Rakshith B Hombegowda master thesis final

A Master Thesis submitted to the Universität Duisburg Essen in partial fulfilment of the
requirements for the degree of Master of Science in Computational Mechanics
Chair of Turbomachinery Department of Energy
Universität Duisburg Essen Fraunhofer UMSICHT
Gebäude MF Osterfield Str. 3
47057 Duisburg 46047 Oberhausen
Germany Germany
Validation and Optimization of the flow in
Laval Nozzles for steam applications
Master Thesis
by
Rakshith Byaladakere Hombegowda
Matr.Nr.: ES03014421
31/05/2016
University Supervisors: Fraunhofer UMSICHT Supervisor
Prof. Dr. –Ing. Fedrich-Karl Benra Dr.-Ing. Björn Bülten
Dr. –Ing Sebastian Schuster

i
Declaration of Authorship
Declaration of Authorship
I, Rakshith Byaladakere Hombegowda, declare that this Master Thesis titled, “Validation and
Optimization of the flow in Laval Nozzles for Steam Applications” and the work presented in
it are my own. I affirm that:
 This work is wholly or mainly in candidature for a Master’s degree in Computational
Mechanics at Universität Duisburg Essen.
 I have consulted the published work of others, this is always clearly attributed.
 I have quoted from the work of others, the source is always given. With exception of
such quotations, this Thesis is entirely my own work.
 I have acknowledged all main sources of help. The thesis is based on work done by
myself jointly with others. I have made clear exactly what was done by others and
what I have contributed myself.
Signed : ____________________________
Date : ____________________________

Acknowledgement ii
Rakshith Byaladakere Hombegowda Master of Science Thesis
Acknowledgements
I would like to express my deepest gratitude to all those who helped me in
accomplishing my Master Thesis. I would like to express my whole hearted thanks to my
supervisor at Fraunhofer UMSICHT Dr.-Ing. Björn Bülten for his excellent guidance,
patience and providing me an comfortable atmosphere for doing my Master Thesis.
I would like to thank my University Prof. Dr. –Ing. Fedrich-Karl Benra, Turbomachinery
Department, University of Duisburg-Essen, for accepting and providing me guidance
throughout my Master Thesis. Also, I would like to thank Dr. –Ing Sebastian Schuster
for his help, professionalism and valuable guidance throughout this project.
Finally, I must express my very profound gratitude to my parents Hombegowda B.E and
Bhagya S.J. Also many thanks to Mahesh Kashappa who always stood by me in difficult
times like a brother and to all my dear friends for providing me unbiased support and
continuous encouragement throughout my years of study and through the process of
reaching and writing this thesis. This accomplishment would not have been possible
without them.

Abstract iii
Abstract
In this work the simulation tool called the ANSYS CFX is utilized to validate and
furthermore optimize through parameterization the flow in Laval nozzles for steam
applications. Condensation in Laval nozzles leads to deterioration of the mechanical
components which results in the loss of efficiency wherein prime reason being the
formation of droplets at the throat. It is of great importance to control the condensation
and thereby controlling the droplet size in order to obtain better efficiency. Hence the
main objective is to validate and resolve different nozzle geometries for high-pressure
nozzle experiments conducted by Gyarmathy (2005). Furthermore, in this validation the
Euler-Euler method is enforced in which both gas and liquid phase are calculated by
solving the Navier-Stokes equations.
At first, suitable meshes with refined walls were selected to numerically verify the
results obtained from ANSYS CFX simulation with that of the experimental results
obtained from Gyarmathy (2005). The credibility of sensitivity analysis through various
model parameters such as Turbulence model, Nucleation Bulk Tension Factor (NBTF)
and Nusselt Number Correlation was introduced to observe changes and their influence
on the existing simulation hence, validating the experimental results. It is evident that
changing the NBTF shifted the Wilson point, furthermore change in Nusselt number
correlation led to the changes in the droplet diameter.
As a final step, the numerical model of the validated nozzle was used to investigate the
own geometry. With parameter changes in the geometry an optimum efficiency with
1µm as the maximum allowable droplet diameter size and preferably uniform flow at
outlet is achieved for short length nozzle having high curvature change to avoid shock at
the throat. As a result of this study it is found that this validated and parameterized study
with the Euler-Euler method approach in ANSYS CFX is applicable to other high
pressure nozzles and the results too would be in nearly good agreement with the
experimental results.

List of contents iv
Contents
1. Introduction.................................................................................................................. 1
1.1. Motivation and Purpose ..................................................................................... 6
1.2. Task Description ................................................................................................ 7
1.3. Thesis Outline .................................................................................................... 8
2. Experiments and Numerical Simulations on low pressure Laval nozzles................... 9
2.1. State of the Art ................................................................................................. 10
2.2. Condensation in nozzle .................................................................................... 11
2.3. Modelling of Multiphase flows........................................................................ 14
2.4. Euler-Euler and Euler-Lagrange approach for multiphase flows .................... 14
2.5. Condensation Modelling.................................................................................. 15
2.5.1. Evolution of Nucleation Theory .......................................................... 15
2.5.2. Homogeneous vs. Heterogeneous Nucleation...................................... 17
2.5.3. Steam Chemistry Influence.................................................................. 18
2.5.4. Droplet Growth Theory........................................................................ 19
3. Numerical Modelling................................................................................................. 24
3.1. The Reynold Averaged Navier-Stokes Equation............................................. 25
3.2. Turbulent Flow................................................................................................. 27
3.2.1. Turbulence Models............................................................................... 28
3.2.2. RANS Model........................................................................................ 30
3.3. Two-Equation Turbulence models................................................................... 31
3.3.1. Turbulence model ...................................................................... 32
3.3.2. Turbulence model ..................................................................... 33
3.3.3. SST-Turbulence Model........................................................................ 34
3.4. Boundary Layer Approximation ...................................................................... 34

List of contents v
3.4.1. Wall function ................................................................................ 35
3.5. The Governing Equations ................................................................................ 37
3.5.1. Conservation of mass........................................................................... 38
3.5.2. Conservation of momentum................................................................. 38
3.5.3. Conservation of energy ........................................................................ 39
3.5.4. Conservation equations for liquid phase.............................................. 40
3.6. Condensation modelling in ANSYS CFX ....................................................... 41
3.6.1. Wall condensation model..................................................................... 41
3.6.2. Equilibrium phase change model......................................................... 42
3.6.3. Droplet condensation model ................................................................ 44
3.7. Character and Structure of IAPWS-IF97......................................................... 45
4. Experiments of High pressure Nozzles & Setup of Numerical Simulation............... 48
4.1. Numerical setup and mesh generation ............................................................. 51
4.1.1. Calculation of Efficiency ..................................................................... 53
4.1.2. Calculation of Nusselt Number............................................................ 53
5. Results of 2/M and 5/B Nozzle.................................................................................. 55
5.1. Numerical verification of 2/M and 5/B Nozzle................................................ 55
5.1.1. Mesh Density Study............................................................................. 55
5.1.2. Superheated case analysis for 2/M and 5/B Nozzles ........................... 58
5.1.3. Wall Refinement .................................................................................. 60
5.1.4. 3-D Effect and Single Precision........................................................... 62
5.1.5. Discussion ............................................................................................ 63
5.2. Validation of 2/M and 5/B Nozzle................................................................... 65
5.2.1. Turbulence model (SST vs ) ...................................................... 65
5.2.2. NBTF Correction ................................................................................. 67
5.2.3. Nusselt Number Correlations............................................................... 70
5.2.4. Discussion ............................................................................................ 74

List of contents vi
6. Parameter investigation to optimize .......................................................................... 75
6.1. Geometry Parametrization ............................................................................... 75
6.2. Task Description .............................................................................................. 75
6.3. Geometry and Mesh setup................................................................................ 76
6.4. Losses in Steam turbine ................................................................................... 78
6.4.1. Frictional losses.................................................................................... 78
6.4.2. Condensation losses ............................................................................. 78
6.4.3. Shock wave losses................................................................................ 78
6.5. Results79
6.5.1. Efficiency of parametrically optimized nozzle .................................... 79
6.5.2. Droplet diameter investigation of parametrized nozzle ....................... 83
6.5.3. Discussion ............................................................................................ 92
7. Conclusion and Scope for Future .............................................................................. 93
7.1. Validation......................................................................................................... 93
7.2. Parametrization .................................................Error! Bookmark not defined.
Appendix........................................................................................................................ 100
Bibliography .................................................................................................................... 96

List of figures vii
List of Figures
Figure 1.1: Impulse turbine vs. Reaction turbine (Chaplin, 2009) .................................... 2
Figure 1.2: Convergent Divergent Nozzle with Mach, Temperature and Pressure
(Lavante, 2014).................................................................................................................. 4
Figure 1.3: Distribution of losses in Low Pressure turbine (Jonas, 1995)......................... 5
Figure 2.1: Axial pressure distribution with spontaneous condensation in the nozzle
(Mohsin & Majid, 2008).................................................................................................. 12
Figure 2.2 : State line for expanding steam with spontaneous condensation. (Mohsin &
Majid, 2008) .................................................................................................................... 13
Figure 2.3: Free energy for nucleation vs. number of water molecules (Jonas, 1995).... 18
Figure 2.4: The Langmuir model and distribution of temperature around the growing
droplet (Fakhari, 2010) .................................................................................................... 22
Figure 3.1: Boundary Layer over a Flat plate (Kempf, 2014)......................................... 28
Figure 3.2: Turbulent models flow chart ......................................................................... 29
Figure 3.3: Statistical Modelling Flow chart ................................................................... 30
Figure 3.4: Velocity profiles subdivisions of near wall region (Salim & Cheah, 2009) . 36
Figure 3.5: Temperature Entropy diagram for liquid vapour mixture (CFX Theory
Guide, 2015). ................................................................................................................... 43
Figure 3.6: Regions and equations of IAPWS-IF97 (Wagner & Kruse, 1998)............... 46
Figure 3.7: Table Generation in ANSYS CFX for IAPWS............................................. 47
Figure 4.1: Nozzle shapes used in (Gyarmathy, 2005).................................................... 49
Figure 4.2: Nozzle 2/M Experimental results (Gyarmathy, 2005) .................................. 50
Figure 4.3: Nozzle 5/B Experimental results (Gyarmathy, 2005)................................... 50
Figure 4.4: 2/M nozzle Geometry with Boundaries ........................................................ 52
Figure 4.5: Meshing for 5/B Laval Nozzle...................................................................... 52
Figure 5.1: Mesh comparison of static pressure and droplet profiles obtained from CFD
simulations along the 2/M nozzle axis with the experimental data reported by
Gyarmathy(2005)............................................................................................................. 56
Figure 5.2: Mesh comparison of static pressure and droplet profiles obtained from CFD
simulations along the 5/B nozzle axis with the experimental data reported by
Gyarmathy(2005)............................................................................................................. 57

