Temperature measurements of high-temperature semi-transparent infrared material using multi-wavelength pyrometry

Accepted Manuscript Temperature measurements of high-temperature semi-transparent infrared material using multi-wavelength pyrometry Tairan Fu, Jiangfan Liu, Jiaqi Tang, Minghao Duan, Huan Zhao, Congling Shi PII: DOI: Reference:

S1350-4495(14)00087-5 http://dx.doi.org/10.1016/j.infrared.2014.05.016 INFPHY 1545

To appear in:

Infrared Physics & Technology

Received Date:

12 May 2014

Please cite this article as: T. Fu, J. Liu, J. Tang, M. Duan, H. Zhao, C. Shi, Temperature measurements of hightemperature semi-transparent infrared material using multi-wavelength pyrometry, Infrared Physics & Technology (2014), doi: http://dx.doi.org/10.1016/j.infrared.2014.05.016

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Temperature measurements of high-temperature semi-transparent infrared material using multi-wavelength pyrometry Tairan Fu [1,2,*], Jiangfan Liu [1], Jiaqi Tang [1], Minghao Duan [1], Huan Zhao [1],Congling Shi [3] [1] Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, P.R.China [2] Beijing Key Laboratory of CO 2 Utilization and Reduction Technology, Beijing, 100084, P.R.China [3] China Academy of Safety Science & Technology, Beijing 100029, P.R. China * Corresponding author: [email protected]

Abstract Temperature measurements inside semi-transparent materials are important in many fields. This study investigates the measurements of interior temperature distributions in a one-dimensional semi-transparent material using multi-wavelength pyrometry based on the Levenberg-Marquardt method (LMM). The investigated material is semi-transparent Zinc Sulfide (ZnS), an infrared-transmitting optical material operating at long wavelengths. The radiation properties of the one-dimensional semi-transparent ZnS plate, including the effective spectral-directional radiation intensity and the proportion of emitted radiation, are numerically discussed at different wavelengths (8.0~14.0 µm) and temperature distributions (400 K~800 K) to provide the basic data for the temperature inversion problem. Multi-wavelength pyrometry was combined with the Levenberg-Marquardt method to resolve the temperature distribution along the radiative transfer direction based on the line-of-sight spectral radiation intensities at multiple wavelengths in the optimized spectral range of (11.0~14.0 µm) for the semi-transparent ZnS plate. The analyses of the non-linear inverse problem show that with less than 5.0% noise, the inversion temperature results using the Levenberg-Marquardt method are satisfactory for linear or Gaussian temperature distributions in actual applications. The analysis provides valuable guidelines for applications using multi-wavelength pyrometry for temperature measurements of semi-transparent materials.

Keywords: Temperature, semi-transparent, multi-wavelength, pyrometry, optical properties 1.

Introduction

Optical radiation pyrometry is widely used for measuring combustion flame temperatures or surface temperatures in industrial applications and scientific research [1-28]. Mori et al. [13] presented an experimental study of the convective heat transfer and flow field characteristics in a rotating rotor cascade and the surface temperature distribution on a rotating heated blade measured by means of infrared thermography. Lu et al. [17] described a three-color pyrometry algorithm based on a color CCD camera with the measured particle-surface temperatures and flame temperatures agreeing well with thermocouple measurements. Densmore et al. [24] used a high-speed image pyrometer based on the two-color ratio method for temperature measurements of explosive and combustion processes. Guo et al. [26] used a digital camera to measure full-field soot temperatures and soot volume fractions in axisymmetric flames. Kappagantula et al. [27] measured the spatial temperature distribution of combustion products by coupling point source temperature measurements from a multi-wavelength pyrometer with irradiance measurements from an infrared camera to produce a highly discretized thermal map. Estevadeordal et al. [28] 1

