Emissivity range constraints algorithm for multi-wavelength pyrometer (MWP)

Jian Xing; R.S. Rana; Weihong Gu

doi:10.1364/OE.24.019185

1. Introduction

Radiation thermometry has been widely used during recent years since they provide an important tool for characterizing a large number of high temperature and ultra-high temperature situations [1–5]. The measurement range is from middle-low temperature zone (above 700K) to ultra high temperature zone (above 3000K). Radiation thermometry techniques commonly include the lightness temperature method, color temperature method, total radiation temperature method and multi-wavelength pyrometer (MWP) measurement method [6–10]. Among these techniques, the multi-wavelength radiation temperature measurement technique is probably the most attractive one because it can obtain the true temperature and spectral emissivity simultaneously by an appropriate inversion calculation of spectral intensity data from multi-wavelength channels [11–15]. With the urgent demand for high temperature testing, the multi-wavelength radiation temperature measurement technique has grown with rapid developments and broad potential applications.

The spectral emissivity of tested high temperature target, which remained yet unknown, is a huge obstacle for true temperature inversion by multi-wavelength pyrometer (MWP). Previously studies mostly focused on how to assume an appropriate relationship between emissivity and wavelength using Least-Squares (LS) [16–18]. This often leads to a large error if the assumption does not match the practice emissivity distribution. An algorithm giving a possible solution to overcome it is proposed in our paper published in [19]. It is based on the continuous temperature data processing algorithm of MWP with no effect from emissivity. Though its advantages have been already demonstrated, the limitations of a continuous temperature data processing method can be highlighted as followed:

(i) As for the estimated value of the initial temperature, as long as the difference is within ± 200K between the initial temperature and the real temperature, the calculated results of the true temperature and spectral emissivity are in good agreement with the real true temperature and spectral emissivity of the measured target.
(ii) As for the search range of the spectral emissivity, the narrower the search range of the spectral emissivity, the more precise are the calculated results of the true temperature and spectral emissivity.
(iii)The processing method of multi-wavelength pyrometer data for continuous temperature measurements presented here cannot be applied to in-line data processing.

So an optimized data processing method based on continuous output signal of MWP is studied to improve the retrieval efficiency to meet in-line needs. Simulation of six materials with typical spectral emissivity distribution and solid rocket plume temperature experiment are used to verify the reliability of proposed method.

2. Theory of emissivity range constraints

If a multi-wavelength pyrometer has n spectral channels, according to Planck Law, the output signal V_i of each spectral channel i can be expressed as:

V_{i} = A_{λ i} ε (λ_{i}, T) C_{1} λ_{i}^{- 5} 1 / [\exp (C_{2} / λ_{i} T) - 1], (i = 1, 2, ..., n)

where A_λi is instrument characteristics of pyrometer; ε(λ_i,T) is spectral emissivity at temperature T; C₁ is first radiation constant, its value is 3.74 × 10⁻¹² W/cm²; C₂ is second radiation constant, its value is 1.44 cm·K.

Generally, Wien approximation is used as follows:

V_{i} = A_{λ i} ε (λ_{i}, T) C_{1} λ_{i}^{- 5} \exp (- C_{2} / λ_{i} T), (i = 1, 2, ..., n)

When the pyrometer is calibrated at a blackbody reference temperature T´, the output signal V_i′ at reference temperature T´ can be expressed as:

{V^{'}}_{i} = A_{λ i} C_{1} λ_{i}^{- 5} \exp (- C_{2} / λ_{i} T^{'}), (i = 1, 2, ..., n)

The ratio of Eqs. (2) and (3) is as follows:

V_{i} / {V^{'}}_{i} = ε (λ_{i}, T) \exp (- C_{2} / λ_{i} T) \exp (C_{2} / λ_{i} {T^{'}}_{i}), i = (1, 2, ..., n)

From Eq. (4), we can see N equations that include N + 1 unknowns, that is, unknown ε (λ_i,T) from N channels and true temperature T are present. So we should search for a method to solve the underdetermined equations problem. A solid relationship of emissivity vs. wavelength is set in advance by previously studies. If the assumption is acceptable; otherwise, there is huge error. So with this background, continuous temperature method was proposed in [19], its core is the proposed assumption that the wavelength and temperature have approximately a similar linear trend for two consecutive time temperature points. The assumption may be expressed as:

ε_{i}^{(2)} = ε_{i}^{(1)} [1 + k (T_{i}^{(2)} - T_{i}^{(1)})]

where ε_i⁽¹⁾ is the spectral emissivity at wavelength λ_i and true temperature T₁, ε_i⁽²⁾ is the spectral emissivity at wavelength λ_i and true temperature T₂; T₁ and T₂ represent two consecutive time temperature points; and k is the proportionality coefficient of the linear trend. In practice, emissivity change with temperature occurs objectively for consecutive time temperature points.

