AdvancedTopic 73 of 104Engineering Mathematics
Heat equation
u_t = α u_xx describes diffusion of temperature through a material.
In depth
The 1D heat equation u_t=αu_xx says temperature changes in proportion to curvature. Solutions with fixed ends are sums of sine modes decaying as e^{-α(nπ/L)²t}, so high spatial frequencies vanish first. Thermal diffusivity α=k/(ρc).
Key points
- Steady state is linear between boundaries
- Fourier number Fo=αt/L²
- Smoothing nature of diffusion
ExampleA hot rod's sharp temperature peak smooths out quickly while the long-wavelength mode lingers.
What is Heat equation?
u_t = α u_xx describes diffusion of temperature through a material.
Why is it important to understand heat equation?
The heat equation governs diffusion and explains how temperature spreads in walls, electronic packages and machined parts.
What does the diffusivity alpha represent?
It measures how fast heat spreads through a material. High alpha means rapid temperature equalisation, as in metals, and low alpha means slow spreading, as in insulators.
Why does the heat equation smooth out sharp temperature changes?
Diffusion always moves heat from hot to cold, so peaks flatten and the solution becomes smoother as time passes.
How is Heat equation used in practice?
Typical applications include estimating how long a steel bar takes to cool to safe handling temperature, checking the thermal response of a PCB during a power pulse and designing insulation thickness for a building wall.
What should I learn before Heat equation?
Study "Partial differential equations" first (topic 72 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Heat equation?
Steady state is linear between boundaries. Fourier number Fo=αt/L². Smoothing nature of diffusion.
Can you give an example of Heat equation?
A hot rod's sharp temperature peak smooths out quickly while the long-wavelength mode lingers.
What comes after Heat equation?
The next topic is "Wave equation", which builds directly on this one.
How difficult is Heat equation?
It is classed as Advanced in Engineering Mathematics (topic 73 of 104). It is an advanced topic. It assumes you are comfortable with the intermediate topics in Engineering Mathematics.