A cylindrical uranium fuel rod of radius 5 mm in a nuclear reactor is generating heat at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The rod is cooled by a liquid (convective heat transfer coefficient $$1000W/{m^2}$$-$$K$$) at $${25^ \circ }C$$. At steady state, the surface temperature (in $$K$$) of the rod is

A $$10$$ $$mm$$ diameter electrical conductor is covered by an insulation of $$2$$ $$mm$$ thickness. The conductivity of the insulation is $$0.08$$ $$W/m$$-$$K$$ and the convection coefficient at the insulation surface is $$10$$ $$W/{m^2}$$ -$$K.$$ Addition of further insulation of the same material will

A plane wall has a thermal conductivity of $$1.15$$ $$W/m.K.$$ If the inner surface is at $${1100^ \circ }C$$ and the outer surface is at $${350^ \circ }C$$, then the design thickness (in meter) of the wall to maintain a steady heat flux of $$2500\,\,W/{m^2}$$ should be ____________.

A material $$P$$ of thickness $$1$$ $$mm$$ is sandwiched between two steel slabs, as shown in the figure below. $$A$$ heat flux $$10\,kW/{m^2}$$ is supplied to one of the steel slabs as shown. The boundary temperatures of the slabs are indicated in the figure. Assume thermal conductivity of this steel is $$10W/m.K.$$. Considering one-dimensional steady state heat conduction for the configuration, the thermal conductivity ($$k$$, in $$W/m.K$$) of material $$P$$ is _____________