Page 138 - Biomedical Engineering and Design Handbook Volume 2, Applications

DESIGN OF RESPIRATORY DEVICES 117

A

B

FIGURE 4.3 Representations of laminar (a) and turbulent (b) flow
through a straight tube. With laminar flow, the pressure drop across a length
of tube is proportional to the flow. With turbulence, the pressure approaches
proportionality with the square of flow. The velocity profile under laminar
flow conditions is parabolic, with higher flows near the center of the tube
than near the boundary.
The pressure required to move gas along a straight tube under conditions of laminar flow is given
by the Poiseuille equation:

V8η l
ΔP = (4.4)
r π 4
where ΔP = pressure drop
V = flow
η= gas viscosity
l = length of tube
r = radius of tube

Thus, for a given straight tube, the Poiseuille equation predicts a linear relationship between pres-
sure and flow and gives rise to the pneumatic analog of Ohm’s law:

ΔP = VR (comparewith ΔV = IR) (4.5)
4
where R is resistance, given by 8ηl/πr .
This Ohm’s law representation of the pressure-flow relationship forms the basis for measurements
of airways resistance and is the principle on which several flow-measuring devices are based. The
pressure required to move gas under conditions of turbulence is always greater than that required to
achieve the same flow under laminar conditions, as additional pressure (or energy) is required to
accelerate molecules in directions other than the direction of bulk flow. This will be manifested as an
upward curve and deviation from linearity on a plot of flow (x axis) versus pressure drop (y axis).
Turbulence in flow through a straight tube may be estimated by calculating the Reynold’s number:

2 srρ
N R = (4.6)
η
where N = Reynold’s number
R
s = linear velocity of flow
ρ= gas density
and other variables are as shown above.