the temperature difference slightly decreases with increasing thickness of the reference coating, for the emissivity 0.9 it is opposite. 5. Conclusions Sample temperatures during high temperature spectral emissivity measurement are obtained from sensitivity analysis of the measurement method. Dimensionless physical quantities concerning the emissivity measurement process have been established utilizing dimensionless analysis technique. Experimental mathematical model have been compiled from these dimensionless quantities as a generalization of sensitivity analysis results. The equation describing the temperature difference DHmc-rc/mc, ave, between measured coating surface and interface of reference and measured coating, dependent on dimensionless process parameters has been evaluated. However, it is not possible to evaluate one equation comprising the effect of all observed parameters. Experimental mathematical model in the first form includes linear dependence to the emissivity of measured coating and polynomial dependencies of second order to the dimensionless sample temperature at heated side and dimensionless thicknesses of measured and reference coating. The model in this form is valid for low value range of measured coating emissivities emc = 0.1 up to 0.5 and for all thicknesses of measured and reference coatings including all measurement temperature levels. Experimental mathematical model in the second form comprises polynomial dependency of sixth order to the dimensionless sample temperature at heated side and polynomial dependency of second order to the dimensionless thicknesses of reference coating. The model in this form is used for high measured coating emissivity emc = 0.9. For each independent thickness of measured coating, the set of experimental model parameters has been derived for all thicknesses of reference coating including all measurement temperature levels. Absolute temperature differences DTmc-rc/mc,ave computed from the experimental mathematical models for observed thicknesses of measured and reference coatings, emissivity of measured coating and sample temperature level are in the range from negative values in the order of units up to positive values in the order of few tens. The emissivity of measured coating has the largest impact on the temperature difference, sample temperature level has lower effect and the thicknesses of the coatings have the minor influence. All evaluated experimental mathematical models have very good agreement with computer model results from sensitivity analysis of high temperature spectral emissivity measurement method. Experimental mathematical models are used for generalization of sensitivity analysis results and for computation of spectral emissivity total uncertainty that is subject of subsequent work. Acknowledgments The result was developed within the CENTEM project, reg. no. CZ.1.05/2.1.00/03.0088, cofunded by the ERDF as part of the
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Ministry of Education, Youth and Sports OP RDI programme and, in the follow-up sustainability stage, supported through CENTEM PLUS (LO1402) by financial means from the Ministry of Education, Youth and Sports under the ”National Sustainability Programme I.”. References [1] H. Watanabe, J. Ishii, H. Wakabayashi, T. Kumano, L. Hanssen, Chapter 9 – spectral emissivity measurements, Exp. Methods Phys. Sci. 46 (2014) 333–366. [2] J. Manara, M. Arduini-Schuster, M. Keller, Infrared-optical characteristics of ceramics at elevated temperatures, Infrared Phys. Technol. 54 (2011) 395–402. [3] G. Neuer, G. Jaroma-Weiland, Spectral and total emissivity of hightemperature materials, Int. J. Thermophys. 19 (1998) 917–929. [4] R. Brandt, C. Bird, G. Neuer, Emissivity reference paints for high temperature applications, Measurement 41 (2008) 731–736. [5] J. Manara, M. Arduini-Schuster, L. Hanssen, Integrating sphere reflectance and transmittance intercomparison measurements for evaluating the accuracies of the achieved results, High Temp. – High Press 38 (2009) 259–276. [6] L.M. Hanssen, C.P. Cagran, A.V. Prokhorov, S.N. Mekhontsev, V.B. Khromchenko, Use of a high-temperature integrating sphere reflectometer for surfacetemperature measurements, Int. J. Thermophys. 28 (2007) 566–580. [7] J. Hameury, B. Hay, J. Filtz, Measurement of infrared spectral directional hemispherical reflectance and emissivity at BNM-LNE, Int. J. Thermophys. 26 (2005) 1973–1983. [8] L. del Campo, R.B. Pérez-Sáez, X. Esquisabel, I. Fernández, M.J. Tello, New experimental device for infrared spectral directional emissivity measurements in a controlled environment, Rev. Sci. Instrum. 77 (2006) 113111-1–113111-8. [9] C.H. Monte, J. Hollandt, The measurement of directional spectral emissivity in the temperature range from 80 °C to 500 °C at the Physikalisch-Technische Bundesanstalt, High Temp. – High Press 39 (2010) 151–164. [10] B. Hay, J. Hameury, N. Fleurence, P. Lacipiere, M. Grelard, V. Scoarnec, G. Davee, New facilities for the measurements of high temperature thermophysical properties at LNE, Int. J. Thermophys. 35 (2014) 1712–1724.
[11] M. Honner, P. Honnerová, Survey of emissivity measurement by radiometric methods, Appl. Opt. 54 (2015) 669–683. [12] R. Siegel, J. Howel, Thermal Radiation Heat Transfer, Taylor & Francis, Washington, 2001. [13] L.R. Koirala, FTIR-spectroscopic Measurement of Directional Spectral Emissivities of Microstructured Surfaces Ph.D. Thesis, University of the Federal Armed Forces, Hamburg, 2004. [14] P. Honnerová, J. Martan, M. Kucˇera, M. Honner, J. Hameury, New experimental device for high-temperature normal spectral emissivity measurements of coatings, Meas. Sci. Technol. 25 (2014) 095501. [15] J.F. Sacadura, Measurement techniques for thermal radiation properties, in: Proc. 9th International Heat Transfer Conference, 1990, pp. 207–222. [16] D.P. DeWitt, G.D. Nutter, Theory and Practice of Radiation Thermometry, John Wiley, New York, 1988. [17] Z. Vesely´, P. Honnerová, J. Martan, M. Honner, Sensitivity analysis of high temperature spectral emissivity measurement method, Infrared Phys. Technol. 71 (2015) 217–222. [18] T. Szirtes, Applied Dimensional Analysis and Modeling, McGraw-Hill, New York, 1998. [19] J.E. Baker, P.S. Westine, F.T. Dodge, Similarity Methods in Engineering Dynamics, Elsevier, Amsterdam, 1991. [20] G.I. Barenblatt, Dimensional Analysis, Gordon and Breach Science Publisher, New York, 1987. [21] P.W. Bridgman, Dimensional Analysis, AMS Press, New York, 1978. [22] H.E. Huntley, Dimensional Analysis, Dover Publications, New York, 1967. [23] R.C. Pankhurst, Dimensional Analysis and Scale Factors, Chapman and Hall, London, 1964. [24] E. Szücs, Similitude and Modelling, Elsevier, Amsterdam, 1980. [25] J. Zierep, Similarity Laws and Modeling, Marcel Dekker, New York, 1971. [26] M. Mauer, P. Kalenda, M. Honner, P. Vacíková, Composite fillers and their influence on emissivity, Phys. Procedia 44 (2013) 262–269.