Problem 45 - College Physics

Question

(a) Calculate the mass flow rate (in grams per second) of blood \(\left(\rho=1.0 \mathrm{~g} / \mathrm{cm}^{3}\right)\) in an aorta with a crosssectional area of \(2.0 \mathrm{~cm}^{2}\) if the flow speed is \(40 \mathrm{~cm} / \mathrm{s}\). (b) Assume that the aorta branches to form a large number of capillaries with a combined cross-sectional area of \(3.0 \times 10^{3} \mathrm{~cm}^{2}\). What is the flow speed in the capillaries?

Step-by-Step Solution

Answer

The mass flow rate of blood in the aorta is calculated to be \( \dot{m} \) grams per second (from Step 2), and the flow speed in the capillaries is \( v \) cm/s (from Step 3).

Step 1: Calculation of the volume flow rate

To find the mass flow rate, first calculate the volume flow rate using the formula \( Q = A \cdot v \) where \( A = 2.0 \, cm^2 \) is the cross-sectional area and \( v = 40 \, cm/s \) is the velocity. Multiply the area and velocity to get the volume flow rate.

Step 2: Calculation of the mass flow rate

Next, calculate the mass flow rate, using the formula \( \dot{m} = \rho \cdot Q \), where \( \rho = 1.0 \, g/cm^3 \) is the density of the blood and \( Q \) is the volume flow rate calculated in the previous step. Multiply the density and volume flow rate to get the mass flow rate.

Step 3: Calculate the flow speed in the capillaries

For the second part of the problem, use the formula \( Q = A \cdot v \) again to find the flow speed in the capillaries. Here, \( Q \) is the same as in the first part (presuming the flow rate is constant along the course), and \( A = 3.0 \times 10^{3} \, cm^{2} \) is the combined cross-sectional area of the capillaries. Solve the equation for \( v \), the flow speed in the capillaries, by dividing the volume flow rate by the cross-sectional area.

Key Concepts

Mass Flow Rate

Mass flow rate is a crucial concept in fluid dynamics, as it helps us understand how much fluid passes through a given area in a set time period. It is commonly used in many fields, including blood flow dynamics. To find the mass flow rate, you need to multiply the density of the fluid by the volume flow rate. In our exercise, the formula used is \( \dot{m} = \rho \cdot Q \).

• \( \dot{m} \) represents the mass flow rate (in grams per second in this case).
• \( \rho \) is the density of the fluid (blood in this problem), given as \( 1.0 \, g/cm^3 \).
• \( Q \) is the volume flow rate.

Calculating the mass flow rate helps us quantify blood flow, ensuring that medical devices like pumps and artificial hearts function correctly.

Volume Flow Rate

Volume flow rate, often denoted as \( Q \), is the measure of how much volume of fluid flows through a cross-section of a passage per unit time. It is especially important when analyzing systems like blood flow in vessels. The formula we use is \( Q = A \cdot v \), where:

• \( A \) is the cross-sectional area of the vessel (in square centimeters).
• \( v \) is the flow velocity (in centimeters per second).

In the exercise, it's used to determine how quickly blood moves through the aorta and capillaries. Knowing the volume flow rate helps medical professionals calculate how efficiently blood is transported through the body, which is vital for patient health.

Cross-Sectional Area

Cross-sectional area plays a vital role in determining the flow characteristics in a vessel or pipe. It is essentially the area of the slice you would see if you cut through the vessel. This is mathematically straightforward, but its implications in fluid dynamics are significant. In the exercise,

• we have an aorta with a cross-sectional area of \( 2.0 \, cm^2 \)
• and capillaries with a combined area of \( 3.0 \times 10^3 \, cm^2 \).

The larger the cross-sectional area, the lower the velocity of the fluid for the same volume flow rate, which is summarized by the continuity equation. By carefully analyzing the cross-sectional area, it is possible to ensure that fluids like blood circulate effectively through various channels without causing undue stress on the walls of these passages.

Blood Flow

Blood flow refers to the movement of blood through the circulatory system and is essential for delivering oxygen and nutrients to cells while removing waste products. In our exercise, the flow of blood is examined through different segments of the circulatory system. Blood flow analysis involves several factors, including velocity, volume, and mass flow rate, all of which interdependently affect how blood circulates.
Important points to consider in blood flow:

• The aorta has a rapid blood flow due to its smaller cross-sectional area compared to capillaries.
• Increased cross-sectional area in capillaries results in a reduced flow speed, balancing pressure and ensuring efficient nutrient exchange.

Successfully understanding blood flow can help address cardiovascular health issues, contributing to more effective treatments and interventions.

Velocity

Velocity is a key component in fluid dynamics, referring to the speed and direction of fluid movement. It’s defined as the distance that the fluid travels per unit time. In the context of our blood flow problem, it affects how quickly blood moves through the aorta and capillaries.

• Initial velocity in the aorta is given as \( 40 \, cm/s \).
• When recalculating, changes in cross-sectional area affect this velocity in the capillaries.

By applying the volume flow rate calculation, which is constant, you can find changes in velocity as aorta branches into numerous capillaries. This aids in maintaining the delicate balance of pressures and nutrient exchange within the body. Mastery over these concepts allows for better planning and management in medical scenarios.