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University of Calgary
PRISM: University of Calgary's Digital Repository
Graduate Studies The Vault: Electronic Theses and Dissertations
2016
Depressurization Dynamic Modeling and Effect on
Flare Flame Distortion
Shafaghat, Ali
Shafaghat, A. (2016). Depressurization Dynamic Modeling and Effect on Flare Flame Distortion
(Unpublished master's thesis). University of Calgary, Calgary, AB. doi:10.11575/PRISM/25558
http://hdl.handle.net/11023/2905
master thesis
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UNIVERSITY OF CALGARY
Depressurization Dynamic Modeling and Effect on Flare Flame Distortion
by
Ali Shafaghat
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF MASTER OF ENGINEERING
GRADUATE PROGRAM IN CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
APRIL, 2016
© Ali Shafaghat 2016
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Abstract
The aim of equipment depressurization in an upset operation is to maintain the internal pressure
of the vessels and piping below the rupture pressure, as the material ultimate tensile strength
decreases with temperature tolerance beyond the acceptable limit. Depressurizing also causes to
decrease the extent and duration of leaks that may occur as a result of mechanical failure. In the
case of ignition and fire, a depressurization system can limit the fuel supply to the fire. In addition,
the main objective of the pressure relief facilities is to keep personnel safe as well as equipment
exposed to overpressure conditions that happen during process upsets. This thesis is intended to
examine the modeling of depressurization in different scenarios, evaluate flare flame distortion
and heat radiation as the most important consequences of this event, and consequently offer some
recommendations for the design of any gas plant that has potential for overpressure.
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Acknowledgements
First and foremost, I would like to declare deep appreciation to my supervisor professor Alex De
Visscher, for his support, value knowledge, and criticism during the course of my Master study
and research. His extensive knowledge, and his value book helped me from the first point to the
end of my research for developing and preparation of my thesis.
I am extremely thankful to my dearest wife, Samaneh Golpich, for her love, kindness and
motivation and encouragement in many respects through my studies. Besides, I would like to
express deep thankful to my parents and my brothers for their support and encouragement.
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Dedication
Dedicated to my gentle-hearted and beloved wife, Samaneh Golpich.
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Table of Contents
Abstract ............................................................................................................................... ii Acknowledgements ............................................................................................................ iii Dedication .......................................................................................................................... iv Table of Contents .................................................................................................................v List of Tables .................................................................................................................... vii
List of Figures .................................................................................................................. viii List of Symbols and Abbreviations.................................................................................... xi 1 Chapter One: Introduction ........................................................................................17 2 Chapter Two: Literature Review ..............................................................................21
2.1 MAJOR SOURCE OF OVERPRESSURE AND RELIEF SCENARIOS ...............21
2.2 PRESSURE RELIEVING DEVICES.......................................................................22
2.2.1 Pressure relief valves .......................................................................................22 2.2.2 Rupture Disks...................................................................................................23
2.3 DEPRESSURIZATION SYSTEM ...........................................................................23
2.4 FLOW AND LEVEL CONTROLS ..........................................................................24 3 Chapter Three: Dynamic Modeling of Depressurization ..........................................27
3.1 MATERIAL AND ENERGY BALANCES .............................................................27
3.1.1 Modeling of High Pressurized Gas Discharges Across the Orifice .................29 3.1.2 Calculation of the vessel wall temperature ......................................................32
3.2 VALVE CORRELATIONS AND EQUATIONS USING FOR DYNAMIC
CALCULATION ......................................................................................................33 3.2.1 General valve equation ....................................................................................33
3.2.2 Supersonic valve equation ...............................................................................35 3.2.3 Subsonic valve equation ..................................................................................35
3.2.4 Masoneilan valve equation ..............................................................................36 3.2.5 Fisher / Universal gas sizing equation .............................................................37
3.3 CHARACTERIZATION OF DISCHARGES OF LIQUEFIED PRESSURIZED
GASES ......................................................................................................................37
3.3.1 Numerical Process to Determine Discharge Flow Type from the Vessel .......38 3.3.2 Modeling Discharge Flow of LPG Across the Vessel Holes ...........................47
3.4 PREDICTION OF DISCHARGE GAS FROM PIPELINES DUE TO RUPTURE 49
3.5 RELIEF SCENARIOS ..............................................................................................52 3.5.1 Fire case ...........................................................................................................52 3.5.2 Adiabatic case or cold depressurization ...........................................................54
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4 Chapter four: Flare Flame Distortion and Heat Radiation Modeling .......................56
4.1 PROCESS FLARES .................................................................................................56 4.1.1 Warm Flare System..........................................................................................56 4.1.2 Cold Flare System ............................................................................................56 4.1.3 Storage Tank Area Flare System .....................................................................57 4.1.4 Sour Gas Flare System .....................................................................................57
4.2 MODELING OF THE FLARE FLAME DISTORTION .........................................57 4.2.1 Calculation of Flare Diameter, Stack Height, and Flare Flame Distortion ......57
4.3 MODELING OF HEAT RADIATION FROM FLARE FLAME DURING
DEPRESSURIZATION ............................................................................................61 4.3.1 Determination of Jet Flare Flames at Flare Tip ...............................................65 4.3.2 Determination of the View Factors ..................................................................76
4.3.3 Mudan Model for Determination of View Factor ............................................77 4.3.4 Determining the view factors with consideration of the crosswind way .........78
5 Chapter Five: Depressurization Dynamic Calculation .............................................80
5.1 DEPRESSURIZATION DYNAMIC CALCULATION ..........................................80 5.1.1 Adiabatic Depressurization Study/Cold Depressurization ...............................81
5.1.2 Adiabatic case ..................................................................................................84
5.2 FIRE CASE DEPRESSURIZATION .......................................................................87
5.2.1 Adiabatic and fire case with different composition .........................................90 5.2.2 Adiabatic and fire case for liquid Methane ......................................................95
5.2.3 Dynamic calculation for determination of flare flame distortion and heat radiation
101
5.3 A CASE STUDY FOR PIPELINE .........................................................................108 6 Conclusion and Future Work ..................................................................................111
6.1 CONCLUSION .......................................................................................................111
6.2 FUTURE WORK ....................................................................................................114 7 References ...............................................................................................................115 Appendices .......................................................................................................................122
Appendix A: Determination of the Darcy Friction Factor fD ......................................122 Appendix B: Auxiliary Equation Buoyance flux parameter De Visscher Air Dispersion
modeling book ......................................................................................................124
Appendix C: Excess Data and Figures ........................................................................125 Appendix D: Fort McMurray Historical Wind Speed .................................................126 Appendix E: Mollier diagram ......................................................................................127
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List of Tables
Table 5-1 : Temperature data between years\ 1981 and 2010 in Fort McMurray, Alberta.81
Table 5-2 Assumption for adiabatic depressurization for methane gas .............................83
Table 5-3 Table 5-3: Assumptions for fire case depressurization for methane gas ...........86
Table 5-4: Assumption for adiabatic and fire case depressurization .................................89
Table 5-5: Depressurization in 15 min considering Fire API, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................90
Table 5-6: Depressurization in 15 min considering Adiabatic, Cv 16.88 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................91
Table 5-7: Depressurization in 30 min considering Adiabatic, Cv 16.88 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................92
Table 5-8: Assumptions for adiabatic and fire depressurization for liquid methane .........94
Table 5-9: Results for depressurization liquid methane at saturated pressure in 30 min Fire
API521 case by masoneilan valve .....................................................................................95
Table 5-10: results for depressurization, 100% opening by universal gas sizing/Fisher valve for
adiabatic case 30 min.. .......................................................................................................96
Table 5-11: Assumptions for flare flame distortion calculation ......................................100
Table 5-12: Flare tip exit velocity ....................................................................................103
Table 5-13: Results for flare flame length .......................................................................107
Table 5-14: Assumptions for Pipeline case study ............................................................107
Table C.1: Critical properties of some component (Reid et al., 1987) ............................124
Table D.1: Wind speed data represent the FortMcmurray ,Alberta .................................125
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List of Figures
Fig 2-1: Usual level control ...............................................................................................24
Fig 3-1: Discharge coefficient ...........................................................................................34
Fig 3-2: Schematic view of level blow up in a depressurizing vessel filled with liquefied
pressurized gas [Wilday, 1992] ..........................................................................................37
Fig 3-3: Tow phase flow pattern separately schematic ......................................................39
Figure 3-4: Flow regime transition criterion for upward two-phase flow in vertical tube 39
Fig 3-5: Adiabatic process [https://en.wikipedia.org/wiki/Adiabatic_process] .................54
Fix 4-1: Flare Stack And Flame Distortion Geometrical Factors[API521] .......................58
Fig 4-2: Absorption factors for water vapour ....................................................................63
Fig 4-3: Carbon-dioxide absorption factor ........................................................................64
Fig 4-4: quadrilateral flare flame shape (Chamberlain, 1987) ..........................................68
Fig 4-5: Geometrical factors for determination of lifted-off flare fames ...........................75
Fig 5-1: Depressing in 15 min adiabatic considering Masoneilan valve fully open ..........84
Fig 5-2: Depressing in 30 min adiabatic considering Masoneilan valve fully open ..........84
Fig 5-3: Depressurization in 15 min considering Fire API 521 and Masoneilan valve fully open
............................................................................................................................................87
Fig 5-4: Depressurization in 30 min considering Fire API 521 and Masoneilan valve fully open
............................................................................................................................................87
Fig5-5: Depressurization in 15 min considering Fire API, CV 16.8 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................90
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Fig5-6: Depressurization in 15 min considering Adiabatic, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................91
Fig 5-8: Depressurization in 30 min considering Adiabatic, Cv 16.88 USGPM (60F, 1psi) and
Masoneilan valve ...............................................................................................................92
Fig 5-9: Depressurization liquid methane at saturated pressure in 30 min Fire API521 case by
masoneilan valve ................................................................................................................95
Fig 5-10: Depressurization, 100% opening by universal gas sizing/Fisher valve for adiabatic case
30 min with CV equal to 21.48 USGPM (60F, 1psi).........................................................96
Fig 5-11: Depressurization 30 min adiabatic liquid methane by universal gas sizing/fisher valve
with 20% opening with CV 21.48 [USGPM (60F, 1psi ....................................................97
Fig 5-12: Depressurization 15 min adiabatic liquid methane by universal gas sizing/fisher valve
with 20% opening with CV 21.48 USGPM (60F, 1psi) ....................................................95
Fig 5-13: Depressurization 15 min adiabatic liquid CH4 by fisher valve with 30% opening and
CV21.48 [USGPM (60F, 1psi)]. ........................................................................................98
Fig 5-14: Mach number versus time in steady state (blue) and dynamic (red) calculation103
Fig 5-15: Flare exit velocity versus time. ........................................................................101
Fig 5-16: Heat radiation versus time at the time of depressurization by Chamberlin method.
