SCUOLA DI INGEGNERIA INDISTRIALE E
DELL’INFORMAZIONE
Laurea Magistrale in Ingegneria Meccanica
Development of a CFD model to simulate the
compressible flow within a pressure-relief safety valve
Relatore: Prof. Giacomo PERSICO
Correlatore: Dott. Ing. Andrea CONSONNI
Riccardo Attilio BANFI
878390
Anno Accademico 2018/2019
Ringraziamenti
Intendo ringraziare sentitamente:
Professor Giacomo Persico per avermi seguito ed aiutato con pazienza e disponibilità
durante tutto lo svolgimento del lavoro
Dott. Ing. Andrea Consonni e l’azienda AST S.p.A. per l’opportunità che mi hanno concesso
Ing. Alessia Incanti e Ing. Christian Garavaglia per le conoscenze che mi hanno trasmesso
durante l’attività di stage
I membri dell’ufficio tecnico e il personale di AST che ho avuto il piacere di conoscere
La mia famiglia, che comprende i miei genitori, mio fratello Davide, i miei zii Mauro e
Nuccia e mia cugina Paola, per il supporto agli studi e per essermi stati vicino nei momenti
difficili
Gli amici, i compagni di corso, le persone che hanno fatto o fanno tuttora parte della mia
vita, e in generale coloro con cui ho avuto la fortuna di condividere una parte del cammino
che mi ha reso la persona che sono
Contents
1 Introduction 1
1.1 AST S.p.A. Company ..................................................................................... 2
1.2 OpenFOAM ................................................................................................... 3
2 Pressure-relief Devices 5
2.1 Terms and Definitions .................................................................................. 6
2.1.1 Dimensional characteristics .............................................................. 6
2.1.2 Device characteristics ........................................................................ 6
2.2 Spring-loaded Pressure-relief Valves ............................................................ 8
2.2.1 Conventional pressure-relief valves .................................................. 8
2.2.2 Working principle ............................................................................. 11
2.2.3 Effect of the backpressure ................................................................. 13
2.2.4 Balanced pressure-relief valves ......................................................... 14
2.3 Sizing of pressure-relief Valves ..................................................................... 17
2.3.1 Converging-diverging nozzle ............................................................. 17
2.3.2 Sizing for vapor or gas ....................................................................... 20
2.3.3 Sizing for liquid ................................................................................. 21
3 Fluid Dynamics 23
3.1 Conservation Laws ........................................................................................ 23
3.1.1 Continuity equation ........................................................................... 25
3.1.2 Momentum equation ......................................................................... 25
3.1.3 Energy equation ................................................................................ 26
3.2 Turbulence .................................................................................................... 27
3.2.1 Reynolds Averaged Navier-Stokes Equations ................................... 27
3.2.2 Boussinesq’s hypothesis .................................................................... 29
3.3 Boundary Layer Theory ................................................................................ 29
3.4 Turbulence Models ....................................................................................... 32
3.4.1 Spalart-Allmaras model .................................................................... 32
3.4.2 − model .......................................................................................... 32
3.4.3 − model ......................................................................................... 33
3.4.4 − SST model .................................................................................. 33
4 CFD Introduction 35
4.1 Finite Volume Method .................................................................................. 37
4.1.1 Discretization of the convection term ............................................... 38
4.1.2 Discretization of the gradient term ................................................... 41
4.1.3 Discretization of the diffusion term .................................................. 42
4.1.4 Discretization of the transient term .................................................. 43
4.2 Solution of the Linear Equation System ....................................................... 44
4.3 Pressure-velocity Coupling Algorithms ........................................................ 45
4.3.1 SIMPLE algorithm ............................................................................ 46
4.3.2 PISO algorithm .................................................................................. 46
4.3.3 PIMPLE algorithm ............................................................................ 48
5 Physical Model 49
5.1 Description of the Valve ................................................................................ 49
5.2 3D Model ....................................................................................................... 53
5.2.1 Assembly modeling ........................................................................... 53
5.2.2 STL manipulation .............................................................................. 57
5.3 Mesh ............................................................................................................ 60
5.3.1 Background grid ................................................................................ 60
5.3.2 Geometry of the valve ........................................................................ 63
5.3.3 Mesh quality control ......................................................................... 72
5.3.4 Parallel meshing ................................................................................ 79
5.3.5 Patch definition ................................................................................. 80
6 Preprocessing 85
6.1 Thermophysical Properties ........................................................................... 85
6.2 Turbulence Model ......................................................................................... 88
6.3 Boundary Conditions .................................................................................... 89
6.3.1 Pressure ............................................................................................. 91
6.3.2 Velocity .............................................................................................. 94
6.3.3 Temperature ...................................................................................... 96
6.3.4 Turbulent quantities .......................................................................... 98
6.4 Solver Selection and Algorithm Control ....................................................... 105
6.5 Discretization Schemes ................................................................................. 109
6.6 Time Step and Data I/O Control ................................................................... 111
6.7 Solution Monitoring ...................................................................................... 114
7 Results 116
7.1 Reference Case .............................................................................................. 117
7.1.1 Flow development ............................................................................. 117
7.1.2 Description of the flow field .............................................................. 119
7.1.3 Converging-diverging nozzle ............................................................. 122
7.1.4 Total pressure, total temperature and entropy fields ....................... 124
7.1.5 Mesh clustering ................................................................................. 127
7.1.6 Convergence history .......................................................................... 127
7.1.7 Force acting on the disc ..................................................................... 130
7.1.8 Actual coefficient of discharge .......................................................... 131
7.2 Experimental Results .................................................................................... 133
7.2.1 Actual coefficient of discharge .......................................................... 133
7.2.2 Force acting on the disc ..................................................................... 134
7.3 Influence of the Turbulence Model ............................................................... 136
7.4 Influence of the Numerical Schemes ............................................................ 139
8 Conclusions 145
A Post-processing Script 148
B Residual Plotting Script 157
C Allrun Script 160
D Function Objects 163
Bibliography 169
List of Figures
1.1 AST facility in Cornaredo .............................................................................. 2
1.2 Overall structure of OpenFOAM ................................................................... 3
1.3 Case directory structure ................................................................................ 4
2.1 Non-reclosing pressure-relief devices: rupture discs and pin actuated
devices ........................................................................................................... 5
2.2 Reclosing pressure-relief devices: spring-loaded, pilot operated, power
actuated ......................................................................................................... 6
2.3 Coventional and balanced pressure-relief valves .......................................... 8
2.4 Conventional pressure-relief valve ............................................................... 10
2.5 Sectional cut of a PRV with adjusting ring ................................................... 11
2.6 Pressure on the disc at valve closed (A), initial opening (B) and fully
opening (C) .................................................................................................... 12
2.7 Lift of disc and inlet pressure relationship ................................................... 13
2.8 Effect of superimposed backpressure in a conventional PRV ...................... 14
2.9 Effect of superimposed backpressure in a balanced PRV ............................. 15
2.10 Balanced pressure-relief valve ...................................................................... 16
2.11 Trend of mass flow rate, pressure and temperature as a function of Mach
number .......................................................................................................... 18
2.12 Converging-diverging duct ............................................................................ 19
2.13 Mass flow rate trend as a function of expansion ratio .................................. 19
3.1 Control volume .............................................................................................. 24
3.2 Velocity profiles of a laminar (dashed line) and a turbulent (solid line) BL
in a channel ................................................................................................... 30
3.3 Universal velocity profile .............................................................................. 31
4.1 Structured grid .............................................................................................. 36
4.2 Generic control volume ................................................................................. 37
4.3 Upwind interpolation .................................................................................... 39
4.4 Linear interpolation ...................................................................................... 40
4.5 Linear upwind interpolation ......................................................................... 40
4.6 Sweby diagram .............................................................................................. 41
4.7 Skew cells ...................................................................................................... 42
4.8 Non-orthogonal correction ........................................................................... 43
4.9 SIMPLE algorithm ........................................................................................ 46
4.10 PISO algorithm ............................................................................................. 47
4.11 PIMPLE algorithm ........................................................................................ 48
5.1 Blowdown ring .............................................................................................. 49
5.2 Inlet nozzle .................................................................................................... 50
5.3 Disc and disc holder ...................................................................................... 50
