Average Temperature of a Sphere as a Function of Time

by Lee Farrell

Submitted : Spring 2009

The problem given was to find the average temperature of a radial position in a solid sphere that is being heated as a function of time. This is something that might be applied in many different fields of engineering. Using the formulas given in the problem, one simply plugs the solution for the temperature into the equation for the average, takes the integral indicated, solves for it between the two values given, and then divides. That solves the first part of the problem. The second part is a bit trickier, requiring the re-defining of many of the terms to fall under the umbrella provided by the new time constant. Once this is reconfigured, one does the same process as before, substituting in the value of the temperature, integrating, and dividing. Once this new equation is reached, one simply plugs in the values for the time constant given and solves.  This project was exceedingly difficult for me for the longest time; I spent several weeks staring at it, willing it to make sense in my mind. It was not until very late in the project that, looking at it in a slightly different light, it mostly clicked and came together. This project challenged me in new ways, and ways that make me thankful that we have computers to do much of this heavy lifting for us in these modern times. However, it is always good to know how to do these on one’s own, as computers are only as good as the data that is fed into them and the program that they are running.

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