Problem 108 - University Physics with Modern Physics

Question

One experimental method of measuring an insulating material's thermal conductivity is to construct a box of the material and measure the power input to an electric heater inside the box that maintains the interior at a measured temperature above the outside surface. Suppose that in such an apparatus a power input of 180 \(\mathrm{W}\) is required to keep the interior surface of the box 65.0 \(\mathrm{C}^{\circ}\) (about 120 \(\mathrm{F}^{\circ}\) ) above the temperature of the outer surface. The total area of the box is 2.18 \(\mathrm{m}^{2}\) , and the wall thickness is 3.90 \(\mathrm{cm}\) . Find the thermal conductivity of the material in SI units.

Step-by-Step Solution

Answer

The thermal conductivity of the material is approximately 0.049 W/m⋅K.

Step 1: Identify Known Values and Formula

To find the thermal conductivity (\( k \)), we'll use the formula for heat transfer rate: \[ Q = \frac{k \cdot A \cdot \Delta T}{d} \]where the heat transfer rate \( Q = 180 \, \text{W} \), \( A = 2.18 \, \text{m}^2 \), \( \Delta T = 65.0^{\circ} \text{C} \), and the wall thickness \( d = 3.90 \, \text{cm} = 0.039 \, \text{m} \).

Step 2: Rearrange Formula to Solve for Thermal Conductivity

Rearrange the heat transfer formula to solve for thermal conductivity \( k \):\[ k = \frac{Q \cdot d}{A \cdot \Delta T} \]

Step 3: Substitute Known Values into the Formula

Substitute the known values into the formula:\[ k = \frac{180 \, \text{W} \cdot 0.039 \, \text{m}}{2.18 \, \text{m}^2 \cdot 65.0 \,^{\circ}\text{C}} \]

Step 4: Calculate the Thermal Conductivity

Perform the calculation:\[ k = \frac{180 \times 0.039}{2.18 \times 65.0} \approx 0.049 \frac{\text{W}}{\text{m} \cdot \text{K}} \]

Step 5: Interpret the Result

The calculated thermal conductivity of the material is \( 0.049 \, \frac{\text{W}}{\text{m} \cdot \text{K}} \), indicating the insulating properties of the material.

Key Concepts

Heat Transfer

In simple terms, heat transfer refers to the movement of thermal energy from one place to another due to a temperature difference. This concept is essential in understanding thermal conductivity. Heat can be transferred in three main ways: conduction, convection, and radiation.
Conduction is the process where heat moves through a solid material without the movement of the material itself. A good way to imagine this is heating one end of a metal rod; the temperature increase travels through the rod to the other side.
Convection involves the movement of heat by the physical movement of fluid (like air or water) from one place to another. This is why the air in a room can feel cooler near an open window.
Radiation occurs when heat is transferred through electromagnetic waves, such as the heat from the sun warming your skin.

• Conduction - Heat moves through materials.
• Convection - Heat carried by fluids.
• Radiation - Heat transferred through waves.

Understanding heat transfer is vital for insulating materials, as they typically work by reducing the amount of heat transferred through conduction.

Insulating Materials

Insulating materials are designed to slow down the heat transfer process. They are commonly used to maintain temperature differences in various applications, like keeping homes warm in winter or cool in summer.
These materials have low thermal conductivity, meaning they don't allow heat to pass through them easily. This quality makes them effective at resisting heat flow. Examples include materials like fiberglass, foam, and even certain textiles.

• Low thermal conductivity: Insulators resist heat flow.
• Common types include fiberglass and foam.
• Used to manage thermal environments in buildings.

In the given problem, the insulating material's effectiveness is demonstrated by how much power is needed to maintain a temperature difference inside the insulated box.

SI Units

SI Units, or the International System of Units, are the global standard for measuring physical quantities. They ensure consistency in scientific communication across the world. For thermal conductivity, the SI unit is watts per meter per Kelvin, written as \( ext{W/m} \cdot \text{K}\).
Using consistent units is crucial in calculations involving thermal conductivity, as seen in the problem where power (180 W) and area (2.18 my) are given in SI units.

• Standard unit for thermal conductivity: \(\text{W/m} \cdot \text{K}\).
• Ensures consistency in international scientific communication.
• Allows clear comparisons of insulating effectiveness across materials.

Proper use of SI units helps accurately convey the results of thermal measurements and predictions.

Thermal Insulation Measurement

Thermal insulation measurement is crucial to understanding how well a material can prevent heat transfer. In the exercise, thermal conductivity was measured by calculating how much energy was required to maintain a desired temperature inside a box made of the insulating material.
The key formula used was \(Q = \frac{k \cdot A \cdot \Delta T}{d}\), where:

• Q is the rate of heat transfer or power in watts.
• A is the area through which heat is being transferred in square meters.
• \(\Delta T\) is the temperature difference across the material in degrees Celsius.
• d is the thickness of the material in meters.

This formula allows scientists and engineers to compare different materials' insulating capabilities by calculating their thermal conductivity. A lower thermal conductivity value indicates better insulating properties.