SIMPLIFIED AND RAPID METHOD FOR DETERMINING FLOW CHARACTERISTICS
OF EVERY GAS-LIFT VALVE (GLV)
by
MEHDI ABBASZADEH SHAHRI, M.S.
A DISSERTATION
IN
PETROLEUM ENGINEERING
Submitted to the Graduate Faculty
Of Texas Tech University in
Partial Fulfillment of
The Requirements for
The Degree of
DOCTOR OF PHILISOPHY
IN
PETROLEUM ENGINEERING
Approved
Herald W. Winkler
Chairperson of the Committee
Lloyd R. Heinze
Co-Chair of the Committee
Waylon V. House
George B. Asquith
Javad Hashemi
Accepted
Peggy G. Miller
Dean of the Graduate School
August, 2011
ACKNOWLEDGEMENTS
First and foremost I would love to extent my gratitude and appreciation to my sincere mentor Dr. Herald
W. Winkler whom was there when I needed his guidance, his sharpness, unbelievable understanding, and
exceptional capabilities throughout this work.
I would like to thank Dr. Lloyd R. Heinze serving as co-chair of my committee and supported me
through this work with providing required instrumentations and technical notes.
I would greatly thank invaluable help of Dr. Waylon V. House with his generous guidance in exploring
new analysis methods to interpret results better.
I would greatly thank Dr. George B. Asquith and Dr. Javad Hashemi serving as my committee
members.
I would like to thank endless help of Dr. Masoud Zabet and Ms. Zahra Mizani for their incredible
moral support, warm company, friendship and kindness.
I would like to extent my gratitude to all Petroleum engineering faculty members at Texas Tech
University whom helped me in the meantime and all the students who supported me.
Last but not least; Special thanks to Dr. Mohamed Y. Soliman (Department Head) with his kindness and
encouragements.
ii
This work is dedicated:
To my unique uncle,
Hossein A. Shahri
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
ii
DEDICATION
iii
TABLE OF CONTENTS
iv
ABSTRACT
vii
LIST OF FIGURES
viii
LIST OF TABLES
xii
NOMENCLATURE
xiv
CHAPTER 1
CHAPTER 2
INTRODUCTION
Objectives
1
Dissertation Overview
2
LITERATURE REVIEW
Gas Lift
3
Gas Lift Valve (GLV)
8
Flow Behavior
12
GLV Performance Models
16
Valve Temperature
24
iv
CHAPTER 3
CHAPTER 4
CHAPTER 5
TESTING PROCEDURES
Static Testing Procedure
25
Probe Testing Procedure
25
Benchmark Valve Testing
27
Hydraulic Stabilization (Aging)
28
Blow-Down Test
31
API (ISO) Testing Procedure
32
BLOW-DOWN TEST
Volumetric Calculations
34
Discharge Coefficient Calculation
37
Flow Area Calculation
38
RESULTS & DISCUSSIONS
Flow Through Ports & Flow Through Ports
inside GLV
CHAPTER 6
43
The Gas Leak Rate
50
Justifying Thornhill-Craver Equation
50
CONCLUSIONS
Conclusions
v
56
CHAPTER 7
RECOMMENDATIONS
Recommendations
Transducer Calibration Using Dead-Weight
57
APPENDIX
A
APPENDIX
B
APPENDIX
C
Data Acquisition System (DAQ)
96
APPENDIX
D
Relevancy of LR change with Pbt
104
Tester
Measurement of Discharge Coefficient, Cd,
Using Benchmark Valve Testing
REFERENCES
58
61
111
vi
ABSTRACT
The current API testing method requires quite amount of time to complete a Gas Lift Valve (GLV)
test. The API method was developed for the GLV manufacture rather than the producer. There is a need
for a method of testing oriented toward the producer. In the proposed method of testing which is based on
the concept of blow-down; the valve is tested in a few seconds. The modified Thornhill-Craver equation
(TC) has been corrected for the discharge coefficient value. Since TC equation primarily developed for
the chokes and liquid passage through chokes, some gas dynamics readjustments needed for gas flow.
This method can easily be applied for the GLVs with check valve on them as well as cross-over seat
valves and all different GLVs with different structural architectures. It can be applied to tubing retrievable
as well as wireline retrievable GLVs. The current proposed industry instrument is not capable of
measuring the performance of cross-over seat valves but this method can perform the test on that
smoothly.
This method is feasible only with help of fast Analogue to Digital date acquisitions. The sample rate in
this method varies between 100-50000 samples per second to achieve the highest possible accuracy for
the measurement of pressure points as the time passes on. This method is aimed to be accurate in the
critical flow region where there is no effect of downstream pressure on the flowrate. The effect of
temperature on the valve opening and closing pressure has been investigated as well. This method will let
the user to evaluate tapered seat orifices as well as sharp-edged. Tapered seat can pass more gas than
sharp-edged seat at the same ball distance from the seat at rest. This method is capable of measurement of
the performance of cross-over seat valves, and GLVs with check valves.
The development of such testing method is for the favor of the producer. Testing GLVs with this
simple, rapid, and very inexpensive method before well installation will confide the producer of having a
well-set and well-handled GLV before each well installation.
In this experimental work, several hundred flow tests have been ran through different GLVs with
various port and ball sizes to quantify the flow behavior at critical flow conditions. It has been found that
the discharge coefficient is changing based on flow velocity profile (Reynolds number), upstream
pressure, flow condition (critical or sub-critical) and the orifice size. In orifice sizes smaller than 3/16
inch, the value is greater whereas the value stays almost constant for the greater orifice sizes when the gas
is flowing through orifice plate and or the ball is very far from the seat. The existence of the GLV body
impacts the discharge coefficient as well. It lowers the value of discharge coefficient by 1%. Applying a
constant value for discharge coefficient in different scenarios is not recommended and will result in up to
10% overestimating when TC equation used.
vii
LIST OF FIGURES
Page
Fig. 2.1—Schematic of a Gas Lift Well
3
Fig. 2.2—Setting GLVs Depth
4
Fig.2.3—Gas Lift Schematic with Instability
5
Fig.2.4—Orifice Valve Performance as its Size Varies
6
Fig.2.5—Gas Lift Unloading- Kick-off
6
Fig.2.6—Positioning the First GLV at Depth
7
Fig.2.7—GLV String Design to Unload a Well
8
Fig.2.8—Schematic of IPO and PPO GLVs
8
Fig. 2.9—Schematic of a Typical Tubing Retrievable IPO GLV
10
Fig. 2.10—Schematic of Cross-over Seats GLV
11
Fig. 2.11—Typical Isentropic Flow Pressure Ratio Responses to Flowrate in IPO GLVs
11
Fig. 2.12—Schematic of Ball/Stem at different Positions
12
Fig. 2.13—Schematic of Fluid Flow in GLV (Constant Upstream with Variable Flowing
Area)
Fig. 2.14—Determining rcritical for 1-1/2‖ J-20 Camco GLV with 5/16‖ Port ID (The
Critical Pressure Ratio is 0.52 at (qimax-qi)/(Piod-Ppd) = 0 for a 5/16‖ Port)
17
20
Fig. 2.15—Variability of Cd with Orifice Type
23
Fig. 3.1—Schematic of the Probe Tester
26
Fig. 3.2—Sample Plot of Changing Pressure with Stem Travel
26
Fig. 3.3—Bellows Assembly Load Rate Curve for 1‖ & 1-1/2‖ GLV
27
Fig. 3.4—Schematic of the Benchmark Valve
28
viii
Fig. 3.5—Hydraulic Stabilizer (Valve Hydro-tester or Ager)
29
Fig. 3.6—Schematic of Blow-Down Dynamic Test Facility
31
Fig. 4.1—Schematic of the Ball – Seat Position
38
Fig. 5.1—Plot of Pressure vs. Time for 1/4‖ Monel, 1-1/2‖ J-20 Camco GLV
43
Fig. 5.2—Calculated Equivalent Port Area Based on Polynomial Regression Analysis (1st
Trial)
Fig. 5.3—Calculated Equivalent Port Area Based on Polynomial Regression Analysis
(2nd Trial)
Fig. 5.4—Plot of Pressure vs. Time, Flowrate, and Apparent Port Size Open to Flow in a
3/16‖ Monel Sharp-Edged Seat
Fig. 5.5—Calculated Equivalent Port Area Based on Exponential Regression Analysis
Fig. 5.6—Calculated Equivalent Port Area Based on Previous Exponential Regression
Analysis
Fig. 5.7—Calculated Equivalent Port Area Based on Measured Raw Data
Fig. 5.8—Effect of Slight Tapered Seat on the Gas Passage in the 1-1/2‖ J-20 Camco
GLV
Fig. 5.9—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in
3/16‖ Monel Sharp-edged Seat
Fig. 5.10—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only
in 1/4‖ Monel Sharp-edged Seat
Fig. 5.11—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only
in 5/16‖ Monel Sharp-edged Seat
Fig. 5.12—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only
in 3/8‖ Monel Sharp-edged Seat
Fig. 5.13—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only
in 1/2‖ Monel Sharp-edged Seat
ix
45
45
46
47
48
49
50
52
52
53
53
54
Fig. 5.14—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only
in all Monel Sharp-edged Seat Port Size
54
Fig. A.1—Plot of Pressure vs. Output mili-volt for 0-500 psi Transducer
59
Fig. A.2—Plot of Pressure vs. Output mili-volt for 0-1000 psi Transducer
60
Fig. B.1—Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV
with 5/16‖ Port Size when the Ball is at 1/4 Fully Open Travel Position
Fig. B.2—Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV
with 5/16‖ Port Size when the Ball is at 1/2 Fully Open Travel Position
Fig. B.3—Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV
with 5/16‖ Port Size when the Ball is at 3/4 Fully Open Travel Position
Fig. B.4— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV
with 5/16‖ Port Size when the Ball is at Fully Open Travel Position
63
64
64
65
Fig. B.5—Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV
with 5/16‖ Port Size when the Ball is at 1-1/2 Fully Open Travel Position
65
(Beyond Fully Open)
Fig. B.6—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First
Second
Fig. B.7—Plot of Pressure rate Against Ball Position in 5/16‖ Monel J-20 Camco GLV
Fig. B.8—Combined Plot of Pressure vs. Time in Benchmark Valve Testing in 3/16‖
Monel Port Size for the First Second
Fig. B.9—Change of Pressure vs. Time relative to Ball Position in 3/16‖ Monel Port
Fig. B.10—Combined Plot of Pressure vs. Time in Benchmark Valve Testing in 1/4‖
Monel Port for the First Second
Fig. B.11—Change of Pressure vs. Time relative to Ball Position in 1/4‖ Monel Port
Fig. B.12—Combined Plot of Pressure vs. Time in Benchmark Valve Testing in 3/8‖
Monel Port for the First Second
x
67
68
68
69
69
70
70
Fig. B.13—Change of Pressure vs. Time relative to Ball Position in 3/8‖ Monel Port
Fig. B.14—Sensitivity of Cd to Pressure and the Valve Body in 5/16‖ Monel Sharpedged Seat at T= 73 oF
Fig. B.15—Ball-Seat Relevancy Due to Angle, And Distance
Fig. C.1—NI 9237 with 4 Channel, ±25mV/V, 24 Bit Resolution with Max. Speed Rate
of 50,000 Samples per Second per Channel [28]
71
72
73
96
Fig. C.2—NI USB-9162 Chassis
97
Fig.C.3—Display Shot of MAX with NI USB-9237 Device
98
Fig. C.4—Front Panel View of the Developed Program
99
Fig.C.5—Block Diagram of the Developed Program
100
Fig. C.6—DAQ Assistant Setup
101
Fig. C.7—Empirical Measurement of Minimum Value for Pressure Drop Increment
102
Fig.D.1—Actual Probe Tester to Measure the Linear Stem Travel
104
Fig. D.2—Probe Test Results for 1/2‖ Monel Port in 1-1/2‖ J-20 GLV at set Pbt = 149
psig
Fig. D.3—Probe Test Results for 1/2‖ Monel Port in 1-1/2‖ J-20 GLV at set Pbt = 444
psig
Fig. D.4—Probe Test Results for 1/2‖ Monel Port in 1-1/2‖ J-20 GLV at set Pbt = 517
psig
xi
106
108
110
LIST OF TABLES
Page
Table 4-1—CP /CV for different Gases
36
Table 4-2—Technical Specifications of Cylinders
37
Table 4-3—Area Open to Flow at Different Ball-seat Positions
39
Table 5-1—Curve-fit Values for 1/4‖ Monel, 1-1/2‖ J-20 Camco GLV
44
Table 5-2—Regression Exponential Analysis to fit the Data in 5/16‖ Port 1-1/2‖ J-20 GLV
47
Table A-1—Pressure vs. Output Voltage in 0- 500 psi Sensotec Transducer
58
Table A-2—Pressure vs. Output Voltage in 0-1000 psi Sensotec Transducer
59
Table B-1—Set Positions of Ball/Stem in 5/16‖ Sharp-Edged Monel Seat
62
Table B-2—1-1/2‖ OD GLV with Ab = 0.77 in2, Sharp-Edged Monel Seat
62
Table B-3—Extracted Empirical Values for Gas Throughput from Benchmark Valve
Testing for 5/16‖ Port
Table B-4—Cd Sensitivity to Upstream Pressure and the GLV Body
Table B-5—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
67
72
76
Table B-6—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
77
Table B-7—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
78
Table B-8—Cd Calculations for 3/16 inch Port Size Using Orifice Port Only and Orifice
79
Port Only Inside the Body of GLV
Table B-9—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
80
Table B-10—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
81
Table B-11—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
82
xii
Table B-12—Cd Calculations for 1/4 inch Port Size Using Orifice Port Only and Orifice
83
Port Only Inside the Body of GLV
Table B-13—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
84
Table B-14—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
85
Table B-15—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
86
Table B-16—Cd Calculations for 5/16 inch Port Size Using Orifice Port Only and Orifice
Port Only Inside the Body of GLV
87
Table B-17—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
88
Table B-18—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
89
Table B-19—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
90
Table B-20—Cd Calculations for 3/8 inch Port Size Using Orifice Port Only and Orifice
Port Only Inside the Body of GLV
91
Table B-21—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
92
Table B-22—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
93
Table B-23—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
94
Table B-24—Cd Calculations for 1/2 inch Port Size Using Orifice Port Only and Orifice
Port Only Inside the Body of GLV
95
Table D.1—Probe Test Results for 1/2‖ Monel Port, 1-1/2‖ J-20 GLV at Pbt= 149 psig
105
Table D.2—Probe Test Results for 1/2‖ Monel Port, 1-1/2‖ J-20 GLV at Pbt= 444 psig
107
Table D.3—Probe Test Results for 1/2‖ Monel Port, 1-1/2‖ J-20 GLV at Pbt= 517 psig
109
xiii
NOMENCLATURE
Symbol
Denote
̅
Minimum mean effective bellows-charged pressure to move the ball off seat, psi
̅
Spring load rate, psi/in
A
Area, in2
Ab
Area of the bellows, in2
Aeff
Effective gas flowing area, in2
Ap=Av
Area of the port, in2
API
American petroleum institute
BHP
Bottom hole pressure, psig
BLR
Bellows load rate, psi/in
Cd
Discharge coefficient
Cv
Flow coefficient
D
Apparent diameter of the Upstream area, in
d
Port diameter (Flowing diameter ≤ Port diameter), in
Fl
Liquid Pressure recovery factor
g
Gravitational acceleration, lb/sec2
GLV
Gas lift valve
H
Dynamic travel of the ball from the seat, in
h
Height, ft
IPO
Injection pressure operated (Gas-Lift Valve)
ISO
International standard Organization
xiv
k = Cp/Cv
Mscf/d
Ratio of gas specific heat at constant pressure to specific heat at constant volume
1000 standard cubic feet per day
Pbt
Bellows charged pressure at temperature, psig
Pid
Injection pressure at depth, psig
Ppd
Production pressure at depth, psig
PPO
Psc
PTRO
Production pressure operated (Gas-Lift Valve)
Pressure at standard condition = 14.696 psig
Test rack opening pressure at standard conditions, psig
Pvc
GLV closing pressure at ambient conditions, psig
Pvo
GLV opening pressure at ambient conditions, psig
qgimax
Maximum injection-gas flowrate, Mscf/D
qgsc
Injection-gas flowrate at standard conditions, Mscf/D
qmass
Mass flowrate, lb/sec
r
Ratio of the upstream pressure to downstream pressure
r
Radius of the port, in
rcritical
Critical pressure ratio
S
SGg
Slant side of the frustum, in
Gas specific gravity
T
Upstream absolute temperature, oR
Tb
Bellows charged temperature, oF
TC
Thornhill-Craver
Tinj
Injection gas temperature, oF
xv
Tsc
Temperature at standard condition = 520 oR
v
Velocity, ft/sec2
Y
Gas expansion factor
Z
Gas compressibility factor
β
Ratio of flow area to inlet port area
θ
Ball-port angle, Rad
μ
Gas viscosity, cp
ρg
Density of gas, lb/ft3
ρgup
Upstream gas density, lb/ft3
xvi
Chapter 1
Introduction
Objectives
The main objective of this study is to develop a testing method to assure each GLV performance at
lower cost and less time. The development of such method is mainly for screening purposes of GLVs
before well installations. Not all GLVs manufactured the same way and some of them may behave totally
different under same conditions. Another critical issue with GLVs is the handling. If each GLV does not
get handled properly, its internal settings may change due to the GLV internal architecture and moving
parts.(due to existence of high viscous dampening fluid inside each GLV)
The current methodology which API is recommending is very time consuming while the proposed
method in this dissertation just takes a few seconds (API method is done at steady-state conditions
whereas this method is done under transient conditions). This method is practically useful with critical
flow patterns or when the flow regime is supersonic. In other word; this method was developed mostly for
high production wells. This method of testing does not substitute the API method and is recommended for
high production wells. The effect of temperature on the GLV performance has been studied and included
in testing as well although in the laboratory scale the temperature effects were negligible. In the
experimental setup, the corresponding time lag due to gas wave travel-time between the valve and the
transducer has been minimized by adding some extra transducers or relocating the transducers. With all
the known characteristics of the working gas (nitrogen) the equivalent GLV port size has been determined
by applying a ―blow-down‖ method. The testing system allows the operator to monitor the difference in
gas passage through a sharp-edged orifice and tapered orifice. The discharge coefficient has been
reestablished for each port size. The overall aims of such test are:
1- To assure the operator of achieving the desired production flowrate with the installed gas lift
system.
2- To reduce or eliminate the costs to retrieve the GLVs from Off-shore and/or on-shore wells.
Another objective of this method is to quantify the ball movement in each GLV which results in gas
passage over its entire range. This is to eliminate the need for probe testing in order to calculate bellows
LR and maximum linear stem travel. Also, since this method of testing uses gas, it has the capability of
measuring the performance characteristics of cross-over seat GLV. With the current probe testing facility
in the oil industry, cross-over seat GLVs cannot be quantified simply because the depth micrometer in the
probe device can not touch the tip of the ball directly in that kind of GLV architecture.
1
Dissertation Overview
This study tried to focus on the systematics of the method.
Chapters 1 & 2 give the overall aim of this work along with all the past done works GLV flow
measurements and installation.
Chapter 3 over views the required testings to understand each GLV behavior more constructively. These
testing procedures are approved by API and ISO as acceptable practice routines. In this chapter, the blowdown procedure which aims to facilitate the extreme flow measurement has been introduced.
Chapter 4 addresses the blow-down method in detail. Blow-down method is acceptable through API
and ISO but this new method has its own uniqueness. Some challenges in furnishing experiments through
this method have been introduced. Some of the main challenges are defining the correct discharge
coefficient value for each flow system as well as the right flowing area in GLV at each flow condition.
Chapter 5 discusses the experimental results. The variation of discharge coefficient with upstream
pressure and flowing area, the variation of flowing area as the stem dynamically moving, the effect of
gas-lift valve body on limiting the flow, the location of the ball during valve operation and the
architecture of the seat in passing gas throughput has been discussed.
Chapter 6 details the conclusions.
Chapter 7 speaks about the recommendations to completely address this problem for future works in a
greater detail.
There are some Appendixes for further clarifications of each measurement and calculation steps. At the
end, the dissertation has been wrapped up with references.
2
Chapter 2
Literature Review
Gas Lift
Gas lift, one of artificial methods of lifting fluid, has been applied extensively for several decades (started
in 1800’s). As an artificial lift method, Gas lift can be applied to wells as deep as 15,000 ft and can lift
fluid at rate of 50,000 STB/D. Gas lift aims to increase the flow rate by reducing the flowing gradient of
the flowing fluid. In other words; adding supplement amount of gas (from an external source) to increase
the gas-liquid ratio (GLR) to reduce the flowing fluid density (or gradient). Gas lift is the only form of
artificial lift that does not acquire downhole pump. Comparing to the other forms of artificial lift methods,
gas lift is simpler, more flexible, and has the ability to operate at vast ranges of fluid production which
makes it a good candidate for offshore applications as well. Unlike the pump-based methods, gas lift is
incapable of reducing the Bottom hole pressure (BHP) very low, requires high pressure gas to operate and
may encounter some production instabilities due to variations in gas injection rate and the injection depth.
Figure 2-1 depicts a schematic of a gas lift well. Gas lift can be continuous or intermittent. In this
dissertation, we deal with continuous gas lift installation.
Fig. 2-1—Schematic of a Gas Lift Well [1]
3
When the well is dead or non-productive, it means that the fluid gradient is high or the (GLR) is low. In
order to force the fluid to flow, the easiest and simplest ways is to inject supplemental amount of gas from
and external source. Fig. 2-2 demonstrates the fluid gradient profiles and how to determine the point of
gas injection to unload a well.
Fig.2-2—Setting GLVs Depth [2]
One of the limitations of each gas lift system is the minimum BHP. The minimum pressure gradient is
around 0.22 psi/ft and rarely go below 0.15 psi/ft [1] therefore gas lift is a good candidate for waterflood
projects where the BHP is maintained although water break through will limit the tubing performance. In
setting the gas lift valve strings the deeper the injection point, the lower the BHP can be forced because of
availability of more gas in solution. The optimum gas injection rate has to achieve to avoid reduction in
net performance due to friction (which is greater than density reduction). Fig. 2-3 demonstrates the
single-injection gas lift installation with some possible instability. The main instabilities in the gas lift
process can occur due to changes in tubing pressure when the injection pressure is not high enough. If the
injection gas pressure reaches so high that the flow becomes critical, the gas lift operation stays stable
regardless of changes in the tubing pressure.
4
Fig. 2-3—Gas Lift Schematic with Instability [3]
Selection of the right orifice port size is very critical and fundamental to the gas lift stability [3]. In the
example shown in Fig. 2-4, the two orifice valve performances intersect the tubing performance at 2.75
MMscf/D injection rate. However, in this rate, the larger orifice is performing instable whereas the
smaller orifice is stable.
5
Fig. 2-4—Orifice Valve Performance as its Size Varies [3]
In each gas lift steady state design, the points of injection has to be determined. The first point of injection
has to be designed for kick-off. It means that at early time, when the tubing is full of liquid and the
annulus are is charged with high pressure gas, the gas pushes the liquid out of tubing through U-tubing.Utube effect means that high injection gas pressure is required to force the gas into the tubing. The required
pressure is calculated based on the gas density inside the annulus and the density of the fluid inside the
tubing at depth of valve. In Fig. 2-5, the required injection pressure to kick-off the well is around 3500
psig. Once the well is kicked off, the operating pressure will reduce as the fluid mixes with lift gas. We
may need to employ another compressor to kick-off the well.
Fig.2-5—Gas Lift Unloading- Kick-off [3]
6
Gas lift installation may vary much. At some high production wells, we may not need to install any GLV
and only a large orifice choke will pass required amount of gas to lift the liquid form the wellbore.
Knowing the mechanism of functioning of as lift will help utilizing such scenarios rather than spending
lots of fund resulting less fluid production. Gas lift is a single point injection but at different depths. The
lower the GLV or orifice check valve can be set, the higher the drawdown can be achieved. Each the
drawdown is higher, the BHP is lower and consequently in high productivity index reservoirs, more fluid
can be lifted.
The installation of GLVs at depth is critical. Wrong order of installation, wrong opening pressure set, etc.
will result in failure in such design. In gas lift, as we go deeper, the set opening pressure of the valves
decreases although the weight of gas column above each GLV increases. This decrease in set valve
opening pressure will cause the upper valves to close as we start to unload the lower valve and so on. Fig.
2-6 and Fig. 2-7 demonstrate the valve depth determination with respect to the flowing tubing pressure (if
injection is through casing), injection gas gradient, and formation fluid gradient. As it has been shown in
Fig. 2-7, the lowest GLV is just an orifice check valve. For high production wells, the lifting gas in
injected from the tubing and lift the fluid from casing area.
Fig.2-6—Positioning the First GLV at Depth [3]
7
Fig. 2-7—GLV String Design to Unload a Well [3]
Gas Lift Valve (GLV)
Gas lift is a closed rotative system that requires free gas. In each gas lift system, there is a compressing
unit to increase the gas pressure as designed, GLV, and the tubulars. GLVs can operate either with
injection gas (injection pressure operated, IPO) or production fluid operated (production pressure
operated, PPO). The operation mechanism of either type of GLVs is the same. In this dissertation, all the
calculations are based on IPO GLVs. Fig.2-8 differentiates the IPO GLV from PPO GLV.
Fig. 2-8—Schematic of IPO GLV (on the left) and PPO GLV (on the right)
8
The first bellows-charged GLV was invented by King [4] in 1940. Prior to introducing bellowscharged GLVs, spring loaded GLVs were common with passage of time, better designs for better
understanding of unloading wells helped developing GLVs. Combining gas-lift with other artificial lift
methods were proposed in the industry as early as 1930’s. The King’s valve was designed to lift a low
volume of liquid. Although some changes have been done on the first design, the main architecture
preserved. In the Middle East gas lift has been primarily adopted for lifting water for waterflood projects
in oil industry. Selecting gas lift as the main lift system is vital, and depends on the availability of a high
pressure and sufficient lean gas sources.
Designing the most suitable and optimum system for each application, off-shore or on-shore, is the
most important part gas lift design. Gas lift system is a closed rotative gas system which demands a high
pressure source of gas, compressors, and gas lift valves (GLVs). GLVs are the heart of each gas lift
design. The GLV has to be selected accordingly. Sizing of compressor and tubulars is interconnected with
the available source of gas and the application type. Assuring the operator of getting the right and
predicted amount of fluid is critical. In this regard, GLVs should be tested based on their performance to
assure of passing the right amount of gas to lift the predicted volume of liquid. Any failure in sizing the
GLV will result in low to no fluid production. Increase in casing or tubing pressure or overloading the
compressors are such examples of possible failure in gas lift design. Gas lift can handle abrasive sand in
low productivity, deviated, and high GOR wells. Gas lift process [5, 6] is limited to the BHP and is less
effective with scale formation, corrosion and existence of paraffin which increases the friction in the
tubular.
GLV by analogy is a mechanical back pressure regulator [2]. In other words, the inlet injection gas
pressure (Pid) and the available production pressure (Ppd) have to pass the pre-determined opening
pressure of each GLV to let it function. The mechanics of GLV is solely based on pressure balance across
the valve itself. Each GLV has a dome section which is charged with gas (usually nitrogen) at a certain
pressure and has dome seal [7] at one end of it for charging and discharging purposes. The dome section
is attached to the bellows assembly. Bellows acts as a piston that can be sealed. Bellows are attached to
the stem which ends to the ball. All the mentioned sections are moving as a single unit in each GLV.
When the GLV is closed, the ball is seated on its sized port area. As a rule, in each GLV the ball is 1/16‖
larger in diameter than each port size. On the downstream side of the port a check valve does not allow
the back flow from either tubing or casing to interfere with each other. Figure 2-9 shows a simple
schematic of a typical tubing retrievable GLV. Fig. 2-10 demonstrates a cross-over seat GLV. Cross-over
seat GLVs are designed to switch from tubing injection to casing injection (vice versa) without rig up for
pulling tubing or running wireline. There is a modification in the structure of cross-over seat GLVs as
depicted in Fig. 2-10 comparing with Fig.2-9.
9
Depending of the position of the ball with respect to the port, the gas flow regime may change.
Theoretically when the area to flow is equal to the port area, the valve is fully open and expected to pass
the maximum gas. This situation is so called orifice flow. In orifice flow, the minimum area is the port
area. Orifice flow performance can be divided into two distinct regions: critical and subcritical. In case of
critical flow, dropping downstream pressure has no effect on the upstream flow rate. When the Pid is not
sufficient to overcome the bellows-charged pressure (Pbt), the flowrate reaches a maximum and then
drops to zero value at some positive production. This flow regime is known as throttling flow. In
throttling flow regime, the open area to flow is smaller than the port area. At this case, the downstream
pressure affects the production flowrate. Fig. 2-11 exhibits different flow regimes in an IPO GLV. There
is another flow regime in between these two main flow regimes which is known as transition. Transition
flow regime is similar to throttling performance except the final production rate is not zero when the
downstream pressure is atmospheric pressure. Transitional flow rarely occurs.
Pi
Fig. 2-9--Schematic of a Typical Bellows-charged Tubing Retrievable IPO GLV
10
Fig. 2-10— Schematic of Cross-over Seats GLV
Fig. 2-11—Typical Isentropic Flow Pressure Ratio Responses to Flowrate in IPO GLVs
11
Flow Behavior
When the ball is seated on the port area, its tip is lowered to ―X‖ inside the port. Fig.2-12 shows the
situation clearly. As the balls rises from the seat, the GLV starts to bet initially open. The injection gas
pressure should be sufficient enough to lift the ball against the bellows pressure which tends to push the
ball down.
r
2r
X
Fig. 2-12-- Schematic of Ball/Stem at different Positions
When the ball distance to the seat is equal to ―X‖, the tubing pressure plays the main role in GLV
closing (characteristics of throttling flow regime). Calculating the value of ―X‖ based on the fact that we
know (in Camco GLVs) the size of the ball is 1/16‖ inch larger than the port size can be done with Eq. 21.
(
)
√
(
)
where, r = radius of the port size, inch
The value of ―X‖ varies with the ball and port size and ranges 0.0423 to 0.1524 for 3/16 to 1/2 inch
port diameter. When the ball is in the ―X‖ range, the flow behavior is throttling. In other words, the GLV
behavior is sensitive to casing and tubing (upstream/downstream) pressures.
12
Eq. 2-2 reveals the throttling pressure range. Bellows assembly load rate plays a critical role in this
regard as well which is related to the bellows charged pressure. Each the bellows charged pressure is set
higher, the corresponding bellows LR would be greater.
where, LR= Bellows assembly load rate, psi/inch
At throttling flow, when the production pressure at depth (Ppd) is approaching the bellows charged
pressure at temperature (Pbt), the ball is close to the seat. When the downstream pressure drops more than
a certain value, the GLV closes. Dynamic tubing sensitivity factor has to be defined to model this
phenomenon. This sensitivity factor is easily related to the ratio of open area to flow to bellows area.
In this work, the equivalent Cd has been measured applying the benchmark valve testing method
explicitly. The value of Cd is changing by the flow’s Reynolds number. At high Reynolds number, the
assumption of uniformity of velocity profile is correct [8]. Cd corrects the velocity profile (Reynolds
