Question

How long will it take to form a thickness of 10 cm of ice on the surface of a lake when the air temperature is -15°C? The thermal conductivity K of ice is 4 x 10^-3 cal/s*cm*°C and its density is 0.92 g/cm^3

Answer 1

Thickness of ice= 10 cm

air temperature = -15°C

thermal conductivity K of ice= 4 x 10^-3 cal/s.cm.^oC

density= 0.92 g/cm³

To solve this problem let us represent the thickness of the ice layer at any instant be represented by y.

When we transfer a small amount of heat to the body the next layer of thickness will form and let us represent the thickness size as,dy water to air through the ice at dt seconds.

So the heat lost by the freezing water, is

dQ = A dy L

where A=area of ice formed,

=density of ice,

L=latent heat of water.

So will get the heat transmitted to the layer of ice already formed,

dQ = [−K A (T2 – T1)dt]/y

where T1 and T2 are the temperatures of the water (near the ice) and air, respectively and K is a constant.

A dy L = {[−K A [(T2 – T1)]/y}dt

or

dt = [L/{k(T2 – T1)}]ydy

Integrating the above equation from y = 0 to y = 10 cm, we get

to integratingdt = [L/{k(T2 – T1)] 10 int ydy = [L/{k(T2 – T1)}][(1/2)(y2)] ¹⁰ocm

where t = time it takes for the ice to grow 10 cm thick.

t = [{−L×20 cm²}/{k(T2 – T1)}]

= (20 cm² * 0.92 gr/cm³ * 80 cal/gr)/(4 * 10⁻³ cal/s ^oC deg∙cm * -15 C deg)

= 24.53 × 10³ s

= 408.88 min

= 6 hr 48 min.