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Aerosol Characterization and Analytical Modeling of Concentric Pneumatic and Flow Focusing Nebulizers for Sample Introduction
by
Arash Kashani
A thesis submitted in conformity with the requirements for the degree of PhD
Mechanical and Industrial Engineering Department University of Toronto
© Copyright by Arash Kashani, 2010
ii
Aerosol Characterization and Analytical Modeling of Concentric
Pneumatic and Flow Focusing Nebulizers for Sample Introduction
Arash Kashani
PhD
Mechanical and Industrial Engineering Department
University of Toronto
2010
Abstract
A concentric pneumatic nebulizer (CPN) and a custom designed flow focusing nebulizer (FFN)
are characterized. As will be shown, the classical Nukiyama-Tanasawa and Rizk-Lefebvre
models lead to erroneous size prediction for the concentric nebulizer under typical operating
conditions due to its specific design, geometry, dimension and different flow regimes. The
models are then modified to improve the agreement with the experimental results. The size
prediction of the modified models together with the spray velocity characterization are used to
determine the overall nebulizer efficiency and also employed as input to a new Maximum
Entropy Principle (MEP) based model to predict joint size-velocity distribution analytically. The
new MEP model is exploited to study the local variation of size-velocity distribution in contrast
to the classical models where MEP is applied globally to the entire spray cross section. As will
be demonstrated, the velocity distribution of the classical MEP models shows poor agreement
with experiments for the cases under study. Modifications to the original MEP modeling are
proposed to overcome this deficiency. In addition, the new joint size-velocity distribution agrees
better with our general understanding of the drag law and yields realistic results.
iii
Acknowledgments
I am deeply grateful to my supervisor, Professor Mostaghmi for his five years of invaluable
guidance, encouragement, support and also for giving me the luxury of experimenting and doing
the project my way. I would also like to thank Professor Coyle for attending my PhD defense and
his careful review of my work. I owe my supervising committee, Professors Sullivan, Chandra
and Ashgriz for their continued support and insightful comments. My special thanks goes to
Professor Ashgriz and his graduate student, Amirreza Amighi, who kindly let us use their lab
facilities and assisted me with the experimental part of the project. I appreciate Professor
Tanner’s group from the chemical department, particularly Mr. Vorobiev for designing the
nebulizer prototypes and Dr. Bandura for reviewing my papers, his great vision and involvement
in the project.
I am thankful to my colleagues at the Centre for Advanced Coating Technologies (CACT),
especially Dr. Hanif Montazeri for his exceptional talent and our fruitful discussion along the
way. I also owe Dr. Ala Moradian. His experience and emotional support helped me during the
tough days. I was lucky to have the company of two great friends, Araz Sarchami and Babak
Samareh in the past years.
The understanding, hard work and support of my beloved wife, Zhinous, and my great parents
made this journey possible for me. To each of them I am sincerely thankful.
v
Table of Contents
Contents
Acknowledgments .......................................................................................................................... iii
Table of Contents ............................................................................................................................ v
List of Tables ................................................................................................................................ vii
List of Figures .............................................................................................................................. viii
List of Appendices ........................................................................................................................ xii
Chapter 1 Introduction .................................................................................................................... 1
1.1 Overview of components and processes in ICP-MS ........................................................... 1
1.2 Sample introduction in ICP-MS .......................................................................................... 3
1.3 Concentric Pneumatic Nebulizer (CPN) – Design and Fundamentals ............................... 5
1.4 Microsample Introduction ................................................................................................. 12
1.5 Objectives ......................................................................................................................... 19
1.6 Summary ........................................................................................................................... 20
Chapter 2 Aerosol Size Characterization of Concentric Pneumatic Nebulizer ............................ 22
2.1 Experiment Setup .............................................................................................................. 22
2.2 Nukiyama–Tanasawa Correlation ..................................................................................... 24
2.3 Rizk–Lefebvre Correlation ............................................................................................... 43
2.4 Variation of Characteristic Mean Drop Sizes ................................................................... 51
2.5 Nebulization Efficiency .................................................................................................... 55
2.6 Contribution ...................................................................................................................... 57
Chapter 3 Aerosol Size Characterization of Flow Focusing Nebulizer ........................................ 59
3.1 Nozzle Design ................................................................................................................... 59
3.2 Theoretical Background .................................................................................................... 63
vi
3.3 Droplet Size Modeling and Variation of Characteristic Mean Drop Sizes ....................... 70
3.4 Nebulizer Performance ..................................................................................................... 76
3.5 Contribution ...................................................................................................................... 83
Chapter 4 Aerosol Velocity Characterization ............................................................................... 85
4.1 General Considerations ..................................................................................................... 85
4.2 Aerosol Velocity Modeling ............................................................................................... 92
4.3 Contribution ...................................................................................................................... 98
Chapter 5 Maximum Entropy Principle - Application on Aerosol Size and Velocity Modeling . 99
5.1 The Need for Statistical Measures and Maximum Entropy Principle .............................. 99
5.2 MEP Formulation ............................................................................................................ 101
5.3 Number or Volume Based Probability Distribution Function? ...................................... 104
5.4 Global and Local Implementation of MEP ..................................................................... 110
5.5 Numerical Solution ......................................................................................................... 113
5.6 MEP Results and Discussion .......................................................................................... 115
5.7 Contribution .................................................................................................................... 127
Chapter 6 Concluding Remarks and Future Works .................................................................... 129
6.1 Contribution .................................................................................................................... 129
6.2 Future Works .................................................................................................................. 131
References ................................................................................................................................... 133
Appendices .................................................................................................................................. 146
vii
List of Tables
Table 1-1: Critical dimensions of different nebulizers used in ICP-MS, * Pressures required to
reach 0.25 and 0.6 (l/min) gas flow rates respectively. ................................................................. 14
Table 2-1: Operating conditions for nebulizers and the measurement devices exploited in
experiments ................................................................................................................................... 25
Table 2-2: Coefficients and exponent of original NT, modified NT and the fitted NT models. .. 38
Table 2-3: Comparison between the model (NT, FNT) and experimental values of D32 (µm). ... 44
Table 2-4: Coefficients and exponents of different RL type correlations, RL=original Rizk-
Lefebvre model, MRL-G=Gras’ modified model, MRL-K= Kahen et al.’s modified model,
FRL= present fitted RL model ...................................................................................................... 48
viii
List of Figures
Figure 1-1: Schematic major components in ICP-MS: i- Sample introduction system ii- Plasma
and iii- Mass Spectrometer. ............................................................................................................ 2
Figure 1-2: (a) A conventional CPN with its critical dimensions and (b) a view of nebulizer tip
under microscope, model: Meinhard TR30-C3. ............................................................................. 7
Figure 1-3: Different design and nebulizer tip of Meinhard concentric pneumatic nebulizers,
(Courtesy of Meinhard Glass Products). ......................................................................................... 8
Figure 1-4: Schematic of processes taking place at the exit of CPN. ........................................... 11
Figure 1-5: Successive stages in an idealized sheet Breakup ....................................................... 11
Figure 1-6: Close view of micronebulizer tip, (a) HEN, (b) MMN, (c) MCN and (d) conventional
CPN ............................................................................................................................................... 16
Figure 1-7: Schematic design of DIHEN and its coupling with plasma torch, (35). .................... 18
Figure 2-1: Experiment setup, dashed line represents makeup gas line and is only used for FFN.
....................................................................................................................................................... 26
Figure 2-2: Schematic of PDPA and fiber optics. ......................................................................... 27
Figure 2-3: Sauter mean diameter versus gas flow rate for distilled water and methanol from
experiment and the original NT model at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s). . 32
Figure 2-4: Error between experiment and the original NT model grows larger by increasing the
liquid flow rate for (a) distilled water and (b) methanol. .............................................................. 33
Figure 2-5: Contribution of first and second term in Nukiyama – Tanasawa equation. Δ: Ql=1
(µl/s), ○: Ql=5 (µl/s) and ◊: Ql=10 (µl/s). Dashed and solid lines represent first and second term
of Nukiyama - Tanasawa equation respectively. .......................................................................... 34
Figure 2-6: Typical size distribution with TR30-C3 CPN at Ql=5 (µl/s) and Qg= 500 (sccm),
liquid: distilled water. D32=20.8 (µm), Dpeak=9.4 (µm). ............................................................... 37
ix
Figure 2-7: Sauter mean diameter versus gas flow rate from experiment, original NT model and
Kahen et al’s MNT model for distilled water at Ql=10 (µl/s). ..................................................... 39
Figure 2-8: Sauter mean diameter versus gas flow rate for distilled water and methanol from
experiment and the FNT model at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s). ............ 41
Figure 2-9: Ratio of calculated to measured Sauter mean diameter versus gas flow rate for
distilled water. ............................................................................................................................... 42
Figure 2-10: Measured versus calculated Sauter mean diameter from the original NT and fitted
NT (FNT) models. ........................................................................................................................ 43
Figure 2-11: Sauter mean diameter versus gas flow rate for distilled water from experiment and
different RL type models at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s). ...................... 50
Figure 2-12: Variation of (a) D30/D32 and (b) D30/D-10 versus the normalized gas low rates at the
downstream axial location of z=10 (mm). .................................................................................... 53
Figure 2-13: Variation of characteristic moment ratio with axial location (a) D30/D-10 and (b)
D30/D32 at Ql=5 (μl/s). ................................................................................................................... 54
Figure 2-14: Nebulization efficiency versus gas flow rates for distilled water measure at z=10
(mm). ............................................................................................................................................. 57
Figure 3-1: Schematic design of the first FFN. ............................................................................ 60
Figure 3-2: The actual prototype of the first custom designed FFN. ............................................ 62
Figure 3-3: Schematic design of the second FFN. ........................................................................ 63
Figure 3-4: Figure 18- Photographs taken from inside and outside of FFN (40) showing (a)-
Capillary and liquid filament, (b)-The liquid jet exiting the orifice and (c)- Unstable wave growth
on the filament surface, breakup and droplet generation .............................................................. 64
Figure 3-5: View of orifice hole and the emitted micro jet of the first FFN prototype. Orifice
diameter do=150 (µm), ΔPg=70 (Kpag) and Ql=1.66 (µl/s). Predicted jet diameter dj= 13.4 (µm).
....................................................................................................................................................... 68
x
Figure 3-6: Distribution curves for flow conditions given in Figure 3-5 (a) number and volume
distribution. (b) cumulative number and volume distribution. ..................................................... 69
Figure 3-7: Comparison between drop size models and experiments at different liquid flow rates
for FFN at: (a) 0.1, (b) 0.2, (c) 0.4, (d) 0.6 and (e) 1.0 (µl/s). ..................................................... 74
Figure 3-8: Variation of (a) D30/D32 and (b) D30/D-10 versus the normalized gas low rates at the
downstream axial location of z=10 (mm). .................................................................................... 75
Figure 3-9: Variation of characteristic moments of the primary size distribution with the jet based
Weber number meared at z=10 (mm). .......................................................................................... 77
Figure 3-10: Standard deviation of the primary size distribution of the FFN versus the jet based
Weber number measured at z=10 (mm). ....................................................................................... 78
Figure 3-11: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=0.2 (µl/s), Qg= 150 (milt/min), D10/dj=2.26 and Wedj=4.5, (point 1 of Figure 3-
9). .................................................................................................................................................. 80
Figure 3-12: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=0.6 (µl/s), Qg= 180 (milt/min), D10/dj=1.5 and Wedj=9.6 (point 2 of Figure 3-
9). .................................................................................................................................................. 81
Figure 3-13: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=1.0 (µl/s), Qg= 320 (ml/min), D10/dj=0.69 and Wedj=20.0 (point 3 of Figure 3-
9). .................................................................................................................................................. 82
Figure 3-14: Comparison between FFN and CPN running at comparable flow conditions Ql=1.0
(μl/s) and Qg~320-370 (ml/min). ................................................................................................... 83
Figure 4-1: Gas in a free turbulent jet exiting a 260 (µm) nebulizer operating at sonic flow and
droplet velocity for 1, 10, 20, 50 and 100 (µm) droplet diameter. ................................................ 91
Figure 4-2: Droplet Mean and Root Mean Square (rms) speed and unseeded gas velocity versus
gas flow rate, measured at z=10 (mm) for (a) CPN and (b) FFN. ................................................ 95
Figure 4-3: Span of (a) velocity and (b) size distribution measured at z=10 (mm) for CPN. ....... 96
xi
Figure 4-4: Span of (a) velocity and (b) size distribution measured at z=10 (mm) for FFN. ....... 97
Figure 5-1: Size and velocity space and probability distribution function of aerosol ................ 102
Figure 5-2: Sellens and Brzustowski’s control volume for MEP modeling. .............................. 106
Figure 5-3: Li and Tankin’s control volume for MEP modeling. ............................................... 110
Figure 5-4: Primary aerosol size distribution measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min), D30=16.2 (µm) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), D30=14.4 (µm).
Error bars represent standard deviation.. .................................................................................... 117
Figure 5-5: Primary aerosol velocity distribution measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min), Uref=47.2 (m/s) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Uref=17.6 (m/s).
Error bars represent standard deviation.. .................................................................................... 121
Figure 5-6: Mean velocity versus droplet diameter measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Error bars represent standard
deviation.. .................................................................................................................................... 124
Figure 5-7: Root mean square velocity versus droplet diameter measured at z=10 (mm) for (a)
Ql=5 (µl/s) at Qg=500 (ml/min) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Error bars represent
standard deviation. ...................................................................................................................... 126
xii
List of Appendices
Appendix A: Generation of Ripples by Wind Blowing over a Viscous Fluid…………………146
Appendix B: PDPA Calibration and Measurement…………………………………….………148
Appendix C: Axial and Radial Variation of D30/D32 and D30/D-10 Ratios of the FFN…………153
Appendix D: Spatial Variation of Mean Droplet Velocity Moments………………………..…161
Appendix E: Empirical Probability Distribution Functions ...…………………………………165
Appendix F: Derivation of Shannon entropy ………………………………………………….169
Appendix G: Bayesian and Shannon entropy …………………………………………………172
1
Chapter 1 Introduction
Overview of components and processes in ICP-MS 1.1
Radio frequency (RF) inductively coupled plasma (ICP) discharges are commonly used as
excitation and ionization sources in atomic spectrometry. Inductively Coupled Plasma Mass
Spectrometry (ICP-MS) is a well established method of trace and ultra-trace elemental and
isotopic analysis (1) and is extensively covered in the literature (2).
A typical ICP-MS device is composed of three major parts (i) Sample Introduction system, (ii)
Plasma and (iii) Mass Spectrometer as shown in Figure 1-1. First an analytical ICP is formed in a
stream of gas flowing through an assembly of three concentric quartz tubes (outer tube, inner
tube and injector tube) known as the plasma torch (3). An induction coil placed around the
plasma torch is initially triggered by an ignition circuit and forms a radio-frequency
electromagnetic field. The electromagnetic field in turn accelerates electrons and transfers energy
from the coil to the plasma in inelastic collisions with gas atoms. The physical properties of the
plasma such as ionization energy and thermal conductivity strongly depend upon the carrier gas
(4). Rare gases are usually used to generate plasma because they emit only atomic spectra in
emission spectrometry and relatively simple spectra in mass spectrometry. Among rare gases, Ar
is generally preferable due to its availability, low cost and higher kinetic energy in contrast to He
and Ne, however its low thermal conductivity (in comparison to the other two) requires a larger
sample residence time in the plasma (4).
Modern ICP-MS instruments typically operate at 1.5-2 (kW) with frequencies of 27 or 40 (MHz)
to generate gas kinetic temperature of 4000-7000 (K) and 0.1 percent ionization degree for
Argon. These large temperatures would increase the number of free electrons and also the
viscosity up to a factor of 10 for Ar in comparison to the room condition. The increasing viscous
effects would normally resist the sample introduction into the plasma. However due to the skin
effect phenomenon in the HF (high frequency) field, the energy is mainly deposited at the
periphery of the plasma, i.e. temperature and viscosity are lower along the axis of the plasma.
2
Hence the central-axial zone of the plasma facilitates sample introduction through penetration of
a carrier gas having a sufficient speed. The skin effect phenomenon explains the success of ICP
over other forms of plasmas such as microwave-induced plasma or direct-current plasma (4). As
proven the skin depth, that is the penetration of the energy at the periphery of plasma, and
coupling efficiency, which relates to the ratio of the torch radius to the skin depth is determined
by the operating frequencies (5). It has been claimed that the radio frequency of 40 (MHz) leads
to lower electron number density and gas kinetic temperature at the axis which as a result
facilitates the sample introduction (6), (7) and (8). To have an efficient spectral analysis, the
liquid sample must be completely desolvated, vaporized, atomized and ionized in the ICP torch
before entering the mass spectrometer of Figure 1-1. This is usually achieved by increasing the
surface area of the liquid sample, i.e. an aerosol is formed to enhance the rate of heat and mass
transfer. Although there are several methods for aerosol formation, pneumatic nebulization to
this day is the most widely used method for sample introduction in ICP-MS. Since the plasma
itself is produced from the argon gas stream, it would be logical to energize and employ the same
gas stream for aerosol generation (4). This is the basis of pneumatic nebulization which will be
further discussed in the next sections.
ArICP Torch
Waste
Spray Chamber Pneumatic Nebulizer
Liquid SamplePump
Mass Spectrometer
Plasma
Sample Introduction
Figure 1-1: Schematic major components in ICP-MS: i- Sample introduction system ii- Plasma
and iii- Mass Spectrometer.
3
Sample introduction in ICP-MS 1.2
Sample Introduction in ICP-MS includes two major separate processes in general: First, aerosol
generation usually by means of pneumatic nebulization and second, aerosol modification or
filtration in a spray chamber.
Although there are a variety of pneumatic nebulizers in ICP-MS, each suitable for a particular
application, they all must meet some general requirements to produce an ideal stream of aerosol.
For instance Kahen et al. mentioned that an ideal aerosol for ICP-MS must contain small and
slow droplets with uniform size and velocity (9). In fact an ideal aerosol should contain droplets
smaller than 10 (μm) for complete desolvation (10), since large droplets are not consumable by
the plasma and may halt spectrometry by cooling a surprisingly large (1-2 mm wide) volume of
the plasma (11) whereas droplets larger than 25 (μm) are not desolvated at all. Todoli and
Mermet (4) state the ideal aerosol must also have uniform droplet number density, uniform
spatial droplet diameters and have similar characteristics irrespective of the sample composition,
i.e. the physical properties should not affect aerosol characteristics. Mclean et al. (12) also add
that the key properties of the plasma like gas temperature, electron temperature and number
density should not be altered by the aerosol. As well, the presence of the aerosol shouldn’t affect
the optimum sampling depth of ions in the analysis of different samples and furthermore the
aerosol must not contribute a solvent load to the plasma. According to the ideal aerosol
properties mentioned above and some other considerations an ideal nebulizer is defined by (12)
and (13) as a device that:
i. consumes small quantities of sample and reagents (small consumption rate),
ii. is able to nebulize wide ranges of solution,
iii. provides 100 percent transport efficiency,
iv. nebulizes solutions containing high solid concentration without clogging or premature
failure,
v. contains no dead volume,
4
vi. contributes no adverse solvent load effects,
vii. its aerosol properties can be predicted by simple models,
viii. generates fine and monodisperse droplets,
ix. creates a narrow plume,
x. is rugged, inexpensive and easy to use.
The currently available nebulizers, as will be discussed, are far from ideal. Even the most
advanced ones only address some of the aforementioned points. For instance, many of the
common pneumatic nebulizers generate relatively coarse and fast aerosol with wide size and
velocity distributions, in other words they exhibit aerosol qualities not suitable for plasma. Thus
the produced aerosols have to be modified and drops must be selectively removed in spray
chambers through a process which is highly inefficient because only 2 percent of the original
aerosol generated by conventional pneumatic nebulizers finds its way into the plasma after
filtration (13). Moreover spray chambers add more complexity to the sample introduction system
that is not desired. In some other instances, the new designs compromise between some of the
main features of the nebulizer and in some cases even sacrifice one for the others, like improving
aerosol generation by modifying nebulizer geometry at the cost of increasing the probability of
nebulizer tip clogging.
Although ICP is considered a routine and mature technique for elemental analysis, its sample
introduction system has remained the weakest point of the instrument (4). Despite the significant
number of publications on sample introduction in ICPs, the commercially available ICP-MS
systems still employ the 70’s nebulizer technology and spray chamber configuration. Todoli and
Mermet (4) attribute this trend to the fact that the majority of studies are or more or less
modifications or improvements rather than a radical step change in terms of sensitivity, precision
and non-spectral interferences. Hence the necessity of comprehensive research on the optimal
design and improvement of sample introduction components to address key problems of ideal
nebulization and spray modification is widely recognized because the future development of
ICP-MS is strongly linked to improvement of the sample introduction system. Reaching that goal
is still left a challenge for the researchers in this field.
5
Concentric Pneumatic Nebulizer (CPN) – Design and 1.3Fundamentals
Today’s modern CPN does not differ fundamentally from the nebulizer described by Gouy (14)
at the end of the 19th
century and the technique has remained the most widely used for sample
introduction in ICPs due to its reliability, robustness, ease of operation (13) and simplicity
because it has no moving or electric parts (15), besides its versatility of aerosol production
allows researchers to adapt new designs for special needs such as high dissolved solid
nebulization, micronebulization and etc. Furthermore there is no better alternative available so
far that demonstrates a better compromise between quality of the results, nebulizer robustness
and ease of operation (4). The small cost of CPN is another important factor, currently as of
March 2009, conventional CPNs are listed between 280 to 530 USD in the manufacturer’s
catalogue (16) in comparison to Direct Injection High Efficiency (DIHEN) from the same vendor
with a price range of 1600 to 5000 USD.
Figure 1-2 shows a conventional Meinhard all glass type-C concentric nebulizer (17) together
with its dimensions. As can be seen, liquid is fed horizontally from the left hand side and travels
along a capillary tube while the gas is introduced from the bottom and moves concentrically with
the liquid, nevertheless the interaction of the two phases is restricted to the area close to the tip of
the nebulizer. The Meinhard CPN has three different designs, Types A, C and K. Type A is the
first commercially available nebulizer from Meinhard and as can be seen from Figure 1-3, the
nebulizer tip and capillary are both coplanar while the capillary for types C and K is recessed.
Tip recession is specially recommended to avoid nebulizer tip blocking (18). Therefore the type
A CPN is designed for general introduction purposes while type C and K are more suitable for
introduction of high concentration solids. The tip recession can be easily recognized in Figure
1-2b where the capillary walls are defocused in contrast to the gas annulus area. The major
difference between types C and K are in the final finishing of the nebulizer tip, while type C has
a flame polish tip; type K tip surface is ground flat and square as type A (Figure 1-3).
The CPN configuration allows the liquid to be freely aspirated without any need for a delivery
device (e.g. peristaltic pumps) due to Venturi effect, it should also be added that the free
aspiration is a unique characteristics of CPNs. Based on the CPN design, type A is expected to
6
have higher suction effect due its larger pressure difference beyond the capillary because types C
and K are recessed and the gas energy and speed is higher inside than outside of the nebulizer.
However, a free uptake rate enhancement of 1.5-2 folds has been reported for the recessed
capillaries (19). All the CPNs may utilize a peristaltic pump for liquid injection at a desired rate.
Using such a device removes the effect of liquid viscosity, an important parameter in self
aspiration, but at the same time introduces periodic noise in aerosol generation due to the
pulsating nature of the pumps. It has been reported that the CPNs have their best performance if
the injection rate is close to the free liquid aspiration rate (4).
A comprehensive study of aerosol generation for CPN and ICP-MS nebulizers in general is a
difficult task because the process is rather fast and happens in a very short timescale (in the order
of several milliseconds), besides there are several fairly complicated processes involved in
droplet generation which are poorly understood. In fact most of the studies aimed at aerosol
production have been carried out in different fields of engineering and aeronautics where the
nozzles and flow conditions are not comparable to ICP-MS applications.
The pneumatic aerosol generation can be divided into two separate processes (i) wave generation
on the liquid surface and (ii) growing of instabilities and disintegration of waves to form
droplets. Under the regular conditions of nebulization in ICP-MS applications, waves with
wavelengths of the order of several tens of micrometers are formed on the liquid surface due to
transfer of energy from the gas phase. However, the gas and liquid velocity at the contact surface
is only a percent of the gas stream because only tangential velocity components are acting on the
surface; as a result a small fraction of gas energy is used for wave generation.
Once the waves are formed, the degree of interaction between the gas and liquid increases. The
magnitude of the force acting on a single wave depends on the relative velocity between the two
phases, wave dimension and drag coefficient. Note that the presence of turbulence increases the
penetration of gas in the liquid bulk and promotes wave generation and growth until the wave
becomes unstable and is fragmented into droplets (4). The wave destruction can be attributed to
three separate mechanisms (20):
1- Filament or sheet formation followed by a collapse into droplets under the combined
action of the gas and the surface tension of the liquid.
7
2- Direct boundary layer stripping of the wave crests.
3- Removal of the surface disturbances through “Taylor instabilities” (21).
The first two processes are principally derived from tangential stresses on the liquid boundary
layer and generally produce fine droplets while the Taylor instability mechanism is due to the
action of normal forces and may lead to large droplet generation, especially if the liquid feed rate
Figure 1-2: (a) A conventional CPN with its critical dimensions and (b) a view of nebulizer tip
under microscope, model: Meinhard TR30-C3.
8
exceeds the rate of liquid removal by the other two processes. It’s worth mentioning that the
surface tension forces resist surface deformation and wave disintegration; as well the liquid
viscosity strongly damps the short wavelengths. Thus low surface tension and viscosity favors
fine aerosol generation as confirmed by (12) and (20).
Figure 1-4 and Figure 1-5 exhibits the wave formation and droplet generation for CPN
schematically. As can be seen the liquid discharging from the centered capillaries is pulled
toward the gas exit due to smaller local pressure. Liquid stretching would in turn decrease the
thickness of formed film and consequently enhances the gas and liquid interaction (20). This
process is called prefilming and is in fact one of the unique features of concentric nebulizers.
Figure 1-3: Different design and nebulizer tip of Meinhard concentric pneumatic nebulizers,
(Courtesy of Meinhard Glass Products).
9
At this stage, it would be interesting to have some qualitative measure for the wavelengths
perturbing the liquid film and from there some raw estimation on the size of resulted droplets.
In reference (22), Squire studied the instability of an infinitely wide moving liquid film with
constant thickness and negligible viscosity. Although the details of his mathematical
formulation are beyond the scope of this chapter, some important conclusions may be drawn.