5.4 Main body ..................................................................................................... 51
5.5 Top lid ........................................................................................................... 51
5.6 Ring and disc position: closed valve and full-up ring (A) vs actual
positioning (B) .............................................................................................. 52
5.7 Valve assembly .............................................................................................. 52
5.8 Mach number field in a 2D simulation ......................................................... 54
5.9 Original profile without (red-marked) adjustments ..................................... 55
5.10 Modeled valve with adjustments .................................................................. 55
5.11 Detail of small gap filling and edge rounding off .......................................... 56
5.12 Effect of the deviation tolerance and angle tolerance on the output data .... 57
5.13 Disc and ring, with different colors for different surfaces they have been
split into ........................................................................................................ 58
5.14 Final assembly of the rotated valve: each color denotes a different
component .................................................................................................... 59
5.15 Bounding box and orientation of the axes .................................................... 60
5.16 Mesh bounding box (white) compared to valve model ................................. 62
5.17 Example of STL surface, castellated, snapped and layered mesh ................ 63
5.18 Feature edges of the adjusting ring ............................................................... 67
5.19 Explanatory depiction of the set of cylinders ................................................ 68
5.20 Detail of the refined area with the set of cylinders (castellated mesh) ......... 68
5.21 Castellated mesh of the valve and the relief box ........................................... 69
5.22 Detail of the refined area with the set of cylinders (snapped mesh) ............ 70
5.23 Detail of the edges of ring and inlet nozzle after snapping ........................... 71
5.24 Detail of the edges of the disc after snapping ............................................... 71
5.25 Face area vector and cell centroid vector ...................................................... 72
5.26 Cell pyramid volume ..................................................................................... 73
5.27 Face concavity ............................................................................................... 73
5.28 Cell-to-tetrahedron decomposition .............................................................. 73
5.29 Face triangular decomposition ..................................................................... 74
5.30 Face weight calculation ................................................................................. 75
5.31 Edges of the ring incorrectly reproduced with activated mesh quality
controls .......................................................................................................... 76
5.32 Set of highly skew faces (colored red) with disabled MQC and detail of
a skew cell ...................................................................................................... 76
5.33 Edges of the ring reproduced with deactivated mesh quality controls ......... 77
5.34 Close-up of the nozzle region for the first-trial and improved mesh ............ 78
5.35 Domain decomposition for parallel meshing ............................................... 79
5.36 External faces of the disc after splitting with autoPatch .............................. 80
5.37 Front, back and lateral surface of the disc .................................................... 81
5.38 Body of the valve (blue, wall-type), boundaries of the relief box
(red, patch-type) ........................................................................................... 82
5.39 Detail of the outlet section ............................................................................ 84
6.1 Recap of the patches to be assigned boundary conditions ........................... 90
6.2 Relieving pressure field assigned in the nozzle region up to the throat ....... 93
6.3 zeroGradient condition at outlet .................................................................. 95
6.4 inletOutlet condition at outlet ....................................................................... 100
6.5 Number of iterations for Ux calculation ....................................................... 107
6.6 PIMPLE algorithm flowchart ........................................................................ 108
7.1 Flow development in consecutive instants ................................................... 118
7.2 Mach number field ........................................................................................ 119
7.3 Supersonic flow regions (colored red) .......................................................... 119
7.4 Detail of the flow recirculation at the right side of the disc .......................... 120
7.5 3D velocity vector plot .................................................................................. 120
7.6 Streamlines of the flow field ......................................................................... 121
7.7 Pressure field ................................................................................................. 121
7.8 Detail of Mach number field in the converging-diverging nozzle ................. 122
7.9 Supersonic flow in the converging-diverging nozzle .................................... 122
7.10 Velocity vector plot in the converging-diverging nozzle ............................... 123
7.11 Pressure distribution on the upper surface of the disc holder ...................... 124
7.12 Total pressure field ....................................................................................... 124
7.13 Total temperature field ................................................................................. 125
7.14 Entropy field ................................................................................................. 125
7.15 Excessive total pressure and negative entropy cells ..................................... 126
7.16 y+ on walls ..................................................................................................... 127
7.17 Trend of residuals for flow and turbulent quantities .................................... 128
7.18 Inlet (red) and outlet (blue) mass flow rate and mass imbalance
(colored green) .............................................................................................. 128
7.19 Minimum (blue) and maximum (red) pressure and velocity in the domain 129
7.20 Area-averaged pressure and temperature at inlet (red) and outlet (blue) ... 129
7.21 Forces acting on the patches of the disc ....................................................... 130
7.22 Measured Force acting on the disc vs Time and Relieving pressure ............ 134
7.23 Representation of pressures and forces acting on the real component ........ 134
7.24 Mach number field for − (left figure) and − SST (right figure) models 136
7.25 Turbulent kinetic energy field for − (left) and − SST (right) models .... 136
7.26 Area averaged p and T for − (dotted line) and − SST (solid line)
models ........................................................................................................... 137
7.27 Forces on the disc for − (dotted line) and − SST (solid line) models .... 137
7.28 Mass flow rates for − (dotted line) and − SST (solid line) models ........ 138
7.29 Residuals for turbulent quantities of − and − SST models ................... 138
7.30 Mach number fields for upwind (left side) and limited linear (right side)
schemes ......................................................................................................... 139
7.31 Supersonic Mach number for upwind (left) and limited linear (right)
schemes ......................................................................................................... 139
7.32 Close-up of the nozzle region for upwind and limited linear schemes ......... 140
7.33 Turbulent kinetic energy field for upwind and limited linear schemes ........ 140
7.34 Static pressure field for upwind and limited linear schemes ........................ 141
7.35 Entropy field for upwind and limited linear schemes .................................. 141
7.36 Averaged p and T for upwind (dotted line) and limited linear (solid line)
schemes ......................................................................................................... 142
7.37 Trend of residuals for upwind (left side) and limited linear (right side)
schemes ......................................................................................................... 142
7.38 Maximum p and U for upwind (dotted line) and limited linear (solid line)
schemes ......................................................................................................... 143
7.39 Forces on the disc for upwind (dotted line) and limited linear (solid line)
schemes ......................................................................................................... 143
7.40 Mass flow rates for upwind (dotted line) and limited linear (solid line)
schemes ......................................................................................................... 144
List of Tables
4.1 Flow speed classification ............................................................................... 44
6.1 Properties of dry air ...................................................................................... 86
6.2 Boundary conditions ..................................................................................... 89
6.3 Mode of operation of totalPressure condition .............................................. 92
7.1 Statistics of the calculation ........................................................................... 116
7.2 Forces acting on the disc ............................................................................... 130
7.3 Mass flow rate ............................................................................................... 131
7.4 Experimental mass flow rates ....................................................................... 133
7.5 Main averaged test data ................................................................................ 133
7.6 Force calculation for − and − SST models ............................................. 137
7.7 Actual coefficient of discharge for − and − SST models ........................ 138
7.8 Force calculation for upwind and limited linear schemes ............................ 143
7.9 Actual coefficient of discharge for upwind and limited linear schemes ....... 144
7.10 Summary table of the outcomes of the analyses ........................................... 144
Abstract
Pressure-relief valves are safety devices widely used in industrial applications to
prevent that the working-fluid operating pressure exceeds the safety levels. Due to their
significance they have been addressed by several studies and publications aimed at
analyzing the flow field during relieving conditions. In those studies, based on CFD
simulations, the difficulty in rendering the complex internal shape of the valves has been
bypassed by means of two-dimensional axisymmetric meshes, or three-dimensional
unstructured grids generated using commercial codes. The current thesis work, based on an
internship at AST S.p.A. Company, pursues the goal of realizing for the first time in scientific
literature a three-dimensional model of a spring-loaded pressure-relief safety valve on a
structured grid for CFD analyses run with an open source code.
The work is carried out using OpenFOAM, an open-source toolbox comprising utilities
for mesh generation, calculation setup and solution. Following the creation of the three-
dimensional model of the valve to be analyzed, attention is being focused on the complexity
of generating a three-dimensional structured mesh and how some obstacles have been taken
on exploiting the Company’s experience and the tools offered by OpenFOAM. After
generating a satisfactory grid, a calculation for the transient simulation of a compressible
flow of air is set up. Not only do the outcomes of the analysis highlight the supersonic nature
of the flow within the valve, but also allow to study the behavior of an open source code for
industrial applications, proving its suitability. Further simulations examine in depth the
effect of the turbulence model and the numerical schemes on the solution, pointing out that
the best accuracy is reached with second-order schemes and - turbulence model.
Eventually, the numerical results of the simulation have been validated by experiments that
prove their reliability.
