Literature studyAtomized spray properties in singlehole injectionVlad Petrescu

Delft

Unive

rsity

ofTe

chno

logy

Literature studyAtomized spray properties in single hole

injection

by

Vlad Petrescu4365119

A research project of the Honours Programme

Faculty of Aerospace Engineering,Delft University of Technology

Supervisor: Ir. B.T.C. Zandbergen

Contents

List of Symbols vi1 Introduction 12 Single hole injector 3

2.1 Exit and jet velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Orifice geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Pressure losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Spray properties 73.1 Jet disintegration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Droplet formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Literaturemodels 94.1 Droplet size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 Spray angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Experiments and equipment 135.1 Laser diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Image based processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3 Patternator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Software packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

6 Conclusion 17A Appendix: SMD and spray anglemodels 19Bibliography 23

iii

List of Symbols

Constants

g Gravitational constant 9.81 m s−2

Roman symbols

A Nozzle geometry constant −

Ap Surface area of the particle m2

Cd Discharge coefficient −

d Orifice nozzle diameter m

d0 Initial droplet diameter m

d32 Sauter mean diameter m

E Bulk elasticity modulus Pa

l Length of the nozzle orifice m

p Pressure Pa

∆P Pressure drop Pa

pV Vapor pressure Pa

r Jet radius m

Re Reynolds number −

V Velocity m s−1

Ve Nozzle exit velocity m s−1

V j Jet velocity m s−1

Vp Volume of the particle m3

W e Weber number −

z Downstream of the nozzle distance m

v

vi

Greek symbols

λ Wavelength m

µ Absolute (dynamic) viscosity m2s−1

ν Kinematic viscosity m2s−1

ρ Density kg m3

σ Surface tension kg s−2

θ Spray angle °

1Introduction

Liquid rocket engines (LRE) are largely used in space missions, where the lack of oxygen becomes a difficultyfor the solid engines. Carrying oxidizer on board of a bi-propellant rocket has became an usual proceduresince the launch of the first liquid propelled rocket in 1926 [1]. Moreover, thrust throttling and motor re-usage after an engine shutdown represent a main advantage of LRE that is essential nowadays.

The injector of a LRE has the purpose of atomizing the propellant that is being burned in the combustionchamber in order to produce thrust. In the case of a bi-propellant engine, the injector also provides the mix-ture between the fuel and oxidizer. The design phase of such an injector is extremely important for attaininga stable combustion and a highly efficient engine [2].

Showerhead injectors consist of multiple orifices, axially directed and represent one of the most simplesttypes of elements. It was used in the German V-2 rocket [3] and it has been replaced later by coaxial, impinging[4], or pintle elements for a better mixture and throttling. However, injection through single axial orifices hasbeen used in automotive industry as well. As a result, research has been done in order to understand thecharacteristics of a sprayed fuel [5]. Sizing the droplets is also important for medicine [6]. The penetration ofa spray is important to heal the targeted area with the minimum amount of liquid. The analogy can be madewith rocket engines. If the droplets are too small, they do not penetrate in the combustion chamber. If theyare too big, the resulted combustion is inefficient. This is why the goal of the current paper is to understandthe principles of a simple injection.

Liquid atomization is extremely nonlinear and complex. Thus, the need for understanding the basics of liq-uid brake-up into droplets has became more and more important in the last few years. Recent discoveriesin numerical simulations based on CFD [7], [8] are in close agreement with the experimental data, but thelatter still represents a requirement for validating the models. This is why, in the current paper, methods ofmeasuring the cone spray angle, sizing the droplets and their distribution are discussed. The behavior of asingle hole injector is chosen as a study material in order to compare it with the existing fundamental modelsthat characterize its performance. Avoiding the complexity of the pintle or the coaxial injectors will enablethe reader to understand how the simplest injector works, providing the basics for researching on complexbehaviors in the future.

The following chapter provides more information on the different types of injectors and their working princi-ples, with the focus on the single hole. In Chapter 3 the different properties of a spray are discussed, includingthe atomization phenomena as a main interest. Chapter 4 presents the existing literature models developedin the last decades by observing the different jet characteristics. A comparison between them is illustratedas well, as a function of the influencing parameters. Chapter 5 states the methodologies that can be usedin order to analyze the spray properties, while the final chapter concludes this literature study and presentsrecommendations for the future research.