List of figures viii
Figure 5.3: Comparison of superheated static pressure obtained from CFD simulations
along the 2/M nozzle axis with the experimental data reported by Gyarmathy(2005) ... 59
Figure 5.4: Comparison of superheated static pressure obtained from CFD simulations
along the 5/B nozzle axis with the experimental data reported by Gyarmathy(2005) .... 59
Figure 5.5: Wall Refinement comparison of static pressure profiles and droplet profile
for wall refinement from CFD simulations along the 2/M nozzle axis ........................... 60
Figure 5.6: Wall Refinement comparison of static pressure profiles and droplet profile
for wall refinement from CFD simulations along the 5/B nozzle axis............................ 61
Figure 5.7: Comparison droplet and pressure profiles of and SST turbulence
models for 2/M Nozzle (NBTF 1.0) ................................................................................ 66
Figure 5.8: Comparison droplet and pressure profiles of and SST turbulence
models for 5/B Nozzle (NBTF 1.0) ................................................................................. 66
Figure 5.9: NBTF influence on 2/M nozzle …………………………………………69
Figure 5.10: NBTF influence on 5/B nozzle ................................................................... 69
Figure 5.11: Nusselt number influence on 2/M nozzle (for NBTF=1)............................ 72
Figure 5.12: Nusselt number influence on 5/B nozzle (for NBTF=1)............................. 72
Figure 6.1: Parametrized geometry.................................................................................. 76
Figure 6.2: Efficiency comparison for superheated case................................................. 80
Figure 6.3: Efficiency comparison for saturated steam case ........................................... 82
Figure 6.4: Superheated steam droplet diameter for different divergent length.............. 84
Figure 6.5: Pressure gradient for Radius 2 mm divergent length 50 mm........................ 85
Figure 6.6: Pressure profile for Radius 2 mm divergent length 50 mm .......................... 86
Figure 6.7: Particle diameter for Radius 2 mm divergent length 50 mm. ....................... 86
Figure 6.8: Flow in parametrized radius 5mm................................................................. 87
Figure 6.9: Flow in parametrized radius 10mm............................................................... 87
Figure 6.10: Flow in parametrized radius 5mm with ellipse length divergent section.... 88
Figure 6.11: Saturated steam droplet diameter for different divergent length. ............... 90
Figure 6.12: Superheated radius 2mm length 30mm....................................................... 91
Figure 6.13: Saturated radius 2mm length 30mm ........................................................... 91
Figure 1.A: Euler-Lagrange vs Euler-Euler Superheated case comparison .................. 100

List of tables ix
List of Tables
Table 1: Specifications of supersonic nozzles used in (Gyarmathy, 2005)..................... 48
Table 2: Specifications of Validating Nozzles ................................................................ 50
Table 3: Boundary conditions for 2/M and 5/B nozzle ................................................... 51
Table 4: 2/M Mesh density efficiency comparison. ........................................................ 57
Table 5: 5/B Mesh density efficiency comparison .......................................................... 57
Table 6: 2/M Wall Refinement efficiency comparison ................................................... 61
Table 7: 5/B Wall Refinement efficiency comparison .................................................... 61
Table 8: Comparison of efficiency and mass flow at outlet for and SST models in
2/M Nozzle. ..................................................................................................................... 67
Table 9: Comparison of efficiency and mass flow at outlet for and SST models in
2/M Nozzle. ..................................................................................................................... 67
Table 10: NBTF Efficiency for 2/M nozzle .................................................................... 70
Table 11: NBTF Efficiency for 5/B nozzle ..................................................................... 70
Table 12: Nusselt Efficiency for 2/M nozzle................................................................... 73
Table 13: Nusselt Efficiency for 5/B nozzle.................................................................... 73
Table 14: Boundary conditions for parametrized nozzle................................................. 77

Nomenclature x
Nomenclature
Nozzle channel width mm
c Area averaged velocity
Specific heat at constant temperature
d Diameter mm
h Specific enthalpy
g Gravity
H Total enthalpy
k Thermal conductivity
Kn Knudsen number -
̅ Mean free path mm
L Length mm
m Mass
̇ Mass flow rate
̇ Mass transfer rate
Mass transfer coefficient -
n Number count per unit mass -
Nu Nusselt number -
P Pressure bar
Pr Prandtl number -
̇ Heat transfer rate
Droplet radius mean value mm
r Radius mm
R Gas constant
Re Reynolds number -
Energy source term -
Mass Source term -

Nomenclature xi
t Time s
T Temperature
Saturation temperature
Sub-cooling temperature (
u Velocity
Velocity of the flow field
W Molecular weight -
x,y,z Spatial dimensions -
u,v,w Velocity dimensions -
Greek
Source term -
Mass density
Wavelength of light nm
Axial coordinate ( ) -
Dynamic viscosity
Delta -
Correction factor for Nusselt number correlation -
Surface tension
Area averaged efficiency %

Nomenclature xii
Subscripts
g Gas phase
p Liquid particle phase
sat Saturation
eff
w
s
i,j
Effective
wall
Isentropic condition
Tensor notations
in Inlet
out Outlet
mix Mixture
Superscripts
* Dimensionless value
´ Fluctuating component
¯ Averaged value

Introduction 1
Chapter 1
This chapter gives an insight on description and importance of nozzles, the purpose
and function of stator and rotor blades and also why condensation occurs in the nozzle,
followed by the motivation and purpose of the present work. Furthermore the outline of
this Master Thesis concludes the chapter.
1. Introduction
There are research and development going on every day to find a new technology and
bring about new innovative ideas in the field of engineering which helps in day to day
activities. These research and development not only helps to improve the quality of the
product with a cost constraint in mind but also utilize them effectively with lesser effort.
Likewise, Sir Charles Parsons invented pressure compound steam turbines which are
devices performing mechanical work on a rotating output shaft by extracting thermal
energy from pressurized steam. This was based on the invention of impulse steam
turbine designed by Gustaf de Laval which was subjected to high centrifugal forces
having limited output due to the strengths of material available in those days. Nozzles
are vital parts in a steam turbine to generate power hence, it is important to device the
components of a steam turbine to obtain better performance effectively. Steam turbines
are used in many industrial applications, often used to generate electricity.
Impulse vs. Reaction Turbine:
There are sophisticated methods to accurately harness the steam power and this has
given rise to two primary turbines called the impulse turbine and reaction turbine. These
two turbines having different designs engage the steam in different method so as to turn
the rotor and generate power.
In an impulse turbine all the pressure energy is converted into kinetic energy by the
nozzle and this helps the jet of fluid to strike the runner blades. In comparison to the
impulse turbine, only some of the available pressure energy in reaction turbine is
converted into kinetic energy before the fluid enters the runner blades. The degree of

Introduction 2
reaction in an impulse turbine is zero however, in a reaction turbine the degree of
reaction is more than zero and less than or equal to one.
IMPULSE TURBINE REACTION TURBINE
Figure 1.1: Impulse turbine vs. Reaction turbine (Chaplin, 2009)
Figure 1.1 shows different stages of fixed and moving blades of an impulse and reaction
turbine respectively. In the graph pressure represents the heat energy and the kinetic
energy is represented by the absolute velocity. As it can be seen from the graph, pressure
remains constant in the moving blades region of the impulse turbine. In contrast to this
there is a pressure drop in the moving blade region for the reaction turbine. Hence the
main difference between the impulse and the reaction turbine is that the pressure drop in
the impulse turbine is only across the fixed blades where as in the reaction turbine the
pressure drop occurs both in fixed as well as in the rotating blades. This results in lower
velocity of steam leaving the fixed blades in reaction turbine (Chaplin, 2009).
The shapes of the moving blades is different for both impulse and reaction turbine. There
is no change in the flow area for an impulse turbine where as in the reaction turbine has
a change in flow area. As a result of this the velocity of steam remains constant although
there is a change in direction.

Introduction 3
Curtis and De Laval steam turbines are examples of turbines which operate at high
pressure ratio. The main principle behind these steam turbines is to achieve high work
output with high efficiency, so that their application in both steam and rocket propulsion
would be enticing (Stratford & Sansome, 1959).There is tremendous amount of research
and development carried out on turbine nozzles for decades as majority of world’s
electricity demand is met with the help of steam operated turbines. To get a high steam
cycle efficiency the enthalpy drop in the turbine was increased (e.g. by lowering the
exhaust pressure) and therefore the steam turbines are operated with condensation.
Likewise, there are many constrains to look for as the boundary conditions such as
temperatures at inlet, outlet and also the Mach number which play a principle part in
designing a nozzle.
Nozzles play a vital role in a steam turbine. The main feature of the nozzle is to modify
the fluid flow wherein they increase the kinetic energy of the fluid flow in accordance
with the pressure. If high enthalpy drops have to be utilized in one stage of the turbine it
is beneficial to use convergent-divergent nozzles to create supersonic flows (Mach
number more than 1). The convergent divergent nozzles have wide applications and
hence can be used in jet engines for rocket propulsion other than to generate electricity.
The fluid flow in the Laval nozzle which is a convergent divergent nozzle undergoes
condensation if the flow is expanded into the two-phase region. It is crucial to analyse
the rate of condensation and control the droplet growth to yield better performance from
the nozzle.
Nucleation can be defined as the occurrence of density concentration in a small volume
of a supersaturated system which undergoes decomposition into two phases in local
equilibrium. To accurately assess and reduce the condensation and frictional losses it is
vital to know the thermodynamic and kinetic conditions at the nucleation onset and
furthermore successive droplet growth must be accurately acknowledged (Jonas, 1995).
The Figure 1.2 illustrates a Laval nozzle which creates supersonic speeds at the outlet.
There exists a change in area between the inlet and outlet of the nozzle in a Laval nozzle.

