Dynamics: Setting the System in Motion

Flow Through Tubes: Basic Principles

Resistance ↑

Whenever a fluid flows through a tube, it encounters resistance. To create and sustain flow, a driving pressure must be present to provide the energy to move the fluid and to overcome resistive forces. Think back for a moment to your study of electricity: Ohm's law governed the flow of electrons in a circuit.

When considering the flow of gas in a tube, one can make an analogy to the electrical circuit. The potential difference is the driving pressure, that is, the pressure change from one end of the tube to the other. The current in the circuit is analogous to the flow in the tube, and the electrical resistance of the circuit is similar to the resistance of the tube to flow. Thus, Ohm's law is transformed into the following relationship that describes flow through tubes.

Note: In respiratory physiology, we indicate flow by writing a V with a dot over it, , indicating volume per time (Table 4-1).

To double the flow for a given resistance, one must double the driving pressure. Viewed from another perspective, a doubling of resistance causes a doubling of the pressure decrease across the length of the tube for any given flow.

Laminar Versus Turbulent Flow ↑

As the molecules of a fluid travel through a tube, they may be aligned in a variety of ways relative to the direction of flow. The particular alignment makes it easier or more difficult for the fluid to flow through the tube or, stated differently, depending on the alignment of the molecules' movements, different amounts of energy are needed to maintain a given flow.

In laminar flow, all of the molecules are moving in a direction that is parallel to the long axis of the tube. Fluid traveling near the center of the tube moves faster, that is, it has a greater velocity, than fluid in or wall of the tube. Under conditions of laminar flow, the change in pressure between two points within the tube is proportional to the flow of the fluid within the tube.

In contrast, under conditions of turbulent flow, some of the molecules in the fluid travel parallel to the long axis of the tube and other molecules travel in a variety of other directions, including perpendicular to the long axis.

To achieve a given flow when turbulent conditions are present, one must have a greater driving pressure. With turbulent flow, the change in pressure between two points in the tube is proportional to the square of the flow.

If there is a greater pressure decrease over the same length of the tube to achieve a given flow, this means more energy is needed (compared with a smaller pressure decrease) to maintain that flow (Fig. 4-1).

During resting breathing, flow is turbulent in the trachea and laminar in the small peripheral airways. At branch points, one often sees a type of flow that manifests some characteristics of both laminar and turbulent flow. This is called marginally laminar or transitional flow because it occurs at transitions between laminar and turbulent flow. In these locations, flow is generally laminar, but there are also eddies generated at angles to the long axis of the tube.

Whether a tube through which a gas is flowing has laminar or turbulent flow depends on a number of factors, including the viscosity of the gas, the density of the gas, the radius of the tube, and the velocity of the molecules in the tube (note: velocity, which has units of distance/time is not the same as flow, which has units of volume/time). The impact of these factors on the type of flow is expressed by Reynolds number.

The larger Reynolds number is, the greater the likelihood that flow will be turbulent. For example, flow is likely to be turbulent if the radius of the tube is large and the velocity of the gas is high. In a rapidly branching system such as the lung, fully laminar flow probably only occurs in the small airways. The very low velocity of the gas in the small airways near the periphery of the lung and the small radii of the individual airways in the periphery contribute to conditions that produce laminar flow.

In a healthy individual, most turbulent flow is found in the central airways, particularly the trachea. During exercise, when the velocity of the gas increases as ventilation increases. Turbulent flow extends more deeply into the lung, perhaps as far as the fifth or sixth generation of airways. Between the central and the peripheral airways, transitional flow is present.

As you take care of patients in the clinical setting, you will never have to calculate Reynolds number. The case described in Thought Question 4-2, however, demonstrates the importance of the principles underlying turbulent and laminar flow.

Airway Size and the Distribution of Resistance in the Lungs ↑

As we continue to examine the factors that determine resistance in the airways, we must now consider another principle of physics, Poiseuille's law. According to this law, the flow of gas, under laminar conditions, is proportional to the change in pressure and the radius of the tube to the fourth power.

Recall our formula that states ΔPressure = Flow x Resistance (or restating, Flow = ΔPressure divided by resistance).

Resistance, which varies inversely with flow, is thus proportional to the length of the tube and inversely proportional to the radius of the tube to the fourth power.

Let us consider the implications of this relationship. All other things being equal, a longer tube has a greater resistance than a shorter tube, and if one reduces the radius of a tube by one half, the resistance increases by a factor of 16!

Where is resistance highest in the lungs? As air moves from the mouth to the alveoli, the radius of the airways gets smaller and smaller. Given the principles of Poiseuille's law, you might conclude that the highest resistance in the lungs resides in the periphery. Another important concept must be considered, however, before you can answer this question. As you travel from the trachea to the alveolus, you witness the branching of succeeding generations of airways (see Chapter 2). Each single airway branches into two or more parallel airways with a combined cross-sectional area greater than the parent airway. In a sense, the radius of the combined smaller airways is greater than the radius of the larger parent airway; thus, the resistance in actually less.

