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Is there a material whose thermal conductivity can be modeled by the function

$$k(T) = \begin{cases}k_1, & T<T_0 \\ k_2, & T\geq T_0\end{cases}$$

where $k_1$, $k_2$ are constants. I believe water would behave like this, considering the conductivity of ice and liquid water. But then I would ask: when modeling the steady state heat equation, since there is a phase change, would one need to add a special boundary condition at the interface?

Follow up question: Is there a material that behaves like that without changing phase?

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    $\begingroup$ Sounds like the Stefan problem. $\endgroup$
    – Chemomechanics
    Commented Jan 10, 2019 at 16:54
  • $\begingroup$ @Chemomechanics I'm thinking of the steady state equation. $\endgroup$
    – Ivan
    Commented Jan 10, 2019 at 16:58
  • $\begingroup$ One way to avoid the interface being "special" is to reformulate the problem using enthalpy instead of temperature. Of course the mesh needs to be fine enough so that the results are not unrealistically "smeared out" where the phase change occurs. $\endgroup$
    – alephzero
    Commented Jan 10, 2019 at 17:19
  • $\begingroup$ If you are trying to model a realistic situation with water and ice, don't forget that water transports heat by convection, as well as by conduction! $\endgroup$
    – alephzero
    Commented Jan 10, 2019 at 17:22
  • $\begingroup$ @alephzero Right. Well, about the first question... is there a material that changes conductivity but not phase? $\endgroup$
    – Ivan
    Commented Jan 10, 2019 at 17:24

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It would not be a steady state situation because the boundary would be moving. Even so, although the temperature at the interface would be continuous, the temperature gradient at the interface would not be continuous. This is because there would be a jump change in heat flux across the interface, equal to the heat of fusion times the interface velocity times density: $$\left[-k\frac{\partial T}{\partial x}\right]^--\left[-k\frac{\partial T}{\partial x}\right]^+=\rho v \lambda$$ This equation says that the heat flux into the interface minus the heat flux out of the interface would be equal to the rate of heat supply required to do the melting.

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  • $\begingroup$ I understand. Thank you. Is there a material that changes conductivity abruptly with temperature, without changing phase? $\endgroup$
    – Ivan
    Commented Jan 10, 2019 at 17:25
  • $\begingroup$ I don't know of any. But, of course, you can have two different materials joined together at a given location. $\endgroup$
    – Chet Miller
    Commented Jan 10, 2019 at 17:27
  • $\begingroup$ Of course. But then conductivity would not be a function of temperature. $\endgroup$
    – Ivan
    Commented Jan 10, 2019 at 17:29
  • $\begingroup$ Good point. Then I can't really think of a material that would behave like this. Sorry. $\endgroup$
    – Chet Miller
    Commented Jan 10, 2019 at 17:37
  • $\begingroup$ Note that temperature would likely not be continuous at the interface, although the discontinuity might be very small to measure. $\endgroup$
    – untreated_paramediensis_karnik
    Commented Jan 10, 2019 at 21:28

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