Problem 54 - Physics

Question

The aorta carries blood away from the heart at a speed of about \(40 \mathrm{~cm} / \mathrm{s}\) and has a radius of approximately \(1.1 \mathrm{~cm}\). The aorta branches eventually into a large number of tiny capillaries that distribute the blood to the various body organs. In a capillary, the blood speed is approximately \(0.07 \mathrm{~cm} / \mathrm{s},\) and the radius is about \(6 \times 10^{-4} \mathrm{~cm} .\) Treat the blood as an incompressible fluid, and use these data to determine the approximate number of capillaries in the human body.

Step-by-Step Solution

Answer

There are approximately \(1.71 \times 10^9\) capillaries in the human body.

Step 1: Calculate the Cross-Sectional Area of the Aorta

The cross-sectional area of a circle is given by the formula \( A = \pi r^2 \). For the aorta, with a radius \( r = 1.1 \mathrm{~cm} \), the area is \( A_{aorta} = \pi (1.1)^2 \mathrm{~cm}^2 \).

Step 2: Calculate the Volume Flow Rate in the Aorta

The flow rate (also known as volume flow rate) is obtained by multiplying the cross-sectional area by the velocity of the blood. Thus, for the aorta, the flow rate is \( Q_{aorta} = A_{aorta} \times v_{aorta} = \pi (1.1)^2 \times 40 \mathrm{~cm/s}\).

Step 3: Calculate the Cross-Sectional Area of a Single Capillary

Using the same formula for the area of a circle \( A = \pi r^2 \), for a capillary with radius \( r = 6 \times 10^{-4} \mathrm{~cm} \), the area is \( A_{capillary} = \pi (6 \times 10^{-4})^2 \mathrm{~cm}^2 \).

Step 4: Set up the Equation for Total Capillary Flow Rate

Since the blood is incompressible, the total flow rate in the aorta equals the sum of flow rates through all capillaries. If there are \( n \) capillaries, the total flow rate through the capillaries is \( n \times A_{capillary} \times v_{capillary} \).

Step 5: Solve for the Number of Capillaries

Set the total aorta flow rate equal to the sum of capillary flow rates and solve for \( n \):\[ \pi (1.1)^2 \times 40 = n \times \pi (6 \times 10^{-4})^2 \times 0.07 \]Solve for \( n \) to find the number of capillaries:\[ n = \frac{\pi (1.1)^2 \times 40}{\pi (6 \times 10^{-4})^2 \times 0.07} \]

Step 6: Calculate the Numerical Solution

Calculate the value of \( n \) using the equation from Step 5. Use a calculator to simplify the values:\[ n \approx \frac{3.799 \times 40}{1.272 \times 10^{-7} \times 0.07} \]\[ n \approx \frac{151.96}{8.904 \times 10^{-9}} \]\( n \approx 1.71 \times 10^9 \).

Key Concepts

Blood Flow

Blood flow is an essential component of the circulatory system, where the heart pumps blood through vessels, delivering oxygen and nutrients to tissues throughout the body. Understanding how blood moves through this system helps us appreciate the dynamics of circulation.
Blood is driven by a pressure gradient initiated by the heart and travels through a network of vessels, including arteries, veins, and capillaries. These vessels vary in size and structure, facilitating different roles in the transport and exchange of substances.
Blood flow can be described using terms such as velocity and volume, both of which play a crucial role in calculating how blood is distributed among various vessels, such as the aorta and numerous capillaries.

Aorta

The aorta is the primary artery in the human body, responsible for carrying oxygenated blood from the heart to various regions. Its large size and robust walls allow it to efficiently handle high-pressure blood directly pumped from the heart.
The cross-sectional area of the aorta plays a critical role in determining the speed and volume of blood flow. Using the formula for the area of a circle, the area of the aorta can be calculated with the radius information provided.

• Speed through the aorta is relatively high compared to other vessels due to its larger size.
• Blood leaves the aorta and flows into smaller arteries, eventually reaching capillaries where exchange occurs.

Understanding the aorta's function is crucial for insight into how blood movement is initiated within the circulatory system.

Capillaries

Capillaries are the smallest blood vessels in the circulatory system, where the exchange of oxygen, carbon dioxide, nutrients, and waste products occurs. These tiny structures form extensive networks, reaching almost every cell in the body.
Blood flows slower through capillaries compared to the aorta, which allows time for exchange processes to take place efficiently. The tiny radius of capillaries results in a small cross-sectional area.
Capillaries maintain a balance between the incoming and outgoing volume of blood, thanks to their ability to form extensive parallel networks that equate the pressure differentials from larger vessels like the aorta. This ensures efficient circulation throughout the body.

Incompressible Fluid

In fluid dynamics, blood is often treated as an incompressible fluid due to its consistent density and volume within the circulatory system. This simplifies the mathematical modeling of blood flow.
An incompressible fluid maintains constant volume, meaning that what enters a vessel must exit at an equivalent rate. This principle is essential for solving problems that involve the distribution of blood between large vessels like the aorta and small ones like capillaries.
This concept is fundamental in equations that calculate flow rates, as it allows us to equate the flow rates of blood moving through different sections of the circulatory system, such as the aorta and capillaries.

Volume Flow Rate

Volume flow rate, often denoted as Q, is a measure of how much blood passes through a cross-section of a vessel in a given unit of time. It is calculated using the formula: \[ Q = A \times v \] where \( A \) is the cross-sectional area and \( v \) is the velocity of blood.
This measure is crucial for understanding how blood moves between different parts of the circulatory system. For example, the flow rate in the aorta must equal the combined flow rate through all capillaries, maintaining a consistent and balanced circulation pattern.

• Volume flow rate helps determine how many capillaries are required to distribute blood effectively.
• Analyzing this rate provides insights into the efficiency and functionality of the circulatory system as a whole.

Understanding volume flow rate is essential in physiologically modeling blood distribution in the human body.