## Michael Farnsworth and Faculty Mentor: Brian Iverson, Mechanical Engineering

### Introduction

There are three major types of heat transfer: conduction, convection, and radiation. In many cases, radiation is ignored due to the fact that the amount of heat transferred by this method tend to be small compared to conduction and convection. However, in space and in some settings on earth, radiation is dominant and important. Absorptivity is a measure of an object’s ability to absorb radiation, and apparent absorptivity is a measure of how the shape of an object affects the amount of radiation that it absorbs. For example, if you were to shine a flashlight on a mirror, the light will hit the mirror once and bounce off. The tiny fraction of the light that the mirror absorbed would be the absorptivity. Now imagine that you had two parallel mirrors such as the mirrors that are located in the sealing room of an LDS temple. If you then were to shine a flashlight at these mirrors, it is possible that the light would bounce back and forth many times, with the mirrors absorbing a little bit of light each time. This greater fraction of light absorbed because of how the mirrors are arranged would be the apparent absorptivity. In the past, researchers have come up with ways to calculate the apparent absorptivity of very simple shapes, like the V-groove shown in Figure 1. However, with more complicated shapes like the origami fold called the Miura fold, coming up with an exact equation would be impossible. However, computer programs can help solve for the apparent absorptivity of surfaces using mathematical techniques called numerical methods, which can help scientists and engineers design satellites that will not overheat or freeze in space.

### Methodology

I used a computer program called APEX to help me determine the apparent absorptivity of different shapes. The first step is to build the shapes you want to test in the computer software. You then tell the program where radiation (light) is coming from, and the absorptivity of each surface in your shape. You then have the program do something called a ray trace, in which the computer keeps track of thousands or even millions of rays. The computer tracks the path of each ray, and whether the ray is absorbed by your shape or not. You can then calculate the apparent absorptivity by dividing how many rays were absorbed by the total number of rays that hit the surface of your object. In my project, I started by making sure that this approach worked by comparing the results I got using a V-groove to equations that scientists have come up with to describe the apparent absorptivity of this simple shape. After this, I then saw how different the apparent absorptivity was when using V-grooves that were not as long when compared to analytically derived equations, which assume infinitely long grooves. Finally, I found the apparent absorptivity of a modified V-groove, and the Miura fold, including how changing different aspects of the shapes affected the apparent absorptivity.

### Results

As shown in Figure 3, the APEX calculated values matched up very closely with the equations that had been found by previous researchers decades ago. Figure 4 shows how using shorter V-grooves differs from infinite solutions. At smaller angles, the error is greatest, possibly because narrow grooves meant that the rays on average were reflected a greater number of times. Shorter grooves gave results that were greater than longer grooves, because longer grooves better approximated an infinite groove. Although not pictured here, the higher the absorptivity of the surface itself, the greater the error was as well. Figure 5 shows how the apparent absorptivity of the modified V-groove changes for different inner lengths as it unfolds from the fully folded “T” shape to a “V” shape, and how they compare to a V-groove that unfolds to the same angle. Though not shown here, using different absorptivities followed similar trends between different inner length ratios. The beta angle of the Miura fold was shown to have the greatest effect on the path that the apparent absorptivity took as it folded and unfolded, and is shown in Figure 6.

### Outcomes

My results were delivered at the Utah Conference for Undergraduate Research. In addition, a first draft of a journal article has been finished, and will be published to the International Journal of Heat and Mass Transfer during 2016. A small portion of my results was also used on a poster presented at ASME IMECE 2015.