Introduction 4
As the fluid enters the nozzle it accelerates as it passes throat region which is considered
to be a subsonic region having high pressures and temperature. At the throat the fluid
flow matches the speed of the sound where the Mach number is 1 and then exceeds it,
becoming a supersonic flow with high velocity at outlet and reduced temperature and
pressure due to expansion of the fluid.
Figure 1.2: Convergent Divergent Nozzle with Mach, Temperature and Pressure (Lavante, 2014)

Introduction 5
Figure 1.3 shows the distribution of losses in a low pressure turbine which shows more
than 25% losses are due to condensation.
Figure 1.3: Distribution of losses in Low Pressure turbine (Jonas, 1995).
The above introduction gives a brief notion on the demand for the improvisation of
nozzles in steam turbines in order to decrease or to keep a check on the condensational
effects so as to increase the performance of the turbine with the aid of nozzle design. A
high performance computational fluid dynamics tool called ANSYS CFX is used in this
master thesis in order to accomplish the desired objectives which are stated in the
following section. Investigation is made to check whether the results obtained are
reliable and accurate solutions promptly for similar steam applications involved in CFD.
This tool has helped many researchers and investigators in saving cost, natural resources,
time and energy in solving fluid flow problems.
It is vital to find an appropriate geometry and generate a mesh for that geometry
satisfying all the given boundary conditions such as temperature, pressure or Mach
number in ANSYS CFX so that the condensation that occurs near the throat region is
thoroughly simulated to yield best results. Selecting a mesh should be in such a way as
to not waste the time on simulating excess undesirable cells in the geometry.

Introduction 6
1.1. Motivation and Purpose
Researchers are working on to improve the overall efficiency of high pressure nozzles in
steam turbines by controlling condensation. Condensation is generally defined as a phase
change from vapour to liquid water state. In the past numerous experiments were
conducted on condensation in low pressure nozzles by Gerber & Kermani (2003)
Hegazy, et al. (2015) and many more, whereas only fewer scientists made progress in
conducting experiments for high pressure nozzles for the occurrence of condensation.
One such experiment was conducted by Gyarmathy (2005) where the superheated steam
in high pressure was considered in Laval nozzles (Gyarmathy, 2005). Investigation was
made on experiments based on the numerical calculation approaches which were carried
out on the work of Gyarmathy (Guo, et al., 2014). Researchers have concluded that
condensation which is caused due to the homogeneous nucleation leads to abrasion and
corrosion of rotor blades, furthermore decreasing the isentropic efficiency (Lamanna,
2000). Therefore it is essential to predict and control the droplet size during
condensation and understand the significance of nucleation in the Laval nozzle. This
experience has appealed to many researchers and engineers in understanding the
fundamental process which leads to various losses in steam turbine during the
condensation process in a multiphase medium and hence, help the steam turbine
manufactures with optimized designs.
This present Master Thesis stands on the above mentioned grounds on validating high
pressure nozzles and optimization through parametrizing a very own nozzle to improve
the overall efficiency and control the droplet size. Here in this work we want to design
and optimize high pressure Laval nozzles where the condensation occurs. But prior to
designing a very own Laval-nozzle it is of utmost importance to be sure that the design
program (here ANSYS CFX) is giving reliable results. Therefore it is essential to
validate the numerical results of a high pressure Laval nozzle before designing it. The
validation is done based on the experimental results of Gyarmathy (2005) after which
designing of own nozzle is made on the grounds of the results analysed during
validation.

Introduction 7
1.2. Task Description
The following tasks are dealt in this Master Thesis for Non-Equilibrium rapidly
expanding supersonic nozzles:
 Literature review on Nucleation of steam, droplet growth theory and influence of
condensation in a high pressure nozzle.
 Modelling of the fluid flow with ANSYS CFX and Recalculation of two high-
pressure nozzles (Gyarmathy, 2005).
 Comparison of experimental and numerical results including a sensitivity
analysis in the numerical simulation.
 Optimization through parameterization of a very own Nozzle geometry with the
objective to achieve maximum allowable droplet size, optimum efficiency and
preferably uniform flow at outlet for two representative cases (one case with
superheated steam at the inlet and one with saturated steam).
 Analysing and calculating the droplet size and efficiency for each case.
 Giving a firm conclusion for the very own geometry based on the grounds of
validated results within the defined boundary conditions.

Introduction 8
1.3. Thesis Outline
To achieve the final task which is design optimization through parameterizing a very
own nozzle, with the aid of available theoretical literatures and also implementing the
results observed on validating the Laval Nozzles taken from Gyarmathy (2005).
Subsequently this validation gives a set of conclusions considering the influences of all
the various parametrical changes on two different Laval nozzles from the paper.
The content of the chapters are as following:
 Chapter 2 presents literature study on low and high pressure nozzle experiments
in the state of the art of this Master thesis with the spotlight being the influence
of condensation in high pressure Laval nozzles.
 Chapter 3 presents the numerical modelling approach carried out in this work
along with the methods and equations used to validate the experimental results.
 Chapter 4 presents the experiments on high pressure nozzle and numerical setup
for validation of selected nozzles.
 Chapter 5 presents the validated results and discussion for the selected two
nozzles from Gyarmathy (2005) paper.
 Chapter 6 presents the results for the optimized new geometry by parameterizing
the radius and the length of the divergent section which is based on the results of
validated nozzles.
 Chapter 7 presents the conclusion of the thesis and scope for the future.

State of the Art 9
Chapter 2
This chapter gives the insight on the State of the Art carried out in the present
work. This is achieved by an extensive literature survey concerning the reasons for the
formation of condensation in a high pressure nozzle. The principle goal of this chapter is
to give the reader a comprehensive insight to the factors which influence the nucleation
and critical aspects of condensation process. The brief outlay of various experiments
and the theories concerning the condensation conducted by engineers and researchers is
portrayed in this chapter.
2. Experiments and Numerical Simulations on low pressure Laval
nozzles
Nozzles are one of the essential parts for industrial applications. Supersonic flow in a
Laval nozzle acts as a fundamental phenomenon which influences a large variety of
industrial application. During the rapid expansion of steam there will be occurrence of
condensation process after the throat section and the expansion process near the
divergent section of the throat causes nucleation of water droplets.
Many experiments were conducted for the flow of fluid in a low pressure nozzle (Moore
& Sieverding, 1976). Gerber & Kermani (2003) studied pressure based Euler-Euler
multiphase model for non-equilibrium condensation. The water droplet distribution in
low and high pressure nozzle was predicted with the aid of equations. Furthermore,
numerical analysis of spontaneously condensing phenomenon in the nozzle of steam jet
vacuum pump was introduced by Wang, et al., (2012) Viscous calculations for steady
flow were made by Simpson & White (1997) where, it indicated that the growth of the
boundary layer had significant impact on the predicted pressure distribution and also on
droplet diameter. Numerical simulations were made for the low pressure nozzle where
prominent , and SST models were considered. The main aim of the
numerical simulation was to predict the flow characteristic of wet steam and validate the
results with the experimental date which were available. One such numerical analysis
was made by (Hegazy, et al., 2015).

State of the Art 10
2.1. State of the Art
A number of literature work is available for the modelling of non-equilibrium
condensing flow. In the present work, one of the primary focus is on validating the
numerical results conducted for high pressure Laval nozzles using a high performance
computational fluid dynamics tool called ANSYS CFX. For this concern, major part of
the literature study was based on the experimental results and conclusions obtained on
high pressure nozzles using various theories concerning the condensation in the past.
The nozzle is an important part of the steam turbine as it accelerates the high pressure
steam passing through it which results in giving a supersonic and low pressure steam
flow. From the thermodynamic temperature entropy (T-S) diagram, water has a
negative-slope saturated vapour line which endorses that an isentropic expansion of the
fluid would possibly induce condensation which would directly hinder the performance
of steam turbine (Rajput, 1993). There have been many attempts to simulate steam
condensation which occurs in the nozzle either by theoretical methods or by numerical
methods. Modelling of condensing flow in a low pressure steam turbine was performed
by various researchers. Wang, et al. (2012) & Zehng, et al. (2011) simulated the Moore
nozzle using CFD tool which was theoretically analysed by Giordano, et al (2010) .
However, very few experiments were conducted on high pressure nozzles. One such
experiment was conducted by Gyarmathy (2005).
From the theoretical background it is clear that considering steam as an ideal gas would
not provide results concerning the condensation. Hence industrial fluid IAPWS-IF97
equation of state which is pre-defined in ANSYS CFX allows researchers to directly
select them for the simulations. Here the IAPWS-IF97 properties have been tested for
extrapolation into metastable regions which can be used effectively for solving non-
equilibrium problems. Brief description on IAPWS-IF97 has been made in later
chapters.

