My thermodynamics teacher solved one exercise but I'm not convinced of his solution. The exercise asks this:
A lake is covered by an ice sheet of thickness $D$ given. Ice latent fusion heat $\lambda$, ice density $\rho$ and thermal conductivity coefficient $\kappa$ are also given. The external air temperature remains constant at $T_e=0° C$, the ice sheet melts in a certain time $\Delta t$. The problem requires to find the temperature of the lake water $T_l$. The water temperature and the air temperature are kept constant in space and time as boundary conditions, they are temperature sources.
This is my teacher's solution: $\delta Q$ required to melt a $dx$ thick quantity of ice is: $$\delta Q= \lambda \rho A dx$$ where A is the area of the lake. The heat pass through the ice by conduction so in steady state holds: $$\delta Q = - \kappa A \frac{T_l-T_e}{x} dt$$ Equating the two $\delta Q$ expressions and integrating: $$\int_D^0 xdx =\int_0^{\Delta t} \kappa \frac{T_l-T_e}{\lambda \rho}dt$$ So plugging in the numerical values: $$T_l=T_c+\frac{D^2\lambda\rho}{2\kappa\Delta t}=3.44°C$$
What doesn't convince me of this solution is that the heat, instead of passing through the ice sheet to the air, contributes to melt the nearer to the water portion of the ice sheet and stops there, no part of the heat reaches the air. So how can we talk about heat conduction though the entire ice sheet to the air ad use the above formulas? And moreover if there was a steady state heat conduction through the ice sheet this would require the existence of gradient of temperature in the entire ice sheet from $0$ to $3.44°C$, but this absurd because this is over the ice fusion temperature. And in the end if the heat from the water just melts the nearer ice how can the melting speed be influenced by the thickness of the above ice?