Problem 84 - Fundamentals of Physics

Question

When you cough, you expel air at high speed through the trachca and upper bronchi so that the air will remove cxcess mucus lining the pathway. You produce the high speed by this procedure: You brcathe in a large amount of air, trap it by closing the glottis (the narrow opening in the larynx ), increase the air pressure by contracting the lungs, partially collapse the trachca and upper bronchi to narrow the pathway, and then expel the air through the pathway by suddenly reopening the glottis. Assume that during the expulsion the volume flow rate is \(7.0 \times 10^{-7} \mathrm{~m}^{3} / \mathrm{s}\). What multiple of \(343 \mathrm{~m} / \mathrm{s}\) (the specd of sound \(v_{s}\) ) is the airspeed through the trachea if the trachea diameter (a) remains its normal value of \(14 \mathrm{~mm}\) and (b) contracts to \(5.2 \mathrm{~mm}\) ?

Step-by-Step Solution

Answer

For 14 mm diameter, the airspeed is approximately 1.01 times the speed of sound. For 5.2 mm diameter, it's approximately 2.7 times the speed of sound.

Step 1: Understand the Problem

We need to find the airspeed in the trachea when the diameter is either 14 mm or 5.2 mm. The goal is to express the airspeed as a multiple of the speed of sound, which is 343 m/s.

Step 2: Use the Continuity Equation

The volume flow rate (Q) can be related to the speed (v) and cross-sectional area (A) of the trachea: \ \( Q = A \cdot v \). We have the volume flow rate \( Q = 7.0 \times 10^{-7} \, \mathrm{m}^3/\mathrm{s} \).

Step 3: Calculate Cross-Sectional Area for 14 mm Diameter

For a circular cross-section, the area \( A \) is \( \pi r^2 \), where \( r \) is the radius. If the diameter is 14 mm (or 0.014 m), then the radius \( r = 0.007 \) m. So, \ \( A = \pi (0.007)^2 \).

Step 4: Calculate Airspeed for 14 mm Diameter

Rearrange \( Q = A \cdot v \) to solve for \( v \): \ \( v = \frac{Q}{A} \). Substitute values: \ \( v = \frac{7.0 \times 10^{-7}}{\pi (0.007)^2} \).

Step 5: Express Airspeed as a Multiple of Speed of Sound for 14 mm

Calculate \( \frac{v}{343} \) to express the airspeed as a multiple of the speed of sound.

Step 6: Calculate Cross-Sectional Area for 5.2 mm Diameter

For a diameter of 5.2 mm (or 0.0052 m), the radius \( r = 0.0026 \) m. Thus, \ \( A = \pi (0.0026)^2 \).

Step 7: Calculate Airspeed for 5.2 mm Diameter

Rearrange \( Q = A \cdot v \) to solve for \( v \): \ \( v = \frac{Q}{A} \). Substitute values: \ \( v = \frac{7.0 \times 10^{-7}}{\pi (0.0026)^2} \).

Step 8: Express Airspeed as a Multiple of Speed of Sound for 5.2 mm

Calculate \( \frac{v}{343} \) to express the airspeed as a multiple of the speed of sound.

Key Concepts

Lung Function

The human respiratory system is a fascinating combination of various physiological processes that facilitate breathing and effective gas exchange. One crucial aspect of lung function involves the ability to expel air forcefully, as during a cough. Coughing is vital for clearing excess mucus or foreign particles from the trachea and bronchi, keeping airways clear.

• During a cough, you first take a deep breath to fill your lungs with air.
• The epiglottis then closes, trapping the air inside the lungs.
• Next, the respiratory muscles contract, increasing the pressure on the trapped air.
• By abruptly opening the glottis, this compressed air is rapidly expelled.

This mechanism ensures that any blockage is pushed out along with the fast-moving air, helping in respiratory sanitation. Understanding how lung function contributes to air expulsion during a cough can help us optimize breathing exercises and treatments for respiratory conditions.

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics. It states that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another if the pipe is closed and no fluid is added or removed. Mathematically, it is represented as:
\[ A_1 v_1 = A_2 v_2 \] where:

• \(A_1\) and \(A_2\) are the cross-sectional areas at the two points, and
• \(v_1\) and \(v_2\) are the fluid velocities at these points.

For the coughing scenario:

• The equation is simplified to \( Q = A \cdot v \) because air density does not change significantly over short distances, allowing us to focus on volumetric flow rate \( Q \).
• This formula helps us calculate the velocity of air through the trachea using the known flow rate and trachea diameter.

Understanding the continuity equation is essential for solving problems involving varying speeds and cross-sectional areas in fluid flow.

Airspeed Calculation

Calculating the airspeed in scenarios like this involves using the cross-sectional area of the trachea and the known volumetric flow rate. The procedure to calculate airspeed involves several steps:

• First, determine the cross-sectional area of the trachea. For example, if the diameter is 14 mm, the radius will be half of that, 0.007 meters. The area then becomes \( \pi r^2 \).
• With the area determined, rearrange the continuity equation \( Q = A \cdot v \) to solve for airspeed \( v \).
• Substitute the values into \( v = \frac{Q}{A} \).
• Given a volumetric flow rate of \( 7.0 \times 10^{-7} \, \text{m}^3/\text{s} \), plug this into the equation to get the airspeed.

This step-by-step calculation is crucial in determining how fast air moves through constricted or dilated passages, providing insights into the efficiency of coughing as a clearing mechanism.

Speed of Sound

The speed of sound is a fundamental concept in physics, describing how fast sound waves travel through a medium. In air at room temperature, this speed is approximately 343 meters per second. The speed of sound depends on the medium's properties, such as temperature and density. In the context of the exercise, we're interested in comparing the airspeed in the trachea to this benchmark:

• The calculated airspeed during a cough is compared to the speed of sound by finding the ratio \( \frac{v}{v_s} \). This tells us how many times faster or slower the expelled air is compared to sound.
• Airspeed faster than sound would suggest supersonic flow, often leading to shock waves, but typically, air expelled during a cough is subsonic.

Understanding how to relate fluid speeds to the speed of sound is essential in biofluid mechanics and has applications in respiratory health studies.