State of the Art 11
The main Objective of the Gyarmathy experiments on Nucleation of steam on High
Pressure Nozzle are listed as follows (Gyarmathy, 2005) :
 Phase equilibrium could be established by determining the amount of sub cooling
that occurs in fast adiabatic expansion of dry steam before nucleation.
 To determine the average size of the droplet along with its specific number
count.
 To estimate the influence of pressure level and quantify the influence of
expansion rate.
The Wilson points were simply detected by providing static pressure taps in the upper
wall slot. The formation and the growth of the droplet size were easily measured along
the flow axis. The optical measurements were based on the attenuation of the red
monochromatic light beam of λ = 632.8nm of a Helium-Neon laser. For this matter,
major importance was given in understanding the nucleation theory and various growth
models which were developed in the past and hence modelling of condensing flow was
necessary to understand the occurrence of condensation in the nozzles. Hill (1966)
analysed the condensation data on supersonic nozzle and correlated the results with the
nucleation and droplet growth theories. Furthermore, he was the first to introduce the
droplet growth theory for precise prediction of theoretical data.
2.2. Condensation in nozzle
For the validation of nucleation and droplet growth theory majority of engineering
investigations were made on convergent-divergent nozzles carrying steam. Figure 2.1
illustrates an expansion of steam in a convergent-divergent nozzle. The whole
condensation process can be conveniently depicted along the length of the nozzle where
the experiments are conducted under steady state condition.

State of the Art 12
Figure 2.1 : Axial pressure distribution with spontaneous condensation in the nozzle (Mohsin &
Majid, 2008)
It was made easy to determine the onset of nucleation from the measurement of pressure
in nozzle experiments, rather than relying on the visual observation of the fog. It was
found that the effects of some of the undesirable heterogeneous nucleation could be
neglected, as the rapid expansion that occurs in the nozzle allows very little time for the
heat transfer between the apparatus and the working fluid weakens its effect. Figure 2.2
illustrates the expansion of steam on an h-s diagram (Hasini, et al., 2012).
At point (1) the steam enters the nozzle as dry superheated vapour. It undergoes
expansion as it passes along the length of the nozzle and the expansion to the sonic
condition is represented by point (2).

State of the Art 13
Figure 2.2 : State line for expanding steam with spontaneous condensation. (Mohsin & Majid, 2008)
Comparing Figure 2.1 and Figure 2.2 point (3) represents the beginning of dry super
cooled region where liquid droplet start to form and grow as vapour when the saturation
line is crossed which may occur before or after the throat region. Here during the initial
stages of droplet growth the nucleation rate associated are so low that the steam further
expands as a dry single phase vapour in a metastable, super cooled or supersaturated
state which can be seen in dry super cooled region in Figure 2.2. Depending on the inlet
conditions and rate of expansion, the nucleation rate increases dramatically and reaches
its maximum point near the Wilson line which is point (4).This region is termed as the
nucleating zone and is terminated by the Wilson point, a point which represents the
maximum super cooling and can be defined as:
( )
Where is the sub cooling temperature, is the saturation temperature and is the
static temperature at vapour static pressure . As the fluid progresses to the
downstream of Wilson point, nucleation effectively terminates and the number of
droplets in the flow remains constant. There is a rapid growth of droplet nuclei between
the points (4) and (5), thus restoring the thermodynamic equilibrium in the system which
is achieved by exchanging the heat and mass with the surrounding liquid. There is a
gradual increase in pressure from point (4) to point (5) due to the conduction of latent
heat, which is released at the droplet surface. This is known as condensation shock

State of the Art 14
which is rather misleading as the changes of flow properties between points (4) and (5)
are continuous, as a result of which there is deceleration of supersonic flow. From Figure
2.2 it is evident that there is a steep increase in entropy as well as in enthalpy from point
(4) to (5), stating that the process is irreversible since the heat transfer in the vapour
occurs through finite temperature difference between the phases. This in thermodynamic
aspect is termed as Thermodynamic Nucleation Losses. Furthermore there is more
expansion in the flow of steam between the points (5) to (6) which is the wet equilibrium
region where the enthalpy decreases.
2.3. Modelling of Multiphase flows
The high pressure steam undergoes expansion within the nozzle which results in the
nucleation of microscopic water droplets. This nucleation grows further by condensation
in the nozzle contributing to wetness losses in the whole system (Fakhari, 2010). Several
modelling approaches with more sophisticated measurement techniques were performed
by various researchers and investigators. In addition to these modelling and investigation
in real world, the performance of computer technology has encouraged implementing
accurate models.
2.4.Euler-Euler and Euler-Lagrange approach for multiphase flows
Significant developments in the field of computational fluid mechanics have given
further insight in the dynamics of multiphase flows. At present there are two popular
approaches for the numerical calculation of multiphase flows: the Euler-Euler approach
and the Euler-Lagrange approach.
The numerical simulation of droplets with the aid of Lagrangian approach tracks the
trajectories and velocities of each individual particle. It also helps in tracking the mass
and temperature associated with each individual particle. The Lagrangian approach is
applicable for both dense and dilute particulate multiphase flows. However, in Eulerian
approach the particle cloud or the droplets in mixture are assumed to be denser and can
hence be classified as a continuum. If both the phases are fluid then, the Euler-Euler
approach is referred to as two-fluid approach. The nature of the flow and the required

State of the Art 15
accuracy determines whether Eularian approach is advantageous or Lagrangian approach
is better suited. In condensation flows, one can expect wide range of droplet sizes,
velocity, temperature and pressure distribution which has to be represented in a
numerical calculation for more realistic calculation of wetness losses. The Euler-Euler
method needs more time than the Euler-Lagrange method for calculations in solving heat
transfer along with four additional equations for one representative droplet size (Fakhari,
2010). Furthermore, enhanced modelling strategies are required to implement more
complicated droplet models in the Eulerian approach. Grid dependency plays a major
role in the Eulerian approach, as the velocity and temperature which is associated with
the gas phase and also the time and spatial scales associated with the droplet nucleation
and growth has to be resolved on a very fine grid. The motion of the particle in the flow
field having significant variation in velocity and temperature fields can be analysed
better using the Lagrangian approach, as the Lagrangian time frame can be adapted. One
of the major advantages of Lagrangian approach over the Eularian approach is that a
direct framework can be achieved for implementing highly nonlinear droplet models.
2.5. Condensation Modelling
In this section a brief explanation of physical modelling is introduced. Evolution of
nucleation theory is discussed followed with the droplet growth theory which is
distinctive part of condensation process. Difference between the homogeneous and
heterogeneous nucleation are also discussed so as to get a better idea on their influence
for condensation in supersonic nozzles.
2.5.1. Evolution of Nucleation Theory
Nucleation may be exemplified as the first irreversible formation of a nucleus in an
equilibrium phase. Volmer and Weber started the development of nucleation theory
(Volmer & Weber, 1926). Their nucleation theory was based on Stefan Boltzman
distribution law which states that the number of molecular clusters of critical size was
related to number of monomers which are capable of bonding to form long chain
molecular cluster in a system. It was possible to obtain an expression for nucleation rate

State of the Art 16
based on assumption that the growth and decay of the droplet had equal probability
considering the rate of molecular collision.
Furthermore by treating the nucleation and considering the kinetics of molecular
interaction as quasi-steady process an expression was obtained for the nucleation rate
that was consistent with the Volmer and Weber’s result. Based on these results many
other researchers and investigators contributed for the development considering the
thermodynamic aspects of the problem. The nucleation theory just described is
commonly known as classical nucleation theory.
There was tremendous progress and efforts to remove some of the uncertainties which
sabotage the classical nucleation theory. Hence another approach called the statistical
mechanical approach was extensively used to study the nucleation. Few of the
mentionable uncertainties in the classical theory are the condensation coefficient and
also the surface tension of small clusters. In this statistical mechanical approach the
nucleation process was thoroughly analysed at the microscopic level so as to find better
results. The complexities of this approach will not be discussed in this work however the
comprehensive treatment on this topic is given by many researchers and investigators
who include (Gyarmathy, 2005) (Wagner & Kruse, 1998) (Hill, 1966) & (Gerber &
Kermani, 2003).
After a series of investigation Volmer, Weber, Becker & Doring found that the classical
theory oversees some of the vital terms in the free energy of formation in the clusters. It
was found that along with the individual molecules, the cluster of molecules as a whole
also possessed degrees of freedom. These degrees of freedom were associated with the
rotation and translation of the clusters. This degree of freedom which was associated
with the included free energy terms in the expression for free energy of formation of
molecular clusters, yielded a high nucleation rate than the previous study (Lothe &
Pound, 1962).

State of the Art 17
Engineers have given emphasis in using the above nucleation process using convergent-
divergent nozzle in contrast to the study of nucleation process in cloud chambers by
scientists. These two approaches will be explained briefly. An electric field was used to
remove the ions in the cloud chamber which was similar to Volmer (Volmer & Weber,
1926), which might cause heterogeneous nucleation and reported good agreement with
that of the predictions made by Becker-Doring equations (Lothe & Pound, 1962). There
were a number of problems in the unsteady nature of piston cloud chamber experiments
and this was mainly because the time associated with each condition is limited as the
supersaturation changes swiftly during this experiment. Hence for this reason it has
proved difficult to differentiate between the homogeneous nucleation and heterogeneous
nucleation. Furthermore the temperature of the vapour drops below the temperature of
the vessel walls due to expansion which results in heat transfer and creation of
temperature and pressure waves within the vapour. Various other developments and
experiments were made to study the homogeneous nucleation by various researchers and
scientists. Investigations with the help of diffusion cloud chamber, which was used to
study the homogeneous nucleation in several substances including water vapour reported
better agreement with the classical nucleation theory.
2.5.2. Homogeneous vs. Heterogeneous Nucleation
Nucleation can be defined as clustering of molecules during a change of phase from
liquid to gaseous form or vice versa accompanied by a release of latent heat. It is
essential to differentiate between homogeneous and heterogeneous nucleation as it
becomes very sensitive in the presence of impurities. Homogeneous nucleation on a
simple note can be defined as the nucleation process that occurs away from the surface
whereas heterogeneous nucleation is one that takes place on the surface of a liquid phase
in a gas phase hence requires lesser free energy for nucleation (Jonas, 1995). During the
expansion phase in the steam turbine it has been assumed that the moisture nucleation
undergoes homogeneous process neglecting the steam impurities. The principle reason
for neglecting these impurities is because of the notion that steam is highly pure fluid.
However, many researchers and investigators have found that even the pure form of
steam contains some of the impurities which provide nucleation seeds which are both

