In a 10 metre deep lake- the bottom is at a constant temperature of 4-C- The air temperature is constant at -4-C- The thermal conductivity of ice is 3 times that water- Neglecting the expansion of water on freezing- the maximum thickness of ice will be

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Standard IX

Physics

Thrust

In a 10 met...

Question

In a $10$ metre deep lake, the bottom is at a constant temperature of $4^{\circ} C$ . The air temperature is constant at $- 4^{\circ} C$ . The thermal conductivity of ice is $3$ times that water. Neglecting the expansion of water on freezing, the maximum thickness of ice will be

7.5 m

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6 m

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5 m

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2.5 m

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Solution

The correct option is A $7.5 m$
The rate of heat flow is the same through water and ice in the steady state.
$I = k A \frac{(4 - 0)}{10 - x} = 3 K a \frac{0 - (- 4)}{x}$
$x = (10 - x) 3$
or $x = 7.5 m$ .
Hence (A) is correct.

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Q. Consider the situation of the previous problem. Assume that the temperature of the water at the bottom of the lake remains constant at 4°C as the ice forms on the surface (the heat required to maintain the temperature of the bottom layer may come from the bed of the lake). The depth of the lake is 1.0 m. Show that the thickness of the ice formed attains a steady state maximum value. Find this value. The thermal conductivity of water = 0.50 W m⁻¹°C⁻¹. Take other relevant data from the previous problem.

Water is densest at 4 $^{\circ}$ C; this is the reason lakes do not completely freeze during extreme winters. Say, the atmosphere is at - $θ^{\circ} C (θ > 0)$ ,now the temperature of the lake's water starts to fall, and the denser water from the top sinks, getting the bottom layers up, and cooling them until the temperature reaches 4 $^{\circ}$ C at the upper surface. Now, further reduction in temperature actually makes the water less dense! Hence the colder water stays on top, starts to freeze, until the top layer of ice reaches - $θ^{\circ} C$ and the bottom layer at 0 $^{\circ}$ C . And this layer of ice is the only area through which the water below loses heat (by conduction), slowly increasing the thickness of the ice (because ice is a bad conduction), and for a very long time the water at the bottom of the lake remains at 4 $^{\circ}$ C! Now, Let the thickness of the layer of ice be $y_{1}$ , at an instant.How much time t, will it take to increase to thickness $y_{2}$ ?

$ρ \to$ density of the ice

K $\to$ thermal conductivity of ice.

L $\to$ Latent heat of fusion

$θ \to$ temperature of surface (as mentioned earlier)

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In a 10 metre deep lake, the bottom is at a constant temperature of 4∘C. The air temperature is constant at −4∘C. The thermal conductivity of ice is 3 times that water. Neglecting the expansion of water on freezing, the maximum thickness of ice will be

The correct option is A 7.5 mThe rate of heat flow is the same through water and ice in the steady state.I=kA(4−0)10−x=3Ka0−(−4)xx=(10−x)3or x=7.5 m.Hence (A) is correct.

In a $10$ metre deep lake, the bottom is at a constant temperature of $4^{\circ} C$ . The air temperature is constant at $- 4^{\circ} C$ . The thermal conductivity of ice is $3$ times that water. Neglecting the expansion of water on freezing, the maximum thickness of ice will be

The correct option is A $7.5 m$
The rate of heat flow is the same through water and ice in the steady state.
$I = k A \frac{(4 - 0)}{10 - x} = 3 K a \frac{0 - (- 4)}{x}$
$x = (10 - x) 3$
or $x = 7.5 m$ .
Hence (A) is correct.