Pressure Dependence of Thermal Diffusion in Aluminum

Pressure Dependence of Thermal Diffusion in Aluminum

Thomas Woosley and Brenner Hohenstein

Abstract

The purpose of this experiment was to determine the diffusion of heat across aluminum sheets under various pressures, with and without a thermal grease on the sheets. The diffusion was determined using a thermal imaging camera to observe the temperature change across the sheet as a function of time. A high heat source was put on the back of the sheet, and then a video of the heat spread was taken. After taking the video, frames from the video were used to calculate the diffusion constant. From a still frame, a single pixel slice through the center of the heat source and fit to a Gaussian curve. This was done several for several frames in a single frame multiple times in order to find the the diffusion independent of the time. This process was repeated for steadily increasing pressures both with and without the thermal grease added.

Overall, the data at low pressures was as expected but as the pressure increased the diffusion did not behave as expected. It was expected that the diffusion would increase with pressure linearly and then begin to taper off at a maximum for both sets, the set with the grease and the set without grease. Additionally, the set with the grease should have less diffusion when compared to the metal to metal contact. The latter prediction was mostly correct but the former was mostly inaccurate. Initially the diffusion increased but at high pressures there was an uncharacteristic drop-off of the diffusion.

Background

The transfer of heat has been known as a physical property for many centuries and was developed by the greats of science: Galileo, Newton, Hooke, Franklin, Fourier, and Black. Throughout the years, the development of thermometers and defining materials as conductors and non-conductors has led to the search for more efficient heat transfer. Finding specific heats and the transmission coefficients of different materials and compounds has allowed for development and design in an effort for better heat transfer. (1) The first known investigation into thermal conduction was done by Thompson, or more commonly known as Rumford, in 1777 when he conducted a series of tests on many different metallics, non-metallics, liquids, gases, clothing, stews, and various other foods. (1) This investigation led to the development of modern day conductivity of materials. These values are well known and well defined for most modern-day materials, as well as their behaviors in specific conditions.

The purpose of this experiment is to not define new materials and their characteristics, but to combine materials and barriers to observe the behaviors. The behaviors that we are most interested in is if contact pressure and varying thermal interface materials change the thermal diffusion in an aluminum-aluminum system. This is not a new concept, but it is one of great importance, from computing to technological advancements. In current silicon-based computing, excess heat buildup has been shown to cause the degradation of electrical device lifespans and a decrease in performance. One study by J. Schleeh et al. found that the inability to remove phonons from a transistor on its own dramatically decreased the performance of it. In the article, Schleeh uses varying thermal dissipation systems to improve data using a low noise amplifier. (4) One of the main limitations that designers and engineers encounter is the amount of heat that can be transferred from their system and how to do it most effectively while staying cost aware. Specifically, this concept is being used in a wide variety of computational systems, from a home computer to a massive data center. The biggest issue in both scenarios is the limited amount of heat that can be transferred from the processing units to the air and away from the system.

Aluminum has a specific temperature of 0.90 J/K and a thermal diffusivity of 97.1 mm²/s at room temperature (6), which makes it a good material to use in testing. Also, note that these two properties of aluminum depend on the purity of the Aluminum used. Heat spread also relies on a materials thickness and its conductivity coefficient. This relation also describes a thermal interface material, such as thermal grease, which aids in the spread of heat by filling gaps and imperfections on the surfaces of the metals and should require less pressure to get the same amount of thermal transfer. The thermal grease is not as efficient as a direct aluminum-aluminum contact, but it is useful in low pressure applications such as computers or electronics.

Theory

Thermal Diffusion can be derived from a general heat equation that essentially states that the rate of change of an object’s temperature, with respect to time, is proportional to the curvature of the temperature. This curvature is essentially the rate at which the temperature is changing. The constant of proportionality α, is the thermal diffusion which relies on the properties of the material.