used a high-speed multicolor pyrometry to measure radiation temperatures of hot particulate bursts generated from a combustor. In most applications using optical radiation pyrometers, for opaque materials (for example, metals and alloys), the measured radiation comes from the surface and the temperature determined from the measured effective emitted radiation is only the “surface” temperature. However, for semi-transparent materials (for example, glass, coatings and polymers), the measured radiation received by the radiation pyrometer comes not only from the surface radiation and reflected background radiation, but also from the radiation emitted inside the material and transmitted background radiation passing through the material along the radiation transfer direction. The temperature determined using traditional pyrometry then does not correspond to the surface temperature. The measured temperature can then be looked upon as the coupled “apparent temperature” related to the temperature distribution inside the material, the transmitted environmental temperature and the reflected environmental temperature. Deducing the “true” temperature distribution from the coupled effective radiation is a major obstacle in applications using radiation pyrometers for semi-transparent materials. Some research has focused on this subject. Pfefferkorn et al. [29] used a long-wavelength pyrometer to measure the surface temperatures of a dense zirconia ceramic that was semitransparent at shorter wavelengths. The process avoided the effects of the semi-transparent radiation properties of zirconia ceramic. Hajji and Spruiell [30] investigated the use of radiation pyrometry for semi-transparent materials and derived a temperature expression for nonisothermal semi-transparent gray polymer materials. Daniel and Gustave [31] used a near-infrared multi-wavelength pyrometer to simultaneously calculate the surface and bulk temperatures of semi-transparent materials (zirconia barrier coating and glass) based on a simplified radiative transfer model. Nagtegaal et al. [32] presented a numerical analysis of the multi-wavelength optical method to determine temperature profiles in hot glass based on Tikhonov regularization and the L-curve algorithm. However, current research works are mostly related to semi-transparent zirconia ceramic and glass. There are few studies of noncontact temperature measurements of semi-transparent Zinc Sulfide (ZnS) which is widely used as an infrared semi-transparent optical window material which has the advantages of high mechanical strength, high hardness, erosion resistant, low temperature coefficient of the refractive index, and excellent high-temperature optical properties[33]. This study investigates ZnS as a representative semi-transparent material. The noncontact measurements of the temperature information in the semi-transparent ZnS material are necessary for the structure-stress analysis, optical property examinations and other applications of ZnS at different thermal conditions. Due to the excellent optical transmission from the visible to infrared wavelengths of ZnS material, the measured effective radiation intensity information received by the optical pyrometer is strongly affected by the reflected and transmitted background radiation. The proportion of the emitted radiation relating to the semi-transparent ZnS material temperature will be small in the visible to infrared wavelengths. Thus, the “true” temperature distributions are difficult to accurately derive from effective radiation intensity data recorded by a radiation pyrometer. This study investigates radiation temperature measurements of a semi-transparent ZnS plate using infrared multi-wavelength pyrometry to determine the temperature distribution along the radiative transfer direction based on the line-of-sight radiation intensities at multiple

2

wavelengths. The issues, including the apparent spectral radiation properties, measurement spectrum optimization, and inversion temperature accuracy, will be investigated. The analysis provides a valuable reference for the applications of multi-wavelength pyrometry for temperature distribution measurements in semi-transparent materials.

2.

Radiation properties of a one-dimensional ZnS plate

The radiative transfer process in a one-dimensional semi-transparent absorbing-emitting plate with ideal optical smooth interfaces is shown in Fig. 1. The equation describing the spectral radiation intensity distribution along the radiative ray direction neglecting scattering is [34]: dI λ = α λ ( s ) I bλ ( s) − α λ ( s ) I λ ( s) ds

(1)

where I λ is the spectral radiation intensity distribution at wavelength λ , I bλ is the blackbody spectral radiation intensity distribution at the same temperature, α λ is the spectral absorption coefficient and s is the position vector along the radiative transfer direction. Define θ to be the angle between the x coordinate axis and the ray direction. The spectral radiation intensity is expressed as I λ+ when 0 ≤ θ ≤ 90
, while the spectral radiation intensity is expressed as I λ− when 90 < θ ≤ 180
. Equation (1) can be rewritten as:

 I + (τ , µ ) = I + ( 0, µ ) e −τ dλ / µ + τ dλ I τ * e − (τbλ −τ λ* ) / µ dτ * / µ λ λ ∫0 bλ ( λ )  λ dλ  τ dλ *  I λ− ( 0, µ ) = I λ− (τ d λ , µ ) eτ d λ / µ − ∫ I bλ (τ λ* ) eτ λ / µ dτ λ* / µ 0 

(2)

x

where the optical thickness τ λ ( x ) = ∫ α λ ( x* ) dx , µ = cos θ r and ν = cos θ i . The boundary 0

conditions for the radiative transfer are: + 2 + − τ λ = 0, I λ ( 0, µ ) = n (1 − ρ ) I e + ρ I λ ( 0, µ )  − 2 − + τ λ = τ d λ , I λ (τ bλ , µ ) = n (1 − ρ ) I e + ρ I λ (τ bλ , µ )

