In the laminar flow of air $$(Pr=0.7)$$ over a heated plate, if $$\delta $$ and $${\delta _T}$$ denote, respectively the hydrodynamic and thermal boundary layer thickness, then

The total emissive power of a surface is $$500$$ $$W/{m^2}$$ at a temperature $${T_1}$$ and $$1200$$ $$W/{m^2}$$ at a temperature $${T_2}$$, where the temperatures are in Kelvin. Assuming the emissivity of the surface to be constant, the ratio of the temperatures $${{{T_1}} \over {{T_2}}}$$ is

A balanced counter-flow heat exchanger has a surface area of $$20\,\,{m^2}$$ and overall heat transfer coefficient of $$20$$ $$W/{m^2}$$-$$K$$. Air $$\left( {{C_p} = 1000J/kg - K} \right)$$ entering at $$0.4$$ $$kg/s$$ and $$280$$ $$K$$ is to preheated by the air leaving the system at $$0.4$$ $$kg/s$$ and $$300$$ $$K.$$ The outlet temperature (in $$K$$) of the preheated air is

If a foam insulation is added to a $$4cm$$ outer diameter pipe as shown as figure, the critical radius of insulation (in $$cm$$) is ________.