number), contraction geometry, and net expansion factor in orifice flow. If the geometry is constant, flow
coefficient can be used instead of Cd. The only difference between these two is the combination of
velocity profile with Cd in flow coefficient. Eq. 2-3 bears the flow coefficient formula. Note that flow
coefficient is just valid to be used for fixed geometry devices. In this dissertation, I used benchmark valve
to measure Cd, therefore at each ball-stem setting, the flowing area was hold constant and the concept of
flow coefficient is valid.
√
where, Cd = Discharge coefficient, dimensionless
d = Port diameter, in
D = Upstream flowing diameter, in
Adiyodi et al. [9] has claimed that TC equation under predicts the GLV size at small orifice sizes and
over predict it at large orifice port sizes. The results of this work reveals that TC equation just
underpredicts the flow at 3/16 inch port size and over predicts the flow for the bigger orifice sizes. If the
port area is assumed as the flowing area with no correction for the ball position, the results are higher than
what actual gas throughput is. When the casing pressure is close to the bellows charged pressure, the
GLV will throttle.
Each GLV can be modeled based solely on its response to pressure and flowrate. The response mainly
depends on mechanical, thermodynamical and frictional factor. Governing equations are conservation of
13
mass, momentum, and energy as well as heat transfer related equations. Modeling the geometrical shape
of the gas passage conduit is another concern. The TC equation was developed for the gas passage in bean
chokes from 1/8‖ to 3/4‖ and not gas-lift industry based on converging nozzle theory. Neely et al. [10]
modeled the GLV as a converging-diverging nozzle in which the pressure at the throat is the minimum
(the velocity is maximum). In his simulation, the cross section of the throat is changing with ball
movement and the position of the stem.
Turzo [11] developed a computational fluid dynamic, CFD, based analytical-numerical solution for the
GLV behavior modeling. He generated the same results as API [12]. In their approach, they solved 5 sets
of equations including Conservation of mass, energy, Navier-Stokes equation, state of the fluid which in
compressible and the enthalpy changes due to change in internal energy at each position. The main errors
associated with such approach come to play in the ball-tip section when the programmer wants to assign
the correct pattern of pressure distribution on that area.
Decker [13] tried to solve the GLV mechanics analytically. He proposed the term of ―Bellows Load
Rate‖ and derived the analytical relationship between the bellows functioning with the acting pressure on
the ball and stem. He tried to locate the ball based on the effective pressure acting on the ball and bellows
area. His work was on the spring-loaded GLVs that never got fully open since the upstream and opening
pressures were very close together. This behavior puts the GLV behavior in the ―throttling‖ mode in
which a small change in downstream pressure would affect the upstream pressure and cause the valve to
wobble. The wobbling initiates corrosion in the ball-tip and seat contact areas which is destructive even in
short term use. The findings in this research [13] revealed that prediction the accurate performance of
each GLV requires knowing:
1- The pressure distribution through the valve
2- Ball position at each stage as a function of mean effective pressure acting on the bellows area;
which is called ―pressure response‖
3- Corresponding flow area regarding to the ball position in the GLV
The relationship between ball position and flow explicitly depends on the ball size and port geometry,
and a general relationship is hard to satisfy all the requirements. The force balance in each GLV is a
delicate function of two independent factors. The mechanical effect which is incorporating with the
bellows behavior and the thermodynamic effect which deals with the dome charged gas pressure. The
overall pressure response is given in Eq. 2-4 as follows:
̅
14
where, ̅ is the Mean effective pressure on the ball at each position, psi
dx: is the distance of the ball movement from the seat, inch
Load rate, LR, is defined as the pressure requires to move the ball off seat by the amount necessary to
obtain an orifice flow regime. LR is a characteristic of bellow in each GLV and directly depends on the
bellows architecture and coil size. LR can be approximated with Eq.2-5 as well as the departing from
opening pressure from closing pressure.
̅
̅
(
)
∫ (
)
where, ̅ is the Mean effective pressure on the ball at each position, psi
K: is the spring load rate, psi/inch
: is the minimum mean effective pressure on the dome to move the ball off seat, psi
Mechanical behavior of bellows can be determined easily. On the other hand, determination of
thermodynamical behavior of the dome is more complex as it depends on the pressure, temperature, dome
volume, gas properties, and bellows area. The viscous effects are negligible comparing with mechanical
and thermodynamical effects.
It has been tested empirically and analytically that the effect of gas compressibility must be included in
analysis of dome behavior because all the gases are not ideal and the real gas behaviors are different. The
frictional non-linearity can be smoothed in any fractional dome volume change and put into a linear
equation as Eq. 2-6.
∫ (
)
Substituting this approximation into the previous equation will lead to Eq. 2-7.
̅
̅
[
]
Eq. 2-7 counts for effect of gas compressibility with including compressibility factor.
15
GLV Performance Models
In order to model each GLV (bellows-charged or spring-loaded), we need to have a broad knowledge of
mechanical behavior of each GLV as well as gas dynamics. As Fig. 2-1 clearly shows,
dome
2
pressure(Pbt, psig) is acting on the dome area (Ab, in ) whereas injection gas pressure(Pid, psig) is acting
on the bellows area less port area (Ab-Ap, in2) and the production pressure (Ppd, psig) is acting on the port
area (Ap=Av). In other words, the GLV stays closed when the opening force (which is the P id acting on
(Ab-Ap) +Ppd acting on Ap) is equal to the closing force (which is Pbt acting on Ab). Eq.2-8 and Eq. 29show the state of the GLV, using a simple force balance, in PPO and IPO GLVs respectively. In this
research, all the calculations are derived based on IPO GLVs.
(
(
)
)
Therefore, IPO GLV stays closed till Eq. 2-10 holds.
(
)
If the GLV is open, it will stay open while the condition in Eq. 2-11 is correct.
To understand and illustrate a better vision of GLV performance behavior, a detailed dynamic force
balance approach is needed to quantify the factors affecting the behavior of GLV at each condition.
The Bernoulli equation has to apply for the fluid element. Eq. 2-12 to 2-15 represents Bernoulli equation
combined with Euler equation which is energy conservation for a fluid element.
Since the height on gravitational field is negligible, the potential energy coming from that source is
negligible. The second term in Eq. 2-12 represent the kinetic energy of the fluid (density replaces mass)
and the last term is the pressure at that element. Applying Eq. 2-12 for two different elements at a pipe
showing in Fig. 2-13 will result in Eq. 2-13. Because the flowing fluid in this case is gas which is
compressible; the value of its density is a function of pressure, temperature and type of gas.
16
D
Upstream Area
P1, ν1
P2, ν2
d
Flowing Area
Fig. 2-13—Schematic of Fluid Flow in GLV (Constant upstream, variable flowing area)
In Fig. 2-13, when the flowing area is equal or greater than assigned port area, we need to use the
value of port area. Because at early stages when the GLV is not fully open, the minimum flowing area is
not the port area and using that value will ended to erroneous results.
Applying mass conservation theory as Eq. 2-14 says that the mass on either sides of flow (upstream and
downstream, Fig. 2-6) has to stay constant.
where, qmass= Mass flowrate inside the pipe
A1 andA2 = Upstream and downstream cross sectional areas
Squaring both sides of Eq. 2-14 will result in Eq. 2-15.if Eq. 2-14 now be substituted in Eq. 2-12, with
some rearrangements, Eq. 2-16 will be written.
(
)
√
√
Substituting for the areas by the pipe diameter will result in final form of Eq. 2-17 and Eq. 2.18. In this
work, the density of the working fluid, which is gas, stays constant for the short term of test time. So the
density term can get cancel out.
√
√
17
where, β=d/D: Ratio of flowing diameter to upstream diameter
qgi= Volumetric gas flowrate (not mass flowrate)
qgsc= Volumetric gas flowrate at standard conditions
Psc , Tsc = Pressure and Temperature at standard conditions
Because there is a difference between what analytically can pass through an opening and what really will
pass, discharge coefficient has to be introduced. This coefficient regulates the ideal flowrate with the
actual. On the other hand, since the flowing fluid in this setup is gas, the expansion coefficient has to be
considered although all of testing in this setup is in critical condition and that value stays constant. Eq. 219 holds all these affecting parameters. Eq. 2-19 is very similar to ISO-5167 [14, 15].
√
√
(( )
[
where,
(
( ) )
√
( ) )]
√
⁄
⁄
Re = Reynolds number for upstream flow = ρ1*v1*D/μ1
μ = Upstream flow viscosity, cp
Y = Gas expansion coefficient = 1-(0.41+.35*β4)*(P1-P2) / (k*P1); pressures in absolute
Eq. 2-19 is not always good because it assumes that with all variations of flowing area, the flow if fully
developed in the port area which is not correct. In each GLV flow system, we need to find the minimum
flow area at each time. In this dissertation, I developed and modified the available TC equation for
different orifice port sizes as well as discharge coefficients. Note that in Eq. 2-19, d/D ratio has to be in
the range of 0.15 to 0.7 and Reynolds number has to be at least 1000. This is one of the main limitations
of such formula to be applied for flow through GLV because the d/D ratio is zero when GLV is closed or
about to initially open and is equal to one when the GLV is fully open. The value of Reynolds number has
been measured to be greater than 35000 which means the flow is turbulent and the velocity profile is
uniform [8].
18
Dynamic force balance of each GLV is used to regulate and understand the actual behavior of each GLV
at different conditions. This sort of behavior has to be understood and addresses accordingly. Dynamic
force balance calculation has the same basis as static force balance but with involvement of other valve
characteristics. One of main factors affecting such force balance is bellows Load Rate (LR, psi/inch)
which is the bellows specific characteristic. LR is the force required to apply to the bellows to displace
the ball off seat for one inch. LR is a mechanical characteristic of each nitrogen-charged bellows GLV.
There is also gas dynamic factor affecting dynamic force balance. This factor is called discharge
coefficient (Cd). Cd is the ratio of measured mass flowrate in each GLV to the theoretical mass flowrate.
One of the adopted formulas, widely accepted for gas flow through chokes, is the Thornhill-Craver (TC)
equation [1]. Since TC equation was selected for the gas throughput calculations of the GLVs and seats,
this equation and its inherent coefficients need to be checked for accuracy. Some tests have been run in
this regard with various port sizes and seats. It has been monitored (and measured) that sharp-edged
monel seat would pass less gas than slight tapered entry tungsten carbide seats. The applied TC equation
[16] shown in Eq. 2-20. TC equation originally has been developed for a 6 inch bean choke with rounded
entrance [17]. If the entrance changes to sharp-edged, the value of Cd will drop. Shahri [18] found that
the Cd values for sharp-edged orifice seat is around 0.85 and is not a constant value. The value of Cd
changes as the orifice size changes but the changes are not much.
√
√
where, qgsc= Volumetric gas flowrate at standard condition, Mscf/D
Aeff = Effective flowing area, in2
Cd = Discharge coefficient (experimental)
Pup = Upstream pressure, psig
g = Gravitational acceleration, 32.174 ft/sec2
k = CP/CV = Ratio of specific heat at constant pressure to specific heat at constant volume
r = ratio of downstream pressure to upstream pressure (Ppd / Pid)
SGg = Specific gravity of gas at valve (air = 1)
T = Injection gas temperature at inlet of the valve, oR
Z = Compressibility factor at valve conditions
19
In this dissertation nitrogen has been used as the primary gas injected and the gas to charge the
bellows. If we substitute Cd= 0.865 (widely accepted value) while assuming the upstream temperature (T)
is the same as temperature at standard condition (T =60 oF = 520 oR) into Eq. 2-20 at critical condition,
we will end up with a simpler form of that equation which is illustrated in Eqs. 2-21, 22 and Eq. 2-23.
(Using Nitrogen, SGg =0.9672, k=1.4)
(2-21)
(Using Air, SGg =1, k=1.4)
(2-22)
(Using Methane, SGg =0.6, k=1.32)
(2-23)
Therefore at critical conditions, using nitrogen, Eq. 2-20 gives us the close approximate answer for the
gas flowrate and/or Aeff. Discharge coefficient is not a fixed number although its variation is not much.
Eq. 2-24 reveals the initial condition that determines if the GLV is in critical condition or not.
k = ratio of gas specific heat at constant pressure to gas specific heat at constant volume, (Cp / Cv)
We can find the critical ratio [12], rcritical, by plotting (qgimax – qgi) / (Piod – Ppd) against (Ppd / Piod) and look
for the point that the data are getting off of abscissa. That point represents the critical ratio. Fig. 2-14 is a
demonstration of the test with real data.
50
(qgimax-qgi)/(Piod-Ppd)
45
40
35
30
25
20
rcritic
15
10
5
0
0
0.2
0.4
0.6
0.8
1
Ppd/Piod
Fig.2-14—Determining rcritical for 1-1/2” J-20 Camco GLV with 5/16” Port ID (The Critical Pressure Ratio is
0.52 at (qimax-qi)/(Piod-Ppd) = 0 for a 5/16” Port)
20
The best curve fit to the pressure decay data during the dynamic test has been found in a very good
(identical) agreement with API [12]. Curve-fit formula can be used for better forecasting the behavior of
pressure points as the time vanishes and this method has been approved by API. The only different
between the approach in this dissertation and what API proposed is that this method is to be done in
transient mode while API’s method is in steady state mode. Therefore this method saves a lot of time
while producing the same outputs.
Along applying TC equation for GLV performance tests, Decker K.L. [19] did a study in continuous
gas lift operation claiming that TC equation can overestimate the flowrate up to 30% higher than actual
flow capacity. It is advantageous to have near steady state flowrate for small changes in pressure
differential across the orifice. If the injection and production pressures are stable, the best pressure
differential would be the minimum one. Choosing to have a large pressure differential is a wise choice
when the well has a history of wide fluctuation, slugging and surging. Because unstable flow can affect
sand and water production, it is better to have stabilized flowrate. Regarding to this claim, some apparatus
has been setup and some GLVs with different port sizes has been tested. Shahri [18] showed that TC
equation overestimates the results up to 5% on the ports. The reason behind that is the Cd value based on
TC has been developed based on rounded entries orifice bean and not sharp-edged.
Poblano et al. and Beggs [1, 17] used Eq. 2-25 as the basis of measurement of injection gas flowrate.
Eq. 2-25 has the same basis and fundamental as TC equation whereas the conversion coefficients are
different.
√
√
where,
, sc: standard condition
Cs = Coefficient based on system of units
d = ID or bore opening to gas flow, in2
Numerous studies [20, 21, 22, 23, 24, 25 and 26] have been carried out by Tulsa University Artificial
Lift Projects (TUALP) aiming to solve the GLV performance issues without using the concept of LR.
Almost all of the researches that have been carried out at TUALP bear numerous empirical coefficients in
each flow system because the nature of their developments which was empirical. In some circumstances,
the resulting values for orifice flow were less than throttling flow of the same conditions which brings
uncertainty into the system. TUALP researches revealed that the end portion of recorded data has more
error that the early recorded data simply because of low rate and entering throttling flow regime. In case
of high rates for 1.5‖ Camco GLVs, the projected error never exceeded 13% but in low rates the error
21
recorded as high as 93%. This dissertation found a maximum of 2% tolerance in error in this research
which is in the reported range of TUALP. One of the main reasons of getting a lower value of error is
using high speed analogue/digital (A to D) recorders that make such experiment possible.
TUALP method of investigating each GLV performance is very close to what API [12] has been
proposed as a recommended practice. Both of these methods dealt with flow capacity, Cv, rather than Cd.
The main reason of picking Cv over Cd is because Cv’s variations are not much and it does not require
variable upstream area into account whereas Cd does. So, the average value of Cv in critical flow regime
is constant.
Rahmeyer [27] studied the pressure recovery factor as a parameter to calculate the maximum flow
through a valve knowing Pid, Pvo and the internal geometry of each GLV. In other word, pressure recovery
factor [28] is a measure of the ability of the valve to convert the kinetic energy of the downstream side to
downstream pressure, Ppd. In order to calculate this factor, the fluid has to be assumed as incompressible
or with the constant mean density at critical or choking condition. Due to the cavitation in choke flow this
factor limits the design and operation in general and precautions in that regard (sizing valves) have to be
taken. Pressure recovery factor has an effect on flow capacity and needs to be determined for each valve.
Pressure recovery factor is the ratio of the theoretical pressure drop across the valve to the actual pressure
drop across the valve at maximum (critical, choke or flash) flow conditions. This factor does not describe
the pressure recovery but describes the effects of choking flow on the theoretical flow conditions, unlike
Cv, the pressure recovery factor, Fl, stays fairly constant for similar valves (geometrically) of different
sizes. However, the cavitation changes with velocity to the 7th power [29]. It is worth noting that in Eq. 226, the pressure recovery due to gas expansion [28] has not been considered.
The orifice flow regime can be formulated by Eq. 2-26.
√
where, Y = Expansion factor
Pid= Injection pressure at depth, psia
Ppd = Production pressure at depth, psia
A minimum of two experimentally determined orifice flow curves are needed to calculate Cd*Y for each
port size. The two injection pressures have to be large enough to sit in the critical flow region. Fig. 2-3
demonstrates different flow regimes clearly. In order to calculate Cd*Y at each point, we need to rearrange Eq. 2-26 to Eq. 2-27.
22
⁄√
where, Pid , Ppd are in psia.
In the next step, for each port size, plot Cd*Y against the dimensionless pressure ratio, (Pid-Ppd) / (Pid*k).
Then draw the best fit straight line through the data and obtain the slope (a) and intercept (c). Then the
Cd*Y can be calculated using Eq. 2-28.
Transition flow can be found when the injection pressure held constant and as the production pressure is
reducing toward zero, the gas flow rate increases, reaches a maximum, then decreases and stays constant.
The flowrate never ceases even though the production pressure reaches atmospheric pressure. Prediction
temperature at each pressure is vital in design of each gas-lift installation. The proposed algorithm [30] is
sophisticated and gives out really good results in practical applications.
WAVE [31] in its computational flow dynamic (CFD) program introduces the Cd based on the type of
orifice. For example, the value of Cd=1 represents a bellmouth smooth entry with no vena contracta. If the
angle changes to sharp-edged, the value of Cd=0.8. Eq. 2-29 defines the relationship of each Cd value
with the placement of the orifice which is shown in Fig. 2-15.
*
[
] +
(
(
(
) ))
Fig. 2-15—Variability of Cd with orifice type
23
Valve Temperature
GLV performance is so linked to nitrogen-charged pressure at temperature. The GLV Pvo, Pvc are so
dependent on the Pbt. The dome pressure is dependent on the dome temperature. Research [24] has shown
that a variation of ±2.5 oF would result in up to 30% change in gas flowrate in throttling flow regime.
In the field, the GLV is positioned inside the side pocket mandrel and is exposed to injection gas
temperature internally and the production fluid temperature externally. The bellows temperature can get
changed accordingly [31]. Eq. 2-29 gives an approximate dome charged pressure at these circumstances.
where, Tb = Bellows charged temperature, oF
Tinj = Injection gas temperature, oF
If the actual gravity differs from 0.65, a second correlation should be applied [2]. An approximate
correction for gas passage can be calculated using Eq. 2-30 and Eq. 2-31.
√
where, CgT = Approximate gas gravity and temperature correction factor for choke charts, dimensionless
TgD = Gas temperature at valve depth, oR
qga = Actual volumetric gas rate, Mscf/D
qgc = Chart volumetric gas rate, Mscf/D
24
Chapter 3
Testing Procedures
Static Testing Procedure
This testing procedure aims to set the test rack or valve opening, PTRO or Pvo, pressure. This procedure has
to be done with probe testing because in this procedure, the dome charge pressure is to be set at higher
values and the pressure shall be adjusted while applying probe testing.
In this test, the GLV is connected to the high pressure source. The simple procedure of such test without
Aging the GLV is as follows:
1- Remove the tail plug
2- Overcharge the dome with at least 50 psi or more
3- Insert the GLV in the tester and correct the Pvo.
Note that there is a stem valve at the tail-plug which has to be pushed down to exit some nitrogen
and lower the dome charged pressure and consequently the Pvo
4- Install the tail plug
Probe Testing Procedure
This testing procedure is to increase the certainty of the operator with the capability of gas passage
through the GLV. In this testing procedure, the charged GLV, Probe, Depth Micrometer, and a Multitester are required. The stepwise procedure for running such test is as follows;
1- At rest, adjust the depth micrometer to the point that the Multi-tester is reading zero impedance
showing the continuity is on or the circuit is open. We need to write down the reading number
and the corresponding pressure.
2- Increase the pressure stepwise and in each step, turn the micrometer knob to hit the ball and
record the distance of ball moving due to the inserted pressure.
3- Keep increasing the pressure and reading the corresponding movement to the pressure till the ball
movement doesn’t change.
4- When there is no ball movement, no change in micrometer reading duo to pressure increase, the
operator may stop the test.
5- Plot the distance reading versus the pressure on a Cartesian paper coordinate.
6- Run the same procedure when the pressure is decreasing. Hence that the operator has to move
back the stem to avoid stem helical induction which causes not quality readings then after.
7- Plot both stem movement with pressure for the increasing and decreasing scenarios and compare
the results. Based on API RP11V2, the best fit would be the line which fits the average.
8- The operator need to draw the line based on the based average point fit.
9- Out of the two plots, the increasing pressure reading is always placed above the decreasing
pressure reading and if the operator’s results is not agree with this, the test need to be redone.
The schematic of the connections, plot results are shown in Fig. 3-1, 3-2 respectively.
25
Pressure, psig
Fig. 3-1-- Schematic of the Probe Tester [16]
Pressure vs Stem Travel, 1-1/2" J-20 Camco GLV, Pbt = 596 psig
640
620
600
580
560
540
520
500
480
460
440
420
400
380
Max. Linear Travel = 0.16 inch
Load Rate = 250 psi/inch
dPLinear = 40 psi
Min. Travel for Fully Open = .2246 inch
Inreasing
Pressure
Decreasing
Pressure
0
0.05
0.1
0.15
Stem Travel, inch
0.2
0.25
Fig. 3-2-- Sample Plot of Changing Pressure with Stem Travel for 1/2” Monel Port
Load rate, LR, or bellow is defined as the pressure required moving the ball for a distance of one inch.
The size and length of bellows has a direct proportion to the LR. The LR [18] in 1‖ GLV is very higher
26
than the 1-1/2‖. Fig. 3-3 shows the difference. The amount of LR is calculated in the region of linear stem
travel and is the pressure over the stem travel.
Maximum linear travel is the ultimate travel of the stem before bellows start to stack. After this point, the
LR will change till it gets its final value.
Fig. 3-3-- Bellows Assembly Load Rate Curve for 1” & 1-1/2” GLV [16]
Benchmark Valve Testing
This testing procedure is to calculate the discharge coefficient. The procedure associated with this test is
to install the benchmark valve which is a GLV without bellow assembly and an adjustable stem to set the
ball positions. In this testing system, the ball has been set in 5 different positions. The GLV will be fully
closed, 20% open, 40% open, 60% open, 80% open, and fully open. The discharge coefficient then after
will be calculated due to the gas passage though at each setting comparing with the theoretical gas
passage through. The schematic of benchmark valve has been depicted in Fig. 3-4. The results of these
testing can be found in Appendix B. These results have been tabulated for each port size in at least 5
different positions. There will be a plot of the gas flow behaviors in all of the conditions as well.
27
Fig. 3-4-- Schematic of the Benchmark Valve [16]
Hydraulic Stabilization (Aging)
This test has to be done and applied to reduce the hysteresis effect associated with the bellows assembly.
The apparatus consists of a diaphragm pump, a GLV holder (chamber), a compressor, and some pop
joints. The compressor need to increase the air pressure on one side of diaphragm pump whereas the
diaphragm pump pressure will reach 5000 psig on the outlet which is hook up to the GLV. The picture of
this apparatus can be found in Fig. 3-5. This tester has been named as Valve Hydro Tester as well.
28
Relief valve
Pressure Gage
Low Pressure
Gas inlet
GLV HydroTester
Chamber
Drain Line
Water inlet
High Pressure water
outlet
Sprague Pump
Poutlet (water) / Pinlet (gas)= 160
Fig. 3-5-- Hydraulic Stabilizer (Valve Hydro-Tester or Ager)
29
The apparatus needed to Age each GLV is as follows:
Test Rack: This equipment is used to measure the Pvo or PTRO of each GLV. There are two general
types in use: the ―donut‖ tester and the ―encapsulated‖ tester. In this research, the encapsulated form
of tester has been applied.
Water bath: This is a water container set at predetermined temperature of 60 oF to immerse each
GLV and set the GLV at that temperature. If the temperature of the water bath is different from 60 oF,
the corresponding pressure has to be corrected. This device is absolutely essential for Nitrogen
charged bellows assembly GLVs while bellows mechanics is so temperature dependent. This device
is not necessary for spring loaded GLVs since spring intensity coefficient is temperature insensitive.
Ager (Valve Hydro Tester or Hydraulic Stabilizer): This device is a water filled chamber of
minimum 5000 psig. The GLVs in the Ager are subjected to predetermined pressure at preset
temperature in different time cycles. The purpose of such test is to reduce the effect of hysteresis
associated with each GLV.
Probe: This device as explained earlier in this chapter is a depth micrometer to measure the
ball/stem movement at each pressure applied on the GLV.
The procedure of aging each pressure charged GLV based on API RP11V1 is as follows:
1- Remove the tail plug
2- Over charge the dome for another 50 psi
3- Put the GLV into the 60oF water bath for at least 15 minutes
4- Remove the GLV from water bath insert it in the Ager
5- Do not hold the GLV from the end because of heat transfer purposes which results in faulty set
pressure
6- Apply gas to open the GLV
7- Adjust the dome pressure to correct Pvo
8- If adjusting for the correct Pvo took more than 30 seconds, remove the GLV from the Probe and
insert it in the Water bath for temperature set assurance
9- Install the tail plug and insert the GLV in the Ager
10- Increase the pressure on the chamber up to 5000 psig for a minimum of 15 minutes
11- Release the pressure and cycle the pressure to 5000 psig for at least three times without pausing
12- Remove the GLV from the Ager chamber and return it to the water bath for at least 15 minutes
for temperature stabilization purposes
13- Remove the GLV from water bath and insert it in the probe device and check the Pvo
30
14- If the Pvo has been changed 5 psi or more, repeat steps 6 through 13 until the pressure does not
change 5 psi or more
Dynamic Testing Procedure (Blow-Down Test)
This testing procedure is so called pressure decay as well. The methodology behind this technique is
simply discharging a certain volume of gas at a certain time till the upstream pressure reaches the final
downstream pressure which is ambient pressure. The initial pressure is very greater than the P vc to assure
the operator of fully open GLV stand point. The detailed of this procedure will be discussed in the next
chapter. This method is aimed to bypass the probe testing whereas the time donated to this method is
tremendously shorter than the conventional techniques. This method will assure the operator at the
wellsite in matter of seconds that if the GLV would pass the required and claimed amount of gas to lift the
certain amount of fluid or production or not. Note that this method won’t substitute the current API RP11
V2 [8] but will raise the certainty and assurance on the operator of having the scheduled production.
The apparatus for this test includes some compartments such as: source of high pressure Nitrogen gas,
upstream and downstream regulators attached to the high-pressured source of gas, an extra empty volume
with known internal capacity, an encapsulated vessel which holds the GLV, the GLV, high-speed I/O
pressure recorder, high speed temperature recorder, and a data-acquisition system (DAQ).
A simple plot of the apparatus diagram has been presented in Fig. 3-6.
Gas Flow
Encapsulated
Vessel
GLV
Downstream
Valve
to DAQ
Known Working Volume
Fig.3-6-- Schematic of Blow-Down Dynamic Test Facility
31
High Pressure
Nitrogen Source
The procedure for running this test is as follows:
1- Knowing the GLV Pvc
2- Set the Pup >> Pvc (better to be about 50-150 psi higher)
3- Shut-in the main feeding valve on the main high-pressure source of gas
4- Wait till Pup stabilized (usually 20-30 seconds)
5- Record Temperature
6- Kick the downstream valve open (open it as fast as possible)
7- Record the P vs. Time
8- Record Temperature
It worth to note that it’s better to start the DAQ recording prior to kicking the downstream valve open.
This is because of the importance of the earliest data. Then after, we can find the starting point. This can
be done through a simple programmed module as well. Usually in large port sizes, if the pressure
differential is greater than 2 psi, the system start to record and for the smaller port sizes, the pressure
differential has to drop to one psi. Real data confirms this pressure differential picks well.
API Testing Procedure
The API testing procedure [11] to test the performance of each GLV is based on Decker K.L. [19]
procedure. In this procedure instead of dynamic stem travel they used a static force balance for the
calculation of the stem travel. The API procedure is to be executed based on constant injection pressure
test (CIPT at steady state) and can be outlined as following steps:
1- Determine initial stem position and dynamic stem position using Eq. 3-1 and Eq. 3-2
respectively.
(
[
(
)
)
]
2- Determine coefficient of flow and critical pressure ratio using dynamic stem travel through Eq. 33 and Eq. 3-4 respectively.
32
3- Compute the flowrate applying the Eq. 3-5.
√
33
Chapter 4
Blow-Down Test
This method is primarily based on discharging a certain volume of gas at the time. Knowing the
capacity of the working gas and its initial pressure, depletion time, and the final pressure (or pressure drop
within the length of time) will allow us to calculate the mass and volumetric flowrate which are time
