EXPERIMENTAL AND MODELING STUDY OF CONDENSATION
IN SUPERSONIC NOZZLES
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Somnath Sinha, B.E., M.A.S., M.S.
*****
The Ohio State University
2008
Dissertation Committee:
Approved by
Professor Barbara Wyslouzil, Adviser
Professor Igor Adamovich
Professor Isamu Kusaka
_________________________________
Adviser
Graduate Program in
Chemical Engineering
ABSTRACT
The formation of aerosols by condensation from the vapor phase proceeds through
nucleation, growth and aging stages. Of these, nucleation is the least well understood,
hardest to predict and hardest to measure. Most of the experimental work on nucleation
has been carried out using materials, for example water, that interact via complex
intermolecular potentials. Recently, approaches based on density functional theory,
molecular dynamics and Monte Carlo simulations, are becoming increasingly important
in advancing nucleation theory. Most of these studies focus on noble gas condensation,
usually Argon, because the intermolecular potentials for these species are relatively
simple.
The primary goal of this work is to develop a new cryogenic supersonic nozzle
apparatus to investigate Argon nucleation. The experimental data on Ar nucleation
measured with the new setup is compared to the data available in the literature and the
predictions of Classical Nucleation Theory (CNT). The results presented here resolve the
prior inconsistency between the data from nucleation pulse chambers and supersonic
nozzles at high pressures and temperatures. Comparing the data with the widely used
CNT indicates that the predictions of this theory are about 12 orders of magnitude below
the measured nucleation rates. Combining the current results with those of the cryogenic
ii
nucleation pulse chamber yields the size of the critical cluster n*. The experimental
values of n* are 30 – 55% smaller than predicted by the Gibbs-Thomson equation and
this may be the primary reason behind the failure of CNT. Since n* is not predicted
correctly, correcting CNT by multiplying with a temperature dependent factor based on
results from one device would not help in predicting nucleation rates for other
experimental devices, where onset occurs at significantly different supersaturations.
In order to understand the entire condensation process, it is also important to
understand growth and aging. In most devices, nucleation and growth occur
simultaneously. Thus, in order to test the available growth laws it is necessary to develop
models that incorporate both nucleation and growth. The second goal of this work is to
modify and use a steady state 1-D model to examine the formation and growth of
H2O/D2O droplets in a supersonic nozzle. The model incorporates a nucleation rate
expression and one of five different growth models and predicts both the properties of the
flow, i.e. pressure, temperature, density, etc, as well as the properties of the aerosol, i.e.
the number density, average size, etc.. The flow properties are compared to the
complimentary pressure trace measurements and the aerosol properties are compared to
those derived from small angle x- ray scattering experiments. Contrary to expectations,
the physically more realistic nonisothermal growth rate expressions over predict the
number densities and, therefore, underpredict the average aerosol size. Surprisingly,
under certain conditions, the isothermal growth rate expressions provide results closer to
iii
the experimental data. The failure of the nonisothermal growth expressions may be due to
an overestimation of the mean droplet temperatures. Subtle effects related to changes in
the boundary layer, not accounted for in a 1-D model, may also play a role in how
nucleation is terminated.
iv
Dedicated to Almighty God and my beloved parents
v
ACKNOWLEDGMENTS
This work would not have been possible with out the motivation, guidance and
challenges presented by my adviser Dr. Barbara Wyslouzil. I have learnt a lot from her
over the period of past few years that would go a long way in making me a better and
stronger person in my future endeavors. I would also like to thank our collaborator Dr.
Gerald Wilemski for his advice and suggestions on my modeling project. I wish to
express my gratitude to Dr. Igor Adamovich and Dr. Isamu Kusaka for being a part of my
dissertation committee and providing constructive criticism and insight on compressible
flow and nucleation theory.
I am grateful to Dr. Shinobu Tanimura for his important suggestions and advice
that helped me to learn carrying out my experiments meticulously. It was a privilege to
work with highly motivated colleagues like Mr. Hartawan Laksmono, Dr. David Ghosh,
Ms. Yun Wu, Ms. Kelley Distel, Mr. Dirk Bergmann, and Mr. Ashutosh Bhabhe who
provided a great work environment. I am especially thankful to Mr. Hartawan Laksmono
and Mr. Ashutosh Bhabhe for helping me carry out my experiments, even at odd hours
and providing valuable inputs. It was great to have discussions with Mr. Hartawan
laksmono and Dr. David Ghosh, while he visited our lab. Their insight and suggestions
were invaluable in helping me getting started during the initial stages of my research.
vi
I am grateful to Prof. Reinhard Strey, University of Koeln, for providing me the
opportunity to visit his lab in Germany. I thank all the group members in Prof. Strey’s lab
for sharing their research experiences and warm hospitality.
I would also like to thank First year Engineering Program at The Ohio State
University and particularly Dr. John Merrill for providing financial support and teaching
opportunity during the last few years of my graduate studies.
I express my gratitude towards Mr. Vikas Khanna, Mr. Thashvin Mukatira, Mr.
Ninad Nargundkar, Ms. Ruchika Srinivasan, Mr. Manish Talreja, Mr. Jay Visaria, and
my circle of friends who have never let me miss my family. I would also like to thank my
room-mates Mr. Vikas Khanna, Mr. Sharath Kumar, Mr. Thashvin Mukatira, Mr. Ninad
Nargundkar and Mr. Shunahshep Shukla for being so understanding and for making me
feel at home.
Finally, I would like to thank my parents, Mrs. Nupur Sinha and Dr. Mahamaya
Prasad Sinha for their unconditional love and faith in me. Without their support and
motivation, I could have never reached this far.
vii
VITA
June 8th, 1981. . . . . . . . . . . . . . . . . . . . . . ……….. Born – Asansol, India
June 2003. . . . . . . . . . . . . . . . . . . . . . …………… Bachelor of Chemical Engineering
University of Pune.
Pune, India
September 2003 – December 2006 . . . . . . . . . . . . Graduate Research Associate
Department of Chemical Engineering
The Ohio State University.
Columbus, Ohio
January 2007 – June 2008 . . . . . . . . . . . . ……… Graduate Research Associate
Department of Chemical Engineering
Graduate Teaching Associate
First year Engineering Program
The Ohio State University,
Columbus, Ohio.
March 2007……………………………………… Master of Applied Statistics
The Ohio State University.
Columbus, Ohio
March 2008……………………………………… Master of Science
(Chemical Engineering)
The Ohio State University.
Columbus, Ohio
June 2008 – present…… . . . . . . . . . . . . ………… Graduate Research Associate
Department of Chemical Engineering
The Ohio State University.
Columbus, Ohio
viii
FIELDS OF STUDY
Major Field: Chemical Engineering
Studies in: Nucleation and Condensation
Aerosol Science
Compressible Flow
ix
TABLE OF CONTENTS
Page
Abstract……………………………………………………………………………..
Dedication…………………………………………………………………………..
Acknowledgments………………………………………………………………….
Vita…………………………………………………………………………………
ii
v
vi
viii
List of Tables……………………………………………………………………….
xiii
List of Figures………………………………………………………………………
xiv
Chapters:
Introduction………………………………………………………………...
1
1.1. Overview…………………………………………………………………..
1.2. Motivation……………………………........................................................
1.3. Argon nucleation literature………………………......................................
1.4. Objectives and Research Contribution……………………………………
1.5. Outline…………………………………………………………………….
1
3
7
12
14
2. A cryogenic supersonic nozzle apparatus to study homogeneous nucleation
of Ar and other simple molecules………………………………………………
16
Abstract…………………………………………………………………………
2.1. Introduction………………………………………………………………..
2.2. Description of the apparatus………………………………………………
2.3. Pressure trace measurements and data analysis…………………………...
2.4. Sample Measurements…………………………………………………….
2.5. Conclusions and Outlook………………………………………………….
Acknowledgements…………………………………………………………….
References……………………………………………………………………...
17
18
21
28
31
35
36
37
3. Argon nucleation in a cryogenic supersonic nozzle apparatus………………...
40
Abstract………………………………………………………………………...
3.1. Introduction…………………………………………..…………………...
41
42
1.
x
3.2. Experimental……………………………………………………………...
3.2.1.
3.2.2.
3.2.3.
3.2.4.
Chemicals………………………………………………………….
The cryogenic supersonic nozzle…………………...……………...
Pressure Trace Measurements……….…………………………….
Characterizing Condensation and Nucleation……………….…….
3.3. Experimental Results…………………….……………………………….
45
45
47
##
48
##
50
The effective area ratio…………………………………………….
Condensing flow experiments……………………...……………...
Comparison with Literature Values……………………………….
Comparison with Classical nucleation theory….………………….
Estimating the critical nucleus size………………………………..
50
53
57
62
67
##
3.4. Conclusions……………………….………………………………………
References……………………………………………………………………..
72
75
3.3.1.
3.3.2.
3.3.3.
3.3.4.
3.3.5.
4.
45
Modeling of H2O/D2O Condensation in Supersonic Nozzles……………
Abstract………………………………………………………………………..
4.1. Introduction……………………..………………………………………...
4.2. Model Description………………………………………………………..
4.3. Choice of Nucleation Rate………………………………………………..
4.4. Choice of Growth Law……………………………………………………
4.5. Results and Discussions…………………………………………………..
4.6. Conclusions and Further work……………………………………………
Acknowledgements……………………………………………………………
Glossary of Symbols…………………………………………………………..
References……………………………………………………………………..
79
80
81
85
88
91
95
111
112
113
116
5. Conclusions and Future work……………..……………………………………
121
5.1. Conclusions…………………………………………….………………….
5.2. Future Work……………………………………………………………….
121
123
xi
Appendices
A. Appendix to Chapter 3 : Thermo-Physical Properties………………....
B. Appendix to Chapter 4 : Diabatic Flow Equations, Growth Law
Expressions and Thermo-Physical Properties………………………….
C. Previous Experiments with Argon in Nozzle without Cooling
Jacket…………………………………………………………………...
D. FORTRAN Code for Data Inversion…………………………………..
E. FORTRAN Code to predict Condensation in the nozzle…..…………..
F. FORTRAN Code to compare Growth rates and Droplet temperatures
predicted by the Non-isothermal growth laws………………………….
G. FORTRAN Code to compute Characteristic time of nucleation,∆tJmax,
and pJmax and TJmax………………………………………………………..
H. FORTRAN Code to compute pressure, p, for a given temperature
range and Constant nucleation rate, JBD, using The Classical
Nucleation Theory……………………………………………………..
I. Drawings of Various Components of Cryogenic Supersonic Nozzle
Apparatus………………………………………………………………
126
Bibliography………………………………………………………………………..
258
xii
128
137
144
163
209
226
233
240
LIST OF TABLES
Page
Table
3.1
The results for pressure trace experiments for Argon. y0 is initial
mole fraction of Ar. p0 and T0 are the stagnation pressure and
temperature , pon and Ton are the onset pressure and temperature,
and pJmax , TJmax , and SJmax are the pressure, temperature, and
supersaturation corresponding to the maximum nucleation rate
Jmax, respectively…………………………………….…………..
xiii
74
LIST OF FIGURES
Page
Figure
1.1
1.2
1.3
1.4
1.5
1.6
2.1
2.2
2.3
The range of nucleation rates for various experimental setups
(Iland et al. 2004) ………………………………………………..
4
Schematic representation of Cryo-Kinetic cleaning system (FSI
International)……………………………………………………...
5
Schematic Layout of Gas Cluster Ion Beam setup Network
(Epion)…………………………………………………….....…….
7
J vs. S plot for n-pentanol along with CNT prediction (Garibeh et
al. 2005)……………………………………………… …………..
8
(a) Wilson plot for n-pentanol The temperature and pressure
correspond to the location of peak nucleation rate (b) Wilson plot
for Argon ………..
(a) Hale plot for n-pentanol (Garibeh et al. 2005) (b) Hale plot for
Argon …………………………………………………………….
12
A schematic diagram of the cryogenic supersonic nozzle
apparatus that illustrates the flow of gases and liquid cryogens
through the experimental setup…………………………………..
21
A cross sectional view of the cryogenic supersonic nozzle
apparatus. Most components are to scale but some of the smaller
parts, for example the nozzle, static pressure probe, and fittings,
have been enlarged for clarity. The key components of the
experiment include (1) the insulated outer box, (2) the insulated
inner box or plenum, (3) the stagnation pressure tap, (4) the roller
bearing slides, (5) the LN2 cooled heat exchanger, (6) the static
mixer, (7) the LN2 cooled supersonic nozzle, (8) the pressure
probe, (9) the RTD, (10) the nozzle exit assembly, (11) the probe
gripper, (12) the linear translation stage, and (13) the turning
wheel………………………………………………………………
23
(left) An exploded view of the nozzle assembly. (right) A detailed
xiv
9
2.4
2.5
2.6
3.1
cross-sectional view (AA section plane) of the nozzle assembly…
The changes in area ratio as a function of position downstream of
the throat, x, (black solid lines) were determined from pressure
trace measurements starting from p0 = 36 kPa and the indicated
temperatures. The room temperature trace is well fit by a bi-linear
function for x between 0.5 and 6.4 cm with slopes of 0.098 cm-1
for x < 4 cm and 0.084 cm-1 for x > 4 cm. The region downstream
of ~6.4 cm also appears to be linear with a slightly lower slope.
At low temperature, the bi-linear fit is valid for x between 0.5 and
5.5 cm and the slopes are 0.118 cm-1 for x < 4 cm and 0.104 cm-1
for x > 4 cm. The area ratios differ drastically because the gas
density in the room temperature experiment is much lower than in
the T0 = 120 K experiment, and, thus, the boundary layers grow
more rapidly when T0 = 298 K. The deviation from the expected
straight line behavior in the low temperature experiment is due to
condensation………………………………………………………
31
The measured pressure ratio profiles as a function the stagnation
conditions. The grey solid line corresponds to the measurement at
room temperature, while the black solid line represents the
expected isentrope for the low temperature experiments. The
broken lines correspond to the experiments carried out at lower
temperatures. For the latter, the rapid increase in pressure above
the isentropic expansion is due to heat addition to the flow by the
growing droplets and is the signature of condensation……………
32
The onset pressures and temperatures measured in the current
cryogenic supersonic nozzle are compared to the supersonic
nozzle measurements of Stein and Wu et al., as well as the NPC
measurements of Iland et al. and Fladerer and Strey……………..
34
A schematic diagram of the cryogenic supersonic nozzle
apparatus. Ar and He are the condensible and carrier gases,
respectively, while independent liquid N2 streams serve as the
working fluid in the heat exchanger and to cool the side walls of
the nozzle, respectively. Condensation is detected by measuring
the static pressure as a function of position. (Sinha et al. 2008a)…
47
xv
26
3.2
3.3
3.4
3.5
The change in area ratio as a function of position downstream of
the throat, x, (black solid line) was determined from pressure trace
measurement starting from p0 = 36 kPa and T0 = 120 K using a bilinear fit valid for x between 0.5 and 5.5 cm with slopes 0.118 cm1
for x < 4 cm and 0.104 cm-1 for x > 4 cm. The straight solid line
indicates the area ratio from a room temperature experiment with
p0 = 90 kPa. The area ratio profile corresponding to the fit and the
room temperature experiment agree closely. ……………………..
52
(a) The measured pressure ratio profiles as a function of the
stagnation conditions. The black solid line represents the expected
isentrope from scaled room temperature experiment. The broken
lines correspond to the experiments carried out at lower
temperatures. For the latter, the rapid increase in pressure above
the isentropic expansion is due to heat addition to the flow by the
growing droplets and is the signature of condensation. (b)
Pressure ratio profiles from experiments in the absence of cooled
nozzle sidewalls. The solid line represents the room temperature
isentrope. The dashed line represents the low temperature
experiment with pure Ar and deviated gently from the isentrope.
The low temperature experiment with Ar-He mixture is
represented by the dash-dotted line………………..………………
54
The pressures and temperatures corresponding to the maximum
nucleation rate. The black diamonds correspond to experiments
with He as the carrier gas while grey diamonds correspond to pure
Ar experiments. The fit function given by Eq. (12) goes through
most of the data points. The solid line below and to the right of
the data is the extrapolated gas/liquid equilibrium line, and the
dashed line is the gas/solid binodal………………………………..
56
Phase diagram of argon along with the onset conditions of argon
nucleation measured in various devices. The solid line represents
gas/liquid bimodal, while the dashed line corresponds to the
gas/solid binodal. The onset conditions are shown by the various
symbols; this work (gray diamonds), Matthew and Steinwandel
(down triangles), Zahoransky et al. (data from 1995: squares, data
from 1999: empty diamonds), Wu et al. (hexagons), Stein (gray
triangles), Fladerer and Strey (empty circles) and Iland et al.
xvi
3.6
3.7
3.8
3.9
(filled circles)……………………………………………………...
A Hale plot for argon shows the supersonic nozzle data measured
in this work (diamonds), by Stein (triangles) , and by Wu et al.
(hexagons) as well as the NPC data of Fladerer and Strey (empty
circles) and Iland et al. (filled circles). The dashed line at 45°
corresponds to perfect agreement. The error bars on the data
represent the uncertainty in estimates of the nucleation rates, 2
orders of magnitude for the nucleation pulse chamber data and
one order of magnitude for the measurements in supersonic
nozzles. The arrows indicate that the data from Fladerer and Strey
extend out to scaled supersaturations of 40, while the Stein data
extend out to 48. The measurements from this work are in good
agreement with the nucleation pulse chamber data of Iland et al.
and the supersonic nozzle data of Wu et al………………………..
61
The data from this work (diamonds) and Iland et al. (filled circles)
along with the gas-liquid binodal (solid line) are plotted as a
function of inverse temperature. The predictions of the CNT for
constant nucleation rates JCNT of 10-20, 10-10and 100 cm-3 s-1 are
shown by the dashed lines. The region shown by the empty circles
represents the JCNT ranging from 105-109 cm-3 s-1 corresponding to
the measuring window of the nucleation pulse chamber
experiments by Iland et al.; while the dotted region represents
JCNT ranging from 1016-1018 cm-3 s-1corresponding to the
measuring window of our experiments……………………………
63
The ratio of the estimated nucleation rates for our experiments,
Jexp of 10-17 cm-3s-1and the rate predictions of CNT, JCNT decreases
slightly temperature decreases (gray diamonds). The error bars
correspond to the range of the nucleation rates typically observed
in nozzles-1017±1 cm-3s-1. The solid line represents the fit to
Jexp/JBD…………………………………………………………….
65
Onset data of argon from this work (diamonds) and Iland et al.
(circles)along with lines of constant critical cluster size for n*GT =
10 (uppermost line), 20, 40, 60, 80, 100, 120(lowermost line)
predicted by the Gibbs-Thompson equation. The dashed line
represents the gas-liquid equilibrium line…..……………………..
68
xvii
59
3.10
3.11
4.1
4.2
4.3
Data from this work (diamonds) and Iland et al. (circles) are
shown in J vs. S plot at six temperatures (42 -52 K). The vertical
error bars represent the error in the estimated nucleation rates in
respective devices. The dashed lines represent the predictions of
the classical nucleation rate at constant temperature, when
multiplied with the temperature dependent correction factor
(Eq.17). The solid lines represent the straight line isotherms
connecting the data from nozzle and nucleation pulse chamber
experiments.…………………….………………………………….
69
Values of n* derived using Eq. 19, plotted against the predicted
critical cluster size, n*GT from the Gibbs-Thomson equation. The
vertical error bars correspond to the uncertainty in the estimated
nucleation rates in supersonic nozzles and nucleation pulse
chambers. The horizontal lines represent the uncertainty in the
supersaturations corresponding to the straight-line isotherms in
Fig.. The left end of the horizontal dotted line represents the n*GT
for nozzle and the right end represents n*GT for the experiments in
nucleation pulse chambers. The solid line corresponds to perfect
agreement. The dashed –dotted and the dashed line represent the 30% and the -55% error lines respectively………………………...
71
The D2O nucleation rates measured in supersonic nozzles are
compared to the predictions of Hale’s scaled nucleation theory
Eq. 11 (solid lines) and the Wölk-Strey expressions Eq. 10
(dashed lines). The ratio Jexp/JHALE ranges from 0.2 at low
temperature to ~8 at high temperatures……………………………..
91
The predicted differences between the droplet temperatures Td and
the gas temperature T∞ for the nonisothermal (dotted) and HKS
(dashed) growth laws are shown as a function droplet size. The
gas phase conditions are typical of those found in the nucleation
zone. The solid line is the difference between the two droplet
temperatures...………………………………………………………………..
96
The temperature difference between the droplets and the
surrounding gas is shown as a function of the droplet radius when
the HKS growth law is used in the model to predict condensation
xviii
4.4
4.5
4.6
4.7
in nozzle H2 with a condensible flow rate of 5.0 g min-1 H2O. The
circles denote the mean droplet size at each location…………...…...
97
The mass fraction of condensate in the flow was determined by
following the depletion of D2O from the gas phase using TDLAS
(Tanimura et al.2005). In this experiment, the mass flow rate of
D2O was 4.155 g min-1 corresponding to an initial mass fraction
of 0.018. The partial pressure of D2O at the inlet to nozzle H is
denoted by pc0………………………………………………………...
99
Pressure ratios as a function of nozzle position are shown for D2O
mass flow rate of (a) 11.2g min-1 and (b) 2.2 g min-1 .The gray
dash-dotted line represents the pressure ratio for the carrier gas
flow in the nozzle, while the gray solid line represents the
pressure ratio for the condensing flow from pressure trace
measurements. The pressure ratios for the condensing flow
predicted by the various models are shown by the different black
lines. The partial pressure of D2O at the inlet to nozzle C2 is
denoted by pc0. The Hertz-Knudsen growth model calculations are
not shown because the results overlap those of the Isothermal
growth model……………………....................................................
101
The experimental values of the aerosol (a) mean radius (b)
number density and (c) polydispersity are compared to the model
predictions as a function of the mass flow rate of D2O (nozzle
C2). All five growth rate expressions are considered. The ±20%
error bars on the number density reflect the uncertainty in the
(multiplicative) calibration factor required to place the SAXS data
on an absolute intensity scale. We note that the uncertainty in the
calibration factor affects all of the data equally, i.e., any change in
the calibration factor would shift the values of N by the same
multiplicative factor……………………………………………….
103
The experimental values of the aerosol (a) mean radius and (b)
number density in nozzle H2 and (c) mean radius and (d) number
density in nozzle C2 are compared to the model predictions as a
function of the flow rate of H2O entering the nozzle. As discussed
in Fig. 4, the large error bars on N are due to uncertainty in the
calibration factor used to place the SAXS spectra on an absolute
xix
intensity scale. …………………………………………………….
The experimental values of the aerosol (a) mean radius and (b)
number density at a flow rate of 5.0 g min-1 H2O in nozzle H2 are
compared to the model predictions as a function of nozzle
position. The location of the experimentally determined peak
nucleation rate is indicated by the arrow in (a). The model
calculations are indicated by the lines and the lines start from the
position corresponding to the maximum nucleation rate, Jmax ,
predicted by the model. The modified HKS GL, discussed in more
detail in the text, predicts the mean droplet radius and number
density when the temperature difference between the droplet and
the surrounding gas estimated by the HKS growth law is
arbitrarily reduced by 50%. The large error bars on N are due to
uncertainty in the calibration factor used to place the SAXS
spectra on an absolute intensity scale………………………….….
107
A.1
Pressure profiles and height along the nozzle…………………….
139
A.2
Wilson plot for Ar. The area to the right of and below the long
dashed lines corresponds to the accessible starting conditions for
our planned experiments……………………………………….….
140
I.1
Sketch of the front plate of the Plenum…………………………...
241
I.2
Sketch of the back plate of the Plenum…………………………...
242
I.3
Sketch of the bottom plate of the Plenum………………………...
243
I.4
Sketch of the right plate of the Plenum…………………………...
244
I.5
Sketch of the left plate of the Plenum…………………………….
245
I.6
Sketch of the lip of the Plenum……………………………………
246
I.7
Sketch of Nozzle S………………………………………………..
247
I.8
Sketch of flange for the Tee joint…………………………………
248
I.10
Sketch of Outer box rear plate…………………………………….
250
4.8
xx
109
I.11
Sketch of Outer box base plate……………………………………
251
I.12
Sketch of Outer box right-side plate………………………………
252
I.13
Sketch of Outer box left-side plate……………………………….
253
I.14
Sketch of Outer box lip…………………………………………...
254
I.15
Outer box assembly………………………………………………
255
I.16
Instrument flange for Outer box………………………………….
256
I.17
Retaining ring at Gas Inlet……………………………………….
257
xxi
CHAPTER 1
INTRODUCTION
1.1
Overview
Processes like condensation, crystallization, melting and evaporation are
examples of first order phase transition. These processes are of fundamental importance
in many fields such as atmospheric physics, astrophysics, nuclear reactor technology,
material science and aeronautical sciences. They must be considered in aerosol
formation, weather prediction, metallurgical techniques, high speed wind tunnel design,
steam- turbine nozzle design etc. The three stages of any phase transformation are
nucleation, growth and aging. Of these, nucleation is the least well understood and often
the most difficult to predict.
Nucleation is the mechanism underlying the formation of fragments of a new
phase within a metastable mother phase. In general, the transformation of a substance
from one phase to another is initiated by a change in the state of the system including, for
example, an increase in partial pressure or a decrease in temperature. If the new state is in
a different region of the phase diagram than the initial state, and the changes occurred
rapidly enough, the system is initially metastable. For vapor to liquid phase
transformations, for instance, the vapor can be brought to a metastable state by increasing
the partial pressure or decreasing the temperature of the system. Here, the ratio of the
1
partial pressure of the vapor to the equilibrium vapor pressure at the same temperature –
the supersaturation – is a measure of the degree of metastability. Once the vapor is
supersaturated, its free energy is higher than the corresponding liquid phase. Since every
system tries to minimize its Gibbs free energy, a metastable vapor will try to transform to
the liquid phase by condensing on an available surface (heterogeneous nucleation) or by
the forming a liquid cluster (homogeneous nucleation). Often the cluster is assumed to
be a spherical droplet with the same properties as the bulk material, and this assumption
is known as the capillarity approximation.
In homogeneous nucleation, the formation of the new phase is not spontaneous
because there is a barrier associated with the formation of the clusters. The net Gibbs free
energy change, or cost of forming a cluster, ∆G , is the summation of two counteracting
terms; the decrease of the bulk free energy due to the transition to a state with lower free
energy and the increase in free energy due to formation of a surface between the cluster
and the mother phase. When the cluster is small, the surface term overrides the bulk term
resulting in an energy barrier for transformation into the new phase. Eventually, however,
∆G reaches a maximum denoted by ∆G*, and the corresponding cluster is called the
critical cluster. Once a cluster crosses the barrier, ∆G decreases monotonically and the
clusters grow spontaneously to form the new liquid phase. The population of the clusters
is characterized by a Boltzmann like distribution with the cluster concentration
proportional to exp (-∆G/kT). As the supersaturation of the vapor is increased, both the
size of the critical cluster r* and the barrier height ∆G* are reduced, thereby increasing
the probability that density fluctuations will send some clusters over the activation barrier
in a given time. This is the thermodynamic aspect of nucleation (McDonald 1962).
2
The kinetic mechanism for cluster formation is based on Szilard’s analogy to gas
phase reaction kinetics. According to this model cluster growth occurs by monomer
addition and removal. The net rate, Jg at which a cluster consisting of g-1 molecules
becomes a cluster consisting of g molecules, is the difference between the rates at which
a g-mer is formed due to addition of a monomer to a (g-1)-mer and the rate of loss of gmers by evaporation of single molecules from a (g-1)-mer.
In homogeneous nucleation, the pseudo-steady state assumption states that the
rate of formation of the g-mers is equal to the rate of formation of g+1- mers and so on.
Hence J becomes independent of g. Classical Nucleation Theory (CNT) used the
assumptions outlined above, and mathematical manipulations to develop a predictive
formula for the nucleation rate (McDonald 1963).
1.2
Motivation
Vapor – liquid nucleation has been investigated extensively using a wide variety
of apparatuses. Figure 1.1 summarizes the devices currently available for generating
nucleation rate measurements. They include thermal diffusion cloud chambers, laminar
flow tube reactors, expansion cloud chambers, laminar flow reactors, expansion cloud
chambers, shock tubes, nucleation pulse chambers and supersonic nozzles. Together
these setups cover more than 20 orders of magnitude, starting from nucleation rates J=103
cm-3s-1, measured by thermal diffusion cloud chamber, up to J=1018 cm-3s-1 reachable
with supersonic nozzles.
3
Figure.1.1: The range of nucleation rates for various experimental setups
(Iland et al. 2004).
In general, experimentalists have worked with rather complex materials, like
water, alcohols, and alkanes, and found that CNT predicts the supersaturation dependence
of the measured rates reasonably well but the temperature dependence is too strong.
Theorists have tried to improve nucleation rate predictions by approaching the problem
from a molecular viewpoint. In this case, calculations most often focus on noble gases
like Argon due to their simple intermolecular potentials. The data required to distinguish
between competing theories is currently very limited. Furthermore, many of the available
data sets are inconsistent with each other or even inconsistent with homogeneous
nucleation. Ar nucleation experiments carried out in supersonic nozzles could help
4
resolve this inconsistency. Experimental data for Ar in supersonic nozzles is extremely
important because they occur at relatively high vapor densities and supersaturations, and,
as illustrated in Figure 1.1, the nucleation rates are ~ 10 orders of magnitude higher than
those that can be reached in other devices. Maximizing vapor densities and nucleation
rates is critical for bridging the gap between experiment and molecular theories because
these conditions will reduce the computational burden inherent in molecular simulation
of nucleation.
In addition to its importance to developing better nucleation theories, Ar
nucleation plays a critical role in a number of industrial applications. One of the
prominent applications is in the cleaning of semiconductor chips. FSI International
introduced the argon/nitrogen aerosol cleaning process in its Cryo-Kinetic cleaning
system. This method is a non-chemical, non-damaging process that cleans the wafer
surface with thousands of microscopic argon/nitrogen crystals.
Figure 1.2: Schematic representation of Cryo-Kinetic cleaning system (FSI International)
Figure 1.2 is a schematic representation of the system. A mixture of argon and
nitrogen gas is pre-cooled to form a liquid/gas mixture. This mixture flows into a tube
5
and is injected through tiny holes into the vacuum chamber. The flow is directed toward
the wafer surface. As the liquid/gas mixture is injected into the vacuum chamber, the
liquid portion expands and breaks up into small fragments. The small fragments undergo
evaporative cooling, which causes them to freeze into solid crystals. Contaminants are
dislodged from the wafer surface mainly through momentum transfer from the ice crystal
to the contaminant particle (FSI International).
Another industrial application of Argon nucleation is in infusion technology; the
proprietary method employed by Epion to drastically change, in a variety of ways, the top
few layers of atoms of a surface, while leaving the underlying bulk of the material
completely unaffected. The technology has many applications within the semiconductor
industry, such as ultra-shallow doping, amorphous SiGe deposition, etching and pore
sealing of low dielectric materials.
The infusion process utilizes a Gas Cluster Ion Beam (GCIB) source to deliver
highly energetic clusters of weakly-bound atoms. The layout of the system is shown in
Figure 1.3. Clusters of atoms are formed by the condensation of individual gas atoms (or
molecules) during the adiabatic expansion of high pressure gas from a nozzle into a
vacuum. Ar is one of the gases used to produce the beam of clusters. After a cluster beam
of neutral atoms is formed, the clusters are ionized by bombardment with energetic
electrons, accelerated by a high voltage potential, and shaped through beam optics to
form an ion beam (Epion). A consistent set of onset data on Argon nucleation will help to
better understand and improve these industrial applications in future.
6
Figure 1.3: Schematic Layout of Gas Cluster Ion Beam setup (Epion).
If condensation in nozzles can be accurately modeled, then such models could be
used to predict operating conditions for future experiments or to optimize industrial
processes. In devices such as supersonic nozzles, nucleation and growth are strongly
coupled because it is the heat release and vapor depletion by growth that quench
nucleation. Accurate nucleation rate expression and growth laws are, therefore, critical. If
a reliable nucleation rate expression is available, then modeling condensation in nozzles
enables us to test growth laws under conditions far from equilibrium and where nonisothermal growth should be important.
1.3
Argon nucleation literature
Only one substance, n-pentanol, has had nucleation rates measured at one
temperature using enough nucleation devices to cover the entire accessible range. The
results of this co-operative experimental effort are illustrated in Figure 1.4. It is amazing
how well the nucleation rate data line up in the J (Nucleation rate) vs. S (super saturation)
plot. The experimental data line up well with a line corresponding to 500XCNT (Classical
7
Nucleation Theory line), where the constant accounts for the incorrect temperature
dependence predicted by the Classical Nucleation Theory.
1018
1015
1012
J / cm-3 s-1
T= 260 K
Hruby et al.
Rudek et al.
Luijten et al.
Gharibeh et al.
Lihavainen et al.
Strey et al.
500xCNT
CNT
109
106
103
100
10-3
1
10
100
Supersaturation (S)
Figure 1.4: J vs. S plot for n-pentanol along with CNT prediction (Gharibeh et al. 2005).
Such plots are not available for Ar in part because working at the low
temperatures required to condense Ar from the vapor phase is extremely challenging. In
fact a true nucleation rate has yet to be measured for Argon. In the absence of true rate
measurements, experimental data can be compared using either a Wilson plot (p vs. T
diagram) or a Hale plot.
8
105
10.000
Gharibeh et al.
Strey et al.
Luijten et al.
Strey at al.
Hruby et al.
Rudek chamber I
Rudek Chamber II
Lihavainen
Vapor Liquid Equilibrium
104
p / Pa
pJmax / kPa
1.000
0.100
Fladerer (2002)
Matthew & Steinwandel
Zahoransky et al. (1995)
Wu et al.
Stein
Zahoransky et al. (1999)
Iland (2004)
103
102
0.010
0.001
200
220
240
260
280
300
101
20
320
TJmax / K
30
40
50
60
70
80
90
T/K
Figure 1.5: (a) Wilson plot for n-pentanol. The temperature and pressure correspond to
the location of peak nucleation rate (b) Wilson plot for Argon
Figure 1.5 (a) is the Wilson plot for n-pentanol. From the plot we can see that the
experimental data are consistent and line up well with each other. The onset points from
the data obtained by Gharibeh et.al, 2005 (represented by squares in Figure 1.5 (a)) lie at
a lower temperature for the same pressure when compared to other experimental setups.
This is expected because the data was collected using a supersonic nozzle setup which
gives higher nucleation rates than the other experimental methods used to obtain
nucleation data.
The inconsistency in the available Argon data is evident in the Wilson plot
illustrated in Figure 1.5 (b). Here, the dashed line corresponds to the vapor-solid
equilibrium, the solid line corresponds to the extrapolated vapor-liquid equilibrium line,
and the onset points for Argon determined by various groups are shown by symbols. Of
all the data, the most plausible data are those obtained by Iland et al. (2007) in a
9
nucleation pulse chamber, the grey circles. The data set has the correct trend when
compared to the binodal and has less scatter.
The unfilled circles in Figure 1.5(b) represent the data collected by Fladerer & Strey
(2006) using nucleation pulse chamber and they agree very well to Iland et. al.data for
higher temperature but drift away for lower temperature. This is because the expansion
chamber used by Fladerer & Stery was smaller than the one used by Iland et al. and the
low temperature experiments were restricted by the depth of the adiabatic expansion.
Thus the lower temperature Fladerer & Strey data were at lower expansion rates that
correspond to lower nucleation rates.
The onset points at lower temperature in Fladerer et al. data set were close to the
lower limit of the temperature range the apparatus could reach and therefore are less
accurate.
The hexagonal markers in Figure 1.5(b) represent the data from the study
conducted by Wu et. al.(1978), where supersonic nozzle was used to condense Ar in
Helium. The data has some scatter but is in agreement with Iland et al. data because for
the same onset pressure, the onset points from Wu et. al. data lies at a lower temperature
than ones from Iland et al. data set. This is because higher nucleation rates and higher
super saturations are achieved in supersonic nozzles when compared to nucleation pulse
chambers. The Wu et.al. data set, however, consists of onset points with different onset
pressures for the same onset temperatures resulting in onsets occurring at
supersaturations that differ by about 2-3 orders of magnitude for the same temperature.
Stein (1974) examined nucleation of Argon in a faster nozzle than Wu et al. and
determined the onset points using light scattering. If we compare the onset points of Stein
10
(shown in Figure 1.5 (b) by grey upward triangles) and Iland et al., we see that for the
same onset pressure Stein’s data has a higher onset temperature. This is not correct, since
higher super saturations are achieved in supersonic nozzles than in nucleation pulse
chambers.
The data from Matthew and Steinwandel’s experiments (Matthew & Steinwandel
1983) (shown in Figure 1.5 (b) by inverted triangles) using cryogenic shock tube to
carryout Ar nucleation lie very close to the binodal and, hence, we suspect heterogeneous
nucleation taking place in these experiments instead of homogeneous nucleation.
The data points from Zahoransky et al. (1995) work are represented in the Wilson
plot by squares and look fine for higher onset temperatures but for lower temperatures the
onset points drift towards the binodal. Since Zahoransky et al. used cryogenic shock
tubes for nucleation which give about the same nucleation rates as those in case of
nucleation pulse chambers, the data at lower temperatures should be close to Iland’s data
set. The onset points from Zahoransky et al. (1999) are also scattered all over the Wilson
plot for Argon.
Figure 1.6 (a) and (b) represent the Hale plot for n-pentanol and Argon
respectively. C0 is the association parameter and Ω is the surface entropy per molecule.
Hale’s scaling formalism (Hale 1992) provides a basis to compare experimental
nucleation rates of any magnitude measured at arbitrary temperature and pressure. A
consistent set of nucleation data should lie along a straight line in the Hale plot. From
Fig.2.4 we see that almost all the n-pentanol data lie along a straight line in the Hale plot,
while in case of Argon Hale plot (Fig.2.5), except for Iland’s and Wu et al. data, all other
points are off the straight line.
11
30
30
Hruby et al.
Strey et al.
Luijten et al.
Gharibeh et al.
Rudek Chamber I
Rudek chamber II
Strey et al.
Lihavainen
-log10(J/1026)
20
25
20
-log10(J/1026)
25
Fladerer (2002)
Zahoransky et al. (1999)
Zahoransky et al. (1995)
Matthew & Steinwandel
Wu et al.
Stein
Iland (2004)
15
10
15
10
n-pentanol
C0= 50, Ω= 1.9
5
5
0
0
0
5
10
15
20
3
C0[Tc/T-1] /(ln S)
25
30
2
0
20
40
60
3
C0[Tc/T-1] /(ln S)
80
100
2
Figure 1.6: (a) Hale plot for n-pentanol (Gharibeh et al. 2005) (b) Hale plot for Argon
Thus it is critical to generate more consistent experimental data for Ar so that it can be
used to compare with molecular simulations that are based on simple intermolecular
potentials.
1.4
Objectives and Research Contribution
As discussed above, it is very important to have more accurate and
consistent data for onset of argon. One of the goals of this research was to provide
accurate experimental data, directly useful to test molecular level theories of nucleation.
The phase transition considered in this work is condensation of argon. Supersonic nozzles
are used to conduct these experiments because they provide higher cooling rates, higher
12
super saturations and much higher nucleation rates than all the techniques discussed. The
critical clusters are very small – on the order of about 5-10 molecules for water, and,
thus, the nucleation kinetics is sensitive to the formation of the smallest cluster (Kim et
al. 2004). The results of this work are expected to resolve the inconsistency between the
nucleation pulse chamber data and the supersonic nozzles. If the data from the two
devices agree, then our data would complement the expansion cloud chamber data for
argon, generated by our collaborator Dr.Reinhard Strey at the Institut für Physikalische
Chemie in Cologne, Germany.
Ar condensation experiments in nozzles are extremely challenging due to the low
temperatures associated with the experiments. Heat exchange and leaks at cryogenic
conditions make these experiments difficult. The first task of this project was to build a
cryogenic nozzle apparatus, and to carry out Ar nucleation experiments either with pure
Ar or in presence of He carrier gas. Although, the set up was built with Ar experiments in
view, it can also be used for other simple molecules such as N2. Our experiments with Ar
have yielded onset results that are consistent with Nucleation Pulse Chamber (NPC) data
and results from the two devices have been used to evaluate the theoretical predictions
from CNT.
A second aim of this work was to modify and use a 1-D model to predict
condensation in supersonic nozzles so that future experiments could be better planned for
experiments that are both expensive and difficult. However, every model needs to be
tested by comparing with experiments before we can use them. In order to put our model
into stringent tests, we tested our model against experimental measurements for
(H2O/D2O) condensation conducted in our research group that include axially resolved
13
pressure measurements, axially resolved gas phase composition measurements, and a
series of small angle x-ray scattering experiments (Tanimura et al. 2005; Wyslouzil et al.
2007).
Such a model would also enable the testing of various growth theories by
comparing the model results with small angle x-ray scattering experiments for H2O/D2O
condensation in nozzles.
1.5
Outline
The thesis explores two main areas: experimental investigation of Ar in a
supersonic nozzle and modeling condensation of H2O/D2O in a series of supersonic
nozzles. Chapters 2 and 3 describe the details of Argon nucleation experiments. Chapter
2 details the construction aspects of our cryogenic supersonic nozzle apparatus and
reports on some preliminary experiments on Ar nucleation. The importance of
characterizing the expansion for experiments at lower temperatures is emphasized along
with the need to keep the nozzle side walls close to the stagnation temperatures. In
Chapter 3, the results of all our Argon nucleation experiments, ranging from pure argon
to Ar - He mixtures, is presented. The results of our experiments are compared with the
data in the literature and are found to be consistent with the measurements in nucleation
pulse chambers. The results are compared with CNT and the size of the critical clusters is
estimated using the first nucleation theorem and the Gibbs-Thomson equation. The data
from this work can be used in conjunction with the nucleation pulse chamber in
stringently test and distinguish between competing nucleation theories and simulations.
Chapter 4 of this thesis presents the results of our 1-D model for H2O and D2O
and compares the model results with experimental results from small angle x-ray
14
scattering
(SAXS)
experiments.
Experiments
enable
us
to
measure
aerosol
characteristics, in particular the average radius 〈r〉 and number density, N, of the droplets
formed during condensation. The modeling results are compared with the experiments
based on these aerosol characteristics. Five different growth laws were used in the model
along with the Hale’s scaled nucleation model. Since the predictions of the nucleation
model is in good agreement with the nucleation rate measurements in nozzles, the growth
laws could be tested by comparing results of our model and experiments. The results are
discussed in detail in Chapter 4.
In Chapter 5 the conclusions of the work presented in this thesis are summarized
along with some proposed future work.
15
CHAPTER 2
A cryogenic supersonic nozzle apparatus to study homogeneous
nucleation of Ar and other simple molecules
Somnath Sinha, Hartawan Laksmono and Barbara E. Wyslouzil
Department of Chemical and Biomolecular Engineering,
The Ohio State University,
Columbus, OH-43210
16
Abstract
We present a supersonic nozzle apparatus to study homogeneous nucleation of argon and
other simple molecules. Experiments can be conducted with pure condensible gas or with
condensible – carrier gas mixtures. The flow through the nozzle is continuous, and
expansions typically start at temperatures T0 in the range of 100 < T0 /K < 120, and
pressures p0 in the range of 30 < p0 / kPa < 36. The gas mixture is cooled using a tube and
fin heat exchanger with evaporating liquid nitrogen on the tube side. The nozzle side
walls are also cooled with liquid nitrogen to maintain them at a temperature ~20 K higher
than the stagnation temperature of the condensing vapor. Static pressure measurements
detect the onset of condensation, and the other properties of the flow are derived by
integrating the diabatic flow equations. We present sample experimental results for pure
argon where at the onset of condensation temperatures range from 47.5 < Ton /K < 49.5
and pressures range from 4.2 < p0 / kPa < 4.9 kPa.
17
2.1
Introduction
Nucleation, the initial step in first order phase transitions, is the process by which
stable fragments of the new phase form within the metastable mother phase. Since the
development of Classical Nucleation Theory (CNT) by Becker and Döring in 1935, there
have been many attempts to describe nucleation theoretically. To date, however, there is
still no theoretical model that can quantitatively describe the nucleation rates of all
substances over the full range of supersaturation and temperature of interest. Despite the
basic conceptual limitations, CNT and its variants are still widely used to predict
nucleation rates.
More recently, a numbers of researchers have focused on improving nucleation
rate theories by approaching the problem from a molecular viewpoint using Density
Functional Theory (DFT) (Evans 1979; Zeng & Oxtoby 1991; Talanquer & Oxtoby 1994;
Granasy et al. 2000), Monte Carlo simulations (Garcia Garcia & Soler Torroja 1981;
Weakliem & Reiss 1993; Senger et al. 1999; Kusaka 2003), and molecular dynamics
simulations (ten Wolde 1998; Yasuoka & Matsumoto 1998; Laasonen et al. 2000;
Toxvaerd 2001; Kraska 2006). In these studies the Lennard-Jones potential is used to
describe the intermolecular potentials because this choice makes the simulations more
reasonable to implement. Unfortunately, the Lennard-Jones potential only adequately
describes the molecular interactions for a relatively restricted class of compounds, for
example, simple molecules like the noble gases. In contrast, most of the experimental
data (Heist and He 1994) have been generated for species including water, alcohols, or
alkanes, whose behavior can only be described by more complex intermolecular
potentials. In order to validate these molecular theories and simulations, reliable
18
experimental data on simple molecules, in particular the noble gas Ar, is essential.
Furthermore, to bridge the gap between theory and experiments more readily,
experiments should be conducted at the highest possible vapor densities, supersaturations,
and nucleation rates.
Experimental investigation of Ar nucleation is challenging. The existing
experimental devices, including shock tubes (Matthew & Steinwandel 1983; Zahoransky
et al. 1995, 1999), the nucleation pulse chamber (Fladerer & Strey 2006; Iland et al.
2007), and supersonic nozzles (Stein 1974; Wu et al. 1978), are all based on cooling the
gas via an adiabatic expansion. The rate of cooling determines the degree to which the
vapor can be supersaturated before spontaneous formation of the liquid occurs. Of these
three devices, the supersonic nozzle has the highest cooling rates and nucleation rates
~1017±1 cm-3s-1. The cooling rates in the shock tube and the nucleation pulse chamber are
comparable and the nucleation rates are ~107±2 cm-3s-1. To reach temperatures low
enough to observe condensation, the expansions must all start at or near the boiling point
of liquid nitrogen, and, under these conditions preventing leaks, eliminating
contamination, and controlling heat transfer become extremely challenging. Heat transfer
is especially difficult to control for supersonic nozzles that must maintain a steady and
high flow rate of cold gas.
Given these challenges it is not surprising that, with the exception of the recent
NPC measurements of Iland et al. and Fladerer and Strey, the experimental data reported
in the literature are frequently scattered and inconsistent. Much of the data generated in
shock tubes (Matthew & Steinwandel 1983; Zahoransky et al. 1995, 1999), for example,
lies very close to the supercooled liquid – vapor binodal, suggesting that heterogeneous
19
nucleation dominated the phase transition. Only at higher temperatures do the shock tube
data give the expected agreement with the nucleation pulse chamber. The limited
supersonic nozzle data, in turn, are either very scattered (Wu et al. 1978) or lie at lower
supersaturations than the NPC data (Stein 1974) and are, therefore, inconsistent with
them.
Further analysis of the supersonic nozzle experiments suggests several potential
problems with the apparatus used by Stein and coworkers (Stein 1974; Wu et al. 1978).
These experiments used an increase in the light scattering at the exit of the nozzle to
detect condensation. To calculate the corresponding pressure and temperature at the
nozzle exit requires the effective shape of the expansion. Stein used the physical
geometry to calculate the conditions at the exit. Wu et al. used the effective shape of the
expansion based on a pressure trace measurement made with Ar at a stagnation
temperature T0 = 295 and stagnation pressure p0 = 101 kPa (Wu et al. 1978). Given that
the effective expansion is a function of the initial gas density and composition, the
assumptions made by Stein and Wu et al. may be difficult to justify. Next, the nozzles
were built of plastic, yet to prevent the addition or removal of the energy from the flow,
the temperature of the nozzle sidewalls should be close to the stagnation temperature.
Although Stein reported that the nozzles did not break, it is possible that they deformed
and changed the expansion as they cooled. Alternatively, if the wall temperature was
significantly different than T0, heat transfer could have modified the flow. The nozzles
used by Stein and Wu et al. had pressure taps near the nozzle exit but, since the values of
the exit pressure are not reported, it is not possible to determine whether the assumptions
regarding the expansion are accurate or not.
20
In this chapter, we present a new cryogenic supersonic nozzle apparatus to
investigate nucleation of noble gases, or other simple molecules that only condense at
very low temperatures. Pressure measurements are used to follow the condensation
process directly, and let us account for changes in the effective expansion as the inlet
conditions vary. The nozzle sidewalls are actively cooled to ensure that heat is neither
added nor removed from the supersonic flow. Finally, preliminary experimental data for
pure Ar condensation are presented and compared to the data available in the literature.
2.2
Description of the apparatus
P - Pressure
T - Temperature
exhaust
Liquid
Nitrogen
vacuum
pumps
P
Mass flow
controller
Heater
Nozzle
Regulator
Argon
Heater
pressure probe
Gas
Mass flow
controller
Heat Exchanger
P
Plenum
Regulator
Outer box
He
T
T
P
Liquid
Nitrogen
Figure 2.1: A schematic diagram of the cryogenic supersonic nozzle apparatus that
illustrates the flow of gases and liquid cryogens through the experimental setup.
21
Figure 2.1 is a schematic diagram of the overall experimental setup that illustrates
the general layout and the flow of condensible vapor, carrier gas, and cryogenic liquid
through the system. After describing the flow of material through the system, we will
discuss the details of the components critical to the success of the experiments.
In the case of Ar condensation, Ar is drawn from the gas side of a high pressure
(2.4 MPa) liquid argon Dewar and warmed to room temperature by an inline heater
(McIlrath, 1000-A). The pressure is controlled by a pressure regulator (Concoa, 2062001)
and the flow by a 300 SLM mass flow controller (MKS Instruments, Type 1559A)
calibrated for Ar. If He is used as a carrier gas, it is supplied by a high pressure (13.8
MPa) gas cylinder and also flows through a heater, regulator, and a 400 SLM mass flow
controller (MKS Instruments, Type 1559A) calibrated for He. Careful flow control
stabilizes the pressure at the inlet of the nozzle and defines the composition of incoming
gas stream. The two gas streams are combined, mixed using an inline static mixer, and
enter the plenum. The gases are cooled by a fin and tube heat exchanger and mixed again
before entering the nozzle. At the exit of the nozzle the gas flows into the outer box and
is removed and vented to atmosphere by a vacuum pump. Liquid nitrogen Dewars supply
the LN2 required to cool the condensible gas mixture and the sidewalls of the nozzle.
Cryogenic pressure regulators (Cash Acme, Type B, 70-205 kPa) at the outlet of the
Dewars improve the liquid flow control.
22
1
2
9
5
7
3
8
10
11
12
13
6
4
Figure 2.2: A cross sectional view of the cryogenic supersonic nozzle apparatus. Most
components are to scale but some of the smaller parts, for example the nozzle, static
pressure probe, and fittings, have been enlarged for clarity. The key components of the
experiment include (1) the insulated outer box, (2) the insulated inner box or plenum, (3)
the stagnation pressure tap, (4) the roller bearing slides, (5) the LN2 cooled heat
exchanger, (6) the static mixer, (7) the LN2 cooled supersonic nozzle, (8) the pressure
probe, (9) the RTD, (10) the nozzle exit assembly, (11) the probe gripper, (12) the linear
translation stage, and (13) the turning wheel.
Figure 2.2 is a cross sectional view illustrating the key components of the
cryogenic supersonic nozzle apparatus. The 0.81×1×0.66 m3 outer box (1), that isolates
the plenum, nozzle, and associate plumbing from the environment, is made of 2.5 cm
thick 6061 aluminum walls covered by 3.8 cm thick insulation (not shown) to reduce heat
transfer to the surroundings. The lid is fastened to the box by twelve 50 mm
perpendicular pull action toggle clamps (not shown) and a custom O-ring, fabricated
from 6.4 mm OD silicon, ensures a tight seal. Custom flanges and fittings on the box
23
sidewalls provide access for the plumbing as well as feedthroughs for the condensible
vapor, liquid nitrogen lines, plenum pressure measurement line, and temperature probes.
The box also acts as a ballast chamber as the gases exiting the nozzle enter the outer box
and are pumped out by the vacuum pumps. The pressure in the outer box is measured
using a 670 kPa capacitance manometer (MKS Instruments, Type 690A Baratron).
During an experiment, the pressure in the outer box is maintained below the pressure at
the exit of the nozzle, to avoid shocks in the nozzle.
The inner 0.56×0.30×0.46 m3 vacuum tight box (2), is constructed of 1.25 cm
thick Al and the lid is sealed to the box by fourteen 34 mm perpendicular pull action
toggle clamps and a custom O-ring fabricated from 6.4 mm OD silicon. This box is also
insulated and serves as the plenum for our system. The cross sectional area of the plenum
is much larger than that of the nozzle, and the gas flowing through the plenum is ~ 2 m/s.
This corresponds to a M ≈ 0.01, thus the gas is essentially at rest compared to the flow
achieved in the nozzle. To measure the stagnation pressure in the plenum, a 9.5 mm
OD×300 mm long piece of flexible stainless steel tubing connects a pressure tap (3) on
the downstream face of the plenum to a feedthrough in a flange on the side wall of the
outer box. A second tube connects the feedthrough to a 134 kPa capacitance manometer
(MKS Intruments, Type 120AA Baratron) located outside of the outer box. The plenum
rests on two 30 cm long × 5 cm wide aluminum roller bearing slides (4) to ease assembly
of the equipment when, for example, the nozzle is changed.
The LN2 cooled heat exchanger (Super Radiator Coils, B50-A424) (5) is a fin and
tube design with LN2 on the tube side and the condensible gas mixture on the fin side.
LN2 enters the bottom of the heat exchanger and, as it vaporizes, it cools the condensible
24
gas mixture from room temperature to temperatures as low as ~100 K without imposing a
large pressure drop on the flowing gas. The fins of the radiator are 0.2 mm thick vertical
0.3 ×0.11 m2 aluminum sheets, spaced 2.5 mm apart while the copper tubes are 9.5 mm
OD×0.4 mm wall thickness. There are a total of 120 fins and 12 loops of copper tubing
supported in a 0.38×0.13×0.3 m3 galvanized steel casing. The total tube surface area
available for heat exchange is 0.1 m2 and the rated cooling capacity of the heat exchanger
is ~300 kW. The heat exchanger is fastened to the upstream side of the plenum and a 25
mm thick Teflon gasket provides a leak tight seal. Because the temperature increases
from the bottom to the top of the heat exchanger, gas is sampled near the bottom of the
plenum through a tube and mixed using a static mixer (6) before entering the nozzle. The
pressure drop across the sampling tube is estimated to be ~ 0.2 kPa, and is negligible
compared to the stagnation pressure.
A detailed view of the supersonic Laval nozzle (7), is shown in Figure 2.3. The
nozzle is a two piece design and each side, including the end flanges, is machined from a
single piece of aluminum. One side has the nozzle profile machined into it, the second is
flat. Both side walls have pockets made by machining out the metal and welding a plate
on top. Liquid nitrogen flows through these pockets at a rate controlled to maintain the
temperature of the nozzle side walls about 20 K above that of the incoming gas. After
applying a thin layer of vacuum grease to ensure a good seal, the two halves are joined
together using dow pins and further tightened using aluminum socket head screws and
spring washers. A wire-mesh, that provides support for the static pressure probe (8), is
attached to a holder and held between the two sides of the nozzle just upstream of the
converging section.
25
Figure 2.3 (left) An exploded view of the nozzle assembly. (right) A detailed crosssectional view (AA section plane) of the nozzle assembly.
The straight section of the nozzle has a small hole for the 2mm O.D. Resistance
Temperature Detector (RTD) (9). After aligning the sensing element of the RTD with the
center of the flow, the RTD is sealed in place with a high strength epoxy. A surface RTD
(not shown) is fastened to the upper nozzle wall to measure the wall temperature. The
wires from the RTDs exit the outer box via a feedthrough in the sidewall to reach the
readout (Omega, DP 41. RTD.MDSS ). The inlet of the nozzle is attached to the plenum
using six stainless steel socket head screws and spring washers. A 3 mm thick plate with
033 Teflon O-rings on both sides provides the sealing between the nozzle and the
plenum.
26
The nozzle is designed to ensure that the flow is one dimensional. CFD modeling
studies of nozzles with similar throat area, geometry and inlet densities confirm that this
assumption is reasonable (Karlsson 2006). In our nozzle, the entrance region is 7.6 cm
long, the converging section is 3.8 cm long and the diverging section is 11.5 cm long. At
the throat, the minimum cross-sectional area A* is 5 mm × 5 mm, while at the exit the
area is 5mm × 13 mm. This yields an exit area ratio (Ae/A*) of 2.6, an opening half angle
α of 2o and an exit mach number M of 2.8 when the heat capacity ratio, γ, is 1.67.
The downstream end of the nozzle is attached to an assembly (10), consisting of a
7.5 cm OD copper Tee soldered to two 12.5 mm × 280 mm × 120 mm brass plates, that
provides additional support to the nozzle. The upstream plate has a 25 mm × 12.5 mm
opening for the gas flow that contains a second wire mesh support for the static pressure
probe. The downstream plate seals against the inner face of insulation box wall using a
234-Viton O-ring and also contains the feedthrough for the static pressure probe.
The static pressure probe (8) used in our experiments is a stainless steel tube, 600
mm long × 0.9 mm diameter. The upstream end of the probe is sealed with a sharp metal
tip to help guide the probe through the two wire mesh supports, and there are two 0.5-mm
diameter holes located 150 mm from the tip to measure the static pressure. The
downstream end is connected to a 134 kPa capacitance manometer (MKS Instruments,
Type 120AA Baratron) via a flexible tube. The feedthrough for the probe consists of a
silicon rubber gasket held, and lightly compressed, by a 6 mm Swagelok male connector.
The downstream end of the static pressure probe is held by a gripper (11) attached to a
linear translation stage (12) that is fastened to the exterior wall of the outer box. The
central rail of the translation stage is rotated by turning a wheel (13), and the central shaft
27
of the wheel is also connected to an optical counter (not shown) and an accumulator
tracks the position of the probe. This arrangement provides a resolution of better than 0.1
mm in the relative location of the nozzle probe. When the experiment is running, a heat
gun warms the translation stage and the silicon rubber seal to prevent icing and ensure
easy movement.
We use a 612 Stokes rotary piston vacuum pump (BOC Edwards) in our
experiments. This pump is comprised of two 412 Stokes vacuum pumps working in
parallel, and delivers a combined pumping speed of 0.25 m3/s. The high pumping speed
of the pumps enables us to use He as a carrier gas and to start the expansions at relatively
low pressures.
All temperature, pressure, and flow rate measurements made during an
experiment are recorded using a data acquisition system that consists of a computer with
a Keithley Instruments KPCI-3101 board. The data acquisition system can record up to 8
differential ended analog signal inputs with 12 bits resolution at a maximum sampling
rate of 225 kHz. At each axial position in the nozzle, we typically take 20 samples at a
sampling rate of 100 kHz and calculate the average and the standard deviation for each
measurement. The results are saved to a text file for further analysis.
2.3
Pressure Trace Measurements and Data Analysis
As the gas mixture flows through the nozzle it undergoes an isentropic expansion.
The cooling produced by the expansion rapidly increases the supersaturation of the
condensible until particle formation and growth deplete the vapor, increase the
temperature, and decrease the supersaturation enough to shut off nucleation. The onset of
28
condensation is detected by measuring the static pressure as a function of position along
the nozzle axis with a resolution of 0.01 cm near the throat and a resolution of 0.1 cm
further downstream.
In the absence of condensation, the pressure decreases monotonically and we can
determine the effective area ratio of the nozzle and the other state variables using the
isentropic relationships. When condensation occurs in the nozzle, we use the effective
area ratio A/A* derived from a non-condensing flow and the measured pressure ratio p/p0
for the condensing flow as input to the diabatic flow equations and calculate the
condensate mass fraction, g, the temperature T, density ρ, and gas velocity u (Wyslouzil
et al. 2000) as a function of the distance downstream of the throat, x. We define the onset
of condensation as the point where the temperature of the condensing flow deviates from
the expected isentropic temperature profile by +0.5 K.
We note that accurately determining the expected isentropic expansion is a critical
step in the analysis. In our preliminary experiments, conducted without actively cooling
of the nozzle sidewalls, we observed deviations from the expected isentropic pressure
ratio that were inconsistent with condensation but were consistent with external heat
addition. In the nozzle used here, the temperature of the nozzle walls is monitored by a
surface mounted RTD and maintained ~ 20 K higher than the stagnation temperature of
the gas mixture to minimize the transfer of energy to or from the flowing gas.
Unlike our experiments with dilute gas mixtures (Kim et al. 2004), measuring an
appropriate isentrope is not entirely straight forward in the current experiments since, for
pure Ar condensation, there is no carrier gas. One solution is to use the pressure trace for
a flow where condensation occurs further downstream than the condensing flow of
29
interest, as the isentrope. This may be done by measuring the static pressure profile at a
slightly reduced pressure or elevated temperature relative to the condensing flow curve.
As long as the initial density of the gas for the two experiments is close, boundary layer
development will not differ significantly between the two expansions. This approach will
yield accurate values for the onset conditions for the experiment with the higher onset
temperature, but the state variables such as g, T and p will only be valid in the region
upstream of the onset of condensation for the reference pressure trace. Furthermore, the
onset conditions associated with reference pressure trace cannot be determined.
A second approach takes advantage of the nozzle design. Ideally, the nozzle area
ratio should increase linearly from a short distance downstream of the throat to the exit of
the nozzle. As illustrated in Figure 2.4, however, careful analysis of A/A* derived from
pressure measurements starting at p0 = 36 kPa and T0 = 298 K (lower solid black line)
indicates that for positions between 0.5 and ~6.4 cm downstream of the throat, the nozzle
is better fit by a bi-linear profile, i.e. the slope of the profile changes slightly near x = 4
cm. This is also true for the data corresponding to p0 = 36 kPa and T0 = 120 K (upper
solid black line) when x is between 0.5 and ~5.5 cm, and where the deviation from
straight line behavior further downstream is due to condensation. Thus, if we fit the
reference pressure trace to a bi-linear function upstream of condensation, we can find
A/A* for the entire nozzle and fabricate a reference isentrope. In Figure 2.4, a fit to the
low temperature data yielded an effective expansion rate of d(A/A*)/dx = 0.118 cm-1 for
x < 4.0 cm (dash dotted line), and d(A/A*)/dx = 0.104 cm-1 downstream of this location
(dashed line). A similar change in the area ratios was observed for the room temperature
data.
30
1.8
1.6
1.4
T0 = 298 K
A/A
*
T0 = 120 K
1.2
1.0
0
2
4
6
8
x (cm)
Figure 2.4: The changes in area ratio as a function of position downstream of the throat,
x, (black solid lines) were determined from pressure trace measurements starting from p0
= 36 kPa and the indicated temperatures. The room temperature trace is well fit by a bilinear function for x between 0.5 and 6.4 cm with slopes of 0.098 cm-1 for x < 4 cm and
0.084 cm-1 for x > 4 cm. The region downstream of ~6.4 cm also appears to be linear with
a slightly lower slope. At low temperature, the bi-linear fit is valid for x between 0.5 and
5.5 cm and the slopes are 0.118 cm-1 for x < 4 cm and 0.104 cm-1 for x > 4 cm. The area
ratios differ drastically because the gas density in the room temperature experiment is
much lower than in the T0 = 120 K experiment, and, thus, the boundary layers grow more
rapidly when T0 = 298 K. The deviation from the expected straight line behavior in the
low temperature experiment is due to condensation.
2.4
Sample Measurements
Our sample measurements were carried out using pure Ar. Figure 2.5 illustrates
the pressure ratios measured for a range of stagnation pressures and temperatures. The
31
grey line is the isentropic expansion of Ar from p0 = 36 kPa and T0 = 298 K. The black
line is the isentropic pressure ratio based on the bi-linear fit to A/A* shown in Figure 2.4.
For the condensation experiments, the stagnation pressure in our experiments ranged
from 30.6 kPa to 36 kPa, and the stagnation temperatures ranged from 104.6 K to 115 K.
0.5
p0 = 30.6 kPa T0 = 104.6 K
p0 = 36 kPa T0 = 111.2 K
0.4
p0 = 36 kPa T0 = 115 K
p0 = 36 kPa T0 = 298 K
Expected isentrope
p/p0
0.3
0.2
0.1
0.0
0
2
4
6
8
10
x (cm)
Figure 2.5: The measured pressure ratio profiles as a function the stagnation conditions.
The grey solid line corresponds to the measurement at room temperature, while the black
solid line represents the expected isentrope for the low temperature experiments. The
broken lines correspond to the experiments carried out at lower temperatures. For the
latter, the rapid increase in pressure above the isentropic expansion is due to heat addition
to the flow by the growing droplets and is the signature of condensation.
The results in Figure 2.5 again emphasize that the pressure trace for the room
temperature experiment differs significantly from those for the lower temperature
32
experiments. At a fixed p0, the density of the gas in the expansion starting from T0 = 298
K is much lower than when T0 = 120 K, and, thus, the boundary layers are thicker and
A/A*, as already illustrated in Figure 2.4, is smaller at a given nozzle position.
The pressure data were analyzed to determine the pressures and temperatures
corresponding to the onset of condensation and Figure 2.6 summarizes the results in a
Wilson plot. The black solid line is the extrapolated vapor-liquid binodal, the dash-dotted
line represents the solid-liquid equilibrium binodal, and the dashed line represents the
vapor – solid binodal. The filled diamonds are the current results, the filled and empty
circles are the onset points measured in a nucleation pulse chamber by Iland et al (2007)
and Fladerer and Strey (2006), respectively, the open triangles are the onset data for pure
Ar measured in supersonic nozzle by Stein (1974) and the open hexagons are the onset
data for Ar or Ar/He mixtures of Wu et al.
Since the nucleation rates in supersonic nozzles are typically 8 – 10 orders of
magnitude higher than in the NPC, at a fixed onset pressure, the onset temperature in
supersonic nozzle must be lower than that observed in the nucleation pulse chamber. Our
data are clearly consistent with the NPC results, while those of Stein’s are not. Although
our experiments started from values of p0 and T0 that are close to some of Stein’s
experiments, our values of Ton are ~ 9 K lower for about the same onset pressure. Finally,
although they do not overlap in pressure or temperature our data appear quite consistent
with those of Wu et al.
We note that the stagnation pressures used in these sample experiments are higher
than the vapor pressure of Ar at the boiling point of N2, 29.2 kPa. Furthermore, changes
33
in the mass flow rate of Ar with temperature suggest that there was some condensation of
Ar in the plenum.
liquid
105
solid or supercooled liquid
gas
p / Pa
104
liquid / gas
solid / gas
solid / liquid
Stein
Wu et al.
Fladerer & Strey
Iland et al.
this work
103
102
101
20
30
40
50
60
70
80
90
100 110
T/K
Figure 2.6: The onset pressures and temperatures measured in the current cryogenic
supersonic nozzle are compared to the supersonic nozzle measurements of Stein and Wu
et al., as well as the NPC measurements of Iland et al. and Fladerer and Strey.
Given the high surface area of the heat exchanger and the low gas velocity (< 2
m/s) over the fins we do not believe that particle entrainment is influencing our results.
Furthermore, modeling studies by Heiler (Heiler 1999) suggest that particle number
densities on the order of 108 cm-3 are required to significantly perturb homogeneous
nucleation.
34
2.5
Conclusions and Outlook
We have developed a cryogenic supersonic nozzle setup to follow the
condensation of argon and other simple molecules. The incoming gas is cooled in a low
pressure drop fin and tube heat exchanger by the vaporization of LN2 flowing through the
tubes. With this heat exchanger, the lowest stagnation temperature we achieve for pure
Ar condensation is ~104 K.
Cooling the sidewalls of the nozzle with LN2 proved to be a crucial step in
reliably observing condensation. By monitoring the surface temperature of the nozzle and
adjusting the flow of LN2 through the sidewalls, we are able to easily maintain the nozzle
surface temperature about 20 K above the stagnation temperature and reduce heat transfer
between the surroundings and the flowing gas.
The current nozzle was designed to have an exit Mach number of 2.8 for gases
such as argon. For the pure Ar we readily observed condensation for temperatures in the
range 47.5 K < Ton< 49.5 K, and pressures in the range of 4.2 kPa< pon< 4.9 kPa in this
nozzle. Our onset points are consistent with the data from nucleation pulse chambers and
of the supersonic nozzle Wu et al., but correspond to significantly higher supersaturations
than the previous pure Ar of Stein. If He is added as a carrier gas, one can measure onset
at lower temperatures and Ar partial pressures while maintaining a high enough overall
gas density to maintain the same effective expansion. Changing the nozzle is a relatively
straightforward process and using a faster nozzle will let us probe the meta-stable region
even further. To go to higher Ar pressures requires a redesigned heat exchanger and a
different working fluid. One option is to use Ar both as the heat exchange fluid and the
35
working fluid. The relatively open, modular aspect of our experiment also makes this
change relatively straightforward.
To determine the onset of condensation requires an isentrope that is consistent
with the expansion during the low temperature experiments. Isentropes starting at the
same stagnation pressure and room temperature cannot be used because the boundary
layers grow more quickly at these reduced gas densities. We are currently investigating
under what conditions room temperature pressure traces at higher pressure are acceptable.
Our experiments clearly illustrate the importance of characterizing the expansion
throughout the nozzle in the absence of condensation under conditions that are close to
those under which condensation occurs.
Acknowledgements
This work was supported by the National Science Foundation under Grant number
CHE-0518042 We thank P. Green for machining the components of our apparatus, R.
Shial of Super Radiator Coils for design and fabrication the heat exchanger, and A.
Bhabhe for help with our experiments.
36
References
Becker, R. & Döring, W. 1935, "Kinetische Behandlung der Keimbildung in
üebersättigten Dämpfen", Annalen der Physik, vol. 24, pp. 719-752.
Evans, R. 1979, "The nature of the liquid vapor interface and other topics in the statistical
mechanics of non-uniform, classical fluids", Advances in Physics, vol. 28, no. 2, pp.
143.
Fladerer, A. & Strey, R. 2006, "Homogeneous nucleation and droplet growth in
supersaturated argon vapor: The cryogenic nucleation pulse chamber", Journal of
Chemical Physics, vol. 124, pp. 164710.
Garcia Garcia, N. & Soler Torroja, J.M. 1981, "Monte Carlo calculation of argon clusters
in homogeneous nucleation", Physical Review Letters, vol. 47, no. 3, pp. 186.
Granasy, L., Jurek, Z. & Oxtoby, D.W. 2000, "Analytical density functional theory of
homogeneous vapor condensation", Physical Review E: Statistical Physics, Plasmas,
Fluids, and Related Interdisciplinary Topics, vol. 62, no. 5-B, pp. 7486.
Heiler, M. 1999, Instationäre Phänomene in homogen/heterogen kondensierenden
Düsen- und Turbinenströmungen., Dissertation Fakultät für Maschinenbau,
Universität Karlsruhe (TH).
Heist, R.H. & He, J.H. 1994, "Review of vapor to liquid homogeneous nucleation
experiments from 1968 to 1992.", Journal of Chemical Physics Reference Data, vol.
23, pp. 781.
Iland, K., Wölk, J., Strey, R. & Kashchiev, D. 2007, "Argon nucleation in a cryogenic
pulse chamber", Journal of Chemical Physics, vol. 127, pp. 154506.
Karlsson, O.M. 2006, Nucleation and condensation in a stationary supersonic flow.
Design, modeling and test of an experiment based on a transparent Laval nozzle.,
Eidgenössische Technische Hochschule ETH Zürich.
Kim, Y.J., Wyslouzil, B.E., Wilemski, G., Wölk, J. & Strey, R. 2004, "Isothermal
nucleation rates in supersonic nozzles and the properties of small water clusters",
Journal of Physical Chemistry A, vol. 108, pp. 4365-4377.
Kraska, T. 2006, " Molecular-dynamics simulation of argon nucleation from
supersaturated vapor in the NVE ensemble", Journal of Chemical Physics, vol. 124,
pp. 054507.
37
Kusaka, I. 2003, "System size dependence of teh free energy surface in cluster simulation
of nucleation", Journal of Chemical Physics, vol. 119, pp. 3820.
Laasonen, K., Wonczak, S., Strey, R. & Laaksonen, A. 2000, "Molecular dynamics
simulations of gas-liquid nucleation of Lennard-Jones fluid", Journal of Chemical
Physics, vol. 113, no. 21, pp. 9741.
Matthew, M.W. & Steinwandel, J. 1983, "
An experimental study of argon condensation in cryogenic shock tubes", Journal of
Aerosol Science, vol. 14, pp. 755.
Senger, B., Schaaf, P., Corti, D.S., Bowles, R., Pointu, D., Voegel, J.C. & Reiss, H. 1999,
"A molecular theory of the homogeneous nucleation rate. II. Application to argon
vapor", Journal of Chemical Physics, vol. 110, no. 13, pp. 6438.
Stein, G.D. 1974, Argon Nucleation in a Supersonic Nozzle, Report to Office of Naval
Research available from National Technical Information Service Number: ADA007357/7GI.
Talanquer, V. & Oxtoby, D.W. 1994, "Dynamic density functional theory of gas-liquid
nucleation", Journal of Chemical Physics, vol. 100, no. 7, pp. 5190.
ten Wolde, P.R. & Frenkel, D. 1998, "Computer simulation study of gas-liquid nucleation
in a Lennard-Jones system", Journal of Chemical Physics, vol. 109, no. 22, pp. 9901.
Toxvaerd, S. 2001, "Molecular-dynamics simulation of homogeneous nucleation in the
vapor phase", Journal of Chemical Physics, vol. 115, no. 19, pp. 8913.
Weakliem, C.L. & Reiss, H. 1993, "Toward a molecular theory of vapor-phase
nucleation. III. Thermodynamic properties of argon clusters from Monte Carlo
simulations and a modified liquid drop theory", Journal of Chemical Physics, vol.
99, no. 7, pp. 5374.
Wedekind, J., Wölk, J., Reguera, D. & Strey, R. 2007, Journal of Chemical Physics, vol.
127, pp. 154515.
Wu, B. J. C., Wagner, P.E. & Stein, G.D. 1978, "Condensation of sulfur hexafluoride in
steady supersonic nozzle flow", Journal of Chemical Physics, vol. 68, pp. 308.
Wu, B. J. C., Wegener, P.P. & Stein, G.D. 1978, "Homogeneous nucleation of argon
carried in helium in supersonic nozzle flow", Journal of Chemical Physics, vol. 69,
pp. 1776.
38
Wyslouzil, B.E., Heath, C.H., Cheung, J.L. & Wilemski, G. 2000, "Binary condensation
in a supersonic nozzle", Journal of Chemical Physics, vol. 113, pp. 7317.
Yasuoka, K. & Matsumoto, M. 1998, "Molecular dynamics of homogeneous nucleation
in the vapor phase. I. Lennard-Jones fluid", Journal of Chemical Physics, vol. 109,
no. 19, pp. 8451.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1999, "
Homogeneous nucleation of argon in an unsteady hypersonic flow field", Journal of
Chemical Physics, vol. 110, pp. 8842.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1995, "Formation of argon clusters by
homogeneous nucleation in supersonic shock tube flow", Journal of Chemical
Physics, vol. 103, pp. 9038.
Zeng, X.C. & Oxtoby, D.W. 1991, "Gas-liquid nucleation in Lennard-Jones fluids",
Journal of Chemical Physics, vol. 94, pp. 4472.
39
CHAPTER 3
Argon nucleation in a cryogenic supersonic nozzle apparatus
Somnath Sinha, Ashutosh Bhabhe, Hartawan Laksmono and Barbara E. Wyslouzil
Department of Chemical and Biomolecular Engineering,
The Ohio State University,
Columbus, OH-43210
40
Abstract
Homogeneous nucleation of argon droplets measured in our newly designed cryogenic
supersonic nozzle apparatus (Sinha et al. 2008a) is presented in this work. The onset
conditions along with conditions corresponding to maximum nucleation rate are reported.
We observed temperatures ranging between 40 K to 53 K and pressures between 1.3 kPa
-8.1 kPa, corresponding to the maximum nucleation rate. From the characteristic time of
nucleation and typical number densities observed in nozzles, we estimated the nucleation
rates for our experiments as J = 1017±1 cm-3 s-1. Our data was found to be consistent with
the onset measurements in nucleation pulse chambers (Fladerer & Strey 2006; Iland et al.
2007). A comparison with classical nucleation theory (CNT) showed that the theory
under predicts the nucleation rate for our experiments by 11-13 orders of magnitude. The
experimental critical cluster sizes were obtained by using data from this work and Iland et
al. along with the first nucleation theorem. Our results indicate that the Gibbs-Thomson
equation over predicts the critical cluster sizes by 30-55 % and is the prime reason for
failure of CNT. We show that a temperature dependent correction function would only
correct the CNT locally.
41
3.1
Introduction
Nucleation from the vapor phase has been an active area of both experimental and
theoretical research for over a century. The first theoretical treatment of nucleation dates
back to 1935, when Becker and Döring put forth the Classical nucleation theory (CNT).
In the classical model, clusters of the new phase are viewed as stationary droplets whose
free energy is the sum of volume and surface contributions. Calculating a nucleation rate
using this “capillarity approximation” requires only measurable bulk property data as
input, and the simplicity of the model has been one of its most attractive features.
CNT does do a reasonable job of predicting the critical supersaturations for a
wide variety of substances (Pound 1972). In some cases, in particular water (Wölk &
Strey 2001), rates are even predicted quantitatively over a limited range of temperature –
and a simple empirical adjustment can make it work over a wide range of temperature T
and supersaturation S (Wölk et al. 2002).
There is also often astonishingly good
agreement between the measured values of n* and the predictions of n*GT based on the
capillarity approximation for substances as diverse as water (Kim et al. 2004), alcohols
(Viisanen & Strey 1994; Iland et al. 2004;),
and nonane (Luijten et al. 1999).
Nevertheless, the capillarity approximation remains physically unrealistic, and the
eventual goal of many research groups is to develop a truly molecular theory of
nucleation. (Evans 1979; Garcia Garcia & Soler Torroja 1981; Zeng & Oxtoby 1991;
Weakliem & Reiss 1993; Talanquer & Oxtoby 1994; ten Wolde 1998; Yasuoka &
Matsumoto 1998;
Senger et al. 1999; Granasy et al. 2000; Laasonen et al. 2000;
Toxvaerd 2001; Kusaka 2003; Kraska 2006)
42
The molecular theories developed over the past 30 years have been based on
density functional theory (DFT) (Evans 1979; Zeng & Oxtoby 1991; Talanquer &
Oxtoby 1994; Granasy et al. 2000), Monte Carlo simulations (Garcia Garcia & Soler
Torroja 1981; Weakliem & Reiss 1993; Senger et al. 1999; Kusaka 2003), and Molecular
dynamic simulations (ten Wolde 1998; Yasuoka & Matsumoto 1998; Laasonen et al.
2000; Toxvaerd 2001; Kraska 2006). In these approaches the intermolecular potential that
governs the interactions between the molecules must be specified. Most of the work to
date uses the Lennard-Jones potential because of computational ease although some
efforts in water simulations (Yasuaoka & Matsumoto 1998) use more complex potentials.
The Lennard -Jones potential is, however, only valid for simple molecules, such as the
noble gases, and so much of the emphasis has been on simulating Argon condensation.
Although Argon data do exist (Stein 1974; Wu et al. 1978; Matthew & Steinwandel
1983; Zahoransky et al. 1995, 1999; Fladerer & Strey 2006; Iland et al. 2007) they are
limited due to the difficulties associated with the conducting experiments at the extremely
low temperatures required to observe Ar condensation. Experiments have been carried
out in nucleation pulse chambers (NPC), shock tubes and supersonic nozzles but many of
the data are inconsistent with each other making it difficult to compare theories and
simulations with these experiments. For example, many of the shock tube data are more
consistent with heterogeneous than homogeneous nucleation. Furthermore, some of the
supersonic nozzle data lie at lower supersaturations than the recent nucleation pulse
chamber data even though the characteristic nucleation rates are orders of magnitude
higher in the nozzle than in the NPC.
43
Surprisingly, Ar is one substance for which CNT fails dramatically. The recent
measurements by Iland et al. made in a cryogenic nucleation pulse chamber (Iland 2004;
Iland et al. 2007) correspond to nucleation rates of ~107 cm-3s-1. They found that their
rates were roughly 20 orders of magnitude higher than those predicted by CNT. To better
understand why CNT fails, more than one reliable set of data is necessary, preferably
corresponding to distinctly different nucleation rates. With two sets of experiments we
can begin to estimate the critical cluster size, perhaps determine why CNT fails so
dismally, and evaluate whether the molecular theories can do a better job. In this article,
we present our first set of data for Argon condensation in a supersonic nozzle in the
temperature range of 40-53 K. Our data appear to be highly consistent with those of Iland
et al. as well as those of Wu et al. Perhaps the most striking result of this work is that by
combining the nozzle and NPC results we find that the critical clusters contain 30 – 50%
smaller than predicted by the Gibbs-Thomson equation.
44
3.2
Experimental
3.2.1
Chemicals
The gases used in these experiments include Ar and He and their physical properties are
summarized in the Appendix A.
3.2.2
The cryogenic supersonic nozzle
The experiments were carried out using the cryogenic supersonic nozzle apparatus
illustrated in Figure 3.1. This apparatus has been described in detail in Sinha et al. (Sinha
et al. 2008a) and we will only briefly summarize it here. The crucial components in the
apparatus include the insulated plenum that houses the LN2 cooled heat exchanger, and
the supersonic nozzle with its LN2 cooled sidewalls. These components are all contained
in a large, insulated outer box that separates the cold components from the warm room
air. The outer box also acts as a ballast volume because the gases exiting the nozzle first
flow into outer box before being removed from it by the vacuum pumps.
To run an experiment, we draw Ar from the gas side of a high pressure liquid Ar
tank, warm it to room temperature using a 1000 W inline heater, regulate the pressure,
and control the flow with a 300 SLM mass flow controller (MKS 1559A) calibrated for
Ar to 1% accuracy. If He is used, we take this from a high pressure gas bottle, warm it to
room temperature, regulate the pressure and control the flow with a 400 SLM mass flow
controller (MKS 1559A) calibrated for He to 1% accuracy. The gases flow into a
common line that contains a static mixer, and enter the plenum. In the plenum, the gas
mixture flows over the fin side of a fin and tube heat exchanger that cools the gas to
stagnation temperatures T0 as low as 100 K by evaporating liquid N2 on the tube side.
45
Because there is a vertical temperature gradient across the heat exchanger, a short tube
containing a second static mixer directs the gas from the bottom of the heat exchanger to
the entrance of the nozzle. Although the stagnation pressure p0 is measured on the front
face of the plenum, i.e. before the gas flows through the sampling tube, we estimate that
the pressure losses, due primarily to the bend in the sampling tube and the contraction at
the entrance to the nozzle, are on the order of 0.2 kPa. These losses are less than 1% of
the lowest values of p0 and are, therefore, ignored.
The gas then enters the nozzle and flows through a region with constant area
where the temperature is measured using a 2mm diameter RTD. A surface mounted RTD
located on top of the nozzle monitors the outer wall temperature. As the gas flows
through the contracting and diverging sections of the nozzle, the static pressure is
determined as a function of position using a static pressure probe. At the exit of the
nozzle, the gas flows in the outer box before it is pumped to atmospheric pressure by a
Stokes 612 vacuum pump and vented to atmosphere.
The nozzle used in this work differs significantly from those used in our
experiments that start close to room temperature. In particular, to reduce the potential for
leaks it is a two piece design with the nozzle shape machined into one sidewall. Both
sidewalls have hollow pockets through which we flow liquid N2 to maintain the sidewall
temperature about 15 – 20 K above the stagnation temperature of the gas. This
temperature difference is comparable to that in many of our “room temperature”
experiments and significantly reduces heat transfer to or from the flowing gas. At 5 mm ×
5mm the throat of the nozzle is also significantly smaller than our standard nozzle design
46
and the exit Mach number, M ≈ 2.8, is faster than our usual nozzles. A more detailed
discussion of the nozzle and drawings are available in Sinha et al. 2008a
P - Pressure
T - Temperature
exhaust
Liquid
Nitrogen
vacuum
pumps
P
Mass flow
controller
Heater
Nozzle
Regulator
Argon
Heater
Mass flow
controller
pressure probe
Heat
Exchanger
P
Plenum
Regulator
Outer box
He
T
T
P
Liquid
Nitrogen
Figure 3.1: A schematic diagram of the cryogenic supersonic nozzle apparatus. Ar and
He are the condensible and carrier gases, respectively, while independent liquid N2
streams serve as the working fluid in the heat exchanger and to cool the side walls of the
nozzle, respectively. Condensation is detected by measuring the static pressure as a
function of position. (Sinha et al. 2008a)
3.2.3
Pressure Trace Measurements
We follow the condensation process in the nozzle by measuring the static pressure
along the nozzle axis. When condensation does not occur in the nozzle, the pressure
decreases monotonically as a function of position, and we can determine the effective
area ratio of the nozzle and the other state variables using the isentropic flow
relationships (Wyslouzil et al. 2000). When condensation does occur in the nozzle, the
pressure initially follows the expected isentropic expansion until a point is reached where
47
heat addition due to particle formation and growth increase the pressure, temperature and
density of the gas noticeably above the isentropic values.
To determine the other properties of the flow – including temperature T, density
ρ, velocity u, and mass fraction condensate g – we integrate the diabatic flow equations
(Wyslouzil et al. 2000) using the effective area ratio, the measured condensing flow
profile, and the stagnation conditions, i.e. p0, T0, and composition, as input. One of the
challenges we face in this work is defining an appropriate effective area ratio in the
absence of carrier gas. As detailed in Sinha et al. (2008a), the current nozzle does not
exhibit a strictly linear increase in area ratio, A/A*, with distance x downstream of the
throat. Rather, there is a distinct change in the slope near x ~ 4 cm that is present both
when T0 is close to room temperature and at low stagnation temperatures. We found,
however, that by choosing a reference pressure trace – generally the one where
condensation occurs the furthest downstream – we can construct an appropriate effective
A(x)/A* relationship by fitting the area ratio derived from the reference pressure trace to
two straight lines. In this chapter, we show that a room temperature pressure trace
measured under the correct stagnation conditions also works well and we present scaling
arguments to demonstrate how to make this choice.
3.2.4
Characterizing Condensation and Nucleation
Once the flow properties have been determined we can find the conditions that
correspond to the onset of condensation or to the maximum nucleation rate. We
arbitrarily define the onset of condensation as that point in the flow where the
temperature of the condensing flow is 0.5 K higher than the expected isentropic
48
temperature. To define the conditions corresponding to the maximum nucleation rate, we
first calculate the supersaturation S profile in the nozzle as
p v ( x)
S ( x) =
∞
p ( x)  g ( x ) 

