A steel ball of diameter $$60$$ $$mm$$ is initially in thermal equilibrium at $${1030^ \circ }C$$ in a furnace. It is suddenly removed from the furnace and cooled in ambient air at $${30^ \circ }C$$ with convective heat transfer coefficient $$h = 20\,W/{m^2}K.$$ The thermo-physical properties of steel are: density $$\rho = 7800\,\,kg/{m^3},$$ , conductivity $$k = 40 W/mK$$ and specific heat $$c = 600 J/kgK.$$ The time required in seconds to cool the steel ball in air from $${1030^ \circ }C$$ to $${430^ \circ }C$$ is

Two large diffuse gray parallel plates, separated by a small distance, have surface temperatures of $$400$$ $$K$$ and $$300$$ $$K.$$ If the emissivities of the surfaces are $$0.8$$ and the Stefan-Boltzmann constant is $$5.67 \times {10^{ - 8}}$$ $$W/{m^2}{K^4},$$ the net radiation heat exchange rate in $$kW/{m^2}$$ between the two plates is

Water (specific heat, $${c_p} = 4.18\,\,kJ/kgK$$ ) enters a pipe at a rate of $$0.01$$ $$kg/s$$ and temperature of $${20^ \circ }C.$$ The pipe, of diameter $$50$$ $$mm$$ and length $$3$$ $$m,$$ is subjected to a wall heat flux $${q_w}$$ in $$W/{m^2}:$$

If $${q_w}$$ $$=5000$$ and the convection heat transfer coefficient at the pipe outlet is $$1000$$ $$W/{m^2}K,$$ the temperature in $$^ \circ C$$ at the inner surface of the pipe at the outlet is

Water (specific heat, $${c_p} = 4.18\,\,kJ/kgK$$ ) enters a pipe at a rate of $$0.01$$ $$kg/s$$ and temperature of $${20^ \circ }C.$$ The pipe, of diameter $$50$$ $$mm$$ and length $$3$$ $$m,$$ is subjected to a wall heat flux $${q_w}$$ in $$W/{m^2}:$$

If $${q_w}$$ $$=2500x,$$ where $$x$$ is $$m$$ and in the direction of flow ($$x=0$$ at the inlet), the bulk mean temperature of the water leaving the pipe in $$^ \circ C$$ is