Restricted Flow
and Orifice Studies
© by Ed Seykota,
1999
In order to build a computer model of the Levitator, and thereby
further validate the theory of Radial Momentum, I conduct various tests to
determine formulas for the basic behavior of air as it flows through a
central orifice and out between the table and the plate.
In addition to determining the behavior of air, these tests also
provide practice calibrating and using the flow and pressure
instrumentation.
In these studies, I measure the mass flux rate (kg/sec) that escapes
through a round orifice in a plenum, as a function of the pressure in the
plenum. The flux turns out to be proportional to the square root of
pressure for escape velocities less than the speed of sound. For higher
velocities, it increases directly with pressure. 

Schematic of device to measure flow through a plate
valve. Air from the plenum at the left at [P, T] flows through the round
orifice with diameter [D] and then through a gap between the plates of
width [h]. The bottleneck valve is a ring of diameter [D] and height
[h]. The question mark [?] indicates the flux, to be determined. In
these trials D = 1/8".

The Device to Measure Flux as a Function of Pressure

Air flows through the flow meter (left), past the pressure meters, into
a plenum above the test table and out through the orifice in the table. I
use two different tables, with orifice sizes of 1/16" and 1/8".
I vary the plenum pressure by adjusting the pump (under the flow meter and
not shown). I measure the pressure and mass flux with the gauges. The
function turns out to have two different forms, depending on the velocity
of flow through the orifice. 
Detail of Plenum and Orifice

View from Under the table
Air enters the plenum through the hose from the pressure gauge and
exits through the orifice in the test table. The plenum is a cylinder
about an inch long and 1/4" in diameter. As the pressure varies from
zero to about 60 psig the mass flux through the orifice increases. The
device is made from smooth acrylic. 
Schematic of Plenum and Orifice

In this schematic, air enters the plenum from the left, with pressure
[P] and temperature [T]. I assume velocity and momentum in the plenum to
be negligible. The air evacuates through the round orifice with diameter
[D] into ambient condition [P0, T0]. I use two tables, one with D =
1/8" and one with D = 1/16". The question mark [?] represents
the unknown, which in this case is the mass flux (kg/sec). 
As
a continuation of the basic tests on orifices, I place a plate at
various distances from the orifice as a valve and measure the flux
versus pressure. Except at low pressures, the flux turns out to be
monotonic with the orifice area.


The Basic Result
As the pressure in the plenum increases, it pushes air through the
orifice at a higher rate so the mass flux rate (kg/sec) increases. The
mass flux through at 1/8" is substantially higher than the mass flux
through at 1/16". The mass flux for both orifices rises quickly at
lower pressures, then less rapidly, indicating a change in function. 

Flux Proportional to Area
This chart shows the ratio of the flux at 1/8" versus the flux at
1/16" at various pressures. This ratio stays in proportion at about
4:1. This is also the ratio of the areas of the two orifices. This
indicates flux is proportional to area. 
In order to derive reasonable formulas to plug in to the simulation
model to represent the central valve process, I use basic physics, logic
and curve fitting. The goal at this point is not to reinvent orifice flow
physics so much as to derive reasonably useful formulas.
For low pressures, the air flows gently and smoothly through the
orifice. From basic energy balance, the energy loss from the pressure drop
through the orifice accelerates the air to a higher kinetic energy level.
Since kinetic energy = 1/2 mv^{2}, the velocity, then, is
proportional to the square root of the pressure drop. Mass flux is
velocity * density * cross section. The effective orifice is somewhat
smaller than the actual orifice since the airflow bends and narrows in the
orifice to create a "vena contracta" effect of about 85%
efficiency. The combined equation for flux, then, is:
Flux = sqrt(2 * pressure_drop / density) * orifice_area
* exit_density * 0.85
As the air exit velocity reaches the speed of sound, the relationship
between pressure and flux changes. Air does not generally flow in open
systems at speeds greater than the speed of sound. It compacts and forms a
shock wave. As the pressure increases, the exit air speed remains at 343
meters / second while the density of the air squeezing through the orifice
continues to increase. Therefore, the flux continues to increase directly
with the pressure, per:
Flux = orifice_area * exit_density * speed_of_sound * (
1 + pressure_drop / p0) * .62
Where p0 is about 100 kpa. That and a 62% vena contracta coefficient
for compacted flow reconciles the data.
The combination of these two equations seems to predict the data fairly
well over the entire range of observations. Again, the goal here is not to
develop defensible orifice physics so much as to derive reasonable
formulas to plug into the simulation model. 

Two Formulas for Flux
The actual data is shown in blue. At pressures below 100 kpa, flux is
proportional to the square root of pressure (equation shown in green). At
higher pressures, it increases directly with pressure (equation shown in
red). 