..........................................................................................................................................102
Fig 5-17: Heat radiation versus time at the time of depressurization by API method. ... 103
Fig 5-18: Allowable design limit for flare system modeling and calculation [API 521] 104
Fig 5-19 Flare flame distortion from top view at normal condition, Chamberlain method105
Fig 5-20: Flare flame distortion from top view at normal condition API method ...........105
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Fig 5-21: Flare flame distortion from top view at the time of depressurization Chamberlain
method..............................................................................................................................106
Fig 5-22: Flare flame distortion from top view at the time of depressurization by API method
..........................................................................................................................................106
Fig 5-23: Mach number versus distance in pipeline ........................................................108
Fig 5-24: Temperature versus distance in pipeline ..........................................................108
Fig 5-25: Pressure versus distance in pipeline .................................................................109
Fig A.1 Friction factor for flow in pipes by Moody chart ...............................................122
Table C.1: Critical properties of some component (Reid et al., 1987) .................................124
Fig C.2: Characteristics of control valve flow with piping losses ...................................124
Table D.1: Wind speed data represent the FortMcmurray ,Alberta .................................125
Fig E.1: Mollier diagram, an enthalpy–entropy versus pressure (GPSA 12 edition) ......126
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List of Symbols and Abbreviations
Symbol Definition
Ae Outlet cross sectional area [m2]
Af Jet cross section after vaporizing [m2]
Ah Hole cross sectional area [m2]
Ap Pipe cross sectional area [m2]
Ar Area ratio [unitless]
CAr Volume ratio [unitless]
Cd Discharge coefficient [unitless]
Cds Droplet size constant [unitless]
Cf Friction coefficient [unitless]
Cp Specific heat at constant pressure [J/kg⋅K]
Cv Specific heat at constant volume [J/(mol⋅K]
CV Valve capacity [USGPM(60F,1psi)]
CΦv Auxiliary variable [m]
dh Hole diameter [m]
dp Internal pipe diameter [m]
dv Vessel diameter [m]
D Diffusion coefficient [m2/s]
fD Darcy friction factor
lp Pipe length [m]
p Pressure ratio [unitless]
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pcr Ratio of critical pressure to atmospheric pressure[unitless]
Pc Critical pressure [pa]
qS,e Exit flow rate [kg/s]
qS,0 initial flow rate [kg/s]
QL Mass of liquid [kg]
Q0 Inventory mass [kg]
QV Vapour mass [kg]
QV,0 Initial vapour mass [kg]
tB Time constant [s]
tE Maximum time validity [s]
u Fluid velocity [m/s]
ua Wind speed [m/s]
ug Gas velocity [m/s]
ue Fluid velocity at exit [m/s]
uf Fluid velocity after flashing [m/s]
uj Fluid velocity after evaporation droplets [m/s]
us Speed of sound [m/s]
us,L Speed of sound in liquid [m/s]
us,V Speed of sound in vapour [m/s]
uVR Dimensionless superficial velocity [unitless]
U Internal energy of the gas [J/mol]
VL,E Volume of liquid after depressurization in the vessel [m3]
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VL,0 Initial volume of liquid in the vessel [m3]
Vp Pipeline volume [m3]
ξ Liquid fraction in vessel [unitless]
ρ Density [kg/m3]
ρF Average density [kg/m3]
ρL Liquid density [kg/m3]
ρtp Two phase density [kg/m3]
ρV Vapour density [kg/m3]
ρe Density at exit [kg/m3]
ρf Density after flashing [kg/m3]
τcr sonic discharge time [unitless]
τs subsonic discharge time [unitless]
υa Air kinematic viscosity [m2/s]
υL Kinematic viscosity of liquid[m2/s]
υV kinematic viscosity of gas[m2/s]
Φm,f Final gas mass fraction [kg/kg]
Φv Void fraction [m3/m3]
Φv,av Median void fraction [m3/m3]
Chapter 3
L1 Lift off height of the flare flame [m]
L2 Flame length [m]
Lb0 Flame length, in low speed air [m]
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Lf Quadrilateral length [m]
Mj Jet Mach number [unitless]
α Angle between flare tip and flame [°]
αc Absorption factor for CO2 [unitless]
αw Absorption factor for H2O [ [unitless]
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Abbreviations
I/O Input/output
ESD Emergency shutdown valve
PRV Pressure relief valve
PSHH Pressure switch high high
PA Pressure alarm
FC Failed to close
PT Pressure transmitter
Barg Bar gauge
HP High pressure
LP Low pressure
PFP Passive fire protective insulation
TERV Thermal expansion relief valves
DIERS Design Institute of Emergency Relief Systems
CV [USGPM(60F,1psi)] The flow coefficient of valve, which represent to gallons per
minute of water that can pass through the valve in 60° F and 1psi pressure drop through the
valve.
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1 Chapter One: Introduction
Process upsets happen when the pressure, temperature, level, and flow controls are not maintained
within the operating limit. These controls can affect each other and a minor upset of one control
loop can cause a major process upset. Therefore these process variables must be kept within
acceptable operating limits. When process variables enter the action limit and there is no action
taken in a short time by the operators, operating malfunction takes place. Any malfunctions of
these parameters can impact on each other and the plant production quality, which can be noxious
for the plant economy and environment. Operating mistakes can activate interlocking systems and
mechanical safety devices. Interlocking and logical systems including programmable logic
controller [PLC], fieldbus control system [FCS], distributed control system [DCS], Emergency
Shutdown System [ESD] and mechanical safety devices protect the equipment, personnel and
environment. However activation of some of these devices will mostly lead to production losses.
The process upsets or operating mistakes can lead to decreasing equipment efficiency and product
quality, unnecessary shutdowns which leads to production losses by releasing them to the
atmosphere through flare systems and in some plants such as polyethylene may lead to
decomposition.
On the other side the most important consequences of depressurization event is to evaluate the
flare flame distortion. Flare flame distortion and heat radiation from flare flames at the event of
depressurization depends on wind velocity, flare tip velocity, flow rate, stack height, stack
diameter, gas composition and other meteorological and stack variables. Flame distortion, caused
by lateral wind acting on the gas jet leaving the flare stack, is an important aspect of heat radiation
because it leads to higher ground-level radiation from the flames. Besides, many factors have to
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be taken into account, which are very important for evaluation of flame length, and flame
distortion, which include radiation exposure from gases burning, radiant heat fraction, flame length
and centre and flame careen. Following API alone is good for flare flame distortion and heat
radiation evaluation but for deressurization event without consideration of dynamic simulation,
depressurization event may face with event escalation. Therefore, for the depressurization
evaluation and fluid thermodynamic behavior, it is recommend doing dynamic simulation.
Consequently, dynamic calculations for determination of the depressurization procedure and the
evaluation of flare flame distortion and heat radiation have been carried out which gave some
important and accurate information to avoid event escalation. Since a depressurization event is the
most important factor for the safety of employees, equipment and plant operation, data taken by
dynamic calculation presented in this thesis offer a solution for determination of depressurization
procedures, depressurization consequences on equipment and flare flame, and how to avoid event
escalation.
All of the gas plants can become an unsafe by suddenly increasing pressure. Overpressure can take
place due to different reasons such as instrument failure, utility failure, and power loss. If pressure
rises suddenly, a depressurization system must take action in place to depressurize to a safe
operation. The pressure must be controlled through an appropriate valve, up to the time when it
reaches a safe range. The high pressure excess gas can be sent to a pressure vessel or directly to
the flare, and appropriate valves have to be considered for these vessels to prevent overpressure as
well. On oil wells, the major gas depressurization valves are installed for this purpose. When the
wells are not in operation, the pressure can rise. Consequently, a depressurization valve releases
the high pressure excess gas to the flare, in which the hydrocarbons must be burned before being
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vented into the atmosphere due to environmentally hazardous consequences [4, 38] Dynamic
modeling for relief systems can lead to a substantial decrease in capital cost, while concurrently
improving plant safety. This thesis takes to account the importance of dynamic analysis to major
areas of each gas plant and description of how upset events can have an effect on flare flame
distortion, the shape of the flame in a crosswind. Additionally, since radiation of flare flame is an
important safety issue as it can cause equipment damage, injury and even death, modeling is
discussed for finding the amount of heat radiation. Moreover, the important parameters will be
assessed to determine how dynamic modeling can help designers and operators to control this
event appropriately in an acceptable time. Besides, precisely specifying relief loads and metal
temperatures and material selections can maintain safety of the plant and provide decision support
of capital disbursement. Detailed depressurization dynamic modeling and calculation is a key
factor of the safety evaluation of oil and gas plants and other high pressure equipment. Rapid
depressurization not just determines the load amount on the pressure relief system such as the flare
network. Rather most importantly, it can lead to considerable reduced temperatures of the vessel
walls, which can cause to breakable and high thermal stresses. Very little research has been done
for evaluating of the depressurization process. Haque et al. (1990) accomplished an analytical and
experimental research for fast depressurization during 100 seconds to evaluate the change of the
fluid and the inner wall temperatures during that time. Another research for pressure vessel
contains methane, conducted through an experimental study with method of slow depressurization
by Yadigaroglu-Wieland (1991), with consideration of four different situations. The
depressurization process takes 4-18 hours and the temperature and pressure data indicated
accordingly. Yadigaroglu-Wieland, numerically indicated the depressurization process and got
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satisfactory results through the experimental data. Other studies have been carried out by, Botros
et al. (1978) (1989), for calculating time for gas pipeline depressurization, and by Failer and
Breslouer (1988), for modeling the depressurization and venting of a fire case containment
chamber, by Picard and Bishnoi (1989), for evaluating the effect of real fluid conditions throughout
the fast dense decompression, natural gas component, and by Kim (1986), to review the process
of storage tank depressurization. This thesis considered three major areas of gas plants with subject
to depressurization in the case of a process upset. Vessel and piping, pipeline in case of puncture
on the wall or total rupture for liquefied pressurized gas and high pressure gas has been modeled
and consequently flare flame distortion which leads to abnormal heat radiation in this case has
been modeled and simulated.
This thesis consists of five chapters. Chapter 2 is the literature review. It will introduce the
definitions of the technical terms used in this thesis, description of major sources of overpressure,
modeling of some possible cases, which cause depressurization, instructions for required relieving
rates according to specified situations, pressure relieving devices and relief scenarios. Chapter 3
explains a general & detailed description of flare flame distortion and relevant equations. Chapter
4 provides a detailed overview of depressurization dynamic modeling, methodology &
assumptions, experiment data, calculation results, equations description, and calculation
procedures will be discussed. Accordingly, Chapter 5 consists of the conclusions.
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2 Chapter Two: Literature Review
2.1 MAJOR SOURCE OF OVERPRESSURE AND RELIEF SCENARIOS
Pressure vessels, towers, heat exchangers, compressors and other operating facilities and piping
should be sized to maintain the pressure of the system. Sizing of equipment should be based on
some important factors that are subject to the design pressure and design temperatures. In addition,
the consequences of process upsets which can happen at operating conditions, and effect of process
variables on each other in upset conditions that cause overpressure are other considerations.
Besides, some natural disasters such as earthquake also can be recognized as a source of
overpressure, as they may cause cracking or rupture of pipe/vessels, which has to be evaluated
based on safety integrity level of plant. At the design stage of any gas plant the minimum relief
load must be calculated to prevent the impact of overpressure in other equipment beyond the
maximum permissible pressure.
Different scenarios have to be considered for calculation of relief loads as well as designing safety
relieve valves and the analysis should be based on the contingencies outlined in [API 521]. One
example that cause to overpressure and process upsets is gas blowby. When the liquid level in a
two-phase separator falls down too low, gas blowby will occur. In this condition gas comes out
through the liquid outlet nozzle because of level control malfunction. This will lead to gas from
the high pressure separator going into the low pressure facilities downstream of the liquid level
control valve. Gas relieving loads have to be calculated by use of the control valve sizing data. The
gas blowby load will then be calculated from the gas valve sizing equation by use of the CV of the
relevant valve which is on the fully open situation. Although enough information is not available
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at this point, engineering assumptions have to be taken into account which is very important. These
assumptions should be considered to minimize gas blowby relief load where possible.
2.2 Pressure relieving devices
Systems that are subject to internal pressure must be equipped with overpressure protection. A
safety device can be activated by inlet static pressure. They have to be configured to open in an
emergency or upset situation to protect equipment and plant against the system overpressure or
increase of equipment internal pressure beyond the specified design value. There are different
kinds of devices that can be utilized in a gas plant, including pressure relief valve, a non-reclosing
pressure relief device, and a vacuum relief valve [API 520].
2.2.1 Pressure relief valves
The safety relief valve is a kind of relief valve which can be utilized to control or bound the buildup
pressure in a vessel or system at the time of process upset which can occur because of instrument
or mechanical failure, or in case of fire. The pressure is alleviated by enabling the pressurized gas
or liquid to gush out from the system by a specific route. The safety relief valve has to be set to
open at a preordained pressure.
An excess fluid which can be liquid, gas or liquid–gas mixture is typically transferred through a
piping system named a flare header to an elevated gas flare where gases are typically burned and
the combusted gases are discharged to the environment [API 520].
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2.2.2 Rupture Disks
These devices are non-reclosing pressure relief devices utilized to protect tanks/vessels which
have potential for overpressure during normal operation condition, as well as protect piping and
other pressurized equipment from excessive pressure and vacuum. Rupture disks are utilized in
single and multiple relief device installations. Besides, they can be used as redundant pressure
relief devices [API 520].
2.3 Depressurization System
The layout of specific valves, piping and instrumentation logical facilities in order to control rapid
reduction of pressure in pressurized equipment by releasing gases has to be taken into account at
the design stage of any gas plant. Monitored depressurization of the vessel can decrease stress in
the vessel walls.
The modeling of depressurization systems is intended to examine the specific parameters that have
to be taken into account to better control the process upset or fire case. All of these parameters are
important key factors for the depressurization model and consequently the model of flare flame
distortion and radiation of the flame that can have hazardous consequences on equipment and
environment at the event of a process upset. Because pressure build up is the key factor of
depressurization systems, many factors such as heat radiation, flare flame distortion at this event
has to be evaluated, since it directly depends on discharge gas flow rate and discharge time. Hence,
dynamic modeling can evaluate the most important variables that affect flare peak load, and as a
consequence flare flame distortion. Logic facilities that can be included a distribution control
systems (DCS), an interlocks, emergency shutdown systems (ESD) and programmable logic
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controllers (PLC) should be considered to do an appropriate actions when they receive adequate
signals from pressure transmitters to shutoff the process line and open the bypass line to reroute
gas to Flare. Some logical systems can be installed as an fieldbus or non-fieldbus, which means an
action can be done automatically in the field or an action can be done by operator whether at the
control room or in the field.
