## Abstract

Temperature measurements are ubiquitous in combustion systems. However, the accuracy of surface temperature measurements of critical components operating in a harsh combustion gases environment is greatly affected by reﬂection and combustion gas radiation. In this paper, an analytical two-color pyrometry model was used to quantitatively analyze the temperature errors caused by the combination of reflection and H_{2}O-CO_{2}-CO-N_{2} mixture radiation. As the results indicate, the most signiﬁcant contributors to the measurement errors are found to be the error arising from H_{2}O-CO_{2}-CO-N_{2} mixture absorption and emission for two-color pyrometer operating at long wavebands. The errors due to reflection predominate over the measurement errors measured at short wavebands. In a combustor where reflected radiation from high-temperature surrounding and hot/cool combustion gas is present, two-color pyrometry is practically inoperative as a consequence of its unacceptably large measurement error and high measurement sensitivity. When the intervening gas is isothermal and the optical distance from surface to detector is considered optically thin, the temperature error has linear growth with both the H_{2}O-CO_{2}-CO-N_{2} mixture concentration and viewing path length increasing. This linear change provides us a method of linear extrapolation to eliminate the effect of uncertain gaseous absorption and emission. The results of this work can be used as a theoretical support for the design and application of a two-color pyrometer in a gas-fired furnace.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Accurate surface temperature measurement in industrial processes have profound scientiﬁc importance and practical value from the viewpoints of efficient energy management, low manufacturing costs, high productivity and high product quality [1,2]. According to Kerr and Ivey that precise knowledge of the surface temperatures of turbine blades is quite vital as far as their overheating protection is concerned and is required for efﬁciency improvements, significant fuel consumption reductions, and financial savings [3]. The principal problem in the application of radiation pyrometry in combustion chamber is the large error caused by reflected ambient radiation and absorption and emission of radiation by interfering combustion gas.

Two-color pyrometry is an advantageous method of measuring surface temperature in industrial processes where unwanted radiation attenuation caused by intervening participating media presents [4,5]. In this method, the ratio of the radiant flux received from two separate spectral bands is used to develop a signal proportional to surface temperature. As long as the factors affecting the thermal radiation received affect both wavebands equally, the temperature inferred by this method remains unaffected [6]. Hence, two-color pyrometry has widely been used for the measurement of the temperature and concentration of soot or gas present in combustion flames [7,8]. Practical surface measurements by two-color pyrometry, however, are still prone to a number of uncertainties associated with environmental effects such as reflected ambient radiation and absorption and emission of radiation by interfering hot gases. Moreover, the environmental effect during practical measurement process is always different from that of calibration process. Therefore, it is quite important to understand these effects to achieve reliable and accurate temperature measurements.

The search for methods of modeling and eliminating the influences of emissivity and reflected radiation from extraneous sources has long challenged the field of radiation temperature measurement [9–12]. Saunders [13] developed a reflection model from which the errors and uncertainties for two-color pyrometers were analyzed and a mean of correcting for reflection error was presented. Araújo et al. [14] used Monte Carlo technique to estimate the reflection uncertainties for two-color pyrometry. Daniel et al. [15] proposed a reflection model to correct the reﬂection errors in surface temperature measurement of gas turbine blade using two-color pyrometry, taking the effects of target surface emissivity, view factor and ambient temperature into account. Nonetheless, a major problem with these aforementioned models is that they did not simultaneously consider the effect of intervening gas when dealing with reflection problems. Since pyrometer views the target through a column of gas, reflected radiation must transmit over a certain distance through the intervening gas before received by pyrometer. Any interaction between the intervening gas and the reflected radiation will change the intensity of the initial reﬂected radiation. Thus, a comprehensive analysis simultaneously taking both reflected ambient radiation and intervening gas radiation into account is required to precisely predict reflection error in two-color pyrometry.

Few researchers have attempted to compensate for the temperature error in two-color pyrometry due to absorption and emission of radiation by the furnace atmosphere or flue gas, in contrast with considerable investigations focusing on reflection problems. A possible reason is that most commercially radiation thermometers are designed to avoid significantly absorbing bands of flue gas. Even so, there is usually some residual overlap between the spectral responsivity of the thermometer and the absorption lines of the gas species [16]. Measurements of tube surface temperature in a large reformer furnace with a commonly used radiation thermometer operating on 1.0 µm which corresponds to the atmospheric window wavebands, undertaken by Saunders [16], indicated that the errors due to gaseous absorption and emission approach 6 °C per meter. In such situation, a more sophisticated understanding the mechanism of the measurement error caused by combustion gases is essential to the proper selection and application of radiometric temperature measurement techniques. Recently, Huang et al. [4,17] put forward a novel wavelength selection method for two-color pyrometry working in the presence of participating medium and investigated the effects of high-temperature H_{2}O-CO_{2}-N_{2} mixture radiation on two-color pyrometry. Despite this work is signiﬁcant achievement, similarly, the shortfall is the fact that it solely focused on the inﬂuence of gas without simultaneously considering reflected ambient radiation effect.