State of the Art 18
solid and liquid particles on surface which acts as source for a heterogeneous process
(Jonas, 1995). Figure 2.3 illustrates the schematic representation of the effect of
nucleation seeds which catalyses nucleation against the free energy ∆G for nucleation.
Figure 2.3 : Free energy for nucleation vs. number of water molecules (Jonas, 1995).
There are few criteria’s that has to be satisfied for significant heterogeneous nucleation
in a steam turbine which can be listed as follows:
 Nucleation seeds or nucleation surfaces must be available
 There must be enough time space and time for the seeds and water molecules to
collide resulting in growth of droplets.
 The energy balance has to favour the heterogeneous nucleation process.
2.5.3. Steam Chemistry Influence
Significant loss of energy occurs during the phase transition of condensing steam
turbines, resulting in reduction of overall efficiency. Steam chemistry influences the
condensation by:
 Changing surface tension
 Providing the nucleation seeds
 Providing energy to droplets
Modification of the steam chemistry may improve turbine efficiency in:

State of the Art 19
 Reducing the deposition on blades which is one of the aspects for efficiency loss.
 Improving the heterogeneous nucleation to promote earlier droplet formation and
to reduce the energy losses
At present, there is lack of detailed understanding of hetero-homogeneous nucleation
and condensation mechanisms which also includes the mathematical simulation of these
processes. There are no additives available for its use in the steam cycles which could
improve the condensation process to something closer to the theoretical thermodynamic
equilibrium. In addition to these there are also insufficient technologies which diminish
the formation of harmful deposits on the blade surface or to remove the deposits.
High quality research is needed for the investigation of formation of chemical clusters
that occur in the high pressure steam turbines leading to nucleation effects. Also with the
aim to reduce the thermodynamic losses associated to the phase transition leading to
nucleation has to be investigated at highest importance (Jonas, 1995).
2.5.4. Droplet Growth Theory
The condensation is initiated by the nucleation process which is described in the
previous sections. The small clusters called the embryos having critical size of liquid
may grow in the supercooled vapour as the vapour molecules condense further on their
surface. In this process there is liberation of latent heat which causes the temperature in
the droplet to rise above the vapour. The vapour temperature starts to incline as there is
no other surface than the vapour itself to conduct the heat liberated. This initial stage
where there is growth in the vapour pressure is known as the condensation shock. Hence,
growth rate of a droplet is a function of both heat transfer rate between the droplet and
the vapour, also it strongly depends on the rate at which the heat is conducted away from
the droplets (Lamanna, 2000).
It is essential to consider the coupling between the mass and energy to formulate the rate
at which the droplet growths. The energy balance around the spherical droplet having
radius r is done and yields the following expression

State of the Art 20
̇ (1)
( ) ̇ ̇ (2)
Where ̇ is the mass transfer rate and ̇ is the heat transfer rate from the droplet. In the
equation (2), the term on the right hand side comprises of latent heat energy which is the
first term of the equation to be eliminated and heat transfer rate ̇ . The term on the left
hand side of the equation (2) is called sensible heating which is usually very small and is
neglected. The term in equation (1) refers to the mass of a spherical droplet which is
given by
(3)
And the rate of heat transfer by conduction is given as,
̇ (4)
Substituting equations (1) and (4) in equation (2) by assuming the liquid phase to be
incompressible it becomes,
( ) (5)
Where refers to mass condensation rate over the surface of the droplet, )
refers to the local latent heat per unit mass and the left hand side of the above equation
( ) is the rate at which the latent heat is to be removed from the droplet.
While some part of the latent heat is used to rise the droplet temperature remaining heat
is converted to the vapour. In the above expression, represents the temperature of the
liquid particle and refers to the temperature of the gas phase (Lamanna, 2000).

State of the Art 21
Simplifying equation (5) by substituting mass of spherical droplet from equation (3) ,the
droplet growth rate can be expressed as
( )
(6)
Where is the surface heat transfer coefficient for which a solution can be provided by
Laplace equation in the spherical coordinates for hot sphere in cold gas, known as the
conduction theory. Thus can be expressed as:
(7)
In the above expression for , is the thermal conductivity of the vapour. This relation
is applied only when the mean free path of the vapour molecules is smaller than the
particle size of the vapour and hence, fulfils the continuum condition. The validity of
this continuum condition is determined by the droplet Knudsen number Kn (Livesey,
1998).
The heat carrying medium is considered as the continuum in a heat transfer process. This
interpretation cannot be made when the heat transfer to small droplets are considered,
because the molecular structure becomes noticeable. The Knudsen number determines
whether the vapour behaves with regard to a droplet as continuum or as a free molecular
gas. Hence, Knudsen number can be defined as the ratio of mean free path of the vapour
molecules to the droplet size expressed in diameter (Moore & Sieverding, 1976).
Knudsen number is expressed as:
̅
(8)
̅ √ (9)
Where is the dynamic viscosity of the vapour. The Knudsen number plays an
important role in the heat transfer coefficient due to the existence of wide range of

State of the Art 22
droplet radius which is formed during the condensation process. The following
differentiation can be applied for the Knudsen number (Lamanna, 2000):
 For Kn 1 kinetic theory is applicable and this process is governed by Hertz-
Knudsen model.
 For Kn 1 Continuum hypothesis is applicable and the transfer process is
governed by diffusion.
Although, a large amount of literature with different levels of complexities on
growth models were established for different values of Knudsen number, a
universally applicable growth model has still not been formulated. It was necessary
to postulate a model in the realistic description which was applicable to the
continuum condition case and also the kinetic gas theory of transfer processes within
the approximated mean free path for the droplet.
Langmuir model is one of the most significant of these postulated models which
takes into account of both continuum and rarefied gas effects (Fakhari, 2010). The
Langmuir model describing the droplet growth is as shown Figure 2.4. The
continuum regime separated from the free molecular regime can be illustrated in the
Knudsen layer at a radius where is an arbitrary constant of order 1. The
temperature at the interface is denoted as .The detailed derivation of the growth
rate can be found in the paper by Fakhari (2010).
Figure 2.4 : The Langmuir model and distribution of temperature around the growing droplet
(Fakhari, 2010).

State of the Art 23
However, for the small droplets which are generated due to homogeneous nucleation,
the heat transfer coefficient has to be modified to account for the Knudsen number
(Kn). This dependency of heat transfer coefficient for steam was formulated by
Gyarmathy (2005), Moore and Sieverding (1976), which is expressed as
(10)
Where c is an empirical factor set to 3.18
Nu is the Nusselt number and is defined as
(11)
The Nusselt number can also be interpreted as convective to conductive heat transfer
ratio.

Experiments and Setup 24
Chapter 3
This chapter gives the insight on description and importance of mathematical
characters and the governing equations concerning the fluid flow in a Laval Nozzle. This
is followed with the description of suitable numerical methods for discretization of the
governing equation of the gas dynamics along with the turbomachinery boundary
conditions.
3. Numerical Modelling
The Droplet growth theory and the Nucleation theory presented in the previous chapter
yields a set of equations which describes the flow field. Even for the assumption of
perfect gas in the flow field, there are only a limited number of analytical solutions for
these equations. These analytical solutions become uncompromising when steam as a
real gas is combined with the generalized boundary condition. Hence, this ensures that
the numerical solutions have to be developed for the real gases same as the numerical
solutions which were developed for ideal gases (Fakhari, 2010). Thus, a greater
emphasis must be made on the mathematical aspects of the equations which are
admissible for developing numerical algorithm for the solutions.
As discussed in chapter 2 many researchers and investigators have used Euler-Euler
approach and Euler-Lagrange method to accurately model the condensing flows in a
nozzle. In Euler-Lagrange method although the individual particles are tracked using the
Lagrangian approach, the mass, energy and momentum equations were solved using the
Eularian approach. The mass and momentum equations for numerical modelling are
based on the Reynolds Averaged Navier-Stokes Equations (RANS) for a 3-D turbulent
flow in a medium, also it requires a turbulence model to represent some of the terms
concerning in the flow field.

3.1. The Reynold Averaged Navier-Stokes Equation
For the numerical modelling of a fluid motion, foundations are provided by a set of
Navier-Stokes equations and also continuity equations. The Newton law of motion
which is applicable to solid is also applicable for all matters including gases and liquids.
However, there exists a prominent difference between the fluids and solids as fluids tend
to distort without limit unlike solids which stay intact. Like for example when a force is
applied on a fluid, the layers of fluid particle will undergo a shear, tensile or
compression stresses based on the type of force applied and the particles will not return
to their original position due to the relative motion between the layers of the fluid when
the applied force is stopped. If a force is applied to a particle be it a fluid or a solid, its
acceleration will be in such a way that is governed by the Newton second law stating
that “the rate of change of momentum in a body is directly proportional to the
unbalanced force acting upon it and takes place in the direction of the force applied on
it”.
Assuming the linear relation between the shear stress and shear rate in a fluid and also
considering it to be a laminar flow, famous physicist Claude-Louis Navier and George
Gabriel Stokes derived equations concerning the motion for viscous fluid from laminar
consideration popularly recognized as the Navier-Stokes equation. For Turbulent flows it
is important to time average this Navier-Stokes equation along with the continuity
equations for which a flow field can be described with mean values. Besides a viscous
part in the Navier-Stokes equation an additional term has been added to the total shear
stress which has been resulted from the time averaging of the Navier-Stokes Equation.
This term is called as Reynolds stresses as it appears only due to Reynolds averaging.
Hence Reynolds Averaged Navier Stokes (RANS) is a time averaged equation of motion
for the fluid flow.
The general form of Navier-Stokes Equation is given as (Kempf, 2014):
( ) * ( ) + (12)