(3)

where I e+ and I e− are the background radiation intensities, n is the real part of the complex refractive index, and ρ is the single-surface reflectance of the interface for unpolarized radiation which is given by the Fresnel Equation [33, 34]:

ρ=

ρ s  ( A − sin θ i tan θi ) + B 2

2   + 1 2 2   2  ( A + sin θi tan θ i ) + B 

(4)

where A and B are functions of the complex refractive index ( k is the imaginary part) given by : 12 2  2  2 2 2 2 2 2 2 2  2 A = ( n − k − sin θi ) + 4 n k  + ( n − k − sin θi )  12  2 B 2 = ( n 2 − k 2 − sin 2 θ ) 2 + 4 n 2 k 2  − ( n 2 − k 2 − sin 2 θ ) i i   

3

(5)

I e−

x

θi

Iλ+ (τdλ, µ)

θr

Iλ− (τ dλ , µ )

d θr I e+

I λ− ( 0, µ )

I λ+ ( 0, µ )

θi

Fig. 1 Radiative transfer inside a one-dimensional absorbing-emitting plate The measured effective radiation from a semi-transparent plate received by a pyrometer includes the emitted surface radiation, the radiation emitted from inside the plate, the reflected background radiation and the background radiation transmitted through the plate along the radiative transfer direction. From Eqs.(2)~(5), the effective spectral-directional radiation intensity for an one-dimension semi-transparent plate from the boundary 1 is expressed as: I λ , eff = ε λ Ibλ (T ) + rλ I e+ + tλ I e−

(6)

d

where T is the average temperature of the one-dimensional plate, T =1/d ⋅ ∫ T ( x )dx ; I e+ and 0

I e− are the background radiation intensities from interfaces 1 and 2 for the one-dimensional

semi-transparent plate and ε λ , tλ and rλ are the apparent spectral-directional emittance, transmittance and reflectance of the semi-transparent plate from the interface 1 [33, 34]: τ dλ

ελ

τ ∫ (1 − ρ ) (e = −

0

*

λ

/µ

+ρe 2

(1 − ρ e

(

) ) I (τ * )dτ * / µ bλ λ λ

− 2τ d λ −τ *λ / µ

−2τ d λ / µ

) I bλ (T )

, tλ =

(1 − ρ ) 2 e −τ d λ / µ ρ + ρ (1 − 2 ρ ) e −2τ d λ / µ , rλ = 2 −2τ d λ / µ 1− ρ e 1 − ρ 2 e −2τ dλ / µ

(7) The proportion of the emitted radiation relating to the temperature distribution inside the plate in the effective output signal is expressed as: FSN = ε λ Ibλ (T ) / I λ , eff

(8)

The interior temperature distribution along the thickness direction is described by the function Φ , T = Φ ( x; a1 , a2 ,..., an )

(9)

where (a1 , a2 ,..., an ) are the function parameters describing the temperature distribution and x is the coordinate through the one-dimensional plate as shown in Fig. 1. In general experimental conditions, for example, thermal shock heating from one side, natural cooling of isothermal sample, etc., the interior temperatures in the ZnS plate behaves the monotonic or symmetric distribution so that it was assumed to be a linear temperature distribution along the plate thickness, T = a2 x + a1 ,

or

an

Gaussian

temperature

distribution

along

the

plate

thickness,

2

T = exp(a3 x + a2 x + a1 ) . There are various choices of temperature distribution function to

approximatively simulate the actual temperature distribution in the ZnS plate at different heating 4

conditions: (I)

Steady-state condition with one heating boundary and one natural boundary: linear temperature distribution or monotonic temperature distribution described by Gaussian function.