Some researches in recent years have began to focus on the individual effects of reflection and combustion gaseous absorption and emission on single-color pyrometry [18,19]. However, few investigations have reported on the coupled effects of reflection and combustion gases on surface temperature measurement using two-color pyrometry. Motivated by these investigations, the purpose of this study is to quantify the coupled effect of reflected ambient radiation and gas absorption and emission on two-color pyrometry using an analytical two-color pyrometry model. The accuracy of this model was further verified by comparing the calculated heat flux distribution with the results of other numerical solutions using two non-isothermal and inhomogeneous combustion scenarios. Subsequently, the model was utilized to characterize the measurement error caused by the coupled influence of reflected ambient radiation and combustion gaseous absorption and emission. Finally, uncertainty and sensitivity analysis of the calculated temperature error to variations in the model’s input parameters were investigated.

## 2. Theoretical background

According to Saunders [13] that in many cases that the ambient temperature is not uniform, it is possible to characterise the background by a single effective temperature. Following Saunders [13] the planar geometry applied to the proposed temperature measurement system is composed of two boundless, parallel and diffusely emitting and reﬂecting boundaries which encompass an emitting and absorbing intervening gas layer, as depicted in Fig. 1. One boundary stands for the surroundings of practical measurement system, *S _{2}*, the other boundary represents target surface

*S*. Because the effect of emissivity on surface temperature measurement is not the main point of this study, the emissivity of target surface is assumed to be a constant less than 1.0.

_{1}Suppose the two-color pyrometer is placed on surroundings plane *S _{2.}* The radiant flux sensed by the pyrometer includes components of emitted flux from

*S*(

_{1}*E*shown in Fig. 1) and flux originating from

_{emitted}*S*reflected by

_{2}*S*(

_{1}*E*shown in Fig. 1), both influenced by intervening combustion gas. Obviously, the value of the radiant flux is related to target surface temperature

_{reflected}*T*, target surface emissivity

_{1}*ɛ*, ambient temperature

_{1}*T*, surroundings surface emissivity

_{2}*ɛ*, gas layer thickness L, gas temperature

_{2}*T*and concentration

_{g}*X*, which can be determined by solving the governing equations of radiative transfer in intervening combustion gas and its boundary conditions. Further details regarding the solution could be found in our previous studies [20,21].

_{g}Gas radiation properties are strong spectral dependency, which makes the spectral integration of transport equation computationally expensive even in one-dimension. The exact solution can only be given by line-by-line calculation, which is very complex and time consuming. As the practical aim of this paper is to measure the target surface temperature using two-color radiation thermometer which use a narrow spectral pass-band, a statistical narrow-band (SNB) model as described following is favored as a feasible alternative to line-by-line approach in these presented spectral radiative transfer calculations.

For an isothermal and homogeneous column of path-length *l* containing an individual absorbing gas at a mole fraction *X* and total pressure *p*, the SNB model with an exponential-tailed-inverse line-strength distribution provides the narrow band averaged transmissivity for Lorentz line profiles given as:

*x*and

_{i}*x*):

_{i+1}*i*stands for the spatial discretization, ${\bar{\tau }_{\Delta \lambda }}({l_m})$ represents narrow-band averaged transmissivity, and the local mean path-length

*l*for the planar slab is taken to be 1.8(

_{m}*x*-

_{i+1}*x*). Since ${\bar{\tau }_{\Delta \lambda }}({l_m})$ and

_{i}*l*are local properties of a combustion gas layer, this formulation in general yields spatially varying narrowband average absorption coefficient. It is expected that the grey-band approximation along with this local absorption coefficient is able to capture characteristics of non-grey radiation encountered in non-isothermal and inhomogeneous media since the local conditions are incorporated into this formulation.

