APPENDIX 5: Flow Meters, Bernoulli's Principle, Laminar
and Turbulent
Flow
The equations governing the motion of fluids are expressions of
Newton's second law, F = ma. Forces associated with fluids fall into three major
categories: (1) gravity, (2) pressure, and (3) friction. In the example using manometers,
the gravitational force per unit volume of fluid is simply ρg, acting in the
vertical direction. Pressure forces are actually the result of differences in pressure
from one point to another and are expressed mathematically as the negative of the
pressure gradient. (A pressure gradient is a vector in the direction of the maximal
rate of pressure increase, with magnitude equal to the pressure derivative in that
direction.) Friction is proportional to viscosity, the physical property of a fluid
that relates shear stress to rate of strain.
P0
= p + ½ρU2
+
ρgz (1)
Equation 1 shows the relationship between velocity and pressure of a fluid in a flow
that meets the conditions described. For flows in tubes, the manometer technique
provides an easy method of measuring mean pressure. Therefore, the simplest flow
meters apply a combination of these two principles to a tube of changing cross-sectional
diameter. For example, the Venturi flow meter shown in Figure
30-18
consists of a tube of varying cross-sectional area that has two ports
for measurement of pressure. The Bernoulli equation for points 1 and 2 in the figure
becomes
P1
+ ½ρU1
2
=
P2
+ ½ρU2
2
(2)
Here, the gravity terms have canceled out because the tube is horizontal, but these
terms are usually negligible for gas flows in any direction.
The volume of the fluid flow (Q) (also called flux) at both locations
must be the same because no fluid is entering or leaving through the tube walls.
The dimensions and SI units for the volume of fluid flow are 13
/t and
m3
/sec, respectively. This volume is determined at each cross section
of the tube by multiplying the average velocity (U) by the cross-sectional area (A):
Q = U1
A1
= U2
A2
(3)
Assuming that A1
, A2
, P1
, and P2
are
known, we now have two equations for the two unknowns U1
and U2
.
Solving these for the velocity U1
produces
To find the volume of the flow (Q), we merely multiply this result by A1
.
Note that velocity is proportional to the square root of the pressure drop or that
the pressure change varies as velocity squared. For a given U1
, or the
magnitude of flow velocity, the pressure drop varies as the square of the ratio of
the areas, or the fourth power of the ratio of the diameters. If we choose an A2
greater than A1
, Equation 4 implies that P2
is greater than
P1
. In this case, the pressure increases in the direction of flow, a
change that initially seems contrary to intuition.
The bobbin flow meters (also called variable-orifice flow meters)
in anesthesia machines use a similar principle. These devices consist of a slightly
tapered vertical tube and a bobbin or ball that fits inside the tube (see Fig.
30-20
). The cross-sectional area of the ring-shaped gap between the bobbin
and the tube wall is proportional to the height of the bobbin. Because changes in
the cross-sectional area of flow are abrupt rather than gradual (as in Fig.
30-20
), the Bernoulli equation does not accurately describe this type of
flow. The flow above the bobbin (i.e., downstream) is highly turbulent, and turbulence
is a condition that dissipates kinetic energy into heat. However, introduction of
the empirical constant Cd
enables one to use the same formulation as in
Equation 4:
and
P1
− P2
= (½ρQ2
)/(Cd
2
A2
)
(6)
where Cd
is a dimensionless constant called the discharge coefficient.
This constant varies with the shape of the orifice and with the value for another
dimensionless parameter, the Reynolds number (Re). The Reynolds number, the overall
ratio of inertial forces to viscous forces in a particular flow, is determined as
follows:
Re = ρUL/μ (7)
where U is mean flow velocity, L is a characteristic length for the flow (in our
flow meter, L is the diameter of the tube), and μ is the viscosity of the fluid.
The dimension for viscosity is M/LT. The value for the Reynolds number is important
to any fluid flow because it determines some of the most important characteristics
of the flow. For example, the transition from laminar, or "smooth," flow to turbulent
flow is determined by the shape of the flow and the Reynolds number. Flow in a long,
straight, smooth-walled tube becomes turbulent at an Re value of approximately 2100.
On the other hand, flow through an abrupt orifice, such as that of the flow meter
in Figure 30-20
, becomes
turbulent at an Re value of less than 100.
Returning now to the function of the flow meter, one can see that
as gas flows upward through the tapered tube, the bobbin begins to rise. As the
bobbin rises, the cross-sectional area of the orifice (A) increases because of the
taper of the tube; therefore, the drop in pressure (P1
−
P2
) decreases. The bobbin reaches an equilibrium position
for a given volume of flow (Q) when the pressure lifting the bobbin is exactly equal
to the weight of the bobbin. In this type of flow meter, the pressure difference
is fixed by the bobbin weight, and the area of the orifice varies with the volume
of the flow, hence the name variable-orifice flow meter. Equations 5 and 6 show
that calibration of these flow meters depends on both the density and the viscosity
of the gas: density (ρ) appears explicitly, and viscosity (μ) appears in
the dependence of Cd
on the Reynolds number. If we use the wrong gas
in a particular flow meter, Equations 5 and 6 and the viscosity and density of the
new gas enable us to predict the change in calibration.
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