Numerical study of electrohydrodynamic atomization by openfoam

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VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

MAI NGOC LUAN

NUMERICAL STUDY OF ELECTROHYDRODYNAMIC ATOMIZATION BY OPENFOAM

Major: AEROSPACE ENGINEERING Major code : 8520120

MASTER’S THESIS

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THIS THESIS IS COMPLETED AT

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY – VNU-HCM Academic Supervisor: Assoc Prof Dr Ngo Khanh Hieu

Examiner 1: Assoc Prof Dr Le Tuan Phuong Nam

Examiner 2: Dr Pham Minh Vuong

This thesis is defended at Ho Chi Minh City University of Technology, VNUHCM on July 15th, 2023

Master's Thesis Committee:

1 Assoc Prof Dr Vu Ngoc Anh Chairman

2 Dr Vuong Thi Hong Nhi Secretary

3 Assoc Prof Dr Le Tuan Phuong Nam Examiner 1

4 Dr Pham Minh Vuong Examiner 2

5 Dr Nguyen Song Thanh Thao Member

Approval of the Chairman of Master's Thesis Committee and the Dean of Faculty of Transportation Engineering after the thesis being corrected

CHAIRMAN OF THESIS DEAN OF FACULTY OF

COMMITTEE TRANSPORTATION ENGINEERING

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VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

SOCIALIST REPUBLIC OF VIETNAM Independence – Freedom – Happiness

MASTER THESIS ASSIGNMENT

Student name: MAI NGOC LUAN Student ID: 2170729

Date of birth: 08/11/1997 Place of birth: Ho Chi Minh City

Major: Aerospace Engineering Major code: 8520120

I THESIS’S TITLE:

Numerical study of electrohydrodynamic atomization by OpenFOAM

(Phân tích hiện tượng phun tĩnh điện bằng phương pháp số sử dụng phần mềm OpenFOAM)

II THESIS ASSIGNMENT:

This thesis aims to develop a electrohydrodynamic solver based on the open-source software OpenFOAM to investigate the single cone-jet mode of electrospray The solver is physically verified and validated with preceding literature and with experiment data under the consideration of liquid’s contact angle condition Additionally, the solver is enhanced to investigate the Taylor cone formulation process under the effects of corona discharge The outcome of this thesis will serve as the basis for future numerical analyses of electrospray

III DATE OF ASSIGNMENT: 14/02/2023 IV DATE OF COMPLETION: 09/06/2023

V SUPERVISOR’S FULL NAME: Associate Professor Dr Ngo Khanh Hieu

Ho Chi Minh City,……/……./2023

SUPERVISOR

(Sign and write full name)

HEAD OF DEPARTMENT

(Sign and write full name)

DEAN OF FACULTY OF TRANSPORTATION ENGINEERING

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ACKNOWLEDGEMENT

This work is accomplished under a collaboration between VNU-HCM Key Lab for Internal Combustion Engine, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam and School of Engineering and Built Environment, Griffith University, Queensland, Australia

I gratefully acknowledge Associate Professor Dr Ngo Khanh Hieu, Dr Dau Thanh Van, Dr Tran Canh Dung, Dr Dinh Xuan Thien, Mr Vu Trung Hieu and Mr Vu Hoai Duc for helping achieve this state of my thesis

I would like to thank Dr Nguyen Tan Hoi and CFD team at TechnoStar Vietnam for supporting me during my Master course

Additional thanks to Ms Truong Van Ngoc for proofreading and grammatical corrections

Ho Chi Minh City, July 2023

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ABSTRACT

Electrospray, or Electrohydrodynamic Atomization (EHDA) operates on the principles of electrohydrodynamics which deal with the motion of fluids placed inside an electrical field When a fluid is subjected to an adequately strong electrical field, its surface can be deformed, creating a meniscus from whose apex thin jets is induced Eventually, these jets are destabilized and disintegrated into microscale or nanoscale charged droplets Among the known operating regimes of electrospray, the stable single cone-jet mode is the most desired and applicable because of its stability, controllability, and high yield rate in comparison to other regimes

In this thesis, we program an electrohydrodynamic solver to simulate the cone-jet mode based on the Taylor-Melcher leaky-dielectric model The solution for the electrostatic governing equations is additionally developed, coupling with Open-FOAM’s interFOAM to model incompressible time-dependent multiphase fluid flow The solver is physically verified and validated with preceding literature as well as with experiment data under the further consideration of liquid’s contact angle condition, followed by analyses on the effects of electrical conductivity, voltage, surface tension, flow rate, and fluid viscosity on spray current and jet diameter Numerical results are in reasonable agreement with experiments and consistent with preceding literature Additional studies on different contact angles are performed, suggesting potentially major impacts of this factor on the cone-jet mode in high voltage and low flow rate circumstances Furthermore, the electrohydrodynamic solver is enhanced to investigate the Taylor cone formulation process under the effects of corona discharge Electrospray-corona simulation and contrasting experiment with high-speed camera show significant improvement of the numerical prediction for Taylor cone formation, implying the crucial role of liquid wetting to the Taylor cone formation in numerical electrospray-corona discharge studies

Keywords: capillary nozzle, CFD, cone-jet, corona discharge, electrospray, liquid

