Question

In winter, a lake of water is covered initially by a
uniform layer of ice of thickness 1 cm. The air temperature at the surface of the ice is −0.5degree C. Estimate the rate at which the layer of ice begins to thicken, assuming that the temperature of the water just below the ice is 0 degree C. You can also assume steady-state conditions and ignore convection. The temperature of the water at the bottom of the lake, depth 1 m, is maintained at 2degree C. Find
the thickness of ice that will eventually be formed. The thermal conductivity of ice is 2.3Wm^-1K^-1 and of water is 0.56Wm^-1K^-1.

Answer 1

Brrrrrrr !
@IrishBoy Just scanned this one .. looks pretty juicy

Answer 2

if i were to have a crack at this, i'd start here and look at the start position and also the ultimate end steady state when \(\dfrac{Q}{t} = \dot Q\) is the same from the bottom of the pond all the way to the outside air......for a volume element of unit cross section area, as there's a ton of symmetry on here.



So just like modelling heat flow on 2 solid, connected rods.

In terms of the time calculation, i think we should look at the latent heat for ice forming.
The idea would be to say that

\(\dot Q = \dot m L\)

As is explained here....



such that \(\dot m = \dot x \cdot 1 = \dot x\) as we are using a unit x-sect area in our little physical model

This resource looks a good refresher....

https://en.wikipedia.org/wiki/Latent_heat

you could go at this with the continuity equation and Fourier's heat law but that's just nuts, IMHO. this approach compresses it into 1-D and it looks linear and do-able, ....though i haven't tried it myself :-|

Answer 3

ooops

\(\dot m =\color{red}{\rho} ( \dot x \cdot 1) = \color{red}{\rho} \dot x\), maybe assuming constant density as it's an estimate...

:$