1

2Single hole injector

The single hole injector is categorized as a non-impinging element. It is considered one of the simplest ele-ments as it consists of an orifice that emerges the liquid or propellant normal to the face of the injector. Inthe automotive industry, engines were relying on this type of injectors in the past [5]. In rocket engines how-ever, a multitude of such orifices form a ’showerhead’ injector and this is used to atomize the propellant andthe oxidizer towards the combustion. An example where the showerhead was used is the V-2 rocket [3], butrecently, more complex hybrid elements such as pintle were introduced in order to achieve a better mixture.The showerhead relies on turbulence and diffusion to achieve mixing [9], but it requires a rather long cham-ber to achieve a complete mixture. Nevertheless, it is still highly used near the chamber walls for film coolingwith propellant.

In order to understand the basics of spray formation and atomization phenomena, this chapter is limited tothe simplest type of injector, namely the single hole. The geometry of its orifice has a high impact on the sprayproperties such as the cone angle or droplet distribution, therefore the importance of the design parametersare discussed here.

2.1. Exit and jet velocity

Spray distributions are functions of orifice size and pressure drop. The latter can be derived from BernoulliEquation (2.1), assuming incompressible flow and a vertical jet. As illustrated in Figure 2.1, an ideal caseis considered with a plain orifice. In Section 2.3 the eventual losses are presented and the velocity is cor-rected.

Figure 2.1: Pressurised liquid issued from a plain orifice

3

4 2. Single hole injector

pr es + 12ρV 2r es −ρg z0 = pV +

1

2ρV 2j −ρg z1 (2.1)

Where pr es represents the pressure in the reservoir and pV is the pressure in the pulverized jet, noted hereas the vapor pressure. As an example, the latter has a value of 2.3kPa for water at 20°. Since the pressurizedliquid in the reservoir is assumed to be static, its velocity Vr es is equal to zero. Also, z0 is the location of thenozzle, the the coordinate system is attached according to Figure 2.1. Therefore z0 is zero as well and the finalexpression of jet velocity can be written as:

V j =√

2(∆P +ρg z1)ρ

(2.2)

Where ∆P is introduced to be the drop pressure i.e. the difference between the reservoir pressure and thevapour pressure of the emerged liquid. Taking a closer look at the exit of the nozzle, the gravitational effectscan be ignored because the value z goes to zero. As a consequence, the idealized exit velocity Ve is introducedby Equation (2.3).

Ve =√

2∆P

ρ(2.3)

2.2. Orifice geometry

The orifice, or the emerging nozzle can take different shapes. The main parameter is the ratio between thelength and the diameter of the orifice (l/d). This ratio is used by empirical models to predict the spray angleor the droplet size, but more details are discussed in Chapter 4. Different types of nozzles are categorized inFigure 2.2. Typically, the holes are between 200 and 600µm in diameter, and the l /d is in range of 2-8, for theshowerhead injector case [10].

(a) Sharp-edged orifice (b) Sharp-edged cone

(c) Short tube with conical entrance (d) Sort tube with rounded entrance

Figure 2.2: Comparison between the different types of orifices [9]

Since the emerging flow is turbulent, the behaviour of the liquid is extremely non-linear. Little research hasbeen made on numerical models that predict this behaviour as a function of the nozzle geometry. Therefore,in the present the experiments presented in Chapter 4 are the main source of investigation.

2.3. Pressure losses 5

2.3. Pressure lossesIn Section 2.1 the exit velocity of a liquid emerging from a plain orifice is expressed as a function of pressuredrop and liquid density by Equation (2.3). In this section, the transition from an idealized plain orifice to thedifferent types of nozzles in Section 2.2 is explained.

Because of the particular shape of different orifices, experiments [9] have shown that the Bernoulli Equa-tion (2.1) equation needs to be corrected. As a consequence, the discharge coefficient Cd has been introduceddue to the pressure losses caused by the specific shape of the nozzle. A few values of the correction factor orthe discharge coefficient of the injectors illustrated in Figure 2.2 are presented in Table 2.1.

