Is the Negative Sign Correct in Freezing Lake Fourier's Law Application?

[WWCY]

Homework Statement

Homework Equations

dQ/dt = -kA(dT/dx)

The Attempt at a Solution

I tried to use Fourier's law of Conduction on this one. I subbed dT for (Θ₀ - Θ₁), and l(t) (function for thickness of ice against time) for dx, reason being that the sheet of ice should get thicker.

I then substituted dQ = dmL_f = ρAL_fdl (reason being that the infinitesimal energy dQ lost from the water should cause it to change phase / freeze by a volume of A(area) x dl) before arriving at:

dl/dt = -k/(ρL_f) x (Θ₀ - Θ₁) / l(t) and getting stuck,

help would be greatly appreciated.

 

[Orodruin]

Exactly what about the differential equation are you stuck with? It is separable so solving it should be a simple matter of integration.

 

[Chestermiller]

I agree with Orodruin. You have already correctly completed the hard part of correctly formulating the differential equation. Solving the differential equation is supposed to be the easy part. If you can't figure out how to solve the differential equation, take their answer and differentiate it with respect to time; then compare the result with your own differential equation.

 

[WWCY]

I managed to get the final equation after some work.

However I realized that i had to change dQ = dmLf = ρALfdl to dQ = -dmLf = -ρALfdl

I rationalised that this negative sign was down to the infinitesimal volume losing energy, is this the right way to think about it?

Thank you both for your responses.