Problem 60 - College Physics

Question

The speed of blood in a major artery of diameter \(1.0 \mathrm{~cm}\) is \(4.5 \mathrm{~cm} / \mathrm{s}\). (a) What is the flow rate in the artery? (b) If the capillary system has a total cross-sectional area of \(2500 \mathrm{~cm}^{2}\), the average speed of blood through the capillaries is what percentage of that through the major artery? (c) Why must blood flow at low speed through the capillaries?

Step-by-Step Solution

Answer

(a) Flow rate is approximately 3.54 cm³/s. (b) Speed in capillaries is about 0.0315% of that in the artery. (c) Slow speed allows efficient nutrient and gas exchange.

Step 1: Understand Flow Rate Equation

To find the flow rate (Q), we'll use the equation: \[ Q = A \cdot v \] where \( A \) is the cross-sectional area of the artery and \( v \) is the speed of blood.

Step 2: Calculate Cross-Sectional Area of Artery

The cross-sectional area \( A \) for a circular artery is given by \[ A = \pi r^2 \] where \( r \) is the radius of the artery. Since the diameter is \( 1.0 \, \text{cm} \), the radius is \( 0.5 \, \text{cm} \). Thus, \[ A = \pi \cdot (0.5)^2 = \frac{\pi}{4} \, \text{cm}^2 \]

Step 3: Calculate Flow Rate in the Artery

Using the area from Step 2, the flow rate \( Q \) is calculated by \[ Q = \frac{\pi}{4} \cdot 4.5 \, \text{cm/s} = \frac{4.5\pi}{4} \, \text{cm}^3/\text{s} \approx 3.54 \, \text{cm}^3/\text{s} \]

Step 4: Relate Speeds in Capillaries to Artery

Given that the flow rate should be consistent, we find the average speed of blood in the capillaries using the equation for flow rate: \( Q = A \cdot v \). The area of the capillaries \( A_c = 2500 \text{ cm}^2 \). The speed in capillaries \( v_c = \frac{Q}{A_c} \).

Step 5: Calculate the Speed in Capillaries

Substitute values to find \( v_c \) in capillaries:\[ v_c = \frac{3.54}{2500} \approx 0.001416 \, \text{cm/s} \]

Step 6: Find the Percentage Speed in Capillaries Relative to Artery

The percentage speed in capillaries compared to the artery is:\[ \text{Percentage} = \left( \frac{v_c}{v_a} \right) \times 100 = \left( \frac{0.001416}{4.5} \right) \times 100 \approx 0.0315\% \]

Step 7: Discuss Why Blood Flows Slow in Capillaries

Blood flow must be slow in capillaries to allow for adequate exchange of nutrients, gases, and waste between blood and tissues. This slow rate maximizes diffusion efficiency across the capillary walls.

Key Concepts

Blood Speed in Arteries

Blood speed in arteries is an important concept in understanding cardiovascular function. Arteries are the vessels that carry oxygen-rich blood away from the heart to the rest of the body. Because they are situated closer to the heart, blood moves through arteries with a fairly high speed.
The speed of blood (\( v \) ) in an artery depends on how much the heart pumps per minute, known as cardiac output, and the diameter of the artery.
In simpler terms:

• Higher cardiac output: Faster blood speed
• Larger diameter: Lower speed, due to wider space for the blood to flow through

Knowing the speed, we can calculate the flow rate, which is the volume of blood flowing through per second. This is crucial for ensuring tissues receive enough oxygen and nutrients.

Cross-Sectional Area Calculation

To understand blood flow properly, we also need to grasp how to calculate the cross-sectional area of blood vessels. For arteries and most blood vessels, this area can be calculated assuming a circular shape.
The formula for the cross-sectional area (\( A \) ) is:\[ A = \pi r^2 \]Where \( r \) is the radius of the artery. If you know the diameter (\( d \) ), simply divide it by two to get the radius.

• For example, if the diameter is 1.0 cm, the radius is 0.5 cm, making \( A = \pi (0.5)^2 \).
• This formula is key for finding flow rate, since \( Q = A \cdot v \). With area and speed known, the flow rate can be derived.

Blood Flow in Capillaries

Capillaries are tiny blood vessels where the actual exchange of gases, nutrients, and waste takes place. Unlike arteries, blood speed in capillaries is much slower.
The entirety of capillary blood flow calculations rely on the principle of consistent flow rate throughout the system, despite the vessels’ different sizes.
Capillaries have the largest total cross-sectional area in the circulatory system.
This means that, even though the same volume of blood is passing through, the speed must decrease significantly in the capillaries:

• Higher total area = Slower speed (since \( v = \frac{Q}{A} \) )
• This ensures sufficient time for exchange of materials, optimizing physiological function.

Diffusion Efficiency in Capillaries

Efficiency in the capillary network is all about optimizing the diffusion process— the movement of molecules from high to low concentration areas.
Slow blood flow in capillaries increases the efficiency of this crucial exchange. Here’s why:

• At a slower speed, red blood cells have maximum contact time with capillary walls.
• Longer exchange time allows more oxygen and nutrients to diffuse into tissues, and waste products to diffuse into the blood.
• The thin walls of capillaries support rapid and effective diffusion, ensuring cells sustain life processes.

This diffusion efficiency is vital for the organism's overall health, supporting energy production, and waste elimination processes.