Problem 3 - Physics

Question

The amount of heat per second conducted from the blood capillaries beneath the skin to the surface is \(240 \mathrm{~J} / \mathrm{s}\). The energy is transferred a distance of \(2.0 \times 10^{-3} \mathrm{~m}\) through a body whose surface area is \(1.6 \mathrm{~m}^{2}\). Assuming that the thermal conductivity is that of body fat, determine the temperature difference between the capillaries and the surface of the skin.

Step-by-Step Solution

Answer

The temperature difference is 1.5°C.

Step 1: Understand the Formula for Heat Conduction

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 transferred per unit time (240 J/s), \(k\) is the thermal conductivity of body fat, \(A\) is the area through which the heat is conducted (1.6 m²), \(\Delta T\) is the temperature difference, and \(d\) is the thickness (2.0 \times 10^{-3} m). We are given all values except for \(\Delta T\) and \(k\). We need to find \(\Delta T\) using the known value of \(k\).

Step 2: Find the Thermal Conductivity of Body Fat

The thermal conductivity \(k\) for body fat is approximately \(0.2 \text{ W/m} \cdot \text{°C}\). This value is typically provided in tables of thermal properties. We'll use this value to calculate the temperature difference.

Step 3: Rearrange the Formula to Solve for Temperature Difference

Rearrange the formula to find \(\Delta T\):\[ \Delta T = \frac{Q \cdot d}{k \cdot A} \]Substitute the known values to solve for \(\Delta T\).

Step 4: Substitute Known Values

Now, substitute the known values into the rearranged formula:\[ \Delta T = \frac{240 \, \text{J/s} \cdot 2.0 \times 10^{-3} \, \text{m}}{0.2 \, \text{W/m°C} \cdot 1.6 \, \text{m}^2} \]

Step 5: Calculate the Temperature Difference

Perform the calculation:\[ \Delta T = \frac{240 \, \text{J/s} \cdot 2.0 \times 10^{-3} \, \text{m}}{0.2 \, \text{W/m°C} \cdot 1.6 \, \text{m}^2} = \frac{0.48}{0.32} = 1.5 \, \text{°C} \]The temperature difference between the capillaries and the skin surface is 1.5°C.

Key Concepts

Thermal Conductivity

Thermal conductivity is a fundamental property that measures how well a material conducts heat. It tells us how easily heat can pass through a substance. In our scenario, we are dealing with body fat, which has a relatively low thermal conductivity of roughly 0.2 W/m°C. This low value indicates that body fat is not an efficient conductor of heat, making it useful for insulation in the human body.
Understanding the thermal conductivity allows us to predict how a substance will respond to different thermal conditions, like heating or cooling. It helps us calculate other important values in heat transfer processes, like the temperature difference between two points. Knowing the thermal conductivity of materials can help in designing clothing, buildings, or even assessing how well our bodies regulate temperature.
Typically, tables listing thermal conductivities for different substances are provided, which aids in conducting various engineering or physiological calculations.

Fourier's Law

Fourier's Law is a key principle used to describe the process of heat conduction. It is expressed by the formula: \[ Q = \frac{k \cdot A \cdot \Delta T}{d} \]Where:- \(Q\) is the heat transfer per unit time, measured in Joules per second (J/s).- \(k\) is the thermal conductivity of the material.- \(A\) is the cross-sectional area through which heat is being transferred.- \(\Delta T\) is the temperature difference across the material.- \(d\) is the thickness of the material.
In our example, we applied Fourier's Law to calculate the temperature difference between the blood capillaries and the skin surface by using known values of heat transfer rate, area, thickness, and thermal conductivity. This law is essential for understanding how heat moves through different substances in our environment.

Temperature Difference

The temperature difference, denoted as \( \Delta T \), is crucial in determining how heat moves between two points. It represents the driving force behind heat conduction, determining how fast or slow this process occurs. In our example, we calculated the temperature difference between skin layers using Fourier's Law, ending with an understanding of how heat moves from blood capillaries to the skin surface.
To find \( \Delta T \), we rearranged and used the formula from Fourier's Law: \[ \Delta T = \frac{Q \cdot d}{k \cdot A} \]We substituted values, resulting in \( \Delta T = 1.5^{\circ}C \), which shows a minor temperature change from capillaries to the skin. This indicates efficient regulation of body temperature, highlighting the body's protective layers that help maintain a stable internal environment.

Body Fat

Body fat plays a crucial role in the human body, acting as more than just a reserve of energy. It serves as insulation, regulating heat transfer between our bodies and the environment. Thanks to its low thermal conductivity, body fat helps slow down heat loss, maintaining body temperature despite external changes.
This insulating property means that body fat is efficient at keeping the body's core temperature stable, especially in colder environments. Understanding the role of body fat in thermal regulation can be vital for studying human physiology and health as well as understanding how various factors, such as body composition, can affect thermal comfort and energy balance. Overall, body fat contributes significantly to the body's ability to adapt to different environments by helping to control how quickly it loses or gains heat.