2.4 Flow and Level Controls
Several methods exist to specify the flow rate of liquids and gases. All of these methods are based
on some specified flow detectors that have to be installed on specific locations. The most common
flow measurement methods used in gas plants are based on creating a pressure drop in a pipe. As
the flow in the pipe is transport through a reduced area, the pressure before the flow meter is higher
than afterwards or downstream. In the constrains, the velocity of the fluid increases, as the same
quantity of flow must pass before and in the constrain. By increasing the velocity a differential
pressure will be created through the flow meter as a result of the Bernoulli effect. Accordingly, by
measuring the pressure differential through the flow meter one can compute the flow rate. In fact,
due to increasing the differential pressure relatively to the square of the flow rate, so ΔP ∝ Q2 .
In other words, Q ∝ √𝛥𝑃 .
Where: Q = the volumetric flow rate [m3/s]
𝛥𝑝 = differential pressure
On the other hand, the term of Level is used for measuring the amount of liquid. In the vessel
containing liquid, the pressure is directly rely on the liquid height in the vessel named as
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hydrostatic pressure. Figure 2-1 shows, by increasing the level in the vessel, the pressure affected
by the liquid will raise linearly in which the equation is as follows:
P = ρ.g.H Equation 2-1
Where: P = pressure [Pa]
H = height of liquid column [m]
ρ = liquid Density [kg/m3]
g = acceleration due to gravity [9.81 m/s2]
Finally, the liquid level in a closed vessel can be calculated by the pressure reading if density of
the liquid is constant.
Fig 2-1 Usual level control, basic instrumentation measuring devices and basic PID control
Technical Training Group, (2003)
So the formula would be as follows:
P high = P gas + ρ.g.H Equation 2-2
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P low = P gas
ΔP = P high – P low = ρ.g.H Equation 2-3
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3 Chapter Three: Dynamic Modeling of Depressurization
3.1 Material and Energy Balances
For gas phase energy balance in a vessel can be written as follows:
2.1
2
dU VsH q Qsoutdt [W] Equation 3-1
In addition, mass balance can be written as follows:
dmqs
dt [kg/s] Equation 3-2
We have:
1U m u [J] Equation 3-3
.H q hsout out Equation 3-4
If well mixed then:
.H q hsout [W] Equation 3-5
Hence, the energy balance becomes:
2( )
2
Vd m u sq h q Qs sdt
[W] Equation 3-6
The left –hand side can be expanded as:
dm duu m
dt dt Equation 3-7
Substitution of the mass flow rate qs leads to:
duu q ms
dt Equation 3-8
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28
Hence, the energy balance becomes:
21(( ) ( ) ( )
2
du Vu q q h q Qs s s
dt m Equation 3-9
Which we have:
Q U A T [W] Equation 3-10
And
T = T Tout in [C ] Equation 3-11
Where:
IU = Internal energy in vessel [J]
U= heat transfer coefficient [W/ (m2.K)]
u= Internal energy per kg in vessel [J/kg]
h= enthalpy per kg [J/kg]
A= wall area [m2]
T = temperature difference between outside and tank [ C ]
m= mass in vessel
At any time, density is calculated as m
V. Then, ρ and U are used to calculate temperature (T) and
pressure (P) based on an equation of state. T and P are used to calculate h which can be shows as:
H= u+pv Equation 3-12
sq is calculated based on a valve equation (section 3.2).
Across the valve, assuming adiabatic conditions, we have:
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22,
2 2
VV s outsidesh+ houtside
Equation 3-13
When the left term is known, we can solve for the first right term at P= 1 atm. Hence,
temperature of gas leaving the choke can be calculated.
Where:
m= mass in vessel [kg]
.H out = enthalpy in exit stream [J]
Vs= velocity of gas at outlet [m/s]
hout = enthalpy per kg leaving can be = h if well-mixed [J/kg]
3.1.1 Modeling of High Pressurized Gas Discharges Across the Orifice
The discharge flow of gases from holes and pipes, and the dynamic behaviour of an adiabatic
depressurization of a high-pressure gas in the vessel, will be described in this section. Due to any
leakage in a pressurized vessel, the left over liquids in the vessel will be quickly depressurized and
expanded which causes low temperature. Since vessels contain gas mixtures, in some cases low
volatile elements might condensate (Haque, 1990). Taking in to account the first law of
thermodynamics, and with consideration of expanding gas which delivers volumetric work, and
by implementing an equations for non-ideal gases, we can determine the reduction of pressure and
temperature throughout the discharge of the compressed gas (Haque et al. 1992).
For adiabatic flow, we can use the following equations to determine pressure [32-43, 63]:
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2( 1) ( 1)0.5
RT (2 2)1 A 20P P 1 t0 2 V M 1
[bar] Equation 3-14
( 1)( 1)(2 2) 2RT 2 P0Q A
0 M 1 P0
[Kg/s] Equation 3-15
Where:
A= cross section area of the orifice [m2]
M= gas molecular weight [kg/Kmol]
R= universal gas constant [J/kg.mol.K]
T= gas absolute temperature [k]
Q= mass flow from the vessel [kg/s]
P= pressure in the vessel [bar]
V= vessel volume [m3]
=C pCv
gas specific heat ratio [unitless]
= gas density [kg/m3]
P0= vessel pressure [bar]
0 = initial gas density [kg/m3]
t= step time [s]
Considering isentropic depressurization for adiabatic case and equation 2-48 the energy balance
would be:
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31
Hi =Hc+
2
2
s Equation 3-16
Hi = Hc+2
c
c s
P
Equation 3-17
Hi= enthalpy of gas in the vessel [J/kg]
Hc= enthalpy of gas passing the orifice [J/kg]
Pc= gas pressure passing the orifice [bar]
c = gas density passing the orifice [kg/m3]
Consequently, the mass flow can be calculated by:
PcQ C A cdc
[kg/s] Equation 3-18
Where:
Cd = discharge coefficient
The discharge coefficient can be calculated by two items, contraction and friction according to the
following formula:
Cd = Cf × Cc Equation 3-19
where
Cf = friction coefficient
Cc = contraction coefficient
When the fluid in the vessel is expanding from all directions, components will have such velocity
vertical to the axis of the expansion. The flowing fluid has to be made curved in the way parallel
to the hole axis. The fluid’s inertia leads to the smallest cross sectional area, without radial
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32
acceleration, which is smaller than the area of the expansion. The recommended value for the
discharge coefficient orifice contraction, where friction is small is 0.95 0.99Cd . [Beek,
1974].
It should be noted that the discharge flow is critical or choked when:
( )1 1
( )2
Pd
Pu
Equation 3-20
Pd = initial gas pressure [bar]
Pu = upstream gas pressure [bar]
3.1.2 Calculation of the vessel wall temperature
The following equations for calculation of the vessel wall temperature can be used [S M
Richardson et al. 1991].
1 11 2 20 0 1 0
xn n nT F T F T qr rk
Equation 3-21
1 1 11 21 1
n n n nT F T F T Tr ri i i i
Equation 3-22
( 1 to n)i
1 11 2 211
xn n nT F T F T qr ri i i k
Equation 3-23
2
tFr
x
Equation 3-24
Where:
nT = temperature in any step time n [C]
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n= step time
0q = heat flux from the medium in the vessel to the inner vessel wall [W/m2]
1q = heat flux from the medium in the vessel to the inner vessel wall [W/m2]
x = thickness of the vessel represent to each side of the vessel, inner and outer layer [mm]
Fr = Fourier number
k = thermal conductivity [W/(m.K)]
wall thermal diffusivity [mm2/s]
3.2 Valve correlations and equations using for dynamic calculation
Five types of valves are common for controlling depressurization and can be selected in a
calculation. Industrial approach allows users to choose any of these valves. This study is based on
two most common and major depressurization valves, known as Masoneilan and Fisher universal
gas sizing [37, 39].
3.2.1 General valve equation
This model can be used if the valve effective throat area is known at the beginning stage. The
model creates restricting assumptions regarding the features of the orifice. The valve equation is
as follows:
2Discharge Flow= C 43200 ( )1
CK G P Kup upterm C Equation 3-25
Where:
Gc = 1 in SI units [2kg.m / N.S ] and 32.17 in imperial units [
2lb.ft / lb . f S ]
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34
Pup= upstream pressure
up = upstream density
C1 can be determined based on geometry of the valve, and at the time of sizing orifices, it will be
the same as the orifice coefficient discharge.
( 1)
2 2 ( 1)
( 1)Kterm
Equation 3-26
Where:
C p
Cv Equation 3-27
The model for this valve also can be shown as follows:
0.5Valve rate=C ( )Area K P Density Kuptermd Equation 3-28
C = Discharge coefficientd
Also Cd can be determined by Fig 3-1.
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35
Fig 3-1: Discharge coefficient(C ) for square-edged circular orifices with corner taps.
[Tuve and Sprenkle, Instruments, 6,201(1933).]
d
3.2.2 Supersonic valve equation
When there is not enough information available regarding the valve, this model can be considered
for calculation. The valve equation is as follows:
Flow rate= C ( )Area P Densityupd Equation 3-29
3.2.3 Subsonic valve equation
Subsonic model can be considered just at the time when the flow over the valve is anticipated to
be entirely subsonic. In general this condition will exist if the pressure is lower than twice of the
backpressure valve at the upstream side. The related equation which has been used for modeling
of this kind of valve is as follow:
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36
(P )( )Discharge Flow= C
1
P P Pup up upback back
Pup
Equation 3-30
Where:
Pup = upstream pressure
Pback
= back pressure
up = upstream density
3.2.4 Masoneilan valve equation
This model can be considered as a general depressurization valves model by the following
equation:
Flow rate= C ( )1
C C Y Pv up upf f Equation 3-31
3Y 0.148f y y Equation 3-32
cp1
3 p1.4
yK
T
Equation 3-33
Where :
K= gas specific heat ratio
pc = calculated pressure drop
pT = total pressure drop
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37
where is expansion factory , Cf is critical flow factor and C1 can be determined based on geometry
of the valve. The expansion is ratio of flow coefficient for a gas to that for a liquid at the same
Reynolds number.
3.2.5 Fisher / Universal gas sizing equation
Fisher / universal gas sizing/Fisher is another type of valve with the following equation.
1
1
52059.64SCFH v
PQ C P
P GT
Equation 3-34
G = Specific Gravity
T= Temperature [˚C]
P1= inlet pressure [bar]
3.3 Characterization of Discharges of Liquefied Pressurized Gases
When liquefied pressurized gas is depressurized, it will lead to the creation of bubbles in the liquid.
Hence, expansion of the liquid at boiling point is a quite complex physical process. When a
liquefied pressurized gas exist in the vessel, a rapid depressurization leads to a flash of the liquid
inside the vessel, that means, because of the rapid depressurization, the liquid, vaporizes quickly
until the out coming vapour/mixture is cooled under the boiling point at the last pressure.
Consequently, raise of gas bubbles in the liquid phase will occur and if the expanded liquid
develops over the hole in the vessel, a two-phase flow will be obvious through the hole in the
vessel. The level blow up is shown in Figure 3-2 [9-10, 42, 51, 53-54, 63, 71]
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Fig 3-2: Schematic view of level blow up in a depressurizing vessel filled with liquefied
pressurized gas [Wilday, 1992]
3.3.1 Numerical Process to Determine Discharge Flow Type from the Vessel
In this procedure, at the beginning the criteria regarding the type of fluid within the vessel and
raising up of liquid level model will be discussed. Then, the initial and end situation of the
numerical process will be introduced accordingly. Subsequently, the model in the form of a
numerical process will be discussed.
Any flow type depends on to its variables of the basic process. The numerical process should be
continued until an acceptable end situation is reached. An accurate analytical procedure for
determining of the flow type for two-phase flow, within a vertical vessel at the time of
depressurization has been investigated by the design institute of Emergency Relief Systems of
AIChE . [DIERS, 1986 and Melhem, 1993].
