Question

A freezer compartment is covered with a 2-mm-thick layer of frost at the time it malfunctions. If the compartment is in ambient air at 20 °C and a coefficient of h 2 W/(m2 K) characterizes heat transfer by natural convection from the exposed surface of the layer, estimate the time required to completely melt the frost. The frost may be assumed to have a mass density of 700 kg/m3 and a latent heat of 334 kJ/kg.

Answer 1

Answer:

The time required to melt the frost is 3.25 hours.

Explanation:

The time required to melt the frost dependes on the latent heat of the frost and the amount of heat it is transfered by convection to the air .

The heat transferred per unit area can be expressed as:

$q=h_c*A*\Delta T\\\\q/A=h_c*\Delta T$

being hc the convective heat transfer coefficient (2 Wm^-2K^-1) and ΔT the difference of temperature (20-0=20 °C or K).

$q/A=h_c*\Delta T=2\frac{W}{m^2K}*20K=40\frac{W}{m^2}$

If we take 1 m^2 of ice, with 2 mm of thickness, we have this volume

$V=T*A = 0.002 m * 1 m^2=0.002m^3$

The mass of the frost can be estimated as

$M=\rho * V=700\frac{kg}{m^3}*0.002m^3= 1.4 kg$

Then,  the amount of heat needed to melt this surface (1 m²) of frost is

$Q=L*M=334\frac{kJ}{kg}*1.4kg= 467.6kJ$

The time needed to melt the frost can be calculated as

$t=\frac{Q}{(q/A)}=\frac{467.6kJ/m2}{40W/m2} = 11.69\frac{kJ}{W}*\frac{1W*s}{1J}*\frac{1000J}{1kJ}= 11690s=3.25h$