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TĨM TẮT LUẬN VĂN

Cơng nghệ phun sương bằng lực tĩnh điện được sử dụng để tạo ra sơn khí từ điện áp lớn Công nghệ này hoạt động dựa trên các nguyên tắc của điện thủy động lực học dùng để giải quyết các vấn đề liên quan đến chuyển động của lưu chất trong điện trường Khi chất lỏng được đặt trong một trường điện trường đủ lớn, bề mặt của nó bị biến dạng tạo ra một cấu trúc có dạng hình nón và từ đó tạo ra những tia chất lỏng bắn ra từ đỉnh của hình nón này Sau đó những tia chất lỏng bị phân tách thành những vi hạt mang điện tích Trong những chế độ hoạt động của công nghệ này, chế độ đơn tia có khả năng ứng dụng cao nhất bởi vì tính ổn định, khả năng điều chỉnh cao và phun hiệu quả của nó

Trong luận văn này, tác giả phát triển một công cụ mô phỏng kết hợp tĩnh điện và cơ lưu chất dựa trên mô hình Taylor-Melcher leaky-dielectric để nghiên cứu chế độ đơn tia Bộ giải các phương trình tĩnh điện được phát triển thêm, kết hợp với bộ giải interFoam có sẵn của OpenFOAM để mô phỏng dòng chuyển động đa pha, không nén, và phụ thuộc vào thời gian Độ tin cây của công cụ mô phỏng này được minh chứng bằng cách tái tạo các hiện tượng vật lý cơ bản và so sánh với các nghiên cứu trước, cũng như là với thí nghiệm có tính tới sự ảnh hưởng của góc tiếp xúc của lưu chất Tiếp đó, sự ảnh hưởng của các yếu như điện dẫn, điện áp, sức căng bề mặt, lưu lượng, độ nhớt lưu chất lên dịng điện phun và đường kính tia phun cũng được phân tích Kết quả mơ phỏng cho thấy sự đồng nhất hợp lý với các dữ liệu so sánh Thêm vào đó, phân tích các góc tiếp xúc lưu chất khác nhau thể hiện sự ảnh hưởng lớn của yếu tố này trong trường hợp điện áp cao và lưu lượng thấp Cuối cùng, công cụ mơ phỏng được cải tiến để xem xét q trình tạo thành của nón Taylor dưới sự ảnh hưởng của hiện tượng phóng điện Các kết quả mơ phỏng cùng với dữ liệu thực nghiệm cho thấy sự cải tiến rõ rệt trong việc dự đốn q trình tạo thành của nón Taylor, từ đó thể hiện vai trị tiềm năng của của tính dính ướt của lưu chất trong mơ phỏng sự hình thành của nón Taylor dưới ảnh hưởng của hiện tượng phóng điện

Từ khóa: ống mao dẫn, phương pháp số động lực học lưu chất, đơn tia, hiện tượng

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THE COMMITMENT

I hereby commit that:

- This master thesis outline is done by me with guidance from Assoc Prof Ngo Khanh Hieu, Dr Dau Thanh Van, Dr Dinh Xuan Thien, and with the support of Dr Canh-Dung Tran, Mr Vu Trung Hieu and Mr Vu Hoai Duc - Design of the experiment apparatus and supporting experiments are carried out by the research team at the School of Engineering and Built Environment, Griffith University, Australia led by Dr Dau Thanh Van The author’s contributions include program development, all disclosed numerical simulations, data curation and visualization, and scientific discussions presented in this thesis

- The data, numbers, results in this work except for specialized experiments are done by me at Ho Chi Minh City University of Technology Any publication or article reusing the content of this work is dominantly authored by me and explicitly declared in the “Publications” section of this thesis - All of the references used in this work are cited fully and clearly in

information: name of the author(s), title, date of publication, place of publication with highest precision in my knowledge

Author,

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vi TABLE OF CONTENTS THESIS ASSIGNMENT iACKNOWLEDGEMENT iiABSTRACT iiiTÓM TẮT LUẬN VĂN ivTHE COMMITMENT vTABLE OF CONTENTS viList of Tables ixList of Figures xNomenclature xiv

Chapter 1 Thesis introduction 1

1.1 Motivation 1

1.2 Objective(s) of the study 3

1.3 Investigation subject and scope of study 4

1.4 Literature review 6

Chapter 2 Background theories 14

2.1 Electrostatics 14

2.1.1 Electric charge and electric field 14

2.1.2 Coulomb’s law 15

2.1.3 Gauss’s law 16

2.1.4 Conservation of charge 17

2.1.5 Electrostatic force density 18

2.2 Computational Fluid Dynamics 18

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2.2.2 OpenFOAM 19

2.2.3 InterFOAM solver 20

2.2.3.1 Pressure-velocity coupling 20

2.2.3.2 Volume of Fluid interface tracking method 25

2.2.3.3 Contact angle correction 28

Chapter 3 Electrohydrodynamic coupling procedure 30

3.1 The Taylor-Melcher leaky-dielectric model 30

3.1.1 Fluidic field 32

3.1.2 Electrostatic field 32

3.1.3 Corona discharge 33

3.2 Structure and solving process of interElectroFoam 34

Chapter 4 Results and discussion 36

4.1 Code validation 36

4.1.1 Physical verification of interElectroFoam 36

4.1.2 Validation with previous literature 41

4.1.3 Validation with experiment results 47

4.2 Dimensionless analyses 53

4.3 Contact angle effects on Taylor cone 60

4.4 Simulation of corona discharge effects in electrospray 63

4.4.1 Corona discharge condition assumptions 64

4.4.2 Simulation results on Taylor cone formulation 64

Chapter 5 Conclusion and prospective future research 71

Publications 73

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Appendix A Experimental apparatus 83

Appendix B Additional experiment results 85

Appendix C Contact angle correction formulation 87

Appendix D Additional simulation results 90

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List of Tables

Table 4-1 Air and Ethanol properties used in simulations 37

Table 4-2 Boundary conditions for the computational domain in physical parame-ters 37

