Question

A cylinder of radius `R` and length l is made uo of a substnace whose thermal conductivity `K` varies with the distance x from the axis as `K=K_(1)x+K_(2)` Detemine the effective thermal conductivity between the flat faces of the cylinder
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Answer 1

Let us subdivide the entire cylinder into a number of coaxial cylindrical shells of infinitesimally small thickness dx Cross sectional area of the shell is `2pi x (dx)` Using the experession for the effective thermal conductivity
`K_(eff) =(sumA_(i)K_(i))/(sumA_(i)),K_(eff)=(1)/(sumA_(i))underset(0)overset(R)intK(2pixdx)`
`=(1)/(piR^(2)) underset(0)overset(R)int(K_(1)x+K_(2))2pix(dx)`
`=(2)/(R^(2)) underset(0)overset(R)int(K_(1)x^(2)+K_(2)x)dx=(1)/(3)(2K_(1)R+3K_(2))` .