_{m}After the detected radiant flux in waveband Δλ was obtained, for two-color pyrometry, target surface temperature can be derived by using bisection method to solve the following equation if the slope of instrumental emissivity set to 1.0:

*h*=6.626×10

^{−34}J•s is known as Plank’s constant,

*c*=2.998×10

_{0}^{8}m/s is known as speed of light in vacuum,

*k*=1.3806×10

^{−23}J/K is known as Boltzmann’s constant,

*n*is refractive index.

## 3. Model validation

The main radiation from sufficient burning products of most natural gas and petroleum fuels consists of the vibration–rotation infrared spectrum of carbon dioxide, carbon monoxide and water vapor. Hence, a H_{2}O-CO_{2}-CO-N_{2} mixture was considered in the present study. In order to verify the accuracies of the detected heat flux by the two-color pyrometer in the proposed temperature measurement system, numerical calculations were performed using two complicated combustion scenarios which have been previously chosen as test cases by Liu et al. [22], Chu et al. [23], Howell et al. [24], Zhou et al. [25], Bordbar et al. [26] and G. Krishnamoorthy [27] using different gas radiative property models and molecular spectroscopic databases. The first scenario considered is a typical configuration of temperature and CO_{2} and H_{2}O concentration distributions encountered in a counter-flow methane-air diffusion flame. Scenario 2 is representative of an oxy- ﬁred diffusion ﬂame with ﬂue gas recirculation and consists of the same proﬁles in Scenario 1 with the spatial distribution of CO_{2} mole fractions increased by 0.6 relative to that of the air ﬂame. The distribution of temperature and radiating gas concentrations is shown in Fig. 2, which was given in detail by Liu et al. [22] and Chu et al. [23]. In the two scenarios, the radiation calculations were performed in a one-dimensional parallel-plates enclosure containing a H_{2}O-CO_{2}-N_{2} mixture at 1 atm total pressure and the two plates surfaces are assumed to be black. The separation distance is 0.5 m and two bounding plates are at 300 K. The abscissa x of Fig. 2 represents the distance from one of the plates. x = 0 and x = 0.5 correspond to the locations of one plate and the other plate, respectively. The spectral range usd to calculate radiative heat flux in the present model is 50-11250 cm^{-1}. These combustion scenarios provide testing platform to assess the effects of non-isothermal and inhomogeneous combustion gas distributions on the radiative heat flux and was chosen here so that the results obtained from this work can be compared to other published results.

Figures 3 and 4 compare the net radiative heat flux distributions predicted by present model and other gas radiative property models using different spectroscopic databases for the air combustion scenario and oxy-fuel combustion scenario. Among of those distributions, the results of LBL calculation based on HITEMP 2010, indicated by red lines in Figs. 3 and 4, are used as the benchmark solution to evaluate the present model. As shown in Figs. 3 and 4, the proposed model, which is easier to formulate and therefore much quicker computationally, predicts accurate radiative heat flux distributions for the oxy-fuel combustion scenario compared to the air combustion scenario. For the air combustion scenario, the present model is in very close/excellent agreement with benchmark solutions in the middle of the enclosure and slightly over-predicts radiative heat flux by about 10% near the walls. The slight discrepancies are attributed to different angular discretization. Kim et al. [28] pointed out that the small differences would disappear, if the absorption coefficient shown in Eq. (2) is based on the actual lengths for each angle through the medium instead of on the mean beam length. The Comparison of results of the present model with benchmark solutions shows that the proposed model can predict fairly accurate radiative heat flux for one-dimensional non-isothermal and inhomogeneous media.

## 4. Results and discussion

#### 4.1 Temperature errors due to reflection and combustion gases radiation

In order to figure out the temperature errors caused by reflection and combustion gases radiation, six measurement cases, depending on the relative target temperature, ambient temperature and gas temperature, were applied in this study. The two operating wavebands for two-color pyrometry model in this section are chosen as a two adjacent narrow-bands with a bandwidth of 25 cm^{-1}. *ΔT* signifies the discrepancy between the calculated temperature using two-color pyrometry model and the assumed real target temperature.

Figures 5–9 indicate the *ΔT* for two-color pyrometer working on the wavenumbers in the spectrum of 50-11250 cm^{-1} with different ambient temperature and gas temperature. The part input parameters used to carry out these calculations are given in Table 1. The blue scatter points in Figs. 5–9 represent the total temperature errors caused by the combined influence of intervening gas H_{2}O-CO_{2}-CO-N_{2} and reflection, while the black one shows the temperature errors due to individual reflection with X_{H2O}=X_{CO2}=X_{CO}=X_{N2}=0, and the red one stands for the temperature errors produced by gases mixture alone with *T _{2}*=

*T*. The values of

_{1}*ΔT*above 1000 K over all studied wavebands are not covered in Figs. 5–9.