1 −
g ∞ 
p (T ( x)) p0 
pv0
=
∞
p (T ( x))
(1)
where pv is the partial pressure of Ar, p∞(T) is the equilibrium vapor pressure of Ar, and
g∞ is the initial mass fraction of the condensing species. We then calculate the nucleation
rate as a function of position using the experimentally derived temperature and
supersaturation profiles and the classical nucleation rate expression of Becker and Döring
(Becker & Döring 1935). This nucleation rate, JBD, is given by
J BD
2
 − 16πv m 2σ 3 
 pv 
v m   exp 
=
 ,
πµ v  kT 
 3( kT )3 ( lnS ) 2 
2σ
(2)
where σ is the surface tension of the macroscopic fluid-vapor interface, vm is the
molecular volume, and k is the Boltzmann constant.
The values of SJmax and TJmax that correspond to the maximum nucleation rate are
easily determined from the nucleation rate profile in the nozzle, as is the corresponding
pressure of the condensible vapor, pvJmax. Furthermore, we can also use the nucleation
rate profile to determine the characteristic time associated with the peak nucleation rate
by evaluating
∆t J max =
∫ J (S , T )dt
J max
(3)
.
Since previous studies have shown that ∆tJmax calculated this way is relatively insensitive
to the choice of the nucleation rate expression, the assumption is made that this
49
relationship also holds for the experimental nucleation rates. To evaluate Eq.3 we use
Eq.2 as the nucleation rate expression.
3.3
Experimental Results
3.3.1
The effective area ratio
One way of obtaining an appropriate isentrope is to use a bi-linear fit function to
the area ratio (A/A*) from a reference isentrope, upstream of condensation. An alternative
is to use scaling to determine conditions at room temperature that will give the same
boundary layer development as at the low temperature, but where condensation will not
occur. We start by assuming that the displacement thickness δ* along the walls scales like
flow over a flat plate, i.e.
 1 
 .
 Re 
δ * ∝ 
(4)
Here Re is the Reynolds number and Re = ρux/µ where x is the distance downstream
from the throat of the nozzle and µ is the dynamic viscosity of the gas mixture. Thus for
two experiments to have same effective expansion, the Re for the flows must be the
same, i.e. (Re) RT = (Re) LT , or
 ρux 
 ρux 


= 

 µ  RT  µ  LT
(5)
50
The subscripts RT and LT denote room temperature and low temperature respectively.
The density, velocity and viscosity can be written in terms of the temperature, pressure,
molecular weight MW, and Mach number M as
 pM W 

 RT 
(6)
(
(7)
ρ =
u = M γRT
)
µ =C T
(8)
where C is a constant, given in Table 3.1 for He and Ar, that depends on the composition
of the mixture, R is the universal gas constant, and γ is the heat capacity ratio. From Eqs.
(5)- (8) it follows that,

γRT 

M
MW 
 pM W
x

RT
C
T





 RT
=

γRT 

M
MW 
 pM W
x

RT
C
'
T





 LT
(9)
The Mach number, M is strictly a function of A/A* and γ. In our experiments we only use
pure Ar or mixtures of Ar and He, and in either case γ is 1.67. Thus, if δ* is the same,
then A/A* for the two flows is the same, and MRT = MLT. At a given nozzle position, the
effective expansion for two experiments is therefore the same if
 p MW

 TC

 p MW

 =


 RT  TC '



 LT
(10)
51
If we conduct the experiments with the same gas composition, we conclude that to obtain
an appropriate isentrope for the low temperature condensing flows, we should run
experiments at a stagnation pressure corresponding to
( p0 )RT
p
=  0
 T0