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39
For calculation of discharge gas flow rate,s
q , equation 3-56 in section 3.3.2 are applicable to
determine the discharge flow rate. Then the median superficial velocity of vapour within the vessel
can be calculated by:
, ( )
qsuV m A
V L
[m/s] Equation 3-35
And bubble increase velocity can be calculated by:
( )41, 2
gL Vu C
Db iL
[m/s] Equation 3-36
Where as per Wallis the limit can be:
1.181
CD
bubbly flow
1.531
CD
churn flow
σ = surface tension [N/m]
Surface tension play an important role at the time of depressurization, at boiling point and also
viscosity is important that is an important factor for determination of flow type. The surface tension
by implementing Walden’s law Pe[ rry,1973] can be calculated as follow:
( )C L TV B L
[N/m] Equation 3-37
And
1
76.56 10C
[m] Equation 3-38
Where:
C Wlden’s law constant
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40
bubbly slug churn wispy annular annular
Fig 3-3: Tow phase flow pattern separately schematic
(http://www-thd.mech.eng.osaka-u.ac.jp/mpe04002/page/research05.htm)
Fig 3-4: Flow regime transition criterion for upward two-phase flow in vertical tube
([http://hmf.enseeiht.fr/travaux/bei/beiep/content/g19/types-flow-pattern)
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41
Where:
VsL = liquid superficial velocity
VsG = gas superficial velocity
In Figure, 3-3 and 3-4 the bubble flow area and the churn turbulent flow area are indicated. The
ratio of superficial vapour velocity, which is dimensionless, can be calculated by:
uVu
VR ub
[unitless] Equation 3-39
Where:
u =VR
superficial velocity [unitless]
u =b
bubble velocity [m/s]
u =V
median superficial velocity [m/s]
Then we need to calculate dimensionless superficial velocity for usual two-phase flow forms. It
should be noted that the dimensionless superficial velocity for bubbly flow has to be higher than
u VR,bf , and for churn flow it has to be higher than u VR,cf with appropriate formulas as follows:
2(1 )
, 3((1 ) (1 ))2
uVR bf
CV VD
[unitless] Equation 3-40
u = VR,bf
bubbly flow minimum superficial velocity [unitless]
And
2
,cf (1 )2
VVR C
VD
[unitless] Equation 3-41
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42
=VR,cf
u churn flow minimum superficial velocity [unitless]
In which D2C = 1.2 for bubbly flow and D2C = 1.5 for churn flow.
Then we need to identify if two-phase flow is occurring in the vessel by the following rules:
,u uVR VR cf
two phase churn flow
,u uVR VR bf
two phase bubbly flow
,u uVR VR bf
, ,
u uVR VR cf
gas flow
After a rapid depressurization, a liquid flash inside the vessel will occur, and because of the
existence of vapour bubbles in the vessel, expansion of liquid will happen consequently. Therefore,
liquid level at the tank or vessel will decrease. Mayinger theory can be utilized for computation of
the void fraction in the liquid state in vessels. Additionally, the process can be utilized to calculate
the increasing of the liquid level due to discharge flow over the sidewall of the vessel, to determine
the state of the discharge flow. The void fraction at the time of depressurization can be computed
by the following process for the liquid phase (Belore 1986). Determination of the superficial
velocity at the beginning time is important factor that can be calculated according to the following
formula:
,0,0
,0
qs
uV A
LV
[m/s] Equation 3-42
And the vapour phase that hast to be released would be as follow:
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43
1(1 ) ,1,
1 ,1
1 ,
d
n U CDg m etop top b iQ
Vtop m e LVm e gtop
Equation 3-43
For two-phase flow, the nature of the top discharge ,m e can be evaluated by the following
equations for churn and bubbly flow form respectively [1, 50, 53-54, 75]:
U1,
,1
2
CD Vb i LV Q Vg gd
m eVtop LV gtop
Equation 3-44
U1,
(1 )1
,
11
CD Vtopb i Ltop top V Q Vg gtopd
m eVtop LV gtop
Equation 3-45
Where:
top = void fraction after depressurization
U,b i
= increase of bubble velocity
VL
= liquid specific volume [m3/kg]
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44
V g = gas specific volume [m3/kg]
Qd
= discharge flow rate [kg/s]
While is the top void fraction, in should be considered that the median void fraction ,V aV
within the vessel before the process upset and depressurization can be calculated by:
1,V aV
[m3/m3] Equation 3-46
= filling degree of vessel [m3/m3]
,V aV = average void fraction in the vessel [m3/m3]
The liquid kinematic viscosity proportion to the gas kinematic viscosity at boiling point should be
determined by utilizing Arrhenius’s correlations for liquid viscosity (Perry 1973) and Arnold’s
relations for gas viscosity (Perry 1973), which can be introduced by the following equation:
3637 ( 10 ) ( 1.47 )
27 36( )
u T Ti V BL
TV L
Equation 3-47
Where :
C =AA
constant 1 6 × kmolm
[ K]
By definition of assistant equation:
( ( ))C
V gL V
[m] Equation 3-48
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45
When the pressure goes down, velocity goes up and this can be leads to shear stress impact to the
liquid droplets which can be sufficient large to entrain liquid droplets inside the vapor phase.
The void fraction after depressurization with consideration of viscous losses can be calculated by:
2,0 0.376 0.176 0.585 0.2560.73 ( ) ( ) ( ) ( )
( ) ( )
u CV V L Ltop g C d
V V L V V
[m3/m3]
Equation 3-49
Two-phase flow in a pipe has different flow forms. Due to the transfers of mass, momentum and
energy among the phases, identifying the flow regime is very important in the numerical modeling
of two-phase flow. Takes to account in mist flow regime, due to the gas velocity is too high and
there is too much small liquid droplets scattered in continuous gas phase that might be stripped
through the wall, this flow regime is not shown in Figures 3-3 and 3-4.
The liquid and bubbles, which expanded to a new volume, can be explained as:
,0L, 1
VL
VE
top
[m3] Equation 3-50
So, for a vertical cylindrical vessel we have:
AL
=Abase
[m2]
V = A ×hL L L [m3] Equation 3-51
For horizontal cylindrical vessel:
A =2 L (2 )L
r h hL L
[m2] Equation 3-52
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46
12V =L cos (2 r h )
L
hLr r h h
L L Lr
[m3] Equation 3-53
For spherical vessel:
2 2( )A r r hL L
[m2] Equation 3-54
2 (3 )3
V h f r hL L L
[m3] Equation 3-55
Where:
A =L
liquid surface [m2]
Abase= vertical cylinder base [m2]
HL= heigh of liquid[m]
r = sphere radius [m]
Φv = void fraction [m3/m3]
CΦv= auxiliary variable [m]
qS = exit flow rate [kg/s]
AL = vessel liquid surface [m2]
dv = vessel diameter [m]
T = liquid/vapour temperature [k]
μi = molecular weight [kg/mol]
ρL = liquid density [kg/m3]
ρV = vapour density [kg/m3]
Lv[ΤΒ] = enthalpy/heat of vaporisation at boiling temperture[J/kg]
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47
TB = boiling point [k]
F= function of pressure
UV,0 = superficial vapour velocity [m/s]
σ = surface tension [N/m]
g = 9.8 [m/s2]
υL = kinematic viscosity of liquid [m2/s]
υV = kinematic viscosity of vapour [m2/s]
L,EV = final liquid volume after expansion [m3]
L,0V = initial liquid volume [m3]
V =L
liquid specific volume [m3/kg]
3.3.2 Modeling Discharge Flow of LPG Across the Vessel Holes
When the medium is considered as a two-phase flow, the related formula for calculation of
volumetric flow rate can be as follows [1, 42, 50, 53-54, 63]:
2 VP PaQ AC ghLd
[m3/s] Equation 3-56
In which:
Pa= atmospheric pressure [pa]
Pv= initial gas pressure [pa]
liquid density [kg/m3]
Ah= hole area [m2]
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48
Cd= discharge coefficient
g= 9.8 [m/s2]
hL= height of liquid [m]
Some recommended values of discharge coefficient as per Beek (1975) can be considered for
some types of orifices, which are sharp orifices, straight orifices, rounded orifices, and for a pipe
rupturewith value of 0.62, 0.82, 0.96, 1 respectively.
1
(1 )( )
F g g
v L
[kg/m3] Equation 3-57
Where g = mass fraction of gas in the two-phase flow [unitless]
F= median fluid density [kg/m3]
When the gas and liquid are not in equilibrium condition pressure decrement can be calculated as
follow (Abuaf et al., 1983):
8
1.5 8 0.81 2.2 10111.1 10
1L
TQ
depTcP P Ts a
gTc
[bar] Equation 3-58
In addition, mass flow rate across the orifice can be calculated by:
2 LQ C A P Pu xd [kg/s] Equation 3-59
Where:
Px= orifice pressure [bar]
Pu= upstream pressure [bar]
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49
Pa= absolute pressure [Pa]
Ps= saturated vapour pressure
T = upstream orifice absolute liquid temperature [K]
TC = upstream orifice absolute liquid critical temperature [K]
Qdep
= depressurization rate [Pas]
g gas density [kg/m3]
L liquid density [kg/m3]
= liquid surface tension [N/m]
3.4 Prediction of Discharge Gas from Pipelines Due To Rupture
At the time of an unexpected rupture in the pipeline, a backpressure can start going up in opposite
way of the speed of sound. This means that, the impose of the backpressure will be on upstream
pressure. The Wilson model of the gas discharge flow over the pipelines can be used to predict the
mass flow rate that is depend on the initial conditions over a periodic of time. At the time of
pipeline, mass flow rate can be calculated by the following semi-experimental model introduced
by Wilson that shows as follow [10, 41, 47, 52-54, 61, 75]:
0
,0( )
2,0( ( ) )( )1
0 0( ) e e
,0 ,0
qs
q ts qst t t
BQ QtQ Bt q t qB Bs s
[kg/s] Equation 3-60
Where:
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50
qS,0 = initial flow rate [kg/s]
Q0 = inventory mass [kg]
tB = time constant [s]
t = time after rupture [s]
The inventory Q0 in the pipeline can be computed as follows:
0 0Q A lp p [kg] Equation 3-61
Where pipe cross section area can be determined by:
2
4
d pAp
Equation 3-62
dp= pipe diameter
And the discharge flow rate at the beginning stage qS,0 will be computed by equation 2-41, with
consideration of Ap instead on Ah.
For total rupture in the pipeline the value of 1.0Cd can be considered as the worst case.
The gas sonic velocity s , with considering adiabatic expansion ( 0S ) can be calculated with
the following formula:
( )dp
s sd
[m/s] Equation 3-63
By consideration and implementing the equations of state, we have:
( )
d
dP P
[s2/m2] Equation 3-64
0z R T
sM
[m/s] Equation 3-65
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51
Where =M molecular weight [kg/mol]
The Colebrook White equation for determination of Darcy friction factor, and estimated for high
Reynolds value, can be determined by the following formula: [detail in appendix A]
2
101/ ( 2 log( / (3.715 )f d pD Equation 3-66
The time constant tB is introduced with the following:
2 / 3 / ( / )t l f l dsB D Equation 3-67
Eventually, the mass flow rate can be calculated at any time t later on the pipeline rupture and by
implementing Wilson model introduced by the mentioned equations above. For ensuring the
applicability of the model, at the time of the backpressure moving upstream and arrives at the
opposite way of the pipeline the following correlation can be used to determine the final
depressurization time through the pipeline.
ltE
s
[S] Equation 3-68
The definition of symbols used in this section, which have not been described before:
pd = Pipe diameter [m]
lP = pipe length [m]
Q =0
inventory mass in pipeline [kg]
P0= initial pressure [Pa]
q =S,0
initial discharge flow [kg/s]
T0= initial temperature [k]
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52
tE= max allowable time [s]
fD = darcy friction factor [unitless]
ε = wall roughness [m]
ζ = constant [unitless]
ρ =0
initial gas density [kg/m3]
ρ = gas density [kg/m3]
3.5 Relief Scenarios
There are different scenarios that need a depressurization to avoid escalation of the event. The
assumption typically made in each scenario are included as well.
3.5.1 Fire case
Basically, the production and processing equipment should be unintegrated into fire regions, by
means of fire walls, plated decks, or edge of the plant. Throughout a fire in one of the fire regions,
all facilities within that region are presumed to be fully exposed to the fire. It is presumed that
throughout a fire there is no feed flow to downstream or product from an affected system, and all
normal heat sources have stopped. Fire relief loads should be computed for pool fire and jet fire
events where relevant. Depressurization is assumed to be isenthalpic as a conservative estimate.
In practice, the depressurization will be between isenthalpic and isentropic (appendix F). Takes to
account the isenthalpic case leads to the largest required release flow rate, and is therefore the
worst case scenario.
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53
3.5.1.1 Heat absorption to liquids through the vessel wall
When vessels containing vapor or/and liquid the total heat absorbed by the insulated vessel can
be calculated by the following equations according to API 521:
Q = C1FAws 0.82 Equation 3-69
Where: Q = total heat absorption to the wetted wall, [Btu/hr]
C1= constant [43 200 in SI units, [21 000 in USC units]];
F = environment factor
Aws= total wetted surface, [ft2]
The terms, Aws 0.82 is the area exposure element or ratio. This proportion determines the concept
that big vessels are less probably than small ones to be entirely exposed to an open fire.
3.5.1.2 Heat absorption from the surface of a vessel covered by water film
Water films can absorb a significant amount of radiation in equipment exposed to fire flame and
it can protect the metal surface. The credibility of water film in contact with the vessel wall relies
on many items. Cold weather, winds, water supply quality including PH, surface tension, and
vessel surface situations that can keep consistent water distribution [API 521].
