A cylinderor radius R and lengthl ismade up of a substance, whosethermalconductivityK varieswith thedistancex from theaxisas K = K_(1)x + K_(2) . Determineof the effective thermalconductivitybetweenthe flatfaces of the cylinder .

A cylinderor radius R and lengthl ismade up of a substance, whosetherm

1.	A cylinderor radius R and lengthl ismade up of a substance, whosethermalconductivityK varieswith thedistancex from theaxisas K = K_(1)x + K_(2) . Determineof the effective thermalconductivitybetweenthe flatfaces of the cylinder .
Answer» Solution :Let us subdividel the entirecylinderinto a numberof COAXIAL cylindricalshells of infinitensimally smallthickness DX . Cross sectionalarea 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))`