 × (To )RT
 LT
(11)
1.8
P0 = 90 kPa
T0 = 298 K
P0 = 36 kPa
T0 = 120 K
1.4
A/A
*
1.6
1.2
1.0
0
2
4
6
8
x (cm)
Figure 3.2: The change in area ratio as a function of position downstream of the throat, x,
(black solid line) was determined from pressure trace measurement starting from p0 = 36
kPa and T0 = 120 K using a bi-linear fit valid for x between 0.5 and 5.5 cm with slopes
0.118 cm-1 for x < 4 cm and 0.104 cm-1 for x > 4 cm. The straight solid line indicates the
area ratio from a room temperature experiment with p0 = 90 kPa. The area ratio profile
corresponding to the fit and the room temperature experiment agree closely.
52
Figure 3.2 illustrates that this approach works very well and that the high pressure, room
temperature pressure trace agrees very well with the bi-linear fit in the region of overlap.
In the current work we used the scaling approach to obtain appropriate isentropes by
running experiments at room temperature but at higher pressures than the corresponding
low temperature condensing flow experiments.
3.3.2
Condensing flow experiments
We conducted experiments with vapor compositions ranging from pure Ar to Ar –
He carrier gas mixtures with mole fractions between 0.43 and 0.84. Figure 3.3(a)
illustrates typical pressure traces corresponding to Ar condensation that were measured
for three Ar-He mixtures. This figure also includes the pressure trace for the expected
isentropic condensation that was found by the method described in Section III.A, i.e. by
using a room temperature experiment run at higher pressure. As illustrated in Figure
3.3(a), the condensing flow curves follow the isentropic curve closely until heat addition
due to rapid particle growth leads to an abrupt increase in pressure. A few cm
downstream, the pressure begins to decrease again as the supersaturation is reduced and
particle growth slows. Comparing these pressure traces we see that when the partial
pressure of Ar at the inlet increases, condensation occurs further upstream in the nozzle
and the deviation from the isentropic expansion is greater. In contrast, Figure 3.3(b)
illustrates pressure deviations that we attribute to external heat addition – i.e. heat transfer
from the surrounding. In these experiments, conducted with pure Ar in an M= 1.96
nozzle whose sidewalls were not cooled by liquid N2, the dashed line, deviates from the
isentrope, solid black line, rather gently. When we maintained the initial partial pressure
53
of Ar but added He as a carrier gas and reran the experiment at the same T0, dot-dash line,
the deviation vanishes. If the deviation between the dashed and solid lines were
condensation, we should still observe it even in the presence of He. Another indication
that heat addition is playing a role is that the pure Ar pressure trace at T0 = 123 K follows
the T0 = 294 rather closely even though the initial density in the former case is much
higher than in the latter case. These experiments clearly demonstrate the importance of
measuring at least one state variable as a function of position in the nozzle rather than
relying solely on the nozzle design.
0.5
0.5
(b)
P0 = 72 kPa T0=298 K y0 = 0.79
T0 = 123 K
P0 = 25.3 kPa T0 =105.7 K y0 = 0.79
0.4
T0 = 123 K
0.4
P0 = 25.3 kPa T0 =103.7 K y0 = 0.63
T0 = 294 K
P0 = 30.6 kPa T0 =101.6 K y0 = 0.43
0.3
p/p0
p/p0
0.3
0.2
0.2
0.1
0.1
34.6 kPa Ar
34.6 kPa Ar
28 kPa He
(a)
0.0
0.0
0
2
4
6
8
10
0
x (cm)
2
4
6
8
10
x (cm)
Figure 3.3(a) The measured pressure ratio profiles as a function of the stagnation
conditions. The black solid line represents the expected isentrope from scaled room
temperature experiment. The broken lines correspond to the experiments carried out at
lower temperatures. For the latter, the rapid increase in pressure above the isentropic
expansion is due to heat addition to the flow by the growing droplets and is the signature
of condensation. (b) Pressure ratio profiles from experiments in the absence of cooled
nozzle sidewalls. The solid line represents the room temperature isentrope. The dashed
line represents the low temperature experiment with pure Ar and deviates gently from the
isentrope. The low temperature experiment with Ar-He mixture is represented by the
dash-dotted line.
54
We then analyzed the condensing flow curves to determine both the values
corresponding to onset, and those corresponding to the maximum nucleation rate. We
report the onset conditions with the understanding that our definition is not necessarily
the same as that reported by experiments that rely on light scattering, although the results
should be close. Only the conditions corresponding to the maximum nucleation rate are
used in further analysis. Both sets of results are summarized in Table 3.1, and values of
pJmax and TJmax are illustrated in Figure 3.4 along with the vapor-liquid (solid line), and
the vapor-solid (dashed line) equilibrium lines. The pure Ar experiments are represented
by the grey diamond, while the Ar–He experiments correspond to the black diamonds.
Overall, we were able to vary the partial pressure of Ar at the nozzle inlet pv0 between
13.3 kPa and 53.2 kPa, while the stagnation temperature, T0, ranged from 101.5 K to 116
K. Given the challenges involved in these experiments, not all combinations of pv0 and T0
were possible. Nevertheless, the observed TJmax values for Ar condensation cover a 13 K
temperature range, 40 < TJmax / K < 53, and the vapor pressures, pJmax, increase by a factor
of ~7, 1.3 < pJmax/ kPa < 8.1.
The measurement window of our experiments is currently restricted to this range
because the minimum T0 that can be reached using our heat exchanger is ~100 K. On the
one hand, with the present nozzle and T0 = 100 K we do not observe condensation in the
accessible region of the nozzle when pv0 is lower than 13.3 kPa. On the other hand
working at pressures higher than 60 kPa increases the flow of gas through the system to
the point that the current heat exchanger cannot cool the gas to a value of T0 low enough
to observe condensation. Although, experiments carried out at higher pressures and
55
temperatures are both desirable and conceptually feasible, it will require some redesign of
the system. To reach lower pressures and temperatures we will build a faster nozzle.
Figure 3.4 also shows a solid line running through the onset data points that is a
fit function for our measured onset data given by:
p J max = exp(1.38 + 0.1452TJ max )
(Pa)
(12)
Although the data scatter around this fit function, the experimental values of pJmax are all
within 12 % of the calculated values. It is noteworthy that the data corresponding to the
experiments with He as the carrier gas are in line with those performed using pure Ar.
105
p / Pa
104
103
102
gas/liquid
gas/solid
101
30
40
50
60
70
T/K
Figure 3.4: The pressures and temperatures corresponding to the maximum nucleation
rate. The black diamonds correspond to experiments with He as the carrier gas while grey
diamonds correspond to pure Ar experiments. The fit function given by Eq.12 goes
through most of the data points. The solid line below and to the right of the data is the
extrapolated gas/liquid equilibrium line, and the dashed line is the gas/solid binodal.
56
As a gas mixture expands in a supersonic nozzle, the supersaturation increases
rapidly downstream of the throat until particle formation and growth increase the
temperature, reduce the monomer concentration, and shut off nucleation. Thus, in a
monotonically expanding nozzle there is a distinct nucleation pulse, and we can
characterize the length of this pulse using Eq. 3. In these experiments we find that the
characteristic time ∆tJmax is on the order of ~ 1 × 10-5 s. In small angle x-ray or neutron
scattering experiments with other species, e.g. water, alcohols and alkanes, in similar
nozzles, we typically observe particle number densities, N on the order of 1012±1 cm-3.
Since the experimental nucleation rate can be found from N and ∆tJmax using the
expression
J exp =
N
∆t J max
,
(13)
these estimates suggest that the nucleation rates for our experiments are on the order of
1017±1 cm-3s-1.
3.3.3
Comparison with Literature Values
Figure 3.5 compares the current data set with the data available in the literature.
The grey diamonds are our data points and the grey circles are the nucleation pulse
chamber (NPC) data of Iland et al. Typically, the nucleation rates reached in the NPC are
about ten orders of magnitude lower than those observed in supersonic nozzles. Relative
to the NPC data, the pressure-temperature data from nozzles should, therefore, lie at
57
higher pressures for the same temperature or at lower temperatures for the same pressure.
From Figure 3.5, it is clear that the data from our nozzle is consistent with the Iland et al.
data as well as the measurements of Fladerer and Strey (Fladerer & Strey 2006)(empty
circles) that were measured in an earlier version of the NPC. The open triangles are the
earliest onset measurements in supersonic nozzles, determined by Stein(Stein 1974) for a
M = 1.85 nozzle. For the same onset pressures, these data lie at significantly higher
temperatures, relative to both our data and the NPC data. The Stein data are not
consistent with the NPC data or our data set even though our stagnation conditions
overlap in part with his. In fact, in our initial measurements in a similar (M=1.96) nozzle,
we were never able to detect condensation although we did observe changes in the flow
due to external heat addition. The other nozzle data, the gray hexagons, are from Wu et
al.(Wu et al. 1978), who used a significantly faster nozzle (M = 3.3) than we did. The
only data point from the Wu et al experiments that overlap our temperature range
corresponds to a pure Ar experiment and the onset is at a pressure about 45% lower than
our measurements.
58
105
p / Pa
104
gas/liquid
gas/solid
Matthew & Steinwandel
Zahoransky et al. (1995)
Zahoransky et al. (1999)
Wu et al.
Stein
Fladerer & Strey
Iland et al.
this work
103
102
101
20
30
40
50
60
70
80
T/K
Figure 3.5: Phase diagram of argon along with the onset conditions of argon nucleation
measured in various devices. The solid line represents gas/liquid bimodal, while the
dashed line corresponds to the gas/solid binodal. The onset conditions are shown by the
various symbols; this work (gray diamonds), Matthew and Steinwandel (down triangles),
Zahoransky et al. (data from 1995: squares, data from 1999: empty diamonds), Wu et al.
(hexagons), Stein (gray triangles), Fladerer and Strey (empty circles) and Iland et al.
(filled circles).
There are a number of potential problems associated with the earlier nozzle
experiments. In both, Stein’s and Wu et al.’s experiments, light scattering at the exit of a
plastic nozzle was used to detect condensation, and the corresponding temperature and
pressure at the exit was found from the effective shape of expansion. The effective shape
of the expansion was based solely on geometry (Stein) or a single pressure trace
measurement made with Ar starting from room temperature and 1 atm, even though the
59
initial densities for the condensing experiments at low temperatures were as much as a
factor of 2 higher than the nozzle calibration data. As discussed in Section III.A, the
effective area ratio is a strong function of density and even small changes can lead to
fairly substantial errors in interpreting the conditions corresponding to the onset of
condensation. Furthermore, we have also seen that heat addition to the flow can be
significant when experiments start at such low temperatures.
The data from Zahoransky et al. (from 1995: squares; from 1999: empty diamonds) were
measured in shock tubes and also correspond to nucleation rates about ten orders of
magnitude lower nucleation rates than those found in nozzles. Thus, our data are
consistent with Zahoransky et al.’s data from 1999 and the higher temperature points
from 1995. Zahoransky’s 1995 data taken at lower temperatures lie too close to the
gas/liquid binodal and suggest heterogeneous rather than homogeneous nucleation.
Similarly, Matthew and Steinwandel’s (Matthew & Steinwandel 1983) data (downward
facing triangles) were also measured in shock tubes and lie right on the binodal,
indicating condensation due to heterogeneous nucleation.
To better compare the consistency between the data sets, while accounting for the
differences in temperature and supersaturation between the experiments, we summarized
the NPC and supersonic nozzle data in Figure 3.6 in the form of a Hale plot. The error
bars correspond to the uncertainty in the nucleation rates for each device noted earlier.
The scaled supersaturation, C0(TC/T-1)3/(ln S)2, in Hale’s scaling formalism (Hale 1992)
provides a basis to compare experimental nucleation rates of any magnitude measured at
arbitrary temperature and pressure. A consistent set of nucleation data should lie close to
60
a straight line with slope equal to 1 when an appropriate value of C0 is applied to all of
the data.
30
Stein
Wu et al.
Fladerer & Strey
Iland et al.
this work
25
to 40
-log10(J/1026)
20
15
to 48
10
Argon
C0=20, Ω=1.4
5
0
0
5
10
15
20
25
30
C0[Tc/T-1]3/(ln S)2
Figure 3.6: A Hale plot for argon shows the supersonic nozzle data measured in this work
(diamonds), by Stein (triangles) , and by Wu et al. (hexagons) as well as the NPC data
of Fladerer and Strey (empty circles) and Iland et al. (filled circles). The dashed line at
45° corresponds to perfect agreement. The error bars on the data represent the uncertainty
in estimates of the nucleation rates, 2 orders of magnitude for the nucleation pulse
chamber data and one order of magnitude for the measurements in supersonic nozzles.
The arrows indicate that the data from Fladerer and Strey extend out to scaled
supersaturations of 40, while the Stein data extend out to 48. The measurements from this
work are in good agreement with the nucleation pulse chamber data of Iland et al and the
supersonic nozzle data of Wu et al.
From Figure 3.6, it is apparent that the data from this work scale well with the
nucleation pulse chamber data of Iland et al. and the high temperature data of Fladerer
61
and Strey. In contrast, none of the onset data measured by Stein reach the 45º line
although the data from Wu et al. look entirely consistent with our data. Physically, Ω can
be thought of as the surface entropy per molecule and is found from C0 using
 3C

Ω =  0 × ln 10 
 16π

1/ 3
(14)
Based on the value of C0=20 found here, for Ar Ω equals 1.4. This value is significantly
lower that the value, Ω= 2.1 that Hale suggested for Ar (Hale 1996) and other simple
liquids. In fact it is lower than that observed for substances such as the alcohols (Ghosh
2007) (1.73 < Ω <1.94), alkanes (Ghosh 2007) (2.31 < Ω < 2.55)– and rather close to that
found for either isotope of water (Sinha et al 2008b).
3.3.4
Comparison with Classical nucleation theory
The classical nucleation theory assumes that the critical clusters, the first
fragments of the new stable phase in the metastable vapor, are compact spherical object
with the same physical properties as the bulk material. Although these assumptions are
difficult to justify for the small clusters that play a crucial role in nucleation process, in
many cases CNT does a reasonable job of predicting nucleation rates over a limited
temperature window.
At other temperatures, however, the disagreement between
experiment and theory is many orders of magnitude because CNT exhibits stronger
temperature dependence than is observed experimentally. In Figure 3.7, we compare the
predictions of CNT with the data from this work (filled diamonds) and from Iland et al.
(filled circles). The pressures are plotted as a function of inverse temperature, because in
62
this form lines of constant nucleation rate are reasonably straight over the temperature
range of interest.
106
105
JCNT /cm-3s-1 =
pon / Pa
1018
104
1016
109
103
100
105
10-10
102
101
1.4
gas/liquid
binodal
1.6
1.8
10-20
2.0
2.2
2.4
2.6
2.8
3.0
100 T-1 / K-1
Figure 3.7: The data from this work (diamonds) and Iland et al. (filled circles) along with
the gas-liquid binodal (solid line) are plotted as a function of inverse temperature. The
predictions of the CNT for constant nucleation rates JCNT of 10-20, 10-10and 100 cm-3 s-1
are shown by the dashed lines. The region shown by the empty circles represents the JCNT
ranging from 105-109 cm-3 s-1 corresponding to the measuring window of the nucleation
pulse chamber experiments by Iland et al.; while the dotted region represents JCNT
ranging from 1016-1018 cm-3 s-1corresponding to the measuring window of our
experiments.
The dashed lines in Figure 3.7, are the predictions of CNT at constant nucleation rates,
JBD, of 10-20,10-10 and 1 cm-3 s-1, the open circles are JBD ranging from 105-107 cm-3 s-1,
and the dots are for JBD ranging from 1016-1018 cm-3 s-1. The open circles correspond to
63
the rates estimated for the NPC, while the dots correspond to the measuring window of
our supersonic nozzle. From Figure 3.7, it is clear that the disagreement between CNT
and the experiments is many orders of magnitude in rate. Alternatively, CNT predicts
values of p that are 4 to 19 times higher than the pJmax observed in our experiments and
the discrepancy increases with decreasing temperatures. Our results confirm the
observations of Iland et al. where the p predicted by CNT was 2 to 13 times higher than
their experimental data. One difference between the two data sets is that our data move
toward higher rates as temperature decreases, while Iland et al’s move toward lower rate.
This difference most likely reflects that fact neither experiment is truly measuring a
constant rate. In the supersonic nozzles, we frequently observe an increase in rate with
decreasing TJmax. This was especially noticeable in the work of Ghosh et al. (Ghosh et al.
2007) for a series of n-alkanes. Likewise, Iland et al. stated that the disagreement
between their data and the earlier work of Fladerer and Strey at low temperatures was due
to lower expansion rates at the deepest expansions and that lower expansion rates would
correspond to lower nucleation rates.
To explore how well CNT predicts the temperature dependence of the supersonic
nozzle rates we calculated the ratio of the estimated nucleation rate for our experiments
(J= 1017±1 cm-3s-1) to the predictions of JBD and plotted this as a function of the inverse
temperature in Figure 3.8. Over the measurement range, the experimental rates are about
12 orders of magnitude higher than the predictions of CNT. If the error bars are taken
into account, the temperature dependence of CNT almost matches that observed for our
data, especially if we assume that the rates may increase as T decreases. The line in
Figure 3.8 is a fit to these data that is given by
64
ln
J exp
J BD
= 39.9 −
578.1
T
(15)
where we have used ln rather than log so that this function can be used to correct
classical nucleation theory in the form first suggested by Wölk and Strey (Wölk & Strey
2001), i.e.
J = J BD × exp(39.9 −
578.1
).
T
(16)
16
log(Jexp/JCNT)
14
12
10
8
1.8
2.0
2.2
100 T
-1
2.4
/K
2.6
-1
Figure 3.8: The ratio of the estimated nucleation rates for our experiments, Jexp of 10-17
cm-3s-1and the rate predictions of CNT, JCNT decreases slightly temperature decreases
(gray diamonds). The error bars correspond to the range of the nucleation rates typically
observed in nozzles-1017±1 cm-3s-1. The solid line represents the fit to Jexp/JBD.
65
In contrast, Iland et al. found the difference between the experiments and
CNT was about 16 orders of magnitude at high temperatures (58 K), and about 26 orders
of magnitude for the lower temperatures (42K). They reported the following fit to their
data scaled by JBD (Iland et al. 2007)
ln
J exp
J BD
= −27 +
3630
T
.
(17)
The difference between the two fit functions is striking. One reason for this difference
may simply be that the predictions of CNT are more sensitive to the onset pressures in the
measurement range of the NPC data than in our range. This is evident in Fig. 7, where at
lower pressures; the lines of constant JCNT are much closer than at higher pressures, for
the same temperature. For example, an increase in p from 500 to 800 Pa (a factor of 1.6
increase) at 45 K, results in CNT rate increase by 10 orders of magnitude, while a
proportionate increase in p from 3000 to 5000 Pa in our data range at the same
temperature results in an increase of only four orders of magnitude in the predicted
nucleation rate. This suggests that even if our data were parallel to the Iland et al. data,
we would not see such a large discrepancy between JBD and the experimental nucleation
rates, nor would we see as strong a deviation from the temperature dependence JBD than
Iland et al. do.
The large difference between the correction functions, Eqs. 15 and 17 also implies
that for Ar these correction functions are local rather than global. In other words, we
cannot use Eq. 16, and expect to predict the Iland et al data nor can we use Eq. 1 together
with Eq. 17 to match the nozzle data. This was not the case for D2O, where Kim et al.
66
showed that the correction function developed using the NPC data alone did an excellent
job of predicting the experimental nucleation rates measured in nozzle experiments.
3.3.5
Estimating the critical nucleus size
Earlier studies with H2O, and D2O found that despite the use of capillarity
approximation, the critical cluster sizes measured by the experiments were generally
within ±20% of those predicted by the Gibbs-Thomson equation,
32π vl σ 3
,
3 (kT ln S ) 3
2
n * GT =
(18)
where the experimental estimates of n* were derived from nucleation rate isotherms
using the first nucleation theorem (Kashchiev 1982; Viisanen et al. 1993; Oxtoby &
Kashchiev 1994),
 d (ln J ) 
 ≅ n * .

 d (ln S )  T
(19)
Because it was not possible to apply Eq. 19 to their data set, Iland et al. used the
Gibbs-Thomson equation to predict the critical nucleus size as a function of pressure and
temperature and overlaid their data on a plot containing lines of constant n*. Figure 3.9
reproduces this calculation and includes both the Iland et al. data and the current nozzle
data. From the figure we see that for the supersonic nozzle Eq.18 predicts a critical
cluster size of a little over 10 atoms at lower temperatures (40 K) up to 30 atoms at higher
temperatures (53 K). The Iland et al data correspond to larger critical clusters, with sizes
predicted to range between 40 and 80.The predicted critical cluster sizes for both data
67
sets increase with an increase in temperature and are consistent with the expectations
from thermodynamic considerations. (Iland 2004).
105
*
nGT = 10
gas/liquid
Iland et al.
this work
nGT* = 20
104
pon / Pa
nGT* = 120
103
102
101
35
40
45
50
55
60
65
T/K
Figure 3.9: Onset data of argon from this work (diamonds) and Iland et al. (circles)along
with lines of constant critical cluster size for n*GT = 10 (uppermost line), 20, 40, 60, 80,
100, 120(lowermost line) predicted by the Gibbs-Thompson equation. The dashed line
represents the gas-liquid equilibrium line.
To determine the critical cluster size experimentally, one should, ideally, have
isothermal nucleation rate measurements as a function of supersaturation made in the
same device. Unfortunately, there are no true nucleation rate measurements for Ar in any
device to date and, therefore, it is not possible to use such an approach to find n*. We can,
however, combine the estimated nucleation rates of argon in the nucleation pulse
68
chamber and the supersonic nozzle to estimate the size of the critical cluster using the
first nucleation theorem, albeit over a bigger range of rate and supersaturation than is
normally used.
1019
1018
1017
1016
1015
J / cm s
-3 -1
1014
1013
1012
1011
1010
109
108
42 K
107
106
10
Iland et al.
this work
52 K
5
104
10
100
1000
S
Figure 3.10: Data from this work (diamonds) and Iland et al. (circles) are shown in J vs.
S plot at six temperatures (42 -52 K). The vertical error bars represent the error in the
estimated nucleation rates in respective devices. The dashed lines represent the
predictions of the classical nucleation rate at constant temperature, when multiplied with
the temperature dependent correction factor (Eq.17). The solid lines represent the straight
line isotherms connecting the data from nozzle and nucleation pulse chamber
experiments.
69
Figure 3.10 shows our data (diamonds) along with those of Iland et al. (circles) at
six different temperatures (42-52 K in steps of 2 K). The vertical error bars on the data
correspond to the uncertainty in the estimated nucleation rates associated with each
experiment. The dashed lines are the nucleation rate isotherms predicted by Eq. 1 after
multiplying by Iland et al.’s correction function, Eq. 17 The temperatures corresponding
to these isotherms are chosen to be the same as the data i.e. 42 K (extreme right) to 52 K
(extreme left) in steps of 2 K. All of the data points are interpolated using the fit
functions. i.e. Eq. 12 and Eq. 2 from Iland et al.(Iland et al. 2007). It is clear from Figure
3.10 that the isotherms predict that nucleation rates in nozzles should peak at much lower
supersaturations than is observed. Clearly a simple temperature correction will not work
to describe the data from both devices in this case.
The solid lines in Figure 3.10 are the straight lines connecting the data from the
two devices at the same temperatures. The slope of this line is an estimate of n* at an
intermediate value of ln(S). The supersaturations corresponding to the midpoint of these
straight-line isotherms were used to calculate the critical cluster size, n*GT, using the
Gibbs-Thomson equation. Figure 3.11 compares the experimental values of n* to the
values of n*GT calculated using Eq. 18. The vertical error bars correspond to the
uncertainty in nucleation rates observed in the two devices, while the horizontal error
bars correspond to a 5% error in logarithm of the S at the midpoint of the straight-line
isotherms. The left ends of the dotted lines represent the value of n*GT calculated for
conditions present in the nozzle while the right ends represent the equivalent for the NPC.
Although we do not expect the actual nucleation rate isotherms to be straight over this
wide range of S, when we applied the same calculation procedure to n*and n*GT using the
70
D2O data of Kim et al.(Kim et al. 2004), we found that the n* values were about 20%
higher than the corresponding n*GT. Thus, our approach of finding n* is most likely an
upper bound on the actual experimental value.
40
n
*
30
20
10
0
0
10
20
30
40
50
60
n*GT
Figure 3.11: Values of n* derived using Eq. 19, plotted against the predicted
critical cluster size, n*GT from the Gibbs-Thomson equation. The vertical error bars
correspond to the uncertainty in the estimated nucleation rates in supersonic nozzles and
nucleation pulse chambers. The horizontal lines represent the uncertainty in the
supersaturations corresponding to the straight-line isotherms in Fig.. The left end of the
horizontal dotted line represents the n*GT for nozzle and the right end represents n*GT for
the experiments in nucleation pulse chambers. The solid line corresponds to perfect
agreement. The dashed –dotted and the dashed line represent the -30% and the -55%
error lines respectively.
In Figure 3.11 we see that the values of n* are all at least 30% lower than the
prediction of the Gibbs-Thomson equation, and the discrepancy increases for smaller
cluster sizes. This indicates that one of the prime reasons for the failure of CNT for Ar is
71
that the theory predicts a much larger critical cluster size and, therefore, a much lower
nucleation rates. When we correct CNT with a single, temperature dependent function,
the size of the critical cluster is not affected. Thus, if CNT does a good job of predicting
the critical cluster sizes, a simple correction to the temperature dependence of the rate,
that is valid over a wide range of supersaturations, is possible. It is remarkable that the
Gibbs-Thomson equation predicts such large critical cluster sizes for Argon even though
it works well for many substances.
3.4
Conclusions
We used the cryogenic supersonic nozzle apparatus presented in our earlier paper
to measure the conditions corresponding to the maximum nucleation rate of Argon in a
supersonic nozzle. Experiments were run with pure Ar and also in the presence of He
carrier gas. From estimates of characteristic time of nucleation and number densities
typically observed in nozzles we expect the nucleation rates to be 1017±1 cm-3s-1. As
demonstrated in a Wilson plot and a Hale plot, our results are consistent with the
measurements in nucleation pulse chambers. Classical nucleation theory under predicts
our estimated nucleation rates by about 11-13 orders of magnitude lower. Unlike Iland et
al., however, we do not see a very strong temperature dependence of the CNT for our
data. In fact the temperature dependence of our data follows that predicted by CNT quite
well. Our results show that the prime reason for failure of the CNT is that it over
estimates the critical cluster sizes. The experimental n* values are 30 – 55% lower than
the n*GT values predicted by the Gibbs-Thomson equation. Finally, we conclude that
since a purely temperature dependent correction factor does not affect the critical cluster
72
size, and the experimental n* are significantly smaller than those predicted by the GibbsThomson equation, this approach can only correct JBD locally, but not globally. True
nucleation rate measurements for argon are desirable in order to provide much more
accurate results and insight into the physics of argon nucleation. Small angle x-ray
scattering experiments to characterize the aerosol at the nozzle exit are extremely
challenging, but offer one possible solution.
73
T on
(K)
po
(kPa)
53.32
T0
(K)
108.64
yo
1.00
9.5185
54.84
8.17841
52.96
49.42
108.62
1.00
8.3089
53.53
7.45707
52.14
46.61
109.45
1.00
6.5048
50.10
6.45431
49.99
44.06
110.03
1.00
5.9125
49.55
5.96541
49.56
41.39
111.41
1.00
5.2820
49.19
5.35499
49.21
39.99
113.16
1.00
4.8930
49.15
4.84231
49.03
36.01
116.21
1.00
4.2263
49.62
4.03676
48.93
36.01
111.16
1.00
4.5462
48.86
4.61276
48.90
35.99
112.53
1.00
4.4282
48.97
4.42273
48.95
35.98
114.96
1.00
4.1400
48.70
4.15190
48.70
35.95
107.33
1.00
4.7840
48.19
4.85618
48.16
34.71
106.88
1.00
4.6905
48.29
4.69872
48.28
33.28
105.73
1.00
4.5961
48.19
4.60261
48.18
30.78
102.38
0.48
1.4879
41.12
1.50648
41.13
30.76
101.57
0.43
1.3068
40.25
1.33348
40.30
30.66
104.58
1.00
4.1385
47.23
4.20007
47.23
25.51
103.74
0.63
1.6492
41.84
1.65940
41.77
25.51
103.60
0.74
2.1142
43.38
2.13591
43.43
25.47
105.73
0.79
2.1536
43.39
2.22364
43.27
25.44
103.15
0.68
1.8683
42.42
1.90609
42.20
25.39
102.51
0.84
2.6335
44.56
2.66007
44.50
25.32
108.79
1.00
2.9880
46.45
3.02565
46.40
p on
(kPa)
p Jmax
(kPa)
TJmax
(K)
Table 3.1: The results for pressure trace experiments for Argon. y0 is initial mole fraction
of Ar. p0 and T0 are the stagnation pressure and temperature , pon and Ton are the onset
pressure and temperature, and pJmax , TJmax , and SJmax are the pressure, temperature, and
supersaturation corresponding to the maximum nucleation rate Jmax, respectively.
74
References
Becker, R. & Döring, W. 1935, "Kinetische Behandlung der Keimbildung in üebersättigten
Dämpfen", Annalen der Physik, vol. 24, pp. 719-752.
Evans, R. 1979, "The nature of the liquid vapor interface and other topics in the statistical
mechanics of non-uniform, classical fluids", Advances in Physics, vol. 28, no. 2, pp. 143.
Fladerer, A. & Strey, R. 2006, "Homogeneous nucleation and droplet growth in
supersaturated argon vapor: The cryogenic nucleation pulse chamber", Journal of
Chemical Physics, vol. 124, pp. 164710.
Garcia Garcia, N. & Soler Torroja, J.M. 1981, "Monte Carlo calculation of argon clusters in
homogeneous nucleation", Physical Review Letters, vol. 47, no. 3, pp. 186.
Ghosh, D. 2007, Pressure trace measurements and the first small angle X-ray scattering
experiments in a supersonic nozzle, Universität zu Köln.
Gladun, C. 1971, Cryogenics, vol. 11, pp. 205.
Granasy, L., Jurek, Z. & Oxtoby, D.W. 2000, "Analytical density functional theory of
homogeneous vapor condensation", Physical Review E: Statistical Physics, Plasmas,
Fluids, and Related Interdisciplinary Topics, vol. 62, no. 5-B, pp. 7486.
Hale, B. 1992, "The scaling of nucleation rates", Metallurgical Transactions A, vol. 23A, pp.
1863-1868.
Hale, B.N. 1996, "Monte Carlo calculations of effective surface tension for small clusters",
Australian Journal of Physics, vol. 49, pp. 425.
Haynes, W.M. 1978, Cryogenics, vol. 18, pp. 621.
Holleman, A.F., Wiberg, E. & Wiberg, N. 1985,
Lehrbuch der Anorganischen Chemie, Walter de Gruyter, Berlin.
Iland, K. 2004, PhD Dissertation, Universität zu Köln.
Iland, K., Wedekind, J. & Wölk, J. 2004, "Homogeneous nucleation rates of 1-Pentanol",
Journal of Chemical Physics, vol. 121, no. 12259.
Iland, K., Wölk, J., Strey, R. & Kashchiev, D. 2007, "Argon nucleation in a cryogenic pulse
chamber", Journal of Chemical Physics, vol. 127, pp. 154506.
75
Kashchiev, D. 1982, "
On the relation between nucleation work, nucleus size, and nucleation rate", Journal of
Chemical Physics, vol. 76, pp. 5098.
Kim, Y.J., Wyslouzil, B.E., Wilemski, G., Wölk, J. & Strey, R. 2004, "Isothermal nucleation
rates in supersonic nozzles and the properties of small water clusters", Journal of
Physical Chemistry A, vol. 108, pp. 4365-4377.
Kraska, T. 2006, " Molecular-dynamics simulation of argon nucleation from supersaturated
vapor in the NVE ensemble", Journal of Chemical Physics, vol. 124, pp. 054507.
Kusaka, I. 2003, "System size dependence of teh free energy surface in cluster simulation of
nucleation", Journal of Chemical Physics, vol. 119, pp. 3820.
Laasonen, K., Wonczak, S., Strey, R. & Laaksonen, A. 2000, "Molecular dynamics
simulations of gas-liquid nucleation of Lennard-Jones fluid", Journal of Chemical
Physics, vol. 113, no. 21, pp. 9741.
Luijten, C. C. M., Peeters, P. & van Dongen, M. E. H. 1999, "Nucleation at high pressure. II.
Wave tube data and analysis", Journal of Chemical Physics, vol. 111, pp. 8535.
Matthew, M.W. & Steinwandel, J. 1983, "
An experimental study of argon condensation in cryogenic shock tubes", Journal of
Aerosol Science, vol. 14, pp. 755.
Oxtoby, D.W. & Kashchiev, D. 1994, Journal of Chemical Physics, vol. 100, pp. 7665.
Pound, G.M. 1972, "Selected values of critical supersaturations for nucleation of liquids from
the vapor", Journal of Physical Chemistry Reference Data, vol. 1, pp. 119.
Reid, R.C., Prausnitz, J.M. & Poling, B.E. 1987, The properties of gases and liquids, 4th edn,
McGraw-Hill, New York.
Senger, B., Schaaf, P., Corti, D.S., Bowles, R., Pointu, D., Voegel, J.C. & Reiss, H. 1999, "A
molecular theory of the homogeneous nucleation rate. II. Application to argon vapor",
Journal of Chemical Physics, vol. 110, no. 13, pp. 6438.
Sinha, S., Laksmono, H. & Wyslouzil, B.E. 2008a, "A cryogenic supersonic nozzle apparatus
to study homogeneous nucleation of Ar and other simple molecules: Manuscript under
preparation".
Sinha, S., Wyslouzil, B.E. & Wilemski, G. 2008b, "Modeling of H2O/D2O condensation in
supersonic nozzles", Submitted to Aerosol Science and Technology.
76
Sprow, F.B. & Prausnitz, J.M. 1966, Transactions of Faraday Society, vol. 62, pp. 1097.
Stein, G.D. 1974, Argon Nucleation in a Supersonic Nozzle, Report to Office of Naval
Research available from National Technical Information Service Number: ADA007357/7GI.
Steward, R.B. & Jacobsen, R.T. 1989, Journal of Physical Chemistry Reference Data, vol. 18,
pp. 639.
Talanquer, V. & Oxtoby, D.W. 1994, "Dynamic density functional theory of gas-liquid
nucleation", Journal of Chemical Physics, vol. 100, no. 7, pp. 5190.
ten Wolde, P.R. & Frenkel, D. 1998, "Computer simulation study of gas-liquid nucleation in a
Lennard-Jones system", Journal of Chemical Physics, vol. 109, no. 22, pp. 9901.
Toxvaerd, S. 2001, "Molecular-dynamics simulation of homogeneous nucleation in the vapor
phase", Journal of Chemical Physics, vol. 115, no. 19, pp. 8913.
Viisanen, Y. & Strey, R. 1994, "Homogeneous nucleation for n-Butanol", Journal of
Chemical Physics, vol. 101, pp. 7835.
Viisanen, Y., Strey, R. & Reiss, H. 1993, "Homogeneous nucleation rates for water", Journal
of Chemical Physics, vol. 99, pp. 4680.
Voronel, A.V., Gorbunova, V.G., Smirnov, V.A., Shmakov, N.G. & Shchekochikhina, V.V.
1973, Soviet Physics JETP, vol. 36, pp. 505.
Wagner, W. 1973, Cryogenics, vol. 13, pp. 470.
Weakliem, C.L. & Reiss, H. 1993, "Toward a molecular theory of vapor-phase nucleation. III.
Thermodynamic properties of argon clusters from Monte Carlo simulations and a
modified liquid drop theory", Journal of Chemical Physics, vol. 99, no. 7, pp. 5374.
Wölk, J. & Strey, R. 2001, "Homogeneous nucleation of H2O and D2O in comparison: The
isotope effect", Journal of Physical Chemistry B, vol. 105, pp. 11683-11701.
Wölk, J., Strey, R., Heath, C.H. & Wyslouzil, B.E. 2002, "
Empirical function for homogeneous water nucleation rates", Journal of Chemical
Physics, vol. 117, pp. 4954.
Wu, B. J. C., Wegener, P.P. & Stein, G.D. 1978, "Homogeneous nucleation of argon carried
in helium in supersonic nozzle flow", Journal of Chemical Physics, vol. 69, pp. 1776.
77
Wyslouzil, B.E., Heath, C.H., Cheung, J.L. & Wilemski, G. 2000, "Binary condensation in a
supersonic nozzle", Journal of Chemical Physics, vol. 113, pp. 7317.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1999, "
Homogeneous nucleation of argon in an unsteady hypersonic flow field", Journal of
Chemical Physics, vol. 110, pp. 8842.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1995, "Formation of argon clusters by
homogeneous nucleation in supersonic shock tube flow", Journal of Chemical Physics,
vol. 103, pp. 9038.
78
CHAPTER 4
Modeling of H2O/D2O Condensation in Supersonic Nozzles
Somnath Sinha and Barbara E. Wyslouzil
Department of Chemical and Biomolecular Engineering,
The Ohio State University,
Columbus, OH-43210
and
Gerald Wilemski
Department of Physics,
Missouri University of Science and Technology,
Rolla, MO-65409
Submitted to Aerosol Science and Technology (May 2008)
79
Abstract: We have developed a steady state 1-D model to examine the formation and
growth of H2O/D2O droplets in a supersonic nozzle. The particle formation rate is
predicted using Hale’s scaled nucleation model. Droplet growth is modeled with five
different growth laws. Both isothermal and nonisothermal growth laws are considered.
We compared the predicted droplet sizes and number densities, to the values determined
by in situ small angle x-ray scattering experiments (SAXS) conducted under similar
conditions. Contrary to our expectations, the isothermal calculations are closer to the
experimental results than expected. Nonisothermal droplet growth does not quench
nucleation rapidly enough and almost always overpredicts the number density and,
therefore, underpredicts the droplet sizes.
80
4.1
Introduction:
Supersonic nozzles have been used extensively to investigate particle formation and
growth (Stein and Wegener 1967; Wegener 1969; Wegener et al. 1972; Moses and Stein
1978; Wyslouzil et al. 2000; Heath et al. 2002; Khan et al. 2003; Kim et al. 2004;
Tanimura et al. 2005; Wyslouzil et al. 2007), and, to a limited extent, to investigate
nanodroplet structure (Wyslouzil et al. 2006). Complementary modeling studies
(Ostwatitsch 1942; Wegener et al. 1972; Moses and Stein 1978; Young 1993; Wyslouzil
et al. 1994; Lamanna 2000; Fladerer et al. 2002; Streletzky et al. 2002) have been
conducted for many years in order to better understand particle formation in these devices
(Ostwatitsch 1942), to test existing nucleation or droplet growth models (Wegener et al.
1972; Moses and Stein 1978; Young 1993; Lamanna 2000; Fladerer et al. 2002), or to
predict appropriate operating conditions for novel nozzle designs (Streletzky et al. 2002).
In monotonically expanding nozzles under steady flow conditions, particle formation
is a self-quenching process. Nucleation produces the initial stable fragments of the new
phase, and the growth of these nuclei to nanometer sized droplets rapidly depletes the
vapor, shutting off nucleation. The heat released to the flow by the phase transition is
often large enough to increase the static pressure p, density ρ, and temperature T, above
that expected for the isentropic expansion of the gas mixture. The spatial variation of
these state variables has been followed using static pressure probes (Stein and Wegener
1967; Wyslouzil et al. 2000; Heath et al. 2002; Streletzky et al. 2002; Khan et al. 2003;
Kim et al. 2004), interferometry (Wyslouzil et al. 1994; Lamanna 2000), or spectroscopy
81
(Tanimura et al. 1995; Tanimura et al. 1996; Tanimura et al. 1997; Paci et al. 2004;
Tanimura et al. 2005, Tanimura et al. 2007), respectively. Alternatively, the depletion of
vapor from the gas phase, or the appearance of the condensate has been directly measured
using spectroscopy (Tanimura et al. 1995; Paci et al. 2004; Tanimura et al. 2005;
Tanimura et al. 2007). Finally, visible light scattering has been used to detect the
appearance of the aerosol, and follow, at least qualitatively, its continued growth (Stein
and Wegener 1967; Streletzky et al. 2002; Karlsson et al. 2007). Despite this extensive
experimental work, none of these measurements provides detailed quantitative
information on the aerosol itself, i.e., the size distribution of the particles formed during
condensation. Yet particle size and number density, the quantities most sensitive to the
competition between nucleation and growth, will provide the most stringent tests for
models of these processes.
In modeling conventional data sets, those that report the state variables or condensate
mass fraction, for example, it is difficult to uniquely determine the combination of
nucleation and growth models that best fit the available experimental data because
nucleation and droplet growth are such strongly coupled processes. Even with the total
flow rate of condensible vapor entering the nozzle fixed, changing the rate of either
process will change the predicted location of condensation in the nozzle. One can,
therefore, often compensate for the perceived inadequacy of one rate expression by
adjusting the other. The size distributions predicted by the different combinations of
nucleation and growth rates, are, however quite sensitive to the details of the calculation.
On the one hand, if droplet growth is overpredicted, nucleation may be terminated
prematurely and the resulting aerosol will have fewer but larger drops. On the other hand,
82
if droplet growth is underpredicted, many more particles are formed and these will be
significantly smaller.
In this work, we take advantage of the recent experimental measurements for water
(D2O/H2O) condensation conducted in our research group that include axially resolved
pressure measurements (Wyslouzil et al. 2007), a limited number of axially resolved gas
phase composition measurements (Tanimura et al. 2005), a series of small angle x-ray
scattering (SAXS) experiments (Wyslouzil et al. 2007), and direct nucleation rate
measurements (Khan et al. 2003; Kim et al. 2004; Wyslouzil et al. 2007). The SAXS
experiments are particularly important because they provide direct estimates of the
aerosol size distribution parameters including the average droplet size 〈r〉, the width of
the size distribution, ξ, and the total droplet number concentration, N, using only a weak
assumption about the general shape of the distribution.
The work reported here loosely follows that of Moses and Stein (1978), who
investigated steam condensation using both axially resolved pressure and light scattering
measurements. They then compared their experimental results to the predictions of a 1-D
model in order to evaluate the merits of a number of growth laws. Although they found
reasonable agreement between their data and two of the nonisothermal growth laws, they
only presented a detailed comparison for one experimental condition, rather than making
predictions over a wide range of conditions. Moreover, because the scattered light
intensity depends on the combination N〈r6〉, they could not distinguish inadequacies in
the models due to droplet size from those due to number density. Their analysis was
limited, both by the lack of size distribution measurements as well as by the absence of
stringent constraints on acceptable nucleation rate expressions as discussed in more detail
83
below. Such constraints are now available from recent advances in nucleation rate
measurements in supersonic nozzles, and they play an essential role in our analysis.
More recently, Lamanna (2000) used a 2-D flow model to predict condensation in
unsteady nozzle flows, and compared the results to experiments made in a Ludwig tube.
The experimentally determined quantities were the oscillation frequency and the droplet
size and number density. The aerosol properties were determined by multi-wavelength
white light extinction, a technique that yields accurate results only for droplets with radii
greater than ~100 nm. The modeling study tested both isothermal and nonisothermal
droplet growth laws, in conjunction with the Internally Consistent Classical Nucleation
Theory (Girshick and Chiu 1990), while aerosol evolution was followed by tracking a
number of moments of the particle size distribution. Lamanna found that both the
oscillation frequency and the droplet size were best matched when nonisothermal droplet
growth was taken into account.
Finally, other related droplet growth studies, such as those conducted in nucleation
pulse chambers or shock tubes (Peters and Paikert 1994; Fladerer et al. 2002; Luo et al.
2006), only study particle growth after nucleation is completed. Since these studies rely
on visible light scattering, the smallest particles observed, 〈r〉 ~ 0.5 µm, are much larger
than the 〈r〉 ~10 nm droplets formed in supersonic nozzles, and because these droplets
were produced by a nucleation pulse that is very short compared to the subsequent
growth time, they are essentially monodisperse. Again, the best agreement between the
data and the models requires nonisothermal droplet growth.
In contrast, we have found that over a rather wide range of experimental conditions,
the nonisothermal growth laws that worked well for larger droplets significantly
84
underpredict the growth rates of the smallest droplets. As a consequence nucleation is not
quenched rapidly enough, and the nonisothermal models predict the formation of an
aerosol that has a much higher number density of smaller droplets than is observed
experimentally.
4.2
Model Description:
In our model we assume that the flow in the nozzle is both steady and one-
dimensional, and that the effective area ratio of the nozzle A/A* is known. At the low
stagnation pressures used in the experiments, boundary layers grow along the nozzle
sidewalls, and these can be compressed as heat is added to the flow by condensation.
Ideally, the area ratio for the condensing flow of interest (A/A*)wet should be used as input
to the model, but this information is only available in a few cases. For most of the
calculations presented here we characterize the expansion using the slope d(A/A*)dry/dx
where (A/A*)dry is the area ratio obtained from measuring the pressure in the absence of
condensation and x measures distance along the flow direction of the nozzle.
At each point in the nozzle, the formation of new particles is determined by a
stationary homogeneous nucleation rate, while the growth of existing droplets is
determined by a droplet growth law. As droplet growth depletes the vapor, the decrease
in supersaturation quenches nucleation. Eventually, some of the smaller droplets become
sub-critical and evaporate, while the larger droplets continue to grow – an effect known
as Ostwald ripening. To account for the addition (removal) of heat to (from) the flowing
gas by the growing (shrinking) droplets, we numerically integrate the diabatic flow
equations (Wyslouzil 1994; Wyslouzil et al. 2000), described in Appendix B, from a
point just downstream of the throat to the nozzle exit. The flow path downstream of the
85
throat is divided into a large number of small steps of length ∆x. For each step, the model
determines all flow variables such as the density ρ, velocity u, pressure p, temperature T,
and the condensate mass fraction g, while simultaneously tracking the aerosol size
distribution as well.
In our model new droplet formation is determined by the species continuity
equation, (Ostwatitsch 1942; Miller 1988; Wilemski 1992) given here in finite difference
form as,
∆(n(xi)/ρ) = J(xi)∆x/(ρu)
(1)
Here, n is a particle number density and ρ, J, and u are, respectively, the total mass
density of the flow, the steady state nucleation rate, and the flow velocity, all at position
xi. The quantity ∆(n(xi)/ρ) is the number of new particles formed per mass of flow in the
interval ∆x at position xi, a quantity conserved at subsequent downstream positions
because we neglect coagulation. As such, it always represents the number of particles
(per mass of flow) that were formed with an initial size set by the critical radius ri* at the
local thermodynamic conditions of position xi. This critical radius is given by the Kelvin
equation,
ri* =
2σ vl
kT ln S
(2)
where σ is the surface tension of the condensable species, vl is the molecular volume in
the condensed phase, k is the Boltzmann constant and S is the supersaturation of the
condensable.
86
After the clusters are formed, their size is assumed to change only by monomer
addition and evaporation, i.e. coagulation is ignored. As the particles formed at xi move
downstream, their size is updated step by step. At position xj > xi, the current radius of
particles born at xi is given simply by
j
∑ ∆r
r ( x j ; xi ) = ri* +
k
,
(3)
k =i +1
where ∆rk is the change of radius due to growth or evaporation at position xk,
 dr  ∆x
.
(4)
∆rk =  
 dt  xk u ( xk )
The growth rate of the droplets, dr/dt, is calculated using one of the growth laws
described later. Since droplets of size ri* have equal probability of growing or
evaporating, we increase the value of ri* by 1% in the code to ensure that the initial
growth rate is not zero.
At any point in the flow, the aerosol consists of a superposition of particles of
different sizes formed at all upstream locations. With the current sizes of these particles
available, it is a simple matter to compute the mass fraction of condensed vapor at any
point in the flow using
g(x j ) =
4πρl
3
j
∑r
3
( x j ; xi )∆ (n( xi ) / ρ )
(5)
i =1
Finally, the model calculates the intensity of the expected scattering signal as a
function of the momentum transfer vector q, where q = (4π/λ)sin(θ/2), λ is the
wavelength of the radiation, and θ is the scattering angle. Because the aerosol is
polydisperse, the intensity I(q) is determined as (Pederson 1997)
87
I (q ) ∝ ∆ρb2 ∫ N ′(r ) F 2 (q, r )dr
(6)
where N ′(r ) is the number distribution of the droplets as a function of r, F(q,r) is the
form factor and ∆ρb is the difference in the scattering length densities of the droplet and
the surrounding medium, a value that depends on the materials involved and the nature of
the scattering radiation. The distribution function N ′(r ) , needed in Eq.(6) to compute
I(q) at position x, is obtained by binning the values of ∆(n(xi)/ρ) ρ(x) according to their
respective particle sizes r(x;xi). For homogeneous spheres with sharp interfaces, the form
factor for droplets of radius r is given as
F (q, r ) =
4π 3 3[sin( qr ) − qr cos( qr )]
r
.
3
(qr ) 3
(7)
To obtain the values of N, 〈r〉, and ξ from the model in a way that is consistent with the
experiments (Wyslouzil et al. 2007), we fit the predicted scattering function to a Schultz
polydisperse distribution of spheres (Kotlarchyk and Chen 1983).
4.3
Choice of Nucleation Rate:
Various theoretical expressions for calculating nucleation rates exist in the
literature (Hale 1992; Wölk and Strey 2001). The first theoretical treatment of
homogeneous nucleation, that is still widely used today, is the classical nucleation theory.
Becker and Döring (1935) derived the following expression for the nucleation rate JBD
J BD
2
 − 16πvl 2σ 3 
2σ  p v 
=
vl   exp 

πmv  kT 
 3(kT ) 3 (ln S ) 2 
(8)
Despite its continued use, it is well known that classical nucleation theory has stronger
temperature dependence than is observed experimentally (Hale 1992; Wölk and Strey
2001). Wölk and Strey (2001) therefore, developed an empirical correction function to
88
bring the predictions of classical nucleation theory into quantitative agreement with their
nucleation rate measurements. For the two isotopes of water they found,

