SIMPLIFIED AND RAPID METHOD FOR DETERMINING FLOW

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

ii

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.

iii

This work is dedicated:

To my unique uncle,

Hossein A. Shahri

iv

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 INTRODUCTION

Objectives 1

Dissertation Overview 2

CHAPTER 2 LITERATURE REVIEW

Gas Lift 3

Gas Lift Valve (GLV) 8

Flow Behavior 12

GLV Performance Models 16

Valve Temperature 24

v

CHAPTER 3 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

CHAPTER 4 BLOW-DOWN TEST

Volumetric Calculations 34

Discharge Coefficient Calculation 37

Flow Area Calculation 38

CHAPTER 5 RESULTS & DISCUSSIONS

Flow Through Ports & Flow Through Ports

inside GLV 43

The Gas Leak Rate 50

Justifying Thornhill-Craver Equation 50

CHAPTER 6 CONCLUSIONS

Conclusions 56

vi

CHAPTER 7 RECOMMENDATIONS

Recommendations 57

APPENDIX A Transducer Calibration Using Dead-Weight

Tester 58

APPENDIX B Measurement of Discharge Coefficient, Cd,

Using Benchmark Valve Testing 61

APPENDIX C Data Acquisition System (DAQ) 96

APPENDIX D Relevancy of LR change with Pbt 104

REFERENCES 111

vii

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.

viii

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) 17

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

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

ix

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) 45

Fig. 5.3—Calculated Equivalent Port Area Based on Polynomial Regression Analysis

(2nd

Trial) 45

Fig. 5.4—Plot of Pressure vs. Time, Flowrate, and Apparent Port Size Open to Flow in a

3/16‖ Monel Sharp-Edged Seat 46

Fig. 5.5—Calculated Equivalent Port Area Based on Exponential Regression Analysis 47

Fig. 5.6—Calculated Equivalent Port Area Based on Previous Exponential Regression

Analysis 48

Fig. 5.7—Calculated Equivalent Port Area Based on Measured Raw Data 49

Fig. 5.8—Effect of Slight Tapered Seat on the Gas Passage in the 1-1/2‖ J-20 Camco

GLV 50

Fig. 5.9—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in

3/16‖ Monel Sharp-edged Seat 52

Fig. 5.10—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only

in 1/4‖ Monel Sharp-edged Seat 52







x


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 63





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 65


with 5/16‖ Port Size when the Ball is at 1-1/2 Fully Open Travel Position

(Beyond Fully Open)

65

Fig. B.6—Combined Plot of Pressure vs. Time in Benchmark Valve Testing for the First

Second 67

Fig. B.7—Plot of Pressure rate Against Ball Position in 5/16‖ Monel J-20 Camco GLV 68

Fig. B.8—Combined Plot of Pressure vs. Time in Benchmark Valve Testing in 3/16‖

Monel Port Size for the First Second 68

Fig. B.9—Change of Pressure vs. Time relative to Ball Position in 3/16‖ Monel Port 69


Monel Port for the First Second 69



Monel Port for the First Second 70

xi


Fig. B.14—Sensitivity of Cd to Pressure and the Valve Body in 5/16‖ Monel Sharp-

edged Seat at T= 73 oF

72

Fig. B.15—Ball-Seat Relevancy Due to Angle, And Distance 73

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] 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 106


psig 108


psig 110

xii

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 67

Table B-4—Cd Sensitivity to Upstream Pressure and the GLV Body 72

Table B-5—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions 76



Table B-8—Cd Calculations for 3/16 inch Port Size Using Orifice Port Only and Orifice

Port Only Inside the Body of GLV

79




xiii


Port Only Inside the Body of GLV

83





Port Only Inside the Body of GLV 87











Table D.1—Probe Test Results for 1/2‖ Monel Port, 1-1/2‖ J-20 GLV at Pbt= 149 psig 105



xiv

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

xv

k = Cp/Cv Ratio of gas specific heat at constant pressure to specific heat at constant volume

Mscf/d 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 Production pressure operated (Gas-Lift Valve)

Psc Pressure at standard condition = 14.696 psig

PTRO 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 Slant side of the frustum, in

SGg Gas specific gravity

T Upstream absolute temperature, oR

Tb Bellows charged temperature, oF

TC Thornhill-Craver

Tinj Injection gas temperature, oF

xvi

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

1

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.

2

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 blow-

down 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.

3

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]

4

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.

5

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.

6

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.U-

tube 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]

7

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]

8

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)

9

The first bellows-charged GLV was invented by King [4] in 1940. Prior to introducing bellows-

charged 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.

10

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.

Fig. 2-9--Schematic of a Typical Bellows-charged Tubing Retrievable IPO GLV

Pi

11

Fig. 2-10— Schematic of Cross-over Seats GLV

Fig. 2-11—Typical Isentropic Flow Pressure Ratio Responses to Flowrate in IPO GLVs

12

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.

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. 2-

1.

(

) √

(

)

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.

X

2r r

13

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

14

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:

15

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.

16

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

pressure(Pbt, psig) is acting on the dome area (Ab, in2) 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 Pid 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. 2-

9show 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.

17

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.

√ √

Upstream Area P1, ν1 D P2, ν2 d

Flowing Area

18

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. 2-

19 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].

19

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

20

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.

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)

05

101520253035404550

0 0.2 0.4 0.6 0.8 1

(qgim

ax-q

gi)/(

Pio

d-P

pd)

Ppd/Piod

rcritic

21

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

22

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. 2-

26, 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 re-

arrange Eq. 2-26 to Eq. 2-27.

23

√

⁄

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

24

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

25

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 Multi-

tester 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.

26

Fig. 3-1-- Schematic of the Probe Tester [16]

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

380

400

420

440

460

480

500

520

540

560

580

600

620

640

0 0.05 0.1 0.15 0.2 0.25

Pres

sure

, psi

g

Stem Travel, inch

Pressure vs Stem Travel, 1-1/2" J-20 Camco GLV, Pbt = 596 psig

InreasingPressure

DecreasingPressure

Max. Linear Travel = 0.16 inch Load Rate = 250 psi/inch dPLinear = 40 psi Min. Travel for Fully Open = .2246 inch

27

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.

28

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.

29

Fig. 3-5-- Hydraulic Stabilizer (Valve Hydro-Tester or Ager)

Water inlet

High Pressure water

outlet

Low Pressure

Gas inlet

Drain Line

Sprague Pump

Poutlet (water) / Pinlet (gas)= 160

GLV Hydro-

Tester

Chamber

Pressure Gage

Relief valve

30

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 60oF,

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

31

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 Pvc 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.

Fig.3-6-- Schematic of Blow-Down Dynamic Test Facility

to DAQ

Gas Flow

Encapsulated

Vessel

GLV

Downstream

Valve Known Working Volume

High Pressure

Nitrogen Source

32

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. 3-3 and Eq. 3-4 respectively.

33

3- Compute the flowrate applying the Eq. 3-5.

√

34

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

35

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.

36

Table 4-1- CP /CV for different Gases

Gas Ratio of Specific Heats Acetylene 1.3

Air, Standard 1.4

Ammonia 1.32

Carbon Dioxide 1.28

Carbon Monoxide 1.4

Chlorine 1.33

Ethane 1.18

Helium 1.66

Helium 1.66

Hydrogen 1.41

Methane 1.32

Natural Gas (Methane) 1.32

Nitrogen 1.4

Oxygen 1.4

Propane 1.12

Steam 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 ft3. In other words, all the

hoses, connectors, valves, and so on hold for less than 2% of the overall internal volume.

37

Table 4-2--Technical Specifications of Cylinders [33]

Cylinder Size

Nominal Size Diameter X

Height (inches)

Nominal Tare

Weight (lbs.)

Water Capacity

(lbs.)

Internal Volume @ 70°F (21°C), 1

ATM (liters/cubic feet)

US DOT Specifications

K 9.25 X 60 135 110 49.9 / 1.76 3AA2400

A 9 X 56 115 96 43.8 / 1.55 3AA2015

B 8.5 X 31 60 37.9 17.2 / 0.61 3AA2015

C 6 X 24 27 15.2 6.88 / 0.24 3AA2015

D 4 X 18 12 4.9 2.24 / 0.08 3AA2015

AL 8 X 53 52 64.8 29.5 / 1.04 3AL2015

BL 7.25 X 39 33 34.6 15.7 / 0.55 3AL2216

CL 6.9 X 21 19 13 5.9 / 0.21 3AL2216

XL 14.5 X 50 75 238 108 / 3.83 4BA240

SSB 8 X 37 95 41.6 18.9 / 0.67 3A1800

10S 4 X 31 21 8.3 3.8 / 0.13 3A1800

LB 2 X 15 4 1 0.44 / 0.016 3E1800

XF 12 X 46 180

60.9 / 2.15 8AL

XG 15 X 56 149 278 126.3 / 4.46 4AA480

XM 10 X 49 90 120 54.3 / 1.92 3A480

XP 10 X 55 55 124 55.7 / 1.98 4BA300

QT

3 X 14 includes

4.5 inches for valve

2.5 includes

1.5 lbs for valve

2 0.900 / 0.0318 4B-240ET

LP5 12.25 X 18.25

18.5 47.7 21.68 / 0.76 4BW240

Medical E 4 x 26

excludes valve and cap

14 excludes valve and

cap

4.5 / 0.16 3AA2015

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.

38

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

Fig. 4.1--Schematic of the Ball – Seat Position

H

a

r

θ

H

r

R

Y

39

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

40

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.

41

[ ( ) ]

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.

42

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.

43

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.

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.

0

100

200

300

400

500

600

700

0 2 4 6 8 10 12

P iod

, psi

g

Time, sec

Pressure vs. Time for 1/4" Monel, J-20 Camco , 1-1/2" GLV

Real Data

ExpDecay-2Fit

44

Table 5.1—Curve-fit Values for 1/4” Monel, 1-1/2” J-20 Camco GLV

Exponential Decay-2 Fit Model ExpDecay2

Equation y = y0 + A1*Exp(-(x-x0)/t1) + A2*Exp(-(x-x0)/t2) Adj. R-Square 0.99999

Value

Constant = y0 -11.856 Constant = x0 0.03469 Constant = A1 230.354 Constant = t1 5.14445 Constant = A2 385.045 Constant = 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 Pup

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.

45

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)

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

1515.215.415.615.8

1616.216.416.616.8

17

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.864th

in E

quva

lent

Por

t Siz

e

Time, sec

64th of inch Port Equivalent 1/4" Port Size, 1-1/2" J-20 GLV

46

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.

47

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 ExpDec1

Equation y = A1*Exp(-x/t1) + y0 Adj. R-Square 0.99976

Value Standard Error

Coefficients y0 554.25927 3.16E-01

Coefficients A1 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 10th 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.

7

10

13

16

19

22

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.864th

in E

quiv

alen

t Por

t Siz

e

Time, Sec

64th of Inch Port Equivalent 5/16" Monel Port Size, 1-1/2" J-20 GLV

48

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.

P = -411.53t + 709.81 R² = 0.9986

670

675

680

685

690

695

700

705

710

715

0 0.02 0.04 0.06 0.08 0.1

Pres

sure

, psi

g

Time, sec

Pressure vs. Time (Based on Curve-fit Data) 5/16" Monel Port Size, 1-1/2" J-20 GLV

49

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.

50

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.

0

100

200

300

400

500

600

0 2 4 6 8 10

P, p

sig

Time, sec

Plot of Pressure vs.Time for 5/16" Monel and Slight Tapered TC Port

5/16" Port,TungstenCarbide withslight Bevel

5/16" Port,Monel

51

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.

52

Fig. 5-9—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 3/16” Monel Sharp-

edged Seat


edged Seat

0.981

0.928

0.889

0.847

0.876 0.884

0.895

0.886

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1 2 3 4 5 6 7 8

Cd

Valu

es

Dimensionless Ball Position

Benchmark Valve Testing for 3/16" Port ID

0.857 0.852 0.853

0.792

0.836 0.838

0.854

0.848

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1 2 3 4 5 6 7 8

Cd

Valu

es


Benchmark Valve Testing for 1/4"" Port ID

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


53

Fig. 5-11—Variation of Cd with Relative Ball-to-Seat Position and in Orifice-Port Only in 5/16” Monel

Sharp-edged Seat


edged Seat

0.880

0.804 0.827

0.787

0.814 0.844 0.842

0.848

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1 2 3 4 5 6 7 8

Cd

Valu

es



0.826 0.806

0.832

0.775 0.788

0.810 0.826

0.837

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1 2 3 4 5 6 7 8

Cd

Valu

es





54


edged Seat

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

0.822

0.779

0.852

0.771

0.813 0.823 0.846

0.843

0.6

0.65

0.7

0.75

0.8

0.85

0.9

1 2 3 4 5 6 7 8

Cd

Valu

es



0.7

0.8

0.9

1

1 2 3 4 5 6 7 8

Cd

Dimensionless Ball (Stem) Position

3/16"

1/4"

5/16"

3/8"

1/2"

Average

Average(Over all)



55

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.

56

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)

57

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.

58

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 16.4

500 20.4

59

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

0

100

200

300

400

500

600

0 5 10 15 20 25

Pres

sure

, psi

mili Volt

500 psi Sensotec Transducer

60

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)


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:


Pressure = 251600 * volt – 12.598 (A-3)


Pressure = 502300 * volt – 15.379 (A-4)

The following results found for Honeywell transducer.


Pressure = 334850 * volt – 9.0552 (A-5)

0

200

400

600

800

1000

1200

0 5 10 15 20 25

Pres

sure

, psi

mili Volt

1000 psi Snesotec Transducer

61

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.

62

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) Remarks

Temperature, oF

(Up/Down)

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


Valve is %75 Fully Open 75 74.5


Valve is fully open 73.5 73


Valve is at 1-1/4 Fully Open 73.5 72.5


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 / Ab 1-(As/Ab)

Fp Production

Pressure Factor As/(Ab-As)

Geometric Fully-Open Stem Travel

inch

3/16 0.0276 0.036 0.964 0.037 0.0714

1/4 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

63

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.

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

0

50

100

150

200

250

300

350

0 5 10 15 20 25 30

Pres

sure

, psi

g

Time, sec

Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel Ball at 1/4 Fully Open Travel Position

RealData

Exp-Decay-2Fit

64

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


when the Ball is at 3/4 Fully Open Travel Position

0

50

100

150

200

250

300

350

0 2 4 6 8 10 12 14 16

Pres

sure

, psi

g

Time, sec


RealData

Exp-Decay-2 Fit

0

50

100

150

200

250

300

350

0 2 4 6 8 10

Pres

sure

, psi

g

Time, sec


RealData

Exp-Decay-2Fit

65


when the Ball is at Fully Open Travel Position


when the Ball is at 1-1/2 Fully Open Travel Position (Beyond Fully Open)

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8 9

Pres

sure

, psi

g

Time, sec

Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel Ball at Fully Open Travel Position

RealData

Exp-Decay-2 Fit

0

50

100

150

200

250

300

350

0 1 2 3 4 5 6 7 8

Pres

sure

, psi

g

Time, sec

Pressure vs. Time, Benchmark Valve Testing, 5/16" Monel Ball at 1-1/2 Fully Open Travel Position

RealData

Exp-Decay-2 Fit

66

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.

