Problem 69 - University Physics with Modern Physics

Question

The celling of a room has an area of 125 \(\mathrm{ft}^{2}\) . The ceiling is insulated to an \(R\) value of 30 (in units of \(\mathrm{ft}^{2} \cdot \mathrm{F}^{\circ} \cdot \mathrm{h} / \mathrm{Btu} )\) . The surface in the room is maintained at \(69^{\circ} \mathrm{F}\) , and the surface in the attic has a temperature of \(35^{\circ} \mathrm{F}\) . What is the heat flow through the ceiling into the attic in 5.0 \(\mathrm{h} ?\) Express your answer in Btu and in joules.

Step-by-Step Solution

Answer

Heat flow is 708.333 Btu or 747,306.15 Joules.

Step 1: Understand the Heat Transfer Formula

The rate of heat transfer through a material can be calculated using the formula: \[ Q = \frac{A \times \Delta T \times t}{R} \] where \( A \) is the area (in \( \mathrm{ft}^2 \)), \( \Delta T \) is the temperature difference (in \(^{ ext{o}}\mathrm{F}\)), \( t \) is the time (in hours), and \( R \) is the thermal resistance (\( \mathrm{ft}^2 \cdot \mathrm{F}^{\circ} \cdot \mathrm{h} / \mathrm{Btu} \)).

Step 2: Identify Given Values

From the problem, we identify: \( A = 125 \, \mathrm{ft}^2 \), \( R = 30 \, \mathrm{ft}^2 \cdot \mathrm{F}^{\circ} \cdot \mathrm{h} / \mathrm{Btu} \), the temperature inside the room (\( T_{\text{inside}} = 69^{\circ}\mathrm{F} \)), the temperature in the attic (\( T_{\text{attic}} = 35^{\circ}\mathrm{F} \)), and the time \( t = 5 \, \mathrm{h} \).

Step 3: Calculate the Temperature Difference

The temperature difference \( \Delta T \) is calculated as: \[ \Delta T = T_{\text{inside}} - T_{\text{attic}} = 69^{\circ}\mathrm{F} - 35^{\circ}\mathrm{F} = 34^{\circ}\mathrm{F} \]

Step 4: Calculate the Heat Flow in Btu

Using the heat transfer formula: \[ Q = \frac{125 \, \mathrm{ft}^2 \times 34^{\circ}\mathrm{F} \times 5 \, \mathrm{h}}{30 \, \mathrm{ft}^2 \cdot \mathrm{F}^{\circ} \cdot \mathrm{h} / \mathrm{Btu}} \] Simplifying this gives: \[ Q = \frac{125 \times 34 \times 5}{30} = \frac{21250}{30} = 708.333 \; \mathrm{Btu} \]

Step 5: Convert Btu to Joules

1 Btu is equivalent to 1055 Joules. Thus, to convert 708.333 Btu to Joules: \[ 708.333 \; \mathrm{Btu} \times 1055 \; \mathrm{J/Btu} = 747306.15 \; \mathrm{J} \]

Key Concepts

Thermal Resistance

Thermal resistance is like the armor for your home, protecting it from losing heat. It tells us how well a material resists the flow of heat. The higher the thermal resistance, the less heat escapes. It's usually referred to as the 'R-value'. The R-value is a measure of how well a layer of insulation material, like that in your ceiling, keeps the heat inside your home. In our case, the ceiling has an R-value of 30, meaning it is fairly effective at insulating the room. The unit you might see is

• ft² · °F · h/Btu

This unit essentially means how much heat is restricted from passing through a unit area over a period of one hour, per degree of temperature difference. This concept is crucial when calculating how much energy your home is losing, which in turn affects your heating bills.

Temperature Difference

Temperature difference, or delta T , is the gap in temperature between two areas. It is one of the key drivers of heat flow. In our exercise, we look at the room's temperature and the attic's temperature. The greater this difference, the faster heat flows through your insulation. We've calculated it as:

• delta T = 69°F (inside) - 35°F (attic) = 34°F

This tells us there's a 34°F gap driving the heat transfer. Imagine this as water flowing from a higher point to a lower one. Similarly, heat moves from a warmer area to a cooler one until temperatures even out. Managing this difference effectively is critical for energy conservation and comfort.

Heat Flow Calculation

Once you have thermal resistance and temperature difference, you can calculate heat flow. This calculation tells you how much heat energy moves through your ceiling over a set time, affecting how much energy you need to keep your home warm. The formula:

• Q = \( \frac{A \times \Delta T \times t}{R} \)

where:

• A is the area (125 ft²),
• \( \Delta T \) is the temperature difference (34°F),
• t is the time (5 hours),
• R is the thermal resistance (30 ft²·°F·h/Btu)

When you plug in the numbers, you get:

• Q = \( \frac{125 \times 34 \times 5}{30} \) = 708.333 Btu

This value can be converted to joules, a unit of energy used in physics, by multiplying by 1055 (since 1 Btu equals 1055 joules). Thus, 708.333 Btu converts to approximately 747306.15 joules. This calculation is fundamental in understanding how much energy you are using or wasting, guiding decisions on improving home insulation.