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

A company has two factories $${S_1},$$ $${S_2}$$ and two warehouses $${D_1},$$ $${D_2}$$ . the supplies from $${S_1}$$ and $${S_2}$$ are $$50$$ and $$40$$ units respectively. Warehouse $${D_1},$$ requires a minimum of $$20$$ units and a maximum of $$40$$ units. Warehouse $${D_2},$$ requires a minimum of $$20$$ units and, over and above, it can take as much as can be supplied. A balanced transport-ation problem is to be formulated for the above situation. The number of supply points, the number of demand points, and the total supply (or total demand) in the balanced transportation problem respectively are