Heat is generated uniformly in a long solid cylindrical rod ( diameter $$ = 10\,\,mm$$) at the rate of $$4 \times {10^7}\,\,W/{m^3}.$$ The thermal conductivity of the rod material is $$25$$ $$W/m.K.$$ Under steady state conditions, the temperature difference between the center and the surface of the rod is ________________ $${}^ \circ C.$$

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 brick wall $$\,\left( {k = 0.9{W \over {m.K}}} \right)$$ of thickness $$0.18$$ $$m$$ separates the warm air in a room from the cold ambient air. On a particular winter day, the outside air temperature is −$${5^ \circ }C.\,$$ and the room needs to be maintained at $${27^ \circ }C.\,$$ The heat transfer coefficient associated with outside air is $$20\,\,W/{m^2}K.$$ Neglecting the convective resistance of the air inside the room, the heat loss, in $$\left( {{W \over {{m^2}}}} \right)$$ is