F20Thermal_Imaging

Thermal Imaging Method to

Find Heat Diffusivity

Tommy Kuan & Zhong Yi Jiang

University of Minnesota

School of Physics and Astronomy

Abstract

The purpose of this experiment is to measure the thermal diffusivity α, of brass, copper, aluminum, and unidirectional-woven carbon fiber using the thermal imaging method [1]. The experiment is done by putting the tip of a heated rod in contact with a thin metal sheet and the record the heat diffusion process with an infrared (IR) camera [2]. The recorded temperature data at different time steps were fitted to the heat equation in a Gaussian form and the σ² of each Gaussian fit was plotted against time to obtain a slope of 8 times α. The α for brass was measured as 34.77 ± 0.55 mm²/s, aluminum was 85.15 ± 1.71 mm²/s, and copper to be 117.35 ± 11.39 mm²/s. In the future, an IR camera with a higher frame rate should be used to increase accuracy for materials with higher α.

Introduction

Thermal diffusivity is a direct consequence of the second law of thermodynamics, where heat has to flow from a higher to a lower temperature. Many natural sciences and engineering disciplines rely heavily on accurate knowledge of a material's thermal diffusivity. This knowledge helps the person of interest learn how fast heat diffuses through the material of their interest. For example, a material of high thermal diffusivity, α, is preferred when it comes to keeping quantum computers cooled in their operational range. It is possible to find α of material if we know the precise density, thermal conductivity, and specific heat capacity of the material. In this experiment, we use the thermal imaging method that records an infrared video of heat diffusing, obtaining α from the output data. The IR camera uses an array of infrared detectors (microbolometers) that are sensitive to the infrared radiations to determine the temperature at each pixel of the data [3].

Theory

The heat equation describes the rate of heat diffusion, we modeled our experiment to a 2-dimensional problem:

Where U is temperature, t is time, and α is the thermal diffusivity.

If we applied Fourier’s theorem and convert the Cartesian to polar coordinates, where r is the radius from the heat source, and A is the amplitude of the Gaussian. The full derivation can be found in [4], which result is in a Gaussian form:

The width of the Gaussian depends on the thermal diffusivity and time. In the experiment, we fit our experimental results in the above equation and extract different widths of the Gaussian at different times. Then a linear fit was used to find the α from the slope of the fit.

Experimental Setup

The experiment consisted of a steel rod, torch, thin sheet of metal plates, IR camera, and tripod clamps. The IR camera laid flat on the tabletop and faced flat upward and toward the black-spraypainted side of the sample metal plate. The illustrated setup can be seen in the figure below.

Calibration video data were recorded prior to the recording of the heat diffusion. The calibration data was treated as an average background temperature data and was subtracted from the actual data later on. The steel rod was heated with a torch for about 25 seconds, and then had short contact with the center of the sample plate while the IR camera records. The output data from the IR camera is 60x80 cells of temperature data that is then analyzed in the sections below.

Data

The figure on the left of each material is the Gaussian fit at different times. One thing to notice is that the actual frame rate couldn't be directly controlled by the IR camera software. We had to manually calculate the frame rate of each recording.

Brass

Aluminum

Copper

Carbon Fiber

Results

Squared of widths of the Gaussian fits (b²) at different times were plotted against time and linearly fitted to obtain α for each material. We only managed to make use of the data from the first 4 time steps due to copper having a relatively high α compared to the other samples and the IR camera did not have a higher capturing speed to capture the entire heat diffusion process. The copper data was also especially noisy and contributed a larger error in the determination of α. The carbon fiber data was interesting to visualize and the linear fits showed that α in the y-axis is ≈ 3.5 times its α in the x-axis. This is because we used a unidirectional woven carbon fiber with the alignment in the figure below.

Unidirectionally-woven carbon fiber

Conclusion

We have established a method that uses the IR camera to determine the thermal diffusivity constant, α, of the sample plates with extraordinary accuracy. We found that α for brass to be 34.77 ± 0.55 mm²/s which differed from the accepted value of 33.67 mm²/s by 2σ, α for aluminum to be 85.15 ± 1.71 mm²/s which differed from the accepted value of 87.61 mm²/s by 1.44σ, and α for copper to be 117.35 ± 11.39 mm²/s which differed from the accepted value of 111.32 mm²/s by 0.53σ. There is a trend where higher values of α generate higher uncertainties. We suspect that the IR camera used was not fast enough at capturing enough frames per second. We suggest using an IR camera that can capture more frames per second for future experiments.

Acknowledgments

We want to thank Professor Elias Puchner for guiding us through this project, thank Kurt and Kevin for helping us with the difficulties we face in the lab.

References

[1] Tim Gfroerer, Ryan Phillips, Peter Rossi. Thermal diffusivity imaging. Am. J. Phys., Vol. 83, No. 11, November 2015.

[2] FLIR C2 Compact Thermal Camera Product Page, https://www.flir.com/products/c2/.

[3] Optotherm, Inc. Optotherm Thermal Imaging, Microbolometers. http://www.optotherm.com/microbolometers.htm.

[4] F. Cernuschi, A. Russo, L. Lorenzoni, A. Figari (2001). ”In-plane thermal diffusivity evaluation by infrared thermography”. Rev. Sci. Instrum. 72, 3988-3995.