A Measurement Technique for Infrared Emissivity of

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Typical microwave absorbing materials consist of a dielectric epoxy material impregnated with a lossy material such as iron or carbon. We study a novel ...
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REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < emissivity measurements. In Section III we discuss the experimental setup. Section IV presents the results and provides some brief discussion of the findings. Finally, Section V draws conclusions from the study. II. THEORETICAL APPROACH Emissivity describes the radiant energy emitted from a body, compared to an ideal blackbody at the same temperature. Emissivity can be wavelength dependent, called spectral emissivity, or can be taken as an average over a spectral response across a specific bandwidth. We are concerned with the whole thermal infrared spectrum and use a sensor that has sensitivity across this band from 7.5 µm to 13 µm. The spectrally averaged emissivity can be expressed as [17] 𝜀=

𝜆 1 𝜆 ∫𝜆 2 𝑠(𝜆)𝐿𝐵𝐵 (𝜆,𝑇)𝑑𝜆 1

∫𝜆 2 𝑠(𝜆)𝐿𝑒𝑚𝑖𝑡 (𝜆,𝑇)𝑑𝜆

=

𝑅𝑒𝑚𝑖𝑡 (𝑇) 𝑅𝐵𝐵 (𝑇)

,

𝑅𝑚𝑒𝑎𝑠 (𝑇) = 𝑅𝑒𝑚𝑖𝑡 (𝑇) + 𝑅𝑟𝑒𝑓𝑙 ,

𝑛−1

𝑢𝜀,𝑇𝑦𝑝𝑒 𝐴 = √

(2)

where 𝑅𝑚𝑒𝑎𝑠 is the radiance of a true object measured by an infrared detector, and 𝑅𝑟𝑒𝑓𝑙 is the contribution of reflections to the measured radiance. We assume that 𝑅𝑟𝑒𝑓𝑙 is independent of the sample temperature, a constant dependent only on the background environment and a constant detector temperature. By solving (2) for 𝑅𝑒𝑚𝑖𝑡 , substituting into (1), and rearranging, we arrive at the expression 𝑅𝑚𝑒𝑎𝑠 (𝑇) = 𝑅𝐵𝐵 (𝑇 ′ )𝜀 + 𝑅𝑟𝑒𝑓𝑙 ,

interpolation along a fourth-order polynomial fit. This form is chosen because the relationship is expected to approximately follow the Stefan-Boltzmann law. After this offset correction, we have a set of 10 pairs of 𝑅𝑚𝑒𝑎𝑠 and 𝑅𝐵𝐵 at the same temperatures to be fit to (3). At extreme temperatures, the infrared emissivity may no longer be independent of temperature and (3) would no longer be valid. Traditional least squares linear fitting only considers possible errors in one dependent variable. In our case, both parameters 𝑅𝑚𝑒𝑎𝑠 and 𝑅𝐵𝐵 are measured values with associated measurement errors. We use the Deming method [18], a special case of the total least squares method, for the linear fit of the two uncertain parameters. The standard error of the slope from this linear fitting gives the type A uncertainty 𝑢𝜀,𝑇𝑦𝑝𝑒 𝐴 also known as the statistical or random error. We use the jackknife method [18] to estimate the type A uncertainty as,

(1)

where 𝜀 is the spectrally-averaged emissivity, 𝑠(𝜆) is the spectral response function of the detector, 𝐿𝑒𝑚𝑖𝑡 is the spectral exitance of the true surface, 𝐿𝐵𝐵 is the spectral radiance of a blackbody as described by Planck’s law of radiation, 𝜆1 and 𝜆2 are wavelength endpoints of detector sensitivity, 𝑅𝑒𝑚𝑖𝑡 is the spectrally-averaged emitted radiance from the surface, and 𝑅𝐵𝐵 is the spectrally-averaged blackbody radiance. The term 𝑅𝑒𝑚𝑖𝑡 is typically not directly measurable because in real-world situations there is also a radiance contribution from reflected radiation from the background and from the detector itself. The spectrally-averaged radiance measured by an infrared detector can be expressed as

(3)

which is a linear combination, in slope-intercept form, of measurable parameters and the desired quantity, emissivity. By measuring both the radiance of an infrared blackbody and the radiance of the sample of interest at multiple temperatures, we can perform a linear fit to obtain the desired emissivity and, incidentally, also quantify the reflected radiance. The 𝑇 ′ in (3) refers to the fact that the equilibrium temperatures for the blackbody and sample are slightly different for the same water circulator set-temperatures due to differences in heat transfer coefficients. To correct for this offset, the blackbody radiance is interpolated to the sample temperatures by use of

2

𝑛

∑𝑛𝑖=1(𝜀̃𝑖 − 𝜀̃)̅ 2 ,

(4)

where 𝑛 is the number of observations (𝑛=10 in our case), 𝜀̃𝑖 refers to the Deming fit emissivity calculated by omitting the 𝑖th observation, and 𝜀̃ ̅ refers the mean of the 𝑛 calculated 𝜀̃𝑖 values. The type B, or systematic uncertainty 𝑢𝜀,𝑇𝑦𝑝𝑒 𝐵 is derived from the manufacturer specification of the infrared sensor and the calibration accuracy of the radiance measurements. A differential uncertainty analysis [19] results in 𝑢𝜀,𝑇𝑦𝑝𝑒 𝐵 = √(

