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Heat flow -- Dynamics of a 1D rod

Developed by L. Engelhardt

In introductory physics, students learn two equations that involve heat: Qstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">mstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; 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margin-bottom: 0px; margin-left: 0px; line-height: normal">Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, describes the amount of heat transferred in calorimetry; and Qstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; 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line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; 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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, rstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 4)
  • Set initial values in a 2D array – Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> – to represent a 1D rod at time tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(Exercise 5)
  • Use the hyperbolic tangent function to model the increasing temperature of a frying pan, and use this to set boundary conditions for Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 6)
  • Convert temperatures between Fahrenheit and Celsius (Exercise 7)
  • Write code to compute Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> for all xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercises 8 and 9)
  • Plot Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> for various values of tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 10)
  • Test for convergence in Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 11)
  • Create animations of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 12)
  • Create contour plots of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> (Exercise 13)
  • Interpret simulated data of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">,style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> from plots, animations, and contour plots (Exercise 14)
  • Carry out the analysis described above for multiple materials (Exercise 15)
Time to Complete 120 min
Exercise 1-3

In the exercises below, you will simulate heat transfer along a Lstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">15style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> cm long frying pan handle for a frying pan that is heated on a stove for 10 minutes.

Exercise 1: Mathematical Derivation

Derive an equation for the temperature of slice number Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> at the end of a one-dimensional rod, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">. Your derivation, and the resulting equation, will be very similar to the derivation and result for the other slices of the rod (for istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">1style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">). The only difference is that there is no heat flow in/out of the right side of this slice.

Exercise 2: x and t

Write lines of code to generate two one-dimensional arrays: one for the discrete values of position, xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, and one for the discrete values of time, tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">.

Exercise 3: Material Properties

You will simulate the handle of a frying pan, using handles made of (1) stainless steel and (2) Bakelite. Look up the relevant material properties for both stainless steel and Bakelite. What are the numerical values for each of these properties for each material?

Exercise 4-6

Exercise 4: Dimensionless constant, r

Compute the constant “rstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">” for stainless steel. What value do you obtain? If it is greater than 0.5, you will need to adjust your discretization of the position and/or the time to decrease the values of rstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">.

Exercise 5: 2D array and initial temperature

Generate a 2D array that will be used to store temperature as a function of both position and time, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">.

The temperature of the frying pan handle will begin at a room temperature of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">72style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> F. Store this initial (jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">) value in the 2D array for all slices of the rod (for all istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">).

Exercise 6: Temperature of the pan

You will need to generate an array of temperatures for the frying pan (at the left edge of the handle) as a function of time, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">. The frying pan will start out at room temperature at time tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">. A typical frying pan will then heat up quicky for the first minute or two after a stove is turned on, and then it will reach a constant temperature of around 350 to 400style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> F. This behavior can be reproduced using the equation

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border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">τstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; width: 668px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">

with Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">rstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">ostyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">ostyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">mstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">72style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> F, Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">sstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">ostyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">vstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">estyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">300style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> F, and τstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">60style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> seconds. “tanh” is the “hyperbolic tangent” function, which is one of the standard built-in functions for any computer math library.

Use this equation to compute an array of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> values, and then plot Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> versus tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> to verify that the temperature of you frying pan agrees with the following plot:

Once you have verified that Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> is correct, use the 1D array of Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> values to set the temperature of the left (istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">=style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">) edge of the handle in the 2D array, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, for all values of time (for all jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">).

Exercise 7-9

Exercise 7: Conversions

You will need to convert the 2D array, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> from units of Fahrenheit to Celsius before beginning the simulation. Then convert back from Celsius to Fahrenheit after the simulation. What will you need to do in order to perform both of these conversions?

Exercise 8: Computing for many i and j

Complete the line of code to compute Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> for all positions, 1style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">istyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">1style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, and all times, jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">>style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">.

Exercise 9: End of the handle

Complete the line of code to compute Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">Nstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> at the end of the handle for all times, jstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">>style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">. (You derived this equation in Exercise 1.)

Exercise 10-12

Exercise 10: Plotting

Plot the temperature of the rod versus position, Tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">(style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, for a few different values of time, tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">. Make sure that your results seem reasonable. Discuss if/why the results seem reasonable.

Note: It is likely that your program won’t work right away! If your code generates errors, or if the results just don’t seem physically reasonable, you will need to debug your code.