∂T/∂t=α∇² T

The heat equation is essentially stating that the rate at which temperature changes is proportional to the temperature gradient, multiplied by a diffusivity constant. This then is changed into polar coordinates which makes it reliant upon only the distance from the center of the heat. A number of assumptions are made to simplify this equation. One of which is to assume the sheets have no thickness and thus can be treated as a two dimensional sheet. For our calculations the sheet is uniform, and holds a constant temperature prior to adding heat. Using these assumptions the equation above can be rewritten in terms of Temperature, time, the diffusivity constant and radius.

∂T/∂t=α(∂²T)/(∂r² )

From these a series of calculations can be done to acquire the temperature at any given point and time. Using the Fourier analysis, the final distribution equation is modeled as:

𝑇(𝑥,𝑡)=𝑡^(−1/2) 𝑒^{(−𝑥^2)/4𝛼𝑡}

Using this result, the data obtained can be fitted using a Gaussian curve. This curve provides the unknown term x, which is the thermal diffusion for the selected range. The full calculation for this result is done by Gfroerer (2)

Set-up

To get a change in pressure, a fixture using 2, 1/2 inch aluminum plates with a 3 inch window were used to sandwich the test plates. In these plates, there were 8 bolts arranged around the outside, shown in figure 1 and coated in a matte paint to mitigate the infrared reflections. Using a known torque to pressure conversion of the bolts, the apparent pressure applied to the samples could be adjusted. This fixture was placed horizontally on two flat bars with a thermal camera below to capture the diffusion. The heat was supplied through a 1/4 inch steel rod which was heated with a blow torch until slightly glowing orange. Each trial had a video taken, which 6-7 consecutive frames were taken and exported to data sheets. When the thermal grease was applied, it was spread to cover the space that would be seen through the window.

Using the MatLab code found in the appendix, a horizontal slice of the data was taken through the hottest point, fitted to the Gaussian curve (figure 2), and record the value of the width of the curve. Once all of the frames were evaluated, they were then fitted to a logarithmic curve using the time elapsed per frame. This curve then gave a final result for the diffusion of the system at a given pressure.

Graph 1: The final data for both systems, with and without thermal grease. The apparent pressure is given because the known pressure was at the bolts surrounding the sample. It was assumed that the apparent pressure at the center is directly related to the pressure applied to the outside of the fixture.

The final data collected was compiled and is found in Graph 1: Diffusivity as a Function of Pressure for both with and without Thermal Grease. This graph shows the variation obtained through the different pressures. Starting without TIM, the data seems to follow an underlying trend that would give the result that is desired. At the highest pressures however, this trend drops off in an uncharacteristic fashion, which lead to further evaluation. With TIM, the points follow a similar trend at lower pressures, moderate pressures show a drop off in the diffusivity, and then at the highest pressures the values peak, but then show a similar decay as the other tests. With both tests having the same abnormalities, this lead us to believe that our tests were flawed in a sense, but the data can still be used to draw conclusions.

For the dip at higher pressures, the best explanation for this would be that the fixture was applying pressure to the side of the plates, creating a splitting in the plates at the center. The apparent pressure at the center was assumed to be equal or off by a simple factor of the pressure applied to the bolts. This splitting would result in a lower diffusivity which corresponds to the drop. This is a point that can be improved upon in future experiments by using different fixtures or ways to apply the pressures to the materials.

The overall evaluation of the data can be interpreted in a multitude of ways. Firstly, taking a first or second order trend of the data could tell a general upwards trend, but the error bars present and the error in the fit can result in a trend of zero, meaning that there is no correlation between pressure and the diffusivity. The best fit of the data would be a logarithmic function, where at lower pressures, small pressure shifts would give large changes in the diffusivity. This model follows the data and agrees with the low pressure reading but doesn’t help explain the high-pressure readings. The variation and seemingly random results at the high pressures could fit within the fits bounds, but further testing is needed at different pressures and using a more reliable fixture to find if the results are scattered or do follow a deeper trend.