Table 4-3 Convergence criteria of the physical parameters 38

Table 4-4 Description of validation test cases used in Singh [28] 41

Table 4-5 Description of validation test cases used in Huh [38] 44

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List of Figures

Figure 1-1 Different forms of the Taylor cone [15] 5

Figure 1-2 Captured developing stages of the Taylor cone [16] 5

Figure 1-3 Corona discharge captured in electrospray [18] 6

Figure 2-1 Illustration of charged particles in space and their electric field [51] 14Figure 2-2 Directory structure of an OpenFOAM simulation 20

Figure 2-3 Solving procedure of the PISO algorithm 23

Figure 2-4 Solving procedure of the SIMPLE algorithm 24

Figure 3-1 Flowchart of the present interElectroFoam 34

Figure 4-1 (a) The dimensions of the computational domain and; (b) The mesh de-scription of physical verification simulation 36

Figure 4-2 Phase fraction results of Ethanol (a) t = 300 ms, 0 V and (b) t = 15 ms 4 0 0 0 V 38

Figure 4-3 Electric field intensity variation with time of Ethanol electrospray 39

Figure 4-4 Additional simulation results of Ethanol electrospray: (a) Charge densi-ty accumulation of liquid interface (contour: charge densidensi-ty; black line: liquid surface), (b) Backflow near the apex of the cone, (c) Vectors of total electrostatic force acting on the interface at t = 0.3 ms 40

Figure 4-5 Comparison of jet diameter variation with flow rate between Singh [28], our interElectroFoam and scaling law for 2-μm nozzle 43

Figure 4-6 Comparison of jet diameter variation with flow rate between Singh [28], our interElectroFoam and scaling law for 10-μm nozzle 43

Figure 4-7 Phase fraction simulation results in different applied voltage: (a) Huh’s results [38], (b) interElectroFoam results 44

Figure 4-8 Electric field plotted along axial coordinate y = 8 μm of Huh [38] and interElectroFoam 46

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Figure 4-10 The dimensions of the computational domains for two nozzle config-urations (a) Nozzle 1 and (b) Nozzle 2 47Figure 4-11 The axisymmetric hybrid mesh model with (i) Feeding nozzle and (ii) Ring electrode The inset figure shows a close-up view of the mesh resolution nearby the nozzle The nozzle’s sharp edges are filleted at r fil 1.5%o.d. 48Figure 4-12 The Taylor cone’s shape produced by different grid resolutions 49Figure 4-13 (a) Experimental measurement of contact angle of PEG-200 on stain-less steel plate and simulation of static droplet on wall boundary with different contact angle conditions; (iii) is overlapped with (i) to demonstrate appearance conformity (b) Simulation shows different fluid propagation schemes due to different contact angles on nozzle; (vi) No angle represents zero-gradient condition

/ n 0.

   In both (a) and (b), the gravitational acceleration vector is downward 50Figure 4-14 (a) Comparison between Taylor cone’s shape of experiment and simu-lation of N1 nozzle configuration; (b) Images of the (i) Taylor cone in experiment; and (ii) 3-D rendered phase fraction from simulation Cone length lcis the distance from nozzle tip (L = 0 mm) to the intersection point of the black lines; cone angle

c

 is approximated by the angle formed by the black lines 51Figure 4-15 Comparison between Taylor cone’s shape of experiment and simu-lation of N2 nozzle configuration; (b) Images of the Taylor cone in experiment using e 5400 V; and (b) 3-D rendered phase fraction from simulation using

6500 V.

s

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Figure 4-31 High-frequency captured Taylor cone’s shape at six consecutive time-steps [t1 - t6] from (a) Simulation neglecting corona discharge; and (b) Simulation involving corona discharge (6700 V), and (c) experiment Electric field intensity is rendered as background contour Three stages of liquid progression annotated (i) Propagating liquid: liquid is advancing from the inner edge to the outer edge of the nozzle; (ii) Edges covered: liquid reached the outer edge and obstructs the corona discharge; (iii) Jet forming: increased electric field is inducing jet at the tip of the Taylor cone 65Figure 4-32 Taylor cone’s shape, ionic wind velocity contour and charge cloud from corona at t2 and t5 Background contours illustrate (a) Ionic wind velocity; and (b) Charge density from corona, vector field represents ionic wind field, streamline denotes electric field Electrodes annotated (i) Nozzle, (ii) Ring electrode Inset shows a glowing region from nearby the outer edge of dry nozzle in high voltage, which is an indicator of strong electric field 67Figure 4-33 Ionic wind velocity and maximum electric field intensity at the noz-zle’s tip variation with discharge current with time to first jet induction tj annotated 68Figure 4-34 The illustration of charge cloud from corona for different discharge currents The boundary of the charge cloud is determined by a selected value of

3

0.75 mC/m

c

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Nomenclature

Regular letters:

q charge C

D electric flux density C/m2

encl

Q enclosed charge C

T viscous stress per unit volume N/m3

fp

f , fb body force per unit volume N/m3

F face flux m3/s

aP, aN discretized coefficients of u

( )