Table 2.1: Discharge coefficients of the different types of orifices [9]

Orifice type Diameter [mm] Cd

Sharped-edge orificeabove 2.50 0.61below 2.50 0.65

Sharped-edge cone1.00 0.69-0.701.60 0.72

Short tube with rounded entrance1.00 (l/d >3.0) 0.881.60 (l/d >3.0) 0.901.00 (l/d ∼ 1.0) 0.70

Short tube with conical entrance

0.50 0.701.00 0.821.60 0.762.50 0.80-0.843.20 0.78-0.84

The corrected exit velocity can now be expressed as follows:

Ve = Cd√

2∆P

ρ(2.4)

Moreover, studies have shown that not only the shape of the orifice governs the characteristics of the spray,but also the manufacturing techniques that influence the surface roughness. A slender tactile sensor mea-sures the roughness of typical diesel injectors to be in between 300 and 600nm [11].

2.4. UsageThe debut of rocketry is based a showerhead injector, which is in fact, a cluster of orifices. The V-2 rocket wassuccessfully launched in 1942 for the first time with a thrust provided by a showerhead injector [3]. The mainX15’s rocket engine was also based on this type of injector. On one hand, the showerhead is associated withlow thermal loads on the injector face and a simple manufacturing process. On the other hand, the mixture israther poor compared with the like or unlike impinging injectors, and this has an impact on the combustionefficiency [12]. However, the multitude of orifice elements normal to the injector plate are still used nowadaysfor film cooling.

The importance of injection through an orifice is widely distributed outside the aerospace field. In agri-culture, chemicals are being pulverised using a simple nozzle [13]. In medicine, nasal sprays are designedcarefully such that the drug penetrates into the right spot [6]. Internal combustion engines were relying aswell on the single hole injector in the past [5]. All these examples confirm the important role that single holeinjectors had and still has on the atomization process.

3Spray properties

The combustion process in engines represents a very important step in achieving high performance with theminimum resources. In order to optimize the fuel consumption, nowadays Computational Fluid Dynamics(CFD) methods are being implemented in order to simulate and design advanced engines. Although thesenumerical solutions have the potential to estimate with a rather small error the behaviour of such complexprocesses, the CFD algorithms are dependent on boundary conditions.

To simulate the heat dissipation, the produced thrust, the pressure or the efficiency of a combustion chambervia a CFD method, the spray characteristics of the emerged fuel through the injector are to be set as boundaryconditions. This means that the velocity distributions, the spray angle, or the droplet size need to be com-puted prior to the CFD code. In this chapter, the latter properties of a spray are presented starting form thebreakup of the liquid sheet to the primary and secondary atomization phenomena.

3.1. Jet disintegration

When issuing from a nozzle, the liquid is found to be in the form of a relatively thin sheet compared withthe radius of the orifice [14, 15]. This sheet is described by unstable waves that will eventually breakup intoligaments as shown in Figure 3.1. Finally, the primary atomization phase is defined when small droplets startto divide from the ligaments. When the turbulence and collision start to occur, the distribution of the dropletswill be affected, and this phase is known as the secondary breakup zone [16].

Figure 3.1: Illustration of a liquid emerged form a 2D nozzle and its sheet breakup. Adapted from [17]

7

8 3. Spray properties

3.2. Droplet formationNumerous models presented in Chapter 4 use the terminology of Sauter mean diameter (SMD), or d32. It isdefined by to be proportional with the ratio of volume and surface area of the particle of interest, in this case,the droplet [18]. This is introduced by Equation (3.1) where Ap and VP represent the area and the volumerespectively of the particle.

d32 =d 3vd 2s

= 6Vp /πAp /π

= 6 VpAp

(3.1)

The Weber number (We) is described by Equation (3.2) and represents the ratio between the inertial forceand the surface tension force. When the kinetic energy of the liquid becomes sufficiently large to overcomethe surface energy, i.e. when the velocity increases, a continuous jet is formed [19]. This is often used in theexisting models to describe the droplets.

W e =ρd0V 2jσ

(3.2)

The density ρ and the surface tension σ values can be found as a function of temperature, for the specificliquid and d0 is the initial diameter of the droplet.

As a preliminary size of the droplet, the optimum wavelength λopt of an inviscid liquid jet can be used for anidealized breakup [19]. A schematic of this phenomena can be observed in Figure 3.2 and the expression asa function of the jet radius r is showed by Equation (3.3). This is referred to as the "Rayleigh breakup" and isfurthered explained by Lord Rayleigh [20] and Plateau [21].

λopt = 2p

2πr (3.3)

The dynamics of the droplets is not as obvious as it might seem at the first look. That is because the maximumvelocity is attained at the centre of the spray, whereas towards the periphery of the cone, the velocity of thedroplets is decreasing [17].

