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

https://de.wikipedia.org/wiki/Universit%C3%A4t

https://www.uni-due.de/verwaltung/orientierung_mf.php

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 : ____________________________

https://de.wikipedia.org/wiki/Universit%C3%A4t

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. Results 79

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

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/po

Axial Coordinate 𝜉 /10-3 m

2M_40E_Dry Superheated2M_40E_Pressure2M_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

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5B_23C_Dry Superheated5B_23C_Pressure5B_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)



Figure 4.4: 2/M nozzle Geometry with Boundaries

The above Figure 4.4 shows the 2/M nozzle with the boundary conditions which is also

similar for 5/B nozzle. For defining the mesh it was made sure that the corner angles

were above 25° with the expansion ratios well below 5 and aspect ratio below 1000.

Furthermore, a minimum of 15 layers is made sure to exist near the wall region with a

first layer thickness of having a growth rate of 1.3. Meshes of Laval nozzle is

as shown in Figure 4.5 below.

Figure 4.5: Meshing for 5/B Laval Nozzle

SYMMETRY

INLE

T

OU

TLET

UPPER WALL SYMMETRY

NOZZLE



4.1.1. Calculation of Efficiency

Calculating the efficiency of a Laval nozzle is the principle objective of this thesis work.

The area average efficiency can be defined as the ratio of square of the area averaged

velocity at outlet to the square of the isentropic velocity at outlet.

Mathematically it is given as,

(50)

√ (51)

Where, is the isentropic velocity at the outlet of the nozzle.

Furthermore, cannot directly be calculated from ANSYS CFX and hence it is

necessary to find enthalpy difference between inlet and the static enthalpy at outlet given

as.

(52)

(53)

Here refers to the isentropic enthalpy at outlet which is a function of area averaged

outlet pressure and entropy at inlet. Hence the area average efficiency can be

simplified as:

(54)

4.1.2. Calculation of Nusselt Number

Nusselt number correlation is used to introduce the heat transfer interphase in the

mixture model. Nusselt number is a dimensionless number which is often used to

express the heat transfer coefficient. This dimensionless number can be directly

specified under the two resistance model which is applicable to both particle and mixture

models (CFX Theory Guide, 2015). In the present case the Nusselt correlation used is



shown in Equation (55) below which is taken from the paper (Moore & Sieverding,

1976).

(55)

And accordingly a dimensionless Knudsen number which is given by the relation in

Equation (56)

√

(56)

Prandtl number is another dimensionless number expressed in Equation (57)

(57)

Where, is empirical correction factor set to 0 or 0.5

is dynamic viscosity of gas

is Temperature of gas

R is the real gas constant (461.5

is the pressure of gas

is the particle diameter

is the specific heat of gas

is the gas thermal conductivity

The default Nusselt correlation used in ANSYS CFX is

(58)

Where c is an empirical factor set to be 3.18 (CFX Theory Guide, 2015) also Kn is taken

from Equation (56)

Parameter investigation 55


Chapter 5

5. Results of 2/M and 5/B Nozzle

In this chapter both the numerical verification and validation of 2/M and 5/B nozzle

from Gyarmathy (2005) is performed. The mesh density study, superheated case

analysis, wall refinement and the 2D effects with single precision have been numerically

verified. These verified results are considered for the sensitivity analysis using the NBTF

and Nusselt number correlation are performed after the validation of SST turbulence

model.

5.1. Numerical verification of 2/M and 5/B Nozzle

5.1.1. Mesh Density Study

The mesh density study of 2/M and 5/B nozzle is obtained by comparing the static

pressure and droplet diameter results from numerical simulations made in ANSYS CFX

with that of the experimental data reported by Gyarmathy (2005)as shown in Figure 5.1

and Figure 5.2. It was necessary to analyze the right mesh which is considered for future

numerical verification and validation of 2/M and 5/B nozzle.

5.1.1.1. Pressure and Droplet Plot for 2/M and 5/B nozzles

The non-dimensional static pressure which is obtained by the pressure normalized by

total inlet pressure is considered as a function of the axial co-ordinate. The inlet is

located at = -30 mm with the throat at = 0 mm and outlet at = 100mm for the 2/M

nozzle and = -20 mm with the throat at = 0 mm and outlet at = 70mm for 5/B

nozzle respectively, however the numerical simulation results have been measured from

= -4 mm with the throat at = 0 mm and outlet at = 14 mm for the pressure profile in

5/B nozzle. For both nozzles the pressure drops continuously and when = 10 mm there

is slight increase in pressure before it again continues to drop. This increase in pressure

is often called as the pressure bump which is due to the condensation process after the

throat of the nozzle.



From close observation in the pressure plots in both Figure 5.1 and Figure 5.2 , there

exists an offset in the results obtained from the ANSYS CFX simulation and also the

pressure bump is captured in ANSYS CFX. There is no noticeable difference in the

pressure plots as the meshes overlap on each other in both 2/M and 5/B nozzle. In nozzle

5/B the pressure bump has been enlarged. The pressure bump in 2/M nozzle is captured

at a static pressure value of 0.48 where as in the 5/B nozzle it is captured at a static

pressure of 0.4 , however the droplet mean radius of 2/M nozzle is observed to be 1.43E-

8m in contrast to 5.6E-8m observed in the 5/B nozzle. Furthermore, it was clear that the

results obtained from the medium mesh and fine mesh for both the nozzle overlapped

with each other compared to the coarse mesh having same droplet mean radius and

pressure bumps at identical location. It was necessary to compare the mass flow and

efficiency influence for the different mesh densities to consider the right mesh for further

simulations.

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).

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

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Coarse Mesh

Medium Mesh

Fine Mesh

Droplet Coarse Mesh

Droplet Medium Mesh

Droplet Fine Mesh



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).

5.1.1.2. Mass Flow and Efficiency for 2/M and 5/B nozzles

Table 4: 2/M Mesh density efficiency comparison.

2/M_High Pressure

Nozzle

Mass flow

Area Average

Efficiency

Mesh Cell Divisions in

XYZ direction

Coarse mesh (1) 1.55 96.830 % 310X73X5

Medium mesh (2) 1.55 96.848 % 620X146X5

Fine mesh (4) 1.55 96.849 % 1240X292X5

Table 5: 5/B Mesh density efficiency comparison

5/B_High Pressure

Nozzle

Mass flow

Area Average

Efficiency

Mesh Cell Divisions in

XYZ direction

Coarse mesh (1) 0.028 93.941 % 390X50X3

Medium mesh (2) 0.028 93.957 % 780X100X3

Fine mesh (4) 0.028 93.959 % 1560X200X3

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

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/po

Axial Coordinate 𝜉 /m m

Coarse MeshMedium MeshFine MeshDroplet Coarse MeshDroplet Medium MeshDroplet Fine Mesh



Table 4 and Table 5 gives a detailed view for the mass flow and also the overall area

average efficiency for different mesh sizes. Medium mesh with 620X146X5 mesh

divisions in X,Y,Z direction respectively for 2/M nozzle had negligible difference in

mass flow, however the area averaged efficiency proved to be better than coarse mesh

and nearly equal to the fine mesh. Furthermore, in the 5/B nozzle too it was found that

the medium mesh having mesh cell divisions of 780X100X3 in X,Y,Z direction proved

to have better area averaged efficiency than the coarse and fine meshes. Hence for

further analysis medium mesh was selcted to be as the ideal case to find solution having

the optimum mesh elements which aids for faster convergence and better results than the

latter meshes.

