Q35 of 67 Page 98

A hollow metallic sphere of radius 20 cm surrounds a concentric metallic sphere of radius 5 cm. The space between the two spheres in filled with a nonmetallic material. The inner and outer spheres are maintained at 50°C and 10°C respectively and it is found that 100 J of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.

Let’s redraw the circuit –


w17.PNG


Let-
Radius of the inner sphere =a
Radius of the outer sphere = b


Given –


a = r1 = 5cm = 0.05m


b = r2 = 20cm = 0.2m


θ1 = T1 = 50°C


θ2 = T2 = 10°



Consider an imaginary shell of radii r and thickness dr.


Area,


Now, rate of flow of heat –


=-


= is change in temperature.


A= Area of cross section of the tube


K = thermal conductivity of the tube


dr = change in length


Here, the negative sign is for decrease in temperature with increase in radius.


q = -


Taking integral on both sides –


=


Solving above integral –


q= = k( ) = 100 (given)


Substituting the values,


K = = 2.8 = 3 W m-1°C-1


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A rod of negligible heat capacity has length 20 cm, area of cross-section 1.0 cm2 and thermal conductivity 200 W m–1 °C–1. The temperature of one end is maintained at 0°C and that of the other end is slowly and linearly varied from 0°C to 60°C in 10 minutes. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these 10 minutes.

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