Q. 1.59 - An Introduction to Thermal Physics

Question

Make a rough estimate of the total rate of conductive heat loss through the windows, walls, floor, and roof of a typical house in a cold climate. Then estimate the cost of replacing this lost energy over a month. If possible, compare your estimate to a real utility bill. (Utility companies measure electricity by the kilowatt-hour, a unit equal to  MJ. In the United States, natural gas is billed in terms, where 1 therm = 10⁵ Btu. Utility rates vary by region; I currently pay about 7 cents per kilowatt-hour for electricity and 50 cents per therm for natural gas.)

Step-by-Step Solution

Answer

The cost of replacing the lost energy we get,

Cost $101.5

Step 1: Calculating the natural gas

Let's say that the typical house has the size of A =64 m². 

If we visual the simplest house model , it would have 4 outdoor walls, a floor, and a roof.

Every wall will be the same length if there are walls of equal length, where

L=8m long. 

We can assume the walls are 3 meters high. 

So, the area of the total wall is equal to:

$$\begin{gathered} A_{\text{wall }}=4\times L\times h \\ =4\times 8\times 3 \\ =96{\mathrm{m}}^2 \end{gathered}$$

We can discover resistance to heat passage through the wall in the characteristics table:

$R=0.2\frac{{\mathrm{Km}}^2}{\mathrm{W}}$

The weather is cold outside, as the problem stated. Let's say the difference in temperature between outside and inside air is:

$\mathrm{\Delta }T=25\mathrm{C}$

energy loss through the walls is equal to:

$$\begin{gathered} {\dot{Q}}_w=\frac{1}{R}\cdot A_{\text{wall }} \\ \cdot \mathrm{\Delta }T=\frac{1}{0.2}\times 96\times 25 \\ =143\frac{\mathrm{kW}}{\mathrm{h}} \end{gathered}$$

In the simple model we can say that floor and roof are the same size:

$A_{\text{roof }}=A_{\text{floor }}=64{\mathrm{m}}^2$

Calculating energy loss through the roof:

$$\begin{gathered} {\dot{Q}}_R=\frac{1}{R}\times A_{\text{roof }} \\ \cdot \mathrm{\Delta }T=\frac{1}{0.2}\times 64\times 25 \\ =95.35\frac{\mathrm{kW}}{\mathrm{h}} \end{gathered}$$

Step 2: Cost Estimation

Total energy loss is equal to:

$$\begin{gathered} \dot{Q}=143+2\times 95.35 \\ =333.7\frac{\mathrm{kW}}{\mathrm{h}} \end{gathered}$$

We are given that cost is 0.007 $\frac{\mathrm{kW}}{\mathrm{h}}$.

Now let's see what is the cost for $\dot{Q}=333.7\frac{\mathrm{kW}}{\mathrm{h}}$:

Cost$$\begin{gathered} =333.7 \times 0.07 \\ =\mathrm{\$}23.4 \end{gathered}$$

If we look cost per therm of natural gas:

1 therm = 10⁵ Btu.
Now let's see what is total energy loss in terms of therm:

$\dot{Q}=333.7\frac{\mathrm{kW}}{\mathrm{h}}=1.14\cdot {10}^7\mathrm{Btu}=203\text{ therms }$

The cost of natural gas per therm is presented to us in the problem as

1 therm = 50 cents 

 Calculating the cost of natural gas:

Cost$$\begin{gathered} =203\times 0.5 \\ =\mathrm{\$}101.5 \end{gathered}$$

Therefore,

Cost = $101.5