5.1.2. Superheated case analysis for 2/M and 5/B Nozzles

Superheated Steam case have been conducted for both 2/M and 5/B nozzles to verify the

results with the experimental values from Gyarmathy (2005). Inlet temperatures of

505°C have been introduced with the inlet pressure values of 108.88 and 100.70 bar for

2/M and 5/B nozzles respectively to analyse the superheated steam case for both nozzles.

As we can see from Figure 5.3 and Figure 5.4 below it is evident that the superheated

case does not match the experimental results. There exists an offset in both the nozzles

and persists throughout the simulation. An offset value of 4mm along the axial

coordinate of 2/M nozzle and an offset value of 0.5mm for the 5/B nozzle along the axial

coordinate is observed. Furthermore, numerical verifications are made with wall

refinement and other sensitivity tests are conducted to find a better agreement for the

existing offset in both the nozzles,



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).

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).

0

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2M_40E_EXP

2M_40E_Dry_Superheated

Superheated Pressure

4mm

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5B_23C_EXP

5B 23C Dry Superheated

Superheated Pressure

0.5mm



5.1.3. Wall Refinement


Figure 5.5 and Figure 5.6 show the comparison between wall refinement and no wall

refinement for static pressure profiles and the droplet diameter profile obtained from

ANSYS CFX simulations along the 2/M and 5/B nozzle axis. As we can see from the

graph, there is negligible difference in the pressure plot in both the nozzles where the

results with wall refinement and without wall refinement overlap each other along the

axis of the nozzle. There exists an offset having a value of 4mm in the axial coordinate

of 2/M nozzle and persist throughout the simulation. The offset for the 5/B nozzle is

lesser compared to 2/M nozzle which is about 0.5mm in the axial coordinate. The figure

also shows the droplet size comparison with refined wall and with no wall refinement.

As we can see from the plot there is negligible difference on the influence of wall

refinement on the droplet diameter on both 2/M and 5/B nozzles.

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

0.0E+0

2.0E-8

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6.0E-8

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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.


2/M_High Pressure Nozzle Mass flow Area Average Efficiency

2/M_No Wall Refinement 1.557 99.96%

2/M_Wall Refined 1.548 97.36%

Table 6: 2/M Wall Refinement efficiency comparison

5/B_High Pressure Nozzle Mass flow ) Area Average Efficiency

5/B_No Wall Refinement 0.029 99.94%

5/B_Wall Refined 0.028 93.95%

Table 7: 5/B Wall Refinement efficiency comparison

0.0E+0

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Table 6 and Table 7 give a detailed view of the influence of wall refinement on the

efficiency and mass flow in 2/M and 5/B nozzles respectively. It was found that the

efficiency decreases with the introduction of no slip condition where the fluid flow will

be subjected to frictional losses near the wall due to the refined mesh for both the

nozzles. In contrast to the refined mesh the fluid flow will have efficiency of almost

100% for the mesh where there is no wall refinement meaning that there are no frictional

losses in the flow and hence a free slip condition. Also, the mass flow has decreased

with the wall refinement solely because of the no slip condition near the wall leading to

frictional losses.

5.1.4. 3-D Effect and Single Precision

It was important to check for simulations for 2/M and 5/B nozzles with 3-D models and

influence of double precision to single precision. The simulation with these parameters

are rather time consuming and impact of these in the current work had to be monitored

in order to save all the computational time and efforts that would involve in fluid

simulation.

5.1.4.1. 3-D Effect

The present work involves the simulation of a 2-D Laval nozzle where the opposite wall

and the symmetrical part of the geometry are considered to be both symmetric for 2/M

and 5/B nozzles so that the effect on extending to any width has no impact on the

simulation. However, it was essential to check for the 3-D effect with opposite wall

being considered as a wall instead of symmetry. Consideration of the 3-D effect increase

the computational time for simulation as there exists a no slip condition on the opposite

wall which will have high frictional losses. From the previous studies on the wall

refinement and without wall refinement it was found that there will be a noticeable loss

in efficiency to frictional losses at the wall however, it was found that considering the

opposite wall to be a wall or a symmetry boundary condition had no impact on the

efficiency , pressure or even the droplet diameter. Henceforth, for effective utilization of

time the 3-D effect was ignored by considering symmetry on both sides of the geometry

as it had no influence on the pressure or droplet plots.



5.1.4.2. Single Precision

Precision in fluid dynamics refers to how accurately the solver runs in order to give a

complete converged solution. Single precision runs will consume less time to give a

complete converged solution whereas, the double precision mode for ANSYS CFX

solver consumes time to yield a converged solution compared to single precision

method. It is important to check if there is an influence on the resulting converged

solution as the double precision takes a detailed evaluation of the flow field thereby

consuming more time and space during the analysis. In the present thesis work as the

geometry is not so complex it was found that the double precision had no impact on

achieving the main objectives. Furthermore the pressure and the droplet plot had no

impact with the use of double precision with respect to single precision and furthermore

for future computations single precisions were considered so as to get better fast

converged solutions. There was no noticeable influence of double precision on the

pressure plot, droplet plots and also the efficiency remained the same.

5.1.5. Discussion

The Numerical verification results presented in the previous section shows the numerical

setup implementation matches the experimental setup from Gyaramathy (2005). The

mesh density study was made and considering the efficiency difference it was found that

medium mesh was more suited for further simulations as the fine mesh had not much

difference in the efficiency and also proved to consume more computational time for

solving as it contained more number of nodes. ANSYS CFX was capable of capturing

the pressure bump and for the superheated case there was noticeable offset with respect

to the experimental results, however the mean droplet radius is in correct order of

magnitude in both 2/M and 5/B nozzles. Furthermore, the wall refinement analysis

yielded an offset of 4mm along the axial coordinate for the 2/M nozzle and about 0.5mm

along the axial coordinate for the 5/B nozzle which were the same even for the

superheated case. It was found that there is no influence of double precision and 3D

effect and hence for future calculations the symmetry side and the opposite wall of both



2/M and 5/B nozzles were considered to be symmetric in boundary condition having a

nozzle width of 0.5mm making it a 2D simulation with optimum number of nodes and

elements for faster convergence with computational time. For future validation, mesh

with a minimum of 15 layers is made sure to exist near the wall region having a first

layer thickness of with a growth rate of 1.3.