dependent. Calculating the speed of gas passing though the orifice and comparing with the sound velocity
will yield us to the situation of the experiment in which the state of test is in critical conditions or
subcritical. If there is a gas leak in the system, it has to be measured and deducted from the results. Since
the ratio of Pdown / Pup has to be less than 0.528 (the critical value for Nitrogen) to have the critical or
entrainment velocity of gas, the test is mostly in critical condition and the correspondent flow regime is
orifice flow rather than throttling flow or transition.
The effect of temperature has been studied throughout the test as well. Since the testing time is so
short, the temperature changes are not much. Cd has been calculated through volumetric measurements
and theoretical calculations. The adopted TC equation has been modified for the value of Cd to fit the
expectations as well. Primary measurements and calculations yield an overestimating of gas passage
through GLV using TC equation. This overestimating tendency has been reported several times in the
literatures [12, 19]. This is in case of taking a pre-set constant value for Cd. The nominal value for this
variable is 0.865 and is dimensionless.
In order to rectify and correct the associated error and over estimating of the gas passage through Cd
has to be corrected. One of the main reasons of such over estimating is assuming the Cd does not changing
and keep this value constant while it is not. Benchmark Valve testing has been developed primarily to
correct this value.
Volumetric Calculations
This section goes over the volumetric base calculation that has been used in this dissertation. The basis
of all the volumetric calculations is the real gas law. In this regard, all the active parameters in the
formulation has been identified and set in the formula to match the final results. There are some constant
inputs used in this analysis which are dominantly depend on the location and testing facility such as the
ambient atmospheric pressure and temperature, the gas constant, ratio of specific heats of the active gas,
the capacity of the storage facility, and the specific gravity of the working gas. On the other hand, there
are some values that has to be measured like the pressure as the gas is venting from the system and the
34
corresponding temperature with time. The rest of this chapter will contain all the factors (parameters) and
their involvement in the test results.
Atmospheric Pressure & Temperature Determination
Atmospheric pressure has been read from a mercury barometer in mmHg and recalculated to psia. The
basis of standard pressure is set to 760 mmHg and 14.696 psia.
Atmospheric temperature has been read on thermometer as well as electronic laser gun thermometer.
Working Gas Pressure, Temperature, corresponding compressibility factor, and Specific
Gravity
Gas pressure (upstream) has been read with an analog dial gauge as well as digital Data Acquisition
System (DAQ) empowered by NI and got setup for this experiment.
Gas temperature (upstream) was assumed the same as the gas tank and has been read with the
electronic laser thermometer. On the downstream side, the temperature is equal to the atmospheric
temperature.
Gas compressibility factor, or Z-factor, has been calculated at each pressure and temperature based on
available correlations.
Specific gravity of the known gas, Nitrogen, is simply is the ratio of the gas molecular weight to the
air molecular weight (which is known).
Ratio of Specific Heat Capacities
There are two types of processes of specific heat capacities for each gas. The first process is happening
while the volume of the gas at the process is constant. This process relates the internal energy to the
temperature thru the value of heat capacity. This is called specific heat capacity for the gas in a constant
volume process, Cv. The second type of process is based on the constant process pressure. This process
relates the enthalpy to the temperature via heat capacity value and is called; specific heat capacity for the
gas in a constant pressure process, Cp. The ratio of Cp/Cv is a constant number for each gas. Table 4-1
contains some values for different types of gases. Since this ratio is dimensionless, it is not changing in
different universal systems. The yellow-colored row is the type of gas used in this experiment.
35
Table 4-1- CP /CV for different Gases
Gas
Ratio of Specific Heats
Acetylene
Air, Standard
Ammonia
Carbon Dioxide
Carbon Monoxide
Chlorine
Ethane
Helium
Helium
Hydrogen
Methane
Natural Gas (Methane)
Nitrogen
Oxygen
Propane
Steam
1.3
1.4
1.32
1.28
1.4
1.33
1.18
1.66
1.66
1.41
1.32
1.32
1.4
1.4
1.12
1.28
Sulphur dioxide
1.26
Internal Gas Storage Capacity Determination
This apparatus has been included some hoses, valves, Gas-Lift Holder (encapsulated vessel), Gas
Cylinder, and some junctions and nipples. The overall internal capacity of the system is the summation of
the total volumes that gas is passing through. The internal volume of the hoses has been calculated based
each conduits and vessels geometrical shape (mathematically) or got from manufactures but the capacity
of the gas (Nitrogen) tank has been looked up through manufacture.
Table 4-2 delivers the nominal values for each cylinder. The yellow-colored row is the type of
cylinder used in this experiment. This cylinder has been chosen because it was small but could sustain
high pressure gas (rated for 2400 psig). In this case, we can examine high pressure testing while
consuming very less volume of gas. The overall internal volume came to 0.56 ft 3. In other words, all the
hoses, connectors, valves, and so on hold for less than 2% of the overall internal volume.
36
Table 4-2--Technical Specifications of Cylinders [33]
Cylinder
Size
K
A
B
C
D
AL
BL
CL
XL
SSB
10S
LB
XF
XG
XM
XP
QT
LP5
Medical E
Nominal Size
Diameter X
Height
(inches)
Nominal
Tare
Weight
(lbs.)
9.25 X 60
9 X 56
8.5 X 31
6 X 24
4 X 18
8 X 53
7.25 X 39
6.9 X 21
14.5 X 50
8 X 37
4 X 31
2 X 15
12 X 46
15 X 56
10 X 49
10 X 55
3 X 14
includes
4.5 inches for
valve
12.25 X
18.25
135
115
60
27
12
52
33
19
75
95
21
4
180
149
90
55
2.5
includes
1.5 lbs for
valve
110
96
37.9
15.2
4.9
64.8
34.6
13
238
41.6
8.3
1
18.5
4 x 26
excludes
valve and cap
Internal Volume @
70°F (21°C), 1
ATM
(liters/cubic feet)
US DOT
Specifications
278
120
124
49.9 / 1.76
43.8 / 1.55
17.2 / 0.61
6.88 / 0.24
2.24 / 0.08
29.5 / 1.04
15.7 / 0.55
5.9 / 0.21
108 / 3.83
18.9 / 0.67
3.8 / 0.13
0.44 / 0.016
60.9 / 2.15
126.3 / 4.46
54.3 / 1.92
55.7 / 1.98
3AA2400
3AA2015
3AA2015
3AA2015
3AA2015
3AL2015
3AL2216
3AL2216
4BA240
3A1800
3A1800
3E1800
8AL
4AA480
3A480
4BA300
2
0.900 / 0.0318
4B-240ET
47.7
21.68 / 0.76
4BW240
4.5 / 0.16
3AA2015
Water
Capacity
(lbs.)
14
excludes
valve and
cap
Discharge Coefficient Calculation
This is another factor which has to be determined prior to the further calculations. Benchmark valve
testing has been employed to measure discharge coefficient in different stem travel positions when the
volumetric gas rate is known with the same real GLV stem, port, and ball/seat structure. Further
information regarding to this factor can be found in Appendix B.
37
Critical Pressure Ratio
This experiment initialized based on the fact that the pressure ratio of downstream to upstream
pressure falls in the supersonic region. In other words, at critical flow, the flowrate is constant regardless
of lowering the downstream pressure. Chapter 2 discusses this issue more in detail.
Calculating the Flow Area
The flow area which is the frustum of a right circular cone (in case of sharp edged-seat) is constantly
changing in this testing system. At the beginning, based on the maximum linear steam travel, the flowing
area can be calculated. If the value of maximum linear steam travel is less than the minimum value
required for fully open flow, the GLV will not get open fully. Consequently, the flowing area is restricted
and the effect of the ball in the flow path should not be ignored. Eq. 4-1 is a general form to calculate the
frustum of a circular cone. Equalizing this value by the port area will give us the minimum steam travel
required for having a fully open flow. Fig. 4-1 represents the ball-seat position and gas area to flow.
S=
√
where, S= Area of the Frustum
R = Ball radius = (32R+1)/32, inch
r= Port radius, inch
H = Ball distance from the seat, inch
a= Radius of the top section of Incomplete Frustum,inch
a
θ
H
r
R
Y
H
r
Fig. 4.1--Schematic of the Ball – Seat Position
38
Eq. 4-2 derived based on the equality of the frustum area to the port area that gas is flowing through.
√
[
(
)
(
(
√(
)
)
)]
And the area open to flow can be calculated based on Eq. 4-3. In this equation, all the other variables
have been calculated based on known constant values of port size and ball size. Table 4.1 shows the open
area to flow relative to the ball distance from the seat at rest in each GLV with different port sizes.
(
(
(
(
√
(
))
√
(
))
)
)
Table 4-3—Area Open to Flow at Different Ball-seat Positions
Area Open to Flow, in2
Orifice
Size
inch
Ball
Radius
inch
Port
Radius
inch
Minimum
Theoretical
Fully Open
inch
1/4
Fully
Open
1/2
Fully
Open
3/4
Fully
Open
Fully
Open
1-1/4
Fully
Open
1-1/2
Fully
Open
0.25
0.5
0.75
1
1.25
1.5
3/16
0.125
0.0938
0.0714
0.0070
0.0140
0.0209
0.0276
0.0276
0.0276
4/16
0.15625
0.1250
0.1003
0.0121
0.0245
0.0369
0.0491
0.0491
0.0491
5/16
0.1875
0.1563
0.1302
0.0185
0.0378
0.0574
0.0767
0.0767
0.0767
6/16
0.21875
0.1875
0.1609
0.0260
0.0538
0.0822
0.1104
0.1104
0.1104
7/16
0.25
0.2188
0.1925
0.0347
0.0726
0.1115
0.1504
0.1504
0.1504
8/16
0.28125
0.2500
0.2246
0.0445
0.0940
0.1451
0.1965
0.1965
0.1965
39
The findings tabulated in Table 4-3 is along with Kulkarni [34] reported. The value of ―Y‖ shown in Fig.
4-1 stays constant while ―H‖ is changing. At rest, when the ball seats on the seat, as the ball size gets
larger, the center-ball-angle with respect to the seat-base line decreases. This means that as the ball gets
larger, it goes deeper inside the seat at rest. For example 3/16 inch port size makes a 42 degree angle
(angle between seat base-line and center of the ball) whereas this angle for 1/2 inch port size drops to 27
degree.
Obviously, each ball position with respect to the seat denotes an angle that is keep changing. It has been
found from the test measurements that the minimum ball distance from the seat in which the effect of ball
in the flow is ignored would be 1.25 times more than theoretical fully open for sharp-edged orifices. This
value strongly depends on the ball size and architecture of the seat (sharp-edged or beveled).
The effects of bellows LR has to be incorporated while coupling it with the acting forces on the GLV
at each stage (based on the pressure regimes in upstream and downstream of the GLV). In other words,
LR has a tremendous effect on the maximum linear steam travel and consequently, maximum ball
movement. The partial effect of upstream and downstream pressures on the ball has been studied as well.
Eq. 4-4 has been written based on force balance including the effect of LR and partial pressure
distribution on the ball.
(
)
where, Pbt = Dome charged pressure at temperature, psig
Pup = Upstream pressure = Pc = Casing pressure, psig
Pdown = Downstream pressure = Pt = Tubing pressure, psig
f = Fraction of pressure acting on the port area = 1- H/Hmax
Hmax = Maximum ball movement such that there is no effect of downstream pressure on the ball,
inch
Solving Eq. 4-4 for the value of dx will result in Eq. 4-5.
(
)
If Hmax > H then Hmax = H and Eq. 4-5 will changes to Eq. 3-2.
40
[
(
)
]
Based on several measurements on the gas passage through GLV with benchmark valve, the ultimate
measured value of Hmeasured = 1.25 Hmax and beyond. Ultimate ball-seat distance, Hultimate = Hmeasured, is the
distance that the measured gas passage through is equal to the gas passage when there is port-only in the
GLV. In other words, there is no pressure loss due to tortuous convoluted flow path. The same set of
experiments showed that the ball affects the flow path based on the theoretical minimum ball movement
for fully open and measured minimum distant proved the claim. Based on measurements, the minimum
stem travel for fully open GLV is between 25-50% beyond theoretical distance. Measurements showed
that existence of GLV body on the way of fluid flow will cause a drop of 1% in gas throughput in each
GLV.
In order to calculate the value of Cd, we need to measure the gas throughput in a different way then
compare it with what TC equation is proposing. The value of Cd then can be calculated. Note that since all
the experiments in this research have been carried out in critical flow conditions with compressible fluid
flow, expansion factor has to be incorporated. Expansion factor is a constant value when the fluid flow is
sonic because the pressure ratios stay constant. Following steps yield the values needed for calculating the
value of Cd.
Phase I
The volumetric calculations start with the known equation of state (EOS) for real gases as written in Eq.
4-5.
where, dPup= Change in Upstream Pressure as the Gas is Discharging from the System, psi
V= The Capacity of the system including the Tank, Connections, Hoses, and fittings in ft3
dn = Number of moles of Gas discharged from the system under pressure drop of dP
The number of moles of gas at standard condition is known as well. Knowing that each mole of gas at
standard conditions occupies an equivalent volume of 379.73 ft3 will help to convert the drained number
of moles at certain pressure differential to the volume. Having the time of drainage will enable the
operator to calculate the volumetric flowrate. So the volumetric flowrate of the system is known.
41
Eq. 4-6 is a better way of understanding the blow-down situation. As the equation shows, the changes
in Pup (while Pdown is constant) are in direct relationship with the number of gas-moles drained out of
system. Therefore, the rate of mole drainage from the system can be calculated. This rate at high pressures
is higher and as the pressure decays, the rate drops.
⁄
Reynolds number (NRe) can be calculated at each pressure. The NRe values of beyond 4000 represent
turbulent flow conditions. In these experiments, the NRe values have been measured to be greater than
35000. Therefore the flow was fully turbulent. The value of gas viscosity has been corrected for each
pressure and temperature. More detail in this regard can be found in [8]. The charts in [8] are insufficient
for these experiments because of limitation in NRe although the procedure is valid.
Phase II
Applying Eq. 2-20 at this stage (knowing the flowrate and the port area from Phase I, either from
benchmark valve testing or pressure decay test with set GLV) with incorporating the value of expansion
factor can lead to calculate the value of Cd. Therefore, this equation can be solved for the value of Cd.
The calculated Cd value then can be used in the test to verify the effective area open to flow for different
GLVs. With this method of testing, the operator can make sure that if the GLV passes the required
volume of gas to lift the expected volume of liquid.
Phase III
In this phase, the new value of Cd has to get compared with its original value reported in TC equation. TC
equation has been derived with rounded edges rather than sharp-edged seat therefore; using that constant
value for applications with sharp-edged seat will overestimate the results. Cd is much more pressure
sensitive than temperature and flowing area. The value of Cd will vary for different flowing areas,
different port sizes, and different ball positions. More detail on this calculation (measurements) procedure
can be found in Appendix B. torturous path of the flowing gas will affect the value of Cd but most
important is the upstream pressure. The value of Cd is basically the volumetric flowrate of gas through a
torturous path to the value of the volumetric flowrate into the same area conduit with no tortuosity.
42
Chapter 5
Results & Discussions
All the findings in this type of testing will be shown in this chapter. The second series of testing has been
run aiming to make sure the Cd values are correct. Besides, this dissertation tries to find a relationship
between the Pid at each moment and the ball position as well as the bellows LR.
Fig.5-1 is a plot of 1/4‖ Monel seat, 1-1/2‖ J-20 Camco GLV. The initial Pid has been set on 610 psig.
Regression curve-fit has been employed on the data and the best fit found.
Pressure vs. Time for 1/4" Monel, J-20 Camco , 1-1/2" GLV
700
600
500
Piod, psig
Real Data
400
Exp
Decay-2
Fit
300
200
100
0
0
2
4
6
Time, sec
8
10
12
Fig. 5.1—Plot of Upstream Pressure vs. Time for 1/4” Monel, 1-1/2” J-20 Camco GLV
The best fit for Fig. 5-1 was found to be a dual exponential fit. This method of fitting has been
mentioned in API [12] as an acceptable method of curve fitting. Table 5-1 contains the values for this
curve fit as well as the accuracy.
43
Table 5.1—Curve-fit Values for 1/4” Monel, 1-1/2” J-20 Camco GLV
Exponential Decay-2 Fit
Model
Equation
Adj. R-Square
Constant =
Constant =
Constant =
Constant =
Constant =
Constant =
ExpDecay2
y = y0 + A1*Exp(-(x-x0)/t1) + A2*Exp(-(x-x0)/t2)
0.99999
Value
y0
-11.856
x0
0.03469
A1
230.354
t1
5.14445
A2
385.045
t2
1.72123
As the data in Table 5-1 represent, the accuracy is very high. Applying this formula with known
coefficient will result in having a value of 0.852 for Cd and back calculate for the effective port size of
15.994 / 64‖. It means that the testing system method is working fine for this port size. Noticing that, the
measured Cd value is less than the referenced value which is 0.865. This difference in Cd value will result
in 1.5% less upstream injection gas needed to lift the known column of fluid in the wellbore which is
better although the changes are not much significant.
In order to make sure that other regression analysis method (mostly polynomial fit) is as accurate as
exponential fit, another experiment with the same port size as previous test has been setup. The initial P up
was 727 psig for this test. Using the found Cd value in this test revealed a value of 15.981 /64‖ for the
first 10th of a second of the test which is in a good agreement but is not recommended. Fig. 5-2
demonstrates an odd variation of equivalent port areas calculated. This odd variation is just because of the
nature of polynomial curve-fit formula. If the aim of the test is just to quantify the maximum performance
of the GLV, polynomial curve-fit will give the approximate answer otherwise, this method of curve-fit is
not suggested. Comparing the collected data between two tests in Fig. 5-2, and Fig. 5-3 reveals that
importance of two main factors affecting the analysis; first, how fast the downstream valve opens and
data is getting collected and second, what type of regression analysis has been employed for analysis.
Because this method of testing is in transient mode (Not steady state like what API is recommending), the
start-time of recording data is very important and deterministic. It has been found empirically that for
small ports (3/16‖, and 1/4‖), the pressure has to be recorded when a drop of at least one psi is seen
whereas at least three psi for medium ports (5/16‖, and 3/8‖), and at least four psi for large ports (7/16‖,
and 1/2‖). The theory behind increment pressure drop selection is entirely empirical but with this method
of start, the effect of slow pressure drop due to speed of ball-valve opening at the outlet of fixture is
minimized. From Fig. 5-2 and Fig. 5-3 It is obvious that TC equation is overestimating 5% at the most.
44
64th in Equvalent Port Size
64th of inch Port Equivalent
1/4" Port Size, 1-1/2" J-20 GLV
17
16.8
16.6
16.4
16.2
16
15.8
15.6
15.4
15.2
15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Time, sec
Fig. 5.2—Calculated Equivalent Port Area Based on Polynomial Regression Analysis (1 st Trial)
Fig. 5.3—Calculated Equivalent Port Area Based on Polynomial Regression Analysis (2 nd Trial)
What can infer from previous analysis is depending on the case of action (maximum GLV
performance or determining the ball position with change in upstream pressure or quantifying the LR of
bellows). The curve-fit regression analysis may vary due to LR and upstream pressure. It is shown in both
Fig. 5-2 and Fig. 5-3 that polynomial-fit is not recommended for entire data fitting. Even this regression
analysis is not consistent at the early collected data. The results of applying polynomial-fit showing that
45
this form of regression analysis can be applied but there is a range of up to 8% error in the analysis which
should be considered. On the other hand, exponential curve-fit do a nice job either. Having exponential
regression applied for the early data is recommended although it bears the little effect of gas expansion
and gas flow through tortuous path which causes lower readings of the apparent port size. Therefore, if
the overall aim is just to make sure of the fact that the GLV can get fully open and pass the required
volume of gas to lift the pre-determined flowrate, No regression method is required but pressure points
has to be recorded in a way that the effect of downstream valve opening is minimized.
Fig. 5-4 demonstrate the blow-down test ran through a 1/4‖ post size at starting pressure of beyond
500 psia. This analysis has been done with specific mathematical software known as origin™. The
software is very user-friendly and capable of handling large sets of data points. This test has been done
with over 35000 pressure points in which Microsoft Excel cannot handle. (max. 32000 points)
Fig. 5.4—Plot of Pressure vs. Time, Flowrate, and Apparent Port Size Open to Flow in a 3/16” Monel
Sharp-Edged Seat
As the GLV port size is getting bigger, the pressure decays faster. Therefore, the accuracy has to come
higher. As it has been suggested, there should be a set point for collecting data based on the port size.
Fig. 5-5 reveals the importance of having a set point in collecting pressure point for better monitoring the
GLV performance.
46
64th in Equivalent Port Size
64th of Inch Port Equivalent
5/16" Monel Port Size, 1-1/2" J-20 GLV
22
19
16
13
10
7
0
0.1
0.2
0.3
0.4
Time, Sec
0.5
0.6
0.7
0.8
Fig. 5.5—Calculated Equivalent Port Area Based on Exponential Regression Analysis
Table 5-2 contains the exponential-fit equation for data in Fig. 5-5. The fit equation shows a very
good agreement fit.
Table 5.2—Regression Exponential Analysis to fit the Data in 5/16” Port 1-1/2” J-20 GLV
Model
Equation
Adj. R-Square
Coefficients
Coefficients
y0
A1
ExpDec1
y = A1*Exp(-x/t1) + y0
0.99976
Value
Standard Error
554.25927
3.16E-01
156.31182
2.87E-01
Coefficients
t1
0.33248
1.92E-03
In order to calculate the equivalent port size, a plot of pressure versus time for the first 10 th of a second
is recommended. Fitting the simplest form of fit on the data would facilitate the calculations. In this
regard, linear fir or second-order polynomial fit is recommended. Fig. 5-6 demonstrates a sample fit.
47
Pressure vs. Time (Based on Curve-fit Data)
5/16" Monel Port Size, 1-1/2" J-20 GLV
715
710
Pressure, psig
705
700
P = -411.53t + 709.81
R² = 0.9986
695
690
685
680
675
670
0
0.02
0.04
Time, sec
0.06
0.08
0.1
Fig. 5.6—Calculated Equivalent Port Area Based on Previous Exponential Regression Analysis
Using the slope of the 1st-order linear regression as shown in Fig. 5-6, will give the volume of gas
vented from the system which simply means flowrate. Knowing the value of Cd (≈ 0.844) as well as the
flowrate will result in calculating the effective area open to flow which is related to the ball location. If
the found value is within 5% off from the known nominated port size, the value should be considered
good since it has been mentioned that the TC equation is overestimating the results by 5%. In this test (as
depicted in Fig. 5-6) the result of calculation shows a value of 2.35% error which is in the margin of the
test. Remember that this method of testing is an approximate. Fig. 5-7 shows the behavior of the real data
collected with Lab View DAQ. A simple comparison of Fig. 5-6 and Fig. 5-7 brings the differences up.
The difference is about 2%. The difference between the two methods of analyzing data saying that the
entire data curve-fit is not required but is a bonus toward relating the pressure decay to ball position and
bellow’s LR. In other word, if the purpose of the test is just verifying the GLV injection-gas throughput,
sophisticated curve-fit is not required.
48
Fig. 5.7—Calculated Equivalent Port Area Based on Measured Raw Data
Some pressure-decay tests have been performed mainly to see the possible effect of tapered-seat
against sharp-edged seat. All the Monel-based tests are based on the assumption of sharp-edged seat.
Tungsten-carbide seat has been used as they are slightly tapered. The difference between measured
effective areas open to flow in both cases has been depicted in Fig. 5-8. These tests done on the seats only
and the seats were not in the GLVs. The calculated results based on Fig. 5-8 reveals that a slight tapered
in the seat can increase the gas passage from 13.49 MSCFD to 13.67 MSCFD. In other words, the gas
passage may increase 1.3% with constant stem travel. If the angle of tapered seat changes, the gas passage
will change with constant stem travel.
49
Plot of Pressure vs.Time for 5/16" Monel and Slight Tapered TC Port
600
500
5/16" Port,
Tungsten
Carbide with
slight Bevel
P, psig
400
300
200
5/16" Port,
Monel
100
0
0
2
4
Time, sec
6
8
10
Fig. 5.8—Effect of Slight Tapered Seat Compared to Sharp-edged Seat on the Gas Passage in the 1-1/2” J-20
Camco GLV
The Gas Leak Rate
Gas leak is another consideration in this design. According to API RP11V1 [34], If the leak rate is
more than 35 scf/D the ball and seat shall be rejected. Although in practical field, the gas leak always
exists and is inevitable to get stopped, but in this experiment sets, the leak rate has been measured. The
amount of gas leak in this testing system has been equivalent to 0.18 of 64th of inch. In this type of testing,
the leak does not affect the results because we are not waiting too long for the pressure to get stabilized
like what API is doing other words, this value has to be deducted from all the gas passage through
measurements. The leak comes into considerations when the GLV is closed and not through testing
system. Therefore, the value of the leak has not been an issue in the testing results.
Justifying TC Equation
Original TC equation has been developed for the flow through chokes. In other words, TC equation
has been developed in the pipes rather than GLV. In GLV since the flowing are is do dependent on the
upstream and downstream pressures, the flowing are does not stay constant. The equivalent flowing area
shown in Fig. 4.2 is developed to justify the flow pattern. Whenever the flow area upstream the port is
greater than the port area, the areas has been equalized to the value of port area therefore the maximum
flowing area cannot exceed the port area. The TC equation has to be corrected for GLV performance.
50
Modeling the behavior of GLV is so interconnected with the bellows assembly and its functionality. The
effect of dome charged pressure on the bellows load rate and maximum linear travel has to be addressed
in each testing measurement otherwise, all the results are faulty. Appendix D contains some data in this
regard. The following questions have to be answered while analyzing the pressure decay results.
1. What is the Dome charged pressure?
2. What is the initial upstream pressure in pressure decay test?
3. What is the GLV closing pressure?
4. What is the bellows LR at that charged pressure?
5. What is the maximum steam travel and maximum linear stem travel?
As it has been emphasized, LR has a strong dependency to dome charged pressure and the maximum
linear and ultimate stem travel (includes the stem travel when bellows getting to stack). Because the
blow-down test results are all look alike, if the operator does not know the Piod, Port Size, Valve Size,
Type of gas used, and Temperature, it is almost impossible to be able to analyze the results.
When the GLV is closed, the gas flowing throughput is zero. As the GLV starts to open, the gas
passage increases till the GLV is beyond theoretical fully open. At the theoretical fully open, the GLV
does not pass the equivalent amount of gas that has to pass due to the existence of the ball in the way
which is an obstruction or limitation to flow. Tests shown that when stem travel is 1.25 times more than
minimum fully open travel, the GLV acts like an orifice with no ball limiting the flow.
The value of Cd is changing with pressure as well. If the set initial pressure changes, the corresponding
Cd value will get affected. Results have been confirmed that at higher pressures, the Cd value is higher
although there is a cap. Table B-4 contains some values of such claim. Based on those data, if the initial
pressure gets double, the Cd will increase 8%.
Discharge coefficient values have been measured for different port sizes at 6 different positions using
benchmark valve. These values have been plotted against findings of port-only and port-only inside the
GLV. In cases of port-only and port-only inside the GLV body, there is no ball. Fig. 5-9 through 5-13
depicts the final Cd values measured. All these values have been measured at Pid = 345 psig at 79± 2 oF.
Fig. 5-14 presents all the Cd values in one graph for the comparing purposes. It was expected to see an
increasing trend as the area open to flow increase but at some point and some port sizes, some
measurement faults hit the results.
51
Benchmark Valve Testing for 3/16" Port ID
1
0.981
0.95
0.928
Cd Values
0.9
0.895
0.889
0.876
0.85
0.75
0.7
0.65
0.6
1
2
3
0.886
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
0.847
0.8
0.884
4
5
Dimensionless Ball Position
6
7
8
Fig. 5-9—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 3/16” Monel Sharpedged Seat
Benchmark Valve Testing for 1/4"" Port ID
0.9
Cd Values
0.854
0.853
0.852
0.857
0.85
0.8
0.848
0.792
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
0.75
0.7
0.65
0.6
0.838
0.836
1
2
3
4
5
Dimensionless Ball Position
6
7
8
Fig. 5-10—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 1/4” Monel Sharpedged Seat
52
Benchmark Valve Testing for 5/16" Port ID
0.9
0.880
0.848
Cd Values
0.85
0.8
0.814
0.827
0.804
0.842
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
0.787
0.75
0.844
0.7
0.65
0.6
1
2
3
4
5
Dimensionless Ball Position
6
7
8
Fig. 5-11—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 5/16” Monel
Sharp-edged Seat
Benchmark Valve Testing for 3/8" Port ID
0.9
0.837
0.85
0.832
Cd Values
0.826
0.810
0.806
0.8
0.788
0.775
0.75
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
0.7
0.65
0.6
0.826
1
2
3
4
5
6
7
8
Dimensionless Ball Position
Fig. 5-12—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 3/8” Monel Sharpedged Seat
53
Benchmark Valve Testing for 1/2" Port ID
0.9
0.852
0.85
0.843
0.813
Cd Values
0.822
0.8
0.771
0.779
0.7
0.65
1
2
3
4
0.846
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
0.75
0.6
0.823
5
6
7
8
Dimensionless Ball Position
Fig. 5-13—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 1/2” Monel Sharpedged Seat
1: 1/4 Fully Theoretical Open
2: 1/2 Fully Theoretical Open
3: 3/4 Fully Theoretical Open
4: Fully Theoretical Open
5: 1-1/4 Fully Theoretical Open
6: 1-1/2 Fully Theoretical Open
7: Orifice Port inside GLV
8: Orifice Port Only
1
0.9
3/16"
1/4"
5/16"
Cd
3/8"
1/2"
0.8
Average
Average
(Over all)
0.7
1
2
3
4
5
6
7
8
Dimensionless Ball (Stem) Position
Fig. 5-14—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in all Monel Sharpedged Seat Port Size
54
The published results are based on the calculated equivalent flowing area at each set-ball position in the
benchmark valve. The ball-seat angle in different ball-seat distance (dimensionless distance relative to
theoretically fully open) can be calculated using tabulated data in Table 4-3. As the final results showing
in Fig. 5-14, the average value of Cd is 0.8403 rather than 0.865. This value has been used for other
equivalent port size measurement and the overall error was less than 10%. This value is along with claims
regarding to overestimating the volumetric flowrate using TC equation.
The effect of pressure on the Cd is considerable as well. Tests on the same port size at different set
pressure showed that as the upstream pressure increases, the value of Cd increases. The same test results
revealed that the final value for Cd at pressures beyond 900 psig would be 0.865. so the Cd value used in
TC equation is valid for pressures higher than 900 psig when 5/16 inch port size is used. The results for
this claim have been published in Fig.B-14.
55
Chapter 6
Conclusions