Exercise 11: Convergence

Decrease the values of Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> and Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">, and re-run the simulation to check for convergence. If the temperature results change when you change the discretization, that means that Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">xstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> and/or Δstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">tstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> are too big, and you need to make them smaller. (But make sure that rstyle="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"><style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">0.5.style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">)style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">

Exercise 12: Animating

Create an animation showing temperature versus position at many different moments in time. Again, make sure that the results seem reasonable, and check for convergence. Discuss.

Exercise 13-15

Exercise 13: Contour plots

Create a contour plot that shows temperature as a function of both position and time in a single static plot. Again, make sure that the results seem reasonable, and check for convergence. Discuss.

Exercise 14: Analysis

At a temperature of 150style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal">style="padding-top: 0px; padding-right: 0px; padding-bottom: 0px; padding-left: 0px; border-top-style: none; border-top-width: 0px; border-right-style: none; border-right-width: 0px; border-bottom-style: none; border-bottom-width: 0px; border-left-style: none; border-left-width: 0px; display: inline; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; line-height: normal"> F, an object will quickly cause a burn if touched. (You can also get burned by cooler objects, but not immediately – only if you continue to touch the object for an extended period of time.) Based on the results of your simulations, discuss where you would – and would not – be able to safely touch the handle of the pan at different moments in time.

Exercise 15: Repeat for other material

Repeat the exercises above for a frying pan handle that is made of Bakelite rather than stainless steel. Discuss your results.

Dynamics of a 1D rod.py

# # Author: Larry Engelhardt (July, 2016)

# Material properties for a few different materials are listed below

#

# 1. Thermal conductivity: `k_t`

# 2. Specific heat capacity: `c`

# 3. Density: `rho`

#

# Stainless Steel

k_t = 16 # W / (m C)

c = 500 # J / (kg C)

rho = 8000 # kg / m^3

# Bakelite

k_t = 0.2 # W / (m C)

c = 920 # J / (kg C)

rho = 1300 # kg / m^3

# Oak

k_t = 0.17 # W / (m C)

c = 2000 # J / (kg C)

rho = 700 # kg / m^3

# Fiberglass

k_t = 0.04 # W / (m C)

c = 700 # J / (kg C)

rho = 2000 # kg / m^3

# BEGINNING THE PROGRAM:

from pylab import *

# Discretizing space

L = 0.15 # Length of handle in meters

Nx = 76 # Number of positions in space

x = linspace(0, L, Nx) # Array of positions

dx = L / (Nx - 1) # Values of delta x (meters)

# Discretizing time

tMax = 600 # Maximum time in seconds

Nt = 1801 # Number of time steps

t = linspace(0, tMax, Nt) # Array of time values

dt = tMax / (Nt - 1) # Size of time step (seconds)

T0 = 72 # Initial temperature (Farenheit)

dT_stove = 300 # Temperature increase of the stove (F)

# Stainless Steel's material properties

k_t = 16 # W / (m C)

c = 500 # J / (kg C)

rho = 8000 # kg / m^3

r = k_t * dt / (c * rho * dx**2) ## THE PARAMETER! (dimensionless) ##

print('r:', r) # Check to make sure that r < 0.5

# Preparing 2D array of temperatures

T = zeros((Nt, Nx)) # 2D array for temperature. NOTE THE ORDER: [time, space]

T[0, :] = T0 # Initial values (time t = 0)

tau = 60 # Used in tanh function, units of seconds

T[:, 0] = T0 + dT_stove * tanh(t/tau) # Edge of rod (at x=0) in deg. F

T = (T - 32) / 1.8 # Convert all temperatures to Celsius

# Verifying that \(T(t)\) for the pan (\(x=0\)) looks correct:

Tpan = T0 + dT_stove * tanh(t/tau)

plot(t, Tpan, lw=2)

grid(True)

xlabel('Time (sec)')

ylabel('Pan Temperature (F)')

grid(True); show()

# Computing the temperature of the handle for all `x` and `t`:

for j in range(1, Nt): # Loop thru TIME

for i in range(1, Nx-1): # Loop thru SPACE

T[j,i] = r*T[j-1, i+1] + (1 - 2*r)*T[j-1,i] + r*T[j-1, i-1]