Sensitivity analysis by introducing NBTF, Nusselt correlation and Turbulence models

are made to check if there is a better agreement to the experimental solution and their

implications on pressure and droplet plots are presented in the following chapters for

both 2/M and 5/B nozzles.



5.2. Validation of 2/M and 5/B Nozzle

5.2.1. Turbulence model (SST vs )

It was necessary to validate the turbulence models to check their influence on efficiency,

pressure and droplet diameter which is the main objective. Necessary steps have been

taken during the simulation of SST model to keep a check on the mesh density as it

influences the value near the wall. The value for the SST model should be less

than 1 in contrast to the value where the value has to be below 30.


The pressure profile obtained from numerical simulations is shown in Figure 5.7 and

Figure 5.8 of nozzles 2/M and 5/B respectively. It can be clearly seen that there is no

noticeable change in the pressure plots obtained from the k model setup and for SST

turbulence model setup in both the nozzles. For better understanding of overlapping the

x and y co-ordinate for SST model are shortened to show the overlap. From previous

simulation results it was found that for non-equilibrium, medium mesh with wall

refinement yield reliable computational results. Hence, all the other numerical

parameters were kept identical considering an identical table range to obtain better

convergence and proper validation on both the turbulence model. Like the results

obtained in previous section with the introduction of SST turbulence model, there persist

an offset throughout the flow which is also followed by the pressure bump at the exact

same position as in model which is in the coordinates (5, 0.6) mm for the 2/M

nozzle and (3, 0.4) mm for 5/B nozzle in (x, y) coordinate respectively. This proved that

the turbulence model had no influence on the pressure profile. The figure also shows the

droplet diameter comparison for SST model in contrast to model. Here with the

default settings for a non-equilibrium case with medium mesh which is wall refined is

selected from previous studies to get better results.

From the figures it can be seen that there is less than 1% that is 0.75% difference in the

droplet size for 2/M nozzle and 0.8% for 5/B nozzle, however the growth of the droplet

size along the axial co-ordinate is simultaneous in both the turbulence models. In SST



model it can be seen that as there is condensation occurrence the droplet size increases

faintly in comparison to model and has a droplet diameter slightly below 1.6e-7 m.

However there still exists an offset to the experimental results.

Figure 5.7: Comparison droplet and pressure profiles of and SST turbulence models for 2/M

Nozzle (NBTF 1.0)

Figure 5.8: Comparison droplet and pressure profiles of and SST turbulence models for 5/B

Nozzle (NBTF 1.0)

0.0E+0

2.0E-8

4.0E-8

6.0E-8

8.0E-8

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1.2E-7

1.4E-7

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Axial Coordinate in mm




2/M Nozzle Model Efficiency Difference Mass flow )

96.83% - 0.077

SST 96.57% 0.26% 0.077

Table 8: Comparison of efficiency and mass flow at outlet for and SST models in 2/M Nozzle.

5/B nozzleModel Efficiency Difference Mass flow )

93.95% - 0.028

SST 93.69% 0.27% 0.028

Table 9: Comparison of efficiency and mass flow at outlet for and SST models in 2/M Nozzle.

Table 8 and Table 9 gives a clear notion on the influence of Turbulence models on the

area average efficiency and mass flow at outlet. From the table above it is evident that

the efficiency in is 0.26% better than the efficiency obtained in the SST model for

2/M nozzle and 0.27% for 5/B nozzle. From the pressure profiles it was found that both

the turbulence models have identical plots however they differ in the droplet plots.

Although the droplet size begins to grow after condensation at the same time the two

models end up deviating as the droplet sizes increase along the axial co-ordinate.

5.2.2. NBTF Correction

NBTF plays a very important role for condensing flows in a steam turbine. NBTF is an

abbreviation for Nucleation Bulk Tension Factor which is used to investigate the

correction of the surface tension in water droplet formed during condensation. In the

present work simulations were performed by changing the NBTF values from 0.9, 1 and

1.1 respectively. The sensitivity analysis with the introduction of NBTF along with their

influence on pressure and droplet size with respect to pressure and droplet plots are

shown in the Figure 5.9 and Figure 5.10 for both 2/M and 5/B nozzles.



5.2.2.1. Pressure and Droplet Plot for 2/M and 5/B nozzle

Both the Figure 5.9 and Figure 5.10 shows the influence of different NBTF values on the

pressure profile for the selected mesh density with respect to the experimental results. It

was found that NBTF 1.1 prolonged the pressure bump for about 0.025 units in the y

axis and 0.5 units along the x axis in comparison to NBTF 1 and the same values of shift

in pressure bump was maintained for NBTF 1 with respect to 0.9 in 2/M nozzle.

Furthermore, in the 5/B nozzle there was a difference of (0.1, 0.02) units in the (x, y)

coordinates of different NBTF values. Although the introduction of NBTF has not

brought a change in the offset of the resulting simulation pressure plot in both the

nozzles, it certainly has an influence on condensation effect where there NBTF is

sensitive to wetness. The figures below give a clear picture on the influence of NBTF on

droplet size. It can be seen that NBTF which has a value 1.0 has nearly the same droplet

diameter as 0.9 and 1.1 respectively. As we can see along the axial co-ordinate and

compare it with the pressure profile the NBTF 0.9 starts early followed by NBTF 1 and

1.1. However there is negligible difference in the droplet diameters with NBTF values of

0.9, 1.0 and 1.1 for both 2/M and 5/B nozzles.



Figure 5.9: NBTF influence on 2/M nozzle

Figure 5.10: NBTF influence on 5/B nozzle

0.0E+0

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4.0E-8

6.0E-8

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5.2.2.2. NBTF Correction for 2/M and 5/B nozzle

Table 10: NBTF Efficiency for 2/M nozzle

NBTF 2/M Nozzle Efficiency Difference Mass flow )

0.9 96.86% 0.03% 0.077

1.0 96.83% - 0.077

1.1 96.82% -0.01% 0.077

Table 11: NBTF Efficiency for 5/B nozzle

NBTF 5/B Nozzle Efficiency Difference Mass flow )

0.9 94.04% 0.01% 0.028

1.0 93.94% - 0.028

1.1 93.83% -0.012% 0.028

Table 10 and Table 11 give the detailed view of the influence of NBTF on efficiency and

mass flow for 2/M and 5/B nozzles respectively. As we can see from the table it is

evident that as the NBTF value increases the efficiency decreases slightly. The

difference in efficiency in percentage is calculated for the default setup where the NBTF

value is 1.0 and as we can see the mass flow rate remains constant for all NBTF values.