This testing procedure is fast, easy, user friendly, and inexpensive. This method has been developed
to benefit the oil producer rather than the GLV manufacture.

The proposed technique proved that the value of Cd is dynamically changing due to dome set
pressure (which affects the initial opening pressure of the GLV) and the port size. The Cd is prone to
change much more due to the pressure than port size and temperature. Applying TC equation with
constant Cd value (0.865) for all GLVs at all dynamic conditions is not recommended due to above
10% overestimating in gas passage through.

TC equation has been developed based on flow through converging nozzle in which the edges are
rounded; therefore the value of Cd is higher than the case of sharp-edged seat. The reason of using
round edges rather than the sharp-edged is the repeatability with less uncertainty.

The range of Cd values has been found to be from 0.76 to 0.98. So, applying a constant value is not
recommended although the error margin would not be greater than 10%.

This testing method proved that there is a dampening effect in each GLV. Dampening is due to
presence of a viscous silicone-based fluid (with the viscosity of 500 centi-stokes) in the GLV to
prevents GLV chatter and wear the seat and ball. Due to the existence of such fluid, this testing
method is not recommended for Pvc measurements if the gas exits the system at higher rate than the
bellows getting stretched.

Although the gas temperature in the lab scale did not show any impact on the results but the
temperature has to get monitored for each test. Temperature has to be monitored at the upstream side
of the flow.

TC is optimistic in flow through chokes and when the ball is very far from the seat base-line in GLVs.

Cd values for smaller orifice sizes found to be greater. It has been found that when the orifice size is
lower than 1/4 inch, the Cd values are up to 8% higher. The Cd values for flow through chokes found
to be 6% greater than larger orifice sizes.

Blow-down technique used in this research has been tried in transient state rather than steady state. In
this regard, the user has to be alert of taking the right data into calculation otherwise, the results will
be erroneous.(due to the short margin of testing time)
56
Chapter 7
Recommendations

Normalizing the results of this research for a unified equation which describes all flow regimes
(for orifice flow at critical condition) is highly recommended.

All the tests have been done using sharp-edged seat and slight tapered seat. It is recommended
that the same test setup on deep tapered seats with different angles. The magnificence of the
answers may attract the GLV manufactures to start manufacturing the GLV in that way. The main
reason behind that is the lesser of the linear stem travel requirement.

Re-designing the developed apparatus in a way that the downstream pressure in the system can be
controllable is recommended. This is mainly because in the real cases, the tubing pressure is not
the same as the atmospheric pressure.

Using bigger ball sizes rather than the regular sized ones is good to be practiced for the sensitivity
analysis of the gas passage through with respect to the ball size.

The same tests need to be run at non-critical flow conditions as well as throttling flow behavior.
The flow behavior in throttling flow can be entirely different from what has been practiced in this
research.

