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I have a question on melting ice using solar energy. Assuming solar energy transfer in ice obeys Beer's law, i.e.

$$R(z)=R_0e^{-z/d}$$ then the intensity could be expressed by

$$I(z)=\frac{dR}{dz}$$

Assume ice at location $z$ with thickness $dz$, then the solar energyreceived by this volume over time $t$ could be expressed by

$$\int_{z}^{z+dz}I(z) dz \Delta t$$

to melt ice of this thickness $dz$, the solar energy needed is

$$\lambda \rho dz$$

where $\lambda$ is latent heat of fushion per unit mass and $\rho$ is density of ice, then to see how long is needed to melt ice of thickness $dz$, solve

$$\int_{z}^{z+dz}I(z) dz \Delta t=\lambda \rho dz$$ $$ \Delta t=\frac{\lambda \rho dz}{\int_{z}^{z+dz}I(z) dz}$$ if we take $dz \rightarrow 0$, i.e. trying to find the limit of the right hand side then we'll notice that $\Delta t$ is not $0$ (You can check this).

This seem strange to me since it means that if $t < \Delta t$ then the ice of thickness $dz$ receives energy however no melt happens since it needs at least time $\Delta t $ to melt. So where does the energy go before time $\Delta t$? Any help is appreciated.

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    $\begingroup$ You used latent heat in your computations as if you were familiar with it., but, correct me if wrong, it seems that you are asking about latent heat itself. Latent heat is the amount of energy needed in a first order phase transition. During the phase transition there is no temperature change. That energy goes on breaking molecular bonds. $\endgroup$
    – SomeUser
    Jun 9 at 2:37
  • $\begingroup$ What is R in your first equation? $\endgroup$
    – nasu
    Jun 9 at 3:51
  • $\begingroup$ @MarcoCiafa Yes during melting temperature of ice remains $0$ degree and if ice of $0$ degree receives solar energy it must be used to melt ice, but as I stated in the question when $dz \rightarrow 0$ the latent heat that used to melt it is not $0$ and when it received energy less then this amount, nothing melts and I'm wondering what this part of energy goes. $\endgroup$
    – Maskoff
    Jun 9 at 15:52
  • $\begingroup$ @nasu R_0 is solar energy flux at surface of ice, $W/m^2$ $\endgroup$
    – Maskoff
    Jun 9 at 15:52
  • $\begingroup$ I don't see why you are integrating over $dz$ if intensity is per unit of area and $dz$ it's just the thikness. I would integrate $I$ wrt $dS$, write the energy needed for melting the ice as $\lambda \epsilon S$ (where $\epsilon$ is the thickness) and, then, take the limit, $S \to 0$. That's just formalism. It will still give a finite $\Delta t$, becuase that's how first order transitions work. $\endgroup$
    – SomeUser
    Jun 9 at 17:19

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