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Researchers should cite this work as follows:

• Berresford, G. (2023). 1981-Geoffrey_Berresford-Differential Equations and Root Cellars. SIMIODE, QUBES Educational Resources. doi:10.25334/P6WB-MT85

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Berresford, Geoffrey C. 1981. Differential Equations and Root Cellars.  UMAP Unit 554. 23 pp. 23 pp. Available from http://www.comap.com.

This is a classic module from UMAP in which the heat equation in one dimension is fully developed by using the standard technique of measuring the heat flow in and out of a small element of mass and equating them at equilibrium. The solution is then used to determine how deep a root cellar is to be to be warmest in winter and coldest in summer, in other words have the heat on the surface and at depth be half a year off from each other. Diffusivity constants for various types of soils, reasonable assumptions, and good modeling formulation support students as they work their way through the requirements of the module.

We quote from the module:

Target Audience: Courses in ordinary or partial differential equations, applied mathematics  or mathematical physics .

Abstract: The description of heat flow through matter is one of the classical accomplishments of applied mathematics. This unit 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 penetrate the surface of the earth and to find the best depth for a root cellar. Students use the specific heats, thermal conductivities. and thermal diffusivities of standard materials to calculate seasonal temperature variations in walls and floors of underground structures.

Prerequisites:    A knowledge of how to solve the ordinary differential equations u'=  cu and u" =  cu and some familiarity with the arithmetic of complex numbers.