### 4.1.1 Case 1 (*T*_{g} <T_{1})

_{g}<T

_{1}

The ΔT produced by low-temperature intervening gas and high-temperature environment (*T _{g}*=400 K,

*T*=1600 K) and by low-temperature intervening gas and environment (

_{2}*T*=800 K,

_{2}*T*=800 K) are shown in Figs. 5 and 6, respectively. Different from single-color thermometry [21], the values of

_{g}*ΔT*for two-color pyrometer caused by intervening gas may larger or smaller than zero, depending on the operating wavelengths of two-color pyrometer. Roughly speaking, as Figs. 5 and 6 indicate, for two-color pyrometer operating at short wavebands, the temperature error caused by reflection predominates over the total errors, while the temperature error due to the gaseous absorption and emission takes an overwhelming component over the total error measured at long wavebands. The similar phenomenon occurs in single-color pyrometer [21]. The results suggest that a short-waveband thermometer operator should pay more attention on the effect of reflection compared to that of gas mixture, particularly when ambient temperature is high.

In a furnace where reflected radiation from high-temperature surrounding and low-temperature gas emission are always present, a two-color pyrometer is virtually useless because of the unacceptably large errors in the temperature measurement as shown in Fig. 5(a). As indicated in Fig. 6(a), the calculated temperature error due to reflection of low-temperature surrounding is insignificant, relative to that caused by low-temperature gaseous effects. As a result, the total error measured at a low ambient temperature and a low combustion gas temperature is dominated by the contribution due to the gas mixture absorption and emission effect.

Low-temperature gas absorption result in the underestimate of the target radiance, the opposite situation appears with the reflection of high-temperature environment, because it can cause an increase in the apparent target radiance. Hence, it is quite natural to imagine that the temperature error due to hot gas mixture can be properly compensated by adopting a low-temperature surrounding. Note that this method is suitable for single-color thermometry, but it is not necessarily suitable for two-color thermometry. Because two-color thermometers infer temperature from the ratio of the measured radiance in two different wavebands, the reduction or addition in signal of each waveband do not necessarily cause the increase or decrease in this ratio.

### 4.1.2 Case 2 (*T*_{g} >T_{1})

_{g}>T

_{1}

The ΔT caused by high-temperature intervening gas and environment (*T _{g}*=1600 K,

*T*=1600 K) and by high-temperature intervening gas and low-temperature environment (

_{2}*T*=1600 K,

_{g}*T*=800 K) are depicted in Figs. 7 and 8, respectively. The emission from gas will significantly increase not only the emission of the target but also the reflection from surrounding, resulting in an immense error in the radiometric temperature measurement. For measurement systems consisted of high-temperature combustion gas and high-temperature surrounding, large uncertainties error of two-color pyrometer making them unsuitable for engineering applications.

_{2}When the high-temperature gas emission occurs in combination with the low-temperature surrounding, the total error is dominated by the contribution due to the gas mixture absorption and emission. Therefore, like the aforementioned treatment on the total error measured at a low ambient temperature and a low combustion gas temperature, the measured error in this case also can be minimized by employing atmospheric window wavebands with the condition that the high-temperature H_{2}O-CO_{2}-CO-N_{2} mixture present in low concentration.

### 4.1.3 Case 3 (*T*_{g} =T_{1})

_{g}=T

_{1}

Figure 9 shows the ΔT as a function of operating wavenumber with *T _{1}*=1200 K,

*T*=1200 K and ambient temperature

_{g}*T*set as several values around target temperature. As the results indicate, the gaseous absorption and emission effects still greatly affects two-color radiation thermometer readings especially for long wavebands, even though

_{2}*T*=

_{g}*T*. This is because the reflected ambient radiation absorbed by intervening gas is not equal to the gaseous emission terms when ambient temperature

_{1}*T*is not equal to gas temperature. Accordingly, not accounting for gaseous absorption and emission effects will lead to the misjudgement of reflection errors.

_{2}#### 4.2 Measurement uncertainty analysis

From a theoretical standpoint, the present model can be directly applied to correct for the coupled effects of reflection and combustion gas absorption and emission. From a practical perspective, however, the input parameters of the model, including ambient temperature *T _{2}*, target surface emissivity

*ɛ*, intervening gas concentration

_{1}*X*and temperature

_{g}*T*, are often nearly impossible to measure accurately. Analyzing and summarizing the influence of measurement uncertainty of these parameters on the accuracy of radiometric temperature measurement is conducive to the recognition, understanding, and avoidance of measuring temperature errors.