H u discretized coefficients matrix

S face area normal vector m2

A area of discharge surface m2

d

p new static pressure, p g x Pa

x position vector of a control volume

e

Ca dimensionless electric capillary number

j

d jet diameter m

E electric field,   V/m

E electric field intensity V/m

on

E onset electric field V/m

z

E electric field component in the perpendicular direction to the outlet

V/m

e

f electrostatic force, N/m3



f surface tension force N/m3

g gravitational acceleration m/s2

ref

l characteristic length m

d

I discharge current A

i.d inner diameter mm

cond

I conductive current A

conv

I convective current A

i.dr ring inner diameter mm

J current density A/m2

e

J corona current density A/m2

0

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c

n corrected interface normal vector

w

n boundary wall normal vector

o.d outer diameter mm

p pressure Pa

Q liquid flow rate ml/h; m3/s

R e Reynold number

e

r interelectrode distance mm

fil

d nozzle sharp edges fillet diameter mm

S outlet area m2

t time s

u fluid velocity m/s

r

u artificial compression term

z

U liquid velocity component in the perpendicular direction to the outlet

m/s

c

l cone length mm

i

v ionic wind velocity m/s

j

t time to first jet induction s

Greek letters:

p

 pressure relaxation factor

 phase fraction of liquid

 permittivity F/m

0

 permittivity of free space F/m

r

 dielectric constant

 fluid kinematic viscosity m2/s

 fluid viscosity Pa.s

 contact angle

0

 angle between uncorrected interface normal vector and wall’s normal vector

 mean curvature of free surface m-1

e

 electrical conductivity of fluid S/m

e

 mobility of charge of gas m2/Vs

 fluid density kg/m3

c

 corona volumetric charge C/m3

e

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xvi  surface tension N/m  electric potential V c cone angle a

 actual contact angle between liquid’s interface and nozzle’s wall

e

 magnetic permittivity H/m

e

 electric characteristic time,  o/ e

m

 magnetic characteristic time, 20

eee refl

  

M

σ electrostatic Maxwell stress tensor N/m2

Abbreviations:

AC alternating current

CF carbon fibre

CSF continuum surface force

EHDA electrohydrodynamic atomization N1 nozzle 1

N2 nozzle 2

VOF volume of fluid

Subscripts:

e experiment

s simulation

l liquid

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Chapter 1 Thesis introduction

1.1 Motivation

Electrohydrodynamic Atomization (EHDA), or electrospray operates on the principles of electrohydrodynamic which deals with the motion of fluids placed inside an electrical field Fluid surface can be controllably deformed, elongated, finally broken up and dispersed into much smaller droplets compared to the size feeding tube (nozzle’s diameter in the case of electrospray) which is useful for a wide range of applications First recorded by William Gilbert [1], the phenomenon together with its underlying science have been studied intensively and developed into technologies that enrich people’s life in different ways Along with the advances of micro-/nanotechnologies, electrospray, due to its great potential, has been found to be useful in chemical, biological, pharmaceutical, internal combustion, propulsion, and manufacturing applications In particular

 For chemical/biological/pharmaceutical applications:

o Mass spectrometry: technique to measure mass-to-charge ratio of ions, producing results called mass spectra which are used to determine the elemental or isotopic signature of a sample, the masses of particles and of molecules and to illuminate the chemical identity or structure of molecules and other chemical compounds [2]

o Food industry: electrospray is used to encapsulate bioactive compounds, enhance aromatic properties, manufacture smart packaging, edible films, and coatings [3-5]

o Inhalable drug/antigen delivery: micro-/nanoparticles produced by electrospray provide better efficacy-to-safety ratio compared to other methods EHDA has been used in delivering treatments for bronchial

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asthma, lung sicknesses and cancer, influenza virus, etc [6] Electrospray can produce drug-loaded particles with a core-shell structure to improve drug protection and drug release accuracy [7] Additionally, ionic wind cooperated with electrospray device is employed to reduce the cumulative charge in the particles and transport them to a target in front of the nozzle [8, 9]

 Internal combustion applications: electrospray atomizes fuels to generate sprays, monodisperse droplets with micro/nanoscale diameters to enhance combustion stability and emissions [10, 11]

 Propulsion applications: colloid electrospray thrusters produce thrust through electrostatic acceleration of charged liquid droplets First successful orbit mission of electrospray thruster was the LISA Pathfinder spacecraft [12]

 Manufacturing applications:

o Electrohydrodynamic jet printing (e-jet printing): the employment of e-jet printing can produce sub-micrometer resolution for patterning or to fabricate devices in electronics or other areas of technology by use of functional or sacrificial inks [13]

o Powder technology: multiple electrosprays system enabled more energy efficient controlled production of particles Produced powder particles can be used as components for the fabrication of other materials, for example, paints, emulsions, as ingredients used in food, cosmetics or surface coating [14]

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electrospray, hereinafter it will be the sole research subject and briefly referred to as cone-jet or electrospray or EHDA

The popularity of EHDA, or specifically the single cone-jet mode, arises the need for developing numerical modelling to reproduce, predict, characterize prominent properties as well as optimize the operational characteristics of this electrohydrodynamic phenomenon The ultimate goal of this development is to produce a numerical method that is able to satisfy industrial demands and replace intricate experiments which often involve complex apparatus construction and consume a large amount of time Furthermore, from an academic perspective, if computational simulations are proven physically accurate, they can also provide vivid explanations for empirical manifestations This could potentially help researchers to achieve a deeper insight into the discovered phenomenon, which would play as the catalyst for new ideas as well as the precursor for novel advancements

1.2 Objective(s) of the study

Understanding the aforementioned challenges, this study introduces the development of a numerical method coupling electrostatic system and fluid dynamic system to simulate the electrospray’s single cone-jet mode This newly programmed solver will then be verified and validated by contrasting with established scaling laws, preceding literatures and experiments provided by the research team from Griffith University, Australia (GU) The detailed procedure of this study is listed as follow

