Problem 50 - College Physics

Question

\(\cdot\) A slab of a thermal insulator with a cross-sectional area of 100 \(\mathrm{cm}^{2}\) is 3.00 cm thick. Its thermal conductivity is 0.075 \(\mathrm{W} /(\mathrm{m} \cdot \mathrm{K}) .\) If the temperature difference between opposite faces is \(80 \mathrm{C}^{\circ},\) how much heat flows the slab in 1 day?

Step-by-Step Solution

Answer

The heat flow through the slab in 1 day is 259,200 J.

Step 1: Identify Given Parameters

We are given the following values:- Cross-sectional area, \( A = 100 \text{ cm}^2 = 0.01 \text{ m}^2 \) (converted to m²)- Thickness \( d = 3.00 \text{ cm} = 0.03 \text{ m} \)- Thermal conductivity \( k = 0.075 \text{ W/mK} \)- Temperature difference \( \Delta T = 80 \text{ °C} \)- Duration \( t = 1 \text{ day} = 86400 \text{ seconds} \).

Step 2: Recall the Heat Transfer Formula

The formula to calculate heat transfer due to conduction is:\[Q = \frac{kA\Delta T t}{d}\]where \( Q \) is the amount of heat transferred, \( k \) is the thermal conductivity, \( A \) is the cross-sectional area, \( \Delta T \) is the temperature difference, \( t \) is the time, and \( d \) is the thickness of the slab.

Step 3: Substitute Given Values into the Formula

Substitute the numbers into the heat transfer formula:\[Q = \frac{0.075 \times 0.01 \times 80 \times 86400}{0.03}\]

Step 4: Calculate the Heat Transfer

Perform the calculation:\[Q = \frac{0.075 \times 0.01 \times 80 \times 86400}{0.03} = \frac{5184}{0.03} = 2.592 \times 10^5 \text{ J}\]Thus, the heat flow through the slab in one day is \( 259,200 \text{ J} \).

Key Concepts

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object to another. This process occurs when there is a temperature difference between bodies or areas. Heat always moves from warmer regions to cooler ones, striving to reach thermal equilibrium. There are three main methods of heat transfer:

• Conduction, which is the direct transfer through materials, like when heat moves through a pan to cook food.
• Convection, where heat is carried by fluid movement, such as currents in water or air.
• Radiation, the transfer through electromagnetic waves, like sunlight warming the Earth.

In the context of our exercise, heat transfer occurs through conduction. This is because the slab of thermal insulator transfers heat between its surfaces due to the temperature difference.

Thermal Insulators

Thermal insulators are materials that resist the flow of heat. These materials are designed to slow down heat transfer, making them vital for maintaining temperature in homes or cooking equipment. Insulators have low thermal conductivity, meaning they are less efficient at conducting thermal energy. Common thermal insulators include:

• Wood
• Styrofoam
• Fiberglass

In the exercise problem, the slab has a thermal conductivity of 0.075 W/mK, indicating it restricts heat transfer to a moderate degree. This property is crucial for applications wanting to control heat movement, such as in building insulation.

Temperature Difference

Temperature difference is the driving force behind heat transfer. It is the variation in temperature between two points or surfaces. A greater temperature difference results in a more substantial heat flow. This is because the thermal energy naturally flows from areas of higher temperatures to those of lower temperatures as the system tries to reach equilibrium. In our exercise, a temperature difference of 80 °C exists between the slab's faces. This significant difference forces heat to flow through the slab, making it a critical factor in calculating how much heat is transferred in a given time period. Mathematically, this is incorporated into the heat transfer equation that we used to find the heat flow, emphasizing its importance in such calculations.