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Transient cooling

When the Biot number is small, a body's temperature follows an exponential approach toward the surrounding temperature over time, governed by a time constant that depends on mass, heat capacity, heat transfer coefficient and area. The formula can be inverted to find the time needed to reach a given temperature.

T(t)−T∞=(T0−T∞)e−t/τT(t) - T_\infty = (T_0 - T_\infty)e^{-t/\tau}lumped capacitance, transient temperature
τ=ρVchA\tau = \dfrac{\rho Vc}{hA}time constant
t=−τln⁡ ⁣(T−T∞T0−T∞)t = -\tau\ln\!\left(\dfrac{T-T_\infty}{T_0-T_\infty}\right)time to a given temperature

Symbols

τ\tautime constants
T0T_0initial temperatureK
T∞T_\inftysurrounding temperatureK

Example

A body with τ=90\tau = 90 s cools from 260260 °C in air at 2222 °C. Time to reach 6060 °C: t=−90ln⁡(38/238)≈165t = -90\ln(38/238) \approx 165 s.

°C can be used directly in the differences T−T∞T-T_\infty in the transient formula, since the 273.15273.15 cancels in the subtraction.
Practise radiation and transient heat for free →

← Combined convection and radiation

Part of Heat Transfer: Radiation and transient heat.