Question

Please include a clear picture showing the variables in the problem. Thanks

OO0Kg? on salbly CafTy 2. Do you like ice skating? A 0.3 m (approximately 1ft) thick sheet of ice covers a lake. The air temperature above the ice is -15 °C (5F) and the water temperature is essentially 0 °C (32F), Assume that the heat flow through the ice freezes a thin layer of water onto the bottom of the sheet. How thick a layer is created in 5 min under these circumstances? Assume the layer is thin enough that it doesn't affect the overal thickness of the sheet. You need to look up the heat conduction of ice and the heat of freezing for water, which you both find in chapter 12 of our book. You might wonder if you need to know the size of the lake. Hmm, can you argue why you do not need it! Think the following. Does a particular area of the lake know how large the lake is while it is freezing? ENG 1119PM

Answer 1

Consider a lake of area, $A$, covered by an ice sheet of thickness, $l$, forms a thin layer of ice on the bottom of the sheet of thickness, $x$, and mass, $m_x$. Let the temperature of air outside be, $T_a$, and the water be $T_w$.

Let $\rho$ be the density of water, $k$ be the thermal conductivity of the ice and $L_{ice}$ be the Latent heat of fusion of ice, Then,

Then heat transferred during the conduction process at the bottom layer is equal to the heat required to form the thin ice layer, i.e.,

$\frac{kA\Delta T}{l}t = m_xL_{ice}$

$\frac{kA(T_w-T_a)}{l}t = \rho AxL_{ice}$

$\Rightarrow x = \frac{k(T_w-T_a)}{l \times \rho L_{ice}}t$

$\\\text{Given that, } \\k = 1.6 \hspace{5pt}Wm^{-1}K^{-1} \\L_{ice} = 3.33 \times10^5 J\hspace{4pt} kg^{-1} \\\rho = 10^{3} kg\hspace{4pt} m^{-3} \\l = 0.3 m \\(T_w-T_a) = 15 K \\t = 5 \hspace{5pt}\text{min} = 300 \text{ s}$

$x \approx 7.207 \times10^{-5} m$

No particular area "knows" how large the lake is because the conduction process is perpendicular to the plane of the area of the lake.