S16_ThermalDiffusionImaging

Thermal Diffusivity Imaging and Modeling of Heat Flux in Various Metals

Jacob Hergenrader & Joseph Ponzillo

Introduction

How heat diffuses in materials is of great importance in many industries. In classical thermodynamics heat flow, including diffusion is modeled by the heat equation. In the heat equation there is a constant that heavily governs how the heat diffuses based on the material, called the thermal diffusivity, of the material. There exists two methods to measure the thermal diffusivity of a material. One way involves measuring the density, thermal conductivity, and the specific heat of the material. The other way takes advantage of specific geometric configurations to solve the heat equation.

For our experiment we used the second method as the firs involves numerous measurements and the second method only requires a way to measure temperature gradients over time. In our experiment we used a Flir C2 thermal imaging camera to observe temperature gradients in brass, copper, and aluminum. Afterwards the videos of the temperature gradients were collected and used to model how heat diffused in each metal.

Theory

The temperature distribution of a given system is described by the heat equation

Where T is the temperature, t is the time, and a is the thermal diffusivity coefficient. For our system with azumithal symmetry we'll be modeling our equation in cylindrical coordinates. Our boundary condition for this system is providing an input of instantaneous contact by a heated metal rod, to a sheet of finite thickness. Mathematically this is modeled as[1]

However, when one simplifies down to an infinitesimally thin sheet we can see a profile that is similar to the one detailed below[2]

Experimental Apparatus and Analysis

Our experiment involved attaching metal sheets to a set of ring stands such that the were perpendicular to our table, in front of a Flir C2 thermal imaging camera, with a pixel resolution of 2.19mm². The sheets were painted black to prevent any environmental infrared radiation from reflecting off of the sheet and confounding our measurements. A metal rod was positioned behind it on another ring stand, such that it could be moved into contact with the metal sheet, and then quickly moved away. A blowtorch was used to heat the rod , which was then quickly brought into contact with the metal sheet, and then removed. Shown below in Figure 1 is a schematic drawing of our experimental apparatus

Figure 1. Diagram of experimental setup used in the experiment.

The data was recorded by a computer at 4 frames per second, and extracted with FLIRtools. The Temperature gradient was measured at each individual frame while temperature was diffusing in the medium. Mapping each point to the spherical Gaussian fit, the parameter in the denominator of the Gaussian model can be rewritten such that b² (t)= R² +8αt. Using b² as a Gaussian fit parameter, like seen in figure 2

Figure 2: Fit of b² for a frame of Brass

We then found b² at each frame and used a least squares fit to determine α for a given metal.

Results and Conclusion

After collecting the data and preforming the least squares fit we got the following fits

Figure 3: Graph of b² vs t for aluminum. The slope of Figure 4: Graph of b² vs t for brass. The slope of

the least squares fit line is y=(3.49 + 0.69)x + 2.63 the least squares fit is y=(2.21 + 0.32)x + 4.72

Figure 5: Graph of b² vs t for copper. The slope of

the least squares fit is y=(8.7 + 2.1)x + 3.6

Using this method we calculated α in aluminum to be (43.6 + 8.6)x10^-6m²/s, which differed from a common alloy of aluminium (64x10^-6 m² /s) [3] by a factor of 2.4σ, α in brass to be (27.7 + 4.1)x10^-6m²/s, which differed from the literature value of the thermal diffusivity coefficient of 34. 12x10^-6 m² /s by a factor of 1.57σ and α in copper to be (109 + 27)x10^-6m²/s[3], which differed from literature values of elemental copper (111x10^-6 m²/s) by .07σ[3]. It can be noted that for higher values of α the uncertainty in b² increases as well. This is most likely due to the fact that materials with higher α diffuse heat quicker while we were only able to record video at 4 frames per second. In spite of this the geometric method was successful in calculating α for each material, however for applications that need a higher degree of precision a more accurate thermal imaging camera should be used.

References

[1]F. Cernuschi, A. Russo, L. Lorenzoni, and A. Figari, “ In-plane thermal diffusivity evaluation by infrared thermography,” Rev. Sci. Instrum.72, 3988–3995 (2001).http://dx.doi.org/10.1063/1.1400151

[2] Am. J. Phys. 83, 923 (2015); http://dx.doi.org/10.1119/1.4928277

[3]Brown; Marco (1958). Introduction to Heat Transfer (3rd ed.). McGraw-Hill. and Eckert; Drake (1959). Heat and Mass Transfer. McGraw-Hill. ISBN 0-89116-553-3. cited in Holman, J.P. (2002). Heat Transfer (9th ed.).