T[j, -1] = (1-r)*T[j-1,i] + r*T[j-1,i-1] # End of rod (x=L)

T = 1.8*T + 32 # Convert back to F from C

x = x * 100 # Convert from meters to cm

# Plotting \(T(x)\) for a few moments in time

jValues = [30, 180, 540, -1] # A few values of the time index (-1 for last element)

for j in jValues:

plot(x, T[j,:], linewidth=2, label='Time (seconds): '+str(t[j]))

legend()

title('Stainless Steel')

xlim([0, 15])

xlabel('Position (cm)')

ylabel('Temperature (\(^{\circ}\)F)')

grid(True);

savefig('StainlessSteel_T_vs_x.png')

show()

# Creating a contour plot of \(T(x, t)\)

colorvals = linspace(70, 160, 91) # colors for contour plot (min, max, num)

contourf(x, t, T, colorvals) # (x-axis, y-axis, z(x,y))

title('Stainless Steel')

xlabel('x (cm)')

ylabel('Time (sec)')

colorbar()

grid(True)

savefig('StainlessSteel_contour.png')

show()

# Repeating using Bakelite

# Discretizing space

L = 0.15 # Length of handle in meters

Nx = 76 # Number of positions in space

x = linspace(0, L, Nx) # Array of positions

dx = L / (Nx - 1) # Values of delta x (meters)

# Discretizing time

tMax = 600 # Maximum time in seconds

Nt = 1801 # Number of time steps

t = linspace(0, tMax, Nt) # Array of time values

dt = tMax / (Nt - 1) # Size of time step (seconds)

T0 = 72 # Initial temperature (Farenheit)

dT_stove = 300 # Temperature increase of the stove (F)

# Bakelite's material properties

k_t = 0.2 # W / (m C)

c = 920 # J / (kg C)

rho = 1300 # kg / m^3

r = k_t * dt / (c * rho * dx**2) ## THE PARAMETER! (dimensionless) ##

print('r:', r) # Check to make sure that r < 0.5

# Preparing 2D array of temperatures

T = zeros((Nt, Nx)) # 2D array for temperature. NOTE THE ORDER: [time, space]

T[0, :] = T0 # Initial values (time t = 0)

tau = 60 # Used in tanh function, units of seconds

T[:, 0] = T0 + dT_stove * tanh(t/tau) # Edge of rod (at x=0) in deg. F

T = (T - 32) / 1.8 # Convert all temperatures to Celsius

for j in range(1, Nt): # Loop thru TIME

for i in range(1, Nx-1): # Loop thru SPACE

T[j,i] = r*T[j-1, i+1] + (1 - 2*r)*T[j-1,i] + r*T[j-1, i-1]

T[j, -1] = (1-r)*T[j-1,i] + r*T[j-1,i-1] # End of rod (x=L)

T = 1.8*T + 32 # Convert back to F from C

x = x * 100 # Convert from meters to cm

jValues = [30, 180, 540, -1] # A few values of the time index (-1 for last element)

for j in jValues:

plot(x, T[j,:], linewidth=2, label='Time (seconds): '+str(t[j]))

legend()

title('Bakelite')

xlim([0, 15])

xlabel('Position (cm)')

ylabel('Temperature (\(^{\circ}\)F)')

grid(True);

savefig('StainlessSteel_T_vs_x.png')

show()

colorvals = linspace(70, 160, 91) # colors for contour plot (min, max, num)

contourf(x, t, T, colorvals) # (x-axis, y-axis, z(x,y))

title('Bakelite')

xlabel('x (cm)')

ylabel('Time (sec)')

colorbar()

grid(True)

savefig('Bakelite_contour.png')

show()

 

For Teachers

Translations

Software Requirements

SoftwareRequirements

Android iOS Windows MacOS
with best with Chrome Chrome Chrome Chrome
support fullscreen? Yes. Chrome/Opera No. Firefox/ Sumsung Internet Not yet Yes Yes
cannot work on some mobile browser that don't understand JavaScript such as..... cannot work on Internet Explorer 9 and below

 

Credits

Fremont Teng; Loo Kang Wee

end faq

 

Version

  1. https://www.compadre.org/PICUP/exercises/exercise.cfm?I=138&A=heatflow_1d 
  2. http://weelookang.blogspot.com/2018/06/heat-flow-dynamics-of-1d-rod.html 

end faq

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