Question

A \(2.5 \mathrm{~m}\)-high, 4-m-wide, and 20 -cm-thick wall of a house has a thermal resistance of \(0.0125^{\circ} \mathrm{C} / \mathrm{W}\). The thermal conductivity of the wall is (a) \(0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (b) \(1.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (c) \(1.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (d) \(16 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) (e) \(32 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\)

Short answer

a) 0.8 W/(m.K) b) 1.2 W/(m.K) c) 1.6 W/(m.K) d) 2.0 W/(m.K) Answer: c) 1.6 W/(m.K)

Step-by-step solution

Identify the given values

We are given: - Height of the wall (h) : \(2.5 \mathrm{~m}\) - Width of the wall (w) : \(4.0 \mathrm{~m}\) - Thickness of the wall (d) : \(0.20 \mathrm{~m}\) - Thermal resistance (R) : \(0.0125 \mathrm{~C}/\mathrm{W} \)

Calculate the area of the wall

To find the area of the wall (A), multiply the height by the width: \(A = h \times w = 2.5 \mathrm{~m} \times 4.0 \mathrm{~m} = 10 \mathrm{~m^2}\)

Rearrange the thermal resistance formula to find thermal conductivity

To find the thermal conductivity (k), we can rearrange the formula for thermal resistance (R) as follows: \(k = \frac{d}{RA}\)

Plug the values into the formula and solve for thermal conductivity

Now, we can plug the values of d, R, and A into the formula and solve for k: \(k = \frac{0.20 \mathrm{~m}}{0.0125 \mathrm{~C}/\mathrm{W} \times 10 \mathrm{~m^2}} = 1.6 \mathrm{~W}/(\mathrm{m} \cdot \mathrm{K})\) Hence, the correct option is (c) \(1.6 \mathrm{~W}/(\mathrm{m} \cdot \mathrm{K})\).

Key concepts

Thermal Resistance

Understanding thermal resistance is crucial when studying heat transfer in materials. It quantifies how well a material resists the flow of heat. Much like electrical resistance in circuits, thermal resistance offers a way to measure how insulation or other barriers slow down the energy transfer between two points. It involves variables such as material thickness, surface area, and temperature difference. In practical scenarios, such as in the given exercise where a wall's thermal resistance is known, one could determine the wall's effectiveness in insulating a space against temperature changes. The lower the thermal resistance, the better the material conducts heat, resulting in more efficient heat transfer.

Heat Transfer

Heat transfer is a fundamental concept in thermal engineering, involving the movement of thermal energy from one place to another due to temperature differences. There are three primary modes of heat transfer: conduction, convection, and radiation. In the context of the exercise, we are primarily dealing with conduction through the wall. Conduction is the transfer of heat through a material without the actual movement of the substance; this process is significantly influenced by the material's thermal conductivity. The understanding of heat transfer is essential for designing energy-efficient buildings, maintaining comfortable indoor environments, and numerous engineering applications.

Material Properties

The properties of materials, such as thermal conductivity, density, and specific heat, play a pivotal role in how they behave under thermal stress. Thermal conductivity, the property highlighted in the exercise, indicates a material's ability to conduct heat. It fundamentally affects how quickly heat can pass through a material when exposed to a temperature gradient. The selection of building materials for walls, insulation, and other construction elements is heavily dependent on these thermal properties. Engineers and architects must understand these properties to make informed decisions that affect the safety, comfort, and energy efficiency of buildings.

Thermal Engineering Education

Thermal engineering education involves the study of energy conversion and heat transfer which are central to designing systems such as heating, cooling, and refrigeration units. The curriculum bridges practical problems, like the one found in this exercise, with theoretical concepts, aiming to develop an intuitive understanding alongside rigorous calculus-based learning. Effective education in this field typically emphasizes real-world applications, such as building efficient thermal barriers or creating temperature control mechanisms, to prepare students for the challenges they will face in their professional roles as engineers or designers.