Where, is the Kronecker delta function which has a function 1 if it possesses same
variable and 0 if they are not equal;
{ (13)
Where,
 Viscous term with Stress tensor ( )
 Accumulative term =
 Convection term = ( )
 Pressure term =
 Gravitational term =
Furthermore, with the continuity equation it can be transformed as following:
The Reynolds Averaged Navier Stokes equations are time averaged and for a stationary,
incompressible Newtonian fluid it is given as:
( ̅ ̅) ̅ * (
̅ ̅
) ̅ ̅̅̅̅̅̅̅+ (15)
The left hand side in equation (15) indicates the change in the mean momentum of the
fluid element which is subjected to the unsteadiness in the mean flow and also the
convection by mean flow. Comparing the Navier-Stokes equation (12) with the
Reynolds time averaged equation (15), there is an additional term ( ̅̅̅̅̅̅̅) besides
the viscous part. This resulting term obtained by Reynolds time averaging is called as
Reynolds stresses having a velocity field of average flow.
( ) * ( ) +
(14)

3.2. Turbulent Flow
Turbulent flow in fluid dynamics can be defined as a type of flow which is administered
by changing fluid velocities resulting in continuous fluctuation in both magnitude and
direction and formation of fluxes called eddies in the flow, in contrast to laminar flow
where the fluid flows with a uniform velocity in the form of layers. Turbulent flow is
generally associated to a non-dimensional quantity called the Reynolds number which is
given as:
(16)
Where is the density of the flowing fluid, is the velocity of the flow, L is the length
of the wall through which the fluid is flowing and is the dynamic viscosity of the fluid.
The Reynolds number which represents the ratio between the inertial forces and viscous
forces as seen from the equation above helps to determine whether the flow is laminar,
transient or turbulent in nature (Kempf, 2014). The flow on Reynolds number is
characterised as following in a pipe:
 Laminar when Re < 2300
 Transient when 2300 < Re < 4200
 Turbulent when Re > 4200
Turbulence regime in the region where there are viscous effects which is close to the
solid boundaries called the boundary layer. It is near this region where the flow gives
rise to a flow structure which is primarily characterised by large-scale eddies. In a pipe
flow the boundary layer grows steadily (Celik, 1999). When considering external flows,
such as flow over an aircraft wing or an automobile, the boundary layer is more confined
to a narrow region which is close to the walls. It is said to be inviscid flow for the flows
away from the wall as the viscous effects are negligible (Kempf, 2014).
For better understanding of how the boundary layer forms in a flow regime can be
imagined with a flow having a free stream velocity, approaching a flat plate which is
as shown in the Figure 3.1 . Due to the presence of friction near the walls of the pipe, the
flow will have zero velocity near the wall and this is called the no slip condition. The
flow velocity will be, at a distance far away from the wall and as the flow

approaches the wall, a boundary layer is formed where the flow varies from zero at the
wall to, far away from the wall.
Figure 3.1: Boundary Layer over a Flat plate (Kempf, 2014).
From the figure above the boundary layer starts as a laminar flow where the Reynolds
number is low indicating that the inertial forces are small compared to the viscous
forces. However, as the length x increases the Reynolds number which is directly
proportional to the length L also increases, thus resulting in the inertial forces to
dominate over the viscous forces creating instability in the boundary layer. This results
in the formation of transition zone until the flow completely develops into a turbulent
flow possessing large eddies. There is always a small laminar layer beneath the turbulent
boundary layer which is called as the laminar sub layer below the buffer layer.
Modelling these turbulent flows has always been an area of interest for various
researchers and scientists as most of the flows are turbulent flows in nature. Based on the
turbulent flow and how to model these turbulent flow researchers have formulated
various turbulent models which are discussed in the imminent chapters.
3.2.1. Turbulence Models
A flow field which is said to be turbulent is characterized by the velocity fluctuation in
all direction furthermore, it will be having an infinite number of degrees of freedom.

Hence, solving a Navier Stokes equation for a turbulent flow looks seemingly impossible
because the equations are elliptic, coupled and non-linear. The flow is chaotic being
three dimensional, diffusive, dissipative and also intermittent. The significant
characteristic of a turbulent flow is that it possesses infinite number of scales so that a
full numerical resolution of the flow requires the construction of grid with a large
number of nodes which is proportional to ⁄
(Celik, 1999). The construction of grid
is achieved by Reynolds decomposition where it reduces the number of scales be it from
infinity to 1 or 2. However, by using the Reynolds decomposition, there are new
unknowns that were introduced in the form of turbulent stresses and turbulent fluxes.
Hence the Reynolds Averaged Navier Stokes Equation (RANS) which is described in
the previous chapters gives an open set of equations. This need for additional equations
to model the new unknowns is known as Turbulence modelling (Gröner, 2014).
Figure 3.2: Turbulent models flow chart
EDDY MODELLING
DNS
Direct Numerical
Simulation
RANS
Reynolds Averaged Navier Stokes
DES
Direct Eddy
Simulation
LES
Large Eddy
Simulation
EVM
Eddy Viscosity
Model
ASM
Algebraic Stress
Model
RSM
Reynolds Stress
Model
NO
MODELLING
STATISTICAL MODELLING
TURBULENT MODELS

3.2.2. RANS Model
The principle objective for the turbulence models is to determine the Reynolds stresses
in a RANS equation. The RANS modelling can be classified further as following under
the Statistical modelling. Here we solve one or more equations, algebraic or transport
equations which are in the form of potential differential equation to determine the eddy
viscosity. RANS modelling gives steady state solutions for many applications due to the
quality of grid it utilizes thus, providing the required accuracy. It helps in modelling the
effect of turbulence on the mean flow (Gröner, 2014).
Figure 3.3: Statistical Modelling Flow chart
1. 1-Equation model (1-transport equation)
 0-Equation model (Algebraic models)
 Baldwin-Lomax model
 Cebeci-Smith model
2. 1-Equation model (1-transport equation)
 Kolmogorov-Prandtl model (k)
 Spallart-Almaras model (𝝑)
3. 2-Equation model (2-transport equations)
 k-𝝐 model
 k-𝝎 model
 k-𝝎 – SST model
4. n-equation model (n transport models)
RANS-Reynolds Averaged Navier Stokes
Statistical Modelling

3.3. Two-Equation Turbulence models
Nowadays it has been the Two-equation turbulence models which have been prominent
and trendy models for a wide range of engineering applications in the field of research
and analysis. These models contribute the independent transport equations for both the
turbulent length scales with the stipulation of providing two variables completing the
two-equation models. This encourages the engineers to apply them in various flow
scenarios as no additional information is necessary to use this model. The two-equation
model is however limited to some flows for which the fundamental assumptions are not
suited. The fundamental assumption includes the assumption that the scales of
turbulence are proportional to the scales of the mean flow hence, there will be some
percentage of error for these two-equation models when applied to the non-equilibrium
flows. Some of the two-equation models hold good near the wall like the low Reynolds
number models and few are compelling for the flow outside the inner region of the
boundary layer for instance the high Reynolds number models. However, two-equation
models are very popular and yield results well within the engineering accuracy when
utilized appropriately.
The two-equation models will have one equation for the kinetic energy and other
equation is based on the two additional variables and . The variable is defined as
turbulent dissipation term and which is defined as rate at which the turbulent kinetic
energy (TKE) is dissipated or specific dissipation rate. These two additional variables
are related to each other and also to the length scale which is also been associated with
the zero-equation models and one-equation models (Kempf, 2014). The mathematical
expression for specific dissipation rate in terms of the turbulent dissipation term and
length scale l is given as follows,
⁄
(17)
Where, c is a constant and is the characteristic length scale.

3.3.1. Turbulence model
The Turbulence model is one of the most commonly utilized simulation
techniques in analysis of a fluid flow. The Turbulence model contains one
equation for Turbulent kinetic energy defined as the mean kinetic energy per unit mass
which is associated with the eddies in a turbulent flow and the second equation for
which is the turbulent dissipation making it a two equation model (Kempf, 2014).
Mathematically the turbulent kinetic energy can be written as
̅̅̅̅̅̅̅ (18)
The turbulent dissipation is defined as the rate at which the turbulence kinetic energy is
converted into thermal internal energy. Mathematically it is given as
⁄
(√ ) (19)
With being the turbulence Reynolds number which is a dimensionless quantity is
given by
√
. It is assumed that the ratio between the Reynolds stress and mean
flow rate of deformation is same in all directions. For a standard turbulence model
the transport equation for turbulent Kinetic Energy is given by
̅ [ ( ) ] (20)
The turbulent dissipation is given by,
̅ * ( ) + (21)

3.3.2. Turbulence model
turbulence model is another popular two-equation model. In contrast to the
turbulence model which solves for the turbulent dissipation rate with , the
turbulence model solves only for the rate at which the dissipation occurs. Similar to
the model there are two equations out of which one equation is for kinetic energy
k and the second equation is for the specific dissipation rate .The model reduces
the turbulent length scale automatically and has high accuracy in predicting the flows
near the wall, however the flow away from the wall is more accurate in model.
Mathematically the relation between the specific dissipation rates with the dissipation
rate is given as,
(22)
Where the coefficient of molecular viscosity and the eddy viscosity is is calculated
with an expression .
. For a standard turbulence model the transport equation for turbulent kinetic
energy (k) is given by
̅ * + (23)
The transport equation for specific dissipation ( is given by
̅ * + (24)
Where the model constants are given as:

3.3.3. SST-Turbulence Model
In practice, the turbulence model is generally more accurate in shear type flows
and is well behaved in the far field (away from the walls). In contrast to the
turbulence model, turbulence model is more accurate and much more
numerically stable in the wall region. The Shear Stress Turbulence model (SST-Model)
is a combination and models and hence, behave better in the far field and
also yields better results near the wall region.
The SST formulation switches to the behavioural stream and avoids the
complication that arises in the model. By using SST and model one can get
better results in the pressure gradient and separating flow. The SST model
produces a bit too large turbulence levels in the regions with large normal strains and
acceleration occurs. This tendency is much less produced in the normal model.
3.4. Boundary Layer Approximation
The Newtonian fluids can be described sufficiently with the aid of the Navier-Stokes
equations which appear in both hydrodynamics and also in aerodynamics. As discussed
in the previous chapters, finding solutions for these equations are tedious processes
through computational means despite supercomputers are available these days. However,
these equations in large parts of the flow domain contains terms that can be neglected.
Furthermore, this allows solving the equations with reduced efforts by simplifications.
Viscous equations are of high importance to be solved near the boundary layer as they
examine the viscous shear stresses near the wall, however non-viscous equations can be
utilized for the flows away from the boundary layer (Veldman, 2012).
It is necessary to derive equations near the boundary layer and wakes which describe the
flow in shear layers. For this considering Navier-Stokes equation is the fundamental step
for a steady, incompressible and two-dimensional flow where the density is assumed
to be constant. These equations are formulated in the Cartesian co-ordinate system ( )
having velocity components as ( ) corresponding to the Cartesian system.

