Heat Conduction Simulator
Answer a shop-floor safety question with the heat equation: solve it with Pdepe, prove it against the exact Fourier solution, and find how long a hot steel bar takes to become safe to handle.
The brief
You are the process engineer, heat treatment shop
Carbon-steel bars 0.4 m long leave the furnace at a uniform 500 °C. They are clamped at both ends in water-cooled jaws held at 25 °C, and a ceramic sleeve insulates their sides, so heat can only leave through the ends.
The shop's safety rule says a bar may be handled only when its hottest point is below 60 °C. The supervisor wants to know how long to wait, and what happens to the wait if one of the jaws is out of service and that end is left insulated.
You will turn the physics into the heat equation, solve it with Reineur Solver's PDE solver, prove the numerical solution against the exact one, and answer both questions in minutes. The finished engine drives a 3D simulator of temperature along the bar over time.
- Bar length
- 0.4 m, carbon steel
- Steel properties
- k = 45 W/m·K, ρ = 7,850 kg/m³, cp = 475 J/kg·K
- Initial temperature
- 500 °C, uniform
- Jaw temperature
- 25 °C
- Safe to handle
- hottest point below 60 °C
What you will be able to do
- Derive thermal diffusivity and the diffusion time scale of a part
- Set up and solve the transient heat equation with Pdepe
- Verify a numerical solution against an exact Fourier series
- Extract an engineering decision from a space–time solution
- Change a boundary condition and quantify its effect