The introduction of NBTF has however has not brought a change in the offset persisting

in the pressure plot for both nozzles, henceforth for further numerical simulations NBTF

1.0 is considered as it does not make a difference on the efficiency and droplet diameter

with respect to the experimental results although there is an influence on the pressure

plot for 2/M nozzle and 5/B Nozzle with an offset continuing as 4mm in 2/M nozzle and

0.5mm for the 5/B nozzle along the axial coordinates.

5.2.3. Nusselt Number Correlations

Nusselt number correlation is used to introduce the heat transfer interphase in the

mixture model. In the present work there are three types of Nusselt numbers which are

implemented namely:

Default Nusselt Correlation



Nusselt correlation with = 0

Nusselt Correlation with = 0.5

The detailed formula for the implementation of Nusselt correlation for sensitivity

analysis can be found in Chapter 4.1.2 on page 53.

5.2.3.1. Pressure and Droplet Plot for 2/M and 5/B nozzle

Figure 5.11and Figure 5.12 show the influence of different Nusselt number correlation

on the pressure profile for the selected mesh density and NBTF value of 1.0 with respect

to the experimental results for both 2/M and 5/B nozzles. The Nusselt empirical factor

was set to 0.5 from the default value described in Equation (58) and it was found

that the droplet diameter size was reduced in both the nozzles which was more evident in

5/B nozzle where there existed a mean radius droplet difference of 0.3E-8m for default

and 0.5. Same difference existed for 0.0 with respect to 0.5 in 5/B nozzle.

In contrast to the variation of NBTF in pressure profile it looks clear that the

introduction of Nusselt correlation has no influence on the pressure profile for both the

nozzles. The figure also gives an idea on the influence of Nusselt correlation factor on

droplet size. The Nusselt empirical correlation factor was set to 0.5 from the default

value and it was found that the droplet diameter size was reduced which can be noticed

in the 5/B nozzle. Furthermore, the factor was changed to 0 the droplet size

increased slightly (0.3E-8m for 5/B nozzle). There is no much difference with the

introduction of Nusselt number on the droplet plot for 2/M nozzle however slight

difference can be figured out in 5/B nozzle. The pressure plot also remains unaffected by

the introduction of Nusselt correlation.



Figure 5.11: Nusselt number influence on 2/M nozzle (for NBTF=1)

Figure 5.12: Nusselt number influence on 5/B nozzle (for NBTF=1)

0.0E+0

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5.2.3.2. Nusselt Correlation for 2/M and 5/B nozzle

Table 12 Nusselt Efficiency for 2/M nozzle

for 2/M Nozzle NBTF Efficiency Mass flow )

Default 1.0 96.83% 0.077

0 1.0 96.61% 0.077

0.5 1.0 96.65% 0.077

Table 13: Nusselt Efficiency for 5/B nozzle

for 5/B Nozzle NBTF Efficiency Mass flow )

Default 1.0 94.02% 0.028

0 1.0 93.93% 0.028

0.5 1.0 93.95% 0.028

Table 12 and Table 13 gives the detailed view of the influence of Nusselt correlation on

efficiency of 2/M and 5/B nozzle along with the mass flow. There is increase in

efficiency in the default case in comparison to the results obtained by introducing = 0

or 0.5 in the Nusselt value having same NBTF value of 1.0 for both the nozzles. The

Nusselt correlation with default value used in ANSYS CFX stated in equation (58)

having NBTF 1.0 is considered to be ideal as there is noticeable difference in the area

average efficiency and mass flow. The introduction Nusselt values have not brought any

change in the offset which is persisting in the pressure plot in 2/M as well as in 5/B

nozzles.



5.2.4. Discussion

The Numerical validation results in the form of sensitivity analysis presented in the

previous section. In the previous chapters the numerical verifications where analysis of

mesh density study with the wall refinement analysis still yielded an offset in the

pressure and droplet profile. Furthermore, the influence of 3D effect was also verified.

The results did not match the experimental results with all the verified setup and hence it

was necessary to do a sensitivity test by introducing the Turbulence models, NBTF and

the Nusselt correlations.

With the same geometrical setup including the mesh density, wall refinement and single

precision settings in 2/M and 5/B nozzles, sensitivity analysis by introducing NBTF and

Nusselt correlation are made to check if they have a better agreement to the experimental

solution. However, the results have proved that there is an impact on the pressure and

droplet diameters with the introduction of NBTF values and no influence to the pressure

plot with the introduction of Nusselt’s correlation but a noticeable impact on the droplet

plots. Since the existence of small difference in the efficiency and the mass flow

however remaining constant we consider the default NBTF value which is 1.0 and the

default nusselt correlation as the main objective of this thesis work is to calculate the

efficiency and the droplet diameter by validating the results with the experimental results

counducted by Gyarmathy (2005) .



Chapter 6

In this chapter investigation on finding an optimized solution from the defined

parametrical values for an own nozzle is carried out. The design of a very own nozzle is

based on the numerical model performed on Gyarmathy high pressure experiment which

in the previous chapters. The parameters leading to find an optimum solution is

identified and furthermore its influence on efficiency droplet diameter and homogeneous

flow at outlet is investigated using ANSYS CFX.

6. Parameter investigation to optimize

Optimization involves algorithms which help to find solutions to complex problems in

high dimensions. In this present work no algorithms have been utilized to find optimum

solution, however geometry parameters have been introduced to investigate and optimize

the efficiency of the Laval nozzle. Geometry parameter study is made to obtain uniform

flow at outlet maintaining the droplet diameter less than 1µm. The two parameters

introduced are the radii at the throat of the nozzle and the length of the divergent section.

6.1. Geometry Parametrization

Parametrization can be defined as a process of optimizing manually defining the

parameters which are necessary for a complete or a relevant specification of a geometric

model. Parametrization of geometrical model serves to reduce the number of control

variables which are responsible for obtaining the best solution and helps to create a

geometrical design in controlled engineering specifications thereby consuming less

computational time and yielding faster results.It is how a model is parametrized has a

tremendous influence on how the design process is carried out and it also greatly

influences the final outcome of the new geometrical model. In this thesis work only two

geometrical parameters are altered and the best solution considering the efficiency,

homogeneity at outlet and droplet diameters are analysed.

6.2. Task Description

Parametrization acts as a first step of a design cycle where the geometry is to be

developed by identifying the variables which can be used for modifying the design of the



geometry to obtain the best possible solution. The proposed parametrization is carried

out manually by changing two design parameters in ANSYS CFX are listed as tasks

below:

Creating an own geometry with same numerical model used for validation of

high pressure nozzles.

Maintaining constant inlet, outlet and throat height for the designed nozzle

with the inlet boundary conditions.

Defining and altering two parameters, radius near the throat and the length

of the nozzle in the divergent section.