Figure 3.2: Droplet formation as a result of the liquid breakup [19]

4Literature models

In this chapter, different models are presented to characterize the droplet size (SMD) in Section 4.1 and thecone angle formed by the emerged spray in Section 4.2. At the end of each section, the models are comparedfor a better insight, using the script from Appendix A. In order to do this, numerical values are taken as anexample. It is considered that water is emerging from a nozzle with an orifice diameter of 300µm, an ori-fice length of 1mm and a discharge coefficient of 0.65. Moreover, the geometry constant A is expressed inEquation (4.1) as a function of this parameters. The water and air (ambient) properties at 20°are presentedin Table 4.1, where the index L stands for liquid (water in this case) and index A stands for air. It is importantto be mentioned that the density of water is changing as a function of the pressure in the reservoir. This issupported by Equation (4.2)

A = 3+0.28 ld

(4.1)

ρL = ρL01−∆P/EL

(4.2)

Table 4.1: Air and water properties at 20°

Parameter Value Unit

EL 2.1200·109 PaνL 1.0034·10−6 m2s−1µL 1.0020·10−3 kgm−1s−1ρL0 9.9821·102 kgm−3σ 7.2800·10−2 kgs−2pV 2.3388·103 PaµA 1.8460·10−5 kgm−1s−1ρA 1.2041 kgm−3

4.1. Droplet size

Many researchers have succeed in trying to create a numerical model based on the experiments performedwith a specific injector. In this section, the focus is pointed on the plain orifice injector, i.e the simplest case ofcharacterizing the behaviour of the jet. The Sauter mean diameter is calculated here by different researchersand the results are compared at the end of this section. Note that in Equation (4.5) ∆P is in bar and d32 is inµm.

9

10 4. Literature models

4.1.1. Harmon [22]

d32 = 3330d 0.3µ0.07L ρ−0.648L σ−0.15V −0.55e µ0.78A ρ−0.052A (4.3)

4.1.2. Hiroyasu et al. [23]

d32 = 4.12d Re0.12W e−0.75(µLµA

)0.54( ρLρA

)0.18(4.4)

4.1.3. Elkotb [24]

d32 = 6156ν0.385L (σρL)0.737ρ0.06A ∆P−0.54 (4.5)

4.1.4. Varde [25]

d32 = 8.7d(ReW e)−0.28 (4.6)

4.1.5. Merrington and Richardson [26]

d32 =500d 1.2ν0.2L

Ve(4.7)

4.1.6. Comparison between different models

Figure 4.1: The Sauter mean diameter of the droplets as a function of the pressure in the reservoir

4.2. Spray angle 11

4.2. Spray angle

The cone angle of the spray plays a major role in the injector design. When the angle θ increases, the at-omization is finer as the liquid is exposed to the ambient. Therefore the liquid tends to breakup faster intosmaller droplets. θ is defined as the angle between the line normal to the face of the injector and the linefrom the centre of the nozzle to to the periphery of the cone at the distance of 60d from the orifice. THis isbetter visualised in Figure 4.2. In this section, different models from the literature that describe the angle θare presented. Note that Equation (4.11) is in radians. At the end of this section, all models are compared inFigure 4.3.

Figure 4.2: Schematic of the spray angle

4.2.1. Abramovich [27]

θ = tan−1(0.13

(1+ ρA

ρL

))(4.8)

4.2.2. Reitz and Bracco [28] simplified by Heywood [29]

θ = tan−1(π

3A

√ρA

ρL

)(4.9)

4.2.3. Ruiz and Chigier [30]

θ = tan−1[π

3A

√ρA

ρL

( ReW e

)−0.25](4.10)

4.2.4. Arai [31]

θ = 0.025(ρL∆Pd 2

µ2L

)(4.11)

12 4. Literature models

4.2.5. Arregle et al. [32]

θ = tan−1(d 0.508P 0.00943r es ρ

0.335A

)(4.12)

4.2.6. Comparison between different models

Figure 4.3: The angle of the spray as a function of the pressure in the reservoir

5Experiments and equipment

Several experimental techniques are introduced in this chapter. In order to determine the spray propertiesdiscussed in Chapter 3 and validate the empirical models presented in Chapter 4, different methods can beadopted. The determination of the SMD, spray angle and spray distribution are explained here.

5.1. Laser diffraction

Fraunhofer diffraction theory states that the particle’s diameter is proportional to the light intensity scatteredby the particle itself. This is the main principle found at the basics of laser diffraction and an illustration isprovided by Figure 5.1.