Using a bigger surge capacity will help dampening the pressure reading fluctuations faster which
helps in recording higher quality pressure points.
57
Appendix A
Transducer Calibration Using Dead-Weight Tester
In this setup, a constant power supply module, an electronic multi-meter, Transducer, and dead-weight
tester are needed. The power supply output voltage is set based on the transducer needed excitation
voltage and kept constant (in this experiment setup the excitation voltage is 10 volt). The multi-tester is
hooked up to the transducer to read the output of the system while applying weight on the dead-weight
tester which forces the inside fluid to build some pressure and consequently some output, mili-volt. The
result of the calibration setup is as follows. Remember that the slope of the line has to be linear for a good
working transducer and all the coefficients and constants has to be plugged in the NI-DAQ program, so
this preliminary calibration has to be constructed prior to any further steps of the experiment.
Table A-1 and Table A-2 contain the results of the 500 psi and 1000 psi Sensotec transducers
respectively. The readings are based on the inserted pressure by dead-weights vs the output reading
voltage in mili-volt.
Fig. A-1 and Fig. A-2 depict the variations of the output voltage based on change in the inserted
pressure for both 500 psi and 1000 psi transducers respectively.
Table A-1 Pressure vs. Output Voltage in 0- 500 psi Sensotec Transducer
500 psi Transducer
Pressure
psi
mV
0
0.6
50
2.5
100
4.5
150
6.6
200
8.4
250
10.7
300
12.4
400
500
16.4
20.4
58
500 psi Sensotec Transducer
600
Pressure, psi
500
400
300
200
100
0
0
5
10
15
20
25
mili Volt
Fig. A-1—Plot of Pressure vs. Output mili-Volt for 0-500 psi Transducer
Table A-2 Pressure vs. Output Voltage in 0-1000 psi Sensotec Transducer
1000 psi Transducer
Pressure
psi
mV
0
0.3
50
1.3
100
2.3
200
4.3
300
6.3
400
8.3
500
10.2
600
12.2
800
16.3
1000
20.2
59
1000 psi Snesotec Transducer
Pressure, psi
1200
1000
800
600
400
200
0
0
5
10
15
20
25
mili Volt
Fig. A-2—Plot of Pressure vs. Output mili-Volt for 0-1000 psi Transducer
Plugging the results of the Table A-1 and Table A-2 in some curve fitting program will result in the
following equations showing the variation of these two sets of variables with each other.
For 500 psi Transducer:
Pressure = 25.16 * mili-volt – 12.598
(A-1)
For 1000 psi Transducer:
Pressure = 50.23 * mili-volt – 15.379
(A-2)
Since these results are based on mili-volt as the output, we need to convert it into volt for the DAQ
program besides the results are good for 10 volt excitation range and should be multiplied by 10 as well.
So, the equations used in the measurements are as follows:
For 500 psi Transducer:
Pressure = 251600 * volt – 12.598
(A-3)
Pressure = 502300 * volt – 15.379
(A-4)
For 1000 psi Transducer:
The following results found for Honeywell transducer.
For 1000 psi Transducer:
Pressure = 334850 * volt – 9.0552
60
(A-5)
Appendix B
Measuring the Discharge Coefficient, Cd, Through Benchmark Valve Testing
The purpose of implementing benchmark valve testing is to assure the correct relationship between the
practical and theoretical gas passage through the GLV. In this regard, the benchmark valve has been set in
6 different pre-known positions plus 2 positions for the gas-passage through the port with and without
presence of the benchmark valve body. The positions and related ball-seat distance has been reflected in
Table B-1.
The steps to set the benchmark valve and run the blowdown test to gather the gas passage throughput data
are as follows:
1. Insert the relevant ball, stem, and seat in the benchmark valve and make sure they are correctly
tightly installed
2. Adjust the ball-seat position in the close position and assure that using depth micrometer
including the multi-meter for the continuity test measurements
3. Extracting the minimum required travel for a ball to put the GLV in fully theoretical open
position from Table B-2.
4. Find the Micrometer setting for each position knowing the fully closed position and the minimum
distance for a GLV for fully open
5. Un-screw the benchmark valve and adjust the micrometer to the set number and re-screw the
benchmark valve till the tip of the depth micrometer hits the ball
6. Detach the depth micrometer from benchmark valve and place benchmark valve in to the
encapsulated tester for the blowdown test
7. Adjust the test upstream pressure through the main regulator and ensure the correct pressure
reading values with either analogue meter as well as pre-programmed LabView software
8. Have the temperature reader handy
9. Close the main pressure valve from the main high pressure Nitrogen bottle
10. Record the temperature and pressure as soon as the downstream valve gets open and record the
temperature when the upstream pressure reached the atmospheric pressure
11. Save the recorded data in separate file and store it in a related path for future calculations
12. Take the benchmark valve out of encapsulated chamber and re-adjust the ball position regarding
to the sea and follow steps 4-11.
Simple calculations and measurements have been carried out for 5/16‖ port size and the corresponding
data has been tabulated in Table B-1.
61
Table B-1—Set Positions of Ball/Stem in 5/16” Sharp-Edged Monel Seat
Benchmark Valve Testing, 5/16” Port Size
Minimum Travel Required for Fully Open = 0.1302 inch
Micrometer reading
Positions
Reading (inch)
Temperature,
o
F
(Up/Down)
Remarks
Position 0
0.698
inch
The Benchmark Valve is fully Closed
Position 1
0.73055
inch
0.698 + (.1302/4) = 0.73055 inch
Valve is %25 Fully Open
76.5
75.5
Position 2
0.7631
inch
0.698+ 2*(.1302/4) = 0.7631 inch
Valve is %50 Fully open
75.5
75
Position 3
0.79565
inch
0.698+ 3*(.1302/4) = 0.79565 inch
Valve is %75 Fully Open
75
74.5
Position 4
0.8282
inch
0.698+ 4*(.1302/4) = 0.8282 inch
Valve is fully open
73.5
73
Position 5
0.86075
inch
0.698+ 5*(.1302/4) = 0.86075 inch
Valve is at 1-1/4 Fully Open
73.5
72.5
Position 6
0.8933
inch
0.698+ 6*(.1302/4) = 0.8933 inch
Valve is at 1-1/2 Fully Open
72.5
71.5
Port Only
with Benchmark Valve
Ball is at its Max distance from Seat
inside Benchmark Valve Body
Port Only
without Benchmark
Valve
Port Only without Benchmark Valve
Body
Table B-2— 1-1/2” OD GLV with Ab = 0.77 in2, Sharp-Edged Monel Seat
1-1/2-inch OD Gas-Lift Valves with Ab= 0.77 in2 for Sharp-Edged Seat
Port Size
(Bore)
ID
inch
Ap
Area of
Port
AS=Ap
in2
As / A b
3/16
0.0276
1/4
1-(As/Ab)
Fp
Production
Pressure Factor
As/(Ab-As)
Geometric
Fully-Open
Stem Travel
inch
0.036
0.964
0.037
0.0714
0.0491
0.064
0.936
0.068
0.1002
5/16
0.0767
0.1
0.9
0.111
0.1302
3/8
0.1104
0.0143
0.9857
0.167
0.161
7/16
0.1503
0.195
0.805
0.243
0.1925
1/2
0.1963
0.255
0.745
0.342
0.2246
62
Sample results for 5/16‖ sharp-edged Monel seat are shown in Figures B-1 to B-5. These results yield to
calculation of discharge coefficient, Cd. The method behind the next series of calculations is
mathematical based and measurement comparisons. In other word, knowing the exact position of the ball
and its relevant distance from the seat gives us all the numbers required to calculate the frustum of a
circular cone. On the other hand, the practical gas passage through the valve has already been measured.
The ratio of the practical value to the theoretical number will give us the value of Cd for that testing
environment. The value of Cd may change since the upstream area to flow varies. When the upstream
flow area expands, the effectiveness of the ball on the flow is reduced therefore the gap between
theoretical flow rate and measured flowrate will vary. These variations are shown themselves in the Cd
number.
Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel
Ball at 1/4 Fully Open Travel Position
350
Pressure, psig
300
250
Real
Data
200
150
ExpDecay-2
Fit
100
50
0
0
5
10
15
20
25
30
Time, sec
Fig. B-1— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV with 5/16” Port Size
when the Ball is at 1/4 Fully Open Travel Position
63
Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel
Ball at 1/2 Fully Open Travel Position
350
300
Pressure, psig
250
Real
Data
200
150
ExpDecay2 Fit
100
50
0
0
2
4
6
8
10
12
14
16
Time, sec
Fig. B-2— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV with 5/16” Port Size
when the Ball is at 1/2 Fully Open Travel Position
Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel
Ball at 3/4 Fully Open Travel Position
350
Pressure, psig
300
250
Real
Data
200
150
100
ExpDecay-2
Fit
50
0
0
2
4
6
8
10
Time, sec
Fig. B-3— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV with 5/16” Port Size
when the Ball is at 3/4 Fully Open Travel Position
64
Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel
Ball at Fully Open Travel Position
350
Pressure, psig
300
250
Real
Data
200
150
ExpDecay2 Fit
100
50
0
0
1
2
3
4
5
6
7
8
9
Time, sec
Fig. B-4— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV with 5/16” Port Size
when the Ball is at Fully Open Travel Position
Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel
Ball at 1-1/2 Fully Open Travel Position
350
Pressure, psig
300
250
Real
Data
200
150
ExpDecay2 Fit
100
50
0
0
1
2
3
4
5
6
7
8
Time, sec
Fig. B-5— Gas Passage Through Results for Benchmark Valve Testing for J-20 GLV with 5/16” Port Size
when the Ball is at 1-1/2 Fully Open Travel Position (Beyond Fully Open)
65
The best curve fit formula follows the following pattern:
(
)
(
)
where,
Yo, A0, X0, B0, A1, and B1 are all constant
Y= is the pressure as getting depleted from the system
X= is the time constant as the pressure exiting the system
ISO [35] and API [36] referred Eq. B-1 as a standard ramp method which is exponential decline-based. In
this method, the time constant has be established for each system and in order to minimize the pressure
fluctuations, a larger surge tank capacity is preferred. For example in Fig. B-5, one time-constant to reach
63.2% of the final value would be reached at pressure of 129 psig that is corresponding to 1.63 second.
Time constants are the time needed for a system to reach 63.2%, 85.6%, 95%, 98% and 99% of its final
value.
Since the pressure is decaying, the best possible fit should be exponential although in some cases the
polynomial fitted better, but the behavior of polynomial fit is limited to the data series and is not reliable
if extrapolation of the results is aimed.
In all plots in Fig. B-1 to B-5, the real early reading is a bit off of the fit trend and this is due to the speed
of opening of the exit valve. The real early data has been dismissed to such error. It is so obvious as the
area open to flow increases, the depletion time drops.
Fig. B-6 is the combined form of 8 plots. As the plot demonstrates, as the balls moves up, the slope gets
larger (exponential decline). In other words, the rate of pressure drop increases, therefore the discharge
coefficient increases relatively.
66
Pressure, psig
Benchmark Valve Testing in 5/16" Monel Sharp-edged Seat
330
320
310
300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
0
0.1
0.2
1/4 Open
1 1/4 Open
0.3
0.4
0.5
Time, Sec
1/2 Open
1 1/2 Open
0.6
0.7
3/4 Open
Port Only, Benchmark
0.8
0.9
1
Full Open
Port Only
Fig. B-6—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First Second
If the line-slopes in Fig. B-6 plotted against the ball position, Fig. B-7 will get generated. The aim of such
plot is to find a relationship between the slope and the LR. Since the plot is clearly based on the slope
(dP/ dt) and ball position, this test will reveal the required tool usable in blowdown test. All the required
data for plotting Fig. B-7 is tabulated in Table B-3.
Table B-3—Extracted Empirical Values for Gas Throughput from Benchmark Valve Testing for 5/16” Port
Ball Position relative to
fully open
Ball Position from
seat for 5/16‖ port
(inch)
Initial Pressure
(psig)
Exponential Slope
(dP/dt)
0
0 = Closed
0
0
1/4
0.03255
330
-0.177
1/2
0.0651
330
-0.336
3/4
0.09765
329
-0.494
1
0.1302
329
-0.616
1 1/4
0.16275
328
-0.653
1 1/2
0.1953
328
-0.676
Port in Benchmark Valve
328
-0.682
Port Only
328
-0.684
67
Plot of (dP/dt) vs. Ball Position for 5/16" Monel Sharp-edged seat
0.7
0.6
dP/dTime
0.5
0.4
0.3
0.2
0.1
0
0
0.25
0.5
0.75
1
1.25
1.5
Ball Position relative to Theoretical Fully Open
Fig. B-7—Plot of Pressure rate Against Ball Position in 5/16” Monel J-20 Camco GLV
Fig. B-8 through B-13 covers the variation of pressure decay with time for different seat and ball sizes.
Benchmark Valve Testing in 3/16" Monel Sharp-edged Seat
330
320
Pressure, psig
310
300
290
280
270
260
250
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time, sec
1/4 Open
Full Open
Port Only
1/2 Open
1-1/4 Open
Port Only- Benchmark Valve
3/14 Open
1-1/2 Open
Fig. B-8—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First Second
68
1
Pressure Change with Time at Relative Port Positions in 3/16" Monel
Seat
0.3
dP/dTime
0.25
0.2
0.15
0.1
0.05
0
0
0.25
0.5
0.75
1
1.25
1.5
Ball Position Relative to Theoretical Minimum Fully-Open
Fig. B-9—Change of Pressure vs. Time relative to Ball Position in 3/16” Monel Port
Benchmark Valve Testing in 1/4" Monel Sharp-edged Seat
330
320
310
Pressure, psig
300
290
280
270
260
250
240
230
220
210
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time, sec
1/4 Open
1/2 Open
3/4 Open
1-1/4 Open
1-1/2 Open
Port Only
Full Open
Fig. B-10—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First Second
69
1
Pressure Change with Time at Relative Port Positions in 1/4" Monel
Seat
0.5
Slope (dP/ dTime)
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0.25
0.5
0.75
1
1.25
1.5
Ball Position to Fully Open Position
Fig. B-11—Change of Pressure vs. Time relative to Ball Position in 1/4” Monel Port
Pressure, psig
Benchmark Testing in 3/8" Monel Sharp-edged Seat
330
320
310
300
290
280
270
260
250
240
230
220
210
200
190
180
170
160
150
140
130
120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time, sec
1/4 Open
1/2 Open
3/4 Open
1-1/4 Open
1-1/2 Open
Port Only
Full Open
Fig. B-12—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First Second
70
1
Pressure Change with Time at Relative Port Positions in 3/8" Monel
Seat
1
Slope (dP/dTime)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.25
0.5
0.75
1
1.25
1.5
Ball Position to Fully Open Position
Fig. B-13—Change of Pressure vs. Time relative to Ball Position in 3/8” Monel Port
With a close look at the Fig. B6, Fig. B8, Fig. B10, Fig. B12 for the first 10th of second and lower will
reveal that the flow behavior for the flow through an orifice port is achieved when the ball is at least 1.25
times more than the minimum theoretical fully travel position. Knowing this fact based on the
measurements would help to analyze and couple the results with actual GLV performances.
There is a very close similarity among Fig. B7, Fig. B9, Fig. B11, Fig. B13 based on the flow behavior.
As is clearly obvious, the pressure decays faster at the bigger port and this can be found using Bernoulli
equation and continuity rule.
In order to quantify the behavior of each Nitrogen-charged GLV effectively, Cd has be determined
precisely or with less degree of uncertainty. Benchmark valve has been adopted to measure the flowrate
through each GLV at different ball positions. Having the flowrate at the depletion time (decay time) will
result in calculating the volumetric flowrate and back calculate the flowing area. Comparing the two
flowing areas with each other will release the value for Cd. Plotting the Cd values based on the ball
position will explore the accuracy of TC equation.
To verify the measured values of Cd and its sensitivity issues to the upstream injection pressure, three
different sets of experiments had been setup at different initial pressures on the port-only case and portonly inside the GLV body. Port-only case is the case that the orifice port only has been set inside the
71
encapsulated vessel with no GLV body and such limiting the flow. This test aimed to clarify the
sensitivity of the measured Cd to pressure as well as effect of tortuosity on the flow. Fig. B14 following
by Table B-4 Show how the changes are happening. The value of Cd at each set pressure does not vary
but there is a trend in different set pressures.
Table B-4—Cd Sensitivity to Upstream Pressure and the GLV Body in 5/16” Orifice Port
Cd
150
Pressure
300
450
600
750
900
0.72691 0.79527 0.83215 0.85332 0.8564 0.86002
0.7375 0.79957 0.83151 0.84464 0.85426 0.85917
Port Only
Port inside BV
As results in Table B-4 represent, the Valve body does not have strong effect on the Cd. The variation
due to the existence of the valve body on the flow stream is less than 0.5% which is practically ignorable.
It worth note that Temperature has effect on the measurements but its effect is ignorable comparing to
pressure effects.
0.88
0.86
0.84
Cd
0.82
0.8
0.78
0.76
0.74
0.72
150
300
450
600
Pressure, psig
Port Only
750
900
Port inside BV
Fig. B-14—Sensitivity of Cd to Pressure and the Valve Body in 5/16” Monel Sharp-edged Seat at T= 73 oF
72
Assuming the Fig. B-15 representing the ball-seat position when the GLV is open. ―H‖ represents the ball
distance from the seat from the closed position to any other ball-set distance.
R
Y
S
H
θ
r
Fig. B-15—Ball-Seat Relevancy Due to Angle, And Distance
Knowing the ball-seat distances based on benchmark valve settings and the fact that the port area can get
calculated directly knowing the port diameter makes the angle calculations easier. Based on what has
been represented in Fig. B-15 the calculation step procedure can be followed through Eq. B-2 through B6. These formulas will help to calculate the effective flowing area at each ball-stem position as the ball is
dynamically moving inside the GLV.
√
(
)
73
(
(
(
(
√
(
√
(
))
))
)
)
In order to calculate the Cd at each pressure and ball-seat condition, some volumetric flow measurements
has to be done. The results of such measurements have to get compared with TC equation. The ratio of
these two findings including the expansion factor will be resulting in the actual value of Cd. Therefore the
following steps needed to be taken and followed:
1. Recording the pressure points with respect to time using blow-down test
2. Calculate the decline rate based on pressure-decay at each two consecutive pressure readings
3. Calculate the decline rate based on mole-decay by knowing the gas properties, the internal
capacity of the testing vessel, and standard temperature value at each two consecutive readings
4. Calculate the gas velocity knowing that each mole of gas occupying certain volume at standard
condition
5. Converting the units of volumetric flowrate to MSCF/D
6. Calculate the critical pressure ratio. This value stays constant because the flow is at critical
conditions
7. Calculate the constants values related to the specific heat capacity ratios
8. Include the gas expansion coefficient which stays constant because at critical flow the critical
pressure ratio stays constant
9. Calculate the flowrate based on TC equation while the actual flowing area value should be used
with no value for Cd
10. The ratio of the square root of the two flowrates will result in the value of Cd. Note that the units
of the two flowrates has to be consistent
Table B-5 through B-24 demonstrates some measurements where resulted in some Cd calculations. The
Cd values then got applied to calculate the maximum equivalent port size (not knowing the port size) and
74
very satisfactory results collected. All the pressure points tabulated in the following tables have been
normalized consistently.
75
Table B-5—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P
psig
345
344.7
344.41
344.11
343.82
343.52
343.22
342.93
342.63
342.34
342.05
341.75
341.46
341.16
340.87
340.58
slope
dP/dt
29.65725
29.63175
29.60628
29.58083
29.5554
29.52999
29.50461
29.47925
29.4539
29.42858
29.40329
29.37801
29.35276
29.32752
29.30231
29.27712
mole
dn/dt
0.286579
0.286333
0.286087
0.285841
0.285595
0.285349
0.285104
0.284859
0.284614
0.284369
0.284125
0.283881
0.283637
0.283393
0.283149
0.282906
flowrate
dV/dt
108.900033
108.80642
108.712886
108.619433
108.526061
108.432769
108.339556
108.246425
108.153373
108.060401
107.967509
107.874697
107.781964
107.689312
107.596739
107.504245
1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1.08900033 94.08963
94.00875
1.0880642
1.08712886 93.92793
1.08619433 93.84719
1.08526061 93.76652
1.08432769 93.68591
1.08339556 93.60538
1.08246425 93.52491
1.08153373 93.44451
1.08060401 93.36419
1.07967509 93.28393
1.07874697 93.20374
1.07781964 93.12362
1.07689312 93.04357
1.07596739 92.96358
1.07504245 92.88367
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
43.493
43.475
43.457
43.439
43.421
43.403
43.385
43.367
43.349
43.331
43.313
43.296
43.278
43.26
43.242
43.224
Cd
0.9805
0.9803
0.9801
0.9799
0.9797
0.9795
0.9792
0.979
0.9788
0.9786
0.9784
0.9781
0.9779
0.9777
0.9775
0.9773
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P
psig
345
344.47
343.94
343.41
342.88
342.35
341.83
341.3
340.78
340.25
339.73
339.2
338.68
338.16
337.64
337.12
slope
dP/dt
53.08911
53.00742
52.92585
52.8444
52.76309
52.68189
52.60083
52.51988
52.43907
52.35837
52.2778
52.19736
52.11703
52.03684
51.95676
51.87681
mole
dn/dt
0.513002
0.512213
0.511424
0.510637
0.509852
0.509067
0.508284
0.507502
0.506721
0.505941
0.505162
0.504385
0.503609
0.502834
0.50206
0.501287
flowrate
dV/dt
194.940759
194.640781
194.341265
194.04221
193.743615
193.445479
193.147802
192.850584
192.553822
192.257518
191.961669
191.666275
191.371336
191.076851
190.78282
190.48924
1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1.94940759 168.4288
1.94640781 168.1696
1.94341265 167.9109
167.6525
1.9404221
1.93743615 167.3945
1.93445479 167.1369
1.93147802 166.8797
1.92850584 166.6229
1.92553822 166.3665
1.92257518 166.1105
1.91961669 165.8549
1.91666275 165.5997
1.91371336 165.3448
1.91076851 165.0904
164.8364
1.9078282
164.5827
1.9048924
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
86.923
86.859
86.794
86.73
86.666
86.602
86.537
86.473
86.409
86.345
86.281
86.217
86.154
86.09
86.026
85.962
Cd
0.928
0.9276
0.9273
0.9269
0.9265
0.9262
0.9258
0.9254
0.925
0.9247
0.9243
0.9239
0.9236
0.9232
0.9228
0.9225
76
Table B-6—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
344.27
343.55
342.82
342.1
341.38
340.66
339.94
339.23
338.51
337.8
337.08
336.37
335.67
334.96
334.25
P,
psig
345
344.13
343.26
342.39
341.53
340.66
339.8
338.94
338.09
337.23
336.38
335.53
334.68
333.84
332.99
332.15
slope
mole
flowrate
dP/dt
72.71826
72.56498
72.41203
72.2594
72.1071
71.95511
71.80345
71.6521
71.50107
71.35036
71.19997
71.0499
70.90014
70.7507
70.60158
70.45276
dn/dt
0.702679
0.701198
0.69972
0.698245
0.696773
0.695305
0.693839
0.692377
0.690917
0.689461
0.688008
0.686558
0.685111
0.683667
0.682226
0.680788
dV/dt
267.018068
266.455254
265.893626
265.333182
264.77392
264.215836
263.658928
263.103194
262.548632
261.995238
261.443011
260.891948
260.342046
259.793304
259.245718
258.699286
slope
mole
flowrate
dP/dt
87.17468
86.9544
86.73469
86.51553
86.29692
86.07886
85.86136
85.6444
85.428
85.21214
84.99682
84.78205
84.56783
84.35414
84.14099
83.92839
dn/dt
0.842372
0.840244
0.83812
0.836003
0.83389
0.831783
0.829681
0.827585
0.825494
0.823408
0.821327
0.819252
0.817182
0.815117
0.813057
0.811003
dV/dt
320.101382
319.292549
318.48576
317.68101
316.878292
316.077604
315.278938
314.482291
313.687656
312.895029
312.104405
311.315779
310.529146
309.7445
308.961837
308.181152
3/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
2.67018068 230.7036
2.66455254 230.2173
2.65893626 229.7321
2.65333182 229.2479
228.7647
2.6477392
2.64215836 228.2825
2.63658928 227.8013
2.63103194 227.3212
226.842
2.62548632
2.61995238 226.3639
2.61443011 225.8868
2.60891948 225.4106
2.60342046 224.9355
2.59793304 224.4614
2.59245718 223.9883
2.58699286 223.5162
Fully Theoretically Open
3
ft /sec
MSCF/D
dV/dt
3.20101382
3.19292549
3.1848576
3.1768101
3.16878292
3.16077604
3.15278938
3.14482291
3.13687656
3.12895029
3.12104405
3.11315779
3.10529146
3.097445
3.08961837
3.08181152
77
dV/dt
276.5676
275.8688
275.1717
274.4764
273.7828
273.091
272.401
271.7127
271.0261
270.3413
269.6582
268.9768
268.2972
267.6192
266.943
266.2685
Constants
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
129.66
129.52
129.39
129.26
129.13
129
128.87
128.74
128.61
128.48
128.35
128.22
128.09
127.96
127.83
127.7
Cd
0.8893
0.8888
0.8883
0.8878
0.8873
0.8869
0.8864
0.8859
0.8854
0.8849
0.8844
0.8839
0.8835
0.883
0.8825
0.882
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
171.46
171.25
171.04
170.83
170.63
170.42
170.21
170.01
169.8
169.59
169.39
169.18
168.97
168.77
168.56
168.36
Cd
0.8467
0.8461
0.8456
0.845
0.8445
0.8439
0.8434
0.8428
0.8423
0.8417
0.8412
0.8406
0.8401
0.8395
0.839
0.8384
Constants
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-7—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
344.07
343.14
342.21
341.28
340.36
339.44
338.52
337.6
336.69
