Q13 of 67 Page 98

Figure shows water in a container having 2.0 mm thick walls made of a material of thermal conductivity 0.50 W m–1 °C–1. The container is kept in a melting ice bath at 0°C. The total surface area in contact with water is 0.05 m2. A wheel is clamped inside the water and is coupled to a block of mass M as shown in the figure. As the block goes down, the wheel rotates. It is found that after some time a steady state is reached in which the block goes down with a constant speed of 10 cm s–1 and the temperature of the water remains constant at 1.0°C. Find the mass M of the block. Assume that the heat flows out of the water only through the walls in contact. Take g = 10 m s–2.


Given:
Thickness of the container: x = 2 mm = 0.002 m
Thermal conductivity of the container: K= 0.50 W m–1 °C–1.
Temperature of the water: T1 = 1 °C
Temperature of the ice bath: T2 = 0 °C
Surface are in contact with the water: 0.05 m2
Speed of the block: v = 10 cm/s = 0.1 m/s
g = 10 m/s2
Formula used:
Rate of amount of heat flowing or heat current is given as:

Here, Δθ is the amount of heat transferred, ΔT is the temperature difference, K is the thermal conductivity of the material, A is the area of cross section of the material and x is the thickness or length of the material.
The effect of the block going down and the heat transfer has one identity in common, which is power.
Power due to a block of mass M moving with constant velocity is given as


Here W is the work done by the block.
 ∴ P = (mg).v
 ∴ P = M× 10× 0.1
 ∴ P = M× 1
Also in terms of Heat Energy: P= Energy per unit time




Hence, the mass of the block is 12.5 kg.


More from this chapter

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11

The ends of a metre stick are maintained at 100°C and 0°C. One end of a rod is maintained at 25°C. Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?

 12

A cubical box of volume 216 cm3 is made up to 0.1 cm thick wood. The inside is heated electrically by a 100 W heater. It is found that the temperature difference between the inside and the outside surface is 5°C in steady state. Assuming that the entire electrical energy spent appears as heat, find the thermal conductivity of the material of the box.

 14

On a winter day when the atmospheric temperature drops to –10°C, ice forms on the surface of a lake.

(a) Calculate the rate of increase of thickness of the ice when 10 cm of ice is already formed.


(b) Calculate the total time taken in forming 10 cm of ice. Assume that the temperature of the entire water reaches 0°C before the ice starts forming. Density of water = 1000 kg m–3, latent heat of fusion of ice = 3.36 × 105 J kg–1 and thermal conductivity of ice = 1.7 W m–1 °C–1. Neglect the expansion of water on freezing.


 15

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 Wm–1 °C–1. Take other relevant data from the previous problem.