Analysing and calculating the droplet size, efficiency and to have

homogeneous flow at the outlet for both saturated and superheated steam

case.

6.3. Geometry and Mesh setup

Importance has been given to the design parameters for developing a very own nozzle.

In this work two design parameters such as the radius near the throat and the length of

the divergent section are changed with the inlet, throat and outlet height being constant.

The Figure 6.1 below shows the design of a very own nozzle with radius 5mm and

ellipse length of the divergent section being 30mm is shown.

Figure 6.1: Parametrized geometry



Parameter variation:

i. Radius 2mm with straight length of 30/50/70 mm in divergent section.

ii. Radius 5mm with straight length of 30/50/70 mm in divergent section.

iii. Radius 10mm with straight length of 30/50/70 mm in divergent section.

iv. Radius 5mm with ellipse length of 30/50/70 mm in divergent section.

The mesh and boundary setup in the very own nozzle is similar to that of the validated

case in 5/B nozzle which has a high expansion ratio from Gyarmathy (2005). For

defining the mesh it was made sure that the corner angles were above 25° with the

expansion ratios well below 5 and aspect ratio below 900. Minimum of 15 layers is

made sure to exist near the wall region having no slip condition for all the cases with a

first layer thickness of having a growth rate of 1.3. The boundary condition for

very own nozzle shown in Table 14 below.

Entity Parametrized nozzle

Condition Non-Equilibrium

Turbulence Model model

Inlet Subsonic, Superheated

Total Temperature : 300

Total Pressure : 28 bar

Inlet Subsonic, Saturated

Total Temperature : 235

Total Pressure : 28 bar

Outlet Supersonic

Upper Wall Boundary type : Wall

Condition : No Slip

Nozzle Boundary type : Wall

Condition : No Slip

Nozzle Inlet height 15 mm

Nozzle Throat height 2 mm

Nozzle Outlet height 10 mm

Table 14: Boundary conditions for parametrized nozzle



6.4. Losses in Steam turbine

The following losses were encountered during the analysis of flow in the parametrized

nozzle namely:

Frictional losses

Condensational losses

Shockwave losses

6.4.1. Frictional losses

Fluid frictional losses are the most significant loss of all the loss sources in a turbine.

There is friction as the high velocity steam passes through the nozzle due to the presence

of turbulence within the steam which is caused due to boundary layer where there is no

slip condition for the fluid flow. The roughness of the nozzle surface is also one of the

reasons for the cause of frictional losses in a steam turbine (Chaplin, 2009).

6.4.2. Condensation losses

Condensational losses are the losses caused due to the presence of moisture in the flow

medium. Thermodynamic wetness loss is observed as condensation starts after the throat

region in a steam turbine due to an irreversible heat transfer between the liquid phase

and the vapor during the thermal relaxation back to an equilibrium phase (Strazmann, et

al., 2012). Hence, the liquid particles obstruct the flow of vapor particles in the form of

shear stress between the two and losses a part of kinetic energy. If the dryness fraction of

steam falls below 0.88, there will be corrosion and erosion of blades in a steam turbine

(Rajput, 2010).

6.4.3. Shock wave losses

Shock wave losses are caused by entropy raise across a shock wave which is generated

during a fluid flow. Shock waves are found in supersonic flow where homogeneity in the

flow field is obstructed leading to high pressure due to non-isentropic process and Mach

number at the outlet.



6.5. Results

6.5.1. Efficiency of parametrically optimized nozzle

Calculating the efficiency of a very own Laval nozzle is the principle objective. The area

average efficiency can be defined as the ratio of square of the area averaged velocity at

outlet to the square of the isentropic velocity at outlet which can be found using the

equation (56) in chapter 4.1.1.

6.5.1.1. Efficiency comparison for superheated case

The efficiency plot for different divergent section length has been plotted in the figures

below. The divergent section length of the parametrized nozzle geometry measured from

coordinate ( ) in mm is plotted along the x-axis with the efficiencies in % along the

y-axis. The Figure 6.2: Efficiency comparison for superheated case shows the area

averaged efficiency comparison between the equilibrium (Equi) and non-equilibrium

(Non_Equi) conditions for a superheated steam having an inlet temperature of 300

and an inlet pressure of 28 bars for two parameter changes such as radius and divergent

length of nozzle. R2, R5, R10 implies radius 2mm, 5mm , 10mm with straight divergent

length respectively and R5_Ellipse referring to ellipse length in the divergent section.

a) Radius 2mm straight length divergent section b) Radius 5mm straight length divergent section

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Equi_R2

Non_Equi_R2

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Equi_R5

Non-Equi_R5



c) Radius 5mm straight length divergent section d) Radius 5mm ellipse length divergent section

Figure 6.2: Efficiency comparison for superheated case

Comparing figures 6.2 a, b, c and d it is evident that the efficiency losses in the

equilibrium for divergent length L30 for different radius are almost equal to 97% which

is due to the losses in the no free slip condition in contrast to the non-equilibrium

condition where the efficiency is close to 96% for divergent length 30mm. As it can be

seen there exists a difference in efficiency between equilibrium and non-equilibrium

conditions. The difference in the efficiencies in non-equilibrium and equilibrium

condition for length 30, 50 and 70mm are consistently decreasing as the length is

increased. The efficiency decreases are because of higher subcooling for nozzles with

shorter length, hence higher expansion rates. The sub cooling temperature for

divergent section straight length of 30, 50 and 70mm respectively were 29, 27.5 and 26

with the decrease in 0.2 degrees for every increase in parametrized radii (2 , 5, 10 mm).

For the divergent section having ellipse length the sub cooling temperature was

found to be 25.

The efficiency drop in the non-equilibrium case from the equilibrium case is due to the

condensation process and the losses due to this condensation are termed as

condensational losses. Shock waves were found during the simulation of nozzles with

radius 2mm and 5mm near the throat for all divergent lengths and this additional loss

source was termed as shock losses which in association with the condensation losses

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Equi_R10

Non-Equi_R10

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Equi_R5_Ellipse

Non-Equi_R5_Ellipse



decrease the efficiency of the nozzle. The above figures of different parameterized radius

and divergent lengths show that the difference in condensational loss during the non-

equilibrium case was as high as 1 percent point (p.p.) for 30mm divergent length which

is the shortest length in the parameterized geometry.

The difference in efficiency however gradually decreased from 0.7 p.p. for 50mm

divergent length to 0.5 p.p. for the maximum parameterized length i.e. 70mm.

Introduction of ellipse length at outlet however decreased the loss source from the shock

however, it proved that the shock losses were negligible compared to the condensational

losses. Introduction of ellipse divergent section has brought no changes in the difference

in efficiency losses indicating that the shock losses found in the latter nozzles are

negligible even for superheated steam case.