The setup is composed of a light source and a detector. Being placed perpendicular to the measuring zone,the laser beam exposes the particle. The diffraction pattern generated by the latter is received by a detector bymeans of a lens. Consisting of multiple ring shaped elements that measure the light distribution and intensity,the detector sends to the data acquisition system (DAQ) an analogical signal. An example of a laser diffractionsystem is the Helos by Sympatec.

The advantage of such a system is reinforced by the simplicity of a ready-to-run system. No calibration isrequired by this apparatus and the accuracy of the measured diameter of the particles can go up to the levelof nm. However, it is advised by the manufacturer to simply check the system with a known diameter of aparticle [33].

Figure 5.1: Illustration of the diffraction pattern resulted by exposing a particle [33]

13

14 5. Experiments and equipment

5.2. Image based processing

When a picture is taken to a subject, the exposure time or the shutter speed, the aperture and the surroundinglight play a major role. Depending on the dynamics of the subject, the exposure time needs to be decreasedas the velocity of the particle increases. As a consequence, the blur motion is calculated as the object velocitytimes the shutter speed. When the speed of an object is too high and the exposure time is set on a relativelyhigh period, a picture similar to Figure 5.2a can be taken. Sometimes, in art photography this is desired, but ifthe diameter of a droplet is to be determined, the picture has to be sharp, as in Figure 5.2b where the breakupphenomena can be seen.

Between the two photos presented in Figure 5.2, one can notice the fact that the frozen picture is less bright.This is because the small exposure time has to be compensated with more light. To do this several parameterscan be changed such as: a larger aperture, an increase in ISO which has the drawback of decreasing theresolution, the use of a flash, or a combination of all of these. In high speed photography, the flash is the mostimportant component, as it is presented later in this section.

(a) Blurred picture. Settings: shutter speed: 1/8 s,aperture: f/32, ISO:100, no flash

(b) Frozen picture. Settings: shutter speed: 1/4000 s,aperture: f/5.6, ISO:800, no flash

Figure 5.2: Example of a blurred and frozen pictures taken in a park

5.2.1. Shadowgraphy

When a picture is taken with the flash behind the camera, blooming regions make their way to the particleof interest, creating a nonhomogeneous area. This can make post processing very difficult since the edgesof the object are not so clear anymore. This is why the principle of shadowgraphy is introduced. It consistsof a camera facing a flash through a diffuser plate, with the particles travelling in between them as shown inFigure 5.4. In this way, the edges of the particles can be better determined, creating a higher contrast betweenthem and the background (almost white, due to the flash), illustrated in Figure 5.3.

As mentioned previously, the light source i.e. the flash or a laser, is a very important component of the setup.Because of the advancements in technology, a laser pulse can have a rather high energy in a very short amountof time, in orders of ns. Because it is mechanically and electronically easier to have a short pulse of light thana high shutter speed inside the camera, many of the high speed photographers chose to take pictures indifferent way. The environment needs to be completely dark, the camera has to start taking a long picture(with an exposure time sometimes even higher than 10s) and when the image needs to be frozen, the lightsource is triggered for a split of a moment. Even though the camera is set to take a single picture for 10s, theonly moment when the image sensor inside the camera will receive information will be during the pulse ofthe flash or laser.

5.3. Patternator 15

Figure 5.3: A droplet imaged usingshadowgraphy principle [34]

Figure 5.4: Shadowgraphy setup example [34]

5.3. PatternatorIn order to achieve a high efficiency of the combustion and low pollutant emissions, the spray is to have auniform dispersion. In order to check this image processing techiques can be implemented [34]. However,the volumetric or mass distribution of droplets can be better measured using a patternator. A schematic isprovided in Figure 5.5. A radial patternator consists of small tubes radially and equidistantly placed down-stream of the spray flow. The volume of liquid gathered in each tube can be later plotted as a function of sprayangle.

The symmetry is not only important for the efficiency, but also for the cycle life of the combustor wall. A defectinjector that produces an asymmetric spray can damage the combustion chamber by overheating certainareas [35].

Figure 5.5: Radial spray distribution instrument [35]

5.4. Software packagesTo conclude this chapter, it is very important to emphasise the importance of software. Neither image basedsizing nor laser diffraction measurements methods can be performed without post processing algorithms.Some of the most popular image toolboxes used in the field are: Python Scikit-image, Image-J with a GUI orMatlab Image Processing as used in [34].

