Problem 84 - Fundamentals of Physics

Question

When you cough, you expel air at high speed through the trachea and upper bronchi so that the air will remove excess mucus lining the pathway. You produce the high speed by this procedure: You breathe 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 trachea 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 \(7.0 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}\). What multiple of \(343 \mathrm{~m} / \mathrm{s}\) (the speed of sound \(v_{s}\) ) is the airspeed through the trachea if the trachea diameter (a) remains its normal value of \(14 \mathrm{~mm}\) and \((\mathrm{b})\) contracts to \(5.2 \mathrm{~mm}\) ?

Step-by-Step Solution

Answer

(a) Airspeed is 0.132 times the speed of sound. (b) Airspeed is 0.963 times the speed of sound.

Step 1: Understand the Problem

The problem involves calculating the airspeed through the trachea during a cough. We are given the volume flow rate and need to find the velocity of air using the cross-sectional area of the trachea, which changes in parts (a) and (b). The result will be expressed as a multiple of the speed of sound.

Step 2: Calculate Cross-Sectional Area

For the trachea with diameter \(d\), the cross-sectional area \(A\) is given by \[ A = \pi \left(\frac{d}{2}\right)^2 \]. Start with the normal diameter: \(d = 14\, \mathrm{mm} = 0.014\, \mathrm{m}\).Thus, \[ A = \pi \left(\frac{0.014}{2}\right)^2 = 1.54 \times 10^{-4} \mathrm{~m}^2. \]

Step 3: Calculate Airspeed for Normal Diameter (a)

Using the volume flow rate \( Q = 7.0 \times 10^{-3} \mathrm{~m}^3/s \), the airspeed \( v \) is \[ v = \frac{Q}{A} = \frac{7.0 \times 10^{-3}}{1.54 \times 10^{-4}} \approx 45.45 \mathrm{~m/s}. \]The multiple of the speed of sound \( v_s = 343 \mathrm{~m/s} \) is \[ \frac{v}{v_s} = \frac{45.45}{343} \approx 0.132. \]

Step 4: Calculate Cross-Sectional Area for Contracted Diameter (b)

With the contracted diameter \(d = 5.2\, \mathrm{mm} = 0.0052\, \mathrm{m}\),We calculate the new cross-sectional area:\[ A = \pi \left(\frac{0.0052}{2}\right)^2 = 2.12 \times 10^{-5} \mathrm{~m}^2. \]

Step 5: Calculate Airspeed for Contracted Diameter (b)

Using the same volume flow rate,\[ v = \frac{Q}{A} = \frac{7.0 \times 10^{-3}}{2.12 \times 10^{-5}} \approx 330.19 \mathrm{~m/s}. \]The multiple of the speed of sound \( v_s = 343 \mathrm{~m/s} \) is \[ \frac{v}{v_s} = \frac{330.19}{343} \approx 0.963. \]

Key Concepts

Volume Flow Rate

Volume flow rate is a fundamental concept in fluid dynamics. It refers to the quantity of fluid that flows through a given surface in a specific time. In the context of a cough, it is the amount of air moving out of the trachea.

The standard unit of volume flow rate is cubic meters per second (\( m^3/s \)). With a given volume flow rate, as in the exercise (\( Q = 7.0 \times 10^{-3} \ m^3/s \)), we can determine how fast air needs to travel through a specific cross-sectional area of the trachea.

It is a useful measure for understanding various fluid processes like ventilation, plumbing, and any system involving fluid movement. Its calculation provides insights into the efficiency and velocity of fluids in motion.

Cross-Sectional Area

The cross-sectional area is the area of the slice or section of a three-dimensional object, perpendicular to the direction of flow. In this scenario, it's the area of the airway (trachea), which a fluid, in this case, air, passes through.

To calculate this area for the trachea, use the formula for the area of a circle: \[ A = \pi \left(\frac{d}{2}\right)^2 \]

For example, when the diameter of the trachea is 14 mm (or 0.014 m),
the cross-sectional area becomes \( 1.54 \times 10^{-4} \ m^2 \).

This concept is crucial because changes in the diameter of the passage influence the velocity of air. As seen in the exercise, reducing the diameter increases the airspeed.

Speed of Sound

The speed of sound is the speed at which sound waves travel through a medium, such as air. In standard conditions at sea level, it is approximately 343 m/s in air.

This figure is a reference point for understanding other speeds, such as the velocity of air expelled during a cough.

In this exercise, we expressed the speed of expelled air as a multiple of the speed of sound to frame the problem in familiar terms. An expelled airspeed of 45.45 m/s, for instance, is approximately 0.132 times the speed of sound. It helps in visualizing how fast air moves relative to sound.

Trachea

The trachea, often referred to as the windpipe, is a vital part of the respiratory system. It connects the pharynx and larynx to the lungs, allowing passage of air.

Structurally, it's a tube supported by cartilage, facilitating the airflow during breathing processes, including coughing. During a cough, the trachea's diameter may compress to propel air at high speeds, clearing any obstructive mucus.

Understanding its function and changes, like narrowing during a cough, is crucial for analyzing airflow dynamics. Changes in the trachea affect airspeed, as it influences the cross-sectional area through which air must pass.