Keywords: CFD; safety valve; 3D; transient; supersonic; OpenFOAM
Sommario
Le valvole di scarico sono dispositivi di sicurezza ampiamente utilizzati in ambito
industriale per evitare che la pressione di esercizio superi i livelli di sicurezza. Data la loro
importanza sono state oggetto di svariati studi e pubblicazioni mirati ad analizzare il campo
di moto durante le condizioni di funzionamento. In questi studi, basati su simulazioni CFD,
la difficoltà di riprodurre la complessa geometria interna delle valvole è stata aggirata
tramite mesh bidimensionali con geometrie assialsimmetriche, o griglie tridimensionali
non strutturate generate tramite codici commerciali. Il presente lavoro di tesi, basato su
un’esperienza di tirocinio per conto dell’azienda AST S.p.A., si pone come obiettivo quello
di realizzare per la prima volta nella letteratura scientifica un modello tridimensionale di
una valvola di sicurezza a molla su griglia strutturata per analisi CFD eseguite tramite un
codice open source.
Il lavoro è svolto utilizzando OpenFOAM, un pacchetto software open-source che offre
strumenti per generare mesh, impostare calcoli CFD e risolverli. Dopo aver riprodotto il
modello tridimensionale contenente la geometria della valvola da analizzare, l’attenzione è
focalizzata sulla complessità di realizzare una mesh strutturata tridimensionale e su come
talune criticità siano state affrontate sfruttando l’esperienza dell’azienda e gli strumenti
messi a disposizione da OpenFOAM. A seguito della generazione di una griglia
soddisfacente, viene messo a punto un modello di calcolo per la simulazione transiente di
un flusso comprimibile di aria. I risultati dell’analisi non solo evidenziano la natura
supersonica del flusso all’interno della valvola, ma permettono anche di studiare il
comportamento di un codice open source per un’applicazione industriale confermandone
l’adeguatezza. Ulteriori simulazioni approfondiscono l’effetto del modello di turbolenza e
degli schemi numerici sulla soluzione del calcolo, sottolineando come la migliore
accuratezza si ottenga con schemi del secondo ordine e modello di turbolenza -. Infine, i
risultati numerici dell’analisi sono stati validati tramite prove sperimentali che ne
confermano l’affidabilità.
Parole chiave: CFD; valvola di sicurezza; 3D, transiente; supersonico; OpenFOAM
1
Chapter 1
Introduction
The focus of this work is the development of a Computational Fluid Dynamic (CFD)
model capable of simulating the working conditions of a spring-loaded pressure-relief valve
(PRV) operating with air. The currently available alternatives regarding computational fluid
dynamics analyses rely on commercial codes and open source codes, where the main
difference between the two consists in the possibility of customizing the source code so as
to adapt it to the user needs. If popular commercial codes such as Ansys-CFX provide user-
friendly interfaces and trustworthy features that facilitate the simulation setup and generate
reliable outcomes due to the grounded experience of countless test cases, they lack the
opportunity to inspect and modify the source code of the software. Thus, they are mainly
used in companies whose main concern is getting dependable results in the shortest
possible time. On the other hand, open source software packages allow the user to access
their codes and adapt them to his/her own requirements, guaranteeing a higher level of
flexibility in exchange for enhanced complexity during the setup phase. For this reason, they
are exploited in the academic context and in research and development departments of
companies. A remarkable advantage of open source codes is that they are free.
This Thesis comes from a work experience in AST S.p.A. technical office. The Company
needed to develop for the first time a CFD model to simulate the behavior of its products in
order not to rely on experimental tests only for their commercialization. Not only would the
development of a CFD model expand the knowledge of the technical office regarding the
topic of CFD, but also it would entail significant savings since experimental tests would be
cut down. The Company was oriented towards the usage of an open source software both to
exploit the advantage of tweaking the code based on its needs and to save costs associated
with paying a yearly license. For this purpose, OpenFOAM was chosen as eligible tool to
carry on the study.
While the initial work dealt with the understanding of the working principles of safety
valves and the fluid dynamic aspects involved, the following task consisted in combining the
assimilated knowledge with the features offered by OpenFOAM to render the operating
conditions of the object of the study, a spring-loaded pressure-relief valve. The core of this
work consisted in generating the mesh for the CFD analysis and setting up the case: the
preprocessing phase was the most crucial for the success of the analysis, therefore particular
attention was payed to the choice of the appropriate solver as well as boundary conditions
and solution parameters. Once guaranteed that the analysis did successfully run, the
reliability of its outcomes was questioned and eventually proved relying on experimental
tests and further researches.
1.1 AST S.p.A. Company 2
1.1 AST S.p.A. Company
Founded in 1951, AST S.p.A. is one of the first Italian manufacturer of spring-loaded
safety relief valves and change-over valves. Since the very early days, it has distinguished
for technology and the degree of customization of its products, which has led the Company
to enter the control valve market in the following years to fulfill the most demanding
requirements of its customers. AST line of production includes a complete range of safety
relief valves and pilot operated safety valves conceived for gas, steam, liquid and cryogenic
applications, as well as the control valve series which comprises valves manufactured from
solid nickel alloy forgings for demanding applications such as melamine and urea services.
In 2006 AST S.p.A. starts the manufacturing of on/off valves, ball valves, gate valves and
rotary control valves. A major business reorganization plan in 2012 results into the
establishment of HIT Valve S.p.A. which takes charge of all the activities related to the
production of on/off, ball, gate and rotary control valves.
Currently AST group consists of more than 250 people. The dedicated facility for safety
and control valves in Cornaredo consists of both an office block and a manufacturing site.
The technical department closely cooperates with two of the most advanced Universities
and Research Centers in the northern Italy, while the workshop is equipped with flexible
machining centers which allow the production capacity to be rapidly increased with the
market requirements and guarantee that quality standards and agreed delivery times are
respected. In addition to the usual equipment needed for production testing, AST can
perform steam testing, high and low temperature fugitive emissions test (ISO 15848),
cryogenic test and high-pressure gas test up to 20.000 psi ([3]).
With regard to quality standards, AST has, for many years, been working with a Quality
Assurance System that received UNI EN ISO 9001 certification as far back as 1993. Since
then AST gained several management systems and product related certifications, among
which: OHSAS 18001 Occupational Health and Safety Assessment, ISO 14001
Environmental Management, Pressure Equipment Directive 2014/68/EU, ATEX
2014/34/EU, ASME and National Board certifications, AQSIQ/SELO for Chinese market
and the approval for installation and exportation to Russia (CU TR).
Figure 1.1: AST facility in Cornaredo
1.2 OpenFOAM 3
1.2 OpenFOAM
OpenFOAM (acronym for Open Source Field Operation and Manipulation) is a free,
open source software for computational fluid dynamics, owned by the OpenFOAM
Foundation and distributed under the General Public Licence, which gives to the users the
freedom to modify and redistribute the software allowing for multiple variants being
released. Its structure comprises a set of proper and third-party libraries arranged in a
hierarchical organization of directories: a collection of over 100 C++ libraries makes up an
assortment of about 250 applications, each of which performs a specific task within a CFD
workflow. A variety of applications is available for meshing generation, conversion and
manipulation, case simulation, result processing and more. In particular, solver
applications have a syntax that closely resembles the partial differential equations being
solved, thanks to the object-oriented features of the C++ programming language such as
inheritance, polymorphism, virtual functions, template classes and operator overloading.
Object orientation is a property that allows to recognize main objects from a numerical
modeling viewpoint, such as data and functions that operate on these data. This gives rise
to a layered programming structure, where the general bottom layers are progressively
specialized to build up the code functionalities (e.g. basic objects such as scalars and vectors
are used to define the finite volume method or the finite elements method).
OpenFOAM simulations are made up of several plain text input files located across the
following directories:
• a constant directory, that contains a full description of the case mesh and files
specifying physical properties
• a system directory, where parameters associated with the solution procedures are set
up
• time directories, each of which named after the simulated time at which the data are
written, containing individual files of data for particular fields
Each text file can be inspected and modified by the user if necessary, even while the
simulation is running.
Figure 1.2: Overall structure of OpenFOAM
1.2 OpenFOAM 4
Figure 1.3: Case directory structure
5
Chapter 2
Pressure-relief Devices
Pressure-relief devices are used in the daily operation of equipment and plants to avoid
that the maximum allowable working pressure increases beyond the safe levels. Despite the
inherently safe design, plenty of causes in everyday operations could lead the working
pressure to rise above its maximum allowable value: operator errors, blocked discharge,
thermal expansion, component failures, fire exposure and so on. A potential line of defense
against these hazards consists in the passive design, but this road is usually exceedingly
expensive, thus pressure-relief devices come into play. They are actuated by the inlet static
pressure and are designed to activate during emergency or abnormal conditions to prevent
a rise of internal fluid in excess of a specified design value.