As you put tubes together in various combinations and consider the resulting resistance, you must pay particular attention to the arrangement of the tubes. Are they in series or in parallel? For tubes in series, the total resistance is equal to the sum of the resistances of the individual tubes:

The total resistance is greater than the resistance of any individual tube.

In contrast, for airways in parallel, the total resistance is less than the resistance of any of the individual tubes:

As you move from the mouth to the alveoli, you go from a single airway (the trachea) to millions of airways (terminal and respiratory bronchioles) in parallel. Consequently, the resistance of the lungs is much less in the small airways in the periphery than in the central airways. Within the mid-sized central airways, one must also consider the effect of the smooth muscle in the medium-term bronchi. The radius of these airways may vary considerably depending on the tone of the muscles and whether they are actively contracting as a result of a change in the balance of the sympathetic and parasympathetic nervous systems and disease states such as asthma.

Under normal conditions, the bulk of resistance in the lung resides in the first six to seven generations of airways (Fig. 4-2). Experimental data indicate that the peak airway resistance also occurs in the fifth to seventh generation of airways. The reason or reasons for this is not entirely clear, but it may be the consequence of the geometry of the airways at the initial branchpoints in the tracheobronchial tree. As emphasized previously, the cross-sectional area of the airways in parallel gradually increases as one goes from the trachea to the periphery of the lung, a finding that accounts for the fact that most of the resistance in the lungs is attributable to the central airways. Diseases of the lung and central airways can, therefore, have a significant impact on total resistance in the more rigorous, consistent, clear, easily detected by tests that assess flow or resistance. In contrast, diseases of the small airways have relatively little impact on lung resistance and can be more difficult to detect with standard tests of pulmonary function. For this reason, the region of the lung containing the small airways has been called the silent zone of the lungs.

Let us revisit the concepts of laminar and turbulent flow in the context of changes airway size. Where are you more likely to encounter turbulent flow, in the central or the peripheral airways? Consider what happens as gas is exhaled from the alveoli. As you go from the millions of airways in the periphery to a diminishing cross-sectional area in the central airways, the velocity of the gas must increase because flow is constant. Imagine four single-lane roads arranged in parallel that are now merging into a two-lane highway. Assume that the flow of cars (number of cars/min) is constant from the single-lane roads to the two-lane highway. In order to accommodate all those cars in the two-lane highway and maintain the flow, the velocity of each car must increase.

Use Animated Figure 4-3 to observe two or four single-lane roads merging and compare the flow and velocity (shown as speed) of the cars for these scenarios. Notice in particular how the cars must speed up to maintain flow in the case where four single-lane roads are merging into two. As already discussed, this is analogous to the movement of gas during exhalation from the many small airways in the periphery of the lung to the lesser cross-sectional area of the central airways. Before reading on, think about how this change in velocity of gas affects the type of flow (laminar versus turbulent) in the peripheral and central airways.

According to Reynolds number, gas moving through a tube at a higher velocity is more likely to demonstrate turbulent flow. Thus, whereas flow in the central airways is more likely to be turbulent, flow in the small bronchioles is laminar (Fig. 4-4).

During normal resting breathing (at lung volumes approximating functional residual capacity), 80% of the resistance to flow in the airways greater than 2 mm in diameter, and nearly half of the total resistance of the lungs is in the central airways.

Bernoulli's Principle ↑

What makes airplanes fly? As the thrust from the engines propels the airplane down the runway, air travels above and below the wing. An airplane wing is shaped so that the distance across the surface from the front to the back of the wing is longer on the top than on the bottom of the wing. Because the flow of air must be the same above and below the wing, the velocity of the gas traveling over the top of the wing must be greater than that beneath the wing. The increased velocity results in a lesser pressure exerted by the gas. In other words, the air above the wing exerts less pressure on the wing than does the air below the wing; the result is "lift," and the plane rises off the runway. This relationship between the velocity of a gas and pressure is Bernoulli's principle, and it is also applicable to the airways of the lungs, as we will see shortly (Fig. 4-5).

First, let us apply Bernoulli's principle to fluids moving through tubes. Imagine a fluid moving at a given flow through a tube with a wide diameter. The molecules in the fluid have a certain velocity and exert a pressure within the tube. The tube abruptly narrows. If we stipulate that the flow remains constant throughout the system, the velocity of the molecules of the fluid must now increase. As with the example of the airplane wing, to move the molecules at a greater velocity, keeping flow constant, requires work to impart energy to the molecules. This can only occur if there is greater pressure in the wider part of the tube. The difference in pressure between the wide and narrow sections of the tube reflects the force necessary to increase the velocity of the molecules. The greater the increase in velocity, the greater the decrease in pressure (Fig. 4-6).

If an external pressure is exerted against the outer walls of the tubes, the pressure inside the small tube may now be less than the pressure outside the tube. This principle plays an important role in the tendency of airways to collapse during a forced exhalation.