_{g}### 4.2.1 Ambient temperature *T*_{2}

_{2}

The ΔT for two-color thermometry at various ambient temperatures *T _{2}* were calculated with

*T*=1200 K,

_{1}*X*=0.2,

_{H2O}*X*=0.2,

_{CO2}*X*=0.2,

_{CO}*X*=0.4,

_{N2}*L*=100 cm,

*ɛ*=0.85,

_{1}*ɛ*=0.9, p=1 atm. Herein, the operating wavebands for two-color pyrometry are chosen as 0.9 um, 1.0 um, 1.25 um, and 3.9 µm, because these bands are the atmospheric window wavebands and are often used to design commercially radiation thermometers for special applications like measuring surface temperature in combustion chamber. The wavenumber combination of the wavebands mentioned above is shown in Table 2. Given the gas temperature has a vital effect on the measurement errors depending on whether the gas temperature is higher or lower than the target temperature, four different gas temperature are considered as shown in Fig. 10.

_{2}The results presented in Fig. 10 indicate that when ambient temperature is less than target temperature, the effect of ambient temperature on measuring temperature error is negligible except for the 3.9 um wavebands with the interfering gas at a high temperature. When ambient temperature is large than target temperature, the error increases nearly exponentially with ambient temperature. The rate of the increase is related with the operating waveband of two-color pyrometer. The shorter the waveband, the faster the rate.

### 4.2.2 Target emissivity *ɛ*_{1}

_{1}

Figure 11 represents the ΔT as a function of operating wavenumber with *T _{1}*=1200 K,

*X*=0.2,

_{H2O}*X*=0.2,

_{CO2}*X*=0.2,

_{CO}*X*=0.4,

_{N2}*L*=100 cm,

*ɛ*

_{2}_{ }= 0.9, p=1 atm and the target surface emissivity at various values. Herein, four different combinations of ambient temperature and gas temperature are considered as shown in Fig. 11(a), 11(b), 11(c), and 11(d). It is noted that the measuring temperature error decreases with the target emissivity approaching 1.0. When the measurement system is at a low ambient temperature, the longer the waveband, the faster the rate of the increase of ΔT. On the contrary, for the high ambient temperature, the shorter the waveband, the faster the rate, which means the measuring temperature error is more sensitive to the change in target emissivity under this circumstance. As a result of the effect of high-temperature combustion gas, the measured error is a value not equal to zero when the target emissivity is equal to 1.0.

For the measurement system composed of high-temperature interfering gas and low-temperature surrounding, the measured error due to low target emissivity is much greater than the measured error of other measurement systems. Therefore, we should pay special attention to the measurement accuracy of target emissivity in this measurement system. When the ambient temperature is larger than target temperature, the shorter the operating waveband, the larger the measured error. In contrast, the longer the operating waveband, the larger the measured error for low ambient temperature.

### 4.2.3 Gas temperature *T*_{g}

_{g}

Figure 12 depicts the effect of gas temperature *T _{g}* on the ΔT for two-color thermometry. Herein, four different ambient temperatures are considered, the measurement uncertainty caused by the change in gas temperature less than 1200 K is insignificant except for 0.9 um wavebands with the surrounding at a high temperature. For gas temperature larger than target temperature, the value of ΔT increases nearly exponentially with gas temperature. The ΔT has the slowest growth rate at 1.0 µm waveband, followed by1.25 µm and 0.9 µm waveband, and the growth rate of ΔT at 3.9 µm band is the fastest. Consequently, two-color pyrometer working on 3.9 µm waveband will be more susceptible to the measurement uncertainty of hot gas temperature.

