A cylinder of radius R and length l is made up of substance whose thermal conductivity K varies with the distance x from the axis as `K=K_1x+K_2`. Determine the the effective thermal conductivity between the flat faces of the cylinder.

Explanation

Let us subdivide the entire cylinder into a number of coaxial cylindrical shells of infinitesimally small thickness. Consider one such shell of radius x and thickness dx. Cross-sectional area of the shell `2pixdx`.
Using, the expression for the effective thermal conductivity.
`K_("eff")=(sumA_iK_i)/(sumA_i)`
We have `K_("eff")=(1)/(sumA_i)int_0^RK(2pixdx)`
`=(1)/(piR^2)int_0^R(K_1x+K_2)2pixdx`
`(2)/(R^2)int_0^R(K_1x^2+K_2x)dx`
`=(2)/(R^2)[K_1(x^3)/(3)+K_2(x^2)/(2)]_0^R=(1)/(3)(2K_1R+3K_2)`.