- #1
unscientific
- 1,734
- 13
Homework Statement
Part (a): Derive Clausius-Clapeyron Equation. Find latent heat of fusion of ice.
Part (b): Find rate of formation of ice
Part (c): What is the maximum thickness of ice formed?
Homework Equations
The Attempt at a Solution
Part (a)
I have derived the relation. Using the values, I found that the latent heat of fusion of ice is ##L = 3.44 \times 10^5 J kg^{-1}## which seems right.
Part (b)
Using heat flux ##J = k (\frac{\partial T}{\partial z})##, consider in time ##\delta t## amount of ice ##\Delta m## is formed.
[tex]\Delta Q = (\Delta m)L = (JA) \Delta t[/tex]
[tex] \rho (\Delta z)L = \frac{k(T_2-T_1)}{z} \Delta t[/tex]
[tex]\frac{dz}{dt} = \frac{k(T_2-T_1)}{\rho L z}[/tex]
Taking ##k = 2.3##, ##T_2-T_1 = 0.5##, ##\rho = 1000##, ##L = 3.44 \times 10^5##:
I find that ##\frac{dz}{dt} = 3.34 \times 10^{-7} m s^{-1}##, so ice forms awfully slow.
Part (c)
In steady state, ## \frac{dQ_1}{dt} = \frac{dQ_2}{dt}##.
Therefore, we equate the heat fluxes:
[tex]k_{water} \left(\frac{2}{1-z_f}\right)A = k_{ice}\frac{0.5}{z_f}A[/tex]
[tex]z_f = \frac{k_{ice}}{4k_{water} + k_{ice}} = 0.51 m [/tex]
It's hard to believe that over half the lake would be frozen, when the rate of formation of ice above (without the interference of the bottom of the lake) is merely ##3.34 \times 10^{-7} m s^{-1}##.