Heat flow -- Dynamics of a 1D rod

Developed by L. Engelhardt

In introductory physics, students learn two equations that involve heat: Q=mcΔT, describes the amount of heat transferred in calorimetry; and QΔt=ktAΔTΔx describes the rate of steady state heat flow through a window. In this Exercise Set we combine these two equations to explore the dynamics of heat flow and temperature change in one dimension. Specifically, we will explore the temperature of a frying pan handle as a function of both position and time, and we will see how this temperature profile depends on the material properties of the handle.

Subject Area Thermal & Statistical Physics
Levels First Year and Beyond the First Year
Available Implementations IPython/Jupyter Notebook and Python
Learning Objectives

Students will be able to:

  • Mathematically derive an equation for the small slice of a 1D rod at the end of the rod (Exercise 1)
  • Create 1D arrays to store the discretized values of both position and time (Exercise 2)
  • Look up the relevant material properties for heat transfer through a rod (Exercise 3)
  • Compute the heat transfer constant, r (Exercise 4)
  • Set initial values in a 2D array – T(x,t) – to represent a 1D rod at time t=0(Exercise 5)
  • Use the hyperbolic tangent function to model the increasing temperature of a frying pan, and use this to set boundary conditions for T(x=0,t) (Exercise 6)
  • Convert temperatures between Fahrenheit and Celsius (Exercise 7)
  • Write code to compute T(x,t) for all x, t (Exercises 8 and 9)
  • Plot T(x) for various values of t (Exercise 10)
  • Test for convergence in T(x,t) (Exercise 11)
  • Create animations of T(x,t) (Exercise 12)
  • Create contour plots of T(x,t) (Exercise 13)
  • Interpret simulated data of T(x,t) from plots, animations, and contour plots (Exercise 14)
  • Carry out the analysis described above for multiple materials (Exercise 15)
Time to Complete 120 min