3.5.1.3 Environmental factor for depressurizing and emptying facilities
Controlled depressurization of the vessel decreases inner pressure and possibility of rupture of the
vessel walls. For designing of depressurization systems some important factors has to be taken into
account to achieve an appropriate objective. The environmental factor, define as a correction
coefficient which is depends on vessel thickness, has an effect on total heat absorption through the
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54
vessel wall and therefore leads to more liquid vaporization in the vessel and consequently higher
depressurization rate. Accordingly, the environmental factor has to be considered at the maximum
value of one with consideration of following factors according to API521 manual.
First, manual controls close to the vessel may not be accessible at the time of fire. Second, failure
status such as fails to close [FC] or fails to open [FO] of the depressurization valve should be
determined to prevent increasing of the fire duration and flare load at the time of an instrument air
failure, to prevent environmental hazardous consequences. Third, ahead of time initiation of
depressurization is suitable to minimize vessel stress to satisfactory values appropriate for the
vessel wall temperature which may result from a fire. Moreover, the last one, a safe location has
to be provided for releasing the excess gas from the system flaring. These factors can have an
effect on environmental factors and the worst case should be considered in these events. [API 521]
Q = C2·F·Aws0.82 Equation 3-70
where C2 is a constant [70 900 in SI units [34 500 in USC units]].
3.5.2 Adiabatic case or cold depressurization
It is often essential to depressurize, and this process can be slow or fast. Depressurization can lead
to a significant reduction of the temperatures for fluid inside the vessel as well as the vessel wall,
because of significant heat transfer to the fluid. This can be hazardous when the wall temperature
goes below the rupture stress temperature of the vessel. As a consequence, it is essential to know
conditions of the vessel during depressurization.
In an adiabatic process, there is no heat transfer between a system and its environment. The
adiabatic process follows the thermodynamics’ first law and energy is changed just as work. This
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55
is the isentropic case, which is the worst case for cold depressurization, because it leads to the
lowest temperature (Appendix F).
Rapid chemical and physical process can be explained by the adiabatic process, which means there
is not sufficient time for energy as heat transfer from the system [3-6, 49].
Fig 3-5: Adiabatic process
(https://commons.wikimedia.org/wiki/File:Adiabatic.svg)
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4 Chapter four: Flare Flame Distortion and Heat Radiation Modeling
4.1 Process Flares
The purpose of the flare system is to safely collect and to dispose all hazardous and flammable
fluids discharged from equipment and systems of the process plants to a safe place. For this reason
different types of flare systems should be considered at the time of designing of the gas plant. It is
not necessary to implement all equipment in all gas plants.
4.1.1 Warm Flare System
To dispose gases with depressurizing temperatures above 0° C, a warm flare collecting system can
be established. This warm flare system should pass the gases including gases with considerable
contents of water released from pressure safety devices to the flare tip.
4.1.2 Cold Flare System
Gases and liquids with depressurizing temperatures below 0°C have to be released through a cold
flare system. The cold flare system comprises of a collecting system for cold gases and another
system for liquids includes the cold blowdown drum and a system to heat the cold flare gases.
Condensing hydrocarbons should be collected in the cold blowdown drum. The gases are routing
through the blowdown drum and are passing to the flare line. Warm flare gases and warmed up
cold flare gases are passed in a common flare line and from there to the flare stack.
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57
4.1.3 Storage Tank Area Flare System
The low-pressure storage tanks should be linked to the tank flare. The flare should be established
near the tank area. Gases discharged from pressure safety devices should be routed simultaneously
directly to the flare stack.
4.1.4 Sour Gas Flare System
For the sour gases coming from the treatment units, a separate collection system should be
established. The gases should be routed to the sour gas flare where the combustible components
of the sour gases are converted in the high temperature flame near the front of the burner.
4.2 Modeling of the Flare Flame Distortion
Modeling of distortion and geometrical shape of the flare flames factors are necessary to determine
the flame lift off and its angle concerning the object. Besides, this calculation can be used for
determination of the view factor. In addition, the flame geometry can be used to determine the
surface area of the flame.
4.2.1 Calculation of Flare Diameter, Stack Height, and Flare Flame Distortion
Two different methods have been introduced, API521 and Chamberlain (1987) models known as
Thornton model in the following sections.
When gas leaves the relief outlet piping, quick changes happen on velocity and density. Different
procedures for computing the required size of outlet piping have been formulated.
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The API manual recommends the calculation of the outlet Mach number as follows:
0.5.53.23*10
2 2.2
q Z TmMaMP d
Equation 4-1
where
qm = gas mass flow rate, [kg/s]
Z = gas compressibility factor,
T = absolute temperature, [K]
P2= absolute pressure on the flare tip during flaring, [kPa]
M = relative molecular weight of gas.
Takes to account the flow can be sonic in some parts of the system and the critical pressure at the
exit condition will be calculated by considering 2
Ma = 1.0, as a sonic flow.
Takes to account having the Mach number and sonic velocity we can calculate flare tip exit
velocity which can be introduced as follow:
Flare tip exit velocity = Jet Mach Number × Sonic Velocity [m/s]
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59
Fig 4-1: Flare Stack and Flame Distortion Geometrical Factors [API 521]
Another approach for calculation of Mach number can be introduced by equation 4-2, which
widely can be used for dispersion modeling at the time of depressurization. Dynamic calculation
can give us an accurate data to implement on this model to determine when and where system has
potential to violate the regulatory. Therefore, Mach number in other form can be defined as follows
and can be calculated in any pressure p [3-6, 12-17, 24-27, 32].
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60
1
2
( 1) 1 ( 1) 11
1 ( )21 1
2 2
aM
p p
p pa a
p p
p pa a
Equation 4-2
Where:
Pa= ambient pressure
C p
Cv
And the effective area of the jet in related to the following Mach number can be calculated by:
(1 ) 1
2
pA
E paA
M a
[m2] Equation 4-3
Where:
AE = exit cross section area
The velocity at flare tip by API method can be calculated by the following formula [API 521,
2008]:
2 / 4
qU j
d [m/s] Equation 4-4
q= actual vapour volume flow rate [m3/s]
d= stack diameter [m]
Where q is volume flow rate of the gas and d is the flare stack diameter.
Finally, the maximum heat radiated can be calculated by the following equation [API 521].
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2 4D kF
Q
Equation 4-5
Where:
F= heat fraction, radiated
Q = heat released [kw]
K= heat radiation (maximum allowable heat radiation from flare flame) [kw/m2]
τ= radiated heat fraction transmitted through the atmosphere
4.3 Modeling of Heat Radiation from Flare Flame during Depressurization
Fire heat transfer lead to a heat flux with the possibility of damage to objects located in its
surroundings. At the time of fire by hydrocarbons, the flame can be composed of high temperature
burning components with a radiation temperature at 800 to 1600 ˚K. By determination of the
temperature and radiation range of the flame, we can calculate the heat flux. Generally, combustion
energy in the flames can be exchanged by three fundamental methods of heat transfer known as
heat convection, heat radiation, and heat conduction. Stefan-Boltzmann theory of the heat transfer
rate caused by a flame-radiating surface can be introduced by the following equations [7-8, 11-18,
27-31, 51, 74-76]:
4 4( )SEP T Taf in which 0 < ε < 1
2[J/(m .s)] Equation 4-6
Where:
SEP = Surface Emissive Power, in [J/[m2⋅s]]
ε = emissivity factor
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σ = constant of Stefan-Boltzmann, 85.6703 10 [J/[m2⋅S⋅T4]]
Tf = temperature of the radiator surface of the flame, in [K]
Ta = ambient temperature, in [K]
It should be noted, use of the Stefan- Boltzmann equation is restricted to computation of the
flames’ surface emissive power. Consequently, the well-known solid flame theory can be utilized,
that means, a portion of the burning heat is radiated by the observable flame surface area of the
flame.
The Surface Emissive Powers [SEPs] describes the heat flux as an emission from a two
dimensional surface as an approximation of a complex three-dimensional process [Crowley, 1991].
Theoretical Surface Emissive Powers [SEPs] can be calculated by the following combustion
energy per second and the equation can be introduced as follows:
theoreticaSEP
l
'Q
A Equation 4-7
Where:
A =Surface area of the flame, in [m2]
Q' = Combustion energy per second, in [J/s]
The calculation of Q' can be driven forward by the following formula:
' ' [J/s]Q m Hc Equation 4-8
Where:
m´= mass flow rate [kg/s]
ΔH = c heat of combustion [j/kg]
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Maximum Surface Emissive Powers [SEPs] can be calculated from TheoreticalSEP and the fraction of
the heat radiated from the flare flame that will be generated, can be determined by:
SEP =F ×SEPmax s theoretical Equation 4-9
where
Fs = combustion fraction energy radiated from the flare flame
For fire case, equation 4-10 and 4-11 either can be used for computation of the combustion fraction
energy radiated from the flare flame.
0.32( )6
F C Ps sv Equation 4-10
in which:
C6 = 0.00325 [Pa]0.32
Psv = saturated vapour pressure before depressurization in [Pa]
The correction factor which can have an effect on surface emission power can be used to
determine SEP in large scale as follow:
0.21 exp 0.00323 0.14jF us Equation 4-11
For calculation of the outlet velocity of the expanding jet, u j we can follow the semi-empirical
model in Section 3.3.1.
The heat flux q at a specific distance from the fire that is considered with the receiver per unit
area can be computed as follows:
q SEP F aviewMax Equation 4-12
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64
Where:
q = heat flux at a specific length, in [J/[m2⋅s]]
Fview = View factor
τa = Atmospheric transmissivity
For determination of atmospheric transmissivity, the following formula and figures can be utilized.
Fig 4-2: Water vapour absorption factor (Raj 1977)
Through this figure the absorption factor for water vapour αw for median flame temperature of Tf
at a distance x from the flame can be determined. Water partial vapour pressure Pw and relative
humidity [RH] can be computed by:
P RH Pw w [Pa] Equation 4-13
= P w saturated water vapour pressure [Pa]
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65
Besides, for the calculating the absorption constant factor for carbon dioxide αc at distance x
from the flame surface for a median flame temperature of Tf [K] we can use Fig 4-3.
Fig 4-3: Carbon-dioxide absorption factor (Raj 1977)
Then atmospheric transmissivity can be determined by:
1 wa c Equation 4-14
If the amount of Pw × X is between 104 and 105 N/m then we can use the following formula:
1
0.082.02 ( )a
P xw
Equation 4-15
4.3.1 Determination of Jet Flare Flames at Flare Tip
A methodology that intends to model the consequence of flare flame geometries on the radiated
heat flux form the flare flame by calculation the radiation source is another procedure that can be
used to calculate the flame distortion and heat radiation. In fact, Chamberlain model (1987) has
been carried out by experimental and practical study while API more depend on practical for flare
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flame study. Chamberlain model (1987) which was developed in several years of a research, well
known as Thornton-model, can determine the flame geometry and the radiation area of the flare
flame from the stacks and flare tip. This model has been confirmed by wind tunnel experiments
and practical evaluation in the field, for onshore plant and offshore plant. The principal theory
for the behaviour of the flare flame shape introduced by Becker et al(1981). Comparison
of experimental data taken in laboratory and data taken from a practical examination on the field
has illustrated that flare flames shape prediction can be determined with subject to a wider range
of ambient and flow conditions. The factors for the buoyancy force and the wind force needs
a usual flare flame length scale over the forces direction. The procedure to calculate the heat
flux at a specific distance of a jet flame can be done by determining some important factors. First,
determination of the outlet velocity of the expanding jet and dimension of the flame. Then, surface
emissive power [SEP], view factor and the last one is determination of the heat flux at a specific
distance. These steps individually will be described in this section. The outlet velocity of a flare
tip is a key factor for determination of the flare flame length, flare flame lift-off and the widths of
the flare flame throat. [18, 30-31, 43-47, 70-76].
For determination of the outlet velocity of the flare tip we have:
0.5( )Tj
u M Rj j Wg Equation 4-16
Where:
R = gas constant 8.314 [J/[mol⋅K]]
Tj = gas jet temperature [K]
Wg = gas molecular weight [kg/mol]
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67
Mj = Mach number
= Specific heat ratio
In which:
Jet Mach number for choked flow can be calculated by:
1
( 1) 2
1
Pc
Pair
Mj
Equation 4-17
And for unchoked flow we have:
2T
1 m' j1 2( 1) 1
20.000036233 d0
1
Wg
M j
Equation 4-18
Where :
In addition for determination of jet momentum flux we have:
2 2
4
u dj j jG
[N] Equation 4-19
m´= mass flow rate [kg/s]
d0= hole diameter [m]
d j = jet diameter [m]
And PC = static pressure at the outlet can be calculated by:
2ln( ) ln( ) ln
1 1P Pc init
[Pa ] Equation 4-20
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68
And the jet gas temperature at flare tip can be calculated by:
1ln( ) ln( ) ( ) ln( )
Patmospheric
T Tj initial P
initial
[k] Equation 4-21
Nevertheless, for high-pressure gas, it would be essential to compute specific heat ratio more
accurately:
Cp
CV
in which for ideal gas R
C CV P W g
[J/[kg⋅˚K]] Equation 4-22
The Thornton model introduces the following rough estimation of the mass fraction of flammable
substances in a stoichiometric mixture with air, W, for use in next section.