335.78
334.87
333.96
333.06
332.16
331.26
P,
psig
345
344.05
343.1
342.16
341.21
340.27
339.33
338.4
337.47
336.54
335.61
334.68
333.76
332.84
331.92
331.01
slope
mole
flowrate
dP/dt
93.36843
93.11574
92.86374
92.61242
92.36178
92.11182
91.86253
91.61392
91.36599
91.11872
90.87212
90.62619
90.38093
90.13633
89.89239
89.64911
dn/dt
0.902222
0.899781
0.897346
0.894917
0.892495
0.89008
0.887671
0.885269
0.882873
0.880483
0.878101
0.875724
0.873354
0.870991
0.868633
0.866283
dV/dt
342.84455
341.916699
340.991359
340.068523
339.148185
338.230338
337.314974
336.402088
335.491673
334.583721
333.678227
332.775183
331.874583
330.976421
330.080689
329.187381
slope
mole
flowrate
dP/dt
95.08872
94.82663
94.56527
94.30463
94.04471
93.78551
93.52701
93.26924
93.01217
92.75581
92.50016
92.24521
91.99096
91.73742
91.48457
91.23242
dn/dt
0.918846
0.916313
0.913788
0.911269
0.908757
0.906253
0.903755
0.901264
0.89878
0.896303
0.893832
0.891369
0.888912
0.886462
0.884019
0.881582
dV/dt
349.16137
348.199013
347.239309
346.282249
345.327828
344.376037
343.42687
342.480319
341.536376
340.595036
339.656289
338.72013
337.786552
336.855546
335.927107
335.001226
1-1/4 Fully Open
3
ft /sec
MSCF/D
Constants
dV/dt
dV/dt
296.2177
3.4284455
295.416
3.41916699
3.40991359 294.6165
3.40068523 293.8192
293.024
3.39148185
292.231
3.38230338
3.37314974 291.4401
3.36402088 290.6514
3.35491673 289.8648
3.34583721 289.0803
288.298
3.33678227
3.32775183 287.5178
3.31874583 286.7396
3.30976421 285.9636
3.30080689 285.1897
3.29187381 284.4179
1-1/2 Fully Open
3
ft /sec
MSCF/D
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
3.4916137
3.48199013
3.47239309
3.46282249
3.45327828
3.44376037
3.4342687
3.42480319
3.41536376
3.40595036
3.39656289
3.3872013
3.37786552
3.36855546
3.35927107
3.35001226
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
78
dV/dt
301.6754
300.8439
300.0148
299.1879
298.3632
297.5409
296.7208
295.903
295.0874
294.2741
293.463
292.6542
291.8476
291.0432
290.241
289.4411
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
171.45
171.23
171.01
170.78
170.56
170.34
170.12
169.89
169.67
169.45
169.23
169.01
168.79
168.57
168.35
168.13
Cd
0.8763
0.8757
0.875
0.8744
0.8738
0.8732
0.8726
0.872
0.8714
0.8708
0.8701
0.8695
0.8689
0.8683
0.8677
0.8671
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
171.45
171.22
171
170.77
170.54
170.32
170.09
169.86
169.64
169.41
169.19
168.96
168.74
168.52
168.29
168.07
Cd
0.8843
0.8837
0.8831
0.8824
0.8818
0.8812
0.8805
0.8799
0.8793
0.8786
0.878
0.8774
0.8768
0.8761
0.8755
0.8749
Constants
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-8—Cd Calculations for 3/16 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
344.05
343.09
342.14
341.2
340.25
339.31
338.37
337.44
336.51
335.57
334.65
333.72
332.8
331.88
330.96
P,
psig
345
344.03
343.05
342.08
341.12
340.15
339.19
338.23
337.28
336.32
335.37
334.43
333.48
332.54
331.6
330.66
slope
mole
flowrate
dP/dt
95.43276
95.16878
94.90553
94.643
94.38121
94.12013
93.85978
93.60015
93.34123
93.08304
92.82555
92.56878
92.31272
92.05737
91.80272
91.54878
dn/dt
0.92217
0.919619
0.917076
0.914539
0.912009
0.909486
0.90697
0.904462
0.90196
0.899465
0.896977
0.894495
0.892021
0.889554
0.887093
0.884639
dV/dt
350.424696
349.455362
348.488711
347.524733
346.563421
345.604769
344.648768
343.695412
342.744693
341.796604
340.851138
339.908286
338.968043
338.030401
337.095352
336.16289
slope
mole
dP/dt
97.49698
97.22145
96.9467
96.67273
96.39953
96.12711
95.85545
95.58457
95.31444
95.04509
94.77649
94.50865
94.24157
93.97524
93.70967
93.44485
dn/dt
0.942117
0.939454
0.936799
0.934152
0.931512
0.92888
0.926255
0.923637
0.921027
0.918424
0.915829
0.913241
0.91066
0.908086
0.90552
0.902961
P-BV
3
ft /sec
MSCF/D
dV/dt
302.7669
301.9294
301.0942
300.2614
299.4308
298.6025
297.7765
296.9528
296.1314
295.3123
294.4954
293.6808
292.8684
292.0583
291.2504
290.4447
flowrate
dV/dt
3.50424696
3.49455362
3.48488711
3.47524733
3.46563421
3.45604769
3.44648768
3.43695412
3.42744693
3.41796604
3.40851138
3.39908286
3.38968043
3.38030401
3.37095352
3.3616289
P
3
ft /sec
dV/dt
358.004387
356.992667
355.983806
354.977796
353.974629
352.974297
351.976792
350.982106
349.99023
349.001158
348.014881
347.031391
346.050681
345.072742
344.097567
343.125147
dV/dt
3.58004387
3.56992667
3.55983806
3.54977796
3.53974629
3.52974297
3.51976792
3.50982106
3.4999023
3.49001158
3.48014881
3.47031391
3.46050681
3.45072742
3.44097567
3.43125147
dV/dt
309.3158
308.4417
307.57
306.7008
305.8341
304.9698
304.1079
303.2485
302.3916
301.537
300.6849
299.8351
298.9878
298.1428
297.3003
296.4601
79
Constants
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
MSCF/D
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
171.45
171.22
170.99
170.77
170.54
170.31
170.08
169.86
169.63
169.41
169.18
168.96
168.73
168.51
168.28
168.06
Cd
0.8859
0.8853
0.8846
0.884
0.8834
0.8827
0.8821
0.8815
0.8808
0.8802
0.8796
0.8789
0.8783
0.8777
0.877
0.8764
CK4
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
0.0762
Q
171.45
171.21
170.98
170.75
170.52
170.28
170.05
169.82
169.59
169.36
169.13
168.9
168.67
168.44
168.21
167.98
Cd
0.8955
0.8948
0.8941
0.8935
0.8928
0.8922
0.8915
0.8909
0.8902
0.8896
0.8889
0.8883
0.8876
0.8869
0.8863
0.8856
Constants
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-9—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
344.6069
344.2143
343.8221
343.4304
343.0391
342.6483
342.2579
341.8679
341.4784
341.0893
340.7007
340.3125
339.9248
339.5375
339.1507
slope
dP/dt
39.30759
39.26281
39.21807
39.17339
39.12876
39.08417
39.03964
38.99516
38.95073
38.90636
38.86203
38.81775
38.77352
38.72935
38.68522
38.64114
mole
dn/dt
0.379831
0.379398
0.378966
0.378534
0.378103
0.377672
0.377241
0.376812
0.376382
0.375954
0.375525
0.375097
0.37467
0.374243
0.373817
0.373391
flowrate
dV/dt
144.3357
144.1712
144.0069
143.8429
143.679
143.5153
143.3518
143.1884
143.0253
142.8623
142.6996
142.537
142.3746
142.2124
142.0503
141.8885
P, psig
345
344.2143
343.4304
342.6483
341.8679
341.0893
340.3125
339.5375
338.7642
337.9927
337.223
336.455
335.6888
334.9243
334.1615
333.4005
slope
dP/dt
78.5704
78.39146
78.21293
78.03481
77.85709
77.67978
77.50287
77.32637
77.15026
76.97456
76.79926
76.62436
76.44985
76.27574
76.10203
75.92872
mole
dn/dt
0.759229
0.7575
0.755774
0.754053
0.752336
0.750623
0.748913
0.747207
0.745506
0.743808
0.742114
0.740424
0.738738
0.737055
0.735377
0.733702
flowrate
dV/dt
288.5069
287.8498
287.1943
286.5402
285.8876
285.2366
284.587
283.9388
283.2922
282.647
282.0033
281.3611
280.7203
280.081
279.4431
278.8067
1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
124.706
1.443357
1.441712 124.5639
124.422
1.440069
1.438429 124.2802
124.1386
1.43679
1.435153 123.9972
1.433518 123.8559
1.431884 123.7148
1.430253 123.5739
1.428623 123.4331
1.426996 123.2924
123.152
1.42537
1.423746 123.0116
1.422124 122.8715
1.420503 122.7315
1.418885 122.5917
1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
2.885069 249.2699
2.878498 248.7022
2.871943 248.1358
2.865402 247.5707
2.858876 247.0069
2.852366 246.4444
245.8831
2.84587
2.839388 245.3232
2.832922 244.7645
244.207
2.82647
2.820033 243.6509
243.096
2.813611
2.807203 242.5424
241.99
2.80081
2.794431 241.4389
240.889
2.788067
80
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
75.476
75.435
75.393
75.352
75.311
75.269
75.228
75.187
75.146
75.104
75.063
75.022
74.981
74.94
74.899
74.858
Cd
0.8569
0.8567
0.8564
0.8562
0.8559
0.8557
0.8554
0.8552
0.8549
0.8547
0.8544
0.8542
0.8539
0.8536
0.8534
0.8531
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
152.64
152.47
152.31
152.14
151.97
151.81
151.64
151.47
151.31
151.14
150.98
150.81
150.64
150.48
150.31
150.15
Cd
0.8519
0.8514
0.8509
0.8504
0.8499
0.8494
0.8489
0.8484
0.8479
0.8474
0.8469
0.8464
0.8459
0.8454
0.8449
0.8444
Table B-10—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
343.8152
342.6345
341.4579
340.2853
339.1167
337.9522
336.7916
335.6351
334.4825
333.3338
332.1891
331.0483
329.9115
328.7785
327.6495
slope
dP/dt
118.4761
118.0692
117.6638
117.2597
116.857
116.4557
116.0558
115.6573
115.2601
114.8643
114.4698
114.0767
113.685
113.2946
112.9055
112.5178
mole
dn/dt
1.144839
1.140907
1.136989
1.133085
1.129194
1.125316
1.121452
1.1176
1.113762
1.109938
1.106126
1.102328
1.098542
1.09477
1.09101
1.087263
flowrate
dV/dt
435.0388
433.5448
432.056
430.5723
429.0936
427.6201
426.1516
424.6882
423.2297
421.7763
420.3279
418.8845
417.446
416.0124
414.5838
413.1601
P, psig
345
343.6434
342.2921
340.9461
339.6054
338.27
336.9398
335.6149
334.2952
332.9807
331.6713
330.3671
329.068
327.774
326.4851
325.2013
slope
dP/dt
135.6626
135.1291
134.5977
134.0685
133.5413
133.0162
132.4931
131.9721
131.4532
130.9363
130.4214
129.9085
129.3977
128.8889
128.3821
127.8772
mole
dn/dt
1.310912
1.305757
1.300623
1.295509
1.290414
1.28534
1.280286
1.275251
1.270237
1.265242
1.260267
1.255311
1.250375
1.245458
1.240561
1.235682
flowrate
dV/dt
498.1467
496.1878
494.2367
492.2932
490.3574
488.4292
486.5086
484.5955
482.69
480.7919
478.9013
477.0182
475.1424
473.274
471.413
469.5593
3/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
4.350388 375.8735
4.335448 374.5827
373.2964
4.32056
4.305723 372.0144
4.290936 370.7369
4.276201 369.4638
368.195
4.261516
4.246882 366.9306
4.232297 365.6705
4.217763 364.4148
4.203279 363.1633
4.188845 361.9162
360.6733
4.17446
4.160124 359.4347
4.145838 358.2004
4.131601 356.9703
Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
4.981467 430.3987
4.961878 428.7063
4.942367 427.0205
4.922932 425.3414
4.903574 423.6688
4.884292 422.0028
4.865086 420.3434
4.845955 418.6905
417.0441
4.8269
4.807919 415.4042
4.789013 413.7707
4.770182 412.1437
410.523
4.751424
408.9088
4.73274
407.3008
4.71413
4.695593 405.6992
81
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
229.47
229.09
228.71
228.34
227.96
227.58
227.21
226.83
226.45
226.08
225.71
225.33
224.96
224.59
224.22
223.85
Cd
0.8532
0.8525
0.8517
0.8509
0.8502
0.8494
0.8487
0.8479
0.8472
0.8464
0.8456
0.8449
0.8441
0.8434
0.8426
0.8419
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
305.15
304.57
304
303.42
302.84
302.27
301.7
301.13
300.56
299.99
299.42
298.85
298.29
297.73
297.16
296.6
Cd
0.7917
0.7909
0.7901
0.7893
0.7885
0.7877
0.7869
0.7861
0.7853
0.7845
0.7837
0.7829
0.7821
0.7813
0.7805
0.7797
Table B-11—Cd Calculations for 1/4 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
343.4888
341.9842
340.4861
338.9947
337.5098
336.0313
334.5594
333.0939
331.6348
330.1821
328.7358
327.2958
325.8621
324.4347
323.0136
slope
dP/dt
151.123
150.4611
149.802
149.1458
148.4925
147.842
147.1944
146.5497
145.9077
145.2686
144.6323
143.9987
143.3679
142.7399
142.1147
141.4922
mole
dn/dt
1.460307
1.453911
1.447542
1.441201
1.434888
1.428603
1.422345
1.416115
1.409912
1.403736
1.397587
1.391465
1.38537
1.379301
1.373259
1.367244
flowrate
dV/dt
554.9168
552.4861
550.066
547.6565
545.2575
542.8691
540.4911
538.1236
535.7664
533.4195
531.083
528.7566
526.4405
524.1344
521.8385
519.5527
P, psig
345
343.4819
341.9705
340.4657
338.9676
337.476
335.991
334.5126
333.0406
331.5751
330.1161
328.6635
327.2173
325.7774
324.3439
322.9167
slope
dP/dt
151.81
151.142
150.4769
149.8148
149.1556
148.4992
147.8458
147.1952
146.5475
145.9027
145.2607
144.6215
143.9851
143.3515
142.7207
142.0927
mole
dn/dt
1.466946
1.460491
1.454064
1.447666
1.441296
1.434953
1.428639
1.422353
1.416094
1.409863
1.403659
1.397483
1.391333
1.385211
1.379116
1.373047
flowrate
dV/dt
557.4393
554.9864
552.5443
550.113
547.6923
545.2823
542.8829
540.4941
538.1158
535.7479
533.3904
531.0434
528.7066
526.3802
524.0639
521.7579
1-1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
5.549168 479.4481
477.348
5.524861
475.257
5.50066
5.476565 473.1752
5.452575 471.1025
5.428691 469.0389
5.404911 466.9843
5.381236 464.9388
5.357664 462.9022
5.334195 460.8745
458.8557
5.31083
5.287566 456.8457
5.264405 454.8446
5.241344 452.8522
5.218385 450.8685
5.195527 448.8935
1-1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
5.574393 481.6276
5.549864 479.5083
5.525443 477.3983
475.2976
5.50113
5.476923 473.2062
5.452823 471.1239
5.428829 469.0509
5.404941 466.9869
464.932
5.381158
5.357479 462.8862
5.333904 460.8493
5.310434 458.8215
5.287066 456.8025
5.263802 454.7925
5.240639 452.7912
5.217579 450.7988
82
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
305.12
304.47
303.83
303.19
302.55
301.91
301.27
300.64
300.01
299.37
298.74
298.11
297.48
296.86
296.23
295.61
Cd
0.8357
0.8347
0.8338
0.8328
0.8319
0.8309
0.83
0.8291
0.8281
0.8272
0.8262
0.8253
0.8243
0.8234
0.8225
0.8215
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
305.12
304.47
303.82
303.18
302.54
301.9
301.26
300.62
299.98
299.35
298.71
298.08
297.45
296.82
296.19
295.56
Cd
0.8376
0.8366
0.8357
0.8347
0.8338
0.8328
0.8319
0.8309
0.83
0.829
0.8281
0.8271
0.8262
0.8252
0.8243
0.8233
Table B-12—Cd Calculations for 1/4 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
343.4441
341.8953
340.3534
338.8184
337.2904
335.7693
334.2551
332.7477
331.247
329.7532
328.2661
326.7856
325.3119
323.8448
322.3843
slope
dP/dt
155.5881
154.8864
154.1879
153.4926
152.8004
152.1113
151.4253
150.7424
150.0625
149.3858
148.7121
148.0414
147.3738
146.7092
146.0475
145.3889
mole
dn/dt
1.503453
1.496673
1.489924
1.483204
1.476515
1.469857
1.463228
1.456629
1.45006
1.44352
1.43701
1.43053
1.424078
1.417656
1.411263
1.404898
flowrate
dV/dt
571.3123
568.7358
566.1709
563.6176
561.0758
558.5455
556.0265
553.519
551.0227
548.5377
546.0639
543.6013
541.1497
538.7093
536.2798
533.8613
P, psig
345
343.4201
341.8474
340.2819
338.7236
337.1724
335.6283
334.0913
332.5614
331.0384
329.5224
328.0134
326.5113
325.016
323.5276
322.046
slope
dP/dt
157.9921
157.2686
156.5484
155.8315
155.1179
154.4075
153.7004
152.9965
152.2959
151.5984
150.9042
150.2131
149.5252
148.8405
148.1589
147.4804
mole
dn/dt
1.526684
1.519692
1.512733
1.505805
1.49891
1.492045
1.485212
1.478411
1.471641
1.464901
1.458193
1.451515
1.444868
1.438251
1.431665
1.425108
flowrate
dV/dt
580.1398
577.4831
574.8385
572.206
569.5856
566.9772
564.3807
561.7962
559.2234
556.6625
554.1133
551.5757
549.0498
546.5354
544.0326
541.5412
P-BV
3
ft /sec
MSCF/D
dV/dt
dV/dt
5.713123 493.6138
5.687358 491.3877
5.661709 489.1717
5.636176 486.9656
5.610758 484.7695
5.585455 482.5833
5.560265 480.4069
478.2404
5.53519
5.510227 476.0836
5.485377 473.9366
5.460639 471.7992
5.436013 469.6715
5.411497 467.5534
5.387093 465.4448
5.362798 463.3457
5.338613 461.2562
P
3
ft /sec
MSCF/D
dV/dt
dV/dt
5.801398 501.2408
5.774831 498.9454
5.748385 496.6604
494.386
5.72206
492.122
5.695856
5.669772 489.8683
487.625
5.643807
5.617962 485.3919
483.169
5.592234
5.566625 480.9564
5.541133 478.7539
5.515757 476.5614
474.379
5.490498
5.465354 472.2066
5.440326 470.0441
5.415412 467.8916
83
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
305.11
304.44
303.78
303.12
302.46
301.81
301.15
300.5
299.85
299.2
298.55
297.9
297.25
296.61
295.96
295.32
Cd
0.848
0.847
0.846
0.845
0.844
0.843
0.842
0.841
0.84
0.8391
0.8381
0.8371
0.8361
0.8351
0.8341
0.8332
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
305.1
304.43
303.76
303.09
302.42
301.75
301.09
300.42
299.76
299.1
298.44
297.78
297.13
296.47
295.82
295.17
Cd
0.8545
0.8535
0.8525
0.8514
0.8504
0.8494
0.8484
0.8474
0.8464
0.8454
0.8444
0.8434
0.8424
0.8414
0.8404
0.8394
Table B-13—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
344.3692
343.7396
343.1111
342.4838
341.8576
341.2326
340.6087
339.986
339.3644
338.7439
338.1246
337.5064
336.8893
336.2734
335.6585
slope
dP/dt
63.07727
62.96194
62.84683
62.73192
62.61723
62.50274
62.38847
62.2744
62.16054
62.04689
61.93345
61.82022
61.70719
61.59437
61.48175
61.36934
mole
dn/dt
0.609518
0.608404
0.607291
0.606181
0.605073
0.603966
0.602862
0.60176
0.60066
0.599561
0.598465
0.597371
0.596279
0.595189
0.5941
0.593014
flowrate
dV/dt
231.616805
231.193333
230.770637
230.348712
229.92756
229.507177
229.087563
228.668716
228.250635
227.833318
227.416765
227.000973
226.585941
226.171668
225.758152
225.345392
1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
2.31616805 200.11692
2.31193333 199.75104
2.30770637 199.38583
2.30348712 199.02129
198.65741
2.2992756
198.2942
2.29507177
2.29087563 197.93165
2.28668716 197.56977
2.28250635 197.20855
2.27833318 196.84799
2.27416765 196.48808
2.27000973 196.12884
2.26585941 195.77025
2.26171668 195.41232
2.25758152 195.05504
2.25345392 194.69842
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
114.882
114.781
114.68
114.579
114.478
114.378
114.277
114.176
114.076
113.975
113.875
113.775
113.675
113.575
113.475
113.375
Cd
0.8799
0.8795
0.879
0.8786
0.8782
0.8778
0.8774
0.877
0.8765
0.8761
0.8757
0.8753
0.8749
0.8745
0.8741
0.8736
P, psig
345
343.9218
342.847
341.7756
340.7075
339.6428
338.5814
337.5233
336.4685
335.417
334.3687
333.3238
332.2821
331.2437
330.2085
329.1766
slope
dP/dt
107.8162
107.4792
107.1434
106.8085
106.4747
106.142
105.8103
105.4796
105.15
104.8214
104.4938
104.1672
103.8417
103.5172
103.1937
102.8712
mole
dn/dt
1.041832
1.038576
1.03533
1.032095
1.028869
1.025654
1.022449
1.019253
1.016068
1.012893
1.009727
1.006572
1.003426
1.00029
0.997164
0.994048
flowrate
dV/dt
395.896023
394.658806
393.425455
392.195958
390.970304
389.74848
388.530474
387.316275
386.105871
384.899249
383.696397
382.497305
381.30196
380.110351
378.922466
377.738293
1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
3.95896023 342.05416
3.94658806 340.98521
3.93425455 339.91959
3.92195958 338.85731
3.90970304 337.79834
336.74269
3.8974848
3.88530474 335.69033
3.87316275 334.64126
3.86105871 333.59547
3.84899249 332.55295
3.83696397 331.51369
3.82497305 330.47767
329.44489
3.8130196
3.80110351 328.41534
3.78922466 327.38901
3.77738293 326.36588
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
235.143
234.788
234.435
234.082
233.73
233.378
233.027
232.676
232.326
231.977
231.628
231.28
230.932
230.585
230.238
229.892
Cd
0.8041
0.8034
0.8028
0.8021
0.8015
0.8008
0.8002
0.7995
0.7989
0.7982
0.7976
0.7969
0.7963
0.7956
0.795
0.7943
84
Table B-14—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
343.2724
341.5535
339.8432
338.1415
336.4483
334.7635
333.0872
331.4193
329.7598
328.1085
326.4655
324.8308
323.2042
321.5858
319.9755
slope
dP/dt
172.756
171.891
171.0302
170.1738
169.3217
168.4738
167.6302
166.7908
165.9556
165.1246
164.2977
163.475
162.6564
161.842
161.0315
160.2252
mole
dn/dt
1.669348
1.660989
1.652671
1.644396
1.636161
1.627969
1.619817
1.611705
1.603635
1.595605
1.587615
1.579665
1.571755
1.563885
1.556054
1.548262
flowrate
dV/dt
634.35211
631.175642
628.01508
624.870344
621.741356
618.628035
615.530304
612.448085
609.381299
606.329871
603.293722
600.272776
597.266958
594.276191
591.3004
588.33951
3/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
548.08022
6.3435211
545.33575
6.31175642
542.60503
6.2801508
539.88798
6.24870344
537.18453
6.21741356
534.49462
6.18628035
531.81818
6.15530304
529.15515
6.12448085
526.50544
6.09381299
523.86901
6.06329871
521.24578
6.03293722
518.63568
6.00272776
516.03865
5.97266958
513.45463
5.94276191
510.88355
5.913004
508.32534
5.8833951
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
356.23
355.37
354.513
353.657
352.804
351.954
351.105
350.259
349.414
348.572
347.732
346.895
346.059
345.226
344.395
343.566
Cd
0.8269
0.8258
0.8248
0.8237
0.8226
0.8216
0.8205
0.8194
0.8184
0.8173
0.8162
0.8152
0.8141
0.813
0.812
0.8109
P, psig
345
342.9088
340.8302
338.7642
336.7108
334.6698
332.6412
330.6249
328.6208
326.6288
324.6489
322.6811
320.7251
318.781
316.8487
314.9281
slope
dP/dt
209.1236
207.856
206.5961
205.3438
204.0991
202.8619
201.6323
200.4101
199.1953
197.9878
196.7877
195.5949
194.4093
193.2308
192.0596
190.8954
mole
dn/dt
2.020769
2.00852
1.996345
1.984244
1.972217
1.960262
1.94838
1.93657
1.924831
1.913164
1.901567
1.89004
1.878584
1.867197
1.855879
1.844629
flowrate
dV/dt
767.892256
763.237635
758.611229
754.012866
749.442376
744.899591
740.384341
735.896462
731.435785
727.002148
722.595385
718.215334
713.861832
709.53472
705.233837
700.959024
Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
663.45891
7.67892256
659.43732
7.63237635
655.4401
7.58611229
651.46712
7.54012866
647.51821
7.49442376
643.59325
7.44899591
639.69207
7.40384341
635.81454
7.35896462
631.96052
7.31435785
628.12986
7.27002148
624.32241
7.22595385
620.53805
7.18215334
616.77662
7.13861832
613.038
7.0953472
609.32204
7.05233837
605.6286
7.00959024
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
476.195
474.803
473.416
472.033
470.655
469.281
467.911
466.545
465.184
463.827
462.475
461.126
459.782
458.443
457.107
455.776
Cd
0.7869
0.7857
0.7844
0.7832
0.782
0.7807
0.7795
0.7783
0.777
0.7758
0.7746
0.7734
0.7721
0.7709
0.7697
0.7685
85
Table B-15—Cd Calculations for 5/16 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
342.7648
340.544
338.3377
336.1456
333.9677
331.804
329.6543
327.5185
325.3965
323.2883
321.1937
319.1127
317.0452
314.9911
312.9503
slope
dP/dt
223.5228
222.0746
220.6358
219.2063
217.7861
216.3751
214.9732
213.5804
212.1966
210.8218
209.4559
208.0989
206.7506
205.4111
204.0802
202.758
mole
dn/dt
2.159909
2.145915
2.132011
2.118198
2.104475
2.09084
2.077294
2.063835
2.050464
2.037179
2.02398
2.010867
1.997839
1.984895
1.972035
1.959258
flowrate
dV/dt
820.765247
815.447574
810.164354
804.915363
799.70038
794.519185
789.371558
784.257282
779.176142
774.127921
769.112408
764.129389
759.178656
754.259997
749.373207
744.518077
1-1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
709.14117
8.20765247
704.5467
8.15447574
699.982
8.10164354
695.44687
8.04915363
690.94113
7.9970038
686.46458
7.94519185
682.01703
7.89371558
677.59829
7.84257282
673.20819
7.79176142
668.84652
7.74127921
664.51312
7.69112408
660.20779
7.64129389
655.93036
7.59178656
651.68064
7.54259997
647.45845
7.49373207
643.26362
7.44518077
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
476.147
474.659
473.177
471.7
470.227
468.76
467.297
465.84
464.387
462.939
461.497
460.059
458.625
457.197
455.774
454.355
Cd
0.8136
0.8122
0.8108
0.8095
0.8081
0.8068
0.8054
0.804
0.8027
0.8013
0.8
0.7986
0.7973
0.7959
0.7946
0.7932
P, psig
345
342.5969
340.2105
337.8407
335.4874
333.1505
330.8299
328.5255
326.2371
323.9647
321.708
319.4671
317.2419
315.0321
312.8377
310.6586
slope
dP/dt
240.3141
238.6402
236.9779
235.3272
233.688
232.0602
230.4438
228.8386
227.2446
225.6617
224.0898
222.5289
220.9788
219.4396
217.911
216.3932
mole
dn/dt
2.322164
2.305989
2.289926
2.273975
2.258136
2.242406
2.226787
2.211276
2.195873
2.180577
2.165388
2.150305
2.135326
2.120453
2.105682
2.091015
flowrate
dV/dt
882.422342
876.275718
870.171908
864.110615
858.091543
852.114398
846.178887
840.284721
834.431611
828.619272
822.847419
817.115771
811.424048
805.77197
800.159264
794.585653
1-1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
762.4129
8.82422342
757.10222
8.76275718
751.82853
8.70171908
746.59157
8.64110615
741.39109
8.58091543
736.22684
8.52114398
731.09856
8.46178887
726.006
8.40284721
720.94891
8.34431611
715.92705
8.28619272
710.94017
8.22847419
705.98803
8.17115771
701.07038
8.11424048
696.18698
8.0577197
691.3376
8.00159264
686.522
7.94585653
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
476.091
474.492
472.898
471.311
469.729
468.153
466.583
465.018
463.459
461.906
460.358
458.816
457.28
455.749
454.223
452.704
Cd
0.8436
0.8421
0.8406
0.8391
0.8375
0.836
0.8345
0.833
0.8315
0.83
0.8285
0.827
0.8255
0.824
0.8225
0.821
86
Table B-16—Cd Calculations for 5/16 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
342.5729
340.1628
337.7697
335.3935
333.0339
330.691
328.3645
326.0545
323.7606
321.4829
319.2212
316.9755
314.7455
312.5312
310.3325
slope
dP/dt
242.7122
241.0047
239.3092
237.6256
235.9539
234.2939
232.6456
231.0089
229.3838
227.77
226.1676
224.5765
222.9966
221.4278
219.87
218.3232
mole
dn/dt
2.345337
2.328837
2.312453
2.296185
2.280031
2.263991
2.248063
2.232248
2.216544
2.20095
2.185466
2.170091
2.154824
2.139665
2.124612
2.109665
flowrate
dV/dt
891.228033
884.958122
878.73232
872.550318
866.411807
860.316482
854.264038
848.254173
842.286589
836.360988
830.477074
824.634554
818.833137
813.072534
807.352458
801.672623
P-BV
3
ft /sec
MSCF/D
dV/dt
dV/dt
770.02102
8.91228033
764.60382
8.84958122
759.22472
8.7873232
753.88347
8.72550318
748.5798
8.66411807
743.31344
8.60316482
738.08413
8.54264038
732.89161
8.48254173
727.73561
8.42286589
722.61589
8.36360988
717.53219
8.30477074
712.48425
8.24634554
707.47183
8.18833137
702.49467
8.13072534
697.55252
8.07352458
692.64515
8.01672623
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
476.083
474.468
472.858
471.255
469.658
468.066
466.481
464.901
463.327
461.758
460.196
458.639
457.088
455.542
454.002
452.468
Cd
0.8478
0.8463
0.8448
0.8432
0.8417
0.8401
0.8386
0.837
0.8355
0.834
0.8325
0.8309
0.8294
0.8279
0.8264
0.8248
P, psig
345
342.6037
340.2241
337.8609
335.5142
333.1838
330.8696
328.5715
326.2893
324.023
321.7724
319.5374
317.318
315.114
312.9253
310.7518
slope
