An un-insulated air conditioning duct of rectangular cross section $$1\,\,m \times 0.5\,m,$$ carrying air at $${20^ \circ }C$$ with a velocity of $$10 m/s,$$ is exposed to an ambient of $${30^ \circ }C$$. Neglect the effect of duct construction material. For air in the range of $${20-30^ \circ }C,$$ data are as follows: thermal conductivity $$=0.025 W/m.K;$$ viscosity $$ = 18\mu Pa.s;$$ Prandtl number $$=0.73;$$ density $$= 1.2$$ $$kg/{m^3}.$$. The laminar flow Nusselt number is $$3.4$$ for constant wall temperature conditions and, for turbulent flow, $$Nu = 0.023\,\,R{e^{0.8}}\,{\Pr ^{0.33}}.$$

The Reynolds number for the flow is

A small copper ball of $$5$$ $$mm$$ diameter at $$500$$ $$K$$ is dropped into an oil bath whose temperature is $$300$$ $$K.$$ the thermal conductivity of copper is $$400$$ $$W/m.$$ $$K,$$ its density $$9000\,\,kg/{m^3}$$ and its specific heat $$385\,J/kg.\,\,K.$$ if the heat transfer coefficient is $$250\,\,W/{m^2}K$$ and lumped analysis is assumed to be valid, the rate of fall of the temperature of the ball at the beginning of cooling will be, in $$K/s.$$

Heat flows through a composite slab, as shown below. The depth of the slab is $$1$$ $$m.$$ the $$k$$ values are in $$W/m. K.$$ The overall thermal resistance in $$K/W$$ is

In case of one dimensional heat conduction in a medium with constant properties, $$T$$ is the temperature at position $$x,$$ at time $$t.$$ Then $${{\partial T} \over {\partial t}}$$ is proportional to