 Firstly, we develop a solver based on the open-source software OpenFOAM, in which the newly implemented electrostatic solution is combined with the solver interFoam already integrated inside the OpenFOAM package The resulting solver is referred to as interElectroFoam

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step, we will also quantitatively contrast the achieved results with some selected publications to consolidate the reliability of the interElectroFoam solver

 Thirdly, we run simulations whose settings (injecting fluids’ properties, device’s sizes, etc.) are consistent with GU’s results and then visually compares the two sets of data to further validate the solution Simultaneously, we carry out numerical investigations on the impacts of different conditions such as voltage, flow rate, fluid properties to the operation of electrospray and contrast the outcome with similarly reported results from literature Moreover, brief analyses on the impacts of contact angle on the characteristics of the cone-jet mode is presented

 Fourthly, as the preceding steps are completed, we continue to enhance the solver to analyze the corona discharge and ionic wind generation and their qualitative correlation with electrospray The definition of corona discharge and ionic will be presented briefly in the next section

Overall, this thesis aims to develop a simplified OpenFOAM-based solver to simulate the electrospray’s single cone-jet mode with reasonably acceptable accuracy This solver does not consider intensive break-up model for ejecting droplets, thermophysical effects and any evaporation model Validating test cases and further analyses including the impact of corona discharge are performed in simplified axisymmetric 2-D simulations, which will be discussed in detail later in this thesis

1.3 Investigation subject and scope of study

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apex’s center point and disintegrated into extremely small droplets Finally, the conical shape and the jet collapse He concluded in this study that the interfacial forces are in equilibrium condition if the semi-vertical angle of the conical shape stays at approximately 49.3̊ The conical shape and its behaviors described by Taylor is later commonly known as the Taylor cone Different forms of the Taylor cone and the captured development of the Taylor cone reported by Taylor are presented in Fig 1-1 and Fig 1-2, respectively

Figure 1-1 Different forms of the Taylor cone [15]

Figure 1-2 Captured developing stages of the Taylor cone [16]

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electrospray, when high electric potential is applied, the air in the vicinity of the nozzle can be ionized, resulting in various forms of electrical discharge from either the nozzle tip or the liquid surface Under the influence of electric field in the interelectrode region and surrounding ionized gas, a stream of charge, corona discharge, can be created flowing down the potential gradient This movement of ions transfers momentum into surrounding neutral molecules, giving rise to ionic wind, around the nozzle which can affect electrospray processes both positively and negatively [17] Corona discharge or ionic wind can be considered as an entirely different research field due to its complexity and broad scope of applications In this work, we only aim to qualitatively investigate the ionic wind phenomenon in electrospray by simplified numerical method We expect the outcome of our work would adequately represent characteristics of ionic wind in electrospray as observed in experiments, and therefore, provide a reliable tool for future quantitative research An illustration of corona discharge in electrospray is provided in Fig 1-3

This figure is taken from Pongrác et al [18] where the effect of water’s conductivity

and DC corona discharge on some electrospray mode has been investigated In this case, the intense discharge indicated by purple glow can be seen originating from the sharp edge of the nozzle

Figure 1-3 Corona discharge captured in electrospray [18]

1.4 Literature review

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fundamental aspects of this electrohydrodynamic phenomenon both experimentally and theoretically Taylor’s work [16] was so pioneering that the electrospray’s common conical shape was named after him Shortly after this publication, Taylor and Melcher [21] introduced a mathematic model, named leaky-dielectric, to explain the interrelationship between the physical aspects involved and approximate the force exerted on the surface of the liquid In its most elementary form, the leaky-dielectric model comprises Stokes equations to account for fluid motion, Gauss’s Law, an expression of current conservation considering Ohmic conductivity with magnetic effects neglected, and expressions for calculating the Maxwell’s stress tensor on the fluid-fluid interface from the Coulombic force and the polarization force [1, 21] This approach has been utilized in various succeeding analyses to numerically represent the Taylor cone and its behaviors Researchers may use different methods to solve the related electrohydrodynamic equations, yet the basic principles remain unchanged

Among early researches, Lastow and Balachandran superseded the heat transfer equation by electrostatic system in the commercial Ansys’ CFX module to develop a numerical tool to simulate the liquid cone formation and atomization [22] and later used the program to investigate a double-layer spray nozzle used for atomizing water and weak saline solutions in the low-voltage stable cone jet mode [23] Both works compared droplets’ diameter variation with flow rate of CFD result, experiment data and also scaling law and reported good agreement The numerical model in these works did not include a breakup model, so the authors can only

estimate the droplets’ diameter from jets’ size Similarly, Sen et al [24] integrated

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was proposed by Kim et al [25] where a non-conductive tip was inserted to the

electrospray nozzle to reduce both the required voltage and the jets’ diameter in electrohydrodynamic jet printing This work employed Ansys’s Fluent with a user-defined function (UDF) to investigate the performance of the new configuration Achieved results suggested that the patterns’ thickness and required voltage were lessened by more than 50% by implementing the tip In this work, the authors neglected the polarization force and only considered Coulombic force in their electrostatic equation, which is a simplification of the original leaky-dielectric

model Around the same period, Lim et al [26] proposed simulation approach for

better prediction on the electrical potential in a large simulation domain considering all electrodes and in a small simulation domain focusing the vicinity of the nozzle tip The proposed method is only loosely based on the leaky-dielectric model but instead assumed constant distribution of interfacial charge and divided the domain to calculate electric potential separately from the Taylor cone’s region The formation of Taylor cone, liquid jet, and the droplet in the EHDA process was successfully replicated numerically with good agreement with experiments However, the accumulation of charge on the liquid–gas interface had to be declared as simulation input Noteworthy early approaches to cone-jet mode simulation also

includes the work of Herrada et al [27] which demonstrated a fairly complicated

formulation of an electrohydrodynamic simulation tool based on Matlab A new interface tracking method was proposed in this work and compared with the Volume-of-Fluid solver they developed in prior works The authors validated their solution by comparing simulated shape of the cone with the one captured experimentally which agreed well with each other, except for the cone-to-jet transition region