67

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

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pres

sure

, psi

g

Time, Sec

Benchmark Valve Testing in 5/16" Monel Sharp-edged Seat

1/4 Open 1/2 Open 3/4 Open Full Open

1 1/4 Open 1 1/2 Open Port Only, Benchmark Port Only

68

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.


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.25 0.5 0.75 1 1.25 1.5

dP/d

Tim

e

Ball Position relative to Theoretical Fully Open

Plot of (dP/dt) vs. Ball Position for 5/16" Monel Sharp-edged seat

250

260

270

280

290

300

310

320

330

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pres

sure

, psi

g

Time, sec


1/4 Open 1/2 Open 3/14 Open

Full Open 1-1/4 Open 1-1/2 Open

Port Only Port Only- Benchmark Valve

69

Fig. B-9—Change of Pressure vs. Time relative to Ball Position in 3/16” Monel Port


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.25 0.5 0.75 1 1.25 1.5

dP/d

Tim

e

Ball Position Relative to Theoretical Minimum Fully-Open

Pressure Change with Time at Relative Port Positions in 3/16" Monel Seat

210

220

230

240

250

260

270

280

290

300

310

320

330

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pres

sure

, psi

g

Time, sec



1-1/4 Open 1-1/2 Open Port Only

70



0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 0.25 0.5 0.75 1 1.25 1.5

Slop

e (d

P/ d

Tim

e)

Ball Position to Fully Open Position


120

130

140

150

160

170

180

190

200

210

220

230

240

250

260

270

280

290

300

310

320

330

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pres

sure

, psi

g

Time, sec

Benchmark Testing in 3/8" Monel Sharp-edged Seat


1-1/4 Open 1-1/2 Open Port Only

71


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 port-

only inside the GLV body. Port-only case is the case that the orifice port only has been set inside the

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.25 0.5 0.75 1 1.25 1.5

Slop

e (d

P/dT

ime)

Ball Position to Fully Open Position


72

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 Pressure 150 300 450 600 750 900

Port Only 0.72691 0.79527 0.83215 0.85332 0.8564 0.86002 Port inside BV 0.7375 0.79957 0.83151 0.84464 0.85426 0.85917

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.

Fig. B-14—Sensitivity of Cd to Pressure and the Valve Body in 5/16” Monel Sharp-edged Seat at T= 73 oF

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

150 300 450 600 750 900

Cd

Pressure, psig

Port Only Port inside BV

73

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.

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 B-

6. These formulas will help to calculate the effective flowing area at each ball-stem position as the ball is

dynamically moving inside the GLV.

√

(

)

θ H

R Y

S

r

74

(

( ( √

))

)

( ( ( √

)) )

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

75

very satisfactory results collected. All the pressure points tabulated in the following tables have been

normalized consistently.

76

Table B-5—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions 1/4 Fully Open

Time, sec

P psig

slope mole flowrate ft3/sec MSCF/D Constants

dP/dt dn/dt dV/dt dV/dt dV/dt CK1 CK2 CK3 CK4 Q Cd 0 345 29.65725 0.286579 108.900033 1.08900033 94.08963 2333.63 1.4286 1.7143 0.0762 43.493 0.9805

0.01 344.7 29.63175 0.286333 108.80642 1.0880642 94.00875 2333.63 1.4286 1.7143 0.0762 43.475 0.9803 0.02 344.41 29.60628 0.286087 108.712886 1.08712886 93.92793 2333.63 1.4286 1.7143 0.0762 43.457 0.9801 0.03 344.11 29.58083 0.285841 108.619433 1.08619433 93.84719 2333.63 1.4286 1.7143 0.0762 43.439 0.9799 0.04 343.82 29.5554 0.285595 108.526061 1.08526061 93.76652 2333.63 1.4286 1.7143 0.0762 43.421 0.9797 0.05 343.52 29.52999 0.285349 108.432769 1.08432769 93.68591 2333.63 1.4286 1.7143 0.0762 43.403 0.9795 0.06 343.22 29.50461 0.285104 108.339556 1.08339556 93.60538 2333.63 1.4286 1.7143 0.0762 43.385 0.9792 0.07 342.93 29.47925 0.284859 108.246425 1.08246425 93.52491 2333.63 1.4286 1.7143 0.0762 43.367 0.979 0.08 342.63 29.4539 0.284614 108.153373 1.08153373 93.44451 2333.63 1.4286 1.7143 0.0762 43.349 0.9788 0.09 342.34 29.42858 0.284369 108.060401 1.08060401 93.36419 2333.63 1.4286 1.7143 0.0762 43.331 0.9786 0.1 342.05 29.40329 0.284125 107.967509 1.07967509 93.28393 2333.63 1.4286 1.7143 0.0762 43.313 0.9784

0.11 341.75 29.37801 0.283881 107.874697 1.07874697 93.20374 2333.63 1.4286 1.7143 0.0762 43.296 0.9781 0.12 341.46 29.35276 0.283637 107.781964 1.07781964 93.12362 2333.63 1.4286 1.7143 0.0762 43.278 0.9779 0.13 341.16 29.32752 0.283393 107.689312 1.07689312 93.04357 2333.63 1.4286 1.7143 0.0762 43.26 0.9777 0.14 340.87 29.30231 0.283149 107.596739 1.07596739 92.96358 2333.63 1.4286 1.7143 0.0762 43.242 0.9775 0.15 340.58 29.27712 0.282906 107.504245 1.07504245 92.88367 2333.63 1.4286 1.7143 0.0762 43.224 0.9773

1/2 Fully Open

Time, sec

P psig


dP/dt dn/dt dV/dt dV/dt dV/dt CK1 CK2 CK3 CK4 Q Cd 0 345 53.08911 0.513002 194.940759 1.94940759 168.4288 2333.63 1.4286 1.7143 0.0762 86.923 0.928

0.01 344.47 53.00742 0.512213 194.640781 1.94640781 168.1696 2333.63 1.4286 1.7143 0.0762 86.859 0.9276 0.02 343.94 52.92585 0.511424 194.341265 1.94341265 167.9109 2333.63 1.4286 1.7143 0.0762 86.794 0.9273 0.03 343.41 52.8444 0.510637 194.04221 1.9404221 167.6525 2333.63 1.4286 1.7143 0.0762 86.73 0.9269 0.04 342.88 52.76309 0.509852 193.743615 1.93743615 167.3945 2333.63 1.4286 1.7143 0.0762 86.666 0.9265 0.05 342.35 52.68189 0.509067 193.445479 1.93445479 167.1369 2333.63 1.4286 1.7143 0.0762 86.602 0.9262 0.06 341.83 52.60083 0.508284 193.147802 1.93147802 166.8797 2333.63 1.4286 1.7143 0.0762 86.537 0.9258 0.07 341.3 52.51988 0.507502 192.850584 1.92850584 166.6229 2333.63 1.4286 1.7143 0.0762 86.473 0.9254 0.08 340.78 52.43907 0.506721 192.553822 1.92553822 166.3665 2333.63 1.4286 1.7143 0.0762 86.409 0.925 0.09 340.25 52.35837 0.505941 192.257518 1.92257518 166.1105 2333.63 1.4286 1.7143 0.0762 86.345 0.9247 0.1 339.73 52.2778 0.505162 191.961669 1.91961669 165.8549 2333.63 1.4286 1.7143 0.0762 86.281 0.9243

0.11 339.2 52.19736 0.504385 191.666275 1.91666275 165.5997 2333.63 1.4286 1.7143 0.0762 86.217 0.9239 0.12 338.68 52.11703 0.503609 191.371336 1.91371336 165.3448 2333.63 1.4286 1.7143 0.0762 86.154 0.9236 0.13 338.16 52.03684 0.502834 191.076851 1.91076851 165.0904 2333.63 1.4286 1.7143 0.0762 86.09 0.9232 0.14 337.64 51.95676 0.50206 190.78282 1.9078282 164.8364 2333.63 1.4286 1.7143 0.0762 86.026 0.9228 0.15 337.12 51.87681 0.501287 190.48924 1.9048924 164.5827 2333.63 1.4286 1.7143 0.0762 85.962 0.9225

77


Time, sec

slope mole flowrate ft

3/sec MSCF/D Constants

P, psig dP/dt dn/dt dV/dt dV/dt dV/dt CK1 CK2 CK3 CK4 Q Cd

0 345 72.71826 0.702679 267.018068 2.67018068 230.7036 2333.63 1.4286 1.7143 0.0762 129.66 0.8893 0.01 344.27 72.56498 0.701198 266.455254 2.66455254 230.2173 2333.63 1.4286 1.7143 0.0762 129.52 0.8888 0.02 343.55 72.41203 0.69972 265.893626 2.65893626 229.7321 2333.63 1.4286 1.7143 0.0762 129.39 0.8883 0.03 342.82 72.2594 0.698245 265.333182 2.65333182 229.2479 2333.63 1.4286 1.7143 0.0762 129.26 0.8878 0.04 342.1 72.1071 0.696773 264.77392 2.6477392 228.7647 2333.63 1.4286 1.7143 0.0762 129.13 0.8873 0.05 341.38 71.95511 0.695305 264.215836 2.64215836 228.2825 2333.63 1.4286 1.7143 0.0762 129 0.8869 0.06 340.66 71.80345 0.693839 263.658928 2.63658928 227.8013 2333.63 1.4286 1.7143 0.0762 128.87 0.8864 0.07 339.94 71.6521 0.692377 263.103194 2.63103194 227.3212 2333.63 1.4286 1.7143 0.0762 128.74 0.8859 0.08 339.23 71.50107 0.690917 262.548632 2.62548632 226.842 2333.63 1.4286 1.7143 0.0762 128.61 0.8854 0.09 338.51 71.35036 0.689461 261.995238 2.61995238 226.3639 2333.63 1.4286 1.7143 0.0762 128.48 0.8849 0.1 337.8 71.19997 0.688008 261.443011 2.61443011 225.8868 2333.63 1.4286 1.7143 0.0762 128.35 0.8844

0.11 337.08 71.0499 0.686558 260.891948 2.60891948 225.4106 2333.63 1.4286 1.7143 0.0762 128.22 0.8839 0.12 336.37 70.90014 0.685111 260.342046 2.60342046 224.9355 2333.63 1.4286 1.7143 0.0762 128.09 0.8835 0.13 335.67 70.7507 0.683667 259.793304 2.59793304 224.4614 2333.63 1.4286 1.7143 0.0762 127.96 0.883 0.14 334.96 70.60158 0.682226 259.245718 2.59245718 223.9883 2333.63 1.4286 1.7143 0.0762 127.83 0.8825 0.15 334.25 70.45276 0.680788 258.699286 2.58699286 223.5162 2333.63 1.4286 1.7143 0.0762 127.7 0.882

Fully Theoretically Open

Time, sec




0 345 87.17468 0.842372 320.101382 3.20101382 276.5676 2333.63 1.4286 1.7143 0.0762 171.46 0.8467 0.01 344.13 86.9544 0.840244 319.292549 3.19292549 275.8688 2333.63 1.4286 1.7143 0.0762 171.25 0.8461 0.02 343.26 86.73469 0.83812 318.48576 3.1848576 275.1717 2333.63 1.4286 1.7143 0.0762 171.04 0.8456 0.03 342.39 86.51553 0.836003 317.68101 3.1768101 274.4764 2333.63 1.4286 1.7143 0.0762 170.83 0.845 0.04 341.53 86.29692 0.83389 316.878292 3.16878292 273.7828 2333.63 1.4286 1.7143 0.0762 170.63 0.8445 0.05 340.66 86.07886 0.831783 316.077604 3.16077604 273.091 2333.63 1.4286 1.7143 0.0762 170.42 0.8439 0.06 339.8 85.86136 0.829681 315.278938 3.15278938 272.401 2333.63 1.4286 1.7143 0.0762 170.21 0.8434 0.07 338.94 85.6444 0.827585 314.482291 3.14482291 271.7127 2333.63 1.4286 1.7143 0.0762 170.01 0.8428 0.08 338.09 85.428 0.825494 313.687656 3.13687656 271.0261 2333.63 1.4286 1.7143 0.0762 169.8 0.8423 0.09 337.23 85.21214 0.823408 312.895029 3.12895029 270.3413 2333.63 1.4286 1.7143 0.0762 169.59 0.8417 0.1 336.38 84.99682 0.821327 312.104405 3.12104405 269.6582 2333.63 1.4286 1.7143 0.0762 169.39 0.8412

0.11 335.53 84.78205 0.819252 311.315779 3.11315779 268.9768 2333.63 1.4286 1.7143 0.0762 169.18 0.8406 0.12 334.68 84.56783 0.817182 310.529146 3.10529146 268.2972 2333.63 1.4286 1.7143 0.0762 168.97 0.8401 0.13 333.84 84.35414 0.815117 309.7445 3.097445 267.6192 2333.63 1.4286 1.7143 0.0762 168.77 0.8395 0.14 332.99 84.14099 0.813057 308.961837 3.08961837 266.943 2333.63 1.4286 1.7143 0.0762 168.56 0.839 0.15 332.15 83.92839 0.811003 308.181152 3.08181152 266.2685 2333.63 1.4286 1.7143 0.0762 168.36 0.8384

78

Table B-7—Cd Calculations for 3/16 inch Port Size at Different Set Ball Positions 1-1/4 Fully Open

Time, sec




0 345 93.36843 0.902222 342.84455 3.4284455 296.2177 2333.63 1.4286 1.7143 0.0762 171.45 0.8763 0.01 344.07 93.11574 0.899781 341.916699 3.41916699 295.416 2333.63 1.4286 1.7143 0.0762 171.23 0.8757 0.02 343.14 92.86374 0.897346 340.991359 3.40991359 294.6165 2333.63 1.4286 1.7143 0.0762 171.01 0.875 0.03 342.21 92.61242 0.894917 340.068523 3.40068523 293.8192 2333.63 1.4286 1.7143 0.0762 170.78 0.8744 0.04 341.28 92.36178 0.892495 339.148185 3.39148185 293.024 2333.63 1.4286 1.7143 0.0762 170.56 0.8738 0.05 340.36 92.11182 0.89008 338.230338 3.38230338 292.231 2333.63 1.4286 1.7143 0.0762 170.34 0.8732 0.06 339.44 91.86253 0.887671 337.314974 3.37314974 291.4401 2333.63 1.4286 1.7143 0.0762 170.12 0.8726 0.07 338.52 91.61392 0.885269 336.402088 3.36402088 290.6514 2333.63 1.4286 1.7143 0.0762 169.89 0.872 0.08 337.6 91.36599 0.882873 335.491673 3.35491673 289.8648 2333.63 1.4286 1.7143 0.0762 169.67 0.8714 0.09 336.69 91.11872 0.880483 334.583721 3.34583721 289.0803 2333.63 1.4286 1.7143 0.0762 169.45 0.8708 0.1 335.78 90.87212 0.878101 333.678227 3.33678227 288.298 2333.63 1.4286 1.7143 0.0762 169.23 0.8701