First the minimum wavelength for an unstable film was given as:
(1-1)
here λmin, ζ, ρg and UR are the wavelength, surface tension, gas density and relative velocity
between liquid and gas velocity respectively. Assuming argon is used close to the sonic
condition and recalling that liquid velocity is negligible compared to gas velocity, then for
UR=ug=276.1 (m/s), ρg=2.217 (kg/m3) and ζ=0.072 (N/m), λmin=2.67 (μm). Hence Equation (1-1)
suggests any wave whose wavelength is shorter than the characteristic value of 2.67 (μm), will
eventually be damped in the flow and does not contribute to the droplet generation. Furthermore,
Squire calculated the optimum wavelength (Figure 1-5) which maximizes the growth rate of
perturbations and produces the most probable drop size by:
(1-2)
Plugging the same values as in Equation (1-1), the optimum wavelength would be λopt=5.35(μm).
Knowing that the drop sizes must be of the same order of magnitude as their generating
wavelength, 5.4 (μm) is the modal characteristic length for ICP-MS nebulizers and as will be
seen in the next chapters, the most probable droplet size is in the same order as predicted by
Equation (1-2). One important feature of Equation (1-2) is its independence from the film
thickness, whilst the thickness of the prefilmed liquid of Figure 1-4 is continuously decreasing.
Rizk and Lefebvre (23) found that for all prefilming type of airblast atomizers, the thickness of
the liquid film at the atomizing jet is mainly governed by the liquid viscosity, the air velocity and
the relative mass flow rates of liquid and air. The film width is also finite and the liquid may not
necessarily have negligible viscosity which would make some deviation from our calculation.
10
Thus the Squire analysis is not intended to give exact value or model a realistic flow condition
but rather present some rough estimate of the length scales here.
Taylor (24) also investigated the problem of ripple formation induced by wind blowing over the
viscous fluid surface and expressed the optimum wavelength as a rather complicated function:
( )
(1-3)
where ηl is the liquid viscosity. Again for the same flow conditions of Equation (1-2) and the
liquid viscosity of ηl=0.001 (Pa.s), θ will be 31.4 and the corresponding wavelength would be
4.7 (μm). Refer to the Appendix A for the shape of the θ function and more details.
Finally, Merrington and Richardson (25) showed a free body of liquid is unstable for Weber
numbers (We= ρgUR2L/ζ>10) where L is a characteristic length. Substituting with appropriate
values once again a characteristic length of 4.3 (μm) is obtained. Thus the characteristic
dimension (optimum wavelength) is of the order of 4-6 (μm) for nominal ICP-MS operating
conditions, although the models (22), (24) and (25) do not necessarily represent the actual
physics of the problem.
When the high velocity gas exits the nebulizer (Figure 1-4), gas streams are entrained from both
sides of the jet due to the pressure drop. Nevertheless in the region surrounded by the high
momentum gas, the entrainment must be supplied by the trapped gas itself which would cause
the formation of toroidal shape vortices. Therefore the liquid surface at the capillary end is
spread out and forms a meniscus from which a series of ligaments as large as the capillary inner
diameter are generated and finally disintegrate into fine droplets. However the frequent
coalescence of these ligaments may promote the generation of coarse droplets that is not desired.
The two gas streams finally recombine with each other and transport the aerosol drops
downstream while normally expanding. The recombination is believed to occur at a location
about half the outer capillary diameter along the axis of the nebulizer (20).
It’s been claimed in that pneumatic aerosol generation, the liquid core is unaltered up to 5 times
the capillary diameters. For TR30-C3 CPN for example, the average capillary is about 250 (μm),
11
Figure 1-4: Schematic of processes taking place at the exit of CPN.
Figure 1-5: Successive stages in an idealized sheet Breakup
L = 5 x AA
Ligament and drop formation
Prefilming
Renublization
Toroidal gas vortex
Spray Plume
Recombination region
Gas exit
12
that would give a length of 1250 (μm) approximately. Over such distance, gas would normally
lose a great deal of its kinetic energy required for liquid breakup due to expansion and become a
source for droplet acceleration or transport. For an efficient nebulization, the gas-liquid
interaction must be as efficient as possible but as Figure 1-4 shows, only one side of the gas jet is
in direct contact with the liquid and is used for droplet production. Thus the conventional
concentric designs are not the best choice for fine aerosol production (4), (18) and (20) in ICP-
MS, hence the CPN must be coupled with aerosol modification devices, i.e. spray chambers or
desolvation systems, to remove the coarse droplets. The need for improved or alternative sample
introduction methods is clear, recalling that the combination of the CPN and spray chambers
results in very poor transport efficiency (the ratio of aerosol reaching the atomization cell to the
total mass sprayed).
Microsample Introduction 1.4
Microsample introduction in ICP-MS has been the subject of many studies in the past 15 years to
several reasons:
(i) in some particular fields (e.g., forensic, biological and clinical analysis, etc.) the available
sample volume may be significantly lower than 1 (ml). (ii) several interferences like polyatomic
ones in ICP-MS can be positively reduced when working at low liquid flow rates. (iii) toxic and
radioactive wastes must be minimized in some applications and (iv) the transport efficiency is
improved at small liquid sample consumption rates.
Conventional CPNs operate at solution feed rate on the order of 0.5-2 (ml/min), this would
require a sample volume of 1-10 (ml) for roughly 5 minutes of analysis. Recall that the
conventional CPNs are not efficient nebulizing devices and their employment for microsample
introduction requires a rather large volume of liquid which is neither always practical nor
affordable, in addition coupling CPNs with spray chambers usually results in very poor transport
efficiency. Exploiting the conventional CPNs at microsample conditions (Ql=10-300 μl/min)
may also lead to dramatic loss of sensitivity and an increase in washout times (4), besides Todoli
and Mermet (18) and Mora et al. (13) claim the critical dimensions of CPNs are not suitable
microsample introduction. As stated before the CPNs have their best performance close to the
aspiration rate; lowering the liquid feed rate below 300 (μl/min) has been reported to cause
13
unstable aerosol generation (26). In the author’s experience with Meinhard TR30-C3 CPN, a
stable aerosol was observed at a flow rate as low as 60 (μl/min).
Therefore microsample introduction requires its own micronebulizer design. In the past decade
several micronebulizers have been developed and demonstrated better performance in terms of
better aerosol generation, higher ICP sensitivities and lower limits of detection at low liquid flow
rates. The micronebulizers have more or less followed the original concentric design and the
nebulizer miniaturization is mainly done through lowering the capillary diameter, wall thickness
and in some cases by reducing the gas-exit cross sectional area. Table 1-1 (taken from (4) and
(18)) compares the critical dimensions of conventional nebulizers to some of their miniaturized
counterparts.
Note in the table, HEN, MMN, MCN, DIN, DIHEN, LB-DIHEN stand for High-Efficiency
Nebulizer, MicroMist Nebulizer, Microconcentric Nebulizer, Direct-Injection Nebulizer, Direct-
Injection High-Efficiency Nebulizer and Large Bore Direct-Injection High-Efficiency Nebulizer
respectively and PFA or PFAN is a special micronebulizer made of tetrafluoroethylene-per-
fluoroalkylvinyl ether copolymer.
Several important conclusions can be drawn from Table 1-1 that may account for better
performance of micronebulizers (18):
i. The length of unaltered liquid core is shorter for micronebulizers. As stated in section
1-3, this length is about 5 times the capillary diameter. According to Table 1-1 for a
conventional CPN, the liquid core extends approximately 2000 (μm) and around 400-
500 (μm) for HEN. Therefore liquid disintegration occurs closer to the nebulizer tip
where gas has higher kinetic energy and a finer aerosol is expected.
ii. The area of gas-liquid interaction is modified for micronebulizer. This area is defined
by multiplying the distance L, along which gas is able to generate droplets, by the the
perimeter of the sample capillary. The distance L is said to be 5 times the annulus
width of the gas exit. Thus for CPN and a 20-30 (μm) wide annulus; the length L
would be 100-150 (μm) and for a sample capillary perimeter of 1.63 (mm) the
resulting interaction area is 0.16-0.25 (mm2). Similar calculations for HEN, MCN and
14
Nebulizer
Gas exit cross
sectional area
(mm2)
Liquid
capillary
inner
diameter (μm)
Liquid
capillary wall
thickness
(μm)
Gas back
pressure at 1
(l/min) argon
(psig)
Nebulizer
dead volume
(μl)
Conventional nebulizers (optimum for liquid flow rates ~ 0.5-1 (ml/min)
Concentric
nebulizer ~ 0.028 400 60 30-40 ~ 100
Cross-flow
nebulizer 0.02 500 200 30-40
Micronebulizers (suitable for liquid flow rates < 100-200 (μl/min)
HEN 0.007-0.008 80-100 30 150
MMN 0.018 140 50 50
PFA (PFAN) 0.021 270 40
MCN 0.017 100 30 50 0.64
DIN 60 30 45/70* < 1,2 (pl)
DIHEN 0.0094 104 20 155 10-55
LB-DIHEN 0.0371 318 16 36
Table 1-1: Critical dimensions of different nebulizers used in ICP-MS, * Pressures required to
reach 0.25 and 0.6 (l/min) gas flow rates respectively.
MMN give an interaction area of 0.03 to 0.07 (mm2) which is 3-4 times smaller than
CPNs and again favor small droplet production.
iii. The prefilming process explained in the previous section would decrease the film
thickness pulled outward from the nebulizer capillary. At given gas and liquid flow
rates, the film thickness is larger for thin capillaries. From this perspective, the gas
and liquid interaction is less efficient for micronebulization.
iv. The kinetic energy of the expanding gas depends on the gas exit cross sectional area
irrespective of the nebulizer type (20). For some micronebulizers, this area is
modified compared to the CPNs (Table 1-1), hence a finer aerosol is expected.
Figure 1-6a shows a close tip view of HEN. HEN is an adaptation of Meinhard type A nebulizer
which is made entirely of glass. The nebulizer cost is similar to conventional CPNs (16), but its
15
reduced gas exit cross sectional area requires an external additional gas cylinder and
consequently using special high pressure adapters and lines for gas streams (18). HEN is
reported to achieve transport efficiency between 90 and 95 percent for liquid flow rates of
Ql=10-1200 (μl/min) (27). The tertiary aerosol (aerosol leaving the spray chamber) of HEN and
CPN has similar velocity but the velocity distribution is considerably narrower for HEN which
leads to better ICP-MS short term signal precision (28). HEN also benefits from a similar droplet
number density for the tertiary aerosol as conventional CPN but at liquid flow rates 100 times
smaller (27). Aside from all the mentioned benefits, HEN suffers from some shortcomings. For
example, due to its small capillary diameter, tip blockage is a frequent problem and avoiding it
needs precise sample filtration even for clean aqueous solutions. In addition, HEN is a very
fragile device and may be easily broken if the nebulizer cleaning is not done carefully (26).
Furthermore the irreproducibility or variability of results is another problem that differs from one
HEN to another.
Another commercially available micronebulizer is the MicroConcentric Nebulizer (MCN) that is
made of polyamide. As can be seen from Figure 1-6c the capillary is extended outside of the
nebulizer tip that shows tolerance to high solid content solutions (29). In fact this is a drawback
for the nebulizer design since gas rapidly loses a fraction of its kinetic energy by expansion.
Besides an extended capillary may deteriorate in long term use and influence aerosol generation.
Thus MCN is considered a rather fragile nebulizer (18), but gives rise to limits of detection close
to or slightly higher than conventional CPNs operated at liquid flow rates more than 10 times
higher in ICP-AES (Inductively Coupled Plasma- Atomic Emission Spectrometry) (30) and
higher ICP-MS sensitivities.
The MicroMist Nebulizer (Figure 1-6b) is a glass modified version of common concentric
nebulizer with reduced dimensions. The major difference between MMN and other
micronebulizers is the tip recession that makes it a suitable option for introduction of high salt
content sample with peace of mind from tip blockage. The capillary inner diameter in this
nebulizer is tapered; that has probably caused the result irreproducibility between MMNs (18).
16
Since the gas exit cross sectional area is smaller for HEN, it would naturally need higher back
pressure to discharge gas at the same rate of flow. From Table 1-1, it’s noticed that the required
back pressure for injecting 1 (l/min) argon for HEN>MMN, MCN> PFA implying the kinetic
energy for aerosol generation is larger for HEN as verified by Todoli and Mermet (4) and (31).
However it should be noted again, there is currently no ultimate nebulizer for all applications.
Each nebulizer has its own advantages and drawbacks and is suited for a particular application in
sample introduction.
Although the discussed micronebulilzers (HEN, MMN, MCN and PFA) produce superior aerosol
in comparison to the conventional CPNs, they still need to be coupled with spray chambers or
desolvation systems. However these instruments are complex and have problems of their own (4)
, to name a few we can mention: (i) existence of memory effects, (ii) intensification of matrix
Figure 1-6: Close view of micronebulizer tip, (a) HEN, (b) MMN, (c) MCN and (d)
conventional CPN
17
effects, (iii) increase of signal noise, (iv) removal of a high proportion of the analyte nebulized
with subsequent loss of sensitivity, (v) wave generation and (vi) postcolumn broadening effects
when separation methods are coupled to ICP techniques.
To avoid these complexities, a new trend is observed toward total aerosol consumption and
direct injection of droplets into the plasma without exploiting spray chambers or desolvation
systems. The last 3 nebulizers in Table 1-1 are of this type. For example, excellent signal
stabilities are reported by a DIN whose capillary diameter was 60 (μm) and operated at 0.2-0.5
(l/min) argon flow and 50-100 (μl/min) liquid flow rate (32) in addition to an external 0.3 (l/min)
makeup gas flow rate to efficiently direct aerosol toward the plasma due to the low rate of the
main argon flow. Like other concentric pneumatic nebulizers, the coarse droplets of DIN are at
the periphery of the spray but the resulting primary aerosol is smaller than the CPNs because of
its reduced dimensions. However in some instances the combination of the CPN and spray
chamber is claimed to produce finer aerosol than DIN (33) but at the cost of very poor transport
efficiency, increased memory effect and poor precision.
DIN is usually placed 1 (mm) below the torch central tube which would increase the nebulizer tip
overheating especially when the HF power increases above 1.3 (kW) (34). Besides, tip blockage
is very common for this nebulizer but it can be avoided by extending the liquid capillary outside
of the nebulizer tip or increasing the make flow rate. Moreover, DIN should only be used for low
liquid flow rates and cannot be used with peristaltic pumps.
DIHEN is an all glass or quartz nebulizer by Meinhard Glass Products (17) that is a cheaper
version of DIN. The nebulizer is very close to HEN in design but about 2.5 times longer, can be
used with peristaltic pumps and is equipped by a supporting tube to reduce the capillary damage
caused by the gas stream-induced oscillations giving it high robustness (Figure 1-7, (35) and
(36)). The critical dimensions of DIHEN are smaller than DIN but may differ from one nebulizer
to another that causing irreproducibility in results. Todoli and Mermet (4) and (37) Paredes et al.
(37) believe high cost, fragility, tip blockage and overheating and the nebulizer sensitivity to
change in operating conditions and sample matrix when used in the plasma have limited the wide
application of this nebulizer for routine analysis despite its advantages. Besides the reported
analytical figures of merit obtained by DIHEN are not as good as expected for two reasons: first,
18
formation of coarse droplet as large as 30 (μm) (38) and second, the rotational motion of the
aerosol (39) which leads to aerosol deposition across the torch. Thus only 30-45 percent of the
droplets find their way into plasma without dispersion and successfully contributed to signals (4).
The tip blocking problem of DIHEN has been overcome by the new design LBDIHEN (Large
Bore DIHEN) which has enlarged capillary and gas annulus area (Table 1-1). The modified
dimensions of LBDIHEN in turn will cause larger aerosol mean diameters and small drop
Figure 1-7: Schematic design of DIHEN and its coupling with plasma torch, (35).
19
velocities, lower ICP-MS sensitivities and more severe matrix effects than DIHEN (40) but the
nebulizer is very suitable for introducing high salt content solutions and slurries.
Objectives 1.5
In this task, a type-C CPN is characterized and will be analytically modeled because not only it is
a good choice for PDPA calibration but it is also a benchmark nebulizer with wide application in
spectrometry. The aerosol size of the nebulizer is characterized in Chapter 2 and the application
of the well known Nukiyama-Tanasawa (NT) and Rizk-Lefebvre (RL) correlations are tested for
the nebulizer under the typical ICP-MS operating conditions. The aerosol velocity of the
nebulizer is then characterized in Chapter 4.
A new direct injection nebulizer will be introduced in Chapter 3 which is not of the prefilming
kind. The new custom-designed nebulizer follows the principle of Flow Focusing Nebulizer
(FFN) first employed by Ganan-Calvo (41). This new class of nebulizers is not commercially
available and has been recently employed for sample introduction in spectrometry (42) and (43) .
The preliminary results of our custom-designed nebulizer are promising although not ideal. The
author believes by resolving some issues of the FFN, it could be an alternative for many of the
current pneumatic nebulizers. The nebulizer is characterized and the fundamentals of the new
custom-designed FFN are discussed in Chapter 3 while the aerosol velocity is characterized in
Chapter 4.
Since characterization results do not represent the actual aerosol, having some statistical measure
of the aerosol is a prerequisite for any numerical or analytical modeling. Thus the other scope of
this task is to present a meaningful and detailed space of aerosol size and velocity (based on
characterization results of chapters 2, 3 and 4) from which physical size and velocity
distributions could be derived. The method of maximum entropy principle (MEP) is used for this
purpose. As will be shown in Chapter 5, the conventional MEP models yield realistic size
distribution while their velocity distribution shows poor agreement with experiments. New
modified MEP models are then proposed to overcome this deficiency and the models will be
tested for both the CPN and the FFN.
20
Summary 1.6
The maximum solvent load and acceptable gas flow rate in ICP-MS put severe constraint on
aerosol generation. For instance, droplets larger than 10 (μm) do not undergo desolvation,
vaporization and ionization processes and are not consumed by plasma. They may even cease
mass spectrometry if they constitute a large fraction of the aerosols. For ICP-MS application,
small and slow aerosol with narrow size and velocity distribution is desired, but common
concentric pneumatic nebulizer design does not fulfill these requirements and the resultant
aerosol stream is generally polydisperse with droplets sometimes as large as 100 (μm). Hence the
CPNs are usually integrated with transported instruments such as spray chambers and
desolvation systems to modify the primary aerosol generated by the nebulizer. Although spray
chamber can reduce the solvent load and remove the coarse droplets, but they add more
complexity to the sample introduction system, besides the combination of spray chambers and
CPNs leads to very poor transport efficiency between 1-2.5 percent. As proven, the critical
dimensions of CPNs such as liquid capillary diameter, wall thickness and gas exit cross sectional
area are not suitable for fine aerosol production. Therefore to improve the atomization and
overcome the low transport efficiency of spray chambers, the CPNs have been miniaturized
while keeping the same fundamentals and principals. The different micronebulizers designs
available, e.g., HEN, MMN, MCN, PFA and others that have all shown better aerosol production
by improving gas-liquid interaction area and benefiting from higher gas kinetic energy at the
nebulizer exit. Since the critical dimension, particularly liquid capillary is reduced for these
nebulizers; the tip blockage has become more problematic. The reduced dimensions also require
higher gas back pressure that needs additional pressure adapters and lines, increase the cost and
bring safety issues as well. In addition to high cost, fragility and results irreproducibility are
some other common problems associated with micro nebulizers.
Total aerosol consumption is a new trend observed for ICP-MS that is designing and employing
nebulizers that can successfully generate fine aerosol below 10 (μm) in size and directly inject
them into plasma without the need for spray chambers or desolvation systems. DIN, DIHEN and
LB-DIHEN are examples of direct injection nebulizers. Although these nebulizers are shown to
have many advantages in terms of aerosol generation and signal quality in MS, but they severely
suffer from some other drawbacks. Their cost is quite significant in comparison to conventional
21
CPNs. The positioning of the nebulizer close to the high temperature plasma often causes tip
overheating. Tip blockage is also repeatedly reported due to their reduced dimensions.
Furthermore, in the case of DIHEN, formation of droplets as large as 30 (μm) and the radial
dispersion of the aerosol increases the droplet deposition in the plasma channel and leads to
general performance not as good as expected.
In a nutshell a nebulizer able to produce fine and narrow aerosol that is not fragile, does not
require a spray chamber, can overcome tip blockage, offers result reproducibility and satisfies
signal quality at a reasonable cost is of high demand in ICP-MS.
22
Chapter 2 Aerosol Size Characterization of Concentric Pneumatic Nebulizer
Experiment Setup 2.1
A conventional TR30-C3 CPN (Figure 1-2) was selected for aerosol characterization because
Meinhard CPNs have been commercially available for a number of years, supplied as standard
ICP-MS sample introduction instruments and are often regarded as a “bench mark” nebulizer to
study sprays and also for comparison as in (33) and (44). In addition to CPN and inspired by
Ganan-Calvo’s flow focusing pneumatic nebulizer (FFPN) (41), a FFN was designed twice in
collaboration with Tanner’s group from the Department of Chemistry at the University of
Toronto (45). The initial design of the FFN was equipped with two CCD cameras to observe
filament formation inside the nebulizer and its disintegration downstream of the orifice. In the
second design, some improvements were carried out to improve nebulizer performance and the
cameras were removed from nebulizer design.
Argon was supplied to the nebulizers (both CPN and FFN) from a pressurized tank (4.8-300SZ,
Linde, Canada) while the nebulizer back pressure was regulated (5126AD, Scott Specialty
Gases, Canada). The volumetric gas flow rate was recorded with a mass flow controller and
readout placed on the gas line (MKS-Type246C, MKS Instruments, MA, USA). For the FFN, an
extra argon line (makeup flow) was used for aerosol transportation and controlled separately
with a Rotameter (PMR1-010537, Cole-Parmer Canada Inc, QC, Canada).
A 5cc-plastic (Becton Dickinson, ON, Canada) and a 1cc-glass (Hamilton Company, NV, USA)
syringe and a computer-controlled model-22 syringe pump (Harvard apparatus, MS, USA) were
exploited for sample injection. Although the C3-CPN can aspirate liquid sample freely at the rate
of 3 (ml/min) or 50 (μl/s), in the experiments the liquid was introduced manually via the syringe
pump system at the rate of 1-10 (μl/s) as in (33) to study the lower end limits of aerosol
production with this nebulizer. In this range the nebulizer is expected to generate finer aerosol
since the available gas kinetic energy per unit volume of liquid is larger than at the free
aspiration rate. Sutton et al. (26) reported unstable aerosol production for liquid flow rates below
23
Ql= 5 (μl/s), however in the author’s experiments the CPN operated robustly down to 1 (μl/s).
Any further decrease in the liquid feed rate caused frequent nebulizer starvation (46). It should
be noted here that microsample measurements are difficult and rather tedious tasks (18) due to
the small available sample volume and the time required for analysis. For example, it takes about
1.67 (min) to spray a sample with a 1 (cc) syringe at a liquid flow rate of 10 (μl/s) whereas the
time scale for good size and velocity characterization is relatively larger meaning that the liquid
spraying had to be stopped regularly to fill up the syringe and renebulize the sample. Although
using a larger size syringe is possible, it would lead to frequent pulsation and unstable spray that
is not desired. In the experiments carried out, distilled water (DW) and methanol, with relatively
similar refractive indices were nebulized. Figure 2-1 exhibits a schematic of the experimental
setup where scale is not preserved for convenience.
A two component Phase Doppler Particle Analyzer (PDPA) manufactured by TSI Inc (MS,
USA) was employed for size and velocity characterization. The PDPA splits a laser into two
equal intensity beams and focuses them to an intersection inside the aerosol spray with a TR60
transceiver probe (Figure 2-2). Droplets passing the measuring probe scatter light independently
from each beam. The scattered light interferes to form a fringe pattern in the plane of the receiver
lenses mounted on a receiver probe. The receiver probe consists of three precisely spaced Photo
Multiplier Tubes (PMTs) which are located 30 degrees off the axis of the transceiver probe. One
of the PMTs is used to determine the temporal frequency of the fringe pattern which is a function
of the droplet velocity, the beam intersection angle and the light wavelength. Therefore, the
droplet velocity is obtained directly from this PMT, but the droplet size is calculated with the aid
of all 3 PMTs by measuring the spatial frequency of the fringe pattern. The spatial frequency is
inversely proportional to the droplet diameter and is also a function of the laser wavelength, the
beam intersection angle, the spacing between different detectors on the receiver and the droplet
refractive index (47). Since methanol and distilled water have very close refractive indices (~
1.33), the setup was not changed for these solutions. In our experiments, a lens with a focal
length of 350 (mm) was used that would allow size measurement as small as 0.5 (μm) up to 115
(μm). Mclean et al. (12) pointed out a good measurement requires at least 5500 counted droplets
in different size classes that contain at least 10 particles/droplets for statistical accuracy.
Therefore, the PDPA was setup to perform 100,000 measurement attempts for each run to make
24
sure these requirements are met. Each single run (size and velocity measurement) was repeated
at least 3 times. The standard deviation between the runs was generally small (less than 1-2%).
Thus the statistical errors are negligible and will not be reported hereafter. The results of the 3
different runs were then combined to assure the experiments were not run-dependent. It is worth
mentioning some difficulties associated with droplet size and velocity measurements of
pneumatic nebulizers working under realistic conditions with PDPA instruments. First, PDPA
requires a high level expertise to acquire good results since it’s rather a complicated optical
device (12). In addition, there must be a high probability of having only one particle in the
sample volume at one time but the pneumatic nebulizers have generally high particle number
densities of the order of 106 particles per cm
-3 (48). Furthermore PDPA measurement is only
useful for spherical droplets. Irregular or deformed droplets yield irregular light scattering
patterns and are not reliably measured, especially close to the nebulizer tip (12). Thus all the
measurements were taken 10 (mm) downstream of the nebulizer tip along the axis to avoid high
rejection rate in the dense spray region and the interference of the laser beam with the
experiment setup. However, numerous measurements were also carried out at different axial and
radial locations to study the local spray characteristics. The PDPA device had to be calibrated to
assure good and reliable spray measurements. The calibration procedure is not included in this
chapter but can be found in Appendix B. Table 2-1 presents a summary of the experiment setup
and the devices exploited in the experiments.
Nukiyama–Tanasawa Correlation 2.2
Atomization is the process of converting a bulk liquid into a multiplicity of small drops in order
to produce a high ratio of surface area to mass in the liquid phase, favoring heat and mass
transfer processes such as evaporation. Conventionally, atomization is accomplished by applying
a high relative velocity between the liquid and surrounding gas phase so that the disruptive
aerodynamic forces overcome the consolidating surface tension forces. This goal is achieved
either by injecting a high velocity liquid into quiescent gas as in the case of plain orifice and
pressure swirl atomizers or in contrary by exploiting a high velocity gas stream to disintegrate a
slow moving liquid flow as in twin-fluid, airblast and air-assist atomizers.
