Physiologic Implications of Artificial Airways: Laminar Flow
Once laminar flow is fully developed, the pressure drop (AP) down the tube can be given by the Hagan-Poiseuille law. which states:
where V is volume flow in L/time, L is length of the tube, r is the tube radius, and p. is fluid viscosity, or
Note that for a tube of fixed dimension, pressure drop is linearly related to volumetric flow rate as measured in liters per unit time (L/time), linearly related to length of the pipe, and inversely proportional to the fourth power of the radius. Resistance to flow (R) can be defined by an equation that is the mechanical equivalent of Ohm s law. Source
for fully developed laminar flow.
Thus, resistance in a tube during laminar flow (fully developed) is directly proportional to the length of the tube and inversely proportional to the fourth power of the radius. It is independent of flow.
The length of tube needed for the development of fully developed laminar flow is defined by the equation:
where μ is fluid viscosity, a is fluid density, K’ is a constant, D is tube diameter, and V is flow rate (ie, L/s).
Reynolds number is a dimensionless number which is the ratio of the inertial forces to the viscous forces in the flowing fluid. It is dependent on tube geometry, physical properties of the gas, and flow rate of fluid. Experimentation has shown that flow is laminar if Re is less than 2,300 and turbulent if it is above 2,500. Flow is transitional if it is in between these ranges of Re.
Since, for laminar flow conditions, Le = K’*Re*D, then the higher the flow rate, the longer the Le. When flow is laminar, the entrance length is approximately 60-70 tube diameters. Thus, for most purposes, the entrance length over which fully developed laminar flow is developed is longer than the endotracheal tube (ETTs are 24-26 cm long). Because fully developed flow is not achieved over the short length of the ETT tube, the Hagan-Foiseuille law underestimates the pressure drop. However, it is a first approximation. Moreover, most quiet breathing is done with transitional or turbulent flow regimens, so the laminar flow equations do not apply.