T_i⁽¹⁾ can be calculated from Eq. (4) as followed:

T_{i}^{(1)} = \frac{1}{\frac{1}{T^{'}} + \frac{λ_{i}}{C_{2}} \ln \frac{ε_{i}^{(1)} V_{i}^{'}}{V_{i}^{(1)}}}

T_i⁽²⁾ is derived from Eq. (5) and Eq. (6):

T_{i}^{(2)} = \frac{1}{\frac{1}{T^{'}} + \frac{λ_{i}}{C_{2}} \ln \frac{ε_{i}^{(1)} [1 + k (T_{i}^{(2)} - T_{i}^{(1)})] V_{i}^{'}}{V_{i}^{(2)}}}

Equation (7) can be calculated by an iterative method: T_i⁽¹⁾ is the input iteration initial value, then ε_i⁽¹⁾ can be obtained from Eq. (6), so the T_i⁽²⁾ can be calculated from Eq. (7). The iterative cut-off condition is true temperature of every channel is determined when spectral emissivity of every channel was real value, that is,

F_{\min} = \sum_{l = 1}^{2} \sum_{i = 1}^{n} {[T_{i}^{(l)} - E (T_{i}^{(l)})]}^{2} \to 0

where

E (T_{i}^{(l)}) = \frac{1}{n} \sum_{i = 1}^{n} T_{i}^{(l)} l = 1, 2

The final purpose of continuous temperature method is searching a group of appropriate spectral emissivity to meet the requirement of Eq. (8), but iteration efficiency is very low because of no limit condition. So the main object of the improved algorithm is exploring a high efficient researching spectral emissivity method to meet requirement of Eq. (8) and obtain true temperature.

According to Eq. (8), true temperature difference F will tend to zero if the emissivity of every channel is correct. So from another angle review, the F value must be changed with the difference of spectral emissivity △ε, and the F value must be increased when △ε gets bigger. Therefore once the key function relation between F and △ε is confirmed, the emissivity searching range will be reduced quickly under the constraint of key function.

Table 1 lists the simulation data employed. Six kinds of spectral emissivity with different fluctuation vs. wavelength at 1800K true temperature point are used. When true temperature is 2000K, ε₂₀₀₀ = ε₁₈₀₀[1 + k (2000-1800)]. Blackbody reference temperature is 1600K. Effective wavelength of 8 channels is 0.4μm, 0.5μm, 0.6μm, 0.7μm, 0.8μm, 0.9μm, 1.0μm, 1.1μm.

Table 1. Spectral Emissivity Model of Various Targets

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Firstly, according to Table 1, we can get V_i/V_i′ at 1800K from Eq. (4); then calculating the true T of every channel when emissivity range is varied from 0 to 1 with 0.01 intervals according to Eq. (4). The inversion temperature according to different emissivity is shown in Fig. 1.

Fig. 1 True temperature inversion value of every wavelength channel when true temperature is 1800K and spectral emissivity range is from 0 to 1, the interval of emissivity is 0.01

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Figure 1 depicts temperature inversion value of 8 spectral channels when spectral emissivity range is varied from 0 to 1 at true temperature 1800K. All calculated temperature variation trends decline with an increase of emissivity although spectral emissivity distribution shape of every target is different. When emissivity is less than the real value, the inversion temperature is higher than real temperature; when emissivity is more than real value, the inversion temperature is lower than real temperature. So we should get the relationship between temperature difference and emissivity difference, where temperature difference is inversion temperature minus 1800K and emissivity difference is the corresponding true emissivity minus the emissivity value at 1800K in Table 1. Retrieved temperatures of every channel are close to the temperature when the emissivity value approaches the true value.

Secondly, we need to determine the function between F and △ε. This was accomplished in a series of steps as follows:

(1) Calculate V_i/V_i′ from Eq. (4) based on model values in Table 1;
(2) Let the initial value of ε be 0.5 and let△ε be varied from 0.01 to 0.4 so that ε∈(0.1,0.9). If the △ε range is wider, then the emissivity will be greater than 1, but this is not reasonable since by definition the emissivity is in the range of 0 and 1.
(3) Calculate F from Eqs. (6)-(8) at different △ε.
(4) Plot the figure of F vs.△ε.
(5) Fit the function of F vs.△ε.