Furthermore, it is assumed that the co-ordinate coincides with the solid boundary. The
axis corresponds to the boundary layer thickness (Gröner, 2014) (Kempf, 2014).
The equations of motion for a steady state 2-D incompressible flow are given as:
(25)
( ) (26)
( ) (27)
For a solid surface the velocity satisfies ( ) = 0, the second condition being
Similarly, for a viscous flow we have at a solid surface.
3.4.1. Wall function
The wall function is a dimensionless wall distance which governs the production of
kinetic energy. The kinetic energy is too high if value is more than 100 which leads to
unrealistic pressure drop and generation of swirl in the flow, which in reality does not
exist. Hence to get more realistic results it is important to know the range for different
turbulence models. In general refers to the mesh size near the wall to analyse the flow
behaviour of the fluid.
Figure 3.4 shows the velocity profiles with the in the x- axis and along the y-axis.
The three important zones which is affected by viscosity namely:
 Viscous sub-layer (
 Buffer layer (
 Log-law region
The above mentioned regions come under the inner layer and have specific values
(Salim & Cheah, 2009)

Figure 3.4: Velocity profiles subdivisions of near wall region (Salim & Cheah, 2009)
Viscous sub-layer:
Near the wall regions of the solid surface the fluid is nearly stationary and the turbulent
eddies must also occur close to the wall. Here the fluid very close to the wall is
dominated by viscous shear in the absence of turbulent shear effects (Salim & Cheah,
2009). Furthermore, it can be assumed that the shear stress is almost equal to the wall
shear stress throughout the viscous sub-layer. This gives a fluid layer which is adjacent
to the wall to have linear relation given as,
(28)
√ (29)
Where, is the shear velocity, is the wall shear stress with fluid density with a
constant . Hence from the above relation, the viscous sub-layer is also called as linear
sub-layer. The lie less than 5 and for SST model should lie below 1 with fine grid
density for reliable results of fluid flow.

Buffer layer:
In the buffer layer the values lie between 5 and 30. For the most popular model
the should be well below 30 wall units and are most desirable for wall functions
(Salim & Cheah, 2009). From Figure 3.4 it is found that before 11 wall units the linear
wall approximation is more precise however, after 11 wall units the logarithmic
approximations are used although neither give accurate values at 11 wall unit.In the
buffer layer we have the relation:
(30)
(31)
Log-law region:
Log-law region is one which exists after the buffer layer region where both the turbulent
effects and the viscous effects are equally important. In this region the ranges
between 30 to 500 , where the shear stress is assumed to be constant and equal to the
wall shear stress which varies gradually away from the wall.
(32)
Here the relationship between and is logarithmic and is given in the form of log-
law as stated in Equation (32) and the layer where takes values ranging between 30
and 500 is called as log-law layer.
3.5. The Governing Equations
It is clear from the concept explained in the previous chapter about homogeneous
nucleation that the condensing steam occurs at significant levels of supercooling when
there endures a very high fluid expansion rates. Although there endures a heterogeneous
droplet formation in the active flow of the fluid, the required droplet surface area for a
reversion to the equilibrium can be achieved by homogeneous nucleation (Gerber &
Kermani, 2003). The classical nucleation theory discussed in the previous chapters helps

the necessity for the modelling of condensing flows with the aid of its properties at
supercooled conditions.
3.5.1. Conservation of mass
The conservation of mass for a vapour phase is expressed with a mass source which
reflects the condensation and vaporization process present in the phase is given as
(33)
In the above expression the gas (vapour) density is and represents the velocity
component in j direction. in Equation Error! Reference source not found.
corresponds to evaporation case as it is positive , consequently is negative for a
condensation process as the gas phase source term is equal and opposite to that of liquid
phase.
3.5.2. Conservation of momentum
The conservation of momentum equations are based on the Reynolds Averaged Navier
Stokes Equation (RANS) for a 3-D turbulent flow and hence, require a turbulence model
to represent the turbulent Reynold’s stress terms. The popular turbulent model is
used as it can be easily adapted for investigation. The eddy viscosity introduces the
influence of turbulence, which in addition with the molecular viscosity helps to obtain an
effective viscosity (Gerber & Kermani, 2003). The momentum equation is thus
given as:
( ) (33)
In the above equation is the source term and contains more smaller terms from the
Reynolds Stress tensor defined in equation (12). In general for

* ( )+ (34)
And the second source term serves as the interphase momentum transfer given as.
(35)
In the above equation is the mass source for the liquid scalar equation having
units . The scalar quantity is obtained from droplet growth rate (Gerber &
Kermani, 2003).
3.5.3. Conservation of energy
The conservation of energy equation consists of source terms one representing the
viscous dissipation ( ) and the other source term which represents the useful viscous
work ( , having dependent variable called the gas total enthalpy ( and is given as;
( ) (37)
Here the total enthalpy is defined as ⁄ and is the temperature
of the gas having an effective thermal conductivity . The total viscous stress energy
contributed by viscous work and viscous dissipation is given as;
( ) (36)
Where, is viscous stress tensor.
is a source term which contains the interphase heat transfer between the gas and
liquid. It can be described by defining a scalar quantity , which is obtained from
droplet growth rate. Thus the vapour energy can be given as:
(37)

Where is the liquid droplet enthalpy (Gerber & Kermani, 2003).
3.5.4. Conservation equations for liquid phase
The conservation equations for the liquid phase are given with the aid of classical
nucleation theory. The conservation of mass fraction for the liquid droplets and the
conservation for the number of droplets N are expressed as following (Blondel, et al.,
2013):
(38)
(39)
Where J is the nucleation rate which is given by the classical nucleation theory and with
C as a non-isothermal correction factor is expressed as;
√ ( ) (40)
In equation (38) and are the interfacial exchange terms which are mathematically
given as:
(41)
(42)
Where, is the droplet growth rate which is defined in equation (6) from droplet growth
theory. Here , which is created due to the nucleation process, is the source term and
is the mass condensation rate of all droplets per unit volume of a multiphase mixture for
homogeneous condensation (Lamanna, 2000).

3.6. Condensation modelling in ANSYS CFX
Different available models for modelling the condensation phenomenon in ANSYS CFX
are discussed in this section. Modelling of multiphase flows is the most important fluid
simulation as the process involves modelling of two or more gases on a microscopic
level. In such flow field it is essential to solve by calculating the velocity and
temperature for each fluid. Here the two phases interact with each other resulting in mass
and heat transfer between the two phases.
A number of approaches are available in ANSYS CFX to model the condensation
phenomenon. They are listed in categories below.
 Wall condensation model
 Equilibrium phase change model
 Droplet condensation model
3.6.1. Wall condensation model
The function of the wall condensation model in ANSYS CFX is that it models
condensation as a mass sink, thereby removing the mass that enters the liquid film from
the fluid domain, however the flow inside the liquid film is not modelled. This model
permits only one condensable component and the change in heat transfer resistance
which is induced by the liquid wall film is considered to be negligible and are not
explicitly modelled (CFX Theory Guide, 2015).
They are further subdivided into two parts based on the turbulent boundary layer
treatment in terms of mass flux at the surface
 Laminar boundary layer model
 Turbulent boundary layer model
The condensation mass flux treatment for laminar flow is as shown in the Equation (43)
( ) (43)

Where the mass transfer coefficient is given as X is the molar fraction and is the
height of the boundary layer. The mass transfer coefficient is calculated in Equation (44)
(44)
Where and is the molecular weight of the condensable B and molecular weight
of the mixture of condensable and non-condensable. Thermal equilibrium is assumed at
the interface when considering for the interface and liquid film. This implies that the
saturation pressure at the given temperature is equal to the partial pressure of the vapour
(CFX Theory Guide, 2015).For turbulent boundary layer the condensable mass flux is
given in Equation (45).
(45)
Where is wall multiplier which is based on the turbulent wall function, is the
mass fraction of the condensable component near the wall and denotes the mass
fraction of the condensable component at the wall.
There is generation of latent heat during condensation and this latent heat is released into
the solid boundary. The effect of this latent heat can be neglected if the wall is
isothermal in nature. In Turbulent boundary layer model, the condensation along the
surface of the solid is treated as a heat source. Using the Equation (45) for condensable
mass flux the heat release can be expressed as
(46)
Here H is the latent heat release during condensation.
3.6.2. Equilibrium phase change model
The equilibrium phase change model is a single fluid, multicomponent model. In this
model thermal equilibrium between the two phases for example water and vapour is
assumed. This model is used for modelling condensing vapours such as wet steams or
refrigerants with small liquid mass fractions. As soon as the saturation temperature for

the given static pressure has been obtained for the water vapour in the flow then it results
in condensation (CFX Theory Guide, 2015).
Figure 3.5 : Temperature Entropy diagram for liquid vapour mixture (CFX Theory Guide, 2015).
The above Figure 3.5 shows two pressure lines of which one is high pressure and the
other being low pressure passes through the saturation region having constant pressure
and temperature. At the subcooled region the entropy is lower than the saturation
entropy and also the mixture is all liquid. However in the superheated region the entropy
is higher than the saturated entropy of the vapour and the mixture is all vapour. In the
saturation region of the dome the mixture is both liquid and vapour hence termed as wet
vapour.
To determine the quality of the flow the ANSYS solver uses the lever rule which is
given as
(47)

