Lumped-Capacitance Time Constant
Lumped-Capacitance Time Constant
A small body with little internal resistance cools exponentially with a time constant set by its heat capacity and surface convection.
The time constant is τ = ρ·V·c/(h·A). where ρ is density, V volume, c specific heat, h the convective coefficient and A the surface area.
Inputs
rho := 8000 kg/m^3=8000 kg/m^3
V := 1e-6 m^3=1e-06 m^3
c := 500 J/(kg·K)=500 m^2/K/s^2
h := 100 W/(m^2·K)=100 kg/K/s^3
A := 0.0006 m^2=6e-04 m^2
Form the capacity-to-convection ratio.
tau := rho·V·c/(h·A)=66.67 s
Results
After one time constant the temperature gap falls to 37% of its start; a smaller body or higher convection responds faster — which is why fine thermocouples react quickly.
Assumptions & limitations
- Lumped capacitance (Biot < 0.1).
- Constant properties and h.
- Uniform body temperature.