(a) When the air temperature is below 0^, the water at the surface of a lake freezes to form an

(a) When the air temperature is below 0^, the water at the...

(a) When the air temperature is below 0$^\circ$C, the water at the surface of a lake freezes to form an ice sheet. Why doesn't freezing occur throughout the entire volume of the lake? (b) Show that the thickness of the ice sheet formed on the surface of a lake is proportional to the square root of the time if the heat of fusion of the water freezing on the underside of the ice sheet is conducted through the sheet. (c) Assuming that the upper surface of the ice sheet is at -10$^\circ$C and the bottom surface is at 0$^\circ$C, calculate the time it will take to form an ice sheet 25 cm thick. (d) If the lake in part (c) is uniformly 40 m deep, how long would it take to freeze all the water in the lake? Is this likely to occur?

Key Concepts

Conduction Heat Transfer

This concept refers to the process by which heat energy is transmitted through a material as a result of a temperature gradient. In the context of freezing lakes, heat is conducted from the warmer water below up through the growing ice sheet to the colder air above, governing the rate at which the ice thickens. Understanding conduction is essential for modeling the heat removal that drives the phase change from water to ice.

Latent Heat and Phase Change

Latent heat of fusion is the amount of energy that must be removed from a substance to change its phase from liquid to solid without changing its temperature. In the freezing of a lake, the removal of this latent heat from the water at the ice-water interface is what enables the phase change to occur. Recognizing the role of latent heat is crucial for setting up the energy balance during the freezing process.

Stefan Problem

This is a class of problems dealing with phase change, where a moving boundary (such as the growing ice-water interface) is tracked over time as heat is conducted through both phases. The Stefan problem provides the mathematical framework for deriving relationships like the thickness of the ice being proportional to the square root of time, accounting for both the conduction through the ice and the latent heat removal at the interface.

Water Density Anomaly

Unlike most substances, water is most dense at around 4°C, and as it cools further, it becomes less dense. This anomaly causes ice to form on the surface rather than at the bottom of a lake. The ice remains at the top, effectively insulating the water below, which is a critical factor in understanding why complete freezing of a lake does not occur simply because of low air temperatures.

Transcript

00:01 So this question starts asking why doesn't freezing occur through the entire volume of a lake? so in this situation, the air above the lake is below zero, and we need that out lake to be at zero degrees and had enough energy to transform into ice.

00:23 And because heat must be conducted from the water to cool it to zero degrees and causes a phase transition, the entire volume of the water is not at phase transition temperature, so only the top part here can transform into ice.

00:45 And that's why the whole body of water doesn't transform, because you can't properly.

00:54 To propagate enough energy through the ice to make this part of the water, to lose enough energy to pass the phase transition becomes ice.

01:07 Now, in part b, they ask you to derive a relation between the thickness of the ice and the time it takes to form this ice.

01:23 So let's walk through how we can do that.

01:29 So the heat that must leave the water in order to freeze must pass through this block of ice.

01:40 So the larger this block is, the harder the heat will have to, the longer the heat will have to travel.

01:49 And worse this process will become.

01:52 So let's consider a section of the ice that has a area, cross -sectional area a.

02:02 So this here will be the cross -section area a.

02:08 And at time t, this has a height of h.

02:19 So it has a function of t.

02:23 In an infinitesimal interval, so d t, we have infinitesimal thickness, that is dh, and the mass of this infinitesimal block will be dm is equal to row dv.

02:49 We assume this density to be constant, so this is row, the cross -section area is the same, the only thing that grows is this th here.

03:03 So this is a relation between the how much mass we have in a infinite small block here with the hive of the block.

03:12 This is just a cube.

03:16 Now the heat must be conducted away from this mass of water to freeze it.

03:22 So dq is equal to dm lf and and we have this infinitesimal mass.

03:35 Lf is the latent heat of the water that is necessary to transform a water to ice.

03:47 So this gives rho a, lf, d, h.

03:54 So here we have a relation between the heat and the height.

04:00 Now, we know how to write the change of heat...