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simulation to predicting the transient ejection of micro/nanoscopic jets from microscale nozzles by COMSOL and investigate effects of applied voltage, liquid’s flow rate and properties on jet ejection dynamics Based on the results, the authors concluded that the ideal conditions for nanoscale printing would be high conductivity liquid, low flow rates, small nozzle’s diameter, and high voltage Du and Chaudhuri [29] also relied on COMSOL to numerically study the formation and breakup of droplets from axisymmetric charged liquid jets in electric fields Interesting physical phenomena in charged droplet breakup and atomization, such as jet instability, necking, the evolution of an unstable jet to droplet breakoff, were observed The research also quantified the effects of electric potential, surface tension, viscosity, and mobility on the droplets’ properties COMSOL utilization might be most common among electro-hydrodynamic jet printing research of various configurations, from conventional single nozzle [30-32], alternating current

(AC) printer [33], and coaxial nozzle [34, 35] Particularly, Zhao et al [34] and Wang et al [35] presented and studied the effects of coaxial nozzles in EHDA jet

printing These works recorded printed structures around 180 times [34] and 120 times [35] smaller than the inner nozzle’s diameter by coaxial nozzles configuration Appeared in a comparably large amount of works, the open-source software OpenFOAM is also a prominent tool for electrohydrodynamic simulations

Ouedraogo et al developed an electrohydrodynamic solver based on the

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experimental gains, demonstrating the reliability of not only OpenFOAM but also the authors’ coupling method The latter research concentrated on electrospray initialization period including the formation of the jet and the characteristics of a few first droplets Simulation results, which agreed well with establish scaling law, suggested the independence of electrospray current on applied voltage and that the charge and size of the atomized droplets would increase after the first ejection Huh and Wirz also discussed some interesting physical aspects of the electrospray cone-jet mode in their analysis for electric thruster extraction [38] and the emission processes for low to moderate conductivity liquids [39] The authors introduced their OpenFOAM-based solver that could capture steady, unsteady, and time-dependent behavior (e.g., start-up, shutdown, thrust changes) as well as details of both cone-jet formation and droplet breakup during the extraction of fluid in an electrospray thruster [38] The dependence of the cone shape on the applied voltage and that of droplet generation frequency on the strength of electrostatic forces were numerically determined, however, this research only presented rough qualitative verifications, therefore possessed inadequate reliability In later publication, Huh and Wirz proposed a solution to improve the stability of their OpenFOAM-based solver in simulating moderately high conductivity fluids [39] The key idea was to modify the interpolating scheme of the electrical properties (permittivity and conductivity) from harmonic averages into a customized function to avoid dramatic change of the mentioned properties in the interface region The authors have successfully stabilized their solution and accurately reproduced the shape of the cone, jet and ejecting droplets by simulations, and suggested the potential of their newly derived scheme in numerically challenging high conductivity cases

Recently, Guan et al reported numerical analyses of high-frequency pulsating

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with different pulse time The numerical model used was validated by contrasting with the experimental duration of the three jetting stages including cone formation, jetting, and meniscus oscillation Excellent agreement with empirical data was reported in both research works, consolidating the reliability of computationally reproduced physics

In addition to the studies previously discussed, Rahmanpour and Reza [42] and

Panahi et al [43] also relied on the leaky-dielectric model to estimate the shape of

the liquid cone and the resulting jet in presence of external electric field [42] and to model the behavior of electrified Newtonian and viscoelastic jet [43] Both works have explored very interesting physical aspects of the formation and operation of the Taylor cone and provided numerical methods well validated with benchmark tests and empirical data Gamero-Castaño and Magnani [44] introduced numerical simulation employing Taylor’s electric potential as a far-field boundary condition to achieve independence from geometry and potential of the electrodes, and the least-squared weighted residual method to the Young–Laplace equation for free surface tracking

Beside the problems revolving around the Taylor cone, the modelling of the deposition of nanoparticles has also been a research interest Here, researchers’ main concern is the trajectory of the particles as well as the density of the droplet deposition, meaning that the Taylor cone simulation along with the leaky-dielectric

model were not required Particularly, Jung et al [45] simulated the deposition

pattern produced by multiple electrohydrodynamic spraying with a capillary–extractor–substrate configuration by the three-dimensional Lagrangian model Newton’s second law was used to relate the force exerted on the particles and their motion The forces considered were the Coulombic force of the external electric field, the drag of air, the particle’s own weight due to gravity, and the Coulombic

force resulted from nearby charged particles Similarly, Wei et al [46] also applied

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accumulated charge on the surface of the mask was the most impactful factor in focusing deposition Another research employing such techniques is the work of

Ondimu et al [47] in which a two-dimensional physical model for determining the

droplet trajectories in cone-jet mode a single nozzle/ring-up configuration was introduced The calculation of the background electric field was performed on COMSOL Multiphysics, and the trajectory of droplets based on Newton’s second law was determined using Matlab The authors assumed and later confirmed that the droplets' dispersion is initiated by the displacement of their center of charge from their center of mass

