Problem 52 - College Physics

Question

The blood plays an important role in removing heat from the body by bringing this heat directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. We shall assume that the blood is brought to the bottom layer of skin at a temperature of \(37^{\circ} \mathrm{C}\) and that the outer surface of the skin is at \(30.0^{\circ} \mathrm{C}\). Skin varies in thickness from \(0.50 \mathrm{~mm}\) to a few millimeters on the palms and soles, so we shall assume an average thickness of \(0.75 \mathrm{~mm}\). A \(165 \mathrm{lb}, 6 \mathrm{ft}\) person has a surface area of about \(2.0 \mathrm{~m}^{2}\) and loses heat at a net rate of \(75 \mathrm{~W}\) while resting. On the basis of our assumptions, what is the thermal conductivity of this person's skin?

Step-by-Step Solution

Answer

The thermal conductivity of the skin is approximately 0.00402 W/m°C.

Step 1: Understanding the Heat Transfer

We are given that the temperature at the bottom layer of the skin is \(37^{\circ} \mathrm{C}\) and the temperature at the outer surface is \(30.0^{\circ} \mathrm{C}\). The skin loses heat at a net rate of \(75 \mathrm{~W}\), and the surface area through which heat transfer occurs is \(2.0 \mathrm{~m}^{2}\). The heat must travel through a skin thickness of \(0.75 \mathrm{~mm}\), which is \(0.00075\) meters.

Step 2: Using Fourier's Law of Heat Conduction

Fourier’s Law of heat conduction states that the rate of heat transfer \(Q\) through a material is proportional to the temperature difference \(\Delta T\) and area \(A\), and inversely proportional to the thickness \(d\). Mathematically, this is expressed as:\[Q = k \cdot A \cdot \frac{\Delta T}{d}\]where \(k\) is the thermal conductivity. We need to solve for \(k\). In this case, \(Q = 75 \mathrm{~W}\), \(A = 2.0 \mathrm{~m}^{2}\), \(\Delta T = 37^{\circ} \mathrm{C} - 30^{\circ} \mathrm{C} = 7^{\circ} \mathrm{C}\), and \(d = 0.00075\) meters.

Step 3: Solving for Thermal Conductivity

Plug in the known values into the equation:\[k = \frac{Q \cdot d}{A \cdot \Delta T} = \frac{75 \mathrm{~W} \cdot 0.00075 \; \mathrm{m}}{2.0 \; \mathrm{m}^2 \cdot 7^{\circ} \mathrm{C}}\]Calculate by substituting the values:\[k = \frac{75 \times 0.00075}{2.0 \times 7}\]\[k = \frac{0.05625}{14} \approx 0.00402 \; \mathrm{W/m \, ^{\circ}C}\]

Step 4: Conclusion and Final Answer

Based on the calculation, the thermal conductivity \(k\) of the skin is approximately \(0.00402 \; \mathrm{W/m \, ^{\circ}C}\). This indicates how efficiently the skin can conduct heat from inside the body to the surface to radiate away.

Key Concepts

Fourier's Law of Heat Conduction

Fourier's Law of Heat Conduction is a fundamental principle in thermal physics that explains how heat flows through a material. According to this law, the rate of heat transfer through a material, often denoted as \(Q\), is directly proportional to the temperature difference across the material and the surface area through which heat flows. On the other hand, it is inversely proportional to the thickness of the material. Hence, the thicker the material, the slower the heat will travel through it. Mathematically, the formula is expressed as:

• \[Q = k \cdot A \cdot \frac{\Delta T}{d}\]

where \(k\) is the thermal conductivity of the material, \(A\) is the cross-sectional area, \(\Delta T\) is the temperature difference, and \(d\) is the material's thickness.
Using this law helps determine how materials conduct heat, which is essential for understanding thermal insulation and energy transfer on a broad scale. It provides insight into the efficiency of different materials in conducting heat and is crucial not only in theoretical physics but in practical applications like designing better-insulating building materials or clothing for extreme temperatures.

Heat Transfer

Heat transfer is the process by which thermal energy moves from one object or substance to another. In this specific exercise, heat is transferred from the body's core to the skin's surface through conduction. This type of heat transfer occurs when there is direct contact between particles, where energy is transferred through collisions and molecular motion.
Transferring heat efficiently is critical for maintaining body temperature, as seen in the example where blood delivers heat to the skin's surface. The skin acts as a conduit through which heat dissipates into the surrounding environment, helping to regulate body temperature.
It's important to differentiate between the three main methods of heat transfer:

• Conduction: Transfer of heat through direct contact, as in the skin's example.
• Convection: Heat transfer through fluids like air or water, often resulting in heat circulation.
• Radiation: Transfer of energy through electromagnetic waves without the need for a medium.

By understanding these distinctions, students can better appreciate how heat flows in different environments and applications.

Skin Thickness

Skin thickness plays a crucial role in determining how effectively the skin conducts heat. In our problem scenario, the skin is treated as a barrier through which thermal energy travels before it can be lost to the air. The average skin thickness assumed here is \(0.75\) mm or \(0.00075\) meters, which significantly influences the overall heat transfer rate because it is a part of the denominator in Fourier's Law.
The variation in skin thickness across different parts of the body means that heat conduction will vary depending on where on the body the measurement is taken. For example, thicker skin areas like the soles and palms will have a different rate of heat transfer than thinner areas.
To account for this variability, average values are often used in calculations. Understanding how skin thickness affects heat flow is valuable not only for physiological studies but also for applications like medical diagnostics and designing apparel intended for temperature regulation.

Temperature Gradient

A temperature gradient is the change in temperature observed over a specific distance. It is a key determinant in heat conduction as it quantifies the driving force behind the flow of thermal energy. In our example, a temperature gradient exists between the blood at the skin's base \(37^{\circ} \mathrm{C}\) and the skin's outer layer \(30.0^{\circ} \mathrm{C}\).
This gradient encourages heat to move from areas of higher temperature to areas of lower temperature, in keeping with the second law of thermodynamics. The steeper the temperature gradient, the faster the rate of heat transfer, following Fourier’s equation.
In applications beyond physiology, understanding the concept of temperature gradients is important for many fields. For instance, when designing thermal insulation, a designer would aim to minimize the temperature gradient across the insulating material to reduce heat loss. Similarly, in meteorology, temperature gradients help explain weather patterns and predict environmental changes.