Lumped Capacitance Time Constant

ρVc is the heat a body stores per kelvin — its thermal capacitance — and hA is the conductance draining it. Their ratio is a time constant in exactly the sense an electrical engineer means, and the analogy is complete: the body is a capacitor, the surface film is a resistor, and the temperature decays exponentially. In one τ the gap to ambient closes by 63%, in three τ by 95%, in five τ by over 99%.

Worked example: an aluminium block, ρ = 2700 kg/m³, V = 100 cm³, c = 900 J/(kg·K), cooling in air at h = 30 W/(m²·K) over A = 0.02 m². τ = 2700 × 0.0001 × 900/(30 × 0.02) = 243/0.6 = 405 s, so it is essentially at room temperature in about twenty minutes. This is the number that sizes thermocouple response — a fine bead responds in milliseconds, a 6 mm thermowell in a stagnant pocket can lag a minute, and a control loop tuned without knowing which one you have will hunt forever. Trap: τ assumes the lumped regime, so check Bi < 0.1 before trusting it, and remember h is not constant during a violent transient — free convection off a hot block starts strong and weakens as the block cools.