### 4.2.4 Gas concentration *Xg* and measuring path length *L*

Figure 13 and Fig. 14 indicate the effect of H_{2}O-CO_{2}-CO-N_{2} concentrations characterized by *X _{H2O}*:

*X*:

_{CO2}*X*=1:1:1 and

_{CO}*X*=1-(

_{N2}*X*+

_{H2O}*X*+

_{CO2}*X*) and measuring path lengths on the

_{CO}*ΔT*for two-color pyrometer with

*T*=1200 K,

_{1}*ɛ*

_{1}_{ }= 0.85,

*ɛ*

_{2}_{ }= 0.9, p=1 atm and four different combinations of ambient temperature and gas temperature, respectively. As shown in Fig. 13 and Fig. 14, the ΔT exhibit a linear relationship with both of gas concentration

*Xg*and measuring path length

*L*increasing. More important is that this linear relationship suggests a simple way to eliminate the effect of uncertain gaseous absorption and emission from the temperature measurement by two-color radiation thermometry. The effect of gaseous absorption and emission is eliminated as measuring path length or gas concentration approach zero. Thus a linear extrapolation of ΔT vs. L or X to L = 0 or X=0 yields a real target temperature immune from gaseous absorption and emission effect. Further detailed investigations on the linear extrapolation method will be summarized in our next study.

For the temperature measurement system composed of low temperature gas, viewing path length and gas mixture concentration have a little effect on the measured temperature error. When the interfering gas is at a high temperature, the ΔT increase as the viewing path length or gas mixture concentration increases except for the error at the 3.9 um wavebands with the surroundings at a high temperature. Similarly, the ΔT has the slowest growth rate at 1.0 µm waveband, which means this waveband is more suitable for radiation temperature measurement in long measuring distance and high gas concentration environment.

#### 4.3 Measurement sensitivity analysis

According to the measurement uncertainty analysis mentioned above, we could conclude that ambient temperature *T _{2}*, target emissivity

*ɛ*, gas temperature

_{1}*T*and concentration

_{g}*X*, and measuring path length L are important parameters that affect the output of the proposed model. To quantitatively understand how the impact factors affect the behavior of the detection system, a sensitivity analysis method was employed herein. Detailed description of the analysis is given in Ref. [17].

_{g}The sensitivity coefficient of each parameter, which defined as the change in the quantity of the output of the proposed model divided by the change in the quantity of input parameter while keeping other parameters set at a given central value, is displayed in Table 3. As the Table shows, comparing with the temperature of low-temperature gas and surrounding, gas concentration and measuring path length, the temperature of hot gas and hot surrounding, and target emissivity have a relatively significant impact on the measured temperature using two-color pyrometry. As a result, the accurate measurement of the temperature of hot gas and hot surrounding, and target emissivity is necessary for the good estimation of real target temperature. Instead, slightly larger measurement uncertainties on the temperature of low-temperature gas and surrounding, gas concentration and measuring path length are permitted, since they have relatively small impact on the estimation of real target temperature.

## 5. Conclusion

This paper performed a theoretical analysis which has as aim to quantify the error committed in the temperature measurement by technique two-color pyrometry. In particular, the temperature error was evaluated as a function of operating wavelength of two-color pyrometry, ambient temperature, target emissivity, H_{2}O-CO_{2}-CO-N_{2} mixture temperature and concentration, and measuring path length. The target surface and background are considered to be a gray body with constant emissivity. Sensitivity analysis of the derived temperature to variations in the model’s input parameters were investigated. The main predictions for the coupling influences of intervening gas mixture and reflected ambient radiation on two-color pyrometry reveal the following.

- (1) For the temperature measurement system consisted of high-temperature surrounding and high-temperature or low-temperature combustion gas, the unacceptably large measurement errors and high measurement sensitivity render two-color pyrometry virtually useless.
- (2) Generally speaking, for two-color ratio thermometer operating at short wavebands, the total errors due to the coupling influences are dominated by the error arising from reflection. The errors due to the gas mixture absorption and emission predominate over the total error measured at long wavebands.
- (3) When the intervening gas is isothermal and the optical distance from the surface to the detector is considered optically thin, the calculated temperature error for two-color pyrometer having a spectral responsivity near 3.9, 1.25, 1.0 and 0.9 µm is the linear growth in magnitude with both gas mixture concentration and viewing path length increasing. This linear change provides us a method of linear extrapolation to eliminate the effect of uncertain gaseous absorption and emission.
- (4) Either in terms of measurement error or measurement sensitivity, the effect of reflection from low-temperature surrounding is insignificant. The absorption and emission from low-temperature combustion gas has a fairly important effect on the temperature measurement for two-color radiation thermometry operating at long wavebands.

## Funding

National Natural Science Foundation of China (51775112, 52005103, 71801046); Guangdong Provincial Applied Science and Technology Research and Development Program (2019B1515120095); Intelligent Manufacturing PHM Innovation Team Program (2018KCXTD029, TDYB2019010); Chongqing Science and Technology Commission (2019jcyj-zdxmX0013).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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