There are two main categories of pressure-relief devices:
• Reclosing, designed to reclose after normal conditions have been restored
• Non-reclosing, that remain open and need manual operation to be closed
Non-reclosing pressure-relief devices are used when losing the content is not an issue. This
category comprises rupture discs and pin actuated devices. A rupture disc is a device that
contains a disc which breaks open when the static differential pressure between the
upstream and the downstream side of the disc reaches a predetermined value. A pin
actuated device is activated by the static pressure acting on a piston held in place by a pin:
when the pressure exceeds a design value, the pin buckles or breaks and the piston moves
to the fully open position.
On the contrary, if the content is toxic or hazardous or when it is needed to return to normal
operations quickly, the usage of reclosing pressure-relief devices is mandatory. Reclosing
pressure-relief devices are known as pressure-relief valves (PRVs), and the most common
types are spring-loaded PRVs, pilot operated PRVs and power actuated PRVs. Each type has
a disc that shuts the passage to the content, and they differ in the way the disc is held in
position: in spring-loaded valves the disc is held closed by a spring, in pilot operated valves
the disc is kept in position by a holding pressure controlled by a pilot valve actuated by the
Figure 2.1: Non-reclosing pressure-relief devices: rupture discs and pin actuated devices
2.1 Terms and Definitions 6
system pressure, power actuated valves instead are actuated by an externally powered
control device.
2.1 Terms and Definitions
For an adequate description of the working principles of pressure-relief devices, the
following definitions (included in API Standard 520 [2]) will be used.
2.1.1 Dimensional characteristics
• actual discharge area: the minimum net area that determines the flow through a
valve
• effective discharge area: a nominal area, provided for a range of sizes in terms of
letter designations from D to T, used with an effective coefficient of discharge to
calculate the relieving capacity of a pressure-relief valve for preliminary sizing
equations
• huddling chamber: an annular chamber located downstream of the seat of a PRV for
the purpose of assisting the valve to achieve lift
• secondary orifice: the annular opening at the outlet of the huddling chamber
2.1.2 Device characteristics
• actual coefficient of discharge: the ratio of the measured mass flow rate in a valve to
that of an ideal nozzle
• backpressure: the pressure at the outlet of a pressure-relief device as a result of the
pressure in the discharge system; it is the sum of the superimposed and built-up
backpressures
Figure 2.2: Reclosing pressure-relief devices: spring-loaded, pilot operated, power actuated
2.1 Terms and Definitions 7
• balanced pressure-relief valve: a spring-loaded PRV that incorporates a bellows or
other means for minimizing the effects of backpressure on the operational
characteristics of the valve
• blowdown: the difference between the set pressure and the closing pressure of a PRV,
expressed as percentage of the set pressure
• built-up backpressure: the increase in pressure at the outlet of a PRV as a result of
flow after the pressure-relief device opens
• chatter: the opening and closing of a PRV at a very high frequency
• closing pressure: the value of decreasing inlet static pressure at which the valve disc
reestablishes contact with the seat
• coefficient of discharge: the ratio of the mass flow rate in a valve to that of an ideal
nozzle
• conventional pressure-relief valve: a spring-loaded PRV whose operational
characteristics are directly affected by changes in the backpressure
• effective coefficient of discharge: a nominal value used with an effective discharge
area to calculate the relieving capacity of a pressure-relief valve for preliminary sizing
equations
• flutter: the abnormal and rapid reciprocating motion of the disc, that anyway does not
contact the seat
• lift: the actual travel of the disc from the closed position when a valve is relieving
• maximum allowable working pressure (MAWP): the maximum gauge pressure
permissible at the top of a completed vessel in its normal operating position at the
designated temperature specified for that pressure
• maximum operating pressure: the maximum pressure expected during normal
system operation
• opening pressure: the value of increasing inlet static pressure at which there is a
measurable lift of the disc
• overpressure: the pressure increase over the set pressure of the relieving device,
expressed in percentage of set pressure
• relieving pressure: the inlet pressure of a pressure-relief device during an
overpressure condition; it is the sum of the set pressure and the overpressure
• set pressure: the inlet gauge pressure at which the pressure-relief device is set to open
under service conditions
• simmer: the audible or visible escape of compressible fluid between the seat and disc
of a PRV that may occur at an inlet static pressure below the set pressure prior to
opening
• superimposed backpressure: the static pressure that exists at the outlet of a pressure-
relief device at the time the device is required to operate as a result of the constant or
variable pressure in the discharge system coming from other sources
• throat area: the minimum cross-sectional flow area of a nozzle in a pressure-relief
valve
2.2 Spring-loaded Pressure-relief Valves 8
2.2 Spring-loaded Pressure-relief Valves
Spring-loaded pressure-relief valves are self-actuated devices whose distinguishing
feature is the presence of a spring that controls the position of the disc. They can be referred
to by three different terms:
• Relief valve: a spring-loaded PRV characterized by gradual opening that is usually
proportional to the increase in pressure; normally used for incompressible fluids
• Safety valve a spring-loaded PRV characterized by rapid opening; normally used to
relieve compressible fluids
• Safety relief valve: a spring-loaded PRV that may be used as either a relief or a safety
valve depending on the application
Another distinction can be made between conventional and balanced PRVs: they differ in
the presence of a bellows, that minimizes the effect of backpressure on the performance
characteristics of the valve.
2.2.1 Conventional pressure-relief valves
A spring-loaded PRV always consists of a 90° angled body with lower inlet and side exit,
which is usually casted. It is flange-connected to the system and the inlet flange is typically
smaller than the outlet one because, if the PRV is designed to work with compressible fluids,
the discharged content always undergoes an expansion with a consequent volume increase.
For the same reason design pressures at the inlet of the valve are always higher than at the
outlet. The bonnet is flange-connected to the body and comprises the disc guide, which
controls the lateral movement of the disc holder, the spring and the cap. The cap is the upper
part and protects the adjusting screw.
The fluid enters the valve through the inlet nozzle, a component thread-connected to
the body which collects the fluid at high pressure and directs it towards the disc. As a matter
of fact, the fluid at high pressure is entirely contained in this component and does not get
in contact with the body. The upper part of the nozzle consists of the seat, an annular surface
that the disc comes into contact with to prevent the flow passage when the valve is fully
closed.
Figure 2.3: Coventional and balanced pressure-relief valves
2.2 Spring-loaded Pressure-relief Valves 9
The disc has a fundamental role in shutting the passage to the fluid. It is generally made
up of two parts: the disc holder and the disc itself. The disc holder contains the disc and can
move axially in the disc guide during the relieving phase. Another solution consists in
entrusting the disc travel guide to a spindle.
The spring is the key element of a spring-loaded PRV: it provides the force to keep the
disc on the nozzle, closing the valve. The choice of the spring must ensure the opening and
closing of the valve in the range of parameters set by regulations, and the property that
influences its performance is the stiffness. Moreover, it is possible to vary the force exerted
by the spring, and consequently alter the set pressure, acting on the adjusting screw
contained in the cap.
Finally, the adjusting ring is a component thread-connected to the nozzle that controls
the opening characteristics of the valve and the blowdown: its position with respect to the
disc can be manually regulated and changes the dimensions of the huddling chamber. A
hole in the body of the valve allows a screwdriver to rotate the toothed ring increasing or
reducing its lift by a fraction of its pitch, thus causing a different pressurization of the
surfaces below the disc, and consequently a different behavior of the valve during both
opening and closing. After the position of the ring has been set, it is held in position by a
screw thread-connected to the body, whose end is positioned between the teeth of the ring.
The set of the following internal parts will be addressed TRIM:
• nozzle
• disc
• disc holder
• adjusting ring
2.2 Spring-loaded Pressure-relief Valves 10
Figure 2.4: Conventional pressure-relief valve
2.2 Spring-loaded Pressure-relief Valves 11
2.2.2 Working principle
The operation of a spring-loaded PRV is based on a force balance. The spring exerts on
the disc holder its elastic force, that is opposed to the force exerted by the fluid. When the
valve is closed during normal operation the system pressure acting against the disc surface
is resisted by the elastic force of the spring. Thus, the disc remains fixed and hinders the
passage of the fluid. As the system pressure approaches the set pressure of the valve, the
seating force between the disc and the nozzle, resulting from the thrust of the fluid and the
elastic force of the spring, approaches zero. Then the fluid can move past the seating surface
into the huddling chamber, where pressure builds up. In this condition pressure acts over a
larger area, then an additional force is available to overcome the spring force, and the valve
opens. As soon as it happens, the sudden flow increase and the restriction to flow through
another annular orifice formed between the inner edge of the disc holder and the outside
diameter of the adjusting ring produces an additional pressure increase, thus again an
additional force. The freshly newborn forces cause the disc to lift substantially at pop.