0.11 334.87 90.62619 0.875724 332.775183 3.32775183 287.5178 2333.63 1.4286 1.7143 0.0762 169.01 0.8695 0.12 333.96 90.38093 0.873354 331.874583 3.31874583 286.7396 2333.63 1.4286 1.7143 0.0762 168.79 0.8689 0.13 333.06 90.13633 0.870991 330.976421 3.30976421 285.9636 2333.63 1.4286 1.7143 0.0762 168.57 0.8683 0.14 332.16 89.89239 0.868633 330.080689 3.30080689 285.1897 2333.63 1.4286 1.7143 0.0762 168.35 0.8677 0.15 331.26 89.64911 0.866283 329.187381 3.29187381 284.4179 2333.63 1.4286 1.7143 0.0762 168.13 0.8671

1-1/2 Fully Open

Time, sec




0 345 95.08872 0.918846 349.16137 3.4916137 301.6754 2333.63 1.4286 1.7143 0.0762 171.45 0.8843 0.01 344.05 94.82663 0.916313 348.199013 3.48199013 300.8439 2333.63 1.4286 1.7143 0.0762 171.22 0.8837 0.02 343.1 94.56527 0.913788 347.239309 3.47239309 300.0148 2333.63 1.4286 1.7143 0.0762 171 0.8831 0.03 342.16 94.30463 0.911269 346.282249 3.46282249 299.1879 2333.63 1.4286 1.7143 0.0762 170.77 0.8824 0.04 341.21 94.04471 0.908757 345.327828 3.45327828 298.3632 2333.63 1.4286 1.7143 0.0762 170.54 0.8818 0.05 340.27 93.78551 0.906253 344.376037 3.44376037 297.5409 2333.63 1.4286 1.7143 0.0762 170.32 0.8812 0.06 339.33 93.52701 0.903755 343.42687 3.4342687 296.7208 2333.63 1.4286 1.7143 0.0762 170.09 0.8805 0.07 338.4 93.26924 0.901264 342.480319 3.42480319 295.903 2333.63 1.4286 1.7143 0.0762 169.86 0.8799 0.08 337.47 93.01217 0.89878 341.536376 3.41536376 295.0874 2333.63 1.4286 1.7143 0.0762 169.64 0.8793 0.09 336.54 92.75581 0.896303 340.595036 3.40595036 294.2741 2333.63 1.4286 1.7143 0.0762 169.41 0.8786 0.1 335.61 92.50016 0.893832 339.656289 3.39656289 293.463 2333.63 1.4286 1.7143 0.0762 169.19 0.878

0.11 334.68 92.24521 0.891369 338.72013 3.3872013 292.6542 2333.63 1.4286 1.7143 0.0762 168.96 0.8774 0.12 333.76 91.99096 0.888912 337.786552 3.37786552 291.8476 2333.63 1.4286 1.7143 0.0762 168.74 0.8768 0.13 332.84 91.73742 0.886462 336.855546 3.36855546 291.0432 2333.63 1.4286 1.7143 0.0762 168.52 0.8761 0.14 331.92 91.48457 0.884019 335.927107 3.35927107 290.241 2333.63 1.4286 1.7143 0.0762 168.29 0.8755 0.15 331.01 91.23242 0.881582 335.001226 3.35001226 289.4411 2333.63 1.4286 1.7143 0.0762 168.07 0.8749

79

Table B-8—Cd Calculations for 3/16 inch Port Size Using Orifice Port Only and Orifice Port Only Inside the Body of GLV P-BV

Time, sec




0 345 95.43276 0.92217 350.424696 3.50424696 302.7669 2333.63 1.4286 1.7143 0.0762 171.45 0.8859 0.01 344.05 95.16878 0.919619 349.455362 3.49455362 301.9294 2333.63 1.4286 1.7143 0.0762 171.22 0.8853 0.02 343.09 94.90553 0.917076 348.488711 3.48488711 301.0942 2333.63 1.4286 1.7143 0.0762 170.99 0.8846 0.03 342.14 94.643 0.914539 347.524733 3.47524733 300.2614 2333.63 1.4286 1.7143 0.0762 170.77 0.884 0.04 341.2 94.38121 0.912009 346.563421 3.46563421 299.4308 2333.63 1.4286 1.7143 0.0762 170.54 0.8834 0.05 340.25 94.12013 0.909486 345.604769 3.45604769 298.6025 2333.63 1.4286 1.7143 0.0762 170.31 0.8827 0.06 339.31 93.85978 0.90697 344.648768 3.44648768 297.7765 2333.63 1.4286 1.7143 0.0762 170.08 0.8821 0.07 338.37 93.60015 0.904462 343.695412 3.43695412 296.9528 2333.63 1.4286 1.7143 0.0762 169.86 0.8815 0.08 337.44 93.34123 0.90196 342.744693 3.42744693 296.1314 2333.63 1.4286 1.7143 0.0762 169.63 0.8808 0.09 336.51 93.08304 0.899465 341.796604 3.41796604 295.3123 2333.63 1.4286 1.7143 0.0762 169.41 0.8802 0.1 335.57 92.82555 0.896977 340.851138 3.40851138 294.4954 2333.63 1.4286 1.7143 0.0762 169.18 0.8796

0.11 334.65 92.56878 0.894495 339.908286 3.39908286 293.6808 2333.63 1.4286 1.7143 0.0762 168.96 0.8789 0.12 333.72 92.31272 0.892021 338.968043 3.38968043 292.8684 2333.63 1.4286 1.7143 0.0762 168.73 0.8783 0.13 332.8 92.05737 0.889554 338.030401 3.38030401 292.0583 2333.63 1.4286 1.7143 0.0762 168.51 0.8777 0.14 331.88 91.80272 0.887093 337.095352 3.37095352 291.2504 2333.63 1.4286 1.7143 0.0762 168.28 0.877 0.15 330.96 91.54878 0.884639 336.16289 3.3616289 290.4447 2333.63 1.4286 1.7143 0.0762 168.06 0.8764

P

Time, sec




0 345 97.49698 0.942117 358.004387 3.58004387 309.3158 2333.63 1.4286 1.7143 0.0762 171.45 0.8955 0.01 344.03 97.22145 0.939454 356.992667 3.56992667 308.4417 2333.63 1.4286 1.7143 0.0762 171.21 0.8948 0.02 343.05 96.9467 0.936799 355.983806 3.55983806 307.57 2333.63 1.4286 1.7143 0.0762 170.98 0.8941 0.03 342.08 96.67273 0.934152 354.977796 3.54977796 306.7008 2333.63 1.4286 1.7143 0.0762 170.75 0.8935 0.04 341.12 96.39953 0.931512 353.974629 3.53974629 305.8341 2333.63 1.4286 1.7143 0.0762 170.52 0.8928 0.05 340.15 96.12711 0.92888 352.974297 3.52974297 304.9698 2333.63 1.4286 1.7143 0.0762 170.28 0.8922 0.06 339.19 95.85545 0.926255 351.976792 3.51976792 304.1079 2333.63 1.4286 1.7143 0.0762 170.05 0.8915 0.07 338.23 95.58457 0.923637 350.982106 3.50982106 303.2485 2333.63 1.4286 1.7143 0.0762 169.82 0.8909 0.08 337.28 95.31444 0.921027 349.99023 3.4999023 302.3916 2333.63 1.4286 1.7143 0.0762 169.59 0.8902 0.09 336.32 95.04509 0.918424 349.001158 3.49001158 301.537 2333.63 1.4286 1.7143 0.0762 169.36 0.8896 0.1 335.37 94.77649 0.915829 348.014881 3.48014881 300.6849 2333.63 1.4286 1.7143 0.0762 169.13 0.8889

0.11 334.43 94.50865 0.913241 347.031391 3.47031391 299.8351 2333.63 1.4286 1.7143 0.0762 168.9 0.8883 0.12 333.48 94.24157 0.91066 346.050681 3.46050681 298.9878 2333.63 1.4286 1.7143 0.0762 168.67 0.8876 0.13 332.54 93.97524 0.908086 345.072742 3.45072742 298.1428 2333.63 1.4286 1.7143 0.0762 168.44 0.8869 0.14 331.6 93.70967 0.90552 344.097567 3.44097567 297.3003 2333.63 1.4286 1.7143 0.0762 168.21 0.8863 0.15 330.66 93.44485 0.902961 343.125147 3.43125147 296.4601 2333.63 1.4286 1.7143 0.0762 167.98 0.8856

80


Time, sec


P, psig dP/dt dn/dt dV/dt dV/dt dV/dt CK1 CK2 CK3 CK4 Q Cd 0 345 39.30759 0.379831 144.3357 1.443357 124.706 2333.63 1.4286 1.7143 0.076171 75.476 0.8569

0.01 344.6069 39.26281 0.379398 144.1712 1.441712 124.5639 2333.63 1.4286 1.7143 0.076171 75.435 0.8567 0.02 344.2143 39.21807 0.378966 144.0069 1.440069 124.422 2333.63 1.4286 1.7143 0.076171 75.393 0.8564 0.03 343.8221 39.17339 0.378534 143.8429 1.438429 124.2802 2333.63 1.4286 1.7143 0.076171 75.352 0.8562 0.04 343.4304 39.12876 0.378103 143.679 1.43679 124.1386 2333.63 1.4286 1.7143 0.076171 75.311 0.8559 0.05 343.0391 39.08417 0.377672 143.5153 1.435153 123.9972 2333.63 1.4286 1.7143 0.076171 75.269 0.8557 0.06 342.6483 39.03964 0.377241 143.3518 1.433518 123.8559 2333.63 1.4286 1.7143 0.076171 75.228 0.8554 0.07 342.2579 38.99516 0.376812 143.1884 1.431884 123.7148 2333.63 1.4286 1.7143 0.076171 75.187 0.8552 0.08 341.8679 38.95073 0.376382 143.0253 1.430253 123.5739 2333.63 1.4286 1.7143 0.076171 75.146 0.8549 0.09 341.4784 38.90636 0.375954 142.8623 1.428623 123.4331 2333.63 1.4286 1.7143 0.076171 75.104 0.8547 0.1 341.0893 38.86203 0.375525 142.6996 1.426996 123.2924 2333.63 1.4286 1.7143 0.076171 75.063 0.8544

0.11 340.7007 38.81775 0.375097 142.537 1.42537 123.152 2333.63 1.4286 1.7143 0.076171 75.022 0.8542 0.12 340.3125 38.77352 0.37467 142.3746 1.423746 123.0116 2333.63 1.4286 1.7143 0.076171 74.981 0.8539 0.13 339.9248 38.72935 0.374243 142.2124 1.422124 122.8715 2333.63 1.4286 1.7143 0.076171 74.94 0.8536 0.14 339.5375 38.68522 0.373817 142.0503 1.420503 122.7315 2333.63 1.4286 1.7143 0.076171 74.899 0.8534 0.15 339.1507 38.64114 0.373391 141.8885 1.418885 122.5917 2333.63 1.4286 1.7143 0.076171 74.858 0.8531

1/2 Fully Open Time, sec



0.01 344.2143 78.39146 0.7575 287.8498 2.878498 248.7022 2333.63 1.4286 1.7143 0.076171 152.47 0.8514 0.02 343.4304 78.21293 0.755774 287.1943 2.871943 248.1358 2333.63 1.4286 1.7143 0.076171 152.31 0.8509 0.03 342.6483 78.03481 0.754053 286.5402 2.865402 247.5707 2333.63 1.4286 1.7143 0.076171 152.14 0.8504 0.04 341.8679 77.85709 0.752336 285.8876 2.858876 247.0069 2333.63 1.4286 1.7143 0.076171 151.97 0.8499 0.05 341.0893 77.67978 0.750623 285.2366 2.852366 246.4444 2333.63 1.4286 1.7143 0.076171 151.81 0.8494 0.06 340.3125 77.50287 0.748913 284.587 2.84587 245.8831 2333.63 1.4286 1.7143 0.076171 151.64 0.8489 0.07 339.5375 77.32637 0.747207 283.9388 2.839388 245.3232 2333.63 1.4286 1.7143 0.076171 151.47 0.8484 0.08 338.7642 77.15026 0.745506 283.2922 2.832922 244.7645 2333.63 1.4286 1.7143 0.076171 151.31 0.8479 0.09 337.9927 76.97456 0.743808 282.647 2.82647 244.207 2333.63 1.4286 1.7143 0.076171 151.14 0.8474 0.1 337.223 76.79926 0.742114 282.0033 2.820033 243.6509 2333.63 1.4286 1.7143 0.076171 150.98 0.8469

0.11 336.455 76.62436 0.740424 281.3611 2.813611 243.096 2333.63 1.4286 1.7143 0.076171 150.81 0.8464 0.12 335.6888 76.44985 0.738738 280.7203 2.807203 242.5424 2333.63 1.4286 1.7143 0.076171 150.64 0.8459 0.13 334.9243 76.27574 0.737055 280.081 2.80081 241.99 2333.63 1.4286 1.7143 0.076171 150.48 0.8454 0.14 334.1615 76.10203 0.735377 279.4431 2.794431 241.4389 2333.63 1.4286 1.7143 0.076171 150.31 0.8449 0.15 333.4005 75.92872 0.733702 278.8067 2.788067 240.889 2333.63 1.4286 1.7143 0.076171 150.15 0.8444

81


Time, sec



0.01 343.8152 118.0692 1.140907 433.5448 4.335448 374.5827 2333.63 1.4286 1.7143 0.076171 229.09 0.8525 0.02 342.6345 117.6638 1.136989 432.056 4.32056 373.2964 2333.63 1.4286 1.7143 0.076171 228.71 0.8517 0.03 341.4579 117.2597 1.133085 430.5723 4.305723 372.0144 2333.63 1.4286 1.7143 0.076171 228.34 0.8509 0.04 340.2853 116.857 1.129194 429.0936 4.290936 370.7369 2333.63 1.4286 1.7143 0.076171 227.96 0.8502 0.05 339.1167 116.4557 1.125316 427.6201 4.276201 369.4638 2333.63 1.4286 1.7143 0.076171 227.58 0.8494 0.06 337.9522 116.0558 1.121452 426.1516 4.261516 368.195 2333.63 1.4286 1.7143 0.076171 227.21 0.8487 0.07 336.7916 115.6573 1.1176 424.6882 4.246882 366.9306 2333.63 1.4286 1.7143 0.076171 226.83 0.8479 0.08 335.6351 115.2601 1.113762 423.2297 4.232297 365.6705 2333.63 1.4286 1.7143 0.076171 226.45 0.8472 0.09 334.4825 114.8643 1.109938 421.7763 4.217763 364.4148 2333.63 1.4286 1.7143 0.076171 226.08 0.8464 0.1 333.3338 114.4698 1.106126 420.3279 4.203279 363.1633 2333.63 1.4286 1.7143 0.076171 225.71 0.8456

0.11 332.1891 114.0767 1.102328 418.8845 4.188845 361.9162 2333.63 1.4286 1.7143 0.076171 225.33 0.8449 0.12 331.0483 113.685 1.098542 417.446 4.17446 360.6733 2333.63 1.4286 1.7143 0.076171 224.96 0.8441 0.13 329.9115 113.2946 1.09477 416.0124 4.160124 359.4347 2333.63 1.4286 1.7143 0.076171 224.59 0.8434 0.14 328.7785 112.9055 1.09101 414.5838 4.145838 358.2004 2333.63 1.4286 1.7143 0.076171 224.22 0.8426 0.15 327.6495 112.5178 1.087263 413.1601 4.131601 356.9703 2333.63 1.4286 1.7143 0.076171 223.85 0.8419

