Problem 67 - University Physics with Modern Physics

Question

A carpenter builds an exterior house wall with a layer of wood 3.0 \(\mathrm{cm}\) thick on the outside and a layer of Styrofoam insulation 2.2 \(\mathrm{cm}\) thick on the inside wall surface. The wood has \(k=0.080 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) , and the Styrofoam has \(k=0.010 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}\) . The interior surface temperature is \(19.0^{\circ} \mathrm{C}\) , and the exterior surface temperature is \(-10.0^{\circ} \mathrm{C}\) (a) What is the temperature at the plane where the wood meets the Styrofoam? (b) What is the rate of heat flow per square meter through this wall?

Step-by-Step Solution

Answer

(a) The temperature at the wood-Styrofoam interface is approximately 14.8°C. (b) The heat flow rate is about 11.3 W/m².

Step 1: Identify Heat Transfer Formula

For materials in series, the rate of heat flow per square meter, \( q \), through the wall can be determined using the formula: \[ q = \frac{(T_1 - T_2)}{R_{\text{total}}} \]where \( T_1 \) is the interior temperature, \( T_2 \) is the exterior temperature, and \( R_{\text{total}} \) is the total thermal resistance of the wall.

Step 2: Calculate Individual Thermal Resistances

Thermal resistance, \( R \), for each layer is calculated using the formula: \[ R = \frac{d}{k} \]where \( d \) is the thickness in meters and \( k \) is the thermal conductivity. - For wood: \[ R_{\text{wood}} = \frac{0.030\, \text{m}}{0.080}\, \text{W/m} \cdot \text{K} = 0.375 \, \text{m}^2\cdot\text{K/W} \] - For Styrofoam: \[ R_{\text{styrofoam}} = \frac{0.022\, \text{m}}{0.010}\, \text{W/m} \cdot \text{K} = 2.2 \, \text{m}^2\cdot\text{K/W} \]

Step 3: Calculate Total Thermal Resistance

The total thermal resistance of the wall is the sum of the resistances of the wood and Styrofoam layers:\[ R_{\text{total}} = R_{\text{wood}} + R_{\text{styrofoam}} = 0.375 + 2.2 = 2.575 \, \text{m}^2\cdot\text{K/W} \]

Step 4: Find Rate of Heat Flow

Using the total thermal resistance in the heat transfer equation, we can find \( q \):\[ q = \frac{(19.0 - (-10.0))}{2.575} = \frac{29.0}{2.575} \approx 11.26 \, \text{W/m}^2 \]This is the rate of heat flow per square meter through the wall.

Step 5: Determine Temperature at Interface of Materials

To find the temperature where the wood meets the Styrofoam, use the heat flow rate and the resistance of only the wood to calculate the temperature difference across the wood:\[ q = \frac{T_i - T_m}{R_{\text{wood}}} \]\[ 11.26 = \frac{19.0 - T_m}{0.375} \]\[ T_m = 19.0 - (11.26 \times 0.375) \approx 14.77\, ^\circ \text{C} \]

Key Concepts

Heat Transfer

Heat transfer is the movement of thermal energy between objects or systems due to a temperature difference. In the context of our exercise, this concept is crucial as it helps us understand how heat flows through the materials of a wall.
Heat transfer usually occurs in three different methods:

• Conduction: This is the process through which heat energy is transferred within a body or from one body to another within solid media. In the carpenter's wall, heat conducts through the wood and Styrofoam.
• Convection: This method involves the movement of fluid (liquid or gas) that carries heat away as it moves. For walls, this is often more relevant on the surface level where air moves along each side.
• Radiation: This mechanism involves the transfer of heat through electromagnetic waves, which can happen even through a vacuum.

In our example, the focus is on conduction, specifically through materials in series, such as inner wood and Styrofoam layers. The resistance each material presents impacts the overall heat flow through the wall.
Calculating the heat flow allows us to determine how effective these materials are in preventing heat from escaping or entering the building.

Thermal Conductivity

Thermal conductivity is a property of a material that indicates its ability to conduct heat. Materials with high thermal conductivity transfer heat effectively, while those with low thermal conductivity do not. In the wall built by the carpenter, there are two materials to consider: wood and Styrofoam. The thermal conductivities for these materials are:

• Wood: 0.080 W/m·K
• Styrofoam: 0.010 W/m·K

These values tell us that wood allows heat to pass through it more easily than Styrofoam does. Styrofoam, with its low thermal conductivity, acts as a better insulator, slowing down the rate of heat transfer.
This property is vital when designing insulation systems, as its knowledge helps to select materials that will maintain desired temperatures within a building by resisting unwanted heat transfer.

Building Insulation

Building insulation is a crucial topic for maintaining energy efficiency and comfort within homes and buildings. Insulation works by reducing the flow of heat into or out of a building, thereby maintaining a pleasant indoor temperature regardless of the extremes outside. In this scenario, the carpenter added layers of wood and Styrofoam as insulation. Though both materials contribute to insulation, their effectiveness varies because of differing thermal resistances. Each material has specific thickness and thermal conductivity that affect the wall's total thermal resistance:

• Wood: Offers moderate resistance due to its thermal property and thickness.
• Styrofoam: Provides higher resistance, being a more effective insulator.

This layered approach in construction helps in achieving better thermal performance, preventing significant heat loss in winter or heat gain in summer. By selecting materials with suitable thermal conductivities and thicknesses, building efficiency can be optimized to save energy and maintain indoor comfort levels more effectively.