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.

Note: The information below was created with the assistance of AI.

1. Level of Mathematics
Target Level:
This article is ideally suited for undergraduate students, especially those enrolled in:

Ordinary Differential Equations (ODEs) courses

Mathematical Modeling

Engineering or Applied Physics programs

Rationale:

The paper centers on solving first-order linear differential equations using Newton’s Law of Cooling.

It explores both continuous and discrete modeling approaches, including parameter sensitivity analysis and nonlinear least-squares fitting, requiring intermediate to advanced math maturity.

2. Application Areas
Primary Applications:

Thermal Physics / Heat Transfer: Estimating the risk of water pipes freezing in cold environments.

Engineering Systems Design: Evaluating insulation and structural heat flow properties.

Environmental Modeling: Predicting responses of interior environments to external weather conditions.

Applied Mathematics: Serving as a hands-on case study in real-world modeling.

Practical Relevance:

The model simulates a real-world scenario (temperature in a cavity with insulated pipes), helping to inform infrastructure resilience planning and HVAC system optimization.

3. Prerequisites
To fully engage with the paper, learners should understand:

Differential equations (formulation and analytical solutions)

Basic thermodynamics (especially Newtonian cooling and conservation of energy)

Linear algebra and numerical methods (for fitting and solving equations)

MATLAB or similar computational tools (used for data fitting, interpolation, and simulations)

Basic statistics (interpreting error norms and sensitivity indices)

4. Subject Matter
Core Topics Explored:

Model derivation using conservation of energy and Newton’s Law of Cooling.

Parameter identification through data fitting with empirical temperature data.

Continuous vs. discrete models:

Continuous model based on linear approximations of temperature profiles.

Discrete model derived from difference equations for irregular sampling times.

Equilibrium temperature estimation for determining the minimum exterior temperature that prevents pipes from freezing.

Parameter sensitivity analysis using relative and absolute sensitivity indices.

5. Correlation to Mathematics Standards
Postsecondary Curriculum (MAA & CRAFTY):

Mathematical modeling: Formulating real-world problems into differential equations.

Quantitative reasoning: Using data to estimate parameters and interpret physical meaning.

Use of technology: Implementing simulations and data analysis using MATLAB.

NCTM Process Standards (for High School to Undergraduate Transitions):

Reasoning and Proof: Justifying assumptions and interpreting the behavior of solutions.

Connections: Applying mathematical theory to engineering and environmental contexts.

Representation: Graphical illustration of model outputs and parameter effects.

Common Core State Standards (CCSSM) – High School:

Functions (F-LE): Modeling with linear functions.

Modeling (MP.4): Applying math to real-world scenarios.

Statistics & Probability (S-ID): Interpreting data patterns, trends, and residuals in model fitting.