Fully Open Time, sec



0.01 343.6434 135.1291 1.305757 496.1878 4.961878 428.7063 2333.63 1.4286 1.7143 0.076171 304.57 0.7909 0.02 342.2921 134.5977 1.300623 494.2367 4.942367 427.0205 2333.63 1.4286 1.7143 0.076171 304 0.7901 0.03 340.9461 134.0685 1.295509 492.2932 4.922932 425.3414 2333.63 1.4286 1.7143 0.076171 303.42 0.7893 0.04 339.6054 133.5413 1.290414 490.3574 4.903574 423.6688 2333.63 1.4286 1.7143 0.076171 302.84 0.7885 0.05 338.27 133.0162 1.28534 488.4292 4.884292 422.0028 2333.63 1.4286 1.7143 0.076171 302.27 0.7877 0.06 336.9398 132.4931 1.280286 486.5086 4.865086 420.3434 2333.63 1.4286 1.7143 0.076171 301.7 0.7869 0.07 335.6149 131.9721 1.275251 484.5955 4.845955 418.6905 2333.63 1.4286 1.7143 0.076171 301.13 0.7861 0.08 334.2952 131.4532 1.270237 482.69 4.8269 417.0441 2333.63 1.4286 1.7143 0.076171 300.56 0.7853 0.09 332.9807 130.9363 1.265242 480.7919 4.807919 415.4042 2333.63 1.4286 1.7143 0.076171 299.99 0.7845 0.1 331.6713 130.4214 1.260267 478.9013 4.789013 413.7707 2333.63 1.4286 1.7143 0.076171 299.42 0.7837

0.11 330.3671 129.9085 1.255311 477.0182 4.770182 412.1437 2333.63 1.4286 1.7143 0.076171 298.85 0.7829 0.12 329.068 129.3977 1.250375 475.1424 4.751424 410.523 2333.63 1.4286 1.7143 0.076171 298.29 0.7821 0.13 327.774 128.8889 1.245458 473.274 4.73274 408.9088 2333.63 1.4286 1.7143 0.076171 297.73 0.7813 0.14 326.4851 128.3821 1.240561 471.413 4.71413 407.3008 2333.63 1.4286 1.7143 0.076171 297.16 0.7805 0.15 325.2013 127.8772 1.235682 469.5593 4.695593 405.6992 2333.63 1.4286 1.7143 0.076171 296.6 0.7797

82


Time, sec



0.01 343.4888 150.4611 1.453911 552.4861 5.524861 477.348 2333.63 1.4286 1.7143 0.076171 304.47 0.8347 0.02 341.9842 149.802 1.447542 550.066 5.50066 475.257 2333.63 1.4286 1.7143 0.076171 303.83 0.8338 0.03 340.4861 149.1458 1.441201 547.6565 5.476565 473.1752 2333.63 1.4286 1.7143 0.076171 303.19 0.8328 0.04 338.9947 148.4925 1.434888 545.2575 5.452575 471.1025 2333.63 1.4286 1.7143 0.076171 302.55 0.8319 0.05 337.5098 147.842 1.428603 542.8691 5.428691 469.0389 2333.63 1.4286 1.7143 0.076171 301.91 0.8309 0.06 336.0313 147.1944 1.422345 540.4911 5.404911 466.9843 2333.63 1.4286 1.7143 0.076171 301.27 0.83 0.07 334.5594 146.5497 1.416115 538.1236 5.381236 464.9388 2333.63 1.4286 1.7143 0.076171 300.64 0.8291 0.08 333.0939 145.9077 1.409912 535.7664 5.357664 462.9022 2333.63 1.4286 1.7143 0.076171 300.01 0.8281 0.09 331.6348 145.2686 1.403736 533.4195 5.334195 460.8745 2333.63 1.4286 1.7143 0.076171 299.37 0.8272 0.1 330.1821 144.6323 1.397587 531.083 5.31083 458.8557 2333.63 1.4286 1.7143 0.076171 298.74 0.8262

0.11 328.7358 143.9987 1.391465 528.7566 5.287566 456.8457 2333.63 1.4286 1.7143 0.076171 298.11 0.8253 0.12 327.2958 143.3679 1.38537 526.4405 5.264405 454.8446 2333.63 1.4286 1.7143 0.076171 297.48 0.8243 0.13 325.8621 142.7399 1.379301 524.1344 5.241344 452.8522 2333.63 1.4286 1.7143 0.076171 296.86 0.8234 0.14 324.4347 142.1147 1.373259 521.8385 5.218385 450.8685 2333.63 1.4286 1.7143 0.076171 296.23 0.8225 0.15 323.0136 141.4922 1.367244 519.5527 5.195527 448.8935 2333.63 1.4286 1.7143 0.076171 295.61 0.8215

1-1/2 Fully Open Time, sec



0.01 343.4819 151.142 1.460491 554.9864 5.549864 479.5083 2333.63 1.4286 1.7143 0.076171 304.47 0.8366 0.02 341.9705 150.4769 1.454064 552.5443 5.525443 477.3983 2333.63 1.4286 1.7143 0.076171 303.82 0.8357 0.03 340.4657 149.8148 1.447666 550.113 5.50113 475.2976 2333.63 1.4286 1.7143 0.076171 303.18 0.8347 0.04 338.9676 149.1556 1.441296 547.6923 5.476923 473.2062 2333.63 1.4286 1.7143 0.076171 302.54 0.8338 0.05 337.476 148.4992 1.434953 545.2823 5.452823 471.1239 2333.63 1.4286 1.7143 0.076171 301.9 0.8328 0.06 335.991 147.8458 1.428639 542.8829 5.428829 469.0509 2333.63 1.4286 1.7143 0.076171 301.26 0.8319 0.07 334.5126 147.1952 1.422353 540.4941 5.404941 466.9869 2333.63 1.4286 1.7143 0.076171 300.62 0.8309 0.08 333.0406 146.5475 1.416094 538.1158 5.381158 464.932 2333.63 1.4286 1.7143 0.076171 299.98 0.83 0.09 331.5751 145.9027 1.409863 535.7479 5.357479 462.8862 2333.63 1.4286 1.7143 0.076171 299.35 0.829 0.1 330.1161 145.2607 1.403659 533.3904 5.333904 460.8493 2333.63 1.4286 1.7143 0.076171 298.71 0.8281

0.11 328.6635 144.6215 1.397483 531.0434 5.310434 458.8215 2333.63 1.4286 1.7143 0.076171 298.08 0.8271 0.12 327.2173 143.9851 1.391333 528.7066 5.287066 456.8025 2333.63 1.4286 1.7143 0.076171 297.45 0.8262 0.13 325.7774 143.3515 1.385211 526.3802 5.263802 454.7925 2333.63 1.4286 1.7143 0.076171 296.82 0.8252 0.14 324.3439 142.7207 1.379116 524.0639 5.240639 452.7912 2333.63 1.4286 1.7143 0.076171 296.19 0.8243 0.15 322.9167 142.0927 1.373047 521.7579 5.217579 450.7988 2333.63 1.4286 1.7143 0.076171 295.56 0.8233

83


Time, sec



0.01 343.4441 154.8864 1.496673 568.7358 5.687358 491.3877 2333.63 1.4286 1.7143 0.076171 304.44 0.847 0.02 341.8953 154.1879 1.489924 566.1709 5.661709 489.1717 2333.63 1.4286 1.7143 0.076171 303.78 0.846 0.03 340.3534 153.4926 1.483204 563.6176 5.636176 486.9656 2333.63 1.4286 1.7143 0.076171 303.12 0.845 0.04 338.8184 152.8004 1.476515 561.0758 5.610758 484.7695 2333.63 1.4286 1.7143 0.076171 302.46 0.844 0.05 337.2904 152.1113 1.469857 558.5455 5.585455 482.5833 2333.63 1.4286 1.7143 0.076171 301.81 0.843 0.06 335.7693 151.4253 1.463228 556.0265 5.560265 480.4069 2333.63 1.4286 1.7143 0.076171 301.15 0.842 0.07 334.2551 150.7424 1.456629 553.519 5.53519 478.2404 2333.63 1.4286 1.7143 0.076171 300.5 0.841 0.08 332.7477 150.0625 1.45006 551.0227 5.510227 476.0836 2333.63 1.4286 1.7143 0.076171 299.85 0.84 0.09 331.247 149.3858 1.44352 548.5377 5.485377 473.9366 2333.63 1.4286 1.7143 0.076171 299.2 0.8391 0.1 329.7532 148.7121 1.43701 546.0639 5.460639 471.7992 2333.63 1.4286 1.7143 0.076171 298.55 0.8381

0.11 328.2661 148.0414 1.43053 543.6013 5.436013 469.6715 2333.63 1.4286 1.7143 0.076171 297.9 0.8371 0.12 326.7856 147.3738 1.424078 541.1497 5.411497 467.5534 2333.63 1.4286 1.7143 0.076171 297.25 0.8361 0.13 325.3119 146.7092 1.417656 538.7093 5.387093 465.4448 2333.63 1.4286 1.7143 0.076171 296.61 0.8351 0.14 323.8448 146.0475 1.411263 536.2798 5.362798 463.3457 2333.63 1.4286 1.7143 0.076171 295.96 0.8341 0.15 322.3843 145.3889 1.404898 533.8613 5.338613 461.2562 2333.63 1.4286 1.7143 0.076171 295.32 0.8332

P Time, sec



0.01 343.4201 157.2686 1.519692 577.4831 5.774831 498.9454 2333.63 1.4286 1.7143 0.076171 304.43 0.8535 0.02 341.8474 156.5484 1.512733 574.8385 5.748385 496.6604 2333.63 1.4286 1.7143 0.076171 303.76 0.8525 0.03 340.2819 155.8315 1.505805 572.206 5.72206 494.386 2333.63 1.4286 1.7143 0.076171 303.09 0.8514 0.04 338.7236 155.1179 1.49891 569.5856 5.695856 492.122 2333.63 1.4286 1.7143 0.076171 302.42 0.8504 0.05 337.1724 154.4075 1.492045 566.9772 5.669772 489.8683 2333.63 1.4286 1.7143 0.076171 301.75 0.8494 0.06 335.6283 153.7004 1.485212 564.3807 5.643807 487.625 2333.63 1.4286 1.7143 0.076171 301.09 0.8484 0.07 334.0913 152.9965 1.478411 561.7962 5.617962 485.3919 2333.63 1.4286 1.7143 0.076171 300.42 0.8474 0.08 332.5614 152.2959 1.471641 559.2234 5.592234 483.169 2333.63 1.4286 1.7143 0.076171 299.76 0.8464 0.09 331.0384 151.5984 1.464901 556.6625 5.566625 480.9564 2333.63 1.4286 1.7143 0.076171 299.1 0.8454 0.1 329.5224 150.9042 1.458193 554.1133 5.541133 478.7539 2333.63 1.4286 1.7143 0.076171 298.44 0.8444

0.11 328.0134 150.2131 1.451515 551.5757 5.515757 476.5614 2333.63 1.4286 1.7143 0.076171 297.78 0.8434 0.12 326.5113 149.5252 1.444868 549.0498 5.490498 474.379 2333.63 1.4286 1.7143 0.076171 297.13 0.8424 0.13 325.016 148.8405 1.438251 546.5354 5.465354 472.2066 2333.63 1.4286 1.7143 0.076171 296.47 0.8414 0.14 323.5276 148.1589 1.431665 544.0326 5.440326 470.0441 2333.63 1.4286 1.7143 0.076171 295.82 0.8404 0.15 322.046 147.4804 1.425108 541.5412 5.415412 467.8916 2333.63 1.4286 1.7143 0.076171 295.17 0.8394

84


Time, sec



0.01 344.3692 62.96194 0.608404 231.193333 2.31193333 199.75104 2333.63 1.4286 1.7143 0.076171 114.781 0.8795 0.02 343.7396 62.84683 0.607291 230.770637 2.30770637 199.38583 2333.63 1.4286 1.7143 0.076171 114.68 0.879 0.03 343.1111 62.73192 0.606181 230.348712 2.30348712 199.02129 2333.63 1.4286 1.7143 0.076171 114.579 0.8786 0.04 342.4838 62.61723 0.605073 229.92756 2.2992756 198.65741 2333.63 1.4286 1.7143 0.076171 114.478 0.8782 0.05 341.8576 62.50274 0.603966 229.507177 2.29507177 198.2942 2333.63 1.4286 1.7143 0.076171 114.378 0.8778 0.06 341.2326 62.38847 0.602862 229.087563 2.29087563 197.93165 2333.63 1.4286 1.7143 0.076171 114.277 0.8774 0.07 340.6087 62.2744 0.60176 228.668716 2.28668716 197.56977 2333.63 1.4286 1.7143 0.076171 114.176 0.877 0.08 339.986 62.16054 0.60066 228.250635 2.28250635 197.20855 2333.63 1.4286 1.7143 0.076171 114.076 0.8765 0.09 339.3644 62.04689 0.599561 227.833318 2.27833318 196.84799 2333.63 1.4286 1.7143 0.076171 113.975 0.8761 0.1 338.7439 61.93345 0.598465 227.416765 2.27416765 196.48808 2333.63 1.4286 1.7143 0.076171 113.875 0.8757

0.11 338.1246 61.82022 0.597371 227.000973 2.27000973 196.12884 2333.63 1.4286 1.7143 0.076171 113.775 0.8753 0.12 337.5064 61.70719 0.596279 226.585941 2.26585941 195.77025 2333.63 1.4286 1.7143 0.076171 113.675 0.8749 0.13 336.8893 61.59437 0.595189 226.171668 2.26171668 195.41232 2333.63 1.4286 1.7143 0.076171 113.575 0.8745 0.14 336.2734 61.48175 0.5941 225.758152 2.25758152 195.05504 2333.63 1.4286 1.7143 0.076171 113.475 0.8741 0.15 335.6585 61.36934 0.593014 225.345392 2.25345392 194.69842 2333.63 1.4286 1.7143 0.076171 113.375 0.8736




0.01 343.9218 107.4792 1.038576 394.658806 3.94658806 340.98521 2333.63 1.4286 1.7143 0.076171 234.788 0.8034 0.02 342.847 107.1434 1.03533 393.425455 3.93425455 339.91959 2333.63 1.4286 1.7143 0.076171 234.435 0.8028 0.03 341.7756 106.8085 1.032095 392.195958 3.92195958 338.85731 2333.63 1.4286 1.7143 0.076171 234.082 0.8021 0.04 340.7075 106.4747 1.028869 390.970304 3.90970304 337.79834 2333.63 1.4286 1.7143 0.076171 233.73 0.8015 0.05 339.6428 106.142 1.025654 389.74848 3.8974848 336.74269 2333.63 1.4286 1.7143 0.076171 233.378 0.8008 0.06 338.5814 105.8103 1.022449 388.530474 3.88530474 335.69033 2333.63 1.4286 1.7143 0.076171 233.027 0.8002 0.07 337.5233 105.4796 1.019253 387.316275 3.87316275 334.64126 2333.63 1.4286 1.7143 0.076171 232.676 0.7995 0.08 336.4685 105.15 1.016068 386.105871 3.86105871 333.59547 2333.63 1.4286 1.7143 0.076171 232.326 0.7989 0.09 335.417 104.8214 1.012893 384.899249 3.84899249 332.55295 2333.63 1.4286 1.7143 0.076171 231.977 0.7982 0.1 334.3687 104.4938 1.009727 383.696397 3.83696397 331.51369 2333.63 1.4286 1.7143 0.076171 231.628 0.7976