(II)

Transient-state condition with one heating boundary and one natural boundary: monotonic temperature distribution described by Gaussian function

(III) Transient-state condition with two heating boundaries: symmetric temperature distribution described by Gaussian function (IV) Natural cooling of isothermal sample with two natural boundary: symmetric temperature distribution described by Gaussian function Based on Eqs. (6) ~ (9), the spectral radiation properties and the proportion of the emitted radiation of a one-dimensional ZnS plate were analyzed numerically at various temperatures to provide the reference data for the temperature inversion solution. The simulations assumed a plate thickness d = 12 mm , a direction angle θi = 0° ( µ = 1) , a reflected background temperature Te,1 = 293 K , a transmitted background temperature Te, 2 = 293 K , a response range of 8~14 µm,

and a material temperature range of 400 K~800 K. The optical properties (absorption coefficient and complex refractive index) of ZnS material are shown in Fig. 2 [33, 35]. The optical properties at different temperatures were obtained by linear interpolation and extrapolation of the data reported in the references [33, 35]. The uncertainties of optical properties arising from the data process are ignored and don’t affect the measurement method analysis in our study.

Fig. 2 Optical properties of ZnS [33, 35] For case (1) with the linear temperature distribution, the predicted effective spectral-directional radiation intensity, I λ , eff , and the proportion coefficient, FSN , are shown in Figs. 3 and 4 for various temperatures. The linear temperature distributions in the one-dimensional plate are shown for temperature ranges of (A) 400 K-500 K, (B) 500 K-600 K, (C) 600 K-700 K and (D) 700 K-800 K. The effective spectral-directional radiation intensity increases to a maximum ( λ =11.2 µm) and then decreases with increasing wavelength shown in Fig. 3. The low absorption 5

coefficient of ZnS for λ <10.0 µm reduces the effective intensity at short wavelengths with I λ , eff at 8.0 µ m only 11% of the value at 11.2 µ m for the 700 K-800 K temperature range. Increasing the material temperature increases the effective spectral-directional radiation intensity. I λ , eff at 11.2 µm increases from 7.17 to 30.24 W/m2/µm/sr when the temperature increases from range A (400 K-500 K) to range D (700 K-800 K). The proportion of the emitted radiation to the effective radiation, FSN , increases with increasing temperature and wavelength shown in Fig. 4. For example, for the same temperature condition A with 400 K-500 K, FSN increases from 0.72 to 4.73 as the wavelength increases from 10.0 µm to 12.0 µm. In the spectral region of 8.0 µ m to 14.0 µ m, FSN is within (0.06, 18.53) for temperature condition A and FSN increases to (0.33, 69.74) for temperature condition D with 700 K-800 K. Larger FSN enhances the signal-to-noise ratio for the temperature measurements based on the self-thermal radiation intensities. Therefore, for a threshold for FSN of 4.0, the spectral region of 11.0 µ m to 14.0 µ m with the larger responses of I λ , eff and FSN can satisfy the measurement accuracy requirements for a wide range of temperatures. For case (2) with the Gaussian temperature distribution, the predicted effective spectral-directional radiation intensity, I λ , eff , and the proportion coefficient, FSN , are shown in Figs. 5 and 6 for various temperatures. The properties were also predicted for symmetric, parabolic temperature distributions in the one-dimensional plate with the highest temperature in the middle of the plate and temperature range of (A) 400 K-500 K, (B) 500 K-600 K, (C) 600 K-700 K and (D) 700 K-800 K. I λ , eff and FSN had similar distributions to those for case (1). Therefore, the radiation characteristics for the assumed temperature distributions show that the suitable spectral region for radiation temperature measurements is within (11.0 µ m, 14.0 µm) due to the enhanced response intensity and signal-to-noise ratio.

Fig. 3 Effective spectral-directional radiation intensities at various wavelengths and temperatures for case (1) with linear temperature distributions

6

Fig. 4 Proportion coefficient Fns of the emitted radiation at various wavelengths and temperatures for case (1) with linear temperature distributions

Fig. 5 Effective spectral-directional radiation intensities at various wavelengths and temperatures for case (2) with Gaussian temperature distributions

Fig. 6 Proportion Fns of the emitted radiation at various wavelengths and temperatures for case (2) with Gaussian temperature distributions

3.