6Conclusion

Starting from the liquid rocket engines, this paper has introduced the reader to the steps of designing andtesting one of the simplest form of injector. The single hole element is presented in Chapter 2 and is arguedwhy the need of such a simple plain orifice is fundamental for understanding the emerged flow characteris-tics. Properties such as the spray angle and the droplet size and distribution are explained in Chapter 3 andthe way to measure them is resented by Chapter 5.

At this phase, little is still known about the non-linear behaviour of the spray, but experiments have shown inthe past that empirical models can be created up to a certain extent. In the future more and more experimentswill be added to the community supporting some models and contradicting others. Chapter 4 contains someof the available models that are in agreement between them and describe the parameters of the spray, theSauter mean diameter and the cone angle respectively.

To conclude this paper, the author recommends further research and investigation not only in the experi-mental field, which creates thousands of data points every year, but also in computational modelling andphysical phenomena that can describe the aforementioned non-linearity.

17

AAppendix: SMD and spray angle models

import matplotlib . pyplot as p l timport numpy as np

##################### DEFINING CONSTANTS AT 20 DEG C #####################

E = 2.1500E9 # Pa − Liquid bulk e l a s t i c i t y modulusnu_L = 1.0034E−6 # m^2/s − Liquid kinematic v i s c o s i t ymu_L = 1.0020E−3 # kg / (ms) − Liquid absolute ( dynamic ) v i s c o s i t yrho_L_0 = 9.9821E2 # kg/m̂ 3 − Liquid densitysigma = 7.2800E−2 # kg/ s^2 − Liquid surface tensionp_V = 2.3388E3 # Pa − Liquid vapor pressuremu_A = 1.8460E−5 # kg (ms) − Air absolute ( dynamic ) v i s c o s i t yrho_A = 1.2041 # kg/m̂ 3 − Air density

############################ NOZZLE GEOMETRY #############################

d_0 = 0.3E−3 # m − O r i f i c e diameterl_0 = 1E−3 # m − Length of the o r i f i c eA = 3+0.28* l_0 /d_0 # − − Geometry constantCd = 0.65 # − − Discharge c o e f f i c i e n t

######################### PARAMETER DEFINITIONS ##########################

# Define the pressure vessel range from 0 to 100 barP_vessel_bar = np . arange (0 ,101 ,1)P_vessel = P_vessel_bar *1e5

# Define pressure drop as the dif ference between the P_vessel and p_Vdelta_P = P_vessel − p_V*np . ones (101)

# Redifine l i qu i d density as a function of pressurerho_L = rho_L_0/(1−delta_P /E)

19

20 A. Appendix: SMD and spray angle models

# Calculate the e x i t v e l o c i t y of the j e tV_e = Cd*np . sqrt (2* delta_P /rho_L )

# Caluclate the volumetric flow rateQ = V_e*np . pi * ( d_0 /2)**2

# Defining the Reynolds numberRe = rho_L *V_e*d_0/mu_L

# Defining the Weber numberWe = rho_L *V_e **2* d_0/sigma

############################### SMD MODELS ###############################

SMD_harmon = 3330*d_0 * * 0 . 3 *mu_L* * 0 . 0 7 * rho_L **( −0.648)* sigma **( −0.15)* \V_e **( −0.55)*mu_A* * 0 . 7 8 * rho_A **( −0.052)*1 e6

# Delta P needs to be in kg/cm^2SMD_hiroyasu = 4.12* d_0*Re * * 0 . 1 2 *We**( −0.75)*(mu_L/mu_A) * * 0 . 5 4 * \

( rho_L/rho_A ) * * 0 . 1 8 * 1 e6

# Delta P needs to be in barSMD_elkotb = 6156*nu_L * * 0 . 3 8 5 * ( sigma* rho_L ) * * 0 . 7 3 7 * rho_A * * 0 . 0 6 * \

( delta_P /1e5 )**( −0.54)

