## Multiple Color Video Pyrometry

### Introduction

The Multiple Color Video Pyrometer (MCVP) was developed to measure the electrode temperatures of the Lithium Lorentz Force Accelerator (LiLFA). Knowledge of the electrode temperature can be used to help determine the efficiency of the anode and cathode, the lifetime of the thruster, and how the arc attaches to the cathode. The two primary reasons for developing th MCVP were to validate the theory of arc attachment within the cathode and to determine the evaporation rate (thus the lifetime) of the cathode.

Measuring the temperature of the electrodes by direct-contact is an extremely difficult task for many reasons:

1. Temperatures exceed that of most direct-contact diagnostics (2500 K).
2. Electromagnetic noise generated by the thruster arc interferes with measurement signals.
3. A detailed temperature profile requires many probes to be placed in the harsh environment.
4. The probe must be electrically insulated from the electrodes and the plasma.

Pyrometry is immune to the problems that plague direct-contact measurement because the thermal radiation from the hot surface is measured at a safe distance from the thruster. The MCVP consists of two components, the Thermal Imager on the right developed by Optical Insights, LLC and the video camera, a Sony XCD-SX900 digital camera.

All objects above zero Kelvin radiate energy in the visible and infrared portions of the electromagnetic spectrum. If the surface is a perfect emitter of radiation (which is rarely the case) the radiation intensity as a function of wavelength and temperature abides by Planck’s Law, $u\left(\lambda,T\right) = \epsilon_\lambda\frac{C_1}{\lambda^5} \left(\frac{1}{\exp\left( \frac{C_2}{T\lambda}\right)-1}\right)$ where $$u$$ is the intensity, $$\epsilon_\lambda$$ is the emissivity as a function of wavelength, $$C_1 = 1.191 \times 10^{16}$$ W nm4/cm2 Sr and $$C_2 = 1.4384\times 10^7$$ nm K are the first and second radiation constants, $$T$$ is the absolute temperature in Kelvin, and $$lambda$$ is the wavelength of the radiation in nm. The intensity as a function of wavelength is plotted in Fig. 1 for three temperatures to illustrate Planck’s Law, with an emissivity of unity.

The MCVP uses a CCD that is sensitive to visible radiation, thus the intensity in the visible spectrum is expanded in the Fig 2. There is a large difference between the intensity radiated within the visible wavelengths as the temperature changes.

### Limitations of Single Color Pyrometry

The principle of single color pyrometry is to determine temperature by measuring the radiation emitted from a surface at one wavelength. The pyrometer is calibrated to convert the measured intensity to a temperature by sighting an obect of known temperature and emisssivity. Fig. 3 shows the relation between intensity at 650 nanometers and temperature with an emissivity of 1. In this case the temperature can be determined very accurately.

What happens when the emissivity of the surface varies between 0.1 and 1? Fig. 4 shows that the relation between measured intensity and temperature is much different. For a given intensity (4 x 10 -3) the temperature is 2200 K for a surface with an emissivity of 1 while a surface with an emissivity 0.1 has a temperature of 2800 K. The surface emissivity must be well known in order to get accurate temperature predictions.

### Why Multiple Colors?

Measuring the intensity at more than one wavelength allows the determination of both temperature and emissivity. If the emissivity is not known or varies along a surface, the temperature cannot be determined accurately via a single intensity measurement. Many factors make it difficult to know the value of the emissivity:

• oxidation
• temperature
• surface roughness
• angle that the surface is viewed from
• coating by contaminants

If the amount that these factors disturb the emissivity value cannot be quantified, it is necessary to determine both the emissivity and temperature. To illustrate how multiple color pyrometry determines the temperature and emissivity, the measurement of thermal radiation emitted by a tungsten surface will be demostrated with the help of Fig. 5 The real emissivity of tungsten reduces the intensity from that of a black body (the black line) to that of the tungsten surface (red line). Four measurements of the intensity (indicated by the vertical colored lines in Fig. 6) supply the data required to determine the temperature and emissivity of the surface. Figure 5 - Difference between the intensity of a tungsten surface and a black body. Figure 6 - Example wavelengths and intensities of measurements of a four-color pyrometer.

The temperature and emissivity are determined by fitting the intensity data to models of intensity and emissivity. The two intensity models commonly used are Planck’s law and and Wien’s approximation.

For most metallic surfaces radiating in the visible and infrared spectrums the emissivity or the natural logarithm can be modelled as linear function of wavelength. The assumption of linear emissivity is supported by published data on metallic surfaces, see Fig. 7.