25
Operating conditions of nebulizers
TR30-C3 CPN
Gas, Gas flow rate Argon, 0.2-0.8 (l/min)
Liquid, Liquid flow rate Distilled Water/Methanol, 1-10 (μl/s)
Applied gas pressure 35-210 (KPag)
Geometrical parameters
Gas annular area (0.03 mm2), Capillary ID (260
μm), Capillary tip recess (500 μm), Tabulation
OD (6mm), Fluid inputs (4mm)
FFN
Gas, Gas flow rate Argon, 0.15-0.21 (l/min)
Liquid, Liquid flow rate Distilled Water/Methanol, 0.1-1 (μl/s)
Applied gas pressure 35-70 (KPag)
Geometrical parameters Capillary ID (175 μm), Capillary OD (1/32”),
Tube OD (3.2 mm), Tube thickness (250 μm),
Capillary tip recess (300 μm/ variable), Orifice
(150 μm)
Mass flow controller and readout MKS-Type246C (MKS Instruments, MA,
USA)
Syringe 5cc-plastic (Becton Dickinson, ON, Canada) -
1cc-glass (Hamilton Company, NV, USA)
Syringe pump Model-22 (Harvard apparatus, MS, USA)
Rotameter Model-PMR1-010537 (Cole-Parmer Canada
Inc, QC, Canada)
PDPA parameters
Off axis angle 30○
Focal length 350 (mm)
Droplet size measurement range 0.5-115 (μm)
Number of measurement attempts 100,000
Table 2-1: Operating conditions for nebulizers and the measurement devices exploited in
experiments
Lefebvre states that airblast and air-assist atomizers have advantages over pressure atomizers
because they require lower liquid pressures and generally produce a finer spray (49). The
physical processes involved in atomization are still not well understood. Hence the majority of
studies in this field suffers from lack of knowledge about the microscopic foundations of aerosol
generation and is empirical in nature. Nevertheless these empirical studies have yielded a
considerable body of information on the atomization phenomenon such as the effect of liquid and
26
Fig
ure
2-1
: E
xper
imen
t se
tup,
das
hed
lin
e re
pre
sents
mak
eup g
as l
ine
and i
s only
use
d f
or
FF
N.
28
gas properties, nozzle geometry, liquid and gas flow ratios, etc. (50).
The first major study on airblast atomization was conducted by Nukiyama and Tansawa in 1939
for characterization of a plain-jet airblast atomizer (51) by collecting droplets on oil-coated glass
slides. The authors investigated different parameters influencing the atomization process and
proposed a correlation for the Sauter mean diameter (D32), a characterstic moment which
represents the mean volume to mean surface area of the spray. For instance, they figured out by
lowering surface tension and liquid viscosity and increasing the liquid density a finer aerosol is
generated. They also showed that the spray Sauter mean diameter can be controlled through the
following dimensionless numbers:
(
√
√
√ ) (2-1)
Here UR, ζ, ρl, do, ηl, Ql and Qg are the relative velocity between the gas and liquid at the
nebulizer exit, surface tension, liquid density, orifice diameter, liquid viscosity, liquid and gas
flow rates respectively. By rearranging Equation (2-1):
(
√
√ √
√ ) (
√ √
) (2-2)
where Wed0 and Ohdo are Weber and Ohnesorge numbers based on the orifice diameter, the two
diemsnionless numbers which usually appear in droplet size characterization studies.
Nevertheless, in the Nukiyama and Tanasawa experiments the gas velocity was kept well below
sonic conditions (20). Thus the gas density is essentially constant. Besides the authors found that
varying the orifice diameter does not significantly affect the Sauter mean diameter. Thus, for
constant gas density and neglecting orifice diameter, Nukiyama and Tanasawa derived the
following correlation from Equation (2-1).
)
(
)
(
√ )
(
)
(2-3)
29
Please note in the absence of nozzle geometrical parameters, i.e. orifice diameter here, Equation
(2-3) is essentially non-dimensionalized meaning that correct units must be used for each
parameter. Here D32, ζ,UR,ρl,ηl, Ql and Qg must be given in (μm), (dynes/cm), (m/s), (g/cm3),
(poise) and (l/min) respectively. In addition Equation (1-3) is defined for a specific ranges of
flow parameters, 0.8< ρl<1.2 (g/cm3), 30<ζ<73 (dynes/cm) and 0.01<ηl<0.8 (poise). The
relative velocity (UR=ug-ul) must be calculated from known liquid and gas velocities. The liquid
velocity is simply calculated by dividing the recorded liquid flow rate by the capillary cross
sectional area but the same procedure may not be used for the gas velocity calculation due to
compressibility effects. Therefore, the gas velocity is estimated from isentropic theory from the
gas back pressure and the atmospheric pressure (52):
(
)
(2-4)
(
)
(2-5)
(
)
(2-6)
√ (2-7)
here Pg,Tg and ρg are the gas exit absolute pressure, absolute temperature and gas density. Exit
pressure has a value of P= 102.9 (KPag) when the flow is not chocked, i.e. when the back gauge
pressure is below 108.3 (KPag). P0, T0 and ρ0 are back or absolute stagnation pressure,
temperature and density where T0 is assumed to be 293 (K) and P0 is adjusted via the pressure
regulator. M is the Mach number whereas u* represents the sonic velocity. k and R are the ratio
of the specific heats of the gas at constant pressure and volume (k=cp/cv) and gas constant
respectively. For Argon k and R are 1.667 and 208.11 (J/kg.K) respectively. It should be noted
that the assumption of one dimensional isentropic flow in an ICP-MS nebulizer may not be true
as the flow irreversibilities and the nebulizer geometry will cause a deviation from the isentropic
condition. However as stated in reference (20), the isentropic flow approximation is valid for
30
nozzles as small as 200 (μm). Hence the isentropic theory provides a good engineering
approximation of the gas exit velocity required in Equation (3-2). Based on the isentropic theory,
the sonic condition is met when the back pressure is 108.3 (KPag). For the CPN nebulizers
however, the sonic condition may be delayed to 175-315 (KPag). This uncertainty has little
effect on the D32 calculation (Equation 1-2) since the second term of Equation (3-2) is dominant.
Figure 2-3a-c compares the Sauter mean diameter variation with gas flow rate from the NT
model and experiment for distilled water (DW) and methanol at different liquid flow rates. As
can be seen, the lower surface tension of methanol has resulted in a decrease of D32 as expected.
The difference between model prediction and experiment is more noticeable at higher liquid
uptake rates. The NT model shows a dropping trend in the Sauter mean diameter with the
increase of gas flow rate although the predicted size is largely overestimated. In our experiments
for instance, the maximum overestimation is up to 4 fold the measured size and this trend is more
obvious for higher liquid flow rates. Similarly, in flame atomic spectrometry although the range
of gas flow is larger than ICP-MS, there are reports of up to 30 folds overestimation for the
organic solvents (53). Furthermore, Figure 2-3a-c and Figure 2-4a-b demonstrate that for each
liquid flow rate, the error in the prediction grows larger with decreasing gas flow rate, i.e. with
smaller liquid to gas flow ratio.
The gas velocity at the nebulizer exit is orders of magnitude larger than the typical liquid
velocities. Thus it is reasonable to assume that the relative velocity is little influenced by the
liquid. Besides, if the isentropic theory (Equations 2-4 to 2-7) is employed to predict gas exit
velocity, the first term of the NT model reaches a minimum plateau when the sonic condition is
met at 108.3 (KPag). Sharp (20) has argued that the adiabatic condition may not be met, at least
for long nebulizers, and the internal heat generation due to friction will cause a deviation from
the isentropic theory that leads to slightly higher exit temperature and gas velocity. Nevertheless,
the deviation may not make a pronounced difference in the exit condition and the contribution of
the first term in the NT model is predetermined.
31
(a)
(b)
0
5
10
15
20
25
30
35
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=1 (µl/s)
Experiment, DW
NT, DW
Experiment, Methanol
NT, Methanol
0
10
20
30
40
50
60
70
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=5 (µl/s)
Experiment, DW
NT, DW
Experiment, Methanol
NT, Methanol
32
(c)
Figure 2-3: Sauter mean diameter versus gas flow rate for distilled water and methanol from
experiment and the original NT model at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s).
Hence the overestimation of size is mainly due to the weight of the second term of the NT model
as reported by Sharp (20), Robles et al. (53) and Canals et al. (54). For instance, Figure 2-5
shows that the contribution of first term increases while at the same time that of the second term
decreases by increasing the gas flow rate. The dominance of the second term is more pronounced
at high liquid flow rates, that is 76-50 percent of the total D32 value at Ql=10 (µl/s), 53-26
percent at Ql=5 (µl/s) and 9-3 percent at Ql=1 (µl/s). It should be noted that as far as the validity
of the experiments is concerned, our experimental results at Ql=10 (µl/s) and variable gas flow
rates with TR30-C3 nebulizer is very similar to those of reference (54) with different Meinhard
concentric nebulizers.
At this stage it would be helpful to discuss the limitations of the Nukiyama – Tanasawa
correlation when applied to atomic and mass spectrometry. First as stated earlier, Nukiyama and
0
20
40
60
80
100
120
140
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=10 (µl/s)
Experiment, DW
NT, DW
Experiment, Methanol
NT, Methanol
33
(a)
(b)
Figure 2-4: Error between experiment and the original NT model grows larger by
increasing the liquid flow rate for (a) distilled water and (b) methanol.
0
50
100
150
200
250
300
0 200 400 600 800 1000
Qg (ml/min)
100×(D32experiment-D32
NT )/ D32
experiment
Ql= 1 µl/s
Ql= 5 µl/s
Ql= 10 µl/s
0
100
200
300
400
500
600
0 100 200 300 400 500 600 700
Qg (ml/min)
100×(D32experiment-D32
NT )/ D32
experiment
Ql= 5 µl/s
Ql= 10 ul/s
34
Tanasawa observed no strong mean diameter dependence on the nozzle geometry (51), but at no
time did they test their correlation for a nozzle with dimensions similar to modern CPNs.
For example, in the Nukiyama and Tansawa experiments, the gas to liquid jet diameter ratio was
between 5 and 17 (liquid jet diameter 0.2-1.0 mm and gas jet diameter 1-5 mm) which is not
comparable to a 250 (µm) liquid capillary and gas annulus thickness of 20-30 (µm). Second, their
nozzle was operated well below the sonic point and as a result, compressibility effects did not
come into the picture. But ICP-MS nebulizers are usually employed at the sonic point or may
even surpass it. Third, back in 1939, Nukiyama and Tanasawa’s measurement technique was
rather primitive. The authors had to collect droplets on oil-coated slides and measure the drop
sizes under a microscope. Therefore it wouldn’t be a surprise to believe that their correlation
could be to some extent biased in favor of larger mean droplet moments. Gretzinger and Marshal
(55) claim that application of Equation (3-2) for the range of 5-30 (µm) is doubtful although the
model is proposed for the range of 15-90 (µm) mean sizes (56).
Figure 2-5: Contribution of first and second term in Nukiyama – Tanasawa equation. Δ: Ql=1
(µl/s), ○: Ql=5 (µl/s) and ◊: Ql=10 (µl/s). Dashed and solid lines represent first and second
term of Nukiyama - Tanasawa equation respectively.
0
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000
Per
cen
t
Qg (ml/min)
Weight of first and second terms in Nukiyama-
Tanasawa equation
Term2/D32NT
Term2/D32NT
Term2/D32NT
Term1/D32NT
Term1/D32NT
Term1/D32NT
35
It should be noted however, that the CPNs normally generate aerosol with Sauter mean diameters
in the range of 10-50 (µm) based on the liquid and gas flow parameters as shown in Figure 2-6.
The Figure 2-6 also demonstrates a peak value of 9.4 (µm) that is of the correct order as
predicted from the optimized wavelength calculation performed in the previous chapter (Refer to
pages 9-10).
Finally and most importantly, liquid and gas flow ratios in the Nukiyama and Tanasawa
experiments are very different from ICP-MS. The typical CPN employs a gas flow rate of up to
1000 (ml/min) with a liquid sample uptake rate of 50 (µl/s). For this range of flow parameters,
the flow ratio is not comparable to the Nukiyama-Tanasawa study (51). The liquid to gas flow
ratio (Ql/Qg) varies between 0.0001 - 0.001 in the 4th
report of the Nukiyama-Tanasawa study
where the NT model was proposed for the first time. This would imply that the ratio (1000Ql/Qg)
will be well below unity after exponentiation in Equation (3-2) and guarantee no severe
overestimation. For instance, at Ql=1 (µl/s) of Figure 2-3a, the gas to liquid flow ratio is between
0.08 - 0.23; the second term constitutes 10 percent of the overall predicted size at most (Figure
2-5) and the overestimation is not severe. By increasing the liquid flow rate to 5 and 10 (µl/s),
i.e. increasing the liquid to gas flow ratio, the predicted size becomes drastically over estimated
and the weight of the second term becomes markedly pronounced (Figure 2-3b-c and Figure
2-5). Furthermore, for any desired liquid flow rate, decreasing the gas flow rate would also lead
to size over prediction. For instance the largest error in Figures (Figure 2-3a-c) is seen at
Qg=260-280 (ml/min) which again signifies the effect of the liquid to gas flow ratio.
Consequently, application of Equation (3-2) to mass and atomic spectrometry in its original form
is under question (54) with only a few exceptions such as (57), (58) and (59). However the
results of these exceptions cannot be generalized because either the experiments were carried out
for a single case or specific range of liquid and gas flow rates that fit into the Nukiyama –
Tanasawa correlation. Another common misuse of the NT model in the atomic and mass
spectrometry is the model implementation for estimating tertiary aerosol, i.e. the aerosol leaving
the spray chamber (57), however the NT model is proposed for primary aerosol restrictively and
36
its employment for modified aerosol is not logical even though it may produce seemingly
acceptable results.
Despite all the criticism of the NT model and its rather restricted boundary conditions (60), the
correlation is still the most widely quoted work in pneumatic atomization (61) and sample
introduction in ICP-MS literature and holds general validity at least in describing the trends of
the physical processes involved in atomization. In the absence of superior models it is still used
for modeling aerosol in MS (57), (62), (63), etc. In addition Montaser and Goligthly (10) believe
that Equation (2-3) shows good agreement between different methods of measurement and
theoretical approaches, thus its merit cannot be easily nullified and a lot of research must be done
before the Nukiyama – Tansawa correlation is regarded as useless in the field of mass and atomic
spectroscopy.
For instance, Kahen et al. (9) modified the NT model, for a DIHEN working at Ql=10-500
(µl/min) and Qg=0.2-1.0 (l/min) with different organic solutions and achieved satisfactory
agreement with experiment. To the author’s knowledge this is the only attempt reported in the
MS literature to modify the NT model for successful application in ICP-MS. Kahen et al.’s
model (MNT) assumes new coefficients for surface tension and viscous terms and also considers
an exponential decay type function for the liquid to gas flow ratio:
) (
)
(
√ )
(
) (2-8)
The coefficients P1 and P2 were obtained through curve fitting and are 86.4 and 105.4
respectively.
To verify the applicability of the MNT model, Equation (2-8) was tested for TR30-C3 CPN.
Figure 2-7 demonstrates a severe underestimation of D32 with the MNT model and the
corresponding curve seems almost flat. This is in part because DIHEN generally produces finer
spray in comparison to type C CPN under similar flow conditions, and at some gas and liquid
flow rates which are specifically attributed to DIHEN, TR30-C3 cannot generate aerosol at all. In
addition, Equation (2-8) has been proposed for a very narrow range of D32, varying from 4.6 to
7.2 (µm). Therefore, we can say that the MNT model is restrictively applicable to the
37
D (um)
dN
/dD
20 40 60 80
0.5
1
1.5
characterized DIHEN and perhaps with some leniency to some other micro nebulizers with
similar dimensions and comparable operating conditions.
Hence, we were motivated to modify the original NT model (Equation 2-3) for TR30-C3 CPN
under conditions mentioned in Table 2-1. We proposed to keep the structure of the original
model and apply the modifications through two coefficients as Kahen et al. (9) and also through
the exponent of the troublesome liquid to gas flow ratio. The new model is called fitted NT
(FNT) to avoid confusion with NT and Kahen et al.’s MNT models.
)
(
)
(
√ )
(
)
(2-9)
The coefficients A, B and C are found by minimizing the quadratic difference between
experimental and calculated values of Sauter diameter from Equation (2-9). The generalized
reduced gradient method embedded in Microsoft Excel Solver, a developed version of the so-
called GRG2 code (64), was employed for this purpose.
Figure 2-6: Typical size distribution with TR30-C3 CPN at Ql=5 (µl/s) and Qg= 500 (sccm),
liquid: distilled water. D32=20.8 (µm), Dpeak=9.4 (µm).
38
Coefficients A, B and exponent C for the fitted model are given in Table 2-2 and can be
compared with those of the NT (Equation 2-3) and the MNT (Equation 2-8) models. As can be
seen the first coefficient has been slightly reduced from NT to ONT models and the modification
is mainly done through the second coefficient and exponent implying that the weight of the first
term in Equation (2-9) has been considerably emphasized. In fact this term now constitutes over
80 percent of the predicted D32 value.
Figure 2-8a-c compares the FNT estimation with the experimental Sauter mean diameter in
which the model shows considerable improvement in comparison to the original NT model of
Figure 2-3a-c. Additionally, an asymptote is seen for the drop size as the nozzle meets the
theoretical sonic conditions, specifically for Ql=5 and 10 (µl/s), which was not seen before.
Figure 2-7 and Figure 2- also illustrate the improved agreement between model and experiment
where the previous 4 fold overestimation is dropped to a maximum of 1.2 times. The agreement
is particularly improved at higher liquid flow rates as the FNT points are populated along the 45
degree slope line in Figure 2- which implies a better prediction comparing to the original model.
It must be added that the proposed FNT model is not intended nor claims to be a universal
optimized substitute for NT but rather a correlation that characterizes a particular nozzle well
under the tested range of flow parameters. As mentioned before, geometrical parameters are
absent
Parameter NT model MNT model FNT model
A 585 86.4 569.9
B 597 105.4 148.7
C 1.5 Exponential decay fitted 0.37
Table 2-2: Coefficients and exponent of original NT, modified NT and the fitted NT models.
39
Figure 2-7: Sauter mean diameter versus gas flow rate from experiment, original NT model
and Kahen et al’s MNT model for distilled water at Ql=10 (µl/s).
in the NT correlation. Further research must be carried out with a wide range of the ICP-MS
nebulizers of different sizes and large range of fluid properties and gas to liquid flow ratios
before claiming to have a universal correlation for application to ICP-MS. However, it would be
interesting at this stage to test the validity of the FNT model (at least qualitatively) with other
studies. Noting that, although there are many published papers on nebulizer characterization for
ICP-MS in the literature, only a few of them have reported all the detailed parameters required
for a comparative study.
For this purpose, Canals et al.’s study was selected (65) in which authors proposed a 6-parameter
model for D32 prediction of 4 different Meinhard CPNs with different solutions under a range of
liquid and gas flow rates.
0
20
40
60
80
100
120
140
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=10 (µl/s)
Experiment
NT
Kahen et al's MNT
40
(a)
(b)
0
5
10
15
20
25
30
35
0 200 400 600 800 1000
D32 (
µm
)
Qg (mlit/min)
Ql=1 (µl/s)
Experiment, DW
FNT, DW
Experiment, Methanol
FNT, Methanol
0
5
10
15
20
25
30
35
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=5 (µl/s)
Experiment, DW
FNT, DW
Experiment, Methanol
FNT, Methanol
41
(c)
Figure 2-8: Sauter mean diameter versus gas flow rate for distilled water and methanol from
experiment and the FNT model at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s).
( ) ( ) [ ( ) ] (2-10)
where coefficients Pij’s depend on nozzle geometry and flow conditions and are tabulated in
(65). Although the study is quite extensive from the point of view of the experiments carried out
and the model has yielded satisfactory results, the author believes it suffers from some issues.
First, it is not a simple and easy correlation to use and the coefficient for each specific
configuration of nebulizer and solution must be taken from a table. Second, the given form of the
correlation (Equation 2-10) certainly does not show the direct dependence of D32 on important
parameters like nozzle geometry and also fluid properties such as surface tension, liquid and gas
densities and viscosities, although all these parameters are embedded in the Pij coefficients.
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
D32 (
µm
)
Qg (ml/min)
Ql=10 (µl/s)
Experiment, DW
FNT, DW
Experiment, Methanol
FNT, Methanol
42
Figure 2-9: Ratio of calculated to measured Sauter mean diameter versus gas flow rate for
distilled water.
Table 2-3 compares Canals et al.’s model for two out of four tested nebulizers (specifications
given (65)) with D32 values from experiment, NT and the new FNT models. As can be seen from
the table, Sauter mean diameter is always larger for the M4 nebulizer which is also predicted by
NT and FNT models. The overestimation by the FNT model is certainly not as severe as the
original NT model. The highlighted areas in Table 2-3 are of particular interest since the flow
conditions are comparable to our experiments and as can be seen the agreement is quite
acceptable while the original NT shows a maximum of 21 fold overestimation! It should be
mentioned that the FNT model was characterized with a different CPN and liquid flow rates
generally smaller than this study and n-Butanol was never sprayed as a sample.
In a nutshell, the modification presented for NT in this work can very well fit into experimental
results in the range given by Table 2-3. Application of FNT to the different nebulizers and
solutions always yields better results than the original NT. The comparison of different
conditions in Table 2-3, also shows the most important parameters for NT type correlations are
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
0 200 400 600 800 1000
D32 c
alc
ula
ted
/D32 m
easu
red
Qg (ml/min)
Ql= 1 µl/s
Ql= 5 µl/s
Ql= 10 µl/s
43
Figure 2-10: Measured versus calculated Sauter mean diameter from the original NT and
fittedd NT (FNT) models.
the gas and liquid flow rates. The new FNT model can provide satisfactory predictions for other
nebulizers and solutions if their ranges of gas and liquid flow rates are comparable to this study.
Rizk–Lefebvre Correlation 2.3
In 1980, Lefebvre published an excellent comprehensive study on airblast prefilming type
nozzles used in gas turbine engines (49). Lefebvre investigated numerous correlations proposed
for different pneumatic atomizer nozzles and drew some general conclusions on the effects of air
and liquid properties on the Sauter mean droplet size. For low viscosity liquids such as water, he
found that the main factors governing D32 are surface tension, air and liquid densities and air
velocity, which clearly confirms the optimization we did in Equation (2-9). On the other hand for
0
20
40
60
80
100
120
140
0 10 20 30 40 50
Pre
dic
ted
D32
(µm
)
Measured D32 (µm)
FNT
NT
44
Nebulizer Ql
(ml/min)
Qg
(l/min) Solvent D32 (65). D32 exp D32 NT D32 FNT
M3 1.90 1.10 Water 10.74 11.1 79.71 25.59
M4 1.90 1.10 Water 11.09 12.6 74.32 20.34
M3 1.90 1.10 Methanol 5.41 5.7 73.59 14.83
M4 1.90 1.10 Methanol 5.61 6.0 73.97 15.3
M3 1.90 1.10 n-Butanol 7.41 7.3 147.12 23.08
M4 1.90 1.10 n-Butanol 7.52 7.9 147.52 23.47
M3 1.90 0.65 Water 17.43 17.0 157.92 31.0
M4 1.90 0.65 Water 18.02 20.5 158.95 32.0
M3 1.90 0.65 Methanol 9.55 10.9 159.28 21.0
M4 1.90 0.65 Methanol 10.04 11.3 159.93 21.6
M3 1.90 0.65 n-Butanol 12.21 12.8 321.08 29.08
M4 1.90 0.65 n-Butanol 12.7 13.2 321.75 29.73
M3 1.90 0.4 Water 37.15 * 320.66 49.05
M4 1.90 0.4 Water 38.09 * 322.34 50.69
M3 1.90 0.4 Methanol 22.53 21.0 325.91 29.88
M4 1.90 0.4 Methanol 23.69 20.9 326.96 30.90
M3 1.90 0.4 n-Butanol 29.50 23.4 660.94 35.7
M4 1.90 0.4 n-Butanol 30.35 22.1 662.03 36.76
M3 0.60 1.10 Water 8.42 9.5 20.12 14.33
M4 0.60 1.10 Water 8.70 12.1 20.73 14.92
M3 0.60 1.10 Methanol 4.05 4.8 17.47 11.26
M4 0.60 1.10 Methanol 4.27 5.3 17.85 11.63
M3 0.60 1.10 n-Butanol 6.02 5.5 30.65 17.68
M4 0.60 1.10 n-Butanol 6.13 6.6 31.05 18.07
M3 0.60 0.65 Water 15.83 15.6 39.93 21.92
M4 0.60 0.65 Water 16.35 16.0 40.97 22.93
M3 0.60 0.65 Methanol 7.74 8.8 35.72 16.25
M4 0.60 0.65 Methanol 8.12 9.9 36.37 16.89
M3 0.60 0.65 n-Butanol 9.61 9.9 64.67 23.78
M4 0.60 0.65 n-Butanol 9.97 11.0 65.34 24.43
M3 0.6 0.4 Water 32.53 27.1 76.26 33.33
M4 0.6 0.4 Water 33.51 * 77.94 34.97
M3 0.6 0.4 Methanol 19.13 15.6 69.95 23.40
M4 0.6 0.4 Methanol 20.04 16.3 71.01 24.43
M3 0.6 0.4 n-Butanol 23.59 17.8 129.79 31.84
M4 0.6 0.4 n-Butanol 24.27 19.2 130.88 32.90
Table 2-3: Comparison between the model (NT, FNT) and experimental values of D32 (µm).
*=inefficient nebulization
highly viscous liquids the air effects become more dominant and D32 is mainly determined from
liquid properties and especially viscosity. In addition Lefebvre (49) states that the viscosity has
45
an effect quite separate from air velocity as experimentally observed by Nukiyama – Tanasawa
(51) and many others. Therefore a correlation for mean drop size should include two terms, one
including air velocity and liquid density and a second term containing liquid viscosity
⟨ ⟩ ⟨ ⟩ (2-11)
For prefilming airblast atomizers, <D32>1 is governed by the Weber number:
⟨ ⟩
(
)
( ) (2-12)
here Lc is a characteristic dimension of the atomizer and represents its linear scale whereas Dp is
the prefilmer lip diameter, and several different sources cited in (49) suggest x~0.5 (Refer to
Equation 2-3 for instance).
