Transient Conduction

Steady-state analysis gives the final distribution and nothing about the journey — yet the journey is usually what matters: quench times, cooking times, how long a steel column survives a fire, how fast a sensor responds.

Two resistances act in series: internal conduction L/kA and external convection 1/hA. Their ratio is Bi = hL/k.

Below Bi = 0.1 the body is treated as one temperature and the equation is first-order with τ = ρVc/hA — thermal mass over surface conductance, structurally identical to RC.

Every intuition from a charging capacitor transfers: 63% of the way in one time constant, 99.3% in five. Since τ ∝ V/A, response time grows roughly with linear size — which is why a fine thermocouple bead responds so much faster than a thick probe.

Above Bi = 0.1 the surface responds while the interior lags, and the governing equation becomes partial — temperature varying with position and time. Solved historically with Heisler charts, now numerically.

That gradient is also a stress: the cooled surface tries to contract around an interior that has not, producing surface tension. Slow cooling avoids cracking a casting, while toughened glass is quenched deliberately so the surface ends in useful compression.

If a body is large enough that the far side never notices, it is treated as semi-infinite and the disturbance penetrates as √(αt) — slowing as it deepens.

That is why ground temperature is stable at two metres while the surface swings 30 °C annually, and why a brief flash scorches a surface without heating what lies behind it.

Thermal diffusivity α = k/ρc governs how fast a temperature change propagates; conductivity governs how much heat flows in steady state. They answer different questions.

A material with high conductivity and high heat capacity conducts well and responds slowly, because every joule passing through must first warm the material it passes through.