Differential Equations and Root Cellars (UMAP)

This module derives the one-dimensional heat equation, proves the uniqueness of its solutions, and finds the solutions by separating the variables and applying the techniques of ordinary differential equations to solve the resulting first and second order equations. The solutions are used to investigate the depth to which annual and daily temperature variations pentrate the surface of the earth, and to find the best depth for a root cellar. Students use the specific heats, thermal conductivities, and thermal diffusitivities of standarad materials to calculate seasonal temperature variations in walls and floors of underground structures.

Table of Contents:

1. INTRODUCTION

2. A QUESTION

3. THE HEAT EQUATION

4. UNIQUENESS AND THE MAXIMAL PRINCIPLE

5. A PERIODIC SOLUTION

6. CONCLUSIONS ABOUT TEMPERATURE BENEATH THE EARTH'S SURFACE

7. ANSWERS TO PROBLEMS

8. REFERENCES