Question

Heat is transferred steadily through a \(0.2-\mathrm{m}\) thick \(8 \mathrm{m} \times 4 \mathrm{m}\) wall at a rate of \(2.4 \mathrm{kW}\). The inner and outer surface temperatures of the wall are measured to be \(15^{\circ} \mathrm{C}\) and \(5^{\circ} \mathrm{C} .\) The average thermal conductivity of the wall is \((a) 0.002 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\) \((b)0.75 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\) \((c) 1.0 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\) \((d) 1.5 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\) \((e) 3.0 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\)

Short answer

Answer: 1.5 W / (m·°C)

Step-by-step solution

Calculate the temperature difference

ΔT of the wall by subtracting the outer surface temperature from the inner surface temperature: ΔT = T1 - T2 = (15 - 5) °C = 10 °C

Find the area A of the wall:

A = length × width = 8 m × 4 m = 32 m²

Calculate the heat flux q:

Heat flux represents the amount of heat transferred per unit area. The formula for heat flux is: q = Q / A where Q is the heat transfer rate (2.4 kW or 2400 W) and A is the area of the wall. Plugging in the values, we get: q = 2400 W / 32 m² = 75 W/m²

Apply Fourier's law of heat conduction:

Fourier's law of heat conduction is given by: q = -k * (ΔT / L) where q is the heat flux, k is the average thermal conductivity (which we need to find), ΔT is the temperature difference, and L is the thickness of the wall. Rearranging the formula, we get: k = -q * L / ΔT We have all the values needed to calculate k: k = (75 W/m²) * (0.2 m) / (10 °C)

Calculate the average thermal conductivity k:

k = (75 W/m²) * (0.2 m) / (10 °C) = 1.5 W / (m·°C) Comparing this result with the options given, our answer is: \((d) 1.5 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\).

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in physics that describes the movement of heat energy from one place to another. It occurs in three primary ways: conduction, convection, and radiation. In the context of our exercise, we are dealing with thermal conduction, where heat is transferred through a solid material, the wall, without any movement of the material itself.

Understanding how rapidly this process occurs in various materials is vital in applications like building insulation, electronic component design, and even clothing manufacturing. The key to heat transfer in solids is the temperature difference: the greater the difference, the more rapidly heat flows from the warmer side to the cooler side. To quantify this rate in a practical situation — like how quickly your room heats up or cools down — we use concepts like thermal conductivity and heat flux.

Fourier's Law of Heat Conduction

Fourier's law of heat conduction is crucial for understanding and calculating heat transfer within materials. This principle tells us that the rate of heat transfer through a material is proportional to the negative gradient of temperatures and the area through which the heat is flowing.

Mathematically, it is expressed as q = -k * (ΔT / L), where q represents the heat flux, k is the thermal conductivity, ΔT is the temperature difference across the material, and L is the thickness of the material. The negative sign indicates that heat flows from higher to lower temperatures. This relationship suggests that, for a given material and thickness, heat flux increases as the temperature difference increases. Consequently, materials with higher thermal conductivity values transfer heat more efficiently, facilitating faster temperature equalization.

Heat Flux

Heat flux is a measure of the rate at which heat energy passes through a surface or material. It's often described in units of W/m², which indicates how many watts of energy flow through one square meter of area. In our exercise, we calculated the heat flux through a wall based on the amount of heat transferred and the wall's area.

Understanding heat flux helps us to design better systems to manage or utilize this heat transfer. For example, high heat flux through an electronic device's case might suggest it needs better cooling systems to avoid overheating. Similarly, low heat flux through a building's insulation could indicate excellent performance in maintaining interior temperatures despite changing exterior conditions. By relating heat flux to material properties and dimensions, as per Fourier's law, we can tailor materials for specific heat management applications.