0.11 333.3238 104.1672 1.006572 382.497305 3.82497305 330.47767 2333.63 1.4286 1.7143 0.076171 231.28 0.7969 0.12 332.2821 103.8417 1.003426 381.30196 3.8130196 329.44489 2333.63 1.4286 1.7143 0.076171 230.932 0.7963 0.13 331.2437 103.5172 1.00029 380.110351 3.80110351 328.41534 2333.63 1.4286 1.7143 0.076171 230.585 0.7956 0.14 330.2085 103.1937 0.997164 378.922466 3.78922466 327.38901 2333.63 1.4286 1.7143 0.076171 230.238 0.795 0.15 329.1766 102.8712 0.994048 377.738293 3.77738293 326.36588 2333.63 1.4286 1.7143 0.076171 229.892 0.7943

85


Time, sec



0.01 343.2724 171.891 1.660989 631.175642 6.31175642 545.33575 2333.63 1.4286 1.7143 0.076171 355.37 0.8258 0.02 341.5535 171.0302 1.652671 628.01508 6.2801508 542.60503 2333.63 1.4286 1.7143 0.076171 354.513 0.8248 0.03 339.8432 170.1738 1.644396 624.870344 6.24870344 539.88798 2333.63 1.4286 1.7143 0.076171 353.657 0.8237 0.04 338.1415 169.3217 1.636161 621.741356 6.21741356 537.18453 2333.63 1.4286 1.7143 0.076171 352.804 0.8226 0.05 336.4483 168.4738 1.627969 618.628035 6.18628035 534.49462 2333.63 1.4286 1.7143 0.076171 351.954 0.8216 0.06 334.7635 167.6302 1.619817 615.530304 6.15530304 531.81818 2333.63 1.4286 1.7143 0.076171 351.105 0.8205 0.07 333.0872 166.7908 1.611705 612.448085 6.12448085 529.15515 2333.63 1.4286 1.7143 0.076171 350.259 0.8194 0.08 331.4193 165.9556 1.603635 609.381299 6.09381299 526.50544 2333.63 1.4286 1.7143 0.076171 349.414 0.8184 0.09 329.7598 165.1246 1.595605 606.329871 6.06329871 523.86901 2333.63 1.4286 1.7143 0.076171 348.572 0.8173 0.1 328.1085 164.2977 1.587615 603.293722 6.03293722 521.24578 2333.63 1.4286 1.7143 0.076171 347.732 0.8162

0.11 326.4655 163.475 1.579665 600.272776 6.00272776 518.63568 2333.63 1.4286 1.7143 0.076171 346.895 0.8152 0.12 324.8308 162.6564 1.571755 597.266958 5.97266958 516.03865 2333.63 1.4286 1.7143 0.076171 346.059 0.8141 0.13 323.2042 161.842 1.563885 594.276191 5.94276191 513.45463 2333.63 1.4286 1.7143 0.076171 345.226 0.813 0.14 321.5858 161.0315 1.556054 591.3004 5.913004 510.88355 2333.63 1.4286 1.7143 0.076171 344.395 0.812 0.15 319.9755 160.2252 1.548262 588.33951 5.8833951 508.32534 2333.63 1.4286 1.7143 0.076171 343.566 0.8109

Fully Open

Time, sec



0.01 342.9088 207.856 2.00852 763.237635 7.63237635 659.43732 2333.63 1.4286 1.7143 0.076171 474.803 0.7857 0.02 340.8302 206.5961 1.996345 758.611229 7.58611229 655.4401 2333.63 1.4286 1.7143 0.076171 473.416 0.7844 0.03 338.7642 205.3438 1.984244 754.012866 7.54012866 651.46712 2333.63 1.4286 1.7143 0.076171 472.033 0.7832 0.04 336.7108 204.0991 1.972217 749.442376 7.49442376 647.51821 2333.63 1.4286 1.7143 0.076171 470.655 0.782 0.05 334.6698 202.8619 1.960262 744.899591 7.44899591 643.59325 2333.63 1.4286 1.7143 0.076171 469.281 0.7807 0.06 332.6412 201.6323 1.94838 740.384341 7.40384341 639.69207 2333.63 1.4286 1.7143 0.076171 467.911 0.7795 0.07 330.6249 200.4101 1.93657 735.896462 7.35896462 635.81454 2333.63 1.4286 1.7143 0.076171 466.545 0.7783 0.08 328.6208 199.1953 1.924831 731.435785 7.31435785 631.96052 2333.63 1.4286 1.7143 0.076171 465.184 0.777 0.09 326.6288 197.9878 1.913164 727.002148 7.27002148 628.12986 2333.63 1.4286 1.7143 0.076171 463.827 0.7758 0.1 324.6489 196.7877 1.901567 722.595385 7.22595385 624.32241 2333.63 1.4286 1.7143 0.076171 462.475 0.7746

0.11 322.6811 195.5949 1.89004 718.215334 7.18215334 620.53805 2333.63 1.4286 1.7143 0.076171 461.126 0.7734 0.12 320.7251 194.4093 1.878584 713.861832 7.13861832 616.77662 2333.63 1.4286 1.7143 0.076171 459.782 0.7721 0.13 318.781 193.2308 1.867197 709.53472 7.0953472 613.038 2333.63 1.4286 1.7143 0.076171 458.443 0.7709 0.14 316.8487 192.0596 1.855879 705.233837 7.05233837 609.32204 2333.63 1.4286 1.7143 0.076171 457.107 0.7697 0.15 314.9281 190.8954 1.844629 700.959024 7.00959024 605.6286 2333.63 1.4286 1.7143 0.076171 455.776 0.7685

86


Time, sec



0.01 342.7648 222.0746 2.145915 815.447574 8.15447574 704.5467 2333.63 1.4286 1.7143 0.076171 474.659 0.8122 0.02 340.544 220.6358 2.132011 810.164354 8.10164354 699.982 2333.63 1.4286 1.7143 0.076171 473.177 0.8108 0.03 338.3377 219.2063 2.118198 804.915363 8.04915363 695.44687 2333.63 1.4286 1.7143 0.076171 471.7 0.8095 0.04 336.1456 217.7861 2.104475 799.70038 7.9970038 690.94113 2333.63 1.4286 1.7143 0.076171 470.227 0.8081 0.05 333.9677 216.3751 2.09084 794.519185 7.94519185 686.46458 2333.63 1.4286 1.7143 0.076171 468.76 0.8068 0.06 331.804 214.9732 2.077294 789.371558 7.89371558 682.01703 2333.63 1.4286 1.7143 0.076171 467.297 0.8054 0.07 329.6543 213.5804 2.063835 784.257282 7.84257282 677.59829 2333.63 1.4286 1.7143 0.076171 465.84 0.804 0.08 327.5185 212.1966 2.050464 779.176142 7.79176142 673.20819 2333.63 1.4286 1.7143 0.076171 464.387 0.8027 0.09 325.3965 210.8218 2.037179 774.127921 7.74127921 668.84652 2333.63 1.4286 1.7143 0.076171 462.939 0.8013 0.1 323.2883 209.4559 2.02398 769.112408 7.69112408 664.51312 2333.63 1.4286 1.7143 0.076171 461.497 0.8

0.11 321.1937 208.0989 2.010867 764.129389 7.64129389 660.20779 2333.63 1.4286 1.7143 0.076171 460.059 0.7986 0.12 319.1127 206.7506 1.997839 759.178656 7.59178656 655.93036 2333.63 1.4286 1.7143 0.076171 458.625 0.7973 0.13 317.0452 205.4111 1.984895 754.259997 7.54259997 651.68064 2333.63 1.4286 1.7143 0.076171 457.197 0.7959 0.14 314.9911 204.0802 1.972035 749.373207 7.49373207 647.45845 2333.63 1.4286 1.7143 0.076171 455.774 0.7946 0.15 312.9503 202.758 1.959258 744.518077 7.44518077 643.26362 2333.63 1.4286 1.7143 0.076171 454.355 0.7932

1-1/2 Fully Open

Time, sec



0.01 342.5969 238.6402 2.305989 876.275718 8.76275718 757.10222 2333.63 1.4286 1.7143 0.076171 474.492 0.8421 0.02 340.2105 236.9779 2.289926 870.171908 8.70171908 751.82853 2333.63 1.4286 1.7143 0.076171 472.898 0.8406 0.03 337.8407 235.3272 2.273975 864.110615 8.64110615 746.59157 2333.63 1.4286 1.7143 0.076171 471.311 0.8391 0.04 335.4874 233.688 2.258136 858.091543 8.58091543 741.39109 2333.63 1.4286 1.7143 0.076171 469.729 0.8375 0.05 333.1505 232.0602 2.242406 852.114398 8.52114398 736.22684 2333.63 1.4286 1.7143 0.076171 468.153 0.836 0.06 330.8299 230.4438 2.226787 846.178887 8.46178887 731.09856 2333.63 1.4286 1.7143 0.076171 466.583 0.8345 0.07 328.5255 228.8386 2.211276 840.284721 8.40284721 726.006 2333.63 1.4286 1.7143 0.076171 465.018 0.833 0.08 326.2371 227.2446 2.195873 834.431611 8.34431611 720.94891 2333.63 1.4286 1.7143 0.076171 463.459 0.8315 0.09 323.9647 225.6617 2.180577 828.619272 8.28619272 715.92705 2333.63 1.4286 1.7143 0.076171 461.906 0.83 0.1 321.708 224.0898 2.165388 822.847419 8.22847419 710.94017 2333.63 1.4286 1.7143 0.076171 460.358 0.8285

0.11 319.4671 222.5289 2.150305 817.115771 8.17115771 705.98803 2333.63 1.4286 1.7143 0.076171 458.816 0.827 0.12 317.2419 220.9788 2.135326 811.424048 8.11424048 701.07038 2333.63 1.4286 1.7143 0.076171 457.28 0.8255 0.13 315.0321 219.4396 2.120453 805.77197 8.0577197 696.18698 2333.63 1.4286 1.7143 0.076171 455.749 0.824 0.14 312.8377 217.911 2.105682 800.159264 8.00159264 691.3376 2333.63 1.4286 1.7143 0.076171 454.223 0.8225 0.15 310.6586 216.3932 2.091015 794.585653 7.94585653 686.522 2333.63 1.4286 1.7143 0.076171 452.704 0.821

87


Time, sec



0.01 342.5729 241.0047 2.328837 884.958122 8.84958122 764.60382 2333.63 1.4286 1.7143 0.076171 474.468 0.8463 0.02 340.1628 239.3092 2.312453 878.73232 8.7873232 759.22472 2333.63 1.4286 1.7143 0.076171 472.858 0.8448 0.03 337.7697 237.6256 2.296185 872.550318 8.72550318 753.88347 2333.63 1.4286 1.7143 0.076171 471.255 0.8432 0.04 335.3935 235.9539 2.280031 866.411807 8.66411807 748.5798 2333.63 1.4286 1.7143 0.076171 469.658 0.8417 0.05 333.0339 234.2939 2.263991 860.316482 8.60316482 743.31344 2333.63 1.4286 1.7143 0.076171 468.066 0.8401 0.06 330.691 232.6456 2.248063 854.264038 8.54264038 738.08413 2333.63 1.4286 1.7143 0.076171 466.481 0.8386 0.07 328.3645 231.0089 2.232248 848.254173 8.48254173 732.89161 2333.63 1.4286 1.7143 0.076171 464.901 0.837 0.08 326.0545 229.3838 2.216544 842.286589 8.42286589 727.73561 2333.63 1.4286 1.7143 0.076171 463.327 0.8355 0.09 323.7606 227.77 2.20095 836.360988 8.36360988 722.61589 2333.63 1.4286 1.7143 0.076171 461.758 0.834 0.1 321.4829 226.1676 2.185466 830.477074 8.30477074 717.53219 2333.63 1.4286 1.7143 0.076171 460.196 0.8325

0.11 319.2212 224.5765 2.170091 824.634554 8.24634554 712.48425 2333.63 1.4286 1.7143 0.076171 458.639 0.8309 0.12 316.9755 222.9966 2.154824 818.833137 8.18833137 707.47183 2333.63 1.4286 1.7143 0.076171 457.088 0.8294 0.13 314.7455 221.4278 2.139665 813.072534 8.13072534 702.49467 2333.63 1.4286 1.7143 0.076171 455.542 0.8279 0.14 312.5312 219.87 2.124612 807.352458 8.07352458 697.55252 2333.63 1.4286 1.7143 0.076171 454.002 0.8264 0.15 310.3325 218.3232 2.109665 801.672623 8.01672623 692.64515 2333.63 1.4286 1.7143 0.076171 452.468 0.8248

P

Time, sec



0.01 342.6037 237.9645 2.29946 873.794694 8.73794694 754.95862 2333.63 1.4286 1.7143 0.076171 474.498 0.8409 0.02 340.2241 236.3117 2.283488 867.725521 8.67725521 749.71485 2333.63 1.4286 1.7143 0.076171 472.91 0.8394 0.03 337.8609 234.6703 2.267628 861.698502 8.61698502 744.50751 2333.63 1.4286 1.7143 0.076171 471.327 0.8379 0.04 335.5142 233.0403 2.251877 855.713346 8.55713346 739.33633 2333.63 1.4286 1.7143 0.076171 469.749 0.8364 0.05 333.1838 231.4217 2.236236 849.769762 8.49769762 734.20107 2333.63 1.4286 1.7143 0.076171 468.178 0.8349 0.06 330.8696 229.8143 2.220704 843.86746 8.4386746 729.10149 2333.63 1.4286 1.7143 0.076171 466.612 0.8333 0.07 328.5715 228.2181 2.205279 838.006154 8.38006154 724.03732 2333.63 1.4286 1.7143 0.076171 465.052 0.8318 0.08 326.2893 226.6329 2.189962 832.18556 8.3218556 719.00832 2333.63 1.4286 1.7143 0.076171 463.497 0.8303 0.09 324.023 225.0588 2.174751 826.405394 8.26405394 714.01426 2333.63 1.4286 1.7143 0.076171 461.948 0.8288 0.1 321.7724 223.4956 2.159646 820.665375 8.20665375 709.05488 2333.63 1.4286 1.7143 0.076171 460.404 0.8273

0.11 319.5374 221.9432 2.144645 814.965226 8.14965226 704.12995 2333.63 1.4286 1.7143 0.076171 458.867 0.8258 0.12 317.318 220.4017 2.129749 809.304668 8.09304668 699.23923 2333.63 1.4286 1.7143 0.076171 457.334 0.8243 0.13 315.114 218.8708 2.114956 803.683427 8.03683427 694.38248 2333.63 1.4286 1.7143 0.076171 455.808 0.8228 0.14 312.9253 217.3506 2.100266 798.10123 7.9810123 689.55946 2333.63 1.4286 1.7143 0.076171 454.287 0.8214 0.15 310.7518 215.8409 2.085678 792.557806 7.92557806 684.76994 2333.63 1.4286 1.7143 0.076171 452.771 0.8199