### Accuracy of Multiple Color Pyrometry

The accuracy of the temperature predicted by any pyrometric method depends on the quality of the calibration and the inherent noise of the detector in the pyrometer (called the errors due to noise and calibration here). Least-squares multiple color pyrometric methods must also be concerned with the uncertainty that results from the fitting of the intensity data to the radiation/emissivity model, which can reduce the accuracy below that of single color pyrometry.
We have accomplished a study of how the accuracy of previously proposed least-squares multiple color pyrometric methods depend on the the errors due to noise and calibration and the the number of colors of the pyrometer. The previously proposed methods are:

• Linear 2-term temperature fitting method
• Linear 2-term emissivity fitting method
• Linear 3-term fitting method
• Nonlinear fitting method

We used a Monte-Carlo simulation to determine the range of temperatures and uncertainties to be expected from the LSMCP methods when the intensity measurements are subject to random noise. The deviation in the simulated intensity from the nominal value was limited to the magnitude of the errors associated with noise and calibration; chosen to be 1%, 3% or 5%. (The standard deviation was 60% of the error.) We determined that the predicted temperatures and uncertainties of the Monte-Carlo simulation change by less than 2% when more than 1000 cases are included, thus we used 1000 cases throughout the analysis.

A Monte-Carlo analysis was conducted for each combination of 4, 6, 8, and 10 color pyrometers and for noise and calibration errors of 1%, 3%, and 5%. The intensity of the radiation from a 2700 K surface was calculated using Planck’s law with an artifcial emissivity model. The values of the coefficients were chosen to be similar to those determined experimentally for many metallic surfaces to ensure that the results of the simulations are applicable to the temperature measurement of metal surfaces.

The dependence of the uncertainty of the predicted temperature on the number of colors of the pyrometer is presented for each fitting method in Figs. 8, 9, and 10; where the plots show the uncertainty resulting from errors associated with noise and calibration of 1%, 3%, and 5%, respectively. For pyrometers of greater than four colors and errors of 1% or 3%, both the linear three-term and nonlinear methods have similar uncertainties in the predicted temperature. 5% errors associated with noise and calibration cause the nonlinear fitting method to predict large uncertainties in temperature, shown in Fig. 10. The linear two-term fitting methods yield smaller uncertainties than that of the linear three-term method in all cases, with the two-term temperature fitting method being the smallest. With the exception of the four-color pyrometer, a general comparison of the uncertainty in the predicted temperature of the LSMCP methods and that of single-color pyrometry can be made. Fig. 8 shows that a 1% error associated with noise and calibration generally results in an uncertainty in predicted temperature of less than 2%, while a single-color pyrometer has approximately a 5% uncertainty for all magnitudes of error. The uncertainty in temperature determined by the LSMCP methods when there is a 3% and 5% error (Figs. 9 and 10) is similar to or larger than that of single-color pyrometry. In order to circumvent the difficulties of lowering the noise of the detector and optics below 3% we developed a method of increasing the accuracy through complementary measurements. Figure 8 - Uncertainty of LSMCP with 1% error associated with noise and calibration. Figure 9 - Uncertainty of LSMCP with 3% error associated with noise and calibration. Figure 10 - Uncertainty of LSMCP with 5% error associated with noise and calibration.

We also determined that the 2-term methods often under-estimated the uncertainty of the predicted temperature. For this reason, we reccomend that researchers use the 3-term linear method or the nonlinear method.

### Complementary Measurements

The uncertainties in the predicted temperature found for the linear three-term and nonlinear fitting methods are larger than those of single-color pyrometry when the errors associated with noise and calibration are greater than 1%, making single-color pyrometry more accurate in many cases. A new method for LSMCP, designated complementary measurements, was developed to reduce the uncertainty below that of single-color pyrometry. We define complementary measurements as additional intensity data measured at each wavelength of the pyrometer. For a pyrometer with a fixed number of colors this implies using adjacent pixels on a CCD array or including measurements taken at successive times.

A Monte-Carlo analysis, similar to that used in section IV, of the linear three-term and nonlinear fitting routines determined the predicted temperature and its uncertainty when complementary measurements are included. The linear two-term fitting methods underestimate the uncertainty of the predicted temperature in the majority of the cases and thus are ignored. The analysis included one to nine complementary measurements, 1%, 3%, and 5% error associated with noise and uncertainty, and 4, 6, 8, and 10-color pyrometers. Figure 11 - Uncertainties associted with the linear 3-term fitting method for 4,6,8, and 10 color pyrometers.

It can be seen in Figs. 11 and 12 that the uncertainty in the predicted temperature is always less than the 5% associated with single-color pyrometry when the number of complementary measurements is greater than three or four. For example, a four-color pyrometer with a 3% error requires two complementary measurements with the nonlinear fitting method and one complementary measurement with the linear fit. The figures also show that the uncertainty in the predicted temperature asymptotes to the same value for each fitting method and error. This yields the minimum uncertainty attainable by that method given an error associated with noise and calibration. Figure 12 - Uncertainties associated with the nonlinear fitting method for 4,6,8, and 10 color pyrometers.

### Results

The Li-LFA was recently fired here at EPPDyL and preliminary temperature data were recorded. The start-up of the Li-LFA was captured by the system at 7.5 frames per second. The intensities measured at the four wavelengths was fit to the gray body formula using a non-linear least squares fit to produce temperature data. No detailed analysis has been done to estimate the error of the temperatures. The resulting movie of the cathode tip temperature is only 8 seconds long because the mirror that was being used to visualize the thruster became coated with lithium which reduced the intensity of the image and could not be calibrated for. A picture of the multi-channel hollow cathode is shown on the left that corresponds with the temperature profile in the movie on the right.

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