Question

Consider a 25-m-long thick-walled concrete duct \((k=\) \(0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of square cross section. The outer dimensions of the duct are \(20 \mathrm{~cm} \times 20 \mathrm{~cm}\), and the thickness of the duct wall is \(2 \mathrm{~cm}\). If the inner and outer surfaces of the duct are at \(100^{\circ} \mathrm{C}\) and \(30^{\circ} \mathrm{C}\), respectively, determine the rate of heat transfer through the walls of the duct. Answer: \(47.1 \mathrm{~kW}\)

Short answer

Answer: The rate of heat transfer through the walls of the concrete duct is 47.1 kW.

Step-by-step solution

Identify the formula for conductive heat transfer through a wall

The formula for conductive heat transfer through a wall is given by: q = (k * A * ΔT) / d where q is the rate of heat transfer (W), k is the thermal conductivity (W/m·K), A is the surface area (m²), ΔT is the temperature difference between the inner and outer surfaces (K), and d is the wall thickness (m).

Calculate the surface area of the duct

The outer dimensions of the duct are given as 20 cm x 20 cm, and its length is 25 m. To calculate the surface area of the duct, we need to convert the dimensions to meters and use the following formula for the surface area of a square cross-sectional duct: A = 4 * L * d where A is the surface area (m²), L is the length of the duct (m), and d is the side of the square cross-section (m). Converting the dimensions and plugging the values into the formula, we get: A = 4 * 25 * 0.2 = 20 m²

Convert the temperature difference to Kelvin

We are given the temperature difference between the inner and outer surfaces of the duct as 100°C and 30°C, respectively. The temperature difference, ΔT, in Kelvin is: ΔT = (100 - 30) K = 70 K

Calculate the rate of heat transfer through the duct walls

Now that we have all the necessary information, we can calculate the rate of heat transfer through the duct walls using the following formula: q = (k * A * ΔT) / d Plugging in the given values for k (0.75 W/m·K), A (20 m²), ΔT (70 K), and d (0.02 m), we obtain: q = (0.75 * 20 * 70) / 0.02 = 47125 W To express the answer in kilowatts, simply divide q by 1,000: q = 47125 / 1000 = 47.1 kW The rate of heat transfer through the walls of the duct is 47.1 kW.

Key concepts

Conductive Heat Transfer

Conductive heat transfer is a mode of thermal energy transportation within materials that occurs at a microscopic scale due to the diffusion of internal energy. Simply put, heat flows from a hotter region to a cooler one when they are in direct contact.

Understanding conductive heat transfer is crucial because it helps in calculating how much heat passes through materials, such as the walls of buildings or, in the case of our exercise, the concrete duct. It involves the transfer of kinetic energy from one molecule to another without any actual motion of the material as a whole. Materials that conduct heat well, like most metals, are described as having high thermal conductivity, while materials that don't, like wood or fiberglass, are known as insulators.

To quantify this heat transfer process, it's helpful to use a formula that takes into account several factors including the temperature difference across the material, the material's thermal conductivity, and the area over which the heat transfer takes place.

Thermal Conductivity

Thermal conductivity, denoted by the symbol 'k', is a material-specific value that measures how effectively heat is conducted through a material. Essentially, it represents the amount of heat (in watts) transferred through a material with a thickness of one meter when there is a temperature difference of one degree Celsius (or one Kelvin) across it.

In our exercise, the thermal conductivity of the concrete is given as 0.75 W/m·K. This relatively low value in comparison to metal, for instance, means that concrete is not as effective at transferring heat. Importantly, known values of thermal conductivity can be used to predict the rate of heat loss or gain through materials, which is a key concern in the design of structures and thermal management systems.

Temperature Difference

The temperature difference, symbolized as ΔT, is a driving force behind conductive heat transfer. It's the amount of thermal energy difference between one side of a material and the other and is usually measured in degrees Celsius (°C) or Kelvin (K).

In our scenario, we calculate the temperature difference between the inner and outer surfaces of the concrete duct. Since the conductive heat transfer formula uses Kelvin for temperature, we have the convenience that a difference of 1°C is equivalent to a difference of 1K. A greater temperature difference means a larger amount of heat will be transferred, assuming all other factors are constant. This concept is keenly observed in real-life applications like thermal insulation where the aim is to maintain a significant temperature difference between the inside and outside of a structure.

Surface Area Calculation

The surface area involved in heat transfer must be calculated accurately to determine the rate of heat moving through a material. In our exercise, we’re dealing with a duct of a square cross section, thus its surface area is critical for determining how much energy is being lost or gained through its walls.

The formula for the surface area of a square cross-sectional shape is given as the perimeter of the square times the length of the duct. In mathematical terms for our example, this equates to the sum of all four sides of the cross-section (which would be four times one side) multiplied by the length of the duct. By calculating the surface area correctly, we can plug this value into our heat transfer formula to find out the amount of heat that passes through the walls over a certain period, which, for practical purposes, serves as the basis for energy efficiency calculations and thermal regulation strategies.