(15.816 0.0395)
WgW
Wg
Equation 4-23
4.3.1.1 Thornton Model for Determination of Flare Flame Geometry
Geometric factors of the flare flame is introduced by Fig 4-4. Calculation of these factors will be
discussed in this section [7-8, 11-18, 27-31, 53-54, 72-76].
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Fig 4-4: quadrilateral flare flame shape used by Thornton model (Chamberlain,
1987, Institute of Chemical Engineering)
The five fundamental factor used to correlate the flare flame form are R , W , W , α and bL 1 2
.
Computation of the quadrilateral which represented to fare flame shape surface area with a median
width can be implemented as an alternative way as follows:
22 2 2 2 1( ) ( ) (
1 2 1 24 2 2
W WA W W W W R
L
[m2] Equation 4-24
A= quadrilateral surface area [m2]
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70
W1 = width of quadrilateral base [m]
W2= throat tip width [m]
RL= throat length [m]
For quadrilateral geometry parameter determination equation 4-25 can be used.
2( ) ( )
1 2 1 2
2 2 2
W W W WA R
L
[m2] Equation 4-25
Where A = cylinder surface area [m2]
For calculation of the quadrilateral width at flare tip:
1 1(0.18 0.31) (1 0.47 )
2 (1.5 ) (25 )W L
b R Rww ee
[m] Equation 4-26
Where
W2 = width of quadrilateral at flare tip [m]
Lb= length of the flame from flare tip [m]
RW=wind speed ratio to the jet speed [unitless]
And for calculation of the quadrilateral base width we have:
'70 ( )1 1
(13.5 1.5) 1 11 6 15
C RwP R Dsair iW D es R Pwe j
[m] Equation 4-27
Where:
1' 1000 0.8(100 )
CRwe
Equation 4-28
Where wind speed to jet speed ratio can be determined by:
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71
uwRW u
j
Equation 4-29
uw = wind velocity, [m/s]
uj = jet velocity, [m/s]
R =i
Richardson number
Moreover, the density ratio among jet and air can be calculated by:
( )
T Wj airair
T Wgj air
Equation 4-30
Where
W1= width of quadrilateral [m]
In addition, air density can be determined by:
P Wairatmosphericair R Tair
[Kg/m3] Equation 4-31
in which:
Pc= static pressure at the hole exit plane [Pa]
Wair = air molecular weight [kg/mol]
Rc = gas constant 8.314 [J/[mol⋅K]]
Tair = air temperature, [K]
We need to determine the Richardson number, which relies on the burning point diameter, which
can be utilized for the computation of the quadrilateral base width:
3( ) ( )4
air gR D Lsi G
Equation 4-32
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72
For calculation of the length of quadrilateral, we can use:
1
2 2 2 2sin ( ) cos( )R L b bL b
Equation 4-33
Where RL = length of quadrilateral, [m]
We can calculate the flame lift-off by implementing equation 4-19 in the following correlation:
0.141b G air Equation 4-34
Where:
α = angle between hole and flame
200.185 0.015
RwK e
Equation 4-35
In tranquil air, α is equal to 0°, and b = 0.2 × L . Additionally, for weak flames pointing straightlyb
when wind speed is too low is equal to 180°.
bb = 0.015 × L .The Richardson number of the flame in tranquil air can be calculated by:
13
( ) 02 20
gR Li L b
D ub s j
Equation 4-36
If 0.05Rw , then the flame is jet dominated. The flame distortion angle can be calculated by:
80001( 90 ) (1 )
(25.6 )( )
0
RwjV R Rwe i L
b
[˚] Equation 4-37
in which:
jv = angle among the hole axis and the wind horizontal direction
Ri[Lb0] = Richardson number based on Lb0
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73
L = b0
flame length in tranquil air [m]
And If Rw > 0.05, then the flame distortion progressively turns to dominated by wind forces
which can be described by the following formula:
12
(134 1726 0.026 )1=( 90 ) (1 )
(25.6 )( )
0
RwjV R Rwe i L
b
[˚] Equation 4-38
Following the modeling, diameter of combustion source can be determined by the following
formula, which corresponding the throat diameter of an illusory nozzle scheme.
12
4 'D
ms
uair j
[m] Equation 4-39
Where:
Ds = hole diameter, [m]
m' = flow rate, [kg/s]
ρair = air density [kg/m3]
uj = jet velocity [m/s]
If the size of the hole is specified, the mass flow rate at the outlet hole should be computed with
the models from Chapter 2 with respect to the computation of the gas discharge rate.
When we deal with choked flow, the actual hole diameter can be computed by:
WgPcj R Tc j
[Kg/m3] Equation 4-40
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74
12
jD ds j
air
[m] Equation 4-41
in which:
Pc = static pressure at the hole outlet [Pa]
dj = diameter of the jet at the outlet hole [m]
ρj = gas jet density [kg/m3]
We can consider that the jet diameter is approximately equal to the diameter of the hole. In the
model, computation of the first excess factor Y by can be driven forward by performing iteration
with the following equation:
2.853 32 2 230.024 0.2 ( ) 032
g Ds Y Y YWu j
Equation 4-42
Where :
Y = dimensionless variable
W = stoichiometric mass fraction (refer to equation 4-23)
The parameter represent the scale among the wind momentum flux in vertical line and jet
momentum flux can be introduced by:
04
air L uairbG
[unitless] Equation 4-43
Or it can be calculated as a linear by the following equation:
tan( )
0.178
[unitless] Equation 4-44
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75
In addition, to compute the jet flame length in tranquil air we can use the following formula:
0L Y Dsb
[m] Equation 4-45
0Lb
= flame length, in tranquil air [m]
And the length of the jet flame from the flare tip to the centre of the exit plane can be calculated
by:
0.51 6.07(( 0.49)) (1 ) ( 90 ))
0 3( 0.4 ) 10L L
b b jVuwe
[m] Equation 3-46
=uw wind velocity [m/s]
The parameters of transformation also determine when jv j :
2 2' ( sin ) ( cos )j jX b X b Equation 4-47
sin' 90 arctan( )cos
j
j
j
b
X b
Equation 3-48
with consideration of 90jv j jv
' 1 2
4
W Wx X
Equation 3-49
Where:
X= flare flame length distance to the object [m]
X' = length distance to the object from the centre of flare flame[m]
Θ' =angle among liftoff flame and plane the bottom centre of the flare flame and object. [°]
Θ =j
angle among hole axis in horizontal direction [°]
W2 = quadrilateral width tip, (refer to Fig 4-4) [m]
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76
W1= quadrilateral base width (refer to Fig 4-4) [m]
Object
Fig 4-5: Geometrical factors for determination of lifted-off flare fames
Knowing the SEP and also by consideration of present correlations, and implementing equations
4-51 to 4-65 we can determine FS and FV and with equation 4-66 we can calculate maximum view
factor and apparently maximum heat flux from equation 4-12 where:
F =v view factor in vertical axis
1 2R=4
W W [m] and
' '
1, ,X X L R Equation 4-50
4.3.2 Determination of the View Factors
Three different radiator shapes can be considered for determination of the view factor. These
shapes can be included a vertical cylindrical radiator, a spherical radiator and a vertical flat
radiator. Nevertheless, the most important for elevated flame has been described in this section.
quadrilateral
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4.3.3 Mudan Model for Determination of View Factor
Mudan in 1987 introduced a theory, which determines the geometrical view factor for horizontal
and vertical surface at a specified point, for cylinder and rectangular shape.
The vertical and horizontal axis view factors can be described by the Atallah (1990) model with
following equations[ 53-54, 72-76]:
2 2( 1) 2 (1 sin ) cos1 1
F tan tan ( )a b b a AD
E D EV AB B C
2 2sin sin1 1
tan ( ) tan ( )ab F F
FC FC
Equation 4-51
2 21 sin sin sin1 1 1
tan ( ) tan ( ) tan (ab F F
Fh D C FC FC
2 2( 1) 2( 1 sin ) 1
tana b b ab AD
AB B
Equation 4-52
Which:
( )LLL bf or
R R R Equation 4-53
Xb
R Equation 4-54
2 2( ( 1) 2 ( 1) sin )A a b a b Equation 4-55
2 2( ( 1) 2 ( 1) sin )B a b a b Equation 4-56
2 21 ( 1) cosC b Equation 4-57
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78
1
1
bD
b
Equation 4-58
cos
sin
aE
b a
Equation 4-59
2( 1)F b Equation 4-60
Lf = the inclined flame distances.
L =b
flame length to exit centre of plane [m]
L =f
quadrilateral flame length [m[
L = average flame height [m]
The angle of incline θ is determined with regard to the vertical specified destination which is placed
across the wind direction from the origin point that can be considered positive amount and it can
be considered negative for specified destination situated upwind from the origin point.
4.3.4 Determining the view factors with consideration of the crosswind way
With consideration of the crosswind way which its direction is perpendicular to the incline angle
the calculation of view factors for an specified point can be calculated as follows: [20, 55, 75]
sin sinsin sin1 1 1 12 2 tan tan tan 2 tan ( )
ab abF F FF D
h I I I I
2 21 11 2 sin 2 sin
( ) tan tana b HD a HD a
G G G
Equation 4-61
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79
2 21 2 sin2
sin cos2 ln
2 2 2 2 22( sin ) 1 2 sin
Fa b a
a bFV Fa b a b a
b
sin sincos 1 1
tan tan
ab ab
F F
I I I
2 2cos 1 2 sin 2 sin1 1tan tan2 2 2sin
ab a b HD a HD a
G G Gb a
1
2 2 2
2 costan
sin
abD
b a
Equation 4-62
Where:
2 2 2 2 2 2(( 1) 4 ( sin ))G a b b a Equation 4-63
2 2( 1)H a b Equation 4-64
2 2( sin )I b Equation 4-65
The maximum view factor is determined as follows:
2 2F ( )max F FV h
Equation 4-66
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5 Chapter Five: Depressurization Dynamic Calculation
5.1 Depressurization Dynamic Calculation
Dynamic calculation has to be implemented at the basic engineering design stage to determine
temporary increasing pressure to compute necessary relief rates and other important variables. In
addition, by dynamic calculation we can understand what occurs at the time of relief. Dynamic
calculation should be carried out with respect to facilities such as distillation towers, compression
units, refrigeration systems with multiple loops, and tube ruptures of heat exchanger [API 521].
Also it is very important for evaluation of flare systems during depressurization. If dynamic
calculation is carried out for distillation tower relief arrangement design, it is required to check the
model to make sure it is conservative with regards to computation the total relief load and in some
cases assumptions have to be made. These assumptions should be examined with accurate analyses
to evaluate their effect on the tower relief load. For instance, many calculation runs need to be
carried out to find out the impact of different parameters on different trays on the relief load.
Besides, it is better to share this dynamic modeling with operating personnel since time to respond
to the situation and controlling depressurization can help plant and personnel for preventing of any
hazardous consequences. Generally, for depressurization calculation two major scenarios are keys
for the estimation of peak flow, relief time, temperature and other important variables. The first
scenario is cold depressurization or well known as the adiabatic case. It is intended to simulate the
actual gas depressurization of vessels and piping containing high pressure fluid. The second one
is the fire case that can be carried out to simulate emergency situations in a plant. The fire is
represented by a heat transfer to the vessel. This scenario is described by API 521. For both cases,
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HYSYS software is used to evaluate the effect of different factors for controlling the conditions.
The depressurization item in HYSYS can be used to evaluate the required time as a function of
pressure and consequently help engineers to recognize material loses at this time. The model
consists of three different stages which are, physical explanation of the pressurized equipment,
thermodynamic conditions of the process, and discharge flow conditions during depressurization
across the valve.
5.1.1 Adiabatic Depressurization Study/Cold Depressurization
Adiabatic depressurization calculation will be evaluated with some assumption data. Since in this
case there is no injection of heat to the system, one important factor for adiabatic calculation is
ambient temperature. The minimum ambient temperature can be taken into account as a safe
margin assumption for the system initial temperature. This assumption is the best engineering
practice for small vessels without insulation. For large vessels, consideration of varying ambient
temperature with some conservatives and delays has to be taken into account for calculation[API
521]. The conservative amount relies on heat amount on the system and maximum heat transfer
rate. An accurate computation should be carried out to determine the minimum temperature in the
system throughout coldest time of the year. Extreme cold can lead to brittleness of construction
materials, and loss of tensile strength and hence to pipe rupture. Table 5-1 shows average minimum
temperature in Fort McMurray in Alberta which can be used to determine the minimum
temperature in worst case at the time of cold depressurization with consideration of other factors.
