Physics

Heat Equation

The heat equation says temperature spreads out over time. Here it plays out in a 3D room: a radiator on one wall warms it, a cold window on the opposite wall chills it, and heat diffuses between them as glowing fog. Paint your own heat or cold on the slice and watch a thermometer settle.

warmer than 18 °Ccolder than 18 °Cthermometer

Drag on the slice to paint heat or cold · drag the thermometer to move it · drag empty space to orbit · scroll to zoom

Room average—

—

Diffusivity α 5.0 × 10⁻³ m²/s

slowfast

Radiator temperature 55 °C

Outside temperature (window) 0 °C

Radiator onWindow openLeaky wallsSofa (does not conduct)

Brush

Show

Plane

Plane offset z = 0.00 m

Fog opacity 1.00

thindense

Thermometer over time

Values

time—

room average—

thermometer—

heat in (radiator)—

heat out (window)—

heat out (walls)—

What to observe

Warmth spreads from the radiator toward the open window. The thermometer rises, then levels off in a steady state: ∇²T = 0, and the heat in equals the heat out.
A hot spot spreads and its peak drops. Its width grows like √(αt), so spreading twice as far takes four times as long.
Open the window in a warm room. Cold pours in and the average falls by about 5 °C, until the radiator and the window balance again.
Lower α. The same final picture emerges, only far more slowly: diffusivity sets the timescale, not the destination.

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