88


Time, sec




0 345 78.5704 0.759229 288.506857 2.88506857 249.269925 2333.63 1.4286 1.7143 0.076171 162.3496 0.8261 0.01 344.21 78.39146 0.7575 287.849811 2.87849811 248.702237 2333.63 1.4286 1.7143 0.076171 162.1715 0.8256 0.02 343.43 78.21293 0.755774 287.194261 2.87194261 248.135842 2333.63 1.4286 1.7143 0.076171 161.9936 0.8251 0.03 342.65 78.03481 0.754053 286.540204 2.86540204 247.570736 2333.63 1.4286 1.7143 0.076171 161.8159 0.8246 0.04 341.87 77.85709 0.752336 285.887637 2.85887637 247.006918 2333.63 1.4286 1.7143 0.076171 161.6385 0.8241 0.05 341.09 77.67978 0.750623 285.236555 2.85236555 246.444384 2333.63 1.4286 1.7143 0.076171 161.4612 0.8236 0.06 340.31 77.50287 0.748913 284.586957 2.84586957 245.883131 2333.63 1.4286 1.7143 0.076171 161.2842 0.8231 0.07 339.54 77.32637 0.747207 283.938838 2.83938838 245.323156 2333.63 1.4286 1.7143 0.076171 161.1073 0.8227 0.08 338.76 77.15026 0.745506 283.292195 2.83292195 244.764456 2333.63 1.4286 1.7143 0.076171 160.9307 0.8222 0.09 337.99 76.97456 0.743808 282.647024 2.82647024 244.207029 2333.63 1.4286 1.7143 0.076171 160.7543 0.8217 0.1 337.22 76.79926 0.742114 282.003323 2.82003323 243.650871 2333.63 1.4286 1.7143 0.076171 160.5781 0.8212

0.11 336.46 76.62436 0.740424 281.361088 2.81361088 243.09598 2333.63 1.4286 1.7143 0.076171 160.4021 0.8207 0.12 335.69 76.44985 0.738738 280.720315 2.80720315 242.542353 2333.63 1.4286 1.7143 0.076171 160.2263 0.8202 0.13 334.92 76.27574 0.737055 280.081002 2.80081002 241.989986 2333.63 1.4286 1.7143 0.076171 160.0507 0.8197 0.14 334.16 76.10203 0.735377 279.443145 2.79443145 241.438877 2333.63 1.4286 1.7143 0.076171 159.8753 0.8193 0.15 333.4 75.92872 0.733702 278.80674 2.7880674 240.889024 2333.63 1.4286 1.7143 0.076171 159.7001 0.8188

1/2 Fully Open

Time, sec




0 345 154.9012 1.496816 568.790079 5.68790079 491.434628 2333.63 1.4286 1.7143 0.076171 335.8411 0.8064 0.01 343.45 154.2057 1.490095 566.236274 5.66236274 489.228141 2333.63 1.4286 1.7143 0.076171 335.1145 0.8055 0.02 341.91 153.5134 1.483405 563.693935 5.63693935 487.03156 2333.63 1.4286 1.7143 0.076171 334.3895 0.8046 0.03 340.37 152.8241 1.476745 561.163011 5.61163011 484.844842 2333.63 1.4286 1.7143 0.076171 333.6662 0.8036 0.04 338.85 152.1379 1.470114 558.643451 5.58643451 482.667942 2333.63 1.4286 1.7143 0.076171 332.9446 0.8027 0.05 337.32 151.4549 1.463514 556.135203 5.56135203 480.500816 2333.63 1.4286 1.7143 0.076171 332.2246 0.8018 0.06 335.81 150.7748 1.456943 553.638217 5.53638217 478.34342 2333.63 1.4286 1.7143 0.076171 331.5064 0.8008 0.07 334.3 150.0979 1.450401 551.152442 5.51152442 476.19571 2333.63 1.4286 1.7143 0.076171 330.7898 0.7999 0.08 332.8 149.4239 1.443889 548.677828 5.48677828 474.057644 2333.63 1.4286 1.7143 0.076171 330.0749 0.7989 0.09 331.31 148.7531 1.437406 546.214325 5.46214325 471.929177 2333.63 1.4286 1.7143 0.076171 329.3617 0.798 0.1 329.82 148.0852 1.430952 543.761883 5.43761883 469.810267 2333.63 1.4286 1.7143 0.076171 328.6501 0.7971

0.11 328.34 147.4203 1.424528 541.320452 5.41320452 467.70087 2333.63 1.4286 1.7143 0.076171 327.9402 0.7962 0.12 326.86 146.7584 1.418132 538.889982 5.38889982 465.600945 2333.63 1.4286 1.7143 0.076171 327.2319 0.7952 0.13 325.4 146.0995 1.411764 536.470426 5.36470426 463.510448 2333.63 1.4286 1.7143 0.076171 326.5253 0.7943 0.14 323.94 145.4435 1.405426 534.061732 5.34061732 461.429337 2333.63 1.4286 1.7143 0.076171 325.8204 0.7934 0.15 322.48 144.7905 1.399115 531.663854 5.31663854 459.35757 2333.63 1.4286 1.7143 0.076171 325.1171 0.7924

89


Time, sec




0 345 251.618 2.431393 923.929487 9.23929487 798.275077 2333.63 1.4286 1.7143 0.076171 512.1812 0.8323 0.01 342.48 249.7828 2.413661 917.191016 9.17191016 792.453038 2333.63 1.4286 1.7143 0.076171 510.3799 0.8307 0.02 339.99 247.9611 2.396057 910.501691 9.10501691 786.673461 2333.63 1.4286 1.7143 0.076171 508.5854 0.8291 0.03 337.51 246.1527 2.378582 903.861152 9.03861152 780.936036 2333.63 1.4286 1.7143 0.076171 506.7977 0.8276 0.04 335.04 244.3574 2.361234 897.269045 8.97269045 775.240455 2333.63 1.4286 1.7143 0.076171 505.0168 0.826 0.05 332.6 242.5752 2.344013 890.725016 8.90725016 769.586414 2333.63 1.4286 1.7143 0.076171 503.2426 0.8244 0.06 330.18 240.8061 2.326918 884.228714 8.84228714 763.973609 2333.63 1.4286 1.7143 0.076171 501.4751 0.8229 0.07 327.77 239.0498 2.309947 877.779792 8.77779792 758.40174 2333.63 1.4286 1.7143 0.076171 499.7144 0.8213 0.08 325.38 237.3063 2.2931 871.377904 8.71377904 752.870509 2333.63 1.4286 1.7143 0.076171 497.9604 0.8197 0.09 323 235.5756 2.276376 865.022706 8.65022706 747.379618 2333.63 1.4286 1.7143 0.076171 496.213 0.8182 0.1 320.65 233.8575 2.259773 858.713858 8.58713858 741.928773 2333.63 1.4286 1.7143 0.076171 494.4722 0.8166

0.11 318.31 232.1519 2.243292 852.451023 8.52451023 736.517683 2333.63 1.4286 1.7143 0.076171 492.7381 0.8151 0.12 315.99 230.4587 2.226931 846.233864 8.46233864 731.146058 2333.63 1.4286 1.7143 0.076171 491.0106 0.8135 0.13 313.68 228.778 2.21069 840.062048 8.40062048 725.81361 2333.63 1.4286 1.7143 0.076171 489.2896 0.812 0.14 311.4 227.1094 2.194566 833.935245 8.33935245 720.520052 2333.63 1.4286 1.7143 0.076171 487.5751 0.8104 0.15 309.12 225.453 2.178561 827.853127 8.27853127 715.265102 2333.63 1.4286 1.7143 0.076171 485.8672 0.8089

Fully Open

Time, sec




0 345 293.0334 2.831593 1076.00522 10.7600522 929.668507 2333.63 1.4286 1.7143 0.076171 687.7832 0.7751 0.01 342.07 290.5445 2.807542 1066.86593 10.6686593 921.77216 2333.63 1.4286 1.7143 0.076171 684.9654 0.7734 0.02 339.16 288.0767 2.783695 1057.80426 10.5780426 913.942883 2333.63 1.4286 1.7143 0.076171 682.16 0.7717 0.03 336.28 285.6298 2.760051 1048.81957 10.4881957 906.180106 2333.63 1.4286 1.7143 0.076171 679.367 0.77 0.04 333.43 283.2038 2.736608 1039.91118 10.3991118 898.483263 2333.63 1.4286 1.7143 0.076171 676.5863 0.7682 0.05 330.6 280.7983 2.713364 1031.07847 10.3107847 890.851795 2333.63 1.4286 1.7143 0.076171 673.818 0.7665 0.06 327.79 278.4133 2.690318 1022.32077 10.2232077 883.285147 2333.63 1.4286 1.7143 0.076171 671.0619 0.7649 0.07 325 276.0485 2.667467 1013.63746 10.1363746 875.782768 2333.63 1.4286 1.7143 0.076171 668.3179 0.7632 0.08 322.24 273.7039 2.64481 1005.02791 10.0502791 868.344112 2333.63 1.4286 1.7143 0.076171 665.5861 0.7615 0.09 319.51 271.3791 2.622346 996.491479 9.96491479 860.968638 2333.63 1.4286 1.7143 0.076171 662.8664 0.7598 0.1 316.79 269.0741 2.600073 988.027557 9.88027557 853.655809 2333.63 1.4286 1.7143 0.076171 660.1588 0.7581

0.11 314.1 266.7886 2.577988 979.635524 9.79635524 846.405093 2333.63 1.4286 1.7143 0.076171 657.4631 0.7564 0.12 311.43 264.5226 2.556092 971.314772 9.71314772 839.215963 2333.63 1.4286 1.7143 0.076171 654.7793 0.7547 0.13 308.79 262.2758 2.534381 963.064693 9.63064693 832.087895 2333.63 1.4286 1.7143 0.076171 652.1075 0.7531 0.14 306.17 260.0481 2.512854 954.884689 9.54884689 825.020371 2333.63 1.4286 1.7143 0.076171 649.4475 0.7514 0.15 303.56 257.8394 2.491511 946.774163 9.46774163 818.012877 2333.63 1.4286 1.7143 0.076171 646.7992 0.7497

90


Time, sec




0 345 302.61 2.924132 1111.17008 11.1117008 960.050953 2333.63 1.4286 1.7143 0.076171 687.737 0.7877 0.01 341.97 299.9558 2.898483 1101.42367 11.0142367 951.630053 2333.63 1.4286 1.7143 0.076171 684.8271 0.7859 0.02 338.97 297.3248 2.87306 1091.76275 10.9176275 943.283015 2333.63 1.4286 1.7143 0.076171 681.9305 0.7841 0.03 336 294.7168 2.847859 1082.18656 10.8218656 935.009191 2333.63 1.4286 1.7143 0.076171 679.0471 0.7823 0.04 333.05 292.1318 2.82288 1072.69437 10.7269437 926.80794 2333.63 1.4286 1.7143 0.076171 676.1768 0.7805 0.05 330.13 289.5694 2.79812 1063.28544 10.6328544 918.678624 2333.63 1.4286 1.7143 0.076171 673.3197 0.7787 0.06 327.24 287.0295 2.773576 1053.95904 10.5395904 910.620613 2333.63 1.4286 1.7143 0.076171 670.4756 0.7769 0.07 324.37 284.5119 2.749249 1044.71445 10.4471445 902.633281 2333.63 1.4286 1.7143 0.076171 667.6445 0.7752 0.08 321.52 282.0163 2.725134 1035.55094 10.3555094 894.716009 2333.63 1.4286 1.7143 0.076171 664.8263 0.7734 0.09 318.7 279.5427 2.701231 1026.4678 10.264678 886.868181 2333.63 1.4286 1.7143 0.076171 662.021 0.7716 0.1 315.91 277.0907 2.677538 1017.46434 10.1746434 879.089189 2333.63 1.4286 1.7143 0.076171 659.2285 0.7699

0.11 313.14 274.6603 2.654052 1008.53985 10.0853985 871.378429 2333.63 1.4286 1.7143 0.076171 656.4488 0.7681 0.12 310.39 272.2512 2.630773 999.693637 9.99693637 863.735303 2333.63 1.4286 1.7143 0.076171 653.6818 0.7663 0.13 307.67 269.8632 2.607697 990.925019 9.90925019 856.159216 2333.63 1.4286 1.7143 0.076171 650.9274 0.7646 0.14 304.97 267.4961 2.584825 982.233313 9.82233313 848.649582 2333.63 1.4286 1.7143 0.076171 648.1857 0.7628 0.15 302.29 265.1498 2.562152 973.617844 9.73617844 841.205817 2333.63 1.4286 1.7143 0.076171 645.4566 0.7611

1-1/2 Fully Open

Time, sec




0 345 319.7045 3.089316 1173.94001 11.7394001 1014.28417 2333.63 1.4286 1.7143 0.076171 687.6547 0.8097 0.01 341.8 316.7418 3.060688 1163.06134 11.6306134 1004.885 2333.63 1.4286 1.7143 0.076171 684.5804 0.8077 0.02 338.64 313.8066 3.032325 1152.28349 11.5228349 995.572937 2333.63 1.4286 1.7143 0.076171 681.521 0.8058 0.03 335.5 310.8987 3.004225 1141.60552 11.4160552 986.347166 2333.63 1.4286 1.7143 0.076171 678.4763 0.8038 0.04 332.39 308.0176 2.976386 1131.02649 11.3102649 977.206888 2333.63 1.4286 1.7143 0.076171 675.4463 0.8019 0.05 329.31 305.1633 2.948804 1120.5455 11.205455 968.151311 2333.63 1.4286 1.7143 0.076171 672.4309 0.7999 0.06 326.26 302.3354 2.921478 1110.16163 11.1016163 959.17965 2333.63 1.4286 1.7143 0.076171 669.43 0.798 0.07 323.23 299.5337 2.894405 1099.87399 10.9987399 950.291128 2333.63 1.4286 1.7143 0.076171 666.4437 0.7961 0.08 320.24 296.758 2.867583 1089.68168 10.8968168 941.484974 2333.63 1.4286 1.7143 0.076171 663.4717 0.7942 0.09 317.27 294.008 2.84101 1079.58382 10.7958382 932.760424 2333.63 1.4286 1.7143 0.076171 660.5141 0.7922 0.1 314.33 291.2835 2.814683 1069.57954 10.6957954 924.116724 2333.63 1.4286 1.7143 0.076171 657.5708 0.7903

0.11 311.42 288.5842 2.7886 1059.66797 10.5966797 915.553122 2333.63 1.4286 1.7143 0.076171 654.6418 0.7884 0.12 308.53 285.91 2.762759 1049.84824 10.4984824 907.068878 2333.63 1.4286 1.7143 0.076171 651.7268 0.7865 0.13 305.67 283.2605 2.737157 1040.11951 10.4011951 898.663256 2333.63 1.4286 1.7143 0.076171 648.826 0.7846 0.14 302.84 280.6356 2.711792 1030.48093 10.3048093 890.335527 2333.63 1.4286 1.7143 0.076171 645.9392 0.7827 0.15 300.03 278.035 2.686662 1020.93168 10.2093168 882.084969 2333.63 1.4286 1.7143 0.076171 643.0664 0.7808