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Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Daily
Average
[°C]
-17.4 -13.3 -6.2 3.3 9.9 14.6 17.1 15.4 9.5 2.3 -8.6 -15.1
Extreme
Wind
Chill[°C]
-58 -60 -57 -46 -21 -6 0 -6 -16 -32 -50 -53
Daily
Maximum
[°C]
-12.2 -7.1 0.6 10 16.9 21.5 23.7 22.2 15.8 7.4 -4.3 -10.1
Daily
Minimum
[°C]
-22.5 -19.5 -12.9 -3.5 2.8 7.7 10.5 8.6 3.2 -2.8 -12.9 -20
Extreme
Minimum
[°C]
-50 -50.6 -44.4 -34.4 -17.3 -4.4 -3.3 -3.1 -15.6 -24.5 -37.8 -47.2
Table 5-1: Temperature data between years 1981 and 2010 in Fort McMurray, Alberta
Since the temperature goes down, the heat flux among the fluid and the vessel is the key important
factor. One example is to evaluate the depressurization of settle out pressure of compressor cycles
at the time of emergency shutdown. The depressurization model needs a factor to determine degree
of degradation for consideration. As a safe conservative, adiabatic depressurization has considered
as an isentropic depressurization while fire case considered as an isenthalpic depressurization.
However, depressurization calculation assumption for implementing this degree can be between
isenthalpic and isentropic process. As seen in to the Mollier diagram (Appendix F) the isentropic
process gives the lowest temperature for adiabatic case while the isenthalpic process gives the
highest peak flow rate for the fire case, which can be used at the design stage.
Some data has been calculated to evaluate different parameters that can effect on flare flame
distortion and heat radiation. Depressurization evaluation has been carried out by Hysys
depressurization utility v8.6 and some figures for flare flame developed by Microsoft Excel 2013.
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A depressurization system has to be taken into account to decrease the vessel stress to a safe level
to avoid of vessel rupture and event escalation. According to API, which is based on experimental
studies, a pressure relief system must be capable to reduce the pressure by 50 % in 15 - 30 minutes.
In the calculation below, the valve capacities (CV), which are used for the restriction orifice size,
needs to reduce the pressure by 50% in 15 or 30 minutes as determined by trial and error and
comparing with few existence experimental data.
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5.1.2 Adiabatic case
Assumption for the related figures are as follows:
Table 5-2 Assumptions for adiabatic depressurization for methane gas
Adiabatic Case
Vessel volume [m3] 64
Vessel thickness [mm] 130
Insulation thickness [mm] 200
Vessel length [m] 9
Vessel diameter [m] 3
Initial pressure [bar] 65
Ambient temperature [C˚] -33.7
Initial temperature [C˚] -27.7
Pipe length [m] 100
Methane [CH4] mole fraction 1
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Fig 5-1: Depressing in 15 min adiabatic considering Masoneilan valve fully open
Fig 5-2: Depressing in 30 min adiabatic considering Masoneilan valve fully open
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A depressurization evaluation has been simulated by Hysys, and Fig 5-1 reveals that with
consideration of reaching 50% of design pressure in 15 minutes, vapor at the outlet can reach a
minimum value of -91.62 ˚C in the range between 515 to 537 seconds. With a maximum allowable
time of 30 minutes, as per API concept, it will reach a minimum value of -87.92 ˚C in the range of
630.5 to 638.5 seconds while vessel wall temperature reaches -29.57 ˚C.
Fig 5-2 considering 30 minutes time for depressurization was obtained with a vapour valve
capacity, CV [USGPM(60F,1psi)] equal to 6.58. It shows that peak flow rate is 6115 kg/h, while
Fig 5-1 in 15 minutes is obtained with a vapour CV [USGPM(60F,1psi)] equal to 12.25, shows
the peak flow rate is 11382 kg/hr. These data are highly dependent to the final pressure setting
with consideration of the point where vessel can be ruptured. While the minimum vessel inner wall
temperature reaches to -29.57 ˚C we can evaluate and change the material of construction right
after vessel outlet nozzles both from top and bottom due to the effect of heat transfer. When
increasing time for depressurization by setting final pressure to 50 % of design pressure, since
there is more time for heat transfer, consequently final temperature will be lower in 15 minutes in
comparison with the 30 minutes case. However, in comparison to set Hysys to find a final pressure,
temperature would be lower due to lower final pressure cause to lower temperature. Consequently,
Since the most important factor at the time of depressurization is to avoid rupture of the vessel and
pipe, especially when historical data does not exist at the operation stage, it is recommend to
consider reduction of pressure to the 50% of design pressure in depressurization case at the
operation stage, while for new projects at the design stage for material selection, it is recommend
to consider Hysys to find a valve capacity (CV) in fire cases as the worst case for 15 and 30 minutes
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time based on maximum material tensile strength, and then implement these values to find a final
pressure in adiabatic depressurization.
5.2 Fire case depressurization
Assumptions for calculation of this case are as follows:
Table 5-3: Assumptions for fire case depressurization for methane gas
API521 Fire Case
Vessel volume [m3] 64
Vessel thickness [mm] 130
Insulation thickness [mm] 200
Vessel length [m] 9
Vessel diameter [m] 3
Initial pressure [bar] 65
Ambient temperature [C˚] -33.7
Initial temperature [C˚] -27.7
Pipe length [m] 100
Methane [CH4] mole fraction 1
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Fig 5-3: Depressurization in 15 min considering Fire API 521 and Masoneilan valve fully open
Fig 5-4: Depressurization in 30 min considering Fire API 521 and Masoneilan valve fully open
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These figures in comparison to the adiabatic case (Fig 5-1 and 5-2), show that heat absorption
through the vessel wall from fire leads to higher flow rate discharge required from the vessel. In
addition, density in this case decreases during the depressurization time while in the adiabatic case
in time between 10-14 minutes we can see increasing density versus time. It happens because of
external heat input into the vessel wall.
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5.2.1 Adiabatic and fire case with different composition
Assumptions for calculation of these cases are as follows:
Table 5-4: Assumptions for adiabatic and fire case depressurization
Adiabatic and Fire Case
Vessel volume [m3] 64
Vessel thickness [mm] 130
Insulation thickness [mm] 200
Vessel length [m] 9
Vessel diameter [m] 3
Initial pressure [bar] 65
Ambient temperature [C˚] -33.7
Initial temperature [C˚] -27.7
Pipe length [m] 100
Methane 0.9060
Ethane 0.0515
Propane 0.0013
i-Butane 0.0013
n-Butane 0.0021
Nitrogen 0.0360
Phase Gas
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Fig 5-5: Depressurization in 15 min considering Fire API, CV 16.8 USGPM (60F, 1psi) and
Masoneilan valve
CV (USGPM) [60F,1psi] 16.8
Final gas Temp. ( C ) -83.4
Inventor Mass (Kg) 4907
Final Gas Mass (kg) 2458
Peak Flow (kg/hr) 1360
Table 5-5: Depressurization in 15 min considering Fire API, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve
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These figures have been taken by setting Hysys, first fire API521 case, as the worst case scenario
for depressurization to reach to the 50% design pressure on 15 minutes and then based on given
CV, adiabatic case has been set to determine final pressure.
Fig 5-6: Depressurization in 15 min considering Adiabatic, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve
CV (USGPM) [60F,1psi] 16.8
Final Gas Temp. at Valve Exit( C ) -86.6
Minimum Gas Temp. at Valve Exit( C ) -95.8
Final Inner Wall Temp. ( C ) -30.6
Minimum Liquid Temp. at Valve Exit( C ) -38.5
Inventor Mass (kg) 4907
Final Gas Mass(kg) 1823
Final Liquid Mass (kg) 25.6
Peak Flow (kg/hr) 19830
Final Pressure (bar) 24
Table 5-6: Depressurization in 15 min considering Adiabatic, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve
Vessel wall temperature yellow is liquid and brown is gas
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Fig 5-8: Depressurization in 30 min considering Adiabatic, CV 16.88 USGPM (60F, 1psi) and
Masoneilan valve
CV (USGPM) [60F,1psi] 16.8
Final Gas Temp. at Valve Exit( C ) -70.37
Minimum Gas Temp. at Valve Exit( C ) -95.87
Final Inner Wall Temp. ( C ) -32.9
Inventor Mass (Kg) 4907
Final Gas Mass(Kg) 657
Final Liquid Mass (kg) 6
Peak Flow (kg/hr) 19830
Final Pressure (bar) 9.64
Table 5-7: Depressurization in 30 min considering Adiabatic, Cv 16.88 USGPM (60F, 1psi) and
Masoneilan valve
Vessel wall temperature yellow is liquid and brown is gas
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Fig 5-6 and 5-8 revealed that the minimum vessel inner wall temperature reaches to -38.56 ˚C for
liquid section and -30.63 in gas section in 15 minutes and by increasing time to 30 minutes vessel
wall temperature for liquid and gas will reach to -48.89 ˚C and -32.9 ˚C respectively. The valve
outlet minimum temperature must be taken into account for material of construction downstream
of the depressurization valve. Two figures shows increasing depressurization time leads to
decreasing vessel metal wall. The study revealed that in some cases following this rule, which is
more realistic for emergency depressurization causes choke flow. Figure 5-6 shows right after
reaching 44% of design pressure that is the limit, based on criteria introduced in equation 3-20,
( )1 10 ( )
2
P
Pa
we will be faced with choke flow. Having the specific heat ratio and pressure
ratio, we can confirm this issue. Increasing time to 30 minutes for the adiabatic case presented by
Fig 5-8, revealed that the final pressure would reach to 9.6 bar, below the critical value for choke
flow with these conditions. Choke flow is hazardous because it can cause vibration on pipes
downstream of depressurization valve and consequently blowout of the flare header pipe. Hence,
determination of allowable time for depressurization must be determined by dynamic calculation
to avoid event escalation.
Takes to account due to in some cases if the final pressure cannot reach to the atmospheric pressure
in a certain time, which is more important for maintenance and plant shutdown, we need to increase
depressurization time by consideration of allowable material tensile strength at the operation stage
or at the design stage increase the downstream pipe size. Engineering cost estimation is
recommend to determine changing material of construction can be beneficial or changing
downstream pipe size.
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5.2.2 Adiabatic and fire case for liquid Methane
Assumptions for this case would be as follows:
Table 5-8: Assumptions for adiabatic and fire depressurization for liquid methane
Adiabatic and Fire Case
Vessel volume [m3] 64
Vessel thickness [mm] 130
Insulation thickness [mm] 200
Vessel length [m] 9
Vessel diameter [m] 3
Initial pressure [bar] 65
Ambient temperature [C˚] -33.7
Initial temperature [C˚] -75.3
Pipe length [m] 100
Methane [mole fraction] 1
Mass density (kg/m3) 211.46
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Fig 5-9: Depressurization liquid methane at saturated pressure in 30 min Fire API521 case by
masoneilan valve
CV (USGPM) [60F,1psi] 21.48
Final Gas Temp. at Valve Exit( C ) 12.45
Minimum Gas Temp. at Valve Exit( C ) -161.6
Peak Flow (kg/hr) 41990
Table 5-9: Results for depressurization liquid methane at saturated pressure in 30 min Fire
API521 case by masoneilan valve
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Fig 5-10: Depressurization, 100% opening by universal gas sizing/Fisher valve for adiabatic case
30 min with CV equal to 21.48 USGPM (60F, 1psi)
CV (USGPM) [60F,1psi] 21.48
Final Gas Temp. at Valve Exit( C ) -161.6
Minimum Gas Temp. at Valve Exit( C ) -161.6
Peak Flow (kg/hr) 68250
Final Pressure (bar) 25.86
Table 5-10: results for depressurization, 100% opening by universal gas sizing/Fisher valve for
adiabatic case 30 min.
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Fig 5-11: Depressurization 30 min adiabatic liquid methane by universal gas sizing/fisher valve
with 20% opening with CV 21.48 [USGPM (60F, 1psi)
Fig 5-12: Depressurization 15 min adiabatic liquid methane by universal gas sizing/fisher valve
with 20% opening with CV 21.48 USGPM (60F, 1psi)
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Fig 5-13: Depressurization 15 min adiabatic liquid CH4 by fisher valve with 30% opening and
CV21.48 [USGPM (60F, 1psi)].