In summary, the contribution of each research is very diverse, from solely providing a novel numerical method [22, 26, 27, 42] to developing an already established tool to investigate a specific problem [28, 31-33, 35-37, 40, 41] or suggesting an improvement to a problem with the aids of simulations [24, 25, 39] In terms of electrohydrodynamic coupling methodology, nearly all reviewed works employed the Taylor-Melcher leaky-dielectric model to simulate the single cone-jet mode of electrospray The differences between mentioned researchers are the foundation on which they additionally programmed an EHD solver and performed their simulations The foundation can be Matlab [27], Ansys [22, 23, 25, 42], Flow3D [24], COMSOL [28-35], OpenFOAM [36-41], etc Additionally, interface tracking method also varies from most widely used Volume-of-Fluid [22-25, 27, 28, 36-42], phase-field method [32-35, 43] and level-set method [29, 30]

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Chapter 2 Background theories

This chapter presents prominent background knowledge related to the physics of electrospray and its computer simulation development

2.1 Electrostatics

Electrostatics, as the name suggests, is the study of stationary electric charges Electrostatic phenomena arise from the forces that electric charges exert on each other [49] These forces are described by Coulomb’s law which quantifies the amount of force between two stationary, electrically charged particles [50]

2.1.1 Electric charge and electric field

Electric charge is a physical property of matter carried by some elementary particles that governs how the particles are affected by an electric or magnetic field Electric charge can be either positive or negative and is neither created nor destroyed [51] On the one hand, two objects carrying net charges of the same type exert a repulsive force on each other when placed relatively close in distance On the other hand, two objects carrying net charges of different types exert an attractive force on each other when placed relatively close in distance Fig 2-1 illustrates two charged particles in space with their corresponding electric field

Figure 2-1 Illustration of charged particles in space and their electric field [51]

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appearance of an amount of charge in a system requires an appearance of an equal amount of opposite charge somewhere else in the same system [52]

Electric charge is measured in the unit of Coulomb (C) Each Coulomb consists

of 18

6.24 10 natural units of electric charge, such as individual electrons or protons which have a net negative charge and positive charge of 1.6021766 4 10319C, respectively

There are different versions of definition for electric field [53], here we consider electric field as an electric property associated with each point in space when charge is present in any form [54] It is a vector quantity and has magnitude, often referred to by the value of E, called the electric field strength or electric field intensity Electric field intensity, however, is not the field itself, just a property of the field [53], 0,q fE (2.1)

where E is the electric field intensity (V/m or N/C), q0is test charge (C) and f is Coulomb force (N) which will be explained more carefully in the next section

Equation (2.1) implies that when a small test charge q0 inside an electric field

E induced by a larger charge, this test charge experiences a force f either pulling or pushing it The electric field intensity associated with a point charge is directed away from positive charge, proportional to the magnitude of the charge, inversely proportional to the permittivity and distance squared [55], as discussed more in the next section

2.1.2 Coulomb’s law

Coulomb’s law was first published in 1785 by French physicist Charles-Augustin de Coulomb [50], used to quantify repulsion or attraction force experienced by two charged particles when positioned relatively close to each other in space The interaction force is confirmed by observations to be proportional to

1 2

q q , inversely proportional to the square of distance between charges 2

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electric permittivity ε (F/m – quantifies the effect of material in determining the electric field intensity in response to charge), with an additional unitless factor of 1/4π [55], thus  1 22,4q qRRf (2.2)

with q1 and q2 are the charge (C) carried by particle 1 and particle 2, respectively,

R is the unit vector pointing from the particle 2 to the particle 1 If the force

perceived by particle 1 is f , the force experienced by particle 2 is equal to  f Additionally, E2 can be represented in terms of electrical field intensity associated with particle 2,

12,

q



fE (2.3)

From Eqs (2.2) and (2.3), we can obtain the electric field intensity associated with the second particle

 222,4qRRE (2.4)

with R is the vector beginning at the particle 2 and ending at the point to be

evaluated

2.1.3 Gauss’s law

Gauss’s law is one of the four fundamental laws of classical electromagnetics, collectively known as Maxwell’s equations [55], stating that the flux of the electric field through a closed surface is equal to the enclosed charge Electric flux density is an alternative to electric field intensity quantifying an electric field

,enclSdSQ D (2.5)

with DE (C/m2) is the electric flux density, S is a closed surface with differential surface normal d S , and Qenclis the enclosed charge

Equation (2.5) is Gauss’s law in integral form, in differential form, we have   e,

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with the rightmost quantity is the volume charge density e (C/m3)

The concept of electric flux density is important in cases where boundaries between media with different permittivities are encountered, as in this thesis’s discussing problem – electrospray Gauss’s law is a method to calculate the electric field in response to a distribution of electric charge in addition to Coulomb’s law as described above

2.1.4 Conservation of charge

The net charge of an isolated system remains constant [56] Two objects exchanging electrons with each other results in net negative charge in the object losing electrons and equivalent net positive charge in the one taking electrons The net charge of the system, however, remains the same as it was before the two objects interacted This principle does not prohibit the creation or destruction of charged particles which is possible on the atomic and nuclear levels Nevertheless, because of charge conservation, charged particles must be created or destroyed only in pairs with equal and opposite charges [56]

In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a transport equation between charge densityeand current density [57]

0,et    J (2.7)

where J is current density (A/m2), which is the summation of ohmic charge conduction and charge convection due to fluid flow, yielding

 0,eeet        uE (2.8)

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2.1.5 Electrostatic force density

