LOW FLOW PRESSURE DROP FLOW RATE INSTABILITIES
IN A COMPRESSIBLE AIR-WATER SYSTEM
by
SUZANNE MARIE BURZYK
Submitted in Partial Fulfillment
of the Requirements for the
Degree of Bachelor of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(MAY 1979)
Signature
of Author..
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Department of Mechanical Engineering, May 11, 1979
Certified by.......
Acceptedby
.....
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Thesis Supervisor
'........
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Chairman,
Archives
MASSACHUSETTSINSTITUTE
OFTECHNOLOGY
JUN 26 1979
LIBRARIES
artmental
Committee on Thesis
2
LOW FLOW PRESSURE DROP FLOW RATE INSTABILITIES
IN A COMPRESSIBLE AIR-WATER SYSTEM
by
SUZANNE MARIE BURZYK
Submitted to the Department of Mechanical Engineering
on May 11, 1979 in partial fulfillment of the requirements
for the Degree of Bachelor
of Science.
ABSTRACT
Experiments were performed to investigate the pressure drop versus
flow rate instability of an air-water flow through a vertical tube with
a large tank of air at the entrance.
An unsteady
region,
occurring
when
the slope of the pressure drop versus flow rate curve was negative, was
found.
In the unsteady region, periodic filling and discharge of the
vertical pipe occurred, leading to large fluctuations in the pressure at
the top of the bottom tank.
Pressure drop fluctuations occurred about
as theoretically predicted.
A minimum flow rate, at which the water flowed back into the bottom
tank, was found in the unsteady region.
The experimentally determined
leakage limit line was compared with the analytically determined leakage
line with fair agreement obtained.
Thesis Supervisor: Peter Griffith
Title: Professor of Mechanical Engineering
3
TABLE OF CONTENTS
TITLE PAGE ............
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ABSTRACT
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LIST OF FIGURES
NOMENCLATURE ...
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EXPERIMENTAL PROCEDURE...
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RESULTS AND DISCUSSION...
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CONCLUSIONS..............
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APPARATUS
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REFERENCES ..............
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ACKNOWLEDGEMENTS ........
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INTRODUCTION
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14
.... 15
APPENDIX A:
THEORY ON THE CALCULATION OF PRESSURE DROPS
USING THE DRIFT FLUX MODEL..................
APPENDIX B:
THEORY ON THE CALCULATION OF THE LEAKAGE LIMIT LINE......29
.. ........ 27
4
LIST OF FIGURES
Figure
Number
Page
16
..................................
1
Schematic of Test Apparatus
2
Steady/Unsteady Flow Regions
3
Plot of Maximum Pressure Drops for Various
Combinations of Superficial Gas and Liquid Velocities ........ 18
4
Plot
5-8
Plots of Maximum Pressure Drops Versus Superficial Gas
Velocities for a Given Superficial Liquid Velocity ........... 20
9-11
Plots of Pressure Fluctuations with Time for
Various Flow Combinations ...................................24
..........................
17
of Experimental and Calculated Leakage Limit Lines..... !9
5
NOMENCLATURE
(ft2 )
A
Total Flow Area
a
Void Fraction (dimensionless)
co
Constant in Equation (A.3)
D
Tube Diameter
f1
Friction Factor corresponding to the Reynolds Number of
Equation (A.5)
g
Gravitational Acceleration (ft/hr2)
9g
Constant
G
(ft.)
