Problem 61 - University Physics with Modern Physics

Question

BIO Conduction Through the Skin. The blood plays an important role in removing heat from the body by bringing this energy directly to the surface where it can radiate away. Nevertheless, this heat must still travel through the skin before it can radiate away. Assume that the blood is brought to the bottom layer of skin at 37.0\(^\circ\)C and that the outer surface of the skin is at 30.0\(^\circ\)C. Skin varies in thickness from 0.50 mm to a few millimeters on the palms and soles, so assume an average thickness of 0.75 mm. A 165-lb, 6-ft-tall person has a surface area of about 2.0 m\(^2\) and loses heat at a net rate of 75 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·K.

Step 1: Understand the Thermal Conduction Formula

The formula for heat conduction is given by Fourier's law: \( Q = \frac{k \cdot A \cdot \Delta T}{d} \), where \( Q \) is the heat transfer rate (in watts), \( k \) is the thermal conductivity (in W/m·K), \( A \) is the area (in m²), \( \Delta T \) is the temperature difference (in K), and \( d \) is the thickness of the material (in meters).

Step 2: Identify the Given Values

From the problem, we have the following:- Heat transfer rate, \( Q = 75 \) Watts- Area, \( A = 2.0 \) m²- Temperature difference, \( \Delta T = 37.0 - 30.0 = 7.0 \) °C (or K, since the Kelvin degree difference is the same as Celsius)- Thickness, \( d = 0.75 \) mm = 0.00075 m.

Step 3: Rearrange the Equation for Thermal Conductivity

We need to solve for \( k \), the thermal conductivity. Rearrange the equation: \[ k = \frac{Q \cdot d}{A \cdot \Delta T} \]

Step 4: Substitute the Values into the Equation

Substitute the given values into the rearranged formula:\[ k = \frac{75 \cdot 0.00075}{2.0 \cdot 7.0} \]

Step 5: Calculate the Thermal Conductivity

Perform the calculation:\[ k = \frac{75 \cdot 0.00075}{14} = \frac{0.05625}{14} \approx 0.00402 \text{ W/m·K} \]

Key Concepts

Fourier's Law

Fourier's Law is a core principle in understanding heat conduction. It describes the transfer of thermal energy through solid materials. The law can be represented by the formula:

• \( Q = \frac{k \cdot A \cdot \Delta T}{d} \)
• Where \( Q \) is the heat transfer rate, \( A \) the area, \( \Delta T \) the temperature difference, \( d \) the thickness, and \( k \) is the thermal conductivity.

It helps us answer how much heat will flow through a material when there is a temperature difference across it.
By understanding Fourier's Law, you can predict how quickly heat will leave the body through the skin, providing insights into how the human body regulates temperature.

Thermal Conductivity

Thermal conductivity is a measure of a material's ability to conduct heat. In the context of skin, it refers to how efficiently heat from the blood is transferred through the skin layers.
A high thermal conductivity means heat transfers quickly, which is crucial for maintaining a stable core body temperature.

• In our example, the calculated thermal conductivity of skin is approximately 0.00402 W/m·K.
• This value represents how heat transfers from the inner layers of skin to the outer environment.

Skin's thermal conductivity helps regulate body temperature and ensures energy balance within the human body.

Skin Physiology

Skin physiology plays a pivotal role in heat regulation and overall thermodynamics of the body. The skin acts as a barrier and a conductor. Understanding its structure helps in comprehending heat conduction.
The skin has several layers, each affecting how heat is transferred:

• The epidermis (outermost layer) and dermis (beneath the epidermis).
• Heat is carried to the bottom layer of the skin by blood at about 37.0\( ^\circ \)C.
• The outer skin surface temperature is typically around 30.0\( ^\circ \)C.

These layers are crucial in the process of radiating heat away from the body, thereby impacting the thermal balance and conduction rates through the skin.

Temperature Difference

Temperature difference is a key factor in the rate of heat transfer across materials. It is the driving force in Fourier’s Law determining the direction and efficiency of heat flow.
In our scenario, the temperature difference across the skin is:

• \( 37.0 \degree\text{C} - 30.0 \degree\text{C} = 7.0 \degree\text{C} \) (or in Kelvin, \( 7.0 \text{K} \)).
• A larger temperature difference would typically result in a faster rate of heat transfer.

Hence, the greater the difference, the more rapidly heat will flow, playing a crucial role in body heat dissipation etiquette.

Biological Thermodynamics

Biological thermodynamics is the study of energy transformation in biological systems, including how organisms regulate their body temperatures.
Heat conduction is a key part of this, as it allows for heat to be effectively transferred through the skin, critical for maintaining homeostasis.

• In humans, this involves blood flow bringing heat to the skin's surface.
• Thermodynamics principles explain how this energy exchange is balanced, allowing for protection against overheating.

Understanding these processes provides insights into how energy conservation mechanisms are employed by living creatures for survival.