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A cylinder of radius R and length l is m...

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.

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Let us subdividel the entire cylinder into a number of coaxial cylindrical shells of infinitensimally small thickness dx . Cross sectional area of the shell ` 2pi x dx `.
`K_("eff") = (sumA_(i)K_(i))/(sumA_(i)) ` we have `K_("eff") = (1)/(sum A_(i)) int_(0)^(R) K (2 pi x dx)`
` = (1)/(piR^(2)) int_(0)^(R) (k_(1)x + k_(2))2pix dx = (2)/(R^(2)) int_(0)^(R) (k_(1)x^(2) + K_(2)x)dx = (2)/(R^(2)) [K_(1).(x^(3))/(3) + K_(2) .(x^(2))/(2)]_(0)^(R) = (1)/(3) (2K_(1) R + 3K_(2))`
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