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If there is a solid immersed in a large (but finite) pool of water, where the solid has temperature $T_s$ and the water has temperature $T_w$, with $T_w>T_s$, how can I calculate $T_s(t)$ and $T_w(t)$? (Suppose that the masses and the material properties of the solid are known.)

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up vote 1 down vote accepted

Typical time dependent heat equation problem with Neumann boundary conditions.

The solution $T_s(x,y,z,t)$ will give you the evolution of temperature at every point of the solid.

The boundary condition is $\nabla T = aT + b$ with $a$, $b$ constants because you will suppose that energy is exchanged by natural convection and conduction (the latter might be neglected depending on the properties of the bodies and the initial conditions) .

In a general case the heat equation can only be solved numerically, there are no analytical solutions. In special cases depending on the geometry (e.g. spherical symmetry) the problem can be simplified. If the heat capacity of the bath is large compared to the one of the solid, further simplifications are possible.

As a summary : the only general method is numerical integration of a partial differential equation of second order with temperature gradients imposed on the boundary.

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