Question

a layer of ice 0⁰C of the thickness x1 is floating on a pond water. L, p and k respectively are the latent heat if fusion of the water , density of the ice and thermal conductivity of ice. If the atmospheric temperature is _T°C the time taken for the thickness of the layer of ice to you increase from X1 to X2 is given by​

Answer 1

Given:

Temperature of ice = 0°C

Thickness of ice = x1

Latent heat of fusion = L

Density of ice = ρ

Thermal  conductivity of ice = k

Atmospheric temperature = -T°C

To find:

Time taken for the thickness of the layer of ice to increase from x1 to x2.

Solution:

As we know that rate of heat flow:

Q/t = k*A*Δt/ Δx

t = time taken

Δt -= temperature difference

Where Δx, thickness of ice formed is the mean distance= (x2-x1)/2

We know that mass of ice = ρ*A*(x2-x1)

Latent heat of fusion = Q/ mass

Q = mass* latent heat of fusion = ρ*A*(x2- x1)*L

Putting the value of Q in the equation we get:

ρ*A*(x2 - x1)*L/ t = 2* k*A*(0-(-T))/ (x2- x1)

t = ρ(x2 - x1)²*L/ 2k*T

Therefore the time taken will be ρL(x2-x1)²/ 2kT.