Figure 2.5: Sectional cut of a PRV with adjusting ring
2.2 Spring-loaded Pressure-relief Valves 12
The operational characteristic of a spring-loaded PRV is expressed in terms of lift of the
disc as a function of pressure acting on it. As soon as the pressure in the inlet nozzle reaches
the set pressure value, the valve pop-opens: the lift exhibits a linear trend at constant
pressure, but that is not sufficient for the valve to reach the full lift condition, thus an
additional pressure contribution, called overpressure, has to be provided. For a
conventional PRV the allowable overpressure is limited at 10% of the set pressure. At full
lift, a further pressure increase until the maximum relieving pressure does not cause any
other change of position of the disc. Then, as long as pressure decreases, the valve begins to
close: for the valve to fully close the closing pressure must be lower than the set pressure
because the disc is now hit by a fluid flow that must be disposed before closing happens.
The difference between the set pressure and the closing pressure is called blowdown. The
adjusting ring is aimed at regulating the blowdown by changing the dimensions of the
huddling chamber, but it necessarily affects the opening phase too. When the ring is down
the force on the disc is lower and thus the lift, consequently a higher pressure is required to
reach the full lift condition with respect to the ring being positioned up. On the contrary, in
this case the higher force exerted on the disc penalizes the closing.
Figure 2.6: Pressure on the disc at valve closed (A), initial opening (B) and fully opening (C)
2.2 Spring-loaded Pressure-relief Valves 13
2.2.3 Effect of the backpressure
The backpressure is a key parameter that affects the operation of pressure-relief valves.
It is defined as the pressure at the outlet of a pressure-relief device as a result of the pressure
in the discharge system, and its magnitude can be split into two components: superimposed
and built-up backpressure. Built-up backpressure generates as a consequence of the fluid
flowing past the disc when the valve is open and filling the body of the valve in its way to the
outlet. Superimposed backpressure on the contrary does not depend on the valve itself and
is present even when the valve is closed: its existence causes a pressure force to be applied
to the valve disc that is additive to the spring force, which increases the pressure at which
the valve will open.
Assuming that the backpressure has the same value in the whole body of the valve and
neglecting the weight of the disc, from a simple equilibrium of forces:
𝐹𝑠 + 𝑝𝐵 𝐴 = 𝑝𝑠𝑒𝑡 𝐴 (2.1)
where in (2.1) FS is the force exerted by the spring, pB is the superimposed backpressure, pset
is the set pressure and A is the area of the inlet nozzle in contact with the disc, i.e. the surface
of the disc where the set pressure is acting before the valve opens, called seat area. It is clear
from the previous considerations that the set pressure must increase to cope with the
backpressure pushing the disc downward.
Figure 2.7: Lift of disc and inlet pressure relationship
2.2 Spring-loaded Pressure-relief Valves 14
2.2.4 Balanced pressure-relief valves
The above-mentioned effects of the backpressure can give rise to several problems.
First, it is evident how the value of the superimposed backpressure affects the set pressure
of the valve: this can be a serious problem when the backpressure is variable. Secondly, an
excessive value of the backpressure can induce fluttering or chattering phenomena,
detrimental for the correct operation of the valve and potentially damaging for the valve
itself. Finally, the main concern during design phase is the compromise between a precise
opening, for safety reasons, and a precise closing, to not waste precious working fluid. For
all these reasons it is advantageous to limit the effect of the backpressure on the
performance characteristics of the valve. Balanced pressure-relief valves achieve this goal
through a specific vented component called bellows. In a balanced pressure-relief valve, the
bellows is attached to the disc holder covering a pressure area approximately equal to the
seating area of the disc. This isolates an area on the disc, approximately equal to the disc
seat area, from the back pressure, exposing it to atmospheric pressure. With the addition of
a bellows, therefore, the set pressure of the pressure-relief valve remains constant in spite
of variations in back pressure.
Assuming again that the backpressure has the same value in the whole body of the valve
and neglecting the weight of the disc, from a simple equilibrium of forces:
𝐹𝑠 = 𝑝𝑠𝑒𝑡 𝐴 (2.2)
where in (2.2) FS is the force exerted by the spring, pB is the superimposed backpressure,
pset is the set pressure and A is the seat area.
Figure 2.8: Effect of superimposed backpressure in a conventional PRV
2.2 Spring-loaded Pressure-relief Valves 15
Another advantage of balanced PRVs is that the bellows prevent the working fluid to
get in touch with the bonnet: this precaution is requested whenever the valve operates with
corrosive fluids that can damage the spring or the other components inside the bonnet. A
balanced pressure-relief valve should be used where the built-up backpressure is too high
for conventional pressure-relief valves or where the superimposed backpressure varies
widely compared to the set pressure. Typically, conventional PRVs are used when the total
backpressure (superimposed plus built-up) does not exceed 10% of the set pressure. Over
this value balanced valves can be applied, up to a total backpressure not exceeding
approximately 50% of the set pressure. When this happens, pilot operated valves are
exploited.
Figure 2.9: Effect of superimposed backpressure in a balanced PRV
2.2 Spring-loaded Pressure-relief Valves 16
Figure 2.10: Balanced pressure-relief valve
2.3 Sizing of Pressure-relief Valves 17
2.3 Sizing of Pressure-relief Valves
Several factors come into play when dealing with pressure-relief valves: the nature of
the working fluid (that can be a gas, a liquid or a two phase liquid/vapor flow), the effect of
the backpressure (as previously discussed) and the type of outflow (critical or subcritical).
The first distinction takes care of the nature of the fluid. In case of a liquid the outflow is
governed by the pressure difference between the upstream plenum and the discharge
system. In this case the fluid dynamic losses, proportional to the square of the flow speed,
assume a key role as they limit the performances of the device. For a gas instead the main
concern is the choke phenomenon, that occurs when the Mach number in the throat section
reaches unitary value and limits the discharged mass flow.
2.3.1 Converging-diverging nozzle
This condition is well-explained by the converging-diverging nozzle theory. The stream
of a perfect gas in a duct obeys the following law, obtained by working out the equation of
state for gases, the continuity equation, the momentum equation and the energy equation:
(𝑀2 − 1)
𝑑𝑈
𝑈=
𝑑𝑆
𝑆−
𝜏𝑐
𝑝𝑆−
𝑑𝑞
𝑐𝑝𝑇 (2.3)
where M is the Mach number, U is the flow velocity, S is the cross section of the duct, τ is
the integral of the viscous stresses acting on the duct perimeter, c is the duct perimeter, p is
the pressure, q is the heat delivered to the system, T is the temperature of the gas and cp its
specific heat at constant pressure. If the flow is adiabatic (dq=0) and isentropic (τ=0), the
equation (2.4) reduces to:
(𝑀2 − 1)
𝑑𝑈
𝑈=
𝑑𝑆
𝑆 (2.4)
which highlights how a convergent duct is required to accelerate a subsonic stream and a
divergent one to decelerate it, while the opposite happens for a supersonic stream. Again,
for an adiabatic and isentropic stream of a perfect gas the following statements apply:
𝜌 =𝑝
𝑅𝑇 (2.5)
𝑀 =
𝑈
√𝑘𝑅𝑇 (2.6)
𝑝𝑇𝑝
= (1 +𝑘 − 1
2𝑀2)
𝑘𝑘−1
(2.7)
𝑇𝑇𝑇
= 1 +𝑘 − 1
2𝑀2 (2.8)
where ρ represents the density, R is the ideal gas constant, k is the specific heat ratio and pT
and TT stand for, respectively, the total pressure and temperature, i.e. the pressure and
2.3 Sizing of Pressure-relief Valves 18
temperature measured when the kinetic head is recovered by an isentropic process. The
mass flow rate flowing through a generic section is defined as:
�̇� = 𝜌𝑈𝑆 (2.9)
Finally, working out equations (2.5), (2.6), (2.7) and (2.8) in (2.9) it is possible to derive the
relationship between the mass flow rate and the thermodynamic quantities of the gas:
�̇� =
𝑝𝑇𝑆
√𝑅𝑇𝑇 𝑓𝑀,𝑘 (2.10)
where in (2.10) fM,k is a parameter depending only on the gas and the Mach number
𝑓𝑀,𝑘 = 𝑀√𝑘 (1 +𝑘 − 1
2𝑀2)
1+𝑘2(1−𝑘)
(2.11)
The (2.10) shows that the mass flow rate depends on the Mach number. In particular, the
mass flow rate has a maximum for M=1. Moreover, given the flow rate, two possible Mach
numbers can be reached in the cross section of the duct, whichever its shape is.
