Find the heat current through the frustum of a cone shown in figure-4.20. Temperature of its two ends are maintained at `T_(1)` and `T_(2)`(T_(2)>T_(1))` respectively and the thermal conductivity of the material is k.

Explanation

Here `r=r_(1)+((r_(2)-r_(1))/(L))x`
As shown in figure we consider an elemental disc at a distance x from left face. Then thermal resistance ofthis elemental disc is given as
`dR_(th)=(1)/(k).(dx)/(pi(r_(1)+(r_(2)-r_(1))/(L)x)^(2))`
Total thermal resistance offrustum is
`R_(th)=intdR_(th)=(1)/(kpi)int_(0)^(1)(dx)/(r_(1)+(r_(2)-r_(1))/(L).x)^(2)`
`=(L)/(kpi(r_(2)-r_(1)))[(1)/((r_(1)+(r_(2)-r_(1))/(L).x))]_(0)^(l)`

`=(L)/(kpi(r_(2)-r_(1)))[(1)/(r_(1))-(1)/(r_(2))]`
`=(L)/(kpi_(1)r_(2))`
Thus heat current through frustum is
`(dQ)/(dt)=(T_(2)-T_(1))/R_(th)=(kpi_(1)r_(2)(T_(2)-T_(1)))/(L)`