Phases have flipped between the vapor and liquid slots in the vessel. Also, a spike or discontinuity
has happened in the dynamic calculation. This happens with supercritical fluid.[refer to Appendix
C]. A spike and discontinuously in pressure in Fig 5-9 reveals that between time 2 to 10 minutes
the system is faced with high pressure drop in a short period across the orifice and cause to
vibration in pipe. In addition, increase velocity/kinetic energy by significant decrease of
pressure/potential energy leads to increasing bubble formation exit from top that cause to valve
rupture.
In terms of characteristics of control valve flow, choosing a quick opening valve for supercritical
fluid cannot be a good procedure for depressurization because the time available to respond by
operators is low, around 4 minutes according to Figure 5-10 and 5-13. Consequently, choosing
equal percentage can be a good suggestion (Appendix C).
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Choosing universal gas sizing/Fisher with setting the opening percentage to 20%, with calculated
CV by fire API 521 case equal to 21.48, in 15 minutes can be a good solution to control
depressurization event of liquid methane. It should be noted setting more than 20% leads to same
problem shown by Figures 5-10 and 5-13, where fluid will reach the critical point and Cv fail to
converge to a value. This means that due to sudden depressurization we may be faced with vacuum
conditions in vessel or pipe before the allowable time.
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5.2.3 Dynamic calculation for determination of flare flame distortion and heat radiation
Flare flame distortion and heat radiation for depressurization case has been evaluated by API and
the Chamberlain method known as the Thornton model and figures developed by Microsoft Excel
2013. Assumptions are as follows:
Table 5-11: Assumptions for flare flame distortion calculation
Normal inventory 550
Total inventory volume [m3] 700
Tip diameter [mm] 600
Normal flow [kg/hr] 123300
Peak flow for depressurization [kg/h 156427
Initial pressure [bar] 65
Ambient temperature [˚C] -37
Initial temperature [˚C] -27
Pipe length [m] 100
Methane [mole fraction] 0.969
Ethane [mole fraction] 0.03
Propane [mole fraction] 0.0001
Butane [mole fraction] 0.0001
Local wind speed [m/s][Refer to Appendix D] 15
Stack height [m] 57
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Fig 5-14: Mach number versus time in steady state (blue) and dynamic (red) calculation
Fig 5-15: Flare exit velocity versus time
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
Mac
h N
um
ber
Time (s)
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
220.00
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
Fla
re e
xit
vel
oci
ty
Time (s)
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Fig 5-14 shows consideration maximum Mach number equal to 0.45 and the steady state
calculation Mach number will violate the criteria, while dynamic results shows that during
depressurization the Mach number may violate the maximum Mach number for only around 1
minute, which is acceptable. Hence, because the process upset is not continually, engineering
judgment for determination of flare header size can help reducing cost of material for construction,
whereas by considering steady state calculation we would conclude that we need to increase flare
header size. In this figure, the blue line represents the steady state calculation and red line represent
to the dynamic results.
Fig 5-16: Heat radiation versus time at the time of depressurization by Chamberlin method
Having the calculation results based on every step time, and by implementing values we can
determine change of Mach number and heat released versus time. As the time to respond to control
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
Hea
t R
adia
tion (
Kw
/m2)
Time (S)
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the event is highly dependent to the historical data taken by calculation, implementing a dynamic
calculation and using data for evaluation of flare flame distortion is the way to meet the safety
regulations while steady state may increase cost of material and pipe size.
Fig 5-17: Heat radiation versus time at the time of depressurization by API method
Velocity at exit Sonic velocity Mach No. at exit
193.300 402.716 0.479
Table 5-12: Flare tip exit velocity
0.000
0.500
1.000
1.500
2.000
2.500
3.000
3.500
4.000
4.500
5.000
0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950
Hea
t R
adia
tion
(K
w/m
2)
Time (s)
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Fig 5-18: Allowable design limit for flare system modeling and calculation [API 521]
Refer to table 5-18, radiation upper than 4.73 needs an immediate action, which has to be watch
over for the design and calculation of the flare at the basic and detail engineering design.
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40 m
85m
Ground
Fig 5-19: Flare flame distortion from top view at normal condition, Chamberlain method
Ground
Fig 5-20: Flare flame distortion from top view at normal condition API method
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Ground
Fig 5-21: Flare flame distortion from top view at the time of depressurization Chamberlain
method
Ground
Fig 5-22: Flare flame distortion from top view at the time of depressurization by API method
43 m
95 m
43 m
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Flame Length (m) Normal operation Depressurization
Chamberlin 40 43
API 85 95
Table 5-13: Results for flare flame lenght
5.3 A Case Study for Pipeline
Assumptions for the evaluation of a 7 km pipeline transferring methane gas where a puncture
occurs at a point close to the end of the pipe are shown in Table 4-12.
Input
Methane [mole fraction] 1
Temperature [˚C] -27
Pressure [bar] 65
Flow rate [kg/hr] 1,700,000
Pipeline length [m] 7000
Pipeline Diameter [in] 40
Elevation Change [m] 1
Pipe Schedule type 40
Table 5-14: Assumptions for Pipeline case study
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Fig 5-23: Mach number versus distance in pipeline
Fig 5-24: Temperature versus distance in pipeline
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Fig 5-25: Pressure versus distance in pipeline
The study assumes maximum inventory, and a puncture or hole close to the destination. The study
revealed that having the Mach number versus distance for any case can give us an important
information for upstream conditions at the time of depressurization. According to Fig 5-23, we can
determine when and where system can violate the criteria then we can install a valve or reroute the
gas to the flare to avoid increasing velocity and Mach number to minimize possible damage to the
pipe. This precaution can prevent escalation of the event.
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6 Conclusion and Future Work
6.1 Conclusion
This thesis consists of three principal topics. Depressurization dynamic modeling of vessel and
piping, pipeline and consequently flare flame distortion and heat radiation evaluation. Dynamic
calculations for depressurization can play a role as a root term model to determine some important
data regards to the quantity of fluids discharge into the environment. As shown in Figures 5-9 to
5-13 depressurization by current industrial approach, considering 100% valve opening causes
problems downstream of the depressurization valve when supercritical fluid needs to be
depressurized. Data shows choosing fisher/universal gas sizing valve by setting in 20% opening
can handle depressurization in 15 minutes. Therefore, by considering an appropriate logical system
we can control this condition. According to data represented in Figures 5-6 and 5-8, set final
pressure as an unknown variable that should be determined by having the valve capacity,
increasing depressurization time leads to decreasing vessel metal wall, while Figures 5-1 and 5-2
shows increasing time by setting final pressure to the 50% of design pressure leads to higher
temperature. Therefore, at the design stage it is recommend to find a final pressure, by
implementing valve capacity (CV) taken by worst case scenario, instead of setting final pressure
to a fixed value to 50% of design pressure. Therefore, following this procedure leads to taking an
accurate data and choosing an appropriate material of construction that causes to avoid vessel and
pipe rupture and event escalation. In addition, it is recommend to takes to account a determination
of possible choke flow far after restriction orifice to consider adequate safety items to avoid event
escalation.
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Dynamic results have been used for evaluation of flare flame distortion and heat radiation by two
principal models, API and Chamberlain known as a Thornton-model. The study by the
Chamberlain model shows that in normal operation flare flame length would be equal to 40 m
while with the API model the flame length is 85 m. At the time of depressurization, the flame
length by Chamberlain model would be equal to 43m while the API method equal to 95m. Besides,
maximum radiation predicted at the time of depressurization by Chamberlain model is equal to
5.15 kw/m2, while the API method predicted 3.984 kw/m2. Comparison of the two methods shows
difference around 52m in flare flame length at the time of depressurization. Using the Chamberlain
method shows that increasing jet velocity causes increasing length of the flame and consequently
increasing heat radiation. On the other side, data taken by using the API method shows that flame
stability highly depends on jet velocity that cause to stability and leads to lower heat radiation,
since it leads to increasing distance from center of the flame to the object as well as the height of
the flame, represented in Fig 4-1. While some regulatory limit the exit velocity, following the API
method shows increasing velocity even though Mach number met the criteria, leads to decreasing
radiation, so radiation limit in the engineering approach following API method has adverse with
any regulation which limit the flare tip exit velocity by considering depressurization case, since
decreasing jet velocity cause to increasing heat radiation by this method.
Taking into attention an extinguished flare flame and considering Fig 5-1 and Fig 5-2, gas density
change versus time also can reveal that increasing density between the time of 6 to 10 minutes for
a maximum 15 minutes and 10 to 14 minutes for a maximum 30 min for the adiabatic case has the
maximum amount. Therefore, with consideration of the equation 2.6 and 2.7 in Air Dispersion
Modeling: Foundations and Applications by De Visscher, the buoyancy flux parameter will have
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a maximum value in this time range and consequently the plume rise has the maximum value at
this time range in terms of plume rise theory viewpoint. The plume rise can be evaluated
dynamically versus time for a detailed review based on air dispersion modeling concept. Three
dimensional numerical model for determination of flare flame distortion can present detailed
information in terms of the momentum flux area in the flare flame, and the velocity that anticipate
a flame geometrical shape which will let them be utilized to anticipate complicated three
dimensional flows. Finally, implementing dynamic calculation can give us accurate results that
can be more practical to evaluate pipe and vessel fluid behaviour at the time of depressurization
both for adiabatic and implementing fire API 521 equation.
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6.2 Future work
Since different theories exist for the determination of flare flame distortion and heat radiation, a
good engineering judgment has to be based on historical and experimental data whether taken by
modern software or by equations and laboratory experiments in different situation. Rapid
depressurization is one of the most important factors that has to be taken into account with high
attention to all consequences and safeguards. A good engineering design can be carried out with
high attention to the safety to keep personnel and plant safe for operation. The current study
emphasises to consider many factors that effect on basic and detailed engineering design. Taking
to account in some conditions of depressurization can take more than the allowable time, in large
volume inventory, many other factors can be evaluated to determine environmental consequences
in terms of biological or air pollution by this event. Consequently, a future study can be focused
on dynamic calculation of air dispersion with special attention to determine the effect of steam
assisted and air assisted flare. Evaluation of depressurization by consideration of small step size
may give us huge paper reports that can be used for flare flame distortion and heat radiation, so a
new programming can be written associate with depressurization, flare flame distortion, heat
radiation and eventually dynamic calculation of air dispersion consequences.
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Appendices
Appendix A: Determination of the Darcy Friction Factor fD
The Darcy friction factor fD can be computed by the Colebrook-White law for the transition zone
as follow:
1 2.51102 log( )3.75 Redf fpD D
[unitless] ........................................ Equation 1
The Reynold’s number subject to the flow regime can be determined by:
Red p
u
[unitless] ................................................................................... Equation 2
where
fD = Darcy friction factor [unitless]
dp = pipe diameter [m]
ε = wall roughness [m]
Re = Reynolds number [unitless]
uf = fluid flow velocity [m/s]
ρf = fluid density [kg/m3]
η = dynamic viscosity [N⋅s/m2]
And the Darcy-Weisbach equation also determines the head loss in the pipe as follow:
2
2
f l upD fhLp g d p
[m] ................................................................................ Equation 3
Where
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g = gravitational acceleration [m/s2]
ΔhLp = head loss [m]
lp = pipe length [m]
The Friction factor can also be determined by the Moody chart for flow in pipes (Figure A.1)
Fig A.1 Friction factor for flow in pipes by Moody chart [Perry's Chemical Engineers' Handbook
1984]
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Appendix B: Auxiliary Equation Buoyance flux parameter De Visscher Air Dispersion
modeling book
2(1 )sF gr ws sb
[m4/S3] ............................................................................ Equation1
1 13 31.6F x
bhu
[m] .................................................................................. Equation 2
s = density of the gas at stack [kg/m3]
= density of the surrounding air [kg/m3]
rs = stack radius [m]
ws = stack gas velocity in vertical direction. [m/s]
x = distance downwind from the source [m]
u = wind speed [m/s ]
h = maximum plume height [m]
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Appendix C: Excess Data and Figures
Table C.1: Critical properties of some component (Reid et al., 1987)
Characteristics of control valve flow
Fig C.2: Characteristics of control valve flow with piping
losses[http://www.flowserveperformance.com/performhelp/sizing_selection:valtek:flow_characteristics]
Solvent
Molecular weight Critical temperature Critical pressure Critical density
g/mol K MPa (bar) g/cm3
Methane (CH4) 16.04 190.4 4.60 (45.4) 0.162
Ethane (C2H6) 30.07 305.3 4.87 (48.1) 0.203
Propane (C3H8) 44.09 369.8 4.25 (41.9) 0.217
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Appendix D: Fort McMurray Historical Wind Speed
Table D.1: Wind speed data represent the FortMcmurray ,Alberta
http://fortmcmurray.weatherstats.ca/metrics/wind_speed.html
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Appendix E: Mollier diagram
Fig E.1: Mollier diagram, an enthalpy–entropy versus pressure (GPSA 12 edition)