Transient Heat Conduction — Time Integration & Thermal Mass
Heat equation time discretization, implicit and explicit schemes, Biot and Fourier numbers, lumped capacitance, and thermal time constants.
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Quick Explainer
What is the lumped capacitance method and when is it valid?
Lumped capacitance treats the entire solid as having uniform temperature — thermal mass (mCp) but no spatial gradient. Valid when Biot number Bi = hL/k < 0.1, meaning convective resistance dominates over conductive resistance. For a small metal part being quenched, it gives exponential temperature decay with time constant tau = mCp/hA.
How do you choose the time step for transient thermal FEM?
For explicit integration, stability requires dt < rho*Cp*L^2/(2k) where L is smallest element size. For implicit methods (unconditionally stable), accuracy drives step size. Rule of thumb: dt ~ tau/20 where tau is the smallest relevant thermal time constant. Too large a step misses the transient response of thin components.