Is this project an undergraduate, graduate, or faculty project?
Undergraduate
Project Type
group
Campus
Daytona Beach
Authors' Class Standing
Alexandria Krol, Sophomore David Cardona, Collin Petrie
Lead Presenter's Name
Alexandria Krol
Lead Presenter's College
DB College of Engineering
Faculty Mentor Name
Dr. Jorge Gonzalez
Abstract
A Differential Equation Approach to Heat Flow in a Thin Rod examines how differential equations can be used to model and understand heat conduction in a fundamental physical system. Heat transfer in solids is a key concept in physics and engineering, particularly in systems where temperature changes over time. A thin rod provides a useful one-dimensional model for studying how heat moves through a material and how temperature varies along the rod as time passes. The primary objective is to develop a mathematical description of this process using differential equations. The analysis begins with physical principles such as conservation of energy and Fourier’s law of heat conduction. The heat equation is derived to represent the relationship between temperature, time, and position along the rod. The equation is then analyzed using the method of separation of variables, which breaks the problem into simpler ordinary differential equations describing the spatial and temporal components of the temperature distribution. These equations are solved under typical boundary and initial conditions to determine how heat spreads through the rod. The resulting solutions show how temperature differences gradually decrease as heat flows from warmer regions to cooler ones, eventually approaching thermal equilibrium. This work highlights how differential equations provide a clear mathematical framework for describing physical processes and predicting the behavior of systems involving heat transfer.
Did this research project receive funding support (Spark, SURF, Research Abroad, Student Internal Grants, Collaborative, Climbing, or Ignite Grants) from the Office of Undergraduate Research?
No
Included in
Heat Transfer, Combustion Commons, Ordinary Differential Equations and Applied Dynamics Commons, Partial Differential Equations Commons
A Differential Equation Approach to Heat Flow in a Thin Rod
A Differential Equation Approach to Heat Flow in a Thin Rod examines how differential equations can be used to model and understand heat conduction in a fundamental physical system. Heat transfer in solids is a key concept in physics and engineering, particularly in systems where temperature changes over time. A thin rod provides a useful one-dimensional model for studying how heat moves through a material and how temperature varies along the rod as time passes. The primary objective is to develop a mathematical description of this process using differential equations. The analysis begins with physical principles such as conservation of energy and Fourier’s law of heat conduction. The heat equation is derived to represent the relationship between temperature, time, and position along the rod. The equation is then analyzed using the method of separation of variables, which breaks the problem into simpler ordinary differential equations describing the spatial and temporal components of the temperature distribution. These equations are solved under typical boundary and initial conditions to determine how heat spreads through the rod. The resulting solutions show how temperature differences gradually decrease as heat flows from warmer regions to cooler ones, eventually approaching thermal equilibrium. This work highlights how differential equations provide a clear mathematical framework for describing physical processes and predicting the behavior of systems involving heat transfer.