From conservation of momentum in the liquid-gas region (66):
( ) (2-13)
or
(
)⁄
(2-14)
Substituting for UR in Equation (2-12) gives:
⟨ ⟩
(
)
(
) (2-15)
The viscous dominant term <D32>2 depends on Ohnesorge number, the ratio of the square root
of the Weber number to the liquid Reynolds number, as:
⟨ ⟩
(
)
( )
(2-16)
46
Substituting Ur from Equation (2-14) gives:
⟨ ⟩
(
)
(
) (2-17)
By combing Equation s (2-17) and (2-15), we’ll have:
(
)
(
) (
)
(
) (2-18)
The characteristic length Lc represents the dimension at the point or surface where the liquid first
meets the gas stream. For prefilming atomizers this would be the prefilmer lip diameter (Dp)
which is often replaced by the liquid capillary diameter for simplification (Lc=Dp=do). In
practice some secondary factors such as liquid Reynolds number and gas Mach number may
influence the atomization process in a manner which is not yet understood (67). To improve the
correlation and account for the secondary effects, the exponent of the first term was reduced
from 0.5 to 0.4 (67). Also if the gas velocity (ug) is replaced by the relative velocity (UR), the
influence of liquid to gas flow ratio must also be adjusted. In reference (67), a value of 0.4 was
suggested for the ratio. Therefore Equation (2-18) becomes:
(
)
(
)
(
)
(
) (2-19)
Equation (2-19) can be rearranged in the following form:
(
)
(
)
( )
(
) (2-20)
where the Weber number (Wedo) and Ohnesorge (Ohdo) numbers based on orifice diameter
appear consistent with Equation (2.2).
Coefficients A and B are atomizer dependent and must be found empirically. Rizk and Lefebvre
(67) found an excellent fit to experiment with values of A=0.48 and B=0.15 for their specific
nozzle.
47
The merit of Equation (2-19) is that unlike the Nukiyama-Tanasawa correlation (Equation 2-3),
it’s presented in nondiemnsionalized form meaning that any consistent units could be used for
calculation. The nozzle dimension is included in the correlation and the gas density is also
considered, although in the original paper (67), the gas velocity was in the range 10-120 (m/s) for
which the gas stream is very weakly compressible. The gas density in our experiment was
calculated from isentropic relation (Equation 2-6) and varies from 1.87 to 3.28 (Kg/m3).
The CPN can be regarded as a converging nozzle although its annular cross section is different
from the circular cross section of converging nozzles. Nevertheless, the parameter which is most
relevant to the flow state in a nozzle is the cross sectional area rather than its length or shape
(20). Therefore, like any other converging nozzle the maximum possible velocity for a CPN is
the sonic velocity. In this respect, the orifice based Weber number (Wedo) in Equation (2-20),
will have its maximum value when the sonic condition is met and remains constant. Similarly the
Nukiyama-Tanasawa correlation (Equation 3-2) suggests that the weight of the first term is
predetermined at the sonic condition and beyond. However, further increase of reservoir tank
pressure beyond 211.3 (Kpa) corresponding to sonic condition, would continuously increase gas
density and mass flow rate while the gas exit velocity remains sonic at the exit plane. As a result,
even at the constant sonic gas exit velocity, an improved atomization is expected. The effect of
liquid to gas flow ratio is only observed in the second term of the Nukiyama-Tanasawa
correlation (Equation 2-3). In contrast, in the Rizk-Lefebvre correlation (Equation 2-20) the ratio
appears in both terms and include the factors involved in atomization more realistically.
The Rizk-Lefebvre (RL) model has recently gained attention in spectrometry. Gras et al. (68)
applied the RL model to 3 nebulizers used for sample introduction in atomic spectrometry for the
first time and reported that while the model is capable of predicting the general trend of
phenomena; it lacks accuracy mainly due to differences in dimensions and operating conditions
between their nozzles and those used in the original study (67). The authors (68) then tried to
overcome the problem by tuning the general RL model (Equation 21-2) by tuning its exponents.
We call this model Gras’ modified RL or (MRL-G) for simplicity. The exponents for this model
are a=0.5, b=0.53 and d=0.49 respectively.
48
Kahen et al. (9) also modified the RL model for modeling the Sauter diameter from a DIHEN
used for micro sampling (MRL-K) but they applied the modifications through both coefficients
and exponents and attained satisfactory results but also state that their modification is very
nebulizer dependent and exhibits tremendous deviation if used on other nebulizers.
As in (9) and (68) and similar to the modification done for the NT model, we also fitted Equation
(2-21) for our specific range of flow parameters (69) but decided to keep the coefficients the
same as the original study (67) and (68). Coefficient B and exponent c are very similar in all the
studies. This is mainly because the term is more dependent on to the liquid viscosity effects and
for low viscosity, the weight of this term is more or less the same (Table 2-4).
(
)
(
)
(
)
(
) (2-21)
As can be seen from Table 2-4, the modification is primarily done through the first term to
account for different operating gas and liquid flow rates (FRL model). Since our ranges of
parameters are closer to Kahen et al.’s study (9), the obtained value for coefficient a is very
similar to their study.
Parameter RL model MRL-G model MRL-K model FRL model
A 0.48 0.48 0.54 0.48
a 0.4 0.5 0.31 0.3
b 0.4 0.53 0.5 0.34
B 0.15 0.15 0.16 0.15
c 0.5 0.49 0.48 0. 5
Table 2-4: Coefficients and exponents of different RL type correlations, RL=original Rizk-
Lefebvre model, MRL-G=Gras’ modified model, MRL-K= Kahen et al.’s modified model,
FRL= present fitted RL model
49
(a)
(b)
0
5
10
15
20
25
30
35
0 200 400 600 800 1000
D32 (
µm
)
Qg (mlit/min)
Ql=1 (µl/s)
RL
MRL-G
MRL-K
FRL
Experiment
0
5
10
15
20
25
30
35
40
0 200 400 600 800 1000
D32 (
µm
)
Qg (mlit/min)
Ql=5 (µl/s)
RL
MRL-G
MRL-K
FRL
Experiment
50
(c)
Figure 2-11: Sauter mean diameter versus gas flow rate for distilled from experiment and
different RL type models at (a) Ql=1 (µl/s), (b) Ql=5 (µl/s) and (c) Ql=10 (µl/s).
The original RL model and the modified models were applied to our flow conditions. The results
of these models together with our fitted model (FRL) are plotted alongside experiment. As can
be seen from Figure 2-11, the original RL model shows underestimation. This behavior was also
reported in (68) and (9). The FRL shows considerably better size prediction in comparison to the
RL and the MRL-G models and is closer to the MRL-K model. This is probably due to the fact
that in the MRL-G model, the range of flow parameters (particularly the liquid flow rate) was
larger than our experiment. In fact as stated in the Chapter 1, we have exploited the CPN not in
its nominal range but rather in its lower limit of aerosol generation in order to produce the finest
possible aerosol size and this range is closer to the micro sampling liquid flow rates of Kahen et
al. (9), thus the MRL-K model prediction is in better agreement with our optimized model.
Despite all the differences, it’s worth mentioning that the RL type correlation predicts size
diameters far better than Nukiyama-Tanasawa correlation (Equation 2-3), particularly at higher
liquid flow rates. Compare Figure 2-11b-c with Figure 2-8b-c for instance.
0
5
10
15
20
25
30
35
40
45
0 200 400 600 800 1000
D32 (
µm
)
Qg (mlit/min)
Ql=10 (µl/s)
RL
MRL-G
MRL-K
FRL
Experiment
51
Variation of Characteristic Mean Drop Sizes 2.4
Although D32 is the most widely used aerosol size characteristic moment used for heat and mass
transfer studies, it doesn’t sufficiently characterize an aerosol. McLean et al. (12) lists 14
different parameters for this purpose and states among them characteristic moment and the size
distribution are the most important ones in dictating the quality of aerosol.
Consider a nebulizer system that can produce perfectly monodisperse aerosol at certain
conditions of 10 (µm) for example, from the definition of Sauter mean diameter, D32 would also
be 10 (µm) and all these droplets are successfully consumed by plasma, that leads to a transport
efficiency of 100 percent. Now imagine another nebulizer that generates the same Sauter mean
diameter at the same flow condition but with quite polydisperse drop size, in this respect the
transport efficiency of the system can be significantly smaller than 100 percent, because the
mass of a single 20,30,50 and 100 (µm) droplet equals 8,27,125 and 1000 times of the 10 (µm)
droplets. Thus prediction of distribution functions is of significant importance particularly in MS
which is the subject of Chapter 5 of the present study.
Customary distribution functions are defined by two parameters: one characteristic moment and
the span or width of the distribution but as will be shown in Chapter 5, a distribution function
can similarly be specified by a number of characteristic moments and have a span comparable to
a distribution function defined by a moment and span directly through the method of Maximum
Entropy Principle (MEP).
Aside from Sauter mean diameter (D32), two other characteristic moments are required for the
analytical study of the aerosols in this study, which are mass mean diameter (D30) and another
nameless characteristic moment (D-10), that is aerosol surface area to its mass ratio, with the
following definition:
∑
(2-22)
∑
(2-23)
52
here ΔNi, Di and Ntot are number of droplets in class diameter Di and total number of droplets.
Therefore, not only each characteristic moment contains some information regarding the spray
individually but also their combination determines the shape of the size distribution function
such as location of the peak diameter, span and its skewness.
Figure 2-12a-b shows variation of (D30/D32) and (D30/D-10) ratios with the normalized gas flow
rate. The gas flow rate here is normalized by the predicted sonic gas flow rate at (Qg=494 ml/min
corresponding to 108.3 Kpag). The Figures show that while the first ratio is an ascending
function with gas flow rate, a reverse trend is observed for the second ratio. Besides, as can be
seen, the ratio is hugely affected by the variation in the normalized gas flow rate from 0.5 to 1.4
whilst the liquid flow rate shows considerably less influence. Thus it seems reasonable to present
the ratios as averaged values over entire liquid flows for each specific gas condition, i.e. only as
a function of normalized gas flow rate as follows:
(
)
(
)
(
)
(2-24)
(
)
(
)
(
)
(2-25)
Equations (2-24) and (2-25) will be used together with FNT (Equation 9-2) and FRL (Equation
2-21) correlations for predicting size distribution of CPN.
Figure 2-13a-b depicts the local variation of D30/D-10 and D30/D32 for CPN with downstream
axial location for the sample liquid flow rate of Ql= 5 (μl/s) respectively. As can be seen at any
axial location the D30/D-10 ratio decreases while D30/D32 ratio increases by growth of the gas flow
rate. Besides, for any gas flow rate, D30/D-10 the ratio drops with the axial location which is more
noticeable at lower gas flow ratios. The opposite trend is observed for D30/D32 ratio and again the
53
(a)
(b)
Figure 2-12: Variation of (a) D30/D32 and (b) D30/D-10 versus the normalized gas low rates at the
downstream axial location of z=10 (mm).
0.65
0.7
0.75
0.8
0.85
0 0.5 1 1.5 2
D30/D
32
Qg/Qgsonic
Ql=1 µl/s
Ql=5 µl/s
Ql=10 µl/s
Average
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
0.4 0.6 0.8 1 1.2 1.4
D30/D
-10
Qg/Qgsonic
Ql=1 µl/s
Ql=5 µl/s
Ql=10 µl/s
Average
54
(a)
(b)
Figure 2-13: Variation of characteristic moment ratio with axial location (a) D30/D-10 and (b)
D30/D32 at Ql=5 (μl/s).
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
0 10 20 30 40 50 60
D30/D
-10
z(mm)
Axial variation of D30/D-10
Qg=266(mli/min)
Qg=317 (ml/min)
Qg=390 (ml/min)
Qg=494 (ml/min)
Qg=590 (ml/min)
Qg=688 (ml/min)
Qg=786 (ml/min)
0.65
0.7
0.75
0.8
0.85
0.9
0.95
0 10 20 30 40 50 60
D30/D
32
z(mm)
Axial variation of D30/D32
Qg=266(mli/min)
Qg=317 (ml/min)
Qg=390 (ml/min)
Qg=494 (ml/min)
Qg=590 (ml/min)
Qg=688 (ml/min)
Qg=786 (ml/min)
55
gradient is a somewhat remarkable for Qg=266-390 (ml/min). Very similar trends are observed at
Ql=1 and 10 (μl/s) which are not shown here.
Nebulization Efficiency 2.5
The role of nebulizers in general is to convert liquid kinetic energy to surface energy, as a result
the surface area of the medium will be increased which in turn promotes the rate of heat and
mass transfer (49). The nebulizer efficiency is thus defined as the ratio of the spray surface
energy to the input kinetic energy of gas and liquid and also the thermal energy of the gas
stream.
The surface energy of droplets per unit time for an ensemble of droplets is:
∑ (2-26)
where n• has the unit (1/s). The consumed thermal energy (THE) and kinetic energy of the liquid
and gas (KE) is:
(2-27)
here ho is the total enthalpy of the gas whereas m•l and Ucap represent liquid flow rate and
velocity in the capillary. Sharp (20) states for short nozzles down to 200 (µm), the adiabatic
condition holds true and the total enthalpy remains constant (Δho=0). Thus by rearranging
Equation (2-27) we’ll have:
(2-28)
Besides, conservation of mass dictates:
∑
(2-29)
Therefore the nebulization efficiency will be:
56
( )
(2-30)
Since efficiency is below 100, we can define a minimum attainable D32 for an ideal nebulizer:
( )
(2-31)
(2-32)
The nebulization efficiency of the CPN under different flow conditions are plotted in Figure
2-14. The figure shows that first the efficiency drops by either decreasing the liquid flow rate or
increasing the gas flow, but more importantly it demonstrates that nebulization is generally a
very poor process as less than 1.2 percent of the input energy is converted to the surface energy.
In fact, nearly all the input energy is consumed for aerosol acceleration or wasted by the gas
expansion. The same trend is also reported for the efficiency of pressure atomizers (70) which
supports this trend. The minimum attainable D32 changes between 0.002-0.4 (µm) for all the
tested flow conditions, If the nebulization was only 10 percent efficient a D32 of 0.02 to 4 (µm)
would be obtained and most likely the entire aerosol would be consumable in the plasma without
perhaps any need for aerosol modification in a spray chamber.
In the derivation of the nebulizer efficiency, the conservation of mass was applied between the
liquid in the capillary and the droplets; however the measurements were taken 10 (mm)
downstream of the nebulizer tip along the centre line and not for the entire spray cross section.
This would introduce some error in efficiency calculation. Nevertheless in this location the spray
is not highly dispersed, i.e. the spray plume is very narrow and the spray cross section is small.
Further downstream where spray is considerably dispersed, D32 calculation along the centre line
may not represent the whole plane of droplets, besides the evaporation effects may also change
the spray size distribution, therefore Equation (30-2) must be used only for the primary aerosol.
57
Figure 2-14: Nebulization efficiency versus gas flow rates for distilled water measure at z=10
(mm).
It must be noted that the nebulization efficiency or transport efficiency, in ICP-MS literature (10)
is usually defined as the volume percent of the aerosol containing in droplets below 10 (µm)
(about 1-5 percent for a conventional CPN) but the definition given here is focused on the
conversion of input kinetic energy to surface energy of the aerosol or in other words the ratio of
minimum attainable to measured Sauter mean diameter.
Contribution 2.6
The CPN was characterized for a specific range of liquid and gas flow conditions. Although the
fundamentals of the nebulizer are very close to air-assist nebulizers, implementing the well
known correlations such as NT and RL lead to erroneous size predictions because the geometry,
dimensions and, more importantly, because the flow conditions are very different in ICP-MS
sample introduction. It was observed that the NT correlation severely overestimates the Sauter
0
0.2
0.4
0.6
0.8
1
1.2
0 200 400 600 800 1000
Eff
icie
ncy
(%
)
Qg (ml/min)
Ql=1 (ul/s)
Ql=5 (ul/s)
Ql=10 (ul/s)
58
mean diameter, while the RL model shows underestimation. The two models were then modified
considering the appropriate ranges of liquid and gas flow rate typical for ICP-MS applications
and a significantly better agreement with experiment was achieved comparing to the original NT
and RL models. Two correlations were proposed for the (D30/D32) and (D30/D-10) ratios which
contain information regarding the shape of distribution functions as will be discussed in the
Chapter 5. Furthermore, variation of these characteristic mean size ratios with axial location was
presented for different gas flow rates. A new nebulization efficiency was derived, introduced and
tested for the CPN under different flow conditions. This new definition of the nebulization
efficiency differs from conventional transport efficiency used in spectrometry. The present
definition focuses on the disintegration of the liquid bulk to the droplets or in the other words on
the conversion of kinetic energy to surface energy for atomization. It was found that from this
efficiency point of view, nebulization is a very poor process as only less than 2 percent of the
input kinetic energy is converted to the aerosol surface energy (for the given conditions). In
addition, a minimum attainable Sauter mean diameter for an ideal nebulizer from the efficiency
definition was obtained which could be useful for aerosol studies.
59
Chapter 3 Aerosol Size Characterization of Flow Focusing Nebulizer
Nozzle Design 3.1
The pneumatic Flow Focusing Nebulizer (FFN) is a relatively novel technology that was first
introduced by Ganan-Calvo (41) and (71). Due to the novelty of the technique, the number of
publications in this field is not comparable to other nebulizers, particularly those used for ICP-
MS sample introduction and despite the unique features of the nebulizer; it’s not commercially
available in the market.
The FFN has a simple design similar to plain-jet atomizers. As claimed in the literature (41), the
nebulizer can operate quite robustly with versatile flows to produce monodisperse spray. The
liquid sample experiences small shear forces associated with other micronebulizers with a solid
contact that may fracture or degrade some molecular species. Besides, as will be explained, in
this method the liquid does not touch the orifice and the necessary pressures are much smaller
than ones required to achieve comparable liquid jets using direct liquid injection leading to
increasingly lower risk of clogging and other operational costs. Furthermore, the nebulizer’s
geometrical parameters have very little influence on the nebulizer performance and the resultant
spray in contrast to concentric nebulizers described in Chapter 1. The simplicity of the
configuration also allows packing and multiplexing with only the restrictions imposed by
manufacturing constraints.
The aforementioned features, makes the nebulizer an excellent choice for ICP-MS applications,
at least theoretically. However there are currently only two papers published that have studied
the possible application of the FFN in spectrometry (42) and (43) where the results showed
superior aerosol generation in comparison to common micronebulizers.
Since the nebulizer was not commercially available, we have designed two prototype custom
designed FFNs in collaboration with Tanner’s group from the Department of Chemistry at the
University of Toronto (45). Figure 3-1 shows the schematic design of first prototype.
61
As can be seen the nebulizer is equipped with two CCD cameras, one to view filament formation
in the pressurized tank section and the other one to observe downstream disintegration of the
filament into droplets whilst the area of observation inside the aperture was lighted by exploiting
an LED. A protective cap was placed under the planar orifice to secure it against displacement.
Inside the cap, there is a 5 (mm) diameter and 35 (mm) long aperture where the filament
disintegration takes place. In this design, a 100 (µm) plain orifice was laser drilled on a 25-50
(µm) stainless steel plate (Part FSS-3/8-DISK-100, Lennox Laser, Glen Arm MD) and was
placed 300 (µm) below the PEEK capillary tube of 360 (µm) OD and 125 (µm) ID (Upchurch
Scientific Inc., Oak Harbor, WA). The liquid was injected through the capillary with the aid of a
syringe and pump system. Inside the tank, pressure was built up by Argon flow feeding from the
nebulizing line. A separate makeup gas line was connected at 45 degree to the aperture and its
flow was directly controlled by a rotameter as shown in Figure 2-1. It’s worth noting that the
makeup gas has no role in aerosol generation and it only avoids aerosol deposition on the
aperture walls and assists in transportation of the aerosol toward plasma.
The prototype nebulizer shown in Figure 3-1 and Figure 3-2 was robust and worked stably down
to liquid flow rates of Ql=1 (µl/s) but, below this rate a stable filament was never formed.
Instead the nebulizer spit the liquid irregularly and frequent starvation was observed. To resolve
this problem and also recalling that this rate of flow is not suitable for microsampling, the
nebulizer was redesigned. The CCD cameras and LED components were removed from the
second design and the heavy steel structure was replaced by a long straight tube and steel
capillary (shown in Figure 3-3) whose critical dimensions are given in Table 2-1. With the
second design, we were able to lower the liquid flow rate to Ql=0. 1 (µl/s) and observe formation
of a stable filament under a microscope.
For characterization of the nebulizer and also the aerosol modeling through the Maximum
Entropy Principle (MEP), only the results of the second prototype were used. It’s worth noting
that the nebulizer makeup gas line was not used during characterization of the primary aerosol,
since its role is limited to the aerosol transportation and to some extent evaporation rather than
bulk liquid disintegration and spray formation.
63
Figure 3-3: Schematic design of the second FFN.
Theoretical Background 3.2
The gas flow used for nebulization builds up a pressure difference (ΔPg) across the nebulizer tip,
i.e. the orifice in Figure 3-3 or Figure 3-4. When this pressure difference is sufficiently large to
overcome the surface tension force of a liquid attached to the mouth of a capillary with diameter
(Dcap), it pulls the liquid toward the orifice (diameter do) and forms a cusp-like shape at a critical
distance from the orifice (H). If the liquid is constantly supplied through the capillary at the rate
(Ql), then a steady state liquid filament with diameter dj is emitted smoothly and extends few
millimeters depending on the flow conditions downstream of the orifice (Figure 3-4a-c).
Ganan-Calvo (41) has shown that the diameter of the liquid jet (dj) can be derived by solving the
one dimensional Navier-Stokes equation in the axial direction (along the nebulizer axis).
Consider the meniscus attached to the capillary in Figure 3-4. The diameter of the filament
changes with the axial
64
Figure 3-4: Photographs taken from inside and outside of FFN (41) showing (a)- Capillary and
liquid filament, (b)-The liquid jet exiting the orifice and (c)- Unstable wave growth on the
filament surface, breakup and droplet generation
location due to the suction effect of the gas expanding from the orifice and the tangential viscous
stresses (ηs) on the jet surface in the axial direction.
(
( )
) ( )
(3-1)
The shear stress term in Equation (3-1) may be neglected in comparison to the kinetic energy of
the fluid and the pressure difference providing that the orifice thickness (L) is smaller or
comparable to its diameter. The capillary equation in the radial direction requires that:
65
( ) (3-2)
In practice, the pressure difference ΔPg is larger than the surface tension forces. Thus, Equation
(3-2) will become:
(
( )
)
(3-3)
Integrating from the mouth of the capillary to the orifice exit, results:
(
) ( ) (3-4)
Since in FFN, the jet diameter is at least an order of magnitude smaller than the capillary and the
orifice diameter (dj<<Dcap & do,), the first term on the left hand side may be neglected and the
resulting jet diameter will be:
(
)
(3-5)
Equation (3-5) shows that first; the jet diameter is only controlled by the applied pressure
difference across the nebulizer tip and the liquid flow rate and second it is independent of liquid
properties like the liquid surface tension and viscosity. Besides, the jet diameter remains constant
up to the breakup point because the gas pressure after the exit remains constant (41).
Equation (3-5) may suggest any desired combination of ΔPg and Ql can be selected to generate a
liquid filament, however when the liquid kinetic energy is not large enough to overcome surface
tension forces no filament is formed. Therefore for any given pressure difference and surface
tension, a minimum sprayable liquid flow rate exists:
(
)
(3-6)
66
The velocity of the filament is calculated from the liquid flow rate and the jet diameter (3-5):
√
(3-7)
Equation (5-3) implies that for any liquid, the jet velocity from the orifice exit down to the
breakup point is controlled not by the liquid flow rate (Ql) but by the applied pressure difference.
In other words for a constant pressure difference, increasing the liquid flow rate will only grow
the jet diameter while keeping the jet velocity untouched.
The physics explained above remains valid as long as the nebulizer is working under the flow
focusing regime. Ganan-Calvo (72) has shown that for the ratios H/do>0.25, the liquid pattern is
flow focusing and the gas flow follows the liquid axially. If the ratio drops below 0.25, the flow
regime will change dramatically. The gas flow becomes radial, i.e. perpendicular to the
symmetry axis. As a result a stagnation point develops between the capillary and the orifice
which causes a portion of the gas to flow upward and mix with the liquid. In such a scenario, the
liquid jet vanishes and is replaced by a plume of unsteady chaotic liquid ligaments.
Once the gas exits the orifice, the cylindrical mixing layer between the gas stream and the
stagnant ambient gas becomes unstable from the classical Kelvin-Helmholtz instability. The
growth rate of the mixing layer depends on the Reynolds number of the flow and the formed ring
vortices of the order (ug/do). Taking a characteristic velocity of 200-300 (m/s) corresponding to
the range of gas flows in our experiments and an orifice diameter of 150 (µm), the resultant
frequency is of the order 1-2 (MHz), which is of the order of the liquid jet breakup frequency (fb~
(ζ/ρl×dj3)0.5
). Thus, the formation of the ring vortices excites the perturbation on the liquid jet
until it disintegrates into a plume of fine droplets (71). By tuning a resonance frequency through
the gas velocity or pressure difference driving the gas stream, one can apply the appropriate
wavelengths on the jet surface to generate drops of desirable sizes.
The breakup process of liquid jets is governed by the Weber number, that is the ratio of
aerodynamic forces to surface tension forces, and the Ohnesorge number, which shows the
relative importance of the stabilizing viscous forces.
67
(3-8)
( ) (3-9)
Please note in pneumatic nebulization, the gas velocity is orders of magnitude larger than the
liquid velocity, thus the relative velocity is substituted by the gas velocity in Equation (3-
8).Besides for the range of parameters in the experiments carried out, the expected jet diameter
changes between 3 and 12 (µm) and the Ohnesorge number is below unity. Thus, the viscosity
has little influence on either jet formation or disintegration (71), making the nebulizer an
excellent choice for introduction of organic based solutions containing cultured cells whose
fragile membranes are prone to rupture (41).
It’s also worth noting that formation of a small micron scale jet from Equation (3-5) does not
necessarily mean the resultant aerosols will have narrow size distribution. Ganan-Calvo (41) and
(71) states as long as the jet based Weber number (Equation 3-8) is kept below a critical value of
40, the aerosol shows monodispersity. At higher values of the Weber number, the time growth of
the non-symmetric perturbations is substantially shortened. Hence, both symmetric and non-
symmetric modes coexist which develops ligaments of different lengths that in turn leads to
higher degree of polydispersity. The jet based Weber number (Equation 3-8) in our experiment
varied in the range 5-30.
Figure 3-5 shows the orifice hole plate of the first prototype together with emitted micro jet for
distilled water at ΔPg=70 (Kpag) and Ql=1.66 (µl/s). The predicted jet diameter from Equation
(3-5) is 13.4 (µm) which is an order of magnitude smaller than orifice diameter of 150 (µm), as
can be noticed from the figure. Besides for the given conditions Equation (3-8) gives a jet based
Weber number of 26.1 that is below the critical value. Hence a relatively monodisperse aerosol
is expected. The number and volume distribution presented in Figure 3-6 shows high degree of
monodispersity for the mentioned flow conditions. For instance the standard deviation of the
number based distribution is 16 percent with mean, mass, Sauter and peak diameters of D10=19,
D30=19.5, D32=20, Dpeak=19.3 (µm) respectively. The close location of these characteristic
68
Figure 3-5: View of orifice hole and the emitted micro jet of the first FFN prototype. Orifice
diameter do=150 (µm), ΔPg=70 (Kpag) and Ql=1.66 (µl/s). Predicted jet diameter dj= 13.4 (µm).
moments on the normalized size axis and the sharp gradient observed in the cumulative
distribution curves are also indications for the generation of a highly monodisperse aerosol.
