Problem 52 - College Physics

Question

Conduction through the skin. 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 \(0.75 \mathrm{mm} . \mathrm{A} 165 \mathrm{lb}, 6 \mathrm{ft}\) 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 0.004 W/m·K.

Step 1: Identify Known Values

We have the following known values:- Inner temperature of the skin, \( T_{ ext{inner}} = 37^{\circ}\, \text{C} \)- Outer temperature of the skin, \( T_{ ext{outer}} = 30^{\circ}\, \text{C} \)- Skin thickness, \( d = 0.75\, \text{mm} = 0.00075\, \text{m} \)- Surface area over which heat is lost, \( A = 2.0\, \text{m}^2 \)- Heat loss rate, \( P = 75\, \text{W} \)

Step 2: Understand Heat Conduction Formula

The heat conduction through a material is described by Fourier's law:\[ P = \frac{k \cdot A \cdot (T_{\text{inner}} - T_{\text{outer}})}{d} \]where \( P \) is the power (heat per time), \( k \) is the thermal conductivity, \( A \) is the area, \( T_{\text{inner}} \) and \( T_{\text{outer}} \) are the temperatures inside and outside the skin respectively, and \( d \) is the thickness of the material.

Step 3: Solve for Thermal Conductivity

Rearrange Fourier's law to solve for the thermal conductivity, \( k \):\[ k = \frac{P \cdot d}{A \cdot (T_{\text{inner}} - T_{\text{outer}})} \]Substitute the known values into the formula:\[ k = \frac{75\, \text{W} \cdot 0.00075\, \text{m}}{2.0\, \text{m}^2 \cdot (37^{\circ} \text{C} - 30^{\circ} \text{C})} \]

Step 4: Calculate the Thermal Conductivity

Calculate the difference in temperature:\[ (T_{\text{inner}} - T_{\text{outer}}) = 37 - 30 = 7^{\circ} \text{C} \]Substitute into the equation:\[ k = \frac{75 \cdot 0.00075}{2.0 \cdot 7} \]Calculate the value:\[ k = \frac{0.05625}{14} \approx 0.004015 \]Thus, the thermal conductivity of the skin is approximately \( 0.004 \text{ W/m} \cdot \text{K} \).

Key Concepts

Understanding Heat Conduction

Heat conduction is a fundamental concept that explains how heat energy transfers through materials. Imagine holding one end of a metal rod while the other end is heated. Eventually, you'll feel the warmth traveling through the rod to your hand. This process occurs because the heat energy moves from the hot part to the cooler part of the rod. In the case of skin, heat moves from the inner body to the outer surface. This transfer occurs because the inner body is warmer than the cool surrounding environment. During this transfer, kinetic energy is passed between molecules, facilitating the movement of heat towards cooler areas. This movement continues until thermal equilibrium is reached, meaning temperatures become even. It's fascinating to see this principle applied to our own bodies as it helps regulate our internal temperature efficiently.

Fourier's Law and Its Role

Fourier's law is an essential tool for understanding heat conduction quantitatively. Jean-Baptiste Joseph Fourier, a French mathematician and physicist, formulated this law. It provides a way to calculate the rate of heat transfer through a material given certain conditions. The law is expressed with the formula: \[P = \frac{k \cdot A \cdot (T_{\text{inner}} - T_{\text{outer}})}{d}\]Here:

• \( P \) represents the rate of heat transfer or power.
• \( k \) is the thermal conductivity of the material.
• \( A \) indicates the area through which heat is transferred.
• \((T_{\text{inner}} - T_{\text{outer}})\) denotes the temperature difference across the material.
• \( d \) is the material's thickness.

By rearranging this formula, we can solve for any unknown variable, such as the thermal conductivity, making Fourier’s law a versatile and powerful tool in thermodynamics.

The Interplay of Biophysics and Heat Transfer

Biophysics merges the principles of physics with biological systems to understand various phenomena such as heat conduction in human skin. Human skin acts as a barrier while also being a key site for heat exchange with the environment. Blood brings heat from the body's core to the skin surface, where it dissipates into the cooler external environment. This regulation is crucial in maintaining a stable internal body temperature and is a fine example of biophysics in action. The skin's ability to conduct heat, determined by its thermal conductivity, is vital in this process. Things like skin thickness and surface area affect how efficiently this heat transfer happens. Understanding these factors helps improve medical strategies for dealing with issues like overheating or hypothermia.

Examining the Temperature Gradient

The temperature gradient is a term used to describe the rate of temperature change across a specific distance. It plays a significant role in heat conduction, especially when applying Fourier’s law. In the context of skin, the gradient is the difference in temperature between the inner and outer layers, divided by the thickness of the skin. For skin:

• Inner temperature: \(37^{\circ}\text{C}\)
• Outer temperature: \(30^{\circ}\text{C}\)
• Temperature gradient: \((37 - 30)^{\circ}\text{C} = 7^{\circ}\text{C}\)

A steeper gradient means more heat is transferred from the body to the environment, which is crucial for effective temperature regulation. Understanding how the gradient influences heat conduction allows for better insights into thermal properties and processes, aiding in practical applications like skin treatment and energy management techniques.