dP/dt
239.6289
237.9645
236.3117
234.6703
233.0403
231.4217
229.8143
228.2181
226.6329
225.0588
223.4956
221.9432
220.4017
218.8708
217.3506
215.8409
mole
dn/dt
2.315543
2.29946
2.283488
2.267628
2.251877
2.236236
2.220704
2.205279
2.189962
2.174751
2.159646
2.144645
2.129749
2.114956
2.100266
2.085678
flowrate
dV/dt
879.906317
873.794694
867.725521
861.698502
855.713346
849.769762
843.86746
838.006154
832.18556
826.405394
820.665375
814.965226
809.304668
803.683427
798.10123
792.557806
P
3
ft /sec
dV/dt
8.79906317
8.73794694
8.67725521
8.61698502
8.55713346
8.49769762
8.4386746
8.38006154
8.3218556
8.26405394
8.20665375
8.14965226
8.09304668
8.03683427
7.9810123
7.92557806
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
476.093
474.498
472.91
471.327
469.749
468.178
466.612
465.052
463.497
461.948
460.404
458.867
457.334
455.808
454.287
452.771
Cd
0.8424
0.8409
0.8394
0.8379
0.8364
0.8349
0.8333
0.8318
0.8303
0.8288
0.8273
0.8258
0.8243
0.8228
0.8214
0.8199
MSCF/D
dV/dt
760.23906
754.95862
749.71485
744.50751
739.33633
734.20107
729.10149
724.03732
719.00832
714.01426
709.05488
704.12995
699.23923
694.38248
689.55946
684.76994
87
Table B-17—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
344.21
343.43
342.65
341.87
341.09
340.31
339.54
338.76
337.99
337.22
336.46
335.69
334.92
334.16
333.4
P,
psig
345
343.45
341.91
340.37
338.85
337.32
335.81
334.3
332.8
331.31
329.82
328.34
326.86
325.4
323.94
322.48
slope
mole
flowrate
dP/dt
78.5704
78.39146
78.21293
78.03481
77.85709
77.67978
77.50287
77.32637
77.15026
76.97456
76.79926
76.62436
76.44985
76.27574
76.10203
75.92872
dn/dt
0.759229
0.7575
0.755774
0.754053
0.752336
0.750623
0.748913
0.747207
0.745506
0.743808
0.742114
0.740424
0.738738
0.737055
0.735377
0.733702
dV/dt
288.506857
287.849811
287.194261
286.540204
285.887637
285.236555
284.586957
283.938838
283.292195
282.647024
282.003323
281.361088
280.720315
280.081002
279.443145
278.80674
slope
mole
flowrate
dP/dt
154.9012
154.2057
153.5134
152.8241
152.1379
151.4549
150.7748
150.0979
149.4239
148.7531
148.0852
147.4203
146.7584
146.0995
145.4435
144.7905
dn/dt
1.496816
1.490095
1.483405
1.476745
1.470114
1.463514
1.456943
1.450401
1.443889
1.437406
1.430952
1.424528
1.418132
1.411764
1.405426
1.399115
dV/dt
568.790079
566.236274
563.693935
561.163011
558.643451
556.135203
553.638217
551.152442
548.677828
546.214325
543.761883
541.320452
538.889982
536.470426
534.061732
531.663854
1/4 Fully Open
3
ft /sec
MSCF/D
Constants
dV/dt
dV/dt
249.269925
2.88506857
248.702237
2.87849811
248.135842
2.87194261
247.570736
2.86540204
247.006918
2.85887637
246.444384
2.85236555
245.883131
2.84586957
245.323156
2.83938838
244.764456
2.83292195
244.207029
2.82647024
243.650871
2.82003323
243.09598
2.81361088
242.542353
2.80720315
241.989986
2.80081002
241.438877
2.79443145
240.889024
2.7880674
1/2 Fully Open
3
ft /sec
MSCF/D
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
5.68790079
5.66236274
5.63693935
5.61163011
5.58643451
5.56135203
5.53638217
5.51152442
5.48677828
5.46214325
5.43761883
5.41320452
5.38889982
5.36470426
5.34061732
5.31663854
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
491.434628
489.228141
487.03156
484.844842
482.667942
480.500816
478.34342
476.19571
474.057644
471.929177
469.810267
467.70087
465.600945
463.510448
461.429337
459.35757
88
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
162.3496
162.1715
161.9936
161.8159
161.6385
161.4612
161.2842
161.1073
160.9307
160.7543
160.5781
160.4021
160.2263
160.0507
159.8753
159.7001
Cd
0.8261
0.8256
0.8251
0.8246
0.8241
0.8236
0.8231
0.8227
0.8222
0.8217
0.8212
0.8207
0.8202
0.8197
0.8193
0.8188
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
335.8411
335.1145
334.3895
333.6662
332.9446
332.2246
331.5064
330.7898
330.0749
329.3617
328.6501
327.9402
327.2319
326.5253
325.8204
325.1171
Cd
0.8064
0.8055
0.8046
0.8036
0.8027
0.8018
0.8008
0.7999
0.7989
0.798
0.7971
0.7962
0.7952
0.7943
0.7934
0.7924
Constants
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-18—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
342.48
339.99
337.51
335.04
332.6
330.18
327.77
325.38
323
320.65
318.31
315.99
313.68
311.4
309.12
P,
psig
345
342.07
339.16
336.28
333.43
330.6
327.79
325
322.24
319.51
316.79
314.1
311.43
308.79
306.17
303.56
slope
mole
flowrate
dP/dt
251.618
249.7828
247.9611
246.1527
244.3574
242.5752
240.8061
239.0498
237.3063
235.5756
233.8575
232.1519
230.4587
228.778
227.1094
225.453
dn/dt
2.431393
2.413661
2.396057
2.378582
2.361234
2.344013
2.326918
2.309947
2.2931
2.276376
2.259773
2.243292
2.226931
2.21069
2.194566
2.178561
dV/dt
923.929487
917.191016
910.501691
903.861152
897.269045
890.725016
884.228714
877.779792
871.377904
865.022706
858.713858
852.451023
846.233864
840.062048
833.935245
827.853127
slope
mole
flowrate
dP/dt
293.0334
290.5445
288.0767
285.6298
283.2038
280.7983
278.4133
276.0485
273.7039
271.3791
269.0741
266.7886
264.5226
262.2758
260.0481
257.8394
dn/dt
2.831593
2.807542
2.783695
2.760051
2.736608
2.713364
2.690318
2.667467
2.64481
2.622346
2.600073
2.577988
2.556092
2.534381
2.512854
2.491511
dV/dt
1076.00522
1066.86593
1057.80426
1048.81957
1039.91118
1031.07847
1022.32077
1013.63746
1005.02791
996.491479
988.027557
979.635524
971.314772
963.064693
954.884689
946.774163
3/4 Fully Open
3
ft /sec
MSCF/D
Constants
dV/dt
dV/dt
798.275077
9.23929487
792.453038
9.17191016
786.673461
9.10501691
780.936036
9.03861152
775.240455
8.97269045
769.586414
8.90725016
763.973609
8.84228714
758.40174
8.77779792
752.870509
8.71377904
747.379618
8.65022706
741.928773
8.58713858
736.517683
8.52451023
731.146058
8.46233864
725.81361
8.40062048
720.520052
8.33935245
715.265102
8.27853127
Fully Open
3
ft /sec
MSCF/D
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
10.7600522
10.6686593
10.5780426
10.4881957
10.3991118
10.3107847
10.2232077
10.1363746
10.0502791
9.96491479
9.88027557
9.79635524
9.71314772
9.63064693
9.54884689
9.46774163
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
929.668507
921.77216
913.942883
906.180106
898.483263
890.851795
883.285147
875.782768
868.344112
860.968638
853.655809
846.405093
839.215963
832.087895
825.020371
818.012877
89
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
512.1812
510.3799
508.5854
506.7977
505.0168
503.2426
501.4751
499.7144
497.9604
496.213
494.4722
492.7381
491.0106
489.2896
487.5751
485.8672
Cd
0.8323
0.8307
0.8291
0.8276
0.826
0.8244
0.8229
0.8213
0.8197
0.8182
0.8166
0.8151
0.8135
0.812
0.8104
0.8089
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
687.7832
684.9654
682.16
679.367
676.5863
673.818
671.0619
668.3179
665.5861
662.8664
660.1588
657.4631
654.7793
652.1075
649.4475
646.7992
Cd
0.7751
0.7734
0.7717
0.77
0.7682
0.7665
0.7649
0.7632
0.7615
0.7598
0.7581
0.7564
0.7547
0.7531
0.7514
0.7497
Constants
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-19—Cd Calculations for 3/8 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
341.97
338.97
336
333.05
330.13
327.24
324.37
321.52
318.7
315.91
313.14
310.39
307.67
304.97
302.29
P,
psig
345
341.8
338.64
335.5
332.39
329.31
326.26
323.23
320.24
317.27
314.33
311.42
308.53
305.67
302.84
300.03
slope
mole
flowrate
dP/dt
302.61
299.9558
297.3248
294.7168
292.1318
289.5694
287.0295
284.5119
282.0163
279.5427
277.0907
274.6603
272.2512
269.8632
267.4961
265.1498
dn/dt
2.924132
2.898483
2.87306
2.847859
2.82288
2.79812
2.773576
2.749249
2.725134
2.701231
2.677538
2.654052
2.630773
2.607697
2.584825
2.562152
dV/dt
1111.17008
1101.42367
1091.76275
1082.18656
1072.69437
1063.28544
1053.95904
1044.71445
1035.55094
1026.4678
1017.46434
1008.53985
999.693637
990.925019
982.233313
973.617844
slope
mole
flowrate
dP/dt
319.7045
316.7418
313.8066
310.8987
308.0176
305.1633
302.3354
299.5337
296.758
294.008
291.2835
288.5842
285.91
283.2605
280.6356
278.035
dn/dt
3.089316
3.060688
3.032325
3.004225
2.976386
2.948804
2.921478
2.894405
2.867583
2.84101
2.814683
2.7886
2.762759
2.737157
2.711792
2.686662
dV/dt
1173.94001
1163.06134
1152.28349
1141.60552
1131.02649
1120.5455
1110.16163
1099.87399
1089.68168
1079.58382
1069.57954
1059.66797
1049.84824
1040.11951
1030.48093
1020.93168
1-1/4 Fully Open
3
ft /sec
MSCF/D
Constants
dV/dt
dV/dt
960.050953
11.1117008
951.630053
11.0142367
943.283015
10.9176275
935.009191
10.8218656
926.80794
10.7269437
918.678624
10.6328544
910.620613
10.5395904
902.633281
10.4471445
894.716009
10.3555094
886.868181
10.264678
879.089189
10.1746434
871.378429
10.0853985
863.735303
9.99693637
856.159216
9.90925019
848.649582
9.82233313
841.205817
9.73617844
1-1/2 Fully Open
3
ft /sec
MSCF/D
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
11.7394001
11.6306134
11.5228349
11.4160552
11.3102649
11.205455
11.1016163
10.9987399
10.8968168
10.7958382
10.6957954
10.5966797
10.4984824
10.4011951
10.3048093
10.2093168
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
dV/dt
1014.28417
1004.885
995.572937
986.347166
977.206888
968.151311
959.17965
950.291128
941.484974
932.760424
924.116724
915.553122
907.068878
898.663256
890.335527
882.084969
90
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
687.737
684.8271
681.9305
679.0471
676.1768
673.3197
670.4756
667.6445
664.8263
662.021
659.2285
656.4488
653.6818
650.9274
648.1857
645.4566
Cd
0.7877
0.7859
0.7841
0.7823
0.7805
0.7787
0.7769
0.7752
0.7734
0.7716
0.7699
0.7681
0.7663
0.7646
0.7628
0.7611
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
687.6547
684.5804
681.521
678.4763
675.4463
672.4309
669.43
666.4437
663.4717
660.5141
657.5708
654.6418
651.7268
648.826
645.9392
643.0664
Cd
0.8097
0.8077
0.8058
0.8038
0.8019
0.7999
0.798
0.7961
0.7942
0.7922
0.7903
0.7884
0.7865
0.7846
0.7827
0.7808
Constants
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-20—Cd Calculations for 3/8 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P,
psig
345
341.67
338.38
335.12
331.88
328.68
325.51
322.37
319.27
316.19
313.14
310.12
307.13
304.17
301.23
298.33
P,
psig
345
341.58
338.2
334.85
331.54
328.26
325.01
321.79
318.6
315.45
312.33
309.23
306.17
303.14
300.14
297.17
3
slope
mole
flowrate
ft /sec
dP/dt
332.6905
329.4823
326.305
323.1584
320.0421
316.9559
313.8994
310.8724
307.8746
304.9057
301.9655
299.0536
296.1697
293.3137
290.4852
287.684
dn/dt
3.2148
3.183799
3.153097
3.122691
3.092579
3.062756
3.033222
3.003972
2.975004
2.946315
2.917903
2.889765
2.861899
2.834301
2.806969
2.779901
dV/dt
1221.62416
1209.84379
1198.17702
1186.62276
1175.17991
1163.84742
1152.6242
1141.50921
1130.5014
1119.59975
1108.80322
1098.11081
1087.5215
1077.03431
1066.64825
1056.36234
dV/dt
12.2162416
12.0984379
11.9817702
11.8662276
11.7517991
11.6384742
11.526242
11.4150921
11.305014
11.1959975
11.0880322
10.9811081
10.875215
10.7703431
10.6664825
10.5636234
slope
mole
flowrate
dP/dt
341.5729
338.1911
334.8427
331.5276
328.2452
324.9954
321.7777
318.5919
315.4376
312.3146
309.2225
306.161
303.1298
300.1286
297.1571
294.2151
dn/dt
3.300631
3.267952
3.235598
3.203563
3.171846
3.140442
3.10935
3.078565
3.048085
3.017907
2.988028
2.958444
2.929154
2.900153
2.87144
2.843011
dV/dt
1254.23973
1241.82192
1229.52706
1217.35393
1205.30132
1193.36804
1181.5529
1169.85475
1158.27241
1146.80475
1135.45062
1124.2089
1113.07849
1102.05828
1091.14717
1080.34409
P-BV
MSCF/D
Constants
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
ft /sec
dV/dt
1055.48328
1045.30504
1035.22495
1025.24206
1015.35545
1005.56417
995.867308
986.263957
976.753213
967.334183
958.005983
948.767737
939.618577
930.557644
921.584087
912.697065
P
MSCF/D
dV/dt
12.5423973
12.4182192
12.2952706
12.1735393
12.0530132
11.9336804
11.815529
11.6985475
11.5827241
11.4680475
11.3545062
11.242089
11.1307849
11.0205828
10.9114717
10.8034409
dV/dt
1083.66312
1072.93414
1062.31138
1051.79379
1041.38034
1031.06998
1020.86171
1010.7545
1000.74736
990.8393
981.029334
971.316494
961.699817
952.178352
942.751155
933.417294
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
3
91
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
687.5921
684.393
681.2099
678.0428
674.8916
671.7562
668.6366
665.5327
662.4443
659.3715
656.3141
653.2721
650.2454
647.2339
644.2376
641.2564
Cd
0.826
0.8239
0.8218
0.8198
0.8177
0.8157
0.8136
0.8116
0.8095
0.8075
0.8054
0.8034
0.8014
0.7994
0.7974
0.7953
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
687.5493
684.2648
680.9972
677.7464
674.5124
671.2951
668.0943
664.9101
661.7423
658.5909
655.4557
652.3367
649.2338
646.147
643.0761
640.0212
Cd
0.837
0.8348
0.8326
0.8305
0.8284
0.8262
0.8241
0.822
0.8198
0.8177
0.8156
0.8135
0.8114
0.8093
0.8072
0.8051
Constants
CK2
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
1.4286
CK3
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
1.7143
Table B-21—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
343.6709
342.3469
341.0279
339.7141
338.4053
337.1016
335.8029
334.5092
333.2205
331.9367
330.6579
329.3841
328.1151
326.851
325.5918
slope
dP/dt
132.9133
132.4013
131.8912
131.3831
130.8769
130.3727
129.8704
129.3701
128.8717
128.3752
127.8806
127.3879
126.8972
126.4083
125.9213
125.4362
mole
dn/dt
1.284346
1.279398
1.274469
1.269559
1.264668
1.259796
1.254942
1.250108
1.245292
1.240494
1.235715
1.230954
1.226212
1.221488
1.216782
1.212094
flowrate
dV/dt
488.051533
486.171285
484.298281
482.432493
480.573893
478.722453
476.878146
475.040945
473.210821
471.387748
469.571699
467.762646
465.960562
464.165421
462.377196
460.59586
1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
421.676524
4.88051533
420.05199
4.86171285
418.433715
4.84298281
416.821674
4.82432493
415.215843
4.80573893
413.616199
4.78722453
412.022718
4.76878146
410.435376
4.75040945
408.854149
4.73210821
407.279014
4.71387748
405.709948
4.69571699
404.146926
4.67762646
402.589926
4.65960562
401.038924
4.64165421
399.493897
4.62377196
397.954823
4.6059586
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
277.3751
276.8601
276.3463
275.8334
275.3216
274.8108
274.301
273.7922
273.2845
272.7778
272.2721
271.7674
271.2637
270.761
270.2593
269.7587
Cd
0.82199
0.82116
0.82034
0.81952
0.8187
0.81788
0.81706
0.81625
0.81543
0.81461
0.8138
0.81298
0.81217
0.81135
0.81054
0.80973
P, psig
345
342.4838
339.986
337.5064
335.0449
332.6013
330.1755
327.7675
325.377
323.0039
320.6482
318.3096
315.9881
313.6835
311.3957
309.1246
slope
dP/dt
251.618
249.7828
247.9611
246.1527
244.3574
242.5752
240.8061
239.0498
237.3063
235.5756
233.8575
232.1519
230.4587
228.778
227.1094
225.453
mole
dn/dt
2.431393
2.413661
2.396057
2.378582
2.361234
2.344013
2.326918
2.309947
2.2931
2.276376
2.259773
2.243292
2.226931
2.21069
2.194566
2.178561
flowrate
dV/dt
923.929487
917.191016
910.501691
903.861152
897.269045
890.725016
884.228714
877.779792
871.377904
865.022706
858.713858
852.451023
846.233864
840.062048
833.935245
827.853127
1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
798.275077
9.23929487
792.453038
9.17191016
786.673461
9.10501691
780.936036
9.03861152
775.240455
8.97269045
769.586414
8.90725016
763.973609
8.84228714
758.40174
8.77779792
752.870509
8.71377904
747.379618
8.65022706
741.928773
8.58713858
736.517683
8.52451023
731.146058
8.46233864
725.81361
8.40062048
720.520052
8.33935245
715.265102
8.27853127
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
585.0157
582.9582
580.9085
578.8666
576.8324
574.806
572.7872
570.7761
568.7726
566.7767
564.7884
562.8077
560.8345
558.8688
556.9105
554.9597
Cd
0.77876
0.77728
0.7758
0.77433
0.77286
0.77139
0.76993
0.76847
0.76701
0.76555
0.76409
0.76264
0.76119
0.75974
0.7583
0.75685
92
Table B-22—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
340.35
335.7626
331.2371
326.7726
322.3682
318.0232
313.7368
309.5081
305.3365
301.2211
297.1611
293.1559
289.2046
285.3066
281.4611
slope
dP/dt
465.0028
458.7354
452.5524
446.4527
440.4353
434.4989
428.6426
422.8652
417.1657
411.543
405.9961
400.5239
395.1255
389.7999
384.546
379.363
mole
dn/dt
4.493339
4.432776
4.37303
4.314089
4.255942
4.198579
4.141989
4.086162
4.031087
3.976755
3.923155
3.870277
3.818112
3.76665
3.715882
3.665798
flowrate
dV/dt
1707.46884
1684.45499
1661.75133
1639.35367
1617.2579
1595.45995
1573.95579
1552.74148
1531.81309
1511.16679
1490.79877
1470.70527
1450.8826
1431.32711
1412.03519
1393.0033
3/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1475.25308
17.0746884
1455.36911
16.8445499
1435.75315
16.6175133
1416.40157
16.3935367
1397.31083
16.172579
1378.47739
15.9545995
1359.8978
15.7395579
1341.56864
15.5274148
1323.48651
15.3181309
1305.64811
15.1116679
1288.05014
14.9079877
1270.68936
14.7070527
1253.56257
14.508826
1236.66662
14.3132711
1219.99841
14.1203519
1203.55485
13.930033
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
902.3635
896.4904
890.6585
884.8676
879.1173
873.4075
867.7377
862.1079
856.5177
850.9669
845.4552
839.9823
834.548
829.152
823.7942
818.4742
Cd
0.85242
0.84942
0.84643
0.84346
0.84049
0.83753
0.83458
0.83164
0.82871
0.82578
0.82287
0.81996
0.81706
0.81418
0.8113
0.80842
P, psig
345
339.85
334.7769
329.7796
324.8568
320.0075
315.2306
310.525
305.8897
301.3235
296.8255
292.3946
288.0299
283.7304
279.495
275.3229
slope
dP/dt
514.9975
507.3099
499.7371
492.2772
484.9288
477.69
470.5593
463.5351
456.6157
449.7996
443.0852
436.4711
429.9557
423.5375
417.2152
410.9872
mole
dn/dt
4.97644
4.902154
4.828977
4.756893
4.685884
4.615936
4.547032
4.479156
4.412294
4.346429
4.281548
4.217636
4.154677
4.092658
4.031565
3.971384
flowrate
dV/dt
1891.04703
1862.81849
1835.01134
1807.61927
1780.6361
1754.05572
1727.87211
1702.07937
1676.67164
1651.64318
1626.98834
1602.70153
1578.77726
1555.21012
1531.99477
1509.12598
Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1633.86463
18.9104703
1609.47518
18.6281849
1585.4498
18.3501134
1561.78305
18.0761927
1538.46959
17.806361
1515.50414
17.5405572
1492.88151
17.2787211
1470.59657
17.0207937
1448.64429
16.7667164
1427.01971
16.5164318
1405.71792
16.2698834
1384.73412
16.0270153
1364.06355
15.7877726
1343.70154
15.5521012
1323.64348
15.3199477
1303.88484
15.0912598
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
1221.27
1212.463
1203.726
1195.056
1186.454
1177.918
1169.45
1161.047
1152.71
1144.438
1136.23
1128.087
1120.007
1111.991
1104.037
1096.145
Cd
0.7711
0.7681
0.76511
0.76212
0.75915
0.75619
0.75324
0.75029
0.74736
0.74444
0.74152
0.73862
0.73573
0.73284
0.72997
0.7271
93
Table B-23—Cd Calculations for 1/2 inch Port Size at Different Set Ball Positions
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
P, psig
345
339.2796
333.654
328.1216
322.6811
317.3307
312.069
306.8946
301.806
296.8018
291.8805
287.0408
282.2814
277.6009
272.998
268.4714
slope
dP/dt
572.0444
562.5593
553.2315
544.0584
535.0374
526.1659
517.4416
508.8619
500.4244
492.1269
483.9669
475.9423
468.0507
460.2899
452.6579
445.1523
mole
dn/dt
5.527686
5.436031
5.345896
5.257256
5.170085
5.08436
5.000056
4.917151
4.835619
4.75544
4.67659
4.599047
4.522791
4.447798
4.374049
4.301523
flowrate
dV/dt
2100.5205
2065.69178
2031.44055
1997.75724
1964.63244
1932.05687
1900.02144
1868.51719
1837.53532
1807.06715
1777.10418
1747.63802
1718.66044
1690.16334
1662.13874
1634.57883
1-1/4 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1814.84971
21.005205
1784.7577
20.6569178
1755.16464
20.3144055
1726.06226
19.9775724
1697.44242
19.6463244
1669.29714
19.3205687
1641.61853
19.0002144
1614.39886
18.6851719
1587.63051
18.3753532
1561.30602
18.0706715
1535.41801
17.7710418
1509.95925
17.4763802
1484.92262
17.1866044
1460.30112
16.9016334
1436.08787
16.6213874
1412.27611
16.3457883
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
1220.78
1210.999
1201.302
1191.689
1182.159
1172.711
1163.346
1154.062
1144.858
1135.734
1126.689
1117.723
1108.835
1100.024
1091.29
1082.632
Cd
0.81285
0.80933
0.80583
0.80233
0.79886
0.79539
0.79194
0.7885
0.78507
0.78165
0.77825
0.77486
0.77148
0.76812
0.76477
0.76143
P, psig
345
339.1371
333.3738
327.7085
322.1394
316.665
311.2836
305.9937
300.7936
295.682
290.6572
285.7177
280.8623
276.0893
271.3975
266.7854
slope
dP/dt
586.2911
576.3277
566.5336
556.906
547.442
538.1388
528.9937
520.004
511.1671
502.4803
493.9412
485.5472
477.2958
469.1847
461.2114
453.3736
mole
dn/dt
5.665352
5.569076
5.474435
5.381403
5.289952
5.200054
5.111685
5.024817
4.939426
4.855485
4.772972
4.69186
4.612127
4.533749
4.456702
4.380965
flowrate
dV/dt
2152.83391
2116.24876
2080.28535
2044.93309
2010.18161
1976.02069
1942.4403
1909.43058
1876.98182
1845.08449
1813.72922
1782.90681
1752.60819
1722.82446
1693.54687
1664.76683
1-1/2 Fully Open
3
ft /sec
MSCF/D
dV/dt
dV/dt
1860.0485
21.5283391
1828.43893
21.1624876
1797.36654
20.8028535
1766.82219
20.4493309
1736.79691
20.1018161
1707.28188
19.7602069
1678.26842
19.424403
1649.74802
19.0943058
1621.71229
18.7698182
1594.153
18.4508449
1567.06205
18.1372922
1540.43148
17.8290681
1514.25347
17.5260819
1488.52033
17.2282446
1463.2245
16.9354687
1438.35854
16.6476683
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
1220.658
1210.633
1200.697
1190.849
1181.088
1171.414
1161.826
1152.322
1142.904
1133.569
1124.318
1115.148
1106.061
1097.054
1088.128
1079.282
Cd
0.82295
0.8193
0.81566
0.81204
0.80843
0.80483
0.80125
0.79768
0.79413
0.79059
0.78706
0.78354
0.78004
0.77655
0.77308
0.76962
94
Table B-24—Cd Calculations for 1/2 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
Time,
sec
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
3
P, psig
345
338.8456
332.8009
326.8641
321.0332
315.3063
309.6815
304.1571
298.7313
293.4022
288.1682
283.0276
277.9787
273.0198
268.1494
263.3659
slope
dP/dt
615.4444
604.4655
593.6825
583.0918
572.69
562.4738
552.4399
542.5849
532.9058
523.3993
514.0624
504.892
495.8853
487.0392
478.3509
469.8176
mole
dn/dt
5.947061
5.840972
5.736775
5.634437
5.533924
5.435205
5.338246
5.243018
5.149488
5.057626
4.967403
4.87879
4.791757
4.706277
4.622322
4.539865
flowrate
dV/dt
2259.8833
2219.56931
2179.97449
2141.08599
2102.89123
2065.37782
2028.53361
1992.34667
1956.80526
1921.89787
1887.6132
1853.94013
1820.86775
1788.38535
1756.4824
1725.14857
ft /sec
dV/dt
22.598833
22.1956931
21.7997449
21.4108599
21.0289123
20.6537782
20.2853361
19.9234667
19.5680526
19.2189787
18.876132
18.5394013
18.2086775
17.8838535
17.564824
17.2514857
P, psig
345
338.8117
332.7343
326.766
320.9048
315.1486
309.4958
303.9443
298.4924
293.1383
287.8802
282.7164
277.6453
272.6651
267.7743
262.9712
slope
dP/dt
618.8327
607.7326
596.8315
586.1261
575.6126
565.2878
555.1481
545.1903
535.4111
525.8074
516.3759
507.1135
498.0173
489.0843
480.3115
471.6961
mole
dn/dt
5.979802
5.872542
5.767205
5.663757
5.562166
5.462396
5.364416
5.268194
5.173697
5.080896
4.989759
4.900256
4.81236
4.726039
4.641268
4.558016
flowrate
dV/dt
2272.32493
2231.56583
2191.53783
2152.22783
2113.62293
2075.7105
2038.47811
2001.91356
1966.00488
1930.7403
1896.10827
1862.09743
1828.69666
1795.895
1763.68171
1732.04624
ft /sec
dV/dt
22.7232493
22.3156583
21.9153783
21.5222783
21.1362293
20.757105
20.3847811
20.0191356
19.6600488
19.307403
18.9610827
18.6209743
18.2869666
17.95895
17.6368171
17.3204624
3
P-BV
MSCF/D
dV/dt
1952.53917
1917.70789
1883.49796
1849.8983
1816.89802
1784.48644
1752.65304
1721.38752
1690.67975
1660.51976
1630.89781
1601.80427
1573.22974
1545.16494
1517.6008
1490.52836
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
1220.407
1209.884
1199.459
1189.13
1178.898
1168.761
1158.719
1148.77
1138.915
1129.151
1119.479
1109.897
1100.405
1091.001
1081.686
1072.459
Cd
0.84325
0.83932
0.83541
0.83151
0.82763
0.82376
0.81991
0.81608
0.81226
0.80845
0.80466
0.80089
0.79713
0.79338
0.78965
0.78594
CK1
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
2333.63
Constants
CK2
CK3
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
1.4286 1.7143
CK4
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
0.076171
Q
1220.378
1209.797
1199.315
1188.931
1178.644
1168.453
1158.358
1148.358
1138.452
1128.638
1118.917
1109.288
1099.749
1090.3
1080.94
1071.668
Cd
0.84558
0.84162
0.83767
0.83374
0.82983
0.82593
0.82205
0.81818
0.81433
0.81049
0.80667
0.80287
0.79908
0.7953
0.79154
0.7878
P
MSCF/D
dV/dt
1963.28874
1928.07288
1893.48869
1859.52484
1826.17021
1793.41387
1761.24509
1729.65332
1698.62822
1668.15962
1638.23754
1608.85218
1579.99391
1551.65328
1523.821
1496.48795
95
Appendix C
Data Acquisition System (DAQ)
National Instrument DAQ has been used for testing. The programmable software is called Lab View
8.5. This version of Lab View is pretty much the latest version available in the academia. The language of
programming is different from ordinary computer programming languages such as basic, C, or
FORTRAN.
Each program in Lab View is called Virtual Instrument, VI. Each VI can be used for different application.
There are some built-in VIs in Lab View to facilitate the programming experience. The DAQ part of the
experiment consists of a high speed USB, NI-9237, which fits inside a Hi-Speed USB carrier chassis, NI9162. Detail about the NI-9237 has been noted as well. Fig. C-1 and Fig. C-2 displays the instruments
respectively. More detail specification can be found at NI website [28].

