Problem 5 - An Introduction to Materials Engineering and Science for Chemical and Materials Engineers

Question

Calculate the thermal resistance of a 2 -mm-thick section of human skin. The thermal conductivity of human skin is \(0.627 \mathrm{~N} / \mathrm{K} \cdot \mathrm{s}\), the specific heat is 3470 \(\mathrm{N} \cdot \mathrm{m} / \mathrm{kg} \cdot \mathrm{K}\), and the density is \(1100 \mathrm{~kg} / \mathrm{m}^{3}\).

Step-by-Step Solution

Answer

The thermal resistance of the skin section is approximately 0.00319 K·m²/W.

Step 1: Identify the formula for thermal resistance

The thermal resistance (R_{th}) of a material can be calculated using the formula: \[ R_{th} = \frac{L}{k \cdot A} \] where \( L \) is the thickness of the material, \( k \) is the thermal conductivity, and \( A \) is the area through which heat is being conducted. Since we are finding the per unit area thermal resistance, \( A \) cancels out, simplifying the formula to \[ R_{th} = \frac{L}{k} \].

Step 2: Convert material thickness from millimeters to meters

The thickness of the skin is given as 2 mm. To use this in the formula, convert it to meters since the thermal conductivity is provided in /m/ units. Thus, \( L = 2 \text{ mm} = 0.002 \text{ m} \).

Step 3: Plug in the values and calculate the thermal resistance

Substitute the values of \( L = 0.002 \text{ m} \) and \( k = 0.627 \text{ W/K} \cdot \text{m} \) into the simplified formula \( R_{th} = \frac{L}{k} \). Calculate: \[ R_{th} = \frac{0.002}{0.627} \approx 0.00319 \text{ K} \cdot \text{m}^2/\text{W} \].

Key Concepts

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. Imagine heat moving through substances just like water flowing through a pipe. Some materials allow heat to pass quickly and efficiently, like metals, which have high thermal conductivity. Others, such as wool or feathers, resist the flow of heat more and have low thermal conductivity. The unit of thermal conductivity is often expressed in watts per meter-kelvin ( W/m·K ) .

For the thermal resistance calculation, the thermal conductivity of human skin is given as 0.627 W/m·K. This means that human skin is moderately good at conducting heat, forming a decent barrier to prevent heat loss or gain. This property is crucial for maintaining body temperature. In problem-solving, including thermal conductivity in calculations helps us understand how effectively a material can transfer heat under certain conditions.

Specific Heat

Specific heat is an important property of materials that defines how much energy is needed to change the temperature of a given mass of the substance by a certain amount. Imagine it as the heat capacity per unit mass. A higher specific heat means the substance requires more energy to heat up or cool down, while a lower specific heat indicates the temperature can be easily changed with less energy.

In this exercise, human skin has a specific heat of 3470 J/kg·K. This is relatively high, suggesting that the skin can absorb quite a lot of heat energy before its temperature changes significantly. Specific heat is a key concept in understanding how different materials respond to heating and cooling, and it's especially important in applications related to thermal regulation, such as designing clothing or heating systems.

Density

Density is a measure of mass per unit volume, essentially describing how tightly packed the molecules in a substance are. High-density materials have molecules that are closely packed together, while low-density materials are more spread out. The density of human skin, given as 1100 kg/m³ in this exercise, means that the skin has a considerable amount of mass contained within a specific volume, allowing it to maintain structural integrity while covering the body.

Understanding density is important in materials engineering and thermal analysis because it affects how heat is conducted through a material. In the context of this problem, density doesn't directly factor into the thermal resistance calculation, but it's a vital property to consider when examining the overall thermal performance and mechanical properties of materials in practical applications.

Materials Engineering

Materials engineering involves studying and developing materials with specific properties to enhance their performance for various applications. It integrates concepts of thermal conductivity, specific heat, and density to tailor materials to specific needs. Whether this is for more efficient heat transfer, improved insulation, or better structural durability, materials engineering strives to optimize material properties.

In this exercise, understanding how human skin behaves in terms of thermal resistance offers insights into how engineers can design materials for clothing, devices, or even biomedical applications to interact effectively with human skin. The interplay of thermal conductivity, specific heat, and density informs engineers in making decisions about material choices, ensuring that their designs meet the physical and thermal requirements necessary in real-world use.