Suppose you have two thermodynamical systems $X_1$ and $X_2$ (for example water and air) with different temperatures ($T_1 > T_2$). Now put them into thermal contact. Is there a formula which describes, how $T_1$ and $T_2$ changes with time?
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In a simple heat conduction theory (no convection) the corresponding time is determined with the lowest eigenvalue of the Sturm-Liouville problem: $\Delta T(t) = A\cdot e^{-t/\tau}$ (regular regime), see, for example, my article. EDIT: For any time $t > 0$ the temperature difference is a series like $\Delta T(t) = \sum_{n=0}^\infty A_n \cdot e^{-t/\tau_n}$ $\tau_{n+1} < \tau_n$, for example, $\tau_n \propto \frac{1}{\pi^2(n+1)^2}$. When $t >> \tau_1$, the fast decaying exponentials $e^{-t/\tau_n}$ are small with respect to the slowest one $e^{-t/\tau_0}$ so only one term remains in this sum. The latter regime is called a regular regime of the heat exchange. For certain one-layer 1D systems $\tau \propto \frac{\rho c L^2}{\kappa \pi^2(n+1)^2}$ Here $\kappa$ is the heat conductivity, $\rho$ is the material density, $c$ is the specific heat capacity, and $L$ is the layer thickness (exact $n$-dependence is determined with the boundary conditions). |
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