Electrostatic force density for incompressible media can be derived from thermodynamic considerations as provided by Melcher [57],

212

e e  

fEE (2.9)

In Eq (2.9), the first term on the right-hand side is known as Coulomb force and is the strongest electrohydrodynamic force acting on a medium containing free electric charge The second term is called dielectrophoretic or polarization force, arising due to the force exerted by an electric field on a non-homogeneous dielectric fluid or the polarization charge at the interface between fluid phases with different dielectric properties [24, 36]

For compressible media, a third term, called electrostrictive force, is additionally considered, accounting for the effect of fluid density on electric permittivity [57] Equation (2.9) is now 221 12 2ee          fEEE , (2.10)

with  is the fluid density (kg/m3)

2.2 Computational Fluid Dynamics 2.2.1 Navier-Stokes equations

All of the knowledges and information in this part are taken as references from [58] The Navier-Stokes equations are derived from Isaac Newton's second law of fluid motion In each equation (they are mainly three equations related to the velocity components ui), the terms are expressing the following: inertial forces, pressure gradient forces, shear forces, and body forces The momentum equation is shown in the general vector form:

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 D

Dt

 u : change in momentum per unit volume of a fluid particle

 p: pressure term which prevents motion due to normal stresses The fluid presses against itself and keeps it from shrinking in volume

 T: stress term which causes motion due to horizontal friction and shear stresses The shear stress causes turbulence and viscous flows Turbulence is the result of the shear stress

 f : force term which is acting on every single fluid particle fp

Although this is the general form of the Navier-Stokes equation, it cannot be applied until it has been more specified Firstly, depending on the type of fluid, an

expression must be determined for the stress tensor T; secondly, if the fluid is

assumed to be incompressible, hence, the continuity equation for steady and incompressible flow is 0.iiux (2.12)

The momentum equation Eq (2.11) can be re-written as

1,jiiijiiiiiuuupuutxxx  xx         (2.13) where xi are the Cartesian coordinates, ui is the velocity component corresponding

i

x-direction, is the density, = ∕ is the kinematic viscosity and t is the representative of time

2.2.2 OpenFOAM

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and utilities, that are designed to perform tasks that involve data manipulation The solvers in OpenFOAM cover a wide range of problems in fluid dynamics

Running a CFD simulation using OpenFOAM is quite different to running a simulation using commercial software packages such as Fluent or CFX The commercial solvers use a Graphical User Interface (GUI) for case setup and data entry whereas for OpenFOAM the case setup is done using text files in a specific folder structure Figure 2-2 shows the basic file structure which must be used for all OpenFOAM simulations

Figure 2-2 Directory structure of an OpenFOAM simulation

In addition, OpenFOAM includes different types of boundary conditions including base, primitive and derived types If the wall function is used for the RANS base model, it is specified in the following directory, depending on the properties that the appropriate wall function selects

2.2.3 InterFOAM solver

InterFOAM solver was first implemented by Ubbink in 1997 [59] Since then, it has undergone many modifications as well as validations in order to probe its usability in a wide variety of applications, ranging from simple floating cylinder simulations to complex multiphase problems such as cavitation or microfluidic capillary flows [60]

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InterFOAM employs the PIMPLE algorithm combining the PISO algorithm (Pressure Implicit with Splitting of Operators) for transient flows and the SIMPLE algorithm (Semi-Implicit Method for Pressure Linked Equations) for steady-state analyses The PISO algorithm was first proposed by Issa [61], and the SIMPLE

algorithm was proposed by Caretto et al [62] Various prominent features of the

interFOAM solver and the underlying PIMPLE algorithm have been outlined in a work by Damián [63] which is utilized as the dominantly referenced document in this section as well as 2.2.3.2

 Discretisation of Navier-Stokes system:

Firstly, we have differential form of the continuity equation and the momentum equation for incompressible flow

0, u (2.14)  p.t      uuuu (2.15)

The non-linear convective term   uu in Eq (2.15) can be linearized using Gauss’s theorem and discretized by a particular scheme for convective term [63], obtaining

 fffPPNN.

fff

Faa

  uu S u u u  u  u (2.16)

Where F, aP and aN are functions of u The fluxes F should satisfy the continuity equation and be also linearized to avoid larger non-linear system Linearization of the convection term implies that an existing velocity flux field that satisfies

continuity will be used to calculate aP and aN Subscript P denotes the value at the computing control volume, N denotes the value at the neighboring control volume, and f indicates face value

 Derivation of the pressure equation

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( )

P P

a u H u p (2.17)

The Hu( ) term consists of two parts: the “transport part” including the matrix

coefficients for all neighbors multiplied by corresponding velocities and the source part including the source part of the transient term and all other source terms apart from the pressure gradient

0( ) NN fat   uH uu (2.18)

The discretised form of the continuity equation is

0.

ff

 uS u (2.19) Use Eq (2.17) to express u:

( ) 1.PPPpaaH u  u (2.20)

The velocity on the cell face is expressed as the face interpolate of Eq (2.20): ( )1.ffPPpaa   H uu (2.21)

This will be used later to calculate the face fluxes When Eq (2.21) is substituted into Eq (2.19), the following form of the pressure equation is obtained:

1( )( ).fPPPfpaaa        H u  SH u  (2.22) The Laplacian on the left-hand side of Eq (2.22) is discretised [63] The final form of the discretised incompressible Navier-Stokes system is:

( )( ) ,PPffa u H u S p (2.23) 1 ( ).ffPffPfpaa              S SH u (2.24)

The face flux F is calculated using Eq (2.21):