Physiologic Implications of Artificial Airways: Turbulent Flow
When, during flow, the fluid molecules no longer flow in parallel, the layers disintegrate and behave in a random fashion. This random behavior of fluid is called turbulent flow and has been found to occur at Re above 2,500. The randomness of the flow results in eddy currents which yield pockets of flow that are in the opposite direction to the main direction of flow. Thus, the pressure required to maintain flow is greater than would be the case for laminar flow. In addition, the smooth parabolic velocity profile becomes flattened. During turbulent flow, the velocity profile becomes fully developed over a shorter entrance length than is the case with laminar flow. The Le for turbulent flow approximates 40 tube diameters and is independent of Re (cf60-70 tube diameters for laminar flow). This is still longer than the length of most endotracheal tubes.
The pressure drop for fully developed turbulent flow is given by the equation:
where f is a friction factor which depends on the roughness of the tube and the Reynolds number. It can be derived from the Moody diagram which is a log-log plot of Re vs the ratio of frictional pressure losses (AP) to kinetic energy in the fluid (Vfe*CT*VVA*) where a is density, V is the volume flow; and A is the area of the cross section of the tube (Fig 3). This ratio:
is called the Moody friction factor (CF). Note that CP is not die same as /, but is related to it. For tubes of different roughness, a family of curves can be drawn.
The Moody diagram demonstrates three flow regions: laminar, transitional, and turbulent (Fig 3). In the laminar region, the slope of the Moody diagram is — 1. Thus, the log of CF vs log Re are inversely related. At very high Reynolds numbers where flow is turbulent, the slope of the Moody diagram is 0. Thus, the log of CP is independent of the log of Re. For transitional flow, the slope is intermediate between laminar and turbulent.
For turbulent flow, from equation 3, the pressure drop is unrelated to fluid viscosity, is related to fluid density, tube length and the square of the volume flow, but inversely related to the fifth power of the diameter of the tube. Contrasted to this is the pressure drop for laminar flow, which is related to viscosity but not density, is related to tube length and volume fluid flow, but inversely to the fourth power of the tube radius. Rohrers equation, intuitively developed, relates pressure drop to flow. Rohrers equation is:
From what preceded, it is apparent that the first term of Rohrers equation relates to laminar flow components and the second term to turbulent flow (equations 1-4).
Figure 3. Schematic Moody diagram. The log of the Reynolds number is plotted against the log of the friction factor (see text).