Modeling Newtonian Cooling

Abstract

We present a case study in which differential equations are applied to a real-world problem: protecting water pipes from freezing under extreme winter weather conditions. The primary objective is to estimate the minimum ambient temperature in an attic crawl space that insulated water pipes can survive without freezing. The problem is modeled using a variant of the standard Newtonian cooling process, formulated as a first-order linear differential equation. Two key parameters, representing heat exchange with the interior and exterior, are determined by fitting the model to empirical temperature data. Parameter sensitivity is analyzed using the local (direct) method. The case study, conducted by students (one of whom is a co-author), is suitable for use in student projects within an ordinary differential equations (ODEs) or mathematical modeling course.