$(30)$ Consider a temperature$-$controlled warehouse with dimensions $50m50m$ and a height of $5$ m $.$ Let us

assume that this warehouse is a simple box shape with no windows and is on the ground, so we can consider

the floor to be insulated.

This warehouse's four walls and ceiling are constructed of four layers. The first interior layer is $2\mathrm{.}0$ cm thick

gypsum board $($drywall$)$ with a thermal conductivity of $k=1\mathrm{.}0\frac{W}{m}*K.$ Next is a $5\mathrm{.}0$ cm insulation layer

with a thermal conductivity of $k=0\mathrm{.}1\frac{W}{m}K.$ The third layer is the brick wall, which is $20\mathrm{.}0$ cm thick with a

thermal conductivity of $k=2\mathrm{.}0\frac{W}{m}*K.$ The outside layer is a $2\mathrm{.}0$ cm thick decorative wood planking

with a thermal conductivity of $k=2\mathrm{.}5\frac{W}{m}*K($Floor is insulated$).$

This space will be maintained at $15C$ using a copper pipe going around the floor that carries steam entering

the pipe as saturated vapor, i$.$e$.x=1\mathrm{.}0$ at $120C.$ The steam condenses as it loses heat into the room, and

we want the length of the pipe such that steam becomes all saturated liquid at $120C$ at the exit. The pipe

is $10\mathrm{.}0$ cm inside and $11\mathrm{.}0$ cm outside in diameter and is made of copper with high thermal conductivity and

having an emissivity of $=0\mathrm{.}8.$ The inside surface of the pipe can be taken at the saturation temperature

of the steam, $120C.$ Due to the high thermal conductivity of copper, we can assume that thermal resistance

due to pipe material is negligible, and the outside surface temperature of the pipe is at $120C.$

Consider a cold day when the ambient temperature is $-10C$ and the combined convective heat transfer

coefficient over all outside surfaces of the warehouse, and the ambient air is

$h=20\frac{W}{m^2}K$

a$.$ What is the total rate of heat loss from the warehouse?

b$.$ What is the temperature of the outside wall surface?

c$.$ What is the rate of steam flow $(K\frac{g}{sec})$ needed to compensate for this heat

loss if steam enters as saturated vapor and leaves as saturated liquid at the

same temperature,

d$.$ Determine the convective heat transfer coefficient $"h"$ and what length of

steam pipe is needed to keep the room at $15C$ if we only have "Natural

Convection" plus "Radiation" from the pipe for heat transfer to the room. Can

the room's radiation temperature be taken at $15C$ as well?

e$.$ Next, we apply forced convection

using small fans blowing air across the steam$-$carrying pipe.

If the air velocity is $V_{\text{a}\text{i}\text{r}}=5\frac{m}{s},$ what is the $"f_{\text{f}\text{o}\text{r}\text{c}\text{e}\text{d}}"$ and

what length of the steam pipe is needed to maintain the

inside temperature at $15C?$ Does radiation change from

Part$-$d$?$

Air: Density $=1\mathrm{.}2k\frac{g}{m^3}$;$k=0\mathrm{.}026\frac{W}{m}K$;

Viscosity $($kinematic$)v=1\mathrm{.}5{10}^{-5}\frac{m^2}{s}$ or $=1\mathrm{.}8{10}^{-5}k\frac{g}{m}sec$