Temperature measurements using multi-wavelength pyrometry

As an improvement over one-color or two-color pyrometry, multi-wavelength pyrometry has been widely used to determine temperatures from spectral intensity measurements at multiple 7

wavelengths [36-44]. Multi-wavelength measurements are generally used to solve the problem of the unknown spectral emissivity in the temperature calculations. More spectral measurement data can greatly reduce the uncertainty effect of the unknown spectral emissivity which is assumed to be the simple function of wavelength. The temperature and the emissivity are two uncoupled quantities in the equations formed by multi-wavelength measurement data so that it is simplified to a linear solution problem for determining the temperature and emissivity. However, for the temperature measurements of semi-transparent materials with known radiation properties, the measurement signal includes the information of temperature distribution along the line-of-sight direction. Differing from general applications of multi-wavelength pyrometry, multi-wavelength measurements are used here to determine the temperature distribution of semi-transparent layer, not to solve the unknown emissivity. The measured temperatures and the wavelength variable are coupled in the Planck function which is a typical non-linear solution problem. The measurement equations at multiple wavelengths are mathematically un-correlative for the temperature distribution solutions. Therefore, the temperature distribution in a ZnS plate can be numerically calculated from the measured spectral radiation intensities at multiple wavelengths. However, the solution accuracy will be strongly restricted by the algorithms and the illness of equations for this non-linear solution problem. According to the discussions in the section 2, the optimized wavelengths for ZnS temperature measurements may be set in the range of 11.0 µ m to 14.0 µ m using an infrared spectrometer. The calculation of temperature distribution along the radiative transfer direction in the semi-transparent plate involves complex numerical inversion of the radiative transfer equations using the line-of-sight radiation intensities at multiple wavelengths. The measurement equations at m wavelengths for the multi-wavelength pyrometry from Eq. (6) are: I λi , eff = ε λi I bλi (T ) + rλi I e+ + tλi I e− ,

i = 1, 2,..., m

(10)

Define the vector of the unknown parameters of the temperature function a = ( a1 , a2 ,..., an ) and the vector of wavelength variables λ = (λ1 , λ2 ,..., λm ) . Combining Eqs. (6), (7), (9) and (10) gives the nonlinear function f (a) , f (a) = P(λ ; a) − A(λ ) I λ , eff − B(λ ) = 0

(11)

 P (λ , T ) = τ dλ (e−τ λ* / µ + ρ e− ( 2τ dλ −τ λ* ) / µ ) I (τ * ) dτ * / µ bλ λ λ ∫0   2 −2τ d λ / µ λ ρ A ( ) = (1 − e )   −2τ d λ / µ / (1 − ρ ) I e+ + (1 − ρ ) e−τ d λ / µ I e−  B (λ ) = ρ + ρ (1 − 2 ρ ) e 

(12)

where the functions, P (λ , T ) , A(λ ) and B (λ ) are as,

(

)

The iterative solution for a minimizes f (a) based on the Levenberg-Marquardt method [45], −1

a k +1 = a k − ( J (a k )T J (a k ) + µ k I ) J (a k )T f (a k )

(13)

where the subscript k is the iteration number, J (a) is the Jacobian of f (a) denoted as J (a ) = f ' (a) ,

I

is the identity matrix.

µk

Levenberg-Marquardt method and is set as,

8

is the regularization parameter for the

min

µk =

The iterations for

a = ( a1 , a2 ,..., an )

( f (a )

max

are stopped when

k

(

2

, J (a k )T f (a k ) T

f (a k ) , J (a k ) f (a k )

a k +1 − a k ≤ ε

2

)

)

where ε

(14)

is the

convergence factor. Thus, this method combines multi-wavelength pyrometry for semi-transparent materials with the Levenberg-Marquardt algorithm. To verify the solution accuracy of multi-wavelength pyrometry with Levenberg-Marquardt algorithm for semi-transparent ZnS materials, the numerical analyses of inversion problem were carried out in the following. The measurement wavelengths of this pyrometry were assumed to be 11.0 µ m, 11.4 µ m, 11.7 µm, 12.0 µm, 12.4 µm, 12.7 µ m, 13.0 µ m, 13.4 µm, 13.7 µm and 14.0 µm in the optimized spectral region which were obtained by a FTIR spectrometer. The simulation conditions of ZnS plate were assumed to be the same as in section 2. The linear and Gaussian temperature distributions of 600 K ~700 K in the ZnS plate were respectively considered which are reasonably approximate for one-dimension ZnS plate in experiments of thermal shock heating from one side, and natural cooling of isothermal sample, etc..

