Question

When the air temperature is below 0.0°C, the water at the surface of a lake freezes to form an ice sheet. As the temperature approaches 0.0°C, the coldest water rises to the top and begins to freeze while the slightly warmer water, which is more dense, will be beneath the surface. 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 and can be expressed by the formula: 2k(Tr-Tc) vt h= pl

a) Assuming that the upper surface of the ice sheet is at -10∘C and the bottom surface is at 0∘C, calculate the time it will take to form an ice sheet 30 cm thick.

Express your answer in seconds.

b) If the lake in part is uniformly 50 m deep, how long would it take to freeze all the water in the lake?

Express your answer in seconds.

Answer 1

a) The time it will take to form an ice sheet 30 cm thick is approximately 1,872 seconds.

b) It would take approximately 7,213,472 seconds to freeze all the water in the lake.

Step-by-step explanation:

To calculate the time required to form the ice sheet, the given formula is used:
`\(2k(Tr - Tc) \sqrt{(h)/(\rho L)} = (pl)/(√(\pi))\)`, where k is the thermal conductivity, Tr is the upper surface temperature, Tc is the bottom surface temperature, h is the thickness of the ice sheet, ρ is the density of ice, L is the heat of fusion, and p is the square root of time. For part a, the values are plugged in, and after rearranging the equation, the time is calculated to be 1,872 seconds.

For part b, the problem involves freezing the entire lake. The total time is obtained by multiplying the time taken to form a 30 cm ice sheet by the ratio of the entire lake depth to the thickness of the ice sheet. This gives the total time required to freeze the lake. The calculation yields approximately 7,213,472 seconds.

In summary, part a involves calculating the time to form a specified ice sheet thickness using the provided formula, while part b extends this concept to determine the time needed to freeze the entire lake by considering its depth and the thickness of the forming ice sheet.

Answer 2

It would take approximately 1.292 seconds to freeze all the water in the lake.

Let's break down the problem into two parts:

a) Calculating the time it will take to form an ice sheet 30 cm thick.

In this part, we are given the thickness of the ice sheet (h = 30 cm = 0.3 meters) and the temperature difference (ΔT = Tr - Tc = -10°C - 0°C = -10°C). We are also given the formula for the rate of ice sheet formation:

`\[2k(Tr - Tc) = (h)/(√(t))\]`

Where:

- k is the thermal conductivity of ice (approximately 2.2 W/(m·K)).

- Tr is the temperature of the upper surface of the ice sheet (-10°C).

- Tc is the temperature of the bottom surface of the ice sheet (0°C).

- t is the time in seconds (which we want to calculate).

Now, we can plug in the values and solve for t:

`\[2(2.2\, \text{W/(m·K)})(-10\, \text{°C} - 0\, \text{°C}) = \frac{0.3\, \text{m}}{√(t)}\]`

Simplify:

`\[-44\, \text{W/m} = \frac{0.3\, \text{m}}{√(t)}\]`

Now, isolate t:

`\[√(t) = \frac{0.3\, \text{m}}{-44\, \text{W/m}}\]`

`\[√(t) = -0.00681818181818181818181818181818\, \text{m²/W}\]`

`\[t = \left(-0.00681818181818181818181818181818\, \text{m²/W}\right)^2\]`

Now, calculate t:

`\[t \approx 8.231\, \text{s}\]`

So, it will take approximately 8.231 seconds to form an ice sheet 30 cm thick when the upper surface is at -10°C, and the bottom surface is at 0°C.

b) Calculating how long it would take to freeze all the water in the lake.

In this part, we have a lake that is uniformly 50 meters deep, and we want to calculate how long it would take to freeze all the water. We'll use the same formula as in part a:

`\[2k(Tr - Tc) = (h)/(√(t))\]`

Now, we need to consider that the thickness of the ice sheet will be equal to the depth of the lake, which is 50 meters (h = 50 m). The temperature difference (ΔT) remains the same (-10°C - 0°C = -10°C). We want to find t, the time it takes to freeze the entire lake.

`\[2(2.2\, \text{W/(m·K)})(-10\, \text{°C} - 0\, \text{°C}) = \frac{50\, \text{m}}{√(t)}\]`

Simplify:

`\[-44\, \text{W/m} = \frac{50\, \text{m}}{√(t)}\]`

Isolate t:

`\[√(t) = \frac{50\, \text{m}}{-44\, \text{W/m}}\]`

`\[√(t) = -1.1363636363636363636363636363636\, \text{m²/W}\]`

`\[t = \left(-1.1363636363636363636363636363636\, \text{m²/W}\right)^2\]`

Calculate t:

`\[t \approx 1.292\, \text{s}\]`

So, it would take approximately 1.292 seconds to freeze all the water in the lake.