The Fit
Another way to view the fit is to plot the model flux against the
actual flux. A perfect fit would be a straight line through equal values.
The combination of the two formulas appears to provide a fairly good fit
for most of the data. 
The Data
Column 
Description of Columns 
Pressure 
Gauge psig (pounds per square inch, gauge)  reading directly from the
pressure gauge. 
Pressure 
Gauge kpa (kilopascals)  gauge pressure psig times 6895 kpa/psi. 
Density 
Density  1.2 kg/m^{3} times (11.6/14.6 + gauge kpa / 78.91).
At Lake Tahoe, the location of the tests, the ambient pressure is 11.6 psi
or 78.91 kpa. At sea level, the presure is 14.6 psi and the density of
air is 1.2 kg/m^{3} . 
Flux1/16 
1/16" diameter numerical reading directly from flow meter for
1/16" diameter orifice. 
Flux1/8 
1/8"  same as above. 
CF 
Conversion Factor  from MEM Flow Products Company, the flow meter
manufacturer. The factor adjusts gauge readings for different ambient
temperature and pressure. The meter is calibrated for base conditions of
70 degrees F and 100 psig. I ran the tests at 50 degrees F and 11.6 psig
(Lake Tahoe). The formula for the factor is:
Where Pg is the operating pressure + 11.6 psi, Ps is the base pressure
(100 psi) + 11.6 psi, Ts is the base temperature (70F) + 460F and Tg is
the operating Temperature, (50F) + 460F. Example, the factor for 5 psig at
50F is f = sqrt(16.6/111.6 * 530/510) = .39316. 
Flux1/16 
1/16" Base Flux (scfm)  gauge reading times adjuster gives the
flow at base conditions (70 deg. F and 14.6 psi) 
Flux1/8 
1/8"  same as above. 
Flux1/16 
1/16" Flux (kg/sec)  base flux (scfm) times 4.5 nt/lb times
.07849 lb/scfm / 9.81 m/s^{2} / 60 s/min converts scfm to kg/sec. 
Flux1/8 
1/8"  same as above 
Pressure 
Pressure 
Density 
Flux1/16 
Flux1/8 
CF 
Flux1/16 
Flux1/8 
Flux1/16 
Flux1/8 
gauge 
kpa 
kg/m^3 
gauge 
gauge 
factor 
scfm 
scfm 
kg/s 
kg/s 










0.5 
3.4 
1.0 

2.50 
0.3357 

0.8392 

4.75E04 
1.0 
6.9 
1.1 
0.80 
3.60 
0.3425 
0.2740 
1.2331 
1.55E04 
6.98E04 
1.5 
10.3 
1.1 

4.40 
0.3493 

1.5368 

8.70E04 
2.0 
13.8 
1.2 
1.20 
5.00 
0.3559 
0.4270 
1.7793 
2.42E04 
1.01E03 
2.5 
17.2 
1.2 

5.50 
0.3624 

1.9929 

1.13E03 
3.0 
20.7 
1.3 
1.60 
6.00 
0.3687 
0.5900 
2.2123 
3.34E04 
1.25E03 
3.5 
24.1 
1.3 

6.40 
0.3750 

2.3999 

1.36E03 
4.0 
27.6 
1.4 
1.80 
6.70 
0.3811 
0.6860 
2.5536 
3.89E04 
1.45E03 
4.5 
31.0 
1.4 

7.10 
0.3872 

2.7491 

1.56E03 
5.0 
34.5 
1.5 
2.00 
7.40 
0.3932 
0.7863 
2.9094 
4.45E04 
1.65E03 
5.5 
37.9 
1.5 

7.70 
0.3990 

3.0726 

1.74E03 
6.0 
41.4 
1.6 
2.10 
8.10 
0.4048 
0.8502 
3.2792 
4.82E04 
1.86E03 
6.5 
44.8 
1.6 

8.30 
0.4105 

3.4075 

1.93E03 
7.0 
48.3 
1.7 
2.30 
8.60 
0.4162 
0.9572 
3.5791 
5.42E04 
2.03E03 
8.0 
55.2 
1.8 
2.40 

0.4272 
1.0253 

5.81E04 

9.0 
62.1 
1.9 
2.50 

0.4380 
1.0950 

6.20E04 

10.0 
69.0 
2.0 
2.60 

0.4485 
1.1661 

6.60E04 

15.0 
103.4 
2.5 
2.90 

0.4977 
1.4433 

8.18E04 

20.0 
137.9 
3.1 
3.10 

0.5425 
1.6816 

9.52E04 

25.0 
172.4 
3.6 
3.30 

0.5838 
1.9265 

1.09E03 

30.0 
206.9 
4.1 
3.45 

0.6224 
2.1473 

1.22E03 

35.0 
241.3 
4.6 
3.70 

0.6587 
2.4373 

1.38E03 

40.0 
275.8 
5.1 
3.85 

0.6932 
2.6687 

1.51E03 

45.0 
310.3 
5.7 
4.05 

0.7260 
2.9402 

1.67E03 

50.0 
344.8 
6.2 
4.20 

0.7574 
3.1810 

1.80E03 

55.0 
379.2 
6.7 
4.35 

0.7875 
3.4257 

1.94E03 

60.0 
413.7 
7.2 
4.50 

0.8165 
3.6744 

2.08E03 

65.0 
448.2 
7.8 
4.63 

0.8446 
3.9061 

2.21E03 