6.5.1.2. Efficiency comparison for saturated steam case

a) Radius 2mm straight length divergent section b) Radius 5mm straight length divergent section

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Sat Equi_R2

Sat Non_Equi_R2

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Sat Equi_R5

Sat Non_Equi_R5



c) Radius 5mm straight length divergent section d) Radius 5mm ellipse length divergent section

Figure 6.3 : Efficiency comparison for saturated steam case

The Figure 6.3 shows the area averaged efficiency comparison between the equilibrium

and non-equilibrium conditions for saturated steam having an inlet temperature of 235

and an inlet pressure of 28 bars for two parameter changes, radius and divergent length

of nozzle. From the above figures it is clear that the efficiency losses in the equilibrium

for different lengths and radius are almost similar to the efficiency losses in the

superheated case. The efficiency of equilibrium cases are slightly less than 97% and the

efficiency for the non-equilibrium case is lesser than 96% for different divergent lengths

and radii near the throat.

Similar to the superheated case the difference in efficiency in the saturated steam case

between equilibrium and non-equilibrium case decreases as the divergent length of the

nozzle increases due to the sub cooling effect, however it is evident that the frictional

losses increase as the length increases. The difference of 0.25% percent point (p.p.) is

observed in efficiencies for every increase in the parameterized divergent length. For a

divergent length of 30 mm the difference is 1 p.p. and it reduces to 0.75 p.p. for 50mm

and furthermore decreases to 0.5 p.p. for the longest parameterized length of 70mm in

the divergent section. This shows that as the length of the nozzle is increased in the

divergent section there is noticeable decrease in the condensation losses however, the

frictional losses increases with the increase in the length of divergent sections. Similar to

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Sat Equi_R10

Sat Non-Equi_R10

93.0%

93.5%

94.0%

94.5%

95.0%

95.5%

96.0%

96.5%

97.0%

97.5%

30 40 50 60 70

Effi

cie

ncy

, A

rea

Ave

rage

Length [mm]

Sat Equi_R5_Ellipse

Sat Non-Equi_R5_Ellipse



the superheated case introduction of ellipse divergent section has no change in efficiency

due to losses.

6.5.2. Droplet diameter investigation of parameterized nozzle

Achieving maximum allowable droplet size preferably with uniform flow at outlet for

both superheated and saturated steam case has also been an objective of optimization

through parameterization.

6.5.2.1. Droplet diameter comparison for superheated steam case

The droplet size evaluation is made at a distance of 1.75mm from the upper wall of the

nozzle throughout the axial length. The droplet diameter is considered along the y axis

and plotted against the axial length of the parameterized nozzle for different length in the

x axis.

a) Divergent length 30mm

0.0E+0

2.0E-8

4.0E-8

6.0E-8

8.0E-8

1.0E-7

1.2E-7

-0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035

H2

Ol.

Par

ticl

e D

iam

ete

r [m

]

Axial Coordinate [m]

Non_Equi_R5_Ellipse

Non_Equi_R10

Non_Equi_R5

Non_Equi_R2




b) Divergent length 70mm

Figure 6.4: Superheated steam droplet diameter for different divergent length.

The above figures show the droplet diameter for the superheated non-equilibrium case

for different radius parameter for different divergent section length. As we can see from

the above figures droplet radius increases steadily for all parameterized divergent section

length of the nozzle with some exceptions in 30 and 50mm divergent length due to 2D

effects which are discussed later. The droplet diameter plots are similar for length 30mm

and 70 mm. From the above figures it is evident that for straight length at the divergent

0.0E+0

2.0E-8

4.0E-8

6.0E-8

8.0E-8

1.0E-7

1.2E-7

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

H2

Ol.

Par

ticl

e D

iam

ete

r [m

]


Non_Equi_R5_Ellipse

Non_Equi_R10

Non_Equi_R5

Non Equi_R2

0.0E+0

2.0E-8

4.0E-8

6.0E-8

8.0E-8

1.0E-7

1.2E-7

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

H2

Ol.

Par

ticl

e D

iam

ete

r [m

]


Non_Equi_R5_EllipseNon_Equi_R10Non_Equi_R5Non Equi_R2



section having lengths 30 ,50 ,70mm the droplet diameter is maximum for the ellipse

divergent length with 6.42 , 9.0 , 1.1 and minimum for radius 2mm with

2.7 , 8.5 , 6.1 .The droplet diameter increases as the length of the divergent

section increases. The droplet diameters are higher in the ellipse length divergent section

as there is no shock wave propagation compared to the latter parametrized geometries.

High droplet diameters were found in the ellipse case as the area change along the x-axis

in the ellipse near the throat is much smaller compared to the area change for the straight

length divergent section which is explained in the later in 2D effects.

6.5.2.2. 2D Effects for superheated droplet plot

Figure 6.5 shown below shows a pressure gradient contour for a divergent length section

of 50mm. As we can see there exists shock wave generated at the throat due to the

presence of high curvature change resulting in a disturbed flow field.

Figure 6.5: Pressure gradient for Radius 2 mm divergent length 50 mm.

This shock wave producing a non-uniform pressure field in the downstream as shown in

the Figure 6.6 will result in the formation on non-uniform nucleation zone which

culminates in formation of non-uniform droplet diameters.



Figure 6.6: Pressure profile for Radius 2 mm divergent length 50 mm

As the droplets are not evenly distributed the droplet diameter sizes are found to vary at

different regions when measured from 1.75mm below the upper wall of the divergent

section of the nozzle. The streak lines formed due to the generation of shock waves as

shown in figure above also exist for radius 5mm curvature. The droplet growth is early

in these cases and there is formation of low pressure areas just after the throat as shown

in Figure 6.6 which is also a reason for the formation of streaks thus producing an

inhomogeneous flow field at the outlet.

Figure 6.7: Particle diameter for Radius 2 mm divergent length 50 mm.

It was necessary to design the geometry in a better way to avoid the streak line formation

which was caused due to the shock wave in order to have better droplet distribution in

the flow field. Hence the radius parameter was increased to 5mm and furthermore to 10

Droplet export line

1.75mm



mm to note the changes in the flow field. As the radius increased from 5mm to 10mm

the streak lines nearly diminished resulting in more homogeneous flow at the outlet,

however 2D effects still existed which influenced the droplet distribution.

Figure 6.8: Flow in parameterized radius 5mm

Figure 6.9: Flow in parameterized radius 10mm

The above Figure 6.8 and Figure 6.9 show the parameterized radius of 5mm and 10mm

for length 50mm at divergent section. It was found that the divergent section length of

the nozzle had no influence on the streak line formation. From the above figures it can



be said that the flow is more homogeneous with lesser streak line formation as the

parametrized radius is increased from 2mm to 5mm and furthermore to 10mm for all

straight divergent section nozzles.