The previous considerations can be extended to the case of a duct with a particular
converging-diverging shape. A converging-diverging duct is required to accelerate a stream
of compressible fluid above M=1. Starting from a volume (A), a converging duct is always
essential to accelerate the fluid. From equation (2.4) the minimum section (throat) may
have sonic conditions, i.e. M=1, depending on the expansion ratio pT/p: if this happens,
pressure perturbations generated in the outlet section (B) cannot travel upstream to A,
Figure 2.11: Trend of mass flow rate, pressure and temperature as a function of Mach number
2.3 Sizing of Pressure-relief Valves 19
while all pressure perturbations can travel from A to B. The diverging duct then accelerates
the supersonic flow to Mach numbers above 1.
Depending on the conditions upstream and downstream of the duct, two cases are possible.
For constant upstream conditions and variable downstream conditions, the flow rate
increases together with the expansion ratio and thus the Mach number until M=1 sets in the
throat: under these circumstances the pressure variation in section B cannot travel
upstream and the mass flow rate becomes constant. For variable upstream conditions and
constant downstream conditions instead, the pressure increase generated in A can always
travel to B, even when sonic conditions have set in the throat, therefore the mass flow rate
keeps increasing with the expansion ratio.
The first case is typical of pressure-relief valves working with compressible fluids. When
the valve opens, the gas flows from the surface of the disc, where it stopped thus having total
pressure condition, towards the throat section between the different components expanding
and increasing its velocity during this process. Then in the throat sonic conditions may
establish, limiting the discharged flow rate.
Figure 2.12: Converging-diverging duct
Figure 2.13: Mass flow rate trend as a function of expansion ratio
2.3 Sizing of Pressure-relief Valves 20
2.3.2 Sizing for vapor or gas
The sizing equations for vapor or gaseous working fluids are based on the ideal gas laws
and the assumption that the flow is isentropic and one-dimensional, so that it can be
described by the pressure-specific volume relationship:
𝑝𝑉𝑘 = 𝑐𝑜𝑠𝑡 (2.12)
The ideal gas assumption shows good agreement with the real behavior of the fluid when
the compressibility factor Z, defined as:
𝑍 =𝑝𝑉
𝑅𝑇 (2.13)
is in the range between 0.8 and 1.1. In other cases the deviations of the real gas from the
ideal gas laws must be taken into account by empirical corrections on Z. Also, the hypothesis
of one-dimensional flow is slightly forced since the gas, limited by the geometries of the
nozzle, the seat, the disc and the ring, encounters in its motion throughout the valve cross
sections that do not really fit the characteristics of a straight pipe. Moreover, friction and
flow detachment phenomena kick in as well. The following dissertation aims at giving a hint
of what lies behind the equations used to preliminarily size PRVs.
Under the above-mentioned assumptions the pressure ratio β between the absolute
static pressure at the outlet and the absolute total pressure at the inlet of the valve can be
related to the Mach number and the specific heat ratio as in (2.7):
𝛽 = 𝑝𝑠𝑝𝑇
= (1 +𝑘 − 1
2𝑀2)
𝑘1−𝑘
(2.14)
In case of choked flow, i.e. M=1, the equation (2.14) becomes:
𝛽𝐶𝑅 = 𝑝𝐶𝐹𝑝𝑇
= (2
𝑘 + 1)
𝑘𝑘−1
(2.15)
In this condition the velocity of the fluid in the throat section reaches the speed of sound
and the choking phenomenon occurs, limiting the mass flow discharged by the valve. The
pressure ratio in correspondence of sonic velocity is called critical pressure ratio βCR and the
pressure at the valve exit pCF is known as the critical flow pressure. The critical pressure
ratio is a discriminating factor between sonic and subsonic flow through the nozzle of a PRV:
when the pressure ratio is higher than βCR the flow is subsonic, on the contrary if it is lower
than βCR the flow is sonic. The onset of a sonic flow limits the discharged mass flow rate
since downstream variations do not affect upstream quantities, then the flow is no more
controlled by the upstream/downstream pressure ratio, but by the upstream/throat
pressure ratio. Considering air as working fluid (k=1.4), the critical pressure ratio is:
𝛽𝐶𝑅,𝑎𝑖𝑟 = 𝑝𝑠𝑝𝑇
= 0.5283 (2.16)
2.3 Sizing of Pressure-relief Valves 21
The sizing of pressure-relief valves is always performed in accordance to the relevant
standards for safety valves, whose most commonly used are API 520 [2] and EN ISO 4126
part 1 and 4. The sizing equations for PRVs in vapor or gas service fall into two general
categories depending on whether the flow is critical or subcritical. If the pressure
downstream of the nozzle is less than, or equal to, the critical flow pressure pCF, then critical
flow will occur, and the following procedure should be applied.
𝑝𝐶𝐹 = 𝑝1 (2
𝑘 + 1)
𝑘𝑘−1
(2.17)
𝐴 =𝑊
𝐶𝐾𝑑𝑝1𝐾𝑏𝐾𝐶√
𝑇𝑍
𝑀 (2.18)
𝐶 = 0.03948√𝑘 (2
𝑘 + 1)
𝑘+1𝑘−1
(2.19)
where p1 is the absolute upstream relieving pressure in kPa, A is the required effective
discharge area of the device in mm2, W is the required flow through the device in kg/h, C is
a function of the specific heat ratio of the gas at inlet relieving temperature, Kb is a correction
factor that takes into account the effect of backpressure in balanced PRVs only, KC is the
combination corrector factor for installations with a rupture disc upstream of the PRV, Kd
is the effective coefficient of discharge (for preliminary sizing, equal to 0.975), M is the
molecular weight of the gas or vapor at inlet relieving conditions. All the physical quantities
are expressed in SI units.
If the downstream pressure exceeds the critical flow pressure pCF, then subcritical flow
will occur, and the following procedure should be applied.
𝐴 =17.9 𝑊
𝐹2𝐾𝑑𝐾𝐶√
𝑇𝑍
𝑀𝑝1(𝑝1 − 𝑝2) (2.20)
𝐹2 = √(𝑘
𝑘 − 1) 𝑟
2𝑘 [
1 − 𝑟𝑘−1𝑘
1 − 𝑟] (2.21)
where F2 is the coefficient of subcritical flow, r is the ratio of backpressure to upstream
relieving pressure and p2 is the backpressure in kPa. It is noticeable that for subcritical flow
the pressure at the outlet of the valve (the backpressure) affects the discharged mass flow
rate in accordance with the theory.
2.3.3 Sizing for liquid
The sizing equations for liquid service pressure-relief valves are easier to be derived due
to the working fluid assumed incompressible. From the energy conservation equation
between upstream and downstream of a nozzle:
2.3 Sizing of Pressure-relief Valves 22
𝑝1𝜌1
+𝑈1
2
2+ 𝑔𝑧1 =
𝑝2𝜌2
+𝑈2
2
2+ 𝑔𝑧2 (2.22)
Considering the geodetical term gz negligible, null upstream flow velocity and constant
density ρ everywhere, the downstream velocity can be deduced from (2.22):
𝑈2 = √2(𝑝1 − 𝑝2)
𝜌 (2.23)
which, included in the equation (2.9) for the mass flow, gives:
�̇� = 𝑆2√2(𝑝1 − 𝑝2)𝜌 (2.24)
clearly depicting the dependency of the mass flow from the pressure difference between
upstream and downstream.
The sizing of PRVs operating with liquids in accordance with API520 states:
𝐴 =11.78 𝑄
𝐾𝑑𝐾𝑤𝐾𝑣𝐾𝐶√
𝐺
𝑝1 − 𝑝2 (2.25)
where A is the required effective discharge area of the device in mm2, Q is the required flow
rate in L/min, Kd is the rated coefficient of discharge (for preliminary sizing, equal to 0.65),
Kw is the correction factor due to backpressure (equal to 1 if the backpressure is
atmospheric), KC is the combination corrector factor for installations with a rupture disc
upstream of the PRV, Kv is a correction factor due to viscosity dependent on the Reynolds
number, G is the specific gravity of the liquid at the flowing temperature referred to water
at standard conditions, p1 and p2 are respectively the relative upstream relieving pressure
and the relative total backpressure, both in kPa. All physical quantities are expressed in SI
units.
