Q16 of 67 Page 98

Three rods of lengths 20 cm each and area of cross-section 1 cm2 are joined to form a triangle ABC. The conductivities of the rods are KAB = 50 J s–1 m–1 °C–1, KBC = 200 J s–1 m–1 °C–1 and KAC = 400 J s–1 m–1 °C–1. The junctions A, B and C are maintained at 40°C, 80°C and 80°C respectively. Find the rate of heat flowing through the rods AB, AC and BC.


Given:
Length of all the rods: x= AB=BC=AC = 20 cm = 0.2 m
Area of cross section of these rods: A = 1 cm2 = 0.0001 m2.
Thermal conductivity of rod AB : KAB = 50 J s–1 m–1 °C–1
Thermal conductivity of rod BC : KBC = 200 J s–1 m–1 °C–1
Thermal conductivity of rod AC : KAC = 400 J s–1 m–1 °C–1
Temperature at A : T1 = 40 °C
Temperature at B : T2 = 80 °C
Temperature at C : T3 = 80 °C
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.
(1)
Rate of heat flowing in the rod AB is



Hence, rate of heat flowing through the rod Ab is 1 J/s or 1 W
(2)
Rate of heat flowing in the rod BC is



Hence, the rate of heat flowing through the rod BC is 0 as both the ends of the rods are maintained at same temperature.
(3)
Rate of heat flowing in the rod BC is



Hence, the rate of heat flowing through the rod AC is 8 J/s.


More from this chapter

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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.

 17

A semicircular rod is joined at its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Fid the ratio of the heat transferred through a cross-section of the semicircular rod to the heat transferred through a cross-section of the straight rod in a given time.

 18

A metal rod of cross-sectional area 1.0 cm3 is being heated at one end. At one time, the temperature gradient is 5.0°C cm–1 at cross-section A and is 2.5°C cm–3 at cross-section B. calculate the rate at which the temperature is increasing in the part AB of the rod. The heat capacity of the part AB = 0.40 J°C–1, thermal conductivity of the material of the rod = 200 W m–1 °C–1. Neglect any loss of heat to the atmosphere.