Question

a) Calculate the rate at which body heat flows out through the clothing of a skier, given the following data: the body surface area is $1.8{\mathrm{m}}^2$ and the clothing is $1.2\mathrm{cm}$ thick; skin surface temperature is ${33}^{\circ }\mathrm{C}$, whereas the outer surface of the clothing is at ${1.0}^{\circ }\mathrm{C};$ the thermal conductivity of the clothing is $0.040\mathrm{W}/\mathrm{m}\cdot \mathrm{K}.(b)$ How would the answer change if, after a fall, the skier's clothes become soaked with water? Assume that the thermal conductivity of water is $0.60\mathrm{W}/\mathrm{m}\cdot \mathrm{K}$

Short answer

The initial heat flow is 192 W. If the skier's clothes become soaked with water, the heat flow increases to 2880 W.

Step-by-step solution

Calculate the initial heat flow

The initial heat flow can be calculated using the formula $Q=kA\frac{\mathrm{\Delta }T}{d}$. Here, $k=0.040$ W/m·K, $A=1.8$ m², $\mathrm{\Delta }T=33$ C - $1$ C = 32 C, and $d=1.2$ cm = $0.012$ m. Plugging in these values gives $Q=0.040$ W/m·K $\times 1.8$ m² $\times 32$ C / $0.012$ m = 192 W.

Calculate the heat flow when the clothes are soaked

When the clothes are soaked, the thermal conductivity changes. The new heat flow can be calculated using the same formula as above with the new thermal conductivity value $k=0.60$ W/m·K. Plugging in these new values gives $Q=0.60$ W/m·K $\times 1.8$ m² $\times 32$ C / $0.012$ m = 2880 W.

Comparison

Comparing the two calculated values, the heat flow when the skier's clothes become soaked with water (2880 W) is greater than when they are dry (192 W), indicating that the skier would lose heat more quickly when their clothes are wet. This is because water has a higher thermal conductivity than the dry clothing material.

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in physics that describes how heat energy moves from one place to another. This can occur through three primary methods: conduction, convection, and radiation. In the given exercise, we focus on conduction.

Conduction is the process where heat is transferred through a material without the substance itself moving. This happens when there's a temperature difference across the material. For instance, in our skier's clothing, heat from the skier's warm body conducts through the clothing to the cooler outer surface.

Key factors influencing heat transfer through conduction include:

• Material's thermal conductivity ($k$): This is a measure of how well heat passes through a material. Higher values mean better heat transfer.
• Temperature difference ($\mathrm{\Delta }T$): Heat transfers more rapidly when the temperature difference between two surfaces is larger.
• Cross-sectional area ($A$): Larger areas allow more heat to pass through.
• Thickness of the material ($d$): Heat must travel through this distance. Thicker materials slow down the transfer of heat.

Understanding these concepts can help in designing clothing for skiers that minimizes unwanted heat loss.

Thermal Insulation

Thermal insulation refers to materials and methods used to reduce the rate of heat transfer. In our skier's situation, clothing acts as insulation, helping to retain body heat and prevent it from escaping into the cold environment.

Good thermal insulation has a low thermal conductivity value, meaning it transfers heat slowly. The thickness of the insulating material is also crucial. The thicker the clothing, the slower the heat transfer due to an increased distance over which the temperature difference must overcome.

When deciding on materials for thermal insulation, you might consider these factors:

• Material properties: Fabrics like wool and synthetic fibers can trap air, increasing their insulating capabilities.
• Environmental conditions: Wet conditions, like our skier encountered, can drastically degrade insulation. Water fills air spaces and increases thermal conductivity, allowing heat to escape more easily.

Keeping insulated can mean the difference between comfort and discomfort, so choosing the right materials for specific conditions is essential.

Rate of Heat Flow

The rate of heat flow is a quantitative measure of how much heat energy moves through a material over time. It's calculated using the formula $$Q=kA\frac{\mathrm{\Delta }T}{d}$$

In this formula:

• $Q$ represents the rate of heat flow.
• $k$ is the thermal conductivity.
• $A$ is the area through which heat flows.
• $\mathrm{\Delta }T$ is the temperature difference across the material.
• $d$ is the thickness of the material.

In the skier example, with dry clothing, the rate of heat flow was 192 W, which means 192 Joules of heat energy was lost each second.

However, when the clothing was wet, this increased drastically to 2880 W due to water's higher thermal conductivity. This means much more heat escapes quickly, making it harder to maintain warmth.

High rates of heat flow can signify insufficient insulation, crucially impacting comfort and safety in cold conditions. By understanding these rates, we can better control how we conserve heat, helping in decision-making for proper clothing design.