S17_ThermalDiffusivity

Thermal Diffusion Imaging and Modeling of Heat Flux in Various Metals

Charlie Dang & Tia Troy

University of Minnesota

School of Physics and Astronomy

Minneapolis, MN 55455

Abstract

The tip of a cylindrical steel rod was heated and brought into contact at the center of a metal plate. A thermal camera was then used to record the thermal diffusion taking place inside the metal. A computer program called, FLIR TOOLS, was then used to translate the recording, frame by frame, into an Excel Spreadsheet where each cell represents the temperature of each pixel of each frame. These temperatures were then used in an equation called, Gaussian Parameter, that when plotted against time gave a slope proportional to 8 times the thermal diffusivity. The thermal diffusivities for three different test metals were 62.7±1.5 mm²/s for Aluminum Alloy, 34.7±0.7 mm²/s for Brass, and 108.2±0.3 mm²/s for Copper.

Introduction

The concept of thermal conductivity and thermal diffusivity are interrelated and are fundamental to thermal physics. Thermal conductivity is well known by many as the amount of heat a substance can transfer. While thermal diffusivity, the rate of heat being transferred in a substance, is not as popular. Our goal for this experiment were to reproduce the analytical approach of the “Thermal Diffusivity Imaging” [1] by three individuals, from the Department of Physics at Davidson College in North Carolina. Also, we would conclude our experiment with ideas for improvements in the future and any assumptions that were made which could affect the results.

Theory

The heat equation for sheet plates [2] after heating source breaks contact with thermal diffusivity = α, is

Using Fourier, Hankel and inverse Laplace integral [3], the heat equation becomes the Gaussian of the thermal radial profile

Solving the thermal radial profile and letting the Gaussian parameter = b², we get a Gaussian linear fit

Plotting b² vs. t give us the slope proportional to 8 times the thermal diffusivity, α.

Experimental Setup

Our experimental setup was depicted in figure 1, where the thermal camera capable of recording at 3 Hz and at 60x80 thermal pixels [4], steel rod and metal sheet are clamped to three ring stands for stability at a consistent height of 55.5cm from the flat table surface. The recorded side of each metal plate is painted with a thermal non-reflective coat to minimize any thermal reflections cast by the environment. The distance between the thermal camera and the plate was 15 cm after calibration, such that 8mm of the rod’s diameter took up about 8 thermal pixels, that is each thermal pixel ≈ 1mm. We chose the distance between the rod and the metal sheet to be 60 cm (about 2.5 feet) to give us work room when heating the rod and avoid bumping the apparatus.

Figure 1: (left) Top down view of the apparatus set up with the thermal camera connected to a computer via USB cable. The distance D from the camera to the plate is 15 cm and from the sheet to the heated tip, L = 60 cm. After the rod is heated, it was slid to the center of the reflective side of the plate where they made contact. (right) Side view, at an angle, of the real experimental apparatus showing concentric circular thermal diffusivity on the thermal camera as the heated rod touches the reflective side of the brass plate. Original figure.

Procedure

We used the propane hand torch to heat the tip of the steel rod for 25 seconds. This was the time it took to heat the rod from normal lab temperature (20°C) to 150°C. The upper limit of the thermal camera was 150°C, so if we overheat and rod didn’t cool enough when contact was made, we’d get bad data. If we didn’t heat enough and the rod cooled just above lab temperature before contact was made, we’d also get bad data because there wasn’t enough heat to diffuse. Next, we started recording and contacted the heated tip of the rod to the reflective side of the plate. After 3 seconds, we pulled the rod away and allowed the diffusion to continue for additional 15 seconds before ending the recording, as this is the duration for most of its heat to dissipate.

Results

We plotted using LSQ fit to determine the slope of the Gaussian linear fit proportional to 8 times the thermal diffusivity, α. For aluminum alloy, we found the slope to be 502.0±12.2 mm²/s. For brass, we found the slope to be 277.6±5.2 mm²/s For copper, we found the slope to be 865.5±2.3 mm²/s. Dividing each slope by 8 we found α for aluminum alloy was 62.7±1.5 mm²/s, α for brass was 34.7±0.7 mm²/s and α for copper was 108.2±0.3 mm²/s, depicted in figures 2.

Figure 2: (upper left) LSQ fit for 10 data points of aluminum alloy at 502.0±12.2 mm²/s, (upper right) LSQ fit for 8 data points of brass at 277.6±5.2 mm²/s, and (bottom) LSQ fit for 8 data points of copper at 865.5±2.3 mm²/s, with value of trendline slope and intercept displayed. Original figure.

Depicted in figure 3 was the plot of for aluminum alloy, brass and copper with error bars using Matlab. The trend of the data points for all three substances matched well with the expected slope of from literature [5]. Even though brass sat a distance from the line of its expected slope in figure 3, it's still parallel meaning the slope is the same and only the initial radius is affected.

Figure 3: Plot from Matlab for aluminum alloy, brass and copper with error bars for each data point. The straight lines represented textbook slope for their corresponding substance. The color coding of the line, data points and error bars were meant to make the plot easily read. Original figure.

Conclusion

We have found the thermal diffusivity, α, for aluminum alloy to be 62.7±1.5 mm²/s which differed from textbook value [5] of 64 mm²/s by 1.7σ, for α of brass was 34.7±0.7 mm²/s which differed from textbook value of 34.12 mm²/s by 0.2σ, and α for copper was 108.2±0.3 mm²/s which differed from textbook value of 112.34 mm²/s by 1.5σ. For future experiments on thermal diffusivity of this type, an improvement can be made if they can find a thermal camera with better thermal resolution than 60x80 pixels and a faster frame per second rate. During our experiment, we didn’t consider the heat loss to surrounding air by radiation while thermal diffusion was taking place which would’ve made b² larger thus increasing the slope of α slightly bringing us closer to that expected textbook values.

References

1. Tim Gfroerer, Ryan Phillips, and Peter Rossi, “Thermal Diffusivity Imaging,” Am. J. Phys. 83, 923 (2015); doi: 10.1119/1.4928277.

2. Mary L. Boas, Mathematical Methods in the Physical Sciences, 3^rd ed. (John Wiley and Sons, Hoboken, 2006), pp. 628, 525-527.

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

4. FLIR C2 Thermal Imaging System, <http://www.flir.com/instruments/content/?id1⁄466732>.

5. Heat And Mass Transfer Data Book. 1st ed. New Age Publishers. Web. 3 Mar. 2017.