(4.17 x 108
+
Mass Flux, (f
lbm-ft/lbf-hr
2)
(lbm/ft2-hr)
g)
A
jf
Superficial Liquid Velocity (ft/s)
jg
Superficial
Gas Velocity
(ft/s)
g
LTOT
Total Length
Lup
Length
Orf
Liquid Mass Flow Rate, pfjfA (lbm/hr)
mg
Gas Mass Flow Rate, p9jgA
n
Constant
Pf
Absolute Liquid Viscosity (lbm/hr-ft)
Ap
Total Pressure Drop (lbf/in2 )
APfric
Frictional Component of Total Pressure Drop (lbf/in2)
Ap
Gravitational Component of Total Pressure Drop (lbf/in2)
Qf
Fluid Volumetric Flow Rate, jf/A (ft3 /s)
Qg
Gas Volumetric Flow Rate, jg/A (ft3 /s)
Re
Reynolds Number
pf
Liquid Density (lbm/ft3)
grav
of Tube
(ft)
of the Vertical
in Equation
Tube
(ft)
(lbm/hr)
(B.1)
6
pg
Gas Density (bm/ft 3)
V
Tank
v
Average Specific Volume, vf + X(vg - Vf)
Vf
Liquid Specific Volume, 1/pf (ft3 /lbm)
vg
Gas Specific
Vvj
Drift Velocity
(ft/s)
x
Flow Quality,
mg
Volume
(ft3 )
Volume,
f +
1/pg
(ft 3 /lbm)
(ft3/lbm)
(dimensionless)
fi
7
INTRODUCTION
Toward
pressure
the end of the oil and gas well's
drop between
the bottom
productive
life, the
and the top of the well begins
to
increase, due to the increase in the holdup, that is the volumetric
concentration of the oil in the pipe.
A long period dump and refill
process begins to occur in the vertical pipe with a period on the order
of 30 minutes.
This fluctuation in the flow and pressure seems to occur
because of the compressibility of the oil-gas system. After the vertical
pipe is almost filled with liquid, a sudden discharge occurs.
oil-gas mixture expands and the pressure drops.
of the well.
the separators
The
The mixture surges out
The sudden increase in the flow results in overloading
at the top of the well causing
carryover.
The objective of this experiment is to discover what combinations
of flow rates will cause fluctuations in the pressure and the surging
of the oil-gas mixture out of the well.
the gas and oil.
An air-water system will model
A short vertical pipe produces the negative sloping
pressure drop flow rate curve, while a tank provides the compressible
volume.
8
APPARATUS
A schematic diagram of the test apparatus is shown in Figure 1.
The tank has a sight
Shop air flows into a 2.86 cu. ft. tank.
gage to determine the water level (if any).
The air flows out of the
tank through a horizontal plexiglas tube .75 ft. long and .75 inches
I.D. This tubing is joined at 900
to a .25 ft. section of plexiglas
(Clear plexiglas is used, so a visual check of leakage back into the
tank can be made.).
This plexiglas is connected to copper tubing .75
inches I.D., which forms a U measuring 5 ft. in length.
Water flows
into this copper section and mixes with the air from the tank.
rest of the 28 ft. vertical
pipe
is plexiglas
with an I.D. of
The
.75 inches.
The end of the pipe feeds into a second tank which empties the air-water
mixture into the drains.
A pressure gage attached to the top of the tank measures the tank
pressure.
A Stantham pressure transducer is attached to a tee with the
pressure gage.
The transducer is connected to a Sanborn WAM meter.
The meter is connected to an x-y recorder to measure pressure fluctuations
versus time.
The air flow rate is measured by a Fischer and Porter flowmeter
model no. FP-1/2-27-G-10/55.
The water flowrate is measured by a
Fischer and Porter flowmeter model no. FP-1/2-21-G-10/83.
Valves are
located upstream of the flowmeters to control the flowrates.
A Watts
air pressure regulator no. 119-4 sets the supply air pressure at 30 psia.
9
EXPERIMENTAL PROCEDURE
Tests were performed to determine the regions of stability and
instability.
The pressure fluctuations in the tank were noted, as the
water and air flows were varied.
The regions with zero or imperceptible
(less than .2 psi) fluctuations were considered steady.
Experiments were carried out to determine the maximum and minimum
pressure drop in the tank for each combination of fluid and gas flow.
Superficial gas and liquid velocity readings were calculated from the
flowmeters.
The average superficial liquid velocities ranged from
.04 8 <Jf<.2 38 ft/s, while the gas ranged from .791<jg <9.887 ft/s.
Pressure versus time traces were taken to get a good estimate of the
maximum pressure fluctuations for each combination of air and water
flows.
The leakage limit lines were determined at the same time the
pressure drop data was taken.
The superficial gas velocity was fixed
and the lowest superficial liquid velocity was found at which leakage
back into the lower tank occurred.