24-bit resolution, ±25 mV/V analog inputs with RJ50 connectors

4 simultaneously sampled analog inputs; 50 kS/s/ch maximum sampling rate

Programmable half- and full-bridge completion; up to 10 V internal excitation

Smart-sensor (TEDS) compatible

1,000 Vrms transient isolation

-40 to 70 °C operating range
Fig. C-1—NI 9237 with 4 Channel, ±25mV/V, 24 Bit Resolution with Max. Speed Rate of 50,000 Samples per
Second per Channel [37]
96
Fig. C-2—NI USB-9162 Chassis [37]
As it has been noted in the specification of the device pictured in Fig. C-1; this device is capable of taking
reading at the rate of up to 50KS/sec/Channel. In this setup, the device has been set to record from 100
Ks/sec to 10KS/sec. the chassis is compatible with all Windows™ operating systems. The wiring to this
device is very simple. The operator needs to connect the Pressure transducers to the device (NI 9237
inside the chassis) and plug in the chassis to the computer.
Procedure to Install the DAQ and MAX on the Computer
After plugging the chassis into the computer we need to assure that the device is reachable through
Measurement and Automation Explorer (MAX). In order to do that, we need to install MAX on the
computer. It usually comes with the Lab View programs or can be downloaded from NI website. When
the MAX has been opened, in the configuration section, look for the installed instrument which is NI
9237. Fig. C-3 displays the screen shot. You can run ―self-test‖ on your device to assure proper working.
After assuring of setting the proper device and if the device is functioning properly, open the National
Instrument Lab View programming software. You can open the pre-existing Vis or ask for a new VI and
start to generate your own. Each VI has two correspondent screens. One screen is called ―Front Panel‖
that allows the user to navigates the program. The second screen is called ―Block Diagram‖ and is the
heart of the program. The user can switch between screens with hitting ―Ctrl + E‖. Fig. C-4, and C-5
display the front panel and the block diagram of the program developed for this research respectively.
97
Fig.C-3—Display Shot of MAX with NI USB-9237 Device
98
Fig. C-4—Front Panel View of the Developed Program
99
Fig.C-5—Block Diagram of the Developed Program
100
Note that DAQ Assistant from NI has been adopted in the developed program to measure the pressure
points with time (pressure decay with time). This sub-VI will collect the changes in the voltage in the set
transducer with respect to the time. DAQ assistant has to be programmed and verified accordingly to
measure the right pressure at each time segment otherwise the collected data is useless. Double clicking
on this Sub-VI module will open another screen to change the set values. The excitation voltage in this
experiment is 10 volts which is 2.5 volts in full bridge architecture. The DAQ assistant has been set for
continuous sampling at a rate of 10,000 samples/ sec/ channel. In the configuration section of this sub-VI,
the user needs to select the channel in which the data is flowing from. Fig. C-6 displays the DAQ
assistant setup.
Fig. C-6—DAQ Assistant Setup
The output readings of DAQ assistant has to be converted to pressure. This is done using the results
published in Appendix A for each pressure transducer. Knowing the source of reading data, inputting the
right coefficients to convert the raw readings to pressure is vital. This program will constantly record the
101
data points and convert the readings into pressure readings based on the values that the user inserted. The
best way to check if the pressure readings are correct is to have an analogue dial pressure gage along with
this program running and eye proofing the readings at each moment. If the digital reading values are not
quite along with the analog gage, it is recommended that the user change the inputs of pressure transducer
calibration values till a good reading agreement resulted.
Since the test time is very short, the program has been designed in two different stages. In the first stage
(or loop) the DAQ assistant is reading the data point continuously and buffer them till the test stopped by
the user. In the second stage, the time will be added to each pressure reading and all the data will be saved
at a pre-set location. In the front panel of this program, user have to define a path to the recorded data
otherwise, the data is not getting recorded. The path usually is a notepad-file type. Then the user can
export the readings to Microsoft Excel, MATLAB, or Origin™ for further calculations.
If the user is intending to automate the program, the start point and end point need to be known.
Automating the recording is very beneficial to the user. As it has been mentioned, we need to know the
starting base. Some blow down tests has been run trying to quantify the pressure drop needed for each
port size (or a range of port sizes). As it has been demonstrated in Fig. C-7, the pressure drop of 3 psi
should be met in 5/16‖ port size. Each the port size gets larger, the pressure drop goes higher too. For the
port range of 3/16‖ and 1/4‖, a minimum pressure drop of 1.5-2 psi and for 3/8‖ to 1/2‖, 4 psi is needed.
Plot of Pressure vs. Time for 5/16" Port ID
1-1/2" J-20 Camco GLV
720
710
y = -389.28x + 709.24
R² = 0.9979
Pressure, psig
700
Test 2
690
680
670
Test 1
660
650
y = -439.12x + 683.57
R² = 0.9995
640
630
0
0.02
0.04
0.06
0.08
0.1
Time, sec
Fig. C-7—Empirical Measurement of Minimum Value for Pressure Drop Increment
102
Knowing the initial upstream pressure (Pid), the port size, the required increment of the pressure drop to
start, we can start the test and sample continuously.
103
Appendix D
Relevancy of Load Rate and Linear Stem Travel to Dome-Charged Pressure, Pbt
This Appendix tried to show that as the Pbt increases, the LR of the bellows assembly increases
although the maximum linear travel of the stem decreases. In other words, Bellows start to stack sooner at
higher set dome pressure than lower which affects the gas passage through at high pressures. Fig. D-1
depicts the actual probe unit that was used for this experiment.
Pressure Gage
Nylon
Bushing
Depth Micrometer
GLV
Digital Ohm-Meter
Gas
Inlet
Valve
Gas
Outlet
Valve
Fig. D-1—Actual Probe Tester to Measure the Linear Stem Travel
Table D-1 through D-3 contains the probe test data at different set PTRO. Fig. D-2 through D-4 depicts
the variations of the maximum linear travel as well as the LR respectively.
104
Table D-1—Probe Test Results for 1/2” Monel Port, 1-1/2” J-20 GLV at Pbt= 149 psig
Increasing Pressure
Stem
Pressure
Stem
Travel
psig
Reading
inch
142
0
0.59
150
0.034
0.624
155
0.041
0.631
160
0.058
0.648
166
0.096
0.686
170
0.113
0.703
176
0.147
0.737
180
0.16
0.75
185
0.175
0.765
190
0.185
0.775
196
0.194
0.784
200
0.199
0.789
209
0.206
0.796
210
0.207
0.797
216
0.211
0.801
220
0.213
0.803
225
0.2165
0.8065
236
0.221
0.811
Decreasing Pressure
Stem
Pressure
Stem
Travel
psig
Reading
inch
140
0
0.59
145
0.005
0.595
152
0.056
0.646
156
0.09
0.68
164
0.136
0.726
170
0.16
0.75
177
0.18
0.77
184
0.19
0.78
194
0.202
0.792
201
0.2075
0.7975
208
0.212
0.802
215
0.215
0.805
222
0.218
0.808
233
0.221
0.811
105
Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV,
PTRO =200 psig
280
260
240
220
200
Pressure, psig
180
160
Inreasing
Pressure
140
120
Max. Linear Travel = 0.18 inch
Load Rate = 172 psi/inch
dPLinear = 39 psi
Min. Travel for Fully Open = .2246 inch
100
80
Decreasing
Pressure
60
40
20
0
0
0.05
0.1
0.15
0.2
0.25
Stem Travel, inch
Fig. D-2—Probe Test Results for 1/2” Monel Port in 1-1/2” J-20 GLV at set Pbt = 149 psig
106
Table D-2—Probe Test Results for 1/2” Monel Port, 1-1/2” J-20 GLV at Pbt= 444 psig
Increasing Pressure
Pressure
psig
Stem
Travel inch
Stem
Reading
434
444
452
464
473
485
495
506
515
525
535
542
560
568
585
597
0
0.051
0.079
0.118
0.145
0.168
0.187
0.197
0.207
0.214
0.219
0.221
0.222
0.2223
0.2225
0.2228
0.645
0.696
0.724
0.763
0.79
0.813
0.832
0.842
0.852
0.859
0.864
0.866
0.867
0.8673
0.8675
0.8678
Decreasing Pressure
Stem
Pressure
Stem
Travel
psig
Reading
inch
426
0
0.645
437
0.063
0.708
448
0.108
0.753
460
0.147
0.792
469
0.1685
0.8135
480
0.1875
0.8325
493
0.203
0.848
516
0.2175
0.8625
525
0.2195
0.8645
533
0.2215
0.8665
546
0.2225
0.8675
558
0.2225
0.8675
571
0.2225
0.8675
590
0.2228
0.8678
608
0.2228
0.8678
107
Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV,
PTRO = 596 psig
640
620
Max. Linear Travel = 0.16 inch
Load Rate = 250 psi/inch
dPLinear = 40 psi
Min. Travel for Fully Open = .2246 inch
600
580
Inreasing
Pressure
Pressure, psig
560
Decreasing
Pressure
540
520
500
480
460
440
420
400
380
0
0.05
0.1
0.15
0.2
Stem Travel, inch
Fig. D-3—Probe Test Results for 1/2” Monel Port in 1-1/2” J-20 GLV at set Pbt = 444 psig
108
0.25
Table D-3—Probe Test Results for 1/2” Monel Port, 1-1/2” J-20 GLV at Pbt= 517 psig
Increasing Pressure
Decreasing Pressure
Pressure
psig
Stem Travel
inch
Stem
Reading
Pressure
psig
Stem Travel
inch
Stem
Reading
501
507
519
528
540
550
560
569
580
590
600
610
620
631
639
650
660
670
684
692
700
710
0
0
0.004
0.025
0.059
0.0895
0.122
0.1295
0.14
0.1495
0.1565
0.163
0.169
0.174
0.1788
0.184
0.1875
0.1915
0.195
0.1955
0.196
0.196
0.644
0.644
0.648
0.669
0.703
0.7335
0.766
0.7735
0.784
0.7935
0.8005
0.807
0.813
0.818
0.8228
0.828
0.8315
0.8355
0.839
0.8395
0.84
0.84
485
494
515
534
545
553
565
574
583
591
601
615
625
635
644
655
665
674
684
695
705
712
725
0
0
0.003
0.0585
0.094
0.114
0.139
0.1485
0.1575
0.1645
0.1705
0.1705
0.178
0.1823
0.1865
0.1895
0.192
0.1943
0.1953
0.1955
0.196
0.196
0.196
0.644
0.644
0.647
0.7025
0.738
0.758
0.783
0.7925
0.8015
0.8085
0.8145
0.822
0.8263
0.8305
0.8335
0.836
0.8383
0.8393
0.8395
0.84
0.84
0.84
0.84
109
Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV,
PTRO = 694 psig
730
720
Max. Linear Travel = 0.13 inch
Load Rate = 361 psi/inch
dPLinear = 47 psi
Min. Travel for Fully Open = .2246 inch
710
700
690
680
670
660
Pressure, psig
650
Increasing
Pressure
640
630
620
610
600
Decreasing
Pressure
590
580
570
560
550
540
530
520
510
500
0
0.05
0.1
0.15
0.2
0.25
Stem Travel, inch
Fig. D-4—Probe Test Results for 1/2” Monel Port in 1-1/2” J-20 GLV at set Pbt = 517 psig
110
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