Here, is the static enthalpy mixture, and are the saturation
enthalpies of vapour and liquid respectively as a function of pressure. The quality of the
flow can be determined as following:
 When X < 0, then the mixture is 100% subcooled liquid and hence the liquid
properties are selected.
 When X > 0, then the mixture is 100% superheated vapour and hence the vapour
properties are selected.
 When 0 ≤ X ≤ 1, then the mixture contains both liquid and vapour.
A single temperature field can be solved for the mixture since local thermodynamic
equilibrium is assumed. A single velocity field is solved for the mixture as the flow is
homogeneous, thus reducing the computational time needed to obtain the solutions (CFX
Theory Guide, 2015).
3.6.3. Droplet condensation model
Droplet condensation model requires a finite time to reach equilibrium condition. The
droplet condensation model includes the losses that occur due to thermodynamic
irreversibility. This model can be used as homogeneous model or as an inhomogeneous
model depending on the configuration set by the user in ANSYS CFX. Unlike the
equilibrium phase change model additional transport equations have to be solved for the
droplet number and volume fractions for all phases (CFX Theory Guide, 2015).
The droplet condensation model is used where there is rapid pressure reduction in the
flow medium leading to nucleation and droplet formation. A nucleation bulk tension
factor is to be selected as this factor scales the bulk surface tension values. It is
recommended to set the NBTF value to 1.0 if the static pressure is below 1 bar and
furthermore, IAPWS database is used for the water properties. These values can be later
altered to match the experimental results (CFX Theory Guide, 2015).
Depending on the size of the droplet the droplet condensation model is further divided
into two parts namely:

 Small droplets phase change model
 Thermal phase change model
Small droplet phase change model is recommended for water droplets which are less
than 1µm in diameter however, it can be used for droplets of all sizes. To determine the
heat and mass transfer in a fluid medium the droplet size is a prime factor. In the droplet
phase change model the effect of Knudsen number and Nusselt number is considered for
calculating the heat and mass transfer at the interface of the droplets. The relation for
Knudsen and Nusselt number can be found in the chapter Droplet Growth Theory.
3.7. Character and Structure of IAPWS-IF97
IAPWS Industrial Fluid 1997 is an industrial standard having Thermodynamic
Properties of Water and Steam in short abbreviated as IAPWS-IF97. This industrial fluid
significantly improves both accuracy and also the calculation speed of all
thermodynamic properties. This section portrays the general information about the
character and structure of the industrial formulation IAPWS-IF97 which includes the
entire range of its validity and also some remarks about the quality of IAPWS-IF97
concerning its accuracy and consistency all along the boundary regions in a fluid flow
(Wagner & Kruse, 1998).
The industrial Formulation IAPWS-IF97 consists of some set of equations for different
regions:
1. Subcooled water
2. Supercritical water/steam
3. Superheated Steam
4. Saturation data
5. High Temperature steam
Covering the following range of validity:
 0 C T 800 C , p 1000 bar (100 MPa)
 800 C T 2000 C , p 500 bar (50 MPa)

Figure 3.6 : Regions and equations of IAPWS-IF97 (Wagner & Kruse, 1998).
Figure 3.6 shows the five regions into which the entire range of validity of IAPWS-IF97
is divided. The regions 1 and 2 are covered by the fundamental equation for specific
Gibbs free energy . Furthermore, region 4 by a fundamental equation of specific
Helmholtz free energy F( . The saturation curve corresponding to region 4 is given
by saturation-pressure equations . Region 5 is the high temperature region and is
also covered by a region equation. Together all these five equations are called as
basic equations.
Where,
Specific Gibbs free energy: (48)
Specific Helmholtz free energy: (49)
In ANSYS CFX, the properties of equation of state are represented by the generation of
table as shown in Figure 3.7, which will be evaluated efficiently in a CFD calculation.
These IAPWS tables are defined in terms of pressure and temperature as they are a
function of enthalpy and entropy which are also evaluated. From the above figure region
4 involves the evaluation of only saturation data which uses pressure and temperature.

Figure 3.7: Table Generation in ANSYS CFX for IAPWS97

Chapter 4
This chapter deals with the validation of two high pressure nozzles based on the
experimental results which were conducted by Gyarmathy (2005). Modelling of the flow
is made with the assistance of ANSYS CFX thereby comparing numerical results with
that of the already existing experimental results.
4. Experiments of High pressure Nozzles & Setup of Numerical
Simulation
The validation of 2-D Laval nozzles is based on the experiments conducted for a high
pressure nozzle by Gyarmathy (2005). The experiments were conducted for nozzles
which were designed for different expansion rates ranging from 10,000 to 200,000
having the pressure ratios between 0.5 and 5MPa are as shown in Table 1 below.
Nozzle Code Expansion Rate
̇
Effective length Throat height Width
B
2/M 10,000 30+100 10 10
2/B 10,000 30+100 10 20
4/B 50,000 20+30 4 20
5/B 100,000 20+70 2 20
6/B 200,000 10+50 2 20
Table 1: Specifications of supersonic nozzles used in (Gyarmathy, 2005)
The Gyarmathy experiments were evaluated with the IAPWS-IF97 steam tables. The
principle objective in this thesis is to validate the numerical model with different
expansion rates. 2/M and 5/B nozzles from Gyarmathy (2005) are selected for
validation. The nozzle 2/M had a lower expansion rate due to its overall length with a
considerably high throat height, furthermore 5/B nozzle with 10 times more expansion
rate in contrast to the 2/M nozzle with a short throat height of 2mm was utilized.

Figure 4.1 Nozzle shapes used in (Gyarmathy, 2005)
The complete experimental assembly and the specifications of the apparatus used for the
experiments can be found in Gyarmathy (2005). The pressure and mean droplet size
surveys, and ̅ respectively were obtained by axially moving the nozzle with the
aid of a centrally positioned rod which was coupled with a shaft driven by a high-
precision gear. As show in Figure 4.1 like in the 2/M nozzle all other nozzles were
provided with a static pressure taps in the upper slot of the wall to measure the pressure
in the nozzle and sapphire windows having Ø 9mm facing each other, helped for the
measurement of droplet diameter. From the experiments conducted it was analyzed that
the uncertainties were the greatest with 2/B nozzle however, most reliable results were
found for nozzles 2/M and 4/B. The experimental results for 2/M case having run
number 40-E and for the 5/B nozzle with 23-C run number were used in this work. The
inlet conditions from the experimentation are tabulated in Table 2.

Table 2: Specifications of Validating Nozzles
Nozzle Run
number
Stagnation Pressure
(bar)
Stagnation Temperature
( )
2/M 40-E 108.88 346.08
5/B 23-C 100.70 347.55
Figure 4.2: Nozzle 2/M Experimental results (Gyarmathy, 2005)
Figure 4.3: Nozzle 5/B Experimental results (Gyarmathy, 2005)
0.0E+0
2.0E-8
4.0E-8
6.0E-8
8.0E-8
1.0E-7
1.2E-7
1.4E-7
1.6E-7
1.8E-7
2.0E-7
0
0.2
0.4
0.6
0.8
1
-40 10 60 110
FogDropletmeanradiusr/m
Non-Dimension,Staticpressure,p/po
Axial Coordinate 𝜉 /10-3 m
2M_40E_Dry Superheated
2M_40E_Pressure
2M_40E_Droplet
0.0E+0
8.0E-9
1.6E-8
2.4E-8
3.2E-8
4.0E-8
4.8E-8
5.6E-8
6.4E-8
7.2E-8
8.0E-8
0
0.2
0.4
0.6
0.8
1
-20 0 20 40 60
FogDropletmeanradiusr/m
Non-Dimension,Staticpressure,p/po
Axial Coordinate 𝜉 /10-3 m
5B_23C_Dry Superheated
5B_23C_Pressure
5B_23C_Droplet

Figure 4.2 and Figure 4.3 illustrate the experimental results for pressure and droplet
profiles obtained from (Gyarmathy, 2005). As it can be seen for the 5/B nozzle pressure
plot in Figure 4.3 the portion where there is occurrence of a pressure bump for 23C run
has been enlarged for better understanding.
4.1. Numerical setup and mesh generation
The two-dimensional Laval nozzle numerical flow simulations were performed with
ANSYS CFX. In the present work different numerical setup along with the mesh
generation setup are carried out in this chapter. The total pressure and total temperatures
were set at the inlet. The boundary condition for the two nozzles 2/M and 5/B are shown
in the table below .
Table 3: Boundary conditions for 2/M and 5/B nozzle
Entity 2/M : run number 40E 5/B : run number 23C
Condition Non-Equilibrium Non-Equilibrium
Turbulence Model model model
Inlet Subsonic
Total Temperature : 346.08
Total Pressure : 108.88 bar
Subsonic
Total Temperature : 347.55
Total Pressure : 100.70 bar
Outlet Supersonic Supersonic
Symmetry Symmetry Symmetry
Upper Wall Boundary type : Wall
Condition : No Slip
Boundary type : Wall
Condition : No Slip
Nozzle Boundary type : Wall
Condition : No Slip
Boundary type : Wall
Condition : No Slip
NBTF 1.0 (Default) 1.0 (Default)
Nusselt Correlation (Default) (Default)

Rakshith B Hombegowda master thesis final

Rakshith B Hombegowda master thesis final

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