SMD_varde = 8.7* d_0 * ( Re*We)**( −0.28)*1 e6

SMD_merrington = 500*d_0 * * 1 . 2 * nu_L * * 0 . 2 / V_e*1e6

############################## ANGLE MODELS ##############################

theta_abramovich = 180/ pi *np . arctan (0.13*(1+ rho_A/rho_L ) )

t h e t a _ r e i t z = 180/ pi *np . arctan (2* pi / sqrt (3*A ) * ( rho_A/rho_L ) * * 0 . 5 )

theta_ruiz = 180/ pi *np . arctan (2* pi / sqrt (3*A ) * ( rho_A/rho_L ) * * 0 . 5 * \(Re/We) * * ( −0 . 2 5 ) )

theta_arai = 180/ pi * 0 . 0 2 5 * ( rho_A* delta_P /1e5 *d_0 **2/mu_A* * 2 ) * * 0 . 2 5

theta_arregle = 180/ pi *np . arctan ( d_0 **0.508* P_vessel **0.00943* rho_L * * 0 . 3 3 5 )

################################# PLOTS ##################################

f = p l t . f i g u r e ( 1 )p l t . plot ( P_vessel_bar [ 1 1 : ] , SMD_harmon[ 1 1 : ] , color = ’ k ’ , l a be l = ’Harmon’ )p l t . plot ( P_vessel_bar [ 1 1 : ] , SMD_hiroyasu [ 1 1 : ] , ’ k−−’, l ab el = ’ Hiroyasu et a l . ’ )p l t . plot ( P_vessel_bar [ 1 1 : ] , SMD_elkotb [ 1 1 : ] , marker = ’o ’ , markersize =8 ,

markevery=5 , markerfacecolor ="None" , color = ’ k ’ , l ab e l = ’ Elkotb ’ )p l t . plot ( P_vessel_bar [ 1 1 : ] , SMD_varde [ 1 1 : ] , marker = ’ s ’ , markersize =8 ,

markevery=5 , markerfacecolor ="None" , color = ’ k ’ , l ab e l = ’ Varde ’ )

21

p l t . plot ( P_vessel_bar [ 1 1 : ] , SMD_merrington [ 1 1 : ] , marker = ’ v ’ , markersize =8 ,markevery=5 , markerfacecolor ="None" , color = ’ k ’ ,l a be l = ’ Merrington and Richardson ’ )

p l t . x label ( " Reservoir pressure [ bar ] " , fo nts i z e = 25)p l t . y label ( " d$_ {32} $ [ $\mu$m] " , fo nts i ze = 25)p l t . legend ( fonts i z e = 25)p l t . tick_params ( l a b e l s i z e =25)p l t . grid ( True )f . show ( )

g = p l t . f i g u r e ( 2 )p l t . plot ( P_vessel_bar [ 1 1 : ] , theta_abramovich [ 1 1 : ] , marker = ’o ’ ,

markersize =8 , markevery=5 , markerfacecolor ="None" , color = ’ k ’ ,l a be l = ’ Abramovich ’ )

p l t . plot ( P_vessel_bar [ 1 1 : ] , t h e t a _ r e i t z [ 1 1 : ] , color = ’ k ’ ,l a be l = ’ Reitz and Bracco ’ )

p l t . plot ( P_vessel_bar [ 1 1 : ] , theta_arai [ 1 1 : ] , marker = ’ ^ ’ , markersize =8 ,markevery=5 , markerfacecolor ="None" , color = ’ k ’ ,l a be l = ’ Hiroyasu and Arai ’ )

p l t . plot ( P_vessel_bar [ 1 1 : ] , theta_arregle [ 1 1 : ] , marker = ’ s ’ , markersize =8 ,markevery=5 , markerfacecolor ="None" , color = ’ k ’ ,l a be l = ’ Arregle et a l . ’ )

p l t . plot ( P_vessel_bar [ 1 1 : ] , theta_ruiz [ 1 1 : ] , ’ k−−’, l ab el = ’ Ruiz and Chigier ’ )p l t . x label ( " Reservoir pressure [ bar ] " , fo nts i z e = 25)p l t . y label ( " $\\ theta$ [ deg ] " , fo nts i z e = 25)axes = p l t . gca ( )axes . set_ylim ( [ 0 , 3 0 ] )p l t . legend ( loc =2 , fo nts i z e = 25)p l t . tick_params ( l a b e l s i z e =25)p l t . grid ( True )g . show ( )

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List of SymbolsIntroductionSingle hole injectorExit and jet velocityOrifice geometryPressure lossesUsage

Spray propertiesJet disintegrationDroplet formation

Literature modelsDroplet sizeSpray angle

Experiments and equipmentLaser diffractionImage based processingPatternatorSoftware packages

ConclusionAppendix: SMD and spray angle modelsBibliography