(Solved): The 5m5m room is heated by under floor heating systems schematically shown below. The thickne ...

The $$5\mathrm{m}\times 5\mathrm{m}$$ room is heated by under floor heating systems schematically shown below. The thickness of the floor $$(Z)$$ is equal to $$15\mathrm{cm}$$ and the volumetric energy generation rate in the floor $$S=S_0(1+bz)$$, where $${\mathrm{S}}_0\left[\mathrm{W}{\mathrm{m}}^{?3}\right]$$ is the specific energy generation rate and b $$\left[{\mathrm{m}}^{?1}\right]$$ is a proportionality constant. The bottom surface of the floor is perfectly insulated and the top surface is kept at constant temperature of $$298\mathrm{K}$$. a) Select the general form of the energy balance equation in an appropriate coordinate system; b) Simplify the energy balance to describe the heat transfer in the floor justifying all the assumptions, and derive a differential equation and boundary conditions describing the steady state temperature distribution in the floor; c) Solve the resulting equation and sketch temperature distribution. Use the steady state temperature distribution to calculate total amount of heat transferred from the floor to the room; d) The air is flowing through the room at the flow rate of $$15\mathrm{kg}\min ?1$$. Calculate the energy generation rate $${\mathrm{S}}_0$$ necessary to keep the temperature in the room equal to $$20^{?}\mathrm{C}$$. The temperature of the air outside the room is equal to $$5^{?}\mathrm{C}$$. Data: Proportionality constant: $$\mathrm{b}=0.5{\mathrm{m}}^{?1}$$ Properties of the floor. $$\mathrm{K}=1.5\mathrm{W}\mathrm{m}{\mathrm{m}}^{?1},?=2000\mathrm{kg}{\mathrm{m}}^{?3},c_p=850\mathrm{J}{\mathrm{kg}}^{?1}{\mathrm{K}}^{?1}$$ Physical properties of air: from handbook/online data base.