10
RESULTS AND DISCUSSION
Pressure drop readings were taken for various combinations of
superficial gas and liquid velocities to determine the regions of
stability and instability.
These regions are shown in Fig. 2.
Any
Thus, any
fluctuation below .2 psi was considered to be negligible.
combination of flows which caused a fluctuation of less than .2 psi was
This plot was used to determine
considered to define a stable point.
in later tests which combinations of flow rates would cause significant
pressure drops.
Figure
rates.
3 shows the dependence
of the pressure
drops
on the flow
As the superficial liquid velocity, jf, increases and the
superficial gas velocity, jg, decreases, the pressure drop increases.
Notice that the pressure drop is more sensitive to changes in the
superficial gas velocity.
According to the drift flux model (see
Appendix A for further treatment), as jf increases and jg decreases,
the pressure drop should increase.
The experimental results tend to
agree with this.
As jg decreases
drop is reached.
This
and jf increases,
a limit to the maximum
pressure
This maximum pressure drop is approximately 12 psi.
is the pressure
due to a column of water
28 ft. high
that the
air in the tank has to support.
Figure 4 shows the calculated and experimentally determined
leakage limit lines.
Qff
Ag0
>
V/nP
Leakage, theoretically, occurs when
is the liquid volumetric flow rate,
, where Qf is the liquid volumetric flow rate,
11
Pf is the fluid density,
go = 4.17 x 108
g is the gravitational
lbm-ft/lbf-hr 2,
V is the tank volume,
n
acceleration,
Qg is the gas volumetric
1, and P is the average
flow rate,
tank pressure.
(See
Appendix B for the derivation of this inequality.)
There is fair agreement between the experimental and calculated
leakage limit lines.
Deviations of the experimental from the analytic
results could be explained by inaccurate pressure drop measurements,
errors in the superficial gas velocity and superficial liquid velocity
readings, an inaccurate tank volume measurement, or by the presence of
friction in the real system which was not accounted for in the equations
modelling the system.
Experimentally determined pressure drops and superficial gas
velocities are plotted for a specific
See Figures 5-8.
superficial liquid velocity.
Then, using the drift flux model analysis, the pressure
drops are calculated and plotted against the superficial gas velocities
for a given superficial liquid velocity.
Refer again to Figures 5-8.
See Appendix A for the equations used in the drift flux model analysis.
Notice that the pressure drop versus superficial gas velocity curve is
negative. According to Wallis(2), the curve is negative in the region
of unstable flow.
These superficial gas and liquid velocities of
Figures 5-8 fell within the limits of the experimentally determined
unstable region.
In Figures 5-8, there is very little agreement between the
calculated and the experimental results.
calculated using the drift flux model.
The pressure drops were
This could be an explanation
12
for some of the disagreement in the results.
assumes slug flow.
column
of water
The drift flux model
A test run, however, had flows ranging from a solid
28 ft. high
to annular
flow.
In Figures 5-8, both the experimental and calculated pressure
drops tend toward a
p of 2 psi.
The experimental pressure, however,
approaches the asymptote at 2 psi more quickly.
Figures 9-11 show traces of the pressure fluctuation with time for
various flow combinations.
These graphs were used to generate
Figures 2-8.
Figures 9-10 show the fluctuations of the pressure with time.
both figures, notice that there seem to be 2 sets of waveforms.
In
In
determining the maximum pressure drop, the larger of the two was used.
Figure 11 illustrates the slow building up of the pressure and
the rapid blowdown.
13
CONCLUSIONS
The regions of stable and unstable growth were found.
The dependence of pressure drop on superficial gas and liquid
velocity was shown.
The experimental results followed the trend of the
predictions of the drift flow model, that is, as jg decreased and jf
increased, the pressure drop increased.
The experimental leakage limit lines agreed fairly well with the
theoretical calculations.
Further study on this problem could include trying to develop a
better model of the system.
Using the drift flux model to calculate
the pressure drops gave poor results.
However, it will be difficult
to accurately model this system because, in the course of a run, the
flow varied from a solid water column to annular flow.