𝜕𝜀 𝜕𝑅𝑚𝑒𝑎𝑠

2

𝑢𝑅𝑚𝑒𝑎𝑠 ) + (

𝜕𝜀 𝜕𝑅𝐵𝐵

2

𝑢𝑅𝐵𝐵 ) ,

(5)

where 𝑢𝑅𝑚𝑒𝑎𝑠 is the radiance measurement uncertainty in the commercial infrared camera, 𝑢𝑅𝐵𝐵 is the combined blackbody radiance measurement uncertainty, considering contributions from the non-ideality of the IR blackbody, and from the temperature interpolation error and is given by: 𝑢𝑅𝐵𝐵 = √𝑢𝑅𝑚𝑒𝑎𝑠 2 + [(1 − 𝜀𝐵𝐵 ) ∗ 𝑅𝐵𝐵 ]2 + 𝑀𝑆𝐸𝑖𝑛𝑡 , (6) where 𝜀𝐵𝐵 refers to the emissivity of the blackbody and 𝑀𝑆𝐸𝑖𝑛𝑡 is the mean-squared-error of the blackbody radiances relative to the fourth order polynomial fit of sample temperatures versus measured radiances. The second term under the radical in (6) results from the non-ideal infrared emissivity of the blackbody. As discussed in the following section, 𝜀𝐵𝐵 is taken here as 0.995, the ‘worst-case’ value. Because of the small relative contribution of the non-ideal emissivity in our case we take this to be a Gaussian distributed error. Use of a lower emissivity blackbody, or higher uncertainty of this emissivity, could result in a negative bias of the resulting sample emissivity. The partial derivatives in (5) are calculated by solving (3) for emissivity and differentiating. The partial derivatives, or sensitivity terms, depend on the measured radiance values. We use the lowest temperature measurement which results in the highest sensitivity and thus the worst-case uncertainty value.

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < The total uncertainty in the derived emissivity is calculated from the root squared sum of type A and type B uncertainties: 𝑢𝜀 = √𝑢𝜀,𝑇𝑦𝑝𝑒 𝐴 2 + 𝑢𝜀,𝑇𝑦𝑝𝑒 𝐵 2 ,

3

measurements is also identical to that of the sample measurements. The experimental setup with the camera looking into the conical blackbody is shown in Figure 3.

(7)

where 𝑢𝜀 is the overall uncertainty in the emissivity from this method. The following section discusses how the radiance quantities are experimentally measured. III. EXPERIMENTAL SETUP A set of epoxy-based material samples of varying iron carbonyl powder concentrations by volume were cast into waveguide shims for characterization of their microwave material properties, as discussed in [15]. The same samples are used in this study, in which we investigate the thermal infrared emissivity of the materials. The samples are 10.7 mm by 4.3 mm and about 2.5 mm thick. The samples are fastened to a temperature controlled brass plate. The temperature of the plate is controlled with a water circulator of nominal temperature stability ±0.01 °C. A calibrated platinum resistance thermometer (PRT) is attached to the back-side of the plate with thermal paste and copper tape to monitor the physical temperature of the sample. We use a commercial thermal infrared camera to view the samples with a close-up lens. The lens has a field of view of 6 mm by 8 mm with a resolution of 25 µm per pixel at a focal distance of 2 cm. Figure 1 shows the camera viewing the sample attached to the brass plate. The set-temperature of the water circulator is varied to 10 temperatures ranging between 19 °C and 45 °C, while the temperature of the sample is monitored and the radiance is measured. These temperatures were chosen to provide a range of radiances while maintaining negligible temperature gradients between the PRT measured temperature and the radiating surface of the sample. We captured 10 calibrated images of the sample at each set temperature and analyze a small square of 50 by 50 pixels near the center of the sample. The mean radiance over this subset of the image is taken and also averaged over the 10 images. Figure 2 shows an example of the image taken of the sample for the unloaded epoxy at 45 °C. A conical water-bath infrared blackbody known as CASOTS [20] was measured to obtain the 𝑅𝐵𝐵 radiance values. The CASOTS blackbody has been shown to have comparable performance to a National Institute of Standards and Technology (NIST) IR blackbody with IR emissivity of no less than 0.995 [20]. The same PRT sensor was attached to the blackbody cavity surface in the same way it was attached to the backside of the temperature controlled brass sample holder. The temperature of the blackbody was controlled with the same refrigerating-heating water circulator used for the samples and is set to the same 10 temperatures. As stated previously, the same circulator set-temperatures result in slightly different equilibrium temperatures of the samples and blackbody due to their different emissivities and convective heat transfer coefficients. The averaging technique for the blackbody

Water line for temperature control PRT sensor lead

IR camera lens Material sample

Fig. 1. Photograph of experimental setup for sample measurements. W/m2/sr

Fig. 2. Example of measured sample infrared radiance image. The shown image is of the pure epoxy sample at a set temperature of 45 °C. The red box encloses the pixels used in the averaging.

CASOTS blackbody

Infrared camera

Fig. 3. Experimental setup for blackbody measurement. The infrared camera is shown looking into the CASOTS infrared blackbody.

> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT)
REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) < and thermal IR performance are of great importance. The results presented here will allow precise modeling, design, and simulation of present and future calibration sources, leading to improved brightness temperature accuracy and precision. Other industries may also benefit from the results presented here including biomedical, defense, and consumer electronics. REFERENCES [1]

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