Find the rate of heat flow through a cross section of the rod shown in figure `(theta_(2)gttheta_(1))` . Thermal conductivity of the material of the rod is `K`.

Answer

`(KpiR_(1)R_(2)(T_(H)-T_(C)))/(L)`

Explanation

`i_(H) = (T_(H) -T_(C))/(R_(eq))`
`dR = (dx)/(KpiR^(2)) tan theta = (R-R_(1))/(x) = (R_(2)-R_(1))/(L)`

`rArr R = R_(1) +x ((R_(2)-R_(1))/(L))`
`R_(eq)=int dR = int (dx)/(kpi[R_(1)+(x(R_(2)-R_(1)))/(L)]^(2))`
`rArr R_(eq) = (L)/(kpiR_(1)R_(2))`
`i_(H) = (T_(H)-T_(C))/(R_(eq)) = (KpiR_(1)R_(2)(T_(H)-T_(C)))/(L)`