Refer to
Wallis (2) for the equations for the pressure drop calculations for
other flow regimes.
14
ACKNOWLEDGEMENTS
I would like to thank Ted Fischer for coming in on Saturdays.
I would also like to thank Tiny and Fred for building and
maintaining my apparatus.
15
REFERENCES
1)
Bjorge, R. W., "Stability and Phase Distribution of a Cocurrent
Air-Water Flow in an Inverted U Tube Array," B.S. Thesis,
Department of Mechanical Engineering, M.I.T., 1978.
2)
Wallis, G. B., One-Dimensional Two-Phase Flow, McGraw-Hill, New
York, 1969.
3)
Shearer,
J. L., Murphy,
A. T., and Richardson,
H. H., Introduction
to System Dynamics, Addison-Wesley Co., Reading, Mass., 1971.
4)
Hsu, Y. Y., and Graham,
R. W., Transport
Processes
Two-Phase Systems, McGraw-Hill, New York, 1976.
in Boiling
and
16
Upper
Tank
__
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To Drains
YAir-Water Mix
Shop
Air,
Water
-
Air
Presssure
RegulatorL
Supply
Pressure
Water
Flowmeter
---
Flowmeter
Tank
Sight
Gage
To Drains
-
//
Figure
Schematic of Test Apparatus
17
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27
APPENDIX A
THE CALCULATION OF THE MAXIMUM PRESSURE DROP, Ap, WITH VARIOUS
COMBINATIONS OF jg AND
f, USING THE DRIFT FLUX MODELS.
(See References
1 and 2)
The pressure drop in the tank-tube system is a combination of a
pressure drop due to friction and a pressure drop due to gravity, or
(A.1)
Ap =APgra v + APfric
where
( Pg + (l-a)Pf ) t
APgrav
with a = void fraction, pg
LUP
(A.2)
air density (0.0763 lbm/ft3), pf = fluid
=
density (62.4 lbm/ft3 ), g = gravitational acceleration,
go
=
4.17x108 lbm-ft/lbf-hr2 , and L
= length of the vertical tube.
The drift flux model is used to estimate the void fraction (see
Wallis
(2)),
J
= a
- g
-- -(A.3)
Co(jg+ f) + Vvj
where co=1.2, jg is the superficial gas velocity, jf is the superficial
liquid velocity, and Vvj=0.35(gD)0 5 from Hsu (4) with D = tube
diameter
(.75 inches)
and g = gravitational
acceleration.
The frictional pressure drop can be found fromfl G2 v
APfric = LTOT
(A.4)
where LTOT is the total length of the tube, fl is the Moody friction
28
factor for a given Reynolds number
(r?
+
rhf) D
---
Re=
Vf
(A.5)
D
where rg is the mass flowrate, rif the liquid mass flowrate, A the
cross-sectional tube area,
f the absolute liquid viscosity
(3x10-5 lbf-s/ft2 ), G = (g+tf)/A,
and
v
=vf +x(vg-f)
where vfl1/pf,
vg'l/pg, and x=hg/(ig+If).
These equations demonstrate how dependent the pressure drop,
is on the superficial
gas and liquid
velocities.
p,
29
APPENDIX
B
CALCULATION OF LEAKAGE LIMIT LINE
The system can be divided into two parts;
the tank, where the
pressure fluctuation with time is found using the equation (see
(3))
reference
dP
Q-9
dP
= <
dt
where Q
(B.1)
V/nP
is the gas volumetric flowrate, V is the tank volume (2.87 ft3 ),
nIl, and P is the average tank pressure; and the vertical tube, where
the pressure fluctuation is
dP
= P
g
Qf
(B.2)
go A
dt
where Qf is the liquid volumetric flowrate, g is the gravitational
8
acceleration, g=4.17x10
lbm-ft/lbf-hr2 and pf is the fluid density
(62.4 lbm/ft3 ).
Leakage occurs when the pressure in the tube is greater than the
pressure in the tank.
Combine Equations B.1 and B.2 to get the
inequality
2Pfo
(B.3)
>A
V/nP
When the lefthand side of this inequality is greater than the righthand
side, then, theoretically, leakage into the tank will occur.