Conclusion

At low pressures, the data concludes that the diffusivity grows logarithmically, which then becomes scattered at medium and high pressures. This experiment concludes that there is a change in thermal diffusivity due to a pressure being applied, but further tests and more data must be taken to conclude to a final point. Possible improvements upon this experiment include: a different and more efficient pressure fixture, using a vacuum to get lower pressures and eliminate thermal radiation, and taking more data points to find the variation at higher pressures.

References

1. “Notes on the History of the Concept of Thermal Conductivity.” Isis, vol. 20, no. 1, 1933, pp. 246–259.

2. Gfroerer, T., Phillips, R., & Rossi, P. (2015). Thermal diffusivity imaging. American Journal of Physics, 83(11), 923-927.

3. Powell, R. L., & Blanpied, W. A. (1954). Thermal conductivity of metals and alloys at low temperatures: a review of the literature (No. NBS-CIRC-556). National Bureau of Standards Gaithersburg MD.

4. J. Schleeh, et al. “Phonon Black-Body Radiation Limit for Heat Dissipation in Electronics.” Nature Materials, vol. 14, no. 2, 2014, pp. 187–192.

5. Oberg Erik, Jones Franklin D, Horton Holbrook L, & Ryffel Henry H. (2016). Torque and Tension in Fasteners. In Machinery's Handbook (30th ed., p. 5). Industrial Press.

6. Fenech, H., and W. M. Rohsenow. “Prediction of Thermal Conductance of Metallic Surfaces in Contact.” Journal of Heat Transfer, vol. 85, no. 1, 1963, p. 15.

Acknowledgements

Special thanks to Kurt Wick for providing the base for our Matlab code as well as to Kevin Booth and Ke Wang for his help throughout the various parts of the lab.

Appendix

Code that was used in MatLab, both the Gaussian and Exponential fits are supplied and can be inserted directly into MatLab. The exponential fit was created using some features and guidance from Kurt Wick.

%Gaussian Fit

x = [-43:36]; %Index centered around zero, depends on where the origin point of the heating: manually entered. This may need to be converted to distance based off the pixels

T = [];%Temperature matrix also can be manually entered, temperature at each pixel.

gauss=@(p,x) p(3) + p(1)*exp(p(2).*x.^2); %creates a model for the gaussian curve based on the two parameters.

%p(1) is the amplitude of the curve and p(2) is the exponential decay of

%the curve p(3) is the offset.

guess = [100 -.009 297.4]; %arbitrary guesses for the values of p(1), p(2) and p(3)

gauss_model=fitnlm(x,T,gauss, guess) %non-linear fit for the data sets based off the general gaussan curve (line 4) and the guess for the parameters

T_fit=feval(gauss_model, x); %Setting the values of the fit into a matrix so it can be graphed

plot(x,T,'o',x,T_fit,'-') %Plotting the data set and the fit for the temperature. actual values are dotted; solid line is the fit.

shg %show graph

%Exponential Fit

x = []; %The time that a frame was taken, corresponding to the y value, seperated by: ;[space]. (EX: 1; 2; 3; ...)

y = []; %The different P2 values from the Gauss fit go here. Seperated using same notation as x values

sig_y = -Q.*y; %use a least squares algorithm to fit the weighted data to a model. Q is the %error to be modeled

myf=fittype('a/(x+b)')

[fitobj, gof, outp] = fit(x, y,myf, 'Weights',(1./sig_y).^2,'StartPoint',[0.3, 4.58])

%plot the original data vs. the selected model

plot(fitobj, 'predfunc') % add another argument to change the default .95 conf. level

title('Exponential fit with 95% Confidence Bounds');

xlabel('Time (s)')

ylabel('Gaussian Distribution (mm)')

grid on;

hold on

errorbar (x,y, sig_y, 'o' )

hold off

fitobj