91


Time, sec




0 345 332.6905 3.2148 1221.62416 12.2162416 1055.48328 2333.63 1.4286 1.7143 0.076171 687.5921 0.826 0.01 341.67 329.4823 3.183799 1209.84379 12.0984379 1045.30504 2333.63 1.4286 1.7143 0.076171 684.393 0.8239 0.02 338.38 326.305 3.153097 1198.17702 11.9817702 1035.22495 2333.63 1.4286 1.7143 0.076171 681.2099 0.8218 0.03 335.12 323.1584 3.122691 1186.62276 11.8662276 1025.24206 2333.63 1.4286 1.7143 0.076171 678.0428 0.8198 0.04 331.88 320.0421 3.092579 1175.17991 11.7517991 1015.35545 2333.63 1.4286 1.7143 0.076171 674.8916 0.8177 0.05 328.68 316.9559 3.062756 1163.84742 11.6384742 1005.56417 2333.63 1.4286 1.7143 0.076171 671.7562 0.8157 0.06 325.51 313.8994 3.033222 1152.6242 11.526242 995.867308 2333.63 1.4286 1.7143 0.076171 668.6366 0.8136 0.07 322.37 310.8724 3.003972 1141.50921 11.4150921 986.263957 2333.63 1.4286 1.7143 0.076171 665.5327 0.8116 0.08 319.27 307.8746 2.975004 1130.5014 11.305014 976.753213 2333.63 1.4286 1.7143 0.076171 662.4443 0.8095 0.09 316.19 304.9057 2.946315 1119.59975 11.1959975 967.334183 2333.63 1.4286 1.7143 0.076171 659.3715 0.8075 0.1 313.14 301.9655 2.917903 1108.80322 11.0880322 958.005983 2333.63 1.4286 1.7143 0.076171 656.3141 0.8054

0.11 310.12 299.0536 2.889765 1098.11081 10.9811081 948.767737 2333.63 1.4286 1.7143 0.076171 653.2721 0.8034 0.12 307.13 296.1697 2.861899 1087.5215 10.875215 939.618577 2333.63 1.4286 1.7143 0.076171 650.2454 0.8014 0.13 304.17 293.3137 2.834301 1077.03431 10.7703431 930.557644 2333.63 1.4286 1.7143 0.076171 647.2339 0.7994 0.14 301.23 290.4852 2.806969 1066.64825 10.6664825 921.584087 2333.63 1.4286 1.7143 0.076171 644.2376 0.7974 0.15 298.33 287.684 2.779901 1056.36234 10.5636234 912.697065 2333.63 1.4286 1.7143 0.076171 641.2564 0.7953

P

Time, sec




0 345 341.5729 3.300631 1254.23973 12.5423973 1083.66312 2333.63 1.4286 1.7143 0.076171 687.5493 0.837 0.01 341.58 338.1911 3.267952 1241.82192 12.4182192 1072.93414 2333.63 1.4286 1.7143 0.076171 684.2648 0.8348 0.02 338.2 334.8427 3.235598 1229.52706 12.2952706 1062.31138 2333.63 1.4286 1.7143 0.076171 680.9972 0.8326 0.03 334.85 331.5276 3.203563 1217.35393 12.1735393 1051.79379 2333.63 1.4286 1.7143 0.076171 677.7464 0.8305 0.04 331.54 328.2452 3.171846 1205.30132 12.0530132 1041.38034 2333.63 1.4286 1.7143 0.076171 674.5124 0.8284 0.05 328.26 324.9954 3.140442 1193.36804 11.9336804 1031.06998 2333.63 1.4286 1.7143 0.076171 671.2951 0.8262 0.06 325.01 321.7777 3.10935 1181.5529 11.815529 1020.86171 2333.63 1.4286 1.7143 0.076171 668.0943 0.8241 0.07 321.79 318.5919 3.078565 1169.85475 11.6985475 1010.7545 2333.63 1.4286 1.7143 0.076171 664.9101 0.822 0.08 318.6 315.4376 3.048085 1158.27241 11.5827241 1000.74736 2333.63 1.4286 1.7143 0.076171 661.7423 0.8198 0.09 315.45 312.3146 3.017907 1146.80475 11.4680475 990.8393 2333.63 1.4286 1.7143 0.076171 658.5909 0.8177 0.1 312.33 309.2225 2.988028 1135.45062 11.3545062 981.029334 2333.63 1.4286 1.7143 0.076171 655.4557 0.8156

0.11 309.23 306.161 2.958444 1124.2089 11.242089 971.316494 2333.63 1.4286 1.7143 0.076171 652.3367 0.8135 0.12 306.17 303.1298 2.929154 1113.07849 11.1307849 961.699817 2333.63 1.4286 1.7143 0.076171 649.2338 0.8114 0.13 303.14 300.1286 2.900153 1102.05828 11.0205828 952.178352 2333.63 1.4286 1.7143 0.076171 646.147 0.8093 0.14 300.14 297.1571 2.87144 1091.14717 10.9114717 942.751155 2333.63 1.4286 1.7143 0.076171 643.0761 0.8072 0.15 297.17 294.2151 2.843011 1080.34409 10.8034409 933.417294 2333.63 1.4286 1.7143 0.076171 640.0212 0.8051

92


Time, sec



0.01 343.6709 132.4013 1.279398 486.171285 4.86171285 420.05199 2333.63 1.4286 1.7143 0.076171 276.8601 0.82116 0.02 342.3469 131.8912 1.274469 484.298281 4.84298281 418.433715 2333.63 1.4286 1.7143 0.076171 276.3463 0.82034 0.03 341.0279 131.3831 1.269559 482.432493 4.82432493 416.821674 2333.63 1.4286 1.7143 0.076171 275.8334 0.81952 0.04 339.7141 130.8769 1.264668 480.573893 4.80573893 415.215843 2333.63 1.4286 1.7143 0.076171 275.3216 0.8187 0.05 338.4053 130.3727 1.259796 478.722453 4.78722453 413.616199 2333.63 1.4286 1.7143 0.076171 274.8108 0.81788 0.06 337.1016 129.8704 1.254942 476.878146 4.76878146 412.022718 2333.63 1.4286 1.7143 0.076171 274.301 0.81706 0.07 335.8029 129.3701 1.250108 475.040945 4.75040945 410.435376 2333.63 1.4286 1.7143 0.076171 273.7922 0.81625 0.08 334.5092 128.8717 1.245292 473.210821 4.73210821 408.854149 2333.63 1.4286 1.7143 0.076171 273.2845 0.81543 0.09 333.2205 128.3752 1.240494 471.387748 4.71387748 407.279014 2333.63 1.4286 1.7143 0.076171 272.7778 0.81461 0.1 331.9367 127.8806 1.235715 469.571699 4.69571699 405.709948 2333.63 1.4286 1.7143 0.076171 272.2721 0.8138

0.11 330.6579 127.3879 1.230954 467.762646 4.67762646 404.146926 2333.63 1.4286 1.7143 0.076171 271.7674 0.81298 0.12 329.3841 126.8972 1.226212 465.960562 4.65960562 402.589926 2333.63 1.4286 1.7143 0.076171 271.2637 0.81217 0.13 328.1151 126.4083 1.221488 464.165421 4.64165421 401.038924 2333.63 1.4286 1.7143 0.076171 270.761 0.81135 0.14 326.851 125.9213 1.216782 462.377196 4.62377196 399.493897 2333.63 1.4286 1.7143 0.076171 270.2593 0.81054 0.15 325.5918 125.4362 1.212094 460.59586 4.6059586 397.954823 2333.63 1.4286 1.7143 0.076171 269.7587 0.80973




0.01 342.4838 249.7828 2.413661 917.191016 9.17191016 792.453038 2333.63 1.4286 1.7143 0.076171 582.9582 0.77728 0.02 339.986 247.9611 2.396057 910.501691 9.10501691 786.673461 2333.63 1.4286 1.7143 0.076171 580.9085 0.7758 0.03 337.5064 246.1527 2.378582 903.861152 9.03861152 780.936036 2333.63 1.4286 1.7143 0.076171 578.8666 0.77433 0.04 335.0449 244.3574 2.361234 897.269045 8.97269045 775.240455 2333.63 1.4286 1.7143 0.076171 576.8324 0.77286 0.05 332.6013 242.5752 2.344013 890.725016 8.90725016 769.586414 2333.63 1.4286 1.7143 0.076171 574.806 0.77139 0.06 330.1755 240.8061 2.326918 884.228714 8.84228714 763.973609 2333.63 1.4286 1.7143 0.076171 572.7872 0.76993 0.07 327.7675 239.0498 2.309947 877.779792 8.77779792 758.40174 2333.63 1.4286 1.7143 0.076171 570.7761 0.76847 0.08 325.377 237.3063 2.2931 871.377904 8.71377904 752.870509 2333.63 1.4286 1.7143 0.076171 568.7726 0.76701 0.09 323.0039 235.5756 2.276376 865.022706 8.65022706 747.379618 2333.63 1.4286 1.7143 0.076171 566.7767 0.76555 0.1 320.6482 233.8575 2.259773 858.713858 8.58713858 741.928773 2333.63 1.4286 1.7143 0.076171 564.7884 0.76409

0.11 318.3096 232.1519 2.243292 852.451023 8.52451023 736.517683 2333.63 1.4286 1.7143 0.076171 562.8077 0.76264 0.12 315.9881 230.4587 2.226931 846.233864 8.46233864 731.146058 2333.63 1.4286 1.7143 0.076171 560.8345 0.76119 0.13 313.6835 228.778 2.21069 840.062048 8.40062048 725.81361 2333.63 1.4286 1.7143 0.076171 558.8688 0.75974 0.14 311.3957 227.1094 2.194566 833.935245 8.33935245 720.520052 2333.63 1.4286 1.7143 0.076171 556.9105 0.7583 0.15 309.1246 225.453 2.178561 827.853127 8.27853127 715.265102 2333.63 1.4286 1.7143 0.076171 554.9597 0.75685

93


Time, sec



0.01 340.35 458.7354 4.432776 1684.45499 16.8445499 1455.36911 2333.63 1.4286 1.7143 0.076171 896.4904 0.84942 0.02 335.7626 452.5524 4.37303 1661.75133 16.6175133 1435.75315 2333.63 1.4286 1.7143 0.076171 890.6585 0.84643 0.03 331.2371 446.4527 4.314089 1639.35367 16.3935367 1416.40157 2333.63 1.4286 1.7143 0.076171 884.8676 0.84346 0.04 326.7726 440.4353 4.255942 1617.2579 16.172579 1397.31083 2333.63 1.4286 1.7143 0.076171 879.1173 0.84049 0.05 322.3682 434.4989 4.198579 1595.45995 15.9545995 1378.47739 2333.63 1.4286 1.7143 0.076171 873.4075 0.83753 0.06 318.0232 428.6426 4.141989 1573.95579 15.7395579 1359.8978 2333.63 1.4286 1.7143 0.076171 867.7377 0.83458 0.07 313.7368 422.8652 4.086162 1552.74148 15.5274148 1341.56864 2333.63 1.4286 1.7143 0.076171 862.1079 0.83164 0.08 309.5081 417.1657 4.031087 1531.81309 15.3181309 1323.48651 2333.63 1.4286 1.7143 0.076171 856.5177 0.82871 0.09 305.3365 411.543 3.976755 1511.16679 15.1116679 1305.64811 2333.63 1.4286 1.7143 0.076171 850.9669 0.82578 0.1 301.2211 405.9961 3.923155 1490.79877 14.9079877 1288.05014 2333.63 1.4286 1.7143 0.076171 845.4552 0.82287

0.11 297.1611 400.5239 3.870277 1470.70527 14.7070527 1270.68936 2333.63 1.4286 1.7143 0.076171 839.9823 0.81996 0.12 293.1559 395.1255 3.818112 1450.8826 14.508826 1253.56257 2333.63 1.4286 1.7143 0.076171 834.548 0.81706 0.13 289.2046 389.7999 3.76665 1431.32711 14.3132711 1236.66662 2333.63 1.4286 1.7143 0.076171 829.152 0.81418 0.14 285.3066 384.546 3.715882 1412.03519 14.1203519 1219.99841 2333.63 1.4286 1.7143 0.076171 823.7942 0.8113 0.15 281.4611 379.363 3.665798 1393.0033 13.930033 1203.55485 2333.63 1.4286 1.7143 0.076171 818.4742 0.80842

Fully Open

Time, sec



0.01 339.85 507.3099 4.902154 1862.81849 18.6281849 1609.47518 2333.63 1.4286 1.7143 0.076171 1212.463 0.7681 0.02 334.7769 499.7371 4.828977 1835.01134 18.3501134 1585.4498 2333.63 1.4286 1.7143 0.076171 1203.726 0.76511 0.03 329.7796 492.2772 4.756893 1807.61927 18.0761927 1561.78305 2333.63 1.4286 1.7143 0.076171 1195.056 0.76212 0.04 324.8568 484.9288 4.685884 1780.6361 17.806361 1538.46959 2333.63 1.4286 1.7143 0.076171 1186.454 0.75915 0.05 320.0075 477.69 4.615936 1754.05572 17.5405572 1515.50414 2333.63 1.4286 1.7143 0.076171 1177.918 0.75619 0.06 315.2306 470.5593 4.547032 1727.87211 17.2787211 1492.88151 2333.63 1.4286 1.7143 0.076171 1169.45 0.75324 0.07 310.525 463.5351 4.479156 1702.07937 17.0207937 1470.59657 2333.63 1.4286 1.7143 0.076171 1161.047 0.75029 0.08 305.8897 456.6157 4.412294 1676.67164 16.7667164 1448.64429 2333.63 1.4286 1.7143 0.076171 1152.71 0.74736 0.09 301.3235 449.7996 4.346429 1651.64318 16.5164318 1427.01971 2333.63 1.4286 1.7143 0.076171 1144.438 0.74444 0.1 296.8255 443.0852 4.281548 1626.98834 16.2698834 1405.71792 2333.63 1.4286 1.7143 0.076171 1136.23 0.74152

0.11 292.3946 436.4711 4.217636 1602.70153 16.0270153 1384.73412 2333.63 1.4286 1.7143 0.076171 1128.087 0.73862 0.12 288.0299 429.9557 4.154677 1578.77726 15.7877726 1364.06355 2333.63 1.4286 1.7143 0.076171 1120.007 0.73573 0.13 283.7304 423.5375 4.092658 1555.21012 15.5521012 1343.70154 2333.63 1.4286 1.7143 0.076171 1111.991 0.73284 0.14 279.495 417.2152 4.031565 1531.99477 15.3199477 1323.64348 2333.63 1.4286 1.7143 0.076171 1104.037 0.72997 0.15 275.3229 410.9872 3.971384 1509.12598 15.0912598 1303.88484 2333.63 1.4286 1.7143 0.076171 1096.145 0.7271