6.5 × 10 3 

J H 2O = J BD exp − 27.56 +
T


(9)
for H2O, and

8.6 × 10 3 

J D2O = J BD exp − 35.98 +
T


(10)
for D2O.
Finally, Hale (Hale 1992; Hale et al. 2004; Hale 2005) developed a nucleation
model based on scaling arguments that has a temperature dependence that matches the
experimental results quite closely. Here the nucleation rate is calculated as,
−W 
J HALE = J 0 exp 

 kT 
where,
(11)
J 0 = 10 26 cm −3 s −1 ,
W 16π 3 (Tc / T − 1) 3
,
=
Ω
kT
3
(ln( S )) 2
where W is the work of formation of a critical cluster, and Ω is a dimensionless surface
entropy per molecule. The values of Ω can be derived from experimental nucleation rate
data or estimated from the physical properties of the substance of interest. For the two
isotopes of water, we estimated Ω H 2O = 1.44 and Ω D2O = 1.48 based on available
nucleation rate data.
89
One limitation inherent in previous modeling studies was the lack of nucleation
rate measurements under conditions typically found in supersonic nozzles. Thus, there
was no unequivocal way to choose between classical nucleation theory (Wegener et al.
1972; Moses and Stein 1978) and its variants (Lamanna 2000; Luo et al. 2006). Recent
work combining small angle neutron scattering (SANS) and small angle x-ray scattering
(SAXS) with pressure measurements has resulted in the first measurements of nucleation
rates for both isotopes of water in supersonic nozzles (Khan et al 2003; Kim et al. 2004;
Wyslouzil et al. 2007). In Figure 4.1, we compare the measured nucleation rates for D2O
to the predictions of the temperature corrected Wölk-Strey nucleation rate (Wölk and
Strey 2001), JWS, and Hale’s scaled nucleation model (Hale 1992; Hale et al. 2004; Hale
2005), JHALE. Both expressions fit the experimental data quite well over a 20 K
temperature range and for supersaturations ranging from 40 to 150. At the lowest
temperature (210 K), however, JHALE seems to provide a slightly better fit to the
supersonic nozzle data than JWS. Furthermore, Eq. 11 is easier to evaluate and is
independent of physical parameter correlations. Thus, we used JHALE for the nucleation
rate expression in our model.
90
1018
Khan et al.
A
B
J / cm-3s-1
C
}
Kim
et al.
Wyslouzil
1017
T/K
1016
20
30
230
220
40 50
210
100
150
S
Figure 4.1: The D2O nucleation rates measured in supersonic nozzles are compared to the
predictions of Hale’s scaled nucleation theory Eq. 11 (solid lines) and the Wölk-Strey
expressions Eq. 10 (dashed lines). The ratio Jexp/JHALE ranges from 0.2 at low temperature
to ~8 at high temperatures.
4.4
Choice of Growth Law:
As the critical clusters and droplets grow, they exchange mass and heat with the
surrounding mixture of carrier-gas and vapor. The mechanisms of heat and mass transfer
depend on the Knudsen number Kn = l/r, i.e., the ratio of the collision mean free path to
the droplet radius. For small Kn, these transfer rates are governed by diffusion, and the
processes take place in the continuum regime, while for large Kn the growth of droplets
occurs under free molecular flow conditions. For intermediate Kn, heat and mass transfer
takes place in the transitional regime (Lamanna 2000). In our experiments, Kn is always
greater than ~40, and heat and mass transfer is always in the free molecular regime.
91
Nevertheless, to more closely tie our efforts to previous work we will consider growth
laws that were developed to be applicable in all growth regimes (A) as well as those
restricted to the free molecular (FM) regime. In particular we consider the following five
growth laws:
(i)
the Hertz-Knudsen (HK) growth law (Wegener et al. 1972) (FM),
(ii) an isothermal growth law (A),
(iii) a nonisothermal droplet growth law (Peters and Paikert 1994) (A),
(iv) the Hertz-Knudsen-Smolder (HKS) nonisothermal growth law (Smolders 1992;
Lamanna 2000) (FM),
(v) an implicit nonisothermal growth law (Kulmala 1990; Kulmala 1993a; Kulmala et al.
1993b) (A).
The mathematical expressions corresponding to each growth law are summarized in the
Appendix B.
The Hertz-Knudsen growth law (i) is based on the kinetic theory of gases. Here,
the growth rate of the droplets is proportional to the difference between the molecular
impingement rate of the vapor molecules onto the droplet surface and the evaporation
rate from the droplet. This growth law is applicable only in the free molecular regime,
i.e., for large Knudsen numbers, and assumes that the droplet is in thermal equilibrium
with the surrounding medium. The latter is a reasonable assumption if the condensing
vapor is a small fraction of a vapor – inert gas mixture and the droplets are not growing
too quickly.
The isothermal growth law (ii) also assumes that the droplet is in thermal
equilibrium with the surrounding gas, but it is applicable in all three growth regimes, i.e.,
the continuum regime, the transition regime and the free molecular regime. This growth
92
law is the reduced form of the nonisothermal droplet growth law (iii) described below,
with the assumption that the droplet temperature, Td, is equal to the surrounding gas
temperature, T∞.
As the droplets grow, the condensation of vapor molecules on the droplet is
accompanied by the release of the latent heat of condensation that must be transferred to
the surrounding gas. Thus, if the droplets are growing rapidly and the heat transfer rate is
small, the temperature of the droplet will be higher than that of the surrounding gas. In
order to account for this effect, we need growth laws that allow the droplet temperatures
to differ from that of the surrounding gas. The three remaining growth laws account for
temperature differences between a droplet and the surrounding gas with different levels
of approximation and computational convenience.
The first nonisothermal droplet growth law (iii) is the modified version of
Young’s growth law (Young 1993) developed by Peters and Paikert (1994), and it is
applicable in all three growth regimes. Young’s growth law uses the Langmuir approach
and matches both the mass and the energy fluxes in the continuum regime with those in
the free molecular regime in the Knudsen layer surrounding the droplet (Young 1993).
The droplet temperature, Td, is calculated by solving the coupled mass and energy
equations. This nonisothermal growth law was tested extensively by Peters and Paikert
(1994) on both growing and evaporating micron size water droplets, i.e., for small to
moderate Kn (~0.05-0.5). However, this growth law was developed to cover the entire
Kn range, and here we test the growth law on droplets with radii less than 10 nm, i.e. for
large Kn.
93
Although the nonisothermal growth law accounts for the physics involved in
droplet growth, solving the coupled mass and energy equations at every step of the
integration scheme requires significant computing time relative to the isothermal and
Hertz-Knudsen growth laws. Alternate growth laws that require less computational effort
to estimate or incorporate the temperature difference between the droplet and the carrier
gas are, therefore, desirable especially in models that use more sophisticated descriptions
of the fluid flow.
The Hertz-Knudsen-Smolders growth law (iv) was developed by combining the
Hertz-Knudsen growth law with a second-order explicit expression for the droplet
temperature (Smolders 1992; Lamanna 2000). Since the droplet temperature is calculated
explicitly in a single step; the computation is much faster than the nonisothermal growth
law. The restriction to the free molecular regime is not a limitation for our conditions.
This growth law was extensively tested for larger droplet sizes (up to ~150 nm) by
Lamanna (2000) while modeling growth of droplets in unsteady nozzle flows, and the
results were found to be close to Young’s full solution for droplet temperature.
Finally, the growth law (v) developed by Kulmala (1990) is based on an
approach similar to the nonisothermal growth law developed by Fuchs and Sutugin
(1970, 1971). Here, the mass and heat fluxes in the continuum regime are multiplied by
Knudsen number dependent factors to connect the expressions corresponding to the free
molecular and continuum regimes. Kulmala then incorporated a linearized temperature
dependence of the equilibrium vapor pressure into the Fuchs-Sutugin equations thereby
making the mass flux linearly dependent on temperature of the gas T∞. This approach
implicitly allows Td to differ from that of the surrounding gas, but Td is not explicitly
94
determined, although as shown in Appendix B, Td can be calculated. Since the energy
balance has been eliminated and the growth rate involves only the gas temperature, the
computation time is significantly reduced compared to the nonisothermal growth law
(iii). Unlike the HKS approach, the Kulmala growth law is applicable in all regimes.
Because the temperature dependence of the equilibrium vapor pressure has been
linearized, this growth law should be restricted to small or moderate supersaturations.
Prior modeling studies by Fladerer et. al. (2002) of droplet growth in a nucleation pulse
chamber found, however, that this expression yields satisfactory results for water even
when the supersaturation exceeded 12.
4.5
Results and Discussion:
Importance of nonisothermal effects
To demonstrate that nonisothermal effects should be important under conditions
typically found during condensation in the nozzle, we calculated the difference in
temperature ∆T between droplets of size r and the surrounding gas temperature, T∞, as a
function of droplet radius for the nonisothermal and HKS growth laws. Figure 4.2
illustrates the results for gas phase conditions representative of those found in the
nucleation zone during condensation in nozzle H2 when T0 = 298 K, p0 = 30 kPa and the
mass flow rate of H2O is 5.0 g min-1. For this experiment, the peak nucleation rate occurs
1.3 cm downstream of the throat. Under these conditions, ∆T is always positive and all of
the droplets are growing rapidly. Here, the nonisothermal growth law predicts higher
values of ∆T than the HKS growth law, especially for the smaller droplets. For droplets
95
larger than ~4 nm, ∆T approaches ~35 K and the difference in the droplet temperatures
50
3.0
40
2.5
2.0
30
p = 10 kPa
T = 218 K
S = 96
20
1.5
Td_NGL - Td_HKS / K
∆T / K
predicted by the two models is less than 1 K.
(iii) NGL
(iv) HKS
1.0
10
0.5
0
1
2
3
4
5
6
7
8
r / nm
Figure 4.2: The predicted differences between the droplet temperatures Td and the gas
temperature T∞ for the nonisothermal (dotted) and HKS (dashed) growth laws are shown
as a function droplet size. The gas phase conditions are typical of those found in the
nucleation zone. The solid line is the difference between the two droplet temperatures.
Since higher temperature differences lead to higher evaporation rates and hence
lower net growth rates, the nonisothermal droplet growth law will suppress the growth of
the smallest droplets relative to the HKS growth law and both growth laws will
significantly suppress droplet growth with respect to the isothermal growth laws.
96
We then examined ∆T predicted by the HKS growth law further downstream in
nozzle H2 for the same H2O condensation experiment. As illustrated in Figure 4.3, just
downstream of the nucleation zone, x = 2cm downstream of the throat, nonisothermal
effects are far more important than near the nozzle exit. As r decreases from 〈r〉, ∆T
decreases rapidly, and for the smallest droplets ∆T is negative, indicating that these
droplets are evaporating rapidly while the larger droplets are growing rapidly, i.e.,
Ostwald ripening is a significant effect even in the early stages of condensation. Both
figures 4.2 and 4.3 suggest that in order to accurately model condensation in a supersonic
nozzle one should incorporate nonisothermal growth effects.
40
x = 5.5 cm
∆Τ HKS / Κ
20
x =2cm
0
-20
-40
0
2
4
6
8
10
12
14
r / nm
Figure 4.3: The temperature difference between the droplets and the surrounding gas is
shown as a function of the droplet radius when the HKS growth law is used in the model
to predict condensation in nozzle H2 with a condensible flow rate of 5.0 g min-1 H2O.
The circles denote the mean droplet size at each location.
97
Comparison of modeling and experimental results
We first compared our model predictions to the axially resolved Tunable Diode
Laser Absorption Spectroscopy (TDLAS) and static pressure measurements made in
nozzle H for D2O by Tanimura. et al. (2005). This data set is important because both g
and p/p0 were measured; where p0 represents the total pressure at the nozzle inlet. Thus,
these experiments also provide the effective area ratio (A/A*)wet during condensation in
the nozzle. To best match Tanimura et al.’s data we therefore used (A/A*)wet in these
calculations.
The results for the isothermal (ii) and the nonisothermal (iii) growth laws are
shown in Figure 4.4. As noted above, the nucleation rate was calculated using JHALE, and
the combination of JHALE with the isothermal growth law does a good job of predicting
the experimentally observed values of g for this experiment. In contrast, when the
nonisothermal growth law is used, condensation is shifted significantly downstream. In
many earlier modeling studies (Wegener et al. 1972; Moses and Stein 1978), the
nucleation rate expression was adjusted by a temperature independent parameter in order
to bring the calculated quantities into agreement with the experimental data. Following
this approach, we multiplied JHALE by a factor Γ = 70 for the nonisothermal growth law,
and greatly improved the agreement with the experimental curve. The problem with this
approach, however, is that the good agreement between JHALE and the available
experimental nucleation rates makes it difficult to justify rate adjustments of more than a
factor of 10 (~1 order of magnitude). In the end it is difficult to say, based on this
comparison alone, which approach is correct. In particular, more detailed information
98
about the aerosol size distribution is critical to distinguish among the various growth
laws.
0.018
0.016
p0 = 30.1 kPa
pc0 = 0.75 kPa
0.014
0.012
T0 = 298.17 K
TDLAS experiments
IGL : Γ = 1
g
0.010
0.008
NGL : Γ = 1
NGL : Γ = 70
0.006
0.004
0.002
0.000
-0.002
0
1
2
3
4
5
6
7
x /cm
Figure 4.4: The mass fraction of condensate in the flow was determined by following the
depletion of D2O from the gas phase using TDLAS (Tanimura et al.2005). In this
experiment, the mass flow rate of D2O was 4.155 g min-1 corresponding to an initial mass
fraction of 0.018. The partial pressure of D2O at the inlet to nozzle H is denoted by pc0.
Recently, Wyslouzil et al. (2007) made the first quantitative size distribution
measurement on aerosols formed by nucleation and growth in supersonic nozzles using
small angle x-ray scattering (SAXS) measurements. In these experiments the stagnation
pressure and temperature were held constant while the mass flow rate of the condensable
(H2O or D2O) was varied. Two nozzles, H2 and C2, characterized by different effective
expansion rates, respectively, were investigated. The D2O experiments used nozzle C2.
Here, all experiments started from a stagnation pressure p0 = 60 kPa and stagnation
99
temperature T0 = 308 K, and the effective expansion rate was d(A/A*)/dx = 0.085 cm−1.
All H2O condensation experiments were conducted with p0 = 30 kPa. For experiments in
nozzle H2, T0 was 298 K and d(A/A*)/dx = 0.055 cm−1, while for nozzle C2 T0 was 308 K
and the effective expansion rate was d(A/A*)/dx = 0.079 cm−1. Most of the SAXS spectra
were measured ~5.5 cm downstream of the throat of the nozzle, but in one H2O
experiment in nozzle H2, droplets were characterized as a function of the distance
downstream of the throat at a water flow rate of 5.0 g min−1. All of the experimental
spectra were fit to Schulz distributions of polydisperse spheres to determine the average
radius, the width or spread of the distribution, and the number density of droplets as a
function of condensible flow rate. The flow rates chosen for the SAXS experiments were
very close to those used for the complementary pressure measurements.
We first consider the D2O experiments. Because (A/A*)wet is not available, the
model used the effective expansion rate as input. Figure 4.5 illustrates the experimental
and predicted pressure ratios, p/p0, as a function of position in nozzle C2 for two D2O
flow rates: 11.2 g min-1 and 2.2 g min-1. The deviation of the condensing flow pressure
profile from that of the carrier gas indicates the onset of condensation, and, for both D2O
flow rates, all of the growth models predict that onset occurs further downstream than
what is observed in the experiments. At the high D2O flow rate, it is easy to distinguish
the predictions of the different models. In particular, the isothermal growth law and the
Kulmala growth expression (v) predict onset closest to the experimental value, but in the
rapid growth region immediately downstream of onset, the pressure increases more
rapidly than is observed experimentally. The nonisothermal (iii) and the HKS (iv) growth
expressions, predict onset further downstream, and the pressure increases less rapidly
100
than for other growth laws. In contrast, at the low D2O flow rate the pressure profiles
predicted by the various growth models are practically indistinguishable. Finally, we note
that the models always predict a higher pressure than the experiments because the models
do not account for compression of the boundary layer caused by condensation (Tanimura
et al. 2005).
0.6
0.6
p0 = 60 kPa
pc0 = 1.8 kPa
T0 = 308 K
(a)
0.5
(b)
p0 = 60 kPa
pc0 = 0.36 kPa
T0 = 308 K
0.5
p/p0
0.4
p/p0
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
2
4
6
8
0
x /cm
2
4
6
8
x /cm
Condensing flow (expt)
Carrier gas (expt)
(ii) Isothermal GL
(iii) Non-isothermal GL
(iv) HKS GL
(v) Kulmala GL
Figure 4.5: Pressure ratios as a function of nozzle position are shown for D2O mass flow
rate of (a) 11.2g min-1 and (b) 2.2 g min-1 .The gray dash-dotted line represents the
pressure ratio for the carrier gas flow in the nozzle, while the gray solid line represents
the pressure ratio for the condensing flow from pressure trace measurements. The
pressure ratios for the condensing flow predicted by the various models are shown by the
different black lines. The partial pressure of D2O at the inlet to nozzle C2 is denoted by
pc0. The Hertz-Knudsen growth model calculations are not shown because the results
overlap those of the Isothermal growth model.
101
To match the experimental and predicted onset locations for the experiments at
the high D2O flow rate, the nucleation rates would have to be increased by factors of 5170, depending on the growth model. As discussed earlier, adjustments greater than ~10
are difficult to justify. This suggests that the delayed onset predicted by the models is
primarily due to underprediction of the growth rate for the smallest droplets formed at
conditions close to the onset.
In order to compare the aerosol properties predicted by the model to those
determined in the SAXS experiments, we used the stagnation conditions corresponding to
the pressure trace experiments and the appropriate effective area ratios as inputs. For each
case we predicted a scattering spectrum 5.5 cm downstream of the throat, fit each
spectrum assuming a Schultz distribution of polydisperse spheres and determined the
parameters of the aerosol size distribution to avoid systematic bias in our comparisons of
the experimental and model results.
Figure 4.6 illustrates the changes in the aerosol parameters as a function of the
mass flow rate of D2O for experiments conducted in nozzle C2. The symbols correspond
to the experimental values and the lines correspond to the different models. The
nonisothermal and the HKS growth laws both capture the basic trends of the
experimental data correctly i.e. the curvature in the increase in 〈r〉 and decrease in ξ /〈r〉
with mass flow rate. In both cases, however, the predicted values of 〈r〉 are about a factor
of 2 smaller than the experimental values, while the predicted polydispersities and
number densities are significantly higher than those observed experimentally.
102
10
(a)
(b)
1e+13
-3
6
N /cm
<r>/ nm
8
4
2
1e+12
0
0
2
4
6
8
10
0
12
2
4
6
8
10
12
D2O flow rate / gmin-1
D2O flow rate / gmin-1
0.38
0.36
SAXS experiments
(i) Hertz-Knudsen GL
(ii) Isothermal GL
(iii) Non-isothermal GL
(iv) HKS GL
(v) Kulmala GL
ξ / <r>
0.34
0.32
0.30
0.28
0.26
0.24
(c)
0.22
0
2
4
6
8
10
12
D2O flowrate / gmin-1
Figure 4.6: The experimental values of the aerosol (a) mean radius (b) number density
and (c) polydispersity are compared to the model predictions as a function of the mass
flow rate of D2O (nozzle C2). All five growth rate expressions are considered. The
±20% error bars on the number density reflect the uncertainty in the (multiplicative)
calibration factor required to place the SAXS data on an absolute intensity scale. We
note that the uncertainty in the calibration factor affects all of the data equally, i.e., any
change in the calibration factor would shift the values of N by the same multiplicative
factor.
103
Surprisingly, the Isothermal and the Hertz-Knudsen growth expressions do a
somewhat better job of predicting the absolute values of 〈r〉 and N, although they predict
a sharper rise and decline for the 〈r〉 and N curves, respectively. This is observed even for
the highest flow rates where growth rates are high and nonisothermal effects should be
important. Finally, the Kulmala growth law does quite well, even though the
supersaturation ratio in the rapid growth region ranges from 50-250, well outside the
range of applicability for this growth law. Since all of the growth laws predict essentially
the same volume fraction of condensate φ at the position where the SAXS spectra were
measured, and φ = N〈r3〉, changes in N and 〈r〉 are anti-correlated. This means that if
droplet growth does not quench nucleation quickly enough, the number of droplets will
be too high and the droplet size will never be properly predicted. It is also noteworthy
that at the lowest D2O flow rate, all the growth models predict about the same 〈r〉 and N
consistent with the pressure ratio profile plot for the lower D2O flow rate in figure 5(b),
where we see similar results for all the growth models.
The results in Figures 4.4 and 4.5 clearly demonstrate the weakness of only
comparing models to integral quantities like g or p/p0. Figure 4 suggests that increasing
JHALE by a factor of 70 to match the experimental results is a reasonable approach. What
is not apparent in Figure 4.4 is that doing so increases the number density of the aerosol
by a factor of ~ 2 relative to the original nonisothermal calculation, further reducing the
final particle size. Furthermore, Figure 4.5 suggests that the isothermal growth models
result in predictions closest to the experiments although from Figure 4.6 it is apparent
that these growth models predict a much sharper rise and decline for the 〈r〉 and N curves,
respectively.
104
The results in Figure 4.6 also suggest that even the models that best fit the data,
underpredict the droplet size and overpredict the number density as the mass flow rate
decreases. One way the size and polydispersity of the droplets can increase and N can
decrease is by coagulation – an effect that is not included in our model. To estimate the
effect of coagulation, we calculated the expected change in the number density between
the nucleation region and observation point using the simple theory of Smoluchowski
(1916, 1917), but with a coagulation constant appropriate to the free molecular collision
regime (Fuchs, 1964; Seinfeld and Pandis, 1998). For this theory, N is given by
N
1
=
N 0 1 + N 0 Kτ
(12)
where
1/ 2
 3kT r 
K = 4
 .
ρ
l


Here, N0 is the number density predicted by the model after nucleation is complete, K is
the coagulation constant, ρl is the liquid density and τ is the time of flight of the droplets
from the nucleation zone to the SAXS observation point, i.e. ~5.5 cm downstream of the
throat.
We calculated the effect of coagulation for the lowest and highest flow rates using
the state variables and number densities predicted by the model using the Kulmala
growth law. In order to get an upper limit for the effect of coagulation, we calculated the
K values by using <r> predicted by the model, 5.5 cm downstream of the nozzle throat.
The decrease in number density predicted by Eq. 12 for the lowest flow rate, 2.25 g min-1,
is only about 30%. To be consistent with the experimental value, the predicted number
density corresponding to the same flow rate should be reduced by 85%. The calculation
105
corresponding to the highest flow rate also suggests coagulation is relatively unimportant,
with the number density predicted to decrease by only ~10%. Thus, accounting for
coagulation does not explain the large discrepancies for number density and size
observed at lower flow rates and will not change the predictions at high flow rates.
We next considered the SAXS data for H2O condensation. Here the expansions
started from a stagnation pressure p0=30 kPa and stagnation temperatures T0=298 K for
nozzle H2 and T0=308 K for nozzle C2. Decreasing the stagnation pressure by 50%
should enhance nonisothermal droplet growth effects since with less carrier gas the rate
of removal of energy released by the vapor condensing on the droplet is smaller. Thus,
the temperature difference between the droplet and the surrounding gas should increase,
and the growth rates should slow noticeably. Since, as illustrated in Figures 4.5 and 4.6,
the Hertz-Knudsen and the isothermal growth laws are practically indistinguishable, the
Hertz-Knudsen calculations are not reported.
Figure 4.7 illustrates the variation in average radius and number density of the aerosols as
a function of the mass flow rate of H2O for both nozzles. As was the case for D2O
condensation at p0 =60 kPa, using the nonisothermal and the HKS growth laws yields
droplet sizes much smaller and number densities much higher than observed
experimentally, although the general shape of the curves is close to that observed in the
experimental results. As in Figure 6, the Kulmala growth law does a good job of
matching the experimental results even though, as before, the supersaturations in the
rapid growth region are much higher than the range where the growth law should apply.
The isothermal growth law predicts results close to the experiments for the lower flow
106
rates, but for mass flow rates higher than 6.0 g min-1, it predicts a shock in the nozzle due
to excessive heat release that is not observed experimentally.
10
(a)
6
N /cm-3
<r> / nm
(b)
1e+13
8
4
1e+12
2
0
0
10
2
4
6
8
10
12
0
H2O mass flow rate/gmin-1
4
6
8
10
12
H2O mass flow rate/gmin-1
(c)
(d)
1e+13
8
6
N /cm-3
<r> / nm
2
4
1e+12
2
0
0
2
4
6
8
10
H2O mass flow rate/gmin
12
0
-1
2
4
6
8
10
12
H2O mass flow rate/gmin-1
SAXS experiments
(iii) Non-isothermal GL
(v) Kulmala GL
(ii) Isothermal GL
(iv) HKS GL
Figure 4.7: The experimental values of the aerosol (a) mean radius and (b) number
density in nozzle H2 and (c) mean radius and (d) number density in nozzle C2 are
compared to the model predictions as a function of the flow rate of H2O entering the
nozzle. As discussed in Fig. 4, the large error bars on N are due to uncertainty in the
calibration factor used to place the SAXS spectra on an absolute intensity scale.
107
The curvature in the Kulmala growth law curve, relative to the isothermal case, indicates
that the difference between the droplet temperature implicit in this expression and the
surrounding gas temperature is more important than at the higher stagnation pressure
used in the D2O modelling. Finally, for the fastest expansions (nozzle C2), and higher
flow rates, the experimental radii lie between those predicted by the Kulmala and HKS
growth laws, although at lower flow rates, the Kulmala growth law does a better job in
predicting 〈r〉 .
It is also noteworthy that the difference between the predictions of the nonisothermal
and the HKS growth laws is much more significant than in the D2O models at higher
pressure. The stronger nonisothermal effects lead to a bigger disparity in the droplet
temperatures predicted by Smolder’s approximate expression and the full Peters and
Paikert’s solution, and the former are in better agreement with the experimental data than
the latter.
All of the calculations presented so far tried to match the aerosol characteristics,
in particular the droplet size and number density, at a fixed position in the nozzle as the
flow rate of condensable was varied. In contrast, Figure 4.8 compares the one set of
position resolved droplet size and number density measurements made during these
SAXS experiments to the predictions of the same four models considered in Figure 4.7,
as well as a modified version of the HKS growth law that is described and discussed in
detail below. As in Figure 4.7, at this flow rate, the isothermal growth law overpredicts
〈r〉 and underpredicts N, while the nonisothermal, HKS and Kulmala growth expressions
underpredict 〈r〉 and overpredict N. Close to the throat, the agreement is best with the
Kulmala growth law although further downstream, after x~3cm, the experimental radii
108
and number density lie between the predictions of the isothermal and the Kulmala growth
law.
10
(b)
(a)
Jmax exp
6
N/cm-3
<r>/ nm
8
1e+13
4
1e+12
2
0
1
2
3
4
5
1
6
2
x / cm
3
4
5
6
x /cm
SAXS experiments
(ii) IGL
(iii) NGL
(iv) HKS GL
HKS modified GL
(v) Kulmala GL
Figure 4.8: The experimental values of the aerosol (a) mean radius and (b) number
density at a flow rate of 5.0 g min-1 H2O in nozzle H2 are compared to the model
predictions as a function of nozzle position. The location of the experimentally
determined peak nucleation rate is indicated by the arrow in (a). The model calculations
are indicated by the lines and the lines start from the position corresponding to the
maximum nucleation rate, Jmax , predicted by the model. The modified HKS GL,
discussed in more detail in the text, predicts the mean droplet radius and number density
when the temperature difference between the droplet and the surrounding gas estimated
by the HKS growth law is arbitrarily reduced by 50%. The large error bars on N are due
to uncertainty in the calibration factor used to place the SAXS spectra on an absolute
intensity scale.
In Figures 4.6, 4.7, and 4.8, the HKS growth law (iv) always predicts higher
growth rates than the nonisothermal growth law (iii) although both growth laws show
similar trends for the key aerosol parameters. In the limit of equal droplet and carrier gas
109
temperatures, the HKS and nonisothermal growth laws reduce to the Hertz-Knudsen and
the isothermal growth law, respectively, and, as illustrated in Figure 6, these laws predict
very similar growth rates. Thus, the observed difference between the HKS and the
nonisothermal growth law must be due to differences in the droplet temperatures
predicted by the two growth laws rather than any difference in the heat and mass fluxes at
the same droplet temperature. This difference in temperature was illustrated in Figure 2
for conditions representative of the nucleation zone.
We also calculated the droplet temperatures implicit in the Kulmala growth
expression by using Eq. (B14) in Appendix B. Our calculations show that even in cases
for which the implicit droplet temperatures for the Kulmala growth law is higher than
those predicted by the HKS and the nonisothermal growth laws, the Kulmala growth law
predicts higher mass flux and growth rates than the other two growth expressions. This
indicates that under conditions for which nonisothermal effects are not very important,
the Kulmala growth law would predict higher mass flux than the Hertz-Knudsen and
isothermal growth rate expressions. The good performance of this growth law under the
conditions discussed in this paper is, therefore, a fortuitous result.
To test the sensitivity of the predicted droplet size to ∆T, we ran the model using a
modified HKS growth law in which we arbitrarily reduced the value of the predicted ∆T
by 50%. For the position resolved data, Figure 4.8, we found much better agreement
between the predicted and experimental average droplet radius near the exit of the nozzle.
The predicted droplet size near x = 2 cm, however, is now too large, although closer to
the observed value than the isothermal calculation. One interpretation of this observation
is that the correction required for ∆T may be size dependent.
110
One of the limitations of the HKS and the nonisothermal growth laws is that they
predict the average temperature for droplets of given radius, and calculate the growth of
droplets of this size based on the average temperature. For very small droplets, however,
the distribution of droplet temperatures at a fixed size can be quite broad (McGraw and
Laviolette 1995; Wedekind et al. 2007). Moreover, the temperature of the droplets of a
particular size that are most likely to grow must be lower than the average temperature.
Therefore, using average droplet temperature to characterize the growth rate results in
underestimation of the growth rate.
4.6
Conclusions and Further work
The present study clearly demonstrates that comparing model predictions of
integral quantities, for example, the mass fraction of condensate or pressure ratio to
experimental data cannot adequately distinguish between different combinations of
nucleation and droplet growth laws. Aerosol size parameters, in particular the average
particle size and number density, provide much more stringent tests of available
nucleation and growth laws. Our results demonstrate that none of the growth laws tested
here agree with the experimental data under all conditions, although for some flow rates,
the isothermal, and the Hertz-Knudsen growth laws predict droplet sizes close to the
experimental values near the nozzle exit. This result is rather surprising given that the
isothermal growth laws ignore the temperature differences between the droplets and the
surrounding gas, an effect that should be important when the droplets are growing
rapidly. The Kulmala growth law predicts higher droplet growth rates than the HKS and
nonisothermal growth laws even when it predicts higher implicit droplet temperatures
and although it is implicitly nonisothermal, it should not be applicable under the highly
111
supersaturated conditions reached in nozzles. Despite this, it does a very good job overall.
Finally, the nonisothermal and the Hertz-Knudsen-Smolders growth laws capture the
shape of the <r> and N variation with condensible flow rate but underpredict the droplet
sizes and over- predict number density. Our results suggest that this is because the
temperature of the droplets that are growing is overpredicted. Although coagulation is not
considered in the model calculations, our analysis shows that it cannot explain the
observed discrepancy between the model and the experimental results. Finally, we
recognize that the present 1-D model is a simplification of the true flow in the nozzle. A
full 3-D model that tracks enough particle bins to include coagulation in a realistic
manner, rather than simply follow the moments of the size distribution, and that
calculates the effects of boundary compression, may result in better agreement with the
experimental data. A complementary set of experimental data that includes pressure,
spectroscopic temperature and composition measurements, as well as detailed position
resolved SAXS data, is currently being acquired.
Acknowledgements:
This work was supported by the National Science Foundation under Grant numbers CHE0518042. Use of the Advanced Photon Source was supported by the U. S. Department of
Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DEAC02-06CH11357. We thank Dr.Grazia Lamanna for providing her code to enable us to
compare our droplet temperature predictions with those from her code.
112
Glossary of Symbols:
Symbol
a
Meaning
sticking coefficient
A
nozzle flow area
A*
nozzle flow area at the throat
Cp
specific heat capacity
D
binary diffusion coefficient of the vapor-gas mixture
F(q,r)
particle form factor
g
mass fraction condensed (condensate mass/total flow mass)
J
nucleation rate
Jm
mass flux
Jq
heat flux
k
Boltzmann constant
K
coagulation constant
l
mean free path
L
specific heat of condensation
m
molecular mass
n(xi)
number density of droplets formed at position xi
N ′(r )
size distribution of droplets
N
total number density of droplets
N0
total number density of droplets at time zero
p
pressure
p0
stagnation pressure of vapor-gas mixture
113
pe(T)
equilibrium vapor pressure of bulk condensable at temperature T
per(T)
equilibrium vapor pressure of droplet of radius r at temperature T
as
given by the Kelvin or Gibbs-Thomson Equation (3a)
q
momentum transfer vector
r
droplet radius
r*
critical radius for cluster formation
〈r〉
average radius of the droplets
S
supersaturation ratio; pv / pe(T)
t
time
T
absolute temperature
T0
stagnation temperature
∆T
temperature difference between droplet and surrounding gas (Td-
T∞)
u
flow velocity
vl
molecular volume of the liquid
W
work of formation of a critical cluster
x
position in nozzle
Greek Symbols
γ
heat capacity ratio
Γ
multiplicative factor for nucleation rate
η
dynamic viscosity of the vapor-gas mixture
θ
scattering angle
κ
thermal conductivity of the vapor-gas mixture
114
λ
wavelength of radiation
ξ
width of size distribution
ρ
mass density
∆ρ b
scattering
length
density difference
between
droplet
and
surrounding medium
σ
surface tension
τ
time of flight of the droplets from the nucleation zone to the SAXS
observation point, ~5.5 cm downstream of the throat
Ω
dimensionless surface entropy per molecule
Subscripts
b
scattering length
d
droplet
g
carrier gas
l
liquid
0
stagnation conditions
v
condensable vapor
∞
conditions far from droplet surface
Dimensionless Numbers
Kn
Knudsen number (l/r)
Knv
Knudsen number for transfer of vapor (lv/r)
KnT
Knudsen number for heat transfer (lg/r)
Nu
Nusselt number
Pr
Prandtl number (Cp η/κ)
Sc
Schmidt number (η/ρD)
115
References:
Becker, R. & Döring, W. 1935, "Kinetische Behandlung der Keimbildung in
üebersättigten Dämpfen", Annalen der Physik, vol. 24, pp. 719-752.
Brock, J.R. & Hidy, G.M. 1965, "Collision-rate theory and the coagulation of freemolecule aerosols", Journal of Applied Physics, vol. 36, pp. 1857-1862.
Fladerer, A., Kulmala, M. & Strey, R. 2002, "Test of applicability of Kulmala’s
analytical expression for the mass flux of growing droplets in highly supersaturated
systems: growth of homogeneously nucleated water droplets", Journal of Aerosol
Science, vol. 33, pp. 391-399.
Fuchs, N.A. 1964, in The Mechanics of Aerosols The MacMillan Company, New York,
pp. 293.
Fuchs, N.A. & Sutugin, A.G. 1970, Highly Dispersed Aerosols, Ann Arbor Science
Publishers, Ann Arbor, MI.
Girshick, S.L. & Chiu, C.P. 1990, "Kinetic nucleation theory: A new expression for the
rate of homogeneous nucleation from an ideal supersaturated vapor", Journal of
Chemical Physics, vol. 93, pp. 1273-1277.
Hale, B. 1992, "The scaling of nucleation rates", Metallurgical Transactions A, vol. 23A,
pp. 1863-1868.
Hale, B.N. 2005, "Temperature dependence of homogeneous nucleation rates for water:
near equivalence of the empirical fit of Wolk and Strey, and the scaled nucleation
model", Journal of Chemical Physics, vol. 122, pp. 204509-1-204509-3.
Hale, B.N. & DiMattio, D.J. 2004, "Scaling of the nucleation rate and a Monte Carlo
discrete sum approach to water cluster free energies of formation", Journal of
Physical Chemistry B, vol. 108, pp. 19780-19785.
Heath, C.H., Streletzky, K., Wyslouzil, B.E., Wölk, J. & & Strey, R. 2002, "H2O-D2O
condensation in a supersonic nozzle", Journal of Chemical Physics, vol. 117, pp.
6176-6185.
Karlsson, M., Alxneit, I., Rütten, F., Wuillemin, D. & Tschudi, H.R. 2007, "A compact
setup to study homogeneous nucleation and condensation", Review of Scientific
Instruments, vol. 78, pp. 034102/1-034102/7.
116
Kell, G.S. & Whalley, E. 1965, "The PVT properties of water. I. Liquid water at 0 to
1500 and at pressures upto 1 kilobar", Philosophical Transcations of the Royal
Society of London. Series A, vol. 258, pp. 565-614.
Khan, A., Heath, C.H., Dieregsweiler, U.M., Wyslouzil, B.E., Wölk, J. & Strey, R. 2003,
"Homogeneous nucleation rates for D2O in a supersonic Laval nozzle", Journal of
Chemical Physics, vol. 119, pp. 3138-3147.
Kim, Y.J., Wyslouzil, B.E., Wilemski, G., Wölk, J. & Strey, R. 2004, "Isothermal
nucleation rates in supersonic nozzles and the properties of small water clusters",
Journal of Physical Chemistry A, vol. 108, pp. 4365-4377.
Kotlarchyk, M. & Chen, S.H. 1983, "Analysis of small angle neutron scattering spectra
from polydisperse interacting colloids", Journal of Chemical Physics, vol. 79, pp.
2461-2469.
Kulmala, M. 1993a, "Condensational growth and evaporation in the transition regime",
Aerosol Science and Technology, vol. 19, pp. 381-388.
Kulmala, M., Vesala, T. & Wagner, P.E. 1993b, "An analytical expression for the rate of
binary condensational particle growth", Proceedings of the Royal Society of London,
vol. 441, pp. 589-605.
Lamanna, G. 2000, On nucleation and droplet growth in condensing nozzle flows,
Eindhoven University of Technology.
Lide, D.R. 1991, Handbook of Chemistry and Physics, 72nd edn, CRC Press, Boston.
Luijten, C. C. M. 1998, Nucleation and droplet growth at high Pressure, Eindhoven
University of Technology.
Luo, X., Prast, B., Van Dongen, M. E. H., Hoeijmakers, H. W. M. & Yang, J. 2006, "On
phase transition in compressible flows: modeling and validation", Journal of Fluid
Mechanics, vol. 548, pp. 403-430.
McGraw, R. & LaViolette, R.A. 1995, "Fluctuations, temperature and detailed balance in
classical nucleation theory", Journal of Chemical Physics, vol. 102, pp. 8983-8994.
Moses, C.A. & Stein, G.D. 1978, "On the growth of steam droplets formed in a Laval
nozzle using both static pressure and light scattering measurements", Journal of
Fluids Engineering, vol. 100, pp. 311-322.
117
Ostwatitsch, K. 1942, "Kondensationserscheinungen in Überschalldüsen", Zeitschrift für
Angewandte Mathematik, vol. 22, pp. 1-14.
Paci, P., Zvinevich, Y., Tanimura, S., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson,
D. & McManus, B. 2004, "Spatially resolved gas phase composition measurements
in supersonic flows using tunable diode laser absorption spectroscopy", Journal of
Chemical Physics, vol. 121, pp. 9964-9970.
Pederson, J.S. 1997, "Analysis of small- angle scattering data from colloids and polymer
solutions: modeling and least squares fitting", Advances in Colloid and Interface
Science, vol. 70, pp. 171-210.
Peters, F. & Paikert, B. 1994, "Measurement and interpretation of growth and
evaporation of monodispersed droplets in a shock tube", International Journal of
Heat and Mass Transfer, vol. 37, pp. 293-302.
Rigby, M., Smith, E.B., Wakeham, W.A. & Maitland, G.C. 1986, in The Forces between
Molecules Clarendron Press, Oxford, pp. 218.
Sinha, S., Laksmono, H. & Wyslouzil, B.E. 2008a, "A cryogenic supersonic nozzle
apparatus to study homogeneous nucleation of Ar and other simple molecules",
Manuscript under preparation.
Smolders, H.J. 1992, Nonlinear wave phenomena in a gas-vapor mixture with phase
transition, Eindhoven Institute of Technology.
Smoluchowski, M.V. 1917, "Versuch einer mathematischen Theorie der
koagulationskinetik kolloider Lösungen", Zeitschrift für physikalische Chemie, vol.
92, pp. 129-168.
Smoluchowski, M.V. 1916, "Drei Vortrage uber Diffusion, Brownsche Bewegung und
Koagulation von Kolloidteilchen", Physikalische Zeitschrift, vol. 17, pp. 557-599.
Stein, G.D. & Wegener, P.P. 1967, "Experiments on number of particles formed by
homogeneous nucleation in the vapor phase", Journal of Chemical Physics, vol. 46,
pp. 3685-3686.
Streletzky, K.A., Zvinevich, Y. & Wyslouzil, B.E. 2002, "Controlling nucleation and
growth of nanodroplets in supersonic nozzles", Journal of Chemical Physics, vol.
116, pp. 4058-4070.
118
Tanimura, S., Okada, Y. & Takeuchi, K. 1997, "Fourier transform infrared spectroscopy
of UF6 clustering in a supersonic Laval nozzle: Cluster configurations in
supercooled and near-equilibrium states", Journal of Chemical Physics, vol. 107, pp.
7096-7105.
Tanimura, S., Okada, Y. & Takeuchi, K. 1996, "FTIR spectroscopy of UF6 clustering in
a supersonic Laval nozzle", Journal of Physical Chemistry, vol. 100, pp. 2842-2848.
Tanimura, S., Okada, Y. & Takeuchi, K. 1995, "FTIR spectroscopy of UF6 in supersonic
nozzle", Reza Kagaku Kenkyu, vol. 17, pp. 113-115.
Tanimura, S., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson, D. & McManus, B.
2007, "Tunable Diode Laser Absorption Spectroscopy Study of CH3CH2OD/D2O
Binary Condensation in a Supersonic Laval Nozzle", Journal of Chemical Physics,
vol. 127, pp. 034305/1-034305/13.
Tanimura, S., Zvinevich, Y., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson, D. &
McManus, B. 2005, "Temperature and gas-phase composition measurements in
supersonic flows using tunable diode laser absorption spectroscopy: The effect of
condensation on the boundary-layer thickness", Journal of Chemical Physics, vol.
122, pp. 194304/1-194304/11.
Viisanen, Y., Strey, R. & Reiss, H. 1993, "Homogeneous nucleation rates for water",
Journal of Chemical Physics, vol. 99, pp. 4680-4692.
Wedekind, J., Reguera, D. & Strey, R. 2007, "Influence of thermostats and carrier gas on
simulations of nucleation", Journal of Chemical Physics, vol. 127, pp. 16501/116501/12.
Wegener, P.P., Clumpner , J. A. & Wu, B. J. C. 1972, "Homogeneous nucleation and
growth of ethanol drops in supersonic flow", Physics of Fluids, vol. 15, pp. 18691876.
Wölk, J. & Strey, R. 2001, "Homogeneous nucleation of H2O and D2O in comparison:
The isotope effect", Journal of Physical Chemistry B, vol. 105, pp. 11683-11701.
Wyslouzil, B.E., Heath, C.H., Cheung, J.L. & Wilemski, G. 2000, "Binary condensation
in a supersonic nozzle", Journal of Chemical Physics, vol. 113, pp. 7317.
Wyslouzil, B.E., Wilemski, G., Strey, R., Seifert, S. & Winans, R.E. 2007, "Small angle
X-ray scattering measurements probe water nanodroplets evolution under highly
119
non-equilibrium conditions", Physical Chemistry Chemical Physics, vol. 9, no. 5353,
pp. 5358.
Wyslouzil, B.E., Wilemski, G., Beals, M.G. & Frish, M. 1994, "Effect of Carrier Gas
Pressure on Condensation in a Supersonic Nozzle", Physics of Fluids, vol. 6, pp.
2845-2854.
Wyslouzil, B.E., Wilemski, G., Strey, R., Heath, C.H. & Dieregsweiler, U. 2006,
"Experimental evidence for internal structure in aqueous-organic nanodroplets",
Physical Chemistry Chemical Physics, vol. 8, pp. 54-57.
Young, J.B. 1993, "The condensation and evaporation of liquid droplets at arbitrary
Knudsen number in the presence of an inert gas", International Journal of Heat and
Mass Transfer, vol. 36, pp. 2941-2956.
120
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1
Conclusions
In this work we have developed a cryogenic supersonic nozzle apparatus that can
be used to carry out condensation experiments with argon and other simple molecules.
With the present apparatus we are able to cool the gas to stagnation temperatures as low
as 100 K. Our study shows that these experiments must be carried out with extreme care
to prevent erroneous and misleading results. We found that in contrast to the previous
argon experiments in nozzles, a single room temperature experiments cannot be used to
characterize the expansion in the nozzle for the latter. The onset of condensation must be
determined using an isentrope that is consistent with the low temperature experiments.
Such an isentrope may be obtained by fitting a two straight lines to the area ratio profile
from a condensing flow experiment, upstream of condensation. Alternative one can
conduct a pressure trace experiment at room temperature that matches the “flat plate”
Reynolds number Re = ρux/µ where ρ is the gas density, u is the velocity, x distance
downstream of the throat, and µ is the dynamic viscosity. Furthermore, it is crucial to
maintain the nozzle sidewalls at temperatures close to the stagnation temperature, to
prevent heat addition or removal from the flowing gas.
121
In the present setup we were able to detect the onset of condensation of argon by
running experiments with either pure argon or in the presence of He carrier gas. We have
observed onset for temperatures in the range of 40 K < Ton< 53 K and pressures in the
range of 1.3 kPa< pon< 8.1 kPa. Our results are consistent with the measurements in
nucleation pulse chambers, and resolve the prior inconsistency between the nucleation
pulse chamber and nozzle data. From the comparison of our estimated nucleation rates
with the predictions of classical nucleation theory, we found that CNT underpredicts the
nucleation rates for our onset conditions. Nucleation pulse chamber results, however, the
temperature dependency of the CNT is not very strong for our onset conditions. Using
results from our work and Iland et al., and the first nucleation theorem, we found that the
size of the critical clusters predicted by the Gibbs-Thomson equation is overestimated by
factors ranging from 30 % to 55 %. The agreement with the Gibbs-Thomson equation
improves with increasing temperature. We believe that this is the primary reason behind
the low nucleation rates predicted by CNT for conditions corresponding to our data.
Furthermore, in case of argon, multiplying the CNT by an empirical temperature
dependent factor would only correct CNT locally, i.e. for a fixed or small range of
nucleation rates. Such empirical correction factors cannot work globally unless the theory
predicts the right critical cluster size.
We also modified and ran a 1-D model to predict condensation in supersonic
nozzles. Modeling condensation in nozzles requires the use of a nucleation rate
expression along with a growth rate expression. Hale’s scaled nucleation rate expression
was used along with five different growth expressions in our model. Two isothermal
growth laws along with three nonisothermal growth laws were used. The Hale’s scaled
122
nucleation rate model provides nucleation rates that are in good agreement with rates
observed in supersonic nozzles. Our model and the growth laws were tested by
comparing the predictions to the experimental results of small angle x-ray scattering
experiments that yield aerosol size parameters such as mean droplet size and number
density. Our modeling work shows that comparing the modeling results based on integral
quantities such as pressure or mass fraction of vapor condensed cannot adequately
distinguish between the different growth laws. Quantities that are sensitive to the
individual effects of nucleation and growth such as aerosol size parameters must be used
for stringent tests of the modeling results.
Surprisingly, we found that none of the growth laws provide results close to the
experiments under all conditions. Contrary to our expectations, the non-isothermal
growth laws overpredict the number density and, therefore, under predict the droplet size.
The nonisothermal growth laws appear to overpredict the mean droplet temperatures that
results in higher evaporation rates and lower net droplet growth rates than observed
experimentally. Thus, nucleation is not quenched fast enough, too many droplets are
produced, and the final droplet size is too small.
5.2
Future work
In our experiments with argon, we observed onset at temperatures as low as 40 K
and pressures as low as 1.3 kPa. Currently, for the range of our onset conditions, only one
data point from Wu et al. data is at the same onset pressure, while all other data are at
lower temperatures and pressures. To better compare our data with those of Wu et al., we
should run experiments that yield lower onset temperatures and pressures. Considering
123
the flexibility of our apparatus, these experiments can be carried out with relative ease. A
nozzle much faster than our present nozzle, with exit area ratio larger than 3.3, should
help us detect onset of condensation at these lower temperatures and pressures.
Experiments at much higher pressures and temperatures with argon would be extremely
valuable, since the vapor densities in these experiments would be high and therefore
nucleation at these conditions will be easier to model using simulation techniques.
Furthermore, to date, there is no true nucleation rate measurement for argon in any
device. Presently the comparison of experimental results with theory is based on
nucleation rate estimates in the experimental devices. Although such comparisons are
reasonable and provide valuable insight, it would be interesting to measure true
nucleation rates for argon. Nucleation rate measurements for argon in nozzles are
possible with a few modifications to our present setup. If pure Ar is used in our
experiments, then smaller, movable pumps can be used and it would be possible to move
our setup to Argonne National Labs, our x-ray beam source for carrying out x-ray
scattering experiments. One of the requirements of x-ray scattering is to have a window
along the sidewalls of the nozzle for the beam to pass through, which is extremely
challenging at the cryogenic temperatures associated with our experiments. However, if
low temperature expansions are appropriately characterized then the x-ray beam can be
passed through the mixture of gas and condensed particles exiting the nozzle. Such
experiments will enable true nucleation rate measurements.
The cryogenic experimental apparatus presented in this work can also be used to
study other molecules of interest such as N2 for which very limited nozzle data is
available. Using a cooling fluid that is at a higher temperature than liquid nitrogen for our
124
present heat exchanger will also allow us to conduct experiments with molecules such
CO2 in our present apparatus.
The 1-D model presented in this work is a simplification of the actual flow in the
nozzle. In future a 3-D model that tracks the particle bins rather than a few moments, and
accounts for the boundary layer compression, would describe the actual flow better. The
effect of coagulation should also be incorporated in such a model. Recently position
resolved x-ray scattering measurements along with pressure trace and Tunable diode laser
absorption spectroscopy (TDLAS) measurements have been carried out by our research
group. The data from the three techniques provide experimental droplet growth rates and
experimental droplet temperatures. These quantities can be compared with those from the
growth law predictions without the use of a nucleation rate expression to quantify the
overestimation of droplet temperatures by the nonisothermal growth laws. Perhaps an
empirical droplet temperature correction could be used to correct the nonisothermal
droplet temperatures and growth rates and incorporated into a 3-D model for closer
agreement with experiments.
125
APPENDIX A
APPENDIX TO CHAPTER 3:
THERMO-PHYSICAL PROPERTIES
126
Thermo-physical Parameters:
Argon
Mw = 39.948 × 10-3 kg mol-1 (Holleman et al. 1985)
Tc = 150.6633 K (Voronel et al. 1973; Steward & Jacobsen 1989)
pc = 4.860 MPa (Wagner 1973; Steward & Jacobsen 1989)
ρc = 13290 mol m-3 (Voronel et al. 1973; Steward & Jacobsen 1989)
T
 Tc
ε = 1 − 