23
Chapter 3
Fluid Dynamics
Fluid dynamics describes the behavior of a moving fluid, either compressible or
incompressible, involving the calculation of several physical quantities that characterize the
state of the fluid. These quantities can be scalars (such as pressure, temperature, density),
vectors (such as velocity and volume forces) and tensors (such as viscous stresses) and need
to be related to the motion of the fluid. Considering a generical scalar quantity , to
accomplish this task two point of views can be adopted:
• Lagrangian point of view, that gives attention to the fluid particle on its motion thus
adopting a control mass approach; in this way, time becomes the only independent
variable and all the other quantities, as well as the position of the particle, become
time-dependent variables
𝑑𝜙𝐿
𝑑𝑡=
𝜕𝜙𝐿
𝜕𝑡 (3.1)
• Eulerian point of view, that considers a control volume fixed in space without paying
attention to the particles passing through it; in this way, the generic scalar quantity
depends on time and position inside the control volume, the latter dependent on time
as well, thus being its time derivative composed of an unsteady term and an advective
term, which expresses the transport of a fluid property by means of the mean flow
𝑑𝜙𝐸
𝑑𝑡=
𝜕𝜙𝐸
𝜕𝑡+ �⃗⃗� ⋅ 𝛻𝜙𝐸 (3.2)
Both point of views can then be exploited to perform the formulation of conservation laws,
through which derive the fundamental equations of fluid dynamics.
3.1 Conservation Laws
Conservation laws come from the physics and apply to some physical quantities related
to a fluid: they state that the variation of the total amount of certain quantities inside a given
domain is equal to the balance between the amount of that quantity entering and leaving
the considered domain, plus the contribution from eventual sources generating that
quantity. Not all flow quantities obey a conservation law, but some do: mass, momentum
and energy. The set of these three equations is enough to completely solve incompressible
flows due to the density being constant, while for compressible flows dissipations due to the
viscous stresses affect the density, therefore the set of equations need to be solved including
an equation of state and a formulation for the viscosity. A conservation equation can be
written in conservative or non-conservative form, depending on how the terms of the
equation are expressed.
3.1 Conservation Laws 24
To express the variation in time of a generic quantity adopting the Eulerian point of
view, a control volume is required. From the previous considerations a generic quantity
included in this control volume can vary whether due to a flux (𝐹 ) of the above-mentioned
quantity across the domain boundaries or due to surface (𝑄𝑆⃗⃗ ⃗⃗ ) or volume (𝑄𝑉) sources and
sinks. Then, the general form of the conservation equation of a generic quantity inside an
arbitrary control volume contoured by a surface S is:
𝜕
𝜕𝑡∫ 𝜙𝑑Ω Ω
= −∮ 𝐹 𝑆
⋅ 𝑑𝑆 + ∫ 𝑄𝑉𝑑ΩΩ
+ ∮ �⃗� 𝑆𝑆
⋅ 𝑑𝑆 (3.3)
where the minus sign of the first surface integral accounts for the flux being positive when
entering the domain. This is the integral formulation of the conservation equation. Then,
applying the Gauss theorem to (3.3) to transform the surface integrals into integrals over
the volume it is possible to derive the differential formulation of the conservation equation,
valid in any point of the flow:
𝜕𝜙
𝜕𝑡+ ∇⃗⃗ ⋅ 𝐹 = 𝑄𝑉 + ∇⃗⃗ ⋅ �⃗� 𝑆 (3.4)
If the equation can be written grouping the fluxes under the divergence operator, then it is
in conservative form. It is worth saying that might represent either a scalar quantity or a
vector without the conservation equation changing its form, but only the physical meaning
of its terms. Indeed, if is a vector, then the fluxes and the surface source terms become
tensors, whereas the volume source terms become vectors. In (3.3) and (3.4) fluxes grouped
by the 𝐹 term can be generated by two contributions:
Figure 3.1: Control volume
3.1 Conservation Laws 25
• Convective contribution, that represents the amount of that is carried away or
transported by the fluid flow
𝐹 𝐶 = 𝜙�⃗⃗� (3.5)
• Diffusive contribution, that represents the amount of that is carried away or
transported by its gradient
𝐹 𝐷 = −𝑘𝜌∇⃗⃗ 𝜙 (3.6)
The convective flux describes the passive transport of the conserved variable by the flow, it
is proportional to the flow velocity and appears as a first order partial derivative term inside
the conservation equations, while the diffusion flux describes an isotropic diffusion
phenomenon and appears as a second order partial derivative term under the Laplace
operator.
3.1.1 Continuity equation
The continuity equation expresses the conservation of mass within a control volume,
stating that the rate of change of mass per unit volume ρ is null. The integral formulation
is:
𝜕
𝜕𝑡∫ 𝜌𝑑Ω Ω
+ ∮ 𝜌�⃗⃗� 𝑆
⋅ 𝑑𝑆 = 0 (3.7)
while the differential form is:
𝜕𝜌
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌�⃗⃗� ) = 0 (3.8)
where only the convective flux appears, since mass does not diffuse.
3.1.2 Momentum equation
The momentum equation corresponds to Newton’s second law of motion, that relates
the forces acting on an object to its acceleration through its mass. In this case, contributions
to the source terms come from external volume forces (such as gravity or applied forces)
and from internal forces (stress). The integral formulation of the momentum conservation
equation is:
𝜕
𝜕𝑡∫ 𝜌�⃗⃗� 𝑑Ω Ω
+ ∮ (𝜌�⃗⃗� )�⃗⃗� 𝑆
⋅ 𝑑𝑆 = ∫ 𝑓 𝑒𝑑ΩΩ
+ ∮ �̿�𝑆
⋅ 𝑑𝑆 (3.9)
where 𝑓 𝑒 accounts for external volume forces and �̿� expresses the stress, which represents
the internal deformability of a fluid, depends on the position and on the orientation of the
surface it acts on and is represented by a tensor. Assuming that the fluid is Newtonian, stress
can be decomposed into an isotropic component (pressure) and a viscous shear stress tensor
�̿�, that represents the internal friction force of fluid layers against each other.
3.1 Conservation Laws 26
�̿� = −𝑝𝐼 ̿ + �̿� (3.10)
Working out equation (3.10) into (3.9), the differential formulation of the momentum
conservation equation can be derived:
𝜕(𝜌�⃗⃗� )
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌�⃗⃗� ∗ �⃗⃗� ) = 𝜌𝑔 − ∇𝑝 + ∇⃗⃗ ⋅ �̿� (3.11)
The viscous shear stress tensor can then be expressed in terms of the flow velocity adopting
Newton’s law for viscous fluids:
�̿� = 2𝜇�̿� −
2
3𝜇∇(∇⃗⃗ ⋅ �⃗⃗� ) (3.12)
where is the dynamic viscosity of the fluid and �̿� is its strain rate tensor:
�̿� =
1
2(∇�⃗⃗� 𝑇 + ∇�⃗⃗� ) (3.13)
eventually leading to the Navier-Stokes equation:
𝜕(𝜌�⃗⃗� )
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌�⃗⃗� ∗ �⃗⃗� ) = 𝜌𝑔 − ∇𝑝 + 𝜇∇2�⃗⃗� +
1
3𝜇∇(∇⃗⃗ ⋅ �⃗⃗� ) (3.14)
3.1.3 Energy equation
The energy equation states that the total energy et, defined as the sum of fluid internal
energy plus its kinetic energy per unit mass, conserves. In this case both conductive and
diffusive fluxes take part in the equation, while the volume source terms are the work of the
volume forces 𝑓 𝑒 and the heat sources qH, and the surface sources are the result of the work
done on the fluid by the internal shear stress acting on the surface of the control volume.
