$(40.$pts$)$ Consider a temperature$-$controlled warehouse with dimensions $50m50m$ and height $5$ m $.$ Let us

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

can consider the floor to be insulated.

The four walls and the ceiling of this warehouse are constructed of three layers. The first interior layer is

$2\mathrm{.}0$ cm thick gypsum board $($drywall$)$ with a thermal conductivity of $k=1\mathrm{.}0\frac{W}{m}*K.$ The next is the

brick wall that is $20\mathrm{.}0$ cm thick with a thermal conductivity of $k=2\mathrm{.}0\frac{W}{m}*K.$ The outside layer is a

$2\mathrm{.}5$ 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 pipe going

around the floor that carries steam entering the pipe as

saturated vapor, i$.$e$.x=1\mathrm{.}0$ at $120C.$ The pipe is $10\mathrm{.}0$ cm

inside and $11\mathrm{.}0$ cm outside in diameter and is made of a

material with a thermal conductivity of $k=20\frac{W}{m}K.$ The

inside surface of the pipe can be taken as a the saturation

temperature of the steam. i$.$e$.,120C.$

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

$h=15\frac{W}{m^2}K($A little chilly and windy... Brrrr$!..).$

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

b$.$ What is the temperature of the inside 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$.$ If the combined $h$ over the outside surface of the pipe is $h_{\text{c}\text{o}\text{n}\text{v}}=10\frac{W}{m^2}K$ what is the length

of the pipe $($in meters$)$ needed to provide enough heat to keep the inside temperature the same?

The heat of condensation $h_{fg}$ can be found in the link.

e$.$ What is the velocity of steam at the inlet of the pipe?

f$.$ What is the surface temperature of the pipe $(C)?$ If the "Safe" temperature to touch is $T_{s}50C$

what would you recommend as a safety measure?

g$.$ If the cost of energy is $$\frac{0\mathrm{.}20}{k}Wh,$ what would it cost per month

to maintain the warehouse at $15C?$

h$.$ Your boss wants to cut the heat loss by $75\%.$ How much of $R_{th}$

insulation should you add? Where would you place it$?$ If it is foam

material of $k=0\mathrm{.}04\frac{W}{m}K$ how thick of insulation would you

need $(cm)?$