94


Time, sec



0.01 339.2796 562.5593 5.436031 2065.69178 20.6569178 1784.7577 2333.63 1.4286 1.7143 0.076171 1210.999 0.80933 0.02 333.654 553.2315 5.345896 2031.44055 20.3144055 1755.16464 2333.63 1.4286 1.7143 0.076171 1201.302 0.80583 0.03 328.1216 544.0584 5.257256 1997.75724 19.9775724 1726.06226 2333.63 1.4286 1.7143 0.076171 1191.689 0.80233 0.04 322.6811 535.0374 5.170085 1964.63244 19.6463244 1697.44242 2333.63 1.4286 1.7143 0.076171 1182.159 0.79886 0.05 317.3307 526.1659 5.08436 1932.05687 19.3205687 1669.29714 2333.63 1.4286 1.7143 0.076171 1172.711 0.79539 0.06 312.069 517.4416 5.000056 1900.02144 19.0002144 1641.61853 2333.63 1.4286 1.7143 0.076171 1163.346 0.79194 0.07 306.8946 508.8619 4.917151 1868.51719 18.6851719 1614.39886 2333.63 1.4286 1.7143 0.076171 1154.062 0.7885 0.08 301.806 500.4244 4.835619 1837.53532 18.3753532 1587.63051 2333.63 1.4286 1.7143 0.076171 1144.858 0.78507 0.09 296.8018 492.1269 4.75544 1807.06715 18.0706715 1561.30602 2333.63 1.4286 1.7143 0.076171 1135.734 0.78165 0.1 291.8805 483.9669 4.67659 1777.10418 17.7710418 1535.41801 2333.63 1.4286 1.7143 0.076171 1126.689 0.77825

0.11 287.0408 475.9423 4.599047 1747.63802 17.4763802 1509.95925 2333.63 1.4286 1.7143 0.076171 1117.723 0.77486 0.12 282.2814 468.0507 4.522791 1718.66044 17.1866044 1484.92262 2333.63 1.4286 1.7143 0.076171 1108.835 0.77148 0.13 277.6009 460.2899 4.447798 1690.16334 16.9016334 1460.30112 2333.63 1.4286 1.7143 0.076171 1100.024 0.76812 0.14 272.998 452.6579 4.374049 1662.13874 16.6213874 1436.08787 2333.63 1.4286 1.7143 0.076171 1091.29 0.76477 0.15 268.4714 445.1523 4.301523 1634.57883 16.3457883 1412.27611 2333.63 1.4286 1.7143 0.076171 1082.632 0.76143

1-1/2 Fully Open

Time, sec



0.01 339.1371 576.3277 5.569076 2116.24876 21.1624876 1828.43893 2333.63 1.4286 1.7143 0.076171 1210.633 0.8193 0.02 333.3738 566.5336 5.474435 2080.28535 20.8028535 1797.36654 2333.63 1.4286 1.7143 0.076171 1200.697 0.81566 0.03 327.7085 556.906 5.381403 2044.93309 20.4493309 1766.82219 2333.63 1.4286 1.7143 0.076171 1190.849 0.81204 0.04 322.1394 547.442 5.289952 2010.18161 20.1018161 1736.79691 2333.63 1.4286 1.7143 0.076171 1181.088 0.80843 0.05 316.665 538.1388 5.200054 1976.02069 19.7602069 1707.28188 2333.63 1.4286 1.7143 0.076171 1171.414 0.80483 0.06 311.2836 528.9937 5.111685 1942.4403 19.424403 1678.26842 2333.63 1.4286 1.7143 0.076171 1161.826 0.80125 0.07 305.9937 520.004 5.024817 1909.43058 19.0943058 1649.74802 2333.63 1.4286 1.7143 0.076171 1152.322 0.79768 0.08 300.7936 511.1671 4.939426 1876.98182 18.7698182 1621.71229 2333.63 1.4286 1.7143 0.076171 1142.904 0.79413 0.09 295.682 502.4803 4.855485 1845.08449 18.4508449 1594.153 2333.63 1.4286 1.7143 0.076171 1133.569 0.79059 0.1 290.6572 493.9412 4.772972 1813.72922 18.1372922 1567.06205 2333.63 1.4286 1.7143 0.076171 1124.318 0.78706

0.11 285.7177 485.5472 4.69186 1782.90681 17.8290681 1540.43148 2333.63 1.4286 1.7143 0.076171 1115.148 0.78354 0.12 280.8623 477.2958 4.612127 1752.60819 17.5260819 1514.25347 2333.63 1.4286 1.7143 0.076171 1106.061 0.78004 0.13 276.0893 469.1847 4.533749 1722.82446 17.2282446 1488.52033 2333.63 1.4286 1.7143 0.076171 1097.054 0.77655 0.14 271.3975 461.2114 4.456702 1693.54687 16.9354687 1463.2245 2333.63 1.4286 1.7143 0.076171 1088.128 0.77308 0.15 266.7854 453.3736 4.380965 1664.76683 16.6476683 1438.35854 2333.63 1.4286 1.7143 0.076171 1079.282 0.76962

95


Time, sec



0.01 338.8456 604.4655 5.840972 2219.56931 22.1956931 1917.70789 2333.63 1.4286 1.7143 0.076171 1209.884 0.83932 0.02 332.8009 593.6825 5.736775 2179.97449 21.7997449 1883.49796 2333.63 1.4286 1.7143 0.076171 1199.459 0.83541 0.03 326.8641 583.0918 5.634437 2141.08599 21.4108599 1849.8983 2333.63 1.4286 1.7143 0.076171 1189.13 0.83151 0.04 321.0332 572.69 5.533924 2102.89123 21.0289123 1816.89802 2333.63 1.4286 1.7143 0.076171 1178.898 0.82763 0.05 315.3063 562.4738 5.435205 2065.37782 20.6537782 1784.48644 2333.63 1.4286 1.7143 0.076171 1168.761 0.82376 0.06 309.6815 552.4399 5.338246 2028.53361 20.2853361 1752.65304 2333.63 1.4286 1.7143 0.076171 1158.719 0.81991 0.07 304.1571 542.5849 5.243018 1992.34667 19.9234667 1721.38752 2333.63 1.4286 1.7143 0.076171 1148.77 0.81608 0.08 298.7313 532.9058 5.149488 1956.80526 19.5680526 1690.67975 2333.63 1.4286 1.7143 0.076171 1138.915 0.81226 0.09 293.4022 523.3993 5.057626 1921.89787 19.2189787 1660.51976 2333.63 1.4286 1.7143 0.076171 1129.151 0.80845 0.1 288.1682 514.0624 4.967403 1887.6132 18.876132 1630.89781 2333.63 1.4286 1.7143 0.076171 1119.479 0.80466

0.11 283.0276 504.892 4.87879 1853.94013 18.5394013 1601.80427 2333.63 1.4286 1.7143 0.076171 1109.897 0.80089 0.12 277.9787 495.8853 4.791757 1820.86775 18.2086775 1573.22974 2333.63 1.4286 1.7143 0.076171 1100.405 0.79713 0.13 273.0198 487.0392 4.706277 1788.38535 17.8838535 1545.16494 2333.63 1.4286 1.7143 0.076171 1091.001 0.79338 0.14 268.1494 478.3509 4.622322 1756.4824 17.564824 1517.6008 2333.63 1.4286 1.7143 0.076171 1081.686 0.78965 0.15 263.3659 469.8176 4.539865 1725.14857 17.2514857 1490.52836 2333.63 1.4286 1.7143 0.076171 1072.459 0.78594

P

Time, sec



0.01 338.8117 607.7326 5.872542 2231.56583 22.3156583 1928.07288 2333.63 1.4286 1.7143 0.076171 1209.797 0.84162 0.02 332.7343 596.8315 5.767205 2191.53783 21.9153783 1893.48869 2333.63 1.4286 1.7143 0.076171 1199.315 0.83767 0.03 326.766 586.1261 5.663757 2152.22783 21.5222783 1859.52484 2333.63 1.4286 1.7143 0.076171 1188.931 0.83374 0.04 320.9048 575.6126 5.562166 2113.62293 21.1362293 1826.17021 2333.63 1.4286 1.7143 0.076171 1178.644 0.82983 0.05 315.1486 565.2878 5.462396 2075.7105 20.757105 1793.41387 2333.63 1.4286 1.7143 0.076171 1168.453 0.82593 0.06 309.4958 555.1481 5.364416 2038.47811 20.3847811 1761.24509 2333.63 1.4286 1.7143 0.076171 1158.358 0.82205 0.07 303.9443 545.1903 5.268194 2001.91356 20.0191356 1729.65332 2333.63 1.4286 1.7143 0.076171 1148.358 0.81818 0.08 298.4924 535.4111 5.173697 1966.00488 19.6600488 1698.62822 2333.63 1.4286 1.7143 0.076171 1138.452 0.81433 0.09 293.1383 525.8074 5.080896 1930.7403 19.307403 1668.15962 2333.63 1.4286 1.7143 0.076171 1128.638 0.81049 0.1 287.8802 516.3759 4.989759 1896.10827 18.9610827 1638.23754 2333.63 1.4286 1.7143 0.076171 1118.917 0.80667

0.11 282.7164 507.1135 4.900256 1862.09743 18.6209743 1608.85218 2333.63 1.4286 1.7143 0.076171 1109.288 0.80287 0.12 277.6453 498.0173 4.81236 1828.69666 18.2869666 1579.99391 2333.63 1.4286 1.7143 0.076171 1099.749 0.79908 0.13 272.6651 489.0843 4.726039 1795.895 17.95895 1551.65328 2333.63 1.4286 1.7143 0.076171 1090.3 0.7953 0.14 267.7743 480.3115 4.641268 1763.68171 17.6368171 1523.821 2333.63 1.4286 1.7143 0.076171 1080.94 0.79154 0.15 262.9712 471.6961 4.558016 1732.04624 17.3204624 1496.48795 2333.63 1.4286 1.7143 0.076171 1071.668 0.7878

96

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, NI-

9162. 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]

97

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.

98

Fig.C-3—Display Shot of MAX with NI USB-9237 Device

99

Fig. C-4—Front Panel View of the Developed Program

100

Fig.C-5—Block Diagram of the Developed Program

101

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

102

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.

Fig. C-7—Empirical Measurement of Minimum Value for Pressure Drop Increment

y = -439.12x + 683.57 R² = 0.9995

y = -389.28x + 709.24 R² = 0.9979

630

640

650

660

670

680

690

700

710

720

0 0.02 0.04 0.06 0.08 0.1

Pres

sure

, psi

g

Time, sec

Plot of Pressure vs. Time for 5/16" Port ID 1-1/2" J-20 Camco GLV

Test 2

Test 1

103

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.

104

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.

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.

Pressure Gage

GLV

Depth Micrometer

Digital Ohm-Meter

Nylon

Bushing

Gas

Inlet

Valve

Gas

Outlet

Valve

105

Table D-1—Probe Test Results for 1/2” Monel Port, 1-1/2” J-20 GLV at Pbt= 149 psig

Increasing Pressure

Decreasing Pressure

Pressure psig

Stem Travel

inch

Stem Reading

Pressure psig

Stem Travel

inch

Stem Reading

142 0 0.59

140 0 0.59

150 0.034 0.624

145 0.005 0.595

155 0.041 0.631

152 0.056 0.646

160 0.058 0.648

156 0.09 0.68

166 0.096 0.686

164 0.136 0.726

170 0.113 0.703

170 0.16 0.75

176 0.147 0.737

177 0.18 0.77

180 0.16 0.75

184 0.19 0.78

185 0.175 0.765

194 0.202 0.792

190 0.185 0.775

201 0.2075 0.7975

196 0.194 0.784

208 0.212 0.802

200 0.199 0.789

215 0.215 0.805

209 0.206 0.796

222 0.218 0.808

210 0.207 0.797

233 0.221 0.811

216 0.211 0.801

220 0.213 0.803

225 0.2165 0.8065

236 0.221 0.811

106

Fig. D-2—Probe Test Results for 1/2” Monel Port in 1-1/2” J-20 GLV at set Pbt = 149 psig

0

20

40

60

80

100

120

140

160

180

200

220

240

260

280

0 0.05 0.1 0.15 0.2 0.25

Pres

sure

, psi

g

Stem Travel, inch

Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV, PTRO =200 psig

InreasingPressure

DecreasingPressure


107


Increasing Pressure

Decreasing Pressure

Pressure

psig

Stem

Travel inch

Stem

Reading

Pressure

psig

Stem

Travel

inch

Stem

Reading

434 0 0.645

426 0 0.645

444 0.051 0.696

437 0.063 0.708

452 0.079 0.724

448 0.108 0.753

464 0.118 0.763

460 0.147 0.792

473 0.145 0.79

469 0.1685 0.8135

485 0.168 0.813

480 0.1875 0.8325

495 0.187 0.832

493 0.203 0.848

506 0.197 0.842

516 0.2175 0.8625

515 0.207 0.852

525 0.2195 0.8645

525 0.214 0.859

533 0.2215 0.8665

535 0.219 0.864

546 0.2225 0.8675

542 0.221 0.866

558 0.2225 0.8675

560 0.222 0.867

571 0.2225 0.8675

568 0.2223 0.8673

590 0.2228 0.8678

585 0.2225 0.8675

608 0.2228 0.8678

597 0.2228 0.8678

108


380

400

420

440

460

480

500

520

540

560

580

600

620

640

0 0.05 0.1 0.15 0.2 0.25

Pres

sure

, psi

g

Stem Travel, inch

Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV, PTRO = 596 psig

InreasingPressure

DecreasingPressure


109


Increasing Pressure

Decreasing Pressure

Pressure

psig

Stem Travel

inch

Stem

Reading

Pressure

psig

Stem Travel

inch

Stem

Reading

501 0 0.644

485 0 0.644

507 0 0.644

494 0 0.644

519 0.004 0.648

515 0.003 0.647

528 0.025 0.669

534 0.0585 0.7025

540 0.059 0.703

545 0.094 0.738

550 0.0895 0.7335

553 0.114 0.758

560 0.122 0.766

565 0.139 0.783

569 0.1295 0.7735

574 0.1485 0.7925

580 0.14 0.784

583 0.1575 0.8015

590 0.1495 0.7935

591 0.1645 0.8085

600 0.1565 0.8005

601 0.1705 0.8145

610 0.163 0.807

615 0.1705 0.822

620 0.169 0.813

625 0.178 0.8263

631 0.174 0.818

635 0.1823 0.8305

639 0.1788 0.8228

644 0.1865 0.8335

650 0.184 0.828

655 0.1895 0.836

660 0.1875 0.8315

665 0.192 0.8383

670 0.1915 0.8355

674 0.1943 0.8393

684 0.195 0.839

684 0.1953 0.8395

692 0.1955 0.8395

695 0.1955 0.84

700 0.196 0.84

705 0.196 0.84

710 0.196 0.84

712 0.196 0.84

725 0.196 0.84

110


500

510

520

530

540

550

560

570

580

590

600

610

620

630

640

650

660

670

680

690

700

710

720

730

0 0.05 0.1 0.15 0.2 0.25

Pres

sure

, psi

g

Stem Travel, inch

Pressure vs Stem Travel, 1/2" Port, 1.5" J-20 Camco GLV, PTRO = 694 psig

IncreasingPressure

DecreasingPressure


111

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