Problem $1$

Imagine a house with four walls. The base dimensions of the house are $10$ meters in width and $20$ meters in

length. Each wall is $4$ meters high. All four walls have an R$-$value of $2\mathrm{.}31m^2*\frac{K}{W},$ which is a measure of

thermal resistance.

Two of the walls, each measuring $10$ meters by $4$ meters, have no windows.

The third wall also measures $10$ meters by $4$ meters and features five windows. Each window is $1\mathrm{.}2$ meters

wide and $1\mathrm{.}8$ meters high. These windows are made of glass that is $0\mathrm{.}5$ cm thick, with a thermal conductivity

$($k$)$ of $0\mathrm{.}78\frac{W}{m*K}.$

The fourth wall is identical in size to the third wall and also has five windows. However, these windows are

double$-$paned. Each double$-$pane window consists of two $0\mathrm{.}5$ cm thick glass layers with a $1\mathrm{.}5$ cm thick layer

of stagnant air between them. The thermal conductivity $(k)$ of this air layer is $0\mathrm{.}026\frac{W}{m*K}.$

The indoor temperature of the house is regulated at $24C$ by a thermostat. During a heating season that lasts

for seven months, the average external temperature is $8C.$ For this scenario, ignore the effects of heat gain

or loss through the windows due to direct sunlight or radiation. The coefficients for heat transfer on the

interior and exterior surfaces of the house are $7\frac{W}{m^2*K}$ and $18\frac{W}{m^2*K},$ respectively.

a$)$ Sketch the house, and label all the dimensions and parameters.

b$)$ Using your sketch, determine the symbolic expressions for the total thermal resistances for heat

flow through each wall, including inside and outside convection effects for the prescribed

conditions.

c$)$ Determine the average rate of heat transfer through each wall.

d$)$ If the house is electricallv heated and the nrice of electricitv is $$\frac{0\mathrm{.}08}{k}Wh.$ determine the amount