ρC
 1000kg
(Haynes 1978)
+ 24.49248ε 0.35 + 8.155083ε 
3
 m
 mol / dm

ρl = M w 
3
pe

T
= exp  C − 5.90418853ε + 1.12549591ε 1.5 − 0.763257913ε 3 − 1.69733438ε 6 
pC

T
(Wagner 1973; Steward & Jacobsen 1989)
(
)
σ = (0.03778ε 1.277 )
N
 (Sprow 1966)
m
Cp,l = 44.1 J mol-1K-1 (Gladun 1971)
µ = C Ar T (Pa-s)
CAr = 1.346 × 10-6 (Reid et al. 1987)
Helium
Mw = 4.0026 X 10-3 kg mol-1 (Holleman et al. 1985)
Cp,g = 20.7233154 J mol-1K-1 (Fladerer & Strey 2006)
µ = C He T Pa-s
CHe = 8.2056 ×10-7 (Reid et al. 1987)
127
APPENDIX B
APPENDIX TO CHAPTER 4 :
DIABATIC FLOW EQUATIONS,
GROWTH LAW EXPRESSIONS & THERMO-PHYSICAL
PROPERTIES
128
Diabatic flow equations
The governing equations for steady nozzle flow are known as the diabatic flow equations.
(Wegener 1969) These equations are used to model condensation as described below.
The forms of the equations used in our model are easily derived from those given
elsewhere. (Wyslouzil et al. 1994; Wyslouzil et al. 2000) The mass density ρ, velocity u,
pressure p, temperature T, effective flow area A and the condensate mass fraction g are
related through these equations. The asterisk denotes the values of variables at the nozzle
throat, the subscript zero indicates the values at the nozzle inlet, L(T) is the latent specific
heat of condensation, and cp is the specific heat of the flow.
 L(T ) / c p − Tw( g )  dg − h( g )γ * T * ( u / u *) d ln ( A / A *)
d ln ( ρ / ρ 0 ) = 
2
h( g )γ * T * ( u / u *) − T
(A1)
d ( u / u *) = −2 ( u / u *)  d ln ( ρ / ρ0 ) + d ln ( A / A *) 
(A2)
2
2
2
dT =  L(T ) / c p  dg − [ w0 ( g ) − h( g ) ] γ * T * ( u / u *) d ( u / u *)
2
2
(A3)
In Eqs. (A1) – (A3), the quantities w0 (g), w (g), and h are given by
w0 ( g ) = µ / [ µ0 (1 − g )]
(A4)
w( g ) = µ / [ µv (1 − g ) ]
(A5)
h( g ) = w0 ( g ) −
cp0  γ 0 −1 


cp  γ 0 
(A6)
where µ is the mean molecular weight of the gas mixture, µ0 is the value of µ at the
stagnation conditions and µv is the molecular weight of the condensible vapor.
129
In modeling condensation, the stagnation conditions and A/A* are known. At any
point in the nozzle g is calculated using Eq.(5). This enables the new value of dg to be
found by subtraction. The increments dln(ρ/ρ0), d(u/u*)2, and dT are then successively
calculated using Eqs.(A1)—(A3). Thermodynamic conditions at the next downstream
location are then updated, with the pressure determined by the ideal gas equation of state
modified for condensation,( Wyslouzil et al. 1994)
p = ρ RT (1 − g ) / µ .
(A7)
The cycle is repeated until the desired downstream location has been reached.
The values of A/A* used in the model come either from the physical nozzle
geometry or by using a linear fit to the A/A* vs. nozzle position x obtained from
experiments conducted in the nozzle and using the slope d(A/A*)/dx in the model. The
latter approach is generally better, especially for small nozzles or low stagnation
pressures, because boundary layers change the expansion rate significantly from that
based on geometry alone. Ideally the area ratio for the condensing flow of interest (A/A*)
wet
should be used but this information is rarely available and (A/A*)dry is used as an
approximation.
130
Growth law expressions
To use consistent notation throughout this paper, many of the following equations have
been modified from their original forms. Based on prior theoretical and modeling studies
(Wegener et al. 1972; Moses and Stein 1978; Kulmala 1990; Peters and Paikert 1994;
Fladerer et al. 2002), the mass and thermal accommodation coefficients are assumed to
be unity for all the growth laws used in our modeling work.
(i) Hertz-Knudsen Growth law
The Hertz-Knudsen droplet growth rate is given by the expression (Wegener et al. 1972)
dr
−1/ 2
= ( 2π mv kT ) v l [ pv − per (T ) ] ,
dt
(B1)
where T = T∞ and pe,r is given by the Kelvin (or Gibbs-Thomson) equation
 2σvl 
p e ,r (T ) = p e (T ) exp 
.
 kTr 
(B2)
(ii) Isothermal growth law
The isothermal growth law used in our model is the reduced form of the nonisothermal
droplet growth law (iii) described below, with the assumption that Td = T∞.
(iii) Nonisothermal growth law
The droplet growth rate is determined by the mass flux Jm
dr J m
=
,
dt ρ l
(B3)
after solving the energy balance equation
J q = LJ m ,
(B4)
for Td. The expressions for Jm and Jq (Peters and Paikert 1994) are
131
Jm =
Jq =
mv D ( pv − pe,r (Td )) p
(B5)
 32 pD  m 1/ 2 
v
kT∞ pg r 1 +

 

5 pg r  2π kT∞  


κ (Td − T∞ )
 64T κ
∞
r 1 +

5 pr χ

1/ 2
 m 


 2π kT∞ 
(B6)




where
 γ + 1  m 1/ 2 p γ + 1  m 1/ 2 p
g
g
v
χ = v  
+


 γ v − 1  mv  p γ g − 1  mg  p


.


(B7)
(iv) Hertz-Knudsen-Smolders (HKS) growth law
The growth rate for the Hertz-Knudsen-Smolders growth law (Smolders 1992; Lamanna
2000) is given by
dr
−1/ 2
= ( 2π mv k ) vl  pvT∞−1/ 2 − per (Td )Td −1/ 2 
dt
(B8)
The droplet temperature is evaluated from the following expression
 Td

−1
 − 1 = f ( S , Ke)[C1 + C2 ] (1 − δ ) ,
 T∞ 
where
1 2
C1 − C2
δ≈2
f ( S , Ke) ,
(C1 + C2 )2
(B9)
f ( S , Ke) = ln S − Ke ,
Ke = 2σvl / kT∞ r
C1 =

T∞  p

− S  ,
ΘS  p e

132
C2 =
mv L
,
kT∞
and
Θ=
Dmod L Nu M
tr
.
tr
Nu H
The parameter Dmod is defined as
Dmod =
κ
Dp g mv
.
kT∞
In the transition regime, at intermediate Knudsen numbers, the Nusselt number is
expressed as
Nu tr =
Nu ct Nu fm
.
Nu ct + Nu fm
which has the correct asymptotic limit for both large and small values of Kn. The
approximate expressions for the Nusselt numbers in the free molecular limit for mass and
energy transfer, respectively, are
NuM fm =
2m Sc
,
π mv Kn
and
Nu H fm =
2 γ + 1 Pr
π 2γ Kn
In the continuum limit, under the hypothesis that Cpv(T∞-Td)/L and (pv-pe,r)/pg are smaller
than unity, the Nusselt numbers for mass and energy transfer are approximated as
NuMct = NuH ct ≈ 2.
(v) Kulmala growth law
The Kulmala growth law is given by (Kulmala 1990; Kulmala 1993a; Kulmala et al.
1993)
133
−1




p e ,r (T∞ ) L2 mv 
kT∞
dr ( p v − p e ,r (T∞ )) 
+
=
(B10)