Furthermore, the ratio of any characteristic moment to jet diameter is very close to the Rayleigh
breakup regime (D/dj=1.89) as reported in references (41) and (71).
Figure 3-6b exhibits that less than 5 percent of the aerosol volume consists of droplets below 10
(µm). As stated in Chapter 1, drops larger than 10 (µm) are not consumed by the plasma and may
even stop mass spectrometry completely. Although the nebulizer works in a very predictable
69
(a)
(b)
Figure 3-6: Distribution curves for flow conditions given in Figure 3-5 (a) number and volume
distribution. (b) cumulative number and volume distribution.
D/D30
fn=
dN
/dD
,fv
=d
N/d
Vo
l
0.5 1 1.5 2 2.50
1
2
3
4
fn
fv
D (um)
cu
mu
lative
dis
trib
utio
n
0 10 20 30 400
20
40
60
80
100
Cumulative fn
Cumulative fv
70
manner and the results are in satisfactory agreement with literature, the quality of the aerosol at
the mentioned flow condition is not suitable for sample introduction. The atomization
performance may be improved by lowering the liquid flow rate but, unfortunately, the
employment of the first prototype below the liquid flow rate of Ql=1 (µl/s) led to very unstable
jet formation.
Another drawback of the first nebulizer is that the closest location for aerosols measurement was
5 (mm) below the cap and the aperture. By considering the 35 (mm) aperture length,
measurements were actually taken 40 (mm) downstream of the orifice exit. Obviously at this
location, a considerable portion of fine droplets may be lost due to deposition on the walls of the
aperture and some might have undergone secondary effects like collision and evaporation, thus
the distribution curves like those plotted in Figure 3-6a-b do not necessarily represent the
primary distribution of the aerosol. In addition, the gradual droplet deposition on the aperture
walls and their continuous downward motion would cause a liquid film formed at the mouth of
the aperture which would burst from time to time and interrupt the experiments.
In the second design of the FFN, the cap, aperture and CCD cameras were removed enabling us
to measure the primary aerosol as close as possible to the orifice exit and also lowering the liquid
flow rate below 1 (µl/s). From this point forward, all the characterization and modeling carried
out on the FFN are attributed to the second prototype, unless otherwise stated.
Droplet Size Modeling and Variation of Characteristic Mean 3.3Drop Sizes
The second FFN prototype was specifically designed as a direct injection nebulizer for
microsampling at low liquid flow rates (Ql= 0.1-1 µl/s). In all the experiments the nebulizing gas
flow rate was kept in a very narrow range of Qg=0.15-0.21 (l/min) in order to control the small
microjet and fine plume of aerosol. However, even for this narrow range, the Sauter mean
diameter changed between 6.6 (µm) to 21 (µm) corresponding to a jet diameter variation between
3.3 (µm) to 12.3 (µm), respectively. Due to very narrow range of flow parameters, unlike the
CPN, optimization of the NT and RL correlations does not seem reasonable but the two models
71
will be used for comparison. Groom et al. (73), proposed the following correlation for size
characterization of their flow focusing nebulizer:
[
(
)
]
(
) (3-10)
here Dcap is the liquid capillary diameter, and WeDcap and OhDcap are Weber and Ohnesorge
numbers based on the capillary diameter respectively.
The original RL model (Equation 21-2) shows that Weber number and liquid to gas mass flow
ratio acts in opposite direction and their exponents are of the same order of magnitude (Table
2-4). In addition, the exponent of the Ohnesorge number in Equation (21-2) is of the order of 1.
Groom et al. (73) also suggested j=1 for Equation (10-3). Nevertheless, the unknown coefficients
C1, C2 and the exponent m are nebulizer dependent and must be found from drop size
measurements. These parameters are given as C1=0.35, C2=0.25 and m=-0.75 for CPN and
C1=0.4, C2=0.4 and m=-0.6 for flow focusing nebulizer, respectively.
By taking the same approach and fitting Equation (3-10) for drop size measurements of the FFN,
C1=6.3, C2=0.31 and m=-0.84 resulted. The obtained exponent (m) is of the same order as
Groom et al.’s study (73) exhibiting similar drop size dependency to Weber number and liquid to
gas mass flow ratio. While C2 is very similar to the mentioned study, coefficient C1 is one order
larger than the reported numbers. However it should be noted that this coefficient is basically a
scaling factor and the reason for this difference is that first, the orifice and capillary diameter
were 2000 (µm) in the study in comparison to 150 and 175 (µm) here and second the liquid to
mass flow ratio was varied between 0.5-1.5 in (73) while it varies between 0.02 to 0.2 in the
present task.
Figure 3-7a-e demonstrates the predictions of the present model in comparison to the classical
NT and RL models. As for the CPN, the NT and RL show overestimation and underestimation
respectively although the dropping trend with increase of gas flow rate is clearly observed. The
72
(a)
(b)
0
5
10
15
20
25
30
35
100 150 200 250
D32 (
µm
)
Qg (mlit/min)
Ql=0.1 (µl/s)
NT
RL
Model
Experiment
0
5
10
15
20
25
30
35
100 150 200 250
D32 (
µm
)
Qg (mlit/min)
Ql=0.2 (µl/s)
NT
RL
Model
Experiment
73
(c)
(d)
0
5
10
15
20
25
30
35
100 150 200 250
D32 (
µm
)
Qg (mlit/min)
Ql=0.4 (µl/s)
NT
RL
Model
Experiment
0
5
10
15
20
25
30
35
100 150 200 250
D32 (
µm
)
Qg (mlit/min)
Ql=0.6 (µl/s)
NT
RL
Model
Experiment
74
(e)
Figure 3-7: Comparison between drop size models and experiments at different liquid flow rates
for FFN at: (a) 0.1, (b) 0.2, (c) 0.4, (d) 0.6 and (e) 1.0 (µl/s).
present model (Equation 10-3), shows better agreement with the experiments that is noticed from
the figure.
As stated in previous chapter, the MEP modeling of the spray requires two size characteristic
moment ratios, hence as far as the modeling is concerned the variation of the dimensionless
ratios, rather than the characteristic moments, are of prime importance. Figure 3-8 depicts
changes in D30/D32 and D30/D-10 ratios with the normalized gas flow rate. Similar to the CPN
(Figure 2-12), an ascending and descending trend is observed for D30/D32 and D30/D-10
respectively. Although for Ql=0.1 (µl/s) similar trends are observed with the increase of
normalized gas flow rate, but the corresponding curve shows some degree of deviation from the
rest. Figure 3-8 suggests unlike CPN, the ratios are influenced by the liquid flows. In addition, a
close look at the distribution curves proves for any liquid flow rate at low gas flows, a weak
0
5
10
15
20
25
30
35
40
100 150 200 250
D32 (
µm
)
Qg (mlit/min)
Ql=1.0 (µl/s)
NT
RL
Model
Experiment
75
(a)
(b)
Figure 3-8: Variation of (a) D30/D32 and (b) D30/D-10 versus the normalized gas low rates at the
downstream axial location of z=10 (mm).
0.75
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
0.5 0.6 0.7 0.8
D30/D
32
Qg/Qgsonic
Ql=0.1 ul/s
Ql=0.2 ul/s
Ql=0.4 ul/s
Ql=0.6 ul/s
1.5
1.6
1.7
1.8
1.9
2
2.1
0.5 0.6 0.7 0.8
D30/D
-10
Qg/Qgsonic
Ql=0.1 ul/s
Ql=0.2 ul/s
Ql=0.4 ul/s
Ql=0.6 ul/s
76
bimodal behavior is observed which is resolved by increasing the gas flow rate that may account
for these irregularities of Figure 3-8. Please note that the axial and radial variation of the two
ratios is given in Appendix C.
Nebulizer Performance 3.4
The second FFN prototype was specially designed for microsampling applications, i.e. liquid
flow rates below Ql=1 (µl/s). For the range of flow parameters in our experiments, the predicted
jet diameter varies between 3 to 12 (µm) and a jet velocity of the order of 10 (m/s). The liquid
Reynolds number is therefore small (Reliq~30-150) and flow can be considered laminar. On the
other hand, the gas velocity calculated from the isentropic theory (Equations 2-4 to 2-7) is two
order of magnitudes larger than the liquid which leads to gas Reynolds number of (Reg=4000-
7000) based on the orifice diameter indicating strong turbulent behavior for the gas flow.
Although the breakup mechanism depending on relative velocity between the gas and liquid are
essentially the same for both pressure and airassist type atomizers, various flow phenomena such
as turbulence and cavitation are not observed in airassist atomization (50), consequently
atomization is solely depended on the momentum of the atomizing gas and air to liquid mass
flow ratio (Refer to Equations (3-10), (2-9) and (2-21) for instance).
Figure 3-9 shows the variation of characteristic size moments with jet based Weber number. In
this graph, the Rayleigh limit is plotted for comparison. Rayleigh (74) has shown for a laminar
breakup of a liquid jet in quiescent air, the resulted drop diameters are 1.89 times that of the jet
diameter. Weber (75) found that the effect of increased air friction on the jet (i.e. increasing
relative velocity) is to shorten the breakup length optimum wave-length that leads to smaller
drop sizes. For example D10, D30 and D32 are proportional to the Weber number to power -0.88, -
0.93 and -0.97, respectively.
It must be mentioned that the Rayleigh and Weber studies were carried out for laminar gas flows
and only presented the most probable drop sizes corresponding to the optimum wavelength,
different from the characteristic moments given here. Thus it seems reasonable to have
characteristic moments larger than the most probable droplet size particularly for higher order
77
Figure 3-9: Variation of characteristic moments of the primary size distribution with the jet based
Weber number measured at z=10 (mm).
moments like D30 and D32. The area marked in Figure 3-9 illustrates cases whose characteristic
moments are larger than the Rayleigh’s prediction. The jet breakup length in our experiments
was 1 to 3 (mm) while the measurements were taken 10 (mm) downstream of the nebulizer tip.
Therefore, coalescence might be responsible for the overgrowth of the characteristic moments at
this distance, particularly because it has occurred at low Weber numbers.
When the Weber number was increased over 10, the characteristic moments dropped below the
Rayleigh limit. This trend was continued down to moment to jet diameter ratio of 0.5-1
indicating an improved atomization at the Weber numbers around 20-25. Ganan-Calvo (41) and
(71) reported a Raleigh type breakup for Weber numbers below the critical value of 40. In other
words, the points are supposed to be populated around the Rayleigh limit in Figure 3-9, or
perhaps slightly below it (due to the increased drag effects). In addition, a fair comparison
between our results and those in (41) and (71) is not possible, since a similar graph was not
presented there. Besides, the aforementioned authors suffice to state below the critical Weber,
78
the distribution is very narrow and did not report monodispersity in full detail. For example, a
fine but polydisperse aerosol can be seen in Figure 8b-c of (71) and a very narrow one in Figure
8a of the same reference. The standard deviation of the primary size distribution for the FFN
versus the Weber number is plotted in Figure 3-10. As can be seen, in contrast to what is claimed
by Ganan Calvo (41) and (71), no clear relation can be found between the Weber number and the
span of the distribution. A closer look at our results show, the distribution becomes narrower by
increasing the Weber number for any desired liquid flow rate.
Based on these findings, we can say that the Weber number should not be used as a measure of
spray monodispersity/polydispersity, instead it’s the ratio of aerodynamic drag to capillary forces
by definition and merely reflects the degree of atomization.
Figure 3-11 to Figure 3-13 illustrate the number and volume-based size distribution together with
the corresponding cumulative distribution for 3 points of Figure 3-9. One, showing a large
characteristic moment to jet ratio, one close to the Rayleigh line and the last one in the region of
improved atomization, i.e. small moment to jet diameter ratio.
Figure 3-10: Standard deviation of the primary size distribution of the FFN versus the jet based
Weber number measured at z=10 (mm).
20
25
30
35
40
45
50
55
60
65
70
0 5 10 15 20 25
10
0*
Drm
s/D
10
Wedj
79
The number based size distribution shows a secondary peak which disappears by increasing the
Weber numbers until it becomes a complete monomodal distribution in Figure 3-13.
The most probable droplet size for all the distribution curves is located below 10 (µm), but the
portion of the aerosol below this critical number changes drastically from one case to another.
For instance 51 percent of aerosol size and 3 percent of the aerosol volume are contained in
drops below 10 (µm) in Figure 3-11. These numbers grow to 79.4 and 20.5 for Figure 3-12 and
98.1 and 87.7 in Figure 3-13.
The small cumulative percentage in the first case is of little application in ICP-MS while the
second case presents numbers close to CPN. Therefore, at this particular Weber numbers the
FFN may be used for ICP-MS providing that it is coupled with a spray chamber for aerosol
modification. It should be noted however, this similar performance is obtained at much smaller
gas flow rates and tank pressure in comparison to the ranges of gas flow rates required for
aerosol production with CPN.
The most interesting case of nebulization is the last figure where an improved atomization is
observed at large Weber number. Nearly all the droplets are consumable and below 10 (µm)
making it possible to inject droplets directly into the plasma without any possible need for spray
chambers or desolvation systems.
The cumulative aerosol volume distribution of Figure 3-12 and Figure 3-13 is plotted together
with a comparable atomization case for CPN running at similar flow conditions in Figure 3-13.
As can be easily noticed, the FFN shows superior atomization. For instance at 10 (μm), 87, 20.5
of the aerosol volume is contained below this diameter while this number is about only 12
percent for the CPN, hence even at intermediate Weber numbers (Figure 3-12) the FFN shows
better performance. The improved atomization is in part due to orifice reduction as explained in
Chapter 1, but more importantly due to different mechanisms of liquid disintegration. In the
concentric type nebulizers, the prefilming liquid is in direct contact with the high velocity gas
from one side (Figure 1-4), in contrast the micro jet of the FFN is completely surrounded by the
gas stream on all sides (Figure 3-4). Besides the gas to liquid area ratio at the orifice exit for the
nebulization in Figure 3-4 is about Ag/Al=350 in comparison to Ag/Al=0.56 for the Type-A CPN
80
(a)
(b)
Figure 3-11: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=0.2 (µl/s), Qg= 150 (milt/min), D10/dj=2.26 and Wedj=4.5, (point 1 in Figure
3-9).
D (um)
fn=
dP
/dD
,fv
=d
P/d
Vo
l
10 20 30 40
0.2
0.4
0.6
0.8
1
1.2
1.4
fn
fv
D (um)
cu
mu
lative
dis
trib
utio
n
0 10 20 30 400
20
40
60
80
100
Cumulative fn
Cumulative fv
81
(a)
(b)
Figure 3-12: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=0.6 (µl/s), Qg= 180 (milt/min), D10/dj=1.5 and Wedj=9.6 (point 2 in Figure
3-9).
D (um)
fn=
dP
/dD
,fv
=d
P/d
Vo
l
10 20 30 400
0.2
0.4
0.6
0.8
1
1.2
1.4
fn
fv
D (um)
cu
mu
lative
dis
trib
utio
n
0 10 20 30 400
20
40
60
80
100
Cumulative fn
Cumulative fv
82
(a)
(b)
Figure 3-13: (a) Number and volume-based size distribution and (b) Cumulative size and volume
distribution at Ql=1.0 (µl/s), Qg= 320 (ml/min), D10/dj=0.69 and Wedj=20.0 (point 3 in Figure
3-9).
D (um)
fn=
dP
/dD
,fv
=d
P/d
Vo
l
10 20 30 400
0.2
0.4
0.6
0.8
1
fn
fv
D (um)
cu
mu
lative
dis
trib
utio
n
0 10 20 30 400
20
40
60
80
100
Cumulative fn
Cumulative fv
83
Figure 3-14: Comparison between FFN and CPN running at comparable flow conditions Ql=1.0
(μl/s) and Qg~320-370 (ml/min).
of the Figure 1-2 which again may account for this improved atomization. It should also be
noted, even for the cases where atomization is not superior to CPN, the aerosol is formed from
disintegration of a micro jet which is two orders of magnitude smaller than the gas orifice and as
a result the solution leaves the orifice without touching it. This feature alone is a great advantage
for the FFN even if the nebulizer has to be coupled to a spray chamber. Please recall from
Chapter 1 that the micronebulization was mainly done through miniaturization of concentric
nebulizer at the cost of increasing the probability of frequent nebulizer clogging.
Contribution 3.5
In this chapter, the two custom designed FFN prototypes were first described and the
fundamentals behind the flow focusing nebulization were explained and finally the nebulizer was
D (um)
cu
mu
lative
vo
lum
ed
istr
ibu
tio
n
0 10 20 30 400
20
40
60
80
100
CPN @ Ql=1.0 (ul/s) and Qg=320 (ml/min)
FFN @ Ql=0.6 (ul/s) and Qg=180 (ml/min)
FFN @ Ql=0.1 (ul/s) and Qg=370 (ml/min)
84
characterized at different flow conditions and correlation (10-3) was fitted to the experimental
results.
It was shown that for small jet based Weber numbers, the coalescence effects are significant
which would result in mean to jet diameter ratios over the Rayleigh limit of 1.89. Increasing the
Weber number would decrease the ratio below the Rayleigh limit since the wavelength of the
fastest growing wave becomes smaller. Atomization was improved by further increasing the
Weber number beyond 20, where the mean to jet diameter ratio dropped to the values 0.5-1.0. It
was also shown that although Ganan-Calvo (41) and (71) state the Weber number is a measure of
spray polydispersity, a clear correlation between the two could not seen in this work.
The performance of the FFN was then studied and compared to the CPN. It was shown the FFN
generally produces a superior aerosol in comparison to the CPN. It was also shown that if the
flow conditions are tuned properly the FFN can provide aerosol with sufficiently large number of
consumable drops with diameters below 10 (µm), in other words the necessity of coupling the
nebulizer with spray chambers and desolvation systems may be removed and the nebulizer could
be used for direct injection of droplets to the plasma.
Another important feature of the nebulizer is that the improved atomization is not achieved by
miniaturizing the nebulizer dimensions as it is usually accomplished for conventional
micronebulizers and direct injection nebuilizers in the market. Because the mechanism of
atomization for this class of the nebulizers is different and since the sample does not touch the
gas orifice, the chance of the nebulizer clogging is theoretically reduced. This is feature could
have some interesting potentials for injecting slurry solutions. However separate quantitative
study on the nebulizer clogging for slurry solutions must be taken to prove the possible superior
performance of the nebulizer.
Although the FFN cannot be considered an ideal nebulizer at this stage and requires further
development, it has exhibited promising results. It must be noted that we have only presented
nebulizer characterization and aerosol modeling (Chapter 5). Since the technology is relatively
new and there are not many publications on the subject, further studies must be carried out to
investigate the analytical performance of the device integrated with ICP-MS instruments.
85
Chapter 4 Aerosol Velocity Characterization
General Considerations 4.1
The droplet size of the CPN and the FFN under different flow conditions were characterized in
Chapters 2 and 3 respectively. Introducing sample in ICP-MS however, requires controlling both
aerosol size and velocity since fast moving droplets won’t have enough time for adequate
vaporization and are source of noise and signal loss in ICP (38) and (76). In addition, aerosol
with wide ranges of velocities produce emission and mass spectrometric signals at different
locations in the plasma, leading to reduced sensitivity and signal fluctuations (65) and (76).
Therefore, not only size and velocity are of paramount importance, the primary aerosol (that is
the unmodified aerosol generated by the nebulizer) is a determining factor in transport efficiency,
nebulizer design and optimization of the operating conditions (9), (48) and (12). In this chapter,
the aerosol velocity of each nebulizer is characterized and the characteristic velocity moments
are presented. The results of size and velocity characterization will then be employed to model
the primary aerosol distribution through Maximum Entropy Principle (MEP) in Chapter 5.
As mentioned in Chapter 2, only the axial velocity component is studied for characterization
because drops passing along the nebulizer axis are more likely to reach the central channel of
ICP and contribute to MS signal (9). Kahen et al. (38), state that the velocity vectors in the
direction of the nebulizer axis are highly oriented at the central region and add that the residence
time of a droplet in ICP is proportional to the axial velocity whereas the other components has
little effect on this time scale. Other studies like (36) clearly shows for the primary aerosol, the
radial velocity component is smaller and negligible in comparison to the axial component
because small drops tend to lose their radial momentum rapidly upon leaving the origin of the
spray (77). A similar trend was also observed in our experiments except for locations farther
downstream where the spatial dispersion becomes important.
86
As the high velocity gas expands from the orifice in Figure 1-4, gas will be entrained from the
sides due to lower local pressure. The entrained gas in turn, drags small liquid drops from the
outer periphery of the spray inward and may even lead to eventual contraction of the spray (78)
that explains why large droplets are observed at the fringes of spray as reported in (9), (4), (18)
and many others and also our experiments. The magnitude of the contraction depends on such
parameters as the total flow rate, the size and initial velocity of the droplets and the gas density
(78). Schlichting (79) has given the analytical solutions for the flow fields of both laminar and
turbulent free jets issued from an infinitely small diameter and has shown that the governing
equations for both regimes are of similar form, with the assumptions that fluid is incompressible
and that the pressure in the jet is constant and equal to the surrounding pressure. Under these
conditions, the momentum of the jet would remain constant with the mass flow gradually
increasing as more fluid is entrained from the sides and the velocity gradually decreases.
Besides, it was also shown that the width of the jet is directly proportional to the axial location,
while the centerline velocity has inverse proportionality with the position. The entrained
volumetric flow rate in (79) is:
(4-1)
√ (4-2)
where ν is kinematic viscosity (ν=ηg/ρg), z is the downstream axial location from the nebulizer
orifice and θ is the kinetic momentum. The equations show that free jets entrain copious amounts
of gas and the volume doubling as the length doubles. Equation (1-4) surprisingly shows that for
a laminar jet, the entrained volume is independent of the initial momentum. Therefore a higher
velocity jet remains narrower over a larger length than one moving at a lower velocity. For
turbulent jets, which is our case, the entrained volume is proportional to the square root of the
initial momentum, however it is not possible to substitute values directly in Equation (2-4)
because for turbulent flows, knowledge of kinematic viscosity (εT) is a prerequisite ( √ ),
and this parameter unlike its laminar counterpart is not a fluid property but rather a characteristic
of the local flow.
87
Jets of finite width maintain a potential flow core until a point where the mixing zone width
equals that of the jet radius (80). Beyond this point, the whole jet is turbulent and the turbulence
is self-preserving because of the mixing process. For the self-preserving region of the jet, the
ratio of the volumetric flow rate Qz at position z, relative to the original volumetric flow rate
(Qo=ugπdo2/4) can be represented by the empirical correlation (81):
( )
(4-3)
here do is the orifice diameter, z’ is the distance from the apparent origin to the position where
the self-preserving part of the jet begins and A is a constant. The value of z’ depends on the
orifice type and values between z’=-0.5do to z’=-7do are reported in different sources. The
constant A also changes from study to study, but direct measurement entrainment by Rico and
Spalding (82) give a value close to 0.32. At a large distance from the orifice, Equation (4-3)
reduces to Qz/Qo=0.32z/do which is the form quoted in (80). Thus, for example a 100 (µm)
diameter jet in air, taking z’=-3do and A=0.32 at z=0.9 and 5.3 (mm), the ratio is Q/Qo=2 and 16
respectively. Besides, the reduced form of Equation (4-3) may be used to find the centerline axial
gas velocity, taking Qz=ug(z)πδ2/4 and recalling that jet width (δ) is proportional to axial location
(z), Equation (4-3) will then become:
( )
(4-4)
Equation (4-4) is only valid beyond the potential core of the turbulent jet that is about seven jet
diameter (80) when the turbulence becomes self-preserving, however in the presence of the
liquid phase; this length drops to about five times the jet diameter as reported in (31). In this
equation, the exit velocity at the orifice (ug), is calculated from the isentropic relations
(Equations 2-4 to 2-7) and remains unchanged within the potential core. Perry and Chilton (83)
have reported the constant K values of 5 and 6.2 for jet exit velocities of ug=2.4-4.9 (m/s) and
ug=10.0-51.8 (m/s) respectively. Gas jets of pneumatic nebulizers have higher velocities and
therefore a value of 6.4 was proposed for them by Tennekes and Lumley (84). Although this
value is given for air, it should be a reasonable approximation for argon because the magnitude
of the eddy viscosity reflects the state of the flow field rather than the molecular characteristics
88
of the gas (80). It should be noted that since Equation (4-4) is derived from Equation (4-3), it is
empirical in nature, thus other mathematical fits are also plausible. For instance, Pryds et al. (85)
have proposed an exponential decay for the gas velocity of an atomizer starting from the orifice
and neglected the potential core in their study. In the present study we’ll use Equation (4-4) to
predict the axial gas velocity because it is also a good measure for spray mean velocity.
The transverse (radial) distribution of the axial velocity is approximately Gaussian in shape and
may be estimated by (83):
( ( )
( )) (
)
(4-5)
Therefore the approximate axial gas velocity profile will be:
( ) ( (
)
) (4-6)
The effect of enclosing a free jet by either spray chamber as in the CPN or delivery tube for FFN
is very dramatic, since there is no longer an infinite reservoir of fluid available to feed the
entrainment process. The result is that the jet must entrain fluid from itself by forming
undesirable recirculation zones. The auxiliary (makeup) flow required for canceling the
recirculation and produce a uniform forward velocity is about 11.5 times the primary jet flow
(86) which is impractical for ICP-MS due to limitations on the injector flow.
Liu et al. (87) defines three distinct regions for the injection of a turbulent jet (Reynolds number
=3000 to 24000) from a nozzle into a straight pipe, a potential core and a self-preserving region
similar to free turbulent jets and a final region where the jet width equals the pipe diameter. This
region is significantly influenced by the wall effects and flow eventually becomes fully
developed. Hill (88), pointed out that under certain conditions, the mean velocity field in the self-
preserving region of the confined turbulent jet could be predicted from the free jet data, using no
other empirical information. Dealy (89) states the flow pattern of confined jets, in general should
depend on the ratio of the orifice diameter (do) to the pipe diameter (Dp) and the Reynolds
number (Re=ugdo/νg). However, for sufficiently large Reynolds number and with the assumption
89
of self preservation, the Reynolds number dependence can be neglected (87), whereas the
Reynolds number does not dictate the shape of the velocity profile in free turbulent jets (90).