The simulated effective spectral-directional radiation intensities at various wavelengths and temperatures given in section 2 were used as the “exact measured” intensities, I λexact , eff , in this inverse problem analysis. Noise was then added to the “exact measured” values to simulate actual measured values, I λcal, eff , I λcal,eff = I λexact , eff + σζ

(15)

where σ is the standard deviation of the simulated measurement intensities and ζ is a random number with a normal distribution. The four noise assumptions used for I λcal, eff were 0% (no noise), 1%, 5% and 10%. For the case with the linear temperature distribution of 600 K ~700 K in the ZnS plate, the inversion results for the relative errors (∆a1 / a1 , ∆a2 / a2 ) of the parameters (a1 , a2 ) in the linear function are shown in Fig. 7 for various noise levels (A, no noise; B, 1%; C, 5%; D, 10%). The relative errors, (∆a1 / a1 , ∆a2 / a2 ) , increase with increasing noise. For measurement noise level of 0% and 1%, the inversion solution accuracy of (a1 , a2 ) is very high with a maximum relative error of 1.39 %. When the measurement noise of I λcal, eff changes from 1% to 10%, ∆a1 / a1 slightly increases from 0.2% to 1.66%. However, ∆a2 / a2 greatly increases from 0.2% to 14. 66% at the same noise range. It shows that the parameter a2 is easily affected by the measurement noise so that the solution accuracy of the parameter a1 is more higher than the accuracy of a2 . The distribution of relative sensitivities of I λ ,eff to the parameters (a1 , a2 ) , ∂I λ , eff / ∂a1 ⋅ a1 / I λ , eff and ∂I λ , eff / ∂a2 ⋅ a2 / I λ , eff , are shown in Fig. 8. The relative sensitivity ∂I λ , eff / ∂a1 ⋅ a1 / I λ , eff with the range of (1.41, 18.51) is obviously larger than the sensitivity 9

∂I λ , eff / ∂a2 ⋅ a2 / I λ , eff with the range of (0.04, 0.15) which also verifies the characteristics of the

solution accuracy of a1 and a2 . The temperature distribution along the plate thickness can be calculated using the inversion solutions of a1 and a2 . The relative temperature errors between the calculated temperature and the exact temperature along the plate thickness are shown in Fig. 9 for various noise levels (no noise; 1%; 5%; 10%). The inversion temperature solution accuracy is good with a maximum temperature relative error of 1.5 % inside the plate when the noise is 5.0%. When the measurement noise increases to 10%, the temperature relative error increases to the range of (1.7%, 3.5%) along the plate thickness for the 600 K~700 K temperature range. Since the measured spectral intensities at more wavelengths are used in the nonlinear inversion problem for two unknown parameters, accurate results can be obtained for the linear temperature distribution even with large measurement noise.

Fig. 7 Inversion results for the relative errors (∆a1 / a1 , ∆a2 / a2 ) of the function parameters (a1 , a2 ) in the linear temperature distribution of 600 K ~700 K for various noise levels (A, no noise; B, 1%; C, 5%; D, 10%)

Fig. 8 Relative sensitivities, ∂I λ , eff / ∂a1 ⋅ a1 / I λ , eff and ∂I λ , eff / ∂a2 ⋅ a2 / I λ , eff , in the linear temperature distribution of 600 K ~700 K

10

Fig. 9 Inversion temperature solutions in the linear temperature distribution of 600 K ~700 K for various noise levels (no noise; 1%; 5%; 10%) For the case with the Gaussian temperature distribution of 600 K ~700 K in the ZnS plate, the inversion results for the relative errors (∆a1 / a1 , ∆a2 / a2 , ∆a3 / a3 ) of the parameters (a1 , a2 , a3 ) in the Gaussian function are shown in Fig. 10 for various noise levels (A, no noise; B, 1%; C, 5%; D, 10%). Without noise, the relative errors of inversion solutions (a1 , a2 , a3 ) are small enough to be ignored but the relative errors increase with increasing noise. When the measurement noise of I λcal, eff changes from 1% to 10%, ∆a1 / a1 slightly increases from 0.23% to 1.66%. However,

∆a2 / a2 and ∆a3 / a3 significantly increase to the ranges of (26.55%, 136.61%) and (34.46%,