The fitting method is based on the curve shape of F vs. △ε, the shape is similar to the exponential function (Fig. 2), and we choose a function with unknown coefficients as follows:

Fig. 2 F vs. △ε curve and fitting curve

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F = a * \exp (b * Δ ε + c) + d

Using the ‘fit’ function of Matlab, the unknown coefficients are obtained as follows

F = 1.318 * \exp (18 * Δ ε + 3.732) - 4.241

The important aim of an improved algorithm is enhancing the efficiency of continuous temperature data processing method, and changing the limit of start values of temperature and spectral emissivity. Therefore we propose the main idea is confirming △ε vs. F relationship of Eq. (10) of every iteration calculation of Eq. (8), rapidly shrinking the range of emissivity until it meets the cut-off condition. The flow chart is as follows in Fig. 3:

Fig. 3 Flow chart for emissivity range constraints algorithm

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3. Simulation results

We used Matlab R2011a on a HP EliteDesk 800G1 with Inter(R) Core(TM) i5-4590 CPU 3.30GHz.The spectral emissivity initial value is set to 0.5 for every channel. The initial value of k is 0.0005, $k \in (- η, η)$ , $η = 0.001$ . The initial temperature may be set to any positive value, even zero. The iteration cut-off condition F is 30 to meet temperature retrieval error of less than 50K. The initial F value is set to 100000 to make the iteration start and the iteration continues until F is less than 30. Simulation results of 6 kinds of targets are shown in Fig. 4:

Fig. 4 Two graphs for every target, left is true temperature retrieval result from 1500K to 2000K of 6 kinds of target materials from Table 1 with an interval of 50K; right is the retrieval emissivity of targets at 1800K.

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Table 2 compares the calculation times for six kinds of measured objects. The calculation times in the second column were obtained using the inversion algorithm in [19] and the calculation times in the third column were obtained using the improved method presented above. The running time of the fitting program was reduced by 91.4% ~97.5%. Thus the calculation efficiency is significantly improved.

Table 2. Comparison of Run Times with Six Emissivity Targets

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4. Experiment

We used the experimental data from [20], which was performed on a solid propellant rocket engine in order to verify the improved algorithm introduced in this work and compare it with a previous algorithm from [19]. The effective wavelengths and voltages outputs of every channel at the reference temperature T´ (T´ = 2252 K) are shown in Table 3. The 11groups output signal data starting from 6.5s with 5ms intervals is shown in Table 4.

Table 3. Effective Wavelengths of the Pyrometer and Outputs at the Reference Temperature (T´ = 2252 K)

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Table 4. Practical Data Performed on a Solid Propellant Rocket Plume

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Calculations were conducted to retrieve the true temperature and spectral emissivity. The results are presented in Table 5.

Table 5. Experimental Results for True Temperature (K) and Spectral Emissivity Data from [20] Using the Improved Algorithm

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The theoretical true temperature of the engine flame near the nozzle exit indicated by the rocket engine designer is 2490.0K. The retrieval result error using the improved algorithm is within ± 31K, but computing time is only 0.134s. The former algorithm is within ± 20K but its computing time is 5.623s, so the former algorithm cannot follow the rocket plume high-speed temperature change in near-real time. Because rocket plume temperature change is severe and rapid, and so the efficiency is preferred under the premise of satisfying certain precision

5. Conclusions

In the true temperature measurement by MWP, an efficient data processing algorithm based on emissivity range constraints is investigated to meet in-line measurement. The relationship between true temperature deviation and emissivity deviation is derived using six targets with different emissivity distribution models. It can, therefore, quickly shrink emissivity range according to the introduced relationship. It is experimentally demonstrated that the proposed methodology has the advantage of offering high computing efficiency. It is noteworthy that it requires no prior assumptions of the initial spectral emissivity range and initial temperature value as well. Simulations and experiments carried out using proposed algorithm discussed above indicate that fairly reasonable results in different spectral emissivity distribution are obtained even in the case of deviations in the spectral emissivity model. The computation time is substantially reduced in comparison with results obtained using conventional methods. It is suitable particularly in-line true temperature measurement by multi-wavelength pyrometer.