As Liu et al. (87) explain the presence of the recirculation zone implies that the confined jet
feeds the required entrained fluid by itself and thus reduce its momentum which in turn leads to
smaller centerline velocity in comparison to the free turbulent case. The following correlation
was then proposed for do/Dp<<0.25 independent of the orifice shape:
( )
(
) (4-7)
where f is the velocity loss function and Dp is the pipe diameter. Since at the limiting case of
Dp→∞, Equation (7-4) must be reduce to Equation (4-4), a polynomial fit for f(do/Dp) was
assumed in (87):
( ) (
) (
)
(4-8)
Because the ratio do/Dp is generally small, the higher order terms in Equation (4-8) can be
neglected. The final form of correlation is given in (87):
( )
( ) (4-9)
Equation (4-9) can predict at what axial distance the direct injection nebulizer should be placed
so that the droplet velocities (assuming they are of the same order as gas velocity) fall in the
range 5-10 (m/s) to ensure total vaporization and consumption (76). Although the FFN and the
plasma is of very similar configuration, i.e. a nebulizer is placed either before the plasma torch or
inside a delivery tube, Equation (4-9) has been proposed for gas exit velocities of 5 to 66 (m/s)
which is smaller than the typical nebulizer gas exit velocity of ICP-MS (Recall a K value of 6.4
was proposed for turbulent free jet of Equation 4-4). Besides the makeup flow effects are also
not considered in Equation (4-9). Nevertheless, the study might be used as a guideline to
investigate the axial gas velocity of the confined FFN in the future. To the best of the author’s
90
knowledge, there is currently no study that directly relates the axial gas velocity of direct
injection nebulizers to the orifice to pipe diameter ratio and the makeup gas flow.
At this stage, it would be very useful to study the dynamics of a single particle travelling in a free
turbulent jet. Consider 1, 10, 20, 50 and 100 (µm) droplets axially injected at the exit of the
nebulizer with an orifice diameter of 260 (µm) through which argon flows at sonic velocity
ug=276.1 (m/s). Assuming that droplets are travelling along the centerline and do not disperse in
the plane normal to the nebulizer axis, the one-dimensional equation for the droplet motions is:
( ( ) )
(4-10)
here mp is droplet mass (mp=π/6×D3ρl), up is the droplet local velocity, CD is the drag coefficient
and Ac is the droplet cross sectional area (Ac=π/4×D2). Hence:
( ( ) )
(4-11)
The jet centerline velocity, ug(z), can be approximated by Equation (4-4) and isentropic theory
(Equations 2-4 to 2-7) that will remain constant for the first 5-7 orifice diameters until the
potential core is extinguished. Assuming that droplet can be modeled as non-deformable spheres,
calculation proceeds by considering a small axial increment and from the difference between the
gas and particle velocity a particle Reynolds number is calculated.
The drag coefficient is then evaluated by having the Reynolds number from (91):
(4-12)
Equation (4-12) is defined for Reynolds number in the range 0<Re<2×105. Although the
equation ignores droplet deformability, it is quite sufficient for our simplified model.
Droplet acceleration is thus calculated from Equation (4-11). By having the initial droplet
velocity and droplet acceleration/deceleration, the time of flight for the incremental distance is
computed and a new terminal velocity for the end of increment is calculated. The procedure is
then repeated farther downstream locations. The results are shown in Figure 4-1 where the solid
91
Figure 4-1: Gas in a free turbulent jet exiting a 260 (µm) nebulizer operating at sonic flow and
droplet velocity for 1, 10, 20, 50 and 100 (µm) droplet diameter.
line represents the gas velocity decaying as 1/z from a distance of 7do. Please note the graph is
presented in logarithmic scale. It is assumed that droplets have negligible initial velocity in
comparison to the gas exit velocity. As can be seen from the figure, all the drop sizes undergo
sudden acceleration in the potential core region and the lighter the droplets experience larger
acceleration. The gas expansion begins as soon as the potential core is extinguished; from this
point forward the droplet trajectory is very interesting. Equation (4-11) suggests droplet
acceleration/deceleration is inversely proportional to the droplet diameter. The larger droplets
require more time to adapt the gas velocity, whereas a small droplet almost instantly reaches the
gas velocity. This is especially true for 1 (µm) droplet in Figure 4-1. Below an axial distance of
z=30 (mm), droplets in the range 10-20 (µm) have higher velocities than those in the range 50-
z (mm)
U(m
/s)
10-2
10-1
100
101
10210
0
101
102
ug
D=1 (um)
D=10 (um)
D=20 (um)
D=50 (um)
D=100 (um)
z=30 (mm)
92
100 (µm) due to their original larger acceleration in the potential core region, therefore below
this location we’ll have a cloud of fast moving droplets within a cloud of slow moving particles.
For downstream locations beyond 30 (mm) this trend is reversed, because 10-20 (µm) droplets
have already adapted the gas velocity, but larger droplets require more time, therefore we’ll have
a cloud of large droplets passing through a cloud of relatively slow moving droplets. 10 and 20
(µm) droplets require 40 and 70 (mm) to reach the gas velocity, respectively, while 50 and 100
(µm) droplets must travel over 200 (mm) to creep up to the gas velocity. The simplified model
presented here is valid for diluted spray where the drop number density is small, in other words
when the gas phase in not disrupted by the presence of the droplets. Please note that in our model
we have neglected the spatial dispersion and evaporation, nevertheless the model help us
understand some basic physical processes occurs during aerosol transportation.
Aerosol Velocity Modeling 4.2
In the spray jet of pneumatic nebulizers, the momentum is exchanged between the gas and liquid
phases. For instance as Sharp (80) explains, a 1 (l/min) flow of argon at 293 (K) and 1.014×105
(Pa) is equivalent to a mass flow rate of 1.66 (g/min) that would typically nebulize an equivalent
mass of liquid, in other words a mass flow ratio of approximately one. Introducing liquid thus
doubles the mass flow rate and to conserve momentum, the mean velocity of the seeded gas must
drop to one half of its non-seeded value from Equation (4-4). Sharp (80) also adds:” Although
the presence of liquid reduces the gas momentum, it does not necessarily reduce the entrainment.
Moving particles entrain fluid in their wake and in dense clouds may cause the fluid to move at
the mean particulate velocity.”
Equation (4-4) and the simplified model presented in the previous section are valid only when
the spray is diluted, in other words when the liquid mass flow rate is small in comparison to that
of the gas and provided that all the droplets travel along the nebulizer axis. For large liquid to gas
flow ratios, the dynamics of particle and gas motion requires detailed modeling that includes the
transfer of momentum between the two phases and effects of droplet dispersion. Hence not only
the actual gas velocity would be different but also the resultant droplet velocities would deviate
from the model. Our aim here is to present simple correlations for estimating aerosol velocity
93
moments based on known information, such as liquid to gas mass flow ratio, Sauter mean
diameter and unseeded gas velocity.
In our experiments with CPN, the liquid to gas mass flow ratio varies between 0.02 - 1.24. As
mentioned earlier, at mass flow ratios close to unity, the gas velocity drops to about half of its
unseeded value to conserve momentum and as the ratio becomes smaller, the seeded gas and
approaches to the unseeded value of Equation (4-4). In addition, Equation (4-10) states that
acceleration is inversely proportional to the area over volume ratio. In our case, this ratio is
similar to (1/D32), hence the following correlation is proposed for the mean velocity of the CPN’s
primary aerosol (measured at z=10 mm):
( ) (
)
(4-13)
A value of C=0.79 and n=1.62 are then obtained from fitting experimental results to the model.
Equation (4-13) suggests at constant mass flow ratio, the mean velocity becomes smaller by
increasing the Sauter mean diameter and similarly at constant Sauter mean diameter, the mean
velocity also decreases by increasing the mass flow ratio, as confirmed by experiment.
In aerosol generation with the FFN, the liquid to gas mass flow ratio is generally small and varies
in a very narrow range; hence the gas flow is very little influenced by the presence of liquid
phase. As can be seen from Figure 4-1, at this location the droplet velocities larger than the gas
velocity are expected. A similar trend was observed in the measurement and the mean to gas
velocity ratio changes from 1.15 to 2.2 but unlike the CPN, the ratio grows larger by increasing
the Sauter mean diameter for each specific liquid flow rate. This could be justified from Figure
4-1, where increasing diameter leads to a dropping trend for mean velocity at z>30 (mm). It
should be noted; since the orifice diameter of FFN is smaller than CPN, its potential core is also
smaller (1mm in comparison to 2 mm respectively). Hence, the dashed line (z=30 mm) in Figure
4-1, would shift toward smaller values too.
Based on the above justification, the following correlation is suggested for the mean to gas
velocity ratio of FFN’s primary aerosol at z=10 (mm):
94
( )
(4-14)
here constant C and exponent n are 10.87 and 1.31 respectively.
Figure 4-2 exhibits the droplet mean, root mean square (rms) of the primary aerosol together
with the predicted unseeded gas velocity for both CPN and FFN. Both droplet mean and rms
velocity grows by increasing the gas flow rate. The unseeded gas velocity is larger for CPN
comparing to the droplet velocity due to momentum transfer between the two phases as
mentioned earlier, but the difference between the two velocities becomes smaller around 500-
600 (ml/min) where the gas velocity reaches a plateau meeting the sonic condition but the liquid
velocity continues to increase, probably due to the fact that in pneumatic nebulization, the sonic
condition is usually delayed to higher tank back pressures. The gas flow rate was varied in a very
narrow range for the FFN, in fact the maximum droplet mean velocity is about half of the CPN
value. For the range of gas flow rates in the experiment, the mean and rms velocities grow
slightly and have very similar values at different liquid flow rates. The unseeded gas velocity
(hollow line in Figure 4-2) stands below the measured mean velocity for the FFN but
experiences larger gradient.
Figure 4-3 and Figure 4-4 present the changes in rms to mean droplet velocity and size ratios
with the gas flow rate for both CPN and FFN respectively. These ratios represent the span of
velocity and size distribution and are required for MEP modeling. As can be seen for both
nebulizers, the span of size distribution becomes smaller by increasing the gas flow. Besides
Equations (3-10), (2-21) and (2-9) predict smaller droplet diameters by increasing the gas flow
for each constant liquid flow rate.
The span of velocity distribution changes differently for each nebulizer. The CPN shows a
dropping trend for mean to rms velocity ratio at all the liquid flows, implying that the axial
velocity is very oriented along the centerline that is not surprising for such high exit velocities.
For axisymmetric turbulent jets, the width of shear layer is directly proportional to the axial co-
ordinate and the centerline velocity proportional to the inverse of the co-ordinate (92), thus
increasing the gas exit velocity would lead to smaller shear layer thickness and more noticeable
95
(a)
(b)
Figure 4-2: Droplet Mean and Root Mean Square (rms) speed and unseeded gas velocity versus
gas flow rate, measured at z=10 (mm) for (a) CPN and (b) FFN.
0
10
20
30
40
50
60
70
200 300 400 500 600 700 800
U10, U
rms,
ug (
m/s
)
Qg (ml/min)
Velocity curves for CPN
U10, Ql=1 (µl/s)
Urms, Ql=1 (µl/s)
U10, Ql=5 (µl/s)
Urms, Ql=5 (µl/s)
U10, Ql=10 (µl/s)
Urms, Ql=10 (µl/s)
ug
Sonic condition
0
5
10
15
20
25
30
140 160 180 200 220
U1
0, U
rms,
ug (
m/s
)
Qg (ml/min)
Velocity curves for FFN
U10, Ql=0.1 (µl/s)
Urms, Ql=0.1 (µl/s)
U10, Ql=0.2 (µl/s)
Urms, Ql=0.2 (µl/s)
U10, Ql=0.4 (µl/s)
Urms, Ql=0.4 (µl/s)
U10, Ql=0.6 (µl/s)
Urms, Ql=0.6 (µl/s)
ug
96
(a)
(b)
Figure 4-3: Span of (a) velocity and (b) size distribution measured at z=10 (mm) for CPN.
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0 200 400 600 800 1000
Urm
s/U
10
Qg (ml/min)
Velocity distribution span for CPN
Ql= 1 (µl/s)
Ql= 5 (µl/s)
Ql= 10 (µl/s)
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0 200 400 600 800 1000
Drm
s/D
10
Qg (ml/min)
Size distribution span for CPN
Ql= 1 (µl/s)
Ql= 5 (µl/s)
Ql= 10 (µl/s)
97
(a)
(b)
Figure 4-4: Span of (a) velocity and (b) size distribution measured at z=10 (mm) for FFN.
0.3
0.32
0.34
0.36
0.38
0.4
0.42
0.44
0.46
0.48
0.5
100 150 200 250
Urm
s/U
10
Qg (ml/min)
Velocity distribution span for FFN
Ql=0.1 (µl/s)
Ql=0.2 (µl/s)
Ql=0.4 (µl/s)
Ql=0.6 (µl/s)
0.45
0.5
0.55
0.6
0.65
0.7
100 150 200 250
Drm
s/D
10
Qg (sccm)
Size distribution span for FFN
Ql=0.1 (µl/s)
Ql=0.2 (µl/s)
Ql=0.4 (µl/s)
Ql=0.6 (µl/s)
98
aerosol confinement. As a result the droplet radial dispersion would be smaller and more uniform
velocity droplets pass the measuring probe of the PDPA. This trend is not observed for the FFN,
perhaps because gas and droplet velocities are smaller and change in a narrower range compared
to the CPN. The span of velocity distribution grows larger by increasing gas flow rate at Ql=0.1-
0.2 (µl/s) but remains constant or have slight variation at Ql=0.4-0.6 (µl/s). In summary we may
say that finer aerosols at higher velocities with narrower size and velocity distribution are
resulted from increasing the gas flow rate for the CPN. Doing so for the FFN would result in a
finer and narrower size distribution with droplets traveling at slightly higher velocities and
increased velocity dispersion. The axial and radial variations of the mean and rms velocities are
given in Appendix D. These plots together with the size characteristics moments of Appendix C
can be used in MEP modeling (Chapter 5) to generate size and velocity distributions at any
desired location such as the spray fringe or centerline.
Contribution 4.3
Droplets with wide ranges of velocities produce emission and mass spectrometric signals at
different locations in the plasma, leading to reduced sensitivity and signal fluctuations. Besides,
high velocity droplets may not have enough residence time for complete evaporation. Therefore,
controlling the aerosol velocity in ICP-MS is of significant importance. In this chapter, first, the
axial and radial profiles of the centerline velocity for a turbulent free jet were presented and then
the response of different droplet sizes in the flow was studied. It was found that below some
axial distance, the aerosol stream consists of fast and small cloud of droplets moving in a cloud
of slow and large droplets but this would be reversed further downstream where small drops
adapt the gas velocity faster than large droplets. As discussed in the chapter, increasing either the
liquid to gas mass flow ratio or the Sauter mean diameter for CPN reduces the mean aerosol to
gas velocity ratio because of the momentum transfer between the two phases. A correlation was
proposed for the velocity ratio as a function of mass flow ratio and Sauter mean diameter in
Equation (4-13). In microsample nebulization with FFN, since the liquid flow is negligible in
comparison to the gas flow, as discussed in the single droplet model, the mean velocity becomes
larger by increasing the droplet size. Equation (4-14) was proposed to account for this
observation.
99
Chapter 5 Maximum Entropy Principle - Application on Aerosol Size and
Velocity Modeling
The Need for Statistical Measures and Maximum Entropy 5.1Principle
The primary aerosol size of the CPN and FFN was characterized in Chapters 2 and 3
respectively, and the velocity characterization was presented in Chapter 4 where some
correlations was either modified or proposed. The characteristic moments of size and velocity
although contain important information, only represent aerosol characteristics macroscopically
rather than the detailed space of size and velocity. Nonetheless the atomization is in general the
process of fragmenting liquid bulk into multiplicity of droplets with different sizes and
velocities. In fact, in many industrial applications like automotive paint sprays, waste
incineration, pharmaceutical/medical drug administration (93) and agricultural aviation for crop
spray (94) the drop size distribution must have a particular form (narrow, wide, few large drops,
few small drops, etc.) for optimal operation. In atomic and mass spectrometry for instance, the
nebulizer transport efficiency is of crucial importance; that is the fraction of aerosol volume
containing sub 10 (µm) droplets (Refer to Figure 3-11 to Figure 3-14). For given flow conditions,
if two nebulizers have the same Sauter mean diameter for example, the one which has higher
transport efficiency is preferred. On the other hand as mentioned in the previous chapter, if the
aerosol droplets travel fast, the particle residence time in the plasma would not be adequate for
complete vaporization even if the entire aerosol contains sub 10 (µm) droplets. Therefore, as far
as sample introduction in ICP-MS is concerned, some aspects of the aerosol generation require
detailed information on the entire size and velocity space that may not be reflected through
characteristic moments alone.
Although the droplet distribution space can be easily measured by Phase Doppler Anemometry,
the stochastic nature of sprays has drastically restricted the development of mathematical or
numerical modeling of sprays. In fact deterministic methods such as instability analysis up to this
point are not advanced enough and generally incapable of including all the mechanism involved
100
in a simple liquid atomization process. Consequently, all they could offer may not exceed more
than single droplet diameters under very simplified conditions at best. In the absence of such
comprehensive models, many different empirical distributions like Rosin-Rammler (95),
Nukiyama-Tanasawa (51), Lognormal (96), upper-limit (97), Log-hyperbolic (98), etc. have
been proposed, none which offers universal prediction and are only valid in the range they are
proposed and tested. Furthermore, among these distributions only Log-hyperbolic has been
expanded to present joint size and velocity distribution while others can only model size
distribution. Besides, the empirical distributions bring little physical understanding of the
process and even worse they are not founded on a mathematical ground. Thus, all of these
correlations could be considered more or less as curve fitting methods for covering a range of
experimental data. The study of these empirical distributions is beyond the scope of this study;
nonetheless they are given in Appendix E for comparison.
Although CFD has been widely used as a powerful tool in many different applications for
modeling spray dynamics, simulating all the details of the primary atomization and liquid
disintegration in Eulerian frame of reference is very time consuming and has considerable
numerical cost. The alternative practice is to use some statistical measure of primary aerosol
space as input and then study the secondary effects such as breakup, collision, coalescence,
evaporation, impaction, etc. in a coupled Eulerian-Lagrangian frame of reference.
Therefore, from practical point of view, it would be very beneficial to have some statistical
measures of the primary aerosol, i.e. the detailed size and velocity space, with as little
information possible and preferably based on the measured characteristic moments.
The maximum entropy principle (MEP) is a promising approach toward the prediction of the
spray distribution which has a mathematical foundation. The principle was developed by Jaynes
(99) based on Shannon’s entropy concept (100) which is a measure of uncertainty of a system
subjected to some prescribed constraints. Shannon originally defined this entropy to study
communication of information through a noisy channel. Jaynes (99) showed that based on
Shannon’s approach all of the formulae of statistical mechanics could be easily derived.
Nevertheless he stated that the foundation of this definition of entropy is quite independent from
statistical mechanics. Since then Shannon entropy and Jaynes’ maximum entropy principles have
101
been successfully employed in many different disciplines of natural and social sciences such as
transportation, population, brand-switching in marketing and vote-switching in elections, finance
insurance and marketing, image reconstruction, pattern recognition, operation research and
engineering, biological medical and technological problems, non-parametric density estimation
(101) to just name a few.
MEP Formulation 5.2
Atomization is a highly nonlinear and stochastic process and the deterministic methods fail
miserably in predicting the aerosol space. Alternatively, MEP can present the least biased and the
most objective solution even if only small pieces of information about the spray are known
beforehand.
The pioneering application of MEP for spray modeling first appeared in the works of Sellens
(102), Sellens and Brzustowski (103) and (104), and Li and Tankin (105) and (106)
independently. All these tasks investigated the problem of distributing droplets in a size and
velocity space when discrete information was given or known from experimental measurement.
Consider a size and velocity space as shown in Figure 5-1. The droplets are assumed to be
distributed in different classes of normalized size and velocity respectively while the space is
equally segmented to avoid any biased analysis. At this point the reference diameter (Dref) and
velocity (Uref) are not determined but will be specified later. In this figure, the number of
droplets in a particular class of normalized size and velocity is given by ΔNij and the total
number of droplets is Ntot=ΣΣ ΔNij, thus the probability of finding a droplet in a particular class
of normalized size and velocity is Pij= ΔNij/Ntot and the corresponding probability per unit
normalized size and velocity class will be fij.
The trivial constraint for any distribution in this space is that the summation of all the
probabilities must be unity (normalization constraint):
∑∑
∑∑
(5-1)
102
Figure 5-1: Size and velocity space and probability distribution function of aerosol
Nevertheless, the aerosol might be subjected to some other constraints which are often presented
in terms of some characteristic moments (γk) either measured from experiment or known from
theoretical analysis.
∑∑ (
)
∑∑ (
)
(5-2)
The constraints (gk’s) in Equation (5-2) are conventionally derived from first principles or
conservation laws acting on the aerosol. However this is not a necessary condition for the given
constraints and any single or joint function of size and velocity can be used as long as its value is
known beforehand.
If ΔD* and ΔU
* are small, Equations (1-5) and (2-5) can be presented in continuous form:
103
∫ ∫
(5-3)
∫ ∫
( ) (5-4)
There are many possible distributions in the size and velocity space that satisfy the normalization
constraint (5-3) and the additional prescribed information (5-4), however the maximum entropy
principle states out of these possible distributions, the one that maximizes the Shannon entropy is
the least biased and the most objective distribution for the prescribed information. The Shannon
entropy is defined as:
∫ ∫
( ) (5-5)
The derivation of Shannon entropy is beyond the scope of this task but presented in Appendix F.
The method of Lagrangian multipliers is employed to find the least biased distribution:
( )
∑
(5-6)
∫ ∫ ( ( ) ∑
( ))
(5-7)
The integral Equation (5-7) is always true if the integrand is identically zero, thus:
( [∑ ( )
]) (5-8)
The unknown coefficients must be found by solving the integral Equations (5-3) and (5-4) with
the probability distribution function given in Equation (5-8). The lower limit of integration for
104
both the minimum diameter and velocity can be set equal to zero although it’s possible to assume
a non-zero value for minimum size since generating a zero size droplet is physically impractical.
The upper limit of the integration is infinity but for numerical calculations some finite value
must be taken. Van Der Geld and Vermeer (107) have proven the boundaries of integration may
shift the the probability distribution but the question of correct integration limits has never been
addressed in the literature. In this task, the upper limit of integration is taken from the PDPA
optical restrictions, which is Dmax=120 (µm) and Umax=100 (m/s) respectively.
Once the entropy function is determined, the size and velocity distributions are obtained by
taking proper integration:
∫
(5-9)
∫
(5-10)
Number or Volume Based Probability Distribution Function? 5.3
Since the employment of MEP in the field of atomization and sprays, two separate MEP
implementations with a fundamental difference can be recognized in the literature. One is the
Sellens and Brzusowski’s approach and the other is Li and Tankin’s. Before presenting the MEP
modeling, the two approaches must be discussed.
Sellens (102) and Sellens and Brzustowski (103) and (104) used a size and velocity space as
shown in Figure 5-2 to study the joint size and velocity distribution of a pressure swirl atomizer.
The author used a control volume surrounding the sheet breakup plane (Figure 5-2) and
expressed the physical laws acting on the control volume as the conservation constraints of mass,
surface energy, momentum and kinetic energy:
105
∫ ∫
(5-11)
∫ ∫
(5-12)
∫ ∫
(5-13)
∫ ∫
(5-14)
∫ ∫
(5-15)
(
) (5-16)
The choice of control volume dictates the reference values for size and velocity normalization
which are the mass mean diameter (D30) and sheet velocity at the plane of breakup (V)
respectively. In Equations (5-12) to (5-15), Sm, Ss, Smv and Ske are the source terms of
conservation equations that may account for dissipations. Furthermore in Equation (5-13) the
sheet thickness at the breakup plane (η*) must also be known from either experiment or analysis.
Conserving surface and kinetic energy through separate constraints has been argued by Mitra and
Li (108) who stated doing so would violate the physical laws since in practical sprays kinetic
energy is responsible for liquid atomization and increase in the surface area or surface energy of
the spray. On the other hand, Sellens and Brzustowski (103) explained conserving both energies
through one constraint would leave out important information concerning the irreversibility of
certain energy transformations and the prior knowledge of the energy distribution between
various energy modes before breakup. They also added that translational energy is directly
106
Figure 5-2: Sellens and Brzustowski’s control volume for MEP modeling.
transformed into surface energy by sheet stretching and drop drag formation but the reverse
transformation from surface to direct kinetic energy is generally not possible and hence separate
constraint for each mode of energy must be considered.
Sellens’ (102) and Sellens and Brzustowski’s models (103) and (104) are in general agreement
with empirical correlations and particularly Rosin-Rammler distribution except that the
distribution function (Equation 16-5) shows some non-zero population for zero size droplets
which is not supported by experiment. Sellens (109) and (110) later resolved the issue by
introducing a new constraint which he called the “partition of surface energy”. As Sellens
explains the unphysical behavior is due to a deficiency in the conservation moments because the
lowest order moment of size is of second order and the small size droplets have little contribution
to the values of the size moments (Equation 5-12 and 5-13). Sellens added that in limiting the
number of very small drops, the physical process at work is a limitation of the concentration of
surface energy. With fixed values of surface tension, flow velocities, etc. it is unlikely that
sufficient deformation energy will be expanded on a given element of mass to reduce the drop
Control Volume
GasLiquid
Sheet Breakup
Drops
107
size beyond a certain point. Thus the amount of deformation energy that a small element of mass
can absorb must be limited or, in other words, the surface to volume ratio of the drops in the
spray imposes a constraint:
∫ ∫
(5-17)
here Kp expresses the strength of the partition constraint which is equal to the nameless
characteristic moment (D-10) that was introduced by Equation (2-23). If the new constraint is
added to set of Equations (5-11) to (5-15) the resulted distribution function will be:
(
) (5-18)
In 1987 Li and Tankin (105) derived a size distribution through the concept of information
theory and maximum entropy principle and showed the resultant distribution is a special case of
Nukiyama and Tanasawa distribution (51). In this study, Li and Tankin were only interested in
size distribution, therefore the velocity was not included in the solution space. Furthermore,
instead of using size classes, the space was divided by volume classes.
∫
(5-19)
∫
(5-20)
where fv and V* are the volume distribution function and the normalized droplet volume
respectively. Li and Tankin stated that since the probability of finding a droplet in a particular
class of size is essentially the same as probability of finding it in the corresponding volume, thus
Equations (5-19) and (5-20) can be rearranged as:
*
+ *
+ (5-21)
108
( ) (
) ( ) (5-22)
∫
(5-23)
∫
(5-24)
(
) (5-25)
Li and Tankin (70), (106) and (111) developed their model to include the velocity by including
the the momentum and kinetic energy constraints, Nevertheless a single constraint is considered
for the total energy (surface and kinetic energy). In contrast to Sellens and Brzustowski (103)
and (104) the control volume was assumed to be extended from the orifice exit to the plane of
breakup. This choice of control volume dictates D30 and capillary velocity (Ucap) for
normalization of size and velocity respectively as shown in Figure 5-3.
∫ ∫
(5-26)
∫ ∫
(5-27)
∫ ∫
(5-28)
∫ ∫
( (
) ) (5-29)
109
(
( (
) )) (5-30)
An interesting feature of Equation (5-30) is that the probability distribution approaches to zero as
the drop diameter decreases; hence there is no need to prescribe an additional constraint like the
partition of surface energy in Equation (5-17). However Dumouchel (112) believes that this
problem was avoided in Lin and Tankin’s papers (105), (106) and (70) due to mathematical
manipulation in Equation (5-22) which indicates that the number based drop size distribution is
proportional to the square of the drop diameter.
In the absence of any priori information, the least biased distribution must be the uniform one.
This is the case for Sellens and Burzustowski’ number based distribution (Equations 5-11 and 5-
16) when constraints (5-12) to (5-15) are not prescribed. But with normalization constant alone,
Li and Tankin’s approach leads to a constant f’(V*) or according to Equation (5-22) a number
based size distribution that is a function of D*2
. Nonetheless such hypothesis has never been
supported by any experimental evidence or by theoretical study. Dumouchel (112) concluded, as
far as MEP is concerned a change of variable introduces supplementary information and must be
avoided. In Cousin et al.’s excellent study (113), the issue of number and volume distribution
was finally addressed. The authors showed that the number and volume based distribution must
be consistent, meaning that entropy maximization with either size or volume must result in the
same size and volume distributions. Besides it was shown a number based size distribution can
be derived by maximizing Shannon entropy. On the other hand, a correct volume based
distribution can only be calculated if a priori distribution that contains the information related to
the shape of droplets is considered to maximize Bayesian entropy which is the more general
form of Shannon entropy (113). Refer to Appendix G for the definition Bayesian entropy and its
relation with Shannon entropy.
Taking Sellens formulation (Equations 5-11 to 5-18) to find the number based distribution, the
corresponding volume based distribution can be easily calculated by:
( )
(
) ( ) (5-31)
110
Figure 5-3: Li and Tankin’s control volume for MEP modeling.
Global and Local Implementation of MEP 5.4
The classical control volume approaches of Sellens and Brzustowski (103) and Li and Tankin
(106), (111) have considered the conservation laws between two plains, the known upstream
plane and the droplet plane. In deriving conservation of mass constraints in Equations (5-12), (5-
24) and (5-27) it was assumed that the liquid flow rate is the same as the rate of droplet masses
passing the PDPA probe are comparable. However, we have observed a considerable mismatch
between these two parameters which is also supported by Li et al.’s study (114). The authors
attributed this problem to two possible factors: (1) defining the optical probe measurement cross
section and (2) high rate of data rejection. On the other hand, the control volume approach
requires the PDPA measurements to cover the entire spray cross section (111) but the probe area
of the order of 10-2
(mm2) and the spray cross section of 10
2 (mm
2) at the downstream location
makes this task nearly impractical.
The other problem of the control volume approach (global implementation) of MEP is that, the
sheet thickness and velocity must be calculated for Sellens and Brzustowski’s model. Lefebvre
(50) criticized this model stating that although the approach is elegant, such quantities are not
Control Volume
GasLiquid
Sheet BreakupDrops
111
easily measured. In fact, Rizk and Lefebvre (23) have shown the film thickness can change
significantly with the flow parameters (23). From this perspective, Li and Tankin’s control
volume extended from the orifice exit is preferable because it links the breakup plane to the
known upstream conditions. However, such control volume requires careful estimation of the
source terms in Equations (5-27) to (5-29) since the effect of downstream gas and liquid
interaction between the two planes can be quite considerable. Moreover and to the best of the
author’s knowledge, there is no study in the literature that has focused on the details of
prefilming, sheet breakup or any photographic measurement for ICP nebulizers. A comparison of
Figure 1-4 with Figure 5-2 and Figure 5-3 of pressure atomizers reveals that the mechanism of
liquid disintegration is very different which makes adapting an appropriate control volume very
difficult.
Ahmadi and Sellens (115) state that the MEP does not have to be necessarily applied on the
entire spray cross section and added that in general the principle holds true for any local control
volume at any particular time and location. This would mean the MEP application is quite
independent of the choice of control volume and is not merely derived from conservation laws.
In other words, if the mass, surface energy, momentum, kinetic energy and partition of surface
energy are important characteristics of the aerosol stream as the whole, they are equally
important description of the local state of spray. This feature of MEP is of particular importance.
Consider the case that you are only interested in the local distribution of size and velocity at a
particular point in the spray, the global implementation of MEP as in Sellens and Brzustowski’s
model (Equations 5-11 to 5-15) or Li and Tankin’s model (Equations 5-26 to 5-30) is incapable
of offering a solution, because they represent the entire spray plane. Ahmadi and Sellens,
findings have proven that the MEP can be regarded as a tool for statistical inference of the spray
that is not restricted to the conservation laws. Since then a number of MEP-based size
distribution such as (112), (117), (118), (119), (120) and (113) have exploited this approach for
aerosol modeling.
In direct injection of droplets by FFN for example, the fate of droplets traveling along the
nebulizer axis is important rather than the entire spray cross section because these droplets are
more likely to reach the plasma channel (9). In Sample introduction with a conventional CPN,
the small drops are found along the axis while larger ones formed at the fringes of the spray and
112
will be probably removed by the spray chamber. Since we are interested to have a statistical
measure of aerosol along the centerline of the nebulizers, the principle has to be implemented
locally rather than globally. Besides, as both size and velocity distribution are important in ICP-
MS our local MEP implementation must also include the velocity subspace.
Before local MEP formulation is presented, it’s worth mentioning again that the global
implementation of MEP dictates the reference values of size and velocity. In contrast, the local
implementation allows us to choose any arbitrary reference values for parameter normalization.
As mentioned in the previous chapters, the measured droplet velocities are orders of magnitude
larger than the liquid velocity in the capillary. Thus Li and Tankin’s normalization with capillary
velocity does not seem appropriate. On the other hand, the film velocity of Sellens and
Brzustowski’s model cannot be easily measured. We have chosen the local unseeded gas
velocity, ug(z) for velocity normalization because its value is of the same order as the measured
droplet velocity which makes it more relevant for normalization. The value of gas velocity can
be calculated from Chapter 4 and the relations given within. In addition the droplet size is
normalized by mass mean diameter D30 as the other two models.
∫ ∫
(5-32)
∫ ∫
(5-33)
∫ ∫
(5-34)
∫ ∫
(5-35)
113
∫ ∫
(5-36)
∫ ∫
(5-37)
(
) (5-38)
( ) (5-39)
As can be noticed from Equations (5-32) to (5-37) the values of constraint are also given except
for the joint momentum and kinetic energy source terms. The values of D30/D20 and D30/D-10 can
be extracted from Chapters 2 and 3 for CPN and FFN respectively. Equations (5-35) and (5-36)
can be simplified to U10/ug(z) and (U10/ug(z))2 + (Urms/ug(z))
2 if spray monodispersity exists in
either size, velocity or both (116). In our study depending on the degree of spray polydispersity,
the momentum and kinetic energy constraints are up to 1.5 and 2.5 folds their monodisprse
values respectively.
Numerical Solution 5.5
After finding the closed form of probability distribution function through method of Lagrangian
multipliers, the unknown coefficients must be determined by solving the integral Equations (5-
32) to (5-37). Li (121) has proven not only a solution to system of equations exists but it is
unique as well, providing that the constraint values are positive, i.e. Γk>0.
Unfortunately the sets of equations are highly nonlinear that makes finding analytical solution
rather impossible. In the present study, we have used the iterative Newton-Raphson method to
find the unknown coefficients. Iteration begins with an initial guess for the coefficients which in
our case are set to zero. The Jacobian matrix is first constructed by differentiating Equations (5-
32) to (5-37) and distribution function (5-38) and then the matrices of constants and Jacobian are
evaluated for the given initial guess.
114
[
]
⏟
[
]
⏟
[
]
⏟
(5-40)
The increment matrix (Δα) is therefore:
(5-41)
The inverse Jacobian matrix (J-1
) is calculated through Gauss-Jordan elimination method. Once
the increment matrix is obtained, the new coefficients are calculated.
(5-42)
The procedure is repeated with new coefficients until the constant matrix converges to zero, or in
other words:
(5-43)
where epsilon is set to a small value (10-10
). It’s important to keep epsilon small because the
equations are in exponential form and double integration is also involved. Therefore a small
imbalance may cause severe numerical error. The convergence of the numerical scheme depends
on the initial guess and also the stiffness of the Jacobian matrix. A second order Newton-
Raphson method has been proposed in (122) to overcome the stability and restrict requirement
on the initial guess. In our study, the local implementation of MEP and velocity normalization
with the local gas velocity, ug(z), reduces the stiffness of the Jacobian matrix significantly and
ensures the numerical convergence with fewer than 30 iterations at most with the first order
Newton-Raphson method shown by Equations (40-5) to (42-5).
115
MEP Results and Discussion 5.6
Figure 5-4 demonstrates the result of size distribution from the coupled MEP model for two
representative cases of Ql=5 (µl/s), Qg=500 (ml/min) and Ql=0.1 (µl/s), Qg=150 (ml/min)
sprayed with CPN and FFN respectively. As can be seen, the standard deviation (STD) of the
experimental results are small (maximum of 8 and 12% for CPN and FFN respectively) modeled
size distributions are in general agreement with experiment. The Skewness of distribution and
the location of most probable diameter are correctly predicted, although probability value
exhibits some underestimation. Besides span of distribution covers the experiment and the
population of very fine and large droplet approaches to zero at the end tail of the distributions as
producing such droplet sizes are not physically probable.
The majority of studies on MEP modeling have focused mainly on the size distribution. The
velocity distribution has received little or no attention even when the joint size and velocity pdf
are employed. For instance, in Sellens and Brzutowski’s paper (103) although momentum and
kinetic energy moments are prescribed, the velocity distribution is not reported and many other
studies the velocity distributions are presented without comparison to the experiment (104),
(109), (106) and (111).
Figure 5-5 presents the velocity distributions for the same cases of Figure 5-4. Several
deficiencies can be noticed from the figure. First, the end tails population does not agree with
experiment; in the experiment the population of small and large velocity converges to zero while
the model predicts some non-zero population for these drops. Second, the span of the distribution
does not show good agreement and, third the location and population of the most probable
velocity is largely different, particularly for the FFN. The behavior of our results resembles those
of Kim et al.’s study (123), one of the few studies that compared the predicted velocity
distribution with experiment (Refer to Figures 7, 9 and 11 in (123)). This problem had also come
to the attention of Bhatia and Durst (98), (124) and (125), where the authors concluded that the
available MEP models fail to predict the overall pdf correctly and proposed an alternative log-
hyperbolic pdf and added that their model is “the best choice among both one- and two-
dimensional distributions”. Nonetheless the log-hyperbolic model requires extensive numerical
calculation which is not practical, besides the model is justified by purely empirical means.
116
(a) CPN
D*
dN
*/d
D*
0 1 2 3 4 5 6 70
0.5
1
1.5
Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
117
(b) FFN
Figure 5-4: Primary aerosol size distribution measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min), D30=16.2 (µm) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), D30=14.4 (µm).
Error bars represent standard deviation.
We also believe nullifying all the past efforts in MEP modeling does not seem to be fair either.
For example, one could equally argue that the log-hyperbolic may only apply to strongly
monomodal distributions while bimodal distributions are successfully modeled by MEP (126)
and (127). Besides, as far as MEP is concerned the resultant distribution is the most objective
and the least biased distribution for the prescribed information. Kapur (101) states MEP may
D*
dN
*/d
D*
0 2 4 6 80
0.5
1
1.5
Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
118
necessarily lead to correct (physical here) solution, in such situation more information should be
specified, i.e., additional constraints should be used or given in a different form.
As may be noticed from Equations (5-32) to (5-37), the velocity information is only given
through joint moments of momentum and kinetic energy while the size information is prescribed
in both formats, first through single moments (conservation of mass, surface energy and partition
of the surface energy) and second in the joint moments. Thus, one possible solution might be
substituting the joint moments. Since the simplest distribution is the Gaussian distribution that
needs only single moments of mean and rms, we prescribed the same information to the MEP
model. This would imply the following substitution in the equations:
∫ ∫
( )
(5-44)
∫ ∫
( ( )
)
( ( )
)
( ( )
)
(5-45)
(
) (5-46)
but the other equations will remain untouched. We will call this model the replaced constraint
model. The alternative approach would be prescribing the two moments to the seat of constraints
so that like size, velocity information is also given in both single and joint formats (added
constraint model). The new constraints and the revised distribution function would then be:
∫ ∫
( )
(5-47)
∫ ∫
( ( )
)
( ( )
)
(5-48)
(
) (5-49)
119
Figure 5-4 shows that adding or replacing velocity moments do not vary the size distribution
considerably as mentioned in (116). As can be seen the applied changes only dislocated the most
probable size and its probability to a very small degree. Thus we may draw the conclusion that
the information regarding size distribution is mainly embedded in the moments of mass, surface
energy and partition of surface energy.
The velocity distributions resulted from the 3 models are depicted in Figure 5-5. As can be seen
the velocity distribution of the added and replaced constraint models on each other for the CPN.
For the FFN, the results are also similar, but the added constraint model have negative skewness
toward the larger velocities., probably because the experimental results show a degree of
bimodality and larger STD, i.e. two separate peaks in the velocity distribution can be recognized.
Aside from this, the addition and replacement of moments have successfully recovered the
overall shape of distribution to a great extent. The location of the most probable velocity, its
probability, the population of end tail velocities and the span of distribution agree very well with
the experiment in comparison to the original model (Figure 5-5).
Since the velocity distribution of both new models is very similar for both the nebulizers, the
replaced cosntarint model is preferable because it needs less amount of prior information. To
check the validity of this claim, the mean and rms velocity are plotted against size classes
(diameters) in Figure 5-6 and Figure 5-7 respectively, because after all we are looking for a
model that can adequately map size and velocity space. As can be seen, the standard deviation of
both mean and rms velocities are initially small but they grow larger with diameter classes
particularly over 45 and 35 (µm) for the CPN and FFN respectively. However, if we refer to size
distribution graphs (Figure 5-4), it is found that for these droplet classes, i.e. D*>2.78 for CPN
and D*>2.43 for FFN, the probability is close to zero. Figure 5-6 and 5-7 reveal some interesting
features of the MEP models too. For example, the mean and rms trend of the original MEP
model does not agree with the experiment and stands above it; in addition the mean velocity
curves reach a plateau around some particular class of size while the rms ones continuously
decreases with drop diameters for both nebulizers. Similar problems can also be observed in Li et
al.’s study (114). Although a different atomizer under different flow condition has been utilized
(Refer to Figure 11 in (114)), Li et al. related the unphysical behavior to the location of the
measurement point and the region of sheet breakup. The replaced constraint model shows no
120
(a) CPN
U*
dN
*/d
U*
0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
1.2
Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
121
(b) FFN
Figure 5-5: Primary aerosol velocity distribution measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min), Uref=47.2 (m/s) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Uref=17.6 (m/s).
Error bars represent standard deviation.
change in either mean or rms values by increasing diameter. These trends do not agree with our
physical understanding that the lighter drops must be more responsive to the surrounding air flow
due to the drag law. The added constraint model on the other hand, demonstrates an interesting
behavior. The mean velocity shows a gradual decrease for drops larger than 15 (µ) sprayed with
the CPN. In contrast, the mean velocity of FFN aerosol grows larger with the drop diameters
U*
dN
*/d
U*
0 2 40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
122
until it become nearly constant at (36-38 m/s) over 20 (µm). Although the experimental curve
reaches its maximum around 20 (µm), it experiences a further decrease to a velocity range of 28-
30 (m/s). In Chapter 4, we discussed that beyond some downstream location the lighter droplets
have smaller velocities (Figure D-1) providing that the spray load is negligible compared to the
gas flow which is the case for FFN. Alternatively for the CPN, the effect of spray load would
reverse this trend and larger drops will have smaller velocities. The replaced constraint model is
the only joint size and velocity distribution that exhibits such trends reasonably.
Regardless of the mean velocity of a droplet, the rms velocity must be smaller for heavier
droplets because the light droplets can be easily affected by the disturbances in the flow and as a
result experience larger deviation from their mean values. This trend is only observed in the
original and replaced constraint models, but again the replaced constraint model better agrees
with the experiment, particularly for the maximum and the rms value below 20 (µm) diameters.
According to the presented size and velocity distributions together with the mean and rms plots
we may claim to have an realistic aerosol size and velocity space, the velocity moments must be
prescribed in both single moments of U10 and and Urms and the joint moments of the momentum
and kinetic energy. The model exhibits general validity for monomodal and weakly bimodal size
and velocity distributions either skewed or symmetric. In our experiments, the CPN always
generated monomodal size and velocity distributions, but the FFN showed some degree of
bimodality in velocity. When the bimodality effects was increased the MEP model deviated from
the experiment which indicates the bimodality behavior requires a separate treatment, perhaps by
adding or replacing some single or joint moments. This case in still under investigation and the
study continues. The MEP model can be developed to multi dimensional spaces including radial
components of velocity or aerosol temperature in a similar manner depending on the application.
Here only the size and velocity distributions of the primary aerosol were presented but the size
and velocity plots in Appendix C and D can be used as input for MEP modeling to derive the
aerosol space at any desired locations either along the nebulizer axis or at the fringes of the
spray.
Although one of the objective in this task was to present a meaningful aerosol size and velocity
space for two particular nebulizers under a given set of conditions, a comment should be added
123
(a) CPN
D um
U1
0m
/s
20 40 60 800
10
20
30
40
50
60Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
124
(b) FFN
Figure 5-6: Mean velocity versus droplet diameter measured at z=10 (mm) for (a) Ql=5 (µl/s) at
Qg=500 (ml/min and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Error bars represent standard
deviation.
D um
U1
0m
/s
20 40 60 80 100 1200
5
10
15
20
25
30
35
40
45
50
Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
125
(a) CPN
D um
Urm
sm
/s
20 40 60 80 100 1200
5
10
15
20
25Experiment
Original 6 constraint model
Replaced constraint model
Added constraint model
126
(b) FFN
Figure 5-7: Root mean square velocity versus droplet diameter measured at z=10 (mm) for (a)
Ql=5 (µl/s) at Qg=500 (ml/min) and (b) Ql=0.1 (µl/s) at Qg=150 (ml/min), Error bars represent
standard deviation.
D um
Urm
sm
/s
20 40 60 80 100 1200
5
10
15
20
25
Experiment
Original 6 constraint
Replaced constraint model
Added constraint model
127
on the application and limitation of MEP modeling. In general any single or joint size and
velocity moment contains some information about the spray. As shown in this chapter,
combination of these moments generates the overall shape of the distribution functions. In fact
each αk exponent in Equation (5-8) is a function of the known constraint values:
( ) (5-50)
However the closed mathematical forms of these functions are not available; therefore finding
the dependency of each exponent to the constraint values is a tedious task and requires extensive
experiment. If the closed forms of the functions were known to us, we would be able to predict
the variation of distribution functions by changing the flow conditions and nozzle parameters
which are reflected through the constraint values (Г’s). For instance, consider the normal
distribution function in Appendix E (Equation E-1). This simple distribution function can be
derived through MEP by prescribing D10 and Drms. By rearranging the equation, the exponents
(α0- α2) can be easily presented as functions of the prescribed moments. If one had characterized
the nozzle before and correlated D10 and Drms to the nozzle and flow parameters, it would be very
easy for him to predict quantitatively how the distribution function is positioned by
increasing/decreasing the liquid flow rate for example. Unfortunately, we only have these closed
forms for very simple distribution functions. The majority of distribution functions either has
only one independent variable (diameter) or does not maximize the entropy function. Hence
finding the closed mathematical forms for MEP based distributions would off paramount
importance in spray and atomization modeling.
Contribution 5.7
Capturing the details of atomization in Eulerian frame of reference is extremely expensive and
time consuming. Therefore any study or numerical modeling strongly depends on some
statistical measure of the spray at least as a prerequisite. Maximum entropy principle is the only
promising approach up to this day that may offer a solution in the absence of deterministic
methods. The theory has a variety of applications in many branches of science and seeks the least
biased and the most objective distribution based on the available pieces of information by
128
maximizing the Shannon entropy. The principle has been exploited for spray modeling since
1985 by different researchers in the field.
The misunderstanding of number and volume based distributions in MEP application was
explained and correct formulation of MEP was presented. As stated, the principle can be
implemented both globally to cover the entire spray cross section or locally for a single point in
space as long as the constraints are known at that point. Since the global implementation of MEP
requires information which is not easy to obtain and dictates the reference values for size and
velocity normalization, the principle was employed locally. As discussed, the local
implementation is more appropriate to ICP-MS application and also allows us to choose the gas
velocity for normalization that can be easily measured or extracted from the literature. In
addition the gas velocity considerable reduces the stiffness of the Jacobian matrix and facilitates
reaching the numerical solution..
The results of size distribution showed that the characterization information were sufficient for
capturing a reasonable size distribution but the velocity distribution displayed several
deficiencies including the incorrect peak velocity and probability, incorrect end tail population
and span. To resolve this problem, two possible solutions were considered, one replacing the
joint moments of momentum and kinetic energy with mean and rms velocities or prescribing the
two velocities along with the other constraints . The mean and rms velocities were prescribed
because the simplest Gaussian distribution only needs these two parameters. Both replacement
and addition of the single velocity moments resolved the spray velocity distribution. When the
mean and rms velocities were plotted against the droplet diameters, only the results of the added
constraint model agreed with experiment. According to the distribution curves and mean and rms
plots, it was concluded that a detailed size and velocity space can be captured when velocity
moments are given in both single moments of mean and rms velocities and joint moments of
momentum and kinetic energy. The model exhibited general validity for monomodal and weakly
bimodal size and velocity distributions either skewed or symmetric.
129
Chapter 6 Concluding Remarks and Future Works
Contribution 6.1
A conventional and benchmark type-C CPN widely used for sample introduction in spectrometry
was characterized. In addition a new direct injection flow focusing nebulizer (FFN) for total
aerosol consumption was introduced and characterized and the nebulizer performance was also
investigated. The other objective of this thesis is to find a mathematical model that is capable of
presenting the details of aerosol size and velocity space.
The major contribution and original findings of the present task are listed as follows:
The application known Nukiyama-Tanasawa (NT) and Rizk-Lefbvre (RL) correlations
for predicting aerosol size of the CPN under typical ICP-MS flow conditions leads to
erroneous results, even though the general trend of the process is correctly predicted.
The overestimation of the NT correlation is attributed to the weight of its second term.
The NT and RL correlations were modified for the CPN under the given flow conditions.
As a result the overestimation of the NT model and the underestimation of the RL model
were resolved.
The variation of characteristic moment ratios (D30/D-10 and D30/D32) of the CPN with the
normalized gas flow rate was studied and a correlation for each ratio was presented.
The variation of the D30/D-10 and D30/D32 with axial location was presented. It was found
while the first ratio drops downstream of the nebulizer axis the latter grow gradually.
A new nebulization efficiency definition different from the transport efficiency in ICP-
MS was introduced. The definition focuses on the conversion of the bulk input kinetic
energy to the surface energy. A minimum attainable Sauter mean diameter based on the
flow conditions was derived for an ideal nebulizer.
130
The Sauter mean diameter from the modified NT and RL models and were used to plot
efficiency curves of CPN. It was found that when it comes to aerosol generation CPN and
nebulizers are poor devices in general.
The fundamentals of the FFN were discussed. The nebulizer was characterized and
correlation for its Sauter mean diameter was proposed.
The performance of the FFN was investigated. It was found that characteristic moment to
jet ratio drops by increasing the jet based Weber number.
It was that found when the flow condition is carefully tuned; the FFN can produce close
to 100 percent consumable aerosol.
It was shown that due to the sample load, the mean aerosol to gas velocity ratio of the
CPN reduces by either increasing the Sauter mean diameter or the liquid to gas mass flow
ratio. A correlation was then proposed.
The sample load of the FFN was negligible and as shown in Chapter 4, beyond a certain
downstream location the smaller droplets may have larger velocities. A correlation was
proposed for the mean aerosol to gas velocity ratio of the FFN was proposed.
It was argued while MEP is usually implemented globally on the entire spray cross
section, its local implementation is more suitable for ICP nebulizers.
Using the gas local velocity for normalization of the velocity space reduces the stiffness
of the Jacobian matrix and facilitates reaching the numerical solution.
The original MEP model presents unrealistic velocity distributions. The end tail
population, peak location, peak value and the distribution span of the model do not agree
with the experiments for both the CPN and the FFN.
Replacing and adding the single moments of mean and rms velocities resolve the
unrealistic velocity distribution. However only the added constraint model can correctly
131
predict the trend of mean and rms velocities with droplet diameter classes and show good
agreement with the drag law.
Future Works 6.2
In the present task, the NT and RL were fitted to a single CPN at specific flow conditions.
Nevertheless from practical point of view, modifing NT and RL models for a wide range of
different ICP-MS nebulizers and flow conditions would be very beneficial. Although the
necessity of comprehensive models is reflected in the ICP-MS literature, up to this day and the
best of the author’s knowledge such inclusive study does not exist. In fact, most of the efforts are
mainly focused on proposing models or correlations for one or a few specific nebulizer(s) at
most.
The designed FFNs have shown promising results and have great potentials. We believe the
current designed has to be further developed to overcome some undesired instability issues, and
results irreproducibility. For instance, we believe that the shape of the exit orifice may have
some influence on the surface perturbation of the liquid jet and its ultimate breakup. The
bimodality of the velocity distribution of the FFN is another issue that would require separate
attention. Ideally, monomodal and monodisperse size and velocity distributions are preferred.
The size distribution in our experiments was always monomodal and although not perfectly
monodisperse but it was satisfactory enough. The bimodality of velocity distribution should be
overcome either by redesigning the nebulizer or by finding the optimum condition at which this
phenomenon becomes minimal.
The numerical modeling of the FFN is another interesting subject. There is only a few numerical
study published on the subject after almost 10 years from the first invention of the nebulizer.
Perhaps because the combination of turbulent and compressible gas flow with the laminar and
low velocity liquid flow make the numerical simulation very expensive if not impossible. A great
deal of information on the physics on mechanism of jet formation-disintegration and the gas-
liquid can be extracted from numerical modeling that is not possible from direct experiment. We
132
are currently engaged in numerical simulation of the flow focusing problem and the preliminary
results seem very promising although there are many challenges to overcome.
The new MEP model with added constraints for velocity showed satisfactory results for CPN.
The bimodality of velocity in FFN however is problematic and basically implies there is a lack of
information regarding the velocity distribution. From analytical modeling point of view, it would
be interesting to know what new velocity moments may possibly contain bimodality information
and resolve the issue. Finding the close mathematical form for the unknown exponents (in pdf’s)
as a function of constraint values is of great importance because this would generalize the
modeling to all the nebulizers working under different flow conditions. In other words, if the
constraint values of different nebulizers are known from characterization, the generalized model
can easily predict how the shape of distribution function may change by varying the nozzle
parameters and flow conditions. The new MEP models must also be tested and validated with
the results of other ICP-MS nebulizers to assure that it can cover wide ranges of sprays and not
merely the aerosol from a specific nozzle at particular flow conditions.