p v + p e ,r (T∞ )  kβ ( KnT )κT∞ 2 
dt
rρ l


 mv β ( Knv ) D1 +

2p




where β is a function of Knudsen number that interpolates between the free molecular
and continuum regimes,
β ( Kn) =
1 + Kn
1 + (4 / 3 + 0.377) Kn + (4 / 3) Kn 2
(B11)
Knv = lv/r
(B12)
KnT = lg/r
(B13)
where lv and lg are given by Eqs.(5.65) and (5.66), respectively, of Wagner (1982).
The droplet temperature implicit in this growth law can be inferred from Eq. (15) of
Kulmala (1993a). Neglecting the surface radiation term one obtains the equivalent of
Eq.(6) in Kulmala (1993a) for the transition regime, which, in our terminology, appears
as
Td − T∞ =
r ρl L dr
.
β ( KnT )κ dt
(B14)
134
Thermo-Physical Parameters
R (Jmol-1K-1)
8.3145
C p − N (J/Kg)
1039.7
κair (W/m-K)
4.184 × 10 −3 (5.69 + 0.017(T − 273.15))
2
H2 O
Mv (g mol-1)
18.0152
Tc(K)
647.15 (Lide 1991)
Cp(J/Kg)
1865
σ (mN/m)
93.6635 + 0.009133T − 0.000275T 2
(Lide 1991; Viisanen et al.
1993)
ρ(g/cm3)
0.08 tanh( y ) + 0.7415t r
0.33
+ 0.32
(Wölk and Strey 2001)
y = (T-225)/46.2 tr = (Tc-T)/Tc
pe (Pa)
exp(77.34491 − 7235.42465 / T − 8.2 ln T + 0.0057113T )
(Wagner 1981)
L(J/g)
R ( 7235.42465 − 8.2T + 0.0057113T 2 ) / M v
55.521×10 −5 T 1.5
c (1,1) p
D N 2 − H 2O (m2/s)
c (1,1) = exp ( 0.348 − 0.459∆ + 0.095∆ 2 − 0.01∆3 )
(Rigby et al. 1986)
∆ = ln(T / 240.3)
D2O
Mv (g mol-1)
20.02
Tc(K)
643.89 (Hill et al. 1982)
Pc(MPa )
21.66 (Hill et al. 1982)
Cp(J/Kg)
1710
σ (mN/m)
93.6635+0.009133T’-0.000275T’2 (Jasper 1972)
135
where T’=T x 1.022
ρ(g/cm3)
0.09 tanh( y ) + 0.847t r
0.33
+ 0.338 (Wölk and Strey 2001)
y = T – 231/51.5 tr = (Tc-T)/Tc
p 
ln e 
 pc 
(
)
Tc
α 1τ + α 2τ 1.9 + α 3τ 2 + α 4τ 5.5 + α 5τ 10 (Hill et al. 1980; 1982)
T
α1 = -7.81583 α2 = 17.6012 α3 = -18.1747 α4 = -3.92488 α5 = 4.19174
τ = 1- T/Tc
L(J/g)
− RT ( ln ( pe / pc ) + Φ ) / M v
Φ = α 1 + 1.9α 2τ 0.9 + 2α 3τ + 5.5α 4τ 4.5 + 10α 5τ 9
D N 2 − D2O (m2/s)
50.5135 ×10 −5 T 1.5
c (1,1) p
136
APPENDIX C
PREVIOUS EXPERIMENTS WITH ARGON IN
NOZZLE WITHOUT COOLING JACKET
137
We conducted our first experiments using nozzle Ar-1, a nozzle without any
cooling jacket and with side wall temperatures close to room temperature. Our
experiments were run at conditions similar to Stein’s experiments. Most of the
experiments were run with pure Ar while in some cases we ran Ar experiments in the
presence of He carrier gas.
The results of the static pressure measurements (p/p0) for two of our earliest
experiments are shown in Figure A.1(a). The dashed line represents the p/p0 for
stagnation conditions p0 = 34.6 kPa and T0 = 294 K while the solid line represents the
measurements at the same starting pressure but at a much lower starting temperature of
123 K. The p/p0 profile for the room temperature run can be thought of as the isentropic
profile since supersaturation is not high enough at any point in the nozzle to observe
condensation in the nozzle. From Figure A.1(a) we see that the p/p0 measurement for the
experiment with lower starting temperature deflects from the measurement for the higher
temperature. Following our conventional practice of detecting onset, we thought that we
had seen condensation in the nozzle with onset at ~ 6.2 cm from the throat. The plot of
this onset condition on the Wilson plot for Ar is shown in Figure A.2 by the black
diamond. Contrary to our expectations this point was located in the region of data points
from Stein’s experiments. Thus, by the same arguments as in section 1.3, our result
would be inconsistent with the nucleation pulse chamber results.
138
0.45
0.45
(a)
T0=123K
0.40
0.35
0.35
0.30
0.30
p/p0
p/p0
(b)
P0=34.6 kPa
0.40
0.25
0.25
T0=123K
0.20
0.20
Ar 34.6 kPa
Ar 34.6 kPa
He 28 kPa
0.15
0.15
T0=294K
0.10
0.10
0
2
4
6
8
0
10
2
4
1.0
0.45
(d)
P0=36 kPa
0.40
8
10
8
10
x / cm
x / cm
(c)
6
Height along the nozzle
Argon
P0 =36 kpa,T0 =294K
P0 = 36 kPa,T0 =114.5K
P0 =34.6 kPa,T 0=123K
P0 =36 kPa,T0 =108.5K
0.9
0.35
0.8
p/p0
h /cm
0.30
0.25
0.7
0.20
T0=114K
0.6
0.15
T0=108K
0.5
0.10
0
2
4
6
8
0
10
2
4
6
x / cm
x / cm
Figure A.1: Pressure profiles and height along the nozzle
In order to ascertain if what we had seen was really condensation we carried out another
experiment at the same starting temperature (T0 = 123) as the previous low temperature
experiment i.e. T0 = 123 K but with a higher total pressure of p0 = 62.6 kPa. It should be
noted that the two experiments were run at the same partial pressure of Ar (pAr = 34.6
kPa.).The pressure measurements for these two experiments are presented in Figure
139
A.1(b).It is seen that by increasing the total pressure of flow in the nozzle by introducing
carrier gas results in disappearance of the deflection in p/p0 from the isentropic value.
Previous studies have shown that presence of carrier gas does not have a significant effect
on the onset. This indicated that observation in Figure A.1(a) was not due to
condensation but some other effects. The conditions corresponding to exit of the nozzle
for the experiment with the He carrier gas is shown in Figure A.2 by the grey diamond.
100
p/ kPa
10
Wu et al.
Zahoransky et al. (1995)
Zahoransky et al.(1999)
Matthew & Steinwandel
Fladerer (2002)
Stein
(101.24,34.66)
(99.3,30.66)
(97.35,24)
(95.4,18.66)
(91.5,13.33)
1
0.1
0.01
20
40
60
80
100
120
T/ K
Figure A.2: Wilson plot for Ar. The area to the right of and below the long dashed lines
corresponds to the accessible starting conditions for our planned experiments.
The point seems to be away from farther away from the Stein data points relative to the
black diamond for experiment with similar conditions but with carrier gas.
140
Figure A.1(c) reinforces the observation from Figure A.1(b). The solid line and
the dashed line in Figure A.1(c) correspond to experiments run with pure Ar at same
starting pressure of 36 kPa and starting temperatures of 114 K and 108 K respectively. If
the p/p0 for these two experiments were due to condensation, then the p/p0 for the
experiment with lower starting temperature should deflect from the isentropic values
upstream compared to the experiment corresponding to the higher starting temperature.
Thus, the bumps that we saw in p/p0 for experiments run with pure Ar and low starting
temperatures and pressures is not due to condensation.
We have tried to explain the observed results by boundary layer theory. Since the
flow in the nozzles for most part is turbulent, we calculated the flat plate turbulent
boundary layer displacement thickness, δ* for our experiments using expression (1).
δ* =
0.382 x
8 × (Re x )1 / 5
(1)
where Rex is the Reynolds number of flow at position x from the throat.
Figure A.1(d) shows the calculated true heights as a function of the nozzle
position by correcting for the boundary layer displacement thickness. In the absence of
any condensation this profile should be linear as shown by the dash–dotted line in the
figure. In case of condensing flows for species such as H2O or propanol these plots
deviated from the linear trend then again became parallel to the straight line further
downstream. However in case of these experiments with pure Ar under conditions of low
pressures and temperatures, these lines continue to deviate further through the
downstream of the nozzle. This indicated that our results may have been influenced by
heat addition. There was a large temperature gradient between the nozzle side walls and
141
the surrounding resulting in flow with heat addition that modified the boundary layers
and the effective shape of the expansion.
From our experiments and analysis, we concluded that we had not definitively
seen condensation in the nozzle. Although these experiments were unsuccessful, we
learnt that it is necessary to keep the nozzle walls at temperatures close to the stagnation
temperatures. Large temperature gradient can significantly modify boundary layer and
effective shape of expansion, resulting in pressure traces that deviate from traces at
adiabatic conditions and appear to be condensation at first glance. We ran our
experiments at conditions similar to Stein and got results which looked like condensation
when we used room temperature pressure measurements as the reference. Thus, it is
highly likely that Stein’s data are due to scattering from density fluctuations due to flow
with heat addition, rather than condensation. We expected onset in the nozzle to occur in
a region in line with the nucleation pulse chamber measurements. For Stein’s operating
conditions our expected region of onset on the Wilson plot is shown by the elliptical
region on Figure A.2.
Our existing experimental results showed that we had not seen any condensation
in nozzle A-1. Thus, in order to detect condensation we needed to reach higher
supersaturation in the nozzle. In the nozzle A-1, this could be achieved by either
increasing the stagnation partial pressure of Ar or by reducing the starting temperature.
However, we were limited by a maximum partial pressure of ~ 30 kPa, above which Ar
would condense on the coils of the radiator. Moreover, it was extremely difficult to
142
prevent heat exchange between our experimental apparatus and the surroundings and the
minimum starting temperature that had been achieved was 108 K.
Due to the above constraints, we decided to have a faster nozzle so that higher
supersaturations could be obtained in the nozzle for the same staring conditions.
Therefore, we built nozzle A-2 with design M~2.8 to help us achieve condensation of Ar.
We designed the nozzle with liquid nitrogen jacket so that the side walls could be
actively cooled by liquid nitrogen and maintained at temperatures close to T0.
143
APPENDIX D
FORTRAN CODE FOR DATA INVERSION
144
c
!!!!!!!!!!!!!!!!!!!!!!!!!THIS IS FOR ARGON-He mixture
!!!!!!!!!!!!!!!!!!!!!!!!!!! SOM
c
c
c
c
c
this version of the program calculates a "wet" isentrope based on the
measured dry isentrope and corrected for the differences in gamma. it
also starts the wet condensing flow integration on the desired data
point rather than on the wet isentrope to avoid any extraneous extra
shifts.
c smoothes all of the good density data first, then integrates from an
c initial value using finer integration grid (up to 5x)
c modified to take in pressure data instead of density data ....jul97,
jlc
c cleaned up and fixed integrator. now reduces multiple data sets,
c finds the onset conditions, and prints out the t-tisw values for all
c data sets in one file........................................ mar99,
bew
c note: stein used to smooth the integrated values as well... may
consider
c doing this for rough data... not yet implemented but easy to do....
bew
c this version has been modified for Nozzle H on train B with Velmex
(PP)
c RTD probe is calibrated and temperature calibration factor is
included
c Now nu.dat has "tempcal" and this program reads in the value and does
implicit real*8(a-h,o-z)
real*8 fcon
real*8 msq,msqw,mssq,machno
real*8 msq1,msq2,msqint,machsq,msquare
real*8 p0dummy,t0dummy, pc0, rg, pi, avog
real*8 dotm,dotncal,pc10,pc20,pocal,poloss,tocal,zc10
real*8 p0, t0, tempcal,fc10,fc20
real*8 xstart, xthroat
real*8 tt0(1000),fc(1000),g(1000),u(1000),
*rr0(1000),pp0(1000),pp0i(1000),pp0d(1000)
real*8 aratio(1000),wg(1000),wmug(1000),t(1000),tisw(1000),
*
tisd(1000),rm(11)
real*8 xd(1000),xw(1000),x(1000)
real*8 dry(1000),dryf(1000),sdry(1000)
real*8 wet(1000),swet(1000),wetf(1000),sweti(1000)
real*8 po(200),p(200),deltapo(200),deltap(200),to(200),
>
fc1(200),fc2(200) ! SOM now reading the flow meters
real*8 deltadry(1000),deltadryf(1000)
real*8 deltawet(1000),deltawetf(1000),dtemp(1000,20)
character*30 dryfil,wetfil,rlsfil,a
character*8 specie(2)
character*4 title(3,2)
common /xval/ xs(1000)
open(5,file='Arhe_fab.dat',status='old')
open(3,file='nu.out',status='unknown')
145
open(9,file='nua.out',status='unknown')
open(10,file='4pp.out',status='unknown')
open(11,file='wilson.out',status='unknown')
open(12,file='legend1.bat',status='unknown')
open(13,file='dtemp.out',status='unknown')
open(14,file='legend3.bat',status='unknown')
open(15,file='legend4.bat',status='unknown')
call echo
pi=3.14159d0
rg=8.3144d7
avog=6.022d23
c read condensible specie
read(5,41,end=50)specie
c
print 1006, specie
1006 format (2a8)
41 format(2a8)
c read stagnation conditions-temp, pressure, partial pressure of
c condensible--pressures are in mm of hg--note t0 and p0 are calculated
from data files later.
***********************************************************************
**********************************
read(5,*)fab,const1,slope1,brk,const2,slope2 ! If fab is non zero
then use a fabricated isentrope
***********************************************************************
**********************************
read(5,*)t0dummy,p0dummy,poloss,tempcal
!PP02
!RTD probe calibration added
write(*,*)'poloss = ',poloss,'
tempcal = ',tempcal
c convert pressures to dyn/cm**2
pconv = 760.d0/1.01325d6
c read molecular weights of carrier (1) and condensible (2)
read(5,*)wmN,wm2 !April 08
read(5,*)cpN,cp2 !April 08
c read latent heat, and specific heat of condensate
read(5,*)dhc2,cpc2
c read starting value and the number of points in the output
read(5,*) xstart, ilast
c read the integration end points (may be different than ifrst, ilast)
c the number of integrations attempts, and the number of integration
c sub-intervals
read(5,*)istart, ifin ,ni, nint
c read name of dry pressure data file
read(5,1)dryfil
1 format(a30)
c read smoothing parameters: m-order, n-number of points
read(5,*)md,nd
if(fab.GT.0) then
146
goto 7001
endif
open(unit=4,file=dryfil,status='old')
c read total number of values in dry pressure data file
read(4,1)a
read(4,*)idend
write(*,*)idend
do i=1,idend
read(4,*)xd(i),po(i),deltapo(i),p(i),deltap(i)
! xd(i) in
0.01mm,
p(i)= (p(i)*0.998359-0.0580226)
!p(i) from
black baratron (SOM)
po(i)=(po(i)*0.999496 + 0.01759)
!SOM (Calibration in
Som's notebook)
dry(i)=p(i)/po(i)
deltadry(i)=dry(i)
*
*((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5
c
write(*,*) i, xd(i), dry(i), deltadry(i)
!debug
enddo
close(unit=4)
7001 continue
! for fab
read(5,*)ndata, dotncal,pocal, tocal
wm1dry=wmN
cp1dry=cpN
!! Som April 08 !!
write(11,1302)dryfil
do kd = 1,ndata
c read name of wet pressure data file
read(5,1)wetfil
open(unit=4,file=wetfil,status='old')
c read total number of values in wet pressure data file
read(4,1)a
read(4,*)idenw
c read x values and all of the wet data p0, p(x), and the associated
standard deviations.
c correct the pressures i.e. baratron calibrations and pressure loss
due to mesh.
do i=1,idenw
read(4,*)xw(i),po(i),deltapo(i),p(i),deltap(i),to(i),dummy,
> dummy,dummy,fc1(i),dummy,fc2(i),dummy
p(i)= (p(i)*0.998359-0.0580226)
black baratron (SOM)
147
!p(i) from
po(i)=(po(i)*0.999496 + 0.01759)
!SOM
wet(i)=p(i)/po(i)
deltawet(i)=wet(i)
*
*((deltap(i)/p(i))**2.0+(deltapo(i)/po(i))**2.0)**0.5
to(i)=to(i)+tempcal
enddo
close(unit=4)
c figure out the average stagnant pressure and temperature
p0=0.0
t0=0.0
fc10=0.0
fc20=0.0
do i=1,idenw
p0=p0+po(i)
t0=t0+to(i)
fc10=fc10+fc1(i)
fc20=fc20+fc2(i)
enddo
p0=p0/idenw
t0=t0/idenw+273.15
fc10=fc10/idenw !SOM
fc20=fc20/idenw !SOM
pc10=p0*(fc10/(fc10+fc20))
write(*,*) 'The partialpressure of Ar :', pc10,'Torr'
devp0=0.0
do i=1,idenw
devp0=devp0+(po(i)-p0)**2.0
enddo
devp0=(devp0/(idenw-1))**0.5
write(*,*) 'average p0 is
', p0,'torr'
write(*,*) 'p0 std dev is
', devp0,'torr'
write(*,*) 'average t0 is
',t0,'k'
wm1 = wmN ! Som April 08
cp1=cpN
write(*,*) 'tN, wm1,cp1 =
', tN,wm1,cp1
148
if((pc10).lt.1.d-18)then
! Som April 08
write(*,*)'Insufficient condensible vapor!'
write(*,*)'Is this for Dry trace????'
pause !SOM
endif
c convert pressures to dyn/cm**2
p0=p0/pconv
!
pct0=pc10+pc20
pct0=pc10 ! Som April 08
pc10=pc10/pconv
wmav=(wm1*(p0-pc10)+wm2*pc10)/p0
w20=wm2*pc10/p0/wmav
wi=1.d0-w20
wmc=wm2 !som
c also let's save the inital average molecular weight
wmav0=wmav
!
cp0=wi*cp1+w20*cp2
c
cpv=cp2!som
cpc=cpc2 !som
dhc=dhc2 !Som
write(*,*) ' wmav, wi=',wmav,wi
write(*,*) 'dhc=', dhc
gamma=cp1dry/(cp1dry-rg*1.d-7/wm1dry)
!debug
gamma
gamma0=gamma
!
gamma0=cp0/(cp0-rg*1.d-7/wmav)
!initial
mixture gamma
rhog0=p0/rg/t0*wmav
write(*,*)'wmav',' w20','wi',' cp0',' gamma0' !chh061098
write(*,*)wmav, w20,wi,cp0,gamma0,gamma ! som
c calculate various exponents and constants involving gamma
eai
= 2.d0*(gamma-1.d0)/(gamma+1.d0)
eai0
= 2.d0*(gamma0-1.d0)/(gamma0+1.d0)
ep
= -gamma/(gamma-1.d0)
ep0
= -gamma0/(gamma0-1.d0)
erho
= -1.d0/(gamma0-1.d0)
emrho = gamma-1.d0
emrho0 = gamma0-1.d0
eam2
= (gamma+1.d0)/(gamma-1.d0)
eam20 = (gamma0+1.d0)/(gamma0-1.d0)
c1
= 2.d0/(gamma-1.d0)
c10
= 2.d0/(gamma0-1.d0)
149
!Ar
c2
c0
c3
= (gamma0+1.d0)/2.d0
= (gamma0-1.d0)/gamma0
= (gamma-1.d0)/gamma
if (fab.gt.0.0) then
goto 7002
endif
c figure out where the throat is for the dry data
c first figure out the value of pstar/p0=pstp0
pstp0 = (1.d0+ 1.0d0/c1)**ep
write(*,*)'pstp0 ',pstp0
do i=1,idend
write(*,*) i, xd(i), dry(i)
!debug
if((dry(i).gt.pstp0).and.(dry(i+1).le.pstp0))then
c
write(*,*) 'true'
!debug
xthroat=(pstp0-dry(i))/(dry(i+1)-dry(i))*(xd(i+1)-xd(i))+xd(i)
go to 5001
endif
enddo
5001
continue
write(*,*) 'dry throat of ',dryfil,' is at ',xthroat
c now shift all the x and scale so that x(i) is in units of cm.
c find number of unused points before xstart
!chh110698
ixstart=0
!chh110698
do i=1,idend
x(i)=(xd(i)-xthroat)/1000.0 ! in units of cm !Shinobu for
Velmex on Train B
if(x(i) .le. xstart)ixstart=i
!chh110698
enddo
write(*,*) 'ixstart= ',ixstart
!chh110698
c
do i=1,idend
write(9,1201) x(i), dry(i)
enddo
!debug
!debug
!debug
c now do linear interpolation to get fixed x intervals
c
save steps in inner loop by beginning interp. where left off
lasti=ixstart
!chh110698
do j=1,ilast
xs(j)=xstart+(j-1)*0.1
!in intervals of 1 mm
do i=lasti,idend
!chh110698
if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then
dryf(j)=dry(i)+(xs(j)-x(i))*(dry(i+1)-dry(i))/(x(i+1)-x(i))
deltadryf(j)=deltadry(i)
lasti=i
!chh110698
goto 5
endif
150
enddo
write(*,*) 'can not interpolate for point', j
5
continue
enddo
c we now have an array dryf(j) at fixed xs(j) intervals.
c through smoothing routine.
c
c smooth dry density values
c first do points at ends of good data range
do j=1,(nd-1)/2
k0=1
i=j
call smooth(md,nd,i,k0,sval,dryf)
sdry(i)=sval
k0=ilast-nd+1
i=ilast+1-j
call smooth(md,nd,i,k0,sval,dryf)
sdry(i)=sval
enddo
c next do points in good data range
do i=3,(ilast-2)
k0=i-(nd-1)/2
call smooth(md,nd,i,k0,sval,dryf)
sdry(i)=sval
enddo
1201
format(f10.4,f10.4,g14.4,f10.4,f10.4,g14.4)
if(fab.eq.0D0) then
goto 7050
endif
7002 do j=1,ilast
xs(j)=xstart+(j-1)*0.1
if (xs(j).lt.brk) then
aastar=const1+slope1*xs(j)
else
aastar=const2+slope2*xs(j)
endif
if (aastar.lt.1D0) then
aastar=1D0
endif
aastarsq=aastar**2
151
now put
msq1=1D0
msq2=9D0
ibiscnt=0.D0
func1=aastarsq-(1/msq1)*((2/(gamma+1))*(1+((gamma-1)/2)*msq1))
> **((gamma+1)/(gamma-1))
10
func2=aastarsq-(1/msq2)*((2/(gamma+1))*(1+((gamma-1)/2)*msq2))
> **((gamma+1)/(gamma-1))
if(func1*func2.gt.0.0) then
ibiscnt=ibiscnt+1
if(ibiscnt.eq.1000) then
write(*,*)'root must be bracketed for bis',msq1,msq2,func1,func2
stop
endif
if (dabs(func1-func2).le.2.d-15.and.dabs(func2).lt.5.d-15.and.
> dabs(msq1-msq2).lt.1.d-5)then
msq=msq2
GOTO 20
endif
machsq=(msq1*func2-msq2*func1)/(func2-func1)
if(dabs(func2).lt.dabs(func1))then
msq1=msq2
func1=func2
endif
msq2=machsq
goto 10
endif
if(func1.lt.0.) then
rtbis = msq1
dmsq = msq2-msq1
else
rtbis = msq2
dmsq = msq1-msq2
endif
do l = 1,50
dmsq = 0.5*dmsq
152
msqint = rtbis+dmsq
func2=aastarsq-(1/msqint)*((2/(gamma+1))*(1+((gamma1)/2)*msqint))
> **((gamma+1)/(gamma-1))
if(func2.lt.0.0)rtbis = msqint
msquare=rtbis
if(abs(dmsq).lt.0.005.or.func2.eq.0.0)goto 20
enddo
20
!
7050
sdry(j)= (1+(emrho/2)*msquare)**ep
write(*,*) xs(j),sdry(j),msquare
enddo
continue
c figure out where the throat is for the wet data
c first figure out the value of pstarw/p0=pstp0w
pstp0w= (1.d0+ 1.0d0/c10)**ep0
write(*,*)'pstp0w ',pstp0w
c
c
5002
do i=1,idenw
write(*,*) i, xw(i), wet(i)
!debug
if((wet(i).gt.pstp0w).and.(wet(i+1).le.pstp0w))then
write(*,*) 'true'
!debug
xthroat=(pstp0w-wet(i))/(wet(i+1)-wet(i))*(xw(i+1)-xw(i))+xw(i)
go to 5002
endif
enddo
continue
write(*,*) 'wet throat of ',wetfil,' is at ',xthroat
c now shift all the x and scale so that x(i) is in units of cm.
c find the number of unused points before xstart.
ixstart=0
do i=1,idenw
x(i)=(xw(i)-xthroat)/1000.0
153
if(x(i).le.xstart)ixstart=i
enddo
write(*,*) 'ixstart= ',ixstart
write(*,*) 'throat shifted'
!debug
c now do linear interpolation to get fixed x intervals
lasti=ixstart
do j=1,ilast
c xs values have already been assigned in dry data analysis
c
write(*,*) xs(j)
!debug
do i=lasti,idenw
if((x(i).le.xs(j)).and.(x(i+1).gt.xs(j)))then
wetf(j)=wet(i)+(xs(j)-x(i))*(wet(i+1)-wet(i))/(x(i+1)-x(i))
deltawetf(j)=deltawet(i)
lasti=i
goto 6
endif
enddo
6
write(*,*) 'can not interpolate for point', j
continue
enddo
c we now have an array wetf(j) at fixed xs(j) intervals.
c through smoothing routine.
write(*,*) 'put through smoothing'
c
c smooth wet pressure values
c first do points at ends of good data range
do j=1,(nd-1)/2
k0=1
i=j
call smooth(md,nd,i,k0,sval,wetf)
swet(i)=sval
k0=ilast-nd+1
i=ilast+1-j
call smooth(md,nd,i,k0,sval,wetf)
swet(i)=sval
enddo
c next do points in good data range
do i=3,(ilast-2)
k0=i-(nd-1)/2
call smooth(md,nd,i,k0,sval,wetf)
swet(i)=sval
enddo
write(*,*) 'finished interpolating points'
c msq is mach number squared
c calculate the 'wet' isentrope
write(*,*) 'calculate the wet isentrope'
do k = 1, ilast
msq=c1*((1.d0/sdry(k))**c3-1.d0)
154
now put
aratio(k)=dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq)
c
machno = dsqrt(msq)
machno=1.d0
100
c
! March04 Shinobu
rm(1)=dsqrt(c10*(c2*(machno*aratio(k))**eai0-1.d0))
do im=1,10
rm(im+1)=dsqrt(c10*(c2*(rm(im)*aratio(k))**eai0-1.d0))
enddo
machno=rm(11)
write(*,*) 'rm(11) = ', rm(im)
!debug
if(dabs(machno-rm(10)).gt.1.d-12)goto 100
sweti(k)=(1.d0+(1.d0/c10)*machno**2.d0)**ep0
write(9,1203)xs(k),sdry(k),sweti(k),swet(k),dsqrt(msq),
& machno,aratio(k)
enddo
1203 format(4(f9.5),2(f5.2),f6.3)
c use finer integration step size than measured point spacing
c generate interior points by linear interpolation
c nint is the number of subintervals between each pair of original x
values
write(*,*) 'nint= ', nint
c calculated the finer grid, interpolating on the wet condensing and
c wet isentrope data
write(*,*) 'calculate the finer grid'
npts=ifin-istart+1
nnpts=(npts-1)*nint+1
jinit=nnpts+2*nint+istart-1
do i=ifin+1,istart,-1
delx=xs(i)-xs(i-1)
delprd = sdry(i)-sdry(i-1)
delprwi = sweti(i)-sweti(i-1)
delprw = swet(i)-swet(i-1)
jinit=jinit-nint
jp=0
do j=jinit,jinit-nint+1,-1
fint=1.d0*dfloat(jp)/(1.d0*nint)
xs(j)=xs(i)-delx*fint
pp0d(j) = sdry(i)-delprd*fint
pp0i(j) = sweti(i)-delprwi*fint
pp0(j) = swet(i)-delprw *fint
jp=jp+1
enddo
enddo
ifin1=istart+nnpts-1
c calculate flow properties at nozzle throat
155
write(*,*) 'calculate flow properties at nozzle throat'
tstar=t0/c2
rhogst=rhog0*c2**erho
ustar=dsqrt(gamma0*rg*tstar/wmav)
rhoust=rhogst*ustar
do k = 1,ni
c need to calculate at istart-1 so adjust if istart=1
c since there is no good data avaiable before 1
write(*,*) 'start'
write(*,5000) istart
5000
format(3(I3,2x))
if(istart.eq.1)istart=istart+1
rosro0=rhogst/rhog0
tst0=tstar/t0
msq = c10*((1.d0/pp0i(istart-1))**c0-1.d0)
aratio(istart-1)=dsqrt(((c10/eam20*(1.d0+msq/c10))**eam20)/msq)
c note! start the wet condensing flow integration on the desired data
c point (i.e. on the wet curve data) rather than on the wet isentrope
c to avoid any extraneous extra shifts/offsets in t etc.
msqw = c10*((1.d0/pp0(istart-1))**c0-1.d0)
tt0(istart-1)=1.d0/(1.d0+msqw/c10)
rr0(istart-1)=(1.d0+msqw/c10)**erho
msq = c10*((1.d0/pp0i(istart))**c0-1.d0)
msqw = c10*((1.d0/pp0( istart))**c0-1.d0)
tt0(istart)=1.d0/(1.d0+msqw/c10)
rr0(istart)=(1.d0+msqw/c10)**erho
g(istart)=0.d0
g(istart-1)=0.d0
fc(istart)=0.0d0
!fraction condensed
u(istart-1)=rhoust/rhog0/aratio(istart-1)/rr0(istart-1)
1200
write(9,*)ifin1-istart+1
write(9,1200)t0,pc10*pconv,pc20*pconv, dotm
format(f7.0, 2f7.2, f7.3)
write(9,1009)
format('
xs(i)cm
u(i)cm/s
t(i)k g(i)gc/gnc
tis ',
&'t(i)-tis
pp0i(i)
pp0(i)
g(i)/(w20+w30)')
!chh100698
write(10,1010)
!4pp plots
1010 format('@with g1')
!4pp plots
write(10,1011)p0*pconv,devp0,t0-273.15
!4pp plots
1011 format('@title "',f6.2,'+/-',f4.2,'torr ',f6.2,'celsius"')
!4pp plots
write(10,1012)pc10*pconv,specie(1),pc20*pconv,
+specie(2)
!4pp plots
1012 format('@subtitle "',2(f7.4,'torr ',a),'"')
!4pp plots
write(10,1013)
!4pp plots
1009
156
1013
1014
plots
format('@with g3')
!4pp plots
write(10,1014)wetfil,dryfil
!4pp plots
format('@subtitle \"',a13,'with dry trace ',a13,'\"') !4pp
write(10,1015)
!4pp plots
format('#@string def \"xscm um/s t
Aratio
tisw ',
&'
pp0i
pp0
pisp0
tisd g
g_ginf\" ')
!4pp plots
write(12,1016)kd,pc10*pconv,specie(1),pc20*pconv,
+specie(2)
!4pp plots
1016 format('legend string ',i2,' \"',2(f7.4,'torr ',a),'\"')
!4pp plots
write(14,1017)kd, wetfil
!4pp plots
1017 format('legend string ',i2,' \"',a13,'\"')
!4pp plots
write(15,1018)kd-1,p0*pconv,devp0,t0-273.15
!4pp plots
1018 format('legend string ',i2,' \"',f6.2,'+/-',f4.2,'torr ',
& f6.2,'celsius"')
!4pp plots
1015
write(*,*) 'start integration'
write(*,5000)istart,ifin1
do i=istart,ifin1
c calculate local value of effective area ratio, aratio
c msq is local mach number squared, mssq = (u/u*)^2
!wet
!dry
!wet
!wet
msq
= c10*((1.d0/pp0i(i))**c0-1.d0)
tisw(i) = 1.d0/(1.d0+msq/c10)*t0
isentrope t
tisd(i) = pp0d(i)**c0*t0
isentrope t
risr0 = (1.d0+msq/c10)**erho
isentrope density
pisp0 = (1.d0+msq/c10)**ep0
isentrope pressure
aratio(i)= dsqrt(((c10/eam20*(1.d0+msq/c10))**eam20)/msq)
msq
= c10*((1.d0/pp0i(i+1))**c0-1.d0)
aratio(i+1) = dsqrt(((c10/eam20*(1.d0+msq/c10))**eam20)/msq)
u(i)=ustar*rosro0/rr0(i)/aratio(i)
mssq=(u(i)/ustar)**2
dar=dlog(aratio(i+1)/aratio(i-1))/2.d0
dp=(pp0(i+1)-pp0(i-1))/2.d0
c gw2-17-00 update specific heat
!
cp=wi*cp1+(w20+w30-g(i))*cpv+g(i)*cpc
cp=wi*cp1+(w20-g(i))*cpv+g(i)*cpc ! Som
cpr=cp/cp0
c gw2-17-00 update "mu/(1-g)" = wmu, and related factors
wmu=wm1*wmc/(wi*wmc+(w20-g(i))*wm1)!som
wmuu0=wmu/wmav0
wg(i)=wmu/(wmc)
c gw 17-2-00 the following factor is almost one
157
approx1=wmuu0*gamma0-(gamma0-1.d0)/cpr
t(i)=tt0(i)*t0
c gw17-2-00 next eq. is slightly in error. it's from wegener and
c should not be used.
c
dtt0=(1.d0-t(i)/tstar/gamma0/mssq)/rr0(i)*dp+tt0(i)*dar
c gw17-2-00 the next equation is ok.
dr=dp/tst0/gamma0/mssq-rr0(i)*dar
c gw17-2-00 the next equation from wegener is wrong. don't use it
c
dgp=(1.d0-t(i)/tstar/mssq)/gamma0/rr0(i)*dp+tt0(i)*dar
c gw 17-2-00 this is the correct eq.
dgp=(approx1-t(i)/tstar/mssq)/gamma0/rr0(i)*dp+tt0(i)*dar
c gw2-17-00 this eq. is from wegener. it is wrong. don't use it.
c
dg=dgp*cp0*t0/dhc
c gw2-17-00 need wg(i) term to get dg correctly
c gw2-17-00
dg=dgp*cp0*t0/(dhc-cp*t(i)*wg(i))
c
dg=dgp*cp0*t0/(dhc-cp*t(i)*wg(i))
c
dg=dgp*cp0*t0/(fdhc(t(i))-cp*t(i)*wg(i))
!!
This is
wrong SOM 30-apr-08
dg=dgp*cp*t0/(fdhc(t(i))-cp*t(i)*wg(i)) ! This is correct
!
if (xs(i).lt.3.00)then
!
!
write(*,*)xs(i),wmc
!
end if
c gw2-17-00 update dtt0
dtt0=(wmuu0-t(i)/tstar/gamma0/mssq)/rr0(i)*dp+
&
tt0(i)*(dar+wg(i)*dg)
c still need to fix this correction for the binary system for the
pressure method
c if we think it matters!
c calculate dt with alt. eq. and dgp from more general results due to
mod. eos
c gw2-17-00 note: cp2 should be cpv in the following eq. it's ok above.
c
cp=wi*cp1+(w20+w30-g(i))*cp2+g(i)*cpc
c
cpr=cp/cp0
c
c0fctr=cpr*wmug(i)/wmav/c0-1.d0
c
wgfctr=1.d0-wg(i)*cp*t(i)/dhc
c
dgp=(c0fctr*c0*dp/sdrwet(i)
c
& -tt0(i)*cpr*dlog(sdrwet(i+1)/sdrwet(i-1))/2.)/wgfctr
c
dg=dgp*cp0*t0/dhc
tt0(i+1)=tt0(i-1)+2.0d0*dtt0
rr0(i+1)=rr0(i-1)+2.0d0*dr
g(i+1)=g(i-1)+2.0d0*dg
c
c
if((w20).gt.0.0)then
fc(i+1)=g(i+1)*wi/(w20+w30)
fc(i+1)=g(i+1)/(w20+w30)
fc(i+1)=g(i+1)/w20 ! Som
else
fc(i+1)=0.0
158
! March04 Shinobu
end if
write(9,1000)xs(i),u(i),t(i),g(i),tisw(i),t(i)-tisw(i),
&pp0i(i),pp0(i),fc(i)
c
c
***********************************************************************
******************************
write(10,1020)xs(i),u(i)/100,t(i),
! SOM
* aratio(i),tisw(i),pp0i(i),pp0(i),pisp0,tisd(i),
* g(i),fc(i)
!
!
!
!
write(10,1020)xs(i),u(i)/100,t(i),
* g(i),
* tisw(i),pp0i(i),pp0(i),pisp0,tisd(i)
! SOM
1020
format(f8.3,f10.2,f8.2,f8.4,f8.2,3f8.4,f8.2,e12.4,2x,e12.4)
!4pp plots
c
c
write(3,1100) i, xs(i),rr0(i),risr0,rr0(i)/risr0,t(i)/tisw(i),
*pp0(i)/pp0i(i)
write(26,1110) xs(i),aratio(i)
1000
format(e12.3,e12.4,f8.2,e12.3,f8.2,f7.3,2e12.4,e12.4)
1100
format(i5,e12.3,5e12.4)
1110
format(e12.3,e13.5)
enddo
write(*,*)'start search'
c now search for the onset conditions using both t(i)-tisw(i) and t(i)tisd
do i = istart,ifin1-1
dtemp(i,kd) = t(i)-tisw(i)
dt1 = t(i) - tisw(i)
dt2 = t(i+1) - tisw(i+1)
if(dt1.le.0.5d0.and.dt2.gt.0.5d0)then
xon = xs(i)+(0.5-dt1)/(dt2-dt1)*(xs(i+1)-xs(i))
pp0on = pp0(i)+(0.5-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i))
pp0ion = pp0i(i)+(0.5-dt1)/(dt2-dt1)*(pp0i(i+1)-pp0(i))
ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i))
tiswon = tisw(i)+(0.5-dt1)/(dt2-dt1)*(tisw(i+1)-tisw(i))
g_ginfon=(1-fc(i))+(0.5-dt1)/(dt2-dt1)*(fc(i)-fc(i+1))
else
endif
enddo
peq = vappress(ton)
supers=(pp0on*pct0*g_ginfon)*133.33/peq
159
&
1300
write(*,*)'using the t-tisw = 0.5 k'
write(*,1300)xon,pp0on*pct0*g_ginfon,ton,
pp0ion*pct0*g_ginfon,tiswon
format('onset occurs at x =',f8.4,f8.4,f7.2,f8.4,f7.2)
write(11,1301)t0,p0*pconv,pct0,ton,pp0on*pct0*g_ginfon,
&
1301
1302
pp0on*pc10*pconv*g_ginfon,xon,supers,wetfil
format(f8.2,f8.2,1x,f8.4,f8.2,f8.4,f8.4,2x,f7.2,2x,f8.2,1x,a13)
format('@\"t0
p0
pct0
ton
pon
p1on
xon S',
&
7x,a13,'\"')
do i = istart,ifin1-1
dt1 = t(i) - tisd(i)
dt2 = t(i+1) - tisd(i+1)
if(dt1.le.0.5d0.and.dt2.gt.0.5d0)then
xon = xs(i)+(0.5-dt1)/(dt2-dt1)*(xs(i+1)-xs(i))
pp0on = pp0(i)+(0.5-dt1)/(dt2-dt1)*(pp0(i+1)-pp0(i))
pp0ion = pp0i(i)+(0.5-dt1)/(dt2-dt1)*(pp0i(i+1)-pp0(i))
ton = t(i)+(0.5-dt1)/(dt2-dt1)*(t(i+1)-t(i))
tisdon = tisd(i)+(0.5-dt1)/(dt2-dt1)*(tisd(i+1)-tisd(i))
g_ginfon=(1-fc(i))+(0.5-dt1)/(dt2-dt1)*(fc(i)-fc(i+1))
else
endif
enddo
&
write(*,*)'using the criterion t-tisd = 0.5 k'
write(*,1300)xon,pp0on*(pc10+pc20)*pconv*g_ginfon,ton,
pp0ion*(pc10+pc20)*pconv*g_ginfon,tiswon
c
c
c
c
c
if(istart-1.eq.1)then
istart=istart+nint-1
else
istart=istart+nint
endif
write(*,*) 'finished integration'
enddo
enddo
c now write out the dtemp files to dtemp.out
1313
do i = istart,ifin1
write(13,1313)xs(i),(dtemp(i,j),j=1,ndata)
enddo
format(f8.4,20(f8.2))
50 stop
end
c
subroutine smooth(m,n,k,k0,sval,y)
c this subroutine produces smoothed values of a tabulated function y
c based on technique described in ralston, "a first course in num.
anal."
c y values do not have to be equally spaced, but x values must be
supplied
160
c
c
c
c
c
c
c
c
regardless of the spacing
m - order of the highest polynomial used in smoothing
n - number of y points in interval over which smoothing is performed
k - point whose smoothed value is desired
k0 - first point in set of n
sval - smoothed value returned to calling program
real*8 p(-2:5,1:200),b(0:5),omega(0:5),gamma(0:5),beta(-1:5)
*,alpha(0:5),y(200),sval,x
common /xval/ x(1000)
beta(-1)=0.
beta(0)=0.
gamma(0)=n
omega(0)=0.
alpha(1)=0.
do i=k0,(n+k0-1)
omega(0)=omega(0)+y(i)
alpha(1)=alpha(1)+x(i)
p(-2,i)=0.
p(-1,i)=0.
p(0,i)=1.
enddo
b(0)=omega(0)/gamma(0)
alpha(1)=alpha(1)/gamma(0)
sval=b(0)
do j=1,m
gamma(j)=0.
omega(j)=0.
alpha(j+1)=0.
do i=k0,(n+k0-1)
p(j,i)=(x(i)-alpha(j))*p(j-1,i) - beta(j-1)*p(j-2,i)
gamma(j)=gamma(j)+p(j,i)*p(j,i)
alpha(j+1)=alpha(j+1)+x(i)*p(j,i)*p(j,i)
omega(j)=omega(j)+y(i)*p(j,i)
enddo
alpha(j+1)=alpha(j+1)/gamma(j)
beta(j)=gamma(j)/gamma(j-1)
b(j)=omega(j)/gamma(j)
sval=sval+b(j)*p(j,k)
enddo
return
end
subroutine echo
character*100 a
write(9,3)
15 read(5,1,end=99)a
write(3,1)a
write(9,2)a
goto 15
99 continue
rewind 5
return
161
1 format(a100)
2 format(1x,a100)
3 format(1h1,20x,'input file',//)
end
*---------------------------------------------------------------------------------SOM 30 April 08---*
* Equation from Fladerer & Strey JCP 124
*-------------------------------------------------------------------------------------real function fdhc(tk)! SOM
double precision tk! SOM
fdhc = (4.184/39.948)*(1808.61227+1.07629316*tk-(0.0464775453D2)
> *(tk*tk))
return
end
*-----------------------------------------------------------------------------------------------*
***************************** Now adding vapor pressure
*****************************************
*-----------------------------------------------------------------------------------------------*
real*8 function vappress(T)
real*8 epi, P, Pc, T, Tc
c from Iland's paper
Pc = 4.86D6 !Pa
Tc = 150.6633 !K
a1 = 5.90418853
a2 = 1.12549591
a3 = 0.763257913
a4 = 1.69799438
epi = 1-(T/Tc)
>
vappress=Pc*exp((Tc/T)
*(-a1*epi+a2*epi**1.5-a3*epi**3-a4*epi**6)) !in Pa
return
end
162
APPENDIX E
FORTRAN CODE TO PREDICT CONDENSATION IN THE
NOZZLE
163
C THIS CODE IS FOR USING HALE'S NUCLEATION RATE
C THIS CODE CANNOT BE USED FOR KULMALA BUT FOR HKS GL. THIS HAS A
FULL VERSION OF HKS WITHOUT ASSUMING MIXTURE TO BE DILUTE.
C EXPANSION OF water vapor IN EXCESS CARRIER GAS WITH TEMPERATURE
AND
C DENSITY PROFILES OF CARRIER ADJUSTED FOR LATENT HEAT RELEASE BY
INTEGRATING
C DIABATIC GAS DYNAMICS EQUATIONS WITH STEADY STATE NUCLEATION RATE
AND
C DROPLET GROWTH LAW TO DETERMINE PARTICLE SIZE DISTRIBUTION AND
MASS
C FRACTION OF CONDENSATE
C DROPLET GROWTH INCLUDES EVAPORATION TERM
C Modified to calculate SANS intensity from aerosol.....9/96
c modified for d2o values ..............................12/96
C modified for variable cp and WMAV in diabatic flow equations...1812-96
c modified the surface tension and density fits to agree with
Jbell...23-3-99
c added compressibility use the Reiss Katz Kegel or classical rate
... 23-2-00
c cleaned up and changed to take dry pressure trace
input................23-2-00
c hardwired the remaining condensible
parameters.........................24-2-00
c added Kulmala growth rate
.............................................11/02
IMPLICIT REAL*8(A-H,O-Z)
REAL*8
MSQ,MACHNO,MACHNOD,MACHSQ,MACHST,IMEAN,NFRAC,N,I23,NCUM,NC
real*8
density,vappress,surftension,Hvap,compressibility,molweight
real*8 cp1,cp2,cpc
DIMENSION RM(11),BIS(1001),FS(1001),PRDRY(900),XD(900)
real*8 aratiom(900)
CHARACTER*50 DRYFIL,A
CHARACTER*8 SPECIE(2)
CHARACTER*4 TITLE(3,2)
164
COMMON /RANDN/ R(2,0:15000),DELNRO(15000),RN(15000),N(15000)
COMMON /FANDE/ F(8201),DNDR(900),DGDR(900),NCUM(900),GCUM(900)
COMMON /JUSTR/ RI(15000)
COMMON /CUM/ NC(0:15000),GC(0:15000)
COMMON /NONISO/ TD(15000),FLUXM(15000),QTERM(15000)
COMMON /NONISO2/ TDC(900),FLUXMC(900),QTERMC(900)
COMMON /I23A/ I23(1001)
COMMON /ACT/ A1,A2,RG,W1,W2
COMMON /I/ ITHEOR,IACTIV,IVOL
COMMON /R6AND8/ R3(900),R4(900),R6(900),R8(900)
COMMON /SCAT/Q(102),PD(102,900),PS(102,900),SID(102),SIS(102),
& SIDF(102),SISF(102),sisqd(102)
COMMON /properties/ p,p20,grho,SIG,RHO,DHC,S,sr,THK,CP1,GAMMA1
COMMON
/params/wm1,wm2,sigmac,dcoef,ci11,ci22,gamg,CV1,RSTEJ,pc,
>wmav,cp0,GAMMA0,ispec
DATA TITLE/ '
',' REV','ISED','
C','LASS','ICAL'/
C Opened new file for pressure ratio (Som)
OPEN(71,FILE='smolders_temp.out',STATUS='unknown')
OPEN(7,FILE='pressure_ratio.out',STATUS='unknown')
OPEN(5,FILE='dbsec.dat',STATUS='OLD')
OPEN(9,FILE='dbsec.out',STATUS='unknown')
open(4,FILE='dbsec_output',STATUS='unknown')
OPEN(3,FILE='dbsec_plotan.out',STATUS='unknown')
OPEN(2,FILE='dbsec_scatan.out',STATUS='unknown')
OPEN(1,FILE='dbsec_scatafn.out',STATUS='unknown')
CALL ECHO
WRITE(7,4050)
4050 FORMAT(3X,'X Pratio_w PratioISD PratioISW')
write(71,1111)
1111
Format(5X, 'T S r Td ptot')
WRITE(3,4000)
4000
FORMAT(5X,'X
TIME
S
G
165
TISW
TEMP Rho/Rho0 RhoIW/Rho0
*
C
N
J')
G/G0
RMEAN
FMEAN
SCSIG/RHOG')
C
WRITE(3,4001)X,TIME,S,TISD,T,RHOG,RHOGISD,GFRACT,NFRAC,TOTNP,RMEAN
C
*,FMEAN,SCTSIGT/RHOG
C 4001 FORMAT(13(1PE10.3))
4001 FORMAT(F9.4,F6.1,F9.2,2F7.1,2F7.4,3(1PE10.3))
IFILE3=1
PI=3.14159
RG=8.3145D7
AVOG=6.022D23
TWO3RD=2.D0/3.D0
ONE3RD=1.D0/3.D0
two13=2**ONE3RD
NIMAX=0
ISANS=0
DGC=16.*PI*AVOG/3./RG**3
C 102 VALUES OF SCATTERING VECTOR
Q(1)=0.01
do i=2,61
Q(I)=Q(I-1)+0.001D0
enddo
do i=62,76
Q(I)=Q(I-1)+0.002D0
ENDDO
do i=77,102
Q(I)=Q(I-1)+0.01D0
ENDDO
C CONTRAST PARAMATER**2, J.PHYS.CHEM. 99, 13232 (1995) (p. 13236)
C H2O(w), D2O(d) and ethanol(e) values from Reinhard email 17-9-96
C ADDITIONAL FACTOR D-48 NEEDED TO CONVERT UNITS OF PD AND PS(JQ,IR)
CNTPW=(-5.6D9)*(-5.6D9)*16*PI**2*1.D-48
CNTPD=6.37D10*6.37D10*16*PI**2*1.D-48
CNTPE=6.4D10*6.4D10*16*PI**2*1.D-48
DELTE=4.D0
DELESQ=16.D0
TESQ=DELESQ/2/PI
166
C PARTICLE FORM FACTORS FOR DROPS(D) AND SHELLS(S)
DO JQ=1,102
QSQ=Q(jq)*Q(jq)
Q6=QSQ*QSQ*QSQ
DO IR=1,500
r3(ir)=ir**3
r6(ir)=r3(ir)*r3(ir)
r4(ir)=r3(ir)*ir
r8(ir)=r4(ir)*r4(ir)
QR=Q(jq)*IR
PD(JQ,IR)=(DSIN(QR)-QR*DCOS(QR))**2/Q6
PS(JQ,IR)=(IR*DSIN(QR)+Q(jq)*TESQ*DCOS(QR))**2
&
*DEXP(-QSQ*TESQ)*DELESQ/QSQ
ENDDO
ENDDO
READ(5,41,END=50)SPECIE
C
PRINT 1006, SPECIE
1006 FORMAT (2A8)
41 FORMAT(2A8)
C READ PARAMETER CONTROLLING HOW DIMERS ARE TREATED:IDIMER= 0- USUAL
CNT, 1-CORRECTED
C READ PARAMETER CONTROLLING WHICH RATE TO USE:IRATE= 0- CNT, 1DIMER MODIFIED, 2- hale
C READ EMPIRICAL WEGENER GAMMA FACTOR TO MULTIPLY Classical RATE
c read empirical factor RKcor to multiply RKK rate
READ(5,*)IDIMER, IRATE, WGAM, RKcor
C READ STAGNATION CONDITIONS-TEMP, PRESSURE, PARTIAL PRESSURE OF
C CONDENSIBLE--PRESSURES ARE IN MM OF HG
READ(5,*)T0,P0,PC0
c read which species is condensing (1=H2O, 2=D2O)
read(5,*)ispec
write(*,*)'ispec is ',ispec
C CONVERT PRESSURES TO DYN/CM**2
P0=P0/760.*1.01325D6
PC0=PC0/760.*1.01325D6
C READ MOLECULAR WEIGHT OF CARRIER (1) AND CONDENSIBLE (2)
167
READ(5,*)WM1
WM2=molweight(ispec)
C THE SPECIFIC HEATS OF CARRIER GAS AND CONDENSIBLE VAPOR
READ(5,*)CP1
CP2=specheat(ispec)
C THE SPECIFIC HEAT OF CONDENSATE
CPC=specheatc(ispec)
C READ INITIAL X, FINAL X, AND STEP LENGTH DX AND THROAT LOCATION
READ(5,*)XINIT, XFIN, DX, XTHROAT
IF(DX.GT.1.D-3)DX=1.D-3
C RFCTR IS A FACTOR SLIGHTLY >1 THAT GOVERNS INITIAL RADIUS OF
DROPLET
C INITIAL RADIUS = CRITICAL RADIUS TIMES RFCTR
C ALPHA IS THE STICKING COEFFICIENT, USUALLY = 1
READ(5,*)RFCTR, ALPHA
C NB IS THE NUMBER OF POINTS IN THE SMALLEST BIN, POSSIBLE VALUES
ARE
C 1, 2, 4, 5, 10, 20
POINTS
THE LARGER BINS CONTAIN 10*NB AND 100*NB
C IGL IS THE GROWTH LAW PARAMETER: 0=HERTZ, 1=ISOTHERMAL GENERAL,
C
2=NONISOTHERMAL GENERAL
C 3=Kulmala formula that implicitly allows droplet temperature to
differ from gas
READ(5,*)IMAX, IWRIT, NB, IGL
C READ PARAMETERS NEEDED TO COMPUTE GASEOUS DIFFUSION COEFFICIENT
C SIGMAi IS THE COLLISION DIAMETER OF COMPONENT i
C EKB = Epsilon/kB = 240.3 for Water/Nitrogen (LENNARD-JONES WELL
DEPTH)
C EKB=sqrt(ekb1*ekb2)
c TKE = kB*T/Epsilon, CI = collision integral
C read pair of values bracketing temperature midway between highest
and lowest
C temperature reached in expansion after condensation starts- USE
TABLE 5.1-3
C
in Cussler's book, same as Hirschfelder, Curtiss, Bird
C EKB1 is for the carrier gas (for nitrogen EKB1=104.2
READ(5,*)EKB1, EKB, SIGMA1, SIGMA2
C
READ(5,*)TKE1, TKE2, CI1, CI2
C use linear fit to determine collision integral, CIS IS THE SLOPE,
168
C CII IS THE INTERCEPT: CI=CIS*T+CII
C
CIS=(CI2-CI1)/(TKE2-TKE1)/EKB
C
CII=CI1-CIS*EKB*TKE1
SIGMAC=(SIGMA1+SIGMA2)/2.
c ****************************************************************
if(ispec.eq.1) then