Funding

National Natural Science Foundation of China (NSFC) (61405045, 31470714); Science and Technology Innovation Talents Funding of Harbin (No.RC2014QN009026); Talent Introduction Scientific Research Start Funding of NEFU.

References and links

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Target	k	Spectral emissivity at 1800K true temperature
Target	k	0.4μm	0.5μm	0.6μm	0.7μm	0.8μm	0.9μm	1.0μm	1.1μm
A	0.0003	0.85	0.75	0.67	0.60	0.55	0.50	0.48	0.45
B	0.00022	0.45	0.48	0.50	0.55	0.60	0.67	0.75	0.85
C	0.00022	0.45	0.55	0.65	0.75	0.74	0.65	0.55	0.45
D	−0.00022	0.85	0.75	0.65	0.55	0.54	0.65	0.75	0.80
E	0.00022	0.85	0.65	0.55	0.65	0.84	0.65	0.55	0.50
F	−0.00031	0.50	0.55	0.65	0.84	0.65	0.55	0.65	0.85

Object	Former algorithm t₁(s)	Improved algorithm t₂(s)	(t₁-t₂)/t₁ × 100%
A	2.510	0.1200	95.2
B	3.620	0.1320	91.4
C	2.430	0.0970	96.0
D	3.610	0.1910	94.7
E	3.590	0.1640	95.4
F	3.880	0.0980	97.5

Channel i	1	2	3	4	5	6	7	8
λ_i(μm)	0.574	0.592	0.623	0.654	0.698	0.748	0.826	0.914
V_i´(mV)	39.4	139.7	117.5	363.7	345.0	493.9	320.7	406.7

Measurement time(s)	V_i (mV) of 8 channels wavelength (μm)
	1	2	3	4	5	6	7	8
	0.574	0.592	0.623	0.654	0.698	0.748	0.826	0.914
1	46.3	254.1	165.3	481.5	367.8	495.0	273.7	323.5
2	46.3	254.1	170.2	476.6	372.7	500.0	278.6	328.4
3	46.3	244.2	165.3	471.6	362.8	495.0	268.7	323.5
4	46.3	244.2	170.2	481.5	372.7	509.9	283.6	333.4
5	46.3	249.1	160.3	471.6	362.8	490.1	268.7	318.5
6	46.3	244.2	160.3	461.7	352.9	480.2	253.9	303.7
7	41.3	234.3	155.4	456.8	343.0	465.3	248.9	293.8
8	46.3	244.2	160.3	461.7	352.9	475.2	253.9	298.7
9	41.3	239.2	155.4	461.7	343.0	470.3	253.9	303.7
10	41.3	239.2	155.4	456.8	343.0	470.3	248.9	298.7
11	41.3	234.3	155.4	451.8	333.1	460.4	239.0	293.8

Time	Former	Improved	ε ₁	ε ₂	ε ₃	ε ₄	ε ₅	ε ₆	ε ₇	ε ₈
1	2504.1	2483.1	0.420	0.671	0.549	0.535	0.460	0.511	0.407	0.410
2	2506.8	2485.8	0.420	0.671	0.552	0.535	0.457	0.512	0.408	0.411
3	2498.4	2480.2	0.420	0.668	0.550	0.540	0.455	0.511	0.406	0.410
4	2505.0	2487.0	0.420	0.668	0.552	0.535	0.457	0.513	0.410	0.412
5	2499.9	2478.9	0.420	0.669	0.548	0.534	0.455	0.510	0.406	0.409
6	2493.3	2472.0	0.420	0.668	0.548	0.533	0.454	0.509	0.403	0.406
7	2482.3	2462.4	0.416	0.665	0.546	0.532	0.452	0.507	0.402	0.404
8	2490.1	2471.0	0.420	0.668	0.548	0.533	0.454	0.508	0.403	0.405
9	2485.3	2465.8	0.415	0.666	0.546	0.533	0.452	0.507	0.403	0.406
10	2483.8	2464.0	0.415	0.666	0.546	0.532	0.452	0.508	0.402	0.405
11	2481.3	2459.3	0.415	0.665	0.546	0.531	0.451	0.506	0.399	0.404

Emissivity range constraints algorithm for multi-wavelength pyrometer (MWP)

Abstract

1. Introduction

2. Theory of emissivity range constraints

3. Simulation results

4. Experiment

5. Conclusions

Funding

References and links

Cited By

Figures (4)

Tables (5)

Equations (11)

Optics Express