Droplet diameter for superheated steam in ellipse divergent length section has a delayed

condensation hence a delayed droplet growth compared to the straight divergent length

section. The delayed condensation in ellipse length was found due to the altered pressure

change and also the altered area change in the ellipse case. Area change in the ellipse

case near the throat is lesser than the area change for the straight length divergent

section.

Figure 6.10: Flow in parameterized radius 5mm with ellipse length divergent section

Figure 6.10 shows the particle distribution in the parameterized radius 5mm with ellipse

divergent length section. It was observed that the droplet growth in the ellipse case starts

0.15m after the straight length divergent section. The area change in the straight length

divergent section is continuous and hence there is uniform expansion rate. In the ellipse

case the change in area is not the same at a length of 10mm from the throat compared to

the change in area at the same position for straight length divergent section, however

there is sudden high change in area near the exit and hence there exists small pressure

difference resulting in inhomogeneous flow in the outlet.



6.5.2.3. Droplet diameter comparison for saturated steam case

The figures below show the comparison of droplet diameters for different divergent

length section. The droplet diameter is plotted along the y axis against the axial

coordinate in x axis. The non-equilibrium results for saturated steam case were similar to

that of the results of superheated steam case.


b) Divergent length 50mm

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

-0.005 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035

H2

Ol.

Par

ticl

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iam

ete

r [m

]


Sat_Non_Equi_R5_EllipseSat_Non_Equi_R10Sat_Non_Equi_R5Sat_Non_Equi_R2

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

2.2E-7

2.4E-7

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

H2

Ol.

Par

ticl

e D

iam

ete

r [m

]


Sat_Non_Equi_R5_Ellipse

Sat_Non_Equi_R10

Sat_Non_Equi_R5

Sat_Non_Equi_R2



c) Divergent length 70mm

Figure 6.11: Saturated steam droplet diameter for different divergent length.

From the above figure it is evident that the droplet growth starts earlier in the throat

compared to superheated steam case. Similar to the superheated case the droplet

diameters in the ellipse case are much higher compared to the straight length divergent

section. This is caused due to the area change near the throat and the area change in the

straight length is continuous in comparison to the ellipse case. Comparing all the

simulations with the geometrical parameters having 5mm radius and ellipse length in the

divergent section for different lengths we have a constant difference of from

30, 50, 70mm divergent section lengths. For the parametrized geometry having different

straight divergent length section the droplet diameters is nearly identical for the

respective radii. The droplet diameter however is larger in the saturated steam case in

contrast to the superheated steam case.

6.5.2.4. 2D Effects for saturated droplet plot

Unlike the superheated case significant 2D effect is not observed for the saturated steam

case. The boundary conditions are however different compared to the superheated case

with the inlet temperatures being 235 with same pressure at inlet with 28 bar for

0.0E+0

5.0E-8

1.0E-7

1.5E-7

2.0E-7

2.5E-7

3.0E-7

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

H2

Ol.

Par

ticl

e D

iam

ete

r [m

]


Sat_Non_Equi_R5_EllipseSat_Non_Equi_R10Sat_Non_Equi_R5Sat_Non_Equi_R2



saturated steam case. From Figure 6.11 all the parameterized cases for different length

had the droplet size growth simultaneously unlike the superheated case. However, the

droplet diameter was found to be higher for the ellipse length in contrast to straight

length in the divergent section. This is due to area change near the throat region of the

ellipse case which is lesser than the change in area for the straight length case along the

x axis.

Figure 6.12: Superheated radius 2mm length 30mm

Figure 6.13: Saturated radius 2mm length 30mm

Figure 6.12 and Figure 6.13 show the comparison in the nucleation rate for both

superheated case and saturated steam case. As we can see the nucleation rate in the

superheated case there is a split in the nucleation zone which is caused due to the

formation of shocks and hence this leads to the streaks in the divergent section.

However, in the saturated steam case the nucleation occurs before the shock has actually



generated and hence no 2D effect is observed. This behaviour is same for all the

parametrized radii and lengths for the saturated steam case.

6.5.3. Discussion

The results obtained by parametrically optimizing a very own nozzle with the given

boundary conditions are presented in the previous section. The results were based on the

same numerical model which was used for the validation of high pressure nozzles from

Gyarmathy (2005). From the verified and validated results of high pressure nozzles

similar mesh density, − turbulence model, default NBTF and default Nusselt number

correlation are implemented to analyse and calculate the efficiency and droplet diameter

for superheated and saturated steam case. From both the superheated and saturated steam

efficiency comparison it was found that with the increase in length of the nozzle in the

divergent section the frictional losses increases however the condensational losses

decreases for example in parameterised straight divergent section length of 50mm it was

observed that the frictional losses accounted to 4.5 percentage points where as

condensational loss was 1 percentage point. The condensational losses were found to be

high in the shortest length nozzle comparing the results of equilibrium and non-

equilibrium cases however the difference in condensational efficiency loss was nearly

1(p.p.), 0.7(p.p.) and 0.5(p.p.) for both superheated and saturated steam cases. Since the

difference in efficiencies due to condensational losses are smaller compared to the

frictional losses it can be said that the nozzle with less frictional losses are best to adapt

than the nozzle with less condensational losses. Furthermore, shock waves persisted in

nozzles with lesser radii near the throat and this affected the droplet diameter. The

nozzle with radius 10 mm with ellipse divergent section had no shocks compared to the

radius 2mm with straight divergent section. The parameterized nozzle with radius 10mm

and straight divergent length had negligible shocks however, the ellipse divergent

section had better efficiencies with larger droplet diameters. A good compromise can be

thus obtained with high radius of 10mm near the throat section with shortest length of

30mm in the divergent section to have maximum efficiency and considerably uniform

flow at outlet.

Conclusion and Scope for Future 93


Chapter 7

7. Conclusion and Scope for Future

7.1. Validation

ANSYS CFX which uses Euler-Euler method of approach to solve steam flow in

supersonic nozzles has been utilised in the present work. Numerical results for nozzles

2/M and 5/B nozzles taken from Gyarmathy (2005) had close agreement with

experimental results however, there exists an offset in pressure and droplet profiles for

both the high pressure nozzles. It was discussed as how the mesh density, wall

refinement, effect of 3D and single precision influenced the accuracy of numerical

verification and was found that the results had no influence. The numerical model

involved in ANSYS CFX was capable of capturing the pressure bump for both nozzles

although a bit early. Sensitivity studies were made employing Turbulence model, NBTF

and Nusselt number correlation showed that pressure and droplet profile could be

changed. The numerical results for superheated case also had an offset in comparison

with the experimental results. It was difficult to obtain a good inference for all the other

numerical validation when the superheated case had an offset nevertheless, it was

necessary to perform a sensitivity analysis to check if there existed a better agreement.

