Question

Two thin metallic spherical shells of radii `r_(1)` and `r_(2)` `(r_(1)lt r_(2))` are placed with their centres coinciding. A material of thermal conductivity `K` is filled in the space between the shells. The inner shells is maintained at temperature `theta_(1)` and the outer shell at temperature `theta_(2)` `(theta_(1)lttheta_(2))`. Calculate the rate at which heat flows radially through the material.

Answer 1

Let us draw two spherical shells of radii `x` and `x+dx` concentric with the given system. Let the temperature at these shells be `theta` and `theta+d theta` respectively. The amount of heat flowing radially inward through the material between `x` and `x+dx` is
`(DeltaQ)/(Deltat)=(K4pix^(2)d theta)/(dx)`
Thus, `K4piint_(theta_(1))^(theta_(2))d theta=(DeltaQ)/(Deltat)int_(r_(1))^(r_(2))(dx)/(x^(2))`
`K4pi(theta_(2)-theta_(1))=(DeltaQ)/(Deltat)((1)/(r_(1))-(1)/(r_(2)))`
`(DeltaQ)/(Deltat)=(4piKr_(1)r_(2)(theta_(2)-theta_(1)))/(r_(2)-r_(1))`