# time dependence of temperature equalization

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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It depends on what kind of assumptions you want to make. You can start by reading on heat transfer and heat equation. – Marek Mar 29 '11 at 19:37

## 1 Answer

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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No kidding, I gave you an upvote, Vladimir. You could have also mentioned the temperature to which both objects are converging - whose temperature differences from both original temperatures is inversely proportional to the heat capacity ratio of both objects. – Luboš Motl Mar 29 '11 at 22:32
Thanks, Lubosh. Yes, I could have developed my answer better but I am very depressed (ill) and cannot do my best anymore. – Vladimir Kalitvianski Mar 30 '11 at 12:40
What is $\tau$, what is $A$? How is the dependence on the specific heats? – martin Mar 30 '11 at 13:39
Thanks, you write the specific heat capacity... however there are two systems involved, why does only one heat capacity occur? Do you have some references? You mentioned 1D systems. How is the situation for 3D systems (like in my example, water - air)? – martin Mar 30 '11 at 15:06
I mentioned the simplest formulas to give you an idea how heat conduction is described for 1D system. For a multi-layer system all material parameters are involved in $\tau_0$. Roughly, the slowest process (the largest $\tau_0$) will determine the characteristic transient time. I cannot give you more details here because I am not in form. Read, please, sources describing heat conduction (books, web lectures, wikipedia, etc.). – Vladimir Kalitvianski Mar 30 '11 at 15:23