Changing to the SST turbulence model from the default − model, no influence on the

pressure profile was proved however the droplet mean radius was observed to be higher with

nearly same efficiency in SST in contrast to − model. NBTF and Nusselt number

correlation factors had influence on the efficiency, pressure and droplet profiles,

nevertheless it was not possible to get a better agreement with the experimental results. The

default NBTF and Nusselt number correlation was considered as the efficiency change was

not significant and yielded better pressure and droplet profiles in comparison to the

experimental results. Results for the 2/M validated nozzle conducted using the Euler-

Lagrange approach is presented in appendix and is also not in agreement with the

experimental results. For same model setup the droplet diameter using the Euler-Lagrange

approach was in better agreement. It was found that Euler-Euler approach had better



agreement for the pressure plots as the pressure bump was too early in the Euler-Lagrange

approach with the same offset as in the Euler-Euler method.

For the future validation it is necessary to know the exact experimental conditions such

as the roughness factor in the nozzle for the fluid flow. Certain assumptions made in

Gyarmathy experiments such that the droplets were assumed to have uniform size with

no coagulation could be misleading for numerical validation. There are not many

experiments conducted on high pressure nozzle to compare the results of numerical

codes. The results thus obtained from simulations proved that the numerical techniques

used are robust in solving the governing equations for the fluid flow in 2D Laval nozzle.

It is recommended to conduct more experiments on high pressure nozzle and also know

the experimental conditions thoroughly as the factors influence the results in a major

way.

7.2. Parameterization

The same model setup used in validation was utilised to optimize the very own nozzle by

changing the radius near the throat and the nozzle divergent length as parameters. From

the conclusions obtained by validating the numerical results with the experimental

results, default NBTF and Nusselt number correlation were implemented for the

parameterized geometries.

The efficiencies are high in nozzles having short divergent section for both the saturated

and superheated steam case. However, it was found that the shorter length nozzle yielded

higher condensational losses and nozzles with high change in curvature near the throat

generated shock waves which in turn produced a distorted pressure field resulting in a

shear line with high droplet diameter. From both the superheated and saturated steam

case it was found that frictional losses are more significant than the condensational

losses and hence designing shorter nozzle is better to minimise frictional losses.

It was achieved to be well below 1µm due to 2D effects caused by shock wave due to

high change in curvature near the throat. The droplet distribution was inhomogeneous in



most of the parameterized geometries. Nucleation is initiated before the shocks resulting

in homogeneous flow at outlet for saturated cases in contrast to the superheated case. It

is due to the presence of the 2D effects that there is non-uniform pressure field in the

divergent section of the nozzle. Hence, the droplets are not evenly distributed but are

concentrated near the streak lines and this on impinging over the rotor blade causes

erosion.

For the future use it is a good compromise to have a nozzle having the shortest length

with the combination of both ellipse and straight length in the divergent section. The

radius near the throat has to be high and optimum to avoid formation shock waves due to

abrupt change in curvature. Importance has to be given in designing a nozzle to avoid

the shock waves near the nucleation zone. It is necessary to be patient in designing a very

own high pressure nozzle to obtain ideal droplet diameter where there are negligible 2D

effects with uniform flow at outlet.

Bibliography 96


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Appendix 100


Appendix

2/M Nozzle Superheated case analysis with Euler-

Lagrange approach

I would like to express my gratitude to Dr.-Ing. Sebastian Schuster, Uni. Duisburg-Essen

Turbomachinery Department for providing validated results of nozzle 2/M 40E run from

Gyarmathy (2005) using Euler-Lagrange method.

Figure 1.A: Euler-Lagrange vs Euler-Euler Superheated case comparison

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 -20 0 20 40 60 80 100 120 140

Fog

Dro

ple

t m

ean

rad

ius

r /

m

No

n-D

ime

nsi

on

, Sta

tic

pre

ssu

re, p

/po

Axial Coordinate 𝜉 /mm

0,05

0.45e-7

Appendix 101


Figure 1.A gives us the results obtained using Euler-Lagrange method to the validated

case in this thesis work using Euler-Euler method with the aid of ANSYS CFX is with

the same boundary conditions and numerical setup. As it can be seen clearly both Euler-

Lagrange and Euler-Euler models overlap each other and the experimental results of

Gyarmathy (2005) from the inlet and are still not in agreement with the numerical

results. The simulation results are conducted for the default Nusselt number correlation

with NBTF value of 1.0. The static pressure in both cases continuously decreases,

however in the Euler-Lagrange model there is a raise in the static pressure and

furthermore it continues to decline. This raise in the pressure is noticed in the form of

pressure bump which is 0.05 static pressure units early compared to the pressure bump

observed in the Euler-Euler model. The pressure bump is in close agreement with an

offset in Euler-Euler model whereas the droplet mean radius is better in agreement for

the Euler-Lagrange model. The droplet mean radius is found to be close to 1.6

with Euler-Euler model whereas for the Euler-Lagrange model it was in close agreement

to experimental results nearing 1.2 , the difference however is found to be

0.45



EPILOGUE

In this Master Thesis we have encountered different levels of flow modelling. It was

essential to recognize the right model for concrete applications. The better description of

physics can be made simpler by augmenting the efforts to solve the equation concerning

the motion flow. However, it is impossible to describe the physics involved in describing

the flow field. The aid of super computers is insufficient to describe the physics involved

in solving the equations of motion in a fluid flow. There is a desire for compromise

between the physical demands and numerical possibilities involved during the

calculation. This is the main characteristic feature of all numerical flow simulations in a

boundless area of Computational Fluid Dynamics (CFD).

Scale errors

REALITY

SCALE MODEL OF THE OBJECT

PHYSICAL MODEL

Discretization errors

Modelling errors

RESULTS FROM COMPUTATIONS

Solution errors

Measuring errors

RESULTS FROM EXPERIMENTS

VERIFICATION AND

VALIDATION



Computational Fluid Dynamics is depicted in the form of a flow chart as shown above.

The above elements utilized in CFD explains the Computational Physics involved in it in

a superior way as to how the verification and validation of results are made with respect

to the results obtained from both experimental and from mathematical calculations, thus

comforting engineers and investigators understand the behaviour of the fluids better.

Modelling: The physical processes that are considered to be relevant during the

study of a flow problem are adapted into mathematical model.

Discretisation: Following the mathematical model formulation, the consecutive

equations are discretised in time and space to a numerical model.

Solution: At last, the discretised partial differential equations are solved in an

iterative way and are finally approximated.

Verification: It is a process of determining that the implemented model

accurately exemplifies the developer’s visionary description of the model and

also the solution to that model.

Validation: It is a process of determining the accuracy or closeness of the model

which is numerically examined in contrast to the real world model which are an

example of experiments.