Modeling Radial Thermal Diffusion Ryan Phillips, Tim Gfroerer, and Peter Rossi Physics Department, Davidson College, Davidson, NC Abstract Thermal conduction is governed by the thermal diffusivity constant of the conducting material. In our experiment, we use a thermal camera to gather temperature data from metal plates, which are heated at their center by a cylindrical rod. We fit our measured radial thermal profiles to Gaussians, and use the results to obtain the diffusivity via the analytical solution to the heat equation. To confirm the accuracy of our approach, we model the system with a radial finite difference simulation. Our results are further verified by comparison with thermal diffusivity values reported in the literature. [2] Results Conclusion The thermal camera offers 1mK resolution in about 10 5 pixels, allowing us to gather data with unprecedented precision. Moreover, the frame rate of 10Hz minimizes the loss of accuracy due to cooling – we collect all data within three seconds of rod contact. Our experimental design enables us to obtain diffusivity values for Aluminum and Copper alloys with less than 5% deviation from literature values. Using our experimental diffusivities, we simulate the radial temperature profiles with an efficient, one- dimensional finite difference method. We achieve a high degree of accuracy and precision in our analysis without any adjustable parameters. The heat flux across an interface depends on the temperature gradient and the thermal diffusivity α of the medium. If the heat flux into a volume element differs from the heat flux out, the temperature will change in accordance with Fick’s 2 nd law. Given the azimuthal symmetry of the experiment, the heat equation reduces to the cylindrical coordinate form shown above, assuming negligible dependence on the z-coordinate. Acknowledgments Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research. References [1]In-plane thermal diffusivity evaluation by infrared thermography, F. Cernuschi, A. Russo, L. Lorenzoni, and A. Figari, Rev. Sci. Instrum. 72, 3988 (2001) [2]Thermal Diffusivity Imaging, Tim Gfroerer, Ryan Phillips (Davidson '16), and Peter Rossi (Davidson '15), American Journal of Physics (in press). Theory Method of Finite Differences Next Time Step Present Time Step α = Measured Diffusivity Simulation Fick’s 1 st Law: Fick’s 2 nd Law: Obtaining Diffusivities Analytic Solution for Instantaneous Heating: Measured Diffusivity Values Al: Cu: Measured radial profiles in aluminum and copper are compared with simulated results (solid lines). The precision of the analysis is highlighted by the logarithmic scale. Using the diffusivity, the temperature distribution at a given time step can be expressed as a linear transformation of the temperature distribution in the previous time step. Literature Diffusivity Values Al: Cu: A heated cylindrical rod briefly touches the center of a metal sheet. On the other side of the sheet, a thermal camera monitors the temperature of the metal as heat diffuses. Experimental Design Thermal Camera Reflection Shield Metal Sheet Heating Rod Torch r Rod Axis Rod Motion This representative thermal image shows the temperature distribution shortly after contact of the rod. The color scale on the right identifies temperature in degrees Celsius. The analytic solution to the heat equation contains a Gaussian term, which describes the shape of the time-dependent radial profile. [1] We obtain Gaussian fits and equate their parameters to the exponential term in the analytic solution. The slope of the result vs. time can be used to identify experimental diffusivity values.