Chapter 2, Problem 2.63P

A spherical particle of radius $r_1$ experiences uniform thermalgeneration at a rate of $\stackrel{.}{q}.$ The particle is encapsulatedby a spherical shell of outside radius $r_2$ that is cooled byambient air. The thermal conductivities of the particle andshell are $k_1$ and $k_2,$ respectively, where $k_1=2k_2.$

1. By applying the conservation of energy principle tospherical control volume A, which is placed at anarbitrary location within the sphere, determine arelationship between the temperature gradientdT /dr and the local radius r, for $0\le r\le r_1.$ By applying the conservation of energy principleto spherical control volume B, which is placedat an arbitrary location within the spherical shell,determine a relationship between the temperaturegradient dT /dr and the local radius r, for $r_1\le r\le r_2.$ On $T-r$ coordinates, sketch the temperature distributionover the range $0\le r\le r_2.$