DCOEF=1.88D-3*DSQRT(1./WM1+1./WM2)*1.01325D6/SIGMAC**2
endif
c ****************************************************************
if(ispec.eq.2) then
DCOEF=1.7655D-3*DSQRT(1./WM1+1./WM2)*1.01325D6/SIGMAC**2
endif
c ****************************************************************
C READ NAME OF DRY PRESSURE TRACE (smoothed output from data
reduction program!)
READ(5,11)idoption,dryfile
write(*,11)idoption,dryfile
11 FORMAT(I2,A20)
c if idoption=0 , use a constant area ratio instead of the dry trace
if(idoption.eq.0)goto 12
open(21,file = 'dry.dat',status='old')
c read P0,T0 of dry file (for info only)
c and the number of values in the dry pressure file
read(21,*)p0d,t0d,iden
write(*,*)p0d,t0d,iden
read(21,10)a
write(*,10)a
10 format(A50)
do i=1,iden
read(21,*)xd(i),dummy,dummy,dummy,dummy,dummy,dummy,dummy,
&
C
dummy,dummy,dummy,dummy,prdry(i)
write(*,*)'prdry is ', prdry(i) (SOM)
169
enddo
close(21)
C FIND FIRST X
DO I=1,IDEN
IF(XINIT.LT.XD(I))THEN
IONE=I-1
ITWO=I
IF(IONE.EQ.0)STOP
GOTO 12
ENDIF
ENDDO
12 CONTINUE
read(5,*)dadx
C CALCULATE STAGNATION GAS NUMBER DENSITY AND CONDENSIBLE MONOMER
NUMBER
C DENSITY (#/CM**3)
RHOG0=P0/RG/T0*AVOG
F10=PC0/RG/T0*AVOG
C CALCULATE STAGNATION GAS MASS DENSITY AND CONDENSIBLE MONOMER MASS
C DENSITY (g/CM**3)
C W0 IS MASS FRACTION OF CONDENSIBLE VAPOR IN GAS
WMAV=(WM1*(P0-PC0)+WM2*PC0)/P0
WMAV0=WMAV
W0=WM2*PC0/P0/WMAV
WI=1.D0-W0
C CP0 is the mixture specific heat at stagnation conditions
CP0=WI*CP1+W0*CP2
C GAMMA1 is for the dry carrier gas
C GAMMA0 is for the wet flow at stagnation conditions (unless we
make CP1 a
C function of temperature--then GAMMA0 would be at stagnation
composition
C
but at the local T.
GAMMA1=CP1/(CP1-RG*1.D-7/WM1)
GAMMA0=CP0/(CP0-RG*1.D-7/WMAV)
RHOG0M=P0/RG/T0*WMAV
170
RHO20M=WM2*PC0/RG/T0
C CALCULATE VARIOUS EXPONENTS AND CONSTANTS INVOLVING GAMMA
EAI=2.*(GAMMA1-1.D0)/(GAMMA1+1.D0)
EAI0=2.*(GAMMA0-1.D0)/(GAMMA0+1.D0)
EP=-GAMMA1/(GAMMA1-1.D0)
EMRHO=GAMMA1-1.D0
EAM2=(GAMMA1+1.D0)/(GAMMA1-1.D0)
ERHO=-1.D0/(GAMMA1-1.D0)
ERHO0=-1.D0/(GAMMA0-1.D0)
C1=2./(GAMMA1-1.D0)
C2=(GAMMA1+1.D0)/2.
c3=(gamma1 - 1.d0)/gamma1
C0=(GAMMA0-1.D0)/GAMMA0
C10=2./(GAMMA0-1.D0)
C20=(GAMMA0+1.D0)/2.
GAMV=CP2/(CP2-RG*1.D-7/WM2)
CV1=CP1-RG*1.D-7/WM1
GAMG=CP1/CV1
c calculate the area ratios from the dry pressure trace
if(idoption.ne.0)then
do i = 1,iden
msq = c1*((1.d0/prdry(i))**c3-1.d0)
aratiom(i)=dsqrt(((c1/eam2*(1.d0+msq/c1))**eam2)/msq)
enddo
endif
C CALCULATE FLOW PROPERTIES AT NOZZLE THROAT
TSTAR=T0/C20
RHOGST=RHOG0*C20**ERHO0
USTAR=DSQRT(GAMMA0*RG*TSTAR/WMAV)
RHOUST=RHOGST*USTAR
IX=0
C**********************************************************
C INITIALIZE ALL NUMBER DENSITIES AND OTHER QUANTITIES
DO I=1,IMAX+1
RRI=I
171
I23(I)=RRI**TWO3RD
ENDDO
X=XINIT
c find the area ratio at x=xinit
if(idoption.eq.0)then
aratio=1.0+dadx*(x-xthroat)
else
aratiom2 = aratiom(itwo)
aratiom1 = aratiom(ione)
dadx=(aratiom2-aratiom1)/(xd(itwo)-xd(ione))
C
write(*,*)'dadx= ',dadx (Som)
aratio=aratiom1+dadx*(x-xd(ione))
endif
machno = 1.01
C FIRST CALCULATE LOCAL DRY MACH NUMBER
1 RM(1)=DSQRT(C1*(C2*(MACHNO*ARATIO)**EAI-1.D0))
DO IM=1,10
RM(IM+1)=DSQRT(C1*(C2*(RM(IM)*ARATIO)**EAI-1.D0))
ENDDO
MACHNO=RM(11)
IF(DABS(MACHNO-RM(10)).GT.1.D-8)GOTO 1
machnod = machno
C CALCULATE FLOW PROPERTIES AT FIRST X
C USE WET GAMMA VALUE
C FIRST CALCULATE LOCAL MACH NUMBER
4 RM(1)=DSQRT(C10*(C20*(MACHNO*ARATIO)**EAI0-1.D0))
DO IM=1,10
RM(IM+1)=DSQRT(C10*(C20*(RM(IM)*ARATIO)**EAI0-1.D0))
ENDDO
MACHNO=RM(11)
IF(DABS(MACHNO-RM(10)).GT.1.D-8)GOTO 4
C WITH MACHNO COMPLETE CALCULATION
T=T0/(1.D0+MACHNO*MACHNO/C10)
RHOG=RHOG0*(T0/T)**ERHO0
P=P0*(RHOG/RHOG0)**GAMMA0
PC=PC0*P/P0
172
C CHANGE MACHNO TO MACHSQ (SQUARE OF MACH NUMBER BASED ON THROAT
SOUND SPEED)
C MACHSQ IS USED TO INTEGRATE DIABATIC FLOW EQUATIONS
MACHSQ=MACHNO*MACHNO*T/TSTAR
TSTT0=TSTAR/T0
C F1T IS THE TOTAL EFFECTIVE MONOMER CONCENTRATION AT X
F1T=F10*RHOG/RHOG0
C CALCULATE STEADY STATE DISTRIBUTION AT FIRST LOCATION--LOW
SUPERSATURATION
F1=F1T
C Values for surface tension, density and vapor pressure are now in
a function
SIG=surftension(T,ispec)
P20=vappress(T,ispec)
rho=density(T,ispec)
V2=WM2/RHO/AVOG
R1=(0.75D0*V2/PI)**ONE3RD*1.D8
C COMPUTE LOCAL SUPERSATURATION
S=PC/P20
C RAPROX IS THE LOCAL STEADY STATE RATE IN THE CONTINUOUS
APPROXIMATION
C RATE IS THE EXACT STEADY STATE VALUE
DG=DGC*(SIG/T)**3*(WM2/RHO/DLOG(S))**2
RAPROX=WGAM*5.3937D19*(PC/T)**2*DSQRT(SIG*WM2)/RHO*DEXP(-DG)
C USE STEADY STATE DISTRIBUTION FOR LIQUID DROP MODEL TO GET STARTED
DGA=4.*PI*SIG*AVOG/RG/T*(.75*V2/PI)**TWO3RD
ALPH=DSQRT(8.*PI*RG*T/WM2)*(.75*V2/PI)**TWO3RD*F1
C CALCULATE EXACT STEADY STATE RATE
RSTEJ=2.*SIG*WM2/RG/T/RHO*1.D8
RCR=2.*SIG*WM2/RG/T/RHO/DLOG(S)*1.D8
R(1,1)=RCR*RFCTR
R(1,0)=R(1,1)
RCONST=4.*PI*0.6022/3./WM2
RN(1)=RCONST*RHO*R(1,1)**3
C
I3CR=3.*RN
173
C
IMAX=3.*RN
C
IF(IMAX.GT.1000)IMAX=1000
SUM=0.D0
FS(IMAX)=1.D0
DO I=1,IMAX-1
J=IMAX+1-I
BIS(J)=I23(J-1)*DEXP(DGA*(I23(J)-I23(J-1)))/S
FS(J-1)=BIS(J)/I23(J)*FS(J)+1.D0
C
SUM=I23(J-1)/I23(J)*BIS(J)*(1.D0+SUM)
ENDDO
IF(IDIMER.EQ.0)THEN
C THE FOLLOWING CORRECTION TO FS(1) MAKES IT CONSISTENT WITH CNT
ALTHOUGH
C IT NOW VIOLATES MASS ACTION LAW FOR MONOMER-DIMER EQUILIBRIUM
FS(1)=(FS(1)-1.D0)*DEXP(DGA)/S+1.D0
ELSE
DEQUK=RG*T/AVOG/1.01325D6*DEXP(-18.59/1.9872+3.59D3/1.9872/T)
FS(1)=(FS(1)-1.D0)/BIS(2)/F1/DEQUK
ENDIF
C
SUM=SUM*DEXP(DGA)/S
C THIS ADJUSTS BIS(2)
C
BIS(2)=BIS(2)*DEXP(DGA)/S
RJALPH=F1/FS(1)
RATE=ALPH*RJALPH
c calculate Reiss-Katz-Kegel rate, RATERKK.
rate also
ALTRcl is the classical
C computed in the RKK subroutine for comparison.
CALL RKK(T,S,RATERKK,ALTRcl,RKcor,ispec)
C
RATE=ALPH*F1/(1.D0+SUM)
C
DELF=F1T
DO I=2,IMAX
F(I)=FS(I)*RJALPH/I23(I)
C THE FOLLOWING FORMULA DOESN'T WORK AT LARGE I DUE TO ROUNDOFF
PROBLEMS
C
F(I)=(F(I-1)-RJALPH/I23(I-1))/BIS(I)
ENDDO
C CALCULATE RELATIVE LIGHT SCATTERING SIGNAL
174
BASLIN=0.D0
GFRACT=0.D0
C COMMENT OUT
C
DO I=1,IMAX
C
J=IMAX+1-I
C
GFRACT=GFRACT+J*F(J)
C
BASLIN=BASLIN+J*J*F(J)
C
ENDDO
C
GFRACT=GFRACT-F1
C END COMMENT OUT
U=RHOUST/RHOG/ARATIO
IF(IRATE.EQ.0)THEN
DELNRO(1)=RAPROX/U/RHOG
ELSE
IF(IRATE.EQ.1)THEN
DELNRO(1)=RATE/U/RHOG
else
DELNRO(1)=RATERKK/U/rhog
endif
ENDIF
N(1)=DELNRO(1)*RHOG*DX
GCONST=4.*PI*RHO*AVOG*1.D-24*DX/3./WM1
GFRACT=GCONST*RCR*RCR*RCR*DELNRO(1)
ZETJ=1.D8*DSQRT(RG*T/2./PI/WM2)*V2*F1*DX/U
SCTSIG=BASLIN
TIME=(XINIT-XTHROAT)*(1.D0/U+1.D0/USTAR)/2.D-6
PWR=P*760./1.01325D6
PCWR=PC*760./1.01325D6
P20WR=P20*760./1.01325D6
C
C
WRITE(6,1000)X,T,PWR,RHOG,MACHNO,U,S,PCWR,P20WR,RN(1),R(1,1),
&RAPROX,RATE,TIME
WRITE(9,1000)X,T,PWR,RHOG,MACHNO,U,S,PCWR,P20WR,RN(1),R(1,1),
&RAPROX,RATE,RATERKK,WGAM*ALTRcl,TIME
1000 FORMAT(1H1,' AT X=',F7.3,'CM, T=',F7.2,' K, P=',F8.2,' MM HG,
GAS
& DENSITY=',1PE9.2,'/CM**3, MACH #=',0PF6.2,' FLOW
SPEED=',1PE9.2,'
175
&CM/S',/' SUPERSATURATION=',0PF6.2,', P2=',1PE9.2,' MM HG,
P2E=',
&E9.2,' MM HG, CRITICAL NUCLEUS HAS',0PF7.1,' MOLECULES, AND
ITS RA
&DIUS=',F7.2,' A',/' APPROXIMATE STEADY STATE
RATE=',1PE9.2,'/(S-CM
&**3), EXACT STEADY STATE RATE=',E9.2,'/(S-CM**3)',/
&' DILLMANN-MEIER RATE=',E9.2,'/(S-CM**3)',/,
&' ALTERNATE APPROX. CL. RATE*WGAM=',E9.2,'/(S-CM**3)',
&'
TIME=',0PF6.1,
&'us',/)
C
WRITE(6,1010)F1T,SCTSIG,DX,GFRACT
WRITE(9,1010)F1T,SCTSIG,DX,GFRACT
1010 FORMAT(' EFFECTIVE MONOMER NUMBER DENSITY AT
X=',1PE11.4,'/CM**3,
& RELATIVE LIGHT SCATTERING SIGNAL=', E11.4,'
DX=',E10.3,/,
&' CONDENSATE MASS FRACTION=', E11.4)
WRITE(9,1009)GAMMA1, GAMMA0
1009 FORMAT(' DRY GAMMA =',F7.4,'
WET GAMMA =',F7.4,/)
WRITE(9,2011) R(1,1)
WRITE(9,2012) N(1)
2011 FORMAT(' R: ',10(1PE12.4))
2012 FORMAT(' N: ',10(1PE12.4))
C
200 ZET=DX*ALPH*RHOG/RHOUST*ARATIO/2.
ZETN=DX/RHOUST*ARATIO
C FTCREL, FTMREL, FTNREL IS THE TOTAL CLUSTER NUMBER DENSITY AT X
NORMALIZED
C BY RHOG
C CALCULATION OF FTCREL IS BASED ON THE CONTINUOUS STEADY STATE
APPROXIMATION
C CALCULATION OF FTMREL IS BASED ON THE MODIFIED STEADY STATE
APPROXIMATION
GX=GFRACT
GFRACTO=GFRACT
OARATIO=ARATIO
FTCREL=ZETN*RAPROX
FTMREL=ZETN*RATE
176
C advance x
X=X+DX
FLUXM(1)=0.D0
IF(N(1).LT.0.1D0)THEN
JX=0
IXMAX=0
ELSE
JX=1
IXMAX=1
ENDIF
DELTAI=0.D0
if(idoption.ne.0)then
if(x.gt.xd(itwo))then
ione=itwo
itwo=ione+1
aratiom2=aratiom(itwo)
aratiom1=aratiom(ione)
dadx=(aratiom2-aratiom1)/(xd(itwo)-xd(ione))
C
write(*,*)'dadx= ',dadx (Som)
endif
endif
100 CONTINUE
if(idoption.eq.0)then
aratio=1.0+dadx*(x-xthroat)
else
aratio=aratiom1+dadx*(x-xd(ione))
endif
C CALCULATE FLOW PROPERTIES AT SPECIFIED X
ORHOG=RHOG
OWMAV=WMAV
C ADVANCE DIABATIC FLOW EQUATIONS NEXT DELTA TO UPDATE LOCAL
CONDITIONS
WGX=WM1/(WI*WM2+(W0-GX)*WM1)
177
C Latent heat is now also in a function at end of program
DHC=Hvap(T,ispec)
C CALCULATE DT AND DGP FROM MORE GENERAL RESULTS DUE TO MODIFIED EOS
C update WMAV and CP for changes in mixture composition, no change
from before
WMAV=(WM1*(P-PC)+WM2*PC)/P
CP=WI*CP1+(W0-GX)*CP2+GX*CPC
CPR=CP/CP0
C only change is including ratio of average molecular weights
CGX=C0/CPR-(WMAV/WMAV0)/(1.D0-GX)
C GAMMA0 should be GAMMASTAR IF T DEPENDENCE OF CP WERE IMPORTANT
C OR IF CONDENSATION BEGINS UPSTREAM OF THE THROAT
ATG=-T/(TSTAR*GAMMA0*MACHSQ)-CGX
DGX=GX-GFRACTO
DLNAAST=DLOG(ARATIO/OARATIO)
DLNRHOR=((DHC/CP-T*WGX)/(GAMMA0*MACHSQ*TSTAR)*DGX
&+CGX*DLNAAST)/ATG
C DEFINITION:
RHORAT=RHOGM/ORHOGM
GAS MIXTURE MASS DENSITIES
RHORAT=DEXP(DLNRHOR)
C SMALL CORRECTION TO RHOG FOR CHANGE IN AVERAGE MOLECULAR WEIGHT
C STRICTLY VALID IN DIFFERENTIAL LIMIT AS IS EQUATION FOR RHORAT
RHOG=ORHOG*RHORAT*OWMAV/WMAV
C equation for DMSQ is ok.
DMSQ=-2.D0*MACHSQ*(DLNRHOR+DLNAAST)
C equation for DT is ok.
is negligible
GAMMA0 is really GAMMASTAR, but T effect
DT=-C0*TSTAR*GAMMA0*DMSQ/2./CPR+DHC*DGX/CP
OLDT=T
T=T+DT
OLDMSQ=MACHSQ
MACHSQ=MACHSQ+DMSQ
OLDP=P
C NEXT EQUATION IS OK BECAUSE RHOG's ARE # DENSITIES
P=P0*(RHOG/RHOG0)*(T/T0)*(1.D0-GX)
178
C Values for surface tension, density and vapor pressure are now in
a function
SIG=surftension(T,ispec)
P20=vappress(T,ispec)
rho=density(T,ispec)
C this is for testing
c
c
c
Write(*,*) 'from main part',wm1,wm2,sigmac,dcoef,ci11,ci22,
&
gamg,CV1,GAMMA1,RSTEJ,CP1
pause
C WATER-NITROGEN DIFFUSION COEFFICIENT FROM HCB
C LINEAR FIT FOR COLLISION INTEGRAL
c
CI=CIS*T+CII
c (1,1) collision integral; epsilon/k for water-nitrogen is 240.3 c
formula from Rigby, Smith, Wakeham, and Maitland, "The Forces Between
Molecules" pp. 217, 218
DLt=DLOG(T/240.3)
ci11=DEXP(0.348-0.459*DLt+0.095*DLt**2-0.010*DLt**3)
DC=DCOEF/CI11/P*T**1.5
c (2,2) collision integral for nitrogen, epsilon/k = 104.2, also
from Rigby et al.
DLt=DLOG(T/104.2)
ci22=DEXP(0.46649-0.57015*DLt+0.19164*DLt**2-0.03708*DLt**3
& +0.00241*DLt**4)
C THK Thermal conductivity of air is close enough to Nitrogen
THK=4.184*(5.69+0.017*(T-273.15))*1.D-5
C
RHO=0.99
C recalculate V2 and R1 at local ambient temp
V2=WM2/RHO/AVOG
R1=(0.75D0*V2/PI)**ONE3RD*1.D8
C UPDATE CLUSTER SIZES
IX=IX+1
GFRACT=0.D0
PG=P-PC
c mass density of carrier gas kg/m^3 = grho
grho=PG/RG/T*WM1*1.d3
CF1=1.D8*WM2*DC*PC*(P/PG)/RG/T
179
CF2=1.D8*32.*DC*P/5./ALPHA/PG/DSQRT(2.*PI*RG*T/WM2)
BIGGAM=(GAMV+1)/(GAMV-1)*PC/P*DSQRT(WMAV/WM2)+
& (GAMG+1)/(GAMG-1)*PG/P*DSQRT(WMAV/WM1)
C CF3 has to be in J not ergs
CF3=1.D7*(64.*T/5.)/DSQRT(2.*PI*RG*T/WMAV)/P/BIGGAM
newixmax=ixmax
DO I=IXMAX,1,-1
SR=DEXP(RSTEJ/R(1,I))
C use kulmala formula for growth rate
IF(IGL.EQ.4)THEN
radius=R(1,I)
CALL hks(radius,T,fluxms,X)
R(2,I)=R(1,I)+1.D8*fluxms*(DX/U)/RHO
c
if (dabs(x-0.568).lt.1.d-3) then
c
write(*,*)X
c
endif
Else IF(IGL.EQ.0)THEN
R(2,I)=R(1,I)+ZETJ*(1.D0-SR/S)
RHOD=RHO
ELSE IF(IGL.EQ.1)THEN
C USE MORE GENERAL GROWTH LAW WITH POSSIBILITY OF NONISOTHERMAL
GROWTH
C NEED TO COMPUTE DROPLET TEMPERATURE TD OR SET EQUAL TO AMBIENT T
C ########
TD(I)=T
R(2,I)=R(1,I)+1.D8*CF1*(DX/U)*(1.D0-SR/S)/R(1,I)
&/RHO/(1.D0+CF2/R(1,I))
RHOD=RHO
ELSE
ibiscnt=0
QTERM(I)=1.D-8*R(1,I)/THK+CF3
C TO CALC droplet temp, using approx that dhc is at T not TD
c use bisection method to find the root in temperature...
TD1 = T-80.D0
c
TD(I)=T+DHC*FLUXM(I)*QTERM(I)
SIGD=surftension(TD1,ispec)
180
P20D=vappress(TD1,ispec)
rhoD=density(TD1,ispec)
RSTEJD=2.*SIGD*WM2/RG/TD1/RHOD*1.D8
SD=PC/P20D
FLUXM1=CF1*(1.D0-DEXP(RSTEJD/R(1,I))/SD)/R(1,I)/
&(1.D0+CF2/R(1,I))
FUNC1 = (TD1-T)/(DHC*QTERM(I))-FLUXM1
TD2 = T+20.D0
606 SIGD=surftension(TD2,ispec)
P20D=vappress(TD2,ispec)
rhoD=density(TD2,ispec)
RSTEJD=2.*SIGD*WM2/RG/TD2/RHOD*1.D8
SD=PC/P20D
FLUXM2=CF1*(1.D0-DEXP(RSTEJD/R(1,I))/SD)/R(1,I)/
&(1.D0+CF2/R(1,I))
FUNC2 = (TD2-T)/(DHC*QTERM(I))-FLUXM2
if(func1*func2.ge.0.0) then
ibiscnt=ibiscnt+1
if(ibiscnt.eq.1000)then
write(*,*)'root must be bracketed for
bis',i,X,td1,td2,func1,func2
stop
endif
c put in a bit of a risky test to see if we can escape bottlenecks
if(dabs(func1-func2).le.2.d-15.and.dabs(func2).lt.5.d-15.and.
& dabs(TD1-TD2).lt.1.d-5)then
td(i)=TD2
goto 299
endif
temp=(td1*func2-td2*func1)/(func2-func1)
if(dabs(func2).lt.dabs(func1))then
td1=td2
func1=func2
endif
181
td2=temp
goto 606
endif
if(func1.lt.0.)then
rtbis = TD1
dtemp = TD2-TD1
else
rtbis= TD2
dtemp = TD1-TD2
endif
do k = 1,50
dtemp = 0.5*dtemp
TD(I) = rtbis+dtemp
SIGD=surftension(TD(I),ispec)
P20D=vappress(TD(I),ispec)
rhoD=density(TD(I),ispec)
RSTEJD=2.*SIGD*WM2/RG/TD(I)/RHOD*1.D8
SD=PC/P20D
FLUXM2=CF1*(1.D0-DEXP(RSTEJD/R(1,I))/SD)/R(1,I)/
&(1.D0+CF2/R(1,I))
FUNC2 = (TD(I)-T)/(DHC*QTERM(I))-FLUXM2
if(func2.lt.0.0)rtbis = TD(I)
if(abs(dtemp).lt.0.005.or.func2.eq.0.0)goto 299
enddo
299
continue
V2D=WM2/RHOD/AVOG
R1=(0.75D0*V2D/PI)**ONE3RD*1.D8
FLUXM(I)=CF1*(1.D0-DEXP(RSTEJD/R(1,I))/SD)/R(1,I)/
&(1.D0+CF2/R(1,I))
R(2,I)=R(1,I)+1.D8*FLUXM(I)*(DX/U)/RHOD
ENDIF
IF(R(2,I).GT.two13*R1)THEN
182
RCUBED=R(2,I)*R(2,I)*R(2,I)
GFRACT=GFRACT+RCUBED*DELNRO(I)
RN(I)=RCUBED*RCONST*RHOD
R(1,I)=R(2,I)
ELSE
R(1,I)=R1
DELNRO(I)=0.D0
RN(I)=1.D0
newixmax=newixmax-1
ENDIF
ENDDO
ixmax=newixmax
GFRACT=GFRACT*GCONST
C
C F1T IS THE TOTAL EFFECTIVE MONOMER CONCENTRATION AT NEW X
F1T=F10*RHOG/RHOG0
C ADJUST MONOMER CONCENTRATION
F1=F1T-GFRACT*RHOG*WM1/WM2
C COMPUTE LOCAL SUPERSATURATION AT NEW X
C PC IS THE MONOMER PARTIAL PRESSURE AT NEW X
PC=F1*RG*T/AVOG
OLDS=S
S=PC/P20
DGA=4.*PI*SIG*AVOG/RG/T*(.75*V2/PI)**TWO3RD
U=USTAR*DSQRT(MACHSQ)
ZETJ=1.D8*DSQRT(RG*T/2./PI/WM2)*V2*F1*DX/U
RSTEJ=2.*SIG*WM2/RG/T/RHO*1.D8
RCR=RSTEJ/DLOG(S)
RNCR=RCONST*RHO*RCR**3
C SKIP STEADY STATE RATE CALCULATION AFTER RATE BECOMES TOO SMALL
IF(ISKIP.EQ.1)GOTO 400
JX=JX+1
C
RCONST=4.*PI*0.6022/3./WM2
C CLASSICAL STEADY STATE RATE, CONTINUOUS APPROXIMATION
DG=DGC*(SIG/T)**3*(WM2/RHO/DLOG(S))**2
RAPROX=WGAM*5.3937D19*(PC/T)**2*DSQRT(SIG*WM2)/RHO*DEXP(-DG)
183
ALPH=DSQRT(8.*PI*RG*T/WM2)*(.75*V2/PI)**TWO3RD*F1
C CALCULATE MODIFIED STEADY STATE RATE
C USE CURTISS, FRURIP, AND BLANDER RESULTS FOR CORRECT MONOMER-DIMER
EQUILIBR.
DEQUK=RG*T/AVOG/1.01325D6*DEXP(-18.59/1.9872+3.59D3/1.9872/T)
I3CR=3.*RNCR
IF(I3CR.GT.999)I3CR=999
SUM=0.D0
FS(I3CR+1)=1.D0
DO I=1,I3CR
J=I3CR+2-I
BIS(J)=I23(J-1)*DEXP(DGA*(I23(J)-I23(J-1)))/S
FS(J-1)=BIS(J)/I23(J)*FS(J)+1.D0
ENDDO
FS(1)=(FS(1)-1.D0)/BIS(2)/F1/DEQUK
RJALPH=F1/FS(1)
RATE=ALPH*RJALPH
c calculate Reiss-Katz-Kegel rate, RATERKK.
rate also
ALTRcl is the classical
C computed in the RKK subroutine for comparison
CALL RKK(T,S,RATERKK,ALTRcl,RKcor,ispec)
IF(IRATE.EQ.0)THEN
DELNRO(JX)=RAPROX/U/RHOG
ELSE
IF(IRATE.EQ.1)THEN
DELNRO(JX)=RATE/U/RHOG
else
DELNRO(JX)=RATERKK/U/rhog
endif
ENDIF
N(JX)=DELNRO(JX)*RHOG*DX
C
GCONST=4.*PI*RHO*AVOG*1.D-24*DX/3./WM1
GFRACT=GFRACT/GCONST
Cxxxxxxx
IF(N(JX).GE.0.1D0)THEN
GFRACT=GFRACT+RCR**3*DELNRO(JX)
IXMAX=JX
184
R(1,JX)=RCR*RFCTR
RN(JX)=RCONST*RHO*R(1,JX)**3
C
CI=CIS*T+CII
c collision integral from the equation; epsilon/k for waternitrogen is 240.3
tred=T/240.3
DLt=DLOG(tred)
ci=DEXP(0.348-0.459*DLt+0.095*DLt**2-0.010*DLt**3)
DC=DCOEF/CI/P*T**1.5
CF1=1.D8*WM2*DC*PC*(P/PG)/RG/T
CF2=1.D8*32.*DC*P/5./ALPHA/PG/DSQRT(2.*PI*RG*T/WM2)
FLUXM(JX)=CF1*(1.D0-DEXP(RSTEJ/R(1,JX))/S)/R(1,JX)/
&(1.D0+CF2/R(1,JX))
TD(JX)=T
C ARRANGE VALUES INTO DESCENDING ORDER FOR R(1,I) IF R(1,JX) .GE.
R(1,JX-1)
C ALL R(1,I) ARE IN DESCENDING ORDER UP TO I=JX-1
C%%%%
IF(R(1,JX).GE.R(1,JX-1))THEN
C####
DO I=JX-2,1,-1
C!!!!
IF(R(1,JX).LT.R(1,I))THEN
C FIRST CHECK EQUALITY
IF(R(1,JX).EQ.R(1,I+1))THEN
DELNRO(I+1)=DELNRO(I+1)+DELNRO(JX)
IXMAX=IXMAX-1
JX=JX-1
C NO OTHER CHANGES NEEDED
GOTO 350
C****
ELSE
C****
C ELSE R(1,JX) LIES BETWEEN R(1,I) AND R(1,I+1)
TEMPD=DELNRO(JX)
TEMPR=R(1,JX)
TEMPN=RN(JX)
185
TEMPTD=TD(JX)
TEMPFL=FLUXM(JX)
TEMPQT=QTERM(JX)
C SHIFT OTHER VALUES TO OPEN UP SLOT FOR TEMP VALUES
DO IT=JX-1,I+1,-1
DELNRO(IT+1)=DELNRO(IT)
R(1,IT+1)=R(1,IT)
RN(IT+1)=RN(IT)
TD(IT+1)=TD(IT)
FLUXM(IT+1)=FLUXM(IT)
QTERM(IT+1)=QTERM(IT)
ENDDO
C INSERT TEMP VALUES
DELNRO(I+1)=TEMPD
R(1,I+1)=TEMPR
RN(I+1)=TEMPN
TD(I+1)=TEMPTD
FLUXM(I+1)=TEMPFL
QTERM(I+1)=TEMPQT
GOTO 350
ENDIF
C****
ENDIF
C!!!!
ENDDO
C####
ENDIF
C%%%%
C END OF SORT PROCEDURE
Cxxxxxxx
ELSE
JX=0
IF(OLDS.GT.S)ISKIP=1
RAPROX=0.D0
RATE=0.D0
RATERKK=0.d0
186
ALTRcl=0.d0
ENDIF
Cxxxxxxx
350 CONTINUE
GFRACT=GFRACT*GCONST
C ADJUST MONOMER CONCENTRATION
F1=F1T-GFRACT*RHOG*WM1/WM2
C COMPUTE LOCAL SUPERSATURATION AT NEW X
C PC IS THE MONOMER PARTIAL PRESSURE AT NEW X
PC=F1*RG*T/AVOG
S=PC/P20
C UPDATE TOTAL NUMBER DENSITY BASED ON STEADY STATE APPROXIMATION
400 ZETN=DX/RHOG/U
GFRACTO=GX
GX=GFRACT
FTCREL=FTCREL+ZETN*RAPROX
FTC=RHOG*FTCREL
FTMREL=FTMREL+ZETN*RATE
FTM=RHOG*FTMREL
C
TZET=TZET+ZET*2.
500 CONTINUE
OARATIO=ARATIO
C
IF(IX.LT.IWRIT)THEN
X=X+DX
if(idoption.ne.0)then
IF(X.GT.XD(ITWO))THEN
IONE=ITWO
ITWO=IONE+1
aratiom2=aratiom(itwo)
aratiom1=aratiom(ione)
dadx=(aratiom2-aratiom1)/(xd(itwo)-xd(ione))
C
write(*,*)'dadx= ',dadx (Som)
ENDIF
endif
187
C
IF(IMAX.LT.NIMAX)IMAX=NIMAX
GOTO 100
ELSE
IX=0
C DO NOT WRITE IF IXMAX=0, NO PARTICLES CREATED YET
IF(IXMAX.EQ.0)THEN
if(ISANS.eq.1)then
ISANS=0
else
ISANS=1
endif
X=X+DX
c
aratio = 1.0+dadxn*(x-xthroat)
GOTO 100
ENDIF
C WRITE RESULTS AT CURRENT X
C FIRST CREATE DENSITIES AND CUMULATIVE DISTRIBUTIONS
IRMAX=INT(R(1,1)+0.5)
DO IR=1,IRMAX+9
DNDR(IR)=0.D0
DGDR(IR)=0.D0
NCUM(IR)=0.D0
GCUM(IR)=0.D0
ENDDO
C FINE-GRAINED CUMULATIVE DISTRIBUTIONS VERSUS DROPLET RADIUS
NC(0)=0.D0
GC(0)=0.D0
SIGNALT=0.D0
DO I=1,IXMAX
J=IXMAX-I+1
N(J)=DELNRO(J)*RHOG*DX
RI(I)=R(1,J)
NC(I)=NC(I-1)+N(J)
RNTN=RN(J)*N(J)
GC(I)=GC(I-1)+RNTN
SIGNALT=SIGNALT+RN(J)*RNTN
188
ENDDO
JMAX=IXMAX
TOTNP=NC(JMAX)
C*********************
if(x.gt.6.d0-dx.and.x.lt.6.d0+dx)then
C One time calc to check adequacy of binning
C PARTICLE FORM FACTORS FOR DROPS(D) AND SHELLS(S)
DO JQ=1,102
QSQ=Q(jq)*Q(jq)
Q6=QSQ*QSQ*QSQ
DO I=1,IXMAX
J=IXMAX-I+1
QR=Q(jq)*RI(I)
PDQR=(DSIN(QR)-QR*DCOS(QR))**2/Q6
PSQR=(RI(I)*DSIN(QR)+Q(jq)*TESQ*DCOS(QR))**2
&
*DEXP(-QSQ*TESQ)*DELESQ/QSQ
SIDF(jq)=SIDF(jq)+N(J)*PDQR
SISF(jq)=SISF(jq)+N(J)*PSQR
ENDDO
ENDDO
WRITE(1,1050)X,TIME
1050 FORMAT(' Computed with the unbinned PSD',
&/' AT X=',F6.3,'CM, TIME=',0PF6.1,'us',//,
&4x,'q',5x,'I(H2O)',4x,'I(D2O)',3x, 'I(EtOH)',/'
(1/A)',4x,'(1/cm)'
&,4x,'(1/cm)',4x,'(1/cm)')
do jq=1,102
C
CNTPW=(-5.6D9)*(-5.6D9)*16*PI**2*1.D-48
C
CNTPD=6.4D10*6.4D10*16*PI**2*1.D-48
C
CNTPE=6.4D10*6.4D10*16*PI**2*1.D-48
WRITE(1,2060)Q(jq),SIDF(jq)*CNTPW,SIDF(jq)*CNTPD,SISF(jq)*CNTPE
enddo
endif
C**********************************************************
C REGROUP THE FINELY RESOLVED CUMULATIVE MASS AND NUMBER
DISTRIBUTIONS INTO
189
C ONE ANGSTROM WIDE BINS
C START THE MASS AND NUMBER DISTRIBUTIONS AT SMALLEST R WITH NONZERO
NC(I)
C#######
DO I=1,IXMAX
c
IF(RI(I).LE.RCR)NC(I)=0.D0
IF(NC(I).GT.0.D0)THEN
IRMIN=INT(RI(I)+0.5)
JI=I
GOTO 601
ENDIF
ENDDO
601
CONTINUE
IRNEXT=IRMIN
602
GCUM(IRNEXT)=0.D0
NCUM(IRNEXT)=0.D0
RNEXT=1.*IRNEXT+0.5
DO J=JI,JMAX
IF(RNEXT.LT.RI(J))THEN
GCUM(IRNEXT)=GC(J-1)+(GC(J)-GC(J-1))*(RNEXT-RI(J-1))/
&
(RI(J)-RI(J-1))
GCUM(IRNEXT)=GCUM(IRNEXT)/GC(JMAX)
NCUM(IRNEXT)=NC(J-1)+(NC(J)-NC(J-1))*(RNEXT-RI(J-1))/
&
(RI(J)-RI(J-1))
C WE WANT K TO MATCH UP WITH J-1
K=JMAX-J+2
TDC(IRNEXT)=TD(K)
FLUXMC(IRNEXT)=FLUXM(K)
QTERMC(IRNEXT)=QTERM(K)
JI=J
IRNEXT=IRNEXT+1
GOTO 602
ENDIF
ENDDO
IRLAST=IRNEXT
ILAST=((IRLAST+9)/10)*10
DO IN=IRLAST,ILAST
190
DNDR(IN)=0.D0
GCUM(IN)=1.D0
NCUM(IN)=1.D0
TDC(IN)=0.
FLUXMC(IN)=0.
QTERMC(IN)=0.
ENDDO
TDC(IRLAST)=TD(1)
FLUXMC(IRLAST)=FLUXM(1)
QTERMC(IRLAST)=QTERM(1)
NCUM(IRLAST)=NC(JMAX)
IGSTART=((IRMIN-1)/10)*10+1
INSTART=((IRMIN-1)/10)*10+1
DO IG=IGSTART,IRMIN-1
GCUM(IG)=0.D0
ENDDO
DO IN=INSTART,IRMIN-1
NCUM(IN)=0.D0
DNDR(IN)=0.D0
TDC(IN)=0.
FLUXMC(IN)=0.
QTERMC(IN)=0.
ENDDO
NCUM(IRMIN-1)=0.D0
C CONSTRUCT PARTICLE NUMBER DENSITY FUNCTION
C AND NEUTRON SCATTERING INTENSITY
do jq=1,102
SIS(jq)=0.D0
SID(jq)=0.D0
enddo
DO IN=IRMIN,IRLAST
DNDR(IN)=NCUM(IN)-NCUM(IN-1)
do jq=1,102
SID(jq)=SID(jq)+DNDR(IN)*PD(jq,IN)
SIS(jq)=SIS(jq)+DNDR(IN)*PS(jq,IN)
enddo
191
NCUM(IN-1)=NCUM(IN-1)/NCUM(IRLAST)
ENDDO
SUMDNDR=NCUM(IRLAST)
NCUM(IRLAST)=1.D0
C*********
C calculate various mean radii-based on different moments of the
distribution
sumrnav=0.d0
sumrmav=0.d0
sumr3av=0.d0
sumr6av=0.d0
sumr8av=0.d0
do ir=irmin,irlast
sumrnav=sumrnav+ir*dndr(ir)
sumrmav=sumrmav+r4(ir)*dndr(ir)
sumr3av=sumr3av+r3(ir)*dndr(ir)
sumr6av=sumr6av+r6(ir)*dndr(ir)
sumr8av=sumr8av+r8(ir)*dndr(ir)
enddo
rnav=sumrnav/sumdndr
rmav=sumrmav/sumr3av
r86av=dsqrt(sumr8av/sumr6av)
rgav=sqrt(0.6)*r86av
r6av=(sumr6av/sumdndr)**(1.d0/6.d0)
C construct scattering intensity in small q limit
do jq=1,102
sisqd(jq)=sumr6av/9*dexp(-q(jq)*q(jq)*rgav*rgav/3)
enddo
C**********************************************************
NFRAC=GC(JMAX)/F1T
SIGNALT=SIGNALT+F1
IMEAN=SIGNALT/GC(JMAX)
RMEAN=1.D8*(3.*IMEAN*V2/4./PI)**ONE3RD
FMEAN=GC(JMAX)/IMEAN
C
TNP(2)=TNP(2)+TNP(1)
C
TNP(1)=0.
GFRACT=GC(JMAX)*WM2/RHOG/WM1
192
c
SCTSIGT=SIGNALT*SIGFAC
SCTSIGT=SIGNALT
C**********************************************************
C CALCULATE WET ISENTROPIC VALUES (NO CONDENSATE) FOR COMPARISON
C FIRST CALCULATE LOCAL MACH NUMBER USING ITERATIVE SCHEME
2 RM(1)=DSQRT(C10*(C20*(MACHNO*ARATIO)**EAI0-1.D0))
DO IM=1,10
RM(IM+1)=DSQRT(C10*(C20*(RM(IM)*ARATIO)**EAI0-1.D0))
ENDDO
MACHNO=RM(11)
IF(DABS(MACHNO-RM(10)).GT.1.D-8)GOTO 2
C WITH MACHNO COMPLETE CALCULATION
TISW=T0/(1.D0+MACHNO*MACHNO/C10)
RHOGISW=RHOG0*(T0/TISW)**ERHO0
PISW=P0*(RHOGISW/RHOG0)**GAMMA0
UISW=RHOUST/RHOGISW/ARATIO
C**********************************************************
C**********************************************************
C CALCULATE DRY ISENTROPIC VALUES (NO CONDENSIBLE) FOR COMPARISON
C FIRST CALCULATE LOCAL MACH NUMBER USING ITERATIVE SCHEME
3 RM(1)=DSQRT(C1*(C2*(MACHNOD*ARATIO)**EAI-1.D0))
DO IM=1,10
RM(IM+1)=DSQRT(C1*(C2*(RM(IM)*ARATIO)**EAI-1.D0))
ENDDO
MACHNOD=RM(11)
IF(DABS(MACHNOD-RM(10)).GT.1.D-8)GOTO 3
C WITH MACHNO COMPLETE CALCULATION
TISD=T0/(1.D0+MACHNOD*MACHNOD/C1)
RHOGISD=RHOG0*(T0/TISD)**ERHO
PISD=P0*(RHOGISD/RHOG0)**GAMMA1
UISD=RHOUST/RHOGISD/ARATIO
C**********************************************************
C
WRITE(9,1000) all of the desired output
PISW=PISW*760./1.01325D6
193
PISD=PISD*760./1.01325D6
PWR=P*760./1.01325D6
PCWR=PC*760./1.01325D6
P20WR=P20*760./1.01325D6
C
U=RHOUST/RHOG/ARATIO
U=USTAR*DSQRT(MACHSQ)
MACHST=DSQRT(MACHSQ)
TIME=TIME+DX*IWRIT/U*1.D6
Porig=P0*760./1.01325D6
C
write(*,2001)X,PWR/Porig,PISD/Porig,PISW/Porig,rnav(commented by som)
C ********************************************************* (SOM)
WRITE(7,4051)X,PWR/Porig,PISD/Porig,PISW/Porig
4051 FORMAT(F9.4,3F7.4)
C *********************************************************
C
WRITE(6,2000)X,TIME,T,PWR,RHOG,MACHST,U,TISW,PISW,RHOGISW,MACHNO,
C
&UISW,TISD,PISD,RHOGISD,MACHNOD,UISD,S,PCWR,P20WR,RNCR,RCR,RATE,
C
&RAPROX,RATENS,RATEP,ALPH,TZET
C
IF(ISANS.EQ.1)THEN
WRITE(2,2050)X,TIME,rnav,rmav,r86av,rgav,r6av
2050 FORMAT(/' AT X=',F6.3,'CM, TIME=',0PF6.1,'us',2x,'Rn=',f7.2,
&' A, Rm=',f7.2,' A, R8/6=',f7.2,' A, Rg=',f7.2,' A, R6av=',
&f7.2,' A',//,
&4x,'q',5x,'I(H2O)',4x,'I(D2O)',3x, 'I(EtOH)',3x,'I(D2O)-small
q'
&,/' (1/A)',4x,'(1/cm)'
&,4x,'(1/cm)',4x,'(1/cm)',4x,'(1/cm)')
do jq=1,102
C
CNTPW=(-5.6D9)*(-5.6D9)*16*PI**2*1.D-48
C
CNTPD=6.4D10*6.4D10*16*PI**2*1.D-48
C
CNTPE=6.4D10*6.4D10*16*PI**2*1.D-48
WRITE(2,2060)Q(jq),SID(jq)*CNTPW,SID(jq)*CNTPD,SIS(jq)*CNTPE,
&sisqd(jq)*cntpd
enddo
194
2060 FORMAT(F6.3,4(1x,1pE9.2))
ISANS=0
ELSE
ISANS=1
ENDIF
WRITE(9,2000)X,TIME,T,PWR,RHOG,MACHST,U,TISW,PISW,RHOGISW,MACHNO,
&UISW,TISD,PISD,RHOGISD,MACHNOD,UISD,S,PCWR,P20WR,RNCR,RCR,RATE,
&RAPROX,RATERKK,WGAM*ALTRcl,RATENS,RATEP,ALPH,TZET
2000 FORMAT(/' AT X=',F6.3,'CM, TIME=',0PF6.1,'us, T=',F7.2,' K,
P=',
&F8.2,' MM HG, GAS DENSITY=',1PE11.4,'/CM**3, MACH *=',0PF5.2,'
FLO
&W SPEED=',1PE9.2,' CM/S',/'
WET ISENTROPE VALUES:
T=',
&0PF7.2,' K, P=',F8.2,' MM HG, GAS DENSITY=',1PE11.4,'/CM**3,
MACH
&#=',0PF5.2,' FLOW SPEED=',1PE9.2,' CM/S '/'
DRY ISENTROPE
VALU
&ES:
T=',0PF7.2,' K, P=',F8.2,' MM HG, GAS
DENSITY=',1PE11.4,'/
&CM**3, MACH #=',0PF5.2,' FLOW SPEED=',1PE9.2,' CM/S ',/'
SUPERSATU
&RATION=',0PF6.2,', P2=',
&1PE9.2,' MM HG, P2E=',E9.2,' MM HG, CRITICAL NUCLEUS
HAS',0PF7.1,'
& MOLECULES, AND ITS RADIUS=',F7.2,' A',/' MODIFIED STEADY
STATE RA
&TE=',1PE9.2,'/(S-CM**3), APPROXIMATE CLASSICAL STEADY STATE
RATE='
&,1PE9.2,'/(S-CM**3)',/' DILLMANN-MEIER RATE=',E9.2,'/(SCM**3)',/,
&' ALTERNATE APPROX. CL. RATE*WGAM=',E9.2,'/(S-CM**3)',/,
&' NONSTEADY PARTICLE FLUX AT (i*-1)=',1PE9.2
&,'/(S-CM**3), AT (i*)=',1PE9.2,'/(S-CM**3)',/,' MONOMER
COLLISION
&FREQUENCY=',E9.2,'/S, ',0PF6.1,' COLLISIONS BY 1-MER FROM LAST
X')
2001 format(F7.3,F8.4,F8.4,F8.4,f9.4)
C
WRITE(6,3010)F1T,SCTSIGT,DX,DGX,GFRACT,GX,NFRAC,TOTNP,FTN,FTM,FTC
195
C
&,IMEAN,RMEAN,FMEAN,TSUBCR
WRITE(9,3010)F1T,SCTSIGT,DX,DGX,GFRACT,GX,NFRAC,TOTNP,FTN,FTM,FTC
&,IMEAN,RMEAN,FMEAN,TSUBCR
3010 FORMAT(' EFFECTIVE MONOMER NUMBER DENSITY =',1PE11.4,'/CM**3,
REL
&ATIVE LIGHT SCATTERING SIGNAL=', E11.4,'
DX=',E10.3,' DGX=',
&E10.3,/,
&' CONDENSATE MASS FRACTION: (SUM 2:IMAX)=', E11.4,'
(NR**3)=',
&E11.4,'
FRACTION CONDENSED=',0PE10.3,/,
&' PARTICLE DENSITY (sum i*:imax)=',1PE10.3,'/CM**3,
FROM
NONSTEA
&DY FLUX =', E10.3,'/CM**3',/,18X,'FROM MOD. SS RATE =',E10.3,
&'/CM**3,
FROM CLASSICAL SS RATE =',E10.3,'/CM**3',/,'
imean=',
&0PF9.0,' MOLECULES,
FMEAN=',1PE10.3,5X,'SUBCRI
RMEAN=',F7.2,' A,
&TICAL NUMBER DENSITY (2:i*-1)=',E10.3,/)
C
IF(IRATE.EQ.0)RATE=RAPROX
IF(IRATE.EQ.2)RATE=RATERKK
WRITE(3,4001)X,TIME,S,TISW,T,RHOG/RHOG0,RHOGISW/RHOG0,GX,TOTNP,
*RATE
! Changed GFACT to GX by SOM
write(4,4002)x,time,s,PWR,PISW,PISD,PWR/P0*1013250./760.
*,pisd/p0*1013250./760
4002 FORMAT(F9.4,F6.1,F6.2,3F7.1,2F7.4,3(1PE10.3))
C RMEAN,FMEAN,SCTSIGT/RHOG
IFILE3=IFILE3+1
C WRITE CLUSTER NUMBER DENSITIES VERSUS DROPLET RADIUS
C WRITE NUMBER AND MASS DENSITIES AND CUMULATIVE DISTRIBUTIONS
WRITE(9,2020)SUMDNDR,(ILAST-INSTART+1)
2020 FORMAT(' NUMBER DENSITY VERSUS
RADIUS',1PE11.3,'/CM**3',/,1X,I4)
DO IR=INSTART,ILAST-9,10
WRITE(9,2021)IR,(DNDR(I),I=IR,IR+9)
2021 FORMAT(I4,10(1PE12.4))
196
ENDDO
WRITE(9,2023)(ILAST-INSTART+1)
2023 FORMAT(/,' CUMULATIVE NUMBER FRACTION VERSUS RADIUS',/,1X,I4)
DO IR=INSTART,ILAST-9,10
WRITE(9,2021)IR,(NCUM(I),I=IR,IR+9)
ENDDO
WRITE(9,2024)(ILAST-INSTART+1)
2024 FORMAT(/,' CUMULATIVE COND. MASS FRACTION VERSUS
RADIUS',/,1X,I4)
DO IR=IGSTART,ILAST-9,10
WRITE(9,2021)IR,(GCUM(I),I=IR,IR+9)
ENDDO
IF(IGL.EQ.2)THEN
WRITE(9,2025)(ILAST-INSTART+1)
2025 FORMAT(/,' DROPLET TEMPERATURE VERSUS RADIUS',/,1X,I4)
DO IR=INSTART,ILAST-9,10
WRITE(9,2026)IR,(TDC(I),I=IR,IR+9)
2026 FORMAT(I4,10(F7.2))
ENDDO
WRITE(9,2027)(ILAST-INSTART+1)
2027 FORMAT(/,' DROPLET QTERM VALUE VERSUS RADIUS',/,1X,I4)
DO IR=INSTART,ILAST-9,10
WRITE(9,2021)IR,(QTERMC(I),I=IR,IR+9)
ENDDO
WRITE(9,2028)(ILAST-INSTART+1)
2028 FORMAT(/,' DROPLET MASS FLUX VERSUS RADIUS',/,1X,I4)
DO IR=INSTART,ILAST-9,10
WRITE(9,2021)IR,(FLUXMC(I),I=IR,IR+9)
ENDDO
ENDIF
C
300 X=X+DX
if(idoption.ne.0)then
IF(X.GT.XD(ITWO))THEN
IONE=ITWO
ITWO=IONE+1
aratiom2 = aratiom(itwo)
aratiom1 = aratiom(ione)
197
dadx=(aratiom2-aratiom1)/(xd(itwo)-xd(ione))
ENDIF
endif
TZET=0.
ENDIF
C
IF(IMAX.LT.NIMAX)IMAX=NIMAX
IF(X.GE.XFIN)THEN
STOP
ELSE
c
DX=DX/2.D0
c
IWRIT=2*IWRIT
GOTO 100
ENDIF
STOP
50 END
C**********************************************************
C
C SUBROUTINE ECHO USED TO WRITE INPUT FILE AT START OF OUTPUT FILE
C
SUBROUTINE ECHO
CHARACTER*80 A
WRITE(3,3)
WRITE(9,3)
15 READ(5,1,END=99)A
WRITE(3,2)A
WRITE(9,2)A
GOTO 15
99 CONTINUE
REWIND 5
RETURN
1 FORMAT(A80)
2 FORMAT(1X,A80)
3 FORMAT(1H1,20X,'INPUT FILE',//)
END
subroutine RKK(T,S,RJrkk,RJcl,RKcor,ispec)
198
c calculates the theoretical classical and Reiss, Katz, Kegel rates
for
c water or d2o
implicit real*8(a-h,o-z)
real*8 k,M,komp
avo = 6.022D+23
!number/mol
R = 8.3145D0
!J/K*mol
k = 1.381D-23
!J/K
pi = 3.141593D0
d = density(T,ispec)*1000.d0
!now in [kg/m**3]
ps = vappress(T,ispec)*0.1d0
sg = surftension(T,ispec)*0.001d0
!now in [Pa]
!now units in [N/m]
komp = compressibility(T,ispec)
!in units of
c for water
if(ispec.eq.1)then
M = .018016
!kg/mol
endif
c for d2o
if(ispec.eq.2)then
M = 0.02020004
!kg/mol
endif
c nucleation rate according to Reiss,Katz,Kegel
c (straight from Judith's notes...)
p1 = S*ps
v =
M/(d*avo)
rp = 2.0* sg * v/(k*t*dlog(s))
encc = 4.*pi/3.*rp**3/v
c
c
g = -encc*k*t*log(s) + (36.*pi)**(1./3.)*v**(2./3.)*
&
encc**(2./3.)*sg
c analytically combine surface and volume terms in above two lines
dgc=16.*pi*avo/3./R**3
DG=DGC*(SG/T)**3*(M/d/DLOG(S))**2
199
c calculate classical (JBD), gershick-chiu (JGC), Reis Katz Kegel
(JRKK)
Preexp = dsqrt(2.*avo*sg/pi/M)*(p1/(k*t))**2*v
RJBD = Preexp*dexp(-dg)/1000000.
!per cm^3 and s
thetta = (36.*pi)**(1./3)*v**(2./3.)*sg/(k*t)
corr = 1./s*dexp(thetta)
RJGC = RJBD*corr
corr1 = dsqrt(encc*k*t*komp*M/(avo*d))*avo*d/M
c
RJRKK = RJGC/corr1
RJcl=RJBD
c
c
RJdm=RJRKK/RKcor
set RJRKK equal to the empirically corrected value of JBD
c for water
if(ispec.eq.1)then
corr2 = exp(-27.56+6500./t)
endif
c for d2o
if(ispec.eq.2)then
corr2 = exp(-35.98 +8610./t)
endif
c
RJRKK = RJBD*corr2*RKcor
if(ispec.eq.1)then
hale= ((16.*pi)/3)*(1.435**3)*(((647.15/T)-1)**3)/
&
((dlog(S))**2)
endif
if(ispec.eq.2)then
hale= ((16.*pi)/3)*(1.4759**3)*(((643.89/T)-1)**3)/
&
((dlog(S))**2)
endif
hal=(1.0D+26)*dexp(-hale)*RKcor
RJRKK=hal
return
end
200
C THE PHYSICAL PROPERTY FUNCTIONS OF THE CONDENSING SPECIES.
C MOLECULAR WEIGHT, HEAT CAPACITY OF THE GAS, HEAT CAPACITY OF THE
LIQUID,
C SURFACE TENSION, EQUIL. VAPOR PRESS. AND DENSITY OF CONDENSATE
real*8 function molweight(ispec)
c for water
if(ispec.eq.1)then
molweight=18.0152
endif
c for D2O
if(ispec.eq.2)then
molweight=20.02
endif
return
end
real*8 function specheat(ispec)
c for water
if(ispec.eq.1)then
specheat=1.865
endif
c for D2O
if(ispec.eq.2)then
specheat=1.710
endif
return
end
real*8 function specheatc(ispec)
c for water
if(ispec.eq.1)then
specheatc=4.217
endif
c for D2O
if(ispec.eq.2)then
specheatc=4.205
endif
201
return
end
real*8 function surftension(T,ispec)
real*8 T, Tp
c using Judith's correlation for H2O; same as D2o except Tp=T
if(ispec.eq.1)then
surftension=93.6635d0 + 9.133d-3*T - 2.75d-4*T*T
endif
c using Judith Bell's correlation for D2O
if(ispec.eq.2)then
Tp = 1.022*T
surftension=93.6635d0 + 9.133d-3*Tp - 2.75d-4*Tp*Tp
endif
return
end
real*8 function vappress(T,ispec)
real*8 logPratio, P, Pc, T, Tc, tau
real*8 alpha1, alpha2, alpha4, alpha11, alpha20
c for water used Judith Bell's correlation
if(ispec.eq.1)then
a1 = 77.34491296
a2 = 7235.42
a4 = 8.2
a5 = 0.0057113
vappress = 10.*exp(a1-a2/T-a4*dlog(T)+a5*T) !in dynes/cm^2
endif
C
Function to calculate the vapor pressure of D2O based on
C
the equation of P.G.Hill, R.D.Chris McMillan and V. Lee
C
J.Phys.Chem.Ref.Data Vol 11, No 1, p1-14 (1992) (dynes/cm^2)
if(ispec.eq.2)then
Tc=643.89
! in Kelvin
Pc=21.66
! in MPa
alpha1 = -7.81583
alpha2 = 17.6012
202
alpha4 = -18.1747
alpha11 = -3.92488
alpha20 = 4.19174
tau = 1-T/Tc
logPratio=(Tc/T)*(alpha1*tau+alpha2*tau**1.9+alpha4*tau**2
&+alpha11*tau**5.5+alpha20*tau**10)
vappress = 10.*1.e6*Pc*exp(logPratio)
!in dynes/cm^2
endif
return
end
real*8 function density(T,ispec)
real*8 T
c for water from J.Bell. - goes over to LDA phase at low T.
if(ispec.eq.1)then
xj = (T-225.)/46.2
tr = (647.15-T)/647.15
density = 0.08*tanh(xj)+0.7415*tr**0.33 + 0.32
endif
c using Judith Bell's correlation -- assumes an equivalent LDA for
d2o
if(ispec.eq.2)then
xj = (T-231.)/51.5
tr = (643.89-T)/643.89
!643.89K = critical
temperature
density = 0.09*tanh(xj)+0.847*tr**0.33 + 0.338
endif
return
end
function Hvap(T,ispec)
real*8 logPratio, P, Pc, T, Tc, tau, function, Hvap
real*8 alpha1, alpha2, alpha4, alpha11, alpha20
c derivative of Judith's correlation for H2O
R=8.3145
!J/K/mol
if(ispec.eq.1)then
! comment removed by som
a2 = 7235.42
203
a4 = 8.2
a5 = 0.0057113
c THIS IS WRONG 28-5-02 GW
C
Hvap=-R*T*(-a2+a4*T-a5*T*T)
C This is right 28-5-02 GW
Hvap=-R*(-a2+a4*T-a5*T*T)/18.0153
!J/g comment removed by
som
endif
c derivative of the Hill et al vapor pressure equation for d2o
if(ispec.eq.2)then
Tc=643.89
! in Kelvin
Pc=21.66
! in MPa
alpha1 = -7.81583
alpha2 = 17.6012
alpha4 = -18.1747
alpha11 = -3.92488
alpha20 = 4.19174
tau = 1-T/Tc
logPratio=(Tc/T)*(alpha1*tau+alpha2*tau**1.9+alpha4*tau**2
&+alpha11*tau**5.5+alpha20*tau**10)
function=alpha1+1.9*alpha2*tau**0.9+2*alpha4*tau
&+5.5*alpha11*tau**4.5+10*alpha20*tau**9
Hvap=-R*Tc*(T*logPratio/Tc+T*function/Tc)/20.02
endif
return
end
real*8 function compressibility(T,ispec)
real*8 T,a,b,c,d,e,f,tcel
c compressibility for H2O
c using Judith Bell's correlation
if(ispec.eq.1)then
compressibility = 50.9804 - 0.374957*tcel + 7.21324d3*tcel**2
&
- 64.1785d-6*tcel**3 + 0.343024d-6*tcel**4
&
-
0.684212d-9*tcel**5
compressibility = 1.0d-11*compressibility
204
endif
c compressibility for D2O
c using Judith Bell's correlation
if(ispec.eq.2)then
a = 53.5216
b = 0.4536
c = 8.7212d-3
d = 8.5541d-5
e = 5.4089d-7
f = 1.3478d-9
tcel = T-273.15
compressibility = 1.0d-11*(a-b*tcel+c*tcel*tcel
&
-d*tcel**3+e*tcel**4-f*tcel**5)
endif
return
end
c Non-isothermal HK equation for massflux
c sr is DEFINED in MAIN program
c ptot is p in MAIN program
subroutine hks(radius,T,fluxms,X)
implicit real*8(a-h,o-z)
real*8 mratio,lamdav,lamdat,kn,mv,mg,massflux1,ls,knt,
& numdot,nuhdot,nutrm,nutrh,Ke,logPratio
COMMON /properties/ p,p20,grho,SIG,RHO,DHC,S,sr,THK,CP1,GAMMA1
COMMON
/params/wm1,wm2,sigmac,dcoef,ci11,ci22,gamg,CV1,RSTEJ,pc,
>wmav,cp0,GAMMA0,ispec
c convert A to m
a=radius*1.d-10
R=8.3145d0
pi=3.14159d0
pe=p20/10.d0
ptot=p/10.d0
wm1k=wm1*1D-3
wm2k=wm2*1.D-3
205
wmavk=wmav*1.d-3
Pcon=PC/10.d0
pgas=ptot-pcon
c THK units are W/cm/K
eta=THK*1.d2
spechp=CP0*1.d3
c latent heat of condensation in j/kg
ls=1000*DHC
c
dc is in units of cm^2/s.
Here dc has units m^2/s
dc=dcoef/ci11/(ptot)*T**1.5*1.D-5
c NUSSELT # mdot
numdot=DSQRT(2D0*(wmavk**2)/wm2k/pi/R/T)*ptot*a/grho/dc
c NUSSELT hdot
nuhdot=DSQRT(2D0/pi)*((GAMMA0+1D0)/(2D0*GAMMA0))*spechp*ptot*
& a/eta/SQRT(R*T/(wmavk))
c nusselt tr mdot
nutrm=(2D0*numdot)/(numdot+2D0)
c nusselt tr mdot
nutrh=(2D0*nuhdot)/(nuhdot+2D0)
dmod=dc*pgas*wm2k/R/T
thetanew=(dmod*ls*nutrm)/(eta*nutrh)
Ke=RSTEJ/radius
tempfunction=DLOG(S)-Ke
c1=(T/thetanew/S)*((ptot/pe)-S)
c2=ls*wm2k/R/T
delta1=tempfunction*((0.5*(c1**2D0))-c2)/((c1+c2)**2D0)
Tdrop=T*(1D0+(tempfunction*(1D0/(c1+c2)))*(1D0-delta1))
if(ispec.eq.1) then
SIGN=93.6635d0 + 9.133d-3*Tdrop - 2.75d-4*Tdrop*Tdrop
!*****************************************************
a1 = 77.34491296
a2 = 7235.42
a4 = 8.2
a5 = 0.0057113
P20N=10.*exp(a1-a2/Tdrop-a4*dlog(Tdrop)+a5*Tdrop)
206
!****************************************************
xj = (Tdrop-225.)/46.2
tr = (647.15-Tdrop)/647.15
rhoN = 0.08*tanh(xj)+0.7415*tr**0.33 + 0.32
!****************************************************
endif
if(ispec.eq.2) then
Tp = 1.022*Tdrop
SIGN=93.6635d0 + 9.133d-3*Tp - 2.75d-4*Tp*Tp
!*****************************************************
Tc=643.89
! in Kelvin
Pc=21.66
! in MPa
alpha1 = -7.81583
alpha2 = 17.6012
alpha4 = -18.1747
alpha11 = -3.92488
alpha20 = 4.19174
tau = 1-Tdrop/Tc
logPratio=(Tc/Tdrop)*(alpha1*tau+alpha2*tau**1.9+alpha4*tau**2
&+alpha11*tau**5.5+alpha20*tau**10)
P20N = 10.*1.e6*Pc*exp(logPratio)
!in dynes/cm^2
!****************************************************
xj = (T-231.)/51.5
tr = (643.89-Tdrop)/643.89
temperature
!643.89K = critical
rhoN = 0.09*tanh(xj)+0.847*tr**0.33 + 0.338
!****************************************************
endif
RSTEJN=2D0*SIGN*WM2/R/1D7/Tdrop/RHON*1.D8 ! R required here
has a value of 8.314D7 as RG in main body
c
C
C
SD=PC*10d0/P20D
massflux1=DSQRT(wm2k/2D0/pi/R)*pe*
& ((S/DSQRT(T))-(DEXP(RSTEJN/radius)/DSQRT(Tdrop)))
massflux1=DSQRT(wm2k/2D0/pi/R)*
207
& ((S*pe/DSQRT(T))((p20N/10D0)*DEXP(RSTEJN/radius)/DSQRT(Tdrop))) ! THIS ONE IS THE RIGHT
ONE TO USE AS VERIFIED FROM THE WEGENER PAPER
if (dabs(x-2.0).lt.1.d-5) then
c
write(*,*)tempfunction
write(71,1112) T,S,radius,Tdrop,(ptot/133.33)
1112
Format(5F8.2)
endif
c change to g/cm^2/s
fluxms=massflux1/10
return
end
208
APPENDIX F
FORTRAN CODE TO COMPARE GROWTH RATES AND
DROPLET TEMPERATURES PREDICTED BY
NON-ISOTHERMAL GROWTH LAWS
209
C
THIS CODE COMPARES HKS ,KULMALA AND PETERS AND PAIKERT.
C
THE PHYSICAL PROPERTY RELATIONS HAVE BEEN CHANGED TO THOSE IN
MAIN CODE....
C
ALL THE UNITS ARE IN SI
C
LAST BIT *** DIFFUSIVITY HAS ALSO BEEN CHANGED AS IN THE MAIN
CODE FOR PAPER 1
C
ALSO INCLUDING KULMALA
C
This ones only for d2o !!!!!!!!!!!!!!!
C
HKS calculations also as in the main code(MU NOT CALCULATED )
PROGRAM CALCTD
IMPLICIT NONE
DOUBLE PRECISION
>
Rv, Rg, Cp, Cv, L_0, L_1, gmax, gamma,
>
t, radius, s, p, g, Cpv,
>
rhelp, rcrit, deltat, Jm, Jqc,
>
ka, kv, k, diff, diffm,
>
psat, psatr, pv, NuH, NuM, lath,
>
theta, ke, fske, c1, c2, rhol, sigma,
>
A1, A2, D0, D1, D2, D3,pg, tm,
>
tdrop1, td1, td2, ftd1, ftd2, bound,
>
pvap1, pstag, tstag, Cpa,Ra,Cva,
>
Cvv, sat0, tdn, tdrop,
>
gammaa,gammav,func1,func2,tdhks,tdint,
>
rtbis,temp,dtemp,mflux, !Extras added by Som
>
dcoef,tred,Dlt,ci,
>
sr,wm1,wm2,R,pi,DLt2,ci2,mratio,THK,cvel,cgvel,
>
euckenfactor,grho,lamdav,lamdat,kn,knt,
>
beta,betat,d,b,c,Kulmalamf,ic,tdkul,! last
three lines are all for Kulmala
>
time,ptot,pc0,T0,x,pos(200),rad(200),pvnew(200),
>
pnew(200),tempr(200),mfrac(200),time1(200),
>
a,dummy,grhks,grkul,grpp,
>
griso,isomf
INTEGER i,ibiscnt,lineno,i2,j ! added by som
210
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
> /addons/gammaa,gammav!Added by Som
C
N2 and D2O
PARAMETER ( Cpa = 1.041D3,Ra = 2.9681D2, Cva = 7.4357D2,
>
C
Cpv = 1.710D3, Cvv = 1.2947D3)
Specific Gas constant for D2O
Rv = 4.1528D2
Open(1,FILE='info.dat',STATUS='OLD')
Open(2,FILE='model.out',STATUS='unknown')
Read(1,*)ptot,pc0,T0
Read(1,*)lineno
write(*,*)ptot,pc0,T0,lineno
read(1,2000)a
write(*,2000)a
2000 Format(A50)
write(2,1000)
1000 Format(1X, 'X(cm) t(us) RADIUS(A) TEMP
TDHKS
TDNGL
TDKUL',
> 2x, 'JMPETERS JMHKS JMKUL JISO JQ GRPP(m/s)',
> 2x, 'GRHKS(m/s) GRKUL(m/s) GRISO')
pstag =133.32*ptot
pvap1 =133.32*pc0
!
(Pa)
! (Pa)
do i2=1,lineno
Read(1,*)pos(i2),tempr(i2),mfrac(i2),pvnew(i2),pnew(i2),
> dummy,dummy,time1(i2),dummy,rad(i2)
radius = rad(i2)*1D-10 !
p = pnew(i2)*133.32
! Total pressure at location
pv= pvnew(i2)*133.32 ! vapor pressure at loc
t = tempr(i2)
g = mfrac(i2)
211
time=time1(i2)
x=pos(i2)
c
s=sup(i2)
C
gmax1
= (Ra/Rv)*(pvap1/pstag)
gmax = 1D0/(((Rv/Ra)*((pstag/pvap1)-1D0))+1D0)
C
Write(*,*) 'gmaxold, gmaxnew =', gmax1, gmax
Cp=((1D0-gmax)*Cpa+(gmax-g)*Cpv)/(1D0-g)
Cv=((1D0-gmax)*Cva+(gmax-g)*Cvv)/(1D0-g)
gamma
= Cp/Cv
**************************************************
gammaa = Cpa/Cva !Added by Som
gammav = Cpv/Cvv !Added by Som
**************************************************
c
Rg
= (1D0 - gmax)*Ra + gmax*Rv
! This was by grazia
Rg=((1D0-gmax)*Ra+(gmax-g)*Rv)/(1D0-g)
s = pv/psat(t)
pg = p-pv
write(*,*)x,time,t,p,pv,s,radius
IF (radius.LE.rcrit(t,s)) THEN
write(*,*)'radius is < rcrit'