With these premises the integral formulation of the energy equation is:
𝜕
𝜕𝑡∫ 𝜌𝑒𝑡𝑑Ω Ω + ∮ 𝜌𝑒𝑡 �⃗⃗�
𝑆
⋅ 𝑑𝑆 = ∮ 𝑘∇⃗⃗ T𝑆 ⋅ 𝑑𝑆 + ∫ (𝜌𝑓 𝑒 ⋅ �⃗⃗� + 𝑞𝐻)𝑑ΩΩ + ∮ (�̿�𝑆 ⋅ �⃗⃗�
) ⋅ 𝑑𝑆 (3.15)
where k is the thermal conductivity of the fluid, responsible for the presence of the diffusive
flux. Expressing the stress tensor in its isotropic and viscous shear stress component, two
possible expressions of the differential formulation of the energy equation can be written,
one in terms of total energy et and one with respect to total enthalpy ht:
𝜕(𝜌𝑒𝑡)
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌𝑒𝑡�⃗⃗� ) = ∇⃗⃗ ⋅ (𝑘∇⃗⃗ 𝑇) − ∇⃗⃗ ⋅ (𝑝�⃗⃗� ) + ∇⃗⃗ ⋅ (�̿� ⋅ �⃗⃗� ) + 𝜌𝑔 ⋅ �⃗⃗� + 𝑞𝐻 (3.16)
𝜕(𝜌ℎ𝑡)
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌ℎ𝑡 �⃗⃗� ) =
𝜕𝑝
𝜕𝑡+ ∇⃗⃗ ⋅ (𝑘∇⃗⃗ 𝑇) + ∇⃗⃗ ⋅ (�̿� ⋅ �⃗⃗� ) + 𝜌𝑔 ⋅ �⃗⃗� + 𝑞𝐻 (3.17)
3.2 Turbulence 27
3.2 Turbulence
Turbulence is a three-dimensional, unsteady, rotational fluid motion with broad-
banded fluctuations of flow quantities occurring in both time and space. Its origin lies in the
nonlinearity of the Navier-Stokes equations, whose main nonlinearity is provided by the
convection term, and its onset is triggered for sufficiently high values of nonlinearity
parameters (such as the Reynolds number). It is the final state of a transition process
composed by a succession of fluid-dynamic instabilities that break the regular laminar
profiles leading to the formation of unsteady vortices. Turbulence is characterized by:
• Fluctuations of flow quantities
• Unstable vortical structures, called eddies, whose size ranges from the problem length
scale to very small scale, depending on the Reynolds number which represents the
ratio of inertial forces to viscous forces (the larger Re, the smaller the scale)
• Dissipation of energy, that is inviscidly transferred from larger to smaller eddies
through the Kolmogorov cascade, until viscous dissipation occurs at the level of the
smallest eddies
• Enhanced diffusivity due to the chaotic motion, that improves mixing via transport
Due to the complexity and the perceived randomness of turbulent flows, an analytical
description is practically impossible, therefore one might resort to a statistical description
of the flow.
3.2.1 Reynolds Averaged Navier-Stokes Equations
The idea of Reynlods consists in writing each flow quantity as a summation of a mean
and a fluctuating component:
𝜙 = �̅�(𝑥) + 𝜙′ (3.18)
�̅�(𝑥) =1
ΔT∫ 𝜙(𝑥, 𝑡)𝑑𝑡
𝑡+Δ𝑇2
𝑡−ΔT2
(3.19)
The such written terms are put in the Navier-Stokes equations, then the equations
themselves are time averaged. The advantage of this strategy lies in the fact that the
averaging procedure of fluctuating terms nullifies them, however there is also a drawback:
newborn mixed terms, difficult to be evaluated, appear.
Applying the Reynolds procedure to the continuity equation is straightforward. The
mean mass balance equation is formally identical to the instantaneous one:
𝜕𝜌
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌�⃗⃗̅� ) = 0 (3.20)
3.2 Turbulence 28
The mean momentum equation is more complex due to the nonlinear term:
𝜕(𝜌�⃗⃗̅� )
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌 (�⃗⃗̅� + �⃗� ) ∗ (�⃗⃗̅� + �⃗� ))
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅= 𝜌𝑔 − ∇�̅� + 𝜇∇2�⃗⃗̅� +
1
3𝜇∇(∇⃗⃗ ⋅ �⃗⃗̅� ) (3.21)
where �⃗⃗̅� stands for the mean component of the flow velocity while �⃗� is the fluctuating
component. From some algebraic steps:
∇⃗⃗ ⋅ (𝜌 (�⃗⃗̅� + �⃗� ) ∗ (�⃗⃗̅� + �⃗� ))
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅= ∇⃗⃗ ⋅ (𝜌�⃗⃗̅� ∗ �⃗⃗̅� ) + ∇⃗⃗ ⋅ (𝜌�⃗� ∗ �⃗� ̅̅ ̅̅ ̅̅ ̅) (3.22)
then the (3.21) becomes:
𝜕(𝜌�⃗⃗̅� )
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌�⃗⃗̅� ∗ �⃗⃗̅� ) = 𝜌𝑔 − ∇�̅� + 𝜇∇2�⃗⃗̅� +
1
3𝜇∇ (∇⃗⃗ ⋅ �⃗⃗̅� ) − ∇⃗⃗ ⋅ (𝜌�⃗� ∗ �⃗� ̅̅ ̅̅ ̅̅ ̅) (3.23)
The fluctuating component, appearing in divergence form and put on the right-hand side,
acts like a stress added to the viscous molecular one; it is called Reynolds stress and it is a
symmetric tensor combining velocity fluctuations:
�̿� = −(𝜌�⃗� ∗ �⃗� ̅̅ ̅̅ ̅̅ ̅) = −𝜌 [𝑢2̅̅ ̅ 𝑢𝑣̅̅̅̅ 𝑢𝑤̅̅ ̅̅𝑣𝑢̅̅̅̅ 𝑣2̅̅ ̅ 𝑣𝑤̅̅ ̅̅𝑤𝑢̅̅ ̅̅ 𝑤𝑣̅̅ ̅̅ 𝑤2̅̅ ̅̅
] (3.24)
whose trace is:
𝑡𝑟(𝑟)̿ = −𝜌(𝑢2̅̅ ̅ + 𝑣2̅̅ ̅ + 𝑤2̅̅ ̅̅ ) = −2𝜌𝜅 (3.25)
where is the turbulent kinetic energy, while u, v and w are the fluctuating components of
the velocity along the three orthogonal directions x, y and z. Thus, it is clear that turbulence
behaves like an additional viscosity inside the momentum equation. Unfortunately, the
equations do not provide any direct expression for the Reynolds stresses, therefore a closure
problem arises and modeling is required to achieve solution.
Finally, applying Reynolds averaging to the thermal energy conservation equation,
obtained subtracting the mechanical terms from the total energy balance equation, one can
get:
𝜕(𝜌𝑐�̅�)
𝜕𝑡+ ∇⃗⃗ ⋅ (𝜌𝑐�̅��⃗⃗̅� ) = 𝑘∇2�̅� − ∇⃗⃗ ⋅ (𝜌𝑐𝑇′�⃗� ̅̅ ̅̅ ) (3.26)
Again, the nonlinear convective term causes the presence of a combined fluctuating term
analogous to the Reynolds stress, which is called turbulent heat flux, and a closure problem
arises as well.
3.2 Turbulence 29
3.2.2 Boussinesq’s hypothesis
The Reynolds stress tensor is symmetric and can be expressed as sum of an isotropic
and a deviatoric anisotropic component �̿�.
�̿� = −𝜌
2
3𝜅 + �̿� (3.27)
To achieve an effective modeling of the deviatoric component, Boussinesq proposed a purely
formal analogy with the Newton’s stress - strain-rate law, modeling the deviatoric
component and consequently the Reynolds stress tensor as directly proportional to the
mean strain rate tensor through a scalar coefficient T called eddy viscosity:
�̿� = −2𝜇𝑇�̿̅� (3.28)
A similar shortcoming was proposed for the turbulent heat flux, introducing the eddy
diffusivity kT:
−𝜌𝑐𝑇′�⃗⃗� ̅̅ ̅̅̅ = 𝑘𝑇∇�̅� (3.29)
Introducing (3.28) and (3.29) inside, respectively, equations (3.23) and (3.26) it appears
clearly the way that turbulence enhances mixing rates increasing diffusion coefficients, both
thermal and molecular:
𝜕(�⃗⃗̅� )
𝜕𝑡+ ∇⃗⃗ ⋅ (�⃗⃗̅� ∗ �⃗⃗̅� ) = 𝑔 −
∇�̅�
𝜌+
𝜇 + 𝜇𝑇𝜌
∇2�⃗⃗̅� (3.30)
𝜕(�̅�)
𝜕𝑡+ ∇⃗⃗ ⋅ (�̅��⃗⃗̅� ) =
𝑘 + 𝑘𝑇𝑐𝜌
∇2�̅� (3.31)
where in the equations above the properties of the fluid have been assumed constant. The
system is drastically simplified, reducing the non-linear fluctuating terms to diffusive
elements in the equations. Turbulent viscosity and diffusivity are linked through the
turbulent Prandtl number, defined as:
PrT = 𝐶𝜇𝑇𝑘𝑇
(3.32)
However, at least a model is needed to evaluate the eddy viscosity.
3.3 Boundary Layer Theory
Pressure-relief devices are characterized by internal flows: in a PRV the flow is guided
