= Lumped thermal model
A <lumped thermal model> assigns one representative <temperature> and <heat capacity> to each region of an apparatus. For a single body of <heat capacity> $C$ connected to a bath by <thermal conductance> $G$, with supplied <power> $P$, an elementary balance is
$$
C\dot T=P-G(T-T_b).
$$
The model stores <internal energy> in the body and treats exchange through a <thermal resistance>, leaving internal spatial <temperature gradients> unresolved.
A small solid specimen with rapid internal equilibration can approximate this behavior. A constant heating step gives a final rise $P/G$ and a <thermal time constant> $C/G$ when $C$ and $G$ are constant. Comparing thermometers at different positions tests the assumption of nearly uniform specimen <temperature>. If appreciable internal gradients persist, the apparatus needs several coupled thermal regions or a spatial <heat equation> model. Fitting a single exponential is useful only after this physical approximation has been checked.
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