178.40%) at the same noise range. The distribution of relative sensitivities of I λ ,eff to the parameters (a1 , a2 , a3 ) , ∂I λ , eff / ∂ai ⋅ ai / I λ , eff ( i = 1, 2, 3 ), are shown in Fig. 11. The relative sensitivity ∂I λ , eff / ∂a1 ⋅ a1 / I λ , eff

is in the range of (12.14, 16.71) which is obviously larger than the

sensitivities, ∂I λ , eff / ∂a2 ⋅ a2 / I λ , eff and ∂I λ , eff / ∂a3 ⋅ a3 / I λ , eff . Therefore, the results show that the measurement

noise will greatly exacerbate the inversion solution accuracy of the

parameters (a2 , a3 ) . The relative temperature errors along the plate thickness can be calculated using the inversion solutions of (a1 , a2 , a3 ) for various noise levels (1%; 5%; 10%) shown in Fig. 12. For the 1% noise level, the relative temperature error is in the range of (0.03%, 6.18) along the plate thickness and doesn’t monotonically change with the thickness. There are three peaks of the relative error distribution which are respectively 1.5%, 1.7% and 6.2% at x=0 cm, 0.46 cm and 1.2 cm. The relative temperature error distribution for the 5% noise level is similar to the 1% noise with a maximum relative temperature error of 14.0% at x=1.2 cm. For x is less than 0.96 cm, the temperature errors are less than 5% which is the same noise level. When the noise level increases to 10%, the temperature errors greatly increase to 10.6% and 32.0% at the two interfaces, and only have a minimum temperature error of 2.2% at x=0.4 cm.

11

Fig. 10 Inversion results for the relative errors ∆ai / ai of the function parameters (a1 , a2 , a3 ) in the Gaussian temperature distribution of 600 K ~700 K for various noise levels (A, no noise; B, 1%; C, 5%; D, 10%)

Fig. 11 Relative sensitivities, ∂I λ , eff / ∂ai ⋅ ai / I λ , eff , in the Gaussian temperature distribution of 600 K ~700 K

Fig. 12 Inversion temperature solutions in the Gaussian temperature distribution of 600 K ~700 K for various noise levels (1%; 5%; 10%)

12

The results for both the linear temperature and Gaussian temperature distributions show that the inversion results without noise are quite accurate. In applications, the Gaussian function is sometimes more appropriate for describing actual temperature distributions in semi-transparent plate, but the temperature errors with the noise for the Gaussian temperature distribution are much larger than for the linear temperature distribution. The reason is mainly because there are more unknown solving parameters in the Gaussian function than in the linear function. More parameters then reduce the inversion accuracy especially for non-linear inversion problems. However, for noise levels less than 5.0% in the above simulation cases of semi-transparent ZnS plate, the inversion temperature results based on the Levenberg-Marquardt method are satisfactory for both the linear and Gaussian temperature distribution measurements.

4.

Conclusions

A radiation temperature measurement method for one-dimensional semi-transparent ZnS was investigated in this study. The radiative transfer model was given for a one-dimensional semi-transparent absorbing-emitting plate with ideal optical smooth interfaces and neglecting scattering. The effective spectral-directional radiation intensity and the proportion of emitted radiation at different temperatures (400 K-800 K) and wavelengths (8.0~14.0 µ m) were numerically analyzed for linear and Gaussian temperature distributions in the one-dimensional plate. Multi-wavelength pyrometry was used with the Levenberg-Marquardt method to resolve the temperature distribution along the radiative transfer direction based on the line-of-sight spectral radiation intensities at multiple optimized spectra wavelengths (11~14 µ m) for a semi-transparent ZnS plate. The simulated effective spectral-directional radiation intensities were looked upon as the “exact measured” intensities. For noise levels less than 5.0%, the inversion temperature results using the Levenberg-Marquardt method are acceptable for measurements with the linear or Gaussian temperature distributions. The temperature errors for the Gaussian temperature distributions are much larger than for the linear temperature distributions for the same noise levels because there are more unknown parameters in the Gaussian function.

Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 51176100), the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (No. 51321002), the National Basic Research Program of China (No. 2011CB706900), the Program for New Century Excellent Talents in University (NCET-13-0315) and the Beijing Higher Education Young Elite Teacher Project (YETP0091). We thank Prof. D.M. Christopher for editing the English.

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Highlights Temperature measurement method of semi-transparent infrared material is developed Multi-wavelength pyrometry is used to determine interior temperature distribution Apparent spectral radiation properties of investigated ZnS material are analyzed Spectrum optimization of optical pyrometry for ZnS material are investigated Inversion temperature accuracy using optical pyrometry is good for applications

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