write(*,*)'rcritical',rcrit(t,s)
pause
ENDIF
c ***************** new expression for diffusivity as in paper1
code******************
!
dcoef=1.7655D-3*DSQRT(1./WM1./WM2)*1.01325D6/SIGMAC**2 ! CGS
units dont need this as it is constant for a given system
dcoef=50.5135 ! Constant for D2O and N2
tred=t/240.3
Dlt=LOG(tred)
ci=EXP(0.348-0.459*Dlt+0.095*Dlt**2-0.010*Dlt**3)
212
diff= (dcoef/ci/p*t**1.5)*1D-5
! THIS IS IN MKS UNITS i.e.
m2/s
THK
= 4.184*(5.69+0.017*(t-273.15))*1D-3 ! Added by som
AS IN MAIN CODE (W/M-K)
diffm
= diff*pg/(Rv*t)
theta
= diffm*lath(t)*NuM(t,p,radius,pv)/
>
c
(THK*NuH(t,p,radius,pv))
ke
= 2D0*sigma(t)/(rhol(t)*Rv*t*radius)
fske
= LOG(s)-ke
write(*,*) fske
c1
= (t/(theta*s))*((p/psat(t))-s)
c2
= lath(t)/(Rv*t)
deltat
= fske*((5D-1*(c1**2)-c2)/((c1+c2)**2D0))
tdhks
= t*(1D0+ (fske*(1D0/(c1+c2)))*(1D0-deltat))
C**********CALCULATE HKS flux********
mflux= rhol(t)*(1/DSQRT(6.28*20.02D-3*1.3806D-23/6.023D23))*
> (20.02D-3/6.023D23/rhol(t))*
> ((pv/DSQRT(t))-(psatr(tdhks,radius,s)/DSQRT(tdhks)))!! There
are terms specific to species
grhks= mflux/rhol(t)
c
write(*,*)'fluxhks',mflux
sr=psatr(t,radius,s)/psat(t)
wm2=0.02002D0 !ONLY GOOD for D2O
wm1=0.02802D0
R=8.3145D0
pi=3.14159d0
DLt2=LOG(t/104.2)
213
ci2=EXP(0.46649-0.57015*DLt2+0.19164*DLt2**2-0.03708*DLt2**3
>
+0.00241*DLt2**4)
mratio= wm2/wm1 !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
THK = 4.184*(5.69+0.017*(t-273.15))*1D-3 ! Added by som AS IN
MAIN CODE (W/M-K)
cvel=DSQRT(8*R*T/(pi*wm2))
cgvel=DSQRT(8*R*T/(pi*wm1))
euckenfactor=((9*(Cpa/Cva))-5d0)/10.d0
grho=pg/Ra/t ! carrier gas density
lamdav=(32*diff*ci)/(3*pi*(1+mratio)*cvel)
lamdat=64.*ci2/25./pi/euckenfactor/grho/cva/cgvel*THK
kn=lamdav/radius
knt=lamdat/radius
beta=(1+kn)/(1+((4./3)+0.377)*kn+(4./3)*kn**2)
betat=(1+knt)/(1+((4./3.)+0.377)*knt+(4./3.)*knt**2)
c defining terms for mass flux
d=(s+sr)*psat(t)/2./p
b=sr*(lath(t))**2.*wm2/R/betat/THK/t**2
c=(R*t/wm2/beta/diff/psat(t)/(1+d))+b
c massflux units = kg/m^2/s as calculated
Kulmalamf=(s-sr)*1./c/radius
tdkul=(lath(t)*Kulmalamf*radius/THK/betat)+t
grkul=Kulmalamf/rhol(t)
********************************************************************
***************
***************
Isothermal Growth law calculations
*************************
call CONJFLUXES(Jm, Jqc, t, t, s, radius, p, g)
isomf= Jm*(-1D0)
griso = isomf/rhol(t)
C*******************************************************************
***************
214
C
Iteration procedure for the calculation of the exact wet-bulb
eq.
C
ibiscnt=0.D0
td1 = t-80.D0
td2 = t+20.D0
call CONJFLUXES(Jm, Jqc, t, td1, s, radius, p, g)
func1 = Jqc+ lath(t)*Jm
10
call CONJFLUXES(Jm, Jqc, t, td2, s, radius, p, g)
func2 = Jqc+ lath(t)*Jm
if(func1*func2.ge.0.0) then
ibiscnt=ibiscnt+1
if(ibiscnt.eq.1000) then
write(*,*)'root must be bracketed for bis',td1,td2,func1,func2
stop
endif
if (dabs(func1-func2).le.2.d-15.and.dabs(func2).lt.5.d-15.and.
> dabs(td1-td2).lt.1.d-5)then
tdrop=td2
GOTO 20
endif
temp=(td1*func2-td2*func1)/(func2-func1)
if(dabs(func2).lt.dabs(func1))then
td1=td2
func1=func2
endif
td2=temp
goto 10
endif
if(func1.lt.0.) then
rtbis = td1
dtemp = td2-td1
215
else
rtbis = td2
dtemp = td1-td2
endif
do j = 1,50
dtemp = 0.5*dtemp
tdint = rtbis+dtemp
call CONJFLUXES(Jm, Jqc, t, tdint, s, radius, p, g)
func2 = Jqc+ lath(t)*Jm
if(func2.lt.0.0)rtbis = tdint
tdrop=rtbis
if(abs(dtemp).lt.0.005.or.func2.eq.0.0)goto 20
grpp= -1D0*Jm/rhol(tdrop)
enddo
*********************************************************
c
20
20
write(*,*) 'tdhks,tdrop', tdhks,tdrop
write(2,1020)x,time,radius*1D10,t,tdhks,tdrop,tdkul,Jm*-1D0,
>
1020
mflux,Kulmalamf,isomf,Jqc,grpp,grhks,grkul,griso
format(7F9.2,4F9.5,F12.4,4E15.5)
enddo! This ones for the i2 loop
END
C------------------------------------------------------------------C
Equilibrium vapor pressure of D2O
C-------------------------------------------------------------------
DOUBLE PRECISION FUNCTION psat(t)
IMPLICIT NONE
DOUBLE PRECISION
>
t,tc,pc,alpha1,alpha2,alpha4,alpha11,
>
alpha20,tau,logPratio
tc=643.89 ! Kelvin
216
pc=21.66D6 !Pa
alpha1= -7.81583
alpha2= 17.6012
alpha4= -18.1747
alpha11= -3.92488
alpha20= 4.19174
tau= 1-t/tc
logPratio=(tc/t)*(alpha1*tau+alpha2*tau**1.9+alpha4*tau**2
> +alpha11*tau**5.5+alpha20*tau**10)
psat= pc*EXP(logPratio)!FROM WOELK ET AL..
END
c---------------------------------------------------------------c Function psatr calculates the vaporpressure at the surface of
c a droplet.
c----------------------------------------------------------------
DOUBLE PRECISION FUNCTION psatr(t, radius, s)
IMPLICIT NONE
DOUBLE PRECISION t, radius, s, ke, gmax, gamma,
>
psat, sigma, rcrit,
>
rhelp, Rv, Rg, Cp, Cv, L_0, L_1, rhol
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
c
rhelp = rcrit(t,s)*(2D0**(1D0/3D0))
!
IF (radius.LE.rcrit(t,s)) THEN
!
ENDIF
ke
= 2.D0*sigma(t)/(rhol(t)*Rv*t*radius)
psatr = psat(t)*EXP(ke)
END
c-------------------------------------------------------------c Function sigma calculates the surface tension
c--------------------------------------------------------------
217
DOUBLE PRECISION FUNCTION sigma(t)
IMPLICIT NONE
DOUBLE PRECISION
>
C
t,tp
Judith Bell:
tp=1.022*t
sigma = (93.6635D0+9.133D-3*tp-2.75D-4*tp*tp)*1D-3
WOELK ET AL..
! FROM
END
c------------------------------------------------------------c Function NuHct calculates the Nusselt Heat continuum
c-------------------------------------------------------------
DOUBLE PRECISION FUNCTION NuHct(t,p,radius)
IMPLICIT NONE
DOUBLE PRECISION
>
t, radius, p
NuHct = 2D0
END
c------------------------------------------------------------c Function NuMct calculates the Nusselt Mass continuum
c------------------------------------------------------------DOUBLE PRECISION FUNCTION NuMct(t,p,radius)
IMPLICIT NONE
DOUBLE PRECISION
>
t, radius, p
NuMct = 2D0
END
c------------------------------------------------------------c Function NuHfm calculates the Nusselt Heat free-molecular
c-------------------------------------------------------------
DOUBLE PRECISION FUNCTION NuHfm(t,p,radius,pv)
218
IMPLICIT NONE
DOUBLE PRECISION
>
t,
>
l, k, ka, kv, g, pr, pi, kn,
>
radius, p, mua, muv, mu,
D0, D1, D2, D3,
>
Rv, Rg, Cp, Cv, L_0, L_1, gmax, gamma,
>
THK,pv,R,wmavg,wm1,wm2
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
wm2= 0.02002D0 !! ONLY FOR D2O !!!!!
wm1= 0.02802D0
wmavg= wm1*(1-pv/p)+wm2*(pv/p)
R=8.3145D0
pi
= 4D0*DATAN(1D0)
THK
= 4.184*(5.69+0.017*(t-273.15))*1D-3 ! Added by som
AS IN MAIN CODE (W/M-K)
NuHfm
= SQRT(2D0/pi)*((gamma +
1D0)/(2D0*gamma))*Cp*p*radius
> /THK/SQRT(R*T/wmavg)
END
c------------------------------------------------------------c Function NuMfm calculates the Nusselt Mass free-molecular
c-------------------------------------------------------------
DOUBLE PRECISION FUNCTION NuMfm(t,p,radius,pv)
IMPLICIT NONE
DOUBLE PRECISION
>
t,
radius, p, g, mu, mua, muv,
>
diff, l, rho, sc,
219
pi, kn,
>
Rv, Rg, Cp, Cv, L_0, L_1, gmax, gamma,
>
dcoef,tred,Dlt,ci,
>
pv,R,wm2,wm1,wmavg
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
pi
rho
= 4D0*DATAN(1D0)
= p/(Rg*t)
c ***************** new expression for diffusivity as in paper1
code******************
!
dcoef=1.7655D-3*DSQRT(1./WM1./WM2)*1.01325D6/SIGMAC**2 ! CGS
units dont need this as it is constant for a given system
dcoef=50.5135 ! Constant for D2O and N2
tred=t/240.3
Dlt=LOG(tred)
ci=EXP(0.348-0.459*Dlt+0.095*Dlt**2-0.010*Dlt**3)
diff= (dcoef/ci/p*t**1.5)*1D-5
! THIS IS IN MKS UNITS i.e.
m2/s
C
***********************************************************************
*************
wm2= 0.02002D0 !! ONLY FOR D2O !!!!!
wm1= 0.02802D0
wmavg= wm1*(1-pv/p)+wm2*(pv/p)
R=8.3145D0
NuMfm
= SQRT(2D0*(wmavg**2)/wm2/pi/R/T)*p*radius/rho/diff
END
c------------------------------------------------------------c Function NuM calculates the Nusselt Mass
c-------------------------------------------------------------
220
DOUBLE PRECISION FUNCTION NuM(t,p,radius,pv)
IMPLICIT NONE
DOUBLE PRECISION
>
t, radius, p, g,
>
NuMct, NuMfm,pv
NuM
>
= NuMct(t,p,radius)/
(1D0 + NuMct(t,p,radius)/NuMfm(t,p,radius,pv))
END
c------------------------------------------------------------c Function NuH calculates the Nusselt Mass
c------------------------------------------------------------DOUBLE PRECISION FUNCTION NuH(t,p,radius,pv)
IMPLICIT NONE
DOUBLE PRECISION
>
t, radius, p, g,
>
NuHct, NuHfm,pv
NuH
>
= NuHct(t,p,radius)/
(1D0 + NuHct(t,p,radius)/NuHfm(t,p,radius,pv))
END
c------------------------------------------------------------c Subroutine for the conjugate fluxes of cond. of the Peters and
Paikert model
c-------------------------------------------------------------
SUBROUTINE CONJFLUXES(Jm, Jqc, t, td, s, radius, p, g)
IMPLICIT NONE
DOUBLE PRECISION
>
t, radius, s, p, psatr, g, pi, ka, kv,
>
psat, Rv, Rg, Cp, Cv, L_0, L_1, sc,
>
gamma, rhelp, td, gmax, k,
221
>
rhol, rcrit, pv, kn, tm, mua, muv, mu,
>
diff, l, pr, D0, D1, D2, D3, rho,
>
Jm, Jqc,BIGGAMMA,pg,denbracm,denbracq,
>
gammaa,gammav,Ra,THK,
>
!Added by Som
boltz,avag,molavg,massmol,term1,term2,term3,
>
term4,term5,term6,
>
expression
dcoef,tred,Dlt,ci ! Added for new diffusivity
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
> /addons/gammaa,gammav !Added by Som
Ra = 2.9681D2 !THIS WAS MISSING BEFORE AND IS NOT THERE IN
COMMON WHICH CAUSED THE PROBLEM
pi
= 4D0*DATAN(1D0)
rho
= p/(Rg*t)
pv
= s*psat(t)
pg
= p-pv
BIGGAMMA= ((gammav+1)/(gammav-1))*(DSQRT(Rv/Rg))*(Pv/p)+
>
((gammaa+1)/(gammaa-1))*(DSQRT(Ra/Rg))*(Pg/p)
! Added by
Som
c
***********************************************************************
**********
boltz=1.38D-23
avag=6.022D23
molavg=28*(pg/p)+20.02*(pv/p) !Only for D2O in CGS units here
massmol=molavg/avag
c ***************** new expression for diffusivity as in paper1
code******************
!
dcoef=1.7655D-3*DSQRT(1./WM1./WM2)*1.01325D6/SIGMAC**2 ! CGS
units dont need this as it is constant for a given system
dcoef=50.5135 ! Constant for D2O and N2
tred=t/240.3
222
Dlt=LOG(tred)
ci=EXP(0.348-0.459*Dlt+0.095*Dlt**2-0.010*Dlt**3)
diff= (dcoef/ci/p*t**1.5)*1D-5
! THIS IS IN MKS UNITS i.e.
m2/s
c
write(*,*)diff
C
***********************************************************************
*************
THK
= 4.184*(5.69+0.017*(t-273.15))*1D-3 ! Added by som
AS IN MAIN CODE (W/M-K)
c
***********************************************************************
*******************
term1=DSQRT(massmol*1D-3/(2D0*pi*boltz*t))
term2=64*t*THK/(5*p*radius*BIGGAMMA)
term3=1+term1*term2
term4=DSQRT(20.02D-3/avag/2D0/pi/boltz/t)!term specific to
D2O!!!!!!!
term5=32*p*diff/(5*pg*radius)
term6=1+term4*term5
c*******************************************************************
************************
Jm
= diff*((psatr(td,radius,s)-pv)*p*20.02D3/avag)!!species specific 20.02 is molwt
>
/boltz/t/pg/radius/term6
Jqc = THK*(td-t)/radius/term3
END
c------------------------------------------------------------c Function rcrit calculates the critical radius
c------------------------------------------------------------
223
DOUBLE PRECISION FUNCTION rcrit(t, s)
IMPLICIT NONE
DOUBLE PRECISION
>
t, s, sigma, rhol, gmax, gamma,
>
Rv, Rg, Cp, Cv, L_0, L_1
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
rcrit = 2D0*sigma(t)/(rhol(t)*Rv*t*LOG(s))
END
c------------------------------------------------------------c Function rhol calculates the density of D2O
c------------------------------------------------------------
DOUBLE PRECISION FUNCTION rhol(temp)
IMPLICIT NONE
DOUBLE PRECISION
>
xj,tr, temp
xj
= (temp-231.)/51.5
tr
= (643.89-temp)/643.89
rhol = (0.09*tanh(xj)+0.847*tr**0.33+0.338)*1D3
END
c---------------------------------------------------------------
224
c Function Lath calculates the latent heat for a temperature t
c for D2O
c--------------------------------------------------------------
DOUBLE PRECISION FUNCTION lath(t)
IMPLICIT NONE
DOUBLE PRECISION
>
t,
gmax, gamma,
>
Rv, Rg, Cp, Cv, L_0, L_1,
>
R,tc,pc,alpha1,alpha2,alpha4,alpha11,alpha20,
>
tau,logPratio,func
COMMON /vapour/ gmax, gamma, Cp, Cv, Rg, Rv, L_0, L_1
R=8.3145D0
tc=643.89 ! Kelvin
pc=21.66D6 !Pa
alpha1= -7.81583
alpha2= 17.6012
alpha4= -18.1747
alpha11= -3.92488
alpha20= 4.19174
tau= 1-t/tc
logPratio=(tc/t)*(alpha1*tau+alpha2*tau**1.9+alpha4*tau**2
> +alpha11*tau**5.5+alpha20*tau**10)
func=alpha1+1.9*alpha2*tau**0.9+2*alpha4*tau
> +5.5*alpha11*tau**4.5+10*alpha20*tau**9
lath= (-R*tc*(t*logPratio/tc+t*func/tc)/20.02)*1D3
END
225
APPENDIX G
FORTRAN CODE TO COMPUTE CHARACTERISTIC TIME OF
NUCLEATION ,∆tJmax, AND pJMAX AND TJMAX
226
C This program calculates the characteristic time and volume for
nucleation Som 04-23-2008
C calculate the nucleation volume/time from the experimentally
C determined S and T profiles
c It also uses the easier expression for the characteristic time, that
integrates
c with respect to time ie Dt = int(J(S,T))/Jmax(smax,Tmax)
IMPLICIT REAL*8(A-H,O-Z)
real*8 XX(900),molfc(900),SS(900),TT(900),PP0(900),PP0I(900)
real*8 msq,aratio(900),U(900),Time(900),TIS(900),rhorat(900)
real*8 g(900),g_ginf(900)
real*8 molweight
real*8 peq(900), pcx(900)
!real*8 P0,T0,rho0,g0,pc10,pc20,IRATE,idat
character*8 a1
character*13 A2
character*11 A3
character*28 A4
character*30 A6
character*17 A5
PI=3.14159
RG=8.3145D7
AVOG=6.022D23
!write(*,*)'what model? 1=Tcorr x JBD, 2=RKK'
commented out by
som
!
!write(*,*)'number of data sets?'
!read(8,*)idat
iline=641 Now read from 4pp.out
pconv = 760.d0/1013250.d0
open(5,file='4pp.out',status='old')
read(5,*)idat,iline
! these are now read from 4pp.out -put
them in the very first line of 4pp.out
5003
999
OPEN(3,FILE='nrates_vs_supersat.out',STATUS='unknown')
write(3,5003)
format(' X
time
S
J ')
write(12,999)
format('
xrmax time(jmax)
ratemax
& 'srmax
trmax
dtime_nuc
xnucl
& ' rhoNZ/rhovv prmax(kpa) phi_vv')
',
vnucl',
OPEN(10,FILE='nucleationrates.out',STATUS='unknown')
opened by som to get xrmax and ratemax
Write(10,4050)
227
! new file
4050
FORMAT( 5X,'xrmax time(jmax)
ratemax
* dtime_nuc
xnucl
vnucl prmax(kpa) ')
do k = 1,idat
read(5,*)P0,T0,pc10 ! from 4pp.out
srmax
trmax
read(5,10)a !lines that we dont need to read
read(5,10)a
read(5,10)a
read(5,10)a
read(5,10)a
read(5,10)a
10
format(A50)
write(3,5002)pc10*pconv
5002 format('# pc10 =',f10.6,' Torr')
C CONVERT PRESSURES TO DYN/CM**2
P0=P0/pconv
PC10=PC10/pconv
do i=1,iline
read(5,*)XX(i),U(i),TT(i),aratio(i),TIS(i),PP0I(i),PP0(i),dummy,
& dummy,g(i),g_ginf(i) !som
***********************************************************************
*******
pcx(i) = PC10*pp0(i)*(1-g_ginf(i)) ! this needs to be g/ginfi
***********************************************************************
********
T=TT(i)
P10=vappress(T)
peq(i) = P10*10 !in dynes/cm2
SS(i)=pcx(i)/peq(i)
if(ss(i).le.1.0)ss(i)=1.11d0
enddo
time(1) = xx(1)/u(1)/100.d0
do i=2,iline
time(i) = time(i-1) + (xx(i)-xx(i-1))/u(i)/100.d0
enddo
write(11,1999)pc10*pconv
1999 format('# pc10 =',f10.6,' Torr')
do i=1,iline
write (11,2000) XX(i),time(i),pcx(i),peq(i),SS(i),TT(i),pp0(i)
enddo
2000 format(f8.2,e12.4,2x,f8.2,1x,f8.2,1x,f10.2,f8.2,f9.3)
228
C MOLECULAR WEIGHT OF CARRIER (1) AND CONDENSIBLE (2)
WM1=4.0026
WM2=39.948
C THE SPECIFIC HEATS OF CARRIER GAS AND CONDENSIBLE VAPOR
CP1=5.177 !J/g-K
CP2=0.5206
C W0 IS MASS FRACTION OF CONDENSIBLE VAPOR IN GAS
WMAV=(WM1*(P0-PC10)+WM2*PC10)/P0
WMAV0=WMAV
W20=WM2*PC10/P0/WMAV ! mass fraction of argon
WI=1.D0-W20-W30 ! mass fraction of He
C CP0 is the mixture Ar- He specific heat at stagnation conditions
CP0=WI*CP1+W20*CP2
rateintegral=0.0d0
ratemax=0.0d0
do j = 2,iline
X=XX(j)
T=TT(j)
S=SS(j)
arat=aratio(j)
dt=time(j)-time(j-1)
c calculate Reiss-Katz-Kegel rate, RATERKK. ALTRcl is the classical
rate also
C computed in the RKK subroutine for comparison
CALL RKK(T,S,ALTRcl,pc10)
2001 format(2e12.3)
if(x.gt.9.5)then
ALTRcl = 0.0
endif
crate = ALTRcl
write(3,5001)X,time(j),S,crate
5001 format(f10.2,e12.4,f12.2,e12.2)
!
write(*,2001)crate
if(crate.ge.ratemax)then
ratemax = crate
xrmax = x
srmax = s
trmax = t
jmax = j
229
!
write(*,2002)j,x,ratemax,trmax
endif
2002 format(I4,f8.2,E12.3,f8.1)
rateintegral = rateintegral+crate*dt
!
write(*,2003)j,x,xrmax,crate,srmax,trmax,ratemax,
!
&
rateintegral,dt
2003 format(I4,2(f8.2),E12.3,2(f8.2),e12.3,e12.3,e12.3)
enddo
dtime_nuc=rateintegral/ratemax
write(*,*)dtime_nuc,srmax
xnucl = dtime_nuc*u(jmax)*100.d0
vnucl = xnucl*aratio(jmax)
!write(*,2030)xrmax,time(jmax),ratemax, srmax,trmax,
!&
dtime_nuc,xnucl,vnucl,rhoNZ/rhovv,pct0*pp0(jmax)/1.e4,
!&
phivv
&
4051
write(10,4051)xrmax,time(jmax),ratemax, srmax,trmax,
dtime_nuc,xnucl,vnucl,pc10*pp0(jmax)/1.e4
format(f8.3,e12.4,1x,e12.4,2x,
> 2(f10.3,1x),e12.4,1x,2(f8.3,1x),f8.5)
enddo
!write(15,4051)xrmax,time(jmax),ratemax, srmax,trmax,
!
&
dtime_nuc,xnucl,vnucl,rhoNZ/rhovv,pct0*pp0(jmax)/1.e4,
! &
phivv
!4051 format(f8.3,e12.4,1x,e12.4,2(f8.3,1x),e12.4,2(f8.3,1x),2f8.5,
!
&
e12.4)
END
subroutine RKK(T,S,RJcl,pc10)
c calculates the theoretical classical rates for Ar
implicit real*8(a-h,o-z)
real*8 k,M
avo = 6.022D+23
R = 8.3145D0
k = 1.381D-23
pi = 3.141593D0
!number/mol
!J/K*mol
!J/K
c interpolate all the physical properties on a mol fraction basis
230
d = density(T)
! in [kg/m**3]
ps = vappress(T)
sg = surftension(T)
M
= 0.039948
! in [Pa]
!in [N/m]
!kg/mol
c nucleation rate
p1 = S*ps
v = M/(d*avo)
rp = 2.0* sg * v/(k*t*dlog(s))
encc = 4.*pi/3.*rp**3/v
!
g = -encc*k*t*log(s) + (36.*pi)**(1./3.)*v**(2./3.)*
!
&
encc**(2./3.)*sg
g = (16.*pi/3)*(sg**3)*(v**2)/((k*t)**2)/((log(s))**2)
c calculate classical (JBD)
Preexp = dsqrt(2.*avo*sg/pi/M)*((p1/(k*t))**2)*v
RJBD = Preexp*dexp(-g/(k*t))/1000000.
!per cm^3 and s
RJcl=RJBD
c NOW add Iland's empirical correction function!............05-23-08 SS
frm JCP 127,154506(2007)
!
RJcl = RJcl*exp(-27.0+3630./T)
RJcl=RJcl
!for Ar
return
end
C
C THE PHYSICAL PROPERTY FUNCTIONS OF THE CONDENSING Ar.
C SURFACE TENSION, EQUIL. VAPOR PRESS. AND DENSITY OF CONDENSATE
real*8 function surftension(T)
real*8 T,epi,Tc
c using Iland's paper
231
Tc=150.6633 ! K
epi = 1-(T/Tc)
surftension=0.03778*(epi**1.277)
return
end
real*8 function vappress(T)
real*8 epi, P, Pc, T, Tc
c from Iland's paper
Pc = 4.86D6 !Pa
Tc = 150.6633 !K
a1 = 5.90418853
a2 = 1.12549591
a3 = 0.763257913
a4 = 1.69799438
epi = 1-(T/Tc)
>
vappress=Pc*exp((Tc/T)
*(-a1*epi+a2*epi**1.5-a3*epi**3-a4*epi**6)) !in Pa
return
end
real*8 function density(T)
real*8 T,rhoc,M,Tc,epi
Tc = 150.6633 !K
epi=1-(T/Tc)
M = 0.039948
rhoc = 13.290 !mol/dm3
c
for Ar from Iland ref
density = M*1000*(rhoc+24.49248*epi**0.35+8.155083*epi) ! Kg/m3
return
end
232
APPENDIX H
FORTRAN CODE TO COMPUTE PRESSURE, p, FOR A GIVEN
TEMPERATURE RANGE AND CONSTANT NUCLEATION RATE,
JBD, USING THE CLASSICAL NUCLEATION THEORY
233
C THIS PROGRAM CALCULATES THE PRESSURE CORRESPONDING TO A GIVEN
TEMPERATURE AND CONSTANT NUCLEATION RATE
IMPLICIT REAL*8(A-H,O-Z)
real*8 TT(1000)
character*8
a1
character*13 A2
character*11 A3
character*28 A4
character*30 A6
character*17 A5
open(5,file='val.dat',status='old')
OPEN(3,FILE='nrates.out',STATUS='unknown')
Write(3,4050)
4050
FORMAT( 1X,'pon
Ton
Tmod
read(5,*)rateset,iend
read(5,*)g1,g2,tb1
read(5,*)g3,g4,tb2
read(5,*)g5,g6
do i = 1,iend
read(5,*)TT(i)
T=TT(i)
Tmod=100/T
if (T.lt.tb1)then
p1=g1
p2=g2
elseif(T.lt.tb2) then
234
S
J ')
p1=g3
p2=g4
else
p1=g5
p2=g6
endif
write(*,*)p1,p2,T
ibiscnt=0.D0
call CNT(T,p1,RJcl)
func1= rateset-RJcl
call CNT(T,p2,RJcl)
10
func2= rateset-RJcl
if(func1*func2.gt.0.0) then
ibiscnt=ibiscnt+1
if(ibiscnt.eq.1000) then
write(*,*)'root must be bracketed for bis',p1,p2,T,func1,func2
stop
endif
if (dabs(func1-func2).le.2D-15
>.and.dabs(func2).lt.51D-15.and.dabs(p1-p2).lt.1.d-5) then
pon=p2
GOTO 20
endif
presson=(p1*func2-p2*func1)/(func2-func1)
if(dabs(func2).lt.dabs(func1))then
p1=p2
func1=func2
endif
p2=presson
235
goto 10
endif
If(func1.lt.0.) then
rtbis = p1
dp = p2-p1
else
rtbis = p2
dp = p1-p2
endif
do l = 1,500
dp = 0.5*dp
pint = rtbis+dp
CALL CNT(T,pint,RJcl)
func2=rateset-RJcl
if(func2.lt.0.0)rtbis = pint
pon=rtbis
if(abs(dp).lt.0.001.or.func2.eq.0.0)goto 20
enddo
20
write(3,5001)pon,T,Tmod,S,RJcl
5001 format(e10.4,1x,3f10.2,1x,e12.4)
enddo
end
subroutine CNT(T,p,RJcl)
c calculates the theoretical classical rates for Ar
236
implicit real*8(a-h,o-z)
real*8 k,M
avo = 6.022D+23
!number/mol
R = 8.3145D0
!J/K*mol
k = 1.381D-23
!J/K
pi = 3.141593D0
c interpolate all the physical properties on a mol fraction basis
d = density(T)
! in [kg/m**3]
ps = vappress(T)
! in [Pa]
sg = surftension(T)
M
= 0.039948
!in [N/m]
!kg/mol
c nucleation rate
S=p/ps
v =
M/(d*avo)
rp = 2.0* sg * v/(k*t*dlog(s))
encc = 4.*pi/3.*rp**3/v
!
!
g = -encc*k*t*log(s) + (36.*pi)**(1./3.)*v**(2./3.)*
&
encc**(2./3.)*sg
g = (16.*pi/3)*(sg**3)*(v**2)/((k*t)**2)/((log(s))**2)
c calculate classical (JBD)
Preexp = dsqrt(2.*avo*sg/pi/M)*((p/(k*t))**2)*v
RJBD = Preexp*dexp(-g/(k*t))/1000000.
!per cm^3 and s
RJcl=RJBD
c NOW add Iland's empirical correction function!............05-23-08
SS frm JCP 127,154506(2007)
!
RJcl = RJcl*exp(-27.0+3630./T)
!for Ar
RJcl=RJcl
return
end
C
C THE PHYSICAL PROPERTY FUNCTIONS OF THE CONDENSING Ar.
C SURFACE TENSION, EQUIL. VAPOR PRESS. AND DENSITY OF CONDENSATE
237
real*8 function surftension(T)
real*8 T,epi,Tc
c using Iland's paper
Tc=150.6633 ! K
epi = 1-(T/Tc)
surftension=0.03778*(epi**1.277)
return
end
real*8 function vappress(T)
real*8 epi, P, Pc, T, Tc
c from Iland's paper
Pc = 4.86D6 !Pa
Tc = 150.6633 !K
a1 = 5.90418853
a2 = 1.12549591
a3 = 0.763257913
a4 = 1.69799438
epi = 1-(T/Tc)
vappress=Pc*exp((Tc/T)
>
*(-a1*epi+a2*epi**1.5-a3*epi**3-a4*epi**6)) !in Pa
return
end
real*8 function density(T)
real*8 T,rhoc,M,Tc,epi
238
Tc = 150.6633 !K
epi=1-(T/Tc)
M = 0.039948
rhoc = 13.290 !mol/dm3
c
for Ar from Iland ref
density = M*1000*(rhoc+24.49248*epi**0.35+8.155083*epi) !
Kg/m3
return
end
239
APPENDIX I
DRAWINGS OF VARIOUS COMPONENTS OF CRYOGENIC
SUPERSONIC NOZZLE APPARATUS
240
241
241
Figure I.1: Sketch of the front plate of the Plenum
242
Figure I.2: Sketch of the back plate of the Plenum
242
243
Figure I.3: Sketch of the
bottom plate of the Plenum
243
244
Figure I.4: Sketch of the right plate of the Plenum
244
245
Figure I.5: Sketch of the left plate of the Plenum
245
246
246
Figure I.6: Sketch of the lip of the Plenum
247
Figure I.7: Sketch of Nozzle S
247
248
248of flange for the Tee joint
Figure I.8: Sketch
249
249
Figure I.9: Sketch of Outer box front plate
250
250
Figure I.10: Sketch of Outer box rear plate
251
Figure I.11: Sketch 251
of Outer box base plate
252
Figure I.12: Sketch of252
Outer box right-side plate
253
Figure I.13: Sketch of253
Outer box left-side plate
254
254of Outer box lip
Figure I.14: Sketch
255
Figure I.15: Outer
255 box assembly
256
Figure I.16: Instrument flange for Outer box
256
257
Figure I.17: Retaining
ring at Gas Inlet
257
BIBLIOGRAPHY
Atkins, P. & De Paula, J. 1998, Atkin's Physical Chemistry, 6th edn, Oxford University
Press, Oxford.
Becker, R. & Döring, W. 1935, "Kinetische Behandlung der Keimbildung in
üebersättigten Dämpfen", Annalen der Physik, vol. 24, pp. 719-752.
Bird, B.R., Stewart, W.E. & Lightfoot, E.N. 2002, Transport Phenomena, 2nd edn, John
Wiley & Sons.
Brock, J.R. & Hidy, G.M. 1965, "Collision-rate theory and the coagulation of freemolecule aerosols", Journal of Applied Physics, vol. 36, pp. 1857-1862.
Dean, J.A. 1992, Lange's Handbook of Chemistry, 14th edn, McGraw-Hill.
Epion , www.epion.com.
Evans, R. 1979, "The nature of the liquid vapor interface and other topics in the statistical
mechanics of non-uniform, classical fluids", Advances in Physics, vol. 28, no. 2, pp.
143.
Fladerer, A., Kulmala, M. & Strey, R. 2002, "Test of applicability of Kulmala’s
analytical expression for the mass flux of growing droplets in highly supersaturated
systems: growth of homogeneously nucleated water droplets", Journal of Aerosol
Science, vol. 33, pp. 391-399.
Fladerer, A. & Strey, R. 2006, "Homogeneous nucleation and droplet growth in
supersaturated argon vapor: The cryogenic nucleation pulse chamber", Journal of
Chemical Physics, vol. 124, pp. 164710.
FSI International , www.fsi-intl.com.
Fuchs, N.A. 1964, in The Mechanics of Aerosols The MacMillan Company, New York,
pp. 293.
258
Fuchs, N.A. & Sutugin, A.G. 1970, Highly Dispersed Aerosols, Ann Arbor Science
Publishers, Ann Arbor, MI.
Garcia Garcia, N. & Soler Torroja, J.M. 1981, "Monte Carlo calculation of argon clusters
in homogeneous nucleation", Physical Review Letters, vol. 47, no. 3, pp. 186.
Gharibeh, M., Kim, Y.J., Dieregsweiler, U., Wyslouzil, B.E., Ghosh, D. & Strey, R.
2005, "Homogeneous nucleation of n- propanol, n- butanol, and n- pentanol in a
supersonic nozzle", Journal of Chemical Physics, vol. 122, pp. 094512.
Ghosh, D. 2007, Pressure trace measurements and the first small angle X-ray scattering
experiments in a supersonic nozzle, Universität zu Köln.
Girshick, S.L. & Chiu, C.P. 1990, "Kinetic nucleation theory: A new expression for the
rate of homogeneous nucleation from an ideal supersaturated vapor", Journal of
Chemical Physics, vol. 93, pp. 1273-1277.
Gladun, C. 1971, Cryogenics, vol. 11, pp. 205.
Granasy, L., Jurek, Z. & Oxtoby, D.W. 2000, "Analytical density functional theory of
homogeneous vapor condensation", Physical Review E: Statistical Physics, Plasmas,
Fluids, and Related Interdisciplinary Topics, vol. 62, no. 5-B, pp. 7486.
Hale, B.N. 2005, "Temperature dependence of homogeneous nucleation rates for water:
near equivalence of the empirical fit of Wolk and Strey, and the scaled nucleation
model", Journal of Chemical Physics, vol. 122, pp. 204509-1-204509-3.
Hale, B.N. 1996, "Monte Carlo calculations of effective surface tension for small
clusters", Australian Journal of Physics, vol. 49, pp. 425.
Hale, B.N. 1992, "The scaling of nucleation rates", Metallurgical Transactions A, vol.
23A, pp. 1863-1868.
Hale, B.N. & DiMattio, D.J. 2004, "Scaling of the nucleation rate and a Monte Carlo
discrete sum approach to water cluster free energies of formation", Journal of
Physical Chemistry B, vol. 108, pp. 19780-19785.
Haynes, W.M. 1978, Cryogenics, vol. 18, pp. 621.
Heath, C.H., Streletzky, K.A., Wyslouzil, B.E., Wölk, J. & Strey, R. 2002, "H2O-D2O
condensation in a supersonic nozzle", Journal of Chemical Physics, vol. 117, pp.
6176-6185.
259
Heiler, M. 1999, Instationäre Phänomene in homogen/heterogen kondensierenden
Düsen- und Turbinenströmungen., Dissertation Fakultät für Maschinenbau,
Universität Karlsruhe (TH).
Heist, R.H. & He, J.H. 1994, "Review of vapor to liquid homogeneous nucleation
experiments from 1968 to 1992.", Journal of Chemical Physics Reference Data, vol.
23, pp. 781.
Holleman, A.F., Wiberg, E. & Wiberg, N. 1985,
Lehrbuch der Anorganischen Chemie, Walter de Gruyter, Berlin.
Iland, K. 2004, PhD Dissertation, Universität zu Köln.
Iland, K., Wedekind, J. & Wölk, J. 2004, "Homogeneous nucleation rates of 1-Pentanol",
Journal of Chemical Physics, vol. 121, no. 12259.
Iland, K., Wölk, J., Strey, R. & Kashchiev, D. 2007, "Argon nucleation in a cryogenic
pulse chamber", Journal of Chemical Physics, vol. 127, pp. 154506.
Karlsson, M., Alxneit, I., Rütten, F., Wuillemin, D. & Tschudi, H.R. 2007, "A compact
setup to study homogeneous nucleation and condensation", Review of Scientific
Instruments, vol. 78, pp. 034102/1-034102/7.
Karlsson, O.M. 2006, Nucleation and condensation in a stationary supersonic flow.
Design, modeling and test of an experiment based on a transparent Laval nozzle.,
Eidgenössische Technische Hochschule ETH Zürich.
Kashchiev, D. 1982, "
On the relation between nucleation work, nucleus size, and nucleation rate", Journal
of Chemical Physics, vol. 76, pp. 5098.
Kell, G.S. & Whalley, E. 1965, "The PVT properties of water. I. Liquid water at 0 to
1500 and at pressures upto 1 kilobar", Philosophical Transcations of the Royal
Society of London. Series A, vol. 258, pp. 565-614.
Khan, A., Heath, C.H., Dieregsweiler, U., Wyslouzil, B.E., Wölk, J. & Strey, R. 2003,
"Homogeneous nucleation rates for D2O in a supersonic Laval nozzle", Journal of
Chemical Physics, vol. 119, pp. 3138-3147.
Kim, Y.J., Wyslouzil, B.E., Wilemski, G., Wölk, J. & Strey, R. 2004, "Isothermal
nucleation rates in supersonic nozzles and the properties of small water clusters",
Journal of Physical Chemistry A, vol. 108, pp. 4365-4377.
260
Kotlarchyk, M. & Chen, S.H. 1983, "Analysis of small angle neutron scattering spectra
from polydisperse interacting colloids", Journal of Chemical Physics, vol. 79, pp.
2461-2469.
Kraska, T. 2006, " Molecular-dynamics simulation of argon nucleation from
supersaturated vapor in the NVE ensemble", Journal of Chemical Physics, vol. 124,
pp. 054507.
Kulmala, M. 1993a, "Condensational growth and evaporation in the transition regime",
Aerosol Science and Technology, vol. 19, pp. 381-388.
Kulmala, M., Vesala, T. & Wagner, P.E. 1993b, "An analytical expression for the rate of
binary condensational particle growth", Proceedings of the Royal Society of London,
vol. 441, pp. 589-605.
Kusaka, I. 2003, "System size dependence of teh free energy surface in cluster simulation
of nucleation", Journal of Chemical Physics, vol. 119, pp. 3820.
Kusaka, I., Oxtoby, D.W. & Wang, Z.G. 1999, "On the direct evaluation of the
equilibrium distribution of clusters by simulation", Journal of Chemical Physics, vol.
111, no. 22, pp. 9958.
Laasonen, K., Wonczak, S., Strey, R. & Laaksonen, A. 2000, "Molecular dynamics
simulations of gas-liquid nucleation of Lennard-Jones fluid", Journal of Chemical
Physics, vol. 113, no. 21, pp. 9741.
Lamanna, G. 2000, On nucleation and droplet growth in condensing nozzle flows,
Eindhoven University of Technology.
Lide, D.R. 1991, Handbook of Chemistry and Physics, 72nd edn, CRC Press, Boston.
Luijten, C. C. M. 1998, Nucleation and droplet growth at high Pressure, Eindhoven
University of Technology.
Luijten, C. C. M., Peeters, P. & van Dongen, M. E. H. 1999, "Nucleation at high
pressure. II. Wave tube data and analysis", Journal of Chemical Physics, vol. 111,
pp. 8535.
Luo, X., Prast, B., Van Dongen, M. E. H., Hoeijmakers, H. W. M. & Yang, J. 2006, "On
phase transition in compressible flows: modeling and validation", Journal of Fluid
Mechanics, vol. 548, pp. 403-430.
261
Matthew, M.W. & Steinwandel, J. 1983, "
An experimental study of argon condensation in cryogenic shock tubes", Journal of
Aerosol Science, vol. 14, pp. 755.
McDonald, J.E. 1963, "Homogeneous Nucleation of Vapor Condensation. II.Kinetic
Aspects", American Journal of Physics, vol. 31, pp. 31.
McDonald, J.E. 1962, "Homogeneous Nucleation of Vapor Condensation.
I.Thermodynamic Aspects", American Journal of Physics, vol. 30, pp. 870.
McGraw, R. & LaViolette, R.A. 1995, "Fluctuations, temperature and detailed balance in
classical nucleation theory", Journal of Chemical Physics, vol. 102, pp. 8983-8994.
Moses, C.A. & Stein, G.D. 1978, "On the growth of steam droplets formed in a Laval
nozzle using both static pressure and light scattering measurements", Journal of
Fluids Engineering, vol. 100, pp. 311-322.
Ostwatitsch, K. 1942, "Kondensationserscheinungen in Überschalldüsen", Zeitschrift für
Angewandte Mathematik, vol. 22, pp. 1-14.
Oxtoby, D.W. & Kashchiev, D. 1994, Journal of Chemical Physics, vol. 100, pp. 7665.
Paci, P., Zvinevich, Y., Tanimura, S., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson,
D. & McManus, B. 2004, "Spatially resolved gas phase composition measurements
in supersonic flows using tunable diode laser absorption spectroscopy", Journal of
Chemical Physics, vol. 121, pp. 9964-9970.
Pederson, J.S. 1997, "Analysis of small- angle scattering data from colloids and polymer
solutions: modeling and least squares fitting", Advances in Colloid and Interface
Science, vol. 70, pp. 171-210.
Peters, F. & Paikert, B. 1994, "Measurement and interpretation of growth and
evaporation of monodispersed droplets in a shock tube", International Journal of
Heat and Mass Transfer, vol. 37, pp. 293-302.
Pound, G.M. 1972, "Selected values of critical supersaturations for nucleation of liquids
from the vapor", Journal of Physical Chemistry Reference Data, vol. 1, pp. 119.
Reid, R.C., Prausnitz, J.M. & Poling, B.E. 1987, The properties of gases and liquids, 4th
edn, McGraw-Hill, New York.
Rigby, M., Smith, E.B., Wakeham, W.A. & Maitland, G.C. 1986, in The Forces between
Molecules Clarendron Press, Oxford, pp. 218.
262
Seinfeld, J.H. 1986, Atmospheric Chemistry and Physics of AIR POLLUTION, John
Wiley & Sons.
Seinfeld, J.H. & Pandis, S.N. 1998,
Atmospheric Chemistry and Physics, Wiley Publications, New York.
Senger, B., Schaaf, P., Corti, D.S., Bowles, R., Pointu, D., Voegel, J.C. & Reiss, H. 1999,
"A molecular theory of the homogeneous nucleation rate. II. Application to argon
vapor", Journal of Chemical Physics, vol. 110, no. 13, pp. 6438.
Sinha, S., Laksmono, H. & Wyslouzil, B.E. 2008a, "A cryogenic supersonic nozzle
apparatus to study homogeneous nucleation of Ar and other simple molecules",
Manuscript under preparation.
Sinha, S., Wyslouzil, B.E. & Wilemski, G. 2008b, "Modeling of H2O/D2O condensation
in supersonic nozzles", Submitted to Aerosol Science and Technology.
Smith, J.M., Van Ness, H.C. & Abbott, M.M. 1996, Introduction to Chemical
Engineering Thermodynamics, 5th edn, McGraw-Hill.
Smolders, H.J. 1992, Nonlinear wave phenomena in a gas-vapor mixture with phase
transition, Eindhoven Institute of Technology.
Smoluchowski, M.V. 1917, "Versuch einer mathematischen Theorie der
koagulationskinetik kolloider Lösungen", Zeitschrift für physikalische Chemie, vol.
92, pp. 129-168.
Smoluchowski, M.V. 1916, "Drei Vortrage uber Diffusion, Brownsche Bewegung und
Koagulation von Kolloidteilchen", Physikalische Zeitschrift, vol. 17, pp. 557-599.
Sprow, F.B. & Prausnitz, J.M. 1966, Transactions of Faraday Society, vol. 62, pp. 1097.
Stein, G.D. 1974, Argon Nucleation in a Supersonic Nozzle, Report to Office of Naval
Research available from National Technical Information Service Number: ADA007357/7GI.
Stein, G.D. & Wegener, P.P. 1967, "Experiments on number of particles formed by
homogeneous nucleation in the vapor phase", Journal of Chemical Physics, vol. 46,
pp. 3685-3686.
Steward, R.B. & Jacobsen, R.T. 1989, Journal of Physical Chemistry Reference Data,
vol. 18, pp. 639.
263
Streletzky, K.A., Zvinevich, Y. & Wyslouzil, B.E. 2002, "Controlling nucleation and
growth of nanodroplets in supersonic nozzles", Journal of Chemical Physics, vol.
116, pp. 4058-4070.
Talanquer, V. & Oxtoby, D.W. 1994, "Dynamic density functional theory of gas-liquid
nucleation", Journal of Chemical Physics, vol. 100, no. 7, pp. 5190.
Tanimura, S., Okada, Y. & Takeuchi, K. 1997, "Fourier transform infrared spectroscopy
of UF6 clustering in a supersonic Laval nozzle: Cluster configurations in
supercooled and near-equilibrium states", Journal of Chemical Physics, vol. 107, pp.
7096-7105.
Tanimura, S., Okada, Y. & Takeuchi, K. 1996, "FTIR spectroscopy of UF6 clustering in
a supersonic Laval nozzle", Journal of Physical Chemistry, vol. 100, pp. 2842-2848.
Tanimura, S., Okada, Y. & Takeuchi, K. 1995, "FTIR spectroscopy of UF6 in supersonic
nozzle", Reza Kagaku Kenkyu, vol. 17, pp. 113-115.
Tanimura, S., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson, D. & McManus, B.
2007, "Tunable Diode Laser Absorption Spectroscopy Study of CH3CH2OD/D2O
Binary Condensation in a Supersonic Laval Nozzle", Journal of Chemical Physics,
vol. 127, pp. 034305/1-034305/13.
Tanimura, S., Zvinevich, Y., Wyslouzil, B.E., Zahniser, M., Shorter, J., Nelson, D. &
McManus, B. 2005, "Temperature and gas-phase composition measurements in
supersonic flows using tunable diode laser absorption spectroscopy: The effect of
condensation on the boundary-layer thickness", Journal of Chemical Physics, vol.
122, pp. 194304/1-194304/11.
ten Wolde, P.R. & Frenkel, D. 1998, "Computer simulation study of gas-liquid nucleation
in a Lennard-Jones system", Journal of Chemical Physics, vol. 109, no. 22, pp. 9901.
Toxvaerd, S. 2001, "Molecular-dynamics simulation of homogeneous nucleation in the
vapor phase", Journal of Chemical Physics, vol. 115, no. 19, pp. 8913.
Viisanen, Y. & Strey, R. 1994, "Homogeneous nucleation for n-Butanol", Journal of
Chemical Physics, vol. 101, pp. 7835.
Viisanen, Y., Strey, R. & Reiss, H. 1993, "Homogeneous nucleation rates for water",
Journal of Chemical Physics, vol. 99, pp. 4680.
264
Voronel, A.V., Gorbunova, V.G., Smirnov, V.A., Shmakov, N.G. & Shchekochikhina,
V.V. 1973, Soviet Physics JETP, vol. 36, pp. 505.
Wagner, W. 1973, Cryogenics, vol. 13, pp. 470.
Weakliem, C.L. & Reiss, H. 1993, "Toward a molecular theory of vapor-phase
nucleation. III. Thermodynamic properties of argon clusters from Monte Carlo
simulations and a modified liquid drop theory", Journal of Chemical Physics, vol.
99, no. 7, pp. 5374.
Wedekind, J., Reguera, D. & Strey, R. 2007, "Influence of thermostats and carrier gas on
simulations of nucleation", Journal of Chemical Physics, vol. 127, pp. 16501/116501/12.
Wegener, P.P., Clumpner , J. A. & Wu, B. J. C. 1972, "Homogeneous nucleation and
growth of ethanol drops in supersonic flow", Physics of Fluids, vol. 15, pp. 18691876.
Wölk, J. & Strey, R. 2001, "Homogeneous nucleation of H2O and D2O in comparison:
The isotope effect", Journal of Physical Chemistry B, vol. 105, pp. 11683-11701.
Wölk, J., Strey, R., Heath, C.H. & Wyslouzil, B.E. 2002, "
Empirical function for homogeneous water nucleation rates", Journal of Chemical
Physics, vol. 117, pp. 4954.
Wu, B. J. C., Wagner, P.E. & Stein, G.D. 1978, "Condensation of sulfur hexafluoride in
steady supersonic nozzle flow", Journal of Chemical Physics, vol. 68, pp. 308.
Wu, B. J. C., Wegener, P.P. & Stein, G.D. 1978, "Homogeneous nucleation of argon
carried in helium in supersonic nozzle flow", Journal of Chemical Physics, vol. 69,
pp. 1776.
Wyslouzil, B.E., Heath, C.H., Cheung, J.L. & Wilemski, G. 2000, "Binary condensation
in a supersonic nozzle", Journal of Chemical Physics, vol. 113, pp. 7317.
Wyslouzil, B.E., Wilemski, G., Beals, M.G. & Frish, M. 1994, "Effect of Carrier Gas
Pressure on Condensation in a Supersonic Nozzle", Physics of Fluids, vol. 6, pp.
2845-2854.
Wyslouzil, B.E., Wilemski, G., Strey, R., Heath, C.H. & Dieregsweiler, U. 2006,
"Experimental evidence for internal structure in aqueous-organic nanodroplets",
Physical Chemistry Chemical Physics, vol. 8, pp. 54-57.
265
Wyslouzil, B.E., Wilemski, G., Strey, R., Seifert, S. & Winans, R.E. 2007, "Small angle
X-ray scattering measurements probe water nanodroplets evolution under highly
non-equilibrium conditions", Physical Chemistry Chemical Physics, vol. 9, no. 5353,
pp. 5358.
Yasuoka, K. & Matsumoto, M. 1998, "Molecular dynamics of homogeneous nucleation
in the vapor phase. I. Lennard-Jones fluid", Journal of Chemical Physics, vol. 109,
no. 19, pp. 8451.
Young, J.B. 1993, "The condensation and evaporation of liquid droplets at arbitrary
Knudsen number in the presence of an inert gas", International Journal of Heat and
Mass Transfer, vol. 36, pp. 2941-2956.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1999, "
Homogeneous nucleation of argon in an unsteady hypersonic flow field", Journal of
Chemical Physics, vol. 110, pp. 8842.
Zahoransky, R.A., Höschele, J. & Steinwandel, J. 1995, "Formation of argon clusters by
homogeneous nucleation in supersonic shock tube flow", Journal of Chemical
Physics, vol. 103, pp. 9038.
Zeng, X.C. & Oxtoby, D.W. 1991, "Gas-liquid nucleation in Lennard-Jones fluids",
Journal of Chemical Physics, vol. 94, pp. 4472.
266