133
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Appendices
Appendix A: Generation of Ripples by Wind Blowing over a
Viscous Fluid
Taylor (24) studied the problem of the growth of disturbances on a liquid surface and presented
equations in terms of dimensionless groups that express the wavelength of maximum instability
for specified gas and liquid properties. Taylor has shown the growth in a wave is determined by
an exponential growth factor, exp[R(α)t], where R(α) is the real part of the term which must have
dimensions of a frequency. Taylor gives the R(α)=2πνK2s, where ν is the kinematic viscosity and
K is the wave number (K=2π/λ) and s is a number.
Obviously when s is negative, the wave is damped and the damping rate is proportional to the
viscosity and inversely proportional to the wavelength. Taylor considered the situation where the
wave is sustained by the gas action, or in other words when s has a positive value and expressed
growth factor, R(α), in terms of dimensionless group (θ) and a dimensionless length (x).
( ) (
)
(A-1)
in which
(A-2)
and
(A-3)
(A-4)
147
Figure A-1: Plot of the maximum values of the dimensionless length xm versus the
dimensionless variable θ. Taken with permission from (20).
For given values of θ, the maximum values of s/x2 are determined and the corresponding x
values given the notation xm are given against θ in Figure (A-1).
Taylor suggests that the most probable drop size generated should correspond approximately to
the wavelength of maximum instability and presented the optimum wavelength by:
(
)
( ) (A-5)
here A is constant which has a value close to 1. By having the fluid properties and flow
conditions, first θ is calculated from Equation (2-A) and the corresponding dimensionless length
(xm) might be found from Figure (A-1).
148
Appendix B: PDPA Calibration and Measurement
To have reliable size and velocity measurement with PDPA, the device has to be calibrated and
optical settings must be set correctly for each particular application.
The PDPA device used in our experiment was already calibrated by Professor Ashgriz’s group.
Therefore the crossing and overlap of the two laser beams were satisfactory as a high level data
rate was observed during the experiments. However acquiring good size and velocity data for
any nozzle requires iterative measurements and careful setting of different parameters.
A TR60 series transceiver probe was used for focusing the two laser beams. The focal length,
beam separation and laser beam diameter of this probe are 350 (mm), 50 (mm) and 2.65 (mm)
respectively. The focal lengths of the receiver front and back lenses are 300 and 250 (mm) while
the slit aperture is 150 (mm). Wavelengths of 514.5 and 488 (nm) were set for green and blue
Argon-Ion laser beams respectively.
As explained in Chapter 2, droplets passing the measuring probe of the PDPA scatter light in
different directions interfering with a fringe pattern in the plane of the receiver lens. The
temporal frequency of the measured scattered light and the fringe spacing may be used to find
the particle velocity. For our optical configuration, the FlowSizer Software reports a fringe
spacing of 3.6 and 3.4 (µm) for the two channels. One of the two laser beams of the PDPA
device is frequency shifted by a Brag cell to determine if a particle is moving in or opposite the
flow direction. The Brag cell frequency is set to 40 (MHz) as suggested in the software manual
(47). The optical signals received in PMT are Gaussian due to nature of the laser beam intensity,
hence a high pass filter with 20 (MHz) frequency is used to remove the low frequency portion of
the signal.
The final signal is then downmixed to eliminate the initially added 40 (MHz) frequency either
completely or partially. Nevertheless it is always recommended to use the downmixing process
when the flow Doppler frequency is 20 (MHz) or lower (47). A downmix frequency of 36 (MHz)
was considered for our experiments. This would leave 4 (MHz) frequency shift on the measured
frequency, therefore with the 3.6 (µm) fringe spacing, a reversal flow up to 14.4 (m/s) can be
calculated.
149
To take particle velocity, a correct band pass filter must be selected. For this purpose, the
frequency count histogram was monitored as data was being captured in real time. A correctly
chosen band pass filter must give a histogram that is not clipped on both end of the distribution.
The 5-30 (MHz) band pass filter for the CPN running at Ql=5 (µl/s) and Qg=500 (ml/min)
generate a valid frequency count histogram as shown in Figure B-1 which corresponds to mean
frequency and velocity of 14.2 (MHz) and 36.7 (m/s) respectively.
Figure B-1: Frequency count histogram of channel, captured for a CPN running at Ql=5 (µl/s)
and Qg=500 (ml/min).
150
The burst threshold is another important factor in velocity measurement which is the analog
voltage level that a signal must reach before the burst gate in the processor is opened. Since
larger particles scatter more light, they have higher signal amplitudes. Increasing the PMT
voltage will also increase the signal amplitudes. This is why the burst threshold level cannot be
adjusted independently without considering the PMT voltage. Typical values for burst threshold
is from 30 (mV) to 300 (mV). For small particles (<10 μm) the optimum value is 30 (mV) or
slightly higher. In our experiment this value was set to 60 (mV).
The data rate on the system can be improved by increasing the PMT voltage because signals of
the small particles will be large enough to be detected by the processor. However, increasing the
PMT voltage also increases the noise level, therefore beyond a certain point; increasing the
voltage would cause a small signal to noise ratio and reduce the data rate. A good data rate was
obtained at a PMT voltage of 770 (V) while the signal to noise ratio (SNR) was set high to assure
only the best quality bursts pass the validation.
Many of the considerations for velocity measurement are equally important for size
measurement. Nevertheless the PMT voltage selection is more critical for size measurement
because it affects the measurable size of the instrument. In general, the voltage must be large
enough to detect small particles in the flow but low enough to avoid excessive PMT saturation.
The following procedure was taken for selecting the appropriate PMT voltage. First the voltage
was set 350 (V) and the mean diameter value (D10) was recorded. The voltage was then gradually
increased. As a result smaller droplets were detected and the D10 value dropped continuously
until about 770 (V), the D10 value was stabilized. Another criterion for checking the PMT voltage
is the Dmax/3 rule of thumb which states that the in the intensity validation plot, the upper
diameter limit curve must reach 1000 (mV) at approximately Dmax/3. The maximum measurable
diameter is determined from the optical settings that was 115 (µm) for our case. Therefore, at
around 38 (µm) droplet diameter, the upper limit curve should be around 1000 (mV) as can be
seen in Figure B-2. The upper and lower limit intercept were set at 150 and 0 (mV) respectively.
The upper limit intercept of 50-250 (mV) is generally acceptable for size measurement and the
slope of the lower curve is usually set 0.1 times the upper slope.
151
Figure B-2: Intensity validation plot used for choosing the correct PMT voltage.
The final criterion for a good size measurement is that the diameter difference between the two
independent size measurements must be within a certain range. The maximum acceptable
diameter difference from years of study is reported to be 7% (47). If a system is setup correctly
the data points on the diameter difference versus diameter graph must be roughly centered
around the diameter difference of zero otherwise a phase calibration may be required. Figure B-3
clearly shows that the captured data points are within the acceptable range and are approximately
symmetric around zero, thus confirms the validity of the experiments.
153
Appendix C: Axial and Radial Variation of D30/D32 and D30/D-10
Ratios of the FFN
(a)
(b)
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
0 10 20 30 40
D30/D
32
z (mm)
Axial variation of D30/D32
at Ql=0.1 (µl/s)
Qg=215 (ml/min)
Qg=205 (ml/min)
Qg=194 (ml/min)
Qg=182 (ml/min)
Qg=169 (ml/min)
Qg=154 (ml/min)
0.75
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
0 10 20 30 40
D3
0/D
32
z (mm)
Axial variation of D30/D32
at Ql=0.2 (µl/s)
Qg=219 (ml/min)
Qg=206 (ml/min)
Qg=195 (ml/min)
Qg=182 (ml/min)
Qg=170 (ml/min)
Qg=155 (ml/min)
154
(c)
(d)
Figure C-1: D30/D32 variation with axial location at Ql (μl/s) (a) 0.1, (b) 0.2, (c) 0.4 and (d) 0.6
(r=0 mm).
0.75
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
0 10 20 30 40
D30/D
32
z (mm)
Axial variation of D30/D32
at Ql=0.4 (µl/s)
Qg=219(ml/min)
Qg=206 (ml/min)
Qg=195 (ml/min)
Qg=183 (ml/min)
Qg=169 (ml/min)
Qg=155 (ml/min)
0.74
0.75
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0.85
0 10 20 30 40
D30/D
32
z (mm)
Axial variation of D30/D32
at Ql=0.6 (µl/s)
Qg=219 (ml/min)
Qg=206 (ml/min)
Qg=196 (ml/min)
Qg=182 (ml/min)
Qg=169 (ml/min)
Qg=155 (ml/min)
155
(a)
(b)
1.45
1.5
1.55
1.6
1.65
1.7
1.75
1.8
1.85
0 10 20 30 40
D30/D
-10
z (mm)
Axial variation of D30/D-10
at Ql=0.1 (µl/s) Qg=215 (ml/min)
Qg=205 (ml/min)
Qg=194 (ml/min)
Qg=182 (ml/min)
Qg=169 (ml/min)
Qg=154 (ml/min)
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
0 10 20 30 40
D30/D
-10
z (mm)
Axial variation of D30/D-10
at Ql=0.2 (µl/s) Qg=219 (ml/min)
Qg=206 (ml/min)
Qg=195 (ml/min)
Qg=182 (ml/min)
Qg=170 (ml/min)
Qg=155 (ml/min)
156
(c)
(d)
Figure C-2: D30/D-10 variation with axial location at Ql (μl/s) (a) 0.1, (b) 0.2, (c) 0.4 and (d) 0.6
(r=0 mm).
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
0 10 20 30 40
D30/D
-10
z (mm)
Axial variation of D30/D-10
at Ql=0.4 (µl/s) Qg=219 (ml/min)
Qg=206 (ml/min)
Qg=195 (ml/min)
Qg=183 (ml/min)
Qg=169 (ml/min)
Qg=155 (ml/min)
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
0 10 20 30 40
D30/D
-10
z (mm)
Axial variation of D30/D-10
at Ql=0.6 (µl/s) Qg=219 (ml/min)
Qg=206 (ml/min)
Qg=196 (ml/min)
Qg=182(ml/min)
Qg=169 (ml/min)
Qg=155 (ml/min)
157
(a)
(b)
1.18
1.2
1.22
1.24
1.26
1.28
1.3
1.32
1.34
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
-10
r (mm)
Radial variation of D30/D-10
at Ql=0.1 (µl/s)
Qg=212 (ml/min)
Qg=202 (ml/min)
Qg=188 (ml/min)
Qg=177 (ml/min)
Qg=165 (ml/min)
Qg=150 (ml/min)
1.2
1.25
1.3
1.35
1.4
1.45
1.5
0 0.2 0.4 0.6 0.8 1 1.2
D3
0/D
-10
r (mm)
Radial variation of D30/D-10
at Ql=0.2 (µl/s)
Qg=215 (ml/min)
Qg=204 (ml/min)
Qg=191 (ml/min)
Qg=179 (ml/min)
Qg=166 (ml/min)
Qg=151 (ml/min)
158
(c)
(d)
Figure C-3: D30/D-10 variation with radial location at z=10 (mm) and Ql (μl/s) (a) 0.1, (b) 0.2, (c)
0.4 and (d) 0.6
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
-10
r (mm)
Radial variation of D30/D-10
at Ql=0.4 (µl/s)
Qg=215 (ml/min)
Qg=204 (ml/min)
Qg=191 (ml/min)
Qg=180 (ml/min)
Qg=165 (ml/min)
Qg=151 (ml/min)
1.15
1.2
1.25
1.3
1.35
1.4
1.45
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
-10
r (mm)
Radial variation of D30/D-10
at Ql=0.6 (µl/s)
Qg=216 (ml/min)
Qg=206 (ml/min)
Qg=192 (ml/min)
Qg=180 (ml/min)
Qg=167 (ml/min)
Qg=154 (ml/min)
159
(a)
(b)
0.76
0.77
0.78
0.79
0.8
0.81
0.82
0.83
0.84
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
32
r (mm)
Radial variation of D30/D32
at Ql=0.1 (µl/s)
Qg=212 (ml/min)
Qg=202 (ml/min)
Qg=188 (ml/min)
Qg=177 (ml/min)
Qg=165 (ml/min)
Qg=150 (ml/min)
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0 0.2 0.4 0.6 0.8 1 1.2
D3
0/D
32
r (mm)
Radial variation of D30/D32
at Ql=0.2 (µl/s)
Qg=215 (ml/min)
Qg=204 (ml/min)
Qg=191 (ml/min)
Qg=179 (ml/min)
Qg=166 (ml/min)
Qg=151 (ml/min)
160
(c)
(d)
Figure C-4: D30/D32 variation with radial location at z=10 (mm) and Ql (μl/s) (a) 0.1, (b) 0.2, (c)
0.4 and (d) 0.6.
0.67
0.69
0.71
0.73
0.75
0.77
0.79
0.81
0.83
0.85
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
32
r (mm)
Radial variation of D30/D32
at Ql=0.4 (µl/s)
Qg=215 (ml/min)
Qg=204 (ml/min)
Qg=191 (ml/min)
Qg=180 (ml/min)
Qg=165 (ml/min)
Qg=151 (ml/min)
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.84
0.86
0 0.2 0.4 0.6 0.8 1 1.2
D30/D
32
r (mm)
Radial variation of D30/D32
at Ql=0.6 (µl/s)
Qg=216 (ml/min)
Qg=206 (ml/min)
Qg=192 (ml/min)
Qg=180 (ml/min)
Qg=167 (ml/min)
Qg=154 (ml/min)
161
Appendix D: Spatial Variation of Mean Droplet Velocity Moments
(a)
(b)
Figure D-1: Variation of (a) U10/ug and (b) Urms/U10 ratios with axial location for FFN at Ql=0.4
(µl/s) and different gas flow rates.
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
0 10 20 30 40
U10/u
g
z (mm)
Axial variation of U10/ug for FFN
Qg= 215 (ml/min)
Qg= 205 (ml/min)
Qg= 195 (ml/min)
Qg= 185 (ml/min)
Qg= 170 (ml/min)
Qg= 155 (ml/min)
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0 10 20 30 40
Urm
s/U
10
z (mm)
Axial variation of Urms/U10 for FFN
Qg= 215 (ml/min)
Qg= 205 (ml/min)
Qg= 195 (ml/min)
Qg= 185 (ml/min)
Qg= 170 (ml/min)
Qg= 155 (ml/min)
162
(a)
(b)
Figure D-2: Variation of (a) U10/ug and (b) Urms/U10 ratios with axial location for CPN at Ql=5
(µl/s) and different gas flow rates.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 10 20 30 40 50 60
U10/u
g
z (mm)
Axial variation of U10/ug for CPN
Qg=250 (ml/min)
Qg=315 (ml/min)
Qg=390 (ml/min)
Qg=495 (ml/min)
Qg=590 (ml/min)
Qg=690 (ml/min)
Qg=780 (ml/min)
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0 10 20 30 40 50 60
Urm
s/U
10
z (mm)
Axial Variation of Urms/U10 for CPN
Qg=250 (ml/min)
Qg=315 (ml/min)
Qg=390 (ml/min)
Qg=495 (ml/min)
Qg=590 (ml/min)
Qg=690 (ml/min)
Qg=780 (ml/min)
163
(a)
(b)
Figure D-3: Variation of (a) U10/ug and (b) Urms/U10 ratios with axial location for FFN at Ql=0.4
(µl/s) and different gas flow rates.
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
0 10 20 30 40
U10/u
g
z (mm)
Axial variation of U10/ug for FFN
Qg= 215 (ml/min)
Qg= 205 (ml/min)
Qg= 195 (ml/min)
Qg= 185 (ml/min)
Qg= 170 (ml/min)
Qg= 155 (ml/min)
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0 10 20 30 40
Urm
s/U
10
z (mm)
Axial variation of Urms/U10 for FFN
Qg= 215 (ml/min)
Qg= 205 (ml/min)
Qg= 195 (ml/min)
Qg= 185 (ml/min)
Qg= 170 (ml/min)
Qg= 155 (ml/min)
164
(a)
(b)
Figure D-4: Variation of (a) U10/ug and (b) Urms/U10 ratios of the primary aerosol with radial
location for FFN at Ql=0.4 (µl/s) and different gas flow rates, measured at z=10 (mm).
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
0 0.2 0.4 0.6 0.8 1
U10/u
g
r (mm)
Radial variation of U10/ug for FFN
Qg= 215 (ml/min)
Qg= 200 (ml/min)
Qg= 190 (ml/min)
Qg= 175 (ml/min)
Qg= 165 (ml/min)
Qg= 150 (ml/min)
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0 0.2 0.4 0.6 0.8 1
Urm
s/U
10
r (mm)
Radial variation of Urms/U10 for FFN
Qg= 215 (ml/min)
Qg= 200 (ml/min)
Qg= 190 (ml/min)
Qg= 175 (ml/min)
Qg= 165 (ml/min)
Qg= 150 (ml/min)
165
Appendix E: Empirical Probability Distribution Functions
The classical method of modeling drop size distributions is empirical (93): a curve is fit to data
collected for a wide range of atomizer nozzles and operating conditions. Curves appearing
frequently become the basis for standard empirical distributions. Some of the most commonly
used distributions are listed below.
Gauss-Normal distribution:
The normal distribution is a number-based continuous bell shaped function and perhaps the
simplest form of distribution functions. The function is defined by having a mean (D10) and a
variance (ν=Drms/D10) parameter with a peak located at the mean. The skewness of the function
is zero that implies the function is essentially symmetric with respect to the mean value.
( ) √
{
( )
} (E-1)
Log-Normal distribution:
If a variate x is distributed so that the distribution of the transformed variate y=ln(x) is Gaussian,
then the variate x is said to be log normally distributed. Therefore:
( )
( )√ {
( )
( )} (E-2)
Where D’ represents the logarithmic mean of size distribution and ζLN represent the width of the
distribution. It is interesting to note that the log-normal distribution is arises when one considers
the theoretical distribution that is produced by continuous random partitioning of a set of sand
particles that was first noticed by Kolmogoroff (96).
Upper-limit distribution:
This distribution is a modification of the log-normal distribution in that a maximum drop size is
introduced:
166
( )
√ ( ) { (
)} (E-3)
√ ( )
Here ζUL represent the span of distribution, Dmax is the upper drop diameter and D’ is another
representative diameter, noting that the distribution is given in volume-based format. The upper-
limit distribution was introduced by Mugele and Evans (97) who wanted to modify the log-
normal distribution by specifying a maximum drop diameter. The upper-limit distribution
approaches the log normal distribution as the maximum diameter tends to infinity.
Root-Normal Distribution:
This distribution was proposed by Tate and Marshall (128) to express the volume distribution of
drops in sprays.
( )
√ {
[√ √
]
} (E-4)
where D’ and ζRN represent a mean diameter and the width of the distribution respectively. Note
that a number-based distribution cannot be derived from the volume-based distribution because
of the unphysical behavior at the lower end of the distribution that a gradient catastrophe near
zero.
Rosin-Rammler Distribution:
The distribution first appeared in (95) to describe the cumulative volume distribution of the coal
particles. The mathematical simplicity of the function has caused its wide use in the spray
literature despite of its shortcomings.
( ) { (
)
} (E-5)
167
here D’ and q are the mean and with of the distribution. Small values of q are associated with
broad sprays and large values result in narrow sprays. For many droplet generation processes, q
ranges from 1.5 to 4, and for rotary atomization q might be as large as 7 whereas monodisprse
spray production demands a q value of infinity (94).
Nukiyama-Tanasawa Distribution:
The distribution is proposed by Nukiyama and Tanasawa (51) to describe the number-based
distribution of sprays from pneumatic nebulizers and resembles the Rosin-Rammler distribution
mathematically.
( ) ( ) (E-6)
where b, p and q are adjustable parameters that control the location of mean and span of the
distribution. a is a normalizing constant and p is sometimes taken to be two. According to
Paloposki (129), physical meaningful results are produced either if p>1 and q>0 or p<-4 and
q<0.
Log-Hyperbolic Distribution:
This function is first applied to sprays by Bhatia et al. (98), (124) and (125) which has the given
form:
( ) ( ) { √ ( ) ( )} (E-7)
here a is a normalizing constant
√
( √ ) (E-8)
and K1 is the modified Bessel function of the first kind and first order. The constraints on the
parameters are as follows: -∞<D<∞, α>0, |β|< α, δ>0 and -∞<µ<∞. The distribution derives its
name from the fact that the logarithm of the probability distribution function is a hyperbola. δ is
the scale parameter, µ is the location parameter and the remaining two parameters describe the
shape of the pdf.
168
The distribution is one of the most successful empirical distributions as it can be fit to a wide
range of experimental data. But it requires extensive and tedious mathematical calculations (93)
and the parameters of the distribution are mathematically unstable (115) which is highly
undesirable.
169
Appendix F: Derivation of Shannon entropy
Consider a size and velocity space shown in Figure 5-1. If the total number of droplets is given
by Ntot, then the number of possible ways to put ΔN1,1 droplets in the first class of size and
velocity will be:
(
)
( ) (F-1)
Once, the ΔN1,1 droplets are placed in the first classes of size and velocity, we can similarly find
the number of ways that ΔN1,2 (out of Ntot-ΔN1,1 droplets) can be placed in the second class of
velocity and first class of size.
(
)
( )
( ) (F-2)
This procedure can be continued until all the droplets are distributed in size and velocity space.
Then the total number of possible states can be written by:
(
) (
) (
∑ ∑
) (F-3)
∑
Equation (F-3) can be simplified by using Equations (F-1) and (F-2) to:
(F-4)
We are now seeking an entropy function as a measure of disorder of the system based on the
total number of possible states given by W. Chakrabarti (130) have proven on the basis of two
fundamental properties of thermodynamic entropy, the form of entropy function can be derived,
which are:
170
(i) The entropy S(W) of system is a positive increasing function of the disorder W, that
is:
( ) ( ) (F-5)
(ii) The entropy S(W) is assumed to be an additive function of the disorder W, that is, for
any two statistically independent systems with degreed of disorder W1 and W2
respectively, the entropy of the composite system is given by:
( ) ( ) ( ) (F-6)
Proof:
Let’s assume that W>e. This is justified by the fact that the macroscopic system we are
interested in consist of large number of microstates and hence corresponds to a large value of
statistical weight W. For any integer n, we can find an integer m(n) such that:
( ) ( ) (F-7)
or
( )
( )
( )
(F-8)
Consequently:
( )
( ) (F-9)
The entropy function S(W) must satisfy both (i) and (ii) conditions. From the first condition it
draws:
( ( )) ( ) ( ( ) ) (F-10)
again from (ii) and (C-10) we have:
( ) ( ) ( ) ( ( ) ) ( ) (F-11)
171
and as a result:
( )
( )
( ) (F-12)
By comparing (F-9) and (F-12), we get:
( ) ( ) (F-13)
Where S(e)=k is a positive constant that depends on the unit of measurement of entropy. The
positivity follows from the positivity of the entropy function postulated from (i).
Now that the form of entropy function is determined, we may further expand Equation (F-13) by
having the statistical weigh function W given in (F-4):
( ) ( ) { ( ) (∑∑ )} (F-14)
Equation (F-14) can be simplified by Sterling’s approximation ln (x!)=x ln (x) –x to:
( ) { ( ) ∑∑ ( ) ∑∑ } (F-15)
∑∑ (F-16)
Therefore:
( ) ∑∑(
) (
) ∑∑ ( ) (F-17)
and the entropy per unit number of droplet would be:
( ) ( )
∑∑ ( ) (F-18)
172
Appendix G: Bayesian and Shannon entropy
In probability theory and information theory, the Kullback–Leibler divergence (also information
divergence, information gain, or relative entropy) is a measure of the difference between two
probability distributions: from a "true" probability distribution P to an arbitrary probability
distribution Q.
Typically P represents data, observations, or a precise calculated probability distribution. The
measure Q typically represents a theory, a model, a description or an approximation of P.
For distributions P and Q of a continuous random variable the K–L divergence of Q from P is
defined to be:
( ) ∫ ( ) ( ( )
( ))
(G-1)
Equation (1-D) is equivalent to the measure of nearness of two probability density functions.
Minimizing the K-L equation subjected to a set of constraints is also identical to maximizing
Bayesian entropy, which is a measure that takes a priori distributions into account.
Shannon entropy is in fact a special case of the Bayesian entropy when the priori distribution is
uniform one, that is Q(x) =1. In other words, maximizing the Shannon entropy with only the
normalization constraint would result in a uniform distribution but in the absence of any other
constraints, maximization of Bayesian entropy gives distribution Q(x).
173
Nomenclature
Ac Cross sectional area of droplet
CD Drag coefficient
do Orifice diameter
dj Jet diameter
D Droplet diameter
D-10 Nameless characteristic mean size diameter
D30 Mass mean diameter
D32 Sauter mean diameter
Dcap Capillary diameter
Dp Prefilmer lip diameter, pipe diameter
Dref Reference velocity
f Joint size and velocity probability distribution function
fb Breakup frequency
ho Total gas enthalpy
H Distance between capillary and orifice plate in FFN
J Jacobian Matrix
k Specific heat ratio, Constant of entropy
K Constant of axial velocity profile
174
Kp Partition constraint
L Orifice plate thickness in FFN
Lc Characteristic length at the point of surface where gas
and liquid meet
m•g Gas mass flow rate
m•l Liquid mass flow rate
mp Droplet mass
M Mach number
n• Number of droplets generated per unit time
Ntot Total number of droplets
Oh Ohnesorge number
Pg Gas exit pressure at the orifice exit
Pij Probability of finding a droplet in the ith
class of droplet
diameter and jth
class of droplet velocity
Pl Liquid pressure
Po Gas back pressure
Qg Gas volumetric flow rate
Ql Liquid volumetric flow rate
r Radial location
R Gas constant
175
Re Reynolds number
s Shannon entropy
Tg Gas exit temperature at the orifice exit
To Gas back temperature
ug Gas exit velocity at the orifice
uj Liquid jet velocity
ul Liquid exit velocity at the orifice
u* Sonic velocity
U Droplet velocity
U10 Droplet mean velocity
Ucap Liquid capillary velocity
UR Relative velocity between liquid and gas the orifice exit
Uref Reference velocity
Urms Droplet root mean square velocity
up Particle velocity
w Constraint number
We Weber number
z Axial location
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Greek Symbols
γ Characteristic moment
εT Turbulent kinematic viscosity
η Atomization efficiency
ηl Liquid viscosity
λ Wavelength
ρg Gas exit density at the orifice exit
ρl Liquid density
ρo Gas back density
σ Surface tension
η*